Preface to the second edition .............................. xix
Preface to the first edition ............................... xxi
Acknowledgements ........................................... xxv
Figure reproductions ..................................... xxvii
Symbols ................................................... xxix
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 ТЕМ ............. 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 fc-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 и-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 /сс-derived structures and superlattices ....... 439
17.6.2 &cc-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 A15 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 At-phase and the M-, P-, and tf-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 Mössbauer 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
ТiO2 .................................................. 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
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