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Tables of Contents for Fundamentals of Crystallography
Chapter/Section Title
Page #
Page Count
List of contributors
xiii
Symmetry in crystals
1
60
Carmelo Giacovazzo
The crystalline state and isometric operations
1
2
Symmetry elements
3
3
Axes of rotational symmetry
3
2
Axes of rototranslation or screw axes
5
1
Axes of inversion
5
1
Axes of rotoreflection
5
1
Reflection planes with translational component (glide planes)
6
1
Lattices
6
1
The rational properties of lattices
7
2
Crystallographic directions
7
1
Crystallographic planes
8
1
Symmetry restrictions due to the lattice periodicity and vice versa
9
2
Point groups and symmetry classes
11
6
Point groups in one and two dimensions
16
1
The Laue classes
17
1
The seven crystal systems
17
1
The Bravais lattices
18
4
Plane lattices
18
1
Space lattices
19
3
The space groups
22
8
The plane and line groups
30
2
On the matrix representation of symmetry operators
32
3
Appendices
The isometric transformations
35
2
Some combinations of movements
37
4
Wigner-Seitz cells
41
2
The space-group rotation matrices
43
2
Symmetry groups
45
10
Symmetry generalization
55
5
References
60
1
Crystallographic computing
61
80
Carmelo Giacovazzo
Introduction
61
1
The metric matrix
61
2
The reciprocal lattice
63
2
Basic transformations
65
3
Transformation from triclinic to orthonormal axes
68
1
Rotations in Cartesian systems
69
4
Some simple crystallographic calculations
73
4
Torsion angles
73
1
Best plane through a set of points
74
1
Best line through a set of points
75
1
Principal axes of a quadratic form
75
2
Metric considerations on the lattices
77
10
Niggli reduced cell
77
3
Sublattices and superlattices
80
1
Coincidence-site lattices
81
2
Twins
83
4
Calculation of the structure factor
87
1
Calculation of the electron density function
88
2
The method of least squares
90
19
Linear least squares
90
2
Reliability of the parameter estimates
92
1
Linear least squares with constraints
92
1
Non-linear (unconstrained) least squares
93
1
Least-squares refinement of crystal structures
94
4
Practical considerations on crystallographic least squares
98
6
Constraints and restraints in crystallographic least squares
104
4
Alternatives to the method of least squares
108
1
Rietveld refinement
109
8
The basis of the technique
109
3
Some practical aspects of Rietveld refinement
112
5
Analysis of thermal motion
117
3
The effect of thermal motion on bond lengths and angles
120
2
About the accuracy of the calculated parameters
122
2
Appendices
Some metric relations between direct and reciprocal lattices
124
1
Some geometrical calculations concerning directions and planes
125
2
Some transformation matrices
127
1
Reciprocity of F and I lattices
127
1
Transformations of crystallographic quantities in rectilinear spaces
128
2
Derivation of the normal equations
130
1
Derivation of the variance-covariance matrix Mx
131
1
Derivation of the unbiased estimate of Mx
131
1
The FFT algorithm and its crystallographic applications
131
2
Examples of twin laws
133
4
References
137
4
The diffraction of X-rays by crystals
141
88
Carmelo Giacovazzo
Introduction
141
1
Thomson scattering
142
2
Compton scattering
144
1
Interference of scattered waves
144
2
Scattering by atomic electrons
146
1
Scattering by atoms
147
1
The temperature factor
148
2
Scattering by a molecule or by a unit cell
150
1
Diffraction by a crystal
151
3
Bragg's law
154
1
The reflection and the limiting spheres
154
1
Symmetry in reciprocal space
155
6
Friedel law
155
1
Effects of symmetry operators in the reciprocal space
156
1
Determination of the Laue class
156
1
Determination of reflections with restricted phase values
157
2
Systematic absences
159
2
Unequivocal determination of the space group
161
1
Diffraction intensities
161
4
Anomalous dispersion
165
4
The Fourier synthesis and the phase problem
169
2
Modulated crystal structures
171
2
Appendices
Mathematical background
173
12
Scattering and related topics
185
16
Scattering of X-rays by gases, liquids, and amorphous solids
201
15
About electron density mapping
216
5
Modulated structures and quasicrystals
221
5
References
226
3
Experimental methods in X-ray crystallography
229
90
Hugo L. Monaco
X-ray sources
229
16
Conventional generators
229
5
Synchrotron radiation
234
7
Monochromatization, collimation, and focusing of X-rays
241
4
Data collection techniques for single crystals
245
42
The Weissenberg camera
247
7
The precession camera
254
5
The rotation (oscillation) method in macromolecular crystallography
259
9
Densitometry
268
5
The single-crystai diffractometer
273
8
Area detectors
281
6
Data collection techniques for polycrystalline materials
287
14
X-ray diffraction of polycrystalline materials
287
2
Cameras used for polycrystalline materials
289
4
Diffractometers used for polycrystalline materials
293
4
Uses of powder diffraction
297
4
Data reduction
301
11
Lorentz correction
301
2
Polarization correction
303
1
Absorption corrections
304
4
Radiation damage corrections
308
2
Relative scaling
310
2
Appendices
Determination of the number of molecules in the unit cell of a crystal
312
2
References
314
5
Solution and refinement of crystal structures
319
84
Davide Viterbo
Introduction
319
2
Statistical analysis of structure factor amplitudes
321
3
The Patterson function and its use
324
11
The heavy atom method
328
7
Advanced Patterson methods
335
1
Direct methods
335
30
Introduction
335
2
Structure invariants and semi-invariants
337
3
Probability methods
340
6
Fixing the origin and the enantiomorph
346
5
Phase determination procedures
351
14
Completing and refining the structure
365
10
Difference Fourier method
366
1
Least-squares method
367
7
Absolute configuration
374
1
Appendices
Structure factor probability distributions
375
2
Patterson vector methods
377
7
Two examples of deriving phase information from positivity
384
1
Probability formulae for triplet invariants
385
2
Pseudotranslational symmetry
387
1
Magic integers
388
2
New multisolution techniques
390
3
Procedures for completing a partial model
393
4
References
397
6
Ionic crystals
403
62
Fernando Scordari
The structure of the atom
403
3
Atoms with a single electron
403
1
Atoms with more than one electron
404
2
Interactions between ions
406
12
Notes on chemical bonds
406
3
Ionic crystals
409
1
Lattice energy: the contributions of attractive and repulsive terms
410
4
Lattice energy: CFSE contribution
414
3
Applications of lattice energy calculations
417
1
Ionic radius
418
6
Maximum filling principle
424
9
Coordination polyhedra
425
1
Radius ratio rule
425
2
Applications of the concept of ionic radius
427
2
Closest packings
429
4
Pauling's rules
433
3
Pauling's first rule
433
1
Pauling's second rule
433
2
Pauling's third rule
435
1
Pauling's fourth rule
436
1
Pauling's fifth rule
436
1
Ideal and defect structures
436
5
MX structures
437
1
MX2 and M2X structures
438
2
MX3 and M2X structures
440
1
Am, BnXp structures
441
4
On the classification of silicates
445
11
Liebau's crystallochemical classification
447
6
Structural formulae
453
1
Relationship between classification parameters and properties of the cations
453
3
Appendices
Application of the concept of the packing coefficient (ci)
456
3
Structural inferences from crystallochemical parameters
459
4
References
463
2
Molecules and molecular crystals
465
70
Gastone Gilli
Chemistry and X-ray crystallography
465
3
Crystal and molecular structure
465
2
The growth of structural information
467
1
The nature of molecular crystals
468
16
Generalities
468
5
A more detailed analysis of intermolecular forces
473
5
Thermodynamics of molecular crystals
478
2
Free and lattice energy of a crystal from atom-atom potentials
480
2
Polymorphism and the prediction of crystalstructures
482
1
Effect of crystal forces on molecular geometry
483
1
Elements of classical stereochemistry
484
15
Structure: constitution, configuration, and conformation
484
2
Isomerism
486
4
Ring conformations
490
2
Ring conformation and group theory
492
6
Computation of puckering coordinates
498
1
Molecular geometry and the chemical bond
499
12
An overview of bond theories
499
2
The VSEPR theory
501
1
Valence bond (VB) theory
502
2
Hybridization. The machinery
504
2
Molecular mechanics
506
5
Molecular hermeneutics: the interpretation of molecular structures
511
18
Correlative methods in structural analysis
511
1
Some three-centre-four-electron linear systems
512
2
Nucleophilic addition to organometallic compounds
514
1
Nucleophilic addition to the carbonyl group
515
1
A case of conformational rearrangement
516
5
Resonance assisted hydrogen bonding (RAHB)
521
8
References
529
6
Protein crystallography
535
64
Giuseppe Zanotti
Introduction
535
1
Protein crystals
536
4
Principles of protein crystallization
536
2
The solvent content of protein crystals
538
1
Preparation of isomorphous heavy-atom derivatives
538
2
How isomorphous are isomorphous derivatives?
540
1
The solution of the phase problem
540
22
The isomorphous replacement method
540
4
Anomalous scattering: a complementary (or alternative) approach to the solution of the phase problem
544
1
The use of anomalous scattering in the determination of the absolute configuration of the macromolecule
545
1
The treatment of errors
546
3
The refinement of heavy-atom parameters
549
2
Picking up minor heavy-atom sites: the difference-Fourier synthesis
551
1
A third approach to the solution of the phase ambiguity: real-space filtering
551
1
Rotation and translation functions and the molecular replacement method
552
8
Direct methods and the maximum- entropy principle in macromolecular Crystallography
560
2
The interpretation of electron density maps and the refinement of the model
562
10
The interpretation of electron density maps
562
1
Interactive computer graphics and model building
562
1
The refinement of the structure
563
9
Protein structure
572
15
General aspects
573
1
Levels of organization of proteins: secondary structure
574
3
Polypeptide chain description
577
1
Higher levels of organization: tertiary and quaternary structure, domains, and subunits
578
4
Groups other than amino acids
582
1
Thermal parameters and disordered structures
583
1
Solvent structure
583
1
The influence of crystal packing
584
1
Protein classification
585
2
Appendices
Some formulae for isomorphous replacement and anomalous dispersion
587
1
Translation functions
588
2
Macromolecular least-squares refinement and the conjugate-gradient algorithm
590
1
Conventions and symbols for amino acids and peptides
591
3
References and further reading
594
5
Physical properties of crystals
599
43
Michele Catti
Introduction
599
1
Crystal anisotropy and tensors
600
5
Tensorial quantities
600
3
Symmetry of tensorial properties
603
2
Overview of physical properties
605
1
Electrical properties of crystals
605
3
Pyroelectricity
606
1
Dielectric impermeability and optical properties
607
1
Elastic properties of crystals
608
11
Crystal strain
609
2
Inner deformation
611
2
Stress tensor
613
1
Elasticity tensor
614
3
Examples and applications
617
2
Piezoelectricity
619
3
Symmetry properties of the piezoelectric tensor
620
2
Crystal defects
622
1
Experimental methods
623
2
Planar defects
625
3
Line defects: dislocations
628
7
The Burgers circuit
629
1
X-ray topography of dislocations
630
2
Energy of a dislocation
632
1
Motion and interaction of dislocations
633
1
Partial dislocations
634
1
Small-angle grain boundaries
635
1
Point defects
635
1
Thermal distribution of defects
636
1
Diffusion
637
2
Ionic conductivity
639
1
Appendix
Properties of second-rank tensors
640
2
Further reading
642
3
Index
645