L 2000

ENME 421 MATERIALS I LABORATORY #2 - CRYSTALLOGRAPHY AND MOLECULAR STRUCTURE Winter 2010

Department of Mechanical Engineering, University of Calgary

1. INTRODUCTION

In crystalline solids the atoms are arranged in a highly periodic manner (organized) in three dimensions over distances that are very great relative to the distances between them. The periodicity of the atoms affect almost all of the properties of crystals; hence it is

important to describe it quantitatively and in a manner which expresses effectively the significant relationships involved. This laboratory

exercise is designed to help the student gain a better appreciation of some of the quantitative relationships associated with crystalline and molecular structures.

Three lattice translations (vectors) are necessary to define the unit cell (smallest repeating arrangement) in crystal systems. From your text and the figures in your text, it may be noted that there are 14 possible Miller-Bravais crystal lattices. The laboratory will

concentrate on the study of three of these only, viz., the face-centred cubic, FCC,, body-centred cubic, BCC, and hexagonal-close-packed,

HCP, lattices. More than 90% of all crystalline materials solidify into one of these systems.

Much of the information necessary to a fundamental understanding of the material in this laboratory is contained in Chapter 3 of

your text. It is especially important that you understand the basic geometry and geometric relationships relevant to each crystal structure

and that you are able to identify and label crystallographic planes and directions using Miller and Miller-Bravais indexes. Crystallographic

directions are relatively straight forward (see Section 3.8 in your text). To index a plane, the following might be of some help.

1.1 Notation of Planes in a Lattice

It is often convenient to consider a stack of parallel planes passing through a lattice. The designation of these planes is slightly more complicated than the designation of directions. Figure 1.1 shows the projection of a lattice along its c axis which is perpendicular to

the ab plane. Consider a plane that is parallel to the c axis and is represented in an edge view by the heavy line in Figure 1.1. The translation periodicity of the lattice, of course, requires that a parallel plane pass through every lattice point (light lines in Figure 1.1). The

intercepts of the plane first considered are two units along the a axis, three along b, and 4 along c. Since all the parallel planes are exactly

alike, it is convenient always to consider the plane nearest to the origin. In order to avoid using fractions in its designation, the reciprocals

of the intercepts are used instead. The resulting integers are named after their inventor, the Miller indices of the plane.

Figure 1.1 Figure 1.2

The procedure for obtaining the indices of any plane is summarized below for the plane shown by the heavy line in Figure 1.1.

Procedure a b c

1. Determine intercepts 2 3 ∞

2. Note their reciprocals ½ ⅓ 0 3. Clear fractions 3 2 0

It is conventional to represent the indices of a plane by enclosing them in parenthesis: (hkl). The meaning of these indices is that the set of parallel planes (hkl) cuts the a axis into h parts, the b axis into k parts, and the c axis into l parts. As an illustration, the (243) plane nearest

the origin is shown in Figure 1.2.

A special case of indexing arises when a lattice can be described by a unit cell having two equal axes inclined at 120 and a third

axis that is orthogonal to the plane of these two axes (Figure 1.3). As can be seen in Figure 1.4, the plane of the two equal axes contains a third axis that is equal in length to the other two. It will be shown later in this write up that this type of lattice occurs in the hexagonal

crystal system, in which case the three coplanar axes are equivalent by symmetry. There is some advantage to displaying this symmetry

equivalence in the indices. If the four hexagonal axes, or Bravais-Miller axes a1, a2, a3, c, are used, then the corresponding hexagonal indices, or Bravais-Miller indices, are (hkil). It is easy to show that

Figure 1.3 Figure 1.4

the relationship between the three equivalent axis is a1 + a2 = -a3 (5)

and between the corresponding indices h + k = -i (6)

A negative index is written with a bar over it; that is, 1)2(11

means (1,1,-2,1). Since the relationship in equation (6) is easily remembered,

the i is sometimes replaced by a dot when the hexagonal indices (hkAl) are used.

1.2 Geometric Relations in Crystal Lattices

Geometric Formulae in crystallography are often expressed in terms of lattice parameters. Such formulae can be used to determine relative positions of various planes and lines within the lattice. If we denote planes by (hkl) and directions by [uvw] then:

1.2.1 For a direction to be normal to a plane in a cubic lattice h = u; k = v; l = w

1.2.2 For a direction to be parallel to a certain plane in a cubic lattice hu + kv + lw = 0

1.2.3 The angle between two directions in a cubic lattice is:¸•

)w + v + u( )w + v + u(

ww + vv + uu = cos

22

22

22

21

21

21

212121

¸

1.2.4 The interplanar spacing in a cubic lattice with lattice parameter 'a' is¸

l + k + h

a = d

222

¸

while for the hexagonal system,¸•

l )c

a( + )k + hk + h(

3

4

a = d

2222

¸•

2. DEFORMATION MECHANISMS

Perhaps the most outstanding property of metals is plasticity, the ability of a metal to be plastically or permanently deformed. All

shaping and forming operations such as forging, rolling, stamping, drawing, extruding and pressing involve plastic deformation and the mechanisms of deformation are therefore of essential interest.. Plasticity is the tolerance of a metal to adjust, it is the ductility of the metal.

Deformation is the result of certain planes of atoms in the crystal lattice slipping relative to one another or the result of one part of a crystal lattice being reoriented about a 'mirror' or 'twin' plane by the action of shearing stresses. The mechanisms are termed slip and twinning

respectively. Some materials (e.g. FCC metals) deform primarily by the slip mechanism and twin systems only become operative under extreme

loading conditions while other materials (e.g. HCP and to a lesser degree BCC) deform by a combination of slip and twinning. Slip planes are planes where the atoms are touching and slip directions are directions where the touching atoms are in a line.

Note that the opposite to slip or ductility is fracture or cleavage. Cleavage is the cleaving or separation of atomic planes. Thus planes on which slip does not occur are cleavage planes. These will b e the least dense packed planes with no possible slip directions (no atoms

touching).

2.1 Deformation via Slip

2.1.1 Microscopic Appearance of Slip

Slip or strain lines appear as parallel sets of closely spaced lines under the optical microscope and can readily be distinguished from other microscopic features (Figure 2.1). Each line represents a number of slip steps produced by the movement of many dislocations to the

crystal surface. During the slip process, entire planes of atoms do not glide instantaneously over one another but slip is normally accomplished

by localized movements of defects called dislocations. Although it is not intended to describe in detail the dislocation theory of plastic flow, a simplistic look at the theory is necessary for some insight into the generation of slip lines or slip bands. Perhaps the simplest form of a

dislocation is the edge dislocation (Figure 2.2). As the dislocation is caused to move through the crystal lattice to the crystal surface, a step of

one atom distance is generated at the surface. Obviously, the resolving power of the optical microscope is incapable of detecting a step of this magnitude but once one step has been produced, slip proceeds in parallel sets of slip planes until a sufficient number of steps have been generated

to create the resolvable slip lines of Figure 2.1. In a polycrystalline material, the crystal orientation changes from grain to grain and this accounts

for the change in the direction of the slip lines as grain boundaries are crossed. In some of the grains cross-slip is also evident. Cross-slip

normally takes place when a conjugate slip system becomes operative after the initial system has been held up by boundaries or defects or by interaction with other slip systems.

Numerous annealing twins can also be observed in the photomicrograph of Figure 2.1. Although annealing twins are crystallographically identical to deformation twins, the origins of the two are quite different. Annealing twins are described in more detail in a

later section. Again the reorientation of the crystal lattice in these regions is evidenced by the directional changes of the slip lines.

2.1.2 Crystallographic Aspects of Slip

Slip planes are usually planes of highest atomic density and greatest interplanar spacing and slip proceeds along directions of highest linear atomic density. The slip plane and the slip direction together constitute the slip system.

In the face-centred cubic (FCC) crystal class the slip planes are the {111} family of planes and the slip directions are the <110> set of slip directions. Figure 2.3 illustrates the patterns of atoms and the corresponding slip directions for the (111) plane. In this crystal system, for

each arbitrarily selected set of reference axes there are four unique {111} planes, each having 3 <110> type slip directions making a total of 12

slip systems. This accounts for the great plasticity of FCC metals. That is, at least one system is always oriented so that slip initially may progress with ease. FCC materials also exhibit rapid work hardening characteristics due to the interaction of dislocations on one slip system with

dislocations on a second system, each impeding the progress of the other.

The pattern of atoms on the (0001) slip plane of the hexagonal close-packed (HCP) crystal class (Figure 2.4) is identical to the pattern

of atoms on (111) FCC slip plane. The three slip directions illustrated along with the (0001) slip plane constitute the three slip systems that are

operative at ordinary temperatures. However, plasticity is greatly enhanced because deformation by twinning is an important mechanism to this class. Moveover, at slightly elevated temperatures, two additional sets of slip systems become operative (Figure 2.5). It is perhaps of

importance to note that the0)1(10

and1)1(10

slip planes share a common10]2[1

slip direction with the (0001) slip plane.

The body-centred cubic (BCC) crystal class is a more loosely-packed structure than the FCC crystal class and the slip systems are not

as well defined which accounts in part for the limited ductility of BCC metals. The pattern of atoms on the most densely populated planes {110}

is again very similar to the pattern of atoms on the {111} FCC and (0001) HCP slip planes. However, the {110} BCC planes contain only two close-packed <111> directions (Figure 2.6a). Two additional families of planes, the {112}

and {123} planes, although not as densely populated as the (110) plane, share a common]1[11

direction with this plane and these are also

operative slip systems (Figure 2.6b). Therefore, for the BCC crystal class, there are a total of 12 unique slip planes having 4 unique slip directions making a total of 48 slip systems,

In general, the number of slip systems is given by:

2

N N = N

dps

where Ns = number of slip systems, Np = multiplicity of a given slip plane and Nd = number of slip directions.

In cubic system, the multiplicity of planes (including those parallel to each other is):

Plane Type hoo hhh hho hko hhl hkl Multiplicity 6 8 12 24 24 48

2.2 Deformation by Twinning

2.2.1 Microscopic Appearance of Twinning

Deformation twins appear as lenticular or lens-shaped regions (Figure 2.7 and 2.8) in the optical microscope and due to the change in

orientation of the crystal lattice within the twin, etchants attack these regions at a different rate than the untwinned matrix or the twin boundaries

may be preferentially attacked because of the greater energy in the boundary region. Figure 2.7 illustrates that two twin systems are operative in pure bismuth and that twinning is directional with directional deformation. Figure 2.8 illustrates both slip and twinning in pure zinc. Again the

parallel sets of slip lines change direction as twin and grain boundaries are crossed. Notice also the large "kink bands" and the deformation twins

within these bands.

2.2.2 Crystallographic Aspects of Twinning

There are two kinds of twins that are of interest to the metallurgist, deformation or mechanical twins and growth or annealing twins.

Deformation by twinning occurs most readily in HCP metals (magnesium, zinc, etc.) and to a lesser degree in BCC metals (tungsten, ± iron).

Deformation twins rarely occur in FCC metals. However, FCC materials, during the growth of a crystal readily form annealing twins, especially in metals with low stacking fault energies.

Twinning is a type of shear deformation that differs from slip in that twins form instantaneously. It occurs when each layer of atoms

on one side of a certain plane, the twin plane, undergoes a fixed shift with respect to its neighbours. As a result the two parts of the twinned crystal represent mirror images of each other. The total displacement of each atom within the twinned portion of the crystal is proportional to its

distance from the plane of symmetry (Figure 2.8b).

Twinning in itself may be an important mode of deformation or it may serve to rotate slip systems into a more preferred orientation for

slip to proceed. In hexagonal close-packed metals the main twinning system is1]1[10 2)1(10

while in BCC and FCC the systems are

{112} <111> and {111} <112> respectively. Figure 2.8b illustrates that a shearing force in the [112] direction produces the same twinned matrix as growth faults in {111} directions in FCC metals, i.e. deformation and annealing twins are crystallographically identical.

3. PROCEDURE

Crystallographic and Polymeric models have been placed at various stations in the laboratory. There are a total of six (6) stations

each of which have been duplicated. The procedure is to move from station to station and, using the models as visual aides, take the necessary measurements and answer the questions asked about the structures associated with each station. Work in groups of two (preferable) or at most

three. Each student is expected to make his own notes and reference report. Sketches should be neat and to scale. Note, you must choose one

specific atom as an origin when finding planes, a prudent choice will reveal the highlighted planes in the models.

3.1 Station 1: The Unit Cell

Each sphere of which the crystallographic models are composed represents an atom.

a) Measure and record the diameter of the spheres.

b) Measure and record the lattice parameters for each of the unit cells FCC, BCC, and HCP crystal structures. c) Using the value for the ball diameter measure in a) and simple geometry, calculate the lattice parameters. How do they

compare with you answer in b)? Show clearly the basis of your calculation.

d) What is the coordination number for each of the crystal lattices represented by the models? What is meant by coordination number?

e) How many atoms are contained in each of the units cells? Remember that some atoms share or are a part of several such

unit cells and thus only a fraction of them are contained in any one cell. f) From your calculations in c) and assuming that the FCC and BCC structures represent ³ -iron and ± -iron respectively

(atomic weight = 55.8), calculate the density of ³ -iron and ± -iron. If the HCP structure is zinc (atomic weight = 65.37)

what is the density of zinc? g) At • 910oC, ³ -iron changes to ± -iron upon cooling to room temperature. What is the volume change associated with this

transformation?

3.2 Station 2: Important Planes in the FCC System

a) Neatly sketch the pattern of atoms on the {100}, {110} and the {111} planes.

b) Calculate the planar atomic densities on the planes in a) and arrange them in order of increasing density.

c) Measure the spacing between the {100}, {110} and {111} type planes.

d) From your basic measurements at Station 1 and using equation (1.2.4) calculate the spacing between the parallel sets of {100}, 110} and {111} type planes. Compare with the measured values in c).

e) What are the slip planes in the FCC structure? How many unique such planes are there? Show by means of an

appropriate sketch. Index the planes. Show and correctly index the slip directions contained in the)1(11

slip plane.

f) Generally, what are the requirements for a plane and direction to be part of a slip system? g) Locate the octahedral and tetrahedral holes in the FCC crystal lattice. Which is the larger? Of what significance is this

observation? (See Figure 2.9).

3.3 Station 3: Important Planes in the HCP System

a) Sketch the pattern of atoms on the (0001),0)1(10

and1)1(10

planes.

b) Measure the spacing between the1)1(10

planes.

c) Using equation (1.2.4) and your basic measurements at station 1, calculate the spacing between the1}1{10

planes and

compare with your answer in (b).

d) Measure the angle between the1)1(10

plane and the (0001) plane. Calculate this angle from basic principles.

e) How many slip systems are possible in the HCP crystal structure?

3.4 Station 4: Stacking in Close-Packed Structures

Deformation twins and annealing twins give rise to stacking faults in close-packed structures. In the FCC crystal structure the

close-packed-planes {111} are stacked in an ABCABCABC sequence. However, perfect stacking is not always attainted in practise

and stacking sequences of ABCABCAB/ABCABC are often encountered. a) Stack the model planes in an ABABAB sequence. What crystal structure is obtained?

b) Stack the planes in an ABCABC sequence. What crystal structure is obtained? Can you find the unit cell in this

structure? c) Now stack the planes in and ABCABCBABCAB sequence. What is the crystal structure at the stacking fault? What is

the structure away from the fault?

d) Arrange the planes in the sequence ABADAB. What does this do to the basic crystal structure?

3.5 Station 5: The BCC Crystal Structure

a) Neatly sketch the pattern of atoms on the {100}, {110}, {112} and {123} planes. From the basic measurements taken at

station 1, calculate the atomic density on each plane. Arrange them in order of increasing atomic density. Which planes

contain close-packed directions? b) Measure the spacings between {110}, {112} ad {123} parallel planes. Calculate the 'd' spacing from equation (1.2.4) and

the basic measurements taken at station 1. Compare calculated to measured values.

c) Sketch the (110) plane and identify (index) the two specific directions contained in this plane.

d) Why is the (111) plane not a slip plane in this system?

e) Locate the octahedral and tetrahedral holes in this crystal lattice. Which are the larger holes? Of what significance is this

observation? (See Figure 2.9).

3.6 Station 6: Ionic and Molecular Structures

a) Study the NaCl type structure. What is the coordination number for this structure?

b) NaCl and CsCl are both ionic compounds. What restrictions does this place on their structures? What is the difference in structure between the two materials? Sketch and describe their unit cells.

c) The silicates are members of an important class of engineering materials called ceramics. The coordination of silicon by

oxygen is fundamental to all silicate structures. What is this coordination? Give the basic formula for isolated Si-O groups, infinite Si-O chains, and the two-dimensional sheet structures.

d) Name and compare two properties that depend on bond type and two properties that can be accounted for on the basis of

their structures. e) Diatomic hydrogen is a small molecule, it can easily fit in the rings of silicon oxide. If you use glass tubing to transport

hydrogen, what happens?