X-ray photograph of zinc blende (Friedrich, Knipping, and
von Laue, 1912) ............................................... xvi
X-ray photograph of deoxyribonucleic acid (Franklin and
Gosling, 1952) ............................................... xvii
1 Crystals and crystal structures .............................. 1
1.1 The nature of the crystalline state ..................... 1
1.2 Constructing crystals from close-packed hexagonal
layers of atoms ......................................... 5
1.3 Unit cells of the hep and ccp structures ................ 6
1.4 Constructing crystals from square layers of atoms ....... 9
1.5 Constructing body-centred cubic crystals ................ 9
1.6 Interstitial structures ................................ 11
1.7 Some simple ionic and covalent structures .............. 18
1.8 Representing crystals in projection: crystal plans ..... 20
1.9 Stacking faults and twins .............................. 20
1.10 The crystal chemistry of inorganic compounds ........... 27
1.10.1 Bonding in inorganic crystals ................... 28
1.10.2 Representing crystals in terms of coordination
polyhedra ....................................... 30
1.11 Introduction to some more complex crystal structures ... 32
1.11.1 Perovskite (СаТiO3), barium titanate (BaТiO3)
and related structures .......................... 32
1.11.2 Tetrahedral and octahedral structures - silicon
carbide and alumina ............................. 34
1.11.3 The oxides and oxy-hydroxides of iron ........... 36
1.11.4 Silicate structures ............................. 38
1.11.5 The structures of silica, ice and water ......... 44
1.11.6 The structures of carbon ........................ 48
Exercises ................................................... 54
2 Two-dimensional patterns, lattices and symmetry ............. 56
2.1 Approaches to the study of crystal structures .......... 56
2.2 Two-dimensional patterns and lattices .................. 57
2.3 Two-dimensional symmetry elements ...................... 59
2.4 The five plane lattices ................................ 62
2.5 The seventeen plane groups ............................. 65
2.6 One-dimensional symmetry: border or frieze patterns .... 66
2.7 Symmetry in art and design: counterchange patterns ..... 66
2.8 Layer (two-sided) symmetry and examples in woven
textiles ............................................... 74
2.9 Non-periodic patterns and tilings ...................... 78
Exercises ................................................... 83
3 Bravais lattices and crystal systems ........................ 86
3.1 Introduction ........................................... 86
3.2 The fourteen space (Bravais) lattices .................. 86
3.3 The symmetry of the fourteen Bravais lattices:
crystal systems ........................................ 90
3.4 The coordination or environments of Bravais lattice
points: space-filling polyhedra ........................ 92
Exercises ................................................... 97
4 Crystal symmetry: point groups, space groups, symmetry-
related properties and quasiperiodic crystals ............... 99
4.1 Symmetry and crystal habit ............................. 99
4.2 The thirty-two crystal classes ........................ 101
4.3 Centres and inversion axes of symmetry ................ 102
4.4 Crystal symmetry and properties ....................... 106
4.5 Translational symmetry elements ....................... 110
4.6 Space groups .......................................... 113
4.7 Bravais lattices, space groups and crystal
structures ............................................ 120
4.8 The crystal structures and space groups of organic
compounds ............................................. 123
4.8.1 The close packing of organic molecules ......... 124
4.8.2 Long-chain polymer molecules ................... 127
4.9 Quasicrystals (quasiperiodic crystals or
crystalloids) ......................................... 129
Exercises .................................................. 134
5 Describing lattice planes and directions in crystals:
Miller indices and zone axis symbols ....................... 135
5.1 Introduction .......................................... 135
5.2 Indexing lattice directions—zone axis symbols ......... 136
5.3 Indexing lattice planes—Miller indices ................ 137
5.4 Miller indices and zone axis symbols in cubic
crystals .............................................. 140
5.5 Lattice plane spacings, Miller indices and Laue
indices ............................................... 141
5.6 Zones, zone axes and the zone law, the addition rule .. 143
5.6.1 The Weiss zone law or zone equation ............ 143
5.6.2 Zone axis at the intersection of two planes .... 143
5.6.3 Plane parallel to two directions ............... 144
5.6.4 The addition rule .............................. 144
5.7 Indexing in the trigonal and hexagonal systems:
Weber symbols and Miller-Bravais indices .............. 145
5.8 Transforming Miller indices and zone axis symbols ..... 148
5.9 Transformation matrices for trigonal crystals with
rhombohedral lattices ................................. 151
5.10 A simple method for inverting a 3 × 3 matrix .......... 152
Exercises .................................................. 153
6 The reciprocal lattice ..................................... 155
6.1 Introduction .......................................... 155
6.2 Reciprocal lattice vectors ............................ 155
6.3 Reciprocal lattice unit cells ......................... 157
6.4 Reciprocal lattice cells for cubic crystals ........... 161
6.5 Proofs of some geometrical relationships using
reciprocal lattice vectors ............................ 163
6.5.1 Relationships between a, b, с and a*, b*, c* ... 163
6.5.2 The addition rule .............................. 164
6.5.3 The Weiss zone law or zone equation ............ 164
6.5.4 d-spacing of lattice planes (hkl) .............. 165
6.5.5 Angle p between plane normals (h1k1l1) and
{h2k2l2) ........................................ 165
6.5.6 Definition of a*, b*, с* in terms of a, b, с ... 166
6.5.7 Zone axis at intersection of planes (h1k1l1)
and (h2k2l2) .................................... 166
6.5.8 A plane containing two directions [u1v1w1]
and [U2V2W2] ................................... 166
6.6 Lattice planes and reciprocal lattice planes .......... 166
6.7 Summary ............................................... 169
Exercises .................................................. 169
7 The diffraction of light ................................... 170
7.1 Introduction .......................................... 170
7.2 Simple observations of the diffraction of light ....... 172
7.3 The nature of light: coherence, scattering and
interference .......................................... 177
7.4 Analysis of the geometry of diffraction patterns
from gratings and nets ................................ 180
7.5 The resolving power of optical instruments: the
telescope, camera, microscope and the eye ............. 187
Exercises .................................................. 197
8 X-ray diffraction: the contributions of Max von
Laue, W.H. and W.L. Bragg and P.P. Ewald ................... 198
8.1 Introduction .......................................... 198
8.2 Laue's analysis of X-ray diffraction: the three Laue
equations ............................................. 199
8.3 Bragg's analysis of X-ray diffraction: Bragg's law .... 202
8.4 Ewald's synthesis: the reflecting sphere
construction .......................................... 204
Exercises .................................................. 209
9 The diffraction of X-rays .................................. 210
9.1 Introduction .......................................... 210
9.2 The intensities of X-ray diffracted beams: the
structure factor equation and its applications ........ 214
9.3 The broadening of diffracted beams: reciprocal
lattice points and nodes .............................. 223
9.3.1 The Scherrer equation: reciprocal lattice
points and nodes ............................... 223
9.3.2 Integrated intensity and its importance ........ 227
9.3.3 Crystal size and perfection: mosaic structure
and coherence length ........................... 227
9.4 Fixed θ, varying λ X-ray techniques: the Laue method .. 228
9.5 Fixed λ, varying θ X-ray techniques: oscillation,
rotation and precession methods ....................... 231
9.5.1 The oscillation method ......................... 232
9.5.2 The rotation method ............................ 234
9.5.3 The precession method .......................... 235
9.6 X-ray diffraction from single crystal thin films and
multilayers ........................................... 239
9.7 X-ray (and neutron) diffraction from ordered
crystals .............................................. 243
9.8 Practical considerations: X-ray sources and
recording techniques .................................. 246
9.8.1 The generation of X-rays in X-ray tubes ........ 247
9.8.2 Synchrotron X-ray generation ................... 248
9.8.3 X-ray recording techniques ..................... 249
Exercises .................................................. 249
10 X-ray diffraction of polycrystalline materials ............. 252
10.1 Introduction .......................................... 252
10.2 The geometrical basis of polycrystalline (powder)
X-ray diffraction techniques .......................... 253
10.2.1 Intensity measurement in the X-ray
diffractometer ................................. 258
10.2.2 Back reflection and Debye-Scherrer powder
techniques ..................................... 260
10.3 Some applications of X-ray diffraction techniques in
polycrystalline materials ............................. 262
10.3.1 Accurate lattice parameter measurements ........ 262
10.3.2 Identification of unknown phases ............... 263
10.3.3 Measurement of crystal (grain) size ............ 266
10.3.4 Measurement of internal elastic strains ........ 266
10.4 Preferred orientation (texture, fabric) and its
measurement ........................................... 267
10.4.1 Fibre textures ................................. 268
10.4.2 Sheet textures ................................. 269
10.5 X-ray diffraction of DNA: simulation by light
diffraction ........................................... 272
10.6 The Rietveld method for structure refinement .......... 277
Exercises .................................................. 280
11 Electron diffraction and its applications .................. 283
11.1 Introduction .......................................... 283
11.2 The Ewald reflecting sphere construction for
electron diffraction .................................. 284
11.3 The analysis of electron diffraction patterns ......... 288
11.4 Applications of electron diffraction .................. 290
11.4.1 Determining orientation relationships between
crystals ....................................... 290
11.4.2 Identification of polycrystalline materials .... 292
11.4.3 Identification of quasiperiodic crystals
(quasicrystals) ................................ 292
11.5 Kikuchi and electron backscattered diffraction
(EBSD) patterns ....................................... 294
11.5.1 Kikuchi patterns in the ТЕМ .................... 294
11.5.2 Electron backscattered diffraction (EBSD)
patterns in the SEM ............................ 298
11.6 Image formation and resolution in the ТЕМ ............. 300
Exercises .................................................. 304
12 The stereographic projection and its uses .................. 308
12.1 Introduction .......................................... 308
12.2 Construction of the stereographic projection of
a cubic crystal ....................................... 311
12.3 Manipulation of the stereographic projection: use
of the Wulff net ...................................... 314
12.4 Stereographic projections of non-cubic crystals ....... 317
12.5 Applications of the stereographic projection .......... 320
12.5.1 Representation of point group symmetry ......... 320
12.5.2 Representation of orientation relationships .... 322
12.5.3 Representation of preferred orientation
(texture or fabric) ............................ 323
12.5.4 Trace analysis ................................. 325
Exercises .................................................. 328
13 Fourier analysis in diffraction and image formation ........ 329
13.1 Introduction—Fourier series and Fourier transforms .... 329
13.2 Fourier analysis in crystallography ................... 332
13.2.1 X-ray resolution of a crystal structure ....... 337
13.3 The structural analysis of crystals and molecules ..... 338
13.3.1 Trial and error methods ........................ 339
13.3.2 The Patterson function: Patterson or vector
maps ........................................... 340
13.3.3 Interpretation of Patterson maps: heavy atom
and isomorphous replacement techniques ......... 346
13.3.4 Direct methods ................................. 348
13.3.5 Charge flipping ................................ 349
13.4 Analysis of the Fraunhofer diffraction pattern from
a grating ............................................. 350
13.5 Abbe theory of image formation ........................ 356
14 The physical properties of crystals and their
description by tensors ..................................... 362
14.1 Introduction .......................................... 362
14.2 Second rank tensor properties ......................... 363
14.2.1 General expression for a second rank tensor
relating two vectors ........................... 363
14.2.2 Simplification of second rank tensor
equations: principal axes ...................... 366
14.2.3 Representation of second rank tensor
properties: the representation quadric ......... 366
14.3 Neumann's principle ................................... 368
14.3.1 Pyroelectricity and ferroelectricity ........... 369
14.4 Second rank tensors that describe stress and strain ... 369
14.4.1 The stress tensor: principal axes
(eigenvectors) and principal values
(eigenvalues) .................................. 369
14.4.2 The strain tensor, Neumann's principle, and
thermal expansion .............................. 372
14.4.3 Atomic displacement parameters (ADPs) ..........
14.5 The optical properties of crystals .................... 374
14.6 Third rank tensors: piezoelectricity .................. 379
14.7 Fourth rank tensor properties: elasticity ............. 380
Exercises ..................................................... 382
Appendix 1 Computer programs, models and model-building in
crystallography ............................................ 385
Appendix 2 Polyhedra in crystallography ...................... 393
Appendix 3 Biographical notes on crystallographers and
scientists mentioned in the text ........................... 403
Appendix 4 Some useful crystallographic relationships ........ 449
Appendix 5 A simple introduction to vectors and complex
numbers and their use in crystallography ................... 452
Appendix 6 Systematic absences (extinctions) in X-ray
diffraction and double diffraction in electron
diffraction patterns ....................................... 459
Appendix 7 Group theory in crystallography ................... 469
Answers to exercises .......................................... 481
Further Reading ............................................... 497
Index ......................................................... 507
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