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Fundamentals of Crystallography
C. GIACOVAZZO, H. L MONACO, D. VITERBO F. SCORDARI, G. GILLI, G. ZANOTTI, M. CATTI
Edited by
C. GIACOVAZZO Dipartimento Geomineralogico, University of Bari, Italy
and Istituto di Ricerca per lo Sviluppo delle Metodologie Cristallografiche, CNR, Bari, Italy
INTERNATIONAL UNION OF CRYSTALLOGRAPHY OXFORD UNIVERSITY PRESS
1992
Contents
List of contributors xiii
1 Symmetry in crystals 1 Carmelo Giacovazzo
The crystalline State and isometric Operations 1 Symmetry elements 3
Axes of rotational symmetry 3 Axes of rototranslation or screw axes 5 Axes of inversion 5 Axes of rotorefiection 5 Refiection planes with translational component (glide planes) 6
Lattices 6 The rational properties of lattices 7
Crystallographic directions 7 Crystallographic planes 8
Symmetry restrictions due to the lattice periodicity and vice versa 9 Point groups and symmetry classes 11
Point groups in one and two dimensions 16 The Laue classes 17 The seven crystal Systems 17 The Bravais lattices 18
Plane lattices 18 Space lattices 19
The space groups 22 The plane and line groups 30 On the matrix representation of symmetry Operators 32 Appendices: LA The isometric
transformations 35 l.B Some combinations of
movements 37 l.C Wigner-Seitz cells 41
l.D The space-group rotation matrices
l .E Symmetry groups l.F Symmetry
generalization References
Crystallographic Computing Carmelo Giacovazzo
Introduction The metric matrix
The reciprocal lattice Basic transformations Transformation from triclinic to orthonormal axes Rotations in Cartesian Systems Some simple crystallographic calculations
Torsion angles Best plane through a set of points Best line through a set of points Principal axes of a quadratic form
Metric considerations on the lattices Niggli reduced cell Sublattices and superlattices Coincidence-site lattices Twins
Calculation of the structure factor Calculation of the electron density function The method of least Squares
Linear least Squares Reliability of the parameter estimates Linear least Squares with constraints Non-linear (unconstrained) least Squares
43 45
55 60
61
61 61 63 65
68 69
73 73 74 75 75 77 77 80 81 83 87
88 90 90 92 92 93
viii Contents
Least-squares refinement of crystal structures Practical considerations on crystallographic least Squares Constraints and restraints in crystallographic least Squares Alternatives to the method of least Squares
Rietveld refinement The basis of the technique Some practical aspects of Rietveld refinement
Analysis of thermal motion The effect of thermal motion on bond lengths and angles
About the accuracy of the calculated Parameters
Appendices: 2.A Some metric relations between direct and reciprocal lattices
2.B Some geometrical calculations concerning directions and planes
2.C Some transformation matrices
2.D Reciprocity of F and I lattices
2.E Transformations of crystallographic quantities in rectilinear Spaces
2.F Derivation of the normal equations
2.G Derivation of the variance-covariance matrix M^
2.H Derivation of the unbiased estimate of Mx
2.1 The FFT algorithm and its crystallographic applications
2.J Examples of twin laws References
3 The diffraction of X-rays by crystals Carmelo Giacovazzo
94
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104
108 109 109
112 117
120
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130
131
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131 133 137
141
Introduction Thomson scattering Compton scattering Interference of scattered waves Scattering by atomic electrons Scattering by atoms The temperature factor Scattering by a molecule or by a unit cell Diffraction by a crystal Bragg's law
The reflection and the limiting spheres Symmetry in reciprocal space
Friedel law
Effects of symmetry Operators in the reciprocal space Determination of the Laue class Determination of reflections with restricted phase values Systematic absences
Unequivocal determination of the space group
Diffraction intensities Anomalous dispersion The Fourier synthesis and the phase problem Modulated crystal structures Appendices: 3.A Mathematical
background 3.B Scattering and related
topics
3.C Scattering of X-rays by gases, liquids, and amorphous solids
3.D About electron density mapping
3.E Modulated structures and quasicrystals
References
Experimental methods in X-ray crystallography Hugo L. Monaco
X-ray sources Conventional generators Synchrotron radiation
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Contents ix
Monochromatization, collimation, and focusing of X-rays 241
Data collection techniques for Single crystals 245
The Weissenberg camera 247 The precession camera 254 The rotation (oscillation) method in macromolecular crystallography 259 Densitometry 268 The single-crystal diffractometer 273 Area detectors 281
Data collection techniques for polycrystalline materials 287
X-ray diffraction of polycrystalline materials 287 Cameras used for polycrystalline materials 289 Diffractometers used for polycrystalline materials 293 Uses of powder diffraction 297
Data reduction 301 Lorentz correction 301 Polarization correction 303 Absorption corrections 304 Radiation damage corrections 308 Relative scaling 310
Appendices: 4.A Determination of the number of molecules in the unit cell of a crystal 312
References 314
Solution and refinement of crystal structures 319 Davide Viterbo
Introduction 319 Statistical analysis of structure factor amplitudes 321 The Patterson function and its use 324
The heavy atom method 328 Advanced Patterson methods 335
Direct methods 335 Introduction 335 Structure invariants and semi-invariants 337 Probability methods 340 Fixing the origin and the enantiomorph 346
Phase determination procedures
Completing and refining the structure Difference Fourier method
Least-squares method Absolute config uration
Appendices: 5.A Structure factor
5.B
5.C
5.D
5.E
5.F
probability distributions Patterson vector methods Two examples of deriving phase Information from positivity Probability formulae for triplet invariants Pseudotranslational symmetry
Magic integers 5.G New multisolution
5.H
References
techniques
Procedures for completing a partial model
351
365
366 367 374
375
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384
385
387 388
390
393 397
lonic crystals Fernando Scordari
The structure of the atom Atoms with a Single electron
Atoms with more than one electron Interactions between ions
Notes on chemical bonds lonic crystals
Lattice energy: the contributions of attractive and repulsive terms Lattice energy: CFSE contribution Applications of lattice energy calculations
lonic radius Maximum Alling principle
Coordination polyhedra Radius ratio rule
Applications of the concept of ionic radius
403
403 403 404
406
406 409
410 414
417
418 424
425 425
427
x I Contents
Closest packings 429 Pauling's rules 433
Pauling's first rule 433 Pauling's second rule 433 Pauling's third rule 435 Pauling's fourth rule 436
Pauling's fifth rule 436 Ideal and defect structures 436
MX structures 437 MX2 and M2X structures 438 MX3 and M2X3 structures 440
AmB„Xp structures 441 On the Classification of Silicates 445
Liebau's crystallochemical Classification 447 Structural formulae 453 Relationship between Classification Parameters and properties of the cations 453
Appendices: 6.A Application of the concept of the packing coefficient (c,) 456
6.B Structural inferences from crystallochemical Parameters 459
References 463
Ring conformations Ring conformation and group theory Computation of puckering coordinates
Molecular geometry and the chemical bond
An overview of bond theories The VSEPR theory
Valence bond (VB) theory Hybridization. The machinery Molecular mechanics
Molecular hermeneutics: the Interpretation of molecular structures
Correlative methods in structural analysis Some three-centre-four-electron linear Systems Nucleophilic addition to organometallic Compounds
Nucleophilic addition to the carbonyl group A case of conformational rearrangement Resonance assisted hydrogen bonding (RAHB)
References
Molecules and molecular crystals 465 Gastone Gilli
Chemistry and X-ray crystallography 465 Crystal and molecular structure 465 The growth of structural Information 467
The nature of molecular crystals 468 Generalities 468 A more detailed analysis of intermolecular forces 473 Thermodynamics of molecular crystals 478 Free and lattice energy of a crystal from
atom-atom potentials 480
Polymorphism and the prediction of crystal structures 482 Effect of crystal forces on molecular geometry 483
Elements of classical stereochemistry 484 Structure: Constitution, configuration, and conformation 484 Isomerism 486
Protein crystallography Giuseppe Zanotti
Introduction Protein crystals
Principles of protein crystallization The solvent content of protein crystals Preparation of isomorphous heavy-atom derivatives How isomorphous are isomorphous derivatives?
The Solution of the phase problem The isomorphous replacement method Anomalous scattering: a complementary (or alternative) approach to the Solution of the phase problem The use of anomalous scattering in the determination of the absolute configuration of the macromolecule The treatment of errors
Contents I xi
The refinement of heavy-atom Parameters Picking up minor heavy-atom sites: the difference-Fourier synthesis A third approach to the resolution of the phase ambiguity: real-space filtering Rotation and translation functions and the molecular replacement method Direct methods and the maximum-entropy principle in macromolecular crystallography
The Interpretation of electron density maps and the refinement of the model
The interpretation of electron density maps
Interactive Computer graphics and model building The refinement of the structure
Protein structure General aspects
Levels of Organization of proteins: secondary structure Polypeptide chain description Higher levels of Organization: tertiary and quaternary structure, domains, and subunits
Groups other than amino acids Thermal parameters and disordered structures Solvent structure The influence of crystal packing Protein Classification
Appendices: 8.A Some formulae for isomorphous replacement and anomalous dispersion
8.B Translation functions 8.C Macromolecular least-
squares refinement and the conjugate-gradient algorithm
8.D Conventions and symbols for amino acids and peptides
References and further reading 594 549
551
551
552
560
562
562
562 563
572
573
574 577
578 582
583 583 584
585
587 588
590
591
9 Physical properties of crystals Michele Catti
Introduction
Crystal anisotropy and tensors Tensorial quantities
Symmetry of tensorial properties Overview of physical properties Electrical properties of crystals
Pyroelectricity Dielectric impermeability and optical properties
Elastic properties of crystals Crystal strain Inner deformation
Stress tensor Elasticity tensor Examples and applications
Piezoelectricity
Symmetry properties of the piezoelectric tensor
Crystal defects Experimental methods
Planar defects Line defects: dislocations
The Burgers circuit X-ray topography of dislocations Energy of a dislocation
Motion and interaction of dislocations Partial dislocations
Small-angle grain boundaries Point defects Thermal distribution of defects Diffusion Ionic conductivity
Appendix: 9.A Properties of second-rank tensors
Further reading Index
599
599
600 600 603
605 605
606
607 608 609 611 613 614
617 619
620 622
623
625 628 629 630 632 633 634
635 635
636 637
639
640 642
645
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