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

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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

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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

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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

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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

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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

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