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 1 Symmetry in crystals Carmelo Giacovazzo The crystalline State and isometric Operations Symmetry elements Axes of rotational symmetry Axes of rototranslation or screw axes Axes of inversion Axes of rotorefiection Refiection 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 Appendices: L A The isometric transformations l.B Some combinations of movements l.C Wigner-Seitz cells xiii l.D The space-group rotation matrices l.E Symmetry groups l.F Symmetry generalization 1 1 3 3 5 5 5 6 6 7 7 8 9 11 16 17 17 18 18 19 22 30 32 35 37 41 References 43 45 55 60 Crystallographic Computing Carmelo Giacovazzo 61 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 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 98 104 108 109 109 112 117 120 122 124 125 127 127 128 130 131 131 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 131 133 137 Experimental methods in X-ray crystallography Hugo L. Monaco 141 X-ray sources Conventional generators Synchrotron radiation 141 142 144 144 146 147 148 150 151 154 154 155 155 Contents Monochromatization, collimation, and focusing of X-rays Data collection techniques for Single crystals The Weissenberg camera The precession camera The rotation (oscillation) method in macromolecular crystallography Densitometry The single-crystal diffractometer Area detectors Data collection techniques for polycrystalline materials X-ray diffraction of polycrystalline materials Cameras used for polycrystalline materials Diffractometers used for polycrystalline materials Uses of powder diffraction Data reduction Lorentz correction Polarization correction Absorption corrections Radiation damage corrections Relative scaling Appendices: 4.A Determination of the number of molecules in the unit cell of a crystal References Solution and refinement of crystal structures Davide Viterbo Introduction Statistical analysis of structure factor amplitudes The Patterson function and its use The heavy atom method Advanced Patterson methods Direct methods Introduction Structure invariants and semi-invariants Probability methods Fixing the origin and the enantiomorph 241 245 247 254 259 268 273 281 287 287 289 293 297 301 301 303 304 308 310 312 314 319 319 321 324 328 335 335 335 337 340 346 Phase determination procedures Completing and refining the structure Difference Fourier method Least-squares method Absolute config uration Appendices: 5.A Structure factor probability distributions 5.B Patterson vector methods 5.C Two examples of deriving phase Information from positivity 5.D Probability formulae for triplet invariants 5.E Pseudotranslational symmetry 5.F Magic integers 5.G New multisolution techniques 5.H Procedures for completing a partial model References ix 351 365 366 367 374 375 377 384 385 387 388 390 393 397 lonic crystals Fernando Scordari 403 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 403 403 404 406 406 409 410 414 417 418 424 425 425 427 x I Contents Closest packings Pauling's rules Pauling's first rule Pauling's second rule Pauling's third rule Pauling's fourth rule Pauling's fifth rule Ideal and defect structures MX structures MX 2 and M2X structures MX 3 and M 2 X 3 structures A m B„X p structures On the Classification of Silicates Liebau's crystallochemical Classification Structural formulae Relationship between Classification Parameters and properties of the cations Appendices: 6.A Application of the concept of the packing coefficient (c,) 6.B Structural inferences from crystallochemical Parameters References 429 433 433 433 435 436 436 436 437 438 440 441 445 447 453 Molecules and molecular crystals Gastone Gilli 465 Chemistry and X-ray crystallography Crystal and molecular structure The growth of structural Information The nature of molecular crystals Generalities A more detailed analysis of intermolecular forces Thermodynamics of molecular crystals Free and lattice energy of a crystal from atom-atom potentials Polymorphism and the prediction of crystal structures Effect of crystal forces on molecular geometry Elements of classical stereochemistry Structure: Constitution, configuration, and conformation Isomerism 465 465 467 468 468 453 456 459 463 473 478 480 482 483 484 484 486 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 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 maximumentropy 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 leastsquares refinement and the conjugate-gradient algorithm 8.D Conventions and symbols for amino acids and peptides References and further reading 594 Physical properties of crystals Michele Catti 599 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 599 600 600 603 605 605 606 549 551 551 552 560 562 562 562 563 572 573 574 577 578 582 583 583 584 585 587 588 590 591 9 607 608 609 611 613 614 617 619 620 622 623 625 628 629 630 632 633 634 635 635 636 637 639 640 642 645
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