ESSENTIALS 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 vii
1 Bravais Lattices in Two Dimensions 1 (15)
1.1 Introduction 1 (1)
1.2 Development of One and Two 2 (1)
Dimensional Lattices
1.3 Basis and the Crystal Structure 3 (1)
1.4 Choice of a Unit Cell 4 (1)
1.5 Wigner-Seitz Unit Cell 4 (2)
1.6 Primitive Lattice Types and Crystal 6 (2)
Systems
1.7 Centering of Plane Lattices 8 (2)
1 Oblique Lattice 8 (1)
2 Rectangular Lattice 9 (1)
3 Square and Hexagonal Lattices 9 (1)
1.8 Two Dimensional Bravais Lattices 10 (1)
(Plane Lattices)
1.9 Unit Cell Calculations 10 (4)
Distance between Two Lattice Points 10 (2)
(Oblique System)
Linear and Planar Atomic Density 12 (1)
Planar Packing Efficiency 13 (1)
1.10 Summary 14 (1)
1.11 Definitions 14 (2)
Review Questions and Problems 15 (1)
2 Bravais Lattices in Three Dimensions 16 (25)
2.1 Introduction 16 (1)
2.2 Development of Three-dimensional 16 (1)
Lattices
2.3 Choice of Axes and Unit Cells 17 (2)
2.4 Derivation of Seven Primitive 19 (2)
Lattices/Unit Cells
(i) Oblique Lattice 19 (1)
(ii) Rectangular Lattice 19 (1)
(iii) Square Lattice 19 (2)
(iv) Rhombic Lattice 21 (1)
(v) Hexagonal Lattice 21 (1)
2.5 Types of Lattice Centering 21 (4)
1 Body Centering (I) 21 (1)
2 Face Centering (F) 22 (1)
3 Base Centering (A--, B--, C--) 22 (1)
4 Rhombohedral Centering 23 (2)
2.6 Derivation of Non-primitive 25 (5)
(Centered) Lattices
Monoclinic System 26 (1)
Orthorhombic System 27 (2)
Tetragonal System 29 (1)
Cubic System 29 (1)
2.7 Number of Lattice Points Per Unit Cell 30 (1)
2.8 Fractional Coordinates (Oblique 30 (1)
System)
2.9 Unit Cell Calculations 31 (6)
1 Volume of the Unit Cell 31 (1)
2 Distance Between Two Lattice Points 32 (3)
(Oblique System)
3 Linear, Planar and Volume Atomic 35 (1)
Density in Crystals
4 Volume Packing Efficiency 36 (1)
2.10 Interplanar Spacing 37 (2)
2.11 Summary 39 (1)
2.12 Definitions 39 (2)
Review Questions and Problems 39 (2)
3 Symmetry Elements in Two Dimensions 41 (19)
3.1 Introduction 41 (1)
3.2 Symmetry Elements 41 (2)
1 Translation 42 (1)
2 Proper Rotation 42 (1)
3 Reflection (Mirror Line) 43 (1)
3.3 Consistent Combinations of Symmetry 43 (1)
Operations
3.4 Combinations of Macroscopic Symmetry 44 (2)
Operations
(i) Rotation with a Translation t 44 (1)
(ii) Rotation and Reflection (Two 44 (2)
Reflections)
3.5 Point Groups in Two Dimensions 46 (1)
3.6 Plane Lattice Consistent with 47 (7)
Rotational Symmetry
3.7 Plane Lattice Consistent with Mirror 54 (1)
Symmetry
3.8 Combinations of Microscopic Symmetry 55 (1)
Operations
3.9 Space Group in Two Dimensions (Plane 55 (2)
Groups)
3.10 Summary 57 (1)
3.11 Definitions 58 (2)
Review Questions and Problems 59 (1)
4 Symmetry Elements in Three Dimensions 60 (19)
4.1 Introduction 60 (1)
4.2 Symmetry Elements 60 (2)
4.3 Macroscopic Symmetry Elements 62 (1)
4.4 Combinations of Macroscopic Symmetry 62 (1)
Operations
4.5 Rotation at a Point 62 (2)
4.6 Axial Combinations (Two Proper 64 (3)
Rotations)
4.7 Rotation and Reflection 67 (1)
(Rotoreflection)
4.8 Rotation and Inversion (Rotoinversion) 68 (1)
4.9 Proper and Improper Rotations 69 (4)
(a) Monoaxial Combinations 70 (1)
(b) Polyaxial Combinations 70 (2)
(c) Coexistence of Proper and Improper 72 (1)
Polyaxials
4.10 Reflection and Inversion 73 (3)
4.11 Classification of Symmetry Operations 76 (1)
4.12 Summary 76 (1)
4.13 Definitions 77 (2)
Review Questions and Problems 77 (2)
5 Derivation of Point Groups 79 (35)
5.1 Introduction 79 (1)
5.2 Derivation of Point Groups 79 (4)
(Conventional Method)
Rotoreflection Channel 79 (1)
1 Proper Rotation: X 80 (1)
2 Reflection (Mirror Plane): m 80 (1)
3 Rotoreflection (≡ 80 (1)
Rotoinversion): X
4 Axial Combinations: XXX 80 (1)
5 Mirror to Rotation Axis: X/m 80 (1)
6 Mirror || to Rotation Axis: Xm 81 (1)
7 Mirror to Axial Combinations XXX/m 81 (1)
8 Mirror || to Axial combinations: XXXm 81 (1)
Rotoinversion Channel 81 (2)
5.3 Point Group Notations 83 (5)
Schoenflies Notation 85 (2)
Hermann-Mauguin (International) Notation 87 (1)
5.4 Linear Orthogonal Transformation 88 (2)
5.5 Symmetry Operations and Group Theory 90 (2)
Group 90 (1)
Order of the Group 91 (1)
Cyclic Group 92 (1)
Generators of a Finite Group 92 (1)
Subgroups and Super Groups 92 (1)
5.6 Matrix Representation of Symmetry 92 (5)
Operations
(i) Orthogonal Axes 93 (3)
(ii) Crystallographic Axes 96 (1)
5.7 Derivation of Point Group (Matrix 97 (8)
Method)
1 Triclinic Crystal System 97 (1)
2 Monoclinic Crystal System 98 (1)
3 Orthorhombic Crystal System 99 (1)
4 Tetragonal Crystal System 100(2)
5 Trigonal Crystal System 102(2)
6 Hexagonal Crystal System 104(1)
7 Cubic Crystal System 104(1)
5.8 Equivalent Positions in Point Groups 105(2)
5.9 Laue Symmetry 107(1)
5.10 Point Groups, Crystal Classes and 107(1)
Crystal Systems
5.11 Summary 108(1)
5.12 Definitions 109(5)
Review Questions and Problems 110(2)
Appendix 1 112(1)
Appendix 2 112(2)
6 Derivation of Space Groups 114(13)
6.1 Introduction 114(1)
6.2 Microscopic Symmetry Elements 114(1)
6.3 Combination of Microscopic Symmetry 114(3)
Operations
Glide Planes 114(1)
Axial Glide 115(1)
Diagonal Glide 115(1)
Diamond Glide 116(1)
Screw Axes 116(1)
6.4 General Equivalent Positions and 117(2)
Special Positions
6.5 Systematic Absences 119(2)
Systematic Absences Due to Lattice 119(1)
Centering
Systematic Absences Due to Microscopic 119(2)
Symmetries
6.6 Space Groups 121(1)
6.7 Classification of Space Groups 121(1)
The Symmorphic Space Groups 121(1)
The Non-symmorphic Space Groups 122(1)
6.8 Derivation of Space Groups 122(3)
Triclinic Crystal System 123(1)
Monoclinic Crystal System 123(1)
Orthorhombic Crystal System 123(1)
Tetragonal Crystal System 123(2)
Trigonal Crystal System 125(1)
Hexagonal Crystal System 125(1)
Cubic Crystal System 125(1)
6.9 Summary 125(1)
6.10 Definitions 125(2)
Review Questions and Problems 126(1)
7 Crystal Planes, Directions and Projections 127(23)
7.1 Crystal Planes and Zones 127(1)
7.2 Crystal Directions and Zone Axes 128(3)
1 The Zone Law 130(1)
2 Zone Axis at the Intersection of two 130(1)
Planes
3 Plane Parallel to Two Directions 130(1)
4 The Addition Rule 131(1)
7.3 Miller-bravais Indices 131(1)
7.4 Transformation of Indices 132(6)
Transformation of Indices of Crystal 133(1)
Planes (Unit Cell)
(i) Rhombohedral and FCC 133(1)
(ii) Hexagonal and Orthorhombic 134(1)
(iii) Rhombohedral and Hexagonal 135(2)
Transformation of Indices of Direction 137(1)
(Zone Axes)
Hexagonal (4-index System) and Trigonal 137(1)
(3-index System)
7.5 Crystal Projections 138(4)
(i) Projection of Atoms/ions in the 138(1)
Unit Cell
(ii) Projection of Crystal Faces 139(1)
(Miller Planes)
Spherical Projection 140(1)
Gnomonic Projection 141(1)
Stereographic Projection 141(1)
7.6 The Reciprocal Lattice 142(5)
Some Geometrical Relationships 145(1)
1 The Zone Law 145(1)
2 Zone Axis at the Intersection of Two 146(1)
Planes
3 A Plane (hkl) Containing Two 146(1)
Directions [w1v1w1] and [u2v2w2]
7.7 Summary 147(1)
7.8 Definitions 148(2)
Review Questions and Problems 149(1)
8 Experiment and Theory of Crystal Growth 150(38)
8.1 Introduction 150(1)
8.2 Methods of Crystal Growth 150(1)
8.3 Solution Growth 151(6)
(a) Aqueous Solution Method 152(1)
(b) Flux Method 152(2)
(c) Hydrothermal Method 154(3)
8.4 Melt growth 157(4)
(a) Bridgman-Stockbarger Method 157(1)
(b) Czochralski Method 158(2)
(c) Zone Refining Method 160(1)
(d) Float Zone Method 161(1)
8.5 Nucleation 161(1)
8.6 Energy of Formation of a Nucleus 162(8)
(i) Homogeneous Nucleation 162(5)
(ii) Hetrogeneous Nucleation 167(1)
Cap Shaped Nucleus 167(2)
Disc--shaped (Cylinderical) Nucleus 169(1)
8.7 Velocity of Growth 170(2)
8.8 Theories/Model of Crystal Growth 172(1)
8.9 Theories Based on Atomic Model 172(3)
1 Kossel's Theory 172(2)
2 Screw Dislocation Theory 174(1)
8.10 Theories Based on Thermodynamics 175(9)
Considerations
1 The Diffusion Theory 175(3)
2 Bulk Diffusion Model 178(1)
3 BCF Bulk Diffusion Model 179(1)
(i) Hemi-spherical Force Field 180(1)
(ii) Semi-cylinderical Force Field 181(1)
(iii) Plane Force Field 182(2)
8.11 Concentration of Kinks and Mobility 184(1)
of Adsorbed Molecules
8.12 Summary 185(1)
8.13 Definitions 186(2)
Review Questions and Problems 186(2)
9 Crystal Imperfections 188(24)
9.1 Introduction 188(1)
9.2 Concentration of Point Imperfections 188(4)
Schottky Imperfection (Monoatomic Solid) 188(2)
Frenkel Imperfection (Monoatomic Solid) 190(1)
Schottky Imperfection (Ionic solid) 191(1)
9.3 The Geometry of Dislocations 192(1)
9.4 Burgers Vector and Burgers Circuit 193(1)
9.5 Energy of a Dislocation 194(2)
9.6 Slip Planes and Slip Directions 196(1)
9.7 Dislocation Reactions 196(3)
9.8 Density of Dislocations 199(1)
9.9 Observation of Dislocations 200(3)
(a) Method Based on Growth Spirals 200(1)
(b) Method Based on Etch Pits 200(1)
(c) Optical and Electron-optical Methods 201(1)
(d) Decoration Method 201(1)
(e) X-ray Diffraction Topography 201(2)
9.10 Surface Imperfections 203(5)
Grain Boundary 203(1)
Tilt and Twist Boundary 203(2)
Stacking Faults 205(1)
(i) Stacking Faults in FCC Crystals 205(1)
(ii) Stacking Faults in HCP Crystals 206(2)
9.11 Summary 208(1)
9.12 Definitions 208(4)
Review Questions and Problems 210(2)
10 Diffraction Methods 212(46)
10.1 Introduction 212(1)
10.2 Production of X-rays 212(3)
10.3 X-ray Diffraction 215(5)
Bragg's Law215(2)
The Laue Equations 217(3)
10.4 Diffraction Condition and Bragg's Law 220(2)
10.5 The X-ray Diffraction Experiments 222(1)
10.6 The Powder Method 222(5)
Indexing of Powder Lines 223(4)
10.7 The Laue Method 227(3)
Indexing of Laue Photographs 228(2)
10.8 The Rotation/Oscillation Method 230(6)
Interpretation of Rotation/Oscillation 231(2)
Photographs (Formation of Layer Lines)
Indexing of Rotation/Oscillation 233(3)
Photographs
10.9 The Weissenberg Method 236(10)
Equi-inclination Setting 241(1)
Indexing of Weissenberg Photographs 242(4)
10.10 The Precession Method 246(1)
10.11 X-ray Diffractometers 247(3)
Powder X-ray Diffractometer 248(1)
Single Crystal X-ray Diffractometer 248(2)
10.12 Other Diffraction Methods 250(3)
Neutron Diffraction 250(2)
Electron Diffraction 252(1)
10.13 Summary 253(2)
10.14 Definitions 255(3)
Review Questions and Problems 255(3)
11 Factors Affecting X-ray Intensities 258(14)
11.1 Introduction 258(1)
11.2 The Structure Factor 258(2)
11.3 The Lorentz Factor 260(2)
11.4 The Polarization Factor 262(2)
11.5 The Temperature Factor 264(1)
11.6 The Multiplicity Factor 264(1)
11.7 The Absorption Factor 265(1)
11.8 Extinction 266(2)
11.9 The R-factor 268(1)
11.10 Summary 268(1)
11.11 Definitions 269(3)
Review Questions and Problems 270(2)
12 Structure Factors and Fourier Synthesis 272(17)
12.1 Introduction 272(1)
12.2 Waves Motion 272(1)
12.3 Superposition of Waves 273(2)
12.4 Phase of a Wave in Three Dimensions 275(1)
(Unit Cell)
12.5 The Structure Factor of a Crystal 276(1)
12.6 The Fourier Synthesis 277(2)
12.7 Special Structure Factors Due to 279(3)
Symmetry
Case I Center of Symmetry 280(1)
Case II 2-fold Rotation (z-axis) 280(1)
Case III m z-axis 280(1)
Case IV 2-fold Screw Axis Along [001], 281(1)
21
Case V a-glide Oriented Along [010] 281(1)
12.8 Special Structure Factors Due to 282(2)
Lattice Centering
Case /Body Centering in a Cubic Lattice 282(1)
Case II Face Centering in a Cubic 283(1)
Lattice
12.9 Anomalous Scattering 284(1)
12.10 Summary 285(1)
12.11 Definitions 286(3)
Review Questions and Problems 287(2)
13 Crystal Structure Analysis 289(12)
13.1 Introduction 289(1)
13.2 Trial and Error Method 289(1)
13.3 The Patterson Function 290(2)
13.4 The Heavy Atom Method 292(1)
13.5 Isomorphous Replacement 293(1)
13.6 Superposition Method 294(1)
13.7 Direct Methods 294(5)
Inequality Relationship 295(1)
Case I Center of Symmetry 296(1)
Case II 2-fold Axis (|| to z) 297(1)
Statistical Method 297(2)
13.8 Summary 299(1)
13.9 Definitions 299(2)
Review Questions and Problems 300(1)
14 Crystal Structure Refinements 301(10)
14.1 Introduction 301(1)
14.2 Successive Fourier Syntheses 301(1)
14.3 Difference Fourier Synthesis 302(3)
(i) Position Error 303(1)
(ii) Missing Atoms 303(1)
(iii) Errors in the Thermal Parameters 304(1)
14.4 Least-squares Refinement 305(2)
14.5 Constrained Least-squares Refinement 307(1)
14.6 Automation of Structure Analysis 307(2)
14.7 Summary 309(1)
14.8 Definitions 309(2)
Review Questions and Problems 310(1)
15 Groups, Matrices and Representation of 311(20)
Symmetry Operations
15.1 Introduction 311(1)
15.2 Elements of Group Theory 311(2)
Group 312(1)
Cycle Group 312(1)
Order of the Group 312(1)
Finite Group 312(1)
Classes 313(1)
15.3 Construction of Group Multiplication 313(3)
Table
15.4 Elements of Matrices 316(3)
Matrix 316(1)
Diagonal Matrix 317(1)
Unit/Identity Matrix 317(1)
Unitary Matrix 317(1)
Singular Matrix 317(1)
Inverse of a Matrix 317(1)
Skew-symmetric/Anti-symmetric Matrix 318(1)
Transpose of a Matrix 318(1)
Orthogonal Matrix 318(1)
Trace/Character of a Matrix 318(1)
Property of Trace/Character 319(1)
15.5 Representation 319(1)
15.6 Orthogonality Theorem 320(1)
15.7 Properties of Irreducible 321(1)
Representations
15.8 Mulliken Symbols 322(1)
15.9 Construction of Character Tables for 323(4)
Point Groups
15.10 Summary 327(1)
15.11 Definitions 328(3)
Review Questions and Problems 329(2)
Bibliography 331(2)
Subject Index 333
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