1 Materials and material properties 1
1.1 Materials and structure 1
1.2 Organization of the book 2
1.3 About length scales 3
1.4 Wave-particle duality and the de Broglie relationship 7
1.5 What is a material property？ 9
1.5.1 Definition of a material property 9
1.5.2 Directional dependence of properties 10
1.5.3 A first encounter with symmetry 12
1.5.4 A first encounter with magnetic symmetry 15
1.6 So，what is this book all about？ 17
1.7 Chapter summary 19
1.8 Historical notes 20
1.9 Selected problems 21
2 The periodic table of the elements and interatomic bonds 23
2.1 About atoms 23
2.1.1 The electronic structure of the atom 23
2.1.2 The hydrogenic model 24
2.2 The periodic table 26
2.2.1 Layout of the periodic table 28
2.2.2 Trends across the table 31
2.3 Interatomic bonds 34
2.3.1 Quantum chemistry 34
2.3.2 Interactions between atoms 34
2.3.3 The ionic bond 36
2.3.4 The covalent bond 38
2.3.5 The metallic bond 39
2.3.6 The van der Waals bond 40
2.3.7 Mixed bonding 41
2.3.8 Electronic states and symmetry 41
2.3.9 Overview of bond types and material properties 42
2.4 Chapter summary 43
2.5 Historical notes 43
2.6 Selected problems 47
3 What is a crystal structure？ 49
3.1 Periodic arrangements of atoms 49
3.2 The space lattice 51
3.2.1 Basis vectors and translation vectors 51
3.2.2 Some remarks about notation 52
3.2.3 More about lattices 54
3.3 The four 2-D crystal systems 56
3.4 The seven 3-D crystal systems 57
3.5 The five 2-D Bravais nets and fourteen 3-D Bravais lattices 60
3.6 Other ways to define a unit cell 64
3.7 2-D and 3-D magnetic Bravais lattices 66
3.8 Chapter summary 71
3.9 Historical notes 72
3.10 Selected problems 73
4 Crystallographic computations 75
4.1 Directions in the crystal lattice 75
4.2 Distances and angles in a 3-D lattice 76
4.2.1 Distance between two points 76
4.2.2 The metric tensor 78
4.2.3 The dot product in a crystallographic reference frame 80
4.3 Worked examples 82
4.3.1 Computation of the length of a vector 82
4.3.2 Computation of the distance between two atoms 83
4.3.3 Computation of the angle between atomic bonds 84
4.3.4 Computation of the angle between lattice directions 84
4.3.5 An alternative method for the computation of angles 85
4.3.6 Further comments 85
4.4 Chapter summary 86
4.5 Historical notes 87
4.6 Selected problems 89
5 Lattice planes 90
5.1 Miller indices 90
5.2 Families of planes and directions 93
5.3 Special case：the hexagonal system 94
5.4 Crystal forms 96
5.5 Chapter summary 101
5.6 Historical notes 101
5.7 Selected problems 102
6 Reciprocal space 104
6.1 The reciprocal basis vectors 104
6.2 Reciprocal space and lattice planes 108
6.3 The reciprocal metric tensor 110
6.3.1 Computation of the angle between planes 112
6.3.2 Computation of the length of the reciprocal lattice vector 112
6.4 Worked examples 114
6.5 Chapter summary 119
6.6 Historical notes 119
6.7 Selected problems 120
7 Additional crystallographic computations 122
7.1 The stereographic projection 122
7.2 About zones and zone axes 125
7.2.1 The vector cross product 126
7.2.2 About zones and the zone equation 130
7.2.3 The reciprocal lattice and zone equation in the hexagonal system 131
7.3 Relations between direct space and reciprocal space 133
7.4 Coordinate transformations 135
7.4.1 Transformation rules 135
7.4.2 Example of a coordinate transformation 138
7.4.3 Converting vector components into Cartesian coordinates 140
7.5 Examples of stereographic projections 143
7.5.1 Stereographic projection of a cubic crystal 143
7.5.2 Stereographic projection of a monoclinic crystal 146
7.6 Chapter summary 149
7.7 Historical notes 150
7.8 Selected problems 151
8 Symmetry in crystallography 152
8.1 Symmetry of an arbitrary object 152
8.2 Symmetry operations 158
8.2.1 Basic isometric transformations 159
8.2.2 Compatibility of rotational symmetries with crystalline translational periodicity 160
8.2.3 Operations of the first kind：pure rotations 162
8.2.4 Operations of the first kind：pure translations 164
8.2.5 Operations of the second kind：pure reflections 166
8.2.6 Operations of the second kind：inversions 167
8.2.7 Symmetry operations that do not pass through the origin 168
8.3 Magnetic symmetry operations 169
8.3.1 Time-reversal symmetry and axial vectors 169
8.3.2 Time-reversing symmetry operations 173
8.4 Combinations of symmetry operations 175
8.4.1 Combination of rotations with the inversion center 175
8.4.2 Combination of rotations and mirrors 177
8.4.3 Combination of rotations and translations 178
8.4.4 Combination of mirrors and translations 181
8.4.5 Relationships and differences between operations of the first and second kind 183
8.4.6 Combinations of magnetic and regular symmetry operators 184
8.5 Point symmetry 186
8.6 Chapter summary 188
8.7 Historical notes 190
8.8 Selected problems 191
9 Point groups 193
9.1 What is a group？ 193
9.1.1 A simple example 193
9.1.2 Group axioms 194
9.1.3 Principal properties of groups 196
9.2 3-D crystallographic point symmetries 197
9.2.1 Step Ⅰ：the proper rotations 198
9.2.2 Step Ⅱ：combining proper rotations with two-fold rotations 199
9.2.3 Step Ⅲa：combining proper rotations with inversion symmetry 201
9.2.4 Step Ⅲb：combining proper rotations with perpendicular reflection elements 203
9.2.5 Step Ⅳ：combining proper rotations with coinciding reflection elements 204
9.2.6 Step Ⅴa：combining inversion rotations with coinciding reflection elements 204
9.2.7 Step Ⅴb：combining proper rotations with coinciding and perpendicular reflection elements 205
9.2.8 Step Ⅵ：combining proper rotations 206
9.2.9 Step Ⅶ：adding reflection elements to Step Ⅵ 207
9.2.10 General remarks 208
9.3 2-D crystallographic point symmetries 220
9.4 Magnetic point groups 221
9.4.1 Derivation 221
9.4.2 Visualization of the magnetic point groups 223
9.4.3 Color，charge，and time reversal 225
9.5 Chapter summary 227
9.6 Historical notes 228
9.7 Selected problems 228
10 Plane groups and space groups 230
10.1 Combining translations with point group symmetry 230
10.2 Plane groups 231
10.2.1 A simple example 231
10.2.2 A more complex example 233
10.2.3 The 17 plane groups 235
10.3 Space groups 236
10.3.1 A simple example 236
10.3.2 A second simple example 238
10.3.3 A more complex example 239
10.3.4 The symmorphic space groups 240
10.3.5 The non-symmorphic space groups 242
10.3.6 Space group generators 243
10.3.7 General remarks 247
10.4 The International Tables for Crystallography 248
10.5 Magnetic space groups 253
10.6 Chapter summary 255
10.7 Historical notes 256
10.8 Selected problems 257
11 X-ray diffraction：geometry 259
11.1 Properties and generation of X-rays 259
11.1.1 How do we generate X-rays？ 261
11.1.2 Wavelength selection 265
11.2 X-rays and crystal lattices 268
11.2.1 Scattering of X-rays by lattice planes 272
11.2.2 Bragg’s law in reciprocal space 276
11.3 Basic experimental X-ray diffraction techniques 280
11.3.1 The X-ray powder diffractometer 281
11.4 Chapter summary 289
11.5 Historical notes 289
11.6 Selected problems 290
12 X-ray diffraction：intensities 291
12.1 Scattering by electrons，atoms，and unit cells 291
12.1.1 Scattering by a single electron 291
12.1.2 Scattering by a single atom 293
12.1.3 Scattering by a single unit cell 298
12.2 The structure factor 300
12.2.1 Lattice centering and the structure factor 300
12.2.2 Symmetry and the structure factor 304
12.2.3 Systematic absences and the International Tables for Crystallography 307
12.2.4 Examples of structure factor calculations 307
12.3 Intensity calculations for diffracted and measured intensities 309
12.3.1 Description of the correction factors 310
12.3.2 Expressions for the total measured intensity 315
12.4 Chapter summary 317
12.5 Historical notes 317
12.6 Selected problems 318
13 Other diffraction techniques 320
13.1 Introductory remarks 320
13.2 Neutron diffraction 321
13.2.1 Neutrons：generation and properties 323
13.2.2 Neutrons：wavelength selection 325
13.2.3 Neutrons：atomic scattering factors 326
13.2.4 Neutrons：scattering geometry and diffracted intensities 330
13.2.5 Neutrons：example powder pattern 334
13.3 Electron diffraction 335
13.3.1 The electron as a particle and a wave 335
13.3.2 The geometry of electron diffraction 337
13.3.3 The transmission electron microscope 338
13.3.4 Basic observation modes in the TEM 340
13.3.5 Convergent beam electron diffraction 343
13.4 Synchrotron X-ray sources for scattering experiments 347
13.4.1 Synchrotron accelerators 348
13.4.2 Synchrotron radiation：experimental examples 350
13.5 Chapter summary 352
13.6 Historical notes 352
13.7 Selected problems 354
14 About crystal structures and diffraction patterns 356
14.1 Crystal structure descriptions 356
14.1.1 Space group description 356
14.1.2 Graphical representation methods 357
14.2 Crystal structures?powder diffraction patterns 360
14.2.1 The Ni powder pattern，starting from the known structure 361
14.2.2 The NaCl powder pattern，starting from the known structure 365
14.2.3 The Ni structure，starting from the experimental powder diffraction pattern 369
14.2.4 The NaCl structure，starting from the experimental powder diffraction pattern 372
14.2.5 General comments about crystal structure determination 375
14.3 Chapter summary 380
14.4 Historical notes 380
14.5 Selected problems 382
15 Non-crystallographic point groups 383
15.1 Example of a non-crystallographic point group symmetry 383
15.2 Icosahedral and related five-fold symmetry groups 384
15.2.1 The icosahedral point groups 384
15.2.2 Fullerene molecular structures 385
15.2.3 Icosahedral group representations 387
15.2.4 Other non-crystallographic point groups with five-fold symmetries 390
15.2.5 Descents in symmetry：decagonal and pentagonal groups 393
15.3 Non-crystallographic point groups with octagonal symmetry 395
15.4 Chapter summary 400
15.5 Historical notes 400
15.6 Selected problems 402
16 Periodic and aperiodic tilings 403
16.1 2-D plane tilings 403
16.1.1 2-D regular tilings 404
16.1.2 2-D Archimedean tilings 405
16.1.3 k-uniform regular tilings 406
16.1.4 Dual tilings - the Laves tilings 407
16.1.5 Tilings without regular vertices 408
16.2 Color tilings 408
16.3 Quasiperiodic tilings 410
16.4 Regular polyhedra and n-D regular polytopes 411
16.5 Crystals with stacking of 36 tilings 415
16.5.1 Simple close-packed structures：ABC stacking 415
16.5.2 Interstitial sites in close-packed structures 416
16.5.3 Representation of close-packed structures 417
16.5.4 Polytypism and properties of SiC semiconductors 419
16.5.5 36 close-packed tilings of polyhedral faces 420
16.6 Chapter summary 421
16.7 Historical notes 422
16.8 Selected problems 424
17 Metallic structures Ⅰ：simple，derivative，and superlattice structures 425
17.1 Introductory comments 425
17.2 Classification of structures 426
17.2.1 Strukturbericht symbols 426
17.2.2 Pearson symbols 427
17.2.3 Structure descriptions in this book 427
17.3 Parent structures 428
17.3.1 Geometrical calculations for cubic structures 430
17.4 Atomic sizes，bonding，and alloy structure 431
17.4.1 Hume-Rothery rules 432
17.4.2 Bonding in close-packed rare gas and metallic structures 433
17.4.3 Phase diagrams 437
17.5 Superlattices and sublattices：mathematical definition 438
17.6 Derivative structures and superlattice examples 439
17.6.1 fcc-derived structures and superlattices 439
17.6.2 bcc-derived superlattices 444
17.6.3 Diamond cubic derived superlattices 446
17.6.4 Hexagonal close-packed derived superlattices 448
17.7 Elements with alternative stacking sequences or lower symmetry 450
17.7.1 Elements with alternative stacking sequences 450
17.7.2 Elements with lower-symmetry structures 451
17.8 Natural and artificial superlattices 455
17.8.1 Superlattice structures based on the L12 cell 455
17.8.2 Artificial superlattices 457
17.8.3 X-ray scattering from long-period multi-layered systems 459
17.8.4 Incommensurate superlattices 459
17.9 Interstitial alloys 461
17.10 Chapter summary 462
17.11 Historical notes 463
17.12 Selected problems 464
18 Metallic structures Ⅱ：complex geometrically determined structures 466
18.1 Electronic states in metals 466
18.2 Topological close packing 468
18.2.1 The Kasper polyhedra 469
18.2.2 Connectivity of Kasper polyhedra 471
18.2.3 Metallic radii 471
18.3 Frank-Kasper alloy phases 472
18.3.1 A1 5 phases and related structures 472
18.3.2 The Laves phases and related structures 479
18.3.3 The sigma phase 486
18.3.4 The μ-phase and the M-，P-，and R-phases 488
18.4 Quasicrystal approximants 490
18.4.1 Mg32（Al，Zn）49 and α-Al-Mn-Si crystal structures 490
18.4.2 Mg32（Al，Zn）49 and α-Al-Mn-Si shell models 491
18.5 Chapter summary 494
18.6 Historical notes 495
18.7 Selected problems 496
19 Metallic structures Ⅲ：quasicrystals 497
19.1 Introductory remarks 497
19.2 The golden mean and pentagonal symmetry 498
19.3 1-D quasicrystals 501
19.3.1 The Fibonacci sequence and lattice derived by recursion 501
19.3.2 Lattice positions in the Fibonacci lattice 503
19.3.3 Construction of the Fibonacci lattice by the projection method 504
19.3.4 The Fourier transform of the Fibonacci lattice 505
19.4 2-D quasicrystals 507
19.4.1 2-D quasicrystals：Penrose tilings 507
19.4.2 The Penrose tiling derived by projection 512
19.4.3 2-D quasicrystals：other polygonal quasicrystals 514
19.5 3-D quasicrystals 516
19.5.1 3-D Penrose tilings 517
19.5.2 Indexing icosahedral quasicrystal diffraction patterns 519
19.5.3 Icosahedral quasicrystal diffraction patterns and quasilattice constants 521
19.5.4 3-D Penrose tiles：stacking，decoration，and quasilattice constants 522
19.5.5 3-D Penrose tiles：projection method 524
19.6 Multiple twinning and icosahedral glass models 525
19.7 Microscopic observations of quasicrystal morphologies 526
19.8 Chapter summary 528
19.9 Historical notes 528
19.10 Selected problems 530
20 Metallic structures Ⅳ：amorphous metals 531
20.1 Introductory comments 531
20.2 Order in amorphous and nanocrystalline alloys 532
20.3 Atomic positions in amorphous alloys 535
20.4 Atomic volume，packing，and bonding in amorphous solids 536
20.4.1 DRPHS model 537
20.4.2 Binding in clusters：crystalline and icosahedral short-range order 539
20.4.3 Icosahedral short-range order models 539
20.5 Amorphous metal synthesis 540
20.6 Thermodynamic and kinetic criteria for glass formation 542
20.7 Examples of amorphous metal alloy systems 543
20.7.1 Metal-metalloid systems 544
20.7.2 Rare earth-transition metal systems 545
20.7.3 Early transition metal-late transition metal systems 546
20.7.4 Multi-component nanocomposite systems 546
20.7.5 Multi-component bulk amorphous systems 548
20.8 X-ray scattering in amorphous materials 550
20.9 Extended X-ray absorption fine structure （EXAFS） 554
20.10 Mossbauer spectroscopy 557
20.11 Chapter summary 558
20.12 Historical notes 558
20.13 Selected problems 560
21 Ceramic structures Ⅰ：basic structure prototypes 561
21.1 Introductory remarks 561
21.2 Ionic radii 562
21.3 Bonding energetics in ionic structures 565
21.4 Rules for packing and connectivity in ionic crystals 566
21.4.1 Pauling’s rules for ionic structures 566
21.4.2 Radius ratio rules for ionic compounds 567
21.5 Oxides of iron 570
21.6 Halide salt structures：CsCl，NaCl，and CaF2 571
21.7 Close-packed sulfide and oxide structures：ZnS and Al2O3 574
21.8 Perovskite and spinel structures 577
21.8.1 Perovskites：ABO3 577
21.8.2 Spinels：AB2O4 580
21.9 Non-cubic close-packed structures：NiAs，CdI2，and TiO2 584
21.10 Layered structures 585
21.10.1 Magnetoplumbite phases 586
21.10.2 Aurivillius phases 586
21.10.3 Ruddlesden-Popper phases 588
21.10.4 Tungsten bronzes 589
21.10.5 Titanium carbosulfide 591
21.11 Additional remarks 591
21.12 Point defects in ceramics 592
21.13 Chapter summary 594
21.14 Historical notes 594
21.15 Selected problems 596
22 Ceramic structures Ⅱ：high-temperature superconductors 597
22.1 Introductory remarks about superconductivity 597
22.2 High-temperature superconductors：nomenclature 598
22.3 Perovskite-based high-temperature superconductors 599
22.3.1 Single-layer perovskite high-temperature superconductors 599
22.3.2 Triple-layer perovskite-based high-temperature superconductors 601
22.4 BSCCO，TBCCO，HBCCO，and ACBCCO HTSC layered structures 606
22.4.1 The BSCCO double-layer high-temperature superconductors 606
22.4.2 The TBCCO double-layer high-temperature superconductors 608
22.4.3 The TBCCO single-layer high-temperature superconductors 611
22.4.4 The HBCCO high-temperature superconductors 613
22.4.5 The ACBCCO high-temperature superconductors 615
22.4.6 Rutheno-cuprate high-temperature superconductors 615
22.4.7 Infinite-layer high-temperature superconductors 616
22.5 Chapter summary 616
22.6 Historical notes 617
22.7 Selected problems 619
23 Ceramic structures Ⅲ：terrestrial and extraterrestrial minerals 620
23.1 Classification of minerals 620
23.2 Silicates overview 622
23.2.1 Orthosilicates （nesosilicates） 624
23.2.2 Pyrosilicates （sorosilicates） 629
23.2.3 Chains of tetrahedra，metasilicates （inosilicates） 630
23.2.4 Double chains of tetrahedra 633
23.2.5 Sheets of tetrahedra，phyllosilicates 634
23.2.6 Networks of tetrahedra，tectosilicates 635
23.2.7 Random networks of tetrahedra：silicate glasses 639
23.2.8 Mesoporous silicates 641
23.2.9 Sol-gel synthesis of silicate nanostructures 642
23.3 Magnetic minerals on Mars and their biogenic origins 643
23.3.1 Hydroxides 646
23.3.2 Sulfates 649
23.4 Chapter summary 650
23.5 Historical notes 651
23.6 Selected problems 652
24 Molecular solids and biological materials 653
24.1 Introductory remarks 653
24.2 Simple molecular crystals：ice，dry ice，benzene，the clathrates，and self-assembled structures 654
24.2.1 Solid H2O：ice 654
24.2.2 Solid CO2：dry ice 656
24.2.3 Hydrocarbon crystals 657
24.2.4 Clathrates 658
24.2.5 Amphiphiles and micelles 659
24.3 Polymers 660
24.3.1 Polymer classification 661
24.3.2 Polymerization reactions and products 662
24.3.3 Polymer chains：spatial configurations 664
24.3.4 Copolymers and self-assembly 666
24.3.5 Conducting and superconducting polymers 668
24.3.6 Polymeric derivatives of fullerenes 670
24.4 Biological macromolecules 671
24.4.1 DNA and RNA 671
24.4.2 Virus structures 674
24.5 Fullerene-based molecular solids 677
24.5.1 Fullerites 679
24.5.2 Fullerides 681
24.5.3 Carbon nanotubes 681
24.6 Chapter summary 685
24.7 Historical notes 685
24.8 Selected problems 687
References 688
Index 716