# Crystallography and Crystal Defects, 2nd Edition

# Crystallography and Crystal Defects, 2nd Edition

ISBN: 978-1-119-96146-8

Jan 2012

536 pages

## Description

Extensively revised and updated, this new edition of a classic text presents a unified approach to crystallography and to the defects found within crystals. The book combines the classical and exact description of symmetry of a perfect crystal with the possible geometries of the major defects-dislocations, stacking faults, point defects, twins, interfaces and the effects of martensitic transformations. A number of important concepts and exciting new topics have been introduced in this second edition, including piezoelectricity, liquid crystals, nanocrystalline concepts, incommensurate materials and the structure of foamed and amorphous solids. The coverage of quasicrystalline materials has been extended, and the data tables, appendices and references have been fully updated.Reinforcing its unrivalled position as the core text for teaching crystallography and crystal defects, each chapter includes problem sets with brief numerical solutions at the end of the book. Detailed worked solutions, supplementary lecture material and computer programs for crystallographic calculations are provided online (http://booksupport.wiley.com).

*Preface to the Second Edition*xiii**Part I Perfect Crystals 1**

**1 Lattice Geometry 3**

1.1 The Unit Cell 3

1.2 Lattice Plane and Directions 7

1.3 The Weiss Zone Law 11

1.4 Symmetry Elements 14

1.5 Restrictions on Symmetry Elements 16

1.6 Possible Combinations of Rotational Symmetries 21

1.7 Crystal Systems 26

1.8 Space Lattices (Bravais Lattices) 26

Problems 37

Suggestions for Further Reading 40

References 41

**2 Point Groups and Space Groups 43**

2.1 Macroscopic Symmetry Elements 43

2.2 Orthorhombic System 49

2.3 Tetragonal System 52

2.4 Cubic System 53

2.5 Hexagonal System 56

2.6 Trigonal System 59

2.7 Monoclinic System 63

2.8 Triclinic System 65

2.9 Special Forms in the Crystal Classes 67

2.10 Enantiomorphous Crystal Classes 68

2.11 Laue Groups 69

2.12 Space Groups 69

2.13 Nomenclature for Point Groups and Space Groups 78

2.14 Groups, Subgroups and Supergroups 79

2.15 An Example of a Three-Dimensional Space Group 79

Problems 82

Suggestions for Further Reading 84

References 84

**3 Crystal Structures 85**

3.1 Introduction 85

3.2 Common Metallic Structures 86

3.3 Related Metallic Structures 93

3.4 Other Elements and Related Compounds 95

3.5 Simple *MX* and *MX*2 Compounds 98

3.6 Other Inorganic Compounds 104

3.7 Interatomic Distances 110

3.8 Solid Solutions 110

3.9 Polymers 113

3.10 Additional Crystal Structures and their Designation 116

Problems 119

Suggestions for Further Reading 121

References 122

**4 Amorphous Materials and Special Types of Crystal–Solid Aggregate 123**

4.1 Introduction 123

4.2 Amorphous Materials 123

4.3 Liquid Crystals 126

4.4 Geometry of Polyhedra 129

4.5 Icosahedral Packing 134

4.6 Quasicrystals 135

4.7 Incommensurate Structures 137

4.8 Foams, Porous Materials and Cellular Materials 137

Problems 139

Suggestions for Further Reading 139

References 140

**5 Tensors 141**

5.1 Nature of a tensor 141

5.2 Transformation of components of a vector 142

5.3 Dummy Suffix Notation 145

5.4 Transformation of Components of a Second-Rank Tensor 146

5.5 Definition of a Tensor of the Second Rank 148

5.6 Tensor of the Second Rank Referred to Principal Axes 149

5.7 Limitations Imposed by Crystal Symmetry for Second-Rank Tensors 153

5.8 Representation Quadric 155

5.9 Radius–Normal Property of the Representation Quadric 159

5.10 Third- and Fourth-Rank Tensors 161

Problems 161

Suggestions for Further Reading 163

References 163

**6 Strain, Stress, Piezoelectricity and Elasticity 165**

6.1 Strain: Introduction 165

6.2 Infinitesimal Strain 166

6.3 Stress 170

6.4 Piezoelectricity 177

6.5 Elasticity of Crystals 181

Problems 193

Suggestions for Further Reading 196

References 196

**Section II Imperfect Crystals 197**

**7 Glide and Texture 199**

7.1 Translation Glide 199

7.2 Glide Elements 203

7.3 Independent Slip Systems 208

7.4 Large Strains of Single Crystals: The Choice of Glide System 218

7.5 Large Strains: The Change in the Orientation of the Lattice During Glide 222

7.6 Texture 228

Problems 235

Suggestions for Further Reading 237

References 237

**8 Dislocations 241**

8.1 Introduction 241

8.2 Dislocation Motion 247

8.3 The Force on a Dislocation 249

8.4 The Distortion in a Dislocated Crystal 253

8.5 Atom Positions Close to a Dislocation 258

8.6 The Interaction of Dislocations with One Another 261

Problems 265

Suggestions for Further Reading 266

References 267

**9 Dislocations in Crystals 269**

9.1 The Strain Energy of a Dislocation 269

9.2 Stacking Faults and Partial Dislocations 277

9.3 Dislocations in c.c.p. Metals 280

9.4 Dislocations in the Rock Salt Structure 288

9.5 Dislocations in Hexagonal Metals 290

9.6 Dislocations in b.c.c. Crystals 295

9.7 Dislocations in Some Covalent Solids 297

9.8 Dislocations in Other Crystal Structures 301

Problems 301

Suggestions for Further Reading 303

References 303

**10 Point Defects 305**

10.1 Introduction 305

10.2 Point Defects in Ionic Crystals 309

10.3 Point Defect Aggregates 310

10.4 Point Defect Configurations 312

10.5 Experiments on Point Defects in Equilibrium 317

10.6 Experiments on Quenched Metals 321

10.7 Radiation Damage 324

10.8 Anelasticity and Point Defect Symmetry 326

Problems 329

Suggestions for Further Reading 331

References 331

**11 Twinning 335**

11.1 Introduction 335

11.2 Description of Deformation Twinning 337

11.3 Examples of Twin Structures 342

11.4 Twinning Elements 350

11.5 The Morphology of Deformation Twinning 354

Problems 358

Suggestions for Further Reading 360

References 360

**12 Martensitic Transformations 363**

12.1 Introduction 363

12.2 General Crystallographic Features 364

12.3 Transformation in Cobalt 366

12.4 Transformation in Zirconium 369

12.5 Transformation of Indium–Thallium Alloys 374

12.6 Transformations in Steels 379

12.7 Transformations in Copper Alloys 382

12.8 Transformations in Ni–Ti-Based Alloys 383

12.9 Transformations in Nonmetals 384

12.10 Crystallographic Aspects of Nucleation and Growth 385

Problems 387

Suggestions for Further Reading 388

References 389

**13 Crystal Interfaces 391**

13.1 The Structure of Surfaces and Surface Free Energy 391

13.2 Structure and Energy of Grain Boundaries 397

13.3 Interface Junctions 409

13.4 The Shapes of Crystals and Grains 414

13.5 Boundaries between Different Phases 420

13.6 Strained Layer Epitaxy of Semiconductors 424

Problems 429

Suggestions for Further Reading 431

References 431

**Appendix 1 Crystallographic Calculations 435**

A1.1 Vector Algebra 435

A1.2 The Reciprocal Lattice 440

A1.3 Matrices 443

A1.4 Rotation Matrices and Unit Quaternions 448

References 449

**Appendix 2 The Stereographic Projection 451**

A2.1 Principles 451

A2.2 Constructions 455

A2.3 Constructions with the Wulff net 460

A2.4 Proof of the Properties of the Stereographic Projection 465

References 468

**Appendix 3 Planar Spacings and Interplanar Angles 469**

A3.1 Planar Spacings 469

A3.2 Interplanar Angles 472

**Appendix 4 Transformation of Indices Following a Change of Unit Cell 473**

A4.1 Change of Indices of Directions 473

A4.2 Change of Indices of Planes 475

A4.3 Example 1: Interchange of Hexagonal and Orthorhombic Indices for Hexagonal Crystals 476

A4.4 Example 2: Interchange of Rhombohedral and Hexagonal Indices 477

**Appendix 5 Slip Systems in C.C.P. and B.C.C. Crystals 481**

A5.1 Independent Glide Systems in C.C.P. Metals 481

A5.2 Diehl’s Rule and the OILS Rule 483

A5.3 Proof of Diehl’s Rule and the OILS Rule 485

References 486

**Appendix 6 Homogeneous Strain 487**

A6.1 Simple Extension 488

A6.2 Simple Shear 488

A6.3 Pure Shear 489

A6.4 The Relationship between Pure Shear and Simple Shear 489

**Appendix 7 Crystal Structure Data 491**

A7.1 Crystal structures of the Elements, Interatomic Distances and Ionic radii at Room Temperature 491

A7.2 Crystals with the Sodium Chloride Structure 495

A7.3 Crystals with the Caesium Chloride Structure 496

A7.4 Crystals with the Sphalerite Structure 497

A7.5 Crystals with the Wurtzite Structure 497

A7.6 Crystals with the Nickel Arsenide Structure 497

A7.7 Crystals with the Fluorite structure 498

A7.8 Crystals with the Rutile Structure 498

**Appendix 8 Further Resources 499**

A8.1 Useful Web Sites 499

A8.2 Computer Software Packages 499

*Brief Solutions to Selected Problems* 501

*Index 509*

- A completely updated and revised new edition.
- Introduces several new important concepts which have arisen since the original edition; new topics include piezoelectricity, liquid crystals, nanocrystalline concepts, incommensurate materials and the structure of foamed and amorphous materials. The topic of quasicrystalline materials has been completely rewritten, and the data tables and references have been fully updated.

“Summing Up: Recommended. Upper-division undergraduates and above.” (*Choice*, 1 August 2012)

- A uniquely unified approach to crystallography and crystal defects.
- A comprehensive and accessible presentation of lecture material for (senior) undergraduates.
- Contains problems at the end of each chapter and detailed worked solutions.