*289*
*15*
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*English*
*Pages 610*
*Year 2009*

INT E R NAT I ONAL T AB L E S FOR C RYST AL L OGR APHY

International Tables for Crystallography Volume A: Space-Group Symmetry Editor Theo Hahn First Edition 1983, Fourth Edition 1995 Corrected Reprint 1996 Volume B: Reciprocal Space Editor U. Shmueli First Edition 1993, Corrected Reprint 1996 Second Edition 2001 Volume C: Mathematical, Physical and Chemical Tables Editors A. J. C. Wilson and E. Prince First Edition 1992, Corrected Reprint 1995 Second Edition 1999

Forthcoming volumes Volume D: Physical Properties of Crystals Editor A. Authier Volume E: Subperiodic Groups Editors V. Kopsky and D. B. Litvin Volume F: Crystallography of Biological Macromolecules Editors M. G. Rossmann and E. Arnold Volume A1: Maximal Subgroups of Space Groups Editors H. Wondratschek and U. Mu¨ller

INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

Volume B RECIPROCAL SPACE

Edited by U. SHMUELI Second Edition

Published for

T HE I NT E RNAT IONAL UNION OF C RYST AL L OGR APHY by

KL UW E R ACADE MIC PUBLISHERS DORDRE CHT /BOST ON/L ONDON

2001

A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 0-7923-6592-5 (acid-free paper)

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, USA In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands

Technical Editor: N. J. Ashcroft First published in 1993 Second edition 2001 # International Union of Crystallography 2001 Short extracts may be reproduced without formality, provided that the source is acknowledged, but substantial portions may not be reproduced by any process without written permission from the International Union of Crystallography Printed in Great Britain by Alden Press, Oxford

Contributing authors R. E. Marsh: The Beckman Institute–139–74, California Institute of Technology, 1201 East California Blvd, Pasadena, California 91125, USA. [3.2]

E. Arnold: CABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA. [2.3] M. I. Aroyo: Faculty of Physics, University of Soﬁa, bulv. J. Boucher 5, 1164 Soﬁa, Bulgaria. [1.5] A. Authier: Laboratoire de Mine´ralogie-Cristallographie, Universite´ P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France. [5.1] G. Bricogne: MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Baˆtiment 209D, Universite´ Paris-Sud, 91405 Orsay, France. [1.3] P. Coppens: Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 142603000, USA. [1.2] J. M. Cowley: Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA. [2.5.1, 2.5.2, 4.3, 5.2] R. Diamond: MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England. [3.3] D. L. Dorset: ExxonMobil Research and Engineering Co., 1545 Route 22 East, Clinton Township, Annandale, New Jersey 08801, USA. [2.5.7, 4.5.1, 4.5.3] F. Frey: Institut fu¨r Kristallographie und Mineralogie, Universita¨t, Theresienstrasse 41, D-8000 Mu¨nchen 2, Germany. [4.2] C. Giacovazzo: Dipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy. [2.2] J. K. Gjùnnes: Institute of Physics, University of Oslo, PO Box 1048, N-0316 Oslo 3, Norway. [4.3] P. Goodman† [2.5.3, 5.2] R. W. Grosse-Kunstleve: Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4-230, Berkeley, CA 94720, USA. [1.4] J.-P. Guigay: European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France. [5.3] T. Haibach: Laboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland. [4.6] S. R. Hall: Crystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia. [1.4] H. Jagodzinski: Institut fu¨r Kristallographie und Mineralogie, Universita¨t, Theresienstrasse 41, D-8000 Mu¨nchen 2, Germany. [4.2] †

R. P. Millane: Whistler Center for Carbohydrate Research, and Computational Science and Engineering Program, Purdue University, West Lafayette, Indiana 47907-1160, USA. [4.5.1, 4.5.2] A. F. Moodie: Department of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia. [5.2] P. S. Pershan: Division of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA. [4.4] S. Ramaseshan: Raman Research Institute, Bangalore 560 080, India. [2.4] M. G. Rossmann: Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA. [2.3] D. E. Sands: Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA. [3.1] M. Schlenker: Laboratoire Louis Ne´el du CNRS, BP 166, F-38042 Grenoble CEDEX 9, France. [5.3] V. Schomaker† [3.2] U. Shmueli: School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel. [1.1, 1.4, 2.1] W. Steurer: Laboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland. [4.6] B. K. Vainshtein† [2.5.4, 2.5.5, 2.5.6] M. Vijayan: Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India. [2.4] D. E. Williams: Department of Chemistry, University of Louisville, Louisville, Kentucky 40292, USA. [3.4] B. T. M. Willis: Chemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England. [4.1] A. J. C. Wilson† [2.1] H. Wondratschek: Institut fu¨r Kristallographie, Universita¨t, D-76128 Karlsruhe, Germany. [1.5] B. B. Zvyagin: Institute of Ore Mineralogy (IGEM), Academy of Sciences of Russia, Staromonetny 35, 109017 Moscow, Russia. [2.5.4] †

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Contents PAGE

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xxv

Preface to the second edition (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Preface (U. Shmueli)

PART 1. GENERAL RELATIONSHIPS AND TECHNIQUES

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1.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1. Reciprocal space in crystallography (U. Shmueli)

1.1.2. Reciprocal lattice in crystallography 1.1.3. Fundamental relationships 1.1.3.1. 1.1.3.2. 1.1.3.3. 1.1.3.4.

Basis vectors .. .. .. .. Volumes .. .. .. .. .. .. Angular relationships .. .. Matrices of metric tensors

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1.1.5.1. Transformations of coordinates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.1.5.2. Example .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1.6. Some analytical aspects of the reciprocal space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1.6.1. Continuous Fourier transform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.1.6.2. Discrete Fourier transform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.1.6.3. Bloch’s theorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.2. General scattering expression for X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.3. Scattering by a crystal: deﬁnition of a structure factor

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1.2.4. The isolated-atom approximation in X-ray diffraction

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1.2.5. Scattering of thermal neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.5.1. Nuclear scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.5.2. Magnetic scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.1.4. Tensor-algebraic formulation 1.1.4.1. 1.1.4.2. 1.1.4.3. 1.1.4.4.

Conventions .. Transformations Scalar products Examples .. ..

1.1.5. Transformations

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1.2. The structure factor (P. Coppens)

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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

1.2.7.1. Direct-space description of aspherical atoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.7.2. Reciprocal-space description of aspherical atoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines) .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.7.2. Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981) .. .. .. .. Table 1.2.7.3. ‘Kubic Harmonic’ functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.7.4. Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Coppens, 1990) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.8. Fourier transform of orbital products

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1.2.8.1. One-centre orbital products .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.8.2. Two-centre orbital products .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.8.1. Products of complex spherical harmonics as deﬁned by equation (1.2.7.2a). .. .. .. .. .. Table 1.2.8.2. Products of real spherical harmonics as deﬁned by equations (1.2.7.2b) and (1.2.7.2c) .. .. Table 1.2.8.3. Products of two real spherical harmonic functions ylmp in terms of the density functions equation (1.2.7.3b) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.9. The atomic temperature factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation .. .. .. .. .. ..

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1.2.11. Rigid-body analysis

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CONTENTS Table 1.2.11.1. The arrays Gijkl and Hijkl to be used in the observational equations Uij Gijkl Lkl Hijkl Skl Tij [equation (1.2.11.9)] .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.12. Treatment of anharmonicity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.12.1. The Gram–Charlier expansion .. .. .. .. 1.2.12.2. The cumulant expansion .. .. .. .. .. .. 1.2.12.3. The one-particle potential (OPP) model .. 1.2.12.4. Relative merits of the three expansions .. .. Table 1.2.12.1. Some Hermite polynomials (Johnson &

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1.2.14. Conclusion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.3. Fourier transforms in crystallography: theory, algorithms and applications (G. Bricogne) .. .. .. .. .. .. .. .. .. .. ..

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1.3.2. The mathematical theory of the Fourier transformation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.2.13. The generalized structure factor

1.3.1. General introduction

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1.3.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.2. Preliminary notions and notation .. .. .. .. .. .. .. .. .. .. 1.3.2.2.1. Metric and topological notions in Rn .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.2.2. Functions over Rn 1.3.2.2.3. Multi-index notation .. .. .. .. .. .. .. .. .. .. .. 1.3.2.2.4. Integration, Lp spaces .. .. .. .. .. .. .. .. .. .. 1.3.2.2.5. Tensor products. Fubini’s theorem .. .. .. .. .. .. 1.3.2.2.6. Topology in function spaces .. .. .. .. .. .. .. .. 1.3.2.2.6.1. General topology .. .. .. .. .. .. .. .. 1.3.2.2.6.2. Topological vector spaces .. .. .. .. .. 1.3.2.3. Elements of the theory of distributions .. .. .. .. .. .. .. .. 1.3.2.3.1. Origins .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.3.2. Rationale .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.3.3. Test-function spaces .. .. .. .. .. .. .. .. .. .. .. 1.3.2.3.3.1. Topology on E
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m .. .. .. .. 1.3.2.3.4. Deﬁnition of distributions .. .. .. .. .. .. .. .. .. 1.3.2.3.5. First examples of distributions .. .. .. .. .. .. .. .. 1.3.2.3.6. Distributions associated to locally integrable functions 1.3.2.3.7. Support of a distribution .. .. .. .. .. .. .. .. .. .. 1.3.2.3.8. Convergence of distributions .. .. .. .. .. .. .. .. 1.3.2.3.9. Operations on distributions .. .. .. .. .. .. .. .. .. 1.3.2.3.9.1. Differentiation .. .. .. .. .. .. .. .. .. 1.3.2.3.9.2. Integration of distributions in dimension 1 1.3.2.3.9.3. Multiplication of distributions by functions 1.3.2.3.9.4. Division of distributions by functions .. .. 1.3.2.3.9.5. Transformation of coordinates .. .. .. .. 1.3.2.3.9.6. Tensor product of distributions .. .. .. .. 1.3.2.3.9.7. Convolution of distributions .. .. .. .. .. 1.3.2.4. Fourier transforms of functions .. .. .. .. .. .. .. .. .. .. .. 1.3.2.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.4.2. Fourier transforms in L1 .. .. .. .. .. .. .. .. .. .. 1.3.2.4.2.1. Linearity .. .. .. .. .. .. .. .. .. .. .. 1.3.2.4.2.2. Effect of afﬁne coordinate transformations 1.3.2.4.2.3. Conjugate symmetry .. .. .. .. .. .. .. 1.3.2.4.2.4. Tensor product property .. .. .. .. .. .. 1.3.2.4.2.5. Convolution property .. .. .. .. .. .. .. 1.3.2.4.2.6. Reciprocity property .. .. .. .. .. .. .. 1.3.2.4.2.7. Riemann–Lebesgue lemma .. .. .. .. .. 1.3.2.4.2.8. Differentiation .. .. .. .. .. .. .. .. .. 1.3.2.4.2.9. Decrease at inﬁnity .. .. .. .. .. .. .. 1.3.2.4.2.10. The Paley–Wiener theorem .. .. .. .. .. 1.3.2.4.3. Fourier transforms in L2 .. .. .. .. .. .. .. .. .. .. 1.3.2.4.3.1. Invariance of L2 .. .. .. .. .. .. .. .. 1.3.2.4.3.2. Reciprocity .. .. .. .. .. .. .. .. .. ..

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CONTENTS 1.3.2.4.3.3. Isometry .. .. .. .. .. .. .. .. .. .. 1.3.2.4.3.4. Eigenspace decomposition of L2 .. .. .. 1.3.2.4.3.5. The convolution theorem and the isometry 1.3.2.4.4. Fourier transforms in S .. .. .. .. .. .. .. .. .. 1.3.2.4.4.1. Deﬁnition and properties of S .. .. .. 1.3.2.4.4.2. Gaussian functions and Hermite functions 1.3.2.4.4.3. Heisenberg’s inequality, Hardy’s theorem 1.3.2.4.4.4. Symmetry property .. .. .. .. .. .. .. 1.3.2.4.5. Various writings of Fourier transforms .. .. .. .. 1.3.2.4.6. Tables of Fourier transforms .. .. .. .. .. .. ..

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1.3.2.6. Periodic distributions and Fourier series .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.1. Terminology .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.2. Zn -periodic distributions in Rn .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.3. Identiﬁcation with distributions over Rn =Zn .. .. .. .. .. .. .. .. 1.3.2.6.4. Fourier transforms of periodic distributions .. .. .. .. .. .. .. .. 1.3.2.6.5. The case of non-standard period lattices .. .. .. .. .. .. .. .. .. 1.3.2.6.6. Duality between periodization and sampling .. .. .. .. .. .. .. .. 1.3.2.6.7. The Poisson summation formula .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.8. Convolution of Fourier series .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.9. Toeplitz forms, Szego¨’s theorem .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.9.1. Toeplitz forms .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.9.2. The Toeplitz–Carathe´odory–Herglotz theorem .. .. .. 1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms 1.3.2.6.9.4. Consequences of Szego¨’s theorem .. .. .. .. .. .. .. 1.3.2.6.10. Convergence of Fourier series .. .. .. .. .. .. .. .. .. .. .. 1 1.3.2.6.10.1. Classical L theory .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.10.2. Classical L2 theory .. .. .. .. .. .. .. .. .. .. .. 1.3.2.6.10.3. The viewpoint of distribution theory .. .. .. .. .. ..

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1.3.2.7. The discrete Fourier transformation .. .. .. .. .. .. .. .. .. .. .. .. 1.3.2.7.1. Shannon’s sampling theorem and interpolation formula .. .. .. 1.3.2.7.2. Duality between subdivision and decimation of period lattices .. 1.3.2.7.2.1. Geometric description of sublattices .. .. .. .. .. 1.3.2.7.2.2. Sublattice relations for reciprocal lattices .. .. .. .. 1.3.2.7.2.3. Relation between lattice distributions .. .. .. .. .. 1.3.2.7.2.4. Relation between Fourier transforms .. .. .. .. .. 1.3.2.7.2.5. Sublattice relations in terms of periodic distributions 1.3.2.7.3. Discretization of the Fourier transformation .. .. .. .. .. .. .. 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) .. 1.3.2.7.5. Properties of the discrete Fourier transform .. .. .. .. .. .. ..

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49 50 50 51 51 51 52 52 53 53 53

1.3.2.5. Fourier transforms of tempered distributions .. .. .. .. .. 1.3.2.5.1. Introduction .. .. .. .. .. .. .. .. .. .. .. 1.3.2.5.2. S as a test-function space .. .. .. .. .. .. .. 1.3.2.5.3. Deﬁnition and examples of tempered distributions 1.3.2.5.4. Fourier transforms of tempered distributions .. 1.3.2.5.5. Transposition of basic properties .. .. .. .. .. 1.3.2.5.6. Transforms of -functions .. .. .. .. .. .. .. 1.3.2.5.7. Reciprocity theorem .. .. .. .. .. .. .. .. .. 1.3.2.5.8. Multiplication and convolution .. .. .. .. .. .. 1.3.2.5.9. L2 aspects, Sobolev spaces .. .. .. .. .. .. ..

1.3.3. Numerical computation of the discrete Fourier transform 1.3.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. 1.3.3.2. One-dimensional algorithms .. .. .. .. .. .. .. .. 1.3.3.2.1. The Cooley–Tukey algorithm .. .. .. .. 1.3.3.2.2. The Good (or prime factor) algorithm .. 1.3.3.2.2.1. Ring structure on Z=N Z .. .. 1.3.3.2.2.2. The Chinese remainder theorem 1.3.3.2.2.3. The prime factor algorithm .. 1.3.3.2.3. The Rader algorithm .. .. .. .. .. .. .. 1.3.3.2.3.1. N an odd prime .. .. .. .. .. 1.3.3.2.3.2. N a power of an odd prime .. 1.3.3.2.3.3. N a power of 2 .. .. .. .. ..

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CONTENTS 1.3.3.2.4. The Winograd algorithms

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1.3.3.3. Multidimensional algorithms .. .. .. .. .. .. .. .. .. .. .. 1.3.3.3.1. The method of successive one-dimensional transforms .. 1.3.3.3.2. Multidimensional factorization .. .. .. .. .. .. .. .. 1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization 1.3.3.3.2.2. Multidimensional prime factor algorithm .. 1.3.3.3.2.3. Nesting of Winograd small FFTs .. .. .. 1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm .. .. 1.3.3.3.3. Global algorithm design .. .. .. .. .. .. .. .. .. .. 1.3.3.3.3.1. From local pieces to global algorithms .. 1.3.3.3.3.2. Computer architecture considerations .. .. 1.3.3.3.3.3. The Johnson–Burrus family of algorithms .. 1.3.4. Crystallographic applications of Fourier transforms

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54

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55 55 55 55 56 56 57 57 57 58 58

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58

1.3.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2. Crystallographic Fourier transform theory .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.1. Crystal periodicity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors .. .. .. 1.3.4.2.1.2. Structure factors in terms of form factors .. .. .. .. .. .. .. 1.3.4.2.1.3. Fourier series for the electron density and its summation .. .. 1.3.4.2.1.4. Friedel’s law, anomalous scatterers .. .. .. .. .. .. .. .. 1.3.4.2.1.5. Parseval’s identity and other L2 theorems .. .. .. .. .. .. .. 1.3.4.2.1.6. Convolution, correlation and Patterson function .. .. .. .. .. 1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation .. .. 1.3.4.2.1.8. Sections and projections .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.1.9. Differential syntheses .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szego¨’s theorem 1.3.4.2.2. Crystal symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.2.1. Crystallographic groups .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.2.2. Groups and group actions .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.2.3. Classiﬁcation of crystallographic groups .. .. .. .. .. .. .. 1.3.4.2.2.4. Crystallographic group action in real space .. .. .. .. .. .. 1.3.4.2.2.5. Crystallographic group action in reciprocal space .. .. .. .. 1.3.4.2.2.6. Structure-factor calculation .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.2.7. Electron-density calculations .. .. .. .. .. .. .. .. .. .. .. 1.3.4.2.2.8. Parseval’s theorem with crystallographic symmetry .. .. .. .. 1.3.4.2.2.9. Convolution theorems with crystallographic symmetry .. .. .. 1.3.4.2.2.10. Correlation and Patterson functions .. .. .. .. .. .. .. ..

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58 59 59 59 60 60 60 61 61 61 62 63 63 64 64 64 66 67 68 68 69 69 70 70

1.3.4.3. Crystallographic discrete Fourier transform algorithms .. .. .. .. .. .. .. .. .. 1.3.4.3.1. Historical introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.2. Deﬁning relations and symmetry considerations .. .. .. .. .. .. .. .. 1.3.4.3.3. Interaction between symmetry and decomposition .. .. .. .. .. .. .. .. 1.3.4.3.4. Interaction between symmetry and factorization .. .. .. .. .. .. .. .. .. 1.3.4.3.4.1. Multidimensional Cooley–Tukey factorization .. .. .. .. .. 1.3.4.3.4.2. Multidimensional Good factorization .. .. .. .. .. .. .. .. 1.3.4.3.4.3. Crystallographic extension of the Rader/Winograd factorization 1.3.4.3.5. Treatment of conjugate and parity-related symmetry properties .. .. .. .. 1.3.4.3.5.1. Hermitian-symmetric or real-valued transforms .. .. .. .. .. 1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms .. .. .. 1.3.4.3.5.3. Complex symmetric and antisymmetric transforms .. .. .. .. 1.3.4.3.5.4. Real symmetric transforms .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.5.5. Real antisymmetric transforms .. .. .. .. .. .. .. .. .. .. 1.3.4.3.5.6. Generalized multiplexing .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6. Global crystallographic algorithms .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.1. Triclinic groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.2. Monoclinic groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.3. Orthorhombic groups .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups .. .. .. .. .. .. 1.3.4.3.6.5. Cubic groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.6. Treatment of centred lattices .. .. .. .. .. .. .. .. .. .. .. 1.3.4.3.6.7. Programming considerations .. .. .. .. .. .. .. .. .. .. ..

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71 71 72 73 73 74 76 76 79 79 80 80 81 82 82 82 82 82 82 83 83 83 83

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1.3.4.4. Basic crystallographic computations

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CONTENTS 1.3.4.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.4.2. Fourier synthesis of electron-density maps .. .. .. .. .. .. .. .. 1.3.4.4.3. Fourier analysis of modiﬁed electron-density maps .. .. .. .. .. .. 1.3.4.4.3.1. Squaring .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.4.3.2. Other non-linear operations .. .. .. .. .. .. .. .. .. 1.3.4.4.3.3. Solvent ﬂattening .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.4.3.4. Molecular averaging by noncrystallographic symmetries 1.3.4.4.3.5. Molecular-envelope transforms via Green’s theorem .. 1.3.4.4.4. Structure factors from model atomic parameters .. .. .. .. .. .. 1.3.4.4.5. Structure factors via model electron-density maps .. .. .. .. .. .. 1.3.4.4.6. Derivatives for variational phasing techniques .. .. .. .. .. .. .. 1.3.4.4.7. Derivatives for model reﬁnement .. .. .. .. .. .. .. .. .. .. .. 1.3.4.4.7.1. The method of least squares .. .. .. .. .. .. .. .. .. 1.3.4.4.7.2. Booth’s differential Fourier syntheses .. .. .. .. .. .. 1.3.4.4.7.3. Booth’s method of steepest descents .. .. .. .. .. .. 1.3.4.4.7.4. Cochran’s Fourier method .. .. .. .. .. .. .. .. .. 1.3.4.4.7.5. Cruickshank’s modiﬁed Fourier method .. .. .. .. .. 1.3.4.4.7.6. Agarwal’s FFT implementation of the Fourier method .. 1.3.4.4.7.7. Lifchitz’s reformulation .. .. .. .. .. .. .. .. .. .. 1.3.4.4.7.8. A simpliﬁed derivation .. .. .. .. .. .. .. .. .. .. 1.3.4.4.7.9. Discussion of macromolecular reﬁnement techniques .. 1.3.4.4.7.10. Sampling considerations .. .. .. .. .. .. .. .. .. .. 1.3.4.4.8. Miscellaneous correlation functions .. .. .. .. .. .. .. .. .. .. 1.3.4.5. Related applications .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.5.1. Helical diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.3.4.5.1.1. Circular harmonic expansions in polar coordinates .. .. 1.3.4.5.1.2. The Fourier transform in polar coordinates .. .. .. .. 1.3.4.5.1.3. The transform of an axially periodic ﬁbre .. .. .. .. .. 1.3.4.5.1.4. Helical symmetry and associated selection rules .. .. .. 1.3.4.5.2. Application to probability theory and direct methods .. .. .. .. .. 1.3.4.5.2.1. Analytical methods of probability theory .. .. .. .. .. 1.3.4.5.2.2. The statistical theory of phase determination .. .. .. ..

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84 84 84 84 84 84 85 86 86 86 87 88 88 88 89 89 90 90 91 91 92 92 92 93 93 93 93 93 93 94 94 96

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1.4. Symmetry in reciprocal space (U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve) 1.4.1. Introduction (U. Shmueli)

1.4.2. Effects of symmetry on the Fourier image of the crystal (U. Shmueli) 1.4.2.1. 1.4.2.2. 1.4.2.3. 1.4.2.4.

Point-group symmetry of the reciprocal lattice .. .. .. .. .. .. .. .. .. .. .. Relationship between structure factors at symmetry-related points of the reciprocal Symmetry factors for space-group-speciﬁc Fourier summations .. .. .. .. .. .. Symmetry factors for space-group-speciﬁc structure-factor formulae .. .. .. ..

1.4.3. Structure-factor tables (U. Shmueli) 1.4.3.1. 1.4.3.2. 1.4.3.3. 1.4.3.4.

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1.4.4. Symmetry in reciprocal space: space-group tables (U. Shmueli)

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Appendix 1.4.1. Comments on the preparation and usage of the tables (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. Appendix 1.4.2. Space-group symbols for numeric and symbolic computations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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104 104 104 105 105 106

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A1.4.2.1. Introduction (U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve) A1.4.2.2. Explicit symbols (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. A1.4.2.3. Hall symbols (S. R. Hall and R. W. Grosse-Kunstleve) .. .. .. .. A1.4.2.3.1. Default axes .. .. .. .. .. .. .. .. .. .. .. .. .. .. A1.4.2.3.2. Example matrices .. .. .. .. .. .. .. .. .. .. .. .. .. Table A1.4.2.1. Explicit symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Some general remarks .. .. .. .. .. Preparation of the structure-factor tables Symbolic representation of A and B .. Arrangement of the tables .. .. .. ..

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CONTENTS Table Table Table Table Table Table

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112 112 113 113 113 115

Appendix 1.4.3. Structure-factor tables (U. Shmueli) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Table Table Table Table Table Table Table

A1.4.2.2. A1.4.2.3. A1.4.2.4. A1.4.2.5. A1.4.2.6. A1.4.2.7. A1.4.3.1. A1.4.3.2. A1.4.3.3. A1.4.3.4. A1.4.3.5. A1.4.3.6. A1.4.3.7.

Lattice symbol L .. .. .. .. .. .. .. .. Translation symbol T .. .. .. .. .. .. Rotation matrices for principal axes .. .. Rotation matrices for face-diagonal axes Rotation matrix for the body-diagonal axis Hall symbols .. .. .. .. .. .. .. .. .. Plane groups .. .. .. .. .. Triclinic space groups .. .. Monoclinic space groups .. Orthorhombic space groups Tetragonal space groups .. Trigonal and hexagonal space Cubic space groups .. .. ..

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1.5.1. List of symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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1.5.2. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Table A1.4.4.1. Crystallographic space groups in reciprocal space

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Appendix 1.4.4. Crystallographic space groups in reciprocal space (U. Shmueli)

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1.5. Crystallographic viewpoints in the classiﬁcation of space-group representations (M. I. Aroyo and H. Wondratschek)

1.5.3. Basic concepts 1.5.3.1. 1.5.3.2. 1.5.3.3. 1.5.3.4.

Representations of ﬁnite groups .. .. .. .. .. .. .. .. .. .. .. .. .. Space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Representations of the translation group T and the reciprocal lattice .. .. Irreducible representations of space groups and the reciprocal-space group

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1.5.4.1. Fundamental regions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.5.4.2. Minimal domains .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.5.4.3. Wintgen positions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.5.4.1. Conventional coefﬁcients
ki T of k expressed by the adjusted coefﬁcients
kai of IT A for the different Bravais types of lattices in direct space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.5.4.2. Primitive coefﬁcients
kpi T of k from CDML expressed by the adjusted coefﬁcients
kai of IT A for the different Bravais types of lattices in direct space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

165 166 167

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1.5.4. Conventions in the classiﬁcation of space-group irreps

1.5.5. Examples and conclusions

1.5.5.1. Examples .. .. .. .. 1.5.5.2. Results .. .. .. .. .. 1.5.5.3. Parameter ranges .. .. 1.5.5.4. Conclusions .. .. .. Table 1.5.5.1. The k-vector types Table 1.5.5.2. The k-vector types Table 1.5.5.3. The k-vector types Table 1.5.5.4. The k-vector types

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2.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.2. The average intensity of general reﬂections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.3. The average intensity of zones and rows .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.1.3.1. Symmetry elements producing systematic absences .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.3.2. Symmetry elements not producing systematic absences .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.3.3. More than one symmetry element .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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PART 2. RECIPROCAL SPACE IN CRYSTAL-STRUCTURE DETERMINATION .. .. .. .. .. .. .. ..

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Mathematical background .. .. .. .. .. .. .. .. Physical background .. .. .. .. .. .. .. .. .. .. An approximation for organic compounds .. .. .. Effect of centring .. .. .. .. .. .. .. .. .. .. ..

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Appendix 1.5.1. Reciprocal-space groups G .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

2.1.2.1. 2.1.2.2. 2.1.2.3. 2.1.2.4.

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168 169 171 172 168 170 172 174

2.1. Statistical properties of the weighted reciprocal lattice (U. Shmueli and A. J. C. Wilson)

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References

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. and Ia3d for the space groups Im3m .. .. .. .. .. .. .. .. for the space groups Im3 and Ia3 .. .. .. .. .. .. .. .. .. for the space groups I4=mmm, I4=mcm, I41 =amd and I41 =acd for the space groups Fmm2 and Fdd2 .. .. .. .. .. .. .. ..

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CONTENTS Table 2.1.3.1. Intensity-distribution effects of symmetry elements causing systematic absences .. .. .. .. .. .. .. .. .. Table 2.1.3.2. Intensity-distribution effects of symmetry elements not causing systematic absences .. .. .. .. .. .. .. .. Table 2.1.3.3. Average multiples for the 32 point groups (modiﬁed from Rogers, 1950) .. .. .. .. .. .. .. .. .. .. .. 2.1.4. Probability density distributions – mathematical preliminaries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.4.1. 2.1.4.2. 2.1.4.3. 2.1.4.4. 2.1.4.5.

Characteristic functions .. .. .. The cumulant-generating function The central-limit theorem .. .. Conditions of validity .. .. .. .. Non-independent variables .. ..

2.1.5. Ideal probability density distributions

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192 193 194 195 195

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2.1.7.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.7.2. Mathematical background .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.7.3. Application to centric and acentric distributions .. .. .. .. .. .. .. 2.1.7.4. Fourier versus Hermite approximations .. .. .. .. .. .. .. .. .. .. Table 2.1.7.1. Some even absolute moments of the trigonometric structure factor Table 2.1.7.2. Closed expressions for 2k [equation (2.1.7.11)] for space groups of

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2.2. Direct methods (C. Giacovazzo)

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203

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2.1.8. Non-ideal distributions: the Fourier method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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199 199 200 203 201 203

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2.1.8.1. General representations of p.d.f.’s of jEj by Fourier series .. 2.1.8.2. Fourier–Bessel series .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.8.3. Simple examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.8.4. A more complicated example .. .. .. .. .. .. .. .. .. .. 2.1.8.5. Atomic characteristic functions .. .. .. .. .. .. .. .. .. .. 2.1.8.6. Other non-ideal Fourier p.d.f.’s .. .. .. .. .. .. .. .. .. 2.1.8.7. Comparison of the correction-factor and Fourier approaches Table 2.1.8.1. Atomic contributions to characteristic functions for p
jEj

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2.1.7. Non-ideal distributions: the correction-factor approach .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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Distributions of sums and averages Distribution of ratios .. .. .. .. .. Intensities scaled to the local average The use of normal approximations ..

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195 196 196 196 196 196 197

2.1.6.1. 2.1.6.2. 2.1.6.3. 2.1.6.4.

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2.1.6. Distributions of sums, averages and ratios

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2.1.5.1. Ideal acentric distributions .. .. .. .. .. .. .. .. .. .. .. .. 2.1.5.2. Ideal centric distributions .. .. .. .. .. .. .. .. .. .. .. .. 2.1.5.3. Effect of other symmetry elements on the ideal acentric and centric 2.1.5.4. Other ideal distributions .. .. .. .. .. .. .. .. .. .. .. .. .. 2.1.5.5. Relation to distributions of I .. .. .. .. .. .. .. .. .. .. .. 2.1.5.6. Cumulative distribution functions .. .. .. .. .. .. .. .. .. .. Table 2.1.5.1. Some properties of gamma and beta distributions .. .. ..

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191 192 193

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203 204 205 205 206 208 208 207

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2.2.1. List of symbols and abbreviations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2.2. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups .. origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

211 212 214 214

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2.2.3. Origin speciﬁcation Table Table Table Table

2.2.3.1. 2.2.3.2. 2.2.3.3. 2.2.3.4.

Allowed Allowed Allowed Allowed groups

2.2.4. Normalized structure factors

2.2.4.1. Deﬁnition of normalized structure factor .. .. .. .. .. 2.2.4.2. Deﬁnition of quasi-normalized structure factor .. .. .. 2.2.4.3. The calculation of normalized structure factors .. .. .. 2.2.4.4. Probability distributions of normalized structure factors Table 2.2.4.1. Moments of the distributions (2.2.4.4) and (2.2.4.5)

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215 216 216 217 217

2.2.5. Phase-determining formulae .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

217

2.2.5.1. 2.2.5.2. 2.2.5.3. 2.2.5.4.

Inequalities among structure factors .. .. .. .. .. .. Probabilistic phase relationships for structure invariants Triplet relationships .. .. .. .. .. .. .. .. .. .. .. Triplet relationships using structural information .. ..

xiii

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CONTENTS 2.2.5.5. Quartet phase relationships .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.6. Quintet phase relationships .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.7. Determinantal formulae .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.8. Algebraic relationships for structure seminvariants .. .. .. .. .. .. .. 2.2.5.9. Formulae estimating one-phase structure seminvariants of the ﬁrst rank 2.2.5.10. Formulae estimating two-phase structure seminvariants of the ﬁrst rank Table 2.2.5.1. List of quartets symmetry equivalent to 1 in the class mmm

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220 222 223 224 224 225 222

2.2.6. Direct methods in real and reciprocal space: Sayre’s equation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2.8. Other multisolution methods applied to small molecules .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

228

Table 2.2.8.1. Magic-integer sequences for small numbers of phases (n) together with the number of sets produced and the root-mean-square error in the phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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2.2.7. Scheme of procedure for phase determination

2.2.9. Some references to direct-methods packages

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2.2.10. Direct methods in macromolecular crystallography .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.10.1. 2.2.10.2. 2.2.10.3. 2.2.10.4.

Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Ab initio direct phasing of proteins .. .. .. .. .. .. .. .. .. .. .. Integration of direct methods with isomorphous replacement techniques Integration of anomalous-dispersion techniques with direct methods .. 2.2.10.4.1. One-wavelength techniques .. .. .. .. .. .. .. .. .. 2.2.10.4.2. The SIRAS, MIRAS and MAD cases .. .. .. .. .. .. ..

231 231 232 232 233 233

2.3. Patterson and molecular-replacement techniques (M. G. Rossmann and E. Arnold) .. .. .. .. .. .. .. .. .. .. .. .. ..

235

2.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

235

2.3.2. Interpretation of Patterson maps

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235 235 236 237 238 236

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238

2.3.3. Isomorphous replacement difference Pattersons

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2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin .. .. .. .. .. .. .. .. .. 2.3.2.2. Harker sections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.3. Finding heavy atoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.4. Superposition methods. Image detection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies .. Table 2.3.2.1. Coordinates of Patterson peaks for C 2 H 6 Cl2 Cu2 N 2 projection .. .. .. .. .. .. Table 2.3.2.2. Square matrix representation of vector interactions in a Patterson of a crystal asymmetric units each containing N atoms .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.2.3. Position of Harker sections within a Patterson .. .. .. .. .. .. .. .. .. .. .. 2.3.3.1. 2.3.3.2. 2.3.3.3. 2.3.3.4. 2.3.3.5. 2.3.3.6. 2.3.3.7.

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2.3.1.1. Background .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.2. Limits to the number of resolved vectors .. .. .. .. 2.3.1.3. Modiﬁcations: origin removal, sharpening etc. .. .. 2.3.1.4. Homometric structures and the uniqueness of structure 2.3.1.5. The Patterson synthesis of the second kind .. .. .. Table 2.3.1.1. Matrix representation of Patterson peaks .. ..

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238 239 239 240 241 239

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242

Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. Finding heavy atoms with centrosymmetric projections Finding heavy atoms with three-dimensional methods .. Correlation functions .. .. .. .. .. .. .. .. .. .. .. Interpretation of isomorphous difference Pattersons .. Direct structure determination from difference Pattersons Isomorphism and size of the heavy-atom substitution ..

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242 242 243 243 244 245 245

2.3.4. Anomalous dispersion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

246

2.3.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.4.2. The Ps
u function .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.4.3. The position of anomalous scatterers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

246 246 247

2.3.5. Noncrystallographic symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

248

2.3.5.1. Deﬁnitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry .. Table 2.3.5.1. Possible types of vector searches .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular cell .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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250

2.3.6. Rotation functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

250

xiv

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239 240

248 249 250

CONTENTS 2.3.6.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.6.2. Matrix algebra .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.6.3. Symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.6.4. Sampling, background and interpretation .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.6.5. The fast rotation function .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.6.1. Different types of uses for the rotation function .. .. .. .. .. .. .. .. .. Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations Table 2.3.6.3. Numbering of the rotation function space groups .. .. .. .. .. .. .. .. Table 2.3.6.4. Rotation function Eulerian space groups .. .. .. .. .. .. .. .. .. .. ..

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250 252 253 254 255 251 254 254 256

2.3.7. Translation functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

258

2.3.7.1. 2.3.7.2. 2.3.7.3. 2.3.7.4.

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Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Position of a noncrystallographic element relating two unknown structures .. .. .. .. .. Position of a known molecular structure in an unknown unit cell .. .. .. .. .. .. .. .. Position of a noncrystallographic symmetry element in a poorly deﬁned electron-density map

2.3.8. Molecular replacement

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2.3.8.1. Using a known molecular fragment .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.8.2. Using noncrystallographic symmetry for phase improvement .. .. .. .. .. .. .. 2.3.8.3. Equivalence of real- and reciprocal-space molecular replacement .. .. .. .. .. Table 2.3.8.1. Molecular replacement: phase reﬁnement as an iterative process .. .. ..

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260

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260 261 262 261

2.3.9. Conclusions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

262

2.3.9.1. Update

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258 259 259 260

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2.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

264

2.4.2. Isomorphous replacement method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

264

2.4.2.1. Isomorphous replacement and isomorphous addition .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.2.2. Single isomorphous replacement method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.2.3. Multiple isomorphous replacement method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

264 265 265

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265

2.4. Isomorphous replacement and anomalous scattering (M. Vijayan and S. Ramaseshan)

2.4.3. Anomalous-scattering method

2.4.3.1. Dispersion correction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.2. Violation of Friedel’s law .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.3. Friedel and Bijvoet pairs .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.4. Determination of absolute conﬁguration .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.5. Determination of phase angles .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.6. Anomalous scattering without phase change .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.3.7. Treatment of anomalous scattering in structure reﬁnement .. .. .. .. .. .. .. .. .. Table 2.4.3.1. Phase angles of different components of the structure factor in space group P222

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265 266 267 267 268 268 268 267

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269

Protein heavy-atom derivatives .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Determination of heavy-atom parameters .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Reﬁnement of heavy-atom parameters .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Treatment of errors in phase evaluation: Blow and Crick formulation .. .. .. .. .. .. .. .. .. .. .. .. .. .. Use of anomalous scattering in phase evaluation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Estimation of r.m.s. error .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Suggested modiﬁcations to Blow and Crick formulation and the inclusion of phase information from other sources Fourier representation of anomalous scatterers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

269 269 270 271 272 273 274 274

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274

2.4.5.1. Neutron anomalous scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.4.5.2. Anomalous scattering of synchrotron radiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

275 275

2.5. Electron diffraction and electron microscopy in structure determination (J. M. Cowley, P. Goodman, B. K. Vainshtein, B. B. Zvyagin and D. L. Dorset) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

276

2.5.1. Foreword (J. M. Cowley) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

276

2.4.4. Isomorphous replacement and anomalous scattering in protein crystallography 2.4.4.1. 2.4.4.2. 2.4.4.3. 2.4.4.4. 2.4.4.5. 2.4.4.6. 2.4.4.7. 2.4.4.8.

2.4.5. Anomalous scattering of neutrons and synchrotron radiation. The multiwavelength method

2.5.2. Electron diffraction and electron microscopy (J. M. Cowley) 2.5.2.1. 2.5.2.2. 2.5.2.3. 2.5.2.4. 2.5.2.5.

Introduction .. .. .. .. .. .. .. .. .. .. .. The interactions of electrons with matter .. .. .. Recommended sign conventions .. .. .. .. .. Scattering of electrons by crystals; approximations Kinematical diffraction formulae .. .. .. .. ..

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xv

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CONTENTS 2.5.2.6. Imaging with electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.2.7. Imaging of very thin and weakly scattering objects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.2.8. Crystal structure imaging .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.2.9. Image resolution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.2.10. Electron diffraction in electron microscopes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.5.2.1. Standard crystallographic and alternative crystallographic sign conventions for electron diffraction

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282 283 284 284 285 280

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285

2.5.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.1.1. CBED .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.1.2. Zone-axis patterns from CBED .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.2. Background theory and analytical approach .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.2.1. Direct and reciprocity symmetries: types I and II .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.2.2. Reciprocity and Friedel’s law .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.2.3. In-disc symmetries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.2.4. Zero-layer absences .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.3. Pattern observation of individual symmetry elements .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.4. Auxiliary tables .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.5. Space-group analyses of single crystals; experimental procedure and published examples .. .. .. .. .. .. .. .. 2.5.3.5.1. Stages of procedure .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.5.2. Examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.6. Use of CBED in study of crystal defects, twins and non-classical crystallography .. .. .. .. .. .. .. .. .. .. 2.5.3.7. Present limitations and general conclusions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.3.8. Computer programs available .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.5.3.1. Listing of the symmetry elements relating to CBED patterns under the classiﬁcations of ‘vertical’ (I), ‘horizontal’ (II) and combined or roto-inversionary axes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.5.3.2. Diagrammatic illustrations of the actions of ﬁve types of symmetry elements (given in the last column in Volume A diagrammatic symbols) on an asymmetric pattern component, in relation to the centre of the pattern at K00 0, shown as ‘ ’, or in relation to the centre of a diffraction order at K0g 0, shown as ‘+’ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.5.3.3. Diffraction point-group tables, giving whole-pattern and central-beam pattern symmetries in terms of BESR diffraction-group symbols and diperiodic group symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.5.3.4. Tabulation of principal-axis CBED pattern symmetries against relevant space groups given as IT A numbers Table 2.5.3.5. Conditions for observation of GS bands for the 137 space groups exhibiting these extinctions .. .. .. .. ..

285 285 286 286 286 287 287 288 288 289 291 291 292 292 295 295

2.5.4. Electron-diffraction structure analysis (EDSA) (B. K. Vainshtein and B. B. Zvyagin) .. .. .. .. .. .. .. .. .. .. ..

306

2.5.3. Space-group determination by convergent-beam electron diffraction (P. Goodman)

2.5.4.1. 2.5.4.2. 2.5.4.3. 2.5.4.4.

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2.5.7. Direct phase determination in electron crystallography (D. L. Dorset) 2.5.7.1. 2.5.7.2. 2.5.7.3. 2.5.7.4. 2.5.7.5. 2.5.7.6. 2.5.7.7.

Problems with ‘traditional’ phasing techniques .. .. Direct phase determination from electron micrographs Probabilistic estimate of phase invariant sums .. .. The tangent formula .. .. .. .. .. .. .. .. .. .. Density modiﬁcation .. .. .. .. .. .. .. .. .. .. Convolution techniques .. .. .. .. .. .. .. .. .. Maximum entropy and likelihood .. .. .. .. .. ..

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2.5.6. Three-dimensional reconstruction (B. K. Vainshtein) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. case

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310 311 312 312 313

The object and its projection .. .. .. .. .. Orthoaxial projection .. .. .. .. .. .. .. .. Discretization .. .. .. .. .. .. .. .. .. .. Methods of direct reconstruction .. .. .. .. The method of back-projection .. .. .. .. .. The algebraic and iteration methods .. .. .. Reconstruction using Fourier transformation Three-dimensional reconstruction in the general

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2.5.6.1. 2.5.6.2. 2.5.6.3. 2.5.6.4. 2.5.6.5. 2.5.6.6. 2.5.6.7. 2.5.6.8.

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290 296 298

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288

2.5.5. Image reconstruction (B. K. Vainshtein) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. defocus .. .. .. .. .. .. .. .. ..

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286

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Introduction .. .. .. .. .. .. Thin weak phase objects at optimal An account of absorption .. .. Thick crystals .. .. .. .. .. .. Image enhancement .. .. .. ..

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2.5.5.1. 2.5.5.2. 2.5.5.3. 2.5.5.4. 2.5.5.5.

Introduction .. .. .. .. .. The geometry of ED patterns Intensities of diffraction beams Structure analysis .. .. .. ..

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CONTENTS 2.5.7.8. Inﬂuence of multiple scattering on direct electron crystallographic structure analysis

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325

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327

PART 3. DUAL BASES IN CRYSTALLOGRAPHIC COMPUTING .. .. .. .. .. .. .. .. .. .. .. .. .. ..

347

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348

3.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

348

References

3.1. Distances, angles, and their standard uncertainties (D. E. Sands)

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348

3.1.3. Length of a vector .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.5. Vector product .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

349

3.1.2. Scalar product

3.1.4. Angle between two vectors

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3.1.7. Components of vector product .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.8.1. Triple vector product .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.8.2. Scalar product of vector products .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.8.3. Vector product of vector products .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

349 349 349

3.1.9. Planes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.1.10. Variance–covariance matrices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

350

3.1.11. Mean values

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3.1.12. Computation

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3.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

353

3.1.6. Permutation tensors

3.1.8. Some vector relationships

3.2. The least-squares plane (R. E. Marsh and V. Schomaker)

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353

3.2.2.1. Error propagation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.2. The standard uncertainty of the distance from an atom to the plane .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

353 355

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355

3.2.3.1. Formulation and solution of the general Gaussian plane .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.3.2. Concluding remarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

356 358

Appendix 3.2.1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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3.3.1. Graphics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

360

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)

3.3.1.1. Coordinate systems, notation and standards .. .. .. .. .. .. .. 3.3.1.1.1. Cartesian and crystallographic coordinates .. .. .. .. 3.3.1.1.2. Homogeneous coordinates .. .. .. .. .. .. .. .. .. 3.3.1.1.3. Notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.1.4. Standards .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.2. Orthogonal (or rotation) matrices .. .. .. .. .. .. .. .. .. .. 3.3.1.2.1. General form .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.2.2. Measurement of rotations and strains from coordinates 3.3.1.2.3. Orthogonalization of impure rotations .. .. .. .. .. 3.3.1.2.4. Eigenvalues and eigenvectors of orthogonal matrices .. 3.3.1.3. Projection transformations and spaces .. .. .. .. .. .. .. .. 3.3.1.3.1. Deﬁnitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.2. Translation .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.3. Rotation .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.4. Scale .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.5. Windowing and perspective .. .. .. .. .. .. .. .. .. 3.3.1.3.6. Stereoviews .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.7. Viewports .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.1.3.8. Compound transformations .. .. .. .. .. .. .. .. .. 3.3.1.3.9. Inverse transformations .. .. .. .. .. .. .. .. .. .. 3.3.1.3.10. The three-axis joystick .. .. .. .. .. .. .. .. .. .. 3.3.1.3.11. Other useful rotations .. .. .. .. .. .. .. .. .. .. 3.3.1.3.12. Symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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360 360 360 361 361 361 361 364 367 367 367 367 368 368 368 368 370 370 371 372 372 373 373

CONTENTS 3.3.1.4. Modelling transformations .. .. .. .. .. .. .. 3.3.1.4.1. Rotation about a bond .. .. .. .. .. 3.3.1.4.2. Stacked transformations .. .. .. .. .. 3.3.1.5. Drawing techniques .. .. .. .. .. .. .. .. .. 3.3.1.5.1. Types of hardware .. .. .. .. .. .. 3.3.1.5.2. Optimization of line drawings .. .. .. 3.3.1.5.3. Representation of surfaces by lines .. 3.3.1.5.4. Representation of surfaces by dots .. .. 3.3.1.5.5. Representation of surfaces by shading 3.3.1.5.6. Advanced hidden-line and hidden-surface 3.3.2. Molecular modelling, problems and approaches

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373 373 373 374 374 375 375 375 375 376

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3.3.2.1. Connectivity .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.2.1.1. Connectivity tables .. .. .. .. .. .. .. .. 3.3.2.1.2. Implied connectivity .. .. .. .. .. .. .. .. 3.3.2.2. Modelling methods .. .. .. .. .. .. .. .. .. .. .. 3.3.2.2.1. Methods based on conformational variables .. 3.3.2.2.2. Methods based on positional coordinates .. .. 3.3.2.2.3. Approaches to the problem of multiple minima 3.3.3. Implementations

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lattice sums (D. E. Williams) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

385

3.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

385

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.4.2.1. Untreated lattice-sum results for the Coulombic energy (n 1) of sodium chloride (kJ mol 1 ; A˚); the lattice constant is taken as 5.628 A˚ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.4.2.2. Untreated lattice-sum results for the dispersion energy (n 6) of crystalline benzene (kJ mol 1 ; A˚ ) .. .. ..

385

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3.4. Accelerated convergence treatment of R

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3.3.3.1. Systems for the display and modiﬁcation of retrieved data 3.3.3.1.1. ORTEP .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.1.2. Feldmann’s system .. .. .. .. .. .. .. .. 3.3.3.1.3. Lesk & Hardman software .. .. .. .. .. .. 3.3.3.1.4. GRAMPS .. .. .. .. .. .. .. .. .. .. .. 3.3.3.1.5. Takenaka & Sasada’s system .. .. .. .. .. 3.3.3.1.6. MIDAS .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.1.7. Insight .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.1.8. PLUTO .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.1.9. MDKINO .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2. Molecular-modelling systems based on electron density 3.3.3.2.1. CHEMGRAF .. .. .. .. .. .. .. .. .. .. 3.3.3.2.2. GRIP .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.3. Barry, Denson & North’s systems .. .. .. .. 3.3.3.2.4. MMS-X .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.5. Texas A&M University system .. .. .. .. .. 3.3.3.2.6. Bilder .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.7. Frodo .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.8. Guide .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.9. HYDRA .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.2.10. O .. .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.3. Molecular-modelling systems based on other criteria .. 3.3.3.3.1. Molbuild, Rings, PRXBLD and MM2/MMP2 3.3.3.3.2. Script .. .. .. .. .. .. .. .. .. .. .. .. 3.3.3.3.3. CHARMM .. .. .. .. .. .. .. .. .. .. .. 3.3.3.3.4. Commercial systems .. .. .. .. .. .. .. ..

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3.4.2. Deﬁnition and behaviour of the direct-space sum

3.4.3. Preliminary description of the method

n

385 386

sum over lattice points X(d)

386

3.4.5. Extension of the method to a composite lattice .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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389

3.4.4. Preliminary derivation to obtain a formula which accelerates the convergence of an R 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 .. .. .. .. .. .. .. .. .. .. .. .. ..

389

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390

3.4.9. Evaluation of the incomplete gamma function

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CONTENTS 3.4.10. Summation over the asymmetric unit and elimination of intramolecular energy terms .. .. .. .. .. .. .. .. .. ..

390

3.4.11. Reference formulae for particular values of n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

390

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Accelerated-convergence results for the Coulombic sum (n 1) of sodium chloride (kJ mol 1 ; A˚ ): the direct sum plus the constant term .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. The reciprocal-lattice results (kJ mol 1 ; A˚ ) for the Coulombic sum (n 1) of sodium chloride .. .. .. .. Accelerated-convergence results for the dispersion sum (n 6) of crystalline benzene (kJ mol 1 ; A˚); the ﬁgures shown are the direct-lattice sum plus the two constant terms .. .. .. .. .. .. .. .. .. .. .. .. The reciprocal-lattice results (kJ mol 1 ; A˚ ) for the dispersion sum (n 6) of crystalline benzene .. .. .. Approximate time (s) required to evaluate the dispersion sum (n 6) for crystalline benzene within 0:001 kJ mol 1 truncation error .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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PART 4. DIFFUSE SCATTERING AND RELATED TOPICS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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4.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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400

3.4.12. Numerical illustrations Table 3.4.12.1. Table 3.4.12.2. Table 3.4.12.3. Table 3.4.12.4. Table 3.4.12.5. References

4.1. Thermal diffuse scattering of X-rays and neutrons (B. T. M. Willis)

4.1.2. Dynamics of three-dimensional crystals 4.1.2.1. 4.1.2.2. 4.1.2.3. 4.1.2.4.

Equations of motion .. .. .. .. .. .. Quantization of normal modes. Phonons Einstein and Debye models .. .. .. .. Molecular crystals .. .. .. .. .. ..

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4.1.5. Phonon dispersion relations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

405

4.1.5.1. Measurement with X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.1.5.2. Measurement with neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.1.5.3. Interpretation of dispersion relations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

405 405 405

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408

4.2.3. General treatment .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

410

4.1.6. Measurement of elastic constants

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4.2. Disorder diffuse scattering of X-rays and neutrons (H. Jagodzinski and F. Frey) 4.2.1. Scope of this chapter

4.2.2. Summary of basic scattering theory

4.2.3.1. Qualitative interpretation of diffuse scattering 4.2.3.1.1. Fourier transforms .. .. .. .. .. 4.2.3.1.2. Applications .. .. .. .. .. .. .. 4.2.3.2. Guideline to solve a disorder problem .. .. .. 4.2.4. Quantitative interpretation

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392

401 402 402 402

4.1.4. Scattering of neutrons by thermal vibrations

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392 392

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4.1.3. Scattering of X-rays by thermal vibrations

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391 392

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4.2.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.2. One-dimensional disorder of ordered layers .. .. .. .. .. .. .. .. .. 4.2.4.2.1. Stacking disorder in close-packed structures .. .. .. .. .. .. 4.2.4.3. Two-dimensional disorder of chains .. .. .. .. .. .. .. .. .. .. .. 4.2.4.3.1. Scattering by randomly distributed collinear chains .. .. .. 4.2.4.3.2. Disorder within randomly distributed collinear chains .. .. .. 4.2.4.3.2.1. General treatment .. .. .. .. .. .. .. .. .. .. 4.2.4.3.2.2. Orientational disorder .. .. .. .. .. .. .. .. .. 4.2.4.3.2.3. Longitudinal disorder .. .. .. .. .. .. .. .. .. 4.2.4.3.3. Correlations between different almost collinear chains .. .. 4.2.4.4. Disorder with three-dimensional correlations (defects, local ordering and 4.2.4.4.1. General formulation (elastic diffuse scattering) .. .. .. .. .. 4.2.4.4.2. Random distribution .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.4.3. Short-range order in multi-component systems .. .. .. .. .. 4.2.4.4.4. Displacements: general remarks .. .. .. .. .. .. .. .. .. 4.2.4.4.5. Distortions in binary systems .. .. .. .. .. .. .. .. .. .. 4.2.4.4.6. Powder diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.4.7. Small concentrations of defects .. .. .. .. .. .. .. .. .. ..

xix

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410 410 411 418 420 420 421 423 425 425 427 427 427 428 429 429 429 431 432 432 433 435 435

CONTENTS 4.2.4.4.8. Cluster method .. .. .. .. .. .. .. .. .. .. 4.2.4.4.9. Comparison between X-ray and neutron methods 4.2.4.4.10. Dynamic properties of defects .. .. .. .. .. .. 4.2.4.5. Orientational disorder .. .. .. .. .. .. .. .. .. .. .. 4.2.4.5.1. General expressions .. .. .. .. .. .. .. .. .. 4.2.4.5.2. Rotational structure (form) factor .. .. .. .. .. 4.2.4.5.3. Short-range correlations .. .. .. .. .. .. .. ..

.. .. .. .. .. .. ..

435 435 436 436 436 437 438

4.2.5. Measurement of diffuse scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

438

4.3. Diffuse scattering in electron diffraction (J. M. Cowley and J. K. Gjùnnes) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

443

4.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

443

4.3.2. Inelastic scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

444

4.3.3. Kinematical and pseudo-kinematical scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

445

4.3.4. Dynamical scattering: Bragg scattering effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

445

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447

4.3.6. Qualitative interpretation of diffuse scattering of electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

447

4.4. Scattering from mesomorphic structures (P. S. Pershan) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

449

4.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

449

Table 4.4.1.1. Some of the symmetry properties of the series of three-dimensional phases described in Fig. 4.4.1.1 .. .. .. Table 4.4.1.2. The symmetry properties of the two-dimensional hexatic and crystalline phases .. .. .. .. .. .. .. .. ..

449 450

4.4.2. The nematic phase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

451

Table 4.4.2.1. Summary of critical exponents from X-ray scattering studies of the nematic to smectic-A phase transition .. 4.4.3. Smectic-A and smectic-C phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

453 453

4.4.3.1. Homogeneous smectic-A and smectic-C phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.3.2. Modulated smectic-A and smectic-C phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.3.3. Surface effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

453 455 455

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456

4.3.5. Multislice calculations for diffraction and imaging

4.4.4. Phases with in-plane order

4.4.4.1. Hexatic phases in two dimensions .. .. .. 4.4.4.2. Hexatic phases in three dimensions .. .. 4.4.4.2.1. Hexatic-B .. .. .. .. .. .. .. 4.4.4.2.2. Smectic-F, smectic-I .. .. .. .. 4.4.4.3. Crystalline phases with molecular rotation 4.4.4.3.1. Crystal-B .. .. .. .. .. .. .. 4.4.4.3.2. Crystal-G, crystal-J .. .. .. .. 4.4.4.4. Crystalline phases with herringbone packing 4.4.4.4.1. Crystal-E .. .. .. .. .. .. .. 4.4.4.4.2. Crystal-H, crystal-K .. .. .. ..

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457 458 458 458 460 460 462 462 462 463

4.4.5. Discotic phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

463

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463

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464

4.4.7.1. Phases with intermediate molecular tilt: smectic-L, crystalline-M,N .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.7.2. Nematic to smectic-A phase transition .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

464 464

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

466

4.4.7. Notes added in proof to ﬁrst edition

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4.4.6. Other phases

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4.5.1. Overview (R. P. Millane and D. L. Dorset) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

466

4.5.2. X-ray ﬁbre diffraction analysis (R. P. Millane) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

466

4.5.2.1. Introduction .. .. .. .. .. .. .. .. .. 4.5.2.2. Fibre specimens .. .. .. .. .. .. .. .. 4.5.2.3. Diffraction by helical structures .. .. .. 4.5.2.3.1. Helix symmetry .. .. .. .. .. 4.5.2.3.2. Diffraction by helical structures 4.5.2.3.3. Approximate helix symmetry .. 4.5.2.4. Diffraction by ﬁbres .. .. .. .. .. .. .. 4.5.2.4.1. Noncrystalline ﬁbres .. .. .. .. 4.5.2.4.2. Polycrystalline ﬁbres .. .. .. .. 4.5.2.4.3. Random copolymers .. .. .. ..

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466 467 467 467 468 469 469 469 469 470

CONTENTS 4.5.2.4.4. Partially crystalline ﬁbres

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4.5.2.5. Processing diffraction data .. .. .. .. .. .. .. .. .. .. .. .. .. 4.5.2.6. Structure determination .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.5.2.6.1. Overview .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.5.2.6.2. Helix symmetry, cell constants and space-group symmetry 4.5.2.6.3. Patterson functions .. .. .. .. .. .. .. .. .. .. .. .. 4.5.2.6.4. Molecular model building .. .. .. .. .. .. .. .. .. .. 4.5.2.6.5. Difference Fourier synthesis .. .. .. .. .. .. .. .. .. 4.5.2.6.6. Multidimensional isomorphous replacement .. .. .. .. .. 4.5.2.6.7. Other techniques .. .. .. .. .. .. .. .. .. .. .. .. .. 4.5.2.6.8. Reliability .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

472 474 474 475 475 476 477 478 479 480

4.5.3. Electron crystallography of polymers (D. L. Dorset) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

481

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487

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486

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4.6.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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486

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4.6. Reciprocal-space images of aperiodic crystals (W. Steurer and T. Haibach) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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481 481 482 483 483

4.6.2.1. Basic concepts .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.2.2. 1D incommensurately modulated structures .. .. .. .. 4.6.2.3. 1D composite structures .. .. .. .. .. .. .. .. .. .. 4.6.2.4. 1D quasiperiodic structures .. .. .. .. .. .. .. .. .. 4.6.2.5. 1D structures with fractal atomic surfaces .. .. .. .. Table 4.6.2.1. Expansion of the Fibonacci sequence Bn n
L L ! LS .. .. .. .. .. .. .. .. .. .. .. .. ..

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4.6.2. The n-dimensional description of aperiodic crystals

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471

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4.5.3.1. Is polymer electron crystallography possible? .. .. 4.5.3.2. Crystallization and data collection .. .. .. .. .. .. 4.5.3.3. Crystal structure analysis .. .. .. .. .. .. .. .. 4.5.3.4. Examples of crystal structure analyses .. .. .. .. Table 4.5.3.1. Structure analysis of poly- -methyl-l-glutamate

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487 487 489 490 493

4.6.3. Reciprocal-space images .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

494

491

4.6.3.1. Incommensurately modulated structures (IMSs) 4.6.3.1.1. Indexing .. .. .. .. .. .. .. .. 4.6.3.1.2. Diffraction symmetry .. .. .. .. .. 4.6.3.1.3. Structure factor .. .. .. .. .. ..

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494 495 495 496

4.6.3.2. Composite structures (CSs) .. 4.6.3.2.1. Indexing .. .. .. 4.6.3.2.2. Diffraction symmetry 4.6.3.2.3. Structure factor ..

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497 498 498 498

4.6.3.3. Quasiperiodic structures (QSs) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1. 3D structures with 1D quasiperiodic order .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1.1. Indexing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1.2. Diffraction symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1.3. Structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1.4. Intensity statistics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image .. 4.6.3.3.2. Decagonal phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.2.1. Indexing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.2.2. Diffraction symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.2.3. Structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.2.4. Intensity statistics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image .. 4.6.3.3.3. Icosahedral phases .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.3.1. Indexing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.3.2. Diffraction symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.3.3. Structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.3.4. Intensity statistics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image .. Table 4.6.3.1. 3D point groups of order k describing the diffraction symmetry and corresponding 5D decagonal space groups with reﬂection conditions (see Rabson et al., 1991) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.6.3.2. 3D point groups of order k describing the diffraction symmetry and corresponding 6D decagonal space groups with reﬂection conditions (see Levitov & Rhyner, 1988; Rokhsar et al., 1988) .. .. .. .. .. .. ..

498 498 499 499 500 501 501 503 505 505 506 507 508 509 511 512 512 513 514

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507 514

CONTENTS 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.6.4.1. Data-collection strategies .. .. .. .. .. .. .. 4.6.4.2. Commensurability versus incommensurability .. 4.6.4.3. Twinning and nanodomain structures .. .. .. .. Table 4.6.4.1. Intensity statistics of the Fibonacci chain 1 .. .. .. .. .. .. and 0 sin = 2 A˚ References

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516 517 517

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PART 5. DYNAMICAL THEORY AND ITS APPLICATIONS

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516

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534

5.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

534

5.1.2. Fundamentals of plane-wave dynamical theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

534

5.1. Dynamical theory of X-ray diffraction (A. Authier)

5.1.2.1. 5.1.2.2. 5.1.2.3. 5.1.2.4. 5.1.2.5. 5.1.2.6.

Propagation equation .. .. .. .. .. .. .. Waveﬁelds .. .. .. .. .. .. .. .. .. .. Boundary conditions at the entrance surface Fundamental equations of dynamical theory Dispersion surface .. .. .. .. .. .. .. Propagation direction .. .. .. .. .. ..

5.1.3. Solutions of plane-wave dynamical theory 5.1.3.1. 5.1.3.2. 5.1.3.3. 5.1.3.4. 5.1.3.5. 5.1.3.6. 5.1.3.7.

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534 535 536 536 536 537

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538

Departure from Bragg’s law of the incident wave .. .. .. .. .. .. Transmission and reﬂection geometries .. .. .. .. .. .. .. .. .. Middle of the reﬂection domain .. .. .. .. .. .. .. .. .. .. .. .. Deviation parameter .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Pendello¨sung and extinction distances .. .. .. .. .. .. .. .. .. Solution of the dynamical theory .. .. .. .. .. .. .. .. .. .. .. Geometrical interpretation of the solution in the zero-absorption case 5.1.3.7.1. Transmission geometry .. .. .. .. .. .. .. .. .. .. .. 5.1.3.7.2. Reﬂection geometry .. .. .. .. .. .. .. .. .. .. .. ..

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538 538 539 539 539 540 540 540 541

5.1.4. Standing waves .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

541

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541

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541

5.1.6. Intensities of plane waves in transmission geometry

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5.1.5. Anomalous absorption

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5.1.6.1. Absorption coefﬁcient .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.2. Boundary conditions for the amplitudes at the entrance surface – intensities of the reﬂected and refracted waves 5.1.6.3. Boundary conditions at the exit surface .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.3.1. Wavevectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.3.2. Amplitudes – Pendello¨sung .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.4. Reﬂecting power .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.5. Integrated intensity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.5.1. Non-absorbing crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.5.2. Absorbing crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.6.6. Thin crystals – comparison with geometrical theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. ..

5.1.7. Intensity of plane waves in reﬂection geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.7.1. Thick crystals .. .. .. .. .. .. 5.1.7.1.1. Non-absorbing crystals 5.1.7.1.2. Absorbing crystals .. 5.1.7.2. Thin crystals .. .. .. .. .. .. 5.1.7.2.1. Non-absorbing crystals 5.1.7.2.2. Absorbing crystals ..

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545 545 546 546 546 547

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

548

5.1.8.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.8.2. Borrmann triangle .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.1.8.3. Spherical-wave Pendello¨sung .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

548 548 549

Appendix 5.1.1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

550

A5.1.1.1. Dielectric susceptibility – classical derivation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. A5.1.1.2. Maxwell’s equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. A5.1.1.3. Propagation equation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

550 550 551

xxii

.. .. .. .. .. ..

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.. .. .. .. .. ..

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.. .. .. .. .. ..

.. .. .. .. .. ..

.. .. .. .. .. ..

.. .. .. .. .. ..

.. .. .. .. .. ..

.. .. .. .. .. ..

545

.. .. .. .. .. ..

5.1.8. Real waves

.. .. .. .. .. ..

541 542 542 542 543 543 544 544 545 545

CONTENTS A5.1.1.4. Poynting vector .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

551

5.2. Dynamical theory of electron diffraction (A. F. Moodie, J. M. Cowley and P. Goodman) .. .. .. .. .. .. .. .. .. .. .. ..

552

5.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

552

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

552

5.2.3. Forward scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

552

5.2.4. Evolution operator .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

552

5.2.5. Projection approximation – real-space solution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

553

5.2.6. Semi-reciprocal space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

553

5.2.2. The deﬁning equations

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

553

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

554

5.2.9. Translational invariance .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

554

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

555

5.2.7. Two-beam approximation 5.2.8. Eigenvalue approach

5.2.10. Bloch-wave formulations

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

555

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

555

5.2.13. Born series .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

555

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

556

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

557

5.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

557

5.3.2. Comparison between X-rays and neutrons with spin neglected .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

557

5.2.11. Dispersion surfaces 5.2.12. Multislice

5.2.14. Approximations

5.3. Dynamical theory of neutron diffraction (M. Schlenker and J.-P. Guigay)

5.3.2.1. 5.3.2.2. 5.3.2.3. 5.3.2.4. 5.3.2.5.

The neutron and its interactions .. .. .. .. .. .. .. Scattering lengths and refractive index .. .. .. .. .. Absorption .. .. .. .. .. .. .. .. .. .. .. .. .. .. Differences between neutron and X-ray scattering .. .. Translating X-ray dynamical theory into the neutron case

5.3.3. Neutron spin, and diffraction by perfect magnetic crystals 5.3.3.1. 5.3.3.2. 5.3.3.3. 5.3.3.4. 5.3.3.5.

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

557 557 558 558 558

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

558

Polarization of a neutron beam and the Larmor precession in a uniform magnetic ﬁeld .. Magnetic scattering by a single ion having unpaired electrons .. .. .. .. .. .. .. .. Dynamical theory in the case of perfect ferromagnetic or collinear ferrimagnetic crystals The dynamical theory in the case of perfect collinear antiferromagnetic crystals .. .. .. The ﬂipping ratio .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

558 559 560 561 561

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

561

5.3.5. Effect of external ﬁelds on neutron scattering by perfect crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

562

5.3.4. Extinction in neutron diffraction (non-magnetic case)

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

5.3.6. Experimental tests of the dynamical theory of neutron scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

562

5.3.7. Applications of the dynamical theory of neutron scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

563

5.3.7.1. 5.3.7.2. 5.3.7.3. 5.3.7.4.

.. .. .. ..

563 563 563 564

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

565

Author index .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

571

Subject index .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

580

References

Neutron optics .. .. .. .. .. .. .. .. .. .. .. .. .. .. Measurement of scattering lengths by Pendello¨sung effects Neutron interferometry .. .. .. .. .. .. .. .. .. .. .. Neutron diffraction topography and other imaging methods

xxiii

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Preface By Uri Shmueli The purpose of Volume B of International Tables for Crystallography is to provide the user or reader with accounts of some well established topics, of importance to the science of crystallography, which are related in one way or another to the concepts of reciprocal lattice and, more generally, reciprocal space. Efforts have been made to extend the treatment of the various topics to include X-ray, electron, and neutron diffraction techniques, and thereby do some justice to the inclusion of the present Volume in the new series of International Tables for Crystallography. An important crystallographic aspect of symmetry in reciprocal space, space-group-dependent expressions of trigonometric structure factors, already appears in Volume I of International Tables for X-ray Crystallography, and preliminary plans for incorporating this and other crystallographic aspects of reciprocal space in the new edition of International Tables date back to 1972. However, work on a volume of International Tables for Crystallography, largely dedicated to the subject of reciprocal space, began over ten years later. The present structure of Volume B, as determined in the years preceding the 1984 Hamburg congress of the International Union of Crystallography (IUCr), is due to (i) computer-controlled production of concise structure-factor tables, (ii) the ability to introduce many more aspects of reciprocal space – as a result of reducing the effort of producing the above tables, as well as their volume, and (iii) suggestions by the National Committees and individual crystallographers of some additional interesting topics. It should be pointed out that the initial plans for the present Volume and Volume C (Mathematical, Physical and Chemical Tables, edited by Professor A. J. C. Wilson), were formulated and approved during the same period.

The obviously delayed publication of Volume B is due to several reasons. Some minor delays were caused by a requirement that potential contributors should be approved by the Executive Committee prior to issuing relevant invitations. Much more serious delays were caused by authors who failed to deliver their contributions. In fact, some invited contributions had to be excluded from this ﬁrst edition of Volume B. Some of the topics here treated are greatly extended, considerably updated or modern versions of similar topics previously treated in the old Volumes I, II, and IV. Most of the subjects treated in Volume B are new to International Tables. I gratefully thank Professor A. J. C. Wilson, for suggesting that I edit this Volume and for sharing with me his rich editorial experience. I am indebted to those authors of Volume B who took my requests and deadlines seriously, and to the Computing Center of Tel Aviv University for computing facilities and time. Special thanks are due to Mrs Z. Stein (Tel Aviv University) for skilful assistance in numeric and symbolic programming, involved in my contributions to this Volume. I am most grateful to many colleagues–crystallographers for encouragement, advice, and suggestions. In particular, thanks are due to Professors J. M. Cowley, P. Goodman and C. J. Humphreys, who served as Chairmen of the Commission on Electron Diffraction during the preparation of this Volume, for prompt and expert help at all stages of the editing. The kind assistance of Dr J. N. King, the Executive Secretary of the IUCr, is also gratefully acknowledged. Last, but certainly not least, I wish to thank Mr M. H. Dacombe, the Technical Editor of the IUCr, and his staff for the skilful and competent treatment of the variety of drafts and proofs out of which this Volume arose.

Preface to the second edition By Uri Shmueli The ﬁrst edition of Volume B appeared in 1993, and was followed by a corrected reprint in 1996. Although practically all the material for the second edition was available in early 1997, its publication was delayed by the decision to translate all of Volume B, and indeed all the other volumes of International Tables for Crystallography, to Standard Generalized Markup Language (SGML) and thus make them available also in an electronic form suitable for modern publishing procedures. During the preparation of the second edition, most chapters that appeared in the ﬁrst edition have been corrected and/or revised, some were rather extensively updated, and ﬁve new chapters were added. The overall structure of the second edition is outlined below. After an introductory chapter, Part 1 presents the reader with an account of structure-factor formalisms, an extensive treatment of the theory, algorithms and crystallographic applications of Fourier methods, and treatments of symmetry in reciprocal space. These are here enriched with more advanced aspects of representations of space groups in reciprocal space. In Part 2, these general accounts are followed by detailed expositions of 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. The latter topic is here enhanced by applications of direct methods to electron crystallography. Part 3, Dual Bases in Crystallographic Computing, deals with applications of reciprocal space to molecular geometry and ‘best’plane calculations, and contains a treatment of the principles of molecular graphics and modelling and their applications; it concludes with the presentation of a convergence-acceleration method, of importance in the computation of approximate lattice sums. 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. The new additions to this part are treatments of polymer crystallography and of reciprocal-space images of aperiodic crystals. Part 5 contains introductory treatments of the theory of the interaction of radiation with matter, the so-called dynamical theory, as applied to X-ray, electron and neutron diffraction techniques. The chapter on the dynamical theory of neutron diffraction is new. I am deeply grateful to the authors of the new contributions for making their expertise available to Volume B and for their

xxv

PREFACE excellent collaboration. I also take special pleasure in thanking those authors of the ﬁrst edition who revised and updated their contributions in view of recent developments. Last but not least, I wish to thank all the authors for their contributions and their patience, and am grateful to those authors who took my requests seriously. I hope that the updating and revision of future editions will be much easier and more expedient, mainly because of the new format of International Tables. Four friends and greatly respected colleagues who contributed to the second edition of Volume B are no longer with us. These are Professors Arthur J. C. Wilson, Peter Goodman, Verner Schomaker and Boris K. Vainshtein. I asked Professors Michiyoshi Tanaka, John Cowley and Douglas Dorset if they were prepared to answer queries related to the contributions of the late Peter Goodman and Boris K. Vainshtein to Chapter 2.5. I am most grateful for their prompt agreement.

This editorial work was carried out at the School of Chemistry and the Computing Center of Tel Aviv University. The facilities they put at my disposal are gratefully acknowledged on my behalf and on behalf of the IUCr. I wish to thank many colleagues for interesting conversations and advice, and in particular Professor Theo Hahn with whom I discussed at length problems regarding Volume B and International Tables in general. Given all these expert contributions, the publication of this volume would not have been possible without the expertise and devotion of the Technical Editors of the IUCr. My thanks go to Mrs Sue King, for her cooperation during the early stages of the work on the second edition of Volume B, while the material was being collected, and to Dr Nicola Ashcroft, for her collaboration during the ﬁnal stages of the production of the volume, for her most careful and competent treatment of the proofs, and last but not least for her tactful and friendly attitude.

xxvi

International Tables for Crystallography (2006). Vol. B, Chapter 1.1, pp. 2–9.

1.1. Reciprocal space in crystallography BY U. SHMUELI where h, k and l are relatively prime integers (i.e. not having a common factor other than 1 or 1), known as Miller indices of the lattice plane, x, y and z are the coordinates of any point lying in the plane and are expressed as fractions of the magnitudes of the basis vectors a, b and c of the direct lattice, respectively, and n is an integer denoting the serial number of the lattice plane within the family of parallel and equidistant
hkl planes, the interplanar spacing being denoted by dhkl ; the value n 0 corresponds to the
hkl plane passing through the origin. Let r xa yb zc and rL ua vb wc, where u, v, w are any integers, denote the position vectors of the point xyz and a lattice point uvw lying in the plane (1.1.2.3), respectively, and assume that r and rL are different vectors. If the plane normal is denoted by N, where N is proportional to the vector product of two in-plane lattice vectors, the vector form of the equation of the lattice plane becomes

1.1.1. Introduction The purpose of this chapter is to provide an introduction to several aspects of reciprocal space, which are of general importance in crystallography and which appear in the various chapters and sections to follow. We ﬁrst summarize the basic deﬁnitions and brieﬂy inspect some fundamental aspects of crystallography, while recalling that they can be usefully and simply discussed in terms of the concept of the reciprocal lattice. This introductory section is followed by a summary of the basic relationships between the direct and associated reciprocal lattices. We then introduce the elements of tensor-algebraic formulation of such dual relationships, with emphasis on those that are important in many applications of reciprocal space to crystallographic algorithms. We proceed with a section that demonstrates the role of mutually reciprocal bases in transformations of coordinates and conclude with a brief outline of some important analytical aspects of reciprocal space, most of which are further developed in other parts of this volume.

N
r

The notion of mutually reciprocal triads of vectors dates back to the introduction of vector calculus by J. Willard Gibbs in the 1880s (e.g. Wilson, 1901). This concept appeared to be useful in the early interpretations of diffraction from single crystals (Ewald, 1913; Laue, 1914) and its ﬁrst detailed exposition and the recognition of its importance in crystallography can be found in Ewald’s (1921) article. The following free translation of Ewald’s (1921) introduction, presented in a somewhat different notation, may serve the purpose of this section: To the set of ai , there corresponds in the vector calculus a set of ‘reciprocal vectors’ bi , which are deﬁned (by Gibbs) by the following properties:
1121

ai bi 1,

1122

s

h a h,

where i and k may each equal 1, 2 or 3. The ﬁrst equation, (1.1.2.1), says that each vector bk is perpendicular to two vectors ai , as follows from the vanishing scalar products. Equation (1.1.2.2) provides the norm of the vector bi : the length of this vector must be chosen such that the projection of bi on the direction of ai has the length 1ai , where ai is the magnitude of the vector ai . . ..

1125

h b k,

h c l,

1126

where h s s0 is the diffraction vector, and h, k and l are integers corresponding to orders of diffraction from the three-dimensional lattice (Lipson & Cochran, 1966). The diffraction vector thus has to satisfy a condition that is analogous to that imposed on the normal to a lattice plane. The next relevant aspect to be commented on is the Fourier expansion of a function having the periodicity of the crystal lattice. Such functions are e.g. the electron density, the density of nuclear matter and the electrostatic potential in the crystal, which are the operative deﬁnitions of crystal structure in X-ray, neutron and electron-diffraction methods of crystal structure determination. A Fourier expansion of such a periodic function may be thought of as a superposition of waves (e.g. Buerger, 1959), with wavevectors related to the interplanar spacings dhkl , in the crystal lattice. Denoting the wavevector of a Fourier wave by g (a function of hkl), the phase of the Fourier wave at the point r in the crystal is given by 2g r, and the triple Fourier series corresponding to the expansion of the periodic function, say G(r), can be written as G
r C
g exp
2ig r,
1127

The consequences of equations (1.1.2.1) and (1.1.2.2) were elaborated by Ewald (1921) and are very well documented in the subsequent literature, crystallographic as well as other. As is well known, the reciprocal lattice occupies a rather prominent position in crystallography and there are nearly as many accounts of its importance as there are crystallographic texts. It is not intended to review its applications, in any detail, in the present section; this is done in the remaining chapters and sections of the present volume. It seems desirable, however, to mention by way of an introduction some fundamental geometrical, physical and mathematical aspects of crystallography, and try to give a uniﬁed demonstration of the usefulness of mutually reciprocal bases as an interpretive tool. Consider the equation of a lattice plane in the direct lattice. It is shown in standard textbooks (e.g. Buerger, 1941) that this equation is given by

g

1123

where C(g) are the amplitudes of the Fourier waves, or Fourier

2 Copyright 2006 International Union of Crystallography

s0 rL n,

where s0 and s are the wavevectors of the incident and scattered beams, respectively, and n is an arbitrary integer. Since rL ua vb wc, where u, v and w are unrestricted integers, equation (1.1.2.5) is equivalent to the equations of Laue:

and

hx ky lz n,

1124

For equations (1.1.2.3) and (1.1.2.4) to be identical, the plane normal N must satisfy the requirement that N rL n, where n is an (unrestricted) integer. Let us now consider the basic diffraction relations (e.g. Lipson & Cochran, 1966). Suppose a parallel beam of monochromatic radiation, of wavelength , falls on a lattice of identical point scatterers. If it is assumed that the scattering is elastic, i.e. there is no change of the wavelength during this process, the wavevectors of the incident and scattered radiation have the same magnitude, which can conveniently be taken as 1. A consideration of path and phase differences between the waves outgoing from two point scatterers separated by the lattice vector rL (deﬁned as above) shows that the condition for their maximum constructive interference is given by

1.1.2. Reciprocal lattice in crystallography

ai bk 0
for i 6 k

rL 0 or N r N rL

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY We shall, in what follows, abandon all the temporary notation used above and write the reciprocal-lattice vector as

coefﬁcients, which are related to the experimental data. Numerous examples of such expansions appear throughout this volume. The permissible wavevectors in the above expansion are restricted by the periodicity of the function G(r). Since, by deﬁnition, G
r G
r rL , where rL is a direct-lattice vector, the right-hand side of (1.1.2.7) must remain unchanged when r is replaced by r rL . This, however, can be true only if the scalar product g rL is an integer. Each of the above three aspects of crystallography may lead, independently, to a useful introduction of the reciprocal vectors, and there are many examples of this in the literature. It is interesting, however, to consider the representation of the equation v rL n,

h ha kb lc or h h 1 a 1 h2 a 2 h3 a 3

or, in matrix notation, A u
UVW B
abc v n, C w u Aa Ab Ac v n
UVW B a B b B c Ca Cb Cc w

hi ai ,

11213

and denote the direct-lattice vectors by rL ua vb wc, as above, or by r L u 1 a1 u 2 a2 u3 a3

1128

3

ui ai

11214

i1

The representations (1.1.2.13) and (1.1.2.14) are used in the tensoralgebraic formulation of the relationships between mutually reciprocal bases (see Section 1.1.4 below).

1.1.3. Fundamental relationships We now present a brief derivation and a summary of the most important relationships between the direct and the reciprocal bases. The usual conventions of vector algebra are observed and the results are presented in the conventional crystallographic notation. Equations (1.1.2.1) and (1.1.2.2) now become

1129

a b a c b a b c c a c b 0

11210

1131

and a a b b c c 1,

or

3

i1

which is common to all three, in its most convenient form. Obviously, the vector v which stands for the plane normal, the diffraction vector, and the wavevector in a Fourier expansion, may still be referred to any permissible basis and so may rL , by an appropriate transformation. Let v UA V B W C, where A, B and C are linearly independent vectors. Equation (1.1.2.8) can then be written as
UA V B W C
ua vb wc n,

11212

1132

respectively, and the relationships are obtained as follows.
11211 1.1.3.1. Basis vectors

The simplest representation of equation (1.1.2.8) results when the matrix of scalar products in (1.1.2.11) reduces to a unit matrix. This can be achieved (i) by choosing the basis vectors ABC to be orthonormal to the basis vectors abc, while requiring that the components of rL be integers, or (ii) by requiring that the bases ABC and abc coincide with the same orthonormal basis, i.e. expressing both v and rL , in (1.1.2.8), in the same Cartesian system. If we choose the ﬁrst alternative, it is seen that: (1) The components of the vector v, and hence those of N, h and g, are of necessity integers, since u, v and w are already integral. The components of v include Miller indices; in the case of the lattice plane, they coincide with the orders of diffraction from a threedimensional lattice of scatterers, and correspond to the summation indices in the triple Fourier series (1.1.2.7). (2) The basis vectors A, B and C are reciprocal to a, b and c, as can be seen by comparing the scalar products in (1.1.2.11) with those in (1.1.2.1) and (1.1.2.2). In fact, the bases ABC and abc are mutually reciprocal. Since there are no restrictions on the integers U, V and W, the vector v belongs to a lattice which, on account of its basis, is called the reciprocal lattice. It follows that, at least in the present case, algebraic simplicity goes together with ease of interpretation, which certainly accounts for much of the importance of the reciprocal lattice in crystallography. The second alternative of reducing the matrix in (1.1.2.11) to a unit matrix, a transformation of (1.1.2.8) to a Cartesian system, leads to non-integral components of the vectors, which makes any interpretation of v or rL much less transparent. However, transformations to Cartesian systems are often very useful in crystallographic computing and will be discussed below (see also Chapters 2.3 and 3.3 in this volume).

It is seen from (1.1.3.1) that a must be proportional to the vector product of b and c, a K
b c, and, since a a 1, the proportionality constant K equals 1a
b c. The mixed product a
b c can be interpreted as the positive volume of the unit cell in the direct lattice only if a, b and c form a right-handed set. If the above condition is fulﬁlled, we obtain bc ca ab a , b , c
1133 V V V and analogously b c c a a b , b , c ,
1134 a V V V where V and V are the volumes of the unit cells in the associated direct and reciprocal lattices, respectively. Use has been made of the fact that the mixed product, say a
b c, remains unchanged under cyclic rearrangement of the vectors that appear in it. 1.1.3.2. Volumes The reciprocal relationship of V and V follows readily. We have from equations (1.1.3.2), (1.1.3.3) and (1.1.3.4)
a b
a b 1 VV If we make use of the vector identity c c

3

1. GENERAL RELATIONSHIPS AND TECHNIQUES
A B
C D
A C
B D
A D
B C,
1135 aa ab ac G ba bb bc
11311 and equations (1.1.3.1) and (1.1.3.2), it is seen that V 1V . ca cb cc 2 ab cos ac cos a 1.1.3.3. Angular relationships
11312 ba cos bc cos b2 The relationships of the angles , , between the pairs of 2 ca cos cb cos c vectors (b, c), (c, a) and (a, b), respectively, and the angles , , between the corresponding pairs of reciprocal basis vectors, can be obtained by simple vector algebra. For example, we This is the matrix of the metric tensor of the direct basis, or brieﬂy the direct metric. The corresponding reciprocal metric is given by have from (1.1.3.3): (i) b c b c cos , with a a a b a c ca sin ab sin G b a b b b c
11313 b and c V V c a c b c c and (ii) a b cos a c cos a2 b a cos b2 b c cos
11314
c a
a b b c c a cos c b cos c2 V2 If we make use of the identity (1.1.3.5), and compare the two The matrices G and G are of fundamental importance in crystallographic computations and transformations of basis vectors expressions for b c , we readily obtain and coordinates from direct to reciprocal space and vice versa. cos cos cos cos
1136 Examples of applications are presented in Part 3 of this volume and in the remaining sections of this chapter. sin sin It can be shown (e.g. Buerger, 1941) that the determinants of G Similarly, and G equal the squared volumes of the direct and reciprocal unit cells, respectively. Thus, cos cos cos cos
1137 sin sin det
G a
b c2 V 2
11315 and

and cos

cos cos cos sin sin

det
G a
b c 2 V 2 ,

1138

and a direct expansion of the determinants, from (1.1.3.12) and (1.1.3.14), leads to

The expressions for the cosines of the direct angles in terms of those of the reciprocal ones are analogous to (1.1.3.6)–(1.1.3.8). For example, cos

11316

V abc
1

cos cos cos sin sin

cos2

cos2

cos2

2 cos cos cos 12

11317

and V a b c
1

1.1.3.4. Matrices of metric tensors

1139

11310

where

and

x x y , z

cos2
11318

The following algorithm has been found useful in computational applications of the above relationships to calculations in reciprocal space (e.g. data reduction) and in direct space (e.g. crystal geometry). (1) Input the direct unit-cell parameters and construct the matrix of the metric tensor [cf. equation (1.1.3.12)]. (2) Compute the determinant of the matrix G and ﬁnd the inverse matrix, G 1 ; this inverse matrix is just G , the matrix of the metric tensor of the reciprocal basis (see also Section 1.1.4 below). (3) Use the elements of G , and equation (1.1.3.14), to obtain the parameters of the reciprocal unit cell. The direct and reciprocal sets of unit-cell parameters, as well as the corresponding metric tensors, are now available for further calculations. Explicit relations between direct- and reciprocal-lattice parameters, valid for the various crystal systems, are given in most textbooks on crystallography [see also Chapter 1.1 of Volume C (Koch, 1999)].

and can be written in matrix form as jrj xT Gx12 ,

cos2

2 cos cos cos 12

Various computational and algebraic aspects of mutually reciprocal bases are most conveniently expressed in terms of the metric tensors of these bases. The tensors will be treated in some detail in the next section, and only the deﬁnitions of their matrices are given and interpreted below. Consider the length of the vector r xa yb zc. This is given by jrj
xa yb zc
xa yb zc12

cos2

xT
xyz

4

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY where km is the Kronecker symbol which equals 1 when k m and equals zero if k 6 m, and by comparison with (1.1.4.2) we have

1.1.4. Tensor-algebraic formulation The present section summarizes the tensor-algebraic properties of mutually reciprocal sets of basis vectors, which are of importance in the various aspects of crystallography. This is not intended to be a systematic treatment of tensor algebra; for more thorough expositions of the subject the reader is referred to relevant crystallographic texts (e.g. Patterson, 1967; Sands, 1982), and other texts in the physical and mathematical literature that deal with tensor algebra and analysis. Let us ﬁrst recall that symbolic vector and matrix notations, in which basis vectors and coordinates do not appear explicitly, are often helpful in qualitative considerations. If, however, an expression has to be evaluated, the various quantities appearing in it must be presented in component form. One of the best ways to achieve a concise presentation of geometrical expressions in component form, while retaining much of their ‘transparent’ symbolic character, is their tensor-algebraic formulation.

xm x0k Tkm ,

where Tkm a0k am is an element of the required transformation matrix. Of course, the same transformation could have been written as xm Tkm x0k ,

xm xn Tpm Tqn x0p x0q ;

2

Qmn Tpm Tqn Q0pq 1.1.4.3. Scalar products

Subscripted quantities are associated in tensor algebra with covariant, and superscripted with contravariant transformation properties. Thus the basis vectors of the direct lattice are represented as covariant quantities and those of the reciprocal lattice as contravariant ones. (ii) Summation convention: if an index appears twice in an expression, once as subscript and once as superscript, a summation over this index is thereby implied and the summation sign is omitted. For example, i x Tij x j will be written xi Tij x j

The expression for the scalar product of two vectors, say u and v, depends on the bases to which the vectors are referred. If we admit only the covariant and contravariant bases deﬁned above, we have four possible types of expression:
I u ui ai , v vi ai u v ui v j
ai aj ui v j gij , i

II u ui a , v vi a i

III u u ai , v vi a

since both i and j conform to the convention. Such repeating indices are often called dummy indices. The implied summation over repeating indices is also often used even when the indices are at the same level and the coordinate system is Cartesian; there is no distinction between contravariant and covariant quantities in Cartesian frames of reference (see Chapter 3.3). (iii) Components (coordinates) of vectors referred to the covariant basis are written as contravariant quantities, and vice versa. For example,

1149

i

IV u ui a , v v ai u v ui v j
ai aj ui v j ji ui vi

11410

(i) The sets of scalar products gij ai aj (1.1.4.7) and gij a a j (1.1.4.8) are known as the metric tensors of the covariant (direct) and contravariant (reciprocal) bases, respectively; the corresponding matrices are presented in conventional notation in equations (1.1.3.11) and (1.1.3.13). Numerous applications of these tensors to the computation of distances and angles in crystals are given in Chapter 3.1. (ii) Equations (1.1.4.7) to (1.1.4.10) furnish the relationships between the covariant and contravariant components of the same vector. Thus, comparing (1.1.4.7) and (1.1.4.9), we have i

r xa yb zc x1 a1 x2 a2 x3 a3 xi ai h ha kb lc h1 a1 h2 a2 h3 a3 hi ai 1.1.4.2. Transformations A familiar concept but a fundamental one in tensor algebra is the transformation of coordinates. For example, suppose that an atomic position vector is referred to two unit-cell settings as follows:

vi v j gij

11411

Similarly, using (1.1.4.8) and (1.1.4.10) we obtain the inverse relationship

1141

and

vi vj gij r x0k a0k

1148

i

u v ui vj
ai a j ui vj ij ui vi , i

1147

i

u v ui vj
ai a j ui vj gij ,

j

r x k ak

1146

3

a a ,b a ,c a

i

1145

the same transformation law applies to the components of a contravariant tensor of rank two, the components of which are referred to the primed basis and are to be transformed to the unprimed one:

We shall adhere to the following conventions: (i) Notation for direct and reciprocal basis vectors: a a1 , b a2 , c a3 1

1144

where Tkm am a0k . A tensor is a quantity that transforms as the product of coordinates, and the rank of a tensor is the number of transformations involved (Patterson, 1967; Sands, 1982). E.g. the product of two coordinates, as in the above example, transforms from the a0 basis to the a basis as

1.1.4.1. Conventions

1143

1142

11412

The corresponding relationships between covariant and contravariant bases can now be obtained if we refer a vector, say v, to each of the bases

Let us multiply both sides of (1.1.4.1) and (1.1.4.2), on the right, by the vectors am , m = 1, 2, or 3, i.e. by the reciprocal vectors to the basis a1 a2 a3 . We obtain from (1.1.4.1)

v v i ai v k ak ,

xk ak am xk km xm ,

and make use of (1.1.4.11) and (1.1.4.12). Thus, e.g.,

5

1. GENERAL RELATIONSHIPS AND TECHNIQUES vi ai
vk gik ai vk ak Hence ak gik ai

11413

ak gik ai

11414

and, similarly, ij

(iii) The tensors gij and g are symmetric, by deﬁnition. (iv) It follows from (1.1.4.11) and (1.1.4.12) or (1.1.4.13) and (1.1.4.14) that the matrices of the direct and reciprocal metric tensors are mutually inverse, i.e. 1 11 12 13 g11 g12 g13 g g g g21 g22 g23 g21 g22 g23 ,
11415 31 32 33 g31 g32 g33 g g g

and their determinants are mutually reciprocal. 1.1.4.4. Examples

There are numerous applications of tensor notation in crystallographic calculations, and many of them appear in the various chapters of this volume. We shall therefore present only a few examples. (i) The (squared) magnitude of the diffraction vector h hi ai is given by 4 sin2 jhj hi hj gij 2 2

Fig. 1.1.4.1. Derivation of the general expression for the rotation operator. The ﬁgure illustrates schematically the decompositions and other simple geometrical considerations required for the derivation outlined in equations (1.1.4.22)–(1.1.4.28).

This is a typical application of reciprocal space to ordinary directspace computations. (iv) We wish to derive a tensor formulation of the vector product, along similar lines to those of Chapter 3.1. As with the scalar product, there are several such formulations and we choose that which has both vectors, say u and v, and the resulting product, u v, referred to a covariant basis. We have

11416

This concise relationship is a starting point in a derivation of unitcell parameters from experimental data. (ii) The structure factor, including explicitly anisotropic displacement tensors, can be written in symbolic matrix notation as F
h

N

j1

T

T

f
i exp
h
j h exp
2ih r
j ,

u v ui ai v j aj

11417

ui v j
ai aj

If we make use of the relationships (1.1.3.3) between the direct and reciprocal basis vectors, it can be veriﬁed that

where
j is the matrix of the anisotropic displacement tensor of the jth atom. In tensor notation, with the quantities referred to their natural bases, the structure factor can be written as F
h1 h2 h3

N

j1

f
j exp
hi hk
ikj exp
2ihi xi
j ,

ai aj V ekij ak ,

11420

where V is the volume of the unit cell and the antisymmetric tensor ekij equals 1, 1, or 0 according as kij is an even permutation of 123, an odd permutation of 123 or any two of the indices kij have the same value, respectively. We thus have

11418

and similarly concise expressions can be written for the derivatives of the structure factor with respect to the positional and displacement parameters. The summation convention applies only to indices denoting components of vectors and tensors; the atom subscript j in (1.1.4.18) clearly does not qualify, and to indicate this it has been surrounded by parentheses. (iii) Geometrical calculations, such as those described in the chapters of Part 3, may be carried out in any convenient basis but there are often some deﬁnite advantages to computations that are referred to the natural, non-Cartesian bases (see Chapter 3.1). Usually, the output positional parameters from structure reﬁnement are available as contravariant components of the atomic position vectors. If we transform them by (1.1.4.11) to their covariant form, and store these covariant components of the atomic position vectors, the computation of scalar products using equations (1.1.4.9) or (1.1.4.10) is almost as efﬁcient as it would be if the coordinates were referred to a Cartesian system. For example, the right-hand side of the vector identity (1.1.3.5), which is employed in the computation of dihedral angles, can be written as
Ai C i
Bk Dk

11419

u v V ekij ui v j ak Vglk ekij ui v j al , k

lk

11421

since by (1.1.4.13), a g al . (v) The rotation operator. The general formulation of an expression for the rotation operator is of interest in crystal structure determination by Patterson techniques (see Chapter 2.3) and in molecular modelling (see Chapter 3.3), and another well known crystallographic application of this device is the derivation of the translation, libration and screw-motion tensors by the method of Schomaker & Trueblood (1968), discussed in Part 8 of Volume C (IT C, 1999) and in Chapter 1.2 of this volume. A digression on an elementary derivation of the above seems to be worthwhile. Suppose we wish to rotate the vector r, about an axis coinciding with the unit vector k, through the angle and in the positive sense, i.e. an observer looking in the direction of k will see r rotating in the clockwise sense. The vectors r, k and the rotated (target) vector r0 are referred to an origin on the axis of rotation (see Fig. 1.1.4.1). Our purpose is to express r0 in terms of r, k and by a general vector

Ai Di
Bk C k

6

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY If all the vectors are referred to a Cartesian basis, that is three orthogonal unit vectors, the direct and reciprocal metric tensors reduce to a unit tensor, there is no difference between covariant and contravariant quantities, and equation (1.1.4.31) reduces to

formula, and represent the components of the rotated vectors in coordinate systems that might be of interest. Let us decompose the vector r and the (target) vector r0 into their components which are parallel
k and perpendicular
? to the axis of rotation: r rk r?

11422

r0 r0k r0?

11423

R ij ki kj
1

11432

where all the indices have been taken as subscripts, but the summation convention is still observed. The relative simplicity of (1.1.4.32), as compared to (1.1.4.31), often justiﬁes the transformation of all the vector quantities to a Cartesian basis. This is certainly the case for any extensive calculation in which covariances of the structural parameters are not considered.

and

It can be seen from Fig. 1.1.4.1 that the parallel components of r and r0 are rk r0k k
k r

cos ij cos eipj kp sin ,

11424 1.1.5. Transformations

and thus r? r

k
k r

1.1.5.1. Transformations of coordinates

11425

It happens rather frequently that a vector referred to a given basis has to be re-expressed in terms of another basis, and it is then required to ﬁnd the relationship between the components (coordinates) of the vector in the two bases. Such situations have already been indicated in the previous section. The purpose of the present section is to give a general method of ﬁnding such relationships (transformations), and discuss some simpliﬁcations brought about by the use of mutually reciprocal and Cartesian bases. We do not assume anything about the bases, in the general treatment, and hence the tensor formulation of Section 1.1.4 is not appropriate at this stage. Let

Only a suitable expression for r0? is missing. We can ﬁnd this by decomposing r0? into its components (i) parallel to r? and (ii) parallel to k r? . We have, as in (1.1.4.24), r? r? 0 k r? k r? 0 0 r? r r
11426 jr? j jr? j ? jk r? j jk r? j ? We observe, using Fig. 1.1.4.1, that jr0? j jr? j jk r? j and k r? k r,

r

and, further, and

and r

r0?
k r? k
r0? r? jr? j2 sin ,

11427

uk
1Gkl
12 uk
2Gkl
22,

l 1, 2, 3
1153

l 1, 2, 3,

1154

where Gkl
12 ck
1 cl
2 and Gkl
22 ck
2 cl
2. Similarly, if we choose the basis vectors cl
1, l = 1, 2, 3, as the multipliers of (1.1.5.1) and (1.1.5.2), we obtain uk
1Gkl
11 uk
2Gkl
21,

l 1, 2, 3,

1155

where Gkl
11 ck
1 cl
1 and Gkl
21 ck
2 cl
1. Rewriting (1.1.5.4) and (1.1.5.5) in symbolic matrix notation, we have

11429

uT
1G
12 uT
2G
22,

or brieﬂy

1156

leading to
11430

uT
1 uT
2fG
22G
12 1 g

where R ij k i kj
1

1152

or

11428

cos ji x j cos Vgim empj k p x j sin ,

x0i R ij x j ,

uj
2cj
2

uk
1ck
1 cl
2 uk
2ck
2 cl
2,

The above general expression can be written as a linear transformation by referring the vectors to an appropriate basis or bases. We choose here r x j aj , r0 x0i ai and assume that the components of k are available in the direct and reciprocal bases. If we make use of equations (1.1.4.9) and (1.1.4.21), (1.1.4.28) can be written as x0i k i
k j x j
1

3

be the given and required representations of the vector r, respectively. Upon the formation of scalar products of equations (1.1.5.1) and (1.1.5.2) with the vectors of the second basis, and employing again the summation convention, we obtain

and equations (1.1.4.23), (1.1.4.25) and (1.1.4.27) lead to the required result cos r cos
k r sin

1151

j1

since the unit vector k is perpendicular to the plane containing the vectors r? and r0? . Equation (1.1.4.26) now reduces to

r0 k
k r
1

uj
1cj
1

j1

r0? r? jr? j2 cos

r0? r? cos
k r sin

3

and cos ji cos Vgim empj k p sin

11431

uT
2 uT
1fG
12G
22 1 g,

is a matrix element of the rotation operator R which carries the vector r into the vector r0 . Of course, the representation (1.1.4.31) of R depends on our choice of reference bases.

1157

and uT
1G
11 uT
2G
21,

7

1158

1. GENERAL RELATIONSHIPS AND TECHNIQUES X 1
X axis of the Cartesian system thus coincides with a directlattice vector, and the X 2
Y axis is parallel to a vector in the reciprocal lattice. Since the basis in (1.1.5.12) is a Cartesian one, the required transformations are given by equations (1.1.5.10) as

leading to uT
1 uT
2fG
21G
11 1 g and uT
2 uT
1fG
11G
21 1 g
1159 Equations (1.1.5.7) and (1.1.5.9) are symbolic general expressions for the transformation of the coordinates of r from one representation to the other. In the general case, therefore, we require the matrices of scalar products of the basis vectors, G(12) and G(22) or G(11) and G(21) – depending on whether the basis ck
2 or ck
1, k = 1, 2, 3, was chosen to multiply scalarly equations (1.1.5.1) and (1.1.5.2). Note, however, the following simpliﬁcations. (i) If the bases ck
1 and ck
2 are mutually reciprocal, each of the matrices of mixed scalar products, G(12) and G(21), reduces to a unit matrix. In this important special case, the transformation is effected by the matrices of the metric tensors of the bases in question. This can be readily seen from equations (1.1.5.7) and (1.1.5.9), which then reduce to the relationships between the covariant and contravariant components of the same vector [see equations (1.1.4.11) and (1.1.4.12) above]. (ii) If one of the bases, say ck
2, is Cartesian, its metric tensor is by deﬁnition a unit tensor, and the transformations in (1.1.5.7) reduce to uT
1 uT
2G
12

xi X k
T 1 ik and X i xk Tki ,

11513

Tki

where ak ei , k, i = 1, 2, 3, form the matrix of the scalar products. If we make use of the relationships between covariant and contravariant basis vectors, and the tensor formulation of the vector product, given in Section 1.1.4 above (see also Chapter 3.1), we obtain 1 Tk1 gki ui jrL j 1 Tk2 hk
11514 jr j V Tk3 ekip ui gpl hl jrL jjr j Note that the other convenient choice, e1 / r and e2 / rL , interchanges the ﬁrst two columns of the matrix T in (1.1.5.14) and leads to a change of the signs of the elements in the third column. This can be done by writing ekpi instead of ekip , while leaving the rest of Tk3 unchanged.

1

1.1.6. Some analytical aspects of the reciprocal space

and uT
2 uT
1G
12

1.1.6.1. Continuous Fourier transform

11510

Of great interest in crystallographic analyses are Fourier transforms and these are closely associated with the dual bases examined in this chapter. Thus, e.g., the inverse Fourier transform of the electron-density function of the crystal

r exp
2ih r d3 r,
1161 F
h

The transformation matrix is now the mixed matrix of the scalar products, whether or not the basis ck
1, k = 1, 2, 3, is also Cartesian. If, however, both bases are Cartesian, the transformation can also be interpreted as a rigid rotation of the coordinate axes (see Chapter 3.3). It should be noted that the above transformations do not involve any shift of the origin. Transformations involving such shifts, notably the symmetry transformations of the space group, are treated rather extensively in Volume A of International Tables for Crystallography (1995) [see e.g. Part 5 there (Arnold, 1983)].

cell

where
r is the electron-density function at the point r and the integration extends over the volume of a unit cell, is the fundamental model of the contribution of the distribution of crystalline matter to the intensity of the scattered radiation. For the conventional Bragg scattering, the function given by (1.1.6.1), and known as the structure factor, may assume nonzero values only if h can be represented as a reciprocal-lattice vector. Chapter 1.2 is devoted to a discussion of the structure factor of the Bragg reﬂection, while Chapters 4.1, 4.2 and 4.3 discuss circumstances under which the scattering need not be conﬁned to the points of the reciprocal lattice only, and may be represented by reciprocal-space vectors with non-integral components.

1.1.5.2. Example This example deals with the construction of a Cartesian system in a crystal with given basis vectors of its direct lattice. We shall also require that the Cartesian system bears a clear relationship to at least one direction in each of the direct and reciprocal lattices of the crystal; this may be useful in interpreting a physical property which has been measured along a given lattice vector or which is associated with a given lattice plane. For a better consistency of notation, the Cartesian components will be denoted as contravariant. The appropriate version of equations (1.1.5.1) and (1.1.5.2) is now r x i ai

1.1.6.2. Discrete Fourier transform The electron density
r in (1.1.6.1) is one of the most common examples of a function which has the periodicity of the crystal. Thus, for an ideal (inﬁnite) crystal the electron density
r can be written as

11511

r
r ua vb wc,

and r X k ek ,

and, as such, it can be represented by a three-dimensional Fourier series of the form

r C
g exp
2ig r,
1163

11512

1162

where the Cartesian basis vectors are: e1 rL jrL j, e2 r jr j and e3 e1 e2 , and the vectors rL and r are given by

g

rL ui ai and r hk ak ,

where the periodicity requirement (1.1.6.2) enables one to represent all the g vectors in (1.1.6.3) as vectors in the reciprocal lattice (see also Section 1.1.2 above). If we insert the series (1.1.6.3) in the

where ui and hk , i, k = 1, 2, 3, are arbitrary integers. The vectors rL and r must be mutually perpendicular, rL r ui hi 0. The

8

1.1. RECIPROCAL SPACE IN CRYSTALLOGRAPHY integrand of (1.1.6.1), interchange the order of summation and integration and make use of the fact that an integral of a periodic function taken over the entire period must vanish unless the integrand is a constant, equation (1.1.6.3) reduces to the conventional form 1

r F
h exp
2ih r,
1164 V h

the form of a plane wave times a function with the periodicity of the Bravais lattice.

Thus
r exp
ik ru
r,

1165

u
r rL u
r

1166

where

where V is the volume of the unit cell in the direct lattice and the summation ranges over all the reciprocal lattice. Fourier transforms, discrete as well as continuous, are among the most important mathematical tools of crystallography. The discussion of their mathematical principles, the modern algorithms for their computation and their numerous applications in crystallography form the subject matter of Chapter 1.3. Many more examples of applications of Fourier methods in crystallography are scattered throughout this volume and the crystallographic literature in general.

and k is the wavevector. The proof of Bloch’s theorem can be found in most modern texts on solid-state physics (e.g. Ashcroft & Mermin, 1975). If we combine (1.1.6.5) with (1.1.6.6), an alternative form of the Bloch theorem results:
r rL exp
ik rL
r

1167

In the important case where the wavefunction is itself periodic, i.e.
r rL
r, we must have exp
ik rL 1. Of course, this can be so only if the wavevector k equals 2 times a vector in the reciprocal lattice. It is also seen from equation (1.1.6.7) that the wavevector appearing in the phase factor can be reduced to a unit cell in the reciprocal lattice (the basis vectors of which contain the 2 factor), or to the equivalent polyhedron known as the Brillouin zone (e.g. Ziman, 1969). This periodicity in reciprocal space is of prime importance in the theory of solids. Some Brillouin zones are discussed in detail in Chapter 1.5.

1.1.6.3. Bloch’s theorem It is in order to mention brieﬂy the important role of reciprocal space and the reciprocal lattice in the ﬁeld of the theory of solids. At the basis of these applications is the periodicity of the crystal structure and the effect it has on the dynamics (cf. Chapter 4.1) and electronic structure of the crystal. One of the earliest, and still most important, theorems of solid-state physics is due to Bloch (1928) and deals with the representation of the wavefunction of an electron which moves in a periodic potential. Bloch’s theorem states that:

Acknowledgements I wish to thank Professor D. W. J. Cruickshank for bringing to my attention the contribution of M. von Laue (Laue, 1914), who was the ﬁrst to introduce general reciprocal bases to crystallography.

The eigenstates of the one-electron Hamiltonian h
h2 2mr2 U
r, where U(r) is the crystal potential and U
r rL U
r for all rL in the Bravais lattice, can be chosen to have

9

International Tables for Crystallography (2006). Vol. B, Chapter 1.2, pp. 10–24.

1.2. The structure factor BY P. COPPENS 1.2.3. Scattering by a crystal: definition of a structure factor

1.2.1. Introduction The structure factor is the central concept in structure analysis by diffraction methods. Its modulus is called the structure amplitude. The structure amplitude is a function of the indices of the set of scattering planes h, k and l, and is deﬁned as the amplitude of scattering by the contents of the crystallographic unit cell, expressed in units of scattering. For X-ray scattering, that unit is the scattering by a single electron
282 10 15 m, while for neutron scattering by atomic nuclei, the unit of scattering length of 10 14 m is commonly used. The complex form of the structure factor means that the phase of the scattered wave is not simply related to that of the incident wave. However, the observable, which is the scattered intensity, must be real. It is proportional to the square of the scattering amplitude (see, e.g., Lipson & Cochran, 1966). The structure factor is directly related to the distribution of scattering matter in the unit cell which, in the X-ray case, is the electron distribution, time-averaged over the vibrational modes of the solid. In this chapter we will discuss structure-factor expressions for X-ray and neutron scattering, and, in particular, the modelling that is required to obtain an analytical description in terms of the features of the electron distribution and the vibrational displacement parameters of individual atoms. We concentrate on the most basic developments; for further details the reader is referred to the cited literature.

In a crystal of inﬁnite size,
r is a three-dimensional periodic function, as expressed by the convolution crystal
r

A
S Ff
rg unit cell
rgFf
r Ff

pc,
1:2:3:1

mb

pcg,
1:2:3:2

kb

lc :
1:2:3:3

na

which gives unit cell
rg A
S Ff

S h k

ha

l

Expression (1.2.3.3) is valid for a crystal with a very large number of unit cells, in which particle-size broadening is negligible. Furthermore, it does not account for multiple scattering of the beam within the crystal. Because of the appearance of the delta function, (1.2.3.3) implies that S H with H ha kb lc . The ﬁrst factor in (1.2.3.3), the scattering amplitude of one unit cell, is deﬁned as the structure factor F: unit cell
rg F
H Ff

unit cell
r exp
2iH r

dr:
1:2:3:4

1.2.4. The isolated-atom approximation in X-ray diffraction To a reasonable approximation, the unit-cell density can be described as a superposition of isolated, spherical atoms located at rj .
1:2:4:1 unit cell
r atom; j
r
r rj :

n

where Iclassical is the classical Thomson scattering of an X-ray beam by a free electron, which is equal to
e2 =mc2 2
1 cos2 2=2 for an unpolarized beam of unit intensity, is the n-electron spacewavefunction expressed in the 3n coordinates of the electrons located at rj and the integration is over the coordinates of all electrons. S is the scattering vector of length 2 sin =. The coherent elastic component of the scattering, in units of the scattering of a free electron, is given by Icoherent; elastic
S 0 exp
2iS rj j 0 drj2 :
1:2:2:2

j

Substitution in (1.2.3.4) gives F
H

j

If integration is performed over all coordinates but those of the jth electron, one obtains after summation over all electrons Icoherent; elastic
S j
r exp
2iS r drj2 ,
1:2:2:3

atom; j gFf
r Ff j

rj g

j

fj exp
2iH rj
1:2:4:2a

or F
h, k, l

where
r is the electron distribution. The scattering amplitude A
S is then given by A
S
r exp
2iS r dr
1:2:2:4a

j

fj exp 2i
hxj kyj lzj

j

fj fcos 2
hxj kyj lzj

i sin 2
hxj kyj lzj g:

or

1:2:4:2b

fj
S, the spherical atomic scattering factor, or form factor, is the Fourier transform of the spherically averaged atomic density j
r, in which the polar coordinate r is relative to the nuclear position. fj
S can be written as (James, 1982)

1:2:2:4b

10 Copyright 2006 International Union of Crystallography

mb

n m p

The total scattering of X-rays contains both elastic and inelastic components. Within the ﬁrst-order Born approximation (Born, 1926) it has been treated by several authors (e.g. Waller & Hartree, 1929; Feil, 1977) and is given by the expression 2 Itotal
S Iclassical n exp
2iS rj 0 dr ,
1:2:2:1

where F is the Fourier transform operator.

na

n m p

where n, m and p are integers, and is the Dirac delta function. Thus, according to the Fourier convolution theorem,

1.2.2. General scattering expression for X-rays

A
S Ff
rg,

unit cell
r
r

fj
S

1.2. THE STRUCTURE FACTOR

atom

the scattering length is essentially real and independent of the energy of the incoming neutron. In either case, b is independent of the Bragg angle , unlike the X-ray form factor, since the nuclear dimensions are very small relative to the wavelength of thermal neutrons. The scattering length is not the same for different isotopes of an element. A random distribution of isotopes over the sites occupied by that element leads to an incoherent contribution, such that effectively total coherent incoherent . Similarly for nuclei with non-zero spin, a spin incoherent scattering occurs as the spin states are, in general, randomly distributed over the sites of the nuclei. For free or loosely bound nuclei, the scattering length is modiﬁed by bfree M=
m Mb, where M is the mass of the nucleus and m is the mass of the neutron. This effect is of consequence only for the lightest elements. It can, in particular, be of signiﬁcance for hydrogen atoms. With this in mind, the structure-factor expression for elastic scattering can be written as
1:2:4:2d F
H bj; coherent exp 2i
hxj kyj lzj

j
r exp
2iS r dr

2 1 j
r exp
2iSr cos #r2 sin # dr d# d' 0 '0 r0

r

sin 2Sr 4r j
r dr 2Sr 2

0

r

4r2 j
rj0
2Sr dr

0

h j0 i,

1:2:4:3

where j0
2Sr is the zero-order spherical Bessel function. j
r represents either the static or the dynamic density of atom j. In the former case, the effect of thermal motion, treated in Section 1.2.9 and following, is not included in the expression. When scattering is treated in the second-order Born approximation, additional terms occur which are in particular of importance for X-ray wavelengths with energies close to absorption edges of atoms, where the participation of free and bound excited states in the scattering process becomes very important, leading to resonance scattering. Inclusion of such contributions leads to two extra terms, which are both wavelength- and scattering-angledependent: fj
S, fj 0
S fj0
S, ifj00
S, :

j

by analogy to (1.2.4.2b).

1.2.5.2. Magnetic scattering The interaction between the magnetic moments of the neutron and the unpaired electrons in solids leads to magnetic scattering. The total elastic scattering including both the nuclear and magnetic contributions is given by 2, jF
Hj2 jFN
H Q
H j
1:2:5:1a

1:2:4:4

The treatment of resonance effects is beyond the scope of this chapter. We note however (a) that to a reasonable approximation the S-dependence of j0 and j00 can be neglected, (b) that j0 and j00 are not independent, but related through the Kramers–Kronig transformation, and (c) that in an anisotropic environment the atomic scattering factor becomes anisotropic, and accordingly is described as a tensor property. Detailed descriptions and appropriate references can be found in Materlick et al. (1994) and in Section 4.2.6 of IT C (1999). The structure-factor expressions (1.2.4.2) can be simpliﬁed when the crystal class contains non-trivial symmetry elements. For example, when the origin of the unit cell coincides with a centre of symmetry
x, y, z ! x, y, z the sine term in (1.2.4.2b) cancels when the contributions from the symmetry-related atoms are added, leading to the expression N=2

F
H 2

j1

fj cos 2
hxj kyj lzj ,

total

describes the polarization vector for the where the unit vector neutron spin, FN
H is given by (1.2.4.2b) and Q is deﬁned by mc exp
2iH r dr: Q H M
r H
1:2:5:2a eh

M
r is the vector ﬁeld describing the electron-magnetization is a unit vector parallel to H. distribution and H Q is thus proportional to the projection of M onto a direction orthogonal to H in the plane containing M and H. The magnitude of this projection depends on sin , where is the angle between Q and H, which prevents magnetic scattering from being a truly threedimensional probe. If all moments M
r are collinear, as may be achieved in paramagnetic materials by applying an external ﬁeld, and for the maximum signal (H orthogonal to M), (1.2.5.2a) becomes mc Q M
H M
r exp
2iH r dr
1:2:5:2b eh

1:2:4:2c

where the summation is over the unique half of the unit cell only. Further simpliﬁcations occur when other symmetry elements are present. They are treated in Chapter 1.4, which also contains a complete list of symmetry-speciﬁc structure-factor expressions valid in the spherical-atom isotropic-temperature-factor approximation.

and (1.2.5.1a) gives jFj2total jFN
H

M
Hj2

1:2:5:1b

and

1.2.5. Scattering of thermal neutrons

jFj2total jFN
H M
Hj2

1.2.5.1. Nuclear scattering

for neutrons parallel and antiparallel to M
H, respectively.

The scattering of neutrons by atomic nuclei is described by the atomic scattering length b, related to the total cross section total by the expression total 4b2 . At present, there is no theory of nuclear forces which allows calculation of the scattering length, so that experimental values are to be used. Two types of nuclei can be distinguished (Squires, 1978). In the ﬁrst type, the scattering is a resonance phenomenon and is associated with the formation of a compound nucleus (consisting of the original nucleus plus a neutron) with an energy close to that of an excited state. In the second type, the compound nucleus is not near an excited state and

1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism A ﬁrst improvement beyond the isolated-atom formalism is to allow for changes in the radial dependence of the atomic electron distribution.

11

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines)

l 0 1

2

3

4

5

Symbol 00

C* 1

11 11 10

1 1 1

20

1=2

Angular function, clmp † 1 x y z

3z2

1

Normalization for wavefunctions, Mlmp ‡

Normalization for density functions, Llmp §

Expression

Numerical value

Expression

Numerical value

1=2

1=4

0.28209

1=4

0.07958

3=41=2

0.48860

1=

0.31831

5=161=2

0.31539

p 3 3 8

0.20675

15=41=2

1.09255

3=4

0.75

7=161=2

0.37318

10 13

0.24485

21 21 22 22

3 3 6 6

x2

30

1=2

5z3

31 31

3=2 3=2

x5z2 y5z2

1 1

21=321=2

0.45705

32 32

15 15

x2 y2 z 2xyz

105=161=2

1.44531

1

1

33 33

15 15

35=321=2

0.59004

4=3

0.42441

40

1=8

35z4

9=2561=2

0.10579

**

0.06942

41 41

5=2 5=2

x7z3 y7z3

45=321=2

0.66905

42 42

15=2 15=2

x2 y2 7z2 1 2xy7z2 1

45=641=2

0.47309

43 43

105 105

x3 3xy2 z
y3 3x2 yz

315=321=2

1.77013

5=4

1.25

44 44

105 105

x4 6x2 y2 y4 4x3 y 4xy3

315=2561=2

0.62584

15=32

0.46875

50

1=8

63z5

15z

11=2561=2

0.11695

—

0.07674

165=2561=2

0.45295

—

0.32298

1155=641=2

2.39677

—

1.68750

385=5121=2

0.48924

—

0.34515

3465=2561=2

2.07566

—

1.50000

693=5121=2

0.65638

—

0.50930

xz yz y2 =2

xy 3z

x3 3xy2 y3 3x2 y

30z2 3 3z 3z

70z3

51 51

15=8

21z4
21z4

14z2 1x 14z2 1y

52 52

105=2

3z3 z
x2 y2 2xy
3z3 z

53 53

105=2

9z2
9z2

54 54

945

z
x4 6x2 y2 y4 z
4x3 y 4xy3

55 55

945

x5 10x3 y2 5xy4 5x4 y 10x2 y3 y5

1
x3 3xy2 1
3x2 y y3

Such changes may be due to electronegativity differences which lead to the transfer of electrons between the valence shells of different atoms. The electron transfer introduces a change in the screening of the nuclear charge by the electrons and therefore

ar{

14 5

4

735 p 512 7 196 p 105 7 p 4
136 28 7

1

0.32033

0.47400

0.33059

affects the radial dependence of the atomic electron distribution (Coulson, 1961). A change in radial dependence of the density may also occur in a purely covalent bond, as, for example, in the H2 molecule (Ruedenberg, 1962). It can be expressed as

12

1.2. THE STRUCTURE FACTOR Table 1.2.7.1. Real spherical harmonic functions (x, y, z are direction cosines) (cont.)

l 6

7

Symbol

Angular function, clmp †

C*

60

1=16

231z6

61 61

21=8

33z5
33z5

62 62

105=8

33z4 18z2 1
x2 y2 2xy
33z4 18z2 1

63 63

315=2

11z3
11z3

64 64

945=2

11z2 1
x4 6x2 y2 y4
11z2 1
4x3 y 4xy3

65 65

10395

z
x5 10x3 y2 5xy4 z
5x4 y 10x2 y3 y5

66 66

10395

70

1=16

x6

315z4 105z2 30z3 5zx 3 30z 5zy

3z
x3 3xy2 3z
3x2 y 3y

429z7

Expression

Numerical value

0.06357

—

0.04171

273=2561=2

0.58262

—

0.41721

1365=20481=2

0.46060

—

0.32611

1365=5121=2

0.92121

—

0.65132

819=10241=2

0.50457

—

0.36104

9009=5121=2

2.36662

—

1.75000

3003=20481=2

0.68318

—

0.54687

15=10241=2

0.06828

—

0.04480

105=40961=2

0.09033

—

0.06488

315=20481=2

0.22127

—

0.15732

315=40961=2

0.15646

—

0.11092

3465=10241=2

1.03783

—

0.74044

3465=40961=2

0.51892

—

0.37723

45045=20481=2

2.6460

—

2.00000

6435=40961=2

0.70716

—

0.58205

13=1024

693z5 315z3 495z4 135z2 495z4 135z2

Expression

Numerical value

15x4 y2 15x2 y4 y6 6x y 20x3 y3 6xy5

Normalization for density functions, Llmp §

1=2

5

5

Normalization for wavefunctions, Mlmp ‡

35z

71 71

7=16

429z6
429z6

72 72

63=8

143z5 110z3 15z
x2 y2 2xy
143z5 110z3 15z

73 73

315=8

143z4
143z4

74 74

3465=2

13z3 3z
x4 6x2 y2 y4
13z3 3z
4x3 y 4xy3

75 75

10395=2

13z3
13z3

76 76

135135

z
x6 15x4 y2 15x2 y4 y6 z
6x5 y 20x3 y3 6xy5

77 77

135135

x7 21x5 y2 35x3 y4 7xy6 7x6 y 35x4 y3 21x2 y5 y7

5x 5y

66z2 3
x3 3xy2 66z2 3
3x2 y y3

1
x5 10x3 y2 5xy4 1
5x4 y 10x2 y3 y5

cos m' * Common factor such that Clm clmp Pml
cos sin m' : † x sin cos ', y sin sin ', z cos . ‡ As deﬁned by ylmp Mlmp clmp where clmp are Cartesian functions. § Paturle & Coppens (1988), as deﬁned by dlmp Llmp clmp where clmp are Cartesian functions. { ar = arctan (2). p ** Nang f
14A5 14A5 20A3 20A3 6A 6A 2g 1 where A
30 480=701=2 .

0valence
r 3 valence
r

The corresponding structure-factor expression is F
H fPj; core fj; core
H Pj; valence fj; valence
H=g

1:2:6:1

j

(Coppens et al., 1979), where 0 is the modiﬁed density and is an expansion/contraction parameter, which is > 1 for valence-shell contraction and < 1 for expansion. The 3 factor results from the normalization requirement. The valence density is usually deﬁned as the outer electron shell from which charge transfer occurs. The inner or core electrons are much less affected by the change in occupancy of the outer shell and, in a reasonable approximation, retain their radial dependence.

exp
2iH rj ,

1:2:6:2

where Pj; core and Pj; valence are the number of electrons (not necessarily integral) in the core and valence shell, respectively, and the atomic scattering factors fj; core and fj; valence are normalized to one electron. Here and in the following sections, the anomalousscattering contributions are incorporated in the core scattering.

13

1. GENERAL RELATIONSHIPS AND TECHNIQUES summarized by

1.2.7. Beyond the spherical-atom description: the atomcentred spherical harmonic expansion 1.2.7.1. Direct-space description of aspherical atoms Even though the spherical-atom approximation is often adequate, atoms in a crystal are in a non-spherical environment; therefore, an accurate description of the atomic electron density requires nonspherical density functions. In general, such density functions can be written in terms of the three polar coordinates r, and '. Under the assumption that the radial and angular parts can be separated, one obtains for the density function:
r, , ' R
r
, ':

in which the direction of the arrows and the corresponding conversion factors Xlm deﬁne expressions of the type (1.2.7.4). The expressions for clmp with l 4 are listed in Table 1.2.7.1, together with the normalization factors Mlm and Llm . The spherical harmonic functions are mutually orthogonal and form a complete set, which, if taken to sufﬁciently high order, can be used to describe any arbitrary angular function. The spherical harmonic functions are often referred to as multipoles since each represents the components of the charge distribution
r, which gives non-zero contribution to the integral lmp
rclmp rl dr, where lmp is an electrostatic multipole moment. Terms with increasing l are referred to as monopolar
l 0, dipolar
l 1, quadrupolar
l 2, octapolar
l 3, hexadecapolar
l 4, triacontadipolar
l 5 and hexacontatetrapolar
l 6. Site-symmetry restrictions for the real spherical harmonics as given by Kara & Kurki-Suonio (1981) are summarized in Table 1.2.7.2. In cubic space groups, the spherical harmonic functions as deﬁned by equations (1.2.7.2) are no longer linearly independent. The appropriate basis set for this symmetry consists of the ‘Kubic Harmonics’ of Von der Lage & Bethe (1947). Some low-order terms are listed in Table 1.2.7.3. Both wavefunction and densityfunction normalization factors are speciﬁed in Table 1.2.7.3. A related basis set of angular functions has been proposed by Hirshfeld (1977). They are of the form cosn k , where k is the angle with a speciﬁed set of
n 1
n 2=2 polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics with l n, n 2, n 4,
0, 1 for n > 1, as shown elsewhere (Hirshfeld, 1977). The radial functions R
r can be selected in different manners. Several choices may be made, such as

1:2:7:1

The angular functions are based on the spherical harmonic functions Ylm deﬁned by

2l 1
l jmj 1=2 m m Ylm
, '
1 Pl
cos exp
im', 4
l jmj

1:2:7:2a

with l m l, where Pml
cos are the associated Legendre polynomials (see Arfken, 1970).

djmj Pl
x , dxjmj 1 dl Pl
x l l
x2 1l : l2 dx The real spherical harmonic functions ylmp , 0 m l, p or are obtained as a linear combination of Ylm :
2l 1
l jmj 1=2 m ylm
, Pl
cos cos m' 2
1 m0
l jmj Pml
x
1

x2 jmj=2

Nlm Pml
cos cos m'
1m
Ylm Yl; m

1:2:7:2b

and ylm
, Nlm Pml
cos sin m'
1m
Ylm

Yl;

m =2i:

1:2:7:2c

The normalization constants Nlm are deﬁned by the conditions 2 ylmp d 1,
1:2:7:3a

R l
r

which are appropriate for normalization of wavefunctions. An alternative deﬁnition is used for charge-density basis functions: jdlmp j d 2 for l > 0 and jdlmp j d 1 for l 0: The functions ylmp and dlmp differ only in the normalization constants. For the spherically symmetric function d00 , a population parameter equal to one corresponds to the function being populated by one electron. For the non-spherical functions with l > 0, a population parameter equal to one implies that one electron has shifted from the negative to the positive lobes of the function. The functions ylmp and dlmp can be expressed in Cartesian coordinates, such that

R l
r

1:2:7:4a

1:2:7:5a

n1 n r exp
r2 n

Gaussian function
1:2:7:5b

or

and dlmp Llm clmp ,

(Slater type function),

where the coefﬁcient nl may be selected by examination of products of hydrogenic orbitals which give rise to a particular multipole (Hansen & Coppens, 1978). Values for the exponential coefﬁcient l may be taken from energy-optimized coefﬁcients for isolated atoms available in the literature (Clementi & Raimondi, 1963). A standard set has been proposed by Hehre et al. (1969). In the bonded atom, such values are affected by changes in nuclear screening due to migrations of charge, as described in part by equation (1.2.6.1). Other alternatives are:

1:2:7:3b

ylmp Mlm clmp

nl 3 n
l r exp
l r
nl 2

R l
r rl Ln2l2
r exp

1:2:7:4b

where the clmp are Cartesian functions. The relations between the various deﬁnitions of the real spherical harmonic functions are

r
Laguerre function, 2
1:2:7:5c

where L is a Laguerre polynomial of order n and degree
2l 2.

14

1.2. THE STRUCTURE FACTOR

fj
S j
r exp
2iS r dr:

Table 1.2.7.2. Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981)

In order to evaluate the integral, the scattering operator exp
2iS r must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959; Cohen-Tannoudji et al., 1977)

, and j are integers.

Symmetry 1 1 2 m 2=m 222 mm2 mmm 4 4 4=m 422 4mm 42m 4=mmm 3 3 32

3m 3m 6 6 6=m 622 6mm 6m2 6=mmm

Choice of coordinate axes

Indices of allowed ylmp , dlmp All
l, m,
2, m,
l, 2,
l, l 2j,
2, 2,
2, 2, ,
2 1, 2,
l, 2,
2, 2,
l, 4,
2, 4, ,
2 1, 4 2,
2, 4,
2, 4, ,
2 1, 4,
l, 4,
2, 4, ,
2 1, 4 2,
2, 4, ,
2 1, 4 2,
2, 4,
l, 3,
2, 3,
2, 3, ,
2 1, 3,
3 2j, 3, ,
3 2j 1, 3,
l, 3,
l, 6, ,
l, 6 3,
2, 3,
2, 6, ,
2, 6 3,
l, 6,
2, 6, ,
2 1, 6 3,
2, 6,
2, 6, ,
2 1, 6,
l, 6,
2, 6, ,
2 1, 6 3,
2, 6, ,
2 1, 6 3,
2, 6,

Any Any 2kz m?z 2kz, m ? z 2kz, 2ky 2kz, m ? y m ? z, m ? y, m ? x 4kz 4kz 4kz, m ? z 4kz, 2ky 4kz, m ? y 4kz, 2kx m?y 4kz, m ? z, m ? x 3kz 3kz 3kz, 2ky 2kx 3kz, m ? y m?x 3kz, m ? y m?x 6kz 6kz 6kz, m ? z 6kz, 2ky 6kz, mky 6kz, m ? y m?x 6kz, m ? z, m ? y

exp
2iS r 4

l0

03 R l
0 r

l Plmp dlmp
r=r,

m0 p

il jl
2SrYlm
, 'Ylm
, :

1:2:7:7a

The Fourier transform of the product of a complex spherical harmonic function with normalization jYlm j2 d 1 and an arbitrary radial function R l
r follows from the orthonormality properties of the spherical harmonic functions, and is given by Ylm R l
r exp
2iS r d 4il jl
2SrR l
rr2 drYlm
, ,

1:2:7:8a

where jl is the lth-order spherical Bessel function (Arfken, 1970), and and ', and are the angular coordinates of r and S, respectively. For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics: exp
2iS r

1 il jl
2Sr
2 l0

m0
2l 1

Pml
cos Pml
cos cosm

l
l

m
l m m0

,

1:2:7:7b

which leads to ylmp
, 'R l
r exp
2iS r d 4il hjl iylmp
, :
1:2:7:8b

Since ylmp occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions dlmp dlmp
, 'R l
r exp
2iS r d 4il hjl idlmp
, :
1:2:7:8c

In (1.2.7.8b) and (1.2.7.8c), hjl i, the Fourier–Bessel transform, is the radial integral deﬁned as hjl i jl
2SrR l
rr2 dr
1:2:7:9

of which hj0 i in expression (1.2.4.3) is a special case. The functions hjl i for Hartree–Fock valence shells of the atoms are tabulated in scattering-factor tables (IT IV, 1974). Expressions for the evaluation of hjl i using the radial function (1.2.7.5a–c) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closed-form expressions are listed in Table 1.2.7.4. Expressions (1.2.7.8) show that the Fourier transform of a directspace spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant. The scattering factors flmp
S of the aspherical density functions R l
rdlmp
, in the multipole expansion (1.2.7.6) are thus given by

atomic
r Pc core P 3 valence
r l max

1 l

l0 m l

In summary, in the multipole formalism the atomic density is described by

1:2:4:3a

1:2:7:6

in which the leading terms are those of the kappa formalism [expressions (1.2.6.1), (1.2.6.2)]; the subscript p is either + or . The expansion in (1.2.7.6) is frequently truncated at the hexadecapolar
l 4 level. For atoms at positions of high site symmetry the ﬁrst allowed functions may occur at higher l values. For trigonally bonded atoms in organic molecules the l 3 terms are often found to be the most signiﬁcantly populated deformation functions.

flmp
S 4il hjl idlmp
, :

1:2:7:8d

The reciprocal-space spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1, except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S.

1.2.7.2. Reciprocal-space description of aspherical atoms The aspherical-atom form factor is obtained by substitution of (1.2.7.6) in expression (1.2.4.3a):

15

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.7.3. ‘Kubic Harmonic’ functions

2 l (a) Coefﬁcients in the expression Klj kmpj ylmp with normalization 0 0 jKlj j2 sin d d' 1 (Kara & Kurki-Suonio, 1981). mp

Even l

mp

l

j

0+

0

1

1

4

1

1 7 1=2 2 3

1 5 1=2 2 3

0.76376

0.64550

6

1

6

2+

4+

6+

1 1 1=2 2 2

1 7 1=2 2 2

0.35355

0.93541 1 1=2 11 4

2

1

10

0.55902

1 1=2 33 8

1 7 1=2 4 3

1 65 1=2 8 3

0.71807

0.38188

0.58184

1 65 1=2 8 6

1

1 11 4 2

0.41143 10

1=2

1 187 1=2 8 6

0.58630

0.69784

2

1 247 1=2 8 6

1 19 1=2 16 3

1 1=2 85 16

0.80202

0.15729

0.57622

l

j

2

3

1

1

7

1

1 13 1=2 2 6

1 11 1=2 2 16

0.73598

0.41458

9

1

9

2

4

Nlj

8

1 1=2 13 4

0.43301

0.90139

1 17 1=2 2 6

1 7 1=2 2 6

0.84163

0.54006

mp 0+

6

1 1=2 3 4

l (b) Coefﬁcients kmpj and density normalization factors Nlj in the expression Klj Nlj

Even l

10+

1 1=2 5 4

0.82916 8

8+

2+

l cos m' kmpj ulmp where ulm Pml
cos sin m' (Su & Coppens, 1994). mp

4+

l

j

0

1

1=4 0:079577

1

4

1

0.43454

1

1=168

6

1

0.25220

1

1=360

6

2

0.020833

1

6+

1=792

16

8+

10+

1.2. THE STRUCTURE FACTOR Table 1.2.7.3. ‘Kubic Harmonic’ functions (cont.) Even l

Nlj

mp

8

1

0.56292

1

1/5940

1 1 672 5940

10

1

0.36490

1

1/5460

1 1 4320 5460

10

2

0.0095165

1

l

j

3

1

0.066667

1

7

1

0.014612

1

1=1560

9

1

0.0059569

1

1=2520

9

2

0.00014800

1 1 456 43680

1=43680 2

4

6

8

1

1=4080

(c) Density-normalized Kubic harmonics as linear combinations 2 of density-normalized spherical harmonic functions. Coefﬁcients in the expression 00 l dlmp . Density-type normalization is deﬁned as 0 0 jKlj j sin d d' 2 l0 . Klj kmpj mp

Even l

mp

l

j

0+

0

1

1

4

1

0.78245

6

1

0.37790

6

2

l

j

2

3

1

1

7

1

0.73145

2+

4+

6+

8+

10+

0.57939 0.91682 0.83848

0.50000

4

6

8

0.63290

(d) Index rules for cubic symmetries (Kurki-Suonio, 1977; Kara & Kurki-Suonio, 1981).

l

j

0 3 4 6 6 7 8 9 9 10 10

1 1 1 1 2 1 1 1 2 1 2

23 T

m3 Th

432 O

43m Td

m3m Oh

by (Stewart, 1969a)
r ni 2i P '
r'
r,

1.2.8. Fourier transform of orbital products If the wavefunction is written as a sum over normalized Slater determinants, each representing an antisymmetrized combination of occupied molecular orbitals i expressed as linear combinations of atomic orbitals ' , i.e. i ci ' , the electron density is given

i

1:2:8:1

with ni 1 or 2. The coefﬁcients P are the populations of the

17

1. GENERAL RELATIONSHIPS AND TECHNIQUES orbital product density functions
r'
r and are given by
1:2:8:2 P ni ci ci :

ylmp
, 'yl0 m0 p0
, '

i

Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the two-centre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slater-type (Bentley & Stewart, 1973; Avery & Ørmen, 1979) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian- (Stewart, 1969b, 1970; Stewart & Hehre, 1970; Hehre et al., 1970) and Slater-type (Clementi & Roetti, 1974) functions are available for many atoms.

If the atomic basis consists of hydrogenic type s, p, d, f, . . . orbitals, the basis functions may be written as

'
r, , ' R l
rylmp
, ',

1:2:8:3b

P

1.2.8.2. Two-centre orbital products

1.2.8.1. One-centre orbital products

1:2:8:3a

L M

where R LMP MLMP (wavefunction)=LLMP (density function). The normalization constants Mlmp and Llmp are given in Table 1.2.7.1, while the coefﬁcients in the expressions (1.2.8.6) are listed in Table 1.2.8.3.

For a multi-Slater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2) but with non-integer values for the coefﬁcients ni . The summation (1.2.8.1) consists of one- and two-centre terms for which ' and ' are centred on the same or on different nuclei, respectively. The latter represent the overlap density, which is only signiﬁcant if '
r and '
r have an appreciable value in the same region of space.

'
r, , ' R l
rYlm
, '

0 Mmm0 R LMP C Lll0 dLMP
, ',
1:2:8:6

or 1.2.9. The atomic temperature factor Since the crystal is subject to vibrational oscillations, the observed elastic scattering intensity is an average over all normal modes of the crystal. Within the Born–Oppenheimer approximation, the theoretical electron density should be calculated for each set of nuclear coordinates. An average can be obtained by taking into account the statistical weight of each nuclear conﬁguration, which may be expressed by the probability distribution function P
u1 , , uN for a set of displacement coordinates u1 , , uN . In general, if
r, u1 , , uN is the electron density corresponding to the geometry deﬁned by u1 , , uN , the time-averaged electron density is given by h
ri
r, u1 , , uN P
u1 , , uN du1 duN :
1:2:9:1

which gives for corresponding values of the orbital products '
r'
r R l
rR l0
rYlm
, 'Yl0 m0
, '

1:2:8:4a

'
r'
r R l
rR l0
rylmp
, 'yl0 m0 p0
, ',

1:2:8:4b

and

respectively, where it has been assumed that the radial function depends only on l. Because the spherical harmonic functions form a complete set, their products can be expressed as a linear combination of spherical harmonics. The coefﬁcients in this expansion are the Clebsch– Gordan coefﬁcients (Condon & Shortley, 1957), deﬁned by Mmm0 Ylm
, 'Yl0 m0
, ' CLll0 YLM
, '
1:2:8:5a

When the crystal can be considered as consisting of perfectly following rigid entities, which may be molecules or atoms, expression (1.2.9.1) simpliﬁes: hrigid group
ri r:g:; static
r uP
u du r:g:; static P
u:

L M

or the equivalent deﬁnition

1:2:9:2 In the approximation that the atomic electrons perfectly follow the nuclear motion, one obtains

0 2 Mmm sin d d'YLM
, 'Ylm
, 'Yl0 m0
, ':
1:2:8:5b CLll 0

0

0

0

0

Mmm CLll 0

The vanish, unless L l l is even, jl 0 and M m m . The corresponding expression for ylmp is ylmp
, 'yl0 m0 p0
, ' 0

0

lj < L < ll

0 Mmm0 C Lll0 yLMP
, ', L M 0

P

0

0

hatom
ri atom; static
r P
u:

1:2:9:3

The Fourier transform of this convolution is the product of the Fourier transforms of the individual functions:
1:2:8:5c

hf
Hi f
HT
H:

1:2:9:4

Thus T
H, the atomic temperature factor, is the Fourier transform of the probability distribution P
u.

0

with M jm m j and jm m j for p p , and M jm m j 0 0 0 and jm m j for p 0 p and P p p . Values of C and C for l 2 are given in Tables 1.2.8.1 and 1.2.8.2. They are valid for the functions Ylm and ylmp with normalization jYlm j2 d 1 and y2lmp d 1. By using (1.2.8.5a) or (1.2.8.5c), the one-centre orbital products are expressed as a sum of spherical harmonic functions. It follows that the one-centre orbital product density basis set is formally equivalent to the multipole description, both in real and in reciprocal space. To obtain the relation between orbital products and the charge-density functions, the right-hand side of (1.2.8.5c) has to be multiplied by the ratio of the normalization constants, as the wavefunctions ylmp and charge-density functions dlmp are normalized in a different way as described by (1.2.7.3a) and (1.2.7.3b). Thus

1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian, centred at the equilibrium position. For the three-dimensional isotropic harmonic oscillator, the distribution is P
u
2hu2 i 2

3=2

expf juj2 =2hu2 ig,

1:2:10:1

where hu i is the mean-square displacement in any direction. The corresponding trivariate normal distribution to be used for anisotropic harmonic motion is, in tensor notation,

18

1.2. THE STRUCTURE FACTOR Table 1.2.7.4. Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990) hjk i

1 0

rN exp
Zrjk
Kr dr, K 4 sin =:

N k 0

1

2

3

1 K2 Z2

K 2 Z 2 2

2
3Z 2

2Z

1

4 K2

24Z
Z 2

K 2 Z 2 3

2K

8KZ

K 2 Z 2 2

K 2 Z 2 3

2

5

6

K2

24
5Z 2

K2

48KZ
5Z 2

8K 2

48K 2 Z
K 2 Z 2 4

3

48K 2
7Z 2

48K 3

384K 3 Z
K 2 Z 2 5

4

3K 2

48K
35Z 4

K2

384K 2 Z
7Z 2

Z2

42K 2 Z 2 3K 4

3K 2

720
7Z 6

35K 2 Z 4 21K 4 Z 2
K 2 Z 2 7

1920KZ
7Z 4

K6

14K 2 Z 2 3K 4

40320
Z 7

7K 2 Z 5 7K 4 Z 3

5760K
21Z 6

K2

18K 2 Z 2 K 4

11520K 3 Z
3Z 2
K 2 Z 2 7

3840K 4 Z

63K 2 Z 4 27K 4 Z 2

K 2 Z 2 8 11520K 2 Z
21Z 4

K 2 Z 2 7

K 2 Z 2 6

K2

11520K 3
33Z 4

3840K 5

46080K 5 Z

K 2 Z 2 6

K 2 Z 2 7

40680K 5
13Z 2

6

30K 2 Z 2 5K 4

22K 2 Z 2 K 4

K 2 Z 2 8

46080K 4 Z
11Z 2

5

K6

K 2 Z 2 8

3840K 4
11Z 2 K 2
K 2 Z 2 7

K 2 Z 2 6

K 6 Z

K 2 Z 2 8

K 2 Z 2 7 1152K 2
21Z 4

K 2 Z 2 6 384K 3
9Z 2

384K 4
K 2 Z 2 5

8

K 2 Z 2 6

K 2 Z 2 5

K 2 Z 2 4

3Z 2
3K 2

K 2 Z 2 6

K 2 Z 2 5

K 2 Z 2 4

K 2 Z 2 3

240Z
K 2

K 2 Z 2 5

K 2 Z 2 4 8K
5Z 2

10K 2 Z 2 K 4

7

K 2 Z 2 8

K 2 Z 2 8

46080K 6

645120K 6 Z

K 2 Z 2 7

K 2 Z 2 8

7

3K 2

K2

645120K 7
K 2 Z 2 8

P
u

j 1 j1=2
2

3=2

1 1 j k 2 jk
u u g:

expf

r
r Dr

1:2:10:2a

with

P
u

23=2

expf

T 1 1 2
u
ug,

ri Dij rj "ijk k rj

1:2:10:2b

T

T
H expf 2 H Hg:

1:2:11:2

1:2:11:3

where the permutation operator "ijk equals +1 for i, j, k a cyclic permutation of the indices 1, 2, 3, or 1 for a non-cyclic permutation, and zero if two or more indices are equal. For i 1, for example, only the "123 and "132 terms occur. Addition of a translational displacement gives

1:2:10:3a

or 2

2 1 , 0

or in tensor notation, assuming summation over repeated indices,

where the superscript T indicates the transpose. The characteristic function, or Fourier transform, of P
u is T
H expf 22 jk hj hk g

3 0 1

0 D 3 2

Here is the variance–covariance matrix, with covariant components, and j 1 j is the determinant of the inverse of . Summation over repeated indices has been assumed. The corresponding equation in matrix notation is j 1 j1=2

1:2:11:1

ri Dij rj ti :

1:2:11:4

When a rigid body undergoes vibrations the displacements vary with time, so suitable averages must be taken to derive the meansquare displacements. If the librational and translational motions are independent, the cross products between the two terms in (1.2.11.4) average to zero and the elements of the mean-square displacement tensor of atom n, Uijn , are given by

1:2:10:3b

With the change of variable b jk 22 jk , (1.2.10.3a) becomes T
H expf b jk hj hk g:

n U11 L22 r32 L33 r22

1.2.11. Rigid-body analysis

n U22

The treatment of rigid-body motion of molecules or molecular fragments was developed by Cruickshank (1956) and expanded into a general theory by Schomaker & Trueblood (1968). The theory has been described by Johnson (1970b) and by Dunitz (1979). The latter reference forms the basis for the following treatment. The most general motions of a rigid body consist of rotations about three axes, coupled with translations parallel to each of the axes. Such motions correspond to screw rotations. A libration around a vector
1 , 2 , 3 , with length corresponding to the magnitude of the rotation, results in a displacement r, such that

n U33

2 L33 r1

L11 r22

2 L11 r3

L22 r12

n U12 L33 r1 r2 n

2L23 r2 r3 T11 2L13 r1 r3 T22 2L12 r1 r2 T33

L12 r32 L13 r2 r3 L23 r1 r3 T12

U13 L22 r1 r3 L12 r2 r3 n U23 L11 r2 r3 L12 r1 r3

1:2:11:5

2

L13 r2 L23 r1 r2 T13 L13 r1 r2

L23 r12 T23 ,

where the coefﬁcients Lij hi j i and Tij hti tj i are the elements of the 3 3 libration tensor L and the 3 3 translation tensor T,

19

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.8.1. Products of complex spherical harmonics as defined by equation (1.2.7.2a) Y00 Y00 Y10 Y00 Y10 Y10 Y11 Y00 Y11 Y10 Y11 Y11 Y11 Y11 Y20 Y00 Y20 Y10 Y20 Y11 Y20 Y20 Y21 Y00 Y21 Y10 Y21 Y11 Y21 Y11 Y21 Y20 Y21 Y21 Y21 Y21 Y22 Y00 Y22 Y10 Y22 Y11 Y22 Y11 Y22 Y20 Y22 Y21 Y22 Y21 Y22 Y22 Y22 Y22

= = = = = = = = = = = = = = = = = = = = = = = = = = =

Table 1.2.8.2. Products of real spherical harmonics as defined by equations (1.2.7.2b) and (1.2.7.2c)

0.28209479Y00 0.28209479Y10 0.25231325Y20 + 0.28209479Y00 0.28209479Y11 0.21850969Y21 0.30901936Y22 0.12615663Y20 + 0.28209479Y00 0.28209479Y20 0.24776669Y30 + 0.25231325Y10 0.20230066Y31 0.12615663Y11 0.24179554Y40 + 0.18022375Y20 + 0.28209479Y00 0.28209479Y21 0.23359668Y31 + 0.21850969Y11 0.26116903Y32 0.14304817Y30 + 0.21850969Y10 0.22072812Y41 + 0.09011188Y21 0.25489487Y42 + 0.22072812Y22 0.16119702Y40 + 0.09011188Y20 + 0.28209479Y00 0.28209479Y22 0.18467439Y32 0.31986543Y33 0.08258890Y31 + 0.30901936Y11 0.15607835Y42 0.18022375Y22 0.23841361Y43 0.09011188Y41 + 0.22072812Y21 0.33716777Y44 0.04029926Y40 0.18022375Y20 + 0.28209479Y00

y00 y00 y10 y00 y10 y10 y11 y00 y11 y10 y11 y11 y11+ y11 y20 y00 y20 y10 y20 y11 y20 y20 y21 y00 y21 y10 y21 y11 y21 y11 y21 y20 y21 y21 y21+ y21 y22 y00 y22 y10 y22 y11 y22 y11 y22 y20 y22 y21 y22 y21 y22 y22 y22+ y22

respectively. Since pairs of terms such as hti tj i and htj ti i correspond to averages over the same two scalar quantities, the T and L tensors are symmetrical. If a rotation axis is correctly oriented, but incorrectly positioned, an additional translation component perpendicular to the rotation axes is introduced. The rotation angle and the parallel component of the translation are invariant to the position of the axis, but the perpendicular component is not. This implies that the L tensor is unaffected by any assumptions about the position of the libration axes, whereas the T tensor depends on the assumptions made concerning the location of the axes. The quadratic correlation between librational and translational motions can be allowed for by including in (1.2.11.5) cross terms of the type hDik tj i, or, with (1.2.11.3), Uij hDik Djl irk rl hDik tj Dji ti irk hti tj i Aijkl rk rl Bijk rk hti tj i,

unsymmetrical, since hi tj i is different from hj ti i. The terms involving elements of S may be grouped as h3 t1 ir1

U11 hr1 i

h23 ir22

h22 ir32

2h3 t1 ir2

S31 r1

12116

S32 r2
S22

1:2:11:8

S11 r3 :

Uij Gijkl Lkl Hijkl Skl Tij :

1:2:11:9

The arrays Gijkl and Hijkl involve the atomic coordinates
x, y, z
r1 , r2 , r3 , and are listed in Table 1.2.11.1. Equations (1.2.11.9) for each of the atoms in the rigid body form the observational equations, from which the elements of T, L and S can be derived by a linear least-squares procedure. One of the diagonal elements of S must be ﬁxed in advance or some other suitable constraint applied because of the indeterminacy of Tr
S. It is common practice to set Tr
S equal to zero. There are thus eight elements of S to be determined, as well as the six each of L and T, for a total of 20 variables. A shift of origin leaves L invariant, but it intermixes T and S. If the origin is located at a centre of symmetry, for each atom at r with vibration tensor Un there will be an equivalent atom at r with

2h2 t1 ir3 ht12 i,

U12 hr1 r2 i h23 ir1 r2 h1 3 ir2 r3 h2 3 ir1 r3 h1 t1 ir3

h3 t2 i r2 h2 t2 ir3 ht1 t2 i:

h1 t1 ir3

As the diagonal elements occur as differences in this expression, a constant may be added to each of the diagonal terms without changing the observational equations. In other words, the trace of S is indeterminate. In terms of the L, T and S tensors, the observational equations are

2h2 3 ir2 r3

h1 2 ir32 h3 t1 ir1

h3 t2 ir2
h2 t2 i

or

which leads to the explicit expressions such as 2

= 0.28209479y00 = 0.28209479y10 = 0.25231325y20 + 0.28209479y00 = 0.28209479y11 = 0.21850969y21 = 0.21850969y22+ 0.12615663y20 + 0.28209479y00 = 0.21850969y22 = 0.28209479y20 = 0.24776669y30 + 0.25231325y10 = 0.20230066y31 0.12615663y11 = 0.24179554y40 + 0.18022375y20 + 0.28209479y00 = 0.28209479y21 = 0.23359668y31 + 0.21850969y11 = 0.18467439y32+ 0.14304817y30 + 0.21850969y10 = 0.18467469y32 = 0.22072812y41 + 0.09011188y21 = 0.18022375y42+ 0.15607835y22+ 0.16119702y40 + 0.09011188y20 + 0.28209479y00 = 0.18022375y42 + 0.15607835y22 = 0.28209479y22 = 0.18467439y32 = 0.22617901y33+ 0.05839917y31+ + 0.21850969y11+ = 0.22617901y33 0.05839917y31 0.21850969y11 = 0.15607835y42 0.18022375y22 = 0.16858388y43+ 0.06371872y41+ + 0.15607835y21+ = 0.16858388y43 0.06371872y41 0.15607835y21 = 0.23841361y44+ + 0.04029926y40 0.18022375y20 + 0.28209479y00 = 0.23841361y44

1:2:11:7

The products of the type hi tj i are the components of an additional tensor, S, which unlike the tensors T and L is

20

1.2. THE STRUCTURE FACTOR Table 1.2.8.3. Products of two real spherical harmonic functions ylmp in terms of the density functions dlmp defined by equation (1.2.7.3b)

Table 1.2.11.1. The arrays Gijkl and Hijkl to be used in the observational equations Uij Gijkl Lkl Hijkl Skl Tij [equation (1.2.11.9)] Gijkl

y00 y00 y10 y00 y10 y10 y11 y00 y11 y10 y11 y11 y11+ y11 y20 y00 y20 y10 y20 y11 y20 y20 y21 y00 y21 y10 y21 y11 y21 y11 y21 y20 y21 y21 y21+ y21 y22 y00 y22 y10 y22 y11 y22 y11 y22 y20 y22 y21 y22 y21 y22 y22 y22+ y22

= 1.0000d00 = 0.43301d10 = 0.38490d20 + 1.0d00 = 0.43302d11 = 0.31831d21 = 0.31831d22+ 0.19425d20 + 1.0d00 = 0.31831d22 = 0.43033d20 = 0.37762d30 + 0.38730d10 = 0.28864d31 0.19365d11 = 0.36848d40 + 0.27493d20 + 1.0d00 = 0.41094d21 = 0.33329d31 + 0.33541d11 = 0.26691d32+ 0.21802d30 + 0.33541d10 = 0.26691d32 = 0.31155d41 + 0.13127d21 = 0.25791d42+ 0.22736d22+ 0.24565d40 + 0.13747d20 + 1.0d00 = 0.25790d42 + 0.22736d22 = 0.41094d22 = 0.26691d32 = 0.31445d33+ 0.083323d31+ + 0.33541d11+ = 0.31445d33 0.083323d31 0.33541d11 = 0.22335d42 0.26254d22 = 0.23873d43+ 0.089938d41+ + 0.22736d21+ = 0.23873d43 0.089938d41 0.22736d21 = 0.31831d44+ + 0.061413d40 0.27493d20 + 1.0d00 = 0.31831d44

kl ij

11

22

33

11 22 33 23 31 12

0 z2 y2 yz 0 0

z2 0 x2 0 xz 0

y2 x2 0 0 0 xy

23 2yz 0 0

31

12

0

0 0

2xz 0 xy y2 yz

x2 xy xz

2xy xz yz z2

Hijkl kl ij

11

22

33

23

11 22 33 23 31 12

0 0 0 0 y

0 0 0

0 0 0 x

0 0

x z

0 z

2y 2x

0 z 0

y 0

31 0 0 0 0 x

12

32

13

21

0

0 2x 0 0 0 y

0 0 2y z 0 0

2z 0 0 0 x 0

2z 0 y 0 0

symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts i are 1

S23 S32 L22 L33

2

S31 S13 L11 L33

3

S12 S21 ,
1:2:11:10 L11 L22

in which the carets indicate quantities referred to the principal axis system. The description of the averaged motion can be simpliﬁed further by shifting to three generally non-intersecting libration axes, one each for each principal axis of L. Shifts of the L1 axis in the L2 and L3 directions by

the same vibration tensor. When the observational equations for these two atoms are added, the terms involving elements of S disappear since they are linear in the components of r. The other terms, involving elements of the T and L tensors, are simply doubled, like the Un components. The physical meaning of the T and L tensor elements is as follows. Tij li lj is the mean-square amplitude of translational vibration in the direction of the unit vector l with components l1 , l2 , l3 along the Cartesian axes and Lij li lj is the mean-square amplitude of libration about an axis in this direction. The quantity Sij li lj represents the mean correlation between libration about the axis l and translation parallel to this axis. This quantity, like Tij li lj , depends on the choice of origin, although the sum of the two quantities is independent of the origin. The non-symmetrical tensor S can be written as the sum of a symmetric tensor with elements SijS
Sij Sji =2 and a skewsymmetric tensor with elements SijA
Sij Sji =2. Expressed in terms of principal axes, SS consists of three principal screw correlations hI tI i. Positive and negative screw correlations correspond to opposite senses of helicity. Since an arbitrary constant may be added to all three correlation terms, only the differences between them can be determined from the data. The skew-symmetric part SA is equivalent to a vector
t=2 with components
ti =2
j tk k tj =2, involving correlations between a libration and a perpendicular translation. The components of SA can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that

1

2 S13 = L11 and 1 3 S12 = L11 ,

1:2:11:11

respectively, annihilate the S12 and S13 terms of the symmetrized S tensor and simultaneously effect a further reduction in Tr
T (the presuperscript denotes the axis that is shifted, the subscript the direction of the shift component). Analogous equations for displacements of the L2 and L3 axes are obtained by permutation of the indices. If all three axes are appropriately displaced, only the diagonal terms of S remain. Referred to the principal axes of L, they represent screw correlations along these axes and are independent of origin shifts. The elements of the reduced T are r TII TII
SKI 2 = LKK K6I

r

21

TIJ TIJ

SKI SKJ = LKK , K

J 6 I:

1:2:11:12

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions – three screw librations about non-intersecting axes (with screw pitches given by S11 = L11 etc.) and three translations. The parameter set consists of three libration and three translation amplitudes, six

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.2.12.1. Some Hermite polynomials (Johnson & Levy, 1974; Zucker & Schulz, 1982)

where the ﬁrst and second terms have been omitted since they are equivalent to a shift of the mean and a modiﬁcation of the harmonic term only. The permutations of j, k, l here, and in the following sections, include all combinations which produce different terms. The coefﬁcients c, deﬁned by (1.2.12.1) and (1.2.12.2), are known as the quasimoments of the frequency function P
u (Kutznetsov et al., 1960). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958) and are invariant to permutation of indices. There are 10, 15, 21 and 28 components of c for orders 3, 4, 5 and 6, respectively. The multivariate Hermite polynomials are functions of the elements of jk 1 and of uk , and are given in Table 1.2.12.1 for orders 6 (IT IV, 1974; Zucker & Schulz, 1982). The Fourier transform of (1.2.12.3) is given by 4 3 jkl 2 T
H 1 ic hj hk hl 4 c jklm hj hk hl hm 3 3 4 5 jklmn ic hj hk hl hm hn 15 4 6 jklmnp hj hk hl hm hn hp T0
H,
1:2:12:4 c 45

H(u) = 1 Hj(u) = wj Hjk(u) = wjwk pjk Hjkl(u) = wjwkwl (wjpkl + wkplj + wlpjk) = wjwkwl 3w(jpkl) Hjklm(u) = wjwkwlwm 6w(jwkplm) + 3pj(kplm) Hjklmn(u) = wjwkwlwmwn 10w(lwmwnpjk) + 15w(npjkplm) Hjklmnp(u) = wjwkwlwmwnwp 15w(jwkwlwmpjk) + 45w(jwkplmpnp) 15pj(kplmpnp) where wj pjk uk and pjk are the elements of 1 , defined in expression (1.2.10.2). Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that pjk pkj and wj wk wk wj as illustrated for Hjkl .

angles of orientation for the principal axes of L and T, six coordinates of axis displacement, and three screw pitches, one of which has to be chosen arbitrarily, again for a total of 20 variables. Since diagonal elements of S enter into the expression for r TIJ , the indeterminacy of Tr
S introduces a corresponding indeterminacy in r T. The constraint Tr
S 0 is unaffected by the various rotations and translations of the coordinate systems used in the course of the analysis.

where T0
H is the harmonic temperature factor. T
H is a powerseries expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary. 1.2.12.2. The cumulant expansion A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958; Johnson, 1969). It expresses the function P
u as 1 1 jkl P
u exp j Dj jk Dj Dk Dj Dk Dl 2 3

1 jklm Dj Dk Dl Dm P0
u:
1:2:12:5a 4

1.2.12. Treatment of anharmonicity The probability distribution (1.2.10.2) is valid in the case of rectilinear harmonic motion. If the deviations from Gaussian shape are not too large, distributions may be used which are expansions with the Gaussian distribution as the leading term. Three such distributions are discussed in the following sections. 1.2.12.1. The Gram–Charlier expansion

Like the moments of a distribution, the cumulants are descriptive constants. They are related to each other (in the onedimensional case) by the identity 2 t2 r tr 2 t2 r t r exp 1 t 1 1 t : 2 r 2 r

The three-dimensional Gram–Charlier expansion, introduced into thermal-motion treatment by Johnson & Levy (1974), is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958). If Dj is the operator d/du j , 1 1 jkl P
u 1 c j Dj c jk Dj Dk c Dj Dk Dl 2 3 c1 cr
1r
12121 D1 Dr P0
u, r where P0
u is the harmonic distribution, 1 1, 2 or 3, and the operator D1 Dr is the rth partial derivative r =
@u1 @ur . Summation is again implied over repeated indices. The differential operators D may be eliminated by the use of three-dimensional Hermite polynomials H1 2 deﬁned, by analogy with the one-dimensional Hermite polynomials, by the expression D1 Dr exp

1 1 j k 2jk u u

1r H1 r
u exp

1:2:12:5b

When it is substituted for t, (1.2.12.5b) is the characteristic function, or Fourier transform of P
t (Kendall & Stuart, 1958). The ﬁrst two terms in the exponent of (1.2.12.5a) can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by P0
u. The Fourier transform of (1.2.12.5a) is, by analogy with the lefthand part of (1.2.12.5b) (with t replaced by 2ih),
2i3 jkl
2i4 jklm hj hk hl hj hk hl hm T0
H T
H exp 3 4 4 3 jkl 2 exp i hj hk hl 4 jklm hj hk hl hm T0
H, 3 3

1 1 j k 2jk u u ,

1:2:12:2

which gives 1 1 1 P
u 1 c jkl Hjkl
u c jklm Hjklm
u c jklmn Hjklmn
u 3 4 5 1 jklmnp Hjklmnp
u P0
u,
1:2:12:3 c 6

1:2:12:6

where the ﬁrst two terms have been omitted. Expression (1.2.12.6) is similar to (1.2.12.4) except that the entire series is in the exponent. Following Schwarzenbach (1986), (1.2.12.6) can be developed in a Taylor series, which gives

22

1.2. THE STRUCTURE FACTOR 0

in which =kT etc. and the normalization factor N depends on the level of truncation. The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters u j uk ul , for example, are linear combinations of the seven octapoles
l 3 and three dipoles
l 1 (Coppens, 1980). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially conﬁned to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model. The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970a; Scheringer, 1985a) 4 3 0 jkl 2 0 T
H T0
H 1 i jkl G
H 4 jklm G jklm
H 3 3 4 5 0 4 0 i"jklmn G jklmn
H 6 i'jklmnp G jklmnp
H , 15 45

2i3 jkl
2i4 jklm T
H 1 hj hk hl hj hk hl hm 3 4
2i6 jklmp 6 jkl mnp hj hk hl hm hn hp 6 2
32 higher-order terms T0
H:
1:2:12:7 This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4), and corresponds to the probability distribution [analogous to (1.2.12.3)] 1 1 P
u P0
u 1 jkl Hjkl
u jklm Hjklm
u 3 4 1 jklmnp 10 jkl mnp Hjklmnp 6 higher-order terms :
1:2:12:8

1:2:12:13

where T0 is the harmonic temperature factor and G represents the Hermite polynomials in reciprocal space. If the OPP temperature factor is expanded in the coordinate system which diagonalizes jk , simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates u j (Dawson et al., 1967; Coppens, 1980; Tanaka & Marumo, 1983).

The relation between the cumulants jkl and the quasimoments c are apparent from comparison of (1.2.12.8) and (1.2.12.4): jkl

c jkl jkl

c jklm jklm

c jklmn jklmn

c jklmnp jklmnp 10 jkl mnp :

1.2.12.4. Relative merits of the three expansions

1:2:12:9

The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982), Kuhs (1983), and by Scheringer (1985b). In general, the Gram– Charlier expression is found to be preferable because it gives a better ﬁt in the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the oneparticle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real- and reciprocal-space expressions. The terms of the OPP model have a speciﬁc physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969; Coppens, 1980), provided the potential function itself can be assumed to be temperature independent. It has recently been shown that the Edgeworth expansion (1.2.12.5a) always has negative regions (Scheringer, 1985b). This implies that it is not a realistic description of a vibrating atom.

The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant jkl contributes not only to the coefﬁcient of Hjkl , but also to higher-order terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a ﬁnite truncation of (1.2.12.6), the probability distribution cannot be represented by a ﬁnite number of terms. This is a serious difﬁculty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type. 1.2.12.3. The one-particle potential (OPP) model When an atom is considered as an independent oscillator vibrating in a potential well V
u, its distribution may be described by Boltzmann statistics. P
u N expf V
u=kTg,

1:2:12:10

with N, the normalization constant, deﬁned by P
u du 1. The classical expression (1.2.12.10) is valid in the high-temperature limit for which kT V
u. Following Dawson (1967) and Willis (1969), the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates:

1.2.13. The generalized structure factor In the generalized structure-factor formalism developed by Dawson (1975), the complex nature of both the atomic scattering factor and the generalized temperature factor are taken into account. We write for the atomic scattering factor:

V V0 j u j jk u j uk jkl u j uk ul jklm u j uk ul um :

0

fj
H fj; c
H ifj; a
H fj ifj

1:2:12:11

Tj
H Tj; c
H iTj; a
H

The equilibrium condition gives j 0. Substitution into (1.2.12.10) leads to an expression which may be simpliﬁed by the assumption that the leading term is the harmonic component represented by jk : f1

0

j k l

jkl u u u

0

j k l m

jklm u u u u

g,

1:2:13:1a
1:2:13:1b

and F
H A
H iB
H,

0

P
u N expf jk u j uk g

00

1:2:13:2

where the subscripts c and a refer to the centrosymmetric and acentric components, respectively. Substitution in (1.2.4.2) gives for the real and imaginary components A and B of F
H

1:2:12:12

23

1. GENERAL RELATIONSHIPS AND TECHNIQUES A
H

j

0

fj; c fj cos
2H rj Tc

into the formalism and the treatment of thermal motion are interlinked. It is important that the complexities of the thermal probability distribution function can often be reduced by very low temperature experimentation. Results obtained with the multipole formalism for atomic asphericity can be used to derive physical properties and d-orbital populations of transition-metal atoms (IT C, 1999). In such applications, the deconvolution of the charge density and the thermal vibrations is essential. This deconvolution is dependent on the adequacy of the models summarized here.

sin
2H rj Ta

00

fj; a fj cos
2H rj Ta sin
2H rj Tc

1:2:13:3a

and B
H

j

0

fj; c fj cos
2H rj Ta sin
2H rj Tc 00

fj; a fj cos
2H rj Tc

sin
2H rj Ta

1:2:13:3b

Acknowledgements

(McIntyre et al., 1980; Dawson, 1967). Expressions (1.2.13.3) illustrate the relation between valencedensity anisotropy and anisotropy of thermal motion.

The author would like to thank several of his colleagues who gave invaluable criticism of earlier versions of this manuscript. Corrections and additions were made following comments by P. J. Becker, D. Feil, N. K. Hansen, G. McIntyre, E. N. Maslen, S. Ohba, C. Scheringer and D. Schwarzenbach. Z. Su contributed to the revised version of the manuscript. Support of this work by the US National Science Foundation (CHE8711736 and CHE9317770) is gratefully acknowledged.

1.2.14. Conclusion This chapter summarizes mathematical developments of the structure-factor formalism. The introduction of atomic asphericity

24

International Tables for Crystallography (2006). Vol. B, Chapter 1.3, pp. 25–98.

1.3. Fourier transforms in crystallography: theory, algorithms and applications BY G. BRICOGNE which has long been adopted in several applied ﬁelds, in particular electrical engineering, with considerable success; the extra work involved handsomely pays for itself by allowing the three different types of Fourier transformations to be treated together, and by making all properties of the Fourier transform consequences of a single property (the convolution theorem). This is particularly useful in all questions related to the sampling theorem; (ii) the various numerical algorithms have been presented as the consequences of basic algebraic phenomena involving Abelian groups, rings and ﬁnite ﬁelds; this degree of formalization greatly helps the subsequent incorporation of symmetry; (iii) the algebraic nature of space groups has been reemphasized so as to build up a framework which can accommodate both the phenomena used to factor the discrete Fourier transform and those which underlie the existence (and lead to the classiﬁcation) of space groups; this common ground is found in the notion of module over a group ring (i.e. integral representation theory), which is then applied to the formulation of a large number of algorithms, many of which are new; (iv) the survey of the main types of crystallographic computations has tried to highlight the roles played by various properties of the Fourier transformation, and the ways in which a better exploitation of these properties has been the driving force behind the discovery of more powerful methods. In keeping with this philosophy, the theory is presented ﬁrst, followed by the crystallographic applications. There are ‘forward references’ from mathematical results to the applications which later invoke them (thus giving ‘real-life’ examples rather than artiﬁcial ones), and ‘backward references’ as usual. In this way, the internal logic of the mathematical developments – the surest guide to future innovations – can be preserved, whereas the alternative solution of relegating these to appendices tends on the contrary to obscure that logic by subordinating it to that of the applications. It is hoped that this attempt at an overall presentation of the main features of Fourier transforms and of their ubiquitous role in crystallography will be found useful by scientists both within and outside the ﬁeld.

1.3.1. General introduction Since the publication of Volume II of International Tables, most aspects of the theory, computation and applications of Fourier transforms have undergone considerable development, often to the point of being hardly recognizable. The mathematical analysis of the Fourier transformation has been extensively reformulated within the framework of distribution theory, following Schwartz’s work in the early 1950s. The computation of Fourier transforms has been revolutionized by the advent of digital computers and of the Cooley–Tukey algorithm, and progress has been made at an ever-accelerating pace in the design of new types of algorithms and in optimizing their interplay with machine architecture. These advances have transformed both theory and practice in several ﬁelds which rely heavily on Fourier methods; much of electrical engineering, for instance, has become digital signal processing. By contrast, crystallography has remained relatively unaffected by these developments. From the conceptual point of view, oldfashioned Fourier series are still adequate for the quantitative description of X-ray diffraction, as this rarely entails consideration of molecular transforms between reciprocal-lattice points. From the practical point of view, three-dimensional Fourier transforms have mostly been used as a tool for visualizing electron-density maps, so that only moderate urgency was given to trying to achieve ultimate efﬁciency in these relatively infrequent calculations. Recent advances in phasing and reﬁnement methods, however, have placed renewed emphasis on concepts and techniques long used in digital signal processing, e.g. ﬂexible sampling, Shannon interpolation, linear ﬁltering, and interchange between convolution and multiplication. These methods are iterative in nature, and thus generate a strong incentive to design new crystallographic Fourier transform algorithms making the fullest possible use of all available symmetry to save both storage and computation. As a result, need has arisen for a modern and coherent account of Fourier transform methods in crystallography which would provide: (i) a simple and foolproof means of switching between the three different guises in which the Fourier transformation is encountered (Fourier transforms, Fourier series and discrete Fourier transforms), both formally and computationally; (ii) an up-to-date presentation of the most important algorithms for the efﬁcient numerical calculation of discrete Fourier transforms; (iii) a systematic study of the incorporation of symmetry into the calculation of crystallographic discrete Fourier transforms; (iv) a survey of the main types of crystallographic computations based on the Fourier transformation. The rapid pace of progress in these ﬁelds implies that such an account would be struck by quasi-immediate obsolescence if it were written solely for the purpose of compiling a catalogue of results and formulae ‘customized’ for crystallographic use. Instead, the emphasis has been placed on a mode of presentation in which most results and formulae are derived rather than listed. This does entail a substantial mathematical overhead, but has the advantage of preserving in its ‘native’ form the context within which these results are obtained. It is this context, rather than any particular set of results, which constitutes the most fertile source of new ideas and new applications, and as such can have any hope at all of remaining useful in the long run. These conditions have led to the following choices: (i) the mathematical theory of the Fourier transformation has been cast in the language of Schwartz’s theory of distributions

1.3.2. The mathematical theory of the Fourier transformation 1.3.2.1. Introduction The Fourier transformation and the practical applications to which it gives rise occur in three different forms which, although they display a similar range of phenomena, normally require distinct formulations and different proof techniques: (i) Fourier transforms, in which both function and transform depend on continuous variables; (ii) Fourier series, which relate a periodic function to a discrete set of coefﬁcients indexed by n-tuples of integers; (iii) discrete Fourier transforms, which relate ﬁnite-dimensional vectors by linear operations representable by matrices. At the same time, the most useful property of the Fourier transformation – the exchange between multiplication and convolution – is mathematically the most elusive and the one which requires the greatest caution in order to avoid writing down meaningless expressions. It is the unique merit of Schwartz’s theory of distributions (Schwartz, 1966) that it affords complete control over all the troublesome phenomena which had previously forced mathematicians to settle for a piecemeal, fragmented theory of the Fourier transformation. By its ability to handle rigorously highly ‘singular’

25 Copyright 2006 International Union of Crystallography

1. GENERAL RELATIONSHIPS AND TECHNIQUES objects (especially -functions, their derivatives, their tensor products, their products with smooth functions, their translates and lattices of these translates), distribution theory can deal with all the major properties of the Fourier transformation as particular instances of a single basic result (the exchange between multiplication and convolution), and can at the same time accommodate the three previously distinct types of Fourier theories within a unique framework. This brings great simpliﬁcation to matters of central importance in crystallography, such as the relations between (a) periodization, and sampling or decimation; (b) Shannon interpolation, and masking by an indicator function; (c) section, and projection; (d) differentiation, and multiplication by a monomial; (e) translation, and phase shift. All these properties become subsumed under the same theorem. This striking synthesis comes at a slight price, which is the relative complexity of the notion of distribution. It is ﬁrst necessary to establish the notion of topological vector space and to gain sufﬁcient control (or, at least, understanding) over convergence behaviour in certain of these spaces. The key notion of metrizability cannot be circumvented, as it underlies most of the constructs and many of the proof techniques used in distribution theory. Most of Section 1.3.2.2 builds up to the fundamental result at the end of Section 1.3.2.2.6.2, which is basic to the deﬁnition of a distribution in Section 1.3.2.3.4 and to all subsequent developments. The reader mostly interested in applications will probably want to reach this section by starting with his or her favourite topic in Section 1.3.4, and following the backward references to the relevant properties of the Fourier transformation, then to the proof of these properties, and ﬁnally to the deﬁnitions of the objects involved. Hopefully, he or she will then feel inclined to follow the forward references and thus explore the subject from the abstract to the practical. The books by Dieudonne´ (1969) and Lang (1965) are particularly recommended as general references for all aspects of analysis and algebra.

S
x 1 if x 2 S

0 if x 2 = S:

1.3.2.2.1. Metric and topological notions in Rn The set Rn can be endowed with the structure of a vector space of dimension n over R, and can be made into a Euclidean space by treating its standard basis as an orthonormal basis and deﬁning the Euclidean norm: n 1=2 P 2 kxk xi : i1

By misuse of notation, x will sometimes also designate the column vector of coordinates of x 2 Rn ; if these coordinates are referred to an orthonormal basis of Rn , then kxk
xT x1=2 ,

where xT denotes the transpose of x. The distance between two points x and y deﬁned by d
x, y kx yk allows the topological structure of R to be transferred to Rn , making it a metric space. The basic notions in a metric space are those of neighbourhoods, of open and closed sets, of limit, of continuity, and of convergence (see Section 1.3.2.2.6.1). A subset S of Rn is bounded if sup kx yk < 1 as x and y run through S; it is closed if it contains the limits of all convergent sequences with elements in S. A subset K of Rn which is both bounded and closed has the property of being compact, i.e. that whenever K has been covered by a family of open sets, a ﬁnite subfamily can be found which sufﬁces to cover K. Compactness is a very useful topological property for the purpose of proof, since it allows one to reduce the task of examining inﬁnitely many local situations to that of examining only ﬁnitely many of them. 1.3.2.2.2. Functions over Rn

1.3.2.2. Preliminary notions and notation

Let ' be a complex-valued function over Rn . The support of ', denoted Supp ', is the smallest closed subset of Rn outside which ' vanishes identically. If Supp ' is compact, ' is said to have compact support. If t 2 Rn , the translate of ' by t, denoted t ', is deﬁned by

Throughout this text, R will denote the set of real numbers, Z the set of rational (signed) integers and N the set of natural (unsigned) integers. The symbol Rn will denote the Cartesian product of n copies of R: Rn R . . . R
n times, n 1,

t '
x '
x

t:

Its support is the geometric translate of that of ':

so that an element x of Rn is an n-tuple of real numbers:

Supp t ' fx tjx 2 Supp 'g: If A is a non-singular linear transformation in Rn , the image of ' by A, denoted A# ', is deﬁned by

x
x1 , . . . , xn :

Similar meanings will be attached to Zn and Nn . The symbol C will denote the set of complex numbers. If z 2 C, its modulus will be denoted by jzj, its conjugate by z (not z ), and its real and imaginary parts by
z and
z: 1
z 12
z z,
z
z z: 2i If X is a ﬁnite set, then jX j will denote the number of its elements. If mapping f sends an element x of set X to the element f
x of set Y, the notation

A# '
x 'A 1
x:

Its support is the geometric image of Supp ' under A: Supp A# ' fA
xjx 2 Supp 'g: If S is a non-singular afﬁne transformation in Rn of the form S
x A
x b

with A linear, the image of ' by S is S # ' b
A# ', i.e.

f : x 7 ! f
x

S # '
x 'A 1
x

will be used; the plain arrow ! will be reserved for denoting limits, as in x p x lim 1 e : !1 p If X is any set and S is a subset of X, the indicator function s of S is the real-valued function on X deﬁned by

b:

Its support is the geometric image of Supp ' under S: Supp S # ' fS
xjx 2 Supp 'g: It may be helpful to visualize the process of forming the image of a function by a geometric operation as consisting of applying that operation to the graph of that function, which is equivalent to

26

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY applying the inverse transformation to the coordinates x. This use of the inverse later affords the ‘left-representation property’ [see Section 1.3.4.2.2.2(e)] when the geometric operations form a group, which is of fundamental importance in the treatment of crystallographic symmetry (Sections 1.3.4.2.2.4, 1.3.4.2.2.5).

mean: a Cauchy sequence of integrable functions may converge to a non-integrable function. To obtain the property of completeness, which is fundamental in functional analysis, it was necessary to extend the notion of integral. This was accomplished by Lebesgue [see Berberian (1962), Dieudonne´ (1970), or Chapter 1 of Dym & McKean (1972) and the references therein, or Chapter 9 of Sprecher (1970)], and entailed identifying functions which differed only on a subset of zero measure in Rn (such functions are said to be equal ‘almost everywhere’). The vector spaces Lp
Rn consisting of function classes f modulo this identiﬁcation for which !1=p R p n j f
xj d x < 1 kfkp

1.3.2.2.3. Multi-index notation When dealing with functions in n variables and their derivatives, considerable abbreviation of notation can be obtained through the use of multi-indices. A multi-index p 2 Nn is an n-tuple of natural integers: p
p1 , . . . , pn . The length of p is deﬁned as n P pi , jpj

Rn

i1

are then complete for the topology induced by the norm k:kp : the limit of every Cauchy sequence of functions in Lp is itself a function in Lp (Riesz–Fischer theorem). The space L1
Rn consists of those function classes f such that R k f k1 j f
xj dn x < 1

and the following abbreviations will be used:
i
ii
iii
iv
v

vi

xp xp11 . . . xpnn @f Di f @i f @xi

Rn

@ jpj f Dp f Dp11 . . . Dpnn f p1 @x1 . . . @xpnn q p if and only if qi pi for all i 1, . . . , n

p

q
p1

q1 , . . . , pn

which are called summable or absolutely integrable. The convolution product: R
f g
x f
yg
x y dn y

qn

Rn

p! p1 ! . . . pn ! p1 p pn
vii ... : q q1 qn Leibniz’s formula for the repeated differentiation of products then assumes the concise form X p Dp
fg Dp q fDq g, q qp

B B B B
rrT f B B B 2 @ @ f @xn @x1

R

yg
y dn y
g f
x

f
x

Rn

which makes it into a Hilbert space. The Cauchy–Schwarz inequality

In certain sections the notation rf will be used for the gradient vector of f, and the notation
rrT f for the Hessian matrix of its mixed second-order partial derivatives: 0 0 1 1 @ @f B @x1 C B @x1 C B B C C B . C B . C B B C . . r B . C, rf B . C C, B B C C @ @ A @ @f A @2f @x21 .. .

n

is well deﬁned; combined with the vector space structure of L1 , it makes L1 into a (commutative) convolution algebra. However, this algebra has no unit element: there is no f 2 L1 such that f g g for all g 2 L1 ; it has only approximate units, i.e. sequences
f such that f g tends to g in the L1 topology as ! 1. This is one of the starting points of distribution theory. The space L2
Rn of square-integrable functions can be endowed with a scalar product R
f , g f
xg
x dn x

while the Taylor expansion of f to order m about x a reads X 1 f
x Dp f
a
x ap o
kx akm : p! jpjm

@xn 0

R

j
f , gj
f , f
g, g1=2

generalizes the fact that the absolute value of the cosine of an angle is less than or equal to 1. The space L1
Rn is deﬁned as the space of functions f such that !1=p R j f
xjp dn x < 1: k f k1 lim k f kp lim p!1

@xn 1 @2f ... @x1 @xn C C C .. .. C : . . C C C @2f A ... @x2n

p!1

Rn

The quantity k f k1 is called the ‘essential sup norm’ of f, as it is the smallest positive number which j f
xj exceeds only on a subset of zero measure in Rn . A function f 2 L1 is called essentially bounded. 1.3.2.2.5. Tensor products. Fubini’s theorem Let f 2 L1
Rm , g 2 L1
Rn . Then the function f g :
x, y 7 ! f
xg
y

1.3.2.2.4. Integration, Lp spaces The Riemann integral used in elementary calculus suffers from the drawback that vector spaces of Riemann-integrable functions over Rn are not complete for the topology of convergence in the

27

is called the tensor product of f and g, and belongs to L1
Rm Rn . The ﬁnite linear combinations of functions of the form f g span a subspace of L1
Rm Rn called the tensor product of L1
Rm and L1
Rn and denoted L1
Rm L1
Rn .

1. GENERAL RELATIONSHIPS AND TECHNIQUES The integration of a general function over Rm Rn may be accomplished in two steps according to Fubini’s theorem. Given F 2 L1
Rm Rn , the functions R F1 : x 7 ! F
x, y dn y

limit and continuity may be deﬁned by means of sequences. For nonmetrizable topologies, these notions are much more difﬁcult to handle, requiring the use of ‘ﬁlters’ instead of sequences. In some spaces E, a topology may be most naturally deﬁned by a family of pseudo-distances
d 2A , where each d satisﬁes (i) and (iii) but not (ii). Such spaces are called uniformizable. If for every pair
x, y 2 E E there exists 2 A such that d
x, y 6 0, then the separation property can be recovered. If furthermore a countable subfamily of the d sufﬁces to deﬁne the topology of E, the latter can be shown to be metrizable, so that limiting processes in E may be studied by means of sequences.

Rn

F2 : y 7 !

R

F
x, y dm x

Rm

exist for almost all x 2 Rm and almost all y 2 Rn , respectively, are integrable, and R R R F
x, y dm x dn y F1
x dm x F2
y dn y: Rm Rn

Rm

Rn

1.3.2.2.6.2. Topological vector spaces The function spaces E of interest in Fourier analysis have an underlying vector space structure over the ﬁeld C of complex numbers. A topology on E is said to be compatible with a vector space structure on E if vector addition [i.e. the map
x, y 7 ! x y] and scalar multiplication [i.e. the map
, x 7 ! x] are both continuous; E is then called a topological vector space. Such a topology may be deﬁned by specifying a ‘fundamental system S of neighbourhoods of 0’, which can then be translated by vector addition to construct neighbourhoods of other points x 6 0. A norm on a vector space E is a non-negative real-valued function on E E such that

Conversely, if any one of the integrals R jF
x, yj dm x dn y
i Rm Rn

ii
iii

R

R

m

R

n

R

n

R

n

R

R

R

!

jF
x, yj d y dm x

m

m

!

jF
x, yj d x dn y

is ﬁnite, then so are the other two, and the identity above holds. It is then (and only then) permissible to change the order of integrations. Fubini’s theorem is of fundamental importance in the study of tensor products and convolutions of distributions.

i0 0

ii 0

1.3.2.2.6.1. General topology Most topological notions are ﬁrst encountered in the setting of metric spaces. A metric space E is a set equipped with a distance function d from E E to the non-negative reals which satisﬁes: 8x, y 2 E

ii d
x, y 0 iff x y
iii d
x, z d
x, y d
y, z 8x, y, z 2 E

if and only if x 0;

Subsets of E deﬁned by conditions of the form
x r with r > 0 form a fundamental system of neighbourhoods of 0. The corresponding topology makes E a normed space. This topology is metrizable, since it is equivalent to that derived from the translation-invariant distance d
x, y
x y. Normed spaces which are complete, i.e. in which all Cauchy sequences converge, are called Banach spaces; they constitute the natural setting for the study of differential calculus. A semi-norm on a vector space E is a positive real-valued function on E E which satisﬁes (i0 ) and (iii0 ) but not (ii0 ). Given a set of semi-norms on E such that any pair (x, y) in E E is separated by at least one 2 , let B be the set of those subsets ; r of E deﬁned by a condition of the form
x r with 2 and r > 0; and let S be the set of ﬁnite intersections of elements of B. Then there exists a unique topology on E for which S is a fundamental system of neighbourhoods of 0. This topology is uniformizable since it is equivalent to that derived from the family of translation-invariant pseudo-distances
x, y 7 !
x y. It is metrizable if and only if it can be constructed by the above procedure with a countable set of semi-norms. If furthermore E is complete, E is called a Fre´chet space. If E is a topological vector space over C, its dual E is the set of all linear mappings from E to C (which are also called linear forms, or linear functionals, over E). The subspace of E consisting of all linear forms which are continuous for the topology of E is called the topological dual of E and is denoted E0 . If the topology on E is metrizable, then the continuity of a linear form T 2 E0 at f 2 E can be ascertained by means of sequences, i.e. by checking that the sequence T
fj of complex numbers converges to T
f in C whenever the sequence
fj converges to f in E.

Geometric intuition, which often makes ‘obvious’ the topological properties of the real line and of ordinary space, cannot be relied upon in the study of function spaces: the latter are inﬁnitedimensional, and several inequivalent notions of convergence may exist. A careful analysis of topological concepts and of their interrelationship is thus a necessary prerequisite to the study of these spaces. The reader may consult Dieudonne´ (1969, 1970), Friedman (1970), Tre`ves (1967) and Yosida (1965) for detailed expositions.

d
x, y d
y, x

x 0

for all 2 C and x 2 E;

iii
x y
x
y for all x, y 2 E:

1.3.2.2.6. Topology in function spaces

i

x jj
x

(symmetry);

(separation); (triangular inequality).

By means of d, the following notions can be deﬁned: open balls, neighbourhoods; open and closed sets, interior and closure; convergence of sequences, continuity of mappings; Cauchy sequences and completeness; compactness; connectedness. They sufﬁce for the investigation of a great number of questions in analysis and geometry (see e.g. Dieudonne´, 1969). Many of these notions turn out to depend only on the properties of the collection O
E of open subsets of E: two distance functions leading to the same O
E lead to identical topological properties. An axiomatic reformulation of topological notions is thus possible: a topology in E is a collection O
E of subsets of E which satisfy suitable axioms and are deemed open irrespective of the way they are obtained. From the practical standpoint, however, a topology which can be obtained from a distance function (called a metrizable topology) has the very useful property that the notions of closure,

1.3.2.3. Elements of the theory of distributions 1.3.2.3.1. Origins At the end of the 19th century, Heaviside proposed under the name of ‘operational calculus’ a set of rules for solving a class of

28

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY differential, partial differential and integral equations encountered in electrical engineering (today’s ‘signal processing’). These rules worked remarkably well but were devoid of mathematical justiﬁcation (see Whittaker, 1928). In 1926, Dirac introduced his famous -function [see Dirac (1958), pp. 58–61], which was found to be related to Heaviside’s constructs. Other singular objects, together with procedures to handle them, had already appeared in several branches of analysis [Cauchy’s ‘principal values’; Hadamard’s ‘ﬁnite parts’ (Hadamard, 1932, 1952); Riesz’s regularization methods for certain divergent integrals (Riesz, 1938, 1949)] as well as in the theories of Fourier series and integrals (see e.g. Bochner, 1932, 1959). Their very deﬁnition often verged on violating the rigorous rules governing limiting processes in analysis, so that subsequent recourse to limiting processes could lead to erroneous results; ad hoc precautions thus had to be observed to avoid mistakes in handling these objects. In 1945–1950, Laurent Schwartz proposed his theory of distributions (see Schwartz, 1966), which provided a uniﬁed and deﬁnitive treatment of all these questions, with a striking combination of rigour and simplicity. Schwartz’s treatment of Dirac’s -function illustrates his approach in a most direct fashion. Dirac’s original deﬁnition reads:

T : ' 7 ! '
0:

It is the latter functional which constitutes the proper deﬁnition of . The previous paradoxes arose because one insisted on writing down the simple linear operation T in terms of an integral. The essence of Schwartz’s theory of distributions is thus that, rather than try to deﬁne and handle ‘generalized functions’ via sequences such as
f [an approach adopted e.g. by Lighthill (1958) and Erde´lyi (1962)], one should instead look at them as continuous linear functionals over spaces of well behaved functions. There are many books on distribution theory and its applications. The reader may consult in particular Schwartz (1965, 1966), Gel’fand & Shilov (1964), Bremermann (1965), Tre`ves (1967), Challifour (1972), Friedlander (1982), and the relevant chapters of Ho¨rmander (1963) and Yosida (1965). Schwartz (1965) is especially recommended as an introduction. 1.3.2.3.2. Rationale The guiding principle which leads to requiring that the functions ' above (traditionally called ‘test functions’) should be well behaved is that correspondingly ‘wilder’ behaviour can then be accommodated R in the limiting behaviour of the f while still keeping the integrals Rn f ' dn x under control. Thus (i) to minimize restrictions on the limiting behaviour of the f at inﬁnity, the '’s will be chosen to have compact support; (ii) to minimize restrictions on the local behaviour of the f , the '’s will be chosen inﬁnitely differentiable. To ensure further the continuity of functionals such as T with respect to the test function ' as the f go increasingly wild, very strong control will have to be exercised in the way in which a sequence
'j of test functions will be said to converge towards a limiting ': conditions will have to be imposed not only on the values of the functions 'j , but also on those of all their derivatives. Hence, deﬁning a strong enough topology on the space of test functions ' is an essential prerequisite to the development of a satisfactory theory of distributions.

i
x 0 for x 6 0, R
ii Rn
x dn x 1:

These two conditions are irreconcilable with Lebesgue’s theory of integration: by (i), vanishes almost everywhere, so that its integral in (ii) must be 0, not 1. A better deﬁnition consists in specifying that R
iii Rn
x'
x dn x '
0

for any function ' sufﬁciently well behaved near x 0. This is related to the problem of ﬁnding a unit for convolution (Section 1.3.2.2.4). As will now be seen, this deﬁnition is still unsatisfactory. Let the sequence
f in L1
Rn be an approximate convolution unit, e.g. 1=2 f
x exp
12 2 kxk2 : 2

1.3.2.3.3. Test-function spaces With this rationale in mind, the following function spaces will be deﬁned for any open subset of Rn (which may be the whole of Rn ): (a) E
is the space of complex-valued functions over which are indeﬁnitely differentiable; (b) D
is the subspace of E
consisting of functions with (unspeciﬁed) compact support contained in Rn ; (c) DK
is the subspace of D
consisting of functions whose (compact) support is contained within a ﬁxed compact subset K of

. When is unambiguously deﬁned by the context, we will simply write E, D, DK . It sometimes sufﬁces to require the existence of continuous derivatives only up to ﬁnite order m inclusive. The corresponding
m spaces are then denoted E
m , D
m , DK with the convention that if m 0, only continuity is required. The topologies on these spaces constitute the most important ingredients of distribution theory, and will be outlined in some detail.

Then for any well behaved function ' the integrals R f
x'
x dn x Rn

exist, and the sequence of their numerical values tends to '
0. It is tempting to combine this with (iii) to conclude that is the limit of the sequence
f as ! 1. However, lim f
x 0 as ! 1

almost everywhere in Rn and the crux of the problem is that R '
0 lim f
x'
x dn x !1

6

Rh

Rn

Rn

i lim fv
x '
x dn x 0

!1

because the sequence
f does not satisfy the hypotheses of Lebesgue’s dominated convergence theorem. Schwartz’s solution to this problem is deceptively simple: the regular behaviour one is trying to capture is an attribute not of the sequence of functions
f , but of the sequence of continuous linear functionals R T : ' 7 ! f
x'
x dn x

1.3.2.3.3.1. Topology on E
It is deﬁned by the family of semi-norms ' 2 E
7 ! p; K
' sup jDp '
xj,

Rn

x2K

where p is a multi-index and K a compact subset of . A

which has as a limit the continuous functional

29

1. GENERAL RELATIONSHIPS AND TECHNIQUES fundamental system S of neighbourhoods of the origin in E
is given by subsets of E
of the form

1.3.2.3.4. Definition of distributions A distribution T on is a linear form over D
, i.e. a map

V
m, ", K f' 2 E
jjpj m ) p, K
' < "g

T : ' 7 ! hT, 'i

for all natural integers m, positive real ", and compact subset K of . Since a countable family of compact subsets K sufﬁces to cover , and since restricted values of " of the form " 1=N lead to the same topology, S is equivalent to a countable system of neighbourhoods and hence E
is metrizable. Convergence in E may thus be deﬁned by means of sequences. A sequence
' in E will be said to converge to 0 if for any given V
m, ", K there exists 0 such that ' 2 V
m, ", K whenever > 0 ; in other words, if the ' and all their derivatives Dp ' converge to 0 uniformly on any given compact K in .

which associates linearly a complex number hT, 'i to any ' 2 D
, and which is continuous for the topology of that space. In the terminology of Section 1.3.2.2.6.2, T is an element of D0
, the topological dual of D
. Continuity over D is equivalent to continuity over DK for all compact K contained in , and hence to the condition that for any sequence
' in D such that (i) Supp ' is contained in some compact K independent of , (ii) the sequences
jDp ' j converge uniformly to 0 on K for all multi-indices p; then the sequence of complex numbers hT, ' i converges to 0 in C. If the continuity of a distribution T requires (ii) for jpj m only, T may be deﬁned over D
m and thus T 2 D0
m ; T is said to be a distribution of ﬁnite order m. In particular, for m 0, D
0 is the space of continuous functions with compact support, and a distribution T 2 D0
0 is a (Radon) measure as used in the theory of integration. Thus measures are particular cases of distributions. Generally speaking, the larger a space of test functions, the smaller its topological dual:

1.3.2.3.3.2. Topology on Dk
It is deﬁned by the family of semi-norms ' 2 DK
7 ! p
' sup jDp '
xj, x2K

where K is now ﬁxed. The fundamental system S of neighbourhoods of the origin in DK is given by sets of the form V
m, " f' 2 DK
jjpj m ) p
' < "g: It is equivalent to the countable subsystem of the V
m, 1=N, hence DK
is metrizable. Convergence in DK may thus be deﬁned by means of sequences. A sequence
' in DK will be said to converge to 0 if for any given V
m, " there exists 0 such that ' 2 V
m, " whenever > 0 ; in other words, if the ' and all their derivatives Dp ' converge to 0 uniformly in K.

m < n ) D
m D
n ) D0
n D0
m :

This clearly results from the observation that if the '’s are allowed to be less regular, then less wildness can be accommodated in T if the continuity of the map ' 7 ! hT, 'i with respect to ' is to be preserved.

1.3.2.3.3.3. Topology on D
It is deﬁned by the fundamental system of neighbourhoods of the origin consisting of sets of the form V
m,
"

p

1.3.2.3.5. First examples of distributions

' 2 D
jjpj m ) sup jD '
xj < " for all , kxk

where (m) is an increasing sequence
m of integers tending to 1 and (") is a decreasing sequence
" of positive reals tending to 0, as ! 1. This topology is not metrizable, because the sets of sequences (m) and (") are essentially uncountable. It can, however, be shown to be the inductive limit of the topology of the subspaces DK , in the following sense: V is a neighbourhood of the origin in D if and only if its intersection with DK is a neighbourhood of the origin in DK for any given compact K in . A sequence
' in D will thus be said to converge to 0 in D if all the ' belong to some DK (with K a compact subset of independent of ) and if
' converges to 0 in DK . As a result, a complex-valued functional T on D will be said to be continuous for the topology of D if and only if, for any given compact K in , its restriction to DK is continuous for the topology of DK , i.e. maps convergent sequences in DK to convergent sequences in C. This property of D, i.e. having a non-metrizable topology which is the inductive limit of metrizable topologies in its subspaces DK , conditions the whole structure of distribution theory and dictates that of many of its proofs.

(i) The linear map ' 7 ! h, 'i '
0 is a measure (i.e. a zeroth-order distribution) called Dirac’s measure or (improperly) Dirac’s ‘-function’. (ii) The linear map ' 7 ! h
a , 'i '
a is called Dirac’s measure at point a 2 Rn . (iii) The linear map ' 7 !
1p Dp '
a is a distribution of order m jpj > 0, and hence isP not a measure. (iv) The linear map ' 7 ! >0 '
is a distribution of inﬁnite order on R: the order of differentiation is bounded for each ' (because ' has compact support) but is not as ' varies. (v) If
p is a sequence of multi-indices p
p1 , . . . , pn such P that jp j ! 1 as ! 1, then the linear map ' 7 ! >0
Dp '
p is a distribution of inﬁnite order on Rn . 1.3.2.3.6. Distributions associated to locally integrable functions a complex-valued function over such that R Let f be n j f
xj d x exists for any given compact K in ; f is then called K locally integrable. The linear mapping from D
to C deﬁned by R ' 7 ! f
x'
x dn x

may then be shown to be continuous over D
. It thus deﬁnes a distribution Tf 2 D0
: R hTf , 'i f
x'
x dn x:

m

1.3.2.3.3.4. Topologies on E
m , Dk , D
m These are deﬁned similarly, but only involve conditions on derivatives up to order m.

As the continuity of Tf only requires that ' 2 D
0
, Tf is actually a Radon measure.

30

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY It can be shown that two locally integrable functions f and g deﬁne the same distribution, i.e.

1.3.2.3.9. Operations on distributions As a general rule, the deﬁnitions are chosen so that the operations coincide with those on functions whenever a distribution is associated to a function. Most deﬁnitions consist in transferring to a distribution T an operation which is well deﬁned on ' 2 D by ‘transposing’ it in the duality product hT, 'i; this procedure will map T to a new distribution provided the original operation maps D continuously into itself.

hTf , 'i hTK , 'i for all ' 2 D, if and only if they are equal almost everywhere. The classes of locally integrable functions modulo this equivalence form a vector space denoted L1loc
; each element of L1loc
may therefore be identiﬁed with the distribution Tf deﬁned by any one of its representatives f. 1.3.2.3.7. Support of a distribution

1.3.2.3.9.1. Differentiation

0

A distribution T 2 D
is said to vanish on an open subset ! of if it vanishes on all functions in D
!, i.e. if hT, 'i 0 whenever ' 2 D
!. The support of a distribution T, denoted Supp T, is then deﬁned as the complement of the set-theoretic union of those open subsets ! on which T vanishes; or equivalently as the smallest closed subset of outside which T vanishes. When T Tf for f 2 L1loc
, then Supp T Supp f , so that the two notions coincide. Clearly, if Supp T and Supp ' are disjoint subsets of , then hT, 'i 0. It can be shown that any distribution T 2 D0 with compact support may be extended from D to E while remaining continuous, so that T 2 E 0 ; and that conversely, if S 2 E 0 , then its restriction T to D is a distribution with compact support. Thus, the topological dual E 0 of E consists of those distributions in D0 which have compact support. This is intuitively clear since, if the condition of having compact support is fulﬁlled by T, it needs no longer be required of ', which may then roam through E rather than D.

(a) Deﬁnition and elementary properties If T is a distribution on Rn , its partial derivative @i T with respect to xi is deﬁned by [email protected] T, 'i hT, @i 'i for all ' 2 D. This does deﬁne a distribution, because the partial differentiations ' 7 ! @i ' are continuous for the topology of D. Suppose that T Tf with f a locally integrable function such that @i f exists and is almost everywhere continuous. Then integration by parts along the xi axis gives R @i f
xl , . . . , xi , . . . , xn '
xl , . . . , xi , . . . , xn dxi Rn

f '
xl , . . . , 1, . . . , xn
f '
xl , . . . , 1, . . . , xn R f
xl , . . . , xi , . . . , xn @i '
xl , . . . , xi , . . . , xn dxi ; Rn

the integrated term vanishes, since ' has compact support, showing that @i Tf [email protected] f . The test functions ' 2 D are inﬁnitely differentiable. Therefore, transpositions like that used to deﬁne @i T may be repeated, so that any distribution is inﬁnitely differentiable. For instance,

1.3.2.3.8. Convergence of distributions A sequence
Tj of distributions will be said to converge in D0 to a distribution T as j ! 1 if, for any given ' 2 D, the sequence of complex numbers
hTj , 'i converges in C to the complex number hT, 'i. P 0 A series 1 j0 Tj of distributions will be said to converge in D and toPhave distribution S as its sum if the sequence of partial sums Sk kj0 converges to S. These deﬁnitions of convergence in D0 assume that the limits T and S are known in advance, and are distributions. This raises the question of the completeness of D0 : if a sequence
Tj in D0 is such that the sequence
hTj , 'i has a limit in C for all ' 2 D, does the map

[email protected] T, 'i [email protected] T, @i 'i hT, @ij2 'i, hDp T, 'i
1jpj hT, Dp 'i, hT, 'i hT, 'i,

where is the Laplacian operator. The derivatives of Dirac’s distribution are hDp , 'i
1jpj h, Dp 'i
1jpj Dp '
0: It is remarkable that differentiation is a continuous operation for the topology on D0 : if a sequence
Tj of distributions converges to distribution T, then the sequence
Dp Tj of derivatives converges to Dp T for any multi-index p, since as j ! 1

' 7 ! lim hTj , 'i j!1

deﬁne a distribution T 2 D0 ? In other words, does the limiting process preserve continuity with respect to '? It is a remarkable theorem that, because of the strong topology on D, this is actually the case. An analogous statement holds for series. This notion of convergence does not coincide with any of the classical notions used for ordinary functions: for example, the sequence
' with '
x cos x converges to 0 in D0
R, but fails to do so by any of the standard criteria. An example of convergent sequences of distributions is provided by sequences which converge to . If
f is a sequence of locally n summable R functionsnon R such that (i) kxk< b f
x d x ! 1 as ! 1 for all b > 0; R (ii) akxk1=a j f
xj dn x ! 0 as ! 1 for all 0 < a < 1; R (iii) there exists d > 0 and M > 0 such that kxk< d j f
xj dn x < M for all ; then the sequence
Tf of distributions converges to in D0
Rn .

hDp Tj , 'i
1jpj hTj , Dp 'i !
1jpj hT, Dp 'i hDp T, 'i:

An analogous statement holds for series: any convergent series of distributions may be differentiated termwise to all orders. This illustrates how ‘robust’ the constructs of distribution theory are in comparison with those of ordinary function theory, where similar statements are notoriously untrue. (b) Differentiation under the duality bracket Limiting processes and differentiation may also be carried out under the duality bracket h, i as under the integral sign with ordinary functions. Let the function ' '
x, depend on a parameter 2 and a vector x 2 Rn in such a way that all functions ' : x 7 ! '
x,

be in D
Rn for all 2 . Let T 2 D0
Rn be a distribution, let I
hT, ' i

31

1. GENERAL RELATIONSHIPS AND TECHNIQUES and let 0 2 be given parameter value. Suppose that, as runs through a small enough neighbourhood of 0 , (i) all the ' have their supports in a ﬁxed compact subset K of Rn ; (ii) all the derivatives Dp ' have a partial derivative with respect to which is continuous with respect to x and . Under these hypotheses, I
is differentiable (in the usual sense) with respect to near 0 , and its derivative may be obtained by ‘differentiation under the h, i sign’: dI hT, @ ' i: d

Tf Tf
S @ 0
S : The latter result is a statement of Green’s theorem in terms of distributions. It will be used in Section 1.3.4.4.3.5 to calculate the Fourier transform of the indicator function of a molecular envelope. 1.3.2.3.9.2. Integration of distributions in dimension 1 The reverse operation from differentiation, namely calculating the ‘indeﬁnite integral’ of a distribution S, consists in ﬁnding a distribution T such that T 0 S. For all 2 D such that 0 with 2 D, we must have hT, i hS, i:

This condition deﬁnes T in a ‘hyperplane’ H of D, whose equation

(c) Effect of discontinuities When a function f or its derivatives are no longer continuous, the derivatives Dp Tf of the associated distribution Tf may no longer coincide with the distributions associated to the functions Dp f . In dimension 1, the simplest example is Heaviside’s unit step function Y Y
x 0 for x < 0, Y
x 1 for x 0: 1 R 0 h
TY 0 , 'i h
TY , '0 i '
x dx '
0 h, 'i:

h1, i h1, 0 i 0

reﬂects the fact that has compact support. To specify T in the whole of D, it sufﬁces to specify the value of hT, '0 i where '0 2 D is such that h1, '0 i 1: then any ' 2 D may be written uniquely as ' '0

0

with

Hence
TY 0 , a result long used ‘heuristically’ by electrical engineers [see also Dirac (1958)]. Let f be inﬁnitely differentiable for x < 0 and x > 0 but have discontinuous derivatives f
m at x 0 [ f
0 being f itself] with jumps m f
m
0 f
m
0 . Consider the functions: g0 f g1

h1, 'i,

gk gk0

hT, 'i hT, '0 i

1 Y

0

hS, i:

k Y :

1

1.3.2.3.9.3. Multiplication of distributions by functions The product T of a distribution T on Rn by a function over Rn will be deﬁned by transposition: hT, 'i hT, 'i for all ' 2 D:

Tf 0 Tf 0 0

In order that T be a distribution, the mapping ' 7 ! ' must send D
Rn continuously into itself; hence the multipliers must be inﬁnitely differentiable. The product of two general distributions cannot be deﬁned. The need for a careful treatment of multipliers of distributions will become clear when it is later shown (Section 1.3.2.5.8) that the Fourier transformation turns convolutions into multiplications and vice versa. If T is a distribution of order m, then needs only have continuous derivatives up to order m. For instance, is a distribution of order zero, and
0 is a distribution provided is continuous; this relation is of fundamental importance in the theory of sampling and of the properties of the Fourier transformation related to sampling (Sections 1.3.2.6.4, 1.3.2.6.6). More generally, Dp is a distribution of order jpj, and the following formula holds for all 2 D
m with m jpj: X jp qj p p
Dp q
0Dq :
1
D q qp

Tf 00 Tf 00 0 0 1
Tf
m Tf
m 0
m

1

. . . m 1 :

Thus the ‘distributional derivatives’
Tf
m differ from the usual functional derivatives Tf
m by singular terms associated with discontinuities. In dimension n, let f be inﬁnitely differentiable everywhere except on a smooth hypersurface S, across which its partial derivatives show discontinuities. Let 0 and denote the discontinuities of f and its normal derivative @ ' across S (both 0 and are functions of position on S), and let
S and @
S be deﬁned by R h
S , 'i ' dn 1 S S

Integration by parts shows that

Rx
x
t dt,

The freedom in the choice of '0 means that T is deﬁned up to an additive constant.

The gk are continuous, their derivatives gk0 are continuous almost everywhere [which implies that
Tgk 0 Tgk0 and gk0 f
k1 almost everywhere]. This yields immediately:

[email protected]
S , 'i

'0 ,

and T is deﬁned by

0 Y

g00

'

0

R

@ ' dn 1 S:

The derivative of a product is easily shown to be

S

@i
T
@i T
@i T and generally for any multi-index p X p p
Dp q
0Dq T: D
T q qp

@i Tf [email protected] f 0 cos i
S , where i is the angle between the xi axis and the normal to S along which the jump 0 occurs, and that the Laplacian of Tf is given by

32

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY T is called an even distribution if T T, an odd distribution if T T. If A I with > 0, A is called a dilation and

1.3.2.3.9.4. Division of distributions by functions Given a distribution S on Rn and an inﬁnitely differentiable multiplier function , the division problem consists in ﬁnding a distribution T such that T S. If never vanishes, T S= is the unique answer. If n 1, and if has only isolated zeros of ﬁnite order, it can be reduced to a collection of cases where the multiplier is xm , for which the general solution can be shown to be of the form T U

mP1

hA# T, 'i n hT,
A 1 # 'i:

Writing symbolically as
x and A# as
x=, we have:
x= n
x:

If n 1 and f is a function with isolated simple zeros xj , then in the same symbolic notation X 1
xj , f
x j f 0
xj j j

ci
i ,

i0

where U is a particular solution of the division problem xm U S and the ci are arbitrary constants. In dimension n > 1, the problem is much more difﬁcult, but is of fundamental importance in the theory of linear partial differential equations, since the Fourier transformation turns the problem of solving these into a division problem for distributions [see Ho¨rmander (1963)].

where each j 1=j f 0
xj j is analogous to a ‘Lorentz factor’ at zero xj . 1.3.2.3.9.6. Tensor product of distributions The purpose of this construction is to extend Fubini’s theorem to distributions. Following Section 1.3.2.2.5, we may deﬁne the tensor product L1loc
Rm L1loc
Rn as the vector space of ﬁnite linear combinations of functions of the form

1.3.2.3.9.5. Transformation of coordinates Let be a smooth non-singular change of variables in Rn , i.e. an inﬁnitely differentiable mapping from an open subset of Rn to 0 in Rn , whose Jacobian @
x J
det @x

f g :
x, y 7 ! f
xg
y,

where x 2 Rm , y 2 Rn , f 2 L1loc
Rm and g 2 L1loc
Rn . Let Sx and Ty denote the distributions associated to f and g, respectively, the subscripts x and y acting as mnemonics for Rm and Rn . It follows from Fubini’s theorem (Section 1.3.2.2.5) that f g 2 L1loc
Rm Rn , and hence deﬁnes a distribution over Rm Rn ; the rearrangement of integral signs gives

vanishes nowhere in . By the implicit function theorem, the inverse mapping 1 from 0 to is well deﬁned. If f is a locally summable function on , then the function # f deﬁned by

hSx Ty , 'x; y i hSx , hTy , 'x; y ii hTy , hSx , 'x; y ii

# f
x f 1
x

for all 'x; y 2 D
Rm Rn . In particular, if '
x, y u
xv
y with u 2 D
Rm , v 2 D
Rn , then

is a locally summable function on 0 , and for any ' 2 D
0 we may write: R # R
f
x'
x dn x f 1
x'
x dn x

0

hS T, u vi hS, uihT, vi: This construction can be extended to general distributions S 2 D0
Rm and T 2 D0
Rn . Given any test function ' 2 D
Rm Rn , let 'x denote the map y 7 ! '
x, y; let 'y denote the map x 7 ! '
x, y; and deﬁne the two functions
x hT, 'x i and !
y hS, 'y i. Then, by the lemma on differentiation under the h, i sign of Section 1.3.2.3.9.1, 2 D
Rm , ! 2 D
Rn , and there exists a unique distribution S T such that

0

R f
y'
yjJ
j dn y by x
y:

0

In terms of the associated distributions

hT# f , 'i hTf , jJ
j
1 # 'i:

This operation can be extended to an arbitrary distribution T by deﬁning its image # T under coordinate transformation through

hS T, 'i hS, i hT, !i:

S T is called the tensor product of S and T. With the mnemonic introduced above, this deﬁnition reads identically to that given above for distributions associated to locally integrable functions:

h# T, 'i hT, jJ
j
1 # 'i,

which is well deﬁned provided that is proper, i.e. that 1
K is compact whenever K is compact. For instance, if : x 7 ! x a is a translation by a vector a in Rn , then jJ
j 1; # is denoted by a , and the translate a T of a distribution T is deﬁned by

hSx Ty , 'x; y i hSx , hTy , 'x; y ii hTy , hSx , 'x; y ii:

The tensor product of distributions is associative:
R S T R
S T:

ha T, 'i hT, a 'i: Let A : x 7 ! Ax be a linear transformation deﬁned by a nonsingular matrix A. Then J
A det A, and

Derivatives may be calculated by

Dpx Dqy
Sx Ty
Dpx Sx
Dqy Ty :

hA# T, 'i jdet AjhT,
A 1 # 'i:

The support of a tensor product is the Cartesian product of the supports of the two factors.

This formula will be shown later (Sections 1.3.2.6.5, 1.3.4.2.1.1) to be the basis for the deﬁnition of the reciprocal lattice. In particular, if A I, where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting A# ' by ' we have: 'i hT, 'i: hT,

1.3.2.3.9.7. Convolution of distributions The convolution f g of two functions f and g on Rn is deﬁned by R R
f g
x f
yg
x y dn y f
x yg
y dn y Rn

33

Rn

1. GENERAL RELATIONSHIPS AND TECHNIQUES whenever the integral exists. This is the case when f and g are both in L1
Rn ; then f g is also in L1
Rn . Let S, T and W denote the distributions associated to f, g and f g, respectively: a change of variable immediately shows that for any ' 2 D
Rn , R f
xg
y'
x y dn x dn y: hW , 'i

of such functions can be constructed which have compact support and converge to , it follows that any distribution T can be obtained as the limit of inﬁnitely differentiable functions T . In topological jargon: D
Rn is ‘everywhere dense’ in D0
Rn . A standard function in D which is often used for such proofs is deﬁned as follows: put 1 1 for jxj 1,
x exp A 1 x2

Rn Rn

Introducing the map from Rn Rn to Rn deﬁned by
x, y x y, the latter expression may be written: hSx Ty , ' i

0

(where denotes the composition of mappings) or by a slight abuse of notation:

with

hW , 'i hSx Ty , '
x yi:

A difﬁculty arises in extending this deﬁnition to general distributions S and T because the mapping is not proper: if K is compact in Rn , then 1
K is a cylinder with base K and generator the ‘second bisector’ x y 0 in Rn Rn . However, hS T, ' i is deﬁned whenever the intersection between Supp
S T
Supp S
Supp T and 1
Supp ' is compact. We may therefore deﬁne the convolution S T of two distributions S and T on Rn by

Z1

A

1

exp

1 1

x2

dx

(so that is in D and is normalized), and put 1 x "
x in dimension 1, " " n Y "
xj in dimension n: "
x

hS T, 'i hS T, ' i hSx Ty , '
x yi

j1

whenever the following support condition is fulﬁlled: n

for jxj 1,

Another related result, also proved by convolution, is the structure theorem: the restriction of a distribution T 2 D0
Rn to a bounded open set in Rn is a derivative of ﬁnite order of a continuous function. Properties (i) to (iv) are the basis of the symbolic or operational calculus (see Carslaw & Jaeger, 1948; Van der Pol & Bremmer, 1955; Churchill, 1958; Erde´lyi, 1962; Moore, 1971) for solving integro-differential equations with constant coefﬁcients by turning them into convolution equations, then using factorization methods for convolution algebras (Schwartz, 1965).

n

‘the set f
x, yjx 2 A, y 2 B, x y 2 Kg is compact in R R for all K compact in Rn ’.

The latter condition is met, in particular, if S or T has compact support. The support of S T is easily seen to be contained in the closure of the vector sum A B fx yjx 2 A, y 2 Bg:

Convolution by a ﬁxed distribution S is a continuous operation for the topology on D0 : it maps convergent sequences
Tj to convergent sequences
S Tj . Convolution is commutative: S T T S. The convolution of p distributions T1 , . . . , Tp with supports A1 , . . . , Ap can be deﬁned by

1.3.2.4. Fourier transforms of functions 1.3.2.4.1. Introduction Given a complex-valued function f on Rn subject to suitable regularity conditions, its Fourier transform F f and Fourier cotransform F f are deﬁned as follows: R F f
f
x exp
2i x dn x

hT1 . . . Tp , 'i h
T1 x1 . . .
Tp xp , '
x1 . . . xp i whenever the following generalized support condition: ‘the set f
x1 , . . . , xp jx1 2 A1 , . . . , xp 2 Ap , x1 . . . xp 2 Kg is compact in
Rn p for all K compact in Rn ’

Rn

F f

is satisﬁed. It is then associative. Interesting examples of associativity failure, which can be traced back to violations of the support condition, may be found in Bracewell (1986, pp. 436–437). It follows from previous deﬁnitions that, for all distributions T 2 D0 , the following identities hold: (i) T T: is the unit convolution; (ii)
a T a T: translation is a convolution with the corresponding translate of ; (iii)
Dp T Dp T: differentiation is a convolution with the corresponding derivative of ; (iv) translates or derivatives of a convolution may be obtained by translating or differentiating any one of the factors: convolution ‘commutes’ with translation and differentiation, a property used in Section 1.3.4.4.7.7 to speed up least-squares model reﬁnement for macromolecules. The latter property is frequently used for the purpose of regularization: if T is a distribution, an inﬁnitely differentiable function, and at least one of the two has compact support, then T is an inﬁnitely differentiable ordinary function. Since sequences

Pn

R

n

R

f
x exp
2i x dn x,

where x i1 i xi is the ordinary scalar product. The terminology and sign conventions given above are the standard ones in mathematics; those used in crystallography are slightly different (see Section 1.3.4.2.1.1). These transforms enjoy a number of remarkable properties, whose natural settings entail different regularity assumptions on f : for instance, properties relating to convolution are best treated in L1
Rn , while Parseval’s theorem requires the Hilbert space structure of L2
Rn . After a brief review of these classical properties, the Fourier transformation will be examined in a space S
Rn particularly well suited to accommodating the full range of its properties, which will later serve as a space of test functions to extend the Fourier transformation to distributions. There exists an abundant literature on the ‘Fourier integral’. The books by Carslaw (1930), Wiener (1933), Titchmarsh (1948), Katznelson (1968), Sneddon (1951, 1972), and Dym & McKean (1972) are particularly recommended.

34

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 1.3.2.4.2. Fourier transforms in L1

F x; y u v F x u F y v:

1.3.2.4.2.1. Linearity Both transformations F and F are obviously linear maps from L1 to L1 when these spaces are viewed as vector spaces over the ﬁeld C of complex numbers.

Furthermore, if f 2 L1
Rm Rn , then F y f 2 L1
Rm as a function of x and F x f 2 L1
Rn as a function of y, and F x; y f F x F y f F y F x f :

This is easily proved by using Fubini’s theorem and the fact that
,
x, y x y, where x, 2 Rm , y, 2 Rn . This property may be written:

1.3.2.4.2.2. Effect of affine coordinate transformations F and F turn translations into phase shifts:

F x; y F x F y :

F a f
exp
2i aF f
F a f
exp
2i aF f
:

1.3.2.4.2.5. Convolution property If f and g are summable, their convolution f g exists and is summable, and " # R R F f g
f
yg
x y dn y exp
2i x dn x:

Under a general linear change of variable x 7 ! Ax with nonsingular matrix A, the transform of A# f is R F A# f
f
A 1 x exp
2i x dn x Rn

R

R

n

f
y exp
2i
AT yjdet Aj dn y

Rn

With x y z, so that

by x Ay

jdet AjF f
AT

exp
2i x exp
2i y exp
2i z,

and with Fubini’s theorem, rearrangement of the double integral gives:

i.e. F A# f jdet Aj
A 1 T # F f and similarly for F . The matrix
A 1 T is called the contragredient of matrix A. Under an afﬁne change of coordinates x 7 ! S
x Ax b with non-singular matrix A, the transform of S # f is given by #

and similarly

Thus the Fourier transform and cotransform turn convolution into multiplication.

F S f
F b
A f

exp
2i bF A# f

1.3.2.4.2.6. Reciprocity property In general, F f and F f are not summable, and hence cannot be further transformed; however, as they are essentially bounded, their products with the Gaussians Gt
exp
22 kk2 t are summable for all t > 0, and it can be shown that f lim F Gt F f lim F Gt F f ,

T

exp
2i bjdet AjF f
A i by +i.

1.3.2.4.2.3. Conjugate symmetry The kernels of the Fourier transformations F and F satisfy the following identities:

t!0

t!0

where the limit is taken in the topology of the L1 norm k:k1 . Thus F and F are (in a sense) mutually inverse, which justiﬁes the common practice of calling F the ‘inverse Fourier transformation’.

exp
2i x exp 2i
x exp 2i
x: As a result the transformations F and F themselves have the following ‘conjugate symmetry’ properties [where the notation f
x f
x of Section 1.3.2.2.2 will be used]: F f
F f
F f

1.3.2.4.2.7. Riemann–Lebesgue lemma If f 2 L1
Rn , i.e. is summable, then F f and F f exist and are continuous and essentially bounded: kF f k kF f k k f k : 1

F f
F f
:

1

1

In fact one has the much stronger property, whose statement constitutes the Riemann–Lebesgue lemma, that F f
and F f
both tend to zero as k k ! 1.

Therefore, (i) f real , f f , F f F f , F f
F f
: F f is said to possess Hermitian symmetry; (ii) f centrosymmetric , f f , F f F f ; (iii) f real centrosymmetric , f f f , F f F f F f , F f real centrosymmetric. Conjugate symmetry is the basis of Friedel’s law (Section 1.3.4.2.1.4) in crystallography. 1.3.2.4.2.4. Tensor product property Another elementary property of F is its naturality with respect to tensor products. Let u 2 L1
Rm and v 2 L1
Rn , and let F x , F y , F x; y denote the Fourier transformations in L1
Rm , L1
Rn and L1
Rm Rn , respectively. Then

F f g F f F g F f g F f F g:

#

with a similar result for F , replacing

Rn

1.3.2.4.2.8. Differentiation Let us now suppose that n 1 and that f 2 L1
R is differentiable with f 0 2 L1
R. Integration by parts yields 1 R 0 F f 0
f
x exp
2i x dx 1

f
x exp
2i x1 1 1 R 2i f
x exp
2i x dx: 1

0

Since f is summable, f has a limit when x ! 1, and this limit must be 0 since f is summable. Therefore

35

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.2.4.3. Fourier transforms in L2

0

with the bound

F f
2iF f

Let f belong to L2
Rn , i.e. be such that !1=2 R 2 n j f
xj d x < 1: k f k2

k2F f k1 k f 0 k1 so that jF f
j decreases faster than 1=jj ! 1. This result can be easily extended to several dimensions and to any multi-index m: if f is summable and has continuous summable partial derivatives up to order jmj, then

1.3.2.4.3.1. Invariance of L2 F f and F f exist and are functions in L2 , i.e. F L2 L2 , F L2 L2 .

m

m

and

Rn

F D f
2i F f

k
2 m F f k1 kDm f k1 : Similar results hold for F , with 2i replaced by 2i . Thus, the more differentiable f is, with summable derivatives, the faster F f and F f decrease at inﬁnity. The property of turning differentiation into multiplication by a monomial has many important applications in crystallography, for instance differential syntheses (Sections 1.3.4.2.1.9, 1.3.4.4.7.2, 1.3.4.4.7.5) and moment-generating functions [Section 1.3.4.5.2.1(c)].

1.3.2.4.3.2. Reciprocity F F f f and F F f f , equality being taken as ‘almost everywhere’ equality. This again leads to calling F the ‘inverse Fourier transformation’ rather than the Fourier cotransformation. 1.3.2.4.3.3. Isometry F and F preserve the L2 norm: kF f k2 kF f k2 k f k2 (Parseval’s/Plancherel’s theorem):

1.3.2.4.2.9. Decrease at infinity Conversely, assume that f is summable on Rn and that f decreases fast enough at inﬁnity for xm f also to be summable, for some multiindex m. Then the integral deﬁning F f may be subjected to the differential operator Dm , still yielding a convergent integral: therefore Dm F f exists, and

This property, which may be written in terms of the inner product (,) in L2
Rn as
F f , F g
F f , F g
f , g,

with the bound

1.3.2.4.3.4. Eigenspace decomposition of L2 Some light can be shed on the geometric structure of these rotations by the following simple considerations. Note that R F 2 f
x F f
exp
2ix dn

implies that F and F are unitary transformations of L2
Rn into itself, i.e. inﬁnite-dimensional ‘rotations’.

Dm
F f
F
2ixm f

kDm
F f k1 k
2xm f k1 : Similar results hold for F , with 2ix replaced by 2ix. Thus, the faster f decreases at inﬁnity, the more F f and F f are differentiable, with bounded derivatives. This property is the converse of that described in Section 1.3.2.4.2.8, and their combination is fundamental in the deﬁnition of the function space S in Section 1.3.2.4.4.1, of tempered distributions in Section 1.3.2.5, and in the extension of the Fourier transformation to them.

Rn

F F f
x f
x

so that F 4 (and similarly F 4 ) is the identity map. Any eigenvalue of F or F is therefore a fourth root of unity, i.e. 1 or i, and L2
Rn splits into an orthogonal direct sum H0 H1 H 2 H 3 , where F (respectively F ) acts in each subspace Hk
k 0, 1, 2, 3 by multiplication by
ik . Orthonormal bases for these subspaces can be constructed from Hermite functions (cf. Section 1.3.2.4.4.2) This method was used by Wiener (1933, pp. 51–71).

1.3.2.4.2.10. The Paley–Wiener theorem An extreme case of the last instance occurs when f has compact support: then F f and F f are so regular that they may be analytically continued from Rn to Cn where they are entire functions, i.e. have no singularities at ﬁnite distance (Paley & Wiener, 1934). This is easily seen for F f : giving vector 2 Rn a vector 2 Rn of imaginary parts leads to R F f
i f
x exp 2i
i x dn x

1.3.2.4.3.5. The convolution theorem and the isometry property In L2 , the convolution theorem (when applicable) and the Parseval/Plancherel theorem are not independent. Suppose that f, g, f g and f g are all in L2 (without questioning whether these properties are independent). Then f g may be written in terms of the inner product in L2 as follows: R R
f g
x f
x yg
y dn y f
y xg
y dn y,

Rn

F exp
2 xf
,

where the latter transform always exists since exp
2 xf is summable with respect to x for all values of . This analytic continuation forms the basis of the saddlepoint method in probability theory [Section 1.3.4.5.2.1( f )] and leads to the use of maximum-entropy distributions in the statistical theory of direct phase determination [Section 1.3.4.5.2.2(e)]. By Liouville’s theorem, an entire function in Cn cannot vanish identically on the complement of a compact subset of Rn without vanishing everywhere: therefore F f cannot have compact support if f has, and hence D
Rn is not stable by Fourier transformation.

Rn

Rn

i.e.

f g
x
x f , g: Invoking the isometry property, we may rewrite the right-hand side as

36

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY
F x f , F g
exp
2ix F f , F g R
F f F g
x

in dimension n: F G
F G
G
:

n

In other words, G is an eigenfunction of F and F for eigenvalue 1 (Section 1.3.2.4.3.4). A complete system of eigenfunctions may be constructed as follows. In dimension 1, consider the family of functions

R

exp
2ix dn F F f F g,

so that the initial identity yields the convolution theorem. To obtain the converse implication, note that R
f , g f
yg
y dn y
f g
0

Hm

Dm G2 G

m 0,

where D denotes the differentiation operator. The ﬁrst two members of the family

Rn

F F f F g
0 R F f
F g
dn
F f , F g,

H0 G,

H1 2DG,

are such that F H0 H0 , as shown above, and

Rn

DG
x 2xG
x i
2ixG
x iF DG
x,

where conjugate symmetry (Section 1.3.2.4.2.2) has been used. These relations have an important application in the calculation by Fourier transform methods of the derivatives used in the reﬁnement of macromolecular structures (Section 1.3.4.4.7).

hence F H1
iH1 : We may thus take as an induction hypothesis that

1.3.2.4.4. Fourier transforms in S

F Hm
im Hm :

1.3.2.4.4.1. Definition and properties of S The duality established in Sections 1.3.2.4.2.8 and 1.3.2.4.2.9 between the local differentiability of a function and the rate of decrease at inﬁnity of its Fourier transform prompts one to consider the space S
Rn of functions f on Rn which are inﬁnitely differentiable and all of whose derivatives are rapidly decreasing, so that for all multi-indices k and p

The identity m 2 D G Dm1 G2 D G G

DG Dm G2 G G

Hm1
x
DHm
x

2xHm
x,

may be written

xk Dp f
x ! 0 as kxk ! 1:

and the two differentiation theorems give:

The product of f 2 S by any polynomial over Rn is still in S (S is an algebra over the ring of polynomials). Furthermore, S is invariant under translations and differentiation. If f 2 S , then its transforms F f and F f are (i) inﬁnitely differentiable because f is rapidly decreasing; (ii) rapidly decreasing because f is inﬁnitely differentiable; hence F f and F f are in S : S is invariant under F and F . Since L1 S and L2 S , all properties of F and F already encountered above are enjoyed by functions of S , with all restrictions on differentiability and/or integrability lifted. For instance, given two functions f and g in S , then both fg and f g are in S (which was not the case with L1 nor with L2 ) so that the reciprocity theorem inherited from L2 F F f f and F F f f

F DHm
2i F Hm
F 2xHm
iD
F Hm
: Combination of this with the induction hypothesis yields F Hm1
im1
DHm
im1 Hm1
,

2Hm

thus proving that Hm is an eigenfunction of F for eigenvalue
im for all m 0. The same proof holds for F , with eigenvalue im . If these eigenfunctions are normalized as
1m 21=4 H m
x p Hm
x, m!2m m=2

allows one to state the reverse of the convolution theorem ﬁrst established in L1 :

then it can be shown that the collection of Hermite functions fH m
xgm0 constitutes an orthonormal basis of L2
R such that H m is an eigenfunction of F (respectively F ) for eigenvalue
im (respectively im ). In dimension n, the same construction can be extended by tensor product to yield the multivariate Hermite functions

F fg F f F g F fg F f F g: 1.3.2.4.4.2. Gaussian functions and Hermite functions Gaussian functions are particularly important elements of S . In dimension 1, a well known contour integration (Schwartz, 1965, p. 184) yields F exp
x2
F exp
x2
exp
2 ,

H m
x H m1
x1 H m2
x2 . . . H mn
xn (where m 0 is a multi-index). These constitute an orthonormal basis of L2
Rn , with H m an eigenfunction of F (respectively F ) for eigenvalue
ijmj (respectively ijmj ). Thus the subspaces Hk of Section 1.3.2.4.3.4 are spanned by those H m with jmj k mod 4
k 0, 1, 2, 3. General multivariate Gaussians are usually encountered in the non-standard form

which shows that the ‘standard Gaussian’ exp
x2 is invariant under F and F . By a tensor product construction, it follows that the same is true of the standard Gaussian G
x exp
kxk2

GA
x exp

37

1 T 2x

Ax,

1. GENERAL RELATIONSHIPS AND TECHNIQUES This possibility of ‘transposing’ F (and F ) from the left to the right of the duality bracket will be used in Section 1.3.2.5.4 to extend the Fourier transformation to distributions.

where A is a symmetric positive-deﬁnite matrix. Diagonalizing A as ELET with EET the identity matrix, and putting A1=2 EL1=2 ET , we may write " # A 1=2 x GA
x G 2

1.3.2.4.5. Various writings of Fourier transforms Other ways of writing Fourier transforms in Rn exist besides the one used here. All have the form Z 1 F h; ! f
n f
x exp
i! x dn x, h

i.e. GA
2A 1 1=2 # G;

Rn

hence (by Section 1.3.2.4.2.3) 1 1=2

F GA jdet
2A j

where h is real positive and ! real non-zero, with the reciprocity formula written: Z 1 f
x n F h; ! f
exp
i! x dn x k

" ## A 1=2 G, 2

i.e.

Rn

1 1=2

F GA
jdet
2A j

1 1=2

G
2A

,

with k real positive. The consistency condition between h, k and ! is 2 hk : j!j The usual choices are:

i.e. ﬁnally F GA jdet
2A 1 j1=2 G42 A 1 :

This result is widely used in crystallography, e.g. to calculate form factors for anisotropic atoms (Section 1.3.4.2.2.6) and to obtain transforms of derivatives of Gaussian atomic densities (Section 1.3.4.4.7.10).

i

Z

2

j f
xj dx

2

1.3.2.4.6. Tables of Fourier transforms The books by Campbell & Foster (1948), Erde´lyi (1954), and Magnus et al. (1966) contain extensive tables listing pairs of functions and their Fourier transforms. Bracewell (1986) lists those pairs particularly relevant to electrical engineering applications.

,

where, by a beautiful theorem of Hardy (1933), equality can only be attained for f Gaussian. Hardy’s theorem is even stronger: if both f and F f behave at inﬁnity as constant multiples of G, then each of them is everywhere a constant multiple of G; if both f and F f behave at inﬁnity as constant multiples of G monomial, then each of them is a ﬁnite linear combination of Hermite functions. Hardy’s theorem is invoked in Section 1.3.4.4.5 to derive the optimal procedure for spreading atoms on a sampling grid in order to obtain the most accurate structure factors. The search for optimal compromises between the conﬁnement of f to a compact domain in x-space and of F f to a compact domain in -space leads to consideration of prolate spheroidal wavefunctions (Pollack & Slepian, 1961; Landau & Pollack, 1961, 1962).

1.3.2.5. Fourier transforms of tempered distributions 1.3.2.5.1. Introduction It was found in Section 1.3.2.4.2 that the usual space of test functions D is not invariant under F and F . By contrast, the space S of inﬁnitely differentiable rapidly decreasing functions is invariant under F and F , and furthermore transposition formulae such as hF f , gi h f , F gi

hold for all f , g 2 S . It is precisely this type of transposition which was used successfully in Sections 1.3.2.3.9.1 and 1.3.2.3.9.3 to deﬁne the derivatives of distributions and their products with smooth functions. This suggests using S instead of D as a space of test functions ', and deﬁning the Fourier transform F T of a distribution T by

1.3.2.4.4.4. Symmetry property A ﬁnal formal property of the Fourier transform, best established in S , is its symmetry: if f and g are in S , then by Fubini’s theorem ! R R n f
x exp
2i x d x g
dn hF f , gi Rn

R

R

n

hF T, 'i hT, F 'i

whenever T is capable of being extended from D to S while remaining continuous. It is this latter proviso which will be subsumed under the adjective ‘tempered’. As was the case with the construction of D0 , it is the deﬁnition of a sufﬁciently strong topology (i.e. notion of convergence) in S which will play a key role in transferring to the elements of its topological dual S 0 (called tempered distributions) all the properties of the Fourier transformation.

Rn

f
x

R

n

R

n

g
exp
2i x d

!

as here;

ii ! 1, h 1, k 2
in probability theory and in solid-state physics; p
iii ! 1, h k 2
in much of classical analysis: It should be noted that conventions (ii) and (iii) introduce numerical factors of 2 in convolution and Parseval formulae, while (ii) breaks the symmetry between F and F .

1.3.2.4.4.3. Heisenberg’s inequality, Hardy’s theorem The result just obtained, which also holds for F , shows that the ‘peakier’ GA , the ‘broader’ F GA . This is a general property of the Fourier transformation, expressed in dimension 1 by the Heisenberg inequality (Weyl, 1931): Z Z 2 2 2 2 x j f
xj dx jF f
j d 1 162

! 2, h k 1

dn x

hf , F gi:

38

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY are both linear and continuous for the topology of S . In the same way that x and have been used consistently as arguments for ' and F ', respectively, the notation Tx and F T will be used to indicate which variables are involved. When T is a distribution with compact support, its Fourier transform may be written

Besides the general references to distribution theory mentioned in Section 1.3.2.3.1 the reader may consult the books by Zemanian (1965, 1968). Lavoine (1963) contains tables of Fourier transforms of distributions. 1.3.2.5.2. S as a test-function space

F Tx hTx , exp
2i xi

A notion of convergence has to be introduced in S
Rn in order to be able to deﬁne and test the continuity of linear functionals on it. A sequence
'j of functions in S will be said to converge to 0 if, for any given multi-indices k and p, the sequence
xk Dp 'j tends to 0 uniformly on Rn . It can be shown that D
Rn is dense in S
Rn . Translation is continuous P for this topology. For any linear differential operator P
D p ap Dp and any polynomial Q
x over Rn ,
'j ! 0 implies Q
x P
D'j ! 0 in the topology of S . Therefore, differentiation and multiplication by polynomials are continuous for the topology on S . The Fourier transformations F and F are also continuous for the topology of S . Indeed, let
'j converge to 0 for the topology on S . Then, by Section 1.3.2.4.2,

since the function x 7 ! exp
2i x is in E while Tx 2 E 0 . It can be shown, as in Section 1.3.2.4.2, to be analytically continuable into an entire function over Cn . 1.3.2.5.5. Transposition of basic properties The duality between differentiation and multiplication by a monomial extends from S to S 0 by transposition: F Dpx Tx
2i p F Tx

Dp
F Tx F
2ixp Tx :

Analogous formulae hold for F , with i replaced by i. The formulae expressing the duality between translation and phase shift, e.g.

k
2 m Dp
F 'j k1 kDm
2xp 'j k1 : The right-hand side tends to 0 as j ! 1 by deﬁnition of convergence in S , hence k km Dp
F 'j ! 0 uniformly, so that
F 'j ! 0 in S as j ! 1. The same proof applies to F .

F a Tx exp
2ia F Tx

F Tx F exp
2i xTx ;

between a linear change of variable and its contragredient, e.g.

1.3.2.5.3. Definition and examples of tempered distributions

F A# T jdet Aj
A 1 T # F T;

A distribution T 2 D0
Rn is said to be tempered if it can be extended into a continuous linear functional on S . If S 0
Rn is the topological dual of S
Rn , and if S 2 S 0
Rn , then its restriction to D is a tempered distribution; conversely, if T 2 D0 is tempered, then its extension to S is unique (because D is dense in S ), hence it deﬁnes an element S of S 0 . We may therefore identify S 0 and the space of tempered distributions. A distribution with compact support is tempered, i.e. S 0 E 0 . By transposition of the corresponding properties of S , it is readily established that the derivative, translate or product by a polynomial of a tempered distribution is still a tempered distribution. These inclusion relations may be summarized as follows: since S contains D but is contained in E, the reverse inclusions hold for the topological duals, and hence S 0 contains E 0 but is contained in D0 . A locally summable function f on Rn will be said to be of polynomial growth if j f
xj can be majorized by a polynomial in kxk as kxk ! 1. It is easily shown that such a function f deﬁnes a tempered distribution Tf via R hTf , 'i f
x'
x dn x:

are obtained similarly by transposition from the corresponding identities in S . They give a transposition formula for an afﬁne change of variables x 7 ! S
x Ax b with non-singular matrix A: F S # T exp
2i bF A# T

exp
2i bjdet Aj
A 1 T # F T,

with a similar result for F , replacing i by +i. Conjugate symmetry is obtained similarly: F T, F T, F T F T with the same identities for F . The tensor product property also transposes to tempered distributions: if U 2 S 0
Rm , V 2 S 0
Rn , F Ux Vy F U F V F Ux Vy F U F V :

Rn

1.3.2.5.6. Transforms of -functions

In particular, polynomials over Rn deﬁne tempered distributions, and so do functions in S . The latter remark, together with the transposition identity (Section 1.3.2.4.4), invites the extension of F and F from S to S 0 .

Since has compact support, F x hx , exp
2i xi 1 ,

i:e: F 1:

1.3.2.5.4. Fourier transforms of tempered distributions The Fourier transform F T and cotransform F T of a tempered distribution T are deﬁned by

It is instructive to show that conversely F 1 without invoking the reciprocity theorem. Since @j 1 0 for all j 1, . . . , n, it follows from Section 1.3.2.3.9.4 that F 1 c; the constant c can be determined by using the invariance of the standard Gaussian G established in Section 1.3.2.4.3:

hF T, 'i hT, F 'i hF T, 'i hT, F 'i

hF 1x , Gx i h1 , G i 1;

for all test functions ' 2 S . Both F T and F T are themselves tempered distributions, since the maps ' 7 ! F ' and ' 7 ! F '

hence c 1. Thus, F 1 . The basic properties above then read (using multi-indices to denote differentiation):

39

1. GENERAL RELATIONSHIPS AND TECHNIQUES F x
m
2i m ,

F xm
2i

F a exp
2ia ,

The same identities hold for F . Taken together with the reciprocity theorem, these show that F and F establish mutually inverse isomorphisms between O M and O 0C , and exchange multiplication for convolution in S 0 . It may be noticed that most of the basic properties of F and F may be deduced from this theorem and from the properties of . Differentiation operators Dm and translation operators a are convolutions with Dm and a ; they are turned, respectively, into multiplication by monomials
2i m (the transforms of Dm ) or by phase factors exp
2i (the transforms of a ). Another consequence of the convolution theorem is the duality established by the Fourier transformation between sections and projections of a function and its transform. For instance, in R3 , the projection of f
x, y, z on the x, y plane along the z axis may be written

jmj
m ;

F exp
2i x ,

with analogous relations for F , i becoming i. Thus derivatives of are mapped to monomials (and vice versa), while translates of are mapped to ‘phase factors’ (and vice versa). 1.3.2.5.7. Reciprocity theorem The previous results now allow a self-contained and rigorous proof of the reciprocity theorem between F and F to be given, whereas in traditional settings (i.e. in L1 and L2 ) the implicit handling of through a limiting process is always the sticking point. Reciprocity is ﬁrst established in S as follows: R F F '
x F '
exp
2i x dn

x y 1z f ;

Rn

R

n

R

its Fourier transform is then

F x '
dn

1 1 F f ,

h1, F x 'i hF 1, x 'i

which is the section of F f by the plane 0, orthogonal to the z axis used for projection. There are numerous applications of this property in crystallography (Section 1.3.4.2.1.8) and in ﬁbre diffraction (Section 1.3.4.5.1.3).

hx , 'i

'
x

1.3.2.5.9. L2 aspects, Sobolev spaces

and similarly

The special properties of F in the space of square-integrable functions L2
Rn , such as Parseval’s identity, can be accommodated within distribution theory: if u 2 L2
Rn , then Tu is a tempered distribution in S 0 (the map u 7 ! Tu being continuous) and it can be shown that S F Tu is of the form Sv , where u F u is the Fourier transform of u in L2
Rn . By Plancherel’s theorem, kuk2 kvk2 . This embedding of L2 into S 0 can be used to derive the convolution theorem for L2 . If u and v are in L2
Rn , then u v can be shown to be a bounded continuous function; thus u v is not in L2 , but it is in S 0 , so that its Fourier transform is a distribution, and

F F '
x '
x: The reciprocity theorem is then proved in S 0 by transposition: F F T F F T T for all T 2 S 0 :

Thus the Fourier cotransformation F in S 0 may legitimately be called the ‘inverse Fourier transformation’. The method of Section 1.3.2.4.3 may then be used to show that F and F both have period 4 in S 0 . 1.3.2.5.8. Multiplication and convolution Multiplier functions
x for tempered distributions must be inﬁnitely differentiable, as for ordinary distributions; furthermore, they must grow sufﬁciently slowly as kxk ! 1 to ensure that ' 2 S for all ' 2 S and that the map ' 7 ! ' is continuous for the topology of S . This leads to choosing for multipliers the subspace O M consisting of functions 2 E of polynomial growth. It can be shown that if f is in O M , then the associated distribution Tf is in S 0 (i.e. is a tempered distribution); and that conversely if T is in S 0 , T is in O M for all 2 D. Corresponding restrictions must be imposed to deﬁne the space O 0C of those distributions T whose convolution S T with a tempered distribution S is still a tempered distribution: T must be such that, for all ' 2 S ,
x hTy , '
x yi is in S ; and such that the map ' 7 ! be continuous for the topology of S . This implies that S is ‘rapidly decreasing’. It can be shown that if f is in S , then the associated distribution Tf is in O 0C ; and that conversely if T is in O 0C , T is in S for all 2 D. The two spaces O M and O 0C are mapped into each other by the Fourier transformation F
O M F
O M O 0

F u v F u F v: Spaces of tempered distributions related to L2
Rn can be deﬁned as follows. For any real s, deﬁne the Sobolev space Hs
Rn to consist of all tempered distributions S 2 S 0
Rn such that
1 j j2 s=2 F S 2 L2
Rn :

These spaces play a fundamental role in the theory of partial differential equations, and in the mathematical theory of tomographic reconstruction – a subject not unrelated to the crystallographic phase problem (Natterer, 1986). 1.3.2.6. Periodic distributions and Fourier series 1.3.2.6.1. Terminology Let Zn be the subset of Rn consisting of those points with (signed) integer coordinates; it is an n-dimensional lattice, i.e. a free Abelian group on n generators. A particularly simple set of n generators is given by the standard basis of Rn , and hence Zn will be called the standard lattice in Rn . Any other ‘non-standard’ ndimensional lattice in Rn is the image of this standard lattice by a general linear transformation. If we identify any two points in Rn whose coordinates are congruent modulo Zn , i.e. differ by a vector in Zn , we obtain the standard n-torus Rn =Zn . The latter may be viewed as
R=Zn , i.e. as the Cartesian product of n circles. The same identiﬁcation may be carried out modulo a non-standard lattice , yielding a non-

C

F
O 0C F
O 0C O M

and the convolution theorem takes the form F S F F S

S 2 S 0 , 2 O M , F 2 O 0C ;

F S T F S F T S 2 S 0 , T 2 O 0C , F T 2 O M :

40

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY n

standard n-torus R = . The correspondence to crystallographic terminology is that ‘standard’ coordinates over the standard 3-torus R3 =Z3 are called ‘fractional’ coordinates over the unit cell; while Cartesian coordinates, e.g. in a˚ngstro¨ms, constitute a set of nonstandard coordinates. Finally, we will denote by I the unit cube 0, 1n and by C" the subset

presentation, as it is more closely related to the crystallographer’s perception of periodicity (see Section 1.3.4.1). 1.3.2.6.4. Fourier transforms of periodic distributions The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section 1.3.4.2.1.1). Let T r T 0 with r deﬁned as in Section 1.3.2.6.2. Then r 2 S 0 , T 0 2 E 0 hence T 0 2 O 0C , so that T 2 S 0 : Zn -periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section 1.3.2.5.8) is applicable, giving:

C" fx 2 Rn kxj j < " for all j 1, . . . , ng: 1.3.2.6.2. Zn -periodic distributions in Rn A distribution T 2 D0
Rn is called periodic with period lattice Z (or Zn -periodic) if m T T for all m 2 Zn (in crystallography the period lattice is the direct lattice). Given with compact support T 0 2 E 0
Rn , then P a distribution n 0 T m2Zn m T is a Z -periodic P distribution. Note that we may write T r T 0 , where r m2Zn
m consists of Dirac ’s at all nodes of the period lattice Zn . Conversely, any Zn -periodic distribution T may be written as r T 0 for some T 0 2 E 0 . To retrieve such a ‘motif’ T 0 from T, a function will be constructed in such a way that 2 D (hence has compact support) and r 1; then T 0 T. Indicator functions (Section 1.3.2.2) such as 1 or C1=2 cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section 1.3.2.3.9.7) as 0 C" , with " and such that 0
x 1 on C1=2 and 0
x 0 outside C3=4 . Then the function n

P

F T F r F T 0

and similarly for F . Since F
m
exp
2i m, formally P exp
2i m Q, F r m2Zn

say. It P is readily shown that Q is tempered and periodic, so that Q 2Zn
Q, while the periodicity of r implies that exp
2ij

m2Z

m

j 1, . . . , n:

Since the ﬁrst factors have single isolated zeros at j 0 in C3=4 , Q c (see Section 1.3.2.3.9.4) and hence by periodicity Q cr; convoluting with C1 shows that c 1. Thus we have the fundamental result:

0 n

1 Q 0,

0

F r r

has the desired property. The sum in the denominator contains at most 2n non-zero terms at any given point x and acts as a smoothly varying ‘multiplicity correction’.

so that F T r F T 0 ;

1.3.2.6.3. Identification with distributions over Rn =Zn

i.e., according to Section 1.3.2.3.9.3, P F T 0
: F T

n

Throughout this section, ‘periodic’ will mean ‘Z -periodic’. Let s 2 R, and let [s] denote the largest integer s. For x
x1 , . . . , xn 2 Rn , let ~x be the unique vector
~x1 , . . . , ~xn with ~xj xj xj . If x, y 2 Rn , then ~x ~y if and only if x y 2 Zn . The image of the map x 7 ! ~x is thus Rn modulo Zn , or Rn =Zn . If f is a periodic function over Rn , then ~x ~y implies f
x f
y; we may thus deﬁne a function ~f over Rn =Zn by putting ~f
~x f
x for any x 2 Rn such that x ~x 2 Zn . Conversely, if ~f is a function over Rn =Zn , then we may deﬁne a function f over Rn by putting f
x ~f
~x, and f will be periodic. Periodic functions over Rn may thus be identiﬁed with functions over Rn =Zn , and this identiﬁcation preserves the notions of convergence, local summability and differentiability. Given '0 2 D
Rn , we may deﬁne P
m '0
x '
x

2Zn

The right-hand side is a weighted lattice distribution, whose nodes 2 Zn are weighted by the sample values F T 0
of the transform of the motif T 0 at those nodes. Since T 0 2 E 0 , the latter values may be written F T 0
hTx0 , exp
2i xi: By the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), T 0 is a derivative of ﬁnite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, F T 0
grows at most polynomially as kk ! 1 (see also Section 1.3.2.6.10.3 about this property). Conversely, let W P 2Zn w
be a weighted lattice distribution such that the weights w grow at most polynomially as kk ! 1. Then W is a tempered distribution, whose Fourier cotransform Tx P 2Zn w exp
2i x is periodic. If T is now written as r T 0 for some T 0 2 E 0 , then by the reciprocity theorem

m2Zn

since the sum only contains ﬁnitely many non-zero terms; ' is periodic, and '~ 2 D
Rn =Zn . Conversely, if '~ 2 D
Rn =Zn we ~ x, and '0 2 D
Rn may deﬁne ' 2 E
Rn periodic by '
x '
~ 0 by putting ' ' with constructed as above. By transposition, a distribution T~ 2 D0
Rn =Zn deﬁnes a unique ~ 'i; ~ conversely, periodic distribution T 2 D0
Rn by hT, '0 i hT, n 0 ~ T 2 D
R periodic deﬁnes uniquely T 2 D0
Rn =Zn by ~ 'i ~ hT, '0 i. hT, We may therefore identify Zn -periodic distributions over Rn with distributions over Rn =Zn . We will, however, use mostly the former

w F T 0
hTx0 , exp
2i xi:

Although the choice of T 0 is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of T 0 will lead to the same coefﬁcients w because of the periodicity of exp
2i x. The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relations

41

1. GENERAL RELATIONSHIPS AND TECHNIQUES w hTx0 , exp
2i xi P Tx w exp
2i x

i

ii

F T 0
A 1 T jdet AjF t0
, so that

2Zn

F T

are referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section 1.3.4.2.1.1). In other words, any periodic distribution T 2 S 0 may be represented by a Fourier series (ii), whose coefﬁcients are calculated by (i). The convergence of (ii) towards T in S 0 will be investigated later (Section 1.3.2.6.10).

iii
iv

v

vi

R jdet Aj
A 1 T # r: R is a lattice distribution: P P
A 1 T
R 2

associated with the reciprocal lattice whose basis vectors a1 , . . . , an are the columns of
A 1 T . Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of co-factors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section 1.3.4.2.1.1 for the crystallographic case n 3). A distribution T will be called -periodic if T T for all 2 ; as previously, T may be written R T 0 for some motif distribution T 0 with compact support. By Fourier transformation,

2Zn

F T 0
A 1 T
A

2

w htx0 , exp
2i xi, 2 Zn P tx w exp
2i x 2Zn

Let T 0 be a distribution with compact support (the ‘motif’). Its Fourier transform F T 0 is analytic (Section 1.3.2.5.4) and may thus be used as a multiplier. We may rephrase the preceding results as follows: (i) if T 0 is ‘periodized by R’ to give R T 0 , then F T 0 is ‘sampled by R ’ to give jdet Aj 1 R F T 0 ; (ii) if F T 0 is ‘sampled by R ’ to give R F T 0 , then T 0 is ‘periodized by R’ to give jdet AjR T 0 . Thus the Fourier transformation establishes a duality between the periodization of a distribution by a period lattice and the sampling of its transform at the nodes of lattice reciprocal to . This is a particular instance of the convolution theorem of Section 1.3.2.5.8. At this point it is traditional to break the symmetry between F and F which distribution theory has enabled us to preserve even in the presence of periodicity, and to perform two distinct identiﬁcations: (i) a -periodic distribution T will be handled as a distribution T~ on Rn =, was done in Section 1.3.2.6.3; P (ii) a weighted lattice distribution W 2Zn W
A 1 T will be identiﬁed with the collection fW j 2 Zn g of its n-tuply indexed coefﬁcients.

F R jdet Aj 1 R

P

2 L

1.3.2.6.6. Duality between periodization and sampling

which we write:

1

W jdet Aj 1 hTx0 , exp
2i xi, P Tx W exp
2i x

in standard coordinates. It gives an n-dimensional Fourier series representation for any periodic distribution over Rn . The convergence of such series in S 0
Rn will be examined in Section 1.3.2.6.10.

F R jdet Aj 1 F A# r
A 1 T # F r
A 1 T # r,

jdet Aj

in non-standard coordinates, or equivalently:

for any ' 2 S , and hence R jdet Aj 1 A# r. By Fourier transformation, according to Section 1.3.2.5.5,

2

1 T

in standard coordinates. The reciprocity theorem may then be written:

m2Zn

F T jdet Aj 1 R F T 0 P F T 0
jdet Aj 1

F t0
A

2Zn

Let P denote the non-standard lattice consisting of all vectors of the form j1 mj aj , where the mj are rational integers and a1 , . . . , an are n linearly independentPvectors in Rn . Let R be the corresponding lattice distribution: R x2
x . Let A be the non-singular n n matrix whose successive columns are the coordinates of vectors a1 , . . . , an in the standard basis of Rn ; A will be called the period matrix of , and the mapping x 7 ! Ax will be denoted by A. According to Section 1.3.2.3.9.5 we have P hR, 'i '
Am hr,
A 1 # 'i jdet Aj 1 hA# r, 'i

2Zn

2Zn

in non-standard coordinates, while P F t F t0

1.3.2.6.5. The case of non-standard period lattices

with

P

1 T

1.3.2.6.7. The Poisson summation formula

so that F T is a weighted reciprocal-lattice distribution, the weight attached to node 2 being jdet Aj 1 times the value F T 0
of the Fourier transform of the motif T 0 . This result may be further simpliﬁed if T and its motif T 0 are referred to the standard period lattice Zn by deﬁning t and t0 so that T A# t, T 0 A# t0 , t r t0 . Then

Let ' 2 S , so that F ' 2 S . Let R be the lattice distribution associated to lattice , with period matrix A, and let R be associated to the reciprocal lattice . Then we may write:

hence

i.e.

hR, 'i hR, F F 'i hF R, F 'i

jdet Aj 1 hR , F 'i

F T 0
jdet AjF t0
AT ,

42

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY P probability theory (Grenander, 1952) and play an important role '
x jdet Aj 1 F '
: in several direct approaches to the crystallographic phase problem x2

2 This identity, which also holds for F , is called the Poisson [see Sections 1.3.4.2.1.10, 1.3.4.5.2.2(e)]. Many aspects of their summation formula. Its usefulness follows from the fact that the theory and applications are presented in the book by Grenander & speed of decrease at inﬁnity of ' and F ' are inversely related Szego¨ (1958). P

(Section 1.3.2.4.4.3), so that if one of the series (say, the left-hand side) is slowly convergent, the other (say, the right-hand side) will be rapidly convergent. This procedure has been used by Ewald (1921) [see also Bertaut (1952), Born & Huang (1954)] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4 in this volume on convergence acceleration techniques for crystallographic lattice sums). When ' is a multivariate Gaussian '
x GB
x exp

1.3.2.6.9.1. Toeplitz forms Let f 2 L1
R=Z be real-valued, so that its Fourier coefﬁcients satisfy the relations c m
f cm
f . The Hermitian form in n 1 complex variables n P n P Tn f
u u c u 0 0

is called the nth Toeplitz form associated to f. It is a straightforward consequence of the convolution theorem and of Parseval’s identity that Tn f may be written: 2 n R1 P Tn f
u u exp
2ix f
x dx:

1 T 2x Bx,

then F '
jdet
2B 1 j1=2 GB 1
,

0 0

and Poisson’s summation formula for a lattice with period matrix A reads: P GB
Am jdet Aj 1 jdet
2B 1 j1=2

1.3.2.6.9.2. The Toeplitz–Carathe´odory–Herglotz theorem It was shown independently by Toeplitz (1911b), Carathe´odory (1911) and Herglotz (1911) that a function f 2 L1 is almost everywhere non-negative if and only if the Toeplitz forms Tn f associated to f are positive semideﬁnite for all values of n. This is equivalent to the inﬁnite system of determinantal inequalities 0 1 c0 c 1 c n B c1 c0 c 1 C B C B C Dn det B c1 C 0 for all n: @ c 1A cn c1 c0

n

m2Z

P

2Zn

G42 B 1
A 1 T

or equivalently P P GC
m jdet
2C 1 j1=2 G42 C 1
m2Zn

2Zn

with C AT BA:

1.3.2.6.8. Convolution of Fourier series Let S R S 0 and T R T 0 be two -periodic distributions, the motifs S 0 and T 0 having compact support. The convolution S T does not exist, because S and T do not satisfy the support condition (Section 1.3.2.3.9.7). However, the three distributions R, S 0 and T 0 do satisfy the generalized support condition, so that their convolution is deﬁned; then, by associativity and commutativity:

The Dn are called Toeplitz determinants. Their application to the crystallographic phase problem is described in Section 1.3.4.2.1.10. 1.3.2.6.9.3. Asymptotic distribution of eigenvalues of Toeplitz forms The eigenvalues of the Hermitian form Tn f are deﬁned as the n 1 real roots of the characteristic equation det fTn f g 0. They will be denoted by

R S 0 T 0 S T 0 S 0 T: By Fourier transformation and by the convolution theorem:

n

n

n

1 , 2 , . . . , n1 :

R F S 0 T 0
R F S 0 F T 0

It is easily shown that if m f
x M for all x, then m
n M for all n and all 1, . . . , n 1. As n ! 1 these bounds, and the distribution of the
n within these bounds, can be made more precise by introducing two new notions. (i) Essential bounds: deﬁne ess inf f as the largest m such that f
x m except for values of x forming a set of measure 0; and deﬁne ess sup f similarly. (ii) Equal distribution. For each n, consider two sets of n 1 real numbers:

F T 0
R F S 0 :

Let fU g 2 , fV g 2 and fW g 2 be the sets of Fourier coefﬁcients associated to S, T and S T 0
S 0 T, respectively. Identifying the coefﬁcients of for 2 yields the forward version of the convolution theorem for Fourier series: W jdet AjU V :

The backward version of the theorem requires that T be inﬁnitely differentiable. The distribution S T is then well deﬁned and its Fourier coefﬁcients fQ g 2 are given by P Q U V :

n

n

n

a1 , a2 , . . . , an1 ,

n

n

n

and b1 , b2 , . . . , bn1 :

n Assume that for each and each n, ja
n j < K and jb j < K with
n K independent of and n. The sets fa
n g and fb g are said to be equally distributed in K, K if, for any function F over K, K,

2

1.3.2.6.9. Toeplitz forms, Szego¨’s theorem Toeplitz forms were ﬁrst investigated by Toeplitz (1907, 1910, 1911a). They occur in connection with the ‘trigonometric moment problem’ (Shohat & Tamarkin, 1943; Akhiezer, 1965) and

n1 1 X F
a
n n!1 n 1 1

lim

43

F
b
n 0:

1. GENERAL RELATIONSHIPS AND TECHNIQUES We may now state an important theorem of Szego¨ (1915, 1920). Let f 2 L1 , and put m ess inf f , M ess sup f. If m and M are ﬁnite, then for any continuous function F
deﬁned in the interval [m, M] we have 1

cm
f

0

In other words, the eigenvalues
n of the Tn and the values f =
n 2 of f on a regular subdivision of ]0, 1[ are equally distributed. Further investigations into the spectra of Toeplitz matrices may be found in papers by Hartman & Wintner (1950, 1954), Kac et al. (1953), Widom (1965), and in the notes by Hirschman & Hughes (1977).

n lim n!1 n1

may be written, by virtue of the convolution theorem, as Sp
f Dp f , where Dp
x

n

Thus, when f 0, the condition number n1 =1 of Tn f tends towards the ‘essential dynamic range’ M=m of f. (ii) Let F
s where s is a positive integer. Then 1

0

(iii) Let m > 0, so that Then

> 0, and let Dn
f det Tn
f .

Dn
f hence log Dn
f

n1 Q

1

n1 P

1

n ,

lim Dn
f 1=
n1 exp

n!1

R1 0

jmjp

Cp
f

log
n :

Putting F
log , it follows that (

X

exp
2imx

sin
2p 1x sin x

is the Dirichlet kernel. Because Dp comprises numerous slowly decaying oscillations, both positive and negative, Sp
f may not converge towards f in a strong sense as p ! 1. Indeed, spectacular pathologies are known to exist where the partial sums, examined pointwise, diverge everywhere (Zygmund, 1959, Chapter VIII). When f is piecewise continuous, but presents isolated jumps, convergence near these jumps is marred by the Gibbs phenomenon: Sp
f always ‘overshoots the mark’ by about 9%, the area under the spurious peak tending to 0 as p ! 1 but not its height [see Larmor (1934) for the history of this phenomenon]. By contrast, the arithmetic mean of the partial sums, also called the pth Cesa`ro sum,

Z n1 1 X s lim
n f
xs dx: n!1 n 1 1
n

f
x exp
2imx dx

0

jmjp

M ess sup f :
n

R1

is bounded: jcm
f j k f k1 , and by the Riemann–Lebesgue lemma cm
f ! 0 as m ! 1. By the convolution theorem, cm
f g cm
f cm
g. The pth partial sum Sp
f of the Fourier series of f, P Sp
f
x cm
f exp
2imx,

1.3.2.6.9.4. Consequences of Szego¨’s theorem (i) If the ’s are ordered in ascending order, then m ess inf f ,

j f
xj dx < 1:

0

It is a convolution algebra: If f and g are in L1 , then f g is in L1 . The mth Fourier coefﬁcient cm
f of f,

Z n1 1 X
n lim F
F f
x dx: n!1 n 1 1

n lim n!1 1

R1

k f k1

1 S0
f . . . Sp
f , p1

converges to f in the sense of the L1 norm: kCp
f f k1 ! 0 as p ! 1. If furthermore f is continuous, then the convergence is uniform, i.e. the error is bounded everywhere by a quantity which goes to 0 as p ! 1. It may be shown that

)

log f
x dx :

Further terms in this limit were obtained by Szego¨ (1952) and interpreted in probabilistic terms by Kac (1954).

Cp
f Fp f , where

1.3.2.6.10. Convergence of Fourier series

jmj exp
2imx p1 jmjp 1 sin
p 1x 2 p1 sin x

Fp
x

The investigation of the convergence of Fourier series and of more general trigonometric series has been the subject of intense study for over 150 years [see e.g. Zygmund (1976)]. It has been a constant source of new mathematical ideas and theories, being directly responsible for the birth of such ﬁelds as set theory, topology and functional analysis. This section will brieﬂy survey those aspects of the classical results in dimension 1 which are relevant to the practical use of Fourier series in crystallography. The books by Zygmund (1959), Tolstov (1962) and Katznelson (1968) are standard references in the ﬁeld, and Dym & McKean (1972) is recommended as a stimulant.

X

1

is the Feje´r kernel. Fp has over Dp the advantage of being everywhere positive, so that the Cesa`ro sums Cp
f of a positive function f are always positive. The de la Valle´e Poussin kernel Vp
x 2F2p1
x

Fp
x

has a trapezoidal distribution of coefﬁcients and is such that cm
Vp 1 if jmj p 1; therefore Vp f is a trigonometric polynomial with the same Fourier coefﬁcients as f over that range of values of m.

1.3.2.6.10.1. Classical L1 theory The space L1
R=Z consists of (equivalence classes of) complexvalued functions f on the circle which are summable, i.e. for which

44

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY and because the family of functions fexp
2imxgm2Z constitutes an orthonormal Hilbert basis for L2 . The sequence of Fourier coefﬁcients cm
f of f 2 L2 belongs to the space `2
Z of square-summable sequences: P jcm
f j2 < 1:

The Poisson kernel Pr
x 1 2

1 X rm cos 2mx m1

1 r2 2r cos 2mx r2

1

m2Z

Conversely, every element c
cm of `2 is the sequence of Fourier coefﬁcients of a unique function in L2 . The inner product P
c, d c m dm

with 0 r < 1 gives rise to an Abel summation procedure [Tolstov (1962, p. 162); Whittaker & Watson (1927, p. 57)] since P
Pr f
x cm
f rjmj exp
2imx:

m2Z

m2Z

2

makes ` into a Hilbert space, and the map from L2 to `2 established by the Fourier transformation is an isometry (Parseval/Plancherel):

Compared with the other kernels, Pr has the disadvantage of not being a trigonometric polynomial; however, Pr is the real part of the Cauchy kernel (Cartan, 1961; Ahlfors, 1966): 1 r exp
2ix Pr
x 1 r exp
2ix

k f kL2 kc
f k`2 or equivalently:
f , g
c
f , c
g:

This is a useful property in applications, since ( f , g) may be calculated either from f and g themselves, or from their Fourier coefﬁcients c
f and c
g (see Section 1.3.4.4.6) for crystallographic applications). By virtue of the orthogonality of the basis fexp
2imxgm2Z , the partial sum Sp
f is the best mean-square ﬁt to f in the linear subspace of L2 spanned by fexp
2imxgjmjp , and hence (Bessel’s inequality) P P jcm
f j2 k f k22 jcM
f j2 k f k22 :

and hence provides a link between trigonometric series and analytic functions of a complex variable. Other methods of summation involve forming a moving average of f by convolution with other sequences of functions p
x besides Dp of Fp which ‘tend towards ’ as p ! 1. The convolution is performed by multiplying the Fourier coefﬁcients of f by those of p , so that one forms the quantities P Sp0
f
x cm
p cm
f exp
2imx: jmjp

jmjp

For instance the ‘sigma factors’ of Lanczos (Lanczos, 1966, p. 65), deﬁned by

1.3.2.6.10.3. The viewpoint of distribution theory The use of distributions enlarges considerably the range of behaviour which can be accommodated in a Fourier series, even in the case of general dimension n where classical theories meet with even more difﬁculties than in dimension 1. Let fwm gm2Z be a sequence of complex numbers with jwm j growing at most polynomially as jmj ! 1, say jwm j CjmjK . Then the sequence fwm =
2imK2 gm2Z is in `2 and even deﬁnes a continuous function f 2 L2
R=Z and an associated tempered distribution Tf 2 0
R=Z. Differentiation of Tf
K 2 times then yields a tempered distribution whose Fourier transform leads to the original sequence of coefﬁcients. Conversely, by the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), the motif T 0 of a Z-periodic distribution is a derivative of ﬁnite order of a continuous function; hence its Fourier coefﬁcients will grow at most polynomially with jmj as jmj ! 1. Thus distribution theory allows the manipulation of Fourier series whose coefﬁcients exhibit polynomial growth as their order goes to inﬁnity, while those derived from functions had to tend to 0 by virtue of the Riemann–Lebesgue lemma. The distributiontheoretic approach to Fourier series holds even in the case of general dimension n, where classical theories meet with even more difﬁculties (see Ash, 1976) than in dimension 1.

sinm=p , m=p

m

lead to a summation procedure whose behaviour is intermediate between those using the Dirichlet and the Feje´r kernels; it corresponds to forming a moving average of f by convolution with p p

1=
2p; 1=
2p Dp ,

which is itself the convolution of a ‘rectangular pulse’ of width 1=p and of the Dirichlet kernel of order p. A review of the summation problem in crystallography is given in Section 1.3.4.2.1.3. 1.3.2.6.10.2. Classical L2 theory The space L2
R=Z of (equivalence classes of) square-integrable complex-valued functions f on the circle is contained in L1
R=Z, since by the Cauchy–Schwarz inequality !2 R1 2 k f k1 j f
xj 1 dx 0

R1 0

2

j f
xj dx

!

R1 0

1

!

1 dx k f k22 1: 2

1.3.2.7. The discrete Fourier transformation

L2 , 1

Thus all the results derived for L hold for a great simpliﬁcation over the situation in R or Rn where neither L nor L2 was contained in the other. However, more can be proved in L2 , because L2 is a Hilbert space (Section 1.3.2.2.4) for the inner product
f , g

R1 0

jMjp

1.3.2.7.1. Shannon’s sampling theorem and interpolation formula Let ' 2
Rn be such that ' has compact support K. Let ' be sampled at the nodes of a lattice , yielding the lattice distribution R '. The Fourier transform of this sampled version of ' is

f
xg
x dx,

R ' jdet Aj
R ,

45

1. GENERAL RELATIONSHIPS AND TECHNIQUES which is essentially periodized by period lattice
, with period matrix A. Let us assume that is such that the translates of K by different period vectors of are disjoint. Then we may recover from R by masking the contents of a ‘unit cell’ V of (i.e. a fundamental domain for the action of in Rn ) whose boundary does not meet K. If V is the indicator function of V , then

which may be viewed as the n-dimensional equivalent of the Euclidean algorithm for integer division: l is the ‘remainder’ of the division by A of a vector in B , the quotient being the matrix D. 1.3.2.7.2.2. Sublattice relations for reciprocal lattices Let us now consider the two reciprocal lattices A and B . Their period matrices
A 1 T and
B 1 T are related by:
B 1 T
A 1 T NT , where NT is an integer matrix; or equivalently by
B 1 T DT
A 1 T . This shows that the roles are reversed in that B is a sublattice of A , which we may write:

V
R : Transforming both sides by F yields 1 ' F V F R ' , jdet Aj

i

i.e. 1 ' F V
R ' V

ii

since jdet Aj is the volume V of V . This interpolation formula is traditionally credited to Shannon (1949), although it was discovered much earlier by Whittaker (1915). It shows that ' may be recovered from its sample values on (i.e. from R ') provided is sufﬁciently ﬁne that no overlap (or ‘aliasing’) occurs in the periodization of by the dual lattice . The interpolation kernel is the transform of the normalized indicator function of a unit cell of containing the support K of . If K is contained in a sphere of radius 1= and if and are rectangular, the length of each basis vector of must be greater than 2=, and thus the sampling interval must be smaller than =2. This requirement constitutes the Shannon sampling criterion.

iii

A

[

l 2A =B

TB=A and TA=B

l B :

l DT A :

P

l

P

l

l2B =A

l 2A =B

are (ﬁnite) residual-lattice distributions. We may incorporate the factor 1=jdet Dj in (i) and
i into these distributions and deﬁne 1 1 SB=A TB=A , SA=B T : jdet Dj jdet Dj A=B Since jdet Dj B : A A : B , convolution with SB=A and SA=B has the effect of averaging the translates of a distribution under the elements (or ‘cosets’) of the residual lattices B =A and A =B , respectively. This process will be called ‘coset averaging’. Eliminating R A and R B between (i) and (ii), and R A and R B between
i and
ii , we may write:

l2B =A

represents B as the disjoint union of B : A translates of A : B =A is a ﬁnite lattice with B : A elements, called the residual lattice of B modulo A . The two descriptions are connected by the relation B : A det D det N, which follows from a volume calculation. We may also combine (i) and (ii) into

l2B =A

l 2A =B

where

A DB : (ii) Call two vectors in B congruent modulo A if their difference lies in A . Denote the set of congruence classes (or ‘cosets’) by B =A , and the number of these classes by B : A . The ‘coset decomposition’ [ B
l A

[

[

1.3.2.7.2.3. Relation between lattice distributions The above relations between lattices may be rewritten in terms of the corresponding lattice distributions as follows: 1
i RA D# R B jdet Dj
ii R B TB=A R A 1
i R B
DT # R A jdet Dj
ii R A TA=B R B

1.3.2.7.2.1. Geometric description of sublattices Let A be a period lattice in Rn with matrix A, and let A be the lattice reciprocal to A , with period matrix
A 1 T . Let B , B, B be deﬁned similarly, and let us suppose that A is a sublattice of B , i.e. that B A as a set. The relation between A and B may be described in two different fashions: (i) multiplicatively, and (ii) additively. (i) We may write A BN for some non-singular matrix N with integer entries. N may be viewed as the period matrix of the coarser lattice A with respect to the period basis of the ﬁner lattice B . It will be more convenient to write A DB, where D BNB 1 is a rational matrix (with integer determinant since det D det N) in terms of which the two lattices are related by

B

A

The residual lattice A =B is ﬁnite, with A : B det D det N B : A , and we may again combine
i and
ii into

1.3.2.7.2. Duality between subdivision and decimation of period lattices

iii

B DT A

i0

R A D#
SB=A R A

i0

R B
DT #
SA=B R B

ii0

ii0

l DB

R B SB=A
D# R B

R A SA=B
DT # R A :

These identities show that period subdivision by convolution with

46

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY SA=B )

Therefore, the duality between subdivision and decimation may be viewed as another aspect of that between convolution and multiplication. There is clearly a strong analogy between the sampling/ periodization duality of Section 1.3.2.6.6 and the decimation/ subdivision duality, which is viewed most naturally in terms of subgroup relationships: both sampling and decimation involve restricting a function to a discrete additive subgroup of the domain over which it is initially given.

SB=A (respectively on the one hand, and period decimation by ‘dilation’ by D# on the other hand, are mutually inverse operations on R A and R B (respectively R A and R B ). 1.3.2.7.2.4. Relation between Fourier transforms Finally, let us consider the relations between the Fourier transforms of these lattice distributions. Recalling the basic relation of Section 1.3.2.6.5, 1 F R A R jdet Aj A 1 by (ii) T R B jdet DBj A=B 1 1 T R jdet Dj A=B jdet Bj B

1.3.2.7.2.5. Sublattice relations in terms of periodic distributions The usual presentation of this duality is not in terms of lattice distributions, but of periodic distributions obtained by convolving them with a motif. Given T 0 2 E 0
Rn , let us form R A T 0 , then decimate its transform
1=jdet AjR A F T 0 by keeping only its values at the points of the coarser lattice B DT A ; as a result, R A is replaced by
1=jdet DjR B , and the reverse transform then yields 1 by (ii), R B T 0 SB=A
R A T 0 jdet Dj

i.e. F R B F R A SA=B

iv and similarly:

F R B SB=A F R A :

v

Thus R A (respectively R B ), a decimated version of R B (respectively R A ), is transformed by F into a subdivided version of F R B (respectively F R A ). The converse is also true: 1 F R B R jdet Bj B 1 1 by (i)
DT # R A jdet Bj jdet Dj 1 T #
D R jdet Aj A

which is the coset-averaged version of the original R A T 0 . The converse situation is analogous to that of Shannon’s sampling theorem. Let a function ' 2 E
Rn whose transform F ' has compact support be sampled as R B ' at the nodes of B . Then 1 F R B '
R jdet Bj B

is periodic with period lattice B . If the sampling lattice B is decimated to A DB , the inverse transform becomes 1 F R A '
R jdet Dj A
R B by (ii) , SA=B

i.e.
iv0

F R B
DT # F R A

v0

F R A D# F R B :

and similarly

hence becomes periodized more ﬁnely by averaging over the cosets of A =B . With this ﬁner periodization, the various copies of Supp may start to overlap (a phenomenon called ‘aliasing’), indicating that decimation has produced too coarse a sampling of '.

Thus R B (respectively R A ), a subdivided version of R A (respectively R B ) is transformed by F into a decimated version of F R A (respectively F R B ). Therefore, the Fourier transform exchanges subdivision and decimation of period lattices for lattice distributions. Further insight into this phenomenon is provided by applying F to both sides of (iv) and (v) and invoking the convolution theorem:
iv00 R A F S R B

1.3.2.7.3. Discretization of the Fourier transformation Let '0 2 E
Rn be such that 0 F '0 has compact support (' is said to be band-limited). Then ' R A '0 is A -periodic, and F '
1=jdet AjR A 0 is such that only a ﬁnite number of points A of A have a non-zero Fourier coefﬁcient 0
A attached to them. We may therefore ﬁnd a decimation B DT A of A such that the distinct translates of Supp 0 by vectors of B do not intersect. The distribution can be uniquely recovered from R B by the procedure of Section 1.3.2.7.1, and we may write: 1 R B R
R A 0 jdet Aj B 1 R
R B 0 jdet Aj A 1
R B 0 ; R TA=B jdet Aj B 0

A=B

R B F SB=A R A :

v00

These identities show that multiplication by the transform of the period-subdividing distribution SA=B (respectively SB=A ) has the effect of decimating R B to R A (respectively R A to R B ). They clearly imply that, if l 2 B =A and l 2 A =B , then F S
l 1 if l 0
i:e: if l belongs A=B

to the class of A ,

0 if l 6 0; F SB=A
l 1 if l 0
i:e: if l belongs to the class of B ,

have these rearrangements being legitimate because 0 and TA=B compact supports which are intersection-free under the action of B . By virtue of its B -periodicity, this distribution is entirely ~ with respect to : characterized by its ‘motif’ B

0 if l 6 0:

47

1. GENERAL RELATIONSHIPS AND TECHNIQUES

1 T
R B 0 : jdet Aj A=B

l l l l k
N 1 k: 1 T ~ ~ Denoting '
Bk by
k and
A k by
k , the relation between ! and may be written in the equivalent form

Similarly, ' may be uniquely recovered by Shannon interpolation from the distribution sampling its values at the nodes of B D 1 A
B is a subdivision of B ). By virtue of its A -periodicity, this distribution is completely characterized by its motif:

i

' TB=A ' TB=A
R A '0 :

Let l 2 B = A and l 2 A = B , and deﬁne the two sets of coefﬁcients

ii

X 1
k exp 2ik
N 1 k jdet Nj n T n k 2Z =N Z X
k exp2ik
N 1 k,
k
k

k2Zn =NZn

for any A 2 A where the summations are now over ﬁnite residual lattices in
all choices of A give the same ', 0 standard form.
2
l
l B for the unique B (if it exists) Equations (i) and (ii) describe two mutually inverse linear 0 such that l B 2 Supp , transformations F
N and F
N between two vector spaces WN 0 if no such B exists: and WN of dimension jdet Nj. F
N [respectively F
N] is the discrete Fourier (respectively inverse Fourier) transform associated Deﬁne the two distributions to matrix N. P
l ! '
l The vector spaces WN and WN may be viewed from two different l2 B = A standpoints: (1) as vector spaces of weighted residual-lattice distributions, of and the form
xTB=A and
xTA=B ; the canonical basis of WN P
l : (respectively WN ) then consists of the
k for k 2 Zn =NZn

l l 2 A = B [respectively
k for k 2 Zn =NT Zn ]; (2) as vector spaces of weight vectors for the jdet Nj -functions The relation between ! and has two equivalent forms: ); the involved in the expression for TB=A (respectively TA=B (respectively W ) consists of weight vectors canonical basis of W N
i R A ! F R B N uk (respectively vk ) giving weight 1 to element k (respectively k )
ii F R A ! R B : and 0 to the others. These two spaces are said to be ‘isomorphic’ (a relation denoted By (i), R A ! jdet BjR B F . Both sides are weighted lattice distributions concentrated at the nodes of B , and equating ), the isomorphism being given by the one-to-one correspondence: the weights at B l A gives P P X !
k
k $
kuk 1 ~ ~
l exp 2il
l A : '
l k k P P jdet Dj l 2 =

k
k $
k vk : B A

1 '
l

'
l A

k

Since l 2 A , l A is an integer, hence X 1 ~ exp
2il l: ~
l '
l jdet Dj l 2 = A

The second viewpoint will be adopted, as it involves only linear algebra. However, it is most helpful to keep the ﬁrst one in mind and to think of the data or results of a discrete Fourier transform as representing (through their sets of unique weights) two periodic lattice distributions related by the full, distribution-theoretic Fourier transform. We therefore view WN (respectively WN ) as the vector space of complex-valued functions over the ﬁnite residual lattice B =A (respectively A =B ) and write:

B

By (ii), we have 1 1
R B 0 R TA=B F R A !: jdet Aj B jdet Aj

Both sides are weighted lattice distributions concentrated at the nodes of B , and equating the weights at A l B gives P ~ ~ exp2il
l : '
l
l

Since l 2 B , l

WN L
B =A L
Zn =NZn

B

l2B =A

WN L
A =B L
Zn =NT Zn

B is ~

an integer, hence P ~ exp
2il l : '
l
l

since a vector such as is in fact the function k 7 !
k. The two spaces WN and WN may be equipped with the following Hermitian inner products:

l2B =A

Now the decimation/subdivision relations between A and B may be written:

', W

A DB BN,

, W

so that l Bk

l
A 1 T k

for k 2 Zn

P '
k
k k

P
k
k , k

which makes each of them into a Hilbert space. The canonical bases fuk jk 2 Zn =NZn g and fvk jk 2 Zn =NT Zn g and WN and WN are orthonormal for their respective product.

for k 2 Zn

with
A 1 T
B 1 T
N 1 T , hence ﬁnally

k

48

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY A =B , then for all multi-indices p
p1 , p2 , . . . , pn
Dp
k F
N
2ik p C
k

1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) By virtue of deﬁnitions (i) and (ii), 1 X exp 2ik
N 1 kuk F
Nvk jdet Nj k X exp2ik
N 1 kvk F
Nuk

Dp C
k F
N
2ikp
k

or equivalently F
NDp
k
2ik p C
k F
NDp C
k
2ikp
k:

k

(4) Convolution property. Let 2 WN and F 2 WN (respectively and C) be related by the DFT, and deﬁne P
k 0
k k 0
k

so that F
N and F
N may be represented, in the canonical bases of WN and WN , by the following matrices: 1 F
Nkk exp 2ik
N 1 k jdet Nj F
N exp2ik
N 1 k:

F C
k

k k

When N is symmetric, Zn =NZn and Zn =NT Zn may be identiﬁed in a natural manner, and the above matrices are symmetric. When N is diagonal, say N diag
1 , 2 , . . . , n , then the tensor product structure of the full multidimensional Fourier transform (Section 1.3.2.4.2.4)

Then

F
j kj ; kj

2i

j

0

k :

1
F C
k jdet Nj F
NF C
k
k:

Since addition on Zn =NZn and Zn =NT Zn is modular, this type of convolution is called cyclic convolution. (5) Parseval/Plancherel property. If , , F, C are as above, then 1
F
NF, F
NCW
F, CW jdet Nj 1
F
N, F
NW
, W : jdet Nj (6) Period 4. When N is symmetric, so that the ranges of indices k and k can be identiﬁed, it makes sense to speak of powers of F
N and F
N. Then the ‘standardized’ matrices
1=jdet Nj1=2 F
N and
1=jdet Nj1=2 F
N are unitary matrices whose fourth power is the identity matrix (Section 1.3.2.4.3.4); their eigenvalues are therefore 1 and i.

,

and where

F
k C
k

F
N
k

F
N F
1 F
2 . . . F
n , k j k j

k 2Z =NT Z

0

n

and

Let the index vectors k and k be ordered in the same way as the elements in a Fortran array, e.g. for k with k 1 increasing fastest, k 2 next fastest, . . . , k n slowest; then

n

F
N
k F
k C
k

gives rise to a tensor product structure for the DFT matrices. The tensor product of matrices is deﬁned as follows: 0 1 a11 B . . . a1n B B . .. C A B @ .. . A: an1 B . . . ann B

1 exp j

0

P

F
NF C
k jdet Nj
k
k

F x F x1 F x2 . . . F xn

where

k 0 2Zn =NZn

F
N F
1 F
2 . . . F
n ,

1.3.3. Numerical computation of the discrete Fourier transform

k j k j : Fj kj ; kj exp 2i j

1.3.3.1. Introduction The Fourier transformation’s most remarkable property is undoubtedly that of turning convolution into multiplication. As distribution theory has shown, other valuable properties – such as the shift property, the conversion of differentiation into multiplication by monomials, and the duality between periodicity and sampling – are special instances of the convolution theorem. This property is exploited in many areas of applied mathematics and engineering (Campbell & Foster, 1948; Sneddon, 1951; Champeney, 1973; Bracewell, 1986). For example, the passing of a signal through a linear ﬁlter, which results in its being convolved with the response of the ﬁlter to a -function ‘impulse’, may be modelled as a multiplication of the signal’s transform by the transform of the impulse response (also called transfer function). Similarly, the solution of systems of partial differential equations may be turned by Fourier transformation into a division problem for distributions. In both cases, the formulations obtained after Fourier transformation are considerably simpler than the initial ones, and lend themselves to constructive solution techniques.

1.3.2.7.5. Properties of the discrete Fourier transform The DFT inherits most of the properties of the Fourier transforms, but with certain numerical factors (‘Jacobians’) due to the transition from continuous to discrete measure. (1) Linearity is obvious. (2) Shift property. If
a
k
k a and
a
k
k a , where subtraction takes place by modular vector arithmetic in Zn =NZn and Zn =NT Zn , respectively, then the following identities hold: F
Nk
k exp2ik
N 1 kF
N
k F
Nk
k exp 2ik
N 1 kF
N
k:

(3) Differentiation identities. Let vectors and C be constructed from '0 2 E
Rn as in Section 1.3.2.7.3, hence be related by the DFT. If Dp designates the vector of sample values of Dpx '0 at the points of B =A , and Dp C the vector of values of Dp 0 at points of

49

1. GENERAL RELATIONSHIPS AND TECHNIQUES Whenever the functions to which the Fourier transform is applied are band-limited, or can be well approximated by band-limited functions, the discrete Fourier transform (DFT) provides a means of constructing explicit numerical solutions to the problems at hand. A great variety of investigations in physics, engineering and applied mathematics thus lead to DFT calculations, to such a degree that, at the time of writing, about 50% of all supercomputer CPU time is alleged to be spent calculating DFTs. The straightforward use of the deﬁning formulae for the DFT leads to calculations of size N 2 for N sample points, which become unfeasible for any but the smallest problems. Much ingenuity has therefore been exerted on the design and implementation of faster algorithms for calculating the DFT (McClellan & Rader, 1979; Nussbaumer, 1981; Blahut, 1985; Brigham, 1988). The most famous is that of Cooley & Tukey (1965) which heralded the age of digital signal processing. However, it had been preceded by the prime factor algorithm of Good (1958, 1960), which has lately been the basis of many new developments. Recent historical research (Goldstine, 1977, pp. 249–253; Heideman et al., 1984) has shown that Gauss essentially knew the Cooley–Tukey algorithm as early as 1805 (before Fourier’s 1807 work on harmonic analysis!); while it has long been clear that Dirichlet knew of the basis of the prime factor algorithm and used it extensively in his theory of multiplicative characters [see e.g. Chapter I of Ayoub (1963), and Chapters 6 and 8 of Apostol (1976)]. Thus the computation of the DFT, far from being a purely technical and rather narrow piece of specialized numerical analysis, turns out to have very rich connections with such central areas of pure mathematics as number theory (algebraic and analytic), the representation theory of certain Lie groups and coding theory – to list only a few. The interested reader may consult Auslander & Tolimieri (1979); Auslander, Feig & Winograd (1982, 1984); Auslander & Tolimieri (1985); Tolimieri (1985). One-dimensional algorithms are examined ﬁrst. The Sande mixed-radix version of the Cooley–Tukey algorithm only calls upon the additive structure of congruence classes of integers. The prime factor algorithm of Good begins to exploit some of their multiplicative structure, and the use of relatively prime factors leads to a stronger factorization than that of Sande. Fuller use of the multiplicative structure, via the group of units, leads to the Rader algorithm; and the factorization of short convolutions then yields the Winograd algorithms. Multidimensional algorithms are at ﬁrst built as tensor products of one-dimensional elements. The problem of factoring the DFT in several dimensions simultaneously is then examined. The section ends with a survey of attempts at formalizing the interplay between algorithm structure and computer architecture for the purpose of automating the design of optimal DFT code. It was originally intended to incorporate into this section a survey of all the basic notions and results of abstract algebra which are called upon in the course of these developments, but time limitations have made this impossible. This material, however, is adequately covered by the ﬁrst chapter of Tolimieri et al. (1989) in a form tailored for the same purposes. Similarly, the inclusion of numerous detailed examples of the algorithms described here has had to be postponed to a later edition, but an abundant supply of such examples may be found in the signal processing literature, for instance in the books by McClellan & Rader (1979), Blahut (1985), and Tolimieri et al. (1989).

e
t1 t2 e
t1 e
t2

e
t e
t e
t e
t 1 , t 2 Z:

1

Thus e deﬁnes an isomorphism between the additive group R=Z (the reals modulo the integers) and the multiplicative group of complex numbers of modulus 1. It follows that the mapping ` 7 ! e
`=N, where ` 2 Z and N is a positive integer, deﬁnes an isomorphism between the one-dimensional residual lattice Z=N Z and the multiplicative group of Nth roots of unity. The DFT on N points then relates vectors X and X in W and W through the linear transformations: 1 X X
k e
k k=N F
N : X
k N k 2Z=NZ X F
N : X
k X
ke
k k=N: k2Z=NZ

1.3.3.2.1. The Cooley–Tukey algorithm The presentation of Gentleman & Sande (1966) will be followed ﬁrst [see also Cochran et al. (1967)]. It will then be reinterpreted in geometric terms which will prepare the way for the treatment of multidimensional transforms in Section 1.3.3.3. Suppose that the number of sample points N is composite, say N N1 N2 . We may write k to the base N1 and k to the base N2 as follows: k k1 N1 k2

k

k2

k1 N2

k1 2 Z=N1 Z, k1

2 Z=N1 Z,

k2 2 Z=N2 Z

k2 2 Z=N2 Z:

The deﬁning relation for F
N may then be written: X X X
k2 k1 N2 X
k1 N1 k2 k1 2Z=N1 Z k2 2Z=N2 Z

k2 k1 N2
k1 N1 k2 : e N1 N2

The argument of e: may be expanded as k2 k1 k1 k1 k2 k2 k1 k2 , N N1 N2 and the last summand, being an integer, may be dropped: X
k2 k1 N2 ( " #) X k k1 X k k2 2 e X
k1 N1 k2 e 2 N N2 k1 k2 k k1 : e 1 N1 This computation may be decomposed into ﬁve stages, as follows: (i) form the N1 vectors Yk1 of length N2 by the prescription Yk1
k2 X
k1 N1 k2 ,

k1 2 Z=N1 Z,

(ii) calculate the N1 transforms 2 Yk1 , Y F
N k1

1.3.3.2. One-dimensional algorithms

Yk1

k2 2 Z=N2 Z;

on N2 points:

k1 2 Z=N1 Z;

(iii) form the N2 vectors Zk2 of length N1 by the prescription k k1 Zk2
k1 e 2 Yk1
k2 , k1 2 Z=N1 Z, k2 2 Z=N2 Z; N

Throughout this section we will denote by e
t the expression exp
2it, t 2 R. The mapping t 7 ! e
t has the following properties:

50

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY (iv) calculate the N2 transforms Zk on N1 points:

X
k2 Y0
k2 e
k2 =NY1
k2 ,

2

1 Zk , Zk F
N 2 2

k2 2 Z=N2 Z;

Z0
k1 X
k1 X
k1 M,

k1 0, . . . , M

1,

Z1
k1 X
k1

k1 0, . . . , M

1,

X

k1 X
k1 Me , N

X
2k1 Z0
k1 ,

2k1

1

Z1
k1 ,

k1 0, . . . , M k1

0, . . . , M

1; 1:

This version is due to Sande (Gentleman & Sande, 1966), and the process of separately obtaining even-numbered and odd-numbered results has led to its being referred to as ‘decimation in frequency’ (i.e. decimation along the result index k ). By repeated factoring of the number N of sample points, the calculation of F
N and F
N can be reduced to a succession of stages, the smallest of which operate on single prime factors of N. The reader is referred to Gentleman & Sande (1966) for a particularly lucid analysis of the programming considerations which help implement this factorization efﬁciently; see also Singleton (1969). Powers of two are often grouped together into factors of 4 or 8, which are advantageous in that they require fewer complex multiplications than the repeated use of factors of 2. In this approach, large prime factors P are detrimental, since they require a full P2 -size computation according to the deﬁning formula.

Xk1
k X
k if k k1 mod N1 , 0 otherwise: According to (i), Xk1 is related to Yk1 by decimation by N1 and offset by k1 . By Section 1.3.2.7.2, F
NX k1 is related to F
N2 Yk1 by periodization by N2 and phase shift by e
k k1 =N, so that X k k1 X
k e Yk1
k2 , N k

1.3.3.2.2. The Good (or prime factor) algorithm 1.3.3.2.2.1. Ring structure on Z=NZ The set Z=NZ of congruence classes of integers modulo an integer N [see e.g. Apostol (1976), Chapter 5] inherits from Z not only the additive structure used in deriving the Cooley–Tukey factorization, but also a multiplicative structure in which the product of two congruence classes mod N is uniquely deﬁned as the class of the ordinary product (in Z) of representatives of each class. The multiplication can be distributed over addition in the usual way, endowing Z=NZ with the structure of a commutative ring. If N is composite, the ring Z=NZ has zero divisors. For example, let N N1 N2 , let n1 N1 mod N, and let n2 N2 mod N: then n1 n2 0 mod N. In the general case, a product of non-zero elements will be zero whenever these elements collect together all the factors of N. These circumstances give rise to a fundamental theorem in the theory of commutative rings, the Chinese Remainder Theorem (CRT), which will now be stated and proved [see Apostol (1976), Chapter 5; Schroeder (1986), Chapter 16].

1

the periodization by N2 being reﬂected by the fact that Yk1 does not depend on k1 . Writing k k2 k1 N2 and expanding k k1 shows that the phase shift contains both the twiddle factor e
k2 k1 =N and 1 . The Cooley–Tukey algorithm is the kernel e
k1 k1 =N1 of F
N thus naturally associated to the coset decomposition of a lattice modulo a sublattice (Section 1.3.2.7.2). It is readily seen that essentially the same factorization can be obtained for F
N, up to the complex conjugation of the twiddle factors. The normalizing constant 1=N arises from the normalizing constants 1=N1 and 1=N2 in F
N1 and F
N2 , respectively. Factors of 2 are particularly simple to deal with and give rise to a characteristic computational structure called a ‘butterﬂy loop’. If N 2M, then two options exist: (a) using N1 2 and N2 M leads to collecting the evennumbered coordinates of X into Y0 and the odd-numbered coordinates into Y1

k2 0, . . . , M

1:

then obtaining separately the even-numbered and odd-numbered components of X by transforming Z0 and Z1 :

where

Y1
k2 X
2k2 1,

k2 0, . . . , M

e
k2 =NY1
k2 ,

This is the original version of Cooley & Tukey, and the process of formation of Y0 and Y1 is referred to as ‘decimation in time’ (i.e. decimation along the data index k). (b) using N1 M and N2 2 leads to forming

k1

k2 0, . . . , M

1;

X
k2 M Y0
k2

(v) collect X
k2 k1 N2 as Zk
k1 . 2 If the intermediate transforms in stages (ii) and (iv) are performed in place, i.e. with the results overwriting the data, then at stage (v) the result X
k2 k1 N2 will be found at address k1 N1 k2 . This phenomenon is called scrambling by ‘digit reversal’, and stage (v) is accordingly known as unscrambling. has thus been performed as The initial N-point transform F
N N1 transforms F
N2 on N2 points, followed by N2 transforms 1 on N1 points, thereby reducing the arithmetic cost from F
N
N1 N2 2 to N1 N2
N1 N2 . The phase shifts applied at stage (iii) are traditionally called ‘twiddle factors’, and the transposition between k1 and k2 can be performed by the fast recursive technique of Eklundh (1972). Clearly, this procedure can be applied recursively if N1 and N2 are themselves composite, leading to an overall arithmetic cost of order N log N if N has no large prime factors. The Cooley–Tukey factorization may also be derived from a geometric rather than arithmetic argument. The decomposition k k1 N1 k2 is associated to a geometric partition of the residual lattice Z=NZ into N1 copies of Z=N2 Z, each translated by k1 2 Z=N1 Z and ‘blown up’ by a factor N1 . This partition in turn induces a (direct sum) decomposition of X as P X Xk1 ,

Y0
k2 X
2k2 ,

k2 0, . . . , M

1.3.3.2.2.2. The Chinese remainder theorem Let N N1 N2 . . . Nd be factored into a product of pairwise coprime integers, so that g.c.d.
Ni , Nj 1 for i 6 j. Then the system of congruence equations

1, 1,

` `j mod Nj ,

and writing:

51

j 1, . . . , d,

has a unique solution ` mod N. In other words, each ` 2 Z=NZ is

1. GENERAL RELATIONSHIPS AND TECHNIQUES associated in a one-to-one fashion to the d-tuple
`1 , `2 , . . . , `d of its residue classes in Z=N1 Z, Z=N2 Z, . . . , Z=Nd Z. The proof of the CRT goes as follows. Let N Ni : Qj Nj i6j

k Then kk

Since g.c.d.
Nj , Qj 1 there exist integers nj and qj such that nj Nj qj Qj 1,

j 1, . . . , d,

then the integer `

d P

`i qi Qi mod N

` `j qj Qj mod Nj qj Qj 1 mod Nj

mod N, j 1, . . . , d,

so that the qj Qj are mutually orthogonal idempotents in the ring Z=NZ, with properties formally similar to those of mutually orthogonal projectors onto subspaces in linear algebra. The analogy is exact, since by virtue of the CRT the ring Z=N Z may be considered as the direct product

mod N

ki kj Qi qj Qj mod N:

d P

j1 d P

qj Q2j kj kj mod N

1

j1

nj Nj Qj kj kj mod N:

d kk kk X j j mod 1: Nj N j1

Therefore, by the multiplicative property of e
:, O d kj kj k k : e e Nj N j1

via the two mutually inverse mappings: (i) ` 7 !
`1 , `2 , . . . , `d by ` `jPmod Nj for each j; (ii)
`1 , `2 , . . . , `d 7 ! ` by ` di1 `i qi Qi mod N . The mapping deﬁned by (ii) is sometimes called the ‘CRT reconstruction’ of ` from the `j . These two mappings have the property of sending sums to sums and products to products, i.e:

Let X 2 L
Z=NZ be described by a one-dimensional array X
k indexed by k. The index mapping (i) turns X into an element of L
Z=N1 Z . . . Z=Nd Z described by a d-dimensional array X
k1 , .N . . , kd ; by N the latter may be transformed d into a new array X
k1 , k2 , . . . , kd . Finally, 1 . . . F
N F
N the one-dimensional array of results X
k will be obtained by reconstructing k according to (ii). The prime factor algorithm, like the Cooley–Tukey algorithm, reindexes a 1D transform to turn it into d separate transforms, but the use of coprime factors and CRT index mapping leads to the further gain that no twiddle factors need to be applied between the successive transforms (see Good, 1971). This makes up for the cost of the added complexity of the CRT index mapping. The natural factorization of N for the prime factor algorithm is thus its factorization into prime powers: F
N is then the tensor product of separate transforms (one for each prime power factor Nj pj j ) whose results can be reassembled without twiddle factors. The separate factors pj within each Nj must then be dealt with by another algorithm (e.g. Cooley–Tukey, which does require twiddle factors). Thus, the DFT on a prime number of points remains undecomposable.

` `0 7 !
`1 `01 , `2 `02 , . . . , `d `0d

``0 7 !
`1 `01 , `2 `02 , . . . , `d `0d

`1 `01 , `2 `02 , . . . , `d `0d 7 ! ` `0

`1 `01 , `2 `02 , . . . , `d `0d 7 ! ``0

(the last proof requires using the properties of the idempotents qj Qj ). This may be described formally by stating that the CRT establishes a ring isomorphism: Z=NZ
Z=N1 Z . . .
Z=Nd Z: 1.3.3.2.2.3. The prime factor algorithm The CRT will now be used to factor the N-point DFT into a tensor product of d transforms, the jth of length Nj . Let the indices k and k be subjected to the following mappings: (i) k 7 !
k1 , k2 , . . . , kd , kj 2 Z=Nj Z, by kj k mod Nj for each j, with reconstruction formula d P

k j q j Qj

j1

!

and hence

Z=N1 Z Z=N2 Z . . . Z=Nd Z

k

d P

d Qj kk X
1 nj Nj k kj mod 1 N j Qj j N j1 d X 1 nj kj kj mod 1, N j j1

mod N for i 6 j,

qj Qj qj Qj

ii

i; j1

Dividing by N, which may be written as Nj Qj for each j, yields

by the deﬁning relation for qj . It may be noted that

i

d P

because all terms with i 6 j contain Nj as a factor; and

2

ki Qi

i1

kk

qi Qi
qj Qj 0

d P

i1

ki Qi mod N:

Cross terms with i 6 j vanish since they contain all the factors of N, hence

i1

is the solution. Indeed,

d P

1.3.3.2.3. The Rader algorithm ki qi Qi mod N;

The previous two algorithms essentially reduce the calculation of the DFT on N points for N composite to the calculation of smaller DFTs on prime numbers of points, the latter remaining irreducible. However, Rader (1968) showed that the p-point DFT for p an odd

i1
k1 , k2 , . . . , kd , kj

2 Z=Nj Z, by kj qj k mod Nj (ii) k 7 ! for each j, with reconstruction formula

52

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY highly composite. In that case, factoring F
p 1 by means of the Cooley–Tukey or Good methods leads to an algorithm of complex An added bonus is that, because ity p log p rather than p2 for F
p.
p 1=2 g 1, the elements of F
p 1C can be shown to be either purely real or purely imaginary, which halves the number of real multiplications involved.

prime can itself be factored by invoking some extra arithmetic structure present in Z=pZ. 1.3.3.2.3.1. N an odd prime The ring Z=pZ f0, 1, 2, . . . , p 1g has the property that its p 1 non-zero elements, called units, form a multiplicative group U
p. In particular, all units r 2 U
p have a unique multiplicative inverse in Z=pZ, i.e. a unit s 2 U
p such that rs 1 mod p. This endows Z=pZ with the structure of a ﬁnite ﬁeld. Furthermore, U
p is a cyclic group, i.e. consists of the successive powers gm mod p of a generator g called a primitive root mod p (such a g may not be unique, but it always exists). For instance, for p 7, U
7 f1, 2, 3, 4, 5, 6g is generated by g 3, whose successive powers mod 7 are: g0 1,

g1 3,

g2 2,

g3 6,

g4 4,

1.3.3.2.3.2. N a power of an odd prime This idea was extended by Winograd (1976, 1978) to the treatment of prime powers N p , using the cyclic structure of the multiplicative group of units U
p . The latter consists of all those elements of Z=p Z which are not divisible by p, and thus has q p 1
p 1 elements. It is cyclic, and there exist primitive roots g modulo p such that U
p f1, g, g2 , g3 , . . . , gq 1 g:

g5 5

The p 1 elements divisible by p, which are divisors of zero, have to be treated separately just as 0 had to be treated separately for N p. When k 62 U
p , then k pk1 with k1 2 Z=p 1 Z. The results X
pk1 are p-decimated, hence can be obtained via the p 1 -point DFT of the p 1 -periodized data Y: 1 Y
k X
pk F
p

[see Apostol (1976), Chapter 10]. The basis of Rader’s algorithm is to bring to light a hidden regularity in the matrix F
p by permuting the basis vectors uk and vk of L
Z=pZ as follows: u00 u0

u0m uk v00 v0

v0m

1

with k gm ,

m 1, . . . , p

m

with k g ,

vk

1;

m 1, . . . , p

with Y
k1

1;

element
0, m 1 1 for all m 0, . . . p

2,

for all m 0, . . . , p

2:

e
g

m m=p

with Z
k2 X
pk2 ,

Y
0

m0 pP2

X1
gm

mZ
m

m0

Y
0
C Z
m ,

m 0, . . . , p

qP 1 m0

X
gm e
g
m m=p

then carrying out the multiplication by the skew-circulant matrix core as a convolution. Thus the DFT of size p may be reduced to two DFTs of size p 1 (dealing, respectively, with p-decimated results and p-decimated data) and a convolution of size q p 1
p 1. The latter may be ‘diagonalized’ into a multiplication by purely real or purely imaginary numbers (because g
q =2 1) by two DFTs, whose factoring in turn leads to DFTs of size p 1 and p 1. This method, applied recursively, allows the complete decomposition of the DFT on p points into arbitrarily small DFTs.

C
m mY
m 1 C
m

k2 2 Z=p 1 Z

(the p 1 -periodicity follows implicity from the fact that the transform on the right-hand side is independent of k1 2 Z=pZ). Finally, the contribution X1 from all k 2 U
p may be calculated by reindexing by the powers of a primitive root g modulo p , i.e. by writing

k

pP2

X
k1 p 1 k2 :

where X0 contains the contributions from k 2 = U
p and X1 those from k 2 U
p . By a converse of the previous calculation, X0 arises from p-decimated data Z, hence is the p 1 -periodization of the p 1 -point DFT of these data: 1 Z
k2 X0
p 1 k1 k2 F
p

Thus the ‘core’ C
p of matrix F
p, of size
p 1
p 1, formed by the elements with two non-zero indices, has a so-called skew-circulant structure because element
m , m depends only on m m. Simpliﬁcation may now occur because multiplication by C
p is closely related to a cyclic convolution. Introducing the in notation C
m e
gm=p we may write the relation Y F
pY the permuted bases as P Y
0 Y
k Y
m 1 Y
0

k2 2Z=pZ

X
k X0
k X1
k ,

2,

element
m 1, 0 1 for all m 0, . . . , p k k element
m 1, m 1 e p

P

When k 2 U
p , then we may write

where g is a primitive root mod p. With respect to these new bases, the matrix representing F
p will have the following elements: element
0, 0 1

1

2,

where Z is deﬁned by Z
m Y
p m 2, m 0, . . . , p 2. Thus Y may be obtained by cyclic convolution of C and Z, which may for instance be calculated by C Z F
p 1F
p 1C F
p 1Z,

1.3.3.2.3.3. N a power of 2 When N 2 , the same method can be applied, except for a slight modiﬁcation in the calculation of X1 . There is no primitive root modulo 2 for > 2: the group U
2 is the direct product of two cyclic groups, the ﬁrst (of order 2) generated by 1, the second (of order N=4) generated by 3 or 5. One then uses a representation

where denotes the component-wise multiplication of vectors. Since p is odd, p 1 is always divisible by 2 and may even be

53

1. GENERAL RELATIONSHIPS AND TECHNIQUES m1 m2

k
1 5 m1

k
1 5

w0 , w1 , . . . , wN 1 be obtained by cyclic convolution of U and V:

m2

wn

and the reindexed core matrix gives rise to a two-dimensional convolution. The latter may be carried out by means of two 2D DFTs on 2
N=4 points.

NP1

um v n

m,

m0

U
z

The cyclic convolutions generated by Rader’s multiplicative reindexing may be evaluated more economically than through DFTs if they are re-examined within a new algebraic setting, namely the theory of congruence classes of polynomials [see, for instance, Blahut (1985), Chapter 2; Schroeder (1986), Chapter 24]. The set, denoted KX , of polynomials in one variable with coefﬁcients in a given ﬁeld K has many of the formal properties of the set Z of rational integers: it is a ring with no zero divisors and has a Euclidean algorithm on which a theory of divisibility can be built. Given a polynomial P
z, then for every W
z there exist unique polynomials Q
z and R
z such that

V
z W
z

zN

m0

NP1

wn zn

n0

1

d Q

1:

Pi
z,

i1

Ui
z U
z mod Pi
z, Vi
z V
z mod Pi
z,

i 1, . . . , d, i 1, . . . , d;

(ii) compute the d polynomial products Wi
z Ui
zVi
z mod Pi
z,

i 1, . . . , d;

(iii) use the CRT reconstruction formula just proved to recover W
z from the Wi
z: W
z

j 1, . . . , d,

has a unique solution H
z modulo P
z. This solution may be constructed by a procedure similar to that used for integers. Let Qj
z P
z=Pj
z Pi
z:

d P

i1

Si
zWi
z mod
zN

1:

When N is not too large, i.e. for ‘short cyclic convolutions’, the Pi
z are very simple, with coefﬁcients 0 or 1, so that (i) only involves a small number of additions. Furthermore, special techniques have been developed to multiply general polynomials modulo cyclotomic polynomials, thus helping keep the number of multiplications in (ii) and (iii) to a minimum. As a result, cyclic convolutions can be calculated rapidly when N is sufﬁciently composite. It will be recalled that Rader’s multiplicative indexing often gives rise to cyclic convolutions of length p 1 for p an odd prime. Since p 1 is highly composite for all p 50 other than 23 and 47, these cyclic convolutions can be performed more efﬁciently by the above procedure than by DFT. These combined algorithms are due to Winograd (1977, 1978, 1980), and are known collectively as ‘Winograd small FFT algorithms’. Winograd also showed that they can be thought of as bringing the DFT matrix F to the following ‘normal form’:

i6j

Then Pj and Qj are coprime, and the Euclidean algorithm may be used to obtain polynomials pj
z and qj
z such that pj
zPj
z qj
zQj
z 1:

With Si
z qi
zQi
z, the polynomial Si
zHi
z mod P
z

is easily shown to be the desired solution. As with integers, it can be shown that the 1:1 correspondence between H
z and Hj
z sends sums to sums and products to products, i.e. establishes a ring isomorphism: KX mod P
KX mod P1 . . .
KX mod Pd : These results will now be applied to the efﬁcient calculation of cyclic convolutions. Let U
u0 , u1 , . . . , uN 1 and V
v0 , v1 , . . . , vN 1 be two vectors of length N, and let W

vm zm

where the cyclotomics Pi
z are well known (Nussbaumer, 1981; Schroeder, 1986, Chapter 22). We may now invoke the CRT, and exploit the ring isomorphism it establishes to simplify the calculation of W
z from U
z and V
z as follows: (i) compute the d residual polynomials

H1
z H2
z mod P
z: If H
z 0 mod P
z, H
z is said to be divisible by P
z. If H
z only has divisors of degree zero in KX , it is said to be irreducible over K (this notion depends on K). Irreducible polynomials play in KX a role analogous to that of prime numbers in Z, and any polynomial over K has an essentially unique factorization as a product of irreducible polynomials. There exists a Chinese remainder theorem (CRT) for polynomials. Let P
z P1
z . . . Pd
z be factored into a product of pairwise coprime polynomials [i.e. Pi
z and Pj
z have no common factor for i 6 j]. Then the system of congruence equations

i1

NP1

Now the polynomial zN 1 can be factored over the ﬁeld of rational numbers into irreducible factors called cyclotomic polynomials: if d is the number of divisors of N, including 1 and N, then

R
z is called the residue of H
z modulo P
z. Two polynomials H1
z and H2
z having the same residue modulo P
z are said to be congruent modulo P
z, which is denoted by

d P

ul zl

l0

W
z U
zV
z mod
zN

degree
R < degree
P:

H
z

NP1

then the above relation is equivalent to

W
z P
zQ
z R
z

H
z Hj
z mod Pj
z,

1:

The very simple but crucial result is that this cyclic convolution may be carried out by polynomial multiplication modulo
zN 1: if

1.3.3.2.4. The Winograd algorithms

and

n 0, . . . , N

F CBA, where A is an integer matrix with entries 0, 1, deﬁning the ‘preadditions’,

54

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY k 2 Zn =NT Zn :

k 2 Zn =NZn ,

B is a diagonal matrix of multiplications, C is a matrix with entries 0, 1, i, deﬁning the ‘post-additions’. The elements on the diagonal of B can be shown to be either real or pure imaginary, by the same argument as in Section 1.3.3.2.3.1. Matrices A and C may be rectangular rather than square, so that intermediate results may require extra storage space.

1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization Let us now assume that this decimation can be factored into d successive decimations, i.e. that N N1 N2 . . . Nd 1 Nd

1.3.3.3. Multidimensional algorithms and hence

From an algorithmic point of view, the distinction between onedimensional (1D) and multidimensional DFTs is somewhat blurred by the fact that some factoring techniques turn a 1D transform into a multidimensional one. The distinction made here, however, is a practical one and is based on the dimensionality of the indexing sets for data and results. This section will therefore be concerned with the problem of factoring the DFT when the indexing sets for the input data and output results are multidimensional.

NT NTd NTd

T T 1 . . . N2 N1 :

Then the coset decomposition formulae corresponding to these successive decimations (Section 1.3.2.7.1) can be combined as follows: [ Zn
k1 N1 Zn k1

1.3.3.3.1. The method of successive one-dimensional transforms

The DFT was deﬁned in Section 1.3.2.7.4 in an n-dimensional setting and it was shown that when the decimation matrix N is has a diagonal, say N diag
N
1 , N
2 , . . . , N
n , then F
N tensor product structure:
2 . . . F
N
n :
1 F
N F
N F
N

[ k1

(

k1 N1

" [ k2

n

#)

k2 N2 Z

... [ [ . . .
k1 N1 k2 . . . N1 N2 . . . Nd 1 kd NZn k1

kd

with kj 2 Zn =Nj Zn . Therefore, any k 2 Z=NZn may be written uniquely as

This may be rewritten as follows:
1 IN
2 . . . IN
n F
N F
N
2 . . . I
n IN
1 F
N

k k1 N1 k2 . . . N1 N2 . . . Nd 1 kd : Similarly:

N

Zn

...

n , IN
1 IN
2 . . . F
N

[ kd

kd NTd Zn

... [ [ . . .
kd NTd kd

where the I’s are identity matrices and denotes ordinary matrix multiplication. The matrix within each bracket represents a onedimensional DFT along one of the n dimensions, the other dimensions being left untransformed. As these matrices commute, the order in which the successive 1D DFTs are performed is immaterial. This is the most straightforward method for building an ndimensional algorithm from existing 1D algorithms. It is known in crystallography under the name of ‘Beevers–Lipson factorization’ (Section 1.3.4.3.1), and in signal processing as the ‘row–column method’.

kd

k1

1

. . . NTd . . . NT2 k1

NT Zn

so that any k 2 Zn =NT Zn may be written uniquely as k kd NTd kd

1

. . . NTd . . . NT2 k1

with kj 2 Zn =NTj Zn . These decompositions are the vector analogues of the multi-radix number representation systems used in the Cooley–Tukey factorization. We may then write the deﬁnition of F
N with d 2 factors as P P X
k2 NT2 k1 X
k1 N1 k2

1.3.3.3.2. Multidimensional factorization

k1 k2

Substantial reductions in the arithmetic cost, as well as gains in ﬂexibility, can be obtained if the factoring of the DFT is carried out in several dimensions simultaneously. The presentation given here is a generalization of that of Mersereau & Speake (1981), using the abstract setting established independently by Auslander, Tolimieri & Winograd (1982). Let us return to the general n-dimensional setting of Section 1.3.2.7.4, where the DFT was deﬁned for an arbitrary decimation matrix N by the formulae (where jNj denotes jdet Nj): 1 X F
N : X
k X
k e k
N 1 k jNj k X F
N : X
k X
kek
N 1 k

T 1 1 e
kT 2 k1 N2 N2 N1
k1 N1 k2 :

The argument of e(–) may be expanded as k2
N 1 k1 k1
N1 1 k1 k2
N2 1 k2 k1 k2 : The ﬁrst summand may be recognized as a twiddle factor, the 2 , respectively, 1 and F
N second and third as the kernels of F
N while the fourth is an integer which may be dropped. We are thus led to a ‘vector-radix’ version of the Cooley–Tukey algorithm, in which the successive decimations may be introduced in all n dimensions simultaneously by general integer matrices. The computation may be decomposed into ﬁve stages analogous to those of the one-dimensional algorithm of Section 1.3.3.2.1: (i) form the jN1 j vectors Yk1 of shape N2 by

k

Yk1
k2 X
k1 N1 k2 ,

with

55

k1 2 Zn =N1 Zn ,

k2 2 Zn =N2 Zn ;

1. GENERAL RELATIONSHIPS AND TECHNIQUES (ii) calculate the jN1 j transforms Yk1 on jN2 j points: P Yk1
k2 ek2
N2 1 k2 Yk1
k2 , k1 2 Zn =N1 Zn ;

M is reduced to 3M=4 by simultaneous 2 2 factoring, and to 15M=32 by simultaneous 4 4 factoring. The use of a non-diagonal decimating matrix may bring savings in computing time if the spectrum of the band-limited function under study is of such a shape as to pack more compactly in a nonrectangular than in a rectangular lattice (Mersereau, 1979). If, for instance, the support K of the spectrum is contained in a sphere, then a decimation matrix producing a close packing of these spheres will yield an aliasing-free DFT algorithm with fewer sample points than the standard algorithm using a rectangular lattice.

k2

(iii) form the jN2 j vectors Zk2 of shape N1 by Zk2
k1 ek2
N 1 k1 Yk1
k2 ,

k1 2 Zn =N1 Zn ,

k2 2 Zn =NT2 Zn ;

(iv) calculate the jN2 j transforms Zk on jN1 j points: 2 P Zk
k1 ek1
N1 1 k1 Zk2
k1 , k2 2 Zn =NT2 Zn ; 2

k1

1.3.3.3.2.2. Multidimensional prime factor algorithm Suppose that the decimation matrix N is diagonal

(v) collect X
k2 NT2 k1 as Zk
k1 . 2 The initial jNj-point transform F
N can thus be performed as 2 on jN2 j points, followed by jN2 j transforms jN1 j transforms F
N 1 on jN1 j points. This process can be applied successively to all F
N d factors. The same decomposition applies to F
N, up to the complex conjugation of twiddle factors, the normalization factor 1=jNj being obtained as the product of the factors 1=jNj j in the successive partial transforms F
Nj . The geometric interpretation of this factorization in terms of partial transforms on translates of sublattices applies in full to this ndimensional setting; in particular, the twiddle factors are seen to be related to the residual translations which place the sublattices in register within the big lattice. If the intermediate transforms are performed in place, then the quantity

X
kd NTd kd

1

. . . NTd NTd

1

N diag
N
1 , N
2 , . . . , N
n

and let each diagonal element be written in terms of its prime factors: m Q
i; j pj , N
i j1

where m is the total number of distinct prime factors present in the N
i . The CRT may be used to turn each 1D transform along dimension i
i 1, . . . , n into a multidimensional transform with a separate ‘pseudo-dimension’ for each distinct prime factor of N
i ; the number i , of these pseudo-dimensions is equal to the cardinality of the set:

. . . NT2 k1

f j 2 f1, . . . , mgj
i, j > 0 for some ig:

will eventually be found at location

The full P n-dimensional transform thus becomes -dimensional, with ni1 i . We may now permute the pseudo-dimensions so as to bring into contiguous position those corresponding to the same prime factor pj ; the m resulting groups of pseudo-dimensions are said to deﬁne ‘p-primary’ blocks. The initial transform is now written as a tensor product of m p-primary transforms, where transform j is on

k1 N1 k2 . . . N1 N2 . . . Nd 1 kd , so that the ﬁnal results will have to be unscrambled by a process which may be called ‘coset reversal’, the vector equivalent of digit reversal. Factoring by 2 in all n dimensions simultaneously, i.e. taking N 2M, leads to ‘n-dimensional butterﬂies’. Decimation in time corresponds to the choice N1 2I, N2 M, so that k1 2 Zn =2Zn is an n-dimensional parity class; the calculation then proceeds by

1; j

pj

2; j

pj

n; j

. . . pj

points [by convention, dimension i is not transformed if
i, j 0]. These p-primary transforms may be computed, for instance, by multidimensional Cooley–Tukey factorization (Section 1.3.3.3.1), which is faster than the straightforward row–column method. The ﬁnal results may then be obtained by reversing all the permutations used. The extra gain with respect to the multidimensional Cooley– Tukey method is that there are no twiddle factors between pprimary pieces corresponding to different primes p. The case where N is not diagonal has been examined by Guessoum & Mersereau (1986).

Yk1
k2 X
k1 2k2 , k1 2 Zn =2Zn , k2 2 Zn =MZn , Yk1 F
MY k1 2 Zn =2Zn ; k1 , P X
k2 MT k1
1k1 k1 k1 2Zn =2Zn

ek2
N 1 k1 Yk1
k2 :

Decimation in frequency corresponds to the choice N1 M, N2 2I, so that k2 2 Zn =2Zn labels ‘octant’ blocks of shape M; the calculation then proceeds through the following steps: " # P k2 k2 Zk2
k1
1 X
k1 Mk2

1.3.3.3.2.3. Nesting of Winograd small FFTs Suppose that the CRT has been used as above to map an ndimensional DFT to a -dimensional DFT. For each 1, . . . , [ runs over those pairs (i, j) such that
i, j > 0], the Rader/ Winograd procedure may be applied to put the matrix of the th 1D DFT in the CBA normal form of a Winograd small FFT. The full DFT matrix may then be written, up to permutation of data and results, as O
C B A :

k2 2Zn =2Zn

ek2
N 1 k1 , Zk F
MZ k2 , 2

X
k2 2k1 Zk
k1 , 2

n

i.e. the 2 parity classes of results, corresponding to the different k2 2 Zn =2Zn , are obtained separately. When the dimension n is 2 and the decimating matrix is diagonal, this analysis reduces to the ‘vector radix FFT’ algorithms proposed by Rivard (1977) and Harris et al. (1977). These lead to substantial reductions in the number M of multiplications compared to the row–column method:

1

A well known property of the tensor product of matrices allows this to be rewritten as

56

C

1

!

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY ! ! we may write: O O B A X
k1 , k2 Tk1 k2
!k1 1 1 or equivalently

and thus to form a matrix in which the combined pre-addition, multiplication and post-addition matrices have been precomputed. This procedure, called nesting, can be shown to afford a reduction of the arithmetic operation count compared to the row–column method (Morris, 1978). Clearly, the nesting rearrangement need not be applied to all dimensions, but can be restricted to any desired subset of them.

k X k1 , 2 Tk2
!k1 : k1

For N an odd prime p, all non-zero values of k1 are coprime with p so that the p p-point DFT may be calculated as follows: (1) form the polynomials PP Tk2
z X
k1 , k2 zk1 k2 k2 mod P
z

1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm Nussbaumer’s approach views the DFT as the evaluation of certain polynomials constructed from the data (as in Section 1.3.3.2.4). For instance, putting ! e
1=N, the 1D N-point DFT NP1

X
k

k0

may be written

X
k!k

k1 k2

for k2 0, . . . , p 1; (2) evaluate Tk2
!k1 for k1 0, . . . , p 1; (3) put X
k1 , k2 =k1 Tk2
!k1 ; (4) calculate the terms for k1 0 separately by " # P P X
k1 , k2 !k2 k2 : X
0, k2

k

k2

X
k Q
!k ,

Step (1) is a set of p ‘polynomial transforms’ involving no multiplications; step (2) consists of p DFTs on p points each since if P Tk2
z Yk2
k1 zk1

where the polynomial Q is deﬁned by Q
z

NP1 k0

X
kzk :

k1

Let us consider (Nussbaumer & Quandalle, 1979) a 2D transform of size N N: X
k1 , k2

NP1 NP1

then

Tk2
!k1

X
k1 , k2 !k1 k1 k2 k2 :

By introduction of the polynomials P Qk2
z X
k1 , k2 zk1 k1

R k2
z

this may be rewritten:

k2

!k2 k2 Qk2
z,

X
k1 , k2 R k2
!k1

P k2

!k2 k2 Qk2
!k1 :

Let us now suppose that k1 is coprime to N. Then k1 has a unique inverse modulo N (denoted by 1=k1 ), so that multiplication by k1 simply permutes the elements of Z=NZ and hence NP1

f
k2

k2 0

NP1

f
k1 k2

k2 0

Sk1 k2
!k1

where k2

zk k2 Qk2
z:

Since only the value of polynomial Sk
z at z !k1 is involved in the result, the computation of Sk may be carried out modulo the unique cyclotomic polynomial P
z such that P
!k1 0. Thus, if we deﬁne: P Tk
z zk k2 Qk2
z mod P
z

2

1.3.3.3.3.1. From local pieces to global algorithms The mathematical analysis of the structure of DFT computations has brought to light a broad variety of possibilities for reducing or reshaping their arithmetic complexity. All of them are ‘analytic’ in that they break down large transforms into a succession of smaller ones. These results may now be considered from the converse ‘synthetic’ viewpoint as providing a list of procedures for assembling them: (i) the building blocks are one-dimensional p-point algorithms for p a small prime; (ii) the low-level connectors are the multiplicative reindexing methods of Rader and Winograd, or the polynomial transform reindexing method of Nussbaumer and Quandalle, which allow the construction of efﬁcient algorithms for larger primes p, for prime powers p , and for p-primary pieces of shape p . . . p ; (iii) the high-level connectors are the additive reindexing scheme of Cooley–Tukey, the Chinese remainder theorem reindexing, and the tensor product construction; (iv) nesting may be viewed as the ‘glue’ which seals all elements.

k2

P

k1

Yk2
k1 !k1 k1 Yk
k1 ;

1.3.3.3.3. Global algorithm design

for any function f over Z=NZ. We may thus write: P X
k1 , k2 !k1 k2 k2 Qk1 k2
!k1

Sk
z

P

step (3) is a permutation; and step (4) is a p-point DFT. Thus the 2D DFT on p p points, which takes 2p p-point DFTs by the row– column method, involves only
p 1 p-point DFTs; the other DFTs have been replaced by polynomial transforms involving only additions. This procedure can be extended to n dimensions, and reduces the number of 1D p-point DFTs from npn 1 for the row–column method to
pn 1=
p 1, at the cost of introducing extra additions in the polynomial transforms. A similar algorithm has been formulated by Auslander et al. (1983) in terms of Galois theory.

k1 0 k2 0

P

k1

k2

57

1. GENERAL RELATIONSHIPS AND TECHNIQUES the f.p. units, so that complex reindexing schemes may be used without loss of overall efﬁciency. Another major consideration is that of data ﬂow [see e.g. Nawab & McClellan (1979)]. Serial machines only have few registers and few paths connecting them, and allow little or no overlap between computation and data movement. New architectures, on the other hand, comprise banks of vector registers (or ‘cache memory’) besides the usual internal registers, and dedicated ALUs can service data transfers between several of them simultaneously and concurrently with computation. In this new context, the devices described in Sections 1.3.3.2 and 1.3.3.3 for altering the balance between the various types of arithmetic operations, and reshaping the data ﬂow during the computation, are invaluable. The ﬁeld of machine-dependent DFT algorithm design is thriving on them [see e.g. Temperton (1983a,b,c, 1985); Agarwal & Cooley (1986, 1987)]. 1.3.3.3.3.3. The Johnson–Burrus family of algorithms In order to explore systematically all possible algorithms for carrying out a given DFT computation, and to pick the one best suited to a given machine, attempts have been made to develop: (i) a high-level notation of describing all the ingredients of a DFT computation, including data permutation and data ﬂow; (ii) a formal calculus capable of operating on these descriptions so as to represent all possible reorganizations of the computation; (iii) an automatic procedure for evaluating the performance of a given algorithm on a speciﬁc architecture. Task (i) can be accomplished by systematic use of a tensor product notation to represent the various stages into which the DFT can be factored (reindexing, small transforms on subsets of indices, twiddle factors, digit-reversal permutations). Task (ii) may for instance use the Winograd CBA normal form for each small transform, then apply the rules governing the rearrangement of tensor product and ordinary product operations on matrices. The matching of these rearrangements to the architecture of a vector and/or parallel computer can be formalized algebraically [see e.g. Chapter 2 of Tolimieri et al. (1989)]. Task (iii) is a complex search which requires techniques such as dynamic programming (Bellman, 1958). Johnson & Burrus (1983) have proposed and tested such a scheme to identify the optimal trade-offs between prime factor nesting and Winograd nesting of small Winograd transforms. In step (ii), they further decomposed the pre-addition matrix A and post-addition matrix C into several factors, so that the number of design options available becomes very large: the N-point DFT when N has four factors can be calculated in over 1012 distinct ways. This large family of nested algorithms contains the prime factor algorithm and the Winograd algorithms as particular cases, but usually achieves greater efﬁciency than either by reducing the f.p. multiplication count while keeping the number of f.p. additions small. There is little doubt that this systematic approach will be extended so as to incorporate all available methods of restructuring the DFT.

Fig. 1.3.3.1. A few global algorithms for computing a 400-point DFT. CT: Cooley–Tukey factorization. PF: prime factor (or Good) factorization. W: Winograd algorithm.

The simplest DFT may then be carried out into a global algorithm in many different ways. The diagrams in Fig. 1.3.3.1 illustrate a few of the options available to compute a 400-point DFT. They may differ greatly in their arithmetic operation counts. 1.3.3.3.3.2. Computer architecture considerations To obtain a truly useful measure of the computational complexity of a DFT algorithm, its arithmetic operation count must be tempered by computer architecture considerations. Three main types of tradeoffs must be borne in mind: (i) reductions in ﬂoating-point (f.p.) arithmetic count are obtained by reindexing, hence at the cost of an increase in integer arithmetic on addresses, although some shortcuts may be found (Uhrich, 1969; Burrus & Eschenbacher, 1981); (ii) reduction in the f.p. multiplication count usually leads to a large increase in the f.p. addition count (Morris, 1978); (iii) nesting can increase execution speed, but causes a loss of modularity and hence complicates program development (Silverman, 1977; Kolba & Parks, 1977). Many of the mathematical developments above took place in the context of single-processor serial computers, where f.p. addition is substantially cheaper than f.p. multiplication but where integer address arithmetic has to compete with f.p. arithmetic for processor cycles. As a result, the alternatives to the Cooley–Tukey algorithm hardly ever led to particularly favourable trade-offs, thus creating the impression that there was little to gain by switching to more exotic algorithms. The advent of new machine architectures with vector and/or parallel processing features has greatly altered this picture (Pease, 1968; Korn & Lambiotte, 1979; Fornberg, 1981; Swartzrauber, 1984): (i) pipelining equalizes the cost of f.p. addition and f.p. multiplication, and the ideal ‘blend’ of the two types of operations depends solely on the number of adder and multiplier units available in each machine; (ii) integer address arithmetic is delegated to specialized arithmetic and logical units (ALUs) operating concurrently with

1.3.4. Crystallographic applications of Fourier transforms 1.3.4.1. Introduction The central role of the Fourier transformation in X-ray crystallography is a consequence of the kinematic approximation used in the description of the scattering of X-rays by a distribution of electrons (Bragg, 1915; Duane, 1925; Havighurst, 1925a,b; Zachariasen, 1945; James, 1948a, Chapters 1 and 2; Lipson & Cochran, 1953, Chapter 1; Bragg, 1975).

58

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY P 0 P Let
X be the density of electrons in a sample of matter FH
H F F H F
H
H H2 H2 contained in a ﬁnite region V which is being illuminated by a parallel monochromatic X-ray beam with wavevector K0 . Then the distribution, the weight FH far-ﬁeld amplitude scattered in a direction corresponding to and is thus a weighted reciprocal-lattice attached to each node H 2 being the value at H of the transform wavevector K K0 H is proportional to F 0 of the motif 0 . Taken in conjunction with the assumption R 3 F
H
X exp
2iH X d X that the scattering is elastic, i.e. that H only changes the direction V but not the magnitude of the incident wavevector K0 , this result yields the usual forms (Laue or Bragg) of the diffraction conditions: F
H H 2 , and simultaneously H lies on the Ewald sphere. hx , exp
2iH Xi: By the reciprocity theorem, 0 can be recovered if F is known for In certain model calculations, the ‘sample’ may contain not only all H 2 as follows [Section 1.3.2.6.5, e.g. (iv)]: volume charges, but also point, line and surface charges. These 1 X singularities may be accommodated by letting be a distribution, FH exp
2iH X: x V H2 and writing These relations may be rewritten in terms of standard, or F
H F
H hx , exp
2iH Xi: ‘fractional crystallographic’, coordinates by putting F is still a well behaved function (analytic, by Section 1.3.2.4.2.10) because has been assumed to have compact support. X Ax, H
A 1 T h, If the sample is assumed to be an inﬁnite crystal, so that is now 3 3 a periodic distribution, the customary limiting process by which it is so that3 a unit cell of the crystal corresponds to x 2 R =Z , and that 0 shown that F becomes a discrete series of peaks at reciprocal-lattice h 2 Z . Deﬁning and by points (see e.g. von Laue, 1936; Ewald, 1940; James, 1948a p. 9; 1 1 Lipson & Taylor, 1958, pp. 14–27; Ewald, 1962, pp. 82–101; A # , 0 A# 0 V V Warren, 1969, pp. 27–30) is already subsumed under the treatment of Section 1.3.2.6. so that
X d3 X
x d3 x,

1.3.4.2. Crystallographic Fourier transform theory we have

1.3.4.2.1. Crystal periodicity

P F
h
h , F h

1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors Let be the distribution of electrons in a crystal. Then, by deﬁnition of a crystal, is -periodic for some period lattice (Section 1.3.2.6.5) so that there exists a motif distribution 0 with compact support such that

F
h

0

x

R , P

where R x2
X . The lattice is usually taken to be the ﬁnest for which the above representation holds. Let have a basis
a1 , a2 , a3 over the integers, these basis vectors being expressed in terms of a standard orthonormal basis
e1 , e2 , e3 as ak Then the matrix

3 P

0
X d3 X 0
x d3 x,

ajk ej :

j1

0

1 a11 a12 a13 A @ a21 a22 a23 A a31 a32 a33

is the period matrix of (Section 1.3.2.6.5) with respect to the unit lattice with basis
e1 , e2 , e3 , and the volume V of the unit cell is given by V jdet Aj. By Fourier transformation F R F 0 , P where R H2
H is the lattice distribution associated to the reciprocal lattice . The basis vectors
a1 , a2 , a3 have coordinates in
e1 , e2 , e3 given by the columns of
A 1 T , whose expression in terms of the cofactors of A (see Section 1.3.2.6.5) gives the familiar formulae involving the cross product of vectors for n 3. The Hdistribution F of scattered amplitudes may be written

59

h2Z3 h0x ,

R

3

exp
2ih xi

R =Z

P

h2Z3

3

0
x exp
2ih x d3 x if 0 2 L1loc
R3 =Z3 ,

F
h exp
2ih x:

These formulae are valid for an arbitrary motif distribution 0 , provided the convergence of the Fourier series for is considered from the viewpoint of distribution theory (Section 1.3.2.6.10.3). The experienced crystallographer may notice the absence of the familiar factor 1=V from the expression for just given. This is because we use the (mathematically) natural unit for , the electron per unit cell, which matches the dimensionless nature of the crystallographic coordinates x and of the associated volume element d3 x. The traditional factor 1=V was the result of the somewhat inconsistent use of x as an argument but of d3 X as a 3 volume element to obtain in electrons per unit volume (e.g. A ). A fortunate consequence of the present convention is that nuisance factors of V or 1=V , which used to abound in convolution or scalar product formulae, are now absent. It should be noted at this point that the crystallographic terminology regarding F and F differs from the standard mathematical terminology introduced in Section 1.3.2.4.1 and applied to periodic distributions in Section 1.3.2.6.4: F is the inverse Fourier transform of rather than its Fourier transform, and the calculation of is called a Fourier synthesis in crystallography even though it is mathematically a Fourier analysis. The origin of this discrepancy may be traced to the fact that the mathematical theory of the Fourier transformation originated with the study of temporal periodicity, while crystallography deals with spatial periodicity; since the expression for the phase factor of a plane wave is exp2i
t K X, the difference in sign between the

1. GENERAL RELATIONSHIPS AND TECHNIQUES where D is the ‘spherical Dirichlet kernel’ P exp
2ih x: D
x

contributions from time versus spatial displacements makes this conﬂict unavoidable.

k
A 1 T hk

1.3.4.2.1.2. Structure factors in terms of form factors In many cases, 0 is a sum of translates of atomic electrondensity distributions. Assume there are n distinct chemical types of atoms, with Nj identical isotropic atoms of type j described by an electron distribution j about their centre of mass. According to quantum mechanics each j is a smooth rapidly decreasing function of x, i.e. j 2 S , hence 0 2 S and (ignoring the effect of thermal agitation) " # Nj n P P j
x xkj , 0
x

D exhibits numerous negative ripples around its central peak. Thus the ‘series termination errors’ incurred by using S
instead of consist of negative ripples around each atom, and may lead to a Gibbs-like phenomenon (Section 1.3.2.6.10.1) near a molecular boundary. As in one dimension, Cesa`ro sums (arithmetic means of partial sums) have better convergence properties, as they lead to a convolution by a ‘spherical Feje´r kernel’ which is everywhere positive. Thus Cesa`ro summation will always produce positive approximations to a positive electron density. Other positive summation kernels were investigated by Pepinsky (1952) and by Waser & Schomaker (1953).

j1 kj 1

which may be written (Section 1.3.2.5.8) " !# Nj n P P 0 j
xkj :

1.3.4.2.1.4. Friedel’s law, anomalous scatterers If the wavelength of the incident X-rays is far from any absorption edge of the atoms in the crystal, there is a constant phase shift in the scattering, and the electron density may be considered to be real-valued. Then R F
h
x exp
2ih x d3 x

kj 1

j1

By Fourier transformation: ( " #) Nj n P P exp
2ih xkj : F
h F j
h kj 1

j1

R3 =Z3

Deﬁning the form factor fj of atom j as a function of h to be fj
h F j
h

we have F
h

n P

j1

fj
h

"

Nj P

kj 1

R

3

R =Z

3

x exp2i
h x d3 x

F
h since
x
x:

#

exp
2ih xkj :

Thus if F
h jF
hj exp
i'
h,

If X Ax and H
A 1 T h are the real- and reciprocal-space 1 coordinates in A˚ and A , and if j
kXk is the spherically symmetric electron-density function for atom type j, then Z1 sin
2kHkkXk fj
H 4kXk2 j
kXk dkXk: 2kHkkXk

then jF
hj jF
hj

and

'
h '
h:

This is Friedel’s law (Friedel, 1913). The set fFh g of Fourier coefﬁcients is said to have Hermitian symmetry. If is close to some absorption edge(s), the proximity to resonance induces an extra phase shift, whose effect may be represented by letting
x take on complex values. Let

0

More complex expansions are used for electron-density studies (see Chapter 1.2 in this volume). Anisotropic Gaussian atoms may be dealt with through the formulae given in Section 1.3.2.4.4.2.

x R
x iI
x

1.3.4.2.1.3. Fourier series for the electron density and its summation The convergence of the Fourier series for P
x F
h exp
2ih x

and correspondingly, by termwise Fourier transformation F
h F R
h iF I
h:

Since R
x and I
x are both real, F R
h and F I
h are both Hermitian symmetric, hence

h2Z3

is usually examined from the classical point of view (Section 1.3.2.6.10). The summation of multiple Fourier series meets with considerable difﬁculties, because there is no natural order in Zn to play the role of the natural order in Z (Ash, 1976). In crystallography, however, the structure factors F
h are often obtained within spheres kHk 1 for increasing resolution (decreasing ). Therefore, successive estimates of are most naturally calculated as the corresponding partial sums (Section 1.3.2.6.10.1): P S
x F
h exp
2ih x: k
A 1 T hk

1

F
h F R
h iF I
h, while F
h F R
h

iF I
h:

Thus F
h 6 F
h, so that Friedel’s law is violated. The components F R
h and F I
h, which do obey Friedel’s law, may be expressed as:

1

F R
h 12F
h F
h, 1 F I
h F
h F
h: 2i

This may be written

S
x
D
x,

60

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 1.3.4.2.1.5. Parseval’s identity and other L2 theorems By Section 1.3.2.4.3.3 and Section 1.3.2.6.10.2, R R P jF
hj2 j
xj2 d3 x V j
Xj2 d3 X: h2Z3

3

R =Z

3

hence the Fourier series representation of , : P ,
t F
hG
h exp
2ih t: h2Z3

3

R =

Clearly, ,
, , as shown by the fact that permuting F and G changes K
h into its complex conjugate. The auto-correlation of is deﬁned as , and is called the Patterson function of . If consists of point atoms, i.e.

Usually
x is real and positive, hence j
xj
x, but the identity remains valid even when
x is made complex-valued by the presence of anomalous scatterers. If fGh g is the collection of structure factors belonging to another electron density A# with the same period lattice as , then R P F
hG
h
x
x d3 x h2Z3

V

3

X
X d X:

, r

R3 =

Thus, norms and inner products may be evaluated either from structure factors or from ‘maps’.

h2Z3

h2Z3

then

W
h F
hG
h: If either or is inﬁnitely differentiable, then the distribution exists, and if we analyse it as P Y
h exp
2ih x, x 0

H2 ; kHk

h2Z3

then the backward version of the convolution theorem reads: P F
hG
h k: Y
h

The cross correlation , between and is the Z3 -periodic distribution deﬁned by:

33 F
X 4 3 kXk 3
sin u u cos u where u 2 : u

If 0 and 0 are locally integrable, R ,
t 0
x
x t d3 x

By Shannon’s theorem, it sufﬁces to calculate S
on an integral subdivision of the period lattice such that the sampling criterion is satisﬁed (i.e. that the translates of by vectors of do not overlap). Values of S
may then be calculated at an arbitrary point X by the interpolation formula: P I
X YS
Y: S
X

R3

R =Z

Let
t

P

3

h2Z3

3

1

I
X

0 :

R

xk

S
is band-limited, the support of its spectrum being contained in the solid sphere deﬁned by kHk 1 . Let be the indicator function of . The transform of the normalized version of is (see below, Section 1.3.4.4.3.5)

k2Z3

Zj Zk
xj

1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation Shannon’s sampling and interpolation theorem (Section 1.3.2.7.1) takes two different forms, according to whether the property of ﬁnite bandwidth is assumed in real space or in reciprocal space. (1) The most usual setting is in reciprocal space (see Sayre, 1952c). Only a ﬁnite number of diffraction intensities can be recorded and phased, and for physical reasons the cutoff criterion is the resolution 1=kHkmax . Electron-density maps are thus calculated as partial sums (Section 1.3.4.2.1.3), which may be written in Cartesian coordinates as P F
H exp
2iH X: S
X

0

The distribution ! r
is well deﬁned, since the generalized support condition (Section 1.3.2.3.9.7) is satisﬁed. The forward version of the convolution theorem implies that if P !x W
h exp
2ih x,

0

j1 k1

#

and is therefore calculable from the diffraction intensities alone. It was ﬁrst proposed by Patterson (1934, 1935a,b) as an extension to crystals of the radially averaged correlation function used by Warren & Gingrich (1934) in the study of powders.

G
h exp
2ih x: 0

N P N P

h2Z3

h2Z3

P

"

Zj
xj ,

contains information about interatomic vectors. It has the Fourier series representation P ,
t jF
hj2 exp
2ih t,

1.3.4.2.1.6. Convolution, correlation and Patterson function Let r 0 and r 0 be two electron densities referred to crystallographic coordinates, with structure factors fFh gh2Z3 and fGh gh2Z3 , so that P F
h exp
2ih x, x x

j1

then

R3 =Z3

R

N P

0

x
x t d3 x:

K
h exp
2ih t:

Y2

(2) The reverse situation occurs whenever the support of the motif 0 does not ﬁll the whole unit cell, i.e. whenever there exists a region M (the ‘molecular envelope’), strictly smaller than the unit cell, such that the translates of M by vectors of r do not overlap and that

The combined use of the shift property and of the forward convolution theorem then gives immediately: K
h F
hG
h;

61

1. GENERAL RELATIONSHIPS AND TECHNIQUES being related by P
P 1 T in order to preserve duality. This change of basis must be such that one of these matrices (say, P) should have a given integer vector u as its ﬁrst column, u being related to the line or plane deﬁning the section or projection of interest. The problem of constructing a matrix P given u received an erroneous solution in Volume II of International Tables (Patterson, 1959), which was subsequently corrected in 1962. Unfortunately, the solution proposed there is complicated and does not suggest a general approach to the problem. It therefore seems worthwhile to record here an effective procedure which solves this problem in any dimension n (Watson, 1970). Let 0 1 u1 B .. C u@ . A

M 0 0 :

It then follows that r
M : Deﬁning the ‘interference function’ G as the normalized indicator function of M according to 1 G
F M
vol
M we may invoke Shannon’s theorem to calculate the value F 0
at an arbitrary point of reciprocal space from its sample values F
h F 0
h at points of the reciprocal lattice as P G
hF
h: F 0
h2Z3

This aspect of Shannon’s theorem constitutes the mathematical basis of phasing methods based on geometric redundancies created by solvent regions and/or noncrystallographic symmetries (Bricogne, 1974). The connection between Shannon’s theorem and the phase problem was ﬁrst noticed by Sayre (1952b). He pointed out that the Patterson function of , written as , r
0 0 , may be viewed as consisting of a motif 0 0 0 (containing all the internal interatomic vectors) which is periodized by convolution with r. As the translates of 0 by vectors of Z3 do overlap, the sample values of the intensities jF
hj2 at nodes of the reciprocal lattice do not provide enough data to interpolate intensities jF
j2 at arbitrary points of reciprocal space. Thus the loss of phase is intimately related to the impossibility of intensity interpolation, implying in return that any indication of intensity values attached to non-integral points of the reciprocal lattice is a potential source of phase information.

un

be a primitive integral vector, i.e. g.c.d.
u1 , . . . , un 1. Then an n n integral matrix P with det P 1 having u as its ﬁrst column can be constructed by induction as follows. For n 1 the result is trivial. For n 2 it can be solved by means of the Euclidean algorithm, which yields z1 , z2 such that u1 z2 u2 z1 1, so that we z u1 z1 . Note that, if z 1 is a solution, may take P u2 z2 z2 then z mu is another solution for any m 2 Z. For n 3,0write 1 z2 u1 B.C with d g.c.d.
u2 , . . . , un so that both z @ .. A u dz zn u1 and are primitive. By the inductive hypothesis there is an d u1 integral 2 2 matrix V with as its ﬁrst column, and an d integral
n 1
n 1 matrix Z with z as its ﬁrst column, with det V 1 and det Z 1. Now put V 1 P , In 2 Z

1.3.4.2.1.8. Sections and projections It was shown at the end of Section 1.3.2.5.8 that the convolution theorem establishes, under appropriate assumptions, a duality between sectioning a smooth function (viewed as a multiplication by a -function in the sectioning coordinate) and projecting its transform (viewed as a convolution with the function 1 everywhere equal to 1 as a function of the projection coordinate). This duality follows from the fact that F and F map 1xi to xi and xi to 1xi (Section 1.3.2.5.6), and from the tensor product property (Section 1.3.2.5.5). In the case of periodic distributions, projection and section must be performed with respect to directions or subspaces which are integral with respect to the period lattice if the result is to be periodic; furthermore, projections must be performed only on the contents of one repeating unit along the direction of projection, or else the result would diverge. The same relations then hold between principal central sections and projections of the electron density and the dual principal central projections and sections of the weighted reciprocal lattice, e.g. P
x1 , 0, 0 $ F
h1 , h2 , h3 ,

i.e. 0

1 0 B 0 z2 B PB B 0 z3 @: : 0 zn

x1 , x2 , 0 $

1; 2
x3 1
x2 , x3

R

R2 =Z2

R

R=Z

h3

10 0 u1 Bd C CB B C CB 0 A : @ : 0

0 : 0

0 0 1 : 0

: : : : :

1 0 0C C 0C C: :A 1

0

1 u1 B dz2 C B C B : C u, B C @ : A dzn

F
h1 , h2 , h3 ,

x1 , x2 , x3 dx1 dx2 $ F
0, 0, h3 ,

x1 , x2 , x3 dx1

: : : : :

The ﬁrst column of P is

h1 ; h2

P

0 :

$ F
0, h2 , h3

and its determinant is 1, QED. The incremental step from dimension n 1 to dimension n is the construction of 2 2 matrix V, for which there exist inﬁnitely many solutions labelled by an integer mn 1 . Therefore, the collection of matrices P which solve the problem is labelled by n 1 arbitrary integers
m1 , m2 , . . . , mn 1 . This freedom can be used to adjust the shape of the basis B.

etc. When the sections are principal but not central, it sufﬁces to use the shift property of Section 1.3.2.5.5. When the sections or projections are not principal, they can be made principal by changing to new primitive bases B and B for and , respectively, the transition matrices P and P to these new bases

62

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY The converse property is also useful: it relates the derivatives of the continuous transform F 0 to the moments of 0 : @ m1 m2 m3 F 0
H F
2im1 m2 m3 X1m1 X2m2 X3m3 0x
H: @X1m1 @X2m2 @X3m3

Once P has been chosen, the calculation of general sections and projections is transformed into that of principal sections and projections by the changes of coordinates: x Px0 ,

h P h0 ,

and an appeal to the tensor product property. Booth (1945a) made use of the convolution theorem to form the Fourier coefﬁcients of ‘bounded projections’, which provided a compromise between 2D and 3D Fourier syntheses. If it is desired to compute the projection on the (x, y) plane of the electron density lying between the planes z z1 and z z2 , which may be written as

For jmj 2 and H 0, this identity gives the well known relation between the Hessian matrix of the transform F 0 at the origin of reciprocal space and the inertia tensor of the motif 0 . This is a particular case of the moment-generating properties of F , which will be further developed in Section 1.3.4.5.2.

1x 1y z1 ; z2
x y 1z :

1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szego¨’s theorem The classical results presented in Section 1.3.2.6.9 can be readily generalized to the case of triple Fourier series; no new concept is needed, only an obvious extension of the notation. Let be real-valued, so that Friedel’s law holds and F
h F
h. Let H be a ﬁnite set of indices comprising the origin: H fh0 0, h1 , . . . , hn g. Then the Hermitian form in n 1 complex variables n P TH
u F
hj hk uj uk

The transform is then F
h k F z1 ; z2
1h 1k l , giving for coefﬁcient
h, k: X sin l
z1 F
h, k, l expf2il
z1 z2 =2g l l2Z

z2

:

1.3.4.2.1.9. Differential syntheses Another particular instance of the convolution theorem is the duality between differentiation and multiplication by a monomial (Sections 1.3.2.4.2.8, 1.3.2.5.8). In the present context, this result may be written m1 m2 m3 @
H F m1 m2 @X1 @X2 @X3m3

j; k0

is called the Toeplitz form of order H associated to . By the convolution theorem and Parseval’s identity, 2 P R n TH
u
x uj exp
2ihj x d3 x: 3 3 j0 R =Z

2im1 m2 m3 H1m1 H2m2 H3m3 F
AT H

If is almost everywhere non-negative, then for all H the forms TH are positive semi-deﬁnite and therefore all Toeplitz determinants DH are non-negative, where

in Cartesian coordinates, and m1 m2 m3 @
h
2im1 m2 m3 hm1 1 hm2 2 hm3 3 F
h F @xm1 1 @xm2 2 @xm3 3

DH det fF
hj

The Toeplitz–Carathe´odory–Herglotz theorem given in Section 1.3.2.6.9.2 states that the converse is true: if DH 0 for all H, then is almost everywhere non-negative. This result is known in the crystallographic literature through the papers of Karle & Hauptman (1950), MacGillavry (1950), and Goedkoop (1950), following previous work by Harker & Kasper (1948) and Gillis (1948a,b). Szego¨’s study of the asymptotic distribution of the eigenvalues of Toeplitz forms as their order tends to inﬁnity remains valid. Some precautions are needed, however, to deﬁne the notion of a sequence
Hk of ﬁnite subsets of indices tending to inﬁnity: it sufﬁces that the Hk should consist essentially of the reciprocal-lattice points h contained within a domain of the form k (k-fold dilation of ) where is a convex domain in R3 containing the origin (Widom, of the 1960). Under these circumstances, the eigenvalues
n Toeplitz forms THk become equidistributed with the sample
n values 0 of on a grid satisfying the Shannon sampling criterion for the data in Hk (cf. Section 1.3.2.6.9.3). A particular consequence of this equidistribution is that the
n geometric means of the
n and of the 0 are equal, and hence as in Section 1.3.2.6.9.4 ( ) R 1=jHk j 3 lim fDHk g exp log
x d x ,

in crystallographic coordinates. A particular case of the ﬁrst formula is P kHk2 F
AT H exp
2iH X
X, 42 H2

where

3 X @2 j1

@Xj2

is the Laplacian of . The second formula has been used with jmj 1 or 2 to compute ‘differential syntheses’ and reﬁne the location of maxima (or other stationary points) in electron-density maps. Indeed, the values at x of the gradient vector r and Hessian matrix
rrT are readily obtained as P
2ihF
h exp
2ih x,
r
x h2Z3

T

rr
x

P

3

h2Z

42 hhT F
h exp
2ih x,

and a step of Newton iteration towards the nearest stationary point of will proceed by T

hk g:

k!1

1

x 7 ! x f
rr
xg
r
x: The modern use of Fourier transforms to speed up the computation of derivatives for model reﬁnement will be described in Section 1.3.4.4.7.

R3 =Z3

where jHk j denotes the number of reﬂections in Hk . Complementary terms giving a better comparison of the two sides were obtained by Widom (1960, 1975) and Linnik (1975).

63

1. GENERAL RELATIONSHIPS AND TECHNIQUES
iii0 Tg0 1 g2 Tg0 2 Tg0 1

This formula played an important role in the solution of the 2D Ising model by Onsager (1944) (see Montroll et al., 1963). It is also encountered in phasing methods involving the ‘Burg entropy’ (Britten & Collins, 1982; Narayan & Nityananda, 1982; Bricogne, 1982, 1984, 1988).

The essential difference between left and right actions is of course not whether the elements of G are written on the left or right of those of X: it lies in the difference between (iii) and (iii0 ). In a left action the product g1 g2 in G operates on x 2 X by g2 operating ﬁrst, then g1 operating on the result; in a right action, g1 operates ﬁrst, then g2 . This distinction will be of importance in Sections 1.3.4.2.2.4 and 1.3.4.2.2.5. In the sequel, we will use left actions unless otherwise stated.

1.3.4.2.2. Crystal symmetry 1.3.4.2.2.1. Crystallographic groups The description of a crystal given so far has dealt only with its invariance under the action of the (discrete Abelian) group of translations by vectors of its period lattice . Let the crystal now be embedded in Euclidean 3-space, so that it may be acted upon by the group M
3 of rigid (i.e. distancepreserving) motions of that space. The group M
3 contains a normal subgroup T
3 of translations, and the quotient group M
3=T
3 may be identiﬁed with the 3-dimensional orthogonal group O
3. The period lattice of a crystal is a discrete uniform subgroup of T
3. The possible invariance properties of a crystal under the action of M
3 are captured by the following deﬁnition: a crystallographic group is a subgroup of M
3 if (i) \ T
3 , a period lattice and a normal subgroup of ; (ii) the factor group G = is ﬁnite. The two properties are not independent: by a theorem of Bieberbach (1911), they follow from the assumption that is a discrete subgroup of M
3 which operates without accumulation point and with a compact fundamental domain (see Auslander, 1965). These two assumptions imply that G acts on through an integral representation, and this observation leads to a complete enumeration of all distinct ’s. The mathematical theory of these groups is still an active research topic (see, for instance, Farkas, 1981), and has applications to Riemannian geometry (Wolf, 1967). This classiﬁcation of crystallographic groups is described elsewhere in these Tables (Wondratschek, 1995), but it will be surveyed brieﬂy in Section 1.3.4.2.2.3 for the purpose of establishing further terminology and notation, after recalling basic notions and results concerning groups and group actions in Section 1.3.4.2.2.2.

(b) Orbits and isotropy subgroups Let x be a ﬁxed element of X. Two fundamental entities are associated to x: (1) the subset of G consisting of all g such that gx x is a subgroup of G, called the isotropy subgroup of x and denoted Gx ; (2) the subset of X consisting of all elements gx with g running through G is called the orbit of x under G and is denoted Gx. Through these deﬁnitions, the action of G on X can be related to the internal structure of G, as follows. Let G=Gx denote the collection of distinct left cosets of Gx in G, i.e. of distinct subsets of G of the form gGx . Let jGj, jGx j, jGxj and jG=Gx j denote the numbers of elements in the corresponding sets. The number jG=Gx j of distinct cosets of Gx in G is also denoted G : Gx and is called the index of Gx in G; by Lagrange’s theorem G : Gx jG=Gx j

gGx 7 ! gx establishes a one-to-one correspondence between the distinct left cosets of Gx in G and the elements of the orbit of x under G. It follows that the number of distinct elements in the orbit of x is equal to the index of Gx in G: jGxj G : Gx

Gx f xj 2 G=Gx g:

Similar deﬁnitions may be given for a right action of G on X. The set of distinct right cosets Gx g in G, denoted Gx nG, is then in one-toone correspondence with the distinct elements in the orbit xG of x.

(i)
g1 g2 x g1
g2 x for all g1 , g2 2 G and all x 2 X ,

(c) Fundamental domain and orbit decomposition The group properties of G imply that two orbits under G are either disjoint or equal. The set X may thus be written as the disjoint union X Gxi ,

for all x 2 X :

An element g of G thus induces a mapping Tg of X into itself deﬁned by Tg
x gx, with the ‘representation property’: (iii) Tg1 g2 Tg1 Tg2 for all g1 , g2 2 G:

i2I

Since G is a group, every g has an inverse g 1 ; hence every mapping Tg has an inverse Tg 1 , so that each Tg is a permutation of X. Strictly speaking, what has just been deﬁned is a left action. A right action of G on X is deﬁned similarly as a mapping
g, x 7 ! xg such that
i0 x
g1 g2
xg1 g2 0

ii

xe x

jGj , jGx j

and that the elements of the orbit of x may be listed without repetition in the form

(a) Left and right actions Let G be a group with identity element e, and let X be a set. An action of G on X is a mapping from G X to X with the property that, if g x denotes the image of
g, x, then ex x

jGj : jGx j

Now if g1 and g2 are in the same coset of Gx , then g2 g1 g0 with g0 2 Gx , and hence g1 x g2 x; the converse is obviously true. Therefore, the mapping from cosets to orbit elements

1.3.4.2.2.2. Groups and group actions The books by Hall (1959) and Scott (1964) are recommended as reference works on group theory.

(ii)

for all g1 , g2 2 G:

where the xi are elements of distinct orbits and I is an indexing set labelling them. The subset D fxi gi2I is said to constitute a fundamental domain (mathematical terminology) or an asymmetric unit (crystallographic terminology) for the action of G on X: it contains one representative xi of each distinct orbit. Clearly, other fundamental domains may be obtained by choosing different representatives for these orbits. If X is ﬁnite and if f is an arbitrary complex-valued function over X, the ‘integral’ of f over X may be written as a sum of integrals over the distinct orbits, yielding the orbit decomposition formula:

for all g1 , g2 2 G and all x 2 X , for all x 2 X :

The mapping Tg0 deﬁned by Tg0
x xg then has the ‘rightrepresentation’ property:

64

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 0 1 Indeed for any g1 , g2 in G, X X X X X @ A f
x f
yi f
i xi Tg#1 Tg#2 f
x Tg#2 f
Tg1 1 x f Tg2 1 Tg1 1 x x2X i2I i2I yi 2Gxi

i 2G=Gxi ! f
Tg1 Tg2 1 x; X X 1 f
gi xi : since Tg1 Tg2 Tg1 g2 , it follows that jGxi j !

gi 2G

i2I

Tg#1 Tg#2 Tg#1 g2 :

In particular, taking f
x 1 for all x and denoting by jX j the number of elements of X: X X X jGj jX j jGxi j jG=Gxi j : jGxi j i2I i2I i2I

It is clear that the change of variable must involve the action of g 1 (not g) if T # is to deﬁne a left action; using g instead would yield a right action. The linear representation operators Tg# on L
X provide the most natural instrument for stating and exploiting symmetry properties which a function may possess with respect to the action of G. Thus a function f 2 L
X will be called G-invariant if f
gx f
x for all g 2 G and all x 2 X . The value f
x then depends on x only through its orbit Gx, and f is uniquely deﬁned once it is speciﬁed on a fundamental domain D fxi gi2I ; its integral over X is then a weighted sum of its values in D: P P f
x G : Gxi f
xi :

(d) Conjugation, normal subgroups, semi-direct products A group G acts on itself by conjugation, i.e. by associating to g 2 G the mapping Cg deﬁned by Cg
h ghg 1 :

Indeed, Cg
hk Cg
hCg
k and Cg
h 1 Cg 1
h. In particular, Cg operates on the set of subgroups of G, two subgroups H and K being called conjugate if H Cg
K for some g 2 G; for example, it is easily checked that Ggx Cg
Gx . The orbits under this action are the conjugacy classes of subgroups of G, and the isotropy subgroup of H under this action is called the normalizer of H in G. If fHg is a one-element orbit, H is called a self-conjugate or normal subgroup of G; the cosets of H in G then form a group G=H called the factor group of G by H. Let G and H be two groups, and suppose that G acts on H by automorphisms of H, i.e. in such a way that

x2X

The G-invariance of f may be written: Tg# f f

1

g
h
g
h

1

for all g 2 G:

Thus f is invariant under each Tg# , which obviously implies that f is invariant under the linear operator in L
X 1 X # T , AG jGj g2G g which averages an arbitrary function by the action of G. Conversely, if AG f f , then

g
h1 h2 g
h1 g
h2 g
eH eH

i2I

where eH is the identity element of H:

Tg#0 f Tg#0
AG f
Tg#0 AG f AG f f

for all g0 2 G,

so that the two statements of the G-invariance of f are equivalent. The identity

Then the symbols [g, h] with g 2 G, h 2 H form a group K under the product rule:

Tg#0 AG AG for all g0 2 G

g1 , h1 g2 , h2 g1 g2 , h1 g1
h2

{associativity checks; [eG , eH ] is the identity; g, h has inverse g 1 , g 1
h 1 }. The group K is called the semi-direct product of H by G, denoted K H G. The elements g, eH form a subgroup of K isomorphic to G, the elements eG , h form a normal subgroup of K isomorphic to H, and the action of G on H may be represented as an action by conjugation in the sense that

is easily proved by observing that the map g 7 ! g0 g (g0 being any element of G) is a one-to-one map from G into itself, so that P # P # Tg Tg0 g

Cg; eH
eG , h eG , g
h:

AG 2 AG ,

g2G

g2G

as these sums differ only by the order of the terms. The same identity implies that AG is a projector:

A familiar example of semi-direct product is provided by the group of Euclidean motions M
3 (Section 1.3.4.2.2.1). An element S of M
3 may be written S R, t with R 2 O
3, the orthogonal group, and t 2 T
3, the translation group, and the product law

and hence that its eigenvalues are either 0 or 1. In summary, we may say that the invariance of f under G is equivalent to f being an eigenfunction of the associated projector AG for eigenvalue 1.

shows that M
3 T
3 O
3 with O
3 acting on T
3 by rotating the translation vectors.

( f ) Orbit exchange One ﬁnal result about group actions which will be used repeatedly later is concerned with the case when X has the structure of a Cartesian product:

(e) Associated actions in function spaces For every left action Tg of G in X, there is an associated left action Tg# of G on the space L
X of complex-valued functions over X, deﬁned by ‘change of variable’ (Section 1.3.2.3.9.5):

X X1 X2 . . . Xn and when G acts diagonally on X, i.e. acts on each Xj separately:

Tg# f
x f
Tg 1 x f
g 1 x:

Then complete sets (but not usually minimal sets) of representatives

R 1 , t1 R 2 , t2 R 1 R 2 , t1 R 1
t2

gx g
x1 , x2 , . . . , xn
gx1 , gx2 , . . . , gxn :

65

1. GENERAL RELATIONSHIPS AND TECHNIQUES of the distinct orbits for the action of G in X may be obtained in the form Dk X 1 . . . X k

1

k fxik gik 2Ik

Xk1 . . . Xn

for each k 1, 2, . . . , n, i.e. by taking a fundamental domain in Xk and all the elements in Xj with j 6 k. The action of G on each Dk does indeed generate the whole of X: given an arbitrary element y
y1 , y2 , . . . , yn of X, there is an index ik 2 Ik such that yk 2
k
k Gxik and a coset of Gx
k in G such that yk xik for any ik representative of that coset; then
k

which is of the form y dk with dk 2 Dk . The various Dk are related in a simple manner by ‘transposition’ or ‘orbit exchange’ (the latter name is due to J. W. Cooley). For instance, Dj may be obtained from Dk
j 6 k as follows: for each yj 2 Xj there exists g
yj 2 G and ij
yj 2 Ij such that
j yj g
yj xij
yj ; therefore

yj 2Xj

monoclinic

Z=2Z Z=2Z

orthorhombic

Z=3Z,
Z=3Z fg

trigonal

Z=6Z,
Z=6Z fg

hexagonal

Z=4Z,
Z=4Z fg

tetragonal

Z=2Z Z=2Z fS3 g

cubic

and the extension of these groups by a centre of inversion. In this list denotes a semi-direct product [Section 1.3.4.2.2.2(d)], denotes the automorphism g 7 ! g 1 , and S3 (the group of permutations on three letters) operates by permuting the copies of Z=2Z (using the subgroup A3 of cyclic permutations gives the tetrahedral subsystem). Step 2 leads to a list of 73 equivalence classes called arithmetic classes of representations g 7 ! Rg , where Rg is a 3 3 integer matrix, with Rg1 g2 Rg1 Rg2 and Re I3 . This enumeration is more familiar if equivalence is relaxed so as to allow conjugation by rational 3 3 matrices with determinant 1: this leads to the 32 crystal classes. The difference between an arithmetic class and its rational class resides in the choice of a lattice mode
P, A=B=C, I, F or R. Arithmetic classes always refer to a primitive lattice, but may use inequivalent integral representations for a given geometric symmetry element; while crystallographers prefer to change over to a non-primitive lattice, if necessary, in order to preserve the same integral representation for a given geometric symmetry element. The matrices P and Q P 1 describing the changes of basis between primitive and centred lattices are listed in Table 5.1 and illustrated in Figs. 5.3 to 5.9, pp. 76–79, of Volume A of International Tables (Arnold, 1995). Step 3 gives rise to a system of congruences for the systems of non-primitive translations ftg gg2G which may be associated to the matrices fRg gg2G of a given arithmetic class, namely:

y
1 y1 , . . . , 1 yk 1 , xik , 1 yk1 , . . . , 1 yn

Dj

Z=2Z

g
yj 1 Dk ,

since the fundamental domain of Xk is thus expanded to the whole of Xk , while Xj is reduced to its fundamental domain. In other words: orbits are simultaneously collapsed in the jth factor and expanded in the kth. When G operates without ﬁxed points in each Xk (i.e. Gxk feg for all xk 2 Xk ), then each Dk is a fundamental domain for the action of G in X. The existence of ﬁxed points in some or all of the Xk complicates the situation in that for each k and each xk 2 Xk such that Gxk 6 feg the action of G=Gxk on the other factors must be examined. Shenefelt (1988) has made a systematic study of orbit exchange for space group P622 and its subgroups. Orbit exchange will be encountered, in a great diversity of forms, as the basic mechanism by which intermediate results may be rearranged between the successive stages of the computation of crystallographic Fourier transforms (Section 1.3.4.3).

tg1 g2 Rg1 tg2 tg1 mod ,

ﬁrst derived by Frobenius (1911). If equivalence under the action of A
3 is taken into account, 219 classes are found. If equivalence is deﬁned with respect to the action of the subgroup A
3 of A
3 consisting only of transformations with determinant +1, then 230 classes called space-group types are obtained. In particular, associating to each of the 73 arithmetic classes a trivial set of non-primitive translations
tg 0 for all g 2 G yields the 73 symmorphic space groups. This third step may also be treated as an abstract problem concerning group extensions, using cohomological methods [Ascher & Janner (1965); see Janssen (1973) for a summary]; the connection with Frobenius’s approach, as generalized by Zassenhaus (1948), is examined in Ascher & Janner (1968). The ﬁniteness of the number of space-group types in dimension 3 was shown by Bieberbach (1912) to be the case in arbitrary dimension. The reader interested in N-dimensional space-group theory for N > 3 may consult Brown (1969), Brown et al. (1978), Schwarzenberger (1980), and Engel (1986). The standard reference for integral representation theory is Curtis & Reiner (1962). All three-dimensional space groups G have the property of being solvable, i.e. that there exists a chain of subgroups

1.3.4.2.2.3. Classification of crystallographic groups Let be a crystallographic group, the normal subgroup of its lattice translations, and G the ﬁnite factor group =. Then G acts on by conjugation [Section 1.3.4.2.2.2(d)] and this action, being a mapping of a lattice into itself, is representable by matrices with integer entries. The classiﬁcation of crystallographic groups proceeds from this observation in the following three steps: Step 1: ﬁnd all possible ﬁnite abstract groups G which can be represented by 3 3 integer matrices. Step 2: for each such G ﬁnd all its inequivalent representations by 3 3 integer matrices, equivalence being deﬁned by a change of primitive lattice basis (i.e. conjugation by a 3 3 integer matrix with determinant 1). Step 3: for each G and each equivalence class of integral representations of G, ﬁnd all inequivalent extensions of the action of G from to T
3, equivalence being deﬁned by an afﬁne coordinate change [i.e. conjugation by an element of A
3]. Step 1 leads to the following groups, listed in association with the crystal system to which they later give rise:

G Gr > Gr

66

1

> . . . > G1 > G0 feg,

where each Gi 1 is a normal subgroup of G1 and the factor group Gi =Gi 1 is a cyclic group of some order mi
1 i r. This property may be established by inspection, or deduced from a famous theorem of Burnside [see Burnside (1911), pp. 322–323] according to which any group G such that jGj p q , with p and q distinct primes, is solvable; in the case at hand, p 2 and q 3.

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Sg# f
x f Sg 1
x f Rg 1
x

The whole classiﬁcation of 3D space groups can be performed swiftly by a judicious use of the solvability property (L. Auslander, personal communication). Solvability facilitates the indexing of elements of G in terms of generators and relations (Coxeter & Moser, 1972; Magnus et al., 1976) for the purpose of calculation. By deﬁnition of solvability, elements g1 , g2 , . . . , gr may be chosen in such a way that the cyclic factor group Gi =Gi 1 is generated by the coset gi Gi 1 . The set fg1 , g2 , . . . , gr g is then a system of generators for G such that the deﬁning relations [see Brown et al. (1978), pp. 26–27] have the particularly simple form

The operators R # g associated to the purely rotational part of each transformation Sg will also be used. Note the relation: Sg# tg R # g: Let a crystal structure be described by the list of the atoms in its unit cell, indexed by k 2 K. Let the electron-density distribution about the centre of mass of atom k be described by k with respect to the standard coordinates x. Then the motif 0 may be written as a sum of translates: P 0 xk k k2K

g1m1 e, gimi

gi 1 gj 1 gi gj

tg :

a
i; i 1 a
i; i 2 a
i; 1 gi 1 gi 2 . . . g1 b
i; j; j 1 b
i; j; j 2 b
i; j; 1 gj 1 gj 2 . . . g1

and the crystal electron density is r 0 . Suppose that is invariant under . If xk1 and xk2 are in the same orbit, say xk2 Sg
xk1 , then

for 2 i r, for 1 i < j r,

xk2 k2 Sg#
xk1 k1 :

with 0 a
i, h < mh and 0 b
i, j, h < mh . Each element g of G may then be obtained uniquely as an ‘ordered word’:

Therefore if xk is a special position and thus Gxk 6 feg, then Sg#
xk k xk k

g grkr grkr 11 . . . g1k1 ,

This identity implies that

with 0 ki < mi for all i 1, . . . , r, using the algorithm of Ju¨rgensen (1970). Such generating sets and deﬁning relations are tabulated in Brown et al. (1978, pp. 61–76). An alternative list is given in Janssen (1973, Table 4.3, pp. 121–123, and Appendix D, pp. 262–271).

Rg xk tg xk mod (the special position condition), and that k R # g k , i.e. that k must be invariant by the pure rotational part of Gxk . Trueblood (1956) investigated the consequences of this invariance on the thermal vibration tensor of an atom in a special position (see Section 1.3.4.2.2.6 below). Let J be a subset of K such that fxj gj2J contains exactly one atom from each orbit. An orbit decomposition yields an expression for 0 in terms of symmetry-unique atoms: 0 1 P P 0 @ S #
xj j A

1.3.4.2.2.4. Crystallographic group action in real space The action of a crystallographic group may be written in terms of standard coordinates in R3 =Z3 as
g, x 7 ! Sg
x Rg x tg mod ,

g 2 G,

with Sg1 g2 Sg1 Sg2 :

An important characteristic of the representation : g 7 ! Sg is its reducibility, i.e. whether or not it has invariant subspaces other than f0g and the whole of R3 =Z3 . For triclinic, monoclinic and orthorhombic space groups, is reducible to a direct sum of three one-dimensional representations: 0
1 1 0 0 Rg B C 0 A; Rg @ 0 Rg
2 0 0 R
3 g

for trigonal, tetragonal and hexagonal groups, it is reducible to a direct sum of two representations, of dimension 2 and 1, respectively; while for tetrahedral and cubic groups, it is irreducible. By Schur’s lemma (see e.g. Ledermann, 1987), any matrix which commutes with all the matrices Rg for g 2 G must be a scalar multiple of the identity in each invariant subspace. In the reducible cases, the reductions involve changes of basis which will be rational, not integral, for those arithmetic classes corresponding to non-primitive lattices. Thus the simpliﬁcation of having maximally reduced representation has as its counterpart the use of non-primitive lattices. The notions of orbit, isotropy subgroup and fundamental domain (or asymmetric unit) for the action of G on R3 =Z3 are inherited directly from the general setting of Section 1.3.4.2.2.2. Points x for which Gx 6 feg are called special positions, and the various types of isotropy subgroups which may be encountered in crystallographic groups have been labelled by means of Wyckoff symbols. The representation operators Sg# in L
R3 =Z3 have the form:

for all g 2 Gxk :

j2J

j 2G=Gxj

j

or equivalently

8 P< P j R j 1
x 0
x j2J : j 2G=Gx j

t j

If the atoms are assumed to be Gaussian, write Zj j
X jdet Uj j1=2 exp

1 1 T 2X Uj X

9 = xj : ;

in Cartesian A coordinates,

where Zj is the total number of electrons, and where the matrix Uj combines the Gaussian spread of the electrons in atom j at rest with the covariance matrix of the random positional ﬂuctuations of atom j caused by thermal agitation. In crystallographic coordinates: Zj j
x jdet Qj j1=2 exp

1 1 T 2x Qj x

with Qj A 1 Uj
A 1 T :

If atom k is in a special position xk , then the matrix Qk must satisfy the identity Rg Q k R g 1 Q k

67

1. GENERAL RELATIONSHIPS AND TECHNIQUES for all g in the isotropy subgroup of xk . This condition may also be written in Cartesian coordinates as

In the absence of dispersion, Friedel’s law gives rise to the phase restriction:

Tg Uk Tg 1 Uk ,

'h h t mod :

where

The value of the restricted phase is independent of the choice of coset representative . Indeed, if 0 is another choice, then 0 g with g 2 Gh and by the Frobenius congruences t 0 Rg t tg , so that

1

Tg ARg A : This is a condensed form of the symmetry properties derived by Trueblood (1956).

h t 0
RTg h t h tg mod 1: Since g 2 Gh , RTg h h and h tg 0 mod 1 if h is not a systematic absence: thus

1.3.4.2.2.5. Crystallographic group action in reciprocal space An elementary discussion of this topic may be found in Chapter 1.4 of this volume. Having established that the symmetry of a crystal may be most conveniently stated and handled via the left representation g 7 ! Sg# of G given by its action on electron-density distributions, it is natural to transpose this action by the identity of Section 1.3.2.5.5: F S # T F t
R # T

g

g

h t h t mod :

The treatment of centred lattices may be viewed as another instance of the duality between periodization and decimation (Section 1.3.2.7.2): the periodization of the electron density by the non-primitive lattice translations has as its counterpart in reciprocal space the decimation of the transform by the ‘reﬂection conditions’ describing the allowed reﬂections, the decimation and periodization matrices being each other’s contragredient. The reader may consult the papers by Bienenstock & Ewald (1962) and Wells (1965) for earlier approaches to this material.

g

exp
2i tg
Rg 1 T# F T

for any tempered distribution T, i.e. F Sg# T
exp
2i tg F T
RTg

1.3.4.2.2.6. Structure-factor calculation Structure factors may be calculated from a list of symmetryunique atoms by Fourier transformation of the orbit decomposition formula for the motif 0 given in Section 1.3.4.2.2.4:

whenever the transforms are functions. Putting T , a Z3 -periodic distribution, this relation deﬁnes a left action Sg of G on L
Z3 given by
Sg F
h exp
2i tg F
RTg h

which is conjugate to the action F S # S F , g

g

F
h F 0
h 2 0 13 P P S #j
xj j A5
h F 4 @

Sg#

in the sense that i:e: S F S # F : g

g

j2J

Sg#

expressing the G-invariance of is then The identity equivalent to the identity Sg F F between its structure factors, i.e. (Waser, 1955a)

F
h exp
2ih tg F
RTg h:

If G is made to act on Z3 via :

j 2G=Gxj

P P

j2J j 2G=Gxj

P P

j2J j 2G=Gxj

F t j R#

j xj j
h exp
2ih t j

R j 1 T# exp
2i xj F j
h P P exp
2ih t j

g, h 7 !
Rg 1 T h,

the usual notions of orbit, isotropy subgroup (denoted Gh ) and fundamental domain may be attached to this action. The above relation then shows that the spectrum fF
hgh2Z3 is entirely known if it is speciﬁed on a fundamental domain D containing one reciprocal-lattice point from each orbit of this action. A reﬂection h is called special if Gh 6 feg. Then for any g 2 Gh we have RTg h h, and hence

j2J j 2G=Gxj

exp2i
RT j h xj F j
RT j h;

i.e. ﬁnally: F
h

F
h exp
2ih tg F
h,

P

P

j2J j 2G=Gxj

expf2ih S j
xj gF j
RT j h:

In the case of Gaussian atoms, the atomic transforms are

implying that F
h 0 unless h tg 0 mod 1. Special reﬂections h for which h tg 6 0 mod 1 for some g 2 Gh are thus systematically absent. This phenomenon is an instance of the duality between periodization and decimation of Section 1.3.2.7.2: if tg 6 0, the projection of on the direction of h has period
tg h=
h h < 1, hence its transform (which is the portion of F supported by the central line through h) will be decimated, giving rise to the above condition. A reﬂection h is called centric if Gh G
h, i.e. if the orbit of h contains h. Then RT h h for some coset in G=Gh , so that the following relation must hold:

F j
h Zj exp

2 1 T 2h
4 Qj h

or equivalently F j
H Zj exp

2 1 T 2H
4 Uj H:

Two common forms of equivalent temperature factors (incorporating both atomic form and thermal motion) are (i) isotropic B: F j
h Zj exp

jF
hj exp
i'h exp
2ih t jF
hj exp
i' h :

68

T 1 4Bj H H,

so that Uj
Bj =82 I, or Qj
Bj =82 A 1
A 1 T ;

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Grouping the summands for hl and hl yields a real-valued summand P 2F
hl cos2hl S l
x 'hl :

(ii) anisotropic ’s: F j
h Zj exp
hT j h,

so that j 22 Qj 22 A 1 Uj
A 1 T , or Uj
1=22 A j AT . In the ﬁrst case, F j
RT j h does not depend on j , and therefore: P F
h Zj expf 14Bj hT A 1
A 1 T hg

l 2
G=Ghl

Case 2: G
hl 6 Ghl , hl is acentric. The two orbits are then disjoint, and the summands corresponding to hl and hl may be grouped together into a single real-valued summand P 2F
hl cos2hl S l
x 'hl :

j2J

P

j 2G=Gxj

expf2ih S j
xj g:

l 2G=Ghl

In order to reindex the collection of all summands of , put

In the second case, however, no such simpliﬁcation can occur: P P exp hT
R j j RT j h F
h Zj j2J

L Lc [ La ,

where Lc labels the Friedel-unique centric reﬂections in L and La the acentric ones, and let L a stand for a subset of La containing a unique element of each pair fhl , hl g for l 2 La . Then

j 2G=Gxj

expf2ih S j
xj g:

x F
0

These formulae, or special cases of them, were derived by Rollett & Davies (1955), Waser (1955b), and Trueblood (1956). The computation of structure factors by applying the discrete Fourier transform to a set of electron-density values calculated on a grid will be examined in Section 1.3.4.4.5.

1.3.4.2.2.7. Electron-density calculations A formula for the Fourier synthesis of electron-density maps from symmetry-unique structure factors is readily obtained by orbit decomposition: P F
h exp
2ih x
x

"

P

l2L l 2G=Ghl

P

l2L

"

F
hl

F
RT l hl exp P

l 2G=Ghl

c2Lc

P

"

a2L a

P

2F
hc

c 2
G=Ghc

"

2F
ha

P

a 2G=Gha

cos2hc S c
x

'hc

#

#

cos2ha S a
x

'ha :

1.3.4.2.2.8. Parseval’s theorem with crystallographic symmetry The general statement of Parseval’s theorem given in Section 1.3.4.2.1.5 may be rewritten in terms of symmetry-unique structure factors and electron densities by means of orbit decomposition. In reciprocal space, P P P F1
hF2
h F1
RT l hl F2
RT l hl ;

h2Z3

P

P

#

2i
RT l hl x

h2Z3

#

expf 2ihl S l
xg ,

l2L l 2G=Ghl

for each l, the summands corresponding to the various l are equal, so that the left-hand side is equal to

where L is a subset of Z3 such that fhl gl2L contains exactly one point of each orbit for the action :
g, h 7 !
Rg 1 T h of G on Z3 . The physical electron density per cubic a˚ngstro¨m is then 1
X
Ax V 3 with V in A . In the absence of anomalous scatterers in the crystal and of a centre of inversion I in , the spectrum fF
hgh2Z3 has an extra symmetry, namely the Hermitian symmetry expressing Friedel’s law (Section 1.3.4.2.1.4). The action of a centre of inversion may be added to that of to obtain further simpliﬁcation in the above formula: under this extra action, an orbit Ghl with hl 6 0 is either mapped into itself or into the disjoint orbit G
hl ; the terms corresponding to hl and hl may then be grouped within the common orbit in the ﬁrst case, and between the two orbits in the second case. Case 1: G
hl Ghl , hl is centric. The cosets in G=Ghl may be partitioned into two disjoint classes by picking one coset in each of the two-coset orbits of the action of I. Let
G=Ghl denote one such class: then the reduced orbit

F1
0F2
0 P 2j
G=Ghc kF1
hc kF2
hc j cos'1
hc

'2
hc

c2Lc

P

a2L a

2jG=Gha kF1
ha kF2
ha j cos'1
ha

'2
ha :

In real space, the triple integral may be rewritten as R R 1
x2
x d3 x jGj 1
x2
x d3 x R3 =Z3

D

(where D is the asymmetric unit) if 1 and 2 are smooth densities, since the set of special positions has measure zero. If, however, the integral is approximated as a sum over a G-invariant grid deﬁned by decimation matrix N, special positions on this grid must be taken into account: 1 X 1
x2
x jNj 3 3 k2Z =NZ

1 X G : Gx 1
x2
x jNj x2D jGj X 1 1
x2
x, jNj x2D jGx j

fRT l hl j l 2
G=Ghl g contains exactly once the Friedel-unique half of the full orbit Ghl , and thus

where the discrete asymmetric unit D contains exactly one point in each orbit of G in Z3 =NZ3 .

j
G=Ghl j 12jG=Ghl j:

69

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.4.2.2.9. Convolution theorems with crystallographic symmetry The standard convolution theorems derived in the absence of symmetry are readily seen to follow from simple properties of functions e
h, x exp
2ih x (denoted simply e in formulae which are valid for both signs), namely:
i

ii

F1
hF2
h with
x

e
h, x e
k, x e
h k, x,

These relations imply that the families of functions and

in real space

fh 7 ! e
h, xgx2R3 =Z3

in reciprocal space

e
h, x e
h, x e
h,

1
x2
x with F
h

x:

jGh j e
l, tg F1
h jGh RTg
l j jGl j l2D g2G XX

or equivalently

01

Sg# e
h, x eh, Sg 1
x

e
Rg 1 T h, tg e
Rg 1 T h, x:

The kernels of symmetrized Fourier transforms are not the functions e but rather the symmetrized sums P P
h, x e h, Sg
x e h, Sg 1
x g2G

02

g2G

iiG
h, x
h, y

P

g2G

PB P #
1 C S j
x
1 j1 A, @ 1

j1 2J1

0

PB @

j2 2J2

j1

1

j 2G=Gx 1 j1

P

2

j2 2G=Gxj

2

1

2 C S #j
x
2 j2 A, 2

j2

where J1 and J2 label the symmetry-unique atoms placed at
1
2 positions fxj1 gj1 2J1 and fxj2 gj2 2J2 , respectively. To calculate the correlation between 1 and 2 we need the following preliminary formulae, which are easily established: if S
x Rx t and f is an arbitrary function on R3 , then
R # f R # f ,
x f x f , R #
x f Rx f ,

g2G

for which the linearization formulae are readily obtained using (i), (ii) and (iv) as P
iG
h, x
k, x e
k, tg
h RTg k, x,

RTg lF2
l:

1.3.4.2.2.10. Correlation and Patterson functions Consider two model electron densities 1 and 2 with the same period lattice Z3 and the same space group G. Write their motifs in terms of atomic electron densities (Section 1.3.4.2.2.4) as 0 1

Sg# 1 e
h, x eh, Sg
x e
h, tg e
RTg h, x

iv0

X 1 F
h
h, x jGh j h2D

Both formulae are simply orbit decompositions of their symmetryfree counterparts.

When crystallographic symmetry is present, the convolution theorems remain valid in their original form if written out in terms of ‘expanded’ data, but acquire a different form when rewritten in terms of symmetry-unique data only. This rewriting is made possible by the extra relation (Section 1.3.4.2.2.5)
iv

Sg
z2
z:

then

both generate an algebra of functions, i.e. a vector space endowed with an internal multiplication, since (i) and (ii) show how to ‘linearize products’. Friedel’s law (when applicable) on the one hand, and the Fourier relation between intensities and the Patterson function on the other hand, both follow from the property
iii

jGx j 1 x Sg
z j jGz j

The backward convolution theorem is derived similarly. Let X 1 1
x F1
k
k, x, jGk j k2D X 1 2
x F2
l
l, x, jGl j l2D

e
h, x e
h, y e
h, x y:

fx 7 ! e
h, xgh2Z3

1 XX jNj z2D g2G jGx

X 1
x
h, x jG j x x2D

h, x Sg
y,

hence

S #
x f S
x R # f

where the choice of sign in must be the same throughout each formula. Formulae (i)G deﬁning the ‘structure-factor algebra’ associated to G were derived by Bertaut (1955c, 1956b,c, 1959a,b) and Bertaut & Waser (1957) in another context. The forward convolution theorem (in discrete form) then follows. Let X 1 F1
h 1
y
h, y, j jG y y2D X 1 F2
h 2
z
h, z, j jG z z2D

and S #
x f

S
x R

#

f;

and S1# f1 S2# f2 S1# f1
S1 1 S2 # f2 S2#
S2 1 S1 # f1 f2 :

The cross correlation 01 02 between motifs is therefore PPPP #
1
2 01 02 S j
x
1 j1 S #j
x
2 j2 1

j1 j2 j1 j2

then

j1 j2 j1 j2

S

j

2

2

1

xj S j
xj 2

1

1

j2

1

2

j1
R #
R #

j

j j2 1

2

1

2

j1
R # which contains a peak of shape
R #

j

j2 j2 at the
2
1 1 interatomic vector S j2
xj2 S j1
xj1 for each j1 2 J1 , j2 2 J2 ,

j1 2 G=Gx
1 , j2 2 G=Gx
2 . j1

70

PPPP

2

j1

j2

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 01

02

is then performed by selecting the n cosine strips labelled
Aj , hj and the n sine strips labelled
Bj , hj , placing them in register, and adding the tabulated values columnwise. The number 60 was chosen as the l.c.m. of 12 (itself the l.c.m. of the orders of all possible non-primitive translations) and of 10 (for decimal convenience). The limited accuracy imposed by the two-digit tabulation was later improved by Robertson’s sorting board (Robertson, 1936a,b) or by the use of separate strips for each decimal digit of the amplitude (Booth, 1948b), which allowed threedigit tabulation while keeping the set of strips within manageable size. Cochran (1948a) found that, for most structures under study at the time, the numerical inaccuracies of the method were less than the level of error in the experimental data. The sampling rate was subsequently increased from 60 to 120 (Beevers, 1952) to cope with larger unit cells. Further gains in speed and accuracy were sought through the construction of special-purpose mechanical, electro-mechanical, electronic or optical devices. Two striking examples are the mechanical computer RUFUS built by Robertson (1954, 1955, 1961) on the principle of previous strip methods (see also Robertson, 1932) and the electronic analogue computer X-RAC built by Pepinsky, capable of real-time calculation and display of 2D and 3D Fourier syntheses (Pepinsky, 1947; Pepinsky & Sayre, 1948; Pepinsky et al., 1961; see also Suryan, 1957). The optical methods of Lipson & Taylor (1951, 1958) also deserve mention. Many other ingenious devices were invented, whose descriptions may be found in Booth (1948b), Niggli (1961), and Lipson & Cochran (1968). Later, commercial punched-card machines were programmed to carry out Fourier summations or structure-factor calculations (Shaffer et al., 1946a,b; Cox et al., 1947, 1949; Cox & Jeffrey, 1949; Donohue & Schomaker, 1949; Grems & Kasper, 1949; Hodgson et al., 1949; Greenhalgh & Jeffrey, 1950; Kitz & Marchington, 1953). The modern era of digital electronic computation of Fourier series was initiated by the work of Bennett & Kendrew (1952), Mayer & Trueblood (1953), Ahmed & Cruickshank (1953b), Sparks et al. (1956) and Fowweather (1955). Their Fourier-synthesis programs used Beevers–Lipson factorization, the program by Sparks et al. being the ﬁrst 3D Fourier program useable for all space groups (although these were treated as P1 or P1 by data expansion). Ahmed & Barnes (1958) then proposed a general programming technique to allow full use of symmetry elements (orthorhombic or lower) in the 3D Beevers–Lipson factorization process, including multiplicity corrections. Their method was later adopted by Shoemaker & Sly (1961), and by crystallographic program writers at large. The discovery of the FFT algorithm by Cooley & Tukey in 1965, which instantly transformed electrical engineering and several other disciplines, paradoxically failed to have an immediate impact on crystallographic computing. A plausible explanation is that the calculation of large 3D Fourier maps was a relatively infrequent task which was not thought to constitute a bottleneck, as crystallographers had learned to settle most structural questions by means of cheaper 2D sections or projections. It is signiﬁcant in this respect that the ﬁrst use of the FFT in crystallography by Barrett & Zwick (1971) should have occurred as part of an iterative scheme for improving protein phases by density modiﬁcation in real space, which required a much greater number of Fourier transformations than any previous method. Independently, Bondot (1971) had attracted attention to the merits of the FFT algorithm. The FFT program used by Barrett & Zwick had been written for signal-processing applications. It was restricted to sampling rates of the form 2n , and was not designed to take advantage of crystallographic symmetry at any stage of the calculation; Bantz & Zwick (1974) later improved this situation somewhat.

The cross-correlation r between the original electron densities is then obtained by further periodizing by Z3 . Note that these expressions are valid for any choice of ‘atomic’
1
2 density functions j1 and j2 , which may be taken as molecular fragments if desired (see Section 1.3.4.4.8). If G contains elements g such that Rg has an eigenspace E1 with eigenvalue 1 and an invariant complementary subspace E2 , while tg has a non-zero component tg
1 in E1 , then the Patterson function r 0 0 will contain Harker peaks (Harker, 1936) of the form Sg
x

2 x t
1 g
Sg
x

x

[where Sg
s represent the action of g in E2 ] in the translate of E1 by t
1 g . 1.3.4.3. Crystallographic discrete Fourier transform algorithms 1.3.4.3.1. Historical introduction In 1929, W. L. Bragg demonstrated the practical usefulness of the Fourier transform relation between electron density and structure factors by determining the structure of diopside from three principal projections calculated numerically by 2D Fourier summation (Bragg, 1929). It was immediately realized that the systematic use of this powerful method, and of its extension to three dimensions, would entail considerable amounts of numerical computation which had to be organized efﬁciently. As no other branch of applied science had yet needed this type of computation, crystallographers had to invent their own techniques. The ﬁrst step was taken by Beevers & Lipson (1934) who pointed out that a 2D summation could be factored into successive 1D summations. This is essentially the tensor product property of the Fourier transform (Sections 1.3.2.4.2.4, 1.3.3.3.1), although its aspect is rendered somewhat complicated by the use of sines and cosines instead of complex exponentials. Computation is economized to the extent that the cost of an N N transform grows with N as 2N 3 rather than N 4 . Generalization to 3D is immediate, reducing computation size from N 6 to 3N 4 for an N N N transform. The complication introduced by using expressions in terms of sines and cosines is turned to advantage when symmetry is present, as certain families of terms are systematically absent or are simply related to each other; multiplicity corrections must, however, be introduced. The necessary information was tabulated for each space group by Lonsdale (1936), and was later incorporated into Volume I of International Tables. The second step was taken by Beevers & Lipson (1936) and Lipson & Beevers (1936) in the form of the invention of the ‘Beevers–Lipson strips’, a practical device which was to assist a whole generation of crystallographers in the numerical computation of crystallographic Fourier sums. The strips comprise a set of ‘cosine strips’ tabulating the functions 2hm A cos
A 1, 2, . . . , 99; h 1, 2, . . . , 99 60 and a set of ‘sine strips’ tabulating the functions 2hm
B 1, 2, . . . , 99; h 1, 2, . . . , 99 B sin 60 for the 16 arguments m 0, 1, . . . , 15. Function values are rounded to the nearest integer, and those for other arguments m may be obtained by using the symmetry properties of the sine and cosine functions. A Fourier summation of the form n X 2hj m 2hj m Y
m Bj sin Aj cos 60 60 j1

71

1. GENERAL RELATIONSHIPS AND TECHNIQUES X 1 It was the work of Ten Eyck (1973) and Immirzi (1973, 1976) F m exp2ih
N 1 m h which led to the general adoption of the FFT in crystallographic jdet Nj 3 3 m2Z =NZ computing. Immirzi treated all space groups as P1 by data expansion. Ten Eyck based his program on a versatile multi-radix FFT routine (Gentleman & Sande, 1966) coupled with a ﬂexible and P indexing scheme for dealing efﬁciently with multidimensional x Fh exp
2ih x transforms. He also addressed the problems of incorporating hZ3 =NT Z3 symmetry elements of order 2 into the factorization of 1D transforms, and of transposing intermediate results by other or P symmetry elements. He was thus able to show that in a large m Fh exp 2ih
N 1 m: number of space groups (including the 74 space groups having h2Z3 =NT Z3 orthorhombic or lower symmetry) it is possible to calculate only the unique results from the unique data within the logic of the FFT In the presence of symmetry, the unique data are algorithm. Ten Eyck wrote and circulated a package of programs for – fx gx2D or fm gm2D in real space (by abuse of notation, D will computing Fourier maps and re-analysing them into structure denote an asymmetric unit for x or for m indifferently); – fFh gh2D in reciprocal space. factors in some simple space groups (P1, P1, P2, P2/m, P21, P222, The previous summations may then be subjected to orbital P212121, Pmmm). This package was later augmented by a handful of new space-group-speciﬁc programs contributed by other crystal- decomposition, to yield the following ‘crystallographic DFT’ lographers (P21212, I222, P3121, P41212). The writing of such (CDFT) deﬁning relations: " # programs is an undertaking of substantial complexity, which has X P 1 deterred all but the bravest: the usual practice is now to expand data x expf2ih S
xg Fh for a high-symmetry space group to the largest subgroup for which a jdet Nj x2D 2G=Gx speciﬁc FFT program exists in the package, rather than attempt to " # write a new program. Attempts have been made to introduce more 1 X 1 P x expf2ih Sg
xg , modern approaches to the calculation of crystallographic Fourier jdet Nj x2D jGx j g2G transforms (Auslander, Feig & Winograd, 1982; Auslander & " # Shenefelt, 1987; Auslander et al., 1988) but have not gone beyond P P x Fh expf 2ih S
xg the stage of preliminary studies. h2D

2G=Gh The task of fully exploiting the FFT algorithm in crystallographic " # computations is therefore still unﬁnished, and it is the purpose of P 1 P this section to provide a systematic treatment such as that (say) of Fh expf 2ih Sg
xg , jGh j g2G h2D Ahmed & Barnes (1958) for the Beevers–Lipson algorithm. Ten Eyck’s approach, based on the reducibility of certain space groups, is extended by the derivation of a universal transposition with the obvious alternatives in terms of m , m Nx. Our problem formula for intermediate results. It is then shown that space groups is to evaluate the CDFT for a given space group as efﬁciently as which are not completely reducible may nevertheless be treated by possible, in spite of the fact that the group action has spoilt the three-dimensional Cooley–Tukey factorization in such a way that simple tensor-product structure of the ordinary three-dimensional their symmetry may be fully exploited, whatever the shape of their DFT (Section 1.3.3.3.1). Two procedures are available to carry out the 3D summations asymmetric unit. Finally, new factorization methods with built-in symmetries are presented. The unifying concept throughout this involved as a succession of smaller summations: (1) decomposition into successive transforms of fewer dimenpresentation is that of ‘group action’ on indexing sets, and of ‘orbit exchange’ when this action has a composite structure; it affords new sions but on the same number of points along these dimensions. This ways of rationalizing the use of symmetry, or of improving possibility depends on the reducibility of the space group, as deﬁned in Section 1.3.4.2.2.4, and simply invokes the tensor product computational speed, or both. property of the DFT; (2) factorization of the transform into transforms of the same number of dimensions as the original one, but on fewer points along 1.3.4.3.2. Defining relations and symmetry considerations each dimension. This possibility depends on the arithmetic A ﬁnite set of reﬂections fFhl gl2L can be periodized without factorability of the decimation matrix N, as described in Section aliasing by the translations of a suitable sublattice NT of the 1.3.3.3.2. reciprocal lattice ; the converse operation in real space is the Clearly, a symmetry expansion to the largest fully reducible sampling of at points X of a grid of the form N 1 (Section subgroup of the space group will give maximal decomposability, 1.3.2.7.3). In standard coordinates, fFhl gl2L is periodized by NT Z3 , but will require computing more than the unique results from more and is sampled at points x 2 N 1 Z3 . than the unique data. Economy will follow from factoring the In the absence of symmetry, the unique data are transforms in the subspaces within which the space group acts 3 3 – the Fh indexed by h 2 Z =NT Z in reciprocal space; irreducibly. – the x indexed by x 2
N 1 Z3 =Z3 ; or equivalently the m For irreducible subspaces of dimension 1, the group action is indexed by m 2 Z3 =NZ3 , where x N 1 m. readily incorporated into the factorization of the transform, as ﬁrst They are connected by the ordinary DFT relations: shown by Ten Eyck (1973). For irreducible subspaces of dimension 2 or 3, the ease of incorporation of symmetry into the factorization depends on the X 1 Fh x exp
2ih x type of factorization method used. The multidimensional Cooley– jdet Nj Tukey method (Section 1.3.3.3.1) is rather complicated; the x2
N 1 Z3 =Z3 multidimensional Good method (Section 1.3.3.3.2.2) is somewhat simpler; and the Rader/Winograd factorization admits a generalization, based on the arithmetic of certain rings of algebraic or

72

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY integers, which accommodates 2D crystallographic symmetries in a most powerful and pleasing fashion. At each stage of the calculation, it is necessary to keep track of the deﬁnition of the asymmetric unit and of the symmetry properties of the numbers being manipulated. This requirement applies not only to the initial data and to the ﬁnal results, where these are familiar; but also to all the intermediate quantities produced by partial transforms (on subsets of factors, or subsets of dimensions, or both), where they are less familiar. Here, the general formalism of transposition (or ‘orbit exchange’) described in Section 1.3.4.2.2.2 plays a central role.

and hence the symmetry properties of T are expressed by the identity T
Sg0 #
Sg00 T: Applying this relation not to T but to
Sg0 1 #
Se00 T gives
Sg0 1 #
Se00 T
Se0 #
Sg00 T,

i.e. 00

T
Sg0
x0 , h00 exp
2ih00 t00g T
x0 , RgT h00 : If the unique F
h F
h0 , h00 were initially indexed by

1.3.4.3.3. Interaction between symmetry and decomposition

all h0
unique h00

(see Section 1.3.4.2.2.2), this formula allows the reindexing of the intermediate results T
x0 , h00 from the initial form

Suppose that the space-group action is reducible, i.e. that for each g2G 0 0 tg Rg 0 Rg , tg 00 ; 0 R00g tg

all x0
unique h00

to the ﬁnal form
unique x0
all h00 ,

by Schur’s 0 lemma, the decimation matrix must then be of the form N 0 if it is to commute with all the Rg . N 0 N00 0 x0 h Putting x and h , we may deﬁne x00 h00 Sg0
x0 R0g x0 t0g , Sg00
x00

R00g x00

on which the second transform (on h00 ) may now be performed, giving the ﬁnal results
x0 , x00 indexed by
unique x0
all x00 ,

which is an asymmetric unit. An analogous interpretation holds if one is going from to F. The above formula solves the general problem of transposing from one invariant subspace to another, and is the main device for decomposing the CDFT. Particular instances of this formula were derived and used by Ten Eyck (1973); it is useful for orthorhombic groups, and for dihedral groups containing screw axes nm with g.c.d.
m, n 1. For comparison with later uses of orbit exchange, it should be noted that the type of intermediate results just dealt with is obtained after transforming on all factors in one summand. A central piece of information for driving such a decomposition is the deﬁnition of the full asymmetric unit in terms of the asymmetric units in the invariant subspaces. As indicated at the end of Section 1.3.4.2.2.2, this is straightforward when G acts without ﬁxed points, but becomes more involved if ﬁxed points do exist. To this day, no systematic ‘calculus of asymmetric units’ exists which can automatically generate a complete description of the asymmetric unit of an arbitrary space group in a form suitable for directing the orbit exchange process, although Shenefelt (1988) has outlined a procedure for dealing with space group P622 and its subgroups. The asymmetric unit deﬁnitions given in Volume A of International Tables are incomplete in this respect, in that they do not specify the possible residual symmetries which may exist on the boundaries of the domains.

t00g ,

and writeSg Sg0 Sg00 (direct sum) as a shorthand for Sg
x Sg0
x0 : Sg00
x00 0 00 We may also deﬁne the representation operators Sg# and Sg# 0 00 acting on functions of x and0 x , respectively (as in Section 00 1.3.4.2.2.4), and the operators Sg and Sg acting on functions of h0 and h00 , respectively (as in Section 1.3.4.2.2.5). Then we may write Sg#
Sg0 #
Sg00 # and Sg
Sg0
Sg00 in the sense that g acts on f
x f
x0 , x00 by

Sg# f
x0 , x00 f
Sg0 1
x0 ,
Sg00 1
x00

and on
h
h0 , h00 by

Sg
h0 , h00 exp
2ih0 t0g exp
2ih00 t00g 0

00

RgT h0 , RgT h00 :

Thus equipped we may now derive concisely a general identity describing the symmetry properties of intermediate quantities of the form X T
x0 , h00 F
h0 , h00 exp
2ih0 x0

1.3.4.3.4. Interaction between symmetry and factorization Methods for factoring the DFT in the absence of symmetry were examined in Sections 1.3.3.2 and 1.3.3.3. They are based on the observation that the ﬁnite sets which index both data and results are endowed with certain algebraic structures (e.g. are Abelian groups, or rings), and that subsets of indices may be found which are not merely subsets but substructures (e.g. subgroups or subrings). Summation over these substructures leads to partial transforms, and the way in which substructures ﬁt into the global structure indicates how to reassemble the partial results into the ﬁnal results. As a rule, the richer the algebraic structure which is identiﬁed in the indexing set, the more powerful the factoring method.

h0

1 X
x0 , x00 exp
2ih00 x00 , jdet N0 j x00

which arise through partial transformation of F on h0 or of on x00 . The action of g 2 G on these quantities will be (i) through
Sg0 # on the function x0 7 ! T
x0 , h00 , (ii) through
Sg00 on the function h00 7 ! T
x0 , h00 ,

73

1. GENERAL RELATIONSHIPS AND TECHNIQUES The ability of a given factoring method to accommodate crystallographic symmetry will thus be determined by the extent to which the crystallographic group action respects (or fails to respect) the partitioning of the index set into the substructures pertaining to that method. This remark justiﬁes trying to gain an overall view of the algebraic structures involved, and of the possibilities of a crystallographic group acting ‘naturally’ on them. The index sets fmjm 2 Z3 =NZ3 g and fhjh 2 Z3 =NT Z3 g are ﬁnite Abelian groups under component-wise addition. If an iterated addition is viewed as an action of an integer scalar n 2 Z via
n times

nh h h . . . h 0

m1 m

m2 N1 1
m

for n 0,

for n < 0,

g:

then an Abelian group becomes a module over the ring Z (or, for short, a Z-module), a module being analogous to a vector space but with scalars drawn from a ring rather than a ﬁeld. The left actions of a crystallographic group G by m 7 ! Rg m Ntg mod NZ

g:

h 7 !
Rg 1 T h

and by

2 Ntg t
1 g N1 tg ,
2 with t
1 g 2 I1 and tg 2 I2 determined as above. Suppose further that N, N1 and N2 commute with Rg for all g 2 G, i.e. (by Schur’s lemma, Section 1.3.4.2.2.4) that these matrices are integer multiples of the identity in each G-invariant subspace. The action of g on m Nx mod NZ3 leads to

mod NT Z3

can be combined with this Z action as follows: P P ng g : m 7 ! ng
Rg m Ntg mod NZ3 , g2G

P

h7 !

g2G

P

2 NRg N 1
m1 N1 m2 t
1 g N1 tg

mod NT Z3 :

ng
Rg 1 T h

2 Rg m1 t
1 g N1
Rg m2 tg

g2G

This provides a left action, on the indexing sets, of the set ( ) P ZG ng g ng 2 Z for each g 2 G

with Sg
m1 Sg
m Sg
m2 N1 1 fSg
m

g2G

P

ag1 g1

g1 2G

with

cg

P

g2 2G

P

bg2 g2

!

P

Introducing the notation

the two components of Sg
m may be written Sg
m1 Sg
1
m1 ,

ag0 b
g0 1 g,

g0 2G

Sg
m2 Sg
2
m2 2
g, m1 mod N2 Z3 ,

with 2
g, m1 N1 1 f
Rg m1 t
1 g

m m1 N1 m2

with m1 2 Z =N1 Z and m2 2 Z3 =N2 Z3 determined by

mod N2 Z3 :

Sg
2
m2 Rg m2 tg
2 mod N2 Z3 ,

cg g,

g2G

1.3.4.3.4.1. Multidimensional Cooley–Tukey factorization Suppose, as in Section 1.3.3.3.2.1, that the decimation matrix N may be factored as N1 N2 . Then any grid point index m 2 Z3 =NZ3 in real space may be written 3

Sg
m1 g

Sg
1
m1 Rg m1 tg
1 mod N1 Z3 ,

then ZG is a ring, and the action deﬁned above makes the indexing sets into ZG-modules. The ring ZG is called the integral group ring of G (Curtis & Reiner, 1962, p. 44). From the algebraic standpoint, therefore, the interaction between symmetry and factorization can be expected to be favourable whenever the indexing sets of partial transforms are ZGsubmodules of the main ZG-modules.

3

mod N1 Z3

and

and

!

mod NZ3 ,

Sg
m Sg
m1 N1 Sg
m2

of symbolic linear combinations of elements of G with integral coefﬁcients. If addition and multiplication are deﬁned in ZG by ! ! P P P ag1 g1 bg2 g2
ag bg g g2 2G

mod NZ3

which we may decompose as

g2G

g1 2G

mod NZ3

Sg
m NRg
N 1 m Ntg

g2G

ng g :

m 7 ! Sg
m Rg m Ntg mod NZ3

and suppose that N ‘integerizes’ all the non-primitive translations tg so that we may write

3

g:

m1 mod N2 Z3 :

These relations establish a one-to-one correspondence m $
m1 , m2 between I Z3 =NZ3 and the Cartesian product I1 I2 of I1 Z3 =N1 Z3 and I2 Z3 =N2 Z3 , and hence I I1 I2 as a set. However I 6 I1 I2 as an Abelian group, since in general m m0 6 !
m1 m01 , m2 m02 because there can be a ‘carry’ from the addition of the ﬁrst components into the second components; therefore, I 6 I1 I2 as a ZG-module, which shows that the incorporation of symmetry into the Cooley–Tukey algorithm is not a trivial matter. Let g 2 G act on I through

for n > 0,

h h . . . h
jnj times

mod N1 Z3 ,

74

Sg
m1 1 g mod N2 Z3 :

The term 2 is the geometric equivalent of a carry or borrow: it 3 3 arises because Rg m1 t
1 g , calculated as a vector in Z =NZ , may 3 be outside the unit cell N1 0, 1 , and may need to be brought back into it by a ‘large’ translation with a non-zero component in the m2 space; equivalently, the action of g may need to be applied around different permissible origins for different values of m1 , so as to map the unit cell into itself without any recourse to lattice translations. [Readers familiar with the cohomology of groups (see e.g. Hall, 1959; MacLane, 1963) will recognize 2 as the cocycle of the extension of ZG-modules described by the exact sequence 0 ! I2 ! I ! I1 ! 0.]

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY Thus G acts on I in a rather complicated fashion: although g 7 ! Sg
1 does deﬁne a left action in I1 alone, no action can be deﬁned in I2 alone because 2 depends on m1 . However, because Sg , Sg
1 and Sg
2 are left actions, it follows that 2 satisﬁes the identity 2
gg0 , m1 Sg
2 2
g0 , m1 2 g, Sg
1
m1

h1
N2 1 T
h

RTg h RTg h2 NT2 RTg h1 ,

mod N2 Z3 with

1

RTg h1 Rg
1 T h1 1
g, h2 mod NT1 Z3 : T
1 T Here R
2 g , Rg and 1 are deﬁned by

This action will now be used to achieve optimal use of symmetry in the multidimensional Cooley–Tukey algorithm of Section 1.3.3.3.2.1. Let us form an array Y according to

T T R
2 g h2 Rg h

T T R
1 g h1 Rg h

Y
m1 , m2
m1 N1 m2

for all m2 2 I2 but only for the unique m1 under the action Sg
1 of G in I1 . Except in special cases which will be examined later, these vectors contain essentially an asymmetric unit of electron-density data, up to some redundancies on boundaries. We may then compute the partial transform on m2 : X 1 Y
m1 , h2 Y
m1 , m2 eh2
N2 1 m2 : jdet N2 j m2 2I2

Y

1
g, h2
N2 1 T
RTg h2

mod NT1 Z3 T T 3 R
2 g h2 mod N1 Z :

Let us then form an array Z according to

Z
h01 , h02 F
h02 NT2 h01

for all h01 but only for the unique h02 under the action of G in Z3 =NT2 Z3 , and transform on h01 to obtain P Z
h01 , h02 e h01
N1 1 m1 : Z
m1 , h2 h01 2Z3 =NT1 Z3

1

efh2 N2
t
2 g

mod NT2 Z3 ,

and

Using the symmetry of in the form Sg# yields by the procedure of Section 1.3.3.3.2 the transposition formula
Sg
1
m1 , h2

mod NT2 Z3 ,

RTg h2 Rg
2 T h2

2
g 1 , m1 Sg 1 f2 g, Sg 1
m1 g mod N2 Z3 :

h2 mod NT1 Z3 :

We may then write

for all g, g0 in G and all m1 in I1 . In particular, 2
e, m1 0 for all m1 , and
2

mod NT2 Z3 ,

h2 h

2
g, m1 g T Y
m1 , R
2 g h2 :

Putting h0 RTg h and using the symmetry of F in the form F
h0 F
h exp
2ih tg ,

By means of this identity we can transpose intermediate results Y initially indexed by

where
2 h tg
hT2 hT1 N2
N2 1 N1 1
t
1 g N1 tg

unique m1
all h2 ,

h2 tg h2
N1 1 t
1 g mod 1

so as to have them indexed by
all m1
unique h2 :

yields by a straightforward rearrangement

We may then apply twiddle factors to get

T 1 Z
m1 , R
2 g h2 e fh2 tg 1
g, h2
N1 m1 g

Z
m1 , h2 eh2
N 1 m1 Y
m1 , h2

ZfSg
1
m1 , h2 g:

and carry out the second transform X 1 Z
m1 , h2 eh1
N1 1 m1 : Z
h1 , h2 jdet N1 j m1 2I1

This formula allows the transposition of intermediate results Z from an indexing by
all m1
unique h2 to an indexing by

The ﬁnal results are indexed by

unique m1
all h2 :

all h1
unique h2 ,

We may then apply the twiddle factors to obtain

which yield essentially an asymmetric unit of structure factors after unscrambling by:

Y
m1 , h2 e h2
N 1 m1 Z
m1 , h2

F
h2 NT2 h1 Z
h1 , h2 :

and carry out the second transform on h2 P Y
m1 , m2 Y
m1 , h2 e h2
N2 1 m2 :

The transposition formula above applies to intermediate results when going backwards from F to , provided these results are considered after the twiddle-factor stage. A transposition formula applicable before that stage can be obtained by characterizing the action of G on h (including the effects of periodization by NT Z3 ) in a manner similar to that used for m. Let

h2 2Z3 =NT2 Z3

The results, indexed by

unique m1
all m2 yield essentially an asymmetric unit of electron densities by the rearrangement

h h2 NT2 h1 ,

m1 N1 m2 Y
m1 , m2 :

with

75

1. GENERAL RELATIONSHIPS AND TECHNIQUES The equivalence of the two transposition formulae up to the intervening twiddle factors is readily established, using the relation

1.3.4.3.4.2. Multidimensional Good factorization This procedure was described in Section 1.3.3.3.2.2. The main difference with the Cooley–Tukey factorization is that if N N1 N2 . . . Nd 1 Nd , where the different factors are pairwise coprime, then the Chinese remainder theorem reindexing makes Z3 =NZ3 isomorphic to a direct sum.

h2 N2 1 2
g, m1 1
g, h2
N1 1 m1 mod 1 which is itself a straightforward consequence of the identity h N 1 Sg
m h tg
RTg h
N 1 m:

Z3 =NZ3
Z3 =N1 Z3 . . .
Z3 =Nd Z3 ,

where each p-primary piece is endowed with an induced ZGmodule structure by letting G operate in the usual way but with the corresponding modular arithmetic. The situation is thus more favourable than with the Cooley–Tukey method, since there is no interference between the factors (no ‘carry’). In the terminology of Section 1.3.4.2.2.2, G acts diagonally on this direct sum, and results of a partial transform may be transposed by orbit exchange as in Section 1.3.4.3.4.1 but without the extra terms or . The analysis of the symmetry properties of partial transforms also carries over, again without the extra terms. Further simpliﬁcation occurs for all p-primary pieces with p other than 2 or 3, since all non-primitive translations (including those associated to lattice centring) disappear modulo p. Thus the cost of the CRT reindexing is compensated by the computational savings due to the absence of twiddle factors and of other phase shifts associated with non-primitive translations and with geometric ‘carries’. Within each p-primary piece, however, higher powers of p may need to be split up by a Cooley–Tukey factorization, or carried out directly by a suitably adapted Winograd algorithm.

To complete the characterization of the effect of symmetry on the Cooley–Tukey factorization, and of the economy of computation it allows, it remains to consider the possibility that some values of m1 may be invariant under some transformations g 2 G under the action m1 7 ! Sg
1
m1 . Suppose that m1 has a non-trivial isotropy subgroup Gm1 , and let g 2 Gm1 . Then each subarray Ym1 deﬁned by Ym1
m2 Y
m1 , m2
m1 N1 m2 satisﬁes the identity Ym1
m2 YS
1
m1 Sg
2
m2 2
g, m1 g

Ym1 Sg
2
m2 2
g, m1

so that the data for the transform on m2 have residual symmetry properties. In this case the identity satisﬁed by 2 simpliﬁes to 2
gg0 , m1 Sg
2 2
g0 , m1 2
g, m1 mod N2 Z3 , which shows that the mapping g 7 ! 2
g, m1 satisﬁes the Frobenius congruences (Section 1.3.4.2.2.3). Thus the internal symmetry of subarray Ym1 with respect to the action of G on m2 is given by Gm1 acting on Z3 =N2 Z3 via

1.3.4.3.4.3. Crystallographic extension of the Rader/ Winograd factorization As was the case in the absence of symmetry, the two previous classes of algorithms can only factor the global transform into partial transforms on prime numbers of points, but cannot break the latter down any further. Rader’s idea of using the action of the group of units U
p to obtain further factorization of a p-primary transform has been used in ‘scalar’ form by Auslander & Shenefelt (1987), Shenefelt (1988), and Auslander et al. (1988). It will be shown here that it can be adapted to the crystallographic case so as to take advantage also of the possible existence of n-fold cyclic symmetry elements
n 3, 4, 6 in a two-dimensional transform (Bricogne & Tolimieri, 1990). This adaptation entails the use of certain rings of algebraic integers rather than ordinary integers, whose connection with the handling of cyclic symmetry will now be examined. Let G be the group associated with a threefold axis of symmetry: G fe, g, g2 g with g3 e. In a standard trigonal basis, G has matrix representation 1 0 0 1 1 1 I, Rg , Rg2 Re 0 1 1 1 1 0

m2 7 ! Sg
2
m2 2
g, m1 mod N2 Z3 : The transform on m2 needs only be performed for one out of G : Gm1 distinct arrays Ym1 (results for the others being obtainable by the transposition formula), and this transforms is Gm1 symmetric. In other words, the following cases occur: maximum saving in computation
by jGj; m2 -transform has no symmetry: G0 < G saving in computation by a factor of G : G0 ; m2 -transform is G0 -symmetric: G no saving in computation; m2 -transform is G-symmetric:

i

Gm1 feg

ii

Gm1

iii Gm1

The symmetry properties of the m2 -transform may themselves be exploited in a similar way if N2 can be factored as a product of smaller decimation matrices; otherwise, an appropriate symmetrized DFT routine may be provided, using for instance the idea of ‘multiplexing/demultiplexing’ (Section 1.3.4.3.5). We thus have a recursive descent procedure, in which the deeper stages of the recursion deal with transforms on fewer points, or of lower symmetry (usually both). The same analysis applies to the h1 -transforms on the subarrays Zh2 , and leads to a similar descent procedure. In conclusion, crystallographic symmetry can be fully exploited to reduce the amount of computation to the minimum required to obtain the unique results from the unique data. No such analysis was so far available in cases where the asymmetric units in real and reciprocal space are not parallelepipeds. An example of this procedure will be given in Section 1.3.4.3.6.5.

in real space, 1 0 I, Re 0 1

Rg

1 1

1 , 0

Rg2

0 1

1 1

in reciprocal space. Note that Rg2 Rg21 T RTg , and that RTg

76

1

J Rg J,

where J

1 0

0 1

so that Rg and RTg are conjugate in the group of 2 2 unimodular

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY The structure of Zp G depends on whether P
X remains irreducible when considered as a polynomial over Zp . Thus two cases arise: (1) P
X remains irreducible mod p, i.e. there is no nth root of unity in Zp ; (2) P
X factors as
X u
X v, i.e. there are nth roots of unity in Zp . These two cases require different developments. Case 1. Zp G is a ﬁnite ﬁeld with p2 elements. There is essentially (i.e. up to isomorphism) only one such ﬁeld, denoted GF
p2 , and its group of units is a cyclic group with p2 1 elements. If is a generator of this group of units, the input data m with m 6 0 may be reordered as

integer matrices. The group ring ZG is commutative, and has the structure of the polynomial ring ZX with the single relation X 2 X 1 0 corresponding to the minimal polynomial of Rg . In the terminology of Section 1.3.3.2.4, the ring structure of ZG is obtained from that of ZX by carrying out polynomial addition and multiplication modulo X 2 X 1, then replacing X by any generator of G. This type of construction forms the very basis of algebraic number theory [see Artin (1944, Section IIc) for an illustration of this viewpoint], and ZG as just deﬁned is isomorphic to the ring Z! of algebraic integers of the form a b! a, b 2 Z, ! exp
2i=3 under the identiﬁcation X $ !. Addition in this ring is deﬁned component-wise, while multiplication is deﬁned by
a1 b1 !
a2 b2 !
a1 a2

a1

2

m0 , m0 , 2 m0 , 3 m0 , . . . , p

b1 b 2

b1 b2 b1 a2 !: 2

3

m0

by the real-space action of ; while the results Fh with h 6 0 may be reordered as

4

In the case of a fourfold axis, G fe, g, g , g g with g e, and 0 1 Rg , with again RTg J 1 Rg J: Rg 1 0

2

h0 , h0 , 2 h0 , 3 h0 , . . . , p

2

h0

by the reciprocal-space action of , where m0 and h0 are arbitrary non-zero indices. The core Cpp of the DFT matrix, deﬁned by 0 1 1 1 ... 1 B1 C C, Fpp B @ ... A C

ZG is obtained from ZX by carrying out polynomial arithmetic modulo X 2 1. This identiﬁes ZG with the ring Zi of Gaussian integers of the form a bi, in which addition takes place component-wise while multiplication is deﬁned by
a1 b1 i
a2 b2 i
a1 a2

2

b1 b2
a1 b2 b1 a2 i:

pp

In the case of a sixfold axis, G fe, g, g2 , g3 , g4 , g5 g with 6 g e, and 1 1 0 1 , Rg , RTg J 1 Rg J: Rg 1 0 1 1

1

will then have a skew-circulant structure (Section 1.3.3.2.3.1) since j
h0
k m0 h0
jk m0 e
Cpp jk e p p

ZG is isomorphic to Z! under the mapping g $ 1 ! since
1 !6 1. Thus in all cases ZG ZX =P
X where P
X is an irreducible quadratic polynomial with integer coefﬁcients. The actions of G on lattices in real and reciprocal space (Sections 1.3.4.2.2.4, 1.3.4.2.2.5) extend naturally to actions of ZG on Z2 in which an element z a bg of ZG acts via m1 m1 7 ! zm
aI bRg m m2 m2

depends only on j k. Multiplication by Cpp may then be turned into a cyclic convolution of length p2 1, which may be factored by two DFTs (Section 1.3.3.2.3.1) or by Winograd’s techniques (Section 1.3.3.2.4). The latter factorization is always favourable, as it is easily shown that p2 1 is divisible by 24 for any odd prime p 5. This procedure is applicable even if no symmetry is present in the data. Assume now that cyclic symmetry of order n 3, 4 or 6 is present. Since n divides 24 hence divides p2 1, the generator g of
p2 1=n this symmetry is representable as for a suitable generator of the group of units. The reordered data will then be
p2 1=nperiodic rather than simply
p2 1-periodic; hence the reindexed results will be n-decimated (Section 1.3.2.7.2), and the
p2 1=n non-zero results can be calculated by applying the DFT to the
p2 1=n unique input data. In this way, the n-fold symmetry can be used in full to calculate the core contributions from the unique data to the unique results by a DFT of length
p2 1=n. It is a simple matter to incorporate non-primitive translations into this scheme. For example, when going from structure factors to electron densities, reordered data items separated by
p2 1=n are not equal but differ by a phase shift proportional to their index mod p, whose effect is simply to shift the origin of the n-decimated transformed sequence. The same economy of computation can therefore be achieved as in the purely cyclic case. Dihedral symmetry elements, which map g to g 1 (Section 1.3.4.2.2.3), induce extra one-dimensional symmetries of order 2 in the reordered data which can also be fully exploited to reduce computation. Case 2. If p 5, it can be shown that the two roots u and v are always distinct. Then, by the Chinese remainder theorem (CRT) for polynomials (Section 1.3.3.2.4) we have a ring isomorphism

in real space, and via h h1 T 7 ! zh
aI bRg 1 h h2 h2 in reciprocal space. These two actions are related by conjugation, since
aI bRTg J 1
aI bRg J and the following identity (which is fundamental in the sequel) holds:
zh m h
zm for all m, h 2 Z2 :

Let us now consider the calculation of a p p two-dimensional DFT with n-fold cyclic symmetry
n 3, 4, 6 for an odd prime p 5. Denote Z=pZ by Zp . Both the data and the results of the DFT are indexed by Zp Zp : hence the action of ZG on these indices is in fact an action of Zp G, the latter being obtained from ZG by carrying out all integer arithmetic in ZG modulo p. The algebraic structure of Zp G combines the symmetry-carrying ring structure of ZG with the ﬁnite ﬁeld structure of Zp used in Section 1.3.3.2.3.1, and holds the key to a symmetry-adapted factorization of the DFT at hand.

Zp X =P
X fZp X =
X

77

ug fZp X =
X

vg

1. GENERAL RELATIONSHIPS AND TECHNIQUES deﬁned by sending a polynomial Q
X from the left-hand-side ring to its two residue classes modulo X u and X v, respectively. Since the latter are simply the constants Q
u and Q
v, the CRT reindexing has the particularly simple form

m 7 !
aI bRg m 0 becomes 7 ! with Mm, 0

h 7 !
aI bRTg h 0 becomes 7 ! with MJh: 0

a bX 7 !
a bu, a bv
, or equivalently a a mod p, M 7 ! b b

with M

Thus the sets of indices and can be split into four pieces as Zp G itself, according as these indices have none, one or two of their coordinates in U
p. These pieces will be labelled by the same symbols – 0, D1 , D2 and U – as those of Zp G. The scalar product h m may be written in terms of and as

1 u : 1 v

The CRT reconstruction formula similarly simpliﬁes to a M 1 7 ! mod p, b v u 1 1 : with M v u 1 1

h m
M 1 T JM 1 ,

and an elementary calculation shows that the matrix
M 1 T JM 1 is diagonal by virtue of the relation uv constant term in P
X 1:

Therefore, h m 0 if h 2 D1 and 2 D2 or vice versa. We are now in a position to rearrange the DFT matrix Fpp . Clearly, the structure of Fpp is more complex than in case 1, as there are three types of ‘core’ matrices:

The use of the CRT therefore amounts to the simultaneous diagonalization (by M) of all the matrices representing the elements of Zp G in the basis (1, X). A ﬁrst consequence of this diagonalization is that the internal structure of Zp G becomes clearly visible. Indeed, Zp G is mapped isomorphically to a direct product of two copies of Zp , in which arithmetic is carried out component-wise between eigenvalues and . Thus if

type 1: D D
with D D1 or D2 ;

type 2: D U or U D; type 3: U U:

(Submatrices of type D1 D2 and D2 D1 have all their elements equal to 1 by the previous remark.) Let be a generator of U
p. We may reorder the elements in D1 , D2 and U – and hence the data and results indexed by these elements – according to powers of . This requires one exponent in each of D1 and D2 , and two exponents in U. For instance, in the h-index space: ( )

0 j 1 D1 j 1, . . . , p 1 0 0 0 0 ( ) 0 0 j 0 D2 j 1, . . . , p 1 2 0 0 (

0 j1 1 0 j2 1 U j1 1, . . . , p 1; 0 2 0 0 1 j2 1, . . . , p 1

CRT

z a bX !
, , CRT

z0 a0 b0 X !
0 , 0 , then CRT

z z0 !
0 , 0 , CRT

zz0 !
0 , 0 : Taking in particular CRT

z !
, 0 6
0, 0, CRT

z0 !
0, 6
0, 0, we have zz0 0, so that Zp G contains zero divisors; therefore Zp G CRT is not a ﬁeld. On the other hand, if z !
, with 6 0 and 6 0, then and belong to the group of units U
p (Section 1.3.3.2.3.1) and hence have inverses 1 and 1 ; it follows that z is CRT a unit in Zp G, with inverse z 1 !
1 , 1 . Therefore, Zp G consists of four distinct pieces: CRT

0 !f
0, 0g, CRT

D1 !f
, 0j 2 U
pg U
p, CRT

D2 !f
0, j 2 U
pg U
p, CRT

U !f
, j 2 U
p, 2 U
pg U
p U
p: A second consequence of this diagonalization is that the actions of Zp G on indices m and h can themselves be brought to diagonal form by basis changes:

78

and similarly for the index. Since the diagonal matrix D commutes with all the matrices representing the action of , this rearrangement will induce skewcirculant structures in all the core matrices. The corresponding cyclic convolutions may be carried out by Rader’s method, i.e. by diagonalizing them by means of two (p 1)-point one-dimensional DFTs in the D D pieces and of two
p 1
p 1-point twodimensional DFTs in the U U piece (the U D and D U pieces involve extra section and projection operations). In the absence of symmetry, no computational saving is achieved, since the same reordering could have been applied to the initial Zp Zp indexing, without the CRT reindexing. In the presence of n-fold cyclic symmetry, however, the rearranged Fpp lends itself to an n-fold reduction in size. The basic fact is that whenever case 2 occurs, p 1 is divisible by n (i.e. p 1 is divisible by 6 when n 3 or 6, and by 4 when n 4), say

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY
h2 , h1 7 ! N2
h2 h2 , N1
h1 h1
h2 : Hermitian symmetry is traditionally dealt with by factoring by 2, i.e. by assuming N 2M. If N2 2I, then each h2 is invariant under G, so that each partial vector Zh2 (Section 1.3.4.3.4.1) inherits the symmetry internally, with a ‘modulation’ by 1
g, h2 . The ‘multiplexing–demultiplexing’ technique provides an efﬁcient treatment of this singular case.

p 1 nq. If g is a generator of the cyclic symmetry, the generator

of U
p may be chosen in such a way that g q . The action of g is then to increment the j index in D1 and D2 by q, and the
j1 , j2 index in U by (q, q). Since the data items whose indices are related in this way have identical values, the DFTs used to diagonalize the Rader cyclic convolutions will operate on periodized data, hence yield decimated results; and the non-zero results will be obtained from the unique data by DFTs n times smaller than their counterparts in the absence of symmetry. A more thorough analysis is needed to obtain a Winograd factorization into the normal from CBA in the presence of symmetry (see Bricogne & Tolimieri, 1990). Non-primitive translations and dihedral symmetry may also be accommodated within this framework, as in case 1. This reindexing by means of algebraic integers yields larger orbits, hence more efﬁcient algorithms, than that of Auslander et al. (1988) which only uses ordinary integers acting by scalar dilation.

(b) Calculation of structure factors The computation may be summarized as follows: 2 F
N

dec
N1

Most crystallographic Fourier syntheses are real-valued and originate from Hermitian-symmetric collections of Fourier coefﬁcients. Hermitian symmetry is closely related to the action of a centre of inversion in reciprocal space, and thus interacts strongly with all other genuinely crystallographic symmetry elements of order 2. All these symmetry properties are best treated by factoring by 2 and reducing the computation of the initial transform to that of a collection of smaller transforms with less symmetry or none at all.

The transform Y F
MY can then be resolved into the separate transforms Ym0 and Ym00 by using the Hermitian symmetry of the 1 1 latter, which yields the demultiplexing formulae Ym 0
h2 iYm 00
h2 Y
h2

Zh2
h1

Therefore

h2 :

h2 Zh2 M
h1

h1

2

we may group the 2n values of h2 into 2n each pair form the multiplexed vector:

1

pairs
h02 , h002 and for

Z Zh02 iZh002 : the 2n After calculating the 2n 1 transforms Z F
MZ, individual transforms Zh0 and Zh00 can be separated by using for 2 2 each pair the demultiplexing formulae

h2 ,

Zh0
h1 iZh00
h1 Z
h1 2

h2 ,

Zh0
h1 2

hence
h2 mod N1 Zn :

Y M
h2

which can be used to halve the number of F
M necessary to compute them, as follows. Having formed the vectors Zh2 given by 2 3 X
1h2 m2 Zh2
m1 4
m1 Mm2 5eh2
N 1 m1 , n 2 n n m 2Z =2Z

Let m m1 N1 m2 , and hence h h2 N2 h1 . Then

N2
h2

iYm 00
h2 1

The number of partial transforms F
M is thus reduced from 2n to n 1 2 . Once this separation has been achieved, the remaining steps need only be carried out for a unique half of the values of h2 . (ii) Decimation in frequency
N1 M, N2 2I Since h2 2 Zn =2Zn we have h2 h2 and
h2 h2 mod 2Zn . The vectors of decimated and scrambled results Zh2 then obey the symmetry relations

(a) Underlying group action Hermitian symmetry is not a geometric symmetry, but it is deﬁned in terms of the action in reciprocal space of point group G 1, i.e. G fe, eg, where e acts as I (the n n identity matrix) and e acts as I. This group action on Zn =NZn with N N1 N2 will now be characterized by the calculation of the cocycle 1 (Section 1.3.4.3.4.1) under the assumption that N1 and N2 are both diagonal. For this it is convenient to associate to any integer vector 0 purpose 1 v1 B . C v @ .. A in Zn the vector
v whose jth component is vn 0 if vj 0 1 if vj 6 0.

h2

1

Ym 0
h2 1

1.3.4.3.5.1. Hermitian-symmetric or real-valued transforms The computation of a DFT with Hermitian-symmetric or realvalued data can be carried out at a cost of half that of an ordinary transform, essentially by ‘multiplexing’ pairs of special partial transforms into general complex transforms, and then ‘demultiplexing’ the results on the basis of their symmetry properties. The treatment given below is for general dimension n; a subset of cases for n 1 was treated by Ten Eyck (1973).

1
e, h2 N2 1 fN
h2

rev
N2

Y Ym01 iYm001 :

1

h2 mod N2 Zn N2
h2

1 F
N

where dec
N1 is the initial decimation given by Ym1
m2
m1 N1 m2 , TW is the transposition and twiddlefactor stage, and rev
N2 is the ﬁnal unscrambling by coset reversal given by F
h2 N2 h1 Zh2
h1 . (i) Decimation in time
N1 2I, N2 M The decimated vectors Ym1 are real and hence have Hermitian transforms Ym1 . The 2n values of m1 may be grouped into 2n 1 pairs
m01 , m001 and the vectors corresponding to each pair may be multiplexed into a general complex vector

1.3.4.3.5. Treatment of conjugate and parity-related symmetry properties

h2 mod NZn N
h2

TW

7 ! Y 7 ! Y 7 ! Z 7 ! Z 7 ! F

h2 g mod N1 Zn

e acts by

79

h02 iZh00
h1 2

2

h002 Z M
h1

h1

which can be solved recursively. If all pairs are chosen so that they differ only in the jth coordinate
h2 j , the recursion is along
h1 j and can be initiated by introducing the (real) values of Zh0 and Zh00 at 2 2
h1 j 0 and
h1 j Mj , accumulated e.g. while forming Z for that pair. Only points with
h1 j going from 0 to 12 Mj need be resolved,

1. GENERAL RELATIONSHIPS AND TECHNIQUES (ii) Decimation in frequency
N1 2I, N2 M The last transformation F(M) gives the real-valued results , therefore the vectors Ym1 after the twiddle-factor stage each have Hermitian symmetry. A ﬁrst consequence is that the intermediate vectors Zh2 need only be computed for the unique half of the values of h2 , the other half being related by the Hermitian symmetry of Ym1 . A second consequence is that the 2n vectors Ym1 may be condensed into 2n 1 general complex vectors

and they contain the unique half of the Hermitian-symmetric transform F. (c) Calculation of electron densities The computation may be summarized as follows: scr
N2

F
N1

F
N2

TW

nat
N1

F 7 ! Z 7 ! Z 7 ! Y 7 ! Y 7 ! where scr
N2 is the decimation with coset reversal given by Zh2
h1 F
h2 N2 h1 , TW is the transposition and twiddlefactor stage, and nat
N1 is the recovery in natural order given by
m1 N1 m2 Ym1
m2 . (i) Decimation in time
N1 M, N2 2I The last transformation F
2I has a real-valued matrix, and the ﬁnal result is real-valued. It follows that the vectors Ym1 of intermediate results after the twiddle-factor stage are real-valued, hence lend themselves to multiplexing along the real and imaginary components of half as many general complex vectors. Let the 2n initial vectors Zh2 be multiplexed into 2n 1 vectors

Z

Zh0 2

Y Ym0 iYm00 1

m01 , m001 ]

[one for each pair be applied to yield

with Ym01 and Ym001 real-valued. The ﬁnal results can therefore be retrieved by the particularly simple demultiplexing formulae:
m01 2m2 Y
m2 ,

iZh00 2

m001 2m2 Y
m2 : 1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms A vector X fX
kjk 2 Zn =NZn g is said to be Hermitianantisymmetric if

Z Zh02 iZh002 : The real-valuedness of the Ym1 may be used to recover the separate result vectors for h02 and h002 . For this purpose, introduce the abbreviated notation

X
k X
k for all k:

Its transform X then satisﬁes

e h02
N 1 m1
c0 is0
m1 e

1

00

X
k X
k for all k ,

00

N m1
c is
m1 R h2
m1 Ym 1
h2

R0 Rh02 ,

i.e. is purely imaginary. If X is Hermitian-antisymmetric, then F iX is Hermitiansymmetric, with iX real-valued. The treatment of Section 1.3.4.3.5.1 may therefore be adapted, with trivial factors of i or 1, or used as such in conjunction with changes of variable by multiplication by i.

R00 Rh002 :

Then we may write Z
c0 is0 R0 i
c00 is00 R00

1.3.4.3.5.3. Complex symmetric and antisymmetric transforms The matrix I is its own contragredient, and hence (Section 1.3.2.4.2.2) the transform of a symmetric (respectively antisymmetric) function is symmetric (respectively antisymmetric). In this case the group G fe, eg acts in both real and reciprocal space as fI, Ig. If N N1 N2 with both factors diagonal, then e acts by

c0 R0 s00 R00 i
s0 R0 c00 R00

or, equivalently, for each m1 , 0 0 c Z s00 R : Z s0 c00 R 00 Therefore R0 and R00 may be retrieved from Z by the ‘demultiplexing’ formula: 0 00 1 R c s00 Z 0 00 R 00 c0 Z c c s0 s00 s0

m1 , m2 7 ! N1
m1
h2 , h1 7 ! N2
h2

h002

h2 , N1
h1

m2 h1

m1 ,

h2 ,

2
e, m1
m1 mod N2 Zn ,

1

N m1 6 0:

1
e, h2
h2 mod N1 Zn :

The symmetry or antisymmetry properties of X may be written

Demultiplexing fails when
h02

m1 , N2
m2

i.e.

which is valid at all points m1 where c0 c00 s0 s00 6 0, i.e. where cos2
h02

to which a general complex F(M) may

Y Ym01 iYm001

[one for each pair
h02 , h002 ], each of which yields by F(M) a vector

h002

1

X
m "X
m for all m,

h002
N 1 m1 12 mod 1:

If the pairs
h02 , h002 are chosen so that their members differ only in one coordinate (the jth, say), then the exceptional points are at
m1 j 12 Mj and the missing transform values are easily obtained e.g. by accumulation while forming Z . The ﬁnal stage of the calculation is then P
1h2 m2 R h2
m1 :
m1 Mm2

with " 1 for symmetry and " 1 for antisymmetry. The computation will be summarized as dec
N1

2 F
N

TW

1 F
N

rev
N2

X 7 ! Y 7 ! Y 7 ! Z 7 ! Z 7 ! X

with the same indexing as that used for structure-factor calculation. In both cases it will be shown that a transform F
N with N 2M and M diagonal can be computed using only 2n 1 partial transforms F
M instead of 2n .

h2 2Zn =2Zn

80

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY (i) Decimation in time
N1 2I, N2 M Since m1 2 Zn =2Zn we have m1 m1 and
m1 m1 mod 2Zn , so that the symmetry relations for each parity class of data Ym1 read Ym1 M
m2

have an internal symmetry expressed by Ym1 M
m2

This symmetry, however, is different for each m1 so that we may multiplex two such vectors Ym01 and Ym001 into a general real vector

m1 "Ym1
m2

m2

or equivalently

Y Ym01 Ym001 ,

m1 : m1 Ym1 "Y

for each of the 2n 1 pairs
m01 , m001 . The 2n transform vectors

Transforming by F
M, this relation becomes

1

Y Ym01 Ym001 : Putting

Ym0
c0

e h2
M 1 m01
c0 is0
h2

Ym00
c00

where R0 and R00 are real vectors and where the multipliers
c0 and
c00 is00 are the inverse twiddle factors. Therefore,

Ym 0
h2 Ym 00
h2 Y
h2 1

Y
c0

c0 is0
h2 Ym 0
h2
c00 is00
h2 Ym 00
h2

h02 Zh00
h1 2

need only be carried out for the unique half of the range of h2 . (ii) Decimation in frequency
N1 M, N2 2I Similarly, the vectors Zh2 of decimated and scrambled results are real and obey internal symmetries h2 Zh2 "Z h2 which are different for each h2 . For each of the 2n the multiplexed vector

Z
h1

h002 "Z M
h1

i
s0 R0 s00 R00

1

h2

After transforming by F
M, the results Z may be demultiplexed by using the relations

2

is00 R00

The values of R 0h2 and R 00h2 at those points h2 where c0 s00 s0 c00 0 can be evaluated directly while forming Y. This demultiplexing and the ﬁnal stage of the calculation, namely 1 X F
h2 Mh1 n
1h1 m1 R m1
h2 2 m 2Zn =2Zn

Z Zh02 Zh002 :

Zh0
h1

is0

and hence the demultiplexing relation for each h2 : 0 00 1 R s c00 Y : R 00 s0 c 0 Y c0 s00 s0 c00

which are different for each h2 . The vectors Zh2 of intermediate results after the twiddle-factor stage may then be multiplexed in pairs as

Zh00
h1 2

is0 R0
c00

c0 R0 c00 R00

1

h2

which can be solved recursively. Transform values at the exceptional points h2 where demultiplexing fails (i.e. where c0 is0 c00 is00 ) can be accumulated while forming Y. Only the unique half of the values of h2 need to be considered at the demultiplexing stage and at the subsequent TW and F(2I) stages. (ii) Decimation in frequency
N1 M, N2 2I The vectors of ﬁnal results Zh2 for each parity class h2 obey the symmetry relations , h2 Z "Z

Zh0
h1 2

is00 R00 ,

1

we then have the demultiplexing relations for each h2 :

h2

1

is0 R0

1

e h2
M 1 m001
c00 is00
h2

1

Hermitian-symmetric

can then be evaluated by the methods of Section 1.3.4.3.5.1(b) at the cost of only 2n 2 general complex F
M. The demultiplexing relations by which the separate vectors Ym0 1 and Ym00 may be recovered are most simply obtained by observing 1 that the vectors Z after the twiddle-factor stage are real-valued since F(2I) has a real matrix. Thus, as in Section 1.3.4.3.5.1(c)(i),

Each parity class thus obeys a different symmetry relation, so that we may multiplex them in pairs by forming for each pair
m01 , m001 the vector

"Y M
h2

1

Y Ym0 Ym00

e h2
M 1 m1 Ym1 "Ym1 :

1

m1 "Ym1
m2 :

m2

1

pairs
h02 , h002

Z Zh02 Zh002

h1

is a Hermitian-symmetric vector without internal symmetry, and the 2n 1 real vectors

which can be solved recursively as in Section 1.3.4.3.5.1(b)(ii).

Z Zh0 Zh00

1.3.4.3.5.4. Real symmetric transforms Conjugate symmetric (Section 1.3.2.4.2.3) implies that if the data X are real and symmetric [i.e. X
k X
k and X
k X
k], then so are the results X . Thus if contains a centre of symmetry, F is real symmetric. There is no distinction (other than notation) between structure-factor and electron-density calculation; the algorithms will be described in terms of the former. It will be shown that if N 2M, a real symmetric transform can be computed with only 2n 2 partial transforms F
M instead of 2n . (i) Decimation in time
N1 2I, N2 M Since m1 2 Zn =2Zn we have m1 m1 and
m1 m1 mod 2Zn . The decimated vectors Ym1 are not only real, but

2

2

n 2

may be evaluated at the cost of only 2 general complex F
M by the methods of Section 1.3.4.3.5.1(c). The individual transforms Zh02 and Zh002 may then be retrieved via the demultiplexing relations Zh0
h1 2

Zh0
h1 2

Zh00
h1 2

h02 Zh00
h1 2

Z
h1

h002 Z M
h1

h1

which can be solved recursively as described in Section 1.3.4.3.5.1(b)(ii). This yields the unique half of the real symmetric results F.

81

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1.3.4.3.5.5. Real antisymmetric transforms If X is real antisymmetric, then its transform X is purely imaginary and antisymmetric. The double-multiplexing techniques used for real symmetric transforms may therefore be adapted with only minor changes involving signs and factors of i.

The symmetry relations obeyed by and F are as follows: for electron densities
m , m
m N t g, or, after factoring by 2,
m 1 , m2 , m1 , m2

1.3.4.3.5.6. Generalized multiplexing So far the multiplexing technique has been applied to pairs of vectors with similar types of parity-related and/or conjugate symmetry properties, in particular the same value of ". It can be generalized so as to accommodate mixtures of vectors with different symmetry characteristics. For example if X1 is Hermitian-symmetric and X2 is Hermitian-antisymmetric, so that X1 is real-valued while X2 has purely imaginary values, the multiplexing process should obviously form X X1 X2 (instead of X X1 iX2 if both had the same type of symmetry), and demultiplexing consists in separating

2
m , M
m1 1 , m2 t g

F
h , h exp2i
h t g h tg F
h , h

or, after factoring by 2,
2

h2 tg F
h 1 , h2 , h1 , h2
1

F
h 1 , h2 , h1 , h2
2

1h2 tg

where ! is a phase factor (e.g. 1 or i) chosen in such a way that all non-exceptional components of X1 and X2 (or X1 and X2 ) be embedded in the complex plane C along linearly independent directions, thus making multiplexing possible. It is possible to develop a more general form of multiplexing/ demultiplexing for more than two vectors, which can be used to deal with symmetry elements of order 3, 4 or 6. It is based on the theory of group characters (Ledermann, 1987).

X
h1

1.3.4.3.6.1. Triclinic groups Space group P1 is dealt with by the methods of Section 1.3.4.3.5.1 and P1 by those of Section 1.3.4.3.5.4. 1.3.4.3.6.2. Monoclinic groups A general monoclinic transformation is of the form Sg : x 7 ! Rg x tg

X
h 1

with Rg a diagonal matrix whose entries are 1 or 1, and tg a vector whose entries are 0 or 12. We may thus decompose both real and reciprocal space into a direct sum of a subspace Zn where Rg acts as the identity, and a subspace Zn where Rg acts as minus the identity, with n n n 3. All usual entities may be correspondingly written as direct sums, for instance:

h h h ,

h1 h 1 h1 ,

h 1

h 2 , h2 , h1 , h2 :

h2 "X M
h1

h1

h 2 "X M
h1

h 1

with " 1 independent of h 1 . This is the same relation as for the same parity class of data for a Hermitian symmetric
" 1 or antisymmetric
" 1 transform. The same techniques may be used to decrease the number of F
M . This generalizes the procedure described by Ten Eyck (1973) for treating dyad axes, i.e. for the case n 1, t
2 0, and t
2 0 (simple dyad) or g g 6 0 (screw dyad). t
2 g Once F
N is completed, its results have Hermitian symmetry properties with respect to h which can be used to obtain the unique electron densities. Structure factors may be computed by applying the reverse procedures in the reverse order.

M M M ,

2 t
2 tg
2 , g tg

m 2 m 2 m2 , h2 h 2 h2 :

We will use factoring by 2, with decimation in frequency when computing structure factors, and decimation in time when computing electron densities; this corresponds to N N1 N2 with N1 M, N2 2I. The non-primitive translation vector Ntg then belongs to MZn , and thus n t
1 g 0 mod MZ ,

FM
h 1

h2 , h2

with " 1 independent of h1 . This is the same relation as for the same parity class of data for a (complex or real) symmetric
" 1 or antisymmetric
" 1 transform. The same techniques can be used to decrease the number of F
M by multiplexing pairs of such vectors and demultiplexing their transforms. Partial vectors with different values of " may be mixed in this way (Section 1.3.4.3.5.6). Once F
N is completed, its results have Hermitian symmetry with respect to h , and the methods of Section 1.3.4.3.5.1 may be used to obtain the unique electron densities. (ii) Transform on h ﬁrst. The partial vectors deﬁned by Xh ; h2 F
h 1 , h2 , h obey symmetry relations of the form

All the necessary ingredients are now available for calculating the CDFT for any given space group.

m 1 m 1 m1 ,

2

h2 tg

h1

When calculating electron densities, two methods may be used. (i) Transform on h ﬁrst. The partial vectors deﬁned by Xh ; h2 F
h , h1 , h2 obey symmetry relations of the form

1.3.4.3.6. Global crystallographic algorithms

m m m ,

2

h2 tg

F
h 1 , h2 , M
h1

X X1 !X2 ,

1 t
1 t
1 , g tg g

h

with its Friedel counterpart

with Friedel counterpart

tg t g tg ,

m2 , m2 tg
2 ;

F
h , h exp2i
h t g h tg F
h ,

The general multiplexing formula for pairs of vectors may therefore be written

N N N ,

m1

while for structure factors

X1 X X2 i X :

Rg R g Rg ,

N tg

m

1.3.4.3.6.3. Orthorhombic groups Almost all orthorhombic space groups are generated by two monoclinic transformations g1 and g2 of the type described in Section 1.3.4.3.6.2, with the addition of a centre of inversion e for centrosymmetric groups. The only exceptions are Fdd2 and Fddd

n n t
2 g 2 Z =2Z :

82

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY diagonal classes with residual threefold symmetry into a single class; see Section 1.3.4.3.5.6). More generally, factoring by q leads to a reduction from q3 to 13
q3 q q. Each of the remaining transforms then has a symmetry induced from the orthorhombic or tetragonal subgroup, which can be treated as above. No implementation of this procedure is yet available.

which contain diamond glides, in which some non-primitive translations are ‘square roots’ not of primitive lattice translations, but of centring translations. The generic case will be examined ﬁrst. To calculate electron densities, the unique octant of data may ﬁrst be transformed on h (respectively h ) as in Section 1.3.4.3.6.2 using the symmetry pertaining to generator g1 . These intermediate results may then be expanded by generator g2 by the formula of Section 1.3.4.3.3 prior to the ﬁnal transform on h (respectively h ). To calculate structure factors, the reverse operations are applied in the reverse order. The two exceptional groups Fdd2 and Fddd only require a small modiﬁcation. The F-centring causes the systematic absence of parity classes with mixed parities, leaving only (000) and (111). For the former, the phase factors exp2i
h t g h tg in the symmetry relations of Section 1.3.4.3.6.2 become powers of ( 1) so that one is back to the generic case. For the latter, these phase factors are odd powers of i which it is a simple matter to incorporate into a modiﬁed multiplexing/demultiplexing procedure.

1.3.4.3.6.6. Treatment of centred lattices Lattice centring is an instance of the duality between periodization and decimation: the extra translational periodicity of induces a decimation of F fFh g described by the ‘reﬂection conditions’ on h. As was pointed out in Section 1.3.4.2.2.3, nonprimitive lattices are introduced in order to retain the same matrix representation for a given geometric symmetry operation in all the arithmetic classes in which it occurs. From the computational point of view, therefore, the main advantage in using centred lattices is that it maximizes decomposability (Section 1.3.4.2.2.4); reindexing to a primitive lattice would for instance often destroy the diagonal character of the matrix representing a dyad. In the usual procedure involving three successive one-dimensional transforms, the loss of efﬁciency caused by the duplication of densities or the systematic vanishing of certain classes of structure factors may be avoided by using a multiplexing/demultiplexing technique (Ten Eyck, 1973): (i) for base-centred or body-centred lattices, two successive planes of structure factors may be overlaid into a single plane; after transformation, the results belonging to each plane may be separated by parity considerations; (ii) for face-centred lattices the same method applies, using four successive planes (the third and fourth with an origin translation); (iii) for rhombohedral lattices in hexagonal coordinates, three successive planes may be overlaid, and the results may be separated by linear combinations involving cube roots of unity. The three-dimensional factorization technique of Section 1.3.4.3.4.1 is particularly well suited to the treatment of centred lattices: if the decimation matrix of N contains as a factor N1 a matrix which ‘integerizes’ all the non-primitive lattice vectors, then centring is reﬂected by the systematic vanishing of certain classes of vectors of decimated data or results, which can simply be omitted from the calculation. An alternative possibly is to reindex on a primitive lattice and use different representative matrices for the symmetry operations: the loss of decomposability is of little consequence in this three-dimensional scheme, although it substantially complicates the deﬁnition of the cocycles 2 and 1 .

1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups All the symmetries in this class of groups can be handled by the generalized Rader/Winograd algorithms of Section 1.3.4.3.4.3, but no implementation of these is yet available. In groups containing axes of the form nm with g.c.d.
m, n 1
i:e: 31 , 32 , 41 , 43 , 61 , 65 along the c direction, the following procedure may be used (Ten Eyck, 1973): (i) to calculate electron densities, the unique structure factors indexed by unique
h, k
all l

are transformed on l; the results are rearranged by the transposition formula of Section 1.3.4.3.3 so as to be indexed by 1 all
h, k unique th of z n and are ﬁnally transformed on (h, k) to produce an asymmetric unit. For a dihedral group, the extra twofold axis may be used in the transposition to produce a unique
1=2nth of z. (ii) to calculate structure factors, the unique densities in
1=nth of z [or
1=2nth for a dihedral group] are ﬁrst transformed on x and y, then transposed by the formula of Section 1.3.4.3.3 to reindex the intermediate results by unique
h, k
all z;

1.3.4.3.6.7. Programming considerations The preceding sections have been devoted to showing how the raw computational efﬁciency of a crystallographic Fourier transform algorithm can be maximized. This section will brieﬂy discuss another characteristic (besides speed) which a crystallographic Fourier transform program may be required to possess if it is to be useful in various applications: a convenient and versatile mode of presentation of input data or output results. The standard crystallographic FFT programs (Ten Eyck, 1973, 1985) are rather rigid in this respect, and use rather rudimentary data structures (lists of structure-factor values, and two-dimensional arrays containing successive sections of electron-density maps). It is frequently the case that considerable reformatting of these data or results must be carried out before they can be used in other computations; for instance, maps have to be converted from 2D sections to 3D ‘bricks’ before they can be inspected on a computer graphics display. The explicitly three-dimensional approach to the factorization of the DFT and the use of symmetry offers the possibility of richer and more versatile data structures. For instance, the use of ‘decimation in frequency’ in real space and of ‘decimation in time’ in reciprocal

the last transform on z is then carried out.

1.3.4.3.6.5. Cubic groups These are usually treated as their orthorhombic or tetragonal subgroups, as the body-diagonal threefold axis cannot be handled by ordinary methods of decomposition. The three-dimensional factorization technique of Section 1.3.4.3.4.1 allows a complete treatment of cubic symmetry. Factoring by 2 along all three dimensions gives four types (i.e. orbits) of parity classes:
000 with residual threefold symmetry,
100,
010,
001 related by threefold axis,
110,
101,
011 related by threefold axis,
111

with residual threefold symmetry.

Orbit exchange using the threefold axis thus allows one to reduce the number of partial transforms from 8 to 4 (one per orbit). Factoring by 3 leads to a reduction from 27 to 11 (in this case, further reduction to 9 can be gained by multiplexing the three

83

1. GENERAL RELATIONSHIPS AND TECHNIQUES to six decimal places or better in most applications (see Gentleman & Sande, 1966).

space leads to data structures in which real-space coordinates are handled by blocks (thus preserving, at least locally, the threedimensional topological connectivity of the maps) while reciprocalspace indices are handled by parity classes or their generalizations for factors other than 2 (thus making the treatment of centred lattices extremely easy). This global three-dimensional indexing also makes it possible to carry symmetry and multiplicity characteristics for each subvector of intermediate results for the purpose of automating the use of the orbit exchange mechanism. Bru¨nger (1989) has described the use of a similar threedimensional factoring technique in the context of structure-factor calculations for the reﬁnement of macromolecular structures.

1.3.4.4.3. Fourier analysis of modified electron-density maps Various approaches to the phase problem are based on certain modiﬁcations of the electron-density map, followed by Fourier analysis of the modiﬁed map and extraction of phase information from the resulting Fourier coefﬁcients. 1.3.4.4.3.1. Squaring Sayre (1952a) derived his ‘squaring method equation’ for structures consisting of equal, resolved and spherically symmetric atoms by observing that squaring such an electron density is equivalent merely to sharpening each atom into its square. Thus P Fh h Fk Fh k ,

1.3.4.4. Basic crystallographic computations 1.3.4.4.1. Introduction Fourier transform (FT) calculations play an indispensable role in crystallography, because the Fourier transformation is inherent in the diffraction phenomenon itself. Besides this obligatory use, the FT has numerous other applications, motivated more often by its mathematical properties than by direct physical reasoning (although the latter can be supplied after the fact). Typically, many crystallographic computations turn out to be convolutions in disguise, which can be speeded up by orders of magnitude through a judicious use of the FT. Several recent advances in crystallographic computation have been based on this kind of observation.

k

sq

where h f
h=f
h is the ratio between the form factor f
h common to all the atoms and the form factor f sq
h for the squared version of that atom. Most of the central results of direct methods, such as the tangent formula, are an immediate consequence of Sayre’s equation. Phase reﬁnement for a macromolecule by enforcement of the squaring method equation was demonstrated by Sayre (1972, 1974).

1.3.4.4.3.2. Other non-linear operations A category of phase improvement procedures known as ‘density modiﬁcation’ is based on the pointwise application of various quadratic or cubic ‘ﬁlters’ to electron-density maps after removal of negative regions (Hoppe & Gassmann, 1968; Hoppe et al., 1970; Barrett & Zwick, 1971; Gassmann & Zechmeister, 1972; Collins, 1975; Collins et al., 1976; Gassmann, 1976). These operations are claimed to be equivalent to reciprocal-space phase-reﬁnement techniques such as those based on the tangent formula. Indeed the replacement of P
x Fh exp
2ih x

1.3.4.4.2. Fourier synthesis of electron-density maps Bragg (1929) was the ﬁrst to use this type of calculation to assist structure determination. Progress in computing techniques since that time was reviewed in Section 1.3.4.3.1. The usefulness of the maps thus obtained can be adversely affected by three main factors: (i) limited resolution; (ii) errors in the data; (iii) computational errors. Limited resolution causes ‘series-termination errors’ ﬁrst investigated by Bragg & West (1930), who used an optical analogy with the numerical aperture of a microscope. James (1948b) gave a quantitative description of this phenomenon as a convolution with the ‘spherical Dirichlet kernel’ (Section 1.3.4.2.1.3), which reﬂects the truncation of the Fourier spectrum by multiplication with the indicator function of the limiting resolution sphere. Bragg & West (1930) suggested that the resulting ripples might be diminished by applying an artiﬁcial temperature factor to the data, which performs a further convolution with a Gaussian point-spread function. When the electron-density map is to be used for model reﬁnement, van Reijen (1942) suggested using Fourier coefﬁcients calculated from the model when no observation is available, as a means of combating series-termination effects. Errors in the data introduce errors in the electron-density maps, with the same mean-square value by virtue of Parseval’s theorem. Special positions accrue larger errors (Cruickshank & Rollett, 1953; Cruickshank, 1965a). To minimize the mean-square electrondensity error due to large phase uncertainties, Blow & Crick (1959) introduced the ‘best Fourier’ which uses centroid Fourier coefﬁcients; the associated error level in the electron-density map was evaluated by Blow & Crick (1959) and Dickerson et al. (1961a,b). Computational errors used to be a serious concern when Beevers–Lipson strips were used, and Cochran (1948a) carried out a critical evaluation of the accuracy limitations imposed by strip methods. Nowadays, the FFT algorithm implemented on digital computers with a word size of at least 32 bits gives results accurate

h

by P
x, where P is a polynomial yields

P
a0 a1 a2 2 a3 3 . . .

P
x a0 a3

P h

P a 1 Fh a2 Fk Fh

PP k

k

k

l

Fk F l F h

k l

. . . exp
2ih x

and hence gives rise to the convolution-like families of terms encountered in direct methods. This equivalence, however, has been shown to be rather superﬁcial (Bricogne, 1982) because the ‘uncertainty principle’ embodied in Heisenberg’s inequality (Section 1.3.2.4.4.3) imposes severe limitations on the effectiveness of any procedure which operates pointwise in both real and reciprocal space. In applying such methods, sampling considerations must be given close attention. If the spectrum of extends to resolution and if the pointwise non-linear ﬁlter involves a polynomial P of degree n, then P() should be sampled at intervals of at most =2n to accommodate the full bandwidth of its spectrum. 1.3.4.4.3.3. Solvent flattening Crystals of proteins and nucleic acids contain large amounts of mother liquor, often in excess of 50% of the unit-cell volume,

84

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY by an indicator function U in real space, whereas they involve a convolution with F U in reciprocal space. The averaging by noncrystallographic symmetries of an electrondensity map calculated by FFT – hence sampled on a grid which is an integral subdivision of the period lattice – necessarily entails the interpolation of densities at non-integral points of that grid. The effect of interpolation on the structure factors recalculated from an averaged map was examined by Bricogne (1976). This study showed that, if linear interpolation is used, the initial map should be calculated on a ﬁne grid, of size /5 or /6 at resolution (instead of the previously used value of /3). The analysis about to be given applies to all interpolation schemes which consist in a convolution of the sampled density with a ﬁxed interpolation kernel function K. Let be a Z3 -periodic function. Let R K be the interpolation kernel in ‘normalized’ form, i.e. such that R3 K
x d3 x 1 and scaled so as to interpolate between sample values given on a unit grid Z3 ; in the case of linear interpolation, K is the ‘trilinear wedge’

occupying connected channels. The well ordered electron density M
x corresponding to the macromolecule thus occupies only a periodic subregion U of the crystal. Thus M U M ,

implying the convolution identity between structure factors (Main & Woolfson, 1963): X 1 FM
h F U
h kFM
k U k

which is a form of the Shannon interpolation formula (Sections 1.3.2.7.1, 1.3.4.2.1.7; Bricogne, 1974; Colman, 1974). It is often possible to obtain an approximate ‘molecular envelope’ U from a poor electron-density map , either interactively by computer graphics (Bricogne, 1976) or automatically by calculating a moving average of the electron density within a small sphere S. The latter procedure can be implemented in real space (Wang, 1985). However, as it is a convolution of with S , it can be speeded up considerably (Leslie, 1987) by computing the moving average mav as mav
x F F F S
x:

K
x W
xW
yW
z, where W
t 1 0

This remark is identical in substance to Booth’s method of computation of ‘bounded projections’ (Booth, 1945a) described in Section 1.3.4.2.1.8, except that the summation is kept threedimensional. The iterative use of the estimated envelope U for the purpose of phase improvement (Wang, 1985) is a submethod of the previously developed method of molecular averaging, which is described below. Sampling rules for the Fourier analysis of envelopetruncated maps will be given there.

jtj

if jtj 1, if jtj 1:

Let be sampled on a grid G1 N1 1 Z3 , and let IN1 denote the function interpolated from this sampled version of . Then: " # P IN1
N1 1 m
N1 1 # K, m2Z3

where
N1 1 # K
x K
N1 x, so that " # P
NT1 k1 F IN1 F jdet N1 j

1.3.4.4.3.4. Molecular averaging by noncrystallographic symmetries Macromolecules and macromolecular assemblies frequently crystallize with several identical subunits in the asymmetric metric unit, or in several crystal forms containing the same molecule in different arrangements. Rossmann & Blow (1963) recognized that intensity data collected from such structures are redundant (Sayre, 1952b) and that their redundancy could be a source of phase information. The phase constraints implied by the consistency of geometrically redundant intensities were ﬁrst derived by Rossmann & Blow (1963), and were generalized by Main & Rossmann (1966). Crowther (1967, 1969) reformulated them as linear eigenvalue equations between structure factors, for which he proposed an iterative matrix solution method. Although useful in practice (Jack, 1973), this reciprocal-space approach required computations of size / N 2 for N reﬂections, so that N could not exceed a few thousands. The theory was then reformulated in real space (Bricogne, 1974), showing that the most costly step in Crowther’s procedure could be carried out much more economically by averaging the electron densities of all crystallographically independent subunits, then rebuilding the crystal(s) from this averaged subunit, ﬂattening the density in the solvent region(s) by resetting it to its average value. This operation is a projection [by virtue of Section 1.3.4.2.2.2(d)]. The overall complexity was thus reduced from N 2 to N log N. The design and implementation of a general-purpose program package for averaging, reconstructing and solvent-ﬂattening electrondensity maps (Bricogne, 1976) led rapidly to the ﬁrst highresolution determinations of virus structures (Bloomer et al., 1978; Harrison et al., 1978), with N 200 000. The considerable gain in speed is a consequence of the fact that the masking operations used to retrieve the various copies of the common subunit are carried out by simple pointwise multiplication

k1 2Z3

1 T #
N F K jdet N1 j 1 " # P

NT1 k1 F
N1T # F K:

k1 2Z3

The transform of IN1 thus consists of (i) a ‘main band’ corresponding to k1 0, which consists of the true transform F
attenuated by multiplication by the central region of F K
N 1 T ; in the case of linear interpolation, for example, sin 2 sin 2 sin 2 ; F K
, ,

(ii) a series of ‘ghost bands’ corresponding to k1 6 0, which consist of translates of F multiplied by the tail regions of
N1T # F K. Thus IN1 is not band-limited even if is. Supposing, however, that is band-limited and that grid G1 satisﬁes the Shannon sampling criterion, we see that there will be no overlap between the different bands: F may therefore be recovered from the main band by compensating its attenuation, which is approximately a temperature-factor correction. For numerical work, however, IN1 must be resampled onto another grid G2 , which causes its transform to become periodized into (" # ) P P # T jdet N2 j

T

T F
N F K : k2 2Z3

85

N2 k2

k1 2Z3

N1 k1

1

1. GENERAL RELATIONSHIPS AND TECHNIQUES This now causes the main band k1 k2 0 to become contaminated by the ghost bands
k1 6 0 of the translates
k2 6 0 of IN1 . Aliasing errors may be minimized by increasing the sampling rate in grid G1 well beyond the Shannon minimum, which rapidly reduces the r.m.s. content of the ghost bands. The sampling rate in grid G2 needs only exceed the Shannon minimum to the extent required to accommodate the increase in bandwidth due to convolution with F U , which is the reciprocalspace counterpart of envelope truncation (or solvent ﬂattening) in real space.

agitation and their chemical identity (which can be used as a pointer to form-factor tables). Form factors are usually parameterized as sums of Gaussians, and thermal agitation by a Gaussian temperature factor or tensor. The formulae given in Section 1.3.4.2.2.6 for Gaussian atoms are therefore adequate for most purposes. Highresolution electron-density studies use more involved parameterizations. Early calculations were carried out by means of Bragg–Lipson charts (Bragg & Lipson, 1936) which gave a graphical representation of the symmetrized trigonometric sums of Section 1.3.4.2.2.9. The approximation of form factors by Gaussians goes back to the work of Vand et al. (1957) and Forsyth & Wells (1959). Agarwal (1978) gave simpliﬁed expansions suitable for mediumresolution modelling of macromolecular structures. This method of calculating structure factors is expensive because each atom sends contributions of essentially equal magnitude to all structure factors in a resolution shell. The calculation is therefore of size / NN for N atoms and N reﬂections. Since N and N are roughly proportional at a given resolution, this method is very costly for large structures. Two distinct programming strategies are available (Rollett, 1965) according to whether the fast loop is on all atoms for each reﬂection, or on all reﬂections for each atom. The former method was favoured in the early times when computers were unreliable. The latter was shown by Burnett & Nordman (1974) to be more amenable to efﬁcient programming, as no multiplication is required in calculating the arguments of the sine/cosine terms: these can be accumulated by integer addition, and used as subscripts in referencing a trigonometric function table.

1.3.4.4.3.5. Molecular-envelope transforms via Green’s theorem Green’s theorem stated in terms of distributions (Section 1.3.2.3.9.1) is particularly well suited to the calculation of the Fourier transforms F U of indicator functions. Let f be the indicator function U and let S be the boundary of U (assumed to be a smooth surface). The jump 0 in the value of f across S along the outer normal vector is 0 1, the jump in the normal derivative of f across S is 0, and the Laplacian of f as a function is (almost everywhere) 0 so that Tf 0. Green’s theorem then reads:
Tf Tf
S @ 0
S @
S :

The function eH
X exp
2iH X satisﬁes the identity eH 42 kHk2 eH . Therefore, in Cartesian coordinates: U
H hTU , eH i F

1

42 kHk 1 42 kHk 1

2

1.3.4.4.5. Structure factors via model electron-density maps

hTU , eH i

h
TU , eH i 2

Section 1:3:2:3:9:1
a

h @
S , eH i 42 kHk2 1 @ eH d2 S Section 1:3:2:3:9:1
c 42 kHk2 S 1 2iH n exp
2iH X d2 S, 2 2 4 kHk S

i.e. F U
H

1 2ikHk2

H n exp
2iH X d2 S,

S

where n is the outer normal to S. This formula was used by von Laue (1936) for a different purpose, namely to calculate the transforms of crystal shapes (see also Ewald, 1940). If the surface S is given by a triangulation, the surface integral becomes a sum over all faces, since n is constant on each face. If U is a solid sphere with radius R, an integration by parts gives immediately: 1 3 F U
H 3 sin X X cos X vol
U X with X 2kHkR: 1.3.4.4.4. Structure factors from model atomic parameters An atomic model of a crystal structure consists of a list of symmetry-unique atoms described by their positions, their thermal

86

Robertson (1936b) recognized the similarity between the calculation of structure factors by Fourier summation and the calculation of Fourier syntheses, the main difference being of course that atomic coordinates do not usually lie exactly on a grid obtained by integer subdivision of the crystal lattice. He proposed to address this difﬁculty by the use of his sorting board, which could extend the scale of subdivision and thus avoid phase errors. In this way the calculation of structure factors became amenable to Beevers–Lipson strip methods, with considerable gain of speed. Later, Beevers & Lipson (1952) proposed that trigonometric functions attached to atomic positions falling between the grid points on which Beevers–Lipson strips were based should be obtained by linear interpolation from the values found on the strips for the closest grid points. This amounts (Section 1.3.4.4.3.4) to using atoms in the shape of a trilinear wedge, whose form factor was indicated in Section 1.3.4.4.3.4 and gives rise to aliasing effects (see below) not considered by Beevers & Lipson. The correct formulation of this idea came with the work of Sayre (1951), who showed that structure factors could be calculated by Fourier analysis of a sampled electron-density map previously generated on a subdivision N 1 of the crystal lattice . When generating such a map, care must be taken to distribute onto the sample grid not only the electron densities of all the atoms in the asymmetric motif, but also those of their images under space-group symmetries and lattice translations. Considerable savings in computation occur, especially for large structures, because atoms are localized: each atom sends contributions to only a few grid points in real space, rather than to all reciprocal-lattice points. The generation of the sampled electron-density map is still of complexity / NN for N atoms and N reﬂections, but the proportionality constant is smaller than that in Section 1.3.4.4.4 by orders of magnitude; the extra cost of Fourier analysis, proportional to N log N , is negligible.

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY The idea of approximating a Fourier transform by a discrete transform on sampled values had already been used by Whittaker (1948), who tested it on the ﬁrst three odd Hermite functions and did not consider the problem of aliasing errors. By contrast, Sayre gave a lucid analysis of the sampling problems associated to this technique. If the periodic sampled map is written in the form of a weighted lattice distribution (as in Section 1.3.2.7.3) as P s
N 1 m
N 1 m ,

B

then its discrete Fourier transform yields P F s
h F
h NT 2Z3

so that each correct value F
h is corrupted by its aliases F
h NT for 6 0. To cure this aliasing problem, Sayre used ‘hypothetical atoms’ with form factors equal to those of standard atoms within the resolution range of interest, but set to zero outside that range. This amounts to using atomic densities with built-in series-termination errors, which has the detrimental effect of introducing slowly decaying ripples around the atom which require incrementing sample densities at many more grid points per atom. Sayre considered another cure in the form of an artiﬁcial temperature factor B (Bragg & West, 1930) applied to all atoms. This spreads each atom on more grid points in real space but speeds up the decay of its transform in reciprocal space, thus allowing the use of a coarser sampling grid in real space. He discounted it as spoiling the agreement with observed data, but Ten Eyck (1977) pointed out that this agreement could be restored by applying the negative of the artiﬁcial temperature factor to the results. This idea cannot be carried to extremes: if B is chosen too large, the atoms will be so spread out in real space as each to occupy a sizeable fraction of the unit cell and the advantage of atom localization will be lost; furthermore, the form factors will fall off so rapidly that round-off error ampliﬁcation will occur when the results are sharpened back. Clearly, there exists an optimal combination of B and sampling rate yielding the most economical computation for a given accuracy at a given resolution, and a formula will now be given to calculate it. Let us make the simplifying assumption that all atoms are roughly equal and that their common form factor can be represented by an equivalent temperature factor Beq . Let 1=dmax be the resolution to which structure factors are wanted. The Shannon . Let be the oversampling sampling interval is =2 1=2dmax rate, so that the actual sampling interval in the map is =2 1=2dmax : then consecutive copies of the transform are in reciprocal space. Let the artiﬁcial separated by a distance 2dmax temperature factor Bextra be added, and let

2 1
dmax

1.3.4.4.6. Derivatives for variational phasing techniques Some methods of phase determination rely on maximizing a certainR global criterion S involving the electron density, of the form R3 =Z3 K
x d3 x, under constraint of agreement with the observed structure-factor amplitudes, typically measured by a 2 residual C. Several recently proposed methods use for S various measures of entropy deﬁned by taking K
log
= or K
log (Bricogne, 1982; Britten & Collins, 1982; Narayan & Nityananda, 1982; Bryan et al., 1983; Wilkins et al., 1983; Bricogne, 1984; Navaza, 1985; Livesey & Skilling, 1985). Sayre’s use of the squaring method to improve protein phases (Sayre, 1974) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980). These methods differ from the density-modiﬁcation procedures of Section 1.3.4.4.3.2 in that they seek an optimal solution by moving electron densities (or structure factors) jointly rather than pointwise, i.e. by moving along suitably chosen search directions vi
x [or Vi
h]. For computational purposes, these search directions may be handled either as column vectors of sample values fvi
N 1 mgm2Z3 =NZ3 on a grid in real space, or as column vectors of Fourier coefﬁcients fVi
hgh2Z3 =NT Z3 in reciprocal space. These column vectors are the coordinates of the same vector Vi in an abstract vector space V L
Z3 =NZ3 of dimension N jdet Nj over R, but referred to two different bases which are related by the DFT and its inverse (Section 1.3.2.7.3). The problem of ﬁnding the optimum of S for a given value of C amounts to achieving collinearity between the gradients rS and rC of S and of C in V , the scalar ratio between them being a Lagrange multiplier. In order to move towards such a solution from a trial position, the dependence of rS and rC on position in V must be represented. This involves the N N Hessian matrices H(S) and H(C), whose size precludes their use in the whole of V . Restricting the search to a smaller search subspace of dimension n spanned by fVi gi1, ..., n we may build local quadratic models of S and C (Bryan & Skilling, 1980; Burch et al., 1983) with respect to n coordinates X in that subspace:

B Beq Bextra : , where The worst aliasing occurs at the outer resolution limit dmax the ‘signal’ due to an atom is proportional to 2 , exp
B=4
dmax

while the ‘noise’ due to the closest alias is proportional to

S
X S
X0 ST0
X

1dmax 2 g:

Thus the signal-to-noise ratio, or quality factor, Q is expB

deﬁnes B in terms of , dmax and Q. The overall cost of the structure-factor calculation from N atoms is then (i) C1 B2=3 N for density generation, (ii) C2
2dmax 3 log
2dmax 3 for Fourier analysis, where C1 and C2 are constant depending on the speed of the computer used. This overall cost may be minimized with respect to for given dmax and Q, determining the optimal B (and hence Bextra ) in passing by the above relation. Sayre (1951) did observe that applying an artiﬁcial temperature factor in real space would not create series-termination ripples: the resulting atoms would have a smaller effective radius than his hypothetical atoms, so that step (i) would be faster. This optimality of Gaussian smearing is ultimately a consequence of Hardy’s theorem (Section 1.3.2.4.4.3).

m2Z3

expf
B=4
2

log Q

C
X

1
dmax 2 :

If a certain value of Q is desired (e.g. Q 100 for 1% accuracy), then the equation

12
X

X0

X0 T H0
S
X

C
X0 CT0
X X0 12
X X0 T H0
C
X

X0 X0 :

The coefﬁcients of these linear models are given by scalar products:

87

1. GENERAL RELATIONSHIPS AND TECHNIQUES S0 i
Vi , rS

Ahp

C0 i
Vi , rC

@jFhcalc j @up

h jFhcalc j

H0
Sij Vi , H
SVj

H0
Cij Vi , H
CVj

jFh jobs

W diag
Wh with Wh

which, by virtue of Parseval’s theorem, may be evaluated either in real space or in reciprocal space (Bricogne, 1984). In doing so, special positions and reﬂections must be taken into account, as in Section 1.3.4.2.2.8. Scalar products involving S are best evaluated by real-space grid summation, because H(S) is diagonal in this representation; those involving C are best calculated by reciprocalspace summation, because H(C) is at worst 2 2 block-diagonal in this representation. Using these Hessian matrices in the wrong space would lead to prohibitively expensive convolutions instead of scalar (or at worst 2 2 matrix) multiplications.

1
2h obs

:

To calculate the elements of A, write: F jFj exp
i' i ; hence @jFj @ @ cos ' sin ' @u @u @u @F @F exp
i' exp
i' : @u @u

In the simple case of atoms with real-valued form factors and isotropic thermal agitation in space group P1, P Fhcalc gj
h exp
2ih xj ,

1.3.4.4.7. Derivatives for model refinement Since the origins of X-ray crystal structure analysis, the calculation of crystallographic Fourier series has been closely associated with the process of reﬁnement. Fourier coefﬁcients with phases were obtained for all or part of the measured reﬂections in the basis of some trial model for all or part of the structure, and Fourier syntheses were then used to complete and improve this initial model. This approach is clearly described in the classic paper by Bragg & West (1929), and was put into practice in the determination of the structures of topaz (Alston & West, 1929) and diopside (Warren & Bragg, 1929). Later, more systematic methods of arriving at a trial model were provided by the Patterson synthesis (Patterson, 1934, 1935a,b; Harker, 1936) and by isomorphous replacement (Robertson, 1935, 1936c). The role of Fourier syntheses, however, remained essentially unchanged [see Robertson (1937) for a review] until more systematic methods of structure reﬁnement were introduced in the 1940s. A particularly good account of the processes of structure completion and reﬁnement may be found in Chapters 15 and 16 of Stout & Jensen (1968). It is beyond the scope of this section to review the vast topic of reﬁnement methods: rather, it will give an account of those aspects of their development which have sought improved power by exploiting properties of the Fourier transformation. It is of more than historical interest that some recent advances in the crystallographic reﬁnement of macromolecular structures had been anticipated by Cochran and Cruickshank in the early 1950s.

j2J

where

gj
h Zj fj
h exp

2 1 4Bj
dh ,

Zj being a fractional occupancy. Positional derivatives with respect to xj are given by @Fhcalc
2ihgj
h exp
2ih xj @xj @jFhcalc j
2ihgj
h exp
2ih xj exp
i'calc h @xj so that the corresponding 3 1 subvector of the right-hand side of the normal equations reads: X @jFhcalc j calc Wh
jFh j jFh jobs @x j h2 " X gj
h
2ihWh
jFhcalc j jFh jobs h2

exp
i'calc h exp
2ih xj :

The setting up and solution of the normal equations lends itself well to computer programming and has the advantage of providing a thorough analysis of the accuracy of its results (Cruickshank, 1965b, 1970; Rollett, 1970). It is, however, an expensive task, of complexity / n j j2 , which is unaffordable for macromolecules.

1.3.4.4.7.1. The method of least squares Hughes (1941) was the ﬁrst to use the already well established multivariate least-squares method (Whittaker & Robinson, 1944) to reﬁne initial estimates of the parameters describing a model structure. The method gained general acceptance through the programming efforts of Friedlander et al. (1955), Sparks et al. (1956), Busing & Levy (1961), and others. The Fourier relations between and F (Section 1.3.4.2.2.6) are used to derive the ‘observational equations’ connecting the structure parameters fup gp1, ..., n to the observations fjFh jobs ,
2h obs gh2H comprising the amplitudes and their experimental variances for a set H of unique reﬂections. The normal equations giving the corrections u to the parameters are then

1.3.4.4.7.2. Booth’s differential Fourier syntheses It was the use of Fourier syntheses in the completion of trial structures which provided the incentive to ﬁnd methods for computing 2D and 3D syntheses efﬁciently, and led to the Beevers–Lipson strips. The limited accuracy of the latter caused the estimated positions of atoms (identiﬁed as peaks in the maps) to be somewhat in error. Methods were therefore sought to improve the accuracy with which the coordinates of the electron-density maxima could be determined. The naive method of peak-shape analysis from densities recalculated on a 3 3 3 grid using highaccuracy trigonometric tables entailed 27 summations per atom. Booth (1946a) suggested examining the rapidly varying derivatives of the electron density rather than its slowly varying values. If

AT WAu AT W, where

88

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY P
x Fh exp
2ih x coefﬁcients used in Booth’s differential syntheses on the other hand h (see also Booth, 1948a). In doing so he initiated a remarkable 0 sequence of formal and computational developments which are still then the gradient vector rx of at x actively pursued today. P
rx
x0 Fh
2ih exp
2ih x0 Let C
x be the electron-density map corresponding to the h current atomic model, with structure factors jFhcalc j exp
i'calc h ; and let
x be the map calculated from observed moduli and O can be calculated by means of three Fourier summations from the calculated phases, i.e. with coefﬁcients fjFh jobs exp
i'calc 3 1 vector of Fourier coefﬁcients h gh2H . If there are enough data for C to have a resolved peak at each
2ihFh : model atomic position xj , then Similarly, the Hessian matrix of at x0
rx C
xj 0 for each j 2 J; P T 0 2 T 0
rx rx
x Fh
4 hh exp
2ih x while if the calculated phases 'calc are good enough, O will also h h have peaks at each xj : can be calculated by six Fourier summations from the unique
rx O
xj 0 for each j 2 J : elements of the symmetric matrix of Fourier coefﬁcients: 0 2 1 It follows that h hk hl P [email protected] 4 hk k 2 kl AFh : rx
C O
xj
2ih
jFhcalc j jFh jobs exp
i'calc h h hl kl l2 exp
2ih xj The scalar maps giving the components of the gradient and Hessian matrix of will be called differential syntheses of 1st order 0 for each j 2 J, and 2nd order respectively. If x0 is approximately but not exactly a maximum of , then the Newton–Raphson estimate of the true where the summation is over all reﬂections in H or related to H by space-group and Friedel symmetry (overlooking multiplicity maximum x is given by: factors!). This relation is less sensitive to series-termination errors x x0
rx rTx
x0 1 rx
x0 : than either of the previous two, since the spectrum of O could have This calculation requires only nine accurate Fourier summations been extrapolated beyond the data in H by using that of C [as in (instead of 27), and this number is further reduced to four if the peak van Reijen (1942)] without changing its right-hand side. Cochran then used the identity is assumed to be spherically symmetrical. The resulting positions are affected by series-termination errors @Fhcalc
2ihgj
h exp
2ih xj in the differential syntheses. Booth (1945c, 1946c) proposed a @xj ‘back-shift correction’ to eliminate them, and extended this treatment to the acentric case (Booth, 1946b). He cautioned against in the form the use of an artiﬁcial temperature factor to ﬁght series-termination 1 @Fhcalc errors (Brill et al., 1939), as this could be shown to introduce
2ih exp
2ih xj coordinate errors by causing overlap between atoms (Booth, 1946c, gj
h @xj 1947a,b). Cruickshank was able to derive estimates for the standard to rewrite the previous relation as uncertainties of the atomic coordinates obtained in this way (Cox rx
C O
xj & Cruickshank, 1948; Cruickshank, 1949a,b) and to show that they " # calc agreed with those provided by the least-squares method. X 1 @F obs h
jFhcalc j jFh j e exp
i'calc The calculation of differential Fourier syntheses was incorpoh
h @x g j j h rated into the crystallographic programs of Ahmed & Cruickshank (1953b) and of Sparks et al. (1956). X 1 @jFhcalc j
jFhcalc j jFh jobs gj
h @xj h 1.3.4.4.7.3. Booth’s method of steepest descents Having deﬁned the now universally adopted R factors (Booth, 0 for each j 2 J 1945b) as criteria of agreement between observed and calculated amplitudes or intensities, Booth proposed that R should be (the operation [] on the ﬁrst line being neutral because of Friedel minimized with respect to the set of atomic coordinates fxj gj2J symmetry). This is equivalent to the vanishing of the 3 1 by descending along the gradient of R in parameter space (Booth, subvector of the right-hand side of the normal equations associated 1947c,d). This ‘steepest descents’ procedure was compared with to a least-squares reﬁnement in which the weights would be Patterson methods by Cochran (1948d). 1 When calculating the necessary derivatives, Booth (1948a, 1949) Wh : gj
h used the formulae given above in connection with least squares. This method was implemented by Qurashi (1949) and by Vand Cochran concluded that, for equal-atom structures with g
h j (1948, 1951) with parameter-rescaling modiﬁcations which made it g
h for all j, the positions x obtained by Booth’s method applied to j very close to the least-squares method (Cruickshank, 1950; Qurashi the difference map C are such that they minimize the residual O & Vand, 1953; Qurashi, 1953). 1X 1
jFhcalc j jFh jobs 2 1.3.4.4.7.4. Cochran’s Fourier method 2 h g
h Cochran (1948b,c, 1951a) undertook to exploit an algebraic similarity between the right-hand side of the normal equations in the with respect to the atomic positions. If it is desired to minimize the least-squares method on the one hand, and the expression for the residual of the ordinary least-squares method, then the differential

89

1. GENERAL RELATIONSHIPS AND TECHNIQUES Unlike Cochran’s original heuristic argument, this result does not depend on the atoms being resolved. Cruickshank (1952) also considered the elements of the normal matrix, of the form X @jF calc j @jF calc j h h wh @up @uq h

synthesis method should be applied to the weighted difference map X Wh
jFhcalc j jFh jobs exp
i'calc h : g
h h

He went on to show (Cochran, 1951b) that the reﬁnement of temperature factors could also be carried out by inspecting appropriate derivatives of the weighted difference map. This Fourier method was used by Freer et al. (1976) in conjunction with a stereochemical regularization procedure to reﬁne protein structures.

associated with positional parameters. The 3 3 block for parameters xj and xk may be written P wh
hhT
2igj
h exp
2ih xj exp
i'calc h h

1.3.4.4.7.5. Cruickshank’s modified Fourier method Cruickshank consolidated and extended Cochran’s derivations in a series of classic papers (Cruickshank, 1949b , 1950, 1952, 1956). He was able to show that all the coefﬁcients involved in the righthand side and normal matrix of the least-squares method could be calculated by means of suitable differential Fourier syntheses even when the atoms overlap. This remarkable achievement lay essentially dormant until its independent rediscovery by Agarwal in 1978 (Section 1.3.4.4.7.6). To ensure rigorous equivalence between the summations over h 2 H (in the expressions of least-squares right-hand side and normal matrix elements) and genuine Fourier summations, multiplicity-corrected weights were introduced by: 1 wh Wh if h 2 Gh with h 2 H , jGh j wh 0

2igk
h exp
2ih xk exp
i'calc h

which, using the identity
z1
z2 12
z1 z2
z1 z2 , becomes 22

h

fexp 2ih
xj

(Friedel’s symmetry makes redundant on the last line). Cruickshank argued that the ﬁrst term would give a good approximation to the diagonal blocks of the normal matrix and to those off-diagonal blocks for which xj and xk are close. On this basis he was able to justify the ‘n-shift rule’ of Shoemaker et al. (1950). Cruickshank gave this derivation in a general space group, but using a very terse notation which somewhat obscures it. Using the symmetrized trigonometric structure-factor kernel of Section 1.3.4.2.2.9 and its multiplication formula, the above expression is seen to involve the values of a Fourier synthesis at points of the form xj Sg
xk . Cruickshank (1956) showed that this analysis could also be applied to the reﬁnement of temperature factors. These two results made it possible to obtain all coefﬁcients involved in the normal equations by looking up the values of certain differential Fourier syntheses at xj or at xj Sg
xk . At the time this did not confer any superiority over the standard form of the leastsquares procedure, because the accurate computation of Fourier syntheses was an expensive operation. The modiﬁed Fourier method was used by Truter (1954) and by Ahmed & Cruickshank (1953a), and was incorporated into the program system described by Cruickshank et al. (1961). A more recent comparison with the least-squares method was made by Dietrich (1972). There persisted, however, some confusion about the nature of the relationship between Fourier and least-squares methods, caused by the extra factors gj
h which make it necessary to compute a differential synthesis for each type of atom. This led Cruickshank to conclude that ‘in spite of their remarkable similarities the leastsquares and modiﬁed-Fourier methods are fundamentally distinct’.

where Gh denotes the orbit of h and Gh its isotropy subgroup (Section 1.3.4.2.2.5). Similarly, derivatives with respect to parameters of symmetry-unique atoms were expressed, via the chain rule, as sums over the orbits of these atoms. Let p 1, . . . , n be the label of a parameter up belonging to atoms with label j. Then Cruickshank showed that the pth element of the right-hand side of the normal equations can be obtained as Dp; j
xj , where Dp; j is a differential synthesis of the form P Dp; j
x Pp
hgj
hwh
jFhcalc j jFh jobs h

exp
i'calc h exp
2ih x

with Pp
h a polynomial in (h, k, l) depending on the type of parameter p. The correspondence between parameter type and the associated polynomial extends Booth’s original range of differential syntheses, and is recapitulated in the following table. P
h, k, l

x coordinate

2ih

y coordinate

2ik

z coordinate

2il

B isotropic

1 2 4
dh 2

B11 anisotropic

h

B

12

anisotropic

hk

B

13

anisotropic

hl

B

22

anisotropic

k2

B23 anisotropic

kl

B33 anisotropic

l2 :

xk

exp
2i'calc h exp 2ih
xj xk g

otherwise,

Parameter type

P wh
hhT gj
hgk
h

90

1.3.4.4.7.6. Agarwal’s FFT implementation of the Fourier method Agarwal (1978) rederived and completed Cruickshank’s results at a time when the availability of the FFT algorithm made the Fourier method of calculating the coefﬁcients of the normal equations much more economical than the standard method, especially for macromolecules. As obtained by Cruickshank, the modiﬁed Fourier method required a full 3D Fourier synthesis – for each type of parameter, since this determines [via the polynomial Pp
h] the type of differential synthesis to be computed;

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY X X @R – for each type of atom j 2 J, since the coefﬁcients of the @R calc calc R Ah calc Bh Dh Fhcalc : differential synthesis must be multiplied by gj
h. calc @A @B h h h h Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a real-space convolution The operation is superﬂuous because of Friedel symmetry, so between the differential synthesis and j
x, the standard electron that R may be simply written in terms of the Hermitian scalar density j for atom type j (Section 1.3.4.2.1.2) smeared by the product in `2
Z3 : isotropic thermal agitation of that atom. Since j is localized, this convolution involves only a small number of grid points. The R
D, Fcalc : requirement of a distinct differential synthesis for each parameter type, however, continued to hold, and created some difﬁculties at If calc is the transform of Fcalc , we have also by Parseval’s theorem the FFT level because the symmetries of differential syntheses are R
D, calc : more complex than ordinary space-group symmetries. Jack & Levitt (1978) sought to avoid the calculation of difference syntheses by using instead ﬁnite differences calculated from ordinary Fourier or We may therefore write difference Fourier maps. @R D
x calc , In spite of its complication, this return to the Fourier @
x implementation of the least-squares method led to spectacular increases in speed (Isaacs & Agarwal, 1978; Agarwal, 1980; Baker which states that D
x is the functional derivative of R with respect & Dodson, 1980) and quickly gained general acceptance (Dodson, to calc . 1981; Isaacs, 1982a,b, 1984). The right-hand side of the normal equations has @[email protected] for its pth element, and this may be written Z 1.3.4.4.7.7. Lifchitz’s reformulation @R @R @calc
x 2 @calc : d x D, Lifchitz [see Agarwal et al. (1981), Agarwal (1981)] proposed calc
x @up @up @up R3 =Z3 @ that the idea of treating certain multipliers in Cruickshank’s modiﬁed differential Fourier syntheses by means of a convolution If up belongs to atom j, then in real space should be applied not only to gj
h, but also to the @j @calc @
xj j polynomials Pp
h which determine the type of differential ; xj synthesis being calculated. This leads to convoluting @j [email protected] @up @up @up with the same ordinary weighted difference Fourier synthesis, rather than j with the differential synthesis of type p. In this way, a single hence Fourier synthesis, with ordinary (scalar) symmetry properties, @j @R needs be computed; the parameter type and atom type both : D, xj intervene through the function @j [email protected] with which it is convoluted. @up @up This approach has been used as the basis of an efﬁcient generalpurpose least-squares reﬁnement program for macromolecular By the identity of Section 1.3.2.4.3.5, this is identical to Lifchitz’s structures (Tronrud et al., 1987). expression
D @j [email protected]
xj . The present derivation in terms of This rearrangement amounts to using the fact (Section scalar products [see Bru¨nger (1989) for another presentation of it] is 1.3.2.3.9.7) that convolution commutes with differentiation. Let conceptually simpler, since it invokes only the chain rule [other uses of which have been reviewed by Lunin (1985)] and Parseval’s P obs calc calc D
x wh
jFh j jFh j exp
i'h exp
2ih x theorem; economy of computation is obviously related to the good h localization of @calc [email protected] compared to @F calc [email protected] . Convolutions, be the inverse-variance weighted difference map, and let us assume whose meaning is less clear, are no longer involved; they were a that parameter up belongs to atom j. Then the Agarwal form for the legacy of having ﬁrst gone over to reciprocal space via differential syntheses in the 1940s. pth component of the right-hand side of the normal equations is Cast in this form, the calculation of derivatives by FFT methods @D appears as a particular instance of the procedure described in j
xj , connection with variational techniques (Section 1.3.4.4.6) to @up calculate the coefﬁcients of local quadratic models in a search while the Lifchitz form is subspace; this is far from surprising since varying the electron density through a variation of the parameters of an atomic model is @j a particular case of the ‘free’ variations considered by the
xj : D @up variational approach. The latter procedure would accommodate in a very natural fashion the joint consideration of an energetic (Jack & Levitt, 1978; Bru¨nger et al., 1987; Bru¨nger, 1988; Bru¨nger et al., 1989; Kuriyan et al., 1989) or stereochemical (Konnert, 1976; 1.3.4.4.7.8. A simplified derivation Sussman et al., 1977; Konnert & Hendrickson, 1980; Hendrickson A very simple derivation of the previous results will now be & Konnert, 1980; Tronrud et al., 1987) restraint function (which given, which suggests the possibility of many generalizations. The weighted difference map D
x has coefﬁcients Dh which are would play the role of S) and of the crystallographic residual (which would be C). It would even have over the latter the superiority of the gradients of the global residual with respect to each Fhcalc : affording a genuine second-order approximation, albeit only in a @R @R subspace, hence the ability of detecting negative curvature and the Dh calc i calc : resulting bifurcation behaviour (Bricogne, 1984). Current methods @Ah @Bh are unable to do this because they use only ﬁrst-order models, and By the chain rule, a variation of each Fhcalc by Fhcalc will result in a this is known to degrade severely the overall efﬁciency of the variation of R by R with reﬁnement process.

91

1. GENERAL RELATIONSHIPS AND TECHNIQUES Suppose that a crystal contains one or several copies of a molecule M in its asymmetric unit. If
x is the electron density of that molecule in some reference position and orientation, then " # P P # # 0 Sg
Tj ,

1.3.4.4.7.9. Discussion of macromolecular refinement techniques The impossibility of carrying out a full-matrix least-squares reﬁnement of a macromolecular crystal structure, caused by excessive computational cost and by the paucity of observations, led Diamond (1971) to propose a real-space reﬁnement method in which stereochemical knowledge was used to keep the number of free parameters to a minimum. Reﬁnement took place by a leastsquares ﬁt between the ‘observed’ electron-density map and a model density consisting of Gaussian atoms. This procedure, coupled to iterative recalculation of the phases, led to the ﬁrst highly reﬁned protein structures obtained without using full-matrix least squares (Huber et al., 1974; Bode & Schwager, 1975; Deisenhofer & Steigemann, 1975; Takano, 1977a,b). Real-space reﬁnement takes advantage of the localization of atoms (each parameter interacts only with the density near the atom to which it belongs) and gives the most immediate description of stereochemical constraints. A disadvantage is that ﬁtting the ‘observed’ electron density amounts to treating the phases of the structure factors as observed quantities, and to ignoring the experimental error estimates on their moduli. The method is also much more vulnerable to series-termination errors and accidentally missing data than the least-squares method. These objections led to the progressive disuse of Diamond’s method, and to a switch towards reciprocal-space least squares following Agarwal’s work. The connection established above between the Cruickshank– Agarwal modiﬁed Fourier method and the simple use of the chain rule affords a partial refutation to both the premises of Diamond’s method and to the objections made against it: (i) it shows that reﬁnement can be performed through localized computations in real space without having to treat the phases as observed quantities; (ii) at the same time, it shows that measurement errors on the moduli can be fully utilized in real space, via the Fourier synthesis of the functional derivative @[email protected]calc
x or by means of the coefﬁcients of a quadratic model of R in a search subspace.

j2J g2G

where Tj : x 7 ! Cj x dj describes the placement of the jth copy of the molecule with respect to the reference copy. It is assumed that each such copy is in a general position, so that there is no isotropy subgroup. The methods of Section 1.3.4.2.2.9 (with j replaced by Cj# , and xj by dj ) lead to the following expression for the autocorrelation of 0 : PPPP 0 0

Sg2
dj2 sg1
dj1 j1 j2 g1 g2

# #
R #
R # g1 Cj1 g2 Cj2 :

If is unknown, consider the subfamily of terms with j1 j2 j and g1 g2 g: PP # # R g Cj
: g

j

The scalar product
, R # in which R is a variable rotation will have a peak whenever R
R g1 Cj1 1
R g2 Cj2

since two copies of the ‘self-Patterson’ of the molecule will be brought into coincidence. If the interference from terms in the Patterson r 0 0 other than those present in is not too serious, the ‘self-rotation function’
, R # (Rossmann & Blow, 1962; Crowther, 1972) will show the same peaks, from which the rotations fCj gj2J may be determined, either individually or jointly if for instance they form a group. If is known, then its self-Patterson may be calculated, and the Cj may be found by examining the ‘cross-rotation function’ , R #
which will have peaks at R R g Cj , g 2 G, j 2 J. Once the Cj are known, then the various copies Cj# of M may be Fourier-analysed into structure factors:

1.3.4.4.7.10. Sampling considerations The calculation of the inner products
D, @calc [email protected] from a sampled gradient map D requires even more caution than that of structure factors via electron-density maps described in Section 1.3.4.4.5, because the functions @j [email protected] have transforms which extend even further in reciprocal space than the j themselves. Analytically, if the j are Gaussians, the @j [email protected] are ﬁnite sums of multivariate Hermite functions (Section 1.3.2.4.4.2) and hence the same is true of their transforms. The difference map D must therefore be ﬁnely sampled and the relation between error and sampling rate may be investigated as in Section 1.3.4.4.5. An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufﬁcient. Tronrud et al. (1987) propose to relax this requirement by applying an artiﬁcial temperature factor to j (cf. Section 1.3.4.4.5) and the negative of that temperature factor to D, a procedure of questionable validity because the latter ‘sharpening’ operation is ill deﬁned [the function exp
kxk2 does not deﬁne a tempered distribution, so the associativity properties of convolution may be lost]. A more robust procedure would be to compute the scalar product by means of a more sophisticated numerical quadrature formula than a mere grid sum.

Mj
h F Cj#
h: The cross terms with j1 6 j2 , g1 6 g2 in 0 0 then contain ‘motifs’ # #
R #
R # g1 Cj1 g2 Cj2 ,

with Fourier coefﬁcients Mj1
RTg1 h Mj2
RTg2 h, translated by Sg2
dj2 Sg1
dj1 . Therefore the ‘translation functions’ (Crowther & Blow, 1967) P T j1 g1 , j2 g2
s jFh j2 Mj1
RTg1 h h

1.3.4.4.8. Miscellaneous correlation functions

Mj2
RTg2 h exp
2ih s

Certain correlation functions can be useful to detect the presence of multiple copies of the same molecule (known or unknown) in the asymmetric unit of a crystal of unknown structure.

will have peaks at s Sg2
dj2 detection of these motifs.

92

Sg1
dj1 corresponding to the

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY 1.3.4.5. Related applications

called the Hankel transform (see e.g. Titchmarsh, 1922; Sneddon, 1972) of order n.

1.3.4.5.1. Helical diffraction The theory of diffraction by helical structures (Cochran et al., 1952; Klug et al., 1958) has played an important part in the study of polypeptides, of nucleic acids and of tobacco mosaic virus.

1.3.4.5.1.3. The transform of an axially periodic fibre Let be the electron-density distribution in a ﬁbre, which is assumed to have translational periodicity with period 1 along z, and to have compact support with respect to the (x, y) coordinates. Thus may be written " # P x y

k 0 ,

1.3.4.5.1.1. Circular harmonic expansions in polar coordinates Let f f
x, y be a reasonably regular function in twodimensional real space. Going over to polar coordinates

k2Z

x r cos ' y r sin '

0

where
x, y, z is the motif. By the tensor product property, the inverse Fourier transform F F xyz may be written " # P F 0
l F 1 1

and writing, by slight misuse of notation, f
r, ' for f
r cos ', r sin ' we may use the periodicity of f with respect to ' to expand it as a Fourier series (Byerly, 1893): P f
r, ' fn
r exp
in' n2Z

l2Z

with

l2Z

with

Similarly, in reciprocal space, if F F
, and if then

and hence consists of ‘layers’ labelled by l: P F F
, , l
l

1 R2 fn
r f
r, ' exp
in' d': 2 0 R cos

z

0

R1 F
, , l F xy 0
, , z exp
2ilz dz:

R sin

0

F
R, with Fn
R

P

n2Z

Changing to polar coordinates in the (x, y) and
, planes decomposes the calculation of F from into the following steps:

n

i Fn
R exp
in

1 R2R1
r, ', z expi
n' 2lz d' dz 2 0 0 R1 Gnl
R gnl
rJn
2Rr2r dr gnl
r

1 R2 F
R, exp
in d , 2in 0

0

where the phase factor in has been introduced for convenience in the forthcoming step.

F
R, , l

0

we obtain:

n2Z

in Gnl
R exp
in

and the calculation of from F into:

1.3.4.5.1.2. The Fourier transform in polar coordinates The Fourier transform relation between f and F may then be written in terms of fn ’s and Fn ’s. Observing that x y Rr cos
' , and that (Watson, 1944) R2

P

1 R2 F
R, , l exp
in d 2in 0 R1 gnl
r Gnl
RJn
2rR2R dR

Gnl
R

exp
iX cos in d 2in Jn
X ,

0

r, ', z

F
R,

R1 R2 P 0 0

n2Z

gnl
r expi
n'

2lz:

n2Z l2Z

These formulae are seen to involve a 2D Fourier series with respect to the two periodic coordinates ' and z, and Hankel transforms along the radial coordinates. The two periodicities in ' and z are independent, so that all combinations of indices (n, l) occur in the Fourier summations.

fn
r exp
in'

exp2iRr cos
' r dr d' " # P n R1 i fn
rJn
2Rr2r dr exp
in ; n2Z

PP

0

1.3.4.5.1.4. Helical symmetry and associated selection rules Helical symmetry involves a ‘clutching’ between the two (hitherto independent) periodicities in ' (period 2) and z (period 1) which causes a subdivision of the period lattice and hence a decimation (governed by ‘selection rules’) of the Fourier coefﬁcients. Let i and j be the basis vectors along '=2 and z. The integer lattice with basis (i, j) is a period lattice for the
', z dependence of the electron density of an axially periodic ﬁbre considered in Section 1.3.4.5.1.3:

hence, by the uniqueness of the Fourier expansion of F: R1 Fn
R fn
rJn
2Rr2r dr: 0

The inverse Fourier relationship leads to R1 fn
r Fn
RJn
2rR2R dR: 0

The integral transform involved in the previous two equations is

93

1. GENERAL RELATIONSHIPS AND TECHNIQUES
r, ' 2k1 , z k2
r, ', z: Suppose the ﬁbre now has helical symmetry, with u copies of the same molecule in t turns, where g.c.d.
u, t 1. Using the Euclidean algorithm, write u t with and positive integers and < t. The period lattice for the
', z dependence of may be deﬁned in terms of the new basis vectors: I, joining subunit 0 to subunit l in the same turn; J, joining subunit 0 to subunit after wrapping around. In terms of the original basis t 1 I i j, J i j: u u u u If and are coordinates along I and J, respectively, '=2 1 t 1 u z

properties which follow from the exchange between differentiation and multiplication by monomials. When the limit theorems are applied to the calculation of joint probability distributions of structure factors, which are themselves closely related to the Fourier transformation, a remarkable phenomenon occurs, which leads to the saddlepoint approximation and to the maximum-entropy method.

or equivalently

(a) Convolution of probability densities The addition of independent random variables or vectors leads to the convolution of their probability distributions: if X1 and X2 are two n-dimensional random vectors independently distributed with probability densities P1 and P2 , respectively, then their sum X X1 X2 has probability density P given by R P
X P1
X1 P2
X X1 dn X1

1.3.4.5.2.1. Analytical methods of probability theory The material in this section is not intended as an introduction to probability theory [for which the reader is referred to Crame´r (1946), Petrov (1975) or Bhattacharya & Rao (1976)], but only as an illustration of the role played by the Fourier transformation in certain speciﬁc areas which are used in formulating and implementing direct methods of phase determination.

'=2 : 1 t z By Fourier transformation, ' , z ,
n, l 2
, ,
m, p

Rn

with the transformations between indices given by the contragredients of those between coordinates, i.e. 1 m n t p l

i.e.

l tn um, l,

which are two equivalent expressions of the selection rules describing the decimation of the transform. These rules imply that only certain orders n contribute to a given layer l. The 2D Fourier analysis may now be performed by analysing a single subunit referred to coordinates and to obtain R1 R1 00

Rn

Then by the convolution theorem

C
t C1
t C2
t, so that P
X may be evaluated by Fourier inversion of its characteristic function as Z 1 P
X C1
tC2
t exp
it X dn t
2n

r, , exp2i
m p d d

and then reindexing to get only the allowed gnl ’s by gnl
r uh

X2 P2
X2 dn X2

(b) Characteristic functions This convolution can be turned into a simple multiplication by considering the Fourier transforms (called the characteristic functions) of P1 , P2 and P , deﬁned with a slightly different normalization in that there is no factor of 2 in the exponent (see Section 1.3.2.4.5), e.g. R C
t P
X exp
it X dn X:

It follows that

hm; p
r

P1
X

This result can be extended to the case where P1 and P2 are singular measures (distributions of order zero, Section 1.3.2.3.4) and do not have a density with respect to the Lebesgue measure in Rn .

1 m n t 1 : p l u

n up

Rn

P P 1 P2 :

and

or alternatively that

R

Rn

mp; mtp
r:

(see Section 1.3.2.4.5 for the normalization factors). It follows from the differentiation theorem that the partial derivatives of the characteristic function C
t at t 0 are related to the moments of a distribution P by the identities Z r1 r2 ...rn P
XX1r1 X2r2 . . . Xnrn dn X

This is u times faster than analysing u subunits with respect to the
', z coordinates. 1.3.4.5.2. Application to probability theory and direct methods The Fourier transformation plays a central role in the branch of probability theory concerned with the limiting behaviour of sums of large numbers of independent and identically distributed random variables or random vectors. This privileged role is a consequence of the convolution theorem and of the ‘moment-generating’

D

i

94

r1 ...rn

@ r1 ...rn C @t1r1 . . . @tnrn t0

for any n-tuple of non-negative integers
r1 , r2 , . . . , rn .

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY numerical evaluation as the discrete Fourier transform of M N
it. This exact method is practical only for small values of the dimension n. In all other cases some form of approximation must be used in the Fourier inversion of M N
it. For this purpose it is customary (Crame´r, 1946) to expand the cumulant-generating function around t 0 with respect to the carrying variables t:

(c) Moment-generating functions The above relation can be freed from powers of i by deﬁning (at least formally) the moment-generating function: R M
t P
X exp
t X dn X Rn

which is related to C
t by C
t M
it so that the inversion formula reads Z 1 M1
itM2
it exp
it X dn t: P
X
2n

logM N
it

Rn

hXi

M N
it exp

log M log M1 log M2 ,

exp

or equivalently of the coefﬁcients of their Taylor series at t 0, viz: @ r1 ...rn
log M : r1 r2 ...rn @t1r1 . . . @tnrn t0

exp

R

(e) Asymptotic expansions and limit theorems Consider an n-dimensional random vector X of the form X X1 X2 . . . XN ,

where the N summands are independent n-dimensional random vectors identically distributed with probability density P. Then the distribution P of X may be written in closed form as a Fourier transform: Z 1 P
X M N
it exp
it X dn t
2n Rn Z 1 expN log M
it it X dn t,
2n

1 T 2N t Qt

hXj i,

j1

1 U 2Nt Qt 8 < X N r

:jrj3 r!

9 =

itr , ;

monomial in t1 , t2 , . . . , tn ,

1 P
E p exp
det
2Q

1 1 T 2E Q E,

where E

X

hXi p : N

( f ) The saddlepoint approximation A limitation of the Edgeworth series is that it gives an accurate estimate of P
X only in the vicinity of X hXi, i.e. for small values of E. These convergence difﬁculties are easily understood: one is substituting a local approximation to log M (viz a Taylorseries expansion valid near t 0) into an integral, whereas integration is a global process which consults values of log M far from t 0. It is possible, however, to let the point t where log M is expanded as a Taylor series depend on the particular value X of X for which an accurate evaluation of P
X is desired. This is the essence of the saddlepoint method (Fowler, 1936; Khinchin 1949; Daniels, 1954; de Bruijn, 1970; Bleistein & Handelsman, 1986), which uses an analytical continuation of M
t from a function over Rn to a function over Cn (see Section 1.3.2.4.2.10). Putting then t s i, the Cn version of Cauchy’s theorem (Ho¨rmander, 1973) gives rise to the identity

Rn

where R

N P

each of which may now be subjected to a Fourier transformation to yield a Hermite function of t (Section 1.3.2.4.4.2) with coefﬁcients involving the cumulants of P. Taking the transformed terms in natural order gives an asymptotic expansion of P for large N called the Gram–Charlier series of P , while grouping the terms according p to increasing powers of 1= N gives another asymptotic expansion called the Edgeworth series of P . Both expansions comprise a leading Gaussian term which embodies the central-limit theorem:

n

n

itr ,

where Q rrT
log M is the covariance matrix of the multivariate distribution P. Expanding the exponential gives rise to a series of terms of the form

These coefﬁcients are called cumulants, since they add when the independent random vectors to which they belong are added, and log M is called the cumulant-generating function. The inversion formula for P then reads Z 1 P
X explog M1
it log M2
it it X dn t:
2n

R

r!

where hi denotes the mathematical expectation of a random vector. The second-order terms may be grouped separately from the terms of third or higher order to give

t0

(d) Cumulant-generating functions The multiplication of moment-generating functions may be further simpliﬁed into the addition of their logarithms:

M
t

r2N

n

where r
r1 , r2 , . . . , rn is a multi-index (Section 1.3.2.2.3). The ﬁrst-order terms may be eliminated by recentring P around its vector of ﬁrst-order cumulants

The moment-generating function is well deﬁned, in particular, for any probability distribution with compact support, in which case it may be continued analytically from a function over Rn into an entire function of n complex variables by virtue of the Paley–Wiener theorem (Section 1.3.2.4.2.10). Its moment-generating properties are summed up in the following relations: @ r1 ...rn M : r1 r2 ...rn r1 @t . . . @tnrn 1

X Nr

P
Y exp
t Y dn Y

is the moment-generating function common to all the summands. This an exact expression for P , which may be exploited analytically or numerically in certain favourable cases. Supposing for instance that P has compact support, then its characteristic function M
it can be sampled ﬁnely enough to accommodate the bandwidth of the support of P PN (this sampling rate clearly depends on n) so that the above expression for P can be used for its

95

1. GENERAL RELATIONSHIPS AND TECHNIQUES P
X

exp
X
2n exp N log M
is Rn

is

X N

maximization of certain entropy criteria. This connection exhibits most of the properties of the Fourier transform at play simultaneously, and will now be described as a ﬁnal illustration. dn s

(a) Deﬁnitions and conventions Let H be a set of unique non-origin reﬂections h for a crystal with lattice and space group G. Let H contain na acentric and nc centric reﬂections. Structure-factor values attached to all reﬂections in H will comprise n 2na nc real numbers. For h acentric, h and h will be the real and imaginary parts of the complex structure factor; for h centric, h will be the real coordinate of the (possibly complex) structure factor measured along a real axis rotated by one of the two angles h , apart, to which the phase is restricted modulo 2 (Section 1.3.4.2.2.5). These n real coordinates will be arranged as a column vector containing the acentric then the centric data, i.e. in the order

for any 2 Rn . By a convexity argument involving the positivedeﬁniteness of covariance matrix Q, there is a unique value of

such that X r
log Mjt0 i : N At the saddlepoint t 0 i , the modulus of the integrand above is a maximum and its phase is stationary with respect to the integration variable s: as N tends to inﬁnity, all contributions to the integral cancel because of rapid oscillation, except those coming from the immediate vicinity of t where there is no oscillation. A Taylor expansion of log M N to second order with respect to s at t then gives N T log M N
is log M N
is X s Qs 2 and hence Z 1 N explog M

X exp
12sT Qs dn s: P
X
2n

1 , 1 , 2 , 2 , . . . , na , na , 1 , 2 , . . . , nc : (b) Vectors of trigonometric structure-factor expressions Let
x denote the vector of trigonometric structure-factor expressions associated with x 2 D, where D denotes the asymmetric unit. These are deﬁned as follows: h
x i h
x
h, x

h
x exp
ih
h, x for h centric,

Rn

The last integral is elementary and gives the ‘saddlepoint approximation’:

where and where

where

exp
S P SP
X p , det
2Q

T

h, x

2r 1
x hr
x for r 1, . . . , na , 2r
x hr
x for r 1, . . . , na ,

N

Q rr
log M NQ: This approximation scheme amounts to using the ‘conjugate distribution’ (Khinchin, 1949) P
Xj P
Xj

1 X expf2ih Sg
xg: jGx j g2G

According to the convention above, the coordinates of
x in Rn will be arranged in a column vector as follows:

X

S log M N

for h acentric

na r
x hr
x for r na 1, . . . , na nc : (c) Distributions of random atoms and moment-generating functions Let position x in D now become a random vector with probability density m
x. Then
x becomes itself a random vector in Rn , whose distribution p
is the image of distribution m
x through the mapping x !
x just deﬁned. The locus of
x in Rn is a compact algebraic manifold L (the multidimensional analogue of a Lissajous curve), so that p is a singular measure (a distribution of order 0, Section 1.3.2.3.4, concentrated on that manifold) with compact support. The average with respect to p of any function

over Rn which is inﬁnitely differentiable in a neighbourhood of L may be calculated as an average with respect to m over D by the ‘induction formula’: R hp, i m
x
x d3 x:

exp
Xj M

instead of the original distribution P
Xj P 0
Xj for the common distribution of all N random vectors Xj . The exponential modulation results from the analytic continuation of the characteristic (or moment-generating) function into Cn , as in Section 1.3.2.4.2.10. The saddlepoint approximation P SP is only the leading term of an asymptotic expansion (called the saddlepoint expansion) for P , which is actually the Edgeworth expansion associated with PN

. 1.3.4.5.2.2. The statistical theory of phase determination The methods of probability theory just surveyed were applied to various problems formally similar to the crystallographic phase problem [e.g. the ‘problem of the random walk’ of Pearson (1905)] by Rayleigh (1880, 1899, 1905, 1918, 1919) and Kluyver (1906). They became the basis of the statistical theory of communication with the classic papers of Rice (1944, 1945). The Gram–Charlier and Edgeworth series were introduced into crystallography by Bertaut (1955a,b,c, 1956a) and by Klug (1958), respectively, who showed them to constitute the mathematical basis of numerous formulae derived by Hauptman & Karle (1953). The saddlepoint approximation was introduced by Bricogne (1984) and was shown to be related to variational methods involving the

D

In particular, one can calculate the moment-generating function M for distribution p as R M
t hp , exp
t i m
x expt
x d3 x D

96

and hence calculate the moments (respectively cumulants ) of p by differentiation of M (respectively log M) at t 0:

r1 r2 ...rn

1.3. FOURIER TRANSFORMS IN CRYSTALLOGRAPHY the modiﬁed distribution of atoms

m
x r11
x r22
x . . . rnn
x d3 x

q
x m
x

D

r1 r2 ...rn

@ r1 ...rn
M r1 @t1 . . . @tnrn @ r1 ...rn
log M : @t1r1 . . . @tnrn

where, by the induction formula, M
may be written as R M
m
x exp
x d3 x D

r
log M N F :

where

SP3

exp
S P SP
F p , det
2Q S log M N

F

and where Q rrT
log M N NQ:

Finally, the elements of the Hessian matrix Q rrT
log M are just the trigonometric second-order cumulants of distribution p, and hence can be calculated via structure-factor algebra from the Fourier coefﬁcients of q
x. All the quantities involved in the expression for P SP
F are therefore effectively computable from the initial data m
x and F .

I ,

I1

where the N copies I of random vector are independent and have the same distribution p
. The joint probability distribution P
F is then [Section 1.3.4.5.2.1(e)] Z 1 expN log M
it it X dn t: P
X
2n

(e) Maximum-entropy distributions of atoms One of the main results in Bricogne (1984) is that the modiﬁed distribution q
x in (SP1) is the unique distribution which has maximum entropy S m
q relative to m
x, where Z q
x 3 S m
q q
x log d x, m
x

Rn

For low dimensionality n it is possible to carry out the Fourier transformation numerically after discretization, provided M
it is sampled sufﬁciently ﬁnely that no aliasing results from taking its Nth power (Barakat, 1974). This exact approach can also accommodate heterogeneity, and has been used ﬁrst in the ﬁeld of intensity statistics (Shmueli et al., 1984, 1985; Shmueli & Weiss, 1987, 1988), then in the study of the 1 and 2 relations in triclinic space groups (Shmueli & Weiss, 1985, 1986). Some of these applications are described in Chapter 2.1 of this volume. This method could be extended to the construction of any joint probability distribution (j.p.d.) in any space group by using the generic expression for the moment-generating function (m.g.f.) derived by Bricogne (1984). It is, however, limited to small values of n by the necessity to carry out n-dimensional FFTs on large arrays of sample values. The asymptotic expansions of Gram–Charlier and Edgeworth have good convergence properties only if Fh lies in the vicinity of hFh i N F m
h for all h 2 H. Previous work on the j.p.d. of structure factors has used for m
x a uniform distribution, so that hFi 0; as a result, the corresponding expansions are accurate only if all moduli jFh j are small, in which case the j.p.d. contains little phase information. The saddlepoint method [Section 1.3.4.5.2.1( f )] constitutes the method of choice for evaluating the joint probability P
F of structure factors when some of the moduli in F are large. As shown previously, this approximation amounts to using the ‘conjugate distribution’ p
p

SP2

The desired approximation is then

(d) The joint probability distribution of structure factors In the random-atom model of an equal-atom structure, N atoms are placed randomly, independently of each other, in the asymmetric unit D of the crystal with probability density m
x. For point atoms of unit weight, the vector F of structure-factor values for reﬂections h 2 H may be written F

SP1

and where is the unique solution of the saddlepoint equation:

The structure-factor algebra for group G (Section 1.3.4.2.2.9) then allows one to express products of ’s as linear combinations of other ’s, and hence to express all moments and cumulants of distribution p
as linear combinations of real and imaginary parts of Fourier coefﬁcients of the prior distribution of atoms m
x. This plays a key role in the use of non-uniform distributions of atoms.

N P

exp
x , M

D

under the constraint that F be the centroid vector of the corresponding conjugate distribution P
F. The traditional notation of maximum-entropy (ME) theory (Jaynes, 1957, 1968, 1983) is in this case (Bricogne, 1984) exp
x qME
x m
x Z
R Z
m
x exp
x d3 x D

r
log Z N F

ME1
ME2
ME3

so that Z is identical to the m.g.f. M, and the coordinates of the saddlepoint are the Lagrange multipliers for the constraints F . Jaynes’s ME theory also gives an estimate for P
F : P ME
F exp
S ,

where S log Z N

F NS m
qME

is the total entropy and is the counterpart to S under the equivalence just established. P ME is identical to P SP , but lacks the denominator. The latter, which is the normalization factor of a multivariate Gaussian with covariance matrix Q, may easily be seen to arise through Szego¨’s theorem (Sections 1.3.2.6.9.4, 1.3.4.2.1.10) from the extra logarithmic term in Stirling’s formula

exp
M

instead of the original distribution p
p0
for the distribution of random vector . This conjugate distribution p is induced from

log
q! q log q

97

q 12 log
2q

1. GENERAL RELATIONSHIPS AND TECHNIQUES distributions with compact support, and thus gives rise to conjugate families of distributions; (v) Bertaut’s structure-factor algebra (a discrete symmetrized version of the convolution theorem), which allows the calculation of all necessary moments and cumulants when the dimension n is small; (vi) Szego¨’s theorem, which provides an asymptotic approximation of the normalization factor when n is large. This multi-faceted application seems an appropriate point at which to end this description of the Fourier transformation and of its use in crystallography.

(see, for instance, Reif, 1965) beyond the ﬁrst two terms which serve to deﬁne entropy, since 1 log 2qME
x d3 x: log det
2Q n R3 =Z3

The relative effect of this extra normalization factor depends on the ratio n dimension of F over R : N number of atoms The above relation between entropy maximization and the saddlepoint approximation is the basis of a Bayesian statistical approach to the phase problem (Bricogne, 1988) where the assumptions under which joint distributions of structure factors are sought incorporate many new ingredients (such as molecular boundaries, isomorphous substitutions, known fragments, noncrystallographic symmetries, multiple crystal forms) besides trial phase choices for basis reﬂections. The ME criterion intervenes in the construction of qME
x under these assumptions, and the distribution qME
x is a very useful computational intermediate in obtaining the approximate joint probability P SP
F and the associated conditional distributions and likelihood functions.

Acknowledgements Many aspects of the theory of discrete Fourier transform algorithms and of its extension to incorporate crystallographic symmetry have been the focus of a long-standing collaborative effort between Professor Louis Auslander, Professor Richard Tolimieri, their coworkers and the writer. I am most grateful to them for many years of mathematical stimulation and enjoyment, for introducing me to the ‘big picture’ of the discrete Fourier transform which they have elaborated over the past decade, and for letting me describe here some of their unpublished work. In particular, the crystallographic extensions of the Rader/Winograd algorithms presented in Section 1.3.4.3.4.3 were obtained by Richard Tolimieri, in a collaboration partially supported by NIH grant GM 32362 (to the writer). I am indebted to the Editor for many useful and constructive suggestions of possible improvements to the text, only a few of which I have been able to implement. I hope to incorporate many more of them in the future. I also wish to thank Dr D. Sayre for many useful comments on an early draft of the manuscript. This contribution was written during the tenure of a Visiting Fellowship at Trinity College, Cambridge, with partial ﬁnancial support from Trinity College and the MRC Laboratory of Molecular Biology. I am most grateful to both institutions for providing ideal working conditions.

( f ) Role of the Fourier transformation The formal developments presented above make use of the following properties of the Fourier transformation: (i) the convolution theorem, which turns the convolution of probability distributions into the multiplication of their characteristic functions; (ii) the differentiation property, which confers moment-generating properties to characteristic functions; (iii) the reciprocity theorem, which allows the retrieval of a probability distribution from its characteristic or moment-generating function; (iv) the Paley–Wiener theorem, which allows the analytic continuation of characteristic functions associated to probability

98

International Tables for Crystallography (2006). Vol. B, Chapter 1.4, pp. 99–161.

1.4. Symmetry in reciprocal space BY U. SHMUELI WITH

APPENDIX 1.4.2

BY

U. SHMUELI, S. R. HALL AND R. W. GROSSE-KUNSTLEVE

including the set of symbols that were used in the preparation of the present tables.

1.4.1. Introduction Crystallographic symmetry, as reﬂected in functions on reciprocal space, can be considered from two complementary points of view. (1) One can assume the existence of a certain permissible symmetry of the density function of crystalline (scattering) matter, a function which due to its three-dimensional periodicity can be expanded in a triple Fourier series (e.g. Bragg, 1966), and inquire about the effects of this symmetry on the Fourier coefﬁcients – the structure factors. Since there exists a one-to-one correspondence between the triplets of summation indices in the Fourier expansion and vectors in the reciprocal lattice (Ewald, 1921), the above approach leads to consequences of the symmetry of the density function which are relevant to the representation of its Fourier image in reciprocal space. The symmetry properties of these Fourier coefﬁcients, which are closely related to the crystallographic experiment, can then be readily established. This traditional approach, the essentials of which are the basis of Sections 4.5–4.7 of Volume I (IT I, 1952), and which was further developed in the works of Buerger (1949, 1960), Waser (1955), Bertaut (1964) and Wells (1965), is one of the cornerstones of crystallographic practice and will be followed in the present chapter, as far as the basic principles are concerned. (2) The alternative approach, proposed by Bienenstock & Ewald (1962), also presumes a periodic density function in crystal space and its Fourier expansion associated with the reciprocal. However, the argument starts from the Fourier coefﬁcients, taken as a discrete set of complex functions, and linear transformations are sought which leave the magnitudes of these functions unchanged; the variables on which these transformations operate are h, k, l and – the Fourier summation indices (i.e., components of a reciprocallattice vector) and the phase of the Fourier coefﬁcient, respectively. These transformations, or the groups they constitute, are then interpreted in terms of the symmetry of the density function in direct space. This direct analysis of symmetry in reciprocal space will also be discussed. We start the next section with a brief discussion of the pointgroup symmetries of associated direct and reciprocal lattices. The weighted reciprocal lattice is then brieﬂy introduced and the relation between the values of the weight function at symmetry-related points of the weighted reciprocal lattice is discussed in terms of the Fourier expansion of a periodic function in crystal space. The remaining part of Section 1.4.2 is devoted to the formulation of the Fourier series and its coefﬁcients (values of the weight function) in terms of space-group-speciﬁc symmetry factors, an extensive tabulation of which is presented in Appendix 1.4.3. This is a revised version of the structure-factor tables given in Sections 4.5– 4.7 of Volume I (IT I, 1952). Appendix 1.4.4 contains a reciprocalspace representation of the 230 crystallographic space groups and some explanatory material related to these space-group tables is given in Section 1.4.4; the latter are interpreted in terms of the two viewpoints discussed above. The tabular material given in this chapter is compatible with the direct-space symmetry tables given in Volume A (IT A, 1983) with regard to the space-group settings and choices of the origin. Most of the tabular material, the new symmetry-factor tables in Appendix 1.4.3 and the space-group tables in Appendix 1.4.4 have been generated by computer with the aid of a combination of numeric and symbolic programming techniques. The algorithm underlying this procedure is brieﬂy summarized in Appendix 1.4.1. Appendix 1.4.2 deals with computer-adapted space-group symbols,

1.4.2. Effects of symmetry on the Fourier image of the crystal 1.4.2.1. Point-group symmetry of the reciprocal lattice Regarding the reciprocal lattice as a collection of points generated from a given direct lattice, it is fairly easy to see that each of the two associated lattices must have the same point-group symmetry. The set of all the rotations that bring the direct lattice into self-coincidence can be thought of as interchanging equivalent families of lattice planes in all the permissible manners. A family of lattice planes in the direct lattice is characterized by a common normal and a certain interplanar distance, and these two characteristics uniquely deﬁne the direction and magnitude, respectively, of a vector in the reciprocal lattice, as well as the lattice line associated with this vector and passing through the origin. It follows that any symmetry operation on the direct lattice must also bring the reciprocal lattice into self-coincidence, i.e. it must also be a symmetry operation on the reciprocal lattice. The roles of direct and reciprocal lattices in the above argument can of course be interchanged without affecting the conclusion. The above elementary considerations recall that for any point group (not necessarily the full point group of a lattice), the operations which leave the lattice unchanged must also leave unchanged its associated reciprocal. This equivalence of pointgroup symmetries of the associated direct and reciprocal lattices is fundamental to crystallographic symmetry in reciprocal space, in both points of view mentioned in Section 1.4.1. With regard to the effect of any given point-group operation on each of the two associated lattices, we recall that: (i) If P is a point-group rotation operator acting on the direct lattice (e.g. by rotation through the angle about a given axis), the effect of this rotation on the associated reciprocal lattice is that of applying the inverse rotation operator, P 1 (i.e. rotation through about a direction parallel to the direct axis); this is readily found from the requirement that the scalar product hT rL , where h and rL are vectors in the reciprocal and direct lattices, respectively, remains invariant under the application of a point-group operation to the crystal. (ii) If our matrix representation of the rotation operator is such that the point-group operation is applied to the direct-lattice (column) vector by premultiplying it with the matrix P, the corresponding operation on the reciprocal lattice is applied by postmultiplying the (row) vector hT with the point-group rotation matrix. We can thus write, e.g., hT rL hT P 1 PrL P 1 T hT PrL . Note, however, that the orthogonality relationship: P 1 P T is not satisﬁed if P is referred to some oblique crystal systems, higher than the orthorhombic. Detailed descriptions of the 32 crystallographic point groups are presented in the crystallographic and other literature; their complete tabulation is given in Chapter 10 of Volume A (IT A, 1983). 1.4.2.2. Relationship between structure factors at symmetryrelated points of the reciprocal lattice Of main interest in the context of the present chapter are symmetry relationships that concern the values of a function deﬁned at the points of the reciprocal lattice. Such functions, of crystal-

99 Copyright 2006 International Union of Crystallography

1. GENERAL RELATIONSHIPS AND TECHNIQUES lographic interest, are Fourier-transform representations of directspace functions that have the periodicity of the crystal, the structure factor as a Fourier transform of the electron-density function being a representative example (see e.g. Lipson & Taylor, 1958). The value of such a function, attached to a reciprocal-lattice point, is called the weight of this point and the set of all such weighted points is often termed the weighted reciprocal lattice. This section deals with a fundamental relationship between functions (weights) associated with reciprocal-lattice points, which are related by point-group symmetry, the weights here considered being the structure factors of Bragg reﬂections (cf. Chapter 1.2). The electron density, an example of a three-dimensional periodic function with the periodicity of the crystal, can be represented by the Fourier series 1X
r Fh exp2ihT r, 1421 V h where h is a reciprocal-lattice vector, V is the volume of the (direct) unit cell, Fh is the structure factor at the point h and r is a position vector of a point in direct space, at which the density is given. The summation in (1.4.2.1) extends over all the reciprocal lattice. Let r Pr t be a space-group operation on the crystal, where P and t are its rotation and translation parts, respectively, and P must therefore be a point-group operator. We then have, by deﬁnition, r Pr t and the Fourier representation of the electron density, at the equivalent position Pr t, is given by 1X Fh exp2ihT Pr t Pr t V h 1X Fh exp2ihT t V h exp2iP T hT r,

1422

noting that hT P P T hT . Since P is a point-group operator, the vectors PT h in (1.4.2.2) must range over all the reciprocal lattice and a comparison of the functional forms of the equivalent expansions (1.4.2.1) and (1.4.2.2) shows that the coefﬁcients of the exponentials exp2iP T hT r in (1.4.2.2) must be the structure factors at the points P T h in the reciprocal lattice. Thus FP T h Fh exp2ihT t,

1423

wherefrom it follows that the magnitudes of the structure factors at h and P T h are the same:

According to equation (1.4.2.5), the phases of the structure factors of symmetry-related reﬂections differ, in the general case, by a phase shift that depends on the translation part of the spacegroup operation involved. Only when the space group is symmorphic, i.e. it contains no translations other than those of the Bravais lattice, will the distribution of the phases obey the pointgroup symmetry of the crystal. These phase shifts are considered in detail in Section 1.4.4 where their tabulation is presented and the alternative interpretation (Bienenstock & Ewald, 1962) of symmetry in reciprocal space, mentioned in Section 1.4.1, is given. Equation (1.4.2.3) can be usefully applied to a classiﬁcation of all the general systematic absences or – as deﬁned in the space-group tables in the main editions of IT (1935, 1952, 1983, 1987, 1992) – general conditions for possible reﬂections. These systematic absences are associated with special positions in the reciprocal lattice – special with respect to the point-group operations P appearing in the relevant relationships. If, in a given relationship, we have P T h h, equation (1.4.2.3) reduces to Fh Fh exp2ihT t

1426

Of course, Fh may then be nonzero only if cos2hT t equals unity, or the scalar product hT t is an integer. This well known result leads to a ready determination of lattice absences, as well as those produced by screw-axis and glide-plane translations, and is routinely employed in crystallographic computing. An exhaustive classiﬁcation of the general conditions for possible reﬂections is given in the space-group tables (IT, 1952, 1983). It should be noted that since the axes of rotation and planes of reﬂection in the reciprocal lattice are parallel to the corresponding elements in the direct lattice (Buerger, 1960), the component of t that depends on the location of the corresponding space-group symmetry element in direct space does not contribute to the scalar product hT t in (1.4.2.6), and it is only the intrinsic part of the translation t (IT A, 1983) that usually matters. It may, however, be of interest to note that some screw axes in direct space cannot give rise to any systematic absences. For example, the general Wyckoff position No. (10) in the space group Pa3 (No. 205) (IT A, 1983) has the coordinates y, 12 z, 12 x, and corresponds to the space-group operation 1 0 1 0 13 20 1 13 0 1 0 3 P, t P, ti tl [email protected] 0 0 1 A, @ 13 A @ 16 A5, 1 1 1 0 0 3 6

1424

1427

1425 P h h 2h t The relationship (1.4.2.3) between structure factors of symmetryrelated reﬂections was ﬁrst derived by Waser (1955), starting from a representation of the structure factor as a Fourier transform of the electron-density function. It follows that an application of a point-group transformation to the (weighted) reciprocal lattice leaves the moduli of the structure factors unchanged. The distribution of diffracted intensities obeys, in fact, the same point-group symmetry as that of the crystal. If, however, anomalous dispersion is negligibly small, and the point group of the crystal is noncentrosymmetric, the apparent symmetry of the diffraction pattern will also contain a false centre of symmetry and, of course, all the additional elements generated by the inclusion of this centre. Under these circumstances, the diffraction pattern from a single crystal may belong to one of the eleven centrosymmetric point groups, known as Laue groups (IT I, 1952).

where ti and tl are the intrinsic and location-dependent components of the translation part t, and are parallel and perpendicular, respectively, to the threefold axis of rotation represented by the matrix P in (1.4.2.7) (IT A, 1983; Shmueli, 1984). This is clearly a threefold screw axis, parallel to 111. The reciprocal-lattice vectors which remain unchanged, when postmultiplied by P (or premultiplied by its transpose), have the form: hT hhh; this is the special position for the present example. We see that (i) hT tl 0, as expected, and (ii) hT ti h. Since the scalar product hT t is an integer, there are no values of index h for which the structure factor Fhhh must be absent. Other approaches to systematically absent reﬂections include a direct inspection of the structure-factor equation (Lipson & Cochran, 1966), which is of considerable didactical value, and the utilization of transformation properties of direct and reciprocal base vectors and lattice-point coordinates (Buerger, 1942). Finally, the relationship between the phases of symmetry-related reﬂections, given by (1.4.2.5), is of fundamental as well as practical importance in the theories and techniques of crystal structure

FPT h Fh , and their phases are related by T

T

100

1.4. SYMMETRY IN RECIPROCAL SPACE determination which operate in reciprocal space (Part 2 of this volume). 1.4.2.3. Symmetry factors for space-group-specific Fourier summations The weighted reciprocal lattice, with weights taken as the structure factors, is synonymous with the discrete space of the coefﬁcients of a Fourier expansion of the electron density, or the Fourier space (F space) of the latter. Accordingly, the asymmetric unit of the Fourier space can be deﬁned as the subset of structure factors within which the relationship (1.4.2.3) does not hold – except at special positions in the reciprocal lattice. If the point group of the crystal is of order g, this is also the order of the corresponding factor-group representation of the space group (IT A, 1983) and there exist g relationships of the form of (1.4.2.3): F
P Ts h Fh exp2ihT ts

1428

We can thus decompose the summation in (1.4.2.1) into g sums, each extending over an asymmetric unit of the F space. It must be kept in mind, however, that some classes of reciprocal-lattice vectors may be common to more than one asymmetric unit, and thus each reciprocal-lattice point will be assigned an occupancy factor, denoted by qh, such that qh 1 for a general position and qh 1mh for a special one, where mh is the multiplicity – or the order of the point group that leaves h unchanged. Equation (1.4.2.1) can now be rewritten as g 1 XX r qha FPTs ha exp2iPTs ha T r, 1429 V s1 ha where the inner summation in (1.4.2.9) extends over the reference asymmetric unit of the Fourier space, which is associated with the identity operation of the space group. Substituting from (1.4.2.8) for FP Ts ha , and interchanging the order of the summations in (1.4.2.9), we obtain g X 1X qha Fha exp2ihTs Ps r ts 14210 r V ha s1 1X qha Fha Aha iBha , 14211 V ha where Ah

g P

cos2hT P s r ts

14212

metric case, when the space-group origin is chosen at a centre of symmetry, and in the noncentrosymmetric case, when dispersion is neglected. In each of the latter two cases the summation over ha is restricted to reciprocal-lattice vectors that are not related by real or apparent inversion (denoted by ha 0), and we obtain 2X r qha Fha Aha 14214 V h 0 a

and r

2X qha Fha Aha cos ha V h 0 a

Bha sin ha

for the dispersionless centrosymmetric and noncentrosymmetric cases, respectively. 1.4.2.4. Symmetry factors for space-group-specific structurefactor formulae The explicit dependence of structure-factor summations on the space-group symmetry of the crystal can also be expressed in terms of symmetry factors, in an analogous manner to that described for the electron density in the previous section. It must be pointed out that while the above treatment only presumes that the electron density can be represented by a three-dimensional Fourier series, the present one is restricted by the assumption that the atoms are isotropic with regard to their motion and shape (cf. Chapter 1.2). Under the above assumptions, i.e. for isotropically vibrating spherical atoms, the structure factor can be written as

P j

sin2hT Ps r ts

14213

The symmetry factors A and B are well known as geometric or trigonometric structure factors and a considerable part of Volume I of IT (1952) is dedicated to their tabulation. Their formal association with the structure factor – following from direct-space arguments – is closely related to that shown in equation (1.4.2.11) (see Section 1.4.2.4). Simpliﬁed trigonometric expressions for A and B are given in Tables A1.4.3.1–A1.4.3.7 in Appendix 1.4.3 for all the two- and three-dimensional crystallographic space groups, and for all the parities of hkl for which A and B assume different functional forms. These expressions are there given for general reﬂections and can also be used for special ones, provided the occupancy factors qh have been properly accounted for. Equation (1.4.2.11) is quite general and can, of course, be applied to noncentrosymmetric Fourier summations, without neglect of dispersion. Further simpliﬁcations are obtained in the centrosym-

14216

where hT hkl is the diffraction vector, N is the number of atoms in the unit cell, fj is the atomic scattering factor including its temperature factor and depending on the magnitude of h only, and rj is the position vector of the jth atom referred to the origin of the unit cell. If the crystal belongs to a point group of order mp and the multiplicity of its Bravais lattice is mL , there are g mp mL general equivalent positions in the unit cell of the space group (IT A, 1983). We can thus rewrite (1.4.2.16), grouping the contributions of the symmetry-related atoms, as Fh

s1

fj exp2ihT rj ,

j1

and g P

N P

Fh

s1

Bh

14215

fj

g P

exp2ihT P s r ts ,

14217

s1

where P s and ts are the rotation and translation parts of the sth space-group operation respectively. The inner summation in (1.4.2.17) contains the dependence of the structure factor of reﬂection h on the space-group symmetry of the crystal and is known as the (complex) geometric or trigonometric structure factor. Equation (1.4.2.17) can be rewritten as P Fh fj Aj h iBj h, 14218 j

where

Aj h

g P

cos2hT P s rj ts

14219

sin2hT P s rj ts

14220

s1

and

101

Bj h

g P

s1

1. GENERAL RELATIONSHIPS AND TECHNIQUES are the real and imaginary parts of the trigonometric structure factor. Equations (1.4.2.19) and (1.4.2.20) are mathematically identical to equations (1.4.2.11) and (1.4.2.12), respectively, apart from the numerical coefﬁcients which appear in the expressions for A and B, for space groups with centred lattices: while only the order of the point group need be considered in connection with the Fourier expansion of the electron density (see above), the multiplicity of the Bravais lattice must of course appear in (1.4.2.19) and (1.4.2.20). Analogous functional forms are arrived at by considerations of symmetry in direct and reciprocal spaces. These quantities are therefore convenient representations of crystallographic symmetry in its interaction with the diffraction experiment and have been indispensable in all of the early crystallographic computing related to structure determination. Their applications to modern crystallographic computing have been largely superseded by fast Fourier techniques, in reciprocal space, and by direct use of matrix and vector representations of space-group operators, in direct space, especially in cases of low space-group symmetry. It should be noted, however, that the degree of simpliﬁcation of the trigonometric structure factors generally increases with increasing symmetry (see, e.g., Section 1.4.3), and the gain of computing efﬁciency becomes signiﬁcant when problems involving high symmetries are treated with this ‘old-fashioned’ tool. Analytic expressions for the trigonometric structure factors are of course indispensable in studies in which the knowledge of the functional form of the structure factor is required [e.g. in theories of structurefactor statistics and direct methods of phase determination (see Chapters 2.1 and 2.2)]. Equations (1.4.2.19) and (1.4.2.20) are simple but their expansion and simpliﬁcation for all the space groups and relevant hkl subsets can be an extremely tedious undertaking when carried out in the conventional manner. As shown below, this process has been automated by a suitable combination of symbolic and numeric high-level programming procedures. 1.4.3. Structure-factor tables 1.4.3.1. Some general remarks This section is a revised version of the structure-factor tables contained in Sections 4.5 through 4.7 of Volume I (IT I, 1952). As in the previous edition, it is intended to present a comprehensive list of explicit expressions for the real and the imaginary parts of the trigonometric structure factor, for all the 17 plane groups and the 230 space groups, and for the hkl subsets for which the trigonometric structure factor assumes different functional forms. The tables given here are also conﬁned to the case of general Wyckoff positions (IT I, 1952). However, the expressions are presented in a much more concise symbolic form and are amenable to computation just like the explicit trigonometric expressions in Volume I (IT I, 1952). The present tabulation is based on equations (1.4.2.19) and (1.4.2.20), i.e. the numerical coefﬁcients in A and B which appear in Tables A1.4.3.1–A1.4.3.7 in Appendix 1.4.3 are appropriate to space-group-speciﬁc structure-factor formulae. The functional form of A and B is, however, the same when applied to Fourier summations (see Section 1.4.2.3). 1.4.3.2. Preparation of the structure-factor tables The lists of the coordinates of the general equivalent positions, presented in IT A (1983), as well as in earlier editions of the Tables, are sufﬁcient for the expansion of the summations in (1.4.2.19) and (1.4.2.20) and the simpliﬁcation of the resulting expressions can be performed using straightforward algebra and trigonometry (see, e.g., IT I, 1952). As mentioned above, the preparation of the present structure-factor tables has been automated and its stages can be summarized as follows:

(i) Generation of the coordinates of the general positions, starting from a computer-adapted space-group symbol (Shmueli, 1984). (ii) Formation of the scalar products, appearing in (1.4.2.19) and (1.4.2.20), and their separation into components depending on the rotation and translation parts of the space-group operations: hT
P s , ts r hT P s r hT ts

1431

for the space groups which are not associated with a unique axis; the left-hand side of (1.4.3.1) is separated into contributions of the relevant plane group and unique axis for the remaining space groups. (iii) Analysis of the translation-dependent parts of the scalar products and automatic determination of all the parities of hkl for which A and B must be computed and simpliﬁed. (iv) Expansion of equations (1.4.2.19) and (1.4.2.20) and their reduction to trigonometric expressions comparable to those given in the structure-factor tables in Volume I of IT (1952). (v) Representation of the results in terms of a small number of building blocks, of which the expressions were found to be composed. These representations are described in Section 1.4.3.3. All the stages outlined above were carried out with suitably designed computer programs, written in numerically and symbolically oriented languages. A brief summary of the underlying algorithms is presented in Appendix 1.4.1. The computer-adapted space-group symbols used in these computations are described in Section A1.4.2.2 and presented in Table A1.4.2.1. 1.4.3.3. Symbolic representation of A and B We shall ﬁrst discuss the symbols for the space groups that are not associated with a unique axis. These comprise the triclinic, orthorhombic and cubic space groups. The symbols are also used for the seven rhombohedral space groups which are referred to rhombohedral axes (IT I, 1952; IT A, 1983). The abbreviation of triple products of trigonometric functions such as, e.g., denoting cos2hx sin2ky cos2lz by csc, is well known (IT I, 1952), and can be conveniently used in representing A and B for triclinic and orthorhombic space groups. However, the simpliﬁed expressions for A and B in space groups of higher symmetry also possess a high degree of regularity, as is apparent from an examination of the structure-factor tables in Volume I (IT I, 1952), and as conﬁrmed by the preparation of the present tables. An example, illustrating this for the cubic system, is given below. The trigonometric structure factor for the space group Pm3 (No. 200) is given by A 8cos2hx cos2ky cos2lz cos2hy cos2kz cos2lx cos2hz cos2kx cos2ly,

1432

and the sum of the above nine-function block and the following one: 8cos2hx cos2kz cos2ly cos2hz cos2ky cos2lx cos2hy cos2kx cos2lz

1433

is the trigonometric structure factor for the space group Pm3m (No. 221, IT I, 1952, IT A, 1983). It is obvious that the only difference between the nine-function blocks in (1.4.3.2) and (1.4.3.3) is that the permutation of the coordinates xyz is cyclic or even in (1.4.3.2), while it is non-cyclic or odd in (1.4.3.3). It was observed during the generation of the present tables that the expressions for A and B for all the cubic space groups, and all the relevant hkl subsets, can be represented in terms of such ‘even’ and ‘odd’ nine-function blocks. Moreover, it was found that the order of the trigonometric functions in each such block remains the same in

102

1.4. SYMMETRY IN RECIPROCAL SPACE each of its three terms (triple products). This is not surprising since each of the above space groups contains threefold axes of rotation along [111] and related directions, and such permutations of xyz for ﬁxed hkl (or vice versa) are expected. It was therefore possible to introduce two permutation operators and represent A and B in terms of the following two basic blocks:

expressions must be given we make use of the convention of replacing cos2u by cu and sin2u by su. For example, cos2hy kx etc. is given as chy kx etc. The symbols are deﬁned below. Monoclinic space groups (Table A1.4.3.3) The following symbols are used in this system:

Epqr p2hxq2kyr2lz p2hyq2kzr2lx p2hzq2kxr2ly

1434

and Opqr p2hxq2kzr2ly p2hzq2kyr2lx p2hyq2kxr2lz, 1435 where each of p, q and r can be a sine or a cosine, and appears at the same position in each of the three terms of a block. The capital preﬁxes E and O were chosen to represent even and odd permutations of the coordinates xyz, respectively. For example, the trigonometric structure factor for the space group Pa3 (No. 205, IT I, 1952, IT A, 1983) can now be tabulated as follows: A 8Eccc 8Ecss 8Escs 8Essc

B 0 0 0 0

hk even even odd odd

kl even odd even odd

chl cos2hx lz,

chk cos2hx ky

shl sin2hx lz,

shk sin2hx ky

1437

so that any expression for A or B in the monoclinic system has the form Kphlqky or Kphkqlz for the second or ﬁrst setting, respectively, where p and q can each be a sine or a cosine and K is a numerical constant. Tetragonal space groups (Table A1.4.3.5) The most frequently occurring expressions in the summations for A and B in this system are of the form P(pq) p2hxq2ky p2kxq2hy

1438

M(pq) p2hxq2ky p2kxq2hy,

1439

and where p and q can each be a sine or a cosine. These are typical contributions related to square plane groups. Trigonal and hexagonal space groups (Table A1.4.3.6) The contributions of plane hexagonal space groups to the ﬁrst term in (1.4.3.6) are

hl even odd odd even

(cf. Table A1.4.3.7), where the sines and cosines are abbreviated by s and c, respectively. It is interesting to note that the only maximal non-isomorphic subgroup of Pa3, not containing a threefold axis, is the orthorhombic Pbca (see IT A, 1983, p. 621), and this group– subgroup relationship is reﬂected in the functional forms of the trigonometric structure factors; the representation of A and B for Pbca is in fact analogous to that of Pa3, including the parities of hkl and the corresponding forms of the triple products, except that the preﬁx E – associated with the threefold rotation – is absent from Pbca. The expression for A for the space group Pm3m [the sum of (1.4.3.2) and (1.4.3.3)] now simply reads: A 8Eccc Occc. As pointed out above, the permutation operators also apply to rhombohedral space groups that are referred to rhombohedral axes (Table A1.4.3.6), and the corresponding expressions for R3 and R 3 bear the same relationship to those for P1 and P1 (Table A1.4.3.2), respectively, as that shown above for the related Pa3 and Pbca. When in any given standard space-group setting one of the coordinate axes is parallel to a unique axis, the point-group rotation matrices can be partitioned into 2 2 and 1 1 diagonal blocks, the former corresponding to an operation of the plane group resulting from the projection of the space group down the unique axis. If, for example, the unique axis is parallel to c, we can decompose the scalar product in (1.4.2.19) and (1.4.2.20) as follows: t1 x P11 P12 T h P s r ts h k y P21 P22 t2 lP33 z t3 , 1436 where the ﬁrst scalar product on the right-hand side of (1.4.3.6) contains the contribution of a plane group and the second product is the contribution of the unique axis itself. The above decomposition often leads to a convenient factorization of A and B, and is applicable to monoclinic, tetragonal and hexagonal families, the latter including rhombohedral space groups that are referred to hexagonal axes. The symbols used in Tables A1.4.3.3, A1.4.3.5 and A1.4.3.6 are based on such decompositions. In those few cases where explicit

p1 hx ky,

p2 kx iy,

p3 ix hy,

q1 kx hy,

q2 hx iy,

q3 ix ky,

14310

where i h k (IT I, 1952). The symbols which represent the frequently occurring expressions in this family, and given in terms of (1.4.3.10), are Chki cos2p1 cos2p2 cos2p3 Ckhi cos2q1 cos2q2 cos2q3 Shki sin2p1 sin2p2 sin2p3 Skhi sin2q1 sin2q2 sin2q3

14311

and these quite often appear as the following sums and differences: PH(cc) Chki Ckhi,

PH(ss) Shki Skhi

MH(cc) Chki Ckhi,

MH(ss) Shki Skhi 14312

The symbols deﬁned in this section are brieﬂy redeﬁned in the appropriate tables, which also contain the conditions for vanishing symbols. 1.4.3.4. Arrangement of the tables The expressions for A and B are usually presented in terms of the short symbols deﬁned above for all the representations of the plane groups and space groups given in Volume A (IT A, 1983), and are fully consistent with the unit-cell choices and space-group origins employed in that volume. The tables are arranged by crystal families and the expressions appear in the order of the appearance of the corresponding plane and space groups in the space-group tables in IT A (1983). The main items in a table entry, not necessarily in the following order, are: (i) the conventional space-group number, (ii) the short Hermann–Mauguin space-group symbol, (iii) brief remarks on the choice of the space-group origin and setting, where appropriate, (iv) the real (A) and imaginary (B) parts of the trigonometric structure factor, and (v) the parity of the hkl subset to which the expressions

103

1. GENERAL RELATIONSHIPS AND TECHNIQUES for A and B pertain. Full space-group symbols are given in the monoclinic system only, since they are indispensable for the recognition of the settings and glide planes appearing in the various representations of monoclinic space groups given in IT A (1983). 1.4.4. Symmetry in reciprocal space: space-group tables

pn qn rn m denotes 2hpn kqn lrn m,

1442

where the fractions pn m, qn m and rn m are the components of the translation part tn of the nth space-group operation. The phase-shift part of an entry is given only if pn qn rn is not a vector in the direct lattice, since such a vector would give rise to a trivial phase shift (an integer multiple of 2).

1.4.4.1. Introduction The purpose of this section, and the accompanying table, is to provide a representation of the 230 three-dimensional crystallographic space groups in terms of two fundamental quantities that characterize a weighted reciprocal lattice: (i) coordinates of pointsymmetry-related points in the reciprocal lattice, and (ii) phase shifts of the weight functions that are associated with the translation parts of the various space-group operations. Table A1.4.4.1 in Appendix 1.4.4 collects the above information for all the spacegroup settings which are listed in IT A (1983) for the same choice of the space-group origins and following the same numbering scheme used in that volume. Table A1.4.4.1 was generated by computer using the space-group algorithm described by Shmueli (1984) and the space-group symbols given in Table A1.4.2.1 in Appendix 1.4.2. It is shown in a later part of this section that Table A1.4.4.1 can also be regarded as a table of symmetry groups in Fourier space, in the Bienenstock–Ewald (1962) sense which was mentioned in Section 1.4.1. The section is concluded with a brief description of the correspondence between Bravais-lattice types in direct and reciprocal spaces. 1.4.4.2. Arrangement of the space-group tables Table A1.4.4.1 is subdivided into point-group sections and space-group subsections, as outlined below. (i) The point-group heading. This heading contains a short Hermann–Mauguin symbol of a point group, the crystal system and the symbol of the Laue group with which the point group is associated. Each point-group heading is followed by the set of space groups which are isomorphic to the point group indicated, the set being enclosed within a box. (ii) The space-group heading. This heading contains, for each space group listed in Volume A (IT A, 1983), the short Hermann– Mauguin symbol of the space group, its conventional space-group number and (in parentheses) the serial number of its representation in Volume A; this is also the serial number of the explicit spacegroup symbol in Table A1.4.2.1 from which the entry was derived. Additional items are full space-group symbols, given only for the monoclinic space groups in their settings that are given in Volume A (IT, 1983), and self-explanatory comments as required. (iii) The table entry. In the context of the analysis in Section 1.4.2.2, the format of a table entry is: hT P n : hT tn , where Pn , tn is the nth space-group operator, and the phase shift hT tn is expressed in units of 2 [see equations (1.4.2.3) and (1.4.2.5)]. More explicitly, the general format of a table entry is n hn kn ln : pn qn rn m

1441

In (1.4.4.1), n is the serial number of the space-group operation to which this entry pertains and is the same as the number of the general Wyckoff position generated by this operation and given in IT A (1983) for the space group appearing in the space-group heading. The ﬁrst part of an entry, hn kn ln :, contains the coordinates of the reciprocal-lattice vector that was generated from the reference vector (hkl) by the rotation part of the nth space-group operation. These rotation parts of the table entries, for a given space group, thus constitute the set of reciprocal-lattice points that are generated by the corresponding point group (not Laue group). The second part of an entry is an abbreviation of the phase shift which is associated with the nth operation and thus

1.4.4.3. Effect of direct-space transformations The phase shifts given in Table A1.4.4.1 depend on the translation parts of the space-group operations and these translations are determined, all or in part, by the choice of the space-group origin. It is a fairly easy matter to ﬁnd the phase shifts that correspond to a given shift of the space-group origin in direct space, directly from Table A1.4.4.1. Moreover, it is also possible to modify the table entries so that a more general transformation, including a change of crystal axes as well as a shift of the spacegroup origin, can be directly accounted for. We employ here the frequently used concise notation due to Seitz (1935) (see also IT A, 1983). Let the direct-space transformation be given by rnew Trold v,

1443

where T is a non-singular 3 3 matrix describing the change of the coordinate system and v is an origin-shift vector. The components of T and v are referred to the old system, and rnew rold is the position vector of a point in the crystal, referred to the new (old) system, respectively. If we denote a space-group operation referred to the new and old systems by P new , tnew and P old , told , respectively, we have Pnew , tnew T, vPold , told T, v1 1

1

TP old T , v TP old T v Ttold

1444 1445

It follows from (1.4.4.2) and (1.4.4.5) that if the old entry of Table A1.4.4.1 is given by n hT P : hT t, the transformed entry becomes n hT TPT 1 : hT TPT 1 v hT v hT Tt,

1446

and in the important special cases of a pure change of setting v 0 or a pure shift of the space-group origin (T is the unit matrix I), (1.4.4.6) reduces to n hT TPT 1 : hT Tt

1447

n hT P : hT Pv hT v hT t,

1448

or respectively. The rotation matrices P are readily obtained by visual or programmed inspection of the old entries: if, for example, hT P is khl, we must have P21 1, P12 1 and P33 1, the remaining Pij ’s being equal to zero. Similarly, if hT P is kil, where i h k, we have 0 1 0 kil k, h k, l hkl 1 1 0 0 0 1

The rotation matrices can also be obtained by reference to Chapter 7 and Tables 11.2 and 11.3 in Volume A (IT A, 1983). As an example, consider the phase shifts corresponding to the operation No. (16) of the space group P4nmm (No. 129) in its two origins given in Volume A (IT A, 1983). For an Origin 2-to-Origin 1 transformation we ﬁnd there v 14 , 14 , 0 and the old Origin 2

104

1.4. SYMMETRY IN RECIPROCAL SPACE entry in Table A1.4.4.1 is (16) khl (t is zero). The appropriate entry for the Origin 1 description of this operation should therefore be hT Pv hT v k4 h4 h4 k4 h2 k2, as given by (1.4.4.8), or h k2 if a trivial shift of 2 is subtracted. The (new) Origin 1 entry thus becomes: (16) khl: 1102, as listed in Table A1.4.4.1.

1.4.4.4. Symmetry in Fourier space As shown below, Table A1.4.4.1 can also be regarded as a collection of the general equivalent positions of the symmetry groups of Fourier space, in the sense of the treatment by Bienenstock & Ewald (1962). This interpretation of the table is, however, restricted to the underlying periodic function being real and positive (see the latter reference). The symmetry formalism can be treated with the aid of the original 4 4 matrix notation, but it appears that a concise Seitz-type notation suits better the present introductory interpretation. The symmetry dependence of the fundamental relationship (1.4.2.5)

F is the structure factor [cf. equation (1.4.2.4)]. In order to make use of the second requirement in deriving permissible symmetry operators on Fourier space, all the relevant transformations, i.e. those which have rotation operators of the orders 1, 2, 3, 4 and 6, must be individually examined. A comprehensive example, covering most of the tetragonal system, can be found in Bienenstock & Ewald (1962). It is of interest to illustrate the above process for a simple particular instance. Consider an operation, the rotation part of which involves a mirror plane, and assume that it is associated with the monoclinic system, in the second setting (unique axis b). We denote the operator by m, u, where uT uvw, and the permissible values of u, v and w are to be determined. The operation is of order 2, and according to requirement (ii) above we have to evaluate hT : 0m, u2 hT : 0I, mu u hT : hT m Iu hkl : 2hu lw, where

1 m 0 0

hT Pn h 2hT tn is given by a table entry of the form: n hT P : hT t, where the phase shift is given in units of 2, and the structure-dependent phase h is omitted. Deﬁning a combination law analogous to Seitz’s product of two operators of afﬁne transformation: aT : bR, r aT R : aT r b,

1449

where R is a 3 3 matrix, aT is a row vector, r is a column vector and b is a scalar, we can write the general form of a table entry as hT : P, t hT P : hT t ,

14410

where is a constant phase shift which we take as zero. The positions hT : 0 and hT P : hT t are now related by the operation P, t via the combination law (1.4.4.9), which is a shorthand transcription of the 4 4 matrix notation of Bienenstock & Ewald (1962), with the appropriate sign of t. Let us evaluate the result of a successive application of two such operators, say P, t and Q, v to the reference position hT : 0 in Fourier space: hT : 0P, tQ, v hT : 0PQ, Pv t hT PQ : hT Pv hT t,

14411

14413

0 0 1 0 0 1

is the matrix representing the operation of reﬂection and I is the unit matrix. For m, u to be an admissible symmetry operator, the phase-shift part of (1.4.4.13), i.e. 2hu lw, must be an integer (multiple of 2). The smallest non-negative values of u and w which satisfy this are the pairs: u w 0, u 12 and w 0, u 0 and w 12, and u w 12. We have thus obtained four symmetry operators in Fourier space, which are identical (except for the sign of their translational parts) to those of the direct-space monoclinic mirror and glide-plane operations. The fact that the component v cancels out simply means that an arbitrary component of the phase shift can be added along the b axis; this is concurrent with arbitrary direct-space translations that appear in the characterization of individual types of space-group operations [see e.g. Koch & Fischer (1983)]. Each of the 230 space groups, which leaves invariant a (real and non-negative) function with the periodicity of the crystal, thus has its counterpart which determines the symmetry of the Fourier expansion coefﬁcients of this function, with equivalent positions given in Table A1.4.4.1.

and perform an inverse operation: hT P : hT tP, t1 hT P : hT tP1 , P1 t

1.4.4.5. Relationships between direct and reciprocal Bravais lattices

hT PP 1 : hT PP 1 t hT t hT : 0

14412

These equations conﬁrm the validity of the shorthand notation (1.4.4.9) and illustrate the group nature of the operators P, t in the present context. Following Bienenstock & Ewald, the operators P, t are symmetry operators that act on the positions hT : 0 in Fourier space, provided they satisfy the following requirements: (i) the application of such an operator leaves the magnitude of the (generally) complex Fourier coefﬁcient unchanged, and (ii) after g successive applications of an operator, where g is the order of its rotation part, the phase remains unchanged up to a shift by an integer multiple of 2 (a trivial phase shift, corresponding to a translation by a lattice vector in direct space). If our function is the electron density in the crystal, the ﬁrst requirement is obviously satisﬁed since Fh FhT P , where

Centred Bravais lattices in crystal space give rise to systematic absences of certain classes of reﬂections (IT I, 1952; IT A, 1983) and the corresponding points in the reciprocal lattice have accordingly zero weights. These absences are periodic in reciprocal space and their ‘removal’ from the reciprocal lattice results in a lattice which – like the direct one – must belong to one of the fourteen Bravais lattice types. This must be so since the point group of a crystal leaves its lattice – and also the associated reciprocal lattice – unchanged. The magnitudes of the structure factors (the weight functions) are also invariant under the operation of this point group. The correspondence between the types of centring in direct and reciprocal lattices is given in Table 1.4.4.1. Notes: (i) The vectors a , b and c , appearing in the deﬁnition of the multiple unit cell in the reciprocal lattice, deﬁne this lattice prior to

105

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.4.4.1. Correspondence between types of centring in direct and reciprocal lattices Direct lattice

Reciprocal lattice

Lattice type(s)

Centring translations

Lattice type(s)

P, R A B C I

0, 12, 12 1 1 2, 0, 2 1 1 2, 2, 0 1 1 1 2, 2, 2

P, R A B C F

Restriction on hkl

Multiple unit cell

k l 2n h l 2n h k 2n h k l 2n

a , b , c a , 2b , 2c 2a , b , 2c 2a , 2b , c 2a , 2b , 2c

F

0, 12, 12 1 1 2, 0, 2 1 1 2, 2, 0

I

k l 2n h l 2n h k 2n

2a , 2b , 2c

R hex

2 3, 1 3,

R hex

h k l 3n

3a , 3b , 3c

1 3, 2 3,

1 3 2 3

the removal of lattice points with zero weights (absences). All the restrictions on hkl pertain to indexing on a , b and c . (ii) The centring type of the reciprocal lattice refers to the multiple unit cell given in the table. (iii) The centring type denoted by R hex is a representation of the rhombohedral lattice R by a triple hexagonal unit cell, in the obverse setting (IT I, 1952), i.e. according to the transformation a aR bR b bR cR c aR bR cR ,

14414

where aR , bR and cR pertain to a primitive unit cell in the rhombohedral lattice R. The corresponding multiple reciprocal cell, with centring denoted by R hex , contains nine lattice points with coordinates 000, 021, 012, 101, 202, 110, 220, 211 and 122 – indexed on the usual reciprocal to the triple hexagonal unit cell deﬁned by (1.4.4.14). Detailed derivations of these correspondences are given by Buerger (1942), and an elementary proof of the reciprocity of I and F lattices can be found, e.g., in pamphlet No. 4 of the Commission on Crystallographic Teaching (Authier, 1981). Intuitive proofs follow directly from the restrictions on hkl, given in Table 1.4.4.1.

Appendix 1.4.1. Comments on the preparation and usage of the tables (U. SHMUELI) The straightforward but rather extensive calculations and text processing related to Tables A1.4.3.1 through A1.4.3.7 and Table A1.4.4.1 in Appendices 1.4.3 and 1.4.4, respectively, were performed with the aid of a combination of FORTRAN and REDUCE (Hearn, 1973) programs, designed so as to enable the author to produce the table entries directly from a space-group symbol and with a minimum amount of intermediate manual intervention. The ﬁrst stage of the calculation, the generation of a space group (coordinates of the equivalent positions), was accomplished with the program SPGRGEN, the algorithm of

which was described in some detail elsewhere (Shmueli, 1984). A complete list of computer-adapted space-group symbols, processed by SPGRGEN and not given in the latter reference, is presented in Table A1.4.2.1 of Appendix 1.4.2. The generation of the space group is followed by a construction of symbolic expressions for the scalar products hT Pr t; e.g. for position No. (13) in the space group P41 32 (No. 213, IT I, 1952, IT A, 1983), this scalar product is given by h34 y k14 x l14 z. The construction of the various table entries consists of expanding the sines and cosines of these scalar products, performing the required summations, and simplifying the result where possible. The construction of the scalar products in a FORTRAN program is fairly easy and the extremely tedious trigonometric calculations required by equations (1.4.2.19) and (1.4.2.20) can be readily performed with the aid of one of several available computer-algebraic languages (for a review, see Computers in the New Laboratory – a Nature Survey, 1981); the REDUCE language was employed for the above purpose. Since the REDUCE programs required for the summations in (1.4.2.19) and (1.4.2.20) for the various space groups were seen to have much in common, it was decided to construct a FORTRAN interface which would process the space-group input and prepare automatically REDUCE programs for the algebraic work. The least straightforward problem encountered during this work was the need to ‘convince’ the interface to generate hkl parity assignments which are appropriate to the space-group information input. This was solved for all the crystal families except the hexagonal by setting up a ‘basis’ of the form: h2, k2, l2, k l2, . . . , h k l4 and representing the translation parts of the scalar products, hT t, as sums of such ‘basis functions’. A subsequent construction of an automatic parity routine proved to be easy and the interface could thus produce any number of REDUCE programs for the summations in (1.4.2.19) and (1.4.2.20) using a list of spacegroup symbols as the sole input. These included trigonal and hexagonal space groups with translation components of 12. This approach seemed to be too awkward for some space groups containing threefold and sixfold screw axes, and these were treated individually. There is little to say about the REDUCE programs, except that the output they generate is at the same level of trigonometric complexity as the expressions for A and B appearing in Volume I (IT I, 1952). This could have been improved by making use of the pattern-matching capabilities that are incorporated in REDUCE, but

106

1.4. SYMMETRY IN RECIPROCAL SPACE it was found more convenient to construct a FORTRAN interpreter which would detect in the REDUCE output the basic building blocks of the trigonometric structure factors (see Section 1.4.3.3) and perform the required transformations. Tables A1.4.3.1–A1.4.3.7 were thus constructed with the aid of a chain composed of (i) a space-group generating routine, (ii) a FORTRAN interface, which processes the space-group input and ‘writes’ a complete REDUCE program, (iii) execution of the REDUCE program and (iv) a FORTRAN interpreter of the REDUCE output in terms of the abbreviated symbols to be used in the tables. The computation was at a ‘one-group-at-a-time’ basis and the automation of its repetition was performed by means of procedural constructs at the operating-system level. The construction of Table A1.4.4.1 involved only the preliminary stage of the processing of the space-group information by the FORTRAN interface. All the computations were carried out on a Cyber 170-855 at the Tel Aviv University Computation Center. It is of some importance to comment on the recommended usage of the tables included in this chapter in automatic computations. If, for example, we wish to compute the expression: A 8Escs Ossc, use can be made of the facility provided by most versions of FORTRAN of transferring subprogram names as parameters of a FUNCTION. We thus need only two FUNCTIONs for any calculation of A and B for a cubic space group, one FUNCTION for the block of even permutations of x, y and z: FUNCTION E(P, Q, R) EXTERNAL SIN, COS COMMON/TSF/TPH, TPK, TPL, X, Y, Z E PTPH X QTPK Y RTPL Z 1 PTPH Z QTPK X RTPL Y 2 PTPH Y QTPK Z RTPL X RETURN END where TPH, TPK and TPL denote 2h, 2k and 2l, respectively, and a similar FUNCTION, say O(P,Q,R), for the block of odd permutations of x, y and z. The calling statement in the calling (sub)program can thus be: A 8 (E(SIN, COS, SIN) O(SIN, SIN, COS)) A small number of such FUNCTIONs sufﬁces for all the spacegroup-speciﬁc computations that involve trigonometric structure factors.

Appendix 1.4.2. Space-group symbols for numeric and symbolic computations

A1.4.2.1. Introduction (U. SHMUELI, S. R. HALL GROSSE-KUNSTLEVE)

AND

R. W.

This appendix lists two sets of computer-adapted space-group symbols which are implemented in existing crystallographic software and can be employed in the automated generation of space-group representations. The computer generation of spacegroup symmetry information is of well known importance in many

crystallographic calculations, numeric as well as symbolic. A prerequisite for a computer program that generates this information is a set of computer-adapted space-group symbols which are based on the generating elements of the space group to be derived. The sets of symbols to be presented are: (i) Explicit symbols. These symbols are based on the classiﬁcation of crystallographic point groups and space groups by Zachariasen (1945). These symbols are termed explicit because they contain in an explicit manner the rotation and translation parts of the space-group generators of the space group to be derived and used. These computer-adapted explicit symbols were proposed by Shmueli (1984), who also describes in detail their implementation in the program SPGRGEN. This program was used for the automatic preparation of the structure-factor tables for the 17 plane groups and 230 space groups, listed in Appendix 1.4.3, and the 230 space groups in reciprocal space, listed in Appendix 1.4.4. The explicit symbols presented in this appendix are adapted to the 306 representations of the 230 space groups as presented in IT A (1983) with regard to the standard settings and choice of spacegroup origins. The symmetry-generating algorithm underlying the explicit symbols, and their deﬁnition, are given in Section A1.4.2.2 of this appendix and the explicit symbols are listed in Table A1.4.2.1. (ii) Hall symbols. These symbols are based on the implied-origin notation of Hall (1981a,b), who also describes in detail the algorithm implemented in the program SGNAME (Hall, 1981a). In the ﬁrst edition of IT B (1993), the term ‘concise space-group symbols’ was used for this notation. In recent years, however, the term ‘Hall symbols’ has come into use in symmetry papers (Altermatt & Brown, 1987; Grosse-Kunstleve, 1999), software applications (Hovmo¨ller, 1992; Grosse-Kunstleve, 1995; Larine et al., 1995; Dowty, 1997) and data-handling approaches (Bourne et al., 1998). This term has therefore been adopted for the second edition. The main difference in the deﬁnition of the Hall symbols between this edition and the ﬁrst edition of IT B is the generalization of the origin-shift vector to a full change-of-basis matrix. The examples have been expanded to show how this matrix is applied. The notation has also been made more consistent, and a typographical error in a default axis direction has been corrected.* The lattice centring symbol ‘H’ has been added to Table A1.4.2.2. In addition, Hall symbols are now provided for 530 settings to include all settings from Table 4.3.1 of IT A (1983). Namely, all non-standard symbols for the monoclinic and orthorhombic space groups are included. Some of the space-group symbols listed in Table A1.4.2.7 differ from those listed in Table B.6 (p. 119) of the ﬁrst edition of IT B. This is because the symmetry of many space groups can be represented by more than one subset of ‘generator’ elements and these lead to different Hall symbols. The symbols listed in this edition have been selected after ﬁrst sorting the symmetry elements into a strictly prescribed order based on the shape of their Seitz matrices, whereas those in Table B.6 were selected from symmetry elements in the order of IT I (1965). Software for selecting the Hall symbols listed in Table A1.4.2.7 is freely available (Hall, 1997). These symbols and their equivalents in the ﬁrst edition of IT B will generate identical symmetry elements, but the former may be used as a reference table in a strict mapping procedure between different symmetry representations (Hall et al., 2000). The Hall symbols are deﬁned in Section A1.4.2.3 of this appendix and are listed in Table A1.4.2.7.

* The correct default axis direction a b of an N preceded by 3 or 6 replaces a b on p. 117, right-hand column, line 4, in the ﬁrst edition of IT B.

107

1. GENERAL RELATIONSHIPS AND TECHNIQUES 0 1 A1.4.2.2. Explicit symbols (U. SHMUELI) 1 0 0 1 B B C 1A @ 0 1 0 A 2A @ 0 As shown elsewhere (Shmueli, 1984), the set of representative 0 0 1 0 operators of a crystallographic space group [i.e. the set that is listed 0 0 1 for each space group in the symmetry tables of IT A (1983) and 0 1 0 0 automatically regenerated for the purpose of compiling the B B C 2C @ 0 1 0 A 2D @ 1 symmetry tables in the present chapter] may have one of the following forms: 0 0 0 1 0 0 1 1 1 1 0 B B C Q, u , 2F @ 0 1 0 A 2G @ 1 Q, u R, v ,

or

P, t Q, u R, v ,

P, t I, 0, P, t, P, t2 , . . . , P, tg1 ,

A1422

where I is a unit operator and g is the order of the rotation operator P (i.e. Pg = I). The representative operations of the space group are evaluated by expanding the generators into cyclic groups, as in (A1.4.2.2), and forming, as needed, ordered products of the expanded groups as indicated in (A1.4.2.1) and explained in detail in the original article (Shmueli, 1984). The rotation and translation parts of the generators (P, t), (Q, u) and (R, v) presented here were adapted to the settings and choices of origin used in the main symmetry tables of IT A (1983). The general structure of a three-generator symbol, corresponding to the last line of (A1.4.2.1), as represented in Table A1.4.2.1, is LSC$r1 Pt1 t2 t3 $r2 Qu1 u2 u3 $r3 Rv1 v2 v3 ,

A1423

where L – lattice type; can be P, A, B, C, I, F, or R. The symbol R is used only for the seven rhombohedral space groups in their representations in rhombohedral and hexagonal axes [obverse setting (IT I, 1952)]. S – crystal system; can be A (triclinic), M (monoclinic), O (orthorhombic), T (tetragonal), R (trigonal), H (hexagonal) or C (cubic). C – status of centrosymmetry; can be C or N according as the space group is centrosymmetric or noncentrosymmetric, respectively. $ – this character is followed by six characters that deﬁne a generator of the space group. ri – indicator of the type of rotation that follows: ri is P or I according as the rotation part of the ith generator is proper or improper, respectively. P, Q, R – two-character symbols of matrix representations of the point-group rotation operators P, Q and R, respectively (see below). t1 t2 t3 , u1 u2 u3 , v1 v2 v3 – components of the translation parts of the generators, given in units of 121 ; e.g. the translation part (0 12 34) is given in Table A1.4.2.1 as 069. An exception: (0 0 56) is denoted by 005 and not by 0010. The two-character symbols for the matrices of rotation, which appear in the explicit space-group symbols in Table A1.4.2.1, are deﬁned as follows:

0 0

1

C 1 0A 0 1

1 1 0 C 0 0A 0 1 1 0 1 C 0 0A 0 0 1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 0 1 1 0 0 1 0 B B B C C C 3C @ 1 1 0 A 4C @ 1 0 0 A 6C @ 1 0 0 A, 0 0 1 0 0 1 0 0 1

A1421

where P, Q and R are point-group operators, and t, u and v are zero vectors or translations not belonging to the lattice-translations subgroup. Each of the forms in (A1.4.2.1), enclosed in braces, is evaluated as, e.g.,

0 1 0 0 1 B C 1 0 A 2B @ 0 0 1 0 1 0 1 0 0 C B 0 0 A 2E @ 1 0 1 0 1 0 0 0 0 C B 1 0 A 3Q @ 1

where only matrices of proper rotation are given (and required), since the corresponding matrices of improper rotation are created by the program for appropriate value of the ri indicator. The ﬁrst character of a symbol is the order of the axis of rotation and the second character speciﬁes its orientation: in terms of direct-space lattice vectors, we have

A 100, B 010, C 001, D 110, E 110, F 100, G 210 and Q 111

for the standard orientations of the axes of rotation. Note that the axes 2F, 2G, 3C and 6C appear in trigonal and hexagonal space groups. In the above scheme a space group is determined by one, two or at most three generators [see (A1.4.2.1)]. It should be pointed out that a convenient way of achieving a representation of the space group in any setting and relative to any origin is to start from the standard generators in Table A1.4.2.1 and let the computer program perform the appropriate transformation of the generators only, as in equations (1.4.4.4) and (1.4.4.5). The subsequent expansion of the transformed generators and the formation of the required products [see (A1.4.2.1) and (A1.4.2.2)] leads to the new representation of the space group. In order to illustrate an explicit space-group symbol consider, for example, the symbol for the space group Ia3d, as given in Table A1.4.2.1:

ICC$I3Q000$P4C393$P2D933

The ﬁrst three characters tell us that the Bravais lattice of this space group is of type I, that the space group is centrosymmetric and that it belongs to the cubic system. We then see that the generators are (i) an improper threefold axis along [111] (I3Q) with a zero translation part, (ii) a proper fourfold axis along [001] (P4C) with translation part (1/4, 3/4, 1/4) and (iii) a proper twofold axis along [110] (P2D) with translation part (3/4, 1/4, 1/4). If we make use of the above-outlined interpretation of the explicit symbol (A1.4.2.3), the space-group symmetry transformations in direct space, corresponding to these three generators of the space group Ia3d, become

108

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.1. Explicit symbols

Explicit symbols

No.

Short Hermann– Mauguin symbol

P1

PAN$P1A000

15

C2c

C12c1

CMC$I1A000$P2B006

P1

PAC$I1A000

15

C2c

A12n1

AMC$I1A000$P2B606

P121

PMN$P2B000

15

C2c

I12a1

IMC$I1A000$P2B600

P2

P112

PMN$P2C000

15

C2c

A112a

AMC$I1A000$P2C600

4

P21

P121 1

PMN$P2B060

15

C2c

B112n

BMC$I1A000$P2C660

4

P21

P1121

PMN$P2C006

15

C2c

I112b

5

C2

C121

CMN$P2B000

16

P222

PON$P2C000$P2A000

5 5

C2 C2

A121 I121

AMN$P2B000 IMN$P2B000

17 18

P2221 P21 21 2

PON$P2C006$P2A000 PON$P2C000$P2A660

5

C2

A112

AMN$P2C000

19

P21 21 21

PON$P2C606$P2A660

5

C2

B112

BMN$P2C000

20

C2221

CON$P2C006$P2A000

No.

Short Hermann– Mauguin symbol

1 2 3

P2

3

Comments

Comments

Explicit symbols

IMC$I1A000$P2C060

5

C2

I112

IMN$P2C000

21

C222

CON$P2C000$P2A000

6

Pm

P1m1

PMN$I2B000

22

F222

FON$P2C000$P2A000

6

Pm

P11m

PMN$I2C000

23

I222

ION$P2C000$P2A000

7

Pc

P1c1

PMN$I2B006

24

I21 21 21

ION$P2C606$P2A660

7 7

Pc Pc

P1n1 P1a1

PMN$I2B606 PMN$I2B600

25 26

Pmm2 Pmc21

PON$P2C000$I2A000 PON$P2C006$I2A000

7

Pc

P11a

PMN$I2C600

27

Pcc2

PON$P2C000$I2A006

7

Pc

P11n

PMN$I2C660

28

Pma2

PON$P2C000$I2A600

7

Pc

P11b

PMN$I2C060

29

Pca21

PON$P2C006$I2A606

8

Cm

C1m1

CMN$I2B000

30

Pnc2

PON$P2C000$I2A066

8

Cm

A1m1

AMN$I2B000

31

Pmn21

PON$P2C606$I2A000

8

Cm

I1m1

Pba2

PON$P2C000$I2A660

Cm Cm

A11m B11m

IMN$I2B000 AMN$I2C000 BMN$I2C000

32

8 8

33 34

Pna21 Pnn2

PON$P2C006$I2A666 PON$P2C000$I2A666

8

Cm

I11m

IMN$I2C000

35

Cmm2

CON$P2C000$I2A000

9

Cc

C1c1

CMN$I2B006

36

Cmc21

CON$P2C006$I2A000

9

Cc

A1n1

AMN$I2B606

37

Ccc2

CON$P2C000$I2A006

9

Cc

I1a1

IMN$I2B600

38

Amm2

AON$P2C000$I2A000

9

Cc

A11a

AMN$I2C600

39

Abm2

AON$P2C000$I2A060

9

Cc

B11n

BMN$I2C660

40

Ama2

AON$P2C000$I2A600

9 10

Cc P2m

I11b P12m1

IMN$I2C060 PMC$I1A000$P2B000

41 42

Aba2 Fmm2

AON$P2C000$I2A660 FON$P2C000$I2A000

10

FON$P2C000$I2A333

P2m P21 m

P112m P121 m1

PMC$I1A000$P2C000 PMC$I1A000$P2B060

43

Fdd2

11 11

44

Imm2

ION$P2C000$I2A000

P21 m

P1121 m

PMC$I1A000$P2C006

45

Iba2

ION$P2C000$I2A660

12

C2m

C12m1

CMC$I1A000$P2B000

46

Ima2

ION$P2C000$I2A600

12

C2m

A12m1

AMC$I1A000$P2B000

47

Pmmm

12

C2m

I12m1

IMC$I1A000$P2B000

48

Pnnn

Origin 1

POC$I1A666$P2C000$P2A000

12 12

C2m C2m

A112m B112m

AMC$I1A000$P2C000 BMC$I1A000$P2C000

48 49

Pnnn Pccm

Origin 2

POC$I1A000$P2C660$P2A066 POC$I1A000$P2C000$P2A006

12

C2m

I112m

IMC$I1A000$P2C000

50

Pban

Origin 1

POC$I1A660$P2C000$P2A000

13

P2c

P12c1

PMC$I1A000$P2B006

50

Pban

Origin 2

POC$I1A000$P2C660$P2A060

13

P2c

P12n1

PMC$I1A000$P2B606

51

Pmma

POC$I1A000$P2C600$P2A600

13

P2c

P12a1

PMC$I1A000$P2B600

52

Pnna

POC$I1A000$P2C600$P2A066

13

P2c

Pmna

POC$I1A000$P2C606$P2A000

P2c P2c P21 c

PMC$I1A000$P2C600 PMC$I1A000$P2C660

53

13 13 14

P112a P112n

54

Pcca

POC$I1A000$P2C600$P2A606

P112b P121 c1

PMC$I1A000$P2C060 PMC$I1A000$P2B066

55 56

Pbam Pccn

POC$I1A000$P2C000$P2A660 POC$I1A000$P2C660$P2A606 POC$I1A000$P2C006$P2A060

POC$I1A000$P2C000$P2A000

14

P21 c

P121 n1

PMC$I1A000$P2B666

57

Pbcm

14

P21 c

P121 a1

PMC$I1A000$P2B660

58

Pnnm

14

P21 c

P1121 a

PMC$I1A000$P2C606

59

Pmmn

Origin 1

POC$I1A660$P2C000$P2A660

Origin 2

POC$I1A000$P2C660$P2A600

14

P21 c

P1121 n

PMC$I1A000$P2C666

59

Pmmn

14

P21 c

P1121 b

PMC$I1A000$P2C066

60

Pbcn

109

POC$I1A000$P2C000$P2A666

POC$I1A000$P2C666$P2A660

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.2.1. Explicit symbols (cont.)

Explicit symbols

No.

Short Hermann– Mauguin symbol

Pbca

POC$I1A000$P2C606$P2A660

110

I41 cd

ITN$P4C063$I2A660

Pnma Cmcm

POC$I1A000$P2C606$P2A666 COC$I1A000$P2C006$P2A000

111 112

P42m P42c

PTN$I4C000$P2A000 PTN$I4C000$P2A006

64

Cmca

COC$I1A000$P2C066$P2A000

113

P421 m

PTN$I4C000$P2A660

65

Cmmm

COC$I1A000$P2C000$P2A000

114

P421 c

PTN$I4C000$P2A666

66

Cccm

COC$I1A000$P2C000$P2A006

115

P4m2

PTN$I4C000$P2D000

67

Cmma

COC$I1A000$P2C060$P2A000

116

P4c2

PTN$I4C000$P2D006

68

Ccca

Origin 1

COC$I1A066$P2C660$P2A660

117

P4b2

PTN$I4C000$P2D660

68

Ccca

Origin 2

COC$I1A000$P2C600$P2A606

118

P4n2

PTN$I4C000$P2D666

69 70

Fmmm Fddd

Origin 1

FOC$I1A000$P2C000$P2A000 FOC$I1A333$P2C000$P2A000

119 120

I4m2 I4c2

ITN$I4C000$P2D000 ITN$I4C000$P2D006

70

Fddd

Origin 2

FOC$I1A000$P2C990$P2A099

121

I42m

ITN$I4C000$P2A000

71

Immm

IOC$I1A000$P2C000$P2A000

122

I42d

ITN$I4C000$P2A609

72

Ibam

IOC$I1A000$P2C000$P2A660

123

P4mmm

PTC$I1A000$P4C000$P2A000

73

Ibca

IOC$I1A000$P2C606$P2A660

124

P4mcc

74

Imma

IOC$I1A000$P2C060$P2A000

125

P4nbm

Origin 1

PTC$I1A660$P4C000$P2A000

75

P4

125

P4nbm

Origin 2

PTC$I1A000$P4C600$P2A060

76 77

P41 P42

PTN$P4C000 PTN$P4C003 PTN$P4C006

126 126

P4nnc P4nnc

Origin 1 Origin 2

PTC$I1A666$P4C000$P2A000 PTC$I1A000$P4C600$P2A066

78

P43

PTN$P4C009

127

P4mbm

79

I4

ITN$P4C000

128

P4mnc

80

I41

ITN$P4C063

129

P4nmm

Origin 1

PTC$I1A660$P4C660$P2A660

81

P4

PTN$I4C000

129

P4nmm

Origin 2

PTC$I1A000$P4C600$P2A600

82

I4

ITN$I4C000

130

P4ncc

Origin 1

PTC$I1A660$P4C660$P2A666

83

P4m

PTC$I1A000$P4C000

130

Origin 2

84 85

P42 m P4n

PTC$I1A000$P4C006 PTC$I1A660$P4C660

131 132

P4ncc P42 mmc P42 mcm

PTC$I1A000$P4C600$P2A606 PTC$I1A000$P4C006$P2A000 PTC$I1A000$P4C006$P2A006

85

P4n

Origin 2

PTC$I1A000$P4C600

133

P42 nbc

Origin 1

PTC$I1A666$P4C666$P2A006

86

P42 n

Origin 1

PTC$I1A666$P4C666

133

P42 nbc

Origin 2

PTC$I1A000$P4C606$P2A060

86

P42 n

Origin 2

PTC$I1A000$P4C066

134

P42 nnm

Origin 1

PTC$I1A666$P4C666$P2A000

87

I4m

ITC$I1A000$P4C000

134

P42 nnm

Origin 2

PTC$I1A000$P4C606$P2A066

No.

Short Hermann– Mauguin symbol

61 62 63

Comments

Origin 1

Comments

Explicit symbols

PTC$I1A000$P4C000$P2A006

PTC$I1A000$P4C000$P2A660 PTC$I1A000$P4C000$P2A666

88

I41 a

Origin 1

ITC$I1A063$P4C063

135

P42 mbc

PTC$I1A000$P4C006$P2A660

88

I41 a

Origin 2

ITC$I1A000$P4C933

136

P42 mnm

PTC$I1A000$P4C666$P2A666

89 90

P422 P421 2

PTN$P4C000$P2A000 PTN$P4C660$P2A660

137 137

P42 nmc P42 nmc

91

P41 22

PTN$P4C003$P2A006

138

P42 ncm

Origin 1

PTC$I1A666$P4C666$P2A660

92

P41 21 2

PTN$P4C663$P2A669

138

P42 ncm

Origin 2

PTC$I1A000$P4C606$P2A606

93

P42 22

PTN$P4C006$P2A000

139

I4mmm

94

P42 21 2

PTN$P4C666$P2A666

140

I4mcm

95

P43 22

PTN$P4C009$P2A006

141

I41 amd

Origin 1 Origin 2

PTC$I1A666$P4C666$P2A666 PTC$I1A000$P4C606$P2A600

ITC$I1A000$P4C000$P2A000 ITC$I1A000$P4C000$P2A006 Origin 1

ITC$I1A063$P4C063$P2A063

96

P43 21 2

PTN$P4C669$P2A663

141

I41 amd

Origin 2

ITC$I1A000$P4C393$P2A000

97 98

I422 I41 22

ITN$P4C000$P2A000 ITN$P4C063$P2A063

142 142

I41 acd I41 acd

Origin 1 Origin 2

ITC$I1A063$P4C063$P2A069 ITC$I1A000$P4C393$P2A006

99

P4mm

PTN$P4C000$I2A000

143

P3

PRN$P3C000

100

P4bm

PTN$P4C000$I2A660

144

P31

PRN$P3C004

101

P42 cm

PTN$P4C006$I2A006

145

P32

102

P42 nm

PTN$P4C666$I2A666

146

R3

Hexagonal axes

103

P4cc

PTN$P4C000$I2A006

146

R3

Rhombohedral axes PRN$P3Q000

104

P4nc

PTN$P4C000$I2A666

147

P3

PRC$I3C000

105 106

P42 mc P42 bc

PTN$P4C006$I2A000 PTN$P4C006$I2A660

148 148

R3 R3

Hexagonal axes RRC$I3C000 Rhombohedral axes PRC$I3Q000

107

I4mm

ITN$P4C000$I2A000

149

P312

108

I4cm

ITN$P4C000$I2A006

150

P321

PRN$P3C000$P2F000

109

I41 md

ITN$P4C063$I2A666

151

P31 12

PRN$P3C004$P2G000

110

PRN$P3C008 RRN$P3C000

PRN$P3C000$P2G000

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.1. Explicit symbols (cont.)

Explicit symbols

No.

Short Hermann– Mauguin symbol

P31 21

PRN$P3C004$P2F008

192

P6mcc

PHC$I1A000$P6C000$P2F006

P32 12 P32 21

PRN$P3C008$P2G000 PRN$P3C008$P2F004

193 194

P63 mcm P63 mmc

PHC$I1A000$P6C006$P2F006 PHC$I1A000$P6C006$P2F000

RRN$P3C000$P2F000

195

P23

PCN$P3Q000$P2C000$P2A000

Rhombohedral axes PRN$P3Q000$P2E000

196

F23

FCN$P3Q000$P2C000$P2A000

PRN$P3C000$I2F000

197

I23

ICN$P3Q000$P2C000$P2A000

P31m

PRN$P3C000$I2G000

198

P21 3

PCN$P3Q000$P2C606$P2A660

158

P3c1

PRN$P3C000$I2F006

199

I21 3

ICN$P3Q000$P2C606$P2A660

159

P31c

PRN$P3C000$I2G006

200

Pm3

PCC$I3Q000$P2C000$P2A000

160 160

R3m R3m

Hexagonal axes RRN$P3C000$I2F000 Rhombohedral axes PRN$P3Q000$I2E000

201 201

Pn3 Pn3

161

R3c

Hexagonal axes

RRN$P3C000$I2F006

202

Fm3

161

R3c

203

Fd3

Origin 1

FCC$I3Q333$P2C000$P2A000

162 163

P31m P31c

Rhombohedral axes PRN$P3Q000$I2E666 PRC$I3C000$P2G000

203

Fd3

Origin 2

FCC$I3Q000$P2C330$P2A033

PRC$I3C000$P2G006

204

Im3

ICC$I3Q000$P2C000$P2A000

164

P3m1

PRC$I3C000$P2F000

205

Pa3

PCC$I3Q000$P2C606$P2A660

No.

Short Hermann– Mauguin symbol

152 153 154 155

R32

Hexagonal axes

155

R32

156

P3m1

157

Comments

Comments

Origin 1 Origin 2

Explicit symbols

PCC$I3Q666$P2C000$P2A000 PCC$I3Q000$P2C660$P2A066 FCC$I3Q000$P2C000$P2A000

165

P3c1

PRC$I3C000$P2F006

206

Ia3

ICC$I3Q000$P2C606$P2A660

166 166

R3m R3m

Hexagonal axes RRC$I3C000$P2F000 Rhombohedral axes PRC$I3Q000$P2E000

207 208

P432 P42 32

PCN$P3Q000$P4C000$P2D000 PCN$P3Q000$P4C666$P2D666

167

R3c

Hexagonal axes

RRC$I3C000$P2F006

209

F432

FCN$P3Q000$P4C000$P2D000

167

R3c

Rhombohedral axes PRC$I3Q000$P2E666

210

F41 32

FCN$P3Q000$P4C993$P2D939

168

P6

PHN$P6C000

211

I432

ICN$P3Q000$P4C000$P2D000

169

P61

PHN$P6C002

212

P43 32

PCN$P3Q000$P4C939$P2D399

170

P65

PHN$P6C005

213

P41 32

PCN$P3Q000$P4C393$P2D933

171

P62

PHN$P6C004

214

I41 32

ICN$P3Q000$P4C393$P2D933

172 173

P64 P63

PHN$P6C008 PHN$P6C006

215 216

P43m F43m

PCN$P3Q000$I4C000$I2D000 FCN$P3Q000$I4C000$I2D000

174

P6

PHN$I6C000

217

I43m

ICN$P3Q000$I4C000$I2D000

175

P6m

PHC$I1A000$P6C000

218

P43n

PCN$P3Q000$I4C666$I2D666

176

P63 m

PHC$I1A000$P6C006

219

F43c

FCN$P3Q000$I4C666$I2D666

177

P622

PHN$P6C000$P2F000

220

I43d

ICN$P3Q000$I4C939$I2D399

178

P61 22

PHN$P6C002$P2F000

221

Pm3m

179

P65 22

PHN$P6C005$P2F000

222

Pn3n

Origin 1

PCC$I3Q666$P4C000$P2D000

180 181

P62 22 P64 22

PHN$P6C004$P2F000 PHN$P6C008$P2F000

222 223

Pn3n Pm3n

Origin 2

PCC$I3Q000$P4C600$P2D006 PCC$I3Q000$P4C666$P2D666

PCC$I3Q000$P4C000$P2D000

182

P63 22

PHN$P6C006$P2F000

224

Pn3m

Origin 1

PCC$I3Q666$P4C666$P2D666

183

P6mm

PHN$P6C000$I2F000

224

Pn3m

Origin 2

PCC$I3Q000$P4C066$P2D660

184

P6cc

PHN$P6C000$I2F006

225

Fm3m

185

P63 cm

PHN$P6C006$I2F006

226

Fm3c

186

P63 mc

PHN$P6C006$I2F000

227

Fd3m

Origin 1

FCC$I3Q333$P4C993$P2D939

FCC$I3Q000$P4C000$P2D000 FCC$I3Q000$P4C666$P2D666

187

P6m2

PHN$I6C000$P2G000

227

Fd3m

Origin 2

FCC$I3Q000$P4C693$P2D936

188 189

P6c2 P62m

PHN$I6C006$P2G000 PHN$I6C000$P2F000

228 228

Fd3c Fd3c

Origin 1 Origin 2

FCC$I3Q999$P4C993$P2D939 FCC$I3Q000$P4C093$P2D930

190

P62c

PHN$I6C006$P2F000

229

Im3m

ICC$I3Q000$P4C000$P2D000

191

P6mmm

PHC$I1A000$P6C000$P2F000

230

Ia3d

ICC$I3Q000$P4C393$P2D933

111

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.2.2. Lattice symbol L The lattice symbol L implies Seitz matrices for the lattice translations. For noncentrosymmetric lattices the rotation parts of the Seitz matrices are for 1 (see Table A1.4.2.4). For centrosymmetric lattices the rotation parts are 1 and 1. The translation parts in the fourth columns of the Seitz matrices are listed in the last column of the table. The total number of matrices implied by each symbol is given by nS. Noncentrosymmetric

Centrosymmetric

Symbol

nS

Symbol

nS

Implied lattice translation(s)

P A B C I R H F

1 2 2 2 2 3 3 4

P A B C I R H F

2 4 4 4 4 6 6 8

0, 0, 0 0, 0, 0 0, 12 , 12 0, 0, 0 12 , 0, 12 0, 0, 0 12 , 12 , 0 0, 0, 0 12 , 12 , 12 0, 0, 0 23 , 13 , 13 0, 0, 0 23 , 13 , 0 0, 0, 0 0, 12 , 12

1 1 2, 2,0

Table A1.4.2.7 lists space-group notation in several formats. The ﬁrst column of Table A1.4.2.7 lists the space-group numbers with axis codes appended to identify the non-standard settings. The second column lists the Hermann–Mauguin symbols in computerentry format with appended codes to identify the origin and cell choice when there are alternatives. The general forms of the Hall notation are listed in the fourth column and the computer-entry representations of these symbols are listed in the third column. The computer-entry format is the general notation expressed as caseinsensitive ASCII characters with the overline (bar) symbol replaced by a minus sign. The Hall notation has the general form:

10 1 0 13 0 1 0 z x 6B CB C B C7 B C [email protected] 1 0 0 [email protected] y A @ 0 A5 @ x A, 0 z y 0 1 0 1 20 10 1 0 1 13 0 1 x 0 1 0 4 4y C 6B CB C B C7 B [email protected] 1 0 0 [email protected] y A @ 34 A5 @ 34 x A, 2

1 2 2 3, 3, 3 1 2 3, 3,0 1 1 2 , 0, 2

0 0 1

1 1 z z 10 1 0 43 13 0 43 1 0 1 0 x 4 4y CB C B C7 B 6B C [email protected] 1 0 0 [email protected] y A @ 14 A5 @ 14 x A 1 1 0 0 1 z 4 4z

0 0 1

20

The corresponding symmetry transformations in reciprocal space, in the notation of Section 1.4.4, are 0 1 2 0 13 0 0 0 1 4hkl@ 1 0 0 A : hkl@ 0 A5 klh : 0; 0 0 1 0 similarly, khl : 1314 and khl : 3114 are obtained from the second and third generator of Ia3d, respectively. The ﬁrst column of Table A1.4.2.1 lists the conventional spacegroup number. The second column shows the conventional short Hermann–Mauguin or international space-group symbol, and the third column, Comments, shows the full international space-group symbol only for the different settings of the monoclinic space groups that are given in the main space-group tables of IT A (1983). Other comments pertain to the choice of the space-group origin – where there are alternatives – and to axial systems. The fourth column shows the explicit space-group symbols described above for each of the settings considered in IT A (1983).

LNAT 1 . . . NAT p V

A1424

L is the symbol specifying the lattice translational symmetry (see Table A1.4.2.2). The integral translations are implicitly included in the set of generators. If L has a leading minus sign, it also speciﬁes an inversion centre at the origin. NAT n speciﬁes the 4 4 Seitz matrix Sn of a symmetry element in the minimum set which deﬁnes the space-group symmetry (see Tables A1.4.2.3 to A1.4.2.6), and p is the number of elements in the set. V is a change-of-basis operator needed for less common descriptions of the space-group symmetry. Table A1.4.2.3. Translation symbol T The symbol T speciﬁes the translation elements of a Seitz matrix. Alphabetical symbols (given in the ﬁrst column) specify translations along a ﬁxed direction. Numerical symbols (given in the third column) specify translations as a fraction of the rotation order |N| and in the direction of the implied or explicitly deﬁned axis.

A1.4.2.3. Hall symbols (S. R. HALL AND R. W. GROSSEKUNSTLEVE) The explicit-origin space-group notation proposed by Hall (1981a) is based on a subset of the symmetry operations, in the form of Seitz matrices, sufﬁcient to uniquely deﬁne a space group. The concise unambiguous nature of this notation makes it well suited to handling symmetry in computing and database applications.

112

Translation symbol

Translation vector

Subscript symbol

a b c n u v w d

1 2 , 0, 0 0, 12 , 0 0, 0, 12 1 1 1 2, 2, 2 1 4 , 0, 0 0, 14 , 0 0, 0, 14 1 1 1 4, 4, 4

1 2 1 3 1 2 4 5

in 31 in 32 in 41 in 43 in 61 in 62 in 64 in 65

Fractional translation 1 3 2 3 1 4 3 4 1 6 1 3 2 3 5 6

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.4. Rotation matrices for principal axes The 3 3 matrices for proper rotations along the three principal unit-cell directions are given below. The matrices for improper rotations (1, 2, 3, 4 and 6) are identical except that the signs of the elements are reversed. Rotation order Axis

Symbol A

a

x

b

y

c

z

1 0

1 @0 0 0 1 @0 0 0 1 @0 0

2 0

1

1 @0 0 0 1 @0 0 0 1 @0 0

0 0 1 0A 0 1 1 0 0 1 0A 0 1 1 0 0 1 0A 0 1

0 1 0 0 1 0 0 1 0

3

Table A1.4.2.5. Rotation matrices for face-diagonal axes The symbols for face-diagonal twofold rotations are 2 and 2 . The facediagonal axis direction is determined by the axis of the preceding rotation Nx, Ny or Nz. Note that the single prime is the default and may be omitted.

N

x

Ny

Nz

1 @0 0 0 1 @0 1 0 0 @1 0

0 0A 1 1 0 0A 1 1 0 0A 1

The matrix symbol NAT is composed of three parts: N is the symbol denoting the |N|-fold order of the rotation matrix (see Tables A1.4.2.4, A1.4.2.5 and A1.4.2.6), T is a subscript symbol denoting the translation vector (see Table A1.4.2.3) and A is a superscript symbol denoting the axis of rotation. The computer-entry format of the Hall notation contains the rotation-order symbol N as positive integers 1, 2, 3, 4, or 6 for proper rotations and as negative integers 1, 2, 3, 4 or 6 for improper rotations. The T translation symbols 1, 2, 3, 4, 5, 6, a, b, c, n, u, v, w, d are described in Table A1.4.2.3. These translations apply additively [e.g. ad signiﬁes a (34 , 14 , 14) translation]. The A axis symbols x, y, z denote rotations about the axes a, b and c, respectively (see Table A1.4.2.4). The axis symbols and signal rotations about the body-diagonal vectors a + b (or alternatively b + c or c + a) and a b (or alternatively b c or c a) (see Table

Preceding rotation

0

1

0 0 1 0 1 0 1 1 0

4 1

0 1 A 1 1 1 0A 0 1 0 0A 1

0

1 @0 0 0 0 @0 1 0 0 @1 0

6 1

0 0 0 1 A 1 0 1 0 1 1 0A 0 0 1 1 0 0 0A 0 1

0

1 @0 0 0 0 @0 1 0 1 @1 0

0 1 1 0 1 0 1 0 0

1 0 1 A 0 1 1 0A 1 1 0 0A 1

A1.4.2.5). The axis symbol * always refers to a threefold rotation along a + b + c (see Table A1.4.2.6). The change-of-basis operator V has the general form (vx, vy, vz). The vectors vx, vy and vz are speciﬁed by vx r1 1 X r1 2 Y r1 3 Z t1 vy r2 1 X r2 2 Y r2 3 Z t2 , vz r3 1 X r3 2 Y r3 3 Z t3 where ri j and ti are fractions or real numbers. Terms in which ri j or ti are zero need not be speciﬁed. The 4 4 change-of-basis matrix operator V is deﬁned as 1 r1 1 r1 2 r1 3 t1 B r2 1 r2 2 r2 3 t2 C C VB @ r3 1 r3 2 r3 3 t3 A 0 0 0 1 The transformed symmetry operations are derived from the speciﬁed Seitz matrices Sn as Sn V Sn V1

Rotation 2

Axis bc

2

b+c

2

ac

2

a+c

2

ab

2

a+b

Matrix 0

1 @0 0 0 1 @0 0 0 0 @0 1 0 0 @0 1 0 0 @ 1 0 0 0 @1 0

and from the integral translations t(1, 0, 0), t(0, 1, 0) and t(0, 0, 1) as 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0

1

0 1 A 0 1 0 1A 0 1 1 0A 0 1 1 0A 0 1 0 0A 1 1 0 0A 1

tn , 1T V tn , 1T A shorthand form of V may be used when the change-of-basis operator only translates the origin of the basis system. In this form vx, vy and vz are speciﬁed simply as shifts in twelfths, implying the matrix operator Table A1.4.2.6. Rotation matrix for the body-diagonal axis The symbol for the threefold rotation in the a + b + c direction is 3*. Note that for cubic space groups the body-diagonal axis is implied and the asterisk * may be omitted.

113

Axis

Rotation

a+b+c

3*

Matrix 0

1 0 0 1 @1 0 0A 0 1 0

1. GENERAL RELATIONSHIPS AND TECHNIQUES 1 The change-of-basis vector (0 0 1) could also be entered as 1 0 0 vx 12 B 0 1 0 vy 12 C (x, y, z 1/12). C VB The reverse setting of the R-centred lattice (hexagonal axes) is @ 0 0 1 vz 12 A speciﬁed using a change-of-basis transformation applied to the 0 0 0 1 standard obverse setting (see Table A1.4.2.2). The obverse Seitz matrices are In the shorthand form of V, the commas separating the vectors may 0 10 10 1 be omitted. 0 1 0 0 1 0 0 23 1 0 0 13 B 0 1 0 2 C B 0 1 0 1 C B 1 1 0 0 C B C A1.4.2.3.1. Default axes 3C B 3C B R3B C, B C, B C 2A @ 1 A @0 @ A 0 1 0 0 0 1 0 0 1 3 3 For most symbols the rotation axes applicable to each N are implied and an explicit axis symbol A is not needed. The rules for 0 0 0 1 0 0 0 1 0 0 0 1 default axis directions are: (i) the ﬁrst rotation or roto-inversion has an axis direction of c; The reverse-setting Seitz matrices are (ii) the second rotation (if |N| is 2) has an axis direction of a if R 3 x, y, z 0 10 10 1 preceded by an |N| of 2 or 4, ab if preceded by an |N| of 3 or 6; 0 1 0 0 1 0 0 23 1 0 0 13 (iii) the third rotation (if |N| is 3) has an axis direction of B 0 1 0 2 C B 0 1 0 1 C B 1 1 0 0 C a + b + c. B C 3C B 3C B B C, B C, B C @ 0 0 1 13 A @ 0 0 1 23 A @ 0 0 1 0 A A1.4.2.3.2. Example matrices 0 0 0 1 0 0 0 1 0 0 0 1 The following examples show how the notation expands to Seitz The conventional primitive hexagonal lattice may be transmatrices. The notation 2xc represents an improper twofold rotation along a formed to a C-centred orthohexagonal setting using the change-ofbasis operator and a c/2 translation: 01 1 1 0 3 1 0 0 0 2 2 0 0 B 1 1 0 0C B 0 1 0 0C B C C 2xc B P 6 x 12y, 12y, z B 2 2 C @ 0 0 1 1 A @0 0 1 0A 2 0 0 0 1 0 0 0 1 The notation 3 represents a threefold rotation along a + b + c: 0 1 In this case the lattice translation for the C centring is obtained by 0 0 1 0 transforming the integral translation t(0, 1, 0): B1 0 0 0C 10 1 0 B C 3 @ 0 1 12 0 0 0 1 0 0A B 0 1 0 0 CB 1 C 0 0 0 1 B CB C 2 V 0 1 0 1 T B CB C @ 0 0 1 0 [email protected] 0 A The notation 4vw represents a fourfold rotation along c (implied) and translation of b/4 and c/4: 1 0 0 0 1 0 1 T 0 1 0 0 1 1 2 2 0 1 B1 0 0 1 C 4 B C 4vw @ The standard setting of an I-centred tetragonal space group may 0 0 1 14 A be transformed to a primitive setting using the change-of-basis 0 0 0 1 operator 1 0 The notation 61 2 (0 0 1) represents a 61 screw along c, a 0 1 0 0 twofold rotation along a b and an origin shift of c/12. Note that B 0 1 1 0 C C the 61 matrix is unchanged by the shifted origin whereas the 2 I 4 y z, x z, x y B @ 1 1 0 0 A matrix is changed by c/6. 0 0 0 1 61 2 0 0 1 0 10 1 Note that in the primitive setting, the fourfold axis is along a + b. 1 1 0 0 0 1 0 0

B1 B B @0

0

0

0 0

B 0 0C C B 1 , B C 1 16 A @ 0 0 1

0

0

0 0

0

0C C 5C 1 6 A 0 1

114

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.7. Hall symbols The ﬁrst column, n:c, lists the space-group numbers and axis codes separated by a colon. The second column lists the Hermann–Mauguin symbols in computerentry format. The third column lists the Hall symbols in computer-entry format and the fourth column lists the Hall symbols as described in Tables A1.4.2.2– A1.4.2.6. n:c

H–M entry

Hall entry

Hall symbol

n:c

H–M entry

Hall entry

Hall symbol

1 2 3:b 3:c 3:a 4:b 4:c 4:a 5:b1 5:b2 5:b3 5:c1 5:c2 5:c3 5:a1 5:a2 5:a3 6:b 6:c 6:a 7:b1 7:b2 7:b3 7:c1 7:c2 7:c3 7:a1 7:a2 7:a3 8:b1 8:b2 8:b3 8:c1 8:c2 8:c3 8:a1 8:a2 8:a3 9:b1 9:b2 9:b3 9:-b1 9:-b2 9:-b3 9:c1 9:c2 9:c3 9:-c1 9:-c2 9:-c3 9:a1 9:a2 9:a3 9:-a1

P1 P -1 P121 P112 P211 P 1 21 1 P 1 1 21 P 21 1 1 C121 A121 I121 A112 B112 I112 B211 C211 I211 P1m1 P11m Pm11 P1c1 P1n1 P1a1 P11a P11n P11b Pb11 Pn11 Pc11 C1m1 A1m1 I1m1 A11m B11m I11m Bm11 Cm11 Im11 C1c1 A1n1 I1a1 A1a1 C1n1 I1c1 A11a B11n I11b B11b A11n I11a Bb11 Cn11 Ic11 Cc11

p1 -p 1 p 2y p2 p 2x p 2yb p 2c p 2xa c 2y a 2y i 2y a2 b2 i2 b 2x c 2x i 2x p -2y p -2 p -2x p -2yc p -2yac p -2ya p -2a p -2ab p -2b p -2xb p -2xbc p -2xc c -2y a -2y i -2y a -2 b -2 i -2 b -2x c -2x i -2x c -2yc a -2yab i -2ya a -2ya c -2yac i -2yc a -2a b -2ab i -2b b -2b a -2ab i -2a b -2xb c -2xac i -2xc c -2xc

P1 P1 P 2y P2 P 2x y P 2b P 2c P 2xa C 2y A 2y I 2y A2 B2 I2 B 2x C 2x I 2x P2y P2 P2x y P2c y P 2 ac y P2a P2a P 2 ab P2b P 2 xb P 2 xbc P 2 xc C2y A2y I2y A2 B2 I2 B2x C2x I2x y C2c y A 2 ab y I2a y A2a y C 2 ac y I2c A2a B 2 ab I2b B2b A 2 ab I2a B 2 xb C 2 xac I 2 xc C 2 xc

9:-a2 9:-a3 10:b 10:c 10:a 11:b 11:c 11:a 12:b1 12:b2 12:b3 12:c1 12:c2 12:c3 12:a1 12:a2 12:a3 13:b1 13:b2 13:b3 13:c1 13:c2 13:c3 13:a1 13:a2 13:a3 14:b1 14:b2 14:b3 14:c1 14:c2 14:c3 14:a1 14:a2 14:a3 15:b1 15:b2 15:b3 15:-b1 15:-b2 15:-b3 15:c1 15:c2 15:c3 15:-c1 15:-c2 15:-c3 15:a1 15:a2 15:a3 15:-a1 15:-a2 15:-a3 16

Bn11 Ib11 P 1 2/m 1 P 1 1 2/m P 2/m 1 1 P 1 21/m 1 P 1 1 21/m P 21/m 1 1 C 1 2/m 1 A 1 2/m 1 I 1 2/m 1 A 1 1 2/m B 1 1 2/m I 1 1 2/m B 2/m 1 1 C 2/m 1 1 I 2/m 1 1 P 1 2/c 1 P 1 2/n 1 P 1 2/a 1 P 1 1 2/a P 1 1 2/n P 1 1 2/b P 2/b 1 1 P 2/n 1 1 P 2/c 1 1 P 1 21/c 1 P 1 21/n 1 P 1 21/a 1 P 1 1 21/a P 1 1 21/n P 1 1 21/b P 21/b 1 1 P 21/n 1 1 P 21/c 1 1 C 1 2/c 1 A 1 2/n 1 I 1 2/a 1 A 1 2/a 1 C 1 2/n 1 I 1 2/c 1 A 1 1 2/a B 1 1 2/n I 1 1 2/b B 1 1 2/b A 1 1 2/n I 1 1 2/a B 2/b 1 1 C 2/n 1 1 I 2/c 1 1 C 2/c 1 1 B 2/n 1 1 I 2/b 1 1 P222

b -2xab i -2xb -p 2y -p 2 -p 2x -p 2yb -p 2c -p 2xa -c 2y -a 2y -i 2y -a 2 -b 2 -i 2 -b 2x -c 2x -i 2x -p 2yc -p 2yac -p 2ya -p 2a -p 2ab -p 2b -p 2xb -p 2xbc -p 2xc -p 2ybc -p 2yn -p 2yab -p 2ac -p 2n -p 2bc -p 2xab -p 2xn -p 2xac -c 2yc -a 2yab -i 2ya -a 2ya -c 2yac -i 2yc -a 2a -b 2ab -i 2b -b 2b -a 2ab -i 2a -b 2xb -c 2xac -i 2xc -c 2xc -b 2xab -i 2xb p22

B 2 xab I 2 xb P 2y P2 P 2x y P 2b P 2c P 2xa C 2y A 2y I 2y A2 B2 I2 B 2x C 2x I 2x y P 2c y P 2ac y P 2a P 2a P 2ab P 2b P 2xb P 2xbc P 2xc y P 2bc y P 2n y P 2ab P 2ac P 2n P 2bc P 2xab P 2xn P 2xac y C 2c y A 2ab y I 2a y A 2a y C 2ac y I 2c A 2a B 2ab I 2b B 2b A 2ab I 2a B 2xb C 2xac I 2xc C 2xc B 2xab I 2xb P22

115

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.2.7. Hall symbols (cont.) n:c

H–M entry

Hall entry

Hall symbol

n:c

H–M entry

Hall entry

Hall symbol

17 17:cab 17:bca 18 18:cab 18:bca 19 20 20:cab 20:bca 21 21:cab 21:bca 22 23 24 25 25:cab 25:bca 26 26:ba-c 26:cab 26:-cba 26:bca 26:a-cb 27 27:cab 27:bca 28 28:ba-c 28:cab 28:-cba 28:bca 28:a-cb 29 29:ba-c 29:cab 29:-cba 29:bca 29:a-cb 30 30:ba-c 30:cab 30:-cba 30:bca 30:a-cb 31 31:ba-c 31:cab 31:-cba 31:bca 31:a-cb 32 32:cab 32:bca 33

P 2 2 21 P 21 2 2 P 2 21 2 P 21 21 2 P 2 21 21 P 21 2 21 P 21 21 21 C 2 2 21 A 21 2 2 B 2 21 2 C222 A222 B222 F222 I222 I 21 21 21 Pmm2 P2mm Pm2m P m c 21 P c m 21 P 21 m a P 21 a m P b 21 m P m 21 b Pcc2 P2aa Pb2b Pma2 Pbm2 P2mb P2cm Pc2m Pm2a P c a 21 P b c 21 P 21 a b P 21 c a P c 21 b P b 21 a Pnc2 Pcn2 P2na P2an Pb2n Pn2b P m n 21 P n m 21 P 21 m n P 21 n m P n 21 m P m 21 n Pba2 P2cb Pc2a P n a 21

p 2c 2 p 2a 2a p 2 2b p 2 2ab p 2bc 2 p 2ac 2ac p 2ac 2ab c 2c 2 a 2a 2a b 2 2b c22 a22 b22 f22 i22 i 2b 2c p 2 -2 p -2 2 p -2 -2 p 2c -2 p 2c -2c p -2a 2a p -2 2a p -2 -2b p -2b -2 p 2 -2c p -2a 2 p -2b -2b p 2 -2a p 2 -2b p -2b 2 p -2c 2 p -2c -2c p -2a -2a p 2c -2ac p 2c -2b p -2b 2a p -2ac 2a p -2bc -2c p -2a -2ab p 2 -2bc p 2 -2ac p -2ac 2 p -2ab 2 p -2ab -2ab p -2bc -2bc p 2ac -2 p 2bc -2bc p -2ab 2ab p -2 2ac p -2 -2bc p -2ab -2 p 2 -2ab p -2bc 2 p -2ac -2ac p 2c -2n

P 2c 2 P 2a 2a P 2 2b P 2 2ab P 2bc 2 P 2ac 2ac P 2ac 2ab C 2c 2 A 2a 2a B 2 2b C22 A22 B22 F22 I22 I 2b 2c P22 P22 P22 P 2c 2 P 2c 2 c P 2a 2a P 2 2a P 2 2b P 2b 2 P 2 2c P 2a 2 P 2b 2b P 2 2a P 2 2b P 2b 2 P 2c 2 P 2c 2c P 2a 2a P 2c 2ac P 2c 2 b P 2b 2a P 2ac 2a P 2bc 2c P 2a 2ab P 2 2bc P 2 2ac P 2ac 2 P 2ab 2 P 2ab 2ab P 2bc 2bc P 2ac 2 P 2bc 2bc P 2ab 2ab P 2 2ac P 2 2bc P 2ab 2 P 2 2ab P 2bc 2 P 2ac 2ac P 2c 2 n

33:ba-c 33:cab 33:-cba 33:bca 33:a-cb 34 34:cab 34:bca 35 35:cab 35:bca 36 36:ba-c 36:cab 36:-cba 36:bca 36:a-cb 37 37:cab 37:bca 38 38:ba-c 38:cab 38:-cba 38:bca 38:a-cb 39 39:ba-c 39:cab 39:-cba 39:bca 39:a-cb 40 40:ba-c 40:cab 40:-cba 40:bca 40:a-cb 41 41:ba-c 41:cab 41:-cba 41:bca 41:a-cb 42 42:cab 42:bca 43 43:cab 43:bca 44 44:cab 44:bca 45 45:cab 45:bca

P b n 21 P 21 n b P 21 c n P c 21 n P n 21 a Pnn2 P2nn Pn2n Cmm2 A2mm Bm2m C m c 21 C c m 21 A 21 m a A 21 a m B b 21 m B m 21 b Ccc2 A2aa Bb2b Amm2 Bmm2 B2mm C2mm Cm2m Am2m Abm2 Bma2 B2cm C2mb Cm2a Ac2m Ama2 Bbm2 B2mb C2cm Cc2m Am2a Aba2 Bba2 B2cb C2cb Cc2a Ac2a Fmm2 F2mm Fm2m Fdd2 F2dd Fd2d Imm2 I2mm Im2m Iba2 I2cb Ic2a

p 2c -2ab p -2bc 2a p -2n 2a p -2n -2ac p -2ac -2n p 2 -2n p -2n 2 p -2n -2n c 2 -2 a -2 2 b -2 -2 c 2c -2 c 2c -2c a -2a 2a a -2 2a b -2 -2b b -2b -2 c 2 -2c a -2a 2 b -2b -2b a 2 -2 b 2 -2 b -2 2 c -2 2 c -2 -2 a -2 -2 a 2 -2b b 2 -2a b -2a 2 c -2a 2 c -2a -2a a -2b -2b a 2 -2a b 2 -2b b -2b 2 c -2c 2 c -2c -2c a -2a -2a a 2 -2ab b 2 -2ab b -2ab 2 c -2ac 2 c -2ac -2ac a -2ab -2ab f 2 -2 f -2 2 f -2 -2 f 2 -2d f -2d 2 f -2d -2d i 2 -2 i -2 2 i -2 -2 i 2 -2c i -2a 2 i -2b -2b

P 2c 2ab P 2bc 2a P 2n 2a P 2n 2ac P 2 ac 2n P 2 2n P 2n 2 P 2n 2n C22 A22 B22 C 2c 2 C 2c 2c A 2a 2a A 2 2a B 2 2b B 2b 2 C 2 2c A 2a 2 B 2b 2b A22 B22 B22 C22 C22 A22 A 2 2b B 2 2a B 2a 2 C 2a 2 C 2a 2a A 2b 2b A 2 2a B 2 2b B 2b 2 C 2c 2 C 2c 2c A 2a 2a A 2 2ab B 2 2ab B 2ab 2 C 2ac 2 C 2ac 2ac A 2ab 2ab F22 F22 F22 F 2 2d F 2d 2 F 2d 2d I22 I22 I22 I 2 2c I 2a 2 I 2b 2b

116

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.7. Hall symbols (cont.) n:c

H–M entry

Hall entry

Hall symbol

n:c

H–M entry

Hall entry

Hall symbol

46 46:ba-c 46:cab 46:-cba 46:bca 46:a-cb 47 48:1 48:2 49 49:cab 49:bca 50:1 50:2 50:1cab 50:2cab 50:1bca 50:2bca 51 51:ba-c 51:cab 51:-cba 51:bca 51:a-cb 52 52:ba-c 52:cab 52:-cba 52:bca 52:a-cb 53 53:ba-c 53:cab 53:-cba 53:bca 53:a-cb 54 54:ba-c 54:cab 54:-cba 54:bca 54:a-cb 55 55:cab 55:bca 56 56:cab 56:bca 57 57:ba-c 57:cab 57:-cba 57:bca 57:a-cb 58 58:cab

Ima2 Ibm2 I2mb I2cm Ic2m Im2a Pmmm P n n n:1 P n n n:2 Pccm Pmaa Pbmb P b a n:1 P b a n:2 P n c b:1 P n c b:2 P c n a:1 P c n a:2 Pmma Pmmb Pbmm Pcmm Pmcm Pmam Pnna Pnnb Pbnn Pcnn Pncn Pnan Pmna Pnmb Pbmn Pcnm Pncm Pman Pcca Pccb Pbaa Pcaa Pbcb Pbab Pbam Pmcb Pcma Pccn Pnaa Pbnb Pbcm Pcam Pmca Pmab Pbma Pcmb Pnnm Pmnn

i 2 -2a i 2 -2b i -2b 2 i -2c 2 i -2c -2c i -2a -2a -p 2 2 p 2 2 -1n -p 2ab 2bc -p 2 2c -p 2a 2 -p 2b 2b p 2 2 -1ab -p 2ab 2b p 2 2 -1bc -p 2b 2bc p 2 2 -1ac -p 2a 2c -p 2a 2a -p 2b 2 -p 2 2b -p 2c 2c -p 2c 2 -p 2 2a -p 2a 2bc -p 2b 2n -p 2n 2b -p 2ab 2c -p 2ab 2n -p 2n 2bc -p 2ac 2 -p 2bc 2bc -p 2ab 2ab -p 2 2ac -p 2 2bc -p 2ab 2 -p 2a 2ac -p 2b 2c -p 2a 2b -p 2ac 2c -p 2bc 2b -p 2b 2ab -p 2 2ab -p 2bc 2 -p 2ac 2ac -p 2ab 2ac -p 2ac 2bc -p 2bc 2ab -p 2c 2b -p 2c 2ac -p 2ac 2a -p 2b 2a -p 2a 2ab -p 2bc 2c -p 2 2n -p 2n 2

I 2 2a I 2 2b I 2b 2 I 2c 2 I 2c 2c I 2a 2a P22 P 2 2 1n P 2ab 2bc P 2 2c P 2a 2 P 2b 2b P 2 2 1ab P 2ab 2b P 2 2 1bc P 2b 2bc P 2 2 1ac P 2a 2c P 2a 2a P 2b 2 P 2 2b P 2c 2c P 2c 2 P 2 2a P 2a 2bc P 2b 2n P 2n 2b P 2ab 2c P 2ab 2n P 2n 2bc P 2ac 2 P 2bc 2bc P 2ab 2ab P 2 2ac P 2 2bc P 2ab 2 P 2a 2ac P 2b 2c P 2a 2b P 2ac 2c P 2bc 2b P 2b 2ab P 2 2ab P 2bc 2 P 2ac 2ac P 2ab 2ac P 2ac 2bc P 2bc 2ab P 2c 2b P 2c 2ac P 2ac 2a P 2b 2a P 2a 2ab P 2bc 2c P 2 2n P 2n 2

58:bca 59:1 59:2 59:1cab 59:2cab 59:1bca 59:2bca 60 60:ba-c 60:cab 60:-cba 60:bca 60:a-cb 61 61:ba-c 62 62:ba-c 62:cab 62:-cba 62:bca 62:a-cb 63 63:ba-c 63:cab 63:-cba 63:bca 63:a-cb 64 64:ba-c 64:cab 64:-cba 64:bca 64:a-cb 65 65:cab 65:bca 66 66:cab 66:bca 67 67:ba-c 67:cab 67:-cba 67:bca 67:a-cb 68:1 68:2 68:1ba-c 68:2ba-c 68:1cab 68:2cab 68:1-cba 68:2-cba 68:1bca 68:2bca 68:1a-cb

Pnmn P m m n:1 P m m n:2 P n m m:1 P n m m:2 P m n m:1 P m n m:2 Pbcn Pcan Pnca Pnab Pbna Pcnb Pbca Pcab Pnma Pmnb Pbnm Pcmn Pmcn Pnam Cmcm Ccmm Amma Amam Bbmm Bmmb Cmca Ccmb Abma Acam Bbcm Bmab Cmmm Ammm Bmmm Cccm Amaa Bbmb Cmma Cmmb Abmm Acmm Bmcm Bmam C c c a:1 C c c a:2 C c c b:1 C c c b:2 A b a a:1 A b a a:2 A c a a:1 A c a a:2 B b c b:1 B b c b:2 B b a b:1

-p 2n 2n p 2 2ab -1ab -p 2ab 2a p 2bc 2 -1bc -p 2c 2bc p 2ac 2ac -1ac -p 2c 2a -p 2n 2ab -p 2n 2c -p 2a 2n -p 2bc 2n -p 2ac 2b -p 2b 2ac -p 2ac 2ab -p 2bc 2ac -p 2ac 2n -p 2bc 2a -p 2c 2ab -p 2n 2ac -p 2n 2a -p 2c 2n -c 2c 2 -c 2c 2c -a 2a 2a -a 2 2a -b 2 2b -b 2b 2 -c 2ac 2 -c 2ac 2ac -a 2ab 2ab -a 2 2ab -b 2 2ab -b 2ab 2 -c 2 2 -a 2 2 -b 2 2 -c 2 2c -a 2a 2 -b 2b 2b -c 2a 2 -c 2a 2a -a 2b 2b -a 2 2b -b 2 2a -b 2a 2 c 2 2 -1ac -c 2a 2ac c 2 2 -1ac -c 2a 2c a 2 2 -1ab -a 2a 2b a 2 2 -1ab -a 2ab 2b b 2 2 -1ab -b 2ab 2b b 2 2 -1ab

P 2n 2n P 2 2ab 1ab P 2ab 2a P 2bc 2 1bc P 2c 2bc P 2ac 2ac 1ac P 2c 2a P 2n 2ab P 2n 2c P 2a 2n P 2bc 2n P 2ac 2b P 2b 2ac P 2ac 2ab P 2bc 2ac P 2ac 2n P 2bc 2a P 2c 2ab P 2n 2ac P 2n 2a P 2c 2n C 2c 2 C 2c 2c A 2a 2a A 2 2a B 2 2b B 2b 2 C 2ac 2 C 2ac 2ac A 2ab 2ab A 2 2ab B 2 2ab B 2ab 2 C22 A22 B22 C 2 2c A 2a 2 B 2b 2b C 2a 2 C 2a 2a A 2b 2b A 2 2b B 2 2a B 2a 2 C 2 2 1ac C 2a 2ac C 2 2 1ac C 2a 2c A 2 2 1ab A 2a 2b A 2 2 1ab A 2ab 2b B 2 2 1ab B 2ab 2b B 2 2 1ab

117

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.2.7. Hall symbols (cont.) n:c

H–M entry

Hall entry

Hall symbol

n:c

H–M entry

Hall entry

Hall symbol

68:2a-cb 69 70:1 70:2 71 72 72:cab 72:bca 73 73:ba-c 74 74:ba-c 74:cab 74:-cba 74:bca 74:a-cb 75 76 77 78 79 80 81 82 83 84 85:1 85:2 86:1 86:2 87 88:1 88:2 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

B b a b:2 Fmmm F d d d:1 F d d d:2 Immm Ibam Imcb Icma Ibca Icab Imma Immb Ibmm Icmm Imcm Imam P4 P 41 P 42 P 43 I4 I 41 P -4 I -4 P 4/m P 42/m P 4/n:1 P 4/n:2 P 42/n:1 P 42/n:2 I 4/m I 41/a:1 I 41/a:2 P422 P 4 21 2 P 41 2 2 P 41 21 2 P 42 2 2 P 42 21 2 P 43 2 2 P 43 21 2 I422 I 41 2 2 P4mm P4bm P 42 c m P 42 n m P4cc P4nc P 42 m c P 42 b c I4mm I4cm I 41 m d I 41 c d P -4 2 m

-b 2b 2ab -f 2 2 f 2 2 -1d -f 2uv 2vw -i 2 2 -i 2 2c -i 2a 2 -i 2b 2b -i 2b 2c -i 2a 2b -i 2b 2 -i 2a 2a -i 2c 2c -i 2 2b -i 2 2a -i 2c 2 p4 p 4w p 4c p 4cw i4 i 4bw p -4 i -4 -p 4 -p 4c p 4ab -1ab -p 4a p 4n -1n -p 4bc -i 4 i 4bw -1bw -i 4ad p42 p 4ab 2ab p 4w 2c p 4abw 2nw p 4c 2 p 4n 2n p 4cw 2c p 4nw 2abw i42 i 4bw 2bw p 4 -2 p 4 -2ab p 4c -2c p 4n -2n p 4 -2c p 4 -2n p 4c -2 p 4c -2ab i 4 -2 i 4 -2c i 4bw -2 i 4bw -2c p -4 2

B 2b 2ab F22 F 2 2 1d F 2uv 2vw I22 I 2 2c I 2a 2 I 2b 2b I 2b 2c I 2a 2b I 2b 2 I 2a 2a I 2c 2c I 2 2b I 2 2a I 2c 2 P4 P 4w P 4c P 4cw I4 I 4bw P4 I4 P4 P 4c P 4ab 1ab P 4a P 4n 1 n P 4bc I4 I 4bw 1bw I 4ad P42 P 4ab 2ab P 4w 2 c P4abw 2 nw P 4c 2 P 4 n 2n P 4cw 2c P 4 nw 2abw I42 I 4bw 2bw P42 P 4 2ab P 4c 2 c P 4n 2 n P 4 2c P 4 2n P 4c 2 P 4c 2ab I42 I 4 2c I 4bw 2 I 4bw 2c P42

112 113 114 115 116 117 118 119 120 121 122 123 124 125:1 125:2 126:1 126:2 127 128 129:1 129:2 130:1 130:2 131 132 133:1 133:2 134:1 134:2 135 136 137:1 137:2 138:1 138:2 139 140 141:1 141:2 142:1 142:2 143 144 145 146:h 146:r 147 148:h 148:r 149 150 151 152 153 154 155:h

P -4 2 c P -4 21 m P -4 21 c P -4 m 2 P -4 c 2 P -4 b 2 P -4 n 2 I -4 m 2 I -4 c 2 I -4 2 m I -4 2 d P 4/m m m P 4/m c c P 4/n b m:1 P 4/n b m:2 P 4/n n c:1 P 4/n n c:2 P 4/m b m P 4/m n c P 4/n m m:1 P 4/n m m:2 P 4/n c c:1 P 4/n c c:2 P 42/m m c P 42/m c m P 42/n b c:1 P 42/n b c:2 P 42/n n m:1 P 42/n n m:2 P 42/m b c P 42/m n m P 42/n m c:1 P 42/n m c:2 P 42/n c m:1 P 42/n c m:2 I 4/m m m I 4/m c m I 41/a m d:1 I 41/a m d:2 I 41/a c d:1 I 41/a c d:2 P3 P 31 P 32 R 3:h R 3:r P -3 R -3:h R -3:r P312 P321 P 31 1 2 P 31 2 1 P 32 1 2 P 32 2 1 R 3 2:h

p -4 2c p -4 2ab p -4 2n p -4 -2 p -4 -2c p -4 -2ab p -4 -2n i -4 -2 i -4 -2c i -4 2 i -4 2bw -p 4 2 -p 4 2c p 4 2 -1ab -p 4a 2b p 4 2 -1n -p 4a 2bc -p 4 2ab -p 4 2n p 4ab 2ab -1ab -p 4a 2a p 4ab 2n -1ab -p 4a 2ac -p 4c 2 -p 4c 2c p 4n 2c -1n -p 4ac 2b p 4n 2 -1n -p 4ac 2bc -p 4c 2ab -p 4n 2n p 4n 2n -1n -p 4ac 2a p 4n 2ab -1n -p 4ac 2ac -i 4 2 -i 4 2c i 4bw 2bw -1bw -i 4bd 2 i 4bw 2aw -1bw -i 4bd 2c p3 p 31 p 32 r3 p 3* -p 3 -r 3 -p 3* p32 p 3 2" p 31 2 (0 0 4) p 31 2" p 32 2 (0 0 2) p 32 2" r 3 2"

P 4 2c P 4 2ab P 4 2n P42 P 4 2c P 4 2ab P 4 2n I42 I 4 2c I42 I 4 2bw P42 P 4 2c P 4 2 1ab P 4a 2b P 4 2 1n P 4a 2bc P 4 2ab P 4 2n P 4ab 2ab 1ab P 4a 2a P 4ab 2n 1ab P 4a 2ac P 4c 2 P 4c 2c P 4n 2c 1n P 4ac 2b P 4n 2 1n P 4ac 2bc P 4c 2ab P 4n 2n P 4n 2n 1n P 4ac 2a P 4n 2ab 1n P 4ac 2ac I42 I 4 2c I 4bw 2bw 1bw I 4bd 2 I 4bw 2aw 1bw I 4bd 2c P3 P 31 P 32 R3 P 3* P3 R3 P 3* P32 P 3 2" P 31 2 (0 0 4) P 31 2" P 32 2 (0 0 2) P 32 2" R 3 2"

118

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.2.7. Hall symbols (cont.) n:c

H–M entry

Hall entry

Hall symbol

n:c

H–M entry

Hall entry

Hall symbol

155:r 156 157 158 159 160:h 160:r 161:h 161:r 162 163 164 165 166:h 166:r 167:h 167:r 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193

R 3 2:r P3m1 P31m P3c1 P31c R 3 m:h R 3 m:r R 3 c:h R 3 c:r P -3 1 m P -3 1 c P -3 m 1 P -3 c 1 R -3 m:h R -3 m:r R -3 c:h R -3 c:r P6 P 61 P 65 P 62 P 64 P 63 P -6 P 6/m P 63/m P622 P 61 2 2 P 65 2 2 P 62 2 2 P 64 2 2 P 63 2 2 P6mm P6cc P 63 c m P 63 m c P -6 m 2 P -6 c 2 P -6 2 m P -6 2 c P 6/m m m P 6/m c c P 63/m c m

p 3* 2 p 3 -2" p 3 -2 p 3 -2"c p 3 -2c r 3 -2" p 3* -2 r 3 -2"c p 3* -2n -p 3 2 -p 3 2c -p 3 2" -p 3 2"c -r 3 2" -p 3* 2 -r 3 2"c -p 3* 2n p6 p 61 p 65 p 62 p 64 p 6c p -6 -p 6 -p 6c p62 p 61 2 (0 p 65 2 (0 p 62 2 (0 p 64 2 (0 p 6c 2c p 6 -2 p 6 -2c p 6c -2 p 6c -2c p -6 2 p -6c 2 p -6 -2 p -6c -2c -p 6 2 -p 6 2c -p 6c 2

P 3* 2 P 3 2" P32 P 3 2"c P 3 2c R 3 2" P 3* 2 R 3 2"c P 3* 2n P32 P 3 2c P 3 2" P 3 2"c R 3 2" P 3* 2 R 3 2"c P 3* 2n P6 P 61 P 65 P 62 P 64 P 6c P6 P6 P 6c P62 P 61 2 (0 P 65 2 (0 P 62 2 (0 P 64 2 (0 P 6c 2c P62 P 6 2c P 6c 2 P 6c 2 c P62 P 6c 2 P62 P 6c 2c P62 P 6 2c P 6c 2

194 195 196 197 198 199 200 201:1 201:2 202 203:1 203:2 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222:1 222:2 223 224:1 224:2 225 226 227:1 227:2 228:1 228:2 229 230

P 63/m m c P23 F23 I23 P 21 3 I 21 3 P m -3 P n -3:1 P n -3:2 F m -3 F d -3:1 F d -3:2 I m -3 P a -3 I a -3 P432 P 42 3 2 F432 F 41 3 2 I432 P 43 3 2 P 41 3 2 I 41 3 2 P -4 3 m F -4 3 m I -4 3 m P -4 3 n F -4 3 c I -4 3 d P m -3 m P n -3 n:1 P n -3 n:2 P m -3 n P n -3 m:1 P n -3 m:2 F m -3 m F m -3 c F d -3 m:1 F d -3 m:2 F d -3 c:1 F d -3 c:2 I m -3 m I a -3 d

-p 6c 2c p223 f223 i223 p 2ac 2ab 3 i 2b 2c 3 -p 2 2 3 p 2 2 3 -1n -p 2ab 2bc 3 -f 2 2 3 f 2 2 3 -1d -f 2uv 2vw 3 -i 2 2 3 -p 2ac 2ab 3 -i 2b 2c 3 p423 p 4n 2 3 f423 f 4d 2 3 i423 p 4acd 2ab 3 p 4bd 2ab 3 i 4bd 2c 3 p -4 2 3 f -4 2 3 i -4 2 3 p -4n 2 3 f -4a 2 3 i -4bd 2c 3 -p 4 2 3 p 4 2 3 -1n -p 4a 2bc 3 -p 4n 2 3 p 4n 2 3 -1n -p 4bc 2bc 3 -f 4 2 3 -f 4a 2 3 f 4d 2 3 -1d -f 4vw 2vw 3 f 4d 2 3 -1ad -f 4ud 2vw 3 -i 4 2 3 -i 4bd 2c 3

P 6c 2c P223 F223 I223 P 2ac 2ab 3 I 2b 2 c 3 P223 P 2 2 3 1n P 2ab 2bc 3 F223 F 2 2 3 1d F 2uv 2vw 3 I223 P 2ac 2ab 3 I 2b 2c 3 P423 P 4n 2 3 F423 F 4d 2 3 I423 P 4acd 2ab 3 P 4bd 2ab 3 I 4bd 2c 3 P423 F423 I423 P 4n 2 3 F 4a 2 3 I 4bd 2c 3 P423 P 4 2 3 1n P 4a 2bc 3 P 4n 2 3 P 4n 2 3 1n P 4bc 2bc 3 F423 F 4a 2 3 F 4d 2 3 1d F 4vw 2vw 3 F 4d 2 3 1ad F 4ud 2vw 3 I423 I 4bd 2c 3

0 5) 0 1) 0 4) 0 2)

0 5) 0 1) 0 4) 0 2)

The codes appended to the space-group numbers listed in the ﬁrst column identify the relationship between the symmetry elements and the crystal cell. Where no code is given the ﬁrst choice listed below applies. Monoclinic. Code = : unique axis choices [cf. IT A (1983) Table 4.3.1] b, -b, c, -c, a, -a; cell choices [cf. IT A (1983) Table 4.3.1] 1, 2, 3. Orthorhombic. Code = : origin choices 1, 2; setting choices [cf. IT A (1983) Table 4.3.1] abc, ba-c, cab, -cba, bca, a-cb. Tetragonal, cubic. Code = : origin choices 1, 2. Trigonal. Code = : cell choices h (hexagonal), r (rhombohedral).

119

1. GENERAL RELATIONSHIPS AND TECHNIQUES Appendix 1.4.3. Structure-factor tables Table A1.4.3.1. Plane groups The symbols appearing in this table are explained in Section 1.4.3 and in Tables A1.4.3.3 (monoclinic), A1.4.3.5 (tetragonal) and A1.4.3.6 (trigonal and hexagonal). System

No.

Oblique

1 2 3 4

p1 p2 pm pg

5 6 7

cm p2mm p2mg

8

p2gg

9 10 11 12

c2mm p4 p4mm p4gm

13 14 15 16 17

p3 p3m1 p31m p6 p6mm

Rectangular

Square

Hexagonal

Symbol

Parity

k 2n k 2n 1

h 2n h 2n 1 h k 2n h k 2n 1

h k 2n h k 2n 1

A

B

c(hk) 2c(hk) 2c(hx)c(ky) 2c(hx)c(ky) 2s(hx)s(ky) 4c(hx)c(ky) 4c(hx)c(ky) 4c(hx)c(ky) 4s(hx)s(ky) 4c(hx)c(ky) 4s(hx)s(ky) 8c(hx)c(ky) 2[P(cc) M(ss)] 4P(cc) 4P(cc) 4M(ss) C(hki) PH(cc) PH(cc) 2C(hki) 2PH(cc)

s(hk) 0 2c(hx)s(ky) 2c(hx)s(ky) 2s(hx)c(ky) 4c(hx)s(ky) 0 0 0 0 0 0 0 0 0 0 S(hki) MH(ss) PH(ss) 0 0

Table A1.4.3.2. Triclinic space groups For the deﬁnition of the triple products ccc, csc etc., see Table A1.4.3.4. P1 [No. 1] hkl

A

B

All

cos 2(hx ky lz) = ccc css scs ssc

sin 2(hx ky lz) = scc csc ccs sss

hkl

A

B

All

2(ccc css scs ssc)

0

P1 [No. 2]

120

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.3. Monoclinic space groups Each expression for A or B in the monoclinic system and for the space-group settings chosen in IT A is represented in terms of one of the following symbols: c
hlcky cos2hx lz cos2ky, chlsky cos2hx lz sin2ky, shlcky sin2hx lz cos2ky,

chkclz cos2hx ky cos2lz, chkslz cos2hx ky sin2lz, shkclz sin2hx ky cos2lz,

shlsky sin2hx lz sin2ky,

shkslz sin2hx ky sin2lz,

A1431

where the left-hand column of expressions corresponds to space-group representations in the second setting, with b taken as the unique axis, and the right-hand column corresponds to representations in the ﬁrst setting, with c taken as the unique axis. The lattice types in this table are P, A, B, C and I, and are all explicit in the full space-group symbol only (see below). Note that s(hl), s(hk), s(ky) and s(lz) are zero for h = l = 0, h = k = 0, k = 0 and l = 0, respectively. Group symbol No.

Short

Full

3 3 4

P2 P2 P21

P121 P112 P121 1

4

P21

P1121

5 5 5 5 5 5 6 6 7

C2 C2 C2 C2 C2 C2 Pm Pm Pc

C121 A121 I121 A112 B112 I112 P1m1 P11m P1c1

7

Pc

P1n1

7

Pc

P1a1

7

Pc

P11a

7

Pc

P11n

7

Pc

P11b

8 8 8 8 8 8 9

Cm Cm Cm Cm Cm Cm Cc

C1m1 A1m1 I1m1 A11m B11m I11m C1c1

9

Cc

A1n1

9

Cc

I1a1

9

Cc

A11a

9

Cc

B11n

9

Cc

I11b

10

P2m

P12m1

Parity

k 2n k 2n 1 l 2n l 2n 1

l 2n l 2n 1 h l 2n h l 2n 1 h 2n h 2n 1 h 2n h 2n 1 h k 2n h k 2n 1 k 2n k 2n 1

l 2n l 2n 1 h l 2n h l 2n 1 h 2n h 2n 1 h 2n h 2n 1 h k 2n h k 2n 1 k 2n k 2n 1

121

A

B

2c(hl)c(ky) 2c(hk)c(lz) 2c(hl)c(ky) 2s(hl)s(ky) 2c(hk)c(lz) 2s(hk)s(lz) 4c(hl)c(ky) 4c(hl)c(ky) 4c(hl)c(ky) 4c(hk)c(lz) 4c(hk)c(lz) 4c(hk)c(lz) 2c(hl)c(ky) 2c(hk)c(lz) 2c(hl)c(ky) 2s(hl)s(ky) 2c(hl)c(ky) 2s(hl)s(ky) 2c(hl)c(ky) 2s(hl)s(ky) 2c(hk)c(lz) 2s(hk)s(lz) 2c(hk)c(lz) 2s(hk)s(lz) 2c(hk)c(lz) 2s(hk)s(lz) 4c(hl)c(ky) 4c(hl)c(ky) 4c(hl)c(ky) 4c(hk)c(lz) 4c(hk)c(lz) 4c(hk)c(lz) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hl)c(ky)

2c(hl)s(ky) 2c(hk)s(lz) 2c(hl)s(ky) 2s(hl)c(ky) 2c(hk)s(lz) 2s(hk)c(lz) 4c(hl)s(ky) 4c(hl)s(ky) 4c(hl)s(ky) 4c(hk)s(lz) 4c(hk)s(lz) 4c(hk)s(lz) 2s(hl)c(ky) 2s(hk)c(lz) 2s(hl)c(ky) 2c(hl)s(ky) 2s(hl)c(ky) 2c(hl)s(ky) 2s(hl)c(ky) 2c(hl)s(ky) 2s(hk)c(lz) 2c(hk)s(lz) 2s(hk)c(lz) 2c(hk)s(lz) 2s(hk)c(lz) 2c(hk)s(lz) 4s(hl)c(ky) 4s(hl)c(ky) 4s(hl)c(ky) 4s(hk)c(lz) 4s(hk)c(lz) 4s(hk)c(lz) 4s(hl)c(ky) 4c(hl)s(ky) 4s(hl)c(ky) 4c(hl)s(ky) 4s(hl)c(ky) 4c(hl)s(ky) 4s(hk)c(lz) 4c(hk)s(lz) 4s(hk)c(lz) 4c(hk)s(lz) 4s(hk)c(lz) 4c(hk)s(lz) 0

Unique axis b c b c b b b c c c b c b b b c c c b b b c c c b b b c c c b

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.3. Monoclinic space groups (cont.) Group symbol No.

Short

Full

10 11

P2m P21 m

P112m P121 m1

11

P21 m

P1121 m

12 12 12 12 12 12 13

C2m C2m C2m C2m C2m C2m P2c

C12m1 A12m1 I12m1 A112m B112m I112m P12c1

13

P2c

P12n1

13

P2c

P12a1

13

P2c

P112a

13

P2c

P112n

13

P2c

P112b

14

P21 c

P121 c1

14

P21 c

P121 n1

14

P21 c

P121 a1

14

P21 c

P1121 a

14

P21 c

P1121 n

14

P21 c

P1121 b

15

C2c

C12c1

15

C2c

A12n1

15

C2c

I12a1

15

C2c

A112a

15

C2c

B112n

15

C2c

I112b

Parity

A

k 2n k 2n 1 l 2n l 2n 1

l 2n l 2n 1 h l 2n h l 2n 1 h 2n h 2n 1 h 2n h 2n 1 h k 2n h k 2n 1 k 2n k 2n 1 k l 2n k l 2n 1 h k l 2n h k l 2n 1 h k 2n h k 2n 1 h l 2n h l 2n 1 h k l 2n h k l 2n 1 k l 2n k l 2n 1 l 2n l 2n 1 h l 2n h l 2n 1 h 2n h 2n 1 h 2n h 2n 1 h k 2n h k 2n 1 k 2n k 2n 1

122

4c(hk)c(lz) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hk)c(lz) 4s(hk)s(lz) 8c(hl)c(ky) 8c(hl)c(ky) 8c(hl)c(ky) 8c(hk)c(lz) 8c(hk)c(lz) 8c(hk)c(lz) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hl)c(ky) 4s(hl)s(ky) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 4c(hk)c(lz) 4s(hk)s(lz) 8c(hl)c(ky) 8s(hl)s(ky) 8c(hl)c(ky) 8s(hl)s(ky) 8c(hl)c(ky) 8s(hl)s(ky) 8c(hk)c(lz) 8s(hk)s(lz) 8c(hk)c(lz) 8s(hk)s(lz) 8c(hk)c(lz) 8s(hk)s(lz)

Unique axis

B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c b c b b b c c c b b b c c c b b b c c c b b b c c c

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.4. Orthorhombic space groups The expressions for A and B for the orthorhombic space groups in their standard settings [as in IT A (1983)] contain one, two or four terms of the form pqr p2hxq2kyr2lz

A1432

preceded by a signed numerical constant, where p, q and r can each be either a sine or a cosine function, and the arguments of the functions in any product of the form (A1.4.3.2) are ordered as in (A1.4.3.2). These products are given in this table as ccc, ccs, csc, scc, ssc, scs, css and/or sss, where c and s are abbreviations for ‘sin’ and ‘cos’, respectively. Note that pqr vanishes if at least one of p, q and r is a sine, and the corresponding index h, k or l is zero. No.

Symbol

16 17

P222 P2221

18

P21 21 2

19

P21 21 21

20

C2221

21 22 23 24

C222 F222 I222 I21 21 21

25 26

Pmm2 Pmc21

27

Pcc2

28

Pma2

29

Pca21

30

Pnc2

31

Pmn21

32

Pba2

33

Pna21

34

Pnn2

35 36

Cmm2 Cmc21

37

Ccc2

38 39

Amm2 Abm2

40

Ama2

Origin

Parity

A

B 4ccc 4ccc 4css 4ccc 4ssc 4ccc 4css 4scs 4ssc 8ccc 8css 8ccc 16ccc 8ccc 8ccc 8scs 8ssc 8css 4ccc 4ccc 4css 4ccc 4ssc 4ccc 4ssc 4ccc 4scs 4ssc 4css 4ccc 4ssc 4ccc 4css 4ccc 4ssc 4ccc 4scs 4ssc 4css 4ccc 4ssc 8ccc 8ccc 8css 8ccc 8ssc 8ccc 8ccc 8ssc 8ccc

l 2n l 2n 1 h k 2n h k 2n 1 h k 2n; k l 2n h k 2n; k l 2n 1 h k 2n 1; k l 2n h k 2n 1; k l 2n 1 l 2n l 2n 1

h, k, l all even h 2n; k, l 2n 1 k 2n; l, h 2n 1 l 2n; h, k 2n 1 l 2n l 2n 1 l 2n l 2n 1 h 2n h 2n 1 h 2n; l 2n h 2n; l 2n 1 h 2n 1; l 2n h 2n 1; l 2n 1 k l 2n k l 2n 1 h l 2n h l 2n 1 h k 2n h k 2n 1 h k 2n; l 2n h k 2n; l 2n 1 h k 2n 1; l 2n h k 2n 1; l 2n 1 h k l 2n h k l 2n 1 l 2n l 2n 1 l 2n l 2n 1 k 2n k 2n 1 h 2n

123

4sss 4sss 4scc 4sss 4ccs 4sss 4scc 4csc 4ccs 8sss 8scc 8sss 16sss 8sss 8sss 8csc 8ccs 8scc 4ccs 4ccs 4csc 4ccs 4sss 4ccs 4sss 4ccs 4scc 4sss 4csc 4ccs 4sss 4ccs 4csc 4ccs 4sss 4ccs 4scc 4sss 4csc 4ccs 4sss 8ccs 8ccs 8csc 8ccs 8sss 8ccs 8ccs 8sss 8ccs

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.4. Orthorhombic space groups (cont.) No.

Symbol

Origin

41

Aba2

42 43

Fmm2 Fdd2

44 45

Imm2 Iba2

46

Iam2

47 48

Pmmm Pnnn

(1)

48

Pnnn

(2)

49

Pccm

50

Pban

(1)

50

Pban

(2)

51

Pmma

52

Pnna

53

Pmna

54

Pcca

55

Pbam

56

Pccn

57

Pbcm

58

Pnnm

59

Pmmn

(1)

59

Pmmn

(2)

Parity

A

h 2n 1 h k 2n h k 2n 1

8ssc 8ccc 8ssc 16ccc 16ccc 8(ccc ssc ccs sss) 16ssc 8(ccc ssc ccs sss) 8ccc 8ccc 8ssc 8ccc 8ssc 8ccc 8ccc 0 8ccc 8ssc 8css 8scs 8ccc 8ssc 8ccc 0 8ccc 8scs 8css 8ssc 8ccc 8scs 8ccc 8ssc 8css 8scs 8ccc 8css 8ccc 8ssc 8scs 8css 8ccc 8ssc 8ccc 8ssc 8css 8scs 8ccc 8css 8ssc 8scs 8ccc 8ssc 8ccc 0 8ccc 8css

h k l 4n h k l 4n 1 h k l 4n 2 h k l 4n 3 l 2n l 2n 1 h 2n h 2n 1 h k l 2n h k l 2n 1 h k 2n; k l 2n h k 2n; k l 2n 1 h k 2n 1; k l 2n h k 2n 1; k l 2n 1 l 2n l 2n 1 h k 2n h k 2n 1 h 2n; k 2n h 2n; k 2n 1 h 2n 1; k 2n h 2n 1; k 2n 1 h 2n h 2n 1 h 2n; k l 2n h 2n; k l 2n 1 h 2n 1; k l 2n h 2n 1; k l 2n 1 h l 2n h l 2n 1 h 2n; l 2n h 2n; l 2n 1 h 2n 1; l 2n h 2n 1; l 2n 1 h k 2n h k 2n 1 h k 2n; h l 2n h k 2n; h l 2n 1 h k 2n 1; h l 2n h k 2n 1; h l 2n 1 k 2n; l 2n k 2n; l 2n 1 k 2n 1; l 2n k 2n 1; l 2n 1 h k l 2n h k l 2n 1 h k 2n h k 2n 1 h 2n; k 2n h 2n; k 2n 1

124

B 8sss 8ccs 8sss 16ccs 16ccs 8(ccs sss ccc ssc) 16sss 8(ccs sss ccc ssc) 8ccs 8ccs 8sss 8ccs 8sss 0 0 8sss 0 0 0 0 0 0 0 8sss 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8ccs 0 0

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.4. Orthorhombic space groups (cont.) No.

Symbol

Origin

Pbcn

61

Pbca

62

Pnma

63

Cmcm

64

Cmca

65 66

Cmmm Cccm

67

Cmma

68

Ccca

(1)

68

Ccca

(2)

69 70

Fmmm Fddd

(1)

70

Fddd

(2)

Immm Ibam

73

Ibca

74

Imma

A

h 2n 1; k 2n h 2n 1; k 2n 1 h k 2n; l 2n h k 2n; l 2n 1 h k 2n 1; l 2n h k 2n 1; l 2n 1 h k 2n; k l 2n h k 2n; k l 2n 1 h k 2n 1; k l 2n h k 2n 1; k l 2n 1 h l 2n; k 2n h l 2n; k 2n 1 h l 2n 1; k 2n h l 2n 1; k 2n 1 l 2n l 2n 1 k l 2n k l 2n 1

60

71 72

Parity

l 2n l 2n 1 h 2n h 2n 1 h l 2n h l 2n 1 k 2n; l 2n k 2n; l 2n 1 k 2n 1; l 2n k 2n 1; l 2n 1 h k l 4n h k l 4n 1 h k l 4n 2 h k l 4n 3 h k 4n; k l 4n; l h 4n h k 4n; k l 4n 2; l h 4n 2 h k 4n 2; k l 4n; l h 4n 2 h k 4n 2; k l 4n 2; l h 4n h k 4n 2; k l 4n 2; l h 4n 2 h k 4n 2; k l 4n; l h 4n h k 4n; k l 4n 2; l h 4n h k 4n; k l 4n; l h 4n 2 l 2n l 2n 1 h 2n; k 2n h 2n; k 2n 1 h 2n 1; k 2n h 2n 1; k 2n 1 k 2n k 2n 1

125

B 8scs 8ssc 8ccc 8css 8scs 8ssc 8ccc 8css 8scs 8ssc 8ccc 8ssc 8scs 8css 16ccc 16css 16ccc 16css 16ccc 16ccc 16ssc 16ccc 16css 16ccc 0 16ccc 16ssc 16scs 16css 32ccc 32ccc 16(ccc sss) 0 16(ccc sss) 32ccc 32ssc

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16sss 0 0 0 0 0 0 A 32sss A 0 0

32css

0

32scs

0

16(ccc ssc scs css)

0

16(ccc ssc scs css) 16(ccc ssc scs css) 16(ccc ssc scs css) 16ccc 16ccc 16ssc 16ccc 16scs 16ssc 16css 16ccc 16css

0 0 0 0 0 0 0 0 0 0 0 0

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups The symbols appearing in this table are based on the factorization of the scalar product appearing in equations (1.4.2.19) and (1.4.2.20) into its plane-group and unique-axis components. The symbols are P
pq p2hxq2ky p2hyq2kx Mpq p2hxq2ky p2hyq2kx, where p and q can each be a sine or a cosine. Explicit trigonometric functions given in the table follow the convention cu cos2u su sin2u Conditions for vanishing symbols: Pss Mss 0 if h 0 or k 0, Psc Msc 0 if h 0, Pcs Mcs 0 if k 0, Mcc Mss 0 if h k or h k, and any explicit sine function vanishes if all the indices (h and k, or l) appearing in its argument are zero. P4 [No. 75] hkl

A

B

All

2[P(cc) M(ss)]c(lz)

2[P(cc) M(ss)]s(lz)

P41 [No. 76] (enantiomorphous to P43 [No. 78]) l

A

B

4n 4n 1 4n 2 4n 3

2[P(cc) M(ss)]c(lz) 2[s(hx ky)s(lz) s(hy kx)c(lz)] 2[M(cc) P(ss)]c(lz) 2[s(hx ky)s(lz) s(hy kx)c(lz)]

2[P(cc) M(ss)]s(lz) 2[s(hx ky)c(lz) s(hy kx)s(lz)] 2[M(cc) P(ss)]s(lz) 2[s(hx ky)c(lz) s(hy kx)s(lz)]

P42 [No. 77] l

A

B

2n 2n 1

2[P(cc) M(ss)]c(lz) 2[M(cc) P(ss)]c(lz)

2[P(cc) M(ss)]s(lz) 2[M(cc) P(ss)]s(lz)

P43 [No. 78] (enantiomorphous to P41 [No. 76]) l

A

B

4n 4n 1 4n 2 4n 3

2[P(cc) M(ss)]c(lz) 2[s(hx ky)s(lz) s(hy kx)c(lz)] 2[M(cc) P(ss)]c(lz) 2[s(hx ky)s(lz) s(hy kx)c(lz)]

2[P(cc) M(ss)]s(lz) 2[s(hx ky)c(lz) s(hy kx)s(lz)] 2[M(cc) P(ss)]s(lz) 2[s(hx ky)c(lz) s(hy kx)s(lz)]

hkl

A

B

All

4[P(cc) M(ss)]c(lz)

4[P(cc) M(ss)]s(lz)

I4 [No. 79]

I41 [No. 80] 2h l

A

B

4n 4n 1 4n 2 4n 3

4[P(cc) M(ss)]c(lz) 4[c(hx ky)c(lz) c(hy kx)s(lz)] 4[M(cc) P(ss)]c(lz) 4[c(hx ky)c(lz) c(hy kx)s(lz)]

4[P(cc) M(ss)]s(lz) 4[c(hx ky)s(lz) c(hy kx)c(lz)] 4[M(cc) P(ss)]s(lz) 4[c(hx ky)s(lz) c(hy kx)c(lz)]

126

A1433

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.5. Tetragonal space groups (cont.)

P4 [No. 81] hkl

A

B

All

2[P(cc) M(ss)]c(lz)

2[M(cc) P(ss)]s(lz)

I4 [No. 82] hkl

A

B

All

4[P(cc) M(ss)]c(lz)

4[M(cc) P(ss)]s(lz)

hkl

A

B

All

4[P(cc) M(ss)]c(lz)

0

P4m [No. 83]

P42 m [No. 84] (B = 0 for all h, k, l) l

A

2n 2n 1

4[P(cc) M(ss)]c(lz) 4[M(cc) P(ss)]c(lz)

P4n [No. 85, Origin 1] hk

A

B

2n 2n 1

4[P(cc) M(ss)]c(lz) 0

0 4[M(cc) P(ss)]s(lz)

P4n [No. 85, Origin 2] (B = 0 for all h, k, l) h

k

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

4[P(cc) M(ss)]c(lz) 4[P(cs) M(sc)]s(lz) 4[M(cs) P(sc)]s(lz) 4[M(cc) P(ss)]c(lz)

P42 n [No. 86, Origin 1] hkl

A

B

2n 2n 1

4[P(cc) M(ss)]c(lz) 0

0 4[M(cc) P(ss)]s(lz)

P42 n [No. 86, Origin 2] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 1 2n

2n 2n 1 2n 2n 1

4[P(cc) M(ss)]c(lz) 4[M(cc) P(ss)]c(lz) 4[M(cs) P(sc)]s(lz) 4[P(cs) M(sc)]s(lz)

I4m [No. 87] hkl

A

B

All

8[P(cc) M(ss)]c(lz)

0

127

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups (cont.)

I41 a [No. 88, Origin 1] 2k l

A

B

4n 4n 1 4n 2 4n 3

8[P(cc) M(ss)]c(lz) 4[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz) 0 4[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz)

0 A 8[M(cc) P(ss)]s(lz) A

I41 a [No. 88, Origin 2] (B = 0 for all h, k, l) h

k

hkl

A

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

8[P(cc) M(ss)]c(lz) 8[s(hx ky)s(lz) c(hy kx)c(lz)] 8[c(hx ky)c(lz) s(hy kx)s(lz)] 8[M(cs) P(sc)]s(lz) 8[M(cc) P(ss)]c(lz) 8[s(hx ky)s(lz) c(hy kx)c(lz)] 8[c(hx ky)c(lz) s(hy kx)s(lz)] 8[P(cs) M(sc)]s(lz)

P422 [No. 89] hkl

A

B

All

4P(cc)c(lz)

4M(ss)s(lz)

hk

A

B

2n 2n 1

4P(cc)c(lz) 4P(ss)c(lz)

4M(ss)s(lz) 4M(cc)s(lz)

P421 2 [No. 90]

P41 22 [No. 91] (enantiomorphous to P43 22 [No. 95]) l

A

B

4n 4n 1 4n 2 4n 3

4P(cc)c(lz) 4[s(hx)c(ky)s(lz) c(kx)s(hy)c(lz)] 4M(cc)c(lz) 4[s(hx)c(ky)s(lz) c(kx)s(hy)c(lz)]

4M(ss)s(lz) 4[c(hx)s(ky)c(lz) s(kx)c(hy)s(lz)] 4P(ss)s(lz) 4[c(hx)s(ky)c(lz) s(kx)c(hy)s(lz)]

P41 21 2 [No. 92] (enantiomorphous to P43 21 2 [No. 96]) 2h 2k l

A

B

4n 4n 1 4n 2 4n 3

4P(cc)c(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)} 4P(ss)c(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)}

4M(ss)s(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)} 4M(cc)s(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)s(lz)}

l

A

B

2n 2n 1

4P(cc)c(lz) 4M(cc)c(lz)

4M(ss)s(lz) 4P(ss)s(lz)

P42 22 [No. 93]

128

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.5. Tetragonal space groups (cont.)

P42 21 2 [No. 94] hkl

A

B

2n 2n 1

4P(cc)c(lz) 4P(ss)c(lz)

4M(ss)s(lz) 4M(cc)s(lz)

P43 22 [No. 95] (enantiomorphous to P41 22 [No. 91]) l

A

B

4n 4n 1 4n 2 4n 3

4P(cc)c(lz) 4[s(hx)c(ky)s(lz) c(kx)s(hy)c(lz)] 4M(cc)c(lz) 4[s(hx)c(ky)s(lz) c(kx)s(hy)c(lz)]

4M(ss)s(lz) 4[c(hx)s(ky)c(lz) s(kx)c(hy)c(lz)] 4P(ss)s(lz) 4[c(hx)s(ky)c(lz) s(kx)c(hy)c(lz)]

P43 21 2 [No. 96] (enantiomorphous to P41 21 2 [No. 92]) 2h 2k l

A

B

4n 4n 1 4n 2 4n 3

4P(cc)c(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)} 4P(ss)c(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)}

4M(ss)s(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)} 4M(cc)s(lz) 2{[P(sc) P(cs)]c(lz) [M(cs) M(sc)]s(lz)}

hkl

A

B

All

8P(cc)c(lz)

8M(ss)s(lz)

2k l

A

B

4n 4n 1 4n 2 4n 3

8P(cc)c(lz) 4{[P(cc) P(ss)]c(lz) [M(cc) M(ss)]s(lz)} 8P(ss)c(lz) 4{[P(cc) P(ss)]c(lz) [M(cc) M(ss)]s(lz)}

8M(ss)s(lz) 4{[P(cc) P(ss)]c(lz) [M(cc) M(ss)]s(lz)} 8M(cc)s(lz) 4{[P(cc) P(ss)]c(lz) [M(cc) M(ss)]s(lz)}

hkl

A

B

All

4P(cc)c(lz)

4P(cc)s(lz)

hk

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(cc)s(lz) 4M(ss)s(lz)

l

A

B

2n 2n 1

4P(cc)c(lz) 4P(ss)c(lz)

4P(cc)s(lz) 4P(ss)s(lz)

I422 [No. 97]

I41 22 [No. 98]

P4mm [No. 99]

P4bm [No. 100]

P42 cm [No. 101]

129

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups (cont.)

P42 nm [No. 102] hkl

A

B

2n 2n 1

4P(cc)c(lz) 4P(ss)c(lz)

4P(cc)s(lz) 4P(ss)s(lz)

l

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(cc)s(lz) 4M(ss)s(lz)

hkl

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(cc)s(lz) 4M(ss)s(lz)

l

A

B

2n 2n 1

4P(cc)c(lz) 4M(cc)c(lz)

4P(cc)s(lz) 4M(cc)s(lz)

P4cc [No. 103]

P4nc [No. 104]

P42 mc [No. 105]

P42 bc [No. 106] hk

l

A

B

2n 2n 1 2n 2n 1

2n 2n 2n 1 2n 1

4P(cc)c(lz) 4M(ss)c(lz) 4M(cc)c(lz) 4P(ss)c(lz)

4P(cc)s(lz) 4M(ss)s(lz) 4M(cc)s(lz) 4P(ss)s(lz)

I4mm [No. 107] hkl

A

B

All

8P(cc)c(lz)

8P(cc)s(lz)

l

A

B

2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz)

8P(cc)s(lz) 8M(ss)s(lz)

2k l

A

B

4n 4n 1 4n 2 4n 3

8P(cc)c(lz) 8[c(hx)c(ky)c(lz) c(kx)c(hy)s(lz)] 8M(cc)c(lz) 8[c(hx)c(ky)c(lz) c(kx)c(hy)s(lz)]

8P(cc)s(lz) 8[c(hx)c(ky)s(lz) c(kx)c(hy)c(lz)] 8M(cc)s(lz) 8[c(hx)c(ky)s(lz) c(kx)c(hy)c(lz)]

I4cm [No. 108]

I41 md [No. 109]

130

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.5. Tetragonal space groups (cont.)

I41 cd [No. 110] 2k l

A

B

4n 4n 1 4n 2 4n 3

8P(cc)c(lz) 8[s(hx)s(ky)c(lz) s(kx)s(hy)s(lz)] 8M(cc)c(lz) 8[s(hx)s(ky)c(lz) s(kx)s(hy)s(lz)]

8P(cc)s(lz) 8[s(hx)s(ky)s(lz) s(kx)s(hy)c(lz)] 8M(cc)s(lz) 8[s(hx)s(ky)s(lz) s(kx)s(hy)c(lz)]

P42m [No. 111] hkl

A

B

All

4P(cc)c(lz)

4P(ss)s(lz)

P42c [No. 112] l

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(ss)s(lz) 4M(cc)s(lz)

hk

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(ss)s(lz) 4M(cc)s(lz)

P421 m [No. 113]

P421 c [No. 114] hkl

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4P(ss)s(lz) 4M(cc)s(lz)

hkl

A

B

All

4P(cc)c(lz)

4M(cc)s(lz)

l

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4M(cc)s(lz) 4P(ss)s(lz)

hk

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4M(cc)s(lz) 4P(ss)s(lz)

hkl

A

B

2n 2n 1

4P(cc)c(lz) 4M(ss)c(lz)

4M(cc)s(lz) 4P(ss)s(lz)

P4m2 [No. 115]

P4c2 [No. 116]

P4b2 [No. 117]

P4n2 [No. 118]

131

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups (cont.)

I4m2 [No. 119] hkl

A

B

All

8P(cc)c(lz)

8M(cc)s(lz)

l

A

B

2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz)

8M(cc)s(lz) 8P(ss)s(lz)

hkl

A

B

All

8P(cc)c(lz)

8P(ss)s(lz)

2h l

A

B

4n 4n 1 4n 2 4n 3

8P(cc)c(lz) 4{[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz)} 8M(ss)c(lz) 4{[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz)}

8P(ss)s(lz) 4{[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz)} 8M(cc)s(lz) 4{[P(cc) M(ss)]c(lz) [M(cc) P(ss)]s(lz)}

hkl

A

B

All

8P(cc)c(lz)

0

I4c2 [No. 120]

I42m [No. 121]

I42d [No. 122]

P4mmm [No. 123]

P4mcc [No. 124] (B = 0 for all h, k, l) l

A

2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz)

P4nbm [No. 125, Origin 1] hk

A

B

2n 2n 1

8P(cc)c(lz) 0

0 8M(ss)s(lz)

P4nbm [No. 125, Origin 2] (B = 0 for all h, k, l) h

k

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(sc)s(lz) 8M(cs)s(lz) 8P(ss)c(lz)

132

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.5. Tetragonal space groups (cont.)

P4nnc [No. 126, Origin 1] hkl

A

B

2n 2n 1

8P(cc)c(lz) 0

0 8M(ss)s(lz)

P4nnc [No. 126, Origin 2] (B = 0 for all h, k, l) h

k

l

A

2n 2n 2n 2n 2n 1 2n 1 2n 1 2n 1

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz) 8M(sc)s(lz) 8P(cs)s(lz) 8M(cs)s(lz) 8P(sc)s(lz) 8P(ss)c(lz) 8M(cc)c(lz)

P4mbm [No. 127] (B = 0 for all h, k, l) hk

A

2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz)

P4mnc [No. 128] (B = 0 for all h, k, l) hkl

A

2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz)

P4nmm [No. 129, Origin 1] hk

A

B

2n 2n 1

8P(cc)c(lz) 0

0 8M(cc)s(lz)

P4nmm [No. 129, Origin 2] (B = 0 for all h, k, l) h

k

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8P(cs)s(lz) 8P(sc)s(lz) 8P(ss)c(lz)

P4ncc [No. 130, Origin 1] hk

l

A

B

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz) 0 0

0 0 8M(cc)s(lz) 8P(ss)s(lz)

133

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups (cont.)

P4ncc [No. 130, Origin 2] (B = 0 for all h, k, l) h

k

l

A

2n 2n 2n 2n 2n 1 2n 1 2n 1 2n 1

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz) 8P(cs)s(lz) 8M(sc)s(lz) 8P(sc)s(lz) 8M(cs)s(lz) 8P(ss)c(lz) 8M(cc)c(lz)

P42 mmc [No. 131] (B = 0 for all h, k, l) l

A

2n 2n 1

8P(cc)c(lz) 8M(cc)c(lz)

P42 mcm [No. 132] (B = 0 for all h, k, l) l

A

2n 2n 1

8P(cc)c(lz) 8P(ss)c(lz)

P42 nbc [No. 133, Origin 1] hkl

l

A

B

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz) 0 0

0 0 8P(ss)s(lz) 8M(cc)s(lz)

P42 nbc [No. 133, Origin 2] (B = 0 for all h, k, l) h

k

l

A

2n 2n 2n 2n 2n 1 2n 1 2n 1 2n 1

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(cc)c(lz) 8M(sc)s(lz) 8P(sc)s(lz) 8M(cs)s(lz) 8P(cs)s(lz) 8P(ss)c(lz) 8M(ss)c(lz)

P42 nnm [No. 134, Origin 1] hkl

A

B

2n 2n 1

8P(cc)c(lz) 0

0 8P(ss)s(lz)

P42 nnm [No. 134, Origin 2] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 1 2n

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8P(ss)c(lz) 8M(sc)s(lz) 8M(cs)s(lz)

134

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.5. Tetragonal space groups (cont.)

P42 mbc [No. 135] (B = 0 for all h, k, l) hk

l

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(cc)c(lz) 8M(ss)c(lz) 8P(ss)c(lz)

P42 mnm [No. 136] (B = 0 for all h, k, l) hkl

A

2n 2n 1

8P(cc)c(lz) 8P(ss)c(lz)

P42 nmc [No. 137, Origin 1] hkl

A

B

2n 2n 1

8P(cc)c(lz) 0

0 8M(cc)s(lz)

P42 nmc [No. 137, Origin 2] (B = 0 for all h, k, l) h

k

l

A

2n 2n 2n 2n 2n 1 2n 1 2n 1 2n 1

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(cc)c(lz) 8P(cs)s(lz) 8M(cs)s(lz) 8P(sc)s(lz) 8M(sc)s(lz) 8P(ss)c(lz) 8M(ss)c(lz)

P42 ncm [No. 138, Origin 1] hk

l

A

B

2n 2n 1 2n 1 2n

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8M(ss)c(lz) 0 0

0 0 8M(cc)s(lz) 8P(ss)s(lz)

P42 ncm [No. 138, Origin 2] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 1 2n

2n 2n 1 2n 2n 1

8P(cc)c(lz) 8P(ss)c(lz) 8P(cs)s(lz) 8P(sc)s(lz)

I4mmm [No. 139] hkl

A

B

All

16P(cc)c(lz)

0

135

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.5. Tetragonal space groups (cont.)

I4mcm [No. 140] (B = 0 for all h, k, l) l

A

2n 2n 1

16P(cc)c(lz) 16M(ss)c(lz)

I41 amd [No. 141, Origin 1] 2h l

A

B

4n 4n 1 4n 2 4n 3

16P(cc)c(lz) 8[P(cc)c(lz) M(cc)s(lz)] 0 8[P(cc)c(lz) M(cc)s(lz)]

0 A 16M(cc)s(lz) A

I41 amd [No. 141, Origin 2] (B = 0 for all h, k, l) h

k

hkl

A

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

16P(cc)c(lz) 16[c(hx)s(ky)s(lz) c(kx)c(hy)c(lz)] 16[c(hx)c(ky)c(lz) c(kx)s(hy)s(lz)] 16[c(hx)s(ky)s(lz) c(kx)s(hy)s(lz)] 16M(cc)c(lz) 16[c(hx)s(ky)s(lz) c(kx)c(hy)c(lz)] 16[c(hx)c(ky)c(lz) c(kx)s(hy)s(lz)] 16[c(hx)s(ky)s(lz) c(kx)s(hy)s(lz)]

I41 acd [No. 142, Origin 1] 2h l

A

B

4n 4n 1 4n 2 4n 3

16P(cc)c(lz) 8[M(ss)c(lz) P(ss)s(lz)] 0 8[M(ss)c(lz) P(ss)s(lz)]

0 A 16M(cc)s(lz) A

I41 acd [No. 142, Origin 2] (B = 0 for all h, k, l) h

k

hkl

A

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

16P(cc)c(lz) 16[s(hx)c(ky)s(lz) s(kx)s(hy)c(lz)] 16[s(hx)s(ky)c(lz) s(kx)c(hy)s(lz)] 16[c(hx)s(ky)s(lz) c(kx)s(hy)s(lz)] 16M(cc)c(lz) 16[s(hx)c(ky)s(lz) s(kx)s(hy)c(lz)] 16[s(hx)s(ky)c(lz) s(kx)c(hy)s(lz)] 16[c(hx)s(ky)s(lz) c(kx)s(hy)s(lz)]

136

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.6. Trigonal and hexagonal space groups The table lists the expressions for A and B for the space groups belonging to the hexagonal family. For the space groups that are referred to hexagonal axes the expressions are given in terms of symbols related to the decomposition of the scalar products into their plane-group and unique-axis components [cf. equations (1.4.3.10)–(1.4.3.12)]. The symbols for the seven rhombohedral space groups in their rhombohedral-axes representation are the same as those used for the cubic space groups [cf. equations (1.4.3.4) and (1.4.3.5), and the notes at the start of Table A1.4.3.7]. Factors of the forms cos
2x and sin2x are abbreviated by c(x) and s(x), respectively. All the symbols used in this table are repeated below. Most expressions are given in terms of Chki cp1 cp2 cp3 , Ckhi cq1 cq2 cq3 and Shki sp1 sp2 sp3 , Skhi sq1 sq2 sq3 ,

A1434

where p1 hx ky, p2 kx iy, p3 ix hy, q1 kx hy, q2 hx iy, q3 ix ky,

A1435

and the abbreviations PHcc Chki Ckhi, PHss Shki Skhi, MHcc Chki Ckhi and MHss Shki Skhi In addition, the following abbreviations are employed for some space groups: u1 lz, u2 lz 13 and u3 lz 13 Conditons for vanishing symbols: Shki Skhi 0 if h k 0, PHss 0 if h k or k i or i h, MHcc 0 if h k or k i or i h and any explicit sine function vanishes if all the indices (h and k, or l) appearing in its argument are zero. P3 [No. 143] hkl

A

B

All

C(hki)c(lz) S(hki)s(lz)

C(hki)s(lz) S(hki)c(lz)

P31 [No. 144] (enantiomorphous to P32 [No. 145]) l

A

B

3n

as for P3 [No. 143]

3n 1 3n 2

c(p1 u1 ) c(p2 u2 ) c(p3 u3 ) c(p1 u1 ) c(p2 u3 ) c(p3 u2 )

s(p1 u1 ) s(p2 u2 ) s(p3 u3 ) s(p1 u1 ) s(p2 u3 ) s(p3 u2 )

P32 [No. 145] (enantiomorphous to P31 [No. 144]) l

A, B

3n 3n 1 3n 2

as for P3 [No. 143] as for l = 3n 2 in P31 [No. 144] as for l = 3n 1 in P31 [No. 144]

R3 [No. 146] (rhombohedral axes) hkl

A

B

All

c(hx ky lz) c(kx ly hz) c(lx hy kz)

s(hx ky lz) s(kx ly hz) s(lx hy kz)

R3 [No. 146] (hexagonal axes) hkl

A

B

All

3[C(hki)c(lz) S(hki)s(lz)]

3[C(hki)s(lz) S(hki)c(lz)]

137

A1436

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

P3 [No. 147] hkl

A

B

All

2[C(hki)c(lz) S(hki)s(lz)]

0

R3 [No. 148] (rhombohedral axes) hkl

A

B

All

2[c(hx ky lz) c(kx ly hz) c(lx hy kz)]

0

R3 [No. 148] (hexagonal axes) hkl

A

B

All

6[C(hki)c(lz) S(hki)s(lz)]

0

P312 [No. 149] hkl

A

B

All

PH(cc)c(lz) PH(ss)s(lz)

MH(cc)s(lz) MH(ss)c(lz)

P321 [No. 150] hkl

A

B

All

PH(cc)c(lz) MH(ss)s(lz)

PH(ss)c(lz) MH(cc)s(lz)

P31 12 [No. 151] (enantiomorphous to P32 12 [No. 153]) l

A

3n

as for P312 [No. 149]

3n 1

cp1 u1 cp2 u2 cp3 u3 cq1 u2 cq2 u3 cq3 u1 cp1 u1 cp2 u3 cp3 u2 cq1 u3 cq2 u2 cq3 u1

3n 2

B

sp1 u1 sp2 u2 sp3 u3 sq1 u2 sq2 u3 sq3 u1 sp1 u1 sp2 u3 sp3 u2 sq1 u3 sq2 u2 sq3 u1

P31 21 [No. 152] (enantiomorphous to P32 21 [No. 154]) l

A

B

3n

as for P321 [No. 150]

3n 1

cp1 u1 cp2 u2 cp3 u3 cq1 u1 cq2 u2 cq3 u3 cp1 u1 cp2 u3 cp3 u2 cq1 u1 cq2 u3 cq3 u2

3n 2

s(p1 u1 ) s(p2 u2 ) s(p3 u3 ) s(q1 u1 ) sq2 u2 ) s(q3 u3 ) sp1 u1 sp2 u3 sp3 u2 sq1 u1 sq2 u3 sq3 u2

P32 12 [No. 153] (enantiomorphous to P31 12 [No. 151]) l

A, B

3n 3n 1 3n 2

as for P312 [No. 149] as for l = 3n 2 in P31 12 [No. 151] as for l = 3n 1 in P31 12 [No. 151]

P32 21 [No. 154] (enantiomorphous to P31 21 [No. 152]) l

A, B

3n 3n 1 3n 2

as for P321 [No. 150] as for l = 3n 2 in P31 21 [No. 152] as for l = 3n 1 in P31 21 [No. 152]

138

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

R32 [No. 155] (rhombohedral axes) hkl

A

B

All

Eccc Ecss Escs Essc Occc Ocss Oscs Ossc

Escc Ecsc Eccs Esss Oscc Ocsc Occs Osss

R32 [No. 155] (hexagonal axes) hkl

A

B

All

3[PH(cc)c(lz) MH(ss)s(lz)]

3[PH(ss)c(lz) MH(cc)s(lz)]

hkl

A

B

All

PH(cc)c(lz) MH(ss)s(lz)

PH(cc)s(lz) MH(ss)c(lz)

hkl

A

B

All

PH(cc)c(lz) PH(ss)s(lz)

PH(cc)s(lz) PH(ss)c(lz)

P3m1 [No. 156]

P31m [No. 157]

P3c1 [No. 158] l

A

B

2n 2n 1

PH(cc)c(lz) MH(ss)s(lz) MH(cc)c(lz) PH(ss)s(lz)

PH(cc)s(lz) MH(ss)c(lz) PH(ss)c(lz) MH(cc)s(lz)

l

A

B

2n 2n 1

PH(cc)c(lz) PH(ss)s(lz) MH(cc)c(lz) MH(ss)s(lz)

PH(cc)s(lz) PH(ss)c(lz) MH(cc)s(lz) MH(ss)c(lz)

P31c [No. 159]

R3m [No. 160] (rhombohedral axes) hkl

A

B

All

Eccc Ecss Escs Essc Occc Ocss Oscs Ossc

Escc Ecsc Eccs Esss Oscc Ocsc Occs Osss

R3m [No. 160] (hexagonal axes) hkl

A

B

All

3[PH(cc)c(lz) MH(ss)s(lz)]

3[PH(cc)s(lz) MH(ss)c(lz)]

R3c [No. 161] (rhombohedral axes) hkl

A

B

2n 2n 1

Eccc Ecss Escs Essc Occc Ocss Oscs Ossc Eccc Ecss Escs Essc Occc Ocss Oscs Ossc

Escc Ecsc Eccs Esss Oscc Ocsc Occs Osss Escc Ecsc Eccs Esss Oscc Ocsc Occs Osss

R3c [No. 161] (hexagonal axes) l

A

B

2n 2n 1

3[PH(cc)c(lz) MH(ss)s(lz)] 3[MH(cc)c(lz) PH(ss)s(lz)]

3[PH(cc)s(lz) MH(ss)c(lz)] 3[PH(ss)c(lz) MH(cc)s(lz)]

139

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

P31m [No. 162] (B 0 for all h, k, l) A 2[PH(cc)c(lz) PH(ss)s(lz)]

P31c [No. 163] (B 0 for all h, k, l) l

A

2n 2n 1

2[PH(cc)c(lz) PH(ss)s(lz)] 2[MH(cc)c(lz) MH(ss)s(lz)]

P3m1 [No. 164] (B 0 for all h, k, l) A 2[PH(cc)c(lz) MH(ss)s(lz)]

P3c1 [No. 165] (B 0 for all h, k, l) l

A

2n 2n 1

2[PH(cc)c(lz) MH(ss)s(lz)] 2[MH(cc)c(lz) PH(ss)s(lz)]

R3m [No. 166] (rhombohedral axes, B 0 for all h, k, l) A 2(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc)

R3m [No. 166] (hexagonal axes, B 0 for all h, k, l) A 6[PH(cc)c(lz) MH(ss)s(lz)]

R3c [No. 167] (rhombohedral axes, B 0 for all h, k, l) hkl

A

2n 2n 1

2(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc) 2(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc)

R3c [No. 167] (hexagonal axes, B 0 for all h, k, l) l

A

2n 2n 1

6[PH(cc)c(lz) MH(ss)s(lz)] 6[MH(cc)c(lz) PH(ss)s(lz)]

P6 [No. 168] hkl

A

B

All

2C(hki)c(lz)

2C(hki)s(lz)

P61 [No. 169] (enantiomorphous to P65 [No. 170]) l

A

6n

as for P6 [No.168]

6n 6n 6n 6n 6n

1 2 3 4 5

B

2[s(p1 )s(u1 ) s(p2 )s(u2 ) s(p3 )s(u3 )] 2[c(p1 )c(u1 ) c(p2 )c(u3 ) c(p3 )c(u2 )] 2S(hki)s(lz) 2[c(p1 )c(u1 ) c(p2 )c(u2 ) c(p3 )c(u3 )] 2[s(p1 )s(u1 ) s(p2 )s(u3 ) s(p3 )s(u2 )]

2[s(p1 )c(u1 ) 2[c(p1 )s(u1 ) 2S(hki)c(lz) 2[c(p1 )s(u1 ) 2[s(p1 )c(u1 )

140

s(p2 )c(u2 ) s(p3 )c(u3 )] c(p2 )s(u3 ) c(p3 )s(u2 )] c(p2 )s(u2 ) c(p3 )s(u3 )] s(p2 )c(u3 ) s(p3 )c(u2 )]

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

P65 [No. 170] (enantiomorphous to P61 [No. 169]) l 6n 6n 6n 6n 6n 6n

A, B as as as as as as

1 2 3 4 5

for P6 [No. 168] for l = 6n 5 in for l = 6n 4 in for l = 6n 3 in for l = 6n 2 in for l = 6n 1 in

P61 P61 P61 P61 P61

[No. 169] [No. 169] [No. 169] [No. 169] [No. 169]

P62 [No. 171] (enantiomorphous to P64 [No. 172]) l

A, B

3n 3n 1 3n 2

as for P6 [No. 168] as for l = 6n 2 in P61 [No. 169] as for l = 6n 4 in P61 [No. 169]

P64 [No. 172] (enantiomorphous to P62 [No. 171]) l

A, B

3n 3n 1 3n 2

as for P6 [No. 168] as for l = 6n 4 in P61 [No.169] as for l = 6n 2 in P61 [No. 169]

P63 [No. 173] l

A, B

2n 2n 1

as for P6 [No. 168] as for l = 6n 3 in P61 [No. 169]

P6 [No. 174] hkl

A

B

All

2C(hki)c(lz)

2S(hki)c(lz)

P6m [No. 175] hkl

A

B

All

4C(hki)c(lz)

0

l

A

B

2n 2n 1

4C(hki)c(lz) 4S(hki)s(lz)

0 0

hkl

A

B

All

2PH(cc)c(lz)

2MH(cc)s(lz)

P63 m [No. 176]

P622 [No. 177]

141

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

P61 22 [No. 178] (enantiomorphous to P65 22 [No. 179]) l

A

6n

as for P622 [No. 177]

6n 1

2sp1 su1 sp2 su2 sp3 su3 sq1 su3 sq2 su1 sq3 su2 2cp1 cu1 cp2 cu3 cp3 cu2 cq1 cu2 cq2 cu1 cq3 cu3 2MH(ss)s(lz) 2cp1 cu1 cp2 cu2 cp3 cu3 cq1 cu3 cq2 cu1 cq3 cu2 2sp1 su1 sp2 su3 sp3 su2 sq1 su2 sq2 su1 sq3 su3

6n 2 6n 3 6n 4 6n 5

B

2sp1 cu1 sp2 cu2 sp3 cu3 sq1 cu3 sq2 cu1 sq3 cu2 2cp1 su1 cp2 su3 cp3 su2 cq1 su2 cq2 su1 cq3 su3 2PH(ss)c(lz) 2cp1 su1 cp2 su2 cp3 su3 cq1 su3 cq2 su1 cq3 su2 2sp1 cu1 sp2 cu3 sp3 cu2 sq1 cu2 sq2 cu1 sq3 cu3

P65 22 [No. 179] (enantiomorphous to P61 22 [No. 178]) l

A, B

6n 6n 6n 6n 6n 6n

as as as as as as

1 2 3 4 5

for P622 [No. 177] for l = 6n 5 in P61 22 [No. 178] for l = 6n 4 in P61 22 [No. 178] for l = 6n 3 in P61 22 [No. 178] for l = 6n 2 in P61 22 [No. 178] for l = 6n 1 in P61 22 [No. 178]

P62 22 [No. 180] (enantiomorphous to P64 22 [No. 181]) l

A, B

n 3n 1 3n 2

as for P622 [No. 177] as for l = 6n 2 in P61 22 [No. 178] as for l = 6n 4 in P61 22 [No.178]

P64 22 [No. 181] (enantiomorphous to P62 22 [No. 180]) l

A, B

3n 3n 1 3n 2

as for P622 [No. 177] as for l = 6n 4 in P61 22 [No. 178] as for l = 6n 2 in P61 22 [No. 178]

P63 22 [No. 182] l

A, B

2n 2n 1

as for P622 [No. 177] as for l = 6n 3 in P61 22 [No. 178]

P6mm [No. 183] hkl

A

B

All

2PH(cc)c(lz)

2PH(cc)s(lz)

l

A

B

2n 2n 1

2PH(cc)c(lz) 2MH(cc)c(lz)

2PH(cc)s(lz) 2MH(cc)s(lz)

P6cc [No. 184]

142

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.6. Trigonal and hexagonal space groups (cont.)

P63 cm [No. 185] l

A

B

2n 2n 1

2PH(cc)c(lz) 2PH(ss)s(lz)

2PH(cc)s(lz) 2PH(ss)c(lz)

P63 mc [No. 186] l

A

B

2n 2n 1

2PH(cc)c(lz) 2MH(ss)s(lz)

2PH(cc)s(lz) 2MH(ss)c(lz)

P6m2 [No. 187] hkl

A

B

All

2PH(cc)c(lz)

2MH(ss)c(lz)

l

A

B

2n 2n 1

2PH(cc)c(lz) 2PH(ss)s(lz)

2MH(ss)c(lz) 2MH(cc)s(lz)

P6c2 [No. 188]

P62m [No. 189] hkl

A

B

All

2PH(cc)c(lz)

2PH(ss)c(lz)

l

A

B

2n 2n 1

2PH(cc)c(lz) 2MH(ss)s(lz)

2PH(ss)c(lz) 2MH(cc)s(lz)

P62c [No. 190]

P6mmm [No. 191] hkl

A

B

All

4PH(cc)c(lz)

0

P6mcc [No. 192] (B 0 for all h, k, l) l

A

2n 2n 1

4PH(cc)c(lz) 4MH(cc)c(lz)

P63 mcm [No. 193] (B 0 for all h, k, l) l

A

2n 2n 1

4PH(cc)c(lz) 4PH(ss)s(lz)

P63 mmc [No. 194] (B 0 for all h, k, l) l

A

2n 2n 1

4PH(cc)c(lz) 4MH(ss)s(lz)

143

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.7. Cubic space groups The symbols appearing in this table are related to the pqr representation used with the orthorhombic space groups as follows: Each of the symbols deﬁned below is a sum of three pqr terms, where the order of hkl is ﬁxed in each of the three terms and that of xyz is permuted. This table and parts of Table A1.4.3.6 (rhombohedral space groups referred to rhombohedral axes) are given in terms of the following two symbols: Epqr phxqkyrlz phyqkzrlx phzqkxrly

A1437

Opqr phxqkzrly phzqkyrlx phyqkxrlz,

A1438

and

where p, q and r can each be a sine or a cosine, and the factor 2 has been absorbed in the abbreviations (see text). As in Tables A1.4.3.1–A1.4.3.6, cosine and sine are abbreviated by c and s, respectively. The permutation of the coordinates is even in Epqr and odd in Opqr. Conditions for vanishing symbols: Epqr = Opqr = 0 if at least one of p, q, r is a sine and the index h, k or l in its argument is zero, Eccc Occc 0 if h k or k l or l h , Esss Osss 0 if h k or k l or l h , Ecss Ocss Escc Oscc 0 if k l , Escs Oscs Ecsc Ocsc 0 if l h and Essc Ossc Eccs Occs 0 if h k P23 [No. 195] hkl

A

B

All

4Eccc

4Esss

hkl

A

B

All

16Eccc

16Esss

F23 [No. 196]

I23 [No. 197] hkl

A

B

All

8Eccc

8Esss

P21 3 [No. 198] hk

kl

hl

A

B

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n

4Eccc 4Ecss 4Escs 4Essc

4Esss 4Escc 4Ecsc 4Eccs

I21 3 [No. 199] hk

kl

hl

A

B

2n 2n 1 2n 1 2n

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

8Eccc 8Escs 8Essc 8Ecss

8Esss 8Ecsc 8Eccs 8Escc

Pm3 [No. 200] hkl

A

B

All

8Eccc

0

Pn3 (Origin 1) [No. 201] hkl

A

B

2n 2n 1

8Eccc 0

0 8Esss

144

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.7. Cubic space groups (cont.) Pn3 (Origin 2) [No. 201] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n

8Eccc 8Essc 8Ecss 8Escs

Fm3 [No. 202] hkl

A

B

All

32Eccc

0

Fd3 (Origin 1) [No. 203] hkl

A

B

4n 4n 1 4n 2 4n 3

32Eccc 16(Eccc Esss) 0 16(Eccc Esss)

0 A 32Esss A

Fd3 (Origin 2) [No. 203] (B = 0 for all h, k, l) hk

kl

hl

A

4n 4n 4n 2 4n 2 4n 2 4n 2 4n 4n

4n 4n 2 4n 4n 2 4n 2 4n 4n 2 4n

4n 4n 2 4n 2 4n 4n 2 4n 4n 4n 2

32Eccc 32Essc 32Ecss 32Escs 16(Eccc 16(Eccc 16(Eccc 16(Eccc

Ecss Escs Essc) Ecss Escs Essc) Ecss Escs Essc) Ecss Escs Essc)

Im3 [No. 204] hkl

A

B

All

16Eccc

0

Pa3 [No. 205] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n

8Eccc 8Ecss 8Escs 8Essc

Ia3 [No. 206] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n

16Eccc 16Ecss 16Escs 16Essc

P432 [No. 207] hkl

A

B

All

4(Eccc Occc)

4(Esss Osss)

145

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.7. Cubic space groups (cont.) P42 32 [No. 208] hkl

A

B

2n 2n 1

4(Eccc Occc) 4(Eccc Occc)

4(Esss Osss) 4(Esss Osss)

F432 [No. 209] hkl

A

B

All

16(Eccc Occc)

16(Esss Osss)

hkl

A

B

4n 4n 1 4n 2 4n 3

16(Eccc 16(Eccc 16(Eccc 16(Eccc

F41 32 [No. 210]

Occc) Osss) Occc) Osss)

16(Esss Osss) 16(Esss Occc) 16(Esss Osss) 16(Esss Occc)

I432 [No. 211] hkl

A

B

All

8(Eccc Occc)

8(Esss Osss)

P43 32 [No. 212] (enantiomorphous to P41 32 [No. 213]) hk

kl

hl

hkl

A

B

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n

4n 4n 4n 4n 4n 1 4n 1 4n 1 4n 1 4n 2 4n 2 4n 2 4n 2 4n 3 4n 3 4n 3 4n 3

4(Eccc Occc) 4(Ecss Oscs) 4(Escs Ossc) 4(Essc Ocss) 4(Eccc Osss) 4(Ecss Ocsc) 4(Escs Occs) 4(Essc Oscc) 4(Eccc Occc) 4(Ecss Oscs) 4(Escs Ossc) 4(Essc Ocss) 4(Eccc Osss) 4(Ecss Ocsc) 4(Escs Occs) 4(Essc Oscc)

4(Esss Osss) 4(Escc Ocsc) 4(Ecsc Occs) 4(Eccs Oscc) 4(Esss Occc) 4(Escc Oscs) 4(Ecsc Ossc) 4(Eccs Ocss) 4(Esss Osss) 4(Escc Ocsc) 4(Ecsc Occs) 4(Eccs Oscc) 4(Esss Occc) 4(Escc Oscs) 4(Ecsc Ossc) 4(Eccs Ocss)

P41 32 [No. 213] (enantiomorphous to P43 32 [No. 212]) h

k

l

hkl

A

B

2n 2n 2n 1 2n 1 2n 1 2n 2n 1 2n 2n 2n 2n 1

2n 2n 1 2n 2n 1 2n 1 2n 2n 2n 1 2n 2n 1 2n

2n 2n 1 2n 1 2n 2n 1 2n 1 2n 2n 2n 2n 1 2n 1

4n 4n 4n 4n 4n 1 4n 1 4n 1 4n 1 4n 2 4n 2 4n 2

4(Eccc Occc) 4(Escs Ossc) 4(Essc Ocss) 4(Ecss Oscs) 4(Eccc Osss) 4(Ecss Ocsc) 4(Escs Occs) 4(Essc Oscc) 4(Eccc Occc) 4(Escs Ossc) 4(Essc Ocss)

4(Esss Osss) 4(Ecsc Occs) 4(Eccs Oscc) 4(Escc Ocsc) 4(Esss Occc) 4(Escc Oscs) 4(Ecsc Ossc) 4(Eccs Ocss) 4(Esss Osss) 4(Ecsc Occs) 4(Eccs Oscc)

146

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.7. Cubic space groups (cont.) h

k

l

hkl

A

B

2n 1 2n 1 2n 2n 1 2n

2n 1 2n 1 2n 2n 2n 1

2n 2n 1 2n 1 2n 2n

4n 2 4n 3 4n 3 4n 3 4n 3

4(Ecss Oscs) 4(Eccc Osss) 4(Ecss Ocsc) 4(Escs Occs) 4(Essc Oscc)

4(Escc Ocsc) 4(Esss Occc) 4(Escc Oscs) 4(Ecsc Ossc) 4(Eccs Ocss)

I41 32 [No. 214] h

k

l

hkl

A

B

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

8(Eccc Occc) 8(Escs Ossc) 8(Essc Ocss) 8(Ecss Oscs) 8(Eccc Occc) 8(Escs Ossc) 8(Essc Ocss) 8(Ecss Oscs)

8(Esss Osss) 8(Ecsc Occs) 8(Eccs Oscc) 8(Escc Ocsc) 8(Esss Osss) 8(Ecsc Occs) 8(Eccs Oscc) 8(Escc Ocsc)

P43m [No. 215] hkl

A

B

All

4(Eccc Occc)

4(Esss Osss)

F43m [No. 216] hkl

A

B

All

16(Eccc Occc)

16(Esss Osss)

I43m [No. 217] hkl

A

B

All

8(Eccc Occc)

8(Esss Osss)

hkl

A

B

2n 2n 1

4(Eccc Occc) 4(Eccc Occc)

4(Esss Osss) 4(Esss Osss)

hkl

A

B

2n 2n 1

16(Eccc Occc) 16(Eccc Occc)

16(Esss Osss) 16(Esss Osss)

P43n [No. 218]

F43c [No. 219]

I43d [No. 220] h

k

l

hkl

A

B

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

8(Eccc Occc) 8(Escs Ossc) 8(Essc Ocss) 8(Ecss Oscs) 8(Eccc Occc) 8(Escs Ossc) 8(Essc Ocss) 8(Ecss Oscs)

8(Esss Osss) 8(Ecsc Occs) 8(Eccs Oscc) 8(Escc Ocsc) 8(Esss Osss) 8(Ecsc Occs) 8(Eccs Oscc) 8(Escc Ocsc)

147

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.3.7. Cubic space groups (cont.)

Pm3m [No. 221] hkl

A

B

All

8(Eccc Occc)

0

Pn3n (Origin 1) [No. 222] hkl

A

B

2n 2n 1

8(Eccc Occc) 0

0 8(Esss Osss)

Pn3n (Origin 2) [No. 222] (B = 0 for all h, k, l) h

k

l

A

2n 2n 2n 1 2n 1 2n 1 2n 1 2n 2n

2n 2n 1 2n 2n 1 2n 1 2n 2n 1 2n

2n 2n 1 2n 1 2n 2n 1 2n 2n 2n 1

8(Eccc Occc) 8(Ecss Ocss) 8(Escs Oscs) 8(Essc Ossc) 8(Eccc Occc) 8(Ecss Ocss) 8(Escs Oscs) 8(Essc Ossc)

Pm3n [No. 223] (B = 0 for all h, k, l) hkl

A

2n 2n 1

8(Eccc Occc) 8(Eccc Occc)

Pn3m (Origin 1) [No. 224] hkl

A

B

2n 2n 1

8(Eccc Occc) 0

0 8(Esss Osss)

Pn3m (Origin 2) [No. 224] (B = 0 for all h, k, l) hk

kl

hl

A

2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n

8(Eccc Occc) 8(Essc Ossc) 8(Ecss Ocss) 8(Escs Oscs)

Fm3m [No. 225] hkl

A

B

All

32(Eccc Occc)

0

Fm3c [No. 226] (B = 0 for all h, k, l) hkl

A

2n 2n 1

32(Eccc Occc) 32(Eccc Occc)

148

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.3.7. Cubic space groups (cont.)

Fd3m (Origin 1) [No. 227] hkl

A

B

4n 4n 1 4n 2 4n 3

32(Eccc Occc) 16(Eccc Esss Occc Osss) 0 16(Eccc Esss Occc Osss)

0 A 32(Esss Osss) A

Fd3m (Origin 2) [No. 227] (B = 0 for all h, k, l) hk

kl

hl

A

4n 4n 4n 2 4n 2 4n 2 4n 2 4n 4n

4n 4n 2 4n 4n 2 4n 2 4n 4n 2 4n

4n 4n 2 4n 2 4n 4n 2 4n 4n 4n 2

32(Eccc Occc) 32(Essc Ossc) 32(Ecss Ocss) 32(Escs Oscs) 16(Eccc Ecss Escs Essc Occc 16(Eccc Ecss Escs Essc Occc 16(Eccc Ecss Escs Essc Occc 16(Eccc Ecss Escs Essc Occc

Ocss Oscs Ossc) Ocss Oscs Ossc) Ocss Oscs Ossc) Ocss Oscs Ossc)

Fd3c (Origin 1) [No. 228] hkl

A

B

4n 4n 1 4n 2 4n 3

32(Eccc Occc) 16(Eccc Esss Occc Osss) 0 16(Eccc Esss Occc Osss)

0 A 32(Esss Osss) A

Fd3c (Origin 2) [No. 228] (B = 0 for all h, k, l) hk

kl

hl

A

4n 4n 4n 2 4n 2 4n 2 4n 2 4n 4n

4n 4n 2 4n 4n 2 4n 2 4n 4n 2 4n

4n 4n 2 4n 2 4n 4n 2 4n 4n 4n 2

32(Eccc Occc) 32(Essc Ossc) 32(Ecss Ocss) 32(Escs Oscs) 16(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc) 16(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc) 16(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc) 16(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc)

Im3m [No. 229] hkl

A

B

All

16(Eccc Occc)

0

Ia3d [No. 230] (B = 0 for all h, k, l) h

k

l

hkl

A

2n 2n 2n 1 2n 1 2n 2n 2n 1 2n 1

2n 2n 1 2n 2n 1 2n 2n 1 2n 2n 1

2n 2n 1 2n 1 2n 2n 2n 1 2n 1 2n

4n 4n 4n 4n 4n 2 4n 2 4n 2 4n 2

16(Eccc Occc) 16(Escs Ossc) 16(Essc Ocss) 16(Ecss Oscs) 16(Eccc Occc) 16(Escs Ossc) 16(Essc Ocss) 16(Ecss Oscs)

149

1. GENERAL RELATIONSHIPS AND TECHNIQUES Appendix 1.4.4. Crystallographic space groups in reciprocal space Table A1.4.4.1. Crystallographic space groups in reciprocal space The table entries are described in detail in Section 1.4.4.1. The general format of an entry is
n hT Pn : hT tn

or

n hT P n : , according as the phase-shift part of the entry is nonzero or zero modolo 2, respectively. Notes: (1) For centrosymmetric space groups with the centre located at the unit-cell origin only those entries are given which correspond to symmetry operations not related by inversion. If the origin in such space groups is chosen elsewhere, all the entries corresponding to the operations of the point group are presented. (2) For trigonal and hexagonal space groups referred to hexagonal axes the Miller–Bravais indices hkil are employed, and for the rhombohedral space groups referred to rhombohedral axes the indices are denoted by hkl (cf. IT I, 1952). Point group: 1

Triclinic

Point group: m

Laue group: 1

Pm P1m1 (1) hkl:

P1 No. 1 (1) (1) hkl:

Monoclinic

Pm P11m Unique axis c (1) hkl: Point group: 1

Triclinic

No. 6 (14) (2) hkl:

Laue group: 1

P1 No. 2 (2) (1) hkl:

Point group: 2

Laue group: 2/m

Unique axis b No. 6 (13) (2) hkl:

Monoclinic

Pc P1c1 (1) hkl:

Unique axis b No. 7 (15) (2) hkl: 001/2

Pc P1n1 (1) hkl:

Unique axis b No. 7 (16) (2) hkl: 101/2 Unique axis b No. 7 (17) (2) hkl: 100/2

Laue group: 2/m

P2 P121 (1) hkl:

Unique axis b No. 3 (3) (2) hkl:

Pc P1a1 (1) hkl:

P2 P112 (1) hkl:

Unique axis c

Pc P11a Unique axis c (1) hkl:

No. 7 (18) (2) hkl: 100/2

No. 3 (4) (2) hkl:

P21 P121 1 (1) hkl:

Unique axis b

No. 4 (5) (2) hkl: 010/2

Pc P11n Unique axis c (1) hkl:

No. 7 (19) (2) hkl: 110/2

P21 P1121 (1) hkl:

Unique axis c

No. 4 (6) (2) hkl: 001/2

Pc P11b Unique axis c (1) hkl:

No. 7 (20) (2) hkl: 010/2

C2 C121 (1) hkl:

Unique axis b No. 5 (7) (2) hkl:

Cm C1m1 (1) hkl:

Unique axis b

No. 8 (21) (2) hkl:

C2 A121 (1) hkl:

Unique axis b No. 5 (8) (2) hkl:

Cm A1m1 (1) hkl:

Unique axis b

No. 8 (22) (2) hkl:

Unique axis b No. 8 (23) (2) hkl:

C2 I121 Unique axis b (1) hkl:

No. 5 (9) (2) hkl:

Cm I1m1 (1) hkl:

C2 A112 (1) hkl:

Unique axis c

No. 5 (10) (2) hkl:

Cm A11m Unique axis c (1) hkl:

No. 8 (24) (2) hkl:

C2 B112 (1) hkl:

Unique axis c

No. 5 (11) (2) hkl:

Cm B11m Unique axis c (1) hkl:

No. 8 (25) (2) hkl:

C2 I112 Unique axis c (1) hkl:

No. 5 (12) (2) hkl:

Cm I11m Unique axis c (1) hkl:

No. 8 (26) (2) hkl:

150

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) Cc C1c1 (1) hkl:

Unique axis b No. 9 (27) (2) hkl: 001/2

P2c P112a Unique axis c (1) hkl:

No. 13 (46) (2) hkl: 100/2

Cc A1n1 (1) hkl:

Unique axis b

No. 9 (28) (2) hkl: 101/2

P2c P112n Unique axis c (1) hkl:

No. 13 (47) (2) hkl: 110/2

Cc I1a1 (1) hkl:

Unique axis b No. 9 (29) (2) hkl: 100/2

P2c P112b Unique axis c (1) hkl:

No. 13 (48) (2) hkl: 010/2

Cc A11a Unique axis c (1) hkl:

No. 9 (30) (2) hkl: 100/2

P21 c P121 c1 (1) hkl:

Unique axis b No. 14 (49) (2) hkl: 011/2

Cc B11n Unique axis c (1) hkl:

No. 9 (31) (2) hkl: 110/2

P21 c P121 n1 (1) hkl:

Unique axis b

No. 14 (50) (2) hkl: 111/2

Cc I11b Unique axis c (1) hkl:

No. 9 (32) (2) hkl: 010/2

P21 c P121 a1 (1) hkl:

Unique axis b

No. 14 (51) (2) hkl: 110/2

P21 c P1121 a Unique axis c (1) hkl:

No. 14 (52) (2) hkl: 101/2

P21 c P1121 n Unique axis c (1) hkl:

No. 14 (53) (2) hkl: 111/2

P21 c P1121 b Unique axis c (1) hkl:

No. 14 (54) (2) hkl: 011/2

Point group: 2/m

Monoclinic

Laue group: 2/m

Unique axis b No. 10 (33) (2) hkl:

P2m P12m1 (1) hkl:

P2m P112m Unique axis c (1) hkl: P21 m P121 m1 (1) hkl:

No. 10 (34) (2) hkl:

Unique axis b

P21 m P1121 m Unique axis c (1) hkl:

No. 11 (35) (2) hkl: 010/2 No. 11 (36) (2) hkl: 001/2

C2m C12m1 (1) hkl:

Unique axis b

No. 12 (37) (2) hkl:

C2m A12m1 (1) hkl:

Unique axis b No. 12 (38) (2) hkl:

C2m I12m1 (1) hkl:

Unique axis b No. 12 (39) (2) hkl:

C2m A112m Unique axis c (1) hkl:

No. 12 (40) (2) hkl:

C2m B112m Unique axis c (1) hkl:

No. 12 (41) (2) hkl:

C2m I112m Unique axis c (1) hkl:

No. 12 (42) (2) hkl:

C2c C12c1 (1) hkl:

Unique axis b

No. 15 (55) (2) hkl: 001/2

C2c A12n1 (1) hkl:

Unique axis b No. 15 (56) (2) hkl: 101/2

C2c I12a1 (1) hkl:

Unique axis b No. 15 (57) (2) hkl: 100/2

C2c A112a Unique axis c (1) hkl:

No. 15 (58) (2) hkl: 100/2

C2c B112n Unique axis c (1) hkl:

No. 15 (59) (2) hkl: 110/2

C2c I112b Unique axis c (1) hkl:

No. 15 (60) (2) hkl: 010/2

Point group: 222 Orthorhombic

Laue group: mmm

P222 No. 16 (61) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Unique axis b No. 13 (44) (2) hkl: 101/2

P2221 No. 17 (62) (1) hkl: (2) hkl: 001/2

(3) hkl: 001/2

(4) hkl:

Unique axis b No. 13 (45) (2) hkl: 100/2

P21 21 2 (1) hkl:

(3) hkl: 110/2

(4) hkl: 110/2

P2c P12c1 (1) hkl:

Unique axis b

P2c P12n1 (1) hkl: P2c P12a1 (1) hkl:

No. 13 (43) (2) hkl: 001/2

151

No. 18 (63) (2) hkl:

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P21 21 21 (1) hkl:

Ccc2 No. 37 (82) (1) hkl: (2) hkl:

(3) hkl: 001/2

(4) hkl: 001/2

Amm2 No. 38 (83) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

(4) hkl:

Abm2 No. 39 (84) (1) hkl: (2) hkl:

(3) hkl: 010/2

(4) hkl: 010/2

(4) hkl:

Ama2 No. 40 (85) (1) hkl: (2) hkl:

(3) hkl: 100/2

(4) hkl: 100/2

(4) hkl:

Aba2 No. 41 (86) (1) hkl: (2) hkl:

(3) hkl: 110/2

(4) hkl: 110/2

(4) hkl: 110/2

Fmm2 No. 42 (87) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Fdd2 No. 43 (88) (1) hkl: (2) hkl:

(3) hkl: 313/4

(4) hkl: 133/4

Imm2 No. 44 (89) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Iba2 No. 45 (90) (1) hkl: (2) hkl:

(3) hkl: 110/2

(4) hkl: 110/2

Ima2 No. 46 (91) (1) hkl: (2) hkl:

(3) hkl: 100/2

(4) hkl: 100/2

No. 19 (64) (2) hkl: 101/2

(3) hkl: 011/2

(4) hkl: 110/2

C2221 No. 20 (65) (1) hkl: (2) hkl: 001/2

(3) hkl: 001/2

(4) hkl:

C222 No. 21 (66) (1) hkl: (2) hkl: F222 No. 22 (67) (1) hkl: (2) hkl: I222 No. 23 (68) (1) hkl: (2) hkl: I21 21 21 (1) hkl:

No. 24 (69) (2) hkl: 101/2

Point group: mm2

Orthorhombic

Pmm2 No. 25 (70) (1) hkl: (2) hkl: Pmc21 No. 26 (71) (1) hkl: (2) hkl: 001/2 Pcc2 No. 27 (72) (1) hkl: (2) hkl: Pma2 No. 28 (73) (1) hkl: (2) hkl: Pca21 No. 29 (74) (1) hkl: (2) hkl: 001/2 Pnc2 No. 30 (75) (1) hkl: (2) hkl: Pmn21 No. 31 (76) (1) hkl: (2) hkl: 101/2 Pba2 No. 32 (77) (1) hkl: (2) hkl:

(3) hkl:

(3) hkl:

(3) hkl:

(3) hkl: 011/2

Laue group: mmm (3) hkl:

(3) hkl: 001/2

(3) hkl: 001/2

(4) hkl:

(4) hkl:

(4) hkl: 001/2 Point group: mmm

(3) hkl: 100/2

(3) hkl: 100/2

(3) hkl: 011/2

(3) hkl: 101/2

(3) hkl: 110/2

(4) hkl: 100/2

(4) hkl: 101/2

Orthorhombic

Laue group: mmm

Pmmm No. 47 (92) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Pnnn Origin 1 (1) hkl: (5) hkl: 111/2

No. 48 (93) (2) hkl: (6) hkl: 111/2

(3) hkl: (7) hkl: 111/2

(4) hkl: (8) hkl: 111/2

Pnnn Origin 2 (1) hkl:

No. 48 (94) (2) hkl: 110/2

(3) hkl: 101/2

(4) hkl: 011/2

Pccm No. 49 (95) (1) hkl: (2) hkl:

(3) hkl: 001/2

(4) hkl: 001/2

No. 50 (96) (2) hkl: (6) hkl: 110/2

(3) hkl: (7) hkl: 110/2

(4) hkl: (8) hkl: 110/2

No. 50 (97) (2) hkl: 110/2

(3) hkl: 100/2

(4) hkl: 010/2

(4) hkl: 011/2

(4) hkl:

(4) hkl: 110/2

Pna21 No. 33 (78) (1) hkl: (2) hkl: 001/2

(3) hkl: 110/2

(4) hkl: 111/2

Pban Origin 1 (1) hkl: (5) hkl: 110/2

Pnn2 No. 34 (79) (1) hkl: (2) hkl:

(3) hkl: 111/2

(4) hkl: 111/2

Pban Origin 2 (1) hkl:

Cmm2 No. 35 (80) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Pmma No. 51 (98) (1) hkl: (2) hkl: 100/2

(3) hkl:

(4) hkl: 100/2

Cmc21 No. 36 (81) (1) hkl: (2) hkl: 001/2

(3) hkl: 001/2

(4) hkl:

Pnna No. 52 (99) (1) hkl: (2) hkl: 100/2

(3) hkl: 111/2

(4) hkl: 011/2

152

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) Pmna No. 53 (100) (1) hkl: (2) hkl: 101/2

(3) hkl: 101/2

(4) hkl:

Pcca No. 54 (101) (1) hkl: (2) hkl: 100/2

(3) hkl: 001/2

(4) hkl: 101/2

Pbam No. 55 (102) (1) hkl: (2) hkl:

(3) hkl: 110/2

(4) hkl: 110/2

Pccn No. 56 (103) (1) hkl: (2) hkl: 110/2 Pbcm No. 57 (104) (1) hkl: (2) hkl: 001/2 Pnnm No. 58 (105) (1) hkl: (2) hkl:

(3) hkl: 011/2

(3) hkl: 011/2

(3) hkl: 111/2

(4) hkl: 101/2

(4) hkl: 010/2

(4) hkl: 111/2

Pmmn Origin 1 No. 59 (106) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) hkl: 110/2 (7) hkl:

(4) hkl: 110/2 (8) hkl:

Pmmn Origin 2 No. 59 (107) (1) hkl: (2) hkl: 110/2

(3) hkl: 010/2

(4) hkl: 100/2

Pbcn No. 60 (108) (1) hkl: (2) hkl: 111/2 Pbca No. 61 (109) (1) hkl: (2) hkl: 101/2 Pnma No. 62 (110) (1) hkl: (2) hkl: 101/2 Cmcm No. 63 (111) (1) hkl: (2) hkl: 001/2 Cmca No. 64 (112) (1) hkl: (2) hkl: 011/2

(3) hkl: 001/2

(3) hkl: 011/2

(3) hkl: 010/2

(3) hkl: 001/2

(3) hkl: 011/2

(4) hkl: 110/2

(4) hkl: 111/2

(4) hkl:

(4) hkl:

(3) hkl:

(4) hkl:

Cccm No. 66 (114) (1) hkl: (2) hkl:

(3) hkl: 001/2

(4) hkl: 001/2

Cmma No. 67 (115) (1) hkl: (2) hkl: 010/2 Ccca Origin 1 (1) hkl: (5) hkl: 011/2

No. 68 (116) (2) hkl: 110/2 (6) hkl: 101/2

Ccca Origin 2 (1) hkl:

No. 68 (117) (2) hkl: 100/2

(3) hkl: 010/2

(3) hkl: (7) hkl: 011/2

(3) hkl:

(4) hkl:

Fddd Origin 1 (1) hkl: (5) hkl: 111/4

No. 70 (119) (2) hkl: (6) hkl: 111/4

(3) hkl: (7) hkl: 111/4

(4) hkl: (8) hkl: 111/4

Fddd Origin 2 (1) hkl:

No. 70 (120) (2) hkl: 330/4

(3) hkl: 303/4

(4) hkl: 033/4

Immm No. 71 (121) (1) hkl: (2) hkl:

(3) hkl:

(4) hkl:

Ibam No. 72 (122) (1) hkl: (2) hkl:

(3) hkl: 110/2

(4) hkl: 110/2

Ibca No. 73 (123) (1) hkl: (2) hkl: 101/2

(3) hkl: 011/2

(4) hkl: 110/2

Imma No. 74 (124) (1) hkl: (2) hkl: 010/2

(3) hkl: 010/2

(4) hkl:

Point group: 4

(4) hkl: 110/2

Cmmm No. 65 (113) (1) hkl: (2) hkl:

Fmmm No. 69 (118) (1) hkl: (2) hkl:

Tetragonal

(3) khl:

(4) khl:

P41 No. 76 (126) (1) hkl: (2) hkl: 001/2

(3) khl: 001/4

(4) khl: 003/4

P42 No. 77 (127) (1) hkl: (2) hkl:

(3) khl: 001/2

(4) khl: 001/2

P43 No. 78 (128) (1) hkl: (2) hkl: 001/2

(3) khl: 003/4

(4) khl: 001/4

I4 No. 79 (129) (1) hkl: (2) hkl:

(3) khl:

(4) khl:

I41 No. 80 (130) (1) hkl: (2) hkl: 111/2

(3) khl: 021/4

(4) khl: 203/4

Point group: 4

Tetragonal

Laue group: 4/m

P4 No. 81 (131) (1) hkl: (2) hkl:

(3) khl:

(4) khl:

I4 No. 82 (132) (1) hkl: (2) hkl:

(3) khl:

(4) khl:

(4) hkl:

(4) hkl: 110/2 (8) hkl: 101/2 Point group: 4/m

(3) hkl: 001/2

Laue group: 4/m

P4 No. 75 (125) (1) hkl: (2) hkl:

Tetragonal

P4m No. 83 (133) (1) hkl: (2) hkl:

(4) hkl: 101/2

153

Laue group: 4/m (3) khl:

(4) khl:

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P42 m No. 84 (134) (1) hkl: (2) hkl:

(3) khl: 001/2

(4) khl: 001/2

P4n Origin 1 (1) hkl: (5) hkl: 110/2

No. 85 (135) (2) hkl: (6) hkl: 110/2

(3) khl: 110/2 (7) khl:

P4n Origin 2 (1) hkl:

No. 85 (136) (2) hkl: 110/2

(3) khl: 100/2

P43 21 2 No. 96 (149) (1) hkl: (2) hkl: 001/2 (5) hkl: 223/4 (6) hkl: 221/4

(3) khl: 223/4 (7) khl:

(4) khl: 221/4 (8) khl: 001/2

(4) khl: 110/2 (8) khl:

I422 No. 97 (150) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

(4) khl: 010/2

I41 22 No. 98 (151) (1) hkl: (2) hkl: 111/2 (6) hkl: 021/4 (5) hkl: 203/4

(3) khl: 021/4 (7) khl: 111/2

(4) khl: 203/4 (8) khl:

P42 n Origin 1 No. 86 (137) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: 111/2 (7) khl:

(4) khl: 111/2 (8) khl:

P42 n Origin 2 No. 86 (138) (1) hkl: (2) hkl: 110/2

(3) khl: 011/2

(4) khl: 101/2 Point group: 4mm

I4m No. 87 (139) (1) hkl: (2) hkl:

(3) khl:

(4) khl:

I41 a Origin 1 No. 88 (140) (1) hkl: (2) hkl: 111/2 (6) hkl: 203/4 (5) hkl: 021/4

(3) khl: 021/4 (7) khl:

I41 a Origin 2 No. 88 (141) (1) hkl: (2) hkl: 101/2

(3) khl: 311/4

Point group: 422 Tetragonal

Tetragonal

Laue group: 4/mmm

P4mm No. 99 (152) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

(4) khl: 203/4 (8) khl: 111/2

P4bm No. 100 (153) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: (7) khl: 110/2

(4) khl: (8) khl: 110/2

(4) khl: 333/4

P42 cm No. 101 (154) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: 001/2 (7) khl:

(4) khl: 001/2 (8) khl:

Laue group: 4/mmm

P422 No. 89 (142) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P42 nm No. 102 (155) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: 111/2 (7) khl:

(4) khl: 111/2 (8) khl:

P421 2 No. 90 (143) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: 110/2 (7) khl:

(4) khl: 110/2 (8) khl:

P4cc No. 103 (156) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

P41 22 No. 91 (144) (1) hkl: (2) hkl: 001/2 (6) hkl: 001/2 (5) hkl:

(3) khl: 001/4 (7) khl: 003/4

(4) khl: 003/4 (8) khl: 001/4

P4nc No. 104 (157) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: (7) khl: 111/2

(4) khl: (8) khl: 111/2

P41 21 2 No. 92 (145) (1) hkl: (2) hkl: 001/2 (6) hkl: 223/4 (5) hkl: 221/4

(3) khl: 221/4 (7) khl:

(4) khl: 223/4 (8) khl: 001/2

P42 mc No. 105 (158) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: 001/2 (7) khl: 001/2

(4) khl: 001/2 (8) khl: 001/2

P42 22 No. 93 (146) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: 001/2 (7) khl: 001/2

(4) khl: 001/2 (8) khl: 001/2

P42 bc No. 106 (159) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: 001/2 (7) khl: 111/2

(4) khl: 001/2 (8) khl: 111/2

P42 21 2 No. 94 (147) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: 111/2 (7) khl:

(4) khl: 111/2 (8) khl:

I4mm No. 107 (160) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P43 22 No. 95 (148) (1) hkl: (2) hkl: 001/2 (6) hkl: 001/2 (5) hkl:

(3) khl: 003/4 (7) khl: 001/4

(4) khl: 001/4 (8) khl: 003/4

I4cm No. 108 (161) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

154

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) I41 md No. 109 (162) (1) hkl: (2) hkl: 111/2 (6) hkl: 111/2 (5) hkl:

(3) khl: 021/4 (7) khl: 203/4

(4) khl: 203/4 (8) khl: 021/4

I41 cd No. 110 (163) (1) hkl: (2) hkl: 111/2 (6) hkl: 110/2 (5) hkl: 001/2

(3) khl: 021/4 (7) khl: 201/4

(4) khl: 203/4 (8) khl: 023/4

Point group: 42m

Tetragonal

Point group: 4/mmm

Laue group: 4/mmm

Tetragonal

Laue group: 4/mmm

P4mmm No. 123 (176) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P4mcc No. 124 (177) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

P4nbm Origin 1 No. 125 (178) (1) hkl: (2) hkl: (3) khl: (5) hkl: (6) hkl: (7) khl: (9) hkl: 110/2 (10) hkl: 110/2 (11) khl: 110/2 (13) hkl: 110/2 (14) hkl: 110/2 (15) khl: 110/2

(4) khl: (8) khl: (12) khl: 110/2 (16) khl: 110/2

P42m No. 111 (164) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P42c No. 112 (165) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

P421 m No. 113 (166) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

P4nbm Origin 2 No. 125 (179) (1) hkl: (2) hkl: 110/2 (6) hkl: 010/2 (5) hkl: 100/2

(3) khl: 100/2 (7) khl:

(4) khl: 010/2 (8) khl: 110/2

(3) khl: (7) khl: 110/2

(4) khl: (8) khl: 110/2

P421 c No. 114 (167) (1) hkl: (2) hkl: (5) hkl: 111/2 (6) hkl: 111/2

(3) khl: (7) khl: 111/2

(4) khl: (8) khl: 111/2

P4nnc Origin 1 No. 126 (180) (1) hkl: (2) hkl: (6) hkl: (5) hkl: (9) hkl: 111/2 (10) hkl: 111/2 (13) hkl: 111/2 (14) hkl: 111/2

(3) khl: (7) khl: (11) khl: 111/2 (15) khl: 111/2

(4) khl: (8) khl: (12) khl: 111/2 (16) khl: 111/2

P4nnc Origin 2 No. 126 (181) (1) hkl: (2) hkl: 110/2 (6) hkl: 011/2 (5) hkl: 101/2

(3) khl: 100/2 (7) khl: 001/2

(4) khl: 010/2 (8) khl: 111/2

P4mbm No. 127 (182) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: (7) khl: 110/2

(4) khl: (8) khl: 110/2

P4mnc No. 128 (183) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: (7) khl: 111/2

(4) khl: (8) khl: 111/2

P4nmm Origin 1 No. 129 (184) (3) khl: 110/2 (1) hkl: (2) hkl: (6) hkl: 110/2 (7) khl: (5) hkl: 110/2 (9) hkl: 110/2 (10) hkl: 110/2 (11) khl: (14) hkl: (15) khl: 110/2 (13) hkl:

(4) khl: 110/2 (8) khl: (12) khl: (16) khl: 110/2

P4m2 No. 115 (168) (1) hkl: (2) hkl: (6) hkl: (5) hkl: P4c2 No. 116 (169) (1) hkl: (2) hkl: (5) hkl: 001/2 (6) hkl: 001/2 P4b2 No. 117 (170) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: (7) khl:

(3) khl: (7) khl: 001/2

(3) khl: (7) khl: 110/2

(4) khl: (8) khl:

(4) khl: (8) khl: 001/2

(4) khl: (8) khl: 110/2

P4n2 No. 118 (171) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2

(3) khl: (7) khl: 111/2

(4) khl: (8) khl: 111/2

I4m2 No. 119 (172) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

I4c2 No. 120 (173) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

P4nmm Origin 2 No. 129 (185) (1) hkl: (2) hkl: 110/2 (6) hkl: 100/2 (5) hkl: 010/2

(3) khl: 100/2 (7) khl: 110/2

(4) khl: 010/2 (8) khl:

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

I42m No. 121 (174) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P4ncc Origin 1 No. 130 (186) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2 (9) hkl: 110/2 (10) hkl: 110/2 (13) hkl: 001/2 (14) hkl: 001/2

(3) khl: 110/2 (7) khl: 001/2 (11) khl: (15) khl: 111/2

(4) khl: 110/2 (8) khl: 001/2 (12) khl: (16) khl: 111/2

P4ncc Origin 2 No. 130 (187) (1) hkl: (2) hkl: 110/2 (6) hkl: 101/2 (5) hkl: 011/2

(3) khl: 100/2 (7) khl: 111/2

(4) khl: 010/2 (8) khl: 001/2

I42d No. 122 (175) (1) hkl: (2) hkl: (6) hkl: 203/4 (5) hkl: 203/4

(3) khl: (7) khl: 021/4

(4) khl: (8) khl: 021/4

155

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P42 mmc No. 131 (188) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: 001/2 (7) khl: 001/2

(4) khl: 001/2 (8) khl: 001/2

P42 mcm No. 132 (189) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: 001/2 (7) khl:

(4) khl: 001/2 (8) khl:

P42 nbc Origin 1 No. 133 (190) (3) khl: 111/2 (1) hkl: (2) hkl: (6) hkl: 001/2 (7) khl: 110/2 (5) hkl: 001/2 (9) hkl: 111/2 (10) hkl: 111/2 (11) khl: (13) hkl: 110/2 (14) hkl: 110/2 (15) khl: 001/2

(4) khl: 111/2 (8) khl: 110/2 (12) khl: (16) khl: 001/2

P42 nbc Origin 2 No. 133 (191) (1) hkl: (2) hkl: 110/2 (6) hkl: 010/2 (5) hkl: 100/2 P42 nnm Origin (1) hkl: (5) hkl: (9) hkl: 111/2 (13) hkl: 111/2

(3) khl: 101/2 (7) khl: 001/2

(4) khl: 011/2 (8) khl: 111/2

No. 134 (192) (3) khl: 111/2 (2) hkl: (6) hkl: (7) khl: 111/2 (10) hkl: 111/2 (11) khl: (14) hkl: 111/2 (15) khl:

(4) khl: 111/2 (8) khl: 111/2 (12) khl: (16) khl:

I41 amd Origin 1 No. 141 (202) (3) khl: 021/4 (1) hkl: (2) hkl: 111/2 (6) hkl: 021/4 (7) khl: 111/2 (5) hkl: 203/4 (9) hkl: 021/4 (10) hkl: 203/4 (11) khl: (15) khl: 203/4 (13) hkl: 111/2 (14) hkl:

1

I41 amd Origin 2 No. 141 (203) (1) hkl: (2) hkl: 101/2 (6) hkl: (5) hkl: 101/2

(3) khl: 131/4 (7) khl: 131/4

(4) khl: 113/4 (8) khl: 113/4

I41 acd Origin 1 No. 142 (204) (1) hkl: (2) hkl: 111/2 (6) hkl: 023/4 (5) hkl: 201/4 (9) hkl: 021/4 (10) hkl: 203/4 (13) hkl: 110/2 (14) hkl: 001/2

(3) khl: 021/4 (7) khl: 110/2 (11) khl: (15) khl: 201/4

(4) khl: 203/4 (8) khl: 001/2 (12) khl: 111/2 (16) khl: 023/4

I41 acd Origin 2 No. 142 (205) (1) hkl: (2) hkl: 101/2 (6) hkl: 001/2 (5) hkl: 100/2

(3) khl: 131/4 (7) khl: 133/4

(4) khl: 113/4 (8) khl: 111/4

Point group: 3 P42 nnm Origin 2 No. 134 (193) (1) hkl: (2) hkl: 110/2 (6) hkl: 011/2 (5) hkl: 101/2

(3) khl: 101/2 (7) khl:

(4) khl: 011/2 (8) khl: 110/2

P42 mbc No. 135 (194) (1) hkl: (2) hkl: (6) hkl: 110/2 (5) hkl: 110/2

(3) khl: 001/2 (7) khl: 111/2

(4) khl: 001/2 (8) khl: 111/2

P42 mnm No. 136 (195) (1) hkl: (2) hkl: (6) hkl: 111/2 (5) hkl: 111/2 P42 nmc Origin (1) hkl: (5) hkl: 111/2 (9) hkl: 111/2 (13) hkl:

(3) khl: 111/2 (7) khl:

(4) khl: 111/2 (8) khl:

No. 137 (196) (3) khl: 111/2 (2) hkl: (6) hkl: 111/2 (7) khl: (10) hkl: 111/2 (11) khl: (14) hkl: (15) khl: 111/2

(4) khl: 111/2 (8) khl: (12) khl: (16) khl: 111/2

P42 nmc Origin 2 No. 137 (197) (1) hkl: (2) hkl: 110/2 (6) hkl: 100/2 (5) hkl: 010/2 P42 ncm Origin (1) hkl: (5) hkl: 110/2 (9) hkl: 111/2 (13) hkl: 001/2

(3) khl: 101/2 (7) khl: 111/2

(4) khl: 011/2 (8) khl: 001/2

No. 138 (198) (3) khl: 111/2 (2) hkl: (6) hkl: 110/2 (7) khl: 001/2 (10) hkl: 111/2 (11) khl: (14) hkl: 001/2 (15) khl: 110/2

(4) khl: 111/2 (8) khl: 001/2 (12) khl: (16) khl: 110/2

Trigonal

Laue group: 3

P3 No. 143 (206) (1) hkl:

(2) kil:

(3) ihl:

P31 No. 144 (207) (1) hkl:

(2) kil: 001/3

(3) ihl: 002/3

P32 No. 145 (208) (1) hkl:

(2) kil: 002/3

(3) ihl: 001/3

R3 (hexagonal axes) (1) hkl:

1

(4) khl: 203/4 (8) khl: (12) khl: 111/2 (16) khl: 021/4

No. 146 (209) (2) kil:

R3 (rhombohedral axes) No. 146 (210) (1) hkl: (2) klh:

Point group: 3

Trigonal

P3 No. 147 (211) (1) hkl:

(3) ihl:

(3) lhk:

Laue group: 3

(2) kil:

(3) ihl:

1

R3 (hexagonal axes) (1) hkl:

No. 148 (212) (2) kil:

R3 (rhombohedral axes) No. 148 (213) (1) hkl: (2) klh:

(3) ihl:

(3) lhk:

P42 ncm Origin 2 No. 138 (199) (1) hkl: (2) hkl: 110/2 (6) hkl: 101/2 (5) hkl: 011/2

(3) khl: 101/2 (7) khl: 110/2

(4) khl: 011/2 (8) khl:

I4mmm No. 139 (200) (1) hkl: (2) hkl: (6) hkl: (5) hkl:

(3) khl: (7) khl:

(4) khl: (8) khl:

P312 No. 149 (214) (1) hkl: (4) khl:

(2) kil: (5) hil:

(3) ihl: (6) ikl:

I4mcm No. 140 (201) (1) hkl: (2) hkl: (6) hkl: 001/2 (5) hkl: 001/2

(3) khl: (7) khl: 001/2

(4) khl: (8) khl: 001/2

P321 No. 150 (215) (1) hkl: (4) khl:

(2) kil: (5) hil:

(3) ihl: (6) ikl:

Point group: 32

156

Trigonal

Laue group: 3m

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P31 12 No. 151 (216) (1) hkl: (4) khl: 002/3

(2) kil: 001/3 (5) hil: 001/3

(3) ihl: 002/3 (6) ikl:

P31 21 No. 152 (217) (1) hkl: (4) khl:

(2) kil: 001/3 (5) hil: 002/3

(3) ihl: 002/3 (6) ikl: 001/3

P32 12 No. 153 (218) (1) hkl: (4) khl: 001/3

(2) kil: 002/3 (5) hil: 002/3

(3) ihl: 001/3 (6) ikl:

P32 21 No. 154 (219) (1) hkl: (4) khl:

(2) kil: 002/3 (5) hil: 001/3

(3) ihl: 001/3 (6) ikl: 002/3

R32 (hexagonal axes) (1) hkl: (4) khl:

Point group: 3m

No. 155 (220) (2) kil: (5) hil:

R32 (rhombohedral axes) No. 155 (221) (1) hkl: (2) klh: (5) hlk: (4) khl:

Point group: 3m

Trigonal

P3m1 No. 156 (222) (1) hkl: (4) khl:

(3) ihl: (6) ikl:

(3) lhk: (6) lkh:

(3) ihl: (6) ikl:

P31m No. 157 (223) (1) hkl: (4) khl:

(2) kil: (5) hil:

(3) ihl: (6) ikl:

P3c1 No. 158 (224) (1) hkl: (4) khl: 001/2

(2) kil: (5) hil: 001/2

(3) ihl: (6) ikl: 001/2

P31c No. 159 (225) (1) hkl: (4) khl: 001/2 R3m (hexagonal axes) (1) hkl: (4) khl:

(2) kil: (5) hil: 001/2 No. 160 (226) (2) kil: (5) hil:

R3m (rhombohedral axes) No. 160 (227) (1) hkl: (2) klh: (4) khl: (5) hlk: R3c (hexagonal axes) (1) hkl: (4) khl: 001/2

No. 161 (228) (2) kil: (5) hil: 001/2

R3c (rhombohedral axes) No. 161 (229) (1) hkl: (2) klh: (4) khl: 111/2 (5) hlk: 111/2

Laue group: 3m

(2) kil: (5) hil:

(3) ihl: (6) ikl:

P31c No. 163 (231) (1) hkl: (4) khl: 001/2

(2) kil: (5) hil: 001/2

(3) ihl: (6) ikl: 001/2

P3m1 No. 164 (232) (1) hkl: (4) khl:

(2) kil: (5) hil:

(3) ihl: (6) ikl:

P3c1 No. 165 (233) (1) hkl: (4) khl: 001/2

(2) kil: (5) hil: 001/2

(3) ihl: (6) ikl: 001/2

R3m (hexagonal axes) (1) hkl: (4) khl:

Laue group: 3m

(2) kil: (5) hil:

Trigonal

P31m No. 162 (230) (1) hkl: (4) khl:

No. 166 (234) (2) kil: (5) hil:

R3m (rhombohedral axes) No. 166 (235) (1) hkl: (2) klh: (5) hlk: (4) khl:

(3) lhk: (6) lkh:

R3c (hexagonal axes) (1) hkl: (4) khl: 001/2

(3) ihl: (6) ikl: 001/2

No. 167 (236) (2) kil: (5) hil: 001/2

R3c (rhombohedral axes) No. 168 (237) (1) hkl: (2) klh: (5) hlk: 111/2 (4) khl: 111/2

Point group: 6

(3) ihl: (6) ikl: 001/2

(3) ihl: (6) ikl:

(3) lhk: (6) lkh:

(3) ihl: (6) ikl: 001/2

(3) lhk: (6) lkh: 111/2

157

(3) ihl: (6) ikl:

Hexagonal

(3) lhk: (6) lkh: 111/2

Laue group: 6/m

P6 No. 168 (238) (1) hkl: (4) hkl:

(2) kil: (5) kil:

(3) ihl: (6) ihl:

P61 No. 169 (239) (1) hkl: (4) hkl: 001/2

(2) kil: 001/3 (5) kil: 005/6

(3) ihl: 002/3 (6) ihl: 001/6

P65 No. 170 (240) (1) hkl: (4) hkl: 001/2

(2) kil: 002/3 (5) kil: 001/6

(3) ihl: 001/3 (6) ihl: 005/6

P62 No. 171 (241) (1) hkl: (4) hkl:

(2) kil: 002/3 (5) kil: 002/3

(3) ihl: 001/3 (6) ihl: 001/3

P64 No. 172 (242) (1) hkl: (4) hkl:

(2) kil: 001/3 (5) kil: 001/3

(3) ihl: 002/3 (6) ihl: 002/3

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P63 No. 173 (243) (1) hkl: (4) hkl: 001/2

Point group: 6

Point group: 6mm (2) kil: (5) kil: 001/2

Hexagonal

P6 No. 174 (244) (1) hkl: (4) hkl:

Point group: 6/m

Laue group: 6/m

(2) kil: (5) kil:

Hexagonal

P6m No. 175 (245) (1) hkl: (4) hkl:

(3) ihl: (6) ihl:

Laue group: 6/m

(2) kil: (5) kil:

(3) ihl: (6) ihl:

P63 m No. 176 (246) (1) hkl: (2) kil: (5) kil: 001/2 (4) hkl: 001/2

Point group: 622 Hexagonal P622 No. 177 (247) (1) hkl: (4) hkl: (7) khl: (10) khl: P61 22 No. 178 (248) (1) hkl: (4) hkl: 001/2 (7) khl: 001/3 (10) khl: 005/6

(3) ihl: (6) ihl: 001/2

(3) ihl: (6) ihl: 001/2

Laue group: 6/mmm

(2) kil: (5) kil: (8) hil: (11) hil:

(2) kil: 001/3 (5) kil: 005/6 (8) hil: (11) hil: 001/2

(3) ihl: (6) ihl: (9) ikl: (12) ikl:

P65 22 No. 179 (249) (1) hkl: (4) hkl: 001/2 (7) khl: 002/3 (10) khl: 001/6

(2) kil: 002/3 (5) kil: 001/6 (8) hil: (11) hil: 001/2

(3) ihl: 001/3 (6) ihl: 005/6 (9) ikl: 001/3 (12) ikl: 005/6

P62 22 No. 180 (250) (1) hkl: (4) hkl: (7) khl: 002/3 (10) khl: 002/3

(2) kil: 002/3 (5) kil: 002/3 (8) hil: (11) hil:

(3) ihl: 001/3 (6) ihl: 001/3 (9) ikl: 001/3 (12) ikl: 001/3

P64 22 No. 181 (251) (1) hkl: (4) hkl: (7) khl: 001/3 (10) khl: 001/3

(2) kil: 001/3 (5) kil: 001/3 (8) hil: (11) hil:

(3) ihl: 002/3 (6) ihl: 002/3 (9) ikl: 002/3 (12) ikl: 002/3

P63 22 No. 182 (252) (1) hkl: (4) hkl: 001/2 (7) khl: (10) khl: 001/2

(2) kil: (5) kil: 001/2 (8) hil: (11) hil: 001/2

(2) kil: (5) kil: (8) hil: (11) hil:

(3) ihl: (6) ihl: (9) ikl: (12) ikl:

P6cc No. 184 (254) (1) hkl: (4) hkl: (7) khl: 001/2 (10) khl: 001/2

(2) kil: (5) kil: (8) hil: 001/2 (11) hil: 001/2

(3) ihl: (6) ihl: (9) ikl: 001/2 (12) ikl: 001/2

P63 cm No. 185 (255) (1) hkl: (2) kil: (5) kil: 001/2 (4) hkl: 001/2 (8) hil: 001/2 (7) khl: 001/2 (10) khl: (11) hil:

(3) ihl: (6) ihl: 001/2 (9) ikl: 001/2 (12) ikl:

P63 mc No. 186 (256) (1) hkl: (2) kil: (5) kil: 001/2 (4) hkl: 001/2 (8) hil: (7) khl: (10) khl: 001/2 (11) hil: 001/2

(3) ihl: (6) ihl: 001/2 (9) ikl: (12) ikl: 001/2

Point group: 6m2

(3) ihl: 002/3 (6) ihl: 001/6 (9) ikl: 002/3 (12) ikl: 001/6

Hexagonal Laue group: 6/mmm

P6mm No. 183 (253) (1) hkl: (4) hkl: (7) khl: (10) khl:

Hexagonal (2) kil: (5) kil: (8) hil: (11) hil:

(3) ihl: (6) ihl: (9) ikl: (12) ikl:

P6c2 No. 188 (258) (1) hkl: (4) hkl: 001/2 (7) khl: 001/2 (10) khl:

(2) kil: (5) kil: 001/2 (8) hil: 001/2 (11) hil:

(3) ihl: (6) ihl: 001/2 (9) ikl: 001/2 (12) ikl:

P62m No. 189 (259) (1) hkl: (4) hkl: (7) khl: (10) khl:

(2) kil: (5) kil: (8) hil: (11) hil:

(3) ihl: (6) ihl: (9) ikl: (12) ikl:

P62c No. 190 (260) (1) hkl: (4) hkl: 001/2 (7) khl: (10) khl: 001/2

(2) kil: (5) kil: 001/2 (8) hil: (11) hil: 001/2

(3) ihl: (6) ihl: 001/2 (9) ikl: (12) ikl: 001/2

Point group: 6/mmm

Hexagonal

P6mmm No. 191 (261) (1) hkl: (2) kil: (5) kil: (4) hkl: (8) hil: (7) khl: (11) hil: (10) khl:

(3) ihl: (6) ihl: 001/2 (9) ikl: (12) ikl: 001/2

158

Laue group: 6/mmm

P6m2 No. 187 (257) (1) hkl: (4) hkl: (7) khl: (10) khl:

Laue group: 6/mmm (3) ihl: (6) ihl: (9) ikl: (12) ikl:

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) P6mcc No. 192 (262) (1) hkl: (2) kil: (5) kil: (4) hkl: (8) hil: 001/2 (7) khl: 001/2 (11) hil: 001/2 (10) khl: 001/2

(3) ihl: (6) ihl: (9) ikl: 001/2 (12) ikl: 001/2

P63 mcm No. 193 (263) (1) hkl: (2) kil: (5) kil: 001/2 (4) hkl: 001/2 (8) hil: 001/2 (7) khl: 001/2 (11) hil: (10) khl:

(3) ihl: (6) ihl: 001/2 (9) ikl: 001/2 (12) ikl:

P63 mmc No. 194 (264) (1) hkl: (2) kil: (5) kil: 001/2 (4) hkl: 001/2 (8) hil: (7) khl: (11) hil: 001/2 (10) khl: 001/2

(3) ihl: (6) ihl: 001/2 (9) ikl: (12) ikl: 001/2

Point group: 23

Cubic Laue group: m3

P23 No. 195 (265) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

F23 No. 196 (266) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

I23 No. 197 (267) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

P21 3 No. 198 (268) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2

I21 3 No. 199 (269) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2

Point group: m3

Pn3 Origin 2 (1) hkl: (5) klh: (9) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

Pn3 Origin 1 No. 201 (271) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) hkl: 111/2 (14) hkl: 111/2 (17) klh: 111/2 (18) klh: 111/2 (21) lhk: 111/2 (22) lhk: 111/2

(3) hkl: (7) klh: (11) lhk: (15) hkl: 111/2 (19) klh: 111/2 (23) lhk: 111/2

(4) hkl: (8) klh: (12) lhk: (16) hkl: 111/2 (20) klh: 111/2 (24) lhk: 111/2

(3) hkl: 101/2 (7) klh: 110/2 (11) lhk: 011/2

(4) hkl: 011/2 (8) klh: 101/2 (12) lhk: 110/2

Fm3 No. 202 (273) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

Fd3 Origin 1 No. 203 (274) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) hkl: 111/4 (14) hkl: 111/4 (17) klh: 111/4 (18) klh: 111/4 (21) lhk: 111/4 (22) lhk: 111/4

(3) hkl: (7) klh: (11) lhk: (15) hkl: 111/4 (19) klh: 111/4 (23) lhk: 111/4

(4) hkl: (8) klh: (12) lhk: (16) hkl: 111/4 (20) klh: 111/4 (24) lhk: 111/4

Fd3 Origin 2 (1) hkl: (5) klh: (9) lhk:

(3) hkl: 101/4 (7) klh: 110/4 (11) lhk: 011/4

(4) hkl: 011/4 (8) klh: 101/4 (12) lhk: 110/4

Im3 No. 204 (276) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

(3) hkl: (7) klh: (11) lhk:

(4) hkl: (8) klh: (12) lhk:

Pa3 No. 205 (277) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2

Ia3 No. 206 (278) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2

No. 203 (275) (2) hkl: 110/4 (6) klh: 011/4 (10) lhk: 101/4

Point group: 432 Cubic Laue group: m3m

Cubic Laue group: m3

Pm3 No. 200 (270) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk:

No. 201 (272) (2) hkl: 110/2 (6) klh: 011/2 (10) lhk: 101/2

P432 No. 207 (279) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl: (18) hlk: (17) hlk: (22) lkh: (21) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

P42 32 No. 208 (1) hkl: (5) klh: (9) lhk: (13) khl: 111/2 (17) hlk: 111/2 (21) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl:

(4) hkl: (8) klh: (12) lhk: (16) khl:

(280) (2) hkl: (6) klh: (10) lhk: (14) khl: 111/2 (18) hlk: 111/2 (22) lkh: 111/2

F432 No. 209 (281) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl:

159

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) (17) hlk: (21) lkh:

(19) hlk: (23) lkh:

(20) hlk: (24) lkh:

(3) hkl: 110/2 (7) klh: 011/2 (11) lhk: 101/2 (15) khl: 133/4 (19) hlk: 111/4 (23) lkh: 331/4

(4) hkl: 101/2 (8) klh: 110/2 (12) lhk: 011/2 (16) khl: 331/4 (20) hlk: 133/4 (24) lkh: 111/4

I432 No. 211 (283) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl: (18) hlk: (17) hlk: (22) lkh: (21) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

P43 32 No. 212 (1) hkl: (5) klh: (9) lhk: (13) khl: 133/4 (17) hlk: 133/4 (21) lkh: 133/4

(284) (2) hkl: 101/2 (6) klh: 110/2 (10) lhk: 011/2 (14) khl: 111/4 (18) hlk: 313/4 (22) lkh: 331/4

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2 (15) khl: 331/4 (19) hlk: 111/4 (23) lkh: 313/4

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2 (16) khl: 313/4 (20) hlk: 331/4 (24) lkh: 111/4

P41 32 No. 213 (1) hkl: (5) klh: (9) lhk: (13) khl: 311/4 (17) hlk: 311/4 (21) lkh: 311/4

(285) (2) hkl: 101/2 (6) klh: 110/2 (10) lhk: 011/2 (14) khl: 333/4 (18) hlk: 131/4 (22) lkh: 113/4

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2 (15) khl: 113/4 (19) hlk: 333/4 (23) lkh: 131/4

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2 (16) khl: 131/4 (20) hlk: 113/4 (24) lkh: 333/4

F41 32 No. 210 (1) hkl: (5) klh: (9) lhk: (13) khl: 313/4 (17) hlk: 313/4 (21) lkh: 313/4

(18) hlk: (22) lkh: (282) (2) hkl: 011/2 (6) klh: 101/2 (10) lhk: 110/2 (14) khl: 111/4 (18) hlk: 331/4 (22) lkh: 133/4

I41 32 No. 214 (286) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2 (13) khl: 311/4 (14) khl: 333/4 (17) hlk: 311/4 (18) hlk: 131/4 (21) lkh: 311/4 (22) lkh: 113/4

Point group: 43m

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2 (15) khl: 113/4 (19) hlk: 333/4 (23) lkh: 131/4

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

F43m No. 216 (288) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: (14) khl: (17) hlk: (18) hlk: (21) lkh: (22) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

P43n No. 218 (290) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: 111/2 (14) khl: 111/2 (17) hlk: 111/2 (18) hlk: 111/2 (21) lkh: 111/2 (22) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2

F43c No. 219 (291) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: 111/2 (14) khl: 111/2 (17) hlk: 111/2 (18) hlk: 111/2 (21) lkh: 111/2 (22) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2

I43d No. 220 (292) (1) hkl: (2) hkl: 010/2 (5) klh: (6) klh: 001/2 (9) lhk: (10) lhk: 100/2 (13) khl: 111/4 (14) khl: 311/4 (17) hlk: 111/4 (18) hlk: 113/4 (21) lkh: 111/4 (22) lkh: 131/4

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2 (15) khl: 313/4 (19) hlk: 133/4 (23) lkh: 331/4

(4) hkl: 001/2 (8) klh: 100/2 (12) lhk: 010/2 (16) khl: 113/4 (20) hlk: 131/4 (24) lkh: 311/4

Point group: m3m

Cubic Laue group: m3m (3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

Pm3m No. 221 (1) hkl: (5) klh: (9) lhk: (13) khl: (17) hlk: (21) lkh:

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2 (16) khl: 131/4 (20) hlk: 113/4 (24) lkh: 333/4

P43m No. 215 (287) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: (14) khl: (17) hlk: (18) hlk: (21) lkh: (22) lkh:

I43m No. 217 (289) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: (14) khl: (17) hlk: (18) hlk: (21) lkh: (22) lkh:

160

Cubic Laue group: m3m

(293) (2) hkl: (6) klh: (10) lhk: (14) khl: (18) hlk: (22) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

Pn3n Origin 1 No. 222 (294) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl: (18) hlk: (17) hlk: (22) lkh: (21) lkh: (25) hkl: 111/2 (26) hkl: 111/2 (29) klh: 111/2 (30) klh: 111/2 (33) lhk: 111/2 (34) lhk: 111/2 (37) khl: 111/2 (38) khl: 111/2 (41) hlk: 111/2 (42) hlk: 111/2 (45) lkh: 111/2 (46) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh: (27) hkl: 111/2 (31) klh: 111/2 (35) lhk: 111/2 (39) khl: 111/2 (43) hlk: 111/2 (47) lkh: 111/2

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh: (28) hkl: 111/2 (32) klh: 111/2 (36) lhk: 111/2 (40) khl: 111/2 (44) hlk: 111/2 (48) lkh: 111/2

Pn3n Origin 2 No. 222 (295) (1) hkl: (2) hkl: 110/2 (5) klh: (6) klh: 011/2 (9) lhk: (10) lhk: 101/2 (13) khl: 001/2 (14) khl: 111/2

(3) hkl: 101/2 (7) klh: 110/2 (11) lhk: 011/2 (15) khl: 010/2

(4) hkl: 011/2 (8) klh: 101/2 (12) lhk: 110/2 (16) khl: 100/2

1.4. SYMMETRY IN RECIPROCAL SPACE Table A1.4.4.1. Crystallographic space groups in reciprocal space (cont.) (17) hlk: 001/2 (21) lkh: 001/2

(18) hlk: 100/2 (22) lkh: 010/2

(19) hlk: 111/2 (23) lkh: 100/2

(20) hlk: 010/2 (24) lkh: 111/2

Pm3n No. 223 (296) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: 111/2 (14) khl: 111/2 (17) hlk: 111/2 (18) hlk: 111/2 (21) lkh: 111/2 (22) lkh: 111/2

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2

Pn3m Origin 1 (1) hkl: (5) klh: (9) lhk: (13) khl: 111/2 (17) hlk: 111/2 (21) lkh: 111/2 (25) hkl: 111/2 (29) klh: 111/2 (33) lhk: 111/2 (37) khl: (41) hlk: (45) lkh:

No. 224 (297) (2) hkl: (6) klh: (10) lhk: (14) khl: 111/2 (18) hlk: 111/2 (22) lkh: 111/2 (26) hkl: 111/2 (30) klh: 111/2 (34) lhk: 111/2 (38) khl: (42) hlk: (46) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2 (27) hkl: 111/2 (31) klh: 111/2 (35) lhk: 111/2 (39) khl: (43) hlk: (47) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2 (28) hkl: 111/2 (32) klh: 111/2 (36) lhk: 111/2 (40) khl: (44) hlk: (48) lkh:

Pn3m Origin 2 No. 224 (298) (1) hkl: (2) hkl: 110/2 (5) klh: (6) klh: 011/2 (9) lhk: (10) lhk: 101/2 (13) khl: 110/2 (14) khl: (17) hlk: 110/2 (18) hlk: 011/2 (21) lkh: 110/2 (22) lkh: 101/2

(3) hkl: 101/2 (7) klh: 110/2 (11) lhk: 011/2 (15) khl: 101/2 (19) hlk: (23) lkh: 011/2

(4) hkl: 011/2 (8) klh: 101/2 (12) lhk: 110/2 (16) khl: 011/2 (20) hlk: 101/2 (24) lkh:

Fm3m No. 225 (299) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl: (18) hlk: (17) hlk: (22) lkh: (21) lkh: Fm3c No. 226 (300) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (13) khl: 111/2 (14) khl: 111/2 (17) hlk: 111/2 (18) hlk: 111/2 (21) lkh: 111/2 (22) lkh: 111/2 Fd3m Origin 1 No. 227 (301) (1) hkl: (2) hkl: 011/2 (5) klh: (6) klh: 101/2 (9) lhk: (10) lhk: 110/2 (13) khl: 313/4 (14) khl: 111/4 (17) hlk: 313/4 (18) hlk: 331/4

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: 111/2 (19) hlk: 111/2 (23) lkh: 111/2 (3) hkl: 110/2 (7) klh: 011/2 (11) lhk: 101/2 (15) khl: 133/4 (19) hlk: 111/4

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: 111/2 (20) hlk: 111/2 (24) lkh: 111/2 (4) hkl: 101/2 (8) klh: 110/2 (12) lhk: 011/2 (16) khl: 331/4 (20) hlk: 133/4

161

(21) lkh: 313/4 (25) hkl: 111/4 (29) klh: 111/4 (33) lhk: 111/4 (37) khl: 101/2 (41) hlk: 101/2 (45) lkh: 101/2

(22) lkh: 133/4 (26) hkl: 133/4 (30) klh: 313/4 (34) lhk: 331/4 (38) khl: (42) hlk: 110/2 (46) lkh: 011/2

(23) lkh: 331/4 (27) hkl: 331/4 (31) klh: 133/4 (35) lhk: 313/4 (39) khl: 011/2 (43) hlk: (47) lkh: 110/2

(24) lkh: 111/4 (28) hkl: 313/4 (32) klh: 331/4 (36) lhk: 133/4 (40) khl: 110/2 (44) hlk: 011/2 (48) lkh:

Fd3m Origin 2 (1) hkl: (5) klh: (9) lhk: (13) khl: 312/4 (17) hlk: 312/4 (21) lkh: 312/4

No. 227 (302) (2) hkl: 312/4 (6) klh: 231/4 (10) lhk: 123/4 (14) khl: (18) hlk: 231/4 (22) lkh: 123/4

(3) hkl: 123/4 (7) klh: 312/4 (11) lhk: 231/4 (15) khl: 123/4 (19) hlk: (23) lkh: 231/4

(4) hkl: 231/4 (8) klh: 123/4 (12) lhk: 312/4 (16) khl: 231/4 (20) hlk: 123/4 (24) lkh:

Fd3c Origin 1 No. 228 (303) (1) hkl: (2) hkl: 011/2 (5) klh: (6) klh: 101/2 (9) lhk: (10) lhk: 110/2 (13) khl: 313/4 (14) khl: 111/4 (17) hlk: 313/4 (18) hlk: 331/4 (21) lkh: 313/4 (22) lkh: 133/4 (25) hkl: 333/4 (26) hkl: 311/4 (29) klh: 333/4 (30) klh: 131/4 (33) lhk: 333/4 (34) lhk: 113/4 (37) khl: 010/2 (38) khl: 111/2 (41) hlk: 010/2 (42) hlk: 001/2 (45) lkh: 010/2 (46) lkh: 100/2

(3) hkl: 110/2 (7) klh: 011/2 (11) lhk: 101/2 (15) khl: 133/4 (19) hlk: 111/4 (23) lkh: 331/4 (27) hkl: 113/4 (31) klh: 311/4 (35) lhk: 131/4 (39) khl: 100/2 (43) hlk: 111/2 (47) lkh: 001/2

(4) hkl: 101/2 (8) klh: 110/2 (12) lhk: 011/2 (16) khl: 331/4 (20) hlk: 133/4 (24) lkh: 111/4 (28) hkl: 131/4 (32) klh: 113/4 (36) lhk: 311/4 (40) khl: 001/2 (44) hlk: 100/2 (48) lkh: 111/2

Fd3c Origin 2 No. 228 (304) (1) hkl: (2) hkl: 132/4 (5) klh: (6) klh: 213/4 (9) lhk: (10) lhk: 321/4 (13) khl: 310/4 (14) khl: 111/2 (17) hlk: 310/4 (18) hlk: 031/4 (21) lkh: 310/4 (22) lkh: 103/4

(3) hkl: 321/4 (7) klh: 132/4 (11) lhk: 213/4 (15) khl: 103/4 (19) hlk: 111/2 (23) lkh: 031/4

(4) hkl: 213/4 (8) klh: 321/4 (12) lhk: 132/4 (16) khl: 031/4 (20) hlk: 103/4 (24) lkh: 111/2

Im3m No. 229 (305) (1) hkl: (2) hkl: (5) klh: (6) klh: (9) lhk: (10) lhk: (14) khl: (13) khl: (18) hlk: (17) hlk: (22) lkh: (21) lkh:

(3) hkl: (7) klh: (11) lhk: (15) khl: (19) hlk: (23) lkh:

(4) hkl: (8) klh: (12) lhk: (16) khl: (20) hlk: (24) lkh:

Ia3d No. 230 (306) (1) hkl: (2) hkl: 101/2 (5) klh: (6) klh: 110/2 (9) lhk: (10) lhk: 011/2 (13) khl: 311/4 (14) khl: 333/4 (17) hlk: 311/4 (18) hlk: 131/4 (21) lkh: 311/4 (22) lkh: 113/4

(3) hkl: 011/2 (7) klh: 101/2 (11) lhk: 110/2 (15) khl: 113/4 (19) hlk: 333/4 (23) lkh: 131/4

(4) hkl: 110/2 (8) klh: 011/2 (12) lhk: 101/2 (16) khl: 131/4 (20) hlk: 113/4 (24) lkh: 333/4

International Tables for Crystallography (2006). Vol. B, Chapter 1.5, pp. 162–188.

1.5. Crystallographic viewpoints in the classification of space-group representations BY M. I. AROYO 1.5.1. List of symbols G; S G G0 P or G T R, S; W w X x, y, z; xi x L a, b, c or t L a , b , c K k G k G Lk
G

H. WONDRATSCHEK

AND

Group, especially space group; site-symmetry group Element of group G Symmorphic space group Point group of space group G Translation subgroup of space group G Matrix; matrix part of a symmetry operation Column part of a symmetry operation Point of point space Coordinates of a point or coefficients of a vector Column of point coordinates or of vector coefficients Vector lattice of the space group G
ak T Basis vectors or row of basis vectors of the lattice L of G Vector of the lattice L of G Reciprocal lattice of the space group G or
ak Basis vectors or column of basis vectors of the reciprocal lattice L Vector of the reciprocal lattice L Vector of reciprocal space Reciprocal-space group Little co-group of k Little group of k (Matrix) representation of G 1.5.2. Introduction

This new chapter on representations widens the scope of the general topics of reciprocal space treated in this volume. Space-group representations play a growing role in physical applications of crystal symmetry. They are treated in a number of papers and books but comparison of the terms and the listed data is difﬁcult. The main reason for this is the lack of standards in the classiﬁcation and nomenclature of representations. As a result, the reader is confronted with different numbers of types and barely comparable notations used by the different authors, see e.g. Stokes & Hatch (1988), Table 7. The k vectors, which can be described as vectors in reciprocal space, play a decisive role in the description and classiﬁcation of space-group representations. Their symmetry properties are determined by the so-called reciprocal-space group G which is always isomorphic to a symmorphic space group G0 . The different symmetry types of k vectors correspond to the different kinds of point orbits in the symmorphic space groups G0 . The classiﬁcation of point orbits into Wyckoff positions in International Tables for Crystallography Volume A (IT A) (1995) can be used directly to classify the irreducible representations of a space group, abbreviated irreps; the Wyckoff positions of the symmorphic space groups G0 form a basis for a natural classiﬁcation of the irreps. This was ﬁrst discovered by Wintgen (1941). Similar results have been obtained independently by Raghavacharyulu (1961), who introduced the term reciprocal-space group. In this chapter a classiﬁcation of irreps is provided which is based on Wintgen’s idea. Although this idea is now more than 50 years old, it has been utilized only rarely and has not yet found proper recognition in the literature and in the existing tables of space-group irreps. Slater (1962) described the correspondence between the special k vectors

of the Brillouin zone and the Wyckoff positions of space group Pm3m. Similarly, Jan (1972) compared Wyckoff positions with points of the Brillouin zone when describing the symmetry Pm3 of the Fermi surface for the pyrite structure. However, the widespread tables of Miller & Love (1967), Zak et al. (1969), Bradley & Cracknell (1972) (abbreviated as BC), Cracknell et al. (1979) (abbreviated as CDML), and Kovalev (1986) have not made use of this kind of classiﬁcation and its possibilities, and the existing tables are unnecessarily complicated, cf. Boyle (1986). In addition, historical reasons have obscured the classiﬁcation of irreps and impeded their application. The ﬁrst considerations of irreps dealt only with space groups of translation lattices (Bouckaert et al., 1936). Later, other space groups were taken into consideration as well. Instead of treating these (lower) symmetries as such, their irreps were derived and classiﬁed by starting from the irreps of lattice space groups and proceeding to those of lower symmetry. This procedure has two consequences: (1) those k vectors that are special in a lattice space group are also correspondingly listed in the low-symmetry space group even if they have lost their special properties due to the symmetry reduction; (2) during the symmetry reduction unnecessary new types of k vectors and symbols for them are introduced. The use of the reciprocal-space group G avoids both these detours. In this chapter we consider in more detail the reciprocal-spacegroup approach and show that widely used crystallographic conventions can be adopted for the classiﬁcation of space-group representations. Some basic concepts are developed in Section 1.5.3. Possible conventions are discussed in Section 1.5.4. The consequences and advantages of this approach are demonstrated and discussed using examples in Section 1.5.5.

1.5.3. Basic concepts The aim of this section is to give a brief overview of some of the basic concepts related to groups and their representations. Its content should be of some help to readers who wish to refresh their knowledge of space groups and representations, and to familiarize themselves with the kind of description in this chapter. However, it can not serve as an introductory text for these subjects. The interested reader is referred to books dealing with space-group theory, representations of space groups and their applications in solid-state physics: see Bradley & Cracknell (1972) or the forthcoming Chapter 1.2 of IT D (Physical properties of crystals) by Janssen (2001). 1.5.3.1. Representations of finite groups Group theory is the proper tool for studying symmetry in science. The elements of the crystallographic groups are rigid motions (isometries) with regard to performing one after another. The set of all isometries that map an object onto itself always fulﬁls the group postulates and is called the symmetry or the symmetry group of that object; the isometry itself is called a symmetry operation. Symmetry groups of crystals are dealt with in this chapter. In addition, groups of matrices with regard to matrix multiplication (matrix groups) are considered frequently. Such groups will sometimes be called realizations or representations of abstract groups. Many applications of group theory to physical problems are closely related to representation theory, cf. Rosen (1981) and

162 Copyright 2006 International Union of Crystallography

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS references therein. In this section, matrix representations of ﬁnite groups G are considered. The concepts of homomorphism and matrix groups are of essential importance. A group B is a homomorphic image of a group A if there exists a mapping of the elements Ai of A onto the elements Bk of B that preserves the multiplication relation (in general several elements of A are mapped onto one element of B): if Ai ! Bi and Ak ! Bk , then Ai Ak ! Bi Bk holds for all elements of A and B (the image of the product is equal to the product of the images). In the special case of a one-to-one mapping, the homomorphism is called an isomorphism. A matrix group is a group whose elements are non-singular square matrices. The law of combination is matrix multiplication and the group inverse is the inverse matrix. In the following we will be concerned with some basic properties of ﬁnite matrix groups relevant to representations. Let M1 and M2 be two matrix groups whose matrices are of the same dimension. They are said to be equivalent if there exists a (non-singular) matrix S such that M2 S 1 M1 S holds. Equivalence implies isomorphism but the inverse is not true: two matrix groups may be isomorphic without being equivalent. According to the theorem of Schur-Auerbach, every ﬁnite matrix group is equivalent to a unitary matrix group (by a unitary matrix group we understand a matrix group consisting entirely of unitary matrices). A matrix group M is reducible if it is equivalent to a matrix group in which every matrix M is of the form D1 X R , O D2 see e.g. Lomont (1959), p. 47. The group M is completely reducible if it is equivalent to a matrix group in which for all matrices R the submatrices X are O matrices (consisting of zeros only). According to the theorem of Maschke, a ﬁnite matrix group is completely reducible if it is reducible. A matrix group is irreducible if it is not reducible. A (matrix) representation
G of a group G is a homomorphic mapping of G onto a matrix group M
G. In a representation every element G 2 G is associated with a matrix M
G. The dimension of the matrices is called the dimension of the representation. The above-mentioned theorems on ﬁnite matrix groups can be applied directly to representations: we can restrict the considerations to unitary representations only. Further, since every ﬁnite matrix group is either completely reducible into irreducible constituents or irreducible, it follows that the inﬁnite set of all matrix representations of a group is known in principle once the irreducible representations are known. Naturally, the question of how to construct all nonequivalent irreducible representations of a ﬁnite group and how to classify them arises. Linear representations are especially important for applications. In this chapter only linear representations of space groups will be considered. Realizations and representations are homomorphic images of abstract groups, but not all of them are linear. In particular, the action of space groups on point space is a nonlinear realization of the abstract space groups because isometries and thus symmetry operations W of space groups G are nonlinear operations. The same holds for their description by matrix-column pairs (W, w),† by the general position, or by augmented
4 4 matrices, see IT A, Part 8. Therefore, the isomorphic matrix representation of a space group, mostly used by crystallographers and listed in the space-group tables of IT A as the general position, is not linear. { In physics often written as the Seitz symbol
W jw.

1.5.3.2. Space groups In crystallography one deals with real crystals. In many cases the treatment of the crystal is much simpler, but nevertheless describes the crystal and its properties very well, if the real crystal is replaced by an ‘ideal crystal’. The real crystal is then considered to be a ﬁnite piece of an undisturbed, periodic, and thus inﬁnitely extended arrangement of particles or their centres: ideal crystals are periodic objects in three-dimensional point space E3 , also called direct space. Periodicity means that there are translations among the symmetry operations of ideal crystals. The symmetry group of an ideal crystal is called its space group G. Space groups G are of special interest for our problem because: (1) their irreps are the subject of the classiﬁcation to be discussed; (2) this classiﬁcation makes use of the isomorphism of certain groups to the so-called symmorphic space groups G0 . Therefore, space groups are introduced here in a slightly more detailed manner than the other concepts. In doing this we follow the deﬁnitions and symbolism of IT A, Part 8. To each space group G belongs an inﬁnite set T of translations, the translation lattice of G. The lattice T forms an inﬁnite Abelian invariant subgroup of G. For each translation its translation vector is deﬁned. The set of all translation vectors is called the vector lattice L of G. Because of the ﬁnite size of the atoms constituting the real crystal, the lengths of the translation vectors of the ideal crystal cannot be arbitrarily small; rather there is a lower limit 0 for their length in the range of a few A˚. When referred to a coordinate system
O, a1 , a2 , a3 , consisting of an origin O and a basis ak , the elements W, i.e. the symmetry operations of the space group G, are described by matrix-column pairs (W, w) with matrix part W and column part w. The translations of G are represented by pairs
I, ti , where I is the
3 3 unit matrix and t i is the column of coefﬁcients of the translation vector ti 2 L. The basis can always be chosen such that all columns t i and no other columns of translations consist of integers. Such a basis p1 , p2 , p3 is called a primitive basis. For each vector lattice L there exists an inﬁnite number of primitive bases. The space group G can be decomposed into left cosets relative to T: G T [
W 2 , w2 T [ . . . [
W i , wi T [ . . . [
W n , wn T
1531 The coset representatives form the ﬁnite set V f
W v , wv g, v 1, . . . , n, with
W 1 , w1
I, o, where o is the column consisting of zeros only. The factor group GT is in books on isomorphic to the point group P of G (called G representation theory) describing the symmetry of the external shape of the macroscopic crystal and being represented by the matrices W 1 , W 2 , . . . , W n . If V can be chosen such that all wv o, then G is called a symmorphic space group G0 . A symmorphic space group can be recognized easily from its conventional Hermann– Mauguin symbol which does not contain any screw or glide component. In terms of group theory, a symmorphic space group is the semidirect product of T and P, cf. BC, p. 44. In symmorphic space groups G0 (and in no others) there are site-symmetry groups which are isomorphic to the point group P of G0 . Space groups can be classiﬁed into 219 (afﬁne) space-group types either by isomorphism or by afﬁne equivalence; the 230 crystallographic space-group types are obtained by restricting the transformations available for afﬁne equivalence to those with positive determinant, cf. IT A, Section 8.2.1. Many important properties of space groups are shared by all space groups of a type. In such a case one speaks of properties of the type. For example, if a space group is symmorphic, then all space groups of its type are

163

1. GENERAL RELATIONSHIPS AND TECHNIQUES symmorphic, so that one normally speaks of a symmorphic spacegroup type. With the concept of symmorphic space groups one can also deﬁne the arithmetic crystal classes: Let G0 be a symmorphic space group referred to a primitive basis and V f
W v , wv g its set of coset representatives with wv o for all columns. To G0 all those space groups G can be assigned for which a primitive basis can be found such that the matrix parts W v of their sets V are the same as those of G0 , only the columns wv may differ. In this way, to a type of symmorphic space groups G0 , other types of space groups are assigned, i.e. the space-group types are classiﬁed according to the symmorphic space-group types. These classes are called arithmetic crystal classes of space groups or of space-group types. There are 73 arithmetic crystal classes corresponding to the 73 types of symmorphic space groups; between 1 and 16 space-group types belong to an arithmetic crystal class. A matrix-algebraic deﬁnition of arithmetic crystal classes and a proposal for their nomenclature can be found in IT A, Section 8.2.2; see also Section 8.3.4 and Table 8.2. 1.5.3.3. Representations of the translation group T and the reciprocal lattice For representation theory we follow the terminology of BC and CDML. Let G be referred to a primitive basis. For the following, the inﬁnite set of translations, based on discrete cyclic groups of inﬁnite order, will be replaced by a (very large) ﬁnite set in the usual way. One assumes the Born–von Karman boundary conditions
I, tbi Ni
I, Ni
I, o

1532

to hold, where tbi
1, 0, 0, (0, 1, 0) or (0, 0, 1) and Ni is a large integer for i 1, 2 or 3, respectively. Then for any lattice translation (I, t),
I, Nt
I, o

1533

holds, where Nt is the column
N1 t1 , N2 t2 , N3 t3 . If the (inﬁnitely many) translations mapped in this way onto (I, o) form a normal subgroup T 1 of G, then the mapping described by (1.5.3.3) is a homomorphism. There exists a factor group G0 GT 1 of G relative to T 1 with translation subgroup T 0 T T 1 which is ﬁnite and is sometimes called the ﬁnite space group. Only the irreducible representations (irreps) of these ﬁnite space groups will be considered. The deﬁnitions of space-group type, symmorphic space group etc. can be transferred to these groups. Because T is Abelian, T 0 is also Abelian. Replacing the space group G by G0 means that the especially well developed theory of representations of ﬁnite groups can be applied, cf. Lomont (1959), Jansen & Boon (1967). For convenience, the prime 0 will be omitted and the symbol G will be used instead of G0 ; T 0 will be denoted by T in the following. Because T (formerly T 0 ) is Abelian, its irreps
T are onedimensional and consist of (complex) roots of unity. Owing to equations (1.5.3.2) and (1.5.3.3), the irreps q1 q2 q3
I, t of T have the form t1 t2 t3 q1 q2 q3 ,
1534
I, t exp 2i q1 q2 q3 N1 N2 N3 where t is the column
t1 , t2 , t3 , qj 0, 1, 2, . . . , Nj 1, j 1, 2, 3, and tk and qj are integers. Given a primitive basis a1 , a2 , a3 of L, mathematicians and crystallographers deﬁne the basis of the dual or reciprocal lattice L by ai aj ij ,

1535

where a a is the scalar product between the vectors and ij is the unit matrix (see e.g. Chapter 1.1, Section 1.1.3). Texts on the physics of solids redeﬁne the basis a1 , a2 , a3 of the reciprocal lattice L , lengthening each of the basis vectors aj by the factor 2. Therefore, in the physicist’s convention the relation between the bases of direct and reciprocal lattice reads (cf. BC, p. 86): ai aj 2ij

1536

In the present chapter only the physicist’s basis of the reciprocal lattice is employed, and hence the use of aj should not lead to misunderstandings. The set of all vectors K,† K k1 a1 k2 a2 k3 a3 ,

1537

ki integer, is called the lattice reciprocal to L or the reciprocal lattice L .‡ If one adopts the notation of IT A, Part 5, the basis of direct space is denoted by a row
a1 , a2 , a3 T , where
T means transposed. For reciprocal space, the basis is described by a column
a1 , a2 , a3 . To each lattice generated from a basis
ai T a reciprocal lattice is generated from the basis
aj . Both lattices, L and L , can be compared most easily by referring the direct lattice L to its conventional basis
ai T as deﬁned in Chapters 2.1 and 9.1 of IT A. In this case, the lattice L may be primitive or centred. If
ai T forms a primitive basis of L, i.e. if L is primitive, then the basis
aj forms a primitive basis of L . If L is centred, i.e.
ai T is not a primitive basis of L, then there exists a centring matrix P, 0 det
P 1, by which three linearly independent vectors of L with rational coefﬁcients are generated from those with integer coefﬁcients, cf. IT A, Table 5.1. Moreover, P can be chosen such that the set of vectors
p1 , p2 , p3 T
a1 , a2 , a3 T P

1538

forms a primitive basis of L. Then the basis vectors
p1 , p2 , p3 of the lattice reciprocal to the lattice generated by
p1 , p2 , p3 T are determined by

p1 , p2 , p3 P 1
a1 , a2 , a3

1539

and form a primitive basis of L . Because det
P 1 1, not all vectors K of the form (1.5.3.7) belong to L . If k1 , k2 , k3 are the (integer) coefﬁcients of these vectors K referred to
aj and kp1 p1 kp2 p2 kp3 p3 are the vectors of L , then K
kj T
aj
kj T P
pi
kpi T
pi is a vector of L if and only if the coefﬁcients
kp1 , kp2 , kp3 T
k1 , k2 , k3 T P

15310

are integers. In other words,
k1 , k2 , k3 T has to fulﬁl the equation
k1 , k2 , k3 T
kp1 , kp2 , kp3 T P 1

15311

As is well known, the Bravais type of the reciprocal lattice L is not necessarily the same as that of its direct lattice L. If W is the matrix of a (point-) symmetry operation of the direct lattice, referred to its basis
ai T , then W 1 is the matrix of the same symmetry operation of the reciprocal lattice but referred to the dual basis
ai . This does not affect the symmetry because in a (symmetry) group the inverse of each element in the group also belongs to the group. Therefore, the (point) symmetries of a lattice { In crystallography vectors are designated by small bold-faced letters. With K we make an exception in order to follow the tradition of physics. A crystallographic alternative would be t . { The lattice L is often called the direct lattice. These names are historically introduced and cannot be changed, although equations (1.5.3.5) and (1.5.3.6) show that essentially neither of the lattices is preferred: they form a pair of mutually reciprocal lattices.

164

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS k , then the number of arms of the star of k and the and its reciprocal lattice are always the same. However, there may little co-group G be differences in the matrix descriptions due to the different number of k vectors in the fundamental region from the orbit of k is jjG k j. and jG orientations of L and L relative to the symmetry elements of G T due to the reference to the different bases
ai and
ai . For Deﬁnition. The group of all elements
W, w 2 G for which W 2 example, if L has the point symmetry (Hermann–Mauguin symbol) k is called the little group Lk of k. 3m1, then the symbol for the point symmetry of L is 31m and vice G versa. Equation (1.5.3.14) for k resembles the equation x Wx t,

1.5.3.4. Irreducible representations of space groups and the reciprocal-space group Let
ai T be a conventional basis of the lattice L of the space group G. From (1.5.3.6), ki qi Ni and k 3k1 ki ai , equation (1.5.3.4) can be written q1 q2 q3

I, t

k

I, t exp ik t

15312

Equation (1.5.3.12) has the same form if a primitive basis
pi T of L has been chosen. In this case, the vector k is given by k 3i1 kpi pi . Let a primitive basis
pi T be chosen for the lattice L. The set of all vectors k (known as wavevectors) forms a discontinuous array. Consider two wavevectors k and k0 k K, where K is a vector of the reciprocal lattice L . Obviously, k and k0 describe the same irrep of T . Therefore, to determine all irreps of T it is necessary to consider only the wavevectors of a small region of the reciprocal space, where the translation of this region by all vectors of L ﬁlls the reciprocal space without gap or overlap. Such a region is called a fundamental region of L . (The nomenclature in literature is not quite uniform. We follow here widely adopted deﬁnitions.) The fundamental region of L is not uniquely determined. Two types of fundamental regions are of interest in this chapter: (1) The ﬁrst Brillouin zone is that range of k space around o for which jkj jK kj holds for any vector K 2 L (Wigner–Seitz cell or domain of inﬂuence in k space). The Brillouin zone is used in books and articles on irreps of space groups. (2) The crystallographic unit cell in reciprocal space, for short unit cell, is the set of all k vectors with 0 ki 1. It corresponds to the unit cell used in crystallography for the description of crystal structures in direct space. Let k be some vector according to (1.5.3.12) and W be the The following deﬁnitions are useful: matrices of G.

t2L

15315

by which the ﬁxed points of the symmetry operation
W, t of a symmorphic space group G0 are determined. Indeed, the orbits of k deﬁned by (1.5.3.13) correspond to the point orbits of G0 , the little k of k corresponds to the site-symmetry group of that co-group G point X whose coordinates
xi have the same values as the vector coefﬁcients
ki T of k, and the star of k corresponds to a set of representatives of X in G0 . (The analogue of the little group Lk is rarely considered in crystallography.) All symmetry operations of G0 may be obtained as combinations of an operation that leaves the origin ﬁxed with a translation of L, i.e. are of the kind
W, t
I, t
W , o. We now deﬁne the analogous group for the k vectors. Whereas G0 is a realization of the corresponding abstract group in direct (point) space, the group to be deﬁned will be a realization of it in reciprocal (vector) space. Deﬁnition. The group G which is the semidirect product of the and the translation group of the reciprocal lattice L point group G of G is called the reciprocal-space group of G. The elements of G are the operations
W, K
I, K
W , o and K 2 L . In order to emphasize that G is a group with W 2 G acting on reciprocal space and not the inverse of a space group (whatever that may mean) we insert a hyphen ‘-’ between ‘reciprocal’ and ‘space’. From the deﬁnition of G it follows that space groups of the same type deﬁne the same type of reciprocal-space group G . Moreover, as G does not depend on the column parts of the space-group operations, all space groups of the same arithmetic crystal class determine the same type of G ; for arithmetic crystal class see Section 1.5.3.2. Following Wintgen (1941), the types of reciprocalspace groups G are listed for the arithmetic crystal classes of space groups, i.e. for all space groups G, in Appendix 1.5.1.

Deﬁnition. The set of all vectors k0 fulﬁlling the condition K 2 L
15313 k0 kW K, W 2 G, 1.5.4. Conventions in the classification of space-group irreps

is called the orbit of k. for which Deﬁnition. The set of all matrices W 2 G
15314 k forms a group which is called the little co-group G of k. The vector k fIg; otherwise G k fIg and k is called k is called general if G special.

Because of the isomorphism between the reciprocal-space groups G and the symmorphic space groups G0 one can introduce crystallographic conventions in the classiﬁcation of space-group irreps. These conventions will be compared with those which have mainly been used up to now. Illustrative examples to the following more theoretical considerations are discussed in Section 1.5.5.1.

k is a subgroup of the point group G. The little co-group G k relative to G . Consider the coset decomposition of G

1.5.4.1. Fundamental regions

k kW K,

K2L

relative Deﬁnition. If fW m g is a set of coset representatives of G k to G , then the set fkW m g is called the star of k and the vectors kW m are called the arms of the star. of the The number of arms of the star of k is equal to the order jGj k k of point group G divided by the order jG j of the symmetry group G vectors from the orbit of k in k. If k is general, then there are jGj each fundamental region and jGj arms of the star. If k is special with

Different types of regions of reciprocal space may be chosen as fundamental regions, see Section 1.5.3.4. The most frequently used type is the ﬁrst Brillouin zone, which is the Wigner–Seitz cell (or Voronoi region, Dirichlet domain, domain of inﬂuence; cf. IT A, Chapter 9.1) of the reciprocal lattice. It has the property that with each k vector also its star belongs to the Brillouin zone. Such a choice has three advantages: (1) the Brillouin zone is always primitive and it manifests the point symmetry of the reciprocal lattice L of G;

165

1. GENERAL RELATIONSHIPS AND TECHNIQUES (2) only k vectors of the boundary of the Brillouin zone may have little-group representations which are obtained from projective k , see e.g. BC, p. 156; representations of the little co-group G (3) for physical reasons, the Brillouin zone may be the most convenient fundamental region. Of these advantages only the third may be essential. For the classiﬁcation of irreps the minimal domains, see Section 1.5.4.2, are much more important than the fundamental regions. The minimal domain does not display the point-group symmetry anyway and the distinguished k vectors always belong to its boundary however the minimal domain may be chosen. The serious disadvantage of the Brillouin zone is its often complicated shape which, moreover, depends on the lattice parameters of L . The body that represents the Brillouin zone belongs to one of the ﬁve Fedorov polyhedra (more or less distorted versions of the cubic forms cube, rhombdodecahedron or cuboctahedron, of the hexagonal prism, or of the tetragonal elongated rhombdodecahedron). A more detailed description is that by the 24 symmetrische Sorten (Delaunay sorts) of Delaunay (1933), Figs. 11 and 12. According to this classiﬁcation, the Brillouin zone may display three types of polyhedra of cubic, one type of hexagonal, two of rhombohedral, three of tetragonal, six of orthorhombic, six of monoclinic, and three types of triclinic symmetry. For low symmetries the shape of the Brillouin zone is so variable that BC, p. 90 ff. chose a primitive unit cell of L for the fundamental regions of triclinic and monoclinic crystals. This cell also reﬂects the point symmetry of L , it has six faces only, and although its shape varies with the lattice constants all cells are afﬁnely equivalent. For space groups of higher symmetry, BC and most other authors prefer the Brillouin zone. Considering L as a lattice, one can refer it to its conventional crystallographic lattice basis. Referred to this basis, the unit cell of L is always an alternative to the Brillouin zone. With the exception of the hexagonal lattice, the unit cell of L reﬂects the point symmetry, it has only six faces and its shape is always afﬁnely equivalent for varying lattice constants. For a space group G with a primitive lattice, the above-deﬁned conventional unit cell of L is also primitive. If G has a centred lattice, then L also belongs to a type of centred lattice and the conventional cell of L [not to be confused with the cell spanned by the basis
aj dual to the basis
ai T ] is larger than necessary. However, this is not disturbing because in this context the fundamental region is an auxiliary construction only for the deﬁnition of the minimal domain; see Section 1.5.4.2.

Deﬁnition. A simply connected part of the fundamental region which contains exactly one k vector of each orbit of k is called a minimal domain .

In general, in representation theory of space groups the Brillouin zone is taken as the fundamental region and is called a representation domain.† Again, the volume of a representation j of the volume of the Brillouin domain in reciprocal space is 1jG zone. In addition, as the Brillouin zone contains for each k vector all k vectors of the star of k, by application of all symmetry operations to one obtains the Brillouin zone; cf. BC, p. 147. As the W 2G Brillouin zone may change its geometrical type depending on the lattice constants, the type of the representation domain may also vary with varying lattice constants; see examples (3) and (4) in Section 1.5.5.1. The simplest crystal structures are the lattice-like structures that are built up of translationally equivalent points (centres of particles) of the space group G is only. For such a structure the point group G equal to the point group Q of its lattice L. Such point groups are called holohedral, the space group G is called holosymmetric. There are seven holohedral point groups of three dimensions: 1, 2m, mmm, 4mmm, 3m, 6mmm and m3m. For the non-holosym Q holds. metric space groups G, G In books on representation theory of space groups, holosymmetric space groups play a distinguished role. Their representation domains are called basic domains . For holosymmetric space Q holds, groups holds. If G is non-holosymmetric, i.e. G

is deﬁned by Q and is smaller than the representation domain in Q. In the literature by a factor which is equal to the index of G these basic domains are considered to be of primary importance. In Miller & Love (1967) only the irreps for the k vectors of the basic domains are listed. Section 5.5 of BC and Davies & Cracknell (1976) state that such a listing is not sufﬁcient for the nonholosymmetric space groups because . Section 5.5 of BC shows how to overcome this deﬁciency; Chapter 4 of CDML introduces new types of k vectors for the parts of not belonging to

. The crystallographic analogue of the representation domain in direct space is the asymmetric unit, cf. IT A. According to its deﬁnition it is a simply connected smallest part of space from which by application of all symmetry operations of the space group the whole space is exactly ﬁlled. For each space-group type the asymmetric units of IT A belong to the same topological type independent of the lattice constants. They are chosen as ‘simple’ bodies by inspection rather than by applying clearly stated rules. Among the asymmetric units of the 73 symmorphic space-group types G0 there are 31 parallelepipeds, 27 prisms (13 trigonal, 6 tetragonal and 8 pentagonal) for the non-cubic, and 15 pyramids (11 trigonal and 4 tetragonal) for the cubic G0 . The asymmetric units of IT A – transferred to the groups G of reciprocal space – are alternatives for the representation domains of the literature. They are formulated as closed bodies. Therefore, for inner points k, the asymmetric units of IT A fulﬁl the condition that each star of k is represented exactly once. For the surface, however, these conditions either have to be worked out or one gives up the condition of uniqueness and replaces exactly by at least in the deﬁnition of the minimal domain (see preceding footnote). The examples of Section 1.5.5.1 show that the conditions for the boundary of the asymmetric unit and its special points, lines and

The choice of the minimal domain is by no means unique. One of the difﬁculties in comparing the published data on irreps of space groups is due to the different representation domains found in the literature. The number of k vectors of each general k orbit in a fundamental of G; see region is always equal to the order of the point group G Section 1.5.3.4. Therefore, the volume of the minimal domain in j of the volume of the fundamental region. reciprocal space is 1jG Now we can restrict the search for all irreps of G to the k vectors within a minimal domain .

{ From deﬁnition 3.7.1 on p. 147 of BC, it does not follow that a representation domain contains exactly one k vector from each star. The condition ‘The intersection of the representation domain with its symmetrically equivalent domains is empty’ is missing. Lines 14 to 11 from the bottom of p. 149, however, state that such a property of the representation domain is intended. The representation domains of CDML, Figs. 3.15–3.29 contain at least one k vector of each star (Vol. 1, pp. 31, 57 and 65). On pp. 66, 67 a procedure is described for eliminating those k vectors from the representation domain which occur more than once. In the deﬁnition of Altmann (1977), p. 204, the representation domain contains exactly one arm (prong) per star.

1.5.4.2. Minimal domains One can show that all irreps of G can be built up from the irreps of T . Moreover, to ﬁnd all irreps of G it is only necessary to consider one k vector from each orbit of k, cf. CDML, p. 31. k

166

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS Table 1.5.4.1. Conventional coefficients
ki T of k expressed by the adjusted coefficients
kai of IT A for the different Bravais types of lattices in direct space

Table 1.5.4.2. Primitive coefficients
kpi T of k from CDML expressed by the adjusted coefficients
kai of IT A for the different Bravais types of lattices in direct space

Lattice types

k1

k2

k3

Lattice types

kp1

kp2

kp3

aP, mP, oP, tP, cP, rP mA, oA mC, oC oF, cF, oI, cI tI hP hR (hexagonal)

ka1 ka1 2ka1 2ka1 ka1 ka2 ka1 ka2 2ka1 ka2

ka2 2ka2 2ka2 2ka2 ka1 ka2 ka2 ka1 2ka2

ka3 2ka3 ka3 2ka3 2ka3 ka3 3ka3

aP, mP, oP, tP, cP, rP mA, oA mC, oC oF, cF oI, cI tI hP hR (hexagonal)

ka1

ka2

ka3

ka1 ka1 ka2 ka2 ka3 ka1 ka2 ka3 ka1 ka3 ka1 ka2 ka1 ka3

ka2 ka3 ka1 ka2 ka1 ka3 ka1 ka2 ka3 ka1 ka3 ka2 ka1 ka2 ka3

ka2 ka3 ka3 ka1 ka2 ka1 ka2 ka3 ka2 ka3 ka3 ka2 ka3

planes are in many cases much easier to formulate than those for the representation domain. The k-vector coefﬁcients. For each k vector one can derive a set of irreps of the space group G. Different k vectors of a k orbit give rise to equivalent irreps. Thus, for the calculation of the irreps of the space groups it is essential to identify the orbits of k vectors in reciprocal space. This means ﬁnding the sets of all k vectors that are related by the operations of the reciprocal-space group G according to equation (1.5.3.13). The classiﬁcation of these k orbits can be done in analogy to that of the point orbits of the symmorphic space groups, as is apparent from the comparison of equations (1.5.3.14) and (1.5.3.15). The classes of point orbits in direct space under a space group G are well known and are listed in the space-group tables of IT A. They are labelled by Wyckoff letters. The stabilizer S G
X of a point X is called the site-symmetry group of X, and a Wyckoff position consists of all orbits for which the site-symmetry groups are conjugate subgroups of G. Let G be a symmorphic space group G0 . Owing to the isomorphism between the reciprocal-space groups G and the symmorphic space groups G0 , the complete list of the types of special k vectors of G is provided by the Wyckoff positions of k correspond to each other and the G0 . The groups S G0
X and G multiplicity of the Wyckoff position (divided by the number of centring vectors per unit cell for centred lattices) equals the number of arms of the star of k. Let the vectors t of L be referred to the conventional basis
ai T of the space-group tables of IT A, as deﬁned in Chapters 2.1 and 9.1 of IT A. Then, for the construction of the irreducible representations k of T the coefﬁcients of the k vectors must be referred to the basis
aj of reciprocal space dual to
ai T in direct space. These k-vector coefﬁcients may be different from the conventional coordinates of G0 listed in the Wyckoff positions of IT A. Example. Let G be a space group with an I-centred cubic lattice L, conventional basis
ai T . Then L is an F-centred lattice. If referred to the conventional basis
aj with ai aj 2ij , the k vectors with coefﬁcients 1 0 0, 0 1 0 and 0 0 1 do not belong to L due to the ‘extinction laws’ well known in X-ray crystallography. However, in the standard basis of G0 , isomorphic to G , the vectors 1 0 0, 0 1 0 and 0 0 1 point to the vertices of the face-centred cube and thus correspond to 2 0 0, 0 2 0 and 0 0 2 referred to the conventional basis
aj . In the following, three bases and, therefore, three kinds of coefﬁcients of k will be distinguished: (1) Coefﬁcients referred to the conventional basis
aj in reciprocal space, dual to the conventional basis
ai T in direct space. The corresponding k-vector coefﬁcients,
kj T , will be called conventional coefﬁcients. (2) Coefﬁcients of k referred to a primitive basis
api in reciprocal space (which is dual to a primitive basis in direct space).

The corresponding coefﬁcients will be called primitive coefﬁcients
kpi T . For a centred lattice the coefﬁcients
kpi T are different from the conventional coefﬁcients
ki T . In most of the physics literature related to space-group representations these primitive coefﬁcients are used, e.g. by CDML. (3) The coefﬁcients of k referred to the conventional basis of G0 . These coefﬁcients will be called adjusted coefﬁcients
kai T . The relations between conventional and adjusted coefﬁcients are listed for the different Bravais types of reciprocal lattices in Table 1.5.4.1, and those between adjusted and primitive coordinates in Table 1.5.4.2. If adjusted coefﬁcients are used, then IT A is as suitable for dealing with irreps as it is for handling space-group symmetry. 1.5.4.3. Wintgen positions In order to avoid confusion, in the following the analogues to the Wyckoff positions of G0 will be called Wintgen positions of G ; the coordinates of the Wyckoff position are replaced by the k-vector coefﬁcients of the Wintgen position, the Wyckoff letter will be called the Wintgen letter, and the symbols for the site symmetries of k of the k G0 are to be read as the symbols for the little co-groups G vectors in G . The multiplicity of a Wyckoff position is retained in the Wintgen symbol in order to facilitate the use of IT A for the description of symmetry in k space. However, it is equal to the multiplicity of the star of k only in the case of primitive lattices L . In analogy to a Wyckoff position, a Wintgen position is a set of orbits of k vectors. Each orbit as well as each star of k can be represented by any one of its k vectors. The zero, one, two or three parameters in the k-vector coefﬁcients deﬁne points, lines, planes or the full parameter space. The different stars of a Wintgen position are obtained by changing the parameters. Remark. Because reciprocal space is a vector space, there is no origin choice and the Wintgen letters are unique (in contrast to the Wyckoff letters, which may depend on the origin choice). Therefore, the introduction of Wintgen sets in analogy to the Wyckoff sets of IT A, Section 8.3.2 is not necessary. It may be advantageous to describe the different stars belonging to a Wintgen position in a uniform way. For this purpose one can deﬁne: Deﬁnition. Two k vectors of a Wintgen position are uni-arm if one can be obtained from the other by parameter variation. The description of the stars of a Wintgen position is uni-arm if the k vectors representing these stars are uni-arm.

167

1. GENERAL RELATIONSHIPS AND TECHNIQUES and Ia Table 1.5.5.1. The k-vector types for the space groups Im3m 3d Comparison of the k-vector labels and parameters of CDML with the Wyckoff positions of IT A for Fm3m,
O5h , isomorphic to the reciprocal-space group G of m3mI. The parameter ranges in the last column are chosen such that each star of k is represented exactly once. The sign means symmetrically equivalent. The coordinates x, y, z of IT A are related to the k-vector coefﬁcients of CDML by x 12
k2 k3 , y 12
k1 k3 , z 12
k1 k2 . k-vector label, CDML

Wyckoff position, IT A

Parameters (see Fig. 1.5.5.1b), IT A

0, 0, 0

4 a m3m

0, 0, 0

H

1 2,

4 b m3m

1 2 , 0, 0

P

1 1 1 4, 4, 4

8 c 43m

1 1 1 4, 4, 4

1 1 2, 2

N 0, 0,

1 2

24 d m.mm

1 1 4, 4,0

,

,

24 e 4m.m

x, 0, 0 : 0 x 12

32 f 32 f 32 f 32 f 32 f

x, x, x: 0 x 14 1 x, x, x: 0 x 14 2 x, x, x: 14 x 12 x, x, 12 x : 0 x 14 x, x, x: 0 x 12 with x 6 14

, , F 12 , 12 3, 12 F1 (Fig. 1.5.5.1b) F2 (Fig. 1.5.5.1b) [ F1 H2 nP

48 g 2.mm

1 1 4 , 4,

48 h m.m2

x, x, 0: 0 x 14

48 i m.m2

1 2

96 j m..

x, y, 0: 0 y x 12

B , , PH1 N1 (Fig. 1.5.5.1b) C , , J , , PH1 (Fig. 1.5.5.1b) C [ B [ J NN1 H1

96 k 96 k 96 k 96 k 96 k 96 k

1 4

GP , ,

192 l 1

D , ,

1 2

.3m .3m .3m .3m .3m

0, 0, G 12

,

A ,

,

1 2

,

1 2

1 2

z: 0 z 14

x, x, 0: 0 x 14 y

x, x, z: 0 z x 14 1 x, x, z: 0 x 2 x z 12 x, x, z: 0 z x 14 x, y, y: 0 y x 12 y x, x, z: 0 x z 12 x x, x, z: 0 x 14, 0 z 12 with z 6 x, z 6 12 x.

..m ..m ..m ..m ..m ..m

1 4

1 4

x, y, z: 0 z y x 12

For non-holosymmetric space groups the representation domain is a multiple of the basic domain . CDML introduced new letters for stars of k vectors in those parts of which do not belong to . If one can make a new k vector uni-arm to some k vector of the basic domain by an appropriate choice of and , one can extend the parameter range of this k vector of to instead of introducing new letters. It turns out that indeed most of these new letters are unnecessary. This restricts the introduction of new types of k vectors to the few cases where it is indispensible. Extension of the parameter range for k means that the corresponding representations can also be obtained by parameter variation. Such representations can be considered to belong to the same type. In this way a large number of superﬂuous k-vector names, which pretend a greater variety of types of irreps than really exists, can be avoided (Boyle, 1986). For examples see Section 1.5.5.1. 1.5.5. Examples and conclusions 1.5.5.1. Examples In this section, four examples are considered in each of which the crystallographic classiﬁcation scheme for the irreps is compared with the traditional one:† { Corresponding tables and ﬁgures for all space groups are available at http:// www.cryst.ehu.es/cryst/get_kvec.html.

y

(space (1) k-vector types of the arithmetic crystal class m3mI groups Im3m and Ia3d), reciprocal-space group isomorphic to Fm3m; ; see Table 1.5.5.1 and Fig. 1.5.5.1; (2) k-vector types of the arithmetic crystal class m3I (Im3 and Ia3), reciprocal-space group isomorphic to Fm3, ; see Table 1.5.5.2 and Fig. 1.5.5.2; (3) k-vector types of the arithmetic crystal class 4mmmI
I4mmm, I4mcm, I41 amd and I41 acd, reciprocalspace group isomorphic to I4mmm. Here changes for different ratios of the lattice constants a and c; see Table 1.5.5.3 and Fig. 1.5.5.3; (4) k-vector types of the arithmetic crystal class mm2F (Fmm2 and Fdd2), reciprocal-space group isomorphic to Imm2. Here

changes for different ratios of the lattice constants a, b and c; see Table 1.5.5.4 and Fig. 1.5.5.4. The asymmetric units of IT A are displayed in Figs. 1.5.5.1 to 1.5.5.4 by dashed lines. In Tables 1.5.5.1 to 1.5.5.4, the k-vector types of CDML are compared with the Wintgen (Wyckoff) positions of IT A. The parameter ranges are chosen such that each star of k is represented exactly once. Sets of symmetry points, lines or planes of CDML which belong to the same Wintgen position are separated by horizontal lines in Tables 1.5.5.1 to 1.5.5.3. The uniarm description is listed in the last entry of each Wintgen position in Tables 1.5.5.1 and 1.5.5.2. In Table 1.5.5.4, so many k-vector types of CDML belong to each Wintgen position that the latter are used as headings under which the CDML types are listed.

168

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS Examples: (a) In m3mI and m3I there are the and F lines of k vectors k1
, , and k2
12 , 12 3, 12 in CDML, see Tables 1.5.5.1 and 1.5.5.2, Figs. 1.5.5.1 and 1.5.5.2. Do they belong to the same Wintgen position, i.e. do their irreps belong to the same type? There is a twofold rotation 2 x, 14 , 14 which maps k2 onto k02
12 , 12 , 12 2 F1 (the rotation 2 is described in the primitive basis of CDML by k 01 k 3 , k 02 k 1 k 2 k 3 1, k 03 k 1 ). The k vectors k1 and k02 are uni-arm and form the line H2 nP [ F1 [ F which protrudes from the body of the asymmetric unit like a ﬂagpole. This proves that k1 and k2 belong to the same Wintgen position, which is 32 f .3m x, x, x. Owing to the shape of the asymmetric unit of IT A (which is similar here to that of the representation domain in CDML), the line x, x, x is kinked into the parts and F. One may choose even between F1 (uni-arm to ) or F2 (completing the plane C NP). The latter transformation is performed by applying the symmetry operation 3 x, x, x for F ! F2 . Remark. The uni-arm description unmasks those k vectors (e.g. those of line F) which lie on the boundary of the Brillouin zone but belong to a Wintgen position which also contains inner k vectors (line ). Such k vectors cannot give rise to little-group representations obtained from projective representations of the k. little co-group G (b) In Table 1.5.5.1 for m3mI, see also Fig. 1.5.5.1, the k-vector planes B HNP, C NP and J HP of CDML belong to the same Wintgen position 96 k ..m. In the asymmetric unit of IT A (as in the representation domain of CDML) the plane x, x, z is kinked into parts belonging to different arms of the star of k. Transforming, e.g., B and J to the plane of C by 2 14 , y, 14
B ! PN1 H1 and 3 x, x, x
J ! PH1 , one obtains a complete plane ( NN1 H1 for C, B and J) as a uni-arm description of the Wintgen position 96 k ..m. This plane protrudes from the body of the asymmetric unit like a wing. Fig. 1.5.5.1. Symmorphic space group Fm3m (isomorphic to the reciprocalspace group G of m3mI). (a) The asymmetric unit (thick dashed edges) imbedded in the Brillouin zone, which is a cubic rhombdodecahedron. (b) The asymmetric unit HNP, IT A, p. 678. The representation domain NH3 P of CDML is obtained by reﬂecting HNP through the plane of NP. Coordinates of the points: 0, 0, 0; N 14 , 14 , 0 N1 = 14 , 14 , 12; H 12 , 0, 0 H1 0, 0, 12 H2 12 , 12 , 12 H3 0, 12 , 0; P 14 , 14 , 14; the sign means symmetrically equivalent. Lines: P x, x, x; F HP 12 x, x, x F1 PH2 x, x, x F2 PH1 x, x, 12 x; H x, 0, 0; N x, x, 0; D NP 14 , 14 , z; G NH x, 12 x, 0. Planes: A HN x, y, 0; B HNP x, 12 x, z PN1 H1 x, x, z; C NP x, x, z; J HP x, y, y PH1 x, x, z. Large black circles: corners of the asymmetric unit (special points); small open circles: other special points; dashed lines: edges of the asymmetric unit (special lines). For the parameter ranges see Table 1.5.5.1.

1.5.5.2. Results of G, the more (1) The higher the symmetry of the point group G one is restricted in the choice of the boundaries of the minimal domain. This is because a symmetry element (rotation or rotoinversion axis, plane of reﬂection, centre of inversion) cannot occur in the interior of the minimal domain but only on its boundary. However, even for holosymmetric space groups of highest symmetry, the description by Brillouin zone and representation domain is not as concise as possible, cf. CDML.

Remark. One should avoid the term equivalent for the relation between and F or between B, C and J as it is used by Stokes et al. (1993). BC, p. 95 give the deﬁnition: ‘Two k vectors k1 and k2 are equivalent if k1 k2 K, where K 2 L ’. One can also express this by saying: ‘Two k vectors are equivalent if they differ by a vector K of the (reciprocal) lattice.’ We prefer to extend this equivalence by saying: ‘Two k vectors k1 and k2 are equivalent if and only if they belong to the same orbit of k’, i.e. if there is a matrix part W and a vector K 2 L belonging to G such that k2 W k1 K, see equation (1.5.3.13). Alternatively, this can be expressed as: ‘Two k vectors are equivalent if and only if they belong to the same or to translationally equivalent stars of k.’ The k vectors of and F or of B, C and J are not even equivalent under this broader deﬁnition, see Davies & Dirl (1987). If the representatives of the k-vector stars are chosen uni-arm, as in the examples, their non-equivalence is evident. (2) In general two trends can be observed: (a) The lower the symmetry of the crystal system, the more irreps of CDML, recognized by different letters, belong to the same Wintgen position. This trend is due to the splitting of lines and planes into pieces because of the more and more complicated shape of the Brillouin zone. Faces and lines on the surface of the Brillouin zone may appear or disappear depending on the lattice parameters, causing different descriptions of Wintgen positions. This does not happen in unit cells or their asymmetric units; see Sections 1.5.4.1 and 1.5.4.2. Examples: (i) The boundary conditions (parameter ranges) for the special lines and planes of the asymmetric unit and for general k vectors of

169

1. GENERAL RELATIONSHIPS AND TECHNIQUES and Ia Table 1.5.5.2. The k-vector types for the space groups Im3 3 Comparison of the k-vector labels and parameters of CDML with the Wyckoff positions of IT A for Fm3
Th3 , isomorphic to the reciprocal-space group G of m3I. The parameter ranges in Fm3 are obtained by extending those of Fm3m such that each star of k is represented exactly once. The k-vector types of
Fm3m , see Table 1.5.5.1, are also listed. The sign means symmetrically equivalent. Lines in parentheses are not special lines but belong to special planes. As in Table 1.5.5.1, the coordinates x, y, z of IT A are related to the k-vector coefﬁcients of CDML by x 12
k2 k3 , y 12
k1 k3 , z 12
k1 k2 . k-vector label, CDML
Fm3m

Wyckoff position, IT A

Fm3

Parameters (see Fig. 1.5.5.2b), IT A

Fm3 4 a m3.

0, 0, 0 1 2 , 0, 0

H

H

4 b m3.

P

P

8 c 23.

1 1 1 4, 4, 4

N

N

24 d 2m

1 1 4, 4,0

24 e mm2..

x, 0, 0: 0 x 12

F F1 [ F1 H2 nP

F F1 [ F1 H2 nP

32 f 32 f 32 f 32 f

x, x, x: 0 x 14 1 x, x, x: 0 x 14 2 x, x, x: 14 x 12 x, x, x: 0 x 12 with x 6 14

D

D

48 g 2..

1 1 4 , 4 , z:

G A

G A AA , , A [ AA [ [ G

48 h 48 h 48 h 48 h 48 h

m.. m.. m.. m.. m..

x, y, 0 : 0 x y 14 x, y, 0 : 0 y 12 x 14 x, y, 0 : 0 y x 12 y x, y, 0 : 0 12 x y x x, y, 0 : 0 y x 12 [ [ 0 y x 14

C B J GP

GP GP GP GP GP GP

96 i 96 i 96 i 96 i 96 i 96 i

1 1 1 1 1 1

x, y, z : 0 z x y 14 x, y, z : 0 z y 12 x 14 x, y, z : 0 z y x 12 y x, y, z : 0 z y x 12 y x, y, z : 0 z 12 x y x

the reciprocal-space group
F4mmm (setting I4mmm) are listed in Table 1.5.5.3. The main condition of the representation domain is that of the boundary plane x, y, z f1
ca2 1 2
x yg4, which for ca 1 forms the triangle Z0 Z1 P (Figs. 1.5.5.3a,b) but for ca 1 forms the pentagon S1 RPGS (Figs. 1.5.5.3c,d). The inner points of these boundary planes are points of the general position GP with the exception of the line Q x, 12 x, 14, which is a twofold rotation axis. The boundary conditions for the representation domain depend on ca; they are much more complicated than those for the asymmetric unit (for this the boundary condition is simply x, y, 14). (ii) In the reciprocal-space group
Imm2 , see Figs. 1.5.5.4(a) to (c), the lines and Q belong to Wintgen position 2 a mm2; G and H belong to 2 b mm2; and R, and U, A and C, and B and D belong to the general position GP. The decisive boundary plane is xa2 yb2 zc2 d 2 4, where d 2 1a2 1b2 1c2 , or xa2 yb2 zc2 d 2 4, where d 2 a2 b2 c2 . There is no relation of the lattice constants for which all the abovementioned lines are realized on the surface of the representation domain simultaneously, either two or three of them do not appear and the length of the others depends on the boundary plane; see Table 1.5.5.4 and Figs. 1.5.5.4(a) to (c). Again, the boundary conditions for the asymmetric unit are independent of the lattice parameters, all lines mentioned above are present and their parameters run from 0 to 12.

.3. .3. .3. .3.

0 z 14

x, y, z : 0 z y x 12 y [ [ x, y, z : 0 z 12 x y x

(b) The more symmetry a space group has lost compared to its holosymmetric space group, the more letters of irreps are introduced, cf. CDML. In most cases these additional labels can be easily avoided by extension of the parameter range in the kvector space of the holosymmetric group. Example. Extension of the plane A NH, Wintgen position 96 j m.. of
Fm3m , to A [ AA 1 NH in the reciprocal-space group
Fm3 of the arithmetic crystal class m3I, cf. Tables 1.5.5.1 and 1.5.5.2 and Fig. 1.5.5.2. Both planes, A and AA, belong to Wintgen position 48 h m.. of
Fm3 . In addition, in the transition from a holosymmetric space group H to a non-holosymmetric space group G, the order of the little co k . Such a k k of a special k vector of H may be reduced in G group H vector may then be incorporated into a more general Wintgen k and described by an extension of the parameter range. position of G Example. Plane H 1 x, y, 0: In
Fm3m , see Fig. 1.5.5.1, all points
, H, N and lines
, , G of the boundary of the asymmetric unit are special. In
Fm3 , see Fig. 1.5.5.2, the lines and H 1 ( means equivalent) are special but , G and N 1 N belong to the plane
A [ AA. The free parameter range on the line 1 is 12 of the full parameter range of 1 , see Section 1.5.5.3. Therefore, the parameter ranges of
A [ AA [ G [ in x, y, 0 can be taken as: 0 y x 12 for A [ AA [ G and (for ) 0 y x 14.

170

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS Is it easy to recognize those letters of CDML which belong to the same Wintgen position? In
I4mmm , the lines and V (V exists for ca 1 only) are parallel, as are and F, but the lines Y and U are not (F and U exist for ca 1 only). The planes C x, y, 0 and D x, y, 12 (D for ca 1 only) are parallel but the planes A 0, y, z and E x, 12 , z are not. Nevertheless, each of these pairs belongs to one Wintgen position, i.e. describes one type of k vector. 1.5.5.3. Parameter ranges

Fig. 1.5.5.2. Symmorphic space group F 3m (isomorphic to the reciprocalspace group G of m3I). (a) The asymmetric unit (thick dashed edges) half imbedded in and half protruding from the Brillouin zone, which is a cubic rhombdodecahedron (as in Fig. 1.5.5.1). (b) The asymmetric unit H 1 P, IT A, p. 610. The representation domain of CDML is HH3 P. Both bodies have HNP in common; H 1 NP is mapped onto NH3 P by a twofold rotation around NP. The representation domain as the asymmetric unit would be the better choice because it is congruent to the asymmetric unit of IT A and is fully imbedded in the Brillouin zone. Coordinates of the points: 0, 0, 0 1 12 , 12 , 0; P 14 , 14 , 14; H 12 , 0, 0 H1 0, 0, 12 H2 12 , 12 , 12 H3 0, 12 , 0; N 14 , 14 , 0 N1 14 , 14 , 12; the sign means symmetrically equivalent. Lines: P x, x, x P 1 x, x, 12 x; F HP 12 x, x, x F1 PH2 x, x, x F2 PH1 x, x, 12 x; H x, 0, 0 H 1 1 1 1 1 x, 0 and N 2 , y, 0; D PN 4 , 4 , z. (G NH x, 2 x, x, 0 N 1 x, x, 0 are not special lines.) Planes: A HN x, y, 0; AA 1 NH x, y, 0; B HNP x, 12 x, z PN1 H1 x, x, z; C NP x, x, z; J HP x, y, y PH1 x, x, z. (The boundary planes B, C and J are parts of the general position GP.) Large black circles: special points of the asymmetric unit; small black circle: special point 1 ; small open circles: other special points; dashed lines: edges and special line D of the asymmetric unit. The edge 1 is not a special line but is part of the boundary plane A [ AA. For the parameter ranges see Table 1.5.5.2.

For the uni-arm description of a Wintgen position it is easy to check whether the parameter ranges for the general or special constituents of the representation domain or asymmetric unit have been stated correctly. For this purpose one may deﬁne the ﬁeld of k as the parameter space (point, line, plane or space) of a Wintgen position. For the check, one determines that part of the ﬁeld of k k (G k which is inside the unit cell. The order of the little co-group G represents those operations which leave the ﬁeld of k ﬁxed pointwise) is divided by the order of the stabilizer [which is the set of all symmetry operations (modulo integer translations) that leave the ﬁeld invariant as a whole]. The result gives the independent fraction of the above-determined volume of the unit cell or the area of the plane or length of the line. If the description is not uni-arm, the uni-arm parameter range will be split into the parameter ranges of the different arms. The parameter ranges of the different arms are not necessarily equal; see the second of the following examples. Examples: (1) Line [ F1 : In
Fm3m the line x, x, x has stabilizer 3m and k 3m. Therefore, the divisor is 2 and x runs from little co-group G 0 to 12 in 0 x 1. (2) Plane B [ C [ J : In
Fm3m , the stabilizer of x, x, z is generated by m.mm and the centring translation t
12 , 12 , 0 modulo integer translations
mod Tint . They generate a group of order 16; k is ..m of order 2. The fraction of the plane is 2 1 of the area G 16 8 212 a2 , as expressed by the parameter ranges 0 x 14, 0 z 12. There are six arms of the star of x, x, z: x, x, z; x, x, z; x, y, x; x, y, x; x, y, y; x, y, y. Three of them are represented in the boundary of the representation domain: B HNP, C NP and J HP; see Fig. 1.5.5.1. The areas of their parameter ranges are 321 , 321 and 161 , respectively; the sum is 18. The same result holds for
Fm3 : the stabilizer is generated by k j jf1gj 1, the 2m and t
12 , 12 , 0 mod Tint and is of order 8, jG 1 quotient is again 8, the parameter range is the same as for
Fm3m . The planes H 1 P and N 1 P are equivalent to J HP and C NP, and do not contribute to the parameter ranges. (3) Plane x, y, 0: In
Fm3m the stabilizer of plane A is generated k (site-symmetry group) m.., by 4mmm and t
12 , 12 , 0, order 32, G order 2. Consequently, HN is 161 of the unit square a2 : 0 y x 12 y. In
Fm3 , the stabilizer of A [ AA is k . Therefore, mmm. plus t
12 , 12 , 0, order 16, with the same group G 1 2 H 1 is 8 of the unit square a in
Fm3 : 0 y x 12. (4) Line x, x, 0: In
Fm3m the stabilizer is generated by m.mm k is m.2m of order 4. The divisor and t
12 , 12 , 0 mod Tint , order 16, G 1 is 4 and thus 0 x 4. In
Fm3 the stabilizer is generated by k m, order 2; the 2m and t
12 , 12 , 0 mod Tint , order 8, and G divisor is 4 again and 0 x 14 is restricted to the same range.† Data for the independent parameter ranges are essential to make sure that exactly one k vector per orbit is represented in the representation domain or in the asymmetric unit. Such data are { Boyle & Kennedy (1988) propose general rules for the parameter ranges of kvector coefﬁcients referred to a primitive basis. The ranges listed in Tables 1.5.5.1 to 1.5.5.4 possibly do not follow these rules.

171

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.5.5.3. The k-vector types for the space groups I4mmm, I4mcm, I41 amd and I41 acd Comparison of the k-vector labels and parameters of CDML with the Wyckoff positions of IT A for I4mmm
D17 4h , isomorphic to the reciprocal-space group G of 4mmmI. For the asymmetric unit, see Fig. 1.5.5.3. Two ratios of the lattice constants are distinguished for the representation domains of CDML: a c and a c, see Figs. 1.5.5.3(a, b) and (c, d). The sign means symmetrically equivalent. The parameter ranges for the planes and the general position GP refer to the asymmetric unit. The coordinates x, y, z of IT A are related to the k-vector coefﬁcients of CDML by x 12
k1 k2 , y 12
k1 k2 2k3 , z 12
k1 k2 .

Wyckoff position, IT A

k-vector labels, CDML ac

ac

0, 0, 0 M

ac

0, 0, 0

1 1 1 2, 2, 2

Parameters (see Fig. 1.5.5.3), IT A

M 12 , 12 ,

1 2

2 a 4mmm

0, 0, 0

2 b 4mmm

1 1 2, 2,0

a c†

0, 0,

X 0, 0,

1 2

X 0, 0,

1 2

4 c mmm.

0, 12 , 0

P 14 , 14 ,

1 4

P 14 , 14 ,

1 4

4 d 4m2

0, 12 ,

8 f ..2m

1 1 1 4, 4, 4

4 e 4mm 4 e 4mm

1 1 2 , 2 , z:

8 g 2mm.

0,

1 2,

8 h m.2m 8 h m.2m

x, x, 0 : 0 x 12 —

N 0, 12 , 0

N 0, 12 , 0

, , V 12 , 12 , W , ,

1 2

1 2

W , ,

, , —

Q 14

1 2

, , F 12 , 12 ,

0, 0, Y , , —

, , —

1 2

0, 0, 1 2

, 14 ,

Y , , U 12 , 12 , 1 4

Q 14

C , , —

C , , D 12 , 12 ,

B , ,

B , ,

A , , E , , GP , ,

1 2

1 4

1 2

GP , ,

1 2

z: 0 z

z0

0 z 12 —

1 4

0 x s1 x, x, 12 : 0 x s 12

s1

0xr 0, y, 12: 0 y g 12

r

1 2

8 j m2m. 8 j m2m.

x, 12 , 0: 0 x 12 —

16 k ..2

x,

16 l m.. 16 l m..

x, y, 0: 0 x y 12§ —

16 m ..m

x, x, z: 0 x 12, 0 z 14 [ 0 x 14, z 14

16 n .m. 16 n .m.

0, y, z: 0 y 12, 0 z 12¶ x, 12, z: transferred to A 0, y, z

32 o 1

x, y, z: 0 x y 12, 0 z 14 [ 0 x y 12

A , , E , ,

0, 0, z: 0 z z0 ‡ 0 z z1 12

0, y, 0: 0 y

, 14 ,

1 4

8 i m2m. 1 2 1 2

1 2

1 2

x, 14: 0 x 14 — x, y,

1 2

x, z 14

† If the parameter range is different from that for a c. ‡ z0 is a coordinate of point Z0 etc., see Figs. 1.5.5.3(b), (d). § For a c, the parameter range includes the equivalent of D MSG. ¶ The parameter range includes A and the equivalent of E.

much more difﬁcult to calculate for the representation domains and cannot be found in the cited tables of irreps. In the way just described the inner parameter range can be ﬁxed. In addition, the boundaries of the parameter range must be determined: (5) Line x, x, x: In (Fm3m)* and (Fm3)* the points 0, 0, 0; 12 , 12 , 12 (and 14 , 14 , 14) are special points; the parameter ranges are open: 0 < x < 14 , 14 < x < 12. (6) Plane x, x, z: In
Fm3m all corners , N, N1 , H1 and all edges are either special points or special lines. Therefore, the parameter ranges are open: x, x, z: 0 < x < 14, 0 < z < 12, where the lines x, x, x: 0 x 14 and x, x, 12 x: 0 x 14 are special lines and thus excepted. (7) Plane x, y, 0: In both
Fm3m and
Fm3 , 0 < x and 0 < y holds. The k vectors of line x, x, 0 have little co-groups of higher order and belong to another Wintgen position in the representation domain (or asymmetric unit) of
Fm3m . Therefore, x, y, 0 is open at its boundary x, x, 0 in the range 0 < x < 14. In the asymmetric unit

of
Fm3 the line x, x, 0: 0 < x < 14 belongs to the plane, in this range the boundary of plane A is closed. The other range x, x, 0: 1 1 1 4 < x < 2 is equivalent to the range 0 < x < 4 and thus does not belong to the asymmetric unit; here the boundary of AA is open.

1.5.5.4. Conclusions As has been shown, IT A can serve as a basis for the classiﬁcation of irreps of space groups by using the concept of reciprocal-space groups: (a) The asymmetric units of IT A are minimal domains of k space which are in many cases simpler than the representation domains of the Brillouin zones. However, the asymmetric units of IT A are not designed particularly for this use, cf. Section 1.5.4.2. Therefore, it should be checked whether they are the optimal choice for this purpose. Otherwise, other asymmetric units could easily be introduced.

172

1.5. CLASSIFICATION OF SPACE-GROUP REPRESENTATIONS

Fig. 1.5.5.3. (a), (b). Symmorphic space group I4mmm (isomorphic to the reciprocal-space group G of 4mmmI). Diagrams for a c, i.e. c a . In the ﬁgures a 125c, i.e. c 125a . (a) Representation domain (thick lines) and asymmetric unit (thick dashed lines, partly protruding) imbedded in the Brillouin zone, which is a tetragonal elongated rhombdodecahedron. (b) Representation domain MXZ1 PZ0 and asymmetric unit MXTT1 P of I4mmm, IT A, p. 468. The part MXTNZ1 P is common to both bodies; the part TNPZ0 is equivalent to the part NZ1 PT1 by a twofold rotation around the axis Q NP. Coordinates of the points: 0, 0, 0; X 0, 12 , 0; M 12 , 12 , 0; P 0, 12 , 14; N 14 , 14 , 14; T 0, 0, 14 T1 12 , 12 , 14; Z0 0, 0, z0 Z1 12 , 12 , z1 with z0 1
ca2 4; z1 12 z0 ; the sign means symmetrically equivalent. Lines: Z0 0, 0, z; V Z1 M 12 , 12 , z; W XP 0, 12 , z; M x, x, 0; X 0, y, 0; Y XM x, 12 , 0; Q PN x, 12 x, 14. The lines Z0 Z1 , Z1 P and PZ0 have no special symmetry but belong to special planes. Planes: C MX x, y, 0; B Z0 Z1 M x, x, z; A XPZ0 0, y, z; E MXPZ1 x, 12 , z. The plane Z0 Z1 P belongs to the general position GP. Large black circles: special points belonging to the representation domain; small open circles: T T1 and Z0 Z1 belonging to special lines; thick lines: edges of the representation domain and special line Q NP; dashed lines: edges of the asymmetric unit. For the parameter ranges see Table 1.5.5.3. (c), (d). Symmorphic space group I4mmm (isomorphic to the reciprocal-space group G of 4mmmI). Diagrams for c a, i.e. a c . In the ﬁgures c 125a, i.e. a 125c . (c) Representation domain (thick lines) and asymmetric unit (dashed lines, partly protruding) imbedded in the Brillouin zone, which is a tetragonal cuboctahedron. (d) Representation domain S1 RXPMSG and asymmetric unit M2 XTT1 P of I4mmm, IT A, p. 468. The part S1 RXTNP is common to both bodies; the part TNPMSG is equivalent to the part T1 NPM2 S1 R by a twofold rotation around the axis Q NP. Coordinates of the points: 0, 0, 0; X 0, 12 , 0; N 14 , 14 , 14; M 0, 0, 12 M2 12 , 12 , 0; T 0, 0, 14 T1 12 , 12 , 14; P 0, 12 , 14; S s, s, 12 S1 s1 , s1 , 0 with s 1
ac2 4; s1 12 s; R r, 12 , 0 G 0, g, 12 with r
ac2 2; g 12 r; the sign means symmetrically equivalent. Lines: M 0, 0, z; W XP 0, 12 , z; S1 x, x, 0; F MS x, x, 12; X 0, y, 0; Y XR x, 12 , 0; U MG 0, y, 12; Q PN x, 12 x, 14. The lines GS S1 R, SN NS1 and GP PR have no special symmetry but belong to special planes. Planes: C S1 RX x, y, 0; D MSG x, y, 12; B S1 SM x, x, z; A XPGM 0, y, z; E RXP x, 12 , z. The plane S1 RPGS belongs to the general position GP. Large black circles: special points belonging to the representation domain; small open circles: M2 M; the points T T1 , S S1 and G R belong to special lines; thick lines: edges of the representation domain and special line Q NP; dashed lines: edges of the asymmetric unit. For the parameter ranges see Table 1.5.5.3.

173

1. GENERAL RELATIONSHIPS AND TECHNIQUES Table 1.5.5.4. The k-vector types for the space groups Fmm2 and Fdd2 20 Comparison of the k-vector labels and parameters of CDML with the Wyckoff positions of IT A for Imm2
C2h , isomorphic to the reciprocal-space group G of mm2F. For the asymmetric unit see Fig. 1.5.5.4. Four ratios of the lattice constants are distinguished in CDML, Fig. 3.6 for the representation domains: (a) a2 < b2 c2 , b2 < c2 a2 and c2 < a2 b2 (see Fig. 1.5.5.4a); (b) c2 a2 b2 (see Fig. 1.5.5.4b); (c) b2 c2 a2 [not displayed because essentially the same as (d)]; (d) a2 b2 c2 (see Fig. 1.5.5.4c). The vertices of the Brillouin zones of Fig. 3.6(a)–(d) with a variable coordinate are not designated in CDML. In Figs. 1.5.5.4 (a), (b) and (c) they are denoted as follows: the end point of the line is 0 , of line is 0 , of line is 0 , of line A is A0 etc. The variable coordinate of the end point is 0 , 0 , 0 , a0 etc., respectively. The line A0 B0 is called ab etc. The plane (111) is called . It has the equation in the a , b , c basis : a2 x b2 y c2 z d 2 4 with d 2 a2 b2 c2 . From this equation one calculates the variable coordinates of the vertices of the Brillouin zone: 0 0, 0, 0 with 0 d 2 4c2 ; Q0 12 , 12 , q0 with q0 12 0 ; 0 0, 0 , 0 with 0 d 2 4b2 ; R 0 12 , r0 , 12 with r0 12 0 ; 0 0 , 0, 0 with 0 d 2 4a2 ; U0 u0 , 12 , 12 with