Crystallography and Crystal Defects E-Book

Anthony Kelly

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1072

Veröffentlichungsjahr: 2020

Beschreibung

The classic book that presents a unified approach to crystallography and the defects found within crystals, revised and updated

This new edition of Crystallography and Crystal Defects explains the modern concepts of crystallography in a clear, succinct manner and shows how to apply these concepts in the analyses of point, line and planar defects in crystalline materials. 

Fully revised and updated, this book now includes:

• Original source references to key crystallographic terms familiar to materials scientists
• Expanded discussion on the elasticity of cubic materials
• New content on texture that contains more detail on Euler angles, orientation distribution functions and an expanded discussion on examples of textures in engineering materials
• Additional content on dislocations in materials of symmetry lower than cubic
• An expanded discussion of twinning which includes the description and classification of growth twins
• The inclusion and explanation of results from atomistic modelling of twin boundaries
• Problem sets with new questions, detailed worked solutions, supplementary lecture material and online computer programs for crystallographic calculations.

Written by authors with extensive lecturing experience at undergraduate level, Crystallography and Crystal Defects, Third Edition continues to take its place as the core text on the topic and provides the essential resource for students and researchers in metallurgy, materials science, physics, chemistry, electrical, civil and mechanical engineering.

Leseprobe

Table of Contents

Cover

Preface to the Third Edition

Companion Website

Part I: Perfect Crystals

1 Lattice Geometry

1.1 The Unit Cell

1.2 Lattice Planes and Directions

1.3 The Weiss Zone Law

1.4 Symmetry Operators

6

1.5 Restrictions on Symmetry Operations

1.6 Possible Combinations of Rotational Symmetries

1.7 Crystal Systems

1.8 Space Lattices (Bravais

12

Lattices)

Suggestions for Further Reading

References

2 Point Groups and Space Groups

2.1 Macroscopic Symmetry Elements

2.2 Orthorhombic System

2.3 Tetragonal System

2.4 Cubic System

2.5 Hexagonal System

2.6 Trigonal System

2.7 Monoclinic System

2.8 Triclinic System

2.9 Special Forms in the Crystal Classes

2.10 Enantiomorphous Crystal Classes

2.11 Laue Groups

2.12 Space Groups

2.13 The 17 Two‐Dimensional Space Groups

2.14 Nomenclature for Point Groups and Space Groups

2.15 Groups, Subgroups, and Supergroups

2.16 An Example of a Three‐Dimensional Space Group

2.17 Frequency of Space Groups in Inorganic Crystals and Minerals

2.18 Magnetic Groups

Suggestions for Further Reading

References

3 Crystal Structures

3.1 Introduction

3.2 Common Metallic Structures

3.3 Related Metallic Structures

3.4 Other Elements and Related Compounds

3.5 Simple MX and MX2 Compounds

3.6 Other Inorganic Compounds

3.7 Interatomic Distances

3.8 Solid Solutions

3.9 Polymers

3.10 Additional Crystal Structures and their Designation

Suggestions for Further Reading

References

4 Amorphous Materials and Special Types of Crystal–Solid Aggregates

4.1 Introduction

4.2 Amorphous Materials

4.3 Liquid Crystals

4.4 Geometry of Polyhedra

4.5 Icosahedral Packing

4.6 Quasicrystals

4.7 Incommensurate Structures

4.8 Foams, Porous Materials, and Cellular Materials

Suggestions for Further Reading

References

5 Tensors

5.1 Nature of a Tensor

5.2 Transformation of Components of a Vector

5.3 Dummy Suffix Notation

5.4 Transformation of Components of a Second‐Rank Tensor

5.5 Definition of a Tensor of the Second Rank

5.6 Tensor of the Second Rank Referred to Principal Axes

5.7 Limitations Imposed by Crystal Symmetry for Second‐Rank Tensors

5.8 Representation Quadric

5.9 Radius–Normal Property of the Representation Quadric

5.10 Third‐ and Fourth‐Rank Tensors

Suggestions for Further Reading

References

6 Strain, Stress, Piezoelectricity and Elasticity

6.1 Strain: Introduction

6.2 Infinitesimal Strain

6.3 Stress

6.4 Piezoelectricity

6.5 Elasticity of Crystals

6.6 Elasticity of Cubic Crystals

Suggestions for Further Reading

References

Part II: Imperfect Crystals

7 Glide

7.1 Translation Glide

7.2 Glide Elements

7.3 Independent Slip Systems

7.4 Large Strains of Single Crystals: The Choice of Glide System

7.5 Large Strains: The Change in the Orientation of the Lattice During Glide

Suggestions for Further Reading

References

8 Dislocations

8.1 Introduction

8.2 Dislocation Motion

8.3 The Force on a Dislocation

8.4 The Distortion in a Dislocated Crystal

8.5 Atom Positions Close to a Dislocation

8.6 The Interaction of Dislocations with One Another

Problems

Suggestions for Further Reading

References

9 Dislocations in Crystals

9.1 The Strain Energy of a Dislocation

9.2 Stacking Faults and Partial Dislocations

9.3 Dislocations in C.C.P. Metals

9.4 Dislocations in the Rock Salt Structure

9.5 Dislocations in Hexagonal Metals

9.6 Dislocations in B.C.C. Crystals

9.7 Dislocations in Some Covalent Solids

9.8 Dislocations in Low Symmetry Crystal Structures

9.9 Dislocations in Other Crystal Structures

Suggestions for Further Reading

References

10 Point Defects

10.1 Introduction

10.2 Point Defects in Ionic Crystals

10.3 Point Defect Aggregates

10.4 Point Defect Configurations

10.5 Experiments on Point Defects in Equilibrium

10.6 Experiments on Quenched Metals

10.7 Radiation Damage

10.8 Anelasticity and Point Defect Symmetry

Suggestions for Further Reading

References

11 Twinning

11.1 Introduction

11.2 Description of Deformation Twinning

11.3 Examples of Twin Structures

11.4 Twinning Elements

11.5 The Morphology of Deformation Twinning

11.6 Friedel's Classification of (Growth) Twinning

11.7 Atomistic Modelling of Twin Boundaries

Suggestions for Further Reading

References

12 Martensitic Transformations

12.1 Introduction

12.2 General Crystallographic Features

12.3 Transformation in Cobalt

12.4 Transformation in Zirconium

12.5 Transformation in Indium–Thallium Alloys

12.6 Transformations in Steels

12.7 Transformations in Copper Alloys

12.8 Transformations in Ni–Ti‐Based Alloys

12.9 Magnetic Shape Memory Alloys

12.10 Transformations in Non‐metals

12.11 Crystallographic Aspects of Nucleation and Growth

12.12 The Shape Memory Effect and Superelasticity

12.13 Modern Theories of Martensitic Transformations

Suggestions for Further Reading

References

13 Grain Boundaries

13.1 The Structure of Surfaces and Surface Free Energy

13.2 Structure and Energy of Grain Boundaries

13.3 Equivalent Geometrical Descriptions of High‐Angle Grain Boundaries

13.4 Interface Junctions

13.5 The Shapes of Crystals and Grains

Suggestions for Further Reading

References

14 Interphase Boundaries

14.1 Boundaries Between Different Phases

14.2 Interphase Boundaries Between C.C.P. and B.C.C. Phases

14.3 Strained Layer Epitaxy of Semiconductors

Suggestions for Further Reading

References

15 Texture

15.1 Texture

15.2 Euler Angles

15.3 Microtexture

Suggestions for Further Reading

References

Appendix 1: Appendix 1 Crystallographic Calculations

A1.1. Vector Algebra

A1.2. The Reciprocal Lattice

A1.3. Matrices

A1.4. Rotation Matrices and Unit Quaternions

References

Appendix 2: Appendix 2 The Stereographic Projection

A2.1 Principles

A2.2 Constructions

A2.3 Constructions with the Wulff Net

A2.4 Proof of the Properties of the Stereographic Projection

References

Appendix 3: Appendix 3 Interplanar Spacings and Interplanar Angles

A3.1. Interplanar Spacings

A3.2. Interplanar Angles

Appendix 4: Appendix 4 Transformation of Indices Following a Change of Unit Cell

A4.1. Change of Indices of Directions

A4.2. Change of Indices of Planes

A4.3. Example 1: Interchange of Hexagonal and Orthorhombic Indices for Hexagonal Crystals

A4.4. Example 2: Interchange of Rhombohedral and Hexagonal Indices

Appendix 5: Appendix 5 Slip Systems in C.C.P. and B.C.C. Crystals

A5.1. Independent Glide Systems in C.C.P. Metals

A5.2. Diehl's Rule and the OILS Rule

A5.3. Proof of Diehl's Rule and the OILS Rule

References

Appendix 6: Appendix 6 Homogeneous Strain

A6.1 Simple Extension

A6.2 Simple Shear

A6.3 Pure Shear

A6.4 The Relationship Between Pure Shear and Simple Shear

Appendix 7: Appendix 7 Crystal Structure Data

A7.1 Crystal Structures of the Elements, Interatomic Distances and Six‐Fold Coordination‐Number Ionic Radii

A7.2 Crystals with the Sodium Chloride Structure

A7.3 Crystals with the Caesium Chloride Structure

A7.4 Crystals with the Sphalerite Structure

A7.5 Crystals with the Wurtzite Structure

A7.6 Crystals with the Nickel Arsenide Structure

A7.7 Crystals with the Fluorite Structure

A7.8 Crystals with the Rutile Structure

Appendix 8: Appendix 8 Further Resources

A8.1 Useful Web Sites

A8.2 Educational and Information Resources

A8.3 Computer Software Packages

Brief Solutions to Selected Problems

1

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Solutions of Eq. (1.22)

Table 1.2 Permissible combinations of rotation axes in crystals

Table 1.3 The crystal systems

Chapter 2

Table 2.1 Special forms in the crystal classes

Table 2.2 Laue groups

Table 2.3 Screw axes in crystals

Table 2.4 Two‐dimensional lattices, point groups and space groups

Chapter 3

Table 3.1 Axial ratios at room temperature of some metals with a h.c.p. cryst...

Table 3.2 Cell dimensions of As, Sb and Bi.

Table 3.3 Axial ratios at room temperature of some materials with the wurtzit...

Table 3.4 Axial ratios at room temperature of some materials with the nickel ...

Table 3.5 Some examples of superlattice types.

Table 3.6 Crystal structures of some common polymers.

Chapter 5

Table 5.1 Properties represented by second‐rank tensors.

Table 5.2 Number of independent components of physical properties represented...

Chapter 6

Table 6.1 Elastic constants of cubic crystals at room temperature

Table 6.2 Forms of the compliance and stiffness constant arrays

Table 6.3 The equations giving the compliances

s

ij

in terms of the stiffness c...

Table 6.4 Elastic constants of hexagonal crystals at room temperature;

x

3

‐axis...

Table 6.5 Forms of the compliance and stiffness constant arrays for the diffe...

Chapter 7

Table 7.1 Glide elements of crystals (at room temperature and at atmospheric ...

Table 7.2 Predominant slip planes for h.c.p. metals at room temperature. In e...

Table 7.3 Independent slip systems in crystals

Table 7.4 The six slip systems in crystals with the NaCl structure, labelled ...

Chapter 9

Table 9.1 Definitely stable dislocations in some Bravais lattices

Table 9.2 The stacking fault energies of c.c.p. metals

Table 9.3 Dislocations in c.c.p. metals

Chapter 10

Table 10.1 Estimated number of jumps made per second by a point defect having...

Table 10.2 Relaxation displacements around a vacancy expressed as a percentag...

Table 10.3 Fraction of sites vacant at the melting point,

n

/

N

, in equilibrium...

Table 10.4 Energies of formation and migration of vacancy defects in metals,

E

Chapter 11

Table 11.1 The twinning elements of various crystals

Chapter 13

Table 13.1 Surface free energies of solids

Table 13.2 Some coincidence lattices for c.c.p. and b.c.c. crystals [20]

Table 13.3 Disorientation descriptions for compound twins in h.c.p. metals

Table 13.4 Experimentally determined energies of high‐angle grain boundaries

Table 13.5 Twin boundary free energies,

γ

T

, relative to surface energies,...

Chapter 14

Table 14.1 Relative energies of random interphase boundaries

Table 14.2 Epitaxial deposition of c.c.p. metals. (After [7].)

Table 14.3 Lattice parameters and stiffness constants for some common semicon...

Chapter 15

Table 15.1 Some common end or final textures

5

Table A5.1 The twelve slip systems in c.c.p. metal crystals, labelled here fo...

Table A5.2 Slip systems and their corresponding Schmid factors in c.c.p. meta...

List of Illustrations

Chapter 1

Figure 1.1 (a) The arrangement of the atoms in graphene, a single sheet of g...

Figure 1.2 Definition of the smallest separations

a

,

b

and

c

of the lattice ...

Figure 1.3 The numbers give the elevations of the centres of the atoms, alon...

Figure 1.4 A rectangular mesh of a hypothetical two‐dimensional crystal with...

Figure 1.5 Demonstration of the law of constancy of angles between faces of ...

Figure 1.6 A vector

r

written as the sum of translations along the

x

‐,

y

‐ an...

Figure 1.7 Examples of various lattice vectors in a crystal

Figure 1.8 The plane (

hkl

) in a crystal making intercepts of

a

/

h

,

b

/

k

and

c

/

Figure 1.9 Examples of various lattice planes in a crystal. The indices of t...

Figure 1.10 The plane (

hkl

) in a crystal making intercepts of

a

/

h

,

b

/

k

and

c

Figure 1.11 Translation symmetry in a crystal

Figure 1.12 Reflection symmetry

Figure 1.13 Diagram to help determine which rotation axes are consistent wit...

Figure 1.14 The five symmetrical plane lattices or nets. Rotational symmetry...

Figure 1.15 Restrictions placed on two‐dimensional lattices through the impo...

Figure 1.16 The two possible arrangements of nets consistent with mirror sym...

Figure 1.17 Examples of possible allowed combinations of rotational symmetri...

Figure 1.18 Stacking of nets to build up a space lattice. The triplet of vec...

Figure 1.19 Unit cells of the 14 Bravais space lattices. (a) Primitive tricl...

Figure 1.20 Lattice points in the net at height zero are marked as dots, tho...

Figure 1.21 A three‐dimensional view of the staggered arrangement of nets in...

Figure 1.22 Lattice points in the net at height zero are marked with dots. T...

Figure 1.23 The stacking of rhombus nets vertically above one another to for...

Figure 1.24 The three possible stacking sequences of rectangular nets. In th...

Figure 1.25 The staggered stacking of rhombus nets. This form of stacking ge...

Figure 1.26 The stacking of triequiangular nets of points in a staggered seq...

Figure 1.27 Lattice points in the net at level zero are marked with a dot, t...

Figure 1.28 The relationship between a primitive cell of the trigonal lattic...

Figure 1.29 Plan view of the alternative triply primitive hexagonal unit cel...

Figure 1.30 The four 〈111〉 three‐fold axes with acute angles of 70.53° betwe...

Figure 1.31 The relationship between the primitive unit cell and the convent...

Figure 1.32 The relationship between the primitive unit cell and the convent...

Chapter 2

Figure 2.1 The basic operation of repetition by a rotation axis. In this exa...

Figure 2.2 Stereograms representing the operation of one‐, two‐, three‐, fou...

Figure 2.3 The repetition of an object by a mirror plane, (a) and (b), and b...

Figure 2.4 The operation of the twofold rotoinversion axis,

Figure 2.5 The operation of the various rotoinversion axes that can occur in...

Figure 2.6 Stereograms of the poles of equivalent general directions and of ...

Figure 2.7 A stereogram of an orthorhombic crystal of point group 222 centre...

Figure 2.8 (a) General location of a

hk

0 pole on the stereogram of an orthor...

Figure 2.9 The cubic point group of lowest symmetry: 23

Figure 2.10 Stereograms centred on 001 of (a) mirror planes parallel to {100...

Figure 2.11 Stereogram of a cubic crystal.

Figure 2.12 (a) Stereogram of the holosymmetric class of the hexagonal syste...

Figure 2.13 Geometry to show that in Miller−Bravais indices (

hkil

),

i

= −(

h

 ...

Figure 2.14 Indices of various directions in the hexagonal system specified ...

Figure 2.15 Geometry to determine the angle

θ

between the (0001) pole a...

Figure 2.16 A stereogram of a trigonal crystal of class

m

with a rhombohedr...

Figure 2.17 The same crystal as in Figure 2.16 indexed using a hexagonal cel...

Figure 2.18 The relationship between the special forms {10

1} and {01

1} in ...

Figure 2.19 Monoclinic stereogram centred on [001] for the 2nd setting

Figure 2.20 A diagram from which the angle

φ

in Figure 2.19 between 010...

Figure 2.21 Diagrams relevant to drawing stereograms of triclinic crystals. ...

Figure 2.22 (a) A twofold rotation axis. (b) A 2

1

screw axis. (c) A glide pl...

Figure 2.23 Screw axes 3

1

and 3

2

: these are screw axes of opposite hand, as ...

Figure 2.24 The 17 two‐dimensional space groups arranged following the

Inter

...

Figure 2.25 An example of a space group.

Figure 2.26 The 10 black‐and‐white plane lattices: (a) parallelogram (obliqu...

Figure 2.27 The effect of antiferromagnetic coupling on the size of the unit...

Chapter 3

Figure 3.1 (a) The conventional unit cell of the c.c.p. crystal structure. (...

Figure 3.2 Close packing of equal spheres. (a) Cubic close‐packed (c.c.p.). ...

Figure 3.3 Plan view of the ABCABCABC… stacking sequence of (111) planes in ...

Figure 3.4 The stacking of closest‐packed planes in (a) the c.c.p. structure...

Figure 3.5 The largest interstice in the c.c.p. structure: the octahedral in...

Figure 3.6 The second largest interstice in the c.c.p. structure: the tetrah...

Figure 3.7 The unit cell of the h.c.p. structure

Figure 3.8 Interstices in the h.c.p. arrangement. The octahedral interstices...

Figure 3.9 The b.c.c crystal structure

Figure 3.10 Tetrahedral (

X

) and octahedral (

O

) interstices in the b.c.c crys...

Figure 3.11 The crystal structure of mercury, showing the primitive rhombohe...

Figure 3.12 The crystal structure of diamond

Figure 3.13 The stacking of

(111)

planes in the diamond and sphalerite struc...

Figure 3.14 (a) The crystal structure of graphite. (b) The crystal structure...

Figure 3.15 The crystal structure of As, Sb and Bi

Figure 3.16 The crystal structure of sodium chloride, NaCl

Figure 3.17 The crystal structure of caesium chloride, CsCl

Figure 3.18 The crystal structures of (a) sphalerite (

α

‐ZnS) and (b) wu...

Figure 3.19 The crystal structure of nickel arsenide, NiAs

Figure 3.20 The crystal structure of calcium fluoride, CaF

2

Figure 3.21 (a) The crystal structure of rutile, TiO

2

. (b) The rutile crysta...

Figure 3.22 The crystal structure of perovskite,

Pm

m

Figure 3.23 The structure of sapphire (

α

‐Al

2

O

3

or corundum). The large ...

Figure 3.24 One‐eighth of the unit cell of spinel, MgAl

2

O

4

Figure 3.25 Coordination about the oxygen ions (solid black circles) in a ga...

Figure 3.26 The structure of calcite (CaCO

3

). The primitive rhombohedral uni...

Figure 3.27 A c.c.p. crystal showing (a) a substitutional solid solution and...

Figure 3.28 (a) A (111) plane of the disordered form of Cu

3

Au. (b) A (111) p...

Figure 3.29 Structures of ordered solid solutions: (a)

B

2, (b)

D

0

3

, (c)

D

0

19

Figure 3.30 Molecular models of (a) polyethylene (–CH

2

–)

n

and (b) polytetraf...

Figure 3.31 (a) Unit cell of polyethylene viewed along [001]. The unit cell ...

Figure 3.32 Illustration of a continuous polymer chain running through neigh...

Chapter 4

Figure 4.1 (a) Construction of a Voronoi polygon in two dimensions. (b) A po...

Figure 4.2

Para

‐azoxyanisole (PAA)

Figure 4.3 A sketch of the molecular arrangements in the three classes of li...

Figure 4.4 Schematics of the three types of distortion of a nematic mesophas...

Figure 4.5 The five Platonic solids. Left‐hand column: the solids. Centre co...

Figure 4.6 Packing of equal‐sized spheres into an icosahedral arrangement. T...

Figure 4.7 The crystal structure of MoAl

12

. This figure is derived from Figu...

Figure 4.8 The microstructure of a foam made from an Al − 9 wt% Si alloy con...

Chapter 5

Figure 5.1 The relationship between a set of orthonormal axes (

Ox

1

,

Ox

2

,

Ox

3

Figure 5.2 The general tensorial relationship between an electrical field

E

...

Figure 5.3 Derivation of the magnitude of the conductivity in a particular d...

Figure 5.4 The representation ellipsoid.

Figure 5.5 The radius–normal property of a representation ellipsoid.

Chapter 6

Figure 6.1 (a) Definition of extension. (b) Definition of shear

Figure 6.2 Distortion of a body in two dimensions. Points

P

and

Q

move to

P

′...

Figure 6.3 Distortion in two dimensions of a rectangular element at

P

. Two s...

Figure 6.4 Imposition of a rigid body rotation

ω

to enable a relative d...

Figure 6.5 The geometrical interpretation of Eq. (6.9) for

e

12

and

e

21

Figure 6.6 Distortion of a unit cube whose edges are parallel to the princip...

Figure 6.7 The force

f

acting on a small area

δA

in a plane surrounding...

Figure 6.8 Definition of stress components (a) in Cartesian coordinates and ...

Figure 6.9 Calculation of the stress acting on the plane

ABC

Figure 6.10 Stresses acting on one face of a cube of side 2

δ

in a varyi...

Figure 6.11 A shear stress at the point P

Figure 6.12 If σ

3

is the largest principal stress and σ

1

is the smallest, th...

Figure 6.13 Orthonormal ‘old’ and ‘new’ sets of axes related to one another ...

Figure 6.14 Orthonormal ‘old’ and ‘new’ sets of axes related to one another ...

Figure 6.15 Orthonormal ‘old’ and ‘new’ sets of axes related to one another ...

Chapter 7

Figure 7.1 Illustration of the process of glide. Blocks 1, 2 and 3 in a crys...

Figure 7.2 Schematic diagram of glide occurring in the direction

β

in a...

Figure 7.3 Schematic diagram of the fine structure observed in slip lines in...

Figure 7.4 Macroscopic measurement of the amount of glide in a crystal

Figure 7.5 Illustration of the inherent centrosymmetric nature of the simple...

Figure 7.6 Pencil glide, also known as wavy glide. Here, the slip direction

Figure 7.7 Illustration of simple shear in two dimensions: (a) before shear ...

Figure 7.8 Simple shear in three dimensions. Here, a point P relative to O m...

Figure 7.9 A small simple shear deformation

e

ij

is equivalent to a pure stra...

Figure 7.10 The strains produced by the physically different slip systems [

Figure 7.11 Schematic diagram to illustrate that the cubic crystal in (a) sl...

Figure 7.12 Deformation in tension of a single crystal. Glide is presumed to...

Figure 7.13 Standard stereogram of a c.c.p. metal crystal which glides on {1...

Figure 7.14 A unit triangle of the standard stereogram of a hexagonal crysta...

Figure 7.15 The 001–011–

stereographic triangle for a b.c.c. metal crystal ...

Figure 7.16 Nomenclature for the various {111} slip planes for a c.c.p. sing...

Figure 7.17 Change in orientation during a glide operation if there is no la...

Figure 7.18 Changes in orientation of c.c.p. metal crystals during glide. Th...

Figure 7.19 Compression of a single crystal between plates

Chapter 8

Figure 8.1 A screw dislocation in a primitive cubic lattice

Figure 8.2 Primitive cubic lattice after the screw dislocation in Figure 8.1...

Figure 8.3 An edge dislocation in a primitive cubic lattice

Figure 8.4 A mixed dislocation, DD′, in a primitive cubic lattice

Figure 8.5 Atom positions around an edge dislocation in a simple cubic cryst...

Figure 8.6 Schematics of a screw dislocation in a simple cubic crystal (a) l...

Figure 8.7 A Burgers circuit around an edge dislocation in a simple cubic cr...

Figure 8.8 Schematic of a closed dislocation loop in a simple cubic crystal...

Figure 8.9 The motion of a dislocation DD

'

in the direction represented ...

Figure 8.10 The dislocation loop shown in (a) can glide so that its area pro...

Figure 8.11 (a) Production of a prismatic dislocation loop by punching. (b) ...

Figure 8.12 An edge dislocation in a region of general stress

σ

showing...

Figure 8.13 Strain due to a screw dislocation lying along the

z

‐axis

Figure 8.14 Stress due to a screw dislocation along the

z

‐axis acting on a s...

Figure 8.15 The relationship between the

x

1

‐ and

x

2

‐axes,

r

and

θ

for a...

Figure 8.16 Schematic of a screw dislocation in a simple cubic crystal looki...

Figure 8.17 Plot of Eq. (8.28)

Figure 8.18 A screw dislocation of twice the width of the one shown in Figur...

Figure 8.19 Dislocation multiplication

Figure 8.20 Force between parallel edge dislocations on slip planes a distan...

Figure 8.21 Orthogonal screw dislocations, a perpendicular distance

d

apart...

Figure 8.22 A jog QR on a dislocation PQRS where PQ and RS have screw disloc...

Chapter 9

Figure 9.1 An edge dislocation formed by making a cut in a cylinder and then...

Figure 9.2 A dislocation reaction in which two dislocations combine to form ...

Figure 9.3 Forces on a small segment of a dislocation loop in equilibrium

Figure 9.4 A dislocation line being extruded between obstacles a distance

l

...

Figure 9.5 Forces acting on a small curved segment of a dislocation when acc...

Figure 9.6 A perfect dislocation on a (111) plane of a c.c.p. metal which ha...

Figure 9.7 (a) Faulted vacancy loop in a c.c.p. metal, showing traces of the...

Figure 9.8 Thompson's tetrahedron. (After Thompson [15]).

Figure 9.9 The vector relationship between the Burgers vectors of a Shockley...

Figure 9.10 The dissociation of a perfect dislocation CB into two partial di...

Figure 9.11 Reaction of two dislocations on a common slip plane. The disloca...

Figure 9.12 Same reaction as Figure 9.11 but showing splitting into partials...

Figure 9.13 Lomer–Cottrell lock

Figure 9.14 Formation of a stacking fault tetrahedron from a Frank vacancy l...

Figure 9.15 Hirth lock

Figure 9.16 Relation between slip planes and directions in {110}

or {110}

Figure 9.17 Edge dislocation in NaCl. The slip plane and the two sheets of i...

Figure 9.18 Edge dislocation in NaCl viewed in a different section to that s...

Figure 9.19 Atoms and lattice vectors in a hexagonal metal

Figure 9.20 Two possible structures for a vacancy loop in a hexagonal metal...

Figure 9.21 A

slip system

Figure 9.22 Relation between slip planes and directions in

slip

Figure 9.23 A ball model of the {110} planes in a b.c.c. metal

Figure 9.24 Double kink in a dislocation line. The dotted lines represent po...

Figure 9.25 A 60° dislocation in sphalerite. The structure of a 60° dislocat...

Figure 9.26 Stacking fault in a silicon film. Point O lies at the bottom sur...

Figure 9.27 Atoms and lattice vectors in the basal plane of graphite

Figure 9.28 A schematic side view of a split 60° dislocation in graphite. At...

Figure 9.29 Reaction between partial dislocations on adjacent planes in grap...

Chapter 10

Figure 10.1 Crystal surface acting as a vacancy source. In (a) an atom jumps...

Figure 10.2 {100} plane of NaCl, showing the sense of the displacements of t...

Figure 10.3 Hypothetical split interstitial in a c.c.p. metal. The plane of ...

Figure 10.4 Hypothetical split interstitial in a b.c.c. metal. The plane of ...

Figure 10.5 Crowdion in an alkali metal, as postulated by Paneth [23]. The l...

Figure 10.6 H‐centre in a KCl crystal. The plane of the diagram is (001). (N...

Figure 10.7 Path followed by an atom jumping into a vacant nearest‐neighbour...

Figure 10.8 Path followed by an atom jumping into a vacant nearest‐neighbour...

Figure 10.9 Effects of changes in length and lattice parameter with temperat...

Figure 10.10 Schematic conductivity plot for a NaCl crystal containing a sma...

Figure 10.11 Effect of CdBr

2

additions on the electrical conductivity of AgB...

Figure 10.12 Annealing out of the quenched‐in resistivity

ρ

of a metal ...

Figure 10.13 Isochronal recovery of electron‐irradiated copper containing an...

Figure 10.14 Interstitial sites occupied by C or N atoms in ferrite, α‐Fe. T...

Figure 10.15 Effect of relaxation on the strain caused by a constant stress

Figure 10.16 Hypothetical tetravacancy in a c.c.p. metal. The dotted lines s...

Chapter 11

Figure 11.1 Structure of a twin in a c.c.p. metal. The plane

S

in (a) is the...

Figure 11.2 Formation of a twin in a c.c.p. metal by shear. The dotted lines...

Figure 11.3 Displacements produced by a twin lamella. The traces

PQ

and

QR

d...

Figure 11.4 The elements of deformation twinning.

O

is an origin,

K

1

is the ...

Figure 11.5 (a) Type I twin. (b) Type II twin. In (a) the lattice vector

l

3

...

Figure 11.6 Twin in a b.c.c. metal. The scheme of the figure is the same as ...

Figure 11.7 Twin in sphalerite

Figure 11.8 Twin in calcite. The scheme of the figure is the same as that of...

Figure 11.9 The (10

2) twin in zirconium. The scheme of the figure is the sa...

Figure 11.10 The (10

2) twin in zinc

Figure 11.11 Plan view of the (1

00) plane in graphite, showing the structur...

Figure 11.12 The formation of a twin in graphite by a partial dislocation on...

Figure 11.13 Shear and the geometry of deformation twinning: a vector

l

3

par...

Figure 11.14 (a) Twin lamella intersecting a surface AB. (b) Dislocation mod...

Figure 11.15 Dislocation model of a thin twin lamella

Figure 11.16 Emissary dislocations. The dislocations shown by a single line ...

Figure 11.17 Pole mechanism for the growth of a twin

Figure 11.18 (a) Projection of 2 × 2 unit cells of pyrite projected onto (00...

Figure 11.19 Structures for Type II twinning with stable twin modes for devi...

Chapter 12

Figure 12.1 Scratched surface intersected by a martensite plate

MM

′

Figure 12.2 The square lattice within the plate outlined in (a) is strained ...

Figure 12.3 The three possible

〈211〉 vectors in a (111) plane

Figure 12.4 (a) Unit cell of the b.c.c. lattice, drawn with (011) in the

x‐y

...

Figure 12.5 A section through a sphere of zirconium and the ellipsoid develo...

Figure 12.6 Undistorted planes of the strain S′

Figure 12.7 Rotation suffered by the undistorted planes of the strain S′

Figure 12.8 Approximate crystallography of a plate of martensite in titanium...

Figure 12.9 Schematic of a partly transformed In–Tl alloy single crystal. Th...

Figure 12.10 The twin relationship of the lamellae shown in Figure 12.9

Figure 12.11 Three parallel plates of martensite with alternating shear stra...

Figure 12.12 The c.c.p. lattice with a b.c.t. cell picked out of it. (After ...

Figure 12.13 Lattice parameters of austenite and martensite as a function of...

Figure 12.14 Habit plane normals of martensite in various steels plotted on ...

Figure 12.15 The (011) plane of a b.c.c. metal (a) before and (b) after a di...

Figure 12.16 The one‐way shape memory effect. (a) a sample annealed in its a...

Figure 12.17 The two‐way shape memory effect. (a) a sample annealed in its a...

Chapter 13

Figure 13.1 The two alternative {10

0} surfaces of a hexagonal metal. The su...

Figure 13.2 The four alternatives for a surface parallel to (0001) in wurtzi...

Figure 13.3 Surface at a small angle

θ

to a {111} plane of a c.c.p. met...

Figure 13.4 A schematic of energy

E

as a function of angle

θ

away from ...

Figure 13.5 Possible (1

0) section through the

γ‐

plot of a c.c.p....

Figure 13.6 A fine wire with a bamboo‐like grain structure to which a load

W

Figure 13.7 Splitting of a crystal of width

w

with a pre‐existing crack of l...

Figure 13.8 Low‐angle symmetrical tilt boundary in a simple cubic lattice. T...

Figure 13.9 Energy of a tilt boundary as a function of the tilt angle θ. Val...

Figure 13.10 Schematic of a high‐angle tilt boundary of good fit between one...

Figure 13.11 An asymmetrical tilt boundary where the misorientation across t...

Figure 13.12 A low‐angle twist boundary in a simple cubic lattice. The bound...

Figure 13.13 (a) Generation of grains 1 and 2 by opposite rotations of

θ/

...

Figure 13.14 Twist boundary of good fit in a simple cubic lattice. The bound...

Figure 13.15 Part of the CSL produced from a c.c.p. lattice by a rotation of...

Figure 13.16 Twin boundary in a monoclinic lattice. The boundary is normal t...

Figure 13.17 Graphical representation of the total Burgers vector

B

of the d...

Figure 13.18 Maximum disorientation angles as a function of angle/axis descr...

Figure 13.19 Grain boundary groove, seen in cross‐section

Figure 13.20 A segment of an interface,

OE

, held in equilibrium by forces

F

x

Figure 13.21 Two boundaries of the same twin joining at right angles to one ...

Figure 13.22 The junction of the interfaces between three grains. Each inter...

Figure 13.23 Twin boundary grooving, seen in cross‐section

Figure 13.24 The

γ‐

plot of Figure 13.5, showing the equilibrium s...

Figure 13.25 Construction due to Herring [43]

Figure 13.26 The same equilibrium shape as shown in Figure 13.24, arising fr...

Figure 13.27 A surface that has reduced its energy by breaking up into facet...

Figure 13.28 The instability of four interfaces meeting along a line through...

Figure 13.29 (a) Truncated octahedron and (b) distorted truncated octahedron...

Figure 13.30 The Weaire–Phelan foam structure. The individual cells within t...

Chapter 14

Figure 14.1 A particle of a phase B situated at a grain boundary of the phas...

Figure 14.2 Interface between two orthorhombic crystals. The interface is no...

Figure 14.3 Energy of a boundary of the type shown in Figure 14.2, between t...

Figure 14.4 Epitaxy of Ag deposited on (001) of NaCl: (a) observed orientati...

Figure 14.5 Superposition of nets representing atoms in unrelaxed (110) b.c....

Figure 14.6 A strained epitaxial layer in the (001) orientation. The in‐plan...

Figure 14.7 The relaxed region with lateral dimension

mh

around a misfit dis...

Chapter 15

Figure 15.1 The {111} pole figure of electrolytic copper rolled to 96.6% red...

Figure 15.2 (a) Standard stereographic projection in the (110) orientation s...

Figure 15.3 Spatial representation of the half‐maximum density of the ODF re...

Figure 15.4 Definition of the orientation of a crystallite in rolled sheet b...

1

Figure A1.1 The addition of two vectors,

a

and

b

, to produce a third resulta...

Figure A1.2 The components

a

x

,

a

y

and

a

z

of a vector

a

referred to three axe...

Figure A1.3 The vector product

a

×

b

= (|

a

||

b

| sin 

θ

)

of two vectors

Figure A1.4 The definition of the reciprocal lattice vector

a

*: the directio...

Figure A1.5 The plane (

hkl

) in the real crystal making intercepts of

a

/

h

,

b

/

Figure A1.6 An anticlockwise rotation of

θ

about

n

carrying the point

x

2

Figure A2.1 (a) Sphere of projection. (b) The angle between two planes is eq...

Figure A2.2 Projections of poles on the surface of a sphere onto a flat piec...

Figure A2.3 (a) Stereographic projection. (b) A small circle projects as a c...

Figure A2.4 (a) Poles of a cubic crystal. (b) Stereogram of a cubic crystal...

Figure A2.5 Construction of a small circle about the centre of the primitive...

Figure A2.6 Rotation of the sphere of projection in Figure A2.5 by 90° about...

Figure A2.7 Construction of a small circle about a pole within the primitive...

Figure A2.8 Construction of a small circle about a pole on the primitive

Figure A2.9 To find the opposite of a given pole

Figure A2.10 An alternative view of the construction in Figure A2.9 to show ...

Figure A2.11 To find the pole of a great circle

Figure A2.12 To find the angle between two poles

Figure A2.13 (a) Projection of lines of latitude and longitude to make the W...

Figure A2.14 (a) Stereographic projection of two poles

and

. (b) Rotation...

Figure A2.15 To find the trace of a pole using the Wulff net.

Figure A2.16 Rotation of poles about an axis B lying on the primitive.

Figure A2.17 Rotation of poles about an inclined axis.

Figure A2.18 Two‐surface analysis

Figure A2.19 The fundamental projectional geometry of the stereographic proj...

Figure A2.20 Geometry to show that any sphere inverts into a sphere

Figure A2.21 Geometry to show that a stereographic projection is angle true...

4

Figure A4.1 Relationship between the conventional hexagonal unit cell with c...

Figure A4.2 Unit cells in the rhombohedral lattice. (a) Obverse setting of r...

6

Figure A6.1 Illustration of the relationship between a pure shear and a simp...

Guide

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Crystallography and Crystal Defects

Third Edition

ANTHONY KELLY and KEVIN M. KNOWLES

Department of Materials Science & Metallurgy University of Cambridge UK

Copyright

This edition first published 2020

© 2020 John Wiley & Sons Ltd

Edition History

John Wiley & Sons (1e, 2000)

John Wiley & Sons (2e, 2012)

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Library of Congress Cataloging‐in‐Publication Data

Names: Kelly, A. (Anthony), author. | Knowles, Kevin M., author.

Title: Crystallography and crystal defects / Anthony Kelly

(University of Cambridge), Kevin M. Knowles (University of

Cambridge).

Description: Third edition. | Hoboken, NJ : John Wiley & Sons, 2020. |

Includes bibliographical references and index.

Identifiers: LCCN 2019035382 (print) | LCCN 2019035383 (ebook) | ISBN

9781119420170 (hardback) | ISBN 9781119420156 (adobe pdf) | ISBN

9781119420163 (epub)

Subjects: LCSH: Crystallography. | Crystals–Defects.

Classification: LCC QD931 .K4 2020 (print) | LCC QD931 (ebook) | DDC

548/.8–dc23

LC record available at https://lccn.loc.gov/2019035382

LC ebook record available at https://lccn.loc.gov/2019035383

Cover Design: Wiley

Cover Image: Courtesy of Kevin M. Knowles, © oxygen/Getty Images

Preface to the Third Edition

While there are a plethora of books devoted to crystallography and, in particular, X‐ray crystallography and electron crystallography of varying levels of sophistication, ranging from undergraduate primers to advanced research level texts, there is still no other textbook which cover the topics of crystallography and crystal defects in as integrated and complete a manner as previous editions of this book. As a consequence, unless there is one of these previous editions to hand, or, better still, this new edition, students and researchers interested in crystallography and crystal defects will have to have one or more sources for their crystallography, and one or more sources for their understanding of defects in crystalline materials and how the microscopic behaviour of the movement of dislocations or formation of twins leads to macroscopic plasticity in crystalline materials.

In this third edition, two new chapters have been created to extend coverage on texture and interphase boundaries, so that the book can continue to be both a useful learning resource and a source reference for senior undergraduates, graduate students and researchers on core aspects of crystallography and crystal defects of particular relevance to materials science. A number of sections of the book have also been either added or extended, such as the sections on magnetic groups, the elasticity of cubic crystals, dislocations in low symmetry crystal structures, atomistic modelling of twin boundaries, the description and classification of growth twins, modern theories of martensitic transformations and the mechanism of the shape memory effect in alloys which exhibit martensitic transformations. Original source references to key crystallographic terms familiar to materials scientists such as Miller indices, Bravais lattices, the Weiss zone law, and Miller–Bravais indices and Weber indices for hexagonal and trigonal materials have been included for completeness. New questions have been added to the end of each chapter where appropriate. As with the Second Edition, brief solutions to all the questions are given after the appendices, while full worked solutions are available at the password protected Wiley Web page accompanying this book at www.wiley.com/go/kelly/crystallography3e. In addition, a number of optical and electron micrographs illustrating various aspects of the microstructure of materials which follow from the theory outlined in this book have been uploaded to this password protected Wiley Web page.

Sadly, the senior author, Anthony Kelly, died on 3 June 2014, and so the responsibility for preparing this extended third edition has fallen entirely on my shoulders as the junior author. Therefore, any typographical errors either introduced during the preparation of this third edition or missed from the first two editions of this book are ultimately my responsibility. It will be evident from reading this book, and prior editions of this book, that the book is unashamedly not an introductory textbook to crystallography and crystal defects, even though it addresses fundamental topics in materials science. However, the aim in writing this third edition has been to continue to communicate both general background knowledge and specialist research knowledge in as accessible a way as possible to an audience educated to degree level, highlighting specialist textbooks, review papers and original research papers for those interested in a deeper understanding of the various concepts introduced. The contents of this third edition have been successfully used by me for two recent week‐long courses in the GIAN program run by the Ministry of Human Resource Development of the Government of India, one on crystallography for materials scientists aimed at advanced undergraduates and new postgraduates, and a more focused one on interfaces in materials for Ph.D. students and post‐doctoral research workers. Discussions with Professors Dipankar Banerjee at the Indian Institute of Science Bangalore, Rajesh Prasad at the Indian Institute of Technology, Delhi and Anjan Sil at the Indian Institute of Technology, Roorkee arising from these GIAN courses have all been invaluable in deciding what to include in this third edition, and at what level of sophistication. Rajesh Prasad deserves a special mention for highlighting a number of typographical errors in the second edition that I have now corrected in this third edition.

There are two colour images on the front cover of this book, both related to the crystallography of devitrite, Na2Ca3Si6O16. The larger of these is a low magnification photograph of devitrite needles nucleated on the surface of a block of soda–lime–silica float glass after a heat treatment of 17 hours at 850 °C observed in transmitted polarized light with a sensitive tint at 45° to the polarizer and analyzer, which are aligned vertically and horizontally, respectively. The needles are in a thin section cut perpendicular to the surface of the glass block. The edge of the sample is below the bottom of this photograph. I am grateful to Dr Robert P. Thompson from the University of Cambridge, Department of Materials Science and Metallurgy for permission to use this photograph. The second smaller colour image of atomistic modelling of a type II twin boundary in devitrite was produced by Prof. Bin Li, now at the School of Materials Science and Engineering, Sun Yat-sen University, Guangzhou, People's Republic of China. I am also grateful to Prof. Li for permission to use this image for the front cover.

Finally, as with the second edition, it is hoped that the reader whose understanding of crystallography and crystal defects goes well beyond what is described here will nevertheless find parts where his or her knowledge has been enriched.

Kevin M. Knowles

March 2020

Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/kelly/crystallography3e

The Website includes:

Solutions

Computer programs for crystallographic calculations

PPT slides of all figures from the book

Optical and electron micrographs illustrating various aspects of the microstructure of materials

Scan this QR code to visit the companion website.

Part IPerfect Crystals

1Lattice Geometry

1.1 The Unit Cell

Crystals are solid materials in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry operations; these operations determine the symmetry of the physical properties of a crystal. For example, the symmetry operations show in which directions the electrical resistance of a crystal will be the same. Many naturally occurring crystals, such as halite (sodium chloride), quartz (silica), and calcite (calcium carbonate), have very well‐developed external faces. These faces show regular arrangements at a macroscopic level, which indicate the regular arrangements of the atoms at an atomic level. Historically, such crystals are of great importance because the laws of crystal symmetry were deduced from measurements of the interfacial angles in them; measurements were first carried out in the seventeenth century. Even today, the study of such crystals still possesses some heuristic advantages in learning about symmetry.

Nowadays the atomic pattern within a crystal can be studied directly by techniques such as high‐resolution transmission electron microscopy. This atomic pattern is the fundamental pattern described by the symmetry operations and we shall begin with it.

In a crystal of graphite the carbon atoms are joined together in sheets. These sheets are only loosely bound to one another by van der Waals forces. A single sheet of such atoms provides an example of a two‐dimensional crystal; indeed, recent research has shown that such sheets can actually be isolated and their properties examined. These single sheets are now termed ‘graphene’. The arrangement of the atoms within a sheet of graphene is shown in Figure 1.1a. In this representation of the atomic pattern, the centre of each atom is represented by a small dot, and lines joining adjacent dots represent bonds between atoms. All of the atoms in this sheet are identical. Each atom possesses three nearest neighbours. We describe this by saying that the coordination number is 3. In this case the coordination number is the same for all the atoms. It is the same for the two atoms marked A and B. However, atoms A and B have different environments: the orientation of the neighbours is different at A and B. Atoms in a similar situation to those at A are found at N and Q; there is a similar situation to B at M and at P.

Figure 1.1 (a) The arrangement of the atoms in graphene, a single sheet of graphite in which the centre of each atom is represented by a dot, some of which are labelled using capital letters, (b) the lattice of graphene, (c) various choices of primitive unit cells for graphene. In (c), the conventional unit cell is outlined in heavy lines and the corresponding x‐ and y‐axes are marked. In (a), H represents the position of an axis of six-fold rotational symmetry (see Section 1.4.2)

It is obvious that we can describe the whole arrangement of atoms and interatomic bonds shown in Figure 1.1a by choosing a small unit such as OXAY, describing the arrangement of the atoms and bonds within it, then moving the unit so that it occupies the position NQXO and repeating the description and then moving it to ROYS and so on, until we have filled all space with identical units and described the whole pattern. If the repetition of the unit is understood to occur automatically, then to describe the crystal we need only describe the arrangement of the atoms and interatomic bonds within one unit. The unit chosen we would call the ‘unit parallelogram’ in two dimensions (in three dimensions, the ‘unit cell’). In choosing the unit we always choose a parallelogram in two dimensions or a parallelepiped in three dimensions. The reason for this will become clear later.

Having chosen the unit, we describe the positions of the atoms inside it by choosing an origin O and taking axes Ox and Oy parallel to the sides, so that the angle between Ox and Oy is ≥90°. We state the lengths of the sides a and b, taking a equal to the distance OX and b equal to the distance OY (Figure 1.1a), and we give the angle γ between Ox and Oy. In this case a = b = 2.45 Å1 (at 25 °C) and γ = 120°. To describe the positions of the atoms within the unit parallelogram, we note that there is one at each corner and one wholly inside the cell. The atoms at O, X, A and Y all have identical surroundings.2

In describing the positions of the atoms we take the sides of the parallelogram, a and b, as units of length. Then the coordinates of the atom at O are (0, 0); those at X (1, 0); those at Y (0, 1); and those at A (1, 1). The coordinates of the atom at O′ are obtained by drawing lines through O′ parallel to the axes Ox and Oy. The coordinates of O′ are therefore . To describe the contents fully inside the unit parallelogram – that is, to describe the positions of the atoms – we need only give the coordinates of the atom at the origin, (0, 0), and those of the atom at O′. The reason is that the atoms at X, A and Y have identical surroundings to those at O and an atom such as O, X, A or Y is shared between the four cells meeting at these points. The number of atoms contained within the area OXAY is two. O′ is within the area, giving one atom. O, X, A and Y provide four atoms each shared between four unit cells, giving an additional . A second way of arriving at the same result is to move the origin of the unit cell slightly away from the centre of the atom at O so that the coordinate of the O atom is (ε1, ε2) and the coordinate of the O′ atom is , where 0 < ε1, ε2 ≪ 1. Under these circumstances the centres of the atoms at X, A and Y lie outside the unit cell, so that the atom count within the unit cell is simply two. We note that the minimum number of atoms which a unit parallelogram could contain in a sheet of graphene is two, since the atoms at O and O′ have different environments.

To describe the atomic positions in Figure 1.1a we chose OXAY as one unit parallelogram. We could equally well have chosen OXTA. The choice of a particular unit parallelogram or unit cell is arbitrary, subject to the constraint that the unit cell tessellates, that is, it repeats periodically, in this case in two dimensions. Therefore, NQPM is not a permissible choice – although NQPM is a parallelogram, it cannot be repeated to produce the graphene structure because P and M have environments which differ from those at N and Q.

The corners of the unit cell OXAY in Figure 1.1