Today’s objective

We have learnt •How do crystallites arrange in a polycrystalline material

•How to represent polycrystal information in stereographic projection

• To get an overview of diffraction phenomenon,

in general, and X-ray diffraction, in particular

X-Ray Diffraction (XRD): Suitable for the study

of the structure of crystalline materials

Why

The typical interatomic spacing in a crystal is of the

order of Å , the wavelength of X-ray is of the same order

This makes crystals to act as diffraction grating for X-

radiation

X-rays can be conveniently produced in Laboratory

• Interplanar spacing, hence lattice parameter

• Orientation of a single crystal or grain

• Measure the size, shape and internal strain of small

crystalline regions

• Crystal structure of an unknown material

Based on the diffraction principles, the following

can be measured in a crystalline materials:

Diffraction

• Diffraction is essentially a scattering phenomena where

at some particular angle the scattered radiation forms

constructive interface (arises when an electromagnetic

waves interact with the periodic structure )

• Diffraction is basically Reinforced Coherent Scattering

Understanding constructive interference

If two waves A and

B are propagating

in same phase, the

resulting wave C

will have magnitude

of addition of both

A and B

Bragg law is satisfied when the wavelength satisfies; nλ ≤ 2d

A

A

+

B

B

C =

λ

λ

λ

Constructive

interference will

occur when:

λ = AB + BC

AB=BC

n λ = 2AB

sin ɵ =AB/d

AB=d sin ɵ

n λ =2d sin ɵ

λ= 2dhklsin ɵhkl

A

B

C

ɵ z

d

90-

ɵ

90-ɵ

ɵ

λ= 2dhklsinɵhkl

If 2ɵ: Bragg angle, and

λ: X-ray wavelength

Essentially, it gives relationship between the angle of

incidence ,wavelength of the incident radiation and the

spacing between parallel lattice plane of a crystal.

• There are three variables: λ,

ɵ, and d

λ is known (X-ray Source)

ɵ is measured in the

experiment (2ɵ)

• ‘d’ can be calculated using

the Bragg’s relation

• For the planes (hkl), the cell

parameter a can be

calculated

When the diffraction condition is met there will be a

diffracted X-ray beam

2222

2

2

2222

222

222

222

sin)(

sin4

)(

sin4

2

lkh

alkh

lkh

a

lkh

ad

dSin

θ - 2θ Scan The θ - 2θ scan maintains these angles with the sample,

detector and X-ray source

Normal to surface

Only (hkl) planes of atoms that share the surface normal

will be seen in the θ - 2θ Scan

2θ θ

surface

Crystal = Lattice + Motif

Diffraction from a crystal

• The structure of a crystal can be defined as:

• A beam of X-rays directed at a crystal interacts with

the electrons of the atoms in the crystal , undergoes

diffraction and gives rise to intensity distribution in the

diffracted output, which is characteristic of the crystal

structure. The output is known as diffraction pattern.

• Diffraction pattern consists of a set of peaks with

certain height (intensity) and spaced at certain intervals

(not the same interval between each of the peaks)

• Therefore, based on arrangement of atoms in a

crystal, intensities of particular diffraction peak is

modified sometimes the pattern go missing

• Lattice decides the position of the peaks (spacing

between the peaks), while the motif decides the

height of the peaks.

Examples of diffraction from crystals

Lattice = SC No missing reflections 100 missing reflection (F = 0)

Lattice = BCC

Lattice = FCC

100 missing reflection (F = 0)

110 missing reflection (F = 0)

Extinction Rules

• Structure Factor (F):

The resultant wave

scattered by all atoms

of the unit cell

• The Structure Factor is

independent of the

shape and size of the

unit cell; but is

dependent on the

position of the atoms

within the cell

Bravais Lattice Diffraction

Condition

Reflections

necessarily absent

Simple all None

Body centred (h + k + l) even (h + k + l) odd

Face centred h, k and l unmixed h, k and l mixed

h2 + k2 + l2 Simple Cubic Face Centred Cubic Body Centred Cubic

1 100

2 110 110

3 111 111

4 200 200 200

5 210

6 211 211

7

8 220 220 220

9 300, 221

Diffraction Extinction Criteria for different

materials with different crystals structures

Typical X-ray diffraction pattern of a BCC

material (IF steel)

Radiation: Cu K, = 1.54 Å

2θ1 2θ2 2θB 2θB

Why are the peaks broad??

θB: Bragg Angle

If uniform crystallites (no

misorientation within

them, imaginary)

But there is always some misorientation

within the grains (each crystallites always

misorientated with each other , reality)

Full width

at half

maximum

(FWHM)

Uniform

Crystallites

Misorientatio

n of

Crystallites

t = thickness of crystallite

K = constant dependent on crystallite shape (~ 0.89)

l = x-ray wavelength

B = FWHM (full width at half maximum)

ɵB = Bragg Angle

• Scherrer Formula: Relation between the crystallites size and the

FWHM

BB

Kt

cos

• Crystallites Size: Essentially uniform agglomeration of crystals

posses same reflection patterns

Crystallite size <1000 Å

Error >20%

• Limitations of Scherrer equation

Peak broadening can be contributed by other factors, like, size,

strain and instrument

Questions

1. Out of the following, which can be measured using X-ray diffraction:

(a) Interplanar spacing, hence lattice parameter

(b) Orientation of a single crystal or grain

(c) Grain boundary character

(d) Size, shape and internal strain of small crystalline regions

2. Which of the following grain sizes can not be measured using X-ray

diffraction:

(a) 10 m

(b) 0.1 m

(c) 0.01 m

(d) 0.001 m

4. Determine the values of 2ɵ and (hkl) for the first three lines on the powder

patterns of substances with the following structures (Cu Kα=3.14Å) in FCC

unit cell (a = 3.00Å).

5. Calculate the crystallite size for FWHM B for = 10, 45, and 80°.