Table Of Content

INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY 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 THE INTERNATIONAL UNION OF CRYSTALLOGRAPHY by KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON 2001 AC.I.P.Catalogue record for this book isavailable from the Library ofCongress ISBN0-7923-6592-5(acid-free paper) Published byKluwerAcademic Publishers, P.O.Box 17,3300AADordrecht, The Netherlands Soldand distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA02061,USA Inall other countries, sold and distributed by Kluwer Academic Publishers, P.O.Box 322,3300AHDordrecht, The Netherlands Technical Editor: N.J.Ashcroft First published in 1993 Second edition2001 (cid:1) International Union ofCrystallography 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 permissionfrom the International Union ofCrystallography Printed in Great Britain byAlden Press,Oxford Contributing authors E. A(cid:1)(cid:2)(cid:3)(cid:4)(cid:5): (cid:6)(cid:7)(cid:8)(cid:9) (cid:10) (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:1)(cid:16) (cid:17)(cid:2)(cid:18)(cid:19)(cid:15)(cid:1)(cid:16)(cid:18)(cid:13)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24) (cid:25)(cid:3)(cid:15)(cid:16) R. E. M(cid:27)(cid:1)(cid:16),: The Beckman Institute–139–74, Cali- (cid:26)(cid:27)(cid:2)(cid:15)(cid:21) (cid:28)(cid:18)(cid:16)(cid:29)(cid:27)(cid:13)(cid:27)(cid:30)(cid:27)(cid:20)(cid:21) (cid:31)(cid:15)(cid:30) (cid:15)(cid:1)(cid:16)(cid:15)(cid:20) !""#$%#(cid:22)&"(cid:21) (cid:17)’(cid:7)( fornia Institute of Technology, 1201 East California )*(&+ Blvd, Pasadena, California 91125, USA. [3.2] M. I. A(cid:1)(cid:3)(cid:20)(cid:3): Faculty of Physics, University of Sofia, R. P. M(cid:18)(cid:4)(cid:4)(cid:27)(cid:2)(cid:15): Whistler Center for Carbohydrate bulv. J. Boucher 5, 1164 Sofia, Bulgaria. [1.5] Research, and Computational Science and A. A(cid:12)(cid:13),(cid:18)(cid:15)(cid:1): Laboratoire de Mine´ralogie-Cristallo- Engineering Program, Purdue University, West graphie, Universite´ P. et M. Curie, 4 Place Jussieu, Lafayette, Indiana 47907-1160, USA. [4.5.1, 4.5.2] F-75252 Paris CEDEX 05, France. [5.1] A. F. M(cid:3)(cid:3)(cid:5)(cid:18)(cid:15): Department of Applied Physics, Royal G. B(cid:1)(cid:18)(cid:29)(cid:3)(cid:14)(cid:2)(cid:15): MRC Laboratory of Molecular Biology, Melbourne Institute of Technology, 124 La Trobe Hills Road, Cambridge CB2 2QH, England, and Street, Melbourne, Victoria 3000, Australia. [5.2] LURE,Baˆtiment209D,Universite´ Paris-Sud,91405 P. S. P(cid:15)(cid:1)(cid:16),(cid:27)(cid:2): Division of Engineering and Applied Orsay, France. [1.3] Science and The Physics Department, Harvard P. C(cid:3)--(cid:15)(cid:2)(cid:16): Department of Chemistry, Natural University, Cambridge, MA 02138, USA. [4.4] Sciences & Mathematics Complex, State University S. R(cid:27).(cid:27)(cid:16)(cid:15)(cid:16),(cid:27)(cid:2): Raman Research Institute, Bangalore of New York at Buffalo, Buffalo, New York 14260- 560 080, India. [2.4] 3000, USA. [1.2] J. M. C(cid:3)(cid:30)(cid:4)(cid:15)(cid:20): Arizona State University, Box 871504, M. G. R(cid:3)(cid:16)(cid:16).(cid:27)(cid:2)(cid:2): Department of Biological Sciences, Department of Physics and Astronomy, Tempe, AZ Purdue University, West Lafayette, Indiana 47907, 85287-1504, USA. [2.5.1, 2.5.2, 4.3, 5.2] USA. [2.3] R. D(cid:18)(cid:27).(cid:3)(cid:2)(cid:5): MRC Laboratory of Molecular Biology, D. E. S(cid:27)(cid:2)(cid:5)(cid:16): Department of Chemistry, University of Hills Road, Cambridge CB2 2QH, England. [3.3] Kentucky, Chemistry–Physics Building, Lexington, D. L. D(cid:3)(cid:1)(cid:16)(cid:15)(cid:13): ExxonMobil Research and Engineering Kentucky 40506-0055, USA. [3.1] Co., 1545 Route 22 East, Clinton Township, M. S(cid:29),(cid:4)(cid:15)(cid:2)3(cid:15)(cid:1): Laboratoire Louis Neél du CNRS, BP Annandale, New Jersey 08801, USA. [2.5.7, 4.5.1, 166, F-38042 Grenoble CEDEX 9, France. [5.3] 4.5.3] V. S(cid:29),(cid:3).(cid:27)3(cid:15)(cid:1)† [3.2] F. F(cid:1)(cid:15)(cid:20): Institut fu¨r Kristallographie und Mineralogie, U. S,.(cid:12)(cid:15)(cid:4)(cid:18): School of Chemistry, Tel Aviv Univer- Universita¨t, Theresienstrasse 41, D-8000 Mu¨nchen sity, 69 978 Tel Aviv, Israel. [1.1, 1.4, 2.1] 2, Germany. [4.2] C. G(cid:18)(cid:27)(cid:29)(cid:3)(cid:19)(cid:27)//(cid:3): Dipartimento Geomineralogico, W. S(cid:13)(cid:15)(cid:12)(cid:1)(cid:15)(cid:1): Laboratory of Crystallography, Swiss Campus Universitario, I-70125 Bari, Italy. [2.2] Federal Institute of Technology, CH-8092 Zurich, J.K.G01(cid:2)(cid:2)(cid:15)(cid:16):InstituteofPhysics,UniversityofOslo, Switzerland. [4.6] PO Box 1048, N-0316 Oslo 3, Norway. [4.3] B. K. V(cid:27)(cid:18)(cid:2)(cid:16),(cid:13)(cid:15)(cid:18)(cid:2)† [2.5.4, 2.5.5, 2.5.6] P. G(cid:3)(cid:3)(cid:5).(cid:27)(cid:2)† [2.5.3, 5.2] M. V(cid:18)0(cid:27)(cid:20)(cid:27)(cid:2): Molecular Biophysics Unit, Indian Insti- R. W. G(cid:1)(cid:3)(cid:16)(cid:16)(cid:15)-K(cid:12)(cid:2)(cid:16)(cid:13)(cid:4)(cid:15)(cid:19)(cid:15): Lawrence Berkeley tute of Science, Bangalore 560 012, India. [2.4] National Laboratory, 1 Cyclotron Road, Mailstop D. E. W(cid:18)(cid:4)(cid:4)(cid:18)(cid:27).(cid:16): Department of Chemistry, University 4-230, Berkeley, CA 94720, USA. [1.4] of Louisville, Louisville, Kentucky 40292, USA. J.-P. G(cid:12)(cid:18)(cid:14)(cid:27)(cid:20): European Synchrotron Radiation [3.4] Facility, BP 220, F-38043 Grenoble, France. [5.3] B. T. M. W(cid:18)(cid:4)(cid:4)(cid:18)(cid:16): Chemical Crystallography Labora- T. H(cid:27)(cid:18)2(cid:27)(cid:29),: Laboratory of Crystallography, Swiss tory, University of Oxford, 9 Parks Road, Oxford Federal Institute of Technology, CH-8092 Zurich, OX1 3PD, England. [4.1] Switzerland. [4.6] A. J. C. W(cid:18)(cid:4)(cid:16)(cid:3)(cid:2)† [2.1] S. R. H(cid:27)(cid:4)(cid:4): Crystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia. H. W(cid:3)(cid:2)(cid:5)(cid:1)(cid:27)(cid:13)(cid:16)(cid:29),(cid:15)3: Institut fu¨r Kristallographie, [1.4] Universita¨t, D-76128 Karlsruhe, Germany. [1.5] H. J(cid:27)(cid:14)(cid:3)(cid:5)/(cid:18)(cid:2)(cid:16)3(cid:18): Institut fu¨r Kristallographie und B. B. Z(cid:19)(cid:20)(cid:27)(cid:14)(cid:18)(cid:2): Institute of Ore Mineralogy (IGEM), Mineralogie, Universita¨t, Theresienstrasse 41, Academy of Sciences of Russia, Staromonetny 35, D-8000 Mu¨nchen 2, Germany. [4.2] 109017 Moscow, Russia. [2.5.4] † Deceased. † Deceased. v Contents PAGE Preface4(cid:17)(’,.(cid:12)(cid:15)(cid:4)(cid:18)5 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xxv Prefacetothesecondedition4(cid:17)(’,.(cid:12)(cid:15)(cid:4)(cid:18)5 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xxv PART 1. GENERAL RELATIONSHIPS AND TECHNIQUES .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 1.1.Reciprocalspaceincrystallography4(cid:17)(’,.(cid:12)(cid:15)(cid:4)(cid:18)5 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.2.Reciprocallatticeincrystallography .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.3.Fundamentalrelationships .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.1.3.1.Basisvectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.1.3.2.Volumes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.1.3.3.Angularrelationships .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 1.1.3.4.Matricesofmetrictensors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 1.1.4.Tensor-algebraicformulation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.1.4.1.Conventions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.1.4.2.Transformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.1.4.3.Scalarproducts .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.1.4.4.Examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6 1.1.5.Transformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7 1.1.5.1.Transformationsofcoordinates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7 1.1.5.2.Example .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1.1.6.Someanalyticalaspectsofthereciprocalspace .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1.1.6.1.ContinuousFouriertransform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1.1.6.2.DiscreteFouriertransform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1.1.6.3.Bloch’stheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 1.2.Thestructurefactor4(cid:28)((cid:6)(cid:3)--(cid:15)(cid:2)(cid:16)5 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.2.GeneralscatteringexpressionforX-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.3.Scatteringbyacrystal:definitionofastructurefactor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.4.Theisolated-atomapproximationinX-raydiffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.2.5.Scatteringofthermalneutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 1.2.5.1.Nuclearscattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 1.2.5.2.Magneticscattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 1.2.6.Effectofbondingontheatomicelectrondensitywithinthespherical-atomapproximation:thekappaformalism .. 11 1.2.7.Beyondthespherical-atomdescription:theatom-centredsphericalharmonicexpansion .. .. .. .. .. .. .. .. .. 14 1.2.7.1.Direct-spacedescriptionofasphericalatoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 1.2.7.2.Reciprocal-space descriptionofasphericalatoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 15 Table 1.2.7.1.Real sphericalharmonicfunctions(x,y,zaredirectioncosines) .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12 Table 1.2.7.2.Index-pickingrulesofsite-symmetricsphericalharmonics(Kara&Kurki-Suonio,1981) .. .. .. .. .. .. 15 Table 1.2.7.3.‘Kubic Harmonic’functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 16 Table 1.2.7.4.Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens,1990) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19 1.2.8.Fouriertransformoforbitalproducts .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 17 1.2.8.1.One-centreorbitalproducts .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 18 1.2.8.2.Two-centreorbitalproducts .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 18 Table 1.2.8.1.Productsofcomplexsphericalharmonicsasdefinedbyequation(1.2.7.2a). .. .. .. .. .. .. .. .. .. .. 20 Table 1.2.8.2.Productsofrealsphericalharmonicsasdefinedbyequations(1.2.7.2b)and(1.2.7.2c) .. .. .. .. .. .. .. 20 Table 1.2.8.3.Products of two real spherical harmonic functions y in terms of the density functions d defined by lmp lmp equation(1.2.7.3b) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 1.2.9.Theatomictemperaturefactor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 18 1.2.10.ThevibrationalprobabilitydistributionanditsFouriertransformintheharmonicapproximation .. .. .. .. .. .. 18 1.2.11.Rigid-bodyanalysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19 vii CONTENTS Table 1.2.11.1.The arrays G and H to be used in the observational equations U (cid:1)G L (cid:2)H S (cid:2)T [equation ijkl ijkl ij ijkl kl ijkl kl ij (1.2.11.9)] .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 1.2.12.Treatmentofanharmonicity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 22 1.2.12.1.TheGram–Charlierexpansion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 22 1.2.12.2.Thecumulantexpansion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 22 1.2.12.3.Theone-particlepotential(OPP)model .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 23 1.2.12.4.Relative meritsofthethreeexpansions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 23 Table 1.2.12.1.SomeHermitepolynomials(Johnson&Levy,1974;Zucker&Schulz,1982) .. .. .. .. .. .. .. .. .. 22 1.2.13.Thegeneralizedstructurefactor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 23 1.2.14.Conclusion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 24 1.3.Fouriertransformsincrystallography:theory,algorithmsandapplications46((cid:8)(cid:1)(cid:18)(cid:29)(cid:3)(cid:14)(cid:2)(cid:15)5 .. .. .. .. .. .. .. .. .. .. .. 25 1.3.1.Generalintroduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 1.3.2.ThemathematicaltheoryoftheFouriertransformation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 1.3.2.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 1.3.2.2.Preliminarynotionsandnotation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 26 1.3.2.2.1.Metricandtopologicalnotionsin(cid:1)n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 26 1.3.2.2.2.Functionsover(cid:1)n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 26 1.3.2.2.3.Multi-indexnotation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27 1.3.2.2.4.Integration,Lp spaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27 1.3.2.2.5.Tensorproducts.Fubini’stheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27 1.3.2.2.6.Topologyinfunctionspaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 1.3.2.2.6.1.Generaltopology .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 1.3.2.2.6.2.Topologicalvectorspaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 1.3.2.3.Elementsofthetheoryofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 1.3.2.3.1.Origins .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 1.3.2.3.2.Rationale .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 29 1.3.2.3.3.Test-functionspaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 29 1.3.2.3.3.1.Topologyon(cid:1)(cid:3)(cid:1)(cid:4) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 29 1.3.2.3.3.2.Topologyon(cid:2) (cid:3)(cid:1)(cid:4) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 k 1.3.2.3.3.3.Topologyon(cid:2)(cid:3)(cid:1)(cid:4) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 1.3.2.3.3.4.Topologieson(cid:1)(cid:3)m(cid:4)(cid:1)(cid:2)(cid:3)m(cid:4)(cid:1)(cid:2)(cid:3)m(cid:4) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 k 1.3.2.3.4.Definitionofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 1.3.2.3.5.Firstexamplesofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 1.3.2.3.6.Distributionsassociatedtolocallyintegrablefunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 30 1.3.2.3.7.Supportofadistribution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 31 1.3.2.3.8.Convergenceofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 31 1.3.2.3.9.Operationsondistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 31 1.3.2.3.9.1.Differentiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 31 1.3.2.3.9.2.Integrationofdistributionsindimension1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 32 1.3.2.3.9.3.Multiplicationofdistributionsbyfunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 32 1.3.2.3.9.4.Divisionofdistributionsbyfunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 33 1.3.2.3.9.5.Transformationofcoordinates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 33 1.3.2.3.9.6.Tensorproductofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 33 1.3.2.3.9.7.Convolutionofdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 33 1.3.2.4.Fouriertransformsoffunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 34 1.3.2.4.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 34 1.3.2.4.2.FouriertransformsinL1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.1.Linearity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.2.Effectofaffinecoordinatetransformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.3.Conjugatesymmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.4.Tensorproductproperty .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.5.Convolutionproperty .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.6.Reciprocity property .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.7.Riemann–Lebesgue lemma .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.8.Differentiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 1.3.2.4.2.9.Decreaseatinfinity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.2.10.ThePaley–Wienertheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.3.FouriertransformsinL2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.3.1.InvarianceofL2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.3.2.Reciprocity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 viii CONTENTS 1.3.2.4.3.3.Isometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.3.4.EigenspacedecompositionofL2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.3.5.Theconvolutiontheoremandtheisometryproperty .. .. .. .. .. .. .. .. .. .. .. .. .. 36 1.3.2.4.4.Fouriertransformsin(cid:3) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 37 1.3.2.4.4.1.Definitionandpropertiesof(cid:3) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 37 1.3.2.4.4.2.GaussianfunctionsandHermitefunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 37 1.3.2.4.4.3.Heisenberg’sinequality,Hardy’stheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.4.4.4.Symmetryproperty .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.4.5.VariouswritingsofFouriertransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.4.6.TablesofFouriertransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.5.Fouriertransformsoftempereddistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.5.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 1.3.2.5.2.(cid:3) asatest-functionspace .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 1.3.2.5.3.Definitionandexamplesoftempereddistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 1.3.2.5.4.Fouriertransformsoftempereddistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 1.3.2.5.5.Transpositionofbasicproperties .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 1.3.2.5.6.Transformsof(cid:2)-functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 1.3.2.5.7.Reciprocity theorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40 1.3.2.5.8.Multiplicationandconvolution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40 1.3.2.5.9.L2 aspects,Sobolevspaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40 1.3.2.6.PeriodicdistributionsandFourierseries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40 1.3.2.6.1.Terminology .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40 1.3.2.6.2.(cid:2)n-periodicdistributionsin(cid:1)n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 41 1.3.2.6.3.Identificationwithdistributionsover(cid:1)n(cid:3)(cid:2)n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 41 1.3.2.6.4.Fouriertransformsofperiodicdistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 41 1.3.2.6.5.Thecaseofnon-standardperiodlattices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42 1.3.2.6.6.Dualitybetweenperiodizationandsampling .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42 1.3.2.6.7.ThePoissonsummationformula .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42 1.3.2.6.8.ConvolutionofFourierseries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 43 1.3.2.6.9.Toeplitzforms,Szego¨’stheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 43 1.3.2.6.9.1.Toeplitzforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 43 1.3.2.6.9.2.TheToeplitz–Caratheódory–Herglotztheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. 43 1.3.2.6.9.3.AsymptoticdistributionofeigenvaluesofToeplitzforms .. .. .. .. .. .. .. .. .. .. .. 43 1.3.2.6.9.4.ConsequencesofSzego¨’stheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 44 1.3.2.6.10.ConvergenceofFourierseries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 44 1.3.2.6.10.1.ClassicalL1 theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 44 1.3.2.6.10.2.ClassicalL2 theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45 1.3.2.6.10.3.Theviewpointofdistributiontheory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45 1.3.2.7.ThediscreteFouriertransformation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45 1.3.2.7.1.Shannon’ssamplingtheoremandinterpolationformula .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45 1.3.2.7.2.Dualitybetweensubdivisionanddecimationofperiodlattices .. .. .. .. .. .. .. .. .. .. .. .. .. 46 1.3.2.7.2.1.Geometricdescriptionofsublattices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 46 1.3.2.7.2.2.Sublatticerelationsforreciprocallattices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 46 1.3.2.7.2.3.Relation betweenlatticedistributions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 46 1.3.2.7.2.4.Relation betweenFouriertransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 47 1.3.2.7.2.5.Sublatticerelationsintermsofperiodicdistributions .. .. .. .. .. .. .. .. .. .. .. .. 47 1.3.2.7.3.DiscretizationoftheFouriertransformation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 47 1.3.2.7.4.MatrixrepresentationofthediscreteFouriertransform(DFT) .. .. .. .. .. .. .. .. .. .. .. .. .. 49 1.3.2.7.5.PropertiesofthediscreteFouriertransform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49 1.3.3.NumericalcomputationofthediscreteFouriertransform .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49 1.3.3.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49 1.3.3.2.One-dimensionalalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 50 1.3.3.2.1.TheCooley–Tukeyalgorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 50 1.3.3.2.2.TheGood(orprimefactor)algorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51 1.3.3.2.2.1.Ring structureon(cid:2)(cid:3)N(cid:2) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51 1.3.3.2.2.2.TheChineseremaindertheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51 1.3.3.2.2.3.Theprimefactoralgorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 52 1.3.3.2.3.TheRader algorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 52 1.3.3.2.3.1.Nanoddprime .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 53 1.3.3.2.3.2.Napowerofanoddprime .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 53 1.3.3.2.3.3.Napowerof2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 53 ix CONTENTS 1.3.3.2.4.TheWinogradalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 54 1.3.3.3.Multidimensionalalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 1.3.3.3.1.Themethodofsuccessiveone-dimensionaltransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 1.3.3.3.2.Multidimensionalfactorization .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 1.3.3.3.2.1.MultidimensionalCooley–Tukeyfactorization .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 1.3.3.3.2.2.Multidimensionalprimefactoralgorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 56 1.3.3.3.2.3.NestingofWinogradsmallFFTs .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 56 1.3.3.3.2.4.TheNussbaumer–Quandallealgorithm .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 57 1.3.3.3.3.Globalalgorithmdesign .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 57 1.3.3.3.3.1.Fromlocalpiecestoglobalalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 57 1.3.3.3.3.2.Computerarchitectureconsiderations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58 1.3.3.3.3.3.TheJohnson–Burrusfamilyofalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58 1.3.4.CrystallographicapplicationsofFouriertransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58 1.3.4.1.Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58 1.3.4.2.CrystallographicFouriertransformtheory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 59 1.3.4.2.1.Crystalperiodicity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 59 1.3.4.2.1.1.Periodlattice,reciprocallatticeandstructurefactors .. .. .. .. .. .. .. .. .. .. .. .. 59 1.3.4.2.1.2.Structurefactorsintermsofformfactors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 60 1.3.4.2.1.3.Fourierseriesfortheelectrondensityanditssummation .. .. .. .. .. .. .. .. .. .. .. 60 1.3.4.2.1.4.Friedel’slaw,anomalousscatterers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 60 1.3.4.2.1.5.Parseval’sidentityandotherL2 theorems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 61 1.3.4.2.1.6.Convolution,correlationandPattersonfunction .. .. .. .. .. .. .. .. .. .. .. .. .. .. 61 1.3.4.2.1.7.Samplingtheorems,continuoustransforms,interpolation .. .. .. .. .. .. .. .. .. .. .. 61 1.3.4.2.1.8.Sectionsandprojections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 62 1.3.4.2.1.9.Differentialsyntheses .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 63 1.3.4.2.1.10.Toeplitzforms,determinantalinequalitiesandSzego¨’stheorem .. .. .. .. .. .. .. .. .. 63 1.3.4.2.2.Crystalsymmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 64 1.3.4.2.2.1.Crystallographicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 64 1.3.4.2.2.2.Groupsandgroupactions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 64 1.3.4.2.2.3.Classificationofcrystallographicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 66 1.3.4.2.2.4.Crystallographicgroupactioninrealspace .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 67 1.3.4.2.2.5.Crystallographicgroupactioninreciprocalspace .. .. .. .. .. .. .. .. .. .. .. .. .. 68 1.3.4.2.2.6.Structure-factorcalculation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 68 1.3.4.2.2.7.Electron-densitycalculations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 69 1.3.4.2.2.8.Parseval’stheoremwithcrystallographicsymmetry .. .. .. .. .. .. .. .. .. .. .. .. .. 69 1.3.4.2.2.9.Convolutiontheoremswithcrystallographicsymmetry .. .. .. .. .. .. .. .. .. .. .. .. 70 1.3.4.2.2.10.CorrelationandPattersonfunctions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 70 1.3.4.3.CrystallographicdiscreteFouriertransformalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 71 1.3.4.3.1.Historicalintroduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 71 1.3.4.3.2.Definingrelationsandsymmetryconsiderations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 72 1.3.4.3.3.Interactionbetweensymmetryanddecomposition .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 73 1.3.4.3.4.Interactionbetweensymmetryandfactorization .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 73 1.3.4.3.4.1.MultidimensionalCooley–Tukeyfactorization .. .. .. .. .. .. .. .. .. .. .. .. .. .. 74 1.3.4.3.4.2.MultidimensionalGoodfactorization .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 76 1.3.4.3.4.3.CrystallographicextensionoftheRader/Winogradfactorization .. .. .. .. .. .. .. .. .. 76 1.3.4.3.5.Treatmentofconjugateandparity-relatedsymmetryproperties .. .. .. .. .. .. .. .. .. .. .. .. .. 79 1.3.4.3.5.1.Hermitian-symmetricorreal-valuedtransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. 79 1.3.4.3.5.2.Hermitian-antisymmetricorpureimaginarytransforms .. .. .. .. .. .. .. .. .. .. .. .. 80 1.3.4.3.5.3.Complexsymmetricandantisymmetrictransforms .. .. .. .. .. .. .. .. .. .. .. .. .. 80 1.3.4.3.5.4.Real symmetrictransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 81 1.3.4.3.5.5.Real antisymmetrictransforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.5.6.Generalizedmultiplexing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.6.Globalcrystallographicalgorithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.6.1.Triclinicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.6.2.Monoclinicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.6.3.Orthorhombicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 82 1.3.4.3.6.4.Trigonal,tetragonalandhexagonalgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 1.3.4.3.6.5.Cubicgroups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 1.3.4.3.6.6.Treatmentofcentredlattices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 1.3.4.3.6.7.Programmingconsiderations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 1.3.4.4.Basiccrystallographiccomputations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 84 x

International Tables for Crystallography Volume B: Reciprocal space PDF

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