SSENTIALS OF CRYSTALLOGRAPHY presents a comprehensive study of the essential aspects of crystallography. The topics include a detail discussion of geometry and symmetry of crystals, a simplified approach to derive the point groups and space groups, methods of crystal growth and related theories, imperfections in crystalline solids, various diffraction methods, procedures for solving crystal structures and computing methods in crystallography. Keeping in view the diverse nature of readers, the treatments and the mathematics used in the book have been kept as simple as possible. This book will serve as a textbook to any crystallographic course at Postgraduate and M. Phil. level. In addition, this will be helpful for all researchers in physics, chemistry, biology, mineralogy etc. who are working with crystallography related problems.

Preface to the Second Edition v

Preface to the First Edition vll

1. Brava is Lattices in Two Diomensions ’

1.1 Inroduction

1.2 Development of One and Two Dimensional Lattices

1.3 Basis and the Crystal Structure 3

1.4 Choice of a Unit Cell 4

1.5 Wigner-Seitz Unit Cell 4

1.6 Primitive Lattice Types and Crystal Systems 6

1.7 Centering of Plane Lattices 8

      1. Oblique Lattice 8

      2. Rectangular Lattice 9

      3. Square and Hexagonal Lattices 9

1.8 Two Dimensional Bravais Lattices (Plane Lattices) 10

1.9 Unit Cell Calculations J O

      Distance between Two Lattice Points (Oblique System) 10

      Linear and Planar Atomic Density 12

      Planar Packing Efficiency 13

1.10 Summary 14

1.1.1 Definitions 14

Revie-w Questions and Problems 15

2. Bravais Lattices in Three Dimensi o ns 16

2.1 Introduction 16

2.2 Development of Three-dimensional Lattices 16

2.3 Choice of Axes and Unit Cells 17

2.4 Derivation of Seven Primitive Lattices/Unit Cells 19

      (i) Oblique Lattice 19

      (ii) Rectangular Lattice 19

      (iii) Square Lattice 19

      (iv) R11ombic Lattice 21

      (v) Hexagonal Lattice 21

2.5 Types of Lattice Centering 21

      1. Body Centering (I) 21

      2. Face Centering (F) 22

      3. Base Centering (A-, B-, C一） 22

      4. Rhombohedra l Centering 23

2.6 Derivation of Non-primitive (Centered) Lattices 25

      Monoclinic System 26

      Orthorhombic System 27

      Tetragona l System 29

      Cubic System 29

2.7 Number of Lattice Points Per Unit Cell 30

2.8 Fractional Coordinates (Oblique System) 30

2.9 Unit Cell Calculations 31

      1. Volume of the Unit Cell 31

      2. Distance Between Two Lattice Points (Oblique System) 32

      3. Linear, Planar and Volume Atomic Density in Crystals 35

      4. Volume Packing Efficiency 36

2.10 Interplanar Spacing 37

2.11 Summary 39

2.12 Definitions 39

Review Questions αnd Problems 39

3. Symmetry Elements in Two Dimensions 41

3.1 1ntroduction 41

3.2 Symmetry Elements 41

      1. Translation 42

      2. Proper Rotation 42

      3. Reflection (Mirror Line) 43

3.3 Consistent Combinations of Symmetry Operations 43

3.4 Combinations of Macroscopic Symmetry Operations 44

      (i) Rotation with a Translationt 44

      (ii) Rotation and Reflection (Two Reflections) 44

3.5 Point Groups in Two Dimensions 46

3.6 Plane Lattice Consistent with Rotational Symmetry 47

3.7 Plane Lattice Consistent with Mirror Symmetry 54

3.8 Combinations of Microscopic Symmetry Operations 55

3.9 Space Group in Two Dimensions (Plane Groups) 55

3.10 Summary 57

3.11 Definitions 58

Review Questions and Problems 59

4. Symmetry Elements in Three Dimensions 60

4.1 Introduction 60

4.2 Symmetry Elements 60

4.3 Macroscopic Symme盯F Elements 62

4.4 Combinations of Macroscopic Symmetry Operations 62

4.5 Rotation at a Point 62

4.6 Axial Combinations (Two Proper Rotations) 64

4.7 Rotation and Reflection (Rotoreflection) 67

4.8 Rotation and Inversion (Rotoinversion) 68

4.9 Proper and Improper Rotations 69

      (a) Monoaxia l Combinations 70

      (b) Polyaxial Combinations 70

      (c) Coexistence of Proper and Improper Polyaxia ls 72

4.10 Reflection and Inversion 73

4.11 Classification of Symmetry Operations 76

4.12 Summary 76

4.13 Defin itions 77

Review Questions and Problems 77

5. Derivatio n o f po int Gro u ps 79

5.1 Introduction 79

5.2 Derivation of Point Groups (Conventional Method) 79

      Rotoreflection Channel 79

      1. Proper Rotation :X 80

      2. Reflection 仙1irror Plane): m 80

      3. Rotoretlection (= Rotoinversion): X 80

      4. Axial Combinations：XXX 80

      5. Mirror I- to Rotation Axis：一－X/m 80

      6. Mirror II to Rotation Axis：Xm 81

      7. Mirror I to Axial Combinations：XXX/m 81

      8. Mirror II to Axial combinations: XXXm 81

      Rotoinversion Channel 81

5.3 Point Group Notations 83

      Schoenflies Notation 85

      Herrnann-Mauguin (International) Notation 87

5.4 Linear Orthogonal Transformation 88

5.5 Symmetry Operations and Group Theory 90

      Group 90

      Order of the Group 91

      Cyclic Group 92

      Generators of a Finite Group 92

      Subgroups and Super Groups 92

5.6 Matrix Representation of Symmetry Operations 92

      (i) Orthogonal Axes 93

      (ii) Crystallographic Axes 96

5.7 Derivation of Point Group (Matrix Method) 97

      1. Triclinic Crystal System 97

      2. Monoclinic Crystal System 98

      3. Orthorhombic Crystal System 99

      4. Tetragonal Crystal System J OO

      5. Trigonal Crystal System 102

      6. Hexagonal Crystal System 104

      7. Cubic Crystal System 104

5.8 Equivalent Positions in Point Groups 105

5.9 Laue Symmetry 107

5.10 Point Groups, Crystal Classes and Crystal Systems 107

5.11 Summary 108

5.12 Definitions 109

Review Questions and Problems 110

Appendix 1 112

Appendix 2 112

6. Derivation of Space Groups 114

6.1 Introduction 114

6.2 Microscopic Symmetry Elements 114

6.3 Combination of M icroscopic Symmetry Operations 114

      Glide Planes 114

      Axial Glide 115

      Diagonal Glide 115

      Diamond Gl ide 116

      Screw Axes 116

6.4 Genera l Equiva lent Positions and Special Positions 117

6.5 Systematic Absences 119

      Systematic Absences Due to Lattice Centering 119

      Systematic Absences Due to Microscopic Symmetries 119

6.6 Space Groups 121

6.7 Classi fication of Space Groups 121

      The Symmorphic Space Groups 121

      The Non-symmorphic Space Groups 122

6.8 Derivation of Space Groups 122

      Triclinic Crystal System 123

      Monoclinic Crystal System 123

      Orthorhombic Crystal System 123

      Tetragonal Crystal System 123

      Trigonal Crystal System 125

      Hexagonal Crystal System 125

      Cubic Crystal System 125

6.9 Suinmary 125

6.10 Definitions 125

Review Questions αnd Problems 126

7. Crystal Planes, Directio ns and Projecti o ns 127

7.1 Crystal Planes and Zones 127

7.2 Crystal Directions and Zone Axes 128

      1. The Zone Law 130

      2. Zone Axis at the Intersection of two Planes 130

      3. Plane Para llel to Two Directions 130

      4. The Addition Rule 131

7.3 Miller-bravais Indices 131

7.4 Transfornation of Indices 132

      Transfonnation of Lndices of Crystal Planes (Unit Cell) 133

      (i) Rhombohedra l and FCC 133

      (ii) Hexagonal and Orthorhombic 134

      (iii) Rhombohedra l and Hexagonal 135

      Transformation of Indices of Direction (Zone Axes) 137

      Hexagonal (4-index System) and Trigonal (3-index System) 137

7.5 Crystal Projections 138

      (i) Projection of Atoms/ions in the Un it Cell 138

      (ii) Projection of Crystal Faces (Miller Planes) 139

      Spherical Projection 140

      Gnomonic Projection 141

      Stereographic Projection 141

7.6 The Reciprocal Lattice 142

      Some Geometrical Relationships 145

      1. The Zone Law 145

      2. Zone Axis at the Intersection of Two Planes 146

      3. A Plane (hkl) Containing Two Directions [u1v 1w1] and [u2v2w2］ 146

7.7 Summary 147

7.8 Definitions 148

Review Queslions and Problems 149

8. Experiment and Theory of Crystal Growth 150

8.1 Introduction 150

8.2 Methods of Crystal Growth 150

8.3 Solution Growth 151

      (a) Aqueous Solution Method 152

      (b) Flux Method 152

      (c) Hydrotherma l Method 154

8.4 Melt growth 157

      (a) Bridgman-Stockbarger Method 157

      (b) Czocbralslci Method 158

      (c) Zone Refining Method 160

      (d) Float Zone Method 161

8.5 Nucleation 161

8.6 Energy of Formation of a Nucleus 162

      (i) Homogeneous Nucleation 162

      (ii) Hetrogeneous Nucleation 167

      Cap Shaped Nucleus 167

      Disc-shaped (Cylinderical) Nucleus 169

8.7 Velocity of Growth 170

8.8 Theories/Model of Crystal Growth 172

8.9 Theories Based on Atomic Model 172

      1. Kossel’s Theory 172

      2. Screw Dislocation Theory 174

8.10 Theories Based on Thermodynamics Considerations 175

      1. The Diffusion Theory 175

      2. Bulk Diffusion Model 178

      3. BCF Bulk Diffusion Model 179

      (i) Hemi-spherical Force Field 180

      (ii) Semi-cylinderical Force Field 181

      (iii) Plane Force Field 182

8.11 Concentration of Kinks and Mobility of Adsorbed Molecules 184

8.12 Summary 185

8.13 Definitions 186

Review Questions and Problems 186

9. Crystal Imperfections 188

9.1 Introduction 188

9.2 Concentration of Point Imperfections 188

      Schottky Imperfection (Moooatomic Solid) 188

      Frenkel Imperfection (Moooatomic Solid) 190

      Schottky Imperfection (Ionic solid) 191

9.3 The Geometry of Dislocations 192

9.4 Burgers Vector and Burgers Circuit 193

9.5 Energy of a Dislocation 194

9.6 Slip Planes and Slip Directions 196

9.7 Dislocation Reactions 196

9.8 Density of Dislocations 199

9.9 Observation of Dislocations 200

      (a) Method Based on Growth Spirals 200

      (b) Method Based on Etch Pits 200

      (c) Optical and Electron-optical Methods 201

      (d) Decoration Method 201

      (e) X-ray Diffraction Topography 201

9.10 Surface Imperfections 203

      Grain Boundary 203

      Tilt and Twist Boundary 203

      Stacking Faults 205

      (i) Stacking Faults in FCC Crystals 205

      (ii) Stacking Faults in HCP Crystals 206

9.11 Summary 208

9.12 Definitions 208

Review Questions and Problems 210

10. Diffraction Methods 212

10.1 Introduction 212

10.2 Production of X-rays 212

10.3 X-ray Diffraction 215

      Bragg’s Law 215

      The Laue Equations 217

10.4 Diffraction Condition and Bragg’s Law 220

10.5 The X-ray Diffraction Experiments 222

10.6 The Powder Method 222

      Indexing of Powder Lines 223

10.7 The Laue Method 227

      Indexing of Laue Photographs 228

10.8 The Rotation/Oscillation Method 230

      Interpretation of Rotation/Oscillation Photographs (Formation of Layer Lines) 231

      Indexing of Rotation/Oscillation Photographs 233

10.9 The Weissenberg Method 236

      Equi-i nclination Setting 241

      Indexing of Weissenberg Photographs 242

10.10 The Precession Method 246

10.11 X-ray Diffractometer s 247

      Powder X-ray Dif丘actometer 248

      Single Crystal X-ray Diffractometer 248

10.12 Other Diffraction Methods 250

      Neutron Diffraction 250

      Electron Diffraction 252

10.13 Summary 253

10.14 Definitions 255

Review Questions and Problems 255

11. Facto rs Affecti ng X-ray Intensities 258

11.1 Introduction 258

11.2 The Structure Factor 258

11.3 The Lorentz Factor 260

11.4 The Polarization Factor 262

11.5 The Temperature Factor 264

11.6 The Multiplicity Factor 264

11.7 The Absorption Factor 265

11.8 Extinction 266

11.9 The R-factor 268

11.10 Summary 268

11.1 1 Definitions 269

Review Questions and Problems 270

12. Structure Factors and Fourier Synthesis 272

12.1 Introduction 272

12.2 Waves Motion 272

12.3 Superposition of Waves 273

12.4 Phase of a Wave in Three Dimensions (Unit Cell) 275

12.5 The Structure Factor of a Crystal 276

12.6 The Fourier Synthesis 277

12.7 Special Structure Factors Due to S归runetry 279

      Case I: Center of Symme盯f 280

      Case II: 2-fold Rotation (z-axis) 280

      Case III: m L z-axis 280

      Case IV: 2-fold Screw Axis Along [001], 21 281

      Case V：α-glide Oriented Along [010] 281

12.8 Special Structure Factors Due to Lattice Centering 282

      Case I: Body Centering in a Cubic Lattice 282

      Case II: Face Centering in a Cubic Lattice 283

12.9 Anomalous Scattering 284

12.10 Swnmary 285

12.11 Definitions 286

Review Questions and Problems 287

13. Crystal Structure Analysis 289

13.1 Introduction 289

13.2 Trial and Error Method 289

13.3 The Patterson Function 290

13.4 The Heavy Atom Method 292

13.5 Isomorphous Replacement 293

13.6 Superposition Method 294

13.7 Direct Methods 294

      Inequality Relationship 295

      Case I. Center of Symmetry 296

      Case II. 2-fold Axis (II to z) 297

      Statistical Method 297

13.8 Summary 299

13.9 Definitions 299

Review Questions and Problems 300

14. Crystal StructUTe Refinements 301

14.1 Introduction 301

14.2 Successive Fourier Syntheses 301

14.3 DifTerencε Fourier Synthesis 302

      (i) Position Error 303

      (ii) Missinε Atoms 303

      (iii) Errors in the Thermal Parameters 304

14.4 Least-squares Refinement 305

14.5 Constrained Least-squares Refinement 307

14.6 Automation of Structure Analysis 307

14.7 Summary 309

14.8 Definitions 309

Review Questions and Problems 310

15 Groups, Matrices and Representation of Symmetry Operations 311

15.1 Introduction 311

15.2 Elements of Group Theory 311

15.3 Construction of Group Multiptication Table 313

15.4 Elements of Maltices 316

      Transpose of a Matrix 318

      Orthogonal Matrix 318

      Trace/Character of a Matrix 318

      Propterty of Trace/Character 319

15.5 Representation 319

15.6 Orthogonality Theorem 320

15.7 Properties of Irreducible Representations 321

15.8 Mulliken Symbols 322

15.9 Construction of Character Tables for Point Groups 323

15.10 Summary 327

15.11 Definitions 328

Review Questions and Problems 329

Bibliography 331

Subject Index 333

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