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International Tables for Crystallography, Volume B, 3rd Edition, Reciprocal Space

U. Shmueli (Editor)
ISBN: 978-1-4020-8205-4
700 pages
October 2008
International Tables for Crystallography, Volume B, 3rd Edition, Reciprocal Space (1402082053) cover image


International Tables for Crystallography is the definitive resource and reference work for crystallography and structural science.

Each of the eight volumes in the series contains articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials. Emphasis is given to symmetry, diffraction methods and techniques of crystal-structure determination, and the physical and chemical properties of crystals. The data are accompanied by discussions of theory, practical explanations and examples, all of which are useful for teaching.

Volume B provides the reader with competent and useful accounts of the numerous aspects of reciprocal space in crystallographic research. Following an introductory chapter, the volume is divided into five parts:

Part 1: Presents an account of structure factor formalisms, extensive treatment of the theory, algorithms and crystallographic applications of Fourier methods, and fundamental as well as advanced treatments of symmetry in reciprocal space.

Part 2: Discusses crystallographic statistics, the theory of direct methods, Patterson techniques, isomorphous replacement and anomalous scattering, and treatments of the role of electron microscopy and diffraction in crystal- structure determination.

Part 3: Includes applications of reciprocal space to molecular geometry and `best'-plane calculations, and contains a treatment of the principles of molecular graphics and modeling and their applications.

Part 4:Contains treatments of various diffuse-scattering phenomena arising from crystal dynamics, disorder and low dimensionality (liquid crystals), and an exposition of the underlying theories and/or experimental evidence. Polymer crystallography and reciprocal-space images of aperiodic crystals are also treated.

Part 5: Discusses introductory treatments of the theory of the interaction of radiation with matter (dynamical theory) as applied to X-ray, electron and neutron diffraction techniques.

Substantially revised and updated to take into account recent developments, the third edition of volume B includes contributions from seven new authors and a new chapter on extensions of the Ewald method for Coulomb interactions in crystals. There are three new sections on electron diffraction and electron microscopy in structure determination.

Volume B is a vital addition to the library of scientists engaged in crystal-structure determination, crystallographic computing, crystal physics and other fields of crystallographic research. Graduate students specializing in crystallography will find much material suitable for self-study and a rich source of references to the relevant literature.

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Table of Contents

Preface (U. Shmueli).

Preface to the Second Edition (U. Shmueli).

Preface to the Third Edition (U. Shmueli).


1.1 Reciprocal Space in Crystallography (U. Shmueli).

1.1.1 Introduction.

1.1.2 Reciprocal Lattice in Crystallography.

1.1.3 Fundamental Relationships.

1.1.4 Tensor-Algebraic Formulation.

1.1.5 Transformations.

1.1.6 Some Analytical Aspects of the Reciprocal Space.

1.2 The Structure Factor (P. Coppens).

1.2.1 Introduction.

1.2.2 General Scattering Expression for X-Rays.

1.2.3 Scattering by a Crystal: Definition of a Structure Factor.

1.2.4 The Isolated-Atom Approximation in X-Ray Diffraction.

1.2.5 Scattering of Thermal Neutrons.

1.2.6 Effect of Bonding on the Atomic Electron Density within the Spherical-Atom Approximation: The Kappa Formalism.

1.2.7 Beyond the Spherical-Atom Description: The Atom-Centred Spherical Harmonic Expansion.

1.2.8 Fourier Transform of Orbital Products.

1.2.9 The Atomic Temperature Factor.

1.2.10 The Vibrational Probability Distribution and its Fourier Transform in the Harmonic Approximation.

1.2.11 Rigid-Body Analysis.

1.2.12 Treatment of Anharmonicity.

1.2.13 The Generalized Structure Factor.

1.2.14 Conclusion.

1.3 Fourier Transforms in Cystallography: Theory, Algorithms and Applications (G. Bricogne).

1.3.1 General Introduction.

1.3.2 The Mathematical Theory of the Fourier Transformation.

1.3.3 Numerical Computation of the Discrete Fourier Transform.

1.3.4 Crystallographic Applications of Fourier Transforms.

1.4 Symmetry in Reciprocal Space (U. Shmueli).

1.4.1 Introduction.

1.4.2 Effects of Symmetry on the Fourier Image of the Crystal.

1.4.3 Structure-Factor Tables.

1.4.4 Symmetry in Reciprocal Space: Space-Group Tables.

1.5 Crystallographic Viewpoints in the Classification of Space-Group Representations (M.I. Aroyo and H. Wondratschek).

1.5.1 List of Abbreviations and Symbols.

1.5.2 Introduction.

1.5.3 Basic Concepts.

1.5.4 Conventions in the Classification of Space-Group Irreps.

1.5.5 Examples and Discussion.

1.5.6 Conclusions.

Appendix A1.5.1. Reciprocal-space groups (G)* 


2.1 Statistical Properties of the Weighted Reciprocal Lattice (U. Shmueli and A.J.C. Wilson).

2.1.1 Introduction.

2.1.2 The Average Intensity of General Reflections.

2.1.3 The Average Intensity of Zones and Rows.

2.1.4 Probability Density Distributions - Mathematical Preliminaries.

2.1.5 Ideal Probability Density Distributions.

2.1.6 Distributions of Sums, Averages and Ratios.

2.1.7 Non-Ideal Distributions: The Correction-Factor Approach.

2.1.8 Non-Ideal Distributions: The Fourier Method.

2.2 Direct Methods (C. Giacovazzo).

2.2.1 List of Symbols and Abbreviations.

2.2.2 Introduction.

2.2.3 Origin Specification.

2.2.4 Normalized Structure Factors.

2.2.5 Phase-Determining Formulae.

2.2.6 Direct Methods in Real and Reciprocal Space: Sayre's Equation.

2.2.7 Scheme of Procedure for Phase Determination: The Small-Molecule Case.

2.2.8 Other Mutlisolution Methods Applied to Small Molecules.

2.2.9 Some References to Direct-Methods Packages: The Small-Molecule Case.

2.2.10 Direct Methods in Macromolecular Crystallography.

2.3 Patterson and Molecular Replacement Techniques, and the Use of Noncrystallographic Symmetry in Phasing (L. Tong, M.G. Rossmann and E. Arnold).

2.3.1 Introduction.

2.3.2 Interpretation of Patterson Maps.

2.3.3 Isomorphous Replacement Difference Pattersons.

2.3.4 Anomalous Dispersion.

2.3.5 Noncrystallographic Symmetry.

2.3.6 Rotation Functions.

2.3.7 Translation Functions.

2.3.8 Molecular Replacement.

2.3.9 Conclusions.

2.4 Isomorphous Replacement and Anomalous Scattering (M. Vijayan and S. Ramaseshan).

2.4.1 Introduction.

2.4.2 Isomorphous Replacement Method.

2.4.3 Anomalous-Scattering Method.

2.4.4 Isomorphous Replacement and Anomalous Scattering in Protein Crystallography.

2.4.5 Anomalous Scattering of Neutrons and Synchrotron Radiation. The Multiwavelength Method.

2.5 Electron Diffraction and Electron Microscopy in Structure Determination (J.M. Cowley, J.C.H. Spence, M. Tanaka, B.K. Vainshtein, B.B. Zvyagin, P.A. Penczek and D.L. Dorset).

2.5.1 Foreword (J.M. Cowley and J.C.H. Spence).

2.5.2 Electron Diffraction and Electron Microscopy (J.M. Crowley).

2.5.3 Point-Group and Space-Group Determination by Convergent-Beam Electron Diffraction (M. Tanaka).

2.5.4 Electron-Diffraction Structure Analysis (B.K. Vainshtein and B.B. Zvyagin).

2.5.5 Image Reconstruction (B.K. Vainshtein).

2.5.6 Three-Dimensional Reconstruction (B.K. Vainshtein and P.A. Penczek).

2.5.7 Single-Particle Reconstruction (P.A. Penczek).

2.5.8 Direct Phase Determination in Electron Crystallography (D.L. Dorset).


3.1 Distances, Angles and Their Standard Uncertainties (D.E. Sands).

3.1.1 Introduction.

3.1.2 Scalar Product.

3.1.3 Length of a Vector.

3.1.4 Angle Between Two Vectors.

3.1.5 Vector Product.

3.1.6 Permutation Tensors.

3.1.7 Components of Vector Product.

3.1.8 Some Vector Relationships.

3.1.9 Planes.

3.1.10 Variance-Covariance Matrices.

3.1.11 Mean Values.

3.1.12 Computation.

3.2 The Least-Squares Plane (R.E. Marsh and V. Schomaker).

3.2.1 Introduction.

3.2.2 Least-Squares Plane Based on Uncorrelated, Isotropic Weights.

3.2.3 The Proper Least-Squares Plane, with Gaussian Weights.

3.3 Molecular Modelling and Graphics (R. Diamond and L.M.D. Cranswick).

3.3.1 Graphics (R. Diamond).

3.3.2 Molecular Modelling, Problems and Approaches (R. Diamond).

3.3.3 Implementations (R. Diamond).

3.3.4 Graphics Software for the Display of Small and Medium-Sized Molecules (L.M.D. Cranswick).

3.4 Accelerated Convergence Treatment of R—n Lattice Sums (D.E. Williams).

3.4.1 Introduction.

3.4.2 Definition and Behaviour of the Direct-Space Sum.

3.4.3 Preliminary Description of the Method.

3.4.4 Preliminary Derivation to Obtain a Formula which Accelerates the Convergence of an R—n Sum Over Lattice Points X(d).

3.4.5 Extension of the Method to a Composite Lattice.

3.4.6 The Case of n = 1 (Coulombic Lattice Energy).

3.4.7 The Cases of n = 2 and n = 3.

3.4.8 Derivation of the Accelerated Convergence Formula Via the Patterson Function.

3.4.9 Evaluation of the Incomplete Gamma Function.

3.4.10 Summation over the Asymmetric Unit and Elimination of Intramolecular Energy Terms.

3.4.11 Reference Formulae for Particular Values of n.

3.4.12 Numerical Illustrations.

3.5 Extensions of the Ewald Methods for Coulomb Interactions in Crystals (T.A. Darden).

3.5.1 Introduction.

3.5.2 Lattice Sums of Point Charges.

3.5.3 Generalization to Gaussian- and Hermite-Based Continous Charge Distributions.

3.5.4 Computational Efficiency.


4.1 Thermal Diffuse Scattering of X-Rays and Neutrons (B.T.M. Willis).

4.1.1 Introduction.

4.1.2 Dynamics of Three-Dimensional Crystals.

4.1.3 Scattering of X-Rays by Thermal Vibrations.

4.1.4 Scattering Neutrons by Thermal Vibrations.

4.1.5 Phonon Dispersion Relations.

4.1.6 Measurement of Elastic Constants.

4.2 Disorder Diffuse Scattering of X-Rays and Neutrons (F. Frey, H. Boysen and H. Jagodzinski).

4.2.1 Introduction.

4.2.2 Basic Scattering Theory.

4.2.3 Qualitative Treatment of Structural Disorder.

4.2.4 General Guidelines for Analysing a Disorder Problem.

4.2.5 Quantitative Interpretation.

4.2.6 Disorder Diffuse Scattering from Aperiodic Crystals.

4.2.7 Computer Simulations and Modelling.

4.2.8 Experimental Techniques and Data Evaluation.

4.3 Diffuse Scattering in Electron Diffraction (J.M. Cowley and J.K. Gjønnes).

4.3.1 Introduction.

4.3.2 Inelastic Scattering.

4.3.3 Kinematical and Pseudo-Kinematical Scattering.

4.3.4 Dynamcial Scattering: Bragg Scattering Effects.

4.3.5 Multislice Calculations for Diffraction and Imaging.

4.3.6 Qualitative Interpretation of Diffuse Scattering of Electrons.

4.4 Scattering from Mesomorphic Structures (P.S. Pershan).

4.4.1 Introduction.

4.4.2 The Nematic Phase.

4.4.3 Smectic-A and Smectic-C Phases.

4.4.4 Phases with In-Plane Order.

4.4.5 Discotic Phases.

4.4.6 Other Phases.

4.5 Polymer Crystallography (R.P. Millane and D.L. Dorset).

4.5.1 Overview.

4.5.2 X-Ray Fibre Diffraction Analysis.

4.5.3 Electron Crystallography of Polymers.

4.6 Reciprocal-Space Images of Aperiodic Crystals (W. Steurer and T. Haibach).

4.6.1 Introduction.

4.6.2 The n-Dimensional Description of Aperiodic Crystals.

4.6.3 Reciprocal-Space Images.

4.6.4 Experimental Aspects of the Reciprocal-Space Analysis of Aperiodic Crystals.


5.1 Dynamical Theory of X-Ray Diffraction (A. Authier).

5.1.1 Introduction.

5.1.2 Fundamentals of Plane-Wave Dynamical Theory.

5.1.3 Solutions of Plane-Wave Dynamical Theory.

5.1.4 Standing Waves.

5.1.5 Anomalous Absorption.

5.1.6 Intensities of Plane Waves in Transmission Geometry.

5.1.7 Intensity of Plane Waves in Reflection Geometry.

5.1.8 Real Waves.

5.2 Dynamical Theory of Electron Diffraction (A.F. Moodie, J.M. Cowley and P. Goodman).

5.2.1 Introduction.

5.2.2 The Defining Equations.

5.2.3 Forward Scattering.

5.2.4 Evolution Operator.

5.2.5 Projection Approximation – Real-Space Solution.

5.2.6 Semi-Reciprocal Space.

5.2.7 Two-Beam Approximation.

5.2.8 Eigenvalue Approach.

5.2.9 Translational Invariance.

5.2.10 Bloch-Wave Formulations.

5.2.11 Dispersion Surfaces.

5.2.12 Multislice.

5.2.13 Born Series.

5.2.14 Approximations.

5.3 Dynamical Theory of Neutron Diffraction (M. Schlenker and J.-P. Guigay).

5.3.1 Introduction.

5.3.2 Comparison Between X-Rays and Neutrons with Spin Neglected.

5.3.3 Neutron Spin and Diffraction by Perfect Magnetic Crystals.

5.3.4 Extinction in Neutron Diffraction (Nonmagnetic Case).

5.3.5 Effect of External Fields on Neutron Scattering by Perfect Crystals.

5.3.6 Experimental Tests of the Dynamical Theory of Neutron Scattering.

5.3.7 Applications of the Dynamical Theory of Neutron Scattering.

Author Index.

Subject Index.

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