Fundamentals of Crystallography

Transcription

1 Fundamentals of Crystallography Second Edition C. GIACOVAZZO, H. L. MONACO, G. ARTIOLI, D. VITERBO, G. FERRARIS, G. GILLI, G. ZANOTTI, M. CATTI Edited by С GIACOVAZZO Professor of Crystallography, University of Bari, Italy OXPORD UNIVERSITY PRESS

2 List of contributors xxi 1 Symmetry in crystals l Carmelo Giacovazzo 1.1 The crystalline state and isometric operations Symmetry elements Axes of rotational symmetry Axes of rototranslation or screw axes Axes of inversion Axes of rotoreflection Reflection planes with translational component (glide planes) Lattices The rational properties of lattices Crystallographic directions Crystallographic planes Symmetry restrictions due to the lattice periodicity and vice versa Point groups and symmetry classes Point groups in one and two dimensions The Laue classes The seven crystal systems The Bravais lattices Plane lattices Space lattices The space groups The plane and line groups On the matrix representation of symmetry operators 32 Appendices 38 1.A The isometric transformations 38 l.a.l Direct movements 38 I.A.2 Opposite movements 39 LB Some combinations of movements 40 l.c Wigner-Seitz cells 44 l.d The space-group matrices 44 l.e Symmetry groups 47 l.e.l Subgroups 49 1.E.2 Cosets 50 1.E.3 Conjugate classes 51 1.E.4 Conjugate subgroups 52

3 1.E.5 Normal subgroups and factor groups 52 I.E.6 Maximal subgroups and minimal supergroups 54 I.E.7 Maximal subgroups and minimal supergroups for three-dimensional crystallographic point groups 55 1.E.8 Limiting groups in two and three dimensions 56 I.E.9 Representation of a group 57 I.E.10 Opposite movements 58 l.f Symmetry generalization 59 l.f.l The symmetry groups G m 59 1.F.2 The G 1 groups 59 1.F.3 The G 2 groups 59 1.F.4 The G 3 groups 60 1.F.5 The G 4 groups 63 l.f.6 The groups of colour symmetry 63 References 64 2 Crystallographic computing 67 Carmelo Giacovazzo 2.1 Introduction The metric matrix The reciprocal lattice Basis 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 Calculation of the electron density function Calculation of the structure factor The method of least squares Linear least squares Linear least squares with constraints Non-linear (unconstrained) least squares 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 Maximum likelihood refinement Gradient methods 118

4 xi 2.13 Rietveld refinement Preliminary steps The basis of the Rietveld refinement 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 135 Appendices A Some metric relations between direct and reciprocal lattices В Some geometrical calculations concerning directions and planes С Some transformation matrices D Reciprocity of F and I lattices E Transformations of crystallographic quantities in rectilinear spaces F Derivation of the normal equations G Derivation of the variance-covariance matrix M x H Derivation of the unbiased estimate of M x The FFT algorithm and its crystallographic applications 146 References The diffraction of X-rays by crystals 153 Carmelo Giacovazzo 3.1 Introduction Basic properties of X-rays 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 Friedellaw 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 176

5 3.14 Diffraction intensities Anomalous dispersion The Fourier synthesis and the phase problem 185 Appendices A Mathematical background A.1 Dirac delta function A.2 A mathematical model for the lattice A.3 Convolutions A.4 Some properties of convolutions A.5 The Fourier transform A.6 Some examples of Fourier transform A.7 Fourier transform of spherically symmetric functions A.8 Deconvolution of spectra B Scattering and related topics B.1 Compton scattering B.2 The anisotropic temperature factor B.3 Symmetry restrictions on the anisotropic temperature factors B.4 The Renninger effect and experimental phase determination by means of multiple diffraction experiments B.5 Electron diffraction B.6 Neutron scattering C About electron density mapping 220 References Beyond ideal crystals 227 Carmelo Giacovazzo 4.1 Introduction Ordering types Crystal twins Diffuse scattering Thermal diffuse scattering Disorder diffuse scattering Modulated crystal structures Quasicrystals Introductory remarks A mathematical basis Aperiodic tiling and quasicrystals Embedding quasicrystals in higher-dimensional space Liquid crystals (or mesomorphic phases) The paracrystal Amorphous and liquid states Diffraction from a finite statistically homogeneous object Diffraction from a finite statistically homogeneous object with equal atoms 265

6 xiii Diffraction from an isotropic statistically homogeneous object The Debye formula Diffraction by gases Diffraction by liquids and amorphous bodies Small-angle scattering 274 Appendices A Examples of twin laws A. 1 Cubic system A.2 Tetragonal system A.3 Hexagonal and trigonal systems A.4 Orthorhombic system A.5 Monoclinic system A.6 Triclinic system В How to recognize and treat twins С Embedding of modulated structures in higher-dimensional space D About Fibonacci numbers and sequences 289 References 291 Experimental methods in X-ray and neutron crystallography 295 Hugo L. Monaco and Gilberto Artioli 5.1 Introduction X-ray sources Conventional generators Synchrotron radiation X-ray optics: monochromatization, collimation, and focusing of X-rays Neutron sources Nuclear reactors Pulsed neutron sources Neutron optics X-ray and neutron detectors X-ray detectors Neutron detectors Data collection techniques for single crystals The Laue method The single crystal cameras The single crystal diffractometer Data collection techniques for polycrystalline materials Diffraction of polycrystalline materials Cameras used for polycrystalline materials Diffractometers used for polycrystalline materials Applications of powder diffraction 362

7 XIV Contents 5.7 In situ measurements in non-ambient conditions High-temperature polycrystalline diffraction Low-temperature single crystal diffractometry High-pressure experiments Data Reduction Lorentz correction Polarization correction Absorption correction Radiation damage correction Relative scaling 386 Appendices A Determination of the number of molecules in the unit cell of a crystal В The cylindrical film cameras geometry С The precession camera geometry D The rotation method geometry 399 References Solution and refinement of crystal structures 413 Davide Viterbo 6.1 Introduction Statistical analysis of structure-factor amplitudes The Patterson function and its use The heavy atom method Direct methods Introduction Structure invariants and semi-invariants Probability methods Fixing the origin and the enantiomorph Phase determination procedures Completing and refining the structure Difference Fourier method Least-squares method Absolute configuration 475 Appendices A Structure-factor probability distributions В Patterson vector methods С Probability formulae for triplet invariants D Pseudotranslational symmetry E Magic integers F New multisolution techniques G Procedures for completing a partial model G.1 Weights for Fourier syntheses G.2 Syntheses for completing a partial model 495 References 496

8 / Contents xv 7 Mineral and inorganic crystals 503 Giovanni Ferraris 7.1 Introduction Bonding aspects Chemical bond and solid-state properties Melting Cleavage Structure and morphology Morphology and optical properties Representing crystal structures The ionic radii Packing of spheres Coordination polyhedra Interstitial sites Ionic radii and coordination polyhedra Electrostatic bond strength and Pauling's rules Bond strength versus bond length The charge distribution (CD) method Applications of CD and ECoN Bond valence and hydrogen bond Bond valence and hydrates Bond strength of the О О hydrogen bond Polymorphism Solid solutions Solid solutions, order/disorder and crystal-chemical formula Structure types Closest-and close-packing type structures Packing spheres only Filling tetrahedral sites Filling octahedral sites Filling octahedral and tetrahedral sites More cp structures Structures with complex anions Orthosilicates Disilicates and ring silicates Chain silicates (inosilicates) Layer (sheet) silicates (phyllosilicates) Tectosilicates More materials of technological interest Modular structures Polytypism Modelling the structure of OD poly types Identification of long-period polytypes Polysomatic series Modelling structures of polysomes Modular series and homology Intersecting modular series 573

9 7.5.8 Miscellany of modular structures Modulated structures Real structures Symmetry domains Unmixing phenomena 578 References Molecules and molecular crystals 585 Gastone Gilli 8.1 Chemistry and X-ray crystallography Crystal and molecular structure The growth of structural information The nature of molecular crystals Intermolecular forces A more detailed analysis of intermolecular forces Thermodynamics of molecular crystals Free and lattice energy of a crystal from atom-atom potentials Polymorphism The prediction of the crystal structures Elements of classical stereochemistry Structure: constitution, configuration, and conformation Isomerism Ring conformations Ring conformation and group theory Computation of puckering coordinates Molecular structure and chemical bond Introduction Quantum-mechanical methods Qualitative bonding theories The VSEPR theory VB theory Molecular mechanics (MM) Molecular mechanics, force fields, and molecular modelling Molecular hermeneutics: the interpretation of molecular structures Correlation methods in structural analysis Some three-centre-four-electron linear systems Nucleophilic addition to organometallic compounds Nucleophilic addition to the carbonyl group Conformational rearrangements Advances in hydrogen bond theory by structure-correlation methods 650 References 657

10 xvii 9 Protein crystallography 667 Giuseppe Zanotti 9.1 Introduction Biological macromolecules Globular proteins Protein folding: general rules Levels of organization of proteins: secondary structure Representation of the polypetide chain conformation Higher levels of organization: tertiary and quaternary structure, domains, and subunits The influence of the medium Groups other than amino-acids Protein classification Nucleic acids Protein crystals Principles of protein crystallization Crystallization methods Testing the conditions: factorial approaches The solvent content of protein crystals Cryotechniques Preparation of isomorphous heavy-atom derivatives How isomorphous are isomorphous derivatives? The solution of the phase problem The isomorphous replacement method The determination of heavy-atom positions The single isomorphous replacement (SIR) method The classical solution of the problem of phase ambiguity: the MIR technique Anomalous scattering: a complementary (or alternative) approach to the solution of the phase problem The optimal choice of wavelength and the multiple anomalous dispersion (MAD) technique The use of anomalous scattering in the determination of the absolute configuration of the macromolecule The treatment of errors The refinement of heavy atoms parameters Maximum-likelihood and Bayesian estimates: an alternative approach in phase refinement Picking up minor heavy-atom sites: the difference-fourier synthesis Density-modification: how to solve the phase ambiguity and improve the electron density map Rotation and translation functions and the molecular replacement method The first step in molecular replacement: the rotation function The rotation matrix С and the choice of variables Translation functions 720

11 Self-rotation and self-translation functions: improving the electron density maps Practical hints in molecular replacement The direct methods in macromolecular crystallography The interpretation of the electron density maps and the refinement of the model The interpretation of the electron density maps Interactive computer graphics and model building The refinement of the structure Constrained versus restrained least-squares Restrained and constrained least-squares Crystallographic refinement by molecular dynamics The strategy of the refinement of protein structures R factor and R[ Ke : structure validation Thermal parameters and disordered structures The organization of solvent The influence of crystal packing Dynamical studies: time-resolved crystallography 744 Appendices A Some formulae for isomorphous replacement and anomalous dispersion В Translation functions С Macromolecular least-squares refinement and the conjugate-gradient algorithm D Conventions and symbols for amino acids and peptides 751 References Physical properties of crystals: Phenomenology and modelling 759 Michele Catti 10.1 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 783

12 xix 10.7 Modelling of structural and elastic behaviour Atomistic potential functions Athermal equation of state Elastic constants 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 808 Appendices A.1 Properties of second-rank tensors A.2 Eigenvalues and eigenvectors A.3 Representation surfaces and their properties 811 References 813 Index 815