david brandon, wayne d. kaplan(auth.) microstructural characterization of materials, 2nd...

Microstructural Characterization of Materials

2nd Edition

Microstructural Characterization of Materials - 2nd Edition David Brandon and Wayne D. Kaplan© 2008 John Wiley & Sons,Ltd. ISBN: 978-0-470-02784-4

MicrostructuralCharacterization

of Materials

2nd Edition

DAVID BRANDON AND WAYNE D. KAPLAN

Technion, Israel Institute of Technology, Israel

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Library of Congress Cataloging-in-Publication Data

Brandon, D. G.Microstructural Characterization of Materials / David Brandon and Wayne D.

Kaplan. – 2nd ed.p. cm. – (Quantitative software engineering series)

Includes bibliographical references and index.ISBN 978-0-470-02784-4 (cloth) – ISBN 978-0-470-02785-1 (pbk.)

1. Materials–Microscopy. 2. Microstructure. I. Kaplan, Wayne D. II.Title.TA417.23.B73 2008620.1’1299–dc22 2007041704

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978 0 470 02784 4 (cloth)ISBN 978 0 470 02785 1 (paper)

Typeset in 10/12 pt Times by Thomson Digital, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

Contents

Preface to the Second Edition xi

Preface to the First Edition xiii

1 The Concept of Microstructure 1

1.1 Microstructural Features 7

1.1.1 Structure–Property Relationships 7

1.1.2 Microstructural Scale 10

1.1.3 Microstructural Parameters 19

1.2 Crystallography and Crystal Structure 24

1.2.1 Interatomic Bonding in Solids 25

1.2.2 Crystalline and Amorphous Phases 30

1.2.3 The Crystal Lattice 30

Summary 42

Bibliography 46

Worked Examples 46

Problems 51

2 Diffraction Analysis of Crystal Structure 55

2.1 Scattering of Radiation by Crystals 56

2.1.1 The Laue Equations and Bragg’s Law 56

2.1.2 Allowed and Forbidden Reflections 59

2.2 Reciprocal Space 60

2.2.1 The Limiting Sphere Construction 60

2.2.2 Vector Representation of Bragg’s Law 61

2.2.3 The Reciprocal Lattice 61

2.3 X-Ray Diffraction Methods 63

2.3.1 The X-Ray Diffractometer 67

2.3.2 Powder Diffraction–Particles and Polycrystals 73

2.3.3 Single Crystal Laue Diffraction 76

2.3.4 Rotating Single Crystal Methods 78

2.4 Diffraction Analysis 79

2.4.1 Atomic Scattering Factors 80

2.4.2 Scattering by the Unit Cell 81

2.4.3 The Structure Factor in the Complex Plane 83

2.4.4 Interpretation of Diffracted Intensities 84

2.4.5 Errors and Assumptions 85

2.5 Electron Diffraction 90

2.5.1 Wave Properties of Electrons 91

2.5.2 Ring Patterns, Spot Patterns and Laue Zones 94

2.5.3 Kikuchi Patterns and Their Interpretation 96

Summary 98

Bibliography 103

Worked Examples 103

Problems 114

3 Optical Microscopy 123

3.1 Geometrical Optics 125

3.1.1 Optical Image Formation 125

3.1.2 Resolution in the Optical Microscope 130

3.1.3 Depth of Field and Depth of Focus 133

3.2 Construction of The Microscope 134

3.2.1 Light Sources and Condenser Systems 134

3.2.2 The Specimen Stage 136

3.2.3 Selection of Objective Lenses 136

3.2.4 Image Observation and Recording 139

3.3 Specimen Preparation 143

3.3.1 Sampling and Sectioning 143

3.3.2 Mounting and Grinding 144

3.3.3 Polishing and Etching Methods 145

3.4 Image Contrast 148

3.4.1 Reflection and Absorption of Light 149

3.4.2 Bright-Field and Dark-Field Image Contrast 150

3.4.3 Confocal Microscopy 152

3.4.4 Interference Contrast and Interference

Microscopy 152

3.4.5 Optical Anisotropy and Polarized Light 157

3.4.6 Phase Contrast Microscopy 163

3.5 Working with Digital Images 165

3.5.1 Data Collection and The Optical System 165

3.5.2 Data Processing and Analysis 165

3.5.3 Data Storage and Presentation 166

3.5.4 Dynamic Range and Digital Storage 167

3.6 Resolution, Contrast and Image Interpretation 170

Summary 171

Bibliography 173

Worked Examples 173

Problems 176

4 Transmission Electron Microscopy 179

4.1 Basic Principles 185

4.1.1 Wave Properties of Electrons 185

4.1.2 Resolution Limitations and Lens Aberrations 187

4.1.3 Comparative Performance of Transmission and Scanning

Electron Microscopy 192

vi Contents

4.2 Specimen Preparation 194

4.2.1 Mechanical Thinning 195

4.2.2 Electrochemical Thinning 198

4.2.3 Ion Milling 199

4.2.4 Sputter Coating and Carbon Coating 201

4.2.5 Replica Methods 202

4.3 The Origin of Contrast 203

4.3.1 Mass–Thickness Contrast 205

4.3.2 Diffraction Contrast and Crystal Lattice Defects 205

4.3.3 Phase Contrast and Lattice Imaging 207

4.4 Kinematic Interpretation of Diffraction Contrast 213

4.4.1 Kinematic Theory of Electron Diffraction 213

4.4.2 The Amplitude–Phase Diagram 213

4.4.3 Contrast From Lattice Defects 215

4.4.4 Stacking Faults and Anti-Phase Boundaries 216

4.4.5 Edge and Screw Dislocations 218

4.4.6 Point Dilatations and Coherency Strains 219

4.5 Dynamic Diffraction and Absorption Effects 221

4.5.1 Stacking Faults Revisited 227

4.5.2 Quantitative Analysis of Contrast 230

4.6 Lattice Imaging at High Resolution 230

4.6.1 The Lattice Image and the Contrast Transfer Function 230

4.6.2 Computer Simulation of Lattice Images 231

4.6.3 Lattice Image Interpretation 232

4.7 Scanning Transmission Electron Microscopy 234

Summary 236

Bibliography 238

Worked Examples 238

Problems 247

5 Scanning Electron Microscopy 261

5.1 Components of The Scanning Electron Microscope 262

5.2 Electron Beam–Specimen Interactions 264

5.2.1 Beam-Focusing Conditions 265

5.2.2 Inelastic Scattering and Energy Losses 266

5.3 Electron Excitation of X-Rays 269

5.3.1 Characteristic X-Ray Images 271

5.4 Backscattered Electrons 277

5.4.1 Image Contrast in Backscattered Electron Images 279

5.5 Secondary Electron Emission 280

5.5.1 Factors Affecting Secondary Electron Emission 283

5.5.2 Secondary Electron Image Contrast 286

5.6 Alternative Imaging Modes 288

5.6.1 Cathodoluminescence 288

5.6.2 Electron Beam Induced Current 288

5.6.3 Orientation Imaging Microscopy 289

Contents vii

5.6.4 Electron Backscattered Diffraction Patterns 289

5.6.5 OIM Resolution and Sensitivity 291

5.6.6 Localized Preferred Orientation and Residual Stress 292

5.7 Specimen Preparation and Topology 294

5.7.1 Sputter Coating and Contrast Enhancement 295

5.7.2 Fractography and Failure Analysis 295

5.7.3 Stereoscopic Imaging 298

5.7.4 Parallax Measurements 298

5.8 Focused Ion Beam Microscopy 301

5.8.1 Principles of Operation and Microscope Construction 302

5.8.2 Ion Beam–Specimen Interactions 304

5.8.3 Dual-Beam FIB Systems 306

5.8.4 Machining and Deposition 306

5.8.5 TEM Specimen Preparation 310

5.8.6 Serial Sectioning 314

Summary 315

Bibliography 318

Worked Examples 318

Problems 326

6 Microanalysis in Electron Microscopy 333

6.1 X-Ray Microanalysis 334

6.1.1 Excitation of Characteristic X-Rays 334

6.1.2 Detection of Characteristic X-Rays 338

6.1.3 Quantitative Analysis of Composition 343

6.2 Electron Energy Loss Spectroscopy 357

6.2.1 The Electron Energy-Loss Spectrum 360

6.2.2 Limits of Detection and Resolution in EELS 361

6.2.3 Quantitative Electron Energy Loss Analysis 364

6.2.4 Near-Edge Fine Structure Information 365

6.2.5 Far-Edge Fine Structure Information 366

6.2.6 Energy-Filtered Transmission Electron Microscopy 367

Summary 370

Bibliography 375

Worked Examples 375

Problems 386

7 Scanning Probe Microscopy and Related Techniques 391

7.1 Surface Forces and Surface Morphology 392

7.1.1 Surface Forces and Their Origin 392

7.1.2 Surface Force Measurements 396

7.1.3 Surface Morphology: Atomic and Lattice Resolution 397

7.2 Scanning Probe Microscopes 400

7.2.1 Atomic Force Microscopy 403

7.2.2 Scanning Tunnelling Microscopy 410

7.3 Field-Ion Microscopy and Atom Probe Tomography 413

viii Contents

7.3.1 Identifying Atoms by Field Evaporation 414

7.3.2 The Atom Probe and Atom Probe Tomography 416

Summary 417

Bibliography 420

Problems 420

8 Chemical Analysis of Surface Composition 423

8.1 X-Ray Photoelectron Spectroscopy 424

8.1.1 Depth Discrimination 426

8.1.2 Chemical Binding States 428

8.1.3 Instrumental Requirements 429

8.1.4 Applications 431

8.2 Auger Electron Spectroscopy 431

8.2.1 Spatial Resolution and Depth Discrimination 433

8.2.2 Recording and Presentation of Spectra 434

8.2.3 Identification of Chemical Binding States 435

8.2.4 Quantitative Auger Analysis 436

8.2.5 Depth Profiling 437

8.2.6 Auger Imaging 438

8.3 Secondary-Ion Mass Spectrometry 440

8.3.1 Sensitivity and Resolution 442

8.3.2 Calibration and Quantitative Analysis 444

8.3.3 SIMS Imaging 445

Summary 446

Bibliography 448

Worked Examples 448

Problems 453

9 Quantitative and Tomographic Analysis of Microstructure 457

9.1 Basic Stereological Concepts 458

9.1.1 Isotropy and Anisotropy 459

9.1.2 Homogeneity and Inhomogeneity 461

9.1.3 Sampling and Sectioning 463

9.1.4 Statistics and Probability 466

9.2 Accessible and Inaccessible Parameters 467

9.2.1 Accessible Parameters 468

9.2.2 Inaccessible Parameters 476

9.3 Optimizing Accuracy 481

9.3.1 Sample Size and Counting Time 483

9.3.2 Resolution and Detection Errors 485

9.3.3 Sample Thickness Corrections 487

9.3.4 Observer Bias 489

9.3.5 Dislocation Density Revisited 490

9.4 Automated Image Analysis 491

9.4.1 Digital Image Recording 494

9.4.2 Statistical Significance and Microstructural Relevance 495

Contents ix

9.5 Tomography and Three-Dimensional Reconstruction 495

9.5.1 Presentation of Tomographic Data 496

9.5.2 Methods of Serial Sectioning 498

9.5.3 Three-Dimensional Reconstruction 499

Summary 500

Bibliography 503

Worked Examples 503

Problems 514

Appendices 517

Appendix 1: Useful Equations 517

Interplanar Spacings 517

Unit Cell Volumes 518

Interplanar Angles 518

Direction Perpendicular to a Crystal Plane 519

Hexagonal Unit Cells 520

The Zone Axis of Two Planes in the Hexagonal System 521

Appendix 2: Wavelengths 521

Relativistic Electron Wavelengths 521

X-Ray Wavelengths for Typical X-Ray Sources 521

Index 523

x Contents

Preface to the Second Edition

The last decade has seen several major changes in the armoury of tools that are routinely

available to the materials scientist and engineer for microstructural characterization.

Some of these changes reflect continuous technological improvements in the collection,

processing and recording of image data. Several other innovations have been both

dramatic and unexpected, not least in the rapid acceptance these tools have gained in both

the research and industrial development communities.

The present text follows the guidelines laid down for the first edition, exploring the

methodology of materials characterization under the three headings of crystal structure,

microstructural morphology and microanalysis. One additional chapter has been added,

on Scanning Probe Microscopy, a topic that, at the time that the first edition was written,

was very much a subject for active research, but a long way from being commonly

accessible in university and industrial laboratories. Today, atomic force and scanning

tunnelling microscopy have found applications in fields as diverse as optronics and

catalysis, friction and cosmetics.

It has proved necessary to split the chapter on Electron Microscopy into two chapters,

one on Transmission techniques, and the other on Scanning methods. These two

expanded chapters reflect the dramatic improvements in the resolution available for

lattice imaging in transmission, and the revolution in sampling and micro-machining

associated with the introduction of the focused ion beam in scanning technology.

The final chapter, on Quantitative Analysis, has also been expanded, to accommodate

the rapid advances in three-dimensional reconstruction that now enable massive data sets

to be assembled which include both chemical and crystallographic data embedded in a

frame of reference given by microstructural morphology. Not least among the new

innovations are orientation imaging microscopy, which allows the relative crystallo-

graphic orientations of the grains of a polycrystalline sample to be individually mapped,

and atom probe tomography, in which the ions extracted from the surface of a sharp

metallic needle are chemically identified and recorded in three dimensions. This last

instrument is a long way from being widely available, but a number of laboratories

do offer their services commercially, bringing three-dimensional analysis and

characterization well below the nanometre range, surely the ultimate in microstructural

characterization.

It only remains to note the greatest difference between the present text and its

predecessor: digital recording methods have all but replaced photography in every

application that we have considered, and we have therefore included sections on digital

recording, processing and analysis. This ‘digital revolution’ has crept up on us slowly,

following the on-going improvements in the storage capacity and processing speed for

computer hardware and software. Today, massive amounts of digital image data can be

handled rapidly and reliably.

At the same time, it is still up to the microscopist and the engineer to make the critical

decisions associated with the selection of samples for characterization, the preparation of

suitable sections and the choice of characterization methods. This task is just as difficult

today as it always was in the past. Hopefully, this new text will help rather than confuse!

Most of the data in this book are taken from work conducted in collaboration with our

colleagues and students at the Technion. We wish to thank the following for their

contributions: David Seidman, Rik Brydson, Igor Levin, Moshe Eizenberg, Arnon

Siegmann, Menachem Bamberger, Michael Silverstein, Yaron Kauffmann, Christina

Scheu, Gerhard Dehm, Ming Wei, Ludmilla Shepelev, Michal Avinun, George Levi,

Amir Avishai, Tzipi Cohen, Mike Lieberthal, Oren Aharon, Hila Sadan, Mor Baram, Lior

Miller, Adi Karpel, Miri Drozdov, Gali Gluzer, Mike Lieberthal, and Thangadurai

Paramasivam.

D.B.

W.D.K.

xii Preface to the Second Edition

Preface to the First Edition

Most logical decisions rely on providing acceptable answers to precise questions, e.g.

what, why and how? In the realm of scientific and technical investigation, the first

question is typically what is the problem or what is the objective? This is then followed

by a why question which attempts to pinpoint priorities, i.e. the urgency and importance

of finding an acceptable answer. The third type of question, how is usually concerned

with identifying means and methods, and the answers constitute an assessment of the

available resources for resolving a problem or achieving an objective. The spectrum of

problems arising in materials science and technology very often depends critically on

providing adequate answers to these last two questions. The answers may take many

forms, but when materials expertise is involved, they frequently include a need to

characterize the internal microstructure of an engineering material.

This book is an introduction to the expertise involved in assessing the microstructure of

engineering materials and to the experimental methods which are available for this purpose. For

this text to be meaningful, the reader should understand why the investigation of the internal

structure of an engineering material is of interest and appreciate why the microstructural

features of the material are so often of critical engineering importance, This text is intended to

provide a basic grasp of both the methodology and the means available for deriving qualitative

and quantitative microstructural information from a suitable sample.

There are two ways of approaching materials characterization. The first of these is in

terms of the engineering properties of materials, and reflects the need to know the

physical, chemical and mechanical properties of the material before we can design an

engineering system or manufacture its components. The second form of characterization

is that which concerns us in this book, namely the microstructural characterization of the

material. In specifying the internal microstructure of an engineering material we include

the chemistry, the crystallography and the structural morphology, with the term materials

characterization being commonly taken to mean just this specification.

Characterization in terms of the chemistry involves an identification of the chemical

constituents of the material and an analysis of their relative abundance, that is a

determination of the chemical composition and the distribution of the chemical elements

within the material. In this present text, we consider methods which are available for

investigating the chemistry on the microscopic scale, both within the bulk of the material

and at the surface.

Crystallography is the study of atomic order in the crystal structure. A crystallographic

analysis serves to identify the phases which are present in the structure, and to describe

the atomic packing of the various chemical elements within these phases. Most phases

are highly ordered, so that they are crystalline phases in which the atoms are packed

together in a well-ordered, regularly repeated array. Many solid phases possess no such

long-range order, and their structure is said to be amorphous or glassy. Several

quasicrystalline phases have also been discovered in which classical long-range order is

absent, but the material nevertheless possesses well-defined rotational symmetry.

The microstructure of the material also includes those morphological features which

are revealed by a microscopic examination of a suitably prepared specimen sample. A

study of the microstructure may take place on many levels, and will be affected by

various parameters associated with specimen preparation and the operation of the

microscope, as well as by the methods of data reduction used to interpret results.

Nevertheless, all microstructural studies have some features in common. They provide an

image of the internal structure of the material in which the image contrast depends upon

the interaction between the specimen and some incident radiation used to probe the

sample morphology. The image is usually magnified, so that the region of the specimen

being studied is small compared with the size of the specimen. Care must be exercised in

interpreting results as being ‘typical’ of the bulk material. While the specimen is a three-

dimensional object, the image is (with few exceptions) a two-dimensional projection.

Even a qualitative interpretation of the image requires an understanding of the spatial

relationship between the two-dimensional imaged features and the three-dimensional

morphology of the bulk specimen.

Throughout this book we are concerned with the interpretation of the interaction

between the probe and a sample prepared from a given material, and we limit the text to

probes of X-rays, visible light or energetic electrons. In all cases, we include three stages

of investigation, namely specimen preparation, image observation and recording, and the

analysis and interpretation of recorded data. We will see that these three aspects of

materials characterization interact: the microstructural morphology defines the phase

boundaries, and the shape and dimensions of the grains or particles, the crystallography

determines the phases present and the nature of the atomic packing within these phases,

while the microchemistry correlates with both the crystallography of the phases and the

microstructural morphology.

This text is intended to demonstrate the versatility and the limitations of the more

important laboratory tools available for microstructural characterization. It is not a

compendium of all of the possible methods, but rather a teaching outline of the most

useful methods commonly found in student laboratories, university research departments

and industrial development divisions.

Most of the data in this book are taken from work conducted in collaboration with our

colleagues and students at the Technion. We wish to thank the following for their

contributions: Moshe Eizenberg, Arnon Siegmann, Menachem Bamberger, Christina

Scheu, Gerhard Dehm, Ming Wei, Ludmilla Shepelev, Michal Avinun, George Levi,

Mike Lieberthal, and Oren Aharon.

D.B.

W.D.K.

xiv Preface to the First Edition

Figure 1.20

Figure 5.23

Figure 5.25

Figure 5.24


Figure 6.27

Figure 7.18

Figure 7.22

Figure 9.23

1

The Concept of Microstructure

This text provides a basic introduction to the most commonly used methods of microstruc-tural characterization. It is intended for students of science and engineering whose courserequirements (or curiosity) lead them to explore beyond the accepted causal connectionbetween the engineering properties of materials and their microstructural features, andprompt them to ask how the microstructures of diverse materials are characterized in thelaboratory.

Most introductory textbooks for materials science and engineering emphasize that theprocessing route used to manufacture a component (shaping processes, thermal treatment,mechanical working, etc.) effectively determines the microstructural features (Figure 1.1).They note the interrelation between the microstructure and the chemical, physical, and/ormechanical properties of materials, developing expressions for the dependence of theseproperties on such microstructural concepts as grain size or precipitate volume fraction.What they do not usually do is to give details of either the methods used to identifymicrostructural features, or the analysis required to convert a microstructural observationinto a parameter with some useful engineering significance.

This book covers three aspects of microstructural characterization (Table 1.1). First, thedifferent crystallographic phases which are present in the sample are identified. Secondly,themorphology of these phases (their size, shape and spatial distribution) are characterized.Finally, the local chemical composition and variations in chemical composition aredetermined.

In all three cases we will explore the characterization of the microstructure at both thequalitative and the quantitative level. Thus, in terms of crystallography, we will beconcerned not only with qualitative phase identification, but also with the elementaryaspects of applied crystallography used to determine crystal structure, as well as with thequantitative determination of the volume fraction of each phase. As for the microstructure,we will introduce stereological relationships which are needed to convert a qualitativeobservation of morphological features, such as the individual grains seen in a cross-section,into a clearly defined microstructural parameter, the grain size. Similarly, we shall not be


satisfied with the microanalytical identification of the chemical elements present in aspecific microstructural feature, but rather we shall seek to determine the local chemicalcomposition throughmicroanalysis. Throughout the text we shall attempt to determine boththe sensitivity of the methods described (the limits of detection) and their accuracy (thespatial or spectral resolution, or the concentration errors).

In general terms, microstructural characterization is achieved by allowing some form ofprobe to interact with a carefully prepared specimen sample. The most commonly usedprobes are visible light, X-ray radiation and a high energy electron beam. These three typesof probe, taken in the same order, form the basis for optical microscopy, X-ray diffractionand electron microscopy. Once the probe has interacted with the sample, the scattered orexcited signal is collected and processed into a form where it can be interpreted, eitherqualitatively or quantitatively. Thus, in microscopy, a two-dimensional image of thespecimen is obtained, while in microanalysis a spectrum is collected in which the signalintensity is recorded as a function of either its energy orwavelength. In diffraction the signalmay be displayed as either a diffraction pattern or a diffraction spectrum.

All the instrumentation that is used to characterize materials microstructure includescomponents that have five distinct functions (Figure 1.2). First, the probe is generated by a

Figure 1.1 The microstructure of an engineering material is a result of its chemicalcomposition and processing history. The microstructure determines the chemical, physicaland mechanical properties of the material, and hence limits its engineering performance.

Table 1.1 On the qualitative level, microstructural characterization is concerned with theidentification of the phases present, their morphology (size and shape), and theidentification of the chemical constituents in each phase. At the quantitative level, it ispossible to determine the atomic arrangements (applied crystallography), the spatialrelationships between microstructural features (stereology), and the microchemicalcomposition (microanalysis).

Qualitative analysis Phaseidentification

Microstructuralmorphology

Microchemicalidentification

Quantitativeanalysis

Appliedcrystallography

Stereology Microanalysis

2 Microstructural Characterization of Materials

source that is filtered and collimated to provide a well-defined beam of known energy orwavelength. This probe beam then interacts with a prepared sample mounted on a suitableobject stage. The signal generated by the interaction between the probe and the sample thenpasses through an optical system to reach the image plane, where the signal data arecollected and stored. Finally, the stored data are read out, processed and recorded, either as afinal image, or as diffraction data, or as a chemical record (for example, a compositionmap).The results then have to be interpreted!

In all the methods of characterization which we shall discuss, two forms of interactionbetween the probe and the specimen occur (Figure 1.3):

1. Elastic scattering, which is responsible for the intensity peaks in X-ray diffractionspectra that are characteristic of the phases present and their orientation in the sample.Elastic scattering also leads to diffraction contrast in transmission electron microscopy(TEM),where it is directly related to the nature of the crystal lattice defects present in thesample (grain boundaries, dislocations and other microstructural features).

2. Inelastic scattering, in which the energy in the probe is degraded and partially convertedinto other forms of signal. In optical microscopy, microstructural features may be

Figure 1.2 Microstructural characterization relies on the interaction of a material samplewith a probe. The probe is usually visible light, X-rays or a beam of high energy electrons.The resultant signal must be collected and interpreted. If the signal is elastically scatteredan image can be formed by an optical system. If the signal is inelastically scattered, orgenerated by secondary emission the image is formed by a scanning raster (as in a televisionmonitor).

The Concept of Microstructure 3

revealed because they partially absorb some wavelengths of the �visible� light thatilluminates the specimen. Gold and copper have a characteristic colour because theyabsorb some of the shorter wavelengths (blue and green light) but reflect the longerwavelengths (red and yellow). The reflection is an elastic process while absorption is aninelastic process.

In electron microscopy, the high energy electrons interacting with a specimen often loseenergy in well-defined increments. These inelastic energy losses are then characteristic ofthe electron energy levels of the atoms in the sample, and the energy loss spectra can beanalysed to identify the chemical composition of a region of the sample beneath the electronbeam (the �probe�). Certain electron energy losses are accompanied by the emission ofcharacteristic X-rays. These X-rays can also be analysed, by either energy dispersive orwavelength dispersive spectroscopy, to yield accurate information on the distribution of thechemical elements in the sample.

Elastic scattering processes are characteristic of optical or electro-optical systems whichform an image in real space (the three dimensions in which we live), but elastic scattering isalso a characteristic of diffraction phenomena, which are commonly analysed in reciprocalspace. Reciprocal space is used to represent the scattering angles thatwe record in real space(see below). In real space we are primarily concerned with the size, shape and spatialdistribution of the features observed, but in reciprocal space it is the angle throughwhich thesignal is scattered by the sample and the intensity of this signal that are significant. Theseangles are inversely related to the size or separation of the features responsible for thecharacteristic intensity peaks observed in diffraction. The elastically scattered signalscollected in optical imaging and diffraction are compared in Figure 1.4. In optical imagingwe study the spatial distribution of features in the image plane, while in a diffraction patternor diffraction spectrum we study the angular distribution of the signal scattered fromthe sample.

Figure 1.3 An elastically scattered signal may be optically focused, to form an image in realspace (the spatial distribution of microstructural features), or the scattering angles can beanalysed from a diffraction pattern in reciprocal space (the angular distribution of the scatteredsignal). Inelastic scattering processes generate both an energy loss spectra, and secondary,excited signals, especially secondary electrons and characteristic X-rays.


Inelastic scattering processes dominate the contrast in scanning electron imagingsystems (as in a scanning electron microscope; Figure 1.5). In principle it is possible todetect either the loss spectra (the energy distribution in the original probe after it hasinteractedwith the sample) or a secondary signal (the excited particles or photons generatedby the probe as a result of inelastic interaction).

Large numbers of secondary electrons are emitted when an energetic electron beamstrikes a solid sample. It is the detection of this secondary electron signal that makespossible thevery striking high resolution images of surface features that are characteristic ofscanning electron microscopy (SEM).

In what follows we will assume that the student is familiar with those aspects ofmicrostructure and crystallography that are commonly included in introductory courses inmaterials science and engineering: some knowledge of the Bravais lattices and the conceptof crystal symmetry; microstructural concepts associated with polycrystals and polyphasematerials (including the significance of microstructural inhomogeneity and anisotropy);and, finally, the thermodynamic basis of phase stability and phase equilibrium in polyphasematerials.

Throughout this book each chapter will conclude with results obtained from samplesof three materials that are representative of a wide range of common engineering materials.

Object

Optical

System

Focused

Image

(a)

Incident

Radiation

Scattered

Radiation

Object

Diffracted

Radiation

(b)

Scattering Angle

Figure 1.4 Schematic representations of anoptical image (a) and adiffractionpattern (b). In theformer distances in the image are directly proportional to distances in the object, and theconstant of proportionality is equal to the magnification. In the latter the scattering angle for thediffracted radiation is inversely proportional to the scale of the features in the object, sothat distances in a diffraction pattern are inversely proportional to the separation of features inthe object.


We will explore the information that can be obtained for these materials from eachof the methods of microstructural characterization that we discuss. The materials we haveselected are:

. a low alloy steel containing approximately 0.4%C;

. a dense, glass-containing alumina;

. a thin-film microelectronic device based on the Al/TiN/Ti system.

An engineering polymer or a structural composite could equally well have been selectedfor these examples. The principles of characterization would have been the same, eventhough the details of interpretation differed.

Our choice of the methods of microstructural characterization that we describe is asarbitrary as our selection of these �typical�materials. Any number of methods of investiga-tion are used to characterize themicrostructure of engineeringmaterials, but this text is not acompendium of all known techniques. Instead we have chosen to limit ourselves to thoseestablished methods that are commonly found in a well-equipped industrial developmentdepartment or university teaching laboratory. The methods selected include optical andelectron microscopy (both scanning and transmission), X-ray and electron diffraction, andthe commoner techniques of microanalysis (energy dispersive and wavelength dispersiveX-ray analysis, Auger electron spectroscopy, X-ray photospectroscopy and electron energyloss spectroscopy). We also discuss surface probe microscopy (SPM), including the atomicforce microscope and the scanning tunnelling microscope, since one or more versions of

Figure 1.5 A scanning image is formed by scanning a focused probe over the specimen andcollecting a data signal from the sample. The signal is processed and displayed on a fluorescentscreen with the same time-base as that used to scan the probe. The magnification is the ratioof the monitor screen size to the amplitude of the probe scan on the specimen. The signalmay be secondary electrons, characteristic X-rays, or a wide variety of other excitationphenomena.


this instrumentation are now commonly available. We also include a brief account ofthe remarkable chemical and spatial resolution that can be achieved by atom probetomography, even though this equipment is certainly not commonly available at the timeof writing.

In each case a serious attempt is made to describe the physical principles of the method,clarify the experimental limitations, and explore the extent to which each technique can beused to yield quantitative information

1.1 Microstructural Features

When sectioned, polished and suitably etched, nearly all engineering materials will befound to exhibit structural features that are characteristic of the material. These featuresmay be visible to the unaided eye, or theymay require a low-powered optical microscope toreveal the detail. The finest sub-structure will only be visible in the electron microscope.Many of the properties of engineering solids are directly and sensitively related to themicrostructural features of thematerial. Such properties are said to be structure sensitive. Insuch cases, the study of microstructure can reveal a direct causal relationship between aparticularmicrostructural feature and a specific physical, chemical or engineering property.

In what follows we shall explore some of these structure–property relationships andattempt to clarify further the meaning of the term microstructure.

1.1.1 Structure–Property Relationships

It is not enough to state that �materials characterization is important�, since it is usual todistinguish between those properties of a material that are structure-sensitive and those thatare structure-insensitive. Examples of structure-insensitive properties are the elasticconstants, which vary only slowly with composition or grain size. For example, there islittle error involved in assuming that all steels have the same tensile (Young�s) modulus,irrespective of their composition. In fact the variation in the elastic modulus of structuralmaterials with temperature (typically less than 10%) exceeds that associated with alloychemistry, grain size or degree of cold work. The thermal expansion coefficient is anotherexample of a property which is less affected by variations in microstructural morphologythan it is by composition, temperature or pressure. The same is true of the specific gravity (ordensity) of a solid material.

In contrast, the yield strength, which is the stress that marks the onset of plastic flow inengineering alloys, is a sensitive function of several microstructural parameters: the grainsize of the material, the dislocation density, and the distribution and volume fraction ofsecond-phase particles. Thermal conductivity and electrical resistivity are also structure-sensitive properties, and heat treating an alloy may have a large affect on its thermal andelectrical conductivity. This is often because both the thermal and the electrical conductivi-ty are drastically reduced by the presence of alloying elements in solid solution in thematrix. Perhaps the most striking example of a structure-sensitive property is the fracturetoughness of an engineering material, which measures the ability of a structural material toinhibit crack propagation and prevent catastrophic brittle failure. Very small changes in


chemistry and highly localized grain boundary segregation (the migration of an impurity tothe boundary, driven by a reduction in the boundary energy), may cause a catastrophic lossof ductility, reducing the fracture toughness by an order of magnitude. Although suchsegregation effects are indeed an example of extreme structure-sensitivity, they are alsoextremely difficult to detect, since the bulk impurity levels associatedwith segregation needonly be of the order of 10�5 [10 parts per million (ppm)].

A classic example of a structure-sensitive property relation is the Petch equation, whichrelates an engineering property, the yield strength of a steel sy, to a microstructural feature,its grain size D, in terms of two material constants, s0 and ky:

sy ¼ s0þkyD�1=2 ð1:1Þ

This relation presupposes that we are able to determine the grain size of the materialquantitatively and unambiguously. The meaning of the term grain size, is explored in moredetail in Section 1.1.3.1.

The fracture surfaces of engineering components that have failed in service, as well asthose of standard specimens that have failed in a tensile, creep ormechanical fatigue test, arefrequently subjected to microscopic examination in order to characterize the microstruc-tural mechanisms responsible for the fracture (a procedure which is termed fractography).In brittle, polycrystalline samples much of the fracture path often lies along specific low-index crystallographic planes within the crystal lattices of the individual grains. Suchfractures are said to be transgranular or cleavage failures. Since neighbouring grains havedifferent orientations in space, the cleavage surfaces are forced to change direction at thegrain boundaries. The line of intersection of the cleavage plane with the grain boundary inone grain is very unlikely to lie in an allowed cleavage planewithin the second grain, so thatcleavage failures in polycrystalline materials must either propagate on unfavourable crystallattice planes, or else link up by intergranular grain boundary failure, which takes place atthe grain boundaries between the cleavage cracks. A fractographic examination ofthe failure surface reveals the relative extent of intergranular and transgranular failure(Figure 1.6). By determining the three-dimensional nature of brittle crack propagation andits dependence on grain size or grain boundary chemistry we are able to explore criticalaspects of the failure mechanism.

Ductile failures are also three-dimensional. A tensile crack in a ductile material typicallypropagates by the nucleation of small cavities in a region of hydrostatic tensile stress that isgenerated ahead of the crack tip. The nucleation sites are often small, hard inclusions, eitherwithin the grains or at the grain boundaries, and the distribution of the cavities depends onthe spatial distribution of these nucleating sites. The cavities growby plastic shear at the rootof the crack, until they join up to form a cellular, ridged surface, termed a dimpled fracture(Figure 1.7). The complex topology of this dimpled, ductile failuremay not be immediatelyobvious from a two-dimensionalmicrograph of the fracture surface, and in fractography it iscommon practice to image the failure twice, tilting the sample between the recording of thetwo images. This process is equivalent to viewing the surface from two different points ofview, and allows a rough surface to be viewed and analysed stereoscopically, in threedi-mensions. The third dimension is deduced from the changes in horizontal displacement forany two points that lie at different heights with respect to the plane of the primary image, a


Figure 1.6 A scanning electron microscope image showing transgranular and intergranularbrittle failure in a partially porous ceramic, aluminium oxynitride (AlON). In three dimensionssome intergranular failure is always present, since transgranular failure occurs by cleavage onspecific crystallographic planes.

Figure 1.7 Intergranular dimple rupture in a steel specimen resulting from microvoidcoalescence at grain boundaries. From Victor Kerlins, Modes of Fracture, Metals HandbookDesk Edition, 2nd Edition, ASM International, 1998, in ASM Handbooks Online (http://www.asmmaterials.info), ASM International, 2004. Reprinted with permission of ASM International.All rights reserved.


phenomenon termed parallax (Figure 1.8, see Section 4.3.6.3). Our two eyes give us thesame impression of depthwhen the brain superimposes the twoviews of theworldwhichwereceive from each eye separately.

The scale of the microstructure determines many other mechanical properties, just as thegrain size of a steel is related to its yield strength. The fracture strength of a brittle structuralmaterialsf is related to the fracture toughnessKc by the size cof the processingdefects presentin the material (cracks, inclusions or porosity), i.e. sf/Kc /

ffiffiffi

cp

. The contribution of work-hardening to the plastic flow stress (the stress required to continue plastic flow after plasticstrain due to a stress increment above the yield stress, Dsy) depends on both the dislocationdensity r and the elastic shear modulus G, i.e. Dsy/G

ffiffiffi

rp

. Similarly, the effectiveness ofprecipitation hardening by a second phase (the increase in yield stress associated with thenucleation and growth of small second-phase particles, Dsp) is often determined by theaverage separation of the second phase precipitates L, through the relationship Dsp/G/L.

1.1.2 Microstructural Scale

Microstructure is a very general term used to cover a wide range of structural features, fromthose visible to the naked eye down to those corresponding to the interatomic distances inthe crystal lattice. It is good practice to distinguish betweenmacrostructure,mesostructure,microstructure and nanostructure.

Figure1.8 Theprinciple of stereoscopic imaging. The left and right eyes see theworld from twodifferent positions, so that two points at different heights subtend different angular separationswhen viewed by the two eyes.


Macrostructure refers to those features which approach the scale of the engineeringcomponent and are either visible to the naked eye, or detectable by common methods ofnondestructive evaluation (dye-penetrant testing, X-ray radiography, or ultrasonic testing).Examples include major processing defects such as large pores, foreign inclusions, orshrinkage cracks. Nondestructive evaluation and nondestructive testing are beyond thescope of this text.

Mesostructure is a less common term, but is useful to describe those features that are onthe borderline of the visible. This is particularly the case with the properties of compositematerials, which are dominated by the size, shape, spatial distribution and volume fractionof the reinforcement, as well as by any cracking present at the reinforcement interface orwithin the matrix, or other forms of defect (gas bubbles or dewetting defects). Themesoscale is also important in adhesive bonding and other joining processes: the lateraldimensions of an adhesive or a brazed joint, for example, or the heat-affected zone (HAZ)adjacent to a fusion weld.

Microstructure covers the scale of structural phenomena most commonly of concern tothe materials scientist and engineers, namely grain and particle sizes, dislocation densitiesand particle volume fractions, microcracking and microporosity.

Finally, the term nanostructure is restricted to sub-micrometre features: the width ofgrain boundaries, grain-boundary films and interfaces, the early nucleation stages ofprecipitation, regions of local ordering in amorphous (glassy) solids, and very small,nanoparticles whose properties are dominated by the atoms positioned at the particlesurface. Quantum dots come into this category, as do the stable thin films often formed atboundaries, interfaces and free surfaces.

Table 1.2 summarizes these different microstructural scales in terms of themagnificationrange required to observe the features concerned.

1.1.2.1 The Visually Observable. The human eye is a remarkably sensitive data collec-tion and imaging system, but it is limited in four respects:

. the range of electromagnetic wavelengths that the eye can detect;

. the signal intensity needed to trigger image �recognition� in the brain;

. the angular separation of image details that the eye can resolve;

. the integration time over which an image is recorded by the eye.

The eye is sensitive to wavelengths ranging from about 0.4 to 0.7 mm, corresponding to acolour scale from dark red to violet. The peak sensitivity of the eye is in the green, and isusually quoted as 0.56 mm, a characteristic emission peak in the spectrum from a mercuryvapour lamp. As a consequence, optical microscopes are commonly focused using a greenfilter, while the phosphors used for the screens of transmission electron microscopes and inmonitors for image scanning systems often fluoresce in the green.

The integration time of the eye is about 0.1 s, after which the signal on the retina decays.Sufficient photons have to be captured by the retina within this time in order to form animage. In absolute darkness, the eye �sees� isolated flashes of light, that, at low intensities,constitute a background of random noise. At low light levels the eye also requires severalminutes to achieve its maximum sensitivity (a process termed dark adaptation). Neverthe-less, when properly dark-adapted, the eye detects of the order of 50% of the incident �green�photons, and a statistically significant image will be formed if of the order of 100 photons


can contribute to each picture element (or pixel). This is as good as the best availablemilitary night-viewing systems, but these systems can integrate the image signal over amuch longer period of time than the 0.1 s available to the eye, so that they can operateeffectively at much lower light levels.

The ability to identify two separate features that subtend a small angle at the eye is termedthe resolution of the eye, and is a function of the pupil diameter (the aperture of the eye) andthe distance at which the features are viewed. The concept of resolution was defined byRaleigh in terms of the apparent width of a point source. If the point source subtends anangle 2a at the lens, then Abbe showed that its apparent width d in the plane of the sourcewas given by d¼ 1.2l/m sin a, where l is thewavelength of the radiation from the source andm is the index of refraction of the intervening medium. Raleigh assumed that two pointsources could be distinguished when the peak intensity collected from one point sourcecoincided with the first minimum in the intensity collected from the other (Figure 1.9) Thatis, the resolution, defined by this Raleigh criterion, is exactly equal to the apparent diameterof a point source d viewed with the aid of a lens subtending an angle a. Larger objects areblurred in the image, so that anobject havinga dimensiond appears to have a dimensiondþ d.Objects smaller thand can bedetected, but have a reduced intensity – and appear to have a sizestill equal to d. The limit of detection is usually determined by background noise levels, but isalways less than the resolution limit. In general, intensity signals that exceed the backgroundnoise by more than 10% can be detected.

The diameter of the fully dilated pupil (the aperture that controls the amount of lightentering the eye) is about 6mm, while it is impossible to focus on an object if it is too close

Table 1.2 Scale of microstructural features, the magnification required to reveal thesefeatures, and some common techniques available for studying their morphology.

Scale Macrostructure Mesostructure Microstructure Nanostructure

Typicalmagnification

·1 ·102 ·104 ·106

Commontechniques

Visualinspection

Optical microscopy Scanning andtransmissionelectronmicroscopy

X-ray diffraction

X-rayradiography

Scanning electronmicroscopy

Atomic forcemicroscopy

Scanningtunnellingmicroscopy

Ultrasonicinspection

High resolutiontransmissionelectronmicroscopy

Characteristicfeatures

Productiondefects

Grain and particlesizes

Dislocationsubstructure

Crystal andinterfacestructure

Porosity, cracksand inclusions

Phase morphologyand anisotropy

Grain and phaseboundaries

Point defects andpoint defectclusters

Precipitationphenomena


(termed the near point, typically about 150mm). It follows that sin a for the eye is of theorder of 0.04. Using green light at 0.56 mmand taking m¼ 1 (for air), we arrive at an estimatefor deye of just under 0.2mm. That is, the unaided eye can resolve features which are a fewtenths of amillimetre apart. The eye records of the order of 106 image features in the field ofview at any one time, corresponding to an object some 20 cm across at the near point(roughly the size of this page!).

1.1.2.2 �With The Aid of The Optical Microscope�. An image which has been magnifiedby a factorMwill contain resolvable features whose size ranges down to the limit dictated bythe resolvingpowerd of theobjective lens in themicroscope. In the image these �just resolved�featureswill have a separationMd. IfMd<deye then the unaided eyewill not be able to resolveall the features recorded in themagnified image.On the other hand, ifMd>deye then the imagewill appear blurred and fewer resolvable featureswill be present. That is, less informationwillbe available to the observer. It follows that there is an �optimum�magnification, correspondingto the ratio deye/d, at which the eye is just able to resolve all the features present in themagnified image and the density of resolvable image points (pixels) in the field of view is amaximum. Lower magnifications will image a larger area of the specimen, but at the cost ofrestricting the observable resolution. In some cases, for example in high resolution electronmicroscopy, this may actually be desirable. A hand lens is then used to identify regions ofparticular interest, and these regions are then enlarged, usually electronically. Highermagnifications than the optimum are seldom justified, since the image features then appearblurred and no additional information is gained.

The optical microscope uses visible electromagnetic radiation as the probe, and the bestoptical lens systems havevalues of m sin a of the order of unity (employing a high-refractiveindex, inert oil as the medium between the objective lens and the specimen). It follows that

Two PointSources in the

ObjectPlane

Optical System

ImagePlane

ImageIntensity

2α

Figure 1.9 The Raleigh criterion defines the resolution in terms of the separation of twoidentical point sources that results in the centre of the image of one source falling on the firstminimum in the image of the second source.


the best possible resolution is of the order of the wavelength, that is, approximately 0.5 mm.Assuming 0.2mm for the resolution of the eye, this implies that, at amagnification of ·400,the optical microscope should reveal all the detail that it is capable of resolving. Highermagnifications are often employed (why strain your eyes?), but there is no point in seekingmagnifications for the optical microscope greater than ·1000.

Modern, digitized imaging systems are capable of recording image intensity levels at arate of better than�106 pixels s�1, allowing for real-time digital image recording, not onlyin the optical microscope, but also in any other form of spatially resolved signal collectionand processing (see Section 3.5).

1.1.2.3 Electron Microscopy. Attempts to improve the resolution of the optical micro-scope by reducing the wavelength of the electromagnetic radiation used to form the imagehave been marginally successful. Ultraviolet (UV) radiation is invisible to the eye, so thatthe image must be viewed on a fluorescent screen, and special lenses transparent to UVarerequired. The shorter wavelength radiation is also strongly absorbed by many engineeringmaterials, severely limiting the potential applications for a �UV� microscope. Far moresuccess has been achieved by reducing the size of the source to the sub-micrometre rangeand scanning a light probe over the sample in an x–y raster while recording a scattered orexcited photon signal. Such near-field microscopes have found applications, especially inbiology where they are used for the study of living cells. Once again, however, suchinstruments fall outside the scope of this text.

Attempts have also been made to develop an X-ray microscope, focusing a sub-nanometre wavelength X-ray beam by using curved crystals, but it is difficult to findpractical solutions to the immense technical problems. More successful has been the use ofsynchrotron X-radiation at energies as high as 400 keV, using X-ray microtomography togenerate three-dimensional image information at sub-micrometre resolutions. Such facili-ties are not generally available.

Electrons are the only feasible alternative. An electron beam of fixed energy will exhibitwavelike properties, the wavelength l being derived, to a good approximation, from the deBroglie relationship: l¼ h/(2meV)1/2, where h is Planck�s constant, m is the mass of theelectron, e is the electron charge andV is the accelerating voltage.WithV in kilovolts and l innanometres, the constant h/(2me)1/2 is approximately equal to 0.037. For an acceleratingvoltage of only 1 kV this wavelength is much less than the interplanar spacing in a crystallinesolid. However, as we shall see in Section 4.1.2, it is not that easy to focus an electron beam.Electromagnetic lenses are needed, and various lens aberrations limit the acceptablevalues ofthe collection angle a for the scattered electrons to between 10�2 and 10�3 rad (360� ¼2 p rad). At these small angles a� sin a, and the Raleigh resolution criterion reduces tod¼ 1.2l/a. (In the electron microscope m¼ 1, since the electron beam will only propagatewithout energy loss in a vacuum.)

Typical interatomic distances in solids are of the order of 0.2 to 0.5 nm, so that, in principle,atomic resolution in the electronmicroscope ought to be achievable at 100 kV. This is indeedthe case, but transmission electron microscopes require thin samples which may be difficultto prepare, and in practice the optimum operating voltage for achieving consistent resolutionof the atomic arrays in a crystal lattice is between 200 kV and 400kV. Commercialtransmission electron microscopes guarantee sub-nanometre resolutions and are capableof detecting themicrostructural and nanostructural features present in engineeringmaterials.


Scanning electron microscopes, in which the resolution depends on focusing the electronbeam into a fine probe, have an additional, statistical resolution limit. The beam currentavailable in the probe Ip decreases rapidly as the beam diameter d is reduced, according to therelationship I p / d8=3, and the intensity of the signal that can be detected is proportional to thebeamcurrent. For some types of signal (most notably,X-rays,whose characteristicwavelengthsare dependent on the chemical species), the statistical limit is reached at probe diameterswhichare much larger than those set by the potential electromagnetic performance of the probe lens.For thick samples in the scanning electron microscope, the resolution in characteristic X-raymaps of chemical concentration is usually limited, both by the statistics ofX-ray generation andby spreading of the electron beam as it loses energy to the sample. In general, the best spatialresolution for X-ray maps is of the order of 1mm.

A secondary electron image, obtained in the same scanning electron microscope, mayhave a resolution that is only limited by the aperture of the probe-forming lens and thewavelength of the electron beam. Combining the Abbe equation, the Raleigh criterion andthe de Broglie relationship (linking wavelength to accelerating voltage), we can estimatethe approximateminimum size of an electron beamprobe for secondary electron imaging ina scanning electronmicroscope. Since the image is formed from a secondary signal, we alsoneed to consider the volume of material beneath the impinging beam which generates thesignal. As the accelerating voltage is increased, the electrons will penetrate deeper into thesample and will be scattered (both elastically and inelastically) over a wider angle. Itfollows that lower accelerating voltages are desirable in order to limit this spread of thebeam within the sample. However, a better signal-to-background ratio will be obtained athigher accelerating voltages. Most scanning electron microscopes are designed to operatein the range 1–30 kV, the lower accelerating voltages being preferred for low-density (loweratomic number) specimens. Assuming the probe convergence angle a to be 10�2 radand taking 4 kVas a typical lower limit to the accelerating voltage, this yields a potentialresolution of about 2 nm for the secondary electron image in the scanning electronmicroscope.

A great deal will be said in later chapters about the factors limiting the spatial resolutionin the different methods of microscopy used to characterize microstructural morphology.For the time being, it will suffice to note that five factors should be considered:

. the physical characteristics of the probe source;

. the optical properties of the imaging system;

. the nature of the specimen–probe interaction;

. the statistics of imaging data collection and storage;

. image processing, and the display and recording of the final image.

Every technology has its limitations, and methods of microstructural characterizationshould be chosen according to the information required. Modern equipment for micro-structural characterization often combines several techniques on one platform, so thelimitations of a specific technique do not always mean that we have to prepare differentsamples, or search for another piece of equipment.

1.1.2.4 �Seeing Atoms�. A common, and not in the least foolish, question is �can we seeatoms?�The answer is, in a certain sense, �yes�, although the detailed science and technologybehind the imaging of atomic and molecular structure is beyond the scope of this text.


The story starts in Berlin, in 1933, when the electron microscopewas developed by ErnstRuska. Shortly afterwards, in 1937, Erwin Mu..ller, also in Berlin, demonstrated that apolished tungsten needle, mounted in a vacuum chamber, would emit electrons from its tipwhen a positive voltage was applied to the needle. This field emission process was used toimage differences in the work function, the energy required to extract an electron from thesurface of the metal. A field emission image of the dependence of the work function on thecrystallographic orientation was formed by radial projection of the electrons onto afluorescent screen. Some 15 years later Mu..ller and Bahadur admitted a small quantityof gas into the chamber, reversed the voltage on the tungsten needle, and observedionization of the gas at the tip surface. When the low pressure gas was cooled to cryogenictemperatures, the radially projected image on a fluorescent screen showed regular arrays ofbright spots, and the field-ion microscope was born (Figure 1.10). The intensity of thesebright spots reflects the electric field enhancement over individual atoms protruding fromthe surface of the tip.

By pulsing the tip voltage or by using pulsed laser excitation, the protruding surfaceatoms can be ionized and accelerated radially away from the tip. These field-evaporatedatoms can be detected with high efficiency using a time-of-flight mass spectrometer, so thatnot only can the surface of a metal tip be imaged at �atomic� resolution in a field-ionmicroscope, but in many cases a high proportion of the individual atoms can be identifiedwith reasonable certainty in a field-ion atom probe. In recent years further technicaldevelopments have improved this instrumentation to the point where positional time-of-flight detectors can record millions of field-evaporated individual ions and display each of

Figure 1.10 The field-ion microscope was the first successful attempt to image �atoms�: (a) Aschematic diagram of the instrument; (b) a tungsten tip imaged by field-ion microscopy.


the atomic species present in a three-dimensional array, clearly resolving initial stages in thenucleation of phase precipitation and the chemistry of interface segregation, all atresolutions on the atomic scale (Section 7.3.2).

Unfortunately, only engineering materials which possess some electrical conductivitycan be studied, and many of these materials lack the mechanical strength to withstand thehigh electric field strengths needed to obtain controlled field-evaporation of atoms from thevery sharp, sub-micrometre sample tip.

A rather more useful instrument for observing �atoms� is the scanning tunnellingmicroscope. In this technique a sharp needle is used as a probe rather than as a specimen.The probe needle, mounted on a compliant cantilever in vacuum, is brought to within a fewatomic distances of the sample surface and the vertical distance z of the tip of the probe fromthe surface is precisely controlled by using a laser sensor and piezoelectric drives. The tip isusually scanned at constant z across the x–y plane, while monitoring either the tip current atconstant applied voltage, or the tip voltage at constant applied current (Section 7.2.2).

The contrast periodicity observed in a scanning tunnelling image reflects periodicity inthe atomic structure of the surface, and some of the first images published demonstratedunequivocally and dramatically that the equilibrium{111} surface of a silicon single crystalis actually restructured to form a 7 · 7 rhombohedral array (Figure 1.11). By varying thevoltage of the tip with respect to the specimen, the electron density of states in the samplecan be determined at varying distances beneath the surface of the solid. The resolution of thescanning tunnelling microscope depends primarily on the mechanical stability of thesystem, and all the commercial instruments available guarantee �atomic� resolution.

In principle, there is no reason why the same needle should not be used to monitor theforce between the needle and a specimen surface over which the needle is scanned. As theneedle approaches the surface it first experiences a van der Waals attraction (due to

Figure 1.11 The scanning tunnelling microscope provides data on the spatial distribution ofthe density of states in the electron energy levels at and beneath the surface. In this example the7 ·7 rhombohedral unit cell of the restructured {111} surface of a silicon crystal is clearlyresolved. Reprinted from I.H. Wilson, p. 162 in Walls and Smith (eds) Surface ScienceTechniques, Copyright 1994, with permission from Elsevier Science.


polarization forces), which at shorter distances is replaced by a repulsion force, as theneedle makes physical contact with the specimen. Scanning the probe needle over thesurface at constant displacement (constant z) and monitoring the changes in the van derWaals force yields a scanning image inwhich, under suitable conditions, �atomic� resolutionmay also be observed (Figure 1.12). No vacuum is now necessary, so the atomic forcemicroscope (Section 7.2.1) is particularly useful for studying solid surfaces in gaseous orliquid media (something that the field-ion microscope and the scanning tunnellingmicroscope cannot do). The atomic force microscope is a powerful tool for imaging bothorganic membranes and polymers. By vibrating the tip (using a piezoelectric transducer), itis possible tomonitor the elastic compliance of the substrate at atomic resolution, providinginformation on the spatial distribution of the atomic bonding at the surface. Surface forcemicroscopy, developed earlier, primarily by Israelachvili, uses much the same principle, butwith a resolution of the order of micrometres. In this case curved cylindrical surfaces ofcleaved mica are brought together at right angles, eventually generating a circular area ofcontact. The surface forces are monitored as a function of both the separation of thecylinders and the environment, and this instrument has proved to be a particularly powerfultool for fundamental studies of surface-active and lubricating films. For these studies themica surfaces can be coated, for example by physical vapour or chemical vapour deposition,to control the nature of the lubricant substrate.

All of these techniques provide spatially resolved information on the surface structure of asolid and have been grouped together under the heading surface probemicroscopy (SPM). Inall cases, these techniques reveal oneor another aspect of the surface atomicmorphology andthe nature of the interatomic bonding at the surface. In a very real sense, todaywe are indeedable to �see atoms�. The limitations of SPMwill be explored in Chapter 7, while the extent to

Figure 1.12 AFM scan of Ca-rich particles on the basal surface of sapphire: (a) a top viewshowing the faceting of the particle: (b) an inclined view of the same area showing the threedimensional morphology of the particle. Reprinted with permission from A. Avishai, PhD,Thesis, Thin-Equilibrium Amorphous Films at Model Metal–Ceramic Interfaces, 2004,Technion, Israel Institute of Technology; Haifa, p. 75.


which transmission electron microscopy and scanning electron microscopy also allow us to�see atoms� is discussed in the appropriate sections of Chapter 4.

1.1.3 Microstructural Parameters

Microstructural features are commonly described in qualitative terms: for example, thestructure may be reported to be equiaxed, when the structure appears similar in all directions,or the particles of a second phase may be described as acicular or plate-like. Theories ofmaterial properties, on the other hand, frequently attempt to define a quantitative dependenceof a measured property on the microstructure, through the introduction of microstructuralparameters. These parameters, for example grain size, porosity or dislocation density, mustalso bemeasured quantitatively if they are to have any predictive valuewithin the frameworkof a useful theory. A classic example is the Petch relationshipwhich links the yield strength ofa steel sy to its grain size D [see Equation (1.1)].

In many cases there is a chasm of uncertainty between the qualitative microstructuralobservations and their association with a predicted or measured material property. To bridgethis chasm we need to build two support piers. To construct the first we note that many of theengineering properties of interest are stochastic in nature, rather than deterministic. That is,the property is not single-valued, but is best described by a probability function.We know thatnot all men are 180 cm tall, nor do they all weigh 70kg. Rather, there are empirical functionsdescribing the probability that an individual taken fromanywell-defined populationwill havea height or weight that falls within a set interval. Clearly, if an engineering property isstochastic in nature, then it is likely that the material parameters which determine thatproperty will also be stochastic, so that we need to determine the statistical distribution of theparameters used to define these material properties.

The second support pier needed to build our bridge is on the microstructural side of thechasm. It is constructed fromour understanding and knowledge of themechanisms of imageformation and the origins of image contrast, in combination with a quantitative analysis ofthe spatial relationship between microstructural image observations and the bulk structureof the material. This geometry-based, quantitative analysis of microstructure is termedstereology, the science of spatial relationships, and is an important component of this book(see Chapter 9). For the time being, we will only consider the significance of some termswhich are used to define a few common microstructural parameters.

1.1.3.1 Grain Size. Most engineering materials, especially structural materials, arepolycrystalline, that is, they consist of a three-dimensional assembly of individual grains,each ofwhich is a single crystalwhose crystal lattice orientation in space differs from that ofits neighbours. The size and shape of these individual grains are as varied as the grains ofsand on the seashore. If we imagine the polycrystalline aggregate separated out into theseindividual grains, we might legitimately choose to define the grain size as the averageseparation of two parallel tangent planes that touch the surfaces of any randomly orientedgrain. This definition is termed the caliper diameterDC, and it is rather difficult to measure(Figure 1.13). We could also imagine counting the number of grains extracted from a unitvolume of the sample, NV, and then defining an average grain size as DV¼NV

�1/3. Thisdefinition is unambiguous and independent of any anisotropy or inhomogeneity in thematerial. However, most samples for microstructural observation are prepared by taking a


planar section through the solid, three-dimensionalmicrostructure, so it ismore practical tocount the number of grains that have been revealed by etching the polished surface of a two-dimensional section through the sample, and then transform the number of grains inter-cepted per unit area of the section NA into an average �grain size� by writing DA¼NA

�1/2.This is a very common procedure for determining grain size, and may well be the preferredmeasure. However, we could also lay down a set of lines on our �random� polished-and-etched section, and count the number of intercepts which these �test� lines make with thegrain boundary traces seen on the section.

In general, samples are usually sectioned on planes selected in accordance with thespecimen geometry, for example, parallel or perpendicular to a rolling plane, and test linesare also usually drawn parallel or perpendicular to specific directions, such as the rolling

Figure 1.13 Grain size may be defined in several ways that are not directly related to oneanother, for example the mean caliper diameter, the average section diameter on a planarsection, or the average intercept length along a random line.


direction of ametal sheet. However, if both the test line on the surface and the sample sectionitself are truly random, then the test line is a random intercept of the boundary array presentin the bulk material. The number of intercepts per unit length of a test line NL gives us yetanother measure of the grain size, DL the mean linear intercept, where DL¼NL

�1, and isalso a commonly accepted definition for grain size (with or without some factor ofproportionality).

However, this is not quite the end of the story. In industrial quality control a totallydifferent, qualitative �measure� of grain size is commonly used. The sample microstructureis compared with a set of standardmicrostructures (ASTMGrain-size Charts) and assignedan ASTM (American Society for Testing Materials) grain size DASTM on the basis of theobserver�s visual judgement of the �best-fit�with the ASTM chart. This is of course a rathersubjective measure of grain size that depends heavily on the experience of the observer.

What happens if the grains are not of uniform size?Well, any section through the samplewill cut through an individual grain at a position determined by the distance of the centre ofgravity of the grain from the plane of the section, so that grains of identical volumemay havedifferent intercept areas on the plane of the section, depending on how far this plane liesfrom their centre of gravity. This distribution of intercept areas will be convoluted with the�true� grain size distribution, and it is difficult (but not impossible) to derive the grain sizedistribution in the bulkmaterial from an observed distribution of the areas intercepted on thepolished section. If the grains are also elongated (as may happen if the grain boundaryenergy depends on orientation) or if elongated grains are partially aligned (as will be thecase for a plastically deformed ductile metal or for a sample that has been cast from themeltin a temperature gradient), then we shall have to think very carefully indeed about thesignificance of any grain size parameter that has been measured from a planar surfacesection. These problems are treated in more detail in Chapter 9.

1.1.3.2 Dislocations and Dislocation Density. Dislocations control many of the me-chanical properties of engineering materials. A dislocation is a line defect in the crystallattice which generates a local elastic strain field. Dislocations may interact with the freesurface, dispersed particles and internal interfaces (such as grain boundaries), as well aswith each other. Each dislocation line is characterized by a displacement vector, the Burgersvector, which defines the magnitude of the elastic strain field around the dislocation. The anglebetween the Burgers vector and the dislocation line in large part determines the nature of thedislocation strain field, that is the shear, tensile and compressive displacements of the atomsfrom their equilibriumpositions.ABurgers vector parallel to thedislocation linedefines a screwdislocation, while a Burgers vector perpendicular to the dislocation line defines an edgedislocation. All other configurations are referred to as mixed dislocations.

In a thin-film specimen imaged by TEM, the Burgers vector and line-sense of thedislocations can usually be determined unambiguously from the diffraction contrastgenerated in the region of the dislocation by the interaction of the dislocation strain fieldwith the electron beam (Section 4.4.5). However, the determination of the dislocationdensity is often ambiguous. A good theoretical definition of the dislocation density is �thetotal line length of the dislocations per unit volume of the sample�. This definition isunambiguous and independent of the dislocation distribution, and the dislocations mayinteract or be aligned along particular crystallographic directions. However, this definitiondoes not make any allowance for families of dislocations that have different Burgers


vectors. This may not matter if the dislocation population is dominated by one specific typeof Burgers vector, as is commonly the case in cold-worked metals, in which the slipdislocations are usually of only one type. However, it may matter a great deal whenassessing the residual dislocation content in semiconductor single crystals. Moreover, sinceit is the strain fields of the dislocations that are imaged, and contrast is only observed if thestrain field has a component perpendicular to the diffracting planes, dislocations may�disappear� from the image under certain diffraction imaging conditions.

A further problem arises when we seek to extend our definition of the term dislocation.For example, when dislocations interact they may form low energy dislocation networks(Figure 1.14), that separate regions of the crystal having slightly different orientations inspace. The dislocation network is thus a sub-grain boundary. Are we to include the array ofdislocations in the sub-boundary in our count of dislocation density or not? Plasticdeformation frequently leads to the formation of dislocation tangles which form cellstructures, and within the cell walls it is usually impossible to resolve the individualdislocations (Figure 1.15). In addition, small dislocation loops may be formed by thecollapse of point defect clusters that often result from plastic deformation, quenching thesample from high temperatures or radiation damage in a nuclear reactor (Figure 1.16).Should these small loops also be counted as dislocations?

A common alternative definition of dislocation density is �the number of dislocationintersections per unit area of a planar section�. In an anisotropic sample this definitionwouldbe expected to give dislocation densities that show a dependence on the plane of the section,so that the two definitions are not equivalent. Furthermore, a dislocation density determinedfrom observations made using one counting method need not agree with that derived fromother measurements, no matter how carefully the dislocation density is defined. The spatialresolution may differ, and the method of specimen preparation could also affect the results,for example, by permitting dislocations to glide out to the surface and annihilate.

Figure 1.14 A lowenergyarray of dislocations forms a dislocation networkwhich constitutes asub-boundary in the crystal that can interact with slip dislocations, as shown here. Suchboundaries separate the crystal into sub-grains of slightly different orientation.


Figure 1.15 Plastic deformation of a ductile metal often results in poorly resolved dislocationtangles which form cells within the grains.

Figure 1.16 Dark-field transmission electron micrograph taken with a g¼ (200) reflectionshowing small defects due to 100 kV Kr ion irradiation of Cu at 294 K. Reproduced withpermission from R.C. Birtcher, M.A. Kirk, K. Furuya, G.R. Lumpkin and M.O. Ruault, in situTransmission Electron Microscopy Investigation of Radiation Effects, Journal of MaterialsResearch, 20(7), 1654–1683, 2005. Copyright (2005), with permission from MaterialsResearch Society.

1.1.3.3 Phase Volume Fraction. Many engineering materials contain more than onephase, and the size, shape and distribution of a second phase are often dominant factors indetermining the effect of the second phase on the properties. As with grain size, when thesecond phase is present as individual particles there are a number of nonequivalent optionsfor defining the particle size, shape and spatial distribution. These definitions are for themost part analogous to those for grain size, discussed above. However, in many cases thesecond phase forms a continuous, interpenetrating network within the primary phase, inwhich both interphase boundaries and grain boundaries are present. If the secondary phaseparticles have shapes that are in any way convoluted (with regions of both positive andnegative curvature at the boundaries), then the same particle may intersect the surface of asection more than once, making it very difficult to estimate just how many second phaseparticles are present. However, there is onemicrostructural parameter that is independent ofboth the scale, shape and the distribution of the second phase, and that is the phase volumefraction fV. Since this is both a shape and scale-independent parameter, for crystallinephases fV can be determined both conveniently and quickly from diffraction data (see theWorked Examples of Chapter 2), while local values of the phase volume fraction can beextracted from images of a planar section (Figure 1.17). On a random section, the volumefraction of the second phase can be estimated from the areal fraction of the second phaseintercepted by the section,A/A0. At the dawn of metallography (well over 100 years ago!) itwas even accepted practice to cut out the regions of interest from a photographic image andthen weigh them relative to the �weight� of the total area sampled. This areal estimate isactually equivalent to a lineal estimate, determined from a random line placed across theplane of the section. Providing the line and section are both �random�, the length of the linetraversing the second phase relative to the total length of test line L/L0 is also an estimate ofthe phase volume fraction. Finally, a random grid of test points on the sample section canalso provide the same information: the number of points falling on regions of the secondphase divided by the total number of test points P/P0 again estimates the volume of thesecond phase relative to the total volume of the sample V/V0. Thus, for the case of trulyrandom sampling: fV¼A / A0¼ L / L0¼P/P0¼V / V0. For anisotropic samples, these rela-tions do not hold, but useful information on the extent of the microstructural anisotropy canstill be determined by comparing results obtained from different sample sections anddirections in the material. Porous materials can also be analysed as though they contained a�second phase�, although precautions have to be taken when sectioning and polishing aporous sample if artifacts associated with loss of solid material and rounding of the poreedges is to be avoided. It is always a good idea to compare microstructural observations ofporous materials with data on pore size and volume fraction determined from physicalmodels using measurements based on density, porisometry or gas adsorption.

1.2 Crystallography and Crystal Structure

The arrangements of the atoms in engineering materials are determined by the chemicalbonding forces. Some degree of order at the atomic level is always present in solids, even inwhat appears to be a featureless, structureless glass or polymer. In what follows we willbriefly review the nature of the chemical forces and outline the ways in which thesechemical forces are related to the engineering properties. We will then discuss some of the


crystallographic tools needed to describe and understand the commonly observed atomicarrangements in ordered, crystalline solids. The body of knowledge that describes andcharacterizes the structure of crystals is termed crystallography.

1.2.1 Interatomic Bonding in Solids

It is a convenient assumption that atoms in solids are packed together much as one wouldpack table tennis balls into a box. The atoms (or, if they carry an electrical charge, the ions)are assumed to be spherical, and to have a diameter which depends on their atomic number

Figure 1.17 The volume fractionof a second phase can be determined from the areal fraction ofthe phase, seen on a random planar section, or from the fractional length of a random test linewhich intercepts the second phase particles in the section, or from the fraction of points in a testarray which falls within the regions of the second phase.


(the number of electrons surrounding the nucleus), their electrical charge (positive, ifelectrons have been removed to form a cation, or negative if additional electrons have beencaptured to form an anion), and, to a much lesser extent, the number of neighbouring atomssurrounding the atom being considered (the coordination number of the atom or ion).

1.2.1.1 Ionic Bonding. In an ionically bonded solid the outer, valency electron shells ofthe atoms are completed or emptied by either accepting or donating electrons. In cookingsalt, NaCl, the sodium atom donates an electron to the chlorine atom to form an ion pair(a positively charged sodium cation and a negatively charged chlorine anion). Both cationsand anions now have stable outer electron shells, the chlorine ion having a full complementof eight electrons in the outer shell, and the sodium ion having donated its lone excesselectron to the chlorine atom.

Tominimize the electrostatic energy, the negative electrical charge on the cationsmust besurrounded by positively charged anion neighbours and vice versa. At the same time, theouter electron shell of the anions will contract towards the positively charged nucleus as aresult of the excess positive charge on the nucleus, while the excess negative charge on theanions will cause a net expansion of the outer electron shell of the anions. For the case ofNaCl, andmany other ionic crystals, the larger anions form an ordered (and closely packed)array, while the smaller cations occupy the interstices. Two opposing factors will determinethe number of neighbours of opposite charge that surrounds a given ion (the coordinationnumber). First, the electrostatic (Coulombic) attraction between ions of opposite chargetends to maximize the density of the ionic array. At the same time, in order to keepneighbouring ions of similar charge separated, the smaller ion must be larger than theinterstices which it occupies in the packing of the larger ion. The smallest possible numberof neighbours of opposite charge is 3, and boron (Z¼ 5) is a small, highly charged cationwhich often has this low coordination number. Coordination numbers of 4, 6 and 8 are foundfor steadily increasing ratios of the two ionic radii (Figure 1.18), while the maximum

3 4 6

8 12

Figure 1.18 The number of neighbours of an ion, its coordination number, is primarilydetermined by the ratio of the radii of the smaller to the larger ion. The regular coordinationpolyhedra allow for 3, 4, 6, 8 or 12 nearest neighbours.


coordination number, 12, corresponds to cations and anions having approximately the samesize. The anion is not always the larger of the two ions, andwhen the cation has a sufficientlylarge atomic number, it may be the anions that occupy the interstices in a cation array.Zirconia, ZrO2, is a good example of packing anions into a cation array.

Silicate structures and glasses are also dominated by ionic bonding, but in this case thetightly coordinated cations form amolecular ion,most notably the SiO4 silicate tetrahedron.These tetrahedra may carry a negative charge (as in magnesium silicate, Mg2SiO4), or theymay be covalently linked to forma poly-ion, inwhich the coordination tetrahedra share theircorner oxygen atoms (as in quartz) to form oxygen bridges, –(Si–O–Si)–. Borates,phosphates and sulfates can form similar structures, but it is the silicates that dominatein engineering importance. Since only the corners of a silicate tetrahedron may be shared,and not the edges or faces, the silicates form very open, low density structures which canaccommodate awide variety of other cations. In addition, the oxygen corner linkage is quiteflexible, and allows any two linked tetrahedra considerable freedom to change their relativeorientation. The average negative charge on the silicate ion in a glass varies inversely withthe number of oxygen bridges, and this charge is neutralized by the presence of additionalcations (known as modifiers) that occupy the interstices between the tetrahedra. Thetetrahedra themselves do not readily change their dimensions, although some substitutionof the Si4þ ion can occur (most notably byB3þ or Al3þ). Since the oxygen bridges constrainthe distance between neighbouring tetrahedra, the glasses possess well-defined short-rangeorder that may extend up to 2 nm from the centres of the silicate tetrahedra.

1.2.1.2 Covalent Bonding. Many important engineeringmaterials are based on chemicalbonding in which neighbouring atoms share electrons that occupy molecular orbitals. Indiamond. (Figure 1.19), the carbon atoms all have four valency electrons, and by sharing

Figure 1.19 In covalently bonded diamond the carbon atoms are tetrahedrally coordinated toeach of their nearest neighbours by shared molecular orbitals.


each of these with four neighbouring carbon atoms, each atom acquires a full complementof eight electrons for the outer valency shell. It is the strong C�C covalent bond whichensures the chemical stability, not only of diamond, but also of most polymer molecules,which are constructed from covalently linked chains of carbon atoms. The oxygen bridgesin silicate glasses are also, to a large extent, covalent bonds, and the same oxygen bridgesprovide the chain linkage in silicone polymers, �(O�SiHR)�.

It is not always possible to describe a bond as simply covalent or ionic. Consider the seriesNaCl, MgO, AlN, SiC, in which the number of electrons participating in the bond betweenindividual cations and anions increases fromone to four. The first two compounds in the seriesare commonly described as ionic solids. The third, aluminiumnitride, could also be describedas ionically bonded, although the effective charge on the ions is appreciably less than thatpredicted by their trivalent nature. The fourth compound, silicon carbide is tetrahedrallycoordinated, as is diamond.Since both constituents are in the samegroup of the periodic table,one might guess that it is covalently bonded. However, this is not entirely correct, and in factthe physical properties are best simulated by assuming that the silicon atoms still carry somepositive charge, while the carbon atoms carry a corresponding negative charge. In fact, as thevalency increases, so does the contribution to the bond strength from covalent bonding.

1.2.1.3 Metals and Semiconductors. Valence electrons may be shared, not only with anearest neighbour atom, but quite generally, throughout the solid. That is, the molecularorbitals of the electronsmay not be localized to a specific pair of atoms. Electrons which arefree to move throughout the solid are said to occupy a conduction band, and to be freeelectrons. The chemical bonding in such a solid is termed metallic bonding, and ischaracterized by a balance between two opposing forces: the coulombic attraction betweenthe free electrons and the array of positively charged, metallic cations, and the repulsiveforces between the closed shells of these cations. The properties typical of themetallic bondare associated with the mobility of the free electrons (especially the high thermal andelectrical conductivity, and the optical reflectivity), and with the nondirectionality of thisbond (for example, mechanical plasticity or ductility).

In some cases, only small numbers of electrons may be present in the conduction band ofa solid, either as a result of thermal excitation or due to the presence of impurities. Suchmaterials are termed semiconductors, and they play a key role in the manufacture ofelectronic devices for the electronics industry. If electrons are thermally excited to occupy aconduction band, then they leave behind vacant holes, which may also be mobile.Semiconductors in which the negatively charged electrons are the dominant electricalcurrent carriers are termed n-type, while those in which the positively charged holes areresponsible for the electronic properties are termed p-type.

If certain impurities are present in very low concentrations, then the electrons may not befree to move throughout the solid, and cannot confer electrical conductivity, but they willnevertheless occupy localized states. Many cation impurities in ceramics give rise tolocalized states that are easily excited and strongly absorb visible light. They are then said toform colour centres. In precious and semi-precious jewels small quantities of cationimpurities are dissolved in the single crystal jewel stone and impart the characteristic colourtones, for example, chromium in ruby. Such cation additions are also ofmajor importance inthe ceramics industry, and the effects may be either deleterious (discoloration) or advanta-geous (a variety of attractive enamels and glazes, Figure 1.20). Irradiation of transparent,


nonconducting solids creates large concentrations of point defects in the material. Suchradiation damage is also often a cause of colour centres.

1.2.1.4 Polarization Forces. In addition to the three types of chemical bonding discussedabove, many of the properties of engineering solids are determined by secondary, or van derWaals bonding, that is associated withmolecular polarization forces. In its weakest form, thepolarization force arises from the polarizability of an electron orbital. Rare gas atoms willliquefy (and solidify) at cryogenic temperatures as a consequence of the small reduction inpotential energy achieved by polarization of an otherwise symmetrical electron orbital.Manymolecular gases (H2, N2, O2, CH4) behave similarly. The properties of a number ofengineering polymers are dominated by the polarizability of themolecular chain, for examplepolyethylene,�(CH2�CH2)�. The ductility of the polymer, aswell as its softening point andglass transition temperature, are determined by a combination of the molecular weight of the

Figure 1.20 The colour and texture of a porcelain glaze depends both on the presence ofdispersed pigment particles and controlled amounts of impurity cations that introduce colourcentres into the silicate glass. These colour centres may also nucleate localized crystallization.The illustration shows a glass jar from the time of the Roman Empire. Reproduced withpermission of the Corning Museum of Glass. (See colour plate section)


polymer chains and their polarizability. This weak bonding is quite sufficient to ensure thatengineering components manufactured from polymers are mechanically stable, and, undersuitable circumstances, these high molecular weight polymers may partially crystallize.

Stronger polarization forces exist when the molecular species has a lower symmetry andpossesses a permanent dipole moment. A good example is carbon dioxide (CO2) but similarmolecular groupings are also often present in high performance engineering polymers.Organic tissues are largely constructed from giant polar moleculeswith properties dictatedby a combination of the molecular configuration and the position of the polar groups withinthe molecule.

The strongest polarization forces are associated with a dipole moment due to hydrogen,namely the hydrogen bond. Hydrogen in its ionized form is a proton, with no electrons toscreen the nucleus. The ionic radius of hydrogen is therefore the smallest possible, andasymmetric molecular groupings which contain hydrogen can have very high dipolemoments. The two compounds that demonstrate this best are water and ammonia, (H2Oand NH3) and the corresponding molecular groups found in engineering polymers are�(OH) and�(NH2). These groups raise both the tensile strength and the softening point ofthe polymer. The families of polyamides and polyamines (which include the �nylons�,actually a commercial trade name) depend for their strength and stiffness on the stronghydrogen bonding between the polymer chains.

1.2.2 Crystalline and Amorphous Phases

We have outlined the nature of the chemical bonding found in engineering solids andliquids, and the atomic coordination requirements associated with this bonding. In someengineering solids the local atomic packing results in long-range order and the material istermed crystalline, while in others short-range atomic order has no long-range conse-quences, and the (�liquid-like�) material is termed amorphous or glassy.

Single phase polycrystallinematerials aremade up ofmany small crystals or grains. Eachgrain has identical atomic packing to that of its neighbours, although the neighbouringgrains are not in the same relative crystal orientation. In polyphasematerials all the grains ofeach individual phase have the same atomic packing, but this packing generally differs fromthat of the other phases present in thematerial. In thermodynamic equilibrium, the grains ofeach phase also have a unique and fixed composition that depends on the temperature andcomposition of the material, and can usually be determined from the appropriate phasediagram. In general, the solid phases in engineering materials may be either crystalline oramorphous. Amorphous phases may form by several quite distinct routes: rapid coolingfrom the liquid phase, condensation from the gaseous phase, or as the result of a chemicalreaction. A good example of amorphous phases produced by chemical reaction are thehighly protective oxide films formed on aluminium alloy and stainless steel components bysurface reactions in air at room temperature.

1.2.3 The Crystal Lattice

Thewell-developed facets readily observed onmany naturally occurring crystals, as well ason many ionic crystals grown from aqueous solution, prompted the development of thescience of crystallography in the latter part of the nineteenth century. An analysis of the


angles between crystal facets permitted an exact description of the symmetry elements of acrystal, and led to speculation that crystal symmetry was a property of the bulk material.Thiswas finally confirmedwith the discovery, at the beginning of the twentieth century, thatsmall, single crystals would strongly diffract X-rays at very specific, fixed angles, to give asharp diffraction pattern that was characteristic of any crystal of the same material when itwas oriented in the same relation to the incident X-ray beam, irrespective of the crystal sizeand shape.

The interpretation of these sharp X-ray diffraction maxima in terms of a completelyordered and regular atomic array of the chemical constituents of the crystal followed almostimmediately, being pioneered by the father and son team of Lawrence and William Bragg.The concept of the crystal latticewas an integral part of this interpretation. The atoms in acrystal were centred at discrete, essentially fixed distances from one another, and theseinteratomic separations constituted an array of lattice vectors that could be defined in termsof an elementary unit of volume, the unit cell for the crystal, that displayed all the symmetryelements characteristic of the bulk crystal.

1.2.3.1 Unit Cells and Point Lattices. This section introduces basic crystallography, anddescribes how a crystalline structure is interpreted and how crystallographic data can beretrieved from the literature. To understand the structure of crystals, it is convenient initiallyto ignore the positions of the atoms, and just concentrate on the periodicity, using a three-dimensional periodic scaffold within which the atoms can be positioned. Such a scaffold istermed a crystal lattice, and is defined as a set of periodic points in space.A single lattice cell(the unit cell) is a parallelepiped, and the unit cells can be packed periodically by integerdisplacements of the unit cell parameters. The unit cell parameters are the three coordinatelengths (a, b, c) determined by placing the origin of the coordinate system at a lattice point,and the three angles (a, b, g) subtended by the lattice cell axes (Figure 1.21). Thus a unit cellwith a¼ b¼ c and a¼ b¼ g¼ 90� is a cube. The various unit cells are generated from thedifferent values of a, b, c and a,b, g.An analysis of how to fill space periodically with latticepoints shows that only seven different unit cells are required to describe all the possible

a

b

c

αβ

γ

Figure 1.21 Schematic representation of the lattice parameters of a unit cell.


point lattices, and these are termed the seven crystal systems. These crystal systems are, inorder of increasing crystal symmetry: triclinic, monoclinic, orthorhombic, tetragonal,rhombohedral, hexagonal, and cubic.

These seven crystal systems are each defined using primitive unit cells, in which eachprimitive cell only contains a single lattice point placed at the origin of the unit cell.However,more complicated point lattice symmetries are possible, each requiring that everylattice point should have identical surroundings. These permutations were first analysed bythe French crystallographer Bravais in 1848, who described the possible (14) point lattices(the Bravais lattices), which are shown in Figure 1.22.

1.2.3.2 Space Groups. In real crystals each individual lattice point actually representseither a group of atoms or a single atom, and it is these atoms that are packed into the crystal.Various degrees of symmetry are possible in this periodic packing of the atoms and atomgroups. For example, if a crystal is built of individual atoms located only at the lattice points

Figure 1.22 The 14 Bravais lattices derived from the seven crystal systems.


of a unit cell, then it will have the highest possible symmetry within that particular crystalsystem. This symmetry will then be retained if there are symmetrical groups of atomsassociated with each lattice point. However, the atomic groups around a lattice point mightalso pack with a lower symmetry, reducing the symmetry of the crystal, so that it belongs toa different symmetry group within the same crystal class.

Combining the symmetries of the atom groups associated with a lattice point with that ofthe Bravais lattices leads to the definition of space groups, which provide criteria for fillingthe Bravais point lattices with atoms and groups of atoms in a periodic array. It has beenfound that there are a total of 230 different periodic space groups, and that the structure of acrystal can always be described by one (or more) of these space groups. This has beenfound to be the most convenient way to visualize any complex crystal structure.

Let us examine the use of space groups to define a crystal structure using a simpleexample. Assume we wish to know the positions of all the atoms in a copper (Cu) crystal.First we need a literature source which contains the crystallographic data. For materialsscience we use Pearson�s Handbook of Crystallographic Data for Intermetallic Phases(the name is misleading, since pure metals and ceramics are also included). The principledata listed in the handbook for Cu appear in the format shown in Table 1.3.

Following the name of the phase, a structure type is given. This is the name of a realmaterial that serves as an example for this particular crystallographic structure, and in thisexample copper is its own structure type. Next the Pearson symbol and space group arelisted, which refer to the type of lattice cell and the symmetry of the structure. Here, thePearson symbol, cF4, means a cubic (c) face-centred lattice (F) with four atoms per unitcell (that is, one atom for each lattice point, in this case). The symmetry descriptionFm�3mis in this case also the name of the space group and is followed by the lattice parameters.For our example, Cu, a¼ b¼ c¼ 0.36148 nm, and a¼ b¼ g¼ 90�. Since a cubic structureclearly has a¼ b¼ g¼ 90�, these values are not listed. Finally theWyckoff generating sitesare given. These are the sites of specific atoms within the crystal structure, upon whichthe space group symmetry operators act. When combined with the space group, thesymmetry operators will generate the positions of all atoms within the unit cell and specifythe �occupancy� or occupation factor. For Cu, x¼ 000, y¼ 000, z¼ 000 and Occ¼ 100.Hence, x¼ y¼ z¼ 0.0 and the occupancy is 1.00. The last term, the occupancy,indicates the probability that a site is occupied by a particular atom species. In thecase of Cu, all the sites (neglecting vacancy point defects) are occupied by Cu, so theoccupancy is 1.00.

Howdowegenerate the crystal structure?We first need details of the symmetry operatorsfor the space group Fm�3m. These can be found in the International Tables For

Table 1.3 Crystallographic data for Cu

PhaseStructuretype

Pearsonsymbolspace

and groupa, b, c(nm)

a, b, g(�) Atoms

Pointset x y z Occ

Cu Cu cF4 Fm�3m 0.36148 Cu 4a 000 000 000 100


Figure 1.23 The space group Fm�3m which defines the symmetry for Cu. Reproduced bypermission of Kluwer Academic Publishers from International Tables for Crystallography,Volume A, Space Group Symmetry, T. Hahn, ed. (1992).


Crystallography, Volume A, Space Group Symmetry. An example is given in Figure 1.23.The data in the tables provide all the symmetry operators for any specific space group. Inorder to generate atomic positions from the generating site data listed for Cu, we add the (x,y, z) values of the generating site to the values listed in the tables. Returning to our example,Cu has a generating site of type 4a with (x,y,z) equal to (0,0,0). In the tables, the Wyckoffgenerating site for 4a has an operator of (0,0,0). This is our first atom site. Now we activatethe general operators which are also listed. The addition of (0,0,0) to our initial generatingsite of (0,0,0) leaves one atom at the origin, while the addition of (0,1/2,1/2), (1/2,0,1/2) and(1/2,1/2,0) to (0,0,0) places three more atoms, one at the centre of each of the faces of theunit cell adjoining the origin. There are therefore four Cu atoms in our unit cell, as shownschematically in Figure 1.24(a).

Figure 1.24(a) does not look like the �complete� FCC structure sketched in mostelementary texts, since only those atoms which �belong� to a single unit cell are shown.Additional atoms at sites (1,0,0), (1,1,0), (0,1,0), (1,0,1), (0,1,1), (0,0,1), (1,1,1), (1,1/2,1/2),(1/2,1,1/2) and (1/2,1/2,1) actually belong to the neighboring unit cells in the crystal lattice.

Figure 1.24 Schematic drawing of (a) the unit cell of Cu and (b) the same unit cell, but withadditional atoms from neighbouring unit cells in order to demonstrate the face-centred cubic(FCC) packing.


Amore commonly accepted (although nomore correct) drawing of the unit cell for a coppercrystal is given in Figure 1.24(b).

We need a clear definition to decide if an atom �belongs� in a specific unit cell: if0� x< 1, 0� y< 1, and 0� z< 1, then we define the atom as �belonging� to the unit cell. Ifnot, then the atom belongs to a neighbouring cell. This provides a very easymethod to countthe number of atoms per cell.

Why all this effort just to define four atoms in an FCC configuration? For this simplestructure, the formalism is not strictly necessary, but for a structure with 92 atoms per unitcell, use of these tables is the easiest and least error-prone way to define the atomicpositions. It is also convenient when computer-simulating crystal structures or diffractionspectra: instead of having to type in the positions of all the atoms in the unit cell, you can justuse the space group operations.

Now consider an example in which the use of occupation factors is important. ACu�Nisolid solution (an alloy) with 50 atom%Ni. Both Cu and Ni have the simple FCC structure,with complete solid solubility over the entire composition range. The description of thestructure according to Pearson would be very similar to that of Cu, but instead of onegenerating site there are now two, one for nickel and one for copper but bothwith the same x,y, and z values, and each having an occupation factor of 0.5, corresponding to the bulk alloyconcentration of 50 atom%Ni. Thus each of the four sites in the cell is occupied by both Cuand Ni each with the same probability of occupancy, 0.5. Other than some slight changes inthe lattice parameters, this is the only difference between pure Cu and the Cu�Ni randomsolid solution.

1.2.3.3 Miller Indices and Unit Vectors. Crystal planes and crystal directions in thelattice are described by a vector notation based on a coordinate system defined by theaxes and dimensions of the unit cell. A direction is defined by a vector whose origin liesat the origin of the coordinate system and whose length is sufficient to ensure that the x, yand z coordinates of the tip of the vector all correspond to an integer number of unit cellcoordinates (Figure 1.25). The length of such a rational vector always corresponds to an

nc=w

nb=v

na=u

x

y

z

Figure 1.25 The definition of direction indices in a crystal lattice.


interatomic repeat distance in the lattice. In a crystal, two or more lattice directions maybe geometrically equivalent, and it is sometimes useful to distinguish a family of crystaldirections. For example, in a cubic crystal the x-axis is defined by the direction [100] (insquare brackets), but the y and z directions, [010] and [001] are, by symmetry,geometrically equivalent. Angular brackets are used as a shorthand for the family ofh100i directions. Of course, in lower symmetry crystals the two directions [100] and[010] may not be equivalent (that is a 6¼ b in the unit cell), and these two directions donot then belong to the same family. Note that all directions which are parallel in thecrystal lattice are considered equivalent, regardless of their point of origin, and aredenoted by the same direction indices [uvw]. However, a negative index is perfectlylegitimate, ½�uvw� 6¼ ½uvw� and indicates that the u coordinate is negative. Since thedirection indices define the shortest repeat distance in the lattice along the line of thevector, they cannot possess a common factor. It follows that the direction indices [422]and [330] should be written [211] and [110], respectively. Finally, the direction indicesall have the dimension of length, the unit of length being defined by the dimensions ofthe unit cell.

Crystal planes are described in terms of the reciprocal of the intercepts of the plane withthe axes of a coordinate system that is defined by the unit cell (Figure 1.26). If the unit cellparameters are a, b and c, and the crystal plane makes intercepts x*, y* and z* along the axesdefined by this unit cell, then the planar indices, termed Miller indices, are (h¼ na/x*,k¼ nb/y*, l¼ nc/z*), where the integer n is chosen to clear the indices (hkl) of fractions. Forexample, a cube plane only intersects one of the axes of a cubic crystal, so two of the valuesx*, y* and z* must be1 and the Miller indices must be one of the three possibilities (100),

x

y

z

y*=nb/k

z*=nc/l

x*=na/h

Figure 1.26 The definition ofMiller indices describing the orientation of a crystal planewithinthe unit cell of the crystal.


(010) and (001), (always given in round brackets). All planeswhich are parallel in the latticeare described by the same indices, irrespective of the intercepts they make with thecoordinate axes (although different values of n will be required to clear the fractions).Reversing the sign of all three Miller indices does not define a new plane. That isðhklÞ � ðhklÞ. However, ðhklÞ 6¼ ð�hklÞ 6¼ ð�hk�lÞ, and these indices refer to three crystallo-graphically distinct, nonparallel planes in the lattice. While the letters [uvw], with squarebrackets, are used to define a set of parallel direction indices, the letters (hkl), with roundbrackets, are used to define theMiller indices of a parallel set of lattice planes. If a family ofgeometrically equivalent (but nonparallel) planes is intended, then this can be indicated bycurly brackets; that is {hkl}. Note that the dimensions of the Miller indices are those ofinverse length, and that the units are the inverse dimensions of the unit cell. Unlike thedirection indices, Miller indices having a common factor do have a specific meaning: theyrefer to fractional values of the interplanar spacing in the unit cell. Thus the indices (422) aredivisible by 2, and correspond to planes which are parallel to, but have just half the spacingof the (211) planes.

In cubic crystals, but only in cubic crystals, any set of direction indices is always normalto the crystal planes having the same set of Miller indices. That is, the [123] direction in acrystal of cubic symmetry is normal to the (123) plane. This is not generally true in lesssymmetric crystals, even though itmay be true for some symmetry directions. If a number ofcrystallographically distinct crystal planes intersect along a common direction [uvw], thenthat shared direction is said to be the zone axis of these planes, and the planes are said to lieon a common zone. There is a simple way of finding out whether or not a particular plane(hkl) lies on a given zone [uvw]. If it does, then huþ kvþ lw¼ 0. This is true for all crystalsymmetries.

1.2.3.4 The Stereographic Projection. It is a great convenience to be able to plot theprominent crystal planes and directions in a crystal on a two-dimensional projection, similarto the projections familiar to us from geographical mapping. By far the most useful of theseis the stereographic projection. In the stereographic projection the crystal is imagined to bepositioned at the centre of a sphere, the projection sphere, and the crystal directions andnormals to the prominent crystal planes are projected from the centre of this sphere (thecentre of the crystal) to intersect its surface (Figure 1.27). Straight lines are then drawn fromthe south pole of the projection sphere, through the points of intersection of thesecrystallographic directions and crystal plane normals with the sphere surface, until thelines intersect a plane placed tangential to the sphere at its north pole. All points around theequator of the sphere now project onto the tangent plane as a circle of radius equal to thediameter of the projection sphere. All points on the projection sphere that lie in the northernhemisphere will project within this circle, which is termed the stereogram. Points lying inthe southern hemisphere will project outside the circle of the stereogram, but by reversingthe direction of projection (from the north pole to a plane tangential to the south pole) wecan also represent points lying in the southern hemisphere (but using an open circle in orderto distinguish the southern hemisphere points). This avoids having to plot any points outsidethe area of the stereogram.

Any such two-dimensional plot of the crystal plane normals and crystal directionsconstitutes a stereographic projection. It is conventional to choose a prominent symmetryplane (usually one face of the unit cell) for the plane of these stereographic projections.


Figure 1.28 shows the stereographic projection for the least symmetrical, triclinic crystalsystem, with the axes of the unit cell [100], [010] and [001] and the faces of the unit cell(100), (010) and (001) plotted on an (001) projection. The (001) plane contains the [100] and[010] directions, while the [001] zone contains the normals to the (100) and (010) planes.However, as noted above, the crystal directions for this very low symmetry crystal do notcoincide with the plane normals having the same indices.

A stereogram (stereographic projection) for a cubic crystal, with the plane of theprojection parallel to a cube plane is shown in Figure 1.29. Plane normals and crystaldirections with the same indices now coincide, as do the plots of crystal planes and thecorresponding zones. The high symmetry of the cubic systemdivides the stereogram into 24geometrically equivalent unit spherical triangles that are projected onto the stereogramfrom the surface of the projection sphere (we ignore the southern hemisphere, sincereversing the sign of a plane normal does not change the crystal plane). Each of these unit

Figure 1.27 Derivation of the stereographic projection, a two-dimensional representation ofthe angular relationships between crystal planes and directions.


triangles possesses all the symmetry elements of the cubic crystal, and is bounded by thetraces of one plane from each of the families {100}, {110} and {111}.

The zones defined by coplanar directions and plane normals pass through the centre ofthe projection sphere (by definition) and intercept this sphere along circles which havethe diameter of the projection sphere. These circles then project onto the stereogram astraces of larger circles, termed great circles, whose maximum curvature is equal to thatof the stereogram. The minimum curvature of a great circle is zero (that is, it projects as astraight line), and so straight lines on the stereogram correspond to planes whose tracespass through the centre of the stereographic projection. The bounding edges of any unitspherical triangle that defines the symmetry elements of a crystal are always greatcircles.

Another property of the stereographic projection is that the cone of directions that makea fixed angle to any given crystallographic direction or crystal plane normal also projectsas a circle on the plane of the stereogram. Such circles are termed small circles (but beware,

[010]

[101]

[100]

(100)

[011]

[110]

(101)

(111)

(110)

(010)(001) (011)

[111]

(111)

(011)

[110]

[111]

[011]

(111)

[001]

[111] [101]

(111)(101)

[111]

[100]

[110]

[110]

(110)

[010]

α

β

γ

Figure 1.28 An (001) stereographic projection of the lattice of a triclinic crystal showing the(100), (010) and (001) planes and the [100], [010] and [001] directions. The angular unit cellparameters, a,b and g, that define the angles between the axes of the unit cell, are alsomarkedonthe stereogram.Note that the normals to the faces of the unit cell do not coincidewith the axes ofthe unit cell.


the axis of the cone of angles does not project to the centre of the circle on the stereogram;Figure 1.30). It follows that the angular scale of a stereographic projection is stronglydistorted, as can be seen from a standard Wulff net (Figure 1.31), that is used to define theangular scale of the stereogram, both in small circles of latitude and in great circles oflongitude. This Wulff net scale is identical to the angular scale commonly used to map thesurface of the globe.

Finally, some spherical triangles, defined by the intersection of three great circles, havegeometrical properties which are very useful in applied crystallography. The sum of theangles at the intersections that form the corners of the spherical triangle always exceeds 2p,while the sides of the spherical triangles, on a stereographic projection also define angles.Great circles that intersect at 90� must each pass through the pole of the other. If more thanone of the six angular elements of a spherical triangle is a right angle, then at least fourelements of the triangle are right angles and if only two of the angles are given, then all theremaining four angles can be derived. If the angles represented by the sides of the sphericaltriangle are denoted by a, b and c, while the opposing angles areA,B andC, then the relationsin a/sinA¼ sin b/sinB¼ sin c/sinC always holds (Figure 1.32). (Compare the unit trianglefor the triclinic unit cell shown in Figure 1.28.)

Hexagonal crystals present a special problem, since the usual Miller indices and directionindices do not reflect the hexagonal symmetry of the crystal. It is common practice tointroduce an additional, redundant axis into the basal plane of the hexagonal unit cell. Thethree axes a1, a2 and a3 then lie at 120

� to one another, with the c-axis mutually perpendicular

(001) (011)(010)

(111)

(110)

(100)

(101)

(101)

(100)

(111)

(110)

(111)

(110)

(111)

(110)

(010)(011)

Figure 1.29 A stereographic projection of a cubic crystal with a cube plane parallel to theplane of the projection. The projection consists of 24 unit spherical triangles bounded by greatcircles. Each triangle contains all the symmetry elements of the crystal.


(Figure 1.33). A redundant t-axis, drawn parallel to a3, results in a fourthMiller index iwhendefining a plane: (hkil), but the sum hþ kþ i¼ 0. A basal plane stereogram for a hexagonalcrystal (zinc) is given in Figure 1.34. The angular distance between the poles lying within thestereogram and its centre, the c-axis [0001], depends on the axial ratio of the unit cell, c/a.Families of crystal planes in a hexagonal lattice have similar indices in the four Miller indexnotation. That is ð10�10Þ and ð1�100Þ are clearly from the same family, while in the three indexnotation this is not obvious: (100) and ð1�10Þ.

Summary

The termmicrostructure is taken to mean those features of a material, not visible to the eye,that can be revealed by examining a selected sample with a suitable probe. Microstructuralinformation includes the identification of the phases present (crystalline or glassy),

100

100

010010 001

hkl

100

010010 001

100

hkl

Figure 1.30 A cone defining a constant angle with a direction in the crystal projects as a smallcircle on the stereogram. Note that the generating pole of the small circle is not at the centre ofthe projected circle on the stereogram.


the determination of their morphology (the grain or particle sizes and their distribution),and the chemical composition of these phases. Microstructural characterization may beeither qualitative (�what does the microstructure look like?�) or quantitative (�what is thegrain size?�).

The two commonest forms of probe used to characterize microstructure are electromag-netic radiation and energetic electrons. In the case of electromagnetic radiation, the opticalmicroscope and the X-ray diffractometer are the two most important tools. The opticalmicroscope uses radiation in the visible range of wavelengths (0.4–0.7mm) to form an imageof an object in either reflected or transmitted light. In X-ray diffraction the wavelengths used

Figure 1.31 TheWulff net gives the angular scale of the stereogram in terms of small circles oflatitude and great circles of longitude, as in a map of the globe.

CA

Ba

c

bFigure 1.32 All six elements of a spherical triangle represent angles. There is a simpletrigonometrical relationship linking all six angles (see text).


to probe the microstructure are of the order of the interatomic and interplanar spacings incrystals (0.5–0.05 nm). The electron microscope uses a wide range of electron beam energiesto probe the microstructure of a specimen. In TEM energies of several hundred kilovolts arecommon, while in SEM the beam energy may be as low as 1 kV.

The interaction of a probe beamwith the samplemay be either elastic or inelastic.Elasticinteraction involves the scattering of the beam without loss of energy, and is the basis ofdiffraction analysis, either usingX-rays or high energy electrons. Inelastic interactionsmay

aγ

a

t

v

u

γ=120°

a1=a2=a3

a1

γ

a2

c

a3

Figure 1.33 The hexagonal unit cell showing the use of a four-axis coordinate system, u(a1), v(a2), t(a3) and w(c).

(0001)

(1012)

(1011)

(2021)

(1010)

(1012)

(1011)

(2021)

(1010)

(1121)

(1120)

(1121)

(1122)

(1122)

(1120)

(2112)

(2111)

(2110)

(1212) (1211)(1210)

(1212)(1211)(1210)

(1100)

(2201)

(1101)

(1102)

(0112)

(0111)

(0221)

(0110)

(0112)

(0111)

(0221)

(0110)

(1102)

(1101)

(2201)

(1100)(2112)

(2111)

(2110)

Figure 1.34 A basal plane stereogram for zinc [hexagonal close-packed (HCP) structure],illustrating the use of the four Miller index system (hkil) for hexagonal crystal symmetry.


result in contrast in an image formed fromelastically scattered radiation (aswhen one phaseabsorbs light while another reflects or transmits the light). Inelastic interactions can alsobe responsible for the generation of a secondary signal. In the scanning electronmicroscopethe primary, high-energy electron beam generates low energy, secondary electrons that arecollected from a scanned raster to form the image. Inelastic scattering and energyadsorption is also the basis of many microanalytical techniques for the determination oflocal chemical composition. Both the energy lost by the primary beam and that generated inthe secondary signal may be characteristic of the atomic number of the chemical elementspresent in the sample beneath the probe. The energy dependence of this signal (the energyspectrum) provides information that identifies the chemical constituents that are present,while the intensity of the signal can be related to the chemical composition.

Many engineering properties of materials are sensitive to the microstructure, which inturn depends on the processing conditions. That is, the microstructure is affected by theprocessing route, while the structure-sensitive properties of a material (not just themechanical properties) are, in their turn, determined by the microstructure. This includesthe microstructural features noted above (grains and particles) as well as various defects inthe microstructure, for example porosity, microcracks and unwanted inclusions (phasesassociated with contamination).

The ability of any experimental technique to distinguish closely-spaced features istermed the resolution of the method and is usually limited by the wavelength of the proberadiation, the characteristics of the probe interactionwith the specimen and the nature of theimage-forming system. In general, the shorter the wavelength and the wider the acceptanceangle of the imaging system for the signal, then the better will be the resolution.Magnifications of the order of ·1000 are more than enough to reveal all the microstructuralfeatures accessible to the optical microscope. On the other hand, thewavelength associatedwith energetic electrons is very much less than the interplanar spacings in crystals, so thatthe transmission electron microscope is potentially able to resolve the crystal lattice itself.The resolution of the scanning electronmicroscope is usually limited by inelastic scatteringevents that occur in the sample. This resolution is of the order of a few nanometres forsecondary electrons, but only of the order of 1 mm for the characteristic X-rays emitted bythe different chemical species

Some microscopic methods of materials characterization are capable of resolvingindividual atoms, in the sense that the images observed reflect a physical effect associatedwith these atoms. Scanning probe microscopy includes scanning tunnelling and atomicforce microscopy, both of which can probe the nanostructure on the atomic scale.

Many microstructural features may be quantitatively described by microstructuralparameters. Two important examples are the volume fraction of a second phase and thegrain or particle size, both ofwhich usually have amajor effect onmechanical properties. Inmany cases the microstructure within any given sample varies, either with respect todirection (anisotropy), or with respect to position (inhomogeneity).

Crystal structure (or the lack of it, in an amorphous or glassy material) reflects the natureof the chemical bonding, and the four types of chemical bond, covalent, ionic,metallic andpolar (or van der Waals), are responsible for the major properties of the common classes ofengineering materials, namely metals and alloys (metallic bonding), ceramics and glasses(covalent and ionic bonding), polymers and plastics (polar and covalent bonding), andsemiconductors (primarily covalent bonding).


In a crystal structure the arrangement of the atoms, ions or molecules is regularlyrepeated in a characteristic spatial array. The unit cell of this crystal lattice is the smallestunit that contains all the symmetry elements of the bulk crystal, and each cluster ofidentically arranged atoms in this unit cell can be represented by a single lattice point. Thereare 14 possible ways of arranging these lattice points to give distinctly different latticesymmetries, the 14 Bravais lattices.

Characteristic directions in the crystal lattice correspond to a particular atomic sequenceand can be defined by direction indices, while any set of parallel atomic planes can bedefined by the normal to these planes, given as Miller indices. Both the direction and theMiller indices are conveniently plotted in two dimensions by mapping them onto animaginary plane using a stereographic projection. This projection has proved to be themostuseful of the geometrically possible mapping options.

Bibliography

1. Villars, P. and Calvert, L.D. (1985) Pearson�s Handbook of Crystallographic Data forIntermetallic Phases, Volumes 1–3, American Society for Metals, Metals Park, OH.

2. Hahn, T. (ed.), (1992) International Tables for Crystallography, Volume A, SpaceGroup Symmetry, Kluwer Academic, London.

3. Barrett, C. and Massalski, T.B. (1980) Structure of Metals, Pergamon Press, Oxford.4. Callister, W.D. (2006) Materials Science and Engineering: An Introduction, 7th

Edition, John Wiley & Sons, Ltd, Chichester.

Worked Examples

To demonstrate the type of information that can be obtained, we conclude each chapter withexamples of microstructural characterization for three different material systems. In thisfirst chapter we examine the crystallographic structure of the phases to be encountered infuture chapters and examine the basic use of the stereographic projection.

Having seen in this chapter how the literature data can be used to understand the crystalstructure of copper, we now look at two slightly more complicated crystal structures thatwill be considered later. The first is Fe3C, iron carbide or cementite, which exists inequilibrium with a-Fe in most steels (see the Fe–C equilibrium phase diagram). PearsonsHandbook of Crystallographic Data for Intermetallic Phases lists the data according to theformat presented in Table 1.4.

Table 1.4 Crystallographic data for Fe3C, iron carbide or cementite, which exists inequilibrium with a-Fe in most steels

PhaseStructuretype

Pearsonsymbol

space groupa, b, c(nm)

a, b, g(�) Atoms

Pointset x y z Occ

CFe3 CFe3 oP16 Pnma 0.50890 C 4c 890 250 450 1000.67433 Fe1 4c 036 250 852 1000.45235 Fe2 8d 186 063 328 100


Note that the name of the phase is listed as CFe3, not Fe3C, the more accepted chemicaldesignation for cementite. This is because Pearson�s handbook lists the chemical constituentsin alphabetical order. We also note that CFe3 is listed as the structure type for this structure,and that it has a primitive (P) orthorhombic (o) structure with 16 (16) atoms per unit cell. Itbelongs to the space group Pnma, and has three generating sites; one for carbon, whichgenerates a total of four carbon atoms per unit cell, and two for iron (Fe1 and Fe2). The first ofthese iron atoms (Fe1) generates a total of four atoms while the second (Fe2) generates a totalof eight atoms, giving a total of 12 iron atoms per unit cell. Since cementite is a stoichiometricphase, the occupation of each site is constant (there is no significant solid solubility), and theoccupation factor for each generating site is therefore 100%.A schematic drawing of the unitcell, with 16 atoms is given in Figure 1.35(a), and eight unit cells with a more conventionalrepresentation (showing the atoms belonging to the neighbouring cells and reflecting the fullsymmetry of the structure) is shown in Figure 1.35(b).

Our second example is a-Al2O3, known as sapphire in its single crystal form (the samename as the gem stone) and alumina or corundum in the polycrystalline form. This is thethermodynamically stable form of alumina. The data given in �Pearson�s handbook arepresented in Table 1.5.

The crystallographicdata fora-Al2O3 canbequite confusing.Wenote that the structure hasrhombohedral symmetry, with 10 atoms per unit cell (hR10), and a space group of R�3c.However, the lattice parameters are listed for a hexagonal unit cell (a¼ 0.4754 nm andc¼ 1.299 nm). Checking the International Tables for Crystallography under the space group

Figure 1.35 Schematic drawing of the structure of Fe3C (cementite) for (a) a single unit cell and(b) eight unit cells. The red spheres represent ions atoms, and the black spheres represent carbonatoms.

Table 1.5 Crystallographic data for a-Al2O3 (hexagonal unit cell).

PhaseStructure

type

PearsonSymbol

and spacegroup

a, b, c(nm)

a, b, g(�) Atoms

Pointset x y z Occ

Al2O3 Al2O3 hR10 R�3c 0.4754 Al 12c 0000 0000 3523 1001.299 0 18e 3064 0000 2500 100


R�3c, you will note that there are two alternative ways to describe this unit cell, the first basedon a rhombohedral unit cell, with 10 atoms per unit cell, and the second based on a hexagonalrepresentation, with 30 atoms per unit cell. The symmetry of the structure is not changed byusing the hexagonal representation, and it is often more convenient to use the hexagonal unitcell, even though not all of the symmetry operations for the higher symmetry hexagonalstructure are correct for a-Al2O3. Thegenerating sites listed in Pearson�s handbook are for thehexagonal unit cell, and includes 12aluminiumcations and 18oxygenanions, giving a total of30 atoms per hexagonal unit cell. The structure is stoichiometric, so the occupation factors forboth cations and anions are 100%.A schematic drawing of the two alternative rhombohedraland hexagonal unit cells is given in Figure 1.36. The crystallographic data for a rhombohedralunit cell are presented inTable 1.6. Throughout the rest of this bookwewill use the hexagonalunit cell to describe a-Al2O3.

Now let us examine some of the basic uses of the stereographic projection. As describedearlier, a stereographic projection is a map that plots the angles between differentcrystallographic directions and plane normals. Of course, we could also calculate theseangles, and today this is done with computer programs, which use the equations listed in

Figure 1.36 Schematic drawing of the hexagonal (a) and rhombohedral (b) unit cells of a-Al2O3. The red spheres represent oxygen anions, and the black spheres represent aluminiumcations.

Table 1.6 Crystallographic data for a-Al2O3 (rhombohedral unit cell).

PhaseStructuretype

Pearsonsymbol

and Spacegroup

a, b, c(nm)

a, b, g(�) Atoms

Pointset x y z Occ

Al2O3 Al2O3 hR10 R�3c 0.51284 55.28 Al 4c 3520 3520 3520 1000 6e 5560 �0560 2500 100


Appendix I. Nevertheless, the stereographic projection is a useful visual representation ofthe angular relationships between different crystallographic planes and directions.

Titanium metal (Ti) is our example for a stereographic projection. Both body-centredcubic and hexagonal phases exist but the phase a-Ti has the hexagonal structure, with latticeparameters a¼ 0.29504 nm and c¼ 0.46833 nm. The stereographic projection (the angularrelationships for crystal planes and directions) of any cubic unit cell is independent of thelattice parameter, but this is not true for other structures (compare the equations at the end ofthe chapter), and for a-Ti we need the ratio of the lattice parameters c/a in order to plot thestereographic projection accurately. Figure 1.37 shows the (0001) basal plane stereographicprojection for a-Ti. In principle, we could centre the projection on any other convenientcrystallographic direction or plane normal, or even use a coordinate system based on thesample geometry, if this were to prove more convenient. Assume we are interested infinding which planes lie at an angle of 90� to the basal (0001) plane. We draw a great circle(using aWulff Net or our computer program), and for this simple case the great circle is justthe perimeter of the stereogram.We see that all planes having indices (hki0) are at 90� from(0001). Simple!

Now let us find the planeswhich are 90� from the direction½11�22�. Againwe select a greatcircle, but this time with its centre on the direction ½11�22� (Figure 1.38). It is important tonote that the direction ½11�22� and the pole of the plane ð11�22Þ do not coincide on theprojection in Figure 1.38, since ½11�22� is not normal to ð11�22Þ in the hexagonal lattice. Thisnew great circle passes through all the planes which are perpendicular to ½11�22�, and hencecontain the ½11�22� direction.

Of course the stereogram also shows planes whose normals are at angles other than 90�

to a chosen direction or plane normal. An example is given in Figure 1.39: the planes

(0001)

(1012)

(1011)

(1010)

(1012)

(1011)

(1010)

(1121)

(1120)

(1121)

(1122)

(1122)

(1120)

(2112)

(2111)

(2110)

(1212) (1211)(1210)

(1212)(1211)(1210)

(1100)

(1101)

(1102)

(0112)

(0111)

(0110)

(0112)

(0111)

(0110)

(1102)

(1101)

(1100)(2112)

(2111)

(2110)

Figure 1.37 The (0001) stereographic projection for a-Ti (hexagonal).


(0001)

(1012)

(1011)

(1010)

(1012)

(1011)

(1010)

(1121)

(1120)

(1121)

(1122)

(1122)

(1120)

(2112)

(2111)

(2110)

(1212) (1211)(1210)

(1212)(1211)(1210)

(1100)

(1101)

(1102)

(0112)

(0111)

(0110)

(0112)

(0111)

(0110)

(1102)

(1101)

(1100)(2112)

(2111)

(2110)

[1122]

Figure 1.38 The stereographic projection for a-Ti, showing a great circle at 90� to the ½11�22�pole.

(0001)

(1012)

(1011)

(1010)

(1012)

(1011)

(1010)

(1121)

(1120)

(1121)

(1122)

(1122)

(1120)

(2112)

(2111)

(2110)

(1212) (1211)(1210)

(1212)(1211)(1210)

(1100)

(1101)

(1102)

(0112)

(0111)

(0110)

(0112)

(0111)

(0110)

(1102)

(1101)

(1100)(2112)

(2111)

(2110)

Figure 1.39 The (0001) stereographic projection for a-Ti, with a small circle constructionshowing planes at an angle of 60� to the ð11�20Þ plane.


whose poles make an angle of 60� with the normal to the ð11�20Þ plane lie on a smallcircle centred about the normal to ð11�20Þ. Only an arc of the small circle appears withinthe stereographic projection circle, and this arc corresponds to the angles lying in thenorthern hemisphere of the projection sphere. The remainder of the small circle can beback-projected from the north pole of the projection sphere. (Note again that the centre ofany small circle on the stereographic projection is not at the geometric centre of thatcircle.)

Problems

1.1. Give three examples of common microstructural features in a polycrystalline,polyphase material. In each case give one example of a physical or mechanicalproperty sensitive to the presence of the feature.

1.2. Give three examples of processing defects which might be present in a bulkmaterial.

1.3. Distinguish between elastic and inelastic scattering of a probe beam of radiationincident on a solid sample.

1.4. What is meant by the term diffraction spectrum?

1.5. Give three examples of structure-sensitive and three examples of structure-insen-sitive properties of engineering solids.

1.6. What magnification would be needed to make the following features visible to theeye:

(a) a 1mm blowhole in a weld bead;(b) the 10 mm diameter grains in a copper alloy;(c) lattice planes separated by 0.15 nm in a ceramic crystal?

1.7. Why is the resolution attainable in the electronmicroscope somuch better than thatof the optical microscope?

1.8. To what extent can we claim to see the �real� features of any microstructure?

1.9. Give three examples of microstructural parameters and, in each case, suggest oneway in which these parameters are linked quantitatively to material properties.

1.10. Define the terms symmetry, crystal lattice and lattice point.

1.11. What features of a crystal lattice are described by the directional indices [uvw] andthe Miller indices (hkl)? (Note, be very careful and very specific!)

1.12. Using literature data, define the unit cell and give the atomic positions for thefollowing materials:

(a) Al;(b) a-Ti;


(c) a-Fe;(d) TiN.

1.13. From the crystallographic data for aluminium metal, obtained in the previousquestion, calculate the minimum distance between neighbouring atoms in analuminium crystal. Compare this value with the diameter of the aluminium atomlisted in the periodic table. Which crystallographic direction (or plane) did you usefor your calculation and why?

1.14. Find the total number of equivalent planes in the cubic structure (multiplicity factor)belonging to the following families:

(a) {100};(b) {110};(c) {111};(d) {210};(e) {321}.

1.15. Calculate the separation of the ð�1012Þ crystal planes in the a-Al2O3 structure (theird-spacing). Do the same for the ð10�12Þ planes. Are these planes crystallographicallyequivalent? (Hint: remember the symmetry is actually rhombohedral, not hexagonal.)

1.16. Compare the planardensity (number of atoms per unit area) for the following planesin an FCC structure containing one atom per lattice point: (a) {100}; (b) {110};(c) {111}. Which plane has the highest planar density?

1.17. Repeat the previous question for theBCC structure, alsowith one atomper lattice point.

1.18. Retrieve the crystallographic data for lead telluride (PbTe). List the positions of theatoms in the unit cell. Howmany atoms belong to each unit cell?What is the Bravaislattice for this material?

1.19. The compound Mg2Si has a cubic structure. Determine the positions of the atomsand calculate the density of this phase. What is the Bravais lattice? Sketch the unitcell and the positions of the Mg and Si atoms.

1.20. Assume that the initial oxidation rate of a reactive metal depends on the planardensity of atoms at the surface. What would then be the relative rate of oxidationexpected for the ð1�100Þ and ð�12�10Þ polished surfaces of a Mg crystal?

1.21. The phase a-U has an orthorhombic structure. How many atoms are in each unitcell? What are their locations? Determine the d-spacing (distance between planes)for ð1�11Þ, ð1�10Þ, ð101Þ, and ð210Þ in this phase.

1.22. Show that a FCC unit cell could also be described as a primitive rhombohedral unitcell. What is the relationship between the rhombohedral lattice parameters and thecubic lattice parameter?

1.23. Sketch the stacking sequence of the (111) atomic planes for FCCcobalt (one atomperlattice point).Do the same for (0002)planes inHCPcobalt. (Note, there are two atomsper lattice point in theHCP structure.) Compare the planar densities of the two planes.


1.24. The surface of a cubic single crystal is parallel to the (001) plane.

(a) Determine the angle between the (115) plane and the surface.Do the same for the(224) plane.

(b) Determine the angle between the (115) and (224) planes.(c) What is the zone-axis that contains both the (115) and the (224) planes?(d) What is the angle and axis of rotation (the rotation vector) required to bring the

(115) plane parallel to what was originally the (224) plane?


2

Diffraction Analysis of CrystalStructure

Radiation that strikes an objectmay be scattered or absorbed.When the scattering is entirelyelastic, no energy is lost in the process, and the wavelength (energy) of the scatteredradiation remains unchanged. The regular arrays of atoms in a crystal lattice interactelastically with radiation of sufficiently short wavelength, to yield a diffraction spectrum inwhich the intensity of the radiation that is scattered out of the incident beam is plotted as afunction of the scattering angle (Figure 2.1). As we shall see below, the scattering angle istwice the angle of diffraction y. Both the diffraction angles and the intensities in the variousdiffracted beams are a sensitive function of the crystal structure. The diffraction anglesdepend on the Bravais point lattice and the unit cell dimensions, while the diffractedintensities depend on the atomic numbers of the constituent atoms (the chemical species)and their geometrical relation with respect to the lattice points.

A diffraction pattern or spectrum may be analysed at two levels. A crystalline materialmay be identified from its diffraction spectrum by comparing the diffraction angles thatcorrespond to the peaks in the spectrum and their relative intensities with a diffractionstandard (for example, the JCPDS file). In this procedure the diffraction spectrum is treatedas a �fingerprint� of the crystal structure in order to identify the crystalline phases asunambiguously as possible. Alternatively, the diffraction spectrummay be comparedwith acalculated spectrum, derived from some hypothetical model of the crystal structure. Theextent to which the predicted spectrum fits the measured data, the degree of fit, thendetermines the confidence with which the model chosen is judged to represent the crystalstructure.

In general, any measured spectrum is first compared with existing data, but if there areserious discrepancieswith the known standard spectra then itmay be necessary to search fora new model of the crystal lattice in order to explain the results. In recent years, computerprocedures have been developed to aid in interpreting crystallographic data, and much ofthe uncertainty and tedium of earlier procedures has been eliminated.


2.1 Scattering of Radiation by Crystals

The condition for a crystalline material to yield a discrete diffraction pattern is that thewavelength of the radiation should be comparablewith, or less than the interatomic spacingin the lattice. In practice thismeans that eitherX-rays, high energy electrons or neutronsmaybe used to extract structural information on the crystal lattice. Although suitable sources ofneutron radiation are now more readily accessible, they are not generally available. Thepresent text is therefore limited to a discussion of the elastic scattering ofX-ray and electronbeams, although much of the theory is independent of the nature of the radiation.

The required specimen dimensions are dictated by the nature of the radiation employed toobtain the diffraction pattern. All materials are highly transparent to neutrons, and it is quitecommon for neutron diffraction specimens to be several centimetres thick. X-rays,however, especially at the wavelengths normally used (�0.1 nm), are strongly absorbedby engineering materials and X-ray diffraction data are limited to submillimetre surfacelayers, fine powders or small crystals. Electron beams used in transmission electronmicroscopy may have energies of up to a few hundred kilovolts (kV), and at these energiesinelastic scattering dominates when the specimen thickness exceeds a tenth of a micro-metre. Electron diffraction data are therefore limited to submicrometre specimen thick-nesses. Thus, even though neutrons, X-rays and electrons may be diffracted by the samecrystal structure, the data collected will refer to a very different sample volume, withimportant implications for the specimen geometry, data interpretation and the proceduresused to select and prepare specimens.

2.1.1 The Laue Equations and Bragg’s Law

A one-dimensional array of atoms interacting with a parallel beam of radiationof wavelength l, incident at an angle a0 will scatter the beam to an angle a andgenerate a path difference D between the incident and scattered beams (Figure 2.2):

30 40 50 60 70 80 90 1000

500

1000

1500

2000

2θ

Inte

nsit

y (a

.u.)

α-Fe

α -Fe α -Feα -Fe

Fe3C

(deg)

Figure 2.1 Diffraction spectrum from a 0.4%C steel (Cu Ka radiation, 0.154 nm). Most of thepeaks are due to BCC a-Fe, but the asymmetry of the major peak is associated with anoverlapping Fe3C peak.


D¼ (y� x)¼ a(cos a� cos a0), where a is the interatomic spacing. The two beams will bein phase, and hence reinforce each other, if D¼ hl, where h is an integer. Now consider acrystal lattice made up of a three-dimensional array of atoms represented by regularlyspaced lattice points that are set at the corners of a primitive unit cell with lattice parametersa, b, and c. The condition that the scattered (diffracted) beam will be in phase with theincident beam for this three-dimensional array of lattice points is now given by a set of threeequations, known as the Laue equations:

D ¼ aðcosa�cosa0Þ ¼ hl ð2:1aÞ

D ¼ bðcosb�cosb0Þ ¼ kl ð2:1bÞ

D ¼ cðcosg�cosg0Þ ¼ ll ð2:1cÞThe cosines of the angles a, b and g, and a0, b0 and g0 define the directions of the incident

and the diffracted beams with respect to the unit cell of this crystal lattice. The choice of theintegers hkl, which are identical to the notation used forMiller indices (Section 1.2.3.3), is,as we shall see below, by no means fortuitous.

A more convenient, and completely equivalent, form of the geometrical relationdetermining the angular distribution of the peak intensities in the diffraction spectrumfrom a regular crystal lattice is the Bragg equation:

nl ¼ 2 dsin � ð2:2Þwhere n is an integer, l is the wavelength of the radiation, d is the spacing of the crystallattice planes responsible for a particular diffracted beam, and y is the diffraction angle, theangle the incident beammakeswith the planes of lattice points (Figure 2.3). The assumptionmade in deriving the Bragg equation is that the planes of atoms responsible for a diffractionpeak behave as a specula mirror, so that the angle of incidence y is equal to the angle ofreflection. The path difference between the incident beam and the beams reflected from twoconsecutive planes is then (x� y) in Figure 2.3. The scattering angle between the incidentand the reflected beams is 2y, and y¼ x(cos2y). But cos(2y)¼ 1� 2 sin2y, while x(siny)¼d, the interplanar spacing, so that (x� y)¼ 2d siny. The distance d between the lattice

Figure 2.2 When the path difference between the incident and the scattered beams from a rowof equidistant point scatterers is equal to an integral number of wavelengths, then the scatteredbeams are in phase, and the amplitudes scattered by each atom will reinforce each other(see text).

Diffraction Analysis of Crystal Structure 57

planes is a function of the Miller indices of the planes and the lattice parameters of thecrystal lattice. The general equations (for all Bravais lattices) are as follows:

1

d2¼ 1

V2ðS11h2 þ S22k

2 þ S33l2 þ 2S12hkþ 2S23klþ 2S31lhÞ ð2:3Þ

where V is the volume of the unit cell:

V ¼ abcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�cos2a�cos2b�cos2gþ 2cosacosbcosg

pð2:4Þ

and the constants Sij are given by:

S11 ¼ b2c2sin2a ð2:5aÞ

S22 ¼ c2a2sin2b S33 ¼ a2b2sin2g ð2:5bÞ

S12 ¼ abc2ðcosacosb�cosgÞ ð2:5cÞ

S23 ¼ a2bcðcosbcosg�cosaÞ ð2:5dÞ

S31 ¼ ab2cðcosgcosa�cosbÞ ð2:5eÞFor an orthorhombic lattice, for which a¼ b¼ g¼ 90�, these equations reduce to:

1

d2¼ h

a

� �2

þ k

b

� �2

þ l

c

� �2

ð2:6Þ

with V¼abc.In theBragg equation, nl¼ 2d siny, the integer n is referred to as the orderof reflection. A

first-order hkl reflection (n¼ 1) corresponds to a path difference of a single wavelengthbetween the incident and the reflected beams from the (hkl) planes, while a second-orderreflection corresponds to a path difference of two wavelengths. However, from the Braggequation, this path difference of two wavelengths for the second-order dhkl reflection isexactly equivalent to a single wavelength path difference from planes of atoms at one halfthe dhkl spacing, and therefore corresponds to planes with Miller indices (2h 2k 2l). It istherefore common practice to label the nth order reflection as coming from planes having a

x

y

d

Figure 2.3 If each plane of atoms in the crystal behaves as a mirror, so that the angle ofincidence is equal to the angle of reflection, then the condition for the beams reflected fromsuccessive planes to be in phase, and hence reinforce each other, is given by Bragg’s law (seetext).


spacing dhkl/n, and with Miller indices (nh nk nl). The Bragg equation is then writtenl¼ 2dhkl siny, where the subscript hkl is now understood to refer to a specific order n of thehkl reflection. For example d110, d220 and d440 would be the first-, second- and fourth-orderreflecting planes for the 110 reflections.

2.1.2 Allowed and Forbidden Reflections

Body-centred or face-centred Bravais lattices have planes of lattice points that give rise todestructive (out-of-phase) interference for some orders of reflection. In the BCC lattice(Figure2.4), the latticepointat1/21/21/2 scatters inphase forallordersof the110reflections,but will give rise to destructive interference forodd orders of the 100 reflections, that is 100,300,500,etc.It is instructivetolist theallowedreflectionsforprimitive,BCCandFCCBravaislattices as a function of the integer (h2þ k2þ l2), (Table 2.1). For cubic symmetry the Braggequation can be rearranged to giveðh2 þ k2 þ l2Þ ¼ 2a sin�=l, leading to a regular array ofdiffracted beams. As can be seen, some values of h2þ k2þ l2 are always absent, and 7 is thefirst of these. Other integers may correspond to more than one reflection, and both the 221and the 300 planes, corresponding to h2þ k2þ l2¼ 9, will diffract at the same angle y.

Those reflections which are disallowed for a particular lattice are referred to as forbiddenreflections, and there are simple rules to determine which reflections are forbidden. In theFCC lattice theMiller indicesmust be either all odd or all even for a reflection to be allowed,and it is the reflecting planeswithmixed odd and even indices that are forbidden. In theBCClattice the sum hþ kþ lmust be even for an allowed reflection, and if the sum of the Millerindices is odd, then the reflection is forbidden. In some diffraction spectra the sequence ofthe diffraction peaksmay be recognized immediately as due to a specific Bravais lattice. Forexample, the two sets of paired reflections (111 and 200, and then 311 and 222) arecharacteristic of FCC symmetry. More often, careful measurements and calculations areneeded to identify the crystal symmetry responsible for a given diffraction pattern.

In most crystals the crystal lattice points correspond to groups of atoms, rather thanindividual atoms, and the different atomic species will scatter more or less strongly,depending on their atomic number (the number of electrons attached to each atom). In the

bccfcc

110

200

110

200

Figure 2.4 The FCC (a) and BCC (b) Bravais lattices contain additional planes of lattice pointswhich lead to some forbidden reflections that are characteristic of these crystal structures. Thedarker lattice points are to the front of the unit cells.


FCC NaCl structure, (Figure 2.5) the sodium cations are located at the lattice points, whilethe chlorine anions are displaced from the lattice points by a constant lattice vector 1/2 0 0.Two types of lattice plane now exist, those that contain both cations and anions (mixedplanes), such as {200} and {220}, and those that consist of an equi-spaced, alternatingsequence of pure anion and cation planes, such as {111} and {311}. Since the cations andanions in the mixed planes are coplanar, the two species always scatter in-phase, and theintensities of the diffraction peaks are enhanced by the additional scattering of the secondatomic species. However, the alternating planes of cations and anions scatter out of phasefor all the odd-order reflecting planes, and hence reduce the diffracted intensity, while thesame alternating cation and anion planes will scatter in-phase for all even-order reflections,thus enhancing the scattered intensity. The extent to which the second atomic species willenhance or reduce the diffracted intensity will depend on the difference in scattering powerassociated with the difference in the atomic numbers of the two atomic species.

2.2 Reciprocal Space

Bragg’s law indicates that the angles of diffraction are inversely proportional to the spacingof the reflecting planes in the crystal lattice. In order to analyse a diffraction pattern it istherefore helpful to establish a three-dimensional coordinate system in which the axes havethe dimensions of inverse length (nm�1). Such a system of coordinates is referred to asreciprocal space.

2.2.1 The Limiting Sphere Construction

Thevalue of siny is constrained to lie between�1, so that, fromBragg’s law, the value of 1/dmust fall in the range between 0 and 2/l if the parallel planes of atoms are to give rise to adiffracted beam. If the beam of radiation is incident along the x-axis, and a diffracting

Table 2.1 Allowed reflections in crystals of cubic symmetry with one atom per lattice site,listed in order of the sum of the squares of the Miller indices.

h2þ k2þ l2 Primitive cubic Face-centred cubic Body-centred cubic

1 100 – –2 110 – 1103 111 111 –4 200 200 2005 210 – –6 211 – 2117 – – –8 220 220 2209 221/300 – –10 310 – 31011 311 311 –12 222 222 22213 320 – –14 321 – 32115 – – –16 400 400 400


crystal is located at the origin of the coordinate system, then a sphere of radius 2/l, termedthe limiting sphere, will enclose all the allowed values of 1/d in reciprocal space and hencedefine all the planes in the crystal that have the potential to diffract at thewavelength l. Nowimagine a smaller sphere of radius 1/l that lies within this limiting sphere and is placed sothat it just touches the limiting sphere on the x-axis, and hence also touches the position ofthe crystal (Figure 2.6). A line passing through the centre of this second sphere, called thereflecting sphere, which is parallel to the diffracted beam and makes an angle 2y with thex-axis, will intersect the periphery of the reflecting sphere at the point P.

2.2.2 Vector Representation of Bragg’s Law

The two vectors k0 and k in Figure 2.6 define thewave vectors of the incident and diffractedbeams, |k0|¼ |k|¼ 1/l, and, if the reciprocal lattice vector OP is equal to g, then |g|¼ 1/d,and Bragg’s law l¼ 2d siny, can be written as a vector equation: k0þ g¼ k. Note that thereciprocal lattice vector g is perpendicular to the diffracting planes, while the wave vectorsk0 and k are parallel to the incident and diffracted beams, respectively. Aswe shall see later,this vector form of the Bragg equation can be very useful indeed.

2.2.3 The Reciprocal Lattice

Wecan nowdefine a lattice in reciprocal space that is in someways analogous to theBravaislattice in real space (Figure 2.7). The origin of coordinates in reciprocal space is the point

Figure 2.5 In the NaCl structure each lattice point corresponds to one cation and one anion.These different ions are coplanar for the 200 planes, and therefore scatter in-phase, enhancingthe 200 diffraction peak. By contrast, the 111 planes of cations and anions are interleaved, andlead to out of phase interference, reducing the intensity of the 111 diffraction peak. The largeranions (red) are distinguished from the smaller cations (grey).


(000) and lies at the position of the crystal, in the centre of the limiting sphere construction(Figure 2.6). Any reciprocal lattice vector ghkl, drawn from this origin to the reciprocallattice point (hkl), will be normal to the reflecting planes that have the Miller indices (hkl).The distance from 000 to hkl, |ghkl|, is equal to 1/dhkl, so that the successive orders ofreflection, 1 to n, are represented by a series of equidistant reciprocal lattice points that liealong a straight line in reciprocal spacewhose origin is at (000). Note that negative values ofn are perfectly legitimate in this representation.

2θ2θ

DiffractedBeam

P

kg

k0

LimitingSphere

ReflectingSphere

Incident

Beam O

Figure 2.6 The limiting sphere and reflecting sphere constructions (see text for details).

000 100 200Reciprocal lattice points in

reciprocal space

1/d100

1/d200

Lattice planes in real space

d100 d200

Figure 2.7 The definition of reciprocal lattice points in terms of the lattice planes of a crystal,defined by their Miller indices. The example shown here is for the cube planes (h00).


The condition for Bragg’s law to be obeyed is that the reciprocal lattice vector g shouldlie on the reflecting sphere. This can be achieved by varying either l or y, as we shall seelater.

2.2.3.1 The Reciprocal Lattice Unit Cell. The dimensions of the reciprocal lattice unitcell can be defined in terms of the corresponding Bravais lattice unit cell. The generalequations are:

a* ¼ bc

Vsina; b* ¼ ca

Vsinb; c* ¼ ab

Vsing ð2:7Þ

where V is the volume of the unit cell. It is important to note that the reciprocal lattice axesa*, b* and c* of the unit cell in reciprocal space are, in general, not parallel to the axes of theunit cell a, b and c in real space. This is consistent with the previous treatment of directionindices andMiller indices. The axes of the unit cell in real space correspond to the directionindices [100], [010] and [001], while the axes of the reciprocal unit cell correspond to theMiller indices (100), (010) and (001) and are aligned normal to the planes (100), (010) and(001).

It should now begin to be clear just why we define Miller indices in terms of reciprocallengths, and a comparison of the reciprocal lattices for FCC and BCC Bravais lattices mayhelp to clarify this evenmore. Figure 2.8 compares the two reciprocal lattices. In this figureall forbidden reflections have been excluded. Both reciprocal lattice unit cell dimensionsare defined by g vectors of type 200. The FCC reciprocal lattice unit cell contains a body-centred allowed reciprocal lattice point at 111. This reciprocal lattice cell is therefore BCC,while the BCC reciprocal lattice unit cell contains allowed face-centred reciprocal latticepoints of the type 110, and is therefore FCC.Reciprocal lattice points in any direction can bederived by simple vector addition and subtraction. The series 020, 121 and 222 thusconstitute a series of reflections that lie along a line in BCC reciprocal space. Eachreflection is separated from the next in the series by the reciprocal lattice vector 101.Similarly, the sequence of reflections 200, 111, 022 in FCC reciprocal space are separatedby the vector �111. Any two nonparallel reflections g1 and g2 define a plane in reciprocalspace. Their common zone n, and any other reflection also lying in this zone g3 must obeythe rules of vector geometry, that is g1 · g2 ¼ n, and n·g3 ¼ 0.

2.3 X-Ray Diffraction Methods

We now take a closer look at the limiting sphere and reflecting sphere, this time with areciprocal lattice superimposed on this construction (Figure 2.9). Three factors determinewhether or not a particular crystal plane will give a diffraction peak:

1. The wavelength of the X-rays in the incident beam. Reducing the wavelength of the X-rays increases the diameter of the limiting sphere, and therefore places longer reciprocallattice vectors, corresponding to smaller interplanar spacings, within the limiting sphere.

2. The angle of the incident beam with respect to the crystal.As the crystal is rotated aboutits centre (or, equivalently, as the X-ray beam is rotated about the crystal), the reflectingsphere sweeps through the reciprocal lattice points, allowing the different diffractingplanes to obey the Bragg law in turn. By rotating about two axes at right angles all the


X

Y

Z

002 022

202 222

000

020

200220

011

211

101121

110

112

*

*

*

bcc

X

Y

Z

002

(a)

(b)

022

202 222

111

000020

200

220

*

*

*

fcc

Figure 2.8 Allowed reflections of the FCC (a) and BCC (b) Bravais lattices plotted in reciprocalspace. The FCC reciprocal lattice has BCC symmetry, while the BCC reciprocal lattice has FCCsymmetry.


reciprocal lattice points lying within the limiting sphere can be made to diffract, each atthe appropriate Bragg angle yhkl.

3. The effective size of the reciprocal lattice points. If the diffraction condition had to beobeyed exactly, then diffraction would only be observed at the precise Bragg angle. Inpractice, the reciprocal lattice points have a finite size determined by the size andperfection of the crystal. In addition, inelastic absorption of the incident beam energylimits the path length of the radiation over which elastic scattering dominates, and hencelimits the crystal dimensions that can contribute effectively to coherent diffraction

Absorption effects are determined by the value of the absorption coefficient. Typicalabsorption coefficients for X-rays result in path lengths of the order of tens of micrometresbefore the coherent elastic scattering of the beam becomes blurred by inelastic scatteringevents. Since interplanar spacings are usually in the range of tenths of a nanometre, itfollows that, for large, otherwise perfect crystals, the effective diameter of a reciprocallattice point (the uncertainty in the value of g) is of the order of 10�5|g|. As a consequence,the lattice parameters of crystalline phases can bemeasuredwith an error that is typically ofthe order of 10 ppm, quite sufficient to allow for accurate determination of changes incrystal dimensions due to temperature (thermal expansion), alloying additions or appliedstress (in particular, residual stresses associated with processing and assembly). Forsufficiently small crystallites and heavily deformed crystals it is possible to measure therange of y over which diffraction from a particular hkl plane is observed, and to derivequantitative information on the crystal size or degree of perfection.

So far we have assumed that the incident beam ismonochromatic and accurately parallel.Again, the effect of these assumptions is best understood from the reflecting sphereconstruction, now modified as in Figure 2.10. If there is a spread of wavelengths in the

..

..

..

..

..

.. . .

.

...

..

..

..

.

. .. .

. . . . .....

.

......

.

.

.

.

Incidentbeam

000

Reflectingsphere

Limitingsphere

k0

g

2θ

k

Figure 2.9 Superimposing the reciprocal lattice on the reflecting sphere constructiondemonstrates the effects of some experimental variables on diffraction (see text).


incident beam, then the limiting sphere becomes a shell, and the reflecting sphere generatesa �new moon� crescent, within which reciprocal lattice points satisfy the Bragg law. If theincident beam is not strictly parallel, then the reflecting sphere is rotated about the centre ofthe limiting sphere, by an angle equal to the divergence or, equivalently, the convergence

LimitingSphere

Beam Convergenceor Divergence Angle

ReflectingSphere

λ1 > λ2

ReflectingSphere

WavelengthDispersive

LimitingSphere

(a)

(b)

Figure 2.10 The effect of variations in X-ray wavelength (a) or inadequate collimation of thebeam (b) are readily understood from the reflecting sphere construction.


angle of the incident beam, generating two crescent volumes within which the Bragg law issatisfied. Both these effects introduce errors into the determination of lattice spacing byX-ray diffraction. These errors depend on the value of |g| and the angle between g and theincident beam.

2.3.1 The X-Ray Diffractometer

AnX-ray diffractometer comprises a source ofX-rays, theX-ray generator, a diffractometerassembly, a detector assembly and X-ray data collection and processing systems. Thediffractometer assembly controls the alignment of the beam, as well as the position andorientation of both the specimen and the X-ray detector.

The X-rays are generated by accelerating a beam of electrons onto a pure metal targetcontained in a vacuum tube. The high energy electrons eject ground-state electrons from theatoms of the target material, creating holes, and X-rays are emitted during the refilling ofthese ground states. If all the electron energy, usually measured in eV, were to be convertedinto an X-ray quantum, then the frequency n would be given by the quantum relationeV¼ hn, where h is Planck’s constant. The X-ray wavelength l is proportional to thereciprocal of this frequency, l¼ c/n, where c is the velocity of light in the medium throughwhich theX-rays propagate. The condition that all the energy of the exciting electron is usedto create a photon sets an upper limit on the frequency of the X-rays generated, and hence alower limit on the X-ray wavelength. The above relations lead to an inverse dependence ofthis minimum wavelength on the accelerating voltage of the X-ray tube, which is given (invacuum) by: lmin¼ 1.243/V, where l is in nanometres and V is in kilovolts.

Above thisminimumwavelength, there is a continuous distribution ofX-raywavelengthsgenerated by the incident electron beam, whose intensity increases with both incidentelectron energy and beam current, aswell aswith the atomic number of the target (that is, thedensity of electrons in the target material). This continuous distribution of photon energiesand wavelengths in the X-rays emitted from the target is referred to as white radiation orBremsstrahlung (German for �radiation braking� – the slowing down of the electrons by theemission of photons).

Superimposed on the continuous spectrum of white radiation are a series of very narrowand intense peaks, the characteristic radiation of the chemical elements (Figure 2.11). Acharacteristic peak corresponds to the energy released when the hole in an inner electronshell, created by a collision event, is filled by an electronwhich originates in a higher energyshell of the same atom. Thus ejection of an electron from the K-shell excites the atom to anenergy state EK, and if the hole in the K-shell is then filled by an electron from the L-shell,then the energy of the atomwill decay to EL, while the decrease in the energy of the excitedatom, (EK�EL), will appear as an X-ray photon of fixedwavelength that contributes to theKa line of the characteristic target spectrum (Figure 2.12). Filling the hole in the K-shellwith an electron from the M-shell would have reduced the energy state of the atom evenfurther, toEM, leading to aKb photon of shorterwavelength thanKa, and a second line in theK-shell spectrum. That is, the residual energy of the excited atom is lower in the EM statethan it is in the EL state.

Further decay of the energy of the excited atom from the EL and EM energy states willresult in thegeneration ofL andMcharacteristic radiation ofmuch longerwavelength. Thereare many alternative options for the origin of a donor electron to fill a hole in the L- or


Nucleus

Lα

K β

Kα

M-shell

L-shell

K-shell

Figure 2.12 Characteristic X-radiation is generated by electron transitions involving the innershells, and the wavelengths are specific to the atomic species present in the target material.

0.02 0.04 0.06 0.08 0.1 0.12

1

2

3

Wavelength (nm)

Rel

ativ

e in

tens

ity

Brehmsstrahlungcontinuum

CharacteristicX-rays

X-rays from amolybdenum

target at 35 kV

Kα

K β

Figure 2.11 An energetic electron beam striking a solid molybdenum target generates acontinuous spectrum of white X-radiation with a sharp cut off at a minimum wavelength,corresponding to the incident electron energy, together with discontinuous, narrow intensitypeaks, the characteristic X-radiation from the molybdenum K-shell.


M-shells, so that the characteristic L and M spectra consist of several closely spaced lines.Clearly, a characteristic line can only be generated in the target by the incident beam if theelectron energy exceeds the excitation energy of the atom for that line. The excitation energyfor the ejection of an electron from a given inner shell of the atom increases with the atomicnumber of the target material (Figure 2.13), since the electrons in any given shell are moretightly bound to a higher atomic number nucleus. The low atomic number elements in thefirst row of the periodic table only contain electrons in the K-shell, and hence can only giveK-lines, while only the heaviest elements (of high atomic number) have M- and N-lines intheir spectra. These spectra can then be very complex (Figure 2.14).

101

102

103

104

105

106

EK

,L,M

(eV

)

00101

Atomic Number [Z]

K

L M

Figure 2.13 The excitation energy required to eject an electron from an inner shell increaseswith atomic number.

K series

β2β1

β3α1

α2

K

LILIILIII

MIMIIMIIIMIV MVNINII

NIV

NIII

NVNVINVIIOIOIII

OIVOV

OII

LI LII LIII{ { {

MI MIII MIVMII MV

log

ener

gy

L series

M series

1

2

3

4

Figure 2.14 The atomic energy levels and characteristic X-ray spectrum for a uranium atom.After Barratt and Massalski, Structure of Metals, 3rd revised edition, with permission fromPergamon Press.


If elastic scattering of the X-rays is to dominate their interaction with a sample, then weneed to ensure that intensity losses due to inelastic scattering processes are minimized. Amonochromatic X-ray beam traversing a thin sample in the x direction loses intensity I at arate given by dI/dx¼�mI, wherem is the linear absorption coefficient for theX-rays. It is themass ofmaterial traversed by the beamwhich is important, rather than the sample thickness,so that the values tabulated in the literature are generally for the mass absorption coefficientm/r, where r is the density, rather than the linear absorption coefficient. The transmittedintensity is then given by:

I=I0 ¼ exp � mrrx

� �ð2:8Þ

Two plots of the mass absorption coefficient as a function of the X-ray wavelength areshown in Figure 2.15, one for the case of constructional steel and the other for an aluminiumalloy. The lower density aluminium alloy has the lower linear absorption coefficient at anygiven wavelength. All materials show a general increase in the mass absorption coefficientwith wavelength, but with a sequence of step discontinuities. These are referred to asabsorption edges, and correspond to the wavelengths at which the incident X-ray photonpossesses sufficient energy to ionize the atom by ejecting an inner-shell electron from anatom in the specimen, similar to the ejection of an electron by an energetic incident electron.It follows that the absorption edges are the X-ray equivalents of the minimum excitationenergies, involved in the generation of characteristic X-rays, as discussed above. Similar toelectron excitation, short wavelength, high energy X-rays can generate secondary charac-teristic X-rays of longer wavelength in the specimen target, a process termed X-rayfluorescence. Note that in X-ray fluorescent excitation there is no background, and whiteradiation is not generated.

In order to avoid fluorescent radiation andminimize absorption of the incident beam, it isimportant to select radiation forX-ray diffractionmeasurements that has awavelength close

10 -1

10 0

10 1

10 2

10 3

10 4

Mas

s ab

sorp

tion

coe

ffic

ient

0.6 1.20.2 0.4 0.8 1.0

Wavelength (nm)

Fe

Al

Figure 2.15 The expected dependence of the X-raymass absorption coefficient onwavelengthfor iron and aluminium. Note that the linear absorption of the much lighter aluminium will beless than that of iron at all wavelengths.


to an absorption minimum, and on the long wavelength side of the absorption edge for thespecimen. Thus Cu Ka radiation (l¼ 0.154 nm) is a poor choice for diffraction measure-ments on steels and other iron alloys (EFeK¼ 7.109 keV, l¼ 0.17433 nm). However, Co Karadiation (l¼ 0.1789 nm) lies just to the long wavelength side of the KFe edge and willtherefore give sharp diffraction patterns from steel, free of background fluorescence.

Assuming thatCoKa radiation is to be used, then thevalues ofm/r for iron and aluminiumare 46 and67.8, respectively. Inserting the densities of the twometals, 7.88 and 2.70 g.cm�3,wecanderive the thicknessof the sample thatwill reduce the intensityof the incidentbeamto1/e of its initial intensity, namely 27.6 mm for iron and 54.6 mm for aluminium. These valueseffectivelydefine the thicknessof the samplewhichprovides the reflectiondiffraction signalfrom solid samples of each metal using CoKa radiation.

X-ray diffraction experiments require either monochromatic or white radiation. Mono-chromatic radiation is generated by exciting K-radiation from a pure metal target and firstfiltering the beam by interposing a foil that strongly absorbs the b component of the K-radiation without appreciable reduction of the intensity of the a component. This can beaccomplished by choosing a filter which has an absorption edge that falls exactly betweentheKa andKbwavelengths.A good example is the use of a nickel filter (ENi K¼ 0.1488 nm)with a copper target (ECu K¼ 0.138 nm), transmitting the Cu Ka. beam (0.154 nm) but notthe Kb, Figure 2.16(a).

More effective selection of a monochromatic beam can be achieved by interposing asingle crystal monochromator which is oriented to diffract at the characteristic Ka peak.This monochromatic diffracted beam can then be used either as the source of radiation forthe actual sample or to filter the diffracted signal [Figure 2.16(b)]. The monochromatorcrystal can also be bent into an arc of a circle, so that radiation from a line source striking anypoint on the arc of the crystal will satisfy the Bragg condition, focusing a diffracted beamfrom themonochromator to a line at the specimen position [Figure 2.16(c)]. The same effectcan be achieved at the detector when the monochromator is placed in the path of the beamdiffracted from the specimen.

An X-ray spectrum is usually recorded by rotating an X-ray detector about the sample,mounted on the diffractometer goniometer stage. The goniometer allows the sample to berotated about one or more axes (Figure 2.17). In order to make full use of the potentialresolution of the method (determined by the sharpness of the diffraction peaks), thediffractometer must be accurately aligned and calibrated, typically to better than 0.01�. Theaccurate positioning of the sample is very important, especially when using a bentmonochromator in a focusing diffractometer. Any displacement of the plane of the samplewill result in a shift in the apparent Bragg angle (Section 2.4.5). Note that if a given plane inthe sample is to remain perpendicular to a radius of the focusing circle, then the detectormust rotate around the focusing circle at a rate which is twice that of the sample.

A number of X-ray detectors can be used (including photographic film), but thecommonest is the proportional counter, in which an incident photon ionizes a low pressuregas, generating a cloud of charged ions which are then collected as a current pulse. In theproportional counter the charge carried by the current pulse is proportional to the photonenergy, and electronic discrimination can be used to eliminate stray photons whose energydoes not correspond to that of the required signal. The present generation of proportionalcounters have an energy resolution better than 150 eV, and can be used to eliminate most ofthe background noise associatedwith white radiation (although they are not able to separate


the Ka and Kb peaks). There is a dead-time associated with the current pulse generated in aproportional counter, and a second photon arriving at the counterwithin amicrosecond or soof the first will not be counted. This sets an upper limit to the counting rate and means thatpeak intensities recorded at high counting rates may be underestimated.

While the maximum thickness of the sample that can be studied by X-ray diffraction isdictated by themass absorption coefficient for the incident radiation (see above), the lateraldimensions are a function of the diffractometer geometry. For an automated powderdiffractometer with a �Bragg–Brentano� geometry the width irradiated perpendicular tothe incident beam is typically of the order of 10mm, while the length of the illuminatedpatch depends on the angle of incidence, and is typically in the range 1–7mm (Figure 2.18).The size and spacing of Soller slits (used to collimate the beam and also called divergenceslits) determine the area illuminated by the incident beam. For fixed slits, the totalilluminated area decreases as the diffraction angle increases (2y), but for sufficiently thicksamples, the total irradiated volume is almost independent of 2y. Some Bragg–Brentanodiffractometers include an automatic (compensating) divergence slit, which increases thewidth of the incident beam as the diffraction angle increases. The irradiated volume of thesample then increases with increasing diffraction angle and the calculated integrated

Bentmonochromator

Source Specimen

Multi-wavelengthnoncollimated source

θ Bragg

Monochromator

Monochromatic beamat θBragg of the

monochromator crystal

(b)(a)

(c)

0.12 0.14 0.16 0.18

Nickel filter

Kα

Kβ

λ (nm)

Inte

nsit

y m

ass

abso

rpti

on c

oeff

icie

nt

Figure 2.16 Some important spectrometer features: (a) Cu K radiation filtered by a nickel foilto remove Kb; (b) a monochromator crystal allows a specific wavelength to be selected from anX-ray source; (c) a fully focusing spectrometermaximizes the diffracted intensity collected at thedetector.


intensities must then take into account the dependence of integrated intensity on diffractingvolume.

2.3.2 Powder Diffraction–Particles and Polycrystals

The grain size of crystalline engineering materials is generally less than the thickness ofmaterial contributing to an X-ray diffraction signal. This is also true of many powdersamples, whether compacted or dispersed. The general term powder diffraction is used todescribe both the nature of the diffraction pattern formed and the subsequent analysis usedto interpret diffraction results obtained from these polycrystalline samples.

If we assume that the individual grains are both randomly oriented andmuch smaller thanthe incident beam cross-section, then this assumption of random orientation is equivalent toallowing thewave vector of the incident beam k0 to take all possible directions in reciprocalspace. That is, the reciprocal lattice is rotated freely about its origin (Figure 2.19). All grainsthat are oriented for Bragg reflection must have g vectors which touch the surface of thereflecting sphere, and it is these grains that then generate diffraction cones that subtendfixedBragg angles 2ywith the incident beam. The innermost cone corresponds to the latticeplanes with the largest d-spacing, corresponding to the minimum observed value of y fordiffraction. The minimum d-spacing that can be detected is determined by the radius of thelimiting sphere, dmin¼ l/2. As the detector is rotated about an axis normal to the incident

SourceDetector

θ rotation

of specimen 2θ rotation

of detector

Focalcircle

Sample

θ 2θ

Figure 2.17 A samplemounted on a goniometer which can be rotated about one or more axis,and a detector which travels along the focusing circle in the Bragg–Brentano geometry.


beam and passing through the sample, the diffracted intensities are recorded as a function of2y, to give the diffraction spectrum for the sample.

If the specimen is rotated about an axis normal to the plane of diffraction (the planecontaining the incident and diffracted beams) at a constant rate dy/dt, while the detector isrotated about the same axis at twice this rate, d(2y)/dt, then the normal to the diffractingplanes in the crystals which are contributing to the spectrum will remain parallel. This isuseful if the grains in the polycrystalline sample are not randomly oriented. Mechanical

θ

Source

Sample

To detector

w

w1

w2

θ 1 < θ 2

θ 1

θ 2

Sollerslits

Figure 2.18 Influence of y on the exposed surface area for a powder diffractometer using theBragg–Brentano geometry.


working (plastic deformation) and directional solidification are twoprocesseswhich tend toalign the grains along specific directions or in certain planes. Similar alignment effects arecommon in thin film electronic devices prepared by chemical vapour deposition (CVD).Such samples are said to possess crystalline texture, and the grains are said to bepreferentially oriented. If the normal to the diffracting planes in a diffraction experimentcoincides with specific directions in the bulk material (parallel or perpendicular to thedirection of mechanical work, for example), then the diffracted intensities of specificcrystallographic reflections for these sample orientations may be markedly enhanced orreduced with respect to those intensities calculated for a randomly oriented polycrystal.

Preferred orientation plays an important role in many material applications, and isassociated with anisotropy of the physical, chemical or mechanical properties. A classiccase is that of the magnetic hysteresis of silicon iron, which is markedly different in theh100i and h111i directions. Transformer steels are therefore processed to ensure a strong(and favourable) texture, exhibiting low hysteresis losses. In manymechanical applicationstexture is considered undesirable, and structural steel sheet is usually cross-rolled (rolled intwo directions at right angles) to limit the tendency to align the orientation of the grains.

The lattice spacings in polycrystalline samples are dependent on the state of stress ofthe sample. Stresses may be due to the conditions under which the component performs inservice (operating stresses), but they may also result from the way in which the componentis assembled into a system (for example, a bolt is put under tension when a nut istightened). However, residual stresses also result from the processing history: gradients of

Figure 2.19 In powder diffraction, a random, polycrystalline sample is equivalent to freerotation of the reciprocal lattice about the centre of the limiting sphere. Each reciprocal latticevector within the limiting sphere then generates a spherical surface which intersects thereflecting sphere on a cone of allowed reflections from the individual crystals whichsubtend an angle 2y with the incident beam.


plastic work in the component, variations in cooling rate, and some special surfacetreatments (ion implantation, chemical surface changes, or mechanical bombardment withhard particles, known as �shot-peening�). In all cases the residual stresses present in thematerial must be in a state of mechanical equilibrium, even though no external forces arebeing applied.

One way of determining residual stress is by accurate X-ray measurement of latticespacings. It is, of course, the lattice strains within the individual grains which are beingsampled, and these must be converted to stresses through a knowledge of the elasticconstants of the phases present. It is useful to distinguish two types of residual stress,namelymacrostresses andmicrostresses.Macrostresses are present when large numbers ofneighbouring crystals of the same phase experience similar stress levels, and the stressesvary smoothly throughout the sample in order to generate an equilibrium stress state (forexample, compressive stresses in the surface layers which are balanced by tensile stresses inthe bulk). On the other hand, microstresses may also exist, in which the stresses in theindividual grains within any volume element may be widely different and of opposite sign,while the average stress in the component sums to zero. A good example would be stressesdue to anisotropy of thermal expansion in a noncubic polycrystal, leading to constraints onthe contraction of crystals exerted by their neighbours during heating or cooling. Macros-tresses result in a displacement of the diffraction maxima from their equilibrium positions,while microstresses result in a broadening of the diffraction peaks.

Given that lattice spacings are measurable by X-ray diffraction to one part in 105, itfollows that lattice strains are detectable to approximately 10�5. For an aluminium alloywith an elastic modulus of 60GPa, this corresponds to a stress of less than 1MPa. It followsthat accurate determination of residual stress levels that are only a few per cent of the bulkyield stress should be possible. Unfortunately, measurements of residual stress using X-raydiffraction are confined to the surface layers of the sample. They may nevertheless beextremely helpful, for example in controlling the quality of surface coatings.

2.3.3 Single Crystal Laue Diffraction

The powder method depends on measuring the intensity diffracted from a monochromaticincident beam as a function of the Bragg angle, given by the relationship l ¼ 2d sin �, andidentifying the lattice planes responsible for each of the diffraction peaks in the spectrum.An alternative would be to use a beam of white radiation and determine the spatialdistribution of the intensity diffracted by a rigidly mounted sample. A particular set ofdiffracting planes in a crystal will then select a wavelength from the incident beam thatsatisfies theBragg criterion for the angle at which the crystal is oriented. In a polycrystallinesample thiswill result in jumbled sets of reflections for the different crystals, many ofwhichwill overlap with those from other crystals. The result would be a confused pattern thatcould not be interpreted. However, if the sample is a single crystal, then the reflections willform a very distinctive Laue pattern which can be used to determine the orientation of thecrystal with respect to the incident beam.

Initially a photographic film but now, more usually, an areal detector array (a charge-coupled device, CCD) is used to record the single crystal diffraction pattern in a Lauecamera. Two camera configurations are possible (Figure 2.20). If the specimen is thinenough, a Laue pattern may be recorded in transmission on a plane perpendicular to the


incident beam. The Laue reflections from a set of crystal planes that lie on the samesymmetry zone will intersect the plane of the film along an ellipse. More commonly, Lauediffraction patterns are recorded in reflection, since there is then no limitation on thethickness of the diffracting crystal. The beam passes through a hole in the centre ofthe recording plane, which is again normal to the incident beam. The symmetry zones of thereflections from the diffracting planes now intersect the detection plane as arcs ofhyperbolae. Examples of Laue patterns taken in both transmission and reflection are givenin Figure 2.21.

The interpretation of a Laue pattern depends on identifying the symmetry axes of thereflecting zones in order to determine the orientation of the single crystal with respect to anexternal coordinate system. Information on the perfection of the single crystal can also bederived from a Laue image, since the presence of sub-grains or twinned regions gives rise to

Reflectingsphere

λ4 < λ3 < λ2 < λ1

Back-reflectionlaue pattern

Transmissionlaue pattern

Singlecrystal

Incidentbeam

000

Figure 2.20 Reflecting sphere construction for single crystal diffraction using whiteX-radiation and the experimental configurations used to record Laue diffraction patterns.Each reciprocal lattice point diffracts that wavelength which satisfies the Bragg relation.


additional sets of diffracted beamswhich are displaced with respect to those due to themaincrystal.

2.3.4 Rotating Single Crystal Methods

While much crystallographic structure analysis can be achieved using randomly oriented,powder or polycrystalline samples, single crystals are often required to confirm a crystallattice model unambiguously. Monochromatic radiation is used and the crystal is mountedat the exact centre of the spectrometer on a goniometer stage. The goniometer allows thecrystal axes to be oriented accurately with respect to the spectrometer and permits

Figure 2.21 Laue diffraction patterns recorded from a single crystal sample in a symmetricalorientation: (a) in transmission, when the symmetry zones intersect the recording plane onellipses; (b)in reflection, in which case the symmetry zones intersect on hyperbolae.


continuous rotation of the crystal about an axis normal to the plane containing the incidentbeam and the detector. Rotation brings reciprocal lattice points lying on planes parallel tothe plane of the spectrometer into the reflecting condition, generating layer lines ofreflections (Figure 2.22).

The single crystal must be large enough to ensure sufficient resolution for structureanalysis in reciprocal space, but not so large as to result in geometrical blurring of thereflections in real space. Using X-rays, crystals between 0.1mm and 1mm in size aresuitable. The range of the rotation angle is restricted in order to reduce overlap frommultiple reflections, and several spectra must be recorded by rotating the crystal about theprominent symmetry axes.

2.4 Diffraction Analysis

So farwe have only discussed thegeometry of diffraction and shown howa determination ofthe angular distribution of the diffracted beams can be used to identify the crystal symmetryand determine the lattice parameters to a high degree of accuracy. This information isusually sufficient to identify the crystalline phases present in a solid sample unambiguously,but there is a great deal of information present in the relative intensities of the diffractedbeams that we have not yet utilized. Tomake use of this information we examine the factorswhich determine the scattered amplitude that we have measured. These factors include thescattering by individual atoms, the summation of the atomic scattering by the unit cells ofthe crystal lattice, the summation over the individual grains of a polycrystal, and thedetection of the diffracted radiation by the diffractometer assembly.

Incidentbeam

Layer lines

000

Figure 2.22 Rotating a single crystal about an axis perpendicular to the plane of thespectrometer and using monochromatic radiation brings each lattice point in turn into thediffracting condition, generating layer lines of reflections.


2.4.1 Atomic Scattering Factors

We confine our attention to incident X-rays and electrons, which are scattered by electronsin the solid, and we ignore diffraction of incident neutrons, which are scattered by theatomic nuclei. If a is the angle between the scattering direction of the incident beam and thedirection in which an interacting electron is accelerated, then J.J. Thomson showed that thescattered intensity is given by:

I ¼ I0e4

r2m2c4sin2a ð2:9Þ

where e and m are the charge and mass of the electron, c is the velocity of electromagneticradiation and r is the distance of the accelerated electron from the incident beam(Figure 2.23).

For an unpolarized X-ray beam we need to average the effect of the electric fieldcomponents that act on the electromagnetic wave (Figure 2.24). The electric field acts

α

Scatteredphoton

Incidentphoton

Electron

r

Figure 2.23 Scattering of an X-ray beam by an electron (see text).

Figure 2.24 If scattering occurs at an angle 2y in the x–z plane, then the applied electric field isin the y–z plane and the average values of the components Ey and Ez must be equal.


perpendicular to the plane of scattering, x� z, and lies in the y� z plane, with componentsEy andEz. For an unpolarized beam, these two components are, on average, equal, while thevalues of a, the angle between the components ofE and the scattering direction, are given byay¼ p/2 and az¼ (p/2� 2y). Assuming that each component contributes half the totalintensity, the factor sin2a should be replaced by ðsin2ay þ sin2azÞ=2. Substituting for ay andaz leads to the relationship

I ¼ I0e4

r2m2c4:ð1þ cos22�Þ

2ð2:10Þ

where the term ð1þ cos22�Þ=2 is called the Lorentz polarization factor.Each atom in the sample contains Z electrons, where Z is the atomic number. In the

direction of the incident beam all the electrons in the atomwill scatter in phase. If the atomicscattering factor f(y) is defined as the amplitude scattered by a single atom divided by theamplitude scattered by an electron, then it follows that f(0)¼ Z, while for y> 0, f(y)< Z,since at larger scattering angles the electrons around an atomwill scatter increasingly out ofphase. The y dependence of the atomic scattering factors for iron (Fe), aluminium (Al) andtungsten (W) are shown in Figure 2.25.

2.4.2 Scattering by the Unit Cell

The next step is to derive the amplitude scattered by a unit cell of the crystal structure. Anypath difference d between the X-ray beam scattered from an atom at the origin and fromanother atom elsewhere in the unit cell will correspond to a phase difference between thetwo scattered beams which is given by j ¼ 2pd=l. If the position of the second atom isdefined by thevector r in the direction [uvw], such that the coordinates of the atom in the unitcell are (x, y, z) with u¼ x/a, v¼ y/b andw¼ z/c, and the atom lies on the plane (hkl), defined

0 0.2 0.4 0.6 0.8 1.0 1.20.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

f (θ)

Sin(θ)/λ [Å-1]

W

Fe

Al

Figure 2.25 The atomic scattering factor as a function of Z and y for aluminium, iron andtungsten.


by the reciprocal lattice vector g (Figure 2.26), then the phase difference for radiationscattered by these two atoms into the hkl reflection is just:

jhkl ¼ 2pðhuþ kvþ lwÞ ¼ 2pg · r ð2:11ÞEach atomwill scatter an amplitudeA, which depends on the atomic scattering factor fZy forthat atom, andwe can represent both the phase and the amplitude of the scatteredwave fromeach atom by a vector A. Using complex notation, Aeij ¼ Aðcosjþ isinjÞ, and thecontribution to the amplitude scattered into the diffracted beam hkl by an atom at uvwin the unit cell will be given by (Figure 2.27):

Aeij / f exp½2piðhuþ kvþ lwÞ� ¼ f expð2pig · rÞ ð2:12Þ

Figure 2.26 Geometry of atomic positions and scattering planes in the unit cell.A

isin

(φ)

AA

Acos( φ)

A 1

A2

A3ΣAn

isin

(φn)

ΣAncos(φn)φ1

φ2

φ3

Resultantamplitude

Figure 2.27 The amplitude–phase diagram and its use to sum the scattered amplitudescontributing to a particular reflection by all the atoms in the unit cell.


2.4.3 The Structure Factor in the Complex Plane

By ignoring the constant of proportionality, which corresponds to the scattering due to asingle electron, we can define a normalized scattering factor due to a complete unit cell ofthe crystal for an hkl reflection by summing the contributions to this reflection from all theNatoms in the unit cell. This new parameter, the structure factor for the hkl reflection, is thengiven by:

Fhkl ¼XN1

fn exp½2piðhun þ kvn þ lwnÞ� ¼XN1

fn expð2pig · rÞ ð2:13Þ

Before proceeding further, we note that, in the complex plane, the following relationshipshold:

1. eij ¼ cosjþ i sinj, so that the real component of the amplitude is resolved along thex-axis and the imaginary component along the y-axis of phase space.

2. I ¼ jAj·jA*j, that is the intensity scattered by any combination of atoms is derived fromthe phase space vector amplitude bymultiplying the real component of the amplitude byits complex conjugate.

3. enpi ¼ ð�1Þn, so that phase angles corresponding to even and oddmultiples of p have noimaginary component of the amplitude, and simply add or subtract from the totalscattered amplitude.

4. eix þ e�ix ¼ 2cosx, which is a second condition for no imaginary component.

Copper has an FCC unit cell containing four atoms, each atom situated at a Bravaislattice point, so that the values of [uvw] are [0 0 0], [1/2 1/2 0], [1/2 0 1/2] and [0 1/2 1/2]. Itfollows that the structure factors for copper are given by: Fhkl ¼f ½1þ epiðhþ kÞ þ epiðhþ lÞ þ epiðkþ lÞ�. If h, k and l are all odd or all even, then Fhkl¼ 4f, butif h, k and l are mixed integers, then Fhkl¼ f(1þ 1� 2)¼ 0, the same result we have notedpreviously.

In the BCC unit cell, characteristic of a-Fe, the atoms are at the twoBravais lattice points[0 0 0] and [1/2 1/2 1/2], and the structure factors are given by Fhkl ¼ f ½1þ epiðhþ kþ lÞ�. Forhþ kþ l even it follows that Fhkl¼ 2f, while if hþ kþ l is odd, then Fhkl¼ f(1� 1)¼ 0,again as we noted previously.

Cubic diamond has an FCC unit cell in which each lattice point corresponds to twoatoms, one at the site of the lattice point and the other displaced by a vector [1/4 1/4 1/4].This is equivalent to two interpenetrating FCC lattices related to one another by thissame displacement vector. It follows that the structure factors are given byFhkl ¼ f ½1þ epiðh þ kÞ þ epiðhþ lÞ þ epiðkþ lÞ�½1þ epi=2ðhþ kþ lÞ�. There are now three possi-bilities. For hþ kþ l odd, the amplitude vector for the set of four �second� atoms has no realcomponent and this vector points either vertically up or vertically down. In both cases thestructure factor for the allowed FCC reflections is increased by

ffiffiffi2

pover that for a single

atom at the origin. A resultant phase angle of p/4 is introduced. For hþ kþ l even, only thereal component exists, and may be either negative, reducing the structure factor of anallowed FCC reflection to zero, or positive, doubling the structure factor to twice the valuefor a single atom at each lattice point.

Finally, consider the case of common salt (NaCl, with an FCC structure). In this case thecations sit on the Bravais lattice points while the anions occupy a second FCC lattice


displaced by [1/2 1/2 1/2]. The scattering factors of the cations and anions are different, andthe structure factors for the different reflections are now given by the relationshipFhkl ¼ ½1þ epiðhþ kÞ þ epiðhþ lÞ þ epiðkþ lÞ�½ fNa þ fClepiðhþ kþ lÞ�. Even values of hþ kþ lnow result in reinforcement of the intensity for the allowed FCC reflections: Fhkl¼ 4(fNaþ fCl), while odd values of hþ kþ l reduce the intensity Fhkl¼ 4(fNa� fCl).

2.4.4 Interpretation of Diffracted Intensities

Weare now in a position to summarize all the physical factors that determine the intensity ofan observed diffraction peak in a recorded spectrum:

1. The Lorentz polarization factor is associated with scattering of unpolarized electro-magnetic radiation:

1þ cos22�

2

� �ð2:14Þ

The exact form of the polarization factor is dependent on thegeometry of the diffractometer(see 4 below).2. The structure factors for the different reflecting planes in the crystal lattice, which

include the effect of the atomic scattering factors for all the atoms present in thematerial:

Fhkl ¼XN1

fnexp½2piðhun þ kvn þ lwnÞ� ð2:15Þ

3. Themultiplicity of the reflecting planesP, which gives the number of planes belonging toa particular family of Miller indices (determined by the symmetry of the crystal). Forexample, in cubic crystals, planes whose poles fall within the unit triangle have amultiplicity of 24, since there are 24 unit triangles, while thosewhose poles that lie alongthe edges of a unit triangle (and are therefore common to two triangles) have amultiplicity of 12. However, there are four {111} planes in the stereogram, whichcorrespond to the apices of the unit triangles (shared by six triangles) and yield amultiplicity of 4, while the {100} reflections correspond to poles on the coordinate axes,shared by eight triangles, and have a multiplicity of 3.

4. The sampling geometry. In the powder method a collector of finite cross section onlysamples that proportion of the cone of radiation which is diffracted at the Bragg angle.The fraction of radiation collected is given by the Lorentz-polarization factor:

L ¼ 1þ cos22acos22�sin2�cos�ð1þ cos22aÞ ð2:16Þ

where y is the diffracting angle determined from Bragg’s law, and a is the diffractingangle of any monochromator in the diffractometer.

5. Absorption effects that depend on the size of the sample and its geometry. In general,absorption can be expected to increase at large values of the diffraction angle. Theabsorption correction can be written as A0(y), and can be estimated for given geometriesand sample densities, using standard tables of mass absorption coefficients For a thick


specimen in the Bragg–Brentano diffractometer the absorption factor will not be afunction of the diffraction angle and is simply A0 ¼ 1/2 m. Thus this factor cancels whencalculated integrated intensities are normalized.

6. Temperature is also an important factor, since at high temperatures random atomicvibration will reduce the coherence of the scattering from the more closely spacedcrystal planes, corresponding to the higher values of y. This effect increases with siny/land is more significant at small d values and larger values of hkl. Assuming an isotropicbehaviour, its influence on the overall integrated intensity can be expressed by thefactor:

e�2Bsin2�=l2 ð2:17Þwhere

B ¼ 8p2 �U2 ð2:18Þand �U

2is the mean square displacement of each atom.

Summing all the above effects yields a general relationship for the diffracted integratedintensity:

I ¼ kjF2j 1þ cos22acos22�sin2�cos�ð1þ cos22aÞPAð�Þ·exp � 2Bsin2�

l

� �ð2:19Þ

where k represents a scaling factor which includes I0. Equation 2.19 can be used to calculatesimulated integrated intensities for any given structure from an appropriate computerprogram. The net integrated intensity is first calculated for each hkl reflection, and then allthe calculated intensities are normalized with respect to themaximum calculated integratedpeak intensity, which is assigned a value of 100%:

Inhkl ¼IhklImaxhkl

·100 ð2:20Þ

2.4.5 Errors and Assumptions

In the present treatment there is little justification for an exact analysis of the errorsinvolved in X-ray diffraction measurements, but some semi-qualitative discussion isnecessary. It is important to distinguish between errors in the measurement of peakpositions and errors in the determination of peak intensities. It is especially importantto recognize the need for accurate calibration and alignment if small changes in latticeparameter (of the order of 10�5) are to be resolved. X-ray diffraction may only be able tosample a small volume of material, but this can be an advantage in the analysis of thinfilms, coatings and solid state devices. Such applications are critical for a wide range ofsystems where the engineering properties are associated with the surface and near-surface regions, such as microelectronic components, optronic devices, wear parts andmachine tools. The monitoring of surface stress or chemical change at the surface canoften be accomplished by X-ray diffraction, and in situ commercial X-ray systems areavailable for extracting measurements at high temperatures or in a controlledenvironment.


As an example, we summarize the errors in measuring lattice parameters by using adiffractometer in the Bragg–Brentano geometry. The Bragg–Brentano diffractometer is themost common automated diffractometer and has a well-defined optical focusing system.The focal circle (Figure 2.28), is tangential to both the source and the specimen surface.Since the specimen is rotated in increments of y, and the detector in increments of 2y, theradius of the focal circle decreases as y increases, and the detector must remain on the(changing) focal circle in order to minimize the spread of the signal beam. These operatingconditions control the three major errors in peak measurement that limit the accuracy whendetermining the lattice spacings and lattice parameters.

The first is peak broadening, and is due either to incorrect alignment of the diffractome-ter, or tomisplacement of the specimenwithin the diffractometer goniometer. This error can

Source

θ1

Sample

θ1

Source

θ1

Sample

FocalCircle

2θ1

θ1

Figure 2.28 The Bragg–Brentano geometry at two different Bragg angles.


be reduced, either by better alignment or by the use of smaller receiving slits in front of thedetector, but residual broadening will always lead to some error in the measurement of thediffraction angle.

The second and third sources of error are both directly related to the specimen geometry.The diffraction signal originates from a region beneath the specimen surface, not from thesurface itself. This results in an error in the apparent diffraction angle, as illustrated inFigure 2.29. A similar error will result if the specimen is placed either above or below thefocal point on the goniometer axis. Using Figure 2.29 we derive the resulting defocus errorin y from the relationship:

sinD2�h=sin�

¼ sinð180�2�ÞR

ð2:21Þ

and thus:

D2�ffi �2hcos�

Rð2:22Þ

Finally,

Ddd

¼ �D2�tan�

¼ cos�

sin�

2hcos�

R¼ k

cos2�

sin�¼ Da

að2:23Þ

Errors in lattice parameter measurements can be reduced by extrapolating the valuescalculated from each observed diffraction peak to a (hypothetical) value for y¼ 90�.That is,by plotting the lattice parameters as a function of cos2y/siny and extrapolating to 0. Analternative is to spread some powder from a diffraction standard (with known latticeparameters) on the surface of the specimen. The peak positions due to diffraction from thestandard powder can then be used to correct for systematic errors.

θh

R

Surface

θ

θ

∆ 2 θ

∆ 2 θ

Figure 2.29 A diffraction signal from below the focusing plane (sample surface) resulting in anerror in 2y.


Some additional factors should be considered in the measurement of peak intensities,especially those associated with efficient data collection. There are two considerations: thegeometry of the sample in the spectrometer, and the response of the detector to the incidentphotons. Both have been mentioned previously. Some geometrical peak broadening isalways associated with the finite diffracting volume in the sample, and while the detectorresponse should remain constant over long periods it will always bewavelength-dependent.

Measurement of the comparative intensities from different diffraction peaks (expressedas a percentage of the intensity of the strongest diffracted peak) is an important diagnostictool, both for identifying unknown phases in the sample, and for refining crystal structuremodels. Such measurements are sensitive to the presence of preferred orientation. Themeasured height of a diffraction peak in an experimental spectrum is a rather inaccurateestimate of the relative peak intensity, and it is important to determine the integratedpeak intensities, that is the total area under the peak, after subtracting background noise,(Figure 2.30).

In addition to a dependence on preferred orientation, the integrated intensity of adiffraction peak may vary with temperature (see above), or as a result of transmissionlosses (if the specimen thickness is below the characteristic absorption thickness). Thereduction in integrated intensity associated with a thin sample can be corrected bymultiplying the integrated intensity using a factor that takes into account the pathlengthin the sample:

½1�expð�2mt cosec�Þ� ð2:24ÞFor thick specimens, this factor reduces to 1. If the absorption coefficient m is known, then inprinciple the thickness of a thin sample t can be measured (not very accurately) bycomparing the intensities with those from a bulk sample. For random samples, mt is bestdetermined from relative integrated intensities by a refinement procedure based on theentire diffraction spectrum, but we can also compare the intensity ratio of a single pair of

50 55 60 65 700

100

200

300

400

500

600

2θ

Inte

nsit

y (a

.u.)

BackgroundMeasured Intensity

Figure 2.30 A weak diffraction peak, illustrating the calculation of integrated peak intensityfrom the total area beneath the peak after subtracting background noise.


reflections recorded at two Bragg angles y1 and y2, I(t)¼ I1/I2 to I(1) for the bulk sample,using the expression:

Y ¼ I tð ÞI 1ð Þ ¼

1�expð�2mtcosec�1Þ1�expð�2mtcosec�2Þ ð2:25Þ

This method of determining thickness is practicable when mt has values in the range0.01–0.5.

Preferred orientation changes the relative values of the integrated peak intensitiesobserved in the diffraction pattern, and reflects the presence of microstructural featuresthat may strongly affect the material properties. Accurate methods for determiningpreferred orientation, especially the probability of finding a crystal having a specificorientation (the crystallite orientation distribution function) are beyond the scope of thistext. However, there is a fairly simple method which is often used to determine qualitativevalues of texture. This method (termed the Harris method) is based on a relation for thevolume fraction of a phase having a crystal orientation lying within a small solid angle dOabout an angle (a, b, g) in an inverse pole figure (see below):

Pða; b; gÞdO=4p ð2:26ÞwhereP(a,b, g) depends only on a,b, and g. Since every crystalmust have some orientation:

1

4p

Z ZPða; b; gÞdO ¼ 1 ð2:27Þ

For randomly oriented materials P is independent of a, b, and g:

P

4p

Z ZdO ¼ 1 ð2:28Þ

so that P¼ 1 for a random polycrystal. Values of P greater than one then indicate that thecorresponding crystallographic direction has a higher probability than would be found in arandom polycrystal, while values of P less than one indicate that this direction is less likelyto be found than in a random polycrystal. P can be experimentally determined in a Bragg–Brentano diffractometer using:

Pða; b; gÞ ¼ IðhklÞPIðhklÞ

PI0ðhklÞ

I0ðhklÞ ð2:29Þ

where I(hkl) is the measured integrated intensity from the plane (hkl) of the sample, andI0(hkl) is the measured or calculated integrated intensity from the same plane in a randomlyoriented standard of the samematerial. Values ofP are plotted on an inverse pole figure. Thisis a stereographic projection whose coordinate system corresponds to the geometry of thesample and on which contours of relative peak intensity are plotted. The contours ofconstant P then define the degree of texture relative to the specimen geometry for the hklreflecting plane used in the measurements.

The measured integrated intensities not only indicate the degree of texture for anyselected diffracting plane, they can also determine the amount of each phase present in amultiphase material.Quantitative phase analysis is important for determining the effect ofdifferent processing parameters on the phase content. To determine the amount of the phase


a in amixture of a and b, we first redefine the integrated intensity in Equation 2.19, isolatingthe constants and focusing on the significant variables:

Ia ¼ K1camm

ð2:30Þ

where ca is the concentration of phase a, mm is the linear absorption coefficient for themixture of phases, andK1 is a constant. The linear absorption coefficient will depend on therelative amount of each phase present in the mixture:

mmrm

¼ oamara

� �þob

mbrb

!ð2:31Þ

where o is the weight-fraction of each phase and r is its density. Rearranging Equation(2.31) and combining with Equation (2.30) yields:

Ia ¼ K1cacaðma�mbÞþ mb

ð2:32Þ

Comparing Ia from the phase mixture with Ia from a pure sample of the a phase, we obtain:

IaIa;p

¼oa

mara

� �oa

mara� mb

rb

� �þ mb

rb

ð2:33Þ

If we know themass absorption coefficients for each phase, Equation (2.33) is sufficient forquantitative phase analysis, but if the mass absorption coefficients are unknown then weneed to prepare a set of standard specimens (usually mixed powders) and construct acalibration curve. In any case, experimental calibration is usually a good idea, since thevariation of Equation (2.33) withoa is nonlinear, as a result of the very different absorptioncoefficients of the different phases.

ManyotherX-ray diffraction techniques are beyond the scope of this book. These includetheX-ray determination of particle size, X-ray residual stress analysis, structure refinementby spectrum fitting (Rietveld analysis) and thin-filmX-ray techniques (especially importantfor semiconductor device technology). References for some of these methods can be foundin the Bibliography for this chapter.

2.5 Electron Diffraction

The dualwave-particle nature of electrons is expressed by the deBroglie relationship for themomentum of an electron, p¼mv¼ h/l, where m is the mass and v the velocity of theelectron, and l is the electron wavelength. Substituting for the electron energy, inelectronvolts, eV¼ 0.5mv2, yields an equation for the nonrelativistic electron wavelength:

l ¼ hffiffiffiffiffiffiffiffiffiffiffiffi2meV

p ð2:34Þ

At an accelerating voltage of 100 keV, the electronwavelength is 0.0037 nm,much less thanthe interplanar spacing in crystals, so that the Bragg angles for electron diffraction are


always very small when compared with those for X-ray diffraction. That is, the elasticscattering of electrons occurs at very small angles. Electron diffraction is, as we shall see,both amajor source of contrast in thin film electronmicroscopy, and an important analyticaltool in its own right.

2.5.1 Wave Properties of Electrons

The de Broglie relationship is not sufficient to define thewavelength of an electron at highenergies, since relativistic effects become important at these energies. If the rest mass ofthe electron is m0, then the relativistic mass is given by m ¼ m0 þ eV=c2, where c is thevelocity of electromagnetic radiation (the velocity of light). Substituting in the de Broglierelation and rearranging leads us to the relativistic equation for the wavelength of theelectron l in terms of the accelerating voltage V (compare the nonrelativistic equation,above):

l ¼ h

p¼ hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2m0eVð1þ eV=m0c2Þp ð2:35Þ

The relativistic correction is significant at the accelerating voltages used in transmissionelectron microscopy, typically 100–400 keV.

An electron beamcan be focusedwith the help of electromagnetic lenses, but the focusingmechanism is quite different from that used in the optical microscopewhich, as we shall seein Chapter 3, relies on the refractive index of glass and geometrical optics to achieve a sharpfocus. A magnetic field will deflect any electron which has a component of velocityperpendicular to the magnetic field vector, and the deflecting force acts in a direction that isperpendicular to the plane containing both the velocity and the magnetic field vectors. As aconsequence, the electron follows a helical path when passing through a uniform magneticfield.

Themagnetic field generated by an electromagnetic lens is cylindrically symmetric, and adivergent electron beam passing through such a lens will be brought to a focus, providingthe angular divergence is small. We are fortunate that the elastic scattering of electrons islimited to small angles and permits both electron diffraction patterns and electronmicroscope images to be brought sharply into focus in the electron microscope. Assuminga wavelength of 0.0037 nm (the wavelength associated with 100 keV electrons) andinterplanar spacings of the order of 0.2 nm, we expect Bragg scattering angles of lessthan 1�. At these small angles it is common to quote theBragg angle in radians and to use theapproximation siny� y. The Bragg relationship can then be written l¼ 2dy, and we willuse this form of Bragg’s equation to describe the elastic scattering of electrons passingthrough a thin-film specimen in the transmission electron microscope.

Finally, the influence of inelastic scattering is often important, since electrons whichhave lost some energy by inelastic scattering will have a longer wavelength. In such a case,the electron beam will no longer be monochromatic, and cannot therefore be brought to asharp focus. The stopping cross-section for electrons s is defined as s ¼ ð1=NÞðdE=dxÞ,whereN is the number of atoms per unit volume and dE/dx is the rate of energy loss per unitdistance travelled by the electrons. As the energy of the electrons is increased (a higheraccelerating voltage in themicroscope), s decreases, although at very low electron energiesthe innershell electrons will no longer contribute to inelastic scattering of the incident beam


and the scattering cross-section will actually decrease. The general shape of the scatteringcross-section curve is shown in Figure 2.31. The inelastic scattering cross-section increaseswith Z, so that higher atomic number materials can only give sharp electron diffractionpatterns if the sample is very thin. For a 200 keV incident electron beam, tungsten or goldfilms thicker than about 100 nm will absorb the incident beam energy and can only giveelectron diffraction patterns in reflection, when the electron beam is at glancing incidence.The maximum thickness for transmission electron diffraction from steel at 200 kV is of theorder of 120 nm, while silicon and aluminium specimens must be less than about 150 nm inthickness. These values are two orders of magnitude less than the maximum samplethickness for a transmission X-ray diffraction experiment.

2.5.1.1 The Limiting Sphere for an Electron Beam. The limiting and reflecting sphereconstructions that we have used to analyse X-ray diffraction phenomena in reciprocal spaceare equally valid for electron diffraction patterns obtained in transmission electronmicroscopy (TEM), although there are two significant differences.

The first concerns the very short wavelength of the electrons when compared with theinterplanar spacings in crystals, that is |k0|¼ |k|�|g| (Figure 2.32). As a consequence, asnoted above, siny� y. In Figure 2.32 the reflecting sphere construction has been rotated by90�, a symbolic nod to the engineering design of the modern electron microscope, in whichthe beam source (the electron gun) is almost invariably mounted vertically, generating abeam of electrons which penetrates a thin-film specimen mounted in the horizontal plane.This may be compared with the standard design for X-ray diffraction units, in which theX-ray beam is usually generated in the horizontal plane in order to ensure maximummechanical stability.

The second modification concerns the effective size of the reciprocal lattice points inelectron diffraction. The elastic scattering cross-section for electrons is much greater thanthat for X-rays, so that the intensity scattered into the diffracted beam increases rapidlywithsample thickness, to the point at which all the energy in the incident beam may betransferred into the diffracted beam and the diffracted beam starts to be rediffracted backinto the incident beam (Figure 2.33). As we shall see, this process, termed double

103 104 1050

1

2

3

4

5

6

I/N

(dE

/dX

)In

elas

tic

scat

teri

ngcr

oss-

sect

ion

(a.u

.)

Incident electron beam energy (eV)

Figure 2.31 Inelastic scattering cross-section as a function of incident electron beam energy.


Figure 2.33 Double diffraction of the electron beam leads to diffracted intensities whichoscillate with film thickness.

Figure 2.32 The wave vectors in electron diffraction k are very large when compared with thereciprocal lattice vectors g, allowing for some simple geometrical approximations.


diffraction, leads to oscillations in the diffracted intensity with increasing specimenthickness. These oscillations have a periodicity t0, the extinction thickness, which ischaracteristic of the electron energy and the structure factor for the actively diffractingplanes:

t0 ¼ pVc

ljFðhklÞj ð2:36Þ

where Vc is the volume of the lattice unit cell. Typical values for t0 at the acceleratingvoltages used in TEM are less than 100 nm. In electron diffraction, this extinction thicknesslimits the effective size of the reciprocal lattice points in reciprocal space to approximately1/t0, of the order of 10

�2d (compared with values of 10�4d or less in X-ray diffraction). Anadditional �small crystallite� effect may also dominate the Bragg condition in electrondiffraction, increasing the size of the reciprocal lattice points even further in very thin filmsand for very small nanocrystals (see Chapter 4).

In electron diffraction both the radius of the reflecting sphere and the size of thereciprocal lattice points are large when compared with the conditions obtained in X-raydiffraction, relaxing the diffracting condition set by Bragg’s law so that several diffractedbeams of electronsmay be scattered simultaneously from a thin sample that has a symmetryaxis in the plane of the film (Figure 2.34). Such multi-beam diffraction would not bepossible in an X-ray diffraction experiment.

2.5.2 Ring Patterns, Spot Patterns and Laue Zones

The electron diffraction pattern obtained in transmission from a thin film of a single crystalwhich is orientedwith amajor zone axis (a symmetry axis) parallel to the electron beamwillcontain all the reciprocal lattice points which are intersected by the reflecting sphere. This

Figure 2.34 Since the radius of curvature of the reflecting sphere is large and the reciprocallattice points have a finite diameter, Bragg diffraction occurs even though the Bragg condition isnot exactly satisfied.


will include all those reciprocal lattice points which surround the 000 spot (whichcorresponds to the directly transmitted beam), in so far as the (slight) curvature of thereflecting sphere permits them to fulfill the Bragg condition, but the pattern may alsoinclude additional points which lie in a layer of the reciprocal lattice above that containingthe origin at 000 (Figure 2.35). The innermost set of diffracting reciprocal lattice points isreferred to as the zero-order Laue zone, while the subsequent, outer rings of diffracting spotsare termed higher order zones.

Adequate calibration of the electron microscope enables distances in the diffractogram(the electron diffraction pattern) to be accurately interpreted as distances (angles) inreciprocal space, although there are several sources of calibration error. These include notonly the physical limitations of the technique, especially ambiguity associated with thefinite diameter of the reciprocal lattice points, but also experimental limitations associatedwith electromagnetic lens aberrations in the microscope, some inevitable curvature of thethin-film specimen, and the response of the recording medium. In general, it is not possibleto specify lattice spacings derived from electron diffraction measurements to much betterthan 2% of the lattice parameter. This is at least two orders of magnitude worse than can

Figure 2.35 Single crystal electron diffraction from more than one Laue zone: (a) mechanismof formation; (b) a diffraction pattern from a [100] oriented aluminium single crystal film.


be achieved by X-ray diffraction. While this is true for standard electron diffractiontechniques, usually termed selected area diffraction, an alternative technique, convergentbeam electron diffraction, can, for some crystals, be verymuchmore accurate andmay evenbe used to determine localized lattice strains, as well as to solve for the crystal structure.Convergent beam electron diffraction is, however, well beyond the scope of this book andthe interested reader should consult the texts by Williams and Carter, or Spence and Zuo,that are listed in the Bibliography for this chapter.

The electron beam can also be focused electromagnetically to a fine probe, orsmall apertures can be used to limit the diameter of a parallel electron beam, enablingselected area diffraction from very small specimen areas, of the order of 20 nm. Thisvolume is a minute fraction of that which can be usefully sampled by X-rays, and canyield phase information on individual crystallites and precipitates in a polycrystallinematerial.

If the area illuminated by the electron beam includes a large number of crystallites, then apowder pattern is generated, analogous to an X-ray powder pattern. In the electrondiffraction case the fluorescent screen or recording medium is positioned normal to theincident beam, and records successive rings of reflections from each family of reflectingplanes (Figure 2.36). The radiusR of a specific ring on the powder pattern is related to the d-spacing of the reflection and the wavelength of the electron beam l by the relation:

d ¼ 2lL2R

ð2:37Þ

where L is the effective camera length of the electronmicroscopewhen used as a diffractioncamera. Equation (2.37) has been written with a factor of 2 in both the numerator and thedenominator, since it is good practice to measure the distances 2R between two diffractionspots hkl and hkl in an electron diffraction pattern, in order to avoid errors associated withdetermining the position of the directly transmitted 000 beam. The parameter L can bevaried in most microscopes, in order to select a value suitable for the lattice parameters ofthe phases being studied, and good calibration of the microscope should give the term lL,the camera constant of the microscope, with an accuracy of about 1%.

2.5.3 Kikuchi Patterns and Their Interpretation

For moderately thick transmission electron microscope specimens, a proportion of theincident electrons will undergo inelastic scattering. These electrons, having lost someenergy, are deflected out of the path of the incident beam to form a diffuse halo around thecentral spot, but before exiting the specimen the same electrons may also be elasticallyscattered from the crystal lattice planes. If the specimen is a sufficiently perfect singlecrystal, this secondary elastic scattering will lead to a characteristic Kikuchi line pattern,which is usually superimposed on the single crystal spot pattern associated with Braggdiffraction of the primary incident beam. TheKikuchi line pattern arises because the diffuseangular distribution of inelastically scattered electrons falls off rapidlywith angle, typicallyobeying an I ¼ I0cos2a law, where a is now the inelastic scattering angle. The crystal latticeplanes will elastically scatter those diffuse electrons which are incident at the exact Bragg


Figure 2.36 Powder patterns in electron diffraction: (a)mechanismof generation; (b) a powderring pattern from tempered carbon steel (indices in red are from a-Fe and indices in black arefrom Fe3C).


angle. Since very little energy is lost in the initial inelastic scattering event, the Bragg angleis effectively unchanged from that for the primary beam. However, more electrons will beelastically scattered away from the distribution of diffusely scattered electrons in the regioncloser to the incident beam, leading to a dark line in the diffuse scattering pattern close to thecentre of the observed diffraction pattern and simultaneously generating a parallel whiteline at a fixed distance from the dark line, which is determined by the interplanar spacing ofthe diffracting planes (Figure 2.37). These pairs of dark and light Kikuchi lines are actuallysections of hyperbolae, but since the g vectors are so much smaller in magnitude than theelectron wave vectors k, they appear on the diffraction pattern as straight lines. From theirgeometry, the spacing of a dark/light pair of Kikuchi lines projected on the plane ofobservation is proportional to the value of 2y. Since the Bragg angles in electron diffractionare very small, the line bisecting each pair of Kikuchi lines is an accurate trace of thediffracting planes projected onto the plane of observation. Any displacement of this tracefrom the centre of the pattern is therefore an accurate measure of the angle which thereflecting lattice planes makewith the primary incident beam. The Kikuchi pattern offers ameans of calibrating crystal misorientations in the electron microscope extremely accu-rately. Because of the comparatively large size of the reciprocal lattice points in electrondiffraction, spot patterns can only be used to determine crystal orientation to within a fewdegrees.

Note that the Kikuchi pattern is formed by diffraction of the diffusely scattered electronswithin the comparatively thick slice of the sample, while the spot pattern is formed bydiffraction of the very much more intense beam of parallel primary electrons from thoseplanes whose reciprocal lattice points intersect the reflecting sphere on the zero order orhigher order Laue zones.

If the darkKikuchi line passes through the centre 000 spot of the primary beam, then theBragg condition for that reflecting plane is accurately fulfilled, and the brightKikuchi linemust pass through the centre of the diffraction spot corresponding to that plane. Theperpendicular distance from any diffracting spot to the corresponding dark Kikuchi line istherefore an accurate measure of the deviation of the primary beam from the exact Braggcondition for that spot.

When the Kikuchi pattern is symmetrically aligned with respect to the incident beam(Figure 2.38), then the incident beam is accurately parallel to a symmetry zone of thecrystal, and this zone can usually be readily identified from the Kikuchi pattern. A series ofKikuchi patterns, taken by tilting the specimen about two axes at right angles in the plane ofthe specimen, can be used to generate a Kikuchi map (Figure 2.39). Since the Kikuchi mapaccurately reflects the crystal symmetry, it can be used to identify the orientations of anyspecific grain in the electron microscope, almost by inspection. Kikuchi patterns are oftenused to align a crystal exactly on a zone axis, or to shift a crystal off a zone axis by any givenangle. With the help of a Kikuchi map, you can tilt the crystal in a controlled way from onezoneaxis to another.

Summary

The regular arrays of atoms in a crystal scatter short-wavelength radiation elastically (eitherX-rays, or electrons, or neutrons), at well-defined angles to the incident beam. The


θ θ

θ −θ

Angular distribution

of diffuse scatteredintensity

Braggdiffracting

plane

Intensity is Bragg scattered

to form an excess line at an

angle- θ from the projection

of the diffracting plane and

leave a deficit line at θ

Figure 2.37 Mechanism of formation of Kikuchi line diffraction patterns (see text).


scattering angle and the scattered intensity are functions of the radiation, the wavelengthand the crystal structure, and this process is termed diffraction. A diffraction pattern is theangular distribution of the scattered intensity in space, and is recorded by collecting thescattered radiation, for example on a photographic emulsion or charge collection device. Adiffraction spectrum is the intensity of the diffracted radiation collected as a function of thescattering angle.

Laue showed that the allowed angles of scattering for a single crystal were a simplefunction of the lattice parameters of the crystal lattice unit cell, and Bragg simplified thescattering relations to yield Bragg’s law, l¼ 2d siny, relating the angle of scattering 2y tothe direction of the incident beam, the spacing d between the planes of atoms in the crystallattice and thewave length l. of the incident radiation The allowed angles of scattering (theBragg diffraction angles) always correspond to integer values of the Miller indices, but notall possible combinations of the Miller indices give rise to diffraction peaks, and somereflections (termed forbidden reflections) are disallowed, for example, if the Bravais latticecontains face-centred or body-centred lattice points.

The intensities of the diffraction peaks are determined by the atomic number and positionof the atoms associated with each lattice point, that is the spacegroup, theWyckoff positionsof the atoms and the atomic species. Thus some �allowed� peaks may be enhanced, ifthe atoms corresponding to each lattice point are scattering in-phase, while others may bereduced, or even absent if the scattering is out of phase and the interference isdestructive.

(242)

(202)

(202)

(224)

(242)

242022

220202

224

Figure 2.38 Kikuchi diffraction pattern froma [111] oriented aluminiumcrystal superimposedon the single crystal spot pattern.


A convenient representation of the angular positions of the allowed diffraction peaks canbe given in reciprocal space. The diffracting planes are then represented by points(reciprocal lattice points) whose positions are determined by the reciprocal of theinterplanar spacing of the diffracting planes and the direction vector normal to theseplanes. In reciprocal space the Bragg equation defines a sphere (the reflecting sphere) andany reciprocal lattice point which can be made to intersect the surface of this sphere willgive rise to a diffracted beam.

Reducing the wavelength of the incident radiation increases the radius of the reflectingsphere, and will allow more reciprocal lattice points that are further from the origin inreciprocal space (corresponding to smaller interplanar spacings), to diffract when theBraggcondition is fulfilled. Rotating the crystal or reducing the grain size in a polycrystalincreases the probability that a reciprocal lattice point will intersect the sphere and give riseto diffraction.

The volume of the specimen sampled by the incident beamdepends on the radiation used.In neutron diffraction, elastic scattering generally occurs over distances of the order ofcentimetres before energy losses due to inelastic scattering events become significant.However, even very high energy (MeV) electrons will be inelastically scattered if thesample thickness exceeds 1 or 2 mm. X-rays are an intermediate case. Most engineeringmaterials irradiatedwith X-rays whosewavelengths are of atomic dimensions will generatean elastically scattered signal from a region 20–100 mm in depth.

An X-ray diffractometer consists of a source of X-rays, a sample goniometer (whichpositions the sample accurately in space), a detector and a data recording system. Thedetector can be rotated about the sample to select the different diffraction angles. For

Figure 2.39 Kikuchi map of a cubic silicon crystal with the principle reflecting planesidentified.


some applications, a photographic emulsion or a position-sensitive detector array mayreplace the single detector. White X-radiation may also be used, for example to record adiffraction pattern from a single crystal in a Laue camera. Many X-ray diffractionstudies are based on the irradiation of a polycrystalline sample which diffractsmonochromatic radiation to give a powder diffraction spectrum. In diffraction from apowder specimen, fine grains of any crystalline phase will diffract at the different Braggangles in a sequence of diffraction cones which are intercepted in turn by the rotatingdetector. If the crystal grains are not randomly oriented in space, but possess somepreferred orientation (crystalline texture), then the diffraction pattern will show intensityanomalies which can be analysed in a texture goniometer, to derive a pole figure thatplots the distribution of the intensity diffracted by a particular family of crystal planeswith respect to the sample coordinates, or a crystallite orientation distribution function,that plots the probability of finding a crystal having a particular orientation as a functionof the Euler angles which describe the grain orientation with respect to the samplecoordinates.

In many cases, an accurate determination of the diffraction angles recorded in thediffraction pattern is sufficient to deduce the phases which are present and the orientationdistribution of the individual crystals. More information can be derived from a measure-ment of the relative intensities of the diffraction peaks. Each unit cell in a crystal scatters aproportion of the incident beam into each diffraction peak associated with a specific hklreflection. The structure factor gives the relative scattering power of the different hkl planesin the crystal, and can be calculated from a suitable model of the crystal structure to predictthe relative peak intensities diffracted from a random polycrystal. The structure factor isonly one of the parameters that determine the relative diffracted intensities, and diffractionanalysis must also take into account several other effects: the Lorentz polarization factor,the multiplicity of the reflecting planes, the specimen geometry, the angular dependence ofX-ray absorption losses and, at elevated temperatures, the effect of thermal vibration on thediffracted intensities.

Electron diffraction differs fromX-ray diffraction in many significant respects. First, theelectron wavelengths that are of practical importance in an electron microscope are verysmall when compared with the interplanar spacings in crystals. Also, in electron diffractionthe effects of inelastic scattering are pronounced and can lead to the formation of Kikuchipatterns. Useful transmission electron microscope sample thicknesses are always sub-micrometre. The short wavelengths in the high energy electron beam increases the diameterof the reflection sphere, which becomes very large compared with the spacing of thereciprocal lattice points, while the small volume of the sample region responsible forelectron diffraction significantly relaxes the conditions for Bragg diffraction. This lattereffect broadens the diffraction peak width and, equivalently, smears the reciprocal latticepoints over a region of reciprocal space that is no longer small in comparison with thespacing of the reciprocal lattice points.

Despite the comparative diffuseness of the reflecting sphere conditions for electrondiffraction, it is still possible to identify accurately the orientation of a sample and to alignany given crystal in the electron microscope column by using a Kikuchi line pattern. Thesepatterns are formed by diffraction from the halo of electrons that forms around the primaryelectron beam in a thick sample as a result of weak inelastic scattering.


Bibliography

1. C. Barrett, and T.B. Massalski, (1980) Structure of Metals, Pergamon Press, Oxford.2. B.D. Cullity, (1956) Elements of X-Ray Diffraction, Addison-Wesley, London.3. J.B. Cohen, (1966) Diffraction Methods in Materials Science, Macmillan, New York.4. I.D. Noyan, and J.B. Cohen, (1987) Residual Stress: Measurement by Diffraction and

Interpretation, Springer-Verlag, London.5. D.B.Willions, and C.B. Carter, (1996) Transmission ElectronMicroscopy: ATextbook

for materials Science, Plenum Press, London.6. J.C.H. Spence, and J.M. Zuo, (1992) Electron Microdiffraction, Plenum Press,

London.

Worked Examples

Let us now use the techniques we have discussed in this chapter for the characterization ofour selected materials. We start with a simple example: An automated Bragg–Brentanodiffractometer is used to verify the crystal structure of ametal powder andmeasure its latticeparameters. Armed with a good powder diffraction spectrum from a samplewhich has beenaccurately mounted in the diffractometer, we use a literature database, the Joint Committeeof Powder Diffraction Standards (JCPDS, now called the International Centre for Diffrac-tion Data). JCPDS is a database of experimentally observed and calculated diffractionspectra, and lists both d-spacings and relative intensities. These diffraction data can becomparedwith ourmeasured spectrum in order to identify the phases present in our sample.The JCPDS data come in two formats:

1. Tabulated cards for the different spectra, which can be accessed either from the commonnames or chemistry of the compounds, or by the d-spacings of the strongest observedreflections.

2. A computerized database which can be accessed using a computer program whichautomatically compares the major d-spacings derived from a spectrum with those listedin the database. While Tabulated cards for the different spectra still exist in manyuniversity libraries, most laboratories now use the computerized database.

TheX-ray diffraction spectrumof the powderwewish to identify is shown in Figure 2.40.We first generate a table of the d-spacings calculated from the prominent reflections byusing Bragg’s law and the known wavelength for CuKa radiation (l¼ 0.1540598 nm). Weinput these d-spacings into the computerized JCPDS database, and the output identifies forus the possible phases which best match the experimentally observed d-spacings and theirrelative intensities. Nowwe extract the data for each of the selected options from the JCPDSdatabase to compare the standardwith themeasured values of d-spacing and intensity (sincethis is a powder sample, texture should not be a problem). The �unknown� powder is nickel(Ni), whose JCPDS card is shown in Figure 2.41. Of course our sample could havecontained several different phases, but in such cases additional information is usuallyavailable, either as prior knowledge of the phases that might exist, or from the chemicalcomponents expected to be present in the material.


From the same powder diffraction spectrum of Ni powder, we can also determine theexact lattice parameter. Any variations in lattice parameter would indicate either thepresence of residual stress (very unlikely in a powder specimen), or a nickel-based alloy thatcontains one or more alloy additions in solid solution. By careful calibration of thespectrometer, and assuming a linear dependence of the lattice parameter on composition

Figure 2.40 X-ray powder diffraction pattern from an �unknown� sample.

d4-0850

2.03 1.76 1.25 2.034

100 42 21 100I/IS

4-0850

Ni

Nickel

4-0850 MINOR CORRECTION

Rad. CuKα λ 1.5405 Filter NiDia. Cut off Coll.I/It G.C.Diffractometer d corr.abs?Ref. Swanson and Taige,JC FEL. Reports, NBS

Sys. CUBIC S.G.O M5 - FM3M

a0 3.5238 b0 c0 A Cα β γ Z 4Ref. IBID.

SPECTROGRAPHIC ANALYSIS SHOWS <0.01% EACHOF Mg, Si, AND Ca.AT 26CTO REPLACE 1-1258, 1-1260, 1-1266, 1-1272,3-1043, 3-1051

dA I/I1

100422120

74

1415

2.0341.7621.2461.06241.01720.88100.80840.7880

111200220311222400331420

hkl dA I/I1 hkl

Ref.2V Color

SignD,8.907 mp

n α β ξ γξ α

1951

Figure 2.41 The JCPDS card for nickel. Reproduced by permission of the International Centrefor Diffraction Data.


(Vegard’s law), we could also determine the concentration of the alloy from an exactdetermination of the lattice parameter.

Figure 2.42 shows the results for our sample, in which the apparent lattice parameter fornickel, determined from the individual reflections and using the known relationship betweenhkl and the lattice parameter for a cubic structure, has been plotted as a function of cos 2y/siny. The systematic errors are quite small, and the best value for the measured latticeparameter is obtained by extrapolating to y¼ p/2: a¼ 0.3522(6) nm. This is close to thevalue listed both by the JCPDS (a¼ 0.35238 nm) and in Pearson’s Handbook of Crystallo-graphic Data for Intermetallic Phases (a¼ 0.35232 nm). To confirm that a solid solution ispresent we would need to prepare a calibration curve for different alloy concentrations anduse analytical techniques to determine the chemistry of the samples.

The same approach can be used to determine the lattice parameters of a polycrystallinesample of a-Al2O3 (Figure 2.43). For alumina the situation is more complicated, since wehave to refine two lattice parameters for the hexagonal unit cell that commonly defines thestructure. This can be done using a simple computer program. We could also determine acorrection factor to account for any systematic errors in our experimental results by using astandard sample, and then adjusting themeasured lattice parameters for the phases of interest.

Figure 2.44 shows a diffraction spectrum from a two-phase mixture of a-Fe and Fe3C in1040 steel. The very weak reflections from the carbide reflect the low volume fraction ofcarbide in the steel. Some of the carbide reflections also overlap with those of iron and thisfurther complicates the analysis of the spectrum. Careful inspection of the diffractionspectrum, and a comparison with simulated diffraction patterns, ensures that we identifyeach of the reflections correctly.

Figure 2.45 is a diffraction spectrum from a completely different type of sample: a thinpolycrystalline film of aluminium deposited on a thin film of TiN formed on an even thinnerfilm of titanium. These films were deposited sequentially on a single crystal, siliconsubstrate. In order to detect such thin films in a Bragg–Brentano diffractometer, longcounting times are required for each value of 2y, of the order of 20 s in the case of the pattern

-1 -0.5 0 0.5 1 1.5 20.350

0.351

0.352

0.353

0.354

0.355

Extrapolated lattice parameter:a=0.3522 (6)nm

cos(2θ)/sin(θ)

Lat

tice

para

met

er(n

m)

Figure 2.42 The lattice parameter of nickel determined from d-spacings taken fromFigure 2.40, showing the method of extrapolating the data to 2y¼ 180 � to minimize errors.


shown in Figure 2.45. Specialized thin film diffractometers significantly reduce thecounting time required for such specimens, but here we only consider the Bragg–Brentanodiffractometer geometry.

The reflections in Figure. 2.45 have all been indexed. The high intensity peak fromsilicon occurs because the silicon single crystal has been oriented in the diffractometer todiffract from the {400} planes. Even a slight misalignment of the silicon crystal wouldremove this reflection from the spectrum.

Consider only the reflections due to the deposited thin films: just a few reflections aredetectable, reflections that, according to the JCPDS, are not necessarily from the strongestreflecting planes. It follows that the deposited films must have a preferred orientation withrespect to the plane of the silicon substrate crystal. Qualitatively, the titanium film has a

Figure 2.43 X-ray powder diffraction pattern from a-alumina

30 40 50 60 70 80 90 1000

500

1000

1500

2000

2θ

Inte

nsit

y(a

.u.)

α-Fe(011)

α-Fe(002)

α-Fe(121) α-Fe

(022)

Fe3C(031)

Figure 2.44 X-ray powder diffraction pattern from 1040 steel.


texture in which the [0001] direction in the titanium lattice is normal to the silicon substratesurface, while the TiN and aluminium films both have a texture in which a [111] direction isnormal to this surface.

We next consider a calibration curve prepared for quantitative phase analysis of a samplecontaining a phase mixture of alumina and nickel (Figure 2.46). The calibration curve wasprepared by mixing known amounts of the two phases in a powder form. The strongdeviation from linearity is due to differential X-ray absorption and confirms the need forsuch a calibration curve.

Figure 2.45 X-ray powder diffraction pattern from a thin polycrystalline film of Al/Ti/TiNdeposited on a single crystal of Si: (a) has an expanded angular scale to show the details of thespectrum around the {400} Si substrate peak; (b) has an expanded intensity scale that shows thedetails of the reflections from the multilayer coating.


Now let us move on to electron diffraction. As stated earlier, selected area electrondiffraction does not have the precision which is available in X-ray diffraction, but we caneasily obtain diffraction patterns from single grains in a polycrystalline sample, or fromselected regions within any single grain. In Chapter 4, we will use selected area diffractionto correlate the crystallographic orientation of a grainwith the contrast due to lattice defectsobserved in an electron micrograph. We need to solve a selected area diffraction pattern inorder to identify the reflections responsible for contrast in the image, aswell as to determinethe zone axis parallel to the incident electron beam and the principle directions in the planeof the thin film sample.

Our first example is a selected area electron diffraction pattern from a randomly orientedaluminium polycrystal (Figure 2.47). The ring pattern results from the intersection of thediffraction cones from the reciprocal lattices summed over a large number of fine, randomlyoriented grains in the polycrystalline sample with the reflection sphere. The d-spacing foreach ring of this diffraction pattern is determined bymeasuring the diameter of the ring andinserting the known value of lL (the microscope camera constant) in Equation (2.37). ForFigure 2.47, lL was determined and the corresponding d-spacing of each ring is given. Wecan either use the information for aluminium from the JCPDS file to insert the hkl valuesresponsible for each ring, orwe can calculate a list of the d-spacings for every possible hkl inaluminium by using the relationship between the d-spacing and the lattice parameter for acubic crystal. This calculationwill not tell us if the reflection has a non-zero structure factor,but calculation of the structure factor for each reflection is also quite simple, especially if wehave an appropriate computer program.

The first step in solving any selected area diffraction pattern is to calibrate the values oflL for the different camera length settings of the microscope. Although most transmissionelectron microscope monitors display a value of the camera length L, this value is only

Figure 2.46 A calibration curve for quantitative analysis of a two phase mixture of nickel andalumina, showing the ratio of the integrated peak intensities summed for nickel and alumina, asa function of the known nickel content


approximate, and calibration is always necessary using a standard specimen of knownlattice parameter. We use aluminium (which is actually not a very good choice, since thelattice parameter of aluminium is sensitive to dissolved impurities). A single crystal regionof an aluminium foil is oriented perpendicular to a low-index zone axis and a series ofdiffraction patterns is recorded for various values of the camera length L at an acceleratingvoltage of 200 kV (Figure 2.48).

Canwe index an electron diffraction patternwithout knowing either lL or the zone axis ofthe crystal? Actually, for the case of aluminium there is a rather simple solution to thisproblem. Aluminium has a FCC structure, for which:

1

d2¼ h2 þ k2 þ l2

a2ð2:38Þ

0.230.20

0.14

0.12

0.11

0.010.093

0.090.08hkl d (nm)

111

002

220

311

222

004

313

042

422

0.233

0.202

0.143

0.122

0.116

0.101

0.092

0.090

0.082

Figure 2.47 Selected area electron diffraction pattern from a polycrystalline aluminiumspecimen. Since the average aluminium grain size is much smaller than the selected area, a�ring� pattern is formed. The measured d-spacings of the rings are indicated, and a table of d-spacings for different planes in aluminium is given.


Putting (h2þ k2þ l2)¼N2, and using the ratio of the d-spacings for any two reflections wecan write:

d22d21

¼ N21

N22

ð2:39Þ

L = 80 cmL = 100 cm

L = 60 cm L = 40 cm

L = 20 cm

R1 R2

Figure 2.48 Selected area electron diffraction patterns of a single crystal of aluminium,recorded at different nominal camera lengths (L).


To relate this equation between d and N to the values of R, the distance from the centre spotof the pattern to the reflections of interest, we substitute d¼ lL/R into Equation (2.39):

d22d21

¼ N21

N22

¼ R21

R22

ð2:40Þ

This would be difficult to solvewithout a computer, but we can use the data from Table 2.1.We measure the distance from the central spot to each of two diffraction spots of interest(Figure 2.48) and find that R2

1=R22 ¼ 0:5. Simple examination of Table 2.1 immediately

identifies the reflecting planes as (200) and (220). Since the lattice parameter for pure Al isknown (a¼ 0.405 nm), the relevant d-spacings can be calculated: d200¼ 0.202 nm andd220¼ 0.143 nm, and we can now return to our original relation between d and l, d¼ 2lL/2R, to calculate lL. The same procedure is possible for any diffraction pattern taken fromaluminium at any arbitrary camera length, giving an accurate calibration of the cameraconstant lL for the microscope (Figure 2.49).

In principle, standard statistical methods should be used to calculate the errors in lL, soits better to index the complete diffraction pattern.We now know that point 1 in Figure 2.48corresponds to a 200 reflection, while point 2 corresponds to a 220 reflection, and the anglesubtended by these two vectors at the origin (000), is measured to be 45�. We now take thevector cross product1 between the two directions [200] and [220], noting that the directionvectors are always perpendicular to the corresponding planes for the cubic systems. Thisgives the zone axis of the diffraction pattern, [001]. We check this by examining astereographic projection for the cubic crystal structure, and note that the angle between(200) and (220) should be 45�, with both planes on a great circle whose zone axis is [001](Figure 2.50). Now continue to move along this same great circle defined by [001] (theperimeter of the stereographic projection), taking angles between plane normals from the

Figure 2.49 Calibration curve for the true value of lL as a function of the nominal cameralength indicated on the microscope monitor.

1It is an accepted convention that the zone axis points up the microscope column, from the specimen to the electron source, thatis, normal to the emulsion side of a negative recording film. It is important to follow this convention when relatingcrystallographic directions indexed from a diffraction pattern to specific features in electron micrographs (see Chapter 4).


stereogram and correlating them to the measured angles between reflections onthe diffraction pattern. Of course lL and 2R to should also be used to measure the d-spacing for each reflection, and confirm that the indices we assign to each reflection areindeed correct. The result is a fully indexed selected area diffraction pattern (Figure 2.51).

For the FCC structure this procedure is very straightforward, but things getmore complicated for noncubic structures, for example the diffraction pattern shown inFigure 2.52 fromalumina.Remember that directions are not usually perpendicular to planesin noncubic crystals, so to determine a zone axis (that is, a direction) from a set of planes, weneed a stereographic projection which includes both planes and directions (find a computerprogram to generate your own stereographic projections). The equations listed in AppendixI can be used to determine a zone from any two indexed planes lying on a given zone, as wellas the angle between these two planes, but the answer should be checked against a computersimulation. And, of course, you still have to solve your own patterns!

As youmay have noticed, the indices of all the diffraction spots in a single crystal patternsuch as Figure 2.51 are related by a simple rule of vector addition or subtraction. This rule isvalid for electron diffraction patterns corresponding to any lattice symmetry, since thelattice planes are directions in reciprocal space, and the diffraction pattern is just a two-dimensional section through reciprocal space. This vector addition rule should always beused to confirm any solution to a single crystal spot pattern.

(001)

(011)

(010)

(101) (100)

(111)

(110)

(111)

(011)

(010)

(111)

(110)(110)

(111)

(101)(100)

(110)

Figure 2.50 Stereographic projection for a cubic crystal in the cube orientation.


200 400

220020

020

040

040

200400

220

220 220

Figure2.51 A fully indexed selectedareadiffractionpattern fromaluminium for the samezoneaxis as the patterns shown in Figure 2.48.

3030

1120

1210

Figure 2.52 Partially indexed selected area diffraction pattern from a-alumina.


Finally, many electron microscopists use JCPDS data to help them solve their electrondiffraction patterns. This can be misleading. Although the structure factor calculation is thesame forbothX-raydiffractionandelectrondiffraction, theatomic scatteringfactors forX-raysand electrons are very different, while the other coefficients in the calculation of total intensityalso differ. Furthermore, the JCPDS always refers to a randomly oriented powder specimen, inwhich weakly scattering planes may remain undetected. Such a plane may actually diffractstrongly in electrondiffractionwhen its reciprocal latticevector intersects the reflecting sphere.

Problems

2.1. The minimum lattice spacing which can be detected by diffraction of an incidentbeam is just half the wavelength of the incident radiation. Why?

2.2. When a first-order reflection is forbidden (for example, 110 in the FCC lattice), thesecond-order reflection (for example, 220 in FCC) is generally allowed. Why?

2.3. In a primitive cubic (PC) lattice the reflections 221 and 300 diffract at the sameBragg angle. Find another pair of reflections in this lattice that also diffract at thesame Bragg angle.

2.4. What reciprocal lattice vector separates the lattice points 110 and �111 in reciprocalspace?Write down theMiller indices of two other reflections that lie on the same zone.

2.5. In general, reciprocal lattice vectors are parallel to lattice directions having the sameindices only in a cubic crystal. Nevertheless, this equivalence is found for somezones of high symmetry in other noncubic Bravais lattices. Give two examples.

2.6. Name three factors which may relax the exact diffraction condition, so that somediffracted intensity is measured for crystal orientations which deviate slightly fromthe Bragg condition.

2.7. Distinguish between white and characteristic X-rays and give one application foreach type of radiation in X-ray diffraction.

2.8. Define the term mass absorption coefficient. Using literature data, estimate thethickness of an iron foil that will ensure 90% transmission for Cu Ka radiation andFe Ka radiation. How do you account for the large difference in the two calculatedthicknesses, given the small difference in atomic number and the fact that the twowavelengths are quite close?

2.9. Estimate theminimum level of residual macrostress detectable by anX-ray line shiftin steel (elastic modulus 220GPa). Justify any assumptions you make.

2.10. Diamond has an FCC structure, but with additional forbidden reflections. Deter-mine the first three additional forbidden reflections and explain their origin.

2.11. Should themeasurement of lattice spacing bemore accurate for thin films or for bulkspecimens? (Hint: the answer is not quite as straightforward as it may appear.)Based on your answer, suggest a suitable specimen holder for the accuratemeasurement of lattice spacings in a powder.


2.12. What, if any, differences would you expect in the diffracted intensity from the samephase in a thin film as opposed to a bulk specimen (assume you are using a Bragg–Brentano diffractometer).

2.13. Index the diffraction pattern from Ni shown in Figure 2.40, using the JCPDS datafrom Figure 2.41.

2.14. Index the diffraction pattern from alumina shown in Figure 2.43. To help you indexthe pattern, write a computer program to calculate d-spacings for the sequence ofplanes (hkil).

2.15. A fully dense sample of PbTe (cubic, r¼ 8.253 g cm�3) is characterized by X-raydiffraction using CuKa in a Bragg–Brentano diffractometer. Calculate the massabsorption coefficient for PbTe (lCu Ka¼ 0.1540598 nm). Calculate the depth of X-ray penetration as a function of 2y.

2.16. For Mg (hexagonal):

(a) Calculate the general structure factor.(b) Calculate the relative integrated intensities for the six strongest reflections listed

in the JCPDS file.(c) Compare your calculated values with the values listed in the JCPDS file and

explain any differences.

2.17. A cast sample consisting of one phase is characterized byX-ray diffraction inBragg–Brentano geometry. The results are listed together with results from a powder sampleof the same material in Table 2.2. Is there preferred orientation in the cast sample?

2.18. A mixture of Si and Mg powders is characterized by X-ray diffraction(l¼ 0.1540598 nm). The results are summarized in Table 2.3.

(a) Solve the diffraction pattern and identify each reflection.(b) Determine the relative amounts of each phase [(m/r)Si¼ 60.3 cm2 g�1;

(m/r)Mg¼ 40.6 cm2 g�1].

2.19. Calculate the energy of the X-ray photons for the following characteristic X-rayemissions:

(a) Cr (l¼ 0.2291 nm);(b) Co (l¼ 0.1790 nm);(c) W (l¼ 0.0209 nm).

Table 2.2 X–ray diffraction data from a cast sample and a reference powder of the samematerial.

Phase h k l Reference powder I (counts) Cast sample I (counts)

a 1 1 1 2017 968a 2 2 0 2339 1736a 1 3 1 6368 4536a 0 4 0 1255 2380a 3 1 1 2022 880


2.20. What is the minimum d-spacing that can be characterized using thesewavelengths?Why?

2.21. Given a single crystal of Si (diamond structure with a¼ 0.54329 nm):

(a) At which 2y will (004) reflect using Cu Ka (l¼ 0.1540598 nm)?(b) About which tilt axis and at what tilt angle must the sample be inclined to reflect

from (111)?(c) At which value of 2y will the {111} planes reflect?

2.22. For GaAs and l¼ 0.1540598 nm:

(a) Calculate the positions of the atoms in the unit cell.(b) Under which conditions is the structure factor zero?(c)Are there reflections thatwould not appear if all the lattice siteswere occupied by

the same type of atom?(d) Calculate the structure factor for the first two reflections.(e) Compare these structure factors with those of the first two reflections for AlAs.(f) Is there a change in structure factor if the sites of the two types of atom in the

lattice are exchanged?

2.23. A single crystal of silicon (diamond structure) is characterized by X-ray diffraction(l¼ 0.1540598 nm) in a rotating spectrometer.

(a) About which zone axis was the crystal rotated if reflections from both (004) and(111) were detected?

(b) What was the rotation angle needed to acquire a reflection from (111) after firstacquiring a reflection from (004)?

(c) Give two additional allowed reflections that would be expected to appear in thisexperiment.

Table 2.3 X–ray diffraction data from a mixture of Si and Mg powders.

Unmixed powders Mixed powdersPhase h k l d-spacing (A) 2y IP (counts) I (counts) Quantity (%)

28.470 214863 16674732.210 2827 63334.426 3140 70336.648 11869 265847.350 144157 11187547.859 1815 40656.180 85800 6658657.432 2041 45763.126 2244 50367.394 301 6768.706 2262 50769.204 22982 1783570.086 1589 35672.576 315 7176.461 34785 2699577.921 378 85


2.24. Cu3Au has a cubic structure. When fully ordered, the atoms are located in thefollowing sites: Au at (0,0,0) andCu at (1/2,

1/2, 0); (1/2, 0,

1/2); (0,1/2,

1/2). Under certainconditions the structure becomes disordered and occupancy of these sites becomesrandom. Assuming the occupancy factor is given by the atomic concentration of thecompound:

(a) Describe the structure factor for the ordered and disordered states.(b) For which reflections will the structure factor remain unchanged when going

through the order–disorder transition?

2.25. Cu has an FCC structure. Using the data from JCPDS:

(a) Which are the three strongest Bragg reflections when using Cu Ka(l¼ 0.1540598 nm)?

(b) Using I1þ cos22�

sin2�cos�

� �PjF2j, calculate the three strongest Bragg reflections and

compare with the data from JCPDS.

(c) Repeat the above calculation using I a1þ cos2ð2�Þ·cos22asin2�cos�ð1þ cos22aÞ

� �P F2�� Að�Þ and

assuming a graphite monochromator with a¼ 26.4�.

2.26. Nickel and copper both have FCC structures. According to the equilibrium phasediagram, nickel and copper form a solid solution over the entire concentrationrange (that is, they are completely miscible), and the lattice parameter of the solidsolution changes with concentration according to Figure 2.53. An X-ray diffrac-tion pattern from an unknown Cu-Ni alloy, acquired using Zn Ka, is presented inFigure 2.54.

(a) Solve the diffraction pattern given in Figure 2.54.

Figure 2.53 Lattice parameter as a function of Cu content in a Ni–Cu alloy.


(b) Determine the lattice parameter of the alloy by using the relation between thelattice parameter and (cos2y)/(siny).

(c) Estimate the concentration of the alloy.(d) Calculate the mass absorption coefficient of the alloy for Zn Ka.(e) Determine the maximum penetration depth for each reflection in the diffraction

pattern of Figure 2.54, assuming that the background level is 5%.

Use the following data:l(Zn Ka)¼ 0.1436 nmANi¼ 58.7 gmol�1; ACu¼ 63.5 gmol�1

ralloy¼ 8.925 g cm�3

Mass absorption coefficients of Ni and Cu: (m/r)Ni¼ 325 cm2 g�1;(m/r)Cu¼ 42 cm2 g�1.

2.27. The X-ray diffraction pattern (acquired with l¼ 0.1540598 nm) from a purepolycrystalline cubic material A is given in Figure 2.55.

(a) Is the Bravais lattice of material A FCC or BCC?(b) Solve the diffraction pattern and index the reflections.(c) Determine the lattice parameter.(d) If an alloy of the same material (a solid solution of A containing B in

Figure 2.56) is solidified by cooling from T1 to T2 fast enough to preventdiffusion, how would this affect the peak shapes of the reflections in X-raydiffraction?

2.28. A sample of polycrystalline Al (FCC) is coated with a 15 mm thick layer of Cu andcharacterized by X-ray diffraction using l¼ 0.1540598 nm. Given that the latticeparameter of Al is 0.405 nm, the mass absorption coefficient for Cu is 52.7 cm2 g�1,

Figure 2.54 X-ray diffraction pattern from an unknown Ni–Cu alloy.


and the background level in the X-ray diffraction pattern is 5% of the maximumincident intensity, sketch an X-ray diffraction pattern for the sample.

2.29. Figure 2.57 is a selected area electron diffraction pattern of a-Fe. Find lL for thediffraction pattern, index the pattern and determine the zone axis. Mark both thezone axis and the great circle containing the normals to the diffracting planes on astandard stereographic projection for a cubic crystal. Is the zone axis of this patternnormal to this great circle?

2.30. Figure 2.58 is a selected area electron diffraction pattern from a polycrystallineregion, taken from a cast metal block. There is concern that silicon may have been

Figure 2.55 X-ray diffraction spectrum from pure A.

T1

T2

100% A 100% B

α β

L+α

L

L+ β

Figure 2.56 Schematic binary phase diagram of A and B.


Figure 2.57 Selected area diffraction pattern from a crystal of a-Fe.

Figure 2.58 Selected area diffraction pattern from a polycrystalline region of a cast steel block.


introduced into the casting as an impurity. Solve the pattern and index the rings. Isthere any evidence for the presence of crystalline silicon?

2.31. Given a FCC crystal, calculate for the listed pairs of planes:

(a) the zone axis,(b) an additional allowed reflecting plane that belongs to each zone axis.

Figure 2.59 A selected area diffraction pattern from a polycrystalline thin film composed of Aland Fe.

Figure 2.60 Selected area diffraction pattern from a sapphire single crystal.


(c) the angle between the normals to all the reflecting planes for each zone.(110) : ð1�21Þ;ð2�20Þ : ð1�11Þ;(111) : ð�1�11Þ;(122) : (210).

2.32. For a BCC crystal, determine the normal to the planes defined by the following pairsof directions:

(a) ½1�32�½�201�;(b) ½102�½22�1�;(c) ½�121�½�1�11�.

Which of these planes are allowed reflections for the BCC lattice?

2.33. A thin, fine-grained, two-phase polycrystalline film composed of Al (FCC, a0.405 nm) and Fe (BCC, a¼ 0.2867 nm) is characterized by electron diffractionin transmission electron microscopy. A ring pattern acquired from the sample ispresented in Figure 2.59. Determinewhich rings belongs to Al and which to Fe, andindex the diffraction rings.

2.34. A selected area diffraction pattern of a sapphire single crystal is given in Figure 2.60.Solve the pattern, and determine the camera constant for this printed pattern.

2.35. A selected area diffraction pattern from an annealing twin in copper is shown inFigure 2.61. Solve the pattern, determine the camera constant, and define thetwinning relationship

d=0.2087 nm

Figure 2.61 Selected area diffraction pattern from a twin in a copper single crystal.


3

Optical Microscopy

The optical microscope is the primary tool for the morphological characterization ofmicrostructure in science, engineering and medicine. In the medical sciences, thin slices ofbiological tissue and other preparations are prepared for transmission optical microscopy,with or without staining, and frequent use is made of additional image contrast available bythe use of fluorescent dyes, dark-field optical microscopy, differential interference or phasecontrast. The geologist also works primarily in transmission, polishing his mineralogicalspecimens down to a thickness of less than 50 mm and mounting them on transparent glassslides. For the geologist the anisotropy of the sample viewed in polarized light is the mostfrequent source of contrast, and provides information not only on the morphologicalcharacteristics but also on the optical properties and spatial orientation of any crystallinephases which are present in the sample.

Metallurgical samples for metallographic examination were originally prepared by HenrySorby (1864) as thin slices, using the same methodology developed earlier for mineralogi-cal specimens, and his specimens have survived intact and are still available for examination(Figure 3.1). However, the presence of the conduction electrons renders metals opaque tovisible light and all metallurgical samples must be examined in reflection. It follows thatonly the surface of the sample can be imaged, and that it is the surface topology and surfaceoptical properties that are responsible for the contrast seen in specimens of metals and theiralloys examined in the optical microscope. In reflection microscopy the contrast may beeither topological, or due to differential absorption of the incident light, or the result ofoptical effects associated with reflection and optical interference.

Polymers and plastics can be imaged in either reflection or transmission, but theamorphous, glassy phases present give poor contrast. However, crystalline polymer phases,frequently formed by slow cooling from a viscous liquid state, are often studied intransmission by casting thin films of the molten polymer onto a glass slide. In polarizedlight these polymer crystals show contrast that is characteristic of the orientation of theoptically anisotropic crystal lattice of the polymer with respect to the polarization vector ofthe incident beam (Figure 3.2). Filled plastics and polymer matrix composites can also be


Figure 3.1 A �Widmanstatten� microstructure in steel. Specimen prepared by Henry Sorby(ca. 1864). Reproduced with permission from Smith, A History of Metallography, p. 166.Published by the University of Chicago Press.

Figure 3.2 �Spherulites� in polyethylene.Arrays of crystallites growing fromacommonnucleusare readily observed in polarized light. Reproduced by permission of John Wiley & Sons, Inc.


Publisher's Note:Permission to reproduce this imageonline was not granted by thecopyright holder. Readers are kindlyrequested to refer to the printed v ersionof this chapter.

examined in reflection, although the extreme differences inmechanical response between alow elastic modulus polymer and a high modulus filler or reinforcement makes specimenpreparation problematic. Furthermore, the ready availability of scanning electron micro-scopes has reduced the motivation to study the morphology of these materials in the opticalmicroscope. Nevertheless, contrast in the scanning electron microscope is insensitive tomaterial anisotropy, while this is the principle source of contrast in polarized lightmicroscopy. For example, elastomeric (rubbery) polymers exhibit molecular alignmentat high elastic strains that gives rise to optical anisotropy which is readily observable inpolarized light.

Ceramics and semiconductors are usually examined by reflection microscopy, despitetheir obvious similarity to mineralogical samples, even though in some cases it mayactually be easier (and more informative) to prepare a thin slice for transmissionexamination. Poor reflectivity, coupled in some cases with strong absorption of theincident light, makes for poor optical contrast in many ceramic samples when viewed byreflection, while their resistance to chemical attack often makes it difficult to find asuitable etchant to reveal the microstructure of a polished ceramic sample. In many cases,the presence of very small quantities of impurities or dopants alters the response of thesample to surface preparation, often through strong segregation of the dopant to grainboundaries and interfaces.

In this chapter we will emphasize the contrast mechanisms which are typical ofmicrostructural morphologies observed by optical microscopy, explaining, at an ele-mentary level, the interaction between a specimen sample and an incident beam ofvisible light. The emphasis will be on reflection microscopy, that is, the metallurgicalmicroscope, although much of the discussion is equally applicable for transmissionsamples.

3.1 Geometrical Optics

Rapid developments in the physical sciences over the past half-century and the technologi-cal revolution associated with semiconductors, microelectronics and communications,have left little place in the modern school science syllabus for the mundane topic ofgeometrical optics. It follows that the university science student�s understanding of imageformation in either the telescope or the optical microscope can no longer be taken forgranted. Nonetheless, in the context of the present discussion, some appreciation of what ishappening inside an optical microscope is desirable.

3.1.1 Optical Image Formation

The lens of an optical magnifying glass forms an image of an object because the refractiveindex of glass is much greater than that of the atmosphere, and reduces the wavelength ofthe light passing through the glass. A parallel beamof light incident at an angle on a polishedblock of glass is deflected, and the ratio of the angle of incidence on the surface to theangle of transmission through the glass is determined by the refractive index of the glass(Figure 3.3).

Optical Microscopy 125

In the case of a convex glass lens (a lens having positive curvature), the sphericalcurvature of the front and back surfaces of the lens results in the angle of deflection of aparallel beam of light varying with the distance of the beam from the axis of the lens, andbringing the parallel light beam to a point focus at a distance f that, for a givenwavelength, isa characteristic of the lens, and is termed its focal length (Figure 3.4). If the lens curvature isnegative, then the lens is concave and a parallel beam incident on the lens will be made todiverge. The beam of light will now appear to originate at a point in front of the lens: animaginary focus corresponding to a negative focal length -f.

There is no reason why the front and back surfaces of the lens should have the samecurvature, nor even why one surface should not have a curvature of opposite sign to the

Figure 3.3 A beam of parallel light (a planar wave front) is deflected on entering a block ofglass because of the change in wavelength associated with the refractive index m of theglass.


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other. It is the net curvature of the two surfaces taken together which determinewhether thelens is convex or concave. Similarly, there is no reason why the refractive index should bethe same for all the lenses in an optical system, and different grades of optical glass possessdifferent refractive indices. The �lenses� used in the optical microscope are alwaysassemblies of convex and concave lens components with refractive indices selected tooptimize the performance of the lens assembly. Depending on their position in themicroscope, these lens assemblies are referred to as the objective lens, the intermediateor tube lens and the eyepiece. It is also quite common for the medium between the sampleand the near side of an objective lens to be a liquid, rather than air. An immersion lens is onethat is designed to be used with such an inert, high refractive index liquid between thesample and the objective lens.

The assumption that a parallel beam of light will be brought to a point focus at a distancedetermined by the focal length of the lens is only a first approximation. Since the refractiveindex of glass varies with the wavelength of light, the focal length is only constant formonochromatic light (light of fixed wavelength). Visible (white) light is not brought toa single focus, but is rather dispersed, the shorter wavelengths (blue light) being brought toa focus at a greater distance from the lens than the longer wavelengths [red light,

Figure 3.5 (a) The focal length of a thin lens depends on thewavelength of the radiation, givingrise to chromatic aberration. (b)When the diameter of the lens is no longer small comparedwiththe focal length, the outermost regions of the lens have a reduced focal length and result inspherical aberration.

Figure 3.4 A parallel beam transmitted through a convex lens along its axis converges to afocus at a fixed distance from the plane of the lens, the focal length.


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Figure 3.5(a)]. It follows that a parallel beam ofwhite light is not brought to a sharp focus bya glass lens, but rather to a region of finite size (the �disc of least confusion�), a conditionreferred to as chromatic aberration. For thicker, larger diameter, lenses, even monochro-matic light is not fully focused, and the outermost regions of the lens (corresponding to thelargest angles of deflection of the incident light) have a shorter focal length than the regionof the lens near its axis [Figure 3.5(b)]. This again results in a finite size for the optimallyfocused beam, another �disc of least confusion�, rather than a point focus (a condition termedspherical aberration). The lens systems used for the lens assemblies in an opticalmicroscope, especially the objective lenses, correct for these aberrations very successfully,but it is important to check the manufacturer�s recommendations to ensure that the lenscombinations (for example, an eyepiece and objective) are fully compatible and areappropriate for the type of specimen to be examined. Objective lenses for biologicalsamples usually compensate for the thickness of a glass cover slip (often 0.1mm inthickness) that protects the sample from the environment.

As might be expected, there is a direct relationship between the cost of an objective lensand its optical performance, and several technical terms are used to describe this perfor-mance. The most common objectives are achromats, which are fully corrected forchromatic aberration at two wavelengths of light (blue and red), as well as for sphericalaberration at an intermediatewavelength (green light). They achieve their best performancewhen used with monochromatic green light. Plan achromats are more fully corrected toensure that not only the central field of view is in focus but also the periphery. They aredesigned for image recording, rather than simple viewing of the sample. The mostexpensive objectives are plan apochromats, that are fully corrected for the full visiblerange (red, green and blue light) and give an in-focus image over the full field of view.Specialized objectives are available for a wide range of viewing conditions, such as phasecontrast, dark-field microscopy, differential interference and long-working distance con-figurations (see below).

The human eye (Figure 3.6) forms an image of the visible world on a light-sensitivemembrane, the retina, by focusing light transmitted through the lens of the eye.Unlike glasslenses, the curvature of the lens in the eye can be adjusted by a system ofmuscles in order toensure that objects at different distances can be brought into sharp focus on the retina.Unfortunately, this control of the focus deteriorates with age, while for many of us thecontrol is imperfect, even in childhood. Prescription spectacles usually do an excellent jobof correcting the focus of our eyes.

To prevent too much light entering the eye, the iris acts as a variable aperture, reducingthe effective diameter of the lens in bright light. The light-sensitive retina is covered in adense array of optical receptors, the rods and cones, which respond to the incident opticalsignal with remarkable sensitivity. In many animal species, including, of course, man, theretina responds not only to the intensity of the incident light, but also to the wavelength,resulting in colour vision. However, some 20%of the human race lack perfect colour vision,while the response of the eye to colour gradually fades as the intensity is reduced, so that asdusk falls the world becomes grey.

The image formed by a simple lens can be analysed using a ray diagram in a thin lensapproximation (Figure 3.7). A ray of light parallel to the axis and coming from a point in theobject plane off the lens axis and at a distance -u in front of the lens will be deflected by the


lens to pass through the focal point at a distance f behind the lens. On the other hand, a raytravelling from the same point in the object plane but passing through the conjugate focalpoint -f in front of the lens, will be deflected parallel to the axis after passing through thelens. A third ray from the same point in the object plane, but now travelling through the

Figure 3.6 In the human eye the lens focuses an image onto the retina, while the iris acts as avariable aperture to limit the amount of light admitted. The space between the lens and the retinais filled with liquid, so that this is an immersion lens system.

-u -f

f v

Figure 3.7 A ray diagram relates the distance of the lens from the object u to both the focallength of the lens f and the position of the image plane v, and determines themagnification in theimage, M¼ v/u.


centre of the lens will be undeflected. As we see from Figure 3.7, all three rays meet at apoint in the image plane at a distance v behind the lens, to form the image. It is an elementaryexercise in trigonometry to show that the magnification of the image created by the lens isgiven by M¼ v/u, while the object and image distances are related by 1/uþ 1/v¼ 1/f. Itfollows that the magnification achieved by a magnifying glass is controlled by its focallength, f and the distance between the magnifying glass and the object, u.

3.1.2 Resolution in the Optical Microscope

The wavelength of the electromagnetic radiation transmitted through the earth�s atmo-sphere from the sun varies from the infrared (at the long wavelength end of the spectrum) tothe ultraviolet (at the short wavelength end), but the peak intensity for solar radiationreaching the earth is in the green region of visible light, very close to the peak sensitivity ofthe eye (approximately 0.56 mm). The temperature of the sun is about 5500K, appreciablyhigher than that of the tungsten-halide light sources usually used for microscopy (about3200K). �Daylight� filters have been developed that modify the halide spectrum to simulatesunlight and other filters are also used, especially �green� filters for approximatelymonochromatic observation and grey-scale recording.

3.1.2.1 Point Source Abbe Image. The resolution of an optical lens is defined in terms ofthe spatial distribution of the intensity coming from a point source of light situated atinfinity, when the point source is imaged in the focal plane of the lens. The calculatedintensity distribution assumes a parallel beam of light travelling along the axis of a thin lensand brought to a focus at the focal distance, as shown in Figure 3.8, and for the cylindricallysymmetric case, the ratio of the peak intensities for the primary and secondary peaks in thisimage intensity distribution is approximately 9:1. The width of the primary peak for thiscase is given by the Abbe equation:

d ¼ 0:61l

msinað3:1Þ

Intensityα

Figure 3.8 The Abbe equation gives the width of the first intensity peak for the image of apoint object at infinity in terms of the angular aperture of the lens a and the wavelength of theradiation l.


where l is the wavelength of the radiation, a is the aperture (half-angle) of the lens,determined by the ratio of the lens radius to its focal length, and m is the refractive index ofthe medium between the lens and the focal point (m�1 for air).

3.1.2.2 Imaging a Diffraction Grating. A diffraction grating consists of an array ofclosely spaced, parallel lines. When illuminated by a normally incident, parallel beam oflight, a cylindricalwavefront is generated by scattering of the incident light from each of thelines in the grating. These wavefronts interfere to generate both zero-order and diffractedtransmitted beams (Figure 3.9). When the spacing of the grating is large compared with thewavelength, the angle of diffraction for the nth-order beam is given by siny¼ nl/d. Itfollows that a lens can only be used to image a diffraction grating if the angular aperture ofthe lens a is large enough to accept both the zero-order and the first-order beams, that issina� siny¼ l/d.

We can compare this condition both with the Bragg equation for diffraction from a three-dimensional crystal lattice, l¼ 2dsiny, and with the Abbe relationship above, which isrelated to the Raleigh resolution criterion (see below), d¼ 0.61l/msina. In effect, all threeequations define the limiting conditions for transmitting information about an object whenusing electromagnetic radiation as the information carrier. The key parameter in all threecases is the ratio l/d, and all three criteria state that the resolution limit on the microstruc-tural information available is directly proportional to the wavelength of the radiation used.

3.1.2.3 Resolution and Numerical Aperture. Raleigh defined optical resolution in termsof the ability of a lens to distinguish between two point sources at infinity when they areviewed in the image plane.His criterion for resolutionwas that the angular separation of two

Incidentwavefront

Zero orderdiffractedwavefront

First orderdiffractedwavefront

Scatteredwaves

Diffractiongrating

Figure 3.9 The diffraction pattern from a grating generates a series of diffracted beams. Toimage the grating at least the zero- and first-order beams must be admitted to the aperture ofthe lens.


sources of equal intensity should ensure that the maximum of the primary image peak fromone source should fall on the first minimum of the image from the second source. (Think ofusing an astronomical telescope to resolve a pair of identical stars with a small angularseparation.) The Raleigh condition implies that the combined image of the two sources willshow a small (13%, actually), but detectable intensityminimum at the centre (Figure 3.10).It follows that the Raleigh criterion corresponds exactly to thewidth of the primary intensitypeak given by the Abbe equation, d¼ 0.61l/msina. We should note that the Raleighresolution is defined in the focal plane of the lens for an image of a point source at infinity.In the optical microscope, the objective lens magnifies the object, placing the image farfrom the lens while the object itself lies close to the focal plane. For optical microscopeobjectives, it follows that the Abbe equation gives the minimum separation that can bedistinguished in the focal plane of the object. Try not to get confused!

It is important to recognize the fundamental significance of the Abbe relation: for anyimaging system based only on wave optics, no image detail can be transmitted which ismuch below the wavelength used to transmit the information. If we wish to maximize theinformation in the image we should collect as much as possible of the optical signalgenerated by the object – that is, we should maximize the aperture of the objective lens.

Actually, it is possible to make use of the particle properties of phonons in order toachieve a resolution that is below the wave optics limit. This is the basis of near-fieldmicroscopy, in which a sub-micrometre light pipe is scanned across the specimen surface.The resolution is then limited by the diameter of the light pipe and its distance from thesurface and can be appreciably better than that set by the Abbe relation, but this techniquegoes well beyond the present text. Most recently, it has proved possible to beat the

0.61λµsinαd =

Figure 3.10 The Raleigh resolution criterion requires that two point sources at infinity have anangular separation sufficient to place the maximum intensity of the primary image peak of onesource at the position of the first minimum of the second.


diffraction limit on resolution by partially quenching the fluorescent emission fromhistological specimens stained with a fluorescent dye. The fluorescent emission isstimulated by pulsed laser irradiation, but a second, picosecond, laser pulse timedimmediately after the first pulse and of appropriate energy, partially quenches the peripheralfluorescent signal, so that only light emitted from the central region is recorded.Resolutionsof 30–40 nmhave been claimed. Again, this experimental technique shouldmainly be takento demonstrate that it is possible to beat the diffraction limit in some situations.

As noted previously, objective lens aberrations can be corrected and should not limit theperformance of the optical microscope. The parameter msina is termed the numericalaperture (NA) of the lens and is an important characteristic of anyobjective lens system.Themaximum values of NA are of the order of 1.3 for an immersion lens system and 0.95 forlenses operating in air, and the value ofNA is usuallymarked on the side of the objective lensby the manufacturer.

It is important to distinguish between the resolution limit of the objective lens and thedetection limit. As the signal intensity generated by a point source decreases it will becomeincreasingly difficult to detect against the background noise of the system. In the reflectionmicroscope, the smallest objects will scatter light outside the lens aperture, resulting in anintensity deficit in the image field. As the object becomes very small the intensity of thesignal from the object will decrease while the apparent size of the object (the Abbe width)will remain unchanged. At some limiting size, when the signal approaches the backgroundnoise limit of the detection system, the objectwill cease to be detectable. The detection limitis well below the resolution limit dictated by the wavelength of the light and the NA of thelens. The situation is illustrated schematically in Figure 3.11. One reason for using dark-field illumination is that it is easier to detect small features that scatter light into the imagingfield, rather than relying on the small amount of light which is scattered out of the objectiveaperture in a bright-field image.

3.1.3 Depth of Field and Depth of Focus

The resolution available for an objectwhose image is in focus in the image plane is finite andlimited by the NA of the objective lens, so it follows that the object need not be at the exactobject distance from the lens u, but may be displaced from this plane without sacrificing

Figure 3.11 Large objects of diameter d are blurred by the diffraction limit d derived from theAbbe relationship, but objects smaller than the Abbe width are still detectable in themicroscope, although the intensity is reduced, while their apparent width remains thatgiven by the Abbe equation.


resolution (Figure 3.12). The distance over which the object remains in focus is defined asthe depth of field:

d ¼ dtana ð3:2Þwhere a is the half the angle subtended by the objective aperture at the focal point. Similarly,the image will remain in focus if it is displaced from its geometrically defined position at adistance v from the lens. The distance over which the image remains in focus is termed thedepth of focus:

D ¼ M2d ð3:3ÞwhereM is the magnification. Both these expressions are approximate and assume that theobjective can be treated as a �thin lens�, which is not the case in commercial microscopes.Since the resolution is given by d¼ 0.61l/msina¼ 0.61l/NA, it follows that the depth offield decreases as the NA increases. For the highest image resolution, the specimen shouldbe positioned to an accuracy of better than 0.5 mm, and this is an essential requirement whenspecifying the mechanical stability of the specimen stage.

The depth of focus is considerably less critical. Bearing in mind that a magnificationgreater than ·100 may be necessary if all the resolved detail visible in the opticalmicroscope is to be recorded, then displacements in the image plane of the order of amillimetre are acceptable.

3.2 Construction of the Microscope

A simplified design for a reflection optical microscope is shown in Figure 3.13. Themicroscope is an assembly of three separate systems. The illuminating system whichprovides the source of light illuminating the sample, the specimen stage that holds thesample in position and controls the x, y and z coordinates of the area under observation, andthe imaging system, which transfers a magnified and undistorted image to the plane ofobservation and to the recording medium. We will discuss each of these in turn.

3.2.1 Light Sources and Condenser Systems

There are two conflicting requirements for the light source. On the one hand, the area of thespecimen being examined beneath the objective lens needs to be uniformly flooded with

Figure 3.12 Since the resolution is finite, the object need not be in the exact object plane inorder to remain in focus: there is an allowed depth of field d. Similarly, the image may beobserved without loss of resolution if the image plane is slightly displaced: there is an alloweddepth of focus D.


light in order to ensure that all themicrostructural features experience the same illuminatingconditions, but on the other hand the incident light needs to be focused onto the specimen toensure that the reflected intensity is always sufficient for comfortable viewing andrecording.

The source of light should be as bright as possible. Fifty years ago this was achieved bystriking a carbon arc, which gave an excellent, though somewhat unstable, source of whitelight. Alternatively, a mercury arc lamp generated an intense monochromatic emission linein the green (l¼ 0.546 mm), which corresponded well to the peak sensitivity of the humaneye. Today, while some small instruments still use a conventional light bulb, highperformance optical microscopes are now equipped with a tungsten-halide discharge tubethat provides a stable and intense source of white light corresponding to a temperature ofabout 3200K (compare the temperature of the sun, about 5500K). Filters can then be used toselect a narrow band of wavelengths, usually in the green, for monochromatic viewing, or tosimulate sunlight more closely (�daylight� filters).

In addition to the source itself, there are other important components in the illuminatingsystem (Figure 3.13). The condenser lens assembly focuses an image of the source close tothe back focal plane of the objective lens so that the surface of the specimen is uniformlyilluminated by a near-parallel beam of light. The condenser aperture limits the amount oflight from the sourcewhich is admitted into themicroscope by reducing the effective sourcesize. Contrast in the image can often be improved by using a small condenser aperture,

2α

Eyepiece

FirstImagePlane

Half-SilveredMirror

ObjectiveBack Focal

Plane

CondenserLens

Lamp

CondenserApertureVirtual Image

Aperture

Objective

Specimen

Figure 3.13 The principle components of the reflection optical microscope and theirgeometrical relationship to one another.


although at the cost of reducing the image intensity and, if the aperture is too small,introducing image artifacts which are associated with the Abbe diffraction pattern of a�point� source. A second aperture, the objective or virtual image aperture, is placed in thevirtual image plane of the sample (Figure 3.13), so that only light illuminating the areaunder observation is admitted to the microscope. This ensures that light is not internallyreflected within the microscope, leading to unwanted background intensity. The size of thevirtual image aperture should be adjusted to the field of view of the microscope at themagnification used. Both the condenser and the virtual image apertures are continuouslyvariable irises which can be adjusted to the required size.

Many reflection microscopes also permit the illuminating system to be repositioned sothat optically transparent specimens can be viewed in transmission. This is important notonly for the thin tissue samples of biology andmedicine, but also formineralogical samples,partially crystalline polymers and thin-film semiconductor materials. It is also extremelyuseful when monitoring the quality of thin-film samples prepared for transmission electronmicroscopy.

Alternatively, the virtual image aperture may be provided with a central stop that allowsan annulus of light to illuminate the area under observation from the periphery of a specialobjective lens assembly. Such dark-field illumination (see below) may greatly enhance thecontrast.

3.2.2 The Specimen Stage

The primary requirement for the specimen stage is mechanical stability and, given theexpected �0.3 mm resolution for a good optical microscope, it is clearly essential that thepositioning of the specimen be accurate to better than this limit. The accurate positioning ofthe specimen in the x–y plane is only one aspect of the stability required. The image isbrought into focus by adjusting the vertical location of the specimen and the accuracy of thisz-adjustment must be within the depth of field for the largest NA objective lens, typicallyalso �0.3 mm.

The necessary mechanical precision is commonly achieved by coarse and fine micro-metre screws for all three (x, y and z) coordinates, and both the time-dependent drift of thestage and the mechanical �slack� in the system need to be minimized. (The �slack� is thedifference in the micrometre reading when the same feature is brought into position fromopposite directions.)

In general, it is the z-adjustment that presents the most problems, since the necessarystage rigidity implies a fairlymassive and hence heavy construction. Two possibilities exist,depending on whether the specimen is to be placed beneath or above the objective lens. Inthe former andmore usual case (Figure 3.13), the plane of the prepared sample surfacemustbe positioned accurately normal to the microscope axis. This is commonly achieved bysupporting the specimen from below on soft plasticine and applying light pressure with asuitable jig (Figure 3.14).

3.2.3 Selection of Objective Lenses

Averywide range of objective lenses are available, depending on the nature of the specimenand the desired imaging mode. The performance of the objective lens is primarily


dependent on its NA, and this is almost universally to be found inscribed on the side of theobjective lens assembly, together with the magnifying power for that lens. Most objectivelenses are achromatic, that is they are not limited to monochromatic light, but arenevertheless recommended for viewing high resolution monochromatic images in thegreen. While achromatic lenses are corrected for both spherical aberration in the green andchromatic aberration at twowavelengths (red and blue), they only yield a focused image inthe central region of the field of view. Plan achromat objectives ensure that the periphery ofthe field of view is also in focus and they are therefore more suitable for image recording.Apochromatic objectives are free of chromatic aberration for threewavelengths (red, greenand blue), and, correspondingly, plan apochromats are designed for recording image detailin full colour.

Histological examination of soft tissues, which accounts for the major proportion of thework of the optical microscope in the life sciences, requires that the specimen be protectedfrom the environment by mounting a thin tissue slice on a glass slide and then protecting itwith a thin cover slip. Similar techniques are used formanypolymer specimens, particularlythose that are partially crystalline. These materials can be cast onto the slide and thespecimen thickness controlled by spinning or by applying uniform pressure to a cover slip.Objective lenses designed for usewith such specimens are corrected for the refractive indexand thickness (often 0.1mm) of the optically flat cover slip.

Not only the resolution, but also the brightness (the intensity per unit area of the image)depends on theNA of the objective lens. For any given conditions of specimen illumination,the brightness of the image decreases as the square of themagnification.However, largerNAlenses increase the cone acceptance angle for the lens, so thatmore light is collected. TheNAmay vary by an order of magnitude in going from a low-power (lowmagnification) lens to ahigh-power (high magnification) immersion objective. A similar order of magnitudeincrease in the magnification will then be required in order to observe all the image detail.It follows that there is still an overall reduction in brightness by a factor of 102/10¼ 10.

Plunger

Specimen

Plasticine

Base

GlassSupport Slide

Figure 3.14 If the specimen is to be placed beneath the objective lens, then itmust bemountedwith the plane of the specimen surface accurately normal to the microscope axis.


The working distance of the objective lens from the specimen surface also decreasesdramatically as the NA of the objective lens assembly is increased, down to of the orderof 0.1mm for the highest-powered lenses. It is only too easy to damage a lens by drivingthe specimen through focus and into the glass lens, and good lenses are expensive toreplace.

Special long working distance lenses are available that allow high magnificationobservation without having the sample in close proximity to the objective lens. Oneinexpensive design creates an intermediate image at unit magnification by reflection(Figure 3.15) and permits a specimen to be imaged while in a hostile environment, forexample in a corrosive medium or at an elevated or cryogenic temperature. Despite theattraction of in situ experiments, little optical microscopy has been done under suchdynamic conditions. There are difficulties: the dimensional stability of the specimen and itssupport structure is one problem; another is to ensure that the optical path between thespecimen and the objective lens assembly is not obscured by a condensate or by chemicalattack.

Cryomicroscopy is subject to the formation of ice crystals, while a high temperature stagemay form opaque deposits that derive from the heating elements, the specimen or thesupporting structure. To be successful, an in situ stage must combine a rapid response timewith experimental stability. In the case of a heating stage, a compromise is required betweenthe large heat capacity needed to ensure thermal stability, and the small heat capacitynecessary to allow a rapid experimental response.

Many other specialized objective lens and stage assemblies are available. One of themostuseful, for both reflection and transmission work, is the dark-field objective whichilluminates the specimen with a cone of light surrounding the lens aperture. The lightscattered by the specimen into the lens aperture is then used to form a dark-field image inwhich the intensity is the inverse of that observed in normal illumination (Figure 3.16).

Standard4 mm Objective

First Image Plane(unit magnification)

Half-Silvered Plate

Specimen

NormalWorkingDistance

3 mm

12.8 mm

Figure 3.15 A long working distance attachment for studying specimens at high temperaturesor in a hostile environment.


3.2.4 Image Observation and Recording

The image magnification provided by the objective lens assembly is limited, and insuffi-cient if the image is to be fully resolvable by the human eye. There are three optionsavailable. The first is to insert an eyepiece and an additional intermediate or tube lens, inorder to view the image directly at a working magnification that is comfortable for theobserver. Most microscopes now available are designed with a tube lens that allows thesample to be placed in the focal plane of the objective, so that the light returning to themicroscope through the objective is essentially parallel, and only brought to an intermediatefocus by the tube lens. This allows for a wide range of optical accessories to be insertedbetween the objective and the tube lens. The second option is to use the additional lenses tofocus the image onto a light-sensitive, photographic emulsion or charge-coupled device(CCD), usually for subsequent enlargement. For the third option it is possible to scan theimage in a television raster and display it on amonitor. For the recording of dynamic eventsin the microscope this may in fact be the preferred technology. In recent years the improvedavailability of high quality, CCD cameras has made it possible to record a digital imagefrom an objective lens without any additional lenses. The consumer market for digitalcameras has allowed high quality CCD technology to all but replace photographicrecording, while conventional television camera technology is now seldom used, evenfor teaching purposes. Nevertheless, professional photographers still make use of photo-graphic emulsions, since they usually require the highest performance CCD systems, whichare still extremely expensive. More recently, there have been significant advances in a newtechnology, complementary metal oxide semiconductor (CMOS), which essentially placesthe camera on a single chip. For colour recording in both CCD and CMOS devicesindividual pixels (picture elements) can be filtered by red, green and blue dyed photodiodes,but while both CCD and CMOS cameras offer pixel sizes of 6 mm or even less, the CMOScameras are limited to of the order of a million pixels per frame, an order of magnitude lessthan the CCD cameras. To make the most of high resolution digital colour recording it maybe necessary to invest in planapochromatic objective lenses (Section 3.2.3).

3.2.4.1 Monocular and Binocular Viewing. Visual observation is most commonlyperformed with a monocular eyepiece, which enlarges the primary image by a factor of·3 to·15.A typical 0.95NA (nonimmersion) objectivemay have a primarymagnification of·40 and a resolution of 0.4 mm, so that, to ensure that all resolved features are readily visible

Specimen

(b)

Specimen

(a)

Figure 3.16 A dark-field image (a) is formed by collecting the diffusely scattered light into theobjective lens aperture, and is the inverse of the normal, bright-field image (b), where thescattered light falls outside the objective aperture and is lost to the objective lens.


to the eye (0.2mm), some further magnification is required, simply calculated as:(0.2 · 103)/(0.4 · 40)¼ ·12.5.

Most goodmicroscopes have an additional intermediate or tube lens (·4, for example), sothat a·3 or·5 eyepiece should then be sufficient to resolve all image detail. Evenwithout anintermediate lens there is no real reason to use a ·15 eyepiece, since superfluous additionalmagnification reduces the field of view, enlarging the resolved features to the point wherethey appear blurred to the eye of the observer.

Some microscopes are equipped with a beam splitter and a binocular viewer. For thosewho have difficulty viewing comfortably through one eye this is undoubtedly a conve-nience, but the microscopist should be aware that there are some disadvantages. Inparticular, it is unusual for the focal plane of both eyes to be identical, so that one eyepieceof the pair needs to be independently focused. The user first focuses a feature of interest inthe plane of the specimen, using just one eye and a fixed-focus eyepiece. He then adjusts thevariable focus of the second eyepiece (without touching the specimen stage controls) untilthe images seen by both eyes merge into a single, simultaneously focused, image. Thisprocedure of adjusting the binocular settings is completed by adjusting the separation of thetwo eyepieces to match the separation of the observer�s eyes. It is important to note that abinocular eyepiece does not provide stereoscopic (three-dimensional) viewing of thesample, which would require two independent objective lenses focused on the same fieldof view. Stereobinoculars (or stereomicroscopes), with twin objectives, are available, butwithmagnifications limited to about ·50. This limit is dictated by the geometrical problemsassociated with the positioning of the twin objective lenses close to the specimen surface.Stereobinoculars are important tools for inspection in the electronics industry.

3.2.4.2 Photographic Recording. A photographic emulsion and the human eye reactvery differently to light. The emulsions have their maximum sensitivity in the ultraviolet(about 0.35 mm) and both black and white, and colour films rely on dyes to extend thephotosensitivity of silver halide emulsions beyond the green (Figure 3.17).Orthochromatic

0.3 0.4 0.5 0.6 0.7

10–2 20

40

60

80

100

120

Wavelength (µm)

Sens

itiv

ity

Rel

ativ

eV

isib

ility

10–1

1

10

Ultraviolet VisibleRange

Orthochromatic

HumanEye

Figure 3.17 The eye is most sensitive to green light, whereas photographic emulsions havesensitivities which decrease steadily with increasing wavelength.


emulsions are not sensitive to red light, which is a convenience in dark-room processing,and are a common choice for photographic recording of monochromatic microscopeimages in green light. Panchromatic film is a common choice for black and whitephotographic recording in daylight, but here too the sensitivity falls steadilywith increasingwavelength in the visible range. In classical black and white photography the recordingmedium always yields a negative in which the clear areas, corresponding to zero excitation(no silver precipitation) and the opaque (black) areas correspond to maximum lightexcitation. This does not have to be the case, and Polaroid cameras commonly producea positive grey scale image. Colour recording films may also be negative, and are then usedfor colour printing, or positive, for use as slides or transparencies. The convention is to usethe suffix chrome for positive transparencies (Kodachrome, Ektachrome, etc.), but to usethe suffix colour for negative film (Fujicolor, Agfacolor, etc.).

The speed of an emulsion is its response to a fixed radiation dose at a standardwavelengthand depends on three factors: the exposure time, the �grain� of the emulsion and thedevelopment process. A photosensitive silver halide grain will react to subsequentdevelopment only if it can absorb a pair of photons. The time interval between the arrivalof the two photons is important, since the grainmay decay from its initial excited state in theinterim. As the incident intensity decreases, the interval between photon excitations of thesame grain increases, and the response of the emulsion is reduced, a phenomenon termedreciprocity failure. Larger halide grain sizes increase the photon collision cross-section andimprove the photosensitivity (the speed of the emulsion). The price paid is a �grainier� imagewith poorer inherent resolution. It follows that some compromise is usually requiredbetween fine-grained, slow-speed emulsions and coarse-grained, high-speed emulsions.

During development, the grain of silver, which is nucleated at an activated halide crystal,grows into a cluster of silver grains which encompasses a much larger volume than thatassociated with the original halide crystal, so that the resolution in the final recorded imageis affected by the growth of the silver grains during the development process. An emulsiondesigned for photomicroscopy, and developed according to the recommendations of themanufacturer, should have a resolution of the order of 10–20 mm and be capable ofenlargement by a factor of ·10. It should therefore be possible to photograph a highresolution imagewithout any loss of information at an appreciably lowermagnification thanthat required to view the fully resolved microstructure. It follows that a low magnification,high resolution recorded image contains farmore information than is available in the field ofview required for observation at full resolution.

The contrast attainable in a given emulsion is defined in terms of the dose dependence ofthe blackening in the developed emulsion. The dose is the amount of light per unit area Emultiplied by the time of exposure t, while the blackeningD is the logarithm of the ratio ofthe intensity incident on the emulsion I0 to the intensity of light transmitted through theexposed and developed emulsion I:

D ¼ logI0I

� �ð3:4Þ

The contrast g is defined as the maximum slope of the curve of D plotted against log Et.Emulsionswith a high g lose image detail because they tend to register as black orwhitewithfew intermediate grey levels,while low g emulsions lack contrast because the grey levels aretoo close together. A major disadvantage of photographic recording is the nonlinearity of


the response of the emulsion and the difficulty of controlling the many parameters involvedin exposing, developing, enlarging and printing the emulsion. It is very difficult to makequantitative measurements of image intensity or contrast based on photographic recordingand digital recording, using a CCD camera, is therefore preferable.

The range of informationwhich can be recorded, either by a photographic emulsion or bya CCD camera, is never unlimited. This is illustrated in Figure 3.18. At very low values ofthe dose Et background noise will start to become a problem, while at very high doses theresponse of the recording media will saturate. High resolution, negative, black and whitefilm is quite capable of responding to four orders of magnitude of the dose, but positiveprints are limited to about two orders ofmagnitude. CCD cameras usually come somewherein between (see below), but have the distinct advantage that there response is linear over thisrange. The response of photographic emulsions to high energy electrons is also linear indose, since the halide grains are excited by a single electron impact (compare the excitationprocess for visible light, which requires two photons to strike a halide grain within a criticaltime interval). Similar considerations apply to CMOS cameras.

3.2.4.3 Television Cameras and Digital Recording. Television cameras and monitorshave been attached to optical microscopes for some considerable time, primarily to allowpresentation ofmicroscope observations in �real time� to large groups, but also for recordingdynamic events occurringunder themicroscope (for example corrosion studies, or the effectsof heating the sample). However, the number of pixels scanned in a standard televisiondisplay iswell below thenumber of image elements that the eye is capable of resolving acrossa single field of view.Nevertheless, the time-base of a television raster does permit any signalcorresponding to two spatial dimensions to be recorded as a time-dependent analogue signalthat can either be processed and displayed on amonitorwith the same time-base or convertedto a digital signal for further processing and storage or display.Wewill discuss digital signalprocessing for applications in microscopy more fully below (Section 3.5).

–3.0 –2.0 –1.0 0 1.0

Log Exposure (lux s)

0

1.0

1.5

2.0

2.5

3.0

3.5

0.5

Den

sity

Underexposure

Overexposure

Linea

r Range

Figure 3.18 The blackening of an emulsion is only linear over a selected range of exposuredose that depends on the speed and contrast of the emulsion. At very low doses reciprocityfailure reduces the blackening, while at very high doses the blackening saturates.


The CCD and (CMOS) cameras are free of many limitations associated with photo-graphic recording. The CCD camera can �grab� a two-dimensional image frame in a fewseconds and record a digitized image of 107 or evenmore pixel points (pixels are defined inSection 3.5). Furthermore, the response of aCCDorCMOScamera is essentially linear overawide range of exposure dose, so that there is a one-to-one correlation between the recorded�brightness� of any pixel point location and the original intensity of the signal from theobject.

Moreover, the digitized image can be efficiently processed, using standard computerprograms to enhance image contrast, remove background noise, or analysed to extractquantitativemorphological image data, such as grain size. Individual image frames can alsobe combined to extract comparative information from the image data sets, for example inthe study of dynamic changes taking place in the sample during viewing.

Finally, the diameter of the CCD camera is typically 25mm or less, so that an image canbe recorded using an objective lens alone, without the need for any intermediate lenses. Thepast few years have seenmajor changes in the design of optical microscopes, with computercontrol replacing manual control of the focus, and the CCD camera and computer monitorreplacing the photographic system.

3.3 Specimen Preparation

For many students, good specimen preparation is a major obstacle to successful opticalmicroscopy. It is unfortunate that every material presents its own individual and uniqueproblems of specimen preparation. For example, the elastic modulus and the hardness of thematerial usually determine the response of the sample to sectioning, grinding and polishing,while the chemical activity determines the response to electrolytic attack and chemicaletching. In what follows we will generalize as far as possible, while recognizing that eachmetal alloy, every ceramic material and all plastic compositions are almost certain torespond differently.

3.3.1 Sampling and Sectioning

The problem with all microscopes is that they lose the �larger picture� by focusing on thedetails. It is only too easy to lose track of the relation between amicroscope image (recordedfroma particular position and in a specific orientation) and the engineering component fromwhich the image was taken.

Engineering systems are assembled from components which frequently have complexgeometries and come in a wide range of shapes and sizes. They are produced by a variety ofprocessing routes, and are unlikely to be of uniformmicrostructure. They often have a �rightway up�, and the materials from which they are made are often inhomogeneous andanisotropic. Both the chemical composition and the microstructural morphology may varyacross a section, even if it is only the surface layers that are �different�. Preferred orientationmay be restricted to themicrostructuralmorphology (elongated inclusions, aligned fibres orflattened grains), or itmay be associatedwith crystalline texture, inwhich certain directionsin the crystals are preferentially aligned along specific directions in the component (forexample, the axis of a copperwire or the rolling direction in a steel plate). Crystalline texture


may exist in the absence of morphological texture, the grains appearing equiaxed eventhough they all share a common crystallographic axis.

As a consequence, it may not be easy to decide how best to section a component formicroscopic examination. Nevertheless, it is always helpful to define the principle axes ofthe component and to ensure that the plane of any section is alignedwith at least one of theseaxes. In even the simplest cases it is usually desirable to examine two sections, perpendic-ular and parallel to a significant symmetry axis of the component. In the case of rolled sheet,sections taken perpendicular to all three principle directions are desirable: the rollingdirection, the transverse direction and the through-thickness direction. In a large casting themicrostructure will vary, both due to differences in the cooling rate and to the effects ofsegregation. Sections taken from the first portion of a casting to solidify may have verydifferent microstructures from the portion which solidified last.

If a section has been taken perpendicular to a principle direction of the component, then itis also important to identify one other principle direction lying in the plane of the section:the trace of a free surface, or a growth or rolling direction. It is only too easy to confusestructurally significant directions, either during mounting and preparation of the section oras the result of image inversion, either in the microscope or during processing. (Note thatnewspapers frequently publish inverted images showing right-handed individuals engagedin apparently left-handed activities.)

3.3.2 Mounting and Grinding

For convenience during surface preparation many samples need to be mounted for ease ofhandling. A polymer resin ormoulding compound is the commonest form of sample holder,and can be die-cast or hot-pressed around the sample without distorting the sample ordamaging themicrostructure (Figure 3.19). Clearly, the sample may need to be sectioned tofit into the die cavity. Very small samples can be supported in any desired orientation using acoiled spring or other mechanical device.

Once the sample is securelymounted, the surface section can be ground flat and polished.Rough grinding requires some care, since it is easy to remove toomuchmaterial, to overheatthe sample, or to introduce sub-surface mechanical or thermal damage. It is helpful toensure that the surface section is cut planar, even before the sample is mounted. Awide range of grinding media is available, designated as either cemented, metal-bonded

Mouldingcompound

Die

Sample

Base

Figure 3.19 Samples are commonly cast within a moulding compound for ease of handlingduring surface preparation.


or resin-bonded. The commonest grinding media are alumina (corundum), silicon carbideand diamond, and all three are available in a wide range of grit sizes. The grit size is definedby the sieve size which will just collect the grit, and the quoted sieve size refers to thenumber of apertures per inch, so that the grit size is an inverse function of the particle size. A#320 grit has been collected by a #320 sieve, having passed though the larger standard size, a#220 sieve.Beyond #600 grit sieving is no longer practical since the grit particles aggregate,but the same definitions are used and very fine, sub-micrometre grits are available. Ingeneral, a #80 grit is used for coarse grinding, corresponding to particles a few tenths of amillimetre in diameter.

Grinding is actually a machining process in which the sharp edges of the grindingmedium cut parallel to the surface of the sample. The rate of material removal depends onthe number of particle contacts, the depth of cut (a function of the applied pressure and thegrit size) and the shear velocity at the interface between the grinding medium and the workpiece. Heat generated at the interface and the debris resulting frommaterial removal are twomajor obstacles to effective grinding. The former increases the ductility of the work piece,and hence the work required to remove more material, while the latter clogs the cuttingsurfaces. These effects can be inhibited by flushing the surface with a suitable coolant toremove both heat and debris from the grinding zone.

The cutting edges of grinding media particles are blunted during grinding. New cuttingsurfaces can be exposed throughwear of the surroundingmatrix, which releases the bluntedgrit particles as debris and exposes fresh cutting surfaces. This may occur naturally duringgrinding of thework piece, but ismore often accomplished by dressing the grindingwheel –using an alumina �dressing stone� to remove the debris and expose new cutting surfaces. Thechoice of matrix is important. Metal-bonded grits are most frequently used in cutting discsfor sectioning hard and brittle samples. They are also used for many coarse-grindingoperations for which diamond is the preferred medium. Resin-bonded discs give a muchcoarser cut in sectioning operations, but are generally preferred for grinding, since they areless liable to lose cutting efficiency due to the build up of debris.

The grinding directionmay be important. For example, it is usually undesirable to grind aregion near a free surface perpendicular to that surface, since the cutting particles are almostcertain to introduce extensive sub-surface damage as they bite into the edge of the sample.Cutting in the reverse direction, so that the grit particles are exiting from the free surfaceduring grinding, will result in much less damage. The extent of the sub-surface damagedepends on the elastic rigidity and hardness of thematerial, so that soft metals are extremelydifficult to grind. However, brittle materials are prone to sub-surface cracking. It isimportant to recognize that sub-surface damage is always introduced during grinding andto ensure that subsequent polishing fully removes this damaged layer.

3.3.3 Polishing and Etching Methods

The primary aim of polishing is to prepare a surface which is both flat and devoid oftopographical features unrelated to the bulk microstructure of the sample. Each polishingstage is designed to remove a layer of damagedmaterial resulting from the previous stage ofsurface preparation. There are three accepted methods of polishing a sample: mechanical,chemical and electrochemical. Of these three, mechanical polishing is by far the mostimportant.


In mechanical polishing the mechanical damage of the earlier stages of preparation areremoved by resorting to finer and finer grit sizes. The number of polishing steps necessary toreduce topographical roughness to below the wavelength of light and, as far as possible,eliminate sub-surface mechanical damage may be as few as three (for hard materials) or asmany as ten (for very soft samples). The carrier for the polishing grit may be a paperbacking (for a silicon carbide grit, down to perhaps a #600 grit SiC paper), to be followed bya cloth polishing wheel (down to perhaps a 1/4 mm diamond grit).

In a chemical polishing solution the products of chemical attack form a viscous barrierfilm at the surface of the sample which inhibits further attack (Figure 3.20). Regions ofnegative curvature (surface grooves and pits) develop thicker layers of the viscous, semi-protective barrier film, while ridges and protrusions, with positive curvature are covered bymuch thinner layers and are attacked faster. As a result the sample is rapidly smoothed todevelop a topographically flat surface. Electrolytic polishing is somewhat similar, but thesample must be an electrical conductor, and is almost always a metal. Positively chargedcations are dissolved in the electrolyte at the sample surface and form a viscous anodic filmof high electrical resistance. For successful electrolytic polishing most of the voltage dropacross the electrolytic cell should be across this anodic film, so that, as in chemicalpolishing, the rate of attack is again controlled by film thickness. External adjustment of thevoltage across the electropolishing cell ensures better control for electrolytic polishingwhen compared with chemical polishing, and soft materials such as lead alloys, which arevery difficult to prepare by mechanical polishing, can be successfully electropolished.Nevertheless, the development of increasingly sophisticated mechanical polishing meth-ods, suitable for even the softest engineering materials, has decreased the importance ofchemical and electrochemical methods of surface preparation.

Etching of the sample refers to the selective removal of material from the surface in orderto develop surface features which are related to the microstructure of the bulk material. Ifthe different phases present differentially reflect and absorb incident light, then etchingmaybe unnecessary. Most nonmetallic inclusions in engineering alloys are visible withoutetching, since themetallicmatrix reflectsmost of the incident lightwhile the inclusion oftenabsorbs it and appears darker. Optically anisotropic samples observed in polarized light also

Figure 3.20 Both chemical and electrolytic polishing rely on aviscous liquid layer, to enhancethe attack of protruding regions and inhibit attack at grooves and recesses, ultimately forming amirror-like, polished surface.


show contrast without etching, associated with differences in crystal orientation (Section3.4.3).

Etching may also develop the surface topography, for example by grooving grainboundaries or giving rise to differences in the height of neighbouring grain surfaces.Etching may also form thin surface films whose thickness reflects the underlying phase andgrain structure. Such films may either absorb light or give rise to interference effects thatdepend sensitively on the film thickness.

Most etching methods involve some form of chemical attack, which is more pro-nounced in those regions of the surface that have a higher energy (grain boundaries, forexample). Thermal etching is an exception. In thermal etching heating of the sampleallows short-range surface diffusion to occur, reducing the energy of the polished surfacenear a boundary and affecting the local topology. At grain boundaries, thermal grooves areformed. In many materials the surface energy of a single crystal is highly anisotropic, sothat thermal etching reduces the total energy by forming surface facets on some suitablyoriented grains.

The commonest etching procedures make use of chemically active solutions to chemi-cally etch the surface and develop a topology which is visible in the microscope. Thesolvents are commonly one or other of the alcohols, but molten salt baths may also be usedfor less reactive samples, such as ceramics. In most cases the sample is immersed in thesolution at a carefully controlled temperature for a given time, then rinsed thoroughly anddried (typically, using alcohol). In some cases electrolytic etching is used to promote thelocalized attack, for example, on stainless steels.

Chemical staining is sometimes used to form a surface film whose thickness depends onthe surface features of the microstructure. A steel sample will develop such a film whenoxidized in air at moderate temperatures, the different grains appearing in a rainbow ofinterference colours, controlled by the oxidation time and temperature that determine thethickness of the coherent oxide film formed on each grain (Section 3.4.5.4).

3.3.3.1 Steels and Non-Ferrous Alloys. Engineering alloys are perhaps the most com-mon structural materials to be prepared for microscope investigation by mechanicalpolishing. The ferrous alloys cover a wide range of hardness, from brittle martensites tolow yield-strength, transformer steels. There are few polishing problems which cannot besolved using simple rules of thumb. Themost common forms of polishing defect are surfacerelief (differential polishing), the rounding of edges, and scratches and plastic deformation.Assuming that the polishing media are free of contamination (primarily debris from theearlier stages of surface preparation), then scratches and plastic deformation can beprevented by selecting a more compliant support for the polishing media, in order toreduce the forces applied to the individual particles and increase the number of particlecontacts per unit area of the sample surface. Conversely, surface relief and edge roundingcan be inhibited by selecting a less compliant support.

3.3.3.2 Pure Metals and Soft Alloys. The softest materials are the hardest to polishmechanically, because they are so susceptible to mechanical damage. If excessive force isapplied to the polishing media, then wear debris will be embedded in the soft surface of thesample and dragged in the direction of shear. Even if no obvious signs of damage are visibleon the polished surface, subsequent etching (see below) is liable to reveal sub-surface tracesof plastic deformation. Successful preparation of these materials is accomplished by


polishing at low shear velocity and applied pressure, so as to inhibit polishing debris fromadhering to the sample and smearing across the surface.

3.3.3.3 Semiconductors, Ceramics and Intermetallics. Brittle materials are usuallyeasier to polish mechanically than soft materials. In particular, there are unlikely to beproblems associatedwith polishing debris adhering to the surface. However,microcrackingand grain pull-out in these materials is very possible, associated especially with adhesivefailure at interfaces and boundaries between regions of different compliance.

The selection of the polishing medium is important, and its hardness should exceed thehardness of the sample. Neither alumina nor silicon carbide can be successfully polishedwith SiC grit, and diamond is the medium of preference. The same is true, to a lesser extent,for silicon nitride samples. Cubic boron nitride (CBN) has some advantages as a grindingmedium. The hardness exceeds that of SiC but the oxidation resistance is better than that ofdiamond.

An important point to realize is that even the most brittle of materials can deformplastically in the high pressure zone beneath a point of contact with a grit particle. Plasticflow in this region will generate internal stresses which, when the contact stresses areremoved, tend to cause cracking and chipping around the original contact, so that damage tothe surface and near-surface region develops adjacent to the original contact area.Again, thesolution is to limit the applied pressure and select a larger compliance for the carrier of thegrinding media.

3.3.3.4 Composite Materials. Some of the most difficult materials to prepare for opticalmicroscopy are engineering composites inwhich a soft, compliant matrix is reinforcedwitha stiff but brittle fibre. While such materials have been successfully prepared in cross-section, fewer optical micrographs have been published showing the distribution of thereinforcement parallel to the fibres. Attempts to prepare such samples often result in loss offibre adhesion, fracture of loose fibres, and damage to the soft, supporting matrix byfragments of the hard reinforcement.

In addition to an awareness of the problems and artifacts (features which are associatedwith preparation defects) involved in specimen preparation for the optical microscope, it isimportant to recognize that some samples may just be unsuitable for optical microscopy.

3.4 Image Contrast

Image contrast in the optical microscope may be developed by several alternative routes,most of which require careful surface preparation. A brief description of the various etchingprocedures commonly used to develop contrast in alloy and ceramic samples has been givenpreviously (Section 3.3.2). A quantitative definition of contrast is best given in terms of theintensity difference between neighbouring resolved image features,C¼ ln(I1/I2). For smallintensity differences this reduces to:

DC ¼ DI=I ð3:5Þand a comparison with the Raleigh criterion for resolution (Section 3.1.2.3), suggests thatDC should be at least 0.14 if features separated by a distance equal to the resolution are to bevisible.


At larger separations verymuch smaller contrast variations are distinguishable, andmanycomputerized image processing systems operate with 256 grey levels, corresponding tointervals of DC¼ 0.004. However, the largest NA objectives accept light scattered from thesurface at much higher angles than objectives of lower NA, so that the contrast obtained inthe image from any given feature is reduced at highNA. It follows that higher magnificationimages tend to show lower contrast.

3.4.1 Reflection and Absorption of Light

Electromagnetic radiation incident on a polished solid surfacemay be reflected, transmittedor absorbed. Specular (mirror-like) surfaces, highly reflecting to visible light, are charac-teristic of the presence of free conduction electrons in the sample material, and hence ofmetallic materials. However, most metals absorb a significant proportion of the incidentlight. Thus copper and gold absorb in the blue, so that the reflected light appears reddish oryellow. However silver and aluminium reflect over 90% of normally incident visible light,and both these metals are used for mirror surfaces.

The high reflectivity of polished aluminium is not affected by the presence of the thin,amorphous oxide protective film formed on the surface in air, since the thickness of this filmis well below the wavelength of visible light. Indeed most samples, and certainly all metalsand alloys (with the partial exception of gold), are normally covered by some kind of surfacefilm, either as a result of surface preparation (polishing and etching), or due to reactions inthe atmosphere. As long as these films are uniform, coherent and of a thickness less than thewavelength of the incident light, they do not interfere with the reflectivity.

The relation between the fraction of the incident light which is reflected and thattransmitted or absorbed depends on the angle of incidence of the light. The refractive indexof the solid, the ratio of the wavelength in free space to that in the solid, determines thecritical angle (the Brewster angle) beyond which no light can be transmitted, and in dark-field illumination this critical angle may be exceeded, increasing the light signal scatteredinto the objective lens (Figure 3.21). The fraction of the incident light reflected from thesurface is sensibly independent of the sample thickness for all thicknesses exceeding thewavelength, and depends only on the material and the angle of incidence, but the fractionthat is transmitted depends sensitively on the thickness, and decreases exponentially asthe thickness increases, in a manner exactly analogous to the case of X-ray absorption(Section 2.3.1).

Mineralogical samples arecommonlypreparedas thin sectionsandexamined inpolarizedlight (Section 3.4.3). It is important that they should be sufficiently thin, primarily to allowadequate transmission, but also to ensure that resolved features in theobject donot overlap inthe projected image of the slice. In general samples of thickness 50 mm or less are suitable.

Ceramic and polymer samples often transmit or absorb an appreciable fraction of theincident light, giving poor contrast, both because of theweak reflected signal and also due tolight that is scattered back into the objective from sub-surface features which are below thefocal plane in the image. An evaporated or sputtered coating of aluminium high lights thesurface topography of such samples, but at the cost of losing information associated withvariations in reflectivity and absorption. Some of these materials (especially crystallinepolymers and glass-ceramicmaterials) can be studied in transmission, using polarized light.Convincing separation of the topological features from effects uniquely associated with the


bulk microstructure is best accomplished by imaging the same area both before and aftercoating with the aluminium reflecting film.

3.4.2 Bright-Field and Dark-Field Image Contrast

In normal, bright-field illumination only a proportion of the incident light is reflected orscattered back into the objective lens. There are two limiting contrast conditions in brightfield illumination. The first is that discussed previously and is often termed Kohlerillumination. The light source is focused at the back focal plane of the objective lens,so that the light from a point source is incident normally on the sample surface while thatfrom an extended source is incident over a range of angles determined by the size of thesource image in the back focal plane of the objective [Figure 3.22(a)]. If, however, the light

IncidentIntensity

Light which is nottransmitted or reflected

is absorbed

ReflectedLight

TransmittedLight

At the critical angle ofincidence no light is

transmitted

αc

Figure 3.21 The relation between the intensity reflected, transmitted and absorbed, and the�critical� Brewster angle for a specularly reflecting surface.

Sample

Extended SourceImage

(a) (b)

Extended Source Imagedin Sample Plane

Figure3.22 Two limiting conditions for bright-field illumination: (a) the source is imaged in theback focal plane of the objective; (b) the source is imaged in the focal plane of the objective.


source is imaged in the plane of the sample, then the distribution of the intensity over thesample surface will reflect that in the source. If the sample is a specular reflector (a mirror)then an image of the source will be visible in the microscope [Figure 3.22(b)]. For highquality light sources, which emit uniformly, there may be some advantage in increasing theincident intensity by focusing the condenser system so that the source and sample planescoincide, but with most light sources, especially at low magnifications, uniform illumina-tion of the sample is best achieved by focusing the light source at the back focal plane of theobjective.

Topographical features which scatter some of the incident light outside the objectivelens will appear dark in the image. This is true of both steps and grain boundary grooves(Figure 3.23). The features themselves may have dimensions considerably less than thelimiting resolution of the objective, but two features will only be observed in the image ifthey are separated by a distance greater than theRaleigh resolution and give rise to sufficientcontrast. In most cases, the size of a topological feature at the surface reflects the surfacepreparation as much as it does the bulk microstructure. An obvious example is the�grooving� at grain and phase boundaries that is associated with chemical or thermaletching. It follows that considerable care may need to be exercised when measuring amicrostructural parameter, such as particle size or porosity. We will return to this inChapter 9.

Since contrast is mainly determined by comparing the intensity of the signal from somefeature with that of the background, features which appear faint in the normal, bright-fieldimage can often be enhanced by using a dark-field objective (Figure 3.16). In some casestopographical information can be enhanced by deflecting the condenser system, so that thespecimen is illuminated from one side only. Such oblique illumination is sometimesavailable as a standard attachment. The apparent shadowing of topological features inoblique illumination helps to bring out the three-dimensional nature of the surface, butusually with some loss of resolution. Such images may also be somewhat misleading, sincethe apparent identification of �hills� and �valleys� depends on the direction from which thelight comes. In the �real�world light comes fromabove, and arranging the illumination in themicroscope so that the sample is illuminated from �below� appears to turn the �valleys� into�hills�! The contrast obtained from a single feature by using bright-field, dark-field andoblique illumination are compared schematically in Figure 3.24.

Sample

Figure 3.23 In bright-field illumination topographicalmicrostructural features are revealed byscattering of light outside the objective lens aperture.


3.4.3 Confocal Microscopy

Confocal microscopy, which has been developed primarily for the biological and healthsciences, should also be useful in materials science, especially in the study of transparentglasses and polymers. In confocal microscopy a parallel beam of light is brought to a sharpfocus at a designated location in the sample. The light source is usually a laser, althoughhigh intensitywhite light from a xenon arc andmonochromatic (green) light from amercuryarc have also been successfully used. The point probe is scanned across the sample in an x–yplane television raster and generates a signal from a thin slice of the sample located at aspecific depth beneath the surface. The resulting image is termed an optical section.A seriesof such optical sections taken at different depths is remarkably free of background noise. Byfocusing the point source using a cone of light, as in dark-field microscopy, and selecting asuitably excited fluorescent signal, it has proved possible to obtain very high resolutionthree-dimensional images of specific sites in a biological tissue samplewhich correspond todifferent fluorescent �labels�. The use of this technique is beyond the scope of the presenttext.

3.4.4 Interference Contrast and Interference Microscopy

In interferencemicroscopy the light reflected from the sample interfereswith light reflectedfrom an optically flat standard reference surface. In order to achieve this, the two beamsmust be coherent, that is, they must have a fixed phase relation and this is achieved byensuring that both beams originate from the same source, using a beam splitter to firstseparate and then recombine the two signal amplitudes from the sample and referencesurfaces.

3.4.4.1 Two-beam Interference. The simplest arrangement for achieving interferencecontrast is to coat an optical quality glass coverslip with a thin layer of silver or aluminium,

I

Oblique

I

Dark Field

I

Bright Field

Figure 3.24 A schematic comparison of contrast in bright-field, dark-field and obliqueillumination.


such that rather more than half of a monochromatic incident light beam will be transmittedby the cover-slip (see below). The light reflected from the thin metal film constitutes thereference beam, while that transmitted through the reference coating and then reflectedfrom the sample surface is the interfering beam (Figure 3.25). Multiple reflections areignored for the time being, but we should note that these will affect both image resolutionand contrast. It is also assumed that there is no absorption of the light, so that the incidentlight is either reflected or transmitted, but not absorbed.

If the reflection coefficient of the metal film is R, then, in the absence of absorption, thetransmission coefficient will be (1�R). Assuming that the reflection coefficient of thesample surface is 1, then no light is absorbed by the sample and the intensity reflected backfrom the sample will be (1�R). The reflected beam from the sample will then betransmitted back through the metal film, with a transmission coefficient (1�R), and theremaining intensity will be re-reflected. Hence the intensity of the second beam from thesample that is transmitted back through the metal film will now be (1�R)2, as comparedwith the intensity of the reference beam,R reflected by the metal film. For the two beams tointerfere strongly they should be of comparable intensity, R¼ (1�R)2. Solving thisequation, the required value of R is (3� ffiffiffi

5p

)/2, or approximately 0.38.The condition for destructive interference is that the two beams differ in path by

(2nþ 1)l/2, so that the phase difference between the beams is equal to p. But the pathdifference is just twice the separation h of the partially reflecting reference surface from thesample, that is 2h¼ (2nþ 1)l/2, and the first destructive peak occurs when the two surfacesare separated by l/4. Successive interference fringes correspond to height contoursseparated by Dh¼ l/2. Assuming that shifts in an interference fringe are detectable toof the order of 10%of the fringe separation and that thewavelength of the incident light is inthe green, then simple two-beam interferometry should be capable of detecting topologicalheight differences to an accuracy of�20 nm, roughly the same sensitivity as that for phasecontrast (Section 3.5.4). An example of two-beam interference fringes from a system ofgrooved grain boundaries is shown in Figure 3.26. The reference surface is at a slight angle

Figure 3.25 Condition for two-beam interference using a half-silvered reference surface.


to the sample surface, and the separation of the fringes in the image corresponds todifferences in separation between the reference and sample surfaces of l/2.

Interference contrast and sensitivity can be improved by placing a drop of immersion oilbetween the coverslip and the sample, reducing the effective wavelength by a factor m, therefractive index of the oil. The objective lensNA for good two-beam interference is limited,since for an NA greater than about 0.3 the difference in pathlength for light passing throughthe periphery of the lens and along the axis is sufficient to destroy the coherency of thebeam. The condenser system should be focused on the back focal plane of the objective(Kohler illumination) to ensure that the incident light is normal to the sample surface. It mayalso be useful to view the specimen inwhite light, when the orders of the interference fringescorrespond to Newton�s colours. These colours correspond to the subtraction of a singlewavelength from thewhite spectrum, generating the complementary colour. For very smallseparations, the shortest wavelengths interfere in the blue and the corresponding Newton�scolour is yellow. As the separation increases the wavelength that interferes moves towardsthe green and the complementary Newton�s colour is magenta. Finally, the interferingwavelengthmoves into the red, and the complementary interference colour is cyan. For stilllarger separations, the interference conditions move to higher orders of (2nþ 1)l/2 and thecolour sequence is repeated, but with increasingly dull contrast as several differentwavelengths start to contribute to the interference.

A region of the specimen that is in contact with the reference plate then appears white. Itis a sobering experience to see just how fewpoints of contact there are between the referenceand sample surfaces, even though the reference surface is optically flat and the sample hasbeen well-polished.

3.4.4.2 Systems For Interference Microscopy. The Mercedes (or Rolls Royce) of micro-interferometers is that designed by Linnik (Figure 3.27), but it is seldomused. Two identicalobjective lenses ensure that no path differences are introduced by the optical system. Theposition of the reference surface can be adjusted along the optic axis and the reference

Figure 3.26 Two-beam interference pattern due to grain boundary grooving.


surface can be tilted accurately about two axes at right angles and in the plane of thesurface, in order to adjust both the spatial separation and the orientation of the interferencefringes.

The position of the reference surface is adjusted in white light so that, at a small angle oftilt, Newton�s interference colours are observed either side of thewhite contourmarking theline of coincidence of the reference imagewith the image of the sample. Far simpler systemsfor interference microscopy are commercially available as standard attachments, but mosthave a rather limited life, since they rely on half-silvered reflecting surfaces that are easilydamaged if brought into contact with a sample. The disadvantage of a metal-coatedcoverslip (which is cheap and can be regarded as a consumable) is the inability to adjust theseparation between the reference surface and the sample or the distance between theinterference fringes, but for many applications this is not critical.

3.4.4.3 Multi-beam Interference Methods. As noted previously, in simple two-beaminterference a proportion of the light is multiply reflected. With the previous assumption ofno absorption loss and a reflection coefficient of 0.38 for the half-silvered coverslip, thesemultiply reflected intensity losses amount to 24% of the incident light and result inunwanted background. By increasing the reflection coefficient of the reference surface theproportion of light which is multiply reflected can be increased, until the value is close to 1.With the geometry shown in Figure 3.28, the dependence of the total reflected intensity I is

Specimen

MatchedObjective

Objective

BeamSplitter

Eyepiece

Source

Ref

eren

ceSu

rfac

e

Figure 3.27 Optical system for a Linnik microinterferometer.


given by:

I ¼ T2

1�Rð Þ2" #

1

1þ 4R1�Rð Þ2 sin

2 d=2ð Þ

24

35 ð3:6Þ

where TandR are the transmission and reflection coefficients, and the reflection coefficientof the sample is nowassumed equal to that of the reference surface. The parameter d is givenby:

d2¼ ð2phcosyÞ

lð3:7Þ

where h is the separation of the reference and sample surfaces. If, once again, we assumethat there is no absorption in the reference film, so thatTþR¼ 1, and the intensity collectedfalls to zerowhen 2hcosy¼ nl. The interference fringes that are formed are now localized atthe reference surface, and the number of beams which contribute is determined by theangular tilt of the reference surfacewith respect to the optic axis and the separation betweenthe reference surface and the sample. Thewidth of the black interference fringes depends onthe number of beams taking part and this width can be very narrowwhen compared with thecos2 intensity dependence obtained in the two-beam case.

For multiple beam interferometry to be effective, the incident beammust be parallel andthe surfaces separated by no more than a few wavelengths. The best patterns have beenproduced by spin-coating the samplewith a thin plastic film and evaporating silver onto theatomically smooth surface of the plastic. Under these conditions the very sharp interferencefringes can reveal topological changes in the surface at the nanometre level, and it is quitepossible to image growth steps on crystals that are only a few atoms in height (Figure 3.29).Of course, there is a penalty to be paid: since the incident and reflected light must beparallel, andmultiple beam interferometry is only possiblewith a lowNA objective, and thelateral resolution is typically no better than 1 or 2 mm.

3.4.4.4 Surface Topology and Interference Fringes. A few more words are in orderconcerning the information which can be derived using interference microscopy. The smalldepth of field of the optical microscope means that the region of the object in focus is a thinslice of thickness of the order of 1 mm or less. The image is then a planar projection of thisslice of material, and contains information on both the topology and the physical properties

Thin film(reference surface)

Specimen

θ R RT2 R3T2 R5T2

RT R3T R5T

T R2T R4T

t

Figure 3.28 Geometry for multiple beam interference.


of the surface. The lateral resolution is limited by the wavelength of the light and the NA ofthe objective lens, but also by the method of surface preparation and the means used torecord the image. Interference micrographs contain image contrast based on variations inthe phase of the light reflected from the surface. These phase variations may be associatedwith three quite distinct optical effects.

1. They may be associated with surface anisotropy and best revealed by polarized light(Section 3.4.4). The phase shift arises from differences in the wavelength of lightpolarized parallel to the two optic axes of the anisotropic sample surface.

2. If they are due to small topological features or spatial inhomogeneities in the opticalproperties (variations in refractive index), then they may be detected by phase contrastmicroscopy or differential interference (Nomarski) contrast (Section 3.4.5), whichidentifies the sign of the phase shift and its approximate magnitude.

3. Finally, they may be due to variations in surface topology, and then interferencemicroscopy is often the most appropriate tool, quantitatively monitoring the separationbetween an optically flat reference surface and the sample surface.

The vertical resolution for topological features in interferencemicroscopy is much betterthan the usual lateral resolution of the optical microscope, of the order of 20 nm for two-beam interference (or phase contrast) and as little as 2 nm for multiple beam interference.This vertical resolution is only available at the expense of lateral resolution, since therequirements for good interference images limit the NA of the objective lens to of the orderof 0.3. In spite of this, quantitative measurements using interference microscopy haveproved extremely valuable, and applications range from measurements of grain boundaryenergy using thermal grooving at grain boundaries, to the height of facets on the surfaces ofa growing crystal, and the size of the slip steps in ductile materials.

3.4.5 Optical Anisotropy and Polarized Light

Many samples are optically anisotropic, that is the refractive index, and hence thewavelength of light in the material, are a function of the direction of propagation. Crystalsof cubic symmetry are optically isotropic, while those with tetragonal, rhombohedral orhexagonal symmetry are characterized by two refractive indices, parallel and perpendicularto the primary symmetry axis. Crystals of even lower symmetry are characterized by threerefractive indices. If a beam of light is incident on an optically anisotropic, transparent

Figure 3.29 Multiple beam interference fringes from a polished quartz specimen.


crystal, the beam will be transmitted through the crystal as two beams whose electromag-netic vectors are aligned parallel to the principle optical axes of the crystal in the planeperpendicular to the incident beam (Figure 3.30). The two beams correspond to differentwavelengths for the light passing through the crystal. On exiting from an anisotropic crystalthe two polarized beamswill recombine, but any object viewed through a slice of the crystalwill appear doubled, due to the different deflections of the two beams travelling through theslice.

A beam of light is said to be polarized when the electromagnetic wave vectors are notrandomly oriented perpendicular to the direction of propagation, but are instead aligned in aspecific direction. Sunlight reflected at a shallow angle from a car roof is partially polarized,with the direction of polarization in the plane of the roof. �Polaroid� sunglasses only transmitlight which is vertically polarized, so the glare from the (approximately horizontal) car roofis cut out by the sunglasses. If the sunglasses are removed and rotated 90�, the glare will betransmitted, since the plane of polarization of the lens is now parallel to that of the reflectedglare from the car roof.

3.4.5.1 Polarization of Light and Its Analysis. In the polarizing microscope the incidentillumination is plane-polarized by inserting a �polarizer� into the path of the condenserassembly. To avoid any spurious changes in polarization of light during transmissionthrough the optical system the plane of polarization is chosen to be perpendicular to theplane of assembly of the microscope components, A second polarizing element, termed the�analyser�, is placed in the path of the imaging lenses and can be rotated about the optic axisof the microscope so that the plane of polarization of the analyser may be at any anglebetween 0� and 90� to that of the polarizer. When the angle between the planes ofpolarization of the polarizer and analyser is set at 90� the sample is said to be viewedthrough crossed polars, and no light reflected from an isotropic sample can be transmittedthrough the analyser to the final image.

When a beam of plane-polarized light is reflected from an optically anisotropic surface,the components of the electromagnetic wave vector reflected from the surface arethemselves resolved into two components, parallel to the principle optic axes of the surface

Crystal

Isotropic Anisotropic

Figure 3.30 Amonochromatic beamof light incident on a transparent crystalwill propagate astwo beams with different refractive indices (corresponding to two different wavelengths) whenthe crystal is optically anisotropic.


(Figure 3.31). Due to the optical anisotropy, these two components are now no longerexactly in phase, and the combined reflected beam therefore has a wave vector whose tipwill rotate as the beam propagates, varying in amplitude as it does so. The tip of theamplitude vector viewed in the plane normal to the direction of propagation of the beamthen describes an ellipse, and the beam is said to be elliptically polarized. If an analyser witha plane of polarization at 90� to that of the polarizer intercepts this reflected beam, then onlythat component of the amplitude parallel to the plane of polarization of the analyser will betransmitted. This situation is summarized in Figure 3.32.

From Figure 3.32 it is clear that the maximum amplitude reflected from the sample willbe transmitted through the analyser when the principal axes of the sample are set at 45� tothe crossed polars. Rotating the analyzer will result in more light being accepted, and whenthe two polars are set parallel, then all the light reflected will be admitted to the final image.

To maximize the contrast in polarized light the condenser aperture should be stoppeddown and the source focused on the back focal plane of the objective (Kohler illumination),so that nearly all the incident light is normal to the sample surface. The NA of the objectivelens has a pronounced effect on the contrast, and only at low values ofNAwill the path of thereflected light be sensibly normal to the specimen surface. At high values of theNA the lightwill be collected from the specimen surface over a wide range of angles, and the effects ofoptical anisotropy are then considerably reduced.

3.4.5.2 The 45� Optical Wedge. If a thin slice of a transparent, optically active,birefringent crystal, such as quartz, is inserted into the optical path between the polarizerand the analyser, and the optical axes of the quartz slice are set at 45� to those of the crossedpolars, then the amplitude of the two components of the beam travelling through the quartzcrystal will be identicalwhile the phase difference between the twobeams at the exit surfacewill depend on the thickness of the crystal. If the crystal is cut as a wedge and a white lightsource is used, then the two beams exiting the wedge will generate bands of interferencewhenever the phase difference Df is equal to p, or more generally, whenever Df¼(2nþ 1)p. The path difference between the twowaves travelling through the quartz crystal

Plane ofpolarizer

Principle axesof surfaceanisotropy

Incident wavevector

Resolvedcomponents

Rotatingwave vector

(a) (b)

Figure 3.31 (a) Plane-polarized light is resolved into twocomponentswhen it interactswith ananisotropic sample. (b) The out-of-phase component then recombines to form an ellipticallypolarized wave.


Dd is related to the difference in the refractive index in the two principal directions and thewavelength of the incident light. If the refractive index Z is given by Z¼ l0/l¼ c/v, where l0is the wavelength in space, l is that in the crystal, c is the velocity of light in space and v thevelocity of light in the crystal, it follows that:

Dd ¼ t Z1�Z2ð Þ ¼ l02p

D� ð3:8Þ

where t is the thickness of the crystal.For each thickness of crystal in the wedge there will be a specific wavelength in the

incident white light that will interfere destructively and be removed from the transmittedbeam, as discussed previously (Section 3.13). The shortest wavelengths of the visiblespectrum are removed first (violet), leaving the transmitted band of light pale yellow, this isfollowed by �blue� interference, leading to a magenta band of transmitted light, then greenfollowed by red (transmitting cyan). For thick crystals Dd¼ (2nþ 1)l0/2, where n is aninteger, and the same order of �interference� colour bands will be repeated, although withdecreasing brilliance for each successive order, since for the thicker regions the interferencecondition at each thickness can be fulfilled for more than one wavelength in the visiblerange.

Amplitudeaccepted by

analyzer

Plane ofanalyzer

Resolved componentsreflected from the sample

Plane ofpolarizer

Incidentamplitude

Figure 3.32 A plane-polarized beam of light incident on an optically anisotropic surface willbe reflected as two beams with wave vectors parallel to the principal optic axes of the surface.These combine to form an elliptically polarized reflected beam, which is resolved by theanalyser, whose plane of polarization is set at 90� to that of the polariser.


This sequence of interference colours was first noticed by Newton, and is referred toas Newton�s colours. Using parallel, rather than crossed, polars the colours are inverted(Figure 3.33).

3.4.5.3 White Light and The Sensitive Tint Plate. Instead of a quartz wedge it is possibleto insert a thin, transparent slice of quartz of uniform thickness, selected to introduce a phasedifference equivalent to destructive interference of green light, tinting the light acceptedinto the imaging system by the analyser magenta. If the sample is optically isotropic,reflection from the surface introduces no further phase shift and, when viewed under whitelight using this sensitive tint plate between crossed polars, the microstructure appearscoloured mauve or magenta. However, if the surface is anisotropic, and so introduces anadditional phase shift into the reflected beam, then this phase shift will result in a colourshift along the sequence of Newton�s colours. A positive phase shift will increase thewavelength for destructive interference, and that area of the specimen surface will nowappearmore blue or cyan.Any negative phase shiftwill result in destructive interference at ashorter wavelength and a colour shift in the image of the sample towards the yellow.Rotating the sample by 90� will reverse the sign of these phase shifts introduced by opticallyanisotropic features present in the microstructure.

The sensitive tint plate is a very powerful tool for exploring the optical anisotropy of asample viewed in reflection, and is also very important when studying anisotropic

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Violet

Violet-red

Grey blue

Pale blue grey

Verypale blue Verypale yellow

Pale yellow-green

Greenish-yellow

Yellow-green

Greenish-yel

Crossed Polars Parallel Polars

1storder

2ndorder

3rdorder

4thorder

5thorder

6thorder

Opt

ical

path

diff

eren

ce(n

m)

Figure 3.33 Newton�s thickness interference colours in crossed and parallel polars.


crystalline polymers in transmission. In Figure 3.34 nodules of a crystalline polymercontain well-ordered microcrystals. The contrast from each nodule viewed in polarizedmonochromatic light includes a black, �Maltese cross� that is aligned along the axes of thepolarizer and analyser with dimensions that reflect theNA of the objective lens. Between thearms of the crosses the alignment of the microcrystals around the nucleation centre for eachnodule is clearly visible. The high optical anisotropy ensures excellent contrast, even whenworking at high resolution and with large NA objectives.

3.4.5.4 Reflection of Polarized Light. A few further words are necessary concerning theinteraction of polarized light with a reflecting surface. As noted earlier (Section 3.4.4),unpolarized light is partially polarized when it is reflected at an angle, with the direction ofpolarization perpendicular to the plane containing the incident and reflected beams.Furthermore, most surfaces prepared for microscopic examination are covered by a surfacefilm. It follows that linearly polarized light incident on the surface is likely to undergo somephase change and, to some extent, become elliptically polarized. The cause may betopographic, for example, fine surface facets or aligned grooves which impart opticalanisotropy to an otherwise isotropic surface, or it may be associated with the conditions ofillumination (a primary beam incident at a glancing angle, as in dark-field illumination) or itmay result from the conditions of collection of the reflected light (a large NA objectiveaccepting a wide cone angle).

In many cases complete extinction is not obtained in crossed polars, and extinction doesnot necessarily mean that the sample itself is anisotropic. Alternative explanations could beloss of linear polarization resulting from an oblique angle of incidence or the existence of�anisotropy� due to surface topology. Changing the plane of focus of the source by adjustingthe condenser lens setting should give an indication of whether or not the illuminatingconditions are responsible for any apparent anisotropy.

Figure 3.34 Crystalline polymers viewed in transmission by polarized light reveal a wealth ofdetail associatedwith the crystallization process. The arms of the blackcrosses are parallel to theplanes of the polarizer and analyser in the microscope.


3.4.6 Phase Contrast Microscopy

In phase contras microscopy, intensity variations in the image are due to interferencebetween a specularly reflected beam and an elastically scattered beam from the samesample surface. Phase contrast is important in the histological examination of organictissues viewed in transmission, when small differences in the scattering from featuresgiving weak contrast need to be amplified, but the technique is also useful for revealing finetopological features in reflection microscopy.

If light is reflected from two neighbouring regions which differ slightly in height by asmall amount h, that is much less than the wavelength, then the reflected beams will be outof phase by a small phase angle:

df ¼ 2ph

l

� �ð3:9Þ

If h is sufficiently small, the phase shift between the two specularly reflectedwavevectors isequivalent to a small elastically scattered amplitude whose wave vector is sensibly out ofphase with the specularly reflected beams by p/2 (Figure 3.35). The problem is to convertthis very weak scattered signal into an observable intensity difference. This requires boththat the scattered amplitude be comparable with the amplitude of the specularly reflectedsignal and that the phase difference be shifted by an additional p/2, so that constructive ordestructive interference may take place.

One solution is illustrated in Figure 3.36. In this figure the source of light is an annularcondenser aperturewhich ensures that the path length to the specimen is sensibly identicalfor all rays. This annular aperture is focused on the back focal plane of the objective. Anintermediate, auxiliary lens is used to bring the image of the source, reflected by the sample

Beam 1k1 k2 Beam 2

h

h<<λ |k| = 1/λ

Diffractedamplitude

δφk2

k1

Figure 3.35 Phase contrast is based on the light scattered from small features associatedwith adifference in height h. The scattered amplitude is approximately p/2 out-of-phase with thespecularly reflected beams.


surface, into focus on a grooved phase platewhich is placed behind the final imaging lens.The diameter and width of the groove in the phase plate is chosen to exactly match theannular source aperture, so that specularly reflected light only passes through the groove,while the scattered light is transmitted through the ungrooved portion. The depth of thegroove is selected to introduce the additional required phase shift ofp/2, so that the scatteredsignal now either reinforces (þp) or interferes (�p) with the specularly reflected signal.That is �bumps� and �hollows� in the surface topology will appear either brighter or darkerthan the background in the image. To ensure that the amplitude of the specularly reflectedsignal is comparable with that of the scattered signal, the grooved region must be coatedwith an absorbing layer that reduces the directly transmitted amplitude by approximately90%, ensuring that the interference contrast will be clearly visible.

A series of phase plates may be provided, especially for use with transmission samples,having phase shifts of (2nþ 1)p/2. These phase plates are particularly useful in studyingbiological cell and tissue samples in which small differences in refractive index can becharacterized by using the phase contrast microscope. Bearing in mind that the height (orphase) differencesmust be small, and assuming that a reasonable value of h/l to achieve thisis 0.1, it follows that the phase contrast microscope will have no difficulty in picking upvariations in surface topology of the order of 10% of the wavelength, or of the order of50 nm. In practice, surface steps of the order of 20 nm are readily detectable.

3.4.6.1 Normarski or Differential Interference Contrast. Nomarski or differential inter-ference contrast is based on a rather different concept that provides a simple alternative toreflection phase contrast microscopy. A double quartz wedge is inserted into the opticalsystem after the polarizer in the 45� position and illuminates the sample with two slightly

EyepieceSecond Imageof Aperture

Half-SilveredMirror

First Imageof Aperture

CondenserLens

Lamp

Annular Aperture

Objective LensSpecimen

Auxiliary Lens

Phase Plate

Figure 3.36 An optical system for phase contrast reflection microscopy.


displaced beams that are separated in the object plane by a small lateral shift. Afterreflection from the sample the light collected by the objective is brought back into registryby a second double prism and the image is viewed through the analyser. Since the pathlength of the twin beams is identical, any residual difference in phase must be due tomicrostructural features that are on the scale of the separation of the twin images. These areusually height changes, for example, those associated with slip steps or boundary grooves.The contrast obtained inNomarski interference reflects these small topological differences,and the interpretation is very similar to phase contrast.

3.5 Working with Digital Images

In this section we introduce some of the terminology associated with digital imaging, andcompare analogue with digital data. Much of the discussion is quite general, and appliesequally to data focused by a system of lenses in an optical or electronmicroscope, as well asto images derived from raster scans, as in a television camera or scanning electronmicroscope.

3.5.1 Data Collection and The Optical System

In a simple optical or transmission electron microscope the image intensity is a continuousfunction of the x–y coordinates in the image plane, I¼ f(x,y). The number of photons orelectrons contributing to the signal is usually large enough to be able to ignore the statisticsof image formation, although for damage-sensitivematerials this is not necessarily the case,while for some weak signals the signal-to-noise ratio may also be a problem. The signal isdependent on the probe energy (the wavelength of the light used or the accelerating voltagefor the electrons), so that we should really write: Il¼ fl(x, y). For analogue optical imagesobtained in monochromatic light and transmission electron microscope images thisequation is adequate, however recording optical images in colour relies on filtering theintensity through colour filters. These filters may be red, green and blue (RGB) for positiveimages, or yellow, magenta and cyan (YMC) for negative images. That is, the analogueintensity is dependent on three colour parameters in addition to the two spatial parameters.For images recorded in monochromatic light, only one wavelength is involved, but thiswavelength still needs to be specified.

3.5.2 Data Processing and Analysis

We first consider the format of the image that wewish to process and analyse. The variationin intensity with position in the image plane of an optical system I¼ f(x, y) representsinformation from the object (the specimen)which has been transmitted by radiation that haspassed through the optical system. The image intensity is a continuous function in x and y.When recording in an analogue data format, such as a photographic negative, thiscontinuous function is modified by the response of the recording medium, but it is stilla continuous function.

In a digital, rather than analogue, data format, the intensity is no longer a continuousfunction in the x–y image plane, but rather a discrete intensity function. That is, the image


stored in a digital format exists as discrete picture elements, or pixels, in which each pixelstores information on the intensity of the image at a specific x,y location in the image.

The location x,y is the position of a pixel, whose dimensions are determined by the totalarea of the recorded image and the total number of pixels. It is important to question theequivalence between the area sampled by a specific pixel and the site on the sample,propagated through the optical system that generates the signal. There are two factors to beconsidered. The first is the resolution of the microscope and the second is the signalcollection efficiency. To be sure that the full resolution of the microscope is available in adigital data collection system, the pixel size needs to be smaller than the resolutionprojected onto the image plane, namely Dx,Dy<Md, where M is the magnification. Inpractice a 3 · 3 array of pixels is usually selected to represent a feature that is just resolvable,and this ensures that no resolution is lost in the digitized image.

Images are often recorded in a rectangular format (usually for aesthetic reasons), so thenumber of pixels along the x and y axes will not be the same. The recommended aspect ratioof width to height for television and video equipment is 4:3, but CCTV cameras are usually1:1, while awidescreen format (high definition television, HDTV) is 16:9. It follows that animage recorded for one format is often distorted when transferred to another format.

In scanning electron microscopes the magnification on the x and y axes is seldomidentical. The magnification may also vary from the centre to the periphery of the field ofview. A square grid will then appear either �barrelled� or as a �pin-cushion�, depending onwhether themagnification is larger or smaller near the centre of the scanned area. Tilting thesample, which can usually be donewithout losing the focus in SEM, always results in someforeshortening perpendicular to the tilt axis. It follows that scanned images may have adifferent magnification in the x and y directions that requires either calibration or correctionfor quantitative work.

The eye, as we noted earlier, only exercises its full resolution (�0.2mm) over a limitedarea of the retina, but the eye typically scans over distances of the order of 20 cm at the nearpoint. It follows that theminimumnumber of pixels to be recorded for a digital image is 106.In dynamic imaging, the eye is very good at following changes in the image and rather fewerpixels can be used for each frame, but for colour images the digital system relies on dyes toselect theRGBcomponents in neighbouring pixels. Typically, this is done in a 2 · 2 group offour pixels, in which two diagonal components are red and blue (R and B), while the othertwo are both green (G).

3.5.3 Data Storage and Presentation

So farwe have only considered digitizing the signalwith respect to its location in the image,not the digitizing of the intensity recorded from a given pixel, that is, from a given location.

We need to be quite clear about the terminology we are using. The intensity is the signalacquired from a given pixel at a specific location, x,y. The brightness, however, is expressedas intensity per unit area, and is therefore the intensity summed over the total number ofpixels that define an area of interest and divided by the area covered by these pixels. Toidentify differences in contrast, we therefore need to ensure that the pixels sample theintensity at spatial intervals which are appreciably smaller than the distances that separatethe variations in intensity that determine contrast. In other words, a sufficiently high densityof pixels will preserve the spatial frequencies in the original (analogue) image that contain


this information. TheNyquist criterion requires that the pixel sampling interval correspondsto twice the highest spatial frequency we wish to record, while Shannon�s sampling theorystates that the sampling interval should be no more than one-half the size of the resolutionlimit of the microscope optical system. For all practical purposes these two criteria areequivalent, but the reader should also remember that there will be a noise limit that restrictsthe significance of the differences in contrast (see below).

3.5.3.1 Data Storage Systems. Two quite separate factors will limit the useful range ofintensity that can be detected and recorded by any system. The first is the lower limit set,either by background noise or by the failure of the system to retain a signal at low doses,while the second is the upper limit set by the saturation of the detection or recording system.

In photography, reciprocity failure at low doses and precipitation overlap at high dosesare the limiting conditions. In addition, the emulsion excitation process and the definition ofcontrast dictate that the response curve for an emulsion is best plotted on a curve of density(the logarithm of the �blackening�) against the logarithm of the dose (the intensity of thesignal multiplied by the exposure time). The useful range is then the linear portion of thiscurve (Figure 3.18), and the slope of this region determines the contrast response, or g of thefilm.

The analogous curve for digital recording technologies is very similar, but in this case thelow dose limit is determined by signal noise, while the high dose limit is due to saturationand �blooming� of the detector, that is, excitation of neighbouring pixels by charge overflow.Two systems are in common use. CCD detectors collect an electrical charge excited by theimage signal formed in the plane of the detector. When exposure is complete, theaccumulated charge is read off from each pixel in sequence. Pixel sizes may be anythingfrom a few micrometres up to tens of micrometres and the frame size (the total number ofpixels in the array) is also variable. However, a typical frame size would be some 20mmacross and may contain 107 pixels or even more.

The second digital image system is the CMOSdetector. In this system the pixel signals areread out line by line, rather than as one complete frame of data. The CMOS system isessentially a �camera� on a single silicon chip, with the read-out electronics surrounding therectangular pixel array. Both CCDandCMOS detectors are used for optical image recordingand data collection. At the time of writing, the CMOS technology is inherently faster, and isslowly replacing CCD technology in optical image recording, even though the total numberof pixels in a large array is still less by almost an order of magnitude. Certainly for dynamic,real-time recording the CMOS technology is preferred. In electron microscopy the CCDtechnology is almost exclusively used, probably because of radiation damage susceptibilityunder the high energy electron beam. As noted above, the use of dye-filter arrays over thedetectors allows both systems to be used for colour photography.

3.5.4 Dynamic Range and Digital Storage

The eye is capable of distinguishing about 20 intensity levels in a grey scale image, as wellas an extraordinarily wide range of colour tints and shades (many thousands). However, notwo individuals have precisely the same colour response.Moreover, whatwe see in nature isoften impossible to reproduce in any recorded image (thinkof the colours of a peacock�s tail,a butterfly�s wing or the iridescent carapace of a beetle). In recording a �true� image,


especially a colour image, we have an almost impossible task. What we see through theeyepiece of an optical microscopewill always appear more vibrant than what we are able torecord on a positive colour transparency or reproduce on a digital colour monitor. Thepositive colour print is an evenpoorer record than a transparencyofwhatwe saw through theeyepiece of the microscope. Moreover, the tonal values will in each case be different, andwill vary, both from one recording system to another and with the settings employed by theoperator.

In both digital and analogue image colour observation and recording, the range of shadesand tints is limited by the performance of the phosphors, dyes and pigments used. Let usstart with the grey scale. A black and white image is referred to as a 1 bit binary image(2n, where n is the number of bits). Every pixel in the image is registered as either black orwhite, as in a newspaper photograph. If there are four levels of intensity recorded per pixel,then this is a 2 bit image, and 2n grey levels per pixel can be stored in n bits of image data.Most systems storemanymore than the 4 or 5 bits needed to record all the grey levels the eyecan detect (Figure 3.37).

Colour can add a great deal to the recorded image, and usually 8 bit data are stored,corresponding to 256 intensity levels in each of the three colours red, green and blue (RGB).When processing the image it is desirable to manipulate the intensity levels selected for animage in order to make full use of the range of levels that can be detected by the eye. This istrue of both grey scale and colour digital images. Digital image data storage is available in asomewhat confusing variety of computer formats, for example those labelled GIF, TIFF(tagged image file format), BMP (Bitmap) or JPEG (Joint Photographic Experts Group)(Table 3.1). These all differ in their digital data storage algorithms, and correspond tocompressed data files. Usually, microscopists will select the storage format which isconvenient for the software they are using. However, all of the formats compress the dataunless we intentionally disable the compression option during file storage. Compression is

Figure 3.37 Bit depth and grey levels for binary images.


convenient when storing or transferring large files, but we should remember that there is nosuch thing as compression without loss of data (Figure 3.38).

Finally, a few words about printing digital images are in order. Most printers are half-tone, that is they print a dot-matrix array. The tones depend only on the density of dots perunit area of the printed page. Two types of half-tone printers are in common use, ink-jetprinters inwhich a droplet stream is ejected by a piezoelectric actuator from an ink reservoirthat contains either black or a coloured ink. The coloured dyes may be red, green and blue,or yellow, magenta and cyan. The ink droplets are then absorbed and dry on the paper withvery little spreading. Laser printers use a totally different principle, creating a separatecharge pattern on a photoelectric surface and transferring toner of the required colour ontothis surface. Laser printers are faster, butmore expensive, but the inks for ink-jet printers aremore expensive. In both cases it is the cost of the dyes and toners that largely determines theeconomics of the process.

Table 3.1 Pixel dimensions of different image sizes, and their relative file size when storedin different compression modes.

Pixel dimension Grey scale (8 bit) Bitmap (24 bit) JPEG (24 bit) TIFF (24 bit)

16 · 16 2 k 2 k 2 k 2 k64 · 64 6 k 13 k 5 k 13 k

128· 128 18 k 49 k 12 k 49 k256· 256 66 k 193 k 22 k 193 k512· 512 258 k 769 k 52 k 770 k

Figure 3.38 Compression algorithms reduce the file size, butmayalso introduce noise into theimage.


Dye sublimation printers sublimate solid dye from a sourcewhich is located very close toa special paper, followed by heating, which provides a continuous tone print. Dyesublimation printers are expensive and primarily used for commercial applications. Waxprinters use a liquid source of coloured wax to provide a continuous tone print, and costabout the same as a good colour laser printer. As noted earlier, the printed image alwayscompares poorly with what the eye observes directly in themicroscope, or evenwhat can beseen on a computer monitor, for which an array of coloured phosphor pixels is excitedelectronically. Nevertheless, the technology continues to improve, and the gap in imagequality between the observed and the recorded image is shrinking. Photographic recordingis no longer an essential component of microscopic investigation. Digital images arecapable of recording all image detail, and digital data files can be processed either tooptimize the printed presentation of image information or to quantitatively analyse themicrostructural information.

3.6 Resolution, Contrast and Image Interpretation

Finally, we summarize some of the problems and pitfalls associated with the interpretationof microstructure based on optical images.

It should be clear by now that the observed or recorded image of a selected sample onlycontains information on the microstructure of the material if the observer understands thesequence of processes used to obtain the image:

1. preparation of the sample;2. imaging in the microscope;3. observing and recording the image.

The ultimate resolution of the features observed in a recorded image includes contribu-tions from all three of the above processes. The width of grain boundary grooves resultingfrom etchingmay limit the resolution in fine-grained microstructures far more than the NAof the objective lens or thewavelength of the light used. However, so may the grain size of aphotographic emulsion used to record the image and the photographic process used todevelop the negative and print the final image. In digital imaging, the pixel size in relation tothe optical resolution, the number of pixels available and the number of grey levels used torecord image brightness will place limitations on subsequent data processing and imageevaluation.

Contrast in the optical microscope can be due to the surface topology, the opticalproperties of the sample (reflection coefficient and/or optical anisotropy), or the presence ofa surface film. The contrast observed will depend on the wavelength and may be associatedwith variations in reflected amplitude, or result fromphase shifts, as in the changes in colourobserved with a sensitive tint plate. The enhancement of contrast, either by photographic ordigital processing, may increase the g in the field of view, but reduce the number of greylevels and result in a loss of information. Image enhancement using phase contrast ordifferential, Normarski interferencemay result in some loss of lateral resolution. Contrast isgenerally better for images viewedwith a lowerNA objective, but only at the cost of reducedresolution.


Finally, resolution by itself is useless without image contrast, but poor resolution, just forthe sake of good contrast, is equally self-defeating. The eye is probably still the best judge ofwhat constitutes a good compromise, both in image observation and for a recorded analogueor digitized image, but computer-assisted data processing can provide themicroscopistwithreliable and objective assistance.

Summary

The optical microscope is the tool of preference for the microstructural characterizationof engineering materials, both because of the wealth of information available in themagnified image and the ready availability of high quality microscopes, specimenpreparation facilities and inexpensive methods of image recording and data processing.The visual impact of the magnified image is immediate, and its interpretation is in terms ofspatial relationships which are already familiar to the casual observer of the macroscopicworld.

Geometrical optics determines the relationship between the object placed on themicroscope stage and its magnified image. The ability to resolve detail in the image islimited primarily by the wavelength of the light used to form the image, together with theangle subtended by a point in the object plane at the objective aperture. The aperture of theeye and the range of wavelengths associated with visible light limit the resolution of the eyeto approximately 0.2mm. That is, the unaided eye can distinguish between two features at acomfortable reading distance (30 cm) if they are separated by 0.2mm.

The numerical aperture (NA) of an objective lens is the product msina, where m is therefractive index of the medium between the lens and the object and a is the half-anglesubtended at the objective (the angular aperture of the lens). Values of NAvary from of theorder of 0.15 for a lowmagnification objective used in air to about 1.3 for a high-power, oil-immersion objective, leading to a limiting (best possible) resolution of the order of half thewavelength of visible light, about 0.3 mm.

A sharply focused image will only be obtained if the features to be imaged are all in theplane of focus, and highNA lenses require accurate focusing. It follows that specimens to beimaged at the best resolution must be accurately planar. This limited depth of field is theprimary reason why samples of opaque materials must be polished optically flat, whilethose of transparent materials must be prepared as thin, parallel-sided sections.

The components of the optical microscope include the light source and condensersystem, the specimen stage, the imaging optics and the image recording system. Each ofthese components has its own engineering requirements: the intensity and uniformity of thelight source; the mechanical stability and positional accuracy of the specimen stage; theoptical precision and alignment of the imaging system, and the sensitivity and reproduc-ibility of the image data-recording system.While in the past photographic recordingwas theprimary option, the rapid development of charge-coupled device (CCD) and complemen-tary metal oxide semiconductor (CMOS) digital image data systems has led to thedevelopment of a new range of microscopes in which digital recording and computer-aided image enhancement have, to a large extent, made the photographic darkroomobsolete.


A major consideration in the application of the optical microscope is the selection andpreparation of a suitable sample. In the first place, the sample must be representative of thefeatureswhich are to be observed. That is, the sampling proceduremust take account of bothinhomogeneity (spatial variations in the features and their distribution) and morphologicalanisotropy (orientational variations, as in fibrous or lamellar structures). Secondly,preparation of the sample surface must reveal the intersection of bulk features with theplane of the sectionwithout introducing artifacts (such as scratches or stains). Inmost cases,surface preparation is a two-stage process. The first step is to prepare a flat, polished,mirror-like surface,while the second is to develop contrast by the use of suitable chemical etchants,solvents or differential staining agents.

Image contrast is a sensitive function of themode of operation of themicroscope, butmostengineeringmaterials are examinedby reflectionmicroscopy. The contrast then reveals localdifferences in the absorption and scattering of the incident light. Chemical etchantscommonly develop topographic features which scatter the incident beam outside theobjective aperture, but they may also differentially stain the surface, so that some featuresabsorb more light than others. In many cases, the different phases and impurities present(�inclusions�) will also give contrast that is associated with differences in the reflectivity ofthe different phases for the incident light, quite independent of the action of an etchant.

The optical imaging conditions can also be controlled in order to enhance the imagecontrast. A �dark-field� image is formed by collecting the scattered light from the object,rather than the specularly reflected light used to form a �bright-field� image. If the specimenis illuminated with plane-polarized light, then the changes in polarization that accompanythe interaction of the light with the specimen can be analyzed using �crossed polars� thatconvert any rotation of the plane of polarization into variations in image intensity. Anoptical wedge or sensitive tint plate can be used to introduce a controlled phase shift andfurther enhance the image contrast in polarized light, often yielding quantitative informa-tion on the optical properties of the sample and its constituents.

In phase contrast reflection microscopy small (>20 nm) topological differences at thesample surface can be converted into variations in image intensity, while in transmission-microscopy differences in refractive index can be similarly imaged. By combining the lightreflected from the sample surfacewith that reflected from an optically flat reference surfaceit is possible to obtain two-beam optical interference, in which the interference fringesagain reflect the topology of the sample surface. Multiple beam interference allows verysmall differences in the height of surface features to be detected with a sensitivity of a fewnanometre, and this vertical �resolution� is several orders ofmagnitude better than the lateralimage resolution, which is limited by the NA of the objective lens.

The quality of digital and analogue recorded images depends on three independentfactors: the preparation of the sample, the optical imaging in the microscope; and thesystem used to record and process the final image data. The observed image in themicroscope, the image recorded on a transparency, the digitized image viewed on acomputer screen and the printed image reproduced in a published text will all differ inquality, reflecting the different technologies being used. The eye is the best judge of thatcombination of resolution and contrast which yields themost information, but digital imagedata processing software now provides the microscopist with some excellent tools formaking that judgement and ensuring that results will be recorded and presented withminimal loss of image quality.


Bibliography

1. Metals Handbook, Volume 7: Atlas of Microstructures of Industrial Alloys, AmericanSociety for Metals, Metals Park, OH, 1972.

2. J.L. McCall and P.M. French (eds), Metallography in Failure Analysis, Plenum Press,London, 1978.

3. J.B. Wachtman, Characterization of Materials, Butterworth-Heinemann, London, 1993.4. J. Russ, The Image ProcessingHandbook, 4th edn, CRCPress, Boca Raton, FL, 2002.

Worked Examples

Webegin by considering the influence of the polishing process on the surface finish of 1040steel. Small samples are cut from a large block and embedded in a thermoplastic mount.Granules of the thermoplastic are poured around a specimen that has been mounted in ametal die. The granules are then compacted at moderate pressure and temperature.Alternatively, and to avoid any damage to the specimen, a thermosetting resin can be castinto the mould containing the specimen and then cured at room temperature.

Figure 3.39 compares optical micrographs of a 1040 steel surface at different stages ofpolishing. A �good� polish results in a planar surface with no scratches visible under the

Figure 3.39 Optical micrographs of 1040 steel after polishing with a sequence of diamondgrits: (a) Rough-grinding to achieve a planar surface; (b) after polishing with 6 mmdiamond grit;(c) after polishing with 1 mm diamond grit; (d) after polishing with 1/4mm diamond grit.


Figure 3.40 The same 1040 steel fromFigure 3.39 after etching for different lengths of time in avery dilute nitric acid: (a) under – etched; (b) a good etch; (c) over etched.


optical microscope. A final polish with 1/4 mm diamond particles removes most surfaceartifacts, and provides a specimen that is ready for etching [Figure 3.39(d)].

The choice of etchant must be consistent with the information sought from the sample. Agood compendium of etchants for metals and alloys is theMetals Handbook that describesthe experimental sample preparation procedures and gives examples. For our 1040 steel wewish to determine the average grain size and shape, so we chose a very dilute solution ofnitric acid in ethanol (nital). Figure 3.40 shows optical micrographs of the polished steelafter etching in nital for increasing times. The two main variables for a given etchant aretime and temperature, and the higher the temperature the faster the etching rate. Too light anetch gives poor contrast and fails to reveal the microstructure adequately, while over-etching results in pitting of the sample surface and loss of resolution.

Alumina is an inert oxide, and the polished surface of an alumina sample is difficult toprepare for optical microscopy by chemical etching, although some chemical etchants thatare based on molten salts or hot, concentrated acids do exist. Instead, microstructuralcontrast in alumina andmany other ceramics is commonly developed by thermal etching. Inthis process a polished specimen is heated to a temperature at which surface diffusion canoccur. For example, the intersection of a grain boundary with a polished surface isthermodynamically unstable, since the surface tension forces are not in equilibrium [Figure3.41(a)]. If surface diffusion takes place, then groove formation along the line ofintersection of a grain boundary with the free surface can reduce the total surface energy[Figure 3.41(b)].Whenviewed under themicroscope, the �thermally etched�grain boundarygrooves will scatter light outside the objective lens aperture, and provide the contrastneeded to identify the grain boundaries.

GrainA Grain B

φθ

equilibrium

γAγB

γGB

GrainA

(a)

(b)

Grain B

non-equilibrium

γGB

γA γB

Figure 3.41 A grain boundary intersecting a polished surface is not in equilibrium (a). Atelevated temperatures surface diffusion forms a grain boundary groove to balance the surfacetension forces (b).


Mass transfer by surface diffusion is a function of both temperature and time, andinsufficient thermal etching will result in poor contrast for the alumina grain boundaries[Figure 3.42(a)]. Increased thermal etching increases the depth and width of the grooves,improving the grain boundary contrast, but with some loss of resolution [Figure 3.42(b)].The same diffusion processes promote grain boundary migration and grain growth,changing the bulk microstructure, while exaggerated �over-etching� will result in �wide�grain boundaries and a serious loss of resolution.

Problems

3.1. What optical properties of an engineering material are important in determining thepreparation of a sample for optical microscopy?

3.2. Are short-sighted individuals blessed with �better� resolution? Discuss!

3.3. The Raleigh resolution criterion assumes two point sources of light and impliesunlimited contrast in the image. What factors are likely to prevent the attainment ofthe Raleigh resolution?

3.4. The better resolution obtained with a high numerical aperture objective is accompa-nied by reduced contrast and depth of field. Why?

3.5. Mechanical stability is a necessity for the specimen stage of an optical microscope.Compare (quantitatively) the mechanical stability required parallel to the optic axisof the microscope with that required in the plane of focus.

3.6. Photographic recording of the optical image is being replaced by digital recording.Why?

3.7. What sections from the following componentswould you select for examination in anoptical microscope? a. Amultilayer capacitor, b. Steelwire, c. A �nylon� plastic sheet,d. A sea shell

Figure 3.42 Polished alumina thermally etched at 1200 �C for 30min shows poor contrast andfails to reveal all the boundaries (a). Thermal etching for 2 h at the same temperature clearlyreveals all the grain boundaries but with some loss of resolution for the finest grains (b).


3.8. Chemical etching is often used to develop image contrast on a polished samplesurface. Give three examples of contrast developed by chemical etching.

3.9. Give one example of amicrostructurewhere you consider that dark-field illuminationwould givemore information than bright field illumination, and justify your opinion.

3.10. What is the orientation of the principal axes of an optically anisotropic sample withrespect to the axis of polarization of the incident light when the observed intensity is amaximum using crossed polars? Explain!

3.11. What is a sensitive tint plate and when would you consider it useful?

3.12. Two-beam interference images have been used to analyze the shape of grain boundarygrooves formed by thermal etching. Derive an expression to show how the spacing ofthe fringes and the angle they make with the boundary will affect the verticalresolution.


4

Transmission Electron Microscopy

The electron microscope extends the resolution available for morphological studies fromthat dictated by the wavelength of visible light to dimensions which are well into the rangerequired to image the lattice planes in a crystal structure, that is from of the order of 0.3mmto of the order of 0.1 nm. The first attempts to focus a beam of electrons using electrostaticand electromagnetic lenses were made in the 1920s, and the first electron microscopesappeared in the 1930s, pioneered primarily by Ruska, working in Berlin. These weretransmission electron microscopes, intended for samples of powders, thin films andsections prepared frombulkmaterials. Reflection electronmicroscopes, capable of imagingthe surfaces of solid samples at glancing incidence, made their appearance after the SecondWorld War but these were soon superseded by the first scanning electron microscopes(Chapter 5), and these were almost immediately combined with the microanalyticalfacilities available in the microprobe (Chapter 6).

Sub-micrometre resolution was demonstrated on the earliest transmission electronmicroscopes that had been manufactured in Europe, and later in Japan and the USA. Theearly developments in electron microscopy are an international success story: in theimmediate post-war period commercial transmission instruments were manufactured inGermany, Holland, Japan, the UK and the USA. The first scanning instruments were madein the UK, while the first microprobe was a French development. As we will discuss inSection 4.1.2, sub-nanometre and even sub-Angstrom, resolution is currently availablefrom advanced transmission electron microscopes. However, perhaps the most importantcharacteristic of the transmission electron microscope is that it combines information fromobjects in real space at excellent resolution, with information from the same object obtainedin reciprocal space, that is, electron diffraction patterns can be recorded (Chapter 2).Together with microanalytical techniques that can be integrated into the same instrument(Chapter 6), this makes the transmission electron microscope one of the most versatile andpowerful tools available for microstructural characterization.

In this chapter we will outline the basic principles involved in focusing an image with ahigh energy beam of electrons before discussing the factors that limit resolution in electron


microscopy. We will then compare the requirements for transmission electron microscopy(TEM) before discussing in detail the specimen preparation procedures, and the origin andinterpretation of contrast in the transmission electron microscope image. In Chapter 5 wewill extend this discussion to scanning electron microscopy (SEM).

The transmission electron microscope is in many ways analogous to a transmissionopticalmicroscope, but themicroscope is usually ‘upside down’, in the sense that the sourceof the electron beam is at the top of the microscope while the recording system is at thebottom (Figure 4.1). An electron gun replaces the optical light source and is maintained at ahigh voltage with respect to earth (typically 100–400 kV). A number of different electronsources have been developed, but the basic design of these different electron sources issimilar (Figure 4.2). In a thermionic source, electrons are extracted from a heated filamentat a low bias voltage that is applied between the source and a cylindrical polished cap (theWehnelt cylinder). This beam of thermionic electrons is brought to a focus by theelectrostatic field and accelerated by an anode held at earth potential beneath the Wehneltcylinder. The beam that enters the microscope column is characterized by the effectivesource size d, the divergence angle of the beam a0, the energy of the electrons E0 and theenergy spread of the electron beam DE. In general, the temperature of the source limits theenergy spread of the electron beam DE to approximately kT. A smaller source size dimproves the beam coherence, and hence the contrast that can be obtained from phase shiftsdue to interactions of the beam when traversing a thin specimen (Section 4.3.3).

Figure 4.1 As in the transmission optical microscope, the transmission electron microscopeincludes a source, a condenser system, a specimen stage, an objective lens and an imagingsystem, as well as a method for observing and recording the image.


Three electron sources are in common use: a heated tungsten filament is capable ofgenerating electron beam current densities of the order of 5 · 104Am�2, from an effectivesource size, defined by the first cross-over of the electron beam, that is some 50 mm across.Thermionic emission temperatures are high, resulting in an appreciable energy spread of theorder of 3 eV, and the coherencyof the beam is also limited.A lanthanumhexaboride (LaB6)crystal can generate an appreciably higher beam current, about 1 · 106Am�2, at much

Figure 4.2 Schematic drawing of (a) a tungsten filament thermionic electron source and (b) aLaB6 tip for enhanced thermionic emission. A sharp tungsten tip (c) is for a common fieldemission gun source. In thermionic sources the filament or tip is heated to eject electrons, whichare then focused with an electrostatic lens (the Wehnelt cylinder) (d). In field emission guns(e) the electrons are extracted by a high electric field applied to the sharp tip by a counterelectrode aperture, and then focused by an anode to image the source.

Transmission Electron Microscopy 181

lower temperatures. Cerium hexaboride (CeB6), is nowalso in use. The energy spread of thebeam is significantly reduced to about 1.5 eV, although the vacuum requirements are morestringent compared with a tungsten filament source. A ‘cold’ field emission gun, in whichthe electrons ‘tunnel’ out of a sharp tip under the influence of a high electric field, cangenerate current densities of the order of 1 · 1010Am�2. The sharp tip of the tungstenneedle that emits the electrons is nomore than 1mm in diameter, so the effective source sizeis less than 0.01 mmand therefore quite coherent.Moreover the temperature of this source islow, and an energy spread of 0.3 eV is typical. More often, a ‘hot’ field emission sourcereplaces the ‘cold’ source. In this case a tungsten needle is heated to enhance emission byelectron tunnelling, a process termed Schottky emission. ‘Hot’ Schottky sources oftencontain zirconium in order to reduce the work function. They have slightly greater energyspread and a larger effective source size than cold field emission sources, but they are morestable and reliable and have a longer useful life and less stringent vacuum requirements.Recently, the introduction of electron beammonochromators has further reduced the energyspread from an electron source to less than 0.15 eV, although at the expense of theachievable electron current density. When available, the reduction in energy spreadprovided by a monochromator can be very important, both for analytical analysis(Chapter 6) and to improve the information limit in the transmission electron microscope(Section 4.3.3).

The high energy electrons from thegun are focused by an electromagnetic condenser lenssystem,whose focus is adjusted by controlling the lens currents (and not the lens position, aswould be the case in the optical microscope). The specimen stage is mechanically complex.In addition to the x–y controls, the stage allows the specimen to be tilted about two axes atright angles in the plane of the specimen, and the tilt axes to be adjusted (this is termed aeucentric stage). Some z-adjustment along the optic axis is important. It is also oftenpossible to rotate the specimen about the optic axis of the microscope. In TEM the standardspecimen diameter is only 3mm,while only samples less than about 0.1 mmare thin enoughto allow most of the high energy electrons to pass without suffering serious energy loss. InSEM, by contrast (Chapter 5), the signal is collected from the specimen surface and theelectron beam loses energy by inelastic scattering as the electrons penetrate beneath thesample surface. Focusing of the image in the transmission electron microscope is notobtained by adjusting the position of the specimen along the z axis, so as to alter its distancefrom the objective lens, but rather by changing the lens current in order to adjust the focallength of the electromagnetic lens in order to focus a first image from the elasticallyscattered electrons that have been transmitted through the thin film specimen. The finalimaging system also employs electromagnetic lenses, and the final image is observed on afluorescent screen that converts the high energy electron image into an image that is visibleto the eye. Typical electron current densities at the screen are of the order of 10�10–10�11 A/m�2, but they may be even lower when studying damage-sensitive materials or at highmagnifications. Photographic emulsions have been used to record the final image but, as inoptical microscopy, advances in the development of charge-coupled devices (CCDs)combined with computerized image processing (Section 3.5) have nowmade digital imagerecording the technology of choice.

The high energy electron beam has a limited path length in air, so that the whole electronmicroscope column must be kept under vacuum. Specimen contamination under the beam,that is, the development of a carbonaceous layer on the specimen surface, is a serious


problem in electron microscopy, and may restrict viewing time for any selected area of thesample and limit the achievable resolution. In general, the vacuum needs to be better than10�6 Torr, while for the highest resolution a vacuum of 10�7 Torr is desirable. The sourcesof contamination include the sample, the components of the microscope and the pumpingsystem itself, and they should be trapped, usually by cryogenic cooling of the specimen andits surroundings.

For comparison, the scanning electronmicroscope (Figure 4.3) again has a source of highenergy electrons and a condenser system, but now employs a probe lens to focus the electronbeam into a fine probe that impinges on the specimen. The electromagnetic probe lens in thescanning electron microscope fulfills a similar function to the objective lens in thetransmission electron microscope, since it determines the ultimate resolution attainablein the microscope. However, the probe lens is placed above the specimen, and plays no partin collecting the image signal from the specimen. Indeed, in SEM the elastically scatteredelectrons are of no especial importance in providingmorphological information. Rather it isthe inelastic scattering processes which occur when the electron probe interacts with thesample, that provide the microstructural information collected in this instrument. Theelectron energy of the beam in scanning electron microscopy is appreciably less than thatused in transmission, usually of the order of 3–30 keV, althoughmuch lower energies, as lowas 100 eV, may prove useful.

The ‘image’ in SEM is obtained by scanning the focused electron probe across the samplesurface in a television raster, and then collecting an image signal from the surface and

Figure 4.3 In the scanning electron microscope a fine probe of electrons is focused onto thesample surface and then scanned across the surface in a television raster. A signal generated bythe interaction of the probe with the sample is collected, amplified and displayed on a monitorwith the same time base as the raster used to scan the sample.


displaying it, after suitable amplification and processing, on a monitor with the same timebase as that used to scan the probe across the sample (Figure 4.3). Line scans of the sampleare made along the x-axis, and at the end of each line scan the beam is switched back to thezero of x and the y coordinate is incremented byDy. The signal collected I is thus a functionof time I(t), where each value of t corresponds to specific x, y coordinates in the plane of thespecimen surface. Once the signal from a complete set of line scans has been collected,corresponding to the range of y selected, the beam is switched back to the zero of the x, ycoordinates, ready to collect data for another image frame. This is also the principleemployed in the cathode ray tube, where the image is formed by scanning the screen witha modulated beam of electrons. There is no reason why the position of the x-scan shouldnot also be incremented by Dx. In this case the x, y coordinates for each frame scan willconstitute a set of pixels (picture elements) of size Dx, Dy, and the total time requiredto acquire an image frame will equal the total number of points in the frame multiplied bythe dwell time at each pixel position. Of course, the signal intensity for each point still has tobe digitized before full digital image processing and analysis can be done. This meansthat the intensity of the signal collected at each point has to be amplified and binned,typically in an 8 bit (256 grey level) register and the total number of pixels shouldcorrespond approximately to the number of points scanned in the image frame on thesample.

The power of the scanning electron microscope derives from the wide range of signalsthat may result from the interaction of the electron probe with the sample surface. Theseinclude: characteristic X-rays, generated by excitation of inner shell electrons; cathodo-luminescence, excitation in the range of visible light that is associatedwith valency electronexcitation; specimen current passing through the sample due to the net absorption of electriccharge; and backscattered electrons, that are elastically and inelastically scattered out of thesurface from the probe beam. The most commonly used signal is that derived from lowenergy secondary electrons, which are ejected from the surface of the target by the inelasticinteraction of the sample with the primary beam. The secondary electrons are emittedin large numbers from a region that is highly localized at the point of impact of theprobe. They are therefore readily detected and are capable of forming an image whosepotential resolution is limited primarily by the diameter of the focused probe at the samplesurface.

The fundamental difference in the operation of the transmission and the scanningelectron microscopes can be summarized in terms of the two modes of data collectionthat are employed to form the image. In both optical microscopy and in transmissionelectron microscopy information is simultaneously collected over the full, magnified fieldof view and focused by suitable lenses to build up a magnified image as a function of theintegrated data collection time. In the scanning electron microscope the information iscollected sequentially, for each data point in turn, while the focused probe is scanned acrossthe sample field of view. The rate of scan must be restricted in order to ensure that thespecimen signal recorded for each image point is statistically adequate, and the total timerequired to form a scanning image is determined by the minimum scanning speed that willachieve this goal for each pixel multiplied by the number of image pixels that are to becollected. This distinction between an optical image, in which the image data are acquiredfor all image points simultaneously, and a scanning image, in which the image is developedsequentially (that is, one pixel at a time) cannot be overemphasized.


4.1 Basic Principles

The design and construction of an electron microscope are well beyond the scope of thistext, but it is still important to have some appreciation of the basic physical principles thatdetermine the behaviour of electrons in a magnetic field and their interaction with matter.

4.1.1 Wave Properties of Electrons

The focusing of an electron beam is possible because of the dual,wave–particle character ofelectrons. This wave–particle duality is expressed in the de Broglie relationship for thewavelength of any particle:

l ¼ hmv

ð4:1Þ

wherem is themass of the particle, v is its velocity and h is Planck’s constant. Assuming thatthe accelerating voltage in the electron gun is V, then the electron energy is

eV ¼ mv2

2ð4:2Þ

where e is the charge on the electron. It follows that l¼ h/(2meV)1/2, or l¼ (1.5/V)1/2 nmwhen V is in volts. This numerical value is rather approximate, since, at the acceleratingvoltages commonly used in the electron microscope, the rest mass of the electron m0 isappreciably less than the relativistic massm, and a correction term should be included in thede Broglie equation:

l ¼ hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m0eV 1þ eV

2m0c2

� �r ð4:3Þ

where c is the velocity of light. The relativistic correction amounts to about 5% at 100 kV,rising to 30% at 1MV. The electron wavelength at 100 kVis 0.00370 nm, nearly two ordersof magnitude less than the interatomic spacings in the solid state. At 10 kV, typical of manyapplications of SEM, the wavelength is only 0.012 nm, still appreciably less than theinteratomic distances in solids.

4.1.1.1 Electrostatic and Electromagnetic Focusing. Electrons are deflected by bothelectrostatic and magnetic fields, and can be brought to a focus by suitably engineering theelectrostatic or magnetic field geometry. In the region of the electron gun the electron beamis influenced by the electrostatic field created by the anode andWehnelt bias cylinder. Theseusually result in a first focus, ‘virtual’ electron source. With just one exception, to be notedbelow, all subsequent focusing in the electron microscope is electromagnetic and isachieved by electromagnetic lenses equipped with soft iron (essentially having zeromagnetic hysteresis) pole pieces. Unlike optical lenses, made from glass, the focal lengthof an electromagnetic lens is variable and can be controlled by varying the lens current thatflows in the coil surrounding the pole pieces.

An electron travelling in a magnetic field is deflected in a direction at right angles to theplane that contains both the magnetic field vector and the original direction of travel of the


electron (its momentum vector). In a uniformmagnetic field an electron that is travelling off-axis will follow a helical path [Figure 4.4(a)]. To a first approximation, electrons of the sameenergy and travelling in a cone of directions originating from any one point within a uniformmagnetic field will be brought together to a focus at a second point after spiralling along theaxis of the field [Figure 4.4(b)].

The image of an object, formed by focusing electrons using electromagnetic lenses,differs in several important respects from that formed by focusing light using glass lenses. Inthe first place, although the image of an object in a plane perpendicular to the axis of anelectromagnetic lens is also in a plane perpendicular to this same axis, it is rotated about thisaxis, so that focusing of the objective lens by adjusting the lens current is accompanied byrotation of the image about the optic axis. It follows that two images of the same object takenat different magnifications will also be rotatedwith respect to one another. This rotation canbe compensated by reversing themagnetic field vector over a proportion of the optical path,and it is now common practice to design electromagnetic lenses to carry the current in thewindings of the lens coil of the upper and lower halves of the lenses in opposite directions.With such a compensated lens the electrons are not confined to travel in a plane, so that thebehaviour of the electron beampassing through the electromagnetic lens system is still quitedifferent from the behaviour of a light beam in the optical microscope.

In the light opticalmicroscope there is an abrupt change in refractive indexwhen the light isdeflected as it enters a glass lens, but the refractive index is constantwithin theglass lens.With

(b)

Object

RotatedImage

H(a)

Figure 4.4 An electron in a magnetic field is deflected at right angles to both the momentumandmagnetic field vectors. (a) An off-axis electron follows a spiral path. (b) Electrons originatingat a point off the axis are brought to a rotated focus.


an electromagnetic lens the deflection of the electrons is continuous, and the magnetic fieldcreated by the lens pole pieces varies continuously throughout the optic path within the lens.

Finally, the angle subtended by the path of an electron with respect to the optic axis isalways very small (less than 1�) so that the optic path length through the electromagneticfield of the lens is always very long when compared with the angular spread of thebeam perpendicular to the optic axis. This means that the numerical aperture in thetransmission electronmicroscope is always very small. Thismay be comparedwith the caseof the opticalmicroscope, forwhich the numerical aperture of the objectivemay correspondto an acceptance angle for scattered light of between 45� and 90�. The numerical aperture(note that this term is not used in electron microscopy) of an electromagnetic lens neverexceeds 10�2.

4.1.1.2 Thick and Thin Electromagnetic Lenses. The physics of electromagnetic focus-ing means that the simple geometrical optics which we applied to the light opticalmicroscope (Section 3.1) is a very inadequate approximation to the optics of imageformation in the transmission electron microscope. In particular, the simple relationshipsbetween the focal length, the magnification, and the relative positions of the object and theimage along the optic axis no longer hold, because they are based on the assumption thatthe lens is thin when compared with the total optical path between the object and the image.The thin lens approximation is also insufficient for high powered objectives in the lightmicroscope, when it is replaced bymuchmore complex, thick lens calculations. In electronmicroscopy all the electromagnetic lenses are ‘thick lenses’.

Nevertheless, it is still common to illustrate the imaging modes in the electron micro-scope using ray diagrams which are presented in two dimensions, as though the electronswere not rotated, and the electron paths changed direction abruptly at the lens positions, asthough the thin lens approximationwere still valid.No quantitative calculations are possiblein such a qualitative model.

4.1.1.3 Resolution and Focusing. Given that the maximum beam divergence in theelectronmicroscope is less than 1�, the Raleigh criterion for the image of a point source canbe reduced to: d ¼ 0:61l=msina � 0:61l=a> 60l. Inserting the value for thewavelength at100 kV, 0.0037 nm, the potential resolution of the transmission electron microscope shouldbe of the order of 0.2 nm. However, at higher operating voltages significantly betterresolution is possible (see below).

As a gross approximation we can use the light-optical expression for depth of field,d� d/a, so that the thin film specimens used in transmission electronmicroscopy should be ofthe order of 20–200nm in thickness if both top and bottom of the film are to be in focussimultaneously. Similarly, for depth of focus,D¼M2d, so that at a magnificationM of 10 000the expected depth of focus is of the order ofmetres! There is therefore no problem in focusingan image on a fluorescent screen and subsequently recording the same image on a photo-graphic emulsion or CCD detector that is placed some distance beneath the focusing screen.

4.1.2 Resolution Limitations and Lens Aberrations

At this point we should consider the optical performance of the electronmicroscope inmoredetail and the reasons why the angular divergence of the electron beam has to be limited tosuch small values.


4.1.2.1 Diffraction Limited Resolution. The diffraction limit on resolution is, as in lightoptical microscopy, that given by the Raleigh criterion, dd ¼ 0:61l=msina. In vacuum, m¼ 1and at small angles sin a¼ a. Inserting the expression given previously for thewavelength of the electron beam in terms of the accelerating voltage:dd ¼ 0:61l=a ¼ 0:75= a

ffiffiffiffiV

p ð1þ 10�6VÞ� �. It follows that, for a given divergence angle, it

should be possible to improve the resolution by increasing the accelerating voltage. Experi-mental electronmicroscopes have been constructedwith accelerating voltages up to 3MV, butcommercial instruments have been limited to about 1MV. At these voltages many samplesexperience extensive radiation damage, which increases with prolonged exposure to theelectron beam. Most high resolution TEM for imaging crystal lattices is performed at 300 orperhaps 400 kV, close to the threshold for the onset of radiation damage in most nonorganicengineeringmaterials. At thesevoltages a point-to-point resolution in non-crystalline samples,of the order of 0.15nm, is readily and routinely attainable.

4.1.2.2 Spherical Aberration. Analogous to light optics, an electron beam parallel to butat a distance from the optic axis of an electromagnetic lens will be brought to a focus by anelectromagnetic lens at a point on the axis that depends on the distance of the beam from theaxis [Figure 4.5(a)], while a beam further from the optic axis will be focused closer tothe lens, so that the plane of ‘best’ focus will correspond to a disc of least confusion whosesize will depend on the angular spread of the beam. This phenomenon is simplyspherical aberration, as in the optical microscope, and the radius of the disc of leastconfusion constitutes an aberration-dependent limit on the resolution that is given approxi-mately by:

ds � Csa3 ð4:4Þ

r1

r2

VV–∆V

Disc of Least Confusion

(a)

(b)

Figure 4.5 Spherical (a) and chromatic aberration (b) prevent a parallel beam from beingbrought to a point focus. Instead a disc of least confusion is formed in the focal plane of the lens.


whereCs is the spherical aberration coefficient of the electromagnetic lens. By comparison,the diffraction limit on resolution dd is inversely proportional to the angular aperture of theobjective a, while the spherical aberration limit ds is proportional to the third power of theangular aperture (Figure 4.6).

It follows that for any given lens of fixed spherical aberration coefficient there should bean optimum angular aperture at which dd equals ds, corresponding to Csa3 ¼ 0:61l=a, ora4 ¼ 0:61l=Cs. The required angular aperture is therefore a sensitive function of both theaccelerating voltage (the electron wavelength) and the spherical aberration coefficient ofthe lens. Typical values for the spherical aberration coefficient of an electromagnetic lensare somewhat less than 1mm. Inserting a moderate value of 0.6mm for Cs and thewavelength appropriate to 100 kV electrons, 0.0037 nm, we obtain a value of about8 · 10�3 for this optimum value of a. Electromagnetic lens design has improved steadilyover the past half-century, and it has now proved feasible to introduce multipole electro-static spherical aberration correctors into themicroscope column that are able to reduce thespherical aberration coefficient to an arbitrarily small value. These correctors are not yetgenerally available, but there is little doubt that they soon will be. Figure 4.7 shows thearrangement of twin hexapole correctors in the FEI Titan, a top-of-the-range, ultra-highresolution transmission electron microscope, in which a corrected Cs of zero has beenachieved.

4.1.2.3 Chromatic Aberration. Chromatic aberration arises because higher energyelectrons are less deflected by a magnetic field than those of lower energy, so that theyare brought to a focus at a point on the optic axis that is further from the center of the lens,once again giving rise to a disc of least confusion, this time determined by the energy spreadin the electron beam and the chromatic aberration coefficient of the electromagneticobjective lens [Figure 4.5(b)].

sd

Figure 4.6 The diffraction and the spherical aberration limits on resolution have an inversedependence on the angular aperture of the objective lens, so that an optimum value of a exists.


There is more than one source of chromatic aberration, although that due to the kineticspread in beam energy is generally themost important. If the electrons are thermally emitted(as is, to some extent, always the case, even for commercial field emission sources), then therelative energy spreadwill be given byDE/E0¼ kT/eV, where k is Boltzmann’s constant ande is the electronic charge. For T¼ 2000K, a reasonable temperature for a tungsten filamentsource, and 100 kVelectrons, the energy spread DE/E0 is about 1.5 · 10�6. Electrons mayalso lose some energy due to inelastic scattering in the thin sample, adding to the chromaticaberration due to the thermal energy spread of the beam. This may affect an appreciablefraction of the incident electrons if the specimen is thick or of high atomic number, as in aheavy metal sample. Fluctuations in the electromagnetic objective lens current may alsocontribute to chromatic aberration, since they will change the focal length of the lens. Theequation that relates the chromatic aberration limit on resolution to these variations in beamenergy and lens current is:

dc ¼ CcDEE0

a ð4:5Þ

where Cc is the chromatic aberration coefficient of the lens, while DE includes instabilitiesin both the accelerating voltage and the objective lens current. As with spherical aberration,

-

+

-

+ +

-Figure 4.7 Schematic of the influence of a double hexapole electrostatic aberration correctoron the shape of an electron beam. The corrector system results in ray paths through the axis of thecolumn (red circles) that are compensated (white circles) for the residual spherical aberration ofthe objective lens.


the resolution limit increases with a, but this time only linearly. Providing thevoltage stability of the electron gun and the current stability of the electromagneticlenses is adequate, chromatic aberration should be limited only by the temperature ofthe electron source, although it may also be affected by inelastic interactions in thespecimen.

4.1.2.4 Lens Astigmatism. The axial symmetry of the electro-optical system is anextremely important factor limiting the performance of the electron microscope, and theexact alignment of the lens components within the microscope column is a critical factor inoptimizing the performance of the instrument. The objective lens is the component that ismost affected by misalignment. The axial symmetry of this lens is especially sensitive tominor disturbances associated with the geometry, size, position and dielectric properties ofthe sample, as well as to small amounts of carbonaceous contamination that may bedeposited on the sample or the objective aperture.

A basic feature of any loss of axial symmetry is a variation in the focal length as theelectrons spiral about the optic axis. This results in two principal focal positions on this axisthat give two line foci at right angles (Figure 4.8). This condition is termed astigmatism.Astigmatism cannot be prevented, both because of the inherent residual asymmetry in thelens and pole piece construction, and because of the extreme sensitivity of the astigmatismto minor misalignment, specimen asymmetry and contamination in the microscope.However, it can be corrected. Complete twofold astigmatism correction is achieved byintroducing sets of correction coils whosevariablemagnetic fields are at right angles to boththe optic axis and the magnetic field of the main lens coils. The correction coil currents canbe periodically adjusted during operation of the microscope to balance exactly any changesin the magnetic asymmetry that is due either to the build-up of contamination or displ-acement of the sample during viewing. This on-line correction is especially important whenworking with thick samples of high dielectric constant, and for magnetic materials inparticular. A number of geometrical arrangements for the astigmator assembly are possible,for example four pairs of coils forming an ‘octet’ or octopole correction system.

x

y

yfocus

xfocus

Figure 4.8 Astigmatism is the result of axial asymmetry and leads to variations in the focallength about the optic axis, resulting in two principle line foci at right angles along the axis.


4.1.3 Comparative Performance of Transmission and ScanningElectron Microscopy

We now summarize the principal differences between the features and the performance ofthe transmission and scanning electron microscopes.

4.1.3.1 The Optics of Image Formation. As noted previously, the primary image in TEMis obtained by focusing the objective lens.A series of additional imaging lenses then enlargethis image to the final magnification on a fluorescent screen, a photographic emulsion or aCCD recording array. In the scanning electron microscope the image is formed, point bypoint, by collecting a signal that is generated by the interaction of a focused electron beamprobe as it is scanned across the surface of the sample in a television raster.

4.1.3.2 Depth of Field and Depth of Focus. As in the light optical microscope, the depthof field of the transmission electron microscope is limited by the NA of the objective lensand the resolution of the microscope, but since the angular aperture of electromagneticlenses is so small, the depth of field in TEM exceeds the resolution by some two orders ofmagnitude.

In SEM the electron probe is focused by a probe lens,whose operation is analogous to thatof the objective lens in transmission microscopy. However, the inelastic scatteringprocesses that occur during interaction of the probe with the specimen, together with therequirement for an adequate signal current, restrict the effective probe size, and hence theresolution, to the nanometre range. It follows that, with an angular aperture for the probelens of the order of 10�3, the depth of field in the scanning electron microscope (d/a) istypically of the order of micrometres, considerably better than the depth of field that can beachieved in optical microscopy and at a much improved resolution.

Both the light optical and the transmission electron microscopes generate a two-dimensional image of a thin, planar section that has been prepared from the bulk material.By contrast, the image in the scanning electron microscope contains considerable in-focusinformation on the three-dimensional topography from the surface of any solid sample.Furthermore, since data collection in the scanning image is collected point-by-point andline-by-line, the question of depth of focus does not arise: there is now no focused image inthe sense of classical optics.

4.1.3.3 Specimen Shape and Dimensions. The electro-optical image requirements inTEM usually place the specimen within the magnetic field of the objective lens (thusmaking the study of magnetic materials problematic). The space available for the specimenis therefore very restricted, so that in addition to the stringent limitations on samplethickness that are dictated by the onset of inelastic scattering in the electron beam, there arealso limitations on the sample’s lateral dimensions. The standard external specimendiameter is 3mm, but only the central 2mm or so of the sample is actually available forexamination.

By contrast, samples for SEM sit well below the probe lens, and well outside the lensmagnetic field. With a long working distance probe-setting, reasonable resolution isavailable even when the lens–sample separation is over 50mm. In addition, there are nolimitations on lateral dimensions, other than those imposed by the design of the samplechamber.Most samples have lateral dimensions similar to those used for opticalmicroscopy


(20–30mm), butmuch larger assemblies have been inserted into the specimen chamber, andspecimen chambers are available that will accept specimens 10 cm or more across.Complete sections of failed engineering components and complex solid-state devices arecommonly inserted into the scanning electron microscope for detailed evaluation.

4.1.3.4 Vacuum Requirements. The vacuum requirements of all electron microscopes,transmission and scanning, are determined by three factors:

1. The need to avoid scattering of the high energy electrons by residual gas in themicroscope column.

2. The necessity for thermal and chemical stability of the electron gun during microscopeoperation.

3. The need to minimize or eliminate beam-induced contamination of the sample duringobservation.

The least stringent requirement is actually the first, since a vacuum of 10�5 Torr is quitesufficient to ensure that the cross-section for scattering of the high energy electrons byresidual gas is negligible. The second factor is very much more important. A heatedtungsten filament is steadily eroded, primarily by oxidation during operation at 10�5 Torr.Both alternative sources, either a low work-function lanthanum hexaboride (LaB6) crystalor field emission source (both of which are operated at much lower temperatures, and hencegenerate a beamwith lower chromatic aberration), require a much better vacuum: typicallyof the order of 10�7 Torr for LaB6 and down to the 10

�10 Torr range for field emission guns.However, the third factor listed above is equally important, since specimen contamina-

tion is most frequently the result of inelastic interaction between contaminant gasesabsorbed on the sample surface and the incident high energy electron beam. Hydrocarbonsarriving at the sample are both polymerized and pyrolysed by the incident electrons to forman adherent, amorphous, carbonaceous layer on the sample surface. After extended electronirradiation of a specific area during observation, the layer of amorphous ‘carbon’ contami-nation may even obscure all morphological detail.

Contamination can be significantly inhibited by cryogenic cooling of the specimensurroundings, in order to trap the condensable contaminant species, and this is the proceduregenerally adopted for TEM. However, the large specimens employed in the scanningelectron microscope make a cryogenic trap much less effective, while the very high beamcurrent concentrated in the focused electron probe exacerbates the rate of contamination.The only adequate solution is to ensure that the source of contamination is not the specimen,for example, by plasma etching of the sample in an argon and oxygen gasmixture to oxidizecarbon deposits on the surface, and then work with the best possible chamber vacuum.

4.1.3.5 Voltage and Current Stability. While chromatic aberration ought not to be aproblem, in either transmission or scanning microscopy, it is a mistake to assume thatcurrent and voltage instabilities only affect the performance through their influence on theobjective or probe lens. In particular, the scanning electron microscope may be susceptibleto image distortion that arises from electrical instability of the scanning systems. Severalcauses may be responsible, but it is the results that concern us: these include differences inthe effective magnifications for the x- and y-scan directions, possible shear distortion of theimage in the x-direction, drift of the image or a dependence of themagnification on distancefrom the optic axis: barrelling if the central region is enlarged, but a pin-cushion effect if it is


the peripheral region that is enlarged. Many of these imaging defects are a result of thepoint-by-point data collection process and are a consequence of distortions in the scanningraster of the probe x–y coordinates with respect to that of the image. Electrostatic chargingof a sample that is an electrical insulator can also be a major source of instability, but canusually be prevented by a conductive coating or by working with very low beam voltagesand currents.

4.2 Specimen Preparation

It is not always easy to prepare good specimens for examination in the transmission electronmicroscope, but many techniques are now available for doing so. If a good specimen hasbeen prepared, the information that can be obtained by TEM is unique. However, there isnothing more frustrating than attempting to extract useful information from a poorlyprepared transmission specimen, and more than one graduate student has acquired a life-long aversion to the transmission electron microscope solely for this reason.

Successful transmission electronmicroscopy depends on three diverse skills: preparing agood specimen; acquiring good data; and possessing the understanding needed to interpretthe data. In what follows we will try to provide a sound foundation for success in all threedomains.

Awide range of experimental methods and a good choice of commercial equipment existto ‘ease the pain’ of specimen preparation for the transmission electron microscope.Providing adequate facilities are available, there is no real barrier to the preparation of good,thin-film specimens from any engineering material, and no excuse whatsoever for wastingtime on the examination of a poor specimen in the microscope. Good samples forconventional TEM commonly need to be thinned uniformly to less than 100 nm, whilethose for lattice imaging in high resolution TEM or sub-micrometre microanalysis byelectron energy loss spectroscopy (EELS) should be less than 20 nm thick. Preparing suchsamples reliably is not trivial, but, with the help of the tools that are now available, successshould be well within reach of a careful and competent electron microscopist.

In what follows we concentrate on the preparation of thin film sections from a bulksample of an engineering material. We will summarize briefly the methods of samplepreparation that have been developed for soft biological tissues, and those techniques thatare available for the dispersion of particulate samples or the deposition of thin film samplesfrom the gaseous or liquid phases. It should be emphasized that every material is a ‘specialcase’, and that every engineering component to be investigated has to be sectioned and asample selected.

The characterization of powder samples by transmission electron microscopy is animportant task, with applications ranging from ceramics technology to cosmetic prepara-tions. The trick is to prepare a sample that reflects the composition, particle shape andparticle size distribution of the bulk powder without introducing artifacts associated witheither particle agglomeration or fragmentation. This usually involves preparing a stabledispersion in a suitable liquid medium, often with the help of surface-active additives. Adrop of the dispersion placed on a glassy carbon filmmounted on amicroscope grid is oftenused but this is seldom satisfactory, since the particles collect at the meniscus as the dropdries, leaving irregular aggregates on the grid that are difficult to interpret. Far better is to


spray the dispersion over the carbon coated grid using a commercial nebulizer (atomizer).The spray droplets are only a few micrometres in diameter, and, for sufficiently dilutedispersions, each droplet will contain no more than one or two particles, so that there is nodanger of aggregation during drying and the size distribution observed in the electronmicroscope will be that characteristic of the original dispersion.

Soft tissues and many polymer samples can be sectioned using a microtome thatgenerates a sequence of very thin slices from the stub of a suitably prepared sample,rather like sliced bread. By examining an ordered sequence of slices (serial sectioning) it ispossible to build up a picture of the three-dimensional structure of soft biological tissues andcellular structures. Glass knives are still commonly used, with the advance of the specimenstub controlled by the thermal expansion of a mounting rod. More common today, bettercontrol of the slice thickness is achieved using a diamond knife and piezoelectric control forthe advance of the specimen mount.

Thin films deposited from the liquid or vapour phase can often be stripped from a suitablesubstrate and, if they are thin enough, examined directly by TEM. However, it is often thecross-section of a thin film microelectronic device that is of interest, and this requires thatthe sample be rigidly mounted and sectioned at a known location. We discuss thepreparation of such samples below.

4.2.1 Mechanical Thinning

The usual starting point for the preparation of a thin film specimen for transmission electronmicroscopy is a sample taken from a bulk component. This sample is typically a 3mmdiameter disc several hundred micrometres in thickness. The disc may be punched out of aductile metal sheet, trepanned from a brittle ceramic, cut from a bar, or machined from alarger section (Figure 4.9). In all cases it is necessary to select the axis of symmetry and thecentre of the disc with respect to the coordinates of the bulk component, since thesedetermine the location that is sampled and the direction of viewing in the microscope. Atthis stage it is essential to minimize mechanical damage to the material and preserve a flatand smooth surface.

The next task is to reduce the thickness of the disc sample. This is accomplished by thesame procedures of grinding and polishing as were described previously for the preparationof optical microscope specimens (Section 3.3). As in the previous discussion, the stiffness(elastic modulus), hardness and toughness of the material determine the optimum choice ofgrinding and polishing media, and ductile metals, brittle ceramics, reinforced compositesand tough alloys will all respond quite differently.

Four mechanical thinning treatments are possible (Figures 4.10 and 4.11):

1. A polished, parallel-sided disc is prepared and then thinned from one or both sides usinga rigid jig tomaintain a planar geometry.A crystallinewax is used to fix the specimen to aflat, polished support. The wax is easily melted on a hotplate, both when the sample isfirst attached and when it is turned over on the baseplate. As the thickness decreases, soshould the grit size of the grinding and polishingmedia be reduced finishing the thinningprocess with sub-micrometre diamond grit. The sample thicknesses should then be100 mm or less. At this stage, the greatest problem is often the relief of internal residualstresses in thematerial thatmay lead to curvature and buckling of the sample as soon as itis removed from the baseplate support (see below).


2. Once the thickness of the sample is reduced to 100 mmor less and the surfaces have beenpolished to a micrographic finish (Section 3.3.2), the disc is secured on an optically flatbaseplate and ‘dimpled’. In this process a fine grinding medium removes material fromthe central area of the disc sample as it is rotated in contact with a polishing wheel. Thisis, essentially, a lapping process in which particles of the grindingmedia are displaced in

Figure 4.10 Mechanical thinning of a disc may be achieved by several methods: (a) a simple,parallel-sided geometry; (b) dimpling to thin the centre while retaining a thicker peripheralsupport; (c) from a wedge in which the region of interest is at the thinned side of thewedge.

Machine and slice

Grind and trepan

Figure 4.9 Some methods used to section a disc from a bulk component.


the region of contact shear, continuously fracturing to expose new cutting edges. Severalcommercial dimplers are available, and the process has been successfully used to thinsamples of hard and brittle materials to thicknesses of 20 mmor less in the central regionof the ‘dimpled’ disc. ‘Dimpling is an excellent method for avoiding the complications

Figure 4.11 Schematic showing a process to prepare a cross-section transmission electronmicroscope specimen from a thin film on a flat substrate. (a) Rectangular sections are cut fromthewafer. (b) Theyare glued together to form a block greater than 3mm in thickness, fromwhicha 2.8mm diameter rod is trepanned. (c) The rod is inserted and glued into a 3.0mm outerdiameter metal tube. (d) Thin sections are then cut from this assembly. (e) Finally, these sectionsare mechanically thinned (f–g).


of distortion by residual stress, since the thicker rim of the sample now acts as a ‘frame’to constrain the thin central region.

3. For some samples it may be an advantage to prepare a ‘wedge’ rather than a ‘dimple’.This would be the case for an interface, where the microstructure of the interface is thesubject of investigation, and themaximum thin area from the interface region is obtainedwhen the interface is aligned parallel to the edge of the wedge. In the case of thin filmsand multilayer sandwiches of different phases (for example, semiconductor devices andcomposite materials) it is common practice to section the sandwich at right angles, andmount the plane of the sandwich perpendicular to the axis of the wedge. In thisconfiguration, each layer is sectioned as a wedge and the morphology and interfacemicrostructure of each layer can be studied in a single thin-film sample. The wedge isprepared using a rigid jig, for which the wedge angle is pre-selected and is typically lessthan 10�.

4. Cross-sections of microelectronic devices are more generally prepared by either of twotechniques that have become standard for the industry (Figure 4.11). The secondtechnique will only be discussed after we introduce focused ion beam systems inChapter 5. In the first method the device to be sectioned is first diced into squares and thediced slices glued together, with the central slices face-to-face, in order to form a blockmore than 3mm in thickness. A rod is then trepanned from this blockwith the axis of therod in the plane of the face-to-face layers and parallel to the plane of the device. This rodis now glued into a 3mmouter diametermetal tube (oftenmade of copper or brass). Thistube assembly can now be sliced and mechanically thinned to form a series of 100 mmthick sections through the device, each of which can be dimpled. There is little danger offailure at the device interfaces, since the disc is constrained by the metal ring around theperiphery. These dimpled samples are then ion-milled (see below).

4.2.2 Electrochemical Thinning

No mechanical thinning process can avoid introducing some sub-surface mechanicaldamage, either through plastic shear or by microcracking. If the material is a metallicconductor, then it is frequently possible to thin the sample by chemical dissolution ratherthan mechanical abrasion. This is most commonly achieved electrochemically. Thetechniques that have been developed for electrochemical thinning are based on standardelectropolishing solutions. However, the conditions required for electropolishing bulkcomponents often differ markedly from those we are concerned with in the preparation ofspecimens for the transmission electron microscope, primarily because the electricallyconducting area that is to be thinned is so small. Far higher current densities are used thanwould be possible in bulk electropolishing. The problem of dissipating the heat generatedduring chemical attack at high current densities can be solved by passing the current througha jet of the polishing solution that impinges on one or both sides of the disc sample(Figure 4.12).

While thin films of all metals and alloys, as well as many other materials that possess thenecessary electrical conductivity, have been successfully prepared by jet-polishing, each ofthese materials requires its own polishing conditions, especially the composition andtemperature of the solution, and the current density. These ‘recipes’ are available in theliterature or from the manufacturers of jet-polishing units.


Electrochemical thinning of a thin-film specimen is complete as soon as the first holeappears near the centre of the disc sample. Initial hole formation is detected either by eye or,more usually, by using an optical laser signal passing through the hole to automaticallyswitch off the current and the jet. Once the specimen has been rinsed and dried, it is ready forinsertion in the microscope. The regions around the central hole in the sample should betransparent to the electron beam (typically 50–200 nm in thickness). If the areas availablefor investigation are found to be too small, then this is most commonly because the jet-polishing process was allowed to continue after formation of the first hole, leading to rapidattack at the edge of the hole and consequent rounding of the rim. Sometimes these areas arefound to have a roughened, etched appearance, usually because the polishing solution isexhausted, contaminated or over-heated, or possibly because the current density isinsufficient.

4.2.3 Ion Milling

The earliest successes in preparing thin-film specimens of ductile metals and alloys byion milling were pioneered by Raymond Castaing in France, in the mid 1950s. Castaingemployed a beam of energetic inert gas ions to sputter away the surface of thin aluminiumalloy foil samples. This technique was initially superseded by chemical thinningmethods, based on electrolytic polishing, but the development of increasingly sophisti-cated ion milling systems has led to a resurgence of ion milling, which is now thepreferred method for removing the final surface layers from samples intended for TEM(Figure 4.13).

V

Pumpedelectrolyte jet

Pumpedelectrolyte jet

Discspecimen

Figure 4.12 In jet polishing a current is passed through a stream of the polishing solution as itimpinges on the disc sample. Thinning is accomplished by electropolishing at a high currentdensity.


There are several reasons for the current preference for ion milling:

1. Ion milling is a clean, low-pressure, gas-phase process, so that contamination of thesurface is easier to control.

2. Electrochemical methods are largely restricted tometallic conductors, while ionmillingis more generally applicable, for example to ceramics and semiconductors.

3. Although some sub-surface radiation damage is often introduced during ionmilling, thiscan be minimized by suitable choice of the milling parameters, for example the ionenergy and the angle of incidence of the ion beamon the sample, and there is no danger ofmechanical damage.

4. Ion milling also removes surface contaminant films, such as the residual anodic oxidelayers associated with electropolishing. In addition, no changes in surface compositionare expected, since milling is normally performed at temperatures well below those atwhich diffusion occurs in the specimen.

5. Thin-film multilayers of different materials deposited on a substrate can seldom bechemically thinned in cross-section.

6. The sophisticated ion milling units now available are able to ensure that thethinned region of the sample selected for examination can be localized to betterthan 1 mm, and this is often a primary consideration in the study of thin-filmmicroelectronic and optronic devices. Such accuracy in selecting the area ofexamination for thin-film electron microscopy is quite impossible using chemicalmethods.

Of course, ion milling also has its problems. Sputtering is a momentum transfer process.The rate of sputtering is a maximumwhen the ion beam is normal to the sample surface andthe atomic weight of the sputtering ions is close to that of the sample material. However, anion beam incident at right-angles alsomaximizes both the sub-surface radiation damage and

Ion gun

Ion gun

Specimen

Figure 4.13 Precision ion milling permits thin film multilayer assemblies to be thinnedperpendicular to the plane of the assembly. Two ion guns are normally used in order tosputter from both sides of the sample. During the process the sample is rotated about the axisperpendicular to the plane of the assembly.


topographic surface irregularities associated with microstructural features. In addition, theneed to avoid ion-induced chemical reactions usually restricts the choice of sputtering ion tothe inert gases (typically argon).

The sputtering rate can, in principle, be improved by increasing the incident ion energy,but only at the cost of occluding large numbers of the sputtering ions in the sub-surfaceregion. These contribute to a radiation-damaged layer. In practice, the ion energy istherefore limited to a few kilovolts. At these energies, the depth of ion injection issufficiently limited to allowmost of the occluded ions to escape to the surface by diffusion,rather than nucleate sub-surface damage.

In general, the angle of incidence for ion milling is restricted to no more than 15�, and, atan energy of the order of 5 kV, the rates of sputtering are then often nomore than 50 mmh�1.It follows that the preferred specimen for subsequent transmission electron microscopy isone that has already been thinned mechanically (usually by dimpling) or electrochemically(usually by jet-polishing) to a thickness of the order of 20–50 mm before it is thinned in theionmiller. The disc specimen is rotated during ionmilling, in order to ensure that thinning isas uniform as possible. The initial stages of milling are performed from both sides of thesample simultaneously, often at an angle of up to 18� to improve the thinning rate. The angleof incidence of the ions (and hence the rate of milling) is then reduced in the final stages ofthe thinning process tominimize surface roughness. Theminimumangle that will result in aoptically planar surface finish with large, uniformly thinned areas is dictated by the ionbeam geometry. At glancing angles a high proportion of this beam could sputter materialfrom the specimen mounting assembly, and hence contaminate the specimen. Primarily forthis reason, the minimum sputtering angles in the final stages of ion milling are usuallybetween 2� and 6�.

In general, milling is judged to be completewhen the first hole is formed in the sample. Ina precision ion-milling system the area selected for thinning is monitored in situ using alight transmission detector. Milling is terminated as soon as the required region isperforated. Unfortunately, this may not work if one or other of the layers in the sampleis transparent to light.

4.2.4 Sputter Coating and Carbon Coating

The electron beam carries a charge, so electrically insulating specimens will generallyacquire some electrostatic charge during examination. In many instances charging is not aproblem, since the small size of the sample and surface conductivity limit the charge. Ifnecessary, the specimen can be coated with a thin electrically conducting layer. Thepreferred conductingmaterial is carbon, since this element has a low atomic number (6) anddeposits as a uniform, amorphous thin film. Any substructure due to the carbon coating is ofvery low contrast and on a nanometre scale.

The carbon coating may be evaporated onto the surface by passing a high electric currentthrough a point contact between two carbon rods, or sputter-coated by bombarding a carbontarget with inert gas ions and depositing the sputtered material on the sample surface.‘Difficult’ samples, such as ceramics, may require coating on both sides. The nanometre-scale morphology of the thin (5–10 nm) coating is sometimes faintly visible in recordedmicroscope images.


4.2.5 Replica Methods

Instead of preparing a thin slice from a component for direct examination in thetransmission electron microscope, it is possible to take a replica from a surface. This maynot be necessary if the surface can be examined at sufficient resolution in a scanningelectron microscope, but this is not always possible and there are several reasons why areplica for insertion in the transmission electron microscope may be desirable.

1. Nondestructive examination may be necessary. For example, in failure analysis, andoften for legal reasons, there is a reluctance to section the component and destroyevidence. Moreover, taking a replica from the surface can be done quite easily in thefield, far from the laboratory, and without destroying the component. This is especiallytrue of forensic investigations at crime scenes and crash sites, where the evidence mustbe preserved for the court.

2. Selecting one component of a complex sample for investigation. When evidence issought for the presence of a specific phase on the surface, collection on a replica canpreserve both the phase morphology and its distribution. Again, forensic examples areeasy to imagine: gun shot residues recovered from skin, or paint pigment particles at thescene of a crash. In many failure investigations corrosion products can also be isolatedconveniently on a replica. Since analysis of the chemical composition and phase contentof a corrosion product will not be obscured by the composition and structure of thebulk material, there is good reason to identify these products by isolating them on areplica.

3. Extracting specific phases from a polyphase material. Suitable chemical etchants cansometimes be used to isolate a selected phase from the bulk material on a replica, whilestill preserving the original phase distribution in the surface section taken from the bulk.The chemistry, crystallography and morphology of the extracted phase can then bestudied in the transmission electron microscope with no interference from the micro-structural features of the remaining constituents of the bulk matrix. An excellentexample is the study of carbide precipitation in steels, where an extraction replica canreveal the composition, crystal structure and morphology of the carbide phases thatwould normally be obscured by the strong diffraction contrast and X-ray excitation ofthe ferrous alloy matrix.

4. Correlation of microstructures using alternative imaging methods. Finally, it may bedesirable to compare observations made on a replica taken from a solid surface withobservations made on the same surface in scanning electron microscopy. Surfacemarkings associated with mechanical fatigue are one example in which combiningthese two techniques may be an advantage. Dimpled ductile failures can be observed inthe scanning electronmicroscope and the nonmetallic inclusions that are associatedwiththe nucleation of the dimples can be extracted on a replica and then identified in thetransmission electron microscope.

The usual procedure (Figure 4.14) is to obtain a negative replica of the surface on aflexible, soluble plastic. The plastic may be cast in place and allowed to harden, or it may bea plastic sheet that has been softened with a suitable solvent and then pressed onto thesurface before allowing the solvent to evaporate. In some cases it will first be necessary toremove loose contamination by cleaning the surface ultrasonically or by using an initial


‘cleaning’ replica before making the final plastic replica. In other cases it may be preciselythe ‘contamination’ that is the subject of interest, as in the case of gunshot residues.

Once a plastic replica has hardened, it can be peeled away from the surface and thenshadowed with a heavy metal, such as a gold–palladium alloy, to enhance the final contrastin the electron microscope. The shadowing metal is selected for minimum particle size andmaximum scattering power, and the particle cluster size is typically about 3 nm. Aftershadowing, a carbon film100–200 nm in thickness is deposited on the plastic replica and theplastic is then dissolved in a suitable organic solvent. The carbon film retains any particlesremoved by the plastic from the replicated surface, as well as the heavy metal shadow thatreflects the original surface topology. The carbon replica is then rinsed and collected on afine-mesh copper grid for examination in the transmission electron microscope.

There may be no need for a ‘negative’ plastic replica. For example, an alloy sample thathas been polished so that the particles of a second phase stand proud of the surface can becoated with a carbon film that is deposited directly onto the surface andwill adhere stronglyto the particles. Further etching of the matrix will release a carbon extraction replica onwhich are distributed, in their original configuration, the particles of the second phase.

4.3 The Origin of Contrast

The electron beam interacts with a thin-film specimen both elastically and inelastically, butit is the elastic interactions that dominate the contrast observed in the transmission electronmicroscope. On the other hand, it is the inelastic scattering events that contain informationon the chemical composition of the sample, and wewill return to this in Chapter 6 when wediscuss EELS.

Contrast arises from three quite distinct image-forming processes, and these are termedmass–thickness contrast, diffraction contrast and phase contrast, respectively. Figure 4.15illustrates the electron scattering processes schematically. If the sample is amorphous, that

Heavy Metal Shadow

Extracted Particle

Negative Plastic Replica

Sample

Carbon Replica

Plastic Replica

Figure 4.14 A negative plastic replica can be used to prepare a thin film carbon replica of theoriginal surface that contains extracted particles. The replica can be shadowed to reflect theoriginal particle morphology.


is it has a glassy microstructure with no long-range crystalline order, then the elasticscattering gives rise to an envelope of transmitted intensity that varies with the scatteringangle according to an approximate cos2y law. The intensity scattered out of the direct beamfrom such a glassy specimen depends on the energy of the electron beam, the samplethickness and the sample density. The image contrast is then said to be due to variations inmass–thickness. If an aperture is placed in the electro-optical column at an image plane ofthe electron source that lies beneath the plane of the specimen, then the aperture willintercept most of the scattered electrons and the image will be dominated by those directtransmitted electronswhich havenot been scattered in passing through the thin-film sample.Mass–thickness contrast usually dominates the features seen in transmission electronmicroscopy of biological samples taken from soft tissues, as in histological studies.

In a crystalline sample, the electrons are elastically scattered according to Bragg’s lawand generate diffracted beams at discrete angles 2yhkl to the direct transmitted beam whichcorrespond to those crystal planes whose Miller indices hkl satisfy the Bragg condition (or,more exactly, those crystal planes whose reciprocal lattice vectors touch the reflectionsphere) (Section 2.5). An aperture can now be inserted into the optical column at an imageplane of the electron source beneath the specimen, so that this aperture allows either thedirectly transmitted beam to pass into the imaging system, to form a bright-field image, orselects one of the diffracted beams to be accepted, forming a dark-field image. In both casesthe image contrast is determined primarily by the presence of crystal lattice defects thataffect the local diffracted intensity generated near the lattice defect. This imaging mode istermed diffraction contrast. Both mass–thickness and diffraction contrast create what are

IncidentBeam

Specimen

Coherently ScatteredDiffracted Beams

Envelope ofElastic Scattering

Intensity

Envelope ofInelastic Scattering

Intensity

Figure 4.15 Most of the incident beam is elastically scattered by the sample, either randomly(as in a glassyor amorphous specimen) or coherently to discrete angles (as in a crystallinephase).The imagemaybe formed fromeither thedirect transmittedbeam,byadiffractedbeam,or by theinterference of the diffracted beams, both with each other and with the direct transmitted beam(see text).


essentially magnified shadows of microstructural features. This is not too dissimilar fromthe elongated shadow of a tree on the grass that shows the leaves and branches in a two-dimensional projection that depends on the angle of the sun.

Finally, if the resolving power of the microscope is adequate, a larger diameter aperturecan be inserted to admit several diffracted beams simultaneously into the imaging system,with or without the direct transmitted beam. These beams interfere in the image plane toyield a lattice image that reflects the periodicity in the crystal in the plane normal to the opticaxis of the microscope, an effect termed phase contrast. Unlike mass–thickness anddiffraction contrast, the phase contrast image makes use of the elastically scatteredelectrons from several different crystal lattice planes and is an image of the crystalstructure. Since not all diffracted beams can be included, this lattice image is incomplete.

4.3.1 Mass–Thickness Contrast

The probability of an electron being elastically scattered out of the incident beam dependson the atomic scattering factor, which increases monotonically with the atomic number andthe total number of atoms in the path of the beam, that is the total thickness of the film.Mass–thickness contrast thus reflects a combination of variations in specimen thickness andspecimen density, and is therefore similar to the effects of mass absorption discussedpreviously for X-rays (Section 2.3.1).

In the life sciences, mass–thickness contrast almost always dominates the image. Thecontrast of soft tissue, biological samples in the electron microscope is often enhanced by aheavy-metal, tissue-staining procedure. In the natural sciences and in engineering studies,mass–thickness contrast only predominates in noncrystalline materials, such as two-phaseglasses, as well as in replica studies, where the contrast is often due to variations in thethickness of a metal shadow deposited on the replica, or to the presence of an extractedphase (Section 4.2.5).

4.3.2 Diffraction Contrast and Crystal Lattice Defects

In the case of a perfect crystal, the contrast in the microscope is associated with theamplitude scattered from the incident beam into a diffracted beam by diffracting planeswhich have a specific reciprocal lattice vector g. This amplitude can be calculated bysumming the diffracted amplitudes from all the unit cells that lie along the path of thediffracted beam (Figure 4.13). The phase difference f in the amplitude scattered by a unitcell that is at a position rwith respect to the origin in the column of material responsible forthe contrast is given by:

f ¼ 2pðg � rÞ ð4:6Þwhile the amplitude that is scattered by this unit cell can be written:

Aeif ¼ A exp½�2piðg � rÞ� ð4:7ÞThe unit cells in the column of crystal being considered are each separated by thelattice parameter a. For Bragg diffraction, that is scattering in phase, g � r¼ n, where nis an integer. Since the electron wavelength is very much less than the lattice parameter,each unit cell in the crystal scatters independently of the others, so that the amplitudescattered by a single unit cell will be proportional to the vector sum of the atomic scattering


factors and hence to the structure factor for the cell F for the specific diffracting planes(Section 2.4.2).

If the amplitude scattered is small compared with the incident amplitude, we can ignoreany reduction in the amplitude of the direct transmitted beam, so that each unit cell in thecolumn scatters the same amplitude. It follows that, at the Bragg position, the totalamplitude scattered by a column of n unit cells will be a linear function of n:

An ¼Xn

Fnexp½�2piðg � rÞ� ð4:8Þ

If the number of unit cells in a column is sufficiently large, we can replace the sum by anintegral, which for convenience we now take over the thickness t of the thin-film samplemeasured from the mid-thickness, that is the integral is taken over the range �t/2:

At ¼ Fa

Zt=2

�t=2

exp �2piðg � rÞ½ �dr ð4:9Þ

If the angle of incidence of the incident beam deviates slightly from the Bragg condition,then the scattering vector g in reciprocal space must be replaced by gþ s, where s is theangular deviation from the Bragg position measured in reciprocal space. Similarly, if thestructure of the ‘perfect’ crystal lattice is distorted, due to the presence of a lattice defectgenerating a displacement R, then the position of the scattering element at the position r inthe column of crystal is shifted to rþR. The phase angle for the amplitude scattered from aunit cell at the position r in a column of the crystal that is both misoriented and defective istherefore given by:

f ¼ 2pðgþ sÞ � ðrþRÞ ð4:10ÞExpanding the brackets we have:

f ¼ 2pðg � rþ g �Rþ s � rþ s �RÞ ð4:11ÞOf these terms g � r¼ n is an integer and has no effect on the phase of the amplitudescattered. The term s �R is obtained bymultiplying two small vectors together, and so can beneglected. The two remaining terms, g �R and s � r, are additive. They represent the phaseshift in the amplitude scattered at the position r into the diffracted beam g due either todeviations from the exact Bragg condition s � r, or to distortions of the crystal lattice, that is,lattice strains associated with the presence of lattice defects g �R. It follows that inmicrostructural features imaged by diffraction contrast we are observing the summedcontrast effects of both deviations in reciprocal space (changes in the specimen orientationwith respect to the incident beam), as well as displacements in real space (displacements ofthe crystal lattice due to lattice strains).

Complete interpretation of diffraction contrast requires that these two effects be separatedand their origin identified. This requires that the operating reflections, the g vectors, beknown and that the causes of the displacements in both real space R, and reciprocal space s,be identified. This is often a difficult process that necessitates knowledge of the thickness ofthe sample, together with a series of both bright-field and dark-field images taken usingdifferent g reflections. In practice, a complete analysis of diffraction contrast may not be


necessary, and image analysis can be limited to a generic identification of the types of latticedefects present: dislocations, stacking faults and point defect clusters, rather than a completequantitative analysis of a defect that includes, for example, the sign and magnitude of adislocation Burgers vector or the value of a stacking fault vector.

4.3.3 Phase Contrast and Lattice Imaging

The transmission electron microscope is commonly operated either as a microscope(bright-field or dark-field images) or as a diffraction camera (Figure 4.16). When operatedas a microscope, imaging either mass–thickness or diffraction contrast, an objectiveaperture, placed in the back focal plane of the objective lens, limits the angle of acceptancefor the beam to an angle a, which is less than the Bragg angle y for any of the coherentlyscattered electron beams. The image is then a bright-field, shadow projection image of theelectrons coming through the objective aperture. In this image the intensity variations in theimage plane reflect the variations in the electron beam current passed down the microscopecolumn after traversing the different regions in the field of view on the thin-film sample.When the incident beam is tilted by a Bragg angle, so that one or other of the coherentlydiffracted electron beams, rather than the direct transmitted beam, passes down the opticaxis of the microscope, then this diffracted beam is accepted into the imaging system. Theobjective aperture now cuts out both the direct transmitted beam and all other diffractedbeams, so that the bright field image is replaced by a dark field image.

If the objective aperture is removed or replaced by an aperture, that is large enough toaccept both the direct transmitted beam and one or more of the diffracted beams into theimaging system, that is a>2y, then the differences in the path length followed by thedifferent beams will result in an interference pattern in the image plane (Figure 4.17). Inorder for this interference pattern to be observed and interpreted, several parameterscharacteristic of the microscope have to be known. These include the values of thechromatic and spherical aberration coefficients, the exact focusing plane of the image,and the coherence and energy spread in the incident electron beam which is determined bythe electron source.

That phase contrast in the transmission electron microscope corresponds to an interfer-ence pattern is extremely important. Imagine an object represented by a number of pointsources and having a total electron wave distribution in the object plane defined by f (x,y).The corresponding function g(x,y) in the image plane to this object function represents theamplitude and phase of f(x,y) after travelling down the microscope column. Each point inthe image planewill contain contributions from all the beams that have been transmitted bythe objective aperture, so:

gðrÞ ¼Z

f ðr 0Þhðr�r 0Þdr 0 ¼ f ðrÞ � hðr�r 0Þ ð4:12Þ

where the more convenient radial coordinates r have been used instead of Cartesiancoordinates (x,y), and h(r) represents the contribution to the electron wave distributionfunction from each individual object point to any given point in the image. The function h(r)is termed a point-spread function or impulse-response function, and g(r) is the convolutionof f(r) with h(r). Thus the electron wave function of the incident beam is modified by theelectron density distribution in the specimen, but also convoluted with a function which


Diffraction Pattern Image

Specimen

ObjectiveLens

ObjectiveAperture

SADAperture

IntermediateLens

ProjectorLens

Screen

Figure 4.16 The transmission electron microscope can be used to image the specimen byfocusing the final image in the plane of the fluorescent screen, or it can be used to image thediffraction pattern from the specimen. To anexcellent approximation, the imageof the specimenis observed when the imaging system is focused on the front focal plane of the objective (theposition of the specimen), while the diffraction pattern is observed when the imaging system isfocused on the back focal plane of the objective (The back focal plane of the objectivecorresponds to the first image plane for the electron diffraction pattern in the microscope).SAD, selected area diffraction.


describes the response of the microscope column: the electron beam source, the electro-magnetic lenses, the aperture sizes and the lens aberrations.

It is now convenient tomove to reciprocal spacewherewe can represent g(r) by aFouriertransform for which:

gðrÞ ¼Xu

GðuÞexpð2piu � rÞ ð4:13Þ

where u is the reciprocal lattice vector. We define the Fourier transform of f(r) as F(u) andthat of h(r) as H(u), and find them to be related [compare Equation (4.10)] by:

GðuÞ ¼ HðuÞFðuÞ ð4:14Þ

Figure 4.17 If the objective aperture accepts a Bragg-diffracted beam as well as the directtransmitted beam, a> 2y, then an interference pattern will be formed in the image plane as aresult of the difference in path lengths of the two beams.


whereH(u) is termed the contrast transfer function (CTF) characteristic of the microscope.The CTF is dimensionless and is plotted as a function of r�1, that is, as a function of thespatial frequency reaching the image plane. The CTF describes the total influence of all thevarious microscope parameters on the phase shift for an electron wave propagating downthe microscope column. There are three major contributions to the CTF:

1. An aperture function that defines the cut-off limit for spatial frequencies above a criticalfrequency determined by the aperture radius.

2. An envelope function that describes the damping of the spatial frequencies due either tochromatic aberration, or to instabilities in the objective lens current, or to the coherencelimit of the electron source.

3. An aberration function that limits the spatial frequencies available for imaging and isusually dominated by the spherical aberration coefficient of the electromagneticobjective lens.

The aberration function can be given as:

BðuÞ ¼ exp½ixðuÞ� ð4:15Þwhere

xðuÞ ¼ pDf lu2 þ 12pCsl

3u4 ð4:16Þ

and Df is the underfocus of the objective lens, that is the distance between the object planeand the focal plane of the lens, l is the wavelength, and Cs is the spherical aberrationcoefficient of the objective lens.

We now examine the influence of the above three contributions on the form of the CTF.Figure 4.18 shows the CTF for a 300 kV transmission electron microscope with a sphericalaberration coefficient of 0.46mmat an objective lens underfocus of�36 nm.Theparametersused in calculating Figure 4.18 include the relative structure factors for the lattice spacings inaluminium. Figure 4.18(a) shows the coherent CTF and is defined only by Equation (4.16.)Figure 4.18(b) shows the spatial coherence envelope, defined by the aperture function, andFigure 4.18(c) shows the temporal coherence envelope that is defined by the damping of thespatial frequencies due to lens instabilities and the limited coherence of the electron source.Finally, Figure 4.18(a)–(c) is combined in Figure 4.18(d), which therefore corresponds to theCTF for this specific microscope column at a selected objective lens defocus.

From Figure 4.18(d) we see that the CTF of this microscope can only transfer a limitedrange of reciprocal spacings, that is, spatial frequencies, to the image. These frequenciescan be correlated with the lattice spacings in the crystal structure of the specimen that areresponsible for the diffraction pattern observed in themicroscope. In regionswhere theCTFis zero, no information can be transferred down the microscope column to the image. Thus,in the present example, only the {1 1 1}, {0 0 2}, and {2 0 2} planes of aluminium can beresolved in the image, always provided that the crystal is oriented in the correct zone axis!

Since the CTF depends on the defocus of the objective lens, the CTF can be tuned toselected spatial frequencies by adjusting the objective lens current. This is illustrated inFigure 4.19, which uses a different presentation for full CTF. The y-axis is the value ofobjective lens focus, with negative values corresponding to an underfocus, and the x-axis isthe spacing in reciprocal space, that is the spatial frequencies, and the intensity is plotted on


Figure 4.18 A CTF for a high-resolution objective lens shown at Scherzer defocus andincluding the normalized structure factors for some crystallographic planes in aluminium.The functionswere calculated using a 300 kVaccelerating voltage, a focal spread of 10 nm, anda semi-convergence angle of 0.3mrad. (a) The coherent CTF defined only by Equation 4.16. (b)The spatial coherence envelope, defined by the aperture function (c) The temporal coherenceenvelope defined by the damping of the spatial frequencies due to electromagnetic lensinstabilities and the coherence of the electron source. Combining (a), (b), and (c) yields (d),the CTF of the microscope at Scherzer defocus.

Figure 4.19 Absolute values of the CTF plotted using the same values used in Figure 4.18, buthere the relative values of the CTF correlates to the plotted brightness (white corresponds to CTFmaxima,whileblack is aCTFof zero). They-axis is the objective lens defocus. The selectedvalueof objective lens defocus (the Scherzer focus) plotted in Figure 4.18 is indicated in Figure 4.19 asa horizontal red line.


a shaded, grey scale. The value of objective lens defocus used to calculate Figure 4.18 isindicated in Figure 4.19 as a red, horizontal line. In this case only the absolute values of theCTF are plotted, so a ‘white’ intensity corresponds to both positive values of the CTF, andthen, after crossing the zero values (black), negative values.

By controlling the objective lens defocus, that is the lens current, we can shift the CTFand enhance the phase contrast for specific, selected crystallographic planes. In order toget the best performance from the microscope, a calculation of the CTF and somecareful thought about the information that is being sought should precede the microscopesession!

Since no information is transferred to the imagewhen theCTFhas avalue of zero, the bestCTF, that is the best objective lens performance, will have few oscillations about the zerovalues at high spatial frequencies. Under-focusing the objective lens moves the first CTFcrossover (first value of zero) to larger values of nm�1 (that is, smaller d-spacings), and canpartially compensate for the spherical aberration of the lens. The position of optimumcompensation is termed the Scherzer defocus (Scherzer predicted this effect in 1949). TheScherzer defocus can be calculated from:

DfSch ¼ �1:2ffiffiffiffiffiffiffiffiffiffiffiffiðCslÞ

pð4:17Þ

The Scherzer defocus crossover determines the resolution limit of the microscope,usually termed the point resolution, and corresponds to theminimum defocus value atwhichall beams below the first CTF crossover have approximately constant phase. Below theScherzer defocus, crystallographic information will still be available from spacings belowthe point resolution, but complete computer simulation of the image is necessary to interpretthe information. Under such conditions the information transferred to the image by themicroscope is severely damped, so that the contrast available from crystallographic planeswith small d-spacings is limited. This sets a second resolution limit for the microscope,termed the information limit. The distinction between the point resolution and theinformation limit is the reason why lattice images can be readily obtained from crystallattice planes that have d-spacings appreciably less than the point resolution of themicroscope. Point resolution is an important criterion for microscopists working onnoncrystalline biological tissue and cell samples, but for the materials community it isthe information limit that is the more important criterion, since this defines the minimuminterplanar spacings that can be resolved in a lattice image.

The introduction of hexapole electrostatic stigmators for spherical aberration correction(Section 4.1.2.2) had amajor impact on the electronmicroscope community. The correctionsystem can reduce the value of the spherical aberration coefficient, so that the pointresolution and information limit of the microscope coincide. Using a monochromaticelectron source to reduce the influence of the chromatic aberration, resolutions of less than0.07 nm are now available.

In addition to this improved resolution limit, it is also possible to extract phaseinformation from the lattice image, so that the information derived is no longerlimited to a two-dimensional intensity distribution in the x–y plane. Initially this wasachieved by analysis of a through-focus series of images, but, in principle, a single recordedimage can be compared with a virtual image based on the known crystal structure toextract in-depth information. Lattice imaging has brought the electron microscopist


a longway from the ‘shadow’ images based onmass–thickness and diffraction contrast. It isimportant to recognize that the computer interpretation of lattice image contrast is now anessential component of image analysis. The microscopist should never be misled intothinking that the periodicity and contrast observed in the lattice image represent directly theposition and electron density of the atoms in the nanostructure being studied.

4.4 Kinematic Interpretation of Diffraction Contrast

The basic assumption of kinematic diffraction theory, that the amplitude diffracted out ofthe incident beam does not affect the intensity of this beam, is so patently false that it mayseem odd to include this section in the text. However, the use of kinematic arguments, usingamplitude–phase diagrams, drastically simplifies the discussion of diffraction contrastfrom lattice defects and allows the major qualitative features of defect contrast to bedemonstrated, so that the loss of quantitative rigour should be forgiven.

4.4.1 Kinematic Theory of Electron Diffraction

The basic equation for the kinematic theory of electron diffraction has been given inSection 4.10, and the generalized form that we will discuss here is:

At ¼ F=aZ þ t=2

�t=2exp½�2piðg þ sÞ � ðr þ RÞ� ð4:18Þ

We justify the approximations of the kinematic theory by assuming that the intensity ofthe diffracted beam is much less than that of the incident beam, so that the intensity of thedirect transmitted beam approximates that in the original incident beam.

4.4.2 The Amplitude–Phase Diagram

Returning to the ‘column’ approximation, which assumed that each unit cell in a column ofcrystal in the thin-film specimen scattered independentlywhenviewed along the direction ofthe diffracted beam (Figure 4.20) and further assuming a perfect crystal free of lattice strains(R¼ 0), then the phasemismatch between successive unit cells in the columnDf is uniquelydetermined by the deviation in reciprocal space from the Bragg condition s, and is given byDf¼ 2pa, where a is the thickness of a unit cell in the column. Replacing the incrementDfbydf and the sumby the integral, that is, taking df¼ 2psdr, and ignoring the structure factorF, since this will be a constant for any given operating reflection, while noting that, forg � r¼ 1, exp[2pi(g � r)]¼ 1, then we obtain the total amplitude diffracted by the column as:

A ¼Z þ t=2

�t=2exp �2pis � r½ �dr ¼ sin ptsð Þ

psð4:19Þ

which corresponds to a relative diffracted intensity:

I=I0 ¼ sin2ðptsÞðpsÞ2 ð4:20Þ


Although this relation is based on an oversimplified model, it demonstrates the effects ofboth the specimen thickness t and the deviation from the Bragg condition s on the diffractedintensity. This equation is best plotted as an amplitude–phase diagram (Figure 4.21). Theradius of the circle is equal to (2ps)�1 anddecreases rapidly as the specimen is tilted out of the

DirectTransmitted

BeamDiffracted

Beam

Unit Cellat r

IncidentBeam

Column ofCrystal Thin Film

Specimen

Figure 4.20 The unit cells in a column of crystal that corresponds to a Bragg scatteringdirection will each scatter a proportion of the incident beam into the diffracted beam.

2πst

2πs1

t

Resultantamplitude

Top ofthin film

Bottom ofthin film

Figure 4.21 In kinematic theory, the amplitude–phase diagram for the diffracted amplitude is afunction of specimen thickness t and deviation from the Bragg condition s.


Bragg condition (increasing s). As the thickness increases at a fixed value of s, the diffractedamplitude oscillates sinusoidally between two maxima that are proportional to �(ps)�1,while the relative diffracted intensity oscillates between zero and a maximum value that isproportional to (ps)�2. It follows that a tapered thin-film sample, viewed at a fixed crystalorientation which is just off the Bragg condition for a particular reflection, will show a seriesof bright fringes, termed thickness fringes, that are parallel to the edge of the specimen. Thefringeswill get closer together and grow fainter as the sample is tilted further from theBraggposition.However, a bent crystalline filmof uniform thicknesswill also show fringes, termedbend contours, whose intensity and separation decreasewith increasing s, the first extinctioncorresponding to the condition st¼ 1.

Both thickness fringes and bend contours are prominent features of the contrast from thincrystalline films. Although dynamic diffraction and absorption effects (Section 4.5)markedly modify the quantitative analysis of these features, the kinematic theory is quiteadequate for a qualitative understanding of this thickness- and orientation-dependentdiffraction contrast.

4.4.3 Contrast From Lattice Defects

The presence of a lattice defect modifies the amplitude–phase diagram of Figure 4.21 byintroducing a second term into the phase shift, so that the integral determining the amplitudebecomes:

A ¼Z þ t=2

�t=2exp½�2piðs � r þ g �RÞ�dr ð4:21Þ

The additional phase shift due to the displacement field of the defect may either increasethe curvature of the amplitude–phase diagram, so that the crystal lattice is tilted locallytowards theBragg condition, ordecrease this curvature, so that the effect of the lattice defectis to tilt the lattice in this region further from the exact Bragg condition (Figure 4.22). In thefirst case, the radius of the amplitude–phase diagram is decreased and the diagram starts tocollapse in the region of the lattice near the defect. In the second case the diagram isexpanded in the region near the defect, opening up the amplitude–phase diagram. For thesake of convenience, the zero of coordinates in the unit cell column of diffracting crystal isnow shifted to coincidewith the position ofmaximum lattice displacement due to the defect,that is the ‘centre’ of the defect displacement field, rather than themid-thickness of the thin-film sample.

It follows that the displacement field R due to the defect has a result equivalent to eitheramplifying or suppressing the effect of s, by increasing or reducing the effective deviation ofthe crystal from the Bragg condition. An important consequence of this conclusion is thatthe effect of the defectwill be reversed if the sign of s is reversed. Thiswill occurwhenever acrystalline film that contains a defect is tilted in order to image the defect with the incidentbeam on opposite sides of the exact Bragg orientation. The position of the maximum valueof R in the column is also important, and in general maximum diffraction contrast fromdefects is expected to occur when R and s are of opposite sign and when the position ofmaximum R is near the mid-point of the foil. In thicker regions the intensity is expected tooscillate as the defect position moves from the top to the bottom of the foil, reflecting the


oscillations in amplitude that are associated with the diameter of the amplitude–phasediagram for a perfect crystal.

We have concentrated on the intensity in the kinematic diffracted beam, and hence on thedark field image, but in practice samples are usually first viewed in bright field before anydefect analysis is attempted in dark field. In general the kinematic intensity observed inbright field is approximately the inverse of that observed in dark field. That this is not alwaysthe case is due to the non-kinematic, dynamic and absorption effects that become importantfor thicker films and will be discussed later (Section 4.5).

Grain and phase boundaries constitute a special class of lattice defect whose contrast canbe qualitatively explained in terms of thickness extinction fringes. The two crystals on eitherside of the boundary are unlikely to deviate from the nearest Bragg position by the sameamount, so that the crystal with the minimum value of s will dominate the contrast andgenerate a series of thickness extinction fringes at an inclined boundarywhose separation isdictated by the value of s and the tilt angle of the boundarywith respect to the incident beam.The number of fringes will depend on the thickness of the thin-film crystal and the structurefactor for the reflection g. Diffraction from the neighbouring crystal will interfere with thissimple, single crystal thickness contrast. This can occur when the second crystal is also neara Bragg diffraction condition, so that the appropriate value of s for the second crystalapproaches that for the first.

4.4.4 Stacking Faults and Anti-Phase Boundaries

Stacking faults and anti-phase boundaries (APBs) constitute a special case since thedisplacement vector does not vary continuously along the column of the diffracting crystal,

Top ofThin Film

Position ofDefect

Position ofDefect

Bottom ofThin Film

ResultantAmplitudeResultant Amplitude

Figure 4.22 The presence of a lattice defect introduces a displacement field that eithercollapses the amplitude–phase diagram (increases the curvature) or tends to open it out(decreases the curvature). (The origin of the amplitude–phase diagram has been moved tocorrespond to the position of maximum displacement R in the column of crystal considered)


but rather changes discontinuously across the plane of the stacking fault or the APB. Theamplitude–phase diagram for the crystal above the point of intersection of the diffractingcolumn with the fault plane is therefore undisturbed, while that below the fault plane, nowpositioned for convenience at the origin of the diagram coordinates, is rotated by an angleequal to the phase angle associated with the fault vector, 2pg �R (Figure 4.23).

For example, a stacking fault in the lattice of an FCCmetal has a fault vector 1/6ah1 1 2i.This vector is exactly equivalent to a fault vector of 1/3ah1 1 1i in the FCC lattice, since itcan be combined with an appropriate unit lattice vector of 1/2ah1 1 0i, as in the dislocationreaction, a6 1 1 2½ � þ a

2�1 �1 0½ � ¼ a

3�1 �1 1½ �. The phase shift 2pg �R, where R is now the fault

vector, will be zero if g is perpendicular to R. If |R| is 1/3ah1 1 1i and |g| is an allowed 1/2ah1 1 0i reflection, then the phase shift can take one of three values, 0 (when the twovectors are mutually perpendicular) or �120�. The sign of R is determined by the type ofstacking fault, which may be either intrinsic or extrinsic that is the fault may be due to amissing plane of atoms (ABC|BCABC) or to an extra plane (ABCBABC) inserted into thestacking sequence.

In the case of anAPB in the crystal, the fault will only be visible if g �R is a noninteger. Soit is not enough for the reciprocal lattice vector to have a resolved component parallel to thefault vector. The reciprocal lattice vectormust be a partial lattice vector of the parent lattice,termed a superlattice vector. Hence theAPBwill only become visible if the diffracted beamis from a superlattice reflection.

If the sample is thick enough, then the amplitude–phase diagram from a column of crystalcontaining a fault plane will exhibit thickness fringes as the fault position moves from thetop to the bottom of the crystal, similar to those observed in a wedge of crystal or at a grain

Above the fault

Below the fault

FaultPosition

2πg. R

Phase Shift

Figure 4.23 An amplitude–phase diagram for a columnof crystal containing a stacking fault orAPB at the origin of the coordinate system.


boundary. However, the origin of these fringes is very different, since diffraction is nowoccurring in both the regions of the crystal, above and below the fault plane.

4.4.5 Edge and Screw Dislocations

Diffraction contrast from dislocations is dominated by theBurgers vector of the dislocationlineb, since the strain field displacements associatedwith the dislocation are proportional tothe value of b. However, the direction of the dislocation line l with respect to the Burgersvector is also important. Dislocations that lie parallel to the Burgers vector are termed screwdislocations, and, in an isotropic crystal, have a cylindrically symmetric strain field, with nostrain component perpendicular to the Burgers vector. It follows that a reciprocal latticevector perpendicular to a screw dislocation in an isotropic crystal should not result in anycontrast. The condition g �R¼ 0 for no diffraction contrast can therefore be replaced by thecondition g � b¼ 0 for no contrast from a screw dislocation. By observing the contrast due tothe presence of the dislocation in different dark-field images as a function of the diffractingvector g in each case, and finding two values of g for which no contrast is observed, it ispossible to identify the direction of the Burgers vector unambiguously, but derivingthe magnitude of the Burgers vector from diffraction images requires a more completeanalysis.

By contrast, edge dislocations have a residual component of the strain field perpendicularto b that results from the dilatation (volume change) associatedwith the presence of an edgedislocation, so that some contrast is expected even if g � b¼ 0. Even so the weakness ofthe contrast compared with that observed with other reflections, together with someknowledge of the Burgers vectors to be expected in the crystal lattice, usually permitssome tentative conclusions to be drawn.

The vector product b · l gives the normal to the allowed glide plane for the dislocation inthe crystal. This plane is undefined for screw dislocations, which are therefore free to cross-slip. In general, a preferred glide plane exists, even for screw dislocations, suggesting thatscrew dislocations are partially dissociated, even when the separation of the partialdislocations is small and no stacking fault contrast is observable.

The sign of the strain-field associated with a dislocation is reversed when the column ofdiffracting crystal ismoved across the projected position of the dislocation line in the image,so that the sign of the phase shift is also reversed. It follows that on one side of thedislocation the amplitude–phase diagram is expanded, enhancing contrast, while on theother side it collapses, reducing the observed contrast, and this depends on whether or notg �R has the same sign as s � r (Figure 4.22). Thus, when the projection of a dislocation linein the image crosses a bend contour, the position of maximum contrast changes sides, andthe true dislocation position can be inferred from the mid-point between the two maxima(Figure 4.24).

The apparent width of a dislocation depends on the numerical value of g � b and the valueof s. Close to the Bragg position, the contrast will be a maximum and the width of theobserved contrast maxima is typically of the order of 10 nm. This is a very poor resolutionfor transmission electron microscopy and constitutes a major barrier to the study ofdislocation–dislocation interactions in sub-boundaries or during plastic shear. Much betterresolution can be obtained in the dark-field image bymoving away from the Bragg position,that is, by tilting the sample to large values of s. The contrast is then very weak, and may be


difficult to see, but the resolution in this weak beam image is appreciably better, down to ofthe order of 2 nm.

In samples of puremetals and ductile alloys dislocations are often observed tomove in theelectron microscope, and such behaviour has been used to study the glide process. Thephenomenon has been attributed to both thermal stresses, associated with electron beamheating and contrast due to scattering processes. More generally, stresses in the sample areinduced by the build up of a carbonaceous contamination film on the surface. Whendislocationmovement occurs during observation in themicroscope, it is often accompaniedby the appearance of slip traces in the image at the top and bottom of the film, where thedislocation has disrupted a layer of surface contamination. These traces can be used todetermine the glide plane in the crystal or, if the glide plane is known, to derive an accuratevalue for the specimen thickness.

4.4.6 Point Dilatations and Coherency Strains

Point defects, such as vacant lattice sites and impurity or alloy atoms, cannot be resolved intransmission electron microscopy, but remarkably small clusters of such defects willgenerate local lattice strains that give rise to detectable diffraction contrast. Such contrasthas been observed for three specific types of defect:

1. Radiation damage that is associated with injected ions or with lattice atoms that havebeen displaced by an energetic incident particle.

2. Defects associated with the condensation of a supersaturation of lattice vacancies, toform, for example, faulted or unfaulted vacancy loops and more complicated defectassemblies, such as stacking fault tetrahedra.

3. The early stages of precipitation, especially the formation of solute rich clusters,generally termed Guinier–Preston zones.

Dislocation Line

Bend Contour

– R

+R

+s

– s

Figure 4.24 The contrast expected from a dislocation line as it crosses a bend contour.When sand R are of the same sign the amplitude–phase diagram is expanded and strong contrast isexpected. The region of maximum contrast lies to one side of the dislocation line and moves tothe opposite side if the dislocation crosses a bend contour.


In all these cases the dominant lattice displacement is usually a hydrostatic strain fieldwhich can be approximated by a point defect dilatation. In the case of radiation damage, awide range of effects has been observed. These include the condensation of interstitialatoms, generated by long-range knock-on collisions, the formation of small vacancy loopsassociated with the annealing out of vacant lattice sites by diffusion, and the formation ofsmall helium bubbles due to a-particle condensation.

The clustering of vacant lattice sites to form a collapsed dislocation loop may result in aplanar stacking fault in the crystal lattice (for example in nickel, Figure 4.25), but if thestacking fault energy is high, shear of the crystal across the plane of the defect will reducethe total energy and result in a small vacancy loop that is bounded by a unit latticedislocation whose Burgers vector lies at an angle to the plane of the defect. These loops canbe resolved byweak beam contrast when they exceed 2–5 nm in diameter, and their Burgersvectors analysed using the criteria described previously for dislocations, for example, bynoting the change in contrast from inside to outside the loop as the sign of s is changed bytilting the specimen so that a bend contour sweeps across the image.

Unresolved, smaller loops behave only as point dilatations, and typically showup as twinblack and white crescents, as do larger loops when they are viewed edge on. Again the signof this contrast, that is, black/white as opposed to white/black, changes depending on thesign of s. However, as we will see below, the situation is usually rather more complicatedthan can be accounted for by the kinematic contrast approximation. The observed contrastdepends on the position of the defect within the thickness of the foil. Point dilatationsobserved near the top surface of the sample show the same contrast in both bright and darkfield, while those near the bottom surface show a reversed contrast for the bright- and dark-field images. It is only the dark-field image that meets the kinematic criterion discussedpreviously, namely that the contrast is determined by whether or not the phase shiftassociated with the defect reinforces that due to a deviation from the Bragg condition.

In addition to the care required to interpret image contrast resulting from nonkinematicdiffraction, there are also potentially confusing effects associated with rotation orinversion of the image in an electromagnetic imaging system. This is particularly thecasewhen diffraction patterns are to be recorded for comparisonwith bright- or dark-field

Figure 4.25 Faulted dislocation loops in nickel. Reproduced by permission of Plenum Pressfrom Williams and Carter, Electron Microscopy: A Textbook for Materials Science (1996).Original micrograph from Knut Urban


lattice images, since any inversion is then equivalent to a sign reversal of the strain.Distinguishing between vacancy and interstitial defects, that is, tensile as opposed to compres-sive dilatations, may be straightforward in theory, but can be quite confusing in practice.

Small precipitates and Guinier–Preston zones behave qualitatively very like vacancyclusters and radiation damage, but now the displacements are determined by lattice misfitthat is associated with mismatch in the effective size of the solvent and solute atoms. As theprecipitates grow, the strains associated with lattice mismatch accumulate, until thesecoherency strains are eventually relieved, at least in part, by the nucleation of interfacial, orVan der Merwe, dislocations at the phase boundary.

If the orientation of the precipitate and thematrix are uniquely related, so that they have alow-index orientation relationship, usually dictated by interfacial energy terms, then themismatch in lattice constants of thematrix and precipitate give rise to interference effects inthe image which are associated with double diffraction. This doubly diffracted beam willresult in a nonlocalized set of interference fringes, termed moir�e fringes (Figure 4.26),whose spacing depends on themismatch between the two operating g vectors for the matrixand the precipitate phase reflections:

dmoir�e ¼ d1d2

½ðd1�d2Þ2 þ d1d2b2�12

ð4:22Þ

where b is the angle of rotation between the two g vectors and |g|¼ 1/d.It is not difficult to distinguish between the moir�e fringes and the interface, Van der

Merw�e, dislocations. The latter are fully localized and have Burgers vectors which can beanalysed by comparing the dark-field images from different reflections in the diffractionpattern, while the former are nonlocalized in the image plane and have a spacing thatdepends only on the operating Bragg reflection.

Moir�e fringes may be caused by either a difference in lattice spacing, or a rotationbetween the twodiffracting regions. It follows thatmoir�e effectsmay also be observedwhendislocation tilt or twist sub-boundaries are present in the thin crystalline film. Once again,diffraction contrast due to the dislocation strain fields must be distinguished from moir�econtrast due to lattice mismatch either side of the sub-boundary.

4.5 Dynamic Diffraction and Absorption Effects

The kinematic theory of contrast is a poor approximation for all but the thinnest of thin-filmcrystalline specimens. We will not attempt a full description of the theory of dynamicdiffraction, but rather limit ourselves to a simplified physical explanation of its significance.We will also outline some of the more important conclusions that affect diffraction imagecontrast. One consequence of dynamic diffraction has already been noted: the position of asmall defect, equivalent to a point dilatation, within the diffracting column of crystal in thethin foil determines whether or not bright-field image contrast is inverted with respect todark-field contrast. In the following discussion we assume that only one diffracting plane isactive, that is only the direct transmitted beam and one diffracted beam contribute to theimage. This condition is termed two-beam diffraction contrast.

The concept of only one diffracted beam contributing to the image can sometimes bemisleading, and some comments on the geometry for effective two-beam diffraction are in


order. The two-beam condition derives from the analytical solution for dynamic electrondiffraction. In the transmission electron microscope, the two-beam condition can beapproximated by first tilting the crystal so that the incident beam is exactly parallel to alow-index zone axis, and then tilting about an axis perpendicular to this zone axis, so that theKikuchi line from a diffracting plane is brought to intersect the corresponding diffractedspot (Figure 4.27). Under these conditions, s for this reflection will be zero, while s for all

(a)

(b)

(c)

Figure 4.26 Moir�e fringes resulting from the overlap of: (a) two lattices with differentd-spacings of the lattice planes; (b) two lattices with the same d-spacing, but with an angleof rotation (b) between them; (c) two lattices that have both different spacings and an angle ofrotation between the two sets of planes.


(242)

(202)

(202)

(224)

(242)

022

202

(242)

(202)

(202)

(224)

(242)

022

202

Figure 4.27 A Kikuchi pattern from a FCC structure first oriented in a [1 1 1] zone axis, aftertilting the specimen such that s¼ 0 for g ¼ 02�2. A two-beam, dark-field diffraction contrastimage can be formed from the 02�2 reflection, or a two-beam bright-field image can be formedfrom the central spot.


the other reflections will have a nonzero value. The selected planewill now diffract farmorestrongly than the other planes on the same zone-axis. A diffraction contrast image can nowbe formed from either the central beam (bright field) or the diffracted beam for which s¼ 0(dark field). Under these two-beam conditions the solution for the intensity of the diffractedbeam is given by:

Ig ¼ðpt=xgÞ2sin2ðptseffectiveÞ

ðptseffectiveÞ2ð4:23Þ

where

seffective ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þð1=x2gÞ

qð4:24Þ

and xg is the extinction distance for the selected reflection g. The extinction distance xgdepends on the volume of the unit cell of the crystal, V, the angle of the Bragg reflection yB,the wavelength of the electron beam l, and the structure factor Fg for the selectedreflection g:

xg ¼pVcos yB

lFgð4:25Þ

Table 4.1 compares the extinction distances for the first few prominent reflectionscalculated for aluminium and gold at 100 kV. For normalized intensities, the intensity ofthe central spot I0 is 1� Ig.

The phenomenon of dynamic electron diffraction is to some extent analogous to theoptical wave propagation of polarized light in anisotropic materials that was discussedearlier (Section 3.4.3). The incident electron beam entering the thin-film crystal is splitinto two propagating beams of slightly different wavelength while passing through thecrystal lattice. In the case of electrons, it is the electrical potential inside the crystal thataffects thewave vector (thewavelength), and an electron beam incident at the Bragg anglepropagates through the crystal as two waves, one with its probability maxima peaked atthe atomic positions and one with its maxima between these positions. The smalldifference in electrical potential experienced by these two waves results in a slightlydifferent electron wavelength, and hence on a difference in their phase angle thatincreases with increasing path length through the crystal (increasing crystal thickness).For a given reciprocal lattice vector, this phase difference will lead to extinction of thedirect transmitted beam at some critical thickness (Figure 4.28). This critical thickness,the extinction thickness xg for that reflection, corresponds to a phase difference of pbetween the two beams as they exit the crystal. At the same time, the intensity transferred,corresponding to the energy scattered into the diffracted electron beam from the directtransmitted beam, also behaves as twowaves of slightly different wave vector propagatingthrough the thickness of the crystal, and will lead to a diffraction maximum at this sameextinction thickness, thus conserving the total energy in the diffracted and directtransmitted beams that exit the thin film sample. In other words, it is assumed that noinelastic scattering events have occurred.

Within the crystal, solutions for thewave equations for the incident and diffracted beamscorrespond to fourwaves. These are termed the Blochwaves, two of which propagate as the


direct transmitted beam, while the other two propagate as the diffracted beam. These twopairs of waves are approximately p/2 out-of-phase, and between each wave pair there is asmall difference in wavelength, caused by the oscillating periodic potential in the crystallattice. This small difference in wavelength generates a beat pattern of standing wavesinside the crystal that result in a thickness-dependent series of complementary maxima andminima in the amplitudes of the direct and diffracted beams. On leaving the crystal the twopairs of four waves recombine into two electron beams, one corresponding to the directtransmitted beam and the other to the diffracted beam (Figure 4.29).

Typical values for the extinction thickness at 100 kV vary from of the order of 20 to100 nm, depending on the atomic number of the material (the electron density) and thereciprocal lattice vector. The most densely packed planes have the strongest scatteringpower and therefore the shortest extinction distance. In two-beamconditions, and providingthe sample can be assumed to be perfectly flat, so that no bending can change the local valueof s, then s¼ 0 and:

Ig ¼ sin2ðpt=xgÞ ð4:26Þ

Thus Ig¼ 0 at t¼ 0 and at Ig¼ xg. Using values of the extinction distance, the thicknessof the thin-film sample can be estimated from the dark field or bright-field image(Figure 4.30). If the thickness is constant, but bending of the sample results in localchanges of s, then bend contourswill be visible in both the bright-field and dark field images(Figure 4.31).

Intensity Intensity1 001

Thi

ckne

sst

(1/2)ξ g

ξ g

(3/2)ξ g

Ig I0

Figure 4.28 The intensity of the diffracted beam Ig and transmitted beam I0 as a function ofcrystal thickness. The periodicity is determined by the extinction distance xg.


The information that can be derived from two-beam diffraction is limited, and it is oftenan advantage to tilt the sample so that a high symmetry zone axis is parallel to the incidentbeam.When this is done accurately, the crystal is aligned parallel to a low-order Laue zoneand several diffracting planes will generate reflections simultaneously, a condition termedmulti-beam diffraction. Since more than one lattice reflection now contributes to thediffraction contrast from the incident beam, an anomalously low extinction distance willresult, leading to especially strong sensitivity to lattice defects. Such conditions areexceptionally favourable to the analysis of the local strains associated with small defectclusters and the early stages of precipitation (Figure 4.32).

So far we have assumed that all the scattering interactions of the electrons passingthrough the thin-film sample are elastic, but for samples whose thickness approaches andexceeds the extinction thickness this is not strictly true. The probability of an inelasticscattering event depends on which of the Bloch waves in the crystal are being considered.Inelastic scattering is associated with those Bloch waves that have their amplitudemaxima at the atomic positions, thus maximizing the probability of inelastic scattering,while those Bloch waves whose amplitude minima fall at the atomic positions cannotinteract inelastically with the atomic nuclei. It follows that inelastic absorption of theBloch waves is, to a good approximation, confined to just one member of each of the twopairs of Bloch waves travelling through the thin film. Therefore, in a sufficiently thickcrystal, the contrast oscillations associated with thickness gradually fade out, since only

Figure 4.29 The Bloch wave model for dynamic diffraction. Electron energy is transmittedthrough the crystal as two pairs of waves with a small wavelength difference within each pair.The two wave pairs have their amplitude maxima at and between the atomic positions.Amplitude (energy) is transferred from one pair of Bloch waves to the other as the electronspropagate through the crystal. On leaving the crystal each pair of Bloch waves recombines toform the direct beam and the diffracted beam.


the waves travelling ‘between’ the atoms are transmitted. It follows that, for thickersamples, the intensity scattered into the diffracted wave by a defect that is near the centreof the foil depends only on the deviation from the Bragg condition and not on the foilthickness. Hence in thick crystals the oscillations in contrast along a dislocation runningat an angle through the foil are restricted to the regions at the top and bottom surfaces, andare absent near the centre.

4.5.1 Stacking Faults Revisited

We noted previously, when discussing the contrast from a point dilatation, that this contrastdepended on the position of the defect in the foil and whether the contrast was observed inbright or dark field. The case of a stacking fault observed at an inclined angle in a thick foilprovides a clear and instructive example of the way that dynamic effects (the extin-ction thickness) and absorption effects (the removal of one of the Bloch waves from the

Figure 4.30 Schematic representation of the variation in intensity of a dark-field Ig and bright-field I0 image as a function of specimen thickness and extinction distance.


transmitted beam) determine the diffraction contrast observed in the microscope(Figure 4.33).

When the stacking fault is near the bottom of a thick foil the secondBloch wave (2), withits amplitude peak maxima at the atomic sites, in the diffracted beam from the region ofcrystal above the fault is absorbed, so that only the first Bloch wave (1), with its peakmaxima between the atomic sites, is available to be scattered into the diffracted beam belowthe fault (Bloch wave 20). The two beams exiting the crystal are then the first Bloch wavefrom the region above the fault (Bloch wave 1) and the second Bloch wave from the regionbelow the fault (Bloch wave 20). When the stacking fault is near the top of the foil the firstBloch wave above the fault is scattered (Bloch waves 1 and 2), and both beams are thentransmitted, with the appropriate phase shift 2pg �R, through the crystal below the fault(Bloch waves 10 and 20). However, both beams 2 and 20 are now absorbed, since both havetheir maxima at the atomic sites, so that the two beams exiting the crystal are now 1 and 10. Itfollows that the bright-field and dark-field images of a stacking fault will be complementarywhen the fault is near the bottom of the foil (black stripes in the dark-field imagecorresponding to white stripes in the bright-field image), but identical when the fault isat the top of the foil. Similar arguments can be applied to strain contrast from small

Figure 4.31 Bright-field micrograph of a thin film of sapphire. Residual stresses due to theprocess used to form the film cause local elastic bending that generates bend contours in theimage which reflect the symmetry of sapphire.


Figure 4.32 Contrast from precipitate nuclei observed in a specimen of uniform curvature inthe region of a low order Laue zone (a symmetry axis). Reproduced from Hirsch et al., ElectronMicroscopy of Thin Crystals, published by Butterworth.

Figure 4.33 The contrast from a stacking fault in bright- and dark-field images results from acombination of extinction interference and absorption (see text).


precipitates or dislocations and used to rationalize the contrast observed and to identify therelative positions of the lattice defects in the foil thickness.

4.5.2 Quantitative Analysis of Contrast

Enough has been said to make it clear that the qualitative analysis of diffraction contrastrequires some special care just to ensure that the sign of the displacement field associatedwith a defect is correctly identified. It is important to image the same defects in both brightfield and dark field, to obtain images frommore than one reflection, and to check the effectof reversing the sign of s, the deviation from theBragg condition, by tilting the samplewhileunder observation. It is also important to verify the calibration of the electron optics in themicroscope, in particular any inversion or rotation of the image that may occur with respectto the diffraction pattern.

Quantitative diffraction contrast analysis requires considerable additional knowledge,well beyond the scope of this text. However, it should be possible to analyse displacementssemi-quantitatively. This includes the sign of the coherency strains around point defectclusters due to vacancies, interstitial defects or solute atoms, as well as any anisotropyassociated with these strains. It also includes complex dislocation and partial dislocationinteractions. These may be associated with plastic flow mechanisms or observed in sub-grain boundaries and semicoherent interfaces between epitaxially related phases. Ingeneral, insufficient effort is made to interpret diffraction contrast quantitatively, consid-ering the ability of modern computer modelling to check any analysis of the contrast bycomputer simulation.

4.6 Lattice Imaging at High Resolution

Phase contrast imaging of crystal lattices and lattice defects is well within the reach ofcommercial transmission electron microscopes and we have described the main features ofthis mode of image formation, especially its sensitive dependence on the physicalparameters of the microscope, as summarized by the CTF of the microscope. In whatfollows we now concentrate on the relationship between the observed lattice image and thecrystal structure.

4.6.1 The Lattice Image and the Contrast Transfer Function

The CTF summarizes the phase shifts introduced by the imaging system of the microscopeinto thewave function of the electrons forming the image in the plane of image observation:a fluorescent screen, a photographic recording emulsion or a CCD. The phase shifts xassociated with this imaging system is expressed as sinx, and the CTF gives sinx as afunction of the wave number or spatial frequency, which is proportional to the scatteringangle subtended by that frequency at the objective aperture and commonly measured inreciprocal space as 1/d (in units of nm�1) (Figure 4.18). The final phase and amplitude foreach spatial frequency is obtained by convoluting the CTF with the calculated phase andamplitude of the electrons transmitted through the sample. These phase shifts andamplitudes are calculated at the exit plane of the thin-film object using the dynamical


theory of electron diffraction, before being ‘transmitted’ through the imaging system of themicroscope, using the CTF, until they interfere in the image plane. The intensity at anyposition in the image plane is obtained bymultiplying the integrated image amplitude by itscomplex conjugate.

The factors that enter into the CTF include parameters that are associated with both theelectron beam source and the objective lens. The source parameters are its coherency andenergy spread. The relatively large dimensions of a LaB6 crystal electron emitter result inpoor coherency for this source. In contrast, a field emission gun has a very small emittingdiameter with excellent coherency, approximating a point source, as well as a very smallenergy spread. The traditional tungsten filament is inferior on both counts, since theeffective source diameter is quite large and the operating temperature very high. Anadditional parameter is the angular spread (the divergence) of the incident beam, which isdetermined by the condenser system and the beam focusing conditions.

The objective lens parameters include the spherical aberration coefficient, as well as thecurrent stability in the lens. Additional corrections are concerned with second- or third-order lens defects. Astigmatism (a second-order defect) is readily eliminated by a suitablecorrection, andmodern correction systems can also remove coma (a third-order defect). Thesingle most important parameter limiting the point resolution of the transmission electronmicroscope is the spherical aberration coefficient of the objective lens. Commerciallyavailable electrostatic spherical aberration correctors offer excellent hope that this limita-tion on the available resolution will soon be removed. At the time of writing, no completesolution is currently available for correcting chromatic aberration. However, the spread inwavelengths (the energy spread) of the electron beam can be significantly reduced byadding a monochromator beneath the electron source. All other performance-limitingfactors can be corrected to better than this limit.

4.6.2 Computer Simulation of Lattice Images

The lattice image in phase contrast is often recorded at or close to the Scherzer focus, theunder-focus that maximizes the spatial frequency for which the electron microscopeimaging system introduces oscillations into the sign of the phase shift (Section 4.3.3).However, the periodicity in the lattice imagewill be preservedwhen a series of images arerecorded while the focus is changed incrementally, and it is only the intensity at eachimage point that changes. These intensity changes can include contrast reversal at defocusvalues below the Scherzer focus, the doubling of characteristic periodicities in the lattice,and apparent changes in the prominence of specific crystallographic directions. It iscommon to observe rows of bright or dark maxima in one lattice direction being replacedby similar rows in a different direction. Interpretation of such observations is notstraightforward.

There is a temptation to use both prior knowledge of the crystal structure and the observedelectron diffraction patterns, in order to assign specific features of the sample crystallog-raphy directly to prominent features in the lattice image. However, some practicalexperience of the sensitivity of lattice image contrast to the defocus of the objective lensand the sample thickness should quickly convince any scientist of the dangers of‘recognizing’ features that confirm preconceptions, while ascribing any discrepancies toimaging artifacts.


The only acceptable scientific procedure is to employ computer simulation to evaluatethe image, and several software packages are now available for this purpose. For successfulcomputer analysis of image contrast, three conditions must be satisfied:

1. Accurate calibration of the microscope parameters must have been reliably achieved,especially the determination of the spherical and chromatic aberration coefficients andthe beam divergence.

2. A contamination-free, uniformly thinned sample should be available, preferably nothicker than the extinction thickness for the preferred imaging reflections.

3. Accurate alignment of the optic axis of the microscope along a prominent zone axis ofthe sample must be made, with an angular objective aperture that accepts as manyreflections into the imaging system as is compatible with the critical spatial frequencies(compare the Scherzer focus, Section 4.3.3.).

If this information can be supplemented by a reasonably accurate estimate of the filmthickness, then so much the better. In many cases film thickness is assumed to be a variableduring image simulation in order to estimate the error associated with an imperfectknowledge of this parameter.

Amodel for the unit cell of the crystal lattice must be inserted into the computer softwareprogram, including the positions of all atoms and their expected occupancy. Lattice defects,such as stacking faults, APBs and dislocations, can be simulated by inserting suitabledisplacement fields in the model lattice. Boundaries are commonly simulated by placingthem accurately parallel to the optic axis, and can also be inserted in the computer modelusing a transformation matrix for the region of the crystal beyond the boundary plane.

Although visual matching of simulated to recorded images has often been employed todecide the ‘best-fit’ simulation, it is far more reliable to derive a ‘difference’ image, inwhich the simulated intensities for each pixel are compared using a computer program,withthe intensities for a through-focus image series recorded by a CCD camera.

4.6.3 Lattice Image Interpretation

A common assumption is that the lattice image ‘shows you where the atoms are’, but this isincorrect and often very misleading.

In the first place, as in any magnified image, the lattice image is recorded in twodimensions, as a periodic pattern of varying intensity. The amplitudes leading to theseintensity variations can be deduced, but the phase information has been lost. Although thephase information can be partially recovered from one or more images recorded in athrough-focus series, this is still a relatively uncommon procedure. Usually it is a singleimage recorded at or close to Scherzer focus that is published in the literature. A latticeimage interference pattern is not localized in space and, as noted previously, is a sensitivefunction of both the objective lens defocus and the sample thickness, in addition to other,less critical parameters. Provided that the imaging conditions are well-known and that thecomputer-simulated image is a goodmatch to the observed image, then the conclusion that amodel lattice used for computer simulation of atomic positions, occupancies and periodi-cities may be a good description of the microstructural morphology is justified.

However, there are still serious problems, primarily due to the nonlocalized nature of thelattice image. For example, it is common to superimpose a unit cell of the model lattice on a


lattice image and identify ‘channels’ along the optic axis of the microscope with blackpatches while rows of high atomic number atoms along the same axis are assumed to be‘seen’ as white patches. Such interpretations will not be accepted by the knowledgeableexpert. An excellent example is the apparent structure of many phase boundaries. These areusually complicated by the differences in the lattice potentials of the two phases. Thisintroduces a discontinuity in ‘refractive index’ that results in a shift in the interferencepattern of one phase with respect to the other, perpendicular to the boundary plane. Thismakes it extremely difficult to deduce the actual atomic displacements at the boundary, eventhough such displacements may be clearly ‘observed’ in the lattice image!

Nevertheless, themorphological information that can nowbe derived from lattice imagesof both phase and grain boundaries is extraordinary in its atomic detail (Figure 4.34), and theneed to make this information quantitative has lead to remarkable improvements in themethods of computer simulation and evaluation of digitally recorded lattice images. Inmostcases, it is now possible to optimize a model for any crystalline nanostructure observed bylattice imaging in the transmission electron microscope.

Figure 4.34 Some examples of lattice images from grain and phase boundaries: (a) anamorphous glassy layer in equilibrium at the surface of sapphire that had been coated withnickel prior to specimen preparation; (b) a layer of calcium cation segregation at a twinboundary in alumina; (c) a twin boundary in B4C. The inset is a magnified view of theboundary region; (d) The intersection of a stacking fault (SF) with twin boundaries in B4C.


4.7 Scanning Transmission Electron Microscopy

While conventional TEM makes use of a (nearly) parallel electron beam incident approxi-mately normal to the plane of the sample, it is also possible to focus the electron beam, at amuch larger convergence angle, into a focal spot on the sample. We can now collect thetransmitted signal as a function of the beam location when it is rastered across the sample.This mode of operation is termed scanning transmission electron microscopy (STEM). Theray optics for STEM are identical to TEM, and can be easily understood by turning thetransmission electronmicroscope ‘upside down’, so that the electron source is at the bottomand the detector at the top. In Figure 4.35 the wave at point B due to a point source at A isidentical to the wave at A due to a point source at B. The STEM detector replaces the TEMelectron source, and the STEM electron source is placed in the detector plane of thetransmission electronmicroscope. An additional scanning system uses the deflector coils toraster the focused beam across the surface of the specimen.

Figure 4.35 Schematic drawing illustrating the principle of reciprocity in electron optics. Theray diagram for STEM operation is just the inverse of that for TEM.


There are a number of detectors employed for STEM imaging, and these are shownschematically in Figure 4.36. The first is a solid-state, bright-field detector that counts thenumber of electrons per unit time as a function of the position of the focused beam on thesample. The scattering angle is relatively small, y1< 10mrad. Due to the principle ofreciprocity, the diffraction contrast in bright-field TEM and STEM are quite similar. Inaddition, in STEMmode mass–thickness contrast can provide an important contribution tothe image. By adding an annular detector, forward-scattered electrons can be collected overa larger scattering angle, y2> 10–50mrad, and we can acquire dark-field STEM images. Inaddition, electrons scattered to even larger angles (y3> 50mrad) can be detected using ahigh-angle annular dark-field (HAADF) detector. The advantage of HAADF-STEM is thatalmost none of the elastically diffracted electrons reach the HAADF detector, and thecontrast is due to inelastically scattered electrons (see, for example Figure 4.37). Suchimages are often termed Z-contrast images, since there is now a direct correlation betweenthe local contrast and the local mass–thickness, which depends on the value of the atomicnumber, Z. The mass concentration may even be estimated by correlating the localconcentration using a suitable expression for inelastic scattering. If the STEM includesa field emission gun source with a probe size less than �0.3 nm, atomic resolutionZ-contrast is obtainable, with a direct correlation between local intensity and the massconcentration within the atomic columns of the sample.

BFDetector

Converged IncidentElectron Beam

Sample

ADFDetector

HAADFDetector

θ1θ2

θ3

ADFDetector

HAADFDetector

Figure 4.36 Schematic drawing of bright-field (BF), annular dark-field (ADF), and high-angleannular dark-field (HADF) detectors for STEM mode imaging.


Summary

In the transmission electron microscope high energy electrons are elastically scattered asthey penetrate a thin specimen. The transmitted electrons are then focused by electromag-netic lenses to form a well-resolved image that can be viewed on a fluorescent screen orrecorded on a photographic emulsion or, more commonly today, a charge-coupled device.The increasing availability of field emission guns for the source of the electron beam hasgreatly improved the performance of the modern transmission electron microscope.

The wavelength of the energetic electrons used in electron microscopy is well below theinteratomic spacing in solids, so that atomic resolution is, in principle, possible, althoughnot always easy to achieve. The electromagnetic lenses used to focus the electron beam andform the image in the microscope occupy an appreciable proportion of the total optical pathlength and suffer from severe lens defects that limit the divergence angle of the beam for asharp focus to a small fraction of a degree. Themost important electromagnetic lens defectsare spherical and chromatic aberration, and astigmatism. The diffraction limit on theresolution improveswith increasing objective lens aperture angle,while the aberration limit

Figure 4.37 HAADF-STEM micrograph of the interface between a pure gold wire and analuminium pad, used to electrically connect a microelectronic device. The join is formed byapplying heat, pressure, and ultrasonic vibrations, which results in the formation of a 20 atom%Al metastable solid solution adjacent to the pad. Above this a two-phase region forms thatcontains Al–Au intermetallic phases with a mean concentration of 35 atom% Al. Since theatomicnumber ofAu is significantly larger than that ofAl, theAl appears dark and theAuappearsbrighter.


on the resolution deteriorates as the aperture angle is increased. The optimum objectiveaperture for the best compromise between these two limits is typically between 10�2 rad and10�3 rad, but new spherical aberration correctors should certainly increase this value. Theelectronmicroscope requires a vacuum in order to prevent scattering of the electron beam inthe microscope column. A high voltage stability for the electron source is also needed inorder to ensure amonochromatic beam, and an equally good current stability is required foroperation of the electromagnetic lenses, in order to maintain a stable focus.

Transmission electron microscope thin-foil specimens must be less than 100 nm inthickness to minimize inelastic scattering of the transmitted beam as it passes through thespecimen. Good specimen preparation is critical. Combinations of mechanical, chemicaland electrochemical methods are generally used, and the final stage of specimen prepara-tion ismost often by ionmilling, inwhich the surface atomic layers are eroded by sputtering,using an incident beam of inert gas ions. To avoid surface charging of thin foils preparedfrom insulator materials it is often necessary to deposit an electrically conductive coating.The morphology of some specimens may be best studied by making a thin replica of thesample surface and using the replica as the thin-foil transmission specimen.

Contrast in the transmission electron image may be associated with differences in themass–thickness of the sample, coherent diffraction of a portion of the incident beam out ofthe objective aperture, or phase contrast that results from interference in the image planebetween two or more of the coherently diffracted beams, with or without the directlytransmitted beam. In biological samples, which are usually amorphous, mass–thicknesscontrast is the most common source of image information, and a variety of heavy-metalstaining agents have been developed to enhance the contrast in soft tissue sections andcellular samples. Crystal lattice defects are generally imaged by diffraction contrast, sincelattice defects strongly affect the amplitude that is scattered out of the incident beam.Periodic crystal lattices, however, can be successfully imaged by phase contrast if themicroscope can be operated with sufficient resolving power. Changes in lattice periodicityat interfaces can also be detected by phase contrast and interpreted when suitable computermodelling procedures are employed.

A simple, qualitative treatment of diffraction contrast is possible using kinematicdiffraction theory, in which the dynamic effects of multiple scattering are ignored. Thediffracted amplitude of the electron beam in the kinematic theory depends on the sumof justtwo terms. The first describes the effect of deviations from the Bragg condition on theamplitude that is scattered by a unit cell in a thin column of crystal in the foil sample, whilethe second describes the effect of displacements from the ideal lattice position, due to thepresence of the defect, on the diffraction amplitude. These contrast effects are mostconveniently summarized by an amplitude–phase diagram that can be used to explain,qualitatively if not quantitatively, the contrast that is observed from stacking faultsand antiphase boundaries, edge and screw dislocations, as well as from the strain fieldsthat are associated with very small precipitate nuclei and radiation damage clusters of pointdefects.

Dynamic diffraction theory, when combined with the inclusion of effects associated withabsorption (inelastic scattering), allows the observed diffraction contrast to be interpretedin much more detail, but requires a deeper understanding of electron diffraction theory,considerable expertise in the operation of the microscope and the careful interpretation ofthe image data.


The lattice or phase-contrast image is an interference image in which several coherentlydiffracted electron beams are recombined in the image plane. Since these beams havefollowed different paths through the electromagnetic lens fields, they experience phaseshifts that depend on the electro-optical properties of the electron source andthe electromagnetic lenses, and are described by the contrast transfer function for themicroscope. The periodicity in a lattice image is essentially independent of the specimenthickness and the focal plane of the lattice image, but these parameters cause grossvariations in the recorded image intensity, so that an observed image cannot be directlyinterpreted as a projection of the periodic crystal lattice of the thin-film sample. Neverthe-less a slightly under-focused image, taken at the Scherzer focus, compensates quiteeffectively for phase shifts that are associated with the lens system. Images taken fromvery thin specimens at the Scherzer focus therefore correspond approximately to thecontrast to be expected from a direct projection of the crystal lattice onto the image plane.However, much better agreement can be achieved by simulating the contrast from a latticemodel with the expected crystal periodicity. Computer image simulation has been usedsuccessfully to construct models of localized lattice structure to an accuracy that is betterthan a fraction of the interatomic spacing in the bulk crystal.

Bibliography

1. D.B. Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook forMaterials Science, Plenum Press, London, 1996.

2. P. Buseck, J. Cowley and L. Eyring (eds), High-Resolution Transmission ElectronMicroscopy and Associated Techniques, Oxford University Press, Oxford, 1998.

3. J.C.H. Spence, Experimental High-Resolution Transmission Electron Microscopy,Claredon Press, Oxford, 1981.

Worked Examples

We now demonstrate the transmission electron microscope techniques that have beendiscussed, using two quite different material systems: polycrystalline alumina and a thinfilm of aluminium deposited by chemical vapour deposition (CVD) on a TiN/Ti/SiO2-coated silicon substrate.

Our first example is polycrystalline alumina. As always, it is important to define thequestions we wish to ask before preparing specimens or selecting the microstructuralcharacterization techniques. For our alumina wewish to establish that the grain boundariesare free from secondary phases.

To study the detailed morphology of the grain boundaries the most suitable method isthin-foil transmission electron microscopy. For a bulk polycrystalline ceramic sample, theeasiest way to prepare a specimen is by first cutting a thin (600 mm) slice from the bulkspecimen with a diamond saw. TEM specimens are limited in diameter to 3.0mm, so a3.0mm disc must be trepanned from the 600 mm slice with an annular ultrasonic tool.Alternatively, a hollow diamond bit can be used to extract a rod from the bulk specimen. Thecircular disc must then be mechanically thinned to �80 mm by using diamond grinding


media. The region in the centre of the disc should now be thinned to perforation by ionmilling. To ensure that the perforation occurs at the disc centre, it is first ‘dimpled’ byfurther grinding to a thickness of�30 mm.Dimpling from both sides will ensure that a final,electro-optically transparent region will be available near the centre of the disc. Finally,the specimen is ion milled on both the top and bottom surfaces of the specimen forapproximately 60min at 5.0 kV, using argon ions at an incident angle of 6� (Figure 4.38).

Figure 4.38 Schematic drawing of the TEM specimen preparation process for a plan-viewbulkspecimen. After trepanning the sample is mechanically thinned to�100mm. The specimen canbe ‘dimpled’ on both sides, if additional mounting wax is added to support the specimen beforedimpling from the second side.


For TEMexamination the nonconducting alumina specimens are usually coated on one sidewith a �10 nm layer of carbon.

Figure 4.39 is a bright-field, 200 kV TEM micrograph of an alumina thin foil seen indiffraction contrast using a small objective aperture to select the direct transmitted beam.The polycrystalline alumina contains sub-micron SiC particles, and the Bragg contrastvariations are due to the changes in crystallographic orientation for each individual grain, aswell as a dislocationnetworkvisible in one of thegrains. Ifwe useKikuchi diffractionwe canalign a particular grain so that a low index zone axis is parallel to the incident beam. Thebright-field image will then show this grain in dark contrast. An example is given inFigure 4.40 which shows an AlNb2 particle located within an alumina grain. We can nowrecord an image using a selected diffracted beam from the aligned grain, a dark-field image,inwhich the diffracting grainwill appear lightwhen comparedwith the neighbouring grains.

Figure 4.39 Bright-field diffraction contrast TEM micrograph of an alumina polycrystalreinforced with SiC particles.


The detection of secondary phases at a grain boundary may be achieved using severaldifferent methods. In the first, high-magnification bright-field images are used. Secondaryphases having a different chemical composition form the matrix will give mass–thicknesscontrast and appear lighter or darker than the neighbouring regions. Figure 4.41 shows anexample for NbO particles at grain boundaries in polycrystalline alumina. An amorphousphase at the grain boundary or at a grain boundary triple junction can be highlighted in a

Figure 4.40 Bright-field diffraction contrast TEM micrograph of an AlNb2 particle within analumina grain. The AlNb2 particle is aligned along a low index zone axis, and thus has a darkcontrast in the bright-field image.

Figure 4.41 TEMmicrograph of a NbO particle located at a grain boundary in polycrystallinealumina. Phase contrast (lattice fringes) andmass–thickness contrast vary from the alumina grainto the NbO grain.


dark-field image. The dark-field image is recorded using diffusely scattered electrons fromthe amorphous region. When a glass-containing grain boundary is oriented accuratelyparallel to the incident beam, the glassy region will appear lighter than the neighbouringcrystalline material. If the TEM resolution is sufficient to detect the lattice planes ofalumina, we can also use phase contrast to study the grain boundary regions. To form alattice image of any given grain, the grain should be oriented to diffract with a low indexzone axis exactly parallel to the incident beam. To record a lattice image of the grainboundary region or of an interface between two phases, both of the boundary- or interface-forming grains should be aligned along low index zones, and the boundary must be parallelto the incident beam. This is a very rigid condition, and exceedingly difficult to obtain in apolycrystalline or polyphase sample unless a special orientation relationship exists betweenthe grains. Figure 4.42 shows a lattice image of an interface between an alumina grain and anickel particle, in a particle reinforced alumina matrix composite. Both phases are in a lowindex zone axis and the interface is parallel to the incident electron beam. As a result, thethin (0.9 nm) amorphous film that is present at the interface can be detected. In some casesinformation on the interface faceting and particle size of small secondary phases at grainboundaries is obtainable when the interface is parallel to the incident beam and at least oneboundary-forming grain is alignedwith a zone axis parallel to the incident beam.Figure 4.43is a lattice image of a SiC particle located within an alumina grain. A lattice image is visible

Figure 4.42 Lattice image of an interface between an alumina grain and a nickel particle. Bothphases are aligned along a low index zone axis with respect to the incident electron beam, andthe interface is parallel to the beam. A thin (0.9 nm) amorphous film at the interface is clearlyvisible.


for the alumina, while onlymoir�e fringes are visible in the area of the SiC particle, since it isnot oriented in a low index zone axis.

The combination of real-space images and reciprocal space images, that is, diffractionpatterns, is one of the main advantages of TEM. Figure 4.44 shows a bright-field diffractioncontrast micrograph of a gold particle which was equilibrated on the basal (0 0 0 1) surfaceof sapphire. Both the particle size and shape can be determined. By taking selected areadiffraction (SAD) patterns from the particle and the substrate, the relative orientation of thetwo phases can be determined. Since there is a direct relationship between the orientation ofthe SAD pattern and the real space image that is known from calibration, the interfaceplanes observed in the micrograph can be determined.

Nowwe return to the CVDAl system. For this examplewewish to characterize the initialdeposition conditions, when the first aluminium nuclei form on the surface of the TiN film,the morphology of the complete aluminium film, and the morphologies of the underlyingTiN and titanium layers.

TEM cross-sections are not easy to prepare, so we use this opportunity to review thespecimen preparation process. For TEM we need very thin specimens, with as much aspossible of the interface region thin enough for successful TEM investigation. There are anumber of methods available to thin bulk specimens, but for a multilayered materialchemical thinning is not a good idea, since the different layers will have different chemical

Figure 4.43 Lattice image of a SiC particle located within an alumina grain. The alumina isalignedwith a low index zone axis parallel to the incident electron beam, and is the source of theobserved lattice image. A moir�e pattern within the SiC particle is due to overlap between thealumina and SiC lattices in the plane of the thin foil.


potentials and will react differently to a chemical or electrochemical etch. Ion milling isreally the only option.

To ensure that the largest possible area of the interface region is at or near the perforationmade by ion milling, we glue four of our multilayer specimens together; one pair face-to-face, and the second pair face-to-back (as shown schematically in Figure 4.11). We need across-section for the TEM sample that is at least 3mm inwidth, so twomore silicon wafers,with a thickness of 300 mm, are glued to the composite specimen. We now use an ultrasonicannular drill to drill down the length of the cross-section, and then glue the resulting ‘rod’inside a copper tube with an outside diameter of 3mm. Once the glue has set, slices are cutfrom the end of the rodwith a diamondwafering saw. These slices aremechanically thinnedto less than 100 mm, dimpled at their centre to less than 20 mm, and then ion milled toperforation. By gluing the samplewafers face-to-face and face-to-backwe are able to locatethe interfaces of interest close to the centre of the 3mm diameter specimen. Combining ionmilling with dimpling improves the probability that perforation will occur in the centre ofthe disc, producing a specimen thin enough for TEM in the region of interest. It is a lot ofwork for just one specimen, but with some practice the time required to prepare a samplebecomes quite reasonable.

Figure 4.45 is a bright-field, diffraction contrast TEM micrograph of the as-preparedspecimen. The thick aluminium film is clearly visible in cross-section, as are the separategrains within this film. Open and closed voids are easily detected, and statistical analysisof both the grain size and the void density is possible. Higher magnification images(Figure 4.46) show the morphology of the TiN/Ti layers, which have a very small grainsize. The dominant crystal defects observable from Figure 4.46 are the grain boundaries,and this image contains important information on the average grain size of both the TiNand Ti. However, Figure 4.46 is a projection of a thick, three-dimensional slice onto a two-dimensional image, and the boundaries that are visible come from many differentgrains within the film. We discuss this problem of image overlap in some detail inChapter 9.

Figure 4.44 (a) Bright-field TEM micrograph of a gold particle thermally equilibrated on the(0 0 01) surface of sapphire (a-Al2O3). (b) The SAD pattern shows the low index orientationrelationship between the two phases, and allows the lattice planes parallel to the interfacebetween the two phases to be determined.


Figure 4.46 A higher magnification TEM micrograph of the interface region, showing somephase contrast and the morphology of the TiN and Ti layers.

Figure 4.45 Low-magnification bright-field micrograph of a cross-section specimen of Al onTiN/Ti/SiO2/Si. The individual Al grains, as well as open and closed voids in the Al film, areclearly visible.


Figure 4.47 is a high resolution TEMmicrograph of the same aluminium grain recordedafter the grain had been oriented into a [1 1 0] zone axis. The aluminium film has a strongh1 1 0i texture, and the {1 1 1} planes are predominately parallel to the original interface.There are several reasons for the variations in contrast observed in the aluminium grain inFigure 4.47. First, changes in thickness from the interface region to the edge of the specimenresult in changes in contrast that are due to thickness extinction effects. In principle, we canuse a computer program to determine the specimen thickness for any region by comparingan experimental image to a simulated image, providing we know the defocus value of theobjective lens and have a defocus–thickness ‘map’ in order to determine the two indepen-dent variables (Figure 4.48).

Additional sources of contrast variations in the lattice image in Figure 4.47 areassociated with local bending, due to residual stresses, or to the presence of dislocations.Unusually, stacking faults are also visible in the aluminium grain near the interface.These faults lie on the {1 1 1} planes, some of which are parallel to the interface. Thefaults were most likely formed during deposition of the film, and are expected to influencethe electrical properties. Figure 4.49 is a higher magnification high resolution TEM imageof the same interface region shown in Figure 4.46. The lattice image from a [1 1 0] zoneaxis projection of the aluminium lattice is visible together with individual grains from theTiN layer.

Figure 4.47 A very fine lattice image of aluminium aligned along a [110] zone axis.


Problems

4.1. Electromagnetic lens systems employ very small angular apertures, primarilybecause of their large spherical aberration. Explain why the spherical aberrationand other lens defects limit the resolving power of a transmission electronmicroscope.

4.2. Sketch a graph of the dependence of electron wavelength on the accelerating voltagein the electron microscope. What factors prevent the development of ultra-highvoltage (>1MeV) electron microscopes for ultra-high resolution?

4.3. Distinguish between mass–thickness contrast and diffraction contrast in electronmicroscopy.Whenmight you expectmass–thickness to dominate the contrast in thin-film specimens?

4.4. List the experimental and specimen parameters which affect diffraction contrast.Define precisely each parameter given in your list.

Figure 4.48 Simulated defocus–thickness map for aluminium aligned on a [110] zone axis.The map is ‘contrast invariant’, meaning that the contrast of each individual simulation has notbeen changedwhen the imageswere ‘pasted’ together. This results in the thinner regions havinga lower contrast than the thicker regions, an effect that is not due to absorption.


4.5. Sketch the kinematic amplitude–phase diagrams for a stacking fault and for adislocation line in a crystalline thin-film specimen. Show for both cases the effectsobserved when the defect either increases or decreases the phase shifts betweenneighbouring atomic columns of crystal.

4.6. Explain the diffraction conditions under which lattice defects will fail to givediffraction contrast in thin-film electron microscopy. Give examples for (a) stackingfaults in the FCC crystal lattice and (b) screw dislocations in a BCC metal lattice.

4.7. Under what conditionsmight a phase contrast, lattice image be expected tomirror theatomic positions in the unit cell of a crystal? What are the problems associated withmaking such an assumption?

4.8. Outline the steps required to prepare a representative TEM specimen from thefollowing samples: (a) tungsten light bulb filaments; (b) a pinch of talcumpowder; (c)large steel bolts, (d) brazed vacuum seals.

4.9. According to the Rayleigh criterion, how will the resolution in TEM change withaccelerating voltage? (Neglect aberrations and assume a constant convergenceangle.) Explain how the contrast might be expected to change, using the sameassumptions.

Figure 4.49 Lattice image of the Al–TiN interface region.


4.10. Explain the difference between the structure factors for X-rays and electrons.

4.11. What are the physical principles behind the operation of the vacuum pumps used forelectron microscopes? (Try to include rotation, diaphragm, scroll, diffusion, turbo,turbo-drag, and ion pumps in your discussion.)

4.12. Assuming the resolution of a TEM can be expressed as d2Total ¼ d2d þ d2s ; wheredd is the Rayleigh diffraction limit and ds is the resolution limit determinedby the spherical aberration coefficient, calculate the optimal convergence angleof the objective lens a*. Using this value for a*, what is the expected resolu-tion for a 200 kV microscope with a spherical aberration coefficient of1.5mm?

4.13. Compare the point resolution defined by the Rayleigh criterion with the Scherzerresolution. Assume a 200 kV TEM with Cs¼ 1.5mm and the optimal value of theconvergence angle. Repeat your calculation for Cs¼ 0.6mm, and then again for300 kV. Summarize your conclusions.

4.14. Two-beam diffracting conditions are used during the study of dislocations in a FCCmetal structure. Which reflections should give zero contrast for the case of aa=2½�101� dislocation?

4.15. Figure 4.50 is a dark-field TEM micrograph of the edge of an aluminium samplerecorded under two-beam diffracting conditions, and a selected area diffraction(SAD) pattern of the grain which contains the reflection used to form the two-beamimage. Solve the diffraction pattern. Use the thickness fringes in the image and thedata in Table 4.1 to estimate the thickness profile of the sample. How would yourresults be changed if the material was gold rather than aluminium?

4.16. A sample made of pure (99.999%) aluminium was thermally annealed, causinggrain growth. Figure 4.51 shows a SAD pattern from a grain adjacent to a grainboundary in the sample (SAD1), and a sketch of the bright-field image from thegrain boundary region (BF1). After tilting the sample by 18� 2�, a second SADpattern (SAD2) and accompanying bright-field image sketch (BF2) wereacquired. Both sketches have been correctly oriented with respect to the SADpatterns.

(a) Indicate which grain has been aligned with a low-index zone axis parallel to theincident beam.

(b) Solve both SAD patterns.(c) Identify the facet planes that define the grain boundary.(d) Give a rough estimate for the thickness of the sample based on the two sketches

and the data given.

4.17. Figure 4.52 shows a series of diffraction patterns acquired from an aluminium grainwhich contains a single dislocation. After recording the first pattern (SAD1) thesample was tilted 36� 2� to acquire the second pattern (SAD2). The sample wasthen tilted a further 22� 2� to obtain the third pattern (SAD3). On each of these


Figure 4.50 (a) Dark-fieldmicrograph from the edge region of an aluminium thin-film sample.(b) SAD pattern showing the reflection that was used to record the dark-field image under two-beam conditions.

Table 4.1 Extinction distance for prominent lattice planes in aluminium and gold at 100 kV.

Material Extinction distance for reflection hkl (nm)

110 111 200 220 400

Al — 56.3 68.5 47.3 76.4Au — 18.3 20.2 27.8 43.5


three diffraction patterns a reflection used to form a two-beam image is indicated forwhich the dislocation contrast was absent.

(a) Solve all three diffraction patterns.(b) Determine the Burgers vector of the dislocation. What reasonable conclusion

can you reach about the line sense of this dislocation?

4.18. A SAD pattern from a single grain of copper (FCC, a¼ 0.3615 nm) is given inFigure 4.53. It is possible that a twin boundary exists in the same grain. Assume thatthe twin boundary plane is {1 1 1}.

(a) Given that your goniometer can be tilted up to �35�, find the nearest zoneaxis for which the twin boundary will be parallel to the incident electronbeam.

(b) What angle and axis of tilt are required to reach this orientation?(c) Sketch the SAD pattern expected for the zone axis you have chosen.

SAD1(a)

(b) SAD2

BF1

BF2

10 nm

10 nm

Figure 4.51 (a) SAD pattern from a grain that is adjacent to a grain boundary in aluminium(SAD1) and a bright-field image sketch from the grain boundary region (BF1). (b) After tilting thesample by 18� 2�, a second SAD pattern (SAD2) and the accompanying bright-field imagesketch (BF2) were acquired. Both sketches are in the correct orientation with respect to the twodiffraction patterns.


4.19. Five lattice images of aluminium taken froma crystal alignedwith a [0 0 1] zone axisparallel to the incident electron beam are given in Figure 4.54. Each image isrecordedwith a different objective lens defocus value that is indicated in the CTF. Inthe first image the positions of the atoms in a unit cell are marked.

(a) Identify the planes (220) and (200) on the images.(b) Given that the lattice parameter of aluminium is 0.405 nm, determine the image

magnification.(c) Explain why the contrast at the positions of the atoms changes from white to

black across the defocus series.

4.20. A SAD pattern from a twinned region in copper is shown in Figure 4.55. A sketch ofthe area from which the SAD pattern was recorded is shown in the correctorientation in Figure 4.56.

SAD1(a) (b)

(c)

SAD2

SAD3

Figure 4.52 A series of diffraction patterns acquired froman aluminiumgrainwhich containsa single dislocation in the region imaged. After recording the pattern SAD1 (a), the samplewas tilted 36� 2� to acquire the pattern SAD2 (b). The sample was then tilted a further22� 2� to generate the pattern SAD3 (c). On each diffraction pattern the reflection is indi-cated that was used to form a two-beam image in which contrast from the dislocationwas absent.


(a) Solve the diffraction pattern and define the orientation relationship between thetwo grains on either side of the twin boundary. What is the boundary plane?

(b) The sample was tilted into a new zone axis, and the resulting SAD pattern isshown in Figure 4.57.A sketch of the bright-field image is shown in Figure 4.58.Estimate the approximate thickness of this sample, explaining any assumptionsthat you have made.

4.21. A Fe–Ni alloy was investigated by thin-film TEM. A SAD pattern was recordedfrom a g-FeNi grain in the thin film. This grain is indicated in Figure 4.59 on bright-field and dark-field images recorded with the sample aligned on a low-index zoneaxis. The SAD pattern for the grain is shown in Figure 4.60. The structure of g-FeNiis an FCC solid solution with four atoms per unit cell (a¼ 0.35871 nm).

(a) Solve the SAD pattern.(b) Which image is bright field and which image is dark field? Why?(c) Explain the changes in contrast that are observed across the two images.

4.22. In a TEM study of polycrystallinemolybdenum (BCC, a¼ 0.3147 nm), a low-anglegrain boundary was found. The boundary geometry is sketched in Figure 4.61. Inorder to identify the dislocations composing the boundary, two-beam diffraction

Figure 4.53 SAD pattern from a single grain of copper.


images were recorded. The first image was obtained after orienting the sample toalign a [1 0 1] zone axis parallel to the incident electron beam. The SAD pattern isshown in Figure 4.62. Use of reflection g1 resulted in the sub-boundary dislocationsgoing out of contrast. The sample was then tilted by 18.43� into a new zone axis,

Figure 4.54 Five lattice images of aluminium aligned on a [001] zone axis and the CTFat eachdefocus value used to record the images.


200 nm

MatrixTwin

Tw

inbo

unda

ry

Hole

Figure 4.56 Sketch of the image from which the SAD pattern in Figure 4.55 was recorded.

Figure 4.55 SAD pattern of a twinned region in copper.


200 nm

MatrixTwin

Tw

in b

ound

ary

Hole

Figure 4.58 Sketch of the bright-field image recorded after tilting the sample into the newzoneaxis shown in Figure 4.57.

Figure 4.57 SAD pattern of the same sample after tilting into a new zone axis.


Figure 4.59 Bright-field and dark-field images recorded from an Fe–Ni alloy.

Figure 4.60 SAD pattern recorded from a grain of the g-FeNi solid solution and indicated onthe micrographs given in Figure 4.59.


from which a two-beam image was recorded using the reflection g2 shown in thesecond SAD pattern (Figure 4.63). Again, no diffraction contrast was detected fromthe sub-boundary dislocations under these diffracting conditions. The sample wasthen tilted to bring the boundary parallel to the incident electron beam.

Grain A Grain B

h = 15.6 nm

dislocation lines

Figure 4.61 Schematic drawing of the low-angle grain boundary found by TEM.

d = 0.2225 nm

g1

Figure 4.62 SAD pattern recorded from the region adjacent to the grain boundary.


d = 0.1573nm

g2

Figure 4.63 SAD pattern recorded after tilting the sample 18.43�..

Planar defect

Planar(a) (b)defect 1 Planar defect 2

d = 0.2087 nmd = 0.2087 nm

Selected area diffraction fromthe circle around planar defect 1

Selected area diffraction fromthe circle around planar defect 2

Planar defect

Figure4.64 Sketches of two types of planar defect in thin copper films, and their correspondingSADpatterns shown in thecorrect orientation: (a) planar defect 1; (b) planar defect 2. The regionsfrom which the SAD patterns were recorded are indicated by the dashed circles.


(a) Determine the Miller indices for g1 and g2.(b) Determine the Burgers vector of the sub-boundary dislocations.(c) The distance between the dislocations in the sub-boundary (h) depends on the

Burgers vector (b) and the relative misorientation of the two crystals (y):

h ¼ b

2sinðu=2Þ : ð4:27Þ

Sketch the SAD pattern that you would expect from the boundary region whenaligned with a [0 0 1] zone axis parallel to the electron beam.

4.23. During a TEM study of copper (FCC, a¼ 0.3615 nm), two types of planar defectwere identified in bright-field diffraction contrast. Sketches of each type of planardefect and their corresponding SAD patterns are given in Figure 4.64.

(d) Solve the diffraction pattern from the region containing the first type of planardefect and propose an explanation for this type of defect.

(e) Solve the diffraction pattern from the region containing the second type ofplanar defect and propose an explanation for this second type of defect.


5

Scanning Electron Microscopy

The scanning electron microscope provides the microscopist with images that closelyapproximate what the physiology of the eye and brain expect, since the depth of field forresolved detail in the scanning electron microscope is very much greater than the spatialresolution in the field of view. That is precisely how our eyes and the visual cortex of ourbrains have evolved in order to perceive the three dimensions of the ‘real’ world. The‘flatness’ of the topological and morphological detail that is observed in the light optical ortransmission electron microscope is replaced, in the scanning electron microscope, by animage that appears very similar to the play of light and shade over the hills and valleys of thecountryside. These features of the real world look remarkably like the hollows andprotrusions of a three-dimensional object viewed in the scanning microscope (Figure 5.1).Only two additional features are needed to complete this ‘optical illusion’. The first is a truerepresentation of the position of the image detail in depth, normal to the image plane.With alittle extra time and effort, scanning electronmicroscopy (SEM) can also provide this three-dimensional, depth information by recording two images from slightly different view-points; a technique termed stereoscopic imaging (Section 5.7.3). The second featuremissing from the scanning electron microscope image when compared with ‘real world’images is the presence of colour. Again, it is possible to use the capacity of the eye torecognize colour by the introduction of colour coding. Colour may be used to enhancecontrast, for examplewhen comparing different images during data processing. Colour canalso be used to code for crystallographic information in the morphological image, as weshall see in orientation imaging microscopy (OIM) (Section 5.6.3).

This visual impact of scanning electron microscope images and the ability to revealdetails that are displaced along the optic axis, in addition to those resolved in the two-dimensional field of view of the image plane, has led to the application of SEM to allbranches of science and engineering from the time it was first introduced (late 1950s). Wefirst describe the various imaging signals that are detectable in the scanning electron


microscope due to the interaction of a focused beam of high energy electrons with a solidsample. The interpretation of the information that can be derived from these differentsignals is then described, together with some of the special techniques that are available toenhance image contrast and assist in image interpretation.

5.1 Components of The Scanning Electron Microscope

The basic structure of the scanning electron microscope was described in Chapter 4 andcompared with that of the transmission electron microscope. The main components of thescanning microscope (Figure 4.3) include the microscope column, the various signaldetector systems, the computer hardware and software used to process the collected data,and the display and recording systems.

As in transmission electron microscopy (TEM), the microscope column is kept undervacuum, and the vacuum system and specimen air-lock are an integral part of themicroscope system. If a field emission gun is used to provide the electron source, thenthe vacuum requirements for this source are very stringent and a separate vacuum pumpingand degassing system is required for the electron gun. If very large samples are to be insertedinto the specimen chamber, or samples are to be viewed under cryogenic conditions, or in acontrolled atmosphere, then a specialized sample chamber is necessary. The variety ofspecimen stages now available is quite remarkable but the steps that need to be taken toprevent out-size samples or special stages from contaminating the remainder of the columnand causing damage to the electron gun, add considerably to the cost of these ‘extras’.

The electromagnetic probe lens of the scanning electron microscope behaves in manyrespects as an inverted transmission electronmicroscope objective lens. A ‘minified’ imageof the electron source is focused onto the specimen surface, in place of themagnified imageof the specimen that is focused onto the image plane by a transmission electron microscope

Figure 5.1 A secondary electron scanning electron micrograph of a ductile fracture surface inmolybdenum.


objective lens. However, there are some major differences in the design of the probe lens.Most importantly, since three-dimensional objects are often studied in the scanning electronmicroscope, the probe lens must have an appreciable ‘working distance’. The workingdistance is an important parameter in the operation of the probe lens andmay vary from 1 or2mm up to 50mm or more. By contrast, in TEM the thin-film specimen usually sits insidethemagnetic field of the transmission electronmicroscope objective lens. Since the electronbeam probe in the scanning electronmicroscope has to be scanned across the sample in a x–y, raster, electromagnetic scanning coils also have to be included in themicroscope column.These are positioned above the probe lens.

Most of the scanning electron microscope signal detection systems are also built into thecolumn. The only exception is the detector for optical fluorescence of the sample, a ratherunusual accessory. Common signal detectors include those for high energy (backscattered)electrons, low energy (secondary) electrons, excited (characteristic) X-rays, and some othersignals that will be discussed later (Figure 5.2). The detectors for characteristic X-rays maybe either energy-dispersive or wavelength-dispersive. In energy-dispersive spectroscopy(EDS) the excited photons are collected as a function of their energy and the spectrum ofenergy-dependent, photon intensity is analysed to determine the chemical composition ofthe region of the sample excited under the electron beam. (Section 6.1.2.2). In wavelength-dispersive spectroscopy (WDS) the intensity of the excited X-radiation is collected as afunction of the wavelength (Section 6.1.2.1). EDS detectors collect the excited X-rayssimultaneously over a wide energy range, and are therefore highly efficient. However, theyhave a restricted energy resolution that may sometimes result in unacceptable overlap of thecharacteristic peaks in the X-ray signal generated by different chemical constituents in the

AbsorbedElectrons

BackscatteredElectrons

IncidentElectrons

X-Ray(EDS/WDS)

AugerElectrons

Cathodoluminesence

TransmittedElectrons

SecondaryElectrons

Sample

Figure 5.2 Schematic drawing of a scanning electron beam incident on a solid sample,showing some of the signals generated that can be used to help characterize themicrostructure.

Scanning Electron Microscopy 263

sample. However, WDS detectors have to be rotated around the X-ray goniometer in orderto scan across the range of wavelengths collected by the curved crystal detectors. Thisprocess can be quite time-consuming. Moreover, the range of wavelengths covered by anysingle detector crystal is limited. If it were not for the very much better spectral resolutionand detection limit of WDS systems, it is doubtful if these would still be in use.

Modern scanning electronmicroscopes now have digital acquisition and storage systemswhich replace the earlier, high-resolution cathode ray tubes that were mounted in front of acamera. The data processing and display systems are also integrated into the microscopesystem. Large numbers of SEM images need to be recorded at high resolution, so thecomputer hardware and software requirements are not trivial. The necessary software fordata processing has been developed over nearly half a century and is very reliable. Thescanning electron microscope can now routinely integrate morphological data fromsecondary electron images with compositional information from EDS microanalysis andprovides compositional mapping of the chemical constituents in the microstructure. To thismicrochemical capability has now been added the determination of the crystallographicsurface orientation of individual crystalline grains within the microstructure. This isaccomplished by analysing electron backscatter diffraction (EBSD) patterns from eachindividual grain (Section 5.10). EBSD is similar to the Kikuchi diffraction patternsobserved in TEM (Section 2.5.3). This crystallographic information can also be mappedonto the morphological secondary electron image in a mode of operation of SEM termedorientation imaging microscopy (OIM, Section 5.6.3). This ability to combine morpho-logical, chemical, and crystallographic information in the output from a single, highresolution instrument has made an impact on research and industry that is in many wayscomparable with that due to the introduction of the optical microscope at the end of thenineteenth century.

Only a limited number of bulk microstructural parameters can be accurately determinedby two-dimensional stereological analysis of sample sections. Deducing additional three-dimensional information from a single two-dimensional sample section, for example, thegrain-size distribution, is at best inaccurate and may actually be misleading. Serialsectioning is the obvious answer. Successive sections are prepared at known intervals andthe recorded data combined to develop a three-dimensional model of the morphologywhose resolution is limited primarily by the sectioning interval (Section 9.5). One methodfor successful high resolution serial sectioning is now available in an important variant ofSEM, termed the dual-beam focused ion beam (FIB, Section 5.8.6). Practical quantitativethree-dimensional microscopy is increasingly available (Section 9.5.3), and the scanningelectron microscope is a key tool for three-dimensional applications.

5.2 Electron Beam–Specimen Interactions

An energetic electron penetrating into a solid sample undergoes both elastic and inelasticscattering, but for thick specimens in SEM it is the inelastic scattering that will predomi-nate, eventually reducing the energy of the electrons in the beam to the kinetic energy of thespecimen kT. The various processes that occur along the scattering path are complex, butthey are generally well-understood. If we understand the nature of the signals generated bythe various beam–specimen interactions and the science behind the operation of the signal


detection systems, then there is usually little ambiguity about the interpretation of imagecontrast in the scanning electron microscope.

5.2.1 Beam-Focusing Conditions

The probe lens, used to focus the electron beam onto the specimen surface in the scanningelectron microscope, has similar characteristics to the objective lens in TEM. The bestresolution obtainable cannot be better than the focused probe size on the sample surface.The positions of the source, the condenser system and the probe lens in effect invert theelectron path in the scanning electron microscope with respect to that in the transmissionelectron microscope. That is, in SEM the electron source (the ‘gun’) is where the imagewould be in TEM. The SEM condenser system reduces the apparent size of the source,rather than magnifying the image, and the SEM probe lens forms the beam probe in theimage plane, where the source would be in the geometry of the transmission microscope.The electron beam probe is, effectively, a ‘minified’ image of the electron source.

In practice, there are three limitations on the minimum diameter that can be achieved forthe probe beam in the plane of the specimen:

1. The spherical and chromatic aberrations of the probe lens, with the spherical aberrationbeing the more important (as for the objective lens in conventional TEM).

2. The maximum beam current that can be focused into a probe of a given diameter. Thismaximum current is a strong function of the electron gun and a major reason forpreferring a field emission gun, despite the cost.

3. The need to allow sufficient working space beneath the probe lens pole-pieces toaccommodate large, topographically rough specimens. Sample sizes are typically 20mmindiameter,but thesizemay range from2 to50or100mmforvariousapplications.This isa long way from the standard 3mm diameter thin-film foil commonly viewed in TEM.

In practice it is the beam current limitation that may prove the most serious, since thebeam current varies approximately as the third power of the beam diameter, and hence fallsdramatically for a finer probe. Since a reduction in the beam current may result in a poorsignal-to-noise ratio in the image, this can be a serious problem. Field emission sources arecapable of generating electron probes with very high beam current densities that can befocused to finer probes (as little as 1 nm in diameter). The working distance between theprobe and the specimen is also an important factor. Typically, some compromise is needed.The minimum probe diameter is fixed by the required resolution and determines themaximum depth of field. This, in turn, translates into a maximum working distance for thesample beneath the probe lens.

The commercial introduction of field emission guns has drastically reduced theminimumsize of the primary electron source and increased the available current density in the probeby some four orders of magnitude, allowing for either a much reduced probe size (betterultimate resolution) or, alternatively, the use of appreciably lower incident electron beamenergies while maintaining an acceptable signal-to-noise ratio. The best resolution, of theorder of 1 nm, is now available at beam energies down to 200V. This may be comparedwiththe conventional, thermal emission sources that were seldom capable of generating ameaningful signal at beam energies below 5 kV, or achieving a secondary electron imageresolution of better than 20 nm.


The spatial distribution and temporal stability of the electron current within the beamprobe is also important. This distribution may be determined experimentally by moving aknife edge across a Faraday cup collector. The beam diameter is usually defined as thewidth of the experimental current distribution at half themeasuredmaximum beam current,that is, the full width at half-maximum (FWHM) diameter, but thismay be a poorguide to thetotal current. The current distribution commonly includes a long tail that extends from thecentral spot and can result in a high level of background noise in the signal. For manyimaging purposes this may be unimportant, but it does affect X-ray data collectionsignificantly when the quantitative analysis of concentration variations across a phaseboundary is being determined.

Although the first step in reaching the ultimate resolution in SEM is to reduce the size ofthe focused electron probe, we need to remember that the resolution in the image alsodepends on the volume of excited material in the sample that generates the signal beingcollected. For example, if our focused probe is only 2 nm in diameter, but we are forming animage from backscattered electrons that originate in a region 500 nm in diameter within thesample, then it is this larger dimension that defines the resolution of the backscatteredelectron image. Increasing the probe diameter by an order of magnitude can increase thebackscattered electron signal by several orders of magnitude, without sacrificing theresolution in the backscattered electron image. It follows that, while we may characterizethe performance of a scanning electron microscope by determining the probe size using aknife-edge specimen, this is not normally the resolution characteristic of the collectedimage data. The situation is very different from that in TEM, where the resolution isprimarily determined by either the point resolution or the information limit, and these arecharacteristics of the electro-optical system rather than the sample.

A fewwords should be added concerning the scanning system. Since the data are acquiredby scanning the electron probe across the surface of the specimen and collecting one of thesignals generated, the rate of data collection is not only limited by the intensity of the probeand the efficiency of signal collection, but also by the scanning speed. Aweak signal willrequire a slower scanning speed to improve the signal-to-noise ratio of the image. In thecollection of characteristic X-ray data, when both the inelastic scattering cross-section andthe collection efficiency are low, the statistics of data collection (the number ofX-ray countscontributing toan intensitymeasurement)willdetermine theavailabledetection limit.Undertheseconditions, thebeam-current temporal stability, or currentdrift, can significantly affectthe accuracy of analysis.Wewill return to these statistical considerations later (Section 5.4).

5.2.2 Inelastic Scattering and Energy Losses

The calculation of inelastic scattering paths for electrons can be simulated quantitatively byMonte-Carlo methods based on random scattering events. These ignore some crystallo-graphic (orientation-dependent) scattering effects, specifically, the effect of lattice anisot-ropy and channelling processes that scatter the incident high energy electrons into preferredcrystallographic directions. The electrons in the beam propagating through the crystalfollowan irregular scattering path, losing energy as their integrated path length in the crystalincreases (Figure 5.3). It is not possible to calculate the average trajectory for multiplyscattered electrons, but it is possible to define and measure two critical, penetration depthsin the sample, as well as to estimate the envelope that defines the boundaries of the electron


trajectories for electrons whose energy exceeds any given average value. Thus the diffusiondepth xD is defined as that depth beyondwhich the electrons can be assumed to be randomlyscattered, so that an electron at this depth is equally likely to be moving in any direction inthe sample. At depths below xD electrons can continue to diffuse to increasing depths, butwith scattering angles that are independent of direction. If a Faraday cup is used to collectthe electrons in an incident electron beam that penetrate a thin-film sample, then thediffusion depth will correspond approximately to a critical sample thickness that reducesthe transmitted beam current to half its initial value. The penetration depth or range xR of theincident electrons is defined as the depth at which the electron energy is reduced to thethermal energy kT. In terms of the Faraday cup experiment, the penetration depthcorresponds to the film thickness that would reduce the transmitted electron current tozero. In practice the Faraday cup also collects secondary electrons, thus increasing themeasured transmitted current at all film thicknesses, but in general the experiment workswell. Both xD and xR decrease with increasing atomic number Z and decreasing incidentbeam energy E0. Whereas the change in the shape of the envelope of electron paths withbeam energy is more or less similar, that with atomic number is not. The shape of theenvelope that defines the scattered electron paths for electrons having a given average

Figure 5.3 A Monte-Carlo simulation for 200 electron trajectories through aluminiumassuming 30 keV incident-energy electrons. The trajectories in red indicate backscatteredelectrons that eventually escape from the surface of the sample. All other trajectoriesrepresent electrons that eventually reach thermal equilibrium (an average energy of kT) andare absorbed into the sample.


energy, corresponding to a given energy loss, changes markedly with the atomic number.This is primarily because the lateral spread of the beam is roughly proportional to thedifference (xD� xR), while xR is appreciably less sensitive to Z than xD. These effects aresummarized schematically in Figure 5.4.

Incidentbeam

E>kT E=kT

(a)

Increasing beam energy

Increasingatomic number

Z

xR

xR

xR

xD xD

xDxD

xR

(b)

Figure 5.4 (a) The electron beam is inelastically scatteredwithin an envelope bounded by thecondition that the average energy has reached the thermal kinetic value kT. (b) The inelasticscattering envelope for an incident beam of energetic electrons depends on both the incidentenergy and the atomic number of the target, and may be qualitatively approximated by the twoparameters, diffusion depth xD and penetration depth or range xR.


5.3 Electron Excitation of X-Rays

If the incident electron energy exceeds the energy required to eject an electron from an atomin the specimen, ionizing the atom, then therewill be a finite probability that ionizationwilloccur. Ionization of the atom is an inelastic scattering event that reduces the energy of theincident electron by an amount that is characteristic of the ionization energy, and raises theenergy of the atom above its ground state by an equal amount. The energy of the excitedatom can then decay by transition of an electron from some higher energy state into the nowvacant state in the atom.All such transitions are accompanied by the emission of a photon. Ifthe excited state of the atom corresponds to the ejection of an electron from one of the innershells of the atom, then this emitted photon will have an energy that lies in the X-ray regionof the electromagnetic spectrum.

In general, decay of an excited atom from an excited state to the ground state takes placein several successive stages, with the emission of several photons, each having a differentenergy and wavelength and each corresponding to a single stage in the transition of theexcited atom back to its ground state. It follows that, if a particular ionization state is to bereached, then the energy lost by the incident electron in the inelastic event must alwaysexceed the threshold energy for the creation of that ionization state, while the energy ofthe most energetic photon that can be emitted will always be less than this thresholdenergy for excitation. Furthermore, if we consider any specific inner shell of electronssurrounding the atom, for example, the innermost K-shell, then, as the atomic numberincreases, the ionization energy for the electrons that occupy this shell must also increase(since the electrons in any given shell are closer to the nucleus and hence more deeplyembedded in an atom of higher atomic number). This conclusion is illustrated schemati-cally in Figure 5.5.

The X-ray spectrum generated when an energetic electron beam is incident on a solidtarget includes a range of wavelengths, starting from a minimum wavelength that can bederived from the deBroglie relationshipl0 ¼ hc

eV, where h is Planck’s constant, c is the speedof light, e is the charge on the electron and V is the accelerating voltage applied to theelectron source (Section 2.3.1).

The wavelengths corresponding to the characteristic X-ray excitation lines that areemitted from the region of the sample beneath the probe constitute a fingerprint for thechemical elements present in the solid and provide a powerful method of identifyingthese chemical constituents and their spatial distribution. The atomic number depen-dence of these wavelengths is illustrated approximately in Figure 2.13, but, rather thana single line, a group of spectral lines is commonly emitted, and there may be partialoverlap of the lines characteristic of one chemical constituent with those emitted by asecond constituent.

In parallel to the characteristic X-ray emission spectrum, the X-rays generated by theincident electron beam also have an absorption spectrum, namely the high energy, shortwavelength X-ray photons will themselves have a finite probability of exciting the atoms inthe sample to higher energy ionization states. Thus aK photon from a higher atomic numberelement will possess sufficient energy to excite an atom of lower atomic number to the Kstate, resulting in absorption of the higher energy photon. The excited atomwill then decayback towards the ground state, generating a new, lower energy photon that is characteristicof this second atom. This process is termed X-ray fluorescence. X-ray absorption, as noted


previously (Section 2.3.1), is characterized by an absorption coefficientmwhich depends onthe wavelength of the X-rays and the atomic numbers of the chemical constituents. Criticalabsorption edges in theX-ray absorption spectrum correspond to those photon energies thathave the threshold for excitation of fluorescent radiation of one of the chemical constituentsof the solid.

Both the excitation and the absorption spectra of either electrons or X-rays can be used toderive information on the chemical composition of the sample.Wewill return to these topicsin much more detail in Chapter 6, where we also consider the quantitative determination ofchemical composition from these signals in the scanning electron microscope.

Energyof

Atom

Valency Electron BandGround

State

KExcitation

K state

L states

M states

Kα Kβ

Lα

(a)

(b)

Bea

mE

nerg

y

IonizationEnergy

PhotonEnergy

Atomic Number, Z

Figure 5.5 (a) An inelastic electron scattering event involving ionization of an inner shellelectron raises the energy of the atom to the appropriate ionization state. Subsequent decay ofthe atom to a lower energy state is accompanied by photon emission. The energy of the emittedphoton is characteristic of the energy difference between the two energy states for the atom, butmust always be less than that needed for the initial ionization event. (b) The ionization energyrequired to eject an electron from a particular inner shell of the atom increases with its atomicnumber.


5.3.1 Characteristic X-Ray Images

TheX-raysignalgeneratedbeneathafocusedelectronprobecomes fromavolumeelementofthe sample which is defined by the envelope of electron energies that exceed the energyrequired to excite the characteristic radiation from any chemical constituent. As the beamvoltage is reduced, so the sizeof this volumeelement shrinks, improving thepotential spatialresolution for theX-raysignal,butalso reducing the intensityof thisemittedsignal.When theenergyof the incidentbeam,determinedbytheacceleratingvoltage, fallsbelowthethresholdenergy for excitation, no characteristicX-ray radiation can be generated. A compromise hasto be found which will ensure a statistically significant characteristic X-ray signal thatremains spatially localized in the region of interest. The intensity of the X-ray signal that isemitted will at first increase as the electron beam energy is increased above the minimumenergyneeded to excite the characteristic signal.However, if the incident energy exceeds thecritical excitation energy by a factor of about four, then the intensity of the X-ray signalescaping from the specimen starts to decrease. This decrease occurs because of increasingabsorption of the excited X-rays within the sample, since the signal generated is now, onaverage, originating deeper beneath the sample surface. This is a result of the increaseddiffusion depth of the electrons at the higher energies. It follows that an ‘optimum’ electronbeam excitation energy exists which givesmaximum excitation of the characteristic X-raysthat are emitted.This energy is of the order of four times the excitation energy. In practice thebeamenergyof theprobe shouldbe selected for thehighest energywhich isof interest, that is,the shortest characteristic X-ray wavelength that is to be detected.

The efficiency for characteristic X-ray generation is low and the X-rays are emitted at allangles. A high proportion of the X-ray signal is either absorbed within the sample or fails toreach the detection system because the X-ray collection is itself rather inefficient. Energy-dispersive collection of the X-rays uses a solid-state detector. The alternative is a crystalspectrometer that employs a series of different curved crystals for each range of wave-lengths of interest. The advantage of the curved-crystal diffractometer, a wavelength-dispersive spectrometer, is that the spectral resolution is excellent, considerably reducingthe chances of peak overlap ambiguities in ascribing each characteristic X-ray wavelengthto a specific chemical constituent in the sample.

Solid-state X-ray detectors depend on the energy discrimination capability of a cryo-genically cooled semiconductor crystal. The electrical charge that is generated in thedetector by each photon absorbed is proportional to the energy of the incident photon. Thischarge results in a current pulse which is proportional to the energy of the photon that hasbeen captured by the energy-dispersive spectrometer. Signal overlap of characteristic peaksis a frequent problem in EDS, although the problem can usually be solved by selecting oneor more alternative characteristic emission lines for analysis from the X-ray energyspectrum. The detector is commonly shielded from contamination by the microscopeusing a polymer film window that is transparent to all but the longest wavelength, lowestenergy X-rays. The detectors are routinely capable of detecting photons of wavelengthsexceeding 5 nm, corresponding to X-ray emission that is associated with some of thelightest atomic elements (boron, carbon, nitrogen, and oxygen).

All EDS detectors are limited in the rate at which they are able to accept X-ray photon‘counts’. This corresponds to the time required for each individual charge pulse to decay inthe detector. This dead-time is typically somewhat less than 1ms, during this time no further


counts can be reliably recorded. It follows that count rates should not exceed 106 s�1. Somecounts will inevitably be lost, since the photons are randomly generated in time. Theproportion of ‘dead-time’ between counts is registered by the counting system. Acceptabledead-times are of the order of 20%. Lower values of the dead-timewill correspond to lowerrates of data accumulation. The effect of increasing dead-time on the collection efficiencyof a typical solid-state detector is shown schematically in Figure 5.6.

The X-ray signal may be displayed in three distinct formats:

1. An X-ray spectrum. This is used primarily in order to identify the chemical elementspresent from their characteristic X-ray �fingerprints� (Figure 5.7). Such a spectrum maybe collected with the beam stationary, at a specific location on the sample surface (pointanalysis). Alternatively, to reduce the effects of contamination build-up, the spectrummay be collected while the beam is scanned over a selected area (�rastered�). Typicalsignal collection times that are required to ensure EDS detection of all the elements thatare present in concentrations exceeding one or two percent are of the order of 100 s.

2. An X-ray line-scan. In this mode the beam is traversed across a selected region of thesample in discrete steps and the signal prerecorded at each step. The spatial resolution forX-ray analysis in a line-scan depends on the number of steps per unit length of line. Thebest obtainable spatial resolution is limited by the volume element of the sample fromwhich the X-rays are generated while the chemical detection limit will depend on thetime of signal acquisition at each step. The signal may be in the form of an entire EDSspectrum, recorded at each point along the line (full-spectrum mapping). Alternativelyone or more characteristic X-ray energies may be selected. Full-spectrum mappingrequires significantly more digital storage than that needed for individual, selectedenergies, but no a priori knowledge of the local chemical distribution is required. Ineither case, the recorded intensity from the characteristic X-ray peaks are displayed as afunction of the position of the electron beam (Figure 5.8). The selection of the windowsin the energy spectrum for the energies that correspond to thewavelengths of interest are

Emission Rate

Cou

ntin

g R

ate Dead-Time

Figure 5.6 The count rate as a function of the signal incident on a solid-state detector,illustrating the effect of dead-time on the counting efficiency.


set to exclude all other photon energies. The number of counts for each selectedcharacteristic energy peak is displayed for each position along the beam traverse. Thismode of operation is especially useful for determining concentration gradients andsegregation effects at grain boundaries, phase boundaries and interfaces.

3. An X-ray chemical concentration map. In this case, the incident electron beam israstered across a selected area of the sample, and photon counts are collected for one ormore energywindows that are characteristic of the chemical components of interest. Thecounts are recorded as a function of the incident electron beam coordinates. The photonsdetected for a particular characteristic X-ray emission line are then displayed as colour-coded dots in a position on the screen that corresponds to the beam coordinates as it isscanned over the area of the specimen surface chosen for analysis. The range ofwavelengths to be assigned a particular colour code is pre-selected in an energy�window�which corresponds to a characteristic photon energy of the element of interest.Several elements may be detected and displayed simultaneously, corresponding toseparate energy windows that are individually colour coded for the different elements(Figure 5.9). In many SEM instruments these areal X-ray maps can be collectedautomatically from several successive selected areas whose position on the sample

Ta

Ta

TaCu

N

Ta

CuO

Al

TaSi

Ta Cu TaTa0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

220000

240000

260000

280000

0 1 2 3 4 5 6 7 8 9 10

Energy (keV)

Cou

nts

Figure 5.7 EDSpointmeasurement fromaprinted circuit board (seemicrograph in Figure 5.8).


surface is chosen to optimize the statistical significance or to be representative ofdifferent morphological features of the microstructure. The counting times can beadjusted to the required level of statistical significance for the chemical constituents, thatis, adjusted to the detection limit for a low concentration element. If full spectra are

Figure 5.8 EDS line-scan across a printed circuit board. The red line in the secondary electronSEM micrograph indicates where the line-scan was acquired.


recorded from each point, then it is also possible to correct for background andsecondary fluorescent excitation. Quantitative chemical analysis for the selected regionof the specimen surface is then possible. This is a very time-consuming process thatpresupposes sufficient beam and sample stage stability, together with negligiblecontamination of the sample surface. Hence, a combination of high instrumentalstandards with experienced operational competence is required.

In most instruments, all three of the above modes of operation involve extended periodsof electron-probe irradiation of the sample surface and hence high irradiation dosesaccompanying the acquisition of the necessary counting statistics. This frequently resultsin the build-up of a carbonaceous contamination film, on the sample surface. It is alwaysgood practice to limit the detection time to the minimum required to collect a statisticallysignificant signal. Contamination films can affect the X-ray spectra, primarily by adding to

Figure 5.9 EDS elemental maps from a printed circuit board for Ta, Al, Cu, O, and Si. Thebrighter the intensity, the stronger the EDS signal for a specific element. A secondary electron(SE) SEM micrograph is included for comparison.


the background radiation and changing the conditions of analysis as the data are acquired.Contamination build-up is amajor reason for preferring to raster the probe over a small areaof the sample rather than relying on a point-analysis to collect the same total number ofcounts from a fixed coordinate position that is itself subject to mechanical drift of thespecimen stage or electronic drift of the probe. The overriding consideration in recordingX-ray data in the scanning electron microscope is always the achievement of statisticalsignificance for the collected signal. The simplest assumption that can be made is that thecounting statistics at each point on the sample surface obey a Poisson distribution, so thatthe counting error that is associatedwith the characteristic line of each chemical constituentis just 1=

ffiffiffiffi

Np

, whereN is the number of counts recorded for this constituent from the volumeelement on the sample surface that is excited by the electron beam.

As noted above, point analysis and the collection of a complete spectrum can providedata corresponding either to a fixed beam coordinate or a selected scanned area. Formoderate counting times, the statistical significance is usually high and can be determinedafter first correcting for background counts. These background corrections are measured intwo separate windows either side of the selected window for the characteristic line ofinterest. It is common practice to choose the collection width of the window for eachcharacteristic line using the FWHM of the characteristic energy peak, N¼NT�NB, whereNT and NB are the total counts within the selected window and the correction for thebackground counts, respectively. The latter is interpolated from the two backgroundmeasurements in order to correspond to the same energy as the characteristic line.

Line analysis results in some reduced statistical significance, in direct proportion to thesquare root of the number of pixels (picture elements) along the line selected for analysis onthe sample surface. Nevertheless this mode is extremely useful in determining changes inconcentration near surfaces, interfaces and grain boundaries, for example, segregationeffects, as well as diffusion concentration gradients that are associated with the precipita-tion kinetics of a second phase or oxidation of a surface.

Elemental maps require far longer counting times and are in a different classaltogether. They should only be acquired together with a corresponding high resolutionsecondary or backscattered electron image (see below). A good, high resolution,secondary electron image will usually contain more than 106 resolved pixels, while anX-ray elemental map recorded at the same resolution would require a prohibitivecounting time (approximately 106 times longer than that for a single point count).Clearly this is impossible, and not just impractical. Neither the life expectancy of theoperator, nor the stability of the electron beam probe, nor the contamination of thesample are anywhere near sufficient for this.

Is elemental mapping just a ‘good idea’? What methodology for elemental mapping is‘practical’? If we assume a counting time of 100 s, as in the case of a point-analysisspectrum, and we require a statistical significance corresponding to a 95% confidencelevel, then the minimum number of counts for a resolved X-ray pixel in a mapped X-rayimage is about 1000. At a relatively high counting rate of 104 s�1, this corresponds to anacquisition rate for an elemental map of 10 pixels s�1. Hence, to acquire an area on thesample surface of 128� 128 pixels would take about 25min, and it would requirewell overa day to acquire a statistically significant elemental map with a pixel density approachingthat of a secondary electron image. As we will see in Section 5.5, the spatial resolution of asecondary electron image is always better than can be achieved for an elemental X-raymap,


simply due to the differences in size of the volume elements in the sample from which thesecondary electron andX-ray signals are emitted. In fact, trying to generate an elemental X-ray map with the same density of measured points as a secondary electron image makes nosense at all.

Independent of the problems associated with X-ray counting statistics, the dimensions ofthe X-ray emitting region beneath the probe will limit the spatial resolution of bothelemental maps and line-scans. The size of this zone is dictated primarily by the electrondiffusion distance in the sample xD, that is, the diameter of the envelope for electrons withenergies greater than the critical excitation energy. This diffusion distance depends on thebeam voltage and the atomic number, or density of the sample. Typical values of thediffusion distance are between 0.5mm and 2mm for a standard scanning instrument. Bychoosing the typical value of 1mm, and using the previously estimated, maximum practicalnumber of X-ray counts per point, we conclude that a useful elemental map can be obtainedfrom a 100� 100 mm2 region, that is, an area of 104 pixels, in just over 16min. Attempts toobtain elemental maps from a larger region may be successful if the counting times aresignificantly increased, but this is usually impractical. Imaging smaller areas, that isemploying higher magnifications, will simply result in fewer effective image points andincreased ‘blurring’, since the resolution will then be dictated by the dimensions of theexcitation envelope.

Recognizing that the resolution in an elemental map is inherently worse than that in thesecondary or backscattered electron image is the first step to making the best use of the X-ray data. By superimposing the X-ray map onto the high resolution, secondary electron orbackscattered electron image, it is possible to see the detailedmorphology of the sample andits relation to the localized variations in chemical composition. A 128� 128 pixel array ofthe X-ray data can be acquired in less than 30min, with each pixel representing an areadetermined by the excitation envelope forX-rays, typically of the order of 1mm in diameter.Such low resolution elemental maps are an extremely useful guide to local variations incomposition near features recorded at high resolution in a secondary electron image, despitethe orders of magnitude difference in the density of resolved pixels.

5.4 Backscattered Electrons

A fraction of the incident high energy electrons will be scattered by angles greater than pand these electrons have a finite probability of escape from the surface. The fraction of theincident beam backscattered R depends rather sensitively on the mass density, or, moreaccurately, the average atomic number of the specimen �Z, increasing with increasing �Z(Figure 5.10). The backscattered electron signal is therefore able to resolve local variationsin mass density and results in atomic number contrast. The backscattered fraction is muchless dependent on the incident beam energy E0, decreasing as E0 increases. Such atomicnumber contrast can be very useful, since it offers the possibility of distinguishing betweendifferent phases at a far better resolution than can be achieved by X-ray microanalysis.Atomic number contrast is more pronounced at lower voltages. The backscattered electronsoriginate in a surface layer whose thickness corresponds approximately to the diffusiondistance. They come from an area beneath the beam that is proportional to this distance andis significantly less than the range of the inelastically scattered electrons with E > kT.


While the average energy of the backscattered electrons is, of course, less than that ofthe primary incident beam, it is nevertheless of the same order of magnitude and thisbackscattered electron signal can be used to acquire crystallographic information. Thebackscattered electrons are detected over a wide angle in an annular region close to theprobe lens pole pieces. As in TEM, the intensity of the inelastically scattered signal fallsroughly as cos2a, where a is the angle between the incident beam and the backscatteredelectron path. As in the transmission case, the backscattered electrons can be diffractedby a crystalline sample, subtracting intensity from the signal at low values of a when theBragg condition is satisfied and adding intensity at higher angles, to give the backscatterequivalent of the Kikuchi line diffraction observed in the transmission electron micro-scope (Section 2.5.3). These EBSD patterns can be collected and recorded using acharge-coupled device (CCD) system. The patterns can also be analysed by appropriatecomputer software, to give the orientation of the crystal surface at each pixel location(Figure 5.11) Improved collection efficiency for backscattered electrons is obtained bytilting the sample to bring the high energy electron diffusion distance closer to the freesurface. Suitable computer software corrects the recorded image for the foreshorteningassociated with the angle between the microscope axis and the normal to the tiltedsample surface. Analysis of EBSD patterns collected as a function of position for arastered incident electron beam can be colour coded for selected ranges of crystalorientation so that the surface orientation of each grain is recorded in the image. (This ismore fully discussed in Section 5.6.5 on OIM.) Although the collection efficiency forbackscattered electrons is high, the backscattered electron current is only a fraction ofthe incident beam current. Field emission sources greatly improve the data collectionstatistics for backscattered electrons.

η

Z

0.5

Figure 5.10 The fraction of backscattered electrons as a function of the average atomicnumber.


5.4.1 Image Contrast in Backscattered Electron Images

Contrast in a backscattered electron image may arise from either of two sources:

1. Regions of the specimen surface that are tilted towards a backscattered electron detectorwill give an enhanced signal, while the signal will be reduced if the surface is tilted awayfrom the detector. A segmented annular detector can therefore be used to obtaina topographic image of the surface inwhich the signals collected fromdetector segmentspositioned on opposite sides of the specimen surface are subtracted and then amplified.To a good approximation, this enhances the topographic contrast from regions that aretilted in opposite directions, while neutralizing contrast due to differences in density(atomic number).

2. Collecting a backscattered image from a detector that surrounds the probe lens polepieces, or, equivalently, summing all the signals from a segmented detector, minimizesthe contrast associated with surface topography, since the signal is then collected fromall possible azimuthal angles. The contrast from features visible in the image is nowmainly atomic number contrast which reflects variations in specimen density that areeither associated with variations in composition or, in some cases, with regions of fineporosity.

Figure 5.11 An experimental EBSD pattern of gold, overlayed with a simulated patternshowing the projections of the possible diffracting planes in red.


The resolution in the backscattered electron image is usually an order of magnitude betterthan that of an X-ray elemental map, but still cannot compete with that available in asecondary electron image (discussed below). The direct relationship between the back-scattered electron image and the high energy electron diffusion distance in the materialtypically limits the resolution to between 50 nm and 100 nm when working with beamenergies of 10–20 keV. The backscattered electron image provides useful information onthe distribution of the phases visible on the surface of a polished sample, providing thephases differ sufficiently in density (as can be seen in Figure 5.12). Commercial scanningelectronmicroscopes can nowoffer a combination of backscattered electron information ongrain morphology with automated EBSD and X-ray elemental mapping of the same grains.This remarkable achievement is fully appreciated by the materials characterizationcommunity.

5.5 Secondary Electron Emission

Most of the electron current emitted from a sample excited by a high energy incidentelectron beam is due to the release of secondary electrons from the sample surface.The secondary electron emission coefficient, that is the number of secondary electronswhich are released per incident high energy electron, is always much greater than oneand may reach values of several hundred. It is useful to separate the secondary electronsignal into two components: first, those secondary electrons that are generated by the

Figure 5.12 A backscattered electron atomic number contrast image from a polished surfacesection, showing a niobium-rich, intermetallic phase in bright contrast, dispersed in an aluminamatrix in dark contrast. The image was recorded with a 10 kV incident electron beam.


high energy electrons in the incident beam as they enter the surface; and, secondly,those secondary electrons that are generated by high energy backscattered electronsthat have returned to the surface region after several, inelastic scattering events. Theformer signal comes from a surface area of the order of the beam probe cross-section,and contains information with a resolution only limited by the probe diameter. Thesecondary electrons that are generated by backscattered electrons come from a surfacearea which is similar to that responsible for the backscattered electron signal and canonly resolve image detail on a scale comparable with the backscattered electronresolution. We will consider, below, how to separate these two signals in order tooptimize the resolution in the secondary electron image.

A schematic representation of the energy distribution of all the electrons emitted by a SEMsample is given in Figure 5.13, and a comparison between a backscattered and secondaryelectron micrograph of Al2O3 particles on the surface of Ni are shown in Figure 5.14.While the energyof the backscattered electrons is just below that of the incident beam,E0, thesecondary electrons generate a huge peak at the low energy end of the energy spectrum. AnAuger excitation signal also exists, at energies that lie just above the secondary electronenergy peak. As we shall see in Chapter 8 (Section 8.2), Auger electrons are an importantsource of information on surface chemistry. However, collection of an Auger spectrumrequires extreme vacuum and specimen degassing that are beyond the capability of thescanning electron microscope. Secondary electrons may have energies as high as 100 or

Ele

ctro

n yi

eld

Electron energyE0

Secondary electrons

Backscatteredelectrons

Auger electrons

Figure 5.13 Schematic illustration of the electron yield as a function of the emitted electronenergy. The yield of secondary electrons is orders of magnitude larger, and their energies ordersof magnitude less, than that of backscattered electrons. Surface analysis using Auger electronswill be discussed in Chapter 8.


200 eV, but their energies usually fall in the range below 10–50 eV (with an energy spread ofabout 5 eV). The secondary electrons are therefore readily deflected by a low bias voltageand can be collected with very high efficiency (close to 100%). Moreover, their low kineticenergy severely restricts their mean free path in the sample, so that the secondary electronsthat escape to the detector are generated within 1–20 nm of the surface. Consequently thesesecondary electrons are almost unaffected by beam spreading of the probe beneath thesurface. To a good approximation, the secondary electron escape distance is given by:

PS ¼ expð�r=LSÞ ð5:1Þ

Figure 5.14 (a) Backscattered electron and (b) secondary electron SEMmicrographs of smallAl2O3 particles on the surface of a nickel substrate. Both micrographs were recorded using a25 kV incident electron beam.


where PS is the probability of escape, r is the distance the secondary electron must travel inorder to escape from the surface of the sample, and LS is the mean free path of a secondaryelectron generated in the bulk solid. For a high atomic number metal, such as platinum,LS� 2 nm. For a low density ceramic, such as MgO, LS� 23 nm.

5.5.1 Factors Affecting Secondary Electron Emission

Four factors directly affect the secondary electron emission current from a sample surface:

1. Thework function of the surface, that is the energy barrier that has to be overcome by anelectron at the Fermi level in the solid in order to permit it to escape from the sample intothe vacuum. Typical work functions are a few eV. The work function depends on bothcomposition and the atomic packing (crystal structure) at the surface, and is sensitive tosurface adsorption and contamination films. In SEM, surface contamination by acarbonaceous layer is generally sufficient to obscure all effects attributable to the workfunction of the substrate.

2. The incident electron beam energy and beam current. As the beam energy is increased,more secondary electrons are expected to be created beneath the probe, but since a higherenergy beam is inelastically scattered at depths which are further beneath the surface, theproportion of secondary electrons that can escape from the sample is eventually reduced.In practice, the yield of secondary electrons rises rapidly as the probe energy increasesfrom zero up to several kilovolts. Theyield then goes through a shallowmaximum, beforedecreasing slowly above 5–10 kV depending upon the material (Figure 5.15). However,the secondary electron current is directly proportional to the current in the incident beam,which decreases as the accelerating voltage for the incident beam is decreased.

Incident Electron Energy (kV)

Seco

ndar

y E

lect

ron

Yie

ld

1 2 3 4 5 6 7

Figure 5.15 Schematic illustration of the expected secondary electronyield as a function of theincident electron beam energy.


3. The density of the sample also has an influence on the secondary electron yield. Athigher values of the atomic number Z, the backscatter coefficient R is higher, so as Zincreases more secondary electrons are created by the backscattered electrons. At thesame time, the secondary electrons in higher atomic number materials have a smallerdiffusion distance, while, at a given beam intensity, the number of inelastic scatteringevents in high Z materials is higher in the surface region than for low Z materials. Itfollows that, for a given excitation energy, larger numbers of secondary electrons shouldbe collected from higher atomic number samples. The Z dependence of secondaryelectronyield ismore pronounced at lower beamenergies, when the diffusion distance ofthe incident electron probe becomes comparable with the mean free path of thesecondary electrons in the solid.

4. Themost pronounced contrast effects in the secondary electron image are due to surfacetopography, or, more precisely, to the local curvature of the surface and the angle ofincidence of the electron probe. In general, changes in local curvature change theprobability that a secondary electron that has been generated near the surface can escape,while the angle of incidence of the probe determines the path length of the incident highenergy electron within the surface region (Figure 5.16). A region protruding from thesurface, that is, a region having a positive radius of curvature increases the chances ofsecondary electrons escaping, while any recessed region, having a negative radius ofcurvature, will reduce the secondary electron current by local trapping of the secondaryelectrons. The secondary electrons are commonly collected using a bias voltage appliedto the collector, so even though some regions of the sample are out of the line of sight(Figure 5.17), we may conclude that the secondary electron image should providetopographic images of rough surfaces having both high resolution and excellent contrast(Figure 5.18). The introduction of field emission guns has made an additional, dramaticimprovement in secondary electron image resolution. This has been further improved byreducing the working distance of the probe lens to just a few millimetres and placing asecondary electron detector inside the magnetic field of the pole pieces (Figure 5.19).The secondary electrons emitted in the forward direction due to the initial inelastic

Z

r < Ls

r > Ls

r < Ls

r < Ls

Sample

Figure 5.16 A rough region on a sample surface affects the probability of escape for secondaryelectrons. If the distance needed to escape from the surface r is greater than themean free path ofthe secondary electrons, Ls, then the secondary electrons cannot escape from the sample.


scattering events in the primary beam, are then trapped by the magnetic field of the polepieces and collected, while those generated by the backscattered electrons at widerangles cannot reach the in-lens detector placed within the electro-optical column. Thesecondary electron signal detected by this in-lens collector is verymuchweaker than thesecondary electron signal normally detected, but this does not matter at the beamintensities generated by a field emission gun. The in-lens detection system improves thesecondary electron image resolution (from 20 to 50 nm to only 1 or 2 nm).

Specimen

Secondary electrontrajectories

Collector(~+200 V)

Figure 5.17 By applying a small bias voltage, the secondary electrons can be collected withhigh efficiency from regions of the surface that are not in the direct line of sight of the collector.

Figure 5.18 High resolution secondary electron SEM micrograph of nanometre-sized TiCNparticles, recorded with a 5 kV incident electron beam at a working distance of 4mm.


5.5.2 Secondary Electron Image Contrast

Some complications in the interpretation of secondary electron image contrast arise fromthe two major factors affecting signal generation: namely, the initial production of thesecondaries by the primary electrons followed by their subsequent escape from the samplesurface. As long as the sample is reasonably planar, the escape of secondaries is only

Sample

BSE Detector

BSE DetectorSE Detector

BSE Detector

SE1

SE2

SE2SE

1

BSE

2

BSE

1

SE3 ET Detector

Column

BackscatteredElectrons

Figure 5.19 The location of secondary electron (SE) and backscattered electron (BSE)detectors. Secondary electrons of the first type (SE1) generated by the primary incidentbeam, are preferentially acquired by a detector placed within the electro-optical column toform a very high resolution image when comparedwith secondary electrons of the second type(SE2) that are acquired by an Everhart–Thornley (ET ) detector. Similarly, BSE1 electrons, definedas primary beam electrons that have undergone very few inelastic scattering events, can beacquired by a BSE detector placed within the column, to improve the resolution for BSEmicrographswhencomparedwith those acquiredwith a conventional annular BSE detector thatcollects all the multiply scattered backscattered electrons.


restricted by variations in work function, which is why the intensity of the signal fallsrapidly to zero at secondary energies below a few eV (Figure 5.13). The generation ofsecondaries depends on the number of inelastic scattering events that occur in the surfaceregion due to either the incident primary beam or the backscattered signal. In general,reducing the beamvoltagewill enhance the secondary electron image contrast, and this doesnot usually cause any sacrifice of resolution.

High resolution scanning instruments that are equipped with a field emission gun arecapable of 1–2 nm resolution by combining an in-lens detector with electron-probe beamenergies that are as low as 200 eV. These conditions give excellent contrast and resolutionthat reflect either variations in atomic number or surface topography. At these low beamvoltages there is usually no problem of surface conductivity and no need to conductivelycoat a nonconducting sample. The beam penetration into the sample (the diffusion distance)is very limited and any electrostatic charging (Figure 5.20) is neutralized by the large yieldof secondary electrons. For such samples, the features showing contrast may be confidentlyassociated with the sample material, and not with any conducting layer that has beendeposited to avoid electrostatic charging.

Topographic contrast also presents some problems of image interpretation. Protrudingregions of the specimen may trap secondary electrons and screen recessed regions from the

Zero NetCurrent Flow

Incident Electron Energy (kV)

Seco

ndar

y E

lect

ron

Yie

ld

1 2 3 4 5 6 7

Incident PrimaryElectron Current

Escaping SecondaryElectron Current

Figure 5.20 Schematic illustration demonstrating charge compensation by selecting anoptimal accelerating voltage. The total incident current decreases as the acceleratingvoltage is decreased (black curve). This curve intersects the characteristic secondaryelectron yield at two points, where the total net current of electrons to the sample is zeroand electrostatic charging of the surface is prevented.


detector, despite the high secondary electron collector efficiency. More commonly,enhanced emission from a region of high positive curvature may extend some distancebeyond the high curvature region. This enhanced emission occurs over a distance thatcorresponds approximately to the electron range in the sample xR. This is because theinelastically scattered, high energy electrons will generate secondary electrons that are ableto escape whenever a high energy electron approaches the surface. Similarly, a darkenedarea in the image, associated with a feature of negative curvature, may also extend fora similar distance beyond the feature. It follows that there is a ‘shadowing’ of hollows and a‘highlighting’ of protrusions that looks very similar to that produced by sunlight falling onhilly country, but is associated with a completely different mechanism, since the secondaryelectrons are being collected from all directions (Figure 5.16). This can be particularlymisleading when operating at high magnifications and close to the resolution limit.

5.6 Alternative Imaging Modes

In addition to the three imaging signals which we have discussed so far (X-rays, backscattered electrons and secondary electrons), there are some other modes of operation forthe scanning electronmicroscope that have useful applications.We shall briefly discuss justtwo of them, namely cathodoluminescence and electron beam image current (EBIC).

5.6.1 Cathodoluminescence

Many optically-active materials will emit electromagnetic radiation in the visible rangewhen suitably excited. Under an energetic electron beam, optically-active regions of theSEM sample will glow, emitting visible light at a wavelength that is characteristic of theenergy levels at the surface of the sample that have been excited by the incident beam.The fluorescing sample may be observed with an optical microscope, in which case theresolution is limited to that characteristic of the optical microscope objective, or the lightemitted may be collected by a photoelectron detector, and then amplified, recorded anddisplayed using the same time-base as the beam scanning coils. In this case the resolution isnot limited by thewavelength of the emitted radiation but rather by the diffusion distance ofthe electron probe in the sample. Biological samples can be labeled by suitable fluorescing‘stains’, and cathodoluminescence used to image soft tissue structures in which differentfeatures are identified by different stains, each of which fluoresces at a characteristicwavelength in the optical range.

5.6.2 Electron Beam Induced Current

If the specimen is electrically isolated from its surroundings and the current flowing throughthe specimen can be monitored then it is possible to form an image that displays thisspecimen current signal synchronously with the scanning of the beam over the sample.

Several variants of this mode of operation have been used to study defects in semi-conductors and solid-state semiconductor devices. For example, the electrical conductivityof a semiconductor is often a sensitive function of its defect structure and dopant or impurityconcentrations, and this conductivity can be monitored as the beam is scanned across the


surface, eitherwith orwithout a bias voltage applied to the sample. This formof operation ofthe scanning electron microscope is referred to as the electron beam induced currentmode.A complete solid-state devicemay be inserted in the specimen chamber and operated in situwhile under observation using this mode. The electric field that is developed in the variousregions of the device will modify both the secondary electron signal and the specimencurrent flowing through the various components of the device. Devices can be imaged athigh resolution and the method has been used by industry for device development and theanalysis of process defects that are associated with device operation. This topic is wellbeyond the scope of the present text.

5.6.3 Orientation Imaging Microscopy

OIM is a relatively recent development, made feasible by the increasing availability offield emission electron sources and high resolution CCD cameras, together with thecomputer data processing and storage capacity required to handle gigabyte quantities ofimage data. When first developed some 50 years ago, the scanning electron microscopeand the microprobe analyser were separate instruments, the first designed to provide sub-micrometre resolution and high depth of field in secondary electron images, and thesecond to perform highly localized chemical analyses using characteristic X-ray excita-tion under a focused electron probe. Combining the detection of these two separatesignals, the secondary electron and the X-ray, in a single, commercial instrument wasaccomplished as early as 1960, but no one dreamed at that time that it would also bepossible to extract crystallographic information from the same sample area and at thesame time in a single instrument.

Three major problems stood in the way of the successful development of an orientationimaging microscope. The first was the lack of a sufficiently sensitive signal detectionsystem for acquiring the angular distribution of the EBSDpattern from a single crystal grainirradiated with a high energy electron probe. The second was the detection of the very weaksignal in the EBSD pattern that was generated using a conventional, tungsten hairpin,thermionic electron source. The third problemwas the need to analyse themassive amountsof data that were required to determine the crystal orientation for a large number of pixellocations on the surface of a polished, polycrystalline sample, and the conversion of thisinformation into an orientation distribution map that could display regions of similarorientation on the sample surface.

Themost recent publications reporting the application of OIM tomaterials research havenow combined local chemical information, derived from X-ray elemental mapping, withmorphological and crystallographic analysis, and have begun to report the serial sectioningof samples in order to develop a complete, three-dimensional characterization of a samplethat has been taken from a specific location in an engineering component.

5.6.4 Electron Backscattered Diffraction Patterns

The geometry of EBSD is very similar to that of Kikuchi diffraction in TEM (Sec-tion 2.5.3), and is shown schematically in Figure 5.21. When the high energy electronprobe strikes a crystal in a solid sample, inelastic scattering events scatter the incidentelectrons so that a fraction acquire velocity vectors that allow them to escape from the


specimen surface. The angular distribution for these backscattered electrons, most ofwhich have been multiply scattered but nevertheless still have energies close to that of theincident electron beam, is determined by diffraction from the crystal planes. Intensity issubtracted from regions of the diffuse backscattered electron distribution close to theelectro-optical axis of the microscope and intensity is added to the darker, high anglescattering regions. The result is a single pair of light and dark lines in the EBSD pattern for

Specimen

Incidentelectron beam

Patterncentre

+θ

−θ

Figure 5.21 Schematic illustration of the formation of an EBSD pattern from a crystallinesample in the scanning electron microscope.


each reflecting crystal plane. These lines correspond to the intersection of hyperbolaewith the plane of observation of the pattern, but since all the scattering angles are small,the light and dark lines appear sensibly straight and parallel. The distance between thelines is proportional to 2y, where y is the Bragg angle for each family of diffracting planes,and hence inversely proportional to the interplanar spacing, since at small values of y,sin y � y. Analysing the geometry of the crystal surface causing the EBSD pattern istherefore quite straightforward.

The bisector of each pair of parallel light and dark diffraction lines is the projection ofthe diffracting plane. With a little practice, it is possible to recognize prominentsymmetry zones directly from the EBSD pattern, even before the crystallographicorientation has been analysed by the computer software. To improve the intensity ofthe signal we can increase the probability that backscattered electrons will escape fromthe surface by tilting the sample through a large angle, leading to considerable fore-shortening of the backscattered electron image. This foreshortening is proportional to theangle between the optic axis of the microscope and the normal to the surface of thediffracting crystal. Foreshortening is usually corrected automatically when displayingthe image data, but the image contrast is unchanged, and usually reflects the ‘shadowing’of topographical features by the inclined incident beam. The incident electrons in theinclined beam have a longer projected path length through the sample perpendicular tothe axis of tilt, so the distribution of back scattered electrons escaping from the surface isalso elongated in this direction. The resolution in this direction is therefore reduced andgrain boundaries are partially ‘blurred’ in the topological image. It follows thatcrystallographic analysis close to a boundary can be rather unreliable. Nevertheless,EBSD analysis has been successfully applied to individual grains with particle sizesbelow 1 mm, allowing the orientation relationship between small crystalline precipitatesand a matrix phase to be determined.

5.6.5 OIM Resolution and Sensitivity

So far, we have only discussed the determination of crystal orientation beneath a stationaryprobe (as in point microanalysis), that is, a probe positioned on a selected image pixel.When applied to a polycrystal, we usually wish to know the distribution of crystalorientations for the grains intersected by the plane of the sample section. This involvescollecting very large sets of EBSD data and presenting the results both quantitatively and inan image format. The first step is to define a grid of test pixels across the region of interest onthe sample surface. The total number of pixels in the image grid is critical, since itdetermines, with the dwell-time per pixel, the time required to collect the data. Even with afield emission source, the required image scan times are long, both with respect to thestability of the microscope and the patience of the operator. The second step is to bin thecrystallographic data from the solved EBSD patterns acquired from each pixel, usually on astereographic projection in which all the crystal orientations are mapped onto a single unitstereographic triangle (Section 1.2.3.4) (Figure 5.22). This information can be convertedinto an image of the microstructure, based on the change in orientation (or, sometimes,crystal structure) from one image region to another. Colour coding is commonly used forsuch micrographs, where each colour defines either an orientation range for the grains, or avariation in crystallographic structure (Figure 5.23).


5.6.6 Localized Preferred Orientation and Residual Stress

OIM starts to reveal its full potential when preferred orientation is present (Figure 5.24) andselected surface orientations are clearly displayed, both in the orientation stereogram and inthe corresponding colour-coded image, but once this information is available, it alsobecomes possible to answer more complex microstructural questions. For example, it is

1/2 h5 h10 h20 h50 h100 h

[001]

[111]

[101]

Figure 5.22 Orientation of gold particles equilibrated on a ð10�10Þ surface of sapphire as afunction of annealing time.

Figure 5.23 OIMmicrograph of a polycrystalline In2O3 film grown on a (001) MgO substrate.The colour assigned to the In2O3 grains is correlated with their orientation relative to thesubstrate surface. Reproduced from J.K. Farrer and C.B. Carter, Texture in Solid-State Reactions,Journal of Materials Science, 41(16), 5169–5184, 2006. Copyright 2006 with permission fromSpringer Science and Business Media. (See colour plate section)


well-known that certain orientation relationships may be preferred when neighbouringgrains are separated by so-called special boundaries. Using OIM it is possible to detectthese boundaries and explore the statistics of special boundary formation. In some cases, thefraction of special boundaries can be controlled by thermomechanical treatment of thematerial to yield improved engineering performance. A closely related question concernsthe spatial distribution of preferred orientation in a polycrystalline material. X-raydiffraction can only determine the average distribution of grain orientations in a samplewith respect to the sample geometry, but with OIM it is possible to explore the localpreferred orientation and its distribution in the material. These local variations are expectedto be important in controlling plastic flow through a die or during a forging operation.

Although EBSD is considerably less sensitive to changes in lattice spacing than X-raydiffraction, which can detect lattice strains of the order of 10�5, the EBSD data can becombined with finite element analysis based on the known thermal anisotropy and elasticconstants of alumina in order to map the calculated residual stress distribution as the stressinvariant (s11þs22) and maximum principal stress (s11). Figure 5.25 shows the calculatedspatial distributionof themaximumprincipal stress (s11) for the same regionsof the aluminasamples in theOIMimages of Figure5.24.Thestressesarestrongly localizedtotheboundary

Figure 5.24 OIM used to study preferred orientation in polycrystalline alumina. (a) A sampleprepared from conventional alumina powder has nearly random crystal orientations. (b) Asample prepared from aligned platelets is highly oriented. The stereographic colour coding forboth micrographs is the same and is shown in (c). Reprinted with permission from V.R. Vedula,S J. Glass, D.M. Saylor, G.S. Rohrer, W.C. Carter, S.A. Langer and E.R. Fuller, Residual-StressPredictions in Polycrystalline Alumina, Journal of The American Ceramic Society, 84(12),2947–2954, 2001. Copyright (2001), with permission from Blackwell Publishing Ltd. (Seecolour plate section)


regionsanddramaticallyreducedbythepreferredorientationofthesecondsample.While theOIM data make this analysis possible, the analysis is far from straightforward. Directinformationon the tensile strains is not available and the annealinghistoryof the samples hasbeen assumed. Nevertheless, the results clearly illustrate the anisotropic distribution of theresidual thermal stresses and the strong effect that the microstructure can have.

Most OIM results that have been published are from single-phasematerials, where all thegrains possess the same crystal structure. There is no reason, in principle, why othermicrostructural morphologies should not be studied by OIM. For example, two-phaseeutectics might be expected to exhibit coupled growth of the two constituent phases. OIMresearch of such systems would be expected to throw some light on the effect of the rate ofheat transfer and casting additions on the crystallography of such coupled growth.

5.7 Specimen Preparation and Topology

The most important specimen requirement in SEM is that electrostatic charging of thesurface should be avoided, since the associated charge instability will lead to unstable

Figure 5.25 Distributionof themaximumprincipal stress calculated for the same regions of thealumina samples shown in Figure 5.24 by inserting the OIM EBSD data into a finite elementprogram. (a) Note the stress scale and the strong localization of the stress near the boundaries.

(b) In the highly oriented sample the residual stress is dramatically reduced. Reprinted withpermission fromV.R.Vedula, S.J.Glass,D.M. Saylor,G.S. Rohrer,W.C.Carter, S.A. Langer andE.R. Fuller, Residual-Stress Predictions in Polycrystalline Alumina, Journal of The AmericanCeramic Society, 84(12), 2947–2954, 2001. Copyright (2001), with permission from BlackwellPublishing Ltd. (See colour plate section)


secondary emission, destroying both the resolution and the image stability. Charging ofnonconducting samples can be prevented or inhibited, either by operating at low beamvoltages or by coating the samples with a thin layer of an electrically conducting film.

The specimen must also fit into the sample chamber without seriously constraining thegeometrical freedom to manipulate the sample in the microscope column. This is neededboth to select any area of interest for examination and to tilt the specimen surface at anyrequired angle with respect to the optic axis. As noted previously, most SEM specimenchambers readily accept large specimens (some well over 10 cm in diameter).

Apart from these two requirements, the specimens must also be stable in the vacuumsystem and under the electron beam. They should be free of any organic residues, such as oiland grease, which might lead to the build up of carbonaceous contamination, either on thespecimen,or in theelectro-optical system,orwithinawavelengthdispersivespectrometer,orelsewhereinthesamplechamber.Looseparticlesneedtoberemovedfromthesamplesurfacebefore insertion in the microscope, usually by ultrasonic cleaning in a suitable solvent,followed by rinsing and drying in warm air. These precautions are especially important forlowvoltage, high resolutionSEM.Atprobeenergiesbelow1keV,all the secondaryelectronscome from a region that is very close to the sample surface. Electron beam inducedcarbonaceous contamination then becomes the primary source of secondary electrons.

5.7.1 Sputter Coating and Contrast Enhancement

Coating of the samples to enhance contrast and improve electrical conductivity is usuallyperformed in a sputtering unit, as discussed previously for nonconducting transmissionelectron microscope specimens (Section 4.2.4). Two types of coating are commonly used,either a heavymetal or an amorphous carbon film.Heavymetal coatings of a gold–palladiumalloy that have been deposited with a 5 nm particle size on the sample surface only interfereswith the resolution at the highest magnifications and these coatings improve the contrastappreciably. However,metal coatings do interferewith chemicalmicroanalysis, andmay notbe suitable if the best resolution is to be achieved. Carbon coatings can be deposited with amuch smaller particle size (�2 nm); this is usually below the resolution limit of theinstrument. Carbon coatings do not improve the contrast, but they are mandatory fornonconducting samples intended for microanalysis at accelerating voltages above �5 keV.

The best solution to electrostatic charging of the specimen is to reduce the beam voltage,but of course this cannot be done if microanalysis is required and the excitation energy forthe characteristic lines of interest is more than a few keV.

5.7.2 Fractography and Failure Analysis

Amajor area of application for the scanning electronmicroscope is in fractography (that is,the imaging of fracture surfaces) and failure analysis, not only for engineering metals andalloys, but also for plastics and composites (polymer, ceramic or metal matrix), as well asengineering ceramics and semiconductor devices (Figure 5.26). The morphology of manyother classes of materials has also been studied using the same scanning electron micro-scope techniques: natural and artificial foams, textiles and fibres, as well as systems that areunstable at ambient temperature and require cryogenic cooling by liquid nitrogen. Figure5.27 shows a few examples.


Figure 5.26 Some examples of failures in engineering materials imaged by SEM: (a)mechanical fatigue failure in steel (from the Metals Handbook, American Society forMetals); (b) brittle failure in porous TiCN; (c) failure of a fibre-reinforced polymer matrixcomposite (courtesy of A. Siegmann).


Figure 5.27 Some other materials studied by SEM: (a) paper; (b) bone; (c) wood. (FromCellular Solids; Structure and Properties, L.J. Gibson and M.F. Ashby, Pergamon Press).

There are some simple but important guidelines to be followed if the maximuminformation is to be extracted from scanning electron fractography. Although these havebeen covered to some extent in previous sections, they will bear repeating:

1. The sample selected for examination in the microscope must bear a known geometrical,spatial and orientation relation to the original engineering system or component fromwhich it was taken. Without this information it may be difficult to evaluate thesignificance of any fractographic observations.

2. The surface should not be damaged or altered in any way by the specimen preparationprocedure. It is unbelievable how often the two halves of a failed component are �fittedtogether� by an unthinking investigator prior to microstructural examination. The resultis superficial damage to the failure surface and, too often, the destruction of importantevidence of the cause of failure.

3. The specimen should bemounted in themicroscope according to the specimen stage x–ycoordinates and the axes of specimen tilt, for example, parallel and perpendicular to aknown direction of crack propagation. A little forethought can save a lot of frustration,not to mention microscope time, and greatly simplify the process of investigation.

4. Images should be recorded over the full range of magnifications that are found to showany significant microstructural features. It is especially important to be able to relate thesurface topography of the microstructural features to the results of any other observa-tions. If these initial observations were purely visual, then an initial magnification in thescanning microscope of ·20 is not too little. A good procedure is to locate features ofinterest by first scanning rapidly over the sample, experimenting with differentmagnifications, and only then starting to record a series ofmagnified images, identifying


features in the magnified images by their geometrical relation to details that can beobserved with the naked eye. A factor of ·3 between image magnifications ensures thatabout 10% of the surface area that is recorded at the lower magnification will always bevisible in the image taken at the next higher magnification (Figure 5.28).

5.7.3 Stereoscopic Imaging

As noted at the beginning of this chapter, the scanning electron microscope is unique in itsability to focus and resolve detail over a large depth of field, parallel to the optic axis of theincident beam. This information can be extracted by recording a pair of images at the samemagnification and from the same area of the sample, but at different angles of tilt of thesample with respect to the optic axis of the microscope. The two images, a stereo pair,correspond to two observations of the specimen surface taken from two different points ofview. If the angle of tilt is well-chosen, then the geometry is equivalent to the stereoscopic,three-dimensional visual image observed by the superposition of the retinal data transmit-ted to the brain by the left and right eyes (Figure 5.29).

Several commercial systems have been developed for viewing stereoscopic imagesdirectly in the scanning microscope, for example by a lateral displacement of the axis ofscan of the probe, either for two sequential scan sequences or for alternate lines of the x-sweep in a single scan. These commercial systems have never been particularly popular,since it is equally easy to record a pair of images without disturbing the scanning coilsettings, but rather by mechanically tilting the specimen itself.

Given that the microscope screen is commonly viewed at a distance of some 30 cm(a comfortable reading distance) and that the eyes are set some 5 or 6 cm apart, the requiredangle of specimen tilt needed to give an impression of depth identical to that received by thehuman eye is �12�. This is equivalent to setting the depth ‘magnification’ equal to thelateralmagnification in the image. If the tilt angle is less than�12� then this will reduce thesensitivity to depth, and this may be necessary for very rough surfaces. Tilt angles greaterthan�12� will amplify the impression of depth and thismay be useful for recording shallowfeatures. An example of a stereo-pair is given in Figure 5.30.

Pairs of stereo images, having their axis of tilt aligned accurately and placed withcorresponding points on the images at the approximate separation of the observers’ eyes,may, with a little practice, be viewed by most people at the normal reading distance, just byfocusing the eyes on infinity and without additional optical aids. More commonly,commercial stereo viewing systems are available, and can provide striking in-depthinformation that is often unobservable in a two-dimensional recorded image.

5.7.4 Parallax Measurements

In addition to the visual impact of a stereo image, it is also possible to extract quantitativeinformation on thevertical distribution of features along the axis of the incident beam.This isaccomplished by measuring the horizontal displacement, perpendicular to the axis of stereotilt, for the same features recorded perpendicular to the tilt axis in the two images of the stereopairs, (Figure 5.31). This displacement is termed parallax and its value is given by:

xL�xR ¼ 2h cos y ð5:2Þ


Figure 5.28 Ductile failure in molybdenum at increasing magnification reveals both thegeneral topographical features and the fine details of the fracture surface.


where xL and xR are the projected distances in the two images measured from any fixedposition perpendicular to the axis of tilt y, h is the ‘height’ difference between the twofeaturesmeasured along the optic axis of the incident electron beam, and the stereo pair havebeen recorded at a tilt angle of �y with respect to the incident beam normal.

Parallax measurements are useful for checking the thickness of surface films, measuringthe height of growth steps, or determining plastic slip displacements. They have also beenused to estimate the roughness of ground and machined surfaces, and to determine thefractal dimensions of a fracture surface.

Left-EyeEquivalent

Right-EyeEquivalent

2θ

ImagingSystemLeft-Eye

ViewpointRight-EyeViewpoint

2θ

Figure5.29 The twin images observedby the left and right eyes are equivalent to the imagepairrecordedbefore and after tilting a specimenbya knownangle about a direction perpendicular tothe optic axis.

Figure 5.30 A stereo pair of micrographs from the surface of a brittle fracture. Hold the book ata comfortable reading distance and then focus the eyes at a distant object above the page. Itshould be possible to fuse the two images into a single three-dimensional view of the surfacetopography (most observers fail at the first attemptwhile someareneverable toviewa stereopairwithout a suitable viewing system).


5.8 Focused Ion Beam Microscopy

In FIB microscopy the specimen probe is a beam of high energy ions focused onto thespecimen, as the name implies. It follows that this instrument is not really an electronmicroscope. Nevertheless, the instrument has many structural features in common with thescanning electron microscope, so that it is appropriate to discuss the principles andapplications of FIB microscopy.

In addition to not really being an electron microscope, the FIB is also unusual in that ithas, until recently, been primarily a tool for the electronics industry, employed for themicromachining of electronic and optronic device components. The first commercial unitswere introduced in the 1980s, primarily for device development as well as for industrialresearch. Today, FIB facilities are increasingly common in research institutes and someuniversities, where they are able to provide unique materials characterization information.In addition to this role inmicrostructural characterization, they are also able to function asplatforms for micromachining, with tolerances that are in the nanometre range.

The key enabling technology for the FIB is the liquidmetal ion source, which operates byfield evaporation of liquid metal at a tungsten tip that is maintained at a high positivepotential. These sources were originally developed for ion propulsion in space at thebeginning of the 1960s. The theory was developed by Taylor in 1964, who demonstratedthat an electrically conducting liquid drop should be drawn into a sharp cone whose tipradius was small enough to generate the very high electric fields required for fieldevaporation of a liquid metal without thermal activation. As the accelerating voltageapplied to the source is increased, the ion current also increases, increasing the source radiusbut leaving the electric field at the cone tip unchanged and equal to the evaporation field forthe dominant ion species (which are usually doubly charged).

Two other technologies preceded the development of the FIB microscope. In 1955Mueller and his co-workers studied both field ionization and field evaporation andwere able

2θO

xL –xR

h

Tiltangle

Incidentbeam

Surface trace

Surface trace

Figure 5.31 The parallax geometry. The difference in separationof two features seen in a stereopair andprojectedperpendicular to the tilt axis is directly related to thedifference inheight of thetwo features, measured along an axis parallel to the incident beam.


to demonstrate atomic resolution in the field-ion microscope. The source of the imagingions in field ion microscopy was an inert gas (usually helium). The sharpened tip of a metalneedle was the ion source in the case of field evaporation from the solid state. Only 5 yearslater, Castaing and his co-workers, who had previously developed the first electron probemicroanalyser, subsequently demonstrated microanalysis of polished metallographicsections by the mass spectroscopy of ions that were sputtered from a sample surface underbombardment by a focused beam of inert gas ions. The secondary ion mass spectrometer(Section 8.3) has a remarkablemass sensitivity, but this is not easy to calibrate. The field ionmicroscope has since developed into the atom probe tomograph (Section 7.3). The atomprobe has improved both the mass sensitivity and the resolution for microanalytical studiesto nanometric dimensions, but is limited in the range of engineering materials that can bestudied by the need for adequate electrical conductivity and mechanical strength.

5.8.1 Principles of Operation and Microscope Construction

Figure 5.32 shows schematically the geometry of the FIB ion source. The first liquid metalion sources were fed by the flow of the liquid metal through a capillary, but much bettercontrol is achieved by allowing the metal from a heated reservoir to wet a tungsten needleand then be sucked into a cone source of ions at the tip of the charged needle. The ion

Tungsten Needle

Reservoir of

Melted Metal

Liquid Metal Film

Ion Beam

Taylor Cone

Filament

Current

Figure 5.32 Schematic drawing of the FIB liquid metal ion source.


currents from a liquid metal source can far exceed those generated by field ionization of agas at a sharp metal tip, while the continuous supply of liquid ensures that the source sizeremains effectively constant during operation. By comparison, field evaporation from asolid metallic needle would rapidly blunt the tip radius.

The choice for the liquid metal is limited. The most successful sources so far employgallium (atomic number 31, melting point 29 �C). Liquid gallium partially wets tungstenand the comparatively heavy gallium ions sputter most solid surfaces efficiently. With atungsten tip radius of about 10 mm (rather larger than that used for a field emission source),the liquid is drawn into a cone of similar length by the applied electric field.

The design of the microscope column (Figure 5.33) differs substantially from that of aconventional electron microscope. For example, electromagnetic focusing is impracticalbecause of the large mass of the ions, and electrostatic lenses must be used to focus theprobe. Sputtering damage within the microscope also needs to be minimized. This isachieved by a sequence of beam-blanking plates with both variable and fixed apertures.Electrostatic quadrupole and octopole assemblies are used to position and scan the ionbeam. The position and orientation of the sample is accurately controlled on a eucentricstage that permits tilt about all three coordinate axes, as well as full Cartesian, x, y and z,control of the sample positionwith respect to the optical axis of the ion beam. This specimenstage also ensures that the axes of specimen tilt remain perpendicular to, and pass through,

Liquid Ion Source

Suppressor Assembly

Extractor Cap

Beam Acceptance Aperture

Asymmetric Electrostatic Lens

Variable Aperture

Quadrupole

Beam Blanking PlatesBlanking Aperture

Octopole

Asymmetric Electrostatic Lens

Detector

Sample

Figure 5.33 Schematic drawing showing the components of the FIB column.


the optic axis. A wide range of FIB accessories are available, including an option forscanning of the sample by a beam of high energy electrons, as in the scanning electronmicroscope. Microscopes that include both ion beam and electron beam probes are oftentermed ‘dual beam’ instruments (Section 5.8.3). Gas injection systems are also available forchemically-assisted etching of the surface or for deposition of thin films in selected areas(see below).

5.8.2 Ion Beam – Specimen Interactions

The interaction between the focused high energy beam of metal ions and the specimencomes under three headings that are sometimes difficult to separate during operation:

1. Micro-machining by sputtering of matrix atoms from the surface. This process can beused to mill and section the sample to an accuracy of approximately 1 or 2 nm.

2. Secondary excitation, primarily by the generation of secondary electrons by the incidention beam.The secondary electrons can be collected and used to image the sample surfacewith a resolution down to about 2 nm.

3. Sub-surface radiation damage, associated with lattice damage, primarily in the form ofpoint defects (vacancies and self-interstitials), but also due to ions from the ion beam thathave been injected into the solid.

Figure 5.34 summarizes these effects. In general, an impinging high energy ion willgenerate so-called “knock-on” damage when it collides with an atom of the solid. Themaximum kinetic energy E that can be transferred in a Newtonian model to an atom of thematrix is given by:

EE0

¼ 4m1m2

ðm1þm2Þ2ð5:3Þ

whereE0 is the energy of the impinging ion andm1 andm2 are themass of the impinging ionand the struck atom, respectively. In electron irradiation damagem1�m2, and the electronbeam energy must exceed�150 kV before individual atoms can be displaced in a thin-filmspecimen viewed by TEM, even for low atomic number specimens. In a FIB instrumentevery high energy ion generates a cascade of multiple atomic displacements, but only asmall fraction of these result in sputtering of atoms from the specimen surface. If thethreshold for atomic displacement is assumed to be 25 eV (a commonly accepted value),then a 25 kV impinging ion may displace up to 1000 of the sample atoms in a collisioncascade. Only a small proportion of these will be sputtered. The remaining point defectsmay either anneal out or condense as radiation damage. Of course, the original impingingion may also be trapped beneath the surface. As in ion milling (Section 4.2.3), themicrofinish of the surface can be improved by tilting the sample so that the incident beamstrikes at a glancing angle.

Sub-surface radiation damage is a serious problem and cannot be ignored. Frenkeldefects are generated consisting of vacancy–interstitial pairs. Self-interstitials in metalshave very low activation energies for diffusion andmay be expected either to anneal out to afree surface or to annihilatewith vacancies before their concentration becomes high enoughto cause them to condense asmore extended lattice defects. Vacancies are anothermatter: inmost materials these point defects diffuse rather slowly at room temperature and their


concentration can increase rapidly before they condense into immobile clusters or smalldislocation loops. The concentration of injected ions is much less than the number ofFrenkel defects formed, but these injected ions can also condense and form defect clusterswhich are visible as ‘radiation damage’ strain fields in thin-film samples that have beenprepared for the transmission electronmicroscope by FIBmilling. The high energy injectedions incident on a crystalline sample also give rise to an additional phenomenon termed ionchannelling. If the ion beam is aligned along a prominent zone axis of the crystal, then theatomic mass distribution can focus the velocity vector of an incident ion along the crystalzone axis. This mass–lens focusing allows ions to tunnel for distances of several tens ofnanometres along the zone axes, depending on the atomic mass and the elastic constants of

Surface

Figure 5.34 Schematic drawing illustrating the damage collision sequence resulting from aprimary ion incident on a sample surface. Squares indicate vacancies.


the targetmaterial. This results in occlusion of the incident ionswell below the surface of thespecimen, as well as a strong crystallographic dependence for the escape of secondaryelectrons generated by the incident ion beam. Since secondary electrons can only escapeand contribute to a secondary electron signal if they are generated within a few nanometresof the surface, very few secondary electrons will be emitted when the ion beam is alignedwith a prominent zone axis. This leads to very strong crystallographic contrast in thesecondary electron image that has been generated by the primary ion beam. Figure 5.35shows this effect in a gold wire-bond. The secondary electron image in Figure 5.35(a) hasbeen excited by a high energy electron beam, while that in Figure 5.35(b) has been excitedby a FIB.

5.8.3 Dual-Beam FIB Systems

The FIB is a powerful tool for both characterization and micromachining when it iscombined with a high-energy scanning electron beam. In these dual-beam instruments, thescanning electron microscope column is usually mounted vertically above the sample, andthe ion beam source and column are attached at an inclined angle to the sample chamber.Both beams are focused onto the sample surface positioned at a set working distance(Figure 5.36). The electron beam is used to monitor the progress of ion beam milling ordeposition, while the two types of secondary image can be recorded from either the ionbeam probe or the electron probe, to provide a wealth of morphological information.

5.8.4 Machining and Deposition

The power of the FIB technology is best demonstrated when it is used to prepare samples insitu, using one of several techniques that we will list and then discuss in more detail:

1. Micromilling can be used to remove appreciable quantities of material, mimickingconventional mechanical machining in what amounts to turning, milling and trepanning

Figure 5.35 Secondary electron micrographs from a gold wire-bond generated using highenergy incident electrons or ions. (a) A 5 kV electron beam reveals the surface topology androughness at high resolution. (b) A 30 kV gallium ion beam provides crystallographicchannelling contrast from the individual grains of the bond.


operations. In dual-beam systems, the electron beam is used to monitor the millingprocess. Perhaps surprisingly, contamination of the FIB is not a serious problem,primarily because the total volume of material removed during micromachining isactually quite small, even though it may be substantial when compared with the size ofthe microcomponent that is produced by the milling process.

2. Gas-assisted etching is possible by injecting small amounts of an active gas through acapillary needle placed close to the sample surface. During ion beam etching, the gasreacts with the sample surface to form volatile species which are then pumped away bythe vacuum system. This is particularly useful if the sample contains atomic species withlow vapour pressures, since such materials tend to redeposit on or around the sample.Some materials such as copper, tend to ion mill anisotropically resulting in a roughsurface finish. Ion milling can then be improved using an appropriate etchant gas thatyields a planar, smooth surface.

3. Gas deposition of thin layers is performed by admitting a reactant gas through thecapillary needle. On exposure to either the ion or the electron beam the gas reacts at thesample surface (compare chemical vapour deposition, CVD). The deposition onlyoccurs under the electron or ion beam probe, so that quite complicated shapes can beselectively deposited with the help of a computer control program to manipulate thespecimen stage and determine the beam scans. In the course of ion beam assisteddeposition, both the deposit and specimen may be partially milled during operation.Even so, ion beam assisted deposition is significantly faster than electron beam assisteddeposition.

4. Cross-sectioning a sample is when the ion beam is used to prepare a section either forsubsequent SEM characterization, or for transfer to another experimental platform(Figure 5.37).This method, when employed on a dual-beam system, allows the user toscan the surface in SEM mode and select a region for sectioning before examining the

Figure 5.36 Schematic drawing of the dual-beam configuration with respect to the samplestage.


sub-surface morphology, all without removing the sample. This approach can be used toprepare specimens for TEM (as discussed in Section 5.8.5).

5. Serial sectioning can be done by repeatedly micromachining away layers from a solidsample and then assembling a composite three-dimensional image from the two-dimensional image data recorded from each of the layers separately. This process isdiscussed in more detail in Section 5.8.6.

Micromilling using FIB technology is possible to near-nanometre accuracy and has beenespecially successful in preparing sub-micrometre samples for mechanical testing andcomponents for microelectronic mechanical systems (MEMS). Fabrication for microelec-tronic device development is also often dependent on FIB technology. In the microelec-tronics industry, circuit editing is a major task that uses the FIB to cut throughmetallizationlines between the individual device components of a microelectronic circuit, or to depositnew metallic conductor lines (usually tungsten) between devices that were previouslyunconnected. Repair of lithography masks is also often possible in the FIB beam. With thedual-beam FIB, the options for fabrication and testing of sub-micrometre components islimited only by the initiative of the user (Figure 5.38).

Gas-assisted etching and deposition rely on gas injection from a capillary tube sited closeto the sample surface (Figure 5.39). Many systems are designed with two or more capillary

Figure 5.37 Secondary electron scanning microscope micrograph of a cross-section from anion-milled microelectronic device. A �FIB Box� has been cut into the surface of the specimen.


Figure 5.38 Secondary electron scanningmicrographof amicrometre-scale �etching� showingthe authors in a good mood. The specimen was prepared by FIB milling of a platinum-coatedsurface using data from a bit-mapped digital photograph.

Figure 5.39 Scanning iongenerated secondaryelectronmicrographof aTEMspecimenmountplaced adjacent to a gas injector. This assembly is used to admit metallo-organic or etch gasesacross the sample surface. ATEM specimen attached to the end of a tungsten nanomanipulatorneedle is just visible.


gas injectors, allowing CVD (or etching) reactions to be performedwith different gases andhence the deposition of components having different compositions.

The processes described above require assemblies that go beyond the dual-beamcombination of ion beam and electron beam columns. The gas injectors have to be carefullymanoeuvred above the specimen, without coming into contact with the specimen’s surface.Several stages are needed to mill a thin-film transmission electron microscope specimenfrom a sample in the FIB. One option uses a nanomanipulator consisting of a sharp tungstenneedle mounted at the end of a piezoelectric drive (Figure 5.39). Computer control of thespecimen location on the specimen stage is critical. The steppingmotors that are commonlyused for SEM stages are not sufficiently accurate. Piezoelectric-driven sample stages arepreferred, and computer-mapped positioning of the sample, on the stage, is essential toavoid driving the specimen into the very expensive detectors and accessories.

5.8.5 TEM Specimen Preparation

The use of FIB technology to prepare thin-film sections for TEM has proved revolutionaryformany laboratories engaged in high resolutionmicrostructural studies. One needs to keepin mind that, as in conventional ion milling, FIB milling introduces beam damage andgallium ion implantation. The artifacts associated with specimen preparation must beunderstood in order to interpret the microstructure seen in the thin-film section. The majoradvantage of FIB for TEM thin-film preparation is that a specific region can be pre-selectedfrom a component before ion milling the specimen. If a dual-beam system is used, then thequality of the TEM specimen can be checked using the scanning transmission electronmicroscope mode (Section 4.7) in the dual-beam system. Twomethods commonly used forpreparing thin sections are discussed below.

5.8.5.1 TheH-barMethod. Thismethod has the advantage that it does not require a dual-beamFIB system, although it is easier to implement in a dual-beamFIB. The process beginswith the mechanical slicing and polishing of a cross-section of the material of interest.Mechanical reduction of the sample thickness by grinding saves time in the FIB, and isalways a good idea.

After mounting the mechanically thinned specimen in the FIB, a protective coating,such as platinum, should be deposited on the surface in the region to be thinned(Figure 5.40). This coating is to prevent high energy ion beam surface damage and iscritical if the surface region of the specimen is to be characterized by TEM. In general,it is better to first deposit the protective coating using the electron beam, and then onlylater micromill with the ion beam. This will avoid damage to the surface region duringion beam deposition.

The ion beam is now rastered over two regions of the specimen, separated by a �1mmthick region in the centre (the beam of the H-bar) (Figure 5.40). In this way the section isthinned until a �1mm lamella is left at the centre of the chosen region, supported by therelatively thickmaterial at the edges. Final thinning of the lamella is then done using amuchlower ion beam current, until the section is thin enough for TEM. In a standard FIB this isjudged by measurement of the lamella thickness from a secondary electron image madeusing a rapid ion beam scan. In a dual-beam system the milling of the thin section can becompleted while an electron beam is scanned independently over the sample surface.Forwardscattered electron images acquired using a STEM detector (Section 4.7), are then


used to determine if the specimen is sufficiently thin. High-angle annular dark-field(HAADF) STEM (Section 4.7) is an even better indicator of the success of the preparationprocess, since regions of high average atomic number will appear in dark contrast when thespecimen is thick, but then appear brighter when the specimen is thin enough for theelectron beam to be transmitted. A final ion beam ‘polish’ at a low ion energy (2 kV) is aneffective method of removing surface layers that have been damaged by the previousmilling stages and eliminating any surface contamination. An example of an H-barspecimen is shown in Figure 5.41.

5.8.5.2 The Lift-out method. The lift-out method is more precise than the H-bar method,since the exact sample region can be selected from which a thin-film specimen is to beprepared without the need for preliminarymechanical thinning. However, this method doesrequire a dual-beam FIB that is equipped with a nanomanipulator.

The sample is first viewed using the SEM facility in the dual-beam instrument, and aregion of the sample is selected for thinning. As in the H-bar method, a protective coatingshould be deposited on the sample surface over the region fromwhich the specimen is to be

Pt

Incid

ent E

lectro

ns

Incident Ions

Figure 5.40 Schematic illustration of the H-bar method for transmission electron microscopespecimen preparation. Prior to ion milling, a thin platinum coating should be deposited on thesurface of the sample in order to protect the specimen. Two �FIB boxes� are cut either side of thelamella with the FIB. The arrow indicates the optic axis of the incident electron beamwhen thespecimen is subsequently inserted in the transmission electron microscope.


cut, in order to avoid surface damage by the high energy ions. Two depressions, or ‘FIBboxes’, are milled into the sample on either side of the selected region, leaving the boxesseparated by a �1mm lamella (Figure 5.42). The boxes are not cut exactly normal to thesample surface, but have a small taper angle running from the surface to the bottom ofthe lamella. The lamella is then cut almost entirely free of the surface by the ion beam, in theshape of a ‘U’ (Figure 5.43).

The nanomanipulator is now inserted. This consists of a sharp tungsten needle attached topiezoelectric drive motors. The more advanced manipulators are computer-controlled andtheir movement calibrated for the viewing direction on the SEM monitor, so that thetungsten needle can be brought gently into contact with the lamella. A gas injector is nowintroduced, and either platinum or tungsten is deposited by electron or ion beam excitationto reinforce the point of contact of the tungsten needle with the thin-film lamella(Figure 5.44). The lamella is then completely separated from the substrate by the ionbeam, and transferred to a TEM specimenmount that is locatedwithin the vacuum chamberbut elsewhere on the stage.

The TEM specimen mount, (or ‘grid’), is usually of a 3mm diameter, half-ring(Figure 5.45). The lamella is gently moved next to the grid, and platinum or tungsten isnow deposited to join the lamella to the grid. Ion milling is then used to cut the join betweenthe sample and the nanomanipulator. Final ion thinning of the �1mm thick lamella isperformed to reach the required specimen thickness, exactly as for the H-bar technique. The

Figure 5.41 Secondary electron SEMmicrograph of anH-bar specimen prepared using a dual-beam FIB. Prior to ion milling, recognition markers were ion milled into the platinum surface.These markers allow automatic image recognition by the computer control software during themilling process. The highly anisotropic milling adjacent to the lamella is termed the �curtainingeffect�, and is associated with rapid removal of material at high beam energies and incidentangles in these regions.


Pt

Figure 5.42 �FIB boxes� cut into the surface of a sample, in the first stage of the lift-out TEMspecimen preparation procedure.

Pt

U-Cut

Figure 5.43 A �U-Cut� milled into the lamella that almost entirely separates the lamella fromthe substrate.


ability to tilt the TEM specimen towards the ion-beam andmonitor the microstructure usingSTEM imaging in the electron beammakes this rather complex process completely feasible,flexible and efficient.

5.8.6 Serial Sectioning

A final option, that has only recently begun tomake an impact inmaterials characterization,is the serial sectioning of microstructural samples using the dual-beam FIB. The concept ofserial sectioning dates back to the early days of optical metallography (the first half of thetwentieth century). Repeatedmechanical grinding and polishingwas employed to prepare aseries of micrographs as a function of the depth of material removed from the sample. Thisprocess was slow, inefficient and inaccurate. Just as serious, the sectioning interval wasseldom less than 20mm. The results seldom justified the effort. While serial sectioning inthe FIB is essentially the same concept, the sectioning accuracy is in the range of 10–20 nmand the full resolution of the secondary electron scanning image is now available to recordthe results. The ion beam is used to mill and polish the sections, while either the ion orelectron beam is used to acquire scanning electron micrographs of the same area as afunction of depth. The process is still time-consuming and is confined to a region of the

Figure 5.44 The nanomanipulator is attached to the surface of the lamella TEM specimen bydepositing a small amount of platinum or tungsten at the point of contact.


sample surface only a few square micrometres in area, but the information gained from theserial sectioning of a complex, solid-state device can be impressive. The FIB technologyremoves layers from a solid samplewhose thickness can be controlled from several tenths ofa micrometre to just a few nanometres (Figure 5.46) Possible applications in materialsscience and engineering could include:

1. the removal of successive layers from a microelectronic device to reconstruct a three-dimensional image from the two-dimensional projected images;

2. determining the connectivity and contiguity in a mesoporous catalyst substrate;3. quantitative analysis of the branching morphology of dendritic cast structures;4. analysing the interconnectivity of the phases in a spinodal alloy.

Summary

In the scanning electronmicroscope a high energy electron beam is focused into a fine probethat is inelastically scattered when it strikes the surface of a solid sample. The inelasticallyscattered electrons generate several signals from the sample that can be collected andamplified. An image is formed by scanning the probe beam across the sample surface in adigitized television raster and displaying one or more of the collected signals on a monitorthat has the same time-base as the probe scan. The most commonly used signal is fromsecondary electrons, but characteristic X-rays, high energy backscattered electrons, visiblecathodoluminescence and the net specimen current have all been used to acquire micro-structural information from samples examined in the scanning electron microscope. Theincreasing availability of field emission sources for the electron beam has greatly improvedthe performance of both TEM and SEM.

In the scanning electron microscope the beam is focused by an electromagnetic probelens to a diameter thatmay be as little as 2 nm.However, the probe current decreases rapidlyas the probe diameter is reduced, and some signals used to acquire microstructural datarequire much larger probe diameters (up to 1mm). This is especially the case for the

Figure 5.45 Schematic drawing of a 3mm diameter half-ring, used for mounting a thin TEMspecimen lamella in the lift-out method.


characteristic X-rays coming from the individual chemical constituents of a sample. Theseare excited by the incident beam with rather low efficiency, since the excitation cross-section is small. The resolution in anX-ray image is not only limited by the photon countingstatistics, but also by the size of the excitation volume beneath the sample surface.

The characteristic X-ray signal is an important source of information for the quantitativedetermination of the microchemistry of the sample (Chapter 6), but can also be used simplyto demonstrate the presence of one or other constituent in any specific region of the sample.The characteristic X-rays are generated in a volume of material beneath the probe that is ofthe order of 1mm in diameter, corresponding roughly to the depth of penetration of theenergetic electrons into the sample and their lateral dispersion by inelastic scattering. TheX-ray signal can be displayed in three distinct formats:

1. An X-ray spectrum, in which the intensity of the signal from a selected region isdisplayed as a function of either the X-ray energy or its wavelength.

2. An X-ray line-scan, in which the intensity of the characteristic signal from one or moreelements are collected and displayed as a function of probe position along the scan line.(An example would be a scan across an interface or a second-phase particle.)

3. The X-ray data can be viewed as an elemental image map in which all photons arrivingwithin a given energy window are displayed and recorded as colour-coded dotswhose position in the image is correlated with the position of the beam at the timeof detection.

Figure 5.46 Reconstructed three-dimensional morphology of a dendritic microstructure in aPb–Sn alloy, produced by serial sectioning in a FIB. From J. Alkemper and P.W. Voorhees, Three-Dimensional Characterization of Dendritic Microstructures, Acta Materialia, 49(5), 897–902,2001. Copyright (2001), with permission from Elsevier.


High-energy backscattered electrons are useful because the intensity of the backscatteredsignal reflects the mass density or average atomic number of the sample and not just thesurface topology. The brighter regions are therefore a clear indication of denser materialwith a higher average atomic number. Nevertheless, the secondary electron signal is veryoften the most useful of the several signals that may be collected. There are two reasons forthis. First, the number of secondary electrons emitted per incident high energy electronexceeds by orders of magnitude the electron current in the primary beam, and, secondly,these secondary electrons can be collected with close to 100% efficiency. The secondaryelectron signal originates in the surface layers of the specimen, since these low energyelectrons have a very limited mean free path in the solid sample. The secondary electronscan be generated both by the incident beam, as it enters the sample and by backscattered,high energy electrons that are leaving the sample. The secondary electron resolution neednot necessarily be degraded by inelastic scattering of the primary beam and because of thehigh secondary electron flux, there are no significant statistical limitations on the resolutionin the secondary electron image. It follows that the ultimate resolution in a secondaryelectron image should be determined primarily by the ability to focus the probe beam, and istypically�2 nm. If the secondary electrons generated by the flux of backscattered electronsare also collected, then the resolution will be degraded to of the order of 20–50 nm.

Other types of signal are also available in the scanning electron microscope: the beamcurrent to the sample or cathodoluminescence, but these are usually of secondary importance.

Specimen preparation for the scanning electron microscope is straightforward, althoughit must be remembered that less stable specimens, such as polymers and biological tissues,may be degraded by the high energy electron beam and give rise to contamination of boththe sample and themicroscope column. For nonconductive specimens a conductive coatingis commonly required to prevent charging, unless very low beam energies are used. Suchcoatings often enhance the image contrast. In general, contrast in the scanning electronmicroscope may be associated with both surface topology and variations in mass density oratomic number. It is not always easy to separate these two sources of contrast.

Insufficient use ismade of stereoscopic analysis. By tilting the sample about a known axisand recording two images at different tilt angles, a stereo image can be observed in whichthe depth distribution of themicrostructural features is clearly evident. In addition, accuratemeasurements of the lateral displacement of features in the two components of the stereoimage, that is, parallax permits their displacement along the optic axis, normal to the planeof observation, to be determined. Stereo imaging in the scanning electron microscopeallows the surface of rough samples to be viewed in three dimensions.

The backscattered electrons also contain diffraction information on the crystallographyof the sample. This information can be extracted by using a charge-coupled device (CCD)camera and the electron backscatter diffraction (EBSD) pattern interpreted automaticallyusing appropriate computer software. In a polished, polycrystalline sample the orientationsof the different grains can be colour-coded and displayed in a digitized image of the grainmorphology, a mode termed orientation imaging microscopy (OIM).

Finally, a focused ion beam FIB can also be used as the specimen probe in the FIBmicroscope. In this instrument the ionbeamfroma liquid-metal field-evaporation source canbeused, either to generate a secondary electron signal, or to micromachine the specimen surface.

If the incident ion beam is aligned along a zone axis of a crystal, the ions may ‘channel’beneath the surface. Channelling greatly reduces the secondary electron yield from the


surface, resulting in strong crystallographic contrast in the secondary electron imagegenerated by the ion beam that is observed. Such contrast is absent if a high energy electronbeam is used to excite the secondary electron signal.

Dual-beam FIB instruments are available that incorporate, in separate optical col-umns, a field emission electron source in addition to the ion beam source. Such aninstrument can be used to micromachine thin-film transmission samples in situ andexamine them in the same FIB instrument operated in a scanning transmission electronmicroscope mode.

Bibliography

1. J.I. Goldstein and H. Yakowitz(eds), Practical Scanning ElectronMicroscopy, PlenumPress, London, (1975).

2. O.C. Wells, Scanning Electron Microscopy, McGraw-Hill Book Company, London,(1974).

3. J.J. Hren, J.I. Goldstein and D.C. Joy (eds), Introduction to Analytical ElectronMicroscopy, Plenum Press, London, (1979).

4. L. Reimer, Scanning Electron Microscopy: Physics of Image Formation andMicroanalysis, Springer, Berlin, (1998).

5. L. Reimer, Image Formation in Low-Voltage Scanning Electron Microscopy, SPIEOptical Engineering Press, Bellingham, WA, (1993).

Worked Examples

Once again we demonstrate the techniques we have discussed. We use SEM for two quitedifferent material systems: polycrystalline alumina and a thin film of aluminium depositedby CVD on a TiN/Ti/SiO2-coated silicon substrate. We also examine a wire-bond from amicroelectronic device using FIB.

The first example is polycrystalline alumina. As always, it is important to define thequestions we want answered before preparing our specimens or specifying the characteri-zation techniques. For our alumina we wish to check for residual porosity, determine thegrain size, and establish that the grain boundaries are free from secondary phases. Thepresence of residual porosity and grain size measurements can best be determined by SEM.However, it is difficult to detect small secondary phase particles or glass at grain boundariesby SEM, so TEM would be required for this (Chapter 4).

We can prepare SEM specimens from alumina in two ways: either by mechanicalpolishing, down to a sub-micrometre diamond grit polish, followed by thermal etching toform grain boundary grooves on the polished surface, or by breaking a suitable mechanicalspecimen, so that we can examine the fracture surface (and learn something about thefeatures determining the fracture strength). Alumina is an electrical insulator, so we wouldusually coat the surface of the specimenwith a conducting layer prior to SEM.However, wehave access to a lowvoltage SEMwith a field emission gun, sowe can use a lowacceleratingvoltage to minimize charging of the insulating surface.


Figure 5.47 shows SEMmicrographs of a polished and thermally etched alumina sample.Two micrographs of the same region are shown. The first was recorded using secondaryelectrons at an accelerating voltage of 5 kV, while the second was recorded at 20 kV, alsousing secondary electrons. The higher accelerating voltage leads to unstable surfacecharging of the specimen, which affects both the secondary electron emission coefficientand the trajectories of the secondary electrons that are collected. The image is blurred, andunsatisfactory for morphological analysis.When using 5 kVelectrons, the incident electroncurrent is compensated by the exiting secondary and backscattered electrons, so charging isreduced, and the image is free of these distortions.

Secondary electrons are generated throughout the depth of penetration of the incidentbeam, but lowering the accelerating voltage helps to ensure that the secondary electrons areonly generated in regions very close to the surface. These have been termed type 1secondary electrons (SE1). With a field emission gun and a specialized secondary electrondetector (discussed below), it is possible to detect the relatively few SE1 electrons released

Figure 5.47 SEM micrographs of thermally etched, polycrystalline alumina, recorded using5 kVelectrons tominimize charging (a), and then20 kVelectrons (b).Noconductivecoatingwaspresent on the specimen.


at low accelerating voltages (Figure 5.48). The resolution in Figure 5.48 is sufficient toresolve very fine surface facets formed during thermal etching.

Secondary electrons are generated from the near-surface region by the primary beam(SE1 electrons), but they can also escape from the sub-surface if the voltage is morethan a few hundred volts (SE2 electrons). In addition, a secondary electron current isgenerated by the backscattered electrons in a zone well beyond the diameter of theincident probe, as well as by stray electrons striking the microscope chamber, thecolumn or the specimen holder (SE3) electrons. The secondary electron detector isusually located within the chamber, above and to the side of the specimen, so the SE3electrons can contribute appreciably to signal noise. An annular secondary electrondetector can be placed above the specimen and in the microscope column to screen outSE3 electrons, and improve the contrast in the image. However, since the detector isabove the specimen, rather than to the side, we lose some of the three-dimensionalshadow effect that is associated with the surface topology and is observed using theconventional secondary electron detector (Figure 5.49). Some scanning electronmicroscopes allow on-line mixing of signals from both types of detector in order tooptimize both resolution and contrast.

Figure 5.50 shows a 5 kV SE1 micrograph of the thermally etched alumina specimen.Grain boundaries are easily identified, since the grain boundary grooves emit fewersecondary electrons than the groove shoulders. It is straightforward to determine theaverage grain size to any required accuracy (see Chapter 9).

Figure 5.51 shows a fracture surface of the same grade of alumina. Two fracture modesare visible: intergranular (grain boundary) fracture and transgranular (cleavage) fracture.Transgranular fracture is characterized by the appearance of planar, crystallographic

Figure 5.48 SEM micrograph clearly showing the fine surface facets on a thermallyetched alumina. The image was recorded using a field emission gun and a specializedsecondary electron detector. Most of the contrast comes from SE1 electrons generated bythe primary beam.


cleavage planes, seen in sharp contrast. Residual porosity is also readily visible, both at thegrain boundaries, and within the grains.

Nowwe return to the CVD aluminium system. For this examplewewish to characterize:

1. the initial deposition conditions, when the first aluminium nuclei form on the surface ofthe TiN;

2. the morphology of the final aluminium film;3. the morphologies of the underlying TiN and titanium films.

Preparation of a suitable sample for SEM is straightforward, since the sample is a metallicconductor and we only need to investigate the surface of the sample. However, the aluminiumnuclei are very small, requiring the highest possible resolution from a low-voltage scanningelectron microscope, generating secondary electrons only from the surface layer. Contamina-tion of the surface can be a critical factor limiting image contrast and resolution. Figure 5.52shows a secondary electron SEM image of the surface imaged using an in-lens secondary

Figure 5.49 Fracture surface of alumina showing cleavage facets and intergranular failure. Theimages were recorded using both a conventional secondary electron detector (a) and an in-lenssecondary electron detector (b). More detailed features of the fracture surface are visible in theimage from the in-lens detector.


electron detector. The central region of Figure 5.52 was exposed to the electron beam forapproximately 30 s, and then themagnificationwas reduced and themicrographwas recorded.Thus the outer region of Figure 5.52 was recorded immediately after focusing on the area ofinterest, reducing the contamination in this area to aminimum. The build-up of contamination

Figure 5.50 Secondary electron SEM micrograph of a thermally etched alumina, recorded atan accelerating voltage of 5 kV.

Figure 5.51 Secondary electron SEM micrograph of the fracture surface of alumina, showingtransgranular fracture (cleavage), intergranular fracture, and pores, both at grain boundaries andwithin the alumina grains.


in the central region of the image is evident. The reason we need an uncontaminated, cleanspecimen is clear. There are three options commonly used to clean samples:

1. plasma etching before inserting the specimen in the microscope;2. heating the specimen above 100 �C, to drive off absorbed gases and water;3. swabbing the sample with a volatile organic solvent (acetone, ethanol or methanol) to

remove grease and oil. (This last option will only remove soluble contamination.)

Since any specimen heating of the surface by the electron beam might cause changes inthe aluminium morphology, we should acquire the images as quickly as possible tominimize the electron dose.

The final, thick aluminium film has a different morphology, and SEM shows that openvoids have formed during the deposition process (Figure 5.53). These voids are processdefects which change both the electrical and optical properties of the aluminium. SEM canalso be used to examine specimens in cross-section. Figure 5.54 is a secondary electron SEMmicrograph of a cross-section froma cleaved sample. The aluminium film is clearly visible atthe surface of the TiN/Ti/SiO2/Si stack. Open voids are also just visible, as well as possibleclosed voids that could not have been detected from the plan-view of the specimen surface.Further details of the film morphology would require TEM (discussed in Chapter 4).

Now for another example: wire-bonding is a commercial process used to connect asemiconductor device to the main board onwhich it is mounted. Thewires are usually gold,although copper wires are now being commercially introduced. The bonding processcommonly uses a�20mmdiameter wire. Thewire is fed through an alumina capillary tube,and the end of thewire ismelted using a spark generated across an electrodegap. Figure 5.55is an ion-induced secondary electron micrograph taken in a dual beam FIB and shows themelted and solidified end of a gold wire, just prior to bonding to the connecting pad on the

Figure 5.52 Secondary electron (in-lens detector) SEM micrograph of aluminium on TiN,recordedwith a 1 kV incident electron beam. The beamwas first focused on the central region ofthe micrograph for approximately 30 s, and then the magnification was reduced and themicrograph was recorded immediately. The build-up of contamination in the central regionstrongly affects the contrast and resolution of the image.


device. Ion channelling orientation contrast reveals the very different grainmorphologies inthe wire ball and the parent wire. FIB can also be used, to prepare H-bar specimens fromthese thinwires for SEMandTEM (Figure 5.56). The anisotropic shape of the gold grains inthe wire is immediately evident on account of the ion channelling contrast.

Figure 5.53 Secondary electron SEM micrograph of a thick aluminium film, showing arelatively rough surface and some open voids.

Figure 5.54 Secondary electron SEM micrograph of a cross-section prepared by cleaving thesilicon wafer together with the Al/TiN/Ti stack of films. The ductile aluminium showsconsiderable plastic deformation compared with the brittle underlying layers and a void isvisible in the aluminium film.


A more serious challenge is to characterize the gold wire after it has been bonded to theconnecting pad on the device. Before the FIB technology was introduced, the only way tocross-section such a join would have been to encapsulate the entire device (Figure 5.57) in amoulding compound, and then to mechanically grind and polish the mounted sample. Atalented experimentalist with steady hands might hope to stop the polishing process whenthe central region of the �40 mm diameter bond was sectioned. Not a very promising

Figure 5.55 Ion-induced secondary electronmicrographof a goldwire used forwire-bonding,showing orientation contrast associated with ion-channelling.

Figure 5.56 Ion-induced secondary electronmicrograph of anH-bar section ionmilled from agold wire. Note the magnification scale.


procedure!With a dual-beam FIB, the ion beam is used to cut away any remaining length ofwire and then section the bond itself. Figure 5.58 shows an example from a copper wireconnected to an aluminium pad on a device. The bond has been sectioned by FIB for SEMcharacterization.

There is no reason to restrict characterization to SEM, and the FIB can be used to preparea thin-film TEM specimen from the same wire-bond. Figure 5.59 shows a thin lamellasection after twoneighbouring ‘FIB boxes’ have been ionmilled and aU-cutmade to almostfree the lamella from the surface. The sample is then attached to the nanomanipulator, cutfree from the substrate and attached to the TEM specimen mount (Figure 5.60). Afterfreeing the nanomanipulator from the sample, the ion beam can be used to thin the lamellauntil it is transparent to an electron beam (Figure 5.61). At this point, the sample can becharacterized in the dual-beam FIB, using STEM, or transferred to the transmissionelectron microscope.

Problems

5.1. How does the working distance of the probe lens from the sample surface affect theminimum probe size in a scanning microscope?

5.2. Given that the signal collected in a scanning system is determined by inelasticscattering and secondary excitation processes, discuss the effect that these probe–specimen interactions have on the scanning resolution. Compare especially charac-teristic X-ray and secondary electron excitation.

Figure 5.57 Lowmagnification SEMmicrograph of a series ofwire bonds on amicroelectronicdevice.


Figure5.58 (a) Secondaryelectron SEMmicrographof a copperwirebonded to analuminiumpad. In (b) the wire has been cut and removed using the FIB ion beam, and the bond has beenpartially sectioned to expose the interface. In (c) the cross-section of the bond is exposed, andSEM can be used to characterize any defects or intermetallic compounds that are present in theinterface region.


5.3. Both surface topology and local mass density can influence the scanning electronimage contrast. Howwould you expect the beam voltage to affect the atomic number(mass density) contrast in a backscattered electron image?

5.4. The secondary electron image is thatmost commonly used for routine examination inthe scanning electron microscope. Why?

5.5. Discuss someways in which samples that are sensitive to degradation in a vacuum orunder an electron beam could nevertheless be imaged in the scanning electronmicroscope.

5.6. What minimum angle of tilt should be used to distinguish two features by stereo-imagingwhen they are separated by avertical distanceh and the lateral resolution is d?

Figure 5.59 (a) Ion-induced and (b) electron-induced secondary electron micrographs of athin lamella attached to the substrate, prior to lift-out.


5.7. How does the depth of field of an optical microscope comparewith that of a scanningelectron microscope at a magnification of ·200 when the scanning electron micro-scope convergence angle 2a¼ 4 · 10�2 rad? How would a change in the workingdistance affect the depth of field and probe size in the scanning electron microscope?Explain any assumptions that you make.

5.8. What signals are generated from a solid sample by an incident beam of high energyelectrons?Which signals are used in TEM, andwhich are used in SEM?Howdoes theresolution in a SEM micrograph depend on the type of signal that is collected?

5.9. Are secondary electrons or backscattered electrons to be preferred for imaging andanalysing a fracture surface? Explain your reasoning.

5.10. Are secondary electrons or backscattered electrons to be preferred for imaging andanalysing variations in the local chemical distribution on a polished samplecontaining aluminium and gold? Give your reasons.

5.11. Are secondary electrons or backscattered electrons to be preferred for imaging andanalysing the local chemical distribution on a polished sample composed ofaluminium and magnesium? Explain your reasoning.

Figure 5.60 Ion-induced secondary electron micrographs: (a) the nanomanipulator attachedto the lamella with a platinum join; (b) immediately after the lamella has been fully separatedfrom the substrate; (c) the lamella is then moved next to a TEM sample mount; (d) the lamella isattached to the TEM mount by platinum deposition.


5.12. Assuming that the standard scanning electron microscope viewing screen is 10· 10 cm2 and based on 256· 256 pixels, what magnification is required to resolveround particles with a radius of 10 nm? (Assume that reasonable resolution isachieved when an image feature covers 3· 3 pixels.)

5.13. Sketch the intensity distribution for secondary electrons that are collected byscanning the incident beam across a protrusion on a flat surface. Assuming thesame detector geometry and feature size, sketch the intensity distribution expectedwhen the incident beam is scanned across an indentation on the same surface.Discuss the dependence of the contrast on the location of the detector and thedimensions of the features.

Figure 5.61 (a) Secondary electron SEM micrograph (5 kV incident electrons) of the thinnedlamella. (b) Bright-field STEMmicrograph of the microstructure recorded in the dual-beam FIB(30 kV incident electrons).


5.14. Given an aluminium sample that contains precipitates of Al–Au intermetalliccompounds, how should the backscattered electron contrast change when thedetector is located (a) at an angle of 170� and (b) an angle of 105� relative to theincident electron beam?Assume a flat surface normal to the incident electron beam.

5.15. Sketch the expected secondary electron yield as a function of the acceleratingvoltage (electron beam energy) for a nonconducting specimen. Indicate theaccelerating voltages that will minimize electrostatic charging of the specimensurface.

5.16. How is the magnification controlled in SEM? What image distortions might beexpected?

5.17. In SEM, unlike TEM, the image can never rotate when the magnification isincreased. Explain why.

5.18. The secondary electron and backscattered electron SEM micrographs shown inFigure 5.62 were acquired from the same area of an AION sample that contained Y-based oxide particles. Explain the difference in contrast between the twomicrographs.

Figure 5.62 (a) Secondary electron (SE) and (b) backscattered electron (BSE) micrographs of Y-based oxides in an AIN matrix.


6

Microanalysis in Electron Microscopy

Wehave already noted (Section 5.3.2) that a proportion of theX-rays emitted under electronexcitation, together with the corresponding energy losses in the primary electron beam, arecharacteristic of the chemical constituents of a solid sample. These characteristic X-raysmay be selected from the observed spectrum of emitted electromagnetic radiation,according to either their energy or their wavelength, and the signal distribution from thesample can be displayed in a line-scan or elemental map to provide both qualitative andquantitative information on themorphological relationship between themicrostructure andthe chemical composition.

In this chapter we explore several ways in which the qualitative chemical informationcontained in the characteristic X-ray signal can bemade quantitative. In all the methods wepresent, inelastic scattering of the probe (either electrons or X-rays) excites a signal thatdepends on the chemical composition of thematerial beneath the probe, and the challenge isto interpret this signal as quantitatively as possible. The chemical sensitivity of eachmethod(the minimum detectable concentration of a selected constituent) will be different, as willthe analytical accuracy (the errors involved in quantitative analysis). We will also beconcerned with the spatial resolution obtainable for the chemical composition, both in theimage plane of the sample surface and within the depth beneath the surface that is sampledby the incident high-energy probe.

In this chapter we only consider those methods that are commonly available asmicroanalytical facilities, attached to either the transmission or the scanning electronmicroscope, but in Chapter 7 we will also describe some additional methods that may beused to characterize the composition of the near-surface layer, thin surface films andadsorbates. We first discuss qualitative and quantitative microanalysis using the excited,characteristic X-rays before considering the microanalytical information that can bederived from the electron energy loss spectra.


6.1 X-Ray Microanalysis

In X-ray microanalysis, the characteristic X-rays emitted from a sample viewed in theelectronmicroscope (usually the scanning electronmicroscope, but also in the transmissionelectronmicroscope) are analysed both qualitatively and quantitatively in order to relate thelocal chemical composition of the sample to the morphological features visible on thesample surface.

The physical basis for the excitation of X-ray emission by a high energy electron beamthat is incident on a solid target has already been introduced (Section 5.3.2), butwe still needto explore the factors that influence the sensitivity of microanalysis (the minimumdetectable concentration of a constituent) and its accuracy (the cumulative microanalyticalerrors). These factors involve both the properties of the X-ray detection system and thecomputer software that is used to convert the collected X-ray data to a quantitative estimateof composition. They also include the specimen preparation procedures that are necessaryto ensure reproducible and accurate results, the geometry of the sample surface in relation tothe electron beam and detector system, together with some features of the compositiondistribution and microstructural morphology of the sample that can also affect the results.

To simplify the discussion, we will assume that the sample has been polished but notetched, and that the specimen surface is smooth and planar, so that the only significantgeometrical parameters are the angle at which the electron beam strikes the specimensurface and the angle subtended by the detection system at the sample surface. If the surfaceroughness is on a scale that is small comparedwith the diffusion depth for the incident high-energy electrons in the sample (typically<1 mm, at 20–30 kV, even for low atomic numbermaterials), then there will be no significant errors in assuming that a polished and etchedsurface is planar. Fracture surfaces, however, and surfaces that have beenmachined, heavilycorroded or etched are certainly not planar. The computer software programs for quantita-tive X-ray microanalysis are not intended to be used for the analysis of rough surfaces. Thesame applies to powder samples, fibres and grits, and the results of X-ray microanalysis onthese materials must be regarded as only qualitative. In addition, all computer correctionprocedures for quantitativemicroanalysis assume that the composition of the sample in thesurface region being analysed is homogeneous. It follows that quantitative microanalysis isonly reliable for regions that are reasonably far from phase boundaries and in the absence ofstrong concentration gradients. This applies both to phase boundaries that are sensiblyperpendicular to the surface being analysed and also to thin films deposited on a substrate.Even so, it is often possible to extract qualitative information on the elements present in athin, sub-micrometre layer, and this information can often be made semi-quantitative byreducing the energy of the primary beam, so that the diffusion distance for the high energyelectron probe is of the order of the film thickness. Nevertheless, the primary purpose ofquantitative microanalysis is the determination of the bulk composition, albeit in a verysmall, micrometre-sized, volume element of the specimen.

6.1.1 Excitation of Characteristic X-Rays

As we noted in the previous chapter, the characteristic X-ray signals and the correspondingenergy loss spectra (Section 5.2) that are generated by an incident beam constitutefingerprints of the local chemistry. To carry the fingerprint analogy further, qualitative


analysis consists in identifying the origin of the print, as from the index finger or the thumb,while quantitative analysis has as its goal the positive identification of the perpetrator; arather more difficult task.

The characteristic X-rays are generated within a region of the envelope of scatteredelectrons for which the average electron energy exceeds the threshold energy for X-rayexcitation. The volume of this region depends on the incident beam energy, the atomicnumber of the elements present and the mass-density of the sample. The mass-density andthe beam energy determine both the diffusion distance (that is, the spread of the scatteredelectron beam, which is dependent on the rate of energy loss) and the range of the electronsin the sample (the thickness required to reduce the electron energy to the thermal level, kT).For example, a copper sample exposed at high beam energies will generate both K and Lcharacteristic radiation (Figure 6.1). The L radiation will originate from a region thatextends to just within the thermal energy envelope,while theK radiation is generatedwithina much smaller volume. At lower beam energies that are below the threshold for Kexcitation, only the L radiation can be excited, but since the range of these lower energyelectrons is much reduced, the source of the L radiation will have a correspondingly smallervolume. In principle, the spatial resolution for the identification of copper in the sampleshould be better when using the L characteristic lines at low electron beam energies.However, better sensitivity and more accurate quantitative analysis is possible with Kexcitation at higher probe voltages, and in many cases, the spatial resolution may besacrificed to the analytical accuracy.

Increasing the energy of the incident electron beam generally increases the total X-raysignal and ensures that most, if not all, characteristic lines are excited. However, the totalamount of white, background radiation or �Bremsstrahlung�, due to inelastic scatteringevents that do not involve the ejection of an inner shell electron, is also increased. If the

E 0 > E

(a) (b)

K E 0 < E K

E > E K

E > E L

E = k T

E = k T

E > E L

Figure 6.1 Schematic representation of the volume elements beneath an electron probe thatgenerate characteristic X-ray radiation in a copper sample (a) well above and (b) below theenergy threshold for K excitation.

Microanalysis in Electron Microscopy 335

beam energy exceeds approximately four times the threshold energy for X-ray excitation,then the ratio of the characteristic line intensity to that of the background intensity (thesignal-to-noise ratio) starts to decrease. In addition, the size of the volume elementgenerating the signal, dependent on the diffusion distance, is also increased. This reducesthe spatial discrimination, that is, the resolution, and is therefore a good reason for limitingthe energy of the incident beam.

Although reducing the beam energy both improves the spatial resolution and limits thebackground level of white radiation, as the beam energy approaches the critical excitationenergy, the intensity of the characteristic line decreases dramatically. As a general rule,there is no advantage to be gained by reducing the beam energy to less than three times theexcitation energy for the shortest wavelength, that is, the highest energy characteristic lineswhich are of interest. In fact, light element analysis, for which only characteristic lines oflong wavelength are available, is best combined with analysis of the heavier, higher atomicnumber constituents by using the characteristic wavelength from the L or even M shells ofthe high atomic number elements.

Before discussing the statistics of signal collection and the �white� noise background inthe spectrum which is associated with emission of X-rays that are not related to character-istic excitation, we should note the existence of some �false� peaks associated with themethod of detection of the photon energy. The first of these peaks are �escape� peaks, belowthe principal peaks observed in the spectrum. These peaks are due to fluorescence of siliconin the detector that has been excited by high energy impinging photons. The Si-K absorptionedge, at 1.74 keV, can reduce the photon energy of any incident radiation that excites thesilicon in the detector by exactly this amount and can lead to a small �escape peak� that is just1.74 keV below any main, characteristic peak. Most software now available will correctautomatically for the presence of an escape peak by first subtracting the background (seebelow) and then adding the escape counts to those registered for the main peak of theelement in question.

The second type of �false� peak that may be important can arise when two photons arrive�simultaneously� at the detector. This is only observed at very high counting rates, and theavailable software is set to reject �second� counts that arrive before the current pulse due toan impinging photon has been cleared and registered (typically this takes >1 ms). Never-theless, there is a finite probability that two photons may impinge on the detectorsufficiently close in time to be registered as a single count of energy equal to the sumof the energies of the two impinging photons. Such events are rarely a problem.

Finally, characteristicX-rays from lowatomic number (lowZ) constituentsmay appear atan energy that depends slightly on the chemical bond energy of the atom in the material,since the decay process responsible for emission involves an electron from the valenceband. Carbon is an excellent example, and the carbon peaks from diamond, polymers andpyrolytic (amorphous) graphite show a clearly detectable shift in peak position. These arenot �false� peaks but they do make quantitative analysis more difficult.

6.1.1.1 Signal-to-Noise Ratio. Before any quantitative analysis is possible, the back-ground �noise� must be subtracted from the intensity detected at the position of acharacteristic �line� by fitting a function to the �white� background signal (Figure 6.2) andcalculating the integrated intensity difference. The simplest function is a straight line that is


based on background counts that are summed on either side of the selected intensity peakand at a sufficient distance from the peak to ensure that themain peak does not interferewiththese backgroundmeasurements. A far better procedure is to use a fitted curve that includes,to a first-order approximation, corrections for the absorption edges that are associated withexcitation of the main peaks in the spectrum. Such fitted curves are commonly included asthe first step in computer software correction routines.

The available spectral resolution in wavelength-dispersive spectrometry (WDS) is farbetter than that available in energy-dispersive spectrometry (EDS). In EDS the energyresolution is usually determined forMnKa radiation (5.9 keV), and is typically in the regionof 140 eV for this radiation. The resolution is amonotonic function of the photon energy, butthe limited energy resolution of the spectrometer, together with the potential for overlapbetween characteristic intensity peaks, makes it impractical to set the acceptance channelfor a characteristic line to the full energy width of the peak. A satisfactory compromise is toset thewidth of the channel to coincidewith the intensities at those energies either side of themaximum characteristic intensity that correspond to half the observed peak height, acondition termed full- width at half-maximum (FWHM).

There are, unfortunately, many cases for which the characteristic signals from differentelements give spectral peaks that overlap significantly. This is especially common in EDS

Figure 6.2 If a background correction can be calculated, then the intensity in a characteristicsignal can be obtainedby subtracting this background from the total number of counts registeredover the full rangeof the detected intensity peaks. In this figure the background is calculated for aspectrum recorded from NaFe(Si2O6).


spectra, but is by no means impossible for WDS results. In such cases, removing thebackground is only the first step in preparing the data, and it is necessary to deconvolute theoverlapping peaks. The simplest way of achieving this is by assuming that the overlappingpeaks are symmetrical and by using the data from the major peak to estimate the additionalcounts in each channel that are due to theminor peak. The errors can be large and somemoreaccurate software algorithms are available. Peak overlap is a problem that often needs to besolved, but is seldom an insoluble problem.

6.1.1.2 Resolution and Detection Limit. The spatial resolution for microanalysis inscanning electron microscopy (SEM) ranges from 0.2 mm to several micrometres, limitedprimarily by the diffusion distance that determines lateral spreading of the beam and therange of the electrons. In transmission electronmicroscopy (TEM) (see Section 4.1.3.3) thevery limited thickness of the thin-film sample limits lateral spreading of the beam byinelastic scattering, and resolutions of the order of 1nm can be obtained when a fieldemission source is available.

In effect, the chemical spatial resolution and the concentration sensitivity are incompetition, as we noted when discussing the size of the envelope that defines the volumeelement generating the characteristic X-ray signal. If detection sensitivity could beimproved for longer wavelength X-rays, then the spatial resolution for microanalysis atlower beam energies could be better exploited. This is made possible by using a fieldemission gun which increases the beam current in the probe by some two orders ofmagnitude and can be used at incident beam energies down to below 1 kV, localizing theexcitation volume to the initial probe size and film thickness. The analysis of longwavelength radiation (1 kV � 1.24 nm) can also provide information on the nature of thechemical bonding, and we have already noted the shifts in peak position for carbon indifferent bonding states.

The detection limit in X-ray microanalysis at the beam energies more commonly used(5–20 kV) is of the order of 0.5 atom %, but in some cases may be as low as 0.1% for EDS.The correction errors in quantitative analysis for awell-calibrated system are about�2%ofthe measured concentration, providing the concentration exceeds a few atom per cent.However, it is important to be aware of the limitations of quantitative analysis and not to bemisled by results obtained from software procedures that introduce statistical bias bynormalizing the composition or assuming stoichiometry. This is a common procedurewhendata for one component in a compound are unavailable.

6.1.2 Detection of Characteristic X-Rays

The detection of characteristic X-rays requires both good discrimination and high detectionefficiency.

To achieve 100%detection efficiency, every photon emitted by the samplewould have tobe recorded. This is impossible for two reasons: first, the detector always subtends a limitedsolid angle at the sample, and only those photons that reach the detector have any chance ofbeing recorded. Secondly, the detector itself has a limited detection efficiency that dependson the energy (that is, the wavelength) of the incident photon. Figure 6.3(a) shows thestructure of a typical lithium-drifted silicon detector and Figure 6.3(b) and (c) shows thedetection efficiency as a function of the energy of the incident photon and the pulse-processing time. At low energies the photon may be absorbed by a beryllium foil window


that protects the detector form contamination in the microscope. Better detection perfor-mance is obtained using thinner but more fragile, polymer-based windows. At very highphoton energies the photonmay pass through the detector (which is typically�3mm thick)without being absorbed and detected.

Holes Electrons

-500V

Au Contact~20 nm

Au Contact~200 nm

P-Type Li-Drifted N-TypeRegion IntrinsicRegion Region

FE

TP

re-A

mpl

ifie

r

Thi

nW

indo

w

0 2 4 6 8 10

0

50

100

150

200

Res

olut

ion

FW

HM

(eV

)

Energy (keV)

(b)

1.0 1.2 1.4 1.6 1.8 2.0

0

20 000

40 000

60 000

80 000

100 000

Cou

nts

Energy (keV)

Mg Kα Al Kα Si Kα(c)

(a)

Figure 6.3 (a) Schematic drawing of the Si-based Li-doped (drifted) EDS detector. Photonspassing through the thin window and gold contact interact with the silicon, generating bothholes and electrons. The accumulated charge is then measured by a field-effect transistor (FET)pre-amplifier, which feeds the signal into the EDS pulse processor. (b) The energy resolution of atypical EDS detector as a function of energy. (c) Short process times (red) degrade the peakresolution compared with long process times (black).


Perfect discrimination, or energy resolution, would require that the wavelength of everyphoton which is detected should be accurately determined and separated from otheractivation events that are associated with photons of different wavelength. This also isimpossible. The background radiation that overlaps a characteristic peak implies someuncertainty in the source of the photon,while the detector can only identify photon energy toan accuracy that is determined by its spectral resolution. This spectral resolution is roughlyproportional to the square root of the incident photon energy and is usually quoted in theliterature as the pulse width at FWHM for MnKa radiation at 5.895 keV. The energyresolution for any other characteristic line energy is then DEE � DE5:9

ffiffiffiffiffiffiffiffiffiffiffiffiE=5:9

p, where DE

is in eV and E is in keV.

6.1.2.1 Wavelength-Dispersive Spectrometers. Far better discrimination of the X-raysignal emitted by the region excited under the electron probe is achieved in a wavelength-dispersive spectrometer. In this system (Figure 6.4), a series of bent single crystals ofdifferent lattice spacings covers the range of wavelengths that are of interest. Thewavelengths within the range of each spectrometer crystal are scanned by rotating thecrystal to scan the Bragg angle 2y and synchronously moving the detector, while keepingthe position of the bent crystal fixed. Note that the radius of curvature of the bent crystal isset to an �average� diameter for the focusing circle. Since energy discrimination is not anissue, an argon gas proportional counter can be used to collect the photons selected by thediffracting crystal. These gas counters have sufficient energy discrimination to ensure thatsecond- and higher-order reflections (wavelengths that are multiples of that selected) arerejected. At the same time, the width of the current pulse generated by a photon in the gas

R

Specimen

X-RayDetectors

Receiving Slits

Bent and Polished Crystal (radius=2R)

Focal Circle

Figure 6.4 Thewavelength dispersive spectrometer is usually a semi-focusing system inwhicha curved crystal reflects the radiation emitted from the specimen surface over a specific solidcollection angle. The distance R from the source to the crystal, the average take-off angle for theX-rays and the Bragg diffraction angle 2y for the wavelength of interest are adjustable. Note thatthis figure shows a fully focusing geometry.


proportional counter is much narrower than can be achieved in the Si(Li) solid-statedetectors, so that the �dead time� of the gas proportional counter is not a limitation on thecounting rate. In the WDS geometry, the angle at which the X-rays are collected fromthe sample is fixed. The angle subtended at the collecting crystalwill varywith 2y, while thediameter of the focusing circlewill change (hence the system is designated �semi-focusing�,since the radius of the collecting crystal is constant and only �fully focusing� for one specificfocusing circle).

There is an additional reason why the system can only be �semi-focusing�, and that isbecause theX-rays aregeneratedover a finite depthof the sample,whichdependsonboth theincident electron energy and the sample density. This effect is not usually significant;however, it does depend on the position of the plane of the spectrometer with respect to theaxis of themicroscope. The bent diffractometer crystal focusesX-rayswhich originate froman elongated region parallel to the axis of bending of the crystal, and is thereforeperpendicular to the plane of the spectrometer. If the spectrometer is horizontal, that isperpendicular to themicroscopecolumn, then the focuswill notbe sensitive to thepositionofthe sample along the axis, but now the area in focus in the sample planewill be rather small,and this could be a problem for areal analysis at low magnifications. If the spectrometer isvertical (parallel to the microscope column), then the sensitivity to vertical displacement ofthe samplewill bemuch greater, but therewill now be an elongated area of the sample (a fewmillimetres) that is in focus. Some spectrometers aremounted at an angle to themicroscopecolumn to achieve a compromise of relative insensitivity to z-axis displacements of thespecimencombinedwith good areal coverage for analyses performed at lowmagnifications.

An important consequence of the geometrical discrimination provided by the WDSsystem is that all the characteristic emission peaks must be scanned sequentially and theonly way to record more than one characteristic line at a time is by using more than onespectrometer. In practice, awavelength-dispersive spectrometer can be programmed to scanthrough a series of characteristic peaks and settings for background measurement, in orderto maximize the efficiency of data acquisition. Nevertheless, the improved discrimination,usually at least an order ofmagnitude in energy resolution when usingWDS instead of EDScarries a heavy penalty in data collection time associated with the sequential (WDS) ratherthan a parallel (EDS) data collection mode.

6.1.2.2 Energy-Dispersive Spectrometry. In the energy-dispersive spectrometer thepulse height recorded for an incident photon by a detector is directly proportional tothe energy of the photon responsible for the pulse. The detectors used for this purpose arelithium-drifted silicon, Si(Li), solid-state detectors. An incident photon absorbed by thesilicon crystal creates ionization events in the active thickness of the detector. The totalcharge developed is proportional to the incident photon energy and is detected as a currentpulse that is shaped, digitized and counted in a multi-channel analyser.

There are two problems associated with EDS, as opposed to WDS systems. The firstconcerns their relatively poor energy resolution. Good WDS systems have better energyresolution, by at least an order of magnitude, especially for the detection of long-wavelength, low-energy radiation. The better energy resolution is also important in caseswhere the characteristic lines from different elements overlap. WDS systems can alsoresolve the multiple lines of L and M spectra unambiguously, improving the accuracy ofquantitative microanalysis using L and M radiation (Figure 6.5).


The second problem occurs with low-energy (long-wavelength) photon detection(Figure 6.6), that requires either a windowless detector, or a detector protected from therest of the system by only a very thin and fragile window. Care is then needed to ensure thatthe detector retains its long-wavelength sensitivity, and is not degraded nor contaminated inthe vacuum environment of the microscope. All solid–state detectors are cooled by liquid

1.6 keV 2.0 keV

Ta

Mα

Ta

Mβ

Si

Kα

W

Mα

W

Mβ Re

MαRe

Mβ

Figure 6.5 Resolution of M-lines in a wavelength-dispersive spectrum of a super alloy.(Courtesy of Oxford Instruments).

0

2000

4000

6000

Counts

2 4 6Energy (keV)

C

F

BaBa

Ba

Ba

BaBaBa

Figure 6.6 Energy-dispersive spectrum of BaF2 showing the resolution of the characteristiclines of low atomic number elements. The carbon signal is from surface contamination of thesample.


nitrogen, so that the detector is often the coldest region in the system. Cryogeniccondensation of contamination on the surface of the detector is then a serious life-limitingconcern.

6.1.2.3 Detection of Long-Wavelength X-Radiation. The wavelength of X-radiationcommonly used for crystallographic structure analysis is usually <0.2 nm, and it was notuntil electron-probe microanalysis was developed (in the late 1950s) that any commercialneed existed for the detection of longer wavelength, �soft� X-radiation. The past decadeshave seen a steady improvement in the reliability, the sensitivity and the discriminationavailable for the detection of wavelengths longer than 1 nm, corresponding to criticalexcitation energies of 1 keV or less.

There are two major problems to be solved: the first concerns the absorption of soft X-rays in the sample itself. Even for low-density specimens, the absorption coefficient for softX-rays is large. Signal detection is significantly improved when the specimen is inclinedtowards the detector, in order to reduce the absorption path for X-rays in the sample. Strongabsorption also occurs in the detector. Ultra-thin-window or windowless detectors arecommon solutions for the detection of the longest wavelengths.

The secondproblemis the lossofwavelengthdiscriminationat longwavelengths. InWDSsystems, the crystals used for soft X-ray applications have long periodicities, such as thosedeveloped by repeated deposition of long-chainmolecularmonolayers from the surface of aLangmuir trough. In this process the molecules segregate to the surface of a liquid in a self-ordered array that can be collected on a suitable curved substrate. Subsequent ordered layersare then deposited in the sameway, one upon another. TheseWDS crystals are very fragile,easily damaged and may contain defects introduced during deposition. Their ability toresolve the longwavelengthsof the incidentX-radiation isgenerally less satisfactory than forthe more perfect, ionic crystals used to analyse the shorter wavelengths.

In spite of these problems, both EDS and WDS systems are easily capable of detectinglight elements down to boron (Z¼ 5), even for quantitative analysis, as long as the specimenis kept free of carbonaceous contamination. This may be compared with the early years ofmicroanalysis, when quantitative analysis of elements below magnesium (Z¼ 12) wasjudged impossible, and the qualitative detection of the light elements below carbon (Z¼ 6)seemed unrealistic. Today, it is more often specimen contamination from the sampleexposed to the electron beam that restricts quantitative analysis of the light elements, ratherthan any equipment short comings.

6.1.3 Quantitative Analysis of Composition

From the point of view of the microscopist, it is the composition of a given region on thesample that has to be derived from the recorded spectrum from the same region. As wewillsee below, this calculation is an iterative process. Before beginning �quantification�, themeasured characteristic intensities from the constituents of the region to be analysed mustfirst be corrected for �white� background noise and spurious �escape� and other peaks. Anypeak-overlap needs to be deconvoluted before determining the area beneath each peak,using the FWHM criterion.

Once the integrated peak intensities have been determined, the concentration of theregion in the sample can be evaluated using measured (or calculated) intensities from


standard samples of known composition. This is done by taking separate account ofcorrections due to fluorescence, absorption and atomic number effects. These correctionparameters themselves depend on the chemical concentration of the sample, which is to bedetermined. A numerical solution has to be sought. Note that inmaking these corrections tothe measured relative peak intensities, it is the fluorescence (F) correction that should beapplied first, followed by the absorption (A) correction, and then the atomic number (Z, ordensity) correction, before repeating the cycle iteratively until the results numericallyconverge. That is, the order of the corrections is F, then A, then Z. This is just the reverseorder of the effects that were considered during generation of the characteristic X-ray signalfrom the sample. In X-ray generationwe first follow the path of an energetic electron as it isslowed down by the atomic number effect (Z). We then consider the absorption of X-raysbefore they escape from the sample (A), before correcting for secondary fluorescence (F).Fluorescence can be a problem, since the range of X-rays in the sample is an order ofmagnitude greater than that of the incident high energy electrons. The volume of the samplethat may fluoresce is therefore of the order of ·1000 the volume of the region we are tryingto analyse. It follows that, in a polyphase sample, secondary fluorescence may beresponsible for some unexpected effects that cannot readily be removed by a routinemicroanalysis correction procedure.

In all software correction protocols, the relative characteristic intensity is defined as theintegrated intensity beneath the peak minus the background counts and corrected forspurious peaks and peak overlap. These adjusted measured intensities are used as a �firstguess� for the relative concentrations of the corresponding elements in order to estimate themagnitude of the required corrections. This first estimate of the corrections is then used tocalculate a �second guess� in an iterative procedure for estimating the relative concentra-tions. In practice, few iteration cycles are needed before the differences between successiveiterations converge to an approximately constant value, which is taken as the �best� estimateof the relative concentrations. The residual error is typically better than 2 % of thecalculated, corrected concentrations. However, there is no guarantee that convergence ofthe iterative series of correction calculations implies accuracy of analysis, not least becausemorphological features of the sample surrounding the nominal volume element beinganalysed may vitiate the correction procedure, either because of the surface topology orbecause of long-range fluorescence effects.

The computer software available for performing these corrections varies considerably,but all such codes follow the same iterative logic. The geometrical parameters must beentered into the program, including the angle between the specimen surface and the incidentbeam, and the angle subtended by the detector at the specimen surface (termed the take-offangle for the X-ray signal). The accelerating voltage (the incident beam energy) must alsobe entered and the calculated standards for the background and characteristic line intensitiesneed to be adjusted whenever the beam voltage is changed. Finally, the program willgenerally ask the operator if the results are to be normalized, that is whether the calculatedconcentrationvalues should be adjusted so that they sum to 100%.This is not usually a goodidea, since if the sumof the calculated concentrations differs seriously from100%, then thisis a strong indicator of a serious analytical problem: either one element has not beendetected at all, or the sample density differs significantly from the expected value (possiblybecause of sub-surface porosity), or because some other feature of the microstructuralmorphology is affecting the results. Alternatively, the take-off angle, the operating voltage,


or some other operating parameter may have been mis-entered into the correction program.It is best to regard deviations from a summation to 100 % as a measure of the accuracy ofanalysis, and to take a very close look at the measurements and the calculation if the sumdeviates from 100 % by more than a few percentage points.

The computer softwaremay also allowyou to omit data fromone element that is known tobe present. When this element is identified, the concentration of the missing element willthen be calculated by assuming that the total of the estimated concentrations is always100%. This may sometimes be useful for light elements that are hard to detect accurately,especially in nonstoichiometric materials, for example, for lithium in an Al–Li castingalloy.

An additional computer software option that is commonly available is to assumestoichiometry during the calculation for compounds containing an undetected element.Typically, themissing elementwill be oxygen, carbon, boron, hydrogen or nitrogen, that is alow atomic number constituent. This procedure significantly improves the analyticalaccuracy when working with known stoichiometric compounds, such as ceramics, but itmay be misleading when two phases with more than one valency are possible, for exampleTiN (Ti3þ) and TiO2 (Ti

4þ), or where a compound is known not to be stoichiometric, such asFe1-xO.

In the present textwewill describe the various corrections for quantitative analysis, not asFAZ corrections, in the sequence that would be performed in an iterative, quantitativeanalysis, computer correction program, but rather in the order that these corrections affectthe X-ray signal as it is generated by the incident beam and subsequently emitted from thesample (the ZAF sequence). Therefore, we first consider the size of the volume element ofthematerial in which the X-rays are generated, thenwe describe the absorption losses as theX-rays travel through the sample before escaping, and finally we analyse the incidence offluorescent excitation for a characteristic excitation line byX-rays of higher energy (shorterwavelength).

We also include a short discussion of the applications ofX-raymicroanalysis in TEM, forthe case of thin-film samples analysed in transmission. Note that while the incident beamenergy in the scanning electronmicroscope is typically less than 30 kV, andmay be less than3 kV, that in the transmission electron microscope is often in the range 80–300 kV.

6.1.3.1 Atomic Number and Absorption Corrections. For the most part, we shall restrictourselves to K excitation in a two-component alloy, and ignore the more complicatedsituations in which the L and M spectra are excited and analysed, although the excitationprocess does not differ in principle. We first discuss the factors affecting the excitationefficiency, which depend primarily on atomic number, and then treat the subsequent processof X-ray absorption within the volume of the specimen.

Atomic Number Correction The foundations for quantitative microanalysis date back tothe work of Castaing (1951), who first introduced the concept of measuring the ratiobetween the intensity Ii of the characteristic X-rays produced from the element i in aspecimen of unknown composition to the intensity of those produced from the sameelement in a standard sample of known composition Istndi .

In order to understand this relation, we first need to examine the number of atomicionization events n that the incident electrons can induce in the sample for a particular


ionization edge of an element (K, L, orM). Following Scott, Love, and Reed, the number ofionization eventswill be directly proportional to the number of atoms per unit volume of thematerial and can be derived from the following relationship:

dn ¼ NAvrA

� �Q

dE=dxð Þ dE ð6:1Þ

where NAv is Avogadro�s number, r is the density and A is the atomic weight. Q is theionization cross-section for an excitation event, and is a measure of the probability thatan incident electron will ionize a constituent atom in the sample. dE/dx is termed thestopping power, and measures the deceleration of the incident electrons as they penetratethe sample.

Ignoring for the moment backscattered electrons, and assuming a form for the stoppingpower that is based on the velocity of electrons in the sample v relative to the speed oflight c:

�dE=dx ¼ const·r·c

v

� �1:4ð6:2Þ

Then the number of ionization events can be calculated as:

n ¼ �ZEc

E0

NAv

A

� �v

c

� �1:4const·Q·dE ð6:3Þ

where E0 is the energy of the incident electron beam, and Ec is the minimum energyrequired to ionized the electron shell of interest (K, L or M), that is, the critical excitationenergy.

Remembering that absorption and fluorescence are not yet included, n will then beproportional to the generated X-ray intensity I, and we can compare the total number ofionization events in a sample of unknown composition to that occurring in a standardsample examined under identical experimental conditions to yield the concentration of theelement I in the unknown.

Ki ¼ IiIstndi

¼ f Cið Þ ð6:4Þ

where Ki is a �sensitivity� factor and is not a constant. In order to make the analysisquantitative, we must now take into account absorption and fluorescence, which we haveignored so far.

The first step is to understand the depth dependence of X-ray generation in the samplefrom which the photons are collected. The depth distribution of photon generation can becalculated, and has also been measured experimentally. An example is shown in Figure 6.7in which the distribution f(rZ) is plotted as a function of depth using the mass–thicknessparameter, rZ. A number of important conclusions can be derived from this distribution.First, characteristic photon production in the surface region of the sample is large, andgreater than unity on the f(rZ) scale. This is due to backscattered electrons that originatefrom the depth of the sample and generate photons in this surface layer. Second, the initialrise in the curve is associated with a progressive increase in electron scattering as the high


energy electrons penetrate further into the sample and their energy is reduced to the criticalexcitation energy. Finally, fewer photons are generated deep in the sample because fewelectrons reach these depths, and many of those that do have an energy that is below thecritical excitation energy.

Absorption Correction We define the total characteristic X-ray intensity that is generatedin a specimen of pure A in the direction of the spectrometer by integrating fA(rZ):

I0 ¼Z10

fA rZð Þd rZð Þ ð6:5Þ

Since the photons must travel through the specimen in the direction of the detector, some ofthe X-rays will be absorbed before they can escape from the surface of the specimen. Asdiscussed inChapter 2, the reduction in intensity depends on themass absorption coefficientfor each specific wavelength passing through the specimen (m/r) and the path length x ofthese X-rays within the specimen:

I ¼ I0 exp � mr

� �rx

� �ð6:6Þ

The path length depends on the relative position of the detector with respect to the specimensurface, or the �collection angle�, often called the take-off angle (Figure 6.8). We can nowredefine the reduction in intensity due to absorption for our specificmicroanalysis geometryas:

I ¼ I0exp � mr

� �rZcosec að Þ

� �ð6:7Þ

0 5 10

1

2

3φφ(ρρ( Z)

ρZ (10–4 gcm–2)

Al Kα

Figure 6.7 Mass–thickness depth distribution f(rZ) for characteristic X-ray generation in analuminium sample under a 20 kV incident electron beam. The depth is given in terms of themass–thickness (rZ).


where the incident beam is assumed normal to the surface and a is the take-off angle. Theemitted intensity is therefore reduced to:

I ¼Z10

fA rZð Þexp½�ðm=rÞrZcosec a�d rZð Þ ð6:8Þ

The absorption factor is simply the ratio of the number of photons emitted in the direction ofthe detector to the number generated in the sample under the beam:

f wð Þ ¼

R10

fA rZð Þexp½�ðm=rÞrZcosec a�d rZð ÞR10

fA rZð Þd rZð Þð6:9Þ

This ratio of course depends on the energy of the incident electron beam (Figure 6.9).

Specimen

α

x

Incidentelectron beam

ExitingX-ray beam

dZ

Z

Figure 6.8 X-ray path length within the sample x for a take-off angle a of X-rays emitted from asample in the direction of the detector.

0 5 10 15 20 25 300.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

E0 (keV)

α=5°α=10°α=15°α=20°α=25°α=30°

f (χ) EK(Cu)=8.048keV

Figure 6.9 Absorption correction function f(w) as a function of the over-voltage (E0� EK) andthe absorption parameter w¼ (m/r)coseca and for different values of the take-off angle.


An example of the expected effect of the absorption correction on the relation betweenthe K intensity ratio and the concentration for Cu–Au binary alloys is shown in Figure 6.10.As expected, the correction exceeds unity for the lower atomic number copper, but is lessthan unity for the higher atomic number gold. The magnitude of the correction variesinversely as the concentration of the component.

In different software correction packages, the �standard� data employed in calculating thecorrections, as well as the correction algorithm for the calculations, vary appreciably. Inmost cases the supplier of the software package is more than willing to discuss and explainthe assumptions and approximations involved. All commercial software packages shouldbe able to convert the relative characteristic intensity measurements to quantitativeestimates of the elemental concentration to an accuracy of better than 2 % of the trueconcentration, providing the constituents are present at concentration levels of at least a fewpercent and characteristic excitation lines of all the elements are detected. However, thisaccuracy can only be achieved for flat, polished specimens, and providing the beam energyis of the order of 3EK for all the constituents with adequate counting statistics for all themeasured characteristic lines.

6.1.3.2 The Fluorescence Correction. The fluorescence correction is, in a sense, theinverse of the absorption correction, since strong absorption in the sample for the X-rays

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.9

0.8

0.7

Concentration of Au

1.0

k/c

Au

Concentration of Cu

k/c

Cu

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.9

1.0

1.1

1.2

Without Absorption

With Absorption

Without Absorption

WithAbsorption

Figure 6.10 Calculated ratios of k/c for the two constituents in a Cu–Au alloywith andwithoutan absorption correction.

generated by the incident beam implies that secondary excitation of lower energy X-rays isalso taking place. The case of nickel and iron is instructive. Figure 6.11 shows theabsorption coefficient for both elements as a function of the photon energy. Since nickelhas the higher atomic number, its X-ray absorption coefficient is generally higher than that


of iron for any given wavelength, but the excitation threshold for displacing a K-shellelectron in nickel comes at a shorter wavelength (higher energy), so that the K absorptionedge for nickel appears at a lower wavelength. It follows that, for a band of criticalwavelengths between the two absorption edges, iron has a very much larger absorptioncoefficient than nickel. The characteristic K-lines of any element that is fluorescingmust lieto the longer wavelength (lower energy) side of the excitation threshold (the absorptionedge), and in the region of low absorption by the fluorescing element. By contrast, thecharacteristic K-lines for nickel lie in the high absorption region for iron, so that nickelradiation will be strongly absorbed in an iron alloy, and result in fluorescent excitation of theiron matrix, enhancing the characteristic signal for iron.More seriously, in a nickel alloy thestrong excitation of iron will result in a fluorescence correction for iron that may amount to30% of the total signal, and nearly all this fluorescent radiation is generated in a volumewellaway from the region beneath the incident beam in which the primary radiation is excited.

The major problem in the quantitative microanalysis of samples where strong fluores-cence is to be expected arises from this large volume in which the fluorescent excitationoccurs. The penetration depth for X-rays in the sample is typically an order of magnitudegreater than the penetration depth for the incident beam, although it decreases withincreasing X-ray wavelength and sample density. Thus the primary excitation events occurin a volume of the order of 1 mm3 in diameter, but secondary fluorescent excitation willoccur in a volume of the order of 103 mm3 or even larger. The situation is illustratedschematically in Figure 6.12.

Consider as an example, a phase boundary between a nickel-rich and an iron-rich phase,in which the X-ray detection system is placed perpendicular to the boundary, either on the

Mas

s ab

sorp

tion

coe

ffic

ient

Energy (keV)

EK(Fe)=7.109

E(K

α )=

6.39

E(K

β )=

7.04

EK(Ni)=8.329E

(Kα )

=7.

47E

(Kβ )

=8.

26

Figure 6.11 Energy dependence of the mass absorption coefficients of iron and nickel and thepositions of the characteristic Ka and Kb lines for these elements. The absorption edgescorrespond to the critical K-excitation thresholds of the two elements.


side of the nickel-rich phase or on the side of the iron-rich phase. Two quite differentexcitation spectra are to be expected from the boundary region (Figure 6.13):

1. When the boundary is perpendicular to the X-ray take-off direction and the detector ispositioned on the nickel-rich side of the boundary, thus maximizing absorption bynickel, then neither the iron nor the nickel signals will be strongly absorbed. However,when the probe is on the nickel-rich side of the boundary, the nickel radiation penetratinginto the iron-rich region will generate a strong fluorescent iron signal that cannot becorrected, even when the electron beam is positioned up to 10 mm or more from theboundary.

2. Rotating the specimen by 180� so that absorption is for the most part through the iron-rich region, will generate roughly the same fluorescent excitation of iron by nickel whenthe probe is on the nickel-rich side of the boundary. However, now strong absorption ofthe primary nickel radiation will also occur, since the nickel radiation must pass throughthe iron-rich region to reach the detector.

The only sensible course of action with this sample is to position the detector first on oneside of the boundary and then on the other, by rotating the sample through 180�, to check forthe significance of these artifacts. Note the strong dependence of these effects on the X-raytake-off angle. In such a case, only semi-quantitative analysis near the boundary is possible.This may be achieved by determining the apparent concentrations as a function of the take-off angle, tilting the specimen towards the detector, and then extrapolating the results to a

IncidentBeam

PrimaryExcitationEnvelope

SecondaryExcitationEnvelope

SecondaryEvent

FluorescentX-Ray

X-Ray

Figure 6.12 Secondary, fluorescent excitation is expected to occur well outside the envelopeof excitation events for the primary characteristic X-radiation.


p/2 take-off angle. Bearing in mind that these changes in fluorescence may be significant atdistances of 10 mm from a phase boundary, concentration measurements near suchinterfaces should be treated with considerable caution.

6.1.3.3 The ZAF Equation. Let us now return to the quantitative correlation of intensitywith concentration, we first consider a binary solid solution containing elements A and B inwhich we wish to determine the concentration of A. Corrections for the backgroundradiation and for any escape or other spurious peaks are assumed to have been made. Wenow include a correction factor for fluorescence and describe the total emitted X-rayintensity as:

I ¼ f DrZð ÞZ10

f rZð Þd rZð Þf wð Þ 1þgþdð Þ ð6:10Þ

Fe Ni

2 µm 30 mm

IFe1

Detector 1 Detector 2

IFe2 INi1

INi2

Probe scan

Figure 6.13 Iron and nickel intensities perpendicular to a Fe–Ni interface measured by twodifferent detectors; detector 1 to the left of the sample, and detector 2 to the right. Using a 1 mmdiameter probe and ignoring fluorescence, the iron signal will drop rapidly away from theinterface over a distance of approximately 2mm. Fluorescence will result in an iron signal fromthe nickel side of the couple, no matter which detector is used. The spurious �iron� intensity willbe approximately 6 % of the signal from pure iron and decay to zero some 30 mm from theinterface. The nickel signal collected by detector 1will showa sharp decrease due to absorptionby the iron, while the nickel intensity collected by detector 2 will be unaffected by absorption.


where fDrZ corresponds to the emission from a thin layer of mass thickness DrZ, and(1þ gþ d) is the correction that has been inserted for fluorescence. Specifically, g is theratio of the intensity of the fluorescent emission to the primary characteristic X-rayemission, while d is the corresponding ratio for the continuum fluorescence contribution.

This equation for the characteristic intensity generated by element A from our sample ofunknown composition containing the elements A and B, can now be expressed relative tothe corresponding intensity from a standard sample of pure A:

IABAIAA

¼ f DrZð ÞABAf DrZð ÞAA

·

R10

f rZð Þd rZð Þ� �AB

AR10

f rZð Þd rZð Þ� �A

A

·f wð ÞABA 1þgþdð ÞABAf wð ÞAA 1þgþdð ÞAA

ð6:11Þ

Z A FIn this equation we can substitute:

CABA ¼ f DrZð ÞABA

f DrZð ÞAAð6:12Þ

So that:

IABAIAA

¼ CABA ZAFð Þ ð6:13Þ

6.1.3.4 Errors, Detection Limits, and Spatial Resolution. Error analysis for quantitativemicroanalysis using both EDS and WDS follows Poisson statistics, so that if we collectN counts for any specific, characteristic X-ray energy, then the standard deviation for themeasurement, s, is N1/2. The relative standard error, e, is then s/N¼N�1/2. It follows that,for a standard error of�1% some 104 counts are required. This value can be easily reachedin WDS, but may require quite long integration times in EDS.

Asnotedearlier, is important to separate the statisticsof themeasuredpeak fromthatof thebackgroundbeforemaking theFAZ corrections. The intensities are expressed as the net peakcount rate (IP�IB), where IP and IB are the number of counts for the characteristic peak (NP)and the overlapping background (NB), dividedby the signal integration times used to acquirethe two sets of counts, tP and tB. The standard deviation of this net peak count rate is then:

sP�B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNP

t2PþNB

t2B

sð6:14Þ

while the standard error for the net peak count-rate is:

eP�B ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNP

t2PþNB

t2B

s

NP

tP�NB

tB

� � ð6:15Þ

The detection limit, sometimes (redundantly) termed theminimum detection limit, dependsdirectly on the signal-to-noise ratio of the spectrum, that is, the net counts of the peak versus


the net counts in the background. The detection limit corresponding to a 95 % confidencelevel can be expressed as:

CDL;0:95 ¼ ZAFð Þffiffiffiffiffiffiffi2IB

pffiffit

pIStndP �IStndB

� ð6:16Þ

The detection limit for heavy elements in a light element matrix can be as low as a few partsper million when using WDS.

It is always important to understand the source of the measured signal, which depends onthe accelerating voltage of the incident, high energy electron beam, the extent of absorptionfor the characteristic primary radiation and fluorescence or secondary radiation. Themaximum depth in the sample from which the characteristic X-ray signal originates,ignoring fluorescence, can be approximated by:

Zr � 0:033 ð E1:70�E1:7

C

�A

r�Z

� �mm½ � ð6:17Þ

where �A and �Z are the average atomicweight and average atomic number in thevolumeofmaterial being excited by the incident beam.

The maximum diameter of the excited volume generating the X-ray signal is usuallyapproximated by:

D � 0:231

rE1:50 �E1:5

C

� mm½ � ð6:18Þ

6.1.3.5 Microanalysis of Thin Films. The maximum useful specimen thickness in thin-film TEM is often determined by the onset of inelastic scattering processes in the sample,although some inelastic scattering always occurs in addition to the dominant, elastically-scattered, transmitted signal. It follows that a characteristicX-ray signal should be availablein thin-film TEM. Since the specimen is very thin and the electron energy is very highcompared with the beam energies used in SEM, the signal will be weak and very fewcharacteristic X-ray photons are generated. As a consequence, it is not generally possible todetect constituents that are present in concentrations of less than 5%. The signal isimproved if a field emission source can be used to generate a highly focused electronprobe which is scanned over a small area of the thin-film sample.

That said, the excellent spatial resolution of the TEM and the negligible lateral spreadof the electron beam transmitted through the thin-film specimen, has motivated thedevelopment of X-ray detection systems and software for quantitative microanalysisdesigned specifically for TEM. These systems depend heavily on the availability of afield-emission source and are combined with EDS, since only this combination providesan adequate X-ray signal intensity for quantitative analysis at low concentrations. As weshall see below, EDS of X-ray signals generated by the exceptionally small volumes in aTEM thin-film specimen is best accomplished in scanning TEM (STEM) mode, ratherthan by conventional TEM.

The principle factors that have to be considered when attempting quantitative thin-filmmicroanalysis by using characteristic X-rays generated in the transmission electronmicroscope are the following:


1. The high background noise that is associated with stray high energy electrons in themicroscope column. These generate both white radiation and spurious characteristic X-rays, especially from copper, which is the major alloy constituent of the specimen stage.

2. Maximizing the X-ray collection efficiency by ensuring the maximum solid angle fordetection, achieved by placing the solid-state detector as close as possible to the sample.

3. The thickness dependence of the characteristic X-ray signal and the use of thickness-dependent intensity measurements in place of standard correction procedures.

4. The reduction of the TEM accelerating voltage for thin-film microanalysis in order toimprove the X-ray excitation probability and the counting statistics.

5. Correlation of the source of X-ray excitation in the samplewith microstructural featuresthat may be observed under different beam conditions.

The problem of excitation from physical components of the microscope column isexacerbated by the electro-optical limitations on beam-probe focusing in TEM. It is notdifficult to focus a fine probe, of the order of 2 nm diameter at FWHM, by using a standardmagnetic lens condenser system,butmuchof thecurrent in thebeamremains in the tail of thedistribution, outside the central, focused probe, and leads to characteristic X-ray excitationwell beyond theFWHMdiameter (Figure6.14).This effect hasbeen successfully reducedbyaberration correctors placed in the condenser assembly of current STEM systems.

A significant improvement is also possible by manufacturing the components of thespecimen holder and specimen mount from a low atomic number element such as graphiteand especially beryllium. Both these elements are electrical conductors and are free ofelectrostatic charging. Specimens that do not require a support grid are also to be preferred.

Solid-state Si(Li) X-ray detectors are used for energy-dispersive microanalysis in TEM.They can be placed within 10 cm of the specimen in an inclined geometry with a high take-off angle (�30–35�) giving a wide solid angle for data collection. Some systems offer ahorizontal take-off angle, designed for a sample tilted towards the detector. This results in asignificant increase in the available X-ray signal, but such a geometry is problematic if thegoal is to measure the concentration associated with an interface or grain boundary, sincesuch interfacesmust necessarily be aligned parallel to the incident electron beam. Recently,sample holders,made of berylliumhave become available. These are cut to allow theX-raysto reach a horizontal detector. Such holders improve the counting statistics without tiltingthe specimen, which may be critical for planar defect characterization.

Variations in the specimen thickness may be an advantage for some samples: assuming awedge-shaped specimen, the relative intensities of the characteristic lines can be measuredas a function of distance from the edge of the hole in the sample prepared by dimpling andion milling. The measured intensity ratios can then be extrapolated to zero thickness, whenthe absorption correction for the generated X-rays extrapolates to unity. In addition X-rayfluorescence can only occur in a region that is well outside the region of primary excitation,so that, thanks to the thin section of the sample, allX-raymicroanalysismeasurementsmadein the transmission electron microscope are free of fluorescence effects.

As the accelerating voltage decreases, the probability of an inelastic interaction thatgenerates characteristic X-rays will increase, so a decrease in the accelerating voltage of thetransmission electron microscope should improve the yield. Unfortunately, any decrease inthe accelerating voltage also decreases the brightness, that is, the current density of thesource per unit solid angle. In practice the �best�, statistically significant, spectra may be


recorded at the highest available accelerating voltage. Results obtained at the highestaccelerating voltage should also correspond to the highest spatial resolution for micro-analysis, since beam spreading in the thin foil decreases with increasing acceleratingvoltage, reducing the diameter of the region in the thin film from which the X-rays aregenerated.

A primary motivation for using energy-dispersive X-ray microanalysis in thin-filmtransmissionmicroscopy is the improvement in the spatial resolution for analysis, due to thesmaller high energy electron probe size available in TEM or STEM. (In, Figure 6.15 twospectra from two neighbouring points are shown). When working with very small probesizes, it can be difficult to know the exact position of the beam relative to a microstructuralfeature. Both specimen drift and beam drift commonly occur during the long acquisitiontimes that may be required to obtain good counting statistics. This is where a STEM systemhas an advantage over conventional TEM, since the beam can be rastered to produce either aline-scan or elemental map, as in SEM, while bright-field STEM, annular dark-field (ADF)STEM, or high-angle annular dark-field (HAADF) STEM images can be acquiredconcurrently, fully synchronizing the EDS measurements with the features of interest inthe microstructure.

(a)

(b)

Probe

Tail

FWHM

FWHM

Probe

Tail

Figure 6.14 (a) Distribution of the beam current in a focused probe includes a significantcomponent in the tail of the electron spatial distribution. (b)Aberration correctors for condensersystems can reduce the tail and the FWHM, resulting in a much sharper focused probe.


6.2 Electron Energy Loss Spectroscopy

Microanalysis that is based on characteristic X-ray excitation faces two practical problems.The first is the very low collection efficiency for X-rays (no better than 10�3 for WDSsystems and of the order of 10�2 for EDS detectors). The second concerns the inefficientcharacteristic X-ray excitation and poor detection resolution for radiation generated fromthe light elements. Although the characteristic spectra from the elements belowmagnesium(Z¼ 12) are readily detectable down to lithium (Z¼ 3), these elements are difficult toanalyse quantitatively, for three reasons. First, due to absorption by the solid-state detectorwindow and loss of detection efficiency at low photon energies. Second, absorption bysample surface contamination and surface films distorts any quantitative analysis. Finally,only a small fraction of the low energy, primary excitation events lead to characteristic X-ray emission, while the remainder generate Augerelectrons (Section 7.2). For example, it isestimated that the chance of a K-shell excitation of carbon (Z¼ 6), yielding a photon is only1:400, and this yield increases only slowly with atomic number, rising to 1:40 for sodium(Z¼ 11).

Analysis of the energy spectrum of the inelastically forward scattered electrons, over-comes all these limitations, since very high electron collection efficiencies, of better than 50%, are possible at the energy loss detector while maximum analysis sensitivity is achievedprecisely for the low energy losses that are characteristic of the low atomic numberelements. Moreover, the electron energy loss spectrum is usually measured by thin-film,

Figure 6.15 EDS X-ray spectrum from a small titanium carbon nitride (TiCN) particle in analuminium matrix, illustrating the power of the transmission microscope to overcome theresolution limitations for X-ray detection in SEM. Two spectra are shown: (a) from aluminiummetal and (b) from the TiCN particle. Note the change in relative intensities (Al/Ti) between thetwo spectra.


transmission microscopy, using forward scattered electrons, precisely the geometry forwhich characteristic X-raymicroanalysis by EDS ismost difficult, primarily due to the verypoor counting statistics (Section 1.3.5).

Two primary modes of operation are possible for electron energy loss spectroscopy(EELS) in the conventional transmission electron microscope. If an image is focused onthe screen, then the back focal plane of the projector lens will contain a diffraction patternthat can serve as the signal for an EELS spectrometer. The selected area aperture in thisplane determines the origin of the signal and the image formed in the back focal plane ofthe projector lens is the image �seen� by the spectrometer. Alternatively, if a diffractionpattern is focused on the screen, then the back focal plane contains an image that serves asthe signal source for the EELS spectrometer, and the source of the signal is then the area ofthe specimen within the selected area aperture that is illuminated by the incident beam.Focusing the incident beam, to form a convergent-beam diffraction pattern on the viewingscreen can also define the region from which an EELS signal is acquired. If the incidentelectron beam is fully focused to a fine probe, as in STEM mode, then this will beanalogous to a diffraction pattern that is focused on the viewing screen. The STEM probewill then also define the sample area from which the signal is collected. In effect, it is theelectrons focused in the back focal plane of the microscope that determine the source ofthe EELS spectrum.

It follows that in TEM mode we can acquire a spectrum from either an image, or adiffraction pattern, while in STEM mode we will always acquire a spectrum from adiffraction pattern. The advantage of STEMmode is that the electron beam can be rasteredacross the sample, to acquire EELS data as a function of the position of the incident electronprobe, in order to form an EELS line-scan or composition map, just as we discussed forEDS. In addition, when using a field emission gun source, the spatial resolution of the EELSsignal is actually better in STEMmode than it is in TEMmode, and sub-nanometre spatialresolution of EELS results for chemical composition has been demonstrated. If a HAADFor ADF detector is available for the STEM system, then the pixel intensity in the image canbe measured at the same time as the EELS spectrum, and this image intensity can becorrelated directly with the local EELS signal.

The electron energy loss spectrometer is a magnetic prism, positioned beneath the mainmicroscope column (Figure 6.16). The magnetic spectrometer is located below theprimary image plane of the microscope and accepts electrons that pass through anaperture positioned on the optic axis. The electrons deflected by the spectrometer willhave an angular spread, with those electrons of energy E0 that have experienced no energyloss being the least deflected. All those electrons that have experienced an energy loss willbe deflected by an additional angle that is dependent on the extent of this energy loss.Instrumental developments in EELS have now improved the energy resolution to muchbetter than 1 eV, as compared with, perhaps, 50 eV for the light element, characteristicX-ray energy resolution in the best EDS systems. Parallel EELS detectors have replacedserial detection systems, providing simultaneous data collection across a large range of theenergy loss spectrum. By adding an extra set of lenses after the standard EELSspectrometer, it is possible to convert the signal from reciprocal space (the originalenergy spectrum) to real space (to form an image). This allows us to collect imagescorresponding to specific values of energy loss that are characteristic of one or other of thechemical constituents of the sample. Such energy loss imaging filters can now provide


quantitative chemical information with a spatial resolution of the order of 0.5 nm and anabsolute volume detection limit of less than 100 atoms. The development of energy-filtered transmission electron microscopy (EFTEM) (Section 6.2.6) has had a majorimpact on materials research at the nanometre level.

One major caveat is in order. In both crystalline and amorphous thin-film specimens,elastic scattering events are farmore probable than inelastic events andmuch of the inelasticenergy absorption spectrummay be generated by electrons which have first been elasticallyscattered. It follows that inelastic scattering of the electrons in a diffracted beam is acommon occurrence. However, it is the energy spectrum in the direct transmitted beam thatis usually collected, while those electrons in the diffracted beams that have been inelas-tically scattered are excluded because they are prevented by an aperture from entering thespectrometer. Relatively little work has been done to compare the energy loss spectrumfrom the direct transmitted beamwith that from a diffracted beam, that is, the energy lossescorresponding to a dark-field image.

Figure 6.16 Schematic drawing of an EELS spectrometer, positioned below the maintransmission electron microscope column.


6.2.1 The Electron Energy-Loss Spectrum

A typical energy loss spectrum is shown in Figure 6.17. The y-axis is the detected intensityat a given energy, measured in arbitrary units (a.u.). The x-axis is the energy loss (E0�E),measured in eV. Energy losses as high as 2000 eV can be detected, although the count ratesat high energy losses are very low. Note that the scale for the y-axis has been changed by afactor of ·500 above the low-loss peak labelled plasmon. The spectrum consists of a seriesof sharp intensity changes superimposed on an exponential decay in the electron currentwith increasing energy loss.

Four processes contribute to the energy losses of the electron beam, although only two ofthese can be detected and measured by the magnetic spectrometers that are used to analysethe energy loss spectrum:

. Phonon excitations. These excitations result in very small energy losses, typically lessthan the thermal energy spread in the incident beam. These peaks are within the zero-losspeak of the energy loss spectrum, and are not resolvable. Nevertheless, the zero-loss peakcan provide important information on the performance of the electron source and thestability of the high-voltage supply.A field emission gun is capable of limiting the thermal(kT) spread of the primary beam energy to less than 0.3 eV. By comparison, a LaB6 sourcemay have a thermal spread of the order of 1.5 eVand a conventional, thermionic emissiontungsten filament, will increase this spread to about 3 eV.

. Electron transitions. These transitions occur both within and between the differentelectron shells of the atom, and commonly correspond to energy changes in the range1–50 eV. These very low-loss peaks can be detected in the spectrum, and may be used toidentify a phase containing a specific element by a comparison with a known, standard

0 500 1000Energy loss (eV)

Inte

nsit

y(a

.u.)

Zero-losspeak

Energy loss near-edge structure(ELNES)

x500

Gai

nP

lasm

on

valence loss coreloss (>50 eV)

Figure 6.17 Example of an EELS spectrum showing the zero-loss peak, a plasmon peak in thelow energy loss region, and a complex excitation absorption edge in the core loss regionassociated with inelastic interaction with an inner (core) electron shell.


spectrum. Analysis of the low energy loss region, in the 10 eV range, can, in principle, beused to determine the localized dielectric properties of a material.

. Plasmon excitation. The plasmon phenomenon is associated with quantized oscillationsin the conduction band of a metallic conductor. These typically result in energy-losspeaks in the range 5–50 eV. The plasmon effect is roughly analogous to the ripplescreated on the surface when a small stone is dropped into a pond. Plasmon peaks havebeen interpreted in chemical terms, since they are concentration-sensitive, but theinterpretation is controversial. The one or more plasmon peaks from metallic conductorsare quite sharp, while those from nonconductors are rather diffuse. If the mean free pathof the plasmons lp is known, and if the sample is thin enough to ensure that only a singleplasmon peak is excited, then we can use the total intensity in the low-loss region of thespectrum relative to the zero-loss intensity to estimate the specimen thickness t from therelationship t ¼ lpln IT=I0ð Þ, where IT is the total integrated intensity that includes theplasmon and zero-loss peaks, and I0 is the intensity of the zero-loss peak alone. Note thatrelative values of specimen thickness are always proportional to the plasmon losses, andthese can therefore be used to check for any thickness dependence of the EELS analysisresults.

. Higher energy losses. The high energy loss region of the spectrum (DE>50 eV) containsthe inner shell absorption edges that are associated with atomic ionization and areaccessible for chemical analysis in EELS. Because the spectral resolution in EELS is somuch better than that available byEDS (some twoorders ofmagnitude) orWDS (about anorder of magnitude), more detailed information is available in the excitation edgeassociated with each individual atomic species. Moreover, since EELS is particularlysensitive to low energy excitations, this signal contains considerable chemical informa-tion. However, the quantitative analysis of chemical composition by EELS is usually lessaccurate than can be achieved by X-ray microanalysis using EDS orWDS. It follows thatthe major application of EELS is for the detection of the atomic species present on thenanometre scale and in the study of localized chemical bonding states, rather than foraccurate, quantitativemicroanalysis. The remainder of this account is limited to the EELSsignal that is specific to the atomic species present.

6.2.2 Limits of Detection and Resolution in EELS

The energy resolution of themagnetic spectrometers commonly available for EELS is of theorder of 0.1 eV, but it requires a monochromatic field emission source to exploit thisresolution. Inmost cases it is the kinetic energy spread in the beam, not the characteristics ofthe EELS detector, which limits the energy resolution. It is therefore common practice torecord the zero-loss peak, corresponding to the primary Gaussian peak for the beam exitingthe specimen. This primary peak is then used to calibrate both the absolute zero for theenergy loss spectrum and to estimate the available energy resolution, usually from theFWHM of the zero-loss peak.

The intensity of the first plasmon peak relative to the zero-loss peak is also a goodbenchmark for judging the suitability of a thin-film specimen for EELS analysis. As a roughguide, if the intensity in the first plasmon peak is less than one-tenth of the zero-loss peak,then the specimen should be thin enough for EELS. The inelastic signal from thin samples isveryweak, but increases as the thickness increases. However, if the sample is too thick, then


multiply scattered electrons obscure the edge structure of the energy loss spectrum. Inaddition, the analysis of multiple scattering, more than one scattering event per electron,requires an error-prone deconvolution correction for quantitative EELS microanalysis. Inpractice an optimum specimen thickness for EELS exists for which the counting statisticsare adequate, while the probability of multiple scattering is small. Not too surprisingly, theoptimum results are obtained at thin-film thicknesses comparable with the extinctionthickness of the sample for dynamic diffraction. These are typically less than 10 nm, butdepend on atomic number (Z), accelerating voltage (E0) and the diffracting conditions.

The analytical sensitivity of an EELS system can be discussed in terms of either theminimum detectable mass, the minimum signal that can be identified from a givenconstituent, or the minimum mass fraction, the minimum detectable concentration of agiven element in the sample. The analytical sensitivity of EELS is far better than can beachieved by X-ray microanalysis and can correspond to a signal from a few hundred atoms.The detection limit varies with atomic number and the position of the edge in the lossspectrum.

Inmany cases the best spatial resolution for analysis, that corresponding to theminimumfocused probe size, is not required and it is possible toworkwithmuch higher incident beamcurrents, either by increasing the probe size in STEM mode, or by using a larger selectedarea aperture in TEMmode. The sensitivity is then no longer limited by the signal statisticsbut rather by errors of extrapolation in subtracting the background corrections from thespectrum and the relative contributions to the signal from the different element-specific,inner-shell absorption edges. Unfortunately, these errors are of the order of 10 % forK-excitation, while data from L-edges are even less accurate, primarily because the L-edgeismuch broader than theK-edge.M-edge data are useful for confirming that a constituent ispresent, but not for estimating concentration. The energy loss spectra can be displayed forany energy range and at any gain (Figure 6.17). The ability to display change in gain isespecially important, given the exponential decay of the signal with increasing energy loss.

Identification of the constituents responsible for the absorption edges observed in theenergy loss spectrum depends on accurate calibration of the energy scale for the magneticspectrometer. The zero-loss peak defines the origin for zero energy by the centre ofmass forthis Gaussian peak, while, as noted previously, the width of the zero-loss peak at FWHMdefines the available energy resolution. A carbon K-edge (Figure 6.18) is often selected todefine the linear energy scale, since carbonaceous contamination is a common feature ofmany acquired spectra. The carbon edge from either diamond or cementite (Fe3C)will havea different shape from the amorphous carbon contamination peak and will appear in aslightly different position on the energy scale, so one has to be careful when calibrating theenergy scale from this ionization edge. The position of each edge is usually defined as theposition of maximum slope.

Energy absorption events corresponding to an excited state of the atom will also result inhigher energy losses that lie beyond the initial absorption edge and, in principle, extend tothe maximum incident beam energy. The total number of events associated with anyparticular absorption edge can only be estimated by fitting an empirical function to the pre-edge data and then extrapolating this estimated background curve to the high-energy lossregion. The estimated background is then subtracted from themeasured data to yield avaluefor the true number of characteristic absorption events (Figure 6.19). Typical empiricalfunctions for the background are of the form I¼AE�r, whereA and r are empirical constants


that have to be determined separately for each pre-edge energy loss region. The signal fromeach edge decreases rapidly at higher energy loss values and it is neither necessary nordesirable to sum the counts derived for a particular edge over all energies above that edge,because the extrapolation errors above the edge increase with energy. However, it isimportant that the data for all edges should be summed over the same range of energy loss.

280 300 310290

Energy loss (eV)

Inte

nsit

y (a

.u.)

C-K

Figure 6.18 A measured carbon K-edge at 284 eV.

NiO

600 800 1000 1200 1400 16000

10

20

30

40

50

60

70

80

Energy loss (eV)

x10

00

SpectrumBackground fitNi edge after background substraction

O

Ni

Figure 6.19 Determination of the number of excitation events associated with a specificabsorption edge for nickel in NiO. The background is fitted to the energy loss curve before thenickel edge, and then extrapolated to the edge region and beyond before being subtracted.


This �energy loss window� D is typically 50–100 eV, and the same window can be used toderive an absolute mass data scale by applying this window to the zero-loss and low energy,plasmon loss region. The assumption is that the data are collected over a fixed solid angle,the collection semi-angle b, determined by the spectrometer aperture, and that the inelasticscattering cross-section for this solid angle and energy range s(bD) is known, so that goodestimated values for Kexcitation are assumed to be available.When this is the case, then thenumber of atoms per unit area N is given by:

N ¼ IKðbDÞI0ðbDÞ :

1

sKðbDÞ ð6:19Þ

where IK refers to the K-edge signal for a particular element and I0 is the zero-loss and lowenergy loss signal collected for the same energy interval and collection angle.

6.2.3 Quantitative Electron Energy Loss Analysis

Quantitative estimates of the absolute mass of elements present in the excited region for anEELS spectrum are generally less useful than determinations of their relative concentra-tions. The simplest method for obtaining a quantitative estimate of relative concentration isby calculating the ratios of the data derived from each K-edge. In this case, the zero-lossintensity data cancel out:

NA

NB¼ IAK bDð Þ

IBK bDð Þ ·sBK bDð ÞsAK bDð Þ ð6:20Þ

The calculation assumes that good estimates are available for all the relevant values of thepartial inelastic scattering cross-sections s for the constituents.

If two separate edges come within the range D then it is important to subtract theextrapolated energy loss curve for the higher energy loss edge to obtain the excitationintensity for the lower atomic number constituent before subtracting that from the lowerenergy-loss absorption edge. Again, any such overlap will further limit the accuracy of theanalysis.

A major problem in applying EELS to quantitative microanalysis is in selecting anoptimum sampling thickness from a wedge-shaped thin-film specimen, prepared bydimpling and ion milling. Figure 6.20 shows the normalized, calculated intensity ratiosfor three different binary element combinations (B/N, Al/O and Al/Ni) as a function of therelative intensity of the first plasmon, low energy loss peak, to the zero-loss peak. As notedpreviously, the relative height of the first plasmon peak is proportional to the absorptionthickness in the thin-film sample. From Figure 6.20, it is clearly an advantage to choose aregion that is as thin as possible, commensuratewith obtaining adequate counting statistics.Even so, comparing results acquired at different sample thickness is probably the best wayto ensure confidence in the analysis.

Characteristic X-ray microanalysis, using the EDS or WDS method, remains far moreaccurate for quantitative microanalysis, but cannot compete with EELS in mass sensitivity.EELS methods are certainly less valuable for high energy loss peaks, but EELS retains itssuperiority for low atomic number materials that result in low energy loss values. Bothtechniques can overlap over the full range of atomic number in the periodic table. Although


EDS possesses greater analytical accuracy, EELS has far better spatial resolution anddetection limits, while the better spectroscopic energy resolution in EELS can also provideinformation on the nature of the chemical bonding in the sample. This is our next topic.

6.2.4 Near-Edge Fine Structure Information

The 0.1 eVenergy resolution of EELS spectrometers has revealed a wealth of fine structurein the region of the absorption edge that is still only partially understood, even though thegeneral physical principles are clear. This energy loss near edge structure (ELNES) reflectsboth the chemical state of the atom and, to some extent, the atomic coordination andsymmetry with respect to the neighbouring atoms. Changes in the chemical bonding stateresult in energy shifts of both the primary edge and the fine-structure peaks, while peaksplitting results from a reduction in coordination symmetry. In alumina, cation disorder thatis present in the higher symmetry, transient crystal structures generates a few broad,comparatively simple peaks at the adsorption edge, while for the stable, less symmetriccorundum phase several sharp peaks are visible in the absorption-edge spectrum(Figure 6.21). Calculations of the energy shifts that are associated with resonant energyexchange processes in the nearest neighbour coordination sphere are consistent withdifferences in the cation coordination symmetry of the g and a alumina phases (as indicatedfor the aluminium L2,3-edge in Figure 6.22).

This high resolution EELS information is not related to chemical composition, but ratherto the localized atomic bonding in the solid. The ability to distinguish bonding states and

0.01 0.1 1.0

IA/IB

IP/I0

IB(K) /IN(K)

IO(K) /IAl(K)

IAl(K) /INi(L2,3)

Figure 6.20 Intensity ratio for two ionization edges in three different materials as a function ofthe thickness. The results are plotted as the ratio of the intensities of the first plasmon, lowenergyloss peak to the zero-loss peak. (Courtesy of David B. Williams and Philips ElectronicInstruments, Inc.).


coordination symmetry provides a major motivation for expanding existing EELSapplications.

6.2.5 Far-Edge Fine Structure Information

Oscillations in the energy loss signal are also observed at some distance above the positionof an edge and are referred to as extended energy loss fine structure (EXELFS). Thewavelength of these oscillations in the energy spectrum is of the order of 20–50 eV, whilethe amplitude may amount to 5 % of the edge signal. The effect is analogous to an effect inX-ray absorption spectra referred to as extended X-ray absorption fine structure (EXAFS).EXELFS is due to elastic scattering of an inelastically scattered electron by the periodiccrystal structure. The oscillations may persist for several hundred eV above the edge.

The information contained in the EXELFS signal is related to the local atomic densityand could, in principle, be extracted to give the radial distribution function (RDF) for thechemical component responsible for the absorption edge. The RDF records the probability

60

(a)

(b)

70 80 90 100 110 120 130 140

60 70 80 90 100 110 120 130 140

Energy loss (eV)

Energy loss (eV)

Inte

nsit

yIn

tens

ity

Figure 6.21 Aluminium L2,3-edge in (a) the cubic, g-Al2O3 metastable phase and (b) therhombohedral, a-Al2O3 stable phase. (Courtesy of Igor Levin).


that an atom occupies a specific coordination sphere located at any given distance from thesource atom. To extract this RDF information, we must first subtract the background fromthe signal to reveal the oscillations, and then perform a Fourier transform of the oscillationsto convert them into the radial distribution function. In principle, this technique shouldsupply information on the near neighbour atomic environment, dramatically extending theusual concept of microanalytical information. However, the results are rather sensitive tothe correction procedure and are controversial.

6.2.6 Energy-Filtered Transmission Electron Microscopy

The addition of an adjustable aperture followed by an additional set of lenses after the EELSmagnetic spectrometer (Figure 6.23) makes it possible to select a part or all of the energyloss spectra to form an image in real space on a charge-coupled device (CCD) camera. Thissystem configuration is termed energy-filtered transmission electron microscopy (EFTEM).

5 10 15 20 25 30 35 40 45Relative energy loss (eV)

Inte

nsity

OctahedralAlO6

TetrahedralAlO4

Al L2,3

Figure 6.22 Calculated ELNES L2,3-edge spectra for tetrahedrally and octahedrallycoordinated Al cations. (Courtesy of Rik Brydson).


This extension of EELS to EFTEM allows us to filter the electron energy loss signal so as toselect a specific energy loss range to form an image.

The energy loss range selectedcanbe fromanypart of theEELS. In the simplest case, onlythe zero-loss peak of the EELS is selected in order to eliminate the chromatic aberrationassociated with inelastic scattering of the transmitted electron beam. An example of a zero-

Figure 6.23 Additional lenses placed behind an adjustable aperture or slit in a parallel EELSsystem can be used to convert the energy loss signal from reciprocal space to real space, so thatan image can be acquired by a CCD camera.

loss diffractionpattern is comparedwith aconventionalpattern inFigure6.24.The improvedsharpness of the diffraction spots and the readily resolved secondary spots demonstrateclearly the additional diffraction information that is obtained by excluding the inelasticallyscattered electrons present in the conventional diffraction pattern. Similar improved


sharpness is also often observed in the image formed by diffraction contrast when theinelastically scattered electrons are excluded from a bright field micrograph (Figure 6.25).

However, the full potential of energy filtered TEM is only realized when energy losssignals from two or more characteristic absorption edges are used to form the image, bycombining different regions of the energy-loss spectra into a single image. One way topicture this is shown in Figure 6.26, where the energy loss spectra is plotted vertically andthe energy ranges (DEi) characteristic of each chemical species form sequential images.

Images formed from a selected energy loss range can be viewed directly, and the contrastis then associated with areas in the microstructure that have resulted in the selectedcharacteristic energy loss. Alternatively, the background subtraction corrections applied tothe energy loss spectra can also be applied to a series of energy loss images, and localcontrast in the images can then be qualitatively related to local concentrationvariations. Theresultant elemental map is analogous to that obtained by X-ray mapping, but the spatialresolution for concentration differences is verymuch better. As an example, we return to the

Figure 6.24 Comparison of a conventional selected area diffraction pattern (a) with an energy-filtered selected area diffraction pattern (b) of flagellin. Reproduced from K. Yonekura, S. Maki-Yonekura and K. Namba, Quantitative Comparison of Zero-Loss and Conventional ElectronDiffraction from Two-Dimensional and Thin Three-Dimensional Protein Crystals, BiophysicalJournal, 82(5), 2784–2797, 2002. Copyright (2002), with permission from the BiophysicalSociety.

Cu–Co alloy shown in Figure 6.25. Copper and cobalt are immiscible in the solid state, andthe alloy is expected to be a twophase dispersion of these constituents. However, copper andcobalt have similar atomic number and the difference in lattice parameter between the twophases is also small. Neither diffraction contrast in TEM nor STEM imaging can be used to


locate the two phases with any confidence. By using the characteristic energy lossesfor copper and cobalt in the energy loss spectra and following the schema indicated inFigure 6.26, it is possible to build up a composite EFTEM image that clearly displays thelocation and size of the cobalt particles within the copper matrix (Figure 6.27).

Summary

Information on the chemical composition of individual microstructural features is often ofcrucial importance, and considerable effort has been expended on microanalysis tocomplement morphological and crystallographic data. Beyond the primary, qualitativerequirement to identify the chemical elements present in a specific region, microanalysishas two quantitative aspects. The first is the spatial resolution of the analysis in the recordedimage, while the second is the spectral sensitivity in the collected spectrum. Thecharacteristic X-ray signal generated in the scanning electron microscope is the mostfrequently used of the microanalytical spectral tools that are available. This signal comesfrom a volume element of material near the surface of the solid sample. This volume isusually at least 1 mm3. The limiting spatial resolution for microanalysis is therefore of theorder of 1 mm. Much better spatial resolution is available in the transmission electron

Figure 6.25 Bright-field TEMmicrographof a cross-section fromaCu–Co alloy, deposited on asiliconwafer that was coatedwith a composite SiO2/TaN/Cu film.Only the zero-loss peak in theelectron energy loss spectrum was used to form this image.


x

y

∆ E

∆ Ei

Image at ∆ Ei

Figure 6.26 Schematic illustration of the concept used for energy-filtered imaging. A selectedrange DE of the energy loss spectra is used for each image.

Figure 6.27 Energy-filtered image of the micrograph shown in Figure 6.25. Green indicatescobalt, and red indicates copper. Blue indicates oxygen-rich areas located in the SiO2 layer.(See colour plate section)


microscope using the energy loss spectra of the transmitted beam from a thin-film sample.X-ray microanalysis also has a limited spectral sensitivity that is associated with theinherent limitations of the Si(Li) solid-state detector and the background of �white�X-radiation that partially obscures the characteristic signal. Errors in quantitative X-raymicroanalysis are typically of the order of 1 % of the elemental composition, and theabsolute limit of detection for any constituent element is usually no better than 0.2 %.

Characteristic X-ray microanalysis may employ the K-, L- or M-spectra excited in thesample, depending on the atomic number of the constituents and the electron energy in theincident beam probe. In general, the incident beam energy is selected to optimize the spatialresolutionwith respect to the counting statistics. The former is best at lower beam energies,while the latter generally improve at higher beam energies. Two methods are commonlyused to collect the characteristic X-ray spectra, namely wavelength dispersion and energydispersion. In wavelength–dispersive spectroscopy (WDS) a proportion of the X-raysemitted from the sample is collected by a curved crystal and photons that have awavelengthwhich fulfills theBragg condition for diffraction by the crystal are focused onto a detector (agas proportional counter). By rotating both the diffracting crystal and the detector to changethe Bragg angle, it is possible to scan across a range of wavelengths and record the excitedX-ray intensity as a function of wavelength.

In energy-dispersive spectroscopy (EDS) a solid-state, lithium-drifted silicon, Si(Li),detector is used to record theX-ray signal. The detector absorbs the photon energy and givesrise to current pulses whose intensity is proportional to the absorbed photon energy. Theelectrical pulses are first digitized and then counted in amultichannel analyser to develop ahistogram of the number of pulses as a function of the energy of the collected photons.Although wavelength dispersion has better spectral resolution, energy-dispersive systemsare able to record all the photons admitted to the detector simultaneously. However, thepulse count rate in EDS is limited by the response of the detector crystal, and this sets anupper limit to the rate of data collection. Although EDS is the most commonly availabledetection system, both systems are important, depending on the application requirements.Both EDS and WDS systems have limited detection efficiency for the long-wavelengthradiation that is characteristic of low atomic number, light elements. Even so, it is possibleto identify all constituent elements in the sample, including boron (Z¼ 5) or even lithium(Z¼ 3), and it is the problems of sample surface contamination, rather than limitations ofthe spectrometer, that restrict the experimental accuracy for light element microanalysis.

To convert the recorded characteristic intensities into a quantitative analysis of thechemical composition, it is first necessary to subtract the background radiation count fromthe integrated characteristic intensity peaks. Reliable methods for optimizing backgroundsubtraction have been developed. Following background subtraction, the spectrummust becorrected for spurious peaks, most notably �escape� peaks that are associated withfluorescent excitation of SiKa photons by the radiation incident on the detector. In manycases it is not possible to analyse for all elements present.Many software programs allow foradditional chemical information, such as the assumption of chemical stoichiometry, to beintroduced into the quantitative analysis software protocol. The size of the volume elementfrom which an electron beam of a given energy will generate characteristic X-rays in thesample depends on the average atomic number or density, while the number of high energyelectrons in the incident beam that are available for X-ray excitation will depend on theproportion that are lost to backscattering of the primary beam. The atomic number or Z


correction, and the backscatter correction are combined in all software correctionprograms. A high Z element in a region of low average Z will have a higher probabilityof exciting characteristic X-rays than would be expected from the chemical composition,while the reverse is true for a low Z element in a region of high average Z.

TheX-ray photons detected by the spectrometer are generated beneath the sample surface,and these photons may undergo inelastic absorption before they escape from the sample. Ifmicroanalysis is to be made quantitative, then these absorption processes also have to becorrected for, based on our knowledge of theX-ray absorption coefficients of all the elementspresent for the characteristic radiation generated by each element individually. Thecharacteristic line intensities are determined from the full width at half-maximum (FWHM)area of each peak after subtracting the background correction. All the corrections to themeasured relative X-ray intensities have to be applied iteratively, assuming initially that thecomposition of the sample is given approximately by themeasured relative intensities of thebackground-corrected, characteristic lines of the X-ray spectra. The initial corrections foratomic number, backscattering and X-ray absorption are all estimated based on thisassumption. The corrected values for the relative concentrations are now used to re-evaluateeach of the various correction factors, and this second iteration then serves to provide amoreaccurate estimate of the composition. In general, several iterations are required to yield acorrected composition whose value no longer converges. The extent of the microanalyticalerrors can be estimated by summing the calculated concentrations of each elementand determining the deviation of the total from 100 %. In many software programsthe calculated compositions are routinely normalized so that they always sum to 100 %,and these normalized values are presented as a �best estimate� of the composition.This practice is acceptable, providing the microscopist recognizes that a usefulcheckon the accuracy of the correction procedure has been lost. For analyses performedon large, homogeneous particles, well away from phase boundaries, there is usually noproblem, but the morphological information in a secondary electron image only reveals thesurface and near-surface structure, whereas the X-ray data are largely sub-surface, and maycontain data fromphase regions that are not visible in the secondary or backscattered electronimage.

One further intensity correction is important for quantitative microanalysis, that due tofluorescence. If X-rays that have been excited by the incident electron beam are absorbedbefore exiting the sample, then they must give rise to other, additional excitation processes.The most important of these is fluorescent excitation of longer wavelength, characteristicradiation from other constituents that are present in the sample. While it is in principlepossible to correct for fluorescence whenever necessary, it is not always possible toidentify the origin of the fluorescent radiation. The electron excitation of X-raysoccurs in a limited, sub-surface volume beneath the electron probe, but, once generated,the primary X-rays then have amean-free path in the sample that is typically 10mmormore.It follows that any subsequent fluorescent excitation can originate from a volume of thesample that is some three orders of magnitude larger than the source of the original, primaryX-ray signal.

While the most important application of X-ray microanalysis is to provide chemicalinformation in SEM, it is also possible to detect characteristic X-rays that are generated inthe thin-film specimens examined by TEM. In this case, the signal is very much weaker,both because of the small volume of material available for excitation in the thin film and


because the probability of inelastic scattering of the electrons is much reduced at the verymuch higher beam energies used in TEM. Fluorescence effects are negligible in thin films,while absorption effects are much reduced and are proportional to the film thickness. Thefilm thickness also limits spreading of the incident beam passing through the sample, so thatthe spatial resolution for X-ray microanalysis in TEM is greatly improved when comparedwith SEM microanalysis, and is limited primarily by the much poorer counting statistics.A field emission gun greatly improves the intensity of the beam probe and EDS micro-analysis facilities are now often available on transmission electron microscopes.

Much improved counting statistics for microanalysis in TEM can be obtained byrecording the energy loss spectra. In electron energy loss spectroscopy (EELS) the lightelements give the most readily detected signal, since the characteristic absorption edges ofthe light element, low atomic number constituents correspond to low energy losses. In theenergy loss spectrum, the first peak, the zero-loss peak, contains information onthe chromatic spread of the source, while plasmon peaks, adjacent to the zero-loss peakin the spectrum, contain information on the thickness of the sample. Despite the rapid,exponential decay of the energy loss signal with increasing energy loss, the characteristicabsorption edges can also be detected in the high energy loss tail of the spectra, andmanyL-,M- and N-edges can be identified with specific high atomic number constituents in thesample. In addition to the characteristic edges, inelastic processes responsible for the energyloss spectra include photon excitations at the thermal excitation level of the zero-losspeak and electron transitions within the atom (energy losses of 1–150 eV). Theplasmon excitations are associated with resonance in the conduction band of a metallicconductor (energy losses of 5–50 eV), but broad plasmon peaks are also found innonconductors.

There has been a steady improvement in the performance of the �hardware� and thecomputer �software� for microanalytical systems available for data processing and quanti-tative analysis in electron microscopy. This has been especially true for energy lossspectroscopy. The energy resolution in EELS is now an order of magnitude better thancan be achieved using characteristic X-ray microanalysis, while the spatial resolution isnow of the order of 2 nm. Nevertheless, quantitative microanalysis using EELS involveslarge errors, mainly in subtracting background corrections from the rapidly decaying tail ofthe energy loss spectrum.

The remarkable energy resolution for EELS has revealed considerable fine structure inthe absorption edges. This energy loss near-edge structure (ELNES) is associated with thechemical state of the atom, that is, the nature of the chemical bonding and the local, atomiccoordination and symmetry in the solid. The extended energy loss fine structure (EXELFS)observed at energies above the edge is associated with local composition changes in thesuccessive atomic shells surrounding atoms of the element responsible for the absorptionedge. The EXELFS oscillations reflect changes in local atomic order. Microanalysis at thislevel is no longer a simple determination of the local chemical composition, but is beginningto answer questions on the chemical state of the atoms, their local packing symmetry and thedegree of atomic order.

Most recently, reliable energy-filtered transmission electron microscopy (EFTEM) hasbecome commercially available. At the simplest level, all electrons which have lost energycan be excluded from the image-forming region in the electron microscope column. In thiscase, either diffraction patterns or bright-field and dark-field images are formed exclusively


by elastically-scattered electrons in the zero-loss peak, providing a significant improvementin image contrast and revealingminor diffraction reflections. Evenmore remarkable are theimages formed from the energy loss spectra of specific, characteristic absorption edges. Thespatial resolution achieved for qualitative detection by EFTEM of individual elements isbetter than 2 nm, providing the electron microscopist with both morphological andchemical information at the level of a few hundred atoms.

Bibliography

1. V.D. Scott, G. Love and S.J.B. Reed,Quantitative Electron-ProbeMicroanalysis, Ellis Horwood,London, 1995.

2. D.B Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook for MaterialsScience, Plenum Press, London, 1996.

3. J.J. Hren, J.I. Goldstein and D.C. Joy (eds), Introduction to Analytical Electron Microscopy,Plenum Press, London, 1979.

4. R.F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum Press,London, 1986.

Worked Examples

We now demonstrate some of the microanalytical techniques we have discussed byapplying EDS and EELS to samples of polycrystalline alumina and a 1040 (0.4 % C)constructional steel.

We start by using EDS in the scanning electron microscope. Our first specimen ispolycrystalline alumina. Alumina is usually sintered (densified) with small quantities ofadditives (dopants) to prevent excessive grain growth and improve the sintering rate. Smallquantities of impurities are also common in alumina (Ca, Fe or Si for example), and thesemay increase the rate of grain growth during sintering. It is important to be able to detectboth impurities and dopants when they are present. Figure 6.28 shows a SEM micrographfrom a polished and thermally etched alumina specimen, together with an EDS spectrumacquired from the entire region shown in the micrograph. The specimen was prepared frompure alumina, and intentionally doped with magnesium, together with a small amount ofsilicon and calcium. Automatic energy calibration and peak identification was performedby a computer program, and only the principle aluminium and oxygen peaks were detected.A thin-window EDS detector would have allowed detection of light elements, such asoxygen. The absence of Mg, Si, and Ca peaks in the EDS spectrum taken from the regionimaged in Figure 6.28 was expected, since the total concentration of these dopants andimpurities in this rather large sample volume is below the detection limit. However, an EDSspectrum, taken with the probe positioned at a single point located at a grain boundary,clearly shows the presence of both Mg and Si, although the amount of Ca at this grainboundary is still below the detection limit (Figure 6.29).

It is clear from this result that both magnesium and silicon have collected at the aluminagrain boundaries. We do not say that they have segregated, since we do not yet know thesolubility limit of these elements at the sintering temperature. We cannot detect any of


the additives within the grains because the solubility limit of these cations is well below thedetection limit of EDS. However, the presence of Si and Mg at the grain boundaries is animportant finding, which implies that the alumina grains are saturatedwith these impurities.

Figure 6.30 shows aTEMmicrograph of a region taken from the same alumina specimen.Grain boundaries are indicated by arrows. EDS in the TEM can also be used to determine ifsilicon and calcium are present at specific boundaries or triple junctions. Figure 6.31 showsan EDS spectrum taken from the glass-containing triple junction identified in Figure 6.30.Significant peak intensities from both Si and Ca are detected, as well as from Mg. The

Figure 6.28 (a) Secondary electron image from thermally etched alumina. (b) EDS spectrumfrom the region shown in (a). Only Al and O peaks are recorded, together with carbon due tosurface contamination.


1 2 3 4

0

500

1000

1500

Energy (keV)

Counts

Si

O Al

Mg

Figure 6.29 EDS point spectrum recorded from a grain boundary in the alumina specimen. AlandO peaks are of course found, but alsoMg and Si peaks, since these elements have collectedat theboundary. In this particular caseCa,whichwaspresent in the specimen, is either below thedetection limit or absent from this specific grain boundary.

Figure 6.30 TEM micrograph of the alumina shown in Figure 6.28(a). An amorphous phase(labelled glass) was detected at the triple junction.


aluminium signal is at least in part from one or other of the alumina grains adjacent to thetriple junction, while the argon signal is from argon ions that have been trapped in the thinTEM specimen during ion milling.

We can also use parallel electron energy loss spectroscopy (PEELS) to microanalyse ouralumina TEM specimen. Figure 6.32 shows portions of the PEELS spectrum acquired froman alumina grain in the energy loss region of the OK-edge (532 eV) and the Al L2,3-edge(73 eV). Quantitative analysis of the PEELS spectrum from the alumina grain confirms thestoichiometric concentration ofAl2O3, andwe can compare the ELNES of theAl edge fromalumina to that frommetallic aluminium (Figure 6.33). The observed differences in the twoELNES spectra reflect the differences in the nature of the chemical bonding of aluminiumatoms in the ionic solid and the metal.

We noted earlier that we could not specify whether magnesium or other dopants andimpurities had segregated to the grain boundaries, or were enriched at the boundaries. Thedistinction between these two conditions depends on the equilibrium concentration of eachadditive in the bulk grains, and the value of the solubility limit for each species. Below thesolubility limit, the element may segregate to the grain boundary if this will lower the grainboundary energy. If the same element is present at a metastable concentration level thatexceeds the solubility limit, then the element may diffuse to enrich the grain boundaries in

Figure 6.31 EDS spectrum taken from the triple junction shown in Figure 6.30. Mg, Si, and Caare all detected. The Al signal is probably mostly from the neighbouring alumina grains, whilethe Ar signal is an artifact introduced during ionmilling. Oxygen could not be detectedwith theEDS system used to collect this spectrum.


order to lower the bulk concentration and approach the equilibrium solubility limit.Measuring bulk solubility limits at low concentrations is always difficult, and especiallychallenging for dopants and impurities in ceramics, when the solubility limit is oftenexpected to be very low. EDS does not have a low enough detection limit for such a task, andso we will use WDS to measure the solubility limit for magnesium in Al2O3.

Our sample is a polycrystalline Al2O3 that has been sintered for 24 h at 1600 �C, afterdoping with magnesium at a level (�5 atom %), well above the solubility limit. Aftersintering at 1600 �C, the samplewas taken directly from the furnace and quenched in water.

Figure 6.32 (a) Al L2,3 -edge (73 eV) and (b) the O K-edge (532 eV) in alumina.


X-ray diffraction confirmed that two phases were present in the sample (Al2O3 andMgAl2O4). Hence, we are in the two-phase region of the Al2O3–MgO pseudo-binary phase diagram and the magnesium concentration in the alumina grains shouldbe at the solubility limit for 1600 �C. The X-ray diffraction results are confirmed bySEM examination, which shows the MgAl2O4 phase is present in the form of platelets(Figure 6.34).

For this experimentwe use aWDSmounted on the same scanning electronmicroscope asour EDS system, and we must first maximize the WDS count rate with respect to the SEMoperation conditions. We increase the beam current to a maximum and measure this beamcurrent using a Faraday cup located on the sample stage. We now determine the WDSintensity as a function of working distance and magnification for a standard sample that islocated on the same stage as our alumina specimen. Figure 6.35 shows results, recorded at15 kVand using a thallium hydrogen phthalate (TAP) crystal inWDS. Themaximum countrate corresponds to a working distance of �11.5mm and a magnification greater than·5000. The explanation for this result is shown schematically in Figure 6.36, which showsthat the geometric configuration of the sample and detection assembly requires a minimumworking distance and a minimummagnification if we are to maximize the collection anglefor the wavelength-dispersive spectrometer.

Once we have optimized the geometry of the system, we need to determine the intensityof the background, white radiation adjacent to the MgKa peak. This is demonstrated inFigure 6.37,whichwas recorded under the sameworking conditions for SEM thatwere used

Figure 6.33 Al L2,3 -edge in alumina and in aluminiummetal. The position of the peak and thepeak shape are significantly different in the two materials, reflecting the very different atomicbonding of the aluminium.


when examining a pure sapphire crystal standard. This measured background can now beused to subtract the background from the WDS signal recorded from the MgO-saturatedAl2O3.

At 15 kV the depth of penetration of the incident electron beam into alumina isapproximately 1.6 mm.Fromover 300 independentmeasurements giving a 95%confidencelevel, the magnesium detection limit was found to be 4.6 ppm. This is really quiteremarkable when compared with the limit of what could be done using EDS. The averageconcentration of magnesium in the saturated Al2O3 grains was found to be 132� 11 ppm.Assuming that the quenching ratewas fast enough, this then represents the solubility limit ofmagnesium in Al2O3 at 1600

�C.We now examine a 1040 steel (0.4 % C) sample by EDS in the scanning electron

microscope. Quantitative EDS analysis of the carbon content for any steel is almostimpossible, since the carbon content is at the limit of detection for light elements and is oftenconfused with carbon contamination in the system. Figure 6.38 shows an EDS spectrumtaken from a large area of the specimen surface. A carbon peak is visible that would suggesta carbon content much higher than expected for 1040 steel. This is due to carbon

Figure 6.34 (a) Backscattered electron and (b) secondary electron SEM micrographs of thequenched Al2O3 sample. The MgAl2O4 phase is present as thin platelets.


contamination at the surface from the breakdown of volatile hydrocarbons present in thevacuum system of the electron microscope. This signal has nothing to do with the carboncontent of the alloy. However, Fe3C platelets are present in the specimen and this compoundhas a carbon content well above the detection limit for EDS. The carbide platelets areclearly seen in the pearlitic regions of the microstructure (Figure 6.39). The presence of thehigher carbon content in the pearlite can be confirmed from anEDS point spectrum analysistaken with the probe beam located on the pearlite (Figure 6.40), which should be comparedwith Figure 6.38.

In addition to pearlite, small dark areas are visible in the a-Fe grains and in the pearlitemicrostructure. Some of these dark regions may be Fe3C particles that have precipitatedbelow the eutectoid temperature (723 �C) during heat treatment. This can be checked by

Figure 6.35 WDS characteristic peak intensity in the scanning electron microscopes asa function of (a) the working distance for the sample and (b) the magnification of the image.


taking an EDS line-scan across some of the dark regions (Figure 6.41). In Figure 6.41 anX-ray line-scan for the carbon peak crosses two dark regions, one located within a pearliticeutectoid region, and the second at the boundary between the pearlite microstructure and ana-Fe grain. The dark region at the interface between the pearlite and a-Fe has a significantlyhigher carbon content, andwe conclude that this is a carbide particle. The dark regionwithin

Figure 6.36 Schematic drawing of the SEM column geometry demonstrating how, at smallworking distances and low magnifications the backscattered electron (BSE) detector may�shadow� the WDS detector, reducing the intensity of the signal that is acquired.

Figure 6.37 WDS background intensity for the MgKa peak region (recorded from a puresapphire standard crystal).


Figure 6.38 EDS spectrum taken from an a-Fe grain in a 1040 steel. The carbon signal is far toohigh to correspond to the 0.4wt % in the steel and certainly not to residual carbon dissolved inthe a-Fe. This signal is in fact due to surface contamination.

Figure 6.39 Secondary electron SEM micrograph from an etched 1040 steel showing a-Fegrains and pearlitic regions of the eutectoid containing both a-Fe and Fe3C.

Figure 6.40 EDS spectrum taken from a pearlite region in the microstructure of Figure 6.39.Note the increase in the carbon signal relative to the Fe signal when comparedwith Figure 6.38.

Figure 6.41 EDS line-scan from the carbon signal and a secondary electron SEMmicrographofthe interface between pearlite and a-Fe.


the pearlite does not showany higher an EDS carbon count than the background, and ismostlikely a surface pore. The EDS line-scan also shows the presence of an additional carbideparticle (on the left) that was not immediately evident from themicrograph. The presence ofFe3C can therefore be detected quite easily from EDS line-scans, but a comparison ofthe EDS results with the secondary electron image from the same area is necessary tointerpret the microstructurewith any confidence. Quantitative analysis of the carbide phasewould be much more difficult, since the carbon signal is strongly affected by the surfacecontamination.

Problems

6.1. Why is it important to know the chemical composition associated with microstruc-tural features? Give three examples.

6.2. Whywould you expect a fracture surface to be a �difficult� sample for microanalysisin the scanning electron microscope?What purpose might be served by comparingmicroanalysis from a fracture surface with that from a polished and etchedspecimen of the same material?

6.3. Why does the best electron beam energy for microanalysis depend on the density ofthe sample?What are the disadvantages of working with too high or too low a beamenergy?

6.4. What factors determine the width of the detection window for a characteristic X-ray excitation line selected for quantitative analysis? Howwould you optimize thecounting statistics to obtain the maximum signal-to-noise ratio for the selectedline?

6.5. Distinguish between sensitivity and accuracy in microanalysis.

6.6. Discuss the statistical limitations on spatial resolution in microanalysis.

6.7. There is little point in attempting to analyse regionsmuch less than 1 mmin diameterin the scanning electron microscope. Why?

6.8. For quantitative analysis, the characteristic X-ray spectrum excited by an electronprobe should be corrected for background white radiation, atomic number effects,backscatter losses, X-ray absorption and X-ray fluorescence. In what order shouldthese corrections be made?

6.9. An aluminium-copper alloy contains 3.5 % of copper. Would you expect theintensity of the CuKa radiation to be increased or decreased by the atomic numbereffect and the backscatter losses? Why?

6.10. A series of iron–nickel alloys are to be microanalysed. Discuss the relativeimportance of the absorption and fluorescent corrections for iron and nickel as afunction of the alloy composition.


6.11. A polished cross-section from a brazed joint is to be microanalysed. Would yourecommend that the line of intersection of the joint with the polished surface bealigned parallel or normal to the X-ray collection system? Assume that thecomposition of the components being joined and that of the braze may containcopper, silver and zinc. Check the relative characteristic peak positions and theabsorption edges for these constituents.

6.12. Why is electron energy loss spectroscopy now the dominant technique for thin-filmmicroanalysis in the transmission electron microscope?

6.13. How would you expect the limit of detection in energy loss spectroscopy to varywith the atomic number of the constituent?

6.14. What are the advantages of parallel electron energy loss spectroscopy as ananalytical tool in the transmission electron microscope?

6.15. Compare the energy resolution of EELS and WDS systems.

6.16. A joint between Al–4wt % Cu and Cu–10wt% Sn was checked by EDS in SEMusingKa characteristic lines and Figure 6.42 shows the distribution of EDS intensityas a function of distance from the interface.

(a) Explain the expected difference between the relative measured intensity and thereal chemical distribution at the interface.

(b) Estimate the spatial resolution of theEDS line-scan based on themeasured intensityvalues. Explain your assumptions.

(c) What was the approximate accelerating voltage used to acquire the line-scan?Justify your estimate!

Figure 6.42 EDS line-scan of a joint between a Al–4wt%Cu alloy and a Cu–10wt% Sn alloy.


6.17. An EELS analysis of a thin-film Al2O3 sample gave the low loss spectrum shown inFigure 6.43. Estimate the sample thickness.

6.18. The cross-section of a silicon sample is sketched in Figure 6.44. Certain regions ofthe sample are coated with a 500 nm thick film of titanium. The sample wascharacterized by a scanning electron microscope that was equipped with secondaryelectron (SE), backscattered electron (BSE), and EDS detectors that were located attake-off angles of 45, 70, and 30�, respectively. The accelerating voltagewas 20 kV,and the incident electron beam was perpendicular to the sample surface.

(a) Discuss the X-ray emission from the silicon beneath the titanium coating as afunction of position on the sample.

(b) Estimate the percentage of Si Ka photons that can escape from the sample in thedirection of the EDS detector.

Figure 6.43 EELS spectrum from a thin-film Al2O3 TEM sample.

.

Figure 6.44 Schematic drawing of a Si wafer partially coated with Ti.


(c) Sketch the expected, qualitative intensity distributions for all three detectors for aline-scan across the surface of the specimen. Explain the relative changes in theintensities that you predict as a function of the beam position.

To estimate the energy of the electrons that penetrate a thickness H of the titanium use therelationship: E2

H ¼ E20�H·2:387 · 105ðV2Þ, where H is in nm and E is in V.

Given:

Material Atomicnumber

Atomic mass(gmol�1)

Density(g cm�3)

Mass absorptioncoefficient forTi-Ka (cm2 g�1)

Mass absorptioncoefficient forSi Ka (cm2 g�1)

CharacteristicX-ray (Ka)energy (eV)

Ti 22 47.9 4.54 108 1458 4510

Si 14 28 2.33 130 347 1740

6.19. During a study of an AlN sample, it was found, from secondary electron micro-graphs, that the sample contained �10 vol% of a second phase. Qualitative SEM-EDS analysis showed that this phase contained yttrium and, possibly, oxygen andaluminium (Figure 6.45). What techniques would you suggest to determine

Figure 6.45 Backscattered electron (BSE) micrograph of AlN, containing a secondary phase(bright contrast).

Figure 6.46 Schematic drawing of copper-filled vias in a silicon wafer.


quantitatively both the structure and chemistry of the second phase? Includesuggested operating parameters for the methods you select.

6.20. In themicroelectronics industry, silicon-based devicesmust be characterized duringprocessing. Figure 6.46 shows a schematic drawing of vertical contact lines (vias)filled with copper in a silicon wafer.

(a) Which SEM signals would you select to characterize the shape and size of the vias,both before and after filling them with copper? Explain your choice.

(b) Using a 20 kV incident electron beam in SEM, what spatial resolution would youexpect for EDS of the copper vias? Explain quantitatively.

(c) A sample containing copper-filled vias is coated with 100 nm of gold. Should anEDS signal be detectable from the silicon and copper? If so, explain the influence ofthe gold film on the intensity of the measured signal.

Given:

Ka (keV) Atomic number Atomic mass (gmol�1) Density (g cm�3)

Si 1.740 14 28.0855 2.33

Cu 8.046 29 63.55 8.96

Au 9.712 (La) 79 196.97 19.3


7

Scanning Probe Microscopyand Related Techniques

So far, we have limited our discussion to microstructural probes that are based on visiblelight (the optical microscope), X-ray diffraction, or high energy electrons. We havediscussed signals that are generated by both the elastic and the inelastic interaction ofthese probes with a carefully prepared specimen and we have established that microstruc-tural information, characteristic of the sample, can be collected on the microstructuralmorphology, the crystal structure and the chemical composition of the phases that arepresent. Moreover, we have demonstrated that this information can be resolved over a verywide range of dimensions, from the everyday scale of visual observation, down to just a fewatomic diameters, or even to the interplanar spacings present in the grains of individualcrystals.

In this chapter we take the discussion a stage further and describe what can be achievedwhen the probe of electromagnetic radiation or high energy electrons is replaced by a sharp,needle-shaped, solid probe that is brought into close proximity or into contact with thesurface of the sample we wish to study. We shall discover that such a probe can provide uswith additional microstructural information. This new information is beyond the range ofthe techniqueswe have discussed so far, and reveals details of both the surface structure andthe surface properties of engineering materials at resolutions that, under suitable condi-tions, can image individual atoms. The information that is now available from scanningprobe microscopy and some related techniques has proved to be of importance forapplications that range from engineering problems in lubrication and adhesion, to soft-tissue, biological, interactions at membranes and across cell boundaries, as well as toelectro-optical devices and sub-micrometre electronics.


7.1 Surface Forces and Surface Morphology

An observant reader will already have realized that, in both optical and transmissionelectronmicroscopy, the image plane of the source and the image plane of the specimen areinversely related, since the diffraction pattern from the specimen in reciprocal space isfocused in an image plane of the source, while the image from the specimen, focused in realspace, is convoluted by the angular distribution of the probe radiation that is emitted fromthe source.

Over 40 years ago scientists working on field emission, at the National Bureau ofStandards, outside Washington (now the National Institute of Standards and TechnologyNIST), placed a tungsten field emission tip, a �source�, in close proximity to a samplesurface and monitored the changes in the field emission current that were associated withchanges in surface topology, immediately beneath the tip field emitter, as the sample surfacewas scanned perpendicular to the emitter. At that time it proved quite impossible tomaintainthe required dimensional stability, and it was only 10 years later, with the development ofreliable piezoelectric drives, that the thermal and mechanical stability necessary to ensurereproducible results was finally achieved. Today, a wide range of experimental methods areavailable for studying either the nanometre-scale contact forces between two solid objects,or the electrical properties of individual surface contacts, or even the atomic structure andsurface chemistry of small surface areas.

We have placed all these diverse techniques under a single heading, scanning probemicroscopy, even though only two of the instruments, the atomic force microscope and thescanning tunnelling microscope are in fact �scanning probe� instruments. We start with abrief discussion of surface-force measurements, and conclude the chapter with theremarkable results now being reported for atom probe tomography, a technique that isable to dissect suitable field ion microscope samples and reveal the three-dimensionalchemical distribution of the individual atoms, identified by their atomic mass.

7.1.1 Surface Forces and Their Origin

When two solid surfaces approach one another, the interaction between them includes bothattractive and repulsive components. These forces may be either long-range or short-range,and the interaction between the surfaces is strongly affected by the presence of surfaceadsorbates or by a gaseous or liquid environment in the space separating the solid surfaces.

Before considering the multi-atomic interactions occurring when two surfaces arebrought together, we first describe a text book, two-atom model for the individual atomicinteractions. If the long-range forces between the two atoms considered are attractive,whilethe short-range forces are repulsive, then we can develop a simple, two-body model for theinter-atomic bonding of the atoms. This model can predict the qualitative attractive andrepulsive forces between atoms from the known physical and chemical properties: theequilibrium, interatomic separation in the solid, the heat of formation and tensile modulusof the material, and its coefficient of thermal expansion. Figure 7.1 illustrates this simpleatomic forcemodelF(R) and its integrationVðRÞ ¼ R R1 FðRÞdR to give the potential energyof this two-body system as a function of the interatomic spacing E(R). The position of theminimum in the potential energy and the depth of the potential energy well are set equal tothe equilibrium interatomic spacing and the atomic binding energy, respectively, while the


curvature at the bottom of the potential well and the asymmetry of this curvature aremonotonic functions of the elastic modulus of the solid and its thermal expansioncoefficient.

In applying similar modelling concepts to two solid surfaces that are brought into closeproximity, wewill have to account for the multi-body interactions of all the surface atoms

Interatomic separation R

Attractive force FA

Repulsive force FR

Net force FN

Rep

ulsi

onA

ttra

ctio

n

For

ceF

Repulsive energy E R

Attractive energy E A

Net energy E N

Att

ract

ion

Rep

ulsi

on

Pot

enti

al e

nerg

y E

E0

R0

Interatomic separation R

Figure 7.1 Interatomic forces and interatomic potentials. The long-range attractive force andthe short-range repulsive forcebalance at the equilibrium inter-atomic separation and the sumofthe two forces can be integrated over distance as the atoms approach to give the atomicpotential. The minimum of the atomic potential curve is the ground state of the system and thepotential curve itself can be related to the physical properties of the corresponding solid (seetext).

Scanning Probe Microscopy and Related Techniques 393

from both surfaces. Wewill also need to consider the effects of surface roughness or localcurvature, and, especially, the presence of a liquid or gaseous environment. As shownschematically in Figure 7.2, this leads to a three-phase model that includes the two solidsand the inter-phase, environmental region. Again, both long-range and short-range forcesmust be considered, and it is now convenient to distinguish three interaction zones in thepotential energy curve (Figure 7.3). At larger separations, only the long-range forcesare experienced. These are usually attractive, and we refer to these larger separations asthe non-contact region, defined as extending beyond the inflection point on our schematicpotential–distance graph. This inflection point corresponds to a maximum in the netattractive force. At distances less than the separation corresponding to a potential energyequal to zero, we can define a contact region. In this regime the attractive forces arenegligible and the short-range and repulsive forces dominate the interaction between thetwo bodies. In between these two regions, in the zone that includes the equilibriumspacing for the two surfaces and either side of the minimum in potential energy, we can

Distance

AA

AB

CC

AA

AB

Figure 7.2 Possible interactions between atoms on two contacting surfaces (AA), betweensurface atoms and atoms in a gaseous or liquid environment (AB), or between atoms in theenvironment (CC). Reprinted with permission from R.G. Horn, Surface Forces and Their Actionin Ceramic Materials, Journal of The American Ceramic Society, 73(5), 1117–1135, 1990.Copyright (1990), with permission from Blackwell Publishing Limited.


define a semi-contact region for which the attractive and repulsive forces have similarmagnitude.

Aswe shall see, the interactions in these three regions can be separated in the atomic forcemicroscope, and this provides unique information on the nature and distribution of thevarious surface forces. The strongest long-range forces between two solid surfaces are theCoulombic electrostatic forces, which may be either repulsive or attractive, depending onwhether the total charge carried by the two surfaces is of the same or of opposite sign. Atshorter distances, polarization, or van der Waals forces are experienced. These polarizationforces are classified under three headings. The strongest forces are associated with perma-nentmolecular dipolemoments that create local electric fields and lead to ordered packing ofan assemblyof the surface dipoles. The electric field of amolecular dipolemay itself polarizeand attract neighbouring atoms, in proportion to the strength of the dipole and thepolarizability of the interacting atoms. This second form of polarization force is knownas the Debye interaction. Finally, random fluctuations in the electrical fields of anypolarizable atom will also induce localized, attractive interactions with neighbouringpolarizable atoms and generate forces that are termed London, or dispersion forces.

van der Waals forces all decay rather rapidly (usually as d�7). The parameters thatdetermine the strength of the interaction are thevalues of the dipolemoment of the atomic ormolecular entities at the surface and their polarizability, together with their separation.Asymmetric molecular adsorbates, such as carbon dioxide or water vapour, have largedipole moments that lead to strong van derWaals adsorption, while symmetrical molecularassemblies, such as methane, can have appreciable polarizability but no dipole moment.

Rep

ulsi

ve f

orce

Att

ract

ive

forc

e

Contactregion

Non-contactregion

Intermittentcontact region

Distance

Figure 7.3 Three regions can be distinguished on the potential energy curve for the interactionof two solid surfaces: a non-contact region dominated by attractive forces; a contact regiondominated by the repulsive force; an intermediate, semi-contact region that includes theequilibrium separation and inwhich the attractive and repulsive forces are of similarmagnitude.


Polar groups that contain hydrogen, especially OH� and NH2�, establish hydrogen bonds

that are strongly attractive at very short range, and are typical of the polar bonds that arecritical for the existence of life on earth.

The strong, short-range, repulsive forces between atoms are associated with the overlapof the atomic inner electron shells and these dominate the interaction as the solid surfacesare forced together. The repulsive forces may also dominate when soluble additives in aliquid medium, such as polymer molecules, prevent the solid surfaces from coming intodirect contact, an effect termed steric hindrance. Charged surfaces in a nonpolar, liquid orgas environment of low dielectric constant, lead to strong, but rather unpredictable,electrostatic forces. The presence of a polar environment, with a high dielectric constant,leads to a layer of counter-charge adjacent to the charged solid surface of opposite sign. Thisis termed the Debye or double layer. Any surface charge, together with the diffuse layer ofcounter-charge, can be of immense practical importance and is readily investigated bytechniques for surface force measurement.

The most successful theory that has been developed to predict the surface forcesresponsible for the stabilization of colloid dispersions and the wetting of solids, wasestablished some 50 years ago by Derjaguin and Landau in Russia, and, independently, byVerwey and Overbeek in Holland. The theory, usually referred to as the DLVO theory, hasbeen modified over time in order to explain the factors that determine the forces betweensolid surfaces in the presence of surface segregation, adsorbate layers, or liquid solutions.DLVO theory is now used to develop chemical formulations for double layers that are usedto inhibit particle contact by establishing metastable minima at nanometre particleseparations. These adsorbates promote wettability and dispersion stability, as shown inFigure 7.4.

7.1.2 Surface Force Measurements

To fully understand surface forces we need to measure them under controlled conditions.The most successful approach to achieving this was developed by Jacob Israelachvili andhis coworkers some 30 years ago. These workers constructed a sensitive force–balanceinstrument that was based on two curved, cylindrical mica surfaces mounted at right anglesand brought into close proximity (Figure 7.5). The cylinders can be coated, for example byvacuum vapour deposition, in order to modify the surface structure, conductivity andcomposition of either or both substrates. The effective area of contact between the crossedcylinders used in this geometry is a circle. With the help of Israelachvili�s equipment, theforce normal to the two surfaces can bemeasured to an accuracy of about 0.1mNm�1,whilethe spacing between the surfaces in the contact region can be determined to an accuracy ofabout 0.1 nm. One of the most remarkable findings of this work has been the presence ofsurface force oscillations. These may be observed as a function of the surface separation. Ifthe surfaces are separated by a polysaccharide or polyalcohol lubricant film, measurableforce oscillations can be detected out as far as 10 nm (Figure 7.6). The oscillations areassociated with the presence of steric hindrance and molecular forces which favour acomplete, integral number of molecular layers of the boundary lubricant that is dissolved inthe liquid medium between the two surfaces. Awide spectrum of micromechanical surfacephenomena have been observed, dependent on the chemistry and surface crystallography ofthe substrates, and the composition of the liquid or gaseous medium.


7.1.3 Surface Morphology: Atomic and Lattice Resolution

We have established that surface forces normal to the surface can be measured extremelyaccurately, as a function of the separation of two solid surfaces, and that sub-nanometrevertical resolution has been achieved. We now need to consider the lateral resolution thatmight be possiblewhen using a solid probe to explore themorphology of a solid surface andits chemical or physical properties.

A flake of cleavedmica is an atomically smooth and almost featureless substrate, despitethe presence of physically adsorbed moisture from the environment. The few, isolated

Double layer

Total

van der Waals

Separation distance

Ene

rgy

0

(a)

(b)

Double layer

Total

van der Waals

Separation distance

Ene

rgy

0

Secondary minimum

Figure 7.4 Asuitabledouble layer at the surface between two solids in a liquidenvironment canresult in metastable minima in the surface energy of the system and determine the wettability ofsolid surfaces or the stability of particulate dispersions. (a) A weakly polar liquid generates adispersed double layer. (b) A strongly polar liquid can result in a thinner double layer and asecondary energyminimum. Redrawnwith permission fromR.G.Horn, Surface Forces and TheirAction inCeramicMaterials, Journal of TheAmericanCeramic Society,73(5), 1117–1135, 1990.Copyright (1990), with permission from Blackwell Publishing Limited.


cleavage steps on such a surface rarely interferewith surface force measurements (althoughdust particles can interfere a great deal). No detectable structure is normally visible on sucha substrate. However, most other surfaces do contain a variety of morphological featuresthat have varying degrees of structural order and are of varying practical importance.Surface roughness is seldom random in either amplitude or wavelength. Contributions tosurface roughness may come from polishing scratches and oriented grinding or machiningridges, or perhaps from corrosive pitting or chemical etching, which often depends on the

Figure 7.5 Basic geometry for monitoring contact forces. Two curved mica plates are broughtinto close proximity, and the normal force between the plates is measured as a function of theirseparation.

0 2 4 6 8 10

D (nm)

F (mNm–1)

-3

-2

-1

0

1

2

3

Attractive van der WaalsforceStructural force due to molecular packing

Figure 7.6 Schematic illustration of the balance between attractive van der Waals forces andstructural forces associated with steric hindrance. As the surfaces approach each other, themaxima and minima of the structural force define the possible equilibrium thicknesses for themultilayer lubricant film between the two solid surfaces. For small values of the lubricant filmthickness, the attractive van der Waals force will be compensated by the repulsive structural orsteric force. The seven smallest distances at which the attractive and repulsive forces arebalanced are denoted by the black circles.


crystal structure of the surface. The equilibrium surface of a polycrystalline solid maycontain facetted crystal surfaces or grooved grain boundaries. Second phases may also bepresent in thematerial. Such second phasesmay either exist in themicrostructure of the bulkmaterial, or they may have been deposited on the surface as a coating, or result from acorrosion reaction. A second phase on the surface may be present as a continuous thin film,or as isolated, discontinuous or interconnected islands.

If the scanning probe is the tip of a solid needlewhose geometry can be approximated by apolished cone or pyramid, then it will have an effective tip radius that is usually in the rangeof 20 – 200 nm,much greater than the interatomic spacings on any solid sample surface. Thevalue of the probe tip radius can usually be determined unambiguously by examining theprobe in a scanning electron microscope. It is generally assumed that the effective contactarea which is established at the sample surface as the probe approaches the specimen willhave an area of at least 100 nm2. The probe tip radius directly limits the resolution that maybe realized in a scanning probe image. One reason for this is illustrated in Figure 7.7. As theprobe is scanned over the surface, the number of atoms beneath the probe that contributes tothe measured signal will vary with the periodicity of the interatomic spacing on the surface.The relative amplitude of these variations in the force oscillations will decrease as the tipradius increases, but the wavelength of the oscillations will remain that of the atomicstructure. For the large tip radii, these fluctuations will be undetectable, but for tip radii of

Figure 7.7 A nanoscale, solid probe, scanned over a solid surface, will experience anoscillating force that has the periodicity of the atomic spacing of the scanned surface, eventhough the probe tip radius may be an order of magnitude larger than the atomic spacing of thesample.


the order of 10 nm it has sometimes proved possible to resolve characteristic inter-atomiclattice spacings, such as the graphite ring structure on the basal plane (Figure 7.8) eventhough themeasured force is due to a many-body interaction that is on a significantly largerscale. It has been argued that this is not �true� spatial resolution and that only electricalmeasurements, which are dependent on the very sensitive response of field emissiontunnelling to the interatomic separation, are able to resolve the atomic structure of thesurface (Section 7.2). Images such as Figure 7.8 are certainly a good approximation toresolution of the atomic packing in the plane of the sample surface.

7.2 Scanning Probe Microscopes

In all scanning instruments, during the data collection process, the data are collected pointby point (that is, pixel by pixel) and not integrated over time for the field of view seen in theimage. Sequential data collection in scanning electron or scanning probe microscopy istherefore quite different from that used in optical or transmission electron microscopes,where data from all image points (pixels) are collected in parallel. In scanning probemicroscopes, the rate of data collection is very slow and this limitation is exacerbated bymechanical and thermal drift of the sample and the imaging system, so that reliable datacollected for an individual image comes from only a limited number of pixels. Images are

Figure 7.8 The periodicity of the carbon ring structure on the basal plane of graphite can berevealed by the atomic forcemicroscope, even though the probe tip has a much larger radius ofcurvature than the carbon ring repeat distance. Is this ‘true� spatial resolution (see text)?(Reproduced from NanoTech America).


often restricted to 256 · 256 pixels and it is unusual for the data set for each image to exceedmore than 105 pixels when using a solid probe to scan the specimen surface.

We first provide a historical chronology of the different scanning probe and other signalsto be discussed in this chapter and illustrate this chronology with sketches that show thedevelopment over time of the concept of a sharp, solid probe (Figure 7.9).

1936 Field emission of electrons was first demonstrated: a negative potential was appliedin a high vacuum to the sharp tip of a thermally smoothed, refractory metal wire. Theneedle-shaped samplewas rigidlymounted by spot-welding to a heating filament thatwas attached to two electrodes [Figure 7.9(a)]. The field emission current iwas shownto obey the Fowler-Nordheim equation:

i ¼ Að�=wÞ12ð�þwÞ F

2exp �Bw

32

F

!ð7:1Þ

where m is the Fermi energy of the solid,f is thework function of the surface,F is theelectric field strength acting at the surface, and A and B are constants. The projectionimage of the field emission current distribution over the hemispherical tip on aphosphor screen clearly showed the dependence of the work function on the crystalstructure of the tip surface.

1956 Atomic resolution was first demonstrated in the field-ion microscope. Imaging wasachieved by field ionization of gas ions at the surface of a cryogenically cooled andpositively-charged sharp, refractory metal tip [Figure 7.9(b)]. The ions were accel-erated to form a projection image of the crystalline tip surface in which the surfaceatoms at the edges of the regularly packed atom planes were clearly resolved. Byincreasing the electric field strength at the tip surface, it was possible to ionize theatoms at the edges of the crystal planes and remove them from the surface by fieldevaporation, to form symmetric, atomically smooth tip surfaces. The rigid supportelectrodes for the sharp tip were cooled to cryogenic temperatures. The field ionmicroscope was the first instrument that was able to achieve atomic resolution.

1967 In the atom probe, field-evaporated ions from the surface of a field-ion microscopespecimen were chemically analysed by time-of-flight mass spectrometry to givemass-resolved chemical analysis of the sample at sub-nanometre resolutions [Figure7.9(c)]. The metal tip was aligned to bring a selected region over a hole in thefluorescent image screen, so that atoms that were field-evaporated from the selectedregion could pass into the mass spectrometer. Over the following 40 years, arealdetection of the field-evaporated ions over a wider angle became possible, leading tomassive data sets of statistically significant,mass-resolved chemical information thatpossess sub-nanometre resolution and were derived from millions of atoms field-evaporated from the sample needle.

1982 The feasibility of the scanning tunnelling microscope was demonstrated soon afterpiezoelectric control of vertical and lateral displacements was shown to be capable ofthe mechanical and thermal stability needed for the atomic resolution of a scanningprobe instrument [Figure 7.9(d)]. The scanning tunnelling microscope was able toprobe the electronic structure of the surface region down to atomic resolution. Thethermodynamically stable restructuring of the packing of surface atoms and the


V

Tip sample

R~0.1 µm

Screen+

+

+

Time of flightspectrometer

Multichanneldetector

V

Tip sample

R~1 nm

ε ε ε

F~ 0.05Vnm–1

Screen

V

Tip sample

R~0.1 µm

F~ 0.5 Vnm–1

Screen+

++

T=20–80 K

V

Tip probe

R~1 nm

Sample~1 nm

I

Tip probe

R~1 nm

Sample~1 nm

Cantileverforce meter

Scan

(a) (b)

(c) (d)

(e)

Figure 7.9 Fifty years of tip assemblies, 1936–1986 (see text). (a) Field emission tip assembly.(b) a Field-ionmicroscope tip. (c)Atomprobe tip assembly and time-of-flightmass spectrometer.(d) Scanning tunnelling probe assembly. (e)Atomic force probe scanning assemblyandcantileverforce monitor.


existence of several crystal structures that were unique to the surfacewas discovered.The sharp tip was rigidly mounted on a support frame that was attached to amechanically stiff, cylindrically symmetric, piezoelectric actuator that allowed forsub-nanometre control of the separation of the probe tip from the specimen surfaceand the lateral positioning of the probe.

1986 Atomic force microscopy was developed by mounting the solid, needle-like probe atthe end of a flexible cantilever andmonitoring the displacement of the cantilever due tothe surface contact forces [Figure 7.9(e)]. The flexible cantilever probe assembly wascombined with a rigid, piezoelectric x-y-z actuator module and the movement of thecantilever was monitored by a split-beam laser detection system. The atomic forcemicroscope, unlike its predecessors, no longer required a vacuum, but could beoperated under atmospheric conditions or in a controlled gaseous or liquid environ-ment. Resonant vibration of the cantilever allowed the tip to probe a wide range ofelectrical, magnetic and mechanical properties of the surface at lateral resolutions inthe nanometre range. By controlling the separation of the probe at fixed distances fromthe sample surface it proved possible to operate the atomic force microscope in eithercontact, semi-contact or non-contactmodes, revealing details of the dominant surfaceforces for a wide range of environmental conditions and engineering materials,including soft, compliant materials such as adhesives and biological tissues.

7.2.1 Atomic Force Microscopy

The basic components of a scanning probe system for either scanning tunnelling or atomicforce microscopy are shown in Figure 7.10. The table-top instrument is mounted on a rigidbase of high damping capacity, in order to ensure freedom from extraneous mechanicalvibrations. The chamber is fully enclosed, both to provide thermal stability and to allow forenvironmental control. A high vacuum is not necessary for atomic force microscopy. The

Graphic Display

ElectronicsInterface

DigitalSignal

Processor

Z

X,Y

Probe

Sample

Scanner

Microscope

Detector

Anti-Vibration Table

Controller

Computer

Figure 7.10 Basic components of a scanning probe microscope for scanning tunnelling oratomic force microscopy are designed to combine sub-nanometre mechanical and thermalstability with sub-nanometre position control and rapid response times (see text).


probe assembly includes both the probe cantilever mount and the sub-nanometre precision,piezoelectric position control system for the probe tip. The sample stage is adjustable toallow for optical alignment of the sample with respect to the probe and to permit coarseadjustment of the initial probe–sample separation. In atomic force microscopy, the split-beam laser system used to monitor the mechanical deflection, and the amplitude andfrequency of vibration of the probe cantilever are both within the microscope chamber(Figure 7.11) but with external control of the mechanical detection system alignment. Thescanning of the probe and the probe–sample separation are under analogue control. All theparameters that are transmitted by the graphics interfaces to the display and controlmonitors are digitized.

The heart of the atomic forcemicroscope is the cylindrical, piezoelectric position controlsystem. Movement in the z-direction is achieved by applying a voltage across thepiezoelectric tube wall that contracts or expands the axial length of the tube. Movementin the x–y plane is achieved by applying voltages of opposite sign across diagonal segmentsof the tube, causing it to deflect by bending in either the �x or �y directions.

Two other components are no less important to the successful functioning of the atomicforce microscope: These are the design of the cantilever for the probe tip, and the detectionsystem for the tip displacement.

Laser

Primary lens

Tracking lens

Sample

Cantilever holder

Piezo tube scanner

Detector

MirrorMirror

Figure 7.11 In the atomic force microscope, the controls for laser alignment and coarseadjustment of the sample position are external to the environmental chamber, but thephotodiode optical assembly is inside the chamber.


V-shaped cantilevers [Figure 7.12(a)] are flexible about the points of support of the V, butpossess high torsional rigidity. They are intended for scanning the sample along the axis ofthe cantilever. Alternative, simple beam cantilevers [Figure 7.12(b)] are much less flexiblethan the V-cantilever perpendicular to the beam axis, but they are sensitive to torsionaldisplacements about an axis parallel to the cantilever beam. They can therefore be scannedeither parallel to the cantilever beam, as in conventional scanning, or perpendicular to thecantilever, when they provide information on the local frictional forces that act parallel,rather than normal, to the plane of the sample surface.

Many different tip materials have been used for atomic force microscopy, includingdiamond, tungsten and tungsten carbide, but silicon nitride is often preferred, since thismaterial possesses good chemical and physical resistance to tip damage. The high elasticrigidity of silicon nitride reduces hysteresis losses andmechanical phase delays in the signalgenerated, effects that would be associated with the tip rather than the sample. Siliconnitride is therefore the tip material of choice for contact mode imaging (see below), but thetip radii are typically only of the order of 20 – 60 nm,which places atomic resolution beyondthe reach of silicon nitride tips.

One alternative, silicon tips, can be integrated into a single-crystal, silicon cantilever.These are now often prepared by focused ion beam milling (Section 5.8). They providehigher resolution, since the nominal tip radii are typically less than 10 nm. Silicon tips aretherefore preferred for applications in the semi-contact region, usually explored by using avibrating cantilever (the tapping mode of operation, see below).

An additional �enabling technology� for atom force microscopy is the detection systemfor determining the tip displacement as it is scanned over the sample surface. This is usuallybased on a split photodiode detector operating with a solid state laser source (Figure 7.13).As the cantilever is deflected, it reflects a laser beam which then scans across a small gapbetween the two halves of the photodiode, so that the signal generated depends on theproportion of the light falling on each half of the photodiode. The cantilever is gold-coatedtomaximize reflectivity and the detector is tuned by adjusting the tilt of amirror. The lengthof the optical arm controls themaximumdeflection amplitude that can bemeasured for eachsetting. If a piezoelectric drive is used to vibrate the cantilever close to its resonant

Figure 7.12 Cantilever assemblies for the atomic force microscope. (a) A thin, V-shapeddesign provides maximum bending sensitivity. (b) A simple beam that permits both flexure andtorque of the cantilever.


frequency, then the alternating signal from the split photodiode can be compared with thedrive signal in order to measure phase shifts in the signal that are related to nanostructuralfeatures on the sample surface.

Finally, it is also possible to apply a bias voltage to the probe and thenmonitor the electriccurrent between the probe and the sample. This is, of course, the basis of the scanningtunnellingmicroscope, but a bias voltagemay also provide some additional flexibility in theoperation of an atomic force microscope.

7.2.1.1 Data Collection and Interpretation. The principle of operation of the atomicforce microscope may be simple, but it should not be forgotten that the instrument ismonitoring the surface topology and behaviour of materials in a spatial regime that mayextend down to atomic separations. Not surprisingly, these measurements are sensitive tomany artifacts, some of which are still poorly understood. For example, piezoelectricscanners show non linear hysteresis and their response may change with time, especially ina new instrument (a process termed �ageing�). This requires frequent calibration of thesystem. Nonlinearity of the piezoelectric response also leads to distortion of the image,unless this is specifically corrected. Time-dependent drift of the atomic force microscopesystem is very common, leading to further image distortions that must be recognized. Toensure that the deflection of the cantilever is accurately measured, the electronic detectionsystemmust be tuned, both to minimize noise and to ensure a fast electrical response of thesystem in order to follow the cantilever movement. The plane of the sample surface shouldalso be as near as possible coplanar with the x–y scan of the probe tip. Expecting artifacts tobe present is a good defence against being misled into interpreting any imaging defect as afeature of the surface topology.

7.2.1.2 Modes ofOperation in Atomic ForceMicroscopy. The basic structural features ofthe atomic force microscope and the scanning tunnelling microscope are compared inFigure 7.14. In the scanning tunnellingmicroscope, it is the current–voltage characteristics i(V) that are measured as a function of the probe separation z at a site in the x–y plane of the

Figure 7.13 Movement of the cantilever probe support is monitored by a laser beam that isreflected onto a split photodiode and generates a signal that is proportional to the deflection ofthe cantilever.


sample surface [Figure 7.14(a)]. In the atomic force microscope, it is the displacement ofthe probe tip at the end of the cantilever that is measured [Figure 7.14(b)]. Vibrating thecantilever probe by a tuned piezoelectric drive close to its resonant frequency, allows theamplitude, frequency and phase of the oscillations to be collected as additional signals thatprovide spatially resolved data on the properties of the sample surface [Figure 7.14(c)].

A simple, straightforward application of the atomic force microscope is to monitorsurface topology in the contact mode. The probe is lowered onto the sample until a surfacerepulsion is detected as a positive deflection of the cantilever. Two options are then possible:either the changes in the repulsive forcemay bemeasured by the deflection of the cantilever,

Figure 7.14 (a) In scanning tunnellingmicroscopy the current–voltage characteristics i(V) aremeasured as a function of the probe separation z at a chosen site in the x–y plane of the samplesurface. (b) In atomic forcemicroscopy thedisplacement of theprobe tip,mountedon the endofa flexible cantilever beam, is measured. (c) Vibrating the cantilever beam of an atomic forcemicroscope close to its resonant frequency allows the amplitude, frequency and phase of theoscillations to be monitored as additional signals that provide spatially resolved data on theproperties of a sample surface.


while keeping the height of the piezoelectric assembly fixed, or a constant repulsive forcemay be set as a fixed elastic deflection of the cantilever, while the changes in height that areneeded to maintain this constant force are monitored. Under atmospheric conditions, filmsof moisture are often present on the surface and can result in a capillary attraction that is notpresent, either in vacuum or when the chamber is filled with dry nitrogen. Electrostaticcharging of a nonconducting surfacemay also result in unpredictable attractive or repulsiveelectrostatic forces between the probe tip and the sample. Most of these effects can beneutralized when the scanning probe is operated in contact mode, providing the deflectionof the cantilever is positive. To ensure maximum deflection sensitivity, a thin, V-shapedcantilever is used, with a silicon nitride tip to minimize wear damage to the tip. Tip wear isoften a problem in contact mode and wear debris can collect on the probe tip. For soft,compliant samples it is important towork at a constant, pre-set cantilever deflection in orderto limit damage to the sample.

Resonant excitation of the cantilever has proved to be most useful in the semi-contactregime, for which the attractive and repulsive surface forces have similar magnitudes(Figure 7.3). This mode of operation is frequently referred to as the tapping mode, since to afirst approximation the probe tip �taps� the sample at its point of closest approach to thesample surface. A �spring-and-dashpot� model for the viscoelastic, �tapping-mode� behav-iour of the probe–surface interaction is useful (Figure 7.15). In the semi-contact regime theforce on the probe tip due to the presence of the sample changes sign during the deflectioncycle, as the tip moves past the minimum potential energy position. This is shownschematically in Figure 7.3. The resultant �damping� alters both the amplitude of theoscillations and their phase with respect to that of the drive crystal. The �phase� image intapping mode can be extremely sensitive to the elastic properties of the sample and iscapable of revealing details of the nanostructure with excellent spatial resolution(Figure 7.16).

As the probe approaches a sample surface, the initial deflection of the cantilever isnormally attractive. Pre-setting the deflection of the cantilever to a negative value will

Figure 7.15 �Spring-and-dashpot�model for simulating damped oscillations of the cantileverin semi-contact with the sample and interpreting changes in the amplitude or phase of theoscillations.


collect subsequent data on this attractive force in the non-contactmode, essentially withoutany repulsive force interaction. Themajor effect observed is an increase in the amplitude ofthe vibrating probe oscillations that is detected by the split photodiode and can then betranslated into a direct measurement of the changes in the attractive forces as the tip isscanned across the surface.

As noted previously, torque displacements can also be monitored at high resolution byusing a focused ion beam machined, silicon tip integrated into a silicon beam. It hasproved possible to image the basal plane of graphite using this technique (Figure 7.17).At this scale, we should avoid interpreting the results in terms of atomic �surfacetopology�, as though the atoms were packed like billiard balls, and remember that we arereally looking at electrostatic and electrodynamic interactions between many neighbour-ing atoms.

By applying a bias voltage between the sample and the cantilever probe tip, and thenmonitoring the tip current, it has also proved possible to operate the atomic forcemicroscope as a scanning tunnelling microscope. Image features are then associated withthe electrical properties of the surface. These electrical signals are interpreted in terms of thespreading resistance of the contact, the contact potential or the local capacitance across thegap between the probe tip and the sample surface. Finally, magnetic domain structures canalso be imaged by atomic forcemicroscopy. The probe–surface separation is then increasedbeyond the range of the attractive van der Waals forces, leaving only the magnetic forcesbetween a suitable tip and the ferromagnetic sample. Clearly the resolution when imagingmagnetic domains will be limited to the separation distance between the probe and thesample, typically several nanometres.

Figure 7.16 A tapping-mode atomic force image of partially crystalline poly(ethylene oxide)uses the phase shift across crystallite boundaries to reveal the structure. (Reproduced fromNanoTech America).


7.2.2 Scanning Tunnelling Microscopy

In scanning tunnellingmicroscopy the probe tip is scanned over the surface of the sample invacuum at a controlled probe–sample separation and the current–voltage characteristicsdetermined for each pixel point. The interpretation of the results is based primarily on theFowler–Nordheim equation [Equation (7.1)]. It is possible to probe the local density ofelectron states at the sample surface quite accurately and, under suitable circumstances, thesensitivity of the tunnelling current to the probe–surface separation allows individualsurface atoms to be resolved. Unlike atomic force microscopy, scanning tunnellingmicroscopy is limited to materials that are electrically conducting, or at least semicon-ducting, and the microscope chamber has to be maintained under ultra high vacuum toprevent adsorption and surface contamination from the gas phase. As in atomic forcemicroscopy, the mechanical and thermal stability of the scanning probe system oftendominates the performance, and drift of the probe tip does occur, so that measurementscannot be made over extended periods of time. This limits the number of pixels fromwhichdata can be usefully accumulated for any given image field. As noted previously, the totalnumber of pixels in a scanning probe image is always at least an order ofmagnitude less thancan be realized in a secondary electron image acquired in the scanning electronmicroscope.

Although the inherent resolution is better, the modes of operation of the scanningtunnelling microscope are more restricted than those of the atomic force microscope, sinceit is primarily the variations in work function and surface density of states that are beingdetected, rather than the surface topography of the sample. It is important to recognize that

Figure 7.17 Graphite basal plane imaged in tapping mode at the limit of resolution of theatomic force microscope. (Reproduced from NanoTech America).


thework function is a surface property and not a bulk property of the solid. In particular, thework function is not the energy that is required to extract an electron from the top of theconduction band and take it to infinity, whichwould be a bulk property, but rather the energyneeded to take a conduction electron to just beyond the solid surface. The work function istherefore sensitive to the atomic packing of the surface fromwhich it was extracted and canbe dramatically affected by crystal structure, surface contamination, adsorbates and thepresence of segregants.

Since, according to the Fowler–Nordheim [Equation (7.1)], the current passing throughthe tip depends exponentially both on the work function and on the effective distancebetween the tip and the sample, variations in these two variables are superimposed as thetip traverses the surface. In general, as in the atomic force microscope, more than onemode of operation is possible. In the first instance, the scanner assembly ismaintained at aconstant height (constant z in the microscope) as it is scanned in the x–y plane. Thechanges in the tunnelling current are then monitored and interpreted as either changes inprobe–sample separation or work function. Alternatively, a given tunnelling current canbe pre-selected, and the height of the scanning head is then continuously adjusted, in afeedback loop, to keep the electron field emission current constant as the tip is scannedover the sample surface. The changes in height are now interpreted as changes in sampletopography. A third option is to scan the tip bias voltage at each x–y pixel location so as toobtain a local value for the voltage dependence of the tunnelling current i(V) for both agiven tip–sample separation Dz and x–y location. The slope of the i(V) curve taken at aknown location x, y,Dz, can nowbe interpreted in terms of the local density of states. Sincethe local density of states depends strongly on the atomic structure, this mode of operationfrequently results in excellent atomic resolution of the surface structure and atomicpacking.

7.2.2.1 Resolving Surface Morphology: Restructuring. Some examples of atomic reso-lution are shown in Figure 7.18. These include the {1 0 0} and {1 1 1} surfaces of silicon, thebasal plane in graphite and an example from a gallium arsenide detector crystal. In all cases,it is the electronic nanostructure associated with the surface packing of the atoms, theatomic morphology, that has been resolved. The contrast in the images also depends on thebias voltage applied between the probe tip and the sample. On this atomic, sub-nanometrescale, it makes little sense to talk of �surface topology�, since the surfaces are oftenatomically smooth. In such cases, the probe is �responding� to the local modulations in theelectronic structure of the array of surface atoms that are present.

One of the more unexpected discoveries revealed by the scanning tunnelling microscopehas been the extent towhich the atomic separations in the surface layers of a crystal deviatefrom their bulk values. In several cases the new atomic packing results in surface arrays thatdo not even have the same symmetry as the bulk lattice. We have already described thereconstruction of the {1 1 1} planes of silicon into a 7 · 7 two-dimensional unit cell [Section1.1.2.4 and Figure 7.18(b)], but this is by no means a unique case.

Figure 7.19 shows a rather complex reconstruction in which the hexagonal symmetry ofthe (0 0 0 1) plane of the InGaAs parent lattice has been lost and replaced by a 2 · 4rectangular unit cell. It should be no surprise that both the chemical and themicroelectronicindustries have amajor interest in these phenomena, since they have a direct bearing on bothcatalytic activity and electronic surface states.


Surface relaxation and complete restructuring are not the only options for changes inatomic packing at a surface. Some surfaces can become �rumpled� on the atomic scale, if thiswill reduce the surface energy through a less symmetrical configuration. It has also beenestablished that the lattice parameters of colloidal nanoparticles may differ substantiallyfrom those of bulk materials that have the same crystal structure and composition. Theassumption that the atomic packing and morphology at a free surface is derived bysectioning parallel to a defined crystallographic plane and then discarding one half of thecrystal is an inaccurate and even misleading approximation. With the scanning tunnellingmicroscope, the equilibrium surface morphology of metallic conductors and semiconduc-tors can now be explored in atomic detail.

7.2.2.2 Electron Energy Levels: Scanning Tunnelling Microscopy Spectroscopy. Al-though the scanning tunnelling microscope can be used to probe the i(V) characteristic as afunction of the probe separation Dz at any point on the surface, the limited, long-term,thermal and mechanical stability make it difficult to obtain such spectroscopic data from

Figure 7.18 Examples of STM micrographs showing (a) the {1 0 0} surface of silicon at lowmagnification, (b) the reconstructed 7· 7 (see text) {1 1 1} surface of silicon,(c) the basal plane inpyrolitic graphite, and (d) the {1 1 0} surface of gallium arsenide with zinc acceptors (triangularfeatures). (Reproduced from NanoTech America). (See colour plate section)


more than a restricted number of pixel points. Nevertheless, it has proved possible to probethe structures associated with selected microelectronic components, such as a p-n junction,or quantum dots. A spectroscopic data set that has been collected from a predeterminedpixel location and at a given probe separation, can be analysed in terms of the gradient ofthe i(V) characteristic, di/dV, in order to extract information on the local density of surfacestates. The bias voltage applied to the probe can be either positive or negative, while thefrequency response of the current signal may also provide information well intothe megahertz region, offering the possibility of determining local dielectric loss factorsand other data that are related to polarization forces (the dipole moments and polariz-abilities of the surface atoms and molecular assemblies). This subject is well beyond theobjectiveswe have set for the present text, but is nevertheless an exciting area of research forsolid-state device technology and condensed-matter physics.

7.3 Field-Ion Microscopy and Atom Probe Tomography

The evolution of the simple field-ion microscope (Figure 1.10) into the atom probe(Section 7.2) occurred some 40 years ago, but the initial results had limited impact onmaterials characterization for two very good reasons: first, the data sets that were collectedcontained counts from, at most, a few thousand atoms. This was insufficient to provideconvincing statistical evidence that they represented the �real�, nanometric scale chemicalcompositions for the constituent elements of a sample. Secondly, not only was it necessaryfor the minute, sub-micrometre tip of the sample needle to be a metallic conductor, but it

Figure 7.19 A 2· 4 restructuring of the (0 0 0 1) surface on InGaAs. Reproduced from P. Vogt,K. Ludge, M. Zorn, M. Pristovsek, W. Braun, W. Richter and N. Esser, Atomic structure andcomposition of the (2 ·4) Reconstruction of InGaP(0 0 1), Journal of Vacuum Science andTechnology B, 18(4), 2210–2214, 2000.


also had to have sufficient mechanical stability to withstand the gigantic stress exertedby the applied electric field during field-evaporation of the surface atoms from thespecimen tip.

The introduction of time-of-flight mass spectrometry, with sub-nanosecond resolution,provided the technology that made the atom probe possible. Instead of relying on amagnetic field to analyse the mass spectrum of the beam of evaporated ions, the time-of-flight mass spectrometer measured the time required for an ion, evaporated from thespecimen tip surface by a voltage pulse, to reach a detector placed at the end of a 1m longflight tube. This time, typically measured in nanoseconds, was directly proportional to theratio of the electric charge carried by the ion ne, and its isotopic mass m, where n is thecharge on the ion and e is the electronic charge.

It therefore follows that the pure elements give mass spectra that contain several spectrallines, both because the different isotopes of a single element can be well-separated by thetime-of-flight mass spectrometer, and because the field evaporation process may generateions of more than one charge from a single atomic species. A complex but well-resolvedmass spectrum is shown in Figure 7.20. Inmany cases,molecular ions are also collected andare often associated with adsorbed residual gases, such as carbon dioxide andwater vapour,that have been trapped at the sample surface.

Improvements in electronic detection have enabled the mass spectrometer flight tube tobe shortened, while the development of channel-plate charge multipliers and chargecoupled device (CCD) collectors has made it feasible to increase the collection anglefrom the tip and obtain a �field evaporation image� by summing the counts from given ne/melemental spectral peaks over time. The data are accumulated for each pixel of the CCDcollector and for each field evaporation excitation pulse. Finally, there is now no need toapply a voltage pulse to the tip in order to excite field evaporation. Instead, an appliedelectric field can be used to reduce the energy barrier for field evaporation of an atom to anypredetermined level and a pulsed laser then used to trigger the release of ions from the tipsurface. This has resulted in a significant improvement in the time resolution of the massspectrometer. Successive layers of atoms can be field evaporated and the charge-to-massratio of each captured ion can be recorded as a function of both the pixel location on theCCD detector (the x–y plane of the field-ion image at a given time) and the pulse number inthe sequence of laser or voltage excitation pulses (the z-coordinate of the sample tip). Atomprobe tomography can provide three-dimensional atom-by-atom chemical analysis at thenanometre level.

7.3.1 Identifying Atoms by Field Evaporation

The time-of-flight mass spectrometer has a time-scale that depends on the length of theflight tube and the accelerating voltage, but is typically of the order 100 ns. It follows that atime resolution of better than 1 ns is essential for good peak separation and that gigahertzelectronics is necessary to process the results. The capacitance of the sample tip andextraction electrode is a significant factor in determining the mass resolution, as is theresponse of the channel-plate/CCD assembly.

Assuming that the mass peaks are sufficiently sharp for the different values of ne/m to beseparated unambiguously, we still need to know both the charge on the ion and its isotopicmass before we can identify the chemical species. The charge on the ion field-evaporated


from a given sample tip is determined by the electric field required to ionize that particularchemical species. Intuitively, we might expect the ionic charge to be equal to a knownchemical valency of the element, but this is incorrect. All field-evaporated species arepositively charged, since a positive potential is applied to the sample tip, and the energyrequired to field-evaporate an ion Mnþ is approximately equal to the sum of the ionization

Figure 7.20 Local electrode atom probe time-of-flight mass spectrum of an Al–0.1Zr–0.1Ti(atom%) alloy containing Al3(Zr1-xTix) (L12 structure) precipitates. (a) Mass spectrum for theentire analysis, representing 7.67· 105 identified atoms, of which 5.21 · 103 atoms(0.679� 0.009 atom%) are identified as Zr and 2.16 · 103 atoms (0.282� 0.006 atom%) areTi. (b) Mass spectrum indicating the core composition of the Al3(Zr1-xTix) precipitate. Thisspectrum represents 8.88· 103 atoms, of which 1508 atoms (17.00� 0.40 atom.%) are Zr and335 atoms (3.77� 0.20 atom%) are Ti. (c) Mass spectrum indicating the composition of thematrix (a-Al solid solution) surrounding the Al3(Zr1-xTix) (L12 structure) precipitate. This massspectrum represents 2.09· 105 identified atoms, of which 224 atoms (0.107� 0.007 atom.%)are Ti. There is no evidence of Zr (<0.010 atom%), indicating that all detectable Zr atoms havepartitioned to the Al3(Zr1-xTix) precipitate. These data are impressive since they represent soluteanalysis in very dilute (0.1 atom.%) alloys. The isotopes of Ti and Zr are clearly resolved, and theZr atoms are observed in two charge states (2þ and 3þ). Courtesy of David Seidman.


potentials Sn In minus nf, the work function of the sample surface in eVmultiplied by thecharge on the evaporated ion. The minimum field strength required to evaporate an ion ofgiven charge depends on the relative values of the ionization potentials that are needed toexcite an atom to a given charge state. In most cases, the observed field-evaporated ions aredoubly charged, but singly charged ions are frequently found, while some species that areonly ionized at very high field strengths may be triply charged. Although doubly chargedions are expected to dominate, it is quite possible for anygiven element to appear in themassspectra with more than one charge, as seen in Figure 7.20. Excellent mass resolution isavailable in these spectra and isotope peaks from a given chemical species can be identifiedfrom their relative abundance in the earth�s crust.

The isotopic abundance of the chemical elements in the earth�s crust is well-documentedand may be used to resolve ambiguities in the mass spectrum that may arise when two ionshave similar charge-to-mass ratios. This can be quite commonwhenmolecular ionic speciesare present.

7.3.2 The Atom Probe and Atom Probe Tomography

The design of one modern atom probe is shown in Figure 7.21. Several samples can beloaded into the ultra-high vacuum chamber through an airlock at the same time. Thesamples are then moved into position, one at a time, over the field-evaporation electrode.The sample is first aligned optically, before forming a field-ion microscope image using anappropriate image gas (usually helium for refractory metals and alloys, or neon in the case

PreparationChamber

Computer ControlledTiming System

DC HighVoltage

HVPulse

Channel Plate

3DAPDetector

Specimen

Time-Of-FlightMass Spectrometer

Single AtomDetector

EnergyCompensating

Lens

Airlock

Pulsed Laser

Figure 7.21 Atom probe tomography is made possible by combining the high spectralresolution of tune-of-flight mass spectrometry with the atomic resolution of the field-ionmicroscope and pulsed field evaporation. The sub-nanosecond time resolution of therecording system is capable of storing data for millions of ions and identifying both theirchemical species and their relative atomic locations in the sample.


of steels). An image intensifier simplifies both viewing and recording. At this stage, the tipis shaped by pulsed field evaporation to remove absorbed debris and leave an atomicallysmooth, symmetric, tip surface.

Final alignment of the sample ensures that a nanostructural area of interest is centred onthe axis of themass spectrometer collection aperture. Typically, a crystallographic directionof interest will be chosen to lie on the axis of the spectrometer. Data collection can nowbegin. Voltage or laser pulses are used to field-evaporate the atoms at the edges of the close-packed planes of the structure. The data from each evaporation pulse are digitally binned,according to the charge-to-mass ratio of the ion ne/m (defined by the time of arrival of theion at the detector) and the x–y pixel coordinates for the ion on the CCD detector. The z-coordinate is registered as a pulse time. The system records the data from each separatepulse and the time register is then incremented by one step before a further pulse is used togenerate the next ion shower to be detected at the CCD. The complete data set for each ionspecies is recorded in three-dimensions but all three x, y and z dimensions have to becalibrated and this is not trivial. Calibration usually assumes known values for the latticeparameters and uses field-ion images of the lattice planes to determine an x–y scale for theCCD plate coordinates. Since the projection is never a simple point projection (the tip is nota simple hemisphere), the effective magnification varies over the CCD plate and thisinformation must be incorporated into the calibration software. In addition, as fieldevaporation proceeds, the effective tip radius of the sample increases, so that the localx–y magnification has to be continuously recalibrated. Often the tip is approximatelyconical and the changes in magnification are �well-behaved�, but this is not always the case,especially if polyphase samples that contain coarse particles are being analysed.

Calibration of the z-axis scale is determined from the number of pulses required toremove the layers of a single crystal plane with known d-spacing. Typically, less than 100atoms are removed by each evaporation pulse, and up to 100 pulses may be required toremove a single lattice plane of atoms. The ion detection efficiency of the time-of-flightmass spectrometer and CCD collection plate is remarkably high (at least 50% andsometimes better than 80%). The collection efficiency depends mainly on the sampledimensions and the collection system.Although the collection efficiency depends very littleon sample composition, this is not quite true for large, second-phase particles, since suchparticles can affect the surface topography appreciably during field evaporation, especiallywhen they lie off the axis of the sample, as is usually the case.

Figure 7.22 shows a few striking examples of atom probe tomographs from someengineering alloys.When compared with every other available technique, the quality of thesub-nanometre chemical information is quite extraordinary. In each case the recordedcounts are due to individual ions. Although not every ion can be detected, most are, and theatom probe is now providing atomic-scale chemical information on the earliest stages ofprecipitation and segregation in key engineering alloys.

Summary

In previous chapters, microstructural characterization has been discussed in terms of theanalysis of a signal that was generated when a beam of electromagnetic radiation orenergetic electrons interacted with a prepared specimen. The discussion of these


Figure 7.22 Some examples of the applications of atom probe tomography in alloydevelopment. (a) Variations in concentration in a copper (red) and silver (green) nano-


electromagnetic and energetic electron probes is now expanded by considering theinteraction of a solid probe, in the form of a sharp point or needle. The location andseparation of the probe from a solid specimen surface are controlled with sub-nanometreaccuracy.

Some �microscopes� have been developed, in which the sample is itself the source of thesignal, and no separate probe is present. The first such instrument to yield atomic resolutionwas the field-ion microscope. In this instrument, the specimen is the tip of a sharp needle.Such sharp needles are also the electron sources used in field emission guns for high-resolution scanning and transmission electron microscopy, and the same �needle� geometryis used for the solid probe in scanning tunnelling and atomic forcemicroscopy. In the atomicforce microscope the sharp probe tip is mounted on a cantilever beamwhose deflection canbe monitored while the probe is scanned over the surface. The cantilever can also bevibrated during scanning, at approximately its natural frequency, and the changes in eitherthe deflection amplitude of the cantilever, or the phase shift of the tip signal with respect tothe driver can be monitored.

In order to monitor changes in the attractive surface force between the probe and thesample, the probe tip can be scanned in the so-called, non-contact regime, far enough fromthe surface to ensure that attractive forces dominate the interaction. Alternatively, it ispossible to monitor the contact regime, with the tip probe close to the surface, where theinteraction is dominated by repulsive forces. The probe can also be scanned close to theequilibrium, zero-force separation, in the semi-contact regime. In this semi-contact regime,vibrating the probe leads to a tappingmode of operation inwhich the probemoves in and outof contactwith the surface. The �tapping� signal is very sensitive to changes in amplitude andphase, yielding high resolution images of the surface atomic features and surface properties.The tapping mode can provide information derived from changes in the amplitude,frequency or phase of the cantilever vibrations with respect to the phase of the driver.Although the information is, for themost part, qualitative, the nanoscale image contrast canbe linked directly to differences in surface elasticity, for different surface phases, orcrystallographically determined elastic anisotropy of a single phase polycrystal. Thisinformation is not available in any other instrument.

The scanning tunnelling microscope employs a bias voltage between the scanning probetip and the sample surface, in order to monitor the voltage dependence of the tunnellingcurrent as a function of the probe–surface separation. The current–voltage characteristic canprovide information on the local density of states, while the scanned tunnelling image is

composite. [Reprinted from F. Wu, D. Isheim, P. Bellon and D.N. Seidman, NanocompositesStabilized by Elevated-Temperature Ball Milling of Ag50Cu50 Powders: An Atom ProbeTomographic Study, Acta Materialia, 54(10), 2605–2613, 2006]. (b) Segregation ofsilicon to a grain boundary in iron [Reprinted from B.W. Krakauer and D.N. Seidman,Subnanometer Scale Study of Segregation at Grain Boundaries in an Fe(Si) Alloy, ActaMaterialia, 46(17), 6145–6161, 1998]. (c) Evolution of a Second Phase in an Ni-Al-Cr Alloy[Reprinted from C.K. Sudbrack, K.E. Yoon, R.D. Noebe and D.N. Seidman, Temporal Evolutionof the Nanostructure and Phase Compositions in a Model Ni–Al–Cr Alloy, Acta Materialia, 54,12, 3199–3210, 2006]. Copyright (2006), with permission from Elsevier. (See colour platesection)

3


capable of resolving the packing of surface atoms and atomic spacing. The scanningtunnelling microscope has revealed that many crystal surfaces undergo previously unsus-pected rearrangement and restructuring, in order tominimize the surface energy of the solid.

The development of the atom probe as an extension of the field-ion microscope, togetherwith wide-angle, time-of-flight mass spectrometry has made atom probe tomography apractical reality. Individual atoms from the smooth tip of a sharp specimen needle are field-evaporated as ions by a pulsed electric field or pulsed laser. The chemical species of the ionscan be identified, from their charge-to-mass ratio in a time-of-flight mass spectrometer, andthen counted as a function of their location on the tip surface (x–y coordinates) and thenumber of pulses in the field-evaporation sequence (the z-coordinate). Data sets consistingof millions of ions have now provided three-dimensional chemical nano-analysis ofprecursor precipitate clusters, grain and phase boundary segregation, and other featuresof the nanostructure of engineering materials. The sample materials that can be examinedby atom probe tomography are limited to those metals and semiconductors that are capableof withstanding the mechanical stresses generated by the electric fields( MVcm�1) neededfor controlled field evaporation.

Bibliography

1 J. Israelachvili, Intermolecular and Surface Forces, 2nd edn, Academic Press, NewYork, 1991.

2 R.Wiesendanger, Scanning ProbeMicroscopy and Spectroscopy, CambridgeUniversityPress, Cambridge, 1994.

3 M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Kluwer/Plenum,New York, 2000.

4 V.M. Mironov, Fundamentals of Scanning Probe Microscopy, Russian Academy ofSciences, Nizhniy Novgorod, 2004,

5 B. Bhushan, H. Fuchs and S., Hosaka, Applied Scanning Probe Methods, Springer,New York, 2004.

Problems

7.1. In the absence of a specimen in the transmission electron microscope, whatinformation on the microstructure of the electron source ought to be present inthe diffraction image plane?

7.2. In scanning probe microscopes, the probe and the sample form a single system.What physical and mechanical properties of the probe are necessary to ensure thatthe properties and structure of the probe do not affect the results?

7.3. What is meant by the term Hamaker coefficient? Under what circumstances mightyou expect the Hamaker coefficient to be negative?

7.4. When two hydrophobic surfaces that are separated by a thick film of water arebrought together, a puzzling attractive force can sometimes be measured at


distances that greatly exceed those associated with polarization forces. Can yousuggest a possible explanation?

7.5. Discuss how the charge on the counter-ions can affect the structure of a double layerformed at the surface of a solid that is immersed in a polar solution.

7.6. Explain the term steric hindrance. Why should a nonpolar, polymer moleculeprevent dispersed particles from flocculating (aggregating together in thedispersion)?

7.7. Explain the origin ofmultipleminima in a surface force–distance curvemeasured inthe presence of a solution of a long chain alcohol or a polysaccharide.

7.8. Distinguish between the lattice resolution of atomic planes observed by highresolution electron microscopy, and atomic resolution observed on the surface ofa semiconductor by scanning tunnelling microscopy.

7.9. What are we �seeing� when we observe the 7 · 7, restructured {1 1 1} surface ofsilicon in the scanning tunnelling microscope?

7.10. Why does field emission tunnelling have a better potential for atomic resolutionthan that available by monitoring the surface forces experienced at the tip of anatomic force microscope? (Hint: compare the Fowler–Nordheim field emissionequation to the DLVO surface force–distance model.)

7.11. What might you expect to find in the surface composition of an electroplated goldlayer examined by atomic force microscopy in air? One word answers are notacceptable!

7.12. What damage mechanisms account for the rather limited working life of a siliconnitride atomic force microscope cantilever tip?

7.13. An atomic force microscope operated in tapping mode can be used to monitor theelastic response of a soft tissue sample surface. How could this information bequantitatively related to the elastic constants of the material?

7.14. Scanning tunnelling spectroscopy is a powerful tool for probing the density of statesof a semi conductor and exploring the structure of a p-n junction and other featuresof solid-state device technology. What are the limitations of this technique whenapplied to semiconductor surfaces?

7.15. Atom probe tomography has produced some remarkable results and commercialinstruments are available. Nevertheless, relatively few papers have been publishedand the development process has taken over 40 years. Why?


8

Chemical Analysis of SurfaceComposition

Chemical analysis of the composition is an important element in the characterization of anengineering material at all stages of the materials cycle (extraction, manufacture, serviceand, ultimately, disposal or recycling). Nevertheless analytical chemistry is not a topic forthis text, which has been restricted to microstructural aspects of the characterizationprocess. In Chapter 6we exploredmicroanalytical methods of chemical analysis that can becombined with electron microscopy, either for bulk samples, using the scanning electronmicroscope, or for thin-film specimens in the transmission electron microscope. Chapter 6covered X-ray microanalysis by energy-dispersive or wavelength-dispersive spectroscopy,as well as electron energyloss spectroscopy (EEELS).

In the present chapterwe extend our treatment ofmicroanalysis to three further analyticalmethods that, in this case, focus on the chemistry of the surface layers of an engineeringcomponent. These additional tools have major applications in surface engineering,especially for the study of solid-state devices. These applications include opto-electronicand superconducting materials, thin-film devices, and both radiation and chemical detec-tors. Before beginning, we should first list some of the many available methods of chemicalanalysis that will not be covered in this chapter:

. Atomic absorption spectrometry depends on the detection of characteristic absorptionspectra for atomic species. A beam of individual atoms is commonly generated by laservaporization of a sample and the absorption spectra for the constituent species arerecorded using a white light source.

. Optical emission spectroscopy is the inverse of atomic absorption. It measures theintensities of the characteristic emission lines of the different atomic species. In thismethod, a spark source is usually used to generate the signal.

. Infrared spectroscopy is based on absorption spectra that are detected in the infrared rangewhen the sample is placed in the path of a suitable infrared radiation source. The infraredabsorption spectra can beused to extract information on the chemical bonding in the system.


. Raman spectroscopy employs inelastic photon scattering, usually after laser excitation.Photon absorption at thewavelength of the excitation radiation is accompanied by photonemission at longer wavelengths. These Raman emission lines reflect the decay ofexcitation states that are primarily associated with variations in the chemical bondingof the sample material.

. Electron spin resonance is observed in paramagnetic materials subjected to microwavefrequencies. The observed resonances correspond to specific electronic states in thematerial, for example those associated with specific chemical valences.

. Nuclear magnetic resonance (NMR) occurs at radio frequencies and corresponds toresonance of the magnetic moment associated with an atomic nucleus for some isotopesof a given chemical species. Imaging by NMR has had a major impact on noninvasive,diagnostic medicine by medical imaging. In particular, it has vastly extended both theresolution and the sensitivity ofX-ray radiography and ultrasoundmedical imagingmethods.

. Fluorescence spectroscopy makes use of characteristic X-ray excitation generated byexposing the sample to a beam of high-energy, �white�X-rays. In contrast to the spectrumof characteristic X-rays generated by an electron beam, there is no excitation ofbackground Bremmstrahlung radiation, and the characteristic lines from minor traceelements can often be detected.

. Rutherford backscattering employs the simple Newtonian laws of momentum transferthat apply when a beam of MeV energy ions (usually helium) is backscattered aftercolliding with individual atoms in a solid specimen target. The energy and angulardistributions of the backscattered ions reflects their momentum distribution and can becalibrated with excellent depth resolution.

Many other techniques of bulk chemical analysis exist. With a few exceptions, notablyhigh resolution scanning secondary ion mass spectrometry (Section 8.3), microbeamexcitation of Raman spectra and infrared absorption, none of the above techniques hasthe microanalytical capability associated with sub-millimetre spatial resolution.

Table 8.1 summarizes the different excitation probes that can be used to excite a solidsample and the corresponding composition-sensitive signals for all the analytical techni-ques that have been outlined above, as well as those that will be discussed in this chapter.

8.1 X-Ray Photoelectron Spectroscopy

X-ray photoelectron spectroscopy (XPS) is also known as electron spectroscopy forchemical analysis (ESCA).1 This technique is used to probe most of the energy levelsin the atom that are revealed by electron energy loss spectroscopy (Section 6.2), but withoutany spatial resolution. XPS finds its place in this chapter because of its depth sensitivity,which is on the nanometre scale.

The photons in a beam ofmonochromatic, characteristic X-rays that is incident on a solidtarget are absorbed by the atoms in the sample and give rise to secondary electrons that areejected from the target with a kinetic energy Es, equal to the difference between the energyof the incident photon hn and the energy thatwas required to displace the secondary electronfrom the target atom. This excitation energy is the sum of the binding energy Eb that is

1The authors prefer XPS, since this name relates directly to the physics of the technique.


required to raise the excited electron to the Fermi level and the orientation-dependentworkfunction f that is required to bring the electron from the Fermi level into the vacuumjust outside the surface:

ES ¼ hn� Ebþfð Þ ð8:1ÞOn rearranging these terms, the binding energy of the photoelectron is given by:

Eb ¼ hn� ESþfð Þ ð8:2ÞThe electron binding energy in the atom increases as the energy of the emitted photoelectrondecreases. A schematic energy diagram for the secondary emission of a 2p photoelectron fromcopper is shown in Figure 8.1, while Figure 8.2 shows the complete copper photoemissionspectrum. In addition to the lines corresponding to the emission of electrons from energy levelsin the inner electron shells, the emission spectrum also includes Auger electrons that areassociatedwith energy transitions within an atom after excitation by an incident X-ray photon.TheAuger peaks can be separated from the photoelectron spectrumby recording two differentspectra using two characteristic, monochromatic X-ray wavelengths, that is, two differentphoton excitation energies, for the photoelectron excitation process. The Auger peaks willalways appear in the sameposition, but the photoemission lineswill all be shifted by the energydifference between the energies of the two incident X-ray beams used to excite the signal.

The electron binding energies that are of interest for a given sample may exceed 1 keV.The relation between the energy of an X-ray photon and its wavelength is l¼ hc/E¼ 1.24/V nm,where c is the speed of light,E is the energy of the photon andV is in keV. It follows that thatsuitable X-ray excitation wavelengths for photoelectron spectroscopy are of the order of0.1–1 nm.

Table 8.1 The probes and corresponding signals for common methods of analyticalchemistry.a

Name of method Probe used Signal detected

Optical spectroscopy Visible light Absorption and emission inthe visible range

Infrared spectroscopy Thermal excitation Infrared radiationRaman spectroscopy Laser excitation Infrared emissionElectron spin resonance Microwave excitation Chemical bonding statesNuclear magnetic resonance Radiowave excitation Selected isotope resonanceX-ray fluorescence �White� X-rays Characteristic X-raysX-ray photoelectronspectroscopy*

X-rays Secondary electronemission

Auger electron spectroscopy* Usually energetic electrons Secondary electrons fromAuger transitions

X-ray microanalysis* Energetic electrons Characteristic X-raysElectron energy lossspectroscopy*

Energetic electrons Absorption edges

Secondary-ion massspectroscopy*

keV inert ion beam Sputtered target atoms andions

Rutherford backscattering MeV (helium) ion beam Angular dependence ofbackscatter intensities

aAn asterisk indicates the method is treated in this text.

Chemical Analysis of Surface Composition 425

1s

2s

2p

3s

3p

3d

4s

1/2

3/2

h

EF

EVAC

Es V

V

=h -(Eb+φ)

φ

Figure 8.1 A 2p photoelectron is emitted from a copper atom as the result of the excitation ofthe atom by absorption of an X-ray photon.

8.1.1 Depth Discrimination

The background signal in the photoelectron spectrum, for example in Figure 8.2, arises frommultiply scattered, secondary electrons that are generated in the deeper, inner layers of thesample. If secondary electrons are inelastically scattered before they can escape from thesurface, then they can only contribute to the background signal, rather than to thecharacteristic peaks in the spectrum. It follows that the observed peaks are due tophotoelectrons that are generated in the immediate surface layers, at a depth which isless than the mean free path of the secondary electrons excited in the material. Bycomparison, the higher background levels on the high binding energy side of each peak,to the left of the peaks in Figure 8.2, are due to inelastically scattered electrons that originatein the deeper layers of the target and correspond to a lower energy for the collected signal.

In practice, both elastic and inelastic scatteringwill occur, and the depth dependence of theintensity for a specific peak leads to the definition of an attenuation length: a measure of theloss of detected signal intensity, normal to the sample surface, as a function of the excitationdepth (Figure 8.3). The broad minimum for photoelectron energies between 10 eV and500 eV corresponds to emission depths of between 2 and 5 atomic layers and demonstratesthe power of the technique to analyse surface composition and chemical binding.

XPS is extremely sensitive to chemical changes occurring at the surface, and has a depthresolution of just one or two atomic layers. However, as noted previously, no lateralresolution is available so there is no imaging capability.


Binding energy (eV)

Inte

nsit

y (a

.u.)

Cu Al Kα XPS

2s

2p1/2

2p3/2

LMM Auger

x10

3s

3p

valence

05001000

Figure 8.2 Photoelectron spectrum from pure copper includes both the photoelectrons andAuger electrons that are associated with energy transitions occurring within the atom afterexcitation (Section 8.2). Reprinted from G.C. Smith, Analysis of nanometer-sized precipitatesusing atom probe techniques, Material Characterization, 25, 1. Copyright (1990), withpermission from Elsevier.

Au

Al

AgAg

AuAu Ag

Au

Au

Ag

AgMo Be

Be Be

Be

{Ag

Ag WFe

PNi

Electron energy (eV)

Mea

n fr

ee p

ath

(A)

523

200010005001005010

5

10

50

100

Ag

AgAgMoC

AuBe{ Be

CC

Au Mo

W

Figure 8.3 Attenuation length for electrons detected normal to the surface for a variety ofdifferent solids as a function of their energy. Note the broad minimum in the range 10–500 eVthat corresponds to just a few atomic layers. (Courtesy of L.C. Feldman and J.W. Mayer,Fundamentals of Surface and Thin Film Analysis, Elsevier Science Publishing Co.).


8.1.2 Chemical Binding States

The binding energies of the electrons in the outermost shell of an atom are sensitive to thechemical state of the atom; the strength and nature of the chemical bonds. Photoelectronemission from the same atomic species, but in a different coordination or binding state, willgenerate multiple peaks in the photoelectron signal. Figure 8.4 is a particularly elegantexample in which the 1s binding energies for carbon in an organic molecule are easilydistinguished by peak splitting. These energy differences are of the order of a few eV for thedifferent carbon binding states.

Similar effects have been observed for the oxidation states ofmetals. Figure 8.5 shows thedifferent 2p states in a titanium atom. The higher binding energy in the oxide results in alarge peak shift of 4–5 eV.

Binding energy (eV)

280285290295

0

Inte

nsit

y (a

.u.)

CF

O

H

F

F

O

H

H

HH

CCC

Figure 8.4 Binding energy of a 1 s electron in carbon is a clear indication of its chemical bindingstate, with well-resolved peaks for each of the four carbon atoms in an ethyltrifluoroacetatemolecule. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probetechniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.


Binding energy (eV)

Inte

nsit

y (a

.u.)

460 054074

2p1/2

2p1/2

2p3/2 2p3/2

TiO2

TiMetal

Figure 8.5 Oxidation states on metal surfaces are resolvable by XPS. The growth of TiO2 ontitaniummetal is accompanied by 2p peak shifts of 4–5 eV. Reprinted fromG.C. Smith, Analysisof nanometersized precipitates using atom probe techniques,Materials Characterization, 25, 1.Copyright (1990), with permission from Elsevier.

8.1.3 Instrumental Requirements

Since the XPS signal comes from the first few atomic layers of the sample, it is important toensure that the sample surface remains uncontaminated throughout the analysis. An ultra-high vacuum chamber is required. Surface layers of contamination are usually removed byargon-ion sputtering with ion beam energies of a few kV. The arrival rate at the samplesurface of gaseous species from the environment depends on their molecular weight, thetemperature of the gas and the gas pressure. For air at a pressure of 10�6 Torr at roomtemperature, a complete monolayer of the gas molecules will arrive at the surface (but notnecessarily stick) in just 1 s. The time required for monolayer coverage of the sample byadsorption depends linearly on the pressure, so we should conclude that pressures below10�8 Torr are essential in an XPS chamber if reasonable time is to be available to collectthe experimental data before environmental contamination of the clean surface invalidatesthe results.

There is no reason why any system for electron spectroscopy should be limited to onlyone type of excitation probe, and it is possible to purchase commercial ultra-high vacuumchambers that combine an analyser system containing several probe sources, specimenstages and both source- and specimen-exchange carousels. Figure 8.6 illustrates schemati-cally some of these options.

The XPS specimen is cleaned by sputtering under an incident ion beam which isusually argon. The signal is generated by either a monochromatic X-ray or a high energyelectron beam. An electron beam can be scanned over the surface in a fine-focus rasterand a secondary electron image of the surface can be used to position the sample. Thephotoelectron signal is usually analysed electrostatically, in a focusing analyser havingan energy resolution that is better than the inherent peak width (typically 1 eV). Theemitted electrons are focused onto a defining input slit and the electrostatic field in the


analyser then focuses the electrons of a given energy onto an output slit, where theelectrons are detected and the signal subsequently amplified. Scanning the analyservoltage then scans the focus of the spectrometer for the full range of the differentelectron energies. It is the width of the output slit that determines the energy resolutionof the spectrometer.

The provision of a specimen carousel, capable of accommodating a number of differentsamples inside the ultra-high vacuum of the spectrometer, dramatically reduces the averagetime required to change specimens. High vacuum airlocks, that incorporate pre-bakefacilities for degassing the samples prior to admitting them into the spectroscope chamber,further reduce sample exchange times, usually to no more than a few hours, despite thestringent ultra-high vacuum requirements. Several degrees of rotational and positionalfreedom are also providedwith the specimen stage. These allow tilting of the specimenwithrespect to both the monochromatic X-ray or high energy electron beam and the spectrome-ter assembly. This is important, since both the take-off angle of the signal from the surfaceand the incident beam angle can affect the recorded spectrum, either as a result of crystallineanisotropy of the specimen or, more often, because the increased secondary electron pathlength at shallow take-off angles significantly reduces the emission depth, changing thethickness of the sampled surface nanolayer.

X-RaySource

ElectronGun

IonGun

Scintillator/Photomultiplier

Sample

ConcentricHemispherical

Analyser

InputSlit Output

Slit

Detector

Lens

Figure 8.6 Schematic layout for an ultra-high vacuum electron spectrometer that may betailored to incorporate various surface probes and specimen stages, providing facilities for bothAuger analysis and X-ray photon spectroscopy. Reprinted from G.C. Smith, Analysis ofnanometersized precipitates using atom probe techniques, Materials Characterization, 25,1. Copyright (1990), with permission from Elsevier.


8.1.4 Applications

The information available from XPS helps to fill an important gap in our ability tocharacterize surface chemistry. The reaction of solids with the environment takes placeat surfaces and interfaces, and the chemical sensitivity of XPS analysis make it a preferredchoice for studies of gas phase absorption and catalysis that involve partial coverage of lessthan a monolayer on the sample surface. The sensitivity of the technique is often sufficientto detect the energy differences that are associated with surface atoms of the same speciesthat differ only in their coordination numbers; for example, atoms sited on low indexsurfaces, close-packed ledge atoms or atoms at kink sites.

With the rapid development of electronic device technology, especially thin-filmdetectors and electro-optical systems, XPS is proving a powerful method for quantifyingchemical changes that occur at the surface. What XPS cannot do is to analyse the surfacecomposition of complex, multiphase samples or to provide useful, lateral, surface resolu-tion. To meet these requirements, Auger electron spectroscopy (AES) is usually moresuitable.

8.2 Auger Electron Spectroscopy

In terms of materials surface characterization, AES is probably the most useful of the threetechniques that we concentrate on in this chapter. Field emission sources for the electronexcitation beam in Auger systems have now been introduced. This has significantlyincreased the signal intensity and dramatically improved the spatial resolution for Augermapping of an elemental distribution. The combination of Auger mapping and controlledsputtering of the sample surface makes it possible to reconstruct a surface morphology andchemistry in depth. Nevertheless, there still remain problems associatedwith the roughenedtopology of a sputtered surface and blurring of composition gradients due to focused atomiccollision sequences during bombardment by the sputtering ions. Even so, sub-micrometrelateral resolution for Auger mapping is now routinely achieved.

A typical energy scheme for Auger nonradiative emission, that is, electron emissionwith no accompanying photon emission, is shown in Figure 8.7. Excitation of the emittingatommay be by either an incident energetic electron beam or by photon excitation (hencethe Auger peaks in the XPS spectra), but electron excitation is generally preferred. Byscanning the electron beam over the sample surface, a secondary electron image can berecorded, as in the scanning electron microscope. A region for Auger analysis is thenselected. The incident electron beammay be focused and scanned across the surface withsufficient lateral resolution of the Auger information for Auger imaging and mapping,providing the signal intensity is sufficient. This is always the case if the source of the highenergy electron probe is a field emission gun. The characteristic energy of the Augerelectrons is determined by three energies that are a function of the excitation levels in theatom: the energy of the excited state, for example EL3 in the case illustrated in Figure 8.7,the energy released by the M-electron that subsequently fills the vacant L-hole, EM1 , andthe energy that is absorbed in allowing the Auger electron to escape to the vacuum fromits original M-level, EM2;3 with an energy AEL3M 1M 2;3. It follows that (Figure 8.7),AEL3M1M2;3

¼ EL3�EM 1�EM 2;3 .


This is an introductory text and we have simplified the potential complexity of atomicabsorption and emission spectra. In any case, most physical methods of chemicalmicroanalysis seldom resolve the fine-energy structure of either the absorption or emissionprocesses. However, this is not the case in Auger spectroscopy, for which the well-resolvedfine structure of the Auger peaks provides an important tool for analysing the chemicalbonding through the quantitative interpretation of adjacent Auger peaks. The energies of theK-, L- and M-states are always well-separated. Not all quantum excitations may beobserved, but all the possible transitions, and practically all their relative transitionprobabilities, are now both well-known and well-understood. There is therefore littleambiguity in data interpretation, providing that the operator remains aware of the impor-tance of meticulous calibration and the need to refer to the professional literature.

Auger excitation is particularly complex, since three separate energy states contribute tothe energy of the Auger electron. This results in far more Auger energy signals for a givenmaterial than is possible, either in an XPS photoelectron spectrum or when analysing thecharacteristic X-ray signals in X-ray microanalysis. Auger electron excitation is actually incompetition with characteristic X-ray excitation, and the energy absorption edges that areused in EELS analysis with an electron beam �probe� are the adsorption edges that are

EVAC

AE = EL - EM - EM*

L3M1M2,3 3 1 2,3

K

L1

L2

L3

M1

M2,3

VEF

Figure 8.7 AnAuger electron originating from theM-level is emitted from copper as a result ofan atom that has been excited to the L-state decaying to the M-state. The energy of the Augerelectron is determined by the energy of the excited state, the energy released by the M-electronthat fills the vacant L-hole, and the energyabsorbed as the Auger electron escapes to the vacuumfrom its original M-level.


associated with both Auger excitation and the same characteristic X-ray excitation. As wehave already noted, the low atomic number constituents in a sample give weak characteristicX-ray signals,while their corresponding absorption edges are strongly defined. It follows thatthe Auger electron emission signal is strongest for these same low atomic number elements,so that the greatest strength ofAuger spectroscopy is its excellent sensitivity to changes in thechemical concentration and chemical binding of these low atomic number constituents.

8.2.1 Spatial Resolution and Depth Discrimination

The energy range of Auger electrons is the same as the range of energies observed forphotoelectrons, so that the Auger electrons can only escape from the same limited depth belowthe sample surface: typically two to five atomic layers. However, in Auger excitation byelectrons, it is possible to focus the incident electronbeam toa fineprobe and limit the region forAuger analysis to the small surface area immediately beneath the probe. As in the scanningelectron microscope, the total electron current in the probe depends sensitively on the probediameterandfalls rapidlyas thesizeof theprobe is reduced.As inscanningelectronmicroscopy,the ultimate limit on probe size for a given source is set by the electro-optical parameters:electron wavelength (the beam energy) and lens aberrations (the performance of the electro-magnetic lenses).A field emissionelectronsource increases theavailableprobecurrentby sometwo orders ofmagnitude.While the probe size depends on the electro-optical parameters of theinstrument, thevirtual sourceof secondaryelectrons (which includesAuger excitation)dependsonboth theprobe size and thevolume fromwhichback scattered electrons are emitted, since thebackscattered electrons near the surface also generate secondary electrons.

Because the Auger signal is derived from the first few atomic layers at the surface, it isstrongly dependent on the vacuum conditions in the instrument. The rate of contaminationunder the electron beam is a very important factor that may limit performance in AES, farmore so than it was for any of the previous methods of microstructural characterization thatwe have discussed. This includes highresolution lattice imaging in transmission electronmicroscopy, although the thin-film transmission electron microscope specimens are alsovery contamination-susceptible. As inXPS, ultra-high vacuum chambers are essential, withresidual pressures significantly better than 10�9 Torr. To achieve these conditions periodicbake-out of the assembly is required in order to degas both the specimens and thesurrounding surfaces. The optimum electron probe size is a compromise between thelimitations onmechanical tolerances imposed by a bakeable, ultra-high vacuum system, thestatistical limitations of signal detection that are determined by the probability of observinga given Auger electron transition, and the electron optics of probe formation.

With the exception of the ultra-high vacuum condition, these requirements are all similarto those that we have discussed before with respect to both EELS and thin-film X-raymicroanalysis. In the case of AES systems, electron beam energies of several keV aregenerally used, with probe diameters of the order of 50–100 nm. Spatial resolutions forAuger mapping of as little as 10 nm have been reported for high intensity LaB6 or fieldemission electron sources with high performance electron optics, but a more realisticresolution formost sample conditionswould be of the order of 0.1mm due toAuger electronsgenerated by back scattered electrons.

As in scanning electron microscopy, the total secondary electron signal that contains theAuger signal exceeds the incident beam current by at least two orders of magnitude, so that


there is no problem in recording a secondary electron scanning image of the surface at aspatial resolution equivalent to the spatial resolution of Auger analysis.

8.2.2 Recording and Presentation of Spectra

The accepted method for presenting Auger data graphically is as the differentiated signalrather than as the raw data in a collected energy spectrum. In Figure 8.8 we compare thesetwo graphical methods of presentation. The differentiated presentation effectively removesthe background signal and each Auger peak is defined by twin maxima and minima in thedifferentiated signal. These correspond to the maximum slopes of the leading and trailingedges of the Auger peak. In the differentiated signal the position of the Auger peak isdefined by the zero point between the twin peaks of the differentiated signal, while thedistance between the maxima and minima defines the peak width. These two parameters,peak position and peak width, are usually quite sufficient to identify the atomic transitionresponsible for the Auger signal, and hence the corresponding chemical species. This isoften achieved by a simple visual comparison with an atlas of published spectra.

Kinetic energy (eV)

10005000

Inte

nsit

y (a

.u.)

Differentiated

Direct

Figure 8.8 Auger electron spectrum from copper can be presented as either the energydependence of the collected signal intensity (the direct signal) or as the differential of thisdirect signal. The differential form is that usually published in the literature. Reprinted fromG.C.Smith, Analysis of nanometersized precipitates using atom probe techniques, MaterialsCharacterization, 25, 1. Copyright (1990), with permission from Elsevier.


Nevertheless, quantification of the Auger signal, as opposed to simple element identifi-cation, requires the same careful attention to the appropriate correction procedures as forX-raymicroanalysis. In that case, we saw that data collection of the complete characteristicX-ray signal in the region of a peak was necessary in order to derive the total, integratedcharacteristic X-ray intensities (Section 6.2.4).

8.2.3 Identification of Chemical Binding States

Since the Auger electrons, like the photoelectrons, originate from the outer shells of theatoms, they are sensitive to the chemical binding states of the atom. In general, two effectsare possible: either a simple energy shift of amainAuger peak that directly reflects a changein the binding energy of the atom, or a change in the energy loss structure on the low energyside of an Auger peak. An example is given in Figure 8.9 for an aluminium Auger peakoriginating either from aluminium metal or from the stable stoichiometric aluminiumoxide. This low energy loss structure, observed in Figure 8.9, is analogous to the energy lossnear-edge structure (ELNES) (Section 6.2.4), and can provide a wealth of additionalchemical information that is only beginning to be explored.

One difficulty inAuger fine structure analysis is to separate crystallographic effects fromchemical effects. For example, the channelling of high energy electrons in the incidentbeam down a low-index direction may affect both the intensity and the structure of theAuger peaks, independent of any chemical effects, and lead to an orientation dependence ofthe recorded spectra.

Figure 8.9 DifferentiatedAuger peak formetallic aluminiumcomparedwith that for aluminiumoxide, showing an 18eV shift in the peak position and extensive differences in the energy lossstructure. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probetechniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.


8.2.4 Quantitative Auger Analysis

As in the case of quantitative X-ray microanalysis, the role of quantitative analysis in bothXPS and AES is to derive data on the chemical composition from a comparison of themeasured intensities of specific spectral lines. The spectrum recorded from an unknownspecimen is compared with that from a known standard obtained under the sameexperimental conditions. For a simple binary system, and assuming a linear dependenceof composition on the measured intensity IA or IB, this implies an atom fraction of A, XA,that is given to a first approximation by:

XA ¼ IA=I0A

IA=I0AþIB=I

0B

� � ð8:3Þ

The superscript 0 refers to pure standards of A and B. In practice, the standard intensityvalues are now available in the literature as calibrated sensitivity factors S and the atomfraction of A for a multi-component system is approximated by:

XA ¼ IA=SAPni¼1 Ii=Sið Þ ð8:4Þ

This equationmay be satisfactory as a first approximation, but it omits corrections that arisefromdifferences in emission probability. The emission probabilities depend on compositionand are due, among other things, to changes with composition of the backscatter coefficientfor the incident electron beam (compare the corrections for quantitative X-ray microanaly-sis, Section 6.1.3.1).

For the simple binary alloy case, a matrix factor FAB can be included in the correctionequation:

XA ¼ FAB:IA=SAIB=SB

ð8:5Þ

where FAB is assumed to be given by:

FAAB XA!0ð Þ ¼ 1þRA EAð Þ

1þRB EAð Þ� �

aBaA

� �3=2

ð8:6Þ

or

FAAB XA!1ð Þ ¼ 1þRA EBð Þ

1þRB EBð Þ� �

aBaA

� �3=2

ð8:7Þ

where aA and aB are the atomic diameters of the two constituents and RA and RB are theirbackscatter coefficients. As in quantitativeX-raymicroanalysis, the problem is to devise aniteration procedure for a multi-component system. In the multi-component case, the matrixfactors are estimated from the initial, measured intensity ratios, by assuming that theintensity ratio gives an approximate, zero-order composition ratio. The calculation can thenbe reiterated. As in the case of X-ray microanalysis, the concentration data reported in theliterature very often omit details of their correction procedures. This is unfortunate, sincethe reader is then unable to judge the experimental significance and accuracy of thepublished data.


8.2.5 Depth Profiling

TheexcellentdepthdiscriminationinbothXPSandAES,that is theresolutionperpendicular tothe surface, makes them ideal tools for the investigation of thin-film devices, multilayeredstructuresanddistributed interfaces.This includes the analysisofmicrostructural features thatare only revealed after sputtering, andnot necessarily associatedwith theoriginal free surface.

Controlled sputtering of the surface by a chemically inert beam of energetic incident ionsis the preferred method for obtaining a concentration profile. Sputtering rates depend bothon the relative mass of the incident ion with respect to the sputtered species, and on theincident angle at which the ion beam impinges on the sample, as well as on the ion energyand the ion beam intensity. A Newtonian, billiard-ball model has been quite successful inpredicting the main features of the sputtering process. In such a Newtonian model, themaximum energy Emax that can be transferred to the sputtered atom at the surface by anincident ion with kinetic energy E0 is given by:

Emax

E0¼ 4m1m2

m1þm2ð Þ2 ð8:8Þ

wherem1 andm2 are the atomic masses of the struck atom and the incident ion. The primaryassumptionmade here is that the struck atoms react independently of their neighbours, so thatthe reaction time for the collisionwith the sputtered atom is small comparedwith the reactiontimeof its neighbouring atoms.Thiswill certainly be true for highvalues ofE0. For the specialcasem1¼m2, then Emax¼E0. The larger the mismatch in the atomic masses of the incidentand the sputtered particle, the smaller the value of Emax. The choice of argon as the preferredsputtering ion is dictated by its chemical inertness and its atomicmass, not too different fromthat of the major constituents in many engineering materials. The maximum value of energytransferred corresponds to a direct, knock-on collision in the forward direction. Thesputtering process is complicated by multiple collisions in which a primary knock-on eventejects a neighbouring atom. Careful calibration of the sputtering rate is important for depthprofiling by controlled sputtering.Multilayer calibration sampleswith known layer thicknessand composition can be prepared by several standard methods, for example by sputterdeposition. A low-angle, taper section through such a multilayer will show a series ofinterlayer interfaces thatmigrate steadily across the field of view as sputtering proceeds. Thisprovides a reliablemethod for calibrating both the sputtering rate and the thickness removed.

In analysing the depth profile data, it is necessary to allow for sputtering ion damage. Theincident ions not only sputter away the surface atoms, but also inject point defects into thesurface layers. The depth of the damage depends rather sensitively on the angle of incidenceof the sputtering ions, as well as on the nature of the point defects that are created. Focusedcollision sequences due to a primary collision, can result in the injection of defects to somedistance below the position of the primary event. Low angles of incidence for the sputteringbeam reduce the depth of damage, but do not affect radiation-induced diffusion ofconstituents that is associated with the high point defect concentrations generated duringion bombardment. In particular, sharp concentration gradients in multilayer samples areoften blurred during sputtering by inter-diffusion of the chemical constituents.

In depth profiling, Auger spectra are usually recorded at sputtering intervals that corre-spond to the removal of less than an atomic layer. Auger spectroscopy has also provedexceptionally successful in the study of grain boundary embrittlement associated with


impurities in steels and other alloys. The impurity segregant responsible for embrittlement isoften localized within a few atomic layers of the grain boundary surface. For Auger analysis,the embrittled material can be fractured in a jig, mounted under ultra-high vacuum, and thesample is then transferred directly to the Auger analysis chamber without breaking thevacuum.Emissionof occludedgas trapped in the sample occurs at themoment of fracture andmakes it undesirable to fracture the sample within the Auger chamber itself.

Even if an alloy is not embrittled by a segregant that is of interest, it is still possible to studygrain boundary segregation phenomena in considerable detail. The sample is first loadedcathodically with hydrogen, to induce hydrogen embrittlement, and then fractured in thevacuum chamber. An example of results obtained from a nickel alloy is shown in Figure 8.10.The differentiated Auger peaks for phosphorus, boron and carbon are clearly visible, butdecrease rapidly as successive layers of atoms adjacent to the fracture surface are sputteredaway. Accurate quantitative data on the depth distribution of segregation for these elements isobtained, even in the absence of an Auger image of their spatial distribution.

8.2.6 Auger Imaging

As in the case of X-ray mapping (Section 5.3.1), the primary factor limiting acquisition of aresolved Auger image is the low signal intensity and the poor counting statistics, especiallyin the absence of a field emission source for the incident electron beam. The same relationsnoted for X-raymapping apply, linking the number of counts needed per pixel point and thenumber of pixel points required to develop a useful image of the microstructural features.The human factor often plays a determining role, in addition to the mechanical and

Figure8.10 Auger spectrum fromabrittle grainboundary failure inanickel alloy.Carbon,boronand phosphorous are all enhanced at the boundary, indicating boundary segregation of theseelements. Reprinted fromG.C. Smith, Analysis of nanometersized precipitates using atom probetechniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.


electrical stability of the system, and 100 s is close to the limit of data acquisition times that aless dedicated user is prepared to accept for routine applications.

As in the case of characteristicX-raymicroanalysis, there are some further possibilities forimproving the spatial �resolution� in AES, especially in deriving a spectrum from a localizedarea. As inX-raymicroanalysis, thin films can provide information from the region beneath afocused probe that is less than20 nmindiameter, simplybecause, in the thin film, there is verylittle spreading and backscattering of the incident beam by inelastic scattering.

In Auger spectroscopy, the signal is coming from the first few atomic layers of thesample, but it may be generated either by primary inelastic collisions of the incident beam,or by secondary collisions of backscattered electrons. For incident beam energies of theorder of 10–20 kV, the area of emission for backscattered electrons reaches of the order of50–100 nm in diameter, while the backscatter coefficient R increases rapidly with atomicnumber (Section 6.1.3.1). It follows that, while most of the Auger signal from a low atomicnumber matrix will be localized within the diameter of the focused incident beam probe,that from a high atomic number material will be distributed over the diameter of thebackscattered electron distribution. This is illustrated schematically in Figure 8.11.

Increasing average atomic number

Increasingincident beam

energy

Sample

Primary AugerExcitation

Excitation by back-scattered electrons

Sample

SampleSample

Figure 8.11 Schematic drawing showing the influence of atomic number and incident beamenergy on primary Auger excitation and Auger excitation by backscattered electrons.


Providing the incident beam can be focused to a fine, nanometre-scale probe and thesource intensity improved by using a field emission gun, then good resolution should beeasier to obtain in an Auger signal than from a characteristic X-ray map, and very mucheasier in the case of a low atomic number matrix. If we set the upper limit of atomic numberat Z¼ 15, then this criterion will include silicon microelectronics technology, the alumini-um aircraft alloys and all polymers! An example of awell-resolvedAuger image is shown inFigure 8.12. Some care is required to avoid errors of image interpretation that are associatedwith the topography of the sample surface. In Figure 8.13, the edge of a surface step mayshadow the signal from the matrix collected by the detector, while electrons penetrating theedge of a feature on the surface may generate a spurious matrix signal in a nearby zone.These topological effects are analogous to those discussed earlier for topological contrast inthe scanning electron microscope (Sections 5.3.3.1 and 5.3.4.2).

8.3 Secondary-Ion Mass Spectrometry

We have seen how a beam of energetic ions incident on a surface will sputter the surfacelayers, displacing surface atoms into thevacuum, and how ion-beam sputtering (ionmilling)can be used to prepare thin films for transmission electron microscopy (Section 4.2.1.3).This same sputtering process is the basis for the focused ion beam (FIB, Section 5.4.4). Ion-beam sputtering is also used in ultra-high vacuum Auger spectroscopy, both to clean thespecimen surface and to remove successive atomic layers sequentially for depth profiling(Section 8.2.5). The next logical step is to analyse the sputtered ion signal, and this isprecisely the function of the ultra-high vacuum secondary-ion mass spectrometer.

In ion milling, for the preparation of thin-film electron microscope samples, thesputtering rate may be several micrometres per hour, corresponding to at least 50 atomlayers per minute. This is more than an order of magnitude too fast for either sputtercleaning of Auger specimens or for depth profiling in Auger analysis. These atomic-scale

Figure 8.12 Auger images of aluminium conduction lines on silicon. (Courtesy of J.B.Wachtman, Characterization of Materials, Butterworth-Heinemann).


sputtering processes require a controlled sputtering rate of no more than a few layers perminute. The sputtering rates of interest for secondary-ion mass spectrometry, (SIMS)analysis are even lower, reflecting the extreme sensitivity of the SIMS ion detection system,which is capable of measuring atomic concentrations in the parts per billion (10�9) range.

The sputtering ions in a SIMS system are usually positively charged, inert Arþ gas ions,but for some purposes a chemically active sputtering ion is desirable: either Csþ or Ga2þ topromote sputtering of electronegative elements, and O2

þ to promote sputtering of electro-positive constituents. The energies of the sputtering ions are typically 1–30 kV. The initialtranslational energy distribution of the ions sputtered from the sample surface depends ontheir complexity and can extend from a few eV up to as high as 100 eV (Figure 8.14).

To ensuremaximumconversion of neutral sputtered atoms to ions, the region of the samplesurface may be bathed in a flux of low-energy electrons trapped in a cylindrically symmetricmagnetic field. The sputtering yield in SIMS depends on a wide range of factors. Theseinclude the ionization potentials (positive ions) or electron affinities (negative ions), themassof the impinging ion, the chemical activity of the target species, the incident angle of the beamon the surface, the take-off angle for the sputtered ions, and, not least, the composition of thetarget. The yields vary overmany orders ofmagnitude and trace elements in the sample oftenhave a major influence on the yields from other constituents. As would be expected, and as ageneral rule, the yield for positive ions decreases as the ionization potential increases, whilethe yield for negative ions increases with the electron affinity.

Figure 8.13 Schematic influence of �shadowing�, beam-penetration and backscattering on anAuger line-scan from silicon that has been partially covered by an aluminium conduction line.Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probetechniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.


The rather sensitive dependence of the yield for a particular species on the composition ofthe target makes quantitative analysis completely dependent on accurate calibrationstandards. For example, the yield of copper ions from an aluminium alloy containing2% of copper actually exceeds the copper ion yield from a pure copper target that has beenexposed to the same sputtering and ion collection conditions. Thewide variations in the ionyield from pure targets, seen as a function of their atomic number, is related to the cyclicvariations in ionization energy across the periodic table of the elements (Figure 8.15). Inaddition to the sensitivity of the yield for a particular species to the target composition, theimpinging ion species can also have a dramatic effect. In comparing oxygen ion sputteringwith caesium ion sputtering, differences in the yield of four orders of magnitude have beenmeasured. This has been explained by the effect of the electron affinity of implanted oxygenin inhibiting ionization of the target, as opposed to the effect of implanted caesium inreducing the work function at the surface and therefore enhancing ionization.

8.3.1 Sensitivity and Resolution

The ions sputtered from the surface of the target specimen have low kinetic energies andmust be accelerated into the spectrometer,where an electrostatic analyser is used to limit theenergy spread of the collected ion signal before it is admitted to the mass analyser(spectrometer). Three types of mass analyser have been used.

Quadrupole analysers have good mass resolution but are limited to the lighter ions [lessthan 103 amu (atomic mass units)]. The sensitivity is correspondingly low.Magnetic sectoranalysers have rather poor mass resolution but can analyse the full range of particle massesand are far more sensitive. Both of these analysers are sequential instruments, in which the

Energy (V)

Rel

ativ

e in

tens

ity

2015105

M3+

M2+

M1+

Figure 8.14 The translational energy of the sputtered species in SIMS may range up to 100 eV.Molecular sputtered ions (M2, M3) have lower translational energies because energy istransferred to internal, molecular vibrational modes.


signal must be scanned across a detector slit at the entrance to the mass analyser, in order torecord each mass peak separately. For this reason, they have sometimes been replaced bytime-of-flight (ToF) mass spectrometers that were described previously for atom probetomography (Section 7.3).

The ToF spectrometer records the time taken for a sputtered ion of given kinetic energy toreach the detector from the target. The mass resolution can be as good as a quadrupoleanalyser and there is no limitation on the detectablemass range. The sensitivity is excellent,and most particles entering the ToF spectrometer can be detected. Moreover the systemoperates in parallel, and not sequentially, so that ions of all masses that are generated by ashort pulse of incident ions can be detected simultaneously. Indeed, the major disadvantageis this need to pulse the signal entering the spectrometer flight tube. Pulsing the incident ionbeam, by pulsing a magnetic field that deflects the incident ion beam off-axis, is just oneoption. Electrostatic deflection of the sputtered ions outside the detector entrance slit is aviable alternative. Since the flight times in the spectrometer require electronics with sub-nanosecond resolution, care must be taken to ensure that parasitic capacitance effects areminimized. Electrostatic pulsing of the sputtered ions may therefore be preferred, eventhough this means that a large fraction of the sputtered ion signal is lost.

A second disadvantage of the ToF spectrometer is fundamental to the technique: sinceall the sputtered ions are accelerated through the same voltage, ions of higher chargeare recorded at shorter times This means that the spectrum is recorded as a function of the

1009080706050403020100

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Atomic number Z

I/Z

(eV

)Calculation: and

Experiment:

Figure 8.15 Measured variations in the ion yield from pure targets as a function of the atomicnumber. (Courtesy of L.C. Feldman and J.W. Mayer, Fundamentals of Surface and Thin FilmAnalysis, Elsevier Science Publishing Co.).


mass-to-charge ratio rather than the mass itself. Inevitably there are occasional�coincidences� in the spectra and hence some ambiguities of interpretation. However, forelements with more than one isotope, most of these ambiguities can be resolved, since theToF spectrometer is quite capable of resolving to better than 0.1 amu, and knowledge of theisotopic abundance for the elements is usually sufficient to remove the ambiguity.

An example of a ToF spectrum is given in Figure 8.16, in which a silicon wafer wasexamined before and after an organic cleaning treatment. The peak at m/ne¼ 18 in thespectrum from the wafer before cleaning is probably associated with H2O

þ ions. Undersuitable conditions, the sensitivity of the SIMS technique enables surface and sub-surfaceimpurities to be detected at the ppb (parts per billion) level. Accurately calibrated data maybe interpreted quantitatively to between 10 ppb and 100 ppb. This is far better than can beachieved using the alternative techniques of microanalysis that we have discussed.

8.3.2 Calibration and Quantitative Analysis

As noted previously, the only satisfactory method for converting the relative intensities ofthe mass peaks in SIMS into mass concentrations is by recording data from knowncalibration samples. This can be done accurately by injecting high-energy ions of the

Figure 8.16 ToF spectra from a silicon wafer examined before (a) and after (b) an organiccleaning treatment. (Reproduced by permission of R.D. Cormia, Advanced Materials &Processes, 12, 18 (1992) ASM International).


impurity of interest into a pure sample of the matrix. The sub-surface distribution of theimplanted impurity can be calculated fromknowledge of the ionicmass, the incident energyand thematrix density. This is done usingMonte-Carlo computer simulations. These can becompared directly with changes in the SIMS signal for the impurity while the surface of theion-implanted standard is eroded by sputtering. Such carefully prepared calibration curvesshould be satisfactory over several orders of magnitude in concentration, with a limit ofdetection of about 100 ppb.

Quantitative SIMS analysis has proved successful despite the serious calibrationproblems, and is usually based on the use of a calibrated parameter termed the relativesensitivity factor (RSF):

IRCR

¼ RSFE ·IECE

ð8:9Þ

EandR refer to the element to be analysed and a reference element, respectively, while I andC are the measured secondary-ion intensities and the true atomic concentrations of the twospecies concerned. It is usual to choose the major component of the matrix as the referenceelement and, for trace element analysis, it is generally assumed that the matrix compositionremains constant. Replacing the suffix R by the suffix M (for matrix) and rearranging theabove equation we can write:

CE ¼ CMRSFE ·IEIM

ð8:10Þ

For trace element analysis, CM is constant, and the ratio of the measured ion intensities canbe simply multiplied by a value of RSF taken from tabulated literature data. This treatmentis also satisfactory for trace element analysis of semiconductors, based on stoichiometriccompounds such as GaAs or InSb, taking either of the major constituents as the referenceelement.

Quantitative analysis of trace impurity, dopant or alloy concentrations up to a maximumof 1 or 2%, can be satisfactorily treated by the use of calibrated standards, but this is notpossible for higher concentrations, since these are especially susceptible to changes in ionyield as a function of concentration. However, it is precisely in this range of compositionthat other techniques of surface analysis, especially Auger spectroscopy (Section 8.2), arecapable of providing data that is both accurate and unambiguous.

8.3.3 SIMS Imaging

The ion beam impinging on the target can be focused to a probe that has a diameter of as littleas 0.1mm and this probemay be rastered across the target surface, as in the focused ion beam(FIB) microscope. The sputtered ions of the different target species can then be acceleratedand detected to form an image of the target in which only ions of a specific mass (or mass-to-charge ratio) will be present. In quadrupole and magnetic spectrometer systems, once thesignal has been electrostatically filtered, an aperture behind the mass analyser, placed in theback-focal plane of a collecting �lens� can be used to select those ions having the requiredmass. The available spatial resolution is poor, and SIMS imaging has limited applications. Asin scanning electronmicroscopy, the final image that maps the distribution of the elements isviewed on amonitor with the same time-base as the scanning coils of the ion beam probe that


is focused on the target sample surface. Although the lateral resolution is very limited(certainly no better than 1mm), the sensitivity is high, while successive scans can revealchanges in the sample with a depth resolution of better than one monolayer. As in Augerspectroscopy, depth profiling is often used to characterize layered structures, but blurringof thecomposition profile by radiation damage effects is a major problem.

A more rewarding technique is to compromise between the �point� analysis of a staticSIMS analysis and the poor spatial resolution of a dynamic SIMS image byusing a line scan.As in X-raymicroanalysis (Section 6.3.2.1), line analysis gives far better counting statisticsandmay provide a satisfactory insight into the relationship between themicrochemistry andthe microstructural morphology, especially when combined with an alternative method ofimaging the sample, such as the FIB. The range of concentrations that may be detected bySIMS covers some four orders of magnitude, while the depth resolution is better than 1 nm.No competitive technique presently available can achieve this same combination of depthresolution and sensitivity.

Finally, the development of high resolution SIMS, based on FIB technology, is still in itsinfancy. AGaþ ion beam, extracted from a liquid Ga source, provides an exceptionally fineion-probe that could, in principle, have a far better analytical image resolution than iscurrently available in SIMS, perhaps as low as 50 nm.

Summary

The chemical composition is the third class of information, following the crystal structureand microstructural morphology, that is required to complete the microstructural charac-terization of a material. However, the sensitivity of many material properties to the surfacecondition of the sample has led to a more specific requirement, for an assessment of thesurface chemistry (the composition of the first few atomic layers of the sample).

Examples of the importance of surface chemistry range from the segregation ofimpurities on a brittle fracture surface, to the composition of catalytically active layerson an inert substrate, and the chemistry of thin-film, semiconductor components or electro-optical detector devices. The chemical analysis of such surfaces is generally beyond thereach of the microanalytical methods that we have discussed earlier in Chapter 6. Forexample, a high-energy electron probe generates a characteristic X-ray signal that comesfrom a sample thickness of the order of 1 mm or even more, while the electron energy lossspectrum detected in a transmission electron microscope requires a sample thickness of theorder of the extinction distance (typically about 10 nm).

Of the wide range of physical phenomena that are used to derive chemical informationfrom a sample, only three result in a signal which is sufficiently localized at the surface toensure that only the surface layers of atoms contribute to the signal. These three methodsare X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES) andsecondary-ion mass spectrometry (SIMS).

In XPS a secondary photoelectron signal is excited by an incident X-ray beam. Since lowenergy, secondary electrons can only escape from the sample if they are created in the firstfew surface layers of atoms, only these first atomic layers are sampled by the detectedsignal. The energy of the photoelectrons is sensitive to the local work function and thebinding energy of the photoelectron. As a result, the technique is able to distinguish


different chemical binding states, for example those associated with the different valencystates of a polyvalent cation. Although XPS has no useful lateral resolution, the sensitivityof the technique to the chemistry of a single surface layer of atoms and the ability to detectchanges associated with the early stages of adsorption, corresponding to just partialcoverage of the surface by an adsorbate, combine to make this a very useful tool formaterials characterization in surface science.

AES differs fromXPS in that the relaxation process for an atom in an excited energy stateoccurs by the emission of an electron whose kinetic energy is determined by the differencesin energy for specific energy states of the excited atom. The energy of the Auger electron istherefore independent of the electron beam or X-ray photon energy used to excite theemission, unlike the energy of the photoelectrons emitted inXPS. Although the range of theAuger electrons escaping from the sample is similar to that of photoelectrons (and bothoriginate from a depth of two to five atoms beneath the surface), the lateral resolutionachievable inAES is limited only by the counting statistics for theAuger electron signal. Asin microanalysis using characteristic X-radiation, the cross-section for Auger excitation issmall, so that large electron beam currents are needed to excite an Auger signal and provideadequate counting statistics. A conventional electron source requires a minimum electronbeam diameter approaching 1 mm if adequate Auger counting statistics are to be recorded.Far better spatial resolution forAuger analysis, perhaps as low as 10 nm, is achievable using afield emission source for the electron beam.

Auger spectra are conventionally presented as a differentiated intensity signal, in whichthe peak position is determined by the crossover of the differentiated signal from a givenAuger peak. The shape of the differentiated peak is sensitive to the chemical state of theexcited atom, so that the Auger spectrum can be an important diagnostic tool for analysingthe nature of the chemical bonding at the surface of the sample. QuantitativeAuger analysisis possible, based on a comparison of measured, integrated intensities relative to knownstandards. In general, quantitative Auger analysis is not very accurate, partly because of thedifficulties of calibration but also because the collected signal is usually from a compara-tively large, thin area, so that the composition may well vary within the area analysed.

In practice, AES is often combined with ion sputtering of the surface in order to removesuccessive layers of atoms, so that the Auger spectrum is then recorded as a function of thesputtered depth. The application of such depth profiling to research and development forthin-film devices is well-established. Quantitative depth profiling is not easy, primarilybecause it is difficult to ensure a uniform sputtering rate over the area to be analysed. Depthprofiling is a standard procedure in the development of electronic and electro-opticaldevices, and has also proved extremely useful for the study of mechanical embrittlementassociated with trace impurity segregation in structural materials.

A high-intensity, field emission electron source in an Auger system can improve thespatial resolution to of the order of 100 nm or better. It is then possible to record an Augerimage by scanning the incident probe beam over the surface and collecting the Augerelectrons characteristic of any specific element of interest. Statistically significant imagesrequire large numbers of Auger electrons, usually between 100 and 1000 per pixel, and itfollows that the signal intensity must be adequate if Auger imaging is to be justified.

SIMS is far more mass-sensitive than either XPS or AES, and it is usually possibleto detect trace impurities or dopants present at concentrations of the order of 10 ppb(parts per billion, 10�9). However, the ions detected are those sputtered from the surface, so


that this technique is destructive, and removes successive atomic layers during the analysis.The SIMS technique is difficult to calibrate because the yield of a particular mass speciesdepends sensitively on the concentrations of other elements present in the sample. In addition,since it is either the mass or the mass per unit charge that is determined, and most elementshave more than one isotope and, often, more than one valency, a knowledge of the isotopicabundance and ionic charge is usually necessary to eliminate ambiguities. It is difficult tolocalize the signal to better than 1mm, so that, although it is possible to collect secondary-ionimages, in SIMS this is usually at the cost of losing the depth resolution, and alwayswith ratherpoor lateral resolution. Even so, high resolution SIMS has been developed using highlyfocused ion beams scanned across the sample in a raster. Images with a spatial resolution ofapproximately 50 nm have been reported, although not for commercial instrumentation.

It is important to recognize the major differences between the three surface analysistechniques discussed in this chapter and the methods of microanalysis discussed inChapter 6. Neither X-ray microanalysis nor EELS are capable of providing analyticalinformation that is localized to the immediate vicinity of a solid surface (the first few atomiclayers). Although themethods ofmicroanalysis discussed inChapter 6 are influenced by thepresence of thin surface films and contamination (which are therefore to be avoided), theprimary objective of microanalysis is to determine bulk concentration on the microscale.The methods of surface analysis, introduced in this chapter, are tailored to determiningchemical composition and composition changes that are present at, and immediatelyadjacent to, the surface itself.

Bibliography

1. J.M. Chabala, K.K. Soni, J. Li, K.L. Gavrilov, and R. Levi-Setti, High ResolutionChemical Imaging with Scanning Ion Probe SIMS. Int. J. Mass Spectrom. IonProcesses, 143, 191–212, 1995.

2. L.C. Feldman, and J.W. Mayer, Fundamentals of Surface and Thin Film Analysis,Elsevier Science Publishing Co., Inc, London, 1986.

3. J.B. Wachtman, Characterization of Materials, Butterworth-Heinemann, London,1993.

4. J.M. Walls, and R.S. Smith, (eds), Surface Science Techniques, Elsevier Science Ltd,Oxford, 1994.

5. A. Benninghoven, E.G. Rudenauer, and H.W. Werner, Secondary Ion Mass Spectro-scopy, John Wiley & Sons, Ltd, New York, 1987.

Worked Examples

In order to demonstrate some of the techniques discussed in this chapter, wewill focus on thecharacterization of samples of aluminium on a Ti/TiN/SiO2/Si stack prepared by chemicalvapour deposition (CVD).We begin withAuger spectroscopy, which can be used to examinethe surface of the aluminium, as well as the relative thickness of the different layers.

Figure 8.17 shows an uncalibrated Auger electron intensity sputter profile, taken throughthe aluminium layer, the TiN film and terminating in the titanium substrate. The spectrumwas taken from a specimen after deposition of a very thin, discontinuous aluminium film


(islands of aluminium on the TiN). A relatively strong oxygen signal is detected for shortsputtering times (less than 2min). This is due to the native oxide that had formed on thealuminiumwhen it was exposed to air. In addition, a surface layer of carbon is evident, againa contaminant associatedwith exposure to air. Both the oxygen and the carbon signal drop tobackground levels after the initial sputtering, although the oxygen signal persists to a deeperlevel, presumably because of oxygen adsorption on TiN regions which were not covered byaluminium.

Sputter time (min)

Aug

er s

igna

l (pe

ak-t

o-pe

ak)

161412108642

0

2

4

6

C

N

Ti

N+Ti

Figure 8.17 Auger signal sputter profile from a multilayer specimen of aluminium depositedon a TiN layer on a titanium substrate. The spectrum was taken after deposition of a very thin,discontinuous aluminium film (that is, isolated islands of aluminium).

Sputter time (min)

Ato

mic

con

cent

rati

on (

%)

1614121086420

20

40

C

Ti

N

Figure 8.18 The same Auger signal profile as Figure 8.17, but now calibrated for atomicconcentration.


Signals from titanium and nitrogen are also evident, but its impossible to differentiatebetween TiN and titanium, since the Auger peaks overlap and the TiN and titanium films arevery thin. Figure 8.18 is a calibrated Auger profile of the same specimen. An increase in thetitaniumandnitrogen signalswith depth is clearly seenwhile the aluminium signal decreases.

We can also useAuger spectroscopy to differentiate between titaniummetal and titaniumcations in TiN. Figure 8.19 shows standardAuger signals from the various atomic species in

Energy (eV)Energy (eV)

450400350300450400350300

Aug

er s

igna

lTi

Ti1

I385L

Ti2

I420L Ti

N(in Si3N4)

KL23L23

L3M23M23

L3M23V

L3VV

Figure 8.19 Standard Auger signals from the various atomic species present in the multilayersample.

Sputtering time (min)

Ato

mic

con

cent

rati

on (

%)

40363228242016128400

10

20

30

40

50

60

70

80

90

100

C

C

N

Ti

Figure 8.20 Auger signal sputter profile from a similar specimen to Figure 8.17, but with amuch thicker, continuous, aluminium film.


our sample. A clear difference is evident in both the peak position and the peak shape for theTi metal, and the TiN and TiOx ceramics.

Figure 8.20 shows an Auger sputter profile from the same type of specimen, but this timewith a much thicker, continuous aluminium film. A definite drop in the oxygen signal isvisible after sputtering through the initial surface layers. (Note that the increase in theoxygen signal after about 24 min of sputtering corresponds to an SiO2 layer beneath the Al/TiN/Ti sandwich.) In addition to verifying the purity and concentration depth profile of ourfilms, we can also determine the thickness of the individual layers, by calibrating thesputtering rate. This is relatively easy, since we already have transmission electronmicroscopy (TEM) results for these materials (Chapter 4), so we need only compareFigure 8.20 with a cross-section TEM micrograph of the same specimen (Figure 8.21).

Figure 8.21 TEM cross-section micrograph of the specimen from which the Auger sputterprofiles in Figure 8.20wereobtained. Thismicrograph canbeused to calibrate the sputtering rate.

Binding energy (eV)

88

N(E

)/E

687072747678808284860

1

2

3

4

5

6

7

8

9

10

Al2O3

Al

Figure 8.22 XPS spectrum from the surface of the aluminium specimen before sputtering,showing the presence of Al2O3 in addition to metallic aluminium.


Finally we can confirm the oxidation of the multilayer specimen surface using XPS.Figure 8.22 is an XPS spectrum taken before sputtering. A strong signal from Al2O3 (74–75 eV) is clearly evident, in addition to the metallic Al 2p peak (72 eV). There is also clearXPS evidence for surface contamination by carbon, in anXPS spectrum that was taken overa wider energy range, but at a slightly lower energy resolution (Figure 8.23). The carbon 1sXPS peak (285 eV) confirms the previous Auger spectroscopy findings. After sputtering for5 min, the superficial Al2O3 layer is completely removed (Figure 8.24).

Binding energy (eV)

N(E

)/E

0

1

2

3

4

5

6

7

8

9

10

0200400600800100012001400

SiLMM

OK

LL

OK

LL

O1s

C1s

Ar2

p3

Al2

sSi

2pA

l2p

O2s

, Ar3

s

Si2s

Figure 8.23 XPS spectrum taken over a larger energy range than that in Figure 8.22, showingthe presence of carbon contamination on the surface of the sample.

Binding energy (eV)

88

N(E

)/E

687072747678808284860

1

2

3

4

5

6

7

8

9

10

Al2p

Al

Figure 8.24 XPS spectrum After sputtering for 5min. The Al2O3 signal is no longer visible.


Problems

8.1. How �thick� is the �free surface� of a solid? Justify your answer! Is there a distinctionbetween the physical thickness, reflecting the electronic structure, and the chemicalthickness, reflecting composition variations?

8.2. Give three examples for which you would expect that surface analysis of a solidsample would be more significant than microanalysis of a polished section.

8.3. Analytical information can be obtained from a wide range of signals. Whatproperties of photoelectrons, Auger electrons and secondary ions make them thepreferred signals for surface analysis?

8.4. Compare the limits of detection for chemical analysis using photoelectrons, Augerelectrons and secondary ions. What factors restrict our confidence in a quantitativeanalysis of the composition for each method?

8.5. An XPS signal contains information on the chemical binding state of the elementsdetected. Suggest how this type of information might be useful in the study of ametallurgical failure. Distinguish between a purely mechanical failure and oneassociated with environmental attack.

8.6. Auger spectroscopy is often used for depth profiling in thin-film devices. Discussthe sources of error involved in plotting concentrations determined by Augerspectroscopy as a function of sputtered depth. (Consider especially the topographyof the sputtered surface, �knock-on� radiation damage, and the quantitative calibra-tion of the Auger signal.)

8.7. Argon (atomic weight 40) is usually used to sputter-clean surfaces for Augerspectroscopy and to remove successive atomic layers in depth profiling. Comparethe expected sputtering efficiency for aluminium, iron and tungsten (atomicweights 27, 56 and 184) by energetic argon ions and suggest some possiblestrategies that might be used to improve the reliability of depth profiling.

8.8. Secondary-ion mass spectrometry frequently gives a larger intensity signalfor sputtered copper ions in an age-hardened copper alloy than from a purecopper, calibration standard. Suggest some possible physical reasons for thisobservation.

8.9. What are the advantages and disadvantages of a time-of-flightmass spectrometer, asopposed to a quadrupole or magnetic sector analyser, for secondary ions?

8.10. The spatial resolution in conventional secondary-ion mass spectrometry (SIMS) islimited to about 10 mm, but in principle this could be improved by taking a line-scanrather than a TV-raster image. Assuming that the only limitation is in the countingstatistics, estimate the expected resolution of a SIMS line-scan.

8.11. Multilayer films are often sectioned at an inclined angle to the surface prior toAugeranalysis. What are the experimental advantages of such a wedge-shaped specimengeometry?


8.12. To a first approximation, an Auger electron intensity peak can be described by aGaussian intensity distribution about an energy E0 according to:

NðeÞ ¼ N0

ð2ps2Þ1=2exp

�e2

2s2

� �ð8:11Þ

where e¼E-E0.

X

Figure 8.25 Schematic drawing of alternative methods for Auger analysis of a thin film on asubstrate.


(a) Find an expression for the total number of electrons counted in an Augerpeak.

(b) Derive an expression for the peak-to-peak height in a dN/de plot.(c) Sketch a graph of both dN/de and N(e) on the same axis.(d) Assuming that the surface concentration of a specific element is propor-

tional to the number of Auger electrons having an energy E0, when might itbe appropriate to use the differentiated signal, peak-to-peak height toestimate a surface concentration?

8.13. A thin film of thickness x was deposited on a single crystal, silicon substrate.Figure 8.25 shows schematically three ways in which the film could be character-ized by Auger spectroscopy, using an incident electron beam 200 nm in diameter:auger depth profiling using Ar ion sputtering perpendicular to the surface; cleavageof the wafer followed by an Auger line-scan of the cross-section (without sputter-ing); and mechanical polishing at a wedge angle of 10� followed by an Auger line-scan analysis (without sputtering).

(a)What do you think is theminimumfilm thickness that could bemeasured byeach of these three methods?

(b) List the advantages and disadvantages for each of the three methods.(c) Suggest an experimental methodology that would optimize the accuracy of

an investigation of this sample by auger electron spectroscopy.

8.14. A thin film of element Awas deposited on a substrate of element B. Auger analysiswas conducted without sputtering and the Auger signal intensity from element Bwasmeasured. Themean free path of theseAuger electronswas lB. For an uncoatedsubstrate, the Auger electron intensity for this peak was I0.

(a) Assuming that the layer of A has a uniform thickness t, what do you expectthe intensity of the signal to be as a function of the thickness of A? Sketch theexpected graph with a thickness scale.

(b) Assuming that the element A formed discontinuous islands on the substrateduring the deposition process, sketch a graph showing theAuger signal fromB as a function of coverage by A.

(c) Assume you have received a substrate of element B, upon which wasdeposited a thin film of A, but that you do not know if themorphology of thefilm of A is continuous or discontinuous. How could you determine this byAuger electron spectroscopy?


9

Quantitative and TomographicAnalysis of Microstructure

In Chapters 6 and 8 we outlined some of the methods available to convert a variety ofspectroscopic data into quantitative estimates of chemical composition, while in Chapter 7we saw that atom probe tomography was capable of combining both three-dimensionalchemical and morphological information with a resolution on the sub-nanometre scale. InChapter 1 we also noted that parameters derived frommicrostructural observations, such asgrain size, particle size and the volume fraction of a second phase, were directly related tostructure-sensitive engineering properties. However, we have not yet attempted to quantifythe three-dimensional microstructural parameters that can be extracted from routine, two-dimensional projection-image data, nor have we described the methods that are availablefor us to estimate numerical values for the three-dimensional parameters that can be derivedfrom measurements taken from two-dimensional images.

In part, the difference in the treatment of chemical and morphological information is aconsequence of the way in which we usually regard information associated with micro-structural chemistry and microstructural morphology. An image of the microstructure is,for many purposes, seen as an end in itself (and �worth a thousand words�, as Confucius issaid to have expressed it). The image is often discussed as though it were itself the object –pearlite, a eutectic or dendrites, to give but a few examples of terms used to describe bothimage features and microstructural constituents. This is not the case with spectroscopicdata, and it is only in the event that the spectrum is used to identify the simple presence of anelement (provide a �fingerprint� of the element), irrespective of its concentration in thematrix, are we likely to be satisfied with the mere identification of a characteristic intensitypeak in the spectrum.

Crystallographic data are also frequently used to identify the presence of a phase,irrespective of its volume fraction in the sample (the phase concentration) or its distributionin space (the phase morphology). Any relation between the crystal orientations and thegeometrical axes of an engineering component, corresponding to the existence of preferredorientation, is then seen as a topic requiring special investigation. The X-ray determinationof particle size or grain size for powders or crystallites, and the presence of microstresses,


which can both be determined from measurements of X-ray peak width and peak shift, arealso techniques of quantification that, while available when required, are generally seen assubsidiary to the main �business� of diffraction studies, generally considered to be phaseidentification and structure determination.

In this chapter we concentrate on the quantification of microstructural morphology andexamine the options available for three-dimensional reconstruction of information derivedfrom serial sections. A general term often used for the computer-generated imaging ofthree-dimensional data sets is tomography, although this is a partial misnomer, since it isderived from theGreek andmeans a drawing (graphos) of a section (tomo). Inmodern usageit refers to the two-dimensional presentation of the three-dimensional data using a computerprogram that allows the projection to be rotated about any axis, usually in defined stepsof the rotation angle, or to beviewed as a selected two-dimentional section taken through thethree-dimentional data set. The earliest work in this area was for the study of anatomy (themorphology of human and animal physiology) and histology (themorphology of biologicaltissues or isolated, single cells).

The study of the spatial relationships between objects is termed stereology, and the basicstereological concepts have been known and applied for over a century, primarily in theverydiverse fields of medicine (that is, anatomy and its microstructural counterpart, histology)and geology (again, both on the macroscale, as the structure of the earth�s crust, and on themicroscale associated withmineralogy). The availability of ever-increasing computationalspeed andmemory has led to the development of awide range of software programs that aredevoted to one or other of the twokey issues for the quantitative interpretation of image data,namely image data processing and image data analysis. Image data processing is concernedwith �correcting� the raw, digitized data by removing random background, enhancing orreducing contrast, and averaging the measured data over neighbouring pixels (datasmoothing). Image data analysis is concerned with extracting quantitative measurementsof selected parameters from the image and combining data sets, for example, those fromsuccessive serial sections. Image data analysis is the principle topic of this chapter. Weassume that the recorded image has already been �processed� to maximize the informationcontent andminimize background noise and artifacts, andwe focus our discussion solely onimage data analysis.

9.1 Basic Stereological Concepts

The quantitative analysis of two-dimensional image data is verymuch a �one step back, twosteps forward� process. A section is first prepared from a three-dimensional object andimaged in two dimensions (�one step back�). These image data (possibly recorded as afunction of time) are then analysed. The expectation is that the image analysiswill result in aquantitative estimate of microstructural parameters that are relevant to the three-dimen-sional object, that is, the �two steps forward�. In some cases, two-dimensional imageanalysis of a microstructural parameter can be unambiguous, and we refer to suchparameters as being �accessible� from the two-dimensional image, although with varyingdegrees of accuracy. In other cases we can only estimate the three-dimensional microstruc-tural parameters from two-dimensional data on the basis of an assumed model for themicrostructural features. Suchmicrostructural parameters are then said to be �inaccessible�.


Before we discuss these complications in any detail, we first need to examine some of themicrostructural factors that may affect stereological analysis.

9.1.1 Isotropy and Anisotropy

Microstructural anisotropy can take two forms, since the term anisotropy is applied to bothanisotropy of the morphology of visible microstructural features and anisotropy of thecrystallographic orientation of the crystalline phases that are present in the microstructure.In both cases, the reference coordinates are the geometry of the engineering component thatis being studied. If the microstructure is isotropic, with respect to both microstructuralmorphology and crystallographic orientation, then both the diffraction spectra and themicrostructural image will be independent of the plane of the sample section that has beentaken through the component or of any selected direction within the section plane.

Crystallographic anisotropy is best termed preferred orientation, in order to avoidconfusion between morphological anisotropy and crystalline anisotropy. Crystalline an-isotropy is associated with the orientation dependence of the physical properties of thecrystal lattice, for example the variation of the elastic tensile modulus measured along thedifferent crystal directions. Preferred orientation is usually determined from an analysis ofthe orientation dependence of the diffraction peak intensities that have been measured indifferent spatial directions within the coordinate system of the sample (Section 2.4).

Morphological anisotropy implies that one or more microstructural parameters dependon the orientation of the direction or plane with respect to which that parameter has beenmeasured. An obvious example is the elongation of the grains in a metal bar due to plasticelongation of the specimen. In this case, the microstructural change in the ratio of the grainlength to grain width (the aspect ratio) will depend both on the total elongation of the barand the plane of the selectedmicrostructural section. If the plane of the sample is defined bythe direction of tensile elongation and its normal, then the aspect ratio observed for thegrains seen in the plane of the section will be a maximum, while the distribution of thisaspect ratio (maximum andminimum values of the aspect ratio in this longitudinal section)will be related to local, crystallographically determined, variations in ductility andstereological constraints that are imposed by neighbouring grains in the material.

The grain aspect ratio in a given section of a rolledmetal sheet (Figure 9.1)will depend onthemechanical processing history. If the sheet was rolled as a continuous strip, in a series ofindividual passes, then the grains will all be elongated along the direction of rolling, but ifthe reduction in thickness is the result of rolling by equal amounts in two directions at rightangles (cross-rolling), then the grains will be flattened in the plane of the sheet, rather thanelongated. In general, at least two sections will be needed to characterize morphologicalanisotropy in a rolled sheet (taken perpendicular and parallel to the plane of the sheet), sothat the sample sections contain all three of the principle directions in the product. These areusually termed the longitudinal, transverse and through-thickness directions.

A similar example is given by the distribution of nonmetallic inclusions in a metal sheet,but in this case it is the distribution of the inclusions, which constitute a second phase in thesystem, that is anisotropic, and not only the shape of the inclusions (Figure 9.2).

Most composite materials exploit mesostructural anisotropy in order to optimize theirengineering properties and to minimize the weight or the dimensions of a structuralcomponent in an engineering system. Composite lay-ups of resin-bonded sheets of fibre can

Quantitative and Tomographic Analysis 459

be oriented with the fibres aligned at predetermined angles in a variety of geometries orarchitectures, in order to obtain the desired mechanical properties. These properties areusually the strength and stiffness of the composite component. In short-fibre reinforcedcomposites the lengths of fibre are randomly distributed in a principal plane of the product,

Figure 9.2 The same copper sheet, imaged unetched but after annealing. Oxide inclusions arevisible, anisotropically aligned along the original rolling direction. (Courtesy of MetalsHandbook, American Society for Metals).

Figure 9.1 Anisotropic grain shapes in a rolled copper sheet revealed by etching a specimenthat has been sectioned parallel to the rolling direction. (Courtesy of Metals Handbook,American Society for Metals).


but all the reinforcing fibres liewithin that plane. Themechanical properties will exhibit in-plane isotropy, but through-thickness anisotropy.

Finally, we should note that any individual particle in the microstructure may exhibitanisotropy that is associated with either its shape or its crystal structure (or both), but theparticles themselves may still be distributed randomly in the sample. If the morphology isunaffected by the plane of the section, then the sample is considered isotropic, irrespectiveof the shape of the individual grains or particles.

9.1.2 Homogeneity and Inhomogeneity

Processing technology is usually geared to ensuring the homogeneity of a material product,so that any random sample selected from an engineering component will have sensiblyidenticalmicrostructure and properties. This is not always the case. For example, cast ingotsof steel and other alloys may be hot-rolled in stages to produce a homogenous finishedproduct (either sheet, rod or profiled bar), but in the early stages of this hot-working process,a distinction must be made between material that is derived from the top, the bottom or themiddle of the original ingot. The heavier, nonmetallic inclusions in the metal will tend to beconcentrated at the bottom of the ingot, while the higher impurity levels and excess alloyconcentrations are usually to be found, partially segregated, in the last fraction of the liquidmetal to solidify, togetherwith the lighter inclusions, at the top of the original ingot. The endproduct may therefore be significantly inhomogeneous, with respect to both the inclusioncontent and, to some extent, the alloy composition.

We should distinguish between several types of inhomogeneity. In the example of the castingot, we note that the inclusion count and the alloy composition are likely to varyindependently. The inclusion count shows morphological inhomogeneity, while the alloycontent shows chemical inhomogeneity. Crystallographic inhomogeneity may also beobserved in the ingot, as an inhomogeneous distribution of the preferred orientation thatreflects the solidification direction in different regions.

As another example, consider a ductile metal bar that undergoes shear during extrusion(Figure 9.3). The amount of shear is a function of the distance from the axis of the bar, andthe preferred orientation (the deformation texture) will therefore vary across the section,

ExtrusionDirection

Die

Figure 9.3 Schematic representation of the distribution of shear deformation across anextruded rod.


with the maximum shear component in the exterior layers. This crystallographic inhomoge-neity may be retained after the metal has been annealed, and can show up as an annealingtexture (the preferred orientation characteristic of the deformedmetal after recrystallization).

Other mechanical-forming operations have similar effects on the texture and itsdistribution in the final component. Forged components usually retain a marked macro-structural inhomogeneity that is on the scale of the component dimensions. Such inho-mogeneity may be beneficial to the toughness and fatigue resistance of the component.Figure 9.4 is from a forged steel cam shaft that shows lines of macrostructural flow whichinhibit fatigue crack propagation through the load-bearing cross-section.

It follows that there are many instances where the microstructure of an engineeringcomponent cannot be adequately characterized from either a single sample, taken from thebulk, or a single section, taken through the sample. An extreme situation would be a solid-state semiconductor device that consists of a large number of micro-components denselyarrayed on a single silicon chip. For such a case, focused ion beammilling could be used toprepare several cross-sections for transmission electron microscopy (Section 5.2.1), inwhich each section includes several of the interfaces between the active and the passivemicroelectronic components. Similar considerationsmay apply to coated or surface-treatedengineering products, comprising surface layers that vary in both microstructure and

Figure 9.4 Macroscopic development of textured �grain� during a forging operation.(Reproduced from J.L. McCall and P.M. French, Metallography in Failure Analysis, Copyright(1978), with kind permission of Springer Science and Business Media).


composition through the layers. This would be true for electroplated components, chemicalvapour deposited coatings, thin-film devices and ceramic-coated cutting tools.

9.1.3 Sampling and Sectioning

In microstructural characterization, the question most often posed is whether an observedmicrostructure is substantially the same as or significantly different from some othermicrostructure. Since microstructures are examined on a diminishing length scale as themagnification is increased, we need to distinguish between the microstructural variabilitywithin a given selected sample, and the variability observable between a set of differentsamples. Moreover, within any region of a selected sample, the different microstructuralfeatures will each have their own characteristic length scale. Such length scales maycorrespond to either a particle size or a particle separation, or to any other specificmicrostructural feature. For example, a eutectic structure (Figure 9.5) could be character-ized by the size of the eutectic colonies, the separation of the colonies, the shape, spacingand dimensions of the lamellae within a colony, or the volume fraction of each phase in theeutectic. (Remember that three-phase ternary eutectics may also exist in engineering alloysystems!)

We will limit our discussion to three length scales for microstructural �sampling� thatmight affect the statistical significance of any attempt to quantify the data derived from amicrostructural investigation:

1. First, themacroscale over which the samples were selected with respect to the geometryof an engineering component. On this scale, someone will have to decide if a singlesample is sufficient, or whether sections should be taken from different regions of thecomponent or in different orientations, in order to test for inhomogeneity and anisotropy.If several samples are to be investigated, for example, from a set of components that havereceived different heat treatments, then we will need to ask if it is sufficient to take asingle section for each condition, or whether several, nominally �identical� samples need

Figure 9.5 Eutectic colonies in a tin alloy are characterized by the size and shape of thecolonies, the volume fraction of the phases present and the spacing of the lamellae within thecolonies. (Courtesy of Metals Handbook, American Society for Metals).


be taken in order to ensure the reproducibility of the results. As an example, a usefulquality-control test might be to take several representative samples from each produc-tion batch of a component and test for uniformity of the grain size.

The statistics of sampling will require data from a minimum of three samples if someestimate is to bemade of both themean and thevariance of anymeasured parameter. Thisrequirement corresponds to assuming a Student’s-test statistical distribution inwhich thetotal number of �degrees of freedom� for a set of n data points is equal to n-2.

2. For any given microstructural cross-section, it will be important to check the variabilityof themicrostructure across the sample section, that is themesostructuralvariability, andthe magnification selected for any recorded image will determine the diameter of thefield of view in real space. The features of interest must be clearly resolved in the image,but, at the same time, a sufficient number of features must be present in the field of viewin order to ensure adequate sample statistics for each selected image area. For smallfeatures that are widely separated, these conditions may be in conflict and require thatseveral images be recorded at different magnifications.

The statistics of the mesostructure will determine whether or not we can make astatement about the uniformity of the morphology within a sample with any confidence.In cases where inhomogeneity is an important issue, it is usually possible to choose anappropriate magnification and then select a regularly-spaced set for the imaged fields ofview, in order to derive a statistically significant data set. For example, a surface coatingcould be sampled by recording a set of images on a section normal to the coating, butwiththe images at a shallow angle to the intersection of the plane of the coating with thesection (Figure 9.6). A set of three or more such lines will improve statistical estimatesfor the depth dependence of the relevant parameters.

3. Finally, the distribution of themicrostructural featureswithin anygiven field of viewwilldetermine the statistical significance of estimates of the microstructural parameters thatare based on this single image field. In general, estimates taken from other fields of viewon the same section, or from other sections, will give different values for both themeasured parameters and their variance. It is therefore important to distinguish betweenerrors that are associated with the analysis of a single field of view and errors associatedwith a complete data set that comprisesmany fields of view taken from either the same orfrom different samples. We will return to this point later.

Substrate

Coating

Figure 9.6 One possible array of imaging sites (þ) for representative, high-magnificationsampling of a coated specimen section.


It is usually possible to select an effective �field of view� that is larger than anyindividually imaged region, either by recording a �panorama� of overlapping images fromneighbouring regions or by scanning an image �strip� along a selected direction in the planeof the section. Such procedures may be unavoidable if the features to be recorded are bothsmall and widely separated, since it may not be possible to resolve a sufficient number offeatures in any single image to provide sufficient statistical data.

Some balance has to be struck in selecting samples from the different length scales thatwe have chosen: the macrostructural, the mesostructural, the microstructural or, possibly,the nanostructural. Visual inspection (or common sense) may allow us to dispense withformal sampling on the macroscale, and this is perfectly legitimate providing we are awareofwhatwe have done. It ismuch less legitimate to dispensewithmeso sampling. At the veryleast, the different regions of any chosen section should be scanned for visually significantdifferences in the microstructure. Ideally, steps should also be taken to ensure that thevariance of a measured parameter, such as grain size, associated with mesostructuralsampling matches that found for the field of view of any single microstructural sample. Inother words, the statistical errors derived when comparing different regions should exceedsignificantly the errors determined for any one region.

The availability of methods of microstructural characterization that extend down to thenanoscale has resulted in new challenges for the statistical analysis of image data. In theatomic forcemicroscope, the number of pixel points in a given image data set is far less thanwe have grown used to in other instruments. The atomic force microscope field of view islimited and, as in the early days of electron microscopy, it is all too easy for the subjectivemicroscopist to �select� images as being �typical� of a phenomenon to be demonstrated.

One further point, it is good practice to select areas of the microstructure for quantitativeanalysis that lie within an imaged region, leaving a well-resolved, microstructural �border�outside the area to be analysed. This is illustrated in Figure 9.7, and avoids selecting a regionfor analysis adjacent to an obvious defect in the sample (due to either processing orsampling). This procedure also enables the observer to assess the importance of possible

Figure 9.7 The region to be analysed quantitatively should lie within a well-resolved,microstructural �border� (See text).


edge effects that can arise, for example, at grain boundaries and grain boundary junctions.The microstructural �border� should be selected to have an area approximately equal to thearea that will be quantitatively sampled, corresponding to a linear dimension that is

ffiffiffi2

pgreater than the actual analysed area and a border width that is approximately 20% of theanalysed image dimensions.

9.1.4 Statistics and Probability

A number of statistical functions and procedures have proved useful in the quantitativeanalysis of microstructural data, but a detailed description of these functions is beyond thescope of this text. Nevertheless, a very brief account of some statistical tools is in order. Ingeneral, themeasured values of any parameter will be distributed about a statistical mean oraveragevalue, and thewidth of this distribution can be defined andmeasured to describe thespread of the measured values in terms of a �statistical error�. Moreover, any statisticalfunction that is derived from data related to frequency of observation can also be interpretedas a probability distribution. The probability distribution, or frequency function, determinesthe probability that a parameter (in this case, a microstructural parameter) will have anygiven value. If the statistical frequency function is found to be skewed (asymmetric), thenthis also may be described by an appropriate statistical parameter. Different statistical testsmay be applied to the data collected, providing the appropriatemathematical conditions aremet, andmay be used to decide, for example, whether two different data sets come from thesame population, or whether the values of two different parameters can be correlated withone another.

We do need to clarify what we mean by probability since this concept has alwayspresented problems. There is considerable overlap between �probability theory,� the study ofparameters that are related by a stochastic or, equivalently, nondeterministic function, andstatistical analysis, that is the estimation of a parameter based on a frequency analysisperformed by sampling from a well-defined population. In many probability-basedprocesses, subsequent events are linked, so that a single event in a chain determines theprobability that the next event will occur. Such sequences are termedMarkov processes andmay be observed inmore than one dimension (chain-branching in polymerization is just oneexample). A further complication arises when a phenomenon can be interpreted as fractal innature: the process is then fully deterministic, but not amenable to analytical analysis.Weather forecasting is one area in which fractal analysis has resulted in major improve-ments in forecast accuracy. Fractal analysis has been fruitfully applied to the propagation offracture in brittle materials. The interested reader is encouraged to consult other texts.Although these topics are frequently of direct relevance to the quantitative analysis ofmicrostructure, they fall well outside our present mandate.

As a very good first approximation, all measurements of microstructural parameters aresubject to statistical errors that are associated with just three separate factors:

1. The inherent variability of the parameter in the bulkmaterial, for example, the grain sizeof a polycrystal. This means the errors associated with the statistical distribution of theparameter that we would like to determine.

2. The statistical errors that are associated with the methodology of sample selection formicrostructural observation. In particular, errors associated with the number, position,orientation and size of the features that are available for analysis.


3. Any errors that are associated with our methods of observation or data collection andspecimen preparation, as well as the method of measurement we have selected. Underthis heading we include the artifacts due to polishing and etching procedures used toprepare the section, the performance and characteristics of the microscope, theproperties of the recording media or the values of the digitizing parameters (pixeldensity, dynamic range or channel width), as well as any intrinsic error associated withthe definition of the microstructural parameters.

It is not always easy to separate the inherent variability of a bulk material parameter fromerrors that are due to specimen preparation, or themethods of observation and data collectionor recording. The essential requirement is to reduce these secondary effects to a level wherethey no longer affect the significance of the results. For example, if it is not possible toexclude these errors from the measurement of grain size, then clearly it is also impossible toaccurately quantify variability in the grain size of the bulk material.

9.2 Accessible and Inaccessible Parameters

The distinction between accessible and inaccessible bulk microstructural parameters isoften basic to the quantitative interpretation of morphological data collected from a two-dimensional sample section, but is often ignored. There are a very limited number ofaccessible microstructural parameters characteristic of the bulk structure that can bederived directly and unambiguously from a two-dimensional section. The volume fractionof a second phase is one of these accessible parameters. However, the particle size, thevolume per particle and the number of particles per unit volume cannot be derived fromobservations made on a planar section without making considerable assumptions about theparticle shape. (For example, we could assume that all the particles have only positivecurvature with no re-entrant angles.)

Figure 9.8 makes clear why such simplifications are often necessary when modellingmicrostructure. A section through a �doughnut� or any other featurewith a partially concave

Figure 9.8 A feature having regions of both positive and negative curvature may intersect thesection surface to appear as more than one area intercepted on the section.


interface can appear in the plane of the section as two disconnected areas of second phase.Dendritic grains are an excellent example of a class of feature that has both positive andnegative curvatures. The dimensions characteristic of dendritic growth: the number ofgrains and the spacing of the branches, for example, cannot be derived from a singlemetallographic section.

Serial sectioning can provide information perpendicular to the plane of the sections.In serial sections, thin layers of known thickness are removed sequentially, using fiducialmarkers to ensure that image data are recorded from the same area at each stage.Serial sectioning techniques are standard practice in many histological studies ofsoft, biological tissues and cells, and have now been extended to many engineeringapplications for the three-dimensional microstructural characterization of materials(Section 9.5). Providing the features of interest in the microstructure have lineardimensions that exceed the thickness of the layers removed at each stage, then eachfeature observed on the nth section can be related to the cross-section of the same featureon the (n� 1)th and (nþ 1)th sections to derive a three-dimensional reconstruction of thebulk microstructure.

The �resolution� perpendicular to the sample surface will, of course, depend on theseparation of the serial sections. Mechanical polishing can be automated to an accuracy ofbetter than�10 mmwith no special precautions, while focused ion beam (FIB) milling canremove controlled thicknesses of about 10 nm. Clearly, the latter technique has majorpotential for electron microscope investigations and is now being used to scan �through�thin-film devices in microelectronic and electro-optical systems. Microtomes (rigidlymounted knives that can cut thin slices from a sample) have been in use in the biologicalsciences for many years and are used to produce serial sections of histological samples.In the atomic force microscope, diamond-knife microtomes have been used for in situserial sectioning.

All the above techniques have disadvantages: mechanical damage, especially whenmicrotome slices are prepared, and radiation damage generated during ion milling.

9.2.1 Accessible Parameters

Only a few bulk microstructural parameters can be accessed from a two-dimensionalsection without making serious morphological assumptions. Nevertheless, it is possible todefine some parameters so that they become accessible from the two-dimensional image.Two examples are grain size and particle size, as we shall see below. Furthermore, a singleassumption is all that is necessary to make the transition from �inaccessible� to �accessible�for a wide range of other parameters. This simple assumption is that all interfaces areconvex, since no completely convex particle (a particle that possesses only positivecurvature) can intersect a planar section discontinuously. Thus, for convex particles only,every area of a grain or second phase particle that is observed on a planar section represents asingle particle or grain in the volume of the bulk material.

Several other parameters that are characteristic of a planar section can be determinedquantitatively, but these are not bulk parameters. An example would be the number ofparticles of a second phase per unit area of the section. Such two-dimensional, but�accessible�, parameters are of limited value, although they have their uses. For example,maximum allowed inclusion counts, taken parallel and perpendicular to a principle axis of


the component, are often written into industrial acceptance criteria for a product, as aquality control, standard requirement.

9.2.1.1 Phase Volume Fraction. The volume fraction of a second phase fV is a bulkmicrostructural parameter that, in the case of a fully crystalline material, can usually bedetermined by quantitative X-ray diffraction analysis (Section 2.4). However, manymaterials contain noncrystalline phases that cannot be identified with any reliability byX-rays, and there may be other reasons why the microstructural determination of fV from aplanar section is to be preferred to a diffraction analysis.

A remarkable result of stereology (the study of spatial relationships) helps us to select areliable method for measuring fV from a planar sample section:

fV ¼ V

V0¼ A

A0¼ L

L0¼ P

P0ð9:1Þ

where V, A, L, and P are the total volume of the second phase, the area of the phaseintersected on a random planar section, the line length traversing the phase for a randomlyoriented line across the section, and the number of points falling within the phase for arandom array of points superimposed on the same planar section, respectively. Thesubscript 0 refers to the total volume of the sample, the total area of the section, the totalline length examined and the total number of points in the test array.

It follows that the same result is to be expected for fV regardless of whether fVismeasuredby sampling a volume fraction, an areal fraction, a line fraction or a point fraction.

We can devise an experiment that could confirm this result (Figure 9.9): First a unitvolume of the sample is immersed in a suitable medium to dissolve the matrix and theparticles could then be collected, to determine their volume, by weighing, providing theirdensity is known [Figure 9.9(a)]. Secondly, a micrograph of a �random� section of thesample could be recorded and digitized, and the total area of particles intersected per unitarea of the section could be assessed from the total number of pixels associated with thesectioned particles [Figure 9.9(b)]. Thirdly a random array of test lines could be super-imposed on the sample, and the total length of line falling within the particle sections couldbe determined and divided by the total length of the test line [Figure 9.9(c)]. Finally, arandom array of pixels could be selected from the random test section and the proportion ofpixels falling within the areas corresponding to the sectioned particles could be determined[Figure 9.9(d)].

Which of these methods is �best�? Dissolving away thematrix is only likely to be feasiblein exceptional circumstances, and is likely to be both time-consuming and prone toexperimental errors that will be difficult to gauge.

Areal analysis is a natural choice for any digitized image, since all the available data onthe sample section are included and it is only necessary to set intensity thresholds anddetermine the proportion of the pixels scanned whose intensity falls within the set intensitywindow. Of course, the pixel spacing has to be appreciably less than the size of the particlesections (and, preferably, less than the resolution in the observed image). The plane of thesection is important and should be chosen to coincide with the principal axes of thecomponent, while close attention must be paid to any statistically significant variations inthe results that might be associated with microstructural anisotropy or microstructuralinhomogeneity.


Linear analysis could also be very useful, especially if the particle spacing is very muchgreater than the particle size. The most efficient methodology for acquiring digitized datafor linear analysis would then involve spacing the test lines at a distance comparable withthe spacing of the features of interest (the particle sections), while controlling the pixel sizealong the lines to well below the particle section diameter, and preferably less thanthe microscope resolution. Such a strategy would allow a much larger number of particlesto be sampled in the section area of the specimen, butwithout loss of particle size resolution.The disadvantage is that the orientation of the test lines is also a significant variable thatmust be controlled. In a linear analysis, data from apredetermined total number of line scanswould be collected and the number of times a selected, threshold intensity was exceededwould be recorded. Normally, the scan lines are separated by a set interval and theirorientation is determined by the sample orientation with respect to the raster. Anisotropy inthe plane of the section can then be assessed by rotating the sample with respect to thescanning raster (Figure 9.10).

Before the introduction of digitized imaging systems, point analysis was by far themost effective method of assessing volume fraction, since a grid of test points placedover an image could be used by an observer to accumulate numerical data rapidly. Themethod is not entirely obsolete, since maximizing digital data collection from a largenumber of widely separated, small features may still be best achieved by spacing thepixels in a regular array whose separation corresponds roughly to the spacing of features.This procedure reduces the amount of data that needs to be acquired for any given levelof significance.

Plane of Section

Plane of Section

(c) Random lines

(b) Areal analysis of a cross-section(a) Dissolution of the matrix

Plane of Section

(d) Random points

Figure 9.9 Possible methodologies for volume fraction analysis based on Equation (9.1) andusing either (a) volumetric, (b) areal, (c) linear or (d) point analyses.


If visual assessment by an observer is still required, perhaps as a rough check on theresults obtained using a software package, then the perceptual limitations of the humanbrain need to be appreciated. Most observers can assess and record, without actuallycounting, the number of points in a nine-point grid (�tictac toe�) (Figure 9.11) that lie within

x

y

z

Line scan

Figure 9.10 An areal scan will detect anisotropy between sections taken in differentorientations through an engineering component, while a line scan will detect anisotropy inthe plane of a section.

X X X

X

XXX

X X

Figure 9.11 A 3 ·3 grid of test points can be used to visually assess and record the number ofpoints falling on the sectioned second phase particles. The grid image is projected on themicroscope image at the appropriate magnification and the specimen is then displaced by fixedintervals in order to acquire a data set for the proportion of grid points that lie within theboundaries of a selected phase in the microstructure.


the particles of a second phase. A regular array of test points, rather than a random array,actually reduces the statistical error, since the sampling array is then uniform.

From Equation (9.1), the areal, lineal and point-scan methods for assessing the volumefraction of a second phase are all expected to yield the same result for a randomly positionedsampling probe, that is, a random planar section, random test lines or a random grid of testpoints. The results should also be equivalent if the microstructure is itself �random�, whichmay often be approximately true. However, observed differences between these threemethods are themselves useful for assessing microstructural order, both isotropy andhomogeneity, and, with care, this assessment can also be made quantitative. The guidingprinciple can be very simply stated: we require minimum error for minimum effort, and wewill see below (Section 9.3) how this principle can be used to determine a strategy forquantitativemicrostructural analysis. In themeanwhilewe should note that there is no pointin collectingmassive data sets unless some useful procedure for data interpretation has beendevised. The three principle procedures of areal, linear and point analysis can often beusefully combined. For example, areal analysis data at a highmagnification in the scanningelectron microscope may be collected for a set of regularly spaced locations on the samplesection. The averagevalues of a chosen parameter at each location can then be compiled intoa frequency distribution function that corresponds to a measure of homogeneity or isotropyacross the complete data set.

9.2.1.2 Particle Size and Grain Size. Any bulk determination of both grain size orparticle size should be based on an assessment of the area of interface present per unitvolume of the bulk material, just as the measurement of volume fraction of a secondphase was based on the assessment of the volume of the second phase particles per unitvolume. In the case of particle size, it is the area of interphase boundary per unit volumeof the second phase that we wish to know. Defined in this way, we obtain inverseparticle size or inverse grain size, and therefore both these parameters have thedimensions of inverse length. These parameters are measures of the surface to volumeratio. In mathematical terms, for convex particles, both parameters estimate the totalcurvature of the interface, defined as dS/dV. The simplest example is that of a sphericalparticle of radius r, whose volume is 4/3pr3 and whose surface area is 4pr2. It followsthat dS/dV¼ (dS/dr)/(dV/dr)¼ 8pr/4pr2¼ 2/r.

That this is indeed the particle-size parameter most commonly of interest becomes clearif the driving force for reducing the internal energy of the system is considered. The totalsurface energy of the system is gS, where g is the surface energy per unit area, so that thedriving force for a reduction in the internal energy is d(gS)/dV. If the surface energy per unitarea is assumed constant, then this reduces to g(dS/dV). For the case of a stable soap bubble,the surface tension force is balanced by the difference in pressureDP between the inside andthe outside of the bubble, leading to the well-known Laplace equation:

DP ¼ 2gr

ð9:2Þ

For the general case, the driving force for any reduction in total surface energy is given bythe relationship d(gS)/dV. A reduction in g may result either from dopant or impuritysegregation or, in the case of anisotropic systems, by faceting on low energy crystallo-graphic planes, which reduces the driving force by reducing the average interface energy.


If g is assumed constant, then an increase in curvature of a single particle will decreasethe stability of this particle, since small particles are less stable than large ones. In apolycrystalline sample grain growth occurs by reducing the �curvature� through a reductionin total grain boundary area per unit volume, so that dS/dV is then negative.

The grain size is the parameter that is easiest to visualize, and several alternativedefinitions of grain size have been given in the literature. For amaterial withmorphologicalanisotropy, themeasured data are often interpreted as an orientational dependence of eitherthe mean intercept length, or the mean caliper diameter, with maximum and minimumvalues quoted for the plane of a selected section. We prefer to select an anisotropy-independent parameter for the grain size and therefore concentrate on a definition that isbased on the inverse of the total curvature of the boundaries in the sample, dS/dV, eventhough this definition is valid only for convex particles. A networkof second-phase particlesthat is interconnected in three dimensions may have a total curvature which is positive,negative, or even zero (Figure 9.12). Clearly, S/V remains a valid measure of themicrostructural scale, with the dimension of inverse length, even though it is now nolonger directly related to the total curvature.

We startwith a line element of lengthDl on a section containing a test grid of parallel linesthat are separated by a distance d (Figure 9.13). Averaging over all angles between the linesegment and the normal to the grid, the probability of the line element intersecting the grid,p, is given by:

p ¼ 2Dlpd

ð9:3Þ

Figure 9.12 Unit volume from a fully interconnected two-phase microstructure that has zerocurvature. The principal radii at every point on the phase boundary are equal and opposite. Inthis sketch the volume fractions of the two phases are equal, but this is not a necessary conditionfor achieving zero curvature of the interface.


For a total length of line Lmade up of randomly oriented segments, the average number ofintersections �N will be given by:

�N ¼ 2l

pdð9:4Þ

and if the total test area on the section isA, then the total length of test linewill beA/d¼ L, sothat the average number of intersections per unit length of test line is linearly related to thetotal length of intercepted interface per unit area of the section:

�N

L¼ l

A

2

p

� �ð9:5Þ

Now consider the intersection of an irregular particle having a volumeVand a surface area Swith a set of parallel test planes of separation d (Figure 9.14). The average area interceptedin a section, A, is given by:

A ¼ V

dð9:6Þ

while the average length of the intercepted boundary on the section is:

�I ¼ pS4d

ð9:7Þ

It follows that the surface-to-volume ratio is given by:

S=V ¼ 4

p

�I�A

!ð9:8Þ

Adapting the relationship for the length of boundary intercepted by unit area of the section,derived above, to the case where we are interested only in the portion of test grid falling

d

∆l

Figure 9.13 A line element that is projected onto a test grid in the same plane of the samplesection has a fixed probability of intersection.


within the area of the second phase particles, then the average intercepted boundary lengthon the section is �L, and we obtain, for a given phase a.

ðS V= Þa¼ 2�N�L

� �a

ð9:9Þ

So, as long as only the length of the test line falling within the particles sectioned by the testplane is counted, the surface-to-volume ratio for the a phase, is just twice the number ofintercepts made by a unit length of test line with the traces of the a phase interfaces on thesurface section.

In the measurement of grain size, the grid of test lines covers the total area of the section,while each boundary trace is shared by two grains, so that each intercept is shared by theneighbouring grains, and the relationship for the surface-to-volume ratio of a polycrystal-line material, defined by the inverse grain size S/V, becomes:

S=V ¼�N

Lð9:10Þ

In the literature, it is sometimes difficult to know exactly what measure of grain size orparticle size has been used, since very often �correction� factors are introduced in order toconvert ameasuredmean intercept length into an estimated �grain size�. One such correctionfactor commonly used is 4/p (see above). If it is the driving force for grain growth or particlecoarsening that is of interest, then this force is given by d(gS)/dV. As noted previously,the driving force may be reduced if g is reduced by segregation or surface faceting.The driving force is larger for smaller particle sizes equivalent to an increase in totalcurvature, providing the volume fraction of particles remains unchanged. Inmost cases, it isadvisable to use the accessible bulk parameter, the surface-to-volume ratio, which has thedimensions of inverse particle or grain size, rather than any alternative definition of grain orparticle size.

d

Figure 9.14 Sectioning of an irregular particle by test planes having a fixed spacing d.


One further case of major importance occurs when the second phase particles have afinite probability of touching one another, known as contiguity. This concept needs to bedistinguished from continuity, which is a percolation condition corresponding to acontinuous, interconnected path through the second phase in the sample. A good examplewould be the continuity condition for the onset of electrical conductivity when anelectrically insulating matrix contains a conducting second phase. Two types of interfaceare now of interest: grain boundaries between two contacting particles of the same phase:interfaces that are shared between the contacting particles, and the interphase boundariesthat separate the particles from thematrix. The relation for the surface-to-volume ratiomustbe modified and becomes:

ðS V= Þa¼�Naaþ2�Nab� �

�Lað9:11Þ

For approximately spherical particles, a percolation threshold for continuity occurs at avolume fraction close to 0.3, and leads to three distinct morphologies for such a two-phasematerial. When the volume fraction is below 0.3, the minor phase is dispersed in the majorphase, predominantly as isolated particles. For volume fractions between 0.3 and 0.7 thetwo phases form two interconnected, continuous networks that can be idealized by the zero-curvature model shown in Figure 9.12. Above 0.7 (a second percolation point) the phaseroles are reversed: the �second� phase now becomes the matrix while the original �matrix�phase is predominantly present as isolated particles.

Both stress corrosion cracking and comminution (grinding) processes are good examplesof morphologies that exhibit similar, twin percolation thresholds. In stress corrosioncracking, isolated micro cracks join up to create a continuous �leakage� path through thecomponent (the first percolation threshold). Eventually, all grain contacts are lost across astress–corrosion failure surface that corresponds to a second percolation point. Similarly, incomminution an applied pressure generates micro cracks in coarse aggregates that can jointogether at a percolation threshold to form a continuous crack network. A secondpercolation point is reached when intersecting cracks isolate the fragments of the solid(crushing the coarse aggregate). Viewed on a two-dimensional section, only a singlepercolation threshold can be observed, corresponding to the transition point at which the�minor� phase becomes the continuous matrix. It follows that the wide, technologicallyimportant region of compositions in which the microstructure contains two interpenetrat-ing, interconnected and continuous phases can only exist in three dimensions and cannot beaccessed by a two-dimensional section.

9.2.2 Inaccessible Parameters

We have already touched on problems associated withmorphological anisotropy, and havedemonstrated how the problems may sometimes be bypassed by using definitions for grainand particle size that are independent of any anisotropy in the material, and are determinedonly by a surface-to-volume ratio. Nevertheless, some measure of morphological anisot-ropy is often required, and we will therefore discuss the limitations to the determination ofinaccessible parameters from two-dimensional sample sections.

A rather unsatisfactory example of an inaccessible parameter is dislocation density. Inprinciple, dislocation density might be quantitatively evaluated from diffraction-contrast


images in thin-film transmission electron microscopy (TEM), using one of two comple-mentary methods. In the first, the number of intersections that the dislocations appearingin contrast make with the top and bottom surfaces of a foil sample with visible area A isestimated, while in the second the number of intersections made by the dislocationdiffraction contrast images with a superimposed test grid is used. In the first instance thetest area is actually 2A, since intersections that occur with both the top and bottomsurfaces of the sample foil are counted, while in the second method the test area is Ld,where L is the total length of test line in the superimposed test grid and d is the foilthickness.

Neithermethod is particularly useful, since estimates of dislocation density that aremadeby either method are liable to major errors for several distinct reasons:

1. There are serious problems associated with changes in the dislocation contrastas a function of the contrast parameter g�b, where b is the Burgers vector of thedislocation and g is the reciprocal lattice vector of the diffracting planes (Section 4.3.2).

2. The thin-film specimens have a variable thickness and there are many problemsassociated with determining the thickness to any degree of accuracy.

3. Surface image forces experienced during sample preparation are expected to cause somedislocation rearrangement at the surface. The two possible test areas defined above areorthogonal to one another, and are therefore unlikely to give the same values for themeasured dislocation density.

4. Diffraction contrast from dislocations is typically quite diffuse, and, even when thedislocations are imaged away from the Bragg condition, the widths are usually a fewnanometres. It follows that the dislocation images frequently overlap and seriousresolution errors are to be expected.

5. Dislocations normal to the plane of the surface cannot be counted on a superimposed testgrid, since they appear as points, while dislocations parallel to the surface do notintersect the foil surfaces, but will intersect the test grid. It follows that, even if thedislocation density, best defined as dislocation line length per unit volume, could beestimated from the number of intercepts per unit area, then this estimate is liable to bestrongly biased.

Of course, many measurements of dislocation density have been published, based on thin-film electron microscopy, but the results are error-prone. Assuming that good contrast andminimum contrast overlap are required, then reasonable estimates of dislocation density arecertainly achievable for the early stages of work-hardening, corresponding to a dislocationdensity of perhaps 1014m-2, but not for either annealed or heavilywork-hardenedmaterials.

The origin and classification of errors in quantitative microstructural analysis issummarized later (Section 9.3).

9.2.2.1 Aspect Ratios. As we have repeatedly emphasized, the estimation of morpho-logical anisotropy in three dimensions is not readily accessible. In particular, particle andgrain shape are difficult to reduce to a single parameter. However, we have seen that anunambiguous determination of grain and particle size is accessible in terms of the surface-to-volume ratio, equivalent, for convex shapes, to a curvature parameter that is not sensitiveto morphological anisotropy. A common solution is to seek a single measure of shape, even


though it is recognized that any such shape factor involves some serious assumptions thatmay be difficult to justify.

The shape factor usually chosen is the grain or particle aspect ratio, and the simplestassumption is that the particles can be treated as either oblate spheroids, that is lenticularparticles, with an aspect ratio greater than one, or elongated ellipsoids, that is prolatespheroids or acicular particles, with an aspect ratio less than one. In both cases the particlesare assumed to preserve cylindrical symmetry. Although the distribution of apparent par-ticle shapes observed on a planar section will depend on the aspect ratio (Figure 9.15), for asingle section of a particle there is noway of distinguishing between an elongated ellipsoidand an oblate spheroid (Figure 9.16).

Prolate ellipsoid(needle-like)

Oblate ellipsoid

(plate-like)

Log of the aspect ratio0 log(a0/b0)log(a0/b0)

Pro

babi

lity

of in

ters

ecti

on

+-

a=b<c a=b<c

ca

a

c

a a

Figure 9.15 Frequency distributions for the aspect ratio of planar sections through oblate(‘hamburger shaped’ or plate-like) and prolate (‘cigar shaped’ or needle-like) particles.

Prolate

Ellipsoid

Oblate Ellipsoid

Plane of Sectionc

a a

Figure 9.16 An oblate ellipsoid (a plate-like particle) and a prolate ellipsoid (a needle-likeparticle) may give identical sections in the plane of a sample section.


Crystallographic anisotropy is often associated with morphological anisotropy, and, insuch cases the second-phase particles, are usually either plate-like or needle-like, ratherthan ellipsoidal, but once again the same problem exists. The surface section of any oblongfeatures could represent either needles or plates, even though the section probabilitydistributions (Figure 9.15) are quite different. As shown in Figure 9.15, plate-like particlesand oblate spheroids are most likely to be sectioned as elongated, acicular shapes, whileacicular needles and prolate spheroids are most likely to be sectioned as small particles oflow aspect ratio.

The ratios of the maximum to minimum caliper dimensions for particle cross-sectionsobserved on a planar section from the sample (Figure 9.17) can be used to estimate the bulkaspect ratio, by assuming that the maximum value for the experimental aspect ratio on agiven section coincides with themaximumvalue of this parameter in the bulkmaterial. Thiswill only be true if all particles have the same aspect ratio. Also, since the �most probable�shape for the intersection of an acicular particle has a low aspect ratio, an underestimate ofthe true ratio is to be expected.

9.2.2.2 Size and Orientation Distributions. Average values of the accessible bulkmicrostructural parameters can usually be determined to of the order of a few per cent,providing sufficient data are collected from representative sample sections, but thedetermination of a size or orientation distribution for the microstructural features is muchmore difficult.

There are two basic problems. In the first place, the volume fraction of the smallestparticles may be low, but the detection errors associated with their small size will almostcertainly lead to large counting errors. In the limit, when the smallest particle size reachesthe resolution limit, these errors become indeterminate. At the other end of the sizedistribution scale, the contribution to the total volume fraction of a second phase that ismade by the few large particles may be very significant, but the poor counting statisticsassociated with the small number of particles severely limits the accuracy of the data

Dmin

Dmax

Plane of Section

Figure 9.17 The maximum (Dmax)and minimum (Dmin) caliper dimensions of a sectionedparticle can bemeasured, and themaximum value of the aspect ratio in the section then used toestimate the bulk aspect ratio or shape factor.


collected. Thus, for small particles, detection errors restrict the accuracy,while the countingstatistics for the largest particles, will be unfavourable for quantitative analysis.

A third problem is associated with the methodology of the analysis of particle size datafrom a surface section that results in the cumulative propagation of errors from themeasurement of the largest size fractions down to the smallest. We need a sectioningfunction for particles of a given size D, which will describe the probability Pd that theapparent particle size on a planar surface section will be less than d. Assuming sphericalparticles, we obtain the simplest possible sectioning function:

1�Pdð Þ2 ¼ 1�ðd D= Þ2 ð9:12ÞWe can now analyse the raw data for the number of particles observed on the surface sectionin each size group of the section (Figure 9.18), since we know that the largest areas ofparticle sections that are imaged can only be due to the largest particles. We thereforecalculate, using the sectioning function, the contribution of this largest particle fraction tothe smaller intercept sections. The statistical error for these large particles will be large, sowe adjust the size intervals for the measured sizes to optimize the accuracy: d1/d2¼ d2/d3¼ . . .dn/dnþ1, thus avoiding a disproportionately large number of small particles. It is alsocommon practice to choose d1/d2¼ 1/

ffiffiffi2

p, in order to halve the area of the selected particle

sections aswe go to successively smaller sizes. The errors in this analysis propagate throughthe distribution, and will be a maximum for the smallest size fraction, and may often resultin unacceptable, negative values. Bearing in mind that the volume of a particle varies as thecube of the particle size, these errors are concentrated in the smallest sizes and may notprove serious.

Figure 9.18 Micrograph of a low carbon steel, containing a grain boundary (arrowed) thatcannot be detected by a computer program for automatic grain sizemeasurements. (Courtesy ofMetals Handbook, American Society for Metals).


Orientation distributions of microstructural features often have engineering signifi-cance, as, for example, in a forged bar or a directionally solidified ingot. Several methodshave been described for analysing such oriented microstructures. It is important todistinguish between some of the different possibilities. The case of partial alignment inthree dimensions, has been treated by applying the orientational dependence of an interceptanalysis made with a parallel array of test lines. Some symmetrical cases, for example,partial orientation parallel to one symmetry axis of the component, but with a randomdistribution in the plane normal to this axis, can be easily analysed.

The presence of two populations of grains or particles, one randomly oriented and theother partially oriented, may also be important. An example would be a partiallyrecrystallized sample, containing both elongated, cold-worked grains and equiaxed, fullyrecrystallized grains.

Composite materials, containing anisotropically distributed reinforcement, used toachieve highly directional mechanical properties, present their own problems. Suchmaterials may include a variety of woven reinforcement in which the distribution of thefibres within the reinforcing weave is an important variable.

9.3 Optimizing Accuracy

The quantitative analysis of microstructural data used to be a boring, time-consuming andrather unrewarding exercise, but this is no longer the case. First of all, digitized imageprocessing and image analysis have made data collection fast and efficient, and haveall but eliminated photographic recording (though not the need for visual judgement!).Secondly, awide range of reliable software packages are readily available to improve boththe quality of the image and the accuracy of the quantitative analysis. Computer-basedanalysis has dramatically improved the counting statistics and reduced the statisticalerrors.

The accuracy of quantitative microstructural analysis may depend on a wide range offactors, but it is convenient to start by asking three key questions whose answers usuallydetermine the practical significance of quantitative data:

1. Is the sample representative of the object? We have already discussed the problem ofsample selection (Section 3.3.1). In what follows we will assume that this first questionhas been answered positively, and that we need only consider the accuracy of ouranalysis of the selected sample, independent of the original bulk object.

2. Is the image representative of the sample? This is a question that we will try to answerbefore we discuss the possible errors of measurement. Our problem here is to identifyartifacts that may appear in the image, then determine their cause and, finally, minimizetheir incidence in our data.

3. How can we optimize the quantitative analysis of the image?There is actually a two-partanswer to this question. First, we need to know what accuracy is achievable. In Poissonstatistics the error in determining themean of a distribution is equal to 1/

ffiffiffiffiN

p, whereN is

the number of measurements (that is, the size of the data set). Other statistical functionsalso show a similar inverse dependence of the statistical error on the number ofmeasurements that are made. Assuming that each measurement requires the same finitetime, a significant reduction in the statistical errors of analysis can only be achieved by


either devoting more time to data collection, or to improving the rate of collection.Modern computing facilities havemade it possible to collect and analyse enormous datasets that include large numbers of variables within the set.

The second part of the answer brings us back to the need to sample the microstructureof thematerial onmore than one length scale. On themacroscale, themicrostructure of aseries of cast components may depend on the flow rate of molten metal into the mouldand its preheat temperature. We will then require sections taken from several differentcomponents if we wish to detect these macro effects. However, the microstructure of acasting may also vary across the plane of a section prepared for examination, and thesemesoscale variations within a section can be detected by collecting several images fromeach section. However, when the microstructure of a single, individual image isanalysed, it is the microscale which is being sampled, and we must recognize that nosingle image can provide information on mesoscale or macroscale variations in the castcomponents.

In order to optimize the efficiency of quantitative microstructural analysis we mustfirst decidewhether it is necessary to sample on all scales, from themacro to the nano, orwhether it is safe to assume that the microstructure is homogeneous on the macroscale,and then restrict our analysis to a single sample. In such a case, it will be important toensure that the errors associated with our analysis of a single image (the microscale) areless than those involved in comparing a set of images from different regions of thesample (the mesoscale).

At this point we should re-emphasize the importance of good sample preparation.Artifacts in optical micrographs are commonly associated with poor polishing and etchingmethods (Section 3.3) that may result in the following defects:

1. Scratches and the traces of scratches that are revealed by etching. Automated imageanalysis systems often count such features as the traces of �boundaries�.

2. Occluded particles of polishingmedia, embedded in a soft, ductile matrix. Such featuresare interpreted by automated image analysis systems as �inclusions� or �second phaseparticles�.

3. Pull-out of grains from the matrix of the sample during final polishing. This can occur inbrittle, polycrystalline materials and is often interpreted as residual �porosity�.

4. Rounding of edges at second-phase interfaces and pores. This effect is often associatedwith elastic mismatch of a second phase. The rounding increases the apparent arealfraction of the microstructural features.

5. Poor contrast at interfaces and failure to reveal some grain boundaries. This leads toan underestimate of the surface-to-volume ratio and an overestimate of the grain orparticle size.

6. Over-etching of the microstructural features. This results in increased resolution andsectioning errors (see below).

Figure 9.18 shows one example of an artifact. If the specimen is polished and etched usingstandard procedures, some grain boundaries may not be revealed, while individual grainsmay vary in contrast due to orientation-dependent chemical staining. Heat-treating thesample to decorate the grain boundaries with small precipitates has no effect on the grainsize but may sometimes improve the uniformity of the etch response of the boundaries.


Lightly polishing a sample after etching can then remove the chemical staining fromindividual grains, improving the contrast of the grain boundaries for automated quantitativeanalysis.

A great deal depends on our awareness of the microstructural limitations of quantitativeanalysis, and of the partially conflicting requirements for accuracy and rapid dataacquisition. Careful sample selection, good specimen preparation, the acquisition of asufficiently large data set and reliable software algorithms are all important components of asuccessful quantitative microstructural analysis.

We now consider in more detail the optimization of accuracy with respect to minimizingthe effort needed to collect and analyse data.

9.3.1 Sample Size and Counting Time

In any statistical analysis, the larger the size of a data set the smaller the statistical error.Unfortunately, all methods of analysis involve more than one source of statistical error. Forexample, in determining the volume fraction of a second phase, both the number of thesecond-phase particles in the field of view of the image and the number of measurements ofthe particle size that are made for this image are important. At too high a magnification theresolution error may be smaller, because of a higher numerical aperture objective lens, butthe field of view may only contain a very limited number of particles, leading to a largesampling error.

Each measurement requires time, even in a fully computerized system. As the data iscollected, the statistical errors decrease, but the time required for the analysis increases. In afully automated system, this analysis time includes the time needed to select the sample areafor analysis, to adjust the contrast and focus, and to correct for background �noise�. Whenusing a digitized computer system, the time required to collect the data from an image is justa small proportion of the total time of investigation. Data accumulation is therefore aninteractive process, in which the operator cannot rely solely on the computer to make all thedecisions.

We should avoid statistical bias. Data collection should never be started at a nonrandompoint, such as a grain corner. Many microstructural parameters involve a ratio, and it isimportant to predetermine the denominator and not the numerator. For example, in an arealanalysis of volume fraction, the total area sampled should be kept constant from one field ofview to another, not the area of the second phase of interest within the section. In thedetermination of grain or particle size, it is the total length of test grid, not the total numberof intersection points, that should be fixed.

Statistical errors are most conveniently summarized in terms of the coefficient ofvariance (CV) defined by:

CV ¼ s2

x20%

s2

�x2ð9:13Þ

where s is the standard deviation (the true spread of values in the population beingsampled), s is the standard error (the measured spread of values observed in thesample), x0 is the true average value of the property in the population and �x is themeasured average value of the property in the sample. It follows that, s and �x areestimates of s and x0.


In terms of Gaussian statistics (the normal or bell-shaped distribution):

�x ¼ 1

N

XN1

x ð9:14Þwhere N is the number of measurements that have been made, while:

s2 ¼ 1

ðN�1ÞXN1

�x�xð Þ2 ð9:15Þ

While the normal distribution is a good approximation inmany cases, some care is requiredand standard texts on statistics should be consulted for alternative statistical frequencyfunctions. This is especially important when only a small data set is available. In principle,the normal distribution is limited to large data sets in which the variables may take anyrational value (noninteger or integer, positive or negative). For the variables of interest to us,in quantitative microstructural analysis, none may take negative values. In many cases, ourdata only take integral values. A good examplewould be the count of a random distributionof surface features on a section, yielding integers above zero. The appropriate statisticalfunction for this case is a Poisson distribution, for which theCV for a sample section is 1/N.If the section is subjected to an areal analysis in order to determine the volume fraction ofsecond phase particles in the bulk, then the CV that is associated with the sectioning of theparticles (CVA) must also be included in the errors associated with areal analysis:

CVAA ¼ 1þCVAð ÞN

ð9:16Þ

For a random distribution of uniform spheres CVA¼ 0.2.However, for a random point count of the proportion of points falling within any given

area we expect a binomial distribution. If a second phase occupies an areal fraction Aa, thensome of these pointsPawill fallwithin the area of the second phase, and theCV for a randompoint count becomes:

CVPR ¼ 1

Pa 1�Aað Þ ð9:17Þ

This value of CVonly applies to the single feature that has been analysed, and a term needsto be added in order to account for the total number of features in the area sampled. The finalresult then becomes:

CVPR ¼ 1

Pa 1�Aað ÞþP�1ð Þ

P·CVAAð9:18Þ

For a random distribution of the sectioned areas of the second phase we can substitute forCVAA and neglect the factor 1/P, leading to an estimated CV for a point count that is givenby:

CVPR ¼ 1

Pa· 1�Aað Þþ 1

N· 1þCVAð Þ ð9:19Þ

where N is now the total number of second-phase features sectioned by the area analysed.Clearly, optimum efficiency requires that Pa%N, that is, for a minimum error attached to a


given �effort� needed to obtain a given number of counts in the data set, the total number ofcounts collected should be of the order of the total number of visible features in the area thathas been sampled.

This result is, strictly speaking, limited to a randompoint count and a randomdistributionof features. This is therefore a �worst case� scenario. The second-phase features on a two-dimensional section can never overlap, and each additional feature has to occupy a smalleravailable area of the sample. It follows that point counts which are located in an orderedpixel array (so that clusters of points are always excluded), will always give an experimentalCV that is smaller than that estimated above. Nevertheless, the primary conclusion remainsvalid: statistical accuracy is always limited by the number of features sectioned, and there isno advantage in increasing the number of data points counted much beyond this value. Itfollows that the statistical significance of the volume fraction of a second phase determinedfrom a data set collected by interrogating individual image pixels is optimized when thepixel size is of the order of the resolution, but the pixel separation is of the order of theseparation of the second phase particles on the section. The pixel array should cover themaximum area of the sample section chosen, so that as many particles as possible areincluded in the analysis.

Linear analysis of boundary and interface traces can be treated similarly. The grid of linesused to probe amicrostructure should be regularly spaced, and the line separation should beof the order of the grain size or the particle separation. Improved accuracy can be obtainedby interpolating between the pixels to reduce the uncertainty in the position of a boundary orinterface. The statistical accuracy is limited both by the number of interface traces that aresampled, and the number of intercept counts collected. For maximum statistical signifi-cance, these two parameters should be of the same order ofmagnitude. This can be achievedby making the spacing of the test grid approximately equal to the particle spacing or grainsize on the section.

In the absence of an automated system, counting is by far the easiest visual method ofestimating the approximate value of an accessible microstructural parameter:

1. Point features (etch pits, particle density, dislocation intercepts) are best estimated as thenumber of points per unit area.

2. Linear features (dislocation density, grain size, particle size) are best estimated from thenumber of intercepts with a superimposed, regular test grid having a line separationsimilar to the separation of the features in the image.

3. Areal features (volume fraction of precipitates or inclusions) are best estimated using asystematic point count, with the point spacing comparable with the spacing of thefeatures on the section.

9.3.2 Resolution and Detection Errors

While the resolution limit of the microscope constitutes the ultimate limit on theaccuracy with which the coordinates of a microstructural feature can be located, it isseldom the controlling factor. In opticalmicroscopy, sample preparation usually determinesthe detection errors. Chemical etchants develop steps and grooves on the surface of the


polished section that scatter light out of the objective aperture over a region in the image ofthe order of 1 mm in width. Thermally etched ceramic samples show boundary groovingwhose width depends on the heat-treatment conditions (Figure 9.19).

Resolution errors are usually associated with the mechanism of contrast. In diffractioncontrast images of thin crystalline films taken in the transmission electron microscope, thewidth of a dislocation image may be of the order of 10 nm, even though the resolution limitfor the microscope is better than 0.2 nm. In a diffraction contrast image, the parameter g�bplays a dominant role that determines both the width and the apparent position of adislocation (Section 4.4.5), and it is possible to improve the resolution by usingweak-beam,dark-field imaging, in which elastic scattering from the dislocation core region dominatesthe contrast. In this case thewidth of the dislocation image is only of the order of 2 nm,whilethe position of the dislocation image corresponds to the position of the dislocation core.

Finally, image recording may also be a limiting factor that degrades the resolution. Indigitized images the resolution limit is about 3 pixels, since this is the minimum numberneeded to record a contrast variation between two microstructural features. In processingdigitized image data, it is essential to use sufficient pixels to achieve the required resolutionover the full field of view of the image, as well as the required range of contrast. We need toask:what is the total number of pixels necessary to record all the resolved information in theimage? Since there are still severe limitations on the rates of data transmission that broadband and wireless communications systems can support, the answer to this questiondominates the electronic transmission of image data over the Internet or by e-mail.

Figure 9.19 Thermal grooving in a polycrystalline ceramic sample typically limits theresolution in an optical micrograph to over 1 mm, but a secondary electron scanningelectron microscope micrograph, as shown here, can significantly improve this limit.


Compression algorithms are available to optimize the efficiency of image-data transmissionand processing, and we will touch on this later (Section 9.4.1).

The apparent width of a linear feature in the image of a surface section d can be used toestimate the error in the areal fraction of a microstructural feature that is associated withpoor image definition. The areal fraction of the boundary or interface traces on the sectionwill be:

AL ¼ p2·d·

N

Lð9:20Þ

where N is the number of intercepts and L is the total length of the test line. However, theestimated particle size D determined from a linear analysis is given by:

D ¼ 2LaN

ð9:21Þ

where La is the length of the test line lying within the particle sections. Inserting thevolume fraction of the particles fa¼ La/L and assuming the error in fa is given byDfa¼AL,then:

Df af a

¼ pdD

ð9:22Þ

We conclude that the error in determining a volume fraction of a second phase willincrease rapidly as the particle size approaches the effective resolution limit.

It follows that any attempt to improve the counting statistics for small grains and second-phase particles that are close to the limit of microstructural resolution may be unrewarding,since the accuracy may then be limited by image resolution, rather than by the number ofparticles that have been sampled or the size of the data set that has been collected.

9.3.3 Sample Thickness Corrections

Errors associatedwith the sample thickness, either in thin-film electronmicroscopy or, for aplanar section, the sectioning errors in optical microscopy, are a major factor limiting theaccuracy of quantitative image analysis. The resolution and detection errors limit theaccuracy that we can achieve in determining the x–y coordinates of a feature intersected bythe x–y plane of the sample section. Similarly, the sectioning errors limit the accuracyassociated with uncertainty in the location of the image plane along the z-axis. Figure 9.20shows schematically the effect of this section thickness, both for an etched surfaceviewed inreflection and for a thin film viewed in transmission.

There are actually two section thickness corrections required:

1. The increase in the measured area of the section through a second-phase particle thatarises from particle projections that liewithin the slice, rather than just in the plane of thesection.

2. Overlap of particles within the slice that obscures part of the projected area ofneighbouring particles.

Since the sample is viewed normal to the projected section from one direction only, thecontribution from internal surfaces will amount to just half the surface area of the particles


that lies within the slice. This correction is, therefore 1=4ð ÞS=V ·t, where S/V is the surface-to-volume ratio and t is the slice thickness. Using linear intercept analysis to estimate thesurface-to-volume ratio, S/V¼ 2Na/La, gives a first-order correction for this additional area,so that the corrected volume fraction of the second-phase is now:

f a ¼Aa

A�Na

La� t2

ð9:23Þ

This slice thickness correction is particle size dependent, and should be negligible for largeparticles and small slice thicknesses.

As the particle size approaches the slice thickness, particles will either be etched away, asin the case of optical microscope observations made in reflection, or start to overlap in the

Observedareas

Trueintercepted

areast

Projected areas

Trueintercepted

areas

t

Figure 9.20 True and observed intercepted areas of a second phase on a sample section arenever the same, but are determined by the thickness t of the sample slice from the material thatcontributes to the projected image. The corrections required are similar for both a reflectedimage of an etched surface recorded in the optical microscope and in the projected image of athin film seen in transmission electron microscopy, but significant overlap of microstructuralfeatures is only possible in a thin-film, transmission electron microscope image.


projected image, as in thin-film transmission electron microscopy. This overlap correctionwill become important for large volume fractions of a second phase. An approximaterelation that accounts for both the contribution from surfaces within the slice and thecontribution due to particle overlap has been suggested by Hilliard:

1�f a ¼ 1�Aa

A

� �exp

s

4Vt

� �ð9:24Þ

This suggested correction is actually an overestimate, since it does not allow for the�exclusion� volume that surrounds each particle, and which significantly reduces the extentof particle overlap estimated on the basis of random positioning.

9.3.4 Observer Bias

It is remarkable the difference that the experience and training of the observer may playin determining the accuracy of even a fully automated and computerized quantitativeprocedure. While computerized image analysis improves rates of data collection byseveral orders of magnitude, it has little effect on observer bias, since it is still the observerwho selects and prepares the samples and determines the computer settings. It is important torecognize thepossible forms thatobserver biasmay takeand the followingexamplemayhelp.

Let us assume that thegrain size of a ceramic is to be determined from a thermally etchedsample (Figure 9.19). Observer A is worried that some boundaries may not be clearlyvisible, so he increases the annealing time to improve the visibility of the boundary grooves.Observer B considers the resolution error, due to the width of the grooves, to be excessive,and so reduces the annealing time accordingly. ObserverCwishes to improve the contrast inthe image, and to do so coats the thermally etched surface with a thin film of a reflectingmetal. We may confidently predict that each of these three observers, using preciselythe same microscope and digital recording system, as well as the same data analysissoftware procedures, will nevertheless arrive at a different average grain size and grain sizedistribution for the same sample material. Providing sufficient data are collected, thesethree observers may well be able to prove statistically that the �grain size� they have eachdetermined is, to a high degree of probability, not the same as that measured by theircolleagues on precisely the same material!

Let us consider onemore example. Imagine that the same sample has been supplied to thesame three observers, but now the sample has already been prepared. Observer A is carefulto keep the magnification of the microscope to a minimum, in order to ensure that a largenumber of grains are recorded in the field of view to be analysed. Observer B, on thecontrary, wishes to ensure that the best possible resolution of the microscope is utilized,and records a series of high-magnification images. Observer C is much more contrast-conscious, and decides to use a dark-field objective to reduce background intensity in theimage to a minimum and highlight the grain boundary grooves.

All of the above choices are completely rational. They reflect legitimate differences inthe professional judgement of the individual observers. In practice, it is not easy to identifythe precise reasons for the decisions made by an observer. Does the trace of an interface(Figure 9.21) cross and then re-cross a test line, adding two points to the tally of boundaryintercepts, or is it just adjacent to the test line (with no increase to the tally of intercept


points)? The test line is just one pixel wide, but the boundary position is much lesswell-defined. There will always be some observer bias, even if it is now located at theobserver/computer interface.

9.3.5 Dislocation Density Revisited

We have already noted the complications associated with the estimation of dislocationdensities from diffraction-contrast images in TEM (Sections 1.1.3.2 and 9.3.2). In thissection we summarize the problems associated with the quantitative determination ofdislocation density and the methods which have been used to overcome them.

One of the first attempts at quantitative analysis of dislocation substructures was madeusing X-ray line broadening, well over 50 years ago. The line broadening was associatedwith both lattice strain (changes in d-spacings) and decreased particle size (sub-grainformation). The difficulty is that an array of dislocations in a slip plane, that is a dislocationpile-up, introduces strain into the lattice, while an array with the same average spacingin a sub-grain boundary results in misorientation across the sub-boundary. It is notstraightforward to separate the higher energy dislocation pile-up from the lower energyconfiguration of the dislocation sub-boundary.

X-ray measurements of dislocation density were subsequently followed by a series ofetch-pit analyses, pioneered by Gilman working with ionic single crystals. The dislocationarrays were generated by grit particles impacting the surface of the ionic crystal and thesubsequently formed etch-pits corresponded to the intersection of the dislocation lines withthe sample surface. The dislocation etch-pits were shown to have a distribution thataccurately confirmed the predictions of dislocation theory.

Although etch-pitting could reveal dislocation substructure in a wide range of semi-conductors, ionic materials and metals (including iron), the resolution limit of the methodwas only a few micrometres. Thin-film TEM led to serious attempts to define dislocationdensity as a microstructural parameter that could be incorporated into theories of workhardening (the increase in yield strength of a ductile material with increasing plasticdeformation).Amajor prediction fromdislocation theorywas that the increase in the tensile

+

+ +

Boundarytouches test line:

No Crossing Points

Boundarycrosses test line:

Two Crossing Points

Figure 9.21 An interfacemay be judged to cross a test line byone observer, increasing the tallyof boundary intercept points, but not by another who judges the boundary not to be intercepted.


yield strength should depend on the square-root of the dislocation density, Ds / Gbffiffiffir

p,

where G is the shear modulus, b is the Burgers vector of the dislocations and r thedislocation density (the dislocation line length per unit volume).

The density of dislocations thatwas revealed by diffraction contrast in a thin film could beestimated, but it proved difficult to account for the relaxation of the dislocations into densearrays that were poorly resolved by diffraction contrast.

It required some 20 years of research effort before researchers finally accepted that theartifacts involved in estimating dislocation density in transmission electron micrographsreflected a basic problem with the general concept of �dislocation density�.

Today, the importance of determining themorphology of the dislocation substructure andidentifying the dominant dislocation interactions in both structural materials and electronicdevices is still fully recognized, but the concept of dislocation density is used sparingly.

As an alternative to dislocation density, workers in the field of plastic deformation haveevoked a complete library of new concepts to describe the complexmorphologies that resultfrom dislocation interactions during the plastic flowof ductile polycrystallinematerials, forexample kink bands, shear bands, cell structure, sub grains, pile-ups and dipoles. For thepresent, it is quite sufficient that the reader understand why dislocation density is aproblematic parameter.

9.4 Automated Image Analysis

Increased computer data storage and handling capacity, combined with reduced prices, hasplaced automated quantitative image analysis within reach of any research, teaching orindustrial laboratory.

The first automated systems were constructed in-house over 50 years ago. These werethen superseded by commercial systems for image analysis and later, following the digitalcamera revolution, by computer software packages for image analysis that could be adaptedto both optical and electron microscopes using digital charge-coupled device (CCD) orcomplementary metal oxide semiconductor (CMOS) image data output, or used withscanning electron microscope and optical scanning systems.

In principal, there are three options for collecting a digitized data set from a two-dimensional projection of a three-dimensional object. These options correspond to the�scanning� of either the source, the object or the image (Figure 9.22):

1. In the scanning electron microscope the electron beam is focused onto the surface of thesample to form a reduced image of the electron source (the probe). This probe is thenrastered across the surface of the sample. Since it is a focused image of the source that isbeing scanned across the surface, this is a source-scanning system and the final, digitizedimage data set is collected as a function of the electron probe position. The scanningnear-field optical microscope also focuses a light beam onto the specimen with aneffective probe diameter that is less than the wavelength of the light used. With thistechnology it is possible to beat the diffraction limit on imaging with visible light andcapture scanned image data at resolutions that approach 10 nm.

2. Ina fewsystems, it is theobject beneath theprobe that is scanned, for example in the (semi-continuous) automated analysis of a powder sample. Some commercial particle analysers


workon this principle, although theydetect the scatteringof light from individual particlesinstead of the data from a focused image. The particles are carried in a stream of a diluteliquid dispersion. They pass through a capillary tube and traverse an aperture that isilluminated by a laser beam. The scattering of the light by each particle is detected and theanalysisofparticle size isperformed in reciprocal space (Section2.2).Millionsofparticles

Specimen

SourceScan

(a)

Focused Source

ObjectScan

Specimen

(b)

Image

Plane

ImageScan

Specimen

(c)

Figure 9.22 Digital image data collection options may be based on scanning of (a) the source,(b) the object or (c) the image.


canbecountedandsized,accordingtothelightscatteringangles,andthedatapresentedasaparticle size distribution frequency function. Typically the size fractions range from a fewtenths of a micrometre up to some tens of micrometres.

3. Finally, focused images, typical of the optical microscope and the transmission electronmicroscope, can be interrogated in the image plane, by using a suitable digital recordingsystem. A CCD camera is commonly used, although CMOS systems are often found inoptical microscopy (Section 3.5). The CCD and CMOS digital image cameras haveinsufficient response time for dynamic recording systems, and video cameras, whichlack the high resolution but have faster recording speeds, are common for time-dependent observations. Good video cameras can capture over 106 pixel points atframe speeds suitable for real-time, video recording (16 frames s�1). For still photogra-phy, a wide range of both monochrome and colour digital cameras are commerciallyavailable. As described in Section 3.5, colour versions are based on a three colour code,usually red, green and blue – the RGB system of primary colours. This can access mostregions of the chromaticity triangle (Figure 9.23). The chromaticity triangle is aconvenient way of analysing colour. By varying the three intensity levels, any colourwithin the triangle that is defined by the three primary colours can be simulated,including white, over a wide dynamic range of intensities.

In the brief discussion that followswewill assume that scanning is of either the source (asin the scanning electron microscope) or the image (as with a CCD or CMOS camera), and

Figure 9.23 The chromaticity triangle is a convenient representation of colours present in thevisible spectrum. A chosen colour within the triangle can be selected by combining differentintensities of the three primary colours taken from the corners of the triangle. Colour monitorscover a colour range limited by the response of the available blue, red and green phosphors.Colour printers are limited by the absorption response of the four or more pigments used.(See colour plate section)


that we are less interested in collecting digital data in reciprocal space (as in a diffractionpattern or particle size analyser). The emphasis in this section is on the quantitative analysisof the digital data (compare Section 3.5).

9.4.1 Digital Image Recording

Several problems have to be solved if digitized data are to be analysed quantitatively andprovide unbiased estimates of microstructural parameters. The first condition is that thenumerical value assigned to a particular pixel should be linearly related to the physicalprocesses that occur in the section plane of the specimen. In the optical microscope, it is thedistributionof the light intensity in the imageplane thatdetermines thecontrast,while inTEMit is the electron current distribution across the imageplane for the thin-film sample that givesthe contrast. In scanning electronmicroscopy (SEM) the signal recorded in the �image� plane,the signal recordedbythecollectoras theprobebeamrastersacross thespecimensurface,maycorrespond to secondary electron emission, backscattered electrons or characteristic X-rays.

Photographic recording of a light-optical image is a nonlinear process that could onlygive a linear correspondence between the blackening of the emulsion and the incidentintensity over a very limited intensity range (Section 3.2.4.2). High energy irradiation, byeither X-rays or electrons, does give a linear photographic response (the number of silvergrains in the developed emulsion is then linearly dependent on the incident dose), but onlyas long as there is no overlap of the silvergrains that have been developed.Digitized images,obtained by scanning an archived photographic recording, should be treated with caution,although the linear range of response available in a transparency (a photographic �negative�)when viewed in transmission is far better than can be obtained from a printed photographthat has to be viewed in reflection.

The resolution of an image-scanning system is usually quoted in dots per inch (dpi) andindicates the maximum density of pixels that the system can record, irrespective of eitherthe density of pixels in the image to be scanned, or the density of pixels needed to retain thebest resolution of the microscope. For quantitative image analysis, the integer valueassigned to each pixel should be regarded as an estimate of the information present inthe pixelated image. The selected pixel density is determined by three factors:

1. The minimum density of pixels that is required to resolve the features of interest.2. The statistical accuracy desired for the value to be assigned to each pixel, which

determines the dynamic range that is selected.3. The processing speed and storage capacity of the computer and data transmission

facilities. Given the low price of storage media and computer RAM, mostmicroscopists tend to use themaximumpixel density that the imaging systemcan provide.

It is important to recognize that the digitized image, in which the numerical intensityvalue assigned to each pixel is a data point, is only the raw data. Prior to undertaking anyquantitative analysis, the raw data may need to be �cleaned-up�, and a number of imageprocessing algorithms are available for this. The contrast may be adjusted, background canbe subtracted, the data can be �smoothed�, edge effects can be enhanced and variousmathematical functions are available for more complex manipulation of the digitizedraw image data. It is also possible for the operator to �manipulate� the image data bydeleting recognized artifacts, such as scratches, or inserting �missing� features, for


example, unetched grain boundaries. In all such operations there is a significant danger ofoperator bias, often combined with some unjustified wishful thinking!

To summarize:

1. There is no substitute for good sample preparation!2. The digitized image should be recorded so that the digitized values are linearly related to

the physical phenomena that are responsible for image contrast.3. The digitized image data should be processed to improve the information content before

undertaking any quantitative image analysis.

9.4.2 Statistical Significance and Microstructural Relevance

In this chapter we have pointed out repeatedly the difficulties attached to quantifyinginformation contained in a two-dimensional micrograph. The transition that we would liketo make from a value judgement, describing a material as �fine-grained�, to a quantitativestatement, that the material has �a grain size of 0.4 mm�, represents additional information.Any attempt to improve the accuracy ofmicrostructural characterization is to be applauded,but the imaging of a specimen is a complex process. The relation between the mechanismsof contrast formation in the image, the bulkmicrostructural features that are responsible forthis contrast, and the engineering properties of the original component are certainly not self-evident.

The results of a careful quantitative microstructural analysis may be statisticallysignificant, yet prove to be irrelevant for the engineering problem that is being investigated.Excellent examples of this phenomenon are to be found in the study of mechanicalproperties that are associated with the presence of �defects� (fracture strength, ductility,notch impact energy and fracture toughness). These properties are all very sensitive to thesize of the defects and the angle between the plane of the defects and the axis of an appliedtensile load. It is the largest defects that have the greatest effect, and therefore no mea-surement of either the average size or the size distribution of the defects can possibly becorrelated successfully with the mechanical properties. Furthermore, each class of engi-neering defect, microcracks, soft inclusions, hard inclusions or porosity, affects theproperties differently, posing additional problems in establishing the relevance of anyquantitative microstructural analysis of the defect content of the material.

Despite this, quantifying grain or particle size, measuring the volume fraction of a secondphase, and determining the extent ofmorphological anisotropy are still important objectivesof microstructural characterization, certainly in research, but also for material and processdevelopment, the improvement of criteria for materials acceptance and for product qualitycontrol. Microstructural characterization is an essential part of technological development,and the quantitative characterization of microstructure is a legitimate objective, but is onlyjustified if the results are relevant to the investigation.

9.5 Tomography and Three-Dimensional Reconstruction

We have noted that there is a wide range of three-dimensional microstructural data thatcannot be interpreted quantitatively from a two-dimensional section, unless we make very


serious simplifying assumptions about the microstructural geometry of the bulk material.A good example would be the interpenetrating two-phase structures that are often typicalof binary eutectics and some spinodal systems. Interconnecting pore structures that arecharacteristic of the earliest stages of sintering have similar topological properties, as domany materials used as particulate filters, catalyst substrates and porous anodes for solid-state capacitors.

For all such cases, it would be highly desirable to be able to view the microstructure inthree dimensions, rather than to rely on cross-sections that have been selected for two-dimensional microstructural characterization. Biologists have been leaders in developingsectioning and data-processing techniques that are designed to provide insight into thestructure of three-dimensional objects, both for the study of anatomyon themacroscale, andfor histology (the microstructure of soft tissues and cells). A term commonly used for thesemethods of three-dimensional microstructural investigation is tomography, from the Greektomo meaning a section, and graphos meaning a picture. In the present scientific usage ofthe term, tomography is the development of a three-dimensional image data file that can beinterrogated by a computer software program, in order to generate three-dimensionalimages and image sections in any selected projection or orientation and from any selectedslice taken from the three-dimensional data set.

9.5.1 Presentation of Tomographic Data

Tomographic data can be collected from a very wide range of imaging technologies, but inpractice the data that constitute the rawmaterial for tomographic analysis are always relatedto a sequence of two-dimensional projections and constitute a series of consecutive, two-dimensional data sets. A simple case would be the optical imaging of a rough surface inreflection from a through-focus series. The image data are analysed using an algorithm thatfilters the spatial frequencies in each image, and then combines the images. Those regionsthat are out of focus in any one imagewill appear blurred and can only contribute to the lowspatial frequencies of that image. These low frequencies can be removed from a Fouriertransform of the pixelated intensity data for each image. The remaining high-frequency, in-focus portions of each image can be combined, colour coding or contourmapping each dataset to provide a striking, high-contrast image in which the z-resolution perpendicular to thex–y plane is determined by the size of the defocus steps, rather than by the numericalaperture of the optical microscope objective lens (Figure 9.24).

Confocal microscopy, in which a focused and scanned light probe generates ascattered signal from features in a slice within an otherwise transparent sample, can beused to collect a series of data sets, so that each set constitutes an �optical slice� taken fromdifferent depths in the specimen. Soft tissues that have been suitably labelled by afluorescing molecule can be imaged using the fluorescent excitation. Resolutions wellbelow the exciting wavelength have been achieved. Again, the data sets from the series ofconfocal optical slices can be combined into a single three-dimensional data set andprocessed to provide quantitative information on the bulk microstructure. Using a second,picosecond laser the fluorescent emission can be partially quenched, improving theresolution for the excited regions to of the order of 30–40 nm, well below the diffractionlimit and dramatically improving the image resolution available for a histologicalexamination.


The availability of powerful X-ray synchrotron sources has been used to collectradiographic projection data of samples. The data have been translated and rotated togenerate two-dimensional data sets for suitably oriented projections which can be com-bined into a three-dimensional image, using software identical to that employed forcomputer-aided tomographic scans (CAT scans) in medical imaging. Many years ago amicro-focus X-ray source was used to probe the bulk structure of an interpenetratingeutectic, Al–Sn bearing alloy by micro-radiography, but at that time the computer and dataprocessing facilities were not available to extract quantitative three-dimensional informa-tion from the micro-radiograph. Figure 9.25 shows a more recent example of a dendriticstructure taken using a synchrotron source.

Tomographic data can be presented in several ways. Using computer graphics the three-dimensional data set can be rotated, either continuously or in steps, so that a completeprojection sequence is visible to the observer on a monitor screen. Alternatively, slices ofselected thickness, taken from the data in a chosen projection, can be viewed, and the slicedisplaced in steps, either in the x–y plane or in the z direction, perpendicular to the viewingscreen, in order to explore the full range of the data set. It is also possible to �zoom� into thedata set, in order to enlarge a specific volume element that may be of particular interest.

It is not always the intensities of the individual pixels in the three-dimensional data setthat provides the best view of the data, and in some cases the gradient of intensity ishighlighted, by differentiating the intensity over a set of neighbouring pixels in order toderive a smoothed intensity gradient. This may be particularly useful when it is the phaseboundaries in the microstructure, rather than the second-phase particles that are of interest.Thismay also be the case if wewish to �see through� the second-phase particles, and inmanycases it may be useful to make the particles semi-transparent, so that we can clearly

Figure 9.24 Example of a reconstructed optical microscope image from a defocus series of agold pad on a circuit board processed using Fourier filtering. (Courtesy of Syncroscopy).


distinguish the particle volume, while simultaneously placing the particle boundary in highcontrast. This means, combining the readout of the different intensity values for the pixelsthat fall within the particles, together with a high contrast for the pixels at the phaseboundaries, identified by the intensity differences between the two phases.

9.5.2 Methods of Serial Sectioning

Automated serial sectioning has now been applied to just about every possible method ofmicrostructural characterization that has been employed inmaterials science. In the presentsection we will consider these methods for serial sectioning and, especially, the minimumspacing of the slices in each case. We will also discuss the origin of artifacts and theattainable three-dimensional resolution.

Mechanical cutting and grinding have always been the primary tools for sectioningengineering components, while mechanical polishing and chemical etching have been thepreferred methods of surface preparation for optical reflection microscopy. Early attemptsto prepare parallel serial sections by mechanical grinding were handicapped by lack ofcontrol of the thickness removed at each stage. Automated, computer-controlled polishingsystems have resulted inmajor improvements, and the control of thickness removal to of theorder of a few micrometres has been claimed for serial sections taken 10–100 mm apart.Diamond microtomes (�ultra-microtomes�) have been used for many years to prepare serialsections of embedded, histological samples with thicknesses down to about 10 nm, andrecent reports have used the same method to prepare serial sections for atomic forcemicroscopy (albeit, only at low resolutions and for materials of low elastic modulus).

Focused ion beam (FIB) milling (Section 5.4.4) is proving to be a remarkably flexibletechnique, not just for sample preparation, but also for serial sectioning, with fine-tuning of

Figure 9.25 Synchrotron micro-radiograph of a Sn–Bi alloy, recorded in-situ duringsolidification, with a resolution of a few micrometres. Reprinted from B. Li, H.D. Brody, andA. Kazimirov, Synchrotron Microradiography of Temperature Gradient Zone Melting inDirectional Solidification, Metallurgical and Materials Transactions A, 37, 1039–1044,2006, with permission from The Minerals, Metals, & Materials Society.


the slice thickness to of the order of 10 nm. The FIB-milled sections can be viewed in situusing secondary electron scanning images and two-dimensional digital data sets can berecorded from successive layers. In dual-beam FIB instruments, continuous scanningelectron microscopy has been used to monitor the progress of the milling process and,if required, successive slices can be removed from orthogonal planes in the sample. SuchFIB-generated three-dimensional data sets are therefore obtained from separate volumeelements that have been sectioned along different coordinate axes, effectively eliminatingany bias associated with interpolation of two-dimensional data collected perpendicular to asingle axis aligned along the milling direction.

The major disadvantage of ion beam milling is associated with the side effects ofradiation damage. These are primarily due to the injection of point defects that condenseinto clusters and dislocation loops, and can result in diffusion-induced composition andphase changes. The damage also includes displacement of individual atoms by focused-collision sequences over distances of the order of 10 nm. Alloying elements are dispersed,both by enhanced diffusion and by such knock-on collisions. Thin layers that have beendeposited on a substrate may experience considerable blurring of the original, as-depositedconcentration gradients.

Depth profiling in X-ray spectroscopy and Auger electron spectroscopy also relies onsputtering under ion bombardment to remove successive atomic layers (Sections 8.1 and8.2), but only Auger spectroscopy has the spatial resolution needed to collect two-dimensional image data from the individual layers. In principle, adequate calibration ofthe depth removed at each stage would make it possible to record and process a three-dimensional digital-data image file, but this has not yet been accomplished. Surfaceroughness due to the sputtering process and radiation damage induced blurring of theconcentration gradients destroy the planarity of the surface and limit the spatial resolution inthe third dimension. Similar problems have prevented secondary-ion mass spectrometryfrom achieving good three-dimensional resolution.

Atom probe tomography (Section 7.3.2), is the only serial sectioning method of three-dimensional, nanostructural characterization that accurately identifies atomic species andvariations in concentration at the resolutions needed to study boundary segregation and theearliest stages of second-phase nucleation. Unfortunately, the engineering materials thatcan be studied by atomic probe tomography have to possess some electrical conductivityandmust be able towithstand themechanical stresses imposed by thevery high electric fieldstrengths at the specimen tip. It should also be noted that, since the ion collection efficiencyis usually of the order of 50–60%, approximately half of the atoms in the volume sampledby the data set are missing.

9.5.3 Three-Dimensional Reconstruction

Collecting the digitized, two-dimensional image data from a set of serial sections is the firststage in preparing a three-dimensional data set for subsequent viewing and analysis. Thesecond stage is to bin the data in such away that the three-dimensional pixels each representelements of known volume in the material being studied. Since the spacing of the serialsections is often significantly larger than the lateral resolution, it may be necessary tointerpolate �virtual� data points by averaging intensity or counts from sets of pixels inneighbouring sections. For some techniques, fiducial markers are used in order to ensure


that successive x–y sections are accurately positioned perpendicular to the z-axis. Hardnessor micro-hardness indentations are a convenient form of fiducial marker for slicethicknesses down to 1 mm or even less since they are readily located. The depth and widthof the impression allows slice thicknesses to bemonitored,with thicknesses ranging up to aslarge as 0.1mm. Additional indentations can be made as the sectioning proceeds.

Most manipulation of the three-dimensional data sets involves rotation, translation and�zoom� functions (magnification), and these functions can of course be combined. Animportant case is that of the atom probe, and the radius of the sample tip increases as atomsare field-evaporated from the surface, reducing the average magnification. Adjusting theinitial magnification scale of the two-dimensional data sets to the uniformmagnification ofthe three-dimensional grid only takes care of part of the problem, since themagnification ofa projected field-evaporated ion image also reflects variations in radius of curvature over thesample tip that are due to local crystallographic features. The densely packed, low indexplanes at the surface generally result in a larger local radius, that is, lower curvature, than theless densely packed regions. Published tomographic images usually, but not always,incorporate these corrections for local variations in magnification.

The restructured, three-dimensional data set taken from a sequence of two-dimensionaldata sets that has been derived from serial sections can be viewed in several ways. Twoclasses of manipulation are basic to viewing comfort on a computer screen. The firstconcerns the processing of the data set itself. The removal of background, smoothing thedata by averaging the contents of the three-dimensional pixels, or selecting intensitygradients, in order to define the phase or grain boundaries are all basic operations neededwhen interpreting the three-dimensional image. The second form of data manipulationconcerns the viewing of the data on a monitor. Rotation, translation and zoom are basicoperations, but for very large data sets it may also be useful to add perspective to the image,for example by reducing the magnification and intensity for data points that are �distant�from the observer in order towiden the field of view for the more �distant� features, while atthe same time reducing their contrast.

In this final section on three-dimensional reconstruction we have not mentionedquantitative analysis, mostly because very little has been published on the subject. Onegood reason for this is statistical, since even the millions of ions collected in the atom probedo not usually result in accurate statistics for particle size or phase volume fraction. As dual-beam FIB technology becomes more widely available, we are likely to see increasing usemade of this technique for serial sectioning of integrated electro-optical and electronicdevices, and for the microstructural analysis of polyphase materials, especially compositesand cast or welded structures. The �inaccessible� microstructural parameters that weremodelled approximately from the ambiguity of a single two-dimensional section will soonbe visible in three-dimensional images generated by reconstruction.

Summary

The quantitative analysis of image data requires an understanding of the stereologicalrelations between the two-dimensional data, recorded from a projection image andthe three-dimensional microstructure of the bulk sample before sectioning. To thisknowledge of stereology must be added an appreciation of the statistical errors that are


associated with sample selection, sectioning of the sample and digital-image, datacollection. The objective is always to optimize the statistical significance of image datawith respect to both the effort involved in collecting the data and the statistical accuracyneeded for the analysis.

The isotropy and homogeneity of thematerial are important microstructural features. Theproperties of most engineering materials are, to some extent, usually both anisotropic andinhomogeneous, and this is reflected in theirmicrostructure.Crystallographic anisotropy isassociated with the preferred alignment of specific crystallographic planes and directionswith respect to the coordinates of the component, whilemorphological anisotropy refers tothe spatial alignment of morphological features (grains or particles, or ordered arrays ofparticles and inclusions). Inhomogeneitymay be chemical, associated with local variationsin composition, or morphological, associated with local variations in grain or particlesize.

A single, planar section is often insufficient to characterize a material microstructure.The position and the orientation of any chosen sample sectionmust be known, and should beselected with respect to the principal axes of the component. The microstructure may alsovary across the selected section (inhomogeneity), as well as with the angle that this sectionmakes with the principal axes of the component (anisotropy). The results of a statisticalanalysis of anymicrostructural parameter may reflectmacroscopic variability in the bulk ofthe component,mesoscopic variability over any given cross-section, ormicroscopic variabil-ity within the observed field of view visible at a given magnification in the microscope.

Statistical variations in an analysis arise from several sources. That of primary interest isgenerally the inherent variability of the parameter being measured: the spread of grain sizeor the non uniformdistribution of a second phase. Sampling errors, overwhich some controlis possible, are associated with the number of features sampled, the number of recordedmeasurements, the location of the samples selected and the plane chosen for the section.There will also be experimental errors involving the quality of the specimen preparation(polishing and etching procedures), the resolution of the microscope, and the recording ofthe image (the total number of pixels in the image and the dynamic range of their measuredintensity values).

It is important to distinguish between accessible and inaccessible microstructuralparameters. Accessible parameters can be determined unambiguously from a two-dimensional section, without making any stereological assumptions about the microstruc-tural morphology. Inaccessible parameters require some geometrical model in order tointerpret the bulk structure (for example, by assuming spherical particles). The volumefraction of a second phase is a straightforward example of an accessible parameter that canbe determined from a surface section to any required degree of accuracy, limited only by theresolution of the imaging system. Providing grain size and particle size are defined solely interms of surface-to-volume ratio, these parameters are also readily accessible.

The surface-to-volume ratio is an accessible parameter, but it is independent of particleshape. Needles, platelets and dendritic shapes are not distinguished, and measures ofparticle shape, derived from two-dimensional image data, require a stereological model ofthe bulk geometry for their interpretation in three dimensions. The same applies to theinterpretation of particle or grain size distributions, as well as to dislocation arrays. In thecase of second-phase particles, it is sometimes useful to assume convexity, that is, all regionshave positive curvature, since no convex particle can then intercept a planar section more


than once. Composite materials, especially those that include highly textured, wovenreinforcement, can present a serious descriptive problem.

Errors of quantitative analysis that are associated with specimen preparation and therecording of image data can be minimized by careful experimental work that is based on anawareness of the origin of the possible artifacts. Counting errors are usually reduced byensuring that the sample selected is large enough to ensure that any bulk variability in themicrostructure is readily detectable above the level of the random, statistical background.Since both the microscopic variability, within a given area, and the mesoscopic variability,from one area to another, contribute to the statistical errors, it is important to ensure thatseveral areas of the specimen are sampled. For example, the average number of particles orgrains that are observed in a specific area should be comparable with the number of sampleareas selected, so as to ensure the minimum statistical variance for any given countingeffort.

The resolution errors of the method of microstructural investigation will limit theaccuracy if the size of the features of interest approaches the resolution limit of themicroscope. Specimen preparation always introduces a thickness or sectioning error that isassociated with the depth of the surface layer over which the sample contributes contrast tofeatures in the projected image. In optical microscopy this slab thickness is of the order ofthe resolution, since the depth of field of the objective lens is also of the order of theresolution. In both scanning and transmission electron microscopy the depth of field isorders of magnitude greater than the resolution limit of the microscope, and thicknesscorrections for quantitative analysis can be very important.

Observer bias is also a major source of variability in quantitative analysis, and mayreflect more than a varying level of experimental competence. Two observers can arrive atsignificantly different quantitative estimates of the same microstructural parameter, as aresult of entirely justifiable differences in professional judgement.

Digital data collection and automated image analysis have not eliminated observer bias.The settings for any computer program are always based on the professional judgement ofthe operator. Fully computerized systems have increased the rate of data collection byorders of magnitude, making quantitative stereological analysis of microstructural mor-phology readily available, reducing the effort required to collect large data sets and achievestatistical significance. However, the wide choice of computer software programs can leadto some confusion, since some programs fail to explain clearly the stereological assump-tions on which they are based. The ease of data processing may also obscure the stepsinvolved in treating digital data. This may degrade the statistical significance during digitaldata processing and the presentation of the final results of analysis.Major errorsmay be dueto careless specimen selection and preparation, and no amount of subsequent automatedimage analysis can �correct� for such experimental sloppiness.

Three-dimensional imaging is now a reality for a range ofmagnifications and resolutionsthat include simple serial-sectioning, by automated grinding and polishing, ion beammilling, with controlled thickness removal to an accuracy of 10 nm, and culminates in thethree-dimensional reconstruction of sample chemistry on the nanoscale by atom probetomography. The assembly of two-dimensional digital data sets from a series of sections ofknown spacing into a three-dimensional matrix of volume element pixels requiresmanipulation of the raw, two-dimensional data to ensure that the three-dimensional matrixof image data is free of both background noise and spatial distortion. The reconstructed


three-dimensional image must be visualized on a computer screen by employingsoftware tools that are able to rotate, translate and magnify the image in order to allowthe observer to select specific data subsets and to introduce perspective. In three-dimensional imaging it is proving possible to study many features that are interconnectedin the bulk microstructure. When sectioned by a planar sample surface, such featuresbecome inaccessible and, until recently, have been beyond the reach of quantitativemicrostructural analysis.

Bibliography

1. J. C. Russ, The Image Processing Handbook, 2nd Edition, CRC Press, London, 1995.2. J. C. Russ and R. T. Dehoff, Practical Stereology, Kluwer Academic/Plenum, New York, 2000.3. J. E. Hilliard and L. R. Lawson, Stereology and Stochastic Geometry (Computational Imaging

and Vision), Kluwer Academic, New York, 2001.4. A. Baddeley and E. B.Vedel Jensen, Stereology for Statisticians, Chapman &Hall/CRC, Boca

Raton, FL, 2005.

Worked Examples

Three examples of size measurements for microstructural features on differentlength scales taken from different microstructures, will demonstrate the principles ofquantitative measurement at the meso, micro and nano morphological levels. Thefirst example is the size of the alumina grains in a sintered body, expected to be in themicrometre range. For these measurements we employ SEM. The second example isprovided by the much smaller size of aluminium grains developed by nucleation andgrowth during chemical vapour deposition. The final example explores the limit ofdetection of TEM in the measurement of ordered domains in a disordered matrix of Pb(Mg1/3,Nb2/3)O3.

Figure 9.26 shows two micrographs of sintered alumina; the first after sintering at1400 �C for 2 h, and the second after sintering at 1600 �C for 10 h. Sintering of ceramics isalways a compromise: on the one hand, the density of the sintered product should bemaximized, which generally requires a high sintering temperature and long sintering times.On the other hand, grain growth should beminimized, usually by limiting both the sinteringtemperature and the sintering time.Optimizing the sintering process involvesmeasuring theresidual porosity and the grain size as a function of sintering time and temperature.

As far as the grain morphology is concerned, two basic questions need to be answered:how does the average grain size depend on the sintering parameters, and what is the shape(aspect ratio) of the alumina grains? To quantify the grain size we apply the linearintercept method (discussed in Section 9.2.1.2). Figure 9.27 shows the processed imagesof Figure 9.26, and the average values of the grain intercept. As expected, a large increasein grain size has occurred after sintering at 1600 �C for 10 h, as compared with sintering at1400 �C for 2 h. By quantifying the grain size as a function of the process parameters, wecan characterize the sintering mechanisms empirically, and so optimize the sinteringprocess.


We now turn to our chemical vapour deposition aluminium samples. We need todetermine the size of the aluminium grains as a function of the deposition time, in orderto assess the influence of some of the process parameters. SEM micrographs of thealuminium grains formed on two different TiN substrates, as a function of the depositiontime, are shown in Figure 9.28. Figure 9.29 shows an example of a processed image, inwhich the aluminium grains are distinguished from the TiN background by their contrast.The nominal aluminium grain size, measured as the projected area of each grain, is afunction of the deposition time for both TiN substrates and is given in Figure 9.30. As in thecase of the previous, alumina sample, employing a computer program can dramaticallyincrease the sample size, and hence improve the statistical significance of the results.

A final example, from a completely different material system, demonstrates thecombination of quantitative microstructural analysis combined with elemental detectionlimits. The material is a Pb(Mg1/3, Nb2/3)O3 (PMN) sample that has been doped with

Figure 9.26 SEMmicrographs of thermally etched alumina, after sintering for (a) 2 h at 1400 �Cand (b) 10 h at 1600 �C.


varying amounts of lanthanum. PMN has the cubic perovskite structure shown inFigure 9.31. In the ideal structure, cations of type A occupy the corner sites of the unitcell with coordinates 0,0,0, while cations of type B occupy the body-centred position1/2,1/2,1/2, and oxygen anions are located in the face-centred sites of the cubic unit cellwithcoordinates of type 1/2,0,0. In disordered PMN there are two types of cations located at thetype A sites (Mg and Nb), while Pb occupies the type B sites. However, under certainconditions chemical ordering can take place to form a superlattice, in which distinctive{111} planes, containing either Mg or Nb cations, form a new face-centred cubic unit cell

Figure 9.27 Micrographs from Figure 9.26 after image processing for analysis. The results ofthe linear intercept method for grain size measurements are included. Note that the two grain-size distributions are in no way self-similar with respect to either the grain size or the grainshape.


Figure

9.28

SEM

micrograp

hsofaluminium

grainsdep

ositedontw

odifferentTiN

substrates,as

afunctionofthedep

ositiontime.


with a larger lattice parameter of 2a0, where a0 is the lattice parameter of the original,disordered PMN lattice, shown in Figure 9.32.

The difference between the disordered and ordered crystal structures is easily detected inthe transmission electron microscope by selected area diffraction after orienting the

Figure 9.29 (a) Original micrograph and (b) the resultant processed image. After processing,the aluminium grains are clearly distinguished from the TiN background by their contrast.


specimen into a [110] zone axis, as shown in Figure 9.33. If the ordered regionsare sufficiently large, then dark-field diffraction contrast images can be recorded usingthe {111} ordered reflections. Figure 9.34 demonstrates such a dark-field image in whichthe ordered regions, separated by anti-phase boundaries, are clearly visible.

PMN crystals may also form small ordered regions that have length scales of the order ofnanometres. Increased doping levels of lanthanum in these crystals seem to increase the sizeof the nano-ordered regions. To confirm this hypothesis, we can use TEM and imageprocessing in order to analyse the effect quantitatively.

Our first step is to select a method to observe the nano-ordered regions. The minimumsize of these regions is just a fewnanometres, so any good transmission electronmicroscopeshould have sufficient resolution. However, we have to detect and distinguish the orderedPMN regions from the disordered background. We can compare two TEM methods

Figure 9.31 The disordered PMN unit cell has the cubic perovskite structure.

Deposition time (s)

Are

a (n

m2 )

0 10 20 30 40 50 60 701

5.0x104

1.0x105

1.5x105

2.0x105

2.5x105

3.0x105

3.5x105

Integrated

Air exposed

Figure9.30 Nominal aluminiumgrain size,measured as theprojected areaof eachaluminiumgrain, as a function of the deposition time for the two TiN substrates that were used.


quantitatively, namely dark-field, diffraction contrast imaging and high-resolution trans-mission electron microscopy using phase contrast to obtain lattice images.

We have already noted that large, ordered regions can be detected in a dark-field imagetaken with the ordered {111} reflections, and nano-ordered regions can be detected in asimilar way. Micrographs showing the influence of varying lanthanum concentrations onthe scale of the ordered regions are shown in Figure 9.35, in which the bright regions

Figure 9.33 Selected area diffraction patterns, taken from a [110] zone axis, of the (a) orderedand (b) disordered PMN crystal structures.

Figure 9.32 Cation positions in the face-centred cubic unit cell of fully ordered PMN. (Theoxygen ions have been omitted for clarity).


are the ordered domains. To quantify the size of these ordered regions, wemust first processthe image to remove the background due to the disordered matrix. This gives us theimage shown in Figure 9.36. We then determine the projected area of each ordered region.The data can now be summarized as the average projected area of an ordered region andthe corresponding standard deviation, plotted as a function of lanthanum concentration(Figure 9.37).

FromFigure 9.37we immediately note a problemwith the data. The average domain areadoes show a tendency to increase with increasing lanthanum content, but the standarddeviation of the data is too large for the effect to be judged statistically significant. A majorreason for the large standard deviation is that the ordered regions are smaller than thethickness of the TEM specimen, with frequent overlap of the ordered regions, invalidatingany quantitative conclusions. We could collect data only from the thinnest regions of thespecimen, providing we could determine the thickness accurately, but instead we prefer touse high resolutionTEM, and then optimize themicroscope parameters to select the orderedregions preferentially.

What do we mean by preferential selection of the ordered regions? The periodicity andcontrast in high resolution TEM lattice images depends not only on the crystal structure, butalso on the microscope contrast transfer function (CTF). In order to detect and differentiatethe ordered regions, themicroscope operating conditions shouldmaximize the intensities ofthe superlattice reflections. Figure 9.38 shows the CTF for the microscope at the Scherzerdefocus, as well as the relative values of the structure factor for the various crystallographicplanes in the PMNcrystal superlattice.Most of the diffracted beamswhich contribute to the[110] lattice image are from the {202}, {222}, and {004} matrix lattice, while the relativestructure factors of the ordered, superlattice, reflections {111} and {113} are ofmuch lowerintensity than those of the matrix reflections. Thus, although the point resolution isoptimized by setting the objective lens current to the Scherzer defocus value, the detection

Figure 9.34 Dark-field TEM micrograph recorded using the {111} ordered PMNlattice reflection. Large fully ordered regions are separated by clearly visible anti-phaseboundaries.


of the ordered superlattice planes will be extremely difficult if we use this setting. However,we canmodify the CTF by changing the object lens defocus and increase the contribution ofone or more of the superlattice reflections to the image, at the same time suppressing thedominant, disorderedmatrix reflections. This is illustrated in Figure 9.39, in which the CTFhas been optimized for the {111} ordered reflections.

This optimized defocus has been used to record the lattice image in Figure 9.40(a).Although Figure 9.40(a) shows ordered PMNdomains under optimized conditions, it is stilldifficult to differentiate the ordered from the disordered regions, but this may be improvedby Fourier filtering of the image data. Figure 9.40(b) is a Fourier transform (FT) of theimage in Figure 9.40(a). By superimposing a mask to all frequencies in the FT other thanthose corresponding to the periodic reflections from PMN (both ordered and disorderedcrystal regions), we can remove background noise in the image [Figure 9.40(c)]. Now theordered regions are easily visible. If we further mask all frequencies in the FTother than the{111} ordered reflections, the contrast from both the background noise and the disordered

Figure 9.35 Series of dark-field TEMmicrographs of PMN showing the influence of lanthanumconcentration on the size of the nano-ordered regions: (a) 5% La; (b) 3% La; (c) 1% La; (d) 1%Laþ22.5 PT. The ordered regions appear in bright contrast.


matrix is suppressed [Figure 9.40(d)]. From images such as [Figure 9.40(d)]. we canmeasure the nano-ordered, projected area and compare the results with the nano-orderedregions seen in a dark-field diffraction contrast image (Figure 9.37). The final results areshown in Figure 9.41. These oncemore give the average ordered area and standard deviationas a function of lanthanum content, but by optimizing high resolution TEM lattice imaging,we have significantly reduced the standard error and there is now no doubt that thelanthanum doping has increased the size of the ordered regions.

Figure 9.36 A dark-field TEM image of PMN (a) processed in order to remove thebackground due to the disordered matrix (b) and obtain an image suitable for quantitativeanalysis.


The calculations required to determine the optimumdefocus, the Fourier filtering and theimage processing, may appear complex, but many computer programs now exist for suchcalculations, both commercial and �free-ware�. Use of these computer programs is notespecially difficult, and can even be a lot of fun. However, it must be remembered thatcomputerized image processing still requires the microscopist�s understanding of themechanism of image formation.

(atom %) La

Are

a (n

m2 )

8765432100

5

10

15

20

25

30

Figure 9.37 Average projected area of the ordered regions in PMN and their standarddeviation, determined by quantitative image analysis, as a function of the lanthanum dopantconcentration.

(nm–1)

109876543210-1.0

-0.5

0.0

0.5

1.0

111

002

202

113

004

313

224

333

404

315

206

335

444

515

426

008

∆ f=–50 nm

Figure 9.38 TheCTF for the high resolution transmissionelectronmicroscopeused to study thenano-ordered regions in PMN at the Scherzer defocus, as well as the relative values of thestructure factor for the various crystallographic planes in the PMN superlattice.


Problems

9.1. A pixel in a digitized image is assigned specific x,y coordinates, while the resolutionof a printer is commonly given in dots per inch (dpi). What dpi would you choose foryour printer to ensure that the intensity assigned to an image pixel is accurate towithin 10% when the separation of the pixel points in the digitized image is0.2mm?

9.2. Given that the resolution of the human eye is about 0.2mm and that images are mostcomfortably viewed from a distance of about 200mm, estimate the number of pixelsneeded to ensure that a digitized image does not appear �discontinuous� to theeye.

9.3. Distinguish between morphological and crystallographic anisotropy, and describesome experimental tests that can be used to identify both forms of anisotropy.

9.4. Define the term sampling error. The number of microstructural features in a givenfield of view and the number of fields of view selected for quantitative microscopyboth contribute to the statistical errors of measurement. Explain how you would plana quantitative analysis of the microstructure in order to ensure maximum statisticalaccuracy for minimum effort.

9.5. The number of particles of a second phase per unit volume of the sample cannot beestimated from a planar section without making some assumption about the particleshape. What is the assumption usually made and why is it necessary?

9.6. The volume fraction of a second phase and the surface-to-volume ratio are bothdescribed as accessible microstructural parameters. What is meant by this term?

(nm–1)

109876543210-1.0

-0.5

0.0

0.5

1.0

111

002

202

113

004

313

224

333

404

315

206

335

444

515

426

008

∆ f=–65 nm

Figure 9.39 The CTF for the same microscope, adjusted to study the nano-ordered regions inPMN at an optimum defocus for the detection of these regions, as well as the relative values ofthe structure factor for the various crystallographic planes in the PMN superlattice.


9.7. How is the surface-to-volume ratio related to grain size, and what factors maycontribute to the ambiguity of the concept of �grain size�?

9.8. The aspect ratio of a microstructural feature (particle, grain, inclusion, cluster, etc.)on a surface section bears no obvious or simple relationship to the shape of the featurein three dimensions. Give three examples of microstructural features that illustratethis unfortunate fact.

9.9. Sectioning errors occur in the quantitative analysis of both polished sections and thinfilms. If an etched boundary appears twice aswide as the depth of etch, can youmodelthe sectioning error as a function of grain size by assuming a boundary curvatureequal to the grain size? (There is no one �correct� answer to this question, and variousassumptions about what constitutes an intersection of the boundary with a test lineare possible.)

Figure 9.40 (a) Lattice image of the ordered PMN domains under optimized high resolutionTEM imaging conditions. (b) A FTof the lattice image shown in (a). (c) By applying a mask to allfrequencies in the FT, other than the periodic reflections from the PMN lattice (both ordered anddisordered), we can remove all background noise in the image. (d) If we further mask allfrequencies in the FT other than the {111} ordered reflections, the contrast from both thebackground noise and the disordered matrix is suppressed.


9.10. List the errors associated with estimating the density of dislocations observed bydiffraction contrast in a thin-filmTEMsample.Compare the errors associatedwith acount of the intersections of the dislocation images with a test grid, as opposed to acount of the number of dislocation intersections with the top and bottom surfaces ofthe thin film.

(atom%) La

Are

a (n

m2 )

8765432100

5

10

15

20

25

30

Figure 9.41 The average ordered area and standard deviation as a function of lanthanumcontent, measured from optimized high resolution TEM lattice images of PMN.


Appendices

Appendix 1: Useful Equations

We present here some sets of equations relating to crystallography that should be useful tothe reader.

Interplanar Spacings

The following relationships give the interplanar spacing d (sometimes referred to as thed-spacing) of the (hkl) planes in the crystal lattices of different crystal structures:

Cubic1

d2¼ h2 þ k2 þ l2

a2

Tetragonal1

d2¼ h2 þ k2

a2þ l2

c2

Orthorhombic1

d2¼ h2

a2þ k2

b2þ l2

c2

Hexagonal1

d2¼ 4

3

h2 þ hkþ k2

a2

� �þ l2

c2

Rhombohedral1

d2¼ ðh2 þ k2 þ l2Þsin2aþ 2ðhkþ klþ hlÞðcos2 a� cosaÞ

a2ð1� 3 cos2 aþ 2 cos3 aÞ

Monoclinic1

d2¼ 1

sin2bh2

a2þ k2sin2b

b2þ l2

c2� 2hl cos b

ac

� �


Triclinic1

d2¼ 1

V2 ðS11h2 þ S22k2 þ S33l

2 þ 2S12hkþ 2S23klþ S13hlÞ

In the equation for triclinic crystals, V is the cell volume, and:

S11 ¼ b2c2sin2a

S22 ¼ a2c2sin2b

S33 ¼ a2b2sin2g

S12 ¼ abc2ðcos a cos b� cos gÞ

S23 ¼ a2bcðcos b cos g� cos aÞ

S13 ¼ ab2cðcos g cos a� cos bÞ

Unit Cell Volumes

The unit cell volumeV for the various unit cells of the different crystal structures is given bythe following relationships:

Cubic V ¼ a3

Tetragonal V ¼ a2c

Orthorhombic V ¼ abc

Hexagonal V ¼ffiffiffi3

pa2c

2¼ 0:866a2c

Rhombohedral V ¼ a3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3 cos2 aþ 2 cos2 a

pMonoclinic V ¼ abc sin b

Triclinic V ¼ abcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos2 a� cos2 b� cos2 gþ 2 cos a cos b cos g

p

Interplanar Angles

The following equations give the anglef between the pole of the crystal plane (h1k1l1) witha spacing d1 and the crystal plane (h2k2l2) with a spacing d2, where V is the unit cell volume:

Cubic cos� ¼ h1h2 þ k1k2 þ l1l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðh21 þ k21 þ l21Þðh22 þ k22 þ l22Þ

q


Tetragonal cos� ¼h1h2 þ k1k2

a2 þ l1l2c2

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih21 þ k21a2 þ l21

c2

� �h22 þ k22a2 þ l22

c2

� �r

Orthorhombic cos� ¼h1h2a2 þ k1k2

b2þ l1l2

c2

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih21a2 þ

k21b2þ l21

c2

� �h22a2 þ

k22b2þ l22

c2

� �r

Hexagonal cos� ¼ h1h2 þ k1k2 þ 12 h1k2 þ h2k1ð Þþ 3a2

4c2 l1l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih21 þ k21 þ h1k1 þ 3a2

4c2 l21

� �h22 þ k22 þ h2k2 þ 3a2

4c2 l22

� �qRhombohedral

cos�¼ a4d1d2V2

sin2 aðh1h2 þ k1k2 þ l1l2Þþðcos2 a� cos aÞðk1l2 þ k2l1 þ l1h2 þ l2h1 þ h1k2 þ h2k1Þ

" #

Monoclinic cos� ¼ d1d2

sin2 bh1h2a2

þ k1k2sin2b

b2þ l1l2

c2� ðl1h2 þ l2h1Þ

accos b

�

Triclinic cos� ¼ d1d2V2

S11h1h2 þ S22k1k2 þ S33l1l2

þ S23ðk1l2 þ k2l1Þþ S13ðl1h2 þ l2h1Þþ S12ðh1k2 þ h2k1Þ

" #

Direction Perpendicular to a Crystal Plane

The formulae in Table A1 define the conditions that need to be met in order for a crystaldirection [uvw] to be perpendicular to a crystal plane (hkl).

Table A1 Directions perpendicular to a crystal plane for the different crystal systems.

Crystal system [uvw], given (hkl ) (hkl ), given [uvw]

Cubicu

h¼ v

k¼ w

lh

u¼ k

v¼ l

w

Tetragonalu

h¼ v

k¼ w

l

c

a

� �2 h

u¼ k

v¼ l

w

a

c

� �2

Orthorhombicu

ha2 ¼ v

kb2 ¼ w

lc2

h

ua2¼ k

vb2¼ l

wc2

Hexagonalu

2kþ h¼ v

hþ 2k¼ 2wc2

3la2h

2u� v¼ k

2v� u¼ l

2wðc=aÞ2

Rhombohedralu

h sin2aþðkþ lÞðcos2a� cos aÞ ¼v

k sin2aþðlþ hÞðcos2a� cos aÞ ¼w

l sin2aþðhþ kÞðcos2a� cos aÞ

h

uþðvþwÞcos a ¼

k

vþðwþ uÞcos a ¼

l

wþðuþ vÞcos a(Continued)

Appendices 519

Hexagonal Unit Cells

While we prefer to work with four indices when describing directions and planes forhexagonal structures, in order to reflect the hexagonal symmetry, there aremany cases in theliterature that use only three indices. Table A2 summarizes the relationships needed toconvert between three and four indices.

Crystal system [uvw], given (hkl ) (hkl ), given [uvw]

Monoclinic u

hb2c2 � lab2c cos b¼

v

ka2c2 � sin2b¼

w

la2b2 � hab2c cos b

h

ua2 þwca cosb¼

k

vb2¼ l

uca cos bþwc2

Triclinicu

hS11 þ kS12 þ lS13¼

v

hS12 þ kS22 þ lS23¼

w

hS13 þ kS23 þ lS33

h

ua2 þ vab cos g þwca cos b¼

k

uab cos g þ vb2 þwbc cos a¼

l

uca cos bþ vbc cos aþwc2

Table A2 Transforming between three and four crystal indicesfor hexagonal crystal systems.

Three indices Four indices

[UVW] [uvtw](HKL) (hkil)

PlanesHKL h¼H

k¼Kl¼ L

i¼�(HþK)

DirectionsU¼ 2uþ v u¼ 1/3(2U�V)V¼ 2vþ u v¼ 1/3(2V�U)W¼w t¼�1/3(UþV)

w¼W

Table A1 (Continued)


The Zone Axis of Two Planes in the Hexagonal System

The zone axis of two planes in the hexagonal system is given by:

u ¼ l2ð2k1 þ h1Þ� l1ð2k2 þ h2Þ

v ¼ l1ð2h2 þ k2Þ� l2ð2h1 þ k1Þ

t ¼ �ðuþ vÞ

w ¼ 3ðh1k2 � h2k1ÞAppendix 2: Wavelengths

Relativistic Electron Wavelengths

For an electron of energy E (kV) and a wavelength l (nm):

l ¼ h 2meeE 1þ eE

2meC2

� �� 12

¼ 0:03877ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEð1þ 0:9788 · 10� 3EÞ

qand thus:

E (kV) l (nm)

100 0.0037200 0.002507300 0.001968400 0.0016431000 0.0008715

X-Ray Wavelengths for Typical X-Ray Sources

Table A3 provides a list of wavelengths from sources typically used in X-ray powderdiffractometers.

Table A3 Target element and X-ray wavelengths for Ka1 and Ka2(in nm).

Target element Ka1 Ka2

Fe 0.1936042 0.1939980Co 0.1788965 0.1792850Ni 0.1657910 0.1661747Cu 0.1540562 0.1544390Mo 0.0709300 0.0713590

Appendices 521

Index

Page numbers in italics indicate figures. Page numbers in bold indicate tables.

3D imaging see tomography

Abbe equation 130–131, 130, 132

aberrations

electron lenses

chromatic 265

spherical 188–189

correction 189, 190, 212

function 210

optical lenses 127–128

absorption

coefficient 65, 350

edges 70, 350

effects on contrast 228

of visible light 149

of X-rays 70

cubic lattices 65

spectral correction 347–349

accessible parameters 467–468

achromat 128, 137

acicular, definition 19

AFM see atomic force microscopy

alloys

heat treating 7

sample preparation 147–148. see also steel

alumina 106

electron energy loss spectroscopy (EELS) 376–

381

electron microscopy 238–241

energy-dispersive spectrometry (EDS) 375–376,

377

grain size determination 503

optical microscopy 175–176

powder diffraction 105

scanning electron microscope (SEM) 318–321

transmission electron microscope (TEM)

micrograph 240

aluminium

as electron diffraction calibrant 109, 110

electron diffraction pattern 113

grain size determination 504

reflectivity 149

American Society for Testing Materials (ASTM) 21

amorphous phases 30

amplitude-phase diagram 82, 213

electron diffraction 213–215

analyte see sample

anion 26

anisotropy

crystallographic 75, 89, 459–461, 479

morphological 459–461

optical, and polarized light 157–163

annular dark-field detector 235

APB (anti-phase boundary) 217–218

aperture

electron microscope, and diffraction 209

function 210

optical microscope 132–133, 137

and image brightness 137

in interference microscopy 154

and polarized light 159

apochromatic lens 128, 137

applications

atomic force microscopy (AFM) 407–408

scanning probe microscopy (SPM) 391

X-ray photoelectron spectroscopy (XPS) 431

APT (atom probe tomography) 413–414, 416, 418

areal analysis 469, 471

and volume fraction 484

aspect ratio

of grains and particles 477–481

of scanning electron microscope (SEM)

images 166

astigmatism, electron lens 191

ASTM grain size 21

atom probe 401

atom probe tomography (APT) 413–414, 416, 418

atomic absorption spectrometry (AAS) 423

atomic force microscope 18, 403

tapping mode 408

atomic force microscopy (AFM) 409

as scanning tunnelling microscope (STM) 409


atomic force microscopy (AFM) (Continued)

compared with scanning tunnelling microscopy

(STM) 407

development 403

instrument 403–405

atomic number

and Auger excitation 439

and inner shell electron excitation energy 69

and secondary ion yield in secondary ion mass

spectrometry (SIMS) 443

imaging and contrast 277

atomic number correction, X-ray quantitation 345–347

atomic scattering factor 80, 205

Auger electrons 281, 425

effect of atomic number on excitation 439

emission 431–432

Auger electron spectroscopy (AES) 431–440

overview 431–433

worked example 448–452

Auger excitation, potential image resolution 433

Auger imaging 438

automated image analysis 491

background

correction

in electron energy loss spectroscopy 363

electron microscope 15

reduction, image analysis 511

subtraction, wavelength-dispersive spectrometry

(WDS) 380

X-ray analysis 335, 353–354

thin films 355


removal 425. see also noise

backscattered electrons 277–280, 278. see also

Rutherford backscattering

diffraction 278, 289

patterns 290

bend contours

in transmission microscopy 215

binocular viewing 139

biological samples, in transmission electron

microscope (TEM) 237

bit depth (of digital image) 168

Bitmap (BMP) digital images 168

Bloch waves 226–227

and stacking faults 227–230

body-centred cubic (BCC) lattice 59, 64

bonding

conduction 28

covalent 27–28

ionic 26–27, 26

secondary 29–30

two-body model 392–393

boron 26

boundary

grain 241, 378

special 293

particle 175

phase 232–233, 233

Bragg equations 58–59, 94, 204

and diffraction grating 131

vector representation 61

Bragg-Brentano diffractometer 72, 74, 86

counting time 105–106

Bravais lattices 32

scattering 57–59

Brermmstrahlung 67

Brewster angle 149

bright-field electron detector 235

bright-field optical image 150

bright-field transmission image 204

brightness, and intensity in digital images 166–167

Burgers vector 21

of a dislocation 21

calibration

atom probe tomography (APT) 417


secondary ion mass spectrometry (SIMS)

444–445

caliper diameter 19

camera length 96

electron microscope 96

cantilever probes 405

in scanning probe microscopy 405

carbon 381

carbon coating 201

Castaing, Raymond 199

cathodoluminescence 288

cation 26

CCD see charge-coupled device

cementite 46–47

ceramics 125

sample preparation 148, 149

characteristic (spectral peak) intensity 344

characteristic X-ray excitation 67, 269

characteristic X-ray images 271

charge compensation, at low beam voltages

287

charge-coupled device (CCD)

data storage 167

in field-ion microscopy 414

optical microscopy 139

chemical analysis

Auger electron spectrometry 431–440

overview 424–425

secondary ion mass spectrometry 440–446

X-ray spectrometry 343–357, 424–431

spectral corrections 343–353

524 Index

chemical binding sites, and Auger electron

spectroscopy (AES) 435

chemical concentration map 273–275, 276

chromatic aberration

electron lens 189–191

optical lens 127–128

chromaticity triangle 493

CMOS (complementary metal oxide

semiconductor) 139

and data storage 167

coefficient of variance 483

coherence envelopes, electron microscope 210, 211

coherency strains 219–221

colour

in chemical concentration maps 273–275

in digital images 168, 493

filtering 165

in scanning electron microscope (SEM)

images 261

colour centres 28

colour coding, preferred orientation 293

complex plane 83

composites

morphological anisotropy 459–460

sample preparation 148

electron microscopy 198

comparative performance: transmission and scanning

microscopy 92

compression, digital images 168–169

computer aided tomography 497

computer simulation, lattice images 231–232

condenser aperture 135

in optical microscope 135

condenser lens, optical microscope 135

conductive coatings 295

for scanning electron microscope samples 295

confocal microscopy 152

and tomography 496

contact region (of two surfaces) 394, 395

contamination (of sample)

and carbon spectral peaks in elemental

analysis 381

electron microscope 193, 195, 275–276, 317


contrast

electron microscopy

backscatter 279–280, 291

diffraction 204, 205–207, 213–215, 221–227

from dislocations 218–219

from lattice defects 215

point defects 219–221

from stacking faults and anti-phase

boundaries 216–218

enhancement, in scanning electron

microscopy 295

from dislocations 477

mass-thickness 205

overview 203–205

phase 207–213

optical 148–150, 170

bright-field 150–151, 150

interference microscopy 152–157

polarized light 159

contrast transfer function (CTF) 209–211, 211, 212,

510, 513

and lattice imaging 230–231

convergent beam electron diffraction 96

coordination number (of ion) 26

copper

crystals 33, 33–34, 35

unit cell 83

covalent bond in materials 27

critical illumination in optical microscopy 150

cross-section 197

preparation for transmission electron

microscopy 197

cryomicroscopy 138

crystal directions 36

crystallite see grain

crystallography, equations 517–521

crystal orientation distribution function 89

crystal systems 32

crystals

scattering 56–59

and contrast 204

structure

bonding 25–30

determination 31–38

grain see grain

lattice 31–42

defects 205–207

and electron beam phase 215–216

contrast 216–221

definition 31

hexagonal 41–42

preferred direction 36–38, 479–481. see also

anisotropy

space groups and symmetry 33–36

spacings 75

unit cells 47–49

cubic 38

face-centred 35–36, 59–60

curvature, definition 473

dark-field images

electron microscope 215–216

optical microscope 150–151

data

collection, atomic force microscopy (AFM) 406

presentation, tomography 497–498

Index 525

data (Continued)

processing, scanning electron microscope

(SEM) 264

recording, optical microscopy 139

storage 166–167

databases, powder diffraction spectra 103

de Broglie relationship 14, 185, 90–91

Debye interaction 395

defocus error 87

deformation texture 461

depth of field, optical microscope 133–134

depth of focus in optical microscopy 133

depth profiling, Auger electron spectroscopy

(AES) 437

detection limit 133, 353

in microanalysis 353

in optical imaging 133

detector

atomic force microscopy (AFM) 405

energy-dispersive spectrometry (EDS) 342–343

scanning electron microscope (SEM) 263

and secondary electron emission 284–285, 286


induced current 289

X-ray 271

scanning transmission electron microscopy

(STEM) 235


442–443

transmission electron microscope (TEM), X-ray

analysis 355

X-ray 71, 338–339

development (of photographic film) 141–142, 142

diamond 83

and electron energy loss spectroscopy (EELS)

calibration 362

differentiated signal spectrum 434

diffracted intensity 84

diffraction

contrast 203

column approximation 206

from crystal lattice defects 205

electron 90–98, 102

patterns

backscatter 289–291

definition 100

Kikuchi 96–98, 99, 100, 101

ring 94–96, 95


grating (optical) 131

spectra 55, 56

spectrum 55

standard 55

X-ray 31, 102

methods 63–67

patterns 76–79

spectra 79–90

diffractometer

electron microscope as 207–213, 208

X-ray 67–73, 74, 86, 101–102

diffusion depth

for high-energy electrons 267, 277

digital data processing 165

digital data recording 142, 494

digital data storage 166

digital images

data collection 492

dynamic range 167–170

optical 165–170

dimpled feature 8, 9

dimpling

of transmission electron microscope

specimens 196

direction (of crystal lattice) 36–38. see also

preferred orientation

directions perpendicular to crystal planes 519

disc of least confusion 127

electron lens 188

dislocation contrast in transmission electron

microscopy 218

dislocation density determination 21, 477, 490

dislocations 21–24, 22, 23

density 476–477

estimation errors 490–491

DLVO theory of surface forces 396

double diffraction 94

ductile failure 8

dynamic electron diffraction 221

dynamic range

digital images 184

eye 167

EBSD. see electron backscatter diffraction

edge dislocations 218

EDS see energy-dispersive spectrometry

EELS see electron energy loss microscopy

elastic scattering 3, 4, 4, 5

X-rays 70

electrical resistivity 7

electrochemical thinning

of electron microscope samples 198

electromagnetic focusing: thick lens effects 187

electromagnetic lens astigmatism 191

electromagnetic lenses 185

electron see scanning electron microscopy;

transmission electron microscopy

electron back scatter diffraction (EBSD) 264,

277–278

diffraction patterns 289–291

image contrast 279–280, 280

526 Index

electron beam

amplitude, and phase 213–215

diffraction see diffraction, electron

energy

and secondary electron emission 283

and X-ray emission 335–336

focussing 185–186

penetration depth 267


scattering 56

electron beam-induced current 288

electron diffraction, resolution limit 188

electron energy loss spectroscopy (EELS)

detection limit 359

overview 357–359

schematic arrangement 359

schematic diagram 359

summary 374


electron energy transitions

in energy loss spectroscopy 360

electron microscope 4–5

contrast 203–218

as diffractometer 207–213, 208

transmission, schematic diagram 180

see also scanning electron microscope;

transmission electron microscope

electron range in solids 267

electron source

Auger electron spectroscopy (AES) 433

beam coherency, and contrast 231


transmission electron microscope 180–182, 181

electron spectroscopy for chemical analysis

(ESCA) 424

electron spin resonance 424

electrons

Auger 281, 425, 431–432

backscattered 277–280, 278, 286

diffraction see diffraction

emitted in scanning electron microscope (SEM)

use 281

energy levels

in scanning electron microscope (SEM)

281–283, 281

in scanning tunnelling microscopy

(STM) 411–412

free 28

secondary 5, 184, 426

type 1 319–320

in X-ray photoelectron spectroscopy

(XPS) 424–425

wave properties 91–94, 185–187

electron transitions, radiation 68

electron wavelength 91, 185, 521

electrostatic force 395

electrostatic lenses 185

elemental analysis see atomic number; chemical

analysis

elemental maps 273–275, 276

emission probability, and Auger electron

spectroscopy (AES) quantitation 436

energy-dispersive spectrometry (EDS) 271–277,

341–343

sample contamination 275


energy-filtered transmission electron microscopy

(EF-TEM) 367–369, 368, 369, 371

energy loss near-edge structure 360

envelope function 210

equiaxed polycrystal microstructure 19

errors

and resolution 485–487

electron diffraction 95

fractography 297

lens aberrations see aberrations

optical microscopy 133, 144

dislocation density 477, 490–491

grain counting 483–485

observer bias 489–490

statistical 466–467

X-ray analysis 353

diffraction measurements 85–90

ESCA (electron spectroscopy for chemical analysis)

see X-ray photoelectron spectroscopy

etching 146–147, 175

focused ion beam (FIB) microscopy 307, 309

and surface roughness 398–399

excitation energy for characteristic X-rays 69, 335

extended energy loss fine structure 366–367

extinction thickness 94

and electron diffraction 225

eye 11–13, 129

dynamic range 167

image formation 129

optical sensitivity 140

eyepiece, optical microscope 127

fabrication, focused ion beam (FIB)

microscopy 308–310

face-centred cubic (FCC) lattices 35–36, 59–60, 64

failure analysis 8, 295–298, 296, 297, 299

Faraday cup, in determining scanning electron

microscope (SEM) beam

diameter 266

far-edge fine structure in energy loss

spectroscopy 366

ferrous alloy see steel

field-emission gun for electron microscopy 181, 265

field-emission microscope 401

Index 527

field evaporation 414–416

field-ion atom probe 401

field-ion microscope 401, 413–414

film, photographic 141–142

fine structure

in Auger electron spectra 432

in extended energy loss spectra 366–367

fluorescence spectroscopy 424

fluorescence (X-ray spectral) correction 349–352

fluorescent screen 182

focal length, optical microscopy 127

focused ion beam

chemical vapor deposition 307

for cross-sections 307

for serial sectioning 499

for thin film specimen preparation 310

machining 304, 306

radiation damage 304

focused ion beam (FIB) microscopy 301

dual-beam 306

overview 301–302

principle of operation 302–304

focussed ion beam (FIB) milling 498–499

forbidden reflections in diffraction 59

Fourier transform 511–512

Fowler-Nordheim equation 401, 411

fractal analysis 466

fraction analysis 24, 25, 469–472, 470

fracture analysis (fractography) 8, 295–298, 320–

321, 321

fracture toughness 7

and fracture strength 10

free electrons 28

frequency function 466

full width at half-maximum (FWHM)

beam diameter 266

criterion 266, 337

gamma, recording response curve 142

geometrical optics 125

GIF image files 168

goniometer 71, 73

grain

aspect ratios 477

boundaries 241, 378

special 293

shape 477–478

size 19–21, 20, 21

definition 473

determination 472–476


great circle 40

grey levels in digital recording 168

grinding 144–145

grit size for grinding media 145

Guinier-Preston zone 219

Harris method 89

H-bar method (sample preparation) 310–311, 311,

312

hexagonal system, zone axes 521

hexagonal unit cells, alternative indices 520

hexapole aberration correction in transmission

microscopy 190, 212

high-angle annular dark-field (HAADF)

detector 235

homogeneity 461–463

human eye, structure and performance 129

image contrast, back-scattered electrons 279

image contrast in optical microscopy 148

image contrast see contrast

imaging 457


image acquisition 491–494

image recording 494–495


445–446

inaccessible microstructural parameters 467, 476

inaccessible parameters 476–481

inelastic energy losses 266

inelastic scattering 3–4, 4, 5, 268

and Bloch waves 226

and diffraction analysis 91–92, 92

scanning electron microscope (SEM)

266–268

information limit 212

infrared spectrometry 423

inhomogeneity 461–463

integration time, eye 11

intensity (of signal)

and brightness 166–167

and data storage 167

interatomic potentials 393

interfacial dislocation 221

interference

electron microscope 207, 207–213

optical 153–155

and sample surface topology 156–157

intergranular failure 9

intermediate lens 140

optical microscope 127, 140

intermetallics, sample preparation 148

International Centre for Diffraction Data 103

International Tables for Crystallography 34–35,

47–48

interplanar spacings in crystals 517

inverse pole figure in preferred orientation 89

ion channelling 305

ionic bonding 26–27

528 Index

ion milling 199–201, 200, 440–441

ions

coordination number 26

ion source, focused ion beam (FIB)

microscopy 301, 302

ionization

and elemental analysis 346

as a result of electron excitation 269

cross-section 346

energy 269

isotropy 459–461, see also anisotropy

Israelachvilli, Jacob 396

Joint Committee of Powder Diffraction Standards

(JCPDS) 103

JPEG (Joint Photographic Experts Group) digital

images 168

Kikuchi patterns 96–98, 99, 100, 101

electron diffraction 223, 224

kinematic theory

of diffraction contrast in transmission microscopy 213

K-line spectra 67–69

lanthanum hexaboride electron sources 181

lattice defects

diffraction contrast in transmission

microscopy 215

lattice image

alumina 242

aluminium 246

computer simulation 231

in high resolution transmission electron

microscopy 230, 232

lattice vacancy 220

lattice vectors in crystals 36

Laue diffraction 76–78, 78, 95

Laue equations 56–57

Laue zones 95

lens

aberration see aberration

electron 185–191, 210

probe 183, 262–263, 265

eye 128

long working distance in optical microscopy 138

optical 126–128, 127

objective 138

lift-out method (TEM sample preparation) 311–314,

313, 314

limiting sphere 60–61, 62

electron beam 92–94

limit of detection

electron energy loss spectroscopy (EELS) 360

eye 12

in energy loss spectroscopy 361

optical microscope 133

X-ray microanalysis 338

X-ray wave-dispersive analysis 354. see also

resolution

linear analysis 470–472

statistical analysis 485

linear intercept analysis 21, 470

line-scan 272, 274

Linnik interferometer 154–155, 155

lithium-drifted silicon X-ray detectors 339

long-range surface forces 394

long-wavelength radiation detection 343

Lorentz polarization factor 84

macrostructure

definition 11

sample variability 463–464

stress 76

variations 482

magnetic domain structures 409

magnification level

optical microscope 14. see also resolution

Markov process 466

mass-absorption coefficient 70

mass spectrometry

secondary ion (SIMS) 440–446

time-of-flight 414–416

mass-thickness contrast 203

mass-to-charge ratio, in secondary ion mass

spectrometry (SIMS) 443–444

materials, examples 6

matrix dissolution 469

mechanical thinning for thin film preparation 195

mesostructure

definition 11

homogeneity 464

sampling 465

metallic bond 28

metals 28–29

homogeneity 461


reflectivity 149

sample preparation



Metals Handbook 175

mica, as scanning probe microscopy (SPM)

substrate 397–398

microanalysis 3

of thin films 354

micromilling, focused ion beam (FIB)

microscopy 306–307

microscope column



Index 529

microscopy

atomic force 18, 18

electron see electron microscope

field-ion 16

ion beam 301–306

optical 13–14

components 135

confocal 152

eyepiece 139–140

image contrast, overview 148–150

interference

multi-beam 155–156, 156

two-beam 152–155

light source 134–136

overview 123–125

phase contrast 163–165

reflection 123

sample preparation 143–148

specimen stage 136

transmission 123

types 2

UV and X-ray 14

microstress 76

microstructure

definition 10, 11

scale 10–19

Miller indices 36, 37–38, 58

minerals, sample preparation 149

Moire fringes 221, 222, 243

Monte Carlo simulations, scanning electron

microscope (SEM) 266–268, 267

morphological anisotropy 459

morphology, surface see topology

Muller, Erwin 16

multi-beam interference 154

multiplicity (of crystal planes) 84

n-type semiconductors 28

NA see numerical aperture

nanostructure, definition 11

National Institute of Standards and Technology

(NIST) 392

near-edge fine structure

in electron energy loss 365

near-field microscopy 132

needle-like particles 478

neutrons, diffraction 56

Newton’s colours 159–161, 161

nickel, powder diffraction spectrum 104

noise

and digital data storage 167

and digital images, compression artefacts 169

and optical image detection 142

and quantitation 336–337. see also background

in image data 458

non-contact region 394, 395

Normarski contrast 164–165

nuclear magnetic resonance 424

numerical aperture (NA) 132–133, 133

and image brightness 137

interference microscopy 154

Nyquist criterion 167

objective aperture 136

objective lens

dark-field 138

in optical microscopy 136


oblique illumination in optical microscopy

151

observer bias in quantitative analysis 489

occupancy (of crystals) 33, 36

optical anisotropy in crystals 157

optical emission spectrometry 423

optical light sources for microscopy 134

optical microscopy

image formation 125–130

objective lens 136–138

worked examples 173–175

optical wedge in optical microscopy 159

optics 125–130

order of reflection 58

orientation distribution 481

orientation imaging microscopy (OIM) 289

and preferred orientation 292–294

applications 292

resolution 291

orthochromatic emulsion 140

p-type semiconductors 28

panchromatic film 141

parallax measurements, scanning electron microscope

(SEM) 298–301, 300

particle (microscopic)

aspect ratio 459, 477–478, 478–480

boundary 175

size

and yield strength 8. see also grain

determination 472–476, 503–513

in powders 73

peak broadening 86–87

peaks, spectral

diffraction patterns 100

intensities, measurement errors 87–88

Pearson symbol 33

Pearson’s Handbook of Crystallographic Data for

Intermetallic

phase boundary contrast in transmission electron

microscopy 216

phase plate in optical microscopy 164

530 Index

Phases 33, 46–47

photodiode detectors 405

photoelectron attenuation length, energy

dependence 427

photoelectron emission, mechanism 426

photoelectron spectroscopy

applications 431

chemical binding states 428

combined with other techniques 430

contamination effects 429

penetration depth, electron beam 267

percolation 476

Petch equation 8

phase boundaries, lattice images 232–233,

233

phase volume 24, 25, 469–472, 470

and particle counting 479–480

and sample thickness 488

phonon 360

photographic emulsion 139, 140–141

optical sensitivity 140

piezoelectric position control 404

pixels 166

plasmon 361

plate-like particles 478

PMN 504–508, 509

point count 24

point analysis 470, 470–472, 471

point dilatations 219–221

point lattice 31

point resolution 211

polarity, liquids 397

polarization forces 29–30, 395–396, 398

polarized light 158–159

analysis 159, 160, 161–162

Polaroid camera 141

polarized light, reflection 162

polishing 145–146

computer-aided 498–499

and surface roughness 398–399

polishing, chemical, for sample preparation 145

polishing, composite materials 148

polishing, electrolytic, for sample preparation

145

polishing, mechanical, for sample preparation

145

polishing, semiconductors and ceramics 148

polishing, soft metals and alloys 147

polycrystalline materials 19

polyethylene 124

polymers, sample preparation 149

porcelain glazes 29

porous materials 24

position control system, atomic force microscopy

(AFM) 404–405

powders

analysis 491–492

diffraction of X-rays 73


patterns 96, 98

worked examples 103–114

sample preparation, electron microscope 194

preferred orientation (of crystal lattice) 75, 89,

292–294, 459

pressure, operating see vacuum requirements

printers 169–170

probability 466–467

of electron emission 436

probe

atomic force microscopy (AFM) 403–404

cantilever 404–405, 405, 406

tip 405

chemical analysis 425

size, transmission electron microscope (TEM) and

scanning electron microscope

(SEM) 356

types 2

probe lens, scanning electron microscope

(SEM) 183, 262–263, 265

process defects 323

projection (of crystal structure onto 2

dimensions) 38–42, 48–51

proportional counters for X-rays 70

quadrupole mass analyser 442

qualitative analysis 2, 334–335


electron energy loss spectroscopy (EELS) 361,

364–365

of contrast 230

interference microscopy 157

lattice images 233

secondary ion mass spectrometry (SIMS) 445

tomography 500

X-ray spectrometry 343–357

quantitative analysis, statistical significance in

495

quantitative microscopy, accuracy and sources of

error 481, 482, 485

quantitative phase analysis 89

radiation damage

channeling 305

focused ion beam (FIB) microscopy 304–306,

305

in crystals 22

Raleigh 12, 131–132

Raman spectrometry 424

rastering 272

ray diagram 129

Index 531

reciprocal lattice unit cell 63

reciprocal space 60–63

reciprocity failure 141

reflecting sphere 65, 66

electron diffraction 93, 94

reflections

from cubic lattices 59–60, 60, 64

of light 149

of polarized light 162

refractive index in optically transparent

materials 126

refraction 125–126, 126

replica specimens 202–203

residual stress 75, 294

resistance, electrical 7

resolution

and detection errors 485–487

Auger electron spectroscopy (AES) 432, 433

definition 12

electron energy loss spectroscopy (EELS) 358,

361–364

electron microscope 14–15, 187

limitations 188

scanning electron microscope (SEM) 317,

338

imaging 266

elemental map 277

secondary electron emission 287

eye 12

image-scanning systems 494

optical microscope 13, 130–133, 170

orientation imaging microscopy (OIM) 291

Raleigh 13

scanning probe microscope 399–400

scanning tunnelling microscope 397–400


442–443

imaging 445–446

spectral, wavelength-dispersive spectrometry

(WDS) 337, 343

transmission electron microscope (TEM) 179

X-ray photoelectron spectroscope (XPS) 427

ring patterns, electron diffraction 94–96, 95

rotating crystal X-ray diffraction 79

Ruska, Ernst 16

Rutherford backscattering 424

salt (common) 83–84

sample

damage

Auger electron spectroscopy (AES)

sputtering 437


density, and secondary electron emission

284


contamination 193

orientation 89

oxidation, and Auger electron spectra 449

preparation

and image analysis 482

electron microscopy

metals 199–201


318–319

cleaning 323

thin films 321

sputter coating 201

transmission electron microscope (TEM),

using focused ion beam 310–315

failure analysis 297

focused ion beam (FIB) microscopy 304–306

milling, ion beam 306–310

optical microscopy 143–145, 172

polishing 145–148, 175, 176

alumina 176

replicas 202–203

scanning electron microscopy (SEM) 294

sectioning see sectioning


194–203, 238–240, 239

metallic films 243

region selection 463–466


failure analysis 297–298

size

and counting time 483–485


X-ray diffraction 72

stage see specimen stage

thickness

and diffracted electron beam intensity 225

and transmission loss 88

counting error correction 487–489, 488

electron energy loss spectroscopy (EELS) 364


and defocus effects 247

X-ray analysis 355

sampling

stereology 463–466

sampling errors 485

sampling procedures 143, 463

for coatings 464

sapphire 47–48

scale levels, microstructural 12

scanning electron microscope (SEM) 6,

192–193

backscattering 277–280


column 383

532 Index

compared with transmission electron microscope

(TEM) 192–194

components 262–264

electron beam 264–268

image analysis 491

orientation imaging 289

overview 183–184, 261–264


schematic diagram 183

secondary electron imaging 280–288

X-ray spectra 269–277

scanning image 184

scanning probe microscopy (SPM)

overview 391, 400–403

probe tip radius 399

probe types 402

scanning system, scanning electron microscope

(SEM) 266

scanning transmission electron detectors 356

scanning transmission electron microscopy

(STEM) 234–235, 234

and scanning electron microscope (SEM) sample

preparation 311

scanning tunnelling microscopy (STM) 410–416

atomic force microscopy 403–409

overview 410–411

summary 420

scattering

by atoms 80–81

equations, beam angles 57–59

from unit cell 81–82

inelastic 3–4, 4, 5

and diffraction analysis 91–92, 92


Scherzer focus 211–212, 231

Schottky emission 182

secondary electrons 5, 184, 426

collection efficiency 285

detector

Everhart-Thornley 286

in-lens 285

emission 280–285, 330

and image contrast 286–288


319–320

escape distance 282

in focused ion beam (FIB) microscopy 304

in X-ray photoelectron spectroscopy (XPS)

424–425

type 1 319–320

secondary emission coefficient, incident energy

dependence 283

secondary-ion mass spectrometry

calibration 444

imaging 445

mass sensitivity 441, 442

mass spectra 444

resolution 444

secondary ion mass spectrometry (SIMS) 440–446

‘shadowing’ 441

secondary phases, detection 241

sectioning

electrochemical 198–199

focused ion beam (FIB) microscopy 307–308, 316

mechanical 195–198, 196, 197


particles 20–21, 475, 475


serial 195

3D scanning electron microscope (SEM)

imaging 264

and tomography 498–499

using focused ion beam (FIB) microscopy

325–326, 327

sections, limitations 467–468

selected area electron diffraction 108, 109, 507–508

SEM see scanning electron microscopy

semi-contact region 394–395

semiconductors 28–29



electron microscopy 198

sensitive tint plate 161

sensitivity see limit of detection; resolution

serial sectioning 195, 264

and tomography 498–499

shadowing (of detectors) 383, 441

Shannon’s sampling theory 167

shear forces, in extruded metals 461–462, 461

short-range surface forces 395

signal strength

backscattered electrons 278

X-rays in scanning electron microscope

(SEM) 271

signal-to-noise ratio 336

signal-to-background see background

significance (statistical)

and microstructural homogeneity 464

in quantitation 495

of X-ray signal 276

single crystal diffraction 76–79

size distributions 479

slack, definition 136

small circle on a stereogram 40

software

3D imaging 499–500

error correction and quantitation 344–345

image analysis 491–494

quantitation 481–483

tomography 497–498, 513

Index 533

solids, interatomic bonding 25–30

solid-state X-ray detectors 271, 355

Soller slits 72

solubility limits

determining 380–381

WDS 378

space groups 32–36

spatial coherence envelope, electron

microscope 210, 211

spatial frequency, of information in

transmission 210

special boundaries 293

specimen see sample

specimen stage



transmission electron microscopy 182


specimen mounting, optical microscope 144

specimen preparation, transmission electron

microscopy 194

spectra

absorption 269–270


capture, electron energy loss spectroscopy

(EELS) 358

differentiated 434

diffraction 55, 56

X-ray 67–73, 79–90

electron energy loss spectroscopy (EELS)

360–361, 360

fine structure 365–367

emission, Auger 434–435

mass spectral 414–416

spectroscopy, X-ray 269–277

spherical aberration 128

electron lens 188–189

corrector 189, 190, 212

spherical triangle 41

spherulites, in polymers 124

sputtering

and optical image contrast 149

in Auger electron spectroscopy (AES) 437,

449–451

in secondary ion mass spectrometry (SIMS)

440–441

ion milling 199–201


442

sputter coating 201, 295

sputtering yield

in secondary-ion mass spectrometry 441

stability, voltage and current in electron

microscopy 193

stacking fault 216–218, 229

and dynamic diffraction 227–230

steel

energy-dispersive spectrometry (EDS) 381–385

optical image 124

optical microscopy 173–175, 174


STEM see scanning transmission electron microscope

stereographic projection (of crystal structure)

38–42, 39, 44, 48–51

noncubic crystals 112

stereology

definition 458

process overview 458–459

stereoscopic imaging 10

optical 140

scanning electron microscope (SEM) 261, 298,

298–300, 317. see also tomography

steric hindrance 396, 398

stereogram, for crystallographic analysis 39

stochastic properties 19

stopping power, high energy electrons in solids 346

stress 76

and orientation imaging microscopy (OIM)

292–294, 294

effect on crystal lattice 75–76

structure factor 83–84

structure-property relationships 7

sub-grain boundary 22

surface force measurements 396

surface forces 392–396

as function of interatomic separation 393

effect of a double layer 397

surface morphology, restructuring 411

surface morphology see topology

surface probe microscopy (SPM) 18–19

surface to volume ratio, particles 472

symmetry (of crystals) 33–36, see also anisotropy

tapping mode, scanning probe microscopy

(SPM) 419

television camera 142–143

television raster 142

TEM see transmission electron microscope

temperature

and etching rate 175

and optical microscopy 138

effect on diffraction peak intensity 88

temporal coherence envelope, electron

microscope 210, 211

thermal conductivity 7

thermionic electron gun 181

thermionic emission, of electrons 181

thickness extinction fringes, in transmission electron

microscopy 216

thin films 105–107

534 Index

grain size 504



X-ray analysis 354–356

thin lens approximation, in geometrical optics 129

three-dimensional imaging see tomography

three-dimensional reconstruction, of microstructural

morphology 495, 499

TIFF (Tagged Image File Format) 168

time-of-flight mass spectrometer, in secondary ion

mass spectrometry (SIMS) 443

tint plate 161–162

titanium 49–51, 49, 50

tomography 495–500

and serial sections 499–500

data presentation 497–498

topology

topographic contrast 151, 287

and image contrast 287–288

and optical interference fringes 156–157

and phase contrast microscopy 163–164

and scanning probe microscopy 397–400

and secondary electron emission 284–285, 284

and specimen preparation in scanning

electron microscope (SEM)

294–298

measuring with atomic force microscopy

(AFM) 407–408

scanning tunnelling microscopy (STM)

411–412

transgranular failure 9


biological samples 237

compared with scanning electron microscope

(SEM) 192–194

components 180

contrast see contrast

history 179

lens aberrations 187–191

overview 179–185

principle of operation 185–187

sample preparation 238–240, 239

transmission imaging and transmission diffraction

comparison 208

transmission intensity, X-rays 70

two-beam diffraction, electron beam 221–225

two-beam interference, in optical microscopy 152

underfocus, electron lens 210

unit cell

hexagonal 44

in reciprocal lattice 63

of crystal lattice 31

scattering 81–82

volume, and scattering 57–59

unit cell (of crystal),

copper 35–36

unit cell volume

of crystals 518

uranium

X-ray spectra 69

vacuum requirements 193



and Auger electrons 281

X-ray photoelectron spectroscopy (XPS)

429

van der Merwe dislocations 221

van der Waals forces 29, 395–396

and steric hindrance 398

visual spectrum 11–14

voltage stability, electron microscope 193–194

volume fraction 24, 25, 469–472, 479–480,

484

wavelength dispersive spectrometry (WDS)

263–264, 340–341

‘shadowing’ 383

wavelengths for common X-ray sources 521

wavelengths of relativistic electrons 521

wave-particle duality 90–91, 185–187

wedge

optical 159–161

samples 196

white radiation 67, 68

Widdmanstatten structure 124

wire-bonding 323–326, 326, 327

work function (of surface), and secondary electron

emission 283, 425

working distance

in scanning electron microscope 26


Wulff net 41, 43

Wyckoff generating sites 33

X-ray diffractometer 67

X-ray goniometer 71

X-ray line-scan 272

X-ray mapping 273

resolution 277

X-ray microanalysis 334

detection limit 338

resolution 338

X-ray monochromator 71

X-ray photoelectron spectroscopy (XPS)

424–431

chemical binding states 428

instrumental requirements

429–430

Index 535

X-rays

absorption and fluorescence 351

adsorption edges 270

counting times 276

diffraction spectra 79–90

excitation 334–338

in scanning electron microscope (SEM) 263–264,

269–277

long wavelength 343

spectrum 272

wavelength 67, 70–71

yield strength 7

ZAF equation 352–353

zero curvature interface 473

zero loss peak 360

zone axis, electron diffraction 111

536 Index

david brandon, wayne d. kaplan(auth.) microstructural characterization of materials, 2nd...

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