Crystals and Crystal Structures

An authoritative, updated text that offers an introduction to crystals and crystal structure with coverage of crystallography, and microscopy of materials

Written in a friendly, non-mathematical style, the updated second edition of Crystals and Crystal Structures offers a comprehensive exploration of the key elements of crystals and crystal structures. Starting with the basics, it includes information on multiple areas of crystallography, including modulated structures, quasicrystals and protein crystallography, and interdisciplinary applications as diverse as the relationship between physical properties and symmetry.

To enhance comprehension of the material presented, the book contains a variety of problems and exercises. The revised second edition offers new material and updates in the field including:

• An introduction to the use of high intensity X-ray analysis of protein structures
• Advances in imaging, scanning electron microscopy, and cryo-electron microscopy
• The relationship between symmetry and physical properties highlighting new findings and an introduction to tensor notation in describing these relationships in a concise fashion
• Nanoparticles as well as crystallographic aspects, defects, surface defects and the impact of these crystallographic features on properties
• Perovskite structures and their variations and the inclusion of their wide-ranging properties

Written for students ofcrystallography, chemistry, physics, materials science, biosciences and geology, Crystals and Crystal Structures, Second Edition provides an understanding of the subject and enables students to read scientific papers and articles describing a crystal structure or use crystallographic databases.

RICHARD J. D. TILLEY D. Sc, Ph. D, is Emeritus Professor in the School of Engineering at the University of Cardiff, Wales, U.K. He has published extensively in the area of solid-state materials science, including nine textbooks (translated into multiple languages), 23 chapters and encyclopedia entries and more than 200 journal papers.

Preface ix

Chapter 1: Crystals and Crystal Structures 1

1.1 Crystal Families and Crystal Systems 1

1.2 Unit Cells and Miller Indices 4

1.3 The Determination of Crystal Structures 6

1.4 The Description of Crystal Structures 6

1.5 Crystal Structures: Metals 8

1.5.1 The Cubic Close-packed (A1) Structure of Copper 9

1.5.2 The Body-Centred Cubic (A2) Structure of Tungsten 9

1.5.3 The Hexagonal (A3) Structure of Magnesium 10

1.6 Crystal Structures: Binary Compounds 10

1.6.1 The Halite (Rock Salt, NaCl) Structure 10

1.6.2 The Rutile Structure 11

1.6.3 The Fluorite Structure 11

1.7 The Cubic Perovskite Structure 12

1.8 The Structure of Urea 13

1.9 The Density of a Crystal 14

Answers to Introductory Questions 15

Problems and Exercises 16

Chapter 2: Lattices, Planes and Directions 19

2.1 Two-dimensional Lattices 19

2.2 Unit Cells 22

2.3 The Reciprocal Lattice in Two Dimensions 22

2.4 Three-dimensional Lattices 26

2.5 Rhombohedral, Hexagonal and Cubic Lattices 29

2.6 Alternative Unit Cells 30

2.7 The Reciprocal Lattice in Three Dimensions 31

2.8 Lattice Planes and Miller Indices 34

2.9 Hexagonal Lattices and Miller-Bravais Indices 37

2.10 Miller Indices and Planes in Crystals 37

2.11 Directions 39

2.12 Lattice Geometry 41

Answers to Introductory Questions 44

Problems and Exercises 44

Chapter 3: Two-dimensional Patterns and Tiling 49

3.1 The Symmetry of an Isolated Shape: Point Symmetry 49

3.2 Rotation Symmetry of a Plane Lattice 52

3.3 The Symmetry of the Plane Lattices 53

3.4 The Ten Plane Crystallographic Point Symmetry Groups 55

3.5 The Symmetry of Patterns: The 17 Plane Groups 57

3.6 Two-dimensional Crystal Structures 63

3.7 General and Special Positions 66

3.8 Tesselations 68

Answers to Introductory Questions 71

Problems and Exercises 71

Chapter 4: Symmetry in Three Dimensions 75

4.1 The Mirror Plane and Axes of Rotation 75

4.2 Axes of Inversion: Rotoinversion 77

4.3 Axes of Inversion: Rotoreflection 80

4.4 The Hermann-Mauguin Symbols for Point Groups 81

4.5 The Symmetry of the Bravais Lattices 83

4.6 The Crystallographic Point Groups 84

Answers to Introductory Questions 87

Problems and Exercises 88

Chapter 5: Symmetry and Physical Properties 93

5.1 Properties and Symmetry 93

5.2 Point Groups and Physical Properties 94

5.3 Specification of Physical Properties 96

5.4 Refractive Index 97

5.5 Optical Activity 100

5.5.1 Specific Rotation 100

5.5.2 Crystal Symmetry and Optical Activity 101

5.5.3 Optical Activity in Homogeneous Crystals 101

5.5.4 Optical Activity in Crystals Containing Molecules 102

5.5.5 Optical Activity and Chiral Molecules 103

5.5.6 Optical Activity, Chemical Reactivity and Symmetry 103

5.6 The Pyroelectric Effect 104

5.6.1 Pyroelectric and Ferroelectric Crystals 104

5.6.2 Crystallographic Aspects of Pyro- and Ferroelectric Behaviour 106

5.7 Dielectric Properties 109

5.7.1 Dielectrics 109

5.7.2 Isotropic Materials 110

5.7.3 Non-isotropic Materials 110

5.8 Magnetic Point Groups and Colour Symmetry 111

Answers to Introductory Questions 113

Problems and Exercises 114

Chapter 6: Building Crystal Structures from Lattices and Space Groups 117

6.1 Symmetry of Three-dimensional Patterns: Space Groups 117

6.2 The Crystallographic Space Groups 119

6.3 Space Group Symmetry Symbols 121

6.4 The Graphical Representation of the Space Groups 125

6.5 Building a Structure from a Space Group: C_s3P₇ 128

6.6 The Structure of Diopside, MgCaSi₂O₆ 131

6.7 The Structure of Alanine, C₃H₇NO₂ 134

Answers to Introductory Questions 139

Problems and Exercises 139

Chapter 7: Diffraction and Crystal Structure Determination 143

7.1 The Occurrence of Diffracted Beams: Bragg s Law 144

7.2 The Geometry of the Diffraction Pattern 145

7.3 Particle Size 149

7.4 The Intensities of Diffracted Beams 150

7.5 The Atomic Scattering Factor 151

7.6 The Structure Factor 152

7.7 Structure Factors and Intensities 156

7.8 Numerical Evaluation of Structure Factors 158

7.9 Symmetry and Reflection Intensities 159

7.10 The Temperature Factor 161

7.11 Powder X-ray Diffraction 163

7.12 Neutron Diffraction 168

7.13 Structure Determination Using X-ray Diffraction 169

7.14 Solving the Phase Problem 171

7.15 Electron Microscopy 172

7.15.1 Diffraction Patterns and Structure Images 172

7.15.2 Diffraction and Fourier Transforms 177

7.16 Protein Crystallography 178

7.16.1 The Phase Problem 178

7.16.2 The Crystallinity Problem: SFX 182

7.16.3 The Crystallinity Problem: Single Particle Cryo-EM 183

Answers to Introductory Questions 185

Problems and Exercises 186

Chapter 8: The Depiction of Crystal Structures 189

8.1 The Size of Atoms 189

8.2 Sphere Packing 190

8.3 Metallic Radii 193

8.4 Ionic Radii 194

8.5 Covalent Radii 197

8.6 Van der Waals Radii 198

8.7 Ionic Structures and Structure Building Rules 198

8.8 The Bond Valence Model 199

8.9 Structures in Terms of Non-metal (Anion) Packing 202

8.10 Structures in Terms of Metal (Cation) Packing 203

8.11 Cation-Centred Polyhedral Representations of Crystals 204

8.12 Polyhedral Representations of Crystals and Diffusion Paths 207

8.13 Structures as Nets 210

8.14 Organic Structures 212

8.15 Protein Structures 212

8.15.1 Proteins: Primary Structure 212

8.15.2 Proteins: Secondary, Tertiary and Quaternary Structure 214

Answers to Introductory Questions 218

Problems and Exercises 219

Chapter 9: Defects, Modulated Structures and Quasicrystals 223

9.1 Defects and Occupancy Factors 223

9.2 Defects and Unit Cell Parameters 225

9.3 Defects and Density 226

9.4 Modular Structures 227

9.5 Polytypes 231

9.6 Crystallographic Shear (CS) Phases 233

9.7 Planar Intergrowths and Polysomes 237

9.8 Incommensurately Modulated

Structures 242

9.9 Quasicrystals 248

Answers to Introductory Questions 252

Problems and Exercises 253

Appendices 257

Appendix A Vector Addition and Subtraction 257

Appendix B Crystallographic Data for Some Inorganic Crystal Structures 259

Appendix C Schoenflies Symbols 263

Appendix D The 230 Space Groups 267

Appendix E Complex Numbers 271

Appendix F Complex Amplitudes 273

Answers to Problems and Exercises 275

Bibliography 283

Index 289

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