Fundamentals of Crystallography (International Union of Crystallography Book Series, No. 2)

Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface......Page 4
Acknowledgements......Page 5
Contents......Page 6
Epigraph......Page 12
Contributors......Page 14
The crystalline state and isometric operations......Page 16
Axes  of  rotational symmetry......Page 18
Axes of  rotoreflection......Page 20
Lattices......Page 21
Crystallographic directions......Page 22
Crystallographic planes......Page 23
Symmetry restrictions due to the lattice 
periodicity and vice versa......Page 24
Point groups and symmetry classes......Page 26
Point groups in one and two dimensions......Page 31
The seven crystal systems......Page 32
Plane  lattices......Page 33
Space  lattices......Page 34
Cubic  lattices......Page 35
Trigonal lattices......Page 36
The space groups......Page 37
The plane and line groups......Page 45
On  the matrix representation of symmetry 
operators......Page 47
1.A  The  isometric transformations......Page 50
Direct movements......Page 51
1.B  Some combinations of movements......Page 52
1.C  Wigner-Seitz  cells......Page 56
1.D The space-group matrices......Page 58
1.E  Symmetry groups......Page 60
Subgroups......Page 62
Conjugate classes......Page 63
Normal subgroups and factor groups......Page 64
Maximal  subgroups and minimal supergroups......Page 66
Limiting groups in  two  and  three  dimensions......Page 67
Representation of a group......Page 68
Character tables......Page 69
The G2 groups......Page 70
The G3  groups......Page 71
The groups of colour symmetry......Page 72
References......Page 75
The metric matrix......Page 76
The reciprocal lattice......Page 78
Basis  transformations......Page 80
Transformation  from triclinic to orthonormal axes......Page 83
Rotations in Cartesian systems......Page 84
Torsion angles......Page 88
Best plane through a set of  points......Page 89
Principal axes of  a quadratic form......Page 90
Niggli reduced cell......Page 92
Sublattices  and superlattices......Page 95
Coincidence-site lattices......Page 96
Twins......Page 98
Calculation of the structure factor......Page 102
Calculation of the electron density function......Page 103
Linear least squares......Page 105
Linear  least squares with constraints......Page 107
Non-linear (unconstrained)  least squares......Page 108
Least-squares refinement of  crystal structures......Page 109
Practical considerations on crystallographic least squares......Page 113
Constraints and restraints  in crystallographic  least 
squares......Page 119
Rigid body refinemenP1......Page 120
Constraints via Lagrangian multipliers......Page 121
Use of  restraints......Page 122
Alternatives to the method of  least squares......Page 123
The basis of  the technique......Page 124
Some practical aspects of Rietveld refinement......Page 127
Analysis of thermal motion......Page 132
The effect of thermal motion on bond  lengths and 
angles......Page 135
About the accuracy of  the calculated parameters......Page 137
2.A  Some metric relations between direct and 
reciprocal  lattices......Page 139
2.B  Some geometrical calculations concerning
directions and planes......Page 140
2.D  Reciprocity of  F and I lattices......Page 142
2.E  Transformations of  crystallographic quantities  in 
rectilinear spaces......Page 143
2.F  Derivation of  the normal equations......Page 145
2.1  The FFT algorithm and  its crystallographic 
applications......Page 146
2.J  Examples of twin laws......Page 148
Orthorhombic system......Page 149
Triclinic system......Page 150
References......Page 152
Introduction......Page 156
Thomson scattering......Page 157
Interference of scattered waves......Page 159
Scattering by atomic electrons......Page 161
Scattering by atoms......Page 162
The temperature factor......Page 163
Scattering by a molecule or by a unit cell......Page 165
Diffraction by a crystal......Page 166
The reflection and  the limiting spheres......Page 169
Friedel law......Page 170
Determination of  the Laue class......Page 171
Determination of  reflections with  restricted phase values......Page 172
Systematic absences......Page 174
Diffraction intensities......Page 176
Anomalous dispersion......Page 180
The Fourier synthesis and the phase problem......Page 184
Modulated crystal structures......Page 186
The Dirac delta function......Page 188
The Fourier transform......Page 190
Some examples of Fourier transform......Page 191
Fourier transform of  spherically symmmetric functions......Page 195
Convolutions......Page 196
Convolutions involving delta functions......Page 197
Some properties of  convolutions......Page 198
Deconvolution  of  spectra......Page 199
Compton scattering......Page 200
The  anisotropic temperature factor......Page 201
Symmetry restrictions on the anisotropic temperature factors......Page 203
The Renninger effect and experimental phase determination 
by means of multiple diffraction experiments......Page 206
Electron diffraction......Page 210
Neutron scattering......Page 213
Introduction......Page 216
Diffraction from a finite statistically homogeneous object......Page 217
Diffraction from a finite statistically homogeneous object with equal atoms......Page 219
Diffraction from an isotropic statistically homogeneous object......Page 221
The Debye  formula......Page 222
Diffraction by  gases......Page 223
Diffraction by  liquids and amorphous bodies......Page 227
Small-angle scattering......Page 228
3.D  About electron density mapping......Page 231
Embedding  of modulated structures in higher-dimensional 
space......Page 236
Quasicrystals......Page 238
References......Page 241
The origin of X-rays in  the conventional sources......Page 244
Sealed-tube and rotating-anode generators......Page 246
Choice of  the type of  radiation......Page 248
Generation of X-rays in a synchrotron radiation source......Page 249
Comparison of  synchrotron and conventionally generated 
X-rays......Page 254
Filters......Page 256
Crystal monochromators......Page 258
Collimators......Page 259
Data collection techniques for single crystals......Page 260
The cylindrical film rotation camera......Page 262
Zero-level Weissenberg photographs......Page 263
Upper-level Weissenberg photographs......Page 267
The precession motion......Page 269
Isolating a zero-level  reciprocal lattice plane......Page 271
Upper-level precession photographs......Page 272
The  rotation (oscillation) method  in macromolecular 
crystallography......Page 274
Geometry of the rotation method......Page 276
Calculation of  the  film coordinates for a reflection......Page 280
Indexing  a rotation photograph......Page 282
Densitometry......Page 283
Optical density  and  integrated  intensity......Page 284
Film as an X-ray detector......Page 285
Microdensitometers......Page 286
Diffractometer geometry......Page 288
Determination of  the  crystal orientation and unit  cell 
parameters: the orientation matrix......Page 291
Measurement  of  the  integrated intensities......Page 293
Area detectors......Page 296
Principles of operation of  area detectors......Page 297
Data  collection with  an area detector......Page 299
X-ray  diffraction of polycrystalline materials......Page 302
Cameras used for  polycrystalline materials......Page 304
Diffractometers used for polycrystalline materials......Page 308
Uses of  powder diffraction......Page 312
Lorentz correction......Page 316
Polarization correction......Page 318
Absorption corrections......Page 319
Radiation damage corrections......Page 323
Relative scaling......Page 325
4.A  Determination of  the number of molecules in the 
unit cell of  a crystal......Page 327
References......Page 329
Introduction......Page 334
Statistical analysis of structure factor amplitudes......Page 336
The Patterson function and its use......Page 339
The heavy atom method......Page 343
Introduction......Page 350
Structure invariants and semi-invariants......Page 352
Probability methods......Page 355
Fixing the origin and the enantiomorph......Page 361
Setting up phase relationships......Page 366
Definition of  an optimum starting set of  phases......Page 367
Figures of merit......Page 370
Electron density maps  (E-maps)......Page 371
Symbolic addition method......Page 373
Multisolution methods......Page 375
Completing and refining the structure......Page 380
Difference Fourier method......Page 381
Least-squares method......Page 382
Absolute configuration......Page 389
5.A  Structure factor  probability distributions......Page 390
5.B  Patterson vector methods......Page 392
5.C  Two examples of  deriving phase  information from 
positivity......Page 399
5.D  Probability formulae for  triplet  invariants......Page 400
5.E  Pseudotranslational  symmetry......Page 402
5.F  Magic integers......Page 403
5.G  New multisolution techniques......Page 405
Weights for Fourier  syntheses......Page 408
Syntheses  for completing  a partial model......Page 409
References......Page 412
Atoms with a single electron......Page 418
Atoms with more than one electron......Page 419
Notes on chemical bonds......Page 421
lonic crystals......Page 424
Lattice energy:  the contributions of  attractive and 
repulsive terms......Page 425
Lattice energy: CFSE contribution......Page 429
Applications  of  lattice energy calculations......Page 432
Ionic radius......Page 433
Maximum filling principle......Page 439
Radius ratio rule......Page 440
Applications of  the concept of  ionic radius......Page 442
Closest packings......Page 444
Pauling's second  rule......Page 448
Pauling's  third  rule......Page 450
Ideal and defect structures......Page 451
MX structures......Page 452
MX2  and M,X  structures......Page 453
MX3  and M2X3 structures......Page 455
A,B,X,  structures......Page 456
On  the classification of silicates......Page 460
Liebau's crystallochemical classification......Page 462
Relationship between  classification parameters and properties 
of  the cations......Page 468
 
6.A Application of  the concept of  the packing coefficient 
(ci)......Page 471
6.B  Structural  inferences from crystallochemical 
parameters......Page 474
References......Page 478
Crystal and molecular structure......Page 480
The growth of structural information......Page 482
Repulsion or exchange forces......Page 483
Monopolar forces......Page 484
2.  Medium H-bonding......Page 485
(a)  Charge assisted H-bonding......Page 486
Intermolecular forces and crystal packing......Page 487
Dispersion energy......Page 488
Atom-atom  potentials......Page 489
Electrostatic energies......Page 490
4.  Hydrogen bonding......Page 492
Thermodynamics of molecular  crystals......Page 493
Free and lattice energy of  a crystal from atom-atom 
potentials......Page 495
Polymorphism and  the prediction of  crystal structures......Page 497
Effect of  crystal forces on molecular geometry......Page 498
Structure: constitution, configuration, and conformation......Page 499
Configurational  isomerism......Page 501
Ring conformations......Page 505
Ring conformation and group theory......Page 507
Computation of  puckering coordinates......Page 513
2.  Valence  shell electron  pair  repulsion  (VSEPR) theory.[......Page 514
6.  Other qualitative theories.......Page 515
The VSEPR theory......Page 516
Valence bond (VB) theory......Page 517
Hybridization. The machinery......Page 519
Molecular mechanics......Page 521
2.  Bond angle bending......Page 522
6.  Non-bonded interactions.......Page 523
7.  Electrostatic  interactions......Page 524
Correlative methods  in structural analysis......Page 526
Some three-centre-four-electron  linear systems......Page 527
Nucleophilic addition to organometallic compounds......Page 529
Nucleophilic addition to the carbonyl group......Page 530
A  case of  conformational  rearrangement......Page 531
Resonance assisted hydrogen bonding (RAHB)......Page 536
References......Page 544
Introduction......Page 550
Principles of  protein crystallization......Page 551
Preparation of isomorphous heavy-atom derivatives......Page 553
The  isomorphous replacement method......Page 555
The determination of heavy-atom positions......Page 556
The single isomorphous replacement (SIR) method......Page 557
The classical solution of the problem of phase ambiguity: the
MIR  technique......Page 558
Anomalous scattering: a complementary  (or alternative) 
approach to the solution of  the phase problem......Page 559
The use of  anomalous scattering  in the determination of 
the absolute  configuration of  the macromolecule......Page 560
The treatment of  errors......Page 561
The refinement of  heavy-atom parameters......Page 564
A  third approach to the solution of  the phase 
ambiguity:  real-space filtering......Page 566
Rotation and translation  functions  and the molecular 
replacement method......Page 567
The first  step in molecular replacement: the rotation function......Page 568
The rotation matrix C  and  the  choice of variables......Page 570
Translation  functions......Page 571
Self-rotation and  self-translation functions: improving  the 
electron density maps......Page 573
Direct methods and the maximum-entropy principle  in 
macromolecular crystallography......Page 575
Interactive computer graphics and model building......Page 577
The refinement of  the structure......Page 578
Constrained versus restrained least  squares......Page 579
Restrained and  constrained least  squares......Page 583
Crystallographic refinement by molecular dynamics......Page 584
The  strategy of the  refinement of  protein  structures......Page 585
Protein structure......Page 587
General aspects......Page 588
Levels of  organization of  proteins: secondary structure......Page 589
Polypeptide chain description......Page 592
Higher levels of  organization:  tertiary and quaternary 
structure, domains,  and subunits......Page 593
Groups other  than amino  acids......Page 597
Solvent structure......Page 598
The influence of  crystal packing......Page 599
Protein classification......Page 600
Appendix  8.A Some formulae for isomorphous 
replacement and anomalous dispersion......Page 602
Appendix 8.B Translation functions......Page 603
Appendix  8.C Macromolecular  least-squares refinement 
and the conjugate-gradient algorithm......Page 605
Appendix 8.D  Conventions and symbols for  amino acids 
and peptides......Page 606
References......Page 609
Introduction......Page 614
Tensorial quantities......Page 615
Symmetry of  tensorial properties......Page 618
Electrical properties of crystals......Page 620
Pyroelectricity......Page 621
Dielectric  impermeability and optical properties......Page 622
Elastic properties of crystals......Page 623
Crystal strain......Page 624
Inner deformation......Page 626
Stress tensor......Page 628
Elasticity tensor......Page 629
Examples and applications......Page 632
Piezoelectricity......Page 634
Symmetry properties of  the piezoelectric tensor......Page 635
Crystal defects......Page 637
Experimental methods......Page 638
Planar defects......Page 640
Line defects: dislocations......Page 643
The Burgers circuit......Page 644
X-ray topography of dislocations......Page 645
Energy of  a dislocation......Page 647
Motion and interaction of  dislocations......Page 648
Partial dislocations......Page 649
Point defects......Page 650
Thermal distribution of defects......Page 651
Diffusion......Page 652
Ionic conductivity......Page 654
Eigenvalues and  eigenvectors......Page 655
Representation surfaces and  their properties......Page 656
Further reading......Page 657
Subject index......Page 660