International Tables for Crystallography, Mathematical, Physical and Chemical Tables

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INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

i

2 s:\ITFC\prelims.3d (Tables of Crystallography)

International Tables for Crystallography Volume A: Space-Group Symmetry Editor Theo Hahn First Edition 1983, Fifth Edition 2002 Volume B: Reciprocal Space Editor U. Shmueli First Edition 1993, Second Edition 2001 Volume C: Mathematical, Physical and Chemical Tables Editor E. Prince First Edition 1992, Third Edition 2004 Volume D: Physical Properties of Crystals Editor A. Authier First Edition 2003 Volume E: Subperiodic Groups Editors V. KopskyÂ and D. B. Litvin First Edition 2002 Volume F: Crystallography of Biological Macromolecules Editors Michael G. Rossmann and Eddy Arnold First Edition 2001

Forthcoming volumes Volume A1: Symmetry Relations between Space Groups Editors H. Wondratschek and U. MuÈller Volume G: De®nition and Exchange of Crystallographic Data Editors S. R. Hall and B. McMahon

ii


INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

Volume C MATHEMATICAL, PHYSICAL AND CHEMICAL TABLES

Edited by E. PRINCE Third Edition

Published for

THE I NTERNATI ONA L U N I O N O F C R Y S T A L L O G R A P H Y by

KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON

2004 iii


A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 1-4020-1900-9 (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 Editors: S. E. King and N. J. Ashcroft First published in 1992 Reprinted with corrections 1995 Second edition 1999 Third edition 2004 # International Union of Crystallography 1992, 1995, 1999, 2004 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 Denmark by P. J. Schmidt A/S


Contributing Authors A. ALBINATI: Istituto Chimica Farmaceutica, UniversitaÁ di Milano, Viale Abruzzi 42, Milano 20131, Italy. [8.6] N. G. ALEXANDROPOULOS: Department of Physics, University of Ioannina, PO Box 1186, Gr-45110 Ioannina, Greece. [7.4.3] F. H. ALLEN: Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, England. [9.5, 9.6] Y. AMEMIYA: Engineering Research Institute, Department of Applied Physics, Faculty of Engineering, University of Tokyo, 2-11-16 Yayoi, Bunkyo, Tokyo 113, Japan. [7.1.8] I. S. ANDERSON: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [4.4.2] U. W. ARNDT: MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England. [4.2.1, 7.1.6] J. BARUCHEL: Experiment Division, ESRF, BP 220, F-38043 Grenoble CEDEX, France. [2.8] P. J. BECKER: Ecole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 ChaÃtenay Malabry CEDEX, France. [8.7] G. BERGERHOFF: Institut fuÈr Anorganische Chemie der UniversitaÈt Bonn, Gerhard-Domagkstrasse 1, D-53121 Bonn, Germany. [9.4] P. T. BOGGS: Scienti®c Computing Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [8.1] L. BRAMMER: Department of Chemistry, University of Missouri± St Louis, 8001 Natural Bridge Road, St Louis, MO 63121-4499, USA. [9.5, 9.6] K. BRANDENBURG: Institut fuÈr Anorganische Chemie der UniversitaÈt Bonn, Gerhard-Domagkstrasse 1, D-53121 Bonn, Germany. [9.4] P. J. BROWN: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [4.4.5, 6.1.2] yB. BURAS [2.5.1, 7.1.5] J. M. CARPENTER: Intense Pulsed Neutron Source, Building 360, Argonne National Laboratory, Argonne, IL 60439, USA. [4.4.1] J. N. CHAPMAN: Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland. [7.2] P. CHIEUX: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [7.3] J. CHIKAWA: Center for Advanced Science and Technology, Harima Science Park City, Kamigori-cho, Hyogo 678-12, Japan. [7.1.7, 7.1.8] C. COLLIEX, Laboratoire AimeÂ Cotton, CNRS, Campus d'Orsay, BaÃtiment 505, F-91405 Orsay CEDEX, France. [4.3.4] D. M. COLLINS: Laboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 20375-5341, USA. [8.2] P. CONVERT: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [7.3] M. J. COOPER: Department of Physics, University of Warwick, Coventry CV4 7AL, England. [7.4.3] P. COPPENS: 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA. [8.7] J. M. COWLEY: Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA. [2.4.1, 4.3.1, 4.3.2, 4.3.8]

D. C. CREAGH: Division of Health, Design, and Science, University of Canberra, Canberra, ACT 2601, Australia. [4.2.3, 4.2.4, 4.2.5, 4.2.6, 10] J. L. C. DAAMS: Materials Analysis Department, Philips Research Laboratories, Prof. Holstaan 4, 5656 AA Eindhoven, The Netherlands. [9.3] W. I. F. DAVID: ISIS Science Division, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, England. [2.5.2] yR. D. DESLATTES [4.2.2] S. L. DUDAREV: Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, England. [4.3.2] Æ UROVICÆ: Department of Theoretical Chemistry, Slovak S. D Academy of Sciences, DuÂbravskaÂ cesta, 842 36 Bratislava, Slovakia. [9.2.2] L. W. FINGER: Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015-1305, USA. [8.3] M. FINK: Department of Physics, University of Texas at Austin, Austin, TX 78712, USA. [4.3.3] W. FISCHER: Institut fuÈr Mineralogie, Petrologie und Kristallographie, UniversitaÈt Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany. [9.1] H. M. FLOWER: Department of Metallurgy, Imperial College, London SW7, England. [3.5] A. G. FOX: Center for Materials Science and Engineering, Naval Postgraduate School, Monterey, CA 93943-5000, USA. [6.1.1] J. R. FRYER: Department of Chemistry, University of Glasgow, Glasgow G12 8QQ, Scotland. [3.5] E. GAèDECKA: Institute of Low Temperature and Structure Research, Polish Academy of Sciences, PO Box 937, 50-950 Wrocøaw 2, Poland. [5.3] L. GERWARD: Physics Department, Technical University of Denmark, DK-2800 Lyngby, Denmark. [2.5.1, 7.1.5] J. GJéNNES: Department of Physics, University of Oslo, PO Box 1048, Blindern, N-0316 Oslo, Norway. [4.3.7, 8.8] O. GLATTER: Institut fuÈr Physikalische Chemie, UniversitaÈt Graz, Heinrichstrasse 28, A-8010 Graz, Austria. [2.6.1] J. R. HELLIWELL: Department of Chemistry, University of Manchester, Manchester M13 9PL, England. [2.1, 2.2] A. W. HEWAT: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [2.4.2] R. L. HILDERBRANDT: Chemistry Division, Room 1055, The National Science Foundation, 4201 Wilson Blvd, Arlington, VA 22230, USA. [4.3.3] A. HOWIE: Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, England [4.3.6.2] H.-C. HU: China Institute of Atomic Energy, PO Box 275 (18), Beijing 102413, People's Republic of China [6.2] J. H. HUBBELL: Room C314, Radiation Physics Building, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [4.2.4] P. INDELICATO: Laboratoire Kastler-Brossel, Case 74, UniversiteÂ Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France. [4.2.2] A. JANNER: Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands. [9.8] T. JANSSEN: Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands. [9.8]

y Deceased.

y Deceased.

v


CONTRIBUTING AUTHORS A. W. S. JOHNSON: Centre for Microscopy and Microanalysis, University of Western Australia, Nedlands, WA 6009, Australia. [5.4.1] J. D. JORGENSEN: Materials Science Division, Building 223, Argonne National Laboratory, Argonne, IL 60439, USA. [2.5.2] V. L. KAREN: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [9.7] E. G. KESSLER JR: Atomic Physics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [4.2.2] E. KOCH: Institut fuÈr Mineralogie, Petrologie und Kristallographie, UniversitaÈt Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany. [1.1, 1.2, 1.3, 9.1] J. H. KONNERT: Laboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 20375-5000, USA. [8.3] P. KRISHNA: Rajghat Education Center, Krishnamurti Foundation India, Rajghat Fort, Varanasi 221001, India. [9.2.1] G. LANDER: ITU, European Commission, Postfach 2340, D-76125 Karlsruhe, Germany. [4.4.1] A. R. LANG: H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, England. [2.7] J. I. LANGFORD: School of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England. [2.3, 5.2, 6.2, 7.1.2] yE. S. LARSEN JR. [3.3] P. F. LINDLEY: ESRF, Avenue des Martyrs, BP 220, F-38043 Grenoble CEDEX, France. [3.1, 3.2.1, 3.2.3, 3.4] E. LINDROTH, Department of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden. [4.2.2] y H. LIPSON. [6.2] A. LOOIJENGA-VOS: Roland Holstlaan 908, NL-2624 JK Delft, The Netherlands. [9.8] D. F. LYNCH: CSIRO Division of Materials Science & Technology, Private Bag 33, Rosebank MDC, Clayton, Victoria 3169, Australia. [4.3.6.1] C. F. MAJKRZAK: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [2.9] S. MARTINEZ-CARRERA: San Ernesto, 6-Esc. 3, 28002 Madrid, Spain. [10] yE. N. MASLEN. [6.1.1, 6.3] R. P. MAY: Institut Laue±Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France. [2.6.2] yR. MEYROWITZ. [3.3] A. MIGHELL: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [9.7] M. A. O'KEEFE: National Center for Electron Microscopy, Lawrence Berkeley National Laboratory MS-72, University of California, Berkeley, CA 94720, USA. [6.1.1] A. OLSEN: Department of Physics, University of Oslo, PO Box 1048, N-0316 Blindern, Norway. [5.4.2] A. G. ORPEN: School of Chemistry, University of Bristol, Bristol BS8 1TS, England. [9.5, 9.6] D. PANDEY: Physics Department, Banaras Hindu University, Varanasi 221005, India. [9.2.1] yW. PARRISH. [2.3, 5.2, 7.1.2, 7.1.3, 7.1.4] L. M. PENG: Department of Electronics, Peking University, Beijing 100817, People's Republic of China. [4.3.2] E. PRINCE: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [8.1, 8.2, 8.3, 8.4, 8.5]

R. PYNN: LANSCE, MS H805, Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA. [4.4.3] G. REN: Beijing Laboratory of Electron Microscopy, Chinese Academy of Sciences, PO Box 2724, Beijing 100080, People's Republic of China. [4.3.2] F. M. RICHARDS: Department of Molecular Biophysics and Biochemistry, Yale University, 260 Whitney Ave, New Haven, CT 06520-8114, USA. [3.2.2] J. R. RODGERS: National Research Council of Canada, Canada Institute for Scienti®c and Technical Information, Ottawa, Canada K1A 0S2. [9.3] A. W. ROSS: Physics Department, The University of Texas at Austin, Austin, TX 78712, USA. [4.3.3] J. M. ROWE: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. [4.4.3] T. M. SABINE: ANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia. [6.4] È RPF: Physik-Department E13, TU MuÈnchen, JamesO. SCHA Franck-Strasse 1, D-85748 Garching, Germany. [4.4.2] M. SCHLENKER: l'Institut National Polytechnique de Grenoble, Laboratoire Louis NeÂel du CNRS, BP 166, F-38042 Grenoble CEDEX 9, France. [2.8] V. F. SEARS: Atomic Energy of Canada Limited, Chalk River Laboratories, Chalk River, Ontario, Canada K0J 1J0. [4.4.4] G. S. SMITH: Manuel Lujan Jr Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. [2.9] V. H. SMITH JR: Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6. [4.3.3] J. C. H. SPENCE: Department of Physics, Arizona State University, Tempe, AZ 85287, USA. [4.3.8] C. H. SPIEGELMAN: Department of Statistics, Texas A&M University, College Station, TX 77843, USA. [8.4, 8.5] J. W. STEEDS: H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, England. [4.3.7] Z. SU: Digital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA. [8.7] P. SUORTTI: Department of Physics, PO Box 9, University of Helsinki, FIN-00014 Helsinki, Finland. [7.4.4] R. TAYLOR: Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, England. [9.5, 9.6] N. J. TIGHE: 42 Lema Lane, Palm Coast, FL 32137-2417, USA. [3.5] V. VALVODA: Department of Physics of Semiconductors, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic. [4.1] P. VILLARS: Intermetallic Phases Databank, Postal Box 1, CH6354 Vitznau, Switzerland. [9.3] J. WANG: Department of Chemistry, Queen's University, Kingston, Ontario, Canada K7L 3N6. [4.3.3] D. G. WATSON: Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, England. [9.5, 9.6] M. J. WHELAN: Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, England. [4.3.2] B. T. M. WILLIS: Chemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England. [2.5.2, 3.6, 4.4.6, 5.5, 6.1.3, 7.4.2, 8.6] yA. J. C. WILSON. [1.4, 3.3, 5.1, 5.2, 7.5, 9.7] yP. M. DE WOLFF. [7.1.1, 9.8] yB. B. ZVYAGIN. [4.3.5]

y Deceased.

y Deceased.

vi


Contents page PREFACE (A. J. C. Wilson)

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PREFACE TO THE THIRD EDITION (E. Prince)

xxxi

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

xxxi

PART 1: CRYSTAL GEOMETRY AND SYMMETRY .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

1

1.1. Summary of General Formulae (E. Koch) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

2

1.1.1. General relations between direct and reciprocal lattices .. .. .. .. .. .. .. .. .. 1.1.1.1. Primitive crystallographic bases .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.1.1.2. Non-primitive crystallographic bases .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.1.1.1. Direct and reciprocal lattices described with respect to conventional 1.1.2. Lattice vectors, point rows, and net planes .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. ..

2 2 3 3 3

1.1.3. Angles in direct and reciprocal space.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

4

1.1.4. The Miller formulae

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

5

1.2. Application to the Crystal Systems (E. Koch) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

6

1.2.1. Triclinic crystal system.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

6

1.2.2. Monoclinic crystal system .. .. .. 1.2.2.1. Setting with ùnique axis 1.2.2.2. Setting with ùnique axis 1.2.3. Orthorhombic crystal system.. ..

.. .. .. ..

6 6 6 6

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. integers .. .. .. .. .. .. .. .. ..

7 7 7 7 8 8 8 9 9

1.3. Twinning (E. Koch) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

10

1.3.1. General remarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

10

1.3.2. Twin lattices .. .. .. .. .. .. .. .. .. .. .. 1.3.2.1. Examples .. .. .. .. .. .. .. .. .. Table 1.3.2.1. Lattice planes and rows that parameters .. .. .. .. .. .. 1.3.3. Implication of twinning in reciprocal space ..

10 11

.. b' c' ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

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.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. basis systems .. .. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

1.2.4. Tetragonal crystal system .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.4.1. Assignment of integers s 100 to pairs h, k with s h2 k2 .. .. .. .. .. 1.2.5. Trigonal and hexagonal crystal system .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.5.1. Description referred to hexagonal axes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.5.2. Description referred to rhombohedral axes .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.5.1. Assignment of integers s 100 to pairs h, k with s h2 k2 hk .. .. .. Table 1.2.5.2. Assignment of integers s1 50 to triplets h, k, l with s1 h2 k2 l2 s2 hk hl kl .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.2.6. Cubic crystal system .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.2.6.1. Assignment of integers s 100 to triplets h, k, l with s h2 k2 l2 .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. and .. .. .. .. .. ..

.. .. .. .. .. .. to .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. are perpendicular to each other independently of the .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

1.3.4. Twinning by merohedry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.3.4.1. Possible twin operations for twins by merohedry .. .. .. .. .. Table 1.3.4.2. Simulated Laue classes, extinction symbols, simulated `possible true space groups for crystals twinned by merohedry (type 2) .. 1.3.5. Calculation of the twin element .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. space .. .. .. ..

.. .. .. .. .. .. groups', .. .. .. .. .. ..

.. .. .. ..

.. .. .. .. ..

.. .. .. ..

.. .. .. .. .. .. metrical .. .. .. .. .. ..

11 12

.. .. .. .. .. .. possible .. .. .. .. .. ..

13 14

1.4. Arithmetic Crystal Classes and Symmorphic Space Groups (A. J. C. Wilson) .. .. .. .. .. .. .. .. ..

15

1.4.1. Arithmetic crystal classes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.4.1.1. Arithmetic crystal classes in three dimensions.. .. .. .. .. .. .. .. .. .. .. .. 1.4.1.2. Arithmetic crystal classes in one, two and higher dimensions .. .. .. .. .. .. Table 1.4.1.1. The two-dimensional arithmetic crystal classes .. .. .. .. .. .. .. .. .. 1.4.2. Classi®cation of space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1.4.2.1. Symmorphic space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 1.4.2.1. The three-dimensional space groups, arranged by arithmetic crystal class vii

1 s:\ITFC\CONTENTS.3d (Tables of Crystallography)

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12 13

15 15 16 15 20 21 16

CONTENTS 1.4.3. Effect of dispersion on diffraction symmetry .. .. .. .. 1.4.3.1. Symmetry of the Patterson function .. .. .. .. 1.4.3.2. `Laue' symmetry .. .. .. .. .. .. .. .. .. .. .. Table 1.4.3.1. Arithmetic crystal classes classi®ed by the References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. number .. .. ..

.. .. .. .. ..

21 21 21 20 21

PART 2: DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION .. .. .. .. .. ..

23

2.1. Classification of Experimental Techniques (J. R. Helliwell) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

24

Table 2.1.1. Summary of main experimental techniques for structure analysis .. .. .. .. .. .. .. .. .. .. .. .. ..

25

2.2. Single-Crystal X-ray Techniques (J. R. Helliwell).. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

26

2.2.1. Laue geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.1.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.1.2. Crystal setting .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.1.3. Single-order and multiple-order re¯ections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.1.4. Angular distribution of re¯ections in Laue diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.1.5. Gnomonic and stereographic transformations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.2. Monochromatic methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.2.1. Monochromatic still exposure .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.2.2. Crystal setting .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.3. Rotation/oscillation geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.3.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.3.2. Diffraction coordinates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.3.3. Relationship of reciprocal-lattice coordinates to crystal system parameters .. .. .. .. .. .. .. .. .. 2.2.3.4. Maximum oscillation angle without spot overlap .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.3.5. Blind region .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.2.3.1. Glossary of symbols used to specify quantitites on diffraction patterns and in reciprocal space 2.2.4. Weissenberg geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.4.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.4.2. Recording of zero layer .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.4.3. Recording of upper layers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5. Precession geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.2. Crystal setting .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.3. Recording of zero-layer photograph .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.4. Recording of upper-layer photographs .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.5.5. Recording of cone-axis photograph .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.2.5.1. The distance displacement (in mm) measured on the ®lm versus angular setting error of the crystal for a screenless precession ( 5 ) setting photograph .. .. .. .. .. .. .. .. .. .. .. 2.2.6. Diffractometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.6.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.6.2. Normal-beam equatorial geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.6.3. Fixed 45 geometry with area detector .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.7. Practical realization of diffraction geometry: sources, optics, and detectors .. .. .. .. .. .. .. .. .. .. .. .. 2.2.7.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.7.2. Conventional X-ray sources: spectral character, crystal rocking curve, and spot size.. .. .. .. .. .. 2.2.7.3. Synchrotron X-ray sources .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.2.7.4. Geometric effects and distortions associated with area detectors .. .. .. .. .. .. .. .. .. .. .. ..

26 26 27 27 29 29 29 30 30 31 31 31 33 33 34 32 34 34 34 34 35 35 35 35 35 36

2.3. Powder and Related Techniques: X-ray Techniques (W. Parrish and J. I. Langford) .. .. .. .. .. .. ..

42

2.3.1. Focusing diffractometer geometries .. .. .. .. .. .. .. .. 2.3.1.1. Conventional re¯ection specimen, ±2 scan .. .. 2.3.1.1.1. Geometrical instrument parameters .. .. 2.3.1.1.2. Use of monochromators .. .. .. .. .. .. 2.3.1.1.3. Alignment and angular calibration .. .. 2.3.1.1.4. Instrument broadening and aberrations .. 2.3.1.1.5. Focal line and receiving-slit widths .. .. 2.3.1.1.6. Aberrations related to the specimen .. .. viii


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35 36 36 36 37 37 37 37 38 41 43 44 44 46 46 47 48 48

CONTENTS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

50 50 50 52 53 53 54 55 57 58 58 60 60 60 62 62 63 63 63 64 65 66 69 61 61 70 70 70 71 71 71 72 72 73 73 74 74 75 75 75 76 78 72 78 79

2.4. Powder and Related Techniques: Electron and Neutron Techniques .. .. .. .. .. .. .. .. .. .. .. ..

80

2.3.2.

2.3.3.

2.3.4.

2.3.5.

2.3.1.1.7. Axial divergence .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.1.8. Combined aberrations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.2. Transmission specimen, ±2 scan .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.3. Seemann±Bohlin method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.4. Re¯ection specimen, ± scan .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.1.5. Microdiffractometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Parallel-beam geometries, synchrotron radiation .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.1. Monochromatic radiation, ±2 scan .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.2. Cylindrical specimen, 2 scan .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.3. Grazing-incidence diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.2.4. High-resolution energy-dispersive diffraction .. .. .. .. .. .. .. .. .. .. .. Specimen factors, angle, intensity, and pro®le-shape measurement .. .. .. .. .. .. .. 2.3.3.1. Specimen factors.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.1.1. Preferred orientation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.1.2. Crystallite-size effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.2. Problems arising from the K doublet .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.3. Use of peak or centroid for angle de®nition .. .. .. .. .. .. .. .. .. .. .. 2.3.3.4. Rate-meter/strip-chart recording .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.5. Computer-controlled automation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.6. Counting statistics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.7. Peak search .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.8. Pro®le ®tting .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.3.9. Computer graphics for powder patterns .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.3.1. Preferred-orientation data for silicon .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.3.2. R(Bragg) values obtained with different preferred-orientation formulae Powder cameras .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.4.1. Cylindrical cameras (Debye±Scherrer) .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.4.2. Focusing cameras (Guinier) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.4.3. Miscellaneous camera types .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Generation, modi®cations, and measurement of X-ray spectra .. .. .. .. .. .. .. .. 2.3.5.1. X-ray tubes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.1.1. Stability .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.1.2. Spectral purity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.1.3. Source intensity distribution and size .. .. .. .. .. .. .. .. .. .. 2.3.5.1.4. Air and window transmission .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.1.5. Intensity variation with take-off angle .. .. .. .. .. .. .. .. .. .. 2.3.5.2. X-ray spectra .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.2.1. Wavelength selection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.3. Other X-ray sources .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.4. Methods for modifying the spectrum .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.4.1. Crystal monochromators .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.3.5.4.2. Single and balanced ®lters .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.5.1. X-ray tube maximum ratings .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.5.2. ®lters for common target elements .. .. .. .. .. .. .. .. .. .. .. .. Table 2.3.5.3. Calculated thickness of balanced ®lters for common target elements ..

2.4.1. Electron techniques (J. M. Cowley) .. .. .. .. .. .. 2.4.1.1. Powder-pattern geometry.. .. .. .. .. .. .. 2.4.1.2. Diffraction patterns in electron microscopes 2.4.1.3. Preferred orientations .. .. .. .. .. .. .. .. 2.4.1.4. Powder-pattern intensities .. .. .. .. .. .. 2.4.1.5. Crystal-size analysis.. .. .. .. .. .. .. .. .. 2.4.1.6. Unknown-phase identi®cation: databases .. 2.4.2. Neutron techniques (A. W. Hewat) .. .. .. .. .. ..

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80 80 80 80 80 81 81 82

2.5. Energy-Dispersive Techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

84

2.5.1. Techniques for X-rays (B. Buras and L. Gerward) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.5.1.1. Recording powder diffraction spectra .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

84 84


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CONTENTS 2.5.1.2. Incident X-ray beam .. .. .. .. .. .. .. .. 2.5.1.3. Resolution .. .. .. .. .. .. .. .. .. .. .. .. 2.5.1.4. Integrated intensity for powder sample .. .. 2.5.1.5. Corrections .. .. .. .. .. .. .. .. .. .. .. 2.5.1.6. The Rietveld method .. .. .. .. .. .. .. .. 2.5.1.7. Single-crystal diffraction .. .. .. .. .. .. .. 2.5.1.8. Applications .. .. .. .. .. .. .. .. .. .. .. 2.5.2. White-beam and time-of-¯ight neutron diffraction (J. 2.5.2.1. Neutron single-crystal Laue diffraction .. .. 2.5.2.2. Neutron time-of-¯ight powder diffraction ..

.. .. .. .. .. .. .. D. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Jorgensen, .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. W. .. ..

.. .. .. .. .. .. .. I. .. ..

.. .. .. .. .. .. .. F. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. David, and .. .. .. .. .. .. .. ..

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

84 85 85 86 86 86 86 87 87 87

2.6. Small-Angle Techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

89

2.6.1. X-ray techniques (O. Glatter) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.2. General principles .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.3. Monodisperse systems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.3.1. Parameters of a particle .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.3.2. Shape and structure of particles .. .. .. .. .. .. .. .. .. .. 2.6.1.3.2.1. Homogeneous particles .. .. .. .. .. .. .. .. .. 2.6.1.3.2.2. Hollow and inhomogeneous particles.. .. .. .. .. 2.6.1.3.3. Interparticle interference, concentration effects .. .. .. .. .. 2.6.1.4. Polydisperse systems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.5. Instrumentation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.5.1. Small-angle cameras .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.5.2. Detectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.6. Data evaluation and interpretation .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.6.1. Primary data handling .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.6.2. Instrumental broadening ± smearing .. .. .. .. .. .. .. .. .. 2.6.1.6.3. Smoothing, desmearing, and Fourier transformation .. .. .. 2.6.1.6.4. Direct structure analysis .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.6.5. Interpretation of results .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.7. Simulations and model calculations .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.7.1. Simulations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.7.2. Model calculation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.7.3. Calculation of scattering intensities .. .. .. .. .. .. .. .. .. 2.6.1.7.4. Method of ®nite elements .. .. .. .. .. .. .. .. .. .. .. .. 2.6.1.7.5. Calculation of distance-distribution functions .. .. .. .. .. .. 2.6.1.8. Suggestions for further reading .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 2.6.1.1. Formulae for the various parameters for h and m scales .. .. .. 2.6.2. Neutron techniques (R. May) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.1. Relation of X-ray and neutron small-angle scattering .. .. .. .. .. .. 2.6.2.1.1. Wavelength .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.1.2. Geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.1.3. Correction of wavelength, slit, and detector-element effects .. 2.6.2.2. Isotopic composition of the sample .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.2.1. Contrast variation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.2.2. Speci®c isotopic labelling.. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.3. Magnetic properties of the neutron .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.3.1. Spin-contrast variation .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.4. Long wavelengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.5. Sample environment .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.6. Incoherent scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.6.1. Absolute scaling .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.6.2. Detector-response correction .. .. .. .. .. .. .. .. .. .. .. 2.6.2.6.3. Estimation of the incoherent scattering level .. .. .. .. .. .. 2.6.2.6.4. Inner surface area .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.7. Single-particle scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.7.1. Particle shape .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.6.2.7.2. Particle mass .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. x


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.. .. .. .. .. .. .. B. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. T. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. M. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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89 89 90 91 91 93 93 96 97 99 99 99 100 100 100 101 101 103 103 103 103 104 104 104 104 104 92 105 105 105 106 106 106 107 107 107 108 108 108 108 108 109 109 109 110 110 110

CONTENTS 2.6.2.7.3. Real-space considerations .. 2.6.2.7.4. Particle-size distribution .. .. 2.6.2.7.5. Model ®tting .. .. .. .. .. .. 2.6.2.7.6. Label triangulation .. .. .. .. 2.6.2.7.7. Triplet isotropic replacement 2.6.2.8. Dense systems .. .. .. .. .. .. .. ..

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

110 111 111 111 111 112

2.7. Topography (A. R. Lang) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

113

2.7.1. Principles .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

113

2.7.2. Single-crystal techniques .. .. .. 2.7.2.1. Re¯ection topographs .. 2.7.2.2. Transmission topographs 2.7.3. Double-crystal topography .. ..

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114 114 115 117

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119 119 120 121 121 121 122

2.8. Neutron Diffraction Topography (M. Schlenker and J. Baruchel) .. .. .. .. .. .. .. .. .. .. .. .. .. ..

124

2.8.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

124

2.8.2. Implementation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

124

2.8.3. Application to investigations of heavy crystals

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124

2.8.4. Investigation of magnetic domains and magnetic phase transitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

124

2.9. Neutron Reflectometry (G. S. Smith and C. F. Majkrzak) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

126

2.9.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

126

2.9.2. Theory of elastic specular neutron re¯ection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

126

2.9.3. Polarized neutron re¯ectivity

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

127

2.9.4. Surface roughness .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

128

2.9.5. Experimental methodology .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

128

2.9.6. Resolution in real space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

129

2.9.7. Applications of neutron re¯ectometry 2.9.7.1. Self-diffusion .. .. .. .. .. .. 2.9.7.2. Magnetic multilayers .. .. .. 2.9.7.3. Hydrogenous materials .. .. References .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. ..

129 129 130 130 130

PART 3: PREPARATION AND EXAMINATION OF SPECIMENS .. .. .. .. .. .. .. .. .. .. .. ..

147

3.1. Preparation, Selection, and Investigation of Specimens (P. F. Lindley) .. .. .. .. .. .. .. .. .. .. .. ..

148

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3.1.1. Crystallization.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.1.2. Crystal growth .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.1.3. Methods of growing crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.1.4. Factors affecting the solubility of biological macromolecules.. .. .. .. .. .. .. 3.1.1.5. Screening procedures for the crystallization of biological macromolecules .. .. 3.1.1.6. Automated protein crystallization .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.1.1.7. Membrane proteins .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.1.1.1. Survey of crystallization techniques suitable for the crystallization weight organic compounds for X-ray crystallography .. .. .. .. .. .. .. Table 3.1.1.2. Commonly used ionic and organic precipitants .. .. .. .. .. .. .. .. .. Table 3.1.1.3. Crystallization matrix parameters for sparse-matrix sampling .. .. .. .. Table 3.1.1.4. Reservoir solutions for sparse-matrix sampling .. .. .. .. .. .. .. .. .. xi


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2.7.4. Developments with synchrotron radiation .. .. .. .. .. .. .. .. .. .. .. .. .. 2.7.4.1. White-radiation topography .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.7.4.2. Incident-beam monochromatization .. .. .. .. .. .. .. .. .. .. .. .. Table 2.7.4.1. Monolithic monochromator for plane-wave synchrotron-radiation 2.7.5. Some special techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.7.5.1. MoireÂ topography .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.7.5.2. Real-time viewing of topograph images .. .. .. .. .. .. .. .. .. .. ..

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.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. low-molecular.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

148 148 148 148 148 150 150 150 149 150 151 152

CONTENTS 3.1.2. Selection of single crystals .. .. .. .. .. .. .. .. .. .. 3.1.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. 3.1.2.2. Size, shape, and quality .. .. .. .. .. .. .. .. 3.1.2.3. Optical examination .. .. .. .. .. .. .. .. .. 3.1.2.4. Twinning .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.1.2.1. Use of crystal properties for selection optical, and mechanical properties .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. and preliminary .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. crystals; morphological, .. .. .. .. .. .. .. .. ..

151 151 151 154 155

3.2. Determination of the Density of Solids .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

156

3.2.1. Introduction (P. F. Lindley) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.1.1. General precautions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2. Description and discussion of techniques (F. M. Richards) .. .. .. .. .. 3.2.2.1. Gradient tube.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.1.1. Technique .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.1.2. Suitable substances for columns .. .. .. .. .. .. .. .. 3.2.2.1.3. Sensitivity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.2. Flotation method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.3. Pycnometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.4. Method of Archimedes .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.5. Immersion microbalance .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.6. Volumenometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.2.2.7. Other procedures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.2.2.1. Possible substances for use as gradient-column components 3.2.3. Biological macromolecules (P. F. Lindley) .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. study .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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156 156 156 156 156 157 158 158 158 158 158 158 158 157 159

Table 3.2.3.1. Typical calculations of the values of VM and Vsolv for proteins .. .. .. .. .. .. .. .. .. .. ..

159

3.3. Measurement of Refractive Index (E. S. Larsen Jr, R. Meyrowitz, and A. J. C. Wilson) .. .. .. .. .. ..

160

3.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

160

3.3.2. Media for general use .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.3.2.1. Immersion media for general use in the measurement of index of refraction .. .. .. .. .. .. 3.3.3. High-index media .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

160 160 160

3.3.4. Media for organic substances .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.3.4.1. Aqueous solutions for use as immersion media for organic crystals .. .. .. .. .. .. .. .. .. Table 3.3.4.2. Organic immersion media for use with organic crystals of low solubility.. .. .. .. .. .. .. ..

161 160 160

3.4. Mounting and Setting of Specimens for X-ray Crystallographic Studies (P. F. Lindley) .. .. .. .. ..

162

3.4.1. Mounting of specimens .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.2. Polycrystalline specimens .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.2.1. General.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.2.2. Non-ambient conditions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.3. Single crystals (small molecules) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.3.1. General.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.3.2. Non-ambient conditions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.4. Single crystals of biological macromolecules at ambient temperatures .. .. .. 3.4.1.5. Cryogenic studies of biological macromolecules .. .. .. .. .. .. .. .. .. .. .. 3.4.1.5.1. Radiation damage .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.5.2. Cryoprotectants .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.5.3. Crystal mounting and cooling .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.5.4. Cooling devices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.1.5.5. General.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.4.1.1. Single-crystal and powder mounting, capillary tubes and other containers Table 3.4.1.2. Single-crystal mounting ± adhesives .. .. .. .. .. .. .. .. .. .. .. .. .. Table 3.4.1.3. Cryoprotectants commonly used for biological macromolecules .. .. .. .. 3.4.2. Setting of single crystals by X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.2.2. Preliminary considerations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.2.3. Equatorial setting using a rotation camera .. .. .. .. .. .. .. .. .. .. .. .. .. xii


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153

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162 162 162 162 162 163 163 164 165 166 166 166 166 167 167 163 164 166 167 167 168 168

CONTENTS 3.4.2.4. Precession geometry setting with moving-crystal methods.. .. .. .. .. .. .. .. .. 3.4.2.5. Setting and orientation with stationary-crystal methods .. .. .. .. .. .. .. .. .. 3.4.2.5.1. Laue images ± white radiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.2.5.2. `Still' images ± monochromatic radiation .. .. .. .. .. .. .. .. .. .. .. 3.4.2.6. Setting and orientation for crystals with large unit cells using oscillation geometry 3.4.2.7. Diffractometer-setting considerations.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.4.2.8. Crystal setting and data-collection ef®ciency .. .. .. .. .. .. .. .. .. .. .. .. ..

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

168 169 169 169 169 170 170

3.5. Preparation of Specimens for Electron Diffraction and Electron Microscopy (N. J. Tighe, J. R. Fryer, and H. M. Flower) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

171

3.5.1. Ceramics and rock minerals .. .. .. .. .. .. .. .. .. .. .. .. 3.5.1.1. Thin fragments, particles, and ¯akes .. .. .. .. .. .. 3.5.1.2. Thin-section preparation .. .. .. .. .. .. .. .. .. .. 3.5.1.3. Final thinning by argon-ion etching .. .. .. .. .. .. 3.5.1.4. Final thinning by chemical etching .. .. .. .. .. .. 3.5.1.5. Evaporated and sputtered thin ®lms .. .. .. .. .. .. Table 3.5.1.1. Chemical etchants used for preparing thin foils 3.5.2. Metals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.5.2.1. Thin sections .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.5.2.2. Final thinning methods .. .. .. .. .. .. .. .. .. .. 3.5.2.3. Chemical and electrochemical thinning solutions .. .. 3.5.3. Polymers and organic specimens .. .. .. .. .. .. .. .. .. .. 3.5.3.1. Cast ®lms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3.5.3.2. Sublimed ®lms .. .. .. .. .. .. .. .. .. .. .. .. .. 3.5.3.3. Oriented solidi®cation .. .. .. .. .. .. .. .. .. .. ..

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.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. from single-crystal ceramic materials .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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171 171 171 172 173 173 173 173 174 174 175 176 176 176 176

3.6. Specimens for Neutron Diffraction (B. T. M. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

177

References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

177

PART 4: PRODUCTION AND PROPERTIES OF RADIATIONS .. .. .. .. .. .. .. .. .. .. .. .. ..

185

4.1. Radiations used in Crystallography (V. Valvoda) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

186

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

186

4.1.2. Electromagnetic waves and particles .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

186

4.1.3. Most frequently used radiations .. .. .. .. .. .. .. .. .. .. .. Table 4.1.3.1. Average diffraction properties of X-rays, electrons, 4.1.4. Special applications of X-rays, electrons, and neutrons .. .. .. .. 4.1.4.1. X-rays, synchrotron radiation, and -rays .. .. .. .. .. 4.1.4.2. Electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.1.4.3. Neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.1.5. Other radiations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.1.5.1. Atomic and molecular beams .. .. .. .. .. .. .. .. .. 4.1.5.2. Positrons and muons .. .. .. .. .. .. .. .. .. .. .. .. 4.1.5.3. Infrared, visible, and ultraviolet light .. .. .. .. .. .. .. 4.1.5.4. Radiofrequency and microwaves .. .. .. .. .. .. .. ..

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

187 187 189 189 189 189 189 189 189 189 190

4.2. X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

191

.. .. .. .. .. and neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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4.2.1. Generation of X-rays (U. W. Arndt) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.1. The characteristic line spectrum .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.1.1. The intensity of characteristic lines .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.2. The continuous spectrum .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.3. X-ray tubes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.3.1. Power dissipation in the anode.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.4. Radioactive X-ray sources .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.5. Synchrotron-radiation sources .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.6. Plasma X-ray sources .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.1.7. Other sources of X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.1.1. Correspondence between X-ray diagram levels and electron con®gurations .. Table 4.2.1.2. Correspondence between IUPAC and Siegbahn notations for X-ray diagram xiii


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191 191 191 192 193 195 195 196 198 199 191 191

CONTENTS Table Table Table Table Table

4.2.1.3. 4.2.1.4. 4.2.1.5. 4.2.1.6. 4.2.1.7.

Copper-target X-ray tubes and their loading .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Relative permissible loading for different target materials .. .. .. .. .. .. .. .. .. .. .. Radionuclides decaying wholly by electron capture, and yielding little or no -radiation Comparison of storage-ring synchrotron-radiation sources .. .. .. .. .. .. .. .. .. .. Intensity gain with storage rings over conventional sources .. .. .. .. .. .. .. .. .. ..

4.2.2. X-ray wavelengths (R. D. Deslattes, E. G. Kessler Jr, P. Indelicato, and E. Lindroth) 4.2.2.1. Historical introduction.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.2. Known problems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.3. Alternative strategies .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.4. The X-ray wavelength scales, old and new .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.5. K-series reference wavelengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.6. L-series reference wavelengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.7. Absorption-edge locations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.8. Outline of the theoretical procedures .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.9. Evaluation of the uncorrelated energy with Dirac±Fock method .. .. .. .. 4.2.2.10. Correlation and Auger shifts .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.11. QED corrections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.2.12. Structure and format of the summary tables .. .. .. .. .. .. .. .. .. .. .. 4.2.2.13. Availability of a more complete X-ray wavelength table.. .. .. .. .. .. .. 4.2.2.14. Connection with scales used in previous literature .. .. .. .. .. .. .. .. .. Table 4.2.2.1. K-series reference wavelengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.2.2. Directly measured L-series reference wavelengths .. .. .. .. .. .. .. Table 4.2.2.3. Directly measured and emission + binding energies K-absorption edges Table 4.2.2.4. Wavelengths of K-emission lines and K-absorption edges .. .. .. .. .. Table 4.2.2.5. Wavelengths of L-emission lines and L-absorption edges .. .. .. .. .. Table 4.2.2.6. Wavelength conversion factors .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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194 196 196 199 200

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200 200 201 201 201 202 202 202 204 205 205 205 211 212 212 203 204 205 206 209 212

4.2.3. X-ray absorption spectra (D. C. Creagh) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.1.1. De®nitions.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.1.2. Variation of X-ray attenuation coef®cients with photon energy .. .. .. .. .. .. 4.2.3.1.3. Normal attenuation, XAFS, and XANES .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.2. Techniques for the measurement of X-ray attenuation coef®cients .. .. .. .. .. .. .. .. 4.2.3.2.1. Experimental con®gurations.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.2.2. Specimen selection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.2.3. Requirements for the absolute measurement of l or =.. .. .. .. .. .. .. .. 4.2.3.3. Normal attenuation coef®cients .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.4. Attenuation coef®cients in the neighbourhood of an absorption edge .. .. .. .. .. .. .. 4.2.3.4.1. XAFS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.4.1.1. Theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.4.1.2. Techniques of data analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.4.1.3. XAFS experiments .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.4.2. X-ray absorption near edge structure (XANES) .. .. .. .. .. .. .. .. .. .. .. .. 4.2.3.5. Comments .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.3.1. Some synchrotron-radiation facilities providing XAFS databases and analysis utilities

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213 213 213 213 213 214 214 215 215 215 216 216 216 217 218 219 220 219

4.2.4. X-ray absorption (or attenuation) coef®cients (D. C. Creagh and J. H. Hubbell) .. .. .. .. 4.2.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.2. Sources of information .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.2.1. Theoretical photo-effect data: pe .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.4.2.2. Theoretical Rayleigh scattering data: R .. .. .. .. .. .. .. .. .. .. .. 4.2.4.2.3. Theoretical Compton scattering data: C .. .. .. .. .. .. .. .. .. .. .. 4.2.4.3. Comparison between theoretical and experimental data sets .. .. .. .. .. .. .. .. 4.2.4.4. Uncertainty in the data tables .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.4.1. Table of wavelengths and energies for the characteristic radiations 4.2.4.2 and 4.2.4.3 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.4.2. Total photon interaction cross section .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.4.3. Mass attenuation coef®cients .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Tables .. .. .. .. .. .. .. .. ..

220 220 221 221 221 229 229 229

4.2.5. Filters and monochromators (D. C. Creagh) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.5.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.5.2. Mirrors and capillaries.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

229 229 236


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xiv

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221 223 230

CONTENTS 4.2.5.2.1. Mirrors .. .. .. .. .. .. .. .. .. .. .. .. 4.2.5.2.2. Capillaries .. .. .. .. .. .. .. .. .. .. .. 4.2.5.2.3. Quasi-Bragg re¯ectors.. .. .. .. .. .. .. 4.2.5.3. Filters .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.5.4. Monochromators .. .. .. .. .. .. .. .. .. .. .. 4.2.5.4.1. Crystal monochromators .. .. .. .. .. .. 4.2.5.4.2. Laboratory monochromator systems .. .. 4.2.5.4.3. Multiple-re¯ection monochromators for sources .. .. .. .. .. .. .. .. .. .. .. .. 4.2.5.4.4. Polarization .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. use with laboratory and .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. synchrotron-radiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

236 237 237 238 238 238 239 239 240

4.2.6. X-ray dispersion corrections (D. C. Creagh) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

241

4.2.6.1. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.1.1. Rayleigh scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.1.2. Thomson scattering by a free electron .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.1.3. Elastic scattering from electrons bound to atoms: the atomic scattering factor, the atomic form factor, and the dispersion corrections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.2. Theoretical approaches for the calculation of the dispersion corrections.. .. .. .. .. .. .. .. .. .. 4.2.6.2.1. The classical approach .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.2.2. Non-relativistic theories .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.2.3. Relativistic theories .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.2.3.1. Cromer and Liberman: relativistic dipole approach .. .. .. .. .. .. .. .. .. .. 4.2.6.2.3.2. The scattering matrix formalism .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.2.4. Intercomparison of theories .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3. Modern experimental techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3.1. Determination of the real part of the dispersion correction: f 0 !; 0 .. .. .. .. .. .. .. .. 4.2.6.3.2. Determination of the real part of the dispersion correction: f 0 !; D .. .. .. .. .. .. .. .. 4.2.6.3.2.1. Measurements using the dynamical theory of X-ray diffraction .. .. .. .. .. .. 4.2.6.3.2.2. Friedel- and Bijvoet-pair techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3.3. Comparison of theory with experiment .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3.3.1. Measurements in the high-energy limit !=! ! 0.. .. .. .. .. .. .. .. .. .. 4.2.6.3.3.2. Measurements in the vicinity of an absorption edge .. .. .. .. .. .. .. .. .. 4.2.6.3.3.3. Accuracy in the tables of dispersion corrections .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3.3.4. Towards a tensor formalism .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.3.3.5. Summary .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.4. Table of wavelengths, energies, and linewidths used in compiling the tables of the dispersion corrections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.2.6.5. Tables of the dispersion corrections for forward scattering, averaged polarization using the relativistic multipole approach.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.1. Values of Etot =mc2 listed as a function of atomic number Z .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.2(a). Comparison between the S-matrix calculations of Kissel (1977) and the form-factor calculations of Cromer & Liberman (1970, 1981, 1983) and Creagh & McAuley for the noble gases and several common metals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.2(b). A comparison of the real part of the forward-scattering amplitudes computed using different theoretical approaches .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.3. A comparison of the imaginary part of the forward-scattering amplitudes f 00 (!; 0) computed using different theoretical approaches .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.4. Comparison of measurements of the real part of the dispersion correction for LiF, Si, Al and Ge for characteristic wavelengths Ag K1 , Mo K1 and Cu K1 with theoretical predictions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.5. Comparison of measurements of f 0 (!; 0) for C, Si and Cu for characteristic wavelengths Ag K1 , Mo K1 and Cu K1 with theoretical predictions .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.6. Comparison of f 0 (!A ; 0) for copper, nickel, zirconium, and niobium for theoretical and experimental data sets.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.7. List of wavelengths, energies, and linewidths used in compiling the table of dispersion corrections .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.2.6.8. Dispersion corrections for forward scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

242 242 242 242 243 243 244 245 245 246 248 248 248 250 250 251 251 251 252 253 253 258 258 258 246 249 249 250 252 253 254 254 255

4.3. Electron Diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

259

4.3.1. Scattering factors for the diffraction of electrons by crystalline solids (J. M. Cowley) .. .. .. .. .. .. .. ..

259

xv


CONTENTS 4.3.1.1. Elastic scattering from a perfect crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.2. Atomic scattering factors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.3. Approximations of restricted validity.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.4. Relativistic effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.5. Absorption effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.6. Tables of atomic scattering amplitudes for electrons .. .. .. .. .. .. .. .. .. .. .. .. 4.3.1.7. Use of Tables 4.3.1.1 and 4.3.1.2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.1.1. Atomic scattering amplitudes for electrons for neutral atoms .. .. .. .. .. .. .. .. Table 4.3.1.2. Atomic scattering amplitudes for electrons for ionized atoms .. .. .. .. .. .. .. 4.3.2. Parameterizations of electron atomic scattering factors (J. M. Cowley, L. M. Peng, G. Ren, S. L. and M. J. Whelan) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.2.1. Parameters useful in electron diffraction as a function of accelerating voltage .. .. Ê 1 Table 4.3.2.2. Elastic atomic scattering factors of electrons for neutral atoms and s up to 2.0 A Ê 1 Table 4.3.2.3. Elastic atomic scattering factors of electrons for neutral atoms and s up to 6.0 A 4.3.3.

4.3.4.

4.3.5.

4.3.6.

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Dudarev, .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Complex scattering factors for the diffraction of electrons by gases (A. W. Ross, M. Fink, R. Hilderbrandt, J. Wang, and V. H. Smith Jr) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.3.2. Complex atomic scattering factors for electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.3.2.1. Elastic scattering factors for atoms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.3.2.2. Total inelastic scattering factors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.3.2.3. Corrections for defects in the theory of atomic scattering .. .. .. .. .. .. .. .. .. .. .. 4.3.3.3. Molecular scattering factors for electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.3.2. Inelastic scattering factors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Electron energy-loss spectroscopy on solids (C. Colliex) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.1. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.1.1. Use of electron beams .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.1.2. Parameters involved in the description of a single inelastic scattering event .. .. .. .. .. 4.3.4.1.3. Problems associated with multiple scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.1.4. Classi®cation of the different types of excitations contained in an electron energy-loss spectrum .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.2. Instrumentation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.2.1. General instrumental considerations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.2.2. Spectrometers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.2.3. Detection systems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.3. Excitation spectrum of valence electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.3.1. Volume plasmons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.3.2. Dielectric description .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.3.3. Real solids.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.3.4. Surface plasmons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.4. Excitation spectrum of core electrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.4.1. De®nition and classi®cation of core edges .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.4.2. Bethe theory for inelastic scattering by an isolated atom .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.4.3. Solid-state effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.4.4. Applications of core-loss spectroscopy .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.4.5. Conclusions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.4.1. Different possibilities for using EELS information as a function of the different accessible parameters (r, , E) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.4.2. Plasmon energies measured (and calculated) for a few simple metals .. .. .. .. .. .. .. .. Table 4.3.4.3. Experimental and theoretical values for the coef®cient in the plasmon dispersion curve together with estimates of the cut-off wavevector .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.3.4.4. Comparison of measured and calculated values for the halfwidth E1=2 (0) of the plasmon line .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Oriented texture patterns (B. B. Zvyagin) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.5.1. Texture patterns .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.5.2. Lattice plane oriented perpendicular to a direction (lamellar texture) .. .. .. .. .. .. .. .. .. .. 4.3.5.3. Lattice direction oriented parallel to a direction (®bre texture).. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.5.4. Applications to metals and organic materials .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Computation of dynamical wave amplitudes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xvi


259 259 260 260 261 261 261 263 272 262 281 282 284 262 262 262 262 389 390 390 286 378 391 391 391 392 392 393 394 394 395 397 397 397 399 401 403 404 404 406 408 410 411 394 397 398 398 412 412 412 413 414 414

CONTENTS 4.3.6.1. The multislice method (D. F. Lynch).. .. .. .. .. .. 4.3.6.2. The Bloch-wave method (A. Howie) .. .. .. .. .. .. 4.3.7. Measurement of structure factors and determination of crystal and J. W. Steeds) .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.8. Crystal structure determination by high-resolution electron 4.3.8.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. 4.3.8.2. Lattice-fringe images .. .. .. .. .. .. .. .. .. .. 4.3.8.3. Crystal structure images .. .. .. .. .. .. .. .. .. 4.3.8.4. Parameters affecting HREM images .. .. .. .. .. 4.3.8.5. Computing methods .. .. .. .. .. .. .. .. .. .. 4.3.8.6. Resolution and hyper-resolution .. .. .. .. .. .. 4.3.8.7. Alternative methods .. .. .. .. .. .. .. .. .. .. 4.3.8.8. Combined use of HREM and electron diffraction

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. thickness by electron diffraction (J. Gjùnnes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

419 419 421 422 424 425 427 427 428

4.4. Neutron Techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

430

4.4.1. Production of neutrons (J. M. Carpenter and G. Lander) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

430

4.4.2. Beam-de®nition devices (I. S. Anderson and O. SchaÈrpf) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.2. Collimators .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.3. Crystal monochromators .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.4. Mirror re¯ection devices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.4.1. Neutron guides .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.4.2. Focusing mirrors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.4.3. Multilayers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.4.4. Capillary optics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.5. Filters .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.6. Polarizers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.6.1. Single-crystal polarizers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.6.2. Polarizing mirrors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.6.3. Polarizing ®lters .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.6.4. Zeeman polarizer .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.7. Spin-orientation devices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.7.1. Maintaining the direction of polarization .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.7.2. Rotation of the polarization direction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.7.3. Flipping of the polarization direction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.2.8. Mechanical choppers and selectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.2.1. Some important properties of materials used for neutron monochromator crystals .. Table 4.4.2.2. Neutron scattering-length densities, Nbcoh, for some commonly used materials .. .. .. Table 4.4.2.3. Characteristics of some typical elements and isotopes used as neutron ®lters .. .. .. Table 4.4.2.4. Properties of polarizing crystal monochromators.. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.2.5. Scattering-length densities for some typical materials used for polarizing multilayers .. 4.4.3. Resolution functions (R. Pynn and J. M. Rowe) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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431 431 431 432 435 435 436 436 437 438 438 438 440 440 442 442 442 442 442 443 433 435 439 440 441 443

4.4.4. Scattering lengths for neutrons (V. F. Sears) .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.4.1. Scattering lengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.4.2. Scattering and absorption cross sections.. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.4.3. Isotope effects .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.4.4. Correction for electromagnetic interactions .. .. .. .. .. .. .. .. .. .. .. 4.4.4.5. Measurement of scattering lengths .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4.4.4.6. Compilation of scattering lengths and cross sections .. .. .. .. .. .. .. .. Table 4.4.4.1. Bound scattering lengths, b, and cross sections, , of the elements and 4.4.5. Magnetic form factors (P. J. Brown) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.5.1. h j0 i form factors for 3d transition elements and their ions .. .. .. .. Table 4.4.5.2. h j0 i form factors for 4d atoms and their ions .. .. .. .. .. .. .. .. .. Table 4.4.5.3. h j0 i form factors for rare-earth ions .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.5.4. h j0 i form factors for actinide ions .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.5.5. h j2 i form factors for 3d transition elements and their ions .. .. .. .. Table 4.4.5.6. h j2 i form factors for 4d atoms and their ions .. .. .. .. .. .. .. .. .. Table 4.4.5.7. h j2 i form factors for rare-earth ions .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.5.8. h j2 i form factors for actinide ions .. .. .. .. .. .. .. .. .. .. .. .. .. Table 4.4.5.9. h j4 i form factors for 3d atoms and their ions .. .. .. .. .. .. .. .. ..

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444 444 452 452 453 453 453 445 454 454 455 455 455 456 457 457 457 458


(J. .. .. .. .. .. .. .. ..

C. H. Spence .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

416

and J. M. Cowley) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

xvii

microscopy .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

414 415

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CONTENTS Table 4.4.5.10. h j4 i form factors for 4d atoms and their Table 4.4.5.11. h j4 i form factors for rare-earth ions.. .. Table 4.4.5.12. h j4 i form factors for actinide ions .. .. Table 4.4.5.13. h j6 i form factors for rare-earth ions.. .. Table 4.4.5.14. h j6 i form factors for actinide ions .. .. 4.4.6. Absorption coef®cients for neutrons (B. T. M. Willis) .. Table 4.4.6.1. Absorption of the elements for neutrons References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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459 459 459 460 460 461 461 462

PART 5: DETERMINATION OF LATTICE PARAMETERS .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

489

5.1. Introduction (A. J. C. Wilson) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

490

5.2. X-ray Diffraction Methods: Polycrystalline (W. Parrish, A. J. C. Wilson, and J. I. Langford) .. .. ..

491

5.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.1.1. The techniques available .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.1.2. Errors and aberrations: general discussion .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.1.3. Errors of the Bragg angle .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.1.4. Bragg angle: operational de®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 5.2.1.1. Functions of the cell angles in equation (5.2.1.3) for the possible unit cells 5.2.2. Wavelength and related problems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.2.1. Errors and uncertainties in wavelength .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.2.2. Refraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.2.3. Statistical ¯uctuations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.3. Geometrical and physical aberrations.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.3.1. Aberrations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.3.2. Extrapolation, graphical and analytical .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.4. Angle-dispersive diffractometer methods: conventional sources .. .. .. .. .. .. .. .. .. Table 5.2.4.1. Centroid displacement h=i and variance W of certain aberrations of diffractometer .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.5. Angle-dispersive diffractometer methods: synchrotron sources .. .. .. .. .. .. .. .. ..

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491 491 491 491 491 492 492 492 492 492 493 493 493 495

5.2.6. Whole-pattern methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

496

5.2.7. Energy-dispersive techniques .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 5.2.7.1. Centroid displacement hE=Ei and variance W of certain aberrations of an energy-dispersive diffractometer .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.2.8. Camera methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 5.2.8.1. Some geometrical aberrations in the Debye±Scherrer method .. .. .. .. .. .. .. .. .. .. .. 5.2.9. Testing for remanent systematic error .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

496 497 497 498 498

5.2.10. Powder-diffraction standards .. .. .. .. .. .. .. .. .. .. .. .. Table 5.2.10.1. NIST values for silicon standards .. .. .. .. .. Table 5.2.10.2. Re¯ection angles for tungsten, silver, and silicon Table 5.2.10.3. Silicon standard re¯ection angles.. .. .. .. .. .. Table 5.2.10.4. Silicon standard high re¯ection angles .. .. .. .. Table 5.2.10.5. Tungsten re¯ection angles .. .. .. .. .. .. .. .. Table 5.2.10.6. Fluorophlogopite standard re¯ection angles .. .. Table 5.2.10.7. Silver behenate standard re¯ection angles .. .. .. 5.2.11. Intensity standards.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 5.2.11.1. NIST intensity standards, SRM 674 .. .. .. .. .. 5.2.12. Instrumental line-pro®le-shape standards .. .. .. .. .. .. .. ..

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

498 499 499 500 501 502 503 503 500 503 501

5.2.13. Factors determining accuracy .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

501

5.3. X-ray Diffraction Methods: Single Crystal (E. Gal-decka).. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

505

5.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. 5.3.1.1. General remarks.. .. .. .. .. .. .. .. 5.3.1.2. Introduction to single-crystal methods 5.3.2. Photographic methods .. .. .. .. .. .. .. .. .. 5.3.2.1. Introduction .. .. .. .. .. .. .. .. ..

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xviii


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494 495

505 505 506 508 508

CONTENTS 5.3.2.2. The Laue method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.3. Moving-crystal methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.3.1. Rotating-crystal method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.3.2. Moving-®lm methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.3.3. Combined methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.3.4. Accurate and precise lattice-parameter determinations .. .. .. .. .. .. .. 5.3.2.3.5. Photographic cameras for investigation of small lattice-parameter changes .. 5.3.2.4. The Kossel method and divergent-beam techniques.. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.4.1. The principle .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.4.2. Review of methods of accurate lattice-parameter determination .. .. .. .. 5.3.2.4.3. Accuracy and precision .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.2.4.4. Applications .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3. Methods with counter recording .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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508 508 508 509 509 509 510 510 510 512 515 515 516

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. the experiment .. .. .. .. .. .. .. .. .. .. .. .. applications .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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

537

5.3.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.2. Standard diffractometers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.2.1. Four-circle diffractometer .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.2.2. Two-circle diffractometer.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.3. Data processing and optimization of the experiment .. .. .. .. .. .. .. .. .. .. 5.3.3.3.1. Models of the diffraction pro®le .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.3.2. Precision and accuracy of the Bragg-angle determination; optimization of 5.3.3.4. One-crystal spectrometers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.4.1. General characteristics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.4.2. Development of methods based on an asymmetric arrangement and their 5.3.3.4.3. The Bond method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.4.3.1. Description of the method .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.4.3.2. Systematic errors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.4.3.3. Development of the Bond method and its applications .. .. 5.3.3.4.3.4. Advantages and disadvantages of the Bond method .. .. .. 5.3.3.5. Limitations of traditional methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.6. Multiple-diffraction methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.7. Multiple-crystal ± pseudo-non-dispersive techniques .. .. .. .. .. .. .. .. .. .. 5.3.3.7.1. Double-crystal spectrometers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.7.2. Triple-crystal spectrometers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.7.3. Multiple-beam methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.7.4. Combined methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.3.8. Optical and X-ray interferometry ± a non-dispersive technique .. .. .. .. .. .. .. 5.3.3.9. Lattice-parameter and wavelength standards .. .. .. .. .. .. .. .. .. .. .. .. .. 5.3.4. Final remarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.4. Electron-Diffraction Methods

5.4.1. Determination of cell parameters from single-crystal patterns (A. W. S. Johnson) 5.4.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.4.1.2. Zero-zone analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5.4.1.3. Non-zero-zone analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 5.4.1.1. Unit-cell information available for photographic recording .. .. .. 5.4.2. Kikuchi and HOLZ techniques (A. Olsen) .. .. .. .. .. .. .. .. .. .. .. .. .. ..

541

References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

541

PART 6: INTERPRETATION OF DIFFRACTED INTENSITIES .. .. .. .. .. .. .. .. .. .. .. .. ..

553

6.1. Intensity of Diffracted Intensities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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5.5. Neutron Methods (B. T. M. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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537 537 538 538 537 538

A. O'Keefe) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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6.1.1. X-ray scattering (E. N. Maslen, A. G. Fox, and M. 6.1.1.1. Coherent (Rayleigh) scattering .. .. .. .. 6.1.1.2. Incoherent (Compton) scattering .. .. .. 6.1.1.3. Atomic scattering factor .. .. .. .. .. .. 6.1.1.3.1. Scattering-factor interpolation ..

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554 554 554 554 565

CONTENTS 6.1.1.4. Generalized scattering factors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.5. The temperature factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6. The generalized temperature factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.1. Gram±Charlier series .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.2. Fourier-invariant expansions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.3. Cumulant expansion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.4. Curvilinear density functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.5. Model-based curvilinear density functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.6.6. The quasi-Gaussian approximation for curvilinear motion .. .. .. .. .. .. .. .. .. .. .. 6.1.1.7. Structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.1.8. Re¯ecting power of a crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.1. Mean atomic scattering factors in electrons for free atoms .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.2. Spherical bonded hydrogen-atom scattering factors.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.3. Mean atomic scattering factors in electrons for chemically signi®cant ions .. .. .. .. .. .. .. Table 6.1.1.4. Coef®cients for analytical approximation to the scattering factors of Tables 6.1.1.1 and 6.1.1.3 Table 6.1.1.5. Coef®cients for analytical approximation to the scattering factors of Table 6.1.1.1 for Ê 1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. the range 2:0 < (sin )/l < 6:0 A Table 6.1.1.6. Angle dependence of multipole functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.7. Indices allowed by the site symmetry for the real form of the spherical harmonics Ylmp;' .. Table 6.1.1.8. Cubic harmonics R 1 Klj (; ') for cubic site symmetries .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.9. fnl (; S) 0 rn exp( r)jl (Sr)dr .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.1.1.10. Indices nmp allowed by the site symmetry for the functions Hn (z)mp (').. .. .. .. .. .. .. Table 6.1.1.11. Indices nx ; ny ; nz allowed for the basis functions Hnx (Ax)Hny (By)Hnz (Cz).. .. .. .. .. .. .. 6.1.2. Magnetic scattering of neutrons (P. J. Brown).. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.1. Glossary of symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.2. General formulae for the magnetic cross section .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.3. Calculation of magnetic structure factors and cross sections .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.4. The magnetic form factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.5. The scattering cross section for polarized neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.2.6. Rotation of the polarization of the scattered neutrons .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.3. Nuclear scattering of neutrons (B. T. M. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.3.1. Glossary of symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.3.2. Scattering by a single nucleus .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.3.3. Scattering by a single atom .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.1.3.4. Scattering by a single crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

581 583 584 585 586 586 587 590 590 591 591 592 592 593 593 593 593 594 594

6.2. Trigonometric Intensity Factors (H. Lipson, J. I. Langford and H.-C. Hu) .. .. .. .. .. .. .. .. .. .. ..

596

6.2.1. Expressions for intensity of diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 6.2.1.1. Summary of formulae for integrated powers of re¯ection .. .. .. .. .. .. .. .. .. .. .. .. .. 6.2.2. The polarization factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

596 597 596

6.2.3. The angular-velocity factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

596

6.2.4. The Lorentz factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

596

6.2.5. Special factors in the powder method

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596

6.2.6. Some remarks about the integrated re¯ection power ratio formulae for single-crystal slabs .. .. .. .. .. ..

598

6.2.7. Other factors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

598

6.3. X-ray Absorption (E. N. Maslen) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

599

6.3.1. Linear absorption coef®cient .. .. .. .. .. .. .. .. 6.3.1.1. True or photoelectric absorption .. .. .. .. 6.3.1.2. Scattering .. .. .. .. .. .. .. .. .. .. .. .. 6.3.1.3. Extinction .. .. .. .. .. .. .. .. .. .. .. .. 6.3.1.4. Attenuation (mass absorption) coef®cients .. 6.3.2. Dispersion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6.3.3. Absorption corrections.. .. .. .. .. .. 6.3.3.1. Special cases .. .. .. .. .. .. 6.3.3.2. Cylinders and spheres .. .. .. 6.3.3.3. Analytical method for crystals 6.3.3.4. Gaussian integration .. .. ..

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599 599 599 599 600 600

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600 600 600 604 606

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565 584 585 586 586 588 588 589 590 590 590 555 565 566 578

CONTENTS 6.3.3.5. Empirical methods .. .. .. .. .. .. .. .. .. .. 6.3.3.6. Measuring crystals for absorption .. .. .. .. .. Table 6.3.3.1. Transmission coef®cients .. .. .. .. .. .. Table 6.3.3.2. Values of A* for cylinders .. .. .. .. .. Table 6.3.3.3. Values of A* for spheres .. .. .. .. .. .. Table 6.3.3.4. Values of (1/A*)(dA*/dR) for spheres .. Table 6.3.3.5. Coef®cients for interpolation of A* and T

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607 608 601 602 602 603 603

6.4. The Flow of Radiation in a Real Crystal (T. M. Sabine) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

609

6.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

609

6.4.2. The model of a real crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

609

6.4.3. Primary and secondary extinction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

609

6.4.4. Radiation ¯ow

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610

6.4.5. Primary extinction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

610

6.4.6. The ®nite crystal .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

610

6.4.7. Angular variation of E.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

610

6.4.8. The value of x

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610

6.4.9. Secondary extinction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

611

6.4.10. The extinction factor .. .. .. .. .. .. .. 6.4.10.1. The correlated block model .. 6.4.10.2. The uncorrelated block model 6.4.11. Polarization .. .. .. .. .. .. .. .. .. ..

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611 611 611 611

6.4.12. Anisotropy .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

612

6.4.13. Asymptotic behaviour of the integrated intensity .. .. .. .. 6.4.13.1. Non-absorbing crystal, strong primary extinction .. 6.4.13.2. Non-absorbing crystal, strong secondary extinction 6.4.13.3. The absorbing crystal.. .. .. .. .. .. .. .. .. .. .. 6.4.14. Relationship with the dynamical theory .. .. .. .. .. .. ..

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612 612 612 612 612

6.4.15. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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613

PART 7: MEASUREMENT OF INTENSITIES .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

617

7.1. Detectors for X-rays .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

618

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7.1.1. Photographic ®lm (P. M. de Wolff) .. .. .. .. 7.1.1.1. Visual estimation .. .. .. .. .. .. .. 7.1.1.2. Densitometry .. .. .. .. .. .. .. .. .. 7.1.2. Geiger counters (W. Parrish and J. I. Langford)

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618 618 618 618

7.1.3. Proportional counters (W. Parrish) .. .. .. .. .. .. .. .. .. 7.1.3.1. The detector system .. .. .. .. .. .. .. .. .. .. .. 7.1.3.2. Proportional counters .. .. .. .. .. .. .. .. .. .. .. 7.1.3.3. Position-sensitive detectors .. .. .. .. .. .. .. .. .. 7.1.3.4. Resolution, discriminination, ef®ciency .. .. .. .. .. 7.1.4. Scintillation and solid-state detectors (W. Parrish) .. .. .. .. 7.1.4.1. Scintillation counters .. .. .. .. .. .. .. .. .. .. .. 7.1.4.2. Solid-state detectors .. .. .. .. .. .. .. .. .. .. .. 7.1.4.3. Energy resolution and pulse-amplitude discrimination 7.1.4.4. Quantum-counting ef®ciency and linearity .. .. .. .. 7.1.4.5. Escape peaks .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.1.5. Energy-dispersive detectors (B. Buras and L. Gerward) .. ..

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619 619 619 619 619 619 619 620 620 621 622 622

7.1.6. Position-sensitive detectors (U. W. Arndt) .. 7.1.6.1. Choice of detector .. .. .. .. .. .. 7.1.6.1.1. Detection ef®ciency .. .. 7.1.6.1.2. Linearity of response .. .. 7.1.6.1.3. Dynamic range .. .. .. ..

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CONTENTS 7.1.6.1.4. Spatial resolution .. .. .. .. .. .. .. .. .. 7.1.6.1.5. Uniformity of response .. .. .. .. .. .. .. 7.1.6.1.6. Spatial distortion .. .. .. .. .. .. .. .. .. 7.1.6.1.7. Energy discrimination .. .. .. .. .. .. .. .. 7.1.6.1.8. Suitability for dynamic measurements .. .. 7.1.6.1.9. Stability .. .. .. .. .. .. .. .. .. .. .. .. 7.1.6.1.10. Size and weight .. .. .. .. .. .. .. .. .. 7.1.6.2. Gas-®lled counters .. .. .. .. .. .. .. .. .. .. .. .. 7.1.6.2.1. Localization of the detected photon .. .. .. 7.1.6.2.2. Parallel-plate counters.. .. .. .. .. .. .. .. 7.1.6.2.3. Current ionization PSD's .. .. .. .. .. .. .. 7.1.6.3. Semiconductor detectors .. .. .. .. .. .. .. .. .. .. 7.1.6.3.1. X-ray-sensitive semiconductor PSD's.. .. .. 7.1.6.3.2. Light-sensitive semiconductor PSD's .. .. .. 7.1.6.3.3. Electron-sensitive PSD's .. .. .. .. .. .. .. 7.1.6.4. Devices with an X-ray-sensitive photocathode .. .. 7.1.6.5. Television area detectors with external phosphor .. 7.1.6.5.1. X-ray phosphors.. .. .. .. .. .. .. .. .. .. 7.1.6.5.2. Light coupling .. .. .. .. .. .. .. .. .. .. 7.1.6.5.3. Image intensi®ers .. .. .. .. .. .. .. .. .. 7.1.6.5.4. TV camera tubes .. .. .. .. .. .. .. .. .. 7.1.6.6. Some applications .. .. .. .. .. .. .. .. .. .. .. .. Table 7.1.6.1. The importance of some detector properties for Table 7.1.6.2. X-ray phosphors .. .. .. .. .. .. .. .. .. .. 7.1.7. X-ray-sensitive TV cameras (J. Chikawa) .. .. .. .. .. .. .. 7.1.7.1. Signal-to-noise ratio .. .. .. .. .. .. .. .. .. .. .. 7.1.7.2. Imaging system .. .. .. .. .. .. .. .. .. .. .. .. .. 7.1.7.3. Image processing .. .. .. .. .. .. .. .. .. .. .. .. 7.1.8. Storage phosphors (Y. Amemiya and J. Chikawa) .. .. .. ..

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625 625 625 625 626 626 626 626 627 627 628 629 629 630 630 630 630 631 632 632 632 632 624 631 633 633 634 635 635

7.2. Detectors for Electrons (J. N. Chapman) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

639

7.2.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

639

7.2.2. Characterization of detectors

639

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7.2.3. Parallel detectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.3.1. Fluorescent screens .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.3.2. Photographic emulsions .. .. .. .. .. .. .. .. .. .. .. .. 7.2.3.3. Detector systems based on an electron-tube device .. .. .. 7.2.3.4. Electronic detection systems based on solid-state devices .. 7.2.3.5. Imaging plates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.4. Serial detectors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.4.1. Faraday cage .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.4.2. Scintillation detectors .. .. .. .. .. .. .. .. .. .. .. .. .. 7.2.4.3. Semiconductor detectors .. .. .. .. .. .. .. .. .. .. .. .. 7.2.5. Conclusions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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

640 640 640 641 641 641 642 642 642 642 643

7.3. Thermal Neutron Detection (P. Convert and P. Chieux) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

644

7.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

644

7.3.2. Neutron capture .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 7.3.2.1. Neutron capture reactions used in neutron detection .. .. .. 7.3.3. Neutron detection processes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.3.3.1. Detection via gas converter and gas ionization: the gas detector .. 7.3.3.2. Detection via solid converter and gas ionization: the foil detector .. 7.3.3.3. Detection via scintillation .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.3.3.4. Films .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 7.3.3.1. Commonly used detection processes .. .. .. .. .. .. .. .. .. Table 7.3.3.2. A few examples of gas-detector characteristics.. .. .. .. .. .. 7.3.4. Electronic aspects of neutron detection .. .. .. .. .. .. .. .. .. .. .. .. .. 7.3.4.1. The electronic chain .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

644 645 644 644 645 645 646 646 646 648 648

xxii


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CONTENTS 7.3.4.2. Controls and adjustments of the electronics 7.3.5. Typical detection systems .. .. .. .. .. .. .. .. .. .. 7.3.5.1. Single detectors .. .. .. .. .. .. .. .. .. .. 7.3.5.2. Position-sensitive detectors .. .. .. .. .. .. 7.3.5.3. Banks of detectors .. .. .. .. .. .. .. .. .. Table 7.3.5.1. Characteristics of some PSDs .. .. .. 7.3.6. Characteristics of detection systems .. .. .. .. .. ..

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648 649 649 649 650 651 651

7.3.7. Corrections to the intensity measurements 7.3.7.1. Single detector .. .. .. .. .. .. 7.3.7.2. Banks of detectors .. .. .. .. .. 7.3.7.3. Position-sensitive detectors .. ..

on .. .. ..

the detection .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

system .. .. .. .. .. .. .. .. ..

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652 652 652 652

7.4. Correction of Systematic Errors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

653

7.4.1. Absorption.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

653

7.4.2. Thermal diffuse scattering (B. T. M. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.2.1. Glossary of symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.2.2. TDS correction factor for X-rays (single crystals) .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.2.2.1. Evaluation of J(q) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.2.2.2. Calculation of .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.2.3. TDS correction factor for thermal neutrons (single crystals) .. .. .. .. .. .. .. .. .. .. 7.4.2.4. Correction factor for powders .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3. Compton scattering (N. G. Alexandropoulos and M. J. Cooper) .. .. .. .. .. .. .. .. .. .. .. 7.4.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section .. .. .. .. .. .. 7.4.3.2.1. Semi-classical radiation theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.2.2. Thomas±Fermi model .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.2.3. Exact calculations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.3. Relativistic treatment of incoherent scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.4. Plasmon, Raman, and resonant Raman scattering .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.3.5. Magnetic scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 7.4.3.1. The energy transfer in the Compton scattering process for selected X-ray energies Table 7.4.3.2. The incoherent scattering function for elements up to Z = 55 .. .. .. .. .. .. .. Table 7.4.3.3. Compton scattering of Mo K X-radiation through 170 from 2s electrons .. .. .. 7.4.4. White radiation and other sources of backgound (P. Suortti) .. .. .. .. .. .. .. .. .. .. .. .. 7.4.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.4.2. Incident beam and sample .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.4.3. Detecting system .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.4.4.4. Powder diffraction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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653 653 654 654 655 656 657 657 657 657 657 659 659 659 660 661 657 658 659 661 661 661 663 664

7.5. Statistical Fluctuations (A. J. C. Wilson) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

666

7.5.1. Distributions of intensities of diffraction

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666

7.5.2. Counting modes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

666

7.5.3. Fixed-time counting.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

666

7.5.4. Fixed-count timing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

667

7.5.5. Complicating phenomena .. .. .. .. .. .. .. .. .. .. 7.5.5.1. Dead time .. .. .. .. .. .. .. .. .. .. .. .. 7.5.5.2. Voltage ¯uctuations .. .. .. .. .. .. .. .. 7.5.6. Treatment of measured-as-negative (and other weak)

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667 667 667 667

7.5.7. Optimization of counting times .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

667

References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

668

PART 8: REFINEMENT OF STRUCTURAL PARAMETERS .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

677

8.1. Least Squares (E. Prince and P. T. Boggs) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

678

8.1.1. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.1.1.1. Linear algebra .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.1.1.2. Statistics .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

678 678 679

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CONTENTS 8.1.2. Principles of least squares

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8.1.3. Implementation of linear least squares .. .. .. .. .. .. .. 8.1.3.1. Use of the QR factorization .. .. .. .. .. .. .. .. 8.1.3.2. The normal equations .. .. .. .. .. .. .. .. .. .. 8.1.3.3. Conditioning .. .. .. .. .. .. .. .. .. .. .. .. .. 8.1.4. Methods for nonlinear least squares .. .. .. .. .. .. .. .. 8.1.4.1. The Gauss±Newton algorithm .. .. .. .. .. .. .. 8.1.4.2. Trust-region methods ± the Levenberg±Marquardt 8.1.4.3. Quasi-Newton, or secant, methods .. .. .. .. .. 8.1.4.4. Stopping rules .. .. .. .. .. .. .. .. .. .. .. .. 8.1.4.5. Recommendations .. .. .. .. .. .. .. .. .. .. .. 8.1.5. Numerical methods for large-scale problems .. .. .. .. .. 8.1.5.1. Methods for sparse matrices.. .. .. .. .. .. .. .. 8.1.5.2. Conjugate-gradient methods .. .. .. .. .. .. .. .. 8.1.6. Orthogonal distance regression .. .. .. .. .. .. .. .. .. ..

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681 681 682 682 682 683 683 683 684 685 685 685 686 687

8.1.7. Software for least-squares calculations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

688

8.2. Other Refinement Methods (E. Prince and D. M. Collins) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

689

8.2.1. Maximum-likelihood methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

689

8.2.2. Robust/resistant methods .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

689

8.2.3. Entropy maximization .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.2.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.2.3.2. Some examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

691 691 691

8.3. Constraints and Restraints in Refinement (E. Prince, L. W. Finger, and J. H. Konnert)

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693

8.3.1. Constrained models .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.3.1.1. Lagrange undetermined multipliers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.3.1.2. Direct application of constraints .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.3.1.1. Symmetry conditions for second-cumulant tensors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.3.2. Stereochemically restrained least-squares re®nement .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.3.2.1. Stereochemical constraints as observational equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.3.2.1. Coordinates of atoms in standard groups appearing in polypeptides and proteins .. .. .. .. Table 8.3.2.2. Ideal values for distances, torsion angles, etc. for a glycine±alanine dipeptide with a trans peptide bond .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.3.2.3. Typical values of standard deviations for use in determining weights in restrained re®nement of protein structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

693 693 693 695 698 698 699

8.4. Statistical Significance Tests (E. Prince and C. H. Spiegelman) .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

702

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700 701

8.4.1. The v distribution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.4.1.1. Values of 2 = for which the c.d.f. (2 ; ) has the values given in the column headings, for various values of .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.4.2. The F distribution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.4.2.1. Values of the F ratio for which the c.d.f. (F; 1 ; 2 ) has the value 0.95, for various choices of 1 and 2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.4.3. Comparison of different models .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.4.3.1. Values of t for which the c.d.f. (t; ) has the values given in the column headings, for various values of .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.4.4. In¯uence of individual data points.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

704 705

8.5. Detection and Treatment of Systematic Error (E. Prince and C. H. Spiegleman) .. .. .. .. .. .. .. ..

707

8.5.1. Accuracy .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

707

8.5.2. Lack of ®t .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

707

8.5.3. In¯uential data points .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

708

8.5.4. Plausibility of results .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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xxiv


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CONTENTS 8.6. The Rietveld Method (A. Albinati and B. T. M. Willis) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

710

8.6.1. Basic theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

710

8.6.2. Problems with the Rietveld method .. .. 8.6.2.1. Indexing .. .. .. .. .. .. .. .. 8.6.2.2. Peak-shape function (PSF) .. .. 8.6.2.3. Background .. .. .. .. .. .. .. 8.6.2.4. Preferred orientation and texture 8.6.2.5. Statistical validity .. .. .. .. ..

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711 711 711 711 712 712

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713

8.7.1. Outline of this chapter.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

713

8.7.2. Electron densities and the n-particle wavefunction.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

713

8.7.3. Charge densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.2. Modelling of the charge density .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.3. Physical constraints .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.3.1. Electroneutrality constraint .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.3.2. Cusp constraint .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.3.3. Radial constraint .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.3.4. Hellmann±Feynman constraint .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.4. Electrostatic moments and the potential due to a charge distribution.. .. .. .. .. .. .. .. 8.7.3.4.1. Moments of a charge distribution .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.4.1.1. Moments as a function of the atomic multipole expansion .. .. .. .. 8.7.3.4.1.2. Molecular moments based on the deformation density .. .. .. .. .. .. 8.7.3.4.1.3. The effect of an origin shift on the outer moments .. .. .. .. .. .. .. 8.7.3.4.1.4. Total moments as a sum over the pseudoatom moments .. .. .. .. .. 8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation 8.7.3.4.2. The electrostatic potential .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.4.2.1. The electrostatic potential and its derivatives .. .. .. .. .. .. .. .. .. 8.7.3.4.2.2. Electrostatic potential outside a charge distribution.. .. .. .. .. .. .. 8.7.3.4.2.3. Evaluation of the electrostatic functions in direct space .. .. .. .. .. 8.7.3.4.3. Electrostatic functions of crystals by modi®ed Fourier summation.. .. .. .. .. .. 8.7.3.4.4. The total energy of a crystal as a function of the electron density.. .. .. .. .. .. 8.7.3.5. Quantitative comparison with theory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.6. Occupancies of transition-metal valence orbitals from multipole coef®cients .. .. .. .. .. 8.7.3.7. Thermal smearing of theoretical densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.7.1. General considerations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.7.2. Reciprocal-space averaging over external vibrations .. .. .. .. .. .. .. .. .. .. 8.7.3.8. Uncertainties in experimental electron densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.3.9. Uncertainties in derived functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.7.3.1. De®nition of difference density functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 8.7.3.2. Expressions for the shape factors S for a parallelepiped with edges x ; y ; and z .. .. Table 8.7.3.3. The matrix M 1 relating d-orbital occupancies Pij to multipole populations Plm .. .. Table 8.7.3.4. Orbital±multipole relations for square-planar complexes (point group D4h) .. .. .. .. Table 8.7.3.5. Orbital±multipole relations for trigonal complexes .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4. Spin densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.2. Magnetization densities from neutron magnetic elastic scattering .. .. .. .. .. .. .. .. .. 8.7.4.3. Magnetization densities and spin densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.3.1. Spin-only density at zero temperature .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.3.2. Thermally averaged spin-only magnetization density .. .. .. .. .. .. .. .. .. .. 8.7.4.3.3. Spin density for an assembly of localized systems .. .. .. .. .. .. .. .. .. .. .. 8.7.4.3.4. Orbital magnetization density .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.4. Probing spin densities by neutron elastic scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.4.2. Unpolarized neutron scattering .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.4.3. Polarized neutron scattering.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

714 714 714 715 715 715 715 715 716 716 716 717 717 718 718 718 718 720 720 720 721 721 722 723 723 723 724 725 714 719 722 723 723 725 725 725 726 726 726 727 727 727 727 728 728

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8.7. Analysis of Charge and Spin Densities (P. Coppens, Z. Su, and P. J. Becker)

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CONTENTS 8.7.4.4.4. Polarized neutron scattering of centrosymmetric crystals .. .. 8.7.4.4.5. Polarized neutron scattering in the noncentrosymmetric case 8.7.4.4.6. Effect of extinction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.4.7. Error analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5. Modelling the spin density .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5.1. Atom-centred expansion .. .. .. .. .. .. .. .. .. .. .. .. ..

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728 728 728 729 729 729

8.7.4.5.1.1. Spherical-atom model .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5.1.2. Crystal-®eld approximation .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5.1.3. Scaling of the spin density .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5.2. General multipolar expansion .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.5.3. Other types of model .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.6. Orbital contribution to the magnetic scattering .. .. .. .. .. .. .. .. .. .. .. 8.7.4.6.1. The dipolar approximation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.6.2. Beyond the dipolar approximation .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.6.3. Electronic structure of rare-earth elements .. .. .. .. .. .. .. .. .. 8.7.4.7. Properties derivable from spin densities .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.7.1. Vector ®elds .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.7.2. Moments of the magnetization density .. .. .. .. .. .. .. .. .. .. .. 8.7.4.8. Comparison between theory and experiment .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.9. Combined charge- and spin-density analysis .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.10. Magnetic X-ray scattering separation between spin and orbital magnetism .. 8.7.4.10.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8.7.4.10.2. Magnetic X-ray structure factor as a function of photon polarization

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

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

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

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

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

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

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

729 729 730 730 730 730 731 731 731 731 732 732 732 732 733 733 733

8.8. Accurate Structure-Factor Determination with Electron Diffraction (J. Gjùnnes).. .. .. .. .. .. ..

735

References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

738

PART 9: BASIC STRUCTURAL FEATURES .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

745

9.1. Sphere Packings and Packings of Ellipsoids (E. Koch and W. Fischer) .. .. .. .. .. .. .. .. .. .. .. ..

746

9.1.1. Sphere packings and packings of circles .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

746

9.1.1.1. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.1.1.2. Homogeneous packings of circles .. .. .. .. .. .. .. .. 9.1.1.3. Homogeneous sphere packings .. .. .. .. .. .. .. .. .. 9.1.1.4. Applications .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.1.1.5. Interpenetrating sphere packings .. .. .. .. .. .. .. .. Table 9.1.1.1. Types of circle packings in the plane .. .. .. .. Table 9.1.1.2. Examples for sphere packings with high contact contact numbers and low densities .. .. .. .. .. 9.1.2. Packings of ellipses and ellipsoids .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. and with low .. .. .. .. .. .. .. .. .. ..

746 746 746 750 751 747

9.2. Layer Stacking .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

752

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. numbers and .. .. .. .. .. .. .. .. .. ..

9.2.1. Layer stacking in close-packed structures (D. Pandey and P. Krishna) .. 9.2.1.1. Close packing of equal spheres .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.1.1. Close-packed layer .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.1.2. Close-packed structures .. .. .. .. .. .. .. .. .. .. .. 9.2.1.1.3. Notations for close-packed structures .. .. .. .. .. .. 9.2.1.2. Structure of compounds based on close-packed layer stackings .. 9.2.1.2.1. Voids in close packing .. .. .. .. .. .. .. .. .. .. .. 9.2.1.2.2. Structures of SiC and ZnS .. .. .. .. .. .. .. .. .. .. 9.2.1.2.3. Structure of CdI2 .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.2.4. Structure of GaSe .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.3. Symmetry of close-packed layer stackings of equal spheres .. .. 9.2.1.4. Possible lattice types .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.5. Possible space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.6. Crystallographic uses of Zhdanov symbols .. .. .. .. .. .. .. .. 9.2.1.7. Structure determination of close-packed layer stackings .. .. .. 9.2.1.7.1. General considerations .. .. .. .. .. .. .. .. .. .. .. 9.2.1.7.2. Determination of the lattice type .. .. .. .. .. .. .. .. xxvi


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

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

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. high densities .. .. .. .. .. .. .. .. .. ..

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

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

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

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

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

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

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

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

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

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

748 751

752 752 752 752 752 753 753 753 754 754 755 755 755 756 756 756 757

CONTENTS 9.2.1.7.3. Determination of the identity period.. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.7.4. Determination of the stacking sequence of layers .. .. .. .. .. .. .. .. .. 9.2.1.8. Stacking faults in close-packed structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.1.8.1. Structure determination of one-dimensionally disordered crystals .. .. .. .. Table 9.2.1.1. Common close-packed metallic structures .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.2.1.2. List of SiC polytypes with known structures in order of increasing periodicity .. Table 9.2.1.3. Intrinsic fault con®gurations in the 6H (A0B1C2A3C4B5,. . .) structure.. .. .. .. Table 9.2.1.4. Intrinsic fault con®gurations in the 9R (A0B1A2C0A1C2B0C1B2,. . .) structure .. Ï urovicÏ) .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2. Layer stacking in general polytypic structures (S. D 9.2.2.1. The notion of polytypism.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2. Symmetry aspects of polytypism .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.1. Close packing of spheres .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.2. Polytype families and OD groupoid families .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.3. MDO polytypes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.4. Some geometrical properties of OD structures .. .. .. .. .. .. .. .. .. .. 9.2.2.2.5. Diffraction pattern ± structure analysis .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.6. The vicinity condition .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.7. Categories of OD structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.7.1. OD structures of equivalent layers .. .. .. .. .. .. .. .. .. .. 9.2.2.2.7.2. OD structures with more than one kind of layer .. .. .. .. .. .. 9.2.2.2.8. Desymmetrization of OD structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.2.9. Concluding remarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3. Examples of some polytypic structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3.1. Hydrous phyllosilicates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3.1.1. General geometry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3.1.2. Diffraction pattern and identi®cation of individual polytypes .. 9.2.2.3.2. Stibivanite Sb2VO5 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3.3. -Hg3S2Cl2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.3.4. Remarks for authors .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.2.2.4. List of some polytypic structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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

757 757 758 759 753 754 758 759 760 760 761 761 761 762 762 763 763 764 764 765 765 766 766 766 767 769 769 771 772 772

9.3. Typical Interatomic Distances: Metals and Alloys (J. L. C. Daams, J. R. Rodgers, and P. Villars) ..

774

9.3.1. Glossary

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

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

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

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

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

777

9.4. Typical Interatomic Distances: Inorganic Compounds (G. Bergerhoff and K. Brandenburg) .. .. .. ..

778

9.4.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.1. Atomic distances between halogens and main-group elements in their preferred oxidation states .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.2. Atomic distances between halogens and main-group elements in their special oxidation states Table 9.4.1.3. Atomic distances between halogens and transition metals.. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.4. Atomic distances between halogens and lanthanoids .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.5. Atomic distances between halogens and actinoids .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.6. Atomic distances between oxygen and main-group elements in their preferred oxidation states Table 9.4.1.7. Atomic distances between oxygen and main-group elements in their special oxidation states .. Table 9.4.1.8. Atomic distances between oxygen and transition elements in their preferred and special oxidation states .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.9. Atomic distances between oxygen and lanthanoids .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.10. Atomic distances between oxygen and actinoids .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.11. Atomic distances in sul®des and thiometallates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.4.1.12. Contact distances between some negatively charged elements .. .. .. .. .. .. .. .. .. .. .. 9.4.2. The retrieval system .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

778

9.4.3. Interpretation of frequency distributions

779 780 781 784 785 785 786 786 787 788 788 789 778

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

778

9.5. Typical Interatomic Distances: Organic Compounds (F. H. Allen, D. G. Watson, L. Brammer, A. G. Orpen, and R. Taylor) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

790

9.5.1. Introduction .. .. .. .. .. .. .. Table 9.5.1.1. Average lengths Br, Te and I .. 9.5.2. Methodology .. .. .. .. .. .. ..

.. .. .. .. for bonds .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. involving the elements .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xxvii


.. H, .. ..

.. B, .. ..

.. .. .. .. .. .. C, N, O, F, Si, .. .. .. .. .. .. .. .. .. .. .. ..

.. P, .. ..

.. S, .. ..

.. Cl, .. ..

.. .. .. As, Se, .. .. .. .. .. ..

790 796 790

CONTENTS 9.5.2.1. 9.5.2.2. 9.5.2.3. 9.5.2.4.

Selection of crystallographic data Program system .. .. .. .. .. .. Classi®cation of bonds .. .. .. .. Statistics .. .. .. .. .. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

790 790 791 791

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

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.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

.. .. .. ..

791 792 792 793

9.5.4. Discussion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

794

9.6. Typical Interatomic Distances: Organometallic Compounds and Coordination Complexes of the d- and f-Block Metals (A. G. Orpen, L. Brammer, F. H. Allen, D. G. Watson, and R. Taylor) .. .. ..

812

9.6.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

812

9.6.2. Methodology .. .. .. .. .. .. .. .. .. .. 9.6.2.1. Selection of crystallographic data 9.6.2.2. Program system .. .. .. .. .. .. 9.6.2.3. Classi®cation of bonds .. .. .. .. 9.6.2.4. Statistics .. .. .. .. .. .. .. ..

9.5.3. Content and arrangement of the table .. .. .. 9.5.3.1. Ordering of entries: the `Bond' column 9.5.3.2. De®nition of `Substructure' .. .. .. .. 9.5.3.3. Use of the `Note' column.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

812 812 813 813 813

9.6.3. Content and arrangement of table of interatomic distances 9.6.3.1. The `Bond' column .. .. .. .. .. .. .. .. .. .. .. 9.6.3.2. De®nition of `Substructure' .. .. .. .. .. .. .. .. 9.6.3.3. Use of the `Note' column.. .. .. .. .. .. .. .. .. 9.6.3.4. Locating an entry in Table 9.6.3.3 .. .. .. .. .. .. Table 9.6.3.1. Ligand index .. .. .. .. .. .. .. .. .. .. .. Table 9.6.3.2. Numbers of entries in Table 9.6.3.3 .. .. .. Table 9.6.3.3. Interatomic distances .. .. .. .. .. .. .. ..

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

814 815 815 817 818 814 817 818

9.6.4. Discussion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

818

9.7. The Space-Group Distribution of Molecular Organic Structures (A. J. C. Wilson, V. L. Karen, and A. Mighell) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

897

9.7.1. A priori classi®cations of space groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.1.1. Kitajgorodskij's categories .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.1.2. Symmorphism and antimorphism .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.1.3. Comparison of Kitajgorodskij's and Wilson's classi®cations .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.1.4. Relation to structural classes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.7.1.1. Kitajgorodskij's categorization of the triclinic, monoclinic and orthorhombic space groups Table 9.7.1.2. Space groups arranged by arithmetic crystal class and degree of symmorphism .. .. .. ..

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

897 897 897 899 900 898 899

9.7.2. Special positions of given symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.7.2.1. Statistics of the use of Wyckoff positions of speci®ed symmetry G in the homomolecular organic crystals .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

900

9.7.3. Empirical space-group frequencies.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

902

9.7.4. Use of molecular symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.4.1. Positions with symmetry 1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.4.2. Positions with symmetry 1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.4.3. Other symmetries .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.4.4. Positions with the full symmetry of the geometric class .. .. .. .. Table 9.7.4.1. Occurrence of molecules with speci®ed point group in centred groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. symmmorphic and other space .. .. .. .. .. .. .. .. .. .. ..

902 902 902 903 903

9.7.5. Structural classes.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

904

9.7.6. A statistical model .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

904

9.7.7. Molecular packing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.7.1. Relation to sphere packing .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.7.7.2. The hydrogen bond and the de®nition of the packing units .. .. .. .. .. .. .. .. .. .. .. .. .. ..

904 904 906

9.7.8. A priori predictions of molecular crystal structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

906

xxviii


903

905

CONTENTS 9.8. Incommensurate and Commensurate Modulated Structures (T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.1.1. Modulated crystal structures.. .. .. .. .. .. .. .. .. 9.8.1.2. The basic ideas of higher-dimensional crystallography 9.8.1.3. The simple case of a displacively modulated crystal.. 9.8.1.3.1. The diffraction pattern .. .. .. .. .. .. .. 9.8.1.3.2. The symmetry .. .. .. .. .. .. .. .. .. .. 9.8.1.4. Basic symmetry considerations .. .. .. .. .. .. .. .. 9.8.1.4.1. Bravais classes of vector modules .. .. .. .. 9.8.1.4.2. Description in four dimensions .. .. .. .. .. 9.8.1.4.3. Four-dimensional crystallography .. .. .. .. 9.8.1.4.4. Generalized nomenclature .. .. .. .. .. .. 9.8.1.4.5. Four-dimensional space groups.. .. .. .. .. 9.8.1.5. Occupation modulation .. .. .. .. .. .. .. .. .. .. 9.8.2. Outline for a superspace-group determination.. .. .. .. .. ..

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

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

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

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

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

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

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

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

907 907 908 909 909 909 910 910 911 911 912 912 913 913

9.8.3. Introduction to the tables.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.1. Tables of Bravais lattices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.2. Table for geometric and arithmetic crystal classes .. .. .. .. .. .. .. .. .. .. 9.8.3.3. Tables of superspace groups.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.3.1. Symmetry elements.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.3.2. Re¯ection conditions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.4. Guide to the use of the tables .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.5. Examples .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.3.6. Ambiguities in the notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 9.8.3.1(a). (2 1)-Dimensional Bravais classes for incommensurate structures .. Table 9.8.3.1(b). (2 2)-Dimensional Bravais classes for incommensurate structures .. Table 9.8.3.2(a). (3 1)-Dimensional Bravais classes for incommensurate structures .. Table 9.8.3.2(b). (3 1)-Dimensional Bravais classes for commensurate structures .. .. Table 9.8.3.3. (3 1)-Dimensional point groups and arithmetic crystal classes .. .. Table 9.8.3.4(a). (2 1)-Dimensional superspace groups.. .. .. .. .. .. .. .. .. .. .. Table 9.8.3.4(b). (2 2)-Dimensional superspace groups .. .. .. .. .. .. .. .. .. .. Table 9.8.3.5. (3 1)-Dimensional superspace groups .. .. .. .. .. .. .. .. .. .. Table 9.8.3.6. Centring re¯ection conditions for (3 1)-dimensional Bravais classes 9.8.4. Theoretical foundation.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.1. Lattices and metric .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.2. Point groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.2.1. Laue class .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.2.2. Geometric and arithmetic crystal classes .. .. .. .. .. .. .. .. .. .. 9.8.4.3. Systems and Bravais classes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.3.1. Holohedry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.3.2. Crystallographic systems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.3.3. Bravais classes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.4. Superspace groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.4.1. Symmetry elements.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.4.4.2. Equivalent positions and modulation relations .. .. .. .. .. .. .. .. 9.8.4.4.3. Structure factor .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.5. Generalizations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

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

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

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

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

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

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

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

915 915 916 916 916 921 935 936 936 915 916 917 918 919 920 921 922 935 937 937 938 938 939 939 939 940 940 940 940 940 941 941

9.8.5.1. Incommensurate composite crystal structures .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9.8.5.2. The incommensurate versus the commensurate case .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

941 942 945

PART 10: PRECAUTIONS AGAINST RADIATION INJURY (D. C. Creagh and S. Martinez-Carrera) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

957

10.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

958

10.1.1. De®nitions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

958

Table 10.1.1. The relationship between SI and the earlier system of units .. .. .. .. .. .. .. .. .. .. .. ..

958

xxix


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

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

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

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

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

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

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

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

907

CONTENTS Table 10.1.2. Maximum primary-dose limit per quarter .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Table 10.1.3. Quality factors (QF) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

960 960

10.1.2. Objectives of radiation protection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

960

10.1.3. Responsibilities .. .. .. .. .. 10.1.3.1. General .. .. .. .. 10.1.3.2. The radiation safety 10.1.3.3. The worker .. .. .. 10.1.3.4. Primary-dose limits

.. .. .. .. ..

960 960 960 960 961

10.2. Protection from Ionizing Radiation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

962

10.2.1. General .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

962

10.2.2. Sealed sources and radiation-producing apparatus .. .. 10.2.2.1. Enclosed installations.. .. .. .. .. .. .. .. .. 10.2.2.2. Open installations .. .. .. .. .. .. .. .. .. .. 10.2.2.3. Sealed sources .. .. .. .. .. .. .. .. .. .. .. 10.2.2.4. X-ray diffraction and X-ray analysis apparatus 10.2.2.5. Particle accelerators .. .. .. .. .. .. .. .. ..

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

962 962 962 962 962 962

10.2.3. Ionizing-radiation protection ± unsealed radioactive materials .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

963

10.3. Responsible Bodies .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

964

Table 10.3.1. Regulatory authorities

.. .. .. .. .. .. of®cer.. .. .. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

.. .. .. .. ..

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

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

964

References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

967

Author Index .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

968

Subject Index .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

984

xxx


Preface

By A. J. C. Wilson A new volume of the International Tables for Crystallography containing mathematical, physical and chemical tables was discussed by the Executive Committee of the International Union of Crystallography at least as early as August 1979. My own ideas about what has become Volume C began to develop in the course of the Executive Committee meeting held at the Ottawa Congress in August 1981. It was then conceived as an editorial condensation of the old volumes II, III and IV, with obsolete material deleted and tables easily reproduced on a pocket calculator reduced to a skeleton form or omitted altogether. However, it soon became obvious that advances since the old volumes were produced could not be satisfactorily accommodated within such a condensation, and that if Volume C were to be a worthy companion of Volume A (Space-Group Symmetry) and Volume B (Reciprocal Space) it would have to consist largely of new material. Work on Volumes B and C began of®cially on 1 January 1983, and the general outlines of the volumes were circulated to the Executive Committee, the National Committees, and others interested. This circulation generated much constructive criticism and offers of help, particularly from several Commissions of the Union. The Chairmen of certain Commissions were particularly helpful in ®nding quali®ed contributors of specialist sections, and from time to time served as members of the

Commission on International Tables for Crystallography. I often had occasion to lament the lack of a Commission on X-ray Diffraction. The revised outlines of the two volumes were approved by the Executive Committee during the Hamburg Congress in 1984. For various reasons the publication of Volume C has taken longer than expected. A requirement that prospective contributors should be approved by the Executive Committee produced some delays, and more serious delays were caused by authors who failed to deliver their contributions by the agreed date ± or at all. A decision was taken to include in this ®rst edition only what was in the Editor's hands in January 1990, and since that date the timetable has been set by the printers. The present Volume is the result. Readers will ®nd a few sections resulting from the original idea of editorial condensation from Volumes II, III and IV, and some sections from those volumes revised or rewritten by their original authors. Most of Volume C is entirely new. I am indebted to many crystallographers for advice and encouragement, to the authors of contributions that arrived before the deadline, to the Chairmen of various Commissions for their help, and to the Technical Editor for his skill and good humour in dealing with much dif®cult material.

Preface to the third edition By E. Prince

This is the third edition of International Tables for Crystallography Volume C. The purpose of this volume is to provide the mathematical, physical and chemical information needed for experimental studies in structural crystallography. It covers all aspects of experimental techniques, using all three principal radiation types, from the selection and mounting of crystals and production of radiation, through data collection and analysis, to the interpretation of results. As such, it is an essential source of information for all workers using crystallographic techniques in physics, chemistry, metallurgy, earth sciences and molecular biology. Volume C of International Tables for Crystallography is one of the many legacies to crystallographers of the late Professor A. J. C. Wilson, whose death on 1 July 1995 left the preparation of a revised and expanded second edition un®nished. When I was appointed as Professor Wilson's successor as Editor, I realised that although most of the material in the ®rst edition was new, some had been carried over from Volumes II,

III, and IV of the earlier series International Tables for X-ray Crystallography and had become outdated. Moreover, many of the topics covered were changing very rapidly, so needed to be brought up to date. In fact, by the time the second edition was published in 1999, more than half the chapters had been revised or updated and two completely new chapters, on re¯ectometry and neutron topography, had been included. The second edition of Volume C was also the ®rst volume of International Tables to be produced entirely electronically. The authors of the second edition were asked if they wished to submit revisions to their articles for this third edition in August 2001. All revisions were received within the following year. In total, 11 chapters have been revised, corrected or updated, and all known errors in the second edition have been corrected. I hope few new errors have been introduced. I thank all authors, especially those who have submitted revisions, and I particularly thank the Editorial staff in Chester for their continued dilligence.

xxxi

2 s:\ITFC\PREFACE.3d (Tables of Crystallography)

International Tables for Crystallography (2006). Vol. C, Chapter 1.1, pp. 2–5.

1.1. Summary of general formulae By E. Koch

In an ideal crystal structure, the arrangement of atoms is threedimensionally periodic. This periodicity is usually described in terms of point lattices, vector lattices, and translation groups [cf. IT A (1983, Section 8.1.3)].

a

1.1.1. General relations between direct and reciprocal lattices

The vectors a, b, c form a primitive crystallographic basis of the vector lattice L, if each translation vector t 2 L may be expressed as

V a b c a b c 2 a b cos a2 6 4 a b cos b2 a c cos b c cos

t ua vb wc

a b c 1

with u, v, w being integers. A primitive basis de®nes a primitive unit cell for a corresponding point lattice. Its volume V may be calculated as the mixed product (triple scalar product) of the three basis vectors:

abc1

2

cos

2

cos

cos2

cos2

VV 1:

2

cos

1:1:1:4

1:1:1:5

As all relations between direct and reciprocal lattices are symmetrical, one can calculate a; b; c from a ; b ; c : a 1:1:1:1

b

c a ; V

c

a b ; V

9 b c sin > > ; > > V > > > > > a c sin > > b ; > > V > > > > > a b sin > > ; c > = V cos cos cos > cos ;> > > > sin sin > > > > cos cos cos > > cos ;> > > > sin sin > > > > cos cos cos > ; cos :> sin sin

1:1:1:6

1:1:1:7

The unit-cell volumes V and V may also be obtained from: V abc sin sin sin abc sin sin sin

The lengths a , b and c of the reciprocal basis vectors and the angles b ^ c , c ^ a and a ^ b are given by:

abc sin sin sin ; 2

Copyright © 2006 International Union of Crystallography

b c ; V

a

Here a, b and c designate the lengths of the three basis vectors and b ^ c, c ^ a and a ^ b the angles between them. Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice L and a primitive reciprocal basis a , b , c : 9 bc > a or a b a c 0; a a 1; > > > V > > = ca or b a b c 0; b b 1; b 1:1:1:2 > V > > > > ab > or c a c b 0; c c 1: ; c V L fr jr ha kb lc and h; k; l integersg:

cos2

In addition, the following equation holds:

1=2

1:1:1:3

31=2 a c cos 7 b c cos 5 c2

2 cos cos cos 1=2 sin 2a b c sin 2 2 1=2

sin sin : 2 2

31=2 ac cos 7 bc cos 5 c2

2 cos cos cos sin 2abc sin 2 2 1=2 : sin sin 2 2

b

a , b , c de®ne a primitive unit cell in a corresponding reciprocal point lattice. Its volume V may be expressed by analogy with V [equation (1.1.1.1)]:

1.1.1.1. Primitive crystallographic bases

V abc a b c 2 a2 ab cos 6 4 ab cos b2 ac cos bc cos

9 ac sin ab sin > ; c ;> > > V V > > > > > cos cos

cos > > ; cos > = sin sin > cos cos cos > > ; cos > > sin sin > > > > > cos cos cos

> > ; cos : sin sin

bc sin ; V

3 s:\ITFC\CH-1-1.3d (Tables of Crystallography)THIS PAGE HAS COMPUTER MODERN \NU

1:1:1:8

1.1. SUMMARY OF GENERAL FORMULAE Table 1.1.1.1. Direct and reciprocal lattices described with respect to conventional basis systems Reciprocal lattice

Direct lattice ac ; bc ; cc Bravais letter

Centring vectors

ac ; bc ; cc Unit-cell volume Vc

Conditions for reciprocal-lattice vectors Unit-cell hac kbc lcc volume Vc

Bravais letter

A

1 2 bc

12 cc

2V

k l 2n

1 2V

A

B

1 2 ac

12 cc

2V

h l 2n

1 2V

B

C

1 2 ac

12 bc

2V

h k 2n

1 2V

C

12 bc 12 cc

2V

h k l 2n

1 2V

F

1 4V

I

1 3V

R

I

1 2 ac

F

1 2 ac 1 2 ac 1 2 bc

12 bc ; 12 cc ; 12 cc

4V

h k 2n; h l 2n; k l 2n

R

1 3 ac 2 3 ac

23 bc 23 cc , 13 bc 13 cc

3V

h k l 3n

As a direct lattice and its corresponding reciprocal lattice do not necessarily belong to the same type of Bravais lattices [IT A (1987, Section 8.2.4)], the Bravais letter of L is given in the last column of Table 1.1.1.1. Except for P lattices, a conventionally chosen basis for L coincides neither with a ; b ; c nor with ac ; bc ; cc . This third basis, however, is not used in crystallography. The designation of scattering vectors and the indexing of Bragg re¯ections usually refers to ac ; bc ; cc . If the differences with respect to the coef®cients of direct- and reciprocal-lattice vectors are disregarded, all other relations discussed in Part 1 are equally true for primitive bases and for conventional bases.

V a b c sin sin sin a b c sin sin sin a b c sin sin sin :

1:1:1:9

1.1.1.2. Non-primitive crystallographic bases For certain lattice types, it is usual in crystallography to refer to a `conventional' crystallographic basis ac ; bc ; cc instead of a primitive basis a; b; c. In that case, ac , bc ; and cc with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors t 2 L, t t1 ac t2 bc t3 cc ; with at least two of the coef®cients t1 , t2 , t3 being fractional. Such a conventional basis de®nes a conventional or centred unit cell for a corresponding point lattice, the volume Vc of which may be calculated by analogy with V by substituting ac ; bc ; cc for a; b; and c in (1.1.1.1). If m designates the number of centring lattice vectors t with 0 t1 ; t2 ; t3 < 1, Vc may be expressed as a multiple of the primitive unit-cell volume V: Vc mV :

1.1.2. Lattice vectors, point rows, and net planes The length t of a vector t ua vb wc is given by t2 u2 a2 v2 b2 w2 c2 2uvab cos 2uwac cos 2vwbc cos :

Accordingly, the length r of a reciprocal-lattice vector r ha kb lc may be calculated from

1:1:1:10

r 2 h2 a2 k2 b2 l 2 c2 2hka b cos

With the aid of equations (1.1.1.2) and (1.1.1.3), the reciprocal basis ac ; bc ; cc may be derived from ac ; bc ; cc . Again, each reciprocal-lattice vector

2hla c cos 2klb c cos :

is an integral linear combination of the reciprocal basis vectors, but in contrast to the use of a primitive basis only certain triplets h; k; l refer to reciprocal-lattice vectors. Equation (1.1.1.5) also relates Vc to Vc , the reciprocal cell volume referred to ac ; bc ; cc . From this it follows that 1 V : m

1:1:2:2

If the coef®cients u, v, w of a vector t 2 L are coprime, uvw symbolizes the direction parallel to t. In particular, uvw is used to designate a crystal edge, a zone axis, or a point row with that direction. The integer coef®cients h; k; l of a vector r 2 L are also the coordinates of a point of the corresponding reciprocal lattice and designate the Bragg re¯ection with scattering vector r . If h; k; l are coprime, the direction parallel to r is symbolized by hkl . Each vector r is perpendicular to a family of equidistant parallel nets within a corresponding direct point lattice. If the coef®cients h; k; l of r are coprime, the symbol hkl describes that family of nets. The distance dhkl between two neighbouring nets is given by

r hac kbc lcc 2 L

Vc

1:1:2:1

1:1:1:11

Table 1.1.1.1 contains detailed information on `centred lattices' described with respect to conventional basis systems. 3


1. CRYSTAL GEOMETRY AND SYMMETRY a dhkl r : 1:1:2:3 au bv cos cw cos h Parallel to such a family of nets, there may be a face or a b au cos bv cw cos cleavage plane of a crystal. k The net planes hkl obey the equation c au cos bv cos cw: hx ky lz n n integer: 1:1:2:4 l 1

Different values of n distinguish between the individual nets of the family; x; y; z are the coordinates of points on the net planes (not necessarily of lattice points). They are expressed in units a, b, and c, respectively. Similarly, each vector t 2 L with coprime coef®cients u; v; w is perpendicular to a family of equidistant parallel nets within a corresponding reciprocal point lattice. This family of nets may be symbolized uvw . The distance d uvw between two neighbouring nets can be calculated from d uvw t 1 :

1.1.3. Angles in direct and reciprocal space The angles between the normal of a crystal face and the basis vectors a; b; c are called the direction angles of that face. They may be calculated as angles between the corresponding reciprocal-lattice vector r and the basis vectors l r ^ a, r ^ b and r ^ c: 9 h k > cos l dhkl; cos dhkl; > = a b 1:1:3:1 > l > ; cos dhkl: c The three equations can be combined to give 9 h k l > = a:b:c : : cos l cos cos 1:1:3:2 or > ; h : k : l a cos l : b cos : c cos :

1:1:2:5

A layer line on a rotation pattern or a Weissenberg photograph with rotation axis uvw corresponds to one such net of the family uvw of the reciprocal lattice. The nets uvw obey the equation uh vk wl n

n integer:

1:1:2:6

Equations (1.1.2.6) and (1.1.2.4) are essentially the same, but may be interpreted differently. Again, n distinguishes between the individual nets out of the family uvw . h; k; l are the coordinates of the reciprocal-lattice points, expressed in units a , b , c ; respectively. A family of nets hkl and a point row with direction uvw out of the same point lattice are parallel if and only if the following equation is satis®ed: hu kv lw 0:

The ®rst formula gives the ratios between a, b, and c, if for any face of the crystal the indices hkl and the direction angles l, , and are known. Once the axial ratios are known, the indices of any other face can be obtained from its direction angles by using the second formula. Similarly, the angles between a direct-lattice vector t and the reciprocal basis vectors l t ^ a , t ^ b and t ^ c are given by 9 u v cos l d uvw; cos d uvw; > = a b 1:1:3:3 w > ; cos d uvw: c The angle between two direct-lattice vectors t1 and t2 or between two corresponding point rows u1 v1 w1 and u2 v2 w2 may be derived from the scalar product

1:1:2:7

This equation is called the `zone equation' because it must also hold if a face hkl of a crystal belongs to a zone uvw. Two (non-parallel) nets h1 k1 l1 and h2 k2 l2 intersect in a point row with direction uvw if the indices satisfy the condition k1 l1 l1 h1 h1 k1 : u:v:w : : 1:1:2:8 kl l h h k 2 2

2 2

2 2

The same condition must be satis®ed for a zone axis uvw de®ned by the crystal faces h1 k1 l1 and h2 k2 l2 . Three nets h1 k1 l1 , h2 k2 l2 , and h3 k3 l3 intersect in parallel rows, or three faces with these indices belong to one zone if h1 k1 l1 h2 k2 l2 0: 1:1:2:9 h k l 3 3 3

t1 t2 u1 u2 a2 v1 v2 b2 w1 w2 c2 u1 v2 u2 v1 ab cos u1 w2 u2 w1 ac cos v1 w2 v2 w1 bc cos 1:1:3:4 as cos

Two (non-parallel) point rows u1 v1 w1 and u2 v2 w2 in the direct lattice are parallel to a family of nets hkl if v 1 w 1 w 1 u 1 u1 v 1 : h:k:l : : 1:1:2:10 v w w u u v 2

2

2 2

t1 t2 : t1 t2

1:1:3:5

Analogously, the angle ' between two reciprocal-lattice vectors r1 and r2 or between two corresponding point rows h1 k1 l1 and h2 k2 l2 or between the normals of two corresponding crystal faces h1 k1 l1 and h2 k2 l2 may be calculated as

2 2

The same condition holds for a face hkl belonging to two zones u1 v1 w1 and u2 v2 w2 . Three point rows u1 v1 w1 , u2 v2 w2 , and u3 v3 w3 are parallel to a net hkl, or three zones of a crystal with these indices have a common face hkl if u1 v1 w1 u2 v2 w2 0: 1:1:2:11 u v w 3 3

1:1:2:12

cos '

r1 r2 r1 r2

1:1:3:6

with r1 r2 h1 h2 a2 k1 k2 b2 l1 l2 c2 h1 k2 h2 k1 a b cos h1 l2 h2 l1 a c cos

3

k1 l2 k2 l1 b c cos :

A net hkl is perpendicular to a point row uvw if 4


1:1:3:7

1.1. SUMMARY OF GENERAL FORMULAE Finally, the angle ! between a ®rst direction uvw of the direct lattice and a second direction hkl of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r . cos !

t r uh vk wl : tr tr

with

l h vij i i ; lj hj

h k wij i i : hj kj

If all angles between the face normals and also the indices for three of the faces are known, the indices of the fourth face may be calculated. Equation (1.1.4.1) cannot be used if two of the faces are parallel. From the de®nition of uij , vij , and wij , it follows that all fractions in (1.1.4.1) are rational:

1:1:3:8

1.1.4. The Miller formulae Consider four faces of a crystal that belong to the same zone in consecutive order: h1 k1 l1 , h2 k2 l2 , h3 k3 l3 , and h4 k4 l4 . The angles between the ith and the jth face normals are designated 'ij . Then the Miller formulae relate the indices of these faces to the angles 'ij : sin '12 sin '43 u12 u43 v12 v43 w12 w43 sin '13 sin '42 u13 u42 v13 v42 w13 w42

k l uij i i ; kj lj

sin '12 sin '43 p sin '13 sin '42 q

with p; q integers:

Therefore, (1.1.4.1) may be rearranged to p cot '12

q cot '13 p

q cot '14 :

1:1:4:2

This equation allows the determination of one angle if two of the angles and the indices of all four faces are known.

1:1:4:1

5



1.2. Application to the crystal systems By E. Koch

a b2 v c au cw cos au cos cw; h k l

Information on the description and classi®cation of Bravais lattices, their assignment to crystal systems, the choice of basis vectors for reduced or conventional basis systems, and on basis transformations is given in IT A (1983, Parts 5 and 9). In the following, for each crystal system, the metrical conditions for conventionally chosen basis systems and the possible Bravais types of lattices are listed. As some of the general formulae from Chapter 1.1 become simpler when not applied to a lattice with general (triclinic) metric, these simpli®ed formulae are tabulated for all crystal systems (except triclinic). Except for triclinic, monoclinic, and orthorhombic symmetry, tables are given that relate pairs h, k or triplets h; k; l of indices to certain sums s of products of these indices needed in equation (1.1.2.2). Such tables may be useful, for example, for indexing powder diffraction patterns.

2 a2 6 V a b c 4 a b cos 0

2 a2 6 V a b c 4 0 a c cos

0 2

b 0

1:1:1:3a

31=2 a c cos 7 0 5 2 c

a b c sin ; 1 1 1 a ; b ; c a sin b c sin ; 90 ; 180 ;

1:1:1:7a

t2 u2 a2 v2 b2 w2 c2 2uwac cos ;

1:1:2:1a

r2 h2 a2 k2 b2 l2 c2 2hla c cos ;

1:1:2:2a

31=2 7 5 2 c 0 0

1:1:1:4b

9 1 1 1 = ; b ; c ; a sin b sin c ;

; 90 ; 180

1:1:1:7b

t2 u2 a2 v2 b2 w2 c2 2uvab cos ;

1:1:2:1b

r2 h2 a2 k2 b2 l 2 c2 2hka b cos ;

1:1:2:2b

a b c2 w au bv cos au cos bv ; h k l

1:1:2:12b

u1 v2 u2 v1 ab cos ;

1:1:3:4b

r1 r2 h1 h2 a2 k1 k2 b2 l1 l2 c2

h1 k2 h2 k1 a b cos :

1:1:3:7b

1.2.3. Orthorhombic crystal system Metrical conditions: Bravais lattice types: Symmetry of lattice points: 6

Copyright © 2006 International Union of Crystallography 7 s:\ITFC\CH-1-2.3d (Tables of Crystallography)

0

1:1:1:3b

t1 t2 u1 u2 a2 v1 v2 b2 w1 w2 c2

1:1:1:4a 9 ;=

a b cos b2

a

1:1:1:1a 9 1 1 1 ; b ; c ;= a sin b c sin ; 90 ; 180 ;

31=2 0 0 5 abc sin ; c2

a b c sin ;

31=2 ac cos 7 0 5 abc sin ; 2 c

a

ab cos b2 0

9 1 1 1 ; b ; c ; = a sin b sin c ; 90 ; 180

;

a; b; c; arbitrary; 90 mP; mC or mA or mI : 2=m .

b2 0

a; b; c; arbitrary; 90 mP; mB or mA or mI : : 2=m

a

mP; mS 2=m

0

1:1:3:7a

1:1:1:1b

1.2.2.1. Setting with ùnique axis b'

Bravais lattice types: Symmetry of lattice points: Simpli®ed formulae: 2 a2 6 V abc 4 0 ac cos

r1 r2 h1 h2 a2 k1 k2 b2 l1 l2 c2 h1 l2 h2 l1 a c cos :

Bravais lattice types: Symmetry of lattice points: Simpli®ed formulae: 2 a2 4 V abc ab cos 0

1.2.2. Monoclinic crystal system

Metrical conditions:

1:1:3:4a


a; b; c; ; ; arbitrary aP 1

Bravais lattice types: Symmetry of lattice points

t1 t2 u1 u2 a2 v1 v2 b2 w1 w2 c2 u1 w2 u2 w1 ac cos ;

1.2.2.2. Setting with ùnique axis c'

1.2.1. Triclinic crystal system No metrical conditions: Bravais lattice type: Symmetry of lattice points:

1:1:2:12a

a; b; c arbitrary; 90 oP; oS oC; oA, oI; oF mmm

1.2. APPLICATION TO THE CRYSTAL SYSTEMS Simpli®ed formulae:

2 2 a V abc 4 0 0

1 a ; a

0 b2 0

1 b ; b

1 c ; c

2 2 a 6 V a b c 4 0 0

1

Table 1.2.4.1. Assignment of integers s 100 to pairs h, k with s h2 k 2

31=2 0 0 5 abc; c2

1 ; a

b

1 ; b

c

1:1:2:1c

26 29

31=2 7 5 2 c 0 0

0 1

1 ; c

1:1:1:7c

1:1:1:3c

a b c a b c ; a

s 1 2 4 5 8 9 10 13 16 17 18 20 25

90 ;

0 b2

1

Each pair h; k represents all eight pairs which result from permutation and different sign combinations.

1:1:1:1c

1:1:1:4c 90 ;

t2 u2 a2 v2 b2 w2 c2 ; r2 h2 a2 k2 b2 l 2 w2 ;

1:1:2:2c

a2 u b2 v c 2 w ; h k l

1:1:2:12c

h 1 1 2 2 2 3 3 3 4 4 3 4 5 4 5 5

k 0 1 0 1 2 0 1 2 0 1 3 2 0 3 1 2

s 32 34 36 37 40 41 45 49 50 52 53 58 61 64 65

h 4 5 6 6 6 5 6 7 7 5 6 7 7 6 8 8 7

k 4 3 0 1 2 4 3 0 1 5 4 2 3 5 0 1 4

s 68 72 73 74 80 81 82 85 89 90 97 98 100

h 8 6 8 7 8 9 9 9 7 8 9 9 7 10 8

u v c2 w 2 ; h k al

k 2 6 3 5 4 0 1 2 6 5 3 4 7 0 6

1:1:2:12d

t1 t2 u1 u2 a2 v1 v2 b2 w1 w2 c2 ;

1:1:3:4c

t1 t2 u1 u2 v1 v2 a2 w1 w2 c2 ;

1:1:3:4d

r1 r2 h1 h2 a2 k1 k2 b2 l1 l2 c2 :

1:1:3:7c

r1 r2 h1 h2 k1 k2 a2 l1 l2 c2 :

1:1:3:7d

1.2.5. Trigonal and hexagonal crystal system

1.2.4. Tetragonal crystal system Metrical conditions: Bravais lattice types: Symmetry of lattice points: Simpli®ed formulae: 2 2 a 0 V abc 4 0 a2 0 0 1 a b ; a

1 c ; c

1:1:1:1d

90 ;

1:1:1:3d

0 a2 0

c

1 ; c

Bravais lattice types: Symmetry of lattice points: Simpli®ed formulae: 2 2 1 2 a 2a 1 2 4 V abc 2 a a2 0 0

t2 u2 v2 a2 w2 c2 ; r2 h2 k2 a2 l 2 c2 sa2 l 2 c2

1:1:1:7d

p 1 a b 23 3 ; a

1:1:2:1d 1:1:2:2d

c

t2 u2 v2

2

sh k :

1 ; c

90 ;

uva2 w2 c2 ;

r2 h2 k2 hka2 l 2 c2 sa2 l 2 c2

For each value of s 100, all corresponding pairs h; k are listed in Table 1.2.4.1.

with 7

8 s:\ITFC\CH-1-2.3d (Tables of Crystallography)

1:1:1:3e

31=2 2 2 1 2 a 0 2a 7 6 V a b c 4 12 a2 a2 0 5 0 0 c2 p p 12 3 a2 c 23 3 a 2 c 1 ;

with 2

31=2 0 p 0 5 12 3 a2 c; 1:1:1:1e c2

9 p 1 1 3 ; c ;= a c 90 ; 60 ; ;

1:1:1:4d

90 ;

a b; c arbitrary 90 ; 120 hP; hR hR 6=mmm hP; 3m

a b 23

31=2 0 7 0 5 c2

a2 c a 2 c 1 ; 1 ; a


31=2 0 0 5 a2 c; c2

2 2 a 6 V a b c 4 0 0

ab

1.2.5.1. Description referred to hexagonal axes

a b; c arbitrary; 90 tP; tI 4=mmm

1:1:1:4e

120 ; 1:1:1:7e 1:1:2:1e 1:1:2:2e

1. CRYSTAL GEOMETRY AND SYMMETRY Table 1.2.5.1. Assignment of integers s 100 to pairs h, k with s h2 k2 hk

Table 1.2.5.2. Asssignment of integers s1 50 to triplets h, k, l with s1 h2 k2 l 2 and to integers s2 hk hl kl

Each pair h, k represents in addition the pairs k; h k and h k; h; the permutations of these three, and the six corresponding centrosymmetrical pairs.

Each triplet h; k; l represents all twelve triplets resulting from permutation and/or simultaneous change of all signs.

s 1 3 4 7 9 12 13 16 19 21 25 27 28

h 1 1 2 2 3 2 3 4 3 4 5 3 4

k 0 1 0 1 0 2 1 0 2 1 0 3 2

s 31 36 37 39 43 48 49 52 57 61 63 64

h 5 6 4 5 6 4 7 5 6 7 5 6 8

2

2

k 1 0 3 2 1 4 0 3 2 1 4 3 0

s

h k

67 73 75 76 79 81 84 91 93 97 100

7 8 5 6 7 9 8 9 6 7 8 10

2 1 5 4 3 0 2 1 5 4 3 0

s1

s2

h

k

l

s1

s2

h

k

l

s1

s2

h

k

l

1 2

0 1 1 1 3 0 2 2 3 1 5 4 4 4 0

1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 2 2 3 3 3 3 3 2 2 3 3 3 3 3 3 4 3 4 3 4 3 3 4 4 4 3 3 3 3 4 4 4 4 4 4 3 3 3

0 1 1 1 1 0 1 1 1 1 1 2 2 2 0 2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 0 2 1 2 1 2 3 1 1 1 3 3 3 3 2 2 2 2 2 2 3 3 3

0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 2 2 0 0 1 1 1 1 0 2 0 2 0 2 0 1 1 1 0 1 1 1 0 0 1 1 1 1 2 2 2

24

12 4 20 12 0 12 13 11 5 5

4 4 4 4 5 4 4 4 5 5 4 4 5 3 5 5 3 4 5 4 4 5 4 5 5 5 5 4 4 5 4 5 4 5 4 5 4 4 5 4 5 5 5 5 4 6 4 4 6 6

2 2 2 3 0 3 3 3 1 1 3 3 1 3 1 1 3 3 2 3 3 2 3 2 2 2 2 4 4 2 4 2 4 2 4 3 3 3 3 3 3 3 3 3 4 0 4 4 1 1

2 2 2 0 0 0 1 1 0 0 1 1 1 3 1 1 3 2 0 2 2 0 2 1 1 1 1 0 0 2 1 2 1 2 1 0 3 3 0 3 1 1 1 1 2 0 2 2 0 0

38

19 11

5 6 5 6 5 6 5 6 6 5 6 4 6 4 6 6 5 4 5 5 5 5 5 5 5 6 6 6 5 6 5 5 6 5 6 6 6 6 4 4 6 6 7 6 6 5 5 5 7 5 7 5 5

3 1 3 1 3 1 3 2 2 4 2 4 2 4 2 2 4 4 4 4 4 4 3 3 3 2 2 2 4 3 4 4 3 4 3 3 3 3 4 4 3 3 0 3 3 5 4 4 1 4 1 5 4

2 1 2 1 2 1 2 0 0 0 1 3 1 3 1 1 0 3 1 1 1 1 3 3 3 2 2 2 2 0 2 2 0 2 1 1 1 1 4 4 2 2 0 2 2 0 3 3 0 3 0 0 3

3 4 5 6 8 9

s h k hk: 10

For each value of s 100, all corresponding pairs h, k are listed in Table 1.2.5.1. 2u v 2v u c2 w 2 ; 2h 2k al t1 t2 u1 u2 v1 v2 r1

r2

1 2 u1 v2

h1 h2 k1 k2

1 2 h1 k 2

1:1:2:12e

1 2 2 u2 v1 a

11 12

2

w1 w2 c ; 1:1:3:4e

1 2 2 h2 k1 a

13 14

2

l1 l2 c : 1:1:3:7e

1.2.5.2. Description referred to rhombohedral axes Metrical conditions: Bravais lattice type: Symmetry of lattice points: Simpli®ed formulae: 2 a2 6 2 V abc 4 a cos a2 cos

a2 cos a2 a2 cos

31=2 a2 cos 7 a2 cos 5 a2

a3 1 3 cos2 2 cos3 1=2 1=2 3 3 3 2a sin 2 sin ; 2 9

1 > cos cos ;> = 2 2 2 2 cos =2 > 1 > ; ; a b c a sin sin cos

V a b c 2 a2 6 2 4 a cos a2 cos

16 17

a b c; hR 3m

a2 cos a2 a cos 2

18

19

1:1:1:1f

20 21

1:1:1:3f 22

31=2 a2 cos 7 a2 cos 5 a2

a3 1 3 cos2 2 cos3 1=2 1=2 2a3 sin 32 sin3 ; 2

1:1:1:4f 8


8 3 3 5 1 7 4 12 6 6 7 5 1 11 0 8 4 4 16 9 7 1 9 9 3 15 8 8 10 6 2 14 9 3 21

25 26

27

29

30

32 33

19 9 1 11 27 14 10 2 10 26 13 7 3 17 16 16 16 4 8 24

34

35

36

37

15 9 15 33 17 13 7 23 16 0 32 6 6

1

40 41

13 31 12 12 20 16 8 4 20

42

43 44 45

46

48 49

50

40 21 19 11 29 21 9 39 20 4 28 22 18 2 18 38 21 15 9 27 16 48 24 12 0 36 25 23 17 7 7 25 47

9

1 > ;> cos cos cos = 2 2 2 2 cos =2 > 1 > ; abc ; a sin sin

1:1:1:7f

t2 u2 v2 w2 a2 2uv uw vwa2 cos ;

1:1:2:1f

1.2. APPLICATION TO THE CRYSTAL SYSTEMS r

2

2

2

2

Simpli®ed formulae:

2

h k l a 2hk hl kla2 cos s1 a2 2s2 a2 cos

2 2 a V abc 4 0 0

1:1:2:2f

with s1 h2 k 2 l 2

and

s2 hk hl kl:

u vw v uw w uv cos cos cos ; h h k k l l 1:1:2:12f

1:1:1:4g

t1 t2 u1 u2 v1 v2 w1 w2 a2 u1 v2 u2 v1 u1 w2 u2 w1

r1

r2

abc

h1 h2 k1 k2 l1 l2 a h1 k2 h2 k1 h1 l2 h2 l1

1 ; a

90 ;

1:1:1:7g

t2 u2 v2 w2 a2 ;

1:1:2:1g

r2 h2 k2 l 2 a2 sa2

1:1:2:2g

1:1:3:4f

2

with

k1 l2 k2 l1 a2 cos :

s h2 k2 l 2 :

1:1:3:7f

For each value of s 100, all corresponding triplets h; k; l are listed in Table 1.2.6.1. u v w ; 1:1:2:12g h k l

1.2.6. Cubic crystal system a b c; 90 cP; cI; cF m3m

Metrical conditions: Bravais lattice types: Symmetry of lattice points:

1:1:1:1g

1 1:1:1:3g a b c ; 90 ; a 31=2 2 2 a 0 0 V a b c 4 0 a2 0 5 a3 a 3 ; 0 0 a2

For each value of s1 50, all corresponding values of s2 and all triplets h, k, l are listed in Table 1.2.5.2.

v1 w2 v2 w1 a2 cos ;

31=2 0 0 5 a3 ; a2

0 a2 0

t1 t2 u1 u2 v1 v2 w1 w2 a2 ;

1:1:3:4g

r1 r2 h1 h2 k1 k2 l1 l2 a2 :

1:1:3:7g

Table 1.2.6.1. Assignment of integers s 100 to triplets h, k, l with s h2 k2 l 2 Each triplet represents all 48 triplets resulting from permutations and sign combinations. s

hkl

1 2 3 4 5 6 8 9

1 1 1 2 2 2 2 3 2 3 3 2 3 3 4 4 3 4 3 3 4 4 3 4

10 11 12 13 14 16 17 18 19 20 21 22 24

0 1 1 0 1 1 2 0 2 1 1 2 2 2 0 1 2 1 3 3 2 2 3 2

0 0 1 0 0 1 0 0 1 0 1 2 0 1 0 0 2 1 0 1 0 1 2 2

s 25 26 27 29 30 32 33 34 35 36 37 38 40 41

hkl 5 4 5 4 5 3 5 4 5 4 5 4 5 4 5 6 4 6 6 5 6 6 5 4

0 3 1 3 1 3 2 3 2 4 2 4 3 3 3 0 4 1 1 3 2 2 4 4

0 0 0 1 1 3 0 2 1 0 2 1 0 3 1 0 2 0 1 2 0 1 0 3

s 42 43 44 45 46 48 49 50 51 52 53 54 56 57 58

hkl 5 5 6 6 5 6 4 7 6 7 5 5 7 5 6 7 6 7 6 5 6 7 5 7

4 3 2 3 4 3 4 0 3 1 5 4 1 5 4 2 4 2 3 5 4 2 4 3

s 59

1 3 2 0 2 1 4 0 2 0 0 3 1 1 0 0 1 1 3 2 2 2 4 0

61 62 64 65 66 67 68 69 70 72 73

9


hkl 7 5 6 6 7 6 8 8 7 6 8 7 5 7 8 6 8 7 6 8 6 8 6

3 5 5 4 3 5 0 1 4 5 1 4 5 3 2 4 2 4 5 2 6 3 6

1 3 0 3 2 1 0 0 0 2 1 1 4 3 0 4 1 2 3 2 0 0 1

s 74 75 76 77 78 80 81

82 83 84 85 86

hkl 8 7 7 7 5 6 8 6 7 8 9 8 7 6 9 8 9 7 8 9 7 9 7 6

3 5 4 5 5 6 3 5 5 4 0 4 4 6 1 3 1 5 4 2 6 2 6 5

1 0 3 1 5 2 2 4 2 0 0 1 4 3 0 3 1 3 2 0 0 1 1 5

s 88 89

90 91 93 94 96 97 98 99 100

hkl 6 9 8 8 7 9 8 7 9 8 9 7 8 9 6 9 8 7 9 7 7 10 8

6 2 5 4 6 3 5 5 3 5 3 6 4 4 6 4 5 7 3 7 5 0 6

4 2 0 3 2 0 1 4 1 2 2 3 4 0 5 1 3 0 3 1 5 0 0


1. CRYSTAL GEOMETRY AND SYMMETRY

1.3. Twinning By E. Koch

contains an evenfold rotation or screw-rotation axis, an inversion twin cannot be distinguished from a reflection twin with twin plane perpendicular to that axis. (b) If the crystal structure contains a mirror or a glide plane, an inversion twin cannot be distinguished from a rotation twin with a twofold twin axis perpendicular to that plane. c If for a centrosymmetrical crystal structure the normal of a twin plane runs parallel to a lattice vector or a twin axis runs perpendicular to a net plane, the twin may be described equally well as a reflection twin or as a rotation twin. The twin components are grown together in a surface called composition surface, twin interface or twin boundary. In most cases, the composition surfaces are low-energy surfaces with good structural fit. For a reflection twin, it is usually a plane parallel to the twin plane. The composition surface of a rotation twin may either be a plane parallel to the twin axis or be a nonplanar surface with irregular shape. If more than two components are twinned according to the same law, the twin is called a repeated twin or a multiple twin. If all the twin boundaries are parallel planes, it is a polysynthetic twin, otherwise it is called a cyclic twin. If the twin components are related to each other by more than one twin law, the shape and the mutual arrangement of the twin domains may be very irregular. With respect to the formation process, one may distinguish between growth twins, transformation twins, and mechanical (deformation, glide) twins. Transformation twins result from phase transitions, e.g. of ferroelectric or ferromagnetic crystals. The corresponding twin domains are usually small and the number of such domains is high. Mechanical twinning is due to mechanical stress and may often be described in terms of shear of the crystal structure. This includes ferroelasticity. Twins are observable by, for example, macroscopic or microscopic observation of re-entrant angles between crystal faces, by etching, by means of different extinction positions for the twin components between cross polarizers of a polarization microscope, by different rotation angles of the plane of polarization of a beam of plane-polarized light passing through the components of a twin showing optical activity, by a splitting of part of the X-ray diffraction spots (except for twins by merohedry), by means of domain contrast or boundary contrast in an X-ray topogram, or by investigation with a transmission electron microscope. The phenomenon of twinning has frequently been described and discussed in the literature and it is impossible, therefore, to give a complete list of references. Further details may be learned, e.g. from a review article by Cahn (1954) or from appropriate textbooks. A comprehensive survey of X-ray topography of twinned crystals is given by Klapper (1987). The following papers are related to twinning by merohedry or pseudo-merohedry: Catti & Ferraris (1976), Grimmer (1984, 1989a,b), Grimmer & Warrington (1985), Donnay & Donnay (1974), Le Page, Donnay & Donnay (1984), Hahn (1981, 1984), Klapper, Hahn & Chung (1987), Flack (1987).

1.3.1. General remarks A twin consists of two or more single crystals of the same species but in different orientation, its twin components. They are intergrown in such a way that at least some of their lattice directions are parallel. The twin law describes the geometrical relation between the twin components. It specifies a symmetry operation, the twin operation, that brings one of the twin components into parallel orientation with the other. The corresponding symmetry element is called the twin element. There are several kinds of twin laws: (1) Reflection twins. Two twin components are related by reflection through a net plane hkl, the twin plane. All lattice vectors parallel to hkl, i.e. a complete lattice plane, coincide for both twin components, and their crystal faces hkl [and h k l] are parallel. As a consequence, their corresponding zone axes parallel to hkl also coincide. A twin plane cannot run parallel to a mirror or glide plane of the crystal structure, i.e. it cannot run parallel to a mirror plane of the point group of the crystal, because in that case both twin components would have the same orientation. It must be noted that the vector normal to a twin plane need not have rational indices nor be parallel to a lattice vector. (2) Rotation twins. The twin components can be brought into parallel orientation by a rotation about an axis, the twin axis. Two cases may be distinguished: (i) Most frequently, the twin axis runs parallel to a lattice vector with components u, v, w. Then the lattice row uvw coincides for all twin components, i.e. they have the common zone axis uvw. Usually, the twin axis is a twofold axis, and all corresponding crystal faces of the two twin components belonging to that zone are parallel. Less frequently, a three-, four-, or sixfold rotation occurs as the twin operation. A twin axis cannot run parallel to a (screw-) rotation axis of the crystal structure which induces the same rotation angle, i.e. it cannot be parallel to such a rotation axis of the point group of the crystal. For example, a twofold twin axis cannot be parallel to a twofold, fourfold, or sixfold axis, but it may run parallel to a threefold axis; a twin axis with rotation angle 60, 90, or 120 , however, may be parallel to a twofold axis. (ii) In some cases, the direction of the twin axis is not rational, but the twofold twin axis runs perpendicular to a lattice row (zone axis) uvw and parallel to a net plane (crystal face) hkl that belongs to that zone. Then the lattices of the twin components coincide only in one lattice row parallel to uvw, and uvw is the common zone axis of both twin components. The crystal faces hkl and h k l are parallel for both components, but the other faces of the zone uvw are not. Neither in case (i) nor in case (ii) does the plane perpendicular to the twin axis need to be a lattice plane. Therefore, in general, it cannot be described by Miller indices. (3) Inversion twins. The twin components are related by inversion through a centre of symmetry, the twin centre. Only noncentrosymmetrical crystals can form such twins. As all corresponding lattice vectors of the two twin components are antiparallel, their entire vector lattices coincide. As a consequence, all corresponding zone axes and crystal faces of the twin components are parallel. In many cases, there does not exist a unique twin law, but a twin may be described equally well by more than one twin law. (a) If the crystal structure of the twin components

1.3.2. Twin lattices For reflection and rotation twins described in the last section, a special situation arises whenever there exists a lattice vector perpendicular to the twin plane or a lattice plane perpendicular to 10

Copyright © 2006 International Union of Crystallography 11 s:\ITFC\CH-1-3.3d (Tables of Crystallography)

1.3. TWINNING

Lattice plane hkl

Lattice row uvw

Perpendicularity condition

For every twin lattice, its twin index i can be calculated from the Miller indices of the net plane hkl and the coprime coefficients u; v; w of the lattice vector t perpendicular to hkl. Referred to a primitive lattice basis, i is simply related to the modulus of the scalar product j of the two vectors r ha kb lc and t ua vb wc:

±

±

±

j r t hu kv lw;

Monoclinic (unique axis b)

(010)

[010]

±

Monoclinic (unique axis c)

(001)

[001]

±

Orthorhombic

(100) (010) (001)

[100] [010] [001]

± ± ±

Hexagonal= trigonal

hk0

uv0

(001)

[001]

u 2h k; v h 2k ±

Table 1.3.2.1. Lattice planes and rows that are perpendicular to each other independently of the metrical parameters

Basis system Triclinic

Rhombohedral

h; k; h k u; v; u v (111) [111]

i

hk0 (001)

uv0 [001]

u h; v k ±

Cubic

hkl

uvw

u h; v k; w l

n integer:

1.3.2.1. Examples (1) Cubic P lattice: [111] is perpendicular to (111). j hu kv lw 3 odd i j jj 3.

a rational twofold twin axis. Such a situation occurs systematically for all reflection and rotation twins with cubic symmetry and for certain twins with non-cubic symmetry (cf. Table 1.3.2.1). In addition, such a perpendicularity may occur occasionally if equation (1.1.2.12) is satisfied. In the case of a noncentrosymmetric crystal structure, different twins result from a twin axis uvw with a perpendicular lattice plane hkl, or from a twin plane hkl with a perpendicular lattice row uvw: the reflection twin consists of two enantiomorphous twin components whereas the rotation twin is built up from two crystals with the same handedness (cf., for example, Brazil twins and DauphineÂ twins of quartz). With respect to the first twin component, the lattice of the second component has the same orientation in both cases. For a centrosymmetrical crystal structure, both twin laws give rise to the same twin. Whenever a twin plane or twin axis is perpendicular to a lattice vector or a net plane, respectively, the vector lattices of the twin components have a three-dimensional subset in common. This sublattice [derivative lattice, cf. IT A (1983, Chapter 13.2)] is called the twin lattice. It corresponds uniquely to the intersection group of the two translation groups referring to the twin components. The respective subgroup index i is called the twin index. It is equal to the ratio of the volumes of the primitive unit cells for the twin lattice and the crystal structure. If one subdivides the crystal lattice into nets parallel to the twin plane or perpendicular to the twin axis, each ith of these nets belongs to the common twin lattice of the two twin components (cf. Fig. 1.3.2.1). Important examples are cubic twins with [111] as twofold twin axis or (111) as twin plane and rhombohedral twins with [001] as twin axis or (001) as twin plane (hexagonal description). In all these cases, the twin index i equals 3.

(a)

(b) Fig. 1.3.2.1. a Projection of the lattices of the twin components of a monoclinic twinned crystal (unique axis c, 93 ) with twin index 3. The twin may be interpreted either as a rotation twin with twin axis [210] or as a reflection twin with twin plane (110). b Projection of the corresponding reciprocal lattices.

11 12 s:\ITFC\CH-1-3.3d (Tables of Crystallography)

for j 2n 1 for j 2n

The same procedure ± but with modified coefficients ± may be applied to a centred lattice described with respect to a conventionally chosen basis: The coprime Miller indices h, k, l that characterize the net plane have to be replaced by larger noncoprime indices h0 , k0 , l0 , if h, k, l do not refer to a (non-extinct) point of the reciprocal lattice. The integer coefficients u, v, w specifying the lattice vector perpendicular to hkl have to be replaced by smaller non-integer coefficients u0 , v0 , w0 , if the centred lattice contains such a vector in the direction uvw.

u h; v k ±

Tetragonal

j jj j jj=2

1. CRYSTAL GEOMETRY AND SYMMETRY p lattice in hexagonal description with 3a: [310] is perpendicular (4) Rhombohedral p 2 is perpendicular to 111. c 12 3a: 11 2 has to be replaced by 1 1 2. Because of the R centring, 11 333 (i) P lattice (cf. Fig. 1.3.2.2): refers to an èxtinct reflection' of an R lattice, the As 111 j hu kv lw 4 even triplet 111 has to be replaced by 333. i j jj=2 2: j h0 u0 k0 v0 l0 w0 4 even (ii) C lattice (cf. also Fig. 1.3.2.2): i j jj=2 2. Because of the C centring, [310] has to be replaced by 32 12 0. j hu0 kv0 lw0 2 even 1.3.3. Implication of twinning in reciprocal space i j jj=2 1: As shown above, the direct lattices of the components of any twin coincide in at least one row. The same is true for the corresponding reciprocal lattices. They coincide in all rows perpendicular to parallel net planes of the direct lattices. For a reflection twin with twin plane hkl; the reciprocal lattices of the twin components have only the lattice points with coefficients nh, nk, nl in common. For a rotation twin with twofold twin axis uvw, the reciprocal lattices of the twin components coincide in all points of the plane perpendicular to uvw, i:e: in all points with coefficients h, k, l that fulfil the condition hu kv lw 0. For a rotation twin with irrational twin axis parallel to a net plane hkl, only reciprocal-lattice points with coefficients nh, nk, nl are common to both twin components. As the entire direct lattices of the two twin components coincide for an inversion twin, the same must be true for their reciprocal lattices. For a reflection or rotation twin with a twin lattice of index i, the corresponding reciprocal lattices, too, have a sublattice with index i in common (cf : Fig. 1.3.2.1b). In analogy to direct Fig. 1.3.2.2. Projection of the lattices ofpthe twin components of an orthorhombic twinned crystal oP; b 3a with twin index 2. The space, the twin lattice in reciprocal space consists of each ith twin may be interpreted either as a rotation twin with twin axis [310] lattice plane parallel to the twin plane or perpendicular to the or as a reflection twin with twin plane (110). The figure shows, in twin axis. If the twin index equals 1, the entire reciprocal lattices addition, that twin index 1 results if the oP lattice is replaced by an oC of the twin components coincide. lattice in this example (twinning by pseudomerohedry). If for a reflection twin there exists only a lattice row uvw that is almost (but not exactly) perpendicular to the twin plane hkl, (3) Orthorhombic C lattice with b 2a: [210] is perpendicular then the lattices of the two twin components nearly coincide in a to (120) (cf. Fig. 1.3.2.3). As (120) refers to an èxtinct reflection' of a C lattice, the three-dimensional subset of lattice points. The corresponding misfit is described by the quantity !, the twin obliquity. It is the triplet 240 has to be used in the calculation. angle between the lattice row uvw and the direction perpendi0 0 0 j h u k v l w 8 even cular to the twin plane hkl. In an analogous way, the twin i j jj=2 4. obliquity ! is defined for a rotation twin. If hkl is a net plane almost (but not exactly) perpendicular to the twin axis uvw, then ! is the angle between uvw and the direction perpendicular to hkl.

(2) Orthorhombic lattice with b to (110).

1.3.4. Twinning by merohedry A twin is called a twin by merohedry if its twin operation belongs to the point group of its vector lattice, i.e. to the corresponding holohedry. As each lattice is centrosymmetric, an inversion twin is necessarily a twin by merohedry. Only crystals from merohedral (i.e. non-holohedral) point groups may form twins by merohedry; 159 out of the 230 types of space groups belong to merohedral point groups. For a twin by merohedry, the vector lattices of all twin components coincide in direct and in reciprocal space. The twin index is 1. The maximal number of differently oriented twin components equals the subgroup index m of the point group of the crystal with respect to its holohedry. Table 1.3.4.1 displays all possibilities for twinning by merohedry. For each holohedral point group (column 1), the types of Bravais lattices (column 2) and the corresponding merohedral point groups (column 3) are listed. Column 4 gives the subgroup index m of a merohedral point group in its

Fig. 1.3.2.3. Projection of the lattices of the twin components of an orthorhombic twinned crystal oC; b 2a with twin index 4. The twin may be interpreted either as a rotation twin with twin axis [210] or as a reflection twin with twin plane (120).


1.3. TWINNING Table 1.3.4.1. Possible twin operations for twins by merohedry

Table 1.3.4.2. Simulated Laue classes, extinction symbols, simulated `possible space groups', and possible true space groups for crystals twinned by merohedry (type 2)

m is the index of the point group in the corresponding holohedry; point groups allowing twins of type 2 are marked by an asterisk.

Holohedry

Point group

Bravais lattice

m

1

aP

1

2

1

2=m

mP; mS

2 m

2 2

1 1

mmm

oP; oS; oI; oF

222 mm2

2 2

1 1

4=mmm

tP; tI

4 4 4=m 422 4mm 42m= 4m2

4 4 2 2 2 2

:m:; :2: 1; :m:; :2: 1; :m: 1 1 1

3 3 32 3m

4 2 2 2

:m; :2 1; :m 1 1

3

8

3 321=312 3m1=31m 3m1=31m 6 6 6=m 622 6mm 62m= 6m2

4 4 4 2 4 4 2 2 2 2

:m:; :2:; m::; 1; ::m; 2::; ::2 :m:; m::; ::m m::; ::2=:2: 1; m::; ::m=:m: 1; m:: :m:; :2: 1; :m:; ::m 1; :m: 1 1 1

4 2 2 2

::m; ::2 1; ::m 1 1

3m

6=mmm

hR

hP

m3m

cP; cI; cF

23 m3 432 43m

Twinned crystal

Possible twin operations

Twin extinction symbol

4=mmm

P - --

Simulated `possible space groups'

Possible true space groups P4=m P4; P 4;

I41 - I41 =a- -

P422; P4mm; P 42m, P 4m2; P4=mmm P42 22 P41 22; P43 22 P4=nmm ± I422; I4mm; I 42m; I 4m2;I4=mmm I41 22 ±

I41 I41 =a

3m1

P - -P31 - -

P321; P3m1, P 3m1 P31 21; P32 21

P3; P 3 P31 ; P32

31m

P - -P31 - -

P312; P31m, P 31m P31 12; P32 12

P3; P 3 P31 ; P32

3m

R--

R32; R3m; R3m

R3; R3

6=m

P - -P62 - -

P6=m P6; P 6; P62 ; P64

P3; P 3 P31 ; P32

6=mmm

P - --

P622; P6mm; P 6m2, P 62m; P6=mmm

P63 - P62 - -

P63 22 P62 22; P64 22

P61 - P--c

P61 22; P65 22 P63 mc; P 62c; P63 =mmc P63 cm; P 6c2; P63 =mcm

P321, P3; P 3; P312; P3m1, P31m; P 3m1; P 31m; P6; P 6; P6=m P63 ; P63 =m P31 ; P32 , P31 21; P32 21; P31 12; P32 12; P62 ; P64 P61 ; P65 P31c; P31c

P42 - P41 - Pn - P42 =n - I -- -

P-c m3m

holohedry. Column 5 shows m 1 possible twin operations referring to the different twin components. These twin operations are not uniquely defined (except for point group 1), but may be chosen arbitrarily from the corresponding right coset of the crystal point group in its holohedry. It is always possible, however, to choose an inversion, a reflection, or a twofold rotation as twin operation. A twin that is not a twin by merohedry as defined above but, because of metrical specialization, has a twin lattice with twin index 1 is called a twin by pseudo-merohedry. Two kinds of twins by merohedry may be distinguished. Type 1: The twin can be described as an inversion twin. Then, only two twin components exist and the twin operation belongs to the Laue class of the crystal. As a consequence, the reciprocal lattices of the twin components are superimposed so that coinciding lattice points refer to Bragg reflections with the same jFj2 values as long as Friedel's law is valid. In that case, no differences with respect to symmetry, or to reflection conditions, or to relative intensities occur between two sets of Bragg

P - -P42 - Pn - I -- Ia -F -- Fd -P21 =a; b --

P432; P 43m; Pm3m P42 32 Pn3m I432; I 43m; Im3m ± F432; F 43m; Fm3m Fd 3m ±

P42 ; P42 =m P41 ; P43 P4=n P42 =n I4=m I4; I 4;

P3c1; P 3c1 P23; Pm3 P21 3 Pn3 I23; I21 3; Im3 Ia3 F23; Fm3 Fd 3 Pa3

intensities measured from a single crystal on the one hand and from a twin on the other hand (whether or not the twin components differ in their volumes). If anomalous scattering is observed and the twin components differ in size, the intensities of Bragg reflections are changed in comparison with the untwinned crystal but the symmetry of the diffraction pattern is unchanged. For equal volumes of the twin components, however, the diffraction pattern is centrosymmetric again. The occurrence of anomalous scattering does not produce additional difficulties for space-group determination. The change of the 13


Simulated Laue class

Single crystal

1. CRYSTAL GEOMETRY AND SYMMETRY Bragg intensities in comparison with the untwinned crystals, however, makes a structure determination more difficult. Type 2: The twin operation does not belong to the Laue class of the crystal. Such twins can occur only in point groups marked by an asterisk in Table 1.3.4.1, i.e. in 55 out of the 159 types of space groups mentioned above. If the different twin components occur with equal volumes, the corresponding diffraction pattern shows enhanced symmetry. On the contrary, the reflection conditions are unchanged in comparison to those As a consequence, for 51 for a single crystal, except for Pa3. out of the 55 space-group types, the derivation of `possible space groups', as described in IT A (1983, Part 3), gives the combination of incorrect results. For P42 =n, I41 =a and Ia3, the simulated Laue class of the twin and the (unchanged) extinction symbol does not occur for single crystals. Therefore, the symmetry of these twins can be determined uniquely. In the the reflection conditions differ for the two twin case of Pa3, components. [This is because the holohedry of Pa3 is m3m whereas the Laue class of the Euclidean normalizer Ia3 of Pa3 cf. IT A (1987, Part 15).] As a consequence, the is m3; reflection conditions for such a twinned crystal differ from all conditions that may be observed for single crystals (hkl cyclically permutable: 0kl only with k 2n or l 2n; 00l only with l 2n) and, therefore, the true symmetry can be identified without uncertainty. In Table 1.3.4.2, all simulated Laue classes (column 1) are listed that may be observed for twins by merohedry of type 2. Column 2 shows the corresponding extinction symbols. The symbols of the simulated `possible space groups' that follow from IT A (1983, Part 3) are gathered in column 3. The last column displays the symbols of those space groups which may be the true symmetry groups for twins by merohedry showing such diffraction patterns.

The plane defined by a1 and a2 is perpendicular to the plane defined by a and a0 and bisects the angle a ^ a0 . Analogous planes refer to b1 and b2 , and c1 and c2 . Vectors ra , rb , and rc lying within one of these planes may be described as linear combinations of a1 and a2 , b1 and b2 , or c1 and c2 , respectively: ra la a1 a a2 ; rb lb b1 b b2 ; rc lc c1 c c2 : The common intersection line of these three planes is parallel to the twin axis. It may be calculated by solving any of the three equations ra rb ;

Solve the inhomogeneous system of three equations that corresponds to this vector equation for the three variables a , lb , and b . Calculate the vector r a1 a a2 . Its components with respect to a, b, c describe the direction of the twin axis. The angle of the twin rotation may then be calculated by sin 12

sin 12 a sin 12 b sin 12 c sin a sin b sin c

with a r ^ a; b r ^ b; c r ^ c. If the basis a, b, c is orthogonal, may be obtained from cos 12 cos a cos b cos c

1:

If the coefficients of r are rational and equals 180 , then r describes the direction either of the twofold twin axis or of the normal of the twin plane. If r is rational and equals 60, 90 or 120 , r is parallel to the twin axis. If r is irrational, but equals 180 and there exists, in addition, a net plane perpendicular to r, this net plane describes the twin plane. If none of these conditions is fulfilled, one has to repeat the calculations with a differently chosen basis system for one of the twin components. The number of possibilities for this choice depends on the lattice symmetry. The following list gives all equivalent basis systems for all descriptions of Bravais lattices used in IT A (1983): aP:

b0 e21 a e22 b e23 c; c0 e31 a e32 b e33 c:

a, b, c;

mP, mS (unique axis b):

a, b, c;

a, b, c;

mP, mS (unique axis c):

a, b, c;

a, b, c;

oP, oS, oI, oF:

a, b, c;

a, b, c;

a, b, c;

a, b, c;

tP, tI: a, b, c; a, b, c; a, b, c; a, b, c; b, a, c; b, a, c; b, a, c; b, a, c;

The coefficients eij have to be obtained by measurement. Basis a, b, c may be mapped onto a0 , b0 , c0 by a pure rotation that brings a to a0 , b to b0 , and c to c0 . To derive the direction of the rotation axis, calculate the three vectors

hP:

c1 c c0 :

a, b, c; b, a b, c; a b, a, c; b, a, c; a b, b, c; a, a b, c; a, b, c; b, a b, c; a b, a, c; b, a, c; a b, b, c; a, a b, c;

hR (hexagonal description): a b, a, c; b, a, c;

a1 , b1 , c1 bisect the angles a a ^ a0 , b b ^ b0 , and c c ^ c0 , respectively. Calculate three further vectors of arbitrary length a2 ; b2 ; c2 which are perpendicular to the planes defined by a and a0 , b and b0 , and c and c0 , respectively, from the scalar products

a, b, c; b, a b, c; a b, b, c; a, a b, c;

hR (rhombohedral description): b, a, c; a, c, b;

a, b, c; b, c, a; c, a, b; c, b, a;

cP, cI, cF: a, b, c; b, c, a; c, a, b; a, b, c; b, c, a; c, a, b; a, b, c; b, c, a; c, a, b; a, b, c; b, c, a; c, a, b; b, a, c; a, c, b; c, b, a; b, a, c; a, c, b; c, b, a; b, a, c; a, c, b; c, b, a; b, a, c; a, c, b; c, b, a.

a2 a a2 a0 0; b2 b b2 b0 0; c2 c c2 c0 0:


rb rc :

a 1 a a 2 l b b 1 b b 2 :

If the twin element cannot be recognized by direct macroscopic or microscopic inspection, it may be calculated as described below. Given are two analogous bases a, b, c and a0 , b0 , c0 referring to the two twin components. If possible, both basis systems should be chosen with the same handedness. If no such bases exist, the twin is a reflection twin and one of the bases has to be replaced by its centrosymmetrical one, e.g. a0 , b0 , c0 by a0 , b0 , c0 . The relation between the two bases is described by a0 e11 a e12 b e13 c;

b1 b b0 ;

or

ra rb : choose la arbitrarily equal to 1.

1.3.5. Calculation of the twin element

a1 a a0 ;

ra rc ;


1.4. Arithmetic crystal classes and symmorphic space groups By A. J. C. Wilson

symbol for the geometric crystal class and the symbol for the Bravais lattice (de Wolff et al., 1985). For example, in the monoclinic system the geometric crystal classes are 2, m, and 2=m, and the Bravais lattices are monoclinic P and monoclinic C. The six arithmetic crystal classes in the monoclinic system are thus 2P, 2C, mP, mC, 2=mP, and 2=mC. In certain cases (loosely, when the geometric crystal class and the Bravais lattice have unique directions that are not necessarily parallel), the crystal class and the lattice can be combined in two different orientations. The simplest example is the combination of the orthorhombic crystal class* mm with the endcentred lattice C. The intersection of the mirror planes of the crystal class de®nes one unique direction, the C centring of the lattice another. If these directions are placed parallel to one another, the arithmetic class mm2C is obtained; if they are placed perpendicular to one another, a different arithmetic classy 2mmC is obtained. The other combinations exhibiting this phenomenon are lattice P with geometric classes 32, 3m, 3m, 4m, and 6m. By consideration of all possible combinations of geometric class and lattice, one obtains the 73 arithmetic classes listed in Table 1.4.2.1.

1.4.1. Arithmetic crystal classes Arithmetic crystal classes are of great importance in theoretical crystallography, and are treated from that point of view in Volume A of International Tables for Crystallography (Hahn, 1995, p. 719). They have, however, at least four applications in practical crystallography: (1) in the classi®cation of space groups (Section 1.4.2); (2) in forming symbols for certain space groups in higher dimensions (see Chapter 9.8 and the references cited therein); (3) in modelling the frequency of occurrence of space groups (see Chapter 9.7 and the references cited therein); and (4) in establishing èquivalent origins' (Wondratschek, 1995, p. 719). The tabulation of arithmetic crystal classes in Volume A is incomplete, and the relation of the notation used in complete tabulations found elsewhere (for example, in Brown, BuÈlow, NeubuÈser, Wondratschek & Zassenhaus, 1978) to that of International Tables is not immediately obvious. Simple descriptions and complete enumerations of the arithmetic crystal classes in one, two and three dimensions are therefore given here. 1.4.1.1. Arithmetic crystal classes in three dimensions

* Here and in Chapter 9.7, it is convenient to use the `short' symbols mm, 32, instead of mm2, 321, etc., whenever it is desired to 3m, 3m, 4m, and 6m emphasize that no implication about orientation is intended. y In the arithmetic crystal class 2mmC, two conventions concerning the nomenclature of axes con¯ict. The ®rst is that, if only one face of the Bravais lattice is centred, the c axis is chosen perpendicular to that face. The second is that, if there is one axis of symmetry uniquely different from any others, that axis is to be chosen as b in the monoclinic system and as c in the remaining systems. The second convention is usually regarded as the more important, and the `standard setting' of 2mmC is mm2A. Both settings are listed in Table 1.4.2.1.

The 32 geometric crystal classes and the 14 Bravais lattices are familiar in three-dimensional crystallography. The threedimensional arithmetic crystal classes are easily derived in an elementary fashion by enumerating the compatible combinations of geometric crystal class and Bravais lattice; the symbol adopted by the International Union of Crystallography for an arithmetic crystal class is simply the juxtaposition of the

Table 1.4.1.1. The two-dimensional arithmetic crystal classes Crystal class Arithmetic Crystal system

Geometric

Number

Symbol

Space group Number

Symbol

Oblique

1 2

1 2

1p 2p

1 2

p1 p2

Rectangular

m

3

mp

2mm

4 5

mc 2mmp

6

2mmc

3 4 5 6 7 8 9

pm pg cm p2mm p2mg p2gg c2mm

7 8

4p 4mmp

10 11 12

p4 p4mm p4gm

9 10 11 12 13

3p 3m1p 31mp 6p 6mmp

13 14 15 16 17

p3 p3m1 p31m p6 p6mm

Square

4 4mm

Hexagonal

3 3m 6 6mm

15 Copyright © 2006 International Union of Crystallography 16 s:\ITFC\CH-1-4.3d (Tables of Crystallography)

1. CRYSTAL GEOMETRY AND SYMMETRY crystal classes result. The two-dimensional geometric and arithmetic crystal classes are listed in Table 1.4.1.1. The number of arithmetic crystal classes increases rapidly with increasing dimensionality; there are 710 (plus 70 enantiomorphs) in four dimensions (Brown, BuÈlow, NeubuÈser, Wondratschek & Zassenhaus, 1978), but those in dimensions higher than three are not needed in this volume.

1.4.1.2. Arithmetic crystal classes in one, two and higher dimensions In one dimension, there are two geometric crystal classes, 1 and m, and a single Bravais lattice, p. Two arithmetic crystal classes result, p and mp. In two dimensions, there are ten geometric crystal classes, and two Bravais lattices, p and c; 13 arithmetic

Table 1.4.2.1. The three-dimensional space groups, arranged by arithmetic crystal class; in a few geometric crystal classes this differs somewhat from the conventional numerical order; see International Tables Volume A, p. 728 Crystal class Arithmetic Crystal system

Geometric

Number

Symbol

Space group Number

Symbol

Triclinic

1 1

1 2

1P 1P

1 2

Monoclinic

2

3

2P

m

4 5

2C mP

6

mC

7

2=mP

8

2=mC

3 4 5 6 7 8 9 10 11 13 14 12 15

P2 P21 C2 Pm Pc Cm Cc P2=m P21 =m P2=c P21 =c C2=m C2=c

9

222P

10

222C

11 12

222F 222I

13

mm2P

14

mm2C

15

2mmC Amm2

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

P222 P2221 P21 21 2 P21 21 21 C2221 C222 F222 I222 I21 21 21 Pmm2 Pmc21 Pcc2 Pma2 Pca21 Pnc2 Pmn21 Pba2 Pna21 Pnn2 Cmm2 Cmc21 Ccc2 C2mm Amm2 C2me Aem2 C2cm Ama2 C2ce Aea2 Fmm2 Fdd2 Imm2 Iba2 Ima2

2=m

Orthorhombic

222

mm

39 40 41

16

mm2F

17

mm2I

16


42 43 44 45 46

P1 P1

1.4. ARITHMETIC CRYSTAL CLASSES AND SYMMORPHIC SPACE GROUPS Table 1.4.2.1. Three-dimensional space groups (cont.) Crystal class Arithmetic Crystal system Orthorhombic (cont.)

Tetragonal

Geometric mmm

4

4 4=m

422

4mm

Number 18

mmmP

19

mmmC

20

mmmF

21

mmmI

22

4P

23

4I

24 25 26

4P 4I 4=mP

27

4=mI

28

422P

29

422I

30

4mmP

31

4mmI

17


Symbol

Space group Number

Symbol

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

Pmmm Pnnn Pccm Pban Pmma Pnna Pmna Pcca Pbam Pccn Pbcm Pnnm Pmmn Pbcn Pbca Pnma Cmcm Cmce Cmmm Cccm Cmme Ccce Fmmm Fddd Immm Ibam Ibca Imma

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

P4 P41 P42 P43 I4 I41 P4 I 4 P4=m P42 =m P4=n P42 =n I4=m I41 =a P422 P421 2 P41 22 P41 21 2 P42 22 P42 21 2 P43 22 P43 21 2 I422 I41 22 P4mm P4bm P42 cm P42 nm P4cc P4nc P42 mc P42 bc I4mm I4cm I41 md I41 cd

1. CRYSTAL GEOMETRY AND SYMMETRY Table 1.4.2.1. Three-dimensional space groups (cont.) Crystal class Arithmetic Crystal system Tetragonal (cont.)

Geometric 4m

4=mmm

Trigonal

3

3 32

3m

3m

Number 32

42mP

33

4m2P

34

4m2I

35

42mI

36

4=mmmP

37

4=mmmI

38

3P

39 40 41 42

3R 3P 3R 312P

43

321P

44 45

32R 3m1P

46

31mP

47

3mR

48

31mP

49

3m1P

50

3mR

18


Symbol

Space group Number

Symbol

111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142

P42m P42c 1m P 42 1c P 42 P 4m2 P 4c2 P4b2 P4n2 I 4m2 I 4c2 I 42m I 42d P4=mmm P4=mcc P4=nbm P4=nnc P4=mbm P4=mnc P4=nmm P4=ncc P42 =mmc P42 =mcm P42 =nbc P42 =nnm P42 =mbc P42 =mnm P42 =nmc P42 =ncm I4=mmm I4=mcm I41 =amd I41 =acd

143 144 145 146 147 148 149 151 153 150 152 154 155 156 158 157 159 160 161 162 163 164 165 166 167

P3 P31 P32 R3 P3 R3 P312 P31 12 P32 12 P321 P31 21 P32 21 R32 P3m1 P3c1 P31m P31c R3m R3c P 31m P 31c P 3m1 P 3c1 R3m R3c

1.4. ARITHMETIC CRYSTAL CLASSES AND SYMMORPHIC SPACE GROUPS Table 1.4.2.1. Three-dimensional space groups (cont.) Crystal class Arithmetic Crystal system Hexagonal

Cubic

Geometric

Number

6

51

6P

6 6=m

52 53

6P 6=mP

622

54

622P

6mm

55

6mmP

6m

56

6m2P

57

62mP

6=mmm

58

6=mmmP

23

59

23P

60 61

23F 23I

62

m3P

63

m3F

64

m3I

65

432P

66

432F

67

432I

68

43mP

69

43mF

70

43mI

71

m3mP

72

m3mF

73

m3mI

m3

432

43m

m3m

19


Symbol

Space group Number

Symbol

168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194

P6 P61 P65 P62 P64 P63 P6 P6=m P63 =m P622 P61 22 P65 22 P62 22 P64 22 P63 22 P6mm P6cc P63 cm P63 mc P 6m2 P6c2 P62m P 62c P6=mmm P6=mmc P63 =mcm P63 =mmc

195 198 196 197 199 200 201 205 202 203 204 206 207 208 213 212 209 210 211 214 215 218 216 219 217 220 221 222 223 224 225 226 227 228 229 230

P23 P21 3 F23 I23 I21 3 Pm3 Pn3 Pa3 Fm3 Fd 3 Im3 Ia3 P432 P42 32 P41 32 P43 32 F432 F41 32 I432 I41 32 P43m P 43n F 43m F 43c I 43m I 43d Pm3m Pn3n Pm3n Pn3m Fm3m Fm3c Fd 3m Fd 3c Im3m Ia3d

1. CRYSTAL GEOMETRY AND SYMMETRY Table 1.4.3.1. Arithmetic crystal classes classi®ed by the number of space groups that they contain Number of space groups in the class 1

2

3

4

1P

1P 2C 222F 4P 3R 6P 23F

4I 3P

3R

32R

2P 222C 4I 3P 31mP 6=mP 23P 43mP

mP 222I 4=mI 312P 3m1P 6m2P 23I 43mF

mC mm2F 422I 321P 3mR 62mP m3F 43mI

2=mC mmmF 4m2I 3m1P

mm2C 3Py 4P m3P

mm2I 312Py

321Py

2mmC 4=mP 622P m3mP

mmmC 422P 6Py

622Py

8

422Py

4mmP

10

mm2P

16

mmmP 4=mmmP

Enantiomorphs combined.

42mI 31mP

3mR

m3I m3mI

432F

432I

mmmI 42mP 6=mmmP

4m2P

4=mmmI

432P

2=mP 222P 4Py 6P 432Py

6

Symbols of the arithmetic crystal classes

mm2A 4mmI 6mmP m3mF

y Enantiomorphs distinguished.

classes contain only a single space group, whereas two contain 16 each. Certain arithmetic crystal classes (3P; 312P; 321P; 422P; 6P; 622P; 432P) contain enantiomorphous pairs of space groups, so that the number of members of these classes depends on whether the enantiomorphs are combined or distinguished. Such classes occur twice in Table 1.4.3.1, marked with or y, respectively. The space groups in Table 1.4.2.1 are listed in the order of the arithmetic crystal class to which they belong. It will be noticed that arrangement according to the conventional spacegroup numbering would separate members of the same arithmetic crystal class in the geometric classes 2=m, 3m, 432, and 43m. 23, m3, This point is discussed in detail in Volume A of International Tables, p. 728. The symbols of ®ve space groups [C2me (Aem2), C2ce (Aea2), Cmce, Cmme, Ccce] have been conformed to those recommended in the fourth, revised edition of Volume A of International Tables.

1.4.2. Classi®cation of space groups Arithmetic crystal classes may be used to classify space groups on a scale somewhat ®ner than that given by the geometric crystal classes. Space groups are members of the same arithmetic crystal class if they belong to the same geometric crystal class, have the same Bravais lattice, and (when relevant) have the same orientation of the lattice relative to the point group. Each one-dimensional arithmetic crystal class contains a single space group, symbolized by p1 and pm, respectively. Most two-dimensional arithmetic crystal classes contain only a single space group; only 2mmp has as many as three. The space groups belonging to each geometric and arithmetic crystal class in two and three dimensions are indicated in Tables 1.4.1.1 and 1.4.2.1, and some statistics for the three-dimensional classes are given in Table 1.4.3.1. 12 three-dimensional 20


1.4. ARITHMETIC CRYSTAL CLASSES AND SYMMORPHIC SPACE GROUPS 1.4.2.1. Symmorphic space groups

The balance of symmetry elements within the symmorphic space groups is discussed in more detail in Subsection 9.7.1.2.

The 73 space groups known as `symmorphic' are in one-to-one correspondence with the arithmetic crystal classes, and their standard `short' symbols (Bertaut, 1995) are obtained by interchanging the order of the geometric crystal class and the Bravais cell in the symbol for the arithmetic space group. In fact, conventional crystallographic symbolism did not distinguish between arithmetic crystal classes and symmorphic space groups until recently (de Wolff et al., 1985); the symbol of the symmorphic group was used also for the arithmetic class. This relationship between the symbols, and the equivalent rule-of-thumb symmorphic space groups are those whose standard (short) symbols do not contain glide planes or screw axes, reveal nothing fundamental about the nature of symmorphism; they are simply a consequence of the conventions governing the construction of symbols in International Tables for Crystallography.* Although the standard symbols of the symmorphic space groups do not contain screw axes or glide planes, this is a result of the manner in which the space-group symbols have been devised. Most symmorphic space groups do in fact contain screw axes and/or glide planes. This is immediately obvious for the symmorphic space groups based on centred cells; C2 contains equal numbers of diad rotation axes and diad screw axes, and Cm contains equal numbers of re¯ection planes and glide planes. This is recognized in the èxtended' space-group symbols (Bertaut, 1995), but these are clumsy and not commonly used; those for C2 and Cm are C12211 and C1ma 1, respectively. In the more symmetric crystal systems, even symmorphic space groups with primitive cells contain screw axes and/or glide planes; P422 (P42221 ) contains many diad screw axes and P4=mmm (P4=m2=m2=m 21 =g ) contains both screw axes and glide planes.

1.4.3. Effect of dispersion on diffraction symmetry In the absence of dispersion (ànomalous scattering'), the intensities of the re¯ections hkl and h k l are equal (Friedel's law), and statements about the symmetry of the weighted reciprocal lattice and quantities derived from it often rest on the tacit or explicit assumption of this law ± the condition underlying it being forgotten. In particular, if dispersion is appreciable, the symmetry of the Patterson synthesis and the `Laue' symmetry are altered. 1.4.3.1. Symmetry of the Patterson function In Volume A of International Tables, the symmetry of the Patterson synthesis is derived in two stages. First, any glide planes and screw axes are replaced by mirror planes and the corresponding rotation axes, giving a symmorphic space group (Subsection 1.4.2.1). Second, a centre of symmetry is added. This second step involves the tacit assumption of Friedel's law, and should not be taken if any atomic scattering factors have appreciable imaginary components. In such cases, the symmetry of the Patterson synthesis will not be that of one of the 24 centrosymmetric symmorphic space groups, as given in Volume A, but will be that of the symmorphic space group belonging to the arithmetic crystal class to which the space group of the structure belongs. There are thus 73 possible Patterson symmetries. An equivalent description of such symmetries, in terms of 73 of the 1651 dichromatic colour groups, has been given by Fischer & Knop (1987); see also Wilson (1993). 1.4.3.2. `Laue' symmetry

* Three examples of informative de®nitions are: 1. The space group corresponding to the zero solution of the Frobenius congruences is called a symmorphic space group (Engel, 1986, p. 155). 2. A space group F is called symmorphic if one of its ®nite subgroups (and therefore an in®nity of them) is of an order equal to the order of the point group Rr (Opechowski, 1986, p. 255). 3. A space group is called symmorphic if the coset representatives Wj can be chosen in such a way that they leave one common point ®xed (Wondratschek, 1995, p. 717). Even in context, these are pretty opaque.

Similarly, the eleven conventional `Laue' symmetries [International Tables for Crystallography (1995), Volume A, p. 40 and elsewhere] involve the explicit assumption of Friedel's law. If dispersion is appreciable, the `Laue' symmetry may be that of any of the 32 point groups. The point group, in correct orientation, is obtained by dropping the Bravais-lattice symbol from the symbol of the arithmetic crystal class or of the Patterson symmetry.

References Grimmer, H. (1989b). Coincidence orientations of grains in rhombohedral materials. Acta Cryst. A45, 505±523. Grimmer, H. & Warrington, D. H. (1985). Coincidence orientations of grains in hexagonal materials. J. Phys (Paris), 46, C4, 231±236. Hahn, Th. (1981). Meroedrische Zwillinge, Symmetrie, DomaÈnen, Kristallstrukturbestimmung. Z. Kristallogr. 156, 114±115, and private communication. Hahn, Th. (1984). Twin domains and twin boundaries. Acta Cryst. A40, C-117. International Tables for Crystallography (1983). Vol. A. Dordrecht: Reidel. International Tables for Crystallography (1987). Vol. A, second, revised ed. Dordrecht: Kluwer Academic Publishers. Klapper, H. (1987). X-ray topography of twinned crystals. Prog. Cryst. Growth Charact. 14, 367±401.

1.1±1.3 Cahn, R. W. (1954). Twinned crystals. Adv. Phys. 3, 363±445. Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray crystal structure determination. Acta Cryst. A32, 163±165. Donnay, G. & Donnay, J. D. H. (1974). Classi®cation of triperiodic twins. Can. Mineral. 12, 422±425. Flack, H. D. (1987). The derivation of twin laws for (pseudo-) merohedry by coset decomposition. Acta Cryst. A43, 564± 568. Grimmer, H. (1984). The generating function for coincidence site lattices in the cubic system. Acta Cryst. A40, 108±112. Grimmer, H. (1989a). Systematic determination of coincidence orientations for all hexagonal lattices with axial ratios c/a in a given interval. Acta Cryst. A45, 320±325. 21


1. CRYSTAL GEOMETRY AND SYMMETRY Fischer, K. F. & Knop, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions in the lambda technique. Z. Kristallogr. 180, 237-242. Hahn, Th. (1995). Editor. International tables for crystallography, Vol. A. Space-group symmetry, fourth, revised, ed. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (1995). Vol. A. Spacegroup symmetry, fourth, revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North Holland. Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199±206. Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Acta Cryst. A41, 278±280. Wondratschek, H. (1995). Introduction to space-group symmetry. International tables for crystallography, Vol. A, edited by Th. Hahn, pp. 712±735. Dordrecht: Kluwer Academic Publishers.

1.1±1.3 (cont.) Klapper, H., Hahn, Th. & Chung, S. J. (1987). Optical, pyroelectric and X-ray topographic studies of twin domains and twin boundaries in KLiSO4 . Acta Cryst. B43, 147±159. LePage, Y., Donnay, J. D. H. & Donnay, G. (1984). Printing sets of structure factors for coping with orientation ambiguities and possible twinning by merohedry. Acta Cryst. A40, 679±684. 1.4 Bertaut, E. F. (1995). Synoptic tables of space-group symbols. Group±subgroup relations. In International tables for crystallography, Vol. A, edited by Th. Hahn, pp. 50±68. Dordrecht: Kluwer Academic Publishers. Brown, H., BuÈlow, R., NeubuÈser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of fourdimensional space. New York: Wiley. Engel, P. (1986). Geometric crystallography. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)

22


references


2.1. Classification of experimental techniques By J. R. Helliwell

specific atom whose X-ray absorption edge is stimulated; the atom absorbs an X-ray photon and yields up a photoelectron, which can be scattered by neighbouring atoms. Interpretation of EXAFS therefore closely follows low-energy electron-diffraction (LEED) theory. All these methods (Table 2.1.1) can be called methods of structure analysis. Techniques for examining the perfection of crystals are also very important. Defects in crystals represent irregularities in the growth of a perfect crystalline array. There are many types of defect. The experimental technique of X-ray topography (Chapter 2.7) is used to image irregularities in a crystal lattice. X-ray techniques have expanded in the 1970's and 1980's with the utilization of synchrotron radiation. The methods based on the use of neutrons and electrons have developed. Broadly speaking, the diffraction geometry is independent of the nature of the wave and depends only on its state, namely, the wavelength, l, the spectral bandpass, l=l, the convergence/divergence angles, and the beam direction. In what follows, the term monochromatic refers to the case where there is, practically speaking, a very small but finite wavelength spread. Similarly, the term polychromatic refers to the situation where the wavelength spread is of the same order as the mean wavelength. The technical means by which a given beam (of X-rays, neutrons or electrons) is conditioned vary, as do the means of detection. These methods are dealt with in the following pages as far as they relate to the geometry of diffraction. In the previous version of International Tables (IT II, 1959, Part 4), various diffraction geometries were detailed and a variety of numerical tables were given. The numerical tables have mainly been dispensed with since the use of hand-held calculators and computers has rendered them obsolete.

The diffraction of a wave of characteristic length, l, by a crystal sample requires that l is of the same order in size as the interatomic separation. Beams of X-rays, neutrons, and electrons can easily satisfy this requirement; for the latter two, the wavelength is determined by the de Broglie relationship l h=p, where h is Planck's constant and p is the momentum. We can define `diffraction geometry' as the description of the relationship between the beam and the sample orientation and the subsequent interception of the diffracted rays by a detector of given geometry and imaging properties. Each diffracted ray represents successful, constructive interference. The full stimulation of a reflection is achieved either by using a continuum of values of l incident on the crystal, as used originally by Friedrich, Knipping & von Laue (1912) (the Laue method) or alternatively by using a monochromatic beam and rotation or precession of a crystal (moving-crystal methods) or a set of randomly oriented crystallites (the powder method). The analysis of single-crystal reflection intensities allows the three-dimensional architecture of molecules to be determined. However, a single crystal cannot always be obtained. Diffraction from noncrystalline samples, i.e. fibres, amorphous materials or solutions, yields less detailed, but often very valuable, molecular information. Another method, surface diffraction, involves the determination of the organization of atoms deposited on the surface of a crystal substrate; a surface of perfectly repeating identical units, in identical environments, on such a substrate is sometimes referred to as a two-dimensional crystal. Ordered twodimensional arrangements of proteins in membranes are studied by electron diffraction and, more recently, by undulator X-radiation. Another experimental probe of the structure of matter is EXAFS (extended X-ray absorption fine structure). This technique yields details of the local environment of a

24 Copyright © 2006 International Union of Crystallography 25 s:\ITFC\ch-2-1.3d (Tables of Crystallography)

2.1 CLASSIFICATION OF EXPERIMENTAL TECHNIQUES Table 2.1.1. Summary of main experimental techniques for structure analysis Beam Name of technique

Usual type

Spectrum

Sample

Usual detectors

A Single crystal Laue

X-ray or neutron

Polychromatic

Stationary single crystal

Film; image plate or storage phosphor; electronic area detector (e.g. CCD); for neutron case, detector sensitive to time-of-flight

Still

X-ray or neutron or electron

Monochromatic

Stationary single crystal

Film; image plate or storage phosphor; electronic area detector (e.g. MWPC, TV, CCD)

Rotation/oscillation

X-ray

Monochromatic

Single crystal rotating about a single axis (typical angular range per exposure 5±15 for small molecule; 1±2 for protein; 0.25±0.5 for virus)

Film; image plate or storage phosphor; electronic area detector (e.g. MWPC, TV, CCD)

Weissenberg

X-ray

Monochromatic

Single crystal rotating about a single axis (angular range 15 ), coupled with detector translation

Film; image plate or storage phosphor

Precession

X-ray

Monochromatic

Single crystal (the normal to a reciprocal-lattice plane precesses about X-ray beam)

Flat film moving behind a screen coupled with crysal so as to be held parallel to a reciprocal-lattice plane

Diffractometry

X-ray or neutron

Monochromatic

Single crystal rotated over a small angular range

Single counter, linear detector or area detector

Monochromatic powder method

X-ray or neutron or electron

Monochromatic

Powder sample rotated to increase range of orientations presented to beam

Film or image plate; counter; 1D positionsensitive detector (linear or curved)

Energy-dispersive powder method

X-ray or neutron

Polychromatic

Powder sample

Energy-dispersive counter (for neutron case, detector sensitive to time-of-flight)

B Polycrystalline powders

C Fibres, solutions, surfaces, and membranes Fibre method

X-ray or neutron

Monochromatic

Single fibre or a bundle of fibres; preferred orientation in a sample

Film or image plate; electronic area detector (e.g. MWPC or TV); records high-angle or low-angle diffraction data

Solution or `small-angle method'

X-ray or neutron

Monochromatic

Dilute solutions of particles; crystalline defects

Counter or MWPC

Surface diffraction

Electron or X-ray

Monochromatic

Atoms deposited or adsorbed onto a substrate

Phosphor or counter

Membranes

Electron or X-ray

Monochromatic

Naturally occurring 2D ordered membrane protein

Film or image plate; CCD

Notes (1) Monochromatic. Typical value of spectral spread, l=l, on a conventional X-ray source; K1 K2 line separation 2:5 10 3 , K1 line width 10 4 . On a synchrotron source a variable quantity dependent on type of monochromator; typical values 10 3 or 10 4 for the two common monochromator types (see Figs. 2.2.7.2 and 2.2.7.3, respectively). (2) CCD charge-coupled device; MWPC multiwire proportional chamber detector; TV television detector. (3) Image plate is a trade name of Fuji. Storage phosphor is a trade name of Kodak. (4) EXAFS can be performed on all types of sample whether crystalline or noncrystalline. It uses an X-ray beam that is tuned around an absorption edge and the transmitted intensity or the fluorescence emission is measured.

25

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2.2. Single-crystal X-ray techniques By J. R. Helliwell

and computer re®nement of crystal orientation following initial crystal setting. Precession photography allows the isolation of a speci®c zone or plane of re¯ections for which indexing can be performed by inspection, and systematic absences and symmetry are explored. From this, space-group assignment is made. The use of precession photography is usually avoided in small-molecule crystallography where auto-indexing methods are employed on a single-crystal diffractometer. In such a situation, the burden of data collection is not huge and symmetry elements can be determined after data collection. This is also now carried out on electronic area detectors in conjunction with auto-indexing principally at present for macromolecular crystallography but also for chemical crystallography. In the following sections, the geometry of each method is dealt with in an idealized form. The practical realization of each geometry is then dealt with, including the geometric distortions introduced in the image by electronic area detectors. A separate section deals with the common means for beam conditioning, namely mirrors, monochromators, and ®lters. Suf®cient detail is given to establish the magnitude of the wavelength range, spectral spreads, beam divergence and convergence angles, and detector effects. These values can then be utilized along with the formulae given for the calculation of spot bandwidth, spot size, and angular re¯ecting range.

In Chapter 2.1 Classi®cation of experimental techniques there are given the various common approaches to the recording of X-ray crystallographic data, in different geometries, for crystal structure analysis. These are: (a) Laue geometry; (b) monochromatic still exposure; (c) rotation/oscillation geometry; (d) Weissenberg geometry; (e) precession geometry; ( f ) diffractometry. The reasons for the choice of order are as follows. Laue geometry is dealt with ®rst because it was historically the ®rst to be used (Friedrich, Knipping & von Laue, 1912). In addition, the ®rst step that should be carried out with a new crystal, at least of a small molecule, is to take a Laue photograph to make the ®rst assessment of crystal quality. For macromolecules, the monochromatic still serves the same purpose. From consideration of the monochromatic still geometry, we can then describe the cases of single-axis rotation (rotation/oscillation method), singleaxis rotation coupled with detector translation (Weissenberg method), crystal and detector precession (precession method), and ®nally three-axis goniostat and rotatable detector or area detector (diffractometry). Method (a) uses a polychromatic beam of broad wavelength Ê if the bandwidth is restricted bandpass (e.g. 0:2 l 2:5 A); (e.g. to l=l 0:2), then it is sometimes referred to as narrowbandpass Laue geometry. The remaining methods (b)±( f ) use a monochromatic beam. There are textbooks that concentrate on almost every geometry. References to these books are given in the respective sections in the following pages. However, in addition, there are several books that contain details of diffraction geometry. Blundell & Johnson (1976) describe the use of the various diffraction geometries with the examples taken from protein crystallography. There is an extensive discussion and many practical details to be found in the textbooks of Stout & Jensen (1968), Woolfson (1970, 1997), Glusker & Trueblood (1971, 1985), Vainshtein (1981), and McKie & McKie (1986), for example. A collection of early papers on the diffraction of Xrays by crystals involving, inter alia, experimental techniques and diffraction geometry, can be found in Bijvoet, Burgers & HaÈgg (1969, 1972). A collection of papers on, primarily, protein and virus crystal data collection via the rotation-®lm method and diffractometry can be found in Wyckoff, Hirs & Timasheff (1985). Synchrotron instrumentation, methods, and applications are dealt with in the books of Helliwell (1992) and Coppens (1992). Quantitative X-ray crystal structure analysis usually involves methods (c), (d), and ( f ), although (e) has certainly been used. Electronic area detectors or image plates are extensively used now in all methods. Traditionally, Laue photography has been used for crystal orientation, crystal symmetry, and mosaicity tests. Rapid recording of Laue patterns using synchrotron radiation, especially with protein crystals or with small crystals of small molecules, has led to an interest in the use of Laue geometry for quantitative structure analysis. Various fundamental objections made, especially by W. L. Bragg, to the use of Laue geometry have been shown not to be limiting. The monochromatic still photograph is used for orientation setting and mosaicity tests, for protein or virus crystallography,

2.2.1. Laue geometry The main book dealing with Laue geometry is AmoroÂs, Buerger & AmoroÂs (1975). This should be used in conjunction with Henry, Lipson & Wooster (1951), or McKie & McKie (1986); see also Helliwell (1992, chapter 7). There is a synergy between synchrotron and neutron Laue diffraction developments (see Helliewell & Wilkinson, 1994). 2.2.1.1. General The single crystal is bathed in a polychromatic beam of X-rays containing wavelengths between lmin and lmax . A particular crystal plane will pick out a general wavelength l for which constructive interference occurs and re¯ect according to Bragg's law l 2d sin ;

where d is the interplanar spacing and is the angle of re¯ection. A sphere drawn with radius 1=l and with the beam direction as diameter, passing through the origin of the reciprocal lattice (the point O in Fig. 2.2.1.1), will yield a re¯ection in the direction drawn from the centre of the sphere and out through the reciprocal-lattice point (relp) provided the relp in question lies on the surface of the sphere. This sphere is known as the Ewald sphere. Fig. 2.2.1.1 shows the Laue geometry, in which there exists a nest of Ewald spheres of radii between 1=lmax and 1=lmin . An alternative convention is feasible whereby only a single Ewald sphere is drawn of radius 1 reciprocal-lattice unit (r.l.u.). Then each relp is no longer a point but a streak between lmin =d and lmax =d from the origin of reciprocal space (see McKie & McKie, 1986, p. 297). In the following discussions on the Laue approach, this notation is not followed. We use the nest of Ewald spheres of varying radii instead. 26

Copyright © 2006 International Union of Crystallography 27 s:\ITFC\ch-2-2.3d (Tables of Crystallography)

2:2:1:1

2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES aligned protein crystal. For a description of the indexing of a Laue photograph, see Bragg (1928, pp. 28, 29). For a Laue spot at a given , only the ratio l=d is determined, whether it is a single or a multiple relp component spot. If the unit-cell parameters are known from a monochromatic experiment, then a Laue spot at a given yields l since d is then known. Conversely, precise unit-cell lengths cannot be determined from a Laue pattern alone; methods are, however, being developed to determine these (see Carr, Cruickshank & Harding, 1992). The maximum Bragg angle max is given by the equation

Any relp (hkl) lying in the region of reciprocal space between the 1=lmax and 1=lmin Ewald spheres and the resolution sphere 1=dmin will diffract (the shaded area in Fig. 2.2.1.1). This region of reciprocal space is referred to as the accessible or stimulated region. Fig. 2.2.1.2 shows a predicted Laue pattern from a well

max sin 1 lmax =2dmin :

2:2:1:2

2.2.1.2. Crystal setting The main use of Laue photography has in the past been for adjustment of the crystal to a desired orientation. With smallmolecule crystals, the number of diffraction spots on a monochromatic photograph from a stationary crystal is very small. With un®ltered, polychromatic radiation, many more spots are observed and so the Laue photograph serves to give a better idea of the crystal orientation and setting prior to precession photography. With protein crystals, the monochromatic still is used for this purpose before data collection via an area detector. This is because the number of diffraction spots is large on a monochromatic still and in a protein-crystal Laue photograph the stimulated spots from the Bremsstrahlung continuum are generally very weak. Synchrotron-radiation Laue photographs of protein crystals can be recorded with short exposure times. These patterns consist of a large number of diffraction spots. Crystal setting via Laue photography usually involves trying to direct the X-ray beam along a zone axis. Angular mis-setting angles " in the spindle and arc are easily calculated from the formula

Fig. 2.2.1.1. Laue geometry. A polychromatic beam containing wavelengths lmin to lmax impinges on the crystal sample. The 1=dmin is drawn centred at O, the resolution sphere of radius dmax origin of reciprocal space. Any reciprocal-lattice point falling in the shaded region is stimulated. In this diagram, the radius of each Ewald sphere uses the convention 1=l.

" tan 1 =D;

2:2:1:3

where is the distance (resolved into vertical and horizontal) from the beam centre to the centre of a circle of spots de®ning a zone axis and D is the crystal-to-®lm distance. After suitable angular correction to the sample orientation, the Laue photograph will show a pronounced blank region at the centre of the ®lm (see Fig. 2.2.1.2). The radius of the blank region is determined by the minimum wavelength in the beam and the magnitude of the reciprocal-lattice spacing parallel to the X-ray beam (see Jeffery, 1958). For the case, for example, of the X-ray beam perpendicular to the a b plane, then lmin c1

cos 2;

2:2:1:4a

where 2 tan 1 R=D

2:2:1:4b

and R is the radius of the blank region (see Fig. 2.2.1.2), and D is the crystal-to-¯at-®lm distance. If lmin is known then an approximate value of c, for example, can be estimated. The principal zone axes will give the largest radii for the central blank region. 2.2.1.3. Single-order and multiple-order re¯ections In Laue geometry, several relp's can occur in a Laue spot or ray. The number of relp's in a given spot is called the multiplicity of the spot. The number of spots of a given multiplicity can be plotted as a histogram. This is known as the multiplicity distribution. The form of this distribution is dependent on the ratio lmax =lmin . The multiplicity distribution

Fig. 2.2.1.2. A predicted Laue pattern of a protein crystal with a zone axis parallel to the incident, polychromatic X-ray beam. There is a pronounced blank region at the centre of the ®lm (see Subsection 2.2.1.2). The spot marked N is one example of a nodal spot (see Subsection 2.2.1.4).

27


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION 1 1 Q 1 1 1 23 33

in Laue diffraction is considered in detail by Cruickshank, Helliwell & Moffat (1987). Any relp nh, nk, nl (n integer) will be stimulated by a wavelength l=n since dnhnknl dhkl =n, i.e. l d 2 hkl sin : n n

0:832 . . . :

1 ... 53 2:2:1:6

Hence, for a relp where dmin < dhkl < 2dmin there is a very high probability (83.2%) that the Laue spot will be recorded as a single-wavelength spot. Since this region of reciprocal space corresponds to 87.5% (i.e. 7=8) of the volume of reciprocal space within the resolution sphere then 0:875 0:832 72:8% is the probability for a relp to be recorded in a single-wavelength spot. According to W. L. Bragg, all Laue spots should be multiple. He reasoned that for each h, k, l there will always be a 2h, 2k, 2l etc. lying within the same Laue spot. However, as the resolution limit is increased to accommodate this many more relp's are added, for which their hkl's have no common divisor. The above discussion holds for in®nite bandwidth. The effect of a more experimentally realistic bandwidth is to increase the proportion of single-wavelength spots. The number of relp's within the resolution sphere is

2:2:1:5

However, dnhnknl must be > dmin as otherwise the re¯ection is beyond the sample resolution limit. If h, k, l have no common integer divisor and if 2h, 2k, 2l is beyond the resolution limit, then the spot on the Laue diffraction photograph is a single-wavelength spot. The probability that h, k, l have no common integer divisor is

3 4 dmax ; 3 V

2:2:1:7

1=dmin and V is the reciprocal unit-cell volume. where dmax The number of relp's within the wavelength band lmax to lmin , , is (Moffat, Schildkamp, Bilderback & Volz, for lmax < 2=dmax 1986) 4 lmax lmin dmax : V 4

2:2:1:8

Ê wavelength Note that the number of relp's stimulated in a 0.1 A Ê , is the same as that interval, for example between 0.1 and 0.2 A Ê , for example. A large number of relp's between 1.1 and 1.2 A are stimulated at one orientation of the crystal sample. The proportion of relp's within a sphere of small d (i.e. at low resolution) actually stimulated is small. In addition, the probability of them being single is zero in the in®nite-band-width case and small in the ®nite-bandwidth case. However, Laue

Fig. 2.2.1.3. A multiple component spot in Laue geometry. A ray of multiplicity 5 is shown as an example. The inner point A corresponds to d and a wavelength l, the next point, B, is d=2 and wavelength l=2. The outer point E corresponds to d=5 and l=5. Rotation of the sample will either exclude inner points (at the lmax surface) or outer points (at the lmin surface) and so determine the recorded multiplicity.

Fig. 2.2.1.4. The variation with M lmax =lmin of the proportions of relp's lying on single, double, and triple rays for the case lmax < 2=dmax . From Cruickshank, Helliwell & Moffat (1987).

28


2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES ± form a set largely distinct, except at short crystal-to-detector distances, from those involved in spatial overlaps, which are mostly singles (Helliwell, 1985). From a knowledge of the form of the angular distribution, it is possible, e.g. from the gaps bordering conics, to estimate dmax and lmin . However, a development of this involving gnomonic projections can be even more effective (Cruickshank, Carr & Harding, 1992).

geometry is an ef®cient way of measuring a large number of and dmax =2 as single-wavelength spots. relp's between dmax The above is a brief description of the overall multiplicity distribution. For a given relp, even of simple hkl values, lying on a ray of several relp's (multiples of hkl), a suitable choice of crystal orientation can yield a single-wavelength spot. Consider, for example, a spot of multiplicity 5. The outermost relp can be recorded at long wavelength with the inner relp's on the ray excluded since they need l's greater than lmax (Fig. 2.2.1.3). Alternatively, by rotating the sample, the innermost relp can be measured uniquely at short wavelength with the outer relp's excluded (they require l's shorter than lmin ). Hence, in Laue geometry several orientations are needed to recover virtually all relp's as singles. The multiplicity distribution is shown in Fig. 2.2.1.4 as a function of lmax =lmin (with the corresponding values of l=lmean ).

2.2.1.5. Gnomonic and stereographic transformations A useful means of transformation of the ¯at-®lm Laue pattern is the gnomonic projection. This converts the pattern of spots lying on curved arcs to points lying on straight lines. The stereographic projection is also used. Fig. 2.2.1.5 shows the graphical relationships involved [taken from International Tables, Vol. II (Evans & Lonsdale, 1959)], for the case of a Laue pattern recorded on a plane ®lm, between the incidentbeam direction SN, which is perpendicular to a ®lm plane and the Laue spot L and its spherical, stereographic, and gnomonic points Sp , St and G and the stereographic projection Sr of the re¯ected beams. If the radius of the sphere of projection is taken equal to D, the crystal-to-®lm distance, then the planes of the gnomonic projection and of the ®lm coincide. The lines producing the various projection poles for any given crystal plane are coplanar with the incident and re¯ected beams. The transformation equations are

2.2.1.4. Angular distribution of re¯ections in Laue diffraction There is an interesting variation in the angular separations of Laue re¯ections that shows up in the spatial distributions of spots on a detector plane (Cruickshank, Helliwell & Moffat, 1991). There are two main aspects to this distribution, which are general and local. The general aspects refer to the diffraction pattern as a whole and the local aspects to re¯ections in a particular zone of diffraction spots. The general features include the following. The spatial density of spots is everywhere proportional to 1=D2 , where D is the crystal-to-detector distance, and to 1=V , where V is the reciprocal-cell volume. There is also though a substantial variation in spatial density with diffraction angle ; a prominent maximum occurs at c sin

1

lmin dmax =2:

2:2:1:10

PG D cot

2:2:1:11

PS D

2:2:1:9

cos 1 sin

PR D tan :

Local aspects of these patterns particularly include the prominent conics on which Laue re¯ections lie. That is, the local spatial distribution is inherently one-dimensional in character. Between multiple re¯ections (nodals), there is always at least one single and therefore nodals have a larger angular separation from their nearest neighbours. The blank area around a nodal in a Laue pattern (Fig. 2.2.1.2) has been noted by Jeffery (1958). The smallest angular separations, and therefore spatially overlapped cases, are associated with single Laue re¯ections. Thus, the re¯ections involved in energy overlaps ± the multiples

2:2:1:12 2:2:1:13

2.2.2. Monochromatic methods In this section and those that follow, which deal with monochromatic methods, the convention is adopted that the Ewald sphere takes a radius of unity and the magnitude of the reciprocal-lattice vector is l=d. This is not the convention used in the Laue section above. Some historical remarks are useful ®rst before progressing to discuss each monochromatic geometry in detail. The original rotation method (for example, see Bragg, 1949) involved a rotation of a perfectly aligned crystal of 360 . For reasons of relatively poor collimation of the X-ray beam, leading to spot-tospot overlap, and background build-up, Bernal (1927) introduced the oscillation method whereby a repeated, limited, angular range was used to record one pattern and a whole series of contiguous ranges on different ®lm exposures were collected to provide a large angular coverage overall. In a different solution to the same problem, Weissenberg (1924) utilized a layer-line screen to record only one layer line but allowed a full rotation of the crystal but now coupled to translation of the detector, thus avoiding spot-to-spot overlap. Again, several exposures were needed, involving one layer line collected on each exposure. The advent of synchrotron radiation with very high intensity allows small beam sizes at the sample to be practicable, thus simultaneously creating small diffraction spots and minimizing background scatter. The very ®ne collimation of the synchrotron beam keeps the diffraction-spot sizes small as they traverse their path to the detector plane.

Fig. 2.2.1.5. Geometrical principles of the spherical, stereographic, gnomonic, and Laue projections. From Evans & Lonsdale (1959).

29


PL D tan 2

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION The diffraction spots lie on curved arcs where each curve corresponds to the intersection with a ®lm of a cone. With a ¯at ®lm the intersections are conic sections. The curved arcs are obviously recognizable for the protein crystal case where there are a large number of spots.

The terminology used today for different methods is essentially the same as originally used except that the rotation method now tends to mean limited angular ranges (instead of 360 ) per diffraction photograph/image. The Weissenberg method in its modern form now employed at a synchrotron is a screenless technique with limited angular range but still with detector translation coupled to crystal rotation.

2.2.2.2. Crystal setting

2.2.2.1. Monochromatic still exposure

Crystal setting follows the procedure given in Subsection 2.2.1.2 whereby angular mis-setting angles are given by equation (2.2.1.3). When viewed down a zone axis, the pattern on a ¯at ®lm or electronic area detector has the appearance of a series of concentric circles. For example, the ®rst circle corresponds to with the beam parallel to 001, l 1, the second to l 2, etc. The radius of the ®rst circle R is related to the interplanar spacing between the (hk0) and (hk1) planes, i.e. l=c (in this example), through , by the formulae

In a monochromatic still exposure, the crystal is held stationary and a near-zero wavelength-bandpass (e.g. l=l 0:001) beam impinges on it. For a small-molecule crystal, there are few diffraction spots. For a protein crystal, there are many (several hundred), because of the much denser reciprocal lattice. The actual number of stimulated relp's depends on the reciprocal-cell parameters, the size of the mosaic spread of the crystal, the angular beam divergence as well as the small, but ®nite, spectral spread, l=l. Diffraction spots are only partially stimulated instead of fully integrated over wavelength, as in the Laue method, or over an angular rotation (the rocking width) in rotating-crystal monochromatic methods.

tan 2 R=D;

cos 2 1

l=c:

2:2:2:1

Fig. 2.2.3.1. (a) Elevation of the sphere of re¯ection. O is the origin of the reciprocal lattice. C is the centre of the Ewald sphere. The incident beam is shown in the plane. (b) Plan of the sphere of re¯ection. R is the projection of the rotation axis on the equatorial plane. (c) Perspective diagram. P is the relp in the re¯ection position with the cylindrical coordiantes ; ; '. The angular coordinates of the diffracted beam are v, . (d) Stereogram to show the direction of the diffracted beam, v, , with DD0 , normal to the incident beam and in the equatorial plane, as the projection diameter. From Evans & Lonsdale (1959).

30


2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES 2.2.3. Rotation/oscillation geometry

cos 1

The main modern book dealing with the rotation method is that of Arndt & Wonacott (1977).

2 ; 2 cos2

2:2:3:10

(d) 0, ¯at cone

2.2.3.1. General

The purpose of the monochromatic rotation method is to stimulate a re¯ection fully over its rocking width via an angular rotation. Different relp's are rotated successively into the re¯ecting position. The method, therefore, involves rotation of the sample about a single axis, and is used in conjunction with an area detector of some sort, e.g. ®lm, electronic area detector or image plate. The use of a repeated rotation or oscillation, for a given exposure, is simply to average out any time-dependent changes in incident intensity or sample decay. The overall crystal rotation required to record the total accessible region of reciprocal space for a single crystal setting, and a detector picking up all the diffraction spots, is 180 2max . If the crystal has additional symmetry, then a complete asymmetric unit of reciprocal space can be recorded within a smaller angle. There is a blind region close to the rotation axis; this is detailed in Subsection 2.2.3.5.

cos

sin 2 2 p : 2 1 2

2

2:2:3:11

2:2:3:12

In this section, we will concentrate on case (a), the normalbeam rotation method ( 0). First, the case of a plane ®lm or detector is considered.

2.2.3.2. Diffraction coordinates Figs. 2.2.3.1(a) to (d) are taken from IT II (1959, p. 176). They neatly summarize the geometrical principles of re¯ection, of a monochromatic beam, in the reciprocal lattice for the general case of an incident beam inclined at an angle () to the equatorial plane. The diagrams are based on an Ewald sphere of unit radius. With the nomenclature of Table 2.2.3.1: Fig 2.2.3.1(a) gives sin sin :

2:2:3:1

Fig. 2.2.3.1(b) gives, by the cosine rule, cos2 cos2 2 2 cos cos

2:2:3:2

cos2 2 cos2 ; 2 cos

2:2:3:3

cos and cos

and Figs. 2.2.3.1(a) and (b) give 2 2 d 2 4 sin2 :

2:2:3:4

The following special cases commonly occur: (a) 0, normal-beam rotation method, then sin

2:2:3:5

and cos

2 2 p ; 2 1 2

2

2:2:3:6

(b) , equi-inclination (relevant to Weissenberg upperlayer photography), then

2 sin 2 sin

2:2:3:7

2 ; 2 cos2

2:2:3:8

cos 1 (c) , anti-equi-inclination

0

Fig. 2.2.3.2. Geometrical principles of recording the pattern on (a) a plane detector, (b) a V-shaped detector, (c) a cylindrical detector.

2:2:3:9 31


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION The coordinates of a re¯ection on a ¯at ®lm YF ; ZF are related to the cylindrical coordinates of a relp ; [Fig. 2.2.3.2(a)] by

Table 2.2.3.1. Glossary of symbols used to specify quantities on diffraction patterns and in reciprocal space

Bragg angle

2

Angle of deviation of the re¯ected beam with respect to the incident beam

S^ o

Unit vector lying along the direction of the incident beam

S^

Unit vector lying along the direction of the re¯ected beam

s S^

YF D tan ZF D sec tan ; which becomes ZF 2D=2

S^ o The scattering vector of magnitude 2 sin . s is perpendicular to the bisector of the angle between S^ o ^ s is identical to the reciprocal-lattice vector d of and S. magnitude l=d, where d is the interplanar spacing, when d is in the diffraction condition. In this notation, the radius of the Ewald sphere is unity. This convention is adopted because it follows that in Volume II of International Tables (p. 175). Note that in Section 2.2.1 Laue geometry the alternative convention (jd j 1=d) is adopted whereby the radius of each Ewald sphere is 1=l. This allows a nest of Ewald spheres between 1=lmax and 1=lmin to be drawn

d 2 =2:

2:2:3:17

The angle of inclination of S^ o to the equatorial plane

The angle between the projections of S^ o and S^ onto the equatorial plane

The angle of inclination of S^ to the equatorial plane

!; ; '

The crystal setting angles on the four-circle diffractometer (see Fig. 2.2.6.1). The ' used here is not the same as that in the rotation method (Fig. 2.2.3.3). This clash in using the same symbol twice is inevitable because of the widespread use of the rotation camera and four-circle diffractometer.

YF cos =1

This situation also corresponds to the case of ¯at electronic area detector inclined to the incident beam in a similar way. Note that Arndt & Wonacott (1977) use instead of here. We use and so follow IT II (1959). This avoids confusion with the of Table 2.2.3.1. D is the crystal to V distance. In the case of the V cassettes of Enraf±Nonius, is 60 . For the case of a cylindrical ®lm or image plate where the axis of the cylinder is coincident with the rotation axis [Fig. 2.2.3.2(c)] then, for in degrees, 2 D 360

2:2:3:18

ZF D tan ;

2:2:3:19

D ZF p : 1 2

2:2:3:20

YF

which becomes

The angle of rotation from a de®ned datum orientation to bring a relp onto the Ewald sphere in the rotation method (see Fig. 2.2.3.3)

2:2:3:15

2:2:3:16

ZF D

The angular coordinate of P, measured as the angle between and S^ o [see Fig. 2.2.3.1(b)]

'

2 ;

YF D tan =sin cos tan

Radial coordinate of a point P in reciprocal space; that is, the radius of a cylinder having the rotation axis as axis

2

where D is the crystal-to-®lm distance. For the case of a V-shaped cassette with the V axis parallel to the rotation axis and the ®lm making an angle to the beam direction [Fig. 2.2.3.2(b)], then

Coordinate of a point P in reciprocal space parallel to a rotation axis as the axis of cylindrical coordinates relative to the origin of reciprocal space

2:2:3:13 2:2:3:14

Here, D is the radius of curvature of the cylinder assuming that the crystal is at the centre of curvature. In the three geometries mentioned here, detector-misalignment errors have to be considered. These are three orthogonal angular errors, translation of the origin, and error in the crystalto-®lm distance.

The equatorial plane is the plane normal to the rotation axis.

The notation now follows that of Arndt & Wonacott (1977) for the coordinates of a spot on the ®lm or detector. ZF is parallel to the rotation axis and . YF is perpendicular to the rotation axis and the beam. IT II (1959, p. 177) follows the convention of y being parallel and x perpendicular to the rotation-axis direction, i.e. YF ; ZF x; y. The advantage of the YF ; ZF notation is that the x-axis direction is then the same as the X-ray beam direction.

Fig. 2.2.3.3. The rotation method. De®nition of coordinate systems. [Cylindrical coordinates of a relp P (; ; ') are de®ned relative to the axial system X0 Y0 Z0 which rotates with the crystal.] The axial system XYZ is de®ned such that X is parallel to the incident beam and Z is coincident with the rotation axis. From Arndt & Wonacott (1977).

32


2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES 3 2 1 0 0 7 6 2:2:3:26 Ux 4 0 cos 'x sin 'x 5 0 sin 'x cos 'x 2 3 cos 'y 0 sin 'y 6 7 0 1 0 Uy 4 2:2:3:27 5 sin ' 0 cos ' y y 2.2.3.3. Relationship of reciprocal-lattice coordinates to crystal 2 3 system parameters sin 'z 0 cos 'z 6 7 2:2:3:28 cos 'z 0 5; Uz 4 sin 'z The reciprocal-lattice coordinates, ; ; ; , etc. used earlier, refer to an axial system ®xed to the crystal, X0 Y0 Z0 of Fig. 0 0 1 2.2.3.3. Clearly, a given relp needs to be brought into the Ewald sphere by the rotation about the rotation axis. The treatment here where 'x , 'y , and 'z are angles around the X0 , Y0 , and Z0 follows Arndt & Wonacott (1977). axes, respectively. The rotation angle required, ', is with respect to some Hence, the relationship between X0 and h is reference `zero-angle' direction and is determined by the X 0 Uz Uy Ux MAh: 2:2:3:29 particular crystal parameters. It is necessary to de®ne a standard orientation of the crystal (i.e. datum) when ' 0 . If we de®ne an axial system X0 Y0 Z0 ®xed to the crystal and a laboratory axis 2.2.3.4. Maximum oscillation angle without spot overlap system XYZ with X parallel to the beam and Z coincident with the For a given oscillation photograph, there is maximum value of rotation axis then ' 0 corresponds to these axial systems the oscillation range, ', that avoids overlapping of spots on a being coincident (Fig. 2.2.3.3). ®lm. The overlap is most likely to occur in the region of the The angle of the crystal at which a given relp diffracts is diffraction pattern perpendicular to the rotation axis and at the maximum Bragg angle. This is where relp's pass through the Ewald sphere with the greatest velocity. For such a separation 2 2 4 1=2 2y 4y0 4x0 d between successive relp's of a , then the maximum allowable : 2:2:3:21 tan'=2 0 d 2 2x0 rotation angle to avoid spatial overlap is given by a The two solutions correspond to the two rotation angles at which ; 2:2:3:30 'max dmax the relp P cuts the sphere of re¯ection. Note that YF , ZF (Subsection 2.2.3.2) are independent of '. where is the sample re¯ecting range (see Section 2.2.7). The values of x0 and y0 are calculated from the particular ' max is a function of ', even in the case of identical cell crystal system parameters. The relationships between the parameters. This is because it is necessary to consider, for a coordinates x0 , y0 , z0 and and are given orientation, the relevant reciprocal-lattice vector perpen dicular to dmax . In the case where the cell dimensions are quite 1=2 2 2 different in magnitude (excluding the axis parallel to the rotation 2:2:3:22 x0 y0 ; axis), then 'max is a marked function of the orientation. z0 : 2:2:3:23 In rotation photography, as large an angle as possible is used up to 'max . This reduces the number of images that need to be processed and the number of partially stimulated re¯ections per X0 can be related to the crystal parameters by image but at the expense of signal-to-noise ratio for individual spots, which accumulate more background since < 'max . In 2:2:3:24 the case of a CCD detector system, ' is chosen usually to be X0 Ah:

The coordinates YF and ZF are related to ®lm-scanner raster units via a scanner-rotation matrix and translation vector. This is necessary because the ®lm is placed arbitrarily on the scanner drum. Details can be found in Rossmann (1985) or Arndt & Wonacott (1977).

A is a crystal-orientation matrix de®ning the standard datum orientation of the crystal. For example, if, by convention, a is chosen as parallel to the X-ray beam at ' 0 and c is chosen as the rotation axis, then, for the general case, 2

a A4 0 0

b cos b sin 0

3 c cos c sin cos 5: c

2:2:3:25

If the crystal is mounted on the goniometer head differently from this then A can be modi®ed by another matrix, M, say, or the terms permuted. This exercise becomes clear if the reader takes an orthogonal case 90 . For the general case, see Higashi (1989). The crystal will most likely be misaligned (slightly or grossly) from the ideal orientation. To correct for this, the misorientation matrices Ux , Uy , and Uz are introduced, i.e.

Fig. 2.2.3.4. The rotation method. The blind region associated with a single rotation axis. From Arndt & Wonacott (1977).

33


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION 2.2.4.1. General

less than so as to optimize the signal-to-noise ratio of the measurement and to sample the rocking-width pro®le. The value of , the crystal rocking width for a given hkl, depends on the reciprocal-lattice coordinates of the hkl relp (see Section 2.2.7). In the region close to the rotation axis, is large. In the introductory remarks to the monochromatic methods used, it has already been noted that originally the rotation method involved 360 rotations contributing to the diffraction image. Spot overlap led to loss of re¯ection data and encouraged Bernal and Weissenberg to devise improvements. With modern synchrotron techniques, the restriction on 'max (equation 2.2.3.30) can be relaxed for special applications. For example, since the spot overlap that is to be avoided involves relp's from adjacent reciprocal-lattice planes, the different Miller indices hkl and h l, k, l do lead in fact to a small difference in Bragg angle. With good enough collimation, a small spot size exists at the detector plane so that the two spots can be resolved. For a standard-sized detector, this is practical for low-resolution data recording. This can be a useful complement to the Laue method where the low-resolution data are rather sparsely stimulated and also tend to occur in multiple Laue spots. Alternatively, a much larger detector can be contemplated and even medium-resolution data can be recorded without major overlap problems. These techniques are useful in some time-resolved applications. For a discussion see Weisgerber & Helliwell (1993). For regular data collection, however, narrow angular ranges are still generally preferred so as to reduce the background noise in the diffraction images and also to avoid loss of any data because of spot overlap.

The conventional Weissenberg method uses a moving ®lm in conjunction with the rotation of the crystal and a layer-line screen. This allows: (a) A larger rotation range of the crystal to be used (say 200 ), avoiding the problem of overlap of re¯ections (referred to in Subsection 2.2.3.4 on oscillation photography). (b) Indexing of re¯ections on the photograph to be made by inspection. The Weissenberg method is not widely used now. In smallmolecule crystallography, quantitative data collection is usually performed by means of a diffractometer. Weissenberg geometry has been revived as a method for macromolecular data collection (Sakabe, 1983, 1991), exploiting monochromatized synchrotron radiation and the image plate as detector. Here the method is used without a layer-line screen where the total rotation angle is limited to 15 ; this is a signi®cant increase over the rotation method with a stationary ®lm. The use of this effectively avoids the presence of partial re¯ections and reduces the total number of exposures required. Provided the Weissenberg camera has a large radius, the X-ray background accumulated over a single spot is actually not serious. This is because the X-ray background decreases approximately according to the inverse square of the distance from the crystal to the detector. The following Subsections 2.2.4.2 and 2.2.4.3 describe the standard situation where a layer-line screen is used. 2.2.4.2. Recording of zero layer

2.2.3.5. Blind region

Normal-beam geometry (i.e. the X-ray beam perpendicular to the rotation axis) is used to record zero-layer photographs. The ®lm is held in a cylindrical cassette coaxial with the rotation axis. The centre of the gap in a screen is set to coincide with the zerolayer plane. The coordinate of a spot on the ®lm measured parallel (ZF ) and perpendicular (YF ) to the rotation axis is given by

In normal-beam geometry, any relp lying close to the rotation axis will not be stimulated at all. This situation is shown in Fig. 2.2.3.4. The blind region has a radius of min dmax sin max

l2 ; 2 2dmin

2:2:3:31

and is therefore strongly dependent on dmin but can be ameliorated by use of a short l. Shorter l makes the Ewald sphere have a larger radius, i.e. its surface moves closer to the Ê resolution, approximately 5% of rotation axis. At Cu K for 2 A the data lie in the blind region according to this simple geometrical model. However, taking account of the rocking width , a greater percentage of the data than this is not fully sampled except over very large angular ranges. The actual increase in the blind-region volume due to this effect is minimized by use of a collimated beam and a narrow spectral spread (i.e. ®nely monochromatized, synchrotron radiation) if the crystal is not too mosaic. These effects are directly related to the Lorentz factor, L 1=sin2 2

2 1=2 :

2 D 360 ZF '=f ;

YF

2:2:4:2

where ' is the rotation angle of the crystal from its initial setting, f is the coupling constant, which is the ratio of the crystal rotation angle divided by the ®lm cassette translation distance, in min 1 , and D is the camera radius. Generally, the values of f and D are 2 min 1 and 28.65 mm, respectively. 2.2.4.3. Recording of upper layers Upper-layer photographs are usually recorded in equi-inclination geometry [i.e. in equations (2.2.3.7) and (2.2.3.8)]. The X-ray-beam direction is made coincident with the generator of the cone of the diffracted beam for the layer concerned, so that the incident and diffracted beams make equal angles () with the equatorial plane, where

2:2:3:32

It is inadvisable to measure a re¯ection intensity when L is large because different parts of a spot would need a different Lorentz factor. The blind region can be ®lled in by a rotation about another axis. The total angular range that is needed to sample the blind region is 2max in the absence of any symmetry or max in the case of mm symmetry (for example).

sin

1

n =2:

2:2:4:3

The screen has to be moved by an amount s tan ;

2:2:4:4

where s is the screen radius. If the cassette is held in the same position as the zero-layer photograph, then re¯ections produced by the same orientation of the crystal will be displaced

2.2.4. Weissenberg geometry Weissenberg geometry (Weissenberg, 1924) is dealt with in the books by Buerger (1942) and Woolfson (1970), for example.

D tan 34


2:2:4:1

2:2:4:5

2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES 2.2.5.2. Crystal setting

Table 2.2.5.1. The distance displacement (in mm) measured on the ®lm versus angular setting error of the crystal for a screenless 5 ) setting photograph precession ( Angular correction, ", in degrees and minutes 0 150 300 450 600 1 150 1 300 1 450 2

Setting of the crystal for one zone is carried out in two stages. First, a Laue photograph is used for small molecules or a monochromatic still for macromolecules to identify the required zone axis and place it parallel to the X-ray beam. This is done by adjustment to the camera-spindle angle and the goniometer-head arc in the horizontal plane. This procedure is usually accurate to a degree or so. Note that the vertical arc will only rotate the pattern around the X-ray beam. Second, a screenless precession photograph is taken using an angle of 7±10 for small molecules or 2±3 for macromolecules. It is better to use un®ltered radiation, as then the edge of the zero-layer circle is easily visible. Let the difference of the distances from the centre of the pattern to the opposite edges of the trace in the direction of displacement be called D so that for the horizontal goniometer-head arc and the dial: arc xRt xLt and dial yUp yDn (Fig. 2.2.5.1). The corrections " to the arc and camera spindle are given by

Distance displacement (mm) for three crystal-to-®lm distances r.l.u

60 mm

0 0.0175 0.035 0.0526 0.070 0.087 0.105 0.123 0.140

0 1.1 2.1 3.2 4.2 5.2 6.3 7.4 8.4

75 mm 0 1.3 2.6 4.0 5.3 6.5 7.9 9.2 10.5

100 mm 0 1.8 3.5 5.3 7.0 8.7 10.5 12.3 14.0

Alternatively, =D ' sin 4" can be used if " is small [from equation (2.2.5.1)]. Notes (1) A value of of 5 is assumed although there is a negligible variation in " with between 3 (typical for proteins) and 7 (typical for small molecules). (2) Crystal-to-®lm distances on a precession camera are usually settable at the ®xed distance D 60, 75, and 100 mm. (3) This table should be used in conjunction with Fig. 2.2.5.1. (4) Values of " are given in intervals of 50 as this is convenient for various goniometer heads which usually have verniers in 50 , 60 or 100 units. The vernier on the spindle of the precession camera is often in 20 units.

sin 4" cos in r:l:u:; D cos2 2" sin2

where D is the crystal-to-®lm angle. It is possible to measure corresponds to 140 error for 2.2.5.1, based on IT II (1959,

2:2:5:1

distance and is the precession to about 0.3 mm ( 1 mm D 60 mm and ' 7 [Table p. 200)].

2.2.5.3. Recording of zero-layer photograph Before the zero-layer photograph is taken, an Nb ®lter (for Mo K) or an Ni ®lter (for Cu K) is introduced into the X-ray beam path and a screen is placed between the crystal and the ®lm at a distance from the crystal of

relative to the zero-layer photograph. This effect can be eliminated by initial translation of the cassette by D tan .

s rs cot ;

2:2:5:2

where rs is the screen radius. Typical values of would be 20 for a small molecule with Mo K and 12±15 for a protein with Cu K. The annulus width in the screen is chosen usually as 2±3 mm for a small molecule and 1±2 mm for a macromolecule. A clutch slip allows the camera motor to be disengaged and the precession motion can be executed under hand control to check for fouling of the goniometer head, crystal, screen or ®lm cassette; s and rs need to be selected so as to avoid this happening. The zero-layer precession photograph produced has a radius of 2D sin corresponding to a resolution limit The distance between spots A is related to the dmin l=2 sin . reciprocal-cell parameter a by the formula

2.2.5. Precession geometry The main book dealing with the precession method is that of Buerger (1964). 2.2.5.1. General The precession method is used to record an undistorted representation of a single plane of relp's and their associated intensities. In order to achieve this, the crystal is carefully set so that the plane of relp's is perpendicular to the X-ray beam. The normal to this plane, the zone axis, is then precessed about the X-ray-beam axis. A layer-line screen allows only relp's of the plane of interest to pass through to the ®lm. The motion of the crystal, screen, and ®lm are coupled together to maintain the coplanarity of the ®lm, screen, and zone.

a

A : D

2:2:5:3

2.2.5.4. Recording of upper-layer photographs The recording of upper-layer photographs involves isolating the net of relp's at a distance from the zero layer of n nl=b, where b is the case of the b axis antiparallel to the X-ray beam. In order to determine n , it is generally necessary to record a cone-axis photograph. If the cell parameters are known, then the camera settings for the upper-level photograph can be calculated directly without the need for a cone-axis photograph. In the upper-layer precession photograph, the ®lm is advanced towards the crystal by a distance

Fig. 2.2.5.1. The screenless precession setting photograph (schematic) and associated mis-setting angles for a typical orientation error when the crystal has been set previously by a monochromatic still or Laue.

Dn and the screen is placed at a distance 35


2:2:5:4

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION sn rs cot n rs cot cos 1 cos

n :

The purpose of the diffractometer goniostat is to bring a selected re¯ected beam into the detector aperture or a number of re¯ected beams onto an area detector of limited aperture (i.e. an aperture that does not intercept all the available diffraction spots at one setting of the area detector) [see Hamlin (1985, p. 431), for example]. Since the use of electronic area detectors is now increasingly common, the properties of these detectors that relate to the geometric prediction of spot position will be described later. The single-counter diffractometer is primarily used for smallmolecule crystallography. In macromolecular crystallography, many relp's are almost simultaneously in the diffraction condition. The single-counter diffractometer was extended to ®ve separate counters [for a review, see Artymiuk & Phillips (1985)], then subsequently to a multi-element linear detector [for a review, see Wlodawer (1985)]. Area detectors offer an even larger aperture for simultaneous acquisition of re¯ections [Hamlin et al. (1981), and references therein]. Large-area on-line image-plate systems are now available commercially to crystallographers, whereby the problem of the limited aperture of electronic area detectors is circumvented and the need for a goniostat is relaxed so that a single axis of rotation can be used. Systems like the R-AXISIIc (Rigaku Corporation) and the MAR (MAR Research Systems) fall into this category, utilizing IP technology and an on-line scanner. A next generation of device beckons, involving CCD area detectors. These offer a much faster duty cycle and greater sensitivity than IP's. By tiling CCD's together, a larger-area device can be realized. However, it is likely that these will be used in conjunction with a three-axis goniostat again, except in special cases where a complete area coverage can be realized.

2:2:5:5

The resulting upper-layer photograph has outer radius Dsin n sin

2:2:5:6

and an inner blind region of radius Dsin n

sin :

2:2:5:7

2.2.5.5. Recording of cone-axis photograph A cone-axis photograph is recorded by placing a ®lm enclosed in a light-tight envelope in the screen holder and using a small precession angle, e.g. 5 for a small molecule or 1 for a protein. The photograph has the appearance of concentric circles centred on the origin of reciprocal space, provided the crystal is perfectly aligned. The radius of each circle is rn s tan n ;

2:2:5:8

where cos n cos Hence, n cos

n :

2:2:5:9

cos tan 1 rn =s. 2.2.6. Diffractometry

The main book dealing with single-crystal diffractometry is that of Arndt & Willis (1966). Hamilton (1974) gives a detailed treatment of angle settings for four-circle diffractometers. For details of area-detector diffractometry, see Howard, Nielsen & Xuong (1985) and Hamlin (1985). 2.2.6.1. General

2.2.6.2. Normal-beam equatorial geometry

In this section, we describe the following related diffractometer con®gurations: (a) normal-beam equatorial geometry [!; ; ' option or !; ; ' (kappa) option]; (b) ®xed 45 geometry with area detector. (a) is used with single-counter detectors. The kappa option is also used in the television area-detector system of Enraf±Nonius (the FAST). (b) is used with the multiwire proportional chamber, XENTRONICS, system. (FAST is a trade name of Enraf± Nonius; XENTRONICS is a trade name of Siemens.)

In normal-beam equatorial geometry (Fig. 2.2.6.1), the crystal is oriented speci®cally so as to bring the incident and re¯ected beams, for a given relp, into the equatorial plane. In this way, the detector is moved to intercept the re¯ected beam by a single rotation movement about a vertical axis (the 2 axis). The value of is given by Bragg's law as sin 1 (d =2). In order to bring d into the equatorial plane (i.e. the Bragg plane into the meridional plane), suitable angular settings of a three-axis goniostat are necessary. The convention for the sign of the angles given in Fig. 2.2.6.1 is that of Hamilton (1974); his choice of sign of 2 is adhered to despite the fact that it is left-handed, but in any case the signs of !; ; ' are standard right-handed. The

Fig. 2.2.6.2. Diffractometry with normal-beam equatorial geometry and angular motions !; and '. The relp at P is moved to Q via ', from Q to R via , and R to S via !. From Arndt & Willis (1966). In this speci®c example, the ' axis is parallel to the ! axis (i.e. 0 ).

Fig. 2.2.6.1 Normal-beam equatorial geometry: the angles !; ; ', 2 are drawn in the convention of Hamilton (1974).

36


2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES ± collimation; ± monochromators; ± mirrors. An exhaustive survey is not given, since a wide range of con®gurations is feasible. Instead, the commonest arrangements are covered. In addition, conventional X-ray sources are separated from synchrotron X-ray sources. The important difference in the treatment of the two types of source is that on the synchrotron the position and angle of the photon emission from the relativistic charged particles are correlated. One result of this, for example, is that after monochromatization of the synchrotron radiation (SR) the wavelength and angular direction of a photon are correlated. The angular re¯ecting range and diffraction-spot size are determined by the physical state of the beam and the sample. Hence, the idealized situation considered earlier of a point sample and zero-divergence beam will be relaxed. Moreover, the effects of the detector-imaging characteristics are considered, i.e. obliquity, parallax, point-spread factor, and spatial distortions.

speci®c reciprocal-lattice point can be rotated from point P to point Q by the ' rotation, from Q to R via , and R to S via ! (see Fig. 2.2.6.2). In the most commonly used setting, the plane bisects the incident and diffracted beams at the measuring position. Hence, the vector d lies in the plane at the measuring position. However, since it is possible for re¯ection to take place for any orientation of the re¯ecting plane rotated about d , it is feasible therefore that d can make any arbitrary angle " with the plane. It is conventional to refer to the azimuthal angle of the re¯ecting plane as the angle of rotation about d . It is possible with a scan to keep the hkl re¯ection in the diffraction condition and so to measure the sample absorption surface by monitoring the variation in intensity of this re¯ection. This scan is achieved by adjustment of the !; ; ' angles. When 90 , the scan is simply a ' scan and " is 0 . The circle is a relatively bulky object whose thickness can inhibit the measurement of diffracted beams at high . Also, collision of the circle with the collimator or X-ray-tube housing has to be avoided. An alternative is the kappa goniostat geometry. In the kappa diffractometer [for a schematic picture, see Wyckoff (1985, p. 334)], the axis is inclined at 50 to the ! axis and can be rotated about the ! axis; the axis is an alternative to therefore. The ' axis is mounted on the axis. In this way, an unobstructed view of the sample is achieved.

2.2.7.2. Conventional X-ray sources: spectral character, crystal rocking curve, and spot size An extended discussion of instrumentation relating to conventional X-ray sources is given in Arndt & Willis (1966) and Arndt & Wonacott (1977). Witz (1969) has reviewed the use of monochromators for conventional X-ray sources. It is generally the case that the K line has been used for single-crystal data collection via monochromatic methods. The continuum Bremsstrahlung radiation is used for Laue photography at the stage of setting crystals. The emission lines of interest consist of the K1 , K2 doublet and the K line. The intrinsic spectral width of the K1 , or K2 line is 10 4 , their separation (l=l) is 2:5 10 3 , and they are of different relative intensity. The K line is eliminated either by use of a suitable metal ®lter or by a monochromator. A mosaic monochromator such as graphite passes the K1 , K2 doublet in its entirety. The monochromator passes a certain, if small, component of a harmonic of the K1 , K2 line extracted from the Bremsstrahlung. This latter effect only becomes important in circumstances where the attenuated main beam is used for calibration; the process of attenuation enhances the shortwavelength harmonic component to a signi®cant degree. In diffraction experiments, this component is of negligible intensity. The polarization correction is different with and without a monochromator (see Chapter 6.2). The effect of the doublet components of the K emission is to cause a peak broadening at high angles. From Bragg's law, the following relationship holds for a given re¯ection:

2.2.6.3. Fixed 45 geometry with area detector The geometry with ®xed 45 was introduced by Nicolet and is now fairly common in the ®eld. It consists of an ! axis, a ' axis, and ®xed at 45 . The rotation axis is the ! axis. In this con®guration, it is possible to sample a greater number of independent re¯ections per degree of rotation (Xuong, Nielsen, Hamlin & Anderson, 1985) because of the generally random nature of any symmetry axis. An alternative method is to mount the crystal in a precise orientation and to use the ' axis to explore the blind region of the single rotation axis. It is feasible to place the capillary containing the sample in a vertically upright position via a 135 bracket mounted on the goniometer head. The bulk of the data is collected with the ! axis coincident with the capillary axis. This setting is bene®cial to make the effect of capillary absorption symmetrical. At the end of this run, the blind region whose axis is coincident with the ! axis can be ®lled in by rotating around the ' axis by 180 . This renders the capillary axis horizontal and a different crystal axis vertical. Hence by rotation about this new crystal axis by max , the blind region can be sampled. 2.2.7. Practical realization of diffraction geometry: sources, optics, and detectors

2.2.7.1. General The tools required for making the necessary measurement of re¯ection intensities include (a) beam-conditioning items; (b) crystal goniostat; (c) detectors. In this section, we describe the common con®gurations for de®ning precise states of the X-ray beam. The topic of detectors is dealt with in Part 7 (see especially Section 7.1.6). The impact of detector distortions on diffraction geometry is dealt with in Subsection 2.2.7.4. Within the topic of beam conditioning the following subtopics are dealt with:

2:2:7:1

For proteins where is relatively small, the effect of the K1 , K2 separation is not signi®cant. For small molecules, which diffract to higher resolution, this is a signi®cant effect and has to be accounted for at high angles. The width of the rocking curve of a crystal re¯ection is given by (Arndt & Willis, 1966) af l tan 2:2:7:2 s l when the crystal is fully bathed by the X-ray beam, where a is the crystal size, f the X-ray tube focus size (foreshortened), s the 37


l tan : l

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Spot-to-spot spatial resolution can be enhanced by use of focusing mirrors, which is especially important for large-protein and virus crystallography, where long cell axes occur. The effect is achieved by focusing the beam on the detector, thereby changing a divergence from the source into a convergence to the detector. In the absence of absorption, at grazing angles, X-rays up to a certain critical energy are re¯ected. The critical angle c is given by 2 1=2 e N l; 2:2:7:3 c mc2

distance between the X-ray tube focus and the crystal, and the crystal mosaic spread (Fig. 2.2.7.1). In the moving-crystal method, is the minimum angle through which the crystal must be rotated, for a given re¯ection, so that every mosaic block can diffract radiation covering a ®xed wavelength band l from every point on the focal spot. This angle can be controlled to some extent, for the protein case, by collimation. For example, with a collimator entrance slit placed as close to the X-ray tube source and a collimator exit slit placed as close to the sample as possible, the value of a f =s can approximately be replaced by a0 f 0 =s0 , where f 0 is the entrance-slit size, a0 is the exit-slit size, and s0 the distance between them. Clearly, for a0 < a, the whole crystal is no longer bathed by the X-ray beam. In fact, by simply inserting horizontal and vertical adjustable screws at the front and back of the collimator, adjustment to the horizontal and vertical divergence angles can be made. The spot size at the detector can be calculated approximately by multiplying the particular re¯ection rocking angle by the distance from the sample to the spot on the detector. In the case of a single-counter diffractometer, tails on a diffraction spot can be eliminated by use of a detector collimator.

where N is the number of free electrons per unit volume of the re¯ecting material. The higher the atomic number of a given material then the larger is c for a given critical wavelength. The product of mirror aperture with re¯ectivity gives a ®gure of merit for the mirror as an ef®cient optical element. The use of a pair of perpendicular curved mirrors set in the horizontal and vertical planes can focus the X-ray tube source to a small spot at the detector. The angle of the mirror to the incident beam is set to reject the K line (and shorter-wavelength Bremsstrahlung). Hence, spectral purity at the sample and diffraction spot size at the detector are improved simultaneously. There is some loss of intensity (and lengthening of exposure time) but the overall signal-to-noise ratio is improved. The primary reason for doing this, however, is to enhance spot-tospot spatial resolution even with the penalty of the exposure time being lengthened. The rocking width of the sample is not affected in the case of 1:1 focusing (object distance image distance). Typical values are tube focal-spot size, f 0:1 mm, tube-to-mirror and mirror-to-sample distances 200 mm, convergence angle 2 mrad, and focal-spot size at the detector 0:3 mm. To summarize, the con®gurations are (a) beam collimator only; (b) ®lter beam collimator; (c) ®lter beam collimator detector collimator (singlecounter case); (d) graphite monochromator beam collimator; (e) pair of focusing mirrors exit-slit assembly; ( f ) focusing germanium monochromator perpendicular focusing mirror exit-slit assembly. (a) is for Laue mode; (b)±( f ) are for monochromatic mode; ( f ) is a fairly recent development for conventional-source work. 2.2.7.3. Synchrotron X-ray sources In the utilization of synchrotron X-radiation (SR), both Laue and monochromatic modes are important for data collection. The unique geometric and spectral properties of SR renders the treatment of diffraction geometry different from that for a conventional X-ray source. The properties of SR are dealt with in Subsection 4.2.1.5 and elsewhere; see Subject Index. Reviews of instrumentation, methods, and applications of synchrotron radiation in protein crystallography are given by Helliwell (1984, 1992). (a) Laue geometry: sources, optics, sample re¯ection bandwidth, and spot size Laue geometry involves the use of the fully polychromatic SR spectrum as transmitted through the beryllium window that is used to separate the apparatus from the machine vacuum. There is useful intensity down to a wavelength minimum of lc =5, where lc is the critical wavelength of the magnet source. The Ê however, if the crystal maximum wavelength is typically 3 A;

Fig. 2.2.7.1. Re¯ection rocking width for a conventional X-ray source. From Arndt & Wonacott (1977, p. 7). (a) Effect of sample mosaic spread. The relp is replaced by a spherical cap with a centre at the origin of reciprocal space where it subtends an angle . (b) Effect of l=lconv , the conventional source type spectral spread. (c) Effect of a beam divergence angle, . The overall re¯ection rocking width is a combination of these effects.

38


2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES is mounted in a capillary then the glass absorbs the wavelengths Ê beyond 2:6 A. The bandwidth can be limited somewhat under special circumstances. A re¯ecting mirror at grazing incidence can be used for two reasons. First, the minimum wavelength in the beam can be sharply de®ned to aid the accurate de®nition of the Laue-spot multiplicity. Second, the mirror can be used to focus the beam at the sample. The maximum-wavelength limit can be truncated by use of aluminium absorbers of varying thickness or by use of a transmission mirror (Lairson & Bilderback, 1992; Cassetta et al., 1993). The measured intensity of individual Laue diffraction spots depends on the wavelength at which they are stimulated. The problem of wavelength normalization is treated by a variety of methods. These include: (a) use of a monochromatic reference data set; (b) use of symmetry equivalents in the Laue data set recorded at different wavelengths; (c) calibration with a standard sample such as a silicon crystal. Each of these methods produces a `l-curve' describing the relative strength of spots measured at various wavelengths. The methods rely on the incident spectrum being smooth and stable with time. There are discontinuities in the `l-curve' at the bromine and silver K-absorption edges owing to the silver bromide in the photographic emulsion case. The l-curve is therefore usually split up into wavelength regions, i.e. lmin to Ê , 0.49 to 0.92 A Ê , and 0.92 A Ê to lmax . Other detector types 0.49 A have different discontinuities, depending on the material making up the X-ray absorbing medium. [The quanti®cation of conventional-source Laue-diffraction data (Rabinovich & Lourie, 1987; Brooks & Moffat, 1991) requires the elimination of spots recorded near the emission-line wavelengths.] The production and use of narrow-bandpass beams may be of interest, e.g. l=l 0:2. Such bandwidths can be produced by a combination of a re¯ection mirror used in tandem with a transmission mirror. Alternatively, an X-ray undulator of 10±100 periods ideally should yield a bandwidth behind a pinhole of l=l ' 0:1±0:01. In these cases, wavelength normalization is more dif®cult because the actual spectrum over which a re¯ection is integrated is rapidly varying in intensity. The spot bandwidth is determined by the mosaic spread and horizontal beam divergence (since H > V ) as l 2:2:7:4 H cot ; l

LR D sin2 R sec2 2 LT D2 T sin sec 2;

2:2:7:7 2:2:7:8

and

R V cos

H sin

T V sin

H cos ;

2:2:7:9 2:2:7:10

where is the angle between the vertical direction and the radius vector to the spot (see Andrews, Hails, Harding & Cruickshank, 1987). For a crystal that is not too mosaic, the spot size is dominated by Lc . For a mosaic or radiation-damaged crystal, the main effect is a radial streaking arising from , the sample mosaic spread. (b) Monochromatic SR beams: optical con®gurations and sample rocking width A wide variety of perfect-crystal monochromator con®gurations are possible and have been reviewed by various authors (Hart, 1971; Bonse, Materlik & SchroÈder, 1976; Hastings, 1977; Kohra, Ando, Matsushita & Hashizume, 1978). Since the re¯ectivity of perfect silicon and germanium is effectively 100%, multiple-re¯ection monochromators are feasible and permit the tailoring of the shape of the monochromator resolution function, harmonic rejection, and manipulation of the polarization state of the beam. Two basic designs are in common use. These are (a) the bent single-crystal monochromator of triangular shape (Lemonnier, Fourme, Rousseaux & Kahn, 1978) and (b) the double-crystal monochromator.

where sample mosaic spread, assumed to be isotropic, H horizontal cross-®re angle, which in the absence of focusing is xH H =P, where xH is the horizontal sample size and H the horizontal source size, and P is the sample to the tangent-point distance; and similarly for V in the vertical direction. Generally, at SR sources, H is greater than V . When a focusing-mirror element is used, H and/or V are convergence angles determined by the focusing distances and the mirror aperture. The size and shape of the diffraction spots vary across the ®lm. The radial spot length is given by convolution of Gaussians as L2R L2c sec2 21=2

2:2:7:5

L2T L2c 1=2 ;

2:2:7:6

and tangentially by Fig. 2.2.7.2. Single-crystal monochromator illuminated by synchrotron radiation: (a) ¯at crystal, (b) Guinier setting, (c) overbent crystal, (d) effect of source size (shown at the Guinier setting for clarity). From Helliwell (1984).

where Lc is the size of the X-ray beam (assumed circular) at the sample, and 39


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Two types of spectral spread occur with synchrotron and neutron sources. The term l=lconv is the spread that is passed down each incident ray in a divergent or convergent incident beam; the subscript refers to conventional source type. This is because it is similar to the K1 , K2 line widths and separation. At the synchrotron, this component also exists and arises from the monochromator rocking width and ®nite-source-size effects. The term l=lcorr is special to the synchrotron or neutron case. The subscript `corr' refers to the fact that the ray direction can be correlated with the photon or neutron wavelength. Usually, an instrument is set to have l=lcorr 0. In the most general case, for a l=lcorr arising from the horizontal ray direction correlation with photon energy, and the case of a horizontal rotation axis, then the rocking width 'R of an individual re¯ection is given by ( )1=2 2 l d 2 H V2 2"s L; 2:2:7:11 'R L2 l corr

In the case of the single-crystal monochromator, the actual curvature employed is very important in the diffraction geometry. For a point source and a ¯at monochromator crystal, there is a gradual change in the photon wavelength selected from the white beam as the length of the monochromator is traversed [Fig. 2.2.7.2(a)]. For a point source and a curved monochromator crystal, one speci®c curvature can compensate for this variation in incidence angle [Fig. 2.2.7.2(b)]. The re¯ected spectral bandwidth is then at a minimum; this setting is known as the `Guinier position'. If the curvature of the monochromator crystal is increased further, a range of photon wavelengths, l=lcorr , is selected along its length so that the rays converging towards the focus have a correlation of photon wavelength and direction [Fig. 2.2.7.2(c)]. The effect of a ®nite source is to cause a change in incidence angle at the monochromator crystal, so that at the focus there is a photon-wavelength gradient across the width of the focus (for all curvatures) [Fig. 2.2.7.2(d)]. The use of a slit in the focal plane is akin to placing a slit at the tangent point to limit the source size. The double-crystal monochromator with two parallel or nearly parallel perfect crystals of germanium or silicon is a common con®guration. The advantage of this is that the outgoing monochromatic beam is parallel to the incoming beam, although it is slightly displaced vertically by an amount 2d cos , where d is the perpendicular distance between the crystals and the monochromator Bragg angle. The monochromator can be rapidly tuned, since the diffractometer or camera need not be re-aligned signi®cantly in a scan across an absorption edge. Between absorption edges, some vertical adjustment of the diffractometer is required. Since the rocking width of the fundamental is broader than the harmonic re¯ections, the strict parallelism of the pair of crystal planes can be relaxed, i.e. detuned so that the harmonic can be rejected with little loss of the fundamental intensity. The spectral spread in the re¯ected monochromatic beam is determined by the source divergence accepted by the monochromator, the angular size of the source, and the monochromator rocking width (see Fig. 2.2.7.3). The double-crystal monochromator is often used with a toroid focusing mirror; the functions of monochromatization are then separated from the focusing (Hastings, Kincaid & Eisenberger, 1978). The rocking width of a re¯ection depends on the horizontal and vertical beam divergences/convergences (after due account for collimation is taken) H and V , the spectral spreads l=lconv and l=lcorr , and the mosaic spread . We assume that !, where ! is the angular broadening of a relp due to a ®nite sample. In the case of synchrotron radiation, H and V are usually widely asymmetric. On a conventional source, usually

H ' V .

where

2:2:7:12

and L is the Lorentz factor 1=sin2 2 2 1=2 . The Guinier setting of the instrument gives l=lcorr 0. The equation for 'R then reduces to 'R L2 H2 V2 =L2 1=2 2"s

2:2:7:13

(from Greenhough & Helliwell, 1982). For example, for 0,

V 0:2 mrad (0.01 ), 15 , l=lconv 1 10 3 and 0:8 mrad (0.05 ), then 'R 0:08 . But 'R increases as increases [see Greenhough & Helliwell (1982, Table 5)]. In the rotation=oscillation method as applied to protein and virus crystals, a small angular range is used per exposure

Fig. 2.2.7.4. The rocking width of an individual re¯ection for the case of Fig. 2.2.7.2(c) and a vertical rotation axis. 'R is determined by the passage of a spherical volume of radius "s (determined by sample mosaicity and a conventional-source-type spectral spread) through a nest of Ewald spheres of radii set by 12 l=lcorr and the horizontal convergence angle H . From Greenhough & Helliwell (1982).

Fig. 2.2.7.3. Double-crystal monochromator illuminated by synchrotron radiation. The contributions of the source divergence V and angular source size source to the range of energies re¯ected by the monochromator are shown.

40


d cos l "s tan 2 l conv

2.2. SINGLE-CRYSTAL X-RAY TECHNIQUES (Subsection 2.2.3.4). For example, 'max may be 1.5 for a protein, and 0.4 or so for a virus. Many re¯ections will be only partially stimulated over the exposure. It is important, especially in the virus case, to predict the degree of penetration of the relp through the Ewald sphere. This is done by analysing the interaction of a spherical volume for a given relp with the Ewald sphere. The radius of this volume is given by E'

'R 2L

In general, we can take account of obliquity and parallax effects whereby the measured spot width, in the radial direction, is w 00 , where w 00 w sec 2 geff tan 2:

As well as changing the spot size, the spot position, i.e. its centre, is also changed by both obliquity and parallax effects by 1 00 w. The spherical drift-chamber design eliminated the 2 w effects of parallax (Charpak, Demierre, Kahn, Santiard & Sauli, 1977). In the case of a phosphor-based television system, the X-rays are converted into visible light in a thin phosphor layer so that parallax is negligible.

2:2:7:14

(Greenhough & Helliwell, 1982). For discussions, see Harrison, Winkler, Schutt & Durbin (1985) and Rossmann (1985). In Fig. 2.2.7.4, the relevant parameters are shown. The diagram shows l=lcorr 2 in a plane, usually horizontal, with a perpendicular (vertical) rotation axis, whereas the formula for 'R above is for a horizontal axis. This is purely for didactic reasons since the interrelationship of the components is then much clearer. For full details, see Greenhough & Helliwell (1982).

(c) Point-spread factor Even at normal incidence, there will be some spreading of the beam size. This is referred to as the point-spread factor, i.e. a single pencil ray of light results in a ®nite-sized spot. In the TVdetector and image-plate cases, the graininess of the phosphor and the system imaging the visible light contribute to the pointspread factor. In the case of a charge-coupled device (CCD) used in direct-detection mode, i.e. X-rays impinging directly on the silicon chip, the point-spread factor is negligible for a spot of typical size. For example, in Laue mode with a CCD used in this way, a 200 mm diameter spot normally incident on the device is not measurably broadened. The pixel size is 25 mm. The size of such a device is small and it is used in this mode for looking at portions of a pattern.

2.2.7.4. Geometric effects and distortions associated with area detectors Electronic area detectors are real-time image-digitizing devices under computer control. The mechanism by which an X-ray photon is captured is different in the various devices available (i.e. gas chambers, television detectors, chargecoupled devices) and is different speci®cally from ®lm or image plates. Arndt (1986 and Section 7.1.6) has reviewed the various devices available, their properties and performances. Section 7.1.8 deals with storage phosphors/image plates.

(d) Spatial distortions The spot position is affected by spatial distortions. These nonlinear distortions of the predicted diffraction spot positions have to be calibrated for independently; in the worst situations, misindexing would occur if no account were taken of these effects. Calibration involves placing a geometric plate, containing a perfect array of holes, over the detector. The plate is illuminated, for example, with radiation from a radioactive source or scattered from an amorphous material at the sample position. The measured positions of each of the resulting `spots' in detector space (units of pixels) can be related directly to the expected position (in mm). A 2D, non-linear, pixel-to-mm and mm-to-pixel correction curve or look-up table is thus established. These are the special geometric effects associated with the use of electronic area detectors compared with photographic ®lm or the image plate. We have not discussed non-uniformity of response of detectors since this does not affect the geometry. Calibration for non-uniformity of response is discussed in Section 7.1.6.

(a) Obliquity In terms of the geometric reproduction of a diffraction-spot position, size, and shape, photographic ®lm gives a virtually true image of the actual diffraction spot. This is because the emulsion is very thin and, even in the case of double-emulsion ®lm, the thickness, g, is only 0:2 mm. Hence, even for a diffracted ray inclined at 2 45 to the normal to the ®lm plane, the `parallax effect', g tan 2, is very small (see below for details of when this is serious). With ®lm, the spot size is increased owing to oblique or non-normal incidence. The obliquity effect causes a beam, of width w, to be recorded as a spot of width w 0 w sec 2:

2:2:7:15

For example, if w 0:5 mm and 2 45 , then w 0 is 0.7 mm. With an electronic area detector, obliquity effects are also present. In addition, the effects of parallax, point-spread factor, and spatial distortions have to be considered. (b) Parallax In the case of a one-atmosphere xenon-gas chamber of thickness g 10 mm, the g tan 2 parallax effect is dramatic [see Hamlin (1985, p. 435)]. The wavelength of the beam has to Ê is used with such a chamber, the be considered. If a l of 1 A photons have a signi®cant probability of fully traversing such a gap and parallax will be at its worst; the spot is elongated and the spot centre will be different from that predicted from the Ê is used geometric centre of the diffracted beam. If a l of 1.54 A then the penetration depth is reduced and an effective g, i.e. geff , of 3 mm would be appropriate. The use of higher pressure in a chamber increases the photon-capture probability, thus reducing Ê parallax is geff pro rata; at four atmospheres and l 1:54 A, very small.

Acknowledgements I am very grateful to various colleagues at the Universities of York and Manchester for their comments on the text of the ®rst edition. However, special thanks go to Dr T. Higashi who commented extensively on the manuscript and found several errors. Any remaining errors are, of course, my own responsibility. Dr. F. C. Korber is thanked for his comments on the diffractometry section. Dr W. Parrish and Mrs E. J. Dodson are also thanked for discussions. Mrs Y. C. Cook is thanked for typing several versions of the manuscript and Mr A. B. Gebbie is thanked for drawing the diagrams. I am grateful to Miss Julie Holt for secretarial help in the production of the second edition. 41


2:2:7:16


2.3. Powder and related techniques: X-ray techniques By W. Parrish and J. I. Langford

diffraction ®le and made it possible to develop systematic methods of analysis. A major advance in the powder method began in the early 1950's with the introduction of commercial high-resolution diffractometers which greatly expanded the use of the method (Parrish, 1949; Parrish, Hamacher & Lowitzsch, 1954). The replacement of ®lm by the Geiger counter, and soon after by scintillation and proportional counters, made it possible to observe X-ray diffraction in real time and to make precision measurements of the intensities and pro®le shapes. The large space around the specimen permitted the design of various devices to vary the specimen temperature and apply stress as well as other experiments not possible in a powder camera. The much higher resolution, angular accuracy, and pro®le determination led to many advances in the interpretation and applications of the method. Powder diffraction began to be used in a large number of technical disciplines and thousands of papers have been published on material structure characterization in inorganic chemistry, mineralogy, metals and alloys, ceramics, polymers, and organic materials. The following is a partial list of the types of studies that are best performed by the powder method and are widely used: ±identi®cation of crystalline phases ±qualitative and quantitative analysis of mixtures and minor constituents ±distinction between crystalline and amorphous states and devitri®cation ±following solid-state reactions ±identi®cation of solid solutions ±isomorphism, polymorphism, and phase-diagram determination ±lattice-parameter measurement and thermal expansion ±preferred orientation ±microstructure (crystallite size, strain, stacking faults, etc.) from pro®le broadening ±in situ high-/low-temperature and high-pressure studies. The introduction of computers for automation and data reduction and the use of synchrotron radiation are greatly expanding the information that can be obtained from the method. The determination and re®nement of crystal structures from powder data are widely used for materials not available as single crystals. The most comprehensive book on the powder method is that of Klug & Alexander (1974), which contains an extensive bibliography. Peiser, Rooksby & Wilson (1955) edited a book written by specialists on various powder methods (mainly cameras), interpretations, and results in various ®elds. AzaÂroff & Buerger (1958) wrote a comprehensive description of the powder-camera method. Barrett & Massalski (1980), Taylor (1961), and Cullity (1978) wrote comprehensive texts on metallurgical applications. Warren (1969), Guinier (1956, 1963), and Schwartz & Cohen (1987) described the theory and application of powder methods to physical problems such as the use of Fourier methods to study deformed metals and alloys to separate crystallite size and microstrain pro®le broadening, stacking faults, order±disorder, amorphous structures, and temperature effects. A general description of the powder method is given by Lipson & Steeple (1970). A book on the powder method written by a number of authors (Bish & Post, 1989) contains detailed papers and long lists of references and describes recent developments including principles of powder

The X-ray diffraction powder method was developed independently by Debye & Scherrer (1916) and by Hull (1917, 1919) and hence is often named the Debye±Scherrer±Hull method. Their classic papers provide the basis for the powder diffraction method. Debye and Scherrer made a 57 mm diameter cylindrical camera, used two ®lms with each forming a half circle in contact with the camera wall, a light-tight cover, a primary-beam collimator, and a long black paper exit tube attached to the outside of the camera to avoid back scattering. (There were no radiation protection surveys in those days!) The powder specimen was 2 mm in diameter, 10 mm long and the exposure two hours. They worked out a method for determining the crystal structure from the powder diagrams, solved the structure of LiF using X-rays from Cu and Pt targets and found that a powder labelled àmorphous silicon' was crystalline with the diamond structure. Hull described many of the experimental factors. He apparently was the ®rst to use a K ®lter and an intensifying screen; he enclosed the X-ray tube in a lead box, used both ¯at and cylindrical ®lms, and measured the effect of X-ray tube voltage on the intensity of Mo K radiation. He described the importance of using small particle sizes, specimen rotation, and the necessity for random orientation. He also worked out the methods for determining the crystal structure from the powder pattern and solved the structures of eight elements and diamond and graphite. Debye & Scherrer did not explicitly mention the use of the method for identi®cation in their 1916 paper but Hull recognized its importance as shown by the title of his 1919 paper, A new method of chemical analysis, in which he wrote `. . . every crystalline substance gives a pattern; that the same substance always gives the same pattern; and that in a mixture of substances each produces its pattern independently of the others, so that the photograph obtained with a mixture is the superimposed sum of the photographs that would be obtained by exposing each of the components separately for the same length of time. This law applies quantitatively to the intensities of the lines, as well as to their positions, so that the method is capable of development as a quantitative analysis.' In the late 1930's, compilations of X-ray powder data for minerals were published but the most important advance in the practical use of the powder method was made by Hanawalt & Rinn (1936) and Hanawalt, Rinn & Frevel (1938). Their detailed paper, entitled Chemical analysis by X-ray diffraction, contained tabulated d's and relative intensities of 1000 chemical substances. The editor wrote in the prologue Ìndustrial and Engineering Chemistry considers itself fortunate in being able to present herewith a complete, new, workable system of analysis, for it is not often that this is possible in a single issue of any journal.' They devised a scheme for pattern classi®cation based on the d's of the three most intense lines in which the patterns were arranged in 77 groups, each of which contained 77 subgroups. The strongest line determined the group, the second strongest the subgroup, and the third the position within the subgroup. They used Mo K radiation, 8 in radius quadrant cassettes and a direct comparison ®lm intensity scale. These data formed the basis of the early ASTM (later JCPDS and now ICDD) powder 42 Copyright © 2006 International Union of Crystallography 43 s:\ITFC\ch-2-3.3d (Tables of Crystallography)

2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES the specimen set for re¯ection, S(T) for a transmission specimen, and M refers to a focusing re¯ection monochromator. The letters are arranged in order of the beam direction. The detector rotates around the diffractometer axis at twice the speed of the specimen in the ±2 scanning used in (a) to (e). In the Seemann±Bohlin geometry (S-B), the specimen is stationary and the detector rotates around the focusing circle in scanning the pattern ( f ). The line focus of the X-ray tube is used in all cases. The monochromator is either symmetrical, with the lattice planes parallel to the crystal surface, or asymmetric with the lattice planes inclined at a small angle to the surface to shorten one of the focal-length distances. Placing the monochromator in the diffracted beam has the important advantage of eliminating specimen ¯uorescence. It also simpli®es shielding the detector if the specimen is radioactive. The monochromator in the incident beam reduces ¯uorescence and radiation damage to the specimen by removing the continuous X-ray spectrum. When the diffracted beam is de®ned by the receiving slit as in (b) and (c), highly oriented pyrolytic graphite (placed in front of the detector) is generally used to obtain high intensity. In the (d) and (c) geometries, a high-quality bent crystal such as silicon or quartz is necessary to achieve good focusing. (a) S(R): The aperture of the incident divergent beam from the line focus of the X-ray tube F is limited by the entrance slit ES and the re¯ection from the specimen converges (`focuses') on the receiving slit RS. The intensity is determined by the ES and RS and the pro®le width is determined mainly by the RS width. The parallel slits PS in the incident and diffracted beams limit the axial divergence. (b) S(R)=M: Same as (a) with the addition of the symmetrical monochromator (usually graphite) to record only the characteristic radiation. ES and RS have the same roÃle as in (a), and only the incident PS are required. (c) M=S(R): Using an incident-beam monochromator, the slit at F 0 determines the effective source size and divergence of the beam striking the specimen, and RS limits the pro®le width. (d) S(T)=M: The divergent incident beam continues to diverge after diffraction from the transmission specimen and the asymmetric monochromator focuses the beam on the detector.

diffraction (Reynolds, 1989), instrumentation, specimen preparation, pro®le ®tting, synchrotron and neutron methods. A recent book by Jenkins & Snyder (1996) gives a useful comprehensive account of basic methods and practices in powder diffractometry. Papers presented at international symposia on powder diffraction describe advances in the ®eld (Block & Hubbard, 1980; Australian Journal of Physics, 1988; Bojarski & Bol-d, 1979; Bish & Post, 1989; Prince & Stalick, 1992). Papers on the theory and new methods and applications are published in the Journal of Applied Crystallography. Powder Diffraction started in 1986 and contains powder data and papers on instrumentation and methods. Papers presented at the Annual Denver Conference on Applications of X-ray Analysis have been published yearly since 1957 as separate volumes entitled Advances in X-Ray Analysis, by Plenum Press. Volume 37 of the series was published in 1994. These volumes contain papers that are roughly equally divided between X-ray powder diffraction and ¯uorescence analysis. This extensive source describes many types of instrumentation, methods and applications. The Norelco Reporter has been published several times a year since 1954 and contains original articles and reprints of papers on powder diffraction, ¯uorescence analysis, and electron microscopy. There are other `house journals' published by Rigaku, Siemens, and other X-ray companies. A history of the powder method in the USA was written by Parrish (1983). The following description includes only the most frequently used methods. The divergent beam from X-ray tubes is best used with focusing geometries, and synchrotron radiation with parallel-beam optics. 2.3.1. Focusing diffractometer geometries The critical elements in the basic geometries of the principal focusing diffractometers used with X-ray tubes are illustrated schematically in Fig. 2.3.1.1. The six arrangements are described below in paragraphs (a) to ( f ) corresponding to the subdivisions of Fig. 2.3.1.1. The most frequently used methods are illustrated in (a) and (b). The abbreviations used are S(R) for

Fig. 2.3.1.1. Basic arrangements of focusing diffractometer methods. Simpli®ed and not to scale; detailed drawings shown in later ®gures. (a)±(e) operate with ±2 scanning; ( f ) ®xed specimen with detector scanning. F line focus of X-ray tube (normal to plane of drawings), F 0 focus of incident-beam monochromator, PS parallel slits (to limit axial divergence), ES entrance (divergence) slit, ESM entrance slit for monochromator, S specimen, RS receiving slit, AS antiscatter slit, D detector, SFC specimen focusing circle, M focusing monochromator. Other symbols described in text.

43


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION X-ray-transparent substrates. Jenkins (1989) has reviewed the instrumentation and experimental procedures.

M and D rotate around S in ±2 scanning and the pro®le width is determined by the monochromator. Only the forward-re¯ection region can be recorded. (e) M=S(T): This is the diffractometer equivalent of the Guinier camera. A symmetric or asymmetric monochromator is used in the incident beam and the pro®le width is determined by the RS. The incident-beam divergence is limited by ESM. ( f ) S(R),(S±B): The re¯ections are focused on a ®xed-radius circle which measures 4. A linkage moves the detector around the focusing circle and always points it to the ®xed specimen. The angular range is limited (normally 30±240 4) and can be changed by moving the specimen and diffractometer to different positions. The pro®le width is determined by ES and RS. The same geometry is used with an incident- or diffracted-beam focusing monochromator. The interaction of the X-ray beam with the specimen varies in different geometries and this may have important consequences on the results, as will be described later. When a re¯ection specimen is used in ±2 or ± scanning, only those crystallites whose lattice planes are oriented nearly parallel to the specimen surface can re¯ect (Fig. 2.3.1.2) (Parrish, 1974). A transmission specimen in ±2 scanning permits re¯ections only from planes nearly normal to the surface. In the S±B case, re¯ections can occur from planes inclined over a range of about 45 to the surface. Transmission specimens must, of course, be mounted on

2.3.1.1. Conventional re¯ection specimen, ±2 scan The re¯ection specimen with ±2 scanning in the focusing arrangement shown in Fig. 2.3.1.3 is the most widely used powder diffraction method. It is estimated that about 10 000 to 15 000 of these diffractometers have been sold since they were introduced in 1948, which makes it the most widely used X-ray crystallographic instrument. Some authors have called it the Bragg±Brentano parafocusing method (Bragg, 1921; Brentano, 1946), but the X-ray optics (described below) are signi®cantly different from the methods and instruments described by these authors. The X-ray tube spot focus was ®rst used as the source and gave broad re¯ections. A narrow entrance slit improved the resolution but caused a large loss of intensity. Early diffractometers were described by LeGalley (1935), Lindemann & Trost (1940), and Bleeksma, Kloos & DiGiovanni (1948); see Parrish (1983). The use of the line focus with parallel slits to limit axial divergence was developed in the late 1940's and gave much higher resolution. A collection of papers by Parrish and co-workers (Parrish, 1965) and Klug & Alexander (1974) describe details of the instrumentation and method. 2.3.1.1.1. Geometrical instrument parameters The powder diffractometer is basically a single-axis goniometer with a large-diameter precision gear and worm drive. The detector and receiving-slit assembly are mounted on an arm attached to the gear in a radial position. The specimen is mounted in a holder carried by a shaft precisely positioned at the centre of the gear. 2=1 reduction gears drive the specimen post at one-half the speed of the detector. Some diffractometers have two large gears, making it possible to drive only the detector with the specimen ®xed or vice versa, or to use 2=1 scanning. Synchronous motors have been used for continuous scanning for ratemeter recording and stepping motors for step-scanning with computer control. The geometry of the method requires that the axis of rotation of the diffractometer be parallel to the X-ray tube focal line to obtain maximum intensity and resolution. The target is normal to the long axis of the tube; vertically mounted tubes require a diffractometer that scans in the vertical plane while a horizontal tube requires a horizontal diffractometer. The X-ray optics are the same for both. The incident angle and the re¯ection angle 2 are de®ned with respect to the central ray that passes through the diffractometer axis of rotation O. The axis of rotation of the specimen is the central axis of the main gear of the diffractometer, as shown in Fig. 2.3.1.3. The centre of the specimen is equidistant from the source F and receiving slit RS. The instrument radius RDC F O O RS. The radius of commercial instruments is in the range 150 to 250 mm, with 185 mm most common. Changing the radius affects the instrument parameters and a number of the aberrations. Larger radii have been used to obtain higher resolution and better pro®le shapes. For example, the asymmetric broadening caused by axial divergence is decreased because the chord of the diffraction cone intercepted by the receiving slit has less curvature. However, if the same entrance slit is used, moving the specimen further from the source proportionately increases the length of specimen irradiated and decreases the intensity. The imaginary specimen focusing circle SFC passes through F, O and the middle of RS and its radius varies with :

Fig. 2.3.1.2. Specimen orientation for three diffractometer geometries. With ±2 scanning, diffraction is possible only from planes nearly parallel to the re¯ection specimen surface (left), and from planes nearly normal to the transmission specimen surface (middle), and from planes inclined different amounts to the specimen surface in Seemann±Bohlin geometry (right).

Fig. 2.3.1.3. X-ray optics in the focusing plane of a `conventional' diffractometer with re¯ection specimen, diffracted-beam monochromator, and ±2 scanning: take-off angle, DC diffractometer circle, MFC monochromator focusing circle, ES and RS entrance- and receiving-slit apertures, Bragg angle, 2 re¯ection angle, O diffractometer and specimen rotation axis; other symbols listed in Fig. 2.3.1.1.

44


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES RSFC RDC =2 sin :

place. It may also be made in one piece (c) using machinable tungsten (Parrish & Vajda, 1966). A variable ES whose width increases with 2 so that the irradiated length is about the same at all angles has been described by Jenkins & Paolini (1974). It is called a compensating slit in which a pair of semicircular cylinders with a ®xed opening is rotated around the axis of the opening by a linkage attached to the specimen shaft of the diffractometer to vary the aperture continously with . The observed intensities must be corrected to obtain the relative intensities and the angular dependence of the aberrations is different from the ®xed aperture slit. Another way to irradiate constantly the entire specimen length is to use a self-centring slit which acts as an entrance and antiscatter slit (de Wolff, 1957). A 1 mm thick brass plate with rounded edge is mounted above the centre of the specimen and is moved in a plane normal to the specimen surface so that the aperture is proportional to sin . It can only be used for forward re¯ections. Owing to the beam divergence, the geometric centre of the irradiated specimen length shifts a small amount during the scan (see also x2:3:5:1:5. It is generally advisable to centre the beam at the smallest 2 to be scanned. Below about 20 , the irradiated length increases rapidly and it is essential to use small apertures and to align the entrance and antiscatter slits carefully. Failure to do this correctly could cause (a) errors in the relative intensities owing to the primary beam exceeding the specimen area, (b) cutoff by the walls of the specimen holder for low-absorbing thick specimens, and (c) increased background from scattering by the specimen holder or the primary beam entering the detector. The transmission specimen method (Subsection 2.3.1.2) has advantages in measuring large d's. The beam converges after re¯ection on the receiving slit RS, whose width de®nes the re¯ection and pro®le width. Only those rays that are within the ±2 setting are in sharp convergence, i.e. ìn focus'. The re¯ections become broader with increasing distance from the RS, and, therefore, this method is not suited for position-sensitive detectors. The RS aperture

2:3:1:1

The specimen holder is set parallel to the central ray at 0 and the gears drive the RS-detector arm at twice the speed of the specimen to maintain the ±2 relation at all angles. The source F is the line focus of the X-ray tube viewed at a take-off angle . The actual width, Fw0 , is foreshortened to Fw Fw0 sin : Fw0

2:3:1:2

In a typical case, 0:4 mm and, at 5 , Fw 0:03 mm and the projected angular width is 0.025 for R 185 mm: The angular aperture ES of the incident beam in the equatorial (focusing) plane is determined by the entrance slit width ESw (also called the `divergence slit' since it limits the divergence of the beam) and its distance D1 from F: ES 2 arctanESw Fw =2D1 :

2:3:1:3

Because the beam is divergent, the length of specimen irradiated Sl in the direction of the incident beam normal to O varies with : Sl R

D 0 = sin ;

2:3:1:4

where is in radians and D 0 is the distance from F to the crossover point before ES and is given by Fw D1 =Fw ES: The approximate relation Sl R= sin

2:3:1:5

is close enough for practical purposes (Parrish, Mack & Taylor, 1966). The intensity is nearly proportional to ES but the maximum aperture that can be used is determined by Sl and the smallest angle to be scanned 2min , as shown in Fig. 2.3.1.4. The entrance-slit width may be increased to obtain higher intensity at the upper angular range; for example, ES 1 for the forwardre¯ection region and 4 for back-re¯ection. Some slit designs are shown in Fig. 2.3.1.5. The base is machined with a pair of rectangular shoulders whose separation A is the sum of the diameters of the two rods (a) or bar widths (b) and the central spacers on both ends that determine the slit opening. The distance P between the centre of the slit opening and the edge of the slit frame is kept constant for all slits to avoid angular errors when changing slits. The rods may be molybdenum or other highly absorbing metal and are cemented in

RS 2 arctanRSw =2R

is the dominant factor in determining the intensity and resolution. For RSw 0:1 mm and R 185 mm, RS 0:031 . Antiscatter slits AS are slightly wider than the beam and are essential in this and other geometries to make certain the detector

Fig. 2.3.1.5. Slit designs made with (a) rods, (b) bars, and (c) machined from single piece. (d) Parallel (Soller) slits made with spacers or slots cut into the two side pieces (not shown) to position the foils.

Fig. 2.3.1.4. Length of specimen irradiated, Sl , as a function of 2 for various angular apertures. Sl R=sin , R 185 mm.

45


2:3:1:6

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION 2.3.1.1.3. Alignment and angular calibration

can receive X-rays only from the specimen area. They must be carefully aligned to avoid touching the beam. The use of the long X-ray source makes it necessary to reduce the axial divergence, which would cause very large asymmetry. This is done with two sets of thin (25 to 50 mm parallel metallic foils PS (`Soller slits'; Soller, 1924) placed before and after the specimen. If a monochromator is used, the set on the side of the monochromator is not essential because the crystal reduces the divergence. The angular aperture of a set of slits is 2 arctan spacing=length:

It is essential to align and calibrate the diffractometer properly. Failure to do so degrades the performance of the instrument, leading to a loss of intensity and resolution, increased background, incorrect pro®le shapes, and errors that cannot be readily diagnosed. Procedures and devices for this purpose are often provided by the manufacturer. The principles and mechanical devices to aid in making a proper alignment have been described by Parrish & Lowitzsch (1959) and the general procedure by Klug & Alexander (1974, p. 280). The alignment requires setting the diffractometer axis of rotation to the selected X-ray tube take-off angle at a distance equal to the radius of the diffractometer. The long axes of the X-ray tube focal line, entrance, receiving, and antiscatter slits must be centred, be parallel to the axis of rotation, and lie in the same plane when the instrument is at 0 . The slits are made parallel to the axis of rotation in the manufacture of the diffractometer, and these steps require positioning of the instrument with respect to the line focus. The parallel-slit foils must also be normal to the rotation axis. A ¯at ¯uorescent screen made as a specimen to ®t into the diffractometer specimen post is used to centre the primary beam by small movements of the ES and/or diffractometer. The diffracted beam can be centred on the curved monochromator with a narrow slit placed at the centre of the monochromator position (with the monochromator removed). The detector arm is then moved to the highest intensity. The procedure is repeated with the receiving slit in position. This is very close to the 0 position described below. The angular calibration of the diffractometer is usually made by accurately measuring the 0 position to establish a ®ducial point. It assumes that the gear system is accurate and that the receiving-slit arm moves exactly to the angle indicated on the scale at all 2 positions. The determination of the angular precision of the gear train requires special equipment and methods; see, for example, Jenkins & Schreiner (1986). It is

2:3:1:7

The overall width of the set and determine the width of the specimen irradiated in the axial direction, which remains constant at all 2's. The construction is illustrated in Fig. 2.3.1.5(d). The aperture is usually 2 to 5 . Each set of parallel slits reduces the intensity; for example, with 12:5 mm long foils with 1 mm spacings, the intensity is about one-half of that without the parallel slits. The aperture can be selected with any combination of spacings and lengths but the greater the length, the fewer foils are needed, and the less is the intensity loss due to thickness of the metal foils (usually 0:025 mm). These slits can be made as interchangeable units of different apertures. 2.3.1.1.2. Use of monochromators Many diffractometers are equipped with a curved highlyoriented pyrolytic graphite monochromator placed after the receiving slit as shown in Fig. 2.3.1.3. Although graphite has a large mosaic spread 0:35 to 0.6 ), the diffracted beam from the specimen is de®ned by the receiving slit, which determines the pro®le shape and width rather than the monochromator. The same results are obtained whether the monochromator is set in the parallel or antiparallel position with respect to the specimen. The most important advantage of graphite is its high re¯ectivity, which is about 25±50% for Cu K. This is much higher than LiF, Si or quartz monochromators that have been used for powder diffraction. The K ®lter and the parallel slits in the diffracted beam can be eliminated and, since each reduces the K intensity by about a factor of two, the use of a graphite monochromator actually increases the available intensity. The diffracted-beam monochromator eliminates specimen ¯uorescence and the scattered background whose wavelengths are different from that of the monochromator setting. For example, a Cu tube can be used for specimens containing Co, Fe, or other elements with absorption edges at longer wavelengths than Cu K to produce patterns with low background. Several monochromator geometries are described by Lang (1956). A specimen in the re¯ection mode may be used with an incident-beam monochromator and ±2 scanning as shown in Fig. 2.3.1.1(c). One of the principal advantages is that it is possible to adjust the monochromator and slits to remove the K2 component and produce patterns with only K1 peaks. The pro®le symmetry, resolution and instrument function are thus greatly improved; see, for example, Warren (1969), WoÈlfel (1981), GoÈbel (1982) and LoueÈr & Langford (1988). The highquality crystal required causes a large loss of intensity and reduces specimen ¯uorescence but does not eliminate it. However, Soller slits in the incident beam and a ®lter are no longer required and the net loss of intensity can be as low as 20%. Such monochromators can now be provided as standard by diffractometer manufacturers and their use is increasing, but they are not as widely used as the diffracted-beam monochromator.

Fig. 2.3.1.6. Zero-angle calibration. (a) XRT X-ray tube anode, takeoff angle, O axis of rotation, PH pinhole, RS receiving slit. Intensity distribution at right. (b) 0 position is median of two curves recorded with 180 rotation of PH.

46


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES 2.3.1.1.4. Instrument broadening and aberrations

essential that the setting of the worm against the gear wheel be adjusted for smooth operation. In practice, this is a compromise between minimum backlash and jerky movement. The backlash can be avoided by scanning in the same direction and running the diffractometer beyond the starting angle before beginning the data collection. Incremental angle encoders have been used when very high precision is required. The 0 position of the diffractometer scale, 20 , can be determined with a pinhole placed in the specimen post as shown in Fig. 2.3.1.6(a) (Parrish & Lowitzsch, 1959). The receiving slit is step scanned in 0.01 or smaller increments and the midpoints of chords at various heights are used to determine the angle. To avoid mis-centring errors of the pin-hole, two measurements are made with the specimen post rotated 180 between measurements. The median angle of the two plots is the 0 position as shown in Fig. 2.3.1.6(b) and the diffractometer scale is then reset to this position. The shape of the curves is determined by the relative sizes of the pinhole and the receivingslit width. With care, the position can be located to about 0.001 . The 20 position can be corrected by using it as a variable in the least-squares re®nement of the lattice parameter of a standard specimen. Another method measures the peak angles of a number of re¯ections on both sides of 0 , which is equivalent to measuring 4. This method may be mechanically impossible with some diffractometers. The ±2 setting of the specimen post is made with the diffractometer locked in the predetermined 0 position and manually (or with a stepping motor) rotating the post to the maximum intensity. A ¯at plate can be used as illustrated in Fig. 2.3.1.7(a). The setting can be made to a small fraction of a degree. Fig. 2.3.1.7(b) shows the effect of incorrect ± 2 setting, which combines with the ¯at-specimen aberration to cause a marked broadening and decrease of peak height but no apparent shift in peak position (Parrish, 1958). The effect increases with decreasing and could cause systematic errors in the peak intensities as well as incorrect pro®le broadening.

The asymmetric form, broadening and angular shifts of the recorded pro®les arise from the K doublet and geometrical aberrations inherent in the imperfect focusing of the particular diffractometer method used. There are additional causes of distortions such as the time constant and scanning speed in ratemeter strip-chart recording, small crystallite size, strain, disorder stacking and similar properties of the specimen, and very small effects due to refractive index and related physical aberrations. Perfect focusing in the sense of re¯ection from a mirror is never realized in powder diffractometry. The focusing is approximate (sometimes called `parafocusing') and the practical selection of the instrument geometry and slit sizes is a compromise between intensity, resolution, and pro®le shape. Increasing the resolution causes a loss of intensity. When setting up a diffractometer, the effects of the various instrument and specimen factors should be taken into account as well as the required precision of the results so that they can be matched. There is no advantage in using high resolution, which increases the recording time (because of the lower intensity and smaller step increments), if the analysis does not require it. A set of runs to determine the best experimental conditions using the following descriptions as a guide should be helpful in obtaining the most useful results. In the symmetrical geometries where the incident and re¯ected beams make the same angle with the specimen surface, the effect of absorption on the intensity is independent of the angle. This is an important advantage since the relative intensities can be compared directly without corrections. The actual intensities depend on the type of specimen. For a solid block of the material, or a compacted powder specimen, the intensity is proportional to 1 , where is the linear absorption coef®cient of the material. The transparency aberration [equation (2.3.1.13)], however, depends on the effective absorption coef®cient of the composite specimen. The need to correct the experimental data for the various aberrations depends on the nature of the required analysis. For example, simple phase identi®cations can often be made using data in which the uncertainty of the lattice spacing d=d is of the order of 1=1000, corresponding to about 0.025 to 0.05 precision in the useful identi®cation range. This is readily attainable in routine practice if care is taken to minimize specimen displacement and the zero-angle calibration is properly carried out. The experimental data can then be used directly for peak search (Subsection 2.3.3.7) to determine the peak angles and intensities (Subsection 2.3.3.5) and the data entered in the search/match program for phase identi®cation. However, in many of the more advanced aspects of powder diffraction, as in crystal-structure determination and the characterization of materials for solid-state studies, much more detailed and more precise data are required, and this involves attention to the pro®le shapes. The following sections describe the origin of the instrumental factors that contribute to the shapes and shift the peaks from their correct positions. Many of these factors can be handled individually. With the use of computer programs, they can be determined collectively by using a standard sample without pro®le broadening and pro®le-®tting methods to determine the shapes (Subsection 2.3.3.6). The resulting instrument function can then be stored and used to determine the contribution of the specimen to the observed pro®les. A series of papers describing the geometrical and physical aberrations occurring in powder diffractometry has been

Fig. 2.3.1.7. (a) ±2 setting at 0 . Flat plate or long narrow slit is rotated to position of highest intensity. (b) and (c) Pro®les obtained with correct ±2 setting (solid pro®le) and 1 and 2 mis-settings (dashed pro®les) at (b) 21 and c 60 (2).

47


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION re¯ections recorded with these methods. The shapes are modi®ed by changing slit sizes.

published by Wilson (1963, 1974). His work provides the mathematical foundation for understanding the origin and treatment of the various sources of errors. The major aberrations are described in the following and are illustrated with experimental pro®les and plots of computed data for better visualization and interpretation of the effects. The information can be used to correct the experimental data, interpret the pro®le broadening and shifts, and evaluate the precision of the analysis. Chapter 5.2 contains tables listing the centroid displacements and variances of the various aberrations. The magnitudes of the aberrations and their effects are illustrated in Figs. 2.3.1.8(a) and (b), which show the Cu K1 ; K2 spectrum inside the experimental pro®le. At high 2's, the shape of the experimental pro®le is largely determined by the spectral distribution, but at low 2's the aberrations are the principal contributors. The basic experimental high-resolution pro®le shapes from specimens without appreciable broadening effects (NIST silicon powder standard) are shown in Figs. 2.3.1.8(c)±( f ). The solid-line pro®les were obtained with a re¯ection specimen (Fig. 2.3.1.3), and the dashed-line pro®les with transmission-specimen geometry (Fig. 2.3.1.12). The differences in the K1 ; K2 doublet separations are explained in Subsection 2.3.1.2. These pro®les are the basic instrument functions which show the pro®le shapes contained in all

2.3.1.1.5. Focal line and receiving-slit widths The projected source width Fw and receiving-slit width RSw each add a symmetrical broadening to the pro®les that is constant for all angles. Both the pro®le width and the intensity increase with increasing take-off angle (Section 2.3.5). However, the contribution of Fw is small when the line focus is used, Fig. 2.3.1.9(a). The receiving slit can easily be changed and it is one of the most important elements in controlling the pro®le width, intensity, and peak-to-background ratio, as is shown in Figs. 2.3.1.9(a) and (c). Because of the contributions of other broadening factors, RS can be about twice F (line focus) without signi®cant loss of resolution. The projected width of the X-ray tube focus Fw is given in equation (2.3.1.2). The aperture is F 2 arctanFw =2R:

2:3:1:8

Fw0

1 mm, 5 , and For a line focus with actual width R 185 mm, F 0:011 . The receiving-slit aperture is RS 2 arctanRSw =2R:

2:3:1:9

For RSw 0:2 mm and R 185 mm. RS is 0.062 . The FWHM of the pro®les is always greater than the receiving-slit aperture because of the other broadening factors. 2.3.1.1.6. Aberrations related to the specimen The major displacement errors arising from the specimen are (1) displacement of the specimen surface from the axis of rotation, (2) use of a ¯at rather than a curved specimen, and (3) specimen transparency. These are illustrated schematically for the focusing plane in Fig. 2.3.1.10(a). The rays from a highly absorbing or very thin specimen with the same curvature as the focusing circle converge at A without broadening and at the correct 2. The rays from the ¯at surface cause an asymmetric pro®le shifted to B. Penetration of the beam below the surface combined with the ¯at specimen causes additional broadening and a shift to C. The most frequent and usually the largest source of angular errors arises from displacement of the specimen surface from the diffractometer axis of rotation. It is not easy to avoid and may arise from several sources. It is advisable to check the reproducibility of inserting the specimen in the diffractometer by recording an isolated peak at low 2 for each insertion. If only a radial displacement s occurs, the re¯ection is shifted 2rad 2s cos =R;

where R is the diffractometer radius. A plot of equation (2.3.1.10) is shown in Fig. 2.3.1.10(b). The shift is to larger or smaller angles depending on the direction of the displacement and there is no broadening if the displacement is only radial and relatively small. Even a small displacement causes a relatively large shift; for example, if s 0:1 mm and R 185 mm, 2 0:06 at 20 2. This gives rise to a systematic error in the recorded re¯ection angles, which increases with decreasing 2. It could be handled with a cos cot plot, providing it was the only source of error. There are other possible sources of displacement such as (a) if the bearing surface of the specimen post was not machined to lie exactly on the axis of rotation, (b) improper specimen preparation or insertion in which the surface was not exactly coincident with the bearing surface or (c) nonplanar specimen surface, irregularities, large particle sizes, and specimen transparency. Source (a) leads to a constant error

Fig. 2.3.1.8. Diffractometer pro®les. (a) and (b) Spectral pro®les l of Cu K doublet (dashed-line pro®les) inside experimental pro®les R (solid line). (c)±( f ) Experimental pro®les with re¯ection specimen (R) geometry (Fig. 2.3.1.3) with ES 1 and RS 0.046 (solid line pro®les), and transmission specimen (T) (Fig. 2.3.1.12) with ES 2 and receiving axial divergence parallel slits (dotted pro®les). Cu K radiation. (a) Si(531), (b) quartz(100), (c) Si(111), (d) Si(220), (e) Si(311), and ( f ) Si(422).

48


2:3:1:10

2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES broadens the pro®le (Langford & Wilson, 1962). The peak and centroid are shifted to smaller 2 as shown in Fig. 2.3.1.10(e). For the case of a thick absorbing specimen, the centroid is shifted

in all measurements, and errors due to (b) and (c) vary with each specimen. Ideally, the specimen should be in the form of a focusing torus because of the beam divergence in the equatorial and axial planes. The curvatures would have to vary continuously and differently during the scan and it is impracticable to make specimens in such forms. An approximation is to make the specimen in a ¯exible cylindrical form with the radius of curvature increasing with decreasing 2 (Ogilvie, 1963). This requires a very thin specimen (thus reducing the intensity) to avoid cracking and surface irregularities, and also introduces background from the substrate. A compromise uses rigid curved specimens, which match the SFC (Fig. 2.3.1.3) at the smallest 2 angle to be scanned, and this eliminates most of the aberration (Parrish, 1968). A major disadvantage of the curvature is that it is not possible to spin the specimen. In practice, a ¯at specimen is almost always used. The specimen surface departs from the focusing circle by an amount h at a distance l=2 from the specimen centre: h RFC

R2FC

l2 =21=2 :

2rad sin 2=2R and for a thin low-absorbing specimen 2rad t cos =R;

2:3:1:11

2 =6 tan :

2:3:1:12

For 1 and 2 20 , 2 0:016 . The peak shift is about one-third as large as the centroid shift in the forwardre¯ection region. This aberration can be interpreted as a continuous series of specimen-surface displacements, which increase from 0 at the centre of the specimen to a maximum value at the ends. The effect increases with and decreasing 2. The pro®le distortion is magni®ed in the small 2-angle region where the axial divergence also increases and causes similar effects. Typical ¯at-specimen pro®les are shown in Fig. 2.3.1.10(c) and computed centroid shifts in Fig. 2.3.1.10(d). The specimen-transparency aberration is caused by diffraction from below the surface of the specimen which asymmetrically

Fig. 2.3.1.9. (a) Effect of source size on pro®le shape, Cu K, ES 1 , RS 0.05 , Si(111). No. 1 2 3 4

Projected size (mm) 1:6 1:0 (spot) 0:32 10 (line) 0:16 10 (line) 0:32 12 (line)

FWHM ( 2) 0.31 0.11 0.13 0.17.

Effect of receiving-slit aperture RS on pro®les of quartz (b) (100) and (c) (121); peak intensities normalized, Cu K, ES 1 .

49


2:3:1:14

where is the effective linear absorption coef®cient of the specimen used, t the thickness in cm, and R the diffractometer radius in cm. The intermediate absorption case is described by Wilson (1963). A plot of equation (2.3.1.13) for various values of is given in Fig. 2.3.1.10 f . The effect varies with sin 2 and is maximum at 90 and zero at 0 and 180 . For example, if 50 cm 1 , the centroid shift is 0.033 at 90 and falls to 0.012 at 20 2. The observed intensity is reduced by absorption of the incident and diffracted beams in the specimen. The intensity loss is exp 2=xs cosec ), where is the linear absorption coef®cient of the powder sample (it is almost always smaller than the solid material) and xs is the distance below the surface, which may be equal to the thickness in the case of a thin ®lm or low-absorbing material specimen. The thick 1 mm specimen of LiF in Fig. 2.3.1.10(e) had twice the peak intensity of the thin 0:1 mm specimen. The aberration can be avoided by making the sample thin. However, the amount of incident-beam intensity contributing to the re¯ections could then vary with because different amounts are transmitted through the sample and this may require corrections of the experimental data. Because the effective re¯ecting volume of low-absorbing specimens lies below the surface, care must be taken to avoid blocking part of the diffracted beam with the antiscatter slits or the specimen holder, particularly at small 2. There are additional problems related to the specimen such as preferred orientation, particle size, and other factors; these are discussed in Section 2.3.3.

This causes a broadening of the low-2 side of the pro®le and shifts the centroid 2 to lower 2: 2rad

2:3:1:13

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION reduced because the chord length intercepted is a smaller fraction of the longer radius diffraction cone. The construction of parallel (Soller) slits (Soller, 1924) is shown in Fig. 2.3.1.5(d). The calculation of the aberration and the present status is summarized by Wilson (1963, pp: 40±45). The results depend on the aperture of the parallel slits, the length of the entrance and receiving slits, and 2. In the limit of small s, the shift of the centroid is

2.3.1.1.7. Axial divergence Divergence in the axial direction (formerly also called `vertical divergence') causes asymmetric broadening and shifts the re¯ections. The aberration is illustrated in Fig. 2.3.1.11 for a low-2 re¯ection in the transmission-specimen mode (Subsection 2.3.1.2). The narrow pro®le was obtained with 4:4 parallel slits placed between the monochromator and detector, and the broad pro®le with the slits removed. The slits caused a 33% reduction in peak intensity. This problem was recognized in the ®rst design of the diffractometer using the X-ray tube line focus when parallel slits were used in the incident and diffracted beams to limit the effect (Parrish, 1949). Increasing the radius reduces the effect if the slit length is kept constant. The intensity is also

2; rad s=l2 cot 2=6;

2:3:1:15

where s is the spacing and l the length of the foils. The shift becomes very large at small 2's but not in®nite as equation (2.3.1.15) implies. The shift is to smaller 2's in the forwardre¯ection region and to larger 2's in back-re¯ection. However, the mathematical formulation is dif®cult to quantify because in the forward-re¯ection region the axial divergence convolves with the ¯at-specimen aberration to increase the asymmetry. In the back-re¯ection region, the effect is not so obvious because the distortion is smaller and the Lorentz and dispersion factors also stretch the pro®les to higher angles. 2.3.1.1.8. Combined aberrations Additional aberrations are caused by inaccurate instrument set-up and alignment. For example, if the receiving-slit position is incorrect, the pro®les are broadened. If, in addition, the incident beam is mis-centred or the ±2 is incorrect, a peak shift accompanies the broadening because the aberrations convolute, causing larger distortions and peak shifts than the individual aberrations, for example, ¯at-specimen, transparency, and axial divergence. 2.3.1.2. Transmission specimen, ±2 scan Transmission-specimen methods are not as widely used as re¯ection methods but they provide important supplemental data and have advantages in a number of applications. Re¯ections occur from lattice planes oriented normal to the specimen surface rather than parallel. Re¯ection and transmission patterns can be compared to determine texture and preferred-orientation effects. The transmission method is better suited to the measurement of large d's. Smaller specimen volumes are required. The surface `roughness' which may cause large intensity errors due to the microabsorption in re¯ection specimens is largely reduced. The same basic diffractometer is used for both methods but the geometry is different because the diffracted beam continues to

Fig. 2.3.1.10. (a) Origin of specimen-related aberrations in focusing plane of conventional re¯ection specimen diffractometer (Fig. 2.3.1.3). A no aberration from curved specimen; B ¯at specimen; C specimen displacement from 0. (b) Computed angular shifts caused by specimen displacement, R 185 mm. (c) Flat-specimen asymmetric aberration, Si(422), Cu K1 , K2 peak intensities normalized. (d) Computed ¯at-specimen centroid shifts for various apertures; parabola for constant irradiated 2 cm specimen length. (e) Transparency asymmetric aberration, LiF(200) powder re¯ection, Cu K, peak intensities normalized, thin specimen (solid-line pro®le) 0:1 mm thick; thick specimen (dotted-line pro®le) 1:0 mm, ES 1 , RS 0.046 . f Computed transparency centroid shifts for various values of linear absorption coef®cient.

Fig. 2.3.1.11. Effect of axial divergence on pro®le shape. Narrow pro®le recorded with parallel slits (PS), 4:4 between monochromator and detector Fig. (2.3.1.12), and broad pro®le with these parallel slits removed. Faujasite, Cu K, ES 2 .

50


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES where M is the Bragg angle of the monochromator for the selected wavelength and the l's are shown in Fig. 2.3.2.12(a). Because the pro®le shape and the intensity are determined by the monochromator, the crystal quality and the accuracy of the bending are crucial factors in determining the quality of the pattern. A ¯at thin quartz (101) wafer bent with a special device to approximate a section of a logarithmic spiral has been successfully used (de Wolff, 1968b). The curvature can be varied to obtain the sharpest focus. Thin silicon crystals that can be bent are now available, and Johann and Johannsen asymmetric crystals may be used. Pyrolytic graphite monochromators are not applicable; the radii would be longer because graphite is too soft to be cut at an angle, and a receiving slit would be necessary to de®ne the diffracted beam because the monochromator produces a broad re¯ection. A polarization factor is introduced by the monochromator,

diverge after it passes through the specimen and the monochromator is required to refocus the beam, on the detector as shown in Fig. 2.3.1.12 (de Wolff, 1968b; Parrish, 1958). The monochromator can be placed before or after the specimen and the position has different effects on the pattern. Using the monochromator in the diffracted beam, the intensity and width of the pro®les are determined by the X-ray focal line width and the quality of the bent monochromator rather than the receiving slit which serves as an antiscatter slit. This geometrical arrangement places the virtual image VI of the focal line at the intersection of the focusing circles. After re¯ection from the specimen, the divergent beam is again re¯ected by the focusing crystal M and converges on the detector. The pattern is recorded with ±2 scanning with the monochromator and detector both mounted on a rigid arm rotating around the diffractometer axis. A beam stop MS can be translated and moved in and out near the crossover point to prevent the primary beam from entering the detector at small 2's. To avoid long radii, the crystal surface is cut at an angle (about 3 ) to the re¯ecting lattice plane. The distances are related by l1 l2 =l3 sinM =sinM RFC l1 l2 =2 sinM l3 =2 sinM ;

p 1 k cos2 2=1 k;

2:3:1:17

2

where k cos 2M for mosaic crystals and k cos 2M for perfect crystals. The value of k is strongly dependent on the surface ®nish of the crystal and the crystal should be measured to determine the effect. A specimen with accurately known structure factors such as silicon can be used to calibrate the intensities. The K-doublet separation is zero at the 2 angle at which the dispersion of the specimen compensates that of the monochromator, i.e. the 2 at which the monochromator is aligned and also depends on the distances. The K1 and K2 peaks are superposed and appear as a single peak over a small range of 2's. The K2 peak gradually separates with increasing 2 but the separation is less than calculated from the wavelengths and the intensity ratio may not be 2:1 until higher angles are reached as shown in Fig. 2.3.1.8. A larger angular aperture T can be used for transmission than for re¯ection R because the specimen is more nearly normal to than parallel to the primary beam:

2:3:1:16

T =R 2RD =1 R=l2 Ls ;

2:3:1:18

where the diffractometer radius RD l1 . For RD 170 mm, specimen length LS 20 mm and l2 65 mm; T could be 4.7 times larger than R but the monochromator length usually limits it to about 3 . The smallest re¯ection angle that can be measured is 2min T RD l2 =l2 :

2:3:1:19

Ê for Cu K Using T 0:5 , 2min 1:75 and d 50 A radiation. Specimen preparation is not dif®cult and the preparation can be easily tested and changed. The specimen must be X-ray transparent and can be a free-standing ®lm or foil, or a powder cemented to a thin substrate. The substrate selection is important because its pattern is included. If both transmission and re¯ection patterns are to be compared, the substrate should be selected to have a minimal contribution to both. For example, Mylar is a good substrate for transmission but has a strong re¯ection pattern, and although rolled Be foil has a few re¯ections it is often satisfactory for both. The absorption factor is A t= cos exp s= cos ;

Fig. 2.3.1.12. X-ray optics of the transmission specimen with asymmetric focusing monochromator and ±2 scanning. (a) Monochromator in diffracted beam. M Bragg angle of monochromator with surface cut at angle to re¯ecting plane, MS adjustable beam stop, I1 , I2 , and I3 de®ned in text and other symbols listed in Fig. 2.3.1.3. (b) Monochromator in incident beam, equivalent to Guinier focusing camera.

where t is the powder thickness and s is the sum of the products of the absorption coef®cients and thicknesses of the powder and the substrate. The optimum specimen thickness to give the highest intensity is t 1, i.e. the specimen should transmit about 38% of the incident K intensity. The transmission can be 51


2:3:1:20

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION re¯ections caused by inclination of the rays to the ®lm. The diffractometer eliminates the broadening and extends the angular range. Diffractometers designed for this geometry have been described by Wassermann & Wiewiorosky (1953), SegmuÈller (1957), Kunze (1964a,b), Parrish, Mack & Vajda (1967), King, Gillham & Huggins (1970), Feder & Berry (1970), and others. The geometry is shown in Fig. 2.3.1.13(b) (Parrish & Mack, 1967). Re¯ections occur from lattice planes with varying inclinations H to the specimen surface. The re¯ecting position of a plane H is H H , where is the incidence angle and 4H the re¯ection angle. The maximum value of H is about 45 . It is essential to align the specimen tangent to FC. This is a critical adjustment because even a small misalignment causes pro®le broadening and loss of peak intensity. The source may be the line focus of the X-ray tube [F in Fig. 2.3.1.13(b)] or at the focus of a monochromator [ES in Fig. 2.3.1.13(a)]; in the latter case, the entrance slit at F 0 limits the divergent beam reaching the specimen. The source, specimen centre O, and receiving slit RS lie on the specimen focusing circle SFC, which has a ®xed radius r. The incidence angle is given by

easily measured with a standard specimen set to re¯ect the K and the specimen to be measured inserted normal to the diffracted beam in front of the detector. It is not critical to achieve the exact value and a range of 15±20% of the transmission can be tolerated. This minimizes the effect of the absorption change with 2, and corrections of the relative intensities are required only when accurate values are required. The intensity of the incident beam can be measured at 0 in the same geometry and used to scale the relative intensities to àbsolute' values. The ¯at specimen, transparency, and specimen surface displacement aberrations are similar to those in re¯ection specimen geometry except that they vary as sin rather than cos . This is an important factor in the measurement of large-dspacing re¯ections. The ¯at-specimen effect is smaller because the irradiated specimen length is usually smaller. The transparency error is also usually smaller because thin specimens are used. An important advantage of the method is that the specimen displacement can be directly determined by measuring the peak in the normal position and again after rotating the specimen holder 180 . The correct peak position is at one-half the angle between the two values. The axial divergence has the same effect as in re¯ection. The limitations are that only the forwardre¯ection region is accessible, and the intensity is about one-half of the re¯ection method (except at small angles) because smaller specimen volumes are used. An alternative arrangement for the transmission specimen mode is to use an incident-beam monochromator as shown in Fig. 2.3.1.12(b). This is similar to the geometry used in the Guinier powder camera with the detector replacing the ®lm. A high-quality focusing crystal is required. WoÈlfel (1981) used a symmetrical focusing monochromator with 260 mm focal length for quantitative analysis. GoÈbel (1982) used an asymmetric monochromator with a position-sensitive detector for high-speed scanning, see x2.3.5.4.1. By proper selection of the source size and distances, the K2 can be eliminated and the pattern contains only the K1 peaks (Guinier & SeÂbilleau, 1952). This geometry can have high resolution with the FWHM typically about 0.05 to 0.07 . The pro®le widths are narrower for the subtractive setting of the monochromator than for the additive setting. The pattern is recorded with ±2 scanning. The 0 position can be determined by measuring 4, i.e. peaks above and below 0 , or calibration can be made with a standard specimen. A slit after the monochromator limits the size of the beam striking the specimen. The width and intensity of the powder re¯ections are limited by the receiving-slit width. A parallel slit is used between the specimen and detector to limit axial divergence. The full spectrum from the X-ray tube strikes the monochromator and only the monochromatic beam reaches the specimen, so that it is preferred for radiation-sensitive materials. On the other hand, the radiation reaching the specimen may cause ¯uorescence (though considerably less than the full spectrum) which adds to the background.

arcsinb=2r;

2:3:1:21

0

where b is the distance from F or F to O, or 2r sin . The angle determines the angular range that can be recorded with a given r, decreasing decreases 2min . The relationships of specimen position on the focusing circle and the recording range

2.3.1.3. Seemann±Bohlin method The Seemann±Bohlin (S±B) diffractometer has the specimen mounted on a radial arm instead of the axis of rotation and a linkage or servomechanism moves the detector around the circumference of a ®xed-radius focusing circle while keeping it pointed to the stationary specimen. All re¯ections occur simultaneously focused on the focusing circle as shown in Fig. 2.3.1.13(a). The method was originally developed for powder cameras by Seemann (1919) and Bohlin (1920) but was not widely used because of the limited angular range and the broad

Fig. 2.3.1.13. Seemann±Bohlin method. (a) X-ray optics using incidentbeam monochromator. (b) X-ray tube line-focus source showing geometrical relations: mean angle of incident beam, H inclination of re¯ecting plane H to specimen surface, H Bragg angle of H plane, t tangent to focusing circle at O. (c) Diffractometer settings for various angular ranges.

52


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES 40 4 to 62% at 210 , and Cr K from 73 to 23% at the same angles. Some of the advantages of the method include the following: (a) the ®xed specimen makes it possible to simplify the design of specimen environment devices; (b) a large aperture can be used and the intensities are higher than for conventional diffractometers; (c) the ¯at-specimen aberration can be eliminated by a single-curvature specimen; (d) a small angle can be used to increase the path length l in the specimen, and hence the intensity of low-absorbing thin-®lm samples l t= sin and for 5 , l 11:5t); (e) the method is useful in thin-®lm and preferredorientation studies because about a 45 range of lattice-plane orientations can be measured and compared with conventional patterns. The limitations include (a) the more complicated diffractometer and its alignment, (b) limited angular range of about 10 to 110 2 for the forward-re¯ection setting, (c) extreme care required in specimen preparation, and (d) larger aberration errors.

are illustrated in Fig. 2.3.1.13(c). To change the range requires rotation of the X-ray tube axis or the diffractometer around F. The detector must also be repositioned. For forward-re¯ection measurements, is usually 10 . Extreme care must be used in the specimen preparation to avoid errors due to microabsorption (particle-shadowing) effects, which increase with decreasing . The 0 position cannot be measured directly and a standard is used for calibration. The range from 0 to about 15 2 is inaccessible because of mechanical dimensions. At 90 , only the back-re¯ection region can be scanned. The aperture of the beam striking the specimen is SB 2 arctanESw =2a;

2:3:1:22

where ESw is the entrance slit width and a the distance between F or F 0 and the slit. The irradiated specimen length l is constant at all angles, l 2r. A large aperture can be used to increase intensity since the specimen is close to F. However, the selection of is limited if is small, and also because of the large ¯atspecimen aberration. The receiving-slit aperture varies with the distance of the slit to the specimen RS 4 2 arctan RSw =2r sin2

:

2.3.1.4. Re¯ection specimen, ± scan In this geometry, the specimen is ®xed in the horizontal plane and the X-ray tube and detector are synchronously scanned in the vertical plane in opposite directions above the centre of the specimen as shown in Fig. 2.3.1.14. The distances source to S and S to RS are equal to that the angles of incidence and diffraction and a constant d=dt are maintained over the entire angular range. A focusing monochromator can be used in the incident or diffracted beam. High- and low-temperature chambers are simpli®ed because the specimen does not move. The arms carrying the X-ray tube and detector must be counterbalanced because of the unequal weights. The method has advantages in certain applications such as the measurement of liquid scattering without a covering window, high-temperature molten samples, and other applications requiring a stationary horizontal sample (Kaplow & Averbach, 1963; Wagner, 1969).

2:3:1:23

Consequently, the resolution and relative intensity gradually change across the pattern. The S±B has greater widths at the smaller 2's and nearly the same widths at the higher angles compared with the ±2 diffractometer. The aperture can be kept constant by using a special slit with offset sides (to avoid shadowing) and pointing the opening to C while the detector remains pointed to O (Parrish et al., 1967). The slit opening is tangent to FC and inclined to the beam and rotates while scanning. The constant aperture slit has RS 4 2 arctanRSw =2r:

2:3:1:24

The axial divergence is limited by parallel slits as in conventional diffractometry and the effects are about the same. The equatorial aberrations are also similar but larger in magnitude. The specimen-aberration errors are listed in Table 5.2.7.1. The ¯at specimen causes asymmetric broadening; the shift is proportional to 2ES and increases with decreasing . It can be eliminated by making the specimen with the same curvature as r FC. In this case, one curvature satis®es the entire angular range because the focusing circle has a ®xed radius. However, the curvature precludes rotating the specimen. The specimen transparency also causes asymmetric broadening and a peak shift that increases with decreasing . For h ! 0, the geometric term is the same as for specimen displacement (Mack & Parrish, 1967). The diffracted intensity is proportional to I0 AhTB, where I0 is the incident intensity determined by ; , and the axial length L of the incident-beam assembly, Ah is the specimen absorption factor, T the transmission of the air path, and B the length LRS of the diffracted ring intercepted by the slit. The X-rays re¯ected at a depth x below the specimen surface are attenuated by expf x cosec x cosec 2

g;

2.3.1.5. Microdiffractometry There are two types of microdiffraction: (a) only a very small amount of powder is available, and (b) information is required from very small areas of a conventional-size specimen. Smallvolume samples have been analysed with a conventional diffractometer by concentrating the powder over a small spot centred on a single-crystal plate such as silicon (510) or an

2:3:1:25

where is the linear absorption coef®cient. The asymmetric geometry causes the absorption to vary with the re¯ection angle. The air absorption path varies with the distance O to RS and reaches a maximum at 180 2 . The expression for air transmission includes the radius of the X-ray tube RT , which is needed only for the case where the X-ray tube focal line is used as F. In a typical instrument with X-ray tube source F and r 174 mm, the transmission of Cu K decreases from 90% at

Fig. 2.3.1.14. Optics of ± scanning diffractometer. X-ray tube and detector move synchronously in opposite directions (arrows) around ®xed horizontal specimen. A focusing monochromator can be used after the receiving slit.

53


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION parallel slits limit vertical divergence. However, all the methods result in a large loss of intensity compared with conventional focusing. In contrast, the storage ring produces a virtually parallel beam with very small vertical divergence of about 0:1 mrad, and the monochromator is used only to select the wavelength. The rest of this section assumes a synchrotronradiation source. Storage-ring X-ray sources have a number of unique properties that are of great importance for powder diffraction. The advantages of synchrotron powder diffraction have been described by Hastings, Thomlinson & Cox (1984), Parrish & Hart (1987), Parrish (1988), and Finger (1989). Excellent patterns with high resolution and high peak-to-back ground ratio have been reported. These include the orders-of-magnitude higher intensity and nearly uniform spectral distribution compared with X-ray tubes, the wide continuous range of selectable wavelengths, and the single pro®le that avoids the problems caused by K doublets and ®lters. Owing to major differences in the diffractometer geometries, comparisons of intensities with X-ray tube focusing methods cannot be predicted simply from the number of source photons.

AT-cut quartz plate, or on Mylar for transmission. It is essential to rotate the specimen and increase the count time. A Gandol® camera has also been used for very small specimens (see Section 2.3.4). A high-brilliance microfocus X-ray source has been used with a collimator made of 10 to 100 mm internal-diameter capillary tube. An X±Y stage is used with an optical microscope to locate selected areas of the specimen. A microdiffractometer has been designed for microanalysis, Fig. 2.3.1.15 (Rigaku Corporation, 1990). It has been used to determine phases and stress in areas < 104 mm2 (Goldsmith & Walker, 1984). The key to the method is the use of an annularring receiving slit, which transmits the entire diffraction cone to the detector instead of a small chord as in conventional diffractometry, thereby utilizing all the available intensity. The pattern is scanned by translating the ring and detector along the direct-beam path so that 2 arctanRRS =L;

2:3:1:26

where RRS is the radius of the ring slit and L the distance from the ®xed specimen. For RRS 15 mm, L varies from 171 to 9 mm in the transmission range 5 to 60 2; a 50 mm diameter scintillation counter is used. A doughnut-shaped proportional counter (3=4 of a full circle) is used for the 30 to 150 re¯ection specimen mode. The slit width is 0:2 mm and the aperture varies with 2. The intensities fall off at the higher 2's because of the small incidence angles to the slit. An alternative method uses a position-sensitive proportional counter. Steinmeyer (1986) has described applications of microdiffractometry. By using synchrotron radiation (Section 2.3.2), single-crystal data for structure determination can now be obtained from a microcrystal about 5±10 mm in size; see Andrews et al. (1988), Bachmann, Kohler, Schultz & Weber (1985), Harding (1988), Newsam, King & Liang (1989), Cheetham, Harding, Mingos & Powell (1993), Harding & Kariuki (1994), and Harding, Kariuki, Cernik & Cressey (1994).

Fig. 2.3.2.1. Method to obtain parallel beam from X-ray tube for powder diffraction. HPS parallel slits to limit axial divergence, ES entrance slits (can be replaced by pair of ¯at parallel steel bars), S specimen, VPS parallel slits to de®ne diffracted beam, M ¯at monochromator (can be omitted). D detector. See also Fig. 2.3.2.4(a).

2.3.2. Parallel-beam geometries, synchrotron radiation The radiation from the X-ray tube is divergent and various methods can be used to obtain a parallel beam as shown in Fig. 2.3.2.1. Symmetrical re¯ection from a ¯at crystal is the usual method. An asymmetric re¯ecting monochromator with small incidence angle and large exit angle expands the beam, or in reverse condenses it (x2:3:5:4:1: A channel monochromator has the advantage of not changing the beam direction. A receiving slit or preferably Soller slits can be used to de®ne the diffracted beam. A graphite monochromator in the diffracted beam or a solid-state detector eliminates ¯uorescence. The incident-beam

Fig. 2.3.1.15. Rigaku microdiffractometer for microanalysis. C collimator, PC ring proportional counter, RS ring slit with radius r, S specimen, SC scintillation counter, PBS primary beam stop, PH pinhole for alignment, L specimen-to-receiving-slit distance.

Ê synchrotron radiation Fig. 2.3.2.2. Silicon powder pattern with 1 A using method shown in Fig. 2.3.2.4(a). The 444 re¯ection is the limit for Cu K radiation.

54


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES maximum peak-to-background ratio is obtained with a wavelength slightly longer than the Ni K-absorption edge but using a wavelength shorter than the edge (b) causes high Ni K ¯uorescence background. The relative intensities of the peaks in each compound are the same with both wavelengths. However, the large change in the Ni absorption across the edge caused a large difference in the ratio of Ni=ZnO intensities. The Ni(111) decreased by 85% and the intensity ratio Ni(111)=ZnO(102) dropped from 4.2 to 1.3. Modi®ed conventional vertical-scanning diffractometers are used to avoid intensity losses from the strong polarization in the horizontal plane. The six basic powder diffraction methods that have been used are: (a) Monochromatic X-rays with ±2 scanning and ¯at specimen as in conventional X-ray tube methods but using parallel-beam X-ray optics. This is the most widely applicable method for polycrystalline materials. (b) Monochromatic X-rays with ®xed specimen and 2 detector scan, used for analysing texture, preferred orientation, and grazing-incidence diffraction. (c) Monochromatic X-rays with a capillary specimen and scanning receiving slit or position-sensitive detector. (d) Energy-dispersive diffraction using a step-scanned channel monochromator, selectable ®xed ±2 positions, and conventional scintillation counter and electronics. The instrumentation is the same as (a) and may be used in methods that require a stationary specimen. (e) Energy-dispersive diffraction using the white beam, solidstate detector and multichannel analyser, and selected ®xed ±2. This is the method frequently used with synchrotron and X-ray tube sources but it has low pattern resolution (Giessen & Gordon, 1968). f Angle-dispersive or energy-dispersive experiments with an imaging-plate detector, whereby complete Debye±Scherrer rings are recorded simultaneously, as in some ®lm methods (Subsection 2.3.4.1) (e.g. Piltz et al., 1992). This is a particularly useful technique for studies under non-ambient conditions, such as experiments at ultra-high pressure (e.g. McMahon & Nelmes, 1993).

The easy wavelength selection makes it possible to avoid specimen ¯uorescence, to record data on both sides of an absorption edge for anomalous-scattering studies, to select optimum angles and wavelengths for lattice-parameter measurements, and to vary the dispersion. Short-wavelength radiation can be used for uncomplicated patterns without the background occurring in X-ray tube spectra. Fig. 2.3.2.2 shows a silicon Ê X-rays in which there are twice pattern obtained with 1:0 A as many re¯ections as can be recorded with Cu K, and the background remains very low out to the highest 2 angles. The Ê ) are especially useful for samples short wavelengths ( 0:7 A mounted in cryostats, furnaces, and pressure cells. Using an incident-beam tunable monochromator, no continuous radiation reaches the specimen and a wavelength can be selected that gives a high peak-to-background ratio and no specimen ¯uorescence. If the specimen contains different chemical phases, patterns can be recorded using wavelengths on both sides of the absorption edge to enhance one of the patterns as an aid in identi®cation. This is illustrated in Fig. 2.3.2.3 for a mixture of Ni and ZnO powders. A pattern (a) with

2.3.2.1. Monochromatic radiation, ±2 scan The X-ray optics of a plane-wave parallel-beam diffractometer is shown schematically in Fig. 2.3.2.4(a). The primary white beam is limited by slits at C1. A channel monochromator CM is used because it has the important property of maintaining the same direction and position for a wide range of wavelengths. It may be used in the dispersive setting with respect to the specimen or in the parallel setting [Fig. 2.3.2.4(b)]. The monochromatic beam is larger than the entrance slit ES and it is unnecessary to realign the powder diffractometer when changing wavelengths. The monochromator can be mounted on a stripped diffractometer for easy alignment and step scanning. There are no characteristic spectral lines and the wavelength calibration of the monochromator is made by step scanning the monochromator across absorption edges of elements in a specimen or pure element foils placed in the beam. The wavelength accuracy is limited by the uncertainty as to what feature of the edge should be measured and which one was used for the wavelength tables. A standard powder sample such as NIST silicon 640b whose lattice parameter is known with moderately high precision can also be used. An alternative method is to measure the re¯ection angle of a single-crystal plate of ¯oat-zones oxygen-free silicon whose lattice parameter is known to 1 part in 10 7 and to determine the wavelength from

Fig. 2.3.2.3. Synchrotron-radiation patterns of a mixture of Ni and ZnO powders. Diffraction pattern using a wavelength (a) slightly longer than the Ni K-absorption edge and (b) slightly shorter. (c) Highresolution energy-dispersive diffraction (EDD) pattern.

55


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION it causes the same specimen-surface-displacement and transparency errors as the focusing geometries. A set of horizontal parallel (Soller) slits is advantageous because of the much higher intensity and it eliminates the displacement errors. The pro®les of specimens without broadening effects have the same FWHM as the aperture of the slits, equation (2.3.1.7). The FWHM increases as tan due to wavelength dispersion. By increasing the length of the foils and keeping the same spacing, the aperture can be reduced to increase the resolution without large loss of intensity. A set of 365 mm long slits with 0.05 aperture has been used and even smaller apertures are feasible. Longer slits decrease the ¯uorescence intensity (if any) reaching the detector. They must be carefully made and aligned to avoid loss of intensity and should be evacuated or ®lled with He to avoid air-absorption losses. The use of a crystal analyser eliminates ¯uorescence and gives the highest resolution powder pro®les with FWHM 0:02 to 0.05 2, depending on the quality of the crystal (Hastings et al., 1984). The alignment of the crystal is critical and must be done with remote automated control every time the wavelength is changed. Displacement aberrations are eliminated but the intensity is much lower than the HPS because of the small rocking angle and low integrated re¯ectivity of the crystal. The correct orientation of crystalline powder particles for re¯ection is far more restrictive for the parallel beam than the X-ray tube divergent beam. A much smaller number of particles will have the exact orientation for re¯ection, and thus the recorded intensity will be lower and relative intensities less accurate. If the specimen is stationary, the standard deviations of the intensities due to particle size are six to nine times higher than in focusing methods (Parrish, Hart & Huang, 1986). It also becomes more dif®cult to achieve the completely randomly oriented specimens required for structure determination and quantitative analysis and, as in X-ray tube data, a preferredorientation term is included in the structure re®nement. It is, therefore, essential to use small particles < 10 mm and to rotate the specimen. Some investigators prefer to oscillate the specimen over a small angle but this is not as effective as rotation. The pro®les are virtually symmetrical except at small angles where axial divergence causes asymmetry. The pro®les in Fig. 2.3.2.5 show the differences in the shape and resolution obtained with conventional focusing (a) and parallel-beam synchrotron methods (b). The effect of the higher resolution on a mixture of nearly equal volumes of quartz, orthoclase, and feldspar recorded with X-ray tube focusing methods is shown in Fig. 2.3.2.5(c) and with synchrotron radiation in Fig. 2.3.2.5(d). The symmetry and nearly constant simple instrument function make it easier to separate overlapping re¯ections and simplify the pro®le-®tting procedures and the interpretation of specimenbroadening effects. The early crystal-structure studies using Rietveld re®nement were not as successful with X-ray tube focusing methods as they were with neutron diffraction because the complicated instrument function made pro®le ®tting dif®cult and inaccurate. The development of synchrotron powder methods with simple symmetrical instrument function, high resolution, and the use of longer wavelengths to increase the dispersion have made structural studies as successful as with neutrons, and have the advantage of orders-of-magnitude higher intensity. Some examples are described by Att®eld, Cheetham, Cox & Sleight (1988), Lehmann, Christensen, FjellvaÊg, Feidenhans'l & Nielsen (1987), and ab initio structure determinations by

the Bragg equation (Hart, 1981). The accuracy is then limited by the angular accuracy of the diffractometer and the orientation setting. It is necessary to monitor the monochromatic beam intensity I0 , which changes during the recording due to decreasing storage-ring current, orbital shifts or other factors. This can be done by inserting a low-absorbing ionization chamber in the beam or by using a scintillation counter to measure scattering from an inclined thin beryllium foil, kapton or other lowabsorbing material. The data are recorded and used to correct the observed intensities. The monitored counts can also be used as a timer for step scanning if a suf®cient number are recorded for good counting statistics. The entrance slit ES determines the irradiated specimen length, which is equal to ES=sin s . Vertical parallel slits VPS with ' 2 are used to limit the axial divergence. The longer the distance between the specimen and detector, the smaller the asymmetry, and a vacuum path should be used to avoid airabsorption losses. The specimen may be used in either re¯ection or transmission simply by rotating it 90 around the diffractometer axis from its previous position. The diffracted beam can be de®ned by a receiving slit (Parrish, Hart & Huang, 1986), horizontal parallel slits HPS [Fig. 2.3.2.4(a)] (Parrish & Hart, 1985) or a high-quality singlecrystal plate which acts as a very narrow receiving slit [Fig. 2.3.2.4(b)] (Cox, Hastings, Thomlinson & Prewitt, 1983; Hastings et al., 1984). If a receiving slit is used, the intensity, pro®le width and shape are determined by the widths of both ES and RS. If either one is much wider than the other, the pro®le has a ¯at top. Increasing the RS width and keeping ES constant causes symmetrical pro®le broadening and increases the intensity as in conventional focusing diffractometry. There are disadvantages in using a receiving slit because the intensities are low and

Fig. 2.3.2.4. (a) Optics of dispersive parallel-beam method for synchrotron X-rays. C1 primary-beam collimator, D1 diffractometer for channel monochromator CM, C2 antiscatter shield, Be beryllium foil for monitor, SC1 and SC2 scintillation counters, ES entrance slit on powder diffractometer D2, VPS vertical parallel slits to limit axial divergence, HPS horizontal parallel slits, which determine the resolution. (b) CM in nondispersive setting and crystal analyser A used as a narrow receiving slit. (c) Fibre specimen FS with receiving slit RS or with position-sensitive detector (not shown) with RS removed.

56


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES 2.3.2.2. Cylindrical specimen, 2 scan

McCusker (1988), Cernik et al. (1991), Morris, Harrison, Nicol, Wilkinson & Cheetham (1992), and others. Structures have also been solved using a two-stage method in which the integrated intensities are determined by pro®le ®tting the individual re¯ections and used in a powder least-squares re®nement method (POWLS) (Will, Bellotto, Parrish & Hart, 1988). The method was tested with silicon, which gave R(Bragg) 0.7%, and quartz, which gave 1.6%, which is a good test of the high quality of the experimental data and the pro®le-®tting procedure. Fig. 2.3.2.6 shows Fourier maps of orthorhombic Mg2 GeO4 calculated using Fourier coef®cients taken directly from the pro®le-®tting intensities. Other types of powder studies have been carried out successfully. For example, these have been used in anomalousscattering studies (Will, Masciocchi, Hart & Parrish, 1987; Will, Masciocchi, Parrish & Hart, 1987), Warren±Averbach pro®le-broadening analysis (Huang, Hart, Parrish & Masciocchi, 1987), studies of texture in thin ®lms (Hart, Parrish & Masciocchi, 1987), and precision lattice-parameter determination (Hart, Cernik, Parrish & Toraya, 1990).

The ¯at specimen can be replaced by a thin cylindrical [Fig. 2.3.2.4(c)] specimen as used in powder cameras. The powder can be coated on a thin ®bre or reactive materials can be forced into a capillary to avoid contact with air. The intensity is lower than for ¯at specimens because of the smaller beam, and less powder is required. Thompson, Cox & Hastings (1987) used the method to determine the structure of Al2 O3 by Rietveld re®nement. They used a two-crystal incident-beam Si(111) monochromator; the ®rst crystal was ¯at and the second a cylindrically bent triangular plate for sagittal focusing to form a 4 2 mm beam with spectral bandwidth l=l ' 10 3 .

Fig. 2.3.2.5. Comparison of patterns obtained with a conventional focusing diffractometer (a) and (c), and synchrotron parallel-beam method (b) and (d). (a) and (b) quartz powder pro®les; (c) and (d) mixture of equal amounts of quartz, orthoclase, and feldspar.

Fig. 2.3.2.6. (a) and (c) Fourier maps of orthorhombic Mg2 GeO4 calculated directly from pro®le-®tted synchrotron powder data. (b) Fourier section of isostructural Mg2 SiO4 calculated from singlecrystal data for comparison with (a).

57


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION 2.3.2.4. High-resolution energy-dispersive diffraction

The method can also be used with a receiving slit or positionsensitive detectors (Lehmann et al., 1987; Shishiguchi, Minato & Hashizume, 1986). The latter can be a short straight detector, which can be scanned to increase the data-collection speed (GoÈbel, 1982), or a longer curved detector.

By step scanning the channel monochromator instead of the specimen, a different wavelength reaches the specimen at each step and the pattern is a plot of intensity versus wavelength or energy (Parrish & Hart, 1985, 1987). The X-ray optics can be the same as described in Subsection 2.3.2.1 and determines the resolution. A scintillation counter with conventional electronic circuits can be used. As in the conventional white-beam energy-

2.3.2.3. Grazing-incidence diffraction In conventional focusing geometry, the specimen and detector are coupled in ±2 relation at all 2's to avoid defocusing and pro®le broadening. In Seemann±Bohlin geometry, changing the specimen position necessitates realigning the diffractometer and very small incidence angles are inaccessible. In parallel-beam geometry, the specimen and detector positions can be uncoupled without loss of resolution. This freedom makes possible the use of different geometries for new applications. The specimen can be set at any angle from grazing incidence to slightly less than 2, and the detector scanned. Because the incident and exit angles are unequal, the relative intensities may differ by small amounts from those of the ±2 scan due to specimen absorption. The re¯ections occur from differently oriented crystallites whose planes are inclined (rather than parallel) to the specimen surface so that particle statistics becomes an important factor. The method is thus similar to Seemann±Bohlin but without focusing. The method can be used for depth-pro®ling analysis of polycrystalline thin ®lms using grazing-incidence diffraction (GID) (Lim, Parrish, Ortiz, Bellotto & Hart, 1987). If the angle of incidence i is less than the critical angle of total re¯ection c , Ê of the ®lm. diffraction occurs only from the top 35 to 60 A Comparison of the GID pattern with a conventional ±2 pattern in which the penetration is much greater gives structural information for phase identi®cation as a function of ®lm depth. The intrinsic pro®le shapes are the same in the two patterns and broadening may indicate smaller particle sizes. However, if the ®lm is epitaxic or highly oriented, it may not be possible to obtain a GID pattern. For i < c , the penetration depth t0 is (Vineyard, 1982) t0 ' l=2c2

i2 1=2

Fig. 2.3.2.7. Penetration depth t0 as a function of grazing-incidence angle for -Fe2 O3 thin ®lm. The critical angle of total re¯ection c is shown by the vertical arrows for different wavelengths.

2:3:2:1

and, for i > c , t0 ' 2i =;

2:3:2:2

where is the linear absorption coef®cient. The thinnest top layer of the ®lm that can be sampled is determined by the ®lm density, which may be less than the bulk value. As i approaches c , the penetration depth increases rapidly and ®ne control becomes more dif®cult. Fig. 2.3.2.7 shows this relation and the advantage of using longer wavelengths for a wider range of penetration control. For example, for a ®lm with 200 cm 1 , Ê , and i 0:1 , only the top 45 A Ê contribute, and l 1:75 A Ê . The patterns increasing i to 0.35 increases the depth to 130 A have much lower intensity than a ±2 scan because of the smaller diffracting volume. Ê polycrystalline ®lm of Fig. 2.3.2.8 shows patterns of a 5000 A iron oxide deposited on a glass substrate and recorded with (a) ±2 scanning and (b) 0.25 GID. The ®lm has preferred orientation as shown by the numbers above the peaks in (a), which are the relative intensities of a random powder sample. The relative intensities are different because in (a) they come from planes oriented parallel to the surface and in (b) the planes are inclined. The glass scattering that is prominent in (a) is absent in (b) because the beam does not penetrate to the substrate.

Fig. 2.3.2.8. Synchrotron diffraction patterns of annealed 5000 AÊ iron Ê , (a) ±2 scan; relative intensities of random oxide ®lm, l 1:75 A powder sample shown above each re¯ection. (b) Grazing incidence pattern of same ®lm with 0:25 showing only re¯ections from top Ê of ®lm, superstructure peak S.S. and -Fe2 O3 peaks not seen in 60 A (a). Absolute intensity is an order of magnitude lower than (a).

58


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES to 22:7 keV) and four detector 2 settings. At small 2 settings, only the large d's are recorded and the peak separation is large. Increasing the 2 setting decreases the d range and the separation of the peaks as shown in Fig. 2.3.2.9(e). These patterns were recorded with the pulse-height analyser set to discriminate only against scintillation counter noise. For given X-ray optics, the pro®les symmetrically broaden with decreasing X-ray photon energy and with . This type of broadening remains symmetrical if E is increased and 2 decreased, or vice versa, Fig. 2.3.2.9 f . The two pro®les shown have been broadened by the X-ray optics but the intrinsic resolution is far better. The number of points recorded per pro®le thus decreases with decreasing pro®le width since M is constant. At the higher energies, it may be desirable to use smaller M steps to increase the number of points to de®ne better the pro®le. Alternatively, increments in sin steps rather than steps would eliminate this variation. Many electronic solid-state devices use thin ®lms that are purposely prepared to have single-crystal structure (e.g. epitaxic growth), or with a selected lattice plane oriented parallel or normal to the ®lm surface to enhance certain properties. The properties vary with the degree of orientation and textural characterization is essential to make the correct ®lm preparation. Preferred orientation can be detected by comparing the relative intensities of the thin-®lm pattern with those of a random powder. The pattern can be recorded with conventional ±2 scanning (l ®xed) or by EDD. However, this only gives information on the planes oriented parallel to the surface. To study inclined planes requires uncoupling the specimen surface and detector angles. This can be done with the EDD method described above without distorting the pro®les (Hart et al., 1987). The principle of the method is illustrated in Fig. 2.3.2.10. The set of lattice planes (hkl) oriented parallel to the surface has its highest intensity in the symmetric ±2 position. Rotating the specimen by an angle r while keeping 2 ®xed reduces the intensity of (hkl) and brings another set of planes (pqr), which are inclined to the surface, to its symmetrical re¯ecting position. The required rotation is determined by the interplanar angle between (hkl) and (pqr). The angular distribution of any plane can be measured with respect to the ®lm surface by step scanning at small r steps. The specimen is rotated clockwise with the limitation s r < 2. A computer automation program is desirable for large numbers of measurements. Fig. 2.3.2.3(c) shows the appearance of a pattern of a specimen containing elements with absorption edges in the recording range and using electronic discrimination only against electronic noise. Starting at the incident high-energy side, the Zn and Ni K ¯uorescence increases as the energy approaches the edges l3 law), decreases abruptly when the energy crosses each edge, and disappears beyond the Ni K edge. Long-wavelength ¯uorescence is absorbed in the windows and air path.

dispersive diffraction (EDD) described in Section 2.5.1, the specimen and detector remain ®xed at selected angles during the recording. This makes it possible to design special experiments that would not be possible with specimen-scanning methods. It also simpli®es the design of specimen-environment chambers for high and low temperatures. The advantages of the method over conventional EDD are the order-of-magnitude higher resolution that can be controlled by the X-ray optics, the ability to handle high peak count rates with a high-speed scintillation counter and conventional circuits, and much lower count times for good statistical accuracy. The accessible range of d's that can be recorded using a selected wavelength range is determined by the 2 setting of the detector. Changing 2 causes the separation of the peaks to expand or compress in a manner similar to a variation of l in conventional diffractometry. This is illustrated in Figs. 2.3.2.9(a)±(d) for a quartz powder specimen using an Si(111) Ê , 6.1 channel monochromator and M 19 to 5 (2.04 to 0:55 A

Fig. 2.3.2.9. (a)±(d) High-resolution energy-dispersive diffraction patterns of quartz powder sample obtained with 2 settings shown in upper left corners. (e) d range as a function of detector 2 setting Ê . f Effect of 2 setting and E on pro®le widths of for l 0:4 to 2 A quartz. Right: 121 re¯ection, 20 2, Ep 10:45 keV; left: 100 re¯ection, 45 2, ep 8.35; both re¯ections broadened by X-ray optics and peak intensity of 100 twice that of 121.

Fig. 2.3.2.10. Specimen orientation for symmetric re¯ection (a) from (hkl) planes and (b) specimen rotated r for symmetric re¯ection from (pqr) planes.

59


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION and they include bibliographies on special handling problems. Powder diffraction standards for angle and intensity calibration are described in Section 5.2.9.

The method is of doubtful use for structure determination or quantitative analysis. The wide range of wavelengths, continually varying absorption and pro®le widths, and other factors create a major dif®culty in deriving accurate values of the relative intensities. Conventional energy-dispersive diffraction methods using white X-rays and a solid-state detector are described in Chapter 2.5 and Section 5.2.7.

2.3.3.1.1. Preferred orientation Preferred orientation changes the relative intensities from those obtained with a randomly oriented powder sample. It occurs in materials that have good cleavage or a morphology that is platy, acicular or any special shape in which the particles tend to orient themselves in specimen preparation. The micas and clay minerals are examples of materials that exhibit very strong preferred orientation. When they are prepared as re¯ection specimens, the basal re¯ections dominate the pattern. It is common in prepared thin ®lms where preferred orientation occurs frequently or may be purposely induced to enhance certain optical, electrical, or magnetic properties for electronic devices. By comparison of the relative intensities with the random powder pattern, the degree of preferred orientation can be observed. Powder re¯ections take place from crystallites oriented in different ways in the instrument geometries as shown in Fig. 2.3.1.2. In re¯ection specimen geometry with ±2 scanning, re¯ections can occur only from lattice planes parallel to the surface and in the transmission mode they must be normal to the surface. In the Seemann±Bohlin and ®xed specimen with 2 scanning methods, the orientation varies from parallel to about 45 inclination to the surface. The effect of preferred orientation can be seen in diffraction patterns obtained by using the same specimen in the different geometries. The effect is illustrated in Fig. 2.3.3.1 for m-chlorobenzoic acid, C7 H5 ClO2 , with re¯ection and transmission patterns and the pattern calculated from the crystal structure. The degree of preferred orientation is shown by comparing the peak intensities of four re¯ections in the three patterns:

2.3.3. Specimen factors, angle, intensity, and pro®le-shape measurement The basic experimental procedure in powder diffraction is the measurement of intensity as a function of scattering angle. The pro®le shapes and 2 angles are derived from the observed intensities and hence the counting statistical accuracy has an important role. There is a wide range of precision requirements depending on the application and many factors are involved: instrument factors, counting statistics, pro®le shape, and particle-size statistics of the specimen. The quality of the specimen preparation is often the most important factor in determining the precision of powder diffraction data. D. K. Smith and colleagues (see, for example, Borg & Smith, 1969; see also Yvon, Jeitschko & PartheÂ, 1977) developed a method for calculating theoretical powder patterns from well determined single-crystal structures and have made available a Fortran program (Smith, Nichols & Zolensky, 1983). This has important uses in powder diffraction studies because it provides reference data with correct I's and d's, free of sample defects, preferred orientation, statistical errors, and other factors. The data can be displayed as recorded patterns by using plot parameters corresponding to the experimental conditions (Subsection 2.3.3.9). Calculated patterns have been used in a large variety of studies such as identi®cation standards, computing intermediate members of an isomorphous series, testing structure models, ordered and disordered structures, and others. Many experiments can be performed with simulated patterns to plan and guide research. The method must be used with some care because it is based on the small single crystal used in the crystalstructure determination and the large powder samples of minerals and ceramics, for example, may have a different composition. Errors in the structure analysis are magni®ed because the powder intensities are based on the squares of the structure factors. The Lorentz and polarization factors for diffractometry geometry have been discussed by Ladell (1961) and Pike & Ladell (1961). Smith & Snyder (1979) have developed a criterion for rating the quality of powder patterns; see also de Wolff (1968a).

(hkl) Re¯ection Transmission Calculated

(200) 0.6 0.5 6.6

(040) 1.6 0.7 4.0

(121) 2.5 9.3 9.1.

Care is required to make certain the differences are not caused by a few fortuitously oriented large particles. Various methods have been used to minimize preferred orientation in the specimen preparation (Calvert, Sirianni, Gainsford & Hubbard, 1983; Smith & Barrett, 1979; Jenkins et al., 1986; Bish & Reynolds, 1989). These include using small particles, loading the powder from the back or side of the specimen holder, and cutting shallow grooves to roughen the surface. The powder has also been sifted directly on the surface of a microscope slide or single-crystal plate that has been wetted with the binder or petroleum jelly. Another method is to mix the powder with an inert amorphous powder such as Lindemann glass or rice starch, or add gum arabic, which after setting can be reground to obtain irregular particles. Any additive reduces the intensity and the peak-to-background ratio of the pattern. A promising method that requires a considerable amount of powder is to mix it with a binder and to use spray drying to encapsulate the particles into small spheres which are then used to prepare the specimen (Smith, Snyder & Brownell, 1979). Preferred orientation would not cause a serious problem in routine identi®cation providing the reference standard had a similar preferred orientation and both patterns were obtained with the same diffractometer geometry. However, when accurate values of the relative intensities are required, as in crystal-

2.3.3.1. Specimen factors Ideally, the specimen should contain a large number of small equal-sized randomly oriented particles. The surface must be ¯at and smooth to avoid microabsorption effects, i.e. particle interferences which reduce the intensities of the incident and re¯ected beams and can lead to signi®cant errors (Cline & Snyder, 1983). The specimen should be homogeneous, particularly if it is a mixture or if a standard has been added. Low packing density and specimen-surface displacement x2:3:1:1:6 may cause signi®cant errors. It is recommended that the powder and the prepared specimen be examined with a low-power binocular optical microscope. Smith & Barrett (1979), Jenkins, Fawcett, Smith, Visser, Morris & Frevel (1986), and Bish & Reynolds (1989) have surveyed methods of specimen preparation 60


(120) 9.8 5.2 3.0

2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES Table 2.3.3.1. Preferred-orientation data for silicon

structure re®nement and quantitative analysis, it may be the major factor limiting the precision. In practice, it is very dif®cult to prepare specimens that have a completely random orientation. Even materials that do not have good cleavage or special morphological forms, such as quartz and silicon, show small deviations from a completely random orientation. These show up as errors in the structure re®nement and a correction factor is required. An empirical correction factor determined by the acute angle ' between the preferred-orientation plane and the diffracting plane (hkl)

hkl

R(Bragg) (%)

GP

111 220 311

1.86 2.02 2.01

0.11 0.11 0.17

4 0 0

0.86

0.15

3 4 5 5 4 6 5

1.73 2.43 1.36 2.44 1.69 1.25 2.40

0.19 0.04 0.19 0.08 0.19 0.29 0.04

3 2 1 3 4 2 3

1 2 1 1 2 0 3

* Selected preferred orientation plane.

Table 2.3.3.2. R(Bragg) values obtained with different preferred-orientation formulae R(Bragg)

No corrections Gaussian Exponential March/Dollase Preferred-orientation plane

Si

SiO2

3.50 1.65 0.75 0.75 100

2.57 1.60 1.83 1.64 211

Icorr: IhklPhkl'

Mg2 GeO4 12.5 5.71 5.30 4.87 100

2:3:3:1

can be used (Will et al., 1988). Three functions have been used to represent Phkl' and the term GP is the variable re®ned: Phkl' exp GP'2

2:3:3:2

(Rietveld, 1969) for transmission specimens; Phkl' expGP=2

'2

2:3:3:3

for re¯ection specimens; and Phkl' GP2 cos2 ' sin2 '=GP

2:3:3:4

(Dollase, 1986). These functions are quite similar for small amounts of nonrandomness. The preferred-orientation plane is selected by trial and error. For example, a modi®ed fast routine of the powder least-squares re®nement program with only seven cycles of re®nement on each plane for the ®rst dozen allowed Miller indices can be used to ®nd the plane that gives the lowest R(Bragg) value as shown in Table 2.3.3.1. All three functions improve the R(Bragg) value as shown in Table 2.3.3.2 but the evidence is not conclusive as to which is the best. More research is required in this area. Several specimens made of the same material may show different preferred-orientation planes, and in some cases the preferred-orientation plane never occurred in the crystal morphology. A more complicated method examines the polar-axis density distribution using a cubic harmonic expansion to describe the crystallite orientation of a rotating sample (JaÈrvinen, Merisalo, Pesonen & Inkinen, 1970; Ahtee, Nurmela, Suortti & JaÈrvinen, 1989; JaÈrvinen, 1993).

Fig. 2.3.3.1. Differences in relative intensities due to preferred orientation as seen in synchrotron X-ray patterns of m-chlorobenzoic acid obtained with a specimen in re¯ection and transmission compared with calculated pattern. Peaks marked are impurities, O absent in experimental patterns.

61


3=2

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Slow rotation, 1=7 r min 1 , shows the variation of the peak intensity with azimuth angle '. The pattern repeats after 360 rotation and the magnitude of the ¯uctations increases with increasing particle sizes and resolution. There is no correlation between the ¯uctations of different re¯ections, as can be seen by comparing the 111, 220 and 311 re¯ections of the 10±20 mm specimen (lower left side) for which the incident-beam intensity was adjusted to give the same average amplitude. The horizontal lines are 10% of the average. This shows the magnitude of errors that could occur using stationary specimens. Similar particle-size effects were found using the integrated intensities derived from pro®le ®tting. The above discussion and Fig. 2.3.3.2 refer to a continuous scan. If the step-scan mode is used to collect data, it is clearly not necessary to rotate the specimen through more than one revolution at each step. The rotating specimen also averages the in-plane preferred orientation but has virtually no effect on the planes oriented parallel to the specimen surface. The slow rotation method is useful in testing the grinding and sifting stages in specimen preparation. When calibrated with known size fractions, it can be used as a rough qualitative measure of the particle sizes.

2.3.3.1.2. Crystallite-size effects In addition to pro®le broadening, which begins to appear when the crystallite sizes are < 1±2 mm, the sizes have a strong effect on the absolute and relative intensities (de Wolff, Taylor & Parrish, 1959; Parrish & Huang, 1983). The particle sizes have to be less than about 5 mm to achieve 1% reproducible relative intensities from a stationary specimen in conventional diffractometer geometry (Klug & Alexander, 1974). The statistical errors arising from the number of particles irradiated can be greatly reduced by using smaller particles and rotating the specimen around the diffraction vector. This brings many more particles into re¯ecting orientations. The particle-size effect is illustrated in Fig. 2.3.3.2 for specimens of NIST silicon standard powder 640 sifted to different size fractions. The powders were packed in a 1 mm deep cavity in a 25:4 mm diameter Al holder using 5% collodion=amyl acetate binder. They were rotated by a synchronous motor (a stepper motor can also be used) around the axis normal to the centre of the specimen surface with the detector arm ®xed at the peak position and the intensity recorded with a strip-chart. Rapid rotation, 60 r min 1 , gives the average peak intensity for all azimuths of the specimen and the small variations result only from the counting statistics. Scaling the intensities to 111 100% for the 5±10 mm fraction, the 10±20 mm fraction is 94%, 20±30 mm 88% and > 30 mm 59%. The decrease is probably due to lower particle-packing density and increasing interparticle microabsorption. The > 5 mm fraction 95% may be due to the larger ratio of oxide coating around the particles to the mass of the particles.

2.3.3.2. Problems arising from the K doublet A common source of error arises from the K doublet which produces a pair of peaks for each re¯ection. The separation of the Cu K1 , K2 peaks increases from 0.05 at 20 2 to 1.08 at 150 2. The overlapping is also dependent on the instrument resolution and may cause errors in the peak angles and intensities when strip-chart recording or peak-search methods (described below) are used. The K1 wavelength is generally used to calculate all the d's even when the low-angle peaks are unresolved. In the region where the doublet is only slightly resolved, the apparent K1 peak angle is shifted to higher angles because of the overlapping K2 tail and similarly the peak intensities will be in error. The relative peak intensities of a re¯ection with superposed doublet compared to a resolved doublet could have an error as large as 50%. Relative peak intensities are used in the ICDD standards ®le and cause no problem because the unknowns are measured in the same way. The integrated intensity avoids this dif®culty but is impractical to use in routine identi®cation. Rachinger (1948) described a simple graphical procedure for removing K2 peaks. The method causes errors because it makes the incorrect assumption that K2 is the exact half-scale version of K1 . Ladell, Zagofsky & Pearlman (1975) developed an exact algorithm using the actual mathematical shapes observed with

Fig. 2.3.3.2. Effect of specimen rotation and particle size on Si powder intensity using a conventional diffractometer (Fig. 2.3.1.3) and Cu K. Numbers below fast rotation are the average intensities.

Fig. 2.3.3.3. Various measures of pro®le.

62


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES 2.3.3.4. Rate-meter/strip-chart recording

the user's diffractometer but, with line-pro®le-®tting programs now available, the K2 component can be modelled precisely along with the K1 . It is possible to isolate the K1 line when using a high-quality incident-beam focusing monochromator as described in Subsection 2.3.1.2, Fig. 2.3.1.12(b), but there may be a loss of intensity. The source size must be narrow and the focal length long enough to separate the components.

Formerly, the most common method of obtaining diffractometer data was by using a rate-meter and strip-chart recorder with the paper moving synchronously with the constant angular velocity of the scan. This simple analogue method is still used and a large fraction of the JCPDS (ICDD) ®le prior to about 1982 was obtained in this way. The method has several limitations: the data are not in the digital form required for computers, and are distorted; manual measurement of the chart takes a long time and has low accuracy. The output of the strip chart lags behind the input by an amount determined by the product of the scanning speed and the time constant of the rate-meter, including the speed of the recorder pen. The peak height is decreased and shifted in the direction of the scan causing asymmetric broadening with loss of resolution. The pro®le shape, K-doublet separation, and scan direction also contribute to distortion. When the product of the scan speed and time constant have the same value, the pro®le shapes are the same even though the total count is determined by the scan speed, Figs. 2.3.3.4(a) and (b). If the product is large, the distortion is severe (c), and very weak peaks may be lost.

2.3.3.3. Use of peak or centroid for angle de®nition The most obvious and commonly used measure of the re¯ection angle of a pro®le is the position of maximum intensities (Fig. 2.3.3.3). The midpoints of chords at various heights have often been used but their values vary with the pro®le asymmetry. Another method is to connect the midpoints of chords near the top of the pro®le and extrapolate to the peak. The computer methods using derivatives are the most accurate and fastest as described in Subsection 2.3.3.7. A more fundamental measure that uses the entire intensity distribution is the centre of gravity (or centroid) de®ned as .R R I2 d2: 2:3:3:5 h2i 2I2 d2

2.3.3.5. Computer-controlled automation Most diffractomers are now sold with computer automation. Older instruments can be easily upgraded by adding a stepping motor to the gear-drive shaft. A large variety of computers and programs is available, and it is not easy to make the best selection. Continuing improvements in computer technology have been made to handle expanded programs with increased speed and storage capabilities. The collected data are displayed on a VDU screen and/or computer printer and stored on hard disk or diskette for later use and analysis. Microprocessors are often used to select the X-ray-generator operating conditions, shutter control, specimen change, and similar tasks that were formerly performed manually. Aside from the elimination of much of the manual labour, automation provides far better control of the data-collection and data-reduction procedures. However, computers do not preclude the necessity of precise alignment and calibration. Smith (1989) has written a detailed description of computer analysis for phase identi®cation and also includes related programs and their sources. Personal computers are widely used for powder-diffraction automation and a typical arrangement is shown in Fig. 2.3.3.5(a). The automation may provide for step scanning,

The variance (mean-square deviation of the mean) is de®ned as W2 h2 h2i2 i .R R I2 d2: 2 h2i2 I2 d2

2:3:3:6

The use of the centroid and variance has two important advantages: (1) most of the aberrations (x2:3:1:1:6) were derived in terms of the centroid and variance; and (2) they are additive, making it easy to determine the composite effect of a number of aberrations. Mathematically, the integration extends from 1 to 1 but the aberrations have a ®nite range. However, the practical use of these measures causes some dif®culty. If the pro®le shapes are Lorentzian, the tails decay slowly. A very wide range would be required to reach points where the signal could no longer be separated from the background and the pro®les must be truncated for the calculation. Truncation limits that have been used are 90% ordinate heights of K1 (Ladell, Parrish & Taylor, 1959), and equal 2 or l limits from the centroid (Taylor, Mack & Parrish, 1964; Langford, 1982). The limits such as 21 and 22 in Fig. 2.3.3.3 must be carefully chosen to avoid errors and this involves the correct determination of the background level. It is not practical to use centroids for overlapping peak clusters unless the pro®le ®tting can accurately resolve the individuals with their correct positions and intensities. Their use has, therefore, been con®ned to simple patterns with small unit cells in which the pro®les were well separated. The difference between the angle derived from the peak and the centroid depends on the asymmetry of the pro®le, which in turn varies with the K-doublet separation and the aberration broadening. Tournarie (1958) found that the centre of a horizontal chord at 60.6% of the K1 peak height corresponds well to the centroid of that line in fairly well resolved doublets. The number, of course, depends on the pro®le shape. There is also the basic problem that most of the X-ray wavelengths were probably determined from the spectral peaks and, if the centroids are measured for the powder pattern, the Bragg equation becomes nonlinear in the sense that the 1:1 correspondence between l and sin is lost.

Fig. 2.3.3.4. Rate-meter strip-chart recordings. REV: scan direction reversed. Scan speed and time constant shown at top.

63


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION A typical VDU screen menu for diffractometer-operation control is shown in Fig. 2.3.3.5(b). A number of runs can be de®ned with the same or different experimental parameters to run consecutively. The run log number, date, and time are usually automatically entered and together with the comment and parameters are carried forward and recorded on the print-outs and graphics to make certain the runs are completely identi®ed. The menu is designed to prompt the operator to enter all the required information before a run can be started. Error messages appear if omissions or entry mistakes are made. There are, of course, many variations to the one shown.

continuous scanning with read-out on the ¯y, or slewing to selected angles to read particular points. Step scanning is the method most frequently used. It is essential that absolute registration and step tracking be reliably maintained for all experimental conditions. The step size or angular increment 2 and count time t at each step, and the beginning and ending angles are selectable. For a given total time available for the experiment, it usually makes no difference in the counting statistical accuracy if a combination of small or large 2 and t (within reasonable limits) is used. A minimal number of steps of the order of 2 ' 0:1 to 0.2 FWHM is required for pro®le ®tting isolated peaks. It is clear that the greater the number of steps, the better the de®nition of the pro®le shape. The step size becomes important when using pro®le ®tting to resolve patterns containing overlapped re¯ections and to detect closely spaced overlaps from the width and small changes in slopes of the pro®les. A preliminary fast run to determine the nature of the pattern may be made to select the best run conditions for the ®nal pattern. Ê Will et al. (1988) recorded a quartz pattern with 1.28 A synchrotron X-rays and 0.01 steps to test the step-size role. The pro®le ®tting was done using all points and repeated with the omission of every second, third, and fourth point corresponding to 2 0:02, 0.03 and 0.04 . The R(Bragg) values were virtually the same (except for 0.04 where it increased), indicating the experimental time could have been reduced by a factor of three with little loss of precision; see also Hill & Madsen (1984). Patterns with more overlapping would require smaller steps. Ideally, the steps could be larger in the background but this also requires a prior knowledge of the pattern and special programming.

2.3.3.6. Counting statistics X-ray quanta arrive at the detector at random and varying rates and hence the rules of statistics govern the accuracy of the intensity measurements. The general problems in achieving maximum accuracy in minimum time and in assessing the accuracy are described in books on mathematical statistics. Chapter 7.5 reviews the pertinent theory; see also Wilson (1980). In this section, only the ®xed-time method is described because the ®xed-count method takes too long for most practical applications. Let N be the average of N, the number of counts in a given time t, over a very large number of determinations. The spread is given by a Poisson probability distribution (if N is large) with standard deviation N 1=2 :

2:3:3:7

Any individual determination of N or the corresponding counting rate n N=t will be subject to a proportionate error " which is also a function of the con®dence level, i.e. the probability that the result deviates less than a certain percentage from the true value. If Q is the constant determined by the con®dence level, then " Q=N 1=2 ;

2:3:3:8

where Q 0:67 for the probable relative error "50 (50% con®dence level) and Q 1:64 and 2.58 for the 90 and 99% con®dence levels "90 ; "99 ; respectively. For a 1% error, N 4500, 27 000, 67 000 for "50 , "90 , "99 , respectively. Fig. 2.3.3.6 shows various percentage errors as a function of N for several con®dence levels.

Fig. 2.3.3.5. (a) Block diagram of typical computer-controlled diffractometer and electronic circuits. The monitor circuit enclosed by the dashed line is optional. HPIB is the interface bus. (b) A fullscreen menu with some typical entries.

Fig. 2.3.3.6. Percentage error as a function of the total number of counts N for several con®dence levels.

64


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES the intensity and peak-to-background ratio are low, the computing time is much increased. Since powder patterns often contain a number of weak peaks that may not be required for the analysis, computer programs often permit the user to select a minimum peak height (MPH) and a standard deviation (SD) that the peak must exceed to be included in the data reduction. For example, MPH 1 would reject peaks less than 1% of the highest peak in the recorded pattern, and SD 4 requires the intensity to exceed the background adjacent to the peak by 4B1=2 . The number of peaks rejected depends on the intensity and peakto-background ratio as illustrated in Fig. 2.3.3.7, where the cutoff level was set at B 4B 1=2 for two recordings of the same pattern with about a 40 times difference in intensities. All visible peaks are included in the high-intensity recording and several are rejected by the cut-off level selected in the lower-intensity pattern. Before carrying out the computer calculations, it may be desirable to subtract unusual background such as is caused by a glass substrate in a thin-®lm pattern. The following method was developed using computergenerated pro®les having the same shapes as conventional diffractometer (Fig. 2.3.1.3) pro®les and adding random counting statistical noise (Huang & Parrish, 1984; Huang, 1988). The best results were obtained using the ®rst derivative dx=dy 0 of a least-squares-®tted cubic polynomial to locate the peaks, combined with the second derivative d2 y=dx2 minimum of a quadratic/cubic polynomial to resolve overlapped re¯ections (Fig. 2.3.3.8). Overlaps with a separate 0:5 FWHM can be resolved and measured and the accuracy of the peak position is 0.001 for noise-free pro®les. Real pro®les with statistical noise have a precision of 0:003 to 0.02 depending on the noise level. The Savitzky & Golay (1964) method (see also Ateiner, Termonia & Deltour, 1974; Edwards & Willson, 1974) was used for smoothing and differentiation of

In practice, there is usually a background count NB . The net peak count NPB NB NP B is dependent on the P=B radio as well as on NPB and NB separately. The relative error "D of the net peak count is "D

NPB "PB 2 NN "B 2 1=2 ; NP B

2:3:3:9

which shows that "D is similarly in¯uenced by both absolute errors NPB "PB and NB "B . The absolute standard deviation of the net peak height is P

B

2 PB B2 1=2

2:3:3:10

and expressed as the per cent standard deviation is P

B

NPB NB 1=2 100: NP B

2:3:3:11

The accuracy of the net peak measurement decreases rapidly as the peak-to-background ratio falls below 1. For example, with NB 50, the dependence of P B on P=B is P=B 0.1 1 10 100

P B (%) 205 24.5 4.9 1.43.

It is obviously desirable to minimize the background using the best possible experimental methods. 2.3.3.7. Peak search The accurate location of the 2 angle corresponding to the peak of the pro®le has been discussed in many papers (see, for example, Wilson, 1965). Computers are now widely used for data reduction, thereby greatly decreasing the labour, improving the accuracy, and making possible the use of specially designed algorithms. It is not possible to present a description of the large number of private and commercial programs. The peak-search and pro®le-®tting methods described below have been successfully used for a number of years and are representative of the results that can now be obtained. They have greatly improved the results in phase identi®cation, integrated intensity measurement, and analyses requiring precise pro®le-shape determination. It is likely that even better programs and methods will be developed in this rapidly changing ®eld. There are two levels of the types of data reduction that may be done. The easiest and most frequently used method is usually called `peak search'. It computes the 2 angles and intensities of the peaks. The results have good precision for isolated peaks but give the values of the composite overlapping re¯ections as they appear, for example, on a strip-chart recording. The calculation is virtually instantaneous and is often all that is needed for phase identi®cation, lattice-parameter determination, and similar analyses. The second, pro®le ®tting, described below, is a more advanced procedure that can resolve overlapping peaks into individual re¯ections and determines the pro®le shape, width, peak and integrated intensities, and re¯ection angle of each resolved peak. This method requires a prior knowledge of the pro®le-®tting function. It is used to determine the integrated intensities for analyses requiring higher precision such as crystalstructure re®nement and quantitative analysis, and pro®le-shape parameters for small crystallite size, microstrain and similar studies. To measure weak peaks, the counting statistical accuracy must be suf®cient to delineate the peak from the background. When

Fig. 2.3.3.7. Effect of 4 maximum peak height (horizontal line) on dropping weak peaks from inclusion in computer calculation. Step scan with (a) t 5 s and (b) t 0:1 s. Five-compound mixture, Cu K.

65


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION The procedure is based on the least-squares ®tting of theoretical pro®le intensities to the digitized powder pattern. The pro®le intensity at the ith step is calculated by P Y xi calc Bxi Ij Pxi Tj j ; 2:3:3:12

the data by least squares in which the values of the derivatives can be calculated using a set of tabulated integers. The convolution range CR expressed as a multiple of the FWHM of the peak can be selected. A minimum of ®ve points is required. For asymmetric peaks, such as occur at small 2's, a CR ' 0:5 FWHM gives the best precision. The larger the CR the larger the intrinsic error but the smaller the random error, and the smaller the number of peaks identi®ed in overlapping patterns. The larger CR also avoids false peaks in patterns with poor counting statistics. Fig. 2.3.3.8(c) shows the dependence of the accuracy of the peak determination on P=. The computer results list the 2's, d's, absolute and relative intensities (scaled to 100) of the identi®ed peaks. The calculation is made with a selected wavelength such as K1 and the possible K2 peaks are ¯agged.

j

where Bxi is the background intensity, Ij is the integrated intensity of the jth re¯ection, Tj is the peak-maximum position, Px P i j is the pro®le function to represent the pro®le shape, and j is taken over j, in which the Pxj has a ®nite value at xi . Unlike the Rietveld method, a structure model is not used. In the least-squares ®tting, Ij and Tj are re®ned together with background and pro®le shape parameters in Pxj . Smoothing the experimental data is not required because it underestimates the estimated standard deviations for the least-squares parameters, which are based on the counting statistics. The experimental pro®les are a convolution of the X-ray line spectrum l and all the combined instrumental and geometrical

2.3.3.8. Pro®le ®tting Pro®le ®tting has greatly advanced powder diffractometry by making it possible to calculate the intensities, peak positions, widths, and shapes of the re¯ections with a far greater precision than had been possible with manual measurements or visual inspection of the experimental data. The method has better resolution than the original data and the entire scattering distribution is used instead of only a few features such as the peak and width. Individual pro®les and clusters of re¯ections can be ®tted, or the entire pattern as in the Rietveld method (Chapter 8.6).

Fig. 2.3.3.9. (a) Computer-generated symmetrical Lorentzian pro®le L and Gaussian G with equal peak heights, 2 and FWHM. (b) Double Gaussian GG shown as the sum of two Gaussians in which I and FWHM of G1 are twice those of G2 and 2 is constant. (c)± f Pro®le ®tting with different functions. Differences between experimental points and ®tted pro®le shown at one-half height. Synchrotron radiation, Si(111).

Fig. 2.3.3.8. (a) Si(220) Cu K re¯ection. (b) First (circles), second (crosses), and third (triangles) derivatives of a seven-point polynomial of data in (a). (c) Average angular deviations as a function of P= for various derivatives.

66


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES has the best ®t as shown by the difference curve at half-height and the lowest Rp RPF value. The Pearson VII function is de®ned as

aberrations G with the true diffraction effects of the specimen S (Parrish, Huang & Ayers, 1976), i.e.

x l G S background:

2:3:3:13

PxPVII a1 x=b2

The pro®le shapes and resolution differ in the various diffractometer geometries and there is no universal pro®le-®tting function. In conventional X-ray tube focusing methods, the pro®les are asymmetric and the shapes change continually across the scattering-angle range owing to the aberrations and the K doublet. To avoid problems caused by the K doublet, a few authors used the K line, but it has only about 1=7 the intensity. The pro®les obtained with synchrotron radiation are symmetrical and narrower, and the widths increase with increasing 2. The different shapes and rates of decay of the tails make it necessary to ®nd an analytical function that best ®ts the particular experimental pro®le. Langford (1987) and Young & Wiles (1982) have compiled and reviewed various pro®le-®tting functions and several are described below. Howard & Preston (1989) give details of the computations in their review of the method. Early pro®le analyses used Gaussian or Lorentzian (Cauchy) curves. Fig. 2.3.3.9(a) shows that the most obvious difference is the rate of decay of the tails. X-ray synchrotron pro®les lie between the two as shown in Figs. 2.3.3.9(c)± f . The function must ®t the tails as well as the main body and single-element functions are generally unsatisfactory. The Voigt function is a convolution of Lorenzian (L) and Gaussian (G) functions of different widths: R PxV LxGx u du; 2:3:3:14

V

Lx 1

Gx;

2:3:3:15

where is the ratio of Lorentz to Gauss and they have the same widths. The re®ned and width of the full ®tted pro®le can be related by a polynomial expansion (Hastings, Thomlinson & Cox, 1984; David, 1986; Cox, Toby & Eddy, 1988) to the widths of the L and G components of the original Voigt function. It is frequently used to ®t synchrotron-radiation pro®les. In the particular case shown in Figs. 2.3.3.9(c)± f , the pseudo-Voigt 67


;

2:3:3:16

where m is a re®nable parameter based on the G=L content (for m 1, the curve is 100% Lorentzian and for m 1 it is 100% Gaussian), and 1=b 221=m 11=2 =W , where W is the FWHM. The peak asymmetry can be incorporated into the pro®le function in several ways. One is to multiply (or add) the symmetrical pro®le function with an asymmetric function (Rietveld, 1969). Another is to dispose two or three functions asymmetrically (Parrish, Huang & Ayers, 1976). A third is to use a split-type function, consisting of two pro®le functions, each of which de®nes one-half the total peak, i.e. the low- or highangle sides of the peak and each has different pro®le widths and shapes but the same height (Toraya, Yoshimura & Somiya, 1983; Howard & Snyder, 1983). Some other functions that have been used include the double Gaussian [Fig. 2.3.3.9(b)] for low-resolution synchrotron data (Will, Masciocchi, Parrish & Hart, 1987), a Gaussian with shifted Lorentzian component to account for the asymmetry on the low-2 side of the tail (Will, Masciocchi, Parrish & Lutz, 1990), pro®le modelling of single isolated peaks with a rational function, e.g. the ratio of two polynomials (Pyrros & Hubbard, 1983). In contrast to these analytical-type functions, some empirical functions have been developed. They are the `learned' (experimental) peak-shape function (Hepp & Baerlocher, 1988) and the direct ®tting of experimental data represented by Fourier series (Mortier & Constenoble, 1973). The sum of Lorentzians has been used for X-ray tube focusing pro®les (Parrish & Huang, 1980; Taupin, 1973). The instrument function l G is determined (see below) by a sum of Lorentzian curves, three each for K1 and K2 and one for the weak K3 satellite. Three Lorentzians were used to match the asymmetry although a greater or lesser number could be used depending on the pro®le symmetry. Each curve has three parameters (intensity, half-width at half-height, and peak position) and the 21 parameters are adjusted by the computer program to give the best ®t to the experimental data, which may contain 150 to 300 points. This is done only once for each particular instrument set-up. After l G is determined, the pro®le ®tting is easy and fast because only the specimen contribution S must be convoluted with l G. If the specimen has no asymmetric broadening other than l G, S can be approximated by a single symmetrical Lorentzian for each re¯ection; a split Lorentzian can be used if there is asymmetric broadening. A function can be tested using isolated pro®les of a standard specimen such as silicon, tungsten, quartz, and others which have 2:5 A

2.3.5.4.2. Single and balanced ®lters Single ®lters to remove the K lines are also used, but better results are generally obtained with a crystal monochromator. The following description provides the basic information on the use of ®lters if monochromators are not used. A single thin ®lter made of, or containing, an element that has an absorption edge of wavelength just less than that of the K1 , K2 doublet will absorb part of that doublet but much more of the K line and part of the white radiation, as shown in Fig. 2.3.5.3. The relative transmission throughout the spectrum depends on the ®lter element and its thickness. A ®lter may be used to modify the X-ray spectral distribution by suppressing certain radiations for any of several reasons: (1) lines. -line intensity need be reduced only enough to avoid overlaps and dif®culties in identi®cation in powder work. 78


2.3. POWDER AND RELATED TECHNIQUES: X-RAY TECHNIQUES Table 2.3.5.3. Calculated thickness of balanced ®lters for common target elements

usually absorbed in the air path or counter-tube window and, hence, are not observed. When using vacuum or helium-path instruments and low-absorbing detector windows, the longerwavelength ¯uorescence spectra may appear. When specimen ¯uorescence is present, the position of the ®lter may have a marked effect on the background. If placed between the X-ray tube and specimen, the ®lter attenuates a portion of the primary spectrum just below the absorption edges of the elements in the specimen, thereby reducing the intensity of the ¯uorescence. When placed between the specimen and counter tube, the ®lter absorbes some of the ¯uorescence from the specimen. The choice of position will depend on the elements of the X-ray tube target and specimen. If the ®lter is placed after the specimen, it is advisable to place it close to the specimen to minimize the amount of ¯uorescence from the ®lter that reaches the detector. The ¯uorescence intensity decreases by the inverse-square law. Maximizing the distance between the specimen and detector also reduces the specimen ¯uorescence intensity detected for the same reason. If the ®lter is to be placed between the X-ray tube and specimen, the ®lter should be close to the tube to avoid ¯uorescence from the ®lter that might be recorded. It is sometimes useful to place the ®lter over only a portion of the ®lm in powder cameras to facilitate the identi®cation of the lines. If possible, the X-ray tube target element should be chosen so that its ®lter also has a high absorption for the specimen X-ray ¯uorescence. For example, with a Cu target and Cu specimen, the continuum causes a large Cu K ¯uorescence that is transmitted by an Ni ®lter; if a Co target is used instead, the Cu K ¯uorescence is greatly decreased by an Fe K ®lter. A second ®lter may be useful in reducing the ¯uorescence background. For example, with a Ge specimen, the continuum from a Cu target causes strong Ge K ¯uorescence, which an Ni ®lter transmits. Addition of a thin Zn ®lter improves the peak=background ratio P=B of the Cu K with only a small Ê ; Zn K-absorption reduction of peak intensity (Ge K, l 1:25 A Ê edge, l 1:28 A). X-ray background is also caused by scattering of the entire primary spectrum with varying ef®ciency by the specimen. The ®lter reduces the background by an amount dependent on its absorption characteristics. When using pulse-amplitude discrimination and specimens whose X-ray ¯uorescence is weak, the remaining observed background is largely due to characteristic line radiation. The ®lter then usually reduces the background and the K radiation by roughly the same amount and P=B is not changed markedly regardless of the position of the ®lter.

Target material Ag Mo Mo Cu Ni Co Fe Cr

Pd Zr Zr Ni Co Fe Mn V

Mo Sr Y Co Fe Mn Cr Ti

(A) Thickness mm g cm 0.0275 0.0392 0.0392 0.0100 0.0094 0.0098 0.0095 0.0097

3

0.033 0.026 0.026 0.0089 0.0083 0.0077 0.0071 0.0059

(B) Thickness mm g cm 0.039 0.104 0.063 0.0108 0.0113 0.0111 0.0107 0.0146

2

0.040 0.027 0.028 0.0095 0.0089 0.0083 0.0077 0.0066

The ®lter is sometimes used instead of black paper or Al foil to screen out visible and ultraviolet light. Filters in the form of pure thin metal foils are available from a number of metal and chemical companies. They should be checked with a bright light source to make certain they are free of pinholes. The balanced-®lter technique uses two ®lters that have absorption edges just above and just below the K1 , K2 wavelengths (Ross, 1928; Young, 1963). The difference between intensities of X-ray diffractometer or ®lm recordings made with each ®lter arises from the band of wavelengths between the absorption edges, which is essential that of the K1 , K2 wavelengths. The thicknesses of the two ®lters should be selected so that both have the same absorption for the K wavelength. Table 2.3.5.3 lists the calculated thicknesses of ®lter pairs for the common target elements. The (A) ®lter was chosen for a 67% transmission of the incident K intensity, and only pure metal foils are used. Adjustment of the thickness is facilitated if the foil is mounted in a rotatable holder so that the ray-path thickness can be varied by changing the inclination of the foil to the beam. Although the two ®lters can be experimentally adjusted to give the same K intensities, they are not exactly balanced at other wavelengths. The use of pulse-amplitude discrimination to remove most of the continuous radiation is desirable to reduce this effect. The limitations of the method are (a) the dif®culties in adjusting the balance of the ®lters, (b) the band-pass is much wider that that of a crystal monochromator, and (c) it requires two sets of data, one of which has low intensity and consequently poor counting statistics.

79


Filter pair (A) (B)


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION

2.4. Powder and related techniques: electron and neutron techniques By J. M. Cowley and A. W. Hewat

diameter. For these reasons, the methods for phase identi®cation from electron diffraction patterns and the corresponding databases (see Subsection 2.4.1.6) are increasingly concerned with single-crystal spot patterns in addition to powder patterns. Instrument manufacturers usually provide values of camera lengths, L, or camera constants, Ll, for a wide range of designated lens-current settings. It is advisable to check these calibrations with samples of known structure and to determine calibrations for non-standard lens settings. The effective camera length, L, is dependent on the specimen height within the objective-lens pole-piece. If a specimen-height adjustment (a z-lift) is provided, it should be adjusted to give a predetermined lens current, and hence focal length, of the objective lens. In some microscopes, at particular lens settings the projector lenses may introduce a radial distortion of the diffraction pattern. This may be measured with a suitable standard specimen.

2.4.1. Electron techniques (By J. M. Cowley) 2.4.1.1. Powder-pattern geometry The electron wavelengths normally used to obtain powder patterns from thin ®lms of polycrystalline materials lie in Ê (20 to 200 kV accelerating the range 8 10 2 to 2 10 2 A voltages). The maximum scattering angles (2B ) observed are usually less than 10 1 rad. Patterns are usually recorded on ¯at photographic plates or ®lms and a small-angle approximation is applied. For a camera length L, the distance from the specimen to the photographic plate in the absence of any intervening electron lenses, the approximation is made that, for a diffraction ring of radius r, l=d 2 sin ' tan 2 r=L; or the interplanar spacing, d, is given by d Ll=r:

2:4:1:1

For a scattering angle of 10 1 rad, the error in this expression is 0.5%. A better approximation, valid to better than 0.1% at 10 1 rad, is d Ll=r1 3r 2 =8L2 :

2.4.1.3. Preferred orientations The techniques of specimen preparation may result in a strong preferred orientation of the crystallites, resulting in strong arcing of powder-pattern rings, the absence of some rings, and perturbations of relative intensities. For example, small crystals of ¯aky habit deposited on a ¯at supporting ®lm may be oriented with one reciprocal-lattice axis preferentially perpendicular to the plane of the support. A ring pattern obtained with the incident beam perpendicular to the support then shows only those rings for planes in the zone parallel to the preferred axis. Such orientation is detected by the appearance of arcing and additional re¯ections when the supporting ®lm is tilted. Tilted specimens give the so-called oblique texture patterns which provide a rich source of threedimensional diffraction information, used as a basis for crystal structure analysis. A full discussion of the texture patterns resulting from preferred orientations is given in Section 2.5.3 of IT B (1993).

2:4:1:2

The `camera constant' Ll may be obtained by direct measurement of L and the accelerating voltage if there are no electron lenses following the specimen. Direct electronic recording of intensities has great advantages over photographic recording (Tsypursky & Drits, 1977). In recent years, electron diffraction patterns have been obtained most commonly in electron microscopes with three or more post-specimen lenses. The camera-constant values are then best obtained by calibration using samples of known structure. With electron-optical instruments, it is possible to attain collimations of 10 6 rad so that for scattering angles of 10 1 rad an accuracy of 10 5 in d spacings should be possible in principle but is not normally achievable. In practice, accuracies of about 1% are expected. Some factors limiting the accuracy of measurement are mentioned in the following sections. The small-angle-scattering geometry precludes application of any of the special camera geometries used for high-accuracy measurements with X-rays (Chapter 2.3).

2.4.1.4. Powder-pattern intensities In the kinematical approximation, the expression for intensities of electron diffraction follows that for X-ray diffraction with the exception that, because only small angles of diffraction are involved, no polarization factor is involved. Following Vainshtein (1964), the intensity per unit length of a powder line is

2.4.1.2. Diffraction patterns in electron microscopes The specimens used in electron microscopes may be selfsupporting thin ®lms or ®ne powders supported on thin ®lms, usually made of amorphous carbon. Specimen thicknesses Ê in order to avoid perturbations must be less than about 103 A of the diffraction patterns by strong multiple-scattering effects. The selected-area electron-diffraction (SAED) technique [see Section 2.5.1 in IT B (1993)] allows sharply focused Ê diffraction patterns to be obtained from regions 103 to 105 A in diameter. For the smaller ranges of selected-area regions, specimens may give single-crystal patterns or very spotty ring paterns, rather than continuous ring patterns, because the number of crystals present in the ®eld of view is small unless Ê or less. By use of the crystallite size is of the order of 100 A the convergent-beam electron-diffraction (CBED) technique, diffraction patterns can be obtained from regions of diameter Ê [see Section 2.5.2 in IT B (1993)] or, in the case of 100 A Ê in some specialized instruments, regions less than 10 A

2 d2M Ih J0 l2 h V h ;

4Ll

where J0 is the incident-beam intensity, h is the structure factor, is the unit-cell volume, V is the sample volume, and M is the multiplicity factor. The kinematical approximation has limited validity. The deviations from this approximation are given to a ®rst approximation by the two-beam approximation to the dynamical-scattering theory. Because an averaging over all orientations is involved, the many-beam dynamical-diffraction effects are less evident than for single-crystal patterns. 80

Copyright © 2006 International Union of Crystallography 81 s:\ITFC\ch-2-4.3d (Tables of Crystallography)

2:4:1:3

2.4. POWDER AND RELATED TECHNIQUES: ELECTRON AND NEUTRON TECHNIQUES giving very broad rings, it is possible to use the method, commonly applied for diffraction by gases, of performing a Fourier transform to obtain a radial distribution function (Goodman, 1963).

By integrating the two-beam intensity expression over excitation error, Blackman (1939) obtained the expression for the ratio of dynamical to kinematical intensities: Idyn =Ikin Ah 1

RAh 0

J0 2x dx;

2:4:1:4

2.4.1.5. Crystal-size analysis The methods used in X-ray diffraction for the determination of average crystal size or size distributions may be applied to electron diffraction powder patterns. Except in the case of very small crystal dimensions, several factors peculiar to electrons should be taken into consideration. (a) Unless energy ®ltering is used to remove inelastically scattered electrons, a component is added to the rings broadened by the effects of inelastic scattering involving electronic excitations. Since the mean free paths for such processes are Ê and the angular spread of the scattered of the order of 103 A electrons is 10 3 to 10 4 rad, the ring broadening for thick samples may be equivalent to the broadening for a crystal size of Ê. the order of 100 A (b) When a powder sample consists of separated crystallites having faces not predominantly parallel or perpendicular to the incident beam, the diffraction rings may be appreciably broadened by refraction effects. The refractive index for electrons is given, to a ®rst approximation, by

where Jo x is the zero-order Bessel function, Ah Hh with the interaction constant 2mel=h2 , and H is the crystal thickness. Careful measurements on ring patterns from thin aluminium ®lms by Horstmann & Meyer (1962) showed agreement with the `Blackman curve' [from equation (2.4.1.4)] to within about 5% with some notable exceptions. Deviations of up to 40 to 50% from the Blackman curve occurred for several re¯ections, such as 222 and 400, which are second-order re¯ections from strong inner re¯ections. A practical algorithm for implementing Blackman corrections has been published by Dvoryankina & Pinsker (1958). Such deviations result from plural-beam systematic interactions, the coherent multiple scattering between different orders of a strong inner re¯ection. When the Bragg condition is satis®ed for one order, the excitation errors for the other orders are the same for all possible crystal orientations and these other orders contribute systematically to the ring-pattern intensities. A correction for the effects of systematic interactions may be made by use of the Bethe second approximation (Bethe, 1928) (see Chapter 8.8). For non-systematic re¯ections, corresponding to reciprocallattice points not collinear with the origin and the reciprocallattice point of interest, the averaging over all crystal orientations ensures that the powder-pattern intensity calculated from the two-beam formula will not be appreciably affected. Appreciable effects from non-systematic interactions may, however, occur when the averaging is over a limited range of crystal orientations, as in the case of strong preferred orientations. It was shown theoretically by Turner & Cowley (1969) and experimentally by Imamov, Pannhorst, Avilov & Pinsker (1976) that appreciable modi®cations of intensities of oblique-texture patterns may result from non-systematic interactions for particular tilt angles, especially for heavy-atom materials [see also Avilov, Parmon, Semiletov & Sirota (1984)]. The techniques for the measurement of electron diffraction intensities are described in Chapter 7.2. Most commonly electron diffraction powder patterns are recorded by photographic methods and a microdensitometer is used for quantitative intensity measurement. The Grigson scanning method, using a scintillator and photomultiplier to record intensities as the pattern is scanned over a ®ne slit, has considerable advantages in terms of linearity and range of the intensity scale (Grigson, 1962). This method also has the advantage that it may readily be combined with an energy ®lter so that only elastically scattered electrons (or electrons inelastically scattered with a particular energy loss) may be recorded. Small-angle electron diffraction may give useful information in some cases, but must be interpreted carefully because the features may result from multiple scattering or other artefacts. It may give additional details of periodicity (super-periods) and deviations of the real symmetry from the ideal symmetry suggested by other data. Care must be taken with the interpretation of additional re¯ections, as they may relate to the structure of small regions that are not typical of the bulk specimens such as are examined by X-ray diffraction. The techniques for interpretation of electron diffraction powder-pattern intensities follow those for X-ray patterns when the kinematical approximation is valid. For very small crystals,

n 1 0 =2E; where 0 is the mean inner potential of the crystal, typically 10 to 20 V, and E is the accelerating voltage of the incident electron beam. For the beam passing through the two faces of a 90 wedge, each at an angle 45 to the beam, for example, the beam is de¯ected by an amount 0 =E 1:5 10

rad

for an inner potential of 15 V and E 100 kV. The broadening of the ring by such de¯ections can correspond to the broadening Ê. due to a particle size of l=2 ' 120 A For crystallites of regular habit, such as the small cubic crystals of MgO smoke, the ring broadening from this source is strongly dependent on the crystallographic planes involved (Sturkey & Frevel, 1945; Cowley & Rees, 1947; Honjo & Mihama, 1954). For more isometric crystal shapes, this dependence is less marked and the broadening has been estimated (Cowley & Rees, 1947) as equivalent to that due to Ê. a particle size of about 200 A 2.4.1.6. Unknown-phase identi®cation: databases To a limited extent, the compilations of data for X-ray diffraction, such as the ICDD Powder Diffraction File, may be used for the identi®cation of phases from electron diffraction data. The nature of the electron diffraction data and the circumstances of its collection have prompted the compilation of databases speci®cally for use with electron diffraction. Factors taken into consideration include the following. (a) Because of the increasing use of single-crystal patterns obtained in the SAED mode in an electron microscope, the use of single-crystal spot patterns, in addition to powder patterns, must be considered for purposes of identi®cation. Methods for the analysis of single-crystal patterns are summarized in Section 5.4.1. (b) The deviations from kinematical scattering conditions may be large, especially for single-crystal patterns, so that little reliance can be placed on relative intensities, and re¯ections kinematically forbidden may be present. 81


4

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION ¯ux on the sample can be increased with a focusing monochromator, the sample volume by using large sample-detector distances or Soller collimators, and the detector solid angle by using a large multidetector. The focusing monochromator is usually made from horizontal strips of pyrolytic graphite or squashed germanium mounted on a vertically focusing plate 100 to 300 mm high. A large beam can thus be focused on a sample up to 50 mm high. Vertical divergence of 5 or more can be tolerated even for a highresolution machine; the peak width is only increased (and made asymmetric) far from scattering angles of 90 , where most of the peaks occur. The effect is in any case purely geometrical, and can readily be included in the data analysis (Howard, 1982). To increase the wavelength spread l=l, and hence intensity, monochromator mosaic can be large (200 ) even for high resolution, since all wavelengths are focused back into the primary-beam direction at scattering angles equal to the monochromator take-off angle (Fig. 2.4.2.1). Ê ) are favoured, to spread Rather long wavelengths (1.5 to 3 A out the pattern, and to reduce the total number of re¯ections excited (increasing their average intensity). Data must then be collected at large scattering angles with good resolution to obtain suf®ciently small d spacings, and this implies a large take-off angle. A graphite ®lter to remove l=2 and higher-order contamination is a popular choice for a primary wavelength of Ê (Loopstra, 1966). Since germanium re¯ections such as hhl 2.4 A with h, l 2n 1 do not produce l=2 contamination, a ®lter is Ê with not needed for primary wavelengths below about 1.6 A high-take-off-angle geometry, but is still necessary for longer wavelengths. The multidetector can be an array of up to 64 individual detectors and Soller collimators for a high-resolution machine (Hewat & Bailey, 1976; Hewat, 1986a), or a position-sensitive detector (PSD) for a high-¯ux machine (Allemand et al., 1975). Gas-®lled (3 He or BF3 ) detectors are usual, though scintillator and other types of solid-state detector are increasingly used; the PSD may be either a single horizontal wire with positiondetection logic comparing the signals obtained at either end, or an array of vertical wire detectors within a common gas envelope. The vertical aperture of the single-wire detector seriously limits the ef®ciency of what is otherwise a very cheap solution, and of course large angular ranges cannot be covered by a single straight wire. The vertical aperture should match the vertical divergence from the monochromator ( 5 ). Composite detectors can be constructed by stacking elements both vertically to increase the aperture, and horizontally to increase the angular range. Construction of a wide-angle (160 ) multiwire detector is dif®cult and expensive, but a solid angle of more than 0.1 sr may be obtained. The solid angle for a collimated multidetector, even if it covers 160 , may be less than 0.01 sr. The sample volume limits the resolution of the PSD, since the detector resolution 3 (typically 0.2 ) is the mean of the element width and the sample diameter (typically 5 mm) divided by the sample-to-detector distance (typically 1500 mm). For a Soller collimator, 3 can be as little as 50 , and does not depend on the sample volume, which can be large (20 mm diameter) even for high resolution. The PSD also requires special precautions to avoid background from the sample environment, while the collimated machine can handle dif®cult sample environments, especially for scattering near 90 . The de®nition of the detector is the number of data points per degree. For pro®le analysis, unless the peak shape is well known a priori, about ®ve points are needed per re¯ection half-width, which is more than usually available from a multiwire PSD.

(c) Compositional information may be obtained by use of X-ray microanalysis (or electron-energy-loss spectroscopy) performed in the electron microscope and this provides an effective additional guide to identi®cation. (d) Electron diffraction data often extend to smaller d spacings than X-ray data because there is no wavelength limitation. (e) The electron diffraction d-spacing information is rarely more precise than 1% and the uncertainty may be 5% for large d spacings. With these points in mind, databases specially designed for use with electron diffraction have been developed. The NIST/ Sandia/ICDD Electron Diffraction Database follows the design principles of Carr, Chambers, Melgaard, Himes, Stalick & Mighell (1987). The 1993 version contains crystallographic and chemical information on over 81 500 crystalline materials with, in most cases, calculated patterns to ensure that diagnostic highd-spacing re¯ections can be matched. It is available on magnetic tape or ¯oppy disks. The MAX-d index (Anderson & Johnson, 1979) has been expanded to 51 580 NSI-based entries (Mighell, Himes, Anderson & Carr, 1988) in book form for manual searching. 2.4.2. Neutron techniques (By A. W. Hewat) Neutrons have advantages over X-rays for the re®nement of crystal structures from powder data because systematic errors (Wilson, 1963) are smaller, and the absence of a form factor means that information is available at small d spacings. It is also easy to collect data at very low or high temperature; examining the structure as a function of temperature (or pressure) is much more useful than simply obtaining `the' crystal structure at STP (standard temperature and pressure). In some cases, `kinetics' measurements at intervals of only a few seconds are needed to follow chemical reactions. A neutron powder diffractometer need not separate all of the Bragg peaks, since complex patterns can be analysed by Rietveld re®nement (Rietveld, 1969), but high resolution will increase the information content of the pro®le, and permit the re®nement of larger and more complex structures. Doubling the unit-cell volume doubles the number of Bragg peaks, requiring higher resolution, but also halves the average peak intensity. Resolution must not then be obtained at the expense of well de®ned line shape, essential for pro®le analysis, nor at the expense of intensity. Two types of diffractometer are required in practice: a highresolution machine with data-collection times of a few hours (or days) for Rietveld structure re®nement, and a high-¯ux machine with data-collection times of a few seconds (or minutes) for kinetics measurements. In both cases, the data-collection rate depends on the product of the ¯ux on the sample, the sample volume, and the solid angle of the detector (Hewat, 1975). The

Fig. 2.4.2.1. Schematic drawing of the high-resolution neutron powder diffractometer D2B at ILL, Grenoble.

82


2.4. POWDER AND RELATED TECHNIQUES: ELECTRON AND NEUTRON TECHNIQUES For this focusing geometry, the different wavelengths l l, re¯ected at different angles from the monochromator, are brought back parallel to the primary beam, and the line width becomes simply the convolution of the primary and detector collimations 1 and 3 :

However, a PSD may be scanned to increase the pro®le de®nition. The minimum scan angle is clearly the angle between elements, from 0.2 for a PSD to 2.5 or more for a multidetector in steps of from 0.025 to 0.1 . If larger scans are performed, it is most convenient to reduce the data to a single pro®le by averaging the counts from different detector elements at each point in the pro®le, after correcting for relative ef®ciencies and angular separations. The resolution, of either machine, should be no better than really necessary for a particular sample: additional resolution merely reveals problems with sample perfection and line shape, making Rietveld re®nement dif®cult, and of course reducing effective intensity. In any case, resolution is ultimately limited by the powder particle size and strain broadening (Hewat, 1975). Ê is the effective size of As an order of magnitude, if D = 1000 A the perfect crystallites that make up a much larger (1 to 10 mm) Ê , the best powder grain, then for lattice spacing d 1 A resolution that one can hope to obtain is of the order d=d d=D 10 3 , corresponding to a line width of 2 ' 0:1 . A few more perfect materials (usually those for which single crystals can be grown!) will produce higherresolution patterns, but then primary extinction may not be negligible. It is not even necessary to have the best possible resolution for all d spacings: ideally, the resolution should be proportional to the density of lines, and this increases with the surface area of the Ewald sphere of radius 1=d. Then we want d=d d 2 or 2 l2 = sin 2. This has a minimum near 2 90 . In fact, the full width at half-height is (Cagliotti, Paoletti & Ricci, 1958)

2 2 21 23 : The collimators 1 and 3 should then be equal and small. Such ®ne collimators are now made routinely from gadolinium oxidecoated stretched plastic foil (Carlile, Hey & Mack, 1977). The collimator 2 should simply be large enough to pass all wavelengths re¯ected by the mosaic spread of the monochromator, i.e. 2 ' 2 . Neutron crystallographers have been reluctant to use large take-off angles because they seem to imply greatly reduced beam intensity. Indeed, large M means small waveband l=l since l=l d=d M cot M cot M . However, l=l and therefore beam intensity can be recovered simply by increasing . This has no effect on the resolution at focusing, but it does increase the line width at low angles where there are few lines. When large d spacings are needed, for example for magnetic structures, it is best for both resolution and intensity to retain the same high take-off geometry and increase the wavelength to bring these lines closer to the focusing angle. A large take-off angle also gives a large choice of high-index re¯ections and Ê! wavelengths up to 6 A A ®xed take-off angle greatly simpli®es machine design: the multidetector collimation 3 is also necessarily ®xed, but the single primary collimator 1 can readily be changed. It is useful to have a second choice, much larger than 3 , to boost intensity for poor samples or exploratory data collection. The resolution at low angles, largely determined by (or 2 ), is not much affected by increasing 1 . Finally, the machine should be designed around the sample environment, since this is one of the strengths of neutron powder diffraction. There is no point in building a neutron machine with superb resolution and intensity (these can much more readily be obtained with X-rays) if it cannot produce precise results for the kind of experiments of most interest ± those for which it is dif®cult to use any other technique (Hewat, 1986b).

2 2 U tan2 V tan W : The parameters U, V, and W are functions of the monochromator mosaic spread and collimation, from which it follows that the minimum in 2 occurs for scattering angles 2 ' 2M . The monochromator take-off angle should then be at least 90 ; in practice, since 2 2 increases quadratically with tan for angles larger than this focusing angle, the monochromator takeoff should be even greater than 90 . A value of 120 to 135 is recommended.

83




2.5. Energy-dispersive techniques

By B. Buras, W. I. F. David, L. Gerward, J. D. Jorgensen and B. T. M. Willis 2.5.1. Techniques for X-rays (By B. Buras and L. Gerward) X-ray energy-dispersive diffraction, XED, invented in the late sixties (Giessen & Gordon, 1968; Buras, Chwaszczewska, Szarras & Szmid 1968), utilizes a primary X-ray beam of polychromatic (`white') radiation. XED is the analogue of whitebeam and time-of-¯ight neutron diffraction (cf. Section 2.5.2). In the case of powdered crystals, the photon energy (or wavelength) spectrum of the X-rays scattered through a ®xed optimized angle is measured using a semiconductor detector connected to a multichannel pulse-height analyser. Single-crystal methods have also been developed. 2.5.1.1. Recording of powder diffraction spectra In XED powder work, the incident- and scattered-beam directions are determined by slits (Fig. 2.5.1.1). A powder spectrum is shown in Fig. 2.5.1.2. The Bragg equation is 2d sin 0 l hc=E;

2:5:1:1a

where d is the lattice-plane spacing, 0 the Bragg angle, l and E the wavelength and the photon energy, respectively, associated with the Bragg re¯ection, h is Planck's constant and c the velocity of light. In practical units, equation (2.5.1.1a) can be written Ê sin 0 6:199: EkeV dA

2:5:1:1b

The main features of the XED powder method where it differs from standard angle-dispersive methods can be summarized as follows: (a) The incident beam is polychromatic. (b) The scattering angle 20 is ®xed during the measurement but can be optimized for each particular experiment. There is no mechanical movement during the recording. (c) The whole energy spectrum of the diffracted photons is recorded simultaneously using an energy-dispersive detector. The scattering angle is chosen to accommodate an appropriate number of Bragg re¯ections within the available photon-energy range and to avoid overlapping with ¯uorescence lines from the sample and, when using an X-ray tube, characteristic lines from the anode. Overlap can often be avoided because a change in the scattering angle shifts the diffraction lines to new energy positions, whereas the ¯uorescence lines always appear at the same energies. Severe overlap problems may be encountered when the sample contains several heavy elements. The detector aperture usually collects only a small fraction of the Debye±Scherrer cone of diffracted X-rays. The

Fig. 2.5.1.1. Standard and conical diffraction geometries: 20 ®xed scattering angle. At low scattering angles, the lozenge-shaped sample volume is very long compared with the beam cross sections (after HaÈusermann, 1992).

collection of an entire cone of radiation greatly increases the intensities. Also, it makes it possible to overcome crystallite stratistics problems and preferred orientations in very small samples (Holzapfel & May, 1982; HaÈusermann, 1992). 2.5.1.2. Incident X-ray beam (a) Bremsstrahlung from an X-ray tube Bremsstrahlung from an X-ray diffraction tube provides a useful continuous spectrum for XED in the photon-energy range 2±60 keV. However, one has to avoid spectral regions close to the characteristic lines of the anode material. A tungsten anode is suitable because of its high output of white radiation having no characteristic lines in the 12±58 keV range. A drawback of Bremsstrahlung is that its spectral distribution is dif®cult to measure or calculate with accuracy, which is necessary for a structure determination using integrated intensities [see equation (2.5.1.7)]. Bremsstrahlung is strongly polarized for photon energies near the high-energy limit, while the low-energy region has a weak polarization. The direction of polarization is parallel to the direction of the electron beam from the ®lament to the anode in the X-ray tube. Also, the polarization is dif®cult to measure or calculate. (b) Synchrotron radiation Synchrotron radiation emitted by electrons or positrons, when passing the bending magnets or insertion devices, such as wigglers, of a storage ring, provides an intense smooth spectrum for XED. Both the spectral distribution and the polarization of the synchrotron radiation can be calculated from the parameters of the storage ring. Synchrotron radiation is almost fully polarized in the electron or positron orbit plane, i.e. the horizontal plane, and inherently collimated in the vertical plane. Full advantage of these features can be obtained using a vertical scattering plane. However, the mechanical construction of the diffractometer, the placing of furnaces, cryogenic equipment, etc. are easier to handle when the X-ray scattering is recorded in the horizontal plane. Recent XED facilities at synchrotron-radiation sources have been described by Besson & Weill (1992), Clark (1992), HaÈusermann (1992), Olsen (1992), and Otto (1997).

Fig. 2.5.1.2. XED powder spectrum of BaTiO3 recorded with synchrotron radiation from the electron storage ring DORIS at DESY-HASYLAB in Hamburg, Germany. Counting time 1 s. Escape peaks due to the Ge detector are denoted by e (from Buras, Gerward, Glazer, Hidaka & Olsen, 1979).


2.5. ENERGY-DISPERSIVE TECHNIQUES 2.5.1.3. Resolution The momentum resolution in energy-dispersive diffraction is limited by the angular divergence of the incident and diffracted X-ray beams and by the energy resolution of the detector system. The observed pro®le is a convolution of the pro®le due to the angular divergence and the pro®le due to the detector response. For resolution calculations, it is usually assumed that the pro®les are Gaussian, although the real pro®les might exhibit geometrical and physical aberrations (Subsection 2.5.1.5). The relative full width at half-maximum (FWHM) of a diffraction peak in terms of energy is then given by E=E en =E2 5:546F"=E cot 0 0 2 1=2 ;

2:5:1:2

where en is the electronic noise contribution, F the Fano factor, " the energy required for creating an electron±hole pair (cf. Subsection 7.1.5.1), and 0 the overall angular divergence of the X-ray beam, resulting from a convolution of the incident- and the diffracted-beam pro®les. For synchrotron radiation, 0 can usually be replaced by the divergence of the diffracted beam because of the small divergence of the incident beam. Fig. 2.5.1.3 shows E=E as a function of Bragg angle 0 . The curves have been calculated from equations (2.5.1.1) and (2.5.1.2) for two values of the lattice-plane spacing and two values of 0 , typical for Bremsstrahlung and synchrotron radiation, respectively. It is seen that in all cases E=E decreases with decreasing angle (i.e. increasing energy) to a certain minimum and then increases rapidly. It is also seen that the minimum point of the E=E curve is lower for the small d value and shifts towards smaller 0 values for decreasing 0 . Calculations of this kind are valuable for optimizing the Bragg angle for a given sample and other experimental conditions (cf. Fukamachi, Hosoya & Terasaki, 1973; Buras, Niimura & Olsen, 1978). The relative peak width at half-height is typically less than 1% for energies above 30 keV. When the observed peaks can be ®tted with Gaussian functions, one can determine the centroids of the pro®les by a factor of 10±100 better than the E=E value of equation (2.5.1.2) would indicate. Thus, it should be possible to achieve a relative resolution of about 10 4 for high energies. A resolution of this order is required for example in residual-stress measurements. The detector broadening can be eliminated using a technique where the diffraction data are obtained by means of a scanning crystal monochromator and an energy-sensitive detector (Bourdillon, Glazer, Hidaka & Bordas, 1978; Parrish & Hart, 1987). A low-resolution detector is suf®cient because its function (besides recording) is just to discriminate the monochromator harmonics. The Bragg re¯ections are not measured simultaneously as in standard XED. The monochromator-scan method can be useful when both a ®xed scattering angle (e.g. for samples in special environments) and a high resolution are required.

evaluated at the energy of the diffraction peak, V the irradiated sample volume, Nc the number of unit cells per unit volume, j the multiplicity factor, F the structure factor, and Cp E; 0 ) the polarization factor. The latter is given by PE sin2 20 ;

Cp E; 0 12 1 cos2 20

2:5:1:4

where PE is the degree of polarization of the incident beam. The de®nition of PE is PE

i0;p E i0;n E ; i0 E

2:5:1:5

where i0;p E and i0;n E are the parallel and normal components of i0 E with respect to the plane de®ned by the incident- and diffracted-beam directions. Generally, Cp E; 0 has to be calculated from equations (2.5.1.4) and (2.5.1.5). However, the following special cases are sometimes of interest: P 0:

Cp 0 12 1 cos2 20 2

2:5:1:6a

P 1:

Cp 0 cos 20

2:5:1:6b

P

Cp 1:

2:5:1:6c

1:

Equation (2.5.1.6a) can often be used in connection with Bremsstrahlung from an X-ray tube. The primary X-ray beam can be treated as unpolarized for all photon energies when there is an angle of 45 between the plane de®ned by the primary and the diffraced beams and the plane de®ned by the primary beam and the electron beam of the X-ray tube. In standard con®gurations, the corresponding angle is 0 or 90 and equation (2.5.1.6a) is generally not correct. However, for 20 < 20 it is correct to within 2.5% for all photon energies (Olsen, Buras, Jensen, Alstrup, Gerward & Selsmark, 1978). Equations (2.5.1.6b) and (2.5.1.6c) are generally acceptable approximations for synchrotron radiation. Equation (2.5.1.6b) is used when the scattering plane is horizontal and (2.5.1.6c) when the scattering plane is vertical. The diffraction directions appear as generatrices of a circular cone of semi-apex angle 20 about the direction of incidence. Equation (2.5.1.3) represents the total power associated with this

2.5.1.4. Integrated intensity for powder sample The kinematical theory of diffraction and a non-absorbing crystal with a `frozen' lattice are assumed. Corrections for thermal vibrations, absorption, extinction, etc. are discussed in Subsection 2.5.1.5. The total diffracted power, Ph , for a Bragg re¯ection of a powder sample can then be written (Buras & Gerward, 1975; Kalman, 1979) Ph hcre2 VNc2 i0 Ejd 2 jFj2 h Cp E; 0 cos 0 0 ;

2:5:1:3

where h is the diffraction vector, re the classical electron radius, i0 E the intensity per unit energy range of the incident beam

Fig. 2.5.1.3.Relative resolution, E=E, as function of Bragg angle, 0 , Ê and (b) 0.5 A Ê . The for two values of the lattice plane spacing: (a) 1 A full curves have been calculated for 0 10 3 , the broken curves for 0 10 4 .

85


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION cone. Generally, only a small fraction of this power is recorded by the detector. Thus, the useful quantity is the power per unit length of the diffraction circle on the receiving surface, Ph0 . At a distance r from the sample, the circumference of the diffraction circle is 2r sin 20 and one has (constants omitted) Ph0 / r 1 VNc2 i0 Ejd 2 jFj2 h

Cp E; 0 0 : sin 0

2:5:1:7

The peak areas in an XED powder spectrum are directly proportional to the Ph0 of equation (2.5.1.7). Quantitative structural analysis requires the knowledge of i0 E and PE. As mentioned above, these quantities are not known with suf®cient accuracy for Bremsstrahlung. For synchrotron radiation they can be calculated, but they will nevertheless contribute to the total uncertainty in the analysis. Accordingly, XED is used rather for identi®cation of a known or assumed structure than for a full structure determination. 2.5.1.5. Corrections (a) Temperature effects The effect of thermal vibrations on the integrated intensities is expressed by the Debye±Waller factor in the same way as for standard angle-dispersive methods. Notice that sin =l 1=2d irrespective of the method used. The contribution of the thermal diffuse scattering to the measured integrated intensities can be calculated if the elastic constants of the sample are known (Uno & Ishigaki 1975). (b) Absorption The transmission factor AE; 0 ) for a small sample bathed in the incident beam and the factor Ac E; 0 for a large sample intercepting the entire incident beam are the same as for monochromatic methods (Table 6.3.3.1). However, when they are applied to energy-dispersive techniques, one has to note that the absorption corrections are strongly varying with energy. In the special case of a symmetrical re¯ection where the incident and diffracted beams each make angles 0 with the face of a thick sample (powder or imperfect crystal), one has Ac E

1 ; 2E

2:5:1:8

where E is the linear attentuation coef®cient evaluated at the energy associated with the Bragg re¯ection. (c) Extinction and dispersion Extinction and dispersion corrections are applied in the same way as for angle-dispersive monochromatic methods. However, in XED, the energy dependence of the corrections has to be taken into account. (d) Geometrical aberrations These are distortions and displacements of the line pro®le by features of the geometry of the apparatus. Axial aberrations as well as equatorial divergence contribute to the angular range 0 of the Bragg re¯ections. There is a predominance of positive contributions to 0 , so that the diffraction maxima are slightly displaced to the low-energy side, and show more tailing on the low-energy side than the high-energy side (Wilson, 1973). (e) Physical aberrations Displacements due to the energy-dependent absorption and re¯ectivity of the sample tend to cancel each other if the incident intensity, i0 E, can be assumed to be constant within the energy range of Bragg re¯ection. With synchrotron radiation, i0 E

varies rapidly with energy and its in¯uence on the peak positions should be checked. Also, the detector response function will in¯uence the line pro®le. Low-energy line shapes are particularly sensitive to the deadlayer absorption, which may cause tailing on the low-energy side of the peak. Integrated intensities, measured as peak areas in the diffraction spectrum, have to be corrected for detector ef®ciency and intensity losses due to escape peaks. 2.5.1.6. The Rietveld method The Rietveld method (see Chapter 8.6) for re®ning structural variables has only recently been applied to energy-dispersive powder data. The ability to analyse diffraction patterns with overlapping Bragg peaks is particularly important for a lowresolution technique, such as XED (Glazer, Hidaka & Bordas, 1978; Buras, Gerward, Glazer, Hidaka & Olsen, 1979; Neuling & Holzapfel, 1992). In this section, it is assumed that the diffraction peaks are Gaussian in energy. It then follows from equation (2.5.1.7) that the measured pro®le yi of the re¯ection k at energy Ei corresponding to the ith channel of the multichannel analyser can be written yi

Ei 2 =Hk2 g; 2:5:1:9

0

where c is a constant, i0 Ei is evaluated at the energy Ei , and Hk is the full width (in energy) at half-maximum of the diffraction peak. AEi is a factor that accounts for the absorption in the sample and elsewhere in the beam path. The number of overlapping peaks can be determined on the basis of their position and half-width. The full width at half-maximum can be expressed as a linear function of energy: Hk UEk V ;

2:5:1:10

where U and V are the half-width parameters. 2.5.1.7. Single-crystal diffraction Energy-dispersive diffraction is mainly used for powdered crystals. However, it can also be applied to single-crystal diffraction. A two-circle system for single-crystal diffraction in a diamond-anvil high-pressure cell with a polychromatic, synchrotron X-ray beam has been devised by Mao, Jephcoat, Hemley, Finger, Zha, Hazen & Cox (1988). Formulae for single-crystal integrated intensities are well known from the classical Laue method. Adaptations to energydispersive work have been made by Buras, Olsen, Gerward, Selsmark & Lindegaard-Andersen (1975). 2.5.1.8. Applications The unique features of energy-dispersive diffraction make it a complement to rather than a substitute for monochromatic angledispersive diffraction. Both techniques yield quantitative structural information, although XED is seldom used for a full structure determination. Because of the ®xed geometry, energydispersive methods are particularly suited to in situ studies of samples in special environments, e.g. at high or low temperature and/or high pressure. The study of anomalous scattering and forbidden re¯ections is facilitated by the possibility of shifting the diffraction peaks on the energy scale by changing the scattering angle. Other applications are studies of Debye±Waller factors, determinative mineralogy, attenuation-coef®cient measurements, on-stream measurements, particle-size and -strain 86


c0 i E AEi jk dk2 jFk j2 expf 4 ln 2Ek Hk 0 i

2.5. ENERGY-DISPERSIVE TECHNIQUES determination, and texture studies. These and other applications can be found in an annotated bibliography covering the period 1968±1978 (Laine & LaÈhteenmaÈki, 1980). The short counting time and the simultaneous recording of the diffraction spectrum permit the study of the kinetics of structural transformations in time frames of a few seconds or minutes. Energy-dispersive powder diffraction has proved to be of great value for high-pressure structural studies in conjunction with synchrotron radiation. The brightness of the radiation source and the ef®ciency of the detector system permit the recording of a diffraction spectrum with satisfactory counting statistics in a reasonable time (100±1000 s) in spite of the extremely small sample volume (10 3 ±10 5 mm3 ). Reviews have been given by Buras & Gerward (1989) and HaÈusermann (1992). Recently, XED experiments have been performed at pressures above 400 GPa, and pressures near 1 TPa may be attainable in the near future (Ruoff, 1992). At this point, it should be mentioned that XED methods have limited resolution and generally give unreliable peak intensities. The situation has been transformed recently by the introduction of the image-plate area detector, which allows angle-dispersive, monochromatic methods to be used with greatly improved resolution and powder averaging (Nelmes & McMahon, 1994, and references therein). 2.5.2. White-beam and time-of-¯ight neutron diffraction (By J. D. Jorgensen, W. I. F. David, and B. T. M. Willis) 2.5.2.1. Neutron single-crystal Laue diffraction In traditional neutron-diffraction experiments, using a continuous source of neutrons from a nuclear reactor, a narrow wavelength band is selected from the wide spectrum of neutrons emerging from a moderator within the reactor. This monochromatization process is extremely inef®cient in the utilization of the available neutron ¯ux. If the requirement of discriminating between different orders of re¯ection is relaxed, then the entire white beam can be employed to contribute to the diffraction pattern and the count-rate may increase by several orders of magnitude. Further, by recording the scattered neutrons on photographic ®lm or with a position-sensitive detector, it is possible to probe simultaneously many points in reciprocal space. If the experiment is performed using a pulsed neutron beam, the different orders of a given re¯ection may be separated from one another by time-of-¯ight analysis. Consider a short polychromatic burst of neutrons produced within a moderator. The subsequent times-of-¯ight, t, of neutrons with differing wavelengths, l, measured over a total ¯ight path, L, may be discriminated one from another through the de Broglie relationship: mn L=t h=l;

The origins of pulsed neutron diffraction can be traced back to the work of Lowde (1956) and of Buras, Mikke, Lebech & Leciejewicz (1965). Later developments are described by Turber®eld (1970) and Windsor (1981). Although a pulsed beam may be produced at a nuclear reactor using a chopper, the major developments in pulsed neutron diffraction have been associated with pulsed sources derived from particle accelerators. Spallation neutron sources, which are based on proton synchrotrons, allow optimal use of the Laue method because the pulse duration and pulse repetition rate can be matched to the experimental requirements. The neutron Laue method is particularly useful for examining crystals in special environments, where the incident and scattered radiations must penetrate heat shields or other window materials. [A good example is the study of the incommensurate structure of -uranium at low temperature (Marmeggi & Delapalme, 1980).] A typical time-of-¯ight single-crystal instrument has a large area detector. For a given setting of detector and sample, a threedimensional region is viewed in reciprocal space, as shown in Fig. 2.5.2.1. Thus, many Bragg re¯ections can be measured at the same time. For an ideally imperfect crystal, with volume Vs and unit-cell volume vc , the number of neutrons of wavelength l re¯ected at Bragg angle by the planes with structure factor F is given by N i0 ll4 Vs F 2 =2v2c sin2 ;

2:5:2:1

where mn is the neutron mass and h is Planck's constant. Ê , equation Expressing t in microseconds, L in metres and l in A (2.5.2.1) becomes t 252:7784 Ll: Inserting Bragg's law, l 2d=n sin , for the nth order of a Ê gives fundamental re¯ection with spacing d in A t 505:5568=nLd sin :

Fig. 2.5.2.1.Construction in reciprocal space to illustrate the use of multi-wavelength radiation in single-crystal diffraction. The circles with radii kmax 2=lmin and kmin 2=lmax are drawn through the origin. All reciprocal-lattice points within the shaded area may be sampled by a linear position-sensitive detector spanning the scattering angles from 2min to 2max . With a position-sensitive area detector, a three-dimensional portion of reciprocal space may be examined (after Schultz, Srinivasan, Teller, Williams & Lukehart, 1984).

2:5:2:2

Different orders may be measured simply by recording the time taken, following the release of the initial pulse from the moderator, for the neutron to travel to the sample and then to the detector.

where i0 l is the number of incident neutrons per unit wavelength interval. In practice, the intensity in equation (2.5.2.3) must be corrected for wavelength-dependent factors, such as detector ef®ciency, sample absorption and extinction, and the contribution of thermal diffuse scattering. Jauch, Schultz & Schneider (1988) have shown that accurate structural data can be obtained using the single-crystal time-of-¯ight method despite the complexity of these wavelength-dependent corrections. 2.5.2.2. Neutron time-of-¯ight powder diffraction This technique, ®rst developed by Buras & Leciejewicz (1964), has made a unique impact in the study of powders in con®ned environments such as high-pressure cells (Jorgensen & 87


2:5:2:3

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Worlton, 1985). As in single-crystal Laue diffraction, the time of ¯ight is measured as the elapsed time from the emergence of the neutron pulse at the moderator through to its scattering by the sample and to its subsequent detection. This time is given by equation (2.5.2.2). Many Bragg peaks, each separated by time of ¯ight, can be observed at a single ®xed scattering angle, since there is a wide range of wavelengths available in the incident beam. A good approximation to the resolution function of a time-of¯ight powder diffractometer is given by the second-moment relationship d=d t=t2 cot 2 L=L2 1=2 ;

2:5:2:4

where d, t and are, respectively, the uncertainties in the d spacing, time of ¯ight, and Bragg angle associated with a given re¯ection, and L is the uncertainty in the total path length (Jorgensen & Rotella, 1982). Thus, the highest resolution is

obtained in back scattering (large 2) where cot is small. Timeof-¯ight instruments using this concept have been described by Steichele & Arnold (1975) and by Johnson & David (1985). With pulsed neutron sources a large source aperture can be viewed, as no chopper is required of the type used on reactor sources. Hence, long ¯ight paths can be employed and this too [see equation (2.5.2.4)] leads to high resolution. For a well designed moderator the pulse width is approximately proportional to wavelength, so that the resolution is roughly constant across the whole of the diffraction pattern. For an ideal powder sample the number of neutrons diffracted into a complete Debye±Scherrer cone is proportional to N 0 i0 ll4 Vs jF 2 cos =4v2c sin2

(Buras & Gerward, 1975). j is the multiplicity of the re¯ection and the remaining symbols in equation (2.5.2.5) are the same as those in equation (2.5.2.3).

88


2:5:2:5


2.6. Small-angle techniques By O. Glatter and R. May

In addition, there is a loss of information in small-angle scattering experiments caused by the averaging over all orientations in space. The three-dimensional structure is represented by a one-dimensional function ± the dependence of the scattered intensity on the scattering angle. This is also true for powder diffraction. To recover the structure uniquely is therefore impossible. The computation of the scattering function for a known structure is called the solution of the scattering problem. This problem can be solved exactly for many different structures. The inversion, i.e. the estimation of the structure of the scatterer from its scattering functions, is called the inverse scattering problem. This problem cannot be solved uniquely. The description and solution of the scattering problem gives information to the experimenter concerning the scattering functions to be expected in a special situation. In addition, this knowledge is the starting point for the evaluation and interpretation of experimental data (solution of the inverse problem). There are methods that give a rough first-order approximation to the solution of the inverse scattering problem using only a minimum amount of a priori information about the system to obtain an initial model. In order to improve this model, one has to solve the scattering problem. The resulting theoretical model functions are compared with the experimental data. If necessary, model modifications are deduced from the deviations. After some iterations, one obtains the final model. It should be noticed that it is possible to find different models that fit the data within their statistical accuracy. In order to reduce this ambiguity, it is necessary to have additional independent information from other experiments. Incorrect models, however, can be rejected when their scattering functions differ significantly from the experimental data. What type of investigations can be performed with small-angle scattering? It is possible to study monodisperse and polydisperse systems. In the case of monodisperse systems, it is possible to determine size, shape, and, under certain conditions, the internal structure. Monodispersity cannot be deduced from small-angle scattering data and must therefore be assumed or checked by independent methods. For polydisperse systems, a size distribution can be evaluated under the assumption of a certain shape for the particles (particle sizing). All these statements are strictly true for highly diluted systems where the interparticle distances are much larger than the particle dimensions. In the case of semi-dilute systems, the result of a small-angle scattering experiment is influenced by the structure of the particles and by their spatial arrangement. Then the scattering curve is the product of the particle scattering function and of the interparticle interference function. If the scattering function of one particle is known, it is possible to evaluate information about the radial distribution of these particles relative to each other. If the system is dense, i.e. if the volume fraction of the particles (scattering centres) is of the same order of magnitude as the volume fraction of the matrix, it is possible to determine these volume fractions and a characteristic length of the phases. The most important practical applications, however, pertain to dilute systems. How are small-angle scattering experiments related to other scattering experiments? Small-angle scattering uses radiation with a wavelength in the range 10 1 to 100 nm, depending on the

2.6.1. X-ray techniques (By O. Glatter) 2.6.1.1. Introduction The purpose of this section is to introduce small-angle scattering as a method for investigation of nonperiodic systems. It should create an understanding of the crucial points of this method, especially by showing the differences from wide-angle diffraction. The most important concepts will be explained. This article also contains a collection of the most important equations and methods for standard applications. For details and special applications, one must refer to the original literature or to textbooks; the reference list is extensive but, of course, not complete. The physical principles of scattering are the same for wideangle diffraction and small-angle X-ray scattering. The electric field of the incoming wave induces dipole oscillations in the atoms. The energy of X-rays is so high that all electrons are excited. The accelerated charges generate secondary waves that add at large distances (far-field approach) to give the overall scattering amplitude. All secondary waves have the same frequency but may have different phases caused by the different path lengths. Owing to the high frequency, it is only possible to detect the scattering intensity ± the square of the scattering amplitude ± and its dependence on the scattering angle. The angle-dependent scattering amplitude is related to the electron-density distribution of the scatterer by a Fourier transformation. All this holds for both wide-angle diffraction and small-angle X-ray scattering. The main difference is that in the former we have a periodic arrangement of identical scattering centres (particles), i.e. the scattering medium is periodic in all three dimensions with a large number of repetitions, whereas in small-angle scattering these particles, for example proteins, are not ordered periodically. They are embedded with arbitrary orientation and with irregular distances in a matrix, such as water. The scattering centres are limited in size, non-oriented, and nonperiodic, but the number of particles is high and they can be assumed to be identical, as in crystallography. The Fourier transform of a periodic structure in crystallography (crystal diffraction) corresponds to a Fourier series, i.e. a periodic structure is expanded in a periodic function system. The Fourier transform of a non-periodic limited structure (small-angle scattering) corresponds to a Fourier integral. In mathematical terms, it is the expansion of a nonperiodic function by a periodic function system. So the differences between crystallography and small-angle scattering are equivalent to the differences between a Fourier series and a Fourier integral. It may seem foolish to expand a non-periodic function with a periodic function system, but this is how scattering works and we do not have any other powerful physical process to study these structures. The essential effect of these differences is that in small-angle scattering we measure a continuous angle-dependent scattering intensity at discrete points instead of sharp, point-like spots as in crystallography. Another important point is that in small-angle scattering we have a linear increase of the signal (scattered intensity) with the number of particles in the measuring volume since intensities are adding. Amplitudes are adding in crystallography, so we have a quadratic relation between the signal and the number of particles. 89 Copyright © 2006 International Union of Crystallography 90 s:\ITFC\chap2-6.3d (Tables of Crystallography)

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION phase ' is 2=l times the difference between the optical path length of the wave and an arbitrary reference wave (with l being the wavelength). The direction of the incident beam is defined by the unit vector s0 and of the scattered beam by s. The angle between these two unit vectors (scattering angle) is 2. The path difference between the rays through a point P and an arbitrary origin O is r(s s0 ). The phase is ' h r if we define the scattering vector h as

problem and on the source used. This range is similar for X-rays and neutrons, but neutrons interact with the nuclei of the atoms whereas X-rays interact with the electrons. The scattering efficiency increases linearly with the atomic number for X-rays. The dependence is much more complicated for neutrons and does not show a systematic trend. The essential fact in neutron scattering is the pronounced difference in the scattering power between hydrogen and deuterium, which is important for varying the contrast between the particles and the matrix. The wavelength and the scattering efficiency limit the range of small-angle scattering experiments to systems in the size range of a few nanometres up to about 100 nm. Special instruments permit the study of larger particles (Bonse & Hart, 1967; Koch, 1988). These instruments need a high intensity of the primary beam (synchrotron radiation) and are not very common. Particles in the size range from 100 nm up to some micrometres can be investigated by static light-scattering techniques (Glatter, Hofer, Jorde & Eigner, 1985; Glatter & Hofer, 1988a,b; Hofer, Schurz & Glatter, 1989). Particles exceeding this limit can be seen in an optical microscope or can be studied with Fraunhofer diffraction (Bayvel & Jones, 1981). Electron microscopy is in competition with all these scattering methods. It has the marked advantage of giving real pictures with rather high resolution but it has the inherent disadvantage that the preparation may introduce artefacts. Small-angle scattering, on the other hand, is a method to study macromolecules in solution, which is a very important advantage for biological samples and for polymers. Crystallography gives more information about the particle (atomic structure) and can be applied to relatively large systems. It is possible to study particles as large as proteins and viruses if good crystals of these substances are available. The experiment needs synchrotron radiation for large molecules like proteins or viruses, i.e. access to a large research facility is necessary. Small-angle X-ray scattering with conventional generators is a typical next-door technique with the advantage of ready availability. Small-angle scattering has developed into a standard measuring method during recent decades, being most powerful for the investigation of submicrometre particles.

h 2=ls

We see that the amplitude A is the Fourier transform of the electron-density distribution . The intensity Ih of the complex amplitude Ah is the absolute square given by the product of the amplitude and its complex conjugate A , RRR 2 e r exp ih r dV ; 2:6:1:4 Ih AhAh where e2 r is the convolution square (Bracewell, 1986): RRR r1 r1 r dV1 : 2:6:1:5 e2 r The intensity distribution in h or reciprocal space is uniquely determined by the structure in real space. Until now, we have discussed the scattering process of a particle in fixed orientation in vacuum. In most cases of smallangle scattering, the following situation is present: ±The scatterers (particles or inhomogeneities) are statistically isotropic and no long-range order exists, i.e. there is no correlation between points at great spatial distance. ±The scatterers are embedded in a matrix. The matrix is considered to be a homogeneous medium with the electron density 0 . This situation holds for particles in solution or for inhomogeneities in a solid. The electron density in equations (2.6.1.3)±(2.6.1.5) should be replaced by the difference in electron density 0 , which can take positive and negative values. The average over all orientations h i leads to

In this subsection, we are concerned with X-rays only, but all equations may also be applied with slight modifications to neutron or electron diffraction. When a wave of X-rays strikes an object, every electron becomes the source of a scattered wave. All these waves have the same intensity given by the Thomson formula 1 1 cos2 2 ; a2 2

sin hr hr (Debye, 1915) and (2.6.1.4) reduces to the form hexp ih ri Z1

2:6:1:1

Ih 4

where Ip is the primary intensity and a the distance from the object to the detector. The factor Tf is the square of the classical electron radius (e2 =mc2 7:90 10 26 [cm2 ]). The scattering angle 2 is the angle between the primary beam and the scattered beam. The last term in (2.6.1.1) is the polarization factor and is practically equal to 1 for all problems dealt with in this subsection. Ie should appear in all following equations but will be omitted, i.e. the amplitude of the wave scattered by an electron will be taken to be of magnitude 1. Ie is only needed in cases where the absolute intensity is of interest. The amplitudes differ only by their phases ', which depend on the positions of the electrons in space. Incoherent (Compton) scattering can be neglected for small-angle X-ray scattering. The

r 2 e2 r

0

sin hr dr hr

2:6:1:6

2:6:1:7

or, with pr r2 e2 r r 2 V r;

2:6:1:8

to Z1 Ih 4

pr 0

sin hr dr; hr

2:6:1:9

is the so-called correlation function (Debye & Bueche, 1949), or characteristic function (Porod, 1951). The function pr is the so-called pair-distance distribution function PDDF (Guinier & 90

91 s:\ITFC\chap2-6.3d (Tables of Crystallography)

2:6:1:2

This vector bisects the angle between the scattered beam and the incident beam and has length h 4=l sin . We keep in mind that sin may be replaced by in small-angle scattering. We now introduce the electron density r. This is the number of electrons per unit volume at the position r. A volume element dV at r contains (r) dV electrons. The scattering amplitude of the whole irradiated volume V is given by RRR Ah r exp ih r dV : 2:6:1:3

2.6.1.2. General principles

Ie Ip Tf

s0 :

2.6. SMALL-ANGLE TECHNIQUES Fournet, 1955; Glatter, 1979). The inverse transform to (2.6.1.9) is given by Z1 1 2:6:1:10 pr 2 Ihhr sinhr dh 2

I0 I2 V 2 4

or by Z1 0

Ihh2

sin hr dh: hr

0

pr dr;

2:6:1:12

i.e. the scattering intensity at h equal to zero is proportional to the area under the PDDF. From equation (2.6.1.11), we find Z 1 V 0 2 Ihh2 dh V 2 2:6:1:13 2

0

1 V r 2 2

R1

2:6:1:11

(Porod, 1982), i.e. the integral of the intensity times h2 is related to the mean-square fluctuation of the electron density irrespective of the structure. We may modify the shape of a particle, the scattering function Ih might be altered considerably, but the integral (2.6.1.13) must remain invariant (Porod, 1951).

The function pr is directly connected with the measurable scattering intensity and is very important for the solution of the inverse scattering problem. Before working out details, we should first discuss equations (2.6.1.9) and (2.6.1.10). The PDDF can be defined as follows: the function pr gives the number of difference electron pairs with a mutual distance between r and r dr within the particle. For homogeneous particles (constant electron density), this function has a simple and clear geometrical definition. Let us subdivide the particle into a very large number of identical small volume elements. The function pr is proportional to the number of lines with a length between r and r dr which are found in the combination of any volume element i with any other volume element k of the particle (see Fig. 2.6.1.1). For r 0, there is no other volume element, so pr must be zero, increasing with r2 as the number of possible neighbouring volume elements is proportional to the surface of a sphere with radius r. Starting from an arbitrary point in the particle, there is a certain probability that the surface will be reached within the distance r. This will cause the pr function to drop below the r 2 parabola and finally the PDDF will be zero for all r > D, where D is the maximum dimension of the particle. So pr is a distance histogram of the particle. There is no information about the orientation of these lines in pr, because of the spatial averaging. In the case of inhomogeneous particles, we have to weight each line by the product of the difference in electron density , and the differential volume element, dV . This can lead to negative contributions to the PDDF. We can see from equation (2.6.1.9) that every distance r gives a sin(hr)=(hr) contribution with the weight pr to the total scattering intensity. Ih and pr contain the same information, but in most cases it is easier to analyse in terms of distances than in terms of sin(x)=x contributions. The PDDF could be computed exactly with equation (2.6.1.10) if Ih were known for the whole reciprocal space. For h 0, we obtain from equation (2.6.1.9)

Invariant Q

R1 0

Ihh2 dh:

2:6:1:14

2.6.1.3. Monodisperse systems In this subsection, we discuss scattering from monodisperse systems, i.e. all particles in the scattering volume have the same size, shape, and internal structure. These conditions are usually met by biological macromolecules in solution. Furthermore, we assume that these solutions are at infinite dilution, which is taken into account by measuring a series of scattering functions at different concentrations and by extrapolating these data to zero concentration. We continue with the notations defined in the previous subsection, which coincide to a large extent with the notations in the original papers and in the textbooks (Guinier & Fournet, 1955; Glatter & Kratky, 1982). There is a notation created by Luzzati (1960) that is quite different in many details. A comparison of the two notations is given in the Appendix of Pilz, Glatter & Kratky (1980). The particles can be roughly described by some parameters that can be extracted from the scattering function. More information about the shape and structure of the particles can be found by detailed discussion of the scattering functions. At first, this discussion will be about homogeneous particles and will be followed by some aspects for inhomogeneous systems. Finally, we have to discuss the influence of finite concentrations on our results. 2.6.1.3.1. Parameters of a particle Total scattering length. The scattering intensity at h 0 must be equal to the square of the number of excess electrons, as follows from equations (2.6.1.7) and (2.6.1.12): I0 2 V 2 4

R1 0

pr dr:

2:6:1:15

This value is important for the determination of the molecular weight if we perform our experiments on an absolute scale (see below). Radius of gyration. The electronic radius of gyration of the whole particle is defined in analogy to the radius of gyration in mechanics: R ri ri2 dVi R2g VR : 2:6:1:16 ri dVi

Fig. 2.6.1.1. The height of the pr function for a certain value of r is proportional to the number of lines with a length between r and r dr within the particle.

V

It can be obtained from the PDDF by 91


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Table 2.6.1.1. Formulae for the various parameters for h (left) and m (right) scales p

or from the innermost part of the scattering curve [Guinier approximation (Guinier, 1939)]: Ih I0 exp h2 R2g =3:

la p RK tan 2 log Im tan m2

R K tan log Ih tan h2 K

A plot of log[Ih vs h2 (Guinier plot) shows at its innermost part a linear descent with a slope tan , where p Rg K tan

q 3 2:628 log e

p Rc Kc tan logIhh tan h2

(see Table 2.6.1.1). The radius of gyration is related to the geometrical parameters of simple homogeneous triaxial bodies as follows (Mittelbach, 1964):

la p tan 2 logImm tan m2 Rc Kc

q 2 Kc 2:146 log e Rt Kt tan

p

Rt Kt

logIhh2 h2 Kt

q 1 1:517 log e

T M

A

l2 a2 Imm0 2 Qm

Ihh2 0 Q

T

la Imm2 0 Qm 2

I0 a2 K P cdz2

K

Imm0 2K a P l cdz2

Ihh2 0 K a2 P 2 cdz2

Mt

Imm2 0 2K 1 P l2 cdz2

Q a2 K P 22 d

2

parallelepiped (edge lengths A, B, C)

R2g 1=12A2 B2 C 2

elliptic cylinder (semi-axes a, b; height h)

R2g

a 2 b2 h 2 h2 R2c 4 12 12

hollow cylinder (height h and radii r1 , r2 )

R2g

r12 r22 h2 : 2 12

Radius of gyration of the thickness. A similar definition exists for lamellar particles. The one-dimensional radius of gyration of the thickness Rt can be calculated from R1 pt rr 2 dr 0 2 Rt R1 ; 2:6:1:21 2 pt r dr

Qm 4 K P l3 ad

0

or from the innermost part of the scattered intensity of thickness It h: It h It 0 exp h2 R2t ;

with It h Ihh (see Table 2.6.1.1 and x2.6.1.3.2.1). Volume. The volume of a homogeneous particle is given by V 22

R2g

0

2

0

pr dr

2:6:1:17

92


I0 : Q

2:6:1:23

This equation follows from equations (2.6.1.12)±(2.6.1.14). Such volume determinations are subject to errors as they rely on the validity of an extrapolation to zero angle [to obtain I0] and to larger angles (h 4 extrapolation for Q). Scattering functions cannot be measured from h equal to zero to h equal to infinity.

prr 2 dr R1

2:6:1:22

2

m!1

R1

2:6:1:20

with Ic h Ihh (see Table 2.6.1.1).

22 Km Os la Qm K lim Imm4

h!1

R52 R51 R32 R31 R2g 1=5a2 b2 c2

Ic h Ic 0 exp h2 R2c =2;

K 1024 =Ie Ê 3 1024 cm=A K Os Q K lim Ihh4

R2g 3=5

where pc r is the PDDF of the cross section or it can be calculated from the innermost part of the scattering intensity of the cross section Ic h:

1 21:0 Ie NL

Mc

2

hollow sphere (radii R1 and R2 )

0

Ihh0 K a2 P cdz2

Mc Mt

V

Ihh0 Q

A 2

R2g 3=5R2

Radius of gyration of the cross section. In the special case of rod-like particles, the two-dimensional analogue of Rg is called radius of gyration of the cross section Rc . It can be obtained from R1 pc rr 2 dr 0 2 Rc R1 ; 2:6:1:19 2 pc r dr

l3 a3 I0 4 Qm R Qm Imm2 dm

I0 V 22 Q R Q Ihh2 dh

sphere (radius R)

ellipsoid (semi-axes a, b, c)

la p tan 2 logImm2 tan m2

tan

2:6:1:18

2.6. SMALL-ANGLE TECHNIQUES Surface. The surface S of one particle is correlated with the scattering intensity I1 h of this particle by

thickness of the sample, c [g cm 3 ] is the concentration, and NL is Loschmidt's (Avogadro's) number.

2 S: 2:6:1:24 h4 Determination of the absolute intensity can be avoided if we calculate the specific surface Os (Mittelbach & Porod, 1965)

Rod-like particles. The mass per unit length Mc M=L, i.e. the mass related to the cross section of a rod-like particle with length L, is given by a similar equation (Kratky & Porod, 1953):

I1 hjh!1 2

Os S=V

lim Ihh4

h!1

Q

:

Ihhh!0 a2 P z2 dcIe NL Ihhh!0 6:68a2 : P z2 dc

Mc

2:6:1:25

Cross section, thickness, and correlation length. By similar equations, we can find the area A of the cross section of a rodlike particle A 2

Ihhh!0 Q

Flat particles. A similar equation holds for the mass per unit area Mt M=A: Ihh2 h!0 a2 P 2z2 dcIe NL Ihh2 h!0 3:34a2 : P z2 dc

2:6:1:26

Mt

and the thickness T of lamellar particles by Ihh2 h!0 T Q

2:6:1:27

hnm 1 Thm cm 1 nm 1 mcm;

0

Thm 2=la: 2 ' m=a l=2h

2:6:1:36

was used in the early years of small-angle X-ray scattering experiments. The formulae for the various parameters for m and the h scale can be found in Table 2.6.1.1, the formulae for the 2 scale can be found in Glatter & Kratky (1982, p. 158). 2.6.1.3.2. Shape and structure of particles

2:6:1:29

In this subsection, we have to discuss how shape, size, and structure of the scattering particle are reflected in the scattering function Ih and in the PDDF pr. In general, it is easier to discuss features of the PDDF, but some characteristics like symmetry give more pronounced effects in reciprocal space.

depending on the length of the chain (Heine, Kratky & Roppert, 1962). For further details, see Kratky (1982b). Molecular weight. Particles of arbitrary shape. The particle is measured at high dilution in a homogeneous solution and has an isopotential specific volume v02 and z2 mol. electrons per gram, i.e. the molecule contains z2 M electrons if M is the molecular weight. The number of effective mol. electrons per gram is given by

2.6.1.3.2.1. Homogeneous particles Globular particles. Only a few scattering problems can be solved analytically. The most trivial shape is a sphere. Here we have analytical expressions for the scattering intensity 2 sinhR hR coshR Ih 3 2:6:1:37 hR3

2:6:1:30

where 0 is the mean electron density of the solvent. The molecular weight can be determined from the intensity at zero angle I0:

and for the PDDF (Porod, 1948)

I0 a2 M P z2 dcIe NL I0 21:0a2 2:6:1:31 P z2 dc (Kratky, Porod & Kahovec, 1951), where P is the total intensity per unit time irradiating the sample, a [cm] is the distance between the sample and the plane of registration, d [cm] is the

pr 12x2 2

3x x3 x r=2R 1;

2:6:1:38

where R is the radius of the sphere. The graphical representation of scattering functions is usually made with a semi-log plot [log Ih vs h] or with a log±log plot [log Ih vs log h]; the PDDF is shown in a linear plot. In order to compare functions from particles of different shape, it is preferable to keep the scattering intensity at zero angle (area under PDDF) and the radius of 93


2:6:1:35

The angular scale 2 with

Persistence length ap . An important model for polymers in solution is the so-called worm-like chain (Porod, 1949; Kratky & Porod, 1949). The degree of coiling can be characterized by the persistence length ap (Kratky, 1982b). Under the assumption that the persistence length is much larger than the cross section of the polymer, it is possible to find a transition point h in an Ihh2 vs h plot where the function starts to be proportional to h. There is an approximation

v02 0 ;

2:6:1:34

with

The maximum dimension D of a particle would be another important particle parameter, but it cannot be calculated directly from the scattering function and will be discussed later.

z2 z2

2:6:1:33

Abscissa scaling. The various molecular parameters can be evaluated from scattered intensities with different abscissa scaling. The abscissa used in theoretical work is h 4=l sin . The most important experimental scale is m [cm], the distance of the detector from the centre of the primary beam with the distance a [cm] between the sample and the detector plane.

but the experimental accuracy of the limiting values Ihhh!0 and Ihh2 h!0 is usually not very high. The correlation length lc is the mean width of the correlation function r (Porod, 1982) and is given by Z1 lc Ihh dh: 2:6:1:28 Q

h ap ' 2:3;

2:6:1:32

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION experimental errors. In any case, this accuracy will be different for different shapes. Any deviation from spherical symmetry will shift the maximum to smaller r values and the value for D will increase [I0 and Rg constant!]. A comparison of PDDF's for a sphere, an oblate ellipsoid of revolution (axial ratio 1:1:0.2), and a prolate ellipsoid of revolution (1:1:3) is shown in Fig. 2.6.1.4. The more we change from the compact, spherical structure to a two- and one-dimensionally elongated structure, the more the maximum shifts to smaller r values and at the same time we have an increase in D. We see that pr is a very informative function. The interpretation of scattering functions in reciprocal space is hampered by the highly abstract nature of this domain. We can see this problem in Fig. 2.6.1.5, where the scattering functions of the sphere and the ellipsoids in Fig. 2.6.1.4 are plotted. A systematic discussion of the features of pr can be found elsewhere (Glatter, 1979, 1982b).

gyration Rg [slope of the main maximum of Ih or the second moment of pr] constant. The scattering function of a sphere with R 65 is shown in Fig. 2.6.1.2 [dashed line, log I0 normalized to 12]. We see distinct minima which are typical for particles of high symmetry. We can determine the size of the sphere directly from the position of the zeros h01 and h02 (Glatter, 1972). R'

4:493 h01

or

R'

7:725 h02

2:6:1:39

or from the position of the first side maximum (Rg ' 4:5=h1 . The minima are considerably flattened in the case of cubes (full line in Fig. 2.6.1.2). The corresponding differences in real space are not so clear-cut (Fig. 2.6.1.3). The pr function of the sphere has a maximum near r R D=2 x ' 0:525 and drops to zero like every PDDF at r D, where D is the maximum dimension of the particle ± here the diameter. The pr for the cube with the same Rg is zero at r ' 175. The function is very flat in this region. This fact demonstrates the problems of accuracy in this determination of D when we take into account

Rod-like particles. The first example of a particle elongated in one direction (prolate ellipsoid) was given in Figs. 2.6.1.4 and 2.6.1.5. An important class is particles elongated in one

Fig. 2.6.1.4. Comparison of the pr function of a sphere (Ð), a prolate ellipsoid of revolution 1:1:3 (ÐÐÐ), and an oblate ellipsoid of revolution 1:1:0.2 (- - - -) with the same radius of gyration.

Fig. 2.6.1.2. Comparison of the scattering functions of a sphere (- - - -) and a cube (Ð) with same radius of gyration.

Fig. 2.6.1.3. Distance distribution function of a sphere (- - - -) and a cube (Ð) with the same radius of gyration and the same scattering intensity at zero angle.

Fig. 2.6.1.5. Comparison of the Ih functions of a sphere, a prolate, and an oblate ellipsoid (see legend to Fig. 2.6.1.4).

94


2.6. SMALL-ANGLE TECHNIQUES PDDF of the cross section pc r to obtain more information on the cross section (Glatter, 1980a).

direction with a constant cross section of arbitrary shape (long cylinders, parallelepipeds, etc.) The cross section A (with maximum dimension d) should be small in comparison to the length of the whole particle L: dL

L D2

d 2 1=2 ' D:

Flat particles. Flat particles, i.e. particles elongated in two dimensions (discs, flat parallelepipeds), with a constant thickness T much smaller than the overall dimensions D, can be treated in a similar way. The scattering function can be written as

2:6:1:40

The scattering curve of such a particle can be written as Ih L=hIc h;

Ih A

2:6:1:41

where the function Ic h is related only to the cross section and the factor 1=h is characteristic for rod-like particles (Kratky & Porod, 1948; Porod, 1982). The cross-section function Ic h is Ic h L 1 Ihh constant Ihh:

R1 0

pc rJ0 hr dr;

It h A2 1 Ihh constant Ihh2 ;

2:6:1:42

1 2

It h 2

2:6:1:43

0

pt r coshr dr

pt r t r

2:6:1:44

0

1

2:6:1:49

Z1 It h coshr dh 0

t r t r:

(Glatter, 1982a). The function pc r is the PDDF of the cross section with pc r r c r hrc rc i:

R1

and

Z1 Ic hhrJ0 hr dh

2:6:1:48

which can be used for the determination of Rt , T, and Mt . In addition, we have again:

where J0 hr is the zero-order Bessel function and pc r

2:6:1:47

where It h is the so-called thickness factor (Kratky & Porod, 1948) or

This function was used in the previous subsection for the determination of the cross-section parameters Rc , A, and Mc . In addition, we have Ic h 2

2 I h; h2 t

2:6:1:50

PDDF's from flat particles do not show clear features and therefore it is better to study f r pr=r (Glatter, 1979). The corresponding functions for lamellar particles with the same basal plane but different thickness are shown in Fig. 2.6.1.7(b). The marked transition points in Fig. 2.6.1.7(b) can be used to determine the thickness. The PDDF of the thickness pt r can give more information in such cases, especially for inhomogeneous particles (see below).

2:6:1:45

The symbol * stands for the mathematical operation called convolution and the symbol h i means averaging over all directions in the plane of the cross section. Rod-like particles with a constant cross section show a linear descent of pr for r d if D > 2:5d. The slope of this linear part is proportional to the square of the area of the cross section, dp A2 2 : 2:6:1:46 2 dr The PDDF's of parallelepipeds with the same cross section but different length L are shown in Fig. 2.6.1.6. The maximum corresponds to the cross section and the point of inflection ri gives a rough indication for the size of the cross section. This is shown more clearly in Fig. 2.6.1.7, where three parallelepipeds with equal cross section area A but different cross-section dimensions are shown. If we find from the overall PDDF that the particle under investigation is a rod-like particle, we can use the

Ê ) and a Fig. 2.6.1.7. Three parallelepipeds with constant length L (400 A constant cross section but varying length of the edges: Ð Ê ; ÐÐÐ 80 20 A Ê ; - - - - 160 10 A Ê . (a) pr function. 40 40 A (b) f r pr=r.

Fig. 2.6.1.6. Distance distributions from homogeneous parallelepipeds Ê (b) 50 50 250 A; Ê (c) with edge lengths of: (a) 50 50 500 A; Ê 50 50 150 A.

95


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION Details of the technique cannot be discussed here, but it is a fact that we can calculate the radial distribution r from the scattering data assuming that the spherical scatterer is only of finite size. The hollow sphere can be treated either as a homogeneous particle with a special shape or as an inhomogeneous particle with spherical symmetry with a step function as radial distribution. The scattering function and the PDDF of a hollow sphere can be calculated analytically. The pr of a hollow sphere has a triangular shape and the function f r pr=r shows a horizontal plateau (Glatter, 1982b).

Composite structures ± aggregates, subunits. The formation of dimers can be analysed qualitatively with the pr function (Glatter, 1979). For an approximate analysis, it is only necessary to know the PDDF of the monomer. Different types of aggregates will have distinct differences in their PDDF. Higher aggregates generally cannot be classified unambiguously. Additional information from other sources, such as the occurrence of symmetry, can simplify the problem. Particles that consist of aggregates of a relatively large number of identical subunits show, at low resolution, the overall structure of the whole particle. At larger angles (higher resolution), the influence of the individual subunits can be seen. In the special case of globular subunits, it is possible to determine the size of the subunits from the position of the minima of the corresponding shape factors using equation (2.6.1.39) (Glatter, 1972; Pilz, Glatter, Kratky & MoringClaesson, 1972).

Rod-like particles. Radial inhomogeneity. If we assume radial inhomogeneity of a circular cylinder, i.e. is a function of the radius r but not of the angle ' or of the value of z in cylindrical coordinates, we can determine some structural details. We define c as the average excess electron density in the cross section. Then we obtain a PDDF with a linear part for r > d and we have to replace in equation (2.6.1.46) by c with the maximum dimension of the cross section d. The pr function differs from that of a homogeneous cylinder with the same c only in the range 0 < r d. A typical example is shown in Fig. 2.6.1.8. The functions for a homogeneous, a hollow, and an inhomogeneous cylinder with varying density c r are shown.

2.6.1.3.2.2. Hollow and inhomogeneous particles We have learned to classify homogeneous particles in the previous part of this section. It is possible to see from scattering data Ih or pr] whether a particle is globular or elongated, flat or rod-like, etc., but it is impossible to determine uniquely a complicated shape with many parameters. If we allow internal inhomogeneities, we make things more complicated and it is clear that it is impossible to obtain a unique reconstruction of an inhomogeneous three-dimensional structure from its scattering function without additional a priori information. We restrict our considerations to special cases that are important in practical applications and that allow at least a solution in terms of a firstorder approximation. In addition, we have to remember that the pr function is weighted by the number of excess electrons that can be negative. Therefore, a minimum in the PDDF can be caused by a small number of distances, or by the addition of positive and negative contributions.

Rod-like particles. Axial inhomogeneity. This is another special case for rod-like particles, i.e. the density is a function of the z coordinate. In Fig. 2.6.1.9, we compare two cylinders with the same size and diameter. One is a homogeneous cylinder diameter d 48 and length L 480, and the with density , other is an inhomogeneous cylinder of the same size and mean but this cylinder is made from slices with a thickness density , respectively. of 20 and alternating densities of 1.5 and 0.5, The PDDF of the inhomogeneous cylinder has ripples with the periodicity of 40 in the whole linear range. This periodicity leads to reflections in reciprocal space (first and third order in the h range of the figure).

Spherically symmetric particles. In this case, it is possible to describe the particle by a one-dimensional radial excess density function r. For convenience, we omit the sign for excess in the following. As we do not have any angle-dependent terms, we have no loss of information from the averaging over angle. The scattering amplitude is simply the Fourier transform of the radial distribution: Z1 sinhr Ah 4 rr dr 2:6:1:51 h

Flat particles. Cross-sectional inhomogeneity. Lamellar particles with varying electron density perpendicular to the basal plane, where is a function of the distance x from the central plane, show differences from a homogeneous lamella of the same size in the PDDF in the range 0 < r < T , where T is the

0

Ih

2

Ah and 1 r 2 2

Z1 hAh 0

sinhr dh r

2:6:1:52

(Glatter, 1977a). These equations would allow direct analysis if Ah could be measured, but we can measure only Ih. r can be calculated from Ih using equation (2.6.1.10) remembering that this function is the convolution square of r [equations (2.6.1.5) and (2.6.1.8)]. Using a convolution square-root technique, we can calculate r from Ih via the PDDF without having a `phase problem' like that in crystallography; i.e. it is not necessary to calculate scattering amplitudes and phases (Glatter, 1981; Glatter & Hainisch, 1984; Glatter, 1988). This can be done because pr differs from zero only in the limited range 0 < r < D (Hosemann & Bagchi, 1952, 1962). In mathematical terms, it is again the difference between a Fourier series and a Fourier integral.

Ê and an Fig. 2.6.1.8. Circular cylinder with a constant length of 480 A Ê . (a) Homogeneous cylinder, (b) hollow outer diameter of 48 A cylinder, (c) inhomogeneous cylinder. The pr functions are shown on the left, the corresponding electron-density distributions r on the right.

96


2.6. SMALL-ANGLE TECHNIQUES A method for distance determination with X-rays by heavyatom labelling was developed by Kratky & Worthman (1947). These ideas are now used for the determination of distances between deuterated subunits of complex macromolecular structures with neutron scattering.

thickness of the lamella. An example is given in Fig. 2.6.1.10 where we compare a homogeneous lamellar particle (with 13) with an inhomogeneous one, t x being a three-step function alternating between the values 1; 1; 1. Flat particles. In-plane inhomogeneity. Lamellae with a homogeneous cross section but inhomogeneities along the basal plane have a PDDF that deviates from that of a homogeneous lamella in the whole range 0 < r < D. These deviations are a measure of the in-plane inhomogeneites; a general evaluation method does not exist. Even more complicated is the situation that occurs in membranes: these have a pronounced crosssectional structure with additional in-plane inhomogeneities caused by the membrane proteins (Laggner, 1982; Sadler & Worcester, 1982).

High-resolution experiments. A special type of study is the comparison of the structures of the same molecule in the crystal and in solution. This is done to investigate the influence of the crystal field on the polymer structure (Krigbaum & KuÈgler, 1970; Damaschun, Damaschun, MuÈller, Ruckpaul & Zinke, 1974; Heidorn & Trewhella, 1988) or to investigate structural changes (Ruckpaul, Damaschun, Damaschun, Dimitrov, JaÈnig, MuÈller, PuÈrschel & Behlke, 1973; Hubbard, Hodgson & Doniach, 1988). Sometimes such investigations are used to verify biopolymer structures predicted by methods of theoretical physics (MuÈller, Damaschun, Damaschun, Misselwitz, Zirwer & Nothnagel, 1984). In all cases, it is necessary to measure the small-angle scattering curves up to relatively high scattering angles (h ' 30 nm 1 , and more). Techniques for such experiments have been developed during recent years (Damaschun, Gernat, Damaschun, Bychkova & Ptitsyn, 1986; Gernat, Damaschun, KroÈber, Bychkova & Ptitsyn, 1986; I'anson, Bacon, Lambert, Miles, Morris, Wright & Nave, 1987) and need special evaluation methods (MuÈller, Damaschun & Schrauber, 1990).

Contrast variation and labelling. An important method for studying inhomogeneous particles is the method of contrast variation (Stuhrmann, 1982). By changing the contrast of the solvent, we can obtain additional information about the inhomogeneities in the particles. This variation of the contrast is much easier for neutron scattering than for X-ray scattering because hydrogen and deuterium have significantly different scattering cross sections. This technique will therefore be discussed in the section on neutron small-angle scattering.

2.6.1.3.3. Interparticle interference, concentration effects So far, only the scattering of single particles has been treated, though, of course, a great number of these are always present. It has been assumed that the intensities simply add to give the total diffraction pattern. This is true for a very dilute solution, but with increasing concentration interference effects will contribute. Biological samples often require higher concentrations for a sufficient signal strength. We can treat this problem in two different ways: ±We accept the interference terms as additional information about our system under investigation, thus observing the spatial arrangement of the particles. ±We treat the interference effect as a perturbation of our single-particle concept and discuss how to remove it. The first point of view is the more general, but there are many open questions left. For many practical applications, the second point of view is important. The radial distribution function. In order to find a general description, we have to restrict ourselves to an isotropic assembly of monodisperse spheres. This makes it possible to

Fig. 2.6.1.9. Inhomogeneous circular cylinder with periodical changes of the electron density along the cylinder axis compared with a homogeneous cylinder with the same mean electron density. (a) pr function; (b) scattering intensity; Ð inhomogeneous cylinder; - - - - homogeneous cylinder.

Fig. 2.6.1.10. pr function of a lamellar particle. The full line corresponds to an inhomogeneous particle, t x is a three-step function with the values 1; 1; 1. The broken line represents the homogeneous lamella with 13.

97


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION their concentrations. For large h values, these curves are identical. In the low-h range, the curves must be extrapolated to zero concentration. It depends on the problem as to whether a linear fit is sufficient or whether a second-degree polynomial has to be used. The extrapolation can be performed in a standard Ih=c versus h plot or in a Zimm plot Ih=c 1 versus h (Cleemann & Kratky, 1960; Kirste & OberthuÈr, 1982). The Zimm plot should be preferred when working with highly concentrated solutions (Pilz, 1982). As mentioned above, the innermost part of the scattering function is lowered and the apparent radius of gyration decreases with increasing concentration. The length of the linear range of the Guinier plot can be extended by the interference effect for non-spherical particles. Thus, an elongated linear Guinier plot is no guarantee of the completeness of the elimination of the concentration effect. Remaining interparticle interferences cannot be recognized in reciprocal space. The PDDF is affected considerably by interparticle interference (Glatter, 1979). It is lowered with increasing distance r, goes through a negative minimum in the region of the maximum dimension D of the particle, and the oscillations vanish at larger r values. This is shown for the hard-sphere model in Fig. 2.6.1.12. The oscillations disappear when the concentration goes to zero. The same behaviour can be found from experimental data even in the case of non-spherical data (Pilz, Goral, Hoylaerts, Witters & Lontie, 1980; Pilz, 1982). In some cases, it may be impossible to carry out experiments with varying concentrations. This will be the case if the structure of the particles depends on concentration. Under certain circumstances, it is possible to find the particle parameters by neglecting the innermost part of the scattering function influenced by the concentration effect (MuÈller & Glatter, 1982).

describe the situation by introducing a radial interparticle distribution function Pr (Zernicke & Prins, 1927; Debye & Menke, 1930). Each particle has the same surroundings. We consider one central particle and ask for the probability that another particle will be found in the volume element dV at a distance r apart. The mean value is (N=v) dV; any deviation from this may be accounted for by a factor Pr. In the range of impenetrability r < D, we have Pr 0 and in the long range r D Pr 1. So the corresponding equation takes the form 2 3 Z1 N sinhr Ih NI1 h41 dr5: 2:6:1:53 4r 2 Pr 1 V hr 0

The second term contains all interparticle interferences. Its predominant part is the `hole' of radius D, where Pr 1 1. This leads to a decrease of the scattering intensity mainly in the central part, which results in a liquid-type pattern (Fig. 2.6.1.11). This can be explained by the reduction of the contrast caused by the high number of surrounding particles. Even if a size distribution for the spheres is assumed, the effect remains essentially the same (Porod, 1952). Up to now, no exact analytical expressions for Pr exist. The situation is even more complicated if one takes into account attractive or repulsive interactions or non-spherical particle shapes (orientation). If we have a system of spheres with known size D, we can use equation (2.6.1.37) for I1 h in equation (2.6.1.53), divide by this function, and calculate Pr from experimental data by Fourier inversion. The interference term can be used to study particle correlations of charged macromolecular solutions (Chen, Sheu, Kalus & Hoffmann, 1988). If there are attractive forces, there will be a tendency for aggregation. This tendency may, for instance, be introduced by some steps in the procedure of preparation of biological samples. Such aggregation leads to an increase of the intensity in the central part (gas type). In this case, we will finally have a polydisperse system of monomers and oligomers. Again, there exist no methods to analyse such a system uniquely.

Aggregates ± gas type. When the particles show a tendency to aggregation with increasing concentration, we can follow the same procedures as discussed for the liquid type, i.e. perform a concentration series and extrapolate the Ih=c curves to zero concentration. However, in most cases, the tendency to aggregation exists at any concentration, i.e. even at very high dilution we have a certain number of oligomers coexisting with monomers. There is no unique way to find the real particle parameters in these cases. It is not sufficient just to neglect the innermost part of the

Elimination of concentration effects ± liquid type. In most cases, the interference effect is a perturbation of our experiment where we are only interested in the particle scattering function. Any remaining concentration effect would lead to errors in the resulting parameters. As we have seen above, the effect is essential at low h values, thus influencing I0, Rg , and the PDDF at large r values. The problem can be handled for the liquid-like type in the following way. We measure the scattering function Ih at different concentrations (typically from a few mg ml 1 up to about 50 mg ml 1 ). The influence of the concentration can be seen in a common plot of these scattering curves, divided by

Fig. 2.6.1.12. Distance distribution ± hard-sphere interference model. Theoretical pr functions: Ð 0; - - - 0:25; ÐÐÐ 0:5; Ð Ð Ð 1:0. Circles: results from indirect transformaÊ tion: 0:5, h1 R 2:0. 2% statistical noise, Dmax 300 A, Rg 0:5%, I0 1:2%.

Fig. 2.6.1.11. Characteristic types of scattering functions: (a) gas type; (b) particle scattering; (c) liquid type.

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2.6. SMALL-ANGLE TECHNIQUES synchrotron radiation is available only at a few places in the world. Reviews on synchrotron radiation and its application have been published during recent years (Stuhrmann, 1978; Holmes, 1982; Koch, 1988). In these reviews, one can also find some remarks on the general principles of the systems including cameras and special detectors.

scattering function because that leads to an increasing loss of essential information about the particle (monomer) itself. 2.6.1.4. Polydisperse systems In this subsection, we give a short survey of the problem of polydispersity. It is most important that there is no way to decide from small-angle scattering data whether the sample is mono- or polydisperse. Every data set can be evaluated in terms of monodisperse or polydisperse structures. Independent a priori information is necessary to make this decision. It has been shown analytically that a certain size distribution of spheres gives the same scattering function as a monodisperse ellipsoid with axes a, b and c (Mittelbach & Porod, 1962). The scattering function of a polydisperse system is determined by the shape of the particles and by the size distribution. As mentioned above, we can assume a certain size distribution and can determine the shape, or, more frequently, we assume the shape and determine the size distribution. In order to do this we have to assume that the scattered intensity results from an ensemble of particles of the same shape whose size distribution can be described by Dn R, where R is a size parameter and Dn R denotes the number of particles of size R. Let us further assume that there are no interparticle interferences or multiple scattering effects. Then the scattering function Ih is given by Ih cn

R1 0

Dn RR6 i0 hR dR;

2.6.1.5.1. Small-angle cameras General. In any small-angle scattering experiment, it is necessary to illuminate the sample with a well defined flux of X-rays. The ideal condition would be a parallel monochromatic beam of negligible dimension and very high intensity. These theoretical conditions can never be reached in practice (Pessen, Kumosinski & Timasheff, 1973). One of the main reasons is the fact that there are no lenses as in the visible range of electromagnetic radiation. The refractive index of all materials is equal to or very close to unity for X-rays. On the other hand, this fact has some important advantages. It is, for example, possible to use circular capillaries as sample holders without deflecting the beam. There are different ways of constructing a small-angle scattering system. Slit, pinhole, and block systems define a certain area where the X-rays can pass. Any slit or edge will give rise to secondary scattering (parasitic scattering). The special construction of the instrument has to provide at least a subspace in the detector plane (plane of registration) that is free from this parasitic scattering. The crucial point is of course to provide the conditions to measure at very small scattering angles. The other possibility of building a small-angle scattering system is to use monochromator crystals and/or bent mirrors to select a narrow wavelength band from the radiation (important for synchrotron radiation) and to focus the X-ray beam to a narrow spot. These systems require slits in addition to eliminate stray radiation.

2:6:1:54

where cn is a constant, the factor R6 takes into account the fact that the particle volume is proportional to R3 , and i0 hR is the normalized form factor of a particle size R. In many cases, one is interested in the mass distribution Dm R [sometimes called volume distribution Dc R]. In this case, we have Ih cm

R1 0

Dm RR3 i0 hR dR:

2:6:1:55

Block collimation ± Kratky camera. The Kratky (1982a) collimation system consists of an entrance slit (edge) and two blocks ± the U-shaped centre piece and a block called bridge. With this system, the problem of parasitic scattering can be largely removed for the upper half of the plane of registration and the smallest accessible scattering angle is defined by the size of the entrance slit (see Fig. 2.6.1.13). This system can be integrated in an evacuated housing (Kratky compact camera) and fixed on the top of the X-ray tube. It is widely used in many laboratories for different applications. In the Kratky system, the X-ray beam has a rectangular shape, the length being much larger than the width. Instrumental broadening can be corrected by special numerical routines. The advantage is a relatively high primary-beam intensity. The main disadvantage is that it cannot be used in special applications such as oriented systems where

The solution of these integral equations, i.e. the computation of Dn R or Dm R from Ih, needs rather sophisticated numerical or analytical methods and will be discussed later. The problems of interparticle interference and multiple scattering in the case of polydisperse systems cannot be described analytically and have not been investigated in detail up to now. In general, interference effects start to influence data from small-angle scattering experiments much earlier, i.e. at lower concentration, than multiple scattering. Multiple scattering becomes more important with increasing size and contrast and is therefore dominant in light-scattering experiments in higher concentrations. A concentration series and extrapolation to zero concentration as in monodisperse systems should be performed to eliminate these effects. 2.6.1.5. Instrumentation X-ray sources are the same for small-angle scattering as for crystallographic experiments. One can use conventional generators with sealed tubes or rotating anodes for higher power. For the vast majority of applications, an X-ray tube with copper anode is used; the wavelength of its characteristic radiation (Cu K line) is 0.154 nm. Different anode materials emit X-rays of different characteristic wavelengths. X-rays from synchrotrons or storage rings have a continuous wavelength distribution and the actual wavelength for the experiment is selected by a monochromator. The intensity is much higher than for any type of conventional source but

Fig. 2.6.1.13. Schematic drawing of the block collimation (Kratky camera): E edge; B1 centre piece; B2 bridge; P primary-beam profile; PS primary-beam stop; PR plane of registration.

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION the two-dimensional scattering pattern has to be recorded. For such applications, any type of point collimation can be used. Slit and pinhole cameras. The simplest way to build a camera is to use two pairs of slits or pinholes at a certain distance apart (Kratky, 1982a; Holmes, 1982). The narrower the slits and the larger the distance between them, the smaller is the smallest attainable scattering angle (sometimes called the `resolution'). Parasitic scattering and difficult alignment are the main problems for all such systems (Guinier & Fournet, 1955). A slit camera that has been used very successfully is that of Beeman and coworkers (Ritland, Kaesberg & Beeman, 1950; Anderegg, Beeman, Shulman & Kaesberg, 1955). A rather unusual design is adopted in the slit camera of Stasiecki & Stuhrmann (1978), whose overall length is 50 m! A highly developed system is the ORNL 10 m camera at Oak Ridge (Hendricks, 1978). Standard-size cameras for laboratory application are commercially available with different designs from various companies. Bonse±Hart camera. The Bonse±Hart camera (Bonse & Hart, 1965, 1966, 1967) is based on multiple reflections of the primary beam from opposite sides of a groove in an ideal germanium crystal (collimator and monochromator). After penetrating the sample, the scattered beam runs through the groove of a second crystal (analyser). This selects the scattering angle. Rotation of the second crystal allows the measurement of the angledependent scattering function. The appealing feature of this design is that one can measure down to very small angles without a narrow entrance slit. The system is therefore favourable for the investigation of very large particles (D > 350 nm). For smaller particles, one obtains better results with block collimation (Kratky & Leopold, 1970). Camera systems for synchrotron radiation. Small-angle scattering facilities at synchrotrons are built by the local staff and details of the construction are not important for the user in most cases. Descriptions of the instruments are available from the local contacts. These small-angle scattering systems are usually built with crystal monochromators and focusing mirrors (point collimation). All elements have to be operated under remote control for safety reasons. A review of the different instruments was published recently by Koch (1988). 2.6.1.5.2. Detectors In this field, we are facing the same situation as we met for X-ray sources. The detectors for small-angle scattering experiments are the same as or slightly modified from the detectors used in crystallography. Therefore, it is sufficient to give a short summary of the detectors in the following; further details are given in Chapter 7.1. If we are not investigating the special cases of fully or partially oriented systems, we have to measure the dependence of the scattered intensity on the scattering angle, i.e. a one-dimensional function. This can be done with a standard gas-filled proportional counter that is operated in a sequential mode (Leopold, 1982), i.e. a positioning device moves the receiving slit and the detector to the desired angular position and the radiation detector senses the scattered intensity at that position. In order to obtain the whole scattering curve, a series of different angles must be positioned sequentially and the intensity readings at every position must be recorded. The system has a very high dynamic range, but ± as the intensities at different angles are measured at different times ± the stability of the primary beam is of great importance. This drawback is eliminated in the parallel detection mode with the use of position-sensitive detectors. Such systems are in most cases proportional counters with sophisticated and expen-

sive read-out electronics that can evaluate on-line the accurate position where the pulses have been created by the incoming radiation. Two-dimensional position-sensitive detectors are necessary for oriented systems, but they also have advantages in the case of non-oriented samples when circular chambers are used or when integration techniques in square detectors lead to a higher signal at large scattering angles. The simplest and cheapest two-dimensional detector is still film, but films are not used very frequently in small-angle scattering experiments because of limited linearity and dynamic range, and fog intensity. Koch (1988) reviews the one- and two-dimensional detectors actually used in synchrotron small-angle scattering experiments. For a general review of detectors, see Hendrix (1985). 2.6.1.6. Data evaluation and interpretation After having discussed the general principles and the basics of instrumentation in the previous subsections, we can now discuss how to handle measured data. This can only be a very short survey; a detailed description of data treatment and interpretation has been given previously (Glatter, 1982a,b). Every physical investigation consists of three highly correlated parts: theory, experiment, and evaluation of data. The theory predicts a possible experiment, experimental data have to be collected in a way that the evaluation of the information wanted is possible, the experimental situation has to be described theoretically and has to be taken into account in the process of data evaluation etc. This correlation should be remembered at every stage of the investigation. Before we can start any discussion about interpretation, we have to describe the experimental situation carefully. All the theoretical equations in the previous subsections correspond to ideal conditions as mentioned in the subsection on instrumentation. In real experiments, we do not measure with a point-like parallel and strictly monochromatic primary beam and our detector will have non-negligible dimensions. The finite size of the beam, its divergence, the size of the detector, and the wavelength distribution will lead to an instrumental broadening as in most physical investigations. The measured scattering curve is said to be smeared by these effects. So we find ourselves in the following situation. The particle is represented by its PDDF pr. This function is not measured directly. In the scattering process it is Fouriertransformed into a scattering function Ih [equation (2.6.1.9)]. This function is smeared by the broadening effects and the final smeared scattering function Iexp h is measured with a certain experimental error h. In the case of polydisperse systems, the situation is very similar; we start from a size-distribution function DR and have a different transformation [equations (2.6.1.54), (2.6.1.55)], but the smearing problem is the same. 2.6.1.6.1. Primary data handling In order to obtain reliable results, we have to perform a series of experiments. We have to repeat the experiment for every sample, to be able to estimate a mean value and a standard deviation at every scattering angle. This experimentally determined standard deviation is often much higher than the standard deviation simply estimated from counting statistics. A blank experiment (cuvette filled with solvent only) is necessary to be able to subtract background scattering coming from the instrument and from the solvent (or matrix in the case of solid samples). Finally, we have to perform a series of such

100


2.6. SMALL-ANGLE TECHNIQUES experiments at different concentrations to extrapolate to zero concentration (elimination of interparticle interferences). If the scattering efficiency of the sample is low (low contrast, small particles), it may be necessary to measure the outer part of the scattering function with a larger entrance slit and we will have to merge different parts of the scattering function. The intensity of the instrument (primary beam) should be checked before each measurement. This allows correction (normalization) for instabilities. It is therefore necessary to have a so-called primary datahandling routine that performs all these preliminary steps like averaging, subtraction, normalization, overlapping, concentration extrapolation, and graphical representation on a graphics terminal or plotter. In addition, it is helpful to have the possibility of calculating the Guinier radius, Porod extrapolation [equations (2.6.1.24)], invariant, etc. from the raw data. When all these preliminary steps have been performed, we have a smeared particle-scattering function Iexp h with a certain statistical accuracy. From this data set, we want to compute Ih and pr [or DR] and all our particle parameters. In order to do this, we have to smooth and desmear our function Iexp h. The smoothing operation is an absolute necessity because the desmearing process is comparable to a differentiation that is impossible for noisy data. Finally, we have to perform a Fourier transform (or other similar transformation) to invert equations (2.6.1.9) or (2.6.1.54), (2.6.1.55). Before we can discuss the desmearing process (collimation error correction) we have to describe the smearing process. 2.6.1.6.2. Instrumental broadening ± smearing These effects can be separated into three components: the twodimensional geometrical effects and the wavelength effect. The geometrical effects can be separated into a slit-length (or slitheight) effect and a slit-width effect. The slit length is perpendicular to the direction of increasing scattering angle; the corresponding weighting function is usually called Pt. The slit width is measured in the direction of increasing scattering angles and the weighting function is called Qx. If there is a wavelength distribution, we call the weighting function W l0 where l0 l=l0 and l0 is the reference wavelength used in equation (2.6.1.2). When a conventional X-ray source is used, it is sufficient in most cases to correct only for the K contribution. Instead of the weighting function W l0 one only needs the ratio between K and K radiation, which has to be determined experimentally (Zipper, 1969). One or more smearing effects may be negligible, depending on the experimental situation. Each effect can be described separately by an integral equation (Glatter, 1982a). The combined formula reads Z1 Z1 Z1 Iexp h 2 QxPtW l0 1

I

0

m

0

x2 t2 1=2 l0

dl0 dt dx:

2:6:1:56

This threefold integral equation cannot be solved analytically. Numerical methods must be used for its solution. 2.6.1.6.3. Smoothing, desmearing, and Fourier transformation There are many methods published that offer a solution for this problem. Most are referenced and some are reviewed in the textbooks (Glatter, 1982a; Feigin & Svergun, 1987). The indirect transformation method in its original version (Glatter,

1977a,b, 1980a,b) or in modifications for special applications (Moore, 1980; Feigin & Svergun, 1987) is a well established method used in the majority of laboratories for different applications. This procedure solves the problems of smoothing, desmearing, and Fourier transformation [inversion of equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)] in one step. A short description of this technique is given in the following. Indirect transformation methods. The indirect transformation method combines the following demands: single-step procedure, optimized general-function system, weighted least-squares approximation, minimization of termination effect, error propagation, and consideration of the physical smoothing condition given by the maximum intraparticle distance. This smoothing condition requires an estimate Dmax as an upper limit for the largest particle dimension: Dmax D:

For the following, it is not necessary for Dmax to be a perfect estimate, but it must not be smaller than D. As pr 0 for r Dmax , we can use a function system for the representation of pr that is defined only in the subspace 0 r Dmax . A linear combination pA r

N P v1

cv 'v r

2:6:1:58

is used as an approximation to the PDDF. Let N be the number of functions and cv be the unknowns. The functions 'v r are chosen as cubic B splines (Greville, 1969; Schelten & Hossfeld, 1971) as they represent smooth curves with a minimum second derivative. Now we take advantage of two facts. The first is that we know precisely how to calculate a smeared scattering function Ih from Ih [equation (2.6.1.56)] and how pr or DR is transformed into Ih [equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)], but we do not know the inverse transformations. The second fact is that all these transformations are linear, i.e. they can be applied to all terms in a sum like that in equation (2.6.1.58) separately. So it is easy to start with our approximation in real space [equation (2.6.1.58)] taking into account the a priori information Dmax . The approximation IA h to the ideal (unsmeared) scattering function can be written as IA h

N P v1

cv v h;

2:6:1:59

where the functions v h are calculated from 'v r by the transformations (2.6.1.9) or (2.6.1.54), (2.6.1.55), the coefficients cv remain unknown. The final fit in the smeared, experimental space is given by a similar series IA h

N P v1

cv v h;

2:6:1:60

where the v h are functions calculated from v h by the transform (2.6.1.56). Equations (2.6.1.58), (2.6.1.59), and (2.6.1.60) are similar because of the linearity of the transforms. We see that the functions v h are calculated from 'v r in the same way as the data Iexp h were produced by the experiment from pr. Now we can minimize the expression L

M P k1

Iexp hk

IA hk 2 = 2 hk ;

2:6:1:61

where M is the number of experimental points. Such leastsquares problems are in most cases ill conditioned, i.e. additional stabilization routines are necessary to find the best

101


2:6:1:57

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION solution. This problem is far from being trivial, but it can be solved with standard routines (Glatter, 1977a,b; Tikhonov & Arsenin, 1977). The whole process of data evaluation is shown in Fig. 2.6.1.14. Similar routines cannot be used in crystallography (periodic structures) because there exists no estimate for Dmax [equation (2.6.1.57)]. Maximum particle dimension. The sampling theorem of Fourier transformation (Shannon & Weaver, 1949; Bracewell, 1986) gives a clear answer to the question of how the size of the particle D is related to the smallest scattering angle h1 . If the scattering curve is observed at increments h h1 starting from a scattering angle h1 , the scattering data contain, at least theoretically, the full information for all particles with maximum dimension D D =h1 :

2:6:1:62

The first application of this theorem to the problem of data evaluation was given by Damaschun & PuÈrschel (1971a,b). In practice, one should always try to stay below this limit, i.e. h1 < =D

and

h h1 ;

Resolution. There is no clear answer to the question concerning the smallest structural details, i.e. details in the pr function that can be recognized from an experimental scattering function. The limiting factors are the maximum scattering angle h2 , the statistical error h, and the weighting functions Pt; Qx, and W l0 (Glatter, 1982a). The resolution of standard experiments is not better than approximately 10% of the maximum dimension of the particle for a monodisperse system. In the case of polydisperse systems, resolution can be defined as the minimum relative peak distance that can be resolved in a bimodal distribution. We know from simulations that this value is of the order of 25%. Special transforms. The PDDF pr or the size distribution function DR is related to Ih by equations (2.6.1.9) or (2.6.1.54), (2.6.1.55). In the special case of particles elongated in one direction (like cylinders), we can combine equations (2.6.1.41) and (2.6.1.43) and obtain

2:6:1:63

taking into account the loss of information due to counting statistics and smearing effects. An optimum value for h =6D is claimed by Walter, Kranold & Becherer (1974). Information content. The number of independent parameters contained in a small-angle scattering curve is given by Nmax h2 =h1 ;

routine. An example of this problem can be found in Glatter (1980a).

2:6:1:64

with h1 and h2 being the lower and upper limits of h. In practice, this limit certainly depends on the statistical accuracy of the data. It should be noted that the number of functions N in equations (2.6.1.58) to (2.6.1.60) may be larger than Nmax because they are not independent. They are correlated by the stabilization

2

Z1

Ih 2 L

pc r 0

J0 hr dr: h

2:6:1:65

This Hankel transform can be used in the indirect transformation method for the calculation of v h in (2.6.1.59). Doing this, we immediately obtain the PDDF of the cross section pc r from the smeared experimental data. It is not necessary to know the length L of the particle if the results are not needed on an absolute scale. For this application, we only need the information that the scatterers are elongated in one direction with a constant cross section. This information can be found from the overall PDDF of the particle or can be a priori information from other experiments, like electron microscopy. The estimate for the maximum dimension Dmax (2.6.1.57) is related to the cross

Fig. 2.6.1.14. Function systems 'v r; v h; and v h used for the approximation of the scattering data in the indirect transformation method.

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2.6. SMALL-ANGLE TECHNIQUES section in this application, i.e. the maximum dimension of the cross section must not be larger than Dmax . The situation is quite similar for flat particles. If we combine (2.6.1.47) and (2.6.1.49), we obtain Z1 Ih 4A

pt r 0

coshr dr; h2

2:6:1:66

pt r being the distance distribution function of the thickness. We have to check that the particles are flat with a constant thickness with maximum thickness T Dmax . A is the area of the particles and would be needed only for experiments on an absolute scale. 2.6.1.6.4. Direct structure analysis It is impossible to determine the three-dimensional structure r directly from the one-dimensional information Ih or pr. Any direct method needs additional a priori information ± or assumptions ± on the system under investigation. If this information tells us that the structure only depends on one variable, i.e. the structure is in a general sense one dimensional, we have a good chance of recovering the structure from our scattering data. Examples for this case are particles with spherical symmetry, i.e. depends only on the distance r from the centre, or particles with cylindrical or lamellar symmetry where depends only on the distance from the cylinder axis or from the distance from the central plane in the lamella. We will restrict our discussion here to the spherical problem but we keep in mind that similar methods exist for the cylindrical and the lamellar case. Spherical symmetry. This case is described by equations (2.6.1.51) and (2.6.1.52). As already mentioned in x2.6.1.3.2.2, we can solve the problem of the calculation of r from Ih in two different ways. We can calculate r via the distance distribution function pr with a convolution square-root technique (Glatter, 1981; Glatter & Hainisch, 1984). The other way goes through the amplitude function Ah and its Fourier transform. In this case, one has to find the right phases (signs) in the square-root operation {Ah Ih1=2 }.The box-function refinement method by Svergun, Feigin & Schedrin (1984) is an iterative technique for the solution of the phase problem using the a priori information that r is equal to zero for r Rmax Dmax =2. The same restriction is used in the convolution squareroot technique. Under ideal conditions (perfect spherical symmetry), both methods give good results. In the case of deviations from spherical symmetry, one obtains better results with the convolution square-root technique (Glatter, 1988). With this method, the results are less distorted by non-spherical contributions. Multipole expansions. A wide class of homogeneous particles can be represented by a boundary function that can be expanded into a series of spherical harmonics. The coefficients are related to the coefficients of a power series of the scattering function Ih, which are connected with the moments of the PDDF (Stuhrmann, 1970b,c; Stuhrmann, Koch, Parfait, Haas, Ibel & Crichton, 1977). Of course, this expansion cannot be unique, i.e. for a certain scattering function Ih one can find a large variety of possible expansion coefficients and shapes. In any case, additional a priori information is necessary to reduce this number, which in turn influences the convergence of the expansion. Only compact, globular structures can be approximated with a small number of coefficients. This concept is not restricted to the determination of the shape of the particles. Even inhomogeneous particles can be described

using all possible radial terms in a general expansion (Stuhrmann, 1970a). The information content can be increased by contrast variation (Stuhrmann, 1982), but in any event one is left with the problem of how to find additional a priori information in order to reduce the possible structures. Any type of symmetry will lead to a considerable improvement. The case of axial symmetry is a good example. Svergun, Feigin & Schedrin (1982) have shown that the quality of the results can be further improved when upper and lower limits for r can be used. Such limits can come from a known chemical composition. 2.6.1.6.5. Interpretation of results After having used all possible data-evaluation techniques, we end up with a desmeared scattering function Ih, the PDDF pr or the size-distribution function DR, and some special functions discussed in the previous subsections. Together with the particle parameters, we have a data set that can give us at least a rough classification of the substance under investigation. The interpretation can be performed in reciprocal space (scattering function) or in real space (PDDF etc.). Any symmetry can be detected more easily in reciprocal space, but all other structural information can be found more easily in real space (Glatter, 1979, 1982b). When a certain structure is estimated from the data and from a priori information, one has to test the corresponding model. That means one has to find the PDDF and Ih for the model and has to compare it with the experimental data. Every model that fits within the experimental errors can be true, all that do not fit have to be rejected. If the model does not fit, it has to be refined by trial and error. In most cases, this process is much easier in real space than in reciprocal space. Finally, we may end up with a set of possible structures that can be correct. Additional a priori information will be necessary to reduce this number.

2.6.1.7. Simulations and model calculations 2.6.1.7.1. Simulations Simulations can help to find the limits of the method and to estimate the systematic errors introduced by the data-evaluation procedure. Simulations are performed with exactly known model systems (test functions). These systems should be similar to the structures of interest. The model data are transformed according to the special experimental situation (collimation profiles and wavelength distribution) starting from the theoretical PDDF (or scattering function). Èxperimental data points' are generated by sampling in a limited h range and adding statistical noise from a random-number generator. If necessary, a certain amount of background scattering can also be added. This simulated data set is subjected to the data-evaluation procedure and the result is compared with the starting function. Such simulation can reveal the influence of each approximation applied in the various evaluation routines. On the other hand, simulations can also be used for the optimization of the experimental design for a special application. The experiment situation is characterized by several contradictory effects: a large width for the functions Pt, Qx, and W l0 leads to a high statistical accuracy but considerable smearing effects. The quality of the results of the desmearing procedure is increased by high statistical accuracy, but decreased by large smearing effects. Simulations can help to find the optimum for a special application.

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION 2.6.1.7.2. Model calculation In the section on data evaluation and interpretation, we have seen that we obtain a rough estimate for the structure of the particles under investigation directly from the experimental data. For further refinement, we have to compare our results with scattering functions or PDDF's from models 2.6.1.7.3. Calculation of scattering intensities The scattering curves can be calculated semi-analytically for simple triaxial bodies and for models composed of some of these bodies. The scattering amplitude for regular bodies like ellipsoids, parallelepipeds, and cylinders can be calculated analytically for any orientation. The spatial averaging has to be performed numerically. Such calculations have been performed for a large number of different models by Porod (1948), Mittelbach & Porod (1961a,b, 1962), and by Mittelbach (1964). More complicated structures can be described by models composed of several such triaxial bodies, but the computing time necessary for such calculations can be hours on a mainframe computer. Models composed only of spherical subunits can be evaluated with the Debye formula (Debye, 1915): Ih iel h

N X N X i1 k1

i Vi k Vk i hk h

sinhdik ; hdik

2.6.1.7.4. Method of finite elements Models of arbitrary shape can be approximated by a large number of very small homogeneous elements of variable electron density. These elements have to be smaller than the smallest structural detail of interest. Sphere method. In this method, the elements consist of spheres of equal size. The diameter of these spheres must be chosen independently of the distance between nearest neighbours, in such a way that the total volume of the model is represented correctly by the sum of all volume elements (which corresponds to a slight formal overlap between adjacent spheres). The scattering intensity is calculated using the Debye formula (2.6.1.67), with i h k h h. The computing time is mainly controlled by the number of mutual distances between the elements. The computing time can be lowered drastically by the use of approximate dik values in (2.6.1.67). Negligible errors in Ih result if dik values are quantized to Dmax =10000 (Glatter, 1980c). For the practical application (input operation), it is important that a certain number of elements can be combined to form so-called substructures that can be used in different positions with arbitrary weights and orientations to build the model. The sphere method can also be used for the computation of scattering curves for macromolecules from a known crystal structure. The weights of the atoms are given by the effective number of electrons 0 Veff ;

2.6.1.7.5. Calculation of distance-distribution functions The PDDF can be calculated analytically only for a few simple models (Porod, 1948; Goodisman, 1980); in all other cases, we have to use a finite element method with spheres. It is possible to define an analogous equation to the Debye formula (2.6.1.67) in real space (Glatter, 1980c). The PDDF can be expressed as

2:6:1:67

where the spatial average is carried out analytically. Another possibility would be to use spherical harmonics as discussed in the previous section but the problem is how to find the expansion coefficients for a certain given geometrical structure.

Zeff Z

Cube method. This method has been developed independently by Fedorov, Ptitsyn & Voronin (1972, 1974a,b) and by Ninio & Luzzati (1972) mainly for the computation of scattered intensities for macromolecules in solution whose crystal structure is known. In the cube method, the macromolecule is mentally placed in a parallelepiped, which is subdivided into small cubes (with edge Ê . Each cube is examined in order to decide lengths of 0.5±1.5 A whether it belongs to the molecule or to the solvent. Adjacent cubes in the z direction are joined to form parallelepipeds. The total scattering amplitude is the sum over the amplitudes from the parallelepipeds with different positions and lengths. The mathematical background is described by Fedorov, Ptitsyn & Voronin (1974a,b). The modified cube method of Fedorov & Denesyuk (1978) takes into account the possible penetration of the molecule by water molecules.

2:6:1:68

where Veff is the apparent volume of the atom given by Langridge, Marvin, Seeds, Wilson, Cooper, Wilkins & Hamilton (1960).

pr

i1

2i p0 r; Ri

2

NP1

N P

i1 ki1

i k pr; dik ; Ri ; Rk :

2:6:1:69

p0 r; Ri ) is the PDDF of a sphere with radius Ri and electron density equal to unity, pr; dik ; Ri ; Rk is the cross-term distance distribution between the ith and kth spheres (radii Ri and Rk ) with a mutal distance dik . Equation (2.6.1.69) [and (2.6.1.67)] can be used in two different ways for the calculation of model functions. Sometimes, it is possible to approximate a macromolecule as an aggregate of some spheres of well defined size representing different globular subunits (Pilz, Glatter, Kratky & MoringClaesson, 1972). The form factors of the subunits are in such cases real parameters of the model. However, in most cases we have no such possibility and we have to use the method of finite elements, i.e. we fit our model with a large number of sufficiently small spheres of equal size, and, if necessary, different weight. The form factor of the small spheres is now not a real model parameter and introduces a limit of resolution. Fourier transformation [equation (2.6.1.10)] can be used for the computation of the PDDF of any arbitrary model if the scattering function of the model is known over a sufficiently large range of h values. 2.6.1.8. Suggestions for further reading Only a few textbooks exist in the field of small-angle scattering. The classic monograph Small-Angle Scattering of X-rays by Gunier & Fournet (1955) was followed by the proceedings of the conference at Syracuse University, 1965, edited by Brumberger (1967) and by Small-Angle X-ray Scattering edited by Glatter & Kratky (1982). The several sections of this book are written by different authors being experts in the field and representing the state of the art at the beginning of the 1980's. The monograph Structure Analysis by Small-Angle X-ray and Neutron Scattering by Feigin & Svergun (1987) combines X-ray and neutron techniques.

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N P

2.6. SMALL-ANGLE TECHNIQUES 2.6.2. Neutron techniques (By R. May) Symbols used in the text A sample area As inner sample surface coherent scattering length of atom i bi Bi spin-dependent scattering length of atom i C sample concentration in g l 1 c volume fraction occupied by matter d sample thickness D particle dimension DCD double-crystal diffractometer d0 Bragg spacing e, e0 unit vectors along the diffracted and incident beams I nuclear spin IFT indirect Fourier transformation N number of particles in the sample NA Avogadro's number Q momentum transfer 4=l sin r radius of a sphere RG radius of gyration s neutron spin SANS small-angle neutron scattering SAXS small-angle X-ray scattering T transmission TOF time of flight v partial specific volume Vp particle volume sample volume Vs

solid angle subtended by a detection element l wavelength scattering-length density 2 full scattering angle dQ= d scattering cross section per particle and unit solid angle 2.6.2.1. Relation of X-ray and neutron small-angle scattering X-ray and neutron small-angle scattering (SAXS and SANS, respectively) are dealing with the same family of problems, i.e. the investigation of ìnhomogeneities' in matter. These inhomogeneities have dimensions D of the order of 1 to 100 nm, which are larger than interatomic distances, i.e. 0.3 nm. The term inhomogeneities may mean clusters in metals, a small concentration of protonated chains in an otherwise identical deuterated polymer ± or vice versa ± but also particles as well defined as purified proteins in aqueous solution. In most cases, the inhomogeneities are not ordered. This is where small-angle scattering is most useful: many systems are not crystalline, cannot be crystallized, or do not exhibit the same properties if they are. One field, if one may say so, of SANS where samples are ordered is low-resolution crystallography of biological macromolecules. It will not be treated further here. In the case of crystalline order, the scattering of the single particle is observed with an amplification factor of N 2 for N identical particles in the crystal, but only for those scattering vectors observing the Bragg condition nl 2d0 sin . For disordered, randomly oriented particles, the amplification is only N, and the scattering pattern is lacking all information on particle orientation. Moreover, the real-space information on the internal arrangement of atoms within the inhomogeneities is reduced to the `distance distribution function', a sine Fourier transform of the scattering intensity. The mathematical descriptions of SAXS and SANS are either identical or hold with equivalent terms. The reader is referred to

Section 2.6.1 on X-ray small-angle scattering techniques for a general description of low-Q scattering. An abundant treatment of SAXS can be found in the book edited by Glatter & Kratky (1982), and in Guinier & Fournet (1955) and Guinier (1968). A general introduction to SANS is given, for example, by Kostorz (1979) and by Hayter (1985). This section deals mainly with the differences between the techniques. Altogether, neutrons are used for low-Q scattering essentially for the same reasons as for other neutron experiments. These reasons are: (1) neutrons are sensitive to the isotopic composition of the sample; (2) neutrons possess a magnetic moment and, therefore, can be used as a magnetic probe of the sample; and (3) because of their weak interaction with and consequent deep penetration into matter, neutrons allow us to investigate properties of the bulk; (4) for similar reasons, strong transparent materials are available as sample-environment equipment. The fact that the kinetic energies of thermal and cold neutrons are comparable to those of excitations in solids, which is a reason for the use of neutrons for inelastic scattering, is, with the exception of time-of-flight SANS (see x2:6:2:1:1), not of importance for SANS. The information obtained from low-Q scattering is always an average over the irradiated sample volume and over time. This average may be purely static (in the case of solids) or also dynamic (liquids). The limited Q range used does not resolve interatomic scattering contributions. P Thus, a `scattering-length density' can be introduced, bi =V , where bi are the (coherent) scattering lengths of the atoms within a volume V with linear dimensions of at least l=. Inhomogeneities can then be understood as regions where the scattering-length density deviates from the prevailing average value. 2.6.2.1.1. Wavelength In the case of SANS ± as in that of X-rays from synchrotron sources ± the wavelength dependence of the momentum transfer Q; Q 4=l sin , where is half the scattering angle and l is the wavelength, has to be taken into account explicitly. Q corresponds to k, h, and 2s used by other authors. SANS offers an optimal choice of the wavelength: with sufficiently large wavelengths, for example, first-order Bragg scattering (and therefore the contribution of multiple Bragg scattering to small-angle scattering) can be suppressed: The Bragg condition written as l=dmax 2 sin =n < 2 cannot hold for l 2dmax , where dmax is the largest atomic distance in a crystalline sample. For the usually small scattering angles in SANS, even quite small l will not produce first-order peaks. The neutrons produced by the fuel element of a reactor or by a pulsed source are moderated by the (heavy) water surrounding the core. Normally, the neutrons leave the reactor with a thermal velocity distribution. Cold sources, small vessels filled with liquid deuterium in the reactor tank, permit the neutron velocity distribution to be slowed down (`cold' neutrons) and lead to neutron wavelengths (range 0.4 to 2 nm) which are more useful for SANS. At reactors, a narrow wavelength band is usually selected for SANS either by an artificial-multilayer monochromator or ± more frequently, owing to the slow speed of cold neutrons ± by a velocity selector. This is a rotating drum with a large number (about 100) of helical slots at its circumference, situated at the entrance of the neutron guides used for collimation. Only neutrons of the suitable velocity are able to pass through this

105


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION drum. The wavelength resolution l=l of velocity selectors is usually between 5 and 40% (full width at half-maximum, FWHM); 10 and 20% are frequently used values. Alternatively, time-of-flight (TOF) SANS cameras have been developed on pulsed neutron sources (e.g. Hjelm, 1988). These use short bunches (about 100 ms long) of neutrons with a `white' wavelength spectrum produced by a pulsed high-energy proton beam impinging on a target with a repetition rate of the order of 10 ms. The wavelength and, consequently, the Q value of a scattered neutron is determined by its flight time, if the scattering is assumed to be quasi-elastic. The dynamic Q range of TOF SANS instruments is rather large, especially in the high-Q limit, owing to the large number of rapid neutrons in the pulse. The low-Q limit is determined by the pulse-repetition rate of the source because of frame overlap with the following pulse. It can be decreased, if necessary with choppers turning in phase with the pulse production and selecting only every nth pulse. This disadvantage does not exist for reactor-based TOF SANS cameras, where the pulse-repetition rate can be optimally adapted to the chosen maximal and minimal wavelength. A principal problem for TOF SANS exists in the ùpscattering' of cold neutrons, i.e. their gain in energy, by 1 H-rich samples: The background scattering may not arrive simultaneously with the elastic signal, and may thus not be attributed to the correct Q value (Hjelm, 1988). 2.6.2.1.2. Geometry With typical neutron wavelengths, low Q need not necessarily mean small angles: The interesting Q range for an inhomogeneity of dimension D can be estimated as 1=D < Q < 10=D. The scattering angle corresponding to the upper Q limit for D 10 nm is 1.4 for Cu K radiation, but amounts to 9.1 for neutrons of 10 nm wavelength. Consequently, it is preferable to speak of low-Q rather than of small-angle neutron scattering. `Pin-hole'-type cameras are the most frequently used SANS instruments; an example is the SANS camera D11 at the Institut Max von Laue±Paul Langevin in Grenoble, France (Ibel, 1976; Lindner, May & Timmins, 1992), from which some of the numbers below are quoted. Since the cross section of the primary beam is usually chosen to be rather large (e.g. 3 5 cm) for intensity reasons, pin-hole instruments tend to be large. The smallest Q value that can be measured at a given distance is just outside the image of the direct beam on the detector (which either has to be attenuated or is hidden behind a beamstop, a neutronabsorbing plate of several 10 cm2 , e.g. of cadmium). Very small Q values thus require long sample-to-detector distances. The area detector of D11, with a surface of 64 64 cm and resolution elements of 1 cm2 , moves within an evacuated tube of 1.6 m diameter and a length of 40 m. Thus, a Q range of 5 10 3 to 5 nm 1 is covered. The geometrical resolution is determined by the length of the free neutron flight path in front of the sample, moving sections of neutron guide into or out of the beam (`collimation'). In general, the collimation length is chosen roughly equal to the sample-to-detector distance. Thus, the geometrical and wavelength contributions to the Q resolution match at a certain distance of the scattered beam from the directbeam position in the detector plane. In order to resolve scattering patterns with very detailed features, e.g. of particles with high symmetry, longer collimation lengths are sometimes required at the expense of intensity. Much more compact double-crystal neutron diffractometers [described for X-rays by Bonse & Hart (1966)] are being used to reach the very small Q values of some 10 4 nm 1 typical of static light scattering. The sample is placed between two crystals. The

first crystal defines the wavelength and the direction of the incoming beam. The other crystal scans the scattered intensity. The resolution of such an instrument is mainly determined by the Darwin widths of the ideal crystals. This fact is reflected in the low neutron yield. Slit geometry can be used, but not 2D detectors. A recent development is the ellipsoidal-mirror SANS camera. The mirror, which needs to be of very high surface quality, focuses the divergent beam from a small (several mm2 ) source through the sample onto a detector with a resolution of the order of 1 1 mm. Owing to the more compact beam image, all other dimensions of the SANS camera can be reduced drastically (Alefeld, Schwahn & Springer, 1989). Whether or not there is a gain in intensity as compared with pin-hole geometry is strongly determined by the maximal sample dimensions. Long mirror with cameras (e.g. 20 m) are always superior to double-crystal instruments in this respect (Alefeld, Schwahn & Springer, 1989), and can also reach the light-scattering Q domain (Qmin of some 10 4 nm 1 , corresponding to particles of several mm dimension). 2.6.2.1.3. Correction of wavelength, slit, and detectorelement effects Resolution errors affect SANS data in the same way as X-ray scattering data, for which one may find a detailed treatment in an article by Glatter (1982b); there is one exception to this; namely, gravity, which of course only concerns neutron scattering, and only in rare cases (Boothroyd, 1989). Since SANS cameras usually work with pin-hole geometry, the influences of the slit sizes, i.e. the effective source dimensions, on the scattering pattern are small; even less important is, in general, the pixel size of 2D detectors. The preponderant contribution to the resolution of the neutron-scattering pattern is the wavelengthdistribution function after the monochromatizing device, especially at larger angles. The situation is more complicated for TOF SANS (Hjelm, 1988). As has been shown in an analytical treatment of the resolution function by Pedersen, Posselt & Mortensen (1990), who also quote some relevant references, resolution effects have a small influence on the results of the data analysis for scattering patterns with a smooth intensity variation and without sharp features. Therefore, one may assume that a majority of SANS patterns are not subjected to desmearing procedures. Resolution has to be considered for scattering patterns with distinct features, as from spherical latex particles (Wignall, Christen & Ramakrishnan, 1988) or from viruses (Cusack, 1984). Size-distribution and wavelength-smearing effects are similar; it is evident that wavelength effects have to be corrected for if the size distribution is to be obtained. Since measured scattering curves contain errors and have to be smoothed before they can be desmeared, iterative indirect methods are, in general, superior: A guessed solution of the scattering curve is convoluted with known smearing parameters and iteratively fitted to the data by a least-squares procedure. The guessed solution can be a simply parameterized scattering curve, without knowledge of the sample (Schelten & Hossfeld, 1971), but it is of more interest to fit the smeared Fourier transform of the distance-distribution function (Glatter, 1979) or the radial density distribution (e.g. Cusack, Mellema, Krijgsman & Miller, 1981) of a real-space model to the data. 2.6.2.2. Isotopic composition of the sample Unlike X-rays, which `see' the electron clouds of atoms within a sample, neutrons interact with the point-like nuclei. Since their

106


2.6. SMALL-ANGLE TECHNIQUES form factor does not decay like the atomic form factor, an isotropic background from the nuclei is present in all SANS measurements. While X-ray scattering amplitudes increase regularly with the atomic number, neutron coherent-scattering amplitudes that give rise to the interference scattering necessary for structural investigations vary irregularly (see Bacon, 1975). Isotopes of the same element often have considerably different amplitudes owing to their different resonant scattering. The most prominent example of this is the difference of the two stable isotopes of hydrogen, 1 H and 2 H (deuterium). The coherent-scattering length of 2 H is positive and of similar value to that of most other elements in organic matter, whereas that of 1 H is negative, i.e. for 1 H there is a 180 phase shift of the scattered neutrons with respect to other nuclei. This latter difference has been exploited vastly in the fields of polymer science (e.g. Wignall, 1987) and structural molecular biology (e.g. Timmins & Zaccai, 1988), in mainly two complementary respects, contrast variation and specific isotopic labelling. In the metallurgy field, other isotopes are being used frequently for similar purposes, for example the nickel isotope 62 Ni, which has a negative scattering length, and the silver isotopes 107 Ag and 109 Ag (see the review of Kostorz, 1988). 2.6.2.2.1. Contrast variation The easiest way of using the scattering-amplitude difference between 1 H and 2 H is the so-called contrast variation. It was introduced into SANS by Ibel & Stuhrmann (1975) on the basis of X-ray crystallographic (Bragg & Perutz, 1952), SAXS (Stuhrmann & Kirste, 1965), and light-scattering (Benoit & Wippler, 1960) work. Most frequently, contrast variation is carried out with mixtures of light (1 H2 O) and heavy water (2 H2 O), but also with other solvents available in protonated and deuterated form (ethanol, cyclohexane, etc.). The scatteringlength density of H2 O varies between 0:562 1010 cm 2 for normal water, which is nearly pure 1 H2 O, and 6:404 1010 cm 2 for pure heavy water. The scattering-length densities of other molecules, in general, are different from each other and from pure protonated and deuterated solvents and can be matched by 1 H=2 H mixture ratios characteristic for their chemical compositions. This mixture ratio (or the corresponding absolute scattering-length density) is called the scattering-length-density match point, or, semantically incorrect, contrast match point. If a molecule contains noncovalently bound hydrogens, they can be exchanged for solvent hydrogens. This exchange is proportional to the ratio of all labile 1 H and 2 H present; in dilute aqueous solutions, it is dominated by the solvent hydrogens. A plot of the scattering-length density versus the 2 H=(2 H+1 H) ratio in the solvent shows a linear increase if there is exchange; the value of the match point also depends on solvent exchange. The fact that many particles have high contrast with respect to 2 H2 O makes neutrons superior to X-rays for studying small particles at low concentrations. The scattered neutron intensity from N identical particles without long-range interactions in a (very) dilute solution with solvent scattering density s can be written as IQ dQ= d NTAI0 ;

2:6:2:1

with the scattering cross section per particle and unit solid angle D R 2 E dQ= d r s expiQ r dr : 2:6:2:1a

The angle brackets indicate P averaging over all particle orientations. With r bi =Vp and I0 constant

R r s dr 2 , we find that the scattering intensity at zero angle is proportional to P bi =Vp s ; 2:6:2:2 which is called the contrast. The exact meaning of Vp is discussed, for example, by Zaccai & Jacrot (1983), and for X-rays by Luzzati, Tardieu, Mateu & Stuhrmann (1976). The scattering-length density r can be written as a sum r 0 F r;

where 0 is the average scattering-length density of the particle at zero contrast, 0, and F r describes the fluctuations about this mean. IQ can then be written IQ 0

2 2 Ic Q 0

s Ics Q Is Q:

2:6:2:4

Is is the scattering intensity due to the fluctuations at zero contrast. The cross term Ics Q also has to take account of solvent-exchange phenomena in the widest sense (including solvent water molecules bound to the particle surface, which can have a density different from that of bulk water). This extension is mathematically correct, since one can assume that solvent exchange is proportional to . The term Ic is due to the invariant volume inside which the scattering density is independent of the solvent (Luzzati, Tardieu, Mateu & Stuhrmann, 1976). This is usually not the scattering of a homogeneous particle at infinte contrast, if the exchange is not uniform over the whole particle volume, as is often the case, or if the particle can be imaged as a sponge (see Witz, 1983). The method is still very valuable, since it allows calculation of the scattering at any given contrast on the basis of at least three measurements at well chosen 1 H=2 H ratios (including data near, but preferentially not exactly at, the lowest contrasts). It is sometimes limited by 2 H-dependent aggregation effects. 2.6.2.2.2. Specific isotopic labelling Specific isotope labelling is a method that has created unique applications of SANS, especially in the polymer field. Again, it is mainly concerned with the exchange of 1 H by 2 H, this time in the particles to be studied themselves, at hydrogen positions that are not affected by exchange with solvent atoms, for example carbon-bound hydrogen sites. With this technique, isolated polymer chains can be studied in the environment of other polymer chains which are identical except for the hydrogen atoms, which are either 1 H or 2 H. Even if some care has to be taken as far as slightly modified thermodynamics are concerned, there is no other method that could replace neutrons in this field. Inverse contrast variation forms an intermediate between the two methods described above. The contrast with respect to the solvent of a whole particle or of well defined components of a particle, for example a macromolecular complex, is changed by varying its degree of deuteration. That of the solvent remains constant. Since solvent-exchange effects remain practically identical for all samples, the measurements can be more precise than in the classical contrast variation (Knoll, Schmidt & Ibel, 1985). 2.6.2.3. Magnetic properties of the neutron Since the neutron possesses a magnetic moment, it is sensitive to the orientation of spins in the sample [see, for example, Abragam et al. (1982)]. Especially in the absence of any other (isotopic) contrast, an inhomogeneous distribution of spins in the

107


2:6:2:3

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION sample is detectable by neutron low-Q scattering. The neutron spins need not be oriented themselves, although important contributions can be expected from measuring the difference between the scattering of neutron beams with opposite spin orientation. At present, several low-Q instruments are being planned or even built including neutron polarization and polarization analysis. Studies of magnetic SANS without (and rarely with) neutron polarization include dislocations in magnetic crystals and amorphous ferromagnets [see the review of Kostorz (1988)]. Janot & George (1985) have pointed out that it is important to apply contrast variation for suppressing surface-roughness scattering and/or volume scattering in order to isolate magnetic scattering contributions by matching the scattering-length density of the material with that of a mixture of heavy and light water or oil, etc.

Important new fields of low-Q scattering, such as dynamic studies of polymers in a shear gradient and time-resolved studies of samples under periodic stress or under high pressure, have become accessible by neutron scattering because the weak interaction of neutrons with (homogeneous) matter permits the use of relatively thick (several mm) sample container walls, for example of cryostats, Couette-type shearing apparatus (Lindner & OberthuÈr, 1985, 1988), and ovens. Air scattering is not prohibitive, and easy-to-handle standard quartz cells serve as sample containers rather than very thin ones with mica windows in the case of X-rays. Unlike with X-rays, samples can be relatively thick, and nevertheless be studied to low Q values. This is particularly evident for metals, where X-rays are usually restricted to thin foils, but neutrons can easily accept samples 1±10 mm thick. 2.6.2.6. Incoherent scattering

2.6.2.3.1. Spin-contrast variation For a long time, the magnetic properties of the neutron have been neglected as far as `nonmagnetic' matter is concerned. Spin-contrast variation, proposed by Stuhrmann (Stuhrmann et al., 1986; Knop et al., 1986), takes advantage of the different scattering lengths of the hydrogen atoms in its spin-up and spindown states. Normally, these two states are mixed, and the cross section of unpolarized neutrons with the undirected spins gives rise to the usual value of the scattering amplitude of hydrogen. If, however, one is able to orient the spins of a given atom, and especially hydrogen, then the interaction of polarized neutrons with the two different oriented states offers an important contribution to the scattering amplitude: A b 2BI s;

2.6.2.5. Sample environment

2:6:2:5

where b is the isotropic nuclear scattering amplitude, B is the spin-dependent scattering amplitude, s is the neutron spin, and I the nuclear spin. For hydrogen, b 0:374 10 12 cm, B 2:9 10 12 cm. The sample protons are polarized at very low temperatures (order of mK) and high magnetic fields (several tesla) by dynamic nuclear polarization, i.e. by spin±spin coupling with the electron spins of a paramagnetic metallo-organic compound present in the sample, which are polarized by a resonant microwave frequency. It is clear that the principles mentioned above also apply to other than biological and chemical material. 2.6.2.4. Long wavelengths An important aspect of neutron scattering is the ease of using long wavelengths: Long-wavelength X-rays are produced efficiently only by synchrotrons, and therefore their cost is similar to that of neutrons. Unlike neutrons, however, they suffer from their strong interaction with matter. This disadvantage, which is acceptable with the commonly used Cu K radiation, is in most cases prohibitive for wavelengths of the order of 1 nm. Very low Q values are more easily obtained with long wavelengths than with very small angles, as is necessary with X-rays, since the same Q value can be observed further away from the direct beam. Objects of linear dimensions of several 100 nm, e.g. opals, where spherical particles of amorphous silica form a close-packed lattice with cell dimensions of up to several hundreds of nm, can still be investigated easily with neutrons. X-ray double-crystal diffractometers (Bonse & Hart, 1966), which may also reach very low Q, are subject to transmission problems, and neutron DCD's again perform better.

Incoherent scattering is produced by the interaction of neutrons with nuclei that are not in a fixed phase relation with that of other nuclei. It arises, for example, when molecules do not all contain the same isotope of an element (isotopic incoherent scattering). The most important source of incoherent scattering in SANS, however, is the spin-incoherent scattering from protons. It results from the fact that only protons and neutrons with identical spin directions can form an intermediate compound nucleus. The statistical probabilities of the parallel and antiparallel spin orientations, the similarity in size of the scattering lengths for spin up and spin down and their opposite sign result in an extremely large incoherent scattering cross section for 1 H, together with a coherent cross section of normal magnitude (but negative sign). Incoherent scattering contributes a background that can be by orders of magnitude more important than the coherent signal, especially at larger Q. On the other hand, it can be used for the calibration of the incoming intensity and of the detector efficiency (see below). 2.6.2.6.1. Absolute scaling Wignall & Bates (1987) compare many different methods of absolute calibration of SANS data. Since the scattering from a thin water sample is frequently already being used for correcting the detector response [see x2.6.2.6.2], there is an evident advantage for performing the absolute calibration by H2 O scattering. For a purely isotropic scatterer, the intensity scattered into a detector element of surface A spanning a solid angle A=4L2 can be expressed as I I0 1

2:6:2:6

with Ti the transmission of the isotropic scatterer, i.e. the relation of the number of neutrons in the primary beam measured within a time interval t after having passed through the sample, ITr , and the number of neutrons I0 observed within t without the sample. In practice, Ti is measured with an attenuated beam; typical attenuation factors are about 100 to 1000. g is a geometrical factor taking into account the sample surface and the solid angle subtended by the apparent source, i.e. the cross section of the neutron guide exit. Vanadium is an incoherent scatterer frequently used for absolute scaling. Its scattering cross section, however, is more than an order of magnitude lower than that of protons. Moreover, the surface of vanadium samples has to be handled with much care in order to avoid important contributions from

108


Ti g=4;

2.6. SMALL-ANGLE TECHNIQUES surface scattering by scratches. The vanadium sample has to be hermetically sealed to prevent hydrogen incorporation (Wignall & Bates, 1987). The coherent cross sections of the two protons and one oxygen in light water add up to a nearly vanishing coherent-scatteringlength density, whereas the incoherent scattering length of the water molecule remains very high. The (quasi)isotropic incoherent scattering from a thin, i.e. about 1 mm or less, sample of 1 H2 O, therefore, is an ideal means for determining the absolute intensity of the sample scattering (Jacrot, 1976; Stuhrmann et al., 1976), on condition that the sample-to-detector distance L is not too large, i.e. up to about 10 m. A function f t H2 O; l that accounts for deviations from the isotropic behaviour due to inelastic incoherent-scattering contributions of 1 H2 O and for the influence of the wavelength dependence of the detector response has to be multiplied to the right-hand side of equation (2.6.2.6) (May, Ibel & Haas, 1982). f can be determined experimentally and takes values of around 1 for wavelengths around 1 nm. Since the intensity scattered into a solid angle is IQ PQNTs I0 g

P

2 s V ;

bi

2:6:2:7

where PQ is the form factor of the scattering of one particle, and the geometrical factor g can be chosen so that it is the same as that of equation (2.6.2.6) (same sample thickness and surface and identical collimation conditions), we obtain IQ 4PQNTs f t H2 O; l

P

bi

2 s V =1

T H2 O: 2:6:2:8

Note that the scattering intensities mentioned above are scattering intensities corrected for container scattering, electronic and neutron background noise, and, in the case of the sample, for the solvent scattering. 2.6.2.6.2. Detector-response correction For geometrical reasons (e.g. sample absorption), and in the case of 2D detectors also for electronic reasons, the scattering curves cannot be measured with a sensitivity uniform over all the angular region. Therefore, the scattering curve has to be corrected by that of a sample with identical geometrical properties, but scattering the neutrons with the same probability into all angles (at least in the forward direction). As we have seen previously, such samples are vanadium and thin cells filled with light water. Again, water has the advantage of a much higher scattering cross section, and is less influenced by surface effects. At large sample-to-detector distances (more than about 10 m), the scattering from water is not sufficiently strong to enable its use for correcting sample scattering curves obtained with the same settings. Experience shows that it is possible in this case to use a water scattering curve measured at a shorter sample-todetector distance. This should be sufficiently large not to be influenced by the deviation of the (flat) detector surface from the spherical shape of the scattered waves and small enough so that the scattering intensity per detector element is still sufficient, for example about 3 m. It is necessary to know the intensity loss factor due to the different solid angles covered by the detector element and by the apparent source in both cases. This can be determined, for instance, by comparing the global scattering intensity of water on the whole detector for both conditions (after correction for the background scattering) or from the intensity shift of the same sample measured at both detector distances in a plot of the logarithm of the intensity versus Q.

2.6.2.6.3. Estimation of the incoherent scattering level For an exact knowledge of the scattering curve, it is necessary to subtract the level of incoherent scattering from the scattering curve, which is initially a superposition of the (desired) coherent sample scattering, electronic and neutron background noise, and (sometimes dominant) incoherent scattering. A frequently used technique is the subtraction of a reference sample that has the same level of incoherent scattering, but lacks the coherent scattering from the inhomogeneities under study. Although this seems simple in the case of solutions, in practice there are problems: Very often, the 1 H=2 H mixture is made by dialysis, and the last dialysis solution is taken as the reference. The dialysis has to be excessive to obtain really identical levels of 1 H, and in reality there is often a disagreement that is more important the lower the sample concentration is. If the concentration is high, then the incoherent scattering from the sample atoms (protons) themselves becomes important. For dilute aqueous solutions, there is a procedure using the sample and reference transmissions for estimating the incoherent background level (May, Ibel & Haas, 1982): The incoherent scattering level from the sample, Ii;s , can be estimated as Ii;s IH2 O fl 1

T H2 O;

2:6:2:9

where IH2 O is the scattering from a water sample, T H2 O is transmission, Ts that of the sample. fl is a factor depending on the wavelength, the detector sensibility, the solvent composition, and the sample thickness; it can be determined experimentally by plotting Ii;s =IH2 O versus 1 Ts =1 T H2 O for a number of partially deuterated solvent mixtures. This procedure is justified because of the overwhelming contribution of the incoherent scattering of 1 H to the macroscopic scattering cross section of the solution, and therefore to its transmission. The procedure should also be valid for organic solvents. The precision of the estimation is limited by the precision of the transmission measurement, the relative error of which can hardly be much better than about 0.005 for reasonable measuring times and currently available equipment, and by the (usually small) contribution of the coherent cross section to the total cross section of the solution. A modified version of (2.6.2.9) can be used if a solvent with a transmission close to that of a sample has been measured, but the factor fl should not be omitted. An equation similar to (2.6.2.9) holds for systems with a larger volume occupation c of particles in a (protonated) solvent with a scattering level Iinc in a cell with identical pathway (without the particles): Ii;s Iinc 1

1 c Tinc =1

Tinc :

2:6:2:9a

In this approximation, the particles' cross-section contribution is assumed to be zero, i.e. the particles are considered as bubbles. In the case of dilute systems of monodisperse particles, the residual background (after initial corrections) can be quite well estimated from the zero-distance value of the distance-distribution function calculated by the indirect Fourier transformation of Glatter (1979). 2.6.2.6.4. Inner surface area According to Porod (1951, 1982), small-angle scattering curves behave asymptotically like IQ constant As Q 4 for large Q, where As is the inner surface of the sample. Theoretically, fitting a straight line to IQQ4 versus Q4 (`Porod plot') at sufficiently large Q therefore yields a zero intercept, which is proportional to the internal surface; a slope

109


Ts =1

2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION can be interpreted as a residual constant background (including the self-term of the constant nuclear `form factor'), which may be used for slightly correcting the estimated background and consequently improving the quality of the data. For monodispersed particles, a particle surface can be deduced from the overall surface. The value of the surface area so determined depends on the maximal Q to which the scattering curve can be obtained with good statistics. This depends also on the magnitude of the background. At least for weakly scattering particles in mixtures of 1 H2 O and 2 H2 O, and even more in pure 1 H2 O, the incoherent background level often cannot be determined precisely enough for interpreting the tail of the scattering curve in terms of the surface area.

2.6.2.7.2. Particle mass With N CNA Vs =Mr , where NA is Avogadro's number, C is the mass concentration of the solute in g l 1 , and Vs is the sample volume in cm 3 (we assume N identical particles randomly distributed in dilute solution), we find that the relative molecular mass Mr of a particle can be determined from the intensity at zero angle, I0 in equation (2.6.2.10), using the relation (Jacrot & Zaccai, 1981), where the particle mass concentration C (in mg ml 1 ) is omitted: I0=fCIH2 O0g 4 f Ts Mr NA ds 10

Single-particle scattering in this context means scattering from isolated structures (clusters in alloys, isolated polymer chains in a solvent, biological macromolecules, etc.) randomly distributed in space and sufficiently far away from each other so that interparticle contributions to the scattering (see Subsection 2.6.2.8) can be neglected. The tendency of polymerization of single particles, for example the monomer±dimer equilibrium of proteins or the formation of higher aggregates, and long-range (e.g. electrostatic) interactions between the particles disturb single-particle scattering. In the absence of such effects, samples with solute volume fractions below about 1% can be regarded as free of volume-exclusion interparticle effects for most purposes. For (monodispersed) protein samples, for example, this means that concentrations of about 5 mg ml 1 are often a good compromise between sufficient scattering intensity and concentration effects. In many cases, series of scattering measurements with increasing particle concentrations have been used for extrapolating the scattering to zero concentration. In the following, we assume that particle interactions are absent.

bi

s V

Mr

2

1

T H2 O: 2:6:2:11

ds is the sample thickness. Note that bi =Mr may depend on solvent exchange; in a given solvent, especially 1 H2 O, it is rather independent of the exact amino acid composition of proteins (Jacrot & Zaccai, 1981). An alternative presentation of equation (2.6.2.11) is I0=fCIH2 O0g 4 f Ts Mr ds 10 3 v2 =NA 1

T H2 O;

2:6:2:11a

where p s s is the contrast; p is the particle scattering-length density (depending on the scattering-length density s of the solvent, in general) and v is the partial specific volume of the particle. Expression (2.6.2.11a) is of advantage when v, which is a linear function of s , is known for a class of particles. A thermodynamic approach to the particle-size problem, in view of the complementarity of different methods, has been given Zaccai, Wachtel & Eisenberg (1986) on the basis of the theory of Eisenberg (1981). It permits the determination of the molecular mass, of the hydration, and of the amount of bound salts.

2.6.2.7.1. Particle shape

2.6.2.7.3. Real-space considerations

All X-ray and neutron small-angle scattering curves can be approximated by a parabolic fit in a narrow Q range near Q 0 (Porod, 1951): IQ ' I0 1 a2 Q2 =3 . . .). In the case of single-particle scattering, a Gaussian approximation to the scattering curve is even more precise (Guinier & Fournet, 1955) in the zero-angle limit: 2:6:2:10

where RG is the radius of gyration of the particle's excess scattering density. The concept of RG and the validity of the Guinier approximation is discussed in more detail in the SAXS section of this volume (x2.6.1). It might be mentioned here that the frequently used QRG < 1 rule for the validity of the Guinier approximation is no more than an indication and should always be tested by a scattering calculation with the model obtained from the experiment: Spheres yield a deviation of 5% of the Gaussian approximation at QRG 1:3, rods at QRG 0:6; ellipsoids of revolution with an elongation factor of 2 can reach as far as QRG 3. More detailed shape information requires a wider Q range. As indicated before, Fourier transforms may help to distinguish between conflicting models. In many instances (e.g. hollow bodies, cylinders), it is much easier to find the shape of the scattering particle from the distance distribution function than from the scattering curve [see x2.6.2.7.3].

The scattering from a large number of randomly oriented particles at infinite dilution, and as a first approximation that of particles at sufficiently high dilution (see above), is completely determined by a function pr in real space, the distancedistribution function. It describes the probability p of finding a given distance r between any two volume elements within the particle, weighted with the product of the scattering-length densities of the two volume elements. Theoretically, pr can be obtained by an infinite sine Fourier transform of the isolated-particle scattering curve IQ

R1 0

pr=Qr sinQr dr:

2:6:2:12

In practice, the scattering curve can be measured neither to Q 0 (but an extrapolation is possible to this limit), nor to Q ! 1. In fact, neutrons allow us to measure more easily the sample scattering in the range near Q 0; X-rays are superior for large Q values. Indirect iterative methods have been developed that fit the finite Fourier transform IQ

DR max 0

pr=Qr sinQr dr

2:6:2:12a

of a pr function described by a limited number of parameters between r 0 and a maximal chord length Dmax within the particle to the experimental scattering curve. It differs from the pr of Section 2.6.1 by a factor of 4.

110


P

P

2.6.2.7. Single-particle scattering

IQ ' I0 exp Q2 =3R2G ;

3

2.6. SMALL-ANGLE TECHNIQUES This procedure was termed the ìndirect Fourier transformation (IFT)' method by Glatter (1979), who uses equidistant B splines in real space that are correlated by a Lagrange parameter, thus reducing the number of independent parameters to be fitted. Errors in determining a residual flat background only affect the innermost spline at r 0; the intensity at Q 0 and the radius of gyration are not influenced by a (small) flat background. Another IFT method was introduced by Moore (1980), who uses an orthogonal set of sine functions in real space. This procedure is more sensitive to the correct choice of Dmax and to a residual background that might be present in the data. A major advantage of IFT is the ease with which the deconvolution of the scattering intensities with respect to the wavelength distribution and to geometrical smearing due to the primary beam and sample sizes is calculated by smearing the theoretical scattering curve obtained from the real-space model. In fact, it is possible to convolute the scattering curves obtained from the single splines that are calculated only once at the beginning of the fit procedure. The convoluted constituent curves are then iteratively fitted to the experimental scattering curves. With the exception of particle symmetry, which is better seen in the scattering curve, structural features are more easily recognized in the pr function (Glatter, 1982a). Once the pr function is determined, the zero-angle intensity and the radius of gyration can be calculated from its integral and from its second moment, respectively. 2.6.2.7.4. Particle-size distribution Indirect Fourier transformation also facilitates the evaluation of particle-size distributions on the assumption that all particles have the same shape and that the size distribution depends on only one parameter (Glatter, 1980). 2.6.2.7.5. Model fitting As in small-angle X-ray scattering, the scattering curves can be compared with those of simple or more elaborate models. This is rather straightforward in the case of highly symmetric particles like icosahedral viruses that can be regarded as spherical at low resolution. The scattering curves of such viruses are easily adapted by spherical-shell models assigning different scattering-length densities to the different shells (e.g. Cusack, 1984). Neutron constrast variation helps decisively to distinguish between the shells. Fitting complicated models to the scattering curves is more critical because of the averaging effect of small-angle scattering. While it is correct and easy to show that the scattering curve produced by a model body coincides with the measured curve, in general a unique model cannot be deduced from the scattering curve alone. Stuhrmann (1970) has presented a procedure using Lagrange polynomials to calculate low-resolution real-space models directly from the scattering information. It has been applied successfully to the scattering curves from ribosomes (Stuhrmann et al., 1976). 2.6.2.7.6. Label triangulation Triangulation is one of the techniques that make full use of the advantages of neutron scattering. It consists in specifically labelling single components of a multicomponent complex, measuring the scattering curves from (a) particles with two labelled components, (b) and (c) particles with either of the two components labelled, and (d) a (reference) particle that is not labelled at all. The comparison of the scattering from b c

with that from a d yields information on the scattering originating exclusively from vectors combining volume elements in one component with volume elements in the other component. From this scattering difference curve, the distances between the centres of mass of the components are obtained. A table of such distances yields the spatial arrangement of the components. If there are n components in the complex, at least 4n 10 for n > 3 distance values are needed to build this model: Three distances define a basic triangle, three more yield a basic tetrahedron, the handedness of which is arbitrary and has to be determined by independent means. At least four more distances are required to fix a further component in space. More than four distances are needed if the resulting tetrahedron is too flat. Label triangulation is based on a technique developed by Kratky & Worthmann (1947) for determining heavy-metal distances in organometallic compounds by X-ray scattering, and was proposed originally by Hoppe (1972); Engelman & Moore (1972) first saw the advantage of neutrons. The need to mix preparations (a) plus (d) and (b) plus (c) for obtaining the desired scattering difference curve in the case of high concentrations and/or inhomogeneous complexes (consisting of different classes of matter) has been shown by Hoppe (1973). The complete map of all protein positions within the small subunit from E: coli ribosomes has been obtained with this method (Capel et al., 1987). An alternative approach for obtaining the distance information contained in the scattering curves from pairs of proteins by fitting the Fourier transform of `moving splines' to the scattering curves has been presented by May & Nowotny (1989) for data on the large ribosomal subunit. The scattering curves should be measured at the scatteringlength-density matching point of the reference particle for reducing undesired contributions. Naturally inhomogeneous particles can be rendered homogeneous by specific partial deuteration. This technique has been successfully applied for ribosomes (Nierhaus et al., 1983). 2.6.2.7.7. Triple isotropic replacement An elegant way of determining the structure of a component inside a molecular complex has been proposed by Pavlov & Serdyuk (1987). It is based on measuring the scattering curves from three preparations. Two contain the complex to be studied at two different levels of labelling, 1 and 2 , and are mixed together to yield sample 1, the third contains the complex at an intermediate level of labelling, 3 (sample 2). If the condition 3 r 1

2:6:2:13

is satisfied by , the relative concentration of particle 2 in sample 1, then the difference between the scattering from the two samples only contains contributions from the single component. Additionally, the contributions from contamination, aggregation, and interparticle effects are suppressed provided that they are the same in the three samples, i.e. independent of the partialdeuteration states. In the case of small complexes, can be obtained by measuring the scattering curves I1 Q; I2 Q; and I3 Q of the three particles as a function of contrast and by plotting the differences of the zero-angle scattering I1 0 I3 0 and I2 0 I3 0 versus . The two curves intercept at the correct ratio 0 . The method, which can be considered as a special case of a systematic inverse contrast variation of a selected component, holds if the concentrations, the complex occupations, and the aggregation behaviour of the three particles are identical. Mathematically, the difference curve is independent of the

111


1 r 2 r


P P SQ expiQrj rk N;

contrast of the rest of the complex with respect to the solvent. In practice, it would be wise to follow the same considerations as with triangulation.

and of the form factor PQ of the inhomogeneities (as before): IQ PQSQ:

2.6.2.8. Dense systems Especially in the case of polymers, but also in alloys, the scattering from the sample can often no longer be described, as in the previous section, as originating from a sum of isolated particles in different orientations. There may be two reasons for this: either the number concentration c of one of the components is higher than about 0.01, leading to excluded-volume effects, and/or there is an electrostatic interaction between components (for example, in solutions of polyelectrolytes, latex, or micelles). In these cases, it is usually the information about the structure of the sample caused by the interactions that is to be obtained rather than the shape of the inhomogeneities or particles in the sample, unless the interactions can be regarded as a weak disturbance. An excellent introduction to the treatment of dense systems is found in the article of Hayter (1985). A detailed description of the theoretical interpretation of correlations in charged macromolecular and supramolecular solutions has been published by Chen, Sheu, Kalus & Hoffmann (1988). The scattering from densely packed particles can be written as the product of the structure factor or structure function SQ, describing the arrangement of the inhomogeneities with respect to each other, in mathematical terms the interference effects of correlations between particle positions, in the sample,

2:6:2:15

Hayter & Penfold (1981) were the first to describe an analytic structure factor for macro-ion solutions. If PQ can be obtained from a measurement of a dilute solution of the particles under study, then the pure structure factor can be calculated by dividing the high-concentration intensity curve by the low-concentration curve. This procedure requires the form factor not to change with concentration, which is not necessarily the case for loosely arranged particles such as polymers. A technique that avoids this problem is contrast variation (see Subsection 2.6.2.2): introducing a fraction of a deuterated molecule into a bulk of identical protonated molecules (or vice versa, with the advantage of reduced incoherent background) yields the scattering of the ìsolated' labelled particle at high-concentration conditions. Partial structure factors can be obtained from a contrastvariation series of a given system at different volume fractions of the particles. Similarly to equation (2.6.2.4), the structure factor can be decomposed into a quadratic function. In the ternary alloy Al±Ag±Zn, for example, the scattering has been decomposed into the contributions from the two minor species Ag and Zn, and their interference, i.e. the partial structure functions for Zn±Zn, Zn±Ag, and Ag±Ag, by using the scattering from three samples with different silver isotopes, and identical sample treatment (Salva-Ghilarducci, Simon, Guyot & Ansara, 1983).

112


2:6:2:14


2.7. Topography By A. R. Lang

2.7.1. Principles The term diffraction topography covers techniques in which images of crystals are recorded by Bragg-diffracted rays issuing from them. It can be arranged at will for these rays to produce an image of a surface bounding the crystal, or of a thin slice cutting through the crystal, or of the projection of a selected volume of the crystal. The majority of present-day topographic techniques aim for as high a spatial resolution as possible in their point-bypoint recording of intensity in the diffracted rays. In principle, any position-sensitive detector with adequate spatial resolution could be employed for recording the image. Photographic emulsions are most widely used in practice. In the following accounts of the various diffraction geometries developed for topographic experiments, the term `film' will be used to stand for photographic emulsion coated on film or on glass plate, or for any other position-sensitive detector, either integrating or capable of real-time viewing, that could serve instead of photographic emulsion. (Position-sensitive detectors, TV cameras, and storage phosphors are described in Sections 7.1.6, 7.1.7, and 7.1.8.) All diffraction geometries described with reference to an X-ray source could in principle be used with neutron radiation of comparable wavelength (see Chapter 4.4). Two factors, often largely independent and experimentally distinguishable, determine the intensity that reaches each point on the topograph image. The first is simply whether or not the corresponding point in the specimen is oriented so that some rays within the incident beam impinging upon it can satisfy the Bragg condition. The intensity of the Bragg-reflected rays will range between maximum and minimum values depending upon how well that condition is satisfied. The consequent intensity variation from point to point on the image is called orientation contrast, and it can be analysed to provide a map of lattice misorientations in the specimen. The sensitivity of misorientation measurement is controllable over a wide range by appropriate choice of diffraction geometry, as will be explained below. The second factor determining the diffracted intensity is the lattice perfection of the crystal. In this case, physical factors such as X-ray wavelength, specimen absorption, and structure factor of the active Bragg reflection fix the range within which the diffracted intensity can lie. One limit corresponds to the case of the ideally perfect crystal. This is a well defined entity, and its diffraction behaviour is well understood [see IT B (1996, Part 5)]. The other limit is that of the ideally imperfect crystal, a less precisely defined entity, but which, for practical purposes, may be taken as a crystal exhibiting negligible primary and secondary extinction. The magnitude, and sometimes also the sign, of the difference in intensity recorded from volume elements of ideally perfect as opposed to ideally imperfect crystals is to a large degree controllable by the choice of experimental parameters (in particular by choice of wavelength). Contrast on the topograph image arising from point-to-point differences in lattice perfection of the specimen crystal was called extinction contrast in earlier X-ray topographic work, but is now more usually called diffraction contrast to conform with terminology used in transmission electron microscopic observations of lattice defects, experiments which have many analogies with the X-ray case. Figs. 2.7.1.1 and 2.7.1.2, respectively, show in plan view the simplest arrangements for taking a reflection topograph and a transmission topograph. The source of X-rays is shown as being point-like at S. If its wavelength spread is large then the Bragg

condition may be satisfied over the whole length CD for Bragg planes oriented parallel to BB0 , and an image of CD will be formed on F by the Bragg-diffracted rays falling on it. The specimen is mounted on a rotatable axis (the ! axis) perpendicular to the plane of the drawing, which represents the median plane of incidence, in order that the angle of incidence on the planes BB0 can be varied. The specimen is usually adjusted so that the diffraction vector, h, of the Bragg reflection of principal interest is perpendicular to the ! axis. Let the mean source-tospecimen and specimen-to-film distances be a and b, respectively. Suppose the source S is extended a distance s in the axial direction (i.e. perpendicular to the plane of incidence). Then diffracted rays from any point on CD will be spread on F over a distance sb=a in the axial direction. This is the simple expression for the axial resolution of the topograph given by ray optics. Transmission topographs have the value of showing defects within the interior of specimens, which may be optically opaque, but are in practice limited to a specimen thickness, t, such that t is less than a few units ( being the normal linear absorption coefficient) unless the specimen structure and the perfection are such as to allow strong anomalous transmission [the Borrmann effect, see IT B (1996, Part 5)]. If a reflection topograph specimen is a nearly perfect crystal then the volume of crystal contributing to the image is restricted to a depth below the surface given approximately by the X-ray extinction distance, h , of the active Bragg reflection, which may be only a few micrometres, rather than the penetration distance, 1 , of the radiation used. Besides the ratio b=a, other important experimental parameters are the degree of collimation of the incident beam and its wavelength spread. The manner in which X-ray topographs exhibit orientation contrast and diffraction contrast under different choices of these parameters is illustrated schematically in Fig. 2.7.1.3. There, a represents a hypothetical specimen consisting of a matrix of perfect crystal C in which are embedded two islands A and B whose lattices differ from C in the following respects. A has the same mean orientation as C but is a region of

Fig. 2.7.1.1. Surface reflection topography with a point source of diverging continuous radiation.

Fig. 2.7.1.2. Transmission topography with a point source of diverging continuous radiation.

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION high imperfection. (In reflection topographs, imperfect regions will always produce stronger integrated reflections than perfect regions and will also do so in transmission topographs under low-absorption conditions.) The island B is assumed to be as perfect as C, but is slightly misoriented with respect to C. The topograph images sketched in b±e could represent either reflection topographs or transmission topographs from a specimen thin compared with the dimension CD in Fig. 2.7.1.2. [Possible distortion of the images relative to the shape of a is neglected: this matter is considered later.] First, suppose that continuous radiation is emitted from the source S. If the ratio b=a is quite small, the topograph image will resemble b. The island A is detected by diffraction contrast whereas island B will not show any area contrast since by assumption the incident radiation contains wavelengths enabling B to satisfy the Bragg condition just as well as C. The low-angle B±C boundary may show up, however, since it contains dislocations that will produce diffraction contrast and might be individually resolvable with a high-resolution topographic technique. Orientation contrast of B becomes manifest when b is increased, and measurement of the misorientation is then possible from the displacement of the image of B [as shown in c] consequent upon the different direction of Bragg-reflected rays issuing from it compared with those from C. Next, suppose that S emits a limited range of wavelengths, e.g. characteristic K radiation, and let the incident beam be collimated to have an angular spread in the plane of incidence that is smaller than the component in that plane of the misorientation between B and C, but larger than the angular range of reflection of C or A. Then, if the specimen is rotated about the ! axis so that A and C satisfy the Bragg condition, B will not do so and the topograph will resemble d. [Island A shows up in d by diffraction contrast, as in b.]. Appropriate rotation of the specimen will bring B into the Bragg-reflecting orientation, but will eliminate reflection from A and C, as shown in e. The images d and e will not undergo significant change with variation in the ratio b=a, except for loss of resolution as b=a increases. The sensitivity of misorientation measurement will increase as the angular and wavelength spread of the incident beam are reduced, but when the angular range of a monochromatic incident beam is lowered to a value comparable with the angular range of reflection of the perfect crystal (generally only a few seconds of arc), it will not be possible with one angular setting alone to obtain an image that will distinguish between diffraction contrast and orientation contrast in the clear way shown in d. The distinction will require comparison of a series of topographs obtained during a step-wise sweep of the

Fig. 2.7.1.3. Differentiation between orientation contrast and diffraction contrast in types of topograph images, b±e, of a crystal surface a.

angular range of reflection by the specimen. This is the procedure adopted in double-crystal or multicrystal topography, as described in Sections 2.7.3 and Subsection 2.7.4.2. Details concerning diverse variants in diffraction geometry used in X-ray topographic experiments, treatments of the diffraction contrast theory underlying X-ray topographic imaging of lattice defects, and listings of applications of X-ray topography can be found in reviews and monographs of which a selection follows. All aspects of X-ray topography are covered in the survey edited by Tanner & Bowen (1980). Armstrong & Wu (1973), Tanner (1976), and Lang (1978) describe experimental techniques and review their applications. The dynamicaldiffraction theoretical basis of X-ray topography is emphasized by Authier (1970, 1977). Kato (1974) and Pinsker (1978) deal comprehensively with X-ray dynamical diffraction theory, which is also the topic of Part 5 of IT B (1996). Introductions to this theory have been presented by Batterman & Cole (1964), Hart (1971), and Hildebrandt (1982), the latter two being especially relevant to X-ray topography. 2.7.2. Single-crystal techniques 2.7.2.1. Reflection topographs Combining the simple diffraction geometry of Fig. 2.7.1.1 with a laboratory microfocus source of continuous radiation offers a simple yet sensitive technique for mapping misorientation textures of large single crystals (Schulz, 1954). One laboratory X-ray source much used produces an apparent size of S 30 mm in the axial direction and 3 mm in the plane of incidence. Smaller source sizes can be achieved with X-ray tubes employing magnetic focusing of the electron beam. Then b=a ratios between 12 and 1 can be adopted without serious loss of geometric resolution, and, with a 0:3 m typically, misorientation angles of 1000 can be measured on images of the type c in Fig. 2.7.1.3. The technique is most informative when the crystal is divided into well defined subgrains separated by low-angle boundaries, as is often the case with annealed melt-grown crystals. On the other hand, when continuous lattice curvature is present, as is usually the case in all but the simplest cases of plastic deformation, it is difficult to separate the relative contributions of orientation contrast and diffraction contrast on topographs taken by this method. In principle, the separation could be effected by recording a series of exposures with a wide range of values of b, and it becomes practicable to do so when exposures can be brief, as they can be with synchrotron-radiation sources (see Section 2.7.4.). For easier separation of orientation contrast and diffraction contrast, and for quicker exposures with conventional X-ray sources, collimated characteristic radiation is used, as in the Berg±Barrett method. Barrett (1945) improved an arrangement earlier described by Berg (1931) by using fine-grain photographic emulsion and by minimizing the ratio b=a, and achieved high topographic resolution ( 1 mm). The method was further developed by Newkirk (1958, 1959). A typical Berg±Barrett arrangement is sketched in Fig. 2.7.2.1. Usually, the source S is

Fig. 2.7.2.1. Berg±Barrett arrangement for surface-reflection topographs.

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2.7. TOPOGRAPHY the focal spot of a standard X-ray tube, giving an apparent source 1 mm square perpendicular to the incident beam. The openings of the slits S1 and S2 are also 1 mm in the plane of incidence, and the distance S1 ±S2 (which may be identified with the distance a is typically 0.3 m. The specimen is oriented so as to Bragg reflect asymmetrically, as shown. Softer radiations, e.g. Cu K, Co K or Cr K, are employed and higher-angle Bragg reflections are chosen 2B ' 90 is most convenient). Fig. 2.7.2.1 indicates three possible film orientations, F1 ±F3 . (These possibilities apply in most X-ray topographic arrangements.) Choice of orientation is made from the following considerations. If minimum distance b is required over the whole length CD, then position F1 is most appropriate. If a geometrically undistorted image of CD is needed, then position F2 , in which the film plane is parallel to the specimen surface, satisfies this condition. If a thick emulsion is used, it should receive X-rays at normal incidence, and be in orientation F3 . If high-resolution spectroscopic photographic plates are used, in which the emulsion thickness is 1 mm only, then considerable obliquity of incidence of the X-rays is tolerable. But these plates have low X-ray absorption efficiency. Nuclear emulsions (particularly Ilford type L4) are much used in X-ray topographic work. Ilford L4 is a high-density emulsion (halide weight fraction 83%) and hence has high X-ray stopping power. The usual minimum emulsion thickness is 25 mm. Such emulsions should be oriented not more than about 2 off perpendicularity to the X-ray beam if resolution loss due to oblique incidence is not to exceed 1 mm (with correspondingly closer limits on obliquity for thicker emulsions). With 1 mm openings of S1 and S2 , and a 0:3 m, most of the irradiated area of CD will receive an angular range of illumination sufficient to allow both components of the K doublet to Bragg reflect. In these circumstances, the distance b must be everywhere less than 1±2 mm if image spreading due to superimposition of the 1 and 2 images is not to exceed a few micrometres. In order to eliminate this major cause of resolution loss (and, incidentally, gain sensitivity in misorientation measurements), the apertures S1 and S2 should be narrowed and/or a increased so that the angular range of incidence on the specimen is less than the difference in Bragg angle of the 1 and 2 components for the particular radiation and Bragg angle being used. (This condition applies equally in the transmission specimen techniques, described below.). With a narrower beam, the illuminated length of CD is reduced. This disadvantage may be overcome by mounting the specimen and film together on a linear traverse mechanism so that during the exposure all the length of CD of interest is scanned. In this way, surface-reflection X-ray topographs can be recorded for comparison with, say, etch patterns or cathodoluminescence patterns (Lang, 1974).

radiation transmission topograph images. Their minimum b=a ratio was set by the need to avoid overlap of Laue images of the crystal produced by different Bragg planes. Collimated characteristic radiation is used in the methods of `section topographs' (Lang, 1957) and `projection topographs' (Lang, 1959a), the latter being sometimes called `traverse topographs'. Fig. 2.7.2.2 explains both techniques. When taking a section topograph, the specimen CD, usually plate shaped, is stationary (disregard the double-headed arrow in the figure). The ribbon-shaped incident beam issuing from the slit P is Bragg reflected by planes normal, or not far from normal, to the major surfaces of the specimen. As drawn, the Bragg planes make an angle with the normal to the X-ray entrance surface of the specimen, the positive sense of being taken in the same sense as the deviation 2B of the Bragg-reflected rays. If the crystal is sufficiently perfect for multiple scattering to occur within it (with or without loss of coherence), then the multiply scattered rays associated with the Bragg reflection excited will fill the volume of the triangular prism whose base is ORT, the ènergy-flow triangle' or `Borrmann triangle', contained between OT and OR whose directions are parallel to the incident wavevector, K0 , and diffracted wavevector, Kh , respectively. Both the K0 and Kh beams issuing from the X-ray exit surface of the crystal carry information about the lattice defects within the crystal. However, it is usual to record only the Kh beam. This falls on the film, F, in a strip extending normal to the plane of incidence, of height equal to the illuminated height of the specimen multiplied by the axial magnification factor a b=a, and forms the section topograph image. The screen, Q, prevents the K0 beam from blackening the film but has a slot allowing the diffracted beam to fall on F. A diffraction-contrast-producing lattice defect cut by OT at I will generate supplementary rays parallel to Kh and will produce an identifiable image on F at I 0 , the `direct image' or `kinematic image' of the defect. The depth of I within CD can be found via the measurement of I 0 T 0 =R0 T 0 . From a series of section topographs taken with a known translation of the specimen between each topograph, a three-dimensional construction of the trajectory of defect I (e.g. a dislocation line) within the crystal can be built up. To obtain good definition of the spatial width of the ribbon incident beam cutting the crystal, the distance between P and the crystal is kept small. The minimum practicable opening of P is about 10 mm. If diffraction is occurring from planes perpendicular to the X-ray entrance surface of the specimen, i.e. symmetrical Laue case diffraction, the width R0 T 0 of the section topograph image is simply 2t sin B , t being the specimen thickness, and neglecting the contribution from the

2.7.2.2. Transmission topographs The term `X-ray topograph' was introduced by Ramachandran (1944) who took transmission topographs of cleavage plates of diamond using essentially the arrangement shown in Fig. 2.7.1.2. (In this case, S was a 0.3 mm diameter pinhole placed in front of the window of a W-target X-ray tube so as to form a point source of diverging continuous radiation.) Ramachandran adopted a distance a 0:3 m and ratio a=b of about 12, which produced images of about 25 mm geometrical resolution having the characteristics of Fig. 2.7.1.3b, i.e. sensitive to diffraction contrast but not to orientation contrast. For each reflection under study, the film was inclined to the incident beam with that obliquity calculated to produce an undistorted image of the specimen plate. Guinier & Tennevin (1949) studied both diffraction contrast and orientation contrast in continuous-

Fig. 2.7.2.2. Arrangements for section topographs and projection topographs.

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION width of the ribbon incident beam. With asymmetric transmission, as drawn in the figure, R0 T 0 t sec B sin 2B . The distance b is made as small as is permitted by the specimen shape and the need to separate the emerging K0 and Kh beams. Suppose b is 10 mm. Then, with a source S having axial extension 100 mm, the distance a SP should be not less than 0.5 m in order to keep the geometrical resolution in the axial direction better than 2 mm, and should be correspondingly longer with larger source sizes. To take a projection topograph, the specimen CD and the cassette holding the film F are together mounted on an accurate linear traversing mechanism that oscillates back and forth during the exposure so that the whole area of interest in the specimen is scanned by the ribbon beam from P. The screen Q is stationary. If the specimen is plate shaped, the best traverse direction to choose is that parallel to the plate, as indicated by the doubleheaded arrow, for then the diffracted beam will have minimum side-to-side oscillation during the traverse oscillation, the opening of Q can be held to a minimum, and thereby unwanted scattering reaching F kept low. The projection topograph image is an orthographic projection parallel to Kh of the crystal volume and its content of diffraction-contrast-producing lattice defects. If the specimen is plate-like, of length L in the plane of incidence, then, with F normal to Kh , the magnification of the topograph image in the direction parallel to the plane of incidence is L cosB . There will generally be a small change of axial magnification a b=a along L. The loss of three-dimensional information occurring through projection can be recovered by taking stereopairs of projection topographs. The first method (Lang, 1959a,b) used hkl, h k l pairs of topographs as stereopairs. One disadvantage of this method is that the convergence angle is fixed at 2B , which may be unsuitably large for thick specimens. The method of Haruta (1965) obtains two views of the specimen using the same hkl reflection, by making a small rotation of the specimen about the h vector between the two exposures, and has the advantage that this rotation, and hence the stereoscopic sensitivity, can be chosen at will. When taking projection topographs, the slit P can be wider than the narrow opening needed for high-resolution section topographs, but not so wide as to cause unwanted K2 reflection

Fig. 2.7.2.3. Arrangements for limited projection topographs and direct-beam topographs.

to occur. Best use of the X-ray source is made when the width of P is the same as or somewhat greater than S. In certain investigations, the methods of Kh -beam `limited projection topographs' (Lang, 1963) and of K0 -beam section topographs and projection topographs are useful; Fig. 2.7.2.3 shows the arrangement of screens and diffracted-beam slits then adopted. The limited projection topograph technique can be employed with a plate-shaped specimen CDD0 C 0 , as in the following examples. Suppose the surface of the plate contains abrasion damage that cannot be removed but that causes diffraction contrast obscuring the images of interior defects in the crystal. The diffracted-beam slit (equivalent to the opening in Q shown in Fig. 2.7.2.2), which is opened to the setting S1 for a standard projection topograph, may now be closed down to setting S2 so as to cut into the RR0 and TT 0 edges of the Kh beam, and thereby prevent direct images of near-surface defects located between CC 0 and XX 0 , and between YY 0 and DD0 , from reaching F. As another example, it may be desired to receive direct images from a specimen surface and a limited depth below it only (e.g. when correlating surface etch pits with dislocation outcrops). Then, setting S3 of the diffracted-beam slit is adopted and only the direct images from defects lying between depth ZZ 0 and the surface DD0 reach F. To record the K0 beam image, either in a section topograph or a projection topograph, some interception of the K0 beam on the OTT 00 side is needed to avoid intense blackening of the film G by radiation coming from the source, which will generally contain much energy in wavelengths other than those undergoing Bragg diffraction by the crystal. Screen S4 , critically adjusted, performs the required interception. Recording both K0 -beam and Kh -beam images is valuable in some studies of dynamical diffraction phenomena, such as the `first-fringe contrast' in stacking-fault fringe patterns (Jiang & Lang, 1983). Such recording can be done simultaneously, on separate films, normal to K0 and Kh , respectively, when 2B is sufficiently large. When collimated characteristic radiation is used, recording projection topographs of reasonably uniform density becomes difficult when the specimen is bent. To keep the ! axis oriented at the peak of the Bragg reflection while the specimen is scanned, several devices for `Bragg-angle control' have been designed, for example the servo system of Van Mellaert & Schwuttke (1972). The signal is taken from a detector registering the Bragg reflection through the film F, but this precludes use of glassbacked emulsions if X-ray wavelengths such as that of Cu K and softer are used. An alternative approach with thin, large-area transmission specimens is to revert to the geometry of Fig. 2.7.1.2 and deliberately elastically bend the crystal to such radius as will enable its whole length to Bragg diffract a single wavelength diverging from S, similar to a Cauchois focusing transmission monochromator (Wallace & Ward, 1975). No specimen traversing is then needed, but b cannot be made small if the wide K0 and Kh beams are to be spatially separated in the plane of F. Quite simple experimental arrangements can be adopted for taking transmission topographs under high-absorption conditions, when only anomalously transmitted radiation can pass through the crystal. The technique has mainly been used in symmetrical transmission, as shown with the specimen CC 0 D0 D in Fig. 2.7.2.4. When the specimen perfection is sufficiently high for the Borrmann effect to be strongly manifested, and t > 10, say, the energy flow transmitted within the Borrmann triangle is constricted to a narrow fan parallel to the Bragg planes, the fan opening angle being only a small fraction of 2B [see IT B (1996, Part 5)]. Radiation of a given wavelength coming from a small source at S and undergoing Bragg

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2.7. TOPOGRAPHY diffraction in CC 0 D0 D will take the path shown by the heavy line in Fig. 2.7.2.4, simplifying the picture to the case of extreme confinement of energy flow to parallelism with the Bragg planes. At the X-ray exit surface DD0 , splitting into K0 and Kh beams occurs. A slit-less arrangement, as shown in the figure, may suffice. Then, when S is a point-like source of K radiation, and distance a is sufficiently large, films F1 and F2 will each record a pair of narrow images formed by the 1 and 2 wavelengths, respectively. A wider area of specimen can be imaged if a line focus rather than a point focus is placed at S (Barth & Hosemann, 1958), but then the 1 and 2 images will overlap. Under conditions of high anomalous transmission, defects in the crystal cause a reduction in transmitted intensity, which appears similarly in the K0 and Kh images. Thus, it is possible to gain intensity and improve resolution by recording both images superimposed on a film F3 placed in close proximity to the X-ray exit face DD0 (Gerold & Meier, 1959). 2.7.3. Double-crystal topography The foregoing description of single-crystal techniques will have indicated that in order to gain greater sensitivity in orientation contrast there are required incident beams with closer collimation, and limitation of dispersion due to wavelength spread of the characteristic X-ray lines used. It suggests turning to prior reflection of the incident beam by a perfect crystal as a means of meeting these needs. Moreover, the application of crystalreflection-collimated radiation to probe angularly step by step as well as spatially point by point the intensity of Bragg reflection from the vicinity of an individual lattice defect such as a dislocation brings possibilities of new measurements beyond the scope provided by simply recording the local value of the integrated reflection. The X-ray optical principles of doublecrystal X-ray topography are basically those of the doublecrystal spectrometer (Compton & Allison, 1935). The properties of successive Bragg reflection by two or more crystals can be effectively displayed by a Du Mond diagram (Du Mond 1937), and such will now be applied to show how collimation and monochromatization result from successive reflection by two crystals, U and V, arranged as sketched in Fig. 2.7.3.1. They are in the dispersive, antiparallel, ` ' setting, and are assumed to be identical perfect crystals set for the same symmetrical Bragg reflection. Only rays making the same glancing angle with both surfaces will be reflected by both U and V. For example, radiation of shorter wavelength reflected at a smaller glancing angle at U (the ray shown by the dashed line) will impinge at a larger glancing angle on V and not satisfy the Bragg condition. In this setting, with a given angle ! between the Bragg-

reflecting planes of each crystal, U V ! and U V . The Du Mond diagram for the setting, Fig. 2.7.3.2, shows plots of Bragg's law for each crystal, the V curve being a reflection of the U curve in a vertical mirror line and differing by ! from the U curve in its coordinate of intersection with the axis of abscissa, in accord with the equations given above. The small angular range of reflection of a monochromatic ray by each perfect crystal is represented exaggeratedly by the band between the parallel curves. Where the band for crystal U superimposes on the band for V (the shaded area) defines semiquantitatively the divergence and wavelength spread in the rays successively reflected by U and V. (It is taken for granted that 12 ! lies between the maximum and minimum incident glancing angles on U, max and min , afforded by the incident beam, assumed polychromatic.) The reflected beam from U alone contains wavelengths ranging from lmin to lmax . Comparison of these and l ranges with the extent of the shaded area illustrates the efficacy of the arrangement in providing a collimated and monochromatic beam, which can be employed to probe the reflecting properties of a third crystal (Nakayama, Hashizume, Miyoshi, Kikuta & Kohra, 1973). Techniques employing three or more successive Bragg reflections find considerable application when used with synchrotron X-ray sources, and will be considered below, in Section 2.7.4. The most commonly used arrangement for double-crystal topography is shown in Fig. 2.7.3.3, in which U is the `reference' crystal, assumed perfect, and V is the specimen crystal under examination. Crystals U and V are in the parallel, ` ' setting, which is non-dispersive when the Bragg planes of U and V have the same (or closely similar) spacings. Before considering the Du Mond diagram for this arrangement, note that Bragg reflection at the reference crystal U is asymmetric, from planes inclined at angle to its surface. Asymmetric reflections have useful properties, discussed, for example, by Renninger (1961), Kohra (1972), Kuriyama & Boettinger (1976), and Boettinger, Burdette & Kuriyama (1979). The asymmetry factor, b, of magnitude jK0 n=Kh nj, n being the

Fig. 2.7.3.1. Double-crystal setting.

Fig. 2.7.2.4. Topographic techniques using anomalous transmission.

Fig. 2.7.3.2. Du Mond diagram for setting in Fig. 2.7.3.1.

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION crystal-surface normal, is also the ratio of spatial widths of the incoming to the outgoing beams, Win =Wout . In the case of symmetric Bragg reflection, the perfect crystal U would totally reflect (in the zero-absorption case) over a small angular range, wS . In the asymmetric case, the ranges of total reflection are win for the incoming rays and wout for the outgoing. Dynamicaldiffraction theory [IT B (1996, Part 5)] shows that wout bwin b1=2 wS , so that win Win wout Wout (as would be expected from energy conservation). Thus, highly asymmetric reflection from the reference crystal U not only provides a spatially wide beam, able to cover a large area of V without recourse to any mechanical traversing motion of the components S, U or V, but also produces a desirably narrow angular probe for studying the angular breadth of reflection of V. In practice, values of b lower than 0.1 can be used. Du Mond diagrams for the arrangement are shown in Fig. 2.7.3.4a and b. For simplicity, the curves (slope dl=d 2d cos ) are represented by straight lines. In the setting, V U ! and V U . In Fig. 2.7.3.4a, the narrow band labelled U passing through the origin represents the beam of angular width wout leaving U. It is assumed that all of the specimen crystal V has the same interplanar spacing as U but that it contains a slightly misoriented minor region V 0 (which may be located as shown in Fig. 2.7.3.3). When ! differs substantially from zero, the bands corresponding to crystal V and its minor part V 0 lie in positions V1 and V10 , respectively, in Fig. 2.7.3.4a. (Only the relevant part of the latter band is drawn, for simplicity.) The offset along the axis between V1 and V10 is the component ' of the misorientation between V and V 0 that lies in the plane of incidence. If ! is reduced step-wise, a doublecrystal topograph image being obtained at F at each angular setting, ' can be found from film densitometry, which will show at what settings band U is most effectively overlapped by band V or by band V 0 . When ! is reduced to zero, the specimen crystal bands are at V2 and V20 . The drawing shows that V 0 has then passed right through the setting for its Bragg reflection, which occurred at a small positive value of !. Since the U and V bands have identical slopes, their overlap occurs at all wavelengths when ! 0. In practice, only the shaded area is involved, corresponding to the wavelength range lmin to lmax , defined by the range of incidence angles, min to max , on the Bragg planes of crystal U. (The width of band U will generally be negligible compared with the range of allowed by source width and slit collimation system.) One component of ' is found in the procedure just described. The second component

Fig. 2.7.3.3. Double-crystal topographic arrangement, setting. Asymmetric reflection from reference crystal U. Specimen crystal divided into regions V and V 0 .

needed to specify the difference between h-vector directions of the Bragg planes of V and V 0 is obtained by repeating the experiment after rotating V by 90 about h. Next consider the more general case when V 0 differs from V in both orientation and interplanar spacing, and both V 0 and V have slightly different interplanar spacings from U. The difference in orientation between V 0 and V, ', and their difference in interplanar spacing, dV 0 dV , can be distinguished by taking two series of double-crystal topographs, the orientation of the specimen in its own plane (its azimuthal angle, ) being changed by a 180 rotation about its h vector between taking the first and second series. As shown schematically in the Du Mond diagram, Fig. 2.7.3.4b, the U, V, and V 0 bands now all have slightly different slopes. [Reference crystal U is reflecting the same small wavelength band as in Fig. 2.7.3.4a.] The setting represented in the diagram is that putting V at the maximum of its Bragg reflection Let the V 0 band be then at position V00 , for the case when 0 . Assume that, when is changed by 180 , the rotation of the specimen in its own plane can be made about the h vector of V precisely. (This assumption simplifies the diagram.) Then this 180 rotation will not cause any translation of the V band along the axis, but does 0 transfer the V 0 band from V00 to the position V180 . With the sense of increasing ! taken as that translating the specimen bands to the right and ! taken as the difference in readings between peak reflection from V and that from V 0 , the diagram shows that, with 0 , !0 V 0 V ', and, with 180 ,

Fig. 2.7.3.4. Du Mond diagrams for setting in Fig. 2.7.3.3. a Case when specimen region V 0 is misoriented with respect to V, but U, V, and V 0 all have the same interplanar spacing. b Case when V 0 differs from V in both orientation and interplanar spacing, and both differ from U in interplanar spacing.

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2.7. TOPOGRAPHY !180 V 0 V ', from which both ' and the difference in Bragg angles can be found. The interplanar spacing difference is given by dV 0 dV V V 0 d cot , d being the mean interplanar spacing of V and V 0 . In practice, series of topographs are taken with azimuthal angles 0, 90, 180, and 270 , so that the two components needed to specify the misorientation vector between the Bragg-plane normals of V and V 0 can be determined. The Du Mond diagram shows that in this slightly dispersive experiment the range of overlap of the U band with any V band can be restricted by reducing the angular range or wavelength range of rays incident on U. Such reduction can be achieved by use of a small source S far distant from U, such as a synchrotron source. It can also be achieved by methods described in Subsection 2.7.4.2. As regards spatial resolution on double-crystal topographs, relations analogous to those for single-crystal topographs apply. If the reference crystal U unavoidably contains some defects, their images on F can deliberately be made diffuse compared with images of defects in V by making the UV distance relatively large. In a nearly dispersion-free arrangement, if the K1 wavelength is being reflected, then so too will the K2 if S is sufficiently widely extended in the incidence plane, as is usually necessary to image a usefully large area of V. If the distance VF cannot be made sufficiently small to reduce to a tolerable value the resolution loss due to simultaneous registration of the 1 and 2 images, then a source S of small apparent size, together with a collimating slit before U, will be needed. In order to obtain imaging of a large area of V, a linear scanning motion to and fro at an angle to SU in the plane of incidence must be performed by the source and collimator relative to the double-crystal camera. Whether it is the source and collimator or the camera that physically move depends upon their relative portability. When the source is a standard sealed-off X-ray tube, it is not difficult to arrange for it to execute the motion (Milne, 1971). In some applications, it may occur that the specimen is so deformed that only a narrow strip of its surface will reflect at each ! setting. Then, a sequence of images can be superimposed on a single film, changing ! by a small step between each exposure. The `zebra' patterns so obtained define contours of equal èffective misorientation', the latter term describing the combined effect of variations in tilt ' and of Bragg-angle changes due to variations in interplanar spacing (Renninger, 1965; Jacobs & Hart, 1977). Double-crystal topography employing the parallel setting was developed independently by Bond & Andrus (1952) and by Bonse & Kappler (1958), and used by the former workers for studying reflections from surfaces of natural quartz crystals, and by the latter for detecting the strain fields surrounding outcrops of single dislocations at the surfaces of germanium crystals. Since then, the method has been much refined and widely applied. The detection of relative changes in interplanar spacing with a sensitivity of 10 8 is achievable using high-angle

Fig. 2.7.3.5. Transmission double-crystal topography in with spatial limitation of beam leaving reference crystal.

setting

reflections and very perfect crystals. These developments have been reviewed by Hart (1968, 1981). Transmitted Bragg reflection (i.e. the Laue case) can be used for either or both crystals U and V, in both the and settings, if desired. When the reference crystal U is used in transmission, a technique due to Chikawa & Austerman (1968), shown in Fig. 2.7.3.5, can be employed if U is relatively thick and, preferably, not highly absorbing of the radiation used. This technique exploits a property of diffraction by ideally perfect crystals, that, for waves satisfying the Bragg condition exactly, the energy-flow vector (Poynting vector) within the energy-flow triangle (the triangle ORT in Figs. 2.7.2.2 and 2.7.2.3) is parallel to the Bragg planes. (In fact, the energy-flow vectors swing through the triangle ORT as the range of Bragg reflection is swept by the incident wave vector, K0 .) Placing a slit Q as shown in Fig. 2.7.3.5 so as to transmit only those diffracted rays emerging from RT whose energy-flow direction in the crystal ran parallel, or nearly parallel, to the Bragg plane OD has the effect of selecting out from all diffracted rays only those that have zero or very small angular deviation from the exact Bragg condition. The slit Q thus provides an angularly narrower beam for studying the specimen crystal V than would be obtained if all diffracted rays from U were allowed to fall on V. The specimen is shown here in the setting, and also oriented to transmit its diffracted beam to the film F. This specimen arrangement is a likely embodiment of the technique but is incidental to the principle of employing spatial selection of transmitted diffraction rays to gain angular selection, a technique first used by Authier (1961). A practical limitation of this technique arises from angular spreading due to Fraunhofer diffraction by the slit Q: use of too fine an opening of Q will defeat the aim of securing an extremely angularly narrow beam for probing the specimen crystal. 2.7.4. Developments with synchrotron radiation 2.7.4.1. White-radiation topography The generation and properties of synchrotron X-rays are discussed by Arndt in Subsection 4.2.1.5. For reference, his list of important attributes of synchrotron radiation is here repeated as follows: (1) high intensity, (2) continuous spectrum, (3) narrow angular collimation, (4) small source size, (5) polarization, (6) regularly pulsed time structure, and (7) computability of properties. All these items influence the design and scope of X-ray topographic experiments with synchrotron radiation, in some cases profoundly. The high intensity of continuous radiation delivered in comparison with the output of standard X-ray tubes, and hence the rapidity with which X-ray topographs could be produced, was the first attribute to attract attention, through the pioneer experiments of Tuomi, Naukkarinen & Rabe (1974), and of Hart (1975a). They used the simple diffraction geometry of the Ramachandran (Fig. 2.7.1.2) and Schulz (Fig.2.7.1.1) methods, respectively. [Since in the transmissionspecimen case a multiplicity of Laue images can be recorded, it is usual to regard this work as a revival of the Guinier & Tennevin (1949) technique.] Subsequent developments in synchrotron X-ray topography have been reviewed by Tanner (1977) and by Kuriyama, Boettinger & Cohen (1982), and described in several chapters in Tanner & Bowen (1980). Some developments of methods and apparatus that have been stimulated by the advent of synchrotron-radiation sources will be described in this and in the following Subsection 2.7.4.2, the division illustrating two recognizable streams of development, the first exploiting the speed and relative instrumental simplicity

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2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION of white-radiation synchrotron X-ray topography, the second directed towards developing sophisticated `beam conditioners' to extract highly collimated and monochromatic beams from the continuous-wavelength output of the synchrotron source. In both monochromatic and continuous-radiation experiments, the high intensity renders it more practicable than with conventional sources to apply electronic `real-time' imaging systems (discussed in Sections 7.1.6 and 7.1.7, and Subsection 2.7.5.2). At the experimental stations where synchrotron X-ray topography is performed, the distance a from the source (the tangent point on the electron orbit) is never less than some tens of metres, e.g. 40 m at the Deutsches Elektronen-Synchrotron, Hamburg (DESY), a maximum of 80 m at the Synchrotron Radiation Source, Daresbury (SRS), and 140 m at the European Synchrotron Radiation Facility (ESRF). The dimensions of the X-ray source (given by the cross section of the electron beam at the tangent point) vary widely between different installations (see Table 4.2.1.7), but the dimension in the plane of the electron orbit is usually several times that normal to it. If Wx and Wz are the corresponding full widths at half-maximum intensity of the source, then with the simple X-ray optics of a white-radiation topograph the geometrical resolution will be Wx b=a and Wz b=a in the orbit-plane (horizontal) and normal to the orbit-plane (vertical) directions, respectively, independent of the orientation of the plane of incidence of the Bragg reflection concerned. Representative dimensions might be Wx 2 mm, Wz 0:5 mm, and a 50 m. With b 100 mm, the horizontal and vertical resolutions of the topograph image would then be 4 and 1 mm, respectively, comparable with those on a conventional source, but with b 10 mm only. Thus, even under synchrotron-source conditions, it is desirable that b should not exceed some centimetres in order to avoid geometrical factors causing a severer limitation of resolution (at least in one dimension) than other factors [such as photo-electron track lengths in the emulsion and point-by-point statistical fluctuations in absorbed photon dose (Lang, 1978)]. Since synchrotron X-rays are generated at all points along a curved electron trajectory, they spread out in a sheet parallel to the orbit plane. So there is in principle no limit to the specimen dimension in that plane that can be illuminated in a white-radiation topograph. However, increased background due to scattering from air and other sources imposes a practical limit of around 100 mm on the beam width. With electrons circulating in a planar orbit, the divergence of radiation normal to the orbit plane is strongly constricted, significant intensity being contained only within a fan of opening angle ' mc2 =E, e.g. 0:25 mrad with electron energy E 2 GeV, equivalent to a vertical distance 12 mm with a 50 m. This does impose a significant restriction on the area of specimen that can be imaged in a transmission topograph unless recourse be had to beam expansion by an asymmetrically reflecting monochromator crystal. For analysis of the three-dimensional configuration of defects within crystals, it is a useful feature of white-radiation transmission topography that different views of the specimen are presented simultaneously by the assemblage of Laue images, and that when studying reflection from a given Bragg plane there is freedom to vary the glancing angle upon it. When interpreting the diffraction contrast effects observed, the relative contributions of all the diffraction orders superimposed must be considered. However, after taking into account source spectral distribution, specimen structure factors, absorption losses and film efficiency, it is often found that a particular order of reflection is dominant in each Laue image (Tuomi, Naukkarinen & Rabe, 1974; Hart, 1975a). The variation of diffraction

contrast with wavelength follows different trends for different types of defect (Lang, Makepeace, Moore & Machado, 1983), so the ability to vary the wavelength with which a given order of reflection is studied can help in identifying the type of defect. If the orbiting electrons are confined to a plane, then the radiation emitted in that plane is completely linearly polarized with the E vector in that plane. It follows that diffraction with the plane of incidence normal to the orbit plane is in pure -polarization mode (polarization factor P 1), and with plane of incidence parallel to the orbit plane in pure polarization mode P j cos 2B j. The former, vertical plane of incidence is often chosen to avoid vanishing of reflections in the region of 2B 90 . The ability to record patterns with either pure -mode or pure -mode polarization is very helpful in the study of several dynamical diffraction phenomena. To facilitate switching of polarization mode, some diffractometers and cameras built for use with synchrotron sources are rotatable bodily about the incident-beam axis (Bonse & Fischer, 1981; Bowen, Clark, Davies, Nicholson, Roberts, Sherwood & Tanner, 1982; Bowen & Davies, 1983). From the diffractiontheoretical standpoint, it is the section topograph that provides the image of fundamental importance. High-resolution sectiontopograph patterns have been recorded with synchrotron radiation using a portable assembly combining crystal mount and narrow incident-beam slit. With the help of optical methods of alignment, this can be transferred between topograph cameras set up at a conventional source and at the synchrotron source (Lang, 1983). The regularly pulsed time structure of synchrotron radiation can be exploited in stroboscopic X-ray topography. The wavefronts of travelling surface acoustic waves (SAW) on lithium niobate crystals have been imaged, and their perturbation by lattice defects disclosed (Whatmore, Goddard, Tanner & Clark, 1982; Cerva & Graeff, 1984, 1985). The latter workers made detailed studies of the relative contributions to the image made by orientation contrast and by `wavefield deviation contrast' (i.e. contrast arising from deviation of the energy-flow vector in the elastically strained crystal). 2.7.4.2. Incident-beam monochromatization In order to achieve extremely small beam divergences and wavelength pass bands dl=l, and, in particular, to suppress transmission of harmonic wavelengths, arrangements much more complicated than the double-crystal systems shown in Figs. 2.7.3.1 and 2.7.3.3 have been applied in synchrotronradiation topography. The properties of monochromator crystals are discussed in Section 4.2.5. In synchrotron-radiation topographic applications, the majority of monochromators are constructed from perfect silicon, with occasional use of germanium. Damage-free surfaces of optical quality can be prepared in any orientation on silicon, and smooth-walled channels can be milled into silicon monoliths to produce multireflection devices. First, for simpler monochromatization systems, one possibility is to set up a monochromator crystal oriented for Bragg reflection with asymmetry b 1 (i.e. giving Wout =Win 1 to produce a narrow monochromatic beam with which section topographs can be taken (Mai, Mardix & Lang, 1980). The standard double-crystal topography arrangement is frequently used with synchrotron sources, the experimental procedure being as described in Section 2.7.3 and benefiting from the small divergence of the incident beam due to remoteness of the source. An example of a more refined angular probe is that obtainable by employing a pair of silicon crystals in setting to prepare the beam

120


2.7. TOPOGRAPHY Table 2.7.4.1. Monolithic monochromator for plane-wave synchrotron-radiation topography Reflection 1 Reflection 2 Reflection 3 Output wavelength Spectral pass band, dl=l Angular divergence of exit beam Size of exit beam

333 131 13 1 0.12378 nm 7 10 6 1:4 10 6 15 15 mm

incident on the specimen crystal, the three crystals together forming a arrangement (Ishikawa, Kitano & Matsui, 1985). The first monochromator is oriented for asymmetric 111 Bragg reflection, the second for highly asymmetric 553 reflection Wout =Win 64 at l 0:12 nm, resulting in a divergence of only 0:5 10 6 in the beam impinging on the specimen. Multireflection systems, some of which were proposed by Du Mond (1937) but not at that time realizable, have become a practicality through the advent of perfect silicon and germanium. When multiple reflection occurs between the walls of a channel cut in a perfect crystal, the tails of the curve of angular dependence of reflection intensity can be greatly attenuated without much loss of reflectivity at the peak of the curve (Bonse & Hart, 1965a). Beaumont & Hart (1974) described combinations of such `channel-cut' monochromators that were suitable for use with synchrotron sources. One combination, consisting of a pair of contra-rotating channel-cut crystals, with each channel acting as a pair of reflecting surfaces in symmetrical setting, has found much favour as a monochromatizing device producing neither angular deviation nor spatial displacement of the final beam, whatever the wavelength it is set to pass. The properties of monoliths with one or more channels and employing two or more asymmetric reflections in succession have been analysed by Kikuta & Kohra (1970), Kikuta (1971), and Matsushita, Kikuta & Kohra (1971). Symmetric channel-cut monochromators in perfect undistorted crystals transmit harmonic reflections. Several approaches to the problem of harmonic elimination may be taken, such as one of the following procedures (or possibly more than one in combination). (1) Using crystals of slightly different interplanar spacing (e.g. silicon and germanium) in the setting, which then becomes slightly dispersive (Bonse, Materlik & SchroÈder, 1976; Bauspiess, Bonse, Graeff & Rauch, 1977).

Fig. 2.7.4.1. Monolithic multiply reflecting monochromator for planewave topography.

(2) Laue case (transmission) followed by Bragg case (reflection), with deliberate slight misorientation between the diffracting elements (Materlik & Kostroun, 1980). (3) Asymmetric reflection in non-parallel channel walls in a monolith (Hashizume, 1983). (4) Misorientating a multiply reflecting channel, either one wall with respect to the opposite wall, or one length segment with respect to a following length segment (Hart & Rodrigues, 1978; Bonse, Olthoff-MuÈnter & Rumpf, 1983; Hart, Rodrigues & Siddons, 1984). For X-ray topographic applications, it is very desirable to have a spatially wide beam issuing from the multiply reflecting device. This is achieved, together with small angular divergence and spectral window, and without need of mechanical bending, in a monolith design by Hashizume, though it lacks wavelength tunability (Petroff, Sauvage, Riglet & Hashizume, 1980). The configuration of reflecting surfaces of this monolith is shown in Fig. 2.7.4.1. Reflection occurs in succession at surfaces 1, 2, and 3. The monochromator characteristics are listed in Table 2.7.4.1. The wavelength is very suitable in many topographic applications, and this design has proved to be an effective beam conditioner for use in synchrotron-radiation `plane-wave' topography. 2.7.5. Some special techniques 2.7.5.1. MoireÂ topography In X-ray optics, the same basic geometrical interpretation of moireÂ patterns applies as in light and electron optics. Suppose radiation passes successively through two periodic media, (1) and (2), whose reciprocal vectors are h1 and h2 , so as to form a moireÂ pattern. Then, the reciprocal vector of the moireÂ fringes will be H h1 h2 . The magnitude, D, of the moireÂ fringe spacing is jHj 1 and may typically lie in the range 0.1 to 1 mm in the case of X-ray moireÂ patterns. Simple special cases are the `rotation' moireÂ pattern in which jh1 j jh2 j d 1 , but h1 makes a small angle with h2 . Then, the spacing of the moireÂ fringes is d= and the fringes run parallel to the bisector of the small angle . The other special case is the `compression' moireÂ pattern. Here, h1 and h2 are parallel but there is a small difference between their corresponding spacings, d1 and d2 . The spacing D of compression moireÂ fringes is given by D d1 d2 =d1 d2 and the fringes lie parallel to the grating rulings or Bragg planes in (1) and (2). X-ray moireÂ topographs achieve sensitivies of 10 7 to 10 8 in measuring orientation differences or relative differences in interplanar spacing. Moreover, if either periodic medium contains a lattice dislocation, Burgers vector b, for which b h 6 0, then a magnified image of the dislocation will appear in the moireÂ pattern, as one or more fringes terminating at the position of the dislocation, the number of terminating fringes being b h, which is necessarily integral (Hashimoto & Uyeda, 1957). X-ray moireÂ topography has been performed with two quite different arrangements, the Bonse & Hart interferometer, and by superposition of separate crystals (BraÂdler & Lang, 1968). For accounts of the principles and applications of the interferometer, see, for example, Bonse & Hart (1965b, 1966), Hart (1968, 1975b), Bonse & Graeff (1977), Section 4.2.6 and x4.2.6.3.1. Fig. 2.7.5.1 shows the arrangement (Hart, 1968, 1972) for obtaining large-area moireÂ topographs by traversing the interferometer relative to a ribbon incident beam in similar fashion to taking a normal projection topograph (Fig. 2.7.2.2); P is the incident-beam slit, Q is a

121


2. DIFFRACTION GEOMETRY AND ITS PRACTICAL REALIZATION stationary slit selecting the beam that it is desired to record, and film, F, and interferometer, SMA, together traverse to and fro as indicated by the double-headed arrow. In Fig. 2.7.5.1, S, M, and A are the three equally thick wafers of the interferometer that remain upstanding above the base of the monolithic interferometer after the gaps between S and M, and M and A, have been milled away. The elements S, M and A are called the splitter, mirror, and analyser, respectively. The moireÂ pattern is formed between the Bragg planes of A and the standing-wave pattern in the overlapping K0 and Kh beams entering it. Maximum fringe visibility occurs in the emerging beam that the slit Q is shown selecting. A dislocation will appear in the moireÂ pattern whether the lattice dislocation lies in S, M, or A, provided b h 6 0. MoireÂ patterns formed in a number of Bragg reflections whose normals lie in, or not greatly inclined to, the plane of the wafers, can be recorded by appropriate orientation of the monolith. By this means, it is easily discovered in which wafer the dislocation lies, and its Burgers vector can be completely determined, including its sense, the latter being found by a deliberate slight elastic deformation of the interferometer (Hart, 1972). Satisfactory moireÂ topographs have been obtained with an interferometer in a synchrotron beam, despite thermal gradients due to the local intense irradiation (Hart, Sauvage & Siddons, 1980). Fig. 2.7.5.2 shows crystal slices (1), ABCD, and (2), EFGH, superposed and simultaneously Bragg reflecting in the BraÂdler± Lang (1968) method of X-ray moireÂ topography. The slices could have been cut from separate crystals. In the case when the Bragg planes of (1) and (2) are in identical orientation but have a translational mismatch across CD and EF with a component parallel to h, strong scattering occurs towards Z as focus, producing extra intensity at T 0 in the K0 beam TT 00 and at R0 in the Kh beam RR00 . It is usual to record the moireÂ pattern using the Kh beam. Projection moireÂ topographs are obtained by the standard procedure of traversing the crystal pair and film together with respect to the incident beam SO. The special procedure devised for mutually aligning the two crystals so that h1 and h2 coincide within their angular range of reflection is explained by BraÂdler & Lang (1968). This method has been applied to silicon and to natural (Lang, 1968) and synthetic quartz (Lang, 1978).

Fig. 2.7.5.1. Scanning arrangement for moireÂ topography with the Bonse±Hart interferometer.

2.7.5.2. Real-time viewing of topograph images Position-sensitive detectors involving the production of electrons are described in Chapter 7.1, Sections 7.1.6 and 7.1.7, and Arndt (1986, 1990). Those descriptions cover all the image-forming devices that form the core of systems set up for `live' X-ray topography. Here, discussion is limited to remarks on the historical development of techniques designed for making X-ray topographic images directly visible, and on the leading systems that are now sufficiently developed to be acceptable for routine use, in particular on topograph cameras set up at synchrotron X-ray sources. Two types of system became practicalities about the same time, that using direct conversion of X-rays to electronic signals by means of an X-ray-sensitive vidicon television camera tube (Chikawa & Fujimoto, 1968), and the indirect method using an external X-ray phosphor coupled to a multistage electronic image-intensifier tube (Reifsnider & Green, 1968; Lang & Reifsnider, 1969) or to a television-camera tube incorporating an image-intensifier stage (Meieran, Landre & O'Hara, 1969). These two approaches, the direct and the indirect, remain in competition. Developments up to the middle 1970's have been comprehensively reviewed by Hartman (1977). Since that time, Si-based, two-dimensional CCD (chargecoupled device) arrays have come into prominence as radiation detectors. They can be used for direct conversion of low-energy X-rays into electronic charges as well as for recording images of phosphor screens. As illustrated by Allinson (1994), four configurations employing CCD arrays for X-ray imaging can be considered: (i) direct detection by the `naked' device; (ii) detection by phosphor coated directly on the CCD array; (iii) phosphor separate, and optically coupled to the CCD by lens or fibre-optics; and (iv) the addition to (iii) of an image intensifier

Fig. 2.7.5.2. Superposition of crystals (1) and (2) for production of moireÂ topographs. [Reproduced from Diffraction and Imaging Techniques in Material Science, Vol. II. Imaging and Diffraction Techniques, edited by S. Amelinckx, R. Gevers & J. Van Landuyt (1978), Fig. 21, p. 695. Amsterdam, New York, Oxford: NorthHolland.]

122


2.7. TOPOGRAPHY or microchannel plate coupled to the phosphor screen. Consider first configuration (i). X-ray absorption efficiency in the active layer of silicon is near unity for radiations such as Cu K, and is not less than about 20% for Mo K or Ag K. Since about onethird of the absorbed energy goes into electron±hole pair production, an absorbed 8 keV X-ray photon creates about 2000 pairs, a large number compared with a combined darkcurrent plus read-out noise level per photosite of a few tens of electrons r.m.s. Thus, single-photon counting is possible. Moreover, a cooled CCD can integrate the charge accumulated in each pixel for up to 103 s. With pixel sizes in the range 10 to 30 mm square, and 1024 1024 (or 2048 2048 arrays in production, sensitive areas of 50 mm square, or greater, are available, sufficient for the majority of topographic applications. For X-ray-sensitive TV-camera tubes, some major improvements in resolution and sensitivity have taken place since the first applications to X-ray topography of Be-windowed vidicons. Using the more sensitive `Saticon' tube with incorporation of a 20 mm-thick Se±As target that provides good X-ray absorption efficiency, Chikawa, Sato & Fujimoto (1984) achieved a resolution as good as 6 mm at a modulation transfer function (MTF) of 5%. This betters that achieved with indirect systems or with standard-pixel-size CCD arrays used as direct detectors. The newer `Harp' tube, which employs avalanche multiplication produced by a high field ( 108 V m 1 ) applied across a 2 mm Se target to increase light sensitivity at least ten fold compared with the Saticon tube, has also been modified into a direct X-ray detector. Increasing the target thickness to 8 mm and adding an X-ray-transparent window provides satisfactory detector efficiency over a useful wavelength range (and the Se K-absorption edge at 0.098 nm causes absorption efficiencies for Cu K and Mo K to be similar, about 25%). The gain is sufficient for detection of single Cu K photon-absorption events (at least for photons absorbed close to the target front, giving the maximum path for avalanche formation). A limiting resolution of about 25 mm (at MTF of 33%) is exhibited (Sato, Maruyama, Goto, Fujimoto, Shidara, Kawamura, Hirai, Sakai & Chikawa, 1993), not yet as good as the 6 mm achieved with the Saticon tube. The rather small 6 9 mm sensitive areas of these camera tubes (when in standard `23 in' size) restricts their range of topographic applications as direct detectors compared with CCD arrays, but their amorphous Se targets are less likely to be degraded by X-radiation damage than crystalline-silicon CCD arrays. The latter do suffer degradation, but recover after treatment (Allinson, Allsopp, Quayle & Magorrian, 1991). In the case of indirect systems, the lens or fibre-optic plate situated between phosphor and detector automatically protects the latter from radiation damage. Somewhat better resolution can be achieved by lens coupling than by fibre-optic coupling of phosphor to detector, but at the expense of loss of light-collection efficiency generally too great to be acceptable. In principle, magnification or de-magnification of the phosphor-screen image on the detector can be selected according to whether phosphor or detector has the better resolution, in order to maximize the system resolution as a whole. Phosphor resolution can be increased by diminishing its thickness below the value that would

be chosen from consideration of X-ray absorption efficiency alone. Using a phosphor screen of Gd2 O2 S(Tb) only 5 mm thick, and lens-coupling it with tenfold magnification on to the target of a low-light-level television camera, Hartmann achieved a system resolution of about 10 mm (Queisser, Hartmann & Hagen, 1981), as good as any demonstrated so far with indirect systems. The phosphors already used (or potentially usable) in real-time X-ray topography are inorganic compounds containing elements of medium or heavy atomic weight. They include ZnS(Ag), NaI(Tl), CsI(Tl), Y2 O2 S(Tb), Y2 O2 S(Eu), La2 O2 S(Eu) and Gd2 O2 S(Tb). Problems encountered are light loss by light trapping within single-crystal phosphor sheets, and resolution loss by light scattering from grain to grain in phosphor powders. Various ways of reducing lateral light-spreading within phosphor screens by imposing a columnar structure upon them have been tried. Most success has been achieved with CsI. Evaporated layers of this crystal have a natural tendency towards columnar cracking normal to the substrate. Then internal reflection within columns reduces `cross talk' between columns (Stevels & KuÈhl, 1974). However, a columnar structure can be very effectively imposed on CsI films evaporated on to fibre-optic plates by etching away the cladding glass surrounding each fibre core to a depth of 10 mm, say. The evaporated CsI starts growing on the protruding cores, and continues as pillars physically separated and hence to a large degree optically separated from their neighbours (Ito, Yamaguchi & Oba, 1987; Allinson, 1994; Castelli, Allinson, Moon & Watson, 1994). The drive to develop systems for 2D imaging of singlecrystal or fibre diffraction patterns produced by synchrotron radiation that offer spatial resolution better than that within the grasp of position-sensitive multiwire gas proportional counters (say 100±200 mm) has produced several phosphor/fibre-optic/ CCD combinations that with some modifications would be useful for real-time X-ray topography. Diffraction-pattern recording requires a sensitive area not less than about 50 mm in diameter, so most systems incorporate a fibre-optic taper to couple a larger phosphor screen with a small CCD array. Spatial resolution in the X-ray image cannot then be better than CCD pixel size multiplied by the taper ratio. In one system that has been fully described, this product is 20.5 mm 2:6, and a point-spread FWHM of 80 mm on a 51 51 mm input area was realised without benefit from a columnar-structure phosphor (Tate, Eikenberry, Barna, Wall, Lawrance & Gruner, 1995). More appropriate for X-ray topography would be unitmagnification optical coupling of phosphor with a CCD array of not less than 1024 1024 elements and not more than 20 mm square pixel size. With such a combination, a system resolution of 25 mm should be achievable; and at a synchrotron X-ray topography station at least one device offering resolution no worse than this should be available. There is a scope for both high-resolution, small-sensitive-area and lower-resolution, large-sensitive-area imaging systems in real-time X-ray topography. It has been shown possible to incorporate both types in a single topography camera for use with synchrotron radiation (Suzuki, Ando, Hayakawa, Nittono, Hashizume, Kishino & Kohra, 1984).

123



2.8. Neutron diffraction topography By M. Schlenker and J. Baruchel

2.8.1. Introduction Some salient differences between neutron diffraction and X-ray diffraction are that a neutron beams are not available in standard (`home') laboratories, b the available neutron fluxes are small even at a high-flux reactor and even when compared with laboratory X-ray 1993), generators (Scherm & Fak, c absorption is negligible in most materials (see Section 4.4.6), and d magnetic scattering is a strong component (see Section 4.4.5). All these differences have effects on the use of neutrons for diffraction imaging (hereafter called, according to standard usage, neutron topography), while the obvious similarities in scattering amplitude and geometry make such topography possible. The effect of a is that the first attempts at neutron topography occurred late, with the work of Doi, Minakawa, Motohashi & Masaki (1971), Ando & Hosoya (1972), and Schlenker & Shull (1973), and that it is practised at very few places in the world, though one of them, at Institut Laue± Langevin (ILL), is open to external users. 2.8.2. Implementation As a result of b, the resolution of neutron topography is poor. It was estimated to be no better than 60 mm in non-polarized work on the instrument installed at ILL Grenoble, for exposure times of hours, as a result of roughly equal contributions from detector resolution, geometric blurring due to beam divergence, and shot noise, i.e. fluctuation in the number of diffracted neutrons reaching a pixel. The same reason leads to the technique being instrumentally simple because refinements that might lead, for example, to better resolution are discouraged by the increase in exposure time they would imply. Typically, a neutron beam with divergence of the order of 100 is monochromated by a nonperfect crystal (mosaic spread a few minutes of arc), and the monochromatic beam illuminates the sample, which can be either a single crystal or a grain in a polycrystal. It is advantageous, but not mandatory, to use a white beam delivered by a curved neutron guide tube as the divergence is already limited and high-energy parts of the spectrum, which would contribute to unwanted background, as well as -rays, are eliminated. After the specimen is set for a chosen Bragg reflexion with the help of a detector and counter, a neutronsensitive photographic detector (see x7.3.1.2.3) is placed across the diffracted beam, as near the sample as possible to minimize geometric blurring effects while avoiding the direct transmitted beam. Very crude but comparatively fast exposures can be made with Polaroid film and an isotopically enriched 6 LiF (ZnS) phosphor screen. Better topographs are obtained with X-ray film associated with a gadolinium foil (if possible isotopically enriched in 157 Gd) acting as an n ! converter, or with a track-etch plastic foil with an 6 LiF or 10 B4 C foil or layer (n ! converter) (Malgrange, Petroff, Sauvage, Zarka & Englander, 1976). Alternatively, an electronic position-sensitive neutron detector can be used for both setting and imaging (Davidson & Case, 1976; Sillou et al., 1989). Polarized neutrons are extremely useful in the investigation of magnetic domains. The use of a polarizing monochromator and a

crude attachment providing a guide field and the possibility to flip the polarization can provide this possibility as an option because the requirements are much less stringent than in quantitative structural polarized-neutron-diffraction work. It is also possible to use the white beam from a curved guide tube directly (Boeuf, Lagomarsino, Rustichelli, Baruchel & Schlenker, 1975), in the same way as in synchrotron-radiation X-ray topography, that is to say making a Laue diagram, each spot of which is a topograph. The technique is then instrumentally extremely simple, but background is a problem. Because the beam divergence is so much larger than for synchrotron radiation, the resolution is much worse than in the latter case, but it is not expected to differ significantly from the monochromatic beam neutron version. The ability of neutron beams to go through furnaces or cooling devices, one of the advantages in neutron diffraction work in general, is of course retained in topography. It is, however, desirable to retain a small ( 90. Contribution of deformation to the r.m.s. radius (the only parameter of importance to the atomic calculation) is roughly constant (0.11 fm) for Z > 90. There is an unknown region, between Bi and Th (83 < Z < 90), where deformation effects start to be important, but for which they are not known. When experiments are done for a particular isotope, we calculated separately the energies for each isotope. As mentioned in the introduction, there are special dif®culties involved when dealing with atoms with open outer shells (obviously this is the most common case). Computing all energies EJ for total angular momentum J would be both impossible and useless. The Dirac±Fock method circumvents this dif®culty. One can evaluate directly an average energy that corresponds to the barycentre of all EJ with weight (2J 1). There are still a few cases for which the average calculation cannot converge (when the open shells have identical symmetry). In that case, the outer electrons have been rearranged in an identical fashion for all hole states of the atom, to minimize possible shifts due to this procedure. 4.2.2.10. Correlation and Auger shifts Once the Dirac±Fock energy is obtained, many-body effects beyond Dirac±Fock relaxation must be taken into account. These include relaxation beyond the spherical average, correlation (due to both Coulomb and magnetic interaction), and corrections due to the autoionizing nature of hole states (Auger shift). Since the many-body generalization of the Dirac±Fock method, the socalled MCDF (multicon®guration Dirac±Fock), is very inef®cient for hole states, we turned to RMBPT to evaluate those quantities. These many-body effects contribute very signi®cantly to the ®nal value. Coulomb correlation is mostly constant along the Periodic Table (at the level of a few eV). Magnetic correlations are very strong at high Z. Auger shift is very important for p states. The interested reader will ®nd more details of these complicated calculations in the original references (Indelicato & Lindroth, 1992; Mooney et al., 1992; Lindroth & Indelicato, 1993; Indelicato & Lindroth, 1996). As these calculations are very time consuming, they are performed only for selected Z and interpolated. Since the Auger shifts do not always have a smooth Z dependence, care has been taken to evaluate them at as many different Z's as practical to ensure a good reproduction of irregularities. 4.2.2.11. QED corrections The QED corrections originate in the quantum nature of both the electromagnetic and electron ®elds. They can be divided in two categories, radiative and non-radiative. The ®rst one includes self-energy and vacuum polarization, which are the main contributions to the Lamb shift in one-electron atoms. These corrections scale as Z 4 =n3 (n being the principal quantum number) and are thus very important for inner shells and high Z. The second category is composed of corrections to the electron± electron interaction that cannot be accounted for by RMBPT or MCDF. These corrections start at the two-photon interaction and include three-body effects. The two-photon, non-radiative QED contribution has been calculated recently only for the ground

205

206 s:\ITFC\ch-4-2-1.3d (Tables of Crystallography)

4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê see text for explanation of typefaces Table 4.2.2.4. Wavelengths of K-emission lines and K-absorption edges in A; Numbers in parentheses are standard uncertainties in the least signi®cant ®gures. Z

Symbol

10

Ne

11

Na

12

Mg

13

Al

14

Si

15

P

16

S

17

Cl

18

Ar

19

K

20

Ca

21

Sc

22

Ti

23

V

24

Cr

25

Mn

26

Fe

27

Co

28

Ni

29

Cu

30

Zn

31

Ga

32

Ge

33

As

34

Se

35

Br

36

Kr

37

Rb

38

Sr

39

Y

40

Zr

41

Nb

42

Mo

A

K2

K1

14.6020(93) 14.6102(44) 11.9013(59) 11.9103(13) 9.8860(39) 9.89153(10) 8.3372(27) 8.341831(58) 7.1269(19) 7.12801(14) 6.1587(14) 6.1601(15) 5.3742(10) 5.374960(89) 4.72993(80) 4.730693(71) 4.19448(62) 4.194939(23) 3.74352(50) 3.7443932(68) 3.36223(39) 3.361710(44) 3.03479(33) 3.0344010(63) 2.75272(27) 2.7521950(57) 2.50798(23) 2.507430(30) 2.29428(19) 2.2936510(30) 2.10635(16) 2.1058220(30) 1.94043(14) 1.9399730(30) 1.79321(12) 1.7928350(10) 1.66199(10) 1.6617560(10) 1.544324(93) 1.54442740(50) 1.438963(84) 1.439029(12) 1.343987(72) 1.3440260(40) 1.257998(65) 1.258030(13) 1.179921(57) 1.179959(17) 1.108801(52) 1.108830(31) 1.043841(47) 1.043836(30) 0.984347(42) 0.9843590(44) 0.929713(39) 0.929704(15) 0.879444(36) 0.879443(15) 0.833059(32) 0.833063(15) 0.790181(30) 0.7901790(25) 0.750448(28) 0.750451(15) 0.713612(25) 0.713607(12)

14.6006(93) 14.6102(44) 11.8994(59) 11.9103(13) 9.8840(39) 9.889554(88) 8.3349(26) 8.339514(58) 7.1208(19) 7.125588(78) 6.1539(14) 6.1571(15) 5.3701(10) 5.372200(78) 4.72560(77) 4.727818(71) 4.19162(60) 4.191938(23) 3.74055(48) 3.7412838(56) 3.35911(38) 3.358440(44) 3.03129(31) 3.030854(14) 2.74886(26) 2.7485471(57) 2.50383(21) 2.503610(30) 2.29012(18) 2.2897260(30) 2.10210(15) 2.1018540(30) 1.93631(13) 1.9360410(30) 1.78919(11) 1.7889960(10) 1.658049(96) 1.6579300(10) 1.540538(85) 1.54059290(50) 1.435151(74) 1.435184(12) 1.340095(65) 1.3401270(96) 1.254054(58) 1.254073(13) 1.175932(52) 1.17595600(90) 1.104778(47) 1.104780(12) 1.039785(42) 1.039756(30) 0.980267(38) 0.9802670(40) 0.925597(35) 0.925567(13) 0.875298(32) 0.875273(15) 0.828875(29) 0.828852(15) 0.785960(27) 0.7859579(27) 0.746189(25) 0.746211(15) 0.709328(22) 0.70931715(41)

K 3

K 1

14.4522(74)

14.4522(74)

11.5752(30)

11.5752(30)

9.5211(30) 7.9412(49) 7.9601(30) 6.7317(26) 6.7531(15) 5.7834(16) 5.7961(30) 5.0202(12)

9.5211(30)

K I2

7.9601(30)

4.39810(99) 4.40347(44) 3.88506(71) 3.88606(30) 3.45189(69) 3.45395(30) 3.08855(45) 3.08975(30) 2.77919(50) 2.77964(30) 2.51445(43) 2.513960(30) 2.28567(37) 2.284446(30) 2.08702(32) 2.0848810(40) 1.91175(28) 1.9102160(40) 1.75784(25) 1.7566040(40) 1.62166(22) 1.6208260(30) 1.50059(19) 1.5001520(30) 1.39246(17) 1.3922340(60) 1.29544(17) 1.295276(30) 1.20821(13) 1.208390(75) 1.12924(13) 1.12938(13) 1.05774(11) 1.057898(76) 0.992646(96) 0.992689(79) 0.933275(87) 0.933284(74) 0.878967(81) 0.8790110(70) 0.829174(71) 0.829222(44) 0.783413(63) 0.783462(44) 0.741232(58) 0.741271(44) 0.702296(53) 0.7023554(30) 0.666266(49) 0.666350(44) 0.632900(44) 0.632887(13)

206


K II 2

6.7531(15) 5.7914(27) 5.7961(30) 5.0246(15) 5.03168(30) 4.40038(77) 4.40347(44) 3.88486(70) 3.88606(30) 3.45216(58) 3.45395(30) 3.08827(45) 3.08975(30) 2.77809(49) 2.77964(30) 2.51262(45) 2.513960(30) 2.28332(40) 2.284446(30) 2.08478(35) 2.0848810(40) 1.90960(31) 1.9102160(40) 1.75617(27) 1.7566040(40) 1.62039(24) 1.6208260(30) 1.49964(21) 1.5001520(30) 1.39201(18) 1.3922340(60) 1.29506(16) 1.295276(30) 1.20774(14) 1.207930(34) 1.12877(13) 1.128957(30) 1.05724(11) 1.057368(33) 0.992152(95) 0.992189(53) 0.932768(84) 0.932804(30) 0.878495(75) 0.8785220(50) 0.828681(67) 0.828692(30) 0.782911(58) 0.782932(30) 0.740716(53) 0.740731(30) 0.701766(48) 0.7018008(30) 0.665721(44) 0.665770(30) 0.632345(38) 0.632303(13)

1.283739(30) 1.195547(25) 1.196018(30) 1.116387(37) 1.116877(30) 1.044699(56) 1.045016(44) 0.979618(57) 0.979935(74) 0.920344(49) 0.920474(30) 0.866209(36) 0.86611(15) 0.816459(33) 0.816462(44) 0.770774(33) 0.770822(44) 0.728801(27) 0.728651(59) 0.690079(28) 0.689940(59) 0.654328(31) 0.654170(59) 0.621162(35) 0.620999(30)

1.283739(30) 1.196018(30) 1.116877(30) 1.044836(20) 1.045016(44) 0.979716(26) 0.979935(74) 0.920390(28) 0.920474(30) 0.866169(35) 0.86611(15) 0.816408(33) 0.816462(44) 0.770718(20) 0.770822(44) 0.728663(21) 0.728651(59) 0.689895(21) 0.689940(59) 0.654078(22) 0.654170(59) 0.620941(21) 0.620999(30)

K abs. edge 14.2391(26) 14.30201(15) 11.4784(16) 11.5692(15) 9.4479(10) 9.51234(15) 7.89928(67) 7.948249(74) 6.70091(46) 6.7381(15) 5.75537(33) 5.7841(15) 4.99591(24) 5.01858(15) 4.37679(18) 4.39717(15) 3.86552(14) 3.870958(74) 3.42856(11) 3.43655(15) 3.061828(87) 3.07035(15) 2.754176(71) 2.7620(15) 2.490681(59) 2.497377(74) 2.263194(49) 2.269211(21) 2.067898(41) 2.070193(14) 1.892275(36) 1.8964592(58) 1.739918(31) 1.7436170(49) 1.605127(27) 1.6083510(42) 1.485300(24) 1.4881401(36) 1.379448(23) 1.3805971(31) 1.282346(20) 1.2833798(40) 1.194711(18) 1.19582(15) 1.115585(16) 1.116597(74) 1.043925(16) 1.04502(15) 0.978818(15) 0.979755(15) 0.919501(13) 0.92041(15) 0.865324(12) 0.865533(15) 0.815270(12) 0.815552(74) 0.769359(11) 0.769742(74) 0.727270(10) 0.7277514(21) 0.6884893(99) 0.6889591(31) 0.6528690(93) 0.6531341(14) 0.6196481(87) 0.61991006(62)

4.2. X-RAYS Table 4.2.2.4. Wavelengths of K-emission lines and K-absorption edges in AÊ (cont.) Z

Symbol

43

Tc

44

Ru

45

Rh

46

Pd

47

Ag

48

Cd

49

In

50

Sn

51

Sb

52

Te

53

I

54

Xe

55

Cs

56

Ba

57

La

58

Ce

59

Pr

60

Nd

61

Pm

62

Sm

63

Eu

64

Gd

65

Tb

66

Dy

67

Ho

68

Er

69

Tm

70

Yb

71

Lu

72

Hf

73

Ta

74

W

75

Re

A

K2

K1

K 3

K 1

K II 2

K I2

0.679318(24) 0.679330(44) 0.647415(22) 0.6474205(61) 0.617652(21) 0.6176458(61) 0.589822(20) 0.5898351(60) 0.563804(18) 0.5638131(26) 0.539426(18) 0.5394358(46) 0.516551(17) 0.5165572(60) 0.495060(16) 0.4950646(46) 0.474840(15) 0.4748391(45) 0.455795(14) 0.4557908(44) 0.437834(13) 0.437836(10) 0.420879(42) 0.42088103(71) 0.404848(13) 0.4048411(59) 0.389684(12) 0.38968378(74) 0.375320(11) 0.3753186(30) 0.361685(11) 0.3616884(30) 0.348755(10) 0.3487542(30) 0.336473(10) 0.33647921(73) 0.3247982(98) 0.3248079(59) 0.3136913(94) 0.31369830(79) 0.3031139(91) 0.3031225(30) 0.2930400(89) 0.2930424(30) 0.2834212(86) 0.2834273(30) 0.2742462(84) 0.2742511(30) 0.2654851(81) 0.26549088(84) 0.2571059(79) 0.2571133(11) 0.2490952(77) 0.24910095(61) 0.2414274(75) 0.2414276(30) 0.2340857(73) 0.2340845(30) 0.2270507(72) 0.2270274(44) 0.2203039(70) 0.2203083(59) 0.2138327(69) 0.21383304(50) 0.2076150(67) 0.2076141(15)

0.675017(21) 0.675030(44) 0.643088(20) 0.6430994(61) 0.613305(18) 0.6132937(61) 0.585459(18) 0.5854639(46) 0.559420(17) 0.55942178(76) 0.535020(15) 0.5350147(46) 0.512124(15) 0.5121251(46) 0.490612(14) 0.4906115(46) 0.470373(13) 0.4703700(45) 0.451310(13) 0.4513018(44) 0.433330(12) 0.4333245(74) 0.416358(40) 0.4163508(14) 0.400310(11) 0.4002960(59) 0.385129(11) 0.38512464(84) 0.370748(10) 0.3707426(30) 0.3570964(97) 0.3570974(30) 0.3441494(94) 0.3441452(30) 0.3318514(91) 0.33185689(62) 0.3201607(88) 0.3201648(59) 0.3090384(84) 0.30904506(46) 0.2984457(81) 0.2984505(30) 0.2883568(79) 0.2883573(30) 0.2787234(76) 0.2787242(30) 0.2695341(74) 0.2695370(30) 0.2607589(72) 0.2607608(42) 0.2523659(71) 0.25237359(62) 0.2443415(68) 0.24434486(44) 0.2366603(67) 0.2366586(30) 0.2293053(65) 0.2293014(30) 0.2222572(64) 0.2222303(44) 0.2154977(63) 0.2155002(59) 0.2090134(61) 0.20901314(18) 0.2027835(60) 0.2027840(30)

0.601881(40) 0.601889(59) 0.573053(37) 0.5730816(42) 0.546191(34) 0.5462139(42) 0.521117(29) 0.5211363(41) 0.497673(29) 0.4976977(60) 0.475739(27) 0.4757401(71) 0.455178(25) 0.4551966(41) 0.435878(24) 0.4358821(51) 0.417736(22) 0.4177477(41) 0.400664(21) 0.4006650(59) 0.384576(20) 0.3845698(59) 0.369407(40) 0.3694051(13) 0.355067(17) 0.3550553(59) 0.341517(16) 0.3415228(11) 0.328692(16) 0.3286909(59) 0.316507(15) 0.3165248(59) 0.304970(14) 0.3049796(74) 0.294021(13) 0.2940366(40) 0.283620(13) 0.283634(59) 0.273732(12) 0.273764(30) 0.264322(12) 0.2643360(74) 0.255371(11) 0.255344(30) 0.246818(11) 0.246834(30) 0.238671(10) 0.238624(30) 0.230896(10) 0.230834(30) 0.2234662(97) 0.2234766(14) 0.2163665(94) 0.216366(30) 0.2095741(93) 0.20960(15) 0.2030802(88) 0.203093(59) 0.1968603(86) 0.196863(59) 0.1908986(83) 0.1908929(30) 0.1851834(81) 0.18518317(70) 0.1796955(79) 0.1796997(44)

0.601318(35) 0.601309(59) 0.572478(32) 0.5724966(42) 0.545606(29) 0.5456189(42) 0.520514(27) 0.5205333(41) 0.497069(25) 0.4970817(60) 0.475124(23) 0.4751181(71) 0.454552(22) 0.4545616(41) 0.435241(20) 0.4352421(51) 0.417089(19) 0.4170966(31) 0.400008(18) 0.4000010(44) 0.383910(17) 0.3839108(59) 0.368730(38) 0.3687346(13) 0.354385(16) 0.354369(10) 0.340826(15) 0.34082708(75) 0.327993(14) 0.3279879(44) 0.315795(14) 0.3158207(30) 0.304249(13) 0.3042656(59) 0.293290(13) 0.2933086(40) 0.282880(12) 0.282904(44) 0.272984(12) 0.273014(30) 0.263567(11) 0.2635810(74) 0.254610(11) 0.254604(30) 0.246054(11) 0.246084(30) 0.237902(10) 0.237884(30) 0.230122(10) 0.230124(30) 0.2226875(98) 0.22269866(72) 0.2155833(95) 0.21559182(57) 0.2087863(95) 0.208843(30) 0.2022872(90) 0.202313(44) 0.1960622(88) 0.196073(44) 0.1900954(86) 0.1900919(59) 0.1843751(83) 0.1843768(30) 0.1788824(81) 0.1788827(44)

0.590423(40) 0.590249(74) 0.561748(44) 0.561668(44) 0.535110(48) 0.535038(30) 0.510283(46) 0.5102357(59) 0.487060(55) 0.4870393(59) 0.465335(62) 0.465335(10) 0.445014(58) 0.445007(15) 0.426120(12) 0.425921(12) 0.408017(57) 0.4079791(74) 0.391161(56) 0.3911079(89) 0.375286(54) 0.375236(30) 0.360326(72) 0.360265(44) 0.346197(49) 0.346115(30) 0.33285(14) 0.332775(15) 0.32023(13) 0.320122(10) 0.30827(12) 0.308165(15) 0.29694(11) 0.296794(30) 0.28619(10) 0.28610(15) 0.275992(98) 0.27590(15) 0.266459(91) 0.26620(15) 0.257069(85) 0.256927(12) 0.248289(23) 0.248164(44) 0.239916(21) 0.23970(30) 0.231955(12) 0.23170(30) 0.224320(18) 0.22410(30) 0.217046(16) 0.21670(30) 0.210099(15) 0.20980(30) 0.203456(14) 0.20330(30) 0.197101(13) 0.19690(30) 0.191017(13) 0.19080(30) 0.185143(12) 0.185191(13) 0.179595(12) 0.179603(15) 0.174234(11) 0.174253(15)

0.590231(22) 0.590249(74) 0.561587(22) 0.561668(44) 0.534977(22) 0.535038(30) 0.510177(51) 0.5102357(59) 0.487019(38) 0.4870393(59) 0.465346(28) 0.465335(10) 0.445011(27) 0.445007(15) 0.425928(26) 0.425921(12) 0.408004(25) 0.4079791(74) 0.391135(27) 0.3911079(89) 0.375234(29) 0.375236(30) 0.36034(12) 0.360265(44) 0.346102(37) 0.346115(30) 0.332728(12) 0.332775(15) 0.320101(11) 0.320122(10) 0.308131(10) 0.308165(15) 0.2967952(99) 0.296794(30) 0.2860408(94) 0.28610(15) 0.2758335(91) 0.27590(15) 0.2661277(87) 0.26620(15) 0.2569028(81) 0.256927(12) 0.2481186(76) 0.248164(44) 0.2397496(75) 0.23970(30) 0.2318190(53) 0.23170(30) 0.2241536(66) 0.22410(30) 0.2168806(64) 0.21670(30) 0.2099331(62) 0.20980(30) 0.2032912(59) 0.20330(30) 0.1969329(58) 0.19690(30) 0.1908468(56) 0.19080(30) 0.1849702(54) 0.185014(12) 0.1794215(52) 0.179424(10) 0.1740571(51) 0.1740566(89)

207


K abs. edge 0.5889852(84) 0.589069(15) 0.5603122(81) 0.560518(15) 0.5337192(74) 0.5339086(69) 0.5090158(75) 0.5091212(42) 0.4857609(74) 0.4859155(57) 0.4640026(71) 0.4641293(35) 0.4435977(70) 0.4437454(48) 0.4244611(68) 0.4245978(29) 0.4064886(65) 0.4066324(27) 0.3895899(64) 0.389746(15) 0.3736775(61) 0.373816(15) 0.358683(27) 0.35841(74) 0.3444778(59) 0.344515(15) 0.3310639(56) 0.331045(15) 0.3184025(55) 0.318445(74) 0.3065382(54) 0.306485(74) 0.2952418(53) 0.295184(74) 0.2845288(52) 0.284534(74) 0.2743634(53) 0.274314(74) 0.2647027(51) 0.264644(74) 0.2555123(51) 0.255534(15) 0.2467265(48) 0.246814(15) 0.2384335(49) 0.238414(15) 0.2304867(46) 0.230483(15) 0.2229099(45) 0.222913(15) 0.2156719(45) 0.2156801(75) 0.2087587(44) 0.208803(74) 0.2021481(43) 0.202243(74) 0.1957973(42) 0.195853(74) 0.1897176(42) 0.189823(74) 0.1838657(41) 0.183943(15) 0.1783098(41) 0.178373(15) 0.1729509(40) 0.173023(15)

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.2.4. Wavelengths of K-emission lines and K-absorption edges in AÊ (cont.) Z

Symbol

A

76

Os

77

Ir

78

Pt

79

Au

80

Hg

81

Tl

82

Pb

83

Bi

209

84

Po

209

85

At

210

86

Rn

222

87

Fr

223

88

Ra

226

89

Ac

227

90

Th

232

91

Pa

231

92

U

238

93

Np

237

94

Pu

244

95

Am

243

96

Cm

248

97

Bk

249

98

Cf

250

99

Es

251

100

Fm

254

K2

K1

K 3

K 1

K II 2

K I2

0.2016443(66) 0.2016420(30) 0.1959045(65) 0.1959069(30) 0.1903859(61) 0.1903839(59) 0.1850702(64) 0.18507664(61) 0.1799628(61) 0.1799607(44) 0.1750380(60) 0.1750386(30) 0.1702924(59) 0.17029527(56) 0.1657170(58) 0.1657183(20) 0.1613031(58) 0.161302(15) 0.1570444(56) 0.157052(30) 0.1529334(56) 0.152942(44) 0.1489599(56) 0.148962(44) 0.1451209(54) 0.145119(20) 0.1414083(54) 0.141412(30) 0.1378266(53) 0.13782600(31) 0.1343514(52) 0.1343516(29) 0.1309879(52) 0.13099111(78) 0.1277298(51) 0.1277287(39) 0.1245763(50) 0.1245705(25) 0.1215172(50) 0.1215158(24) 0.1185536(49) 0.1185427(23) 0.1156777(49) 0.1156630(54) 0.1128873(48) 0.1128799(82) 0.1101788(47) 0.1102072(98) 0.1075497(47) 0.107514(14)

0.1968007(59) 0.1967970(30) 0.1910492(57) 0.1910499(30) 0.1855187(55) 0.1855138(59) 0.1801914(57) 0.18019780(47) 0.1750720(54) 0.1750706(44) 0.1701355(53) 0.1701386(30) 0.1653781(53) 0.16537816(38) 0.1607911(52) 0.1607903(46) 0.1563656(51) 0.156362(15) 0.1520953(50) 0.152102(30) 0.1479727(49) 0.147982(44) 0.1439878(50) 0.143992(44) 0.1401373(48) 0.140132(19) 0.1364131(47) 0.136419(12) 0.1328194(47) 0.13282021(36) 0.1293324(46) 0.1293302(27) 0.1259572(46) 0.12595977(36) 0.1226871(45) 0.1226882(36) 0.1195212(45) 0.1195140(23) 0.1164501(45) 0.1164463(33) 0.1134742(44) 0.1134635(21) 0.1105860(43) 0.1105745(49) 0.1077832(43) 0.1077793(75) 0.1050620(43) 0.1050554(89) 0.1024201(42) 0.102386(13)

0.1744279(77) 0.1744336(44) 0.1693667(75) 0.1693695(30) 0.1645026(72) 0.1645035(44) 0.1598202(73) 0.1598249(13) 0.1553217(69) 0.1553233(44) 0.1509866(68) 0.1509823(89) 0.1468107(67) 0.1468129(10) 0.1427865(65) 0.142780(11) 0.1389056(63) 0.138922(30) 0.1351623(62) 0.135172(59) 0.1315499(61) 0.131552(74) 0.1280599(60) 0.128072(74) 0.1246890(58) 0.124689(15) 0.1214301(57) 0.121432(30) 0.1182861(56) 0.11828686(78) 0.1152364(55) 0.1152427(21) 0.1122860(53) 0.11228858(66) 0.1094299(52) 0.1094230(39) 0.1066627(51) 0.1066611(18) 0.1039811(51) 0.1039794(17) 0.1013837(50) 0.1013753(17) 0.0988636(48) 0.0988598(55) 0.0964130(47) 0.0963915(83) 0.0940403(46) 0.094036(14) 0.0917379(45) 0.091715(10)

0.1736101(79) 0.1736136(44) 0.1685444(77) 0.1685445(30) 0.1636756(74) 0.1636775(44) 0.1589887(75) 0.15899527(77) 0.1544857(72) 0.1544893(44) 0.1501462(70) 0.1501443(74) 0.1459663(68) 0.14596836(58) 0.1419372(66) 0.1419492(54) 0.1380520(65) 0.138072(30) 0.1343044(63) 0.134322(59) 0.1306882(61) 0.130692(74) 0.1271937(61) 0.127192(74) 0.1238185(59) 0.123815(15) 0.1205554(58) 0.120552(30) 0.1174071(56) 0.11740759(59) 0.1143530(55) 0.1143583(21) 0.1113979(54) 0.11140132(65) 0.1085378(53) 0.1085265(28) 0.1057661(52) 0.1057595(18) 0.1030805(51) 0.1030803(17) 0.1004790(50) 0.1004708(16) 0.0979546(49) 0.0979514(54) 0.0955000(48) 0.0954860(90) 0.0931231(47) 0.093090(14) 0.0908165(46) 0.0907943(98)

0.169085(11) 0.169103(15) 0.164150(11) 0.164152(15) 0.1593872(99) 0.159392(15) 0.1548206(99) 0.154832(30) 0.1504204(94) 0.150402(30) 0.1461874(92) 0.146142(15) 0.1421118(88) 0.142122(30) 0.1381841(87) 0.138172(15) 0.1343966(85) 0.134382(30) 0.1307448(83) 0.130722(59) 0.1272218(79) 0.127192(74) 0.1238183(79) 0.123792(74) 0.1205312(77) 0.120535(14) 0.1173552(73) 0.117322(30) 0.1142910(71) 0.114262(15) 0.1113088(69) 0.111292(30) 0.1084449(67) 0.108372(15) 0.1056621(66) 0.105670(31) 0.1029688(64) 0.1029724(26) 0.1003579(63) 0.1003537(24) 0.0978295(63) 0.0978355(23) 0.0953724(61)

0.1689066(50) 0.1689085(89) 0.1639697(51) 0.163958(10) 0.1592048(46) 0.159202(15) 0.1546363(48) 0.154620(13) 0.1502334(46) 0.150202(30) 0.1459989(77) 0.145952(15) 0.1419201(75) 0.141912(15) 0.1379910(72) 0.137972(15) 0.1342012(69) 0.134182(30) 0.1305470(67) 0.130522(59) 0.1270211(66) 0.126982(74) 0.1236157(63) 0.123582(74) 0.1203271(60) 0.120320(14) 0.1171477(59) 0.117112(30) 0.1140810(57) 0.114042(13) 0.1110964(56) 0.111072(30) 0.1082301(54) 0.108182(15) 0.1054450(53) 0.105457(31) 0.1027494(52) 0.1027429(26) 0.1001364(51) 0.1001357(24) 0.0976059(50) 0.0975952(15) 0.0951469(49) 0.0942501(50) 0.0927593(48) 0.0927508(84) 0.0904543(47)

state of two-electron ions (Blundell, Mohr, Johnson & Sapirstein, 1993; Lindgren, Persson, Salomonson & Labzowsky, 1995) and cannot be evaluated in practice for atoms with more than two or three electrons. The radiative corrections split up into two contributions. The ®rst contribution is composed of one-electron radiative corrections (self-energy and vacuum polarization). For the self-energy and Z > 10, one must use all-order calculations (Mohr, 1974a,b, 1975, 1982, 1992; Mohr & Soff, 1993). Vacuum polarization can be evaluated at the Uehling (1935) and Wichmann & Kroll (1956) level. Higher-order effects

0.0884443(59) 0.0882127(45) 0.0884212(100) 0.0881872(99)

0.1678092(40) 0.167873(15) 0.1628853(39) 0.162922(15) 0.1581346(38) 0.158182(15) 0.1535699(40) 0.1535953(74) 0.1491786(38) 0.149182(15) 0.1449460(37) 0.144952(15) 0.1408707(37) 0.1408821(74) 0.1369439(37) 0.136942(15) 0.1331589(36) 0.1295098(36) 0.1259898(35) 0.1225852(36) 0.1192985(35) 0.1161246(34) 0.1130642(34) 0.113072(15) 0.1101087(34) 0.1072452(33) 0.107232(15) 0.1044744(33) 0.1044605(62) 0.1017982(33) 0.0991999(33) 0.0966801(33) 0.0942405(32) 0.0918695(32) 0.091862(10) 0.0895840(32) 0.0895878(97) 0.0873575(32) 0.0873356(80)

are much smaller than for the self-energy (Soff & Mohr, 1988) and have been neglected. The second contribution is composed of radiative corrections to the electron±electron interaction, and scales as Z 3 =n3 . Ab initio calculations have been performed only for few-electron ions (Indelicato & Mohr, 1990, 1991). Here we use the Welton approximation which has been shown to reproduce very closely ab initio results in all examples that have been calculated (Indelicato, Gorceix & Desclaux 1987; Indelicato & Desclaux 1990; Kim, Baik, Indelicato & Desclaux, 1991; Blundell, 1993a,b).

208


0.0929867(61) 0.0929715(82) 0.0906838(60)

K abs. edge

4.2. X-RAYS Ê see text for explanation of typefaces Table 4.2.2.5. Wavelengths of L-emission lines and L-absorption edges in A; Numbers in parentheses are standard uncertainties in the least signi®cant ®gures. Z

Symbol

20

Ca

21

Sc

22

Ti

23

V

24

Cr

25

Mn

26

Fe

27

Co

28

Ni

29

Cu

30

Zn

31

Ga

32

Ge

33

As

34

Se

35

Br

36

Kr

37

Rb

38

Sr

39

Y

40

Zr

41

Nb

42

Mo

43

Tc

44

Ru

45

Rh

46

Pd

47

Ag

48

Cd

49

In

50

Sn

A

L1

L2 36.331(30) 30.947(46) 31.350(44) 27.215(37) 27.420(30) 24.143(30) 24.250(44) 21.640(24) 21.640(44) 19.390(20) 19.450(15) 17.525(17) 17.590(30) 15.922(14) 15.9722(89) 14.532(12) 14.5612(44) 13.341(10) 13.3362(44) 12.2529(90) 12.2542(44) 11.2916(77) 11.2922(15) 10.4371(68) 10.4363(12) 9.6744(60) 9.6710(12) 8.9914(52) 8.99013(74) 8.3776(46) 8.37473(74) 7.8242(41) 7.82032(13) 7.3226(37) 7.32521(44) 6.8674(33) 6.86980(44) 6.4539(30) 6.45590(44) 6.0766(27) 6.0766(27) 5.7326(24) 5.73199(44) 5.4151(22) 5.41445(12) 5.1228(20) 4.8541(18) 4.85388(10) 4.6055(16) 4.60552(13) 4.3753(15) 4.37595(10) 4.1623(14) 4.163002(74) 3.9644(13) 3.965020(89) 3.7802(12) 3.780787(89) 3.6084(11) 3.606964(59)

36.331(30) 31.350(44) 27.420(30) 24.250(44) 21.490(11) 21.640(44) 19.359(21) 19.450(15) 17.503(17) 17.590(30) 15.905(15) 15.9722(89) 14.520(12) 14.5612(44) 13.336(11) 13.3362(44) 12.2489(90) 12.2542(44) 11.2858(78) 11.2922(15) 10.4306(68) 10.4363(12) 9.6680(60) 9.6710(12) 8.9852(52) 8.99013(74) 8.3715(46) 8.37473(74) 7.8180(41) 7.82032(13) 7.3164(36) 7.31841(30) 6.8610(32) 6.86290(30) 6.4466(29) 6.44890(30) 6.0684(26) 6.070250(79) 5.7226(23) 5.72439(30) 5.4054(21) 5.40663(12) 5.1139(19) 5.11488(44) 4.8449(17) 4.845823(74) 4.5966(16) 4.59750(13) 4.3672(15) 4.367736(74) 4.1541(13) 4.154492(44) 3.9560(12) 3.956409(59) 3.7716(11) 3.771977(59) 3.5997(10) 3.599994(44)

L 1

L 2

35.941(30) 30.587(47) 31.020(30) 26.843(37) 27.050(30) 23.764(30) 23.880(59) 21.276(24) 21.270(15) 19.036(20) 19.110(30) 17.194(17) 17.260(15) 15.610(14) 15.666(12) 14.236(12) 14.2712(89) 13.063(10) 13.0532(44) 11.9819(93) 11.9832(44) 11.0226(78) 11.0232(30) 10.1717(69) 10.1752(15) 9.4126(59) 9.4142(12) 8.7335(52) 8.73593(74) 8.1233(46) 8.12522(74) 7.5736(40) 7.574441(98) 7.0749(36) 7.07601(44) 6.6224(33) 6.62400(44) 6.2110(29) 6.21209(44) 5.8357(26) 5.836214(76) 5.4931(23) 5.49238(44) 5.1778(21) 5.17716(12) 4.8880(19) 4.8874(12) 4.6210(17) 4.620649(44) 4.3744(16) 4.374206(59) 4.1461(14) 4.146282(74) 3.9347(13) 3.934789(44) 3.7382(12) 3.738286(59) 3.5553(11) 3.555363(59) 3.38472(100) 3.384921(44)

* These values are for the unresolved L 2 and L 15 emission lines.

209


LI abs. edge 28.275(32)

19.779(19)

26.953(14) 27.290(15) 23.8561(89)

27.3105(36) 27.290(15) 24.206(10)

17.804(15) 16.70(15) 16.113(19)

21.246(18) 17.90(15) 19.0781(57)

21.5867(49) 20.70(15) 19.4063(43)

14.611(34)

17.2248(92) 17.2023(74) 15.627(14) 15.6182(74) 14.251(23) 14.2422(74) 13.016(14) 13.0142(15) 11.8652(66) 11.8622(15) 10.8414(29) 10.8282(74) 9.9340(27) 9.9241(15) 9.1182(17) 9.1251(15) 8.4105(58) 8.4071(15) 7.7669(35) 7.7531(74) 7.1630(21) 7.1681(15) 6.6449(59) 6.6441(15) 6.17624(70) 6.1731(15) 5.75742(82) 5.7561(15) 5.3773(15) 5.3781(15) 5.03480(63) 5.0311(15) 4.72145(60) 4.7191(15) 4.4368(13) 4.4361(15) 4.17814(78) 4.1801(15) 3.94053(55) 3.94256(74) 3.72251(52) 3.72286(15) 3.51704(26) 3.51645(15) 3.32528(29) 3.32575(15) 3.14784(47) 3.14735(15) 2.98309(56) 2.98234(15)

17.5402(35) 17.5253(74) 15.9290(44) 15.9152(74) 14.5396(57) 14.5252(74) 13.2934(64) 13.2882(15) 12.134(14) 12.1312(15) 11.1040(29) 11.1002(15) 10.1849(46) 10.1872(15) 9.3649(29) 9.3671(15) 8.64459(77) 8.6461(15) 7.9991(30) 7.9841(74) 7.3841(17) 7.3921(15) 6.8643(67) 6.8621(15) 6.38937(84) 6.3871(15) 5.9658(15) 5.9621(15) 5.5816(15) 5.5791(15) 5.23529(98) 5.2301(15) 4.9179(31) 4.9131(15) 4.62991(94) 4.6301(15) 4.36776(32) 4.3691(15) 4.12730(50) 4.12996(74) 3.90655(62) 3.90746(15) 3.69817(53) 3.69996(15) 3.50348(45) 3.50475(15) 3.32322(42) 3.32375(15) 3.15521(70) 3.15575(15)

22.099(24)

13.4000(86) 12.295(13) 11.292(16)

5.23798(44)* 4.91857(74) 4.92327(30)* 4.6341(13) 4.3681(13) 4.37187(30)* 4.1277(12) 4.13106(30)* 3.9088(10) 3.908929(59)* 3.7034(10) 3.703406(44)* 3.51355(97) 3.514133(59)* 3.33796(83) 3.338430(44)* 3.17475(77) 3.175098(44)*

LIII abs. edge 35.7704(68) 35.491(15) 31.109(36)

24.896(15)

5.58638(44)*

LII abs. edge 35.384(40) 35.131(15) 30.718(17)

10.361(12) 13.060(15) 9.518(11) 9.5171(74) 8.775(12) 8.7731(15) 8.092(13) 8.1071(15) 7.498(13) 7.5031(15) 6.958(14) 6.9591(74) 6.4561(41) 6.470(15) 6.0010(11) 6.0081(74) 5.5945(16) 5.5921(74) 5.22968(53) 5.2171(74) 4.89881(41) 4.8791(74) 4.59975(43) 4.5751(74) 4.32423(40) 4.3041(74) 4.0581(74) 3.8443(16) 3.8351(74) 3.6334(17) 3.6291(74) 3.43948(61) 3.4371(15) 3.25639(29) 3.25645(15) 3.08443(17) 3.08495(15) 2.92533(19) 2.92604(15) 2.776792(71) 2.77694(15)

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.2.5. Wavelengths of L-emission lines and L-absorption edges in AÊ (cont.) Z

Symbol

51

Sb

52

Te

53

I

54

Xe

55

Cs

56

Ba

57

La

58

Ce

59

Pr

60

Nd

61

Pm

62

Sm

63

Eu

64

Gd

65

Tb

66

Dy

67

Ho

68

Er

69

Tm

70

Yb

71

Lu

72

Hf

73

Ta

74

W

75

Re

76

Os

77

Ir

78

Pt

79

Au

80

Hg

81

Tl

82

Pb

A

L2 3.44794(99) 3.448452(89) 3.29788(92) 3.29851(13) 3.15734(85) 3.157957(89) 3.02568(78) 3.025940(22) 2.90167(73) 2.90204(30) 2.78522(68) 2.785572(74) 2.67563(64) 2.675383(60) 2.57122(59) 2.57059(18) 2.47329(55) 2.47294(44) 2.38081(51) 2.38079(52) 2.29340(48) 2.29263(59) 2.21054(48) 2.210430(24) 2.13214(42) 2.13156(17) 2.05817(40) 2.05783(30) 1.98699(37) 1.98753(30) 1.91986(35) 1.919939(44) 1.85606(33) 1.856472(15) 1.79537(31) 1.795701(45) 1.73758(29) 1.738003(19) 1.68248(29) 1.682875(74) 1.63031(26) 1.630314(74) 1.58049(25) 1.580484(74) 1.53290(23) 1.532953(30) 1.48748(22) 1.487452(30) 1.44399(21) 1.443982(74) 1.40238(20) 1.402361(74) 1.36252(19) 1.362520(74) 1.32434(18) 1.324340(30) 1.28773(17) 1.287739(44) 1.25261(16) 1.25266(10) 1.21890(15) 1.218768(44) 1.18651(15) 1.186498(74)

L1

L 1

L 2

3.43913(93) 3.439462(59) 3.28894(86) 3.289249(89) 3.14828(81) 3.148647(89) 3.01640(76) 3.016582(15) 2.89237(69) 2.89244(30) 2.77580(64) 2.775992(74) 2.66607(60) 2.665740(74) 2.56108(56) 2.56163(17) 2.46280(52) 2.46304(30) 2.36999(48) 2.370526(16) 2.28227(45) 2.28223(44) 2.19926(42) 2.199873(13) 2.12081(40) 2.120673(95) 2.04670(37) 2.04683(30) 1.97586(35) 1.97653(30) 1.90883(33) 1.908839(44) 1.84511(31) 1.845092(17) 1.78449(29) 1.784481(20) 1.72677(27) 1.7267720(70) 1.67177(26) 1.671915(59) 1.61949(24) 1.619534(44) 1.56959(23) 1.569604(74) 1.52194(22) 1.521993(30) 1.47642(21) 1.4763112(95) 1.43288(19) 1.432922(59) 1.39121(18) 1.391231(74) 1.35130(19) 1.351300(44) 1.31308(17) 1.313060(44) 1.27643(16) 1.276419(44) 1.24126(15) 1.241219(74) 1.20750(14) 1.207408(59) 1.17507(14) 1.175028(30)

3.22551(92) 3.225718(59) 3.07663(85) 3.076816(89) 2.93720(78) 2.937484(89) 2.80659(69) 2.806553(19) 2.68362(66) 2.68374(30) 2.56812(61) 2.568249(74) 2.45941(57) 2.458947(74) 2.35598(53) 2.35580(18) 2.25890(49) 2.25883(44) 2.16724(45) 2.167008(19) 2.08060(42) 2.07973(59) 1.99850(42) 1.998432(30) 1.92080(37) 1.92053(17) 1.84744(34) 1.84683(30) 1.77701(32) 1.77683(44) 1.71052(30) 1.71065(10) 1.64732(28) 1.647484(32) 1.58720(26) 1.587466(86) 1.52995(24) 1.5302410(70) 1.47538(24) 1.475672(74) 1.42361(21) 1.423611(44) 1.37419(20) 1.374121(74) 1.32697(19) 1.327000(44) 1.28188(18) 1.281812(13) 1.23872(17) 1.238599(30) 1.19742(16) 1.197288(74) 1.15786(15) 1.157827(44) 1.11995(14) 1.119917(30) 1.08359(13) 1.083546(44) 1.04869(13) 1.048696(74) 1.01519(12) 1.015145(59) 0.98298(11) 0.982925(44)

3.02325(67) 3.023395(44)* 2.88209(61) 2.88221(12)* 2.75031(54) 2.75057(12)* 2.62740(47)


210


2.51216(47) 2.51184(30)* 2.40421(26) 2.404386(89)* 2.30307(24) 2.303312(98)* 2.20843(21) 2.20900(17)* 2.11936(20) 2.11943(59)* 2.03554(18) 2.035448(88)* 1.95675(18) 1.95593(89)* 1.88225(17) 1.882206(41)* 1.81237(16) 1.81215(17)* 1.74582(14) 1.74553(30)* 1.68377(14) 1.68303(30)* 1.62497(12) 1.62371(10)* 1.56818(11) 1.567168(50)* 1.51486(10) 1.51401(13)* 1.464210(95) 1.46402(30)* 1.416041(89) 1.415521(74)* 1.370061(85) 1.370141(44) 1.326241(78) 1.326410(74) 1.282314(74) 1.284559(30) 1.244447(70) 1.2443048(98) 1.206487(67) 1.206618(59) 1.170095(62) 1.16981(12) 1.135812(72) 1.135337(44) 1.102006(63) 1.102017(44) 1.070479(53) 1.070236(44) 1.039584(51) 1.03977(10) 1.01029(20) 1.010325(44) 0.98221(19) 0.98222(10)

LI abs. edge

LII abs. edge

LIII abs. edge

2.638437(69) 2.63884(15) 2.50998(50) 2.50994(15) 2.38965(37) 2.38804(74) 2.273869(70) 2.27373(15) 2.1676(29) 2.16733(74) 2.0697(15) 2.06783(74) 1.97705(28) 1.97803(74) 1.89320(71) 1.89343(74) 1.81477(33) 1.81413(74) 1.73904(18) 1.73903(15)

2.82990(51) 2.82944(74) 2.687685(87) 2.68794(15) 2.55532(31) 2.55424(74) 2.427862(95) 2.42924(15) 2.3135(17) 2.31393(15) 2.20482(12) 2.20483(15) 2.10317(10) 2.10533(74) 2.01084(14) 2.01243(74) 1.92607(36) 1.92553(74) 1.84373(16) 1.84403(15)

2.99986(66) 3.00035(15) 2.85523(35) 2.85554(15) 2.72067(32) 2.71964(74) 2.590303(89) 2.59264(15) 2.47326(16) 2.47404(15) 2.363082(97) 2.36294(15) 2.25958(20) 2.2610(15) 2.16586(39) 2.1660(15) 2.07945(22) 2.07913(74) 1.99616(19) 1.99673(15)

1.66743(74) 1.60201(12) 1.60022(15) 1.54065(17) 1.53812(15) 1.47922(25) 1.47842(15) 1.42285(98) 1.42232(15) 1.37058(41) 1.36922(15) 1.31957(28) 1.31902(15) 1.27145(14) 1.27062(15) 1.22612(28) 1.22502(15) 1.18266(60) 1.18182(15) 1.14043(22) 1.14022(15) 1.10009(24) 1.1002640(49) 1.06152(30) 1.06132(15) 0.91604(28) 1.024685(74) 0.98968(21) 0.98941(15) 0.95583(36) 0.95581(15) 0.9240(12) 0.92361(15) 0.8933(14) 0.893213(19) 0.86383(45) 0.863683(30) 0.83546(43) 0.83531(15) 0.80795(15) 0.80811(15) 0.78172(24) 0.7818404(49)

1.76763(74) 1.69495(13) 1.69533(15) 1.62830(21) 1.62712(15) 1.56264(23) 1.56322(15) 1.50195(80) 1.50232(15) 1.44500(20) 1.44452(15) 1.39091(27) 1.39052(15) 1.33792(26) 1.33862(15) 1.28942(27) 1.28922(15) 1.243391(70) 1.24282(15) 1.197954(60) 1.19852(15) 1.1550(10) 1.1548587(22) 1.11368(14) 1.11372(15) 1.07431(38) 1.07452(15) 1.03670(20) 1.03712(15) 1.000786(57) 1.00142(15) 0.96675(18) 0.96711(15) 0.93395(27) 0.9341861(21) 0.90263(12) 0.9027409(46) 0.87238(26) 0.87221(15) 0.843512(77) 0.84341(15) 0.81575(18) 0.8157395(16)

1.91913(15) 1.84534(42) 1.84573(15) 1.77767(16) 1.77613(15) 1.71092(21) 1.71173(15) 1.65023(44) 1.64972(15) 1.59241(33) 1.59162(15) 1.53614(34) 1.53682(15) 1.48318(27) 1.48352(15) 1.43366(27) 1.43342(15) 1.3858(10) 1.38622(15) 1.341053(93) 1.34052(15) 1.2972(14) 1.2971383(68) 1.25506(34) 1.25532(15) 1.21543(99) 1.21552(15) 1.17673(27) 1.17732(15) 1.14002(23) 1.14082(15) 1.10535(22) 1.10582(15) 1.07200(36) 1.0722721(19) 1.04009(27) 1.0401625(52) 1.00919(30) 1.00912(15) 0.97953(25) 0.97931(15) 0.95113(22) 0.9511590(22)

4.2. X-RAYS Table 4.2.2.5. Wavelengths of L-emission lines and L-absorption edges in AÊ (cont.) Z

Symbol

83

Bi

84

Po

85

At

86

Rn

87

Fr

88

Ra

89

Ac

90

Th

91

Pa

92

U

93

Np

94

A

L2

L1

L 1

L 2

Pu

209 1.15540(14) 1.155377(15) 209 1.12549(13) 1.125497(74) 210 1.09670(13) 1.096726(74) 222 1.06900(12) 1.069006(74) 223 1.04232(11) 1.042316(74) 226 1.01662(11) 1.016575(74) 227 0.99185(11) 0.991795(74) 232 0.96798(10) 0.9679082(23) 231 0.944896(96) 0.944834(74) 238 0.922622(93) 0.922572(13) 237 0.901230(88) 0.901059(13) 244 0.880355(85)

1.14390(13) 1.143877(30) 1.11393(12) 1.113877(59) 1.08510(12) 1.085016(74) 1.05735(11) 1.057246(74) 1.03063(11) 1.030505(74) 1.00489(10) 1.004745(74) 0.980070(98) 0.979945(74) 0.956154(94) 0.9560826(15) 0.933002(90) 0.932854(74) 0.910674(86) 0.910653(13) 0.889223(83) 0.889141(13) 0.868290(79)

0.95205(11) 0.951992(13) 0.92228(10) 0.92201(30) 0.893639(96) 0.89350(13) 0.866054(91) 0.86606(13) 0.839482(86) 0.83941(13) 0.813866(82) 0.813762(74) 0.789163(78) 0.78904(13) 0.765343(75) 0.7652610(14) 0.742301(71) 0.742331(74) 0.720056(68) 0.719995(12) 0.698624(65) 0.698488(13) 0.677776(60)

0.79354(13) 0.7935516(15) 0.77321(12) 0.77371(15) 0.75462(12) 0.754692(13) 0.73623(11) 0.736241(13) 0.71848(11)

95

Am

243 0.860288(84)

0.848190(81)

0.657686(59)

0.70134(10)

96

Cm

248 0.840918(80)

0.828776(78)

0.638265(56)

0.684815(98)

97

Bk

249 0.822159(76)

0.809987(69)

0.619449(53)

0.668638(94)

98

Cf

250 0.803608(73)

0.791421(66)

0.601005(50)

0.652873(89)

99

Es

251 0.786043(70)

0.773837(63)

0.583354(49)

0.638227(82)

100

Fm

254 0.769077(67) 0.76904(62)

0.756843(60) 0.75674(60)

0.566272(47) 0.56619(34)

0.623826(82) 0.62369(41)

0.95526(18) 0.955194(59) 0.92932(18) 0.929384(74) 0.90444(17)

LI abs. edge 0.75649(58) 0.75711(15) 0.7332(13)

LII abs. edge 0.789102(88) 0.78871(15) 0.76325(13)

LIII abs. edge 0.92387(11) 0.92341(15) 0.897554(85)

0.73868(13)

0.88055(15)

0.71511(13)

0.85751(15) 0.8580(30) 0.83533(16) 0.835383(74) 0.81406(14)

0.69240(13)

0.8251(27)

0.64449(15) 0.64451(15)

0.67077(12) 0.67071(15) 0.64970(13)

0.802768(44) 0.80281(15)

0.60569(11) 0.60591(15) 0.58759(12)

0.62966(11) 0.62991(15) 0.610354(92)

0.760637(99) 0.76071(15) 0.740958(97)

0.569885(39) 0.56951(15)

0.591930(66) 0.59191(15)

0.722319(52) 0.72231(15)

0.55239(34)

0.57368(37)

0.704136(20)

0.53651(15)

0.55721(15)

0.68671(15)

0.49060(49)

0.50851(52)

0.63748(98)

0.476569(92)

0.493804(98)

0.62300(19)

0.44966(13)

0.46534(12)

0.59414(20)


4.2.2.12. Structure and format of the summary tables Table 4.2.2.4 summarizes the theoretical and experimental results for prominent K-series lines and the K-absorption edge. For the emission lines, the upper number (in italics) is the theoretical estimate for this line and the lower number is the experimentally measured value (1) from Table 4.2.2.1 or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to the optically based scale. For the K absorption edge, the upper number (also in italics) was obtained by combining emission lines and photoelectron spectroscopy (see Subsection 4.2.2.7), and the lower number is the experimentally measured value (1) from Table 4.2.2.3 or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. For the experimental emission and absorption entries, bold type is used for wavelengths directly measured on an optically based scale. Ê = The numerical values for wavelengths in angstrom units (1 A 0.1 nm) are given to a number of signi®cant ®gures commensurate with their estimated uncertainties, which appear in parentheses after each theoretical and experimental value. Figure 4.2.2.1 shows plots of the relative deviation between theoretical and experimental values for the K-series lines and the K-absorption edge as a function of Z. The error bars shown in the

®gure are the experimental uncertainties. In general, these plots show good agreement between theory and experiment except in the low-Z and high-Z regions. At the low-Z end of the table, the particular calculational approach used is not optimum, and the experimental data are suprisingly weak. At the high end, experimental data have rather large uncertainties, and thus do not provide an accurate test of the theory. Table 4.2.2.5 summarizes the theoretical and experimental results for prominent L-series lines and the L-absorption edges. The experimental database of high-accuracy emission data is much more limited than was the case for the K series, and there have been very few high-accuracy edge-location measurements. The format of this table is similar to that of Table 4.2.2.4. For the emission lines, the upper number (in italics) is the theoretical estimate for this line, and the lower number is the experimentally measured value. Numbers in bold type were directly measured on the optical scale (see Table 4.2.2.2), and numbers in normal type are from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. For the L-absorption edges, the upper number (also in italics) is obtained by combining emission lines and photoelectron spectroscopy (see Subsection 4.2.2.7) and the lower number is the experimentally measured value. The numbers in bold type

211


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.2.6. Wavelength conversion factors Numbers in parentheses are standard uncertainties in the leastsigni®cant ®gures.

Ê l A Ê l A*) l kxu Ê A Ê A*= Ê kxu=A

Cu K1

Mo K1

W K1

1.54059292(45) 1.540562(3) 1.537400 1.0000201(20) 1.00207683(29)

0.70931713(41) 0.709300(1) 0.707831 1.0000242(22) 1.00209955(58)

0.20901313(18) 0.2090100

data resource to the scienti®c community that can be more easily up-dated and expanded.

4.2.2.14. Connection with scales used in previous literature

1.00001498(86)

are recent measurements by Kraft, StuÈmpel, Becker & Kuetgens (1996), and the numbers in normal type are from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. Figure 4.2.2.2 shows relative deviations between the theoretical and experimental values for most of the tabulated data. The error bars shown in the ®gure are the experimental uncertainties. 4.2.2.13. Availability of a more complete X-ray wavelength table This article and the accompanying X-ray wavelength tables are an up-dated version of the contribution to the International Tables for Crystallography, Volume C, 2nd edition that was published in 1999. This article has been subject to more critical review and analysis and the data are consistent with the most recent adjustment of the fundamental physical constants (Mohr & Taylor, 2000). We believe that these data represent a signi®cant improvement in consistency, coverage and accuracy over previously available resources. The results presented here are a subset of a larger effort that includes all K- and L-series lines connecting the n = 1 to n = 4 shells. The more complete table has been submitted for archival publication and will be made avaialble on the NIST Physical Reference Data web site. Electronic publishing of this resource will provide a convenient

Fig. 4.2.2.1. Relative deviations between theoretical and experimental results for K-series spectra. The topmost data set concerns the K-edge location, while the other data sets, beginning at the bottom, refer to the K2 , K1 , K 3 and K 1 , respectively. The ordinate scales have been displaced for clarity by the indicated multiples of 0.001.

In order to compare historical data for X-ray spectra with the results in the present tabulation, certain conversion factors are needed. As discussed in the introduction, the principal units Ê unit. There is the found in the literature are the xu and the A* additional complication that there were several different de®nitions in use at various times and at the same time in different laboratories. For the convenience of the reader, we summarize in Table 4.2.2.6 the main conversion factors needed. Ê can be converted The numerical values for the wavelengths in A to energies in electron volts by using the conversion factor Ê (Mohr & Taylor, 2000). 12 398.41857 (49) eV A Our current efforts owe their inception to the encouragement of the late A. J. C. Wilson, who persistently communicated the need for an updated wavelength resource for the crystallographic community. The larger effort evolved at NIST with the support of the Standard Reference Data Program as established with the help of the late Jean Gallagher, and sustained by the program's current Director, John Rumble. Early phases of the preparation of this material bene®ted from the efforts of John Schweppe. Cedric Powell supplied valuable advice in the area of electron binding energies. We are particularly grateful to the Editor, E. Prince, for his help and patience in the development of these wavelength tables. Richard Deslattes died between the ®rst publication of this article and this revision. This work would not have been possible without his dedication to this project over more than a decade. The earlier wavelength table of the late J. A. Bearden, under whom one of the present authors (RDD) studied, was a signi®cant in¯uence on this project.

Fig. 4.2.2.2. Comparison of L-series data with experiment for the indicated range of Z. Indicated data, beginning at the bottom, refer to the L2 , L1 , and L 1 emission lines and the LIII , LII , and LI absorption edges. For clarity, the plots have been displaced vertically by multiples of 0.002 for the emission lines and 0.004 for the absorption edges.

212


4.2. X-RAYS 4.2.3. X-ray absorption spectra (by D. C. Creagh) 4.2.3.1. Introduction 4.2.3.1.1. De®nitions This section deals with the manner in which the photon scattering and absorption cross sections of an atom varies with the energy of the incident photon. Further information concerning these cross sections and tables of the X-ray attenuation coef®cients are given in Section 4.2.4. Information concerning the anomalous-dispersion corrections is given in Section 4.2.6. When a highly collimated beam of monoenergetic photons passes through a medium of thickness t, it suffers a decrease in intensity according to the relation I I0 exp l t;

4:2:3:1

where l is the linear attenuation coef®cient. Most tabulations express l in c.g.s. units, l having the units cm 1 . An alternative, often more convenient, way of expressing the decrease in intensity involves the measurement of the mass per unit area mA of the specimen rather than the specimen thickness, in which case equation (4.2.3.1) takes the form I I0 exp =mA ;

The magnitudes of these scattering cross sections depend on the type of atom involved in the interactions and on the energy of the photon with which it interacts. In Fig. 4.2.3.1, the theoretical cross sections for the interaction of photons with a carbon atom are given. Values of pe are from calculations by Scho®eld (1973), and those for Rayleigh and Compton scattering are from tabulations by Hubbell & éverbù (1979) and Hubbell (1969), respectively. Note the sharp discontinuities that occur in the otherwise smooth curves. These correspond to photon energies that correspond to the energies of the K and LI LII LIII shells of the carbon energies. Notice also that pe is the dominant interaction cross section, and that the Rayleigh scattering cross section remains relatively constant for a broad range of photon energies, whilst the Compton scattering peaks at a particular photon energy 100 keV). Other interaction mechanisms exist [e.g. DelbruÈck (Papatzacos & Mort, 1975; Alvarez, Crawford & Stevenson, 1958), pair production, nuclear Thompson], but these do not become signi®cant interaction processes for photon energies less than 1 MeV. This section will not address the interaction of photons with atoms for which the photon energy exceeds 100 keV.

4:2:3:2

where is the density of the material and = is the mass absorption coef®cient. The linear attenuation coef®cient of a medium comprising atoms of different types is related to the mass absorption coef®cients by P l gi =i ; 4:2:3:3 i

where gi is the mass fraction of the atoms of the ith species for which the mass absorption coef®cient is =i . The summation extends over all the atoms comprising the medium. For a crystal having a unit-cell volume of Vc , 1X 4:2:3:4 l i ; Vc where i is the photon scattering and absorption cross section. If i is expressed in terms of barns=atom then Vc must be expressed Ê 3 and l is in cm 1 . (1 barn 10 28 m2 .) in terms of A The mass attenuation coef®cient = is related to the total photon±atom scattering cross section according to cm2 =g NA =M cm2 =atom 4:2:3:5 NA =M 10 24 barns=atom;

4.2.3.1.3. Normal attenuation, XAFS, and XANES The curves shown in Fig. 4.2.3.1 are the result of theoretical calculations of the interactions of an isolated atom with a single photon. Experiments are not usually performed on isolated atoms, however. When experiments are performed on ensembles of atoms, a number of points of difference emerge between the experimental data and the theoretical calculations. These effects arise because the presence of atoms in proximity with one another can in¯uence the scattering process. In short: the total attenuation coef®cient of the system is not the sum of all the individual attenuation coef®cients of the atoms that comprise the system. Perhaps the most obvious manifestation of this occurs when the photon energy is close to an absorption edge of an atom. In Fig. 4.2.3.2, the mass attenuation of several germanium compounds is plotted as a function of photon energy. The

where NA Avogadro's number 6:0221367 36 1023 atoms=gram atom (Cohen & Taylor, 1987) and M atomic weight relative to M12 C 12:0000. 4.2.3.1.2. Variation of X-ray attenuation coef®cients with photon energy When a photon interacts with an atom, a number of different absorption and scattering processes may occur. For an isolated atom at photon energies of less than 100 keV (the limit of most conventional X-ray generators), contributions to the total cross section come from the photo-effect, coherent (Rayleigh) scattering, and incoherent (Compton) scattering. pe R C :

4:2:3:6

The relation between the photo-effect absorption cross section pe and the X-ray anomalous-dispersion corrections will be discussed in Section 4.2.6.

Fig. 4.2.3.1. Theoretical cross sections for photon interactions with carbon showing the contributions of photoelectric, elastic (Rayleigh), inelastic (Compton), and pair-production cross sections to the total cross sections. Also shown are the experimental data (open circles). From Gerstenberg & Hubbell (1982).

213


4. PRODUCTION AND PROPERTIES OF RADIATIONS energy scale measures the distance from the K-shell edge energy of germanium (11.104 keV). These curves are taken from Hubbell, McMaster, Del Grande & Mallett (1974). Not only does the experimental curve depart signi®cantly from the theoretically predicted curve, but there is a marked difference in the complexity of the curves between the various germanium compounds. Far from the absorption edge, the theoretical calculations and the experimental data are in reasonable agreement with what one might expect using the sum rule for the various scattering cross sections and one could say that this region is one in which normal attenuation coef®cients may be found. Closer to the edge, the almost periodic variation of the mass attenuation coef®cient is called the extended X-ray absorption ®ne structure (XAFS). Very close to the edge, more complicated ¯uctuations occur. These are referred to as X-ray absorption near edge ®ne structure (XANES). The boundary of the XAFS and XANES regions is somewhat arbitrary, and the physical basis for making the distinction between the two will be outlined in Subsection 4.2.3.4. Even in the region where normal attenuation may be thought to occur, cooperative effects can exist, which can affect both the Rayleigh and the Compton scattering contributions to the total attenuation cross section. The effect of cooperative Rayleigh scattering has been discussed by Gerward, Thuesen, StibiusJensen & Alstrup (1979), Gerward (1981, 1982, 1983), Creagh & Hubbell (1987), and Creagh (1987a). That the Compton scattering contribution depends on the physical state of the scattering medium has been discussed by Cooper (1985). Care must therefore be taken to consider the physical state of the system under investigation when estimates of the theoretical interaction cross sections are made.

Fig. 4.2.3.2. The dependence of the X-ray attenuation coef®cient on energy for a range of germanium compounds, taken in the neighbourhood of the germanium absorption edge (from IT IV, 1974).

4.2.3.2. Techniques for the measurement of X-ray attenuation coef®cients 4.2.3.2.1. Experimental con®gurations Experimental con®gurations that set out to determine the X-ray linear attenuation coef®cient l or the corresponding mass absorption coef®cients = must have characteristics that re¯ect the underlying assumptions from which equation (4.2.3.1) was derived, namely: (i) the incident and transmitted beams are parallel and there is no divergence in the transmitted beam; (ii) the photons in the incident and transmitted beams have the same energy; (iii) the specimen is of suf®cient thickness. Because of the considerable discrepancies that often exist in X-ray attenuation measurements (see, for example, IT IV, 1974), the IUCr Commission on Crystallographic Apparatus set up a project to determine which, if any, of the many techniques for the measurement of X-ray attenuation coef®cients is most likely to yield correct results. In the project, a number of different experimental con®gurations were used. These are shown in Fig. 4.2.3.3. The con®gurations used ranged in complexity from that

Fig. 4.2.3.3. Schematic representations of experimental apparatus used in the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987; Creagh, 1985). X: characteristic line from sealed X-ray tube; b: Bremsstrahlung from a sealed X-ray tube; r: radioactive source; s: synchrotron-radiation source; : -®lter for characteristic X-rays; S: collimating slits; M: monochromator.

214


4.2. X-RAYS of Fig. 4.2.3.3(a), which uses a slit-collimated beam from a sealed tube and a -®lter to select its characteristic radiation, and a proportional counter and associated electronics to detect the transmitted-beam intensity, to that of Fig. 4.2.3.3( f ), which uses a modi®cation to a commercial X-ray-¯uorescence analyser. Sources of X-rays included conventional sealed X-ray tubes, X-ray-¯uorescence sources, radioisotope sources, and synchrotron-radiation sources. Detectors ranged from simple ionization chambers, which have no capacity for photon energy detection, to solid-state detectors, which provide a relatively high degree of energy discrimination. In a number of cases (Figs. 4.2.3.3c, d, e, and f ), monochromatization of the beam was effected using single Bragg re¯ection from silicon single crystals. In Fig. 4.2.3.3(i), the incident-beam monochromator is using re¯ections from two Bragg re¯ectors tuned so as to eliminate harmonic radiation from the source. The performance of these systems was evaluated for a range of materials that included: (i) highly perfect silicon single crystals (Creagh & Hubbell, 1987); (ii) polycrystalline copper foils that exhibited a high degree of preferred orientation; and (iii) pyrolytic graphite that contained a high density of regular voids. The results of this study are outlined in Section 4.2.3.2.3. 4.2.3.2.2. Specimen selection Although the most important component in the experiment is the specimen itself, examination of the data ®les held at the US National Institute of Standards and Technology (Gerstenberg & Hubbell, 1982; Saloman & Hubbell, 1986; Hubbell, Gerstenberg & Saloman, 1986) has shown that, in general, insuf®cient care has been taken in the past to select an experimental device with characteristics that are appropriate to the specimen chosen. Nor has suf®cient care been taken in the determination of the dimensions, homogeneity, and defect structure of the specimens. To achieve the best results, the following procedures should be followed. (i) The dimensions of the specimen should be determined using at least two different techniques, and sample thicknesses should be chosen such that the Nordfors (1960) criterion, later con®rmed by Sears (1983), that the condition 2 lnI0 =I 4

4:2:3:7

be satis®ed. This enables the best compromise between achieving good counting statistics and avoiding multiple photon scattering within the sample. Wherever possible, different sample thicknesses should be chosen to enable a test of equation (4.2.3.1) to be made. If deviations from equation (4.2.3.1) exist, either the sample material or the experimental con®guration, or both, are not appropriate for the measurement of l . If the attenuation of the material under test falls outside the limits set by the Nordfors criterion and the material is in the form of a powder, the mixing of this powder with one with low attenuation and no absorption edge in the region of interest can be used to bring the total attenuation of the sample within the Nordfors range. (ii) The sample should be examined by as many means as possible to ascertain its regularity, homogeneity, defect structure, and, especially for very thin specimens, freedom from pinholes and cracks. Where a diluent has been used to reduce the attenuation so that the Nordfors criterion is satis®ed, care must be taken to ensure intimate mixing of the two materials and the absence of voids.

Since the theory upon which equation (4.2.3.1) is based envisages that each atom scatters as an individual, it is necessary to be aware of whether such cooperative effects as Laue±Bragg scattering (which may become signi®cant in single-crystal specimens) and small-angle X-ray scattering (SAXS) (which may occur if a distribution of small voids or inclusions exists) occur in polycrystalline and amorphous specimens. Knowledge that cooperative scattering may occur in¯uences the choice of collimation of the beam. (iii) The sample should be mounted normal to the beam. 4.2.3.2.3. Requirements for the absolute measurement of l or = The following prescription should be followed if accurate, absolute measurements of l and = are to be obtained. (i) X-ray source and X-ray monochromatization. The energy of the incident photons should be measured directly using re¯ections from a single-crystal silicon monochromator, and the energy spread of the beam should be measured. Measurements should be made of the state of polarization, since X-raypolarization effects are known to be signi®cant in some measurements (Templeton & Templeton, 1982, 1985, 1986). The results of a survey on X-ray polarization were given by Jennings (1984). If a single-crystal monochromator is employed, it should be placed between the sample and the detector. (ii) Collimation. It is of some advantage if both the incidentbeam- and the transmitted-beam-de®ning slits can be varied in width. Should it be necessary to combat the effects of Laue±Bragg scattering in a single-crystal specimen, an incident beam with a high degree of collimation is required (Gerward, 1981). To counter the effects of small-angle X-ray scattering, it may be necessary to widen the detector aperture (Chipman, 1969). That these effects can be marked has been shown by Parratt, Porteus, Schnopper & Watanabe (1959), who investigated the in¯uence collimator and monochromator con®gurations have on X-ray-attenuation measurements. (iii) Detection. Detectors that give some degree of energy discrimination should be used. Compromise may be necessary between sensitivity and energy resolution, however, and these factors should be taken into account when a choice is being made between proportional and solid-state detectors. Whichever detection system is chosen, it is essential that the system dead-time be determined experimentally. For descriptions of techniques for the determination of system dead-time, see, for example, Bertin (1975).

4.2.3.3. Normal attenuation coef®cients Fig. 4.2.3.1 shows that the X-ray attenuation coef®cients are a smooth function of photon energy over a relatively large range of photon energies, and that discontinuities occur whenever the photon energy corresponds to a resonance in the electron cloud surrounding the nucleus. In Fig. 4.2.3.2, the effect of the interaction of the ejected photoelectron with the electron's neighbouring atoms is shown. Such edge effects (XAFS) can extend 1000 eV from the edge. It is conventional, however, to extrapolate the smooth curve to the edge value, and a curve of normal attenuation coef®cients results. These are taken to be the attenuation coef®cients of the individual atoms. Tables of these normal attenuation coef®cients are given in Section 4.2.4.

215


4. PRODUCTION AND PROPERTIES OF RADIATIONS 4.2.3.4. Attenuation coef®cients in the neighbourhood of an absorption edge 4.2.3.4.1. XAFS Although the existence of XAFS has been known for more than 60 years following experiments by Fricke (1920) and Hertz (1920), it is only in the last decade that a proper theoretical description has been developed. Kronig (1932a) suggested a long-range-order theory based on quantum-mechanical precepts, although later (Kronig, 1932b) he applied a short-range-order (SRO) theory to explain the existence of XAFS in molecular spectra. As time progressed, important suggestions were made by others, notably Kostarev (1941, 1949), who applied this SRO theory to condensed matter, Sawada, Tsutsumi, Shiraiwa, Ishimura & Obashi (1959), who accounted for the lifetime of the excited photoelectron and the core-hole state in terms of a mean free path, and Schmidt (1961a,b, 1963), who showed the in¯uence atomic vibrations have on the phase of the backscattered waves. Nevertheless, neither the experimental data nor the theories were suf®ciently good to enable Azaroff & Pease (1974) to decide which theory was the correct one to use. However, Sayers, Lytle & Stern (1970) produced a theoretical approach based on SRO theory, later extended by Lytle, Stern & Sayers (1975), and this is the foundation upon which all modern work has been built. Since 1970, a great deal of theoretical effort has been expended to improve the theory because of the need to interpret the wealth of data that became available through the increasing use of synchrotron-radiation sources in XAFS experiments. A number of major reviews of XAFS theory and its use for the resolution of experimental data have been published. Contributions have been made by Stern, Sayers & Lytle (1975), Lee, Citrin, Eisenberger & Kincaid (1981), Lee (1981), and Teo (1981). The rapid growth of the use of synchrotron-radiation sources has led to the development of the use of XAFS in a wide variety of research ®elds. The XAFS community has met regularly at conferences, producing conference proceedings that demonstrate the maturation of the technique. The reader is directed to the proceedings edited by Mustre de Leon, Stern, Sayers, Ma & Rehr (1988), Hasnain (1990), and Kuroda, Ohta, Murata, Udagawa & Nomura (1992), and to the papers contained therein. In the following section, a brief, simpli®ed, description will be given of the theory of XAFS and of the application of that theory to the interpretation of XAFS data. 4.2.3.4.1.1. Theory The theory that will be outlined here has evolved through the efforts of many workers over the past decade. The oscillatory part of the X-ray attenuation relative to the `background' absorption may be written as E

l E l0 E ; l0 E

4:2:3:8

where l E is the measured value of the linear attenuation coef®cient at a photon energy E and l0 E is the `background' linear attenuation coef®cient. This is sometimes the extrapolation of the normal attenuation curve to the edge energy, although it is usually found necessary to modify this extrapolation somewhat to improve the matching of the higher-energy data with the XAFS data (Dreier, Rabe, Matzfeld & Niemann, 1984). In most computer programs, the normal attenuation curve is ®tted to the data using cubic spline ®tting routines.

The origin of XAFS lies in the interaction of the ejected photoelectron with electrons in its immediate vicinity. The wavelength of a photoelectron ejected when a photon is absorbed is given by l 2=k, where k 2m=h2 E

4:2:3:9

This outgoing spherical wave can be back-scattered by the electron clouds of neighbouring atoms. This back-scattered wave interferes with the outgoing wave, resulting in the oscillation of the absorption rate that is observed experimentally and called XAFS. Equation (4.2.3.8) was written with the assumption that the absorption rate was directly proportional to the linear absorption coef®cient. It is conventional to express E in terms of the momentum of the ejected electron, and the usual form of the theoretical expression for k is P k Ni =krj2 j fi kj expi2 k2 ri = sin2kri 'i k: i

4:2:3:10 Here the summation extends over the shells of atoms that surround the absorbing atom, Ni representing the number of atoms in the ith shell, which is situated a distance ri from the absorbing atom. The back-scattering amplitude from this shell is fi k for which the associated phase is 'i k. Deviations due to thermal motions of the electrons are incorporated through a Debye±Waller factor, exp i2 k2 , and is the mean free path of the electron. The amplitude function fi k depends only on the type of backscattering atom. The phase, however, contains contributions from both the absorber and the back-scatterer: 'li k 'lj k 'i k

l;

4:2:3:11

where l 1 for K and LI edges, and l 2 or 0 for LII and LIII edges. The phase is sensitive to variations in the energy threshold, the magnitude of the effect being larger for small electron energies than for electrons with considerable kinetic energy, i.e. the effect is more marked in the neighbourhood of the absorption edge. Since the position of the edge varies somewhat for different compounds (Azaroff & Pease, 1974), some impediment to the analysis of experimental data might occur, since the determination of the interatomic distance ri depends upon the precise knowledge of the value of 'i k. In ®tting the experimental data based on an empirical value of threshold energy using theoretically determined phase shifts, the difference between the theoretical and the experimental threshold energies E0 cannot produce a good ®t at an arbitrarily chosen distance ri , since the effect will be seen primarily at low k values 0:3rE0 =k, whereas changing ri affects 'i k at high k values 2kr. This was ®rst demonstrated by Lee & Beni (1977). The signi®cance of the Debye±Waller factor exp i2 k2 should not be underestimated in this type of investigation. In XAFS studies, one is seeking to determine information regarding such properties of the system as nearest- and next-nearestneighbour distances and the number of nearest and next-nearest neighbours. The theory is a short-range-order theory, hence deviations of atoms from their expected positions will in¯uence the analysis signi®cantly. Thus, it is often of value, experimentally, to work at liquid-nitrogen temperatures to reduce the effect of atomic vibrations. Two distinct types of disorder are observed: vibrational, where the atom vibrates about a mean position in the structure, and static, where the atom occupies a position not expected theoretically. These terms can be separated from one

216


E0 1=2 :

4.2. X-RAYS another if the variation of XAFS spectra with temperature is studied, because the two have different temperature dependences. A discussion of the effect of a thermally activated disorder that is large compared with the static order has been given by Sevillano, Meuth & Rehr (1978). For systems with large static disorders, e.g. liquids and amorphous solids, equation (4.2.3.10) has to be modi®ed somewhat. The XAFS equation has to be averaged over the pair distribution function gr for the system: Fk k k

Z/ gr exp 2r= 0

sin2kr 'k dr: r2

4:2:3:12

Other factors that must be taken into account in XAFS analyses include: inelastic scattering (due to multiple scattering in the absorbing atom and excitations of the atoms surrounding the atom from which the photoelectron was ejected) and multiple scattering of the photoelectron. Should multiple scattering be signi®cant, the simple model given in equation (4.2.3.10) is inappropriate, and more complex models such as those proposed by Pendry (1983), Durham (1983), Gurman (1988, 1995), Natoli (1990), and Rehr & Albers (1990) should be used. Several computer programs are now available commercially for use in personal computers (EXCURVE, FEFF5, MSCALC). Readers are referred to scienti®c journals to ®nd how best to contact the suppliers of these programs.

The Fourier spectrum contains peaks indicating that the nearest-neighbour, next-nearest-neighbour, etc. distances will Ê depending differ from the true spacings by between 0.2 to 0.5 A on the elements involved. These position shifts are determined for model systems and then transferred to the unknown systems to predict interatomic spacings. Fig. 4.2.3.4 illustrates the various steps in the Fourier-transform analysis of XAFS data. The technique works best for systems having well separated peaks. Its primary weakness as a technique lies in the fact that the phase functions are not linear functions of k, and the spacing shift will depend on E0 , the other factors including the weighting of data before the Fourier transforms are made, the range of k space transformed, and the Debye±Waller factors of the system. In the curve-®tting technique (CF), least-squares re®nement is used to ®t the spectra in k space using some structural model for the system. Such techniques, however, can only indicate which of several possible choices is more likely to be correct, and do not prove that that structure is the correct structure. It is possible to combine the FF and CF techniques to simplify the data analysis. Also, for data containing single-scatter peaks, the phase and amplitude components can be separated and analysed separately using either theory or model compounds (Stern, Sayers & Lytle, 1975). Each XAFS data set depends on two sets of strongly correlated variables: fFk; ; ; Ng and f'k; E0 ; rg. The elements of each set are not independent of one another. To determine N and ,

4.2.3.4.1.2. Techniques of data analysis Three assumptions must be made if XAFS data are to be used to provide accurate structural and chemical information: (i) XAFS occurs through the interaction of waves singly scattered by neighbouring atoms; (ii) the amplitude function of the atoms is insensitive to the type of chemical bond (the postulate of transferability), which implies that one can use the same amplitude function for a given atom in problems involving compounds of that atom, whatever the nature of its neighbours or the nature of the bond; and (iii) the phase function can be transferred for each pair of absorber±back-scatterer atoms. Of these three assumptions, (ii) is of the most questionable validity. See, for example, Stern, Bunker & Heald (1981). It is usual, when analysing XAFS data, to search the literature for, or make suf®cient measurements of, l0 remote from the absorption edge to produce a curve of l0 E versus E that can be extrapolated to the position of the edge. From equation (4.2.3.8), it is possible to produce a curve of E versus E from which the variation of k with k can be deduced using equation (4.2.3.9). It is also customary to multiply k by some power of k to compensate for the damping of the XAFS amplitudes with increasing k. The power chosen is somewhat arbitrary but k3 is a commonly used weighting function. Two different techniques may be used to analyse the new data set, the Fourier-transform technique or the curve-®tting technique. In the Fourier-transform technique (FF), the Fourier transform of the kn k is determined for that region of momentum space from the smallest, k1 , to the largest, k2 , wavevectors of the photoelectron, yielding the radical distribution function n r 0 in coordinate r 0 space. n r 0

1 21=2

Zk2

kn k expi2kr 0 dk:

k1

4:2:3:13

Fig. 4.2.3.4. Steps in the reduction of data from an XAFS experiment using the Fourier transform technique: (a) after the removal of background k versus k; (b) after multiplication by a weighting function (in this case k3 ); (c) after Fourier transformation to determine r0 .

217


4. PRODUCTION AND PROPERTIES OF RADIATIONS one must know Fk well. To determine r; 'k must be known accurately. Attempts have been made by Teo & Lee (1979) to calculate Fk and 'k from ®rst principles using an electron±atom scattering model. Parametrized versions have been given by Teo, Lee, Simons, Eisenberger & Kincaid (1977) and Lee et al. (1981). Claimed accuracies for r, , and N in XAFS determinations are 0.5, 10, and 20%, respectively. Acceptable methods for data analysis must conform to a number of basic criteria to have any validity. Amongst these are the following: (i) the data analysis must not give rise to systematic error in the sense that it must provide unbiased estimates of parameters; (ii) the assumed (hypothetical) model must be able to describe the data adequately; (iii) the number of parameters used to describe the best ®t of data must not exceed the number of independent data points; (iv) where multiple solutions exist, supplementary information or assumptions used to resolve the ambiguity must conform to the philosophy of choice of the model structure. The techniques for estimation of the parameters must always be given, including all known sources of uncertainty. A complete list of criteria for the correct analysis and presentation of XAFS data is given in the reports of the International Workshops on Standards and Criteria in XAFS (Lytle, Sayers & Stern, 1989; Bunker, Hasnain & Sayers, 1990). 4.2.3.4.1.3. XAFS experiments The variety and number of experiments in which XAFS experiments have been used is so large that it is not possible here to give a comprehensive list. By consulting the papers given in such texts as those edited by Winick & Doniach (1980), Teo & Joy (1981), Bianconi, Incoccia & Stipcich (1983), Mustre de Leon et al. (1988), Hasnain (1990), and Kuroda et al. (1992), the reader may ®nd references to a wide variety of experiments in ®elds of research ranging from archaeology to zoology. In crystallography, XAFS experiments have been used to assist in the solution of crystal structures; the large variations in the atomic scattering factors can be used to help solve the phase problem. Helliwell (1984) reviewed the use of these techniques in protein crystallography. A further discussion of the use of these anomalous-dispersion techniques in crystallography has been given by Creagh (1987b). The relation that exists between the attenuation (related to the imaginary part of the dispersion correction, f 00 ) and intensity (related to the atomic form factor and the real part of the dispersion correction, f 0 ) is discussed by Creagh in Section 4.2.6. Speci®cally, modulations occur in the observed diffracted intensities from a specimen as the incident photon energy is scanned through the absorption edge of an atomic species present in the specimen. This technique, referred to as diffraction anomalous ®ne structure (DAFS) is complementary to XAFS. Because of the dependence of intensity on the geometrical structure factor, and the fact that the structure factor itself depends on the positional coordinates of the absorbing atom, it is possible to discriminate, in some favourable cases, between the anomalous scattering between atoms occupying different sites in the unit cell (Sorenson et al., 1994). In many systems of biological interest, the arrangement of radicals surrounding an active site must be found in order that the role of that site in biochemical processes may be assessed. A study of the XAFS spectrum of the active atom yields structural information that is speci®c to that site. Normal crystallographic techniques yield more general information concerning the crystal structure. An example of the use of XAFS in biological systems is the study of iron±sulfur proteins undertaken by Shulman,

Weisenberger, Teo, Kincaid & Brown (1978). Other, more recent, studies of biological systems include the characterization of the Mn site in the photosynthetic oxygen evolving complexes including hydroxylamine and hydroquinone (Riggs, Mei, Yocum & Penner-Hahn, 1993) and an XAFS study with an in situ electrochemical cell on manganese Schiff-base complexes as a model of a photosystem (Yamaguchi et al., 1993). It must be stressed that the theoretical expression (equation 4.2.3.10) does not take into account the state of polarization of the incident photon. Templeton & Templeton (1986) have shown that polarization effects may be observed in some materials, e.g. sodium bromate. Given that most XAFS experiments are undertaken using the highly polarized radiation from synchrotron-radiation sources, it is of some importance to be aware of the possibility that dichroic effects may occur in some specimens. Because XAFS is a short-range-order phenomenon, it is particularly useful for the structural study of such disordered systems as liquid metals and amorphous solids. The analysis of such disordered systems can be complicated, particularly in those cases where excluded-volume effects occur. Techniques for analysis for these cases have been suggested by Crozier & Seary (1980). Fuoss, Eisenberger, Warburton & Bienenstock (1981) suggested a technique for the investigation of amorphous solids, which they call the differential anomalous X-ray scattering (DAS) technique. This method has some advantages when compared with conventional XAFS methods because it makes more effective use of low-k information, and it does not depend on a knowledge of either the electron phase shifts or the mean free paths. Both the conventional XAFS and DAS techniques may be used for studies of surface effects and catalytic processes such as those investigated by Sinfelt, Via & Lytle (1980), Hida et al. (1985), and Caballero, Villain, Dexpert, Le Peltier & Lynch (1993). It must be stressed that in all the foregoing discussion it has been assumed that the detection of XAFS has been by measurement of the linear attenuation coef®cient of the specimen. However, the process of photon absorption followed by the ejection of a photoelectron has as its consequence both X-ray ¯uorescence and surface XAFS (SEXAFS) and Auger electron emission. All of these techniques are extremely useful in the analysis of dilute systems. SEXAFS techniques are extremely sensitive to surface conditions since the mean free path of electrons is only about Ê . Discussions of the use of SEXAFS techniques have been 20 A given by Citrin, Eisenberger & Hewitt (1978) and Stohr, Denley & Perfettii (1978). A major review of the topic is given in Lee et al. (1981). SEXAFS has the capacity of sensing thin ®lms deposited on the surface of substrates, and has applications in experiments involving epitaxic growth and absorption by catalysts. Fluorescence techniques are important in those systems for which the absorption of the specimen under investigation contributes only very slightly to the total attenuation coef®cient since it detects the ¯uorescence of the absorbing atom directly. Experiments by Hastings, Eisenberger, Lengeler & Perlman (1975) and Marcus, Powers, Storm, Kincaid & Chance (1980) proved the importance of this technique in analysing dilute alloy and biological specimens. Materlik, Bedzyk & Frahm (1984) have demonstrated its use in determining the location of bromine atoms absorbed on single-crystal silicon substrates. Oyanagi, Matsushita, Tanoue, Ishiguro & Kohra (1985) and Oyanagi, Takeda, Matsushita, Ishiguro & Sasaki (1986) have also used ¯uorescence XAFS techniques for the characterization of very

218


4.2. X-RAYS Table 4.2.3.1. Some synchrotron-radiation facilities providing XAFS databases and analysis utilities Country

Synchrotron source

France

LURE

UniversiteÂ Paris-Sud, LURE, 91405 Orsay, France

Italy

Frascati

Laboratori Nationali di Frascati, CP 13, 00044 Frascati, Italy

Japan

Photon Factory

Photon Factory, National Laboratory for High Energy Physics, 1-1 Oho, Tsukuba-gun, Ibaraki 305, Japan

Germany

DESY

DESY, Notkestrasse 85, 2000 Hamburg 52, Germany

United Kingdom

SRC/ Daresbury

Daresbury Laboratory, Daresbury, Warrington WA4 4AD, England

USA

CHESS

CHESS, Cornell University, Ithaca, New York 14853, USA

NSLS

NSLS, Brookhaven National Laboratory, Upton, New York 11973, USA

SPEAR

SSRL, Stanford University, Bin 69, PO Box 4349, Stanford, California 94305, USA

Address

Maher (1985). This matter is discussed more fully in x4.3.4.4.2. A more recent development has been the observation of topographic XAFS (Bowen, Stock, Davies, Pantos, Birnbaum & Chen, 1984). This ®ne structure is observed in white-beam topographs taken using synchrotron-radiation sources. The technique provides the means of simultaneously determining spatially resolved microstructural and spectroscopic information for the specimen under investigation. In all the preceeding discussion, however, the electron was assumed to undergo only single-scattering processes. If multiple scattering occurs, then the theory has to be changed somewhat. x4.2.3.4.2 discusses the effect of multiple scattering. 4.2.3.4.2. X-ray absorption near edge structure (XANES)

thin ®lms. More recently, Oyanagi et al. (1987) have applied the technique to the study of short-range order in high-temperature superconductors. Oyanagi, Martini, Saito & Haga (1995) have studied in detail the performance of a 19-element high-purity Ge solid-state detector array for ¯uorescence X-ray absorption ®ne structure studies. A less-sensitive technique, but one that can be usefully employed for thin-®lm studies, is that in which XAFS modulations are detected in the beam re¯ected from a sample surface. This technique, Re¯EXAFS, has been used by Martens & Rabe (1980) to investigate super®cial regions of copper oxide ®lms by means of re¯ection of the X-rays close to the critical angle for total re¯ection. If a thin ®lm is examined in a transmission electron microscope, the electron beam loses some of its kinetic energy in interactions between the electron beam and the electrons within the ®lm. If the resultant energy loss is analysed using a magnetic analyser, XAFS-like modulations are observed in the electron energy spectrum. These modulations, electron-energyloss ®ne structure (EELS), which were ®rst observed in a conventional transmission electron microscope by Leapman & Cosslet (1976), are now used extensively for microanalyses of light elements incorporated into heavy-element matrices. Most major manufacturers of transmission electron microscopes supply electron-energy-loss spectrometers for their machines. There are more problems in analysing electron-energy-loss spectra than there are for XAFS spectra. Some of the dif®culties encountered in producing reliable techniques for the routine analysis of EELS have been outlined by Joy &

In Fig. 4.2.3.2(c), there appears to be one cycle of strong oscillation in the neighbourhood of the absorption edge before the quasi-periodic variation of the XAFS commences. The electrons that cause this strong modulation of the photoelectric scattering cross section have low k values, and the electron is strongly scattered by neighbouring atoms. It was mentioned in x4.2.3.4.1 that conventional XAFS theory assumes a weak, single-scattering interaction between the ejected photoelectron and its environment. A schematic diagram illustrating the difference between single- and multiple-scattering processes is given in Fig. 4.2.3.5. Evidently, the multiple-scattering process is very complicated and a discussion of the theory of XANES is too complex to be given here. The reader is directed to papers by Pendry (1983), Lee (1981), and Durham (1983). A more recent review of the study of ®ne structure in ionization cross sections and their use in surface science has been given by Woodruff (1986). The data from XANES experiments can be analysed to determine structural information such as coordination geometry, the symmetry of unoccupied valence electronic states, and the effective charge on the absorbing atom (Natoli, Misemer, Doniach & Kutzler, 1980; Kutzler, Natoli, Misemer, Doniach & Hodgson, 1981). XANES experiments have been performed to resolve many problems, inter alia: the origin of white lines (Lengeler, Materlik & MuÈller, 1983); absorption of gases on metal surfaces (Norman, Durham & Pendry, 1983); the effect of local symmetry in 3d elements (Petiau & Calas, 1983); and the determination of valence states in materials (Lereboures, DuÈrr, d'Huysser, Bonelle & Lenglet, 1980).

Fig. 4.2.3.5. Schematic representations of the scattering processes undergone by the ejected photoelectron in the single-scattering (XAFS) case and the full multiple-scattering regime (XANES).

219


4. PRODUCTION AND PROPERTIES OF RADIATIONS 4.2.3.5. Comments For reliable experiments using XAFS and XANES to be undertaken, intense-radiation sources must be used. Synchrotron-radiation sources are such a source of highly intense X-rays. Their ready availability to experimenters and the comparative simplicity of the equipment required to perform the experiments have made experiments involving XAFS and XANES very much easier to perform than has hitherto been the case. At some synchrotron-radiation sources, database and program libraries for the storage and analysis of XAFS and XANES data exist. These are usually part of the general computing facilities (Pantos, 1982). Crystallographers seeking information concerning the nature and extent of these computer facilities can ®nd such information by contacting the computer centre at one of the synchrotron-radiation establishments listed in Table 4.2.3.1. 4.2.4. X-ray absorption (or attenuation) coef®cients (By D. C. Creagh and J. H. Hubbell) 4.2.4.1. Introduction This data set is intended to supersede those data sets given in International Tables for X-ray Crystallography, Vols. III (Koch, MacGillavry & Milledge, 1962) and IV (Hubbell, McMaster, Del Grande & Mallett, 1974). It is not intended here to give a detailed bibliography of experimental data that have been obtained in the past 90 years. This has been the subject of a number of publications, e.g. Saloman & Hubbell (1987), Hubbell, Gerstenberg & Saloman (1986), Saloman & Hubbell (1986), and Saloman, Hubbell & Sco®eld (1988). Further commentary on the validity and the quality of the experimental data in existing tabulations has been given by Creagh & Hubbell (1987) and Creagh (1987). Existing tabulations of X-ray attenuation (or absorption) cross sections fall into three distinct categories: purely theoretical, purely experimental, and an evaluated mixture of theoretical and experimental data. Compilations of the purely theoretically derived data exist for: photo-effect absorption cross sections (Storm & Israel, 1970; Cromer & Liberman, 1970; Sco®eld, 1973; Hubbell, Veigele, Briggs, Brown, Cromer & Howerton, 1975; Band, Kharitonov & Trzhaskovskaya, 1979; Yeh & Lindau, 1985); Compton scattering cross sections (Hubbell et al., 1975); Rayleigh scattering cross sections (Hubbell et al., 1975; Hubbell & éverbù, 1979; Schaupp, Schumacher, Smend, Rullhausen & Hubbell, 1983). Many purely experimental compilations exist, and the crosssection data given in computer programs used in the analysis of results in X-ray-¯uorescence spectroscopy, electron-probe microanalysis, and X-ray diffraction are usually (evaluated) compilations of several of the following compilations: Allen (1935, 1969), Victoreen (1949), Liebhafsky, Pfeiffer, Winslow & Zemany (1960), Koch et al. (1962), Heinrich (1966), Theisen & Vollath (1967), Veigele (1973), Leroux & Thinh (1977), Montenegro, Baptista & Duarte (1978), and Plechaty, Cullen & Howerton (1981). If a comparison is made between these data sets, signi®cant discrepancies are found, and questions must be asked concerning the reliability of the data sets that are compared. Jackson & Hawkes (1981) and Gerward (1986) have produced sets of parametric tables to simplify the application of X-ray attenuation data for the solution of problems in computer-aided tomography and X-ray-¯uorescence analysis.

Compilations by Henke, Lee, Tanaka, Shimambukuro & Fujikawa (1982) and the earlier tables of McMaster, Del Grande, Mallett & Hubbell (1969/1970) are examples of the judicious application of both theoretical and experimental data to produce a comprehensive data set of X-ray interaction cross sections. Because of the discrepancies that appear to exist between experimental data sets, the IUCr Commission on Crystallographic Apparatus set up a project to establish which, if any, of the existing methods for measuring X-ray interaction cross sections (X-ray attenuation coef®cients) and which theoretical calculations could be considered to be the most reliable. A discussion of some of the major results of this project is given in Section 4.2.3. A more detailed description of this project has been given by Creagh & Hubbell (1987, 1990). In this section, tabulations of the total X-ray interaction cross sections and the mass absorption coef®cient m are given for Ê a range of characteristic X-ray wavelengths [Ti K 2.7440 A Ê (or 24:942 keV]. The inter(or 4:509 keV to Ag K 0.4470 A action cross sections are expressed in units of barns=atom (1 barn 10 28 m2 ) whilst the mass absorption coef®cient is given in cm2 g 1 . Table 4.2.4.1 sets out the wavelengths of the characteristic wavelengths used in Tables 4.2.4.2 and 4.2.4.3, which list values of and m , respectively. Users of these tables should be aware of three important facts. (i) The values given in the tables are derived for the case of isolated atoms, and cooperative effects may become important in condensed phases (Section 4.2.3). (ii) The values are based solely on theoretical calculations. (iii) The limits to the reliability of the data when compared with experimental values are shown in Fig. 4.2.4.4. The linear attenuation coef®cient l in units of cm 1 can be de®ned operationally as Io l ln t 4:2:4:1 I from the exponential attenuation relationship I exp l t I0

Fig. 4.2.4.1. Agreement between theory and experiment for oxygen Z 8 in the `soft' X-ray region. The solid line is for the Sco®eld (1973) values without renormalization and the dotted line is for the semi-empirical data of Henke et al. (1982).

220


4:2:4:2

4.2. X-RAYS Table 4.2.4.1. Table of wavelengths and energies for the characteristic radiations used in Tables 4.2.4.2 and 4.2.4.3 Radiation

Ê) l (A

E (keV)

Ag K K 1 Pd K K 1 Rh K K 1 Mo K K 1 Zn K K 1 Cu K K 1 Ni K K 1 Co K K 1 Fe K K 1 Mn K K 1 Cr K K 1 Ti K K 1

0.5608 0.4970 0.5869 0.5205 0.6147 0.5456 0.7107 0.6323 1.4364 1.2952 1.5418 1.3922 1.6591 1.5001 1.7905 1.6208 1.9373 1.7565 2.1031 1.9102 2.2909 2.0848 2.7496 2.5138

22.103 24.942 21.125 23.819 20.169 22.724 17.444 19.608 8.631 9.572 8.041 8.905 7.472 8.265 6.925 7.629 6.400 7.038 5.895 6.490 5.412 5.947 4.509 4.932

in which an idealized plane-parallel slab of material is interposed normally into a parallel beam of monoenergetic X-rays initially of intensity I0 , attenuated by the interposed slab to a reduced intensity I. The linear attenuation coef®cient l for multi-element substances may be obtained in two ways. Through the mass absorption coef®cients, we have P l gi m i ; 4:2:4:3

4.2.4.2. Sources of information 4.2.4.2.1. Theoretical photo-effect data: pe Of the many theoretical data sets in existence, those of Storm & Israel (1970), Cromer & Liberman (1970), and Sco®eld (1973) have often been used as bench marks against which both experimental and theoretical data have been compared. In particular, theoretical data produced using the S-matrix approach have been compared with these values. See, for example, Kissel, Roy & Pratt (1980). Some indication of the extent to which agreement exists between the different theoretical data sets is given in x4.2.6.2.4 (Tables 4.2.6.3 and 4.2.6.5). These tables show that the values of f 0 !; 0, which is proportional to , calculated using modern relativistic quantum mechanics, agree to better than 1%. It has also been demonstrated by Creagh & Hubbell (1987, 1990) in their analysis of the results of the IUCr X-ray Attenuation Project that there appears to be no rational basis for preferring one of these data sets over the other. These tables do not list separately photo-effect cross sections. However, should these be required, the data can be found using Table 4.2.6.8. The cross section in barns=atom is related to f 0 !; 0 expressed in electrons=atom by 5636l f 0 !; 0; where l is expressed in aÊngstroÈms. The values for pe used in this compilation are derived from recent tabulations based on relativistic Hartree±Fock±Dirac± Slater calculations by Creagh. The extent to which this data set differs from other theoretical and experimental data sets has been discussed by Creagh (1990). 4.2.4.2.2. Theoretical Rayleigh scattering data: R If each of the atoms gives rise to scattering in which momentum but not energy changes occur, and if each of the atoms can be considered to scatter as if it were an isolated atom, the cross section may be written as R re2

i

where gi is the mass fraction of the element i for which the mass attenuation coef®cient m i is in units of cm2 g 1 , and is the density of the material in units of g cm 3 . The summation is over all the constituent elements. The mass attenuation coef®cient m is sometimes written as l =: For a crystal with unit-cell volume Vc , 1X ; 4:2:4:4 l Vc i i

R1 1

1 cos2 ' f 2 q; Z dcos ';

4:2:4:6

where re is the classical radius of the electron;

where the summation is over all the atoms in the cell. If i is in Ê 3 , then l is in cm 1 . barns/atom and Vc is in A These tables list total interaction cross sections and mass attenuation coef®cients for isolated atoms calculated for characteristic X-ray photon emissions ranging from Ti K to Ag K . The total interaction cross section is de®ned by pe R C ;

4:2:4:5

where pe is the photo-effect cross section; R is the Rayleigh (unmodi®ed, elastic) cross section; C is the Compton (modi®ed, inelastic) cross section. The reader's attention is drawn to the fact that in the neighbourhood of an absorption edge for aggregations of atoms signi®cant deviations may be found because of cooperative effects (XAFS and XANES). A discussion of these effects is given in Section 4.2.3.

Fig. 4.2.4.2. The total cross section for silicon Z 14 compared with the unrenormalized Sco®eld values. The measured and theoretical attenuation coef®cients show systematic differences of several percent for the photon energy range 10 to 100 keV.

221


4. PRODUCTION AND PROPERTIES OF RADIATIONS ' is the angle of scattering 2 if is the Bragg angle); 2 dcos ' is the solid angle between cones with angles ' and ' d'; f q; Z is the atomic scattering factor as de®ned by Cromer & Waber (1974); q is sin'=2=l; the momentum transfer parameter. Here l is expressed in aÊngstroÈms. Reliable tables of f q; Z exist and have been reviewed recently by Kane, Kissel, Pratt & Roy (1986). The most recent schematic tabulations of f q; Z are those of Hubbell & éverbù

(1979) and Schaupp et al. (1983). The data used in these tables Ê 1, have been derived from the tabulation for q 0:02 to 109 A for all Z's from 1 to 100 by Hubbell & éverbù (1979) based on the exact formula of Pirenne (1946) for H, and relativistic calculations by Doyle & Turner (1968), Cromer & Waber (1974), éverbù (1977, 1978), and high-q extensions using the Bethe±Levinger expression in Levinger (1952). As mentioned in Creagh & Hubbell (1987), the atoms in highly ordered single crystals do not scatter as though they are isolated atoms. Rather, cooperative effects become important. In this case, the Rayleigh scattering cross section must be replaced by two cross sections: the Laue±Bragg cross section LB , and the thermal diffuse scattering cross section TD . That is, R is replaced by LB TD . These effects are discussed elsewhere (Subsection 4.2.3.2). Brie¯y, LB re2 l2 =2NVc

Fig. 4.2.4.3. The total cross section for uranium Z 92: The theoretical values (solid line) are partially obscured by the high density of available measurements. Deviations of the measured values from the theoretical predictions are mostly of the order of 5%, although a few data sets deviate by more than 30%.

P H

Cp mdjFj2 exp 2MH :

4:2:4:7

In equation (4.2.4.7), which is due to De Marco & Suortti (1971), Cp 12 1 cos2 '; dH is the spacing of the (hkl) planes in the crystal; mH is the multiplicity of the hkl Bragg re¯ection; FH is the geometrical structure factor for the crystal structure that contains N atoms in a cell of volume Vc ; exp 2MH is the Debye-Waller temperature factor. It is assumed that the total thermal diffuse scattering is equal to the scattering lost from Laue±Bragg scattering because of thermal vibrations. TD r 2e l2 =2NVc

P H

fCp mdjFj2 1

exp 2MgH : 4:2:4:8

Fig. 4.2.4.4. Comparison between this tabulation and experimental data contained in Saloman & Hubbell (1986). The upper set corresponds to the average percent deviation between the experimental data and this tabulation for the energy range 10 to 100 keV. The lower set corresponds to the energy range 1 to 10 keV. For explanation of symbols see text.

222


4.2. X-RAYS Table 4.2.4.2. Total photon interaction cross section (barns=atom) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K

Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K

Energy (MeV) 2.494E 02 2.382E 02 2.272E 02 2.210E 02 2.112E 02 2.017E 02 1.961E 02 1.744E 02 9.572E 03 8.905E 03 8.631E 03 8.265E 03 8.041E 03 7.649E 03 7.472E 03 7.058E 03 6.925E 03 6.490E 03 6.400E 03 5.947E 03 5.895E 03 5.412E 03 4.932E 03 4.509E 03

1 Hydrogen 6.10E 01 6.10E 01 6.12E 01 6.14E 01 6.16E 01 6.18E 01 6.19E 01 6.24E 01 6.47E 01 6.50E 01 6.51E 01 6.53E 01 6.55E 01 6.58E 01 6.59E 01 6.63E 01 6.64E 01 6.69E 01 6.70E 01 6.77E 01 6.78E 01 6.89E 01 7.04E 01 7.24E 01

2 Helium 1.26E+00 1.26E+00 1.27E+00 1.28E+00 1.29E+00 1.30E+00 1.31E+00 1.34E+00 1.69E+00 1.78E+00 1.82E+00 1.89E+00 1.94E+00 2.04E+00 2.09E+00 2.23E+00 2.28E+00 2.48E+00 2.53E+00 2.83E+00 2.87E+00 3.31E+00 3.94E+00 4.73E+00

3 Lithium 2.01E+00 2.01E+00 2.04E+00 2.06E+00 2.09E+00 2.13E+00 2.16E+00 2.28E+00 4.19E+00 4.74E+00 5.02E+00 5.46E+00 5.76E+00 6.40E+00 6.73E+00 7.65E+00 7.99E+00 9.34E+00 9.67E+00 1.16E+01 1.19E+01 1.50E+01 1.94E+01 2.51E+01

4 Beryllium 2.97E+00 2.99E+00 3.06E+00 3.13E+00 3.23E+00 3.35E+00 3.42E+00 3.83E+00 1.07E+01 1.28E+01 1.38E+01 1.54E+01 1.66E+01 1.90E+01 2.02E+01 2.37E+01 2.50E+01 3.01E+01 3.13E+01 3.88E+01 3.99E+01 5.14E+01 6.82E+01 8.97E+01

5 Boron 4.40E+00 4.44E+00 4.47E+00 4.79E+00 5.05E+00 5.35E+00 5.56E+00 6.61E+00 2.54E+01 3.10E+01 3.39E+01 3.83E+01 4.15E+01 4.80E+01 5.14E+01 6.09E+01 6.45E+01 7.84E+01 8.18E+01 1.02E+02 1.05E+02 1.36E+02 1.81E+02 2.39E+02

6 Carbon 6.59E+00 6.68E+00 6.78E+00 7.45E+00 8.02E+00 8.68E+00 9.14E+00 1.15E+01 5.37E+01 6.64E+01 7.28E+01 8.28E+01 8.99E+01 1.04E+02 1.12E+02 1.33E+02 1.41E+02 1.72E+02 1.79E+02 2.24E+02 2.30E+02 2.99E+02 3.98E+02 5.23E+02

7 Nitrogen 1.00E+01 1.02E+01 1.05E+01 1.17E+01 1.28E+01 1.41E+01 1.50E+01 1.96E+01 1.03E+02 1.27E+02 1.40E+02 1.59E+02 1.73E+02 2.01E+02 2.16E+02 2.57E+02 2.72E+02 3.31E+02 3.46E+02 4.32E+02 4.44E+02 5.75E+02 7.62E+02 1.00E+03

8 Oxygen 1.52E+01 1.55E+01 1.62E+01 1.82E+01 2.02E+01 2.25E+01 2.41E+01 3.25E+01 1.80E+02 2.24E+02 2.46E+02 2.80E+02 3.04E+02 3.54E+02 3.80E+02 4.51E+02 4.78E+02 5.81E+02 6.06E+02 7.56E+02 7.76E+02 1.00E+03 1.33E+03 1.73E+03

2.494E 2.382E 2.272E 2.210E 2.112E 2.017E 1.961E 1.744E 9.572E 8.905E 8.631E 8.265E 8.041E 7.649E 7.472E 7.058E 6.925E 6.490E 6.400E 5.947E 5.895E 5.412E 4.932E 4.509E

9 Fluorine 2.27E+01 2.32E+01 2.50E+01 2.77E+01 3.11E+01 3.50E+01 3.76E+01 5.15E+01 2.95E+02 3.66E+02 4.02E+02 4.58E+02 4.98E+02 5.78E+02 6.20E+02 7.36E+02 7.79E+02 9.46E+02 9.86E+02 1.23E+03 1.26E+03 1.62E+03 2.14E+03 2.79E+03

10 Neon 3.33E+01 3.40E+01 3.62E+01 4.12E+01 4.65E+01 5.26E+01 5.68E+01 7.86E+01 4.57E+02 5.67E+02 6.22E+02 7.08E+02 7.68E+02 8.92E+02 9.56E+02 1.13E+03 1.20E+03 1.45E+03 1.51E+03 1.88E+03 1.93E+03 2.48E+03 3.26E+03 4.23E+03

11 Sodium 4.77E+01 4.88E+01 5.07E+01 5.96E+01 6.75E+01 7.67E+01 5.30E+01 1.16E+02 6.77E+02 8.39E+02 9.20E+02 1.05E+03 1.14E+03 1.32E+03 1.41E+03 1.67E+03 1.76E+03 2.13E+03 2.22E+03 2.75E+03 2.83E+03 3.62E+03 4.74E+03 6.13E+03

12 Magnesium 6.68E+01 6.85E+01 8.26E+01 8.42E+01 9.55E+01 1.09E+02 1.18E+02 1.65E+02 9.67E+02 1.20E+03 1.31E+03 1.49E+03 1.61E+03 1.87E+03 2.00E+03 2.36E+03 2.50E+03 3.02E+03 3.14E+03 3.88E+03 3.98E+03 5.09E+03 6.64E+03 8.57E+03

13 Aluminium 9.16E+01 9.40E+01 1.05E+02 1.16E+02 1.32E+02 1.51E+02 1.63E+02 2.29E+02 1.34E+03 1.65E+03 1.81E+03 2.05E+03 2.22E+03 2.57E+03 2.75E+03 3.24E+03 3.42E+03 4.13E+03 4.29E+03 5.30E+03 5.43E+03 6.93E+03 9.01E+03 1.16E+04

14 Silicon 1.23E+02 1.26E+02 1.40E+02 1.56E+02 1.78E+02 2.03E+02 2.20E+02 3.10E+02 1.79E+03 2.21E+03 2.42E+03 2.75E+03 2.97E+03 3.43E+03 3.67E+03 4.32E+03 4.56E+03 5.49E+03 5.71E+03 7.03E+03 7.20E+03 9.16E+03 1.19E+04 1.52E+04

15 Phosphorus 1.62E+02 1.67E+02 1.85E+02 2.06E+02 2.35E+02 2.69E+02 2.92E+02 4.10E+02 2.36E+03 2.90E+03 3.17E+03 3.59E+03 3.88E+03 4.48E+03 4.78E+03 5.62E+03 5.93E+03 7.12E+03 7.41E+03 9.10E+03 9.33E+03 1.18E+04 1.53E+04 1.95E+04

16 Sulfur 2.10E+02 2.16E+02 2.02E+02 2.67E+02 3.05E+02 3.49E+02 3.78E+02 5.32E+02 3.03E+03 3.72E+03 4.06E+03 4.60E+03 4.97E+03 5.72E+03 6.11E+03 7.17E+03 7.56E+03 9.06E+03 9.42E+03 1.15E+04 1.18E+04 1.50E+04 1.93E+04 2.45E+04

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

223


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.4.2. Total photon interaction cross section (barns=atom) (cont.) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



17 Chlorine 2.68E+02 2.75E+02 3.15E+02 3.41E+02 3.89E+02 4.45E+02 4.83E+02 6.78E+02 3.82E+03 4.68E+03 5.11E+03 5.78E+03 6.24E+03 7.17E+03 7.66E+03 8.97E+03 9.46E+03 1.13E+04 1.18E+04 1.44E+04 1.47E+04 1.86E+04 2.38E+04 3.01E+04

18 Argon 3.36E+02 3.45E+02 3.84E+02 4.29E+02 4.89E+02 5.59E+02 6.06E+02 8.51E+02 4.74E+03 5.80E+03 6.34E+03 7.15E+03 7.72E+03 8.86E+03 9.46E+03 1.11E+04 1.17E+04 1.39E+04 1.44E+04 1.76E+04 1.80E+04 2.27E+04 2.91E+04 3.69E+04

19 Potassium 4.17E+02 4.29E+02 4.84E+02 5.32E+02 6.06E+02 6.93E+02 7.52E+02 1.05E+03 5.80E+03 7.09E+03 7.73E+03 8.72E+03 9.40E+03 1.08E+04 1.15E+04 1.34E+04 1.41E+04 1.69E+04 1.75E+04 2.13E+04 2.18E+04 2.74E+04 3.49E+04 4.41E+04

20 Calcium 5.12E+02 5.26E+02 6.30E+02 6.52E+02 7.42E+02 8.48E+02 9.20E+02 1.29E+03 7.02E+03 8.57E+03 9.34E+03 1.05E+04 1.13E+04 1.30E+04 1.38E+04 1.61E+04 1.69E+04 2.01E+04 2.09E+04 2.54E+04 2.60E+04 3.26E+04 4.15E+04 5.20E+04

21 Scandium 6.20E+02 6.37E+02 7.25E+02 7.89E+02 8.99E+02 1.03E+03 1.11E+03 1.56E+03 8.38E+03 1.02E+04 1.11E+04 1.25E+04 1.35E+04 1.54E+04 1.64E+04 1.91E+04 2.01E+04 2.39E+04 2.47E+04 3.01E+04 3.08E+04 3.85E+04 4.87E+04 6.03E+04

22 Titanium 7.44E+02 7.64E+02 8.63E+02 9.47E+02 1.08E+03 1.23E+03 1.33E+03 1.86E+03 9.93E+03 1.21E+04 1.32E+04 1.48E+04 1.39E+04 1.80E+04 1.91E+04 2.20E+04 2.31E+04 2.74E+04 2.85E+04 3.53E+04 3.63E+04 4.69E+04 6.79E+03 8.68E+03

23 Vanadium 8.85E+02 9.09E+02 9.98E+02 1.12E+03 1.28E+03 1.46E+03 1.58E+03 2.20E+03 1.16E+04 1.41E+04 1.53E+04 1.72E+04 1.85E+04 2.11E+04 2.25E+04 2.62E+04 2.75E+04 3.26E+04 3.37E+04 4.05E+04 4.14E+04 6.32E+03 8.16E+03 1.04E+04

24 Chromium 1.04E+03 1.07E+03 1.19E+03 1.33E+03 1.51E+03 1.72E+03 1.86E+03 2.58E+03 1.33E+04 1.60E+04 1.73E+04 1.96E+04 2.13E+04 2.53E+04 2.74E+04 3.32E+04 3.53E+04 4.15E+04 4.25E+04 5.79E+03 5.93E+03 7.50E+03 9.68E+03 1.24E+04


25 Manganese 1.22E+03 1.25E+03 1.37E+03 1.55E+03 1.76E+03 2.01E+03 2.17E+03 3.02E+03 1.55E+04 1.88E+04 2.05E+04 2.29E+04 2.46E+04 2.80E+04 2.97E+04 3.42E+04 3.58E+04 5.40E+03 5.62E+03 6.87E+03 7.04E+03 8.90E+03 1.15E+04 1.47E+04

26 Iron 1.42E+03 1.46E+03 1.65E+03 1.80E+03 2.04E+03 2.33E+03 2.52E+03 3.49E+03 1.78E+04 2.15E+04 2.34E+04 2.61E+04 2.80E+04 3.17E+04 3.35E+04 5.04E+03 5.31E+03 6.34E+03 6.59E+03 8.06E+03 8.26E+03 1.04E+04 1.35E+04 1.72E+04

27 Cobalt 1.64E+03 1.68E+03 1.89E+03 2.07E+03 2.35E+03 2.68E+03 2.90E+03 4.01E+03 2.02E+04 2.43E+04 2.63E+04 2.93E+04 3.14E+04 4.71E+03 5.02E+03 5.87E+03 6.18E+03 7.39E+03 7.68E+03 9.40E+03 9.62E+03 1.22E+04 1.57E+04 2.00E+04

28 Nickel 1.88E+03 1.93E+03 2.19E+03 2.38E+03 2.70E+03 3.07E+03 3.31E+03 4.57E+03 2.27E+04 2.72E+04 2.94E+04 4.41E+03 4.76E+03 5.46E+03 5.82E+03 6.80E+03 7.16E+03 8.56E+03 8.89E+03 1.09E+04 1.11E+04 1.41E+04 1.81E+04 2.31E+04

29 Copper 2.14E+03 2.20E+03 2.49E+03 2.71E+03 3.07E+03 3.49E+03 3.77E+03 5.18E+03 2.53E+04 4.13E+03 4.50E+03 5.07E+03 5.47E+03 6.27E+03 6.68E+03 7.81E+03 8.23E+03 9.83E+03 1.02E+04 1.25E+04 1.28E+04 1.61E+04 2.08E+04 2.65E+04

30 Zinc 2.43E+03 2.49E+03 2.88E+03 3.07E+03 3.47E+03 3.95E+03 4.26E+03 5.86E+03 3.90E+03 4.75E+03 5.18E+03 5.83E+03 6.29E+03 7.21E+03 7.68E+03 8.98E+03 9.46E+03 1.13E+04 1.17E+04 1.43E+04 1.47E+04 1.85E+04 2.39E+04 3.04E+04

31 Gallium 2.74E+03 2.81E+03 3.21E+03 3.46E+03 3.91E+03 4.44E+03 4.80E+03 6.60E+03 4.46E+03 5.44E+03 5.92E+03 6.67E+03 7.19E+03 8.24E+03 8.79E+03 1.03E+04 1.08E+04 1.29E+04 1.34E+04 1.64E+04 1.68E+04 2.12E+04 2.72E+04 3.46E+04

32 Germanium 3.08E+03 3.16E+03 3.55E+03 3.87E+03 4.38E+03 4.97E+03 5.37E+03 7.38E+03 5.08E+03 6.19E+03 6.75E+03 7.60E+03 8.19E+03 9.38E+03 1.00E+04 1.17E+04 1.23E+04 1.47E+04 1.53E+04 1.86E+04 1,91E+04 2.41E+04 3.09E+04 3.93E+04

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

224


4.2. X-RAYS Table 4.2.4.2. Total photon interaction cross section (barns=atom) (cont.) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



33 Arsenic 3.44E+03 3.53E+03 4.04E+03 4.33E+03 4.89E+03 5.55E+03 5.99E+03 8.22E+03 5.77E+03 7.03E+03 7.65E+03 8.62E+03 9.29E+03 1.06E+04 1.13E+04 1.32E+04 1.39E+04 1.66E+04 1.73E+04 2.11E+04 2.16E+04 2.72E+04 3.50E+04 4.44E+04


41 Niobium 7.41E+03 7.59E+03 8.57E+03 9.22E+03 1.04E+04 1.16E+04 1.25E+04 2.73E+03 1.40E+04 1.70E+04 1.85E+04 2.07E+04 2.23E+04 2.55E+04 2.72E+04 3.17E+04 3.33E+04 3.96E+04 4.12E+04 5.01E+04 5.13E+04 6.42E+04 8.21E+04 1.04E+05

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

34 Selenium 3.84E+03 3.94E+03 4.53E+03 4.82E+03 5.45E+03 6.18E+03 6.66E+03 9.11E+03 6.52E+03 7.94E+03 8.64E+03 9.73E+03 1.05E+04 1.20E+04 1.28E+04 1.49E+04 1.57E+04 1.88E+04 1.95E+04 2.38E+04 2.44E+04 3.07E+04 3.94E+04 5.00E+04

35 Bromine 4.26E+03 4.37E+03 5.00E+03 5.35E+03 6.04E+03 6.83E+03 7.36E+03 1.00E+04 7.34E+03 8.94E+03 9.73E+03 1.10E+04 1.18E+04 1.35E+04 1.44E+04 1.68E+04 1.77E+04 2.11E+04 2.19E+04 2.67E+04 2.74E+04 3.45E+04 4.42E+04 5.61E+04

36 Krypton 4.72E+03 4.84E+03 5.50E+03 5.92E+03 6.68E+03 7.55E+03 8.13E+03 1.10E+04 8.24E+03 1.00E+04 1.09E+04 1.23E+04 1.32E+04 1.52E+04 1.61E+04 1.88E+04 1.98E+04 2.36E+04 2.45E+04 2.99E+04 3.07E+04 3.85E+04 4.94E+04 6.27E+04

37 Rubidium 5.21E+03 5.34E+03 5.98E+03 6.52E+03 7.35E+03 8.30E+03 8.94E+03 1.21E+04 9.21E+03 1.12E+04 1.22E+04 1.37E+04 1.48E+04 1.69E+04 1.80E+04 2.10E+04 2.22E+04 2.64E+04 2.74E+04 3.34E+04 3.42E+04 4.30E+04 5.51E+04 6.98E+04

38 Strontium 5.72E+03 5.86E+03 6.48E+03 7.15E+03 8.06E+03 9.09E+03 9.78E+03 1.32E+04 1.03E+04 1.25E+04 1.36E+04 1.53E+04 1.65E+04 1.89E+04 2.01E+04 2.34E+04 2.47E+04 2.94E+04 3.05E+04 3.72E+04 3.81E+04 4.78E+04 6.12E+04 7.75E+04

39 Yttrium 6.25E+03 6.41E+03 7.27E+03 7.80E+03 8.79E+03 9.90E+03 1.06E+04 1.43E+04 1.14E+04 1.39E+04 1.51E+04 1.70E+04 1.83E+04 2.09E+04 2.23E+04 2.60E+04 2.73E+04 3.26E+04 3.38E+04 4.12E+04 4.22E+04 5.29E+04 6.77E+04 8.56E+04

40 Zirconium 6.79E+03 6.96E+03 7.80E+03 8.47E+03 9.52E+03 1.07E+04 1.15E+04 2.47E+03 1.26E+04 1.54E+04 1.67E+04 1.88E+04 2.03E+04 2.32E+04 2.47E+04 2.87E+04 3.02E+04 3.60E+04 3.74E+04 4.55E+04 4.66E+04 5.84E+04 7.47E+04 9.43E+04

42 43 Molybdenum Technetium 9.36E+03 8.65E+03 9.61E+03 8.86E+03 9.30E+03 9.95E+03 1.15E+04 1.07E+04 1.23E+04 1.20E+04 1.27E+04 2.24E+03 2.19E+03 2.42E+03 3.00E+03 3.32E+03 1.54E+04 1.69E+04 1.87E+04 2.05E+04 2.03E+04 2.23E+04 2.28E+04 2.51E+04 2.46E+04 2.70E+04 2.81E+04 3.08E+04 2.99E+04 3.28E+04 3.48E+04 3.82E+04 3.66E+04 4.02E+04 4.36E+04 4.77E+04 4.52E+04 4.95E+04 5.50E+04 6.02E+04 5.63E+04 6.16E+04 7.05E+04 7.71E+04 8.99E+04 9.83E+04 1.13E+05 1.24E+05

44 Ruthenium 9.33E+03 9.56E+03 1.07E+04 1.92E+03 2.17E+03 2.46E+03 2.65E+03 3.64E+03 1.85E+04 2.25E+04 2.44E+04 2.74E+04 2.95E+04 3.37E+04 3.59E+04 4.18E+04 4.39E+04 5.22E+04 5.42E+04 6.58E+04 6.73E+04 8.41E+04 1.07E+05 1.35E+05

45 Rhodium 1.00E+04 1.03E+04 1.18E+03 2.10E+03 2.38E+03 2.70E+03 2.91E+03 3.99E+03 2.02E+04 2.45E+04 2.67E+04 3.00E+04 3.23E+04 3.68E+04 3.92E+04 4.56E+04 4.79E+04 5.69E+04 5.91E+04 7.17E+04 7.34E+04 9.16E+04 1.17E+05 1.47E+05

46 Palladium 1.00E+04 1.88E+03 2.10E+03 2.30E+03 2.60E+03 2.94E+03 3.18E+03 4.36E+03 2.21E+04 2.67E+04 2.91E+04 3.27E+04 3.52E+04 4.01E+04 4.27E+04 4.96E+04 5.21E+04 6.19E+04 6.42E+04 7.79E+04 7.97E+04 9.94E+04 1.27E+05 1.59E+05

47 Silver 2.00E+03 2.05E+03 2.29E+03 2.51E+03 2.84E+03 3.21E+03 3.47E+03 4.76E+03 2.40E+04 2.91E+04 3.16E+04 3.55E+04 3.82E+04 4.36E+04 4.64E+04 5.39E+04 5.66E+04 6.72E+04 6.97E+04 8.45E+04 8.64E+04 1.08E+05 1.37E+05 1.72E+05

48 Cadmium 2.18E+03 2.23E+03 2.49E+03 2.73E+03 3.09E+03 3.50E+03 3.78E+03 5.18E+03 2.61E+04 3.16E+04 3.44E+04 3.86E+04 4.15E+04 4.73E+04 5.03E+04 5.84E+04 6.14E+04 7.28E+04 7.55E+04 9.15E+04 9.36E+04 1.17E+05 1.48E+05 1.86E+05

225





49 Indium 2.37E+03 2.43E+03 2.64E+03 2.97E+03 3.36E+03 3.81E+03 4.11E+03 5.63E+03 2.83E+04 3.43E+04 3.72E+04 4.18E+04 4.50E+04 5.12E+04 5.45E+04 6.32E+04 6.64E+04 7.87E+04 8.17E+04 9.90E+04 1.01E+05 1.26E+05 1.60E+05 2.00E+05

50 Tin 2.57E+03 2.64E+03 3.00E+03 3.23E+03 3.65E+03 4.13E+03 4.46E+03 6.11E+03 3.06E+04 3.71E+04 4.03E+04 4.52E+04 4.86E+04 5.54E+04 5.89E+04 6.83E+04 7.18E+04 8.50E+04 8.82E+04 1.07E+05 1.09E+05 1.36E+05 1.73E+05 2.15E+05


57 Lanthanum 3.97E+03 4.05E+03 5.10E+03 5.49E+03 6.20E+03 7.03E+03 7.59E+03 1.04E+04 5.10E+04 6.14E+04 6.67E+04 7.47E+04 8.03E+04 9.11E+04 9.68E+04 1.11E+05 1.17E+05 1.38E+05 1.43E+05 1.48E+05 1.50E+05 5.19E+04 6.55E+04 8.19E+04

58 59 60 61 Cerium Praseodymium Neodymium Promethium 4.26E+03 4.56E+03 4.89E+03 5.22E+03 4.82E+03 5.15E+03 5.51E+03 5.90E+03 5.47E+03 5.85E+03 6.25E+03 6.69E+03 5.89E+03 6.29E+03 6.73E+03 7.20E+03 6.63E+03 7.11E+03 7.62E+03 8.14E+03 7.69E+03 8.07E+03 8.62E+03 9.22E+03 8.12E+03 8.70E+03 9.29E+03 9.94E+03 1.11E+04 1.19E+04 1.27E+04 1.36E+04 5.42E+04 5.78E+04 6.15E+04 6.57E+04 6.56E+04 7.00E+04 7.42E+04 7.90E+04 7.12E+04 7.58E+04 8.04E+04 8.44E+04 7.96E+04 8.49E+04 9.00E+04 9.56E+04 8.56E+04 9.12E+04 9.68E+04 1.03E+05 9.70E+04 1.03E+05 1.09E+05 1.16E+05 1.03E+05 1.09E+05 1.16E+05 1.23E+05 1.19E+05 1.26E+05 1.18E+05 1.22E+05 1.24E+05 1.32E+05 1.21E+05 9.63E+04 1.27E+05 1.39E+05 1.05E+05 1.13E+05 1.31E+05 1.02E+05 1.09E+05 4.19E+04 1.15E+05 1.22E+05 4.74E+04 5.01E+04 1.19E+05 4.52E+04 4.86E+04 5.18E+04 5.54E+04 5.57E+04 6.01E+04 6.43E+04 6.98E+04 7.02E+04 7.52E+04 8.11E+04 8.31E+04 8.77E+04 9.51E+04 1.02E+05

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

51 Antimony 2.79E+03 2.86E+03 3.20E+03 3.50E+03 3.95E+03 4.48E+03 4.84E+03 6.62E+03 3.31E+04 4.00E+04 4.35E+04 4.88E+04 5.25E+04 5.98E+04 6.35E+04 7.37E+04 7.74E+04 9.17E+04 9.51E+04 1.15E+05 1.18E+05 1.46E+05 1.85E+05 2.00E+05

52 Tellurium 3.02E+03 3.09E+03 3.50E+03 3.78E+03 4.28E+03 4.85E+03 5.23E+03 7.16E+03 3.57E+04 4.32E+04 4.69E+04 5.25E+04 5.65E+04 6.43E+04 6.84E+04 7.94E+04 8.34E+04 9.88E+04 1.02E+05 1.24E+05 1.27E+05 1.57E+05 1.98E+05 1.60E+05

226


53 Iodine 3.26E+03 3.34E+03 3.71E+03 4.09E+03 4.62E+03 5.24E+03 5.65E+03 7.73E+03 3.84E+04 4.64E+04 5.04E+04 5.65E+04 6.07E+04 6.92E+04 7.35E+04 8.52E+04 8.96E+04 1.06E+05 1.10E+05 1.33E+05 1.36E+05 1.68E+05 2.11E+05 6.17E+04

54 Xenon 3.52E+03 3.61E+03 4.04E+03 4.41E+03 4.98E+03 5.65E+03 6.09E+03 8.34E+03 4.13E+04 4.99E+04 5.42E+04 6.07E+04 6.52E+04 7.42E+04 7.88E+04 9.14E+04 9.60E+04 1.13E+05 1.18E+05 1.42E+05 1.45E+05 1.57E+05 2.48E+05 5.78E+04

55 Caesium 3.79E+03 3.89E+03 4.04E+03 4.75E+03 5.37E+03 6.09E+03 6.57E+03 8.98E+03 4.44E+04 5.36E+04 5.81E+04 6.51E+04 7.00E+04 7.96E+04 8.45E+04 9.77E+04 1.03E+05 1.21E+05 1.26E+05 1.60E+05 1.63E+05 1.77E+05 5.70E+04 7.28E+04

56 Barium 4.08E+03 4.18E+03 4.76E+03 5.11E+03 5.78E+03 6.55E+03 7.06E+03 9.65E+03 4.76E+04 5.74E+04 6.23E+04 6.98E+04 7.50E+04 8.52E+04 9.04E+04 1.04E+05 1.09E+05 1.29E+05 1.34E+05 1.47E+05 1.50E+05 1.34E+05 7.16E+04 7.62E+04

62 Samarium 5.57E+03 6.29E+03 7.14E+03 7.69E+03 8.69E+03 9.84E+03 1.06E+04 1.44E+04 6.97E+04 8.36E+04 9.06E+04 1.01E+05 1.08E+05 1.07E+05 1.14E+05 9.56E+04 9.89E+04 4.39E+04 4.59E+04 5.52E+04 5.64E+04 6.97E+04 8.74E+04 1.09E+05

63 Europium 5.93E+03 6.71E+03 7.59E+03 8.17E+03 9.23E+03 1.05E+04 1.13E+04 1.54E+04 9.93E+04 8.88E+04 9.59E+04 1.07E+05 1.02E+05 1.21E+05 8.70E+04 1.03E+05 4.04E+04 4.67E+04 4.87E+04 5.65E+04 6.03E+04 7.54E+04 9.34E+04 1.16E+05

64 Gadolinium 6.32E+03 7.15E+03 8.09E+03 8.72E+03 9.84E+03 1.11E+04 1.20E+04 1.63E+04 7.83E+04 9.40E+04 1.02E+05 9.84E+04 1.05E+05 8.75E+04 9.29E+04 3.99E+04 4.20E+04 4.93E+04 5.09E+04 6.14E+04 6.29E+04 7.78E+04 9.76E+04 1.22E+05

4.2. X-RAYS Table 4.2.4.2. Total photon interaction cross section (barns=atom) (cont.) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



65 Terbium 6.66E+03 7.52E+03 8.15E+03 9.16E+03 1.03E+04 1.17E+04 1.26E+04 1.72E+04 8.20E+04 9.87E+04 9.29E+04 1.08E+05 8.38E+04 9.40E+04 3.89E+04 4.22E+04 4.43E+04 5.22E+04 5.44E+04 6.46E+04 6.68E+04 8.29E+04 1.05E+05 1.31E+05

66 Dysprosium 7.15E+03 8.07E+03 9.15E+03 9.85E+03 1.11E+04 1.26E+04 1.35E+04 1.84E+04 8.74E+04 9.31E+04 1.02E+05 8.53E+04 9.23E+04 3.72E+04 3.94E+04 4.53E+04 4.75E+04 5.59E+04 5.77E+04 6.93E+04 7.07E+04 8.77E+04 1.11E+05 1.39E+05

67 Holmium 7.59E+03 8.57E+03 9.69E+03 1.01E+04 1.18E+04 1.33E+04 1.43E+04 1.95E+04 9.20E+04 7.17E+04 7.78E+04 8.60E+04 3.53E+04 4.00E+04 4.24E+04 4.87E+04 5.12E+04 6.02E+04 6.24E+04 7.45E+04 7.67E+04 9.51E+04 1.20E+05 1.50E+05

68 Erbium 8.05E+03 9.08E+03 1.03E+04 1.11E+04 1.25E+04 1.41E+04 1.52E+04 2.07E+04 8.53E+04 7.42E+04 7.97E+04 3.42E+04 3.67E+04 4.14E+04 4.39E+04 5.05E+04 5.30E+04 6.22E+04 6.44E+04 7.75E+04 7.92E+04 9.78E+04 1.23E+05 1.54E+05

69 Thulium 8.52E+03 9.62E+03 1.08E+04 1.17E+04 1.32E+04 1.50E+04 1.61E+04 2.19E+04 6.59E+04 8.27E+04 3.28E+04 3.67E+04 3.93E+04 4.46E+04 4.47E+04 5.50E+04 5.78E+04 6.82E+04 7.10E+04 8.55E+04 8.75E+04 1.08E+05 1.39E+05 1.73E+05

70 Ytterbium 9.02E+03 1.02E+04 1.15E+04 1.24E+04 1.40E+04 1.58E+04 1.70E+04 2.31E+04 6.90E+04 3.10E+04 3.36E+04 3.76E+04 4.08E+04 4.57E+04 4.86E+04 5.63E+04 5.92E+04 7.01E+04 7.21E+04 8.73E+04 8.94E+04 1.11E+05 1.41E+05 1.78E+05

71 Lutetium 9.53E+03 1.08E+04 1.22E+04 1.31E+04 1.48E+04 1.67E+04 1.80E+04 2.44E+04 7.26E+04 3.52E+04 3.81E+04 4.24E+04 4.53E+04 5.17E+04 5.49E+04 6.33E+04 6.65E+04 7.84E+04 8.13E+04 9.85E+04 1.01E+05 1.25E+05 1.59E+05 2.00E+05

72 Hafnium 1.01E+04 1.14E+04 1.29E+04 1.38E+04 1.56E+04 1.76E+04 1.90E+04 2.57E+04 7.70E+04 3.56E+04 3.85E+04 4.30E+04 4.59E+04 5.21E+04 5.54E+04 6.40E+04 6.73E+04 7.91E+04 8.21E+04 9.90E+04 1.01E+05 1.26E+05 1.60E+05 2.01E+05


73 Tantalum 1.17E+04 1.20E+04 1.35E+04 1.46E+04 1.65E+04 1.86E+04 2.00E+04 2.72E+04 3.30E+04 3.76E+04 4.05E+04 4.41E+04 4.85E+04 5.39E+04 5.84E+04 6.60E+04 6.92E+04 8.16E+04 8.46E+04 1.01E+05 1.06E+05 1.31E+05 1.64E+05 2.60E+05

74 Tungsten 1.24E+04 1.27E+04 1.45E+04 1.54E+04 1.74E+04 1.96E+04 2.11E+04 2.86E+04 3.29E+04 3.97E+04 4.26E+04 4.80E+04 5.13E+04 5.83E+04 6.19E+04 7.15E+04 7.51E+04 8.82E+04 9.15E+04 1.10E+05 1.12E+05 1.40E+05 1.77E+05 2.18E+05

75 Rhenium 1.30E+04 1.34E+04 1.52E+04 1.62E+04 1.83E+04 2.07E+04 2.22E+04 3.01E+04 3.67E+04 4.43E+04 4.78E+04 5.34E+04 5.72E+04 6.50E+04 6.88E+04 7.95E+04 8.33E+04 9.83E+04 1.02E+05 1.22E+05 1.25E+05 1.55E+05 1.96E+05 2.46E+05

76 Osmium 1.37E+04 1.41E+04 1.59E+04 1.71E+04 1.93E+04 2.17E+04 2.34E+04 3.16E+04 3.73E+04 4.48E+04 4.84E+04 5.41E+04 5.80E+04 6.55E+04 6.97E+04 8.03E+04 8.44E+04 9.94E+04 1.02E+05 1.23E+05 1.27E+05 1.57E+05 1.99E+05 2.49E+05

77 Iridium 1.44E+04 1.48E+04 1.69E+04 1.80E+04 2.02E+04 2.28E+04 2.46E+04 3.31E+04 4.01E+04 4.83E+04 5.20E+04 5.81E+04 6.24E+04 7.09E+04 7.51E+04 8.68E+04 9.11E+04 1.07E+05 1.11E+05 1.34E+05 1.37E+05 1.70E+05 2.15E+05 2.65E+05

78 Platinum 1.52E+04 1.56E+04 1.78E+04 1.89E+04 2.13E+04 2.40E+04 2.58E+04 3.48E+04 4.08E+04 4.89E+04 5.28E+04 5.66E+04 6.34E+04 6.87E+04 7.62E+04 8.45E+04 9.20E+04 1.04E+05 1.12E+05 1.31E+05 1.34E+05 1.66E+05 2.11E+05 2.66E+05

79 Gold 1.60E+04 1.64E+04 1.77E+04 1.99E+04 2.24E+04 2.52E+04 2.71E+04 3.65E+04 4.24E+04 5.08E+04 5.58E+04 6.14E+04 6.69E+04 7.49E+04 8.03E+04 9.17E+04 9.71E+04 1.14E+05 1.18E+05 1.41E+05 1.45E+05 1.79E+05 2.23E+05 2.80E+05

80 Mercury 1.68E+04 1.72E+04 1.94E+04 2.09E+04 2.35E+04 2.65E+04 2.84E+04 3.82E+04 4.29E+04 5.18E+04 5.58E+04 6.20E+04 6.68E+04 6.91E+04 8.02E+04 9.14E+04 9.71E+04 1.09E+05 1.19E+05 1.36E+05 1.42E+05 1.80E+05 2.33E+05 2.98E+05

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

227





81 Thallium 1.76E+04 1.80E+04 1.99E+04 2.19E+04 2.46E+04 2.78E+04 2.98E+04 4.01E+04 4.83E+04 5.81E+04 6.29E+04 7.03E+04 7.54E+04 8.58E+04 9.11E+04 1.05E+05 1.11E+05 1.31E+05 1.36E+05 1.64E+05 1.68E+05 2.22E+05 2.66E+05 3.36E+05

82 Lead 1.84E+04 1.89E+04 2.05E+04 2.29E+04 2.58E+04 2.91E+04 3.12E+04 4.19E+04 5.11E+04 6.15E+04 6.66E+04 7.44E+04 7.98E+04 9.01E+04 9.64E+04 1.12E+05 1.17E+05 1.39E+05 1.48E+05 1.74E+05 1.78E+05 2.22E+05 2.82E+05 3.56E+05

83 Bismuth 1.93E+04 1.98E+04 2.27E+04 2.40E+04 2.70E+04 3.04E+04 3.27E+04 4.38E+04 5.42E+04 6.51E+04 7.04E+04 7.86E+04 8.43E+04 9.57E+04 1.02E+05 1.17E+05 1.23E+05 1.46E+05 1.51E+05 1.82E+05 1.86E+05 2.32E+05 2.94E+05 3.67E+05

84 Polonium 2.02E+04 2.07E+04 2.37E+04 2.51E+04 2.82E+04 3.18E+04 3.41E+04 4.58E+04 5.66E+04 6.80E+04 7.35E+04 8.22E+04 8.81E+04 1.00E+05 1.06E+05 1.22E+05 1.29E+05 1.51E+05 1.57E+05 1.89E+05 1.93E+05 1.99E+05 2.88w+05 3.17E+05

85 Astatine 2.11E+04 2.16E+04 2.49E+04 2.62E+04 2.94E+04 3.31E+04 3.56E+04 4.07E+04 5.56E+04 6.68E+04 7.21E+04 8.04E+04 8.65E+04 9.82E+04 1.04E+05 1.20E+05 1.26E+05 1.49E+05 1.54E+05 1.86E+05 1.91E+05 2.37E+05 2.99E+05 3.78E+05

86 Radon 2.20E+04 2.26E+04 2.55E+04 2.73E+04 3.07E+04 3.45E+04 3.71E+04 3.98E+04 6.22E+04 7.48E+04 8.11E+04 9.06E+04 9.72E+04 1.11E+05 1.17E+05 1.36E+05 1.43E+05 1.69E+05 1.75E+05 2.11E+05 2.16E+05 2.70E+05 3.43E+05 4.33E+05

87 Francium 2.30E+04 2.36E+04 2.60E+04 2.85E+04 3.20E+04 3.60E+04 3.87E+04 3.22E+04 6.55E+04 7.87E+04 8.92E+04 9.94E+04 1.02E+05 1.16E+05 1.23E+05 1.42E+05 1.49E+05 1.77E+05 1.82E+05 2.21E+05 2.23E+05 2.82E+05 3.56E+05 4.49E+05

88 Radium 2.41E+04 2.46E+04 2.68E+04 2.98E+04 3.34E+04 3.76E+04 4.03E+04 3.30E+04 6.56E+04 7.80E+04 8.55E+04 9.53E+04 1.02E+05 1.16E+05 1.23E+05 1.43E+05 1.49E+05 1.76E+05 1.83E+05 2.19E+05 2.25E+05 2.79E+05 3.53E+05 4.99E+05


89 Actinium 2.51E+04 2.57E+04 2.83E+04 3.11E+04 3.48E+04 3.92E+04 3.42E+04 5.40E+04 1.07E+05 1.14E+05 1.16E+05 1.19E+05 1.43E+05 1.50E+05 1.67E+05 1.47E+05 1.74E+05 1.96E+05 2.00E+05 2.32E+05 2.37E+05 2.79E+05 3.32E+05 3.95E+05

90 Thorium 2.62E+04 2.68E+04 3.09E+04 3.23E+04 3.62E+04 4.07E+04 3.80E+04 3.70E+04 6.57E+04 8.70E+04 9.84E+04 1.10E+05 1.18E+05 1.34E+05 1.42E+05 1.54E+05 1.72E+05 1.87E+05 1.96E+05 2.35E+05 2.40E+05 2.96E+05 3.77E+05 4.76E+05

91 Protactinium 2.73E+04 2.79E+04 3.00E+04 3.42E+04 3.77E+04 4.24E+04 4.55E+04 3.87E+04 6.74E+04 8.11E+04 8.78E+04 9.82E+04 1.06E+05 1.19E+05 1.27E+05 1.46E+05 1.53E+05 1.81E+05 1.88E+05 2.27E+05 2.31E+05 2.88E+05 3.83E+05 4.84E+05

92 Uranium 2.84E+04 2.90E+04 3.54E+04 3.50E+04 3.40E+04 2.74E+04 2.96E+04 4.03E+04 7.11E+04 8.54E+04 9.23E+04 1.03E+05 1.12E+05 1.26E+05 1.33E+05 1.54E+05 1.61E+05 1.90E+05 1.97E+05 2.37E+05 2.43E+05 3.03E+05 3.82E+05 4.88E+05

93 Neptunium 2.95E+04 3.02E+04 3.42E+04 2.99E+04 4.08E+04 4.58E+04 4.91E+04 2.57E+04 7.47E+04 8.92E+04 9.67E+04 1.08E+05 1.23E+05 1.32E+05 1.40E+05 1.61E+05 1.69E+05 1.98E+05 2.17E+05 2.48E+05 2.53E+05 3.14E+05 3.97E+05 3.79E+05

94 Plutonium 3.07E+04 3.14E+04 3.03E+04 2.27E+04 4.24E+04 4.76E+04 5.10E+04 1.62E+04 7.29E+04 8.75E+04 9.48E+04 1.06E+05 1.13E+05 1.28E+05 1.36E+05 1.57E+05 1.65E+05 1.94E+05 2.02E+05 2.43E+05 2.48E+05 3.08E+05 3.90E+05 3.65E+05

95 Americium 3.18E+04 3.26E+04 3.34E+04 2.50E+04 4.39E+04 4.93E+04 5.29E+04 1.89E+04 7.63E+04 9.27E+04 1.01E+05 1.14E+05 1.44E+05 1.46E+05 1.48E+05 1.73E+05 1.81E+05 2.15E+05 2.43E+05 2.72E+05 2.78E+05 3.49E+05 4.47E+05 4.26E+05

96 Curium 2.89E+04 2.96E+04 2.36E+04 2.46E+04 4.05E+04 4.57E+04 4.92E+04 2.01E+04 7.95E+04 9.51E+04 1.03E+05 1.15E+05 1.38E+05 1.41E+05 1.47E+05 1.73E+05 1.79E+05 2.11E+05 2.42E+05 2.62E+05 2.68E+05 3.33E+05 4.22E+05 4.03E+05

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

228


4.2. X-RAYS 4.2.4.3. Comparison between theoretical and experimental data sets

Table 4.2.4.2. Total photon interaction cross section (barns=atom) (cont.) Energy (MeV) 2.494E 02 2.382E 02 2.272E 02 2.210E 02 2.112E 02 2.017E 02 1.961E 02 1.744E 02 9.572E 03 8.905E 03 8.631E 03 8.265E 03 8.041E 03 7.649E 03 7.472E 03 7.058E 03 6.925E 03 6.490E 03 6.400E 03 5.947E 03 5.895E 03 5.412E 03 4.932E 03 4.509E 03

Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K

97 Berkelium 2.13E+04 2.18E+04 2.41E+04 2.50E+04 2.98E+04 3.37E+04 3.64E+04 2.01E+04 7.63E+04 9.27E+04 1.01E+05 1.13E+05 1.43E+05 1.46E+05 1.48E+05 1.73E+05 1.82E+05 2.16E+05 2.43E+05 2.72E+05 2.78E+05 3.49E+05 4.47E+05 4.26E+05

98 Californium 3.06E+04 3.44E+04 3.86E+04 2.89E+04 4.62E+04 5.21E+04 5.59E+04 2.09E+04 8.67E+04 1.04E+04 1.13E+05 1.26E+05 1.50E+05 1.52E+05 1.61E+05 1.87E+05 1.96E+05 2.30E+05 2.53E+05 2.86E+05 2.93E+05 3.63E+05 4.59E+05 4.38E+05

This equation is not in a convenient form for computation and the alternative formalism presented by Sano, Ohtaka & Ohtsuki (1969) is often used in calculations. In this formalism, TD 2re2

R1 1

Cp f 2 q; Zf1

exp 2Mqg dcos ': 4:2:4:9

The values of f q; Z are those of Cromer & Waber (1974). Cross sections calculated using equation (4.2.4.8) tend to oscillate at low energy and this corresponds to the inclusion of Bragg peaks in the summation or integration. Eventually, these oscillations abate and TD becomes a smoothly varying function of energy. Creagh & Hubbell (1987) and Creagh (1987) have stressed that, before cross sections are calculated for a given ensemble of atoms, care should be taken to ascertain whether single-atom or single-crystal scattering is appropriate for that ensemble. 4.2.4.2.3. Theoretical Compton scattering data: C by

The bound-electron Compton scattering cross section is given C r 2e

R1 1

1 k1

cos '

f cos2 ' k2 1 1 k1

2

Saloman & Hubbell (1986) and Saloman et al. (1988) have published an extensive comparison of the experimental database with the theoretical values of Sco®eld (1973, 1986) for photon energies between 0.1 and 100 keV. Some examples taken from Saloman & Hubbell (1986) are shown in Figs. 4.2.4.1, 4.2.4.2, and 4.2.4.3. Comparisons between theory and experiment exist for about 80 elements and space does not permit reproduction of all the available information. This information has been summarized in Fig. 4.2.4.4. Superimposed on the Periodic Table of the elements are two sets of data. The upper set corresponds to the average percent deviation between experiment and theory for the photon energy range 10 to 100 keV. The lower set corresponds to the average percent deviation between experiment and theory for the photon energy range 1 to 10 keV. An upwards pointing arrow " means that exp theor > 0. No arrow implies that exp theor 0: A downwards pointing arrow # means that exp theor < 0: An asterisk means no experimental data set was available. For example: for tin Z 50, the experimental data are on average 5% higher than the theoretical predictions for the range of photon energies from 10 to 100 keV. For the range 1 to 10 keV, the experimental data are on average 7% higher than the theoretical predictions. Fig. 4.2.4.4 is given as a rapid means of comparing theory and experiment. For more detailed information, see Saloman & Hubbell (1986), Saloman et al. (1988), and Creagh (1990). 4.2.4.4. Uncertainty in the data tables It is not possible to generalize on the accuracy of the experimental data sets. Creagh & Hubbell (1987) have shown that many experiments for which the precision quoted by the author is high differ from other accurate measurements by a considerable amount. It must be stressed that the experimental apparatus has to be chosen so that it is appropriate for the atomic system being investigated. Details concerning the proper choice of measuring system are given in Section 4.2.3. Within about 200 eV of an absorption edge, deviations of up to 200% may be observed between theory and experiment. This is the region in which XAFS and XANES oscillations occur. With respect to the theoretical data: the detailed agreement between the several methods for calculating the photo-effect cross sections is quite remarkable and it is estimated that the reliability of these data is to within 2% for the energy range considered in this compilation. Some problems may exist, however, close to the absorption edges. Errors in the calculation of the Rayleigh and the Compton scattering cross sections are assessed to be of the order of 5%. Because the greater proportion of total attenuation is photoelectric, the accuracy of the total scattering cross section should be much better than 5% and usually close to 2%.

cos '2

cos ' 1 gIq; z dcos ':

4.2.5. Filters and monochromators (By D. C. Creagh)

4:2:4:10

Here k h!=mc2 and Iq; z is the incoherent scattering intensity expressed in electron units. The other symbols have the meanings de®ned in xx4.2.4.2.1 and 4.2.4.2.2. Values of C incorporated into the tables of total cross section have been computed using the incoherent scattering intensities from the tabulation by Hubbell et al. (1975) based on the calculations by Cromer & Mann (1967) and Cromer (1969).

4.2.5.1. Introduction All sources of X-rays, whether they be produced by conventional sealed tubes, rotating-anode systems, or synchrotron-radiation sources, emit over a broad spectral range. In many cases, this spectral diversity is of concern, and techniques have been developed to minimize the problem. These techniques

229


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.4.3. Mass attenuation coef®cients (cm2 g 1 ) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



1 Hydrogen 3.63E 01 3.65E 01 3.66E 01 3.67E 01 3.68E 01 3.69E 01 3.70E 01 3.73E 01 3.86E 01 3.88E 01 3.89E 01 3.90E 01 3.91E 01 3.93E 01 3.94E 01 3.96E 01 3.97E 01 4.00E 01 4.00E 01 4.05E 01 4.05E 01 4.12E 01 4.21E 01 4.33E 01

2 Helium 1.89E 01 1.90E 01 1.92E 01 1.93E 01 1.94E 01 1.96E 01 1.97E 01 2.02E 01 2.55E 01 2.68E 01 2.74E 01 2.85E 01 2.92E 01 3.07E 01 3.14E 01 3.35E 01 3.43E 01 3.74E 01 3.81E 01 4.25E 01 4.31E 01 4.98E 01 5.92E 01 7.12E 01

3 Lithium 1.72E 01 1.74E 01 1.77E 01 1.79E 01 1.82E 01 1.85E 01 1.87E 01 1.98E 01 3.64E 01 4.12E 01 4.36E 01 4.73E 01 5.00E 01 5.55E 01 5.84E 01 6.63E 01 6.93E 01 8.10E 01 8.39E 01 1.01E+00 1.03E+00 1.30E+00 1.68E+00 2.18E+00

4 Beryllium 1.95E 01 2.00E 01 2.05E 01 2.09E 01 2.16E 01 2.24E 01 2.29E 01 2.56E 01 7.16E 01 8.53E 01 9.23E 01 1.03E+00 1.11E+00 1.27E+00 1.35E+00 1.58E+00 1.67E+00 2.01E+00 2.09E+00 2.59E+00 2.66E+00 3.44E+00 4.56E+00 6.00E+00

5 Boron 2.37E 01 2.47E 01 2.59E 01 2.67E 01 2.81E 01 2.98E 01 3.09E 01 3.68E 01 1.41E+00 1.73E+00 1.89E+00 2.14E+00 2.31E+00 2.67E+00 2.87E+00 3.39E+00 3.59E+00 4.37E+00 4.55E+00 5.69E+00 5.84E+00 7.59E+00 1.01E+01 1.33E+01

6 Carbon 3.15E 01 3.35E 01 3.58E 01 3.74E 01 4.02E 01 4.35E 01 4.58E 01 5.76E 01 2.69E+00 3.33E+00 3.65E+00 4.15E+00 4.51E+00 5.24E+00 5.62E+00 6.68E+00 7.07E+00 8.62E+00 8.99E+00 1.12E+01 1.16E+01 1.50E+01 1.99E+01 2.62E+01

7 Nitrogen 4.04E 01 4.37E 01 4.77E 01 5.03E 01 5.51E 01 6.07E 01 6.45E 01 8.45E 01 4.42E+00 5.48E+00 6.01E+00 6.85E+00 7.44E+00 8.66E+00 9.29E+00 1.10E+01 1.17E+01 1.42E+01 1.49E+01 1.86E+01 1.91E+01 2.47E+01 3.28E+01 4.30E+01

8 Oxygen 3.29E 01 5.82E 01 6.44E 01 6.85E 01 7.60E 01 8.48E 01 9.08E 01 1.22E+00 6.78E+00 8.42E+00 9.25E+00 1.05E+01 1.15E+01 1.33E+01 1.43E+01 1.70E+01 1.80E+01 2.19E+01 2.28E+01 2.84E+01 2.92E+01 3.78E+01 4.99E+01 6.52E+01


9 Fluorine 6.60E 01 7.35E 01 8.22E 01 8.79E 01 9.84E 01 1.11E+00 1.19E+00 1.63E+00 9.35E+00 1.16E+01 1.28E+01 1.45E+01 1.58E+01 1.83E+01 1.97E+01 2.33E+01 2.47E+01 3.00E+01 3.13E+01 3.89E+01 3.99E+01 5.15E+01 6.78E+01 8.84E+01

10 Neon 9.06E 01 1.02E+00 1.15E+00 1.23E+00 1.39E+00 1.57E+00 1.69E+00 2.35E+00 1.36E+01 1.69E+01 1.86E+01 2.11E+01 2.29E+01 2.66E+01 2.85E+01 3.38E+01 3.58E+01 4.34E+01 4.52E+01 5.61E+01 5.76E+01 7.41E+01 9.72E+01 1.26E+02

11 Sodium 1.13E+00 1.28E+00 1.45E+00 1.56E+00 1.77E+00 2.01E+00 2.17E+00 3.03E+00 1.77E+01 2.20E+01 2.41E+01 2.74E+01 2.97E+01 3.45E+01 3.69E+01 4.37E+01 4.62E+01 5.59E+01 5.82E+01 7.21E+01 7.40E+01 9.49E+01 1.24E+02 1.61E+02

12 Magnesium 1.50E+00 1.70E+00 1.93E+00 2.09E+00 2.37E+00 2.70E+00 2.92E+00 4.09E+00 2.40E+01 2.96E+01 3.25E+01 3.69E+01 4.00E+01 4.63E+01 4.96E+01 5.85E+01 6.19E+01 7.47E+01 7.78E+01 9.62E+01 9.87E+01 1.26E+02 1.65E+02 2.12E+02

13 Aluminium 1.85E+00 2.10E+00 2.39E+00 2.59E+00 2.94E+00 3.36E+00 3.64E+00 5.11E+00 2.98E+01 3.68E+01 4.03E+01 4.58E+01 4.96E+01 5.73E+01 6.13E+01 7.23E+01 7.64E+01 9.21E+01 9.59E+01 1.18E+02 1.21E+02 1.55E+02 2.01E+02 2.59E+02

14 Silicon 2.38E+00 2.71E+00 3.09E+00 3.35E+00 3.81E+00 4.36E+00 4.73E+00 6.64E+00 3.85E+01 4.75E+01 5.20E+01 5.89E+01 6.37E+01 7.36E+01 7.87E+01 9.27E+01 9.78E+01 1.18E+02 1.22E+02 1.51E+02 1.54E+02 1.96E+02 2.55E+02 3.27E+02

15 Phosphorus 2.84E+00 3.24E+00 3.70E+00 4.01E+00 4.57E+00 5.23E+00 5.67E+00 7.97E+00 4.58E+01 5.64E+01 6.17E+01 6.98E+01 7.55E+01 8.70E+01 9.30E+01 1.09E+02 1.15E+02 1.39E+02 1.44E+02 1.77E+02 1.81E+02 2.30E+02 2.97E+02 3.79E+02

16 Sulfur 3.55E+00 4.05E+00 4.64E+00 5.02E+00 5.72E+00 6.55E+00 7.11E+00 9.99E+00 5.68E+01 6.98E+01 7.63E+01 8.63E+01 9.33E+01 1.07E+02 1.15E+02 1.35E+02 1.42E+02 1.70E+02 1.77E+02 2.17E+02 2.22E+02 2.81E+02 3.62E+02 4.60E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

230


4.2. X-RAYS Table 4.2.4.3. Mass attenuation coef®cients (cm2 g 1 ) (cont.) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



17 Chlorine 4.09E+00 4.67E+00 5.35E+00 5.79E+00 6.61E+00 7.55E+00 8.20E+00 1.15E+01 6.48E+01 7.95E+01 8.69E+01 9.81E+01 1.06E+02 1.22E+02 1.30E+02 1.52E+02 1.61E+02 1.92E+02 2.00E+02 2.44E+02 2.50E+02 3.16E+02 4.04E+02 5.11E+02

18 Argon 4.56E+00 5.21E+00 5.96E+00 6.46E+00 7.37E+00 8.42E+00 9.14E+00 1.28E+01 7.14E+01 8.75E+01 9.55E+01 1.08E+02 1.16E+02 1.34E+02 1.43E+02 1.67E+02 1.76E+02 2.10E+02 2.18E+02 2.66E+02 2.72E+02 3.42E+02 4.38E+02 5.56E+02

19 Potassium 5.78E+00 6.60E+00 7.56E+00 8.19E+00 9.33E+00 1.07E+01 1.16E+01 1.62E+01 8.94E+01 1.09E+02 1.19E+02 1.34E+02 1.45E+02 1.66E+02 1.77E+02 2.07E+02 2.18E+02 2.60E+02 2.70E+02 3.28E+02 3.36E+02 4.21E+02 5.38E+02 6.80E+02

20 Calcium 6.92E+00 7.90E+00 9.04E+00 9.79E+00 1.12E+01 1.27E+01 1.38E+01 1.93E+01 1.05E+02 1.29E+02 1.40E+02 1.58E+02 1.70E+02 1.95E+02 2.08E+02 2.42E+02 2.55E+02 3.03E+02 3.14E+02 3.82E+02 3.91E+02 4.90E+02 6.24E+02 7.81E+02

21 Scandium 7.47E+00 8.53E+00 9.76E+00 1.06E+01 1.20E+01 1.38E+01 1.49E+01 2.08E+01 1.12E+02 1.37E+02 1.49E+02 1.67E+02 1.80E+02 2.06E+02 2.20E+02 2.56E+02 2.69E+02 3.19E+02 3.32E+02 4.03E+02 4.12E+02 5.16E+02 6.52E+02 8.08E+02

22 Titanium 8.43E+00 9.61E+00 1.10E+01 1.19E+01 1.36E+01 1.55E+01 1.68E+01 2.34E+01 1.25E+02 1.52E+02 1.66E+02 1.86E+02 2.00E+02 2.27E+02 2.40E+02 2.77E+02 2.91E+02 3.45E+02 3.58E+02 4.44E+02 4.57E+02 5.90E+02 8.54E+01 1.09E+02

23 Vanadium 9.42E+00 1.07E+01 1.23E+01 1.33E+01 1.51E+01 1.73E+01 1.87E+01 2.60E+01 1.37E+02 1.66E+02 1.81E+02 2.03E+02 2.19E+02 2.50E+02 2.66E+02 3.09E+02 3.25E+02 3.85E+02 3.99E+02 4.79E+02 4.89E+02 7.47E+01 9.65E+01 1.23E+02

24 Chromium 1.09E+01 1.24E+01 1.42E+01 1.54E+01 1.75E+01 1.99E+01 2.15E+01 2.99E+01 1.55E+02 1.85E+02 2.01E+02 2.27E+02 2.47E+02 2.93E+02 3.18E+02 3.85E+02 4.08E+02 4.80E+02 4.92E+02 6.70E+01 6.86E+01 8.68E+01 1.12E+02 1.43E+02


25 Manganese 1.21E+01 1.37E+01 1.57E+01 1.70E+01 1.93E+01 2.20E+01 2.38E+01 3.31E+01 1.70E+02 2.07E+02 2.24E+02 2.51E+02 2.70E+02 3.06E+02 3.25E+02 3.75E+02 3.93E+02 5.92E+01 6.16E+01 7.53E+01 7.72E+01 9.75E+01 1.26E+02 1.61E+02

26 Iron 1.38E+01 1.57E+01 1.79E+01 1.94E+01 2.20E+01 2.51E+01 2.71E+01 3.76E+01 1.92E+02 2.32E+02 2.52E+02 2.81E+02 3.02E+02 3.42E+02 3.62E+02 5.43E+01 5.72E+01 6.84E+01 7.10E+01 8.69E+01 8.90E+01 1.13E+02 1.45E+02 1.85E+02

27 Cobalt 1.51E+01 1.72E+01 1.96E+01 2.12E+01 2.41E+01 2.74E+01 2.96E+01 4.10E+01 2.06E+02 2.48E+02 2.69E+02 3.00E+02 3.21E+02 4.81E+01 5.13E+01 6.00E+01 6.32E+01 7.55E+01 7.85E+01 9.60E+01 9.83E+01 1.24E+02 1.60E+02 2.04E+02

28 Nickel 1.74E+01 1.98E+01 2.26E+01 2.44E+01 2.77E+01 3.15E+01 3.40E+01 4.69E+01 2.33E+02 2.79E+02 3.02E+02 4.53E+01 4.88E+01 5.60E+01 5.97E+01 6.98E+01 7.35E+01 8.78E+01 9.13E+01 1.12E+02 1.14E+02 1.44E+02 1.86E+02 2.37E+02

29 Copper 1.83E+01 2.08E+01 2.38E+01 2.56E+01 2.91E+01 3.30E+01 3.57E+01 4.91E+01 2.40E+02 3.92E+01 4.27E+01 4.80E+01 5.18E+01 5.94E+01 6.33E+01 7.40E+01 7.80E+01 9.31E+01 9.68E+01 1.18E+02 1.21E+02 1.53E+02 1.97E+02 2.51E+02

30 Zinc 2.02E+01 2.30E+01 2.62E+01 2.82E+01 3.20E+01 3.63E+01 3.93E+01 5.40E+01 3.59E+01 4.38E+01 4.77E+01 5.37E+01 5.79E+01 6.64E+01 7.08E+01 8.27E+01 8.71E+01 1.04E+02 1.08E+02 1.32E+02 1.35E+02 1.71E+02 2.20E+02 2.80E+02

31 Gallium 2.14E+01 2.43E+01 2.77E+01 2.98E+01 3.38E+01 3.84E+01 4.15E+01 5.70E+01 3.85E+01 4.70E+01 5.12E+01 5.76E+01 6.21E+01 7.12E+01 7.59E+01 8.86E+01 9.34E+01 1.11E+02 1.16E+02 1.42E+02 1.45E+02 1.83E+02 2.35E+02 2.99E+02

32 Germanium 2.31E+01 2.62E+01 2.98E+01 3.21E+01 3.64E+01 4.13E+01 4.46E+01 6.12E+01 4.22E+01 5.14E+01 5.59E+01 6.30E+01 6.79E+01 7.78E+01 8.29E+01 9.69E+01 1.02E+02 1.22E+02 1.27E+02 1.55E+02 1.58E+02 1.99E+02 2.56E+02 3.26E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

231


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.4.3. Mass attenuation coef®cients (cm2 g 1 ) (cont.) Radiation Ag K 1 Pd K 1 Rh K 1 Ag K Pd K Rh K Mo K 1 Mo K Zn K 1 Cu K 1 Zn K Ni K 1 Cu K Co K 1 Ni K Fe K 1 Co K Mn K 1 Fe K Cr K 1 Mn K Cr K Ti K 1 Ti K



33 Arsenic 2.50E+01 2.84E+01 3.23E+01 3.48E+01 3.93E+01 4.46E+01 4.82E+01 6.61E+01 4.64E+01 5.65E+01 6.15E+01 6.93E+01 7.47E+01 8.55E+01 9.11E+01 1.06E+02 1.12E+02 1.34E+02 1.39E+02 1.70E+02 1.74E+02 2.19E+02 2.81E+02 3.57E+02


41 Niobium 4.36E+01 4.92E+01 5.56E+01 5.98E+01 6.71E+01 7.55E+01 8.10E+01 1.77E+01 9.04E+01 1.10E+02 1.20E+02 1.34E+02 1.45E+02 1.66E+02 1.76E+02 2.05E+02 2.16E+02 2.57E+02 2.67E+02 3.25E+02 3.32E+02 4.16E+02 5.32E+02 6.71E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

34 Selenium 2.65E+01 3.00E+01 3.41E+01 3.68E+01 4.16E+01 4.71E+01 5.08E+01 6.95E+01 4.97E+01 6.05E+01 6.59E+01 7.42E+01 8.00E+01 9.16E+01 9.76E+01 1.14E+02 1.20E+02 1.43E+02 1.49E+02 1.82E+02 1.86E+02 2.34E+02 3.00E+02 3.81E+02

35 Bromine 2.91E+01 3.29E+01 3.74E+01 4.03E+01 4.55E+01 5.15E+01 5.55E+01 7.56E+01 5.53E+01 6.74E+01 7.33E+01 8.26E+01 8.90E+01 1.02E+02 1.09E+02 1.27E+02 1.33E+02 1.59E+02 1.65E+02 2.02E+02 2.06E+02 2.60E+02 3.33E+02 4.23E+02

36 Krypton 3.07E+01 3.48E+01 3.95E+01 4.25E+01 4.80E+01 5.43E+01 5.84E+01 7.93E+01 5.92E+01 7.21E+01 7.85E+01 8.83E+01 9.52E+01 1.09E+02 1.16E+02 1.35E+02 1.42E+02 1.70E+02 1.76E+02 2.15E+02 2.20E+02 2.77E+02 3.55E+02 4.50E+02

37 Rubidium 3.32E+01 3.76E+01 4.27E+01 4.59E+01 5.18E+01 5.85E+01 6.30E+01 8.51E+01 6.49E+01 7.90E+01 8.60E+01 9.68E+01 1.04E+02 1.19E+02 1.27E+02 1.48E+02 1.56E+02 1.86E+02 1.93E+02 2.36E+02 2.41E+02 3.03E+02 3.88E+02 4.92E+02

38 Strontium 3.56E+01 4.03E+01 4.57E+01 4.91E+01 5.54E+01 6.25E+01 6.72E+01 9.06E+01 7.06E+01 8.59E+01 9.35E+01 1.05E+02 1.13E+02 1.30E+02 1.38E+02 1.61E+02 1.70E+02 2.02E+02 2.10E+02 2.56E+02 2.62E+02 3.28E+02 4.21E+02 5.32E+02

39 Yttrium 3.84E+01 4.34E+01 4.91E+01 5.29E+01 5.95E+01 6.71E+01 7.21E+01 9.70E+01 7.73E+01 9.40E+01 1.02E+02 1.15E+02 1.24E+02 1.42E+02 1.51E+02 1.76E+02 1.85E+02 2.21E+02 2.29E+02 2.79E+02 2.86E+02 3.58E+02 4.59E+02 5.80E+02

40 Zirconium 4.07E+01 4.60E+01 5.20E+01 5.59E+01 6.29E+01 6.25E+01 7.61E+01 1.63E+01 8.35E+01 1.01E+02 1.10E+02 1.24E+02 1.39E+02 1.54E+02 1.63E+02 1.91E+02 2.00E+02 2.38E+02 2.47E+02 3.00E+02 3.08E+02 3.86E+02 4.93E+02 6.22E+02

42 43 Molybdenum Technetium 5.25E+01 4.84E+01 6.03E+01 5.45E+01 6.80E+01 6.15E+01 7.20E+01 6.60E+01 7.71E+01 7.41E+01 7.95E+01 1.38E+01 1.38E+01 1.49E+01 1.88E+01 2.04E+01 9.65E+01 1.04E+02 1.17E+02 1.26E+02 1.27E+02 1.37E+02 1.43E+02 1.54E+02 1.54E+02 1.66E+02 1.76E+02 1.90E+02 1.88E+02 2.02E+02 2.19E+02 2.35E+02 2.30E+02 2.47E+02 2.73E+02 2.94E+02 2.84E+02 3.05E+02 3.45E+02 3.70E+02 3.53E+02 3.79E+02 4.42E+02 4.74E+02 5.65E+02 6.04E+02 7.12E+02 7.61E+02

44 Ruthenium 5.06E+01 5.69E+01 7.00E+01 1.14E+01 1.29E+01 1.47E+01 1.58E+01 2.17E+01 1.10E+02 1.34E+02 1.46E+02 1.63E+02 1.76E+02 2.01E+02 2.14E+02 2.49E+02 2.62E+02 3.11E+02 3.23E+02 3.92E+02 4.01E+02 5.01E+02 6.39E+02 8.04E+02

45 Rhodium 5.35E+01 6.01E+01 1.14E+01 1.23E+01 1.39E+01 1.58E+01 1.70E+01 2.33E+01 1.18E+02 1.44E+02 1.56E+02 1.75E+02 1.89E+02 2.16E+02 2.29E+02 2.67E+02 2.80E+02 3.33E+02 3.46E+02 4.20E+02 4.29E+02 5.36E+02 6.83E+02 8.60E+02

46 Palladium 5.55E+01 1.06E+01 1.21E+01 1.30E+01 1.47E+01 1.67E+01 1.80E+01 2.47E+01 1.25E+02 1.51E+02 1.65E+02 1.85E+02 1.99E+02 2.27E+02 2.41E+02 2.81E+02 2.95E+02 3.50E+02 3.63E+02 4.41E+02 4.51E+02 5.63E+02 7.16E+02 9.01E+02

47 Silver 1.01E+01 1.15E+01 1.30E+01 1.40E+01 1.58E+01 1.79E+01 1.94E+01 2.65E+01 1.34E+02 1.63E+02 1.77E+02 1.98E+02 2.13E+02 2.43E+02 2.59E+02 3.01E+02 3.16E+02 3.75E+02 3.89E+02 4.72E+02 4.83E+02 6.02E+02 7.65E+02 9.61E+02

48 Cadmium 1.06E+01 1.20E+01 1.36E+01 1.46E+01 1.66E+01 1.88E+01 2.02E+01 2.78E+01 1.40E+02 1.69E+02 1.84E+02 2.07E+02 2.22E+02 2.53E+02 2.69E+02 3.13E+02 3.29E+02 3.90E+02 4.05E+02 4.90E+02 5.02E+02 6.26E+02 7.95E+02 9.95E+02

232





49 Indium 1.13E+01 1.27E+01 1.45E+01 1.56E+01 1.76E+01 2.00E+01 2.16E+01 2.95E+01 1.48E+02 1.80E+02 1.95E+02 2.19E+02 2.36E+02 2.69E+02 2.86E+02 3.32E+02 3.49E+02 4.13E+02 4.28E+02 5.19E+02 5.31E+02 6.63E+02 8.41E+02 1.05E+03

50 Tin 1.18E+01 1.34E+01 1.52E+01 1.64E+01 1.85E+01 2.10E+01 2.26E+01 3.10E+01 1.55E+02 1.88E+02 2.04E+02 2.29E+02 2.47E+02 2.81E+02 2.99E+02 3.47E+02 3.64E+02 4.31E+02 4.47E+02 5.42E+02 5.54E+02 6.91E+02 8.76E+02 1.09E+03


57 Lanthanum 1.72E+01 1.95E+01 2.21E+01 2.38E+01 2.69E+01 3.05E+01 3.29E+01 4.49E+01 2.21E+02 2.66E+02 2.89E+02 3.24E+02 3.48E+02 3.95E+02 4.19E+02 4.83E+02 5.07E+02 5.97E+02 6.18E+02 7.44E+02 7.60E+02 2.25E+02 2.84E+02 3.55E+02

58 59 60 61 Cerium Praseodymium Neodymium Promethium 1.83E+01 1.95E+01 2.04E+01 2.17E+01 2.07E+01 2.20E+01 2.30E+01 2.45E+01 2.35E+01 2.50E+01 2.61E+01 2.78E+01 2.53E+01 2.69E+01 2.81E+01 2.99E+01 2.86E+01 3.04E+01 3.18E+01 3.38E+01 3.24E+01 3.45E+01 3.60E+01 3.83E+01 3.49E+01 3.72E+01 3.88E+01 4.13E+01 4.77E+01 5.07E+01 5.30E+01 5.63E+01 2.33E+02 2.47E+02 2.57E+02 2.73E+02 2.82E+02 2.99E+02 3.10E+02 3.28E+02 3.06E+02 3.24E+02 3.36E+02 3.55E+02 3.43E+02 3.63E+02 3.76E+02 3.97E+02 3.68E+02 3.90E+02 4.04E+02 4.26E+02 4.17E+02 4.41E+02 4.57E+02 4.82E+02 4.42E+02 4.68E+02 4.84E+02 5.11E+02 5.10E+02 5.39E+02 4.92E+02 5.88E+02 5.35E+02 5.65E+02 5.05E+02 4.00E+02 5.47E+02 6.16E+02 4.39E+02 4.68E+02 5.61E+02 4.48E+02 4.55E+02 1.94E+02 4.94E+02 1.88E+02 1.98E+02 2.32E+02 5.12E+02 1.93E+02 2.03E+02 2.37E+02 2.38E+02 2.38E+02 2.51E+02 2.94E+02 3.00E+02 3.00E+02 3.14E+02 3.69E+02 3.57E+02 3.75E+02 3.97E+02 4.62E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

51 Antimony 1.25E+01 1.41E+01 1.60E+01 1.73E+01 1.96E+01 2.22E+01 2.39E+01 3.27E+01 1.64E+02 1.98E+02 2.15E+02 2.41E+02 2.59E+02 2.96E+02 3.14E+02 3.65E+02 3.83E+02 4.54E+02 4.71E+02 5.70E+02 5.82E+02 7.23E+02 9.15E+02 9.91E+02

52 Tellurium 1.29E+01 1.46E+01 1.66E+01 1.79E+01 2.02E+01 2.29E+01 2.47E+01 3.38E+01 1.68E+02 2.04E+02 2.21E+02 2.48E+02 2.67E+02 3.04E+02 3.23E+02 3.74E+02 3.94E+02 4.66E+02 4.83E+02 5.85E+02 5.98E+02 7.40E+02 9.32E+02 7.51E+02

233


53 Iodine 1.40E+01 1.59E+01 1.80E+01 1.94E+01 2.19E+01 2.18E+01 2.68E+01 3.67E+01 1.82E+02 2.20E+02 2.39E+02 2.68E+02 2.88E+02 3.30E+02 3.49E+02 4.08E+02 4.25E+02 5.03E+02 5.22E+02 6.31E+02 6.45E+02 7.96E+02 1.00E+03 2.83E+02

54 Xenon 1.46E+01 1.65E+01 1.88E+01 2.02E+01 2.29E+01 2.27E+01 2.80E+01 3.82E+01 1.90E+02 2.29E+02 2.49E+02 2.78E+02 2.99E+02 3.43E+02 3.62E+02 4.22E+02 4.40E+02 5.20E+02 5.40E+02 6.52E+02 6.66E+02 7.21E+02 1.03E+03 2.65E+02

55 Caesium 1.56E+01 1.76E+01 2.00E+01 2.15E+01 2.43E+01 2.42E+01 2.98E+01 4.07E+01 2.01E+02 2.43E+02 2.63E+02 2.95E+02 3.17E+02 3.63E+02 3.83E+02 4.46E+02 4.65E+02 5.49E+02 5.69E+02 6.86E+02 7.00E+02 7.60E+02 2.60E+02 3.30E+02

56 Barium 1.62E+01 1.83E+01 2.08E+01 2.24E+01 2.54E+01 2.52E+01 3.10E+01 4.23E+01 2.09E+02 2.52E+02 2.73E+02 3.06E+02 3.25E+02 3.76E+02 3.96E+02 4.61E+02 4.80E+02 5.66E+02 5.86E+02 6.45E+02 6.60E+02 5.70E+02 3.14E+02 3.34E+02

62 Samarium 2.23E+01 2.52E+01 2.86E+01 3.08E+01 3.48E+01 3.94E+01 4.24E+01 5.78E+01 2.79E+02 3.35E+02 3.63E+02 4.05E+02 4.34E+02 3.54E+02 3.71E+02 1.63E+02 1.76E+02 1.66E+02 2.04E+02 2.21E+02 2.25E+02 2.79E+02 3.50E+02 4.35E+02

63 Europium 2.35E+01 2.66E+01 3.01E+01 3.24E+01 3.66E+01 4.15E+01 4.47E+01 6.09E+01 2.93E+02 3.52E+02 3.80E+02 4.24E+02 4.34E+02 4.80E+02 3.75E+02 4.08E+02 4.19E+02 1.95E+02 2.03E+02 2.44E+02 2.49E+02 3.09E+02 3.90E+02 4.88E+02

64 Gadolinium 2.42E+01 2.74E+01 3.10E+01 3.34E+01 3.77E+01 4.27E+01 4.60E+01 6.26E+01 3.00E+02 3.60E+02 3.89E+02 4.33E+02 4.03E+02 3.35E+02 3.56E+02 1.53E+02 1.61E+02 1.89E+02 1.95E+02 2.35E+02 2.41E+02 2.98E+02 3.74E+02 4.69E+02




65 Terbium 2.55E+01 2.88E+01 3.26E+01 3.51E+01 3.96E+01 4.49E+01 4.83E+01 6.58E+01 3.14E+02 3.76E+02 4.06E+02 4.52E+02 3.21E+02 3.60E+02 1.49E+02 1.71E+02 1.80E+02 2.11E+02 2.19E+02 2.63E+02 2.69E+02 3.32E+02 4.19E+02 5.24E+02

66 Dysprosium 2.65E+01 2.99E+01 3.39E+01 3.65E+01 4.12E+01 4.66E+01 5.02E+01 6.83E+01 3.24E+02 3.87E+02 4.19E+02 3.36E+02 3.62E+02 1.38E+02 1.46E+02 1.68E+02 1.76E+02 2.07E+02 2.14E+02 2.57E+02 2.62E+02 3.25E+02 4.10E+02 5.15E+02

67 Holmium 2.77E+01 3.13E+01 3.54E+01 3.81E+01 4.30E+01 4.87E+01 5.24E+01 7.13E+01 3.36E+02 4.02E+02 3.98E+02 4.44E+02 1.29E+02 1.46E+02 1.55E+02 1.78E+02 1.87E+02 2.20E+02 2.28E+02 2.72E+02 2.80E+02 3.47E+02 4.38E+02 5.47E+02

68 Erbium 2.90E+01 3.27E+01 3.71E+01 3.99E+01 4.50E+01 5.09E+01 5.48E+01 7.44E+01 3.49E+02 4.17E+02 2.87E+02 1.23E+02 1.32E+02 1.49E+02 1.58E+02 1.82E+02 1.91E+02 2.24E+02 2.32E+02 2.78E+02 2.85E+02 3.52E+02 4.43E+02 5.54E+02

69 Thulium 3.04E+01 3.43E+01 3.89E+01 4.18E+01 4.71E+01 5.33E+01 5.74E+01 7.79E+01 3.65E+02 1.08E+02 1.17E+02 1.31E+02 1.40E+02 1.59E+02 1.69E+02 1.96E+02 2.06E+02 2.43E+02 2.53E+02 3.05E+02 3.12E+02 3.86E+02 4.94E+02 6.21E+02

70 Ytterbium 3.14E+01 3.54E+01 4.01E+01 4.32E+01 4.87E+01 5.50E+01 5.93E+01 8.04E+01 3.75E+02 1.08E+02 1.17E+02 1.31E+02 1.42E+02 1.59E+02 1.69E+02 1.96E+02 2.06E+02 2.44E+02 2.51E+02 3.04E+02 3.11E+02 3.87E+02 4.92E+02 6.19E+02

71 Lutetium 3.28E+01 3.71E+01 4.20E+01 4.51E+01 5.09E+01 5.75E+01 6.19E+01 8.40E+01 3.91E+02 1.21E+02 1.31E+02 1.46E+02 1.56E+02 1.78E+02 1.89E+02 2.18E+02 2.29E+02 2.70E+02 2.80E+02 3.39E+02 3.47E+02 4.31E+02 5.47E+02 6.88E+02

72 Hafnium 3.40E+01 3.84E+01 4.35E+01 4.67E+01 5.27E+01 5.95E+01 6.41E+01 8.69E+01 1.00E+02 1.20E+02 1.30E+02 1.45E+02 1.55E+02 1.76E+02 1.87E+02 2.16E+02 2.27E+02 2.67E+02 2.77E+02 3.34E+02 3.41E+02 4.25E+02 5.39E+02 6.78E+02


73 Tantalum 3.54E+01 4.00E+01 4.53E+01 4.87E+01 5.48E+01 6.20E+01 6.67E+01 9.04E+01 1.02E+02 1.22E+02 1.32E+02 1.47E+02 1.58E+02 1.79E+02 1.90E+02 2.20E+02 2.31E+02 2.73E+02 2.83E+02 3.39E+02 3.46E+02 4.32E+02 5.46E+02 6.85E+02

74 Tungsten 3.68E+01 4.15E+01 4.70E+01 5.05E+01 5.69E+01 6.43E+01 6.92E+01 9.38E+01 1.08E+02 1.30E+02 1.41E+02 1.57E+02 1.68E+02 1.91E+02 2.03E+02 2.34E+02 2.46E+02 2.88E+02 3.01E+02 3.61E+02 3.69E+02 4.57E+02 5.79E+02 7.25E+02

75 Rhenium 3.83E+01 4.32E+01 4.89E+01 5.25E+01 5.92E+01 6.69E+01 7.19E+01 9.74E+01 1.19E+02 1.43E+02 1.55E+02 1.72E+02 1.87E+02 2.09E+02 2.22E+02 2.57E+02 2.68E+02 3.16E+02 3.27E+02 3.94E+02 4.05E+02 5.01E+02 6.33E+02 7.94E+02

76 Osmium 3.95E+01 4.45E+01 5.04E+01 5.41E+01 6.10E+01 6.89E+01 7.41E+01 1.00E+02 1.18E+02 1.42E+02 1.54E+02 1.71E+02 1.84E+02 2.09E+02 2.21E+02 2.55E+02 2.68E+02 3.14E+02 3.27E+02 3.92E+02 4.03E+02 4.99E+02 6.31E+02 7.92E+02

77 Iridium 4.11E+01 4.64E+01 5.24E+01 5.63E+01 6.34E+01 7.16E+01 7.70E+01 1.04E+02 1.23E+02 1.48E+02 1.60E+02 1.78E+02 1.91E+02 2.16E+02 2.30E+02 2.65E+02 2.78E+02 3.30E+02 3.40E+02 4.11E+02 4.18E+02 5.20E+02 6.59E+02 8.26E+02

78 Platinum 4.26E+01 4.80E+01 5.43E+01 5.83E+01 6.57E+01 7.41E+01 7.97E+01 1.07E+02 1.21E+02 1.45E+02 1.57E+02 1.75E+02 1.88E+02 2.14E+02 2.27E+02 2.61E+02 2.76E+02 3.25E+02 3.57E+02 4.23E+02 4.34E+02 5.41E+02 6.83E+02 8.19E+02

79 Gold 4.44E+01 5.00E+01 5.65E+01 6.07E+01 6.83E+01 7.71E+01 8.29E+01 1.12E+02 1.30E+02 1.55E+02 1.68E+02 1.88E+02 2.01E+02 2.29E+02 2.43E+02 2.79E+02 2.95E+02 3.48E+02 3.61E+02 4.34E+02 4.45E+02 5.51E+02 6.99E+02 8.76E+02

80 Mercury 4.58E+01 5.16E+01 5.83E+01 6.26E+01 7.04E+01 7.95E+01 8.54E+01 1.15E+02 1.16E+02 1.41E+02 1.54E+02 1.74E+02 1.88E+02 2.16E+02 2.30E+02 2.60E+02 2.73E+02 3.27E+02 3.39E+02 4.16E+02 4.27E+02 5.41E+02 6.99E+02 8.97E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

234





81 Thallium 4.72E+01 5.31E+01 6.00E+01 6.45E+01 7.25E+01 8.18E+01 8.79E+01 1.18E+02 1.45E+02 1.75E+02 1.89E+02 2.11E+02 2.26E+02 2.57E+02 2.71E+02 3.14E+02 3.31E+02 3.90E+02 4.03E+02 4.87E+02 5.00E+02 5.97E+02 7.15E+02 9.89E+02

82 Lead 4.88E+01 5.49E+01 6.20E+01 6.66E+01 7.49E+01 8.45E+01 9.08E+01 1.22E+02 1.51E+02 1.81E+02 1.96E+02 2.16E+02 2.35E+02 2.67E+02 2.83E+02 3.27E+02 3.43E+02 4.06E+02 4.20E+02 5.07E+02 5.18E+02 6.43E+02 8.15E+02 1.03E+03

83 Bismuth 5.06E+01 5.70E+01 6.44E+01 6.91E+01 7.77E+01 8.76E+01 9.41E+01 1.26E+02 1.57E+02 1.88E+02 2.04E+02 2.28E+02 2.44E+02 2.76E+02 2.95E+02 3.39E+02 3.55E+02 4.21E+02 4.34E+02 5.24E+02 5.35E+02 6.66E+02 8.44E+02 1.06E+03

84 Polonium 5.30E+01 5.96E+01 6.73E+01 7.23E+01 8.12E+01 9.15E+01 9.83E+01 1.32E+02 1.63E+02 1.96E+02 2.12E+02 2.37E+02 2.54E+02 2.88E+02 3.05E+02 3.54E+02 3.70E+02 4.35E+02 4.52E+02 5.44E+02 5.58E+02 6.91E+02 8.30E+02 1.10E+03

85 Astatine 5.51E+01 6.20E+01 7.00E+01 7.51E+01 8.43E+01 9.50E+01 1.02E+02 1.17E+02 1.71E+02 1.86E+02 2.07E+02 2.31E+02 2.48E+02 2.82E+02 2.99E+02 3.45E+02 3.63E+02 4.26E+02 4.44E+02 5.33E+02 5.45E+02 6.80E+02 8.60E+02 1.08E+03

86 Radon 5.45E+01 6.12E+01 6.90E+01 7.21E+01 8.32E+01 9.36E+01 1.01E+02 1.08E+02 1.71E+02 2.05E+02 2.23E+02 2.49E+02 2.67E+02 3.04E+02 3.21E+02 3.73E+02 3.92E+02 4.60E+02 4.77E+02 5.76E+02 5.89E+02 7.34E+02 9.32E+02 1.18E+03

87 Francium 5.67E+01 6.37E+01 7.18E+01 7.70E+01 8.64E+01 9.72E+01 1.04E+02 8.70E+01 1.77E+02 2.13E+02 2.30E+02 2.57E+02 2.77E+02 3.12E+02 3.32E+02 3.84E+02 4.03E+02 4.77E+02 4.93E+02 5.97E+02 6.02E+02 7.58E+02 9.61E+02 1.21E+03

88 Radium 5.84E+01 6.56E+01 7.40E+01 7.93E+01 8.90E+01 1.00E+02 1.08E+01 8.80E+01 1.75E+02 2.10E+02 2.28E+02 2.54E+02 2.73E+02 3.10E+02 3.29E+02 3.80E+02 3.98E+02 4.70E+02 4.87E+02 5.85E+02 5.99E+02 7.43E+02 9.41E+02 1.33E+03


89 Actinium 6.07E+01 6.82E+01 7.68E+01 8.24E+01 9.24E+01 1.04E+02 1.10E+02 9.08E+01 2.49E+02 2.85E+02 3.03E+02 3.09E+02 3.17E+02 3.81E+02 3.99E+02 4.44E+02 4.61E+02 5.21E+02 5.30E+02 6.18E+02 6.29E+02 7.39E+02 8.83E+02 1.05E+03

90 Thorium 6.19E+01 6.95E+01 7.82E+01 8.39E+01 9.41E+01 1.06E+02 9.87E+01 9.65E+01 1.70E+02 2.19E+02 2.55E+02 2.85E+02 3.06E+02 3.48E+02 3.69E+02 3.89E+02 4.06E+02 4.46E+02 4.85E+02 5.09E+02 6.23E+02 7.68E+02 9.78E+02 1.23E+03

91 Protactinium 6.48E+01 7.27E+01 8.19E+01 8.78E+01 9.84E+01 1.11E+02 1.19E+02 1.01E+02 1.73E+02 2.08E+02 2.25E+02 2.52E+02 2.71E+02 3.06E+02 3.25E+02 3.75E+02 3.94E+02 4.65E+02 4.82E+02 5.82E+02 5.93E+02 7.38E+02 9.83E+02 1.24E+03

92 Uranium 6.55E+01 7.35E+01 8.27E+01 8.86E+01 9.93E+01 1.12E+02 7.49E+01 1.02E+02 1.85E+02 2.22E+02 2.40E+02 2.68E+02 2.88E+02 3.26E+02 3.47E+02 4.00E+02 4.20E+02 4.96E+02 5.28E+02 6.17E+02 6.32E+02 7.66E+02 9.66E+02 1.23E+03

93 Neptunium 6.84E+01 7.67E+01 8.63E+01 9.25E+01 1.04E+02 1.16E+02 1.25E+02 4.22E+01 1.90E+02 2.27E+02 2.46E+02 2.75E+02 3.14E+02 3.35E+02 3.55E+02 4.10E+02 4.30E+02 5.05E+02 5.52E+02 6.30E+02 6.45E+02 8.00E+02 1.01E+03 9.65E+02

94 Plutonium 7.05E+01 7.91E+01 8.89E+01 5.60E+01 1.07E+02 1.20E+02 1.29E+02 3.99E+01 1.80E+02 2.16E+02 2.34E+02 2.62E+02 2.80E+02 3.17E+02 3.36E+02 3.89E+02 4.08E+02 4.08E+02 4.98E+02 6.00E+02 6.12E+02 7.60E+02 9.62E+02 9.00E+02

95 Americium 7.20E+01 8.08E+01 9.08E+01 5.95E+01 1.09E+02 1.22E+02 1.31E+02 4.81E+01 1.89E+02 2.27E+02 2.41E+02 2.73E+02 3.22E+02 3.33E+02 3.52E+02 4.07E+02 4.26E+02 5.03E+02 5.81E+02 6.27E+02 6.42E+02 7.95E+02 1.03E+03 9.55E+02

96 Curium 7.35E+01 8.24E+01 6.00E+01 6.43E+01 1.11E+02 1.10E+02 1.34E+02 4.90E+01 1.94E+02 2.32E+02 2.51E+02 2.80E+02 3.38E+02 3.43E+02 3.60E+02 4.21E+02 4.37E+02 5.15E+02 5.90E+02 6.40E+02 6.55E+02 8.12E+02 1.03E+03 9.84E+02

02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

235




97 Berkelium 6.66E+01 7.52E+01 8.51E+01 6.10E+01 1.03E+02 1.02E+02 1.25E+02 4.90E+01 1.86E+02 2.26E+02 2.46E+02 2.77E+02 3.52E+02 3.57E+02 3.62E+02 4.22E+02 4.43E+02 5.26E+02 5.92E+02 6.64E+02 6.78E+02 8.52E+02 1.09E+03 1.04E+03

re is the classical radius of the electron, and Nj is the number density of atoms of type j. An angle of total external re¯ection c exists for the material, which is a function of the incident photon energy, since fj !; is a function of photon energy. Thus, a polychromatic beam incident at the critical angle of one of the photon energies E will re¯ect totally those components having energies less than E, and transmit those components with energies greater than E. Fig. 4.2.5.1 shows calculations by Fukumachi, Nakano & Kawamura (1986) for the re¯ectivity of single layers of aluminium, copper and platinum as a function of incident energy for a ®xed angle of incidence (0.2 ). For the aluminium specimen, the re¯ectivity curve shows the rapid decrease in re¯ectivity as the critical angle is exceeded. The re¯ectivity in this region varies as E 2 . The effect of increasing atomic number can be seen: the higher the atomic factor f !; , the greater the energy that can be re¯ected from the surface. Also visible are the effects of the dispersion corrections f 0 !; and f 00 !; on re¯ectivity. For copper, the K shell is excited, and for platinum the LI , LII and LIII shells are excited by the polychromatic beam. Interfaces can therefore be used to act as low-pass energy ®lters. The surface roughness and the existence of impurities and contaminants on the interface will, however, in¯uence the characteristics of the re¯ecting surface, sometimes signi®cantly.

98 Californium 7.35E+01 8.24E+01 9.26E+01 6.92E+01 1.11E+02 1.25E+02 1.34E+02 5.00E+01 2.08E+02 2.49E+02 2.70E+02 3.01E+02 3.60E+02 3.66E+02 3.86E+02 4.48E+02 4.69E+02 5.52E+02 6.07E+02 6.87E+02 7.03E+02 8.71E+02 1.10E+03 1.05E+03

4.2.5.2. Mirrors and capillaries

involve the use of ®lters, mirrors, and Laue and Bragg crystal monochromators, chosen so as to provide the best compromise between ¯ux and spectral purity in a particular experiment. In other chapters, authors have discussed the use of techniques to improve the spectral purity of X-ray sources. This section does not purport to be a comprehensive exposition on the topic of ®lters and monochromators. Rather, it seeks to point the reader towards the information given elsewhere in this volume, and to add complementary information where necessary. A search of the Subject Index will ®nd references to ®lters and monochromators that are not explicitly mentioned in the text of this section. The ability to select photon energies, or bands of energies, depends on the scattering power of the atoms from which the monochromator is made and the arrangement of the atoms within the monochromator. The scattering powers of the atoms and their dependence on the energy of the incident photons were discussed in Sections 4.2.3 and 4.2.4 and are discussed more fully in Section 4.2.6. In brief, the scattering power of the atom, or atomic scattering factor, is de®ned, for a given incident photon energy, as the ratio of the scattering power of the atom to that of a free Thomson electron. The scattering power is denoted by the symbol f !; and is a complex quantity, the real part of which, f 0 !; , is related to the elastic scattering cross section, and the imaginary part of which, f 00 !; , is related directly to the photoelectric scattering cross section and therefore the linear attenuation coef®cient l . At an interface between, say, air and the material from which the monochromator is made, re¯ection and refraction of the incident photons can occur, as dictated by Maxwell's equations. There is an associated refractive index n given by n 1 1=2 ; where

re l2 =

P j

Nj fj !; ;

4:2:5:1

4:2:5:2

Whilst neither of these classes of X-ray optical device is strictly speaking a monochromator, they nevertheless form component parts of monochromator systems in the laboratory and at synchrotron-radiation sources. 4.2.5.2.1. Mirrors In the laboratory, mirrors are used in conjunction with conventional sealed tubes and rotating-anode sources, the emission from which consists of Bremsstrahlung upon which is superimposed the characteristic spectrum of the anode material (Subsection 2.3.5.2). The shape of the Bremsstrahlung spectrum can be signi®cantly modi®ed by mirrors, and the intensity emitted at harmonics of the characteristic wavelength can be

Fig. 4.2.5.1. The variation of specular re¯ectivity with incident photon energy is shown for materials of different atomic number and a constant angle of incidence of 0.2 . (a) Aluminium: note the rapid decrease of re¯ectivity with energy. (b) Copper: the sudden decrease of re¯ectivity is due to the modi®cation of the scattering-length density owing to absorption at the K-absorption edge. (c) Platinum: the three discontinuities in the re¯ectivity curve are due to absorption at the LI -, LII -, and LIII -absorption edges.

236


4.2. X-RAYS signi®cantly reduced. More importantly, the mirrors can be fashioned into shapes that enable the emitted radiation to be brought to a focus. Ellipsoidal, logarithmic spiral, and toroidal mirrors have been manufactured commercially for use with laboratory X-ray sources. Since the X-rays are emitted isotropically from the anode surface, it is important to devise a mirror system that has a maximum angle of acceptance and a relatively long focal length. At synchrotron-radiation sources, the high intensities that are generated over a very broad spectral range give rise to signi®cant heat loading of subsequent monochromators and therefore degrade the performance of these elements. In many systems, mirrors are used as the ®rst optical element in the monochromator, to reduce the heat load on the primary monochromator and to make it easier for the subsequent monochromators to reject harmonics of the chosen radiation. Shaped mirror geometries are often used to focus the beam in the horizontal plane (Subsection 2.2.7.3). A schematic diagram of the optical elements of a typical synchrotron-radiation beamline is shown in Fig. 4.2.5.2. In this, the primary mirror acts as a thermal shunt for the subsequent monochromator, minimizes the high-energy component that may give rise to possible harmonic content in the ®nal beam, and acts as a vertical collimator. The radii of curvature of mirrors can be changed using a mechanical fourpoint bending system (Oshima, Harada & Sakabe, 1986). More recent advances in mirror technology enable the shape of the mirror to be changed through use of the piezoelectric effect (Sussini & Labergerie, 1995). 4.2.5.2.2. Capillaries Capillaries, and bundles of capillaries, are ®nding increasing use in situations where a focused beam is required. The radiation is guided along the capillary by total external re¯ection, and the shape of the capillary determines the overall ¯ux gain and the uniformity of the focused spot. Gains in ¯ux of 100 and better have been reported. There is, however, a degradation in the angular divergence of the outgoing beam. For single capillaries, applications are laboratory-based protein crystallography, microtomography, X-ray microscopy, and micro-X-ray ¯uorescence spectroscopy. The design and construction of capillaries for use in the laboratory and at synchrotron-radiation sources has been discussed by Bilderback, Thiel, Pahl & Brister (1994), Balaic & Nugent (1995), Balaic, Nugent, Barnea, Garrett & Wilkins (1995), Balaic et al. (1996), and Engstrom, Rindby & Vincze (1996). They are usually used after other monochroma-

tors in these applications and their role as a low-pass energy ®lter is not of much signi®cance. Bundles of capillaries are currently being produced commercially to produce focused beams (ellipsoidally shaped bundles) and half-ellipsoidal bundles are used to form beams of large cross section from conventional laboratory sources (Peele et al., 1996; Kumakov & Komarov, 1990). 4.2.5.2.3. Quasi-Bragg re¯ectors For one interface, the re¯ectivity (R) and the transmissivity (T ) of the surface are determined by the Fresnel equations, viz: 2 R 1 2 =1 2 ; 4:2:5:3 and

2 T 21 =1 2 ;

4:2:5:4

where 1 and 2 are the angles between the incident ray and the surface plane and the re¯ected ray and the surface plane, respectively. If a succession of interfaces exists, the possibility of interference between successively re¯ected rays exists. Parameters that de®ne the position of the interference maxima, the line breadths of those maxima, and the line intensity depend inter alia on the regularity in layer thickness, the interface surface roughness, and the existence of surface tilts between successive interfaces. Algorithms for solving this type of problem are incorporated in software currently available from a number of commercial sources (Bede Scienti®c, Siemens, and Philips). The re¯ectivity pro®le of a system having a periodic layer structure is shown in Fig. 4.2.5.3. This is the re¯ectivity pro®le for a multiple-quantum-well structure of alternating aluminium gallium arsenide and indium gallium arsenide layers (Holt, Brown, Creagh & Leon, 1997). Note the interference maxima that are superimposed on the Fresnel re¯ectivity curve. From the full width at half-maximum of these interference lines, it can be inferred that the energy discrimination of the system, E=E, is 2%. The energy range that can be re¯ected by such a multilayer system depends on the interlayer thickness: the higher the photon energy, the thinner the layer thickness. Commercially available multilayer mirrors exist, and hitherto they have been used as monochromators in the soft X-ray region in X-ray ¯uorescence spectrometers. These monochromators are typically made of alternating layers of tungsten and carbon, to maximize the difference in scattering-

Fig. 4.2.5.2. The use of mirrors in a typical synchrotron-radiation beamline. The X-rays are emitted tangentially to the orbit of the stored positron beam. They pass through a beam-de®ning slit onto a mirror that serves three purposes, viz energy discrimination, heat absorption, and focusing, by means of a mechanical four-point bending system. The beam then passes into a double-crystal monochromator, which selects the desired photon energy. The second element of this monochromator is capable of being bent sagittally using a mechanical four-point bending system to focus the beam in the horizontal plane. The beam is then refocused and redirected by a second mirror.

237


4. PRODUCTION AND PROPERTIES OF RADIATIONS length density at the interfaces. Whilst the energy resolution of such systems is not especially good, these monochromators have a good angle of acceptance for the incident beam, and reasonably high photon ¯uxes can be achieved using conventional laboratory sources. A recent development of this, the Goebel mirror system, is supplied as an accessory to a commercially available diffractometer (Siemens, 1996a,b,c; OSMIC, 1996). This system combines the focusing capacity of a curved mirror with the energy selectivity of the multilayer system. The spacing between layers in this class of mirror multilayers can be laterally graded to enhance the incident acceptance angle. These multilayers can be ®xed to mirrors of any ®gure to a precision of 0.30 and can therefore can be used to form parallel beams (parabolic optical elements) as well as focused beams (elliptical optical elements) of high quality. 4.2.5.3. Filters It is usual to consider only the cases where a quasimonochromatic beam is to be extracted from a polychromatic beam. Before discussing this class of usage, mention must be made of two simple forms of ®ltering of radiation. In the ®rst, screening, a thin layer of absorbing foil is used to reduce the effect of specimen ¯uorescence on photon counting, ®lm and imaging-plate detectors. A typical example is the use of aluminium foil in front of a Polaroid camera used in a Laue camera to reduce the K-shell ¯uorescence radiation from a transition-metal crystal when using a conventional sealed molybdenum X-ray source. A 0.1 mm thick foil will reduce the ¯uorescent radiation from the crystal by a factor of about ®ve, and this radiation is emitted isotropically from the specimen. In contrast, the wanted Laue-re¯ected beams are emitted as a nearly parallel beam, and the signal-to-noise ratio in the resulting photograph is much increased. The second case is the ultimate limiting case of ®ltering, shielding. If it is necessary to shield an object completely from a polychromatic incident beam, a suf®cient thickness of absorbing material, calculated using the data in Section 4.2.4, to reduce the beam intensity to the level of the ambient background is inserted in the beam. [The details of how shielding systems are designed are given in reference works such as the Handbook of Radiation Measurement and Protection (Brodsky, 1982).] In general, the use of an absorber of one atomic species will provide insuf®cient shielding. The use of composite absorbers is necessary to achieve a maximum of shielding for a minimum of weight. This is of utmost importance if one is designing, say, the shielding of an X-ray telescope to be carried in a rocket or a balloon (Grey, 1996). To produce shielding that satis®es the requirements of minimum weight, good mechanical rigidity, and ability to be constructed to good levels of mechanical tolerance, shielding must be constructed using a number of layers of different absorbers, chosen such that the highestenergy radiation is just stopped in the ®rst layer, the L-shell ¯uorescent radiation created in the absorption process is stopped in the second, and the lower-energy L and M ¯uorescent radiation is stopped by the next layer, and so on until the desired radiation level is reached. In the usual case involving ®lters, the problem is one of removing as much as possible of the Bremsstrahlung radiation and unwanted characteristic radiation from the spectrum of a laboratory sealed tube or rotating-anode source whilst retaining as much of the wanted radiation as is possible. To give an example, a thin characteristic radiation ®lter of nickel of appropriate thickness almost completely eliminates the

Bremsstrahlung and K radiation from an X-ray source with a copper target, but reduces the intensity in the Cu K doublet by only about a factor of two. For many applications, this is all that is necessary to provide the required degree of monochromatization. If there is a problem with the residual Bremsstrahlung, this problem may be averted by making a second set of measurements with a different ®lter, one having an absorption edge at an energy a little shorter than that of the desired emission line. The difference between the two sets of measurements corresponds to a comparatively small energy range spanning the emission line. This balanced-®lter method is more cumbersome than the single-®lter method, but no special equipment or dif®cult adjustments are required. In general, if the required emission is from an element of atomic number Z, the ®rst foil is made from material having atomic number Z 1 and the second from atomic number Z 1. A better balance can be achieved using three foils (Young, 1963). The use of ®lters is discussed in more detail in §2.3.5.4.2. Data for ®lters for the radiations in common use are given in Tables 2.3.5.2 and 2.3.5.3. The information necessary for choosing ®lter materials and estimating their optimum thicknesses for other radiations is given in Sections 4.2.2, 4.2.3, and 4.2.4. It should be remembered that ®ltration changes the wavelength of the emission line slightly, but this is only of signi®cance for measurements of lattice parameters to high precision (Delf, 1961). 4.2.5.4. Monochromators 4.2.5.4.1. Crystal monochromators Even multifoil balanced ®lters transmit a wide range of photon energies. Strictly monochromatic radiation is impossible, since all atomic energy levels have a ®nite width, and emission from these levels therefore is spread over a ®nite energy range. The corresponding radiative line width is important for the correct evaluation of the dispersion corrections in the neighbourhood of absorption edges (§4.2.6.3.3.2). Even MoÈssbauer lines, originating as they do from nuclear energy levels that are much narrower than atomic energy levels, have a ®nite line width. To achieve line widths comparable to these requires the use of monochromators using carefully selected single-crystal re¯ections.

Fig. 4.2.5.3. The re¯ectivity of a multiple-quantum-well device is shown. This consists of 40 alternating layers of AlGaAs and InGaAs. Shown also, but shifted downwards on the vertical scale for the purpose of clarity, is the theoretical prediction based on standard electromagnetic theory.

238


4.2. X-RAYS Crystal monochromators make use of the periodicity of `perfect' crystals to select the desired photon energy from a range of photon energies. This is described by Bragg's law, 2dhkl sin nl;

4:2:5:5

where dhkl is the spacing between the planes having Miller indices hkl, is the angle of incidence, n is the order of a particular re¯ection (n 1; 2; 3; . . .), and l is the wavelength. If there are wavelength components with values near l=2, l=3, . . ., these will be re¯ected as well as the wanted radiation, and harmonic contamination can result. This can be a dif®culty in spectroscopic experiments, particularly XAFS, XANES and DAFS (Section 4.2.3). Equation (4.2.5.5) neglects the effect of the refractive index of the material. This is usually omitted from Bragg's law, since it is of the order of 10 5 in magnitude. Because the refractive index is a strong function of wavelength, the angles at which the successive harmonics are re¯ected are slightly different from the Bragg angle of the fundamental. This fact can be used in multiple-re¯ection monochromators to minimize harmonic contamination. As can be seen in Fig. 4.2.5.4, each Bragg re¯ection has a ®nite line width, the Darwin width, arising from the interaction of the radiation with the periodic electron charge distribution. [See, for example, Warren (1968) and Subsection 7.2.2.1.] Each Bragg re¯ection therefore contains a spread of photon energies. The higher the Miller indices, the narrower the Darwin width becomes. Thus, for experiments involving the MoÈssbauer effect, extreme back-re¯ection geometry is used at the expense of photon ¯ux. If the beam propagates through the specimen, the geometry is referred to as transmission, or Laue, geometry. If the beam is re¯ected from the surface, the geometry is referred to as re¯ection, or Bragg, geometry. Bragg geometry is the most commonly used in the construction of crystal monochromators. Laue geometry has been used in only a relatively few applications until recently. The need to handle high photon ¯uxes with their associated high power load has led to the use of diamond crystals in Laue con®gurations as one of the ®rst components of X-ray optical systems (Freund, 1993). Phase plates can be created using the Laue geometry (Giles et al., 1994). A schematic diagram of a system used at the European Synchrotron Radiation Facility is shown in Fig. 4.2.5.5. Radiation from an insertion device falls on a Laue-geometry pre-monochromator and then passes through a channel-cut (multiplere¯ection) monochromator. The strong linear polarization from the source and the monochromators can be changed into circular polarization by the asymmetric Laue-geometry polarizer and analysed by a similar Laue-geometry analysing crystal. More will be said about polarization in §4.2.5.4.4 and Section 6.2.2. 4.2.5.4.2. Laboratory monochromator systems Many laboratories use powder diffractometers using the Bragg±Brentano con®guration. For these, a suf®cient degree of monochromatization is achieved through the use of a diffractedbeam monochromator consisting of a curved-graphite monochromator and a detector, both mounted on the 2 arm of the diffractometer. Such a device rejects the unwanted K radiation and ¯uorescence from the sample with little change in the magnitude of the K lines. Incident-beam monochromators are

also used to produce closely monochromatic beams of the desired energy. Single-re¯ection monochromators used for the reduction of spectral energy spread are described in Subsections 2.2.7.2 and 2.3.5.4. For most applications, this simple means of monochromatization is adequate. Increasingly, however, more versatility and accuracy are being demanded of laboratory diffractometer systems. Increased angular accuracy in both the and 2 axes, excellent monochromatization, and parallel-beam geometry are all demands of a user community using improved techniques of data collection and data analysis. The necessity to study thin ®lms has generated a need for accurately collimated beams of small cross section, and there is a need to have well collimated and monchromatic beams for the study of rough surfaces. This, coupled with the need to analyse data using the Rietveld method (Young, 1993), has caused a revolution in the design of commercial diffractometers, with the use of principles long since used in synchrotron-radiation research for the design of laboratory instruments. Monochromators of this type are brie¯y discussed in §4.2.5.4.3. 4.2.5.4.3. Multiple-re¯ection monochromators for use with laboratory and synchrotron-radiation sources Single-re¯ection devices produce re¯ected beams with quite wide, quasi-Lorentzian, tails (Subsection 2.3.3.8), a situation

Fig. 4.2.5.4. In (a), the schematic rocking curve for a silicon crystal in the neighbourhood of the 111 Bragg peak is shown. The full curve is due to the crystal set to the true Bragg angle, and the dotted curve corresponds to a surface tilted at an angle of 200 with respect to the beam prior to the acquisition of the rocking curve. Only the 111 and 333 re¯ections are shown for clarity. The 222 re¯ection is very weak because the geometrical structure factor is small. The separation of the 111 and 333 peaks occurs because the refractive index is different for these re¯ections. In a double-crystal monochromator, white radiation from the source will produce the scattered intensity given by the full curve. If that intensity distribution now falls on the second crystal, which is tilted with respect to the ®rst, an angle of tilt can be found for which the Bragg condition is not ful®lled in the second crystal, and the 333 radiation cannot be re¯ected. The resultant re¯ected intensity is shown in (b). Note that this is an idealized case, and in practice the existence of tails in the re¯ectivity curve can allow the transmission of some harmonic radiation through the doublecrystal monochromator.

239


4. PRODUCTION AND PROPERTIES OF RADIATIONS that is not acceptable, for example, for the study of small-angle scattering (SAXS, Chapter 2.6). The effect of the tails can be reduced signi®cantly through the use of multiple Bragg re¯ections. The use of multiple Bragg re¯ections from a channel cut in a monolithic silicon crystal such that the channel lay parallel to the (111) planes of the crystal was shown by Bonse & Hart (1965) to remove the tails of re¯ections almost completely. This class of device, referred to as a (symmetrical) channel-cut crystal, is the most frequently used form of monochromator produced for modern X-ray laboratory diffractometers and beamlines at synchrotron-radiation sources (Figs. 4.2.5.2, 4.2.5.5). The use of symmetrical and asymmetrical Bragg re¯ections for the production of highly collimated monochromatic beams has been discussed by Beaumont & Hart (1973). This paper contains descriptions of the con®gurations of channel-cut monochromators and combinations of channel-cut monochromators used in modern laboratory diffractometers produced by Philips, Siemens, and Bede Scienti®c. In another paper, Hart (1971) discussed the whole gamut of Bragg re¯ecting X-ray optical devices. Hart & Rodriguez (1978) extended this to include a class of device in which the second wafer of the channel-cut monochromator could be tilted with respect to the ®rst (Fig. 4.2.5.6), thereby providing an offset of the crystal rocking curves with the consequent removal of most of the contaminant harmonic radiation (Fig. 4.2.5.4). The version of monochromator shown here is designed to provide thermal stability for high incident-photon ¯uxes. Berman & Hart (1991) have also devised a class of adaptive X-ray monochromators to be used at high thermal loads where thermal expansion can cause a signi®cant degradation of the rocking curve, and therefore a signi®cant loss of ¯ux and spectral purity. The cooling of Bragggeometry monochromators at high photon ¯uxes presents a dif®cult problem in design. Kikuta & Kohra (1970), Matsushita, Kikuta & Kohra (1971) and Kikuta (1971) have discussed in some detail the performance of asymmetrical channel-cut monochromators. These ®nd application under circumstances in which beam widths need to be condensed or expanded in X-ray tomography or for micro X-ray ¯uorescence spectroscopy. Hashizume (1983) has described the design of asymmetrical monolithic crystal mono-

chromators for the elimination of harmonics from synchrotronradiation beams. Many installations use a system designed by the Kohzu Company as their primary monochromator. This is a separated element design in which the reference crystal is set on the axis of the monochromator and the ®rst crystal is set so as to satisfy the Bragg condition in both elements. One element can be tilted slightly to reduce harmonic contamination. When the wavelength is changed (i.e. is changed), the position of the ®rst wafer is changed either by mechanical linkages or by electronic positioning devices so as to maintain the position of the outgoing beam in the same place as it was initially. This design of a ®xedheight, separated-element monochromator was due initially to Matsushita, Ishikawa & Oyanagi (1986). More recent designs incorporate liquid-nitrogen cooling of the ®rst crystal for use with high-power insertion devices at synchrotron-radiation sources. In many installations, the second crystal can be bent into a cylindrical shape to focus the beam in the horizontal plane. The design of such a sagittally focusing monochromator is discussed by Stephens, Eng & Tse (1992). Creagh & Garrett (1965) have descibed the properties of a monochromator based on a primary monchromator (Berman & Hart, 1991) and a sagittally focusing second monochromator at the Australian National Beamline at the Photon Factory. A recent innovation in X-ray optics has been made at the European Synchrotron Radiation Facility by the group led by Snigirev (1994). This combines Bragg re¯ection of X-rays from a silicon crystal with Fresnel re¯ection from a linear zone-plate structure lithographically etched on its surface. Han¯and et al. (1994) have reported the use of this class of re¯ecting optics for the focusing of 25 to 30 keV photon beams for high-pressure crystallography experiments (Fig. 4.2.5.7). Further discussion on these monochromators is to be found in this volume in Subsection 2.2.7.2, §2.3.5.4.1, Chapter 2.7, and Section 7.4.2. 4.2.5.4.4. Polarization All scattering of X-rays by atoms causes a probable change of polarization in the beam. Jennings (1981) has discussed the effects of monochromators on the polarization state for

Fig. 4.2.5.5. A schematic diagram of a beamline designed to produce circularly polarized light from initially linearly polarized light using Lauecase re¯ections. Radiation from an insertion device is initially monochromated by a cooled diamond crystal, operating in Laue geometry. The outgoing radiation has linear polarization in the horizontal plane. It then passes through a silicon channel-cut monochromator and into a silicon crystal of a thickness chosen so as to produce equal amounts of radiation from the and branches of the dispersion surface. These recombine to produce circularly polarized radiation at the exit surface of the crystal.

240


4.2. X-RAYS conventional diffractometers of that era. For accurate Rietveld modelling or accurate charge-density studies, the theoretical scattered intensity must be known. This is not a problem at synchrotron-radiation sources, where the incident beam is initially almost completely linearly polarized in the plane of the orbit, and is subsequently made more linearly polarized through Bragg re¯ection in the monochromator systems. Rather,

it is a problem in the laboratory-based systems where the source is in general a source of elliptical polarization. It is essential to determine the polarization for the particular monochromator and the source combined to determine the correct form of the polarization factor to use in the formulae used to calculate scattered intensity (Chapter 6.2). 4.2.6. X-ray dispersion corrections (By D. C. Creagh)

Fig. 4.2.5.6. A schematic diagram of a Hart-type tuneable channel-cut monochromator is shown. The monochromator is cut from a single piece of silicon. The re¯ecting surfaces lie parallel to the (111) planes. Cuts are made in the crystal block so as to form a lazy hinge, and the second wafer of the monochromator is able to be de¯ected by a force generated by a current in an electromagnet acting on an iron disc glued to the upper surface of the wafer. Cooling of the primary crystal of the monochromator is by a jet of water falling on the underside of the wafer. This type of system can tolerate incident-beam powers of 500 W mm 2 without signi®cant change to the width of the re¯ectivity curve.

The term ànomalous dispersion' is often used in the literature. It has been dropped here because there is nothing ànomalous' about these corrections. In fact, the scattering is totally predictable. For many years after the theoretical prediction of the dispersion of X-rays by Waller (1928), and the application of this theory to the case of hydrogen-like atoms by HoÈnl (1933a,b), no real use was made by experimentalists of dispersion-correction effects in X-ray scattering experiments. The suggestion by Bijvoet, Peerdeman & Van Bommel (1951) that dispersion effects might be used to resolve the phase problem in the solution of crystal structures stimulated interest in the practical usefulness of this hitherto neglected aspect of the scattering of photons by atoms. In one of the ®rst texts to discuss the problem, James (1955) collated experimental data and discussed both the classical and the non-relativistic theories of the anomalous scattering of X-rays. James's text remained the principal reference work until 1974, when an Inter-Congress Conference of the International Union of Crystallography dedicated to the discussion of the topic produced its proceedings (Ramaseshan & Abrahams, 1975). At that conference, reference was made to a theoretical data set calculated by Cromer & Liberman (1970) using relativistic quantum mechanics. This data set was later used in IT IV (1974) and has been used extensively by crystallographers for more than a decade.

Fig. 4.2.5.7. A schematic diagram of the use of a Bragg±Fresnel lens to focus hard X-rays onto a high-pressure cell. The diameter of the sample in such a cell is typically 10 mm. The insert shows a scanning electron micrograph of the surface of the Bragg±Fresnel lens.

241


4. PRODUCTION AND PROPERTIES OF RADIATIONS The rapid development in computing techniques, improvements in materials of construction and experimental equipment, and the use of synchrotron-radiation sources for X-ray scattering experiments have led to the production of a number of reviews of both the theoretical and the experimental aspects of the anomalous scattering of X-rays. Review articles by Gavrila (1981), Kissel, Pratt, Kane & Roy (1985), and Creagh (1985) discuss both the theoretical and the experimental techniques for the determination of the X-ray dispersion corrections. Creagh (1986) has discussed the use of X-ray anomalous scattering for the characterization of materials, and a review by Helliwell (1984) has described the anomalous scattering by atoms contained in proteins and its use for the solution of the structure of proteins. In a number of papers, Karle (1980, 1984a,b,c, 1985) has recently shown how powerful dispersion techniques can be in the solution of crystal structures. Indeed, the high intensity afforded by synchrotron-radiation sources, together with improvements in specimen-handling techniques, has led to the general use of dispersion techniques for the solution of the phase problem in crystal structures. In particular, the MAD (multiple-wavelength anomalous-dispersion) technique is used extensively for the solution of such macromolecular crystal structures as proteins and the like. The origin of the technique lies in the Bijvoet relations, but the implementation and the development of the technique is due to Hendrickson (1994). In this section, a brief discussion of the physical principles underlying the theoretical tabulations of X-ray scattering will be given. This will be followed by a discussion of experimental techniques for the determination of the dispersion corrections. In the next section, theoretical and experimentally determined values for the dispersion corrections will be compared for a number of elements. Currently, there is some discussion about the nature of the dispersion corrections: are they to be considered to exhibit tensor characteristics? It is clear that in all the theoretical calculations the atoms are considered to be isolated, and, therefore, if there is a tensor associated with the X-ray scattering, it must be associated with the reaction of the atoms with the polarization state of the incident radiation. Since the property of polarization of the X-ray beam is described by a ®rst-rank tensor, it follows that the form factor must be described by a second-rank tensor (Templeton, 1994). Either the detection system of the experimental equipment must be capable of resolving the change in the polarization states of the incident and scattered radiation, or the incident radiation must be plane polarized for this property to be observable. Except for certain diffractometers at synchrotron-radiation sources, or for specially designed conventional laboratory equipment, it is not possible to determine the polarization states before and after scattering. To a very good approximation, therefore, one can describe the form factor as being made up of a number of separate components, the largest of which is a zeroth-rank tensor that corresponds to the conventionally accepted description of the form factor. The magnitudes of any of the higher-order tensor components are small compared with this term. Whether or not they are observable depends on the characteristics of the diffraction system used in the experiment. The form-factor formalism in its zeroth-order mode has been used extremely successfully for the solution of crystal structures, the description of wave®elds in crystals, the determination of the distribution of electron density in crystal structures, etc. as has been shown by Creagh (1993).

It must be stressed that all of the crystal structures solved so far have been solved using the conventionally accepted, zerothorder, form-factor description of X-ray scattering. As well, all the data concerning the distribution of electron density within crystals have used this description. Further discussion of this issue will be given in x4.2.6.3.3.4. 4.2.6.1. De®nitions 4.2.6.1.1. Rayleigh scattering When photons interact with atoms, a number of different scattering processes can occur. The dominant scattering mechanisms are: elastic scattering from the bound electrons (Rayleigh scattering); elastic scattering from the nucleus (nuclear Thomson scattering); virtual pair production in the ®eld of the screened nucleus (DelbruÈck scattering); and inelastic scattering from the bound electrons (Compton scattering). Of the elastic scattering processes, only Rayleigh scattering has a signi®cant amplitude in the range of photon energies used by crystallographers < 100 keV). Compton scattering will be discussed elsewhere (Section 4.2.4). The essential feature of Rayleigh scattering is that the internal energy of the atom remains unchanged in the interaction. The momentum hki and polarization ei of the incident photon may be modi®ed during the process to hkf and ef hki ; ei A ! A hkf ; ef : 4.2.6.1.2. Thomson scattering by a free electron From classical electromagnetic theory, it can be shown that the fraction of incident intensity scattered by a free electron is, at a position r, ' from the scattering electron, I=I0 re =r2 12 1 cos2 ';

where re is the classical radius of the electron 2:817938 10 15 m). The factor 12 1 cos2 ' arises from the assumption that the electromagnetic wave is initially unpolarized. Should the wave be polarized, the factor is necessarily different from that given in equation (4.2.6.1). Equation (4.2.6.1) is the basis on which the scattering power of ensembles of electrons is compared. 4.2.6.1.3. Elastic scattering from electrons bound to atoms: the atomic scattering factor, the atomic form factor, and the dispersion corrections In considering the interaction of a photon with electrons bound in an atom, one assumes that each electron scatters independently of its fellows, and that the total scattering power of the atom is the sum of the contributions from all the electrons in the atom. Assuming that one can de®ne an electron density r for an atom containing a single electron, one can show that the scattering power of that atom relative to the scattering power of a Thomson free electron is R f D r expikf ki r dV ; 4:2:6:2 where D kf

ki change in photon momentum

2jkj sin=2; being the total angle of scattering of the photon. The scattering power for the atom relative to a free electron is referred to as the atomic form factor or the atomic scattering factor of the atom.

242


4:2:6:1

4.2. X-RAYS The result (4.2.6.2), which was derived using purely classical arguments, has been shown by Nelms & Oppenheimer (1955) to be identical to the result gained by quantum mechanics. If it is assumed that the atom has spherical symmetry, Z1 sin r 2 f 4 r 4:2:6:3 r dr: r

supposed to be effectively the same as the well understood problem of the driven damped pendulum system. In this type of problem, the natural amplitude of the system was modi®ed by a correction term (a real number) dependent on the proximity of the impressed frequency to the natural resonant frequency of the system and a loss term (an imaginary number) that was related to the damping factor for the resonant system. Thus the scattering power came to be written in the form

For an atom containing Z electrons, the atomic form factor becomes 1 nZ Z X sin r 2 f 4 n r 4:2:6:4 r dr: r n1

f f0 f 0 if 00 ;

0

0

Exact solutions for the form factor are dif®cult to obtain, and therefore approximations have to be made to enable equation (4.2.6.4) to be evaluated. The two most commonly used approximations are the Thomas±Fermi (Thomas, 1927; Fermi, 1928) and the Hartree±Fock (Hartree, 1928; Fock, 1930) techniques. In the Thomas±Fermi model, the atomic electrons are considered to be a degenerate gas obeying Fermi±Dirac statistics and the Pauli exclusion principle, the ground-state energy of the atom being equal to the zero-point energy of this gas. The average charge density can be written in terms of the radial potential function, V r, which may then be substituted into Poisson's equation, r2 V r r="0 , which can then be solved for V r using the boundary conditions that limr!1 V r 0 and that limr!0 rV r Ze. The Thomas±Fermi charge distributions for different atoms are related to each other. If the form factor is known for a `standard' atom for which the atomic number is Z0 then, for an atom with atomic number Z, fZ D Z=Z0 f0 D0 :

4:2:6:5

4:2:6:7

where f0 is the atomic scattering factor remote from the resonant energy levels, f 0 is the real part of the anomalous-scattering factor, and f 00 is the imaginary part of the anomalous-scattering factor. The nomenclature of (4.2.6.7) has been superseded, but one still encounters it occasionally in modern papers. In what follows, a brief exposition of the various theories for the anomalous scattering of X-rays and descriptions of modern experimental techniques for their determination will be given. Comparisons will be made between the several theoretical and experimental results for a number of atomic species. From these comparisons, conclusions will be drawn as to the validity of the various theories and the relevance of certain experiments. 4.2.6.2. Theoretical approaches for the calculation of the dispersion corrections All the theories that will be discussed here have the following assumptions in common: the elastic scattering is from an isolated neutral atom and that atom is spherically symmetrical. All but the most recent of the theoretical approaches neglect changes in polarization of the incident photon caused by the interaction of the photon with the atom. In the event, few experimental con®gurations are able to detect such changes in polarization, and the only observable for most experiments is the momentum change of the photon.

Here,

4.2.6.2.1. The classical approach 0

Z=Z0

1=3

:

The most accurate calculations of wavefunctions of manyelectron atoms have been made using the self-consistent-®eld (Hartree±Fock) method. In this independent-particle model, each electron is assumed to move in the ®eld of the nucleus and in an average ®eld due to the other electrons. With this approach, the charge distribution can be written as r

nZ P n1

n r

nZ P n1

n r n r;

4:2:6:6

where n r is the charge-density distribution of the nth electron and n r is its wavefunction. The technique has been extended to include the effects of both exchange and correlation. Tables of relativistic Hartree±Fock values have been given by Cromer & Waber (1974). Their notation Fx; Z is related to the notation used earlier as follows: fZ Fx; Z;

In the classical approach, electrons are thought of as occupying energy levels within the atom characterized by an angular frequency !n and a damping factor n . The forced vibration of an electron gives rise to a dipolar radiation ®eld, when the atomic scattering factor can be shown to be f

!2

!2 : !2n in !

If the probability that the electron is to be found in the nth orbit is gn , the real part of the atomic scattering factor may be written as Re f

X n

gn

X gn !2n : !2 !2n n

Re f f0 f 0 ;

jkj sin'=2 : 2 4 In the foregoing discussion, the fact that the electrons occupy de®nite energy levels within the atoms has been ignored: it has been assumed that the energy of the photon is very different from any of these energy levels. The theory for calculating the scattering power of an atom near a resonant energy level was

4:2:6:10

where f0 represents the sum of all the elements of the set of oscillator strengths and is unity for a single-electron atom. The second term may be written as

243


4:2:6:9

The probability gn is referred to as the oscillator strength corresponding to the virtual oscillator having natural frequency !n . Equation (4.2.6.9) may be written as

where x

4:2:6:8

0

Z1

f !i

!0 2 dg =d!0 0 d! !2 !0 2

4:2:6:11

4. PRODUCTION AND PROPERTIES OF RADIATIONS if the atom is assumed to have an in®nite number of energy states. For an atom containing electrons, it is assumed that the overall value of f 0 is the coherent sum of the contribution of each individual electron, whence X Z !0 2 dg =d!0 f d!0 2 02 ! ! 1

0

4:2:6:12

M=NA m 10

dg : ! d! 2

4:2:6:14

An expression linking the real and imaginary parts of the dispersion corrections can now be written: 2X f P 0

Z1 !

! 0 f 00 ! 0 ; 0 0 d! : !2 ! 0 2

4:2:6:15

This is referred to as the Kramers±Kronig transform. Note that the term involving the restoring force has been omitted from this equation. Equations (4.2.6.12), (4.2.6.14), and (4.2.6.15) are the fundamental equations of the classical theory of photon scattering, and it is to these equations that the predictions of other theories are compared. 4.2.6.2.2. Non-relativistic theories The matrix element for Rayleigh scattering from an atom having a radially symmetric charge distribution may be written as M M1 ei ef M2 ei jf ef ji ;

4:2:6:16

where ei and ef represent the initial and ®nal states of the photon. The matrix element M1 represents scattering for polarizations ei and ef perpendicular to the plane of scattering and M2 represents scattering for polarization states lying in the plane of scattering. Averaged over polarization states, the differential scattering cross section takes the form d re2 4:2:6:17 jM1 j2 jM2 j2 : d 2 Here is the photoelectric scattering cross section, which is related to the mass attenuation coef®cient m by

re cA P

1 1 h1jT1 j1i h1jT2 j1i: 4:2:6:20 m m In this equation, the initial and ®nal wavefunctions are designated as h1j and j1i, respectively, and the terms T1 and T2 are given by M ei ef f0

E1

1 e P exp iki r H 0 h! i i

E1

1 e P exp ikf r H 0 h! i f

T1 ef P exp ikf r and T2 ei P exp iki r

where is an in®nitesimal positive quantity. The ®rst term of equation (4.2.6.20) corresponds to the atomic scattering factor and is identical to the value given by classical theory. The terms involving T1 and T2 correspond to the dispersion corrections. Equation (4.2.6.20) contains no terms to account for radiation damping. More complete theories take the effect of the ®nite width of the radiating level into account. It is necessary to realize that the atomic scattering factor depends on both the photon's frequency ! and the momentum vector D. To emphasize this dependence, equation (4.2.6.7) is rewritten as f !; D f0 D f 0 !; D if 00 !; D:

4:2:6:21

In the dipole approximation, it can be shown that 2 f !; 0 P 0

Z1 0

!0 f 00 !0 ; 0 0 d! ; !2 !02

4:2:6:22

which may be compared with equation (4.2.6.15) and ! f 00 !; 0 !; 4:2:6:23 4re c which may be compared with equation (4.2.6.14). There is a direct correspondence between the predictions of the classical theory and the theory using second-order perturbation theory and non-relativistic quantum mechanics. The extension of HoÈnl's (1933a,b) study of the scattering of X-rays by the K shell of atoms to other electron shells has been presented by Wagenfeld (1975). In these calculations, the energy of the photon was assumed to be such that relativistic effects do not occur, nor do transitions within the discrete states of the atom occur. Transitions to continuum states do occur, and, using the analytical expressions for the wavefunctions of the hydrogen-like atom, analytical expressions may be developed for the photoelectric scattering cross sections. By expansion of the retardation factor exp ik r as the power series 1 ik r 12 k r2 . . ., it is possible to determine dipolar, quadrupolar, and higher-order

244


0

where H^ 0 P ihr. After application of the second-order perturbation theory, the matrix element may be deduced to be

is not unity, but the total oscillator strength for the atom must be equal to the total number of electrons in the atom. The imaginary part of the dispersion correction f 00 is associated with the damping of the incident wave by the bound electrons. It is therefore functionally related to the linear absorption coef®cient, l , which can be determined from experimental measurement of the decrease in intensity of the photon beam as it passes through a medium containing the atoms under investigation. It can be shown that the attenuation coef®cient per atom a is related to the density of the oscillator states by 22 e2 dg a ; 4:2:6:13 "0 mc d!

f 00

4:2:6:18

re2 2 A; 4:2:6:19 2 is the Hamiltonian for the unperturbed atom and H^ H^

!

whence

;

where M is the molecular weight and NA is Avogadro's number. Using the vector potential of the wave®eld A, an expression for the perturbed Hamiltonian of a hydrogen-like atom coupled to the radiation ®eld may be written as

!

and the oscillator strength of the th electron Z1 dg d! g d!

24

4.2. X-RAYS terms in the analytical expression for the photoelectric scattering cross section. The values of the cross section so obtained were used to calculate the values of f 0 !; D using the Kramers±Kronig transform [equation (4.2.6.22)] and f 00 !; D using equation (4.2.6.23). The work of Wagenfeld (1975) predicts that the values of f 0 !; D and f 00 !; D are functions of D. Whether or not this is a correct prediction will be discussed in Subsection 4.2.6.3. Wang & Pratt (1983) have drawn attention to the importance of bound±bound transitions in the dispersion relation for the calculation of forward-scattering amplitudes. Their inclusion is especially important for elements with small atomic numbers. In a later paper, Wang (1986) has shown that, for silicon at the wavelengths of Mo K and Ag K1 , values for f 0 !; 0 of 0.084 and 0.055, respectively, are obtained. These values should be compared with those listed in Table 4.2.6.4. 4.2.6.2.3. Relativistic theories 4.2.6.2.3.1. Cromer and Liberman: relativistic dipole approach It is necessary to consider relativistic effects for atoms having all but the smallest atomic numbers. Cromer & Liberman (1970) produced a set of tables based on a relativistic approach to the scattering of photons by isolated atoms that was later reproduced in IT IV (1974). Subsequent experimental determinations drew attention to inaccuracies in these tables in the neighbourhood of absorption edges owing to the poor convergence of the Gaussian integration technique, which was used to evaluate the real part of the dispersion correction. In a later paper, Cromer & Liberman (1981) recalculated 34 instances for which the incident radiation lay close to the absorption edges of atoms using a modi®ed integration procedure. Care should be exercised when using the Cromer & Liberman computer program, especially for calculations of f 0 !; 0 for high atomic weight elements at low photon energies. As Creagh (1990) and Chantler (1994) have shown, incorrect values of f 0 !; 0 can be calculated because an insuf®cient number of values of f 00 !; 0 are calculated prior to performing the Kramers±Kronig transform. In a new tabulation, Chantler (1995) presents the Cromer & Liberman data using a ®ner integrating grid. It should be noted that the relativistic correction is the same as that used in this tabulation. These relativistic calculations are based on the scattering formula developed by Akhiezer & Berestetsky (1957) for the scattering amplitude for photons by a bound electron, viz: Si!f

2i"1 h!1

"2

h!2

4ehc2 f: 2mc2h!1 !2 1=2 4:2:6:24

Here the angular frequencies of the incident and scattered photons are !1 and !2 , respectively, and the initial and ®nal energy states of the atom are "1 and "2 , respectively. The scattering factor f is a complicated expression that includes the initial and ®nal polarization states of the photon, the Dirac velocity operator, and the phase factors expik1 r and expik2 r for the incident and scattered waves, respectively. Summation is over all positive and negative intermediate states except those positive energy states occupied by other atomic electrons. The form of this expression is not easily related to the form-factor formalism that is most widely used by crystallographers, and a number of manipulations of the formula for the scattering factor are necessary to relate it more directly to the crystallographic formalism. In doing so, a number of assump-

tions and simpli®cations were made. Cromer & Liberman restricted their study to coherent, forward scatter in which changes in photon polarization did not occur. With these approximations, and using the electrical dipole approximation expik r 1, they were able to show that Etot if 00 !; 0: 4:2:6:25 mc2 In equation (4.2.6.25), f (0) is the atomic form factor for the case of forward scatter D 0, and the term 53 Etot =mc2 arises from the application of the dipole approximation to determine the contribution of bound electrons to the scattering process. The term f 00 !; 0 is related to the photoelectric scattering cross section expressed as a function of photon energy h! by mc h! h! 4:2:6:26 f 00 !; 0 4he2 and Z1 1 " "1 " "1 P d" : f !; 0 2 2 2 hre c h! " "1 2 f !; 0 f 0 f !; 0 53

mc2

4:2:6:27 These equations may be compared with equations (4.2.6.23) and (4.2.6.22), respectively. But equation (4.2.6.25) differs from equation (4.2.6.21) by the term 53 Etot =mc2 , which is constant for each atomic species, and is related to the total Coulomb energy of the atom. Evidently, to keep the formalism the same, one must write Etot : 4:2:6:28 mc2 In Table 4.2.6.1, values of Etot =mc2 are set out as a function of atomic number for elements ranging in atomic number from 3 to 98. To develop their tables, Cromer & Liberman (1970) used the Brysk & Zerby (1968) computer code for the calculation of photoelectric cross sections, which was based on Dirac±Slater relativistic wavefunctions (Liberman, Waber & Cromer, 1965). They employed a value for the exchange potential of 0.667r1=3 and experimental rather than computed values of the energy eigenvalues for the atoms. The wide use of their tables by crystallographers inevitably meant that criticism of the accuracy of the tables was forthcoming on both theoretical and experimental grounds. Stibius-Jensen (1979) drew attention to the fact that the use of the dipole approximation too early in the argument caused an error of 12 Zh!=mc2 2 in the tabulated values. More recently, Cromer & Liberman (1981) include this term in their calculations. Some experimental de®ciencies of the tabulated values of f 0 !; 0 have been discussed by Cusatis & Hart (1977), Hart & Siddons (1981), Creagh (1980, 1984, 1985, 1986), Deutsch & Hart (1982), Dreier, Rabe, Malzfeldt & Niemann (1984), Bonse & Hartman-Lotsch (1984), and Bonse & Henning (1986). In the latter two cases, the Kramers±Kronig transformation of photoelectric scattering results has been performed without taking into account the term that arises in the relativistic case for the total Coulomb energy of the atom. Although good agreement with the Cromer & Liberman tables is claimed, their failure to include this term is an implied criticism of the Cromer & Liberman tables. That this is unjusti®ed can be seen by references to Fig. 4.2.6.2 taken from Bonse & Henning (1986), which shows that their interferometer results [which measure f 0 !; 0 directly] and the Kramers±Kronig results differ

245


f 0 !; 0 f !; 0 53

4. PRODUCTION AND PROPERTIES OF RADIATIONS from one another by Etot =mc2 in the neighbourhood of the K-absorption edge of niobium in the compound lithium niobate. Further theoretical objections have been made by Creagh (1984) and Smith (1987), who has shown that the Stibius-Jensen correction is not valid, and that, when higher-order multipolar expansions and retardation are considered, the total self-energy correction becomes Etot =mc2 rather than 53 Etot =mc2 . Fig. 4.2.6.1 shows the variation of the self-energy correction with atomic number for the modi®ed form factor (Creagh, 1984; Smith, 1987; Cromer & Liberman, 1970). For the imaginary part of the dispersion correction f 00 !; 0, which depends on the calculation of the photoelectric scattering cross section, better agreement is found between theoretical results and experimental data. Details of this comparison have been given elsewhere (Section 4.2.4). Suf®ce it to say that Creagh & Hubbell (1990), in reporting the results of the IUCr Xray Attenuation Project, could ®nd no rational basis for preferring the Sco®eld (1973) Hartree±Fock calculations to the Cromer & Liberman (1970, 1981) and Storm & Israel (1970) Dirac±Hartree±Fock±Slater calculations. Computer programs based on the Cromer & Liberman program (Cromer & Liberman, 1983) are in use at all the major synchrotron-radiation laboratories. Many other laboratories have also acquired copies of their program. This program must be modi®ed to remove the incorrect Stibius±Jensen correction term, and, as will be seen later, the energy term should be modi®ed to be Etot =mc2 .

Table 4.2.6.1. Values of Etot =mc2 listed as a function of atomic number Z

4.2.6.2.3.2. The scattering matrix formalism Kissel, Pratt & Roy (1980) have developed a computer program based on the second-order S-matrix formalism suggested by Brown, Peierls & Woodward (1955). Their aim was to provide a prescription for the accurate 1%) prediction of the total-atom Rayleigh scattering amplitudes. Their model treats the elastic scattering as the sum of bound electron, nuclear, and DelbruÈck scattering cross sections, and treats the Rayleigh scattering by considering second-order, single-electron transitions from electrons bound in a relativistic, self-consistent, central potential. This potential was a Dirac± Hartree±Fock±Slater potential, and exchange was included by use of the Kohn & Sham (1965) exchange model. They omitted radiative corrections. In principle, the observables in an elastic scattering process are momentum hk and polarization e. The complex polarization vectors e satisfy the conditions

0

e e1;

e k 0:

4:2:6:29

In quantum mechanics, elastic scattering is described in terms of a differential scattering amplitude, M, which is related to the elastic cross section by equation (4.2.6.16). If polarization is not an observable, then the expression for the differential scattering cross section takes the form of equation (4.2.6.17). If polarization is taken into account, as may be the case when a polarizer is used on a beam scattered from a sample irradiated by the linearly polarized beam from a synchrotronradiation source, the full equation, and not equation (4.2.6.17), must be used to compute the differential scattering cross section. The principle of causality implies that the forward-scattering amplitude M!; 0 should be analytic in the upper half of the ! plane, and that the dispersion relation Z1 2!2 Im M!0 ; 0 0 Re M!; 0 d! !0 !02 !2 0

4:2:6:30

Symbol

3 4 5 6 7 8 9 10

Li Be B C N O F Ne

0.0004 0.0006 0.0012 0.0018 0.0030 0.0042 0.0054 0.0066

11 12 13 14 15 16 17 18

Na Mg Al Si P S Cl Ar

0.0084 0.0110 0.0125 0.0158 0.0180 0.0210 0.0250 0.0285

19 20

K Ca

0.0320 0.0362

21 22 23 24 25 26 27 28 29 30

Sc Ti V Cr Mn Fe Co Ni Cu Zn

0.0410 0.0460 0.0510 0.0560 0.0616 0.0680 0.0740 0.0815 0.0878 0.0960

31 32 33 34 35 36

Ga Ge As Se Br Kr

0.104 0.114 0.120 0.132 0.141 0.150

37 38

Rb Sr

0.159 0.171

39 40 41 42 43 44 45 46 47 48

Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

0.180 0.192 0.204 0.216 0.228 0.246 0.258 0.270 0.285 0.300

Z

Symbol

Etot =mc2

49 50 51 52 53 54

In Sn Sb Te I Xe

0.318 0.330 0.348 0.363 0.384 0.396

55 56

Cs Ba

0.414 0.438

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

0.456 0.474 0.492 0.516 0.534 0.558 0.582 0.610 0.624 0.648 0.672 0.696 0.723 0.750 0.780

72 73 74 75 76 77 78 79 80

Hf Ta W Re Os Ir Pt Au Hg

0.804 0.834 0.864 0.900 0.919 0.948 0.984 1.014 1.046

81 82 83 84 85 86

Tl Pb Bi Po At Rn

1.080 1.116 1.149 1.189 1.224 1.260

87 88

Fr Ra

1.296 1.332

89 90 91 92 93 94 95 96 97 98

Ac Th Pa U Np Pu Am Cm Bk Cf

1.374 1.416 1.458 1.470 1.536 1.584 1.626 1.669 1.716 1.764

should hold, with the consequence that Z1 2 Im M!0 ; 0 0 Re M1; 0 d! : !0 0

246


Etot =mc2

Z

4:2:6:31

4.2. X-RAYS This may be rewritten as M!; 0

M1; 0 f 0 !; 0 if 00 !; 0;

4:2:6:32

with the value of f 0 !; 0 de®ned by equation (4.2.6.15). Using the conservation of probability, ! ; 4:2:6:33 Im M!; 0 4re c tot which is to be compared with equation (4.2.6.23). Starting with Furry's extension of the formalism of quantum mechanics proposed by Feynman and Dyson, the total Rayleigh amplitude may be written as X hnjT jpihpjT jni hnjT jpihnjT jpi 1 2 1 2 ; 4:2:6:34 Mn E h! E h! E E n p n p p where T1 a ei expiki r

Because excessive amounts of computer time are required to use these direct techniques for calculating the amplitudes from all the subshells, simpler methods are usually used for calculating outer-shell amplitudes. Kissel & Pratt (1985) used estimates for outer-shell amplitudes based on the predictions of the modi®ed form-factor approach. A tabulation of the modi®ed relativistic form factors has been given by Schaupp, Schumacher, Smend, Rullhusen & Hubbell (1983). Because of the generality of their approach, the computer time required for the calculation of the scattering amplitudes for a particular energy is quite long, so that relatively few calculations have been made. Their approach, however, does not con®ne itself solely to the problem of forward scattering of photons as does the Cromer & Liberman (1970) approach. Using their model, Kissel et al. (1980) have been able to show that it is incorrect to assign a dependence of the dispersion corrections on the scattering vector D. This is at variance with some established crystallographic practices, in which the dispersion corrections are accorded the same dependence on D as f0 D, and also at

and T2 a ef exp ikf r: The jpi are the complete set of bound and continuum states in the external ®eld of the atomic potential. Singularities occur at all photon energies that correspond to transitions between bound jni and bound state jpi. These singularities are removed if the ®nite widths of these states are considered, and the energies E are replaced by iE =2, where is the total (radiative plus nonradiative) width of the state (Gavrila, 1981). By use of the formalism suggested by Brown et al. (1955), it is possible to reduce the numerical problems to one-dimensional radial integrals and differential equations. The required multipole expansions of T1 and the speci®cation of the radial perturbed orbitals that are characterized by angular-momentum quantum numbers have been discussed by Kissel (1977). Ultimately, all the angular dependence on the photon scattering angle is written in terms of the associated Legendre functions, and all the energy dependence is in terms of multipole amplitudes. Solutions are not found for the inhomogeneous radial wave equations, and Kissel (1977) expressed the solution as the linear sum of two solutions of the homogeneous equation, one of which was regular at the origin and the other regular at in®nity.

Fig. 4.2.6.1. The relativistic correction in electrons per atom for: (a) the modi®ed form-factor approach; (b) the relativistic multipole approach; (c) the relativistic dipole approach.

Fig. 4.2.6.2. Measured values of f 0 !; 0 at the K-edge of Nb in LiNbO3 and the Kramers±Kronig transformation of f 00 !; 0. The curve is obtained by transformation and the points are measured by interferometry. For (a), the polarization of the incident radiation is parallel to the hexagonal c axis, and for (b) it is at right angles to the hexagonal c axis. After Bonse & Henning (1986). Note that the distortion of the dispersion curve is due to X-ray absorption near-edge structure (XANES) effects (Section 4.2.4).

247


4. PRODUCTION AND PROPERTIES OF RADIATIONS variance with the predictions of Wagenfeld's (1975) nonrelativistic model. 4.2.6.2.4. Intercomparison of theories A discussion of the validity of the non-relativistic dipole approximation for the calculation of forward Rayleigh scattering amplitudes has been given by Roy & Pratt (1982). They compared their relativistic multipole calculations with the relativistic dipole approximation and with the nonrelativistic dipole approximation for two elements, silver and lead. They concluded that a relativistic correction to the form factor of order Z2 persists in the high-energy limit, and that this constant correction accounts for much of the deviation from the non-relativistic dipole approximation at all energies above threshold. In addition, their results illustrate that cancelling occurs amongst the relativistic, retardation, and higher multipole contributions to the scattering amplitude. This implies that care must be taken in assessing where to terminate the series that describes the multipolarity of the scattering process. In a later paper, Roy, Kissel & Pratt (1983) discussed the elastic photon scattering for small momentum transfers and the validity of the form-factor theories. In this paper, which compares the relativistic modi®ed form factor with experimental results for lead and a relativistic form factor and the tabulation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975), it is shown that the modi®ed relativistic form-factor approach gives better agreement with experiment Ê 1 than the nonfor high momentum transfers < 104 A relativistic, form-factor theories. Kissel et al. (1980) used the S-matrix technique to calculate the real part of the forward-scattering amplitude f 0 !; 0 for the inert gases at the wavelength of Mo K1 . These values are compared with the predictions of the relativistic dipole theory (RDP) and the relativistic multipole theory (RMP) in Table 4.2.6.2(a). In most cases, the agreement between the S matrix and the RMP theory is excellent, considering the differences in the methodology of the two sets of calculations. Table 4.2.6.2(b) shows comparisons of the real part of the forwardscattering amplitude f 00 !; 0 calculated for the atoms aluminium, silicon, zinc, germanium, silver, samarium, tantalum and lead using the approach of Kissel et al. (1980) with that of Cromer & Liberman (1970, 1981), with tabulations by Wagenfeld (1975), and with values taken from the tables in this section. Although reasonably satisfactory agreement exists between the relativistic values, large differences exist between the non-relativistic value (Wagenfeld, 1975) and the relativistic values. The major difference between the relativistic values occurs because of differences in estimation of the selfconsistent-®eld term, which is proportional to Etot =mc2 . The Cromer & Liberman (1970) relativistic dipole value is 53 Etot =mc2 , whereas the tabulation in this section uses the relativistic multipole value of Etot =mc2 . This causes a vertical shift of the curve, but does not alter its shape. Should better estimates of the self-energy term be found, the correction is simply that of adding a constant to each value of f 0 !; 0 for each atomic species. There is a signi®cant discrepancy between the Kissel et al. (1980) result for 62 Sm and the other theoretical values. This is the only major point of difference, however, and the results are better in accord with the relativistic multipole approach than with the relativistic dipole approach. Note that the relativistic multipole approach does not include the Stibius-Jensen correction, which alters the shape of the curve.

In x4.2.6.3.3, some examples are given to illustrate the extent to which predictions of these theories agree with experimental data for f 0 !; 0. That there is little to choose between the different theoretical approaches where the calculation of f 00 !; 0 is concerned is illustrated in Table 4.2.6.3. In most cases, the agreement between the scattering matrix, relativistic dipole, and relativistic multipole values is within 1%. In contrast, there are some signi®cant differences between the relativistic and the non-relativistic values of f 00 !; 0. The extent of the discrepancies is greater the higher the atomic number, as one might expect from the assumptions made in the formulation of the non-relativistic model. Some detailed comparisons of theoretical and experimental data for linear attenuation coef®cients [proportional to f 00 !; 0 have been given by Creagh & Hubbell (1987) for silicon, and for copper and carbon by Gerward (1982, 1983). These tend to con®rm the assertion that, at the 1% level of accuracy, there is little to choose between the various relativistic models for computing scattering cross sections. Further discussion of this is given in x4.2.6.3.3. 4.2.6.3. Modern experimental techniques The atomic scattering factor enters directly into expressions for such macroscopic material properties as the refractive index, n, and the linear attenuation coef®cient, l . The refractive index depends on the dielectric susceptibility through n 1 1=2 ; where

re l2 X N f !; D j j j

4:2:6:36

and Nj is the number density of atoms of type j. The imaginary part of the dispersion correction f 00 !; D for the case where D 0 is related to the atomic scattering cross section through equation (4.2.6.23). Experimental techniques that measure refractive indices or X-ray attenuation coef®cients to determine the dispersion corrections involve measurements for which the scattering vector, D, is zero or close to it. Data from these experiments may be compared directly with data sets such as Cromer & Liberman (1970, 1981). Other techniques measure the intensities of Bragg re¯ections from crystalline materials or the variation of intensities within one particular Laue re¯ection (PendelloÈsung). For these cases, D ghkl , the reciprocal-lattice vector for the re¯ection or re¯ections measured. These techniques can be compared only indirectly with existing relativistic tabulations, since these have been developed for the D 0 case. Data are available for elements having atomic numbers less than 20 in the nonrelativistic case (Wagenfeld, 1975). The following sections will discuss some modern techniques for the measurement of dispersion corrections, and an intercomparison will be made between experimental data and theoretical calculations for a representative selection of atoms and at two extremes of photon energies: near to and remote from an absorption edge of those atoms. 4.2.6.3.1. Determination of the real part of the dispersion correction: f 0 !; 0 X-ray interferometer techniques are now used extensively for the measurement of the refractive index of materials and hence

248


4:2:6:35

4.2. X-RAYS Table 4.2.6.2(a). Comparison between the S-matrix calculations of Kissel (K) (1977) and the form-factor calculations of Cromer & Liberman (C & L) (1970, 1981, 1983) and Creagh & McAuley (C & M) for the noble gases and several common metals; f 0 (!, 0) values are given for two frequently used photon energies Energy (keV) 17.479 (Mo K1 )

22.613 (Ag K1 )

RDP (C & L)

S matrix (K)

Ne Ar Kr Xe

0.021 0.155 0.652 0.684

0.024 0.170 0.478 0.416

0.026 0.174 0.557 0.428

Al Zn Ta Pb

0.032 0.260 0.937 1.910

0.039 0.323 0.375 1.034

0.041 0.324 0.383 1.162

Element

Table 4.2.6.2(b). A comparison of the real part of the forwardscattering amplitudes computed using different theoretical approaches: KPR (Kissel et al., 1980); C & L (Cromer & Liberman, 1970, 1981); W (Wagenfeld, 1975); and C & M (this data set) f 0 !; 0

RMP (C & M)

C&L

Atom 13

KPR

1970

1981

W

C&M

Cr K1 Cu K1 Ag K1

13.320 13.328 13.209 13.204 13.039 13.032

13.316 13.376 13.203 13.235 13.020 13.078

13.326 13.213 13.041

Si

Cr K1 Cu K1 Ag K1

14.333 14.244 14.042

14.354 14.441 14.242 14.282 14.029 14.071

14.365 14.254 14.052

30

Zn

Cr K1 Cu K1 Ag K1

29.161 29.316 28.369 28.388 30.323 30.260

29.314 28.383 30.232

29.383 28.451 30.324

32

Ge

Cr K1 Cu K1 Ag K1

31.538 30.837 32.228

31.538 30.837 32.228

47

Ag

Cu K1

47.075 46.940

46.936

47.131

62

Sm

Ag K1

58.307 56.304

56.299

56.676

14

f 0 !; 0. All the interferometers are transmission-geometry LLL devices (Bonse & Hart, 1965, 1966a,b,c,d, 1970), and initially they were used to measure the X-ray refractive indices of such materials as the alkali halides, beryllium and silicon using the characteristic radiation emitted by sealed X-ray tubes. Measurements were made for such characteristic emissions as Ag K1 , Mo K1 , Cu K1 and Cr K1 by a variety of authors (Creagh & Hart, 1970; Creagh, 1970; Bonse & HellkoÈtter, 1969; Bonse & Materlik, 1972). The ready availability of synchrotron-radiation sources led to the adaptation of the simple LLL interferometers to use this new radiation source. Bonse & Materlik (1975) reported measurements at DESY, Hamburg, made with a temporary adaptation of a diffraction-beam line. Recent advances in X-ray interferometry have led to the establishment of a permanent interferometer station at DESY (Bonse, Hartmann-Lotsch & Lotsch, 1983b). This, and many of the earlier interferometers invented by Bonse, makes its phase measurements by the rotation of a phase-shifting plate in the beams emanating from the ®rst wafer of the interferometer. In contrast, the LLL interferometer designed by Hart (1968) uses the movement of the position of lattice planes in the third wafer of the interferometer relative to the standing-wave ®eld formed by the recombination of two of the diffracted beams within the interferometer. Measurements made with and without the specimen in position enabled both the refractive index and the linear attenuation coef®cient to be determined. The use of energy-dispersive detection meant that these parameters could be determined for harmonics of the fundamental frequency to which the interferometer was tuned (Cusatis & Hart, 1975, 1977). Subsequently, measurements have been made by Siddons & Hart (1983) and Hart & Siddons (1981) for zirconium, niobium, nickel, and molybdenum. Hart (1985) planned to provide detailed dispersion curves for a large number of elements capable of being rolled into thin foils. Both types of interferometers have yielded data of high quality, and accuracies better than 0.2 electrons have been claimed for measurements of f 0 !; 0 in the neighbourhood of the K- and L-absorption edges of a number of elements. The energy window has been claimed to be as low as 0.3 eV in width. However, on the basis of the measured values, it would seem that the width of the energy window is more likely to be about 2 eV for a primary wavelength of 5 keV.

30.20 31.92 32.14

31.614 30.911 32.302

73

Ta

Ag K1

72.625 72.063

71.994

72.617

82

Pb

Ag K1

80.966 80.090

80.012

80.832

Apparently, the aÊngstroÈm-ruler design is the better of the two interferometer types, since the interferometer to be mounted at the EU storage ring is to be of this type (Buras & Tazzari, 1985). Interferometers of this type have the advantage of enabling direct measurements of both refractive index and linear attenuation coef®cients to be made. The determination of the energy scale and the assessment of the energy bandpass of such a system are two factors that may in¯uence the accuracy of this type of interferometer. One of the oldest techniques for determining refractive indices derives from measurement of the deviation produced when a prism of the material under investigation is placed in the photon beam. Recently, a number of groups have used this technique to determine the X-ray refractive index, and hence f 0 !; 0. Deutsch & Hart (1984a,b) have designed a novel doublecrystal transmission spectrometer for which they were able to detect to high accuracy the angular rotation of one element with respect to the other by reference to the PendelloÈsung maxima that are observed in the wave ®eld of the primary wafer. In this second paper, data gained for beryllium and lithium ¯uoride wedges are discussed. Several Japanese groups have used more conventional monochromator systems having Bragg-re¯ecting optics to determine the refractive indices of a number of materials. Hosoya, Kawamure, Hunter & Hakano (1978; cited by Bonse & Hartmann-Lotsch, 1984) made determinations of f 0 !; 0 in the region of the K-absorption edge for copper. More recently, Ishida & Katoh (1982) have described the use of a multiplere¯ection diffractometer for the determination of X-ray refrac-

249


Al

Radiation

_______________________

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.6.3. A comparison of the imaginary part of the forward-scattering amplitudes f 00 (!; 0) computed using different theoretical approaches: KPR (Kissel et al., 1980); C & L (Cromer & Liberman, 1981); W (Wagenfeld, 1975); and C & M (this data set) f 0 !; 0 Atom 13

Al

14

Radiation

KPR

C&L

Cr K1 Cu K1 Ag K1

0.514 0.243 0.031

0.522 0.246 0.031

Si

Cr K1 Cu K1 Ag K1

30

Zn

Cr K1 Cu K1 Ag K1

32

Ge

Cr K1 Cu K1 Ag K1

47

Ag

Cu K1

62

Sm

Cu K1

0.694 0.330 0.043 1.370 0.678 0.932

12.16

C&M 0.512 0.246 0.031

0.70 0.33 0.047

1.373 0.678 0.938 1.786 0.886 1.190

4.242

W

0.692 0.330 0.043 1.371 0.678 0.938

1.84 0.87 1.23

1.784 0.886 1.190

4.282

4.282

12.218

12.218

73

Ta

Ag K1

4.403

4.399

4.399

82

Pb

Ag K1

6.937

6.929

6.929

tive indices. Later, Katoh et al. (1985a,b) described its use for the measurement of f 0 !; 0 for lithium ¯uoride and potassium chloride at a wavelength near that of Mo K1 and for germanium in the neighbourhood of its K-absorption edge. Measurements of the linear attenuation coef®cient l over an extended energy range can be used as a basis for the determination of the real part of the dispersion correction f 0 !; 0 because of the Kramers±Kronig relation, which links f 0 !; 0 and f 00 !; 0. However, as Creagh (1980) has pointed out, even if the integration can be performed accurately [implying the knowledge of f 00 !; 0 over several decades of photon energies and the exact energy at which the absorption edge occurs], there will still be some ambiguity in the result because there still has to be the inclusion of the appropriate relativistic correction term. The experimental procedures that must be adopted to ensure that the linear attenuation coef®cients are measured correctly have been given in Subsection 4.2.3.2. One other problem that must be addressed is the accuracy to which the photon energy can be measured. Accuracy in the energy scale becomes paramount in the neighbourhood of an absorption edge where large variations in f 0 !; 0 occur for very small changes in photon energy h!. Despite these dif®culties, Creagh (1977, 1978, 1982) has used the technique to determine f 0 !; 0 and f 00 !; 0 for several alkali halides and Gerward, Thuesen, Stibius-Jensen & Alstrup (1979) used the technique to measure these dispersion corrections for germanium. More recently, the technique has been used by Dreier et al. (1984) to determine f 0 !; 0 and f 00 !; 0 for a number of transition metals and rare-earth atoms. The experi-

mental con®guration used by them was a conventional XAFS system. Similar techniques have been used by Fuoss & Bienenstock (1981) to study a variety of amorphous materials in the region of an absorption edge. Henke et al. (1982) used the Kramers±Kronig relation to compute the real part of the dispersion correction for most of the atoms in the Periodic Table, given their measured scattering cross sections. This data set was computed speci®cally for the soft X-ray region h! < 1:5 keV). Linear attenuation coef®cient measurements yield f 0 !; 0 directly and f 00 !; 0 indirectly through use of the Kramers± Kronig integral. Data from these experiments do not have the reliability of those from refractive-index measurements because of the uncertainty in knowing the correct value for the relativistic correction term. None of the previous techniques is useful for small photon energies. These photons would experience considerable attenuation in traversing both the specimen and the experimental apparatus. For small photon energies or large atomic numbers, re¯ection techniques are used, the most commonly used technique being that of total external re¯ection. As Henke et al. (1982) have shown, when re¯ection occurs at a smooth (vacuum±material) interface, the refractive index of the re¯ecting material can be written as a single complex constant, and measurement of the angle of total external re¯ection may be related directly to the refractive index and therefore to f 0 !; D. Because the X-ray refractive indices of materials are only slightly less than unity, the scattering wavevector D is small, and the scattering angle is only a few degrees in magnitude. Assuming that there is not a strong dependence of f 0 !; D with D, one may consider that this technique provides an estimate of f 0 !; 0 for a photon energy range that cannot be surveyed using more precise techniques. A recent review of the use of re¯ectometers to determine f 0 !; 0 has been given by Lengeler (1994).

4.2.6.3.2. Determination of the real part of the dispersion correction: f 0 !; D This classi®cation includes those experiments in which measurements of the geometrical structure factors Fhkl for various Bragg re¯ections are undertaken. Into this category fall those techniques for which the period of standing-wave ®elds (PendelloÈsung) and re¯ectivity of perfect crystals in Laue or Bragg re¯ection are measured. Also included are those techniques from which the atomic scattering factors are inferred from measurements of Bijvoet- or Friedel-pair intensity ratios for noncentrosymmetric crystal structures. 4.2.6.3.2.1. Measurements using the dynamical theory of X-ray diffraction The development of the dynamical theory of X-ray diffraction (see, for example, Section 5 in IT B, 1995) and recent advances in techniques for crystal growth have enabled experimentalists to determine the geometrical structure factor Fhkl for a variety of materials by measuring the spacing between minima in the internal standing wave ®elds within the crystal (PendelloÈsung). Two classes of PendelloÈsung experiment exist: those for which the ratio l= cos is kept constant and the thickness of the samples varies; and those for which the specimen thickness remains constant and l= cos is allowed to vary. Of the many experiments performed using the former technique, measurements by Aldred & Hart (1973a,b) for

250


4.2. X-RAYS silicon are thought to be the most accurate determinations of the atomic form factor f !; D for that material. From these data, Price, Maslen & Mair (1978) were able to re®ne values of f 0 !; D for a number of photon energies. Recently, Deutsch & Hart (1985) were able to extend the determination of the form factor to higher values of momentum transfer hD. This technique requires for its success the availability of large, strain-free crystals, which limits the range of materials that can be investigated. A number of experimentalists have attempted to measure PendelloÈsung fringes for parallel-sided specimens illuminated by white radiation, usually from synchrotron-radiation sources. [See, for example, Hashimoto, Kozaki & Ohkawa (1965) and Aristov, Shmytko & Shulakov (1977).] A technique in which the PendelloÈsung fringes are detected using a solid-state detector has been reported by Takama, Kobayashi & Sato (1982). Using this technique, Takama and his co-workers have reported measurements for silicon (Takama, Iwasaki & Sato, 1980), germanium (Takama & Sato, 1984), copper (Takama & Sato, 1982), and aluminium (Takama, Kobayashi & Sato, 1982). A feature of this technique is that it can be used with small crystals, in contrast to the ®rst technique in this section. However, it does not have the precision of that technique. Another technique using the dynamical theory of X-ray diffraction determines the integrated re¯ectivity for a Braggcase re¯ection that uses the expression for integrated re¯ectivity given by Zachariasen (1945). Using this approach, Freund (1975) determined the value of the atomic scattering factor f !; g222 for copper. Measurements of intensity are dif®cult to make, and this method is not capable of yielding results having the precisions of the PendelloÈsung techniques. 4.2.6.3.2.2. Friedel- and Bijvoet-pair techniques The Bijvoet-pair technique (Bijvoet et al., 1951) is used extensively by crystallographers to assist in the resolution of the phase problem in the solution of crystal structures. Measurements of as many as several hundred values for the diffracted intensities Ihkl for a crystal may be made. When these are analysed, the Cole & Stemple (1962) observation that the ratio of the intensities scattered in the Bijvoet or Friedel pair is independent of the state of the crystal is assumed to hold. This is a necessary assumption since in a large number of structure analyses radiation damage occurs during the course of an experiment. For simple crystal structures, Hosoya (1975) has outlined a number of ways in which values of f 0 !; ghkl and f 00 !; ghkl may be extracted from the Friedel-pair ratios. Measurements of these corrections for atoms such as gallium, indium, arsenic and selenium have been made. In more complicated crystal structures for which the positional parameters are known, attempts have been made to determine the anomalous-scattering corrections by leastsquares-re®nement techniques. Measurements of these corrections for a number of atoms have been made, inter alia, by Engel & Sturm (1975), Templeton & Templeton (1978), Philips, Templeton, Templeton & Hodgson (1978), Templeton, Templeton, Philips & Hodgson (1980), Philips & Hodgson (1985), and Chapuis, Templeton & Templeton (1985). There are a number of problems with this approach, not the least of which are the requirement to measure intensities accurately for a large period of time and the assumption that specimen perfection does not affect the intensity ratio. Also, factors such as crystal shape and primary and secondary extinction may adversely affect the ability to measure intensity ratios correctly. One problem that has to be addressed in this type of

determination is the fact that f 0 !; 0 and f 00 !; 0 are related to one another, and cannot be re®ned separately. 4.2.6.3.3. Comparison of theory with experiment In this section, discussion will be focused on (i) the scattering of photons having energies considerably greater than that of the K-absorption edge of the atom from which they are scattered, and (ii) scattering of photons having energies in the neighbourhood of the K-absorption edge of the atom from which they are scattered. 4.2.6.3.3.1. Measurements in the high-energy limit !=! ! 0 In this case, there is some possibility of testing the validity of the relativistic dipole and relativistic multipole theories since, in the high-energy limit, the value of f 0 !; 0 must approach a value related to the total self energy of the atom Etot =mc2 . That there is an atomic number dependent systematic error in the relativistic dipole approach has been demonstrated by Creagh (1984). The question of whether the relativistic multipole approach yields a result in better accord with the experimental data is answered in Table 4.2.6.4, where a comparison of values of f 0 !; 0 is made for three theoretical data sets (this work; Cromer & Liberman, 1981; Wagenfeld, 1975) with a number of experimental results. These include the `direct' measurements using X-ray interferometers (Cusatis & Hart, 1975; Creagh, 1984), the Kramers± Kronig integration of X-ray attenuation data (Gerward et al., 1979), and the angle-of-the-prism data of Deutsch & Hart (1984b). Also included in the table are ìndirect' measurements: those of Price et al. (1978), based on PendelloÈsung measurements, and those of Grimvall & Persson (1969). These latter data estimate f 0 !; ghkl and not f 0 !; 0. Table 4.2.6.4 details values of the real part of the dispersion correction for LiF, Si, Al and Ge for the characteristic wavelengths Ag K1 , Mo K1 and Cu K1 . Of the atomic species listed, the ®rst three are approaching the high-energy limit at Ag K1 , whilst for germanium the K-shell absorption edge lies between Mo K1 and Ag K1 . The high-energy-limit case is considered ®rst: both the relativistic dipole and relativistic multipole theories underestimate f 0 !; 0 for LiF whereas the non-relativistic theory overestimates f 0 !; 0 when compared with the experimental data. For silicon, however, the relativistic multipole yields values in good agreement with experiment. Further, the values derived from the work of Takama et al. (1982), who used a PendelloÈsung technique to measure the atomic form factor of aluminium are in reasonable agreement with the relativistic multipole approach. Also, some relatively imprecise measurements by Creagh (1985) are in better accordance with the relativistic multipole values than with the relativistic dipole values. Further from the high-energy limit (smaller values of !=! , the relativistic multipole approach appears to give better agreement with theory. It must be reported here that measurements by Katoh et al. (1985a) for lithium ¯uoride at a Ê yielded a value of 0.018 in good wavelength of 0.77366 A agreement with the relativistic multipole value 0.017. At still smaller values of !=! , the non-relativistic theory yields values considerably at variance with the experimental data, except for the case of LiF using Cu K1 radiation. The relativistic multipole approach seems, in general, to be a little better than the relativistic approach, although agreement between experiment and theory is not at all good for germanium. Neither

251


4. PRODUCTION AND PROPERTIES OF RADIATIONS of the experiments cited here, however, has claims to high accuracy. In Table 4.2.6.5, a comparison is made of measurements of f 00 !; 0 derived from the results of the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987, 1990) with a number of theoretical predictions. The measurements were made on carbon, silicon and copper specimens at the characteristic wavelengths Cu K1 , Mo K1 and Ag K1 . The principal conclusion that can be drawn from perusual of Table 4.2.6.5 is that only minor, non-systematic differences exist between the predictions of the several relativistic approaches and the experimental results. In contrast, the non-relativistic theory fails for higher values of atomic number. 4.2.6.3.3.2. Measurements in the vicinity of an absorption edge The advent of the synchrotron-radiation source as a routine experimental tool and the deep interest that many crystallographers have in both XAFS and the anomalous-scattering determinations of crystal structures have stimulated considerable interest in the determination of the dispersion corrections in the neighbourhood of absorption edges. In this region, the interaction of the ejected photoelectron with electrons belonging to neighbouring atoms causes the modulations that are referred to as XAFS. Both f 00 !; 0 (which is directly proportional to the X-ray scattering cross section) and f 0 !; 0 [which is linked to f 00 !; 0 through the Kramers±Kronig integral] exhibit these modulations. It is at this point that one must realize that the theoretical tabulations are for the interactions of photons with isolated atoms. At best, a comparison of theory and experiment can show that they follow the same trend. Measurements have been made in the neighbourhood of the absorption edges of a variety of atoms using the `direct' techniques interferometry, Kramers±Kronig, refraction of a prism and critical-angle techniques, and by the ìndirect' re®nement techniques. In Table 4.2.6.6, a comparison is made of experimental values taken at or near the absorption edges of copper, nickel and niobium with theoretical predictions. These have not been adjusted for any energy window that might be thought to exist in any particular experimental con®guration. The theoretical values for niobium have been calculated at the energy at which the experimentalists claimed the experiment was conducted. Despite the considerable experimental dif®culties and the wide variety of experimental apparatus, there appears to be close agreement between the experimental data for each type of atom. There appears to be, however, for both copper and nickel, a large discrepancy between the theoretical values and the experimental values. It must be remembered that the experimental values are averages of the value of f 0 !; 0, the average being taken over the range of photon energies that pass through the device when it is set to a particular energy value. Furthermore, the exact position of the wavelength chosen may be in doubt in absolute terms, especially when synchrotronradiation sources are used. Therefore, to be able to make a more realistic comparison between theory and experiment, the theoretical data gained using the relativistic multipole approach (this work) were averaged over a rectangular energy window of 5 eV width in the region containing the absorption edge. The rectangular shape arises because of the shape of the re¯ectivity curve and 5 eV was chosen as a result of (i) analysis of the characteristics of the interferometers used by Bonse et al. and Hart et al., and (ii) a statement concerning the experimental bandpass of the interferometer used by Bonse & Henning (1986). It must also be borne in mind that mechanical vibrations and

Table 4.2.6.4. Comparison of measurements of the real part of the dispersion correction for LiF, Si, Al and Ge for characteristic wavelengths Ag K1 , Mo K1 and Cu K1 with theoretical predictions; the experimental accuracy claimed for the experiments is shown thus: (10) 10% error f 0 !; 0 Sample LiF

Si

Al

Ge

Theory This work Cromer & Liberman (1981) Wagen®eld (1975) Experiment Creagh (1984) Deutsch & Hart (1984b) Theory This work Cromer & Liberman (1981) Wagenfeld (1975) Experiment Cusatis & Hart (1975) Price et al. (1978) Gerward et al. (1979) Creagh (1984) Deutsch & Hart (1984b) Theory This work Cromer & Liberman (1981) Wagenfeld (1975) Experiment Creagh (1985) Takama et al. (1982) Theory This work Cromer & Liberman (1981) Wagenfeld (1975) Experiment Gerward et al. (1979) Grimvall & Persson (1969)

Cu K1

Mo K1

Ag K1

0.075 0.068

0.017 0.014

0.010 0.006

0.080

0.023

0.015

0.085 (5) 0.020 (10) 0.0217 (1) ±

0.014 (10) 0.0133 (1)

0.254 0.242

0.817 0.071

0.052 0.042

0.282

0.101

0.071

0.0863 (2)

0.0568 (2)

±

± 0.085 (7) 0.244 (7) 0.099 (7)

0.047 (7) 0.070 (7)

0.236 (5) 0.091 (5) ± 0.0847 (1)

0.060 (5) 0.0537 (1)

0.213 0.203

0.0645 0.0486

0.041 0.020

0.235

0.076

0.553

± 0.20 (5)

0.065 (20) 0.07 (5)

0.044 (20) 0.035 (10)

1.089 1.167

0.155 0.062

0.302 0.197

1.80

0.08

0.14

1.04

0.30

0.43

1.79

0.08

0.27

thermal ¯uctuations can broaden the energy window and that 5 eV is not an overestimate of the width of this window. Note that for elements with atomic numbers less than 40 the experimental width is greater than the line width. For the Bonse & Henning (1986) data, two values are listed for each experiment. Their experiment demonstrates the effect the state of polarization of the incoming photon has on the value of f 0 !; 0. Similar X-ray dichroism has been shown for sodium bromate by Templeton & Templeton (1985) and Chapuis et al. (1985). The theoretical values are for averaged polarization in

252


Reference

4.2. X-RAYS Table 4.2.6.5. Comparison of measurements of f 0 (!; 0) for C, Si and Cu for characteristic wavelengths Ag K1 , Mo K1 and Cu K1 with theoretical predictions; the measurements are from the IUCr X-ray Attenuation Project Report (Creagh & Hubbell, 1987, 1990), corrected for the effects of Compton, Laue±Bragg, and small-angle scattering f 0 !; 0 Sample 6

C

14

29

Si

Cu

Reference Theory This work Cromer & Liberman (1981) Wagenfeld (1975) Sco®eld (1973) Storm & Israel (1970) Experiment IUCr Project Theory This work Cromer & Liberman (1981) Wagenfeld (1975) Sco®eld (1973) Storm & Israel (1970) Experiment IUCr Project Theory This work Cromer & Liberman (1981) Sco®eld (1973) Experiment IUCr Project

Cu K1

Mo K1

Ag K1

0.0091 0.0091

0.0016 0.0016

0.0009 0.0009

± 0.0093 0.0090

± 0.0016 0.0016

± 0.0009 0.0009

0.0093

0.0016

0.0009

0.330 0.330

0.070 0.0704

0.043 0.0431

0.330 0.332 0.331

0.071 0.0702 0.0698

0.044 0.0431 0.0429

0.332

0.0696

0.0429

0.588 0.589

1.265 1.265

0.826 0.826

0.586

1.256

0.826

0.588

1.267

0.826

the incident photon beam. Another important feature is the difference of 0.16 electrons between the Kramers±Kronig and the interferometer values. Bonse & Henning (1986) did not add the relativistic correction term to their Kramers±Kronig values. Inclusion of this term would have reduced the quoted values by 0.20, bringing the two data sets into close agreement with one another. Katoh et al. (1985b) have made measurements spanning the K-absorption edge of germanium using the deviation by a prism method, and these data have been shown to be in excellent agreement with the theory on which these tables are based (Creagh, 1993). In contrast, the theoretical approach of Pratt, Kissell & Bergstrom (1994) does not agree so well, especially near to, and at higher photon energies, than the K-edge energy. Also, Chapuis et al. (1985) have measured the dispersion corrections for holmium in [HoNa(edta)] 8H2 O for the characteristic emission lines Cu K1 , Cu K2 , Cu K , and Mo K1 using a re®nement technique. Their results are in reasonable agreement with the relativistic multipole theory, e.g. for f 0 !; D at the wavelength of Cu K1 experiment gives 16:0 0:2 whereas the relative multipole approach yields 15:0. For Cu K2 , experiment yields 13:9 0:3 and theory gives 13:67. The discrepancy between theory and experiment may well be explained by the oxidation state of the holmium ion, which is in the form Ho3 . The oxidation state of an atom affects both the position of the absorption edge and the magnitude of the

relativistic correction. Both of these will have a large in¯uence on the value of f 0 !; D in the neighbourhood of the absorption edge, Another problem that may be of some signi®cance is the natural width of the absorption edge, about 60 eV. What is remarkable is the extent of the agreement between theory and experiment given the nature of the experiment. In these experiments, the intensities of many re¯ections (usually nearly 1000) are analysed and compared. Such a procedure can be followed only if there is no dependence of f 0 !; D on D. It had often been thought that the dispersion corrections should exhibit some functional dependence on scattering angle. Indeed, some texts ascribe to these corrections the same functional dependence on angle of scattering as the form factor. A fundamental dependence was also predicted theoretically on the basis of non-relativistic quantum mechanics (Wagenfeld, 1975). This prediction is not supported by modern approaches using relativistic quantum mechanics [see, for example, Kissel et al. (1980)]. Reference to Tables 4.2.6.4 and 4.2.6.6 shows that the agreement between experimental values derived from diffraction experiments and those derived from `direct' experiments is excellent. They are also in excellent agreement with the recent calculations, using relativistic quantum mechanics, so that it may be inferred that there is indeed no functional dependence of the dispersion corrections on scattering angle. Moreover, Suortti, Hastings & Cox (1985) have recently demonstrated that f 0 !; D was independent of D in a powder-diffraction experiment using a nickel specimen. 4.2.6.3.3.3. Accuracy in the tables of dispersion corrections Experimentalists must be aware of two potential sources of error in the values of f 0 !; 0 listed in Table 4.2.6.5. One is computational, arising from the error in calculating the relativistic correction. Stibius-Jensen (1980) has suggested that this error may be as large as 0:25Etot =mc2 . This means, for example, that the real part of the dispersion correction f 0 !; 0 Ê is 1:168 0:146. for lead at the wavelength of 0.55936 A The effect of this error is to shift the dispersion curve vertically without distorting its shape. Note, however, that the direction of the shift is either up or down for all atoms: the effect of multipole cancellation and retardation will be in the same direction for all atoms. The second possible source of error occurs because the position of the absorption edge varies somewhat depending on the oxidation state of the scattering atom. This has the effect of displacing the dispersion curve laterally. Large discrepancies may occur for those regions in which the dispersion corrections are varying rapidly with photon energy, i.e. near absorption edges. It must also be borne in mind that in the neighbourhood of an absorption edge polarization effects may occur. The tables are valid only for average polarization. 4.2.6.3.3.4. Towards a tensor formalism The question of how best to describe the interaction of X-rays with crystalline materials is quite dif®cult to answer. In the form factor formalism, the atoms are supposed to scatter as though they are isolated atoms situated at ®xed positions in the unit cell. In the vast majority of cases, the polarization on scattering is not detected, and only the scattered intensities are measured. From the scattered intensities, the distribution of the electron density within the unit cell is calculated, and the difference between the form-factor model and that calculated from the intensities is taken as a measure of the nature and location of chemical bonds between atoms in the unit cell.

253


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.6.6. Comparison of f 0 (!A ; 0) for copper, nickel, zirconium, and niobium for theoretical and experimental data sets; in this table: BR Bragg re¯ection; IN interferometer; KK Kramers±Kronig; CA critical angle; and REF re¯ectivity; measurements have been made for the K-absorption edges of copper and nickel and near the K-absorption edges of zirconium and niobium; claimed experimental errors are not worse than 5% f 0 !A ; 0 Reference Experiment Freund (1975) Begum, Hart, Lea & Siddons (1986) Bonse & Materlik (1972) Bonse, Hartmann-Lotsch & Lotsch (1983a) Hart & Siddons (1981) Kawamura & Fukimachi (1978; cited in Bonse & Hartmann-Lotsch, 1984) Dreier et al. (1984) Bonse & Hartmann-Lotsch (1984) Fukamachi et al. (1978; cited in Bonse & Hartmann-Lotsch, 1984) Bonse & Henning (1986) Stanglmeier, Lengeler, Weber, Gobel & Schuster (1992) Creagh (1990, 1993) Theory Cromer & Liberman (1981) This work Averaged values (5 eV) window)

Method

Cu

BR IN IN IN IN KK

8.2 7.84 8.3 9.3

KK IN KK KK CA

8.2 8.3 8.3 8.8 10.0

Nb

Zr

4.396

6.670

7.66 8.1 9.2 7.9

7.83

7.8 8.1 7.7

IN KK REF

8.5

8.1

REF

8.2

7.7

13.50 9.5 9.0

9.45 9.40 7.53

This is the zeroth-order approximation to a solution, but it is in fact the only way crystal structures are solved ab initio. The existence of chemical bonding imposes additional restrictions on the symmetry of lattices, and, if the associated in¯uence this has on the complexity of energy levels is taken into account, signi®cant changes in the scattering factors may occur in the neighbourhood of the absorption edges of the atoms comprising the crystal structure. The magnitudes of the dispersion corrections are sensitive to the chemical state, particularly oxidation state, and phenomena similar to those observed in the XAFS case (Section 4.2.4) are observed. The XAFS interaction arising from the presence of neighbouring atoms is proportional to f 00 !; 0 and therefore is related to f 0 !; 0 through the Kramers±Kronig integral. It is not surprising that these modulations are observed in diffracted intensities in those X-ray diffraction experiments where the photon energy is scanned through the absorption edge of an atomic species in the crystal lattice. Studies of this type are referred to as diffraction absorption ®ne structure (DAFS) experiments. A recent review of work performed using counter techniques has been given by Sorenson (1994). Creagh & Cookson (1995) have described the use of imaging-plate techniques to study the structure and site symmetry using the DAFS technique. This technique has the ability to discriminate between different lattice sites in the unit cell occupied by an atomic species. XAFS cannot make this discrimination. The DAFS modulations are small perturbations to the diffracted intensities. They are, however, signi®cantly larger than the tensor effects described in the following paragraphs. In the case where the excited state lacks high symmetry and is oriented by crystal bonding, the scattering can no longer be

7.37; 7.21;

7.73 7.62 6.8

4.20; 7.39 4.04; 7.23 8.18

6.207 6.056 6.04

Table 4.2.6.7. List of wavelengths, energies, and linewidths used in compiling the table of dispersion corrections (a) Agarwal (1979); (b) Deutsch & Hart (1982) Radiation 79

Au K1 W K1 73 Ta K1 47 Ag K1 42 Mo K1 29 Cu K1 27 Co K1 26 Fe K1 24 Cr K1 22 Ti K1 74

Wavelength Ê) (A 0.180195 0.209010 0.215947 0.559360 0.709260 1.540520 1.788965 1.93597 2.289620 2.748510

Energy (keV) 68.803 59.318 57.412 22.165 17.480 8.04792 6.9302 6.4040 5.4149 4.5108

Linewidth (eV) 46 (a) 43 (a) 42 (a) 7 (a) 4 (a) 2.61 (b) 1.8 1.6 1.5 1.4

described by a scalar scattering factor but must be described by a symmetric second-rank tensor. The consequences of this have been described by Templeton (1994). It follows therefore that material media can be optically active in the X-ray region. Hart (1994) has used his unique polarizing X-ray optical devices to study, for example, Faraday rotation in such materials as iron, in the region of the iron K-absorption edge, and cobalt(III) bromide monohydrate in the region of the cobalt K-absorption edge. The theory of anisotropy in anomalous scattering has been treated extensively by Kirfel (1994), and Morgenroth, Kirfel

254


Ni

4.2. X-RAYS Table 4.2.6.8. Dispersion corrections for forward scattering Ê ) 2.748510 Wavelength (A Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

0

f = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 =

0.0035 0.0013 0.0117 0.0050 0.0263 0.0139 0.0490 0.0313 0.0807 0.0606 0.1213 0.1057 0.1700 0.1710 0.2257 0.2621 0.2801 0.3829 0.3299 0.5365 0.3760 0.7287 0.3921 0.9619 0.3821 1.2423 0.3167 1.5665 0.1832 1.9384 0.0656 2.3670 0.5083 2.8437 1.3666 3.3694 5.4265 4.0017 2.2250 0.5264 1.6269 0.6340 1.2999 0.7569 1.0732 0.8956 0.8901 1.0521 0.7307 1.2272 0.5921 1.4240 0.4430 1.6427 0.3524 1.8861 0.2524 2.1518 0.1549 2.4445 0.0687 2.7627 0.0052 3.1131 0.0592 3.4901 0.1009 3.9083

2.289620

1.935970

1.788965

1.540520

0.709260

0.559360

0.215947

0.209010

0.180195

0.0023 0.0008 0.0083 0.0033 0.0190 0.0094 0.0364 0.0213 0.0606 0.0416 0.0928 0.0731 0.1324 0.1192 0.1793 0.1837 0.2295 0.2699 0.2778 0.3812 0.3260 0.5212 0.3647 0.6921 0.3898 0.8984 0.3899 1.1410 0.3508 1.4222 0.2609 1.7458 0.0914 2.1089 0.1987 2.5138 0.6935 2.9646 1.6394 3.4538 4.4818 0.4575 2.1308 0.5468 1.5980 0.6479 1.2935 0.7620 1.0738 0.8897 0.9005 1.0331 0.7338 1.1930 0.6166 1.3712 0.4989 1.5674 0.3858 1.7841 0.2871 2.0194 0.1919 2.2784 0.1095 2.5578 0.0316 2.8669

0.0015 0.0006 0.0060 0.0023 0.0140 0.0065 0.0273 0.0148 0.0461 0.0293 0.0716 0.0518 0.1037 0.0851 0.1426 0.1318 0.1857 0.1957 0.2309 0.2765 0.2774 0.3807 0.3209 0.5081 0.3592 0.6628 0.3848 0.8457 0.3920 1.0596 0.3696 1.3087 0.3068 1.5888 0.1867 1.9032 0.0120 2.2557 0.3318 2.6425 0.8645 3.0644 1.9210 3.5251 3.5716 0.4798 2.0554 0.5649 1.5743 0.6602 1.2894 0.7671 1.0699 0.8864 0.9134 1.0193 0.7701 1.1663 0.6412 1.3291 0.5260 1.5069 0.4179 1.7027 0.3244 1.9140 0.2303 2.1472

0.0013 0.0005 0.0052 0.0019 0.0121 0.0055 0.0237 0.0125 0.0403 0.0248 0.0630 0.0440 0.0920 0.0725 0.1273 0.1126 0.1670 0.1667 0.2094 0.2373 0.2551 0.3276 0.2979 0.4384 0.3388 0.5731 0.3706 0.7329 0.3892 0.9202 0.3880 1.1388 0.3532 1.3865 0.2782 0.6648 0.1474 1.9774 0.0617 2.3213 0.3871 2.6994 0.9524 3.1130 2.0793 3.5546 3.3307 0.4901 2.0230 0.5731 1.5664 0.6662 1.2789 0.7700 1.0843 0.8857 0.9200 1.0138 0.7781 1.1557 0.6523 1.3109 0.5390 1.4821 0.4363 1.6673 0.3390 1.8713

0.0008 0.0003 0.0038 0.0014 0.0090 0.0039 0.0181 0.0091 0.0311 0.0180 0.0492 0.0322 0.0727 0.0534 0.1019 0.0833 0.1353 0.1239 0.1719 0.1771 0.2130 0.2455 0.2541 0.3302 0.2955 0.4335 0.3331 0.5567 0.3639 0.7018 0.3843 0.8717 0.3868 1.0657 0.3641 1.2855 0.3119 1.5331 0.2191 1.8069 0.0687 2.1097 0.1635 2.4439 0.5299 2.8052 1.1336 3.1974 2.3653 3.6143 3.0029 0.5091 1.9646 0.5888 1.5491 0.6778 1.2846 0.7763 1.0885 0.8855 0.9300 1.0051 0.7943 1.1372 0.6763 1.2805 0.5657 1.4385

0.0003 0.0001 0.0005 0.0002 0.0013 0.0007 0.0033 0.0016 0.0061 0.0033 0.0106 0.0060 0.0171 0.0103 0.0259 0.0164 0.0362 0.0249 0.0486 0.0363 0.0645 0.0514 0.0817 0.0704 0.1023 0.0942 0.1246 0.1234 0.1484 0.1585 0.1743 0.2003 0.2009 0.2494 0.2262 0.3064 0.2519 0.3716 0.2776 0.4457 0.3005 0.5294 0.3209 0.6236 0.3368 0.7283 0.3463 0.8444 0.3494 0.9721 0.3393 1.1124 0.3201 1.2651 0.2839 1.4301 0.2307 1.6083 0.1547 1.8001 0.0499 2.0058 0.0929 2.2259 0.2901 2.4595 0.5574 2.7079

0.0004 0.0000 0.0001 0.0001 0.0004 0.0004 0.0015 0.0009 0.0030 0.0019 0.0056 0.0036 0.0096 0.0061 0.0152 0.0098 0.0218 0.0150 0.0298 0.0220 0.0406 0.0313 0.0522 0.0431 0.0667 0.0580 0.0826 0.0763 0.0998 0.0984 0.1191 0.1249 0.1399 0.1562 0.1611 0.1926 0.1829 0.2348 0.2060 0.2830 0.2276 0.3376 0.2496 0.3992 0.2704 0.4681 0.2886 0.5448 0.3050 0.6296 0.3147 0.7232 0.3240 0.8257 0.3242 0.9375 0.3179 1.0589 0.3016 1.1903 0.2758 1.3314 0.2367 1.4831 0.1811 1.6452 0.1067 1.8192

0.0006 0.0000 0.0005 0.0000 0.0009 0.0000 0.0012 0.0001 0.0020 0.0002 0.0025 0.0004 0.0027 0.0007 0.0025 0.0012 0.0028 0.0019 0.0030 0.0028 0.0020 0.0040 0.0017 0.0056 0.0002 0.0077 0.0015 0.0103 0.0032 0.0134 0.0059 0.0174 0.0089 0.0219 0.0122 0.0273 0.0159 0.0338 0.0212 0.0414 0.0259 0.0500 0.0314 0.0599 0.0377 0.0712 0.0438 0.0840 0.0512 0.0984 0.0563 0.1146 0.0647 0.1326 0.0722 0.1526 0.0800 0.1745 0.0880 0.1987 0.0962 0.2252 0.1047 0.2543 0.1106 0.2858 0.1180 0.3197

0.0006 0.0000 0.0005 0.0000 0.0009 0.0000 0.0013 0.0001 0.0020 0.0002 0.0026 0.0004 0.0028 0.0007 0.0028 0.0011 0.0031 0.0017 0.0034 0.0026 0.0026 0.0037 0.0025 0.0052 0.0012 0.0071 0.0003 0.0096 0.0017 0.0125 0.0041 0.0162 0.0067 0.0204 0.0097 0.0255 0.0130 0.0315 0.0179 0.0387 0.0221 0.0468 0.0272 0.0561 0.0330 0.0666 0.0386 0.0787 0.0454 0.0921 0.0500 0.1074 0.0579 0.1242 0.0648 0.1430 0.0721 0.1636 0.0796 0.1863 0.0873 0.2112 0.0954 0.2386 0.1026 0.2682 0.1082 0.3003

0.0006 0.0000 0.0005 0.0000 0.0010 0.0000 0.0014 0.0001 0.0023 0.0001 0.0030 0.0003 0.0034 0.0005 0.0037 0.0008 0.0044 0.0012 0.0052 0.0018 0.0050 0.0027 0.0055 0.0038 0.0050 0.0052 0.0045 0.0069 0.0042 0.0091 0.0030 0.0118 0.0017 0.0149 0.0002 0.0187 0.0015 0.0231 0.0047 0.0284 0.0070 0.0344 0.0101 0.0413 0.0139 0.0492 0.0173 0.0582 0.0219 0.0682 0.0244 0.0796 0.0298 0.0922 0.0344 0.1063 0.0393 0.1218 0.0445 0.1389 0.0501 0.1576 0.0560 0.1782 0.0613 0.2006 0.0668 0.2251

255


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.2.6.8. Dispersion corrections for forward scattering (cont.) Ê ) 2.748510 Wavelength (A Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

0

f = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f 0= f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 =

0.1056 4.3505 0.1220 4.8946 0.0654 5.4198 0.0304 5.9818 0.1659 6.5803 0.3487 7.2047 0.6073 7.8739 0.9294 8.5988 1.3551 9.3504 1.9086 10.1441 2.5003 10.9916 3.5070 11.9019 5.1325 12.6310 7.5862 13.5168 9.2145 12.7661 11.6068 10.1013 13.9940 3.4071 9.6593 3.7063 8.1342 4.0732 7.2079 4.4110 6.5722 4.7587 6.0641 5.1301 5.6727 5.5091 5.3510 5.9005 5.0783 6.3144 4.8443 6.7524 4.6288 7.2035 4.5094 7.6708 4.3489 8.1882 4.1616 8.6945 4.0280 9.2302 3.9471 9.7921 3.9079 10.3763 3.8890 10.9742

2.289620

1.935970

1.788965

1.540520

0.709260

0.559360

0.215947

0.209010

0.180195

0.0247 3.1954 0.1037 3.6029 0.1263 3.9964 0.1338 4.4226 0.1211 4.8761 0.0801 5.3484 0.0025 5.8597 0.1091 6.4069 0.2630 6.9820 0.4640 7.5938 0.7387 8.2358 1.1086 8.9174 1.5975 9.6290 2.2019 10.3742 3.0637 11.1026 4.2407 11.8079 5.6353 12.6156 8.1899 11.7407 10.3310 12.8551 11.0454 10.0919 12.8190 3.5648 9.3304 3.8433 7.9841 4.1304 7.1451 4.4278 6.5334 4.7422 6.0570 5.0744 5.6630 5.4178 5.3778 5.7756 5.0951 6.1667 4.8149 6.5527 4.5887 6.9619 4.4106 7.3910 4.2698 7.8385 4.1523 8.2969

0.1516 2.3960 0.0489 2.7060 0.0138 3.0054 0.0659 3.3301 0.1072 3.6768 0.1301 4.0388 0.1314 4.4331 0.1220 4.8540 0.0861 5.2985 0.0279 5.7719 0.0700 6.2709 0.2163 6.8017 0.4165 7.3594 0.6686 7.9473 0.9868 8.5620 1.4022 9.2067 1.9032 9.8852 2.6313 10.5776 3.5831 11.2902 4.6472 12.0003 6.3557 12.8927 8.0962 11.8734 10.9279 9.2394 10.5249 9.9814 13.2062 3.6278 9.3497 3.8839 7.9854 4.1498 7.1681 4.4280 6.5583 4.7292 6.0597 5.0280 5.6628 5.3451 5.3448 5.6776 5.0823 6.0249 4.8591 6.3813

0.2535 2.0893 0.1448 2.3614 0.0720 2.6241 0.0066 2.9086 0.0496 3.2133 0.0904 3.5326 0.1164 3.8799 0.1331 4.2509 0.1305 4.6432 0.1128 5.0613 0.0634 5.5027 0.0214 5.9728 0.1473 6.4674 0.3097 6.9896 0.5189 7.5367 0.7914 8.1113 1.1275 8.7159 1.5532 9.3585 2.1433 10.0454 2.7946 10.7091 3.6566 11.4336 4.8792 12.1350 6.7923 12.8653 8.1618 11.9121 10.0720 9.2324 10.2609 9.9412 13.5405 3.6550 9.3863 3.9016 8.0413 4.1674 7.1503 4.4320 6.5338 4.7129 6.0673 5.0074 5.6969 5.3151 5.3940 5.6309

0.4688 1.6079 0.3528 1.8200 0.2670 2.0244 0.1862 2.2449 0.1121 2.4826 0.0483 2.7339 0.0057 3.0049 0.0552 3.2960 0.0927 3.6045 0.1215 3.9337 0.1306 4.2820 0.1185 4.6533 0.0822 5.0449 0.0259 5.4591 0.0562 5.8946 0.1759 6.3531 0.3257 6.8362 0.5179 7.3500 0.7457 7.9052 1.0456 8.4617 1.4094 9.0376 1.8482 9.6596 2.4164 10.2820 3.1807 10.9079 4.0598 11.5523 5.3236 12.2178 8.9294 11.1857 8.8380 11.9157 9.1472 9.1891 9.8046 9.8477 14.9734 3.7046 9.4367 3.9380 8.0393 4.1821 7.2108 4.4329

0.9393 2.9676 1.5307 3.2498 2.7962 3.5667 2.9673 0.5597 2.0727 0.6215 1.6832 0.6857 1.4390 0.7593 1.2594 0.8363 1.1178 0.9187 0.9988 1.0072 0.8971 1.1015 0.8075 1.2024 0.7276 1.3100 0.6537 1.4246 0.5866 1.5461 0.5308 1.6751 0.4742 1.8119 0.4205 1.9578 0.3680 2.1192 0.3244 2.2819 0.2871 2.4523 0.2486 2.6331 0.2180 2.8214 0.1943 3.0179 0.1753 3.2249 0.1638 3.4418 0.1578 3.6682 0.1653 3.9035 0.1723 4.1537 0.1892 4.4098 0.2175 4.6783 0.2586 4.9576 0.3139 5.2483 0.3850 5.5486

0.0068 2.0025 0.1172 2.2025 0.2879 2.4099 0.5364 2.6141 0.8282 2.8404 1.2703 3.0978 2.0087 3.3490 5.3630 3.6506 2.5280 0.5964 1.9556 0.6546 1.6473 0.7167 1.4396 0.7832 1.2843 0.8542 1.1587 0.9299 1.0547 1.0104 0.9710 1.0960 0.8919 1.1868 0.8200 1.2838 0.7527 1.3916 0.6940 1.5004 0.6411 1.6148 0.5890 1.7358 0.5424 1.8624 0.5012 1.9950 0.4626 2.1347 0.4287 2.2815 0.3977 2.4351 0.3741 2.5954 0.3496 2.7654 0.3302 2.9404 0.3168 3.1241 0.3091 3.3158 0.3084 3.5155 0.3157 3.7229

0.1247 0.3561 0.1321 0.3964 0.1380 0.4390 0.1431 0.4852 0.1471 0.5342 0.1487 0.5862 0.1496 0.6424 0.1491 0.7016 0.1445 0.7639 0.1387 0.8302 0.1295 0.9001 0.1171 0.9741 0.1013 1.0519 0.0809 1.1337 0.0559 1.2196 0.0216 1.3095 0.0146 1.4037 0.0565 1.5023 0.1070 1.6058 0.1670 1.7127 0.2363 1.8238 0.3159 1.9398 0.4096 2.0599 0.5194 2.1843 0.6447 2.3143 0.7989 2.4510 0.9903 2.5896 1.2279 2.7304 1.5334 2.8797 1.9594 3.0274 2.6705 3.1799 5.5645 0.6167 2.8957 0.6569 2.4144 0.6994

0.1146 0.3346 0.1219 0.3726 0.1278 0.4128 0.1329 0.4562 0.1371 0.5025 0.1391 0.5517 0.1406 0.6047 0.1409 0.6607 0.1373 0.7195 0.1327 0.7822 0.1251 0.8484 0.1147 0.9185 0.1012 0.9922 0.0839 1.0697 0.0619 1.1512 0.0316 1.2366 0.0001 1.3259 0.0367 1.4195 0.0809 1.5179 0.1335 1.6194 0.1940 1.7250 0.2633 1.8353 0.3443 1.9496 0.4389 2.0679 0.5499 2.1906 0.6734 2.3197 0.8137 2.4526 1.0234 2.5878 1.2583 2.7310 1.5632 2.8733 1.9886 3.0218 2.6932 3.1695 5.6057 0.6192 2.9190 0.6592

0.0717 0.2514 0.0769 0.2805 0.0819 0.3112 0.0863 0.3443 0.0905 0.3797 0.0934 0.4177 0.0960 0.4582 0.0981 0.5014 0.0970 0.5469 0.0959 0.5955 0.0928 0.6469 0.0881 0.7013 0.0816 0.7587 0.0728 0.8192 0.0613 0.8830 0.0435 0.9499 0.0259 1.0201 0.0057 1.0938 0.0194 1.1714 0.0494 1.2517 0.0835 1.3353 0.1222 1.4227 0.1666 1.5136 0.2183 1.6077 0.2776 1.7056 0.3455 1.8069 0.4235 1.9120 0.5140 2.0202 0.6165 2.1330 0.7322 2.2494 0.8709 2.3711 1.0386 2.4949 1.2397 2.6240 1.4909 2.7538

256


4.2. X-RAYS Table 4.2.6.8. Dispersion corrections for forward scattering (cont.) Ê ) 2.748510 Wavelength (A Lu Hf Ta W Re Os Ir Pt Au Hg TI Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf

0

f = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 = f0 = f 00 =

3.9056 11.5787 4.0452 12.2546 4.0905 12.9479 4.1530 13.6643 4.2681 14.3931 4.4183 15.1553 4.5860 15.9558 4.8057 16.7870 5.0625 17.6400 5.4327 18.5241 5.8163 19.4378 6.4779 20.3336 7.0419 21.2196 7.7195 22.1974 8.5994 23.2213 10.2749 24.2613 10.8938 24.3041 12.3462 25.5374 12.3496 25.1363 13.6049 26.2511 14.4639 27.4475 12.3528 30.1725 17.4143 31.7405 18.0862 33.8963 19.7042 37.3716 24.9307 41.4852 32.8492 32.5421 23.6520 21.9334

2.289620

1.935970

1.788965

1.540520

0.709260

0.559360

0.215947

0.209010

0.180195

4.0630 8.7649 4.0564 9.2832 3.9860 9.8171 3.9270 10.3696 3.9052 10.9346 3.9016 11.5251 3.9049 12.1453 3.9435 12.7910 3.9908 13.4551 4.1029 14.1473 4.2233 14.8643 4.4167 15.5987 4.6533 16.3448 4.9604 17.1410 5.3399 17.9390 5.7275 18.7720 6.2180 19.6009 6.7502 20.4389 7.4161 21.3053 8.2118 22.2248 9.4459 23.1548 9.9362 23.1239 11.1080 24.1168 11.4073 23.2960 11.7097 24.5715 10.4100 25.8115 9.2185 29.3028 23.5202 31.2999

4.6707 6.7484 4.4593 7.1518 4.3912 7.5686 4.2486 8.0005 4.1390 8.4435 4.0478 8.9067 3.9606 9.3923 3.8977 9.8985 3.8356 10.4202 3.8228 10.9650 3.8103 11.5300 3.8519 12.1106 3.9228 12.7017 4.0267 13.3329 4.1781 13.9709 4.3331 14.6313 4.5387 15.3016 4.7764 15.9778 5.0617 16.6687 5.3692 17.4018 5.7337 18.1406 6.1485 18.8728 6.6136 19.6379 6.9721 20.1548 7.7881 21.1738 8.6102 21.8880 9.3381 21.9514 9.7799 22.4858

5.1360 5.9574 4.9466 6.3150 4.7389 6.6850 4.5529 7.0688 4.4020 7.4631 4.2711 7.8753 4.1463 8.3074 4.0461 8.7578 3.9461 9.2222 3.8921 9.7076 3.8340 10.2108 3.8236 10.7292 3.8408 11.2575 3.8855 11.8209 3.9706 12.3915 4.0549 12.9815 4.1818 13.5825 4.3309 14.1902 4.5270 14.8096 4.7310 15.4642 4.9639 16.1295 5.2392 16.7952 5.5633 17.4837 5.8130 17.9579 6.2920 18.8618 6.7506 19.5119 7.4293 20.3581 7.8616 20.8536

6.6179 4.6937 6.1794 4.9776 5.7959 5.2718 5.4734 5.5774 5.2083 5.8923 4.9801 6.2216 4.7710 6.5667 4.5932 6.9264 4.4197 7.2980 4.2923 7.6849 4.1627 8.0900 4.0753 8.5060 4.0111 8.9310 3.9670 9.3834 3.9588 9.8433 3.9487 10.3181 3.9689 10.8038 4.0088 11.2969 4.0794 11.7994 4.1491 12.3296 4.2473 12.8681 4.3638 13.4090 4.5053 13.9666 4.6563 14.3729 4.8483 15.0877 5.0611 15.6355 5.3481 16.3190 5.5545 16.7428

0.4720 5.8584 0.5830 6.1852 0.7052 6.5227 0.8490 6.8722 1.0185 7.2310 1.2165 7.6030 1.4442 7.9887 1.7033 8.3905 2.0133 8.8022 2.3894 9.2266 2.8358 9.6688 3.3944 10.1111 4.1077 10.2566 5.1210 11.0496 7.9122 9.9777 8.0659 10.4580 7.2224 7.7847 6.7704 8.1435 6.8494 8.5178 7.2400 8.8979 8.0334 9.2807 9.6767 9.6646 11.4937 4.1493 9.4100 4.3056 7.8986 4.5125 7.3248 4.6980 6.8498 4.9086 6.6561 5.0785

0.3299 3.9377 0.3548 4.1643 0.3831 4.3992 0.4201 4.6430 0.4693 4.8944 0.5280 5.1558 0.5977 5.4269 0.6812 5.7081 0.7638 5.9978 0.8801 6.2989 1.0117 6.6090 1.1676 6.9287 1.3494 7.2566 1.5613 7.5986 1.8039 7.9509 2.0847 8.3112 2.4129 8.6839 2.8081 9.0614 3.2784 9.4502 3.8533 9.8403 4.6067 10.2413 5.7225 10.6428 6.9995 9.5876 13.5905 6.9468 6.7022 7.3108 6.2891 7.6044 6.3438 7.9477 6.4144 8.1930

2.1535 0.7436 1.9785 0.7905 1.8534 0.8392 1.7565 0.8905 1.6799 0.9441 1.6170 1.0001 1.5648 1.0589 1.5228 1.1193 1.4693 1.1833 1.4389 1.2483 1.4111 1.3189 1.3897 1.3909 1.3721 1.4661 1.3584 1.5443 1.3540 1.6260 1.3475 1.7103 1.3404 1.7986 1.3462 1.8891 1.3473 1.9845 1.3524 2.0819 1.3672 2.1835 1.3792 2.2876 1.3941 2.3958 1.4180 2.4979 1.4359 2.6218 1.4655 2.7421 1.4932 2.8653 1.5323 2.9807

2.4402 0.7010 2.1778 0.7454 2.0068 0.7915 1.8819 0.8388 1.7868 0.8907 1.7107 0.9437 1.6486 0.9993 1.5998 1.0565 1.5404 1.1171 1.5055 1.1796 1.4740 1.2456 1.4497 1.3137 1.4290 1.3851 1.4133 1.4592 1.4066 1.5367 1.3982 1.6167 1.3892 1.7004 1.3931 1.7863 1.3922 1.8770 1.3955 1.9695 1.4083 2.0661 1.4184 1.1650 1.4312 2.2679 1.4527 2.3652 1.4684 2.4829 1.4952 2.5974 1.5203 2.7147 1.5562 2.8250

1.8184 2.8890 2.2909 3.0246 3.1639 3.1610 3.8673 0.6433 2.8429 0.6827 2.4688 0.7238 2.2499 0.7669 2.1036 0.8116 1.9775 0.8589 1.8958 0.9080 1.8288 0.9594 1.7773 1.0127 1.7346 1.0685 1.7005 1.1266 1.6784 1.1876 1.6571 1.2504 1.6367 1.3162 1.6299 1.3840 1.6190 1.4553 1.6136 1.5284 1.6170 1.6047 1.6188 1.6831 1.6231 1.7648 1.6351 1.8430 1.6424 1.9358 1.6592 2.0271 1.6746 2.1208 1.6984 2.2102

& Fischer (1994) have extended this to the description of kinematic diffraction intensities in lattices containing anisotropic anomalous scatterers. Their treatment was developed for space groups up to orthorhombic symmetry. All the preceeding treatments apply to scattering in the neighbourhood of an absorption edge, and to a fairly restricted class of crystals for which the local site symmetry of the electron density of states in the excited state is very different from the apparent crystal symmetry.

These approaches seek to treat the scattering from the crystal as though the scattering from each atomic position can be described by a symmetric second-rank tensor whose properties are determined by the point-group symmetries of those sites. Clearly, this procedure cannot be followed unless the structure has been solved by the usual method. The tensor approach can then be used to explain apparent de®ciencies in that model such as the existence of `forbidden' re¯ections, birefringence, and circular dichroism.

257


4. PRODUCTION AND PROPERTIES OF RADIATIONS Scattering of X-rays from the electron spins in antiferromagnetically ordered materials can also be described by imposing a tensor description on the form factor (Blume, 1994). The tensor in this case is a fourth-rank tensor, and the strength of the interaction, even for the favourable case of resonance scattering, is several orders of magnitude lower in intensity than the polarization effects. Nevertheless, studies have been made on holmium and uranium arsenide, and signi®cant magnetic Bragg scattering has been observed. All the cases cited above represent exciting, state-of-the-art, scienti®c studies. However, none of the work will assist in the solution of crystal structures directly. Researchers should avoid the temptation, in the ®rst instance, to ascribe anything but a scalar value to the form factor. 4.2.6.3.3.5. Summary For the imaginary part of the dispersion correction f 00 !; D, the following observations can be made. (i) Measurements of the linear absorption coef®cient l from which f 0 !; 0 is deduced should follow the recommendations set out in Subsection 4.2.3.2. (ii) There is no rational basis for preferring one set of relativistic calculations of atomic scattering cross sections over another, as Creagh & Hubbell (1987, 1990) and Kissel et al. (1980) have shown. (iii) The total scattering cross section for an ensemble of atoms is not simply the sum of the individual scattering cross sections in the neighbourhood of an absorption edge and therefore f 0 !; 0 will ¯uctuate as ! ! ! . (iv) There is no dependence of f 00 !; D and D. For the real part of the dispersion correction f 0 !; D, the following observations can be made. (i) The relativistic multipole values listed here tend to accord better with experiment than the non-relativistic and relativistic dipole values.

(ii) There is no dependence of f 0 !; D on D. (iii) The theoretical tables are calculated for averaged polarizations. (iv) Experimentalists wishing to compare their data with theoretical predictions should take account of the energy bandpass of their system when determining the appropriate theoretical value. They should also be aware of the fact that the position of the absorption edge depends on the oxidation state of the scattering atom, and that there is an inaccuracy in the tables of f 0 !; 0 of either 0:20Etot =mc2 or 0:10Etot =mc2 .

4.2.6.4. Table of wavelengths, energies, and linewidths used in compiling the tables of the dispersion corrections Table 4.2.6.7 lists the characteristic emission wavelengths that are commonly used by crystallographers in their experiments. Also included are the emission energies (since many systems use energy rather than wavelength discrimination) and the line widths (full width at half-maximum) of these lines (Agarwal, 1979; Stearns, 1984; Deutsch & Hart, 1984a,b).

4.2.6.5. Tables of the dispersion corrections for forward scattering, averaged polarization using the relativistic multipole approach See Subsection 4.2.6.3 for comments on the accuracy of these tables. Note also that in the neighbourhood of absorption edges the values for condensed matter may be signi®cantly different from the values in the tables due to XAFS and XANES effects. The values in Table 4.2.6.8 are for scattering by isolated atoms.

258



4.3. Electron diffraction

By C. Colliex, J. M. Cowley, S. L. Dudarev, M. Fink, J. Gjùnnes, R. Hilderbrandt, A. Howie, D. F. Lynch, L. M. Peng, G. Ren, A. W. Ross, V. H. Smith Jr, J. C. H. Spence, J. W. Steeds, J. Wang, M. J. Whelan and B. B. Zvyagin 4.3.1. Scattering factors for the diffraction of electrons by crystalline solids (By J. M. Cowley) 4.3.1.1. Elastic scattering from a perfect crystal The most important interaction of electrons with crystalline matter is the interaction with the electrostatic potential field. The scattering into sharp, Bragg reflections is considered in terms of the interaction of an incident plane wave with a timeindependent, averaged, periodic potential field which may be written 1 X 'r V h expf 2ih rg; 4:3:1:1

h where is the unit-cell volume and the Fourier coefficients, V h, may be referred to as the structure amplitudes corresponding to the reciprocal-lattice vectors h. In conformity with the crystallographic sign convention used thoughout this volume [see also Volume B (IT B, 1992)], we choose a free-electron approximation for the incident electron beam of the form exp ik r and the interaction is represented by inserting the potential (4.3.1.1) in the SchroÈdinger wave equation r 2 r 2kfE 'rg r 0;

4:3:1:2

where eE is the kinetic energy of the incident beam, k 2=l is the magnitude of the wavevector for the incident electrons, and is an ìnteraction constant' defined by 2mel=h2 ;

4:3:1:3

where h is Planck's constant. Relativistic values of m and l are assumed (see Subsection 4.3.1.4). The solution of equation (4.3.1.2), subject to the boundary conditions imposed by the need to fit the waves in the crystal with the incoming and outgoing waves in vacuum at the crystal surfaces, then allows the directions and amplitudes of the diffracted beams to be obtained in terms of the crystal periodicities and the Fourier coefficients, V h, of 'r by the eigenvalue or Bloch-wave method (Bethe, 1928). Alternatively, the scattered amplitudes may be obtained from the integral form of (4.3.1.2), r expf ik0 rg Z expf ikjr r0 jg 0 K 'r r0 dr0 ; jr r0 j

4:3:1:4

where expf ik0 rg represents the incident beam, K =l, and the integral is taken over the space of the variable, r0 . An iterative solution of (4.3.1.4) leads to the Born series,

0

1

2

...;

and

Z n r K

0 n 1 r dr0 ;

4:3:1:5

for n 1. Terms of the series for n 1; 2; . . . may be considered to represent the contributions from single, double and multiple scattering of the incident electron beam. This

where 'n xy is the projection of the potential distribution within the slice in the direction of the incident beam, taken to be the z axis; 'n x; y

zn R z zn

'x; y; z dz:

4:3:1:7

Propagation of the wave between the centres of slices is represented by convolution with a propagation function, pxy, so that the wave entering the n 1th slice may be written n1 xy

n xy

qn xy pn xy:

4:3:1:8

In the small-angle approximation, the function pn xy is given by the usual Fresnel diffraction theory as pxy i=lz expf ikx2 y2 =2zg:

4:3:1:9

In the limit that the slice thickness, z, tends to zero, the iteration of (4.3.1.8) gives an exact account of the diffraction by the crystal. On the basis of the above-mentioned and other related formulations of the diffraction problem, several computing methods have been devised for calculation of the amplitudes and intensities of the many diffracted beams of appreciable intensity that may emerge simultaneously from a crystal (see Section 4.3.6). In this way, a degree of accuracy may be achieved in the calculation of the intensities of spots in diffraction patterns or of the contrast in electron-microscope images of crystals (Section 4.3.8). 4.3.1.2. Atomic scattering factors All such calculations require a knowledge of the potential distribution, 'r, or its Fourier coefficients, V h. It is usually convenient to express the potential distribution in terms of the sum of contributions of individual atoms centred at the positions r ri . Thus: P 'r 'i r ri 4:3:1:10 i

As a first approximation, the functions 'i r may be identified with the potential distributions for individual, isolated atoms or ions, with the usual spreading due to thermal motion. The interatomic binding and the interactions of ions that are thereby neglected may have important effects on diffraction intensities in some cases. In this approximation, the Fourier transforms for individual atoms may be written

259 Copyright © 2006 International Union of Crystallography 2 s:\ITFC\CHAP-4-3.3d (Tables of Crystallography)

4:3:1:6

i

expf ik0 rg

expf ikjr r0 jg 0 'r jr r0 j

qn xy expf i'n xyg;

or, in terms of the Fourier transforms, Vi , of the 'i r P Vi h expf2i h ri g: 4:3:1:11 V h

where 0

method has been applied to the diffraction from crystals by Fujiwara (1959). A further formulation of the scattering problem in integral form is that due to Cowley & Moodie (1957) who considered the progressive modification of an incident plane wave as it passed through successive thin slices of a crystal. The effect of the nth slice on the incident electron wave is that of a phase-grating so that the wavefunction is modified by multiplication by a transmission function,

4. PRODUCTION AND PROPERTIES OF RADIATIONS Vi s

fiB s=K;

4:3:1:12

where s 4l 1 sin jk k0 j and the f B are the Born electron scattering amplitudes, as conventionally defined, in Ê . Here is half the scattering angle and, again, units of A K =l: Some values of f B s listed in the accompanying Tables 4.3.1.1 and 4.3.1.2 are obtained from the atomic potentials '0 r for isolated, spherically symmetrical atoms or ions by the relation Z1

B

f s 4K

r 2 'r

0

sin sr dr: sr

4:3:1:13

By the use of Poisson's equation relating the potential and charge-density distributions, it is possible to derive the Mott± Bethe formula for f B s in terms of the atomic scattering factors for X-rays, fx s: f B s 2

2

me fZ h2 "0

fx sg=s2 ;

4:3:1:14

where "0 is the permittivity of vacuum, or 4:3:1:15 f B s 0:023934 l2 fZ fx sg= sin2 B Ê Ê [for l in A, f s in A, and fx s in electron units]. This was used for the other listed f B s values. 4.3.1.3. Approximations of restricted validity a Kinematical approximation. In the limiting case of a vanishingly weak interaction of the incident electrons with the scattering potential of the crystal, the Born series (4.3.1.5) may be terminated at the term 1 , corresponding to single scattering. Then the diffracted wave is given for a potential 'r as sexp ikR=R, with R s K 'r expfir sg dr ; 4:3:1:16 where R is the distance to the point of observation. For a periodic potential, 'r, the scattering amplitude for the h beam is R h NK 'r expf2ih rg dr ; 4:3:1:17 where the integral is taken over one unit cell and N is the number of unit cells. From (4.3.1.16), it then follows that the scattering amplitude h is proportional to the structure amplitude, V h; h NKV h P NK fel;i h expf2i h ri g: i

4:3:1:18 4:3:1:19

The intensity of the h diffracted beam is then proportional to h h, and so to jV hj2 . Similarly, we may write the differential scattering cross section for the scattering from a single isolated atom as j f B sj2 K 2 jVi sj2 :

4:3:1:20

b Two-beam approximation. For some specific orientations of a crystal of relatively simple structure, the incident beam may be close to the Bragg angle for a strong, inner reflection but not for any other reflection. Then the approximation may be made that only those beams with indices 0 and h have appreciable intensity. The intensities of these beams for a parallel-sided, plate-shaped, centrosymmetric crystal are given in MacGillary's (1940) development of the theory of Bethe (1928) as Ih I0 fV hg2

sin2 fth2 h 2 1=2 g 2 h2 h 2

4:3:1:21

and I0 I0 Ih, where I0 is the incident-beam intensity, t is the crystal thickness, h is the extinction distance given by h =V h, and h is the excitation error which measures the distance of the reciprocal-lattice point h from the Ewald sphere. A formula due to Blackman (1939), obtained by integrating (4.3.1.21) over h , provides a useful first approximation for the intensities of ring or arc patterns given by polycrystalline material (see Section 2.5.2). c Phase-grating approximations. For extremely thin crystals, the scattering can be approximated by that of a twodimensional potential distribution given by projection of the three-dimensional distribution in the beam direction. Then, by analogy with (4.3.1.6), the emerging wave is xy expf i'xyg when 'xy

RH 0

'xyz dz

4:3:1:23

and the diffraction amplitudes are given by the Fourier transform of this expression. For thicker crystals, this approximation applies in the limit of very high electron-accelerating voltage, with the value of Ê , viz appropriate for the Compton wavelength, l 0:024262 A 0:0005068. It may be noted that for the special case of a single layer of atoms the solution of the wave equations (4.3.1.2) or (4.3.1.4), with the real potential (4.3.1.1) inserted, leads to a form equivalent to the Moliere high-energy approximation for the scattering by single atoms, namely Z1 i KV s fexp i'q 1g expfi q sg d2 q; 4:3:1:24 l 1

where q is a two-dimensional vector with components x; y; and R1 'q; z dz; 4:3:1:25 'q 1

and this, in the low-angle approximation, is the same as (4.3.1.23). Then the scattered amplitude can be considered as made up from contributions from individual atoms that are equal (apart from bonding effects) to the complex atomic scattering amplitudes tabulated in connection with the diffraction of electrons by gases. 4.3.1.4. Relativistic effects It has been shown by Fujiwara (1961) that, at least for electron energies up to 1 MeV or so, the relativistic effects on diffraction amplitudes and geometry are adequately described by the use of relativisitically corrected values for the mass and wavelength of the electrons; m m0 1 lh

2

1=2

1=2 eE 1 2em0 E 1 lc 2m0 c2

12:2639=E 0:97845 10

6

4:3:1:26 2 1=2

E2 1=2 ;

4:3:1:27

where m0 is the rest mass, lc is the Compton wavelength Ê if E is in volts. Consequently, =c, and l is given in A varies with the incident electron energy as 1 h2 =m20 c2 l2 1=2 , or

260

3 s:\ITFC\CHAP-4-3.3d (Tables of Crystallography)

4:3:1:22

2=flE1 1

2 1=2 g:

4.3. ELECTRON DIFFRACTION Values of l; l 1 ; m=m0 ; =c, and are listed for various values of the accelerating voltage, E, in Table 4.3.2.1 with l in Ê and E in volts. A 4.3.1.5. Absorption effects Any scattering process, whether elastic or inelastic, which removes energy from the set of diffracted beams being considered, may be said to constitute an absorption process. For example, for a measurement of the intensities of the elastically scattered, sharp Bragg reflections from a crystal, any process which gives diffuse background scattering or results in a detectable loss of energy gives rise to absorption. The diffracted amplitudes in such cases may be calculated, at least as a first approximation, in terms of a complex potential, 'c r, containing an imaginary part 'i r due to an àbsorption function' and a small added real part 'r. Then under the crystallographic sign convention, 'c r 'r i'i r 'r. Correspondingly, for a centrosymmetric crystal, the structure amplitude becomes complex and may be written V h V0 h

iV 0 h V 00 h:

4:3:1:28

Under the appropriate conditions of observation, important contributions to the imaginary and real additions to the structure amplitudes may be given by the excitation of phonons, plasmons, or electron transitions, or by diffuse scattering due to crystal defects or disorder. The additional terms iV 0 h and V 00 h, however, are not invariant properties of the crystal structure but depend on the conditions of the diffraction experiment, such as the accelerating voltage and orientation of the incident beam, the aperture or resolution of the recording system, and the use of energy filtering or discrimination. In spite of this, it may often be convenient to treat them as being produced by phenomenological complex potentials, defined for a limited range of experimental conditions. 4.3.1.6. Tables of atomic scattering amplitudes for electrons Ê for all Tables 4.3.1.1 and 4.3.1.2 list values of f B s in A neutral atoms and most chemically significant ions, respectively. The values have been given by Doyle & Turner (1968) for several cases, denoted by RHF using the relativistic Hartree± Fock atomic potentials of Coulthard (1967). For all other atoms and ions, f B s has been found using the Mott±Bethe formula [equation (4.3.1.15)] for s 6 0, and the X-ray scattering factors of Table 2.2A of IT IV (1974). Thus all other neutral atoms except hydrogen are based on the relativistic Hartree±Fock wavefunctions of Mann (1968). These are designated by *RHF. For H and for ions below Rb, denoted by HF, f B s is ultimately based on the nonrelativistic Hartree±Fock wavefunctions of Mann (1968). For ions above Rb, denoted by *DS, modified relativistic Dirac±Slater wavefunctions calculated by Cromer & Waber (1974) are used. For low values of s, the Mott formula becomes less accurate, since Z fx s tends to zero with s for neutral atoms. Except for the RHF atoms, f B s for s from 0.01 to 0.03 are omitted in Table 4.3.1.1 and for s from 0.04 to 0.11, only two decimal places are given. f B s is then accurate to the figure quoted. For these atoms, f B 0 was found using the formula given by Ibers (1958): f B 0

4me2 Zhr 2 i; 3h2

where hr2 i is the mean-square atomic radius.

4:3:1:29

For ionized atoms, fel 0 1. The values listed at s 0 in Table 4.3.1.2 for RHF atoms were calculated by Doyle & Turner (1968) with 'r in equation (4.3.1.13) replaced by '0 r, where '0 r 'r

4:3:1:30

Here, Z is the ionic charge. This approach omits the Coulomb field due to the excess or deficiency of charge on the nucleus. With the use of these values, the structure factor for forward scattering by a neutral unit cell containing ions may be found in the conventional way. Similar values are not available for other ions because the atomic potential data are lacking. For computer applications, numerical approximations to the f s of these tables have been given by Doyle & Turner (1968) as Ê 1 . An alternative is sums of Gaussians for the range s 0 to 2 A to make Gaussian fits to X-ray scattering factors, then use the Mott formula to derive electron scattering factors. As discussed by Peng & Cowley (1988), this practice may lead to problems for small values of s. An additional problem occurs in highresolution electron-microscopy (HREM) image-simulation programs, where it is usually necessary to have electron scattering Ê 1 . Fox, O'Keefe & Tabbernor factors for the range 0 to 6 A (1989) point out that extrapolation of the Gaussian fits of Doyle Ê 1 can be highly inaccurate. & Turner (1968) to values past 2 A Ê 1 , Fox et al. have used sums of For the range of s from 2 to 6 A polynomials to make accurate fits to the X-ray scattering factors of Doyle & Turner (1968) for many elements (Section 6.1.1), and electron scattering factors can be generated from these data by use of the Mott formula. Recently, Rez, Rez & Grant (1994) have published new tables of X-ray scattering factors obtained using a multiconfiguration Dirac±Fock code and two parameterizations in terms of four Ê 1 Gaussians, one of higher accuracy over the range of about 2 A and the other of lower accuracy over the extended range of about Ê 1 . These authors suggest that electron scattering factors may 6A best be obtained from these X-ray scattering factors by using the Mott formula. They provide a table of values for the electron scattering factor values for zero scattering angle, fel 0, for many elements and ions, which may be of value for the calculation of mean inner potentials. 4.3.1.7. Use of Tables 4.3.1.1 and 4.3.1.2 In order to calculate the Fourier coefficients V h of the potential distribution 'r, for insertion in the formulae used to calculate intensities [such as (4.3.1.6), (4.3.1.20), (4.3.1.21)], or in the numerical methods for dynamical diffraction calculations, use V hin volts 47:87801h= ; where h

P i

fi expf2i h ri g:

4:3:1:31 4:3:1:32

The fi values are obtained from Tables 4.3.1.1 and 4.3.1.2, and Ê 3 . The V h and the fi tabulated are

is the unit-cell volume in A properties of the crystal structure and the isolated atoms, respectively, and are independent of the particular scattering theory assumed. Expressions for the calculation of intensities in the kinematical approximation are given for powder patterns and oblique texture patterns in Section 2.5.3, and for thin crystal plates in Section 2.5.1 of Volume B (IT B, 1992). Since the formulas for kinematical scattering, such as (4.3.1.19) and (4.3.1.20), include the parameter K =l, which varies with the energy of the electron beam through relativistic effects, it may be considered

261


eZ=r:

4. PRODUCTION AND PROPERTIES OF RADIATIONS that the electron scattering factors for kinematical calculations should be multiplied by relativistic factors. For high-energy electrons, the relativistic variations of the electron mass, the electron wavelength and the interaction constant, , become significant. The relations are m m0 1 2 1=2 ; eE l h 2em0 E 1 2m0 c2 lc

1

1=2

2 1=2 ;

4:3:1:33

where m0 is the rest mass, lc is the Compton wavelength, h=m0 c, and v=c. Consequently, varies with the incident electron energy as 2=flE1 1 2e=hc :

2 1=2 g 4:3:1:34

For the calculation of intensities in the kinematical approximation, the values of f B s listed in Tables 4.3.1.1 and 4.3.1.2, which were calculated using m0 , must be multiplied by m=m0 1 2 1=2 for electrons of velocity v. Values of l; 1=l; m=m0 , v=c; and are listed for various values of the accelerating voltage; E, in Table 4.3.2.1. 4.3.2. Parameterizations of electron atomic scattering factors (By J. M. Cowley, L. M. Peng, G. Ren, S. L. Dudarev, and M. J. Whelan) For computer applications, numerical approximations to the f s of Tables 4.3.1.1 or 4.3.1.2 are usually preferred and various approximations as sums of Gaussians have been proposed. The initial Gaussian fits were given by Doyle & Turner (1968) for the Ê 1 . However, for some purposes, as in the range s 0 to 2 A image-simulation programs for high-resolution electron microscopy, atomic scattering factors are needed for higher s values, up Ê 1 , and, as pointed out by Fox, O'Keefe & Tabbernor to 6 A (1989), extrapolation of the Gaussian fits of Doyle & Turner to Ê 1 can be highly inaccurate. values above 2 A An alternative approach to obtaining numerical values for the electron scattering factors is to make use of the polynomial fits to X-ray scattering factors of Fox et al. or the more recent tables of X-ray scattering factors produced by Rez, Rez & Grant (1994), who used a multiconfiguration Dirac±Fock code and two parameterizations in terms of four Gaussians, one of higher Ê 1 and the other of lower accuracy over the range of about 2 A Ê 1 . The electron accuracy over the extended range of about 6 A scattering factors may then be derived from the X-ray scattering factors by use of the Mott formula (4.3.1.14). For small angles of scattering, the determination of electron scattering factors in this way may give problems, since the X-ray scattering factor tends to the atomic number, and both the numerator and denominator of (4.3.1.14) tend to zero. However, the electron scattering factor may be determined for zero scattering angle using equation (4.3.1.29) and Rez, Rez & Grant (1994) listed values of fel 0 for many elements and ions. Recently, Peng, Ren, Dudarev & Whelan (1996) have developed a new algorithm, based on a combined modified simulated-annealing and least-squares method, to parameterize both the elastic and absorptive scattering factors as sums of five Gaussians of the form n P fel s ai exp bi s2 ; 4:3:2:1

where ai and bi are fitting parameters. The values of their fitting parameters for the range of s values from 0 to 2.0 for elastic electron scattering factors for all neutral atoms with atomic numbers up to 98 are given in Table 4.3.2.2 and the values obtained separately for these atoms for the range of s from 0 to Ê 1 are given in Table 4.3.2.3. For Table 4.3.2.2, the fitting 6.0 A was made to the values of f given in Table 4.3.1.1. For Table Ê 1 were 4.3.2.3, the f values in the range of s from 2.0 to 6.0 A those obtained by using the Mott formula to convert the X-ray scattering factors derived from the Dirac±Fock calculations of Rez, Rez & Grant (1994). Similar tables for atomic scattering factors of ions can be found in Peng (1998). As an indication of the accuracy with which the parameterized f values of (4.3.2.1) reproduce the numerical values of the reference f values, Peng et al. (1996) computed values of " 100 =f 0, where is the square root of the mean square deviation, 2 , between the numerical and fitted scattering factors. The values of " are typically in the range 0.02 to 0.05, and are consistently smaller (with a few exceptions) than the corresponding values given for the parameterizations of previous workers (Weickenmeier & Kohl, 1991; Bird & King, 1990; Doyle & Turner, 1968). For the absorptive scattering factors, corresponding to the imaginary parts added to the real elastic scattering factors as a consequence of inelastic scattering processes, Peng et al. (1996) have tabulated values for particular elemental crystals and a selection of crystals of compounds having the zinc-blend structure. The main contribution to the absorptive scattering factors arises from the thermal vibrations of the atoms in the crystals so that the numerical values are not characteristic of the individual atom types but depend on the type of bonding of the atoms in the crystal, as indicated by the Debye±Waller factor, and must be calculated separately for each temperature. The authors offer copies of their computer programs, freely available via electronic mail, from which the parameterization of the absorptive scattering factors can be derived for other materials and temperatures, given the values of the atomic numbers of the elements, the Debye±Waller factor and the electron accelerating voltage. 4.3.3. Complex scattering factors for the diffraction of electrons by gases (By A. W. Ross, M. Fink, R. Hilderbrandt, J. Wang, and V. H. Smith Jr) 4.3.3.1. Introduction This section includes tables of scattering factors of interest for gas-phase electron diffraction from atoms and molecules in the keV energy region. In addition to the tables and a description of their uses, a discussion of the theoretical uncertainties related to the material in the tables is also provided. The tables give scattering factors for elastic and inelastic scattering from free atoms. The theory of molecular scattering based on these atomic quantities is also discussed. 4.3.3.2. Complex atomic scattering factors for electrons 4.3.3.2.1. Elastic scattering factors for atoms It has long been known that the first Born approximation provides an inadequate description at the 4% accuracy level for elastic and total differential cross sections in the 40 keV energy range for atoms heavier than Ne (Schomaker & Glauber, 1952; Glauber & Schomaker, 1953). Results of early experimental work have been confirmed for both atomic and molecular

i1

262


(continued on page 388)

4.3. ELECTRON DIFFRACTION ˚ for electrons for neutral atoms Table 4.3.1.1. Atomic scattering amplitudes (A) Self-consistent field calculations: HF: non-relativistic Hartree±Fock; RHF, *RHF: relativistic Hartree±Fock. H 1 HF

He 2 RHF

Li 3 RHF

Be 4 RHF

B 5 RHF

C 6 RHF

N 7 RHF

O 8 RHF

F 9 RHF

Ne 10 RHF

Na 11 RHF

0.00 0.01 0.02 0.03 0.04 0.05

0.529

0.51 0.51

0.418 0.418 0.417 0.415 0.413 0.410

3.286 3.265 3.200 3.097 2.961 2.800

3.052 3.042 3.011 2.961 2.892 2.807

2.794 2.788 2.768 2.736 2.693 2.638

2.509 2.505 2.492 2.471 2.442 2.406

2.211 2.209 2.201 2.187 2.168 2.144

1.983 1.982 1.976 1.966 1.953 1.937

1.801 1.800 1.796 1.789 1.779 1.767

1.652 1.651 1.648 1.642 1.635 1.626

4.778 4.749 4.663 4.527 4.348 4.138

0.06 0.07 0.08 0.09 0.10

0.50 0.49 0.48 0.47 0.45

0.407 0.404 0.399 0.395 0.390

2.622 2.435 2.245 2.058 1.879

2.710 2.601 2.484 2.362 2.237

2.574 2.502 2.423 2.339 2.250

2.363 2.313 2.259 2.200 2.138

2.116 2.083 2.047 2.007 1.963

1.917 1.893 1.867 1.839 1.808

1.752 1.735 1.716 1.694 1.671

1.615 1.602 1.587 1.570 1.552

3.908 3.667 3.425 3.190 2.967

0.11 0.12 0.13 0.14 0.15

0.44 0.425 0.411 0.396 0.382

0.384 0.378 0.372 0.366 0.359

1.710 1.554 1.411 1.282 1.166

2.111 1.987 1.865 1.748 1.635

2.159 2.067 1.974 1.882 1.791

2.072 2.005 1.936 1.866 1.796

1.918 1.870 1.821 1.770 1.718

1.774 1.739 1.702 1.664 1.625

1.646 1.619 1.591 1.562 1.532

1.533 1.512 1.490 1.467 1.443

2.759 2.569 2.395 2.239 2.099

0.16 0.17 0.18 0.19 0.20

0.366 0.353 0.338 0.324 0.311

0.352 0.345 0.338 0.330 0.323

1.063 0.971 0.889 0.817 0.753

1.528 1.427 1.332 1.243 1.161

1.702 1.616 1.533 1.453 1.377

1.727 1.658 1.591 1.524 1.460

1.666 1.614 1.561 1.510 1.458

1.585 1.545 1.504 1.463 1.422

1.501 1.469 1.436 1.404 1.371

1.418 1.393 1.367 1.340 1.313

1.974 1.863 1.763 1.674 1.594

0.22 0.24 0.25 0.26 0.28 0.30

0.285 0.261 0.249 0.238 0.218 0.199

0.308 0.293 0.286 0.278 0.264 0.250

0.646 0.562 0.526 0.494 0.440 0.396

1.013 0.887 0.832 0.781 0.690 0.614

1.235 1.107 1.048 0.993 0.892 0.803

1.337 1.222 1.168 1.117 1.020 0.932

1.358 1.262 1.216 1.171 1.085 1.006

1.341 1.261 1.222 1.184 1.110 1.040

1.304 1.238 1.206 1.173 1.110 1.049

1.259 1.204 1.176 1.149 1.095 1.043

1.458 1.344 1.295 1.249 1.167 1.095

0.32 0.34 0.35 0.36 0.38 0.40

0.182 0.167 0.160 0.153 0.141 0.130

0.236 0.224 0.217 0.211 0.200 0.189

0.359 0.328 0.314 0.301 0.279 0.259

0.549 0.494 0.469 0.446 0.406 0.371

0.725 0.657 0.625 0.596 0.543 0.497

0.853 0.781 0.748 0.717 0.658 0.606

0.932 0.863 0.831 0.800 0.742 0.689

0.974 0.911 0.881 0.853 0.798 0.747

0.991 0.935 0.908 0.882 0.831 0.784

0.991 0.942 0.918 0.894 0.849 0.805

1.031 0.973 0.946 0.921 0.872 0.827

0.42 0.44 0.45 0.46 0.48 0.50

0.120 0.111 0.107 0.103 0.096 0.089

0.178 0.169 0.164 0.159 0.151 0.143

0.241 0.226 0.219 0.212 0.200 0.188

0.341 0.314 0.302 0.291 0.271 0.253

0.455 0.419 0.402 0.387 0.358 0.333

0.559 0.517 0.497 0.479 0.444 0.413

0.641 0.596 0.575 0.555 0.518 0.484

0.700 0.656 0.635 0.615 0.577 0.542

0.739 0.697 0.677 0.658 0.621 0.586

0.764 0.725 0.706 0.687 0.652 0.619

0.785 0.746 0.727 0.709 0.675 0.642

0.55 0.60 0.65 0.70 0.80 0.90 1.00

0.075 0.064 0.055 0.048 0.037 0.029 0.024

0.125 0.110 0.097 0.086 0.068 0.055 0.046

0.164 0.145 0.128 0.115 0.093 0.077 0.064

0.215 0.186 0.164 0.145 0.117 0.096 0.081

0.280 0.239 0.207 0.182 0.144 0.118 0.098

0.348 0.297 0.256 0.223 0.175 0.141 0.117

0.411 0.353 0.305 0.266 0.208 0.167 0.137

0.466 0.403 0.350 0.307 0.241 0.193 0.159

0.510 0.445 0.390 0.344 0.272 0.219 0.180

0.544 0.479 0.424 0.376 0.300 0.244 0.201

0.569 0.505 0.450 0.403 0.325 0.266 0.221

1.10 1.20 1.30 1.40 1.50

0.020 0.017 0.014 0.012 0.011

0.038 0.032 0.028 0.024 0.021

0.054 0.046 0.040 0.035 0.031

0.069 0.059 0.051 0.045 0.040

0.083 0.072 0.062 0.055 0.048

0.099 0.085 0.073 0.064 0.057

0.115 0.098 0.085 0.074 0.065

0.133 0.113 0.097 0.085 0.074

0.150 0.128 0.110 0.095 0.084

0.168 0.143 0.123 0.106 0.093

0.185 0.158 0.135 0.117 0.103

0.019 0.016 0.015 0.013 0.012

0.028 0.024 0.022 0.019 0.017

0.035 0.031 0.028 0.026 0.023

0.043 0.038 0.035 0.031 0.028

0.051 0.045 0.041 0.037 0.034

0.058 0.052 0.047 0.043 0.039

0.066 0.059 0.053 0.048 0.044

0.074 0.066 0.060 0.054 0.049

0.083 0.074 0.066 0.060 0.054

0.092 0.081 0.073 0.065 0.059

sin =l Ê 1) (A

1.60 1.70 1.80 1.90 2.00

Element Z Method

263


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Mg 12 RHF

Al 13 RHF

Si 14 RHF

P 15 RHF

S 16 RHF

Cl 17 RHF

Ar 18 RHF

K 19 RHF

Ca 20 RHF

Sc 21 RHF

Ti 22 RHF

0.00 0.01 0.02 0.03 0.04 0.05

5.207 5.187 5.124 5.022 4.884 4.717

5.889 5.867 5.800 5.692 5.547 5.371

5.828 5.810 5.759 5.675 5.561 5.421

5.488 5.476 5.439 5.378 5.296 5.192

5.161 5.152 5.124 5.079 5.016 4.938

4.857 4.851 4.830 4.795 4.746 4.685

4.580 4.576 4.559 4.531 4.493 4.444

8.984 8.921 8.731 8.434 8.054 7.619

9.913 9.860 9.699 9.442 9.104 8.703

9.307 9.264 9.134 8.926 8.649 8.318

8.776 8.740 8.631 8.455 8.220 7.937

0.06 0.07 0.08 0.09 0.10

4.527 4.320 4.102 3.879 3.656

5.170 4.949 4.717 4.478 4.237

5.258 5.077 4.882 4.677 4.467

5.071 4.935 4.785 4.625 4.457

4.845 4.740 4.623 4.496 4.362

4.613 4.529 4.436 4.335 4.227

4.386 4.320 4.245 4.163 4.074

7.157 6.691 6.239 5.815 5.426

8.258 7.789 7.312 6.841 6.388

7.946 7.548 7.139 6.729 6.328

7.618 7.274 6.917 6.556 6.199

0.11 0.12 0.13 0.14 0.15

3.437 3.226 3.025 2.835 2.657

3.999 3.767 3.544 3.330 3.128

4.255 4.043 3.835 3.632 3.437

4.285 4.109 3.933 3.758 3.586

4.222 4.078 3.931 3.783 3.635

4.113 3.994 3.871 3.746 3.620

3.980 3.881 3.779 3.674 3.566

5.073 4.756 4.474 4.222 3.997

5.959 5.560 5.192 4.855 4.550

5.944 5.580 5.239 4.924 4.633

5.853 5.522 5.209 4.916 4.643

0.16 0.17 0.18 0.19 0.20

2.492 2.340 2.199 2.071 1.953

2.938 2.760 2.595 2.441 2.299

3.249 3.070 2.900 2.740 2.589

3.417 3.253 3.094 2.942 2.796

3.487 3.342 3.200 3.061 2.927

3.493 3.367 3.242 3.118 2.997

3.458 3.348 3.239 3.130 3.022

3.795 3.612 3.446 3.295 3.154

4.273 4.023 3.797 3.593 3.408

4.366 4.122 3.899 3.695 3.509

4.390 4.157 3.943 3.745 3.564

0.22 0.24 0.25 0.26 0.28 0.30

1.748 1.577 1.502 1.434 1.313 1.211

2.046 1.832 1.737 1.650 1.495 1.363

2.315 2.076 1.969 1.869 1.689 1.534

2.525 2.281 2.169 2.064 1.872 1.702

2.671 2.436 2.326 2.221 2.026 1.851

2.763 2.543 2.438 2.337 2.148 1.974

2.811 2.609 2.512 2.417 2.238 2.070

2.902 2.680 2.578 2.481 2.299 2.134

3.086 2.815 2.695 2.584 2.383 2.206

3.183 2.906 2.783 2.669 2.462 2.281

3.242 2.967 2.844 2.730 2.523 2.341

0.32 0.34 0.35 0.36 0.38 0.40

1.123 1.047 1.013 0.980 0.921 0.868

1.251 1.154 1.111 1.070 0.997 0.932

1.400 1.284 1.231 1.182 1.094 1.017

1.553 1.422 1.362 1.306 1.205 1.115

1.694 1.554 1.490 1.429 1.318 1.218

1.816 1.672 1.606 1.542 1.425 1.319

1.915 1.772 1.705 1.641 1.522 1.412

1.982 1.842 1.776 1.714 1.595 1.487

2.048 1.905 1.838 1.775 1.657 1.548

2.119 1.974 1.906 1.842 1.722 1.612

2.178 2.032 1.964 1.899 1.778 1.668

0.42 0.44 0.45 0.46 0.48 0.50

0.821 0.777 0.757 0.738 0.701 0.667

0.875 0.825 0.801 0.779 0.737 0.700

0.949 0.888 0.861 0.834 0.786 0.743

1.036 0.965 0.933 0.903 0.847 0.797

1.130 1.051 1.014 0.980 0.917 0.860

1.224 1.138 1.098 1.061 0.991 0.928

1.313 1.223 1.181 1.141 1.066 0.998

1.387 1.295 1.252 1.211 1.134 1.064

1.449 1.357 1.314 1.272 1.194 1.123

1.511 1.418 1.374 1.332 1.252 1.179

1.566 1.472 1.428 1.385 1.305 1.230

0.55 0.60 0.65 0.70 0.80 0.90 1.00

0.592 0.528 0.473 0.425 0.347 0.286 0.239

0.618 0.551 0.494 0.445 0.366 0.304 0.255

0.651 0.578 0.517 0.465 0.383 0.320 0.270

0.692 0.610 0.543 0.487 0.401 0.335 0.284

0.741 0.648 0.573 0.513 0.419 0.350 0.298

0.796 0.692 0.609 0.541 0.440 0.366 0.311

0.854 0.740 0.648 0.574 0.462 0.383 0.324

0.912 0.790 0.690 0.609 0.488 0.402 0.339

0.966 0.838 0.733 0.647 0.515 0.422 0.354

1.018 0.885 0.775 0.684 0.544 0.444 0.371

1.067 0.930 0.816 0.721 0.573 0.467 0.389

1.10 1.20 1.30 1.40 1.50

0.202 0.172 0.148 0.129 0.113

0.217 0.185 0.160 0.139 0.123

0.231 0.198 0.172 0.150 0.132

0.243 0.210 0.183 0.160 0.141

0.255 0.221 0.193 0.169 0.150

0.267 0.232 0.202 0.178 0.158

0.278 0.242 0.212 0.187 0.166

0.290 0.252 0.220 0.194 0.174

0.303 0.262 0.230 0.202 0.181

0.316 0.273 0.239 0.211 0.188

0.330 0.285 0.249 0.219 0.195

1.60 1.70 1.80 1.90 2.00

0.100 0.089 0.080 0.072 0.065

0.109 0.096 0.087 0.078 0.070

0.117 0.104 0.093 0.084 0.076

0.125 0.111 0.100 0.090 0.082

0.133 0.119 0.107 0.096 0.087

0.141 0.126 0.113 0.102 0.093

0.148 0.132 0.119 0.108 0.098

0.156 0.138 0.127 0.112 0.101

0.162 0.144 0.132 0.118 0.107

0.169 0.151 0.137 0.124 0.112

0.175 0.157 0.143 0.129 0.117

sin =l Ê 1) (A

Element Z Method

264


4.3. ELECTRON DIFFRACTION Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) V 23 RHF

Cr 24 RHF

Mn 25 RHF

Fe 26 RHF

Co 27 RHF

Ni 28 RHF

Cu 29 RHF

Zn 30 RHF

Ga 31 RHF

Ge 32 RHF

As 33 RHF

0.00 0.01 0.02 0.03 0.04 0.05

8.305 8.274 8.180 8.029 7.826 7.581

6.969 6.945 6.875 6.762 6.610 6.427

7.506 7.484 7.412 7.296 7.140 6.949

7.165 7.145 7.081 6.978 6.839 6.669

6.854 6.836 6.779 6.687 6.562 6.410

6.569 6.552 6.501 6.418 6.306 6.169

5.600 5.587 5.547 5.482 5.395 5.287

6.065 6.051 6.009 5.941 5.849 5.735

7.108 7.088 7.027 6.927 6.792 6.629

7.378 7.359 7.303 7.211 7.088 6.935

7.320 7.306 7.260 7.184 7.081 6.953

0.06 0.07 0.08 0.09 0.10

7.303 7.002 6.686 6.365 6.045

6.221 5.997 5.764 5.527 5.291

6.732 6.493 6.241 5.981 5.719

6.474 6.260 6.032 5.796 5.558

6.234 6.040 5.834 5.619 5.401

6.010 5.834 5.646 5.449 5.249

5.165 5.029 4.886 4.737 4.585

5.603 5.457 5.299 5.133 4.962

6.441 6.236 6.017 5.792 5.564

6.759 6.562 6.351 6.129 5.902

6.803 6.634 6.449 6.253 6.048

0.11 0.12 0.13 0.14 0.15

5.732 5.430 5.142 4.871 4.616

5.061 4.838 4.625 4.423 4.231

5.459 5.206 4.962 4.728 4.506

5.320 5.087 4.861 4.644 4.436

5.182 4.967 4.758 4.555 4.361

5.048 4.848 4.654 4.465 4.283

4.434 4.285 4.139 3.998 3.862

4.790 4.618 4.449 4.283 4.123

5.337 5.113 4.896 4.686 4.486

5.672 5.442 5.217 4.996 4.783

5.838 5.625 5.411 5.200 4.992

0.16 0.17 0.18 0.19 0.20

4.378 4.158 3.953 3.763 3.588

4.051 3.882 3.723 3.574 3.434

4.297 4.100 3.916 3.743 3.583

4.240 4.054 3.880 3.716 3.562

4.177 4.002 3.836 3.681 3.534

4.110 3.944 3.788 3.640 3.500

3.731 3.607 3.488 3.375 3.267

3.969 3.822 3.681 3.547 3.421

4.295 4.114 3.942 3.781 3.629

4.578 4.382 4.195 4.017 3.849

4.789 4.593 4.404 4.222 4.048

0.22 0.24 0.25 0.26 0.28 0.30

3.276 3.006 2.885 2.772 2.568 2.386

3.179 2.953 2.849 2.750 2.568 2.403

3.292 3.039 2.924 2.817 2.620 2.445

3.284 3.039 2.928 2.824 2.632 2.461

3.267 3.032 2.924 2.823 2.637 1.471

3.245 3.018 2.914 2.816 2.636 2.474

3.067 2.885 2.800 2.719 2.568 2.428

3.186 2.977 2.880 2.789 2.620 2.468

3.352 3.108 2.997 2.892 2.701 2.531

3.541 3.268 3.143 3.026 2.813 2.623

3.724 3.433 3.299 3.172 2.940 2.733

0.32 0.34 0.35 0.36 0.38 0.40

2.225 2.079 2.011 1.947 1.826 1.716

2.252 2.114 2.049 1.987 1.870 1.761

2.288 2.146 2.080 2.017 1.899 1.790

2.308 2.168 2.104 2.042 1.925 1.818

2.321 2.184 2.121 2.060 1.946 1.841

2.328 2.195 2.133 2.073 1.962 1.858

2.299 2.180 2.123 2.069 1.965 1.868

2.329 2.203 2.144 2.087 1.980 1.882

2.379 2.242 2.179 2.119 2.006 1.903

2.455 2.304 2.235 2.169 2.048 1.938

2.548 2.384 2.308 2.237 2.105 1.986

0.42 0.44 0.45 0.46 0.48 0.50

1.614 1.520 1.476 1.433 1.352 1.277

1.660 1.567 1.523 1.480 1.399 1.323

1.690 1.597 1.553 1.511 1.431 1.356

1.719 1.628 1.584 1.542 1.462 1.388

1.743 1.653 1.610 1.569 1.490 1.416

1.763 1.674 1.631 1.591 1.513 1.440

1.777 1.691 1.651 1.611 1.535 1.464

1.790 1.704 1.663 1.624 1.549 1.478

1.808 1.720 1.679 1.639 1.563 1.492

1.837 1.745 1.702 1.661 1.583 1.510

1.878 1.780 1.734 1.691 1.608 1.533

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.111 0.973 0.856 0.757 0.602 0.490 0.408

1.155 1.014 0.894 0.792 0.631 0.514 0.427

1.189 1.047 0.927 0.824 0.659 0.538 0.446

1.222 1.080 0.959 0.854 0.686 0.561 0.466

1.251 1.110 0.988 0.883 0.712 0.583 0.485

1.277 1.136 1.015 0.909 0.737 0.605 0.504

1.303 1.163 1.041 0.935 0.761 0.626 0.523

1.319 1.181 1.061 0.955 0.781 0.646 0.541

1.334 1.197 1.078 0.973 0.800 0.665 0.558

1.349 1.212 1.093 0.989 0.817 0.681 0.574

1.367 1.228 1.108 1.004 0.832 0.697 0.589

1.10 1.20 1.30 1.40 1.50

0.345 0.297 0.259 0.228 0.203

0.361 0.310 0.269 0.237 0.210

0.377 0.323 0.280 0.246 0.218

0.393 0.336 0.291 0.255 0.226

0.409 0.350 0.303 0.265 0.235

0.425 0.364 0.315 0.275 0.243

0.442 0.378 0.327 0.285 0.252

0.457 0.391 0.339 0.296 0.261

0.473 0.405 0.350 0.306 0.270

0.488 0.418 0.362 0.317 0.279

0.502 0.431 0.374 0.327 0.288

1.60 1.70 1.80 1.90 2.00

0.182 0.163 0.148 0.134 0.122

0.188 0.169 0.154 0.139 0.127

`0.195 0.175 0.159 0.144 0.132

0.202 0.181 0.165 0.149 0.136

0.209 0.188 0.170 0.154 0.141

0.217 0.194 0.176 0.160 0.146

0.224 0.201 0.182 0.165 0.150

0.232 0.208 0.188 0.170 0.155

0.240 0.215 0.194 0.175 0.160

0.248 0.222 0.200 0.181 0.165

0.256 0.229 0.206 0.187 0.170

sin =l Ê 1) (A

Element Z Method

265


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Se 34 RHF

Br 35 RHF

Kr 36 RHF

Rb 37 RHF

Sr 38 RHF

Y 39 *RHF

Zr 40 *RHF

Nb 41 *RHF

Mo 42 *RHF

Tc 43 *RHF

Ru 44 *RHF

0.00 0.01 0.02 0.03 0.04 0.05

7.205 7.192 7.154 7.090 7.004 6.895

7.060 7.049 7.016 6.962 6.888 6.795

6.897 6.889 6.861 6.814 6.750 6.670

11.778 11.699 11.460 11.088 10.613 10.073

13.109 13.035 12.816 12.468 12.013 11.476

12.674

12.166

10.679

10.856

9.558

11.79 11.34

11.41 11.04

10.13 9.86

10.260 10.230 10.138 9.989 9.790 9.548

10.35 10.10

9.18 8.99

0.06 0.07 0.08 0.09 0.10

6.767 6.621 6.460 6.288 6.105

6.684 6.558 6.418 6.266 6.104

6.574 6.464 6.341 6.207 6.064

9.504 8.934 8.385 7.872 7.402

10.888 10.273 9.655 9.052 8.478

10.84 10.31 9.77 9.23 8.70

10.62 10.15 9.68 9.20 8.72

9.54 9.20 8.85 8.49 8.12

9.272 8.972 8.655 8.330 8.004

9.80 9.48 9.14 8.78 8.42

8.77 8.53 8.27 8.00 7.73

0.11 0.12 0.13 0.14 0.15

5.916 5.722 5.525 5.328 5.132

5.935 5.760 5.580 5.399 5.217

5.913 5.755 5.593 5.428 5.260

6.976 6.593 6.248 5.938 5.658

7.940 7.443 6.988 6.575 6.200

8.20 7.722 7.278 6.865 6.485

8.26 7.818 7.400 7.007 6.640

7.77 7.421 7.090 6.772 6.472

7.680 7.364 7.058 6.763 6.481

8.07 7.720 7.383 7.057 6.746

7.46 7.190 6.928 6.672 6.426

0.16 0.17 0.18 0.19 0.20

4.938 4.749 4.564 4.384 4.211

5.036 4.857 4.680 4.507 4.339

5.092 4.925 4.759 4.595 4.434

5.403 5.170 4.954 4.754 4.566

5.862 5.555 5.278 5.025 4.794

6.136 5.816 5.523 5.254 5.008

6.299 5.983 5.689 5.419 5.168

6.187 5.918 5.665 5.427 5.203

6.213 5.957 5.715 5.486 5.269

6.451 6.171 5.907 5.658 5.423

6.188 5.960 5.741 5.533 5.332

0.22 0.24 0.25 0.26 0.28 0.30

3.884 3.585 3.446 3.314 3.069 2.849

4.017 3.718 3.578 3.443 3.192 2.963

4.123 3.829 3.690 3.556 3.303 3.071

4.224 3.916 3.773 3.636 3.382 3.149

4.387 4.039 3.882 3.735 3.465 3.224

4.570 4.195 4.027 3.869 3.583 3.329

4.721 4.333 4.158 3.995 3.697 3.433

4.792 4.426 4.258 4.099 3.804 3.539

4.868 4.507 4.341 4.182 3.888 3.622

4.994 4.614 4.439 4.273 3.969 3.695

4.959 4.618 4.459 4.306 4.021 3.759

0.32 0.34 0.35 0.36 0.38 0.40

2.651 2.475 2.393 2.316 2.173 2.045

2.757 2.570 2.484 2.402 2.250 2.113

2.858 2.665 2.575 2.490 2.330 2.186

2.936 2.742 2.651 2.564 2.402 2.254

3.007 2.810 2.718 2.630 2.466 2.315

3.101 2.895 2.799 2.708 2.538 2.383

3.196 2.982 2.883 2.789 2.613 2.452

3.298 3.080 2.978 2.880 2.698 2.531

3.379 3.158 3.054 2.955 2.770 2.600

3.448 3.223 3.118 3.018 2.830 2.658

3.518 3.296 3.192 3.092 2.904 2.730

0.42 0.44 0.45 0.46 0.48 0.50

1.929 1.824 1.776 1.729 1.642 1.562

1.989 1.877 1.825 1.775 1.683 1.598

2.055 1.936 1.881 1.828 1.730 1.640

2.119 1.995 1.938 1.883 1.780 1.686

2.178 2.052 1.993 1.936 1.830 1.733

2.241 2.111 2.049 1.991 1.881 1.780

2.305 2.171 2.108 2.047 1.934 1.829

2.379 2.239 2.173 2.110 1.991 1.883

2.444 2.300 2.233 2.168 2.046 1.934

2.500 2.355 2.287 2.221 2.098 1.984

2.570 2.421 2.351 2.284 2.157 2.040

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.389 1.245 1.124 1.019 0.847 0.711 0.603

1.416 1.266 1.141 1.034 0.860 0.725 0.616

1.447 1.290 1.160 1.050 0.873 0.737 0.628

1.483 1.319 1.182 1.068 0.887 0.749 0.640

1.522 1.350 1.208 1.089 0.902 0.762 0.651

1.562 1.383 1.235 1.111 0.918 0.774 0.662

1.603 1.417 1.263 1.135 0.935 0.787 0.673

1.646 1.452 1.292 1.159 0.952 0.800 0.684

1.690 1.490 1.324 1.185 0.971 0.814 0.695

1.734 1.528 1.357 1.214 0.992 0.830 0.707

1.782 1.569 1.391 1.243 1.013 0.845 0.719

1.10 1.20 1.30 1.40 1.50

0.515 0.444 0.385 0.337 0.297

0.528 0.456 0.396 0.347 0.306

0.540 0.467 0.407 0.357 0.315

0.551 0.478 0.417 0.365 0.325

0.562 0.488 0.427 0.375 0.333

0.572 0.498 0.436 0.384 0.341

0.582 0.507 0.445 0.393 0.349

0.591 0.516 0.454 0.401 0.356

0.601 0.525 0.462 0.408 0.364

0.611 0.534 0.470 0.416 0.371

0.621 0.542 0.478 0.423 0.378

1.60 1.70 1.80 1.90 2.00

0.264 0.236 0.212 0.192 0.175

0.272 0.243 0.219 0.198 0.180

0.280 0.250 0.225 0.204 0.185

0.290 0.257 0.233 0.208 0.188

0.297 0.264 0.239 0.214 0.194

0.303 0.272 0.244 0.221 0.201

0.311 0.278 0.251 0.227 0.206

0.318 0.285 0.257 0.233 0.211

0.325 0.291 0.263 0.238 0.216

0.332 0.298 0.269 0.244 0.222

0.338 0.304 0.275 0.249 0.227

sin =l Ê 1) (A

Element Z Method

266


4.3. ELECTRON DIFFRACTION Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Rh 45 *RHF

Pd 46 *RHF

Ag 47 RHF

Cd 48 RHF

In 49 RHF

Sn 50 RHF

Sb 51 RHF

Te 52 *RHF

I 53 RHF

Xe 54 RHF

Cs 55 RHF

0.00 0.01 0.02 0.03 0.04 0.05

9.242

7.583

7.43 7.35

9.232 9.213 9.153 9.057 8.926 8.764

10.434 10.406 10.320 10.181 9.995 9.768

10.859 10.833 10.750 10.615 10.433 10.209

10.974 10.950 10.876 10.755 10.591 10.387

11.003

8.90 8.73

8.671 8.654 8.599 8.510 8.391 8.244

10.65 10.47

10.905 10.887 10.828 10.731 10.599 10.434

10.794 10.777 10.725 10.638 10.520 10.371

16.508 16.391 16.050 15.521 14.855 14.106

0.06 0.07 0.08 0.09 0.10

8.53 8.31 8.01 7.83 7.58

7.26 7.16 7.03 6.91 6.77

8.075 7.888 7.689 7.480 7.267

8.577 8.369 8.144 7.909 7.666

9.509 9.224 8.923 8.612 8.297

9.950 9.664 9.357 9.037 8.709

10.150 9.884 9.596 9.291 8.976

10.25 10.01 9.74 9.46 9.16

10.238 10.017 9.773 9.511 9.235

10.194 9.993 9.771 9.530 9.274

13.326 12.556 11.823 11.145 10.525

0.11 0.12 0.13 0.14 0.15

7.33 7.079 6.836 6.598 6.366

6.62 6.474 6.319 6.162 6.003

7.052 6.837 6.625 6.418 6.215

7.421 7.176 6.933 6.695 6.464

7.983 7.674 7.374 7.084 6.805

8.380 8.053 7.732 7.419 7.118

8.654 8.331 8.010 7.694 7.386

8.85 8.538 8.224 7.914 7.608

8.948 8.654 8.357 8.059 7.764

9.007 8.732 8.451 8.167 7.884

9.965 9.458 9.000 8.583 8.201

0.16 0.17 0.18 0.19 0.20

6.143 5.929 5.722 5.524 5.334

5.843 5.684 5.526 5.369 5.214

6.018 5.827 5.643 5.464 5.293

6.240 6.024 5.817 5.618 5.427

6.539 6.286 6.045 5.817 5.601

6.829 6.552 6.289 6.039 5.803

7.088 6.800 6.524 6.261 6.010

7.309 7.018 6.738 6.467 6.209

7.472 7.186 6.908 6.639 6.379

7.603 7.325 7.053 6.787 6.529

7.848 7.519 7.212 6.922 6.649

0.22 0.24 0.25 0.26 0.28 0.30

4.976 4.648 4.493 4.345 4.066 3.809

4.913 4.626 4.487 4.352 4.093 3.850

4.967 4.665 4.522 4.384 4.122 3.878

5.070 4.745 4.592 4.447 4.173 3.922

5.203 4.846 4.682 4.525 4.236 3.973

5.368 4.979 4.801 4.633 4.323 4.044

5.547 5.131 4.940 4.760 4.428 4.131

5.727 3.291 5.090 4.899 4.548 4.234

5.889 5.442 5.234 5.036 4.670 4.341

6.039 5.586 5.374 5.172 4.795 4.454

6.143 5.684 5.471 5.268 4.890 4.547

0.32 0.34 0.35 0.36 0.38 0.40

3.572 3.353 3.249 3.150 2.962 2.788

3.622 3.408 3.306 3.208 3.022 2.848

3.651 3.440 3.339 3.242 3.058 2.886

3.690 3.476 3.375 3.278 3.093 2.922

3.734 3.515 3.412 3.313 3.127 2.955

3.792 3.564 3.458 3.356 3.165 2.990

3.865 3.625 3.514 3.408 3.210 3.030

3.952 3.700 3.583 3.472 3.265 3.078

4.046 3.780 3.658 3.541 3.325 3.130

4.147 3.870 3.742 3.620 3.394 3.191

4.235 3.953 3.822 3.697 3.465 3.255

0.42 0.44 0.45 0.46 0.48 0.50

2.626 2.477 2.406 2.338 2.210 2.090

2.686 2.535 2.464 2.395 2.264 2.143

2.726 2.576 2.505 2.436 2.306 2.185

2.762 2.613 2.542 2.474 2.344 2.223

2.795 2.646 2.576 2.507 2.378 2.257

2.828 2.678 2.608 2.539 2.409 2.288

2.864 2.712 2.640 2.571 2.440 2.318

2.907 2.750 2.677 2.606 2.473 2.350

2.953 2.791 2.715 2.642 2.506 2.380

3.006 2.838 2.759 2.684 2.543 2.414

3.064 2.890 2.809 2.731 2.586 2.453

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.828 1.609 1.426 1.273 1.035 0.861 0.731

1.875 1.650 1.462 1.304 1.058 0.879 0.745

1.915 1.688 1.497 1.335 1.082 0.897 0.758

1.953 1.724 1.531 1.366 1.107 0.916 0.773

1.987 1.758 1.563 1.397 1.132 0.936 0.789

2.019 1.790 1.594 1.426 1.157 0.956 0.805

2.048 1.819 1.622 1.453 1.181 0.976 0.821

2.077 1.847 1.649 1.479 1.205 0.997 0.838

2.104 1.871 1.673 1.503 1.227 1.016 0.855

2.132 1.897 1.697 1.526 1.248 1.036 0.871

2.163 1.923 1.721 1.548 1.269 1.055 0.888

1.10 1.20 1.30 1.40 1.50

0.631 0.551 0.485 0.431 0.384

0.641 0.559 0.493 0.437 0.391

0.652 0.568 0.500 0.444 0.397

0.664 0.578 0.508 0.451 0.403

0.676 0.587 0.516 0.458 0.409

0.688 0.597 0.525 0.465 0.416

0.701 0.608 0.533 0.472 0.422

0.715 0.619 0.542 0.480 0.428

0.729 0.630 0.551 0.487 0.435

0.743 0.642 0.561 0.495 0.442

0.758 0.654 0.570 0.502 0.450

1.60 1.70 1.80 1.90 2.00

0.345 0.310 0.281 0.255 0.232

0.351 0.316 0.286 0.260 0.237

0.357 0.321 0.291 0.265 0.241

0.362 0.327 0.297 0.270 0.246

0.368 0.332 0.302 0.274 0.250

0.374 0.337 0.307 0.279 0.255

0.379 0.343 0.311 0.284 0.259

0.385 0.348 0.316 0.288 0.264

0.391 0.353 0.321 0.293 0.268

0.397 0.358 0.325 0.297 0.272

0.405 0.363 0.332 0.299 0.272

sin =l Ê 1) (A

Element Z Method

267


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Ba 56 RHF

La 57 *RHF

Ce 58 *RHF

Pr 59 *RMF

Nd 60 *RHF

Pm 61 *RHF

Sm 62 *RHF

Eu 63 RHF

Gd 64 *RHF

Tb 65 *RHF

Dy 66 *RHF

0.00 0.01 0.02 0.03 0.04 0.05

18.267 18.157 17.828 17.309 16.636 15.854

17.805

17.378

16.987

16.606

16.243

15.897

15.266

14.974

14.641

16.45 15.79

16.10 15.46

15.62 14.94

15.30 14.67

14.99 14.39

14.70 14.12

15.563 15.486 15.260 14.898 14.425 13.867

14.30 13.81

13.90 13.37

13.64 13.14

0.06 0.07 0.08 0.09 0.10

15.008 14.138 13.278 12.431 11.675

15.05 14.28 13.51 12.74 12.01

14.77 14.03 13.29 12.56 11.85

14.22 13.47 12.72 11.99 11.29

13.97 13.25 12.52 11.82 11.15

13.72 13.03 12.33 11.65 11.00

13.48 12.81 12.14 11.49 10.86

13.253 12.611 11.963 11.329 10.722

13.27 12.70 12.11 11.52 10.95

12.81 12.22 11.62 11.02 10.45

12.60 12.03 11.44 10.87 10.32

0.11 0.12 0.13 0.14 0.15

10.958 10.302 9.707 9.168 8.682

11.32 10.671 10.072 9.522 9.017

11.19 10.561 9.981 9.448 8.958

10.65 10.052 9.506 9.008 8.556

10.52 9.944 9.412 8.928 8.486

10.40 9.833 9.316 8.843 8.413

10.27 9.722 9.218 8.758 8.336

10.150 9.618 9.128 8.678 8.267

10.39 9.871 9.382 8.926 8.505

9.91 9.407 8.942 8.512 8.121

9.79 9.303 8.848 8.429 8.045

0.16 0.17 0.18 0.19 0.20

8.241 7.840 7.474 7.139 6.829

8.555 8.131 7.742 7.384 7.053

8.507 8.094 7.714 7.365 7.041

8.144 7.768 7.424 7.107 6.815

8.084 7.717 7.380 7.071 6.785

8.020 7.661 7.332 7.029 6.749

7.953 7.602 7.280 6.983 6.710

7.891 7.548 7.232 6.942 6.673

8.114 7.754 7.422 7.114 6.828

7.761 7.430 7.128 6.849 6.591

7.693 7.370 7.073 6.800 6.547

0.22 0.24 0.25 0.26 0.28 0.30

6.275 5.791 5.570 5.361 4.975 4.628

6.462 5.948 5.714 5.495 5.092 4.730

6.462 5.957 5.728 5.312 5.115 4.759

6.291 5.831 5.620 5.421 5.053 4.719

6.272 5.822 5.615 5.421 5.059 4.731

6.247 5.806 5.605 5.413 5.059 4.737

6.218 5.787 5.589 5.402 5.055 4.739

6.191 5.768 5.574 5.390 5.030 4.740

6.316 5.868 5.664 5.472 5.117 4.796

6.127 5.720 5.534 5.358 5.030 4.731

6.092 5.693 5.510 5.337 5.016 4.723

0.32 0.34 0.35 0.36 0.38 0.40

4.313 4.028 3.893 3.769 3.533 3.318

4.405 4.111 3.974 3.844 3.602 3.381

4.438 4.146 4.010 3.881 3.640 3.420

4.414 4.136 4.006 3.882 3.648 3.434

4.432 4.157 4.029 3.906 3.675 3.462

4.443 4.173 4.047 3.925 3.697 3.486

4.450 4.185 4.060 3.940 3.715 3.306

4.456 4.195 4.072 3.954 3.731 3.525

4.504 4.238 4.113 3.993 3.767 3.559

4.457 4.205 4.086 3.971 3.755 3.554

4.454 4.206 4.089 3.976 3.763 3.565

0.42 0.44 0.43 0.46 0.48 0.50

3.123 2.944 2.861 2.781 2.631 2.494

3.180 2.997 2.911 2.829 2.676 2.535

3.219 3.035 2.949 2.866 2.712 2.570

3.238 3.057 2.973 2.891 2.739 2.598

3.267 3.087 3.003 2.922 2.769 2.628

3.292 3.114 3.029 2.948 2.796 2.655

3.314 3.137 3.053 2.973 2.821 2.680

3.335 3.159 3.075 2.995 2.844 2.703

3.367 3.189 3.105 3.025 2.872 2.730

3.368 3.194 3.113 3.034 2.884 2.745

3.380 3.209 3.128 3.050 2.901 2.763

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.197 1.951 1.745 1.570 1.288 1.073 0.904

2.230 1.979 1.770 1.592 1.308 1.090 0.920

2.262 2.008 1.796 1.617 1.329 1.109 0.936

2.291 2.037 1.824 1.643 1.351 1.128 0.953

2.320 2.064 1.849 1.666 1.372 1.146 0.969

2.346 2.089 1.872 1.688 1.391 1.164 0.985

2.371 2.113 1.895 1.709 1.411 1.181 1.000

2.394 2.156 1.917 1.730 1.429 1.198 1.016

2.419 2.138 1.937 1.749 1.446 1.213 1.030

2.457 2.178 1.958 1.770 1.465 1.231 1.045

2.456 2.197 1.977 1.788 1.482 1.246 1.060

1.10 1.20 1.30 1.40 1.50

0.772 0.666 0.580 0.511 0.436

0.785 0.678 0.391 0.521 0.463

0.799 0.690 0.602 0.530 0.470

0.814 0.702 0.612 0.539 0.478

0.828 0.715 0.623 0.548 0.486

0.842 0.727 0.634 0.557 0.494

0.856 0.739 0.644 0.566 0.502

0.870 0.752 0.655 0.575 0.511

0.883 0.763 0.666 0.383 0.519

0.897 0.776 0.676 0.595 0.527

0.910 0.787 0.687 0.604 0.535

1.60 1.70 1.80 1.90 2.00

0.411 0.367 0.337 0.304 0.277

0.415 0.374 0.340 0.310 0.284

0.421 0.380 0.345 0.314 0.288

0.428 0.386 0.350 0.319 0.292

0.435 0.392 0.355 0.324 0.296

0.442 0.398 0.360 0.328 0.301

0.449 0.404 0.366 0.333 0.305

0.457 0.409 0.372 0.337 0.307

0.463 0.416 0.377 0.343 0.313

0.470 0.423 0.382 0.348 0.318

0.478 0.429 0.388 0.353 0.322

sin =l Ê 1) (A

Element Z Method

268


4.3. ELECTRON DIFFRACTION Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Ho 67 *RHF

Er 68 *RHF

Tm 69 *RHF

Yb 70 *RHF

Lu 71 *RHF

Hf 72 *RHF

Ta 73 *RHF

W 74 *RHF

Re 75 *RHF

Os 76 *RHF

Ir 77 *RHF

0.00 0.01 0.02 0.03 0.04 0.05

14.355

14.080

13.814

13.557

13.486

13.177

12.856

12.543

12.263

11.987

11.718

13.57 13.14

13.16 12.70

12.92 12.48

12.70 12.28

12.74 12.38

12.55 12.23

12.31 12.01

12.06 11.80

11.83 11.60

11.59 11.39

11.37 11.18

0.06 0.07 0.08 0.09 0.10

12.66 12.15 11.61 11.08 10.55

12.19 11.66 11.11 10.58 10.06

12.00 11.48 10.96 10.44 9.93

11.81 11.31 10.80 10.29 9.80

11.95 11.50 11.03 10.55 10.08

11.85 11.45 11.02 10.59 10.16

11.69 11.33 10.95 10.55 10.15

11.51 11.18 10.83 10.47 10.10

11.34 11.04 10.73 10.40 10.05

11.15 10.88 10.59 10.29 9.98

10.96 10.72 10.45 10.17 9.88

0.11 0.12 0.13 0.14 0.15

10.05 9.562 9.108 8.681 8.284

9.56 9.095 8.662 8.262 7.895

9.45 8.994 8.571 8.180 7.821

9.33 8.892 8.480 8.098 7.746

9.62 9.180 8.762 8.370 8.001

9.73 9.308 8.907 8.525 8.163

9.75 9.363 8.982 8.616 8.266

9.74 9.369 9.011 8.663 8.327

9.71 9.366 9.028 8.697 8.376

9.65 9.334 9.016 8.702 8.396

9.58 9.281 8.982 8.686 8.395

0.16 0.17 0.18 0.19 0.20

7.917 7.577 7.262 6.971 6.700

7.557 7.247 6.962 6.698 6.454

7.490 7.185 6.905 6.646 6.407

7.421 7.123 6.849 6.595 6.360

7.660 7.343 7.047 6.774 6.520

7.822 7.502 7.202 6.922 6.660

7.933 7.617 7.321 7.040 6.776

8.006 7.699 7.408 7.132 6.870

8.067 7.769 7.485 7.213 6.954

8.099 7.813 7.537 7.272 7.019

8.111 7.836 7.570 7.313 7.067

0.22 0.24 0.25 0.26 0.28 0.30

6.213 5.788 5.595 5.412 5.075 4.771

6.017 5.632 5.457 5.290 4.981 4.699

5.978 5.601 5.428 5.265 4.961 4.685

5.938 5.568 5.398 5.238 4.940 4.669

6.063 5.664 5.483 5.312 4.996 4.712

6.185 5.768 5.578 5.399 5.069 4.772

6.295 5.867 5.672 5.487 5.147 4.840

6.388 5.957 5.759 5.571 5.224 4.910

6.475 6.043 5.843 5.653 5.301 4.981

6.547 6.117 5.917 5.727 5.372 5.049

6.604 6.180 5.982 5.792 5.437 5.113

0.32 0.34 0.35 0.36 0.38 0.40

4.494 4.240 4.121 4.007 3.790 3.591

4.440 4.200 4.087 3.978 3.771 3.579

4.430 4.195 4.084 3.976 3.773 3.583

4.419 4.188 4.078 3.973 3.773 3.586

4.453 4.215 4.103 3.996 3.793 3.604

4.503 4.258 4.143 4.033 3.825 3.632

4.563 4.310 4.191 4.078 3.865 3.668

4.626 4.366 4.245 4.129 3.910 3.709

4.691 4.425 4.301 4.182 3.959 3.753

4.755 4.485 4.359 4.237 4.010 3.800

4.816 4.543 4.415 4.293 4.061 3.848

0.42 0.44 0.45 0.46 0.48 0.50

3.405 3.233 3.151 3.073 2.924 2.785

3.399 3.232 3.153 3.076 2.930 2.793

3.406 3.241 3.162 3.086 2.942 2.806

3.411 3.248 3.170 3.095 2.952 2.818

3.429 3.265 3.187 3.111 2.968 2.834

3.454 3.288 3.209 3.133 2.988 2.853

3.486 3.317 3.237 3.159 3.013 2.876

3.523 3.350 3.269 3.190 3.041 2.902

3.563 3.387 3.304 3.224 3.072 2.930

3.606 3.427 3.342 3.260 3.105 2.961

3.651 3.468 3.382 3.299 3.141 2.994

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.477 2.216 1.995 1.085 1.497 1.260 1.073

2.490 2.232 2.012 1.823 1.515 1.276 1.088

2.505 2.248 2.028 1.839 1.530 1.291 1.101

2.518 2.263 2.043 1.854 1.545 1.305 1.114

2.534 2.278 2.058 1.868 1.558 1.317 1.126

2.551 2.294 2.073 1.882 1.571 1.329 1.138

2.571 2.311 2.089 1.896 1.583 1.341 1.148

2.592 2.330 2.105 1.911 1.596 1.352 1.159

2.616 2.349 2.122 1.926 1.608 1.363 1.169

2.641 2.371 2.140 1.942 1.621 1.374 1.179

2.669 2.394 2.160 1.959 1.634 1.385 1.189

1.10 1.20 1.30 1.40 1.50

0.922 0.799 0.698 0.614 0.544

0.935 0.811 0.708 0.623 0.552

0.948 0.822 0.719 0.632 0.560

0.960 0.833 0.729 0.642 0.569

0.971 0.844 0.739 0.651 0.577

0.982 0.854 0.748 0.660 0.585

0.993 0.864 0.758 0.668 0.593

1.003 0.874 0.767 0.677 0.601

1.012 0.883 0.776 0.685 0.609

1.022 0.892 0.784 0.694 0.617

1.031 0.901 0.793 0.702 0.624

1.60 1.70 1.80 1.90 2.00

0.485 0.436 0.394 0.358 0.327

0.492 0.442 0.399 0.363 0.331

0.500 0.449 0.405 0.368 0.336

0.507 0.455 0.411 0.373 0.341

0.515 0.462 0.417 0.379 0.345

0.522 0.469 0.423 0.384 0.350

0.530 0.475 0.429 0.389 0.355

0.537 0.482 0.435 0.395 0.360

0.544 0.489 0.441 0.400 0.365

0.551 0.495 0.447 0.406 0.370

0.558 0.502 0.453 0.411 0.374

sin =l Ê 1) (A

Element Z Method

269


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Pt 78 *RHF

Au 79 RHF

Hg 80 RHF

Tl 81 *RHF

Pb 82 RHF

Bi 83 RHF

Po 84 *RHF

At 85 *RHF

Rn 86 RHF

Fr 87 *RHF

Ra 88 *RHF

0.00 0.01 0.02 0.03 0.04 0.05

10.813

10.964 10.948 10.897 10.813 10.698 10.555

12.109

13.096 13.070 12.989 12.857 12.678 12.456

13.473

20.561

12.95 12.74

13.09 12.89

13.492 13.470 13.403 13.292 13.139 12.949

18.715

11.71 11.51

12.597 12.573 12.494 12.366 12.193 11.979

13.368

10.55 10.40

10.573 10.559 10.511 10.434 10.328 10.195

17.14 16.41

18.94 18.15

0.06 0.07 0.08 0.09 0.10

10.23 10.03 9.82 9.60 9.37

10.040 9.865 9.673 9.467 9.251

10.387 10.197 9.989 9.766 9.533

11.27 11.00 10.72 10.42 10.12

11.730 11.454 11.155 10.840 10.516

12.197 11.908 11.595 11.264 10.921

12.49 12.21 11.90 11.57 11.22

12.65 12.38 12.08 11.76 11.43

12.724 12.469 12.187 11.884 11.565

15.64 14.87 14.13 13.42 12.77

17.31 16.42 15.54 14.69 13.88

sin =l Ê 1) (A

Element Z Method

0.11 0.12 0.13 0.14 0.15

9.13 8.882 8.636 8.389 8.145

9.028 8.799 8.568 8.337 8.106

9.291 9.045 8.796 8.547 8.299

9.81 9.500 9.195 8.896 8.603

10.186 9.855 9.527 9.203 8.888

10.571 10.219 9.869 9.523 9.186

10.87 10.509 10.153 9.798 9.449

11.08 10.729 10.375 10.021 9.671

11.232 10.892 10.546 10.199 9.854

12.16 11.605 11.093 10.620 10.180

13.12 12.419 11.776 11.187 10.648

0.16 0.17 0.18 0.19 0.20

7.904 7.667 7.436 7.210 6.991

7.877 7.652 7.431 7.214 7.003

8.055 7.815 7.579 7.350 7.128

8.320 8.046 7.781 7.526 7.282

8.581 8.285 7.999 7.724 7.461

8.857 8.539 8.233 7.939 7.658

9.109 8.779 8.459 8.151 7.856

9.328 8.991 8.666 8.350 8.046

9.512 9.177 8.849 8.531 8.223

9.770 9.386 9.023 8.681 8.356

10.155 9.702 9.285 8.899 8.540

0.22 0.24 0.25 0.26 0.28 0.30

6.572 6.181 5.995 5.817 5.478 5.164

6.598 6.216 6.035 5.859 5.525 5.214

6.702 6.305 6.116 5.934 5.591 5.272

6.822 6.399 6.201 6.011 5.654 5.327

6.969 6.520 6.310 6.110 5.736 5.395

7.132 6.654 6.432 6.221 5.828 5.472

7.303 6.800 6.567 6.345 5.933 5.560

3.474 6.952 6.709 6.477 6.047 5.658

7.639 7.102 6.852 6.612 6.166 5.762

7.754 7.208 6.954 6.712 6.261 5.852

7.891 7.318 7.055 6.807 6.347 5.931

0.32 0.34 0.35 0.36 0.38 0.40

4.873 4.603 4.475 4.352 4.120 3.905

4.924 4.654 4.526 4.403 4.169 3.952

4.976 4.702 4.572 4.447 4.211 3.991

5.025 4.746 4.614 4.488 4.249 4.028

5.083 4.797 4.662 4.533 4.290 4.066

5.148 4.852 4.714 4.581 4.333 4.104

5.222 4.915 4.772 4.636 4.380 4.146

5.305 4.987 4.838 4.697 4.433 4.192

5.397 5.065 4.912 4.765 4.492 4.244

5.480 5.141 4.984 4.834 4.555 4.300

5.555 5.212 5.053 4.900 4.616 4.356

0.42 0.44 0.45 0.46 0.48 0.50

3.704 3.518 3.430 3.345 3.184 3.034

3.750 3.562 3.472 3.386 3.223 3.070

3.787 3.597 3.507 3.420 3.256 3.102

3.823 3.632 3.541 3.454 3.288 3.133

3.858 3.665 3.573 3.485 3.318 3.162

3.893 3.698 3.606 3.517 3.348 3.191

3.931 3.732 3.639 3.548 3.378 3.219

3.972 3.769 3.673 3.582 3.408 3.248

4.017 3.808 3.711 3.617 3.441 3.277

4.067 3.854 3.754 3.658 3.477 3.311

4.118 3.901 3.798 3.700 3.516 3.346

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.701 2.420 2.181 1.976 1.647 1.396 1.198

2.732 2.446 2.203 1.995 1.661 1.407 1.208

2.760 2.471 2.225 2.015 1.676 1.419 1.218

2.789 2.497 2.248 2.035 1.692 1.431 1.228

2.816 2.522 2.271 2.055 1.708 1.444 1.239

2.842 2.546 2.293 2.076 1.725 1.457 1.249

2.868 2.570 2.315 2.096 1.742 1.471 1.260

2.893 2.593 2.337 2.116 1.758 1.485 1.272

2.918 2.616 2.358 2.135 1.775 1.499 1.283

2.945 2.639 2.378 2.154 1.791 1.513 1.295

2.974 2.663 2.399 2.173 1.808 1.527 1.307

1.10 1.20 1.30 1.40 1.50

1.040 0.909 0.801 0.709 0.632

1.048 0.918 0.809 0.717 0.639

1.057 0.926 0.816 0.724 0.646

1.066 0.934 0.824 0.731 0.653

1.075 0.942 0.831 0.738 0.659

1.084 0.949 0.838 0.745 0.666

1.093 0.957 0.846 0.752 0.672

1.102 0.965 0.853 0.758 0.678

1.112 0.974 0.860 0.765 0.684

1.122 0.982 0.867 0.771 0.690

1.132 0.990 0.874 0.778 0.696

1.60 1.70 1.80 1.90 2.00

0.565 0.508 0.459 0.416 0.379

0.572 0.514 0.465 0.422 0.384

0.579 0.521 0.471 0.427 0.389

0.585 0.527 0.476 0.432 0.394

0.591 0.533 0.482 0.438 0.399

0.598 0.538 0.488 0.443 0.404

0.603 0.544 0.493 0.448 0.409

0.609 0.550 0.498 0.453 0.413

0.615 0.555 0.503 0.458 0.418

0.621 0.561 0.508 0.463 0.423

0.626 0.566 0.513 0.468 0.427

270


4.3. ELECTRON DIFFRACTION Ê for electrons for neutral atoms (cont.) Table 4.3.1.1. Atomic scattering amplitudes (A) Ac 89 *RHF

Th 90 *RHF

Pa 91 *RHF

U 92 RHF

Np 93 *RHF

Pu 94 *RHF

Am 95 *RHF

Cm 96 *RHF

Bk 97 *RHF

Cf 98 *RHF

0.00 0.01 0.02 0.03 0.04 0.05

20.484

20.115

19.568

18.759

18.191

17.840

17.710

17.406

16.841

19.10 18.41

18.92 18.33

18.37 17.77

19.119 19.047 18.825 18.470 17.999 17.436

17.70 17.16

17.10 16.55

16.80 16.28

16.80 16.33

16.53 16.08

16.28 15.85

0.06 0.07 0.08 0.09 0.10

17.64 16.84 16.01 15.19 14.40

17.66 16.93 16.19 15.43 14.68

17.11 16.39 15.66 14.92 14.20

16.805 16.131 15.436 14.738 14.052

16.55 15.91 15.25 14.58 13.92

15.95 15.31 14.65 14.00 13.37

15.70 15.09 14.47 13.84 13.24

15.80 15.24 14.66 14.06 13.47

15.58 15.04 14.48 13.91 13.33

15.37 14.84 14.30 13.75 13.20

0.11 0.12 0.13 0.14 0.15

13.64 12.923 12.253 11.632 11.058

13.95 13.255 12.594 11.972 11.388

13.51 12.850 12.228 11.646 11.102

13.389 12.756 12.157 11.595 11.069

13.28 12.665 12.085 11.540 11.029

12.76 12.191 11.653 11.149 10.679

12.65 12.095 11.572 11.083 10.626

12.90 12.344 11.817 11.319 10.848

12.78 12.241 11.729 11.243 10.784

12.66 12.135 11.637 11.164 10.716

0.16 0.17 0.18 0.19 0.20

10.528 10.038 9.586 9.168 8.780

10.845 10.339 9.868 9.430 9.022

10.597 10.128 9.691 9.285 8.906

10.579 10.122 9.696 9.299 8.928

10.551 10.104 9.688 9.300 8.936

10.243 9.836 9.457 9.102 8.770

10.200 9.803 9.433 9.086 8.760

10.407 9.993 9.605 9.241 8.900

10.353 9.948 9.568 9.212 8.878

10.294 9.898 9.527 9.178 8.850

0.22 0.24 0.25 0.26 0.28 0.30

8.083 7.474 7.196 6.935 6.455 6.025

8.287 7.645 7.353 7.079 6.578 6.129

8.221 7.617 7.341 7.081 6.600 6.167

8.254 7.659 7.387 7.129 6.652 6.221

8.275 7.689 7.420 7.165 6.694 6.266

8.163 7.619 7.368 7.129 6.683 6.274

8.164 7.631 7.384 7.148 6.708 6.304

8.277 7.721 7.465 7.222 6.770 6.358

8.266 7.720 7.468 7.229 6.784 6.378

8.249 7.713 7.466 7.231 6.793 6.393

0.32 0.34 0.35 0.36 0.38 0.40

5.637 5.285 5.122 4.966 4.675 4.410

5.727 5.364 5.196 5.036 4.738 4.466

5.775 5.418 5.252 5.093 4.796 4.524

5.830 5.473 5.307 5.148 4.850 4.576

5.878 5.523 5.357 5.197 4.899 4.625

5.899 5.553 5.391 5.235 4.940 4.669

5.933 5.591 5.429 5.274 4.981 4.710

5.981 5.635 5.472 5.316 5.021 4.749

6.006 5.664 5.502 5.347 5.055 4.784

6.026 5.687 5.528 5.374 5.084 4.815

0.42 0.44 0.45 0.46 0.48 0.50

4.168 3.946 3.842 3.742 3.554 3.381

4.218 3.992 3.885 3.784 3.592 3.416

4.275 4.046 3.938 3.835 3.641 3.462

4.325 4.094 3.985 3.881 3.685 3.503

4.372 4.140 4.030 3.925 3.727 3.543

4.417 4.185 4.076 3.970 3.771 3.586

4.459 4.226 4.116 4.010 3.810 3.624

4.497 4.263 4.152 4.046 3.844 3.657

4.532 4.299 4.189 4.082 3.880 3.693

4.565 4.333 4.222 4.116 3.914 3.726

0.55 0.60 0.65 0.70 0.80 0.90 1.00

3.003 2.687 2.421 2.193 1.824 1.541 1.318

3.032 2.712 2.442 2.212 1.840 1.554 1.330

3.071 2.744 2.470 2.235 1.857 1.568 1.342

3.106 2.775 2.495 2.257 1.875 1.582 1.353

3.141 2.805 2.522 2.280 1.893 1.597 1.365

3.179 2.839 2.551 2.306 1.912 1.611 1.377

3.213 2.869 2.578 2.330 1.930 1.626 1.389

3.244 2.897 2.603 2.352 1.949 1.641 1.402

3.277 2.927 2.630 2.376 1.968 1.657 1.415

3.309 2.957 2.657 2.400 1.987 1.673 1.427

1.10 1.20 1.30 1.40 1.50

1.142 0.999 0.882 0.784 0.702

1.152 1.007 0.889 0.791 0.708

1.161 1.016 0.896 0.797 0.714

1.171 1.024 0.904 0.803 0.720

1.181 1.033 0.911 0.810 0.725

1.191 1.041 0.918 0.816 0.731

1.201 1.049 0.926 0.823 0.737

1.212 1.058 0.933 0.830 0.743

1.222 1.067 0.941 0.836 0.748

1.233 1.076 0.948 0.843 0.754

1.60 1.70 1.80 1.90 2.00

0.632 0.571 0.518 0.472 0.432

0.637 0.576 0.523 0.477 0.436

0.643 0.581 0.528 0.481 0.440

0.649 0.585 0.534 0.485 0.443

0.653 0.591 0.537 0.490 0.449

0.659 0.596 0.542 0.495 0.453

0.664 0.601 0.547 0.499 0.457

0.669 0.606 0.551 0.503 0.461

0.674 0.611 0.555 0.507 0.465

0.679 0.165 0.560 0.511 0.469

sin =l Ê 1) (A

Element Z Method

271


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for ionized atoms Table 4.3.1.2. Atomic scattering amplitudes (A) A discussion of the values quoted here for s = 0 is given in Subsection 4.3.1.6. Self-consistent field calculations: HF: non-relativistic Hartree Fock; DS: modified Dirac Slater; RHF, *RHF: relativistic Hartree Fock. H1 1 HF

Li1 3 RHF

Be2 4 RHF

F1 9 HF

Na1 11 RHF

Mg2 12 RHF

0.00 0.01 0.02 0.03 0.04 0.05

12.00 6.78

0.157 239.497 59.992 26.750 15.115 9.730

0.082 478.762 119.752 53.268 29.999 19.229

11.74 6.41

12.21 6.85

1.130 240.469 60.963 27.719 16.081 10.692

0.831 479.511 120.500 54.015 30.745 19.972

0.06 0.07 0.08 0.09 0.10

4.03 2.45 1.48 0.87 0.47

6.804 5.040 3.894 3.109 2.546

13.378 9.850 7.560 5.990 4.867

3.55 1.86 0.79 0.09 0.39

3.97 2.25 1.16 0.43 0.08

7.762 5.993 4.841 4.049 3.480

14.119 10.589 8.296 6.722 5.595

0.11 0.12 0.13 0.14 0.15

0.20 0.023 0.095 0.173 0.224

2.130 1.813 1.567 1.370 1.212

4.036 3.404 2.912 2.522 2.207

0.72 0.949 1.107 1.215 1.285

0.43 0.688 0.870 1.000 1.092

3.056 2.731 2.475 2.269 2.100

4.760 4.123 3.626 3.230 2.909

6.56 5.610 4.868 4.280 3.804

0.16 0.17 0.18 0.19 0.20

0.257 0.276 0.286 0.288 0.287

1.082 0.974 0.883 0.806 0.740

1.949 1.735 1.556 1.404 1.274

1.329 1.352 1.359 1.355 1.343

1.157 1.200 1.226 1.239 1.242

1.960 1.841 1.738 1.650 1.571

2.645 2.425 2.239 2.081 1.944

0.22 0.24 0.25 0.26 0.28 0.30

0.276 0.259 0.250 0.240 0.221 0.203

0.634 0.552 0.518 0.487 0.435 0.393

1.066 0.907 0.841 0.783 0.685 0.605

1.300 1.243 1.212 1.179 1.112 1.046

1.228 1.194 1.173 1.150 1.099 1.046

1.440 1.332 1.284 1.240 1.161 1.092

0.32 0.34 0.35 0.36 0.38 0.40

0.186 0.170 0.163 0.156 0.143 0.132

0.357 0.327 0.314 0.301 0.279 0.259

0.539 0.485 0.461 0.439 0.400 0.366

0.981 0.918 0.889 0.860 0.804 0.753

0.992 0.939 0.912 0.887 0.837 0.789

0.42 0.44 0.45 0.46 0.48 0.50

0.122 0.112 0.108 0.104 0.096 0.090

0.242 0.227 0.220 0.213 0.200 0.189

0.337 0.312 0.300 0.290 0.270 0.252

0.704 0.660 0.639 0.618 0.580 0.544

0.55 0.60 0.65 0.70 0.80 0.90 1.00

0.075 0.064 0.055 0.048 0.037 0.029 0.024

0.165 0.145 0.129 0.115 0.093 0.077 0.064

0.216 0.188 0.165 0.146 0.118 0.097 0.081

1.10 1.20 1.30 1.40 1.50

0.020 0.017 0.014 0.012 0.011

0.054 0.046 0.040 0.035 0.031

0.069 0.059 0.052 0.045 0.040

0.027 0.024 0.022 0.020 0.018

0.035 0.032 0.028 0.026 0.023

sin =l Ê 1) (A

1.60 1.70 1.80 1.90 2.00

Element Z Method

O1 8 HF

Si4 14 HF

Cl1 17 RHF

K1 19 RHF

45.52 29.36

60.34 38.80

6.770 232.585 53.125 19.957 8.423 3.162

3.436 242.773 63.260 30.004 18.349 12.939

20.58 15.29 11.85 9.49 7.81

27.10 20.05 15.46 12.32 10.08

0.381 1.219 2.187 2.783 3.147

9.983 8.184 6.999 6.169 5.559

8.41 7.147 6.162 5.379 4.747

3.361 3.472 3.513 3.504 3.461

5.092 4.720 4.416 4.160 3.939

3.413 3.089 2.817 2.585 2.387

4.230 3.800 3.440 3.135 2.873

3.393 3.308 3.211 3.108 3.000

3.745 3.571 3.414 3.269 3.135

1.720 1.546 1.472 1.406 1.290 1.193

2.066 1.819 1.716 1.624 1.466 1.336

2.451 2.129 1.995 1.876 1.674 1.509

2.779 2.563 2.458 2.357 2.165 1.988

2.893 2.676 2.575 2.479 2.300 2.135

1.029 0.972 0.946 0.920 0.872 0.827

1.110 1.038 1.005 0.974 0.917 0.866

1.228 1.136 1.094 1.056 0.987 0.925

1.372 1.257 1.206 1.159 1.075 1.001

1.827 1.680 1.613 1.548 1.429 1.322

1.983 1.843 1.778 1.715 1.596 1.488

0.744 0.702 0.682 0.662 0.625 0.590

0.785 0.746 0.727 0.709 0.675 0.642

0.820 0.777 0.757 0.738 0.701 0.668

0.871 0.822 0.799 0.778 0.737 0.701

0.937 0.880 0.853 0.829 0.783 0.741

1.226 1.139 1.099 1.061 0.991 0.928

1.388 1.296 1.253 1.212 1.135 1.064

0.467 0.403 0.351 0.307 0.241 0.193 0.159

0.512 0.446 0.391 0.345 0.272 0.219 0.180

0.569 0.506 0.451 0.403 0.325 0.266 0.221

0.593 0.529 0.474 0.426 0.347 0.286 0.239

0.620 0.553 0.496 0.447 0.367 0.305 0.256

0.652 0.580 0.519 0.468 0.385 0.321 0.271

0.796 0.691 0.608 0.541 0.439 0.366 0.311

0.912 0.789 0.690 0.609 0.488 0.402 0.338

0.133 0.113 0.097 0.085 0.075

0.150 0.128 0.110 0.095 0.084

0.185 0.157 0.135 0.118 0.103

0.201 0.172 0.148 0.129 0.113

0.217 0.186 0.160 0.140 0.123

0.231 0.198 0.172 0.150 0.132

0.267 0.232 0.202 0.178 0.158

0.290 0.252 0.221 0.195 0.173

0.091 0.081 0.073 0.066 0.060

0.100 0.089 0.080 0.072 0.065

0.141 0.126 0.113 0.102 0.093

0.155 0.139 0.125 0.114 0.103

272


Al3 13 HF

4.3. ELECTRON DIFFRACTION Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A)

sin =l Ê 1) (A

Element Z Method

Ca2 20 RHF

Sc3 21 HF

Ti2 22 HF

Ti3 22 HF

Ti4 22 HF

V2 23 RHF

V3 23 HF

V5 23 HF

47.18 31.02

76.36 49.43

22.23 16.92 13.47 11.10 9.39

34.80 25.98 20.24 16.31 13.49

Cr2 24 HF

Cr3 24 HF

Mn2 25 RHF

32.79 22.00

47.19 31.03

2.846 481.525 122.510 56.018 32.738 21.953

16.13 12.58 10.26 8.67 7.51

22.24 16.93 13.48 11.11 9.41

16.085 12.537 10.225 8.630 7.479

0.00 0.01 0.02 0.03 0.04 0.05

2.711 481.390 122.375 55.883 32.602 21.817

47.08 30.91

32.80 22.01

47.15 30.98

61.67 40.13

2.904 481.582 122.566 56.074 32.791 22.005

0.06 0.07 0.08 0.09 0.10

15.948 12.399 10.085 8.489 7.336

22.13 16.82 13.37 11.00 9.30

16.14 12.59 10.27 8.67 7.52

22.19 16.89 13.44 11.07 9.36

28.42 21.35 16.77 13.62 11.36

16.134 12.583 10.267 8.668 7.514

0.11 0.12 0.13 0.14 0.15

6.473 5.807 5.279 4.850 4.495

8.03 7.057 6.295 5.684 5.185

6.65 5.977 5.444 5.011 4.653

8.09 7.120 6.359 5.747 5.247

9.68 8.400 7.400 6.603 5.954

6.648 5.980 5.449 5.018 4.661

8.13 7.155 6.394 5.782 5.284

11.41 9.815 8.574 7.584 6.784

6.65 5.983 5.455 5.026 4.671

8.14 7.172 6.410 5.800 5.302

6.618 5.954 5.428 5.002 4.650

0.16 0.17 0.18 0.19 0.20

4.196 3.939 3.716 3.519 3.343

4.770 4.421 4.121 3.863 3.637

4.349 4.089 3.863 3.663 3.485

4.832 4.481 4.182 3.923 3.697

5.418 4.971 4.591 4.266 3.984

4.360 4.102 3.877 3.679 3.503

4.868 4.518 4.220 3.961 3.735

6.126 5.577 5.113 4.719 4.378

4.372 4.116 3.894 3.698 3.523

4.888 4.539 4.242 3.984 3.759

4.353 4.100 3.880 3.686 3.514

0.22 0.24 0.25 0.26 0.28 0.30

3.041 2.787 2.674 2.568 2.376 2.204

3.259 2.953 2.821 2.699 2.482 2.294

3.178 2.920 2.806 2.699 2.504 2.331

3.318 3.012 2.879 2.757 2.540 2.352

3.520 3.155 2.998 2.857 2.610 2.399

3.200 2.946 2.833 2.727 2.536 2.365

3.358 3.053 2.921 2.799 2.584 2.396

3.824 3.391 3.209 3.045 2.761 2.524

3.224 2.973 2.862 2.758 2.569 2.401

3.384 3.081 2.950 2.830 2.616 2.430

3.220 2.975 2.865 2.764 2.579 2.415

0.32 0.34 0.35 0.36 0.38 0.40

2.049 1.907 1.842 1.778 1.660 1.551

2.128 1.980 1.911 1.846 1.725 1.614

2.174 2.032 1.966 1.903 1.783 1.673

2.185 2.037 1.968 1.903 1.781 1.670

2.217 2.057 1.984 1.915 1.788 1.673

2.211 2.071 2.005 1.943 1.825 1.716

2.231 2.073 2.015 1.950 1.829 1.718

2.322 2.147 2.068 1.994 1.858 1.736

2.249 2.111 2.046 1.984 1.867 1.759

2.266 2.120 2.053 1.988 1.868 1.758

2.266 2.131 2.068 2.007 1.893 1.787

0.42 0.44 0.45 0.46 0.48 0.50

1.451 1.359 1.316 1.274 1.196 1.124

1.512 1.419 1.375 1.333 1.253 1.180

1.572 1.478 1.433 1.391 1.310 1.235

1.568 1.474 1.429 1.387 1.306 1.232

1.569 1.473 1.428 1.385 1.304 1.229

1.615 1.522 1.477 1.435 1.354 1.279

1.616 1.522 1.477 1.434 1.354 1.279

1.627 1.528 1.481 1.437 1.354 1.277

1.659 1.566 1.522 1.480 1.399 1.324

1.657 1.563 1.519 1.476 1.395 1.320

1.688 1.597 1.553 1.511 1.432 1.357

0.55 0.60 0.65 0.70 0.80 0.90 1.00

0.967 0.838 0.733 0.647 0.515 0.422 0.354

1.019 0.886 0.776 0.685 0.544 0.444 0.371

1.070 0.933 0.818 0.722 0.574 0.467 0.389

1.068 0.931 0.817 0.722 0.574 0.467 0.389

1.066 0.930 0.816 0.721 0.574 0.467 0.389

1.113 0.973 0.856 0.757 0.602 0.490 0.408

1.113 0.974 0.857 0.758 0.603 0.491 0.408

1.110 0.971 0.855 0.756 0.603 0.491 0.408

1.156 1.015 0.895 0.793 0.632 0.515 0.427

1.154 1.013 0.894 0.792 0.632 0.515 0.427

1.190 1.049 0.928 0.824 0.659 0.538 0.446

1.10 1.20 1.30 1.40 1.50

0.302 0.262 0.230 0.203 0.180

0.316 0.273 0.239 0.211 0.188

0.331 0.285 0.249 0.220 0.195

0.331 0.285 0.249 0.220 0.195

0.330 0.285 0.249 0.219 0.195

0.345 0.297 0.259 0.228 0.203

0.346 0.297 0.259 0.228 0.203

0.345 0.297 0.259 0.228 0.202

0.361 0.310 0.270 0.237 0.211

0.361 0.310 0.270 0.237 0.211

0.377 0.323 0.280 0.246 0.218

1.60 1.70 1.80 1.90 2.00

0.161 0.145 0.131 0.119 0.108

0.181 0.163 0.148 0.134 0.123

273


0.195 0.175 0.159 0.144 0.132

4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A)

sin =l Ê 1) (A

Element Z Method

Mn3 25 HF

Mn4 25 HF

Fe2 26 RHF

Fe3 26 RHF

Co2 27 RHF

Co3 27 HF

Ni2 28 RHF

2.298 720.318 181.800 82.070 47.160 30.996

2.754 481.433 122.419 55.928 32.650 21.867

0.00 0.01 0.02 0.03 0.04 0.05

47.18 31.02

61.76 40.22

2.802 481.481 122.467 55.976 32.696 21.913

0.06 0.07 0.08 0.09 0.10

22.23 16.93 13.48 11.11 9.41

28.51 21.44 16.85 13.70 11.45

16.046 12.500 10.189 8.596 7.447

22.210 16.907 13.459 11.089 9.388

16.002 12.457 10.148 8.556 7.409

Ni3 28 HF

Cu1 29 RHF

Cu2 29 HF

Zn2 30 RHF

47.15 30.98

2.703 481.382 122.368 55.878 32.600 21.819

47.12 30.96

3.280 242.618 63.107 29.855 18.206 12.803

32.55 21.77

2.599 481.278 122.265 55.776 32.499 21.719

22.20 16.90 13.45 11.08 9.38

15.955 12.411 10.103 8.513 7.368

22.17 16.87 13.43 11.06 9.36

9.856 8.066 6.893 6.076 5.479

15.90 12.36 10.06 8.47 7.33

15.857 12.316 10.011 8.424 7.282

0.11 0.12 0.13 0.14 0.15

8.14 7.174 6.413 5.804 5.307

9.77 8.492 7.492 6.695 6.047

6.588 5.926 5.403 4.979 4.629

8.124 7.156 6.398 5.790 5.294

6.553 5.893 5.371 4.950 4.603

8.12 7.150 6.393 5.787 5.293

6.513 5.856 5.336 4.917 4.572

8.10 7.132 6.376 5.770 5.277

5.027 4.671 4.383 4.144 3.942

6.47 5.817 5.299 4.883 4.540

6.430 5.776 5.260 4.845 4.504

0.16 0.17 0.18 0.19 0.20

4.894 4.547 4.251 3.995 3.771

5.514 5.068 4.689 4.366 4.086

4.335 4.084 3.867 3.676 3.506

4.884 4.538 4.243 3.989 3.767

4.311 4.063 3.847 3.659 3.492

4.883 4.538 4.245 3.993 3.772

4.283 4.036 3.824 3.638 3.473

4.869 4.526 4.234 3.983 3.764

3.766 3.612 3.474 3.349 3.234

4.253 4.009 3.799 3.615 3.453

4.219 3.976 3.768 3.586 3.426

0.22 0.24 0.25 0.26 0.28 0.30

3.399 3.099 2.969 2.850 2.639 2.455

3.625 3.262 3.108 2.968 2.723 2.516

3.217 2.976 2.869 2.769 2.589 2.428

3.397 3.100 2.972 2.855 2.646 2.466

3.207 2.971 2.866 2.768 2.592 2.434

3.405 3.111 2.984 2.868 2.662 2.484

3.193 2.961 2.858 2.763 2.590 2.436

3.400 3.109 2.984 2.869 2.666 2.490

3.030 2.851 2.769 2.690 2.544 2.410

3.178 2.950 2.850 2.757 2.588 2.438

3.154 2.930 2.831 2.740 2.574 2.428

0.32 0.34 0.35 0.36 0.38 0.40

2.294 2.149 2.083 2.019 1.900 1.791

2.336 2.179 2.107 2.039 1.913 1.799

2.282 2.150 2.088 2.029 1.917 1.813

2.307 2.165 2.099 2.037 1.920 1.813

2.293 2.163 2.103 2.045 1.935 1.833

2.327 2.187 2.123 2.061 1.946 1.841

2.298 2.172 2.113 2.056 1.949 1.849

2.336 2.199 2.135 2.075 1.962 1.858

2.285 2.169 2.114 2.061 1.959 1.864

2.303 2.180 2.123 2.067 1.963 1.866

2.296 2.176 2.120 2.066 1.964 1.869

0.42 0.44 0.45 0.46 0.48 0.50

1.691 1.598 1.554 1.512 1.432 1.357

1.695 1.600 1.555 1.512 1.431 1.355

1.716 1.626 1.583 1.542 1.463 1.389

1.715 1.623 1.580 1.538 1.459 1.385

1.739 1.650 1.608 1.567 1.489 1.416

1.743 1.653 1.610 1.569 1.490 1.417

1.756 1.670 1.628 1.588 1.512 1.440

1.762 1.674 1.631 1.591 1.513 1.441

1.774 1.690 1.649 1.610 1.535 1.464

1.775 1.690 1.649 1.610 1.535 1.464

1.781 1.697 1.658 1.619 1.546 1.476

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.190 1.049 0.928 0.825 0.660 0.538 0.447

1.188 1.047 0.927 0.824 0.660 0.538 0.447

1.223 1.081 0.959 0.855 0.687 0.561 0.466

1.220 1.079 0.958 0.854 0.686 0.561 0.466

1.252 1.111 0.989 0.883 0.713 0.583 0.485

1.252 1.111 0.989 0.884 0.713 0.584 0.486

1.277 1.137 1.015 0.910 0.737 0.605 0.504

1.278 1.138 1.016 0.910 0.738 0.606 0.505

1.303 1.163 1.042 0.935 0.761 0.627 0.523

1.303 1.164 1.043 0.936 0.762 0.628 0.524

1.319 1.182 1.061 0.956 0.782 0.646 0.541

1.10 1.20 1.30 1.40 1.50

0.377 0.323 0.281 0.246 0.219

0.377 0.323 0.281 0.246 0.218

0.393 0.336 0.291 0.256 0.226

0.393 0.336 0.291 0.256 0.226

0.409 0.350 0.303 0.265 0.235

0.410 0.350 0.304 0.266 0.235

0.425 0.364 0.315 0.275 0.243

0.426 0.364 0.315 0.276 0.244

0.441 0.378 0.327 0.286 0.252

0.442 0.378 0.327 0.286 0.253

0.457 0.391 0.339 0.296 0.261

0.202 0.182 0.164 0.149 0.136

0.202 0.182 0.164 0.149 0.136

0.209 0.188 0.170 0.155 0.141

1.60 1.70 1.80 1.90 2.00

274


0.217 0.195 0.176 0.160 0.146

0.224 0.201 0.182 0.165 0.150

0.232 0.208 0.188 0.170 0.155

4.3. ELECTRON DIFFRACTION Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A)

sin =l Ê 1) (A

Element Z Method

Ga3 31 HF

Ge4 32 HF

Br1 35 RHF

Rb1 37 RHF

Sr2 38 RHF

Y3 39 *DS

Zr4 40 *DS

Nb3 41 *DS

Nb5 41 *DS

Mo3 42 *DS

Mo5 42 *DS

5.545 244.880 65.359 32.090 20.419 14.987

4.642 483.320 124.299 57.798 34.505 23.704

48.84 32.67

63.31 41.74

49.33 33.15

77.86 50.92

49.44 33.26

78.04 51.09

23.86 18.54 15.07 12.68 10.95

30.02 22.95 18.34 15.17 12.90

24.34 19.01 15.52 13.12 11.38

36.28 27.45 21.70 17.76 14.92

24.45 19.11 15.63 13.23 11.49

36.45 27.61 21.87 17.92 15.09

0.00 0.01 0.02 0.03 0.04 0.05

47.03 30.87

61.67 40.13

9.357 230.004 50.565 17.431 5.942 0.738

0.06 0.07 0.08 0.09 0.10

22.09 16.78 13.34 10.97 9.28

28.43 21.37 16.78 13.63 11.38

1.978 3.508 4.399 4.917 5.202

12.005 10.176 8.957 8.091 7.442

17.816 14.246 11.907 10.283 9.101

0.11 0.12 0.13 0.14 0.15

8.02 7.057 6.305 5.703 5.213

9.71 8.437 7.442 6.650 5.818

5.335 5.367 5.331 5.248 5.132

6.932 6.516 6.166 5.863 5.595

8.206 7.506 6.942 6.477 6.084

9.66 8.658 7.867 7.227 6.696

11.20 9.898 8.874 8.052 7.378

10.07 9.062 8.257 7.601 7.057

12.82 11.212 9.952 8.945 8.124

10.18 9.170 8.364 7.708 7.163

12.98 11.369 10.105 9.094 8.269

0.16 0.17 0.18 0.19 0.20

4.809 4.470 4.182 3.934 3.719

5.481 5.041 4.669 4.351 4.078

4.996 4.846 4.688 4.527 4.365

5.352 5.130 4.925 4.733 4.552

5.746 5.449 5.186 4.949 4.734

6.249 5.867 5.534 5.242 4.981

6.817 6.343 5.936 5.582 5.273

6.596 6.200 5.853 5.548 5.275

7.444 6.873 6.388 5.969 5.605

6.702 6.304 5.957 5.650 5.376

7.586 7.012 6.523 6.101 5.733

0.22 0.24 0.25 0.26 0.28 0.30

3.364 3.081 2.960 2.849 2.654 2.487

3.631 3.281 3.133 3.000 2.769 2.574

4.046 3.745 3.602 3.465 3.208 2.975

4.218 3.914 3.773 3.638 3.384 3.151

4.352 4.021 3.871 3.729 3.466 3.228

4.535 4.161 3.995 3.841 3.560 3.311

4.751 4.327 4.143 3.972 3.668 3.403

4.805 4.410 4.233 4.069 3.771 3.507

5.002 4.520 4.313 4.124 3.791 3.505

4.903 4.505 4.327 4.162 3.860 3.592

5.123 4.634 4.424 4.232 3.892 3.599

0.32 0.34 0.35 0.36 0.38 0.40

2.340 2.210 2.150 2.093 1.986 1.888

2.407 2.262 2.195 2.133 2.018 1.914

2.765 2.576 2.488 2.405 2.252 2.114

2.938 2.744 2.652 2.565 2.403 2.254

3.012 2.815 2.723 2.635 2.470 2.319

3.088 2.886 2.792 2.702 2.534 2.381

3.168 2.957 2.860 2.768 2.596 2.439

3.269 3.054 2.955 2.859 2.681 2.518

3.255 3.034 2.933 2.837 2.659 2.497

3.351 3.132 3.031 2.934 2.752 2.585

3.344 3.117 3.013 2.915 2.732 2.566

0.42 0.44 0.45 0.46 0.48 0.50

1.798 1.714 1.674 1.635 1.562 1.493

1.819 1.732 1.691 1.652 1.577 1.507

1.990 1.877 1.825 1.775 1.682 1.598

2.119 1.996 1.938 1.883 1.780 1.686

2.180 2.053 1.994 1.937 1.831 1.733

2.240 2.111 2.050 1.992 1.883 1.782

2.295 2.163 2.102 2.042 1.931 1.828

2.369 2.231 2.167 2.105 1.988 1.881

2.350 2.215 2.152 2.092 1.978 1.872

2.432 2.292 2.225 2.162 2.042 1.931

2.414 2.276 2.211 2.148 2.031 1.922

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.337 1.201 1.082 0.977 0.803 0.667 0.559

1.351 1.216 1.098 0.994 0.821 0.684 0.575

1.415 1.266 1.140 1.034 0.860 0.725 0.616

1.483 1.318 1.182 1.068 0.887 0.749 0.640

1.522 1.350 1.208 1.089 0.902 0.761 0.651

1.564 1.385 1.237 1.113 0.919 0.775 0.662

1.604 1.419 1.266 1.137 0.937 0.788 0.673

1.647 1.454 1.295 1.161 0.954 0.801 0.684

1.643 1.453 1.295 1.163 0.955 0.802 0.685

1.690 1.491 1.326 1.188 0.973 0.815 0.696

1.686 1.489 1.326 1.188 0.974 0.816 0.696

1.10 1.20 1.30 1.40 1.50

0.474 0.406 0.351 0.307 0.271

0.489 0.419 0.363 0.317 0.280

0.528 0.456 0.396 0.347 0.306

0.551 0.478 0.417 0.366 0.324

0.562 0.488 0.427 0.376 0.332

0.572 0.498 0.436 0.384 0.340

0.582 0.507 0.445 0.393 0.348

0.591 0.516 0.453 0.401 0.356

0.592 0.516 0.453 0.401 0.356

0.601 0.525 0.462 0.408 0.363

0.601 0.525 0.462 0.408 0.363

0.272 0.243 0.219 0.198 0.180

0.288 0.257 0.232 0.209 0.190

0.296 0.265 0.238 0.215 0.196

0.303 0.271 0.244 0.221 0.201

0.311 0.278 0.251 0.227 0.206

0.318 0.285 0.257 0.232 0.211

0.318 0.285 0.257 0.232 0.211

0.325 0.292 0.263 0.238 0.216

0.325 0.292 0.263 0.238 0.216

1.60 1.70 1.80 1.90 2.00

275


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A) Mo3 42 *DS

Ru3 44 *DS

Ru4 44 *DS

Rh3 45 *DS

Rh4 45 *DS

Pd2 46 *DS

Pd4 46 *DS

Ag1 47 *DS

Ag2 47 *DS

Cd2 48 *DS

In3 49 *DS

0.00 0.01 0.02 0.03 0.04 0.05

92.49 60.17

49.53 33.34

63.83 42.27

49.53 33.35

63.87 42.31

35.30 24.50

63.89 42.32

21.21 15.77

35.23 24.43

35.15 24.36

49.41 33.23

0.06 0.07 0.08 0.09 0.10

42.61 32.01 25.13 20.40 17.02

24.54 19.21 15.73 13.33 11.59

30.54 23.46 18.85 15.68 13.39

24.55 19.22 15.74 13.34 11.61

30.58 23.50 18.89 15.72 13.44

18.61 15.03 12.69 11.06 9.87

30.61 23.52 18.92 15.75 13.46

12.79 10.96 9.73 8.87 8.22

18.54 14.97 12.63 11.00 9.82

18.47 14.90 12.56 10.94 9.76

24.43 19.11 15.65 13.26 11.53

0.11 0.12 0.13 0.14 0.15

14.50 12.585 11.086 9.891 8.919

10.29 9.281 8.480 7.827 7.286

11.69 10.381 9.351 8.521 7.840

10.31 9.301 8.502 7.853 7.313

11.74 10.430 9.399 8.571 7.891

8.97 8.259 7.687 7.214 6.813

11.76 10.458 9.431 8.603 7.924

7.70 7.287 6.935 6.631 6.361

8.92 8.217 7.652 7.184 6.789

8.87 8.169 7.608 7.146 6.756

10.24 9.249 8.463 7.826 7.299

0.16 0.17 0.18 0.19 0.20

8.118 7.449 6.881 6.395 5.975

6.827 6.432 6.088 5.784 5.511

7.271 6.788 6.372 6.011 5.692

6.858 6.465 6.124 5.822 5.553

7.322 6.841 6.426 6.065 5.747

6.467 6.163 5.892 5.647 5.423

7.357 6.877 6.464 6.105 5.788

6.117 5.894 5.687 5.494 5.312

6.448 6.149 5.883 5.643 5.424

6.419 6.126 5.865 5.629 5.416

6.856 6.476 6.147 5.858 5.600

0.22 0.24 0.25 0.26 0.28 0.30

5.283 4.739 4.508 4.297 3.931 3.620

5.042 4.646 4.469 4.304 4.003 3.733

5.153 4.712 4.519 4.340 4.020 3.738

5.087 4.695 4.520 4.356 4.057 3.789

5.211 4.771 4.578 4.400 4.080 3.799

5.026 4.679 4.521 4.370 4.090 3.836

5.255 4.818 4.626 4.449 4.130 3.850

4.975 4.667 4.522 4.383 4.120 3.876

5.036 4.697 4.541 4.394 4.121 3.871

5.036 4.705 4.553 4.410 4.142 3.898

5.158 4.786 4.621 4.466 4.184 3.930

0.32 0.34 0.35 0.36 0.38 0.40

3.352 3.118 3.011 2.910 2.725 2.557

3.490 3.269 3.166 3.067 2.882 2.711

3.488 3.263 3.158 3.058 2.872 2.702

3.547 3.327 3.224 3.125 2.939 2.768

3.548 3.323 3.218 3.118 2.931 2.759

3.601 3.385 3.284 3.185 3.000 2.828

3.601 3.376 3.271 3.171 2.984 2.812

3.649 3.437 3.336 3.239 3.055 2.883

3.640 3.427 3.327 3.230 3.046 2.875

3.672 3.463 3.363 3.268 3.086 2.917

3.701 3.490 3.390 3.295 3.115 2.947

0.42 0.44 0.45 0.46 0.48 0.50

2.406 2.267 2.202 2.140 2.024 1.916

2.553 2.408 2.339 2.273 2.148 2.033

2.545 2.400 2.332 2.266 2.143 2.028

2.609 2.462 2.393 2.326 2.199 2.082

2.601 2.454 2.385 2.319 2.193 2.077

2.668 2.520 2.450 2.382 2.253 2.133

2.653 2.505 2.436 2.369 2.242 2.124

2.723 2.573 2.502 2.434 2.304 2.182

2.716 2.567 2.497 2.429 2.300 2.179

2.759 2.611 2.541 2.473 2.343 2.222

2.790 2.643 2.574 2.506 2.378 2.258

0.55 0.60 0.65 0.70 0.80 0.90 1.00

1.682 1.487 1.325 1.189 0.975 0.817 0.696

1.779 1.568 1.392 1.244 1.014 0.846 0.719

1.776 1.567 1.392 1.244 1.015 0.846 0.720

1.823 1.607 1.426 1.274 1.036 0.863 0.732

1.820 1.606 1.426 1.274 1.037 0.863 0.732

1.869 1.647 1.461 1.304 1.059 0.880 0.745

1.863 1.644 1.460 1.304 1.060 0.880 0.746

1.913 1.687 1.496 1.335 1.083 0.898 0.759

1.912 1.686 1.496 1.335 1.083 0.898 0.759

1.953 1.725 1.531 1.367 1.107 0.917 0.774

1.989 1.760 1.564 1.397 1.132 0.936 0.789

1.10 1.20 1.30 1.40 1.50

0.602 0.525 0.462 0.409 0.363

0.621 0.542 0.477 0.423 0.377

0.621 0.542 0.477 0.423 0.377

0.631 0.551 0.485 0.430 0.384

0.631 0.551 0.485 0.430 0.384

0.642 0.560 0.493 0.437 0.391

0.642 0.560 0.493 0.437 0.391

0.653 0.569 0.501 0.444 0.397

0.653 0.569 0.501 0.444 0.397

0.664 0.578 0.508 0.451 0.403

0.676 0.588 0.516 0.458 0.409

1.60 1.70 1.80 1.90 2.00

0.325 0.292 0.263 0.238 0.216

0.338 0.304 0.275 0.249 0.227

0.338 0.304 0.275 0.249 0.227

0.344 0.310 0.280 0.254 0.232

0.344 0.310 0.280 0.254 0.232

0.350 0.316 0.286 0.260 0.237

0.350 0.316 0.286 0.260 0.237

0.356 0.322 0.291 0.265 0.241

0.356 0.322 0.291 0.265 0.241

0.362 0.327 0.296 0.270 0.246

0.368 0.332 0.301 0.274 0.251

sin =l Ê 1) (A

Element Z Method

276


4.3. ELECTRON DIFFRACTION Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A) Sn2 50 RHF

Sn4 50 RHF

Sb5 51 *DS

I1 53 RHF

Cs1 55 RHF

Ba2 56 *DS

La3 57 *DS

Ce3 58 *DS

Ce4 58 *DS

Pr3 59 *DS

0.00 0.01 0.02 0.03 0.04 0.05

6.144 484.819 125.792 59.280 35.972 25.152

3.971 961.330 243.305 110.331 63.782 42.227

50.16 33.97

78.38 51.44

13.835 225.540 46.145 13.083 1.690 3.399

9.035 248.365 68.827 35.532 23.823 18.344

37.64 26.81

51.70 35.49

51.62 35.42

65.94 44.36

51.53 35.34

0.06 0.07 0.08 0.09 0.10

19.242 15.646 13.280 11.625 10.411

30.510 23.435 18.833 15.668 13.395

25.14 19.81 16.32 13.91 12.16

36.81 27.97 22.22 18.28 15.45

5.981 7.365 8.103 8.462 8.586

15.307 13.414 12.124 11.180 10.448

20.87 17.25 14.86 13.17 11.92

26.65 21.30 17.78 15.34 13.57

26.59 21.23 17.72 15.29 13.51

32.62 25.51 20.87 17.66 15.34

26.51 21.15 17.65 15.22 13.45

0.11 0.12 0.13 0.14 0.15

9.484 8.750 8.152 7.653 7.227

11.703 10.407 9.388 8.571 7.902

10.85 9.825 9.010 8.344 7.790

13.35 11.743 10.486 9.480 8.660

8.560 8.437 8.249 8.021 7.767

9.851 9.345 8.903 8.506 8.143

10.96 10.185 9.547 9.003 8.531

12.22 11.163 10.313 9.610 9.016

12.17 11.119 10.275 9.576 8.987

13.60 12.258 11.185 10.312 9.586

12.11 11.064 10.224 9.531 8.947

0.16 0.17 0.18 0.19 0.20

6.856 6.528 6.234 5.968 5.725

7.345 6.875 6.473 6.124 5.818

7.319 6.911 6.555 6.239 5.955

7.982 7.413 6.928 6.512 6.149

7.500 7.228 6.955 6.685 6.423

7.806 7.491 7.193 6.911 6.643

8.113 7.738 7.395 7.080 6.787

8.505 8.056 7.658 7.300 6.974

8.482 8.038 7.645 7.292 6.970

8.972 8.441 7.979 7.569 7.202

8.446 8.008 7.619 7.270 6.954

0.22 0.24 0.25 0.26 0.28 0.30

5.293 4.918 4.747 4.586 4.289 4.021

5.304 4.886 4.703 4.535 4.232 3.967

5.464 5.050 4.864 4.691 4.375 4.094

5.548 5.067 4.860 4.671 4.336 4.048

5.925 5.468 5.255 5.054 4.681 4.348

6.143 5.688 5.475 5.272 4.893 4.549

6.256 5.785 5.568 5.362 4.980 4.633

6.398 5.900 5.674 5.461 5.069 4.716

6.403 5.912 5.690 5.480 5.094 4.745

6.568 6.033 5.794 5.570 5.163 4.800

6.396 5.914 5.695 5.489 5.109 4.766

0.32 0.34 0.35 0.36 0.38 0.40

3.778 3.556 3.452 3.352 3.164 2.991

3.730 3.516 3.416 3.320 3.140 2.973

3.840 3.611 3.503 3.401 3.208 3.031

3.793 3.567 3.463 3.363 3.177 3.006

4.049 3.782 3.658 3.541 3.325 3.129

4.237 3.954 3.823 3.698 3.466 3.255

4.318 4.032 3.899 3.772 3.536 3.321

4.396 4.106 3.971 3.843 3.602 3.383

4.429 4.142 4.008 3.880 3.640 3.422

4.474 4.179 4.042 3.911 3.667 3.444

4.454 4.170 4.037 3.910 3.673 3.455

0.42 0.44 0.45 0.46 0.48 0.50

2.830 2.680 2.610 2.541 2.412 2.290

2.817 2.672 2.604 2.537 2.410 2.291

2.867 2.716 2.644 2.575 2.444 2.322

2.848 2.702 2.632 2.565 2.438 2.319

2.951 2.789 2.714 2.641 2.505 2.379

3.064 2.890 2.808 2.731 2.586 2.452

3.124 2.946 2.862 2.782 2.632 2.495

3.183 3.000 2.914 2.832 2.679 2.538

3.221 3.038 2.952 2.870 2.715 2.573

3.240 3.054 2.967 2.883 2.726 2.582

3.255 3.072 2.986 2.904 2.749 2.606

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.020 1.791 1.594 1.426 1.157 0.956 0.805

2.023 1.794 1.597 1.428 1.157 0.956 0.804

2.050 1.819 1.622 1.453 1.181 0.976 0.821

2.051 1.822 1.625 1.455 1.182 0.977 0.821

2.103 1.871 1.673 1.503 1.227 1.017 0.855

2.163 1.923 1.721 1.548 1.269 1.055 0.888

2.197 1.951 1.745 1.570 1.288 1.072 0.904

2.232 1.980 1.770 1.592 1.307 1.090 0.920

2.264 2.010 1.797 1.617 1.328 1.108 0.936

2.268 2.011 1.796 1.615 1.326 1.107 0.935

2.295 2.038 1.823 1.641 1.349 1.126 0.952

1.10 1.20 1.30 1.40 1.50

0.688 0.597 0.525 0.465 0.416

0.688 0.597 0.524 0.465 0.416

0.702 0.608 0.534 0.473 0.422

0.702 0.608 0.533 0.473 0.422

0.729 0.630 0.551 0.487 0.435

0.757 0.654 0.571 0.504 0.448

0.771 0.666 0.581 0.512 0.456

0.785 0.678 0.591 0.521 0.463

0.799 0.690 0.602 0.530 0.471

0.799 0.690 0.602 0.530 0.471

0.813 0.702 0.612 0.539 0.478

1.60 1.70 1.80 1.90 2.00

0.374 0.338 0.306 0.279 0.255

0.374 0.338 0.306 0.279 0.255

0.379 0.343 0.311 0.284 0.259

0.379 0.343 0.311 0.284 0.259

0.391 0.353 0.321 0.293 0.268

0.402 0.364 0.330 0.301 0.276

0.409 0.369 0.335 0.306 0.280

0.415 0.374 0.340 0.310 0.284

0.421 0.380 0.345 0.314 0.288

0.421 0.380 0.345 0.314 0.288

0.428 0.386 0.350 0.319 0.292

sin =l Ê 1) (A

Element Z Method

Sb3 51 *DS

277


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A) Pr4 59 *DS

Nd3 60 *DS

Pm3 61 *DS

Sm3 62 *DS

Eu2 63 *DS

Eu3 63 *DS

Gd3 64 *DS

Tb3 65 *DS

Dy3 66 *DS

Ho3 67 *DS

Er3 68 *DS

0.00 0.01 0.02 0.03 0.04 0.05

65.86 44.30

51.44 35.25

51.35 35.15

51.26 35.07

36.99 26.17

51.17 34.97

51.08 34.90

51.04 34.85

50.92 34.72

50.83 34.64

50.74 34.55

0.06 0.07 0.08 0.09 0.10

32.55 25.44 20.81 17.60 15.29

26.42 21.07 17.57 15.14 13.38

26.33 20.98 17.49 15.06 13.30

26.25 20.90 17.41 14.98 13.23

20.26 16.67 14.30 12.65 11.44

26.15 20.81 17.32 14.90 13.15

26.08 20.74 17.25 14.83 13.08

26.02 20.68 17.19 14.77 13.02

25.92 20.58 17.09 14.68 12.94

25.83 20.50 17.01 14.60 12.86

25.74 20.41 16.93 14.53 12.79

0.11 0.12 0.13 0.14 0.15

13.55 12.210 11.141 10.272 9.550

12.04 11.003 10.167 9.480 8.901

11.97 10.937 10.106 9.422 8.849

11.90 10.870 10.044 9.366 8.796

10.51 9.770 9.167 8.663 8.229

11.83 10.802 9.979 9.305 8.740

11.76 10.739 9.919 9.248 8.686

11.71 10.681 9.863 9.194 8.634

11.62 10.604 9.792 9.128 8.574

11.55 10.538 9.728 9.068 8.517

11.48 10.469 9.664 9.007 8.460

0.16 0.17 0.18 0.19 0.20

8.939 8.413 7.955 7.549 7.187

8.405 7.972 7.589 7.245 6.932

8.357 7.930 7.551 7.212 6.904

8.310 7.886 7.512 7.177 6.874

7.850 7.511 7.206 6.927 6.669

8.257 7.838 7.468 7.138 6.839

8.207 7.791 7.425 7.099 6.804

8.158 7.746 7.383 7.059 6.768

8.102 7.694 7.335 7.016 6.729

8.049 7.645 7.290 6.974 6.690

7.995 7.594 7.242 6.930 6.650

0.22 0.24 0.25 0.26 0.28 0.30

6.561 6.033 5.797 5.577 5.176 4.818

6.384 5.910 5.695 5.492 5.118 4.780

6.364 5.899 5.687 5.488 5.121 4.789

6.342 5.884 5.677 5.481 5.120 4.793

6.204 5.792 5.601 5.419 5.080 4.768

6.316 5.865 5.661 5.469 5.115 4.794

6.289 5.845 5.645 5.456 5.107 4.792

6.258 5.822 5.625 5.439 5.096 4.787

6.227 5.798 5.604 5.421 5.085 4.780

6.196 5.773 5.582 5.402 5.071 4.772

6.162 5.745 5.557 5.380 5.055 4.760

0.32 0.34 0.35 0.36 0.38 0.40

4.496 4.205 4.069 3.939 3.697 3.476

4.473 4.193 4.061 3.936 3.700 3.484

4.486 4.210 4.081 3.956 3.724 3.509

4.496 4.224 4.096 3.973 3.743 3.531

4.482 4.218 4.093 3.973 3.748 3.540

4.501 4.233 4.107 3.987 3.759 3.550

4.504 4.240 4.116 3.997 3.773 3.566

4.504 4.244 4.122 4.005 3.784 3.579

4.502 4.247 4.126 4.011 3.793 3.590

4.498 4.247 4.128 4.014 3.799 3.600

4.492 4.244 4.128 4.016 3.804 3.607

0.42 0.44 0.45 0.46 0.48 0.50

3.273 3.087 3.000 2.916 2.759 2.614

3.285 3.103 3.017 2.934 2.779 2.636

3.312 3.130 3.045 2.962 2.807 2.664

3.335 3.155 3.069 2.988 2.833 2.690

3.347 3.168 3.084 3.003 2.850 2.709

3.356 3.176 3.092 3.010 2.856 2.714

3.374 3.196 3.112 3.031 2.878 2.736

3.389 3.213 3.130 3.049 2.897 2.756

3.403 3.228 3.146 3.066 2.915 2.775

3.414 3.241 3.160 3.081 2.931 2.791

3.423 3.253 3.172 3.094 2.945 2.807

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.299 2.040 1.823 1.639 1.347 1.125 0.951

2.324 2.065 1.848 1.664 1.369 1.144 0.968

2.351 2.091 1.872 1.686 1.389 1.162 0.984

2.376 2.115 1.895 1.708 1.408 1.179 0.999

2.397 2.137 1.917 1.730 1.428 1.197 1.015

2.400 2.138 1.917 1.729 1.427 1.196 1.014

2.423 2.160 1.939 1.749 1.445 1.212 1.029

2.444 2.181 1.959 1.769 1.463 1.228 1.044

2.463 2.201 1.978 1.788 1.480 1.244 1.058

2.481 2.220 1.997 1.806 1.497 1.260 1.072

2.498 2.237 2.014 1.823 1.513 1.274 1.086

1.10 1.20 1.30 1.40 1.50

0.813 0.702 0.612 0.539 0.478

0.827 0.714 0.623 0.548 0.486

0.841 0.727 0.634 0.557 0.494

0.855 0.739 0.644 0.567 0.502

0.869 0.751 0.655 0.576 0.510

0.869 0.751 0.655 0.576 0.510

0.882 0.763 0.666 0.585 0.519

0.895 0.775 0.676 0.595 0.527

0.908 0.787 0.687 0.604 0.535

0.921 0.798 0.697 0.613 0.544

0.934 0.810 0.708 0.623 0.552

1.60 1.70 1.80 1.90 2.00

0.428 0.386 0.350 0.319 0.292

0.435 0.392 0.355 0.324 0.296

0.442 0.398 0.360 0.328 0.301

0.449 0.404 0.366 0.333 0.305

0.456 0.410 0.371 0.338 0.309

0.456 0.410 0.371 0.338 0.309

0.463 0.416 0.377 0.343 0.313

0.470 0.423 0.382 0.348 0.318

0.478 0.429 0.388 0.353 0.322

0.485 0.436 0.394 0.358 0.327

0.492 0.442 0.399 0.363 0.331

sin =l Ê 1) (A

Element Z Method

278


4.3. ELECTRON DIFFRACTION Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes (A) Tm3 69 *DS

Yb2 70 *DS

Yb3 70 *DS

Lu3 71 *DS

Hf 4 72 *DS

Ta5 73 *DS

W6 74 *DS

Os4 76 *DS

Ir3 77 *DS

Ir4 77 *DS

Pt2 78 *DS

0.00 0.01 0.02 0.03 0.04 0.05

50.67 34.47

36.30 25.49

50.58 34.40

50.50 34.32

64.91 43.35

79.42 52.47

94.00 61.68

65.56 44.00

51.44 35.25

65.65 44.09

37.41 26.60

0.06 0.07 0.08 0.09 0.10

25.67 20.34 16.86 14.46 12.72

19.61 16.03 13.68 12.05 10.86

25.59 20.27 16.79 14.39 12.65

25.52 20.19 16.72 14.32 12.59

31.63 24.54 19.93 16.76 14.47

37.84 28.99 23.24 19.29 16.45

44.11 33.51 26.62 21.89 18.50

32.26 25.17 20.55 17.36 15.06

26.43 21.09 17.60 15.18 13.42

32.36 25.26 20.64 17.45 15.15

20.68 17.09 14.72 13.06 11.84

0.11 0.12 0.13 0.14 0.15

11.42 10.406 9.603 8.950 8.405

9.95 9.243 8.669 8.191 7.788

11.35 10.343 9.542 8.891 8.349

11.29 10.282 9.484 8.835 8.296

12.77 11.463 10.432 9.600 8.918

14.34 12.727 11.460 10.443 9.613

15.98 14.051 12.545 11.341 10.361

13.34 12.019 10.971 10.122 9.421

12.10 11.071 10.248 9.571 9.006

13.43 12.108 11.061 10.211 9.510

10.91 10.170 9.567 9.058 8.623

0.16 0.17 0.18 0.19 0.20

7.943 7.545 7.196 6.888 6.610

7.436 7.128 6.852 6.601 6.372

7.890 7.495 7.150 6.843 6.569

7.839 7.447 7.104 6.801 6.529

8.348 7.863 7.445 7.081 6.760

8.924 8.343 7.847 7.418 7.042

9.550 8.870 8.292 7.795 7.364

8.833 8.330 7.895 7.512 7.172

8.523 8.104 7.734 7.404 7.107

8.919 8.415 7.978 7.594 7.253

8.241 7.902 7.595 7.314 7.055

0.22 0.24 0.25 0.26 0.28 0.30

6.128 5.717 5.532 5.358 5.038 4.748

5.962 5.601 5.434 5.276 4.980 4.708

6.093 5.688 5.505 5.334 5.019 4.734

6.058 5.659 5.479 5.310 5.000 4.720

6.214 5.765 5.566 5.382 5.049 4.754

6.415 5.907 5.687 5.485 5.124 4.809

6.650 6.080 5.836 5.613 5.220 4.882

6.591 6.107 5.893 5.693 5.331 5.009

6.585 6.138 5.936 5.745 5.395 5.078

6.668 6.181 5.965 5.764 5.398 5.072

6.587 6.173 5.981 5.799 5.458 5.145

0.32 0.34 0.35 0.36 0.38 0.40

4.484 4.240 4.126 4.015 3.806 3.612

4.456 4.221 4.110 4.003 3.800 3.609

4.474 4.234 4.122 4.013 3.807 3.616

4.464 4.228 4.117 4.010 3.807 3.618

4.489 4.247 4.134 4.025 3.821 3.632

4.530 4.279 4.162 4.051 3.843 3.650

4.586 4.323 4.201 4.086 3.872 3.675

4.719 4.456 4.333 4.215 3.994 3.789

4.789 4.524 4.400 4.280 4.055 3.845

4.779 4.512 4.387 4.267 4.042 3.834

4.857 4.590 4.464 4.343 4.114 3.900

0.42 0.44 0.45 0.46 0.48 0.50

3.431 3.262 3.182 3.105 2.958 2.820

3.432 3.266 3.187 3.110 2.965 2.829

3.437 3.271 3.191 3.115 2.969 2.833

3.442 3.277 3.199 3.123 2.979 2.844

3.455 3.291 3.213 3.137 2.993 2.859

3.473 3.307 3.229 3.153 3.009 2.875

3.495 3.327 3.248 3.172 3.027 2.892

3.599 3.423 3.340 3.259 3.106 2.963

3.651 3.471 3.385 3.303 3.146 3.000

3.641 3.462 3.377 3.295 3.140 2.995

3.702 3.518 3.430 3.346 3.186 3.036

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.514 2.253 2.031 1.839 1.529 1.289 1.099

2.526 2.267 2.046 1.855 1.544 1.304 1.113

2.528 2.269 2.047 1.855 1.544 1.303 1.112

2.541 2.283 2.061 1.870 1.558 1.317 1.125

2.557 2.299 2.077 1.885 1.572 1.329 1.137

2.573 2.315 2.092 1.900 1.585 1.341 1.148

2.590 2.331 2.108 1.914 1.598 1.353 1.159

2.646 2.375 2.144 1.945 1.623 1.375 1.179

2.675 2.399 2.164 1.962 1.636 1.386 1.189

2.672 2.398 2.164 1.962 1.636 1.386 1.189

2.704 2.423 2.184 1.979 1.649 1.397 1.199

1.10 1.20 1.30 1.40 1.50

0.946 0.821 0.718 0.632 0.560

0.959 0.833 0.728 0.641 0.569

0.958 0.832 0.728 0.641 0.569

0.970 0.843 0.738 0.650 0.577

0.981 0.854 0.748 0.659 0.585

0.992 0.864 0.757 0.668 0.593

1.002 0.873 0.766 0.677 0.601

1.021 0.892 0.784 0.693 0.616

1.031 0.901 0.792 0.701 0.624

1.031 0.901 0.792 0.701 0.624

1.040 0.909 0.800 0.709 0.631

1.60 1.70 1.80 1.90 2.00

0.500 0.449 0.405 0.368 0.336

0.507 0.455 0.411 0.373 0.341

0.507 0.455 0.411 0.373 0.341

0.515 0.462 0.417 0.379 0.345

0.522 0.469 0.423 0.384 0.350

0.529 0.475 0.429 0.389 0.355

0.537 0.482 0.435 0.395 0.360

0.551 0.495 0.447 0.405 0.370

0.558 0.502 0.453 0.411 0.374

0.558 0.502 0.453 0.411 0.374

0.565 0.508 0.459 0.416 0.379

sin =l Ê 1) (A

Element Z Method

279


4. PRODUCTION AND PROPERTIES OF RADIATIONS Ê for electrons for ionized atoms (cont.) Table 4.3.1.2. Atomic scattering amplitudes A Pt4 78 *DS

Au1 79 *DS

Au3 79 *DS

Hg1 80 *DS

Hg2 80 *DS

Tl1 81 *DS

Tl3 81 *DS

Pb2 82 *DS

Pb4 82 *DS

Bi3 83 *DS

Bi5 83 *DS

0.00 0.01 0.02 0.03 0.04 0.05

65.73 44.16

23.52 18.07

51.50 35.32

23.84 18.38

37.35 26.53

24.11 18.65

51.46 35.29

37.98 27.15

65.83 44.27

52.12 35.91

80.28 53.34

0.06 0.07 0.08 0.09 0.10

32.42 25.33 20.71 17.52 15.22

15.05 13.19 11.94 11.04 10.35

26.49 21.15 17.67 15.25 13.50

15.36 13.49 12.23 11.31 10.60

20.63 17.04 14.67 13.03 11.82

15.62 13.74 12.47 11.54 10.83

26.48 21.15 17.67 15.26 13.51

21.23 17.62 15.24 13.57 12.34

32.54 25.45 20.83 17.65 15.35

27.09 21.74 18.23 15.81 14.04

38.69 29.85 24.09 20.13 17.29

0.11 0.12 0.13 0.14 0.15

13.50 12.178 11.130 10.281 9.579

9.80 9.341 8.946 8.599 8.285

12.18 11.153 10.331 9.660 9.097

10.04 9.565 9.156 8.795 8.466

10.89 10.165 9.569 9.072 8.644

10.26 9.775 9.356 8.983 8.645

12.20 11.178 10.362 9.696 9.139

11.40 10.642 10.024 9.500 9.049

13.64 12.316 11.272 10.427 9.730

12.70 11.663 10.827 10.138 9.559

15.17 13.539 12.262 11.234 10.392

0.16 0.17 0.18 0.19 0.20

8.988 8.484 8.047 7.662 7.320

7.997 7.731 7.480 7.243 7.018

8.617 8.201 7.834 7.506 7.211

8.166 7.887 7.624 7.376 7.141

8.271 7.940 7.640 7.367 7.114

8.335 8.045 7.773 7.516 7.272

8.664 8.253 7.890 7.567 7.275

8.653 8.297 7.975 7.679 7.406

9.144 8.644 8.211 7.830 7.492

9.061 8.629 8.245 7.901 7.589

9.690 9.097 8.587 8.144 7.755

0.22 0.24 0.25 0.26 0.28 0.30

6.735 6.246 6.029 5.827 5.459 5.131

6.598 6.210 6.028 5.852 5.519 5.209

6.693 6.247 6.046 5.855 5.505 5.186

6.703 6.301 6.112 5.930 5.587 5.270

6.658 6.253 6.065 5.886 5.550 5.240

6.817 6.400 6.205 6.017 5.664 5.337

6.765 6.326 6.127 5.939 5.591 5.275

6.912 6.472 6.268 6.075 5.714 5.383

6.913 6.429 6.214 6.014 5.647 5.320

7.040 6.566 6.351 6.148 5.773 5.433

7.100 6.564 6.329 6.111 5.721 5.377

0.32 0.34 0.35 0.36 0.38 0.40

4.385 4.565 4.439 4.318 4.090 3.880

4.921 4.652 4.525 4.402 4.170 3.953

4.895 4.627 4.501 4.379 4.149 3.936

4.975 4.702 4.573 4.448 4.212 3.993

4.953 4.686 4.560 4.437 4.206 3.990

5.035 4.756 4.624 4.497 4.257 4.034

4.985 4.718 4.591 4.469 4.238 4.022

5.078 4.797 4.664 4.536 4.295 4.072

5.023 4.751 4.623 4.501 4.269 4.053

5.123 4.838 4.704 4.576 4.333 4.108

5.069 4.790 4.660 4.535 4.300 4.083

0.42 0.44 0.45 0.46 0.48 0.50

3.684 3.502 3.416 3.333 3.176 3.028

3.752 3.564 3.475 3.389 3.225 3.073

3.737 3.552 3.464 3.379 3.218 3.068

3.789 3.599 3.509 3.422 3.258 3.104

3.788 3.600 3.511 3.424 3.260 3.107

3.827 3.635 3.544 3.456 3.290 3.134

3.821 3.633 3.544 3.457 3.293 3.139

3.864 3.671 3.579 3.491 3.323 3.166

3.851 3.663 3.574 3.488 3.323 3.168

3.899 3.705 3.613 3.524 3.355 3.197

3.881 3.692 3.603 3.516 3.351 3.197

0.55 0.60 0.96 0.70 0.80 0.90 1.00

2.700 2.422 2.184 1.980 1.650 1.397 1.199

2.735 2.449 2.206 1.998 1.663 1.408 1.208

2.733 2.448 2.206 1.998 1.663 1.408 1.209

2.762 2.474 2.227 2.017 1.678 1.420 1.218

2.765 2.476 2.229 2.017 1.678 1.420 1.218

2.790 2.498 2.249 2.036 1.693 1.432 1.229

2.795 2.502 2.252 2.038 1.694 1.432 1.228

2.819 2.524 2.272 2.057 1.709 1.445 1.239

2.823 2.528 2.276 2.059 1.710 1.445 1.239

2.847 2.549 2.295 2.077 1.726 1.458 1.250

2.850 2.554 2.300 2.080 1.727 1.458 1.249

1.10 1.20 1.30 1.40 1.50

1.040 0.909 0.800 0.709 0.631

1.048 0.917 0.808 0.717 0.638

1.048 0.917 0.808 0.716 0.638

1.057 0.926 0.816 0.724 0.645

1.057 0.925 0.816 0.724 0.645

1.066 0.934 0.824 0.731 0.652

1.066 0.933 0.824 0.731 0.652

1.075 0.942 0.831 0.738 0.659

1.075 0.941 0.831 0.738 0.659

1.084 0.950 0.838 0.745 0.665

1.084 0.949 0.838 0.745 0.665

1.60 1.70 1.80 1.90 2.00

0.565 0.508 0.459 0.416 0.379

0.572 0.514 0.465 0.422 0.384

0.572 0.514 0.465 0.422 0.384

0.578 0.520 0.470 0.427 0.389

0.578 0.520 0.470 0.427 0.389

0.585 0.526 0.476 0.432 0.394

0.585 0.527 0.476 0.432 0.394

0.591 0.532 0.482 0.438 0.399

0.591 0.533 0.482 0.438 0.399

0.597 0.538 0.487 0.443 0.404

0.597 0.538 0.487 0.443 0.404

sin =l Ê 1) (A

Element Z Method

280


4.3. ELECTRON DIFFRACTION Ê for electrons for Table 4.3.1.2. Atomic scattering amplitudes (A) ionized atoms (cont.) Element Ra2 Z 88 Method *DS sin =l Ê 1) (A

Ac3 89 *DS

U3 92 *DS

U4 92 *DS

U6 92 *DS

0.00 0.01 0.02 0.03 0.04 0.05

40.04 29.19

54.00 37.78

54.02 37.81

68.15 46.56

96.83 64.49

0.06 0.07 0.08 0.09 0.10

23.23 19.57 17.14 15.42 14.12

28.91 23.53 19.98 17.51 15.70

28.95 23.57 20.03 17.57 15.76

34.80 27.67 23.01 19.78 17.44

46.89 36.26 29.33 24.56 21.12

0.11 0.12 0.13 0.14 0.15

13.11 12.291 11.602 11.008 10.486

14.31 13.217 12.324 11.577 10.939

14.39 13.300 12.416 11.679 11.050

15.67 14.296 13.192 12.287 11.528

18.55 16.573 15.010 13.749 12.709

0.16 0.17 0.18 0.19 0.20

10.018 9.592 9.200 8.836 8.495

10.382 9.889 9.446 9.042 8.671

10.503 10.018 9.583 9.188 8.824

10.879 10.314 9.816 9.371 8.967

11.837 11.093 10.451 9.889 9.391

0.22 0.24 0.25 0.26 0.28 0.30

7.873 7.315 7.057 6.811 6.355 5.940

8.008 7.427 7.161 6.909 6.444 6.022

8.174 7.602 7.340 7.091 6.629 6.208

8.261 7.655 7.380 7.122 6.647 6.219

8.544 7.843 7.534 7.247 6.729 6.273

0.32 0.34 0.35 0.36 0.38 0.40

5.563 5.219 5.059 4.906 4.621 4.360

5.639 5.291 5.128 4.973 4.683 4.417

5.824 5.472 5.307 5.149 4.853 4.580

5.830 5.475 5.309 5.151 4.853 4.580

5.865 5.497 5.327 5.164 4.861 4.584

0.42 0.44 0.45 0.46 0.48 0.50

4.122 3.904 3.801 3.703 3.518 3.348

4.174 3.951 3.847 3.747 3.558 3.385

4.329 4.098 3.989 3.885 3.688 3.506

4.328 4.097 3.988 3.883 3.686 3.504

4.330 4.096 3.987 3.881 3.683 3.500

0.55 0.60 0.65 0.70 0.80 0.90 1.00

2.975 2.664 2.400 2.174 1.808 1.527 1.307

3.005 2.689 2.421 2.193 1.824 1.541 1.319

3.107 2.776 2.496 2.258 1.875 1.583 1.354

3.106 2.774 2.494 2.256 1.874 1.582 1.354

3.100 2.768 2.489 2.252 1.872 1.582 1.354

1.10 1.20 1.30 1.40 1.50

1.132 0.991 0.874 0.778 0.696

1.142 0.999 0.882 0.784 0.702

1.171 1.024 0.904 0.804 0.720

1.172 1.025 0.904 0.804 0.720

1.172 1.025 0.905 0.804 0.720

1.60 1.70 1.80 1.90 2.00

0.626 0.566 0.513 0.467 0.427

0.632 0.571 0.518 0.472 0.431

0.648 0.586 0.533 0.486 0.444

0.648 0.586 0.533 0.486 0.444

0.648 0.586 0.533 0.486 0.444

E (keV)

281


Table 4.3.2.1. Parameters useful in electron diffraction as a function of accelerating voltage, E l

1=l

m=m0

v=c

1 2 3 4 5

0.387629 0.273961 0.223579 0.193530 0.173015

2.57979 3.65016 4.47270 5.16715 5.77986

1.00196 1.00391 1.00587 1.00783 1.00978

0.06247 0.08821 0.10788 0.12439 0.13887

0.0081126 0.0057448 0.0046975 0.0040741 0.0036493

6 7 8 9 10

0.157863 0.146082 0.136581 0.128707 0.122043

6.33460 6.84548 7.32168 7.76958 8.19383

1.01174 1.01370 1.01566 1.01761 1.01957

0.15191 0.16384 0.17490 0.18524 0.19498

0.0033361 0.0030931 0.0028975 0.0027358 0.0025991

15 20 25 30 35

0.099407 0.085882 0.076632 0.069789 0.064459

10.05963 11.64383 13.04940 14.32899 15.51381

1.02935 1.03914 1.04892 1.05871 1.06849

0.23711 0.27186 0.30184 0.32837 0.35227

0.0021374 0.0018641 0.0016790 0.0015433 0.0014386

40 45 50 55 60

0.060153 0.056580 0.053551 0.050941 0.048659

16.62414 17.67403 18.67366 19.63072 20.55115

1.07828 1.08806 1.09784 1.10763 1.11741

0.37406 0.39410 0.41268 0.43000 0.44622

0.0013548 0.0012859 0.0012280 0.0011786 0.0011357

65 70 75 80 85

0.046642 0.044843 0.043223 0.041756 0.040418

21.43968 22.30012 23.13560 23.94874 24.74173

1.12720 1.13698 1.14677 1.15655 1.16634

0.46147 0.47586 0.48948 0.50239 0.51467

0.0010982 0.0010650 0.0010354 0.0010087 0.0009847

90 95 100 120 140

0.039190 0.038060 0.037013 0.033491 0.030739

25.51646 26.27454 27.01738 29.85866 32.53222

1.17612 1.18591 1.19569 1.23483 1.27397

0.52637 0.53754 0.54822 0.58667 0.61956

0.0009628 0.0009428 0.0009244 0.0008638 0.0008180

160 180 200 250 300

0.028509 0.026654 0.025079 0.021986 0.019687

35.07642 37.51759 39.87466 45.48412 50.79517

1.31310 1.35224 1.39138 1.48922 1.58707

0.64810 0.67314 0.69531 0.74101 0.77652

0.0007820 0.0007529 0.0007289 0.0006839 0.0006526

350 400 450 500 550

0.017891 0.016439 0.015233 0.014212 0.013334

55.89295 60.83109 65.64563 70.36195 74.99858

1.68491 1.78276 1.88060 1.97845 2.07629

0.80483 0.82786 0.84691 0.86286 0.87638

0.0006297 0.0006122 0.0005984 0.0005873 0.0005783

600 650 700 750 800

0.012568 0.011893 0.011292 0.010755 0.010269

79.56945 84.08529 88.55452 92.98385 97.37874

2.17414 2.27198 2.36983 2.46767 2.56552

0.88794 0.89793 0.90661 0.91421 0.92091

0.0005707 0.0005644 0.0005590 0.0005543 0.0005503

850 900 950 1000 1100

0.009829 0.009427 0.009058 0.008719 0.008115

101.74364 106.08226 110.39769 114.69256 123.22919

2.66336 2.76121 2.85905 2.95690 3.15259

0.92684 0.93212 0.93684 0.94108 0.94836

0.0005468 0.0005437 0.0005410 0.0005385 0.0005344

1200 1300 1400 1500

0.007593 0.007136 0.006733 0.006374

131.70646 140.13516 148.52355 156.87810

3.34828 3.54397 3.73966 3.93535

0.95436 0.95936 0.96358 0.96718

0.0005310 0.0005282 0.0005259 0.0005240

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.2.2. Elastic atomic scattering factors of electrons for neutral atoms and s up to 2.0 AÊ

1

Element

Z

a1

a2

a3

a4

a5

b1

b2

b3

b4

H He Li Be B

1 2 3 4 5

0.0349 0.0317 0.0750 0.0780 0.0909

0.1201 0.0838 0.2249 0.2210 0.2551

0.1970 0.1526 0.5548 0.6740 0.7738

0.0573 0.1334 1.4954 1.3867 1.2136

0.1195 0.0164 0.9354 0.6925 0.4606

0.5347 0.2507 0.3864 0.3131 0.2995

3.5867 1.4751 2.9383 2.2381 2.1155

12.3471 4.4938 15.3829 10.1517 8.3816

18.9525 12.6646 53.5545 30.9061 24.1292

38.6269 31.1653 138.7337 78.3273 63.1314

C N O F Ne

6 7 8 9 10

0.0893 0.1022 0.0974 0.1083 0.1269

0.2563 0.3219 0.2921 0.3175 0.3535

0.7570 0.7982 0.6910 0.6487 0.5582

1.0487 0.8197 0.6990 0.5846 0.4674

0.3575 0.1715 0.2039 0.1421 0.1460

0.2465 0.2451 0.2067 0.2057 0.2200

1.7100 1.7481 1.3815 1.3439 1.3779

6.4094 6.1925 4.6943 4.2788 4.0203

18.6113 17.3894 12.7105 11.3932 9.4934

50.2523 48.1431 32.4726 28.7881 23.1278

Na Mg Al Si P

11 12 13 14 15

0.2142 0.2314 0.2390 0.2519 0.2548

0.6853 0.6866 0.6573 0.6372 0.6106

0.7692 0.9677 1.2011 1.3795 1.4541

1.6589 2.1882 2.5586 2.5082 2.3204

1.4482 1.1339 1.2312 1.0500 0.8477

0.3334 0.3278 0.3138 0.3075 0.2908

2.3446 2.2720 2.1063 2.0174 1.8740

10.0830 10.9241 10.4163 9.6746 8.5176

48.3037 39.2898 34.4552 29.3744 24.3434

138.2700 101.9748 98.5344 80.4732 63.2996

S Cl Ar K Ca

16 17 18 19 20

0.2497 0.2443 0.2385 0.4115 0.4054

0.5628 0.5397 0.5017 1.4031 1.3880

1.3899 1.3919 1.3428 2.2784 2.1602

2.1865 2.0197 1.8899 2.6742 3.7532

0.7715 0.6621 0.6079 2.2162 2.2063

0.2681 0.2468 0.2289 0.3703 0.3499

1.6711 1.5242 1.3694 3.3874 3.0991

7.0267 6.1537 5.2561 13.1029 11.9608

19.5377 16.6687 14.0928 68.9592 53.9353

50.3888 42.3086 35.5361 194.4329 142.3892

Sc Ti V Cr Mn

21 22 23 24 25

0.3787 0.3825 0.3876 0.4046 0.3796

1.2181 1.2598 1.2750 1.3696 1.2094

2.0594 2.0008 1.9109 1.8941 1.7815

3.2618 3.0617 2.8314 2.0800 2.5420

2.3870 2.0694 1.8979 1.2196 1.5937

0.3133 0.3040 0.2967 0.2986 0.2699

2.5856 2.4863 2.3780 2.3958 2.0455

9.5813 9.2783 8.7981 9.1406 7.4726

41.7688 39.0751 35.9528 37.4701 31.0604

116.7282 109.4583 101.7201 113.7121 91.5622

Fe Co Ni Cu Zn

26 27 28 29 30

0.3946 0.4118 0.3860 0.4314 0.4288

1.2725 1.3161 1.1765 1.3208 1.2646

1.7031 1.6493 1.5451 1.5236 1.4472

2.3140 2.1930 2.0730 1.4671 1.8294

1.4795 1.2830 1.3814 0.8562 1.0934

0.2717 0.2742 0.2478 0.2694 0.2593

2.0443 2.0372 1.7660 1.9223 1.7998

7.6007 7.7205 6.3107 7.3474 6.7500

29.9714 29.9680 25.2204 28.9892 25.5860

86.2265 84.9383 74.3146 90.6246 73.5284

Ga Ge As Se Br

31 32 33 34 35

0.4818 0.4655 0.4517 0.4477 0.4798

1.4032 1.3014 1.2229 1.1678 1.1948

1.6561 1.6088 1.5852 1.5843 1.8695

2.4605 2.6998 2.7958 2.8087 2.6953

1.1054 1.3003 1.2638 1.1956 0.8203

0.2825 0.2647 0.2493 0.2405 0.2504

1.9785 1.7926 1.6436 1.5442 1.5963

8.7546 7.6071 6.8154 6.3231 6.9653

32.5238 26.5541 22.3681 19.4610 19.8492

98.5523 77.5238 62.0390 52.0233 50.3233

Kr Rb Sr Y Zr

36 37 38 39 40

0.4546 1.0160 0.6703 0.6894 0.6719

1.0993 2.8528 1.4926 1.5474 1.4684

1.7696 3.5466 3.3368 3.2450 3.1668

2.7068 -7.7804 4.4600 4.2126 3.9557

0.8672 12.1148 3.1501 2.9764 2.8920

0.2309 0.4853 0.3190 0.3189 0.3036

1.4279 5.0925 2.2287 2.2904 2.1249

5.9449 25.7851 10.3504 10.0062 8.9236

16.6752 130.4515 52.3291 44.0771 36.8458

42.2243 138.6775 151.2216 125.0120 108.2049

Nb Mo Tc Ru Rh

41 42 43 44 45

0.6123 0.6773 0.7082 0.6735 0.6413

1.2677 1.4798 1.6392 1.4934 1.3690

3.0348 3.1788 3.1993 3.0966 2.9854

3.3841 3.0824 3.4327 2.7254 2.6952

2.3683 1.8384 1.8711 1.5597 1.5433

0.2709 0.2920 0.2976 0.2773 0.2580

1.7683 2.0606 2.2106 1.9716 1.7721

7.2489 8.1129 8.5246 7.3249 6.3854

27.9465 30.5336 33.1456 26.6891 23.2549

98.5624 100.0658 96.6377 90.5581 85.1517

Pd Ag Cd In

46 47 48 49

0.5904 0.6377 0.6364 0.6768

1.1775 1.3790 1.4247 1.6589

2.6519 2.8294 2.7802 2.7740

2.2875 2.3631 2.5973 3.1835

0.8689 1.4553 1.7886 2.1326

0.2324 0.2466 0.2407 0.2522

1.5019 1.6974 1.6823 1.8545

5.1591 5.7656 5.6588 6.2936

15.5428 20.0943 20.7219 25.1457

46.8213 76.7372 69.1109 84.5448

282


b5

4.3. ELECTRON DIFFRACTION Table 4.3.2.2. Elastic atomic scattering factors of electrons for neutral atoms and s up to 2.0 AÊ Element

1

(cont.)

Z

a1

a2

a3

a4

a5

b1

b2

b3

Sn Sb Te I Xe Cs

50 51 52 53 54 55

0.7224 0.7106 0.6947 0.7047 0.6737 1.2704

1.9610 1.9247 1.8690 1.9484 1.7908 3.8018

2.7161 2.6149 2.5356 2.5940 2.4129 5.6618

3.5603 3.8322 4.0013 4.1526 4.2100 0.9205

1.8972 1.8899 1.8955 1.5057 1.7058 4.8105

0.2651 0.2562 0.2459 0.2455 0.2305 0.4356

2.0604 1.9646 1.8542 1.8638 1.6890 4.2058

7.3011 6.8852 6.4411 6.7639 5.8218 23.4342

27.5493 24.7648 22.1730 21.8007 18.3928 136.7783

81.3349 68.9168 59.2206 56.4395 47.2496 171.7561

Ba La Ce Pr Nd

56 57 58 59 60

0.9049 0.8405 0.8551 0.9096 0.8807

2.6076 2.3863 2.3915 2.5313 2.4183

4.8498 4.6139 4.5772 4.5266 4.4448

5.1603 5.1514 5.0278 4.6376 4.6858

4.7388 4.7949 4.5118 4.3690 4.1725

0.3066 0.2791 0.2805 0.2939 0.2802

2.4363 2.1410 2.1200 2.2471 2.0836

12.1821 10.3400 10.1808 10.8266 10.0357

54.6135 41.9148 42.0633 48.8842 47.4506

161.9978 132.0204 130.9893 147.6020 146.9976

Pm Sm Eu Gd Tb

61 62 63 64 65

0.9471 0.9699 0.8694 0.9673 0.9325

2.5463 2.5837 2.2413 2.4702 2.3673

4.3523 4.2778 3.9196 4.1148 3.8791

4.4789 4.4575 3.9694 4.4972 3.9674

3.9080 3.5985 4.5498 3.2099 3.7996

0.2977 0.3003 0.2653 0.2909 0.2761

2.2276 2.2447 1.8590 2.1014 1.9511

10.5762 10.6487 8.3998 9.7067 8.9296

49.3619 50.7994 36.7397 43.4270 41.5937

145.3580 146.4179 125.7089 125.9474 131.0122

Dy Ho Er Tm Yb

66 67 68 69 70

0.9505 0.9248 1.0373 1.0075 1.0347

2.3705 2.2428 2.4824 2.3787 2.3911

3.8218 3.6182 3.6558 3.5440 3.4619

4.0471 3.7910 3.8925 3.6932 3.6556

3.4451 3.7912 3.0056 3.1759 3.0052

0.2773 0.2660 0.2944 0.2816 0.2855

1.9469 1.8183 2.0797 1.9486 1.9679

8.8862 7.9655 9.4156 8.7162 8.7619

43.0938 33.1129 45.8056 41.8420 42.3304

133.1396 101.8139 132.7720 125.0320 125.6499

Lu Hf Ta W Re

71 72 73 74 75

0.9927 1.0295 1.0190 0.9853 0.9914

2.2436 2.2911 2.2291 2.1167 2.0858

3.3554 3.4110 3.4097 3.3570 3.4531

3.7813 3.9497 3.9252 3.7981 3.8812

3.0994 2.4925 2.2679 2.2798 1.8526

0.2701 0.2761 0.2694 0.2569 0.2548

1.8073 1.8625 1.7962 1.6745 1.6518

7.8112 8.0961 7.6944 7.0098 6.8845

34.4849 34.2712 31.0942 26.9234 26.7234

103.3526 98.5295 91.1089 81.3910 81.7215

Os Ir Pt Au Hg

76 77 78 79 80

0.9813 1.0194 0.9148 0.9674 1.0033

2.0322 2.0645 1.8096 1.8916 1.9469

3.3665 3.4425 3.2134 3.3993 3.4396

3.6235 3.4914 3.2953 3.0524 3.1548

1.9741 1.6976 1.5754 1.2607 1.4180

0.2487 0.2554 0.2263 0.2358 0.2413

1.5973 1.6475 1.3813 1.4712 1.5298

6.4737 6.5966 5.3243 5.6758 5.8009

23.2817 23.2269 17.5987 18.7119 19.4520

70.9254 70.0272 60.0171 61.5286 60.5753

Tl Pb Bi Po At

81 82 83 84 85

1.0689 1.0891 1.1007 1.1568 1.0909

2.1038 2.1867 2.2306 2.4353 2.1976

3.6039 3.6160 3.5689 3.6459 3.3831

3.4927 3.8031 4.1549 4.4064 4.6700

1.8283 1.8994 2.0382 1.7179 2.1277

0.2540 0.2552 0.2546 0.2648 0.2466

1.6715 1.7174 1.7351 1.8786 1.6707

6.3509 6.5131 6.4948 7.1749 6.0197

23.1531 23.9170 23.6464 25.1766 20.7657

78.7099 74.7039 70.3780 69.2821 57.2663

Rn Fr Ra Ac Th

86 87 88 89 90

1.0756 1.4282 1.3127 1.3128 1.2553

2.1630 3.5081 3.1243 3.1021 2.9178

3.3178 5.6767 5.2988 5.3385 5.0862

4.8852 4.1964 5.3891 5.9611 6.1206

2.0489 3.8946 5.4133 4.7562 4.7122

0.2402 0.3183 0.2887 0.2861 0.2701

1.6169 2.6889 2.2897 2.2509 2.0636

5.7644 13.4816 10.8276 10.5287 9.3051

19.4568 54.3866 43.5389 41.7796 34.5977

52.5009 200.8321 145.6109 128.2973 107.9200

Pa U Np Pu Am

91 92 93 94 95

1.3218 1.3382 1.5193 1.3517 1.2135

3.1444 3.2043 4.0053 3.2937 2.7962

5.4371 5.4558 6.5327 5.3213 4.7545

5.6444 5.4839 -.1402 4.6466 4.5731

4.0107 3.6342 6.7489 3.5714 4.4786

0.2827 0.2838 0.3213 0.2813 0.2483

2.2250 2.2452 2.8206 2.2418 1.8437

10.2454 10.2519 14.8878 9.9952 7.5421

41.1162 41.7251 68.9103 42.7939 29.3841

124.4449 124.9023 81.7257 132.1739 112.4579

Cm Bk Cf

96 97 98

1.2937 1.2915 1.2089

3.1100 3.1023 2.7391

5.0393 4.9309 4.3482

4.7546 4.6009 4.0047

3.5031 3.4661 4.6497

0.2638 0.2611 0.2421

2.0341 2.0023 1.7487

8.7101 8.4377 6.7262

35.2992 34.1559 23.2153

109.4972 105.8911 80.3108

283


b4

b5

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.2.3. Elastic atomic scattering factors of electrons for neutral atoms and s up to 6.0 AÊ

1

Element

Z

a1

a2

a3

a4

a5

b1

b2

b3

b4

H He Li Be B

1 2 3 4 5

0.0088 0.0084 0.0478 0.0423 0.0436

0.0449 0.0443 0.2048 0.1874 0.1898

0.1481 0.1314 0.5253 0.6019 0.6788

0.2356 0.1671 1.5225 1.4311 1.3273

0.0914 0.0666 0.9853 0.7891 0.5544

0.1152 0.0596 0.2258 0.1445 0.1207

1.0867 0.5360 2.1032 1.4180 1.1595

4.9755 2.4274 12.9349 8.1165 6.2474

16.5591 7.7852 50.7501 27.9705 21.0460

43.2743 20.3126 136.6280 74.8684 59.3619

C N O F Ne

6 7 8 9 10

0.0489 0.0267 0.0365 0.0382 0.0380

0.2091 0.1328 0.1729 0.1822 0.1785

0.7537 0.5301 0.5805 0.5972 0.5494

1.1420 1.1020 0.8814 0.7707 0.6942

0.3555 0.4215 0.3121 0.2130 0.1918

0.1140 0.0541 0.0652 0.0613 0.0554

1.0825 0.5165 0.6184 0.5753 0.5087

5.4281 2.8207 2.9449 2.6858 2.2639

17.8811 10.6297 9.6298 8.8214 7.3316

51.1341 34.3764 28.2194 25.6668 21.6912

Na Mg Al Si P

11 12 13 14 15

0.1260 0.1130 0.1165 0.0567 0.1005

0.6442 0.5575 0.5504 0.3365 0.4615

0.8893 0.9046 1.0179 0.8104 1.0663

1.8197 2.1580 2.6295 2.4960 2.5854

1.2988 1.4735 1.5711 2.1186 1.2725

0.1684 0.1356 0.1295 0.0582 0.0977

1.7150 1.3579 1.2619 0.6155 0.9084

8.8386 6.9255 6.8242 3.2522 4.9654

50.8265 32.3165 28.4577 16.7929 18.5471

147.2073 92.1138 88.4750 57.6767 54.3648

S Cl Ar K Ca

16 17 18 19 20

0.0915 0.0799 0.1044 0.2149 0.2355

0.4312 0.3891 0.4551 0.8703 0.9916

1.0847 1.0037 1.4232 2.4999 2.3959

2.4671 2.3332 2.1533 2.3591 3.7252

1.0852 1.0507 0.4459 3.0318 2.5647

0.0838 0.0694 0.0853 0.1660 0.1742

0.7788 0.6443 0.7701 1.6906 1.8329

4.3462 3.5351 4.4684 8.7447 8.8407

15.5846 12.5058 14.5864 46.7825 47.4583

44.6365 35.8633 41.2474 165.6923 134.9613

Sc Ti V Cr Mn

21 22 23 24 25

0.4636 0.2123 0.2369 0.1970 0.1943

2.0802 0.8960 1.0774 0.8228 0.8190

2.9003 2.1765 2.1894 2.0200 1.9296

1.4193 3.0436 3.0825 2.1717 2.4968

2.4323 2.4439 1.7190 1.7516 2.0625

0.3682 0.1399 0.1505 0.1197 0.1135

4.0312 1.4568 1.6392 1.1985 1.1313

22.6493 6.7534 7.5691 5.4097 5.0341

71.8200 33.1168 36.8741 25.2361 24.1798

103.3691 101.8238 107.8517 94.4290 80.5598

Fe Co Ni Cu Zn

26 27 28 29 30

0.1929 0.2186 0.2313 0.3501 0.1780

0.8239 0.9861 1.0657 1.6558 0.8096

1.8689 1.8540 1.8229 1.9582 1.6744

2.3694 2.3258 2.2609 0.2134 1.9499

1.9060 1.4685 1.1883 1.4109 1.4495

0.1087 0.1182 0.1210 0.1867 0.0876

1.0806 1.2300 1.2691 1.9917 0.8650

4.7637 5.4177 5.6870 11.3396 3.8612

22.8500 25.7602 27.0917 53.2619 18.8726

76.7309 80.8542 83.0285 63.2520 64.7016

Ga Ge As Se Br

31 32 33 34 35

0.2135 0.2135 0.2059 0.1574 0.1899

0.9768 0.9761 0.9518 0.7614 0.8983

1.6669 1.6555 1.6372 1.4834 1.6358

2.5662 2.8938 3.0490 3.0016 3.1845

1.6790 1.6356 1.4756 1.7978 1.1518

0.1020 0.0989 0.0926 0.0686 0.0810

1.0219 0.9845 0.9182 0.6808 0.7957

4.6275 4.5527 4.3291 3.1163 3.9054

22.8742 21.5563 19.2996 14.3458 15.7701

80.1535 70.3903 58.9329 44.0455 45.6124

Kr Rb Sr Y Zr

36 37 38 39 40

0.1742 0.3781 0.3723 0.3234 0.2997

0.8447 1.4904 1.4598 1.2737 1.1879

1.5944 3.5753 3.5124 3.2115 3.1075

3.1507 3.0031 4.4612 4.0563 3.9740

1.1338 3.3272 3.3031 3.7962 3.5769

0.0723 0.1557 0.1480 0.1244 0.1121

0.7123 1.5347 1.4643 1.1948 1.0638

3.5192 9.9947 9.2320 7.2756 6.3891

13.7724 51.4251 49.8807 34.1430 28.7081

39.1148 185.9828 148.0937 111.2079 97.4289

Nb Mo Tc Ru Rh

41 42 43 44 45

0.1680 0.3069 0.2928 0.2604 0.2713

0.9370 1.1714 1.1267 1.0442 1.0556

2.7300 3.2293 3.1675 3.0761 3.1416

3.8150 3.4254 3.6619 3.2175 3.0451

3.0053 2.1224 2.5942 1.9448 1.7179

0.0597 0.1101 0.1020 0.0887 0.0907

0.6524 1.0222 0.9481 0.8240 0.8324

4.4317 5.9613 5.4713 4.8278 4.7702

19.5540 25.1965 23.8153 19.8977 19.7862

85.5011 93.5831 82.8991 80.4566 80.2540

Pd Ag Cd In

46 47 48 49

0.2003 0.2739 0.3072 0.3564

0.8779 1.0503 1.1303 1.3011

2.6135 3.1564 3.2046 3.2424

2.8594 2.7543 2.9329 3.4839

1.0258 1.4328 1.6560 2.0459

0.0659 0.0881 0.0966 0.1091

0.6111 0.8028 0.8856 1.0452

3.5563 4.4451 4.6273 5.0900

12.7638 18.7011 20.6789 24.6578

44.4283 79.2633 73.4723 88.0513

284


b5

4.3. ELECTRON DIFFRACTION Table 4.3.2.3. Elastic atomic scattering factors of electrons for neutral atoms and s up to 6.0 AÊ Element

1

(cont.)

Z

a1

a2

a3

a4

a5

b1

b2

b3

b4

Sn Sb Te I Xe Cs

50 51 52 53 54 55

0.2966 0.2725 0.2422 0.2617 0.2334 0.5713

1.1157 1.0651 0.9692 1.0325 0.9496 2.4866

3.0973 2.9940 2.8114 2.8097 2.6381 4.9795

3.8156 4.0697 4.1509 4.4809 4.4680 4.0198

2.5281 2.5682 2.8161 2.3190 2.5020 4.4403

0.0896 0.0809 0.0708 0.0749 0.0655 0.1626

0.8268 0.7488 0.6472 0.6914 0.6050 1.8213

4.2242 3.8710 3.3609 3.4634 3.0389 11.1049

20.6900 18.8800 16.0752 16.3603 14.0809 49.0568

71.3399 60.6499 50.1724 48.2522 41.0005 202.9987

Ba La Ce Pr Nd

56 57 58 59 60

0.5229 0.5461 0.2227 0.5237 0.5368

2.2874 2.3856 1.0760 2.2913 2.3301

4.7243 5.0653 2.9482 4.6161 4.6058

5.0807 5.7601 5.8496 4.7233 4.6621

5.6389 4.0463 7.1834 4.8173 4.4622

0.1434 0.1479 0.0571 0.1360 0.1378

1.6019 1.6552 0.5946 1.5068 1.5140

9.4511 10.0059 3.2022 8.8213 8.8719

42.7685 47.3245 16.4253 41.9536 43.5967

148.4969 145.8464 95.7030 141.2424 141.8065

Pm Sm Eu Gd Tb

61 62 63 64 65

0.5232 0.5162 0.5272 0.9664 0.5110

2.2627 2.2302 2.2844 3.4052 2.1570

4.4552 4.3449 4.3361 5.0803 4.0308

4.4787 4.3598 4.3178 1.4991 3.9936

4.5073 4.4292 4.0908 4.2528 4.2466

0.1317 0.1279 0.1285 0.2641 0.1210

1.4336 1.3811 1.3943 2.6586 1.2704

8.3087 7.9629 8.1081 16.2213 7.1368

40.6010 39.1213 40.9631 80.2060 35.0354

135.9196 132.7846 134.1233 92.5359 123.5062

Dy Ho Er Tm Yb

66 67 68 69 70

0.4974 0.4679 0.5034 0.4839 0.5221

2.1097 1.9693 2.1088 2.0262 2.1695

3.8906 3.7191 3.8232 3.6851 3.7567

3.8100 3.9632 3.7299 3.5874 3.6685

4.3084 4.2432 3.8963 4.0037 3.4274

0.1157 0.1069 0.1141 0.1081 0.1148

1.2108 1.0994 1.1769 1.1012 1.1860

6.7377 5.9769 6.6087 6.1114 6.7520

32.4150 27.1491 33.4332 30.3728 35.6807

116.9225 96.3119 116.4913 110.5988 118.0692

Lu Hf Ta W Re

71 72 73 74 75

0.4680 0.4048 0.3835 0.3661 0.3933

1.9466 1.7370 1.6747 1.6191 1.6973

3.5428 3.3399 3.2986 3.2455 3.4202

3.8490 3.9448 4.0462 4.0856 4.1274

3.6594 3.7293 3.4303 3.2064 2.6158

0.1015 0.0868 0.0810 0.0761 0.0806

1.0195 0.8585 0.8020 0.7543 0.7972

5.6058 4.6378 4.3545 4.0952 4.4237

27.4899 21.6900 19.9644 18.2886 19.5692

95.2846 80.2408 73.6337 68.0967 68.7477

Os Ir Pt Au Hg

76 77 78 79 80

0.3854 0.3510 0.3083 0.3055 0.3593

1.6555 1.5620 1.4158 1.3945 1.5736

3.4129 3.2946 2.9662 2.9617 3.5237

4.1111 4.0615 3.9349 3.8990 3.8109

2.4106 2.4382 2.1709 2.0026 1.6953

0.0787 0.0706 0.0609 0.0596 0.0694

0.7638 0.6904 0.5993 0.5827 0.6758

4.2441 3.8266 3.1921 3.1035 3.8457

18.3700 16.0812 12.5285 11.9693 15.6203

65.1071 58.7638 49.7675 47.9106 56.6614

Tl Pb Bi Po At

81 82 83 84 85

0.3511 0.3540 0.3530 0.3673 0.3547

1.5489 1.5453 1.5258 1.5772 1.5206

3.5676 3.5975 3.5815 3.7079 3.5621

4.0900 4.3152 4.5532 4.8582 5.0184

2.5251 2.7743 3.0714 2.8440 3.0075

0.0672 0.0668 0.0661 0.0678 0.0649

0.6522 0.6465 0.6324 0.6527 0.6188

3.7420 3.6968 3.5906 3.7396 3.4696

15.9791 16.2056 15.9962 17.0668 15.6090

65.1354 61.4909 57.5760 55.9789 49.4818

Rn Fr Ra Ac Th

86 87 88 89 90

0.4586 0.8282 1.4129 0.7169 0.6958

1.7781 2.9941 4.4269 2.5710 2.4936

3.9877 5.6597 7.0460 5.1791 5.1269

5.7273 4.9292 -1.0573 6.3484 6.6988

1.5460 4.2889 8.6430 5.6474 5.0799

0.0831 0.1515 0.2921 0.1263 0.1211

0.7840 1.6163 3.1381 1.2900 1.2247

4.3599 9.7752 19.6767 7.3686 6.9398

20.0128 42.8480 102.0436 32.4490 30.0991

62.1535 190.7366 113.9798 118.0558 105.1960

Pa U Np Pu Am

91 92 93 94 95

1.2502 0.6410 0.6938 0.6902 0.7577

4.2284 2.2643 2.4652 2.4509 2.7264

7.0489 4.8713 5.1227 5.1284 5.4184

1.1390 5.9287 5.5965 5.0339 4.8198

5.8222 5.3935 4.8543 4.8575 4.1013

0.2415 0.1097 0.1171 0.1153 0.1257

2.6442 1.0644 1.1757 1.1545 1.3044

16.3313 5.7907 6.4053 6.2291 7.1035

73.5757 25.0261 27.5217 27.0741 32.4649

91.9401 101.3899 103.0482 111.3150 118.8647

Cm Bk Cf

96 97 98

0.7567 0.7492 0.8100

2.7565 2.7267 3.0001

5.4364 5.3521 5.4635

5.1918 5.0369 4.1756

3.5643 3.5321 3.5066

0.1239 0.1217 0.1310

1.2979 1.2651 1.4038

7.0798 6.8101 7.6057

32.7871 31.6088 34.0186

110.1512 106.4853 90.5226

285


b5

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms H; Z 1 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj 5.3956E 4.8779E 3.7546E 2.6709E 1.8733E 1.3383E 9.8468E 7.4696E 5.8260E 4.6554E 3.7970E 3.1489E 2.6551E 2.2685E 1.9591E 1.7086E 1.5030E 1.3322E 1.1889E 1.0674E 9.6365E 8.7427E 7.9675E 7.2908E 6.6968E 6.1724E 5.7072E 5.2927E 4.9217E 4.5884E 4.2878E 4.0157E 3.7688E 3.5440E 3.3387E 3.1507E 2.9781E 2.8194E 2.6730E 2.5377E 2.4124E 2.2962E 2.1882E 2.0876E 1.9939E 1.9062E 1.8243E 1.7475E 1.6755E 1.6078E 1.5441E 1.4842E 1.4277E 1.3743E 1.3239E 1.2762E 1.2311E 1.1882E 1.1476E 1.1091E 1.0724E

10 keV 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s

2.1835E 2.3657E 2.8959E 3.7159E 4.7280E 5.8258E 6.9249E 7.9743E 8.9502E 9.8462E 1.0665E 1.1414E 1.2100E 1.2732E 1.3316E 1.3858E 1.4364E 1.4838E 1.5283E 1.5703E 1.6100E 1.6477E 1.6836E 1.7178E 1.7505E 1.7818E 1.8118E 1.8407E 1.8685E 1.8952E 1.9211E 1.9460E 1.9702E 1.9936E 2.0162E 2.0382E 2.0596E 2.0804E 2.1006E 2.1202E 2.1394E 2.1581E 2.1763E 2.1941E 2.2115E 2.2284E 2.2451E 2.2613E 2.2771E 2.2927E 2.3080E 2.3229E 2.3376E 2.3519E 2.3660E 2.3798E 2.3934E 2.4068E 2.4199E 2.4327E 2.4454E

j f sj 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01


40 keV 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


j f sj 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

286



60 keV 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


j f sj 03 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01


90 keV 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


03 03 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) He; Z 2 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj 4.2610E 4.0739E 3.5950E 2.9975E 2.4232E 1.9367E 1.5495E 1.2497E 1.0190E 8.4110E 7.0272E 5.9355E 5.0706E 4.3748E 3.8074E 3.3405E 2.9524E 2.6267E 2.3511E 2.1160E 1.9139E 1.7392E 1.5871E 1.4539E 1.3366E 1.2329E 1.1407E 1.0585E 9.8473E 9.1842E 8.5856E 8.0433E 7.5508E 7.1020E 6.6919E 6.3164E 5.9714E 5.6538E 5.3611E 5.0903E 4.8395E 4.6069E 4.3905E 4.1891E 4.0012E 3.8256E 3.6614E 3.5074E 3.3630E 3.2273E 3.0997E 2.9795E 2.8661E 2.7591E 2.6580E 2.5623E 2.4717E 2.3858E 2.3044E 2.2271E 2.1533E

10 keV 01 01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


j f sj 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01


40 keV 01 01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


j f sj 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

287



60 keV 01 01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


j f sj 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01


90 keV 01 01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

s


02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Li; Z 3 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3.3440E+00 2.2942E+00 1.0905E+00 5.7692E 01 3.6834E 01 2.6569E 01 2.0504E 01 1.6450E 01 1.3523E 01 1.1308E 01 9.5794E 02 8.2007E 02 7.0897E 02 6.1809E 02 5.4277E 02 4.7985E 02 4.2685E 02 3.8184E 02 3.4337E 02 3.1025E 02 2.8158E 02 2.5660E 02 2.3473E 02 2.1549E 02 1.9847E 02 1.8336E 02 1.6988E 02 1.5782E 02 1.4698E 02 1.3720E 02 1.2836E 02 1.2034E 02 1.1304E 02 1.0638E 02 1.0029E 02 9.4702E 03 8.9566E 03 8.4834E 03 8.0465E 03 7.6424E 03 7.2678E 03 6.9200E 03 6.5965E 03 6.2950E 03 6.0137E 03 5.7508E 03 5.5047E 03 5.2740E 03 5.0574E 03 4.8540E 03 4.6625E 03 4.4821E 03 4.3120E 03 4.1513E 03 3.9995E 03 3.8558E 03 3.7197E 03 3.5907E 03 3.4683E 03 3.3520E 03 3.2415E 03

s


j f sj 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

40 keV


s


j f sj 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

288


60 keV


s


j f sj 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Be; Z 4 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


s


j f sj 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

40 keV


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

289


60 keV


s


j f sj 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV

3.5638E+00 2.9164E+00 1.8057E+00 1.0515E+00 6.5100E 01 4.4012E 01 3.2108E 01 2.4788E 01 1.9908E 01 1.6443E 01 1.3862E 01 1.1855E 01 1.0269E 01 8.9832E 02 7.9199E 02 7.0308E 02 6.2796E 02 5.6392E 02 5.0890E 02 4.6130E 02 4.1988E 02 3.8364E 02 3.5175E 02 3.2357E 02 2.9856E 02 2.7627E 02 2.5633E 02 2.3843E 02 2.2230E 02 2.0772E 02 1.9451E 02 1.8249E 02 1.7154E 02 1.6153E 02 1.5236E 02 1.4395E 02 1.3620E 02 1.2905E 02 1.2245E 02 1.1633E 02 1.1066E 02 1.0539E 02 1.0049E 02 9.5912E 03 9.1642E 03 8.7648E 03 8.3909E 03 8.0403E 03 7.7110E 03 7.4015E 03 7.1102E 03 6.8356E 03 6.5765E 03 6.3319E 03 6.1006E 03 5.8817E 03 5.6743E 03 5.4776E 03 5.2911E 03 5.1137E 03 4.9449E 03

s


02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) B; Z 5 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

8.0833E 02 9.0430E 02 1.1818E 01 1.6052E 01 2.1145E 01 2.6434E 01 3.1415E 01 3.5851E 01 3.9718E 01 4.3078E 01 4.6047E 01 4.8713E 01 5.1149E 01 5.3411E 01 5.5541E 01 5.7562E 01 5.9496E 01 6.1353E 01 6.3143E 01 6.4873E 01 6.6548E 01 6.8171E 01 6.9746E 01 7.1274E 01 7.2760E 01 7.4202E 01 7.5605E 01 7.6970E 01 7.8296E 01 7.9588E 01 8.0845E 01 8.2069E 01 8.3261E 01 8.4423E 01 8.5555E 01 8.6660E 01 8.7737E 01 8.8788E 01 8.9813E 01 9.0815E 01 9.1794E 01 9.2750E 01 9.3684E 01 9.4598E 01 9.5492E 01 9.6366E 01 9.7223E 01 9.8062E 01 9.8882E 01 9.9687E 01 1.0048E+00 1.0125E+00 1.0201E+00 1.0275E+00 1.0348E+00 1.0420E+00 1.0490E+00 1.0560E+00 1.0627E+00 1.0694E+00 1.0760E+00


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

290


60 keV


s


j f sj 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) C; Z 6 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.0201E 01 1.1089E 01 1.3649E 01 1.7586E 01 2.2496E 01 2.7910E 01 3.3405E 01 3.8662E 01 4.3505E 01 4.7877E 01 5.1788E 01 5.5297E 01 5.8472E 01 6.1369E 01 6.4040E 01 6.6532E 01 6.8876E 01 7.1099E 01 7.3218E 01 7.5251E 01 7.7206E 01 7.9094E 01 8.0920E 01 8.2690E 01 8.4408E 01 8.6077E 01 8.7701E 01 8.9280E 01 9.0820E 01 9.2319E 01 9.3782E 01 9.5207E 01 9.6599E 01 9.7957E 01 9.9283E 01 1.0058E+00 1.0184E+00 1.0308E+00 1.0429E+00 1.0547E+00 1.0663E+00 1.0776E+00 1.0887E+00 1.0995E+00 1.1102E+00 1.1206E+00 1.1308E+00 1.1408E+00 1.1506E+00 1.1602E+00 1.1696E+00 1.1789E+00 1.1880E+00 1.1969E+00 1.2056E+00 1.2142E+00 1.2227E+00 1.2310E+00 1.2392E+00 1.2472E+00 1.2551E+00


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

291


60 keV


s


j f sj 02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV

2.9237E+00 2.6459E+00 2.0345E+00 1.4432E+00 1.0093E+00 7.1786E 01 5.2662E 01 3.9938E 01 3.1277E 01 2.5186E 01 2.0769E 01 1.7452E 01 1.4929E 01 1.2947E 01 1.1350E 01 1.0043E 01 8.9579E 02 8.0441E 02 7.2661E 02 6.5971E 02 6.0171E 02 5.5103E 02 5.0647E 02 4.6707E 02 4.3204E 02 4.0076E 02 3.7270E 02 3.4744E 02 3.2462E 02 3.0394E 02 2.8514E 02 2.6800E 02 2.5233E 02 2.3797E 02 2.2478E 02 2.1263E 02 2.0143E 02 1.9108E 02 1.8149E 02 1.7260E 02 1.6433E 02 1.5663E 02 1.4946E 02 1.4276E 02 1.3649E 02 1.3063E 02 1.2513E 02 1.1996E 02 1.1511E 02 1.1054E 02 1.0624E 02 1.0218E 02 9.8344E 03 9.4719E 03 9.1289E 03 8.8042E 03 8.4964E 03 8.2042E 03 7.9268E 03 7.6632E 03 7.4122E 03

s


02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) N; Z 7 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.2336E 01 1.3129E 01 1.5435E 01 1.9031E 01 2.3626E 01 2.8880E 01 3.4462E 01 4.0074E 01 4.5502E 01 5.0604E 01 5.5319E 01 5.9639E 01 6.3586E 01 6.7197E 01 7.0515E 01 7.3584E 01 7.6443E 01 7.9124E 01 8.1654E 01 8.4056E 01 8.6348E 01 8.8544E 01 9.0656E 01 9.2693E 01 9.4663E 01 9.6572E 01 9.8425E 01 1.0022E+00 1.0198E+00 1.0368E+00 1.0535E+00 1.0697E+00 1.0856E+00 1.1011E+00 1.1162E+00 1.1310E+00 1.1454E+00 1.1596E+00 1.1734E+00 1.1870E+00 1.2003E+00 1.2132E+00 1.2260E+00 1.2385E+00 1.2507E+00 1.2627E+00 1.2745E+00 1.2860E+00 1.2974E+00 1.3085E+00 1.3194E+00 1.3302E+00 1.3407E+00 1.3511E+00 1.3612E+00 1.3712E+00 1.3811E+00 1.3908E+00 1.4003E+00 1.4096E+00 1.4189E+00


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

292


60 keV


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) O; Z 8 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.4403E 01 1.5125E 01 1.7229E 01 2.0544E 01 2.4839E 01 2.9858E 01 3.5335E 01 4.1030E 01 4.6723E 01 5.2266E 01 5.7543E 01 6.2496E 01 6.7117E 01 7.1403E 01 7.5371E 01 7.9052E 01 8.2479E 01 8.5681E 01 8.8688E 01 9.1524E 01 9.4213E 01 9.6773E 01 9.9219E 01 1.0157E+00 1.0382E+00 1.0600E+00 1.0811E+00 1.1015E+00 1.1213E+00 1.1405E+00 1.1593E+00 1.1775E+00 1.1954E+00 1.2127E+00 1.2298E+00 1.2464E+00 1.2626E+00 1.2785E+00 1.2941E+00 1.3093E+00 1.3242E+00 1.3389E+00 1.3532E+00 1.3673E+00 1.3811E+00 1.3947E+00 1.4080E+00 1.4210E+00 1.4339E+00 1.4465E+00 1.4588E+00 1.4710E+00 1.4830E+00 1.4947E+00 1.5063E+00 1.5177E+00 1.5288E+00 1.5398E+00 1.5507E+00 1.5613E+00 1.5719E+00


s


j f sj 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

293


60 keV


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) F; Z 9 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.6411E 01 1.7076E 01 1.9016E 01 2.2094E 01 2.6122E 01 3.0892E 01 3.6194E 01 4.1823E 01 4.7594E 01 5.3352E 01 5.8974E 01 6.4374E 01 6.9511E 01 7.4354E 01 7.8898E 01 8.3153E 01 8.7138E 01 9.0873E 01 9.4383E 01 9.7690E 01 1.0082E+00 1.0378E+00 1.0661E+00 1.0930E+00 1.1189E+00 1.1437E+00 1.1676E+00 1.1907E+00 1.2130E+00 1.2347E+00 1.2557E+00 1.2761E+00 1.2960E+00 1.3154E+00 1.3344E+00 1.3529E+00 1.3709E+00 1.3886E+00 1.4059E+00 1.4229E+00 1.4395E+00 1.4557E+00 1.4717E+00 1.4873E+00 1.5027E+00 1.5178E+00 1.5326E+00 1.5471E+00 1.5614E+00 1.5755E+00 1.5893E+00 1.6028E+00 1.6162E+00 1.6293E+00 1.6422E+00 1.6549E+00 1.6675E+00 1.6798E+00 1.6919E+00 1.7038E+00 1.7156E+00


s


j f sj 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

294


60 keV


s


j f sj 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ne; Z 10 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.8367E 01 1.8982E 01 2.0785E 01 2.3660E 01 2.7450E 01 3.1983E 01 3.7084E 01 4.2582E 01 4.8318E 01 5.4148E 01 5.9953E 01 6.5635E 01 7.1135E 01 7.6404E 01 8.1416E 01 8.6164E 01 9.0650E 01 9.4884E 01 9.8881E 01 1.0266E+00 1.0623E+00 1.0963E+00 1.1285E+00 1.1593E+00 1.1887E+00 1.2168E+00 1.2439E+00 1.2699E+00 1.2951E+00 1.3193E+00 1.3429E+00 1.3657E+00 1.3879E+00 1.4094E+00 1.4304E+00 1.4509E+00 1.4709E+00 1.4905E+00 1.5096E+00 1.5283E+00 1.5466E+00 1.5645E+00 1.5821E+00 1.5994E+00 1.6163E+00 1.6329E+00 1.6492E+00 1.6652E+00 1.6810E+00 1.6965E+00 1.7117E+00 1.7266E+00 1.7414E+00 1.7558E+00 1.7701E+00 1.7841E+00 1.7980E+00 1.8116E+00 1.8250E+00 1.8382E+00 1.8512E+00


s


j f sj 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

295


60 keV


s


j f sj 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Na; Z 11 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

9.9384E 02 1.3450E 01 2.1863E 01 2.9721E 01 3.5796E 01 4.1211E 01 4.6596E 01 5.2136E 01 5.7840E 01 6.3655E 01 6.9509E 01 7.5314E 01 8.1020E 01 8.6564E 01 9.1910E 01 9.7036E 01 1.0193E+00 1.0659E+00 1.1102E+00 1.1523E+00 1.1924E+00 1.2304E+00 1.2667E+00 1.3012E+00 1.3343E+00 1.3659E+00 1.3962E+00 1.4254E+00 1.4535E+00 1.4806E+00 1.5067E+00 1.5321E+00 1.5567E+00 1.5806E+00 1.6038E+00 1.6264E+00 1.6484E+00 1.6699E+00 1.6909E+00 1.7114E+00 1.7315E+00 1.7511E+00 1.7703E+00 1.7892E+00 1.8077E+00 1.8259E+00 1.8437E+00 1.8612E+00 1.8784E+00 1.8953E+00 1.9119E+00 1.9283E+00 1.9444E+00 1.9602E+00 1.9757E+00 1.9911E+00 2.0062E+00 2.0211E+00 2.0357E+00 2.0502E+00 2.0644E+00


j f sj

5.3502E 02 7.2184E 02 1.1663E 01 1.5792E 01 1.8976E 01 2.1809E 01 2.4619E 01 2.7505E 01 3.0474E 01 3.3498E 01 3.6543E 01 3.9564E 01 4.2535E 01 4.5425E 01 4.8214E 01 5.0890E 01 5.3445E 01 5.5879E 01 5.8191E 01 6.0386E 01 6.2470E 01 6.4449E 01 6.6332E 01 6.8125E 01 6.9836E 01 7.1472E 01 7.3038E 01 7.4543E 01 7.5990E 01 7.7384E 01 7.8731E 01 8.0033E 01 8.1295E 01 8.2520E 01 8.3710E 01 8.4867E 01 8.5995E 01 8.7095E 01 8.8168E 01 8.9217E 01 9.0242E 01 9.1245E 01 9.2228E 01 9.3190E 01 9.4134E 01 9.5059E 01 9.5967E 01 9.6858E 01 9.7734E 01 9.8594E 01 9.9439E 01 1.0027E+00 1.0109E+00 1.0189E+00 1.0268E+00 1.0346E+00 1.0423E+00 1.0498E+00 1.0572E+00 1.0646E+00 1.0718E+00


296


60 keV

s

s


j f sj 02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV


s


02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Mg; Z 12 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.1836E 01 1.4577E 01 2.2243E 01 3.1959E 01 4.0638E 01 4.7758E 01 5.3991E 01 5.9897E 01 6.5741E 01 7.1608E 01 7.7492E 01 8.3370E 01 8.9187E 01 9.4903E 01 1.0048E+00 1.0588E+00 1.1110E+00 1.1611E+00 1.2091E+00 1.2550E+00 1.2988E+00 1.3407E+00 1.3808E+00 1.4190E+00 1.4557E+00 1.4908E+00 1.5245E+00 1.5568E+00 1.5880E+00 1.6180E+00 1.6470E+00 1.6750E+00 1.7021E+00 1.7284E+00 1.7540E+00 1.7788E+00 1.8030E+00 1.8265E+00 1.8495E+00 1.8719E+00 1.8938E+00 1.9153E+00 1.9363E+00 1.9568E+00 1.9770E+00 1.9967E+00 2.0161E+00 2.0351E+00 2.0538E+00 2.0721E+00 2.0902E+00 2.1079E+00 2.1253E+00 2.1425E+00 2.1594E+00 2.1761E+00 2.1925E+00 2.2086E+00 2.2245E+00 2.2402E+00 2.2557E+00


j f sj

6.3908E 02 7.8538E 02 1.1921E 01 1.7043E 01 2.1601E 01 2.5331E 01 2.8589E 01 3.1671E 01 3.4716E 01 3.7768E 01 4.0826E 01 4.3882E 01 4.6911E 01 4.9889E 01 5.2795E 01 5.5615E 01 5.8337E 01 6.0953E 01 6.3461E 01 6.5858E 01 6.8148E 01 7.0333E 01 7.2418E 01 7.4409E 01 7.6312E 01 7.8132E 01 7.9876E 01 8.1549E 01 8.3157E 01 8.4705E 01 8.6197E 01 8.7638E 01 8.9031E 01 9.0382E 01 9.1691E 01 9.2963E 01 9.4200E 01 9.5405E 01 9.6579E 01 9.7724E 01 9.8843E 01 9.9937E 01 1.0101E+00 1.0205E+00 1.0308E+00 1.0408E+00 1.0507E+00 1.0604E+00 1.0699E+00 1.0792E+00 1.0884E+00 1.0974E+00 1.1062E+00 1.1149E+00 1.1235E+00 1.1319E+00 1.1402E+00 1.1484E+00 1.1564E+00 1.1644E+00 1.1722E+00


297


60 keV

s

s


j f sj 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

90 keV

6.0792E+00 4.8079E+00 2.9317E+00 1.8547E+00 1.3205E+00 1.0209E+00 8.2385E 01 6.7986E 01 5.6862E 01 4.7999E 01 4.0833E 01 3.4996E 01 3.0236E 01 2.6309E 01 2.3047E 01 2.0325E 01 1.8040E 01 1.6108E 01 1.4466E 01 1.3059E 01 1.1848E 01 1.0799E 01 9.8842E 02 9.0826E 02 8.3762E 02 7.7508E 02 7.1943E 02 6.6970E 02 6.2507E 02 5.8486E 02 5.4849E 02 5.1549E 02 4.8544E 02 4.5799E 02 4.3284E 02 4.0975E 02 3.8849E 02 3.6886E 02 3.5070E 02 3.3386E 02 3.1822E 02 3.0366E 02 2.9009E 02 2.7741E 02 2.6554E 02 2.5443E 02 2.4400E 02 2.3419E 02 2.2497E 02 2.1628E 02 2.0808E 02 2.0034E 02 1.9303E 02 1.8610E 02 1.7954E 02 1.7332E 02 1.6741E 02 1.6180E 02 1.5647E 02 1.5139E 02 1.4655E 02

s


02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Al; Z 13 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.3210E 01 1.5975E 01 2.3638E 01 3.4084E 01 4.4565E 01 5.3577E 01 6.1144E 01 6.7819E 01 7.4063E 01 8.0125E 01 8.6110E 01 9.2064E 01 9.7964E 01 1.0380E+00 1.0953E+00 1.1513E+00 1.2058E+00 1.2586E+00 1.3096E+00 1.3586E+00 1.4058E+00 1.4511E+00 1.4946E+00 1.5363E+00 1.5763E+00 1.6148E+00 1.6517E+00 1.6873E+00 1.7215E+00 1.7544E+00 1.7863E+00 1.8170E+00 1.8468E+00 1.8757E+00 1.9037E+00 1.9308E+00 1.9573E+00 1.9830E+00 2.0080E+00 2.0325E+00 2.0563E+00 2.0796E+00 2.1024E+00 2.1247E+00 2.1466E+00 2.1680E+00 2.1890E+00 2.2096E+00 2.2298E+00 2.2496E+00 2.2691E+00 2.2883E+00 2.3071E+00 2.3257E+00 2.3439E+00 2.3619E+00 2.3796E+00 2.3970E+00 2.4142E+00 2.4311E+00 2.4478E+00


j f sj



298


60 keV

s

90 keV

s

j f sj

6.0318E 02 7.2754E 02 1.0703E 01 1.5340E 01 1.9968E 01 2.3937E 01 2.7264E 01 3.0191E 01 3.2918E 01 3.5562E 01 3.8173E 01 4.0766E 01 4.3339E 01 4.5882E 01 4.8384E 01 5.0833E 01 5.3217E 01 5.5528E 01 5.7762E 01 5.9913E 01 6.1981E 01 6.3965E 01 6.5868E 01 6.7692E 01 6.9440E 01 7.1115E 01 7.2723E 01 7.4266E 01 7.5750E 01 7.7178E 01 7.8553E 01 7.9880E 01 8.1163E 01 8.2403E 01 8.3605E 01 8.4770E 01 8.5902E 01 8.7003E 01 8.8074E 01 8.9118E 01 9.0135E 01 9.1129E 01 9.2100E 01 9.3050E 01 9.3979E 01 9.4888E 01 9.5780E 01 9.6654E 01 9.7511E 01 9.8353E 01 9.9179E 01 9.9991E 01 1.0079E+00 1.0157E+00 1.0235E+00 1.0310E+00 1.0385E+00 1.0459E+00 1.0531E+00 1.0602E+00 1.0673E+00


s


02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Si; Z 14 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.5342E 01 1.7816E 01 2.4867E 01 3.5156E 01 4.6543E 01 5.7164E 01 6.6307E 01 7.4140E 01 8.1102E 01 8.7578E 01 9.3811E 01 9.9907E 01 1.0590E+00 1.1183E+00 1.1768E+00 1.2343E+00 1.2905E+00 1.3454E+00 1.3987E+00 1.4504E+00 1.5003E+00 1.5485E+00 1.5950E+00 1.6398E+00 1.6830E+00 1.7245E+00 1.7645E+00 1.8031E+00 1.8403E+00 1.8762E+00 1.9108E+00 1.9444E+00 1.9768E+00 2.0082E+00 2.0387E+00 2.0683E+00 2.0970E+00 2.1250E+00 2.1522E+00 2.1788E+00 2.2046E+00 2.2299E+00 2.2546E+00 2.2788E+00 2.3024E+00 2.3255E+00 2.3482E+00 2.3704E+00 2.3922E+00 2.4136E+00 2.4346E+00 2.4553E+00 2.4755E+00 2.4955E+00 2.5151E+00 2.5345E+00 2.5535E+00 2.5722E+00 2.5907E+00 2.6088E+00 2.6268E+00


j f sj

8.3536E 02 9.6834E 02 1.3453E 01 1.8911E 01 2.4918E 01 3.0509E 01 3.5311E 01 3.9414E 01 4.3055E 01 4.6432E 01 4.9671E 01 5.2841E 01 5.5962E 01 5.9046E 01 6.2089E 01 6.5082E 01 6.8015E 01 7.0877E 01 7.3662E 01 7.6363E 01 7.8975E 01 8.1497E 01 8.3928E 01 8.6268E 01 8.8521E 01 9.0687E 01 9.2771E 01 9.4777E 01 9.6707E 01 9.8568E 01 1.0036E+00 1.0209E+00 1.0377E+00 1.0538E+00 1.0695E+00 1.0847E+00 1.0994E+00 1.1138E+00 1.1277E+00 1.1412E+00 1.1545E+00 1.1673E+00 1.1799E+00 1.1922E+00 1.2042E+00 1.2160E+00 1.2275E+00 1.2387E+00 1.2498E+00 1.2606E+00 1.2713E+00 1.2817E+00 1.2920E+00 1.3021E+00 1.3120E+00 1.3218E+00 1.3314E+00 1.3408E+00 1.3501E+00 1.3593E+00 1.3683E+00


299


60 keV

s

90 keV

s

j f sj

7.0425E 02 8.1585E 02 1.1325E 01 1.5910E 01 2.0952E 01 2.5643E 01 2.9671E 01 3.3113E 01 3.6166E 01 3.8996E 01 4.1710E 01 4.4366E 01 4.6983E 01 4.9567E 01 5.2117E 01 5.4626E 01 5.7084E 01 5.9484E 01 6.1819E 01 6.4083E 01 6.6274E 01 6.8389E 01 7.0428E 01 7.2390E 01 7.4278E 01 7.6095E 01 7.7842E 01 7.9523E 01 8.1141E 01 8.2699E 01 8.4202E 01 8.5652E 01 8.7053E 01 8.8408E 01 8.9719E 01 9.0990E 01 9.2224E 01 9.3422E 01 9.4587E 01 9.5720E 01 9.6825E 01 9.7902E 01 9.8954E 01 9.9980E 01 1.0099E+00 1.0197E+00 1.0293E+00 1.0387E+00 1.0480E+00 1.0570E+00 1.0659E+00 1.0746E+00 1.0832E+00 1.0916E+00 1.0999E+00 1.1081E+00 1.1161E+00 1.1240E+00 1.1317E+00 1.1394E+00 1.1470E+00


s


02 02 02 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) P; Z 15 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

1.7700E 01 1.9853E 01 2.6134E 01 3.5783E 01 4.7310E 01 5.8970E 01 6.9558E 01 7.8740E 01 8.6737E 01 9.3899E 01 1.0056E+00 1.0692E+00 1.1310E+00 1.1915E+00 1.2511E+00 1.3097E+00 1.3673E+00 1.4238E+00 1.4790E+00 1.5327E+00 1.5849E+00 1.6356E+00 1.6847E+00 1.7322E+00 1.7781E+00 1.8225E+00 1.8653E+00 1.9067E+00 1.9467E+00 1.9854E+00 2.0228E+00 2.0590E+00 2.0940E+00 2.1280E+00 2.1610E+00 2.1930E+00 2.2241E+00 2.2543E+00 2.2838E+00 2.3125E+00 2.3405E+00 2.3678E+00 2.3944E+00 2.4205E+00 2.4459E+00 2.4709E+00 2.4953E+00 2.5192E+00 2.5426E+00 2.5656E+00 2.5882E+00 2.6104E+00 2.6322E+00 2.6536E+00 2.6746E+00 2.6954E+00 2.7157E+00 2.7358E+00 2.7556E+00 2.7750E+00 2.7943E+00


j f sj



300


60 keV

s

90 keV

s

j f sj

8.1757E 02 9.1488E 02 1.1981E 01 1.6299E 01 2.1419E 01 2.6579E 01 3.1258E 01 3.5310E 01 3.8827E 01 4.1964E 01 4.4875E 01 4.7650E 01 5.0338E 01 5.2974E 01 5.5570E 01 5.8128E 01 6.0642E 01 6.3109E 01 6.5522E 01 6.7875E 01 7.0165E 01 7.2388E 01 7.4541E 01 7.6625E 01 7.8638E 01 8.0581E 01 8.2457E 01 8.4267E 01 8.6013E 01 8.7698E 01 8.9324E 01 9.0896E 01 9.2415E 01 9.3884E 01 9.5307E 01 9.6686E 01 9.8023E 01 9.9322E 01 1.0058E+00 1.0181E+00 1.0301E+00 1.0417E+00 1.0531E+00 1.0641E+00 1.0750E+00 1.0855E+00 1.0959E+00 1.1060E+00 1.1160E+00 1.1257E+00 1.1352E+00 1.1446E+00 1.1538E+00 1.1628E+00 1.1717E+00 1.1804E+00 1.1890E+00 1.1974E+00 1.2057E+00 1.2139E+00 1.2220E+00


s

6.9716E 02 7.7888E 02 1.0187E 01 1.3851E 01 1.8196E 01 2.2573E 01 2.6542E 01 2.9979E 01 3.2960E 01 3.5621E 01 3.8088E 01 4.0440E 01 4.2717E 01 4.4951E 01 4.7151E 01 4.9319E 01 5.1450E 01 5.3541E 01 5.5587E 01 5.7582E 01 5.9523E 01 6.1408E 01 6.3234E 01 6.5000E 01 6.6707E 01 6.8355E 01 6.9945E 01 7.1479E 01 7.2959E 01 7.4387E 01 7.5766E 01 7.7097E 01 7.8384E 01 7.9629E 01 8.0834E 01 8.2002E 01 8.3134E 01 8.4234E 01 8.5302E 01 8.6341E 01 8.7353E 01 8.8338E 01 8.9299E 01 9.0237E 01 9.1152E 01 9.2048E 01 9.2924E 01 9.3781E 01 9.4621E 01 9.5444E 01 9.6251E 01 9.7043E 01 9.7821E 01 9.8584E 01 9.9334E 01 1.0007E+00 1.0080E+00 1.0151E+00 1.0221E+00 1.0290E+00 1.0359E+00

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) S; Z 16 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj

1.1006E 01 1.2043E 01 1.5086E 01 1.9859E 01 2.5809E 01 3.2194E 01 3.8340E 01 4.3867E 01 4.8699E 01 5.2946E 01 5.6770E 01 6.0318E 01 6.3679E 01 6.6926E 01 7.0098E 01 7.3211E 01 7.6271E 01 7.9279E 01 8.2231E 01 8.5122E 01 8.7947E 01 9.0702E 01 9.3384E 01 9.5990E 01 9.8519E 01 1.0097E+00 1.0334E+00 1.0564E+00 1.0786E+00 1.1001E+00 1.1210E+00 1.1411E+00 1.1606E+00 1.1795E+00 1.1978E+00 1.2155E+00 1.2328E+00 1.2495E+00 1.2657E+00 1.2816E+00 1.2969E+00 1.3119E+00 1.3266E+00 1.3408E+00 1.3547E+00 1.3683E+00 1.3816E+00 1.3946E+00 1.4074E+00 1.4199E+00 1.4321E+00 1.4441E+00 1.4559E+00 1.4674E+00 1.4788E+00 1.4899E+00 1.5009E+00 1.5117E+00 1.5223E+00 1.5328E+00 1.5430E+00

5.7166E+00 5.1240E+00 3.8669E+00 2.6999E+00 1.8770E+00 1.3494E+00 1.0157E+00 7.9856E 01 6.5066E 01 5.4451E 01 4.6461E 01 4.0179E 01 3.5148E 01 3.1001E 01 2.7513E 01 2.4553E 01 2.2019E 01 1.9834E 01 1.7940E 01 1.6290E 01 1.4847E 01 1.3579E 01 1.2461E 01 1.1472E 01 1.0593E 01 9.8093E 02 9.1084E 02 8.4795E 02 7.9133E 02 7.4019E 02 6.9388E 02 6.5180E 02 6.1347E 02 5.7846E 02 5.4640E 02 5.1697E 02 4.8989E 02 4.6492E 02 4.4184E 02 4.2046E 02 4.0062E 02 3.8218E 02 3.6500E 02 3.4898E 02 3.3400E 02 3.1999E 02 3.0684E 02 2.9451E 02 2.8291E 02 2.7199E 02 2.6171E 02 2.5200E 02 2.4282E 02 2.3415E 02 2.2593E 02 2.1815E 02 2.1076E 02 2.0374E 02 1.9707E 02 1.9073E 02 1.8469E 02

301


60 keV

s

90 keV

s

j f sj

9.2940E 02 1.0162E 01 1.2720E 01 1.6733E 01 2.1731E 01 2.7091E 01 3.2250E 01 3.6889E 01 4.0943E 01 4.4508E 01 4.7711E 01 5.0685E 01 5.3503E 01 5.6223E 01 5.8881E 01 6.1489E 01 6.4054E 01 6.6575E 01 6.9049E 01 7.1473E 01 7.3842E 01 7.6152E 01 7.8401E 01 8.0587E 01 8.2708E 01 8.4763E 01 8.6754E 01 8.8681E 01 9.0546E 01 9.2349E 01 9.4093E 01 9.5782E 01 9.7415E 01 9.8998E 01 1.0053E+00 1.0202E+00 1.0346E+00 1.0486E+00 1.0622E+00 1.0754E+00 1.0883E+00 1.1008E+00 1.1131E+00 1.1250E+00 1.1366E+00 1.1480E+00 1.1591E+00 1.1700E+00 1.1806E+00 1.1911E+00 1.2013E+00 1.2113E+00 1.2211E+00 1.2308E+00 1.2403E+00 1.2496E+00 1.2587E+00 1.2678E+00 1.2766E+00 1.2853E+00 1.2939E+00


s

7.9309E 02 8.6578E 02 1.0821E 01 1.4229E 01 1.8470E 01 2.3017E 01 2.7395E 01 3.1330E 01 3.4771E 01 3.7790E 01 4.0510E 01 4.3031E 01 4.5417E 01 4.7722E 01 4.9973E 01 5.2184E 01 5.4358E 01 5.6494E 01 5.8591E 01 6.0646E 01 6.2654E 01 6.4613E 01 6.6520E 01 6.8373E 01 7.0171E 01 7.1915E 01 7.3603E 01 7.5237E 01 7.6817E 01 7.8346E 01 7.9825E 01 8.1256E 01 8.2640E 01 8.3981E 01 8.5280E 01 8.6539E 01 8.7761E 01 8.8946E 01 9.0098E 01 9.1219E 01 9.2308E 01 9.3370E 01 9.4404E 01 9.5414E 01 9.6398E 01 9.7361E 01 9.8301E 01 9.9221E 01 1.0012E+00 1.0100E+00 1.0187E+00 1.0272E+00 1.0355E+00 1.0436E+00 1.0516E+00 1.0595E+00 1.0673E+00 1.0749E+00 1.0824E+00 1.0898E+00 1.0970E+00

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Cl; Z 17 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.2194E 01 2.3926E 01 2.9056E 01 3.7298E 01 4.7986E 01 6.0057E 01 7.2310E 01 8.3821E 01 9.4156E 01 1.0330E+00 1.1146E+00 1.1890E+00 1.2581E+00 1.3237E+00 1.3870E+00 1.4486E+00 1.5088E+00 1.5679E+00 1.6258E+00 1.6827E+00 1.7383E+00 1.7927E+00 1.8458E+00 1.8976E+00 1.9481E+00 1.9971E+00 2.0448E+00 2.0911E+00 2.1361E+00 2.1797E+00 2.2221E+00 2.2633E+00 2.3033E+00 2.3421E+00 2.3799E+00 2.4166E+00 2.4523E+00 2.4871E+00 2.5210E+00 2.5540E+00 2.5862E+00 2.6176E+00 2.6483E+00 2.6783E+00 2.7076E+00 2.7362E+00 2.7642E+00 2.7917E+00 2.8186E+00 2.8450E+00 2.8708E+00 2.8962E+00 2.9211E+00 2.9456E+00 2.9696E+00 2.9933E+00 3.0165E+00 3.0394E+00 3.0619E+00 3.0840E+00 3.1059E+00


j f sj



302


60 keV

s

90 keV

s

j f sj

1.0397E 01 1.1184E 01 1.3516E 01 1.7238E 01 2.2022E 01 2.7389E 01 3.2826E 01 3.7935E 01 4.2523E 01 4.6570E 01 5.0167E 01 5.3427E 01 5.6442E 01 5.9299E 01 6.2051E 01 6.4729E 01 6.7350E 01 6.9923E 01 7.2450E 01 7.4931E 01 7.7364E 01 7.9746E 01 8.2073E 01 8.4344E 01 8.6556E 01 8.8709E 01 9.0801E 01 9.2833E 01 9.4804E 01 9.6716E 01 9.8570E 01 1.0037E+00 1.0211E+00 1.0380E+00 1.0544E+00 1.0703E+00 1.0858E+00 1.1008E+00 1.1154E+00 1.1296E+00 1.1434E+00 1.1568E+00 1.1700E+00 1.1827E+00 1.1952E+00 1.2074E+00 1.2193E+00 1.2309E+00 1.2423E+00 1.2535E+00 1.2644E+00 1.2751E+00 1.2856E+00 1.2959E+00 1.3060E+00 1.3159E+00 1.3257E+00 1.3353E+00 1.3447E+00 1.3540E+00 1.3631E+00


s

8.8780E 02 9.5354E 02 1.1505E 01 1.4669E 01 1.8727E 01 2.3284E 01 2.7895E 01 3.2233E 01 3.6123E 01 3.9558E 01 4.2608E 01 4.5371E 01 4.7927E 01 5.0349E 01 5.2680E 01 5.4949E 01 5.7170E 01 5.9350E 01 6.1492E 01 6.3595E 01 6.5657E 01 6.7676E 01 6.9649E 01 7.1575E 01 7.3451E 01 7.5276E 01 7.7050E 01 7.8773E 01 8.0444E 01 8.2066E 01 8.3638E 01 8.5162E 01 8.6640E 01 8.8073E 01 8.9463E 01 9.0811E 01 9.2120E 01 9.3392E 01 9.4628E 01 9.5830E 01 9.6999E 01 9.8138E 01 9.9248E 01 1.0033E+00 1.0138E+00 1.0242E+00 1.0342E+00 1.0441E+00 1.0537E+00 1.0631E+00 1.0724E+00 1.0814E+00 1.0903E+00 1.0990E+00 1.1076E+00 1.1160E+00 1.1242E+00 1.1323E+00 1.1403E+00 1.1481E+00 1.1558E+00

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ar; Z 18 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj



j f sj

1.3577E 01 1.4437E 01 1.6988E 01 2.1099E 01 2.6497E 01 3.2755E 01 3.9359E 01 4.5824E 01 5.1825E 01 5.7223E 01 6.2030E 01 6.6343E 01 7.0271E 01 7.3920E 01 7.7376E 01 8.0694E 01 8.3913E 01 8.7058E 01 9.0139E 01 9.3163E 01 9.6132E 01 9.9044E 01 1.0190E+00 1.0469E+00 1.0742E+00 1.1008E+00 1.1268E+00 1.1522E+00 1.1768E+00 1.2007E+00 1.2241E+00 1.2467E+00 1.2687E+00 1.2901E+00 1.3109E+00 1.3311E+00 1.3507E+00 1.3698E+00 1.3884E+00 1.4064E+00 1.4240E+00 1.4412E+00 1.4579E+00 1.4742E+00 1.4902E+00 1.5057E+00 1.5209E+00 1.5358E+00 1.5503E+00 1.5645E+00 1.5785E+00 1.5921E+00 1.6055E+00 1.6186E+00 1.6315E+00 1.6442E+00 1.6566E+00 1.6688E+00 1.6808E+00 1.6926E+00 1.7042E+00


303


60 keV

s

90 keV

s

j f sj

1.1486E 01 1.2206E 01 1.4350E 01 1.7811E 01 2.2348E 01 2.7606E 01 3.3150E 01 3.8581E 01 4.3619E 01 4.8154E 01 5.2188E 01 5.5806E 01 5.9101E 01 6.2160E 01 6.5055E 01 6.7834E 01 7.0531E 01 7.3165E 01 7.5747E 01 7.8281E 01 8.0769E 01 8.3210E 01 8.5603E 01 8.7945E 01 9.0234E 01 9.2469E 01 9.4649E 01 9.6772E 01 9.8839E 01 1.0085E+00 1.0280E+00 1.0470E+00 1.0655E+00 1.0834E+00 1.1008E+00 1.1178E+00 1.1342E+00 1.1502E+00 1.1658E+00 1.1809E+00 1.1956E+00 1.2100E+00 1.2240E+00 1.2377E+00 1.2510E+00 1.2640E+00 1.2767E+00 1.2891E+00 1.3012E+00 1.3131E+00 1.3248E+00 1.3362E+00 1.3474E+00 1.3583E+00 1.3691E+00 1.3796E+00 1.3900E+00 1.4002E+00 1.4102E+00 1.4201E+00 1.4297E+00


s

9.8141E 02 1.0416E 01 1.2223E 01 1.5165E 01 1.9017E 01 2.3480E 01 2.8188E 01 3.2791E 01 3.7073E 01 4.0917E 01 4.4341E 01 4.7411E 01 5.0203E 01 5.2796E 01 5.5248E 01 5.7604E 01 5.9889E 01 6.2120E 01 6.4308E 01 6.6455E 01 6.8564E 01 7.0633E 01 7.2661E 01 7.4647E 01 7.6588E 01 7.8483E 01 8.0331E 01 8.2132E 01 8.3885E 01 8.5589E 01 8.7246E 01 8.8856E 01 9.0421E 01 9.1941E 01 9.3417E 01 9.4852E 01 9.6246E 01 9.7602E 01 9.8920E 01 1.0020E+00 1.0145E+00 1.0267E+00 1.0385E+00 1.0501E+00 1.0614E+00 1.0724E+00 1.0831E+00 1.0936E+00 1.1039E+00 1.1140E+00 1.1238E+00 1.1335E+00 1.1429E+00 1.1522E+00 1.1613E+00 1.1702E+00 1.1790E+00 1.1876E+00 1.1960E+00 1.2044E+00 1.2125E+00

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) K; Z 19 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

1.6059E 01 2.2522E 01 3.4923E 01 4.5491E 01 5.5955E 01 6.7457E 01 7.9761E 01 9.2197E 01 1.0412E+00 1.1516E+00 1.2518E+00 1.3421E+00 1.4245E+00 1.5004E+00 1.5713E+00 1.6387E+00 1.7034E+00 1.7660E+00 1.8271E+00 1.8868E+00 1.9452E+00 2.0025E+00 2.0586E+00 2.1136E+00 2.1674E+00 2.2200E+00 2.2714E+00 2.3216E+00 2.3706E+00 2.4184E+00 2.4650E+00 2.5104E+00 2.5547E+00 2.5979E+00 2.6399E+00 2.6810E+00 2.7210E+00 2.7600E+00 2.7980E+00 2.8352E+00 2.8714E+00 2.9069E+00 2.9415E+00 2.9753E+00 3.0085E+00 3.0408E+00 3.0726E+00 3.1036E+00 3.1340E+00 3.1639E+00 3.1931E+00 3.2218E+00 3.2500E+00 3.2776E+00 3.3048E+00 3.3315E+00 3.3577E+00 3.3835E+00 3.4089E+00 3.4339E+00 3.4585E+00


j f sj



304


60 keV

s

90 keV

s

j f sj

7.6992E 02 1.0697E 01 1.6365E 01 2.1160E 01 2.5879E 01 3.1028E 01 3.6511E 01 4.2049E 01 4.7371E 01 5.2299E 01 5.6774E 01 6.0784E 01 6.4427E 01 6.7758E 01 7.0855E 01 7.3784E 01 7.6592E 01 7.9309E 01 8.1958E 01 8.4549E 01 8.7090E 01 8.9585E 01 9.2034E 01 9.4437E 01 9.6791E 01 9.9096E 01 1.0135E+00 1.0355E+00 1.0570E+00 1.0780E+00 1.0984E+00 1.1184E+00 1.1377E+00 1.1566E+00 1.1750E+00 1.1929E+00 1.2103E+00 1.2272E+00 1.2437E+00 1.2598E+00 1.2754E+00 1.2907E+00 1.3056E+00 1.3201E+00 1.3343E+00 1.3481E+00 1.3616E+00 1.3748E+00 1.3877E+00 1.4003E+00 1.4127E+00 1.4248E+00 1.4367E+00 1.4483E+00 1.4598E+00 1.4710E+00 1.4820E+00 1.4928E+00 1.5034E+00 1.5138E+00 1.5240E+00

1.0427E+01 7.2836E+00 4.4276E+00 3.1230E+00 2.3076E+00 1.7305E+00 1.3205E+00 1.0312E+00 8.2613E 01 6.7839E 01 5.6931E 01 4.8644E 01 4.2234E 01 3.7122E 01 3.2933E 01 2.9441E 01 2.6484E 01 2.3949E 01 2.1753E 01 1.9835E 01 1.8150E 01 1.6662E 01 1.5340E 01 1.4163E 01 1.3110E 01 1.2166E 01 1.1316E 01 1.0549E 01 9.8559E 02 9.2270E 02 8.6551E 02 8.1340E 02 7.6580E 02 7.2223E 02 6.8225E 02 6.4549E 02 6.1162E 02 5.8036E 02 5.5145E 02 5.2466E 02 4.9979E 02 4.7666E 02 4.5512E 02 4.3503E 02 4.1625E 02 3.9868E 02 3.8222E 02 3.6676E 02 3.5224E 02 3.3858E 02 3.2571E 02 3.1356E 02 3.0210E 02 2.9126E 02 2.8100E 02 2.7128E 02 2.6206E 02 2.5332E 02 2.4500E 02 2.3709E 02 2.2957E 02

s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ca; Z 20 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj

9.8165E 02 1.2836E 01 2.0193E 01 2.7615E 01 3.4001E 01 4.0218E 01 4.6659E 01 5.3253E 01 5.9768E 01 6.5975E 01 7.1731E 01 7.6970E 01 8.1739E 01 8.6084E 01 9.0086E 01 9.3822E 01 9.7357E 01 1.0074E+00 1.0401E+00 1.0719E+00 1.1030E+00 1.1335E+00 1.1633E+00 1.1926E+00 1.2214E+00 1.2496E+00 1.2773E+00 1.3044E+00 1.3309E+00 1.3568E+00 1.3821E+00 1.4068E+00 1.4309E+00 1.4545E+00 1.4774E+00 1.4998E+00 1.5216E+00 1.5429E+00 1.5637E+00 1.5839E+00 1.6036E+00 1.6229E+00 1.6417E+00 1.6600E+00 1.6779E+00 1.6954E+00 1.7125E+00 1.7293E+00 1.7456E+00 1.7616E+00 1.7773E+00 1.7927E+00 1.8077E+00 1.8225E+00 1.8370E+00 1.8512E+00 1.8651E+00 1.8788E+00 1.8922E+00 1.9055E+00 1.9185E+00


305


60 keV

s

90 keV

s

j f sj



s

7.1133E 02 9.2808E 02 1.4551E 01 1.9861E 01 2.4426E 01 2.8865E 01 3.3460E 01 3.8162E 01 4.2808E 01 4.7238E 01 5.1345E 01 5.5083E 01 5.8483E 01 6.1577E 01 6.4423E 01 6.7076E 01 6.9586E 01 7.1987E 01 7.4306E 01 7.6563E 01 7.8768E 01 8.0929E 01 8.3050E 01 8.5132E 01 8.7176E 01 8.9182E 01 9.1148E 01 9.3074E 01 9.4959E 01 9.6802E 01 9.8603E 01 1.0036E+00 1.0208E+00 1.0375E+00 1.0539E+00 1.0698E+00 1.0853E+00 1.1004E+00 1.1152E+00 1.1295E+00 1.1435E+00 1.1572E+00 1.1705E+00 1.1836E+00 1.1963E+00 1.2087E+00 1.2208E+00 1.2326E+00 1.2442E+00 1.2556E+00 1.2666E+00 1.2775E+00 1.2881E+00 1.2986E+00 1.3088E+00 1.3188E+00 1.3287E+00 1.3383E+00 1.3478E+00 1.3572E+00 1.3663E+00

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Sc; Z 21 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



306


60 keV

s

90 keV

s

j f sj



s

7.7965E 02 9.8340E 02 1.4879E 01 2.0286E 01 2.5060E 01 2.9609E 01 3.4240E 01 3.8983E 01 4.3729E 01 4.8338E 01 5.2692E 01 5.6717E 01 6.0413E 01 6.3787E 01 6.6881E 01 6.9743E 01 7.2423E 01 7.4961E 01 7.7389E 01 7.9733E 01 8.2009E 01 8.4232E 01 8.6407E 01 8.8540E 01 9.0633E 01 9.2689E 01 9.4706E 01 9.6686E 01 9.8626E 01 1.0053E+00 1.0239E+00 1.0421E+00 1.0599E+00 1.0774E+00 1.0944E+00 1.1110E+00 1.1272E+00 1.1430E+00 1.1585E+00 1.1736E+00 1.1883E+00 1.2027E+00 1.2168E+00 1.2305E+00 1.2439E+00 1.2570E+00 1.2697E+00 1.2823E+00 1.2945E+00 1.3065E+00 1.3182E+00 1.3297E+00 1.3409E+00 1.3519E+00 1.3627E+00 1.3733E+00 1.3837E+00 1.3939E+00 1.4039E+00 1.4138E+00 1.4234E+00

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ti; Z 22 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.0031E 01 2.4904E 01 3.7245E 01 5.1119E 01 6.3742E 01 7.5771E 01 8.7970E 01 1.0050E+00 1.1314E+00 1.2555E+00 1.3743E+00 1.4855E+00 1.5889E+00 1.6844E+00 1.7727E+00 1.8549E+00 1.9320E+00 2.0050E+00 2.0747E+00 2.1416E+00 2.2063E+00 2.2692E+00 2.3304E+00 2.3902E+00 2.4486E+00 2.5059E+00 2.5620E+00 2.6170E+00 2.6708E+00 2.7236E+00 2.7753E+00 2.8260E+00 2.8756E+00 2.9241E+00 2.9716E+00 3.0182E+00 3.0637E+00 3.1083E+00 3.1519E+00 3.1946E+00 3.2364E+00 3.2774E+00 3.3175E+00 3.3568E+00 3.3953E+00 3.4330E+00 3.4700E+00 3.5063E+00 3.5419E+00 3.5768E+00 3.6110E+00 3.6447E+00 3.6777E+00 3.7102E+00 3.7421E+00 3.7734E+00 3.8043E+00 3.8346E+00 3.8644E+00 3.8938E+00 3.9227E+00


j f sj

1.1551E 01 1.4273E 01 2.1068E 01 2.8612E 01 3.5434E 01 4.1900E 01 4.8425E 01 5.5106E 01 6.1846E 01 6.8481E 01 7.4847E 01 8.0821E 01 8.6371E 01 9.1479E 01 9.6176E 01 1.0052E+00 1.0456E+00 1.0837E+00 1.1198E+00 1.1544E+00 1.1878E+00 1.2203E+00 1.2519E+00 1.2828E+00 1.3130E+00 1.3427E+00 1.3718E+00 1.4004E+00 1.4284E+00 1.4559E+00 1.4829E+00 1.5093E+00 1.5352E+00 1.5605E+00 1.5854E+00 1.6096E+00 1.6333E+00 1.6565E+00 1.6792E+00 1.7014E+00 1.7231E+00 1.7443E+00 1.7650E+00 1.7852E+00 1.8051E+00 1.8244E+00 1.8434E+00 1.8619E+00 1.8801E+00 1.8979E+00 1.9153E+00 1.9324E+00 1.9491E+00 1.9655E+00 1.9816E+00 1.9974E+00 2.0129E+00 2.0281E+00 2.0431E+00 2.0577E+00 2.0722E+00


307


60 keV

s

90 keV

s

j f sj

9.8169E 02 1.2118E 01 1.7852E 01 2.4210E 01 2.9955E 01 3.5396E 01 4.0883E 01 4.6498E 01 5.2163E 01 5.7740E 01 6.3094E 01 6.8119E 01 7.2787E 01 7.7082E 01 8.1029E 01 8.4674E 01 8.8068E 01 9.1257E 01 9.4284E 01 9.7183E 01 9.9980E 01 1.0269E+00 1.0534E+00 1.0793E+00 1.1046E+00 1.1295E+00 1.1539E+00 1.1778E+00 1.2013E+00 1.2244E+00 1.2470E+00 1.2692E+00 1.2909E+00 1.3122E+00 1.3330E+00 1.3533E+00 1.3732E+00 1.3927E+00 1.4117E+00 1.4303E+00 1.4485E+00 1.4662E+00 1.4836E+00 1.5006E+00 1.5172E+00 1.5334E+00 1.5493E+00 1.5648E+00 1.5800E+00 1.5949E+00 1.6095E+00 1.6238E+00 1.6378E+00 1.6515E+00 1.6649E+00 1.6781E+00 1.6911E+00 1.7038E+00 1.7163E+00 1.7285E+00 1.7406E+00


s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) V; Z 23 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj



j f sj



308


60 keV

s

90 keV

s

j f sj


9.6240E+00 7.7618E+00 5.0815E+00 3.4805E+00 2.5702E+00 1.9795E+00 1.5589E+00 1.2469E+00 1.0117E+00 8.3302E 01 6.9612E 01 5.8993E 01 5.0721E 01 4.4159E 01 3.8861E 01 3.4527E 01 3.0927E 01 2.7898E 01 2.5315E 01 2.3090E 01 2.1153E 01 1.9454E 01 1.7952E 01 1.6617E 01 1.5423E 01 1.4350E 01 1.3383E 01 1.2508E 01 1.1713E 01 1.0989E 01 1.0329E 01 9.7241E 02 9.1697E 02 8.6602E 02 8.1910E 02 7.7582E 02 7.3581E 02 6.9877E 02 6.6443E 02 6.3253E 02 6.0285E 02 5.7520E 02 5.4940E 02 5.2529E 02 5.0274E 02 4.8161E 02 4.6178E 02 4.4316E 02 4.2565E 02 4.0917E 02 3.9363E 02 3.7897E 02 3.6511E 02 3.5202E 02 3.3962E 02 3.2787E 02 3.1673E 02 3.0615E 02 2.9610E 02 2.8654E 02 2.7744E 02

s

8.9803E 02 1.0876E 01 1.5622E 01 2.1000E 01 2.5942E 01 3.0615E 01 3.5293E 01 4.0073E 01 4.4920E 01 4.9743E 01 5.4436E 01 5.8905E 01 6.3106E 01 6.7008E 01 7.0612E 01 7.3944E 01 7.7037E 01 7.9928E 01 8.2653E 01 8.5243E 01 8.7724E 01 9.0117E 01 9.2438E 01 9.4699E 01 9.6908E 01 9.9072E 01 1.0119E+00 1.0328E+00 1.0532E+00 1.0733E+00 1.0929E+00 1.1123E+00 1.1313E+00 1.1498E+00 1.1681E+00 1.1859E+00 1.2034E+00 1.2205E+00 1.2373E+00 1.2537E+00 1.2697E+00 1.2854E+00 1.3008E+00 1.3159E+00 1.3306E+00 1.3450E+00 1.3591E+00 1.3728E+00 1.3863E+00 1.3996E+00 1.4125E+00 1.4252E+00 1.4376E+00 1.4498E+00 1.4618E+00 1.4735E+00 1.4850E+00 1.4963E+00 1.5074E+00 1.5183E+00 1.5290E+00

4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Cr; Z 24 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



309


60 keV

s

90 keV

s

j f sj



s

9.9555E 02 1.1813E 01 1.6068E 01 2.0721E 01 2.5245E 01 2.9778E 01 3.4439E 01 3.9248E 01 4.4153E 01 4.9066E 01 5.3888E 01 5.8526E 01 6.2931E 01 6.7058E 01 7.0897E 01 7.4460E 01 7.7771E 01 8.0861E 01 8.3763E 01 8.6507E 01 8.9121E 01 9.1629E 01 9.4048E 01 9.6395E 01 9.8679E 01 1.0091E+00 1.0309E+00 1.0523E+00 1.0733E+00 1.0939E+00 1.1141E+00 1.1340E+00 1.1535E+00 1.1726E+00 1.1914E+00 1.2098E+00 1.2278E+00 1.2455E+00 1.2629E+00 1.2799E+00 1.2966E+00 1.3129E+00 1.3288E+00 1.3445E+00 1.3598E+00 1.3748E+00 1.3895E+00 1.4039E+00 1.4180E+00 1.4318E+00 1.4454E+00 1.4587E+00 1.4717E+00 1.4845E+00 1.4970E+00 1.5093E+00 1.5213E+00 1.5332E+00 1.5448E+00 1.5562E+00 1.5674E+00

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Mn; Z 25 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



310


60 keV

s

90 keV

s

j f sj

1.1703E 01 1.3822E 01 1.9151E 01 2.5373E 01 3.1241E 01 3.6799E 01 4.2317E 01 4.7935E 01 5.3664E 01 5.9442E 01 6.5174E 01 7.0758E 01 7.6123E 01 8.1207E 01 8.5979E 01 9.0438E 01 9.4597E 01 9.8482E 01 1.0213E+00 1.0556E+00 1.0882E+00 1.1193E+00 1.1492E+00 1.1780E+00 1.2059E+00 1.2331E+00 1.2596E+00 1.2856E+00 1.3110E+00 1.3360E+00 1.3604E+00 1.3844E+00 1.4080E+00 1.4312E+00 1.4540E+00 1.4763E+00 1.4982E+00 1.5197E+00 1.5408E+00 1.5615E+00 1.5818E+00 1.6017E+00 1.6213E+00 1.6404E+00 1.6592E+00 1.6776E+00 1.6956E+00 1.7133E+00 1.7306E+00 1.7476E+00 1.7643E+00 1.7807E+00 1.7967E+00 1.8125E+00 1.8279E+00 1.8431E+00 1.8580E+00 1.8726E+00 1.8870E+00 1.9011E+00 1.9150E+00


s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Fe; Z 26 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



311


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Co; Z 27 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



312


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ni; Z 28 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



313


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Cu; Z 29 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



314


60 keV

s

90 keV

s

j f sj


6.4098E+00 5.5985E+00 4.2702E+00 3.2913E+00 2.6158E+00 2.1198E+00 1.7392E+00 1.4403E+00 1.2027E+00 1.0125E+00 8.5955E 01 7.3579E 01 6.3556E 01 5.5374E 01 4.8642E 01 4.3066E 01 3.8412E 01 3.4494E 01 3.1169E 01 2.8322E 01 2.5866E 01 2.3731E 01 2.1862E 01 2.0214E 01 1.8752E 01 1.7449E 01 1.6280E 01 1.5227E 01 1.4275E 01 1.3410E 01 1.2622E 01 1.1902E 01 1.1242E 01 1.0634E 01 1.0075E 01 9.5577E 02 9.0792E 02 8.6353E 02 8.2228E 02 7.8388E 02 7.4808E 02 7.1464E 02 6.8338E 02 6.5410E 02 6.2664E 02 6.0086E 02 5.7663E 02 5.5383E 02 5.3234E 02 5.1207E 02 4.9294E 02 4.7485E 02 4.5774E 02 4.4153E 02 4.2618E 02 4.1161E 02 3.9777E 02 3.8462E 02 3.7212E 02 3.6022E 02 3.4888E 02

s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Zn; Z 30 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj



j f sj



315


60 keV

s

90 keV

s

j f sj



s

1.2528E 01 1.4161E 01 1.8338E 01 2.3430E 01 2.8416E 01 3.3166E 01 3.7827E 01 4.2522E 01 4.7308E 01 5.2189E 01 5.7141E 01 6.2120E 01 6.7075E 01 7.1955E 01 7.6713E 01 8.1313E 01 8.5726E 01 8.9938E 01 9.3941E 01 9.7739E 01 1.0134E+00 1.0475E+00 1.0800E+00 1.1109E+00 1.1405E+00 1.1688E+00 1.1960E+00 1.2223E+00 1.2477E+00 1.2724E+00 1.2964E+00 1.3199E+00 1.3428E+00 1.3652E+00 1.3871E+00 1.4087E+00 1.4298E+00 1.4506E+00 1.4710E+00 1.4911E+00 1.5108E+00 1.5302E+00 1.5493E+00 1.5681E+00 1.5865E+00 1.6047E+00 1.6226E+00 1.6402E+00 1.6575E+00 1.6745E+00 1.6913E+00 1.7077E+00 1.7239E+00 1.7399E+00 1.7555E+00 1.7709E+00 1.7861E+00 1.8010E+00 1.8157E+00 1.8302E+00 1.8444E+00

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ga; Z 31 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



316


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ge; Z 32 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



317


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) As; Z 33 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.7657E 01 3.1195E 01 4.1229E 01 5.5772E 01 7.1917E 01 8.7454E 01 1.0169E+00 1.1488E+00 1.2753E+00 1.4000E+00 1.5248E+00 1.6503E+00 1.7758E+00 1.9007E+00 2.0237E+00 2.1441E+00 2.2609E+00 2.3738E+00 2.4826E+00 2.5871E+00 2.6875E+00 2.7841E+00 2.8770E+00 2.9665E+00 3.0530E+00 3.1367E+00 3.2177E+00 3.2964E+00 3.3729E+00 3.4474E+00 3.5201E+00 3.5910E+00 3.6603E+00 3.7281E+00 3.7944E+00 3.8594E+00 3.9231E+00 3.9855E+00 4.0469E+00 4.1071E+00 4.1662E+00 4.2243E+00 4.2815E+00 4.3377E+00 4.3930E+00 4.4474E+00 4.5009E+00 4.5537E+00 4.6057E+00 4.6569E+00 4.7074E+00 4.7571E+00 4.8062E+00 4.8546E+00 4.9023E+00 4.9494E+00 4.9959E+00 5.0417E+00 5.0870E+00 5.1317E+00 5.1758E+00


j f sj



318


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Se; Z 34 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



319


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Br; Z 35 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



320


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Kr; Z 36 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj

2.0669E 01 2.2246E 01 2.6884E 01 3.4187E 01 4.3323E 01 5.3184E 01 6.2809E 01 7.1702E 01 7.9811E 01 8.7311E 01 9.4426E 01 1.0134E+00 1.0815E+00 1.1491E+00 1.2164E+00 1.2834E+00 1.3496E+00 1.4147E+00 1.4785E+00 1.5406E+00 1.6009E+00 1.6591E+00 1.7153E+00 1.7694E+00 1.8214E+00 1.8715E+00 1.9196E+00 1.9660E+00 2.0107E+00 2.0538E+00 2.0955E+00 2.1359E+00 2.1751E+00 2.2132E+00 2.2502E+00 2.2863E+00 2.3215E+00 2.3560E+00 2.3897E+00 2.4226E+00 2.4550E+00 2.4867E+00 2.5179E+00 2.5485E+00 2.5786E+00 2.6082E+00 2.6373E+00 2.6660E+00 2.6942E+00 2.7220E+00 2.7494E+00 2.7764E+00 2.8029E+00 2.8292E+00 2.8550E+00 2.8805E+00 2.9056E+00 2.9304E+00 2.9548E+00 2.9789E+00 3.0027E+00


321


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Rb; Z 37 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



322


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Sr; Z 38 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.2526E 01 3.0252E 01 4.8384E 01 6.6408E 01 8.3296E 01 1.0073E+00 1.1853E+00 1.3580E+00 1.5193E+00 1.6683E+00 1.8072E+00 1.9391E+00 2.0663E+00 2.1908E+00 2.3135E+00 2.4347E+00 2.5542E+00 2.6716E+00 2.7867E+00 2.8990E+00 3.0084E+00 3.1147E+00 3.2178E+00 3.3179E+00 3.4150E+00 3.5092E+00 3.6008E+00 3.6899E+00 3.7767E+00 3.8613E+00 3.9438E+00 4.0245E+00 4.1034E+00 4.1806E+00 4.2561E+00 4.3302E+00 4.4029E+00 4.4742E+00 4.5442E+00 4.6129E+00 4.6805E+00 4.7469E+00 4.8122E+00 4.8764E+00 4.9396E+00 5.0017E+00 5.0630E+00 5.1233E+00 5.1827E+00 5.2412E+00 5.2989E+00 5.3558E+00 5.4119E+00 5.4672E+00 5.5218E+00 5.5757E+00 5.6289E+00 5.6814E+00 5.7333E+00 5.7846E+00 5.8352E+00


j f sj



323


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Y; Z 39 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



324


60 keV

s

90 keV

s

j f sj


1.4482E+01 1.1145E+01 6.9418E+00 4.7031E+00 3.4509E+00 2.6368E+00 2.0730E+00 1.6733E+00 1.3841E+00 1.1691E+00 1.0042E+00 8.7329E 01 7.6771E 01 6.8019E 01 6.0626E 01 5.4321E 01 4.8899E 01 4.4209E 01 4.0131E 01 3.6571E 01 3.3451E 01 3.0707E 01 2.8286E 01 2.6140E 01 2.4232E 01 2.2528E 01 2.1002E 01 1.9629E 01 1.8390E 01 1.7268E 01 1.6248E 01 1.5319E 01 1.4469E 01 1.3689E 01 1.2973E 01 1.2312E 01 1.1702E 01 1.1137E 01 1.0612E 01 1.0125E 01 9.6709E 02 9.2472E 02 8.8512E 02 8.4805E 02 8.1329E 02 7.8066E 02 7.4998E 02 7.2109E 02 6.9387E 02 6.6818E 02 6.4390E 02 6.2094E 02 5.9920E 02 5.7860E 02 5.5905E 02 5.4049E 02 5.2284E 02 5.0606E 02 4.9008E 02 4.7485E 02 4.6033E 02

s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Zr; Z 40 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



325


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Nb; Z 41 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



326


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Mo; Z 42 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



327


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Tc; Z 43 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



328


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ru; Z 44 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



329


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Rh; Z 45 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



330


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pd; Z 46 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



331


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ag; Z 47 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



332


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Cd; Z 48 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



333


60 keV

s

90 keV

s

j f sj


1.0307E+01 9.0722E+00 6.8996E+00 5.1962E+00 4.0051E+00 3.1449E+00 2.5056E+00 2.0245E+00 1.6605E+00 1.3835E+00 1.1707E+00 1.0049E+00 8.7457E 01 7.6999E 01 6.8434E 01 6.1315E 01 5.5309E 01 5.0174E 01 4.5738E 01 4.1870E 01 3.8471E 01 3.5467E 01 3.2798E 01 3.0416E 01 2.8282E 01 2.6363E 01 2.4633E 01 2.3068E 01 2.1648E 01 2.0356E 01 1.9178E 01 1.8101E 01 1.7113E 01 1.6205E 01 1.5369E 01 1.4597E 01 1.3883E 01 1.3221E 01 1.2606E 01 1.2033E 01 1.1500E 01 1.1001E 01 1.0535E 01 1.0098E 01 9.6878E 02 9.3026E 02 8.9404E 02 8.5991E 02 8.2773E 02 7.9736E 02 7.6865E 02 7.4149E 02 7.1577E 02 6.9139E 02 6.6826E 02 6.4630E 02 6.2542E 02 6.0557E 02 5.8666E 02 5.6865E 02 5.5148E 02

s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) In; Z 49 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



334


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Sn; Z 50 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



335


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Sb; Z 51 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



336


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Te; Z 52 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



337


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) I; Z 53 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



338


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Xe; Z 54 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



339


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Cs; Z 55 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



340


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ba; Z 56 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.7398E 01 3.7833E 01 6.0578E 01 8.2674E 01 1.0484E+00 1.2793E+00 1.5068E+00 1.7226E+00 1.9272E+00 2.1246E+00 2.3179E+00 2.5071E+00 2.6915E+00 2.8695E+00 3.0401E+00 3.2030E+00 3.3585E+00 3.5072E+00 3.6501E+00 3.7878E+00 3.9212E+00 4.0508E+00 4.1770E+00 4.3001E+00 4.4203E+00 4.5379E+00 4.6528E+00 4.7653E+00 4.8753E+00 4.9830E+00 5.0885E+00 5.1919E+00 5.2933E+00 5.3928E+00 5.4906E+00 5.5868E+00 5.6814E+00 5.7746E+00 5.8665E+00 5.9572E+00 6.0468E+00 6.1354E+00 6.2230E+00 6.3096E+00 6.3955E+00 6.4805E+00 6.5648E+00 6.6484E+00 6.7312E+00 6.8135E+00 6.8951E+00 6.9760E+00 7.0564E+00 7.1362E+00 7.2154E+00 7.2941E+00 7.3722E+00 7.4498E+00 7.5268E+00 7.6033E+00 7.6792E+00


j f sj



341


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) La; Z 57 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.9216E 01 3.8584E 01 6.0440E 01 8.3544E 01 1.0653E+00 1.3012E+00 1.5342E+00 1.7554E+00 1.9639E+00 2.1634E+00 2.3572E+00 2.5466E+00 2.7314E+00 2.9103E+00 3.0823E+00 3.2470E+00 3.4045E+00 3.5553E+00 3.7002E+00 3.8400E+00 3.9753E+00 4.1068E+00 4.2349E+00 4.3599E+00 4.4820E+00 4.6015E+00 4.7184E+00 4.8329E+00 4.9449E+00 5.0546E+00 5.1621E+00 5.2674E+00 5.3707E+00 5.4721E+00 5.5717E+00 5.6695E+00 5.7657E+00 5.8604E+00 5.9538E+00 6.0459E+00 6.1368E+00 6.2265E+00 3.2019E 02 1.1973E 01 2.0651E 01 2.9241E 01 3.7747E 01 4.6173E 01 5.4525E 01 6.2806E 01 7.1017E 01 7.9159E 01 8.7234E 01 9.5245E 01 1.0319E+00 1.1109E+00 1.1892E+00 1.2670E+00 1.3441E+00 1.4207E+00 1.4968E+00


j f sj



342


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ce; Z 58 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.9574E 01 3.8819E 01 6.0425E 01 8.3296E 01 1.0601E+00 1.2941E+00 1.5271E+00 1.7499E+00 1.9606E+00 2.1618E+00 2.3569E+00 2.5474E+00 2.7333E+00 2.9137E+00 3.0874E+00 3.2540E+00 3.4136E+00 3.5664E+00 3.7133E+00 3.8549E+00 3.9920E+00 4.1252E+00 4.2549E+00 4.3815E+00 4.5051E+00 4.6260E+00 4.7444E+00 4.8604E+00 4.9739E+00 5.0850E+00 5.1939E+00 5.3006E+00 5.4052E+00 5.5079E+00 5.6088E+00 5.7079E+00 5.8053E+00 5.9012E+00 5.9956E+00 6.0887E+00 6.1807E+00 6.2714E+00 7.7888E 02 1.6656E 01 2.5429E 01 3.4114E 01 4.2713E 01 5.1234E 01 5.9680E 01 6.8057E 01 7.6366E 01 8.4607E 01 9.2783E 01 1.0090E+00 1.0895E+00 1.1695E+00 1.2490E+00 1.3278E+00 1.4062E+00 1.4840E+00 1.5612E+00


j f sj



343


60 keV

s

90 keV

s

j f sj


1.9420E+01 1.4750E+01 9.2733E+00 6.3846E+00 4.6891E+00 3.5719E+00 2.8071E+00 2.2688E+00 1.8761E+00 1.5788E+00 1.3467E+00 1.1613E+00 1.0111E+00 8.8800E 01 7.8631E 01 7.0158E 01 6.3038E 01 5.7002E 01 5.1837E 01 4.7379E 01 4.3497E 01 4.0090E 01 3.7080E 01 3.4406E 01 3.2016E 01 2.9871E 01 2.7938E 01 2.6189E 01 2.4602E 01 2.3159E 01 2.1842E 01 2.0638E 01 1.9533E 01 1.8518E 01 1.7581E 01 1.6717E 01 1.5918E 01 1.5177E 01 1.4489E 01 1.3847E 01 1.3249E 01 1.2689E 01 1.2166E 01 1.1675E 01 1.1215E 01 1.0781E 01 1.0372E 01 9.9866E 02 9.6227E 02 9.2789E 02 8.9534E 02 8.6447E 02 8.3517E 02 8.0735E 02 7.8091E 02 7.5580E 02 7.3189E 02 7.0910E 02 6.8736E 02 6.6661E 02 6.4681E 02

s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pr; Z 59 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.8608E 01 3.8583E 01 6.0624E 01 8.2224E 01 1.0364E+00 1.2621E+00 1.4905E+00 1.7116E+00 1.9222E+00 2.1243E+00 2.3206E+00 2.5126E+00 2.7001E+00 2.8822E+00 3.0578E+00 3.2264E+00 3.3880E+00 3.5428E+00 3.6916E+00 3.8350E+00 3.9739E+00 4.1087E+00 4.2401E+00 4.3681E+00 4.4933E+00 4.6157E+00 4.7356E+00 4.8530E+00 4.9680E+00 5.0806E+00 5.1909E+00 5.2990E+00 5.4050E+00 5.5091E+00 5.6113E+00 5.7116E+00 5.8103E+00 5.9074E+00 6.0030E+00 6.0973E+00 6.1903E+00 6.2821E+00 8.9579E 02 1.7926E 01 2.6799E 01 3.5581E 01 4.4278E 01 5.2894E 01 6.1437E 01 6.9912E 01 7.8320E 01 8.6661E 01 9.4937E 01 1.0315E+00 1.1131E+00 1.1942E+00 1.2748E+00 1.3548E+00 1.4342E+00 1.5132E+00 1.5917E+00


j f sj



344


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Nd; Z 60 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.8873E 01 3.8730E 01 6.0589E 01 8.2028E 01 1.0317E+00 1.2549E+00 1.4824E+00 1.7040E+00 1.9158E+00 2.1191E+00 2.3165E+00 2.5094E+00 2.6978E+00 2.8810E+00 3.0581E+00 3.2284E+00 3.3918E+00 3.5485E+00 3.6990E+00 3.8442E+00 3.9848E+00 4.1212E+00 4.2540E+00 4.3836E+00 4.5101E+00 4.6340E+00 4.7553E+00 4.8741E+00 4.9906E+00 5.1046E+00 5.2163E+00 5.3258E+00 5.4332E+00 5.5387E+00 5.6421E+00 5.7438E+00 5.8437E+00 5.9420E+00 6.0387E+00 6.1341E+00 6.2282E+00 3.7860E 02 1.2957E 01 2.2024E 01 3.0993E 01 3.9870E 01 4.8660E 01 5.7369E 01 6.6005E 01 7.4573E 01 8.3073E 01 9.1508E 01 9.9881E 01 1.0819E+00 1.1645E+00 1.2466E+00 1.3282E+00 1.4093E+00 1.4899E+00 1.5700E+00 1.6496E+00


j f sj



345


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pm; Z 61 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

2.9104E 01 3.8845E 01 6.0533E 01 8.1824E 01 1.0271E+00 1.2476E+00 1.4737E+00 1.6955E+00 1.9083E+00 2.1126E+00 2.3109E+00 2.5045E+00 2.6938E+00 2.8781E+00 3.0565E+00 3.2283E+00 3.3934E+00 3.5519E+00 3.7042E+00 3.8511E+00 3.9932E+00 4.1312E+00 4.2655E+00 4.3964E+00 4.5244E+00 4.6497E+00 4.7723E+00 4.8926E+00 5.0104E+00 5.1258E+00 5.2389E+00 5.3498E+00 5.4586E+00 5.5654E+00 5.6702E+00 5.7731E+00 5.8743E+00 5.9738E+00 6.0718E+00 6.1683E+00 6.2635E+00 7.4180E 02 1.6693E 01 2.5860E 01 3.4927E 01 4.3899E 01 5.2783E 01 6.1585E 01 7.0313E 01 7.8972E 01 8.7564E 01 9.6091E 01 1.0456E+00 1.1296E+00 1.2132E+00 1.2963E+00 1.3789E+00 1.4610E+00 1.5426E+00 1.6238E+00 1.7046E+00


j f sj



346


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Sm; Z 62 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.9309E 01 3.8936E 01 6.0457E 01 8.1613E 01 1.0224E+00 1.2402E+00 1.4648E+00 1.6864E+00 1.8998E+00 2.1051E+00 2.3041E+00 2.4984E+00 2.6884E+00 2.8737E+00 3.0533E+00 3.2265E+00 3.3932E+00 3.5533E+00 3.7073E+00 3.8558E+00 3.9995E+00 4.1389E+00 4.2746E+00 4.4070E+00 4.5363E+00 4.6629E+00 4.7870E+00 4.9085E+00 5.0277E+00 5.1445E+00 5.2590E+00 5.3712E+00 5.4814E+00 5.5895E+00 5.6957E+00 5.7999E+00 5.9023E+00 6.0031E+00 6.1023E+00 6.1999E+00 1.3042E 02 1.0804E 01 2.0184E 01 2.9453E 01 3.8619E 01 4.7688E 01 5.6666E 01 6.5561E 01 7.4381E 01 8.3131E 01 9.1814E 01 1.0043E+00 1.0899E+00 1.1749E+00 1.2594E+00 1.3434E+00 1.4270E+00 1.5101E+00 1.5928E+00 1.6751E+00 1.7569E+00


j f sj



347


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Eu; Z 63 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

2.9531E 01 3.9035E 01 6.0373E 01 8.1406E 01 1.0181E+00 1.2334E+00 1.4565E+00 1.6778E+00 1.8918E+00 2.0978E+00 2.2976E+00 2.4925E+00 2.6832E+00 2.8693E+00 3.0500E+00 3.2246E+00 3.3927E+00 3.5544E+00 3.7099E+00 3.8600E+00 4.0052E+00 4.1461E+00 4.2832E+00 4.4169E+00 4.5475E+00 4.6754E+00 4.8007E+00 4.9236E+00 5.0441E+00 5.1622E+00 5.2780E+00 5.3916E+00 5.5031E+00 5.6126E+00 5.7201E+00 5.8256E+00 5.9293E+00 6.0313E+00 6.1317E+00 6.2306E+00 4.4808E 02 1.4091E 01 2.3577E 01 3.2950E 01 4.2216E 01 5.1383E 01 6.0456E 01 6.9445E 01 7.8356E 01 8.7197E 01 9.5970E 01 1.0468E+00 1.1333E+00 1.2192E+00 1.3046E+00 1.3896E+00 1.4741E+00 1.5582E+00 1.6419E+00 1.7251E+00 1.8081E+00


j f sj



348


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Gd; Z 64 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

3.0927E 01 3.9480E 01 5.9894E 01 8.1801E 01 1.0307E+00 1.2497E+00 1.4745E+00 1.6977E+00 1.9136E+00 2.1212E+00 2.3219E+00 2.5173E+00 2.7083E+00 2.8948E+00 3.0761E+00 3.2517E+00 3.4211E+00 3.5842E+00 3.7413E+00 3.8928E+00 4.0394E+00 4.1816E+00 4.3200E+00 4.4549E+00 4.5868E+00 4.7159E+00 4.8425E+00 4.9665E+00 5.0883E+00 5.2076E+00 5.3248E+00 5.4397E+00 5.5525E+00 5.6633E+00 5.7721E+00 5.8789E+00 5.9839E+00 6.0871E+00 6.1887E+00 5.5211E 03 1.0408E 01 2.0128E 01 2.9720E 01 3.9195E 01 4.8560E 01 5.7823E 01 6.6990E 01 7.6071E 01 8.5072E 01 9.3999E 01 1.0286E+00 1.1165E+00 1.2039E+00 1.2907E+00 1.3770E+00 1.4628E+00 1.5482E+00 1.6332E+00 1.7178E+00 1.8021E+00 1.8860E+00


j f sj



349


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Tb; Z 65 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj



j f sj



350


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Dy; Z 66 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj



j f sj



351


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ho; Z 67 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

3.0231E 01 3.9271E 01 5.9908E 01 8.0493E 01 1.0009E+00 1.2061E+00 1.4215E+00 1.6395E+00 1.8537E+00 2.0615E+00 2.2633E+00 2.4601E+00 2.6527E+00 2.8412E+00 3.0252E+00 3.2039E+00 3.3770E+00 3.5440E+00 3.7052E+00 3.8609E+00 4.0115E+00 4.1576E+00 4.2998E+00 4.4384E+00 4.5738E+00 4.7064E+00 4.8365E+00 4.9640E+00 5.0893E+00 5.2123E+00 5.3331E+00 5.4518E+00 5.5684E+00 5.6830E+00 5.7955E+00 5.9062E+00 6.0149E+00 6.1219E+00 6.2272E+00 4.7612E 02 1.4970E 01 2.5032E 01 3.4959E 01 4.4758E 01 5.4438E 01 6.4007E 01 7.3471E 01 8.2841E 01 9.2125E 01 1.0133E+00 1.1046E+00 1.1952E+00 1.2852E+00 1.3747E+00 1.4636E+00 1.5521E+00 1.6402E+00 1.7280E+00 1.8154E+00 1.9025E+00 1.9894E+00


j f sj



352


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Er; Z 68 10 keV

s

j f sj

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 keV

s

j f sj

3.0375E 01 3.9302E 01 5.9766E 01 8.0248E 01 9.9659E 01 1.1994E+00 1.4127E+00 1.6294E+00 1.8433E+00 2.0512E+00 2.2533E+00 2.4504E+00 2.6433E+00 2.8323E+00 3.0168E+00 3.1964E+00 3.3703E+00 3.5385E+00 3.7009E+00 3.8578E+00 4.0097E+00 4.1571E+00 4.3004E+00 4.4401E+00 4.5766E+00 4.7103E+00 4.8414E+00 4.9701E+00 5.0964E+00 5.2205E+00 5.3425E+00 5.4623E+00 5.5801E+00 5.6958E+00 5.8096E+00 5.9215E+00 6.0314E+00 6.1396E+00 6.2461E+00 6.7708E 02 1.7095E 01 2.7272E 01 3.7310E 01 4.7217E 01 5.7003E 01 6.6673E 01 7.6238E 01 8.5704E 01 9.5082E 01 1.0438E+00 1.1360E+00 1.2275E+00 1.3184E+00 1.4087E+00 1.4985E+00 1.5878E+00 1.6768E+00 1.7654E+00 1.8537E+00 1.9417E+00 2.0294E+00


j f sj



353


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Tm; Z 69 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

3.0510E 01 3.9326E 01 5.9615E 01 7.9997E 01 9.9236E 01 1.1928E+00 1.4039E+00 1.6193E+00 1.8326E+00 2.0406E+00 2.2429E+00 2.4402E+00 2.6334E+00 2.8227E+00 3.0078E+00 3.1880E+00 3.3629E+00 3.5321E+00 3.6957E+00 3.8538E+00 4.0068E+00 4.1553E+00 4.2998E+00 4.4406E+00 4.5782E+00 4.7129E+00 4.8450E+00 4.9747E+00 5.1021E+00 5.2273E+00 5.3503E+00 5.4712E+00 5.5901E+00 5.7070E+00 5.8220E+00 5.9350E+00 6.0462E+00 6.1556E+00 6.2632E+00 8.6016E 02 1.9042E 01 2.9331E 01 3.9480E 01 4.9496E 01 5.9386E 01 6.9160E 01 7.8825E 01 8.8388E 01 9.7861E 01 1.0725E+00 1.1656E+00 1.2580E+00 1.3497E+00 1.4409E+00 1.5315E+00 1.6217E+00 1.7115E+00 1.8009E+00 1.8901E+00 1.9790E+00 2.0676E+00


j f sj



354


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Yb; Z 70 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

3.0638E 01 3.9343E 01 5.9457E 01 7.9742E 01 9.8816E 01 1.1863E+00 1.3952E+00 1.6092E+00 1.8219E+00 2.0297E+00 2.2321E+00 2.4297E+00 2.6231E+00 2.8126E+00 2.9981E+00 3.1790E+00 3.3547E+00 3.5249E+00 3.6895E+00 3.8487E+00 4.0029E+00 4.1526E+00 4.2981E+00 4.4399E+00 4.5785E+00 4.7143E+00 4.8474E+00 4.9780E+00 5.1064E+00 5.2326E+00 5.3567E+00 5.4787E+00 5.5987E+00 5.7167E+00 5.8328E+00 5.9470E+00 6.0593E+00 6.1699E+00 6.2787E+00 1.0263E 01 2.0817E 01 3.1219E 01 4.1477E 01 5.1601E 01 6.1597E 01 7.1474E 01 8.1238E 01 9.0899E 01 1.0047E+00 1.0995E+00 1.1935E+00 1.2867E+00 1.3793E+00 1.4713E+00 1.5628E+00 1.6538E+00 1.7444E+00 1.8347E+00 1.9247E+00 2.0144E+00 2.1039E+00


j f sj



355


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Lu; Z 71 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj

3.1759E 01 3.9524E 01 5.8749E 01 8.0077E 01 1.0028E+00 1.2058E+00 1.4161E+00 1.6305E+00 1.8436E+00 2.0521E+00 2.2551E+00 2.4530E+00 2.6465E+00 2.8361E+00 3.0218E+00 3.2030E+00 3.3794E+00 3.5504E+00 3.7160E+00 3.8763E+00 4.0315E+00 4.1822E+00 4.3287E+00 4.4716E+00 4.6112E+00 4.7478E+00 4.8818E+00 5.0134E+00 5.1427E+00 5.2699E+00 5.3949E+00 5.5179E+00 5.6390E+00 5.7581E+00 5.8752E+00 5.9905E+00 6.1040E+00 6.2156E+00 4.2364E 02 1.5062E 01 2.5726E 01 3.6238E 01 4.6604E 01 5.6833E 01 6.6932E 01 7.6908E 01 8.6771E 01 9.6528E 01 1.0619E+00 1.1576E+00 1.2524E+00 1.3465E+00 1.4400E+00 1.5328E+00 1.6251E+00 1.7169E+00 1.8083E+00 1.8993E+00 1.9901E+00 2.0806E+00 2.1709E+00


j f sj



356


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Hf; Z 72 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



357


60 keV

s

90 keV

s

j f sj


1.4316E+01 1.1875E+01 8.2428E+00 5.9438E+00 4.5536E+00 3.6138E+00 2.9320E+00 2.4215E+00 2.0319E+00 1.7291E+00 1.4891E+00 1.2950E+00 1.1367E+00 1.0053E+00 8.9498E 01 8.0166E 01 7.2222E 01 6.5419E 01 5.9559E 01 5.4482E 01 5.0057E 01 4.6176E 01 4.2750E 01 3.9708E 01 3.6992E 01 3.4552E 01 3.2351E 01 3.0355E 01 2.8540E 01 2.6882E 01 2.5365E 01 2.3972E 01 2.2691E 01 2.1509E 01 2.0418E 01 1.9410E 01 1.8475E 01 1.7607E 01 1.6801E 01 1.6051E 01 1.5352E 01 1.4699E 01 1.4090E 01 1.3519E 01 1.2984E 01 1.2483E 01 1.2011E 01 1.1567E 01 1.1148E 01 1.0754E 01 1.0380E 01 1.0027E 01 9.6929E 02 9.3756E 02 9.0743E 02 8.7879E 02 8.5153E 02 8.2556E 02 8.0080E 02 7.7716E 02 7.5458E 02

s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ta; Z 73 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



358


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) W; Z 74 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



359


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Re; Z 75 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



360


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Os; Z 76 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



361


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ir; Z 77 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



362


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pt; Z 78 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



363


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Au; Z 79 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



364


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Hg; Z 80 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



365


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Tl; Z 81 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



366


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pb; Z 82 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



367


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Bi; Z 83 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



368


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Po; Z 84 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



369


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) At; Z 85 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



370


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Rn; Z 86 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



371


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Fr; Z 87 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



372


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ra; Z 88 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



373


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Ac; Z 89 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



374


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Th; Z 90 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



375


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) Pa; Z 91 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



376


60 keV

s

90 keV

s

j f sj



s


4.3. ELECTRON DIFFRACTION Table 4.3.3.1. Partial wave elastic scattering factors for neutral atoms (cont.) U; Z 92 s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j f sj

10 keV


40 keV

s

j f sj



j f sj



377


60 keV

s

90 keV

s

j f sj



s


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.2. Inelastic scattering factors Element Z s

H 1

He 2

Li 3

Be 4

B 5

C 6

N 7

O 8

F 9

0 1 2 3 4 5 6 7 8 9 10

0.00000 0.23712 0.62750 0.85836 0.95050 0.98252 0.99349 0.99740 0.99889 0.99949 0.99976

0.00000 0.20585 0.67665 1.15502 1.50584 1.72314 1.84691 1.91496 1.95208 1.97248 1.98385

0.00000 0.88290 1.29649 1.58907 1.89856 2.17978 2.41052 2.58662 2.71445 2.80427 2.86613

0.00000 1.11410 2.05741 2.33547 2.54347 2.76339 2.97864 3.17603 3.34823 3.49247 3.60942

0.00000 0.97969 2.35860 3.03124 3.31571 3.50570 3.68270 3.85613 4.02198 4.17553 4.31351

0.00000 0.82589 2.30155 3.35857 3.92722 4.23471 4.43840 4.60589 4.75972 4.90533 5.04263

0.00000 0.70270 2.12384 3.37855 4.23128 4.76279 5.09842 5.32971 5.50893 5.66203 5.80060

0.00000 0.64471 2.05154 3.44994 4.52527 5.26209 5.74264 6.05888 6.28032 6.45035 6.59304

0.00000 0.58625 1.93281 3.38314 4.60924 5.53803 6.19823 6.65331 6.96851 7.19556 7.36980

11 12 13 14 15 16 17 18 19 20

0.99988 0.99993 0.99996 0.99998 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000

1.99032 1.99407 1.99629 1.99763 1.99846 1.99898 1.99931 1.99953 1.99968 1.99977

2.90828 2.93687 2.95627 2.96947 2.97851 2.98474 2.98907 2.99209 2.99423 2.99576

3.70189 3.77366 3.82863 3.87036 3.90185 3.92555 3.94336 3.95676 3.96686 3.97448

4.43432 4.53780 4.62485 4.69701 4.75615 4.80418 4.84294 4.87406 4.89897 4.91886

5.17019 5.28663 5.39115 5.48356 5.56419 5.63375 5.69320 5.74363 5.78614 5.82179

5.92915 6.04900 6.16020 6.26247 6.35555 6.43942 6.51429 6.58055 6.63878 6.68961

6.72061 6.83868 6.94951 7.05375 7.15137 7.24214 7.32588 7.40250 7.47210 7.53488

7.51316 7.63830 7.75211 7.85802 7.95762 8.05146 8.13968 8.22220 8.29896 8.36993

21 22 23 24 25 26 27 28 29 30

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

1.99984 1.99988 1.99991 1.99994 1.99995 1.99996 1.99997 1.99998 1.99998 1.99999

2.99685 2.99764 2.99822 2.99865 2.99897 2.99920 2.99938 2.99952 2.99962 2.99970

3.98027 3.98466 3.98802 3.99060 3.99259 3.99413 3.99532 3.99626 3.99699 3.99757

4.93474 4.94741 4.95752 4.96560 4.97207 4.97726 4.98143 4.98479 4.98750 4.98970

5.85157 5.87639 5.89702 5.91415 5.92836 5.94015 5.94992 5.95804 5.96477 5.97038

6.73375 6.77191 6.80478 6.83300 6.85718 6.87785 6.89550 6.91055 6.92339 6.93433

7.59117 7.64136 7.68590 7.72528 7.75997 7.79043 7.81712 7.84046 7.86084 7.87860

8.43516 8.49480 8.54904 8.59816 8.64246 8.68227 8.71793 8.74980 8.77820 8.80347

31 32 33 34 35 36 37 38 39 40

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

1.99999 1.99999 1.99999 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000

2.99976 2.99981 2.99985 2.99988 2.99990 2.99992 2.99993 2.99994 2.99995 2.99996

3.99803 3.99840 3.99869 3.99892 3.99911 3.99927 3.99939 3.99949 3.99958 3.99964

4.99149 4.99294 4.99413 4.99510 4.99590 4.99656 4.99711 4.99756 4.99794 4.99825

5.97504 5.97893 5.98217 5.98489 5.98716 5.98907 5.99068 5.99203 5.99318 5.99414

6.94365 6.95160 6.95838 6.96416 6.96910 6.97332 6.97693 6.98003 6.98268 6.98496

7.89407 7.90753 7.91925 7.92943 7.93829 7.94600 7.95270 7.95854 7.96362 7.96804

8.82591 8.84581 8.86344 8.87904 8.89284 8.90503 8.91581 8.92532 8.93373 8.94116

41 42 43 44 45 46 47 48 49 50

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000

2.99997 2.99997 2.99998 2.99998 2.99998 2.99999 2.99999 2.99999 2.99999 2.99999

3.99970 3.99975 3.99979 3.99982 3.99984 3.99987 3.99989 3.99990 3.99992 3.99993

4.99851 4.99873 4.99891 4.99907 4.99920 4.99931 4.99940 4.99948 4.99955 4.99961

5.99496 5.99566 5.99625 5.99675 5.99719 5.99755 5.99787 5.99814 5.99838 5.99858

6.98692 6.98860 6.99006 6.99131 6.99240 6.99334 6.99415 6.99486 6.99547 6.99601

7.97190 7.97526 7.97820 7.98077 7.98301 7.98498 7.98670 7.98822 7.98954 7.99071

8.94772 8.95352 8.95864 8.96317 8.96718 8.97073 8.97387 8.97666 8.97913 8.98132

51 52 53 54 55 56 57 58 59 60

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000

2.99999 2.99999 3.00000 3.00000 3.00000 3.00000 3.00000 3.00000 3.00000 3.00000

3.99994 3.99995 3.99995 3.99996 3.99996 3.99997 3.99997 3.99998 3.99998 3.99998

4.99966 4.99970 4.99974 4.99977 4.99980 4.99983 4.99985 4.99986 4.99988 4.99989

5.99876 5.99891 5.99904 5.99915 5.99925 5.99934 5.99941 5.99948 5.99954 5.99959

6.99647 6.99688 6.99724 6.99755 6.99783 6.99807 6.99828 6.99846 6.99863 6.99877

7.99174 7.99264 7.99344 7.99415 7.99477 7.99532 7.99581 7.99624 7.99663 7.99697

8.98327 8.98500 8.98654 8.98791 8.98913 8.99021 8.99119 8.99205 8.99283 8.99352

378


4.3. ELECTRON DIFFRACTION Table 4.3.3.2. Inelastic scattering factors (cont.) Element Z s

Ne 10

Na 11

Mg 12

Al 13

Si 14

P 15

S 16

Cl 17

Ar 18

0 1 2 3 4 5 6 7 8 9 10

0.00000 0.53234 1.80053 3.24475 4.54966 5.61925 6.44652 7.06089 7.50742 7.83224 8.07400

0.00000 1.20747 2.28779 3.49193 4.70014 5.77806 6.68058 7.40330 7.96372 8.39028 8.71388

0.00000 1.61544 2.94235 3.94530 5.01729 6.03833 6.94760 7.72396 8.36460 8.87926 9.28563

0.00000 1.72194 3.50721 4.52834 5.47022 6.41063 7.29048 8.07744 8.75902 9.33363 9.80759

0.00000 1.65386 3.78096 5.07436 6.01322 6.88008 7.71078 8.48211 9.17588 9.78380 10.30489

0.00000 1.53700 3.80549 5.40954 6.52767 7.40945 8.20204 8.94372 9.62983 10.25033 10.79947

0.00000 1.49816 3.95432 5.80343 7.06849 7.98702 8.75585 9.46405 10.12957 10.74677 11.30797

0.00000 1.42156 3.94695 6.00685 7.48589 8.53330 9.33634 10.03048 10.67346 11.27732 11.83831

0.00000 1.33265 3.84282 6.03450 7.72422 8.96990 9.88639 10.61498 11.25279 11.84272 12.39562

11 12 13 14 15 16 17 18 19 20

8.26134 8.41386 8.54414 8.65993 8.76576 8.86416 8.95646 9.04331 9.12498 9.20160

8.96233 9.15796 9.31737 9.45219 9.57025 9.67661 9.77440 9.86554 9.95113 10.03176

9.60450 9.85601 10.05748 10.22270 10.36206 10.48302 10.59078 10.68886 10.77955 10.86431

10.19272 10.50348 10.75458 10.95937 11.12908 11.27266 11.39691 11.50688 11.60616 11.69727

10.74365 11.10840 11.40955 11.65803 11.86416 12.03702 12.18418 12.31167 12.42419 12.52524

11.27657 11.68496 12.03077 12.32173 12.56614 12.77207 12.94685 13.09681 13.22723 13.34234

11.80888 12.24905 12.63101 12.95942 13.24013 13.47958 13.68412 13.85970 14.01162 14.14443

12.35087 12.81180 13.22075 13.57972 13.89228 14.16303 14.39702 14.59936 14.77489 14.92805

12.90995 13.38195 13.80922 14.19149 14.53031 14.82851 15.08974 15.31806 15.51761 15.69240

21 22 23 24 25 26 27 28 29 30

9.27319 9.33982 9.40157 9.45854 9.51090 9.55885 9.60262 9.64244 9.67857 9.71128

10.10777 10.17933 10.24653 10.30944 10.36817 10.42279 10.47344 10.52026 10.56342 10.60310

10.94406 11.01935 11.09051 11.15772 11.22111 11.28077 11.33679 11.38925 11.43824 11.48389

11.78195 11.86133 11.93618 12.00698 12.07405 12.13758 12.19771 12.25454 12.30814 12.35861

12.61744 12.70264 12.78217 12.85695 12.92761 12.99459 13.05816 13.11853 13.17583 13.23017

13.44547 13.53918 13.62538 13.70552 13.78062 13.85143 13.91849 13.98216 14.04271 14.10034

14.26190 14.36711 14.46250 14.54998 14.63102 14.70673 14.77793 14.84524 14.90911 14.96988

15.06274 15.18230 15.28952 15.38669 15.47564 15.55783 15.63441 15.70626 15.77405 15.83830

15.84616 15.98223 16.10354 16.21260 16.31151 16.40201 16.48552 16.56318 16.63589 16.70436

31 32 33 34 35 36 37 38 39 40

9.74083 9.76748 9.79146 9.81302 9.83238 9.84974 9.86530 9.87924 9.89171 9.90287

10.63950 10.67281 10.70323 10.73097 10.75622 10.77917 10.80002 10.81893 10.83606 10.85158

11.52631 11.56565 11.60205 11.63566 11.66665 11.69517 11.72137 11.74542 11.76747 11.78765

12.40603 12.45048 12.49206 12.53089 12.56708 12.60075 12.63202 12.66102 12.68789 12.71273

13.28163 13.33028 13.37621 13.41949 13.46020 13.49843 13.53427 13.56783 13.59919 13.62846

14.15517 14.20732 14.25686 14.30387 14.34842 14.39057 14.43041 14.46799 14.50341 14.53673

15.02777 15.08297 15.13561 15.18580 15.23360 15.27911 15.32237 15.36345 15.40243 15.43935

15.89938 15.95759 16.01315 16.06621 16.11691 16.16535 16.21159 16.25573 16.29780 16.33789

16.76915 16.83069 16.88931 16.94527 16.99876 17.04993 17.09890 17.14577 17.19062 17.23350

41 42 43 44 45 46 47 48 49 50

9.91286 9.92179 9.92977 9.93691 9.94330 9.94901 9.95412 9.95870 9.96279 9.96646

10.86562 10.87832 10.88980 10.90018 10.90956 10.91803 10.92569 10.93260 10.93885 10.94449

11.80612 11.82300 11.83842 11.85250 11.86534 11.87705 11.88773 11.89747 11.90634 11.91442

12.73569 12.75688 12.77642 12.79442 12.81100 12.82625 12.84028 12.85317 12.86501 12.87589

13.65576 13.68117 13.70481 13.72678 13.74718 13.76610 13.78364 13.79988 13.81492 13.82884

14.56805 14.59744 14.62500 14.65080 14.67494 14.69750 14.71857 14.73823 14.75656 14.77364

15.47428 15.50730 15.53847 15.56786 15.59554 15.62159 15.64608 15.66909 15.69068 15.71093

16.37603 16.41229 16.44672 16.47938 16.51033 16.53964 16.56736 16.59355 16.61828 16.64161

17.27449 17.31364 17.35099 17.38661 17.42053 17.45281 17.48351 17.51267 17.54035 17.56660

51 52 53 54 55 56 57 58 59 60

9.96974 9.97269 9.97533 9.97770 9.97983 9.98174 9.98346 9.98500 9.98639 9.98765

10.94959 10.95420 10.95837 10.96214 10.96555 10.96863 10.97143 10.97396 10.97625 10.97832

11.92178 11.92849 11.93460 11.94017 11.94525 11.94987 11.95409 11.95793 11.96143 11.96463

12.88588 12.89505 12.90346 12.91118 12.91827 12.92477 12.93074 12.93621 12.94124 12.94585

13.84172 13.85362 13.86462 13.87478 13.88417 13.89284 13.90085 13.90824 13.91507 13.92137

14.78955 14.80435 14.81813 14.83093 14.84284 14.85390 14.86418 14.87373 14.88259 14.89082

15.72991 15.74768 15.76432 15.77989 15.79444 15.80804 15.82075 15.83262 15.84370 15.85405

16.66360 16.68431 16.70381 16.72216 16.73941 16.75562 16.77085 16.78514 16.79856 16.81114

17.59148 17.61504 17.63733 17.65841 17.67833 17.69714 17.71490 17.73166 17.74746 17.76235

379


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.3.2. Inelastic scattering factors (cont.) Element Z s

K 19

Ca 20

Sc 21

Ti 22

V 23

Cr 24

Mn 25

Fe 26

Co 27

0 1 2 3 4 5 6 7 8 9 10

0.00000 1.95409 4.16743 6.28656 8.01589 9.37354 10.40504 11.20105 11.85904 12.44416 12.98647

0.00000 2.48962 4.63722 6.63788 8.34650 9.75686 10.88531 11.76628 12.47072 13.06908 13.60753

0.00000 2.47505 4.74095 6.79142 8.56715 10.05075 11.27230 12.24977 13.02927 13.67229 14.23080

0.00000 2.41147 4.73155 6.81860 8.67221 10.24289 11.56360 12.64663 13.51955 14.23079 14.83118

0.00000 2.34470 4.71014 6.82580 8.74208 10.38420 11.78746 12.96597 13.93468 14.72718 15.38717

0.00000 1.87338 4.31232 6.58097 8.59326 10.33687 11.84876 13.14017 14.22015 15.11304 15.85527

0.00000 2.20760 4.61890 6.75751 8.76139 10.51832 12.05425 13.39278 14.54015 15.50762 16.31906

0.00000 2.15605 4.62675 6.80527 8.86447 10.68060 12.27171 13.66798 14.87959 15.91441 16.78966

0.00000 2.09981 4.60110 6.79526 8.88855 10.75511 12.40221 13.86069 15.14206 16.25102 17.19864

11 12 13 14 15 16 17 18 19 20

13.49465 13.96833 14.40504 14.80303 15.16201 15.48312 15.76851 16.02103 16.24393 16.44057

14.10800 14.57744 15.01608 15.42228 15.79475 16.13321 16.43845 16.71212 16.95643 17.17399

14.73745 15.20828 15.64906 16.06067 16.44246 16.79376 17.11456 17.40559 17.66824 17.90434

15.35997 15.84156 16.28847 16.70589 17.09532 17.45682 17.79031 18.09602 18.37470 18.62758

15.95492 16.46015 16.92150 17.34908 17.74762 18.11901 18.46388 18.78257 19.07558 19.34372

16.48500 17.03426 17.52612 17.97537 18.39057 18.77626 19.13474 19.46719 19.77444 20.05728

17.00553 17.59751 18.12000 18.59107 19.02256 19.42164 19.79241 20.13716 20.45724 20.75361

17.53092 18.16629 18.72120 19.21579 19.66452 20.07700 20.45925 20.81484 21.14592 21.45382

18.00479 18.69422 19.29164 19.81842 20.29119 20.72194 21.11878 21.48693 21.82968 22.14909

21 22 23 24 25 26 27 28 29 30

16.61428 16.76819 16.90520 17.02787 17.13845 17.23887 17.33077 17.41550 17.49419 17.56774

17.36753 17.53980 17.69348 17.83103 17.95471 18.06655 18.16830 18.26148 18.34738 18.42708

18.11606 18.30570 18.47561 18.62806 18.76522 18.88908 19.00144 19.10390 19.19786 19.28454

18.85623 19.06249 19.24834 19.41579 19.56682 19.70334 19.82710 19.93973 20.04268 20.13724

19.58811 19.81018 20.01151 20.19384 20.35893 20.50852 20.64429 20.76783 20.88059 20.98391

20.31668 20.55380 20.77001 20.96677 21.14567 21.30828 21.45620 21.59092 21.71390 21.82646

21.02715 21.27883 21.50974 21.72111 21.91428 22.09062 22.25155 22.39849 22.53279 22.65576

21.73952 22.00392 22.24793 22.47258 22.67895 22.86824 23.04170 23.20059 23.34616 23.47965

22.44651 22.72298 22.97940 23.21667 23.43574 23.63764 23.82343 23.99424 24.15122 24.29549

31 32 33 34 35 36 37 38 39 40

17.63689 17.70222 17.76420 17.82320 17.87951 17.93334 17.98488 18.03427 18.08161 18.12700

18.50150 18.57136 18.63728 18.69974 18.75913 18.81578 18.86994 18.92181 18.97154 19.01926

19.36496 19.44000 19.51040 19.57676 19.63959 19.69928 19.75618 19.81056 19.86264 19.91259

20.22456 20.30560 20.38121 20.45211 20.51889 20.58206 20.64204 20.69918 20.75377 20.80603

21.07897 21.16684 21.24845 21.32459 21.39596 21.46316 21.52669 21.58697 21.64436 21.69915

21.92983 22.02511 22.11329 22.19525 22.27176 22.34348 22.41099 22.47478 22.53529 22.59286

22.76860 22.87245 22.96833 23.05717 23.13979 23.21694 23.28926 23.35731 23.42158 23.48250

23.60223 23.71501 23.81902 23.91523 24.00449 24.08758 24.16522 24.23800 24.30650 24.37117

24.42819 24.55037 24.66306 24.76720 24.86369 24.95334 25.03688 25.11498 25.18823 25.25716

41 42 43 44 45 46 47 48 49 50

18.17051 18.21221 18.25215 18.29039 18.32697 18.36193 18.39533 18.42720 18.45760 18.48657

19.06508 19.10909 19.15135 19.19192 19.23087 19.26822 19.30404 19.33836 19.37122 19.40267

19.96056 20.00667 20.05100 20.09364 20.13466 20.17411 20.21204 20.24849 20.28352 20.31714

20.85616 20.90432 20.95063 20.99521 21.03813 21.07948 21.11931 21.15767 21.19462 21.23020

21.75159 21.80189 21.85021 21.89670 21.94146 21.98460 22.02620 22.06632 22.10502 22.14236

22.64781 22.70038 22.75079 22.79921 22.84580 22.89069 22.93396 22.97572 23.01604 23.05497

23.54043 23.59568 23.64851 23.69914 23.74777 23.79455 23.83961 23.88308 23.92503 23.96556

24.43245 24.49069 24.54620 24.59926 24.65009 24.69889 24.74583 24.79104 24.83464 24.87675

25.32224 25.38389 25.44246 25.49826 25.55156 25.60260 25.65158 25.69868 25.74404 25.78778

51 52 53 54 55 56 57 58 59 60

18.51415 18.54040 18.56535 18.58906 18.61157 18.63294 18.65319 18.67240 18.69058 18.70781

19.43274 19.46147 19.48890 19.51509 19.54005 19.56385 19.58651 19.60808 19.62860 19.64811

20.34941 20.38036 20.41003 20.43845 20.46565 20.49168 20.51656 20.54034 20.56305 20.58472

21.26443 21.29737 21.32904 21.35948 21.38871 21.41677 21.44370 21.46952 21.49427 21.51797

22.17837 22.21309 22.24656 22.27882 22.30989 22.33980 22.36859 22.39628 22.42290 22.44848

23.09258 23.12890 23.16398 23.19785 23.23056 23.26212 23.29257 23.32193 23.35024 23.37752

24.00473 24.04261 24.07923 24.11464 24.14889 24.18200 24.21401 24.24495 24.27484 24.30371

24.91744 24.95678 24.99485 25.03170 25.06736 25.10189 25.13532 25.16768 25.19900 25.22931

25.83002 25.87085 25.91035 25.94858 25.98561 26.02148 26.05624 26.08992 26.12257 26.15420

380



Ni 28

Cu 29

Zn 30

Ga 31

Ge 32

As 33

Se 34

Br 35

Kr 36

0 1 2 3 4 5 6 7 8 9 10

0.00000 2.04471 4.57025 6.78094 8.90504 10.81621 12.51132 14.02217 15.36345 16.53914 17.55583

0.00000 1.69573 4.20291 6.60632 8.78952 10.72987 12.47751 14.05989 15.48041 16.73670 17.83189

0.00000 1.93762 4.48366 6.71087 8.87296 10.85168 12.62492 14.22287 15.66617 16.96032 18.10676

0.00000 2.05186 4.79699 6.99727 9.07164 11.03258 12.81781 14.42883 15.88673 17.20429 18.38602

0.00000 2.02388 4.96356 7.31756 9.37130 11.28885 13.06230 14.67729 16.14378 17.47556 18.68056

0.00000 1.94862 4.96762 7.53254 9.69780 11.60839 13.35790 14.96447 16.43328 17.77364 18.99403

0.00000 1.96187 5.16391 7.87187 10.09282 11.98915 13.70612 15.29281 16.75585 18.09916 19.32906

0.00000 1.92529 5.22242 8.08782 10.44730 12.39473 14.09835 15.66034 17.10934 18.44981 19.68468

0.00000 1.86388 5.17677 8.15809 10.70005 12.77835 14.51642 16.06428 17.49432 18.82545 20.06045

11 12 13 14 15 16 17 18 19 20

18.42792 19.17588 19.82231 20.38840 20.89193 21.34665 21.76253 22.14646 22.50302 22.83520

18.77754 19.59174 20.29556 20.90979 21.45278 21.93949 22.38138 22.78682 23.16168 23.51000

19.11098 19.98462 20.74392 21.40691 21.99101 22.51151 22.98092 23.40889 23.80261 24.16724

19.43536 20.35933 21.16939 21.87992 22.50626 23.06295 23.56269 24.01592 24.43082 24.81353

19.76289 20.72723 21.58126 22.33572 23.00321 23.59669 24.12833 24.60873 25.04664 25.44900

20.09987 21.09536 21.98591 22.77927 23.48524 24.11476 24.67882 25.18761 25.65004 26.07350

20.45124 21.47006 22.38987 23.21633 23.95687 24.62031 25.21613 25.75365 26.24150 26.68718

20.81812 21.85427 22.79705 23.65097 24.42167 25.11610 25.74214 26.30797 26.82157 27.29024

21.20099 22.24990 23.21055 24.08679 24.88315 25.60513 26.25912 26.85209 27.39114 27.88309

21 22 23 24 25 26 27 28 29 30

23.14497 23.43371 23.70247 23.95219 24.18376 24.39808 24.59612 24.77888 24.94740 25.10270

23.83452 24.13716 24.41932 24.68209 24.92646 25.15335 25.36367 25.55837 25.73841 25.90478

24.50642 24.82271 25.11795 25.39353 25.65055 25.88998 26.11272 26.31965 26.51166 26.68965

25.16859 25.49929 25.80799 26.09645 26.36601 26.61776 26.85266 27.07158 27.27537 27.46487

25.82113 26.16701 26.48960 26.79109 27.07314 27.33704 27.58385 27.81449 28.02983 28.23066

26.46391 26.82591 27.16300 27.47786 27.77250 28.04847 28.30702 28.54916 28.77579 28.98774

27.09700 27.47606 27.82837 28.15706 28.46453 28.75264 29.02284 29.27632 29.51404 29.73688

27.72039 28.11740 28.48569 28.82879 29.14946 29.44989 29.73179 29.99650 30.24515 30.47868

28.33419 28.74992 29.13493 29.49305 29.82740 30.14045 30.43417 30.71014 30.96963 31.21369

31 32 33 34 35 36 37 38 39 40

25.24584 25.37784 25.49968 25.61229 25.71658 25.81336 25.90339 25.98737 26.06594 26.13966

26.05845 26.20040 26.33158 26.45293 26.56530 26.66955 26.76643 26.85668 26.94095 27.01985

26.85452 27.00720 27.14857 27.27951 27.40089 27.51352 27.61817 27.71557 27.80641 27.89131

27.64092 27.80439 27.95609 28.09689 28.22758 28.34897 28.46182 28.56685 28.66476 28.75617

28.41779 28.59202 28.75413 28.90492 29.04515 29.17560 29.29699 29.41005 29.51546 29.61385

29.18578 29.37066 29.54314 29.70396 29.85384 29.99353 30.12372 30.24510 30.35836 30.46411

29.94561 30.14099 30.32373 30.49453 30.65409 30.80309 30.94223 31.07215 31.19351 31.30692

30.69790 30.90357 31.09639 31.27706 31.44623 31.60456 31.75270 31.89127 32.02090 32.14220

31.44321 31.65898 31.86171 32.05209 32.23076 32.39834 32.55546 32.70272 32.84072 32.97004

41 42 43 44 45 46 47 48 49 50

26.20905 26.27456 26.33659 26.39551 26.45162 26.50519 26.55647 26.60565 26.65293 26.69844

27.09393 27.16367 27.22952 27.29187 27.35108 27.40746 27.46126 27.51275 27.56211 27.60954

27.97085 28.04555 28.11590 28.18232 28.24521 28.30490 28.36171 28.41590 28.46773 28.51740

28.84170 28.92188 28.99724 29.06822 29.13525 29.19870 29.25892 29.31621 29.37085 29.42307

29.70584 29.79200 29.87284 29.94886 30.02050 30.08816 30.15222 30.21302 30.27084 30.32596

30.56298 30.65553 30.74230 30.82380 30.90049 30.97279 31.04111 31.10580 31.16719 31.22558

31.41300 31.51232 31.60540 31.69278 31.77492 31.85227 31.92524 31.99422 32.05955 32.12156

32.25575 32.36212 32.46184 32.55544 32.64339 32.72615 32.80415 32.87778 32.94742 33.01340

33.09125 33.20489 33.31151 33.41161 33.50568 33.59417 33.67753 33.75616 33.83044 33.90072

51 52 53 54 55 56 57 58 59 60

26.74233 26.78471 26.82567 26.86530 26.90368 26.94086 26.97690 27.01185 27.04575 27.07863

27.65519 27.69920 27.74169 27.78276 27.82249 27.86098 27.89827 27.93444 27.96953 28.00358

28.56509 28.61097 28.65519 28.69787 28.73910 28.77900 28.81764 28.85509 28.89142 28.92667

29.47310 29.52111 29.56728 29.61175 29.65465 29.69610 29.73619 29.77501 29.81264 29.84914

30.37863 30.42906 30.47743 30.52393 30.56870 30.61187 30.65357 30.69389 30.73292 30.77075

31.28122 31.33437 31.38524 31.43403 31.48089 31.52600 31.56948 31.61145 31.65203 31.69130

32.18052 32.23672 32.29038 32.34172 32.39094 32.43820 32.48367 32.52749 32.56977 32.61063

33.07602 33.13559 33.19235 33.24654 33.29838 33.34806 33.39576 33.44162 33.48580 33.52842

33.96734 34.03060 34.09077 34.14811 34.20285 34.25521 34.30537 34.35352 34.39980 34.44437

381



Rb 37

Sr 38

Y 39

Zr 40

Nb 41

Mo 42

Tc 43

Ru 44

Rh 45

0 1 2 3 4 5 6 7 8 9 10

0.00000 2.47108 5.52799 8.42971 10.98758 13.15459 14.95100 16.50508 17.91915 19.23525 20.46323

0.00000 3.04685 5.98984 8.78804 11.30230 13.52254 15.38843 16.96971 18.37743 19.67796 20.89436

0.00000 3.14857 6.23152 9.05817 11.58543 13.85142 15.79296 17.42770 18.85174 20.14726 21.35427

0.00000 3.13536 6.31697 9.20196 11.79126 14.12392 16.15287 17.86266 19.32606 20.63128 21.83462

0.00000 2.71132 6.07762 9.10320 11.80981 14.25176 16.39784 18.21807 19.75964 21.10549 22.32362

0.00000 2.62807 6.02878 9.08596 11.84206 14.35492 16.60221 18.53376 20.16607 21.56834 22.81354

0.00000 2.98330 6.36677 9.36034 12.07333 14.55835 16.83500 18.84164 20.55607 22.02040 23.30155

0.00000 2.57466 6.13781 9.30184 12.11783 14.70247 17.06754 19.15747 20.94877 22.47422 23.79566

0.00000 2.52855 6.12994 9.34697 12.19330 14.80976 17.22869 19.39679 21.27607 22.88053 24.26063

11 12 13 14 15 16 17 18 19 20

21.60452 22.66058 23.63390 24.52760 25.34529 26.09128 26.77063 27.38906 27.95268 28.46765

22.03122 23.08945 24.07058 24.97699 25.81159 26.57772 27.27935 27.92103 28.50783 29.04513

22.48536 23.54310 24.52855 25.44341 26.29007 27.07138 27.79065 28.45165 29.05861 29.61611

22.96037 24.01599 25.00342 25.92401 26.77965 27.57279 28.30631 28.98346 29.60784 30.18338

23.45249 24.50867 25.49806 26.42303 27.28544 28.08753 28.83193 29.52168 30.16004 30.75055

23.95215 25.01107 26.00220 26.93042 27.79829 28.60801 29.36203 30.06306 30.71409 31.31829

24.45652 25.52134 26.51474 27.44542 28.31748 29.13355 29.89596 30.60716 31.26982 31.88675

24.97275 26.04805 27.04651 27.98082 28.85706 29.67848 30.44747 31.16636 31.83769 32.46417

25.47475 26.57052 27.57948 28.51972 29.40064 30.22717 31.00229 31.72846 32.40810 33.04374

21 22 23 24 25 26 27 28 29 30

28.93986 29.37468 29.77685 30.15044 30.49879 30.82469 31.13035 31.41757 31.68781 31.94225

29.53825 29.99232 30.41201 30.80144 31.16418 31.50320 31.82098 32.11952 32.40048 32.66518

30.12888 30.60160 31.03868 31.44418 31.82168 32.17429 32.50463 32.81489 33.10687 33.38206

30.71422 31.20455 31.65846 32.07980 32.47208 32.83842 33.18153 33.50371 33.80689 34.09271

31.29692 31.80291 32.27225 32.70854 33.11511 33.49499 33.85088 34.18512 34.49971 34.79636

31.87899 32.39960 32.88351 33.33406 33.75438 34.14739 34.51575 34.86180 35.18758 35.49488

32.46091 32.99536 33.49318 33.95743 34.39106 34.79684 35.17737 35.53497 35.87171 36.18941

33.04863 33.59396 34.10309 34.57888 35.02411 35.44141 35.83324 36.20184 36.54925 36.87726

33.63805 34.19377 34.71364 35.20037 35.65660 36.08482 36.48738 36.86647 37.22407 37.56196

31 32 33 34 35 36 37 38 39 40

32.18187 32.40754 32.61999 32.81991 33.00795 33.18470 33.35077 33.50674 33.65317 33.79063

32.91474 33.15010 33.37204 33.58129 33.77849 33.96426 34.13916 34.30375 34.45859 34.60420

33.64170 33.88680 34.11825 34.33679 34.54311 34.73782 34.92149 35.09466 35.25788 35.41165

34.36249 34.61738 34.85831 35.08610 35.30147 35.50504 35.69741 35.87911 36.05068 36.21259

35.07650 35.34134 35.59190 35.82904 36.05353 36.26602 36.46712 36.65738 36.83731 37.00740

35.78518 36.05979 36.31977 36.56607 36.79949 37.02072 37.23038 37.42904 37.61721 37.79540

36.48965 36.77377 37.04294 37.29814 37.54022 37.76993 37.98791 38.19475 38.39097 38.57707

37.18746 37.48122 37.75971 38.02398 38.27490 38.51325 38.73969 38.95482 39.15918 39.35328

37.88172 38.18473 38.47222 38.74524 39.00470 39.25140 39.48604 39.70923 39.92152 40.12342

41 42 43 44 45 46 47 48 49 50

33.91965 34.04078 34.15452 34.26139 34.36186 34.45640 34.54543 34.62938 34.70863 34.78356

34.74111 34.86983 34.99086 35.10468 35.21178 35.31260 35.40758 35.49713 35.58165 35.66151

35.55648 35.69286 35.82127 35.94218 36.05605 36.16334 36.26446 36.35984 36.44986 36.53491

36.36536 36.50944 36.64531 36.77342 36.89421 37.00813 37.11559 37.21700 37.31275 37.40322

37.16815 37.31999 37.46339 37.59879 37.72663 37.84731 37.96126 38.06888 38.17056 38.26666

37.96406 38.12364 38.27458 38.41732 38.55226 38.67981 38.80039 38.91438 39.02216 39.12409

38.75351 38.92072 39.07914 39.22917 39.37123 39.50570 39.63297 39.75343 39.86744 39.97536

39.53756 39.71247 39.87842 40.03581 40.18504 40.32650 40.46055 40.58757 40.70792 40.82195

40.31539 40.49786 40.67125 40.83593 40.99231 41.14075 41.28160 41.41524 41.54201 41.66225

51 52 53 54 55 56 57 58 59 60

34.85449 34.92175 34.98564 35.04642 35.10435 35.15964 35.21253 35.26319 35.31180 35.35851

35.73706 35.80863 35.87653 35.94105 36.00244 36.06095 36.11682 36.17023 36.22140 36.27048

36.61534 36.69148 36.76366 36.83217 36.89729 36.95927 37.01836 37.07477 37.12872 37.18038

37.48878 37.56975 37.64647 37.71924 37.78834 37.85405 37.91662 37.97628 38.03324 38.08771

38.35755 38.44358 38.52507 38.60233 38.67566 38.74534 38.81162 38.87475 38.93497 38.99247

39.22055 39.31186 39.39836 39.48037 39.55818 39.63208 39.70233 39.76919 39.83290 39.89368

40.07755 40.17434 40.26607 40.35304 40.43556 40.51392 40.58838 40.65921 40.72666 40.79095

40.93001 41.03243 41.12953 41.22163 41.30903 41.39202 41.47088 41.54587 41.61725 41.68524

41.77630 41.88449 41.98714 42.08455 42.17703 42.26486 42.34833 42.42770 42.50322 42.57515

382



Pd 46

Ag 47

Cd 48

In 49

Sn 50

Sb 51

Te 52

I 53

Xe 54

0 1 2 3 4 5 6 7 8 9 10

0.00000 2.13081 5.95961 9.23632 12.07598 14.75560 17.27178 19.54545 21.52940 23.22845 24.68459

0.00000 2.42343 6.03591 9.33858 12.23784 14.90041 17.40539 19.71426 21.77057 23.55443 25.08751

0.00000 2.70975 6.30459 9.51999 12.46054 15.11768 17.59096 19.90073 22.00282 23.85882 25.46742

0.00000 2.83441 6.63451 9.81909 12.72978 15.37682 17.82417 20.11901 22.24093 24.14878 25.82409

0.00000 2.81535 6.84114 10.16026 13.05475 15.67918 18.10185 20.37444 22.49751 24.43757 26.16687

0.00000 2.73816 6.87653 10.44583 13.41599 16.02031 18.41562 20.66395 22.77720 24.73296 26.50212

0.00000 2.77296 7.11536 10.81525 13.82032 16.40069 18.76463 20.98609 23.08280 25.04189 26.83738

0.00000 2.74654 7.21010 11.09773 14.22622 16.81256 19.14401 21.33615 23.41315 25.36736 27.17764

0.00000 2.68555 7.17883 11.24839 14.58946 17.24286 19.55212 21.71173 23.76633 25.71067 27.52690

11 12 13 14 15 16 17 18 19 20

25.95299 27.08363 28.11340 29.06590 29.95479 30.78766 31.56892 32.30163 32.98838 33.63173

26.41494 27.58563 28.64026 29.60758 30.50579 31.34559 32.13323 32.87265 33.56677 34.21817

26.85884 28.07659 29.16253 30.14943 31.05985 31.90798 32.70240 33.44836 34.14941 34.80832

27.28022 28.55092 29.67532 30.68802 31.61513 32.47438 33.27702 34.03004 34.73796 35.40399

27.68354 29.00920 30.17704 31.22095 32.16920 33.04260 33.85522 34.61612 35.33110 36.00409

28.07182 29.45143 30.66588 31.74576 32.71973 33.61083 34.43575 35.20579 35.92835 36.60835

28.45037 29.87984 31.14154 32.26075 33.26464 34.17721 35.01722 35.79817 36.52924 37.21658

28.82353 30.29667 31.60411 32.76453 33.80183 34.73959 35.59782 36.39189 37.13281 37.82815

29.19578 30.70506 32.05470 33.25645 34.32964 35.29596 36.17566 36.98544 37.73796 38.44232

21 22 23 24 25 26 27 28 29 30

34.23429 34.79871 35.32765 35.82374 36.28950 36.72736 37.13958 37.52826 37.89533 38.24253

34.82936 35.40286 35.94119 36.44685 36.92226 37.36974 37.79149 38.18955 38.56581 38.92198

35.42759 36.00964 36.55686 37.07161 37.55620 38.01286 38.44370 38.85072 39.23576 39.60052

36.03077 36.62069 37.17607 37.69917 38.19222 38.65737 39.09668 39.51207 39.90537 40.27825

36.63803 37.23540 37.79848 38.32947 38.83052 39.30370 39.75100 40.17433 40.57546 40.95605

37.24925 37.85373 38.42413 38.96261 39.47124 39.95205 40.40697 40.83784 41.24642 41.63434

37.86439 38.47574 39.05314 39.59874 40.11461 40.60268 41.06486 41.50293 41.91862 42.31353

38.48306 39.10122 39.68537 40.23781 40.76059 41.25563 41.72476 42.16973 42.59223 42.99385

39.10481 39.72986 40.32066 40.87972 41.40916 41.91088 42.38669 42.83831 43.26737 43.67544

31 32 33 34 35 36 37 38 39 40

38.57142 38.88338 39.17963 39.46123 39.72910 39.98406 40.22683 40.45802 40.67820 40.88788

39.25963 39.58012 39.88470 40.17445 40.45030 40.71310 40.96359 41.20239 41.43009 41.64720

39.94654 40.27521 40.58776 40.88531 41.16881 41.43912 41.69700 41.94311 42.17803 42.40230

40.63222 40.96869 41.28889 41.59393 41.88479 42.16236 42.42739 42.68057 42.92249 43.15369

41.31759 41.66149 41.98898 42.30118 42.59909 42.88359 43.15548 43.41544 43.66408 43.90195

42.00309 42.35407 42.68851 43.00755 43.31218 43.60332 43.88176 44.14821 44.40329 44.64756

42.68916 43.04689 43.38796 43.71351 44.02456 44.32203 44.60672 44.87936 45.14059 45.39097

43.37606 43.74024 44.08765 44.41943 44.73661 45.04011 45.33077 45.60932 45.87642 46.13264

44.06399 44.43437 44.78786 45.12561 45.44865 45.75792 46.05429 46.33848 46.61118 46.87297

41 42 43 44 45 46 47 48 49 50

41.08751 41.27753 41.45833 41.63032 41.79386 41.94930 42.09701 42.23733 42.37059 42.49714

41.85418 42.05147 42.23947 42.41855 42.58909 42.75142 42.90589 43.05284 43.19258 43.32545

42.61638 42.82071 43.01568 43.20167 43.37904 43.54813 43.70925 43.86275 44.00892 44.14808

43.37466 43.58582 43.78758 43.98031 44.16435 44.34004 44.50771 44.66765 44.82017 44.96556

44.12954 44.34730 44.55561 44.75486 44.94538 45.12751 45.30155 45.46779 45.62655 45.77809

44.88151 45.10560 45.32023 45.52577 45.72255 45.91091 46.09115 46.26354 46.42839 46.58596

45.63102 45.86118 46.08187 46.29346 46.49628 46.69066 46.87690 47.05527 47.22605 47.38950

46.37851 46.61449 46.84098 47.05837 47.26699 47.46717 47.65920 47.84334 48.01988 48.18906

47.12440 47.36592 47.59796 47.82090 48.03509 48.24083 48.43843 48.62815 48.81026 48.98499

51 52 53 54 55 56 57 58 59 60

42.61729 42.73137 42.83969 42.94256 43.04027 43.13311 43.22136 43.30529 43.38515 43.46119

43.45175 43.57180 43.68589 43.79434 43.89743 43.99544 44.08864 44.17732 44.26171 44.34207

44.28053 44.40658 44.52650 44.64060 44.74916 44.85245 44.95074 45.04430 45.13338 45.21823

45.10413 45.23615 45.36191 45.48169 45.59576 45.70439 45.80784 45.90638 46.00026 46.08971

45.92270 46.06065 46.19222 46.31767 46.43727 46.55128 46.65995 46.76355 46.86231 46.95647

46.73651 46.88032 47.01765 47.14874 47.27386 47.39326 47.50718 47.61588 47.71958 47.81853

47.54589 47.69546 47.83846 47.97514 48.10575 48.23052 48.34969 48.46350 48.57219 48.67597

48.35114 48.50634 48.65492 48.79711 48.93314 49.06324 49.18764 49.30657 49.42024 49.52889

49.15259 49.31330 49.46733 49.61492 49.75629 49.89165 50.02123 50.14524 50.26389 50.37741

383



Cs 55

Ba 56

La 57

Ce 58

Pr 59

Nd 60

Pm 61

Sm 62

Eu 63

0 1 2 3 4 5 6 7 8 9 10

0.00000 3.26056 7.51043 11.50727 14.94313 17.68653 19.99684 22.12327 24.14779 26.07414 27.88767

0.00000 3.86513 7.93796 11.82172 15.29163 18.13325 20.46851 22.56799 24.55922 26.46226 28.26625

0.00000 3.98173 8.16628 12.07836 15.59334 18.54373 20.94271 23.03816 25.00033 26.87743 28.66647

0.00000 3.82992 7.95853 11.89091 15.41466 18.38275 20.84073 23.01484 25.04584 26.97495 28.80594

0.00000 3.79780 7.92593 11.86936 15.40728 18.42223 20.93672 23.15493 25.21725 27.17051 29.02323

0.00000 3.76602 7.89361 11.85347 15.40977 18.46877 21.03547 23.29454 25.38565 27.36050 29.23184

0.00000 3.73391 7.86055 11.84146 15.42104 18.52393 21.14100 23.43911 25.55681 27.55050 29.43711

0.00000 3.70093 7.81853 11.81226 15.40725 18.54486 21.20679 23.54344 25.69054 27.70729 29.61325

0.00000 3.66574 7.76614 11.76155 15.35978 18.51896 21.21677 23.58933 25.76819 27.81321 29.74482

11 12 13 14 15 16 17 18 19 20

29.57070 31.10881 32.49628 33.73826 34.84870 35.84631 36.75052 37.57867 38.34476 39.05936

29.95527 31.51424 32.93352 34.21247 35.35955 36.38969 37.32054 38.16937 38.95105 39.67740

30.35349 31.92398 33.36671 34.67707 35.85864 36.92200 37.88201 38.75472 39.55512 40.29583

30.53305 32.14682 33.63749 34.99934 36.23339 37.34741 38.35393 39.26759 40.10283 40.87249

30.77376 32.41595 33.94154 35.34403 36.62203 37.78028 38.82863 39.77988 40.64762 41.44464

31.00189 32.66791 34.22365 35.66263 36.98171 38.18301 39.27368 40.26438 41.16740 41.99498

31.22252 32.90744 34.48807 35.95863 37.31495 38.55691 39.68908 40.71987 41.65990 42.52055

31.41694 33.12205 34.72733 36.22826 37.62034 38.90190 40.07535 41.14694 42.12558 43.02149

31.57265 33.30236 34.93493 36.46740 37.89555 39.21679 40.43188 41.54518 42.56396 43.49721

21 22 23 24 25 26 27 28 29 30

39.73002 40.36211 40.95944 41.52485 42.06062 42.56873 43.05095 43.50897 43.94439 44.35873

40.35724 40.99692 41.60099 42.17275 42.71475 43.22905 43.71747 44.18169 44.62325 45.04367

40.98674 41.63525 42.24674 42.82512 43.37330 43.89358 44.38788 44.85790 45.30520 45.73127

41.58717 42.25515 42.88273 43.47463 44.03444 44.56494 45.06842 45.54682 46.00186 46.43513

42.18196 42.86863 43.51172 44.11671 44.68778 45.22821 45.74062 46.22720 46.68983 47.13020

42.75827 43.46675 44.12817 44.74866 45.33303 45.88506 46.40777 46.90364 47.37476 47.82296

43.31265 44.04590 44.72844 45.36699 45.96689 46.53242 47.06702 47.57351 48.05422 48.51119

43.84501 44.60572 45.31205 45.97111 46.58874 47.16970 47.71785 48.23636 48.72785 49.19458

44.35455 45.14531 45.87800 46.56001 47.19760 47.79597 48.35940 48.89142 49.39496 49.87253

31 32 33 34 35 36 37 38 39 40

44.75343 45.12986 45.48926 45.83279 46.16150 46.47636 46.77820 47.06781 47.34587 47.61299

45.44433 45.82660 46.19170 46.54079 46.87495 47.19512 47.50220 47.79696 48.08010 48.35226

46.13749 46.52519 46.89561 47.24991 47.58915 47.91432 48.22631 48.52592 48.81386 49.09079

46.84809 47.24213 47.61852 47.97846 48.32305 48.65331 48.97016 49.27443 49.56689 49.84820

47.54986 47.95022 48.33261 48.69825 49.04827 49.38372 49.70555 50.01462 50.31171 50.59753

48.24990 48.65706 49.04583 49.41747 49.77315 50.11396 50.44089 50.75482 51.05658 51.34690

48.94618 49.36081 49.75652 50.13464 50.49640 50.84292 51.17523 51.49427 51.80089 52.09586

49.63849 50.06129 50.46456 50.84968 51.21796 51.57057 51.90859 52.23300 52.54470 52.84449

50.32626 50.75804 51.16953 51.56224 51.93753 52.29665 52.64073 52.97081 53.28783 53.59264

41 42 43 44 45 46 47 48 49 50

47.86970 48.11650 48.35382 48.58204 48.80151 49.01255 49.21545 49.41048 49.59791 49.77796

48.61399 48.86579 49.10810 49.34133 49.56583 49.78191 49.98988 50.19001 50.38254 50.56773

49.35727 49.61382 49.86090 50.09891 50.32821 50.54913 50.76197 50.96700 51.16446 51.35460

50.11898 50.37975 50.63099 50.87315 51.10658 51.33164 51.54863 51.75783 51.95950 52.15386

50.87271 51.13781 51.39333 51.63972 51.87737 52.10664 52.32785 52.54128 52.74720 52.94584

51.62645 51.89581 52.15551 52.40602 52.64775 52.88109 53.10636 53.32387 53.53388 53.73663

52.37989 52.65358 52.91750 53.17213 53.41794 53.65529 53.88456 54.10604 54.32003 54.52677

53.13312 53.41122 53.67940 53.93818 54.18803 54.42936 54.66254 54.88791 55.10577 55.31637

53.88602 54.16865 54.44116 54.70412 54.95801 55.20329 55.44034 55.66953 55.89117 56.10553

51 52 53 54 55 56 57 58 59 60

49.95088 50.11688 50.27620 50.42903 50.57560 50.71611 50.85077 50.97979 51.10338 51.22173

50.74578 50.91692 51.08137 51.23932 51.39097 51.53654 51.67621 51.81018 51.93866 52.06182

51.53763 51.71377 51.88321 52.04615 52.20279 52.35332 52.49792 52.63677 52.77008 52.89802

52.34116 52.52158 52.69532 52.86259 53.02357 53.17843 53.32736 53.47053 53.60813 53.74033

53.13743 53.32217 53.50027 53.67190 53.83725 53.99650 54.14981 54.29736 54.43931 54.57584

53.93236 54.12127 54.30355 54.47940 54.64898 54.81247 54.97004 55.12184 55.26804 55.40880

54.72650 54.91942 55.10575 55.28566 55.45932 55.62691 55.78858 55.94450 56.09482 56.23968

55.51997 55.71677 55.90699 56.09081 56.26840 56.43994 56.60557 56.76546 56.91976 57.06860

56.31287 56.51342 56.70740 56.89499 57.07636 57.25170 57.42115 57.58487 57.74301 57.89571

384



Gd 64

Tb 65

Dy 66

Ho 67

Er 68

Tm 69

Yb 70

Lu 71

Hf 72

0 1 2 3 4 5 6 7 8 9 10

0.00000 3.71361 7.85846 11.87023 15.51039 18.74647 21.51468 23.91627 26.09535 28.13360 30.06041

0.00000 3.61003 7.71283 11.75719 15.40658 18.64177 21.43284 23.88725 26.12906 28.22255 30.19256

0.00000 3.58129 7.67561 11.73154 15.39347 18.65196 21.47837 23.96877 26.24261 28.36456 30.35998

0.00000 3.55174 7.63634 11.70630 15.38593 18.67051 21.53281 24.05874 26.36312 28.51140 30.52974

0.00000 3.52320 7.60049 11.68736 15.38890 18.70234 21.60137 24.16249 26.49629 28.66909 30.70818

0.00000 3.49352 7.56010 11.65817 15.37564 18.71081 21.64017 24.23254 26.59446 28.79252 30.85451

0.00000 3.46493 7.51513 11.61547 15.33913 18.68517 21.63498 24.25196 26.63899 28.86244 30.94989

0.00000 3.55150 7.64243 11.69634 15.42634 18.82076 21.83614 24.49728 26.90009 29.12578 31.21431

0.00000 3.53769 7.65165 11.69972 15.47770 18.94311 22.03782 24.75775 27.18736 29.41882 31.50686

11 12 13 14 15 16 17 18 19 20

31.88606 33.61615 35.25308 36.79567 38.24058 39.58478 40.82754 41.97127 43.02132 43.98512

32.05250 33.81265 35.47907 37.05282 38.53150 39.91198 41.19272 42.37492 43.46263 44.46218

32.24256 34.02361 35.71101 37.30767 38.81250 40.22273 41.53630 42.75330 43.87646 44.91082

32.43217 34.23101 35.93593 37.55182 39.07905 40.51552 41.85898 43.10864 44.26601 45.33481

32.62811 34.44208 36.16142 37.79297 39.33878 40.79768 42.16747 43.44678 44.63605 45.73781

32.79487 34.62698 36.36324 38.01213 39.57731 41.05867 42.45440 43.76280 44.98352 46.11808

32.91479 34.76985 36.52769 38.19791 39.78550 41.29145 42.71447 44.05283 45.30566 46.47366

33.18232 35.04127 36.80277 38.47694 40.07028 41.58529 43.02162 44.37769 45.65219 46.84497

33.47536 35.33636 37.10023 38.77678 40.37345 41.89416 43.33975 44.70926 46.00131 47.21519

21 22 23 24 25 26 27 28 29 30

44.87117 45.68820 46.44445 47.14738 47.80343 48.41809 48.99594 49.54081 50.05590 50.54390

45.38131 46.22827 47.01119 47.73763 48.41429 49.04700 49.64070 50.19957 50.72711 51.22628

45.86307 46.74079 47.55172 48.30329 49.00232 49.65483 50.26607 50.84048 51.38187 51.89346

46.32062 47.23010 48.07047 48.84887 49.57206 50.24620 50.87671 51.46830 52.02503 52.55037

46.75648 47.69780 48.56826 49.37460 50.12337 50.82069 51.47209 52.08245 52.65604 53.19655

47.16983 48.14358 49.04508 49.88057 50.65631 51.37830 52.05210 52.68272 53.27458 53.83157

47.55927 48.56642 49.50015 50.36614 51.17032 51.91852 52.61627 53.26864 53.88020 54.45500

47.95743 48.99245 49.95417 50.84752 51.67789 52.45076 53.17146 53.84498 54.47593 55.06843

48.35143 49.41196 50.39996 51.31957 52.17554 52.97288 53.71663 54.41164 55.06245 55.67321

31 32 33 34 35 36 37 38 39 40

51.00714 51.44764 51.86716 52.26731 52.64951 53.01508 53.36518 53.70092 54.02325 54.33309

51.69962 52.14932 52.57728 52.98520 53.37461 53.74685 54.10319 54.44473 54.77251 55.08747

52.37799 52.83784 53.27506 53.69148 54.08870 54.46818 54.83122 55.17900 55.51260 55.83299

53.04730 53.51836 53.96579 54.39154 54.79733 55.18470 55.55503 55.90957 56.24946 56.57572

53.70715 54.19060 54.64928 55.08529 55.50047 55.89647 56.27474 56.63663 56.98331 57.31587

54.35706 54.85399 55.32491 55.77207 56.19744 56.60278 56.98964 57.35943 57.71342 58.05275

54.99660 55.50813 55.99230 56.45152 56.88790 57.30331 57.69941 58.07770 58.43952 58.78608

55.62616 56.15239 56.64997 57.12145 57.56904 57.99473 58.40029 58.78729 59.15715 59.51116

56.24769 56.78923 57.30083 57.78512 58.24446 58.68094 59.09641 59.49253 59.87082 60.23260

41 42 43 44 45 46 47 48 49 50

54.63124 54.91843 55.19531 55.46248 55.72046 55.96972 56.21069 56.44373 56.66918 56.88734

55.39043 55.68217 55.96338 56.23467 56.49660 56.74968 56.99434 57.23098 57.45995 57.68157

56.14107 56.43763 56.72340 56.99903 57.26511 57.52217 57.77068 58.01106 58.24368 58.46890

56.88928 57.19099 57.48161 57.76182 58.03226 58.29348 58.54598 58.79021 59.02657 59.25542

57.63530 57.94249 58.23823 58.52327 58.79826 59.06380 59.32041 59.56858 59.80874 60.04126

58.37845 58.69148 58.99268 59.28282 59.56261 59.83267 60.09357 60.34582 60.58989 60.82617

59.11849 59.43774 59.74472 60.04027 60.32511 60.59992 60.86531 61.12181 61.36992 61.61007

59.85047 60.17613 60.48910 60.79022 61.08030 61.36002 61.63005 61.89094 62.14323 62.38738

60.57912 60.91146 61.23063 61.53755 61.83303 62.11783 62.39262 62.65802 62.91457 63.16278

51 52 53 54 55 56 57 58 59 60

57.09846 57.30280 57.50057 57.69196 57.87716 58.05634 58.22965 58.39725 58.55928 58.71589

57.89613 58.10387 58.30503 58.49981 58.68840 58.87098 59.04771 59.21873 59.38420 59.54426

58.68699 58.89824 59.10288 59.30113 59.49319 59.67924 59.85944 60.03396 60.20294 60.36651

59.47708 59.69184 59.89994 60.10163 60.29711 60.48657 60.67018 60.84811 61.02050 61.18749

60.26650 60.48475 60.69629 60.90137 61.10021 61.29301 61.47995 61.66120 61.83691 62.00723

61.05505 61.27683 61.49183 61.70030 61.90249 62.09859 62.28882 62.47333 62.65230 62.82588

61.84266 62.06804 62.28653 62.49842 62.70394 62.90334 63.09682 63.28456 63.46674 63.64352

62.62381 62.85289 63.07498 63.29036 63.49931 63.70208 63.89888 64.08991 64.27535 64.45537

63.40309 63.63591 63.86160 64.08049 64.29285 64.49897 64.69906 64.89334 65.08201 65.26522

385



Ta 73

W 74

Re 75

Os 76

Ir 77

Pt 78

Au 79

Hg 80

Tl 81

0 1 2 3 4 5 6 7 8 9 10

0.00000 3.51110 7.66380 11.69611 15.50242 19.03205 22.21447 25.01132 27.48515 29.73240 31.82262

0.00000 3.47440 7.67506 11.69835 15.51176 19.08462 22.35035 25.23775 27.77632 30.05529 32.15546

0.00000 3.42084 7.62562 11.62131 15.45172 19.08376 22.44359 25.43378 28.05343 30.37962 32.49975

0.00000 3.39710 7.69565 11.73418 15.58139 19.22645 22.62667 25.68056 28.36068 30.72615 32.86379

0.00000 3.35665 7.70093 11.75575 15.62856 19.31307 22.77449 25.90537 28.65719 31.07261 33.23607

0.00000 2.62718 7.28299 11.40374 15.28380 19.11956 22.75224 26.02263 28.88260 31.37642 33.58901

0.00000 2.92007 7.38597 11.54618 15.43449 19.21786 22.84500 26.17202 29.11511 31.68425 33.94846

0.00000 3.21624 7.64856 11.74935 15.65381 19.38841 22.97772 26.32856 29.33911 31.98262 34.30452

0.00000 3.35316 7.98942 12.05736 15.91600 19.60547 23.15315 26.50417 29.56069 32.27085 34.65371

11 12 13 14 15 16 17 18 19 20

33.79048 35.65180 37.41684 39.09480 40.69343 42.21771 43.66969 45.04926 46.35531 47.58687

34.12464 35.98581 37.75117 39.42981 41.02943 42.55571 44.01175 45.39840 46.71511 47.96098

34.47451 36.33640 38.10173 39.78048 41.38044 42.90766 44.36609 45.75753 47.08215 48.33936

34.84275 36.70384 38.46783 40.14591 41.74578 43.27348 44.73342 46.12814 47.45859 48.72466

35.22395 37.08589 38.84850 40.52538 42.12467 43.65234 45.11300 46.50976 47.84426 49.11697

35.60220 37.47454 39.24070 40.91911 42.51974 44.04884 45.51094 46.90936 48.24630 49.52295

35.98954 37.87334 39.64231 41.32016 42.91950 44.44763 45.90946 47.30865 48.64781 49.92857

36.38055 38.28093 40.05533 41.73338 43.33120 44.85771 46.31852 47.71764 49.05803 50.34169

36.77187 38.69506 40.47858 42.15838 43.75506 45.27989 46.73945 48.13811 49.47908 50.76463

21 22 23 24 25 26 27 28 29 30

48.74383 49.82728 50.83952 51.78390 52.66446 53.48571 54.25230 54.96886 55.63980 56.26923

49.13553 50.23921 51.27353 52.24104 53.14507 53.98955 54.77868 55.51676 56.20800 56.85643

49.52858 50.64979 51.70388 52.69265 53.61877 54.48551 55.29660 56.05594 56.76750 57.43512

49.92586 51.06198 52.13342 53.14138 54.08786 54.97556 55.80766 56.58766 57.31920 58.00591

50.32771 51.47623 52.56264 53.58768 54.55277 55.46004 56.31214 57.11212 57.86326 58.56891

50.73978 51.89694 52.99462 54.03338 55.01429 55.93896 56.80952 57.62853 58.39880 59.12335

51.15174 52.31766 53.42649 54.47860 55.47468 56.41593 57.30402 58.14109 58.92956 59.67213

51.56978 52.74286 53.86116 54.92490 55.93453 56.89088 57.79523 58.64929 59.45512 60.21510

51.99627 53.17482 54.30069 55.37411 56.39540 57.36512 58.28421 59.15401 59.97622 60.75286

31 32 33 34 35 36 37 38 39 40

56.86093 57.41832 57.94447 58.44213 58.91374 59.36149 59.78733 60.19301 60.58011 60.95004

57.46579 58.03952 58.58075 59.09230 59.57669 60.03620 60.47288 60.88854 61.28485 61.66329

58.06247 58.65296 59.20973 59.73566 60.23333 60.70509 61.15307 61.57916 61.98508 62.37241

58.65132 59.25877 59.83140 60.37207 60.88343 61.36788 61.82759 62.26453 62.68050 63.07711

59.23243 59.85704 60.44580 61.00158 61.52702 62.02456 62.49642 62.94463 63.37103 63.77731

59.80523 60.44746 61.05295 61.62451 62.16472 62.67603 63.16068 63.62073 64.05808 64.47447

60.37161 61.03082 61.65256 62.23954 62.79433 63.31935 63.81684 64.28890 64.73746 65.16428

60.93174 61.60768 62.24557 62.84799 63.41746 63.95636 64.46693 64.95127 65.41133 65.84891

61.48617 62.17854 62.83242 63.45027 64.03451 64.58745 65.11132 65.60820 66.08005 66.52868

41 42 43 44 45 46 47 48 49 50

61.30409 61.64342 61.96908 62.28204 62.58316 62.87323 63.15296 63.42301 63.68397 63.93635

62.02520 62.37182 62.70424 63.02348 63.33045 63.62598 63.91083 64.18568 64.45116 64.70783

62.74254 63.09676 63.43623 63.76202 64.07507 64.37627 64.66642 64.94624 65.21638 65.47745

63.45584 63.81802 64.16487 64.49750 64.81692 65.12405 65.41973 65.70471 65.97970 66.24532

64.16499 64.53547 64.89001 65.22977 65.55582 65.86912 66.17054 66.46088 66.74089 67.01122

64.87147 65.25055 65.61303 65.96012 66.29295 66.61251 66.91974 67.21549 67.50053 67.77557

65.57101 65.95912 66.33000 66.68491 67.02499 67.35131 67.66483 67.96644 68.25694 68.53709

66.26568 66.66316 67.04278 67.40582 67.75348 68.08686 68.40696 68.71470 69.01093 69.29641

66.95579 67.36294 67.75158 68.12304 68.47856 68.81925 69.14617 69.46027 69.76243 70.05346

51 52 53 54 55 56 57 58 59 60

64.18064 64.41727 64.64663 64.86905 65.08486 65.29432 65.49770 65.69521 65.88707 66.07345

64.95619 65.19670 65.42978 65.65578 65.87506 66.08789 66.29456 66.49531 66.69034 66.87986

65.72997 65.97444 66.21130 66.44094 66.66371 66.87993 67.08989 67.29386 67.49205 67.68468

66.50215 66.75069 66.99142 67.22475 67.45107 67.67072 67.88399 68.09118 68.29252 68.48825

67.27247 67.52519 67.76989 68.00700 68.23692 68.46003 68.67665 68.88708 69.09158 69.29039

68.04124 68.29812 68.54675 68.78759 69.02108 69.24762 69.46754 69.68116 69.88876 70.09061

68.80754 69.06891 69.32177 69.56661 69.80389 70.03403 70.25740 70.47434 70.68515 70.89009

69.57187 69.83793 70.09520 70.34419 70.58540 70.81927 71.04618 71.26651 71.48058 71.68866

70.33409 70.60500 70.86682 71.12009 71.36534 71.60303 71.83358 72.05737 72.27474 72.48602

386



Pb 82

Bi 83

Po 84

At 85

Rn 86

Fr 87

Ra 88

Ac 89

Th 90

0 1 2 3 4 5 6 7 8 9 10

0.00000 3.34621 8.22217 12.41125 16.22970 19.86837 23.37158 26.70653 29.78902 32.55386 34.99597

0.00000 3.27806 8.29008 12.73684 16.58625 20.17179 23.62798 26.93777 30.02992 32.83610 35.33272

0.00000 3.33281 8.55914 13.12943 16.98331 20.51378 23.91920 27.19750 30.28696 33.12152 35.66572

0.00000 3.32115 8.68626 13.45896 17.39841 20.88858 24.23940 27.48284 30.56176 33.41324 35.99589

0.00000 3.26930 8.67890 13.67755 17.80231 21.29218 24.58760 27.79204 30.85496 33.71411 36.32636

0.00000 3.83241 9.01288 13.96293 18.20342 21.72790 24.97453 28.13066 31.16853 34.02674 36.66045

0.00000 4.44464 9.43761 14.28181 18.59738 22.18172 25.39443 28.49894 31.50434 34.35324 36.99990

0.00000 4.60565 9.70196 14.56415 18.95415 22.62985 25.83912 28.89569 31.86395 34.69753 37.34880

0.00000 4.53225 9.73030 14.68382 19.10180 22.83294 26.08504 29.15467 32.12548 34.96690 37.63823

11 12 13 14 15 16 17 18 19 20

37.16069 39.11241 40.90935 42.59319 44.18948 45.71259 47.17058 48.56838 49.90941 51.19617

37.54632 39.53176 41.34687 43.03767 44.63479 46.15634 47.61243 49.00889 50.34947 51.63680

37.92853 39.95218 41.79059 43.49207 45.09182 46.61211 48.06580 49.46012 50.79941 52.08653

38.30468 40.36861 42.23479 43.95125 45.55657 47.07707 48.52881 49.92093 51.25863 52.54511

38.67709 40.78241 42.68028 44.41593 46.02985 47.55215 49.00237 50.39203 51.72758 53.01275

39.04766 41.19310 43.12499 44.88381 46.51011 48.03675 49.48660 50.87389 52.20682 53.49000

39.41742 41.60084 43.56818 45.35352 46.99576 48.52946 49.98052 51.36601 52.69615 53.97677

39.78790 42.00369 44.00554 45.82004 47.48239 49.02718 50.48246 51.86803 53.19634 54.47467

40.11077 42.36942 44.41624 46.27058 47.96308 49.52726 50.99268 52.38179 53.70975 54.98596

21 22 23 24 25 26 27 28 29 30

52.43042 53.61325 54.74527 55.82679 56.85810 57.83958 58.77191 59.65612 60.49357 61.28596

52.87284 54.05891 55.19583 56.28406 57.32388 58.31562 59.25976 60.15705 61.00858 61.81573

53.32354 54.51192 55.65265 56.74634 57.79336 58.79400 59.74862 60.65779 61.52233 62.34333

53.78256 54.97254 56.11619 57.21425 58.26721 59.27537 60.23906 61.15870 62.03492 62.86859

54.24992 55.44075 56.58647 57.68794 58.74575 59.76029 60.73186 61.66081 62.54761 63.39292

54.72616 55.91711 57.06414 58.16821 59.22997 60.24988 61.22825 62.16539 63.06165 63.91754

55.21118 56.40144 57.54895 58.65471 59.71947 60.74374 61.72786 62.67210 63.57676 64.44222

55.70716 56.89625 58.04351 59.15007 60.21675 61.24415 62.23269 63.18267 64.09436 64.96805

56.21594 57.40291 58.54882 59.65498 60.72234 61.75155 62.74312 63.69739 64.61464 65.49516

31 32 33 34 35 36 37 38 39 40

62.03522 62.74351 63.41306 64.04617 64.64514 65.21220 65.74952 66.25915 66.74304 67.20301

62.58016 63.30374 63.98851 64.63658 65.25011 65.83123 66.38202 66.90451 67.40061 67.87214

63.12218 63.86049 64.56004 65.22276 65.85064 66.44569 67.00992 67.54527 68.05362 68.53677

63.66083 64.41300 65.12666 65.80353 66.44543 67.05424 67.63183 68.18009 68.70083 69.19582

64.19764 64.96290 65.69003 66.38054 67.03606 67.65832 68.24908 68.81013 69.34321 69.85004

64.73377 65.51124 66.25108 66.95460 67.62324 68.25858 68.86224 69.43589 69.98121 70.49985

65.26900 66.05784 66.80966 67.52556 68.20682 68.85483 69.47109 70.05715 70.61459 71.14499

65.80418 66.60329 67.36615 68.09365 68.78691 69.44713 70.07566 70.67391 71.24336 71.78549

66.33929 67.14746 67.92027 68.65848 69.36298 70.03482 70.67518 71.28532 71.86659 72.42036

41 42 43 44 45 46 47 48 49 50

67.64078 68.05791 68.45590 68.83610 69.19977 69.54808 69.88210 70.20283 70.51118 70.80798

68.32080 68.74819 69.15581 69.54503 69.91716 70.27337 70.61478 70.94241 71.25721 71.56003

68.99643 69.43420 69.85158 70.24999 70.63072 70.99501 71.34398 71.67868 72.00009 72.30911

69.66673 70.11516 70.54262 70.95052 71.34019 71.71287 72.06972 72.41180 72.74012 73.05561

70.33226 70.79145 71.22912 71.64668 72.04547 72.42674 72.79167 73.14134 73.47678 73.79895

70.99341 71.46344 71.91143 72.33878 72.74685 73.13689 73.51008 73.86754 74.21030 74.53935

71.64990 72.13082 72.58923 73.02652 73.44402 73.84301 74.22466 74.59011 74.94041 75.27655

72.30179 72.79371 73.26267 73.71006 74.13719 74.54533 74.93566 75.30931 75.66736 76.01079

72.94803 73.45099 73.93061 74.38824 74.82516 75.24264 75.64185 76.02393 76.38996 76.74094

51 52 53 54 55 56 57 58 59 60

71.09401 71.36997 71.63652 71.89424 72.14367 72.38531 72.61959 72.84694 73.06770 73.28222

71.85169 72.13291 72.40439 72.66672 72.92050 73.16622 73.40436 73.63536 73.85959 74.07742

72.60656 72.89322 73.16978 73.43689 73.69514 73.94509 74.18721 74.42198 74.64979 74.87103

73.35912 73.65144 73.93331 74.20539 74.46831 74.72264 74.96890 75.20756 75.43905 75.66378

74.10871 74.40690 74.69426 74.97149 75.23925 75.49811 75.74864 75.99132 76.22661 76.45494

74.85557 75.15982 75.45287 75.73544 76.00821 76.27179 76.52674 76.77359 77.01282 77.24487

75.59945 75.90996 76.20890 76.49700 76.77496 77.04342 77.30296 77.55413 77.79743 78.03332

76.34056 76.65754 76.96255 77.25636 77.53968 77.81316 78.07743 78.33304 78.58052 78.82035

77.07783 77.40152 77.71286 78.01262 78.30154 78.58030 78.84953 79.10982 79.36171 79.60571

387



Pa 91

U 92

0 1 2 3 4 5 6 7 8 9 10

0.00000 4.48605 9.64760 14.50305 18.88630 22.67010 26.01979 29.17627 32.20889 35.09984 37.82179

0.00000 4.46901 9.65990 14.53454 18.93052 22.75975 26.15376 29.33734 32.38794 35.29714 38.04410

11 12 13 14 15 16 17 18 19 20

40.35125 42.67134 44.77871 46.68677 48.42205 50.01657 51.50089 52.89990 54.23161 55.50792

40.60907 42.97470 45.13292 47.09059 48.86853 50.49569 52.00222 53.41447 54.75278 56.03139

21 22 23 24 25 26 27 28 29 30

56.73622 57.92097 59.06486 60.16964 61.23646 62.26615 63.25930 64.21633 65.13757 66.02329

57.25961 58.44323 59.58582 60.68962 61.75612 62.78632 63.78093 64.74047 65.66531 66.55575

31 32 33 34 35 36 37 38 39 40

66.87380 67.68946 68.47077 69.21834 69.93291 70.61540 71.26680 71.88825 72.48094 73.04613

67.41207 68.23461 69.02374 69.77999 70.50398 71.19646 71.85830 72.49049 73.09410 73.67027

41 42 43 44 45 46 47 48 49 50

73.58512 74.09919 74.58967 75.05784 75.50493 75.93219 76.34076 76.73178 77.10631 77.46537

74.22018 74.74506 75.24613 75.72462 76.18173 76.61863 77.03649 77.43640 77.81941 78.18654

51 52 53 54 55 56 57 58 59 60

77.80991 78.14084 78.45901 78.76522 79.06021 79.34470 79.61932 79.88468 80.14135 80.38984

78.53875 78.87696 79.20201 79.51472 79.81586 80.10614 80.38622 80.65672 80.91823 81.17129

scattering (Kimura, Schomaker, Smith & Weinstock, 1968; Bartell & Brockway, 1953; Hanson, 1962; Fink & Kessler, 1966; Geiger, 1964; Kessler, 1959; Seip, 1965; SchaÈfer & Seip, 1967; Kohl & Bonham, 1967; Bonham & Cox, 1967; Seip & Stùlevik, 1966; Seip & Seip, 1966; Arnesen & Seip, 1966; McClelland & Fink, 1985; Coffman, Fink & Wellenstein, 1985). New partial wave scattering factors based on relativistic Hartree±Fock fields (Biggs, Mendelsohn & Mann, 1975) are presented here at a number of energies (Table 4.3.3.1). Because of the availability of these partial wave results, first Born approximation results are no longer needed for gas-phase work in this energy range. The scattering of keV electrons from atoms is calculated in the central-field approximation in which the potential of the target is averaged over the angular coordinates and the resulting spherically symmetric potential V r is used in the computation. In order to take the relativistic effects properly into account, the Dirac equation has been used. In addition to the straightforward correction for the electron mass, spin-polarization effects are also included in these calculations. The scattering wavefunctions are four-component spinors that can be reduced to two components as shown by Mott & Massey (1965). This reduction leads to two decoupled second-order differential equations in SchroÈdinger form: d2 gl r ll 1 2 k Ul r gl r 0; dr 2 r2 where Ul r

2 V r 2 V 2 r

n 0 r

3 4

0 2 00 12 ;

and e2 ; hc 1=2 v2

1 c2

1 V r ; ( l 1 for j l 12 n l for j l 12,

where j is the total angular momentum for the lth partial wave including the two spin directions. Asymptotic solutions are available when Ul r is small relative to the centrifugal term, ll 1=r 2 . If this term is taken into account, but Ul r is neglected, gl r approaches gl r jl kr cos l nl kr sin l with a similar limit holding for the l lead directly to the scattering factors

1 solution. These limits

1 X l 1exp 2il 1 2ki lfexp2i l 1 1g Pl cos 1 X f exp 2il g 2ki exp 2i l 1 gPl1 cos f

and the elastic differential scattering cross section 388


4.3. ELECTRON DIFFRACTION d j f j2 jgj2 fg f g d

AB exp i' A B exp i' ; jAj2 jBj2

tanl B0l a jl ka kBl a j0l ka

where k2

ll 1 Ul ri r2

Now l r is computed. Starting with l r0 l 0 0 and l r1 l r 0:2rl1 , the integration of following the Numerov procedure is given by l ri 1 :

This recurrence relation is carried through for a=r steps, where a is the asymptotic limit. At the asymptotic limit gl r is lim gl r Bl r jl kr cos l

Bl rnl kr sin l :

The proportionality factor is eliminated by matching the logarithmic derivative of l r 0 r=r to the same derivative of gl r at r a. From this equality, the partial wave phase shifts are calculated as follows: 1 dl a l a dr fB0l a jl ka kBl a j0l ka cos 1 B0l anl ka kBl an0l ka sin l g Bl a jl ka cos l

Bl anl ka sin l 1 :

Solving for l leads to

where 1 dl a; l a dr ll 1 Bl a k2 a2 !

and 2ll 1 : a3 It is straightforward to calculate the scattering amplitudes by partial wave summation since stable numerical methods are readily available for the spherical Bessel functions, jl kr, the Neumann functions, nl kr, and the Legendre polynomials, Pl cos (Yates, 1971). Particular attention was given to the choices of the integration step size, r, and the matching radius, a. Both were varied to ensure the stability of the scattering factors to 0.1% for light atoms and to 0.3% for heavier atoms and higher incident energies. The results of the sensitivity calculations are summarized elsewhere (Ross & Fink, 1986). Smoothing was carried out by the following procedure: Sixteen data points, quarter s units apart, were least-squares fit to a cubic polynomial and the eighth point was changed to lie on this analytical curve. This procedure was repeated in Ê 1 . The points for running point average mode for s > 10 A 1 Ê s < 10A were left unchanged since no oscillations were seen. Smoothed and unsmoothed data in quarter s units for f and g are available on tape at cost from the authors. B0l a

Total inelastic scattering in the first Born approximation (Bonham & Fink, 1974) is obtained by including all possible excitation processes:

i 0; 1; 2; . . . :

l ri1 2 r 2 Bl ri r4 B2l ri =12l ri

Ssinel S

kn jh k

nj

N P i1

exp is0n ri j

2 0 ij ;

where ri is the nuclear electron vector, kn k2 E0n , s0n k2 kn2 2kkn cos 21=2 , E0n is the energy loss of the incident electron upon excitation of the scatterer to the nth state, is the Bragg angle, S signifies a sum over all bound states and an integration over the continuum, and N is equal to the number of electrons in the atom. The sum is carried out over all states n for which E0n is less than the incident electron energy. The Morse approximation is obtained by making three assumptions: (1) that the incident energy is so high that N 1, i.e. that all states are accessible; (2) that the ratio kn =k is unity for all inelastic processes of any importance; and (3) that s0n may be replaced by its elastic value s. With these approximations, closure may be used to obtain (Morse, 1932; Heisenberg, 1931; Bethe, 1930): Ss Z

Fx2 s

N P N P i

j6i

h

0 j expis

rij j

0 i:

The function Ss is the X-ray incoherent scattering factor (Wang, Sagar, Schmider & Smith, 1993) and is related to the inelastic electron scattering cross section by 389


!Bl a jl ka

4.3.3.2.2. Total inelastic scattering factors

and ri ir;

kBl an0l ka 1

!Bl anl ka ;

where A, B and ' describe the direction and degree of spin polarization of the incoming electrons. The latter term is equal to 0 when unpolarized electrons A B 1 are used in the scattering experiment. The results printed in the tables were obtained in three steps. First, atomic wavefunctions were calculated and transformed into centrosymmetric potentials via Poisson's equation. Second, a sufficient number of phases, l and l 1 , were computed in order to calculate the scattering factors f and g by performing the partial wave sums. Finally, the results were smoothed, because small oscillations were seen between nearest neighbours in the second difference function. These oscillations were only of the order of 0.1% of the data, so smoothing only had an effect in the third or fourth significant figure. For the scattering potentials, we used relativistic Hartree± Fock wavefunctions calculated by Biggs, Mendelsohn & Mann (1975). The wavefunctions were used to calculate the potentials and their derivatives since they are needed for Ul r to solve the appropriate Dirac equation. In order to solve the second-order differential equation, one must take advantage of the known asymptotic solutions. Following the procedure developed by Numerov (Numerov, 1924; Melkanov, Sawada & Raynal, 1966), an auxiliary function, l r is introduced: r 2 B r g r ; l ri 1 12 l i l i

Bl ri

B0l anl ka

4. PRODUCTION AND PROPERTIES OF RADIATIONS inel s 4Ss=a2 s4 : Inelastic scattering factors for X-rays and electrons are given in Table 4.3.3.2 in the Morse (1932) approximation for elements Z 1 to Z 92 with HF wave functions (Bunge, Barrientos & Bunge, 1993; McLean & McLean, 1981). There are two kinds of relativistic correction that can be made on inelastic scattering factors. The first is for relativistic effects on the atomic field and has been neglected. This should not be too serious since HF wavefunctions are used and the corrections are only large for the heavier atoms where the Ê 1 tends to contribution to the total scattering for s > 3 4 A be negligible. The other correction is for effects in the scattering process, which can be significant above 40 keV, but again these corrections tend to be localized to the small-angle Ê 1 (Yates, 1970). Hence the tables of inelastic region s < 3 A scattering factors given here are based on HF atomic fields since these appear to be the most accurate results presently available. The inelastic scattering equations must be modified in order to compare theory with experiment. First, the Morse theory is corrected to ensure that both energy and momentum are conserved in the scattering process. In the description of the elastic scattering process, no transformation is required from the centre-of-mass system (CMS), where the scattering factors are calculated, to the laboratory system (LS), where data are taken, since the nuclei are heavy compared with the incident electrons. In the inelastic channels, a similar argument holds for scattering involving the bound states. However, for ionizing processes, the interaction can be assumed to take place between the incident electron and the ejected electron, so that the CMS is entirely different from the LS. Considering the atomic electrons as free particles and considering only the ionization process, the transformation between the CMS and the LS is possible and leads to the Bethe modification (Tavard & Bonham, 1969) for inelastic scattering. The inelastic cross section can now be given by inel

4 cos 2Ss cos a2 s4 cos4

for < =4 and by inel 0 for > =4. Another modification is necessary because the average energy of inelastically scattered electrons varies with energy and is given from approximate conservation of energy and momentum for a fast incident particle by k2 cos2 2. This Ê 1 at 40 keV the average energy of means that for s > 30A inelastically scattered electrons may be around 30 keV and the fact that the response of the detector may be different for the 40keV inelastically scattered electrons and the elastic ones may have to be considered (Fink, Bonham, Lee & Ng, 1969). In addition to the values given in Table 4.3.3.2, a few calculations of Ss have been carried out with very exact wavefunctions that include more than 85% of the correlation energy (Kohl & Bonham, 1967; Bartell & Gavin, 1964; Peixoto, Bunge & Bonham, 1969; Thakkar & Smith, 1978; Wang, Esquivel, Smith & Bunge, 1995). 4.3.3.2.3. Corrections for defects in the theory of atomic scattering Errors in the inelastic scattering factors from the three approximations made in the Morse theory have been investigated (Tavard & Bonham, 1969; Bonham, 1965b). The Morse theory breaks down at very large scattering

angles > 30 , and is incorrect at small angles. Investigations carried out so far indicate that the small-angle failure is Ê 1 . It must be stressed that these not serious outside s 1 A uncertainties do not introduce important errors into the analysis of molecular structure using theoretical atomic scattering amplitudes. This is mainly because such deviations are smooth compared with molecular features and thus do not interfere with the analysis of molecular structure. 4.3.3.3. Molecular scattering factors for electrons The simplest theory of molecular scattering assumes that a molecule consists of spherical atoms and that each electron is scattered by only one atom in the molecule. If only single scattering is allowed within each atom, the molecular intensity can be written as Is Ia s Im s " X M 4I0 2 4 2 fZi asR i1

i

Z1

j6i

Zi

Fi sZj

Fj s 3

dr Pij r; T sin sr=sr 5;

4:3:3:1

0

where M is the number of constituent atoms in the molecule, Fi s and Si s are the coherent and incoherent X-ray scattering factors, and Pij r; T is the probability of finding atom i at a distance r from atom j at the temperature T (Bonham & Su, 1966; Kelley & Fink, 1982b; Mawhorter, Fink & Archer, 1983; Mawhorter & Fink, 1983; Miller & Fink, 1985; Hilderbrandt & Kohl, 1981; Kohl & Hilderbrandt, 1981). The constant I0 is proportional to the product of the intensities of the electron and molecular beams and R is the distance from the point of scattering to the detector. The single sum is the atomic intensity Ia s and the double sum is the molecular intensity Im s. This expression, referred to here as the independent atom model (IAM), may be improved by replacing the atomic elastic electron scattering factors by their partial wave counterparts. This modification is necessary to explain the failure of the Born approximation observed in molecules containing light and heavy atoms in proximity (Schomaker & Glauber, 1952; Seip, 1965), and may be written as Is Ia s Im s ( M I0 X 2 j fi j2 4Si s=a2 s4 R i1

M X M X i

Z1 0

j6i

j fi j j fj j cos i

j 9 =

dr Pij r; T sin sr=sr : ;

4:3:3:2

This is the most commonly used expression for the interpretation of molecular gas electron-diffraction patterns in the keV energy range. If it is necessary to consider relativistic effects in the scattering intensity, equation (4.3.3.2) becomes (Yates & Bonham, 1969)

390


M X M X

Fi s2 Si sg

4.3. ELECTRON DIFFRACTION Is Ia s Im s ( M I0 X 2 j fi j2 jgi j2 4Si s=a2 s4 R i1

M X M X i

j6i

j fi j j fj j cos fi

9 =

Z1 0

fj jgi j jgj j cos gi

dr Pij r; T sin sr=sr ; ;

gj 4:3:3:3

where jgi j and gi refer to the scattering-factor magnitude and phase for electrons that have changed their electron spin state during the scattering process and j fi j and fi refer to retention of spin orientation. The incident electron beam is assumed to be unpolarized and no attempt has been made to consider relativistic effects on the inelastic scattering cross section, which is usually negligible in the structural s range. If it is necessary to consider binding effects, the first Born approximation may usually be used in describing molecular scattering, since binding effects are largest for molecules containing small atoms where the Born approximation is most valid. The exact expression for Is in the first Born approximation can be written as (Bonham & Fink, 1974; Tavard & Roux, 1965; Tavard, Rouault & Roux, 1965; Iijima, Bonham & Ando, 1963; Bonham, 1967) ( M X 4I0 Is 2 4 2 Z 2i Zi asR i1

M X M X i

2

j6i

M X i1

Z1 Zi Zj *Z

dr Pij r; T sin sr=sr 0

dr r ri sin sr=sr

Zi

*Z

+

dr c rsin sr=sr

+ 9 = vib

;

vib

;

Acknowledgements The authors acknowledge with gratitude the contributions of Kenneth and Lise Hedberg, who made many helpful suggestions regarding the manuscript and carefully checked the numerical results for smoothness and consistency.

where r

N R P i1

dr1 . . .

R

1977; Sasaki, Konaka, Iijima & Kimura, 1982; Shibata, Hirota, Kakuta & Muramatsu, 1980; Horota, Kakuta & Shibata, 1981; Xie, Fink & Kohl, 1984). Further studies using correlated wavefunctions (accounting for up to 60% of the correlation energy) showed that in the elastic channel the binding effects are Ê 1 is only weakly modified; only the maximum at s 8 10 A further reduced. However, strong effects are seen in the inelastic Ê 1 significantly channel, deepening the minimum at s 3 4 A (Breitenstein, Endesfelder, Meyer, Schweig & Zittlau, 1983; Breitenstein, Endesfelder, Meyer & Schweig, 1984; Breitenstein, Mawhorter, Meyer & Schweig, 1984; Wang, Tripathi & Smith, 1994). Detailed calculations on CO2 and H2 O averaging over many internuclear distances and applying the pair distribution functions Pij r showed that vibrational effects do not alter the binding effects (Breitenstein, Mawhorter, Meyer & Schweig, 1986). For CO2 , the calculations have been confirmed in essence by an experimental set of data (McClelland & Fink, 1985). However, more molecules and more detailed analysis will be available in the future. The binding effects make it desirable to avoid the small-angle-scattering range when structural information is the main goal of a diffraction analysis. The problem of intramolecular multiple scattering may necessitate corrections to the molecular intensity when three or more closely spaced heavy atoms are present. This correction (Karle & Karle, 1950; Hoerni, 1956; Bunyan, 1963; Gjùnnes, 1964; Bonham, 1965a, 1966) appears to be more serious for three atoms in a right triangular configuration than for a collinear arrangement of three atoms. A case study by Kohl & Arvedson (1980) on SF6 showed the importance of multiple scattering. However, their approach is too cumbersome to be used in routine structure work. A very good approximate technique is available utilizing the Glauber approximation (Bartell & Miller, 1980; Bartell & Wong, 1972; Wong & Bartell, 1973; Bartell, 1975); Kohl's results are reproduced quite well using the atomic scattering factors only. Several applications of the multiple scattering routines showed that the internuclear distances are rather insensitive to this perturbation, but the mean amplitudes of vibration can easily change by 10% (Miller & Fink, 1981; Kelley & Fink, 1982a; Ketkar & Fink, 1985).

drN j r1 ; . . . ; rN j2 r

ri

and c r

N P N R P i

j6i

dr1 . . .

R

2

drN j r1 ; . . . ; rN j r

4.3.4. Electron energy-loss spectroscopy on solids (By C. Colliex)

ri rj :

The brackets h ivib denote averaging over the vibrational motion, r is the Dirac delta function, and ri ; . . . ; rn is the molecular wavefunction. Binding effects appear to be proportional to the ratio of the number of electrons involved in binding to the total number of electrons in the system (Kohl & Bonham, 1967; Bonham & Iijima, 1965) so that binding effects in molecules containing mainly heavy atoms should be quite small. The intensities, Is, for many small molecules have been calculated based on molecular Hartree±Fock wavefunctions. In most cases, a distinctive minimum has been found at about Ê 1 and a much small maximum at s 8 10 A Ê 1 in s 3 4A the cross-sectional difference curve between the IAM and the molecular HF results (Pulay, Mawhorter, Kohl & Fink, 1983; Kohl & Bartell, 1969; Liu & Smith, 1977; Epstein & Stewart,

4.3.4.1. Definitions 4.3.4.1.1. Use of electron beams Among the different spectroscopies available for investigating the electronic excitation spectrum of solids, inelastic electron scattering experiments are very useful because the range of accessible energy and momentum transfer is very large, as illustrated in Fig. 4.3.4.1 taken from Schnatterly (1979). Absorption measurements with photon beams follow the photon dispersion curve, because it is impossible to vary independently the energy and the momentum of a photon. In a scattering experiment, a quasi-parallel beam of monochromatic particles is incident on the specimen and one measures the changes in energy and momentum that can be attributed to the creation of a given excitation in the target. Inelastic neutron scattering is the most

391


4. PRODUCTION AND PROPERTIES OF RADIATIONS powerful technique for the low-energy range < 0:1 eV. On the other hand, inelastic X-ray scattering is well suited for the study of high momentum and large energy transfers because the energy resolution is limited to 1 eV and the cross section increases with momentum transfer. In the intermediate range, inelastic electron scattering [or electron energy-loss spectroscopy (EELS)] is the most useful technique. For recent reviews on different aspects of the subject, the reader may consult the texts by Schnatterly (1979), Raether (1980), Colliex (1984), Egerton (1986), and Spence (1988). 4.3.4.1.2. Parameters involved in the description of a single inelastic scattering event The importance of inelastic scattering as a function of energy and momentum transfer is governed by a double differential cross section: d2 ; d dE

4:3:4:1

where d corresponds to the solid angle of acceptance of the detector and dE to the energy window transmitted by the spectrometer. The experimental conditions must therefore be defined before any interpretation of the data is possible. Integrations of the cross section over the relevant angular and energy-loss domains correspond to partial or total cross sections, depending on the feature measured. For instance, the total inelastic cross section i corresponds to the probability of suffering any energy loss while being scattered into all solid angles. The discrimination between elastic and inelastic signal is generally defined by the energy resolution of the spectrometer, that is the minimum energy loss that can be unambiguously distinguished from the zero-loss peak; it is therefore very dependent on the instrumentation used. The kinematics of a single inelastic event can be described as shown in Fig. 4.3.4.2. In contrast to the elastic case, there is no simple relation between the scattering angle and the transfer of momentum hq. One has also to take into account the energy loss E. Combining both equations of conservation of momentum and energy,

Fig. 4.3.4.1. Definition of the regions in E; q space that can be investigated with the different primary sources of particles available at present [courtesy of Schnatterly (1979)].

h2 k02 h2 k 2 E ; 2m0 2m0 and q2 k2 k02 one obtains

" 2E0 1 qa0 R 2

1

E E0

2kk0 cos ; #

1=2 cos

4:3:4:3 E ; R

4:3:4:4

where the fundamental units a0 h2 =m0 e2 Bohr radius and R m0 e4 =2h2 Rydberg energy are used to introduce dimensionless quantities. In this kinematical description, one deals only with factors concerning the primary or the scattered particle, without considering specifically the information on the ejected electron. For a core-electron excitation of an atom, one distinguishes q (the momentum exchanged by the incident particle) and v (the momentum gained by the excited electron), the difference being absorbed by the recoil of the target nucleus (Maslen & Rossouw, 1983). 4.3.4.1.3. Problems associated with multiple scattering The strong coupling potential between the primary electron and the solid target is responsible for the occurrence of multiple inelastic events (and of mixed inelastic±elastic events) for thick specimens. To describe the interaction of a primary particle with an assembly of randomly distributed scattering centres (with a density N per unit volume), a useful concept is the mean free path defined as 1=N

4:3:4:5

for the cross section . The ratio t= measures the probability of occurrence of the event associated with the cross section when the incident particle travels a given length t through the material. This is true in the single scattering case, that is when t= 1. For increased thicknesses, one must take into account all multiple scattering events and this contribution begins to be non> a few tens of nanometres. Multiple scattering negligible for t is responsible for a broadening of the angular distribution of the

Fig. 4.3.4.2. A primary electron of energy E0 and wavevector k is inelastically scattered into a state of energy E0 E and wavevector k0 . The energy loss is E and the momentum change is hq. The scattering angle is and the scattered electron is collected within an aperture of solid angle d .

392


4:3:4:2

4.3. ELECTRON DIFFRACTION scattering electrons ± mostly due to single or multiple elastic events ± and of an important fraction of inelastic electrons suffering more than one energy loss. The probability of having n inelastic processes of mean free path is governed by the Poisson distribution: Pn t

t 1t n : exp n!

4:3:4:6

Multiple losses introduce additional peaks in the energy-loss spectrum; they are also responsible for an increased background that complicates the detection of single characteristic core-loss signals. Consequently, when the specimen thickness is not very > 50 nm for 100 keV primary electrons), it is small (i.e. for t necessary to retrieve the single scattering profile that is truly representative of the electronic and chemical properties of the specimen. Deconvolution techniques have therefore been developed to remove the effects of plural scattering from the low-loss spectrum (up to 100 eV) or from ionization edges but very few treatments deal with both angle and energy-loss distributions. Batson & Silcox (1983) have made a detailed study of the inelastic scattering properties of incident 75 keV electrons through a 100 nm thick polycrystalline aluminium film, over Ê 1 and the full range of transferred wavevectors 0 ! 3 A energy losses 0 ! 100 eV. Schattschneider (1983) has proposed a matrix approach that is sufficiently elaborate to handle angle-resolved energy-loss data. Simpler deconvolution schemes have been proposed and used for processing multiple losses without specific consideration of angular truncation effects. They are valid when the data have been recorded over sufficiently large angles of collection so that all appreciable inelastic scattering is included. They are generally based on Fourier transform techniques, except for the iterative approach of Daniels, Festenberg, Raether & Zeppenfeld (1970), which has been used for energy losses up to about 60 eV (Colliex, Gasgnier & Trebbia, 1976). The most accurate current methods are the Fourier-log method of Johnson & Spence (1974) and Spence (1979), and the Fourier-ratio method of Swyt & Leapman (1982), which only applies to the core-loss region. The first is far more complete and accurate and applies to any spectral range when one has recorded a full spectrum in unsaturated conditions.

4.3.4.1.4. Classification of the different types of excitations contained in an electron energy-loss spectrum Figs. 4.3.4.3 and 4.3.4.4 display examples of electron energyloss spectra over large energy domains, typically from 1 to about 2000 eV. With one instrument, all elementary excitations from the near infrared to the X-ray domain can be investigated. Apart from the main beam or zero-loss peak, two major families of electronic transitions can be distinguished in such spectra: a The excitations in the low or moderate energy-loss region 1 < E < 50 eV concern the quasifree (valence and conduction) electron gas. In a pure metal, such as Al, the dominant feature is the intense narrow peak at 15 eV with its multiple satellites at about 30, 45, and 60 eV. One also detects an interband transition at 1.5 eV and a surface plasmon peak at 7 eV. For the more complex mineral specimen, rhodizite, this contribution lies in the intense and broad, but not very specific, peak around 25 eV. All these features are discussed in detail in Subsection 4.3.4.3. b The excitations in the high-energy-loss domain 50 < E < 2000 eV concern excitation and ionization processes from core atomic orbitals: in Al, the L2;3 edge is associated with the creation of holes on the 2p level, L1 is due to the excitation of 2s, and K of 1s electrons. These contributions appear as edges superposed on a regularly decreasing background. In the complex specimen, the succession of these

Fig. 4.3.4.4. Complete electron energy-loss spectrum of a thin rhodizite crystal (thickness 60 nm). Separate spectra from eight significantly overlapping energy ranges were measured and matched. Primary energy 60 keV. Semi-angle of collection 5 mrad. Recording time 300 s (parallel acquisition). Scanned area 30 40 nm. Analysed mass 2 10 15 g [courtesy of Engel, Sauer, Zeitler, Brydson, Williams & Thomas (1988)].

Fig. 4.3.4.3. Excitation spectrum of aluminium from 1 to 250 eV, investigated by EELS on 300 keV primary electrons [courtesy of Schnatterly (1979)].

Fig. 4.3.4.5. Schematic energy-level representation of the origin of a core-loss excitation (from a core level C at energy Ec to an unoccupied state U above the Fermi level Ef ) and its general shape in EELS, as superimposed on a continuously decreasing background.

393


4. PRODUCTION AND PROPERTIES OF RADIATIONS different edges on top of the monotonously decaying background is a signature of the elemental composition, the intensity of the signals being roughly proportional to the relative concentration in the associated element. Core-level EELS spectroscopy therefore investigates transitions from one well defined atomic orbital to a vacant state above the Fermi level: it is a probe of the energy distribution of vacant states in a solid, see Fig. 4.3.4.5. As the excited electron is promoted on a given atomic site, the information involved has two specific characters: it provides the local atomic point of view and it reflects the existence of the hole created, which can be more or less screened by the surrounding population of electrons in the solid. The properties of this family of excitations are the subject of Subsection 4.3.4.4. The non-characteristic background is due to the superposition of several contributions: the high-energy tail of valence-electron scattering, the tails of core losses with lower binding energy, Bremsstrahlung energy losses, plural scattering, etc. It is therefore rather difficult to model its behaviour, although some efforts have been made along this direction using Monte Carlo simulation of multiple scattering (Jouffrey, Sevely, Zanchi & Kihn, 1985). When one monochromatizes the natural energy width of the primary beam to much smaller values (about a few meV) than its natural width, one has access to the infrared part of the electromagnetic spectrum. An example is provided in Fig. 4.3.4.6 for a specimen of germanium in the energy-loss range 0 up to 500 meV. In this case, one can investigate phonon modes, or the bonding states of impurities on surfaces. This field has been much less extensively studied than the higher-energy-loss range [for references see Ibach & Mills (1982)]. Generally, EELS techniques can be applied to a large variety of specimens. However, for the following review to remain of limited size, it is restricted to electron energy-loss spectroscopy on solids and surfaces in transmission and reflection. It omits some important aspects such as electron energy-loss spectroscopy in gases with its associated information on atomic and molecular states. In this domain, a bibliography of inner-shell excitation studies of atoms and molecules by electrons, photons or theory is available from Hitchcock (1982).

Fig. 4.3.4.6. Energy-loss spectrum, in the meV region, of an evaporated germanium film (thickness ' 25 nm). Primary electron energy 25 keV. Scattering angle < 10 4 . One detects the contributions of the phonon excitation, of the Ge O bonding, and of intraband transitions [courtesy of SchroÈder & Geiger (1972)].

Table 4.3.4.1. Different possibilities for using EELS information as a function of the different accessible parameters (r, , E) Selection parameter

Results

Working mode of the spectrometer

1

h

r

Spectrum Ir E

Analyser

2

r

h

Spectrum Ih E

Analyser

3

h

E

Energy-filtered image IE r

Filter

4

r

E

Energy-filtered diffraction pattern IE h

Filter

4.3.4.2. Instrumentation 4.3.4.2.1. General instrumental considerations In a dedicated instrument for electron inelastic scattering studies, one aims at the best momentum and energy resolution with a well collimated and monochromatized primary beam. This is achieved at the cost of poor spatial localization of the incident electrons and one assumes the specimens to be homogeneous over the whole irradiated volume. In a sophisticated instrument such as that built by Fink & Kisker (1980), the energy resolution can be varied from 0.08 to 0.7 eV, and the Ê 1 , but momentum transfer resolution between 0.03 and 0.2 A typical values for the electron-beam diameter are about 0.2 to 1 mm. Nowadays, many energy-analysing devices are coupled with an electron microscope: consequently, an inelastic scattering study involves recording for a primary intensity I0 , the current Ir; h; E scattered or transmitted at the position r on the specimen, in the direction h with respect to the primary beam, and with an energy loss E. Spatial resolution is achieved either with a focused probe or by a selected area method, angular acceptance is defined by an aperture, and energy width is controlled by a detector function after the spectrometer. It is not possible from signal-to-noise considerations to reduce simultaneously all instrumental widths to very small values. One of the parameters r; h or E is chosen for signal integration, another for selection, and the last is the variable. Table 4.3.4.1. classifies these different possibilities for inelastic scattering studies. Because of the great variety of possible EELS experiments, it is impossible to build an optimum spectrometer for all applications. For instance, the design of a spectrometer for low-energy incident electrons and surface studies is different from that for high-energy incident electrons and transmission work. In the latter category, instruments built for dedicated EELS studies (Killat, 1974; Gibbons, Ritsko & Schnatterly, 1975; Fink & Kisker, 1980; etc.) are different from those inserted within an electron-microscope environment, in which case it is possible to investigate the excitation spectrum from a specimen area well characterized in image and diffraction [see the reviews by Colliex (1984) and Egerton (1986)]. The literature on dispersive electron±optical systems (equivalent to optical prisms) is very large. For example, the theory of uniform field magnets, which constitute an important family of analysing devices, has been extensively developed for the components in high-energy particle accelerators (Enge, 1967; Livingood, 1969). As for EELS spectrometers, they can be classified as:

394


Integration parameter

4.3. ELECTRON DIFFRACTION a Monochromators, which filter the incident beam to obtain the smallest primary energy width. The natural width for a heated W filament is about 1 eV, possibly rising to about a few eV as a consequence of stochastic interactions [Boersch (1954) effect, analysed for instance by Rose & Spehr (1980)]. For a low-temperature field-emission source, this energy spread is only 0.3 eV. This constitutes a clear gain but remains insufficient for meV studies. In this case, one has to introduce a filter lens such as the three-electrode design developed by Hartl (1966) or a cylindrical electrostatic deflector before the accelerator [Kuyatt & Simpson (1967) or Gibbons et al. (1975)]. In both cases, an energy resolution of 50 meV has been achieved for electron beams of 50±300 keV at the specimen. b Analysers, which measure the energy distribution of the beam scattered from the specimen. They can be used either strictly as analysers displaying the energy loss from a given specimen volume, or as filters (or selecting devices) that provide 2D images or diffraction patterns with a given energy loss. 4.3.4.2.2. Spectrometers Fig. 4.3.4.7 defines the basic parameters of a `general' energy-loss spectrometer: a region of electrostatic E and/or magnetic B fields transforms a distribution of electrons I0 x0 ; y0 ; t0 ; u0 ; in the object plane of coordinate z0 along the principal trajectory, into a distribution of electrons I1 x1 ; y1 ; t1 ; u1 ; in the object plane of coordinate z1 , coincident with the detector plane (or optically conjugate to it). The transverse coordinates are labelled as x; y, the angular ones as t; u, and p=p E=2E is the relative change in absolute momentum value associated with the energy loss. Common properties of such systems are: a first-order imaging properties or stigmatism, i.e. all electrons leaving x0 ; y0 are focused at the same x1 ; y1 point, independently of their inclination on the optical axis;

Fig. 4.3.4.7. Schematic drawing of a uniform magnetic sector spectrometer with induction B normal to the plane of the figure. Definition of the coordinates used in the text (the object plane at coordinate z0 along the mean trajectory coincides with the specimen, and the image plane at z1 coincides with the dispersion plane and the detector level).

b strong chromatic aberration in order to realize an efficient discrimination between electrons of different . The spectrometer performance can be evaluated with the following parameters: D dispersion beam displacement in the spectrometer image plane for a given momentum change ; it is generally expressed in cm=eV. The higher the dispersion, the easier it is to resolve small energy losses. For a straight-edge 90 magnetic sector, D / 2R=E0 , where R is the curvature radius of the mean trajectory and E0 is the primary energy. Emin energy resolution. This corresponds to the minimumenergy variation that can be resolved by the instrument. It takes into account the width of the image ximage Mr, where M is the spectrometer magnification and r the radius of the spectrometer source, as well as the second- and higher-order angular aberrations. These are responsible for the imperfect focusing of the electrons that enter the spectrometer within a cone of angular acceptance 0 and contribute through a term xaber C 20 . Moreover, one must convolute these terms with the natural width E0 of the primary beam, including AC fields, and with the detection slit width xslit . Combining all these effects, as shown schematically in Fig. 4.3.4.8, one obtains approximately: xtot xslit 2 ximage 2 xaber 2 D E 20 1=2 4:3:4:7 and the corresponding energy resolution is defined as Emin xtot min =D. In many situations, the dominant factor is the second-order aberration term C 20 so that the figure of merit F, defined as F 0 E0 =Emin , is of the order of unity for an uncorrected magnetic spectrometer. From this simplified discussion, one deduces that there is generally competition between large angular acceptance for the inelastic signal, which is very important for obtaining a high signal-to-noise ratio (SNR) for core-level excitations, and good energy resolution. Two solutions have been used to remedy this limitation. The first is to improve spectrometer design, for example by correcting second-order aberrations in a homogeneous magnetic prism (Crewe, 1977a; Parker, Utlaut & Isaacson, 1978; Egerton, 1980b; Krivanek & Swann, 1981; etc). This can enhance the figure of merit by at least a factor of 100. The second possibility is to transform the distribution of electrons to be analysed at the exit surface of the specimen into a more convenient distribution at the spectrometer entrance. This can be done by introducing versatile transfer optics (see Crewe, 1977b; Egerton, 1980a; Johnson, 1980; Craven & Buggy, 1981; etc.). As a final remark on the energy resolution of a spectrometer, it is meaningless to define it without reference to the size and the angular aperture of the analysed beam.

Fig. 4.3.4.8. Different factors contributing to the energy resolution in the dispersion plane [courtesy of Johnson (1979)].

395


4. PRODUCTION AND PROPERTIES OF RADIATIONS Historically, many types of spectrometer have been used since the first suggestion by Wien (1897) that an energy analyser could be designed by employing crossed electric and magnetic fields. Reviews have been published by Klemperer (1965), Metherell (1971), Pearce-Percy (1978), and Egerton (1986). Nowadays, two configurations are mostly used and have become commercially available on modern electron microscopes: these are spectrometers on TEM/STEM instruments and filters on CTEM ones. In the first case, homogeneous magnetic sectors are the simplest and most widely used devices. Recent instrumental developments by Shuman (1980), Krivanek & Swann (1981), and Scheinfein & Isaacson (1984) have given birth to a generation of spectrometers with the following major character-

istics: double focusing, correction for second-order aberrations, dispersion plane perpendicular to the trajectory. This has been made possible by a suitable choice of several parameters, such as the tilt angles and the radius of curvature for the entrance and exit faces and the correct choice of the object source position. As an example, for a 100 keV STEM equipped with a field emission gun, the coupling illustrated in Fig. 4.3.4.9 leads to an energy resolution of 0.35 eV for 0 7:5 mrad on the specimen as visible on the elastic peak, and 0.6 eV for 0 25 mrad as checked on the fine structures on core losses. Krivanek, Manoubi & Colliex (1985) demonstrated a sub-eV energy resolution over the whole range of energy losses up to 1 or 2 keV. A very competitive solution is the Wien filter, which consists of uniform electric and magnetic fields crossed perpendicularly, see Fig. 4.3.4.10. An electron travelling along the axis with a velocity v0 such that jv0 j E=B is not deflected, the net force on it being null. All electrons with different velocities, or at some angle with respect to the optical axis, are deflected. The dispersion of the system is greatly enhanced by decelerating the electrons to about 100 eV within the filter, in which case D ' a few 100 mm=eV. A presently achievable energy resolution of 150 meV at a spectrometer collection half-angle of 12.5 mrad has been demonstrated by Batson (1986, 1989). It allows the study of the detailed shape of the energy distribution of the electrons emitted from a field emission source and the taking of it into account in the investigation of band-gap states in semiconductors (Batson, 1987). Filtering devices have been designed to form an energyfiltered image or diffraction pattern in a CTEM. The first

Fig. 4.3.4.9. Optical coupling of a magnetic sector spectrometer on a STEM column.

Fig. 4.3.4.10. Principle of the Wien filter used as an EELS spectrometer: the trajectories are shown in the two principal (dispersive and focusing) sections.

Fig. 4.3.4.11. Principle of the Castaing & Henry filter made from a magnetic prism and an electrostatic mirror. R1 ; R2 , and R3 are the real conjugate stigmatic points, and V1 ; V2 , and V3 the virtual ones: the dispersion plane coincides with the R3 level and achromatic one with the V3 level.)

396


4.3. ELECTRON DIFFRACTION Table 4.3.4.2. Plasmon energies measured (and calculated) for a few simple metals; most data have been extracted from Raether (1980)

Li Na K Rb Cs

Monovalent

Divalent

Trivalent

Tetravalent

h!p (eV)

h!p (eV)

h!p (eV)

h!p (eV)

Meas.

Calc.

7.1 5.7 3.7 3.4 2.9

(8.0) (5.9) (4.3) (3.9) (3.4)

Be Mg Ca Sr Ba

Meas.

Calc.

18.7 10.4 8.8 8.0 7.2

(18.4) (10.9) (8.0) (7.0) (6.7)

B Al Ga In Sc

solution, reproduced in Fig. 4.3.4.11, is due to Castaing & Henry (1962). It consists of a double magnetic prism and a concave electrostatic mirror biased at the potential of the microscope cathode. The system possesses two pairs of stigmatic points that may coincide with a diffraction plane and an image plane of the electron-microscope column. One of these sets of points is achromatic and can be used for image filtering. The other is strongly chromatic and is used for spectrum analysis. Zanchi, Sevely & Jouffrey (1977) and Rose & Plies (1974) have proposed replacing this system, which requires an extra source of high voltage for the mirror, by a purely magnetic equivalent device. Several solutions, known as the and ! filters, with three or four magnets, have thus been built, both on very high voltage microscopes (Zanchi, Perez & Sevely, 1975) and on more conventional ones (Krahl & Herrmann, 1980), the latest version now being available from one EM manufacturer (Zeiss EM S12). 4.3.4.2.3. Detection systems The final important component in EELS is the detector that measures the electron flux in the dispersion plane of the spectrometer and transfers it through a suitable interface to the data storage device for further computer processing. Until about 1990, all systems were operated in a sequential acquisition mode. The dispersed beam was scanned in front of a narrow slit located in the spectrometer dispersion plane. Electrons were then generally recorded by a combination of scintillator and photomultiplier capable of single electron counting.

Meas.

Calc.

22.7 14.95 13.8 11.4 14.0

(?) (15.8) (14.5) (12.5) (12.9)

C Si Ge Sn Pb

Meas.

Calc.

34.0 16.5 16.0 13.7 (13)

(31) (16.6) (15.6) (14.3) (13.5)

This process is, however, highly inefficient: while the counts are measured in one channel, all information concerning the other channels is lost. These requirements for improved detection efficiency have led to the consideration of possible solutions for parallel detection of the EELS spectrum. They use a multiarray of detectors, the position, the size and the number of which have to be adapted to the spectral distribution delivered by the spectrometer. In most cases with magnetic type devices, auxiliary electron optics has to be introduced between the spectrometer and the detector so that the dispersion matches the size of the individual detection cells. Different systems have been proposed and tested for recording media, the most widely used solutions at present being the photodiode and the chargecoupled diode arrays described by Shuman & Kruit (1985), Krivanek, Ahn & Keeney (1987), Strauss, Naday, Sherman & Zaluzec (1987), Egerton & Crozier (1987), Berger & McMullan (1989), etc. Fig. 4.3.4.12 shows a device, now commercially available from Gatan, that is made of a convenient combination of these different components. This progress in detection has led to significant improvements in many areas of EELS: enhanced detection limits, reduced beam damage in sensitive materials, data of improved quality in terms of both SNR and resolution, and access to time-resolved spectroscopy at the ms time scale (chronospectra). Several of these important consequences are illustrated in the following sections. 4.3.4.3. Excitation spectrum of valence electrons Most inelastic interaction of fast incident electrons is with outer atomic shells in atoms, or in solids with valence electrons (referred to as conduction electrons in metals). These involve excitations in the 0±50 eV range, but, in a few cases, interband transitions from low-binding-energy shells may also contribute. 4.3.4.3.1. Volume plasmons

Fig. 4.3.4.12. A commercial EELS spectrometer designed for parallel detection on a photodiode array. The family of quadrupoles controls the dispersion on the detector level [courtesy of Krivanek et al. (1987))].

The basic concept introduced by the many-body theory in the interacting free electron gas is the volume plasmon. In a condensed material, the assembly of loosely bound electrons behaves as a plasma in which collective oscillations can be induced by a fast external charged particle. These eigenmodes, known as volume plasmons, are longitudinal charge-density fluctuations around the average bulk density in the plasma n ' 1028 e =m3 . Their eigen frequency is given, in the free electron gas, as 2 1=2 ne : 4:3:4:8 !p m"0 The corresponding h!p energy, measured in an energy-loss spectrum (see the famous example of the plasmon in aluminium

397


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.3.4.3. Experimental and theoretical values for the coefficient in the plasmon dispersion curve together with estimates of the cut-off wavevector ( from Raether, 1980)

Li Na K Mg Al In Si

Measured

Calculated

Ê 1) qc (A

0.24 0.24 0.14 0.35 (0.5 (0.5 0.41 0.3

0.35 0.32 0.29 0.39

0.9 0.8 0.8 1.0

0.43

1.3

0.45

1.1

0.2 0.45 0.40 0.66

Ê A Ê A Ê A Ê A

1

) ) 1 ) 1 ) 1

Experimental (eV) Li Na K Rb Cs Al Mg Si Ge

in Fig. 4.3.4.3), is the plasmon energy, for which typical values in a selection of pure solid elements are gathered in Table 4.3.4.2. The accuracies of the measured values depend on several instrumental parameters. Moreover, they are sensitive to the specimen crystalline state and to its degree of purity. Consequently, there exist slight discrepancies between published values. Numbers listed in Table 4.3.4.2 must therefore be accepted with a 0.1 eV confidence. Some specific cases require comments: amorphous boron, when prepared by vacuum evaporation, is not a well defined specimen. Carbon exists in several allotropic varieties. The selection of the diamond type in the table is made for direct comparison with the other tetravalent specimens (Si, Ge, Sn). The results for lead (Pb) are still subject to confirmation. The volumic mass density is an important factor (through n) in governing the value of the plasmon energy. It varies with temperature and may be different in the crystal, in the amorphous, and in the liquid phases. In simple metals, the amorphous state is generally less dense than the crystalline one, so that its plasmon energy shifts to lower energies. The above description applies only to very small scattering vectors q. In fact, the plasmon energy increases with scattering angle (and with momentum transfer hq). This dependence is known as the dispersion relation, in which two distinct behaviours can be described: a For small momentum transfers q < qc , the dispersion curve is parabolic: h!p q h!p 0

h2 2 q: m0

Table 4.3.4.4. Comparison of measured and calculated values for the halfwidth E1=2 (0) of the plasmon line ( from Raether, 1980)

4:3:4:9

Theory (eV)

2.2 0.3 0.25 0.6 1.2 0.53 0.7 3.2 3.1

2.55 0.12 0.15 0.64 0.96 0.43 0.7 5.4 3.9

bending of the experimental curves. Electron±electron correlations have also been considered, which has slightly improved the agreement between calculated and measured values of (Bross, 1978a; b). b For large momentum transfers, there exists a critical wavevector qc , which corresponds to a strong decay of the plasmon mode into single electron±hole pair excitations. This can be calculated using conservation rules in energy and momentum, giving h!p 0

h2 2 h2 qc q2 2qc qF ; m0 2m0 c

4:3:4:11

where qF is the Fermi wavevector. A simple approximation is qc ' !p =vF , vF being the Fermi velocity. Single pair excitations can be created by fast incoming electrons in the domain of scattering conditions contained between the two curves: 9 h2 > 2 Emax q 2qqF > > = 2m0 4:3:4:12 > > h2 2 > q 2qqF ; Emin 2m0 shown in Fig. 4.3.4.13. They bracket the curve E h2 q2 =2m0 corresponding to the transfer of energy and momentum to an isolated free electron. For momentum transfers such as q > qc , the plasmon mode is heavily damped and it is difficult to distinguish its own specific behaviour from the electron±hole continuum. A few studies, e.g. Batson & Silcox (1983), indicate that the plasmon dispersion curve flattens as it enters the

The coefficient has been measured in a number of substances and calculated for the free-electron case in the random phase approximation (Lindhard, 1954); see Table 4.3.4.3 for some data. A simple expression for is 35

EF ; h!p 0

4:3:4:10

where EF is the Fermi energy of the electron gas. More detailed observations indicated that it is not possible to describe the dispersion curve over a large momentum range with a single q2 law. In fact, one has to fit the experiment data with different linear or quadratic slopes as a function of q [see values indicated for Al and In in Table 4.3.4.3, and Hohberger, Otto & Petri (1975)]. Moreover, anisotropy has been found along different q directions in monocrystals (Manzke, 1980). In parallel, refinements have been brought into the calculations by including band-structure effects to deal with the anisotropy of the dispersion relation and with the

Fig. 4.3.4.13. The dispersion curve for the excitation of a plasmon (curve 1) merges into the continuum of individual electron±hole excitations (between curves 2 and 4) for a critical wavevector qc . The intermediate curve (3) corresponds to Compton scattering on a free electron.

398


4.3. ELECTRON DIFFRACTION quasiparticle domain and approaches the centre of the continuum close to the free-electron curve. However, not only is the scatter between measurements fairly high, but a satisfactory theory is not yet available [see Schattschneider (1989) for a compilation of data on the subject]. Plasmon lifetime is inversely proportional to the energy width of the plasmon peak E1=2 . Even for Al, with one of the smallest plasmon energy widths ' 0:5 eV, the lifetime is very short: after about five oscillations, their amplitude is reduced to 1=e. Such a damping demonstrates the strength of the coupling of the collective modes with other processes. Several mechanisms compete for plasmon decay: a For small momentum transfer, it is generally attributed to vertical interband transitions. Table 4.3.4.4, extracted from Raether (1980), compares a few measured values of E1=2 0, with values calculated using band-structure descriptions. b For moderate momentum transfer q, a variation law such as 2

4

E1=2 q E1=2 0 Bq Oq

4:3:4:13

has been measured. The q dependence of E1=2 is mainly accounted for by non-vertical transitions compatible with the band structure, the number of these transitions increasing with q (Sturm, 1982). Other mechanisms have also been suggested, such as phonons, umklapp processes, scattering on surfaces, etc. c For large momentum transfer (i.e. of the order of the critical wavevector qc ), the collective modes decay into the strong electron±hole-pair channels already described giving rise to a clear increase of the damping for values of q > qc . Within this free-electron-gas description, the differential cross section for the excitation of bulk plasmons by incident electrons of velocity v is given by dp Ep 1 ; d

2Na0 m0 v2 2 2E

4:3:4:14

where N is the density of atoms per volume unit and E is the characteristic inelastic angle defined as Ep =2E0 in the non-relativistic description and as Ep = m0 v2 {with

1 v2 =c2 1=2 } in the relativistic case. The angular dependence of the differential cross section for plasmon scattering is shown in Fig. 4.3.4.14. The integral cross section up to an angle 0 is Z 0 p 0 0

dp Ep log 0 =E d : Na0 m0 v2 d

4:3:4:15

The total plasmon cross section is calculated for 0 c qc =k0 . Converted into mean free path, this becomes 1 1 a 0 log c (non-relativistic formula); p E Np E 4:3:4:16 and a m v2 p 0 0 Ep

hqc v log 1:132 h !p

!

1

(relativistic formula): 4:3:4:17

The behaviour of p as a function of the primary electron energy is shown in Fig. 4.3.4.15. 4.3.4.3.2. Dielectric description The description of the bulk plasmon in the free-electron gas can be extended to any type of condensed material by introducing the dielectric response function "q; !, which describes the frequency and wavevector-dependent polarizability of the medium; cf. Daniels et al (1970). One associates, respectively, the "T and "L functions with the propagation of transverse and longitudinal EM modes through matter. In the small-q limit, these tend towards the same value: lim "T q; ! lim "L q; ! "0; !:

q!0

q!0

As transverse dielectric functions are only used for wavevectors close to zero, the T and L indices can be omitted so that: "L q; ! "q; ! and

"T q; ! "0; !:

The transverse solution corresponds to the normal propagation of EM waves in a medium of dielectric coefficient "0; !, i.e. to

Fig. 4.3.4.14. Measured angular dependence of the differential cross section d= d for the 15 eV plasmon loss in Al (dots) compared with a calculated curve by Ferrell (solid curve) and with a sharp cut-off approximation at c (dashed curved). Also shown along the scattering angle axis, E characteristic inelastic angle defined as E=2E0 , R ~ median inelastic angle defined by R ~ c 0 d= d d 1=2R 0 d= d d ,R and average inelastic angle defined by d= d d = d= d d [courtesy of Egerton (1986)].

Fig. 4.3.4.15. Variation of plasmon excitation mean free path p as a function of accelerating voltage V in the case of carbon and aluminium [courtesy of Sevely (1985)].

399


4. PRODUCTION AND PROPERTIES OF RADIATIONS 2 2

qc "0; ! 0: 4:3:4:18 !2 For longitudinal fields, the only solution is "q; ! 0, which is basically the dispersion relation for the bulk plasmon. In the framework of the Maxwell description of wave propagation in matter, it has been shown by several authors [see, for instance, Ritchie (1957)] that the transfer of energy between the beam electron and the electrons in the solid is governed by the magnitude of the energy-loss function Im1="q; !, so that d2 1 1 1 : 4:3:4:19 Im dE d Ne a0 2 q2 "q; ! One can deduce (4.3.4.14) by introducing a function at energy loss !p for the energy-loss function: 1 4:3:4:20 Im !p ! !p : "q; ! 2

As a special case, in an insulator, nf 0 and all the electrons ni n have a binding energy at least equal to the band gap Eg ' h!i , giving !2p Eg =h2 ne2 =m"0 . This description constitutes a satisfactory first step into the world of real solids with a complex system of valence and conduction bands between which there is a strong transition rate of individual electrons under the influence of photon or electron beams. In optical spectroscopy, for instance, this transition rate, which governs the absorption coefficient, can be deduced from the calculation of the factor "2 as "2 !

A jM 0 j2 J 0 !; !2 jj jj

4:3:4:24

where Mjj0 is the matrix element for the transition from the occupied level j in the valence band to the unoccupied level j0 in the conduction band, both with the same k value (which means for a vertical transition). Jjj0 ! is the joint density of states (JDOS) with the energy difference h!. This formula is also valid

As a consequence of the causality principle, a knowledge of the energy-loss function Im1="! over the complete frequency (or energy-loss) range enables one to calculate Re1="! by Kramers±Kronig analysis: Z1 1 2 1 !0 Re 1 PP Im d!0 ; 4:3:4:21 0 0 2 !2 "! "! ! 0

where PP denotes the principal part of the integral. For details of efficient practical evaluation of the above equation, see Johnson (1975). The dielectric functions can be easily calculated for simple descriptions of the electron gas. In the Drude model, i.e. for a free-electron plasma with a relaxation time , the dielectric function at long wavelengths q ! 0 is "! "1 ! i"2 ! 1

! 2p !2 1

1 1=i!;

4:3:4:22

with !2p ne2 =m"0 , as above. The behaviour of the different functions, the real and imaginary terms in ", and the energy-loss function are shown in Fig. 4.3.4.16. The energy-loss term exhibits a sharp Lorentzian profile centred at ! !p and of width 1=. The narrower and more intense this plasmon peak, the more the involved valence electrons behave like free electrons. In the Lorentz model, i.e. for a gas of bound electrons with one or several excitation eigenfrequencies !i , the dielectric function is X n e2 1 i "! 1 ; 4:3:4:23 2 2 m"0 !i ! i!=i i where ni denotes the density of electrons oscillating with the frequency !i and i is the associated relaxation time. The characteristic "1 , "2 , and Im1=" behaviours are displayed in Fig. 4.3.4.17: a typical ìnterband' transition (in solid-state terminology) can be revealed as a maximum in the "2 function, simultaneous with a `plasmon' mode associated with a maximum in the energy-loss function and slightly shifted to higher energies with respect to the annulation conditions of the "1 function. In most practical situations, there coexist a family of nf free electrons (with plasma frequency !2p nf e2 =m"0 and one or several families of ni bound electrons (with eigenfrequencies !i . The influence of bound electrons is to shift the plasma frequency towards lower values if !i > !p and to higher values if !i < !p .

Fig. 4.3.4.16. Dielectric and optical functions calculated in the Drude model of a free-electron gas with h!p 16 eV and 1:64 10 16 s. R is the optical reflection coefficient in normal k2 =n 12 k2 with n and k the incidence, i.e. R n 12p real and imaginary parts of ". The effective numbers neff "2 and neff Im 1=" are defined in Subsection 4.3.4.5 [courtesy of Daniels et al. (1970)].

400


4.3. ELECTRON DIFFRACTION for small-angle-scattering electron inelastic processes. When parabolic bands are used to represent, respectively, the upper part of the valence band and the lower part of the conduction band in a semiconductor, the dominant JDOS term close to the onset of the interband transitions takes the form JDOS / E

Eg 1=2 ;

4:3:4:25

where Eg is the band-gap energy. This concept has been successfully used by Batson (1987) for the detection of gap energy variations between the bulk and the vicinity of a single misfit dislocation in a GaAs specimen. The case of non-vertical transitions involving integration over k-space has also been considered (Fink et al., 1984; Fink & Leising, 1986). 4.3.4.3.3. Real solids The dielectric constants of many solids have been deduced from a number of methods involving either primary photon or electron beams. In optical measurements, one obtains the values of "1 and "2 from a Krakers±Kronig analysis of the optical

absorption and reflection curves, while in electron energy-loss measurements they are deduced from Kramers±Kronig analysis of energy-loss functions. Fig. 4.3.4.18 shows typical behaviours of the dielectric and energy-loss functions. a For a free-electron metal (Al), the Drude model is a satisfactory description with a well defined narrow and intense maximum of Im 1=" corresponding to the collective plasmon excitation together with typical conditions "1 ' "2 ' 0 for this energy h!p . One also notices a weak interband transition below 2 eV. b For transition and noble metals (such as Au), the results strongly deviate from the free-electron gas function as a consequence of intense interband transitions originating mostly from the partially or fully filled d band lying in the vicinity of, or just below, the Fermi level. There is no clear condition for satisfying the criterion of plasma excitation " 0 so that the collective modes are strongly damped. However, the higherlying peak is more generally of a collective nature because it coincides with the exhaustion of all oscillator strengths for interband transitions. c Similar arguments can be developed for a semiconductor (InSb) or an insulator (Xe solid). In the first case, one detects a few interband transitions at small energies that do not prevent the occurrence of a pronounced volume plasmon peak rather similar to the free-electron case. The difference between the gap and the plasma energy is so great that the valence electrons behave collectively as an assembly of free particles. In contrast, for wide gap insulators (alkali halides, oxides, solid rare gases), a number of peaks are seen, owing to different interband transitions and exciton peaks. Excitons are quasi-particles consisting of a conduction-band electron and a valence-band hole bound to each other by Coloumb interaction. One observes the existence of a band gap [no excitation either in "2 or in Im 1=" below a given critical value Eg ] and again the higher-lying peak is generally of a rather collective nature. CÏerenkov radiation is emitted when the velocity v of an electron travelling through a medium exceeds the speed of light for a particular frequency in this medium. The criterion for CÏerenkov emission is "1 ! >

c2 2: v2

4:3:4:26

In an insulator, "1 is positive at low energies and can considerably exceed unity, so that a `radiation peak' can be detected in the corresponding energy-loss range (between 2 and 4 eV in Si, Ge, III±V compounds, diamond, . . .); see Von Festenberg (1968), KroÈger (1970), and Chen & Silcox (1971). The associated scattering angle, ' lel =lph ' 10 5 rad for high-energy electrons, is very small and this contribution can only be detected using a limited forward-scattering angular acceptance. In an anisotropic crystal, the dielectric function has the character of a tensor, so that the energy-loss function is expressed as 0 1 1 B C Im@ P P A: "ij qi qj i

Fig. 4.3.4.17. Same as previous figure, but for a Lorentz model with an oscillator of eigenfrequency h!0 10 eV and relaxation time 0 6:6 10 16 s superposed on the free-electron term [courtesy of Daniels et al. (1970)].

j

If it is transformed to its orthogonal principal axes "11 ; "22 ; "33 , and if the q components in this system are q1 ; q2 ; q3 , the above expression simplifies to

401


4:3:4:27

4. PRODUCTION AND PROPERTIES OF RADIATIONS

Fig. 4.3.4.18. Dielectric coefficients "1 , "2 and Im 1=" from a collection of typical real solids: a aluminium [courtesy of Raether (1965)]; b gold [courtesy of Wehenkel (1975)]; c InSb [courtesy of Zimmermann (1976)]; d solid xenon at ca 5 K [courtesy of Keil (1968)].

402


0

1

1 A : Im@ P "ii q 2i

4.3. ELECTRON DIFFRACTION 4:3:4:28

The corresponding charge-density fluctuation is contained within the x boundary plane, z being normal to the surface: x; z ' cosq x

i

In a uniaxial crystal, such as a graphite, "11 "22 "? and "33 "k (i.e. parallel to the c axis): "q; ! "? sin2 "k cos2 ;

4.3.4.3.4. Surface plasmons Volume plasmons are longitudinal waves of charge density propagating through the bulk of the solid. Similarly, three exist longitudinal waves of charge density travelling along the surface between two media A and B (one may be a vacuum): these are the surface plasmons (Kliewer & Fuchs, 1974). Boundary conditions imply that "A ! "B ! 0:

4:3:4:31

and the associated electrostatic potential oscillates in space and time as

4:3:4:29

where is the angle between q and the c axis. The spectrum depends on the direction of q, either parallel or perpendicular to the c axis, as shown in Fig. 4.3.4.19 from Venghaus (1975). These experimental conditions may be achieved by tilting the graphite layer at 45 with respect to the incident axis, and recording spectra in two directions at E with respect to it (see Fig. 4.3.4.20).

!tz;

'x; z cos q x

!t exp qjzj:

4:3:4:32

The characteristic energy !s of this surface mode is estimated in the free electron case as: In the planar interface case: 9 !p > !s p > > > 2 > > > > (interface metal±vacuum); > > > > !p > > > !s = 1=2 1 "d 4:3:4:33 > (interface metal±dielectric of constant "d ); > > > > > 2 1=2 > > !pA !2pB > > !s > > > 2 > > ; (interface metal A±metal B).

4:3:4:30

In the spherical interface case:

Fig. 4.3.4.19. Dielectic functions in graphite derived from energy losses for E ? c (i.e. the electric field vector being in the layer plane) and for Ekc [from Daniels et al. (1970)]. The dashed line represents data extracted from optical reflectivity measurements [from Taft & Philipp (1965)].

403


4. PRODUCTION AND PROPERTIES OF RADIATIONS !s l

!p 2l 1=l

1=2

4:3:4:34a

(metal sphere in vacuum ± the modes are now quantified following the l quantum number in spherical geometry); !p !s l 4:3:4:34b 2l 1=l 11=2 (spherical void within metal). Thin-film geometry: 1=2 1 exp qt !s !p 1 "d

4:3:4:35

(metal layer of thickness t embedded in dielectric films of constant "d ). The two solutions result from the coupling of the oscillations on the two surfaces, the electric field being symmetric for the !s mode and antisymmetric for the !s . In a real solid, the surface plasmon modes are determined by the roots of the equation "!s 1 for vacuum coating [or "!s "d for dielectric coating]. The probability of surface-loss excitation Ps is mostly governed by the Im{ 1=1 "!} energy-loss function, which is analogous for surface modes to the bulk Im{ 1="!} energy-loss function. In normal incidence, the differential scattering cross section dPs =d is zero in the forward direction, reaches a maximum for E =31=2 , and decreases as 3 at large angles. In non-normal incidence, the angular distribution is asymmetrical, goes through a zero value for momentum transfer hq in a direction perpendicular to the interface, and the total probability increases as Ps '

Ps O ; cos '

Core excitations appear as edges superimposed, from the threshold energy Ec upwards, above a regularly decreasing background. As explained below, the basic matrix element governing the probability of transition is similar for optical absorption spectroscopy and for small-angle-scattering EELS spectroscopy. Consequently, selection rules for dipole transitions define the dominant transitions to be observed, i.e. l0

l l 1

and

j0

j j 0; 1:

4:3:4:37

This major rule has important consequences for the edge shapes to be observed: approximate behaviours are also shown in Fig. 4.3.4.22. A very useful library of core edges can be found in the EELS atlas (Ahn & Krivanek, 1982), from which we have selected the family of edges gathered in Fig. 4.3.4.23. They display the following typical profiles: (i) K edges for low-Z elements 3 Z 14. The carbon K edge occurring at 284 eV is a nice example with a clear hydrogenic or saw-tooth profile and fine structures on threshold depending on the local environment (amorphous, graphite, diamond, organic molecules, . . .); see Isaacson (1972a,b). < (ii) L2;3 edges for medium-Z elements 11 < Z 45. The L2;3 edges exhibit different shapes when the outer occupied shell changes in nature: a delayed profile is observed as long as the first vacant d states are located, along the energy scale, rather above the Fermi level (sulfur case). When these d states coincide with the first accessible levels, sharp peaks, generally known as `white lines', appear at threshold (this is the case for transition elements with the Fermi level inside the d band). These lines are generally split by the spin-orbit term on the initial level into 2p3=2 and 2p1=2 (or L3 and L2 ) terms. For higher-Z elements, the bound d levels are fully occupied, and

4:3:4:36

where ' is the incidence angle between the primary beam and the normal to the surface. As a consequence, the probability of producing one (and several) surface losses increases rapidly for grazing incidences. 4.3.4.4. Excitation spectrum of core electrons 4.3.4.4.1. Definition and classification of core edges As for any core-electron spectroscopy, EELS spectroscopy at higher energy losses mostly deals with the excitation of well defined atomic electrons. When considering solid specimens, both initial and final states in the transition are actually eigenstates in the solid state. However, the initial wavefunction can be considered as purely atomic for core excitations. As a first consequence, one can classify these transitions as a function of the parameters of atomic physics: Z is the atomic number of the element; n, l, and j l s are the quantum numbers describing the subshells from which the electron has been excited. The spectroscopy notation used is shown in Fig. 4.3.4.21. The list of major transitions is displayed as a function of Z and Ec in Fig. 4.3.4.22.

Fig. 4.3.4.20. Geometric conditions for investigating the anisotropic energy-loss function.

Fig. 4.3.4.21. Definition of electron shells and transitions involved in core-loss spectroscopy [from Ahn & Krivanek (1982)].

404


4.3. ELECTRON DIFFRACTION no longer contribute as host orbitals for the excited 2p electrons. One finds again a more traditional hydrogenic profile (such as for the germanium case). < (iii) M4;5 edges for heavier-Z elements 37 < Z 83. A sequence of M4;5 edge profiles, rather similar to L2;3 edges, is observed, the difference being that one then investigates the density of the final f states. White lines can also be detected when the f levels lie in the neighbourhood of the Fermi level, e.g. for rare-earth elements.

The deeper accessible signals, for incident electrons in the range of 100±400 kV primary voltage, lie between 2500 and 3000 eV, which corresponds roughly to the middle of the second row of transition elements (Mo±Ru) for the L2;3 edge and to the very heavy metals (Pb±Bi) for the M4;5 edge. (iv) A final example in Fig. 4.3.4.23 concerns one of these resonant peaks associated with the excitation of levels just below the conduction band. These are features with high intensity of the same order or even superior to that of plasmons of conduction

Fig. 4.3.4.22. Chart of edges encountered in the 50 eV up to 3 keV energy-loss range with symbols identifying the types of shapes [see Ahn & Krivanek (1982) for further comments].

405


4. PRODUCTION AND PROPERTIES OF RADIATIONS band electrons previously described in Subsection 4.3.4.3. It occurs with the M2;3 level for the first transition series, with the N2;3 level for the second series (for example, strontium in Fig. 4.3.4.23) or with the O2;3 level for the third series, including the rare-earth elements. The shape varies gradually from a plasmonlike peak with a short lifetime to an asymmetric Fano-type profile, a consequence of the coupling between discrete and continuum final states of the same energy (Fano, 1961). 4.3.4.4.2. Bethe theory for inelastic scattering by an isolated atom (Bethe, 1930; Inokuti, 1971, 1979) As a consequence of the atomic nature of the excited wavefunction in core-loss spectroscopy, the first step involves deriving a useful theoretical expression for inelastic scattering by an isolated atom. The differential cross section for an electron of wavevector k to be scattered into a final plane wave of vector k0 , while promoting one atomic electron from 0 to n , is given in a one-electron excitation description by 2 0 dn m0 k jh n k0 jV rj 0 kij2 ; 4:3:4:38 d dE k 2h2

see, for instance, Landau & Lifchitz (1966) and Mott & Massey (1952). The potential V r corresponds to the Coulomb interaction with all charges (both in the nucleus and in the electron cloud) of the atom. The momentum change in the scattering event is hq hk k0 . The final-state wavefunction is normalized per unit energy range. The orthogonality between initial- and final-state wavefunctions restricts the inelastic scattering to the only interactions with atomic electrons: dn 4 2 k0 2 4 jE n q; Ej2 : d dE a0 q k

4:3:4:39

The first part of the above expression has the form of Rutherford scattering. is introduced to deal, to a first approximation, with relativistic effects. The ratio k0 =k is generally assumed to be equal to unity. This kinematic scattering factor is modified by the second term, or matrix element, which describes the response of the atomic electrons: P 4:3:4:40 E n q; E exp iq r n j 0 ; j

where the sum extends over all atomic electrons at positions rj . The dimensionless quantity is known as the inelastic form factor. For a more direct comparison with photoabsorption measurements, one introduces the generalized oscillator strength (GOS) as df q; E E jE n q; Ej2 dE R qa0 2

4:3:4:41

for transitions towards final states " in the continuum E is then the energy difference between the core level and the final state of kinetic energy " above the Fermi level, scaled in energy to the Rydberg energy R]. Also, fn q

En jE n qj2 R qa0 2

4:3:4:42

for transition towards bound states. In this case, En is the energy difference between the two states involved. The generalized oscillator strength is a function of both the energy E and the momentum hq transferred to the atom. It is displayed as a three-dimensional surface known as the Bethe surface (Fig. 4.3.4.24), which embodies all information concerning the inelastic scattering of charged particles by atoms. The angular dependence of the cross section is proportional to 1 df q; E q2 dE at a given energy loss E. In the small-angle limit qrc 1, where rc is the average radius of the initial orbital), the GOS reduces to the optical oscillator strength df q; E df 0; E ! dE dE and

E n q; E ! E n 0; E q 2

Fig. 4.3.4.23. A selection of typical profiles K; L2;3 ; M4;5 ; and N2;3 illustrating the most important behaviours encountered on major edges through the Periodic Table. A few edges are displayed prior to and others after background stripping. [Data extracted from Ahn & Krivanek (1982).]

j

2 0 ; 4:3:4:43

where u is the unit vector in the q direction. When one is concerned with a given orbital excitation, the sum over rj reduces to a single term r for this electron. With some elementary calculations, the resulting cross section is

406


P u rj n

4.3. ELECTRON DIFFRACTION 2

d 4 2 R 1 df 0; E : 2 2 2 d dE E k E dE

4:3:4:44

The major angular dependence is contained, as in the low-loss domain, in the Lorentzian factor 2 E2 1 , with the characteristic inelastic angle E being again equal to E= m0 v2 . Over this reduced scattering-angle domain, known as the dipole region, the GOS is approximately constant and the inner-shell EELS spectrum is directly proportional to the photoabsorption cross section opt , whose data can be used to test the results of single-atom calculations. For larger scattering angles, Fig. 4.3.4.24 exhibits two distinct behaviours for energy losses just above the edge df = dE drops regularly to zero), and for energy losses much greater than the core-edge threshold. In the latter case, the oscillator strength is mostly concentrated in the Bethe ridge, the maximum of which occurs for: 9 E 2 > qa0 (non-relativistic formula), > = R 4:3:4:45 2 E E > > ; qa0 2 (relativistic formula): R 2m0 c2 R This contribution at large scattering angles is equivalent to direct knock-on collisions of free electrons, i.e. to the curve E h2 q2 =2m0 lying in the middle of the valence-electron±hole excitations continuum (see Fig. 4.3.4.13). The non-zero width of the Bethe ridge can be used as an electron Compton profile to analyse the momentum distribution of the atomic electrons [see also x4:3:4:4:4c. The energy dependence of the cross section, responsible for the various edge shapes discussed in x4:3:4:4:1, is governed by 1 df q; E ; E dE i.e. it corresponds to sections through the Bethe surface at constant q. Within the general theory described above, various models have been developed for practical calculations of energy differential cross sections.

Fig. 4.3.4.24. Bethe surface for K-shell ionization, calculated using a hydrogenic model. The generalized oscillator strength is zero for energy loss E below the threshold EK . The horizontal coordinate is related to scattering angle through q [from Egerton (1979)].

The hydrogenic model due to Egerton (1979) is an extension of the quantum-mechanical calculations for a hydrogen atom to inner-shell electron excitations in an atom Z by introduction of some useful parametrization (effective nuclear charge, effective threshold energy). It is applied in practice for K and L2;3 shells. In the Hartree±Slater (or Dirac±Slater) description, one calculates the final continuum-state wavefunction in a selfconsistent central field atomic potential (Leapman, Rez & Mayers, 1980; Rez, 1989). The radial dependence of these wavefunctions is given by the solution of a SchroÈdinger equation with an effective potential: Veff r V r

4:3:4:46

where l0 l0 1h2 =2m0 r 2 is the centrifugal potential, which is important for explaining the occurrence of delayed maxima in spectra involving final states of higher l0 . This approach is now useful for any major K, L2;3 , M4;5 ; . . . edge, as illustrated by Ahn & Rez (1985) and more specifically in rare-earth elements by Manoubi, Rez & Colliex (1989). These differential cross sections can be integrated over the relevant angular and energy domains to provide data comparable with experimental measurements. In practice, one records the energy spectral distribution of electrons scattered into all angles up to the acceptance value of the collection aperture. The integration has therefore to be made from qmin ' kE for the zero scattering-angle limit, up to qmax ' k . Fig. 4.3.4.25 shows how such calculated profiles can be used for fitting experimental data. Setting [or equal to an effective upper limit max ' E=E0 1=2 corresponding to the criterion qmax r ' 1, the integral cross section is the total cross section for the excitation of a given core level. These ionization cross sections are required for quantification in all analytical techniques using core-level excitations and de-excitations, such as EELS, Auger electron spectroscopy, and X-ray microanalysis (see Powell, 1976, 1984). A convenient way of comparing total cross sections is to rewrite the Bethe asymptotic cross section as

Fig. 4.3.4.25. A novel technique for simulating an energy-loss spectrum with two distinct edges as a superposition of theoretical contributions (hydrogenic saw-tooth for O K, Lorentzian white lines and delayed continuum for Fe L2;3 calculated with the Hartree±Slater description). The best fit between the experimental and the simulated spectra is shown; it can be used to evaluate the relative concentration of the two elements [see Manoubi et al. (1990)].

407


l0 l0 1 h2 ; 2m0 r 2

4. PRODUCTION AND PROPERTIES OF RADIATIONS nl Enl2 6:51 10

14

Znl bnl

logCnl Unl ; Unl

4:3:4:47

when the result is given in cm2 , nl is the total cross section per atom or molecule or ionization of the nl subshell with edge energy Enl , Znl is the number of electrons on the nl level, and Unl is the overvoltage defined as E0 =Enl . bnl and cnl are two parameters representing phenomenologically the average number of electrons involved in the excitation and their average energy loss (one finds for the major K and L2;3 edges bnl ' 0:6±0:9 and cnl ' 0:5±0:7). These values are in practice estimated from plots of curves nl E 2nl Unl as a function of log Unl , known as Fano plots. From least-squares fits to linear regions, one can evaluate the values of bnl (slope of the curves) and of log cnl (coordinate at the origin) for various elements and shells. However, it has been shown more recently (Powell, 1989) that the interpretation of Fano plots is not always simple, since they typically display two linear regions. It is only in the linear region for the higher incident energies that the plots show the asymptotic Bethe dependence with the slope directly related to the optical data. At lower incident energies, another linear region is found with a slope typically 10±20% greater. Despite great progress over the last two decades, more cross-section data, either theoretical or experimental, are still required to improve to the 1% level the accuracy in all techniques using these signals. 4.3.4.4.3. Solid-state effects The characteristic core edges recorded from solid specimens display complex structures different from those described in atomic terms. Moreover, their detailed spectral distributions depend on the type of compound in which the element is present (Leapman, Grunes & Fejes, 1982; Grunes, Leapman, Wilker, Hoffmann & Kunz, 1982; Colliex, Manoubi, Gasgnier & Brown, 1985). Modifications induced by the local solid-state environment concern (see Fig. 4.3.4.26) the following: a The threshold (or edge itself), which may vary in position, slope, and associated fine structures. From photoelectron spectroscopies (UPS, XPS), an edge displacement along the energy scale is known as a `chemical shift': it is due to a shift in the energy of the initial level as a consequence of the atomic potential modifications induced by valence-electron charge transfer (e.g. from metal to oxide). EELS is actually a twolevel spectroscopy and the observed changes at edge onset concern both initial and final states. Consequently, measured shifts are due to a combination of core-level energy shift with bandgap and exciton creation. Some important shifts have been measured in EELS such as: ± carbon K: 284 to 288 eV from graphite to diamond;

Fig. 4.3.4.26. Definition of the different fine structures visible on a core-loss edge.

± aluminium L2;3 : 73 to 77 eV from metal to Al2 O3 ; ± silicon L2;3 : 99.5 to 106 eV from Si to SiO2 . However, `chemical shift' constitutes a simplified description of the more complex changes that may occur at a given threshold in various compounds. It assumes a rigid translation of the edge, but in most cases the onset changes in shape and there are no simple features to correlate through the different spectra. This remark is more relevant with the increased energy resolution that is now available. With a sub-eV value, extra peaks or splittings can frequently be detected on edges that exhibit simple shapes when recorded at lower resolution. Among others, the L32 white lines in transition metals show different behaviours when involved in various environments: ± crystal-field-induced splitting for each line in the oxides Sc2 O3 , TiO2 when compared with the metal (see Fig. 4.3.4.27). ± relative change in L3 =L2 intensity ratio between different ionic species [most important when the occupancy degree n for the d band is of the order of 5, i.e. around the middle of the transition series, e.g. Mn and Fe oxides; see for instance, Rask, Miner & Buseck (1987) and Rao, Thomas, Williams & Sparrow (1984)]. ± presence of a narrow white line instead of a hydrogenic profile when the electron transfer from the metal to its ligand induces the existence of vacant d states at the Fermi level (CuO compared with Cu, see Fig. 4.3.4.28). b The near-edge fine structures (ELNES), which extend over the first 20 or 30 eV above threshold (Taftù & Zhu, 1982; Colliex et al., 1985). These are very similar to XANES structures in X-ray photoabsorption spectroscopy: they mostly reflect the spectral distribution of vacant accessible levels and are consequently very sensitive to site symmetry and charge transfer. Several approaches have been proposed to interpret them. A molecular-orbital description [e.g. Fischer (1970) or Tossell, Vaughan & Johnson (1974)] classifies the energy levels, both occupied and unoccupied, for clusters comprising the central excited ion and its first shell of neighbours. Its major success lies in the interpretation of level splitting on edges. A one-electron band calculation constitutes a second step with noticeable successes in the case of metals (MuÈller, Jepsen &

Fig. 4.3.4.27. High-energy resolution spectra on the L2;3 titanium edge from two phases (rutile and anatase) of TiO2 . Each atomic line L3 and L2 is split into two components A and B by crystal-field effects. The new level of splitting B1 B2 that distinguishes the two spectra is not yet understood. In Ti metal, the L3 and L2 lines are not split by structural effects [courtesy of Brydson et al. (1989)].

408


4.3. ELECTRON DIFFRACTION Wilkins, 1982). Core-loss spectroscopy, however, imposes specific conditions on the accessible final state: the overlap with the initial core wavefunction involves a projection in space on the site of the core hole, and the dominant dipole selection rules are responsible for angular symmetry selection. When extending the band-structure calculations to energy states rather high above the Fermi level, more elaborate methods, combining the conceptual advantage of the tight-binding method with the accuracy of ab initio pseudopotential calculations, have been developed (Janssen & Sankey, 1987). This self-consistent pseudo-atomic orbital band calculation has been used to describe ELNES structures on different covalent solids (Weng, Rez & Ma, 1989; Weng, Rez & Sankey, 1989). The most promising description at present is the multiple scattering method developed for X-ray absorption spectra by Durham, Pendry & Hodges (1981) and Vvedensky, Saldin & Pendry (1985). It interprets the spectral modulations, in the energy range 10 to 30 eV above the edge, as due to interference effects, on the excited site, between all waves back-scattered by the neighbouring atoms (see Fig. 4.3.4.29). This multiple scattering description in real space should in principle converge towards the local point of view in the solid-state band model, calculated in reciprocal space (Heine, 1980). As an example investigated by EELS, the oxygen and magnesium K edges in MgO have been calculated by Lindner, Sauer, Engel & Kambe (1986) and by Weng & Rez (1989) for increased numbers of coordination shells and different potential models (representing variable ionicities). Fig. 4.3.4.30 shows the comparison of an experimental spectrum with such a calculation. Another useful idea emerging from this model is the simple relation, expressed by Bianconi, Fritsch, Calas & Petiau (1985):

Er

Eb d 2 C;

4:3:4:48

where Er is the energy position of a given resonance peak attributed to multiple scattering from a given shell of neighbours (d is the distance to this shell), and Eb is a reference energy close to the threshold energy. This simple law, advertised as the way of measuring `bond lengths with a ruler' (Stohr, Sette & Jonson, 1984), seems to be quite useful when comparing similar structures (Lytle, Greegor & Panson, 1988). Other effects, generally described as multi-electron contributions, cannot be systematically omitted. They all deal with the presence of a core hole on the excited atom and with its influence on the distribution of accessible electron states. Of particular importance are the intra-atomic configuration interactions for white lines, as explained by Zaanen, Sawatsky, Fink, Speier & Fuggle (1985) for L3 and L2 lines in transition metals and by Thole, van der Laan, Fuggle, Swatsky, Karnatak & Esteva (1985) for M4;5 lines in rare-earth elements. c The extended fine structures (EXELFS) are equivalent to the well known EXAFS oscillations in X-ray absorption spectroscopy (Sayers, Stern & Lytle, 1971; Teo & Joy, 1981). Within the previously described multiscattering theory, it corresponds to the first step, the single scattering regime (see Fig. 4.3.4.29a). These extended oscillations are due to the interference on the excited atom between the outgoing excited

Fig. 4.3.4.29. Illustration of the single and multiple scattering effects used to describe the final wavefunction on the excited site. This theory is very fruitful for understanding and interpreting EXELFS and ELNES features, respectively equivalent to EXAFS and XANES encountered in X-ray absorption spectra.

Fig. 4.3.4.28. The dramatic change in near-edge fine structures on the L3 and L2 lines of Cu, from Cu metal to CuO. The appearance of the intense narrow white lines is due to the existence of vacant d states close to the Fermi level [courtesy of Leapman et al. (1982)].

Fig. 4.3.4.30. Comparison of the experimental O K edge (solid line) with calculated profiles in the multiple scattering approach [courtesy of Weng & Rez (1989)].

409


4. PRODUCTION AND PROPERTIES OF RADIATIONS electron wavefunction and its components reflected on the nearest-neighbour atoms. This interference is destructive or constructive depending on the ratio between the return path length 2ri (where ri is the radial distance with the ith shell of backscattering atoms) and the wavelength of the excited electron. Fourier analysis of EXELFS structures, from 50 eV above the ionization threshold, gives the radial distribution function around this specific site. This is mostly a technique for measuring the local short-range order. Its accuracy has been established to be Ê on nearest-neighbour distances with test better than 0.1 A specimens, but such performance requires correction procedures for phase shifts. The method therefore seems more promising for measuring changes in interatomic distances in specimens of the same chemical composition. The major advantage of EXELFS is its applicability for small specimen volumes that can moreover be characterized by other high-resolution electron-microscopy modes. It is also possible to investigate bond lengths in different directions by selecting the scattering angle of the transmitted electron and the specimen orientation (Disko, Krivanek & Rez, 1982). On the other hand, the major limitations of EXELFS are due to the dose requirements for sufficient SNR and to the fact that the accessible excitation range is limited to edges below 2± 3 keV and to oscillation domains 200 or 300 eV at the maximum. 4.3.4.4.4. Applications for core-loss spectroscopy a Quantitative microanalysis The main field of application of core-loss EELS spectroscopy has been its use for local chemical analysis (Maher, 1979; Colliex, 1984; Egerton, 1986). The occurrence of an edge superimposed on the regularly decreasing background of an EELS spectrum is an indication of the presence of the associated element within the analysed volume. Methods have been developed to extract quantitative composition information from these spectra. The basic idea lies in the linear relationship between the measured signal S and the number N of atoms responsible for it (this is valid in the single core-loss domain for specimen thickness, i.e. up to several micrometres): S I0 N;

4:3:4:49

where I0 is the incident-beam intensity and the relevant excitation cross section in the experimental conditions used, and N is the number of atoms per unit area of specimen. As a satisfactory approximation for taking into account multiple scattering events (either elastic or inelastic in the low-loss region), Egerton (1978) has proposed that equation (4.3.4.49) be rewritten: S ; I0 ; N ; ;

S

Ec

IE

BE dE:

NA SA ; B ; : NB SB ; A ;

This can be used to determine the NA =NB ratio without standards, if the cross-section ratio B =A (also called the kAB factor) is previously known: accuracy at present is limited to 5% for most edges. But it is also possible to extract from this formula the cross-section (or k factor) experimental values for comparison with the calculated ones, if the local stoichiometry of the specimen is satisfactorily known [Hofer, Golob & Brunegger (1988) and Manoubi et al. (1989) for the M4;5 edges]. Improvements have recently been made in order to reduce the different sources of errors. For medium-thickness specimens (i.e. for t ' lp where lP is the mean free path for plasmon excitation), deconvolution techniques are introduced for a safer determination of the signal. When the background extrapolation method cannot be used, i.e. when edges overlap noticeably, new approaches (such as illustrated in Fig. 4.3.4.25) try to determine the best simulated profile over the whole energy-loss range of interest. It requires several contributions, either deduced from previous measurements on standard (Shuman & Somlyo, 1987; Leapman & Swyt, 1988), or from reasonable mathematical models with different contributions for dealing with transitions towards bound states or continuum states (Manoubi, Tence, Walls & Colliex, 1990). b Detection limits This method has been shown to be the most successful of all EM techniques in terms of ultimate mass sensitivity and associated spatial resolution. This is due to the strong probability of excitation for the signals of interest (primary ionization event) and to the good localization of the characteristic even within the irradiated volume of material. Variations in composition have been recorded at a subnanometre level (Scheinfein & Isaacson, 1986; Colliex, 1985; Colliex, Maurice & Ugarte, 1989). In terms of ultimate sensitivity (minimum number of identified atoms), the range of a few tens of atoms (10 21 g) has been reached as early as about 15 years ago in the pioneering work of

4:3:4:51

Fig. 4.3.4.31. The conventional method of background subtraction for the evalulation of the characteristic signals SO K and SFe L2;3 used for quantitative elemental analysis (to be compared with the approach described in Fig. 4.3.4.25).

410


4:3:4:52

4:3:4:50

where all quantities correspond to a limited angle of collection and to a limited integration window (eV) above threshold for signal measurement. A major problem is the evaluation of the signal itself after background subtraction. The method generally used, demonstrated in Fig. 4.3.4.31, involves extrapolating a modelized background profile below the core loss of interest. Following Egerton (1978), the choice of a power law BE AE R is satisfactory in many cases, and the signal is then defined as EcR

Numerical methods have been developed to perform this process with a well controlled analysis of statistical errors (Trebbia, 1988). In many cases, one is interested in elemental ratios; consequently, the useful formula becomes

4.3. ELECTRON DIFFRACTION Isaacson & Johnson (1975). Very recently, a level close to the single-atom identification has been demonstrated (Mory & Colliex, 1989). A major obstacle is then often radiation damage, and consequent specimen modification induced by the very intense primary dose required for obtaining sufficient SNR values. On the other hand, the EELS technique has long been less fruitful for investigating low concentrations of impurities within a matrix. This is a consequence of the very high intrinsic background under the edges of interest: in most applications, the atomic concentration detection limit was in the range 10 3 to 10 2 . The introduction of satisfactory methods for processing the systematic sources of noise in spectra acquired with parallel detection devices (Shuman & Kruit, 1985) has greatly modified this situation. One can now take full benefit from the very high number of counts thus recorded within a reasonable time (106 to 107 counts per channel) and detection of calcium of the order of 10 5 atomic concentration in an organic matrix has been demonstrated by Shuman & Somlyo (1987). c Crystallographic information in EELS Although not particularly suited to solving crystal-structure problems, EELS carries structural information at different levels: In a crystalline specimen, one detects orientation effects on the intensity of core-loss edges. This is a consequence of the channelling of the Bloch standing waves as a function of the crystal orientation This observation requires well collimated angular conditions and inelastic localization better than the lattice spacing responsible for elastic diffraction. When these criteria apply, the changes in core-loss excitations with crystallographic orientation can be used to determine the crystallographic site of specific atoms (Tafto & Krivanek, 1982). An equivalent method, known as ALCHEMI (atom location by channelling enhanced microanalysis), which involves measuring the change of X-ray production as a function of crystal orientation, has been applied to the determination of the preferential site for substitutional impurities in many crystals (Spence & Tafto, 1983). Energy-filtered electron-diffraction patterns of core-loss edges could reveal the symmetry of the local coordination of selected atomic species rather than the symmetry of the crystal as a whole. This type of information should be compared with ELNES data (Spence, 1981). At large scattering angles, and for energy losses far beyond the excitation threshold, the Bethe ridge [or electron Compton profile (see xx4.3.4.3.3 and 4.3.4.4.2)] constitutes a major feature easily observable in energy-filtered diffraction patterns (Reimer & Rennekamp, 1989). The width of this feature is associated with the momentum distribution of the excited electrons (Williams & Bourdillon, 1982). Quantitative analysis of the data is similar to the Fourier method for EXELFS oscillations. After subtracting the background contribution, the spectrum is converted into momentum space and Fourier transformed to obtain the reciprocal form factor Br: it is the autocorrelation of the ground-state wavefunction in a direction specified by the scattering vector q. This technique of data analysis to study electron momentum densities is directly developed from high-energy photon-scattering experiments (Williams, Sparrow & Egerton, 1984).

investigating various aspects of the electronic structure of solids, As a fundamental application, it is now possible to construct a self-consistent set of data for a substance by combination of optical or energy-loss functions over a wide spectral range (Altarelli & Smith, 1974; Shiles, Sazaki, Inokuti & Smith, 1980: Hagemann, Gudat & Kunz, 1975). Sum-rule tests provide useful guidance in selecting the best values from the available measurements. The Thomas±Reiche±Kuhn f-sum rule can be expressed in a number of equivalent forms, which all require the knowledge of a function "2 ; ; Im 1=" describing dissipative processes over all frequencies: 9 Z1 2 > > > !"2 ! d! !p ; > > > 2 > > > 0 > > > 1 > Z 2 = !! d! !p ; 4:3:4:53 4 > > > 0 > > > > Z1 > > 1 2 > > d! !p : > ! > ; "! 2 0

One defines the effective number density neff of electrons contributing to these various absorption processes at an energy h! by the partial f sums: 9 Z! > m0 > 0 0 0 > > neff !j"2 2 2 ! "2 ! d! ; > > 2 e > > > 0 > > > ! > Z = m0 0 0 0 ! ! d! ; neff !j 2 2 4:3:4:54 e > > > 0 > > > > Z! > > m0 1 0 0 > > > d! ! : neff !j 1=" 2 2 > 0 ; 2 e "! 0

As an example, the values of neff ! from the infrared to beyond the K-shell excitation energy for metallic aluminium are shown in Fig. 4.3.4.32. In this case, the conduction and core-electron contributions are well separated. One sees that the excitation of conduction electrons is virtually completed above the plasmon resonance only, but the different behaviour of the integrands below this value is a consequence of the fact that they describe different properties of matter: "2 ! is a measure of the rate of energy dissipation from an electromagnetic wave, ! describes

4.3.4.5. Conclusions Since the early work of Hillier & Baker (1944), EELS spectroscopy has established itself as a prominent technique for

Fig. 4.3.4.32. Values of neff for metallic aluminium based on composite optical data [courtesy of Shiles et al. (1980)].

411


4. PRODUCTION AND PROPERTIES OF RADIATIONS the decrease in amplitude of the wave, and Im " 1 ! is related to the energy loss of a fast electron. The above curve shows some exchange of oscillator strength from core to valence electrons, arising from the Pauli principle, which forbids transitions to occupied states for the deeper electrons. More practically, in the microanalytical domain, the combination of high performance attained by using EELS with parallel detection (i.e. energy resolution below 1 eV, spatial resolution below 1 nm, minimum concentration below 10 3 atom, time resolution below 10 ms) makes it a unique tool for studying local electronic properties in solid specimens.

The latter is defined by the absolute value c and the normal projection cn on the ab plane, with components xn , yn along the axes a, b. In the triclinic case,

The formation of textures in specimens for diffraction experiments is a natural consequence of the tendency for crystals of a highly anisotropic shape to deposit with a preferred orientation. The corresponding diffraction patterns may present some special advantages for the solution of problems of phase and structure analysis. Lamellar textures composed of crystals with the most fully developed face parallel to a plane but randomly rotated about its normal are specially important. The ease of interpretation of patterns of such textures when oriented obliquely to the primary beam (OT patterns) is a valuable property of the electron-diffraction method (Pinsker, 1953; Vainshtein, 1964; Zvyagin, 1967; Zvyagin, Vrublevskaya, Zhukhlistov, Sidorenko, Soboleva & Fedotov, 1979). Texture patterns (T patterns) are also useful in X-ray diffraction (Krinary, 1975; Mamy & Gaultier, 1976; PlancËon, Rousseaux, Tchoubar, Tchoubar, Krinari & Drits, 1982). 4.3.5.2. Lattice plane oriented perpendicular to a direction (lamellar texture) If in the plane of orientation (the texture basis) the crystal has a two-dimensional cell a, b, , the c axis of the reciprocal cell will be the texture axis. Reciprocal-lattice rods parallel to c intersect the plane normal to them (the ab plane of the direct lattice) in the positions hk of a two-dimensional net that has periods 1=a sin and 1=b sin with an angle 0 between them, whatever the direction of the c axis in the direct lattice.

Fig. 4.3.5.1. The relative orientations of the direct and the reciprocal axes and their projections on the plane ab, with indication of the distances Bhk and Dhkl that define the positions of reflections in lamellar texture patterns.

cos cos = sin

4:3:5:1 4:3:5:2

(Zvyagin et al., 1979). The lattice points of each rod with constant hk and integer l are at intervals of c 1=d001 , but their real positions, described by their distances Dhkl from the plane ab, depend on the projections of the axes a and b on c (see Fig. 4.3.5.1), the equations xn

a cos =c

yn

b cos =c

4:3:5:3 4:3:5:4

being satisfied. The reciprocal-space representation of a lamellar texture is formed by the rotation of the reciprocal lattice of a single crystal about the c axis. The rods hk become cylinders and the lattice points become circles lying on the cylinders. In the case of highenergy electron diffraction (HEED), the wavelength of the electrons is very short, and the Ewald sphere, of radius 1=l, is so great that it may be approximated by a plane passing through the origin of reciprocal space and normal to the incident beam. The patterns differ in their geometry, depending on the angle ' through which the specimen is tilted from perpendicularity to the primary beam. At ' 0, the pattern consists of hk rings. When ' 6 0 it contains a two-dimensional set of reflections hkl falling on hk ellipses formed by oblique sections of the hk cylinders. In the limiting case of ' =2, the ellipses degenerate into pairs of parallel straight lines theoretically containing the maximum numbers of reflections. The reflection positions are defined by two kinds of distances: (1) between the straight lines hk (length of the short axes of the ellipses hk): Bhk 1= sin h2 =a2 k2 =b2

2hk cos =ab1=2

4:3:5:5

and (2) from the reflection hkl to the line of the short axes: Dhkl ha cos =c kb cos =c lc hxn

kyn l=d001 :

4:3:5:6 4:3:5:7

In patterns obtained under real conditions 0 < ' < =2, accelerating voltage V proportional to l 2 , distance L between the specimen and the screen), these values are presented in the scale of Ll, Dhkl also being proportional to 1= sin ' with maximum value Dmax Bhk tan ' for the registrable reflections. The values of Bhk and Dhkl , determined by the unit cells and the indices hkl, are the objects of the geometrical analysis of the OT patterns. When the symmetry is higher than triclinic, the expression for Bhk and Dhkl are much simpler. Such OT patterns are very informative, because the regular two-dimensional distribution of the hkl reflections permits definite indexing, cell determination, and intensity measurements. For low-symmetry and fine-grained substances, they present unique advantages for phase identification, polytypism studies, and structure analysis. In the X-ray study of textures, it is impossible to neglect the curvature of the Ewald sphere and the number of reflections recorded is restricted to larger d values. However, there are advantages in that thicker specimens can be used and reflections with small values of Bhk , especially the 00l reflections, can be recorded. Such patterns are obtained in usual powder cameras with the incident beam parallel to the platelets of the oriented aggregate and are recorded on photographic film in the form of hkl reflection sequences along hk lines, as was demonstrated by

412


2

yn c=bcos

4.3.5. Oriented texture patterns (By B. B. Zvyagin) 4.3.5.1. Texture patterns

cos cos = sin2

xn c=acos

4.3. ELECTRON DIFFRACTION Mamy & Gaultier (1976). The hk lines are no longer straight, but have the shapes described by Bernal (1926) for rotation photographs. It is difficult, however, to prepare good specimens. Other arrangements have been developed recently with advantages for precise intensity measurements. The reflections are recorded consecutively by means of a powder diffractometer fitted with a goniometer head. The relation between the angle of tilt ' and the angle of diffraction (twice the Bragg angle) 2 depends on the reciprocal-lattice point to be recorded. If the latter is defined by a vector of length H 2 sin =l and by the angle ! between the vector and the plane of orientation (texture basis), the relation ' ! permits scanning of reciprocal space along any trajectory by proper choice of consecutive values of ! or . In particular, if ! is constant, the trajectory is a straight line passing through the origin at an angle ! to the plane of orientation (Krinary, 1975). Using additional conditions ! arctanD=B, H B2 D2 1=2 , PlancËon et al. (1982) realized the recording and the measurement of intensities along the cylinder-generating hk rods for different shapes of the misorientation function N. In the course of development of electron diffractometry, a deflecting system has been developed that permits scanning the electron diffraction pattern across the fixed detector along any direction over any interval (Fig. 4.3.5.2). The intensities are measured point by point in steps of variable length. This system

is applicable to any kind of two-dimensional intensity pattern, and in particular to texture patterns (Zvyagin, Zhukhlistov & Plotnikev, 1996). Electron diffractometry provides very precise intensity measurements and very reliable structural data (Zhukhlistov et al., 1997). If the effective thickness of the lamellae is very small, of the order of the lattice parameter c, the diffraction pattern generates into a combination of broad but recognizably distinct 00l reflections and broad asymmetrical hk bands (Warren, 1941). The classical treatments of the shape of the bands were given by MeÂring (1949) and Wilson (1949) [for an elementary introduction see Wilson (1962)]. 4.3.5.3. Lattice direction oriented parallel to a direction (fibre texture) A fibre texture occurs when the crystals forming the specimen have a single direction in common. Each point of the reciprocal lattice describes a circle lying in a plane normal to the texture axis. The pattern, considered as plane sections of the reciprocallattice representation, resembles rotation diagrams of single crystals and approximates to the patterns given by cylindrical lattices (characteristic, for example, of tubular crystals). If the a axis is the texture axis, the hk rods are at distances Bhk h cos =a k=b= sin

4:3:5:8

from the texture axis and Dhk h=a

4:3:5:9

from the plane normal to the texture axis (the zero plane b c ). On rotation, they intersect the plane normal to the incident beam and pass through the texture axis in layer lines at distances Dhk from the zero line, while the reflection positions along these lines are defined by their distances from the textures axis (see Fig. 4.3.5.3): Bhkl B2hk hxn

kyn l2 =d 2001 1=2 :

4:3:5:10

If the texture axis forms an angle " with the a axis and " =2 with the projection of a on the plane ab, then Bhk f hsin =a ksin 0 =bg= sin f hcos "=a k cos "=bg= sin

4:3:5:11 4:3:5:12

Dhk fhcos =a kcos 0

4:3:5:13

fhsin

Fig. 4.3.5.2. a Part of the OTED pattern of the clay mineral kaolinite and b the intensity profile of a characteristic quadruplet of reflections recorded with the electron diffractometry system. The scanning direction is indicated in a.

"=a k sin "=bg= sin :

4:3:5:14

Fig. 4.3.5.3. The projections of the reciprocal axes on the plane ab of the direct lattice, with indications of the distances B and D of the hk rows from the fibre-texture axes a or hk:

413


=bg= sin

4. PRODUCTION AND PROPERTIES OF RADIATIONS The relation between the angles , ", and the direction hk of the texture axis is given by the expression cos sin h=a 2

h a

" kcos =b 2

k2 b

2

2hkcos =ab

1=2

:

4:3:5:15

The layer lines with constant h that coincide when " 0 are split when " 6 0 according to the sign of k, since then Dhk 6 Dhk and Bhk and Bhkl defining the reflection positions along the layer line take other values. Such peculiarities have been observed by means of selected-area electron diffraction for tabular particles and linear crystal aggregates of some phyllosilicates in the simple case of =2 (Gritsaenko, Zvyagin, Boyarskaya, Gorshkov, Samotoin & Frolova, 1969). When fibres or linear aggregates are deposited on a film (for example, in specimens for high-resolution electron diffraction) with one direction parallel to a plane, they form a texture that is intermediate between lamellar and fibre. The points of the reciprocal lattice are subject to two rotations: around the fibre axis and around the normal to the plane. The first rotation results in circles, the second in spherical bands of different widths, depending on the position of the initial point relative to the texture axis and the zero plane normal to it. The diffraction patterns correspond to oblique plane sections of reciprocal space, and consist of arcs having intensity maxima near their ends; in some cases, the arcs close to form complete circles. In particular, when the particle elongation is in the a direction, the angular range of the arcs decreases with h and increases with k (Zvyagin, 1967). 4.3.5.4. Applications to metals and organic materials The above treatment, though general, had layer silicates primarily in view. Texture studies are particularly important for metal specimens that have been subjected to cold work or other treatments; the phenomena and their interpretation occupy several chapters of the book by Barrett & Massalski (1980). Similarly, Kakudo & Kasai (1972) devote much space to texture in polymer specimens, and Guinier (1956) gives a good treatment of the whole subject. The mathematical methods for describing and analysing textures of all types have been described by Bunge (1982; the German edition of 1969 was revised in many places and a few errors were corrected for the English translation). 4.3.6. Computation of dynamical wave amplitudes 4.3.6.1. The multislice method (By D. F. Lynch) The calculation of very large numbers of diffracted orders, i.e. more than 100 and often several thousand, requires the multislice procedure. This occurs because, for N diffracted orders, the multislice procedure involves the manipulation of arrays of size N, whereas the scattering matrix or the eigenvalue procedures involve manipulation of arrays of size N by N. The simplest form of the multislice procedure presumes that the specimen is a parallel-sided plate. The surface normal is usually taken to be the z axis and the crystal structure axes are often chosen or transformed such that the c axis is parallel to z and the a and b axes are in the xy plane. This can often lead to rather unconventional choices for the unit-cell parameters. The maximum tilt of the incident beam from the surface normal is restricted to be of the order of 0.1 rad. For the calculation of wave amplitudes for larger tilts, the structure must be reprojected down an axis close to the incident-beam direction.

For simple calculations, other crystal shapes are generally treated by the column approximation, that is the crystal is presumed to consist of columns parallel to the z axis, each column of different height and tilt in order to approximate the desired shape and variation of orientation. The numerical procedure involves calculation of the transmission function through a thin slice, calculation of the vacuum propagation between centres of neighbouring slices, followed by evaluation in a computer of the iterated equation un h; k pn fpn

. . . p3 p2 p1 q1 q2 q3 . . . qn g 4:3:6:1

in order to obtain the scattered wavefunction, un h; k, emitted from slice n, i.e. for crystal thickness H z1 z2 . . . zn ; the symbol indicates the operation `convolution' defined by f1 x f2 x

R1 1

f1 w f2 x

w dw;

and pn exp

i2zn l=2fhh

h00 =a2 kk

k00 =b2 g

is the propagation function in the small-angle approximation between slice n 1 and slice n over the slice spacing zn . For simplicity, the equation is given for orthogonal axes and h00 , k00 are the usually non-integral intercepts of the Laue circle on the reciprocal-space axes in units of 1=a, 1=b. The excitation errors, h; k, can be evaluated using h; k

l=2fhh

h00 =a2 kk

k00 =b2 g:

4:3:6:2

The transmission function for slice n is qn h; k Ffexpi'n x; yzn g;

4:3:6:3

where F denotes Fourier transformation from real to reciprocal space, and 'n x; yzn p 'x; y

zn

R

1 zn

zn

'x; y; z dz

1

and

2 W l 1 1 2 1=2

and v ; c where W is the beam voltage, v is the relativistic velocity of the electron, c is the velocity of light, and l is the relativistic wavelength of the electron. The operation in (4.3.6.1) is most effectively carried out for large N by the use of the convolution theorem of Fourier transformations. This efficiency presumes that there is available an efficient fast-Fourier-transform subroutine that is suitable for crystallographic computing, that is, that contains the usual crystallographic normalization factors and that can deal with a range of values for h, k that go from negative to positive. Then, un h; k FfF 1 un 1 h; kF 1 qn h; kg; where F denotes

4:3:6:4

( " #) ny nx X 1 X hx ky uh; k Ux; y exp 2i ; nx ny x1 y1 n x ny

and F

414


1

1

denotes

Ux; y

nh X

nk X

h nh k nk

4.3. ELECTRON #) hx ky uh; k exp 2i ; ; nx ny (

"

where nh nx =2 1, nk ny =2 1, and nx ; ny are the sampling intervals in the unit cell. The array sizes used in the calculations of the Fourier transforms are commonly powers of 2 as is required by many fast Fourier subroutines. The array for un h; k is usually defined over the central portion of the reserved computer array space in order to avoid oscillation in the Fourier transforms (Gibbs instability). It is usual to carry out a 64 64 beam calculation in an array of 128 128, hence the critical timing interval in a multislice calculation is that interval taken by a fast Fourier transform for 4N coefficients. If the number of beams, N, is such that there is still appreciable intensity being scattered outside the calculation aperture, then it is usually necessary to impose a circular aperture on the calculation in order to prevent the symmetry of the calculation aperture imposing itself on the calculated wavefunction. This is most conveniently achieved by setting all ph; k coefficients outside the desired circular aperture to zero. It is clear that the iterative procedure of (4.3.6.1) means that care must be taken to avoid accumulation of error due to the precision of representation of numbers in the computer that is to be used. Practical experience indicates that a precision of nine significant figures (decimal) is more than adequate for most calculations. A precision of six to seven (decimal) figures (a common 32-bit floating-point representation) is only barely satisfactory. A computer that uses one of the common 64-bit representations (12 to 16 significant figures) is satisfactory even for the largest calculations currently contemplated. The choice of slice thickness depends upon the maximum value of the projected potential within a slice and upon the validity of separation of the calculation into transmission function and propagation function. The second criterion is not severe and in practice sets an upper limit to slice thickness of Ê . The first criterion depends upon the atomic number about 10 A of atoms in the trial structure. In practice, the slice thickness will be too large if two atoms of medium to heavy atomic weight Z 30 are projected onto one another. It is not necessary to take slices less than one atomic diameter for calculations for fast electron (acceleration voltages greater than 50 keV) diffraction or microscopy. If the trial structure is such that the symmetry of the diffraction pattern is not strongly dependent upon the structure of the crystal parallel to the slice normal, then the slices may be all identical and there is no requirement to have a slice thickness related to the periodicity of the structure parallel to the surface normal. This is called the `no upper-layer-line' approximation. If the upper-layer lines are important, then the slice thickness will need to be a discrete fraction of the c axis, and the contents of each slice will need to reflect the actual atomic contents of each slice. Hence, if there were four slices per unit cell, then there would need to be four distinct qh; k, each taken in the appropriate order as the multislice operation proceeds in thickness. The multislice procedure has two checks that can be readily performed during a calculation. The first is applied to the transmission function, qh; k, and involves the evaluation of a unitarity test by calculation of PP qh0 ; k0 q h h0 ; k k0 h; k 4:3:6:5 h0

k0

for all h, k, where q denotes the complex conjugate of q, and h; k is the Kronecker delta function. The second test can be applied to any calculation for which no phenomenological

DIFFRACTION absorption potential has been used in the evaluation of the qh; k. In that case, the sum of intensities of all beams at the final thickness should be no less than 0.9, the incident intensity being taken as 1.0. A value of this sum that is less than 0.9 indicates that the number of beams, N, has been insufficient. In some rare cases, the sum can be greater than 1.0; this is usually an indication that the number of beams has been allowed to come very close to the array size used in the convolution procedure. This last result does not occur if the convolution is carried out directly rather than by use of fast-Fourier-transform methods. A more complete discussion of the multislice procedure can be obtained from Cowley (1975) and Goodman & Moodie (1974). These references are not exhaustive, but rather an indication of particularly useful articles for the novice in this subject. 4.3.6.2. The Bloch-wave method (By A. Howie) Bloch waves, familiar in solid-state valence-band theory, arise as the basic wave solutions for a periodic structure. They are thus always implicit and often explicit in dynamical diffraction calculations, whether applied in perfect crystals, in almost perfect crystals with slowly varying defect strain fields or in more general structures that (see Subsection 4.3.6.1) can always, for computations, be treated by periodic continuation. The SchroÈdinger wave equation in a periodic structure, P 2 2 2 r 4 Ug exp2ig r 0; 4:3:6:6 g

can be applied to high-energy, relativistic electron diffraction, taking l 1 as the relativistically corrected electron wave number (see Subsection 4.3.1.4). The Fourier coefficients in the expression for the periodic potential are defined at reciprocallattice points g by the expression m exp Mg X Ug U g fj sing =l exp 2ig rj ;

m0 j 4:3:6:7 where fj is the Born scattering amplitude (see Subsection 4.3.1.2) of the jth atom at position rj in the unit cell of volume and Mg is the Debye±Waller factor. The simplest solution to (4.3.6.6) is a single Bloch wave, consisting of a linear combination of plane-wave beams coupled by Bragg reflection. P r bk; r Ch exp2ik h r: 4:3:6:8 h

In practice, only a limited number of terms N, corresponding to the most strongly excited Bragg beams, is included in (4.3.6.8). Substitution in (4.3.6.6) then yields N simultaneous equations for the wave amplitudes Cg : P Ug0 Cg g0 0: 4:3:6:9 2 U0 k g2 Cg g0 60

Usually, and the two tangential components kx and ky are fixed by matching to the incident wave at the crystal entrance surface. kz then emerges as a root of the determinant of coefficients appearing in (4.3.6.9). Numerical solution of (4.3.6.9) is considerably simplified (Hirsch, Howie, Nicholson, Pashley & Whelan, 1977) in cases of transmission high-energy electron diffraction where all the important reciprocal-lattice points lie in the zero-order Laue zone gz 0 and 2 jUg j. The equations then reduce to a

415


4. PRODUCTION AND PROPERTIES OF RADIATIONS standard matrix eigenvalue problem (for which efficient subroutines are widely available): P Mgh Ch Cg ; 4:3:6:10 h

where Mgh Ug h =2 sg gh and sg k2 k g2 =2 is the distance, measured in the z direction, of the reciprocal-lattice point g from the Ewald sphere. There will in general be N distinct eigenvalues kz z corresponding to N possible values kz j , j 1; 2; N, each with its eigenfunction defined by N wave amplitudes C0 j ; Cg j ; . . . ; Ch j . The waves are normalized and orthogonal so that P j l P j l Cg Cg jl ; Cg Ch gh : 4:3:6:11

and the behaviour of dispersion surfaces as a function of energy, yields accurate data on scattering amplitudes via the criticalvoltage effect (see Section 4.3.7). Static crystal defects induce elastic scattering transitions k j ! kl on sheets of the same dispersion surface. Transitions between points on dispersion surfaces of different energies occur because of thermal diffuse scattering, generation of electronic excitations or the emission of radiation by the fast electron. The Bloch-wave picture and the dispersion surface are central to any description of these phenomena. For further information and references, the reader may find it helpful to consult Section 5.2.10 of Volume B (IT B, 1996).

j

g

In simple transmission geometry, the complete solution for the total coherent wavefunction r is P P j exp 2q j z Cg j exp2i g r: r j

g

4:3:6:12 Inelastic and thermal-diffuse-scattering processes cause anomalous absorption effects whereby the amplitude of each component Bloch wave decays with depth z in the crystal from its initial value j C0 j . The decay constant is computed using an imaginary optical potential iU 0 r with Fourier coefficients iUg0 iU 0 g (for further details of these see Humphreys & Hirsch, 1968, and Subsection 4.3.1.5 and Section 4.3.2). m X j 0 j C Uh C g h : 4:3:6:13 q j 2 h z g;h g The Bloch-wave, matrix-diagonalization method has been extended to include reciprocal-lattice points in higher-order Laue zones (Jones, Rackham & Steeds, 1977) and, using pseudopotential scattering amplitudes, to the case of low-energy electrons (Pendry, 1974). The Bloch-wave picture may be compared with other variants of dynamical diffraction theory, which, like the multislice method (Subsection 4.3.6.1), for example, employ plane waves whose amplitudes vary with position in real space and are determined by numerical integration of first-order coupled differential equations. For cases with N < 50 beams in perfect crystals or in crystals containing localized defects such as stacking faults or small point-defect clusters, the Bloch-wave method offers many advantages, particularly in thicker crystals Ê For high-resolution image calculations in thin with t > 1000 A. crystals where the periodic continuation process may lead to several hundred diffracted beams, the multislice method is more efficient. For cases of defects with extended strain fields or crystals illuminated at oblique incidence, coupled plane-wave integrations along columns in real space (Howie & Basinski, 1968) can be the most efficient method. The general advantage of the Bloch-wave method, however, is the picture it affords of wave propagation and scattering in both perfect and imperfect crystals. For this purpose, solutions of equations (4.3.6.9) allow dispersion surfaces to be plotted in k space, covering with several sheets j all the wave points k j for a given energy E. Thickness fringes and other interference effects then arise because of interference between waves excited at different points k j . The average current flow at each point is normal to the dispersion surface and anomalous-absorption effects can be understood in terms of the distribution of Blochwave current within the unit cell. Detailed study of these effects,

4.3.7. Measurement of structure factors and determination of crystal thickness by electron diffraction (by J. Gjùnnes and J. W. Steeds) Current advances in quantitative electron diffraction are connected with improved experimental facilities, notably the combination of convergent-beam electron diffraction (CBED) with new detection systems. This is reflected in extended applications of electron diffraction intensities to problems in crystallography, ranging from valence-electron distributions in crystals with small unit cells to structure determination of biological molecules in membranes. The experimental procedures can be seen in relation to the two main principles for measurement of diffracted intensities from crystals: ± rocking curves, i.e. intensity profiles measured as function of deviation, sg , from the Bragg condition, and ± integrated intensities, which form the well known basis for X-ray and neutron diffraction determination of crystal structure. Integrated intensities are not easily defined in the most common type of electron-diffraction pattern, viz the selectedarea (SAD) spot pattern. This is due to the combination of dynamical scattering and the orientation and thickness variations usually present within the typically micrometre-size illuminated area. This combination leads to spot pattern intensities that are poorly defined averages over complicated scattering functions of many structure factors. Convergent-beam electron diffraction is a better alternative for intensity measurements, especially for inorganic structures with small-to-moderate unit cells. In CBED, a fine beam is focused within an area of a few hundred aÊngstroÈms, with a divergence of the order of a tenth of a degree. The diffraction pattern then appears in the form of discs, which are essentially two-dimensional rocking curves from a small illuminated area, within which thickness and orientation can be regarded as constant. These intensity distributions are obtained under well defined conditions and are well suited for comparison with theoretical calculations. The intensity can be recorded either photographically, or with other parallel recording systems, viz YAG screen/CCD camera (Krivanek, Mooney, Fan, Leber & Meyer, 1991) or image plates (Mori, Oikawa & Harada, 1990) ± or sequentially by a scanning system. The inelastic background can be removed by an energy filter (Krahl, PaÈtzold & Swoboda,1990; Krivanek, Gubbens, Dellby & Meyer, 1991). Detailed intensity profiles in one or two dimensions can then be measured with high precision for low-order reflections from simple structures. But there are limitations also with the CBED technique: the crystal should be fairly perfect within the illuminated area and the unit cell relatively small, so that overlap between discs can be avoided. The current development of electron diffraction is therefore characterized by a wide range of techniques, which extend from the traditional spot pattern to two-dimensional, filtered rocking curves, adapted to the

416


4.3. ELECTRON DIFFRACTION structure problems under study and the specimens that are available. Spot-pattern intensities are best for thin samples of crystals with light atoms, especially organic and biological materials. Dorset and co-workers (Dorset, Jap, Ho & Glaeser, 1979; Dorset, 1991) have shown how conventional crystallographic techniques (`direct phasing') can be applied in ab initio structure determination of thin organic crystals from spot intensities in projections. Two main complications were treated by them: bending of the crystal and dynamical scattering. Thin crystals will frequently be bent; this will give some integration of the reflection, but may also produce a slight distortion of the structure, as pointed out by Cowley (1961), who proposed a correction formula. The thickness range for which a kinematical approach to intensities is valid was estimated theoretically by Dorset et al. (1979). For organic crystals, they quoted a few hundred aÊngstroÈms as a limit for kinematical scattering in dense projections at 100 kV. Radiation damage is a problem, but with low-dose and cryotechniques, electron-microscopy methods can be applied to many organic crystals, as shown by several recent investigations. Voigt-Martin, Yan, Gilmore, Shankland & Bricogne (1994) collected electron-diffraction intensities from a beam-sensitive Ê Fourier map by a direct method dione and constructed a 1.4 A based on maximum entropy. Large numbers of electrondiffraction intensities have been collected from biological molecules crystallized in membranes. The structure amplitudes can be combined with phases extracted from high-resolution micrographs, following Unwin & Henderson's (1975) early work. KuÈhlbrandt, Wang & Fujiyoshi (1994) collected about 18 000 amplitudes and 15 000 phases for a protein complex in an electron cryomicroscope operating at 4.2 K (Fujiyoshi et al., 1991). Using these data, they determined the structure from a Ê resolution. three-dimensional Fourier map calculated to 3.4 A The assumption of kinematical scattering in such studies has been investigated by Spargo (1994), who found the amplitudes to be kinematic within 4% but with somewhat larger deviations for phases. For inorganic structures, spot-pattern intensities are less useful because of the stronger dynamical interactions, especially in dense zones. Nevertheless, it may be possible to derive a structure and refine parameters from spot-pattern intensities. Andersson (1975) used experimental intensities from selected projections for comparison with dynamical calculations, including an empirical correction factor for orientation spread, in a structure determination of V14 O8 . Recently, Zou, Sukharev & HovmoÈller (1993) combined spot-pattern intensities read from film by the program ELD with image processing of highresolution micrographs for structure determination of a complex perovskite. A considerable improvement over the spot pattern has been obtained by the elegant double-precession technique devised by Vincent & Midgley (1994). They programmed scanning coils above and below the specimen in the electron microscope so as to achieve simultaneous precession of the focused incident beam and the diffraction pattern around the optical axis. The net effect is equivalent to a precession of the specimen with a stationary incident beam. Integrated intensities can be obtained from reflections out to a Bragg angle equal to the precession angle ' for the zeroth Laue zone. In addition, reflections in the first and second Laue zones appear as broad concentric rings. Dynamical effects are reduced appreciably by this procedure, especially in the non-zero Laue zones. The experimental integrated intensities, Ig , must be multiplied with a geometrical factor analogous to the Lorentz factor in X-ray diffraction, viz

Ig IGexp sin ";

g2

2nkh ; 2kg

4:3:7:1

where nh is the reciprocal spacing between the zeroth and nth layers. The intensities can be used for structure determination by procedures taken over from X-ray crystallography, e.g. the conditional Patterson projections that are used by the Bristol group (Vincent, Bird & Steeds, 1984). The precession method may be seen as intermediate between the spot pattern and the CBED technique. Another intermediate approach was proposed by Goodman (1976) and used later by Olsen, Goodman & Whitfield (1985) in the structure determination of a series of selenides. CBED patterns from thin crystals were taken in dense zones; intensities were measured at corresponding points in the discs, e.g. at the zone-axis position. Structure parameters were determined by fitting the observed intensities to dynamical calculations. Higher precision and more direct comparisons with dynamical scattering calculations are achieved by measurements of intensity distributions within the CBED discs, i.e. one- or twodimensional rocking curves. An up-to-date review of these techniques is found in the recent book by Spence & Zuo (1992), where all aspects of the CBED technique, theory and applications are covered, including determination of lattice constants and strains, crystal symmetry, and fault vectors of defects. Refinement of structure factors in crystals with small unit cells are treated in detail. For determination of bond charges, the structure factors (Fourier potentials) should be determined to an accuracy of a few tenths of a percent; calculations must then be based on many-beam dynamical scattering theory, see Chapter 8.8. Removal of the inelastic background by an energy filter will improve the data considerably; analytical expressions for the inelastic background including multiple-scattering contributions may be an alternative (Marthinsen, Holmestad & Hùier, 1994). Early CBED applications to the determination of structure factors were based on features that can be related to dynamical effects in the two-beam case. Although insufficient for most accurate analyses, the two-beam expression for the intensity profile may be a useful guide. In its standard form, q Ug =k2 2 Ig s 4:3:7:2 sin t s2g Ug =k2 ; s2g Ug =k2 where Ug and sg are Fourier potential and excitation error for the reflection g, k wave number and t thickness. The expression can be rewritten in terms of the eigenvalues i; j that correspond to the two Bloch-wave branches, i, j: Igi; j sg where

Ug =k2 i

2

j

sin2 t i

j ;

4:3:7:3

q i h

i; j 12 s2g s2g Ug =k2 :

Note that the minimum separation between the branches i, j or the gap at the dispersion surface is j

i min Ug =k 1=g ;

4:3:7:4

where g is an extinction distance. The two-beam form is often found to be a good approximation to an intensity profile Ig (sg ) even when other beams are excited, provided an effective potential Ugeff , which corresponds to the gap at the dispersion surface, is substituted for Ug . This is suggested by many features in CBED and Kikuchi patterns and borne out by detailed calculations, see e.g. Hùier (1972). Approximate expressions for

417


cos "

4. PRODUCTION AND PROPERTIES OF RADIATIONS Ugeff have been developed along different lines; the best known is the Bethe potential Ugeff Ug

X Ug h Uh : 2ksh h

4:3:7:5

Other perturbation approaches are based on scattering between Bloch waves, in analogy with the ìnterband scattering' introduced by Howie (1963) for diffuse scattering; the term `Bloch-wave hybridization' was introduced by Buxton (1976). Exact treatment of symmetrical few-beam cases is possible (see Fukuhara, 1966; Kogiso & Takahashi, 1977). The three-beam case (Kambe, 1957; Gjùnnes & Hùier 1971) is described in detail in the book by Spence & Zuo (1992). Many intensity features can be related to the structure of the dispersion surface, as represented by the function (kx ,ky ). The gap [equation (4.3.7.4)] is an important parameter, as in the four-beam symmetrical case in Fig. 4.3.7.1. Intensity measurements along one dimension can then be referred to three groups, according to the width of the gap, viz: small gap ± integrated intensity; large gap ± rocking curve, thickness fringes; zero gap ± critical effects. A small gap at the dispersion surface implies that the twobeam-like rocking curve above approaches a kinematical form and can be represented by an integrated intensity. Within a certain thickness range, this intensity may be proportional to 2 jU eff g j , with an angular width inversely proportional to gt. Several schemes have been proposed for measurement of relative integrated intensities for reflections in the outer, high-angle region, where the lines are narrow and can be easily separated from the background. Steeds (1984) proposed use of the HOLZ (high-order Laue-zone) lines, which appear in CBED patterns taken with the central disc at the zone-axis position. Along a ring that defines the first-order Laue zone (FOLZ), reflections appear

as segments that can be associated with scattering from strongly excited Bloch waves in the central ZOLZ part into the FOLZ reflections. Vincent, Bird & Steeds (1984) proposed an intensity expression Ig j / j" j g j j2 exp 2t

1

exp 2 j t 2 j

4:3:7:6

for integrated intensity for a line segment associated with scattering from (or into) the ZOLZ Bloch wave j. " j is here the excitation coefficient and j the matrix element for scattering between the Bloch wave j and the plane wave g. j and are absorption coefficients for the Bloch wave and plane wave, respectively; t is the thickness. From measurements of a number of such FOLZ (or SOLZ) reflections, they were able to carry out ab initio structure determinations using so-called conditional Patterson projections and coordinate refinement. Tanaka & Tsuda (1990) have refined atomic positions from zone-axis HOLZ intensities. Ratios between HOLZ intensities have been used for determination of the Debye±Waller factor (Holmestad, Weickenmeier, Zuo, Spence & Horita, 1993). Another CBED approach to integrated intensities is due to Taftù & Metzger (1985). They measured a set of high-order reflections along a systematic row with a wide-aperture CBED tilted off symmetrical incidence. A number of high-order reflections are then simultaneously excited in a range where the reflections are narrow and do not overlap. Gjùnnes & Bùe (1994) and Ma, Rùmming, Lebech & Gjùnnes (1992) applied the technique to the refinement of coordinates and thermal parameters in high-Tc superconductors and intermetallic compounds. The validity and limitation of the kinematical approximation and dynamic potentials in this case has been discussed by Gjùnnes & Bùe (1994). Zero gap at the dispersion surface corresponds to zero effective Fourier potential or, to be more exact, an accidental degeneracy, (i) = ( j), in the Bloch-wave solution. This is the basis for the critical-voltage method first shown by Watanabe, Uyeda & Fukuhara (1969). From vanishing contrast of the Kikuchi line corresponding to a second-order reflection 2g, they determined a relation between the structure factors Ug and U2g . Gjùnnes & Hùier (1971) derived the condition for the accidental degeneracy in the general centrosymmetrical three-beam case 0,g,h, expressed in terms of the excitation errors sg;h and Fourier potentials Ug;h;g h , viz 2ksg

Ug Uh2 Ug2 h m ; Uh Ug h m0

2ksh

Uh Ug2 Ug2 h m ; Ug Ug h m0 4:3:7:7

where m and m0 are the relativistic and rest mass of the incident electron. Experimentally, this condition is obtained at a particular voltage and diffraction condition as vanishing line contrast of a Kikuchi or Kossel line ± or as a reversal of a contrast feature. The second-order critical-voltage effect is then obtained as a special case, e.g. by the mass ratio: m=m0 crit

Fig. 4.3.7.1. (a) Dispersion-surface section for the symmetric fourbeam case (0, g, g+h, m), k is a function of kx , referred to (b), where kx =ky =0 corresponds to the exact Bragg condition for all three reflections. The two gaps appear at sg Uh Um =k with widths Ug Ugh =k.

4:3:7:8

Measurements have been carried out for a number of elements and alloy phases; see the review by Fox & Fisher (1988) and later work on alloys by Fox & Tabbernor (1991). Zone-axis critical voltages have been used by Matsuhata & Steeds (1987). For analytical expressions and experimental determination of non-systematic critical voltages, see Matsuhata & Gjùnnes (1994).

418


U2h h2 : 2 Uh2 U2h

4.3. ELECTRON DIFFRACTION Large gaps at the dispersion surface are associated with strong inner reflections ± and a strong dynamical effect of two-beamlike character. The absolute magnitude of the gap ± or its inverse, the extinction distance ± can be obtained in different ways. Early measurements were based on the split of diffraction spots from a wedge, see Lehmpfuhl (1974), or the corresponding fringe periods measured in bright- and dark-field micrographs (Ando, Ichimiya & Uyeda, 1974). The most precise and applicable large-gap methods are based on the refinement of the fringe pattern in CBED discs from strong reflections, as developed by Goodman & Lehmpfuhl (1967) and Voss, Lehmpfuhl & Smith (1980). In recent years, this technique has been developed to high perfection by means of filtered CBED patterns, see Spence & Zuo (1992) and papers referred to therein. See also Chapter 8.8. The gap at the dispersion surface can also be obtained directly from the split observed at the crossing of a weak Kikuchi line with a strong band. Gjùnnes & Hùier (1971) showed how this can be used to determine strong low-order reflections. High voltage may improve the accuracy (Terasaki, Watanabe & Gjùnnes, 1979). The sensitivity of the intersecting Kikuchi-line (IKL) method was further increased by the use of CBED instead of Kikuchi patterns (Matsuhata, Tomokiyo, Watanabe & Eguchi, 1984; Taftù & Gjùnnes, 1985). In a recent development, Hùier, Bakken, Marthinsen & Holmestad (1993) have measured the intensity distribution in the CBED discs around such intersections and have refined the main structure factors involved. Two-dimensional rocking curves collected by CBED patterns around the axis of a dense zone are complicated by extensive many-beam dynamical interactions. The Bristol±Bath group (Saunders, Bird, Midgley & Vincent, 1994) claim that the strong dynamic effects can be exploited to yield high sensitivity in refinement of low-order structure factors. They have also developed procedures for ab initio structure determination based on zone-axis patterns (Bird & Saunders, 1992), see Chapter 8.8. Determination of phase invariants. It has been known for some time (e.g Kambe, 1957) that the dynamical three-beam case contains information about phase. As in the X-ray case, measurement of dynamical effects can be used to determine the value of triplets (Zuo, Hùier & Spence, 1989) and to determine phase angles to better than one tenth of a degree (Zuo, Spence, Downs & Mayer, 1993) which is far better than any X-ray method. Bird (1990) has pointed out that the phase of the absorption potential may differ from the phase of the real potential. Thickness is an important parameter in electron-diffraction experiments. In structure-factor determination based on CBED patterns, thickness is often included in the refinement. Thickness can also be determined directly from profiles connected with large gaps at the dispersion surface (Goodman & Lehmpfuhl, 1967; Blake, Jostsons, Kelly & Napier, 1978; Glazer, Ramesh, Hilton & Sarikaya, 1985). The method is based on the outer part of the fringe profile, which is not so sensitive to the structure factor. The intensity minimum of the ith fringe in the diffracted disc occurs at a position corresponding to the excitation error si and expressed as s2i 1="2g t2 n2i ;

method originally proposed by Ackermann (1948), where si2 is plotted against ni and the thickness is taken from the slope, is more accurate. In both cases, the outer part of the rocking curve is emphasized; exact knowledge of the gap is not necessary for a good determination of thickness, provided the assumption of a two-beam-like rocking curve is valid. 4.3.8. Crystal structure determination by high-resolution electron microscopy (By J. C. H. Spence and J. M. Cowley) 4.3.8.1. Introduction For the crystallographic study of real materials, highresolution electron microscopy (HREM) can provide a great deal of information that is complementary to that obtainable by X-ray and neutron diffraction methods. In contrast to the statistically averaged information that these other methods provide, the great power of HREM lies in its ability to elucidate the detailed atomic arrangements of individual defects and the microcrystalline structure in real crystals. The defects and inhomogeneities of real crystals frequently exert a controlling influence on phase-transition mechanisms and more generally on all the electrical, mechanical, and thermal properties of solids. The real-space images that HREM provides (such as that shown in Fig. 4.3.8.1) can give an immediate and dramatic impression of chemical crystallography processes, unobtainable by other methods. Their atomic structure is of the utmost importance for

4:3:7:9

where ni is a small integer describing the order of the minimum. This equation can be arranged in two ways for graphic determination of thickness. The commonest method appears to be to plot (si =ni )2 against 1=ni 2 and then determine the thickness from the intersection with the ordinate axis (Kelly, Jostsons, Blake & Napier, 1975). Glazer et al. (1985) claim that the

Fig. 4.3.8.1. Atomic resolution image of a tantalum-doped tungsten trioxide crystal (pseudo-cubic structure) showing extended crystallographic shear-plane defects (C), pentagonal-column hexagonaltunnel (PCHT) defects (T), and metallization of the surface due to Ê, oxygen desorption (JEOL 4000EX, crystal thickness less than 200 A 400 kV, Cs 1 mm). Atomic columns are black. [Smith, Bursill & Wood (1985).]

419


4. PRODUCTION AND PROPERTIES OF RADIATIONS an understanding of the properties of real materials. The HREM method has proven powerful for the determination of the structure of such defects and of the submicrometre-sized microcrystals that constitute many polyphase materials. In summary, HREM should be considered the technique of choice where a knowledge of microcrystal size, shape or morphology is required. In addition, it can be used to reveal the presence of line and planar defects, inclusions, grain boundaries and phase boundaries, and, in favourable cases, to determine atomic structure. Surface atomic structure and reconstruction have also been studied by HREM. However, meaningful results in this field require accurately controlled ultra-high-vacuum conditions. The determination of the atomic structure of point defects by HREM so far has proven extremely difficult, but this situation is likely to change in the near future. The following sections are not intended to review the applications of HREM, but rather to provide a summary of the main theoretical results of proven usefulness in the field, a selected bibliography, and recommendations for good experimental practice. At the time of writing (1997), the point Ê. resolution of HREM machines lies between 1 and 2 A The function of the objective lens in an electron microscope is to perform a Fourier synthesis of the Bragg-diffracted electron beams scattered (in transmission) by a thin crystal, in order to produce a real-space electron image in the plane r. This electron image intensity can be written 2 R j rj2 u expf2iu rgPu expfiug du ; 4:3:8:1 where u represents the complex amplitude of the diffracted wave after diffraction in the crystal as a function of the reciprocal-lattice vector u [magnitude 2 sin =l in the plane perpendicular to the beam, so that the wavevector of an incident plane wave is written K0 kz 2u. Following the convention of Section 2.5.1 in IT B (1992), we write jK0 j 2l 1 . The function u is the phase factor for the objective-lens transfer function and Pu describes the effect of the objective aperture: 1 for juj < u0 Pu 0 for juj u0 :

Image formation in the transmission electron microscope is conventionally treated by analogy with the Abbe theory of coherent optical imaging. The overall process is subdivided as follows. a The problem of beam±specimen interaction for a collimated kilovolt electron beam traversing a thin parallel-sided slab of crystal in a given orientation. The solution to this problem gives the elastically scattered dynamical electron wavefunction r, where r is a two-dimensional vector lying in the downstream surface of the slab. Computer algorithms for dynamical scattering are described in Section 4.3.6. b The effects of the objective lens are incorporated by multiplying the Fourier transform of r by a function T u, which describes both the wavefront aberration of the lens and the diffractionlimiting effects of any apertures. The dominant aberrations are spherical aberration, astigmatism, and defect of focus. The image intensity is then formed from the modulus squared of the Fourier transform of this product. c All partial coherence effects may be incorporated by repeating this procedure for each of the component energies and directions that make up the illumination from an extended electron source, and summing the resulting intensities. Because this procedure requires a separate dynamical calculation for each component direction of the incident beam, a number of useful approximations of restricted validity have been developed; these are described in Subsection 4.3.8.4. This treatment of partial coherence assumes that a perfectly incoherent effective source can be identified. For fieldemission HREM instruments, a coherent sum (over directions) of complex image wavefunctions may be required. General treatments of the subject of HREM can be found in the texts by Cowley (1981) and Spence (1988). The sign

For a periodic object, the image wavefunction is given by summing the contributions from the set of reciprocal-lattice points, g, so that 2 P 2 j rj g expf2ig rgPg expfigg : 4:3:8:2 g

Ê 1 , it is apparent that, for all For atomic resolution, with u0 1A but the simplest structures and smallest unit cells, this synthesis will involve many hundreds of Bragg beams. A scattering calculation must involve an even larger number of beams than those that contribute resolvable detail to the image, since, as described in Section 2.5.1 in IT B (1992), all beams interact strongly through multiple coherent scattering. The theoretical basis for HREM image interpretation is therefore the dynamical theory of electron diffraction in the transmission (or Laue) geometry [see Chapter 5.2 in IT B (1992)]. The resolution of HREM images is limited by the aberrations of the objective electron lens (notably spherical aberration) and by electronic instabilities. An intuitive understanding of the complicated effect of these factors on image formation from multiply scattered Bragg beams is generally not possible. To provide a basis for understanding, therefore, the following section treats the simplified case of few-beam `lattice-fringe' images, in order to expose the relationship between the crystal potential, its structure factors, electron-lens aberrations, and the electron image.

Fig. 4.3.8.2. Imaging conditions for few-beam lattice images. For three-beam axial imaging shown in c, the formation of half-period fringes is also shown.

420


4.3. ELECTRON DIFFRACTION conventions used throughout the following are consistent with the standard crystallographic convention of Section 2.5.1 of IT B (1992), which assumes a plane wave of form expf ik r !tg and so is consistent with X-ray usage. 4.3.8.2. Lattice-fringe images We consider few-beam lattice images, in order to understand the effects of instrumental factors on electron images, and to expose the conditions under which they faithfully represent the scattering object. The case of two-beam lattice images is instructive and contains, in simplified form, most of the features seen in more complicated many-beam images. These fringes were first observed by Menter (1956) and further studied in the pioneering work of Komoda (1964) and others [see Spence (1988) for references to early work]. The electron-microscope optic-axis orientation, the electron beam, and the crystal setting are indicated in Fig. 4.3.8.2. If an objective aperture is used that excludes all but the two beams shown from contributing to the image, equation (4.3.8.2) gives the image intensity along direction g for a centrosymmetric crystal of thickness t as Ix; t j 0 tj2 j g tj2 2j 0 jj g j cosf2x=dg ug g t

0 tg: 4:3:8:3

The Bragg-diffracted beams have complex amplitudes g t j g tj expfig tg. The lattice-plane period is dg in direction g [Miller indices hkl]. The lens-aberration phase function, including only the effects of defocus f and spherical aberration (coefficient Cs ), is given by ug 2=lff l2 u2g =2 Cs l4 u4g =4g:

2 1=2

sint1 w g t i1 w2

1=2

1=2

cos ug

sint1 w2 1=2 =g 4:3:8:5

where g is the two-beam extinction distance, Vg =g is a Fourier coefficient of crystal potential, sg is the excitation error (see Fig. 4.3.8.2), w sg g , and the interaction parameter is defined in Section 2.5.1 of IT B (1992). The two-beam image intensity given by equation (4.3.8.3) therefore depends on the parameters of crystal thickness t, orientation sg , structure factor Vg , objective-lens defocus f , and spherical-aberration constant Cs . We consider first the variation of lattice fringes with crystal thickness in the two-beam approximation (Cowley, 1959; Hashimoto, Mannami & Naiki, 1961). At the exact Bragg condition sg 0, equations (4.3.8.5) and (4.3.8.3) give sin2t=g sin 2x=d ug :

4:3:8:6

If we consider a wedge-shaped crystal with the electron beam approximately normal to the wedge surface and edge, and take x and g parallel to the edge, this equation shows that sinusoidal lattice fringes are expected whose contrast falls to zero (and

u0 2x=d g t

0 t:

For a uniformly intense line source subtending a semiangle c , the total lattice-fringe intensity is R Ix 1=c Ix; d: The resulting fringe visibility C Imax Imin =Imax Imin is proportional to C sin = , where 2f c =d. The contrast falls to zero for , so that the range of focus over which fringes are expected is z d=c . This is the approximate depth of field for lattice images due to the effects of the finite source size alone. The case of three-beam fringes in the axial orientation is of more practical importance [see Fig. 4.3.8.2b]. The image intensity for g g and sg s g is Ix; t j 0 j2 2j g j2 2j g j2 cos4x=d 4j 0 jj g j cos2x=d cosug g t

0 t:

4:3:8:7

The lattice image is seen to consist of a constant background plus cosine fringes with the lattice spacing, together with cosine fringes of half this spacing. The contribution of the half-spacing fringes is independent of instrumental parameters (and therefore of electronic instabilities if c 0). These fringes constitute an important HREM image artifact. For kinematic scattering, g t 0 t =2 and only the half-period fringes will then be seen if ug n, or for focus settings Cs l2 u2g =2:

4:3:8:8

Fig. 4.3.8.2c indicates the form of the fringes expected for two focus settings with differing half-period contributions. As in the case of two-beam fringes, dynamical scattering may cause 0 to be severely attenuated at certain thicknesses, resulting also in a strong half-period contribution to the image. Changes of 2 in ug in equation (4.3.8.7) leave Ix; t unchanged. Thus, changes of defocus by amounts ff 2n=lu2g

4:3:8:9

Cs 4n=l3 u4g

4:3:8:10

or changes in Cs by yield identical images. The images are thus periodic in both f and Cs . This is a restricted example of the more general phenomenon of n-beam Fourier imaging discussed in Subsection 4.3.8.3. We note that only a single Fourier period will be seen if ff is less than the depth of field z. This leads to the approximate condition c > l=d, which, when combined with the Bragg law, indicates that a single period only of images will be seen when adjacent diffraction discs just overlap.

421


u0

f nl 1 ug 2

=g g exp isg t

exp isg t;

Ix; t 1

Ix; j 0 j2 j g j2 2j g jj 0 j

4:3:8:4

The effects of astigmatism and higher-order aberrations have been ignored. The defocus, f , is negative for the objective lens weakened (i.e. the focal length increased, giving a bright first Fresnel-edge fringe). The magnitude of the reciprocal-lattice vector ug dg 1 2 sin B =l, where B is the Bragg angle. If these two Bragg beams were the only beams excited in the crystal (a poor approximation for quantitative work), their amplitudes would be given by the `two-beam' dynamical theory of electron diffraction as 0 t fcost1 w2 1=2 =g iw1 w2

reverses sign) at thicknesses of tn ng =2. This apparent abrupt translation of fringes (by d=2 in the direction x) at particular thicknesses is also seen in some experimental many-beam images. The effect of changes in focus (due perhaps to variations in lens current) is seen to result in a translation of the fringes (in direction x), while time-dependent variations in the accelerating voltage have a similar effect. Hence, time-dependent variations of the lens focal length or the accelerating voltage result in reduced image contrast (see below). If the illumination makes a small angle lu0 with the optic axis, the intensity becomes

4. PRODUCTION AND PROPERTIES OF RADIATIONS The axial three-beam fringes will coincide with the lattice planes, and show atom positions as dark if ug 2n 1=2 and 0 t g t =2. This total phase shift of between 0 and the scattered beams is the desirable imaging condition for phase contrast, giving rise to dark atom positions on a bright background. This requires Cs 4n

1=l3 u4g

2f =l2 u2g

as a condition for identical axial three-beam lattice images for n 0; 1; 2; . . .. This family of lines has been plotted in Fig. 4.3.8.3 for the (111) planes of silicon. Dashed lines denote the locus of `white-atom' images (reversed contrast fringes), while the dotted lines indicate half-period images. In practice, the depth of field is limited by the finite illumination aperture c , and few-beam lattice-image contrast will be a maximum at the stationary-phase focus setting, given by f0

Cs l2 u2g :

4:3:8:11

This choice of focus ensures ru 0 for u ug , and thus ensures the most favourable trade-off between increasing c and loss of fringe contrast for lattice planes g. Note that f0 is not equal to the Scherzer focus fs (see below). This focus setting is

also indicated on Fig. 4.3.8.3, and indicates the instrumental conditions which produce the most intense (111) three- (or five-) beam axial fringes in silicon. For three-beam axial fringes of spacing d, it can be shown that the depth of field z is approximately z ln 21=2 d=c :

4:3:8:12

This depth of field, within which strong fringes will be seen, is indicated as a boundary on Fig. 4.3.8.3. Thus, the finer the image detail, the smaller is the focal range over which it may be observed, for a given illumination aperture c . Fig. 4.3.8.4 shows an exact dynamical calculation for the contrast of three-beam axial fringes as a function of f in the neighbourhood of f0 . Both reversed contrast and half-period fringes are noted. The effects of electronic instabilities on lattice images are discussed in Subsection 4.3.8.3. It is assumed above that c is sufficiently small to allow the neglect of any changes in diffraction conditions (Ewald-sphere orientation) within c . Under a similar approximation but without the approximations of transfer theory, Desseaux, Renault & Bourret (1977) have analysed the effect of beam divergence on two-dimensional fivebeam axial lattice fringes. When two-dimensional patterns of fringes are considered, the Fourier imaging conditions become more complex (see Subsection 4.3.8.3), but half-period fringe systems and reversedcontrast images are still seen. For example, in a cubic projection, a focus change of ff =2 results in an image shifted by half a unit cell along the cell diagonal. It is readily shown that expi f expi f ff if ff 2na2 =l 2mb2 =l when n, m are integers and a and b are the two dimensions of any orthogonal unit cell that can be chosen for p x; y. Thus, changes in focus by ff n; m produce identical images in crystals for which such a cell can be chosen, regardless of the number of beams contributing (Cowley & Moodie, 1960). For closed-form expressions for the few-beam (up to 10 beams) two-dimensional dynamical Bragg-beam amplitudes g in orientations of high symmetry, the reader is referred to the work of Fukuhara (1966).

Fig. 4.3.8.3. A summary of three- (or five-) beam axial imaging conditions. Here, ff is the Fourier image period, f0 the stationaryphase focus, Cs 0 the image period in Cs , and a scattering phase of =2 is assumed. The lines are drawn for the (111) planes of silicon at 100 kV with c 1:4 mrad.

Fig. 4.3.8.4. The contrast of few-beam lattice images as a function of focus in the neighbourhood of the stationary-phase focus [see Olsen & Spence (1981)].

4.3.8.3. Crystal structure images We define a crystal structure image as a high-resolution electron micrograph that faithfully represents a projection of a crystal structure to some limited resolution, and which was obtained using instrumental conditions that are independent of the structure, and so require no a priori knowledge of the structure. The resolution of these images is discussed in Subsection 4.3.8.6, and their variation with instrumental parameters in Subsection 4.3.8.4. Equation (4.3.8.2) must now be modified to take account of the finite electron source size used and of the effects of the range of energies present in the electron beam. For a perfect crystal we may write, as in equation (2.5.1.36) in IT B (1992), RR IT r j u0 ; f ; rj2 Gu0 Bf ; u0 du0 df 4:3:8:13a for the total image intensity due to an electron source whose normalized distribution of wavevectors is Gu0 , where u0 has components u1 ; v1 , and which extends over a range of energies corresponding to the distribution of focus Bf ; u. If is also assumed to vary linearly across c and changes in the diffraction conditions over this range are assumed to make only negligible changes in the diffracted-beam amplitude g , the expression for a Fourier coefficient of the total image intensity IT r becomes

422


Ig

P

4.3. ELECTRON DIFFRACTION h expf ihg frh

rh

gg

jh

gj2 2 g;

3=4 dp 0:66 C 1=4 : s l

h

h g

expfih

gg f12 h2

4:3:8:13b

where h and g are the Fourier transforms of Gu0 and Bf ; u0 , respectively. For the imaging of very thin crystals, and particularly for the case of defects in crystals, which are frequently the objects of particular interest, we give here some useful approximations for HREM structure images in terms of the continuous projected crystal potential Rt 'x; y 1=t 'x; y; z dz; 0

where the projection is taken in the electron-beam direction. A brief summary of the use of these approximations is included in Section 2.5.1 of IT B (1992) and computing methods are discussed in Subsection 4.3.8.5 and Section 4.3.4. The projected-charge-density (PCD) approximation (Cowley & Moodie, 1960) gives the HREM image intensity (for the simplified case where Cs 0) as Ix; y 1 f l=2"0 "p x; y;

Ix; y 1 2'p x; y Ffsin u; vPu; vg 1 2'p x; y Sx; y;

4:3:8:14

where F denotes Fourier transform, denotes convolution, and u and v are orthogonal components of the two-dimensional scattering vector u. The function Sx; y is sharply peaked and negative at the `Scherzer focus' f fs 1:2Cs l

1=2

4:3:8:15a

and the optimum objective aperture size 0 1:5l=Cs 1=4 :

The occurrence of appreciable multiple scattering, and therefore of the failure of the WPO approximation, depends on specimen thickness, orientation, and accelerating voltage. Detailed comparisons between accurate multiple-scattering calculations, the PCD approximation, and the WPO approximation can be found in Lynch, Moodie & O'Keefe (1975) and Jap & Glaeser (1978). As a very rough guide, equation (4.3.8.14) can be expected to fail for light elements at 100 keV and thicknesses greater than about 5.0 nm. Multiple-scattering effects have been predicted within single atoms of gold at 100 keV. The WPO approximation may be extended to include the effects of an extended source (partial spatial coherence) and a range of incident electron-beam energies (temporal coherence). General methods for incorporating these effects in the presence of multiple scattering are described in Subsection 4.3.8.5. Under the approximations of linear imaging outlined below, it can be shown (Wade & Frank, 1977; Fejes, 1977) that sin u; vPu; v in equation (4.3.8.14) may be replaced by A0 u Pu expiu exp 2 2 l2 u4 =2 r=2 Pu expiu expi2 2 l2 u4 =2 exp 2 u20 q

4:3:8:13c

where p x; y is the projected charge density for the specimen (including the nuclear contribution) and is related to 'p x; y through Poisson's equation. Here, "0 " is the specimen dielectric constant. This approximation, unlike the weak-phase-object approximation (WPO), includes multiple scattering to all orders of the Born series, within the approximation that the component of the scattering vector is zero in the beam direction (a `flat' Ewald sphere). Contrast is found to be proportional to defocus and to p x; y. The failure conditions of this approximation are discussed by Lynch, Moodie & O'Keefe (1975); briefly, it fails for u0 > =2 (and hence if Cs , f or u0 becomes large) or for large thicknesses t t < 7 nm is suggested for specimens of Ê medium atomic weight and l 0:037 A. The PCD result becomes increasingly accurate with increasing accelerating voltage for small Cs . The WPO approximation has been used extensively in combination with the Scherzer-focus condition (Scherzer, 1949) for the interpretation of structure images (Cowley & Iijima, 1972). This approximation neglects multiple scattering of the beam electron and thereby allows the application of the methods of linear transfer theory from optics. The image intensity is then given, for plane-wave illumination, by

4:3:8:15b

It forms the impulse response of an electron microscope for phase contrast. Contrast is found to be proportional to 'p and to the interaction parameter , which increases very slowly with accelerating voltage above about 500 keV. The point resolution [see Subsection 2.5.1.9 of IT B (1992) and Subsection 4.3.8.6] is conventionally defined from equation (4.3.8.15b) as l=0 , or

4:3:8:17 if astigmatism is absent. Here, u ui vj and juj 2=l u2 v2 1=2 . In addition, u0 is the Fourier transform of the source intensity distribution (assumed Gaussian), so that r=2 is small in regions where the slope of u0 is large, resulting in severe attenuation of these spatial frequencies. If the illuminating beam divergence c is chosen as the angular half width for which the distribution of source intensity falls to half its maximum value, then c lu0 ln 21=2 :

4:3:8:18

The quantity q is defined by q Cs l3 u3 f lu2 T 2; where T 2 expresses a coupling between the effects of partial spatial coherence and temporal coherence. This term can frequently be neglected under HREM conditions [see Wade & Frank (1977) for details]. The damping envelope due to chromatic effects is described by the parameter Cc Q Cc 2 V0 =V 20 4 2 I0 =I 20 1=2 ; 4:3:8:19 2 E0 =E 20 where 2 V0 and 2 I0 are the variances in the statistically independent fluctuations of accelerating voltage V0 and objective-lens current I0 . The r.m.s. value of the high voltage fluctuation is equal to the standard deviation V0 2 V0 1=2 . The full width at half-maximum height of the energy distribution of electrons leaving the filament is E 22 ln 21=2 E0 2:355 2 E0 1=2 :

4:3:8:20

Here, Cc is the chromatic aberration constant of the objective lens. Equations (4.3.8.14) and (4.3.8.17) indicate that under linear imaging conditions the transfer function for HREM contains a chromatic damping envelope more severely attenuating than a Gaussian of width U0 2=l1=2 ; which is present in the absence of any objective aperture Pu. The resulting resolution limit

423


4:3:8:16

4. PRODUCTION AND PROPERTIES OF RADIATIONS di l=21=2

4:3:8:21

is known as the information resolution limit (see Subsection 4.3.8.6) and depends on electronic instabilities and the thermalenergy spread of electrons leaving the filament. The reduction in the contribution of particular diffracted beams to the image due to limited spatial coherence is minimized over those extended regions for which ru is small, called passbands, which occur when fn Cs l8n 3=21=2 :

4:3:8:22

The Scherzer focus fs corresponds to n 0. These passbands become narrower and move to higher u values with increasing n, but are subject also to chromatic damping effects. The passbands occur between spatial frequencies U1 and U2 , where U1;2 Cs

1=4

l

3=4

f8n 2=21=2 1g1=2 :

4:3:8:23

Their use for extracting information beyond the point resolution of an electron microscope is further discussed in Subsection 4.3.8.6. Fig. 4.3.8.5 shows transfer functions for a modern instrument for n 0 and 1. Equations (4.3.8.14) and (4.3.8.17) provide a simple, useful, and popular approach to the interpretation of HREM images and valuable insights into resolution-limiting factors. However, it must be emphasized that these results apply only (amongst other conditions) for 0 g (in crystals) and therefore do not apply to the usual case of strong multiple electron scattering. Equation (4.3.8.13b) does not make this approximation. In real space, for crystals, the alignment of columns of atoms in the beam direction rapidly leads to phase

Fig. 4.3.8.5. a The transfer function for a 400 kV electron microscope Ê at the Scherzer focus; the curve is with a point resolution of 1.7 A based on equation (4.3.8.17). In b is shown a transfer function for similar conditions at the first `passband' focus [n 1 in equation (4.3.8.22)].

changes in the electron wavefunction that exceed =2, leading to the failure of equation (4.3.8.14). Accurate quantitative comparisons of experimental and simulated HREM images must be based on equation (4.3.8.13a), or possibly (4.3.8.13b), with u0 ; f ; r obtained from many-beam dynamical calculations of the type described in Subsection 4.3.8.5. For the structure imaging of specific types of defects and materials, the following references are relevant. (i) For line defects viewed parallel to the line, d'Anterroches & Bourret (1984); viewed normal to the line, Alexander, Spence, Shindo, Gottschalk & Long (1986). (ii) For problems of variable lattice spacing (e.g. spinodal decomposition), Cockayne & Gronsky (1981). (iii) For point defects and their ordering, in tunnel structures, Yagi & Cowley (1978); in semiconductors, Zakharov, Pasemann & Rozhanski (1982); in metals, Fields & Cowley (1978). (iv) For interfaces, see the proceedings reported in Ultramicroscopy (1992), Vol. 40, No. 3. (v) For metals, Lovey, Coene, Van Dyck, Van Tendeloo, Van Landuyt & Amelinckx (1984). (vi) For organic crystals, Kobayashi, Fujiyoshi & Uyeda (1982). (vii) For a general review of applications in solid-state chemistry, see the collection of papers reported in Ultramicroscopy (1985), Vol. 18, Nos. 1±4. (viii) Radiation-damage effects are observed at atomic resolution by Horiuchi (1982).

4.3.8.4. Parameters affecting HREM images The instrumental parameters that affect HREM images include accelerating voltage, astigmatism, optic-axis alignment, focus setting f , spherical-aberration constant Cs , beam divergence c , and chromatic aberration constant Cc . Crystal parameters influencing HREM images include thickness, absorption, ionicity, and the alignment of the crystal zone axis with the beam, in addition to the structure factors and atom positions of the sample. The accurate measurement of electron wavelength or accelerating voltage has been discussed by many workers, including Uyeda, Hoier and others [see Fitzgerald & Johnson (1984) for references]. The measurement of Kikuchiline spacings from crystals of known structure appears to be the most accurate and convenient method for HREM work, and allows an overall accuracy of better than 0.2% in accelerating voltage. Fluctuations in accelerating voltage contribute to the chromatic damping term in equation (4.3.8.19) through the variance 2 V0 . With the trend toward the use of higher accelerating voltages for HREM work, this term has become especially significant for the consideration of the information resolution limit [equation (4.3.8.21)]. Techniques for the accurate measurement of astigmatism and chromatic aberration are described by Spence (1988). The displacement of images of small crystals with beam tilt may be used to measure Cs ; alternatively, the curvature of higher-order Laue-zone lines in CBED patterns has been used. The method of Budinger & Glaeser (1976) uses a similar dark-field imagedisplacement method to provide values for both f and Cs , and appears to be the most convenient and accurate for HREM work. The analysis of optical diffractograms initiated by Thon and coworkers from HREM images of thin amorphous films provides an invaluable diagnostic aid for HREM work; however, the determination of Cs by this method is prone to large errors, especially at small defocus. Diffractograms provide a rapid method for the determination of focus setting (see Krivanek, 1976) and in addition provide a sensitive indicator of specimen movement, astigmatism, and the damping-envelope constants and c .

424


4.3. ELECTRON DIFFRACTION Misalignment of the electron beam, optic axis, and crystal axis in bright-field HREM work becomes increasingly important with increasing resolution and specimen thickness. The first-order effects of optical misalignment are an artifactual translation of spatial frequencies in the direction of misalignment by an amount proportional to the misalignment and to the square of spatial frequency. The corresponding phase shift is not observable in diffractograms. The effects of astigmatism on transfer functions for inclined illumination are discussed in Saxton (1978). The effects of misalignment of the beam with respect to the optic axis are discussed in detail by Smith, Saxton, O'Keefe, Wood & Stobbs (1983), where it is found that all symmetry elements (except a mirror plane along the tilt direction) may be destroyed by misalignment. The maximum allowable misalignment for a given resolution in a specimen of thickness t is proportional to =8t:

4:3:8:24

Misalignment of a crystalline specimen with respect to the beam may be distinguished from misalignment of the optic axis with respect to the beam by the fact that, in very thin crystals, the former does not destroy centres of symmetry in the image. The use of known defect point-group symmetry (for example in stacking faults) to identify a point in a HREM image with a point in the structure and so to resolve the black or white atomic contrast ambiguity has been described (Olsen & Spence, 1981). Structures containing screw or glide elements normal to the beam are particularly sensitive to misalignment, and errors as small as 0.2 mrad may substantially alter the image appearance. A rapid comparison of images of amorphous material with the beam electronically tilted into several directions appears to be the best current method of aligning the beam with the optic axis, while switching to convergent-beam mode appears to be the most effective method of aligning the beam with the crystal axis. However, there is evidence that the angle of incidence of the incident beam is altered by this switching procedure. The effects of misalignment and choice of beam divergence c on HREM images of crystals containing dynamically forbidden reflections are reviewed by Nagakura, Nakamura & Suzuki (1982) and Smith, Bursill & Wood (1985). Here the dramatic example of rutile in the [001] orientation is used to demonstrate how a misalignment of less than 0.2 mrad of the electron beam with respect to the crystal axis can bring up a coarse set of Ê ), which produce an image of incorrect symmetry, fringes (4.6 A since these correspond to structure factors that are forbidden both dynamically and kinematically. Crystal thickness is most accurately determined from images of planar faults in known orientations, or from crystal morphology for small particles. It must otherwise be treated as a refinement parameter. Since small crystals (such as MgO smoke particles, which form as perfect cubes) provide such an independent method of thickness determination, they provide the most convincing test of dynamical imaging theory. The ability to match the contrast reversals and other detailed changes in HREM images as a function of either thickness or focus (or both) where these parameters have been measured by an independent method gives the greatest confidence in image interpretation. This approach, which has been applied in rather few cases [see, for example, O'Keefe, Spence, Hutchinson & Waddington (1985)] is strongly recommended. The tendency for n-beam dynamical HREM images to repeat with increasing thickness in cases where the wavefunction is dominated by just two Bloch waves has been analysed by several workers (Kambe, 1982). Since electron scattering factors are proportional to the difference between atomic number and X-ray scattering factors,

and inversely proportional to the square of the scattering angle (see Section 4.3.1), it has been known for many years that the low-order reflections that contribute to HREM images are extremely sensitive to the distribution of bonding electrons and so to the degree of ionicity of the species imaged. This observation has formed the basis of several charge-density-map determinations by convergent-beam electron diffraction [see, for example, Zuo, Spence & O'Keefe (1988)]. Studies of ionicity effects on HREM imaging can be found in Anstis, Lynch, Moodie & O'Keefe (1973) and Fujiyoshi, Ishizuka, Tsuji, Kobayashi & Uyeda (1983). The depletion of the elastic portion of the dynamical electron wavefunction by inelastic crystal excitations (chiefly phonons, single-electron excitations, and plasmons) may have dramatic effects on the HREM images of thicker crystals (Pirouz, 1974). For image formation by the elastic component, these effects may be described through the use of a complex òptical' potential and the appropriate Debye±Waller factor (see Section 2.5.1). However, existing calculations for the absorption coefficients derived from the imaginary part of this potential are frequently not applicable to lattice images because of the large objective apertures used in HREM work. It has been suggested that HREM images formed from electrons that suffer small energy losses (and so remain ìn focus') but large-angle scattering events (within the objective aperture) due to phonon excitation may contribute high-resolution detail to images (Cowley, 1988). For measurements of the imaginary part of the optical potential by electron diffraction, the reader is referred to the work of Voss, Lehmpfuhl & Smith (1980), and references therein. All evidence suggests, however, that for the crystal thicknesses generally used Ê the effects of àbsorption' are small. for HREM work t < 200A In summary, the general approach to the matching of computed and experimental HREM images proceeds as follows (Wilson, Spargo & Smith, 1982). (i) Values of , c , and Cs are determined by careful measurements under well defined conditions (electron-gun bias setting, illumination aperture size, specimen height as measured by focusing-lens currents, electron-source size, etc). These parameters are then taken as constants for all subsequent work under these instrumental conditions (assuming also continuous monitoring of electronic instabilities). (ii) For a particular structure refinement, the parameters of thickness and focus are then varied, together with the choice of atomic model, in dynamical computer simulations until agreement is obtained. Every effort should be made to match images as a function of thickness and focus. (iii) If agreement cannot be obtained, the effects of small misalignments must be investigated (Smith et al., 1985). Crystals most sensitive to these include those containing reflections that are absent due to the presence of screw or glide elements normal to the beam. 4.3.8.5. Computing methods The general formulations for the dynamical theory of electron diffraction in crystals have been described in Section 5.2 of IT B (1992). In Section 4.3.6, the computing methods used for calculating diffraction-beam amplitudes have been outlined. Given the diffracted-beam amplitudes, g , the image is calculated by use of equations (4.3.8.2), including, when appropriate, the modifications of (4.3.8.13b). The numerical methods that can be employed in relation to crystal-structure imaging make use of algorithms based on (i) matrix diagonalization, (ii) fast Fourier transforms, (iii) realspace convolution (Van Dyck, 1980), (iv) Runge-Kutta (or similar) methods, or (v) power-series evaluation. Two other solutions, the Cowley±Moodie polynomial solution and the

425


4. PRODUCTION AND PROPERTIES OF RADIATIONS Feynman path-integral solution, have not been used extensively for numerical work. Methods (i) and (ii) have proven the most popular, with (ii) (the multislice method) being used most extensively for HREM image simulations. The availability of inexpensive array processors has made this technique highly efficient. A comparison of these two N-beam methods is given by Self, O'Keefe, Buseck & Spargo (1983), who find the multislice method to be faster (time proportional to N log2 N) than the diagonalization method (time proportional to N 2 ) for N > 16. Computing space increases roughly as N 2 for the diagonalization method, and as N for the multislice. The problem of steeply inclined boundary conditions for multislice computations has been discussed by Ishizuka (1982). In the Bloch-wave formulation, the lattice image is given by P P i j i j Ir C 0 C 0 C g C h exp i2 i j t i; j h;g

2g

h r

f ; Cs ; g f ; Cs ; h ; 4:3:8:25

Cgi

where and i are the eigenvector elements and eigenvalues of the structure matrix [see Hirsch, Howie, Nicholson, Pashley & Whelan (1977) and Section 4.3.4]. Using modern personal computers or workstations, it is now possible to build efficient single-user systems that allow interactive dynamical structure-image calculations. Either an image intensifier or a cooled scientific grade charge-coupled device and single-crystal scintillator screen may be used to record the images, which are then transferred into a computer (Daberkow, Herrman, Liu & Rau, 1991). This then allows for the possibility of automated alignment, stigmation and focusing to the level of accuracy needed at 0.1 nm point resolution (Krivanek & Mooney, 1993). An image-matching search through trial structures, thickness and focus parameters can then be completed rapidly. Where large numbers of pixels, large dynamic range and high sensitivity are required, the Image Plate has definite advantages and so should find application in electron holography and biology (Shindo, Hiraga, Oikawa & Mori, 1990). For the calculation of images of defects, the method of periodic continuation has been used extensively (Grinton & Cowley, 1971). Since, for kilovolt electrons traversing thin crystals, the transverse spreading of the dynamical wavefunction is limited (Cowley, 1981), the complex image amplitude at a particular point on the specimen exit face depends only on the crystal potential within a cylinder a few aÊngstroÈms in diameter, erected about that point (Spence, O'Keefe & Iijima, 1978). The width of this cylinder depends on accelerating voltage, specimen thickness, and focus setting (see above references). Thus, small overlapping `patches' of exit-face wavefunction may be calculated in successive computations, and the results combined to form a larger area of image. The size of the àrtificial superlattice' used should be increased until no change is found in the wavefunction over the central region of interest. For most defects, the positions of only a few atoms are important and, since the electron wavefunction is locally determined (for thin specimens at Scherzer focus), it appears that very large calculations are rarely needed for HREM work. The simulation of profile images of crystal surfaces at large defocus settings will, however, frequently be found to require large amounts of storage. A new program should be tested to ensure that a under approximate two-beam conditions the calculated extinction distances for small-unit-cell crystals agree roughly with tabulated values (Hirsch et al., 1977), b the simulated dynamical images

have the correct symmetry, c for small thickness, the Scherzerfocus images agree with the projected potential, and d images and beam intensities agree with those of a program known to be correct. The damping envelope (product representation) [equation (4.3.8.17)] should only be used in a thin crystal with 0 > g ; in general, the effects of partial spatial and temporal coherence must be incorporated using equation (4.3.8.13a) or (4.3.8.13b), depending on whether variations in diffraction conditions over c are important. Thus, a separate multislice dynamical-image calculation for each component plane wave in the incident cone of illumination may be required, followed by an incoherent sum of all resulting images. The outlook for obtaining higher resolution at the time of writing (1997) is broadly as follows. (1) The highest point resolution currently obtainable is close to 0.1 nm, and this has been obtained by taking advantage of the reduction in electron wavelength that occurs at high voltage [equation (4.3.8.16)]. A summary of results from these machines can be found in Ultramicroscopy (1994), Vol. 56, Nos. 1±3, where applications to fullerenes, glasses, quasicrystals, interfaces, ceramics, semiconductors, metals and oxides and other systems may be found. Fig. 4.2.8.6 shows a typical result. High cost, and the effects of radiation damage (particularly at larger thickness where defects with higher free energies are likely to be found), may limit these machines to a few specialized laboratories in the future. The attainment of higher resolution through this approach depends on advances in high-voltage engineering. (2) Aberration coefficients may be reduced if higher magnetic fields can be produced in the pole piece, beyond the saturation flux of the specialized iron alloys currently used. Research into superconducting lenses has therefore continued for many years in a few laboratories. Fluctuations in lens current are also eliminated by this method. (3) Electron holography was originally developed for the purpose of improving electron-microscope resolution, and this approach is reviewed in the following section. (4) Electron±optical correction of aberrations has been under study for many years in work by Scherzer, Crewe, Beck, Krivanek, Lanio, Rose and others ± results of recent experimental tests are described in Haider & Zach (1995) and Krivanek, Dellby, Spence, Camps & Brown (1997). The attainment of 0.1 nm point resolution is considered feasible. Aberration correctors will also provide benefits other than increased resolution, including greater space in the pole piece for increased sample tilt and access to X-ray detectors, etc.

Fig. 4.3.8.6. Structure image of a thin lamella of the 6H polytype of SiC projected along [110] and recorded at 1.2 MeV. Every atomic column (darker dots) is separately resolved at 0.109 nm spacing. The central horizontal strip contains a computer-simulated image; the structure is sketched at the left. [Courtesy of H. Ichinose (1994).]

426


4.3. ELECTRON DIFFRACTION The need for resolution improvement beyond 0.1 nm has been questioned ± the structural information retrievable by a single HREM image is always limited by the fact that a projection is obtained. (This problem is particularly acute for glasses.) Methods for combining different projected images (particularly of defects) from the same region (Downing, Meisheng, Wenk & O'Keefe, 1990) may now be as important as the search for higher resolution. 4.3.8.6. Resolution and hyper-resolution Since the resolution of an instrument is a property of the instrument alone, whereas the ability to distinguish HREM image features due to adjacent atoms depends on the scattering properties of the atoms, the resolution of an electron microscope cannot easily be defined [see Subsection 2.5.1.9 in IT B (1992)]. The Rayleigh criterion was developed for the incoherent imaging of point sources and cannot be applied to coherent phase contrast. Only for very thin specimens of light elements for which it can be assumed that the scattering phase is =2 can the straightforward definition of point resolution dp [equation (4.3.8.16)] be applied. In general, the dynamical wavefunction across the exit face of a crystalline sample bears no simple relationship to the crystal structure, other than to preserve its symmetry and to be determined by the `local' crystal potential. The use of a dynamical `R factor' between computed and experimental images of a known structure has been suggested by several workers as the basis for a more general resolution definition. For weakly scattering specimens, the most satisfactory method of measuring either the point resolution dp or the information limit di [see equation (4.3.8.21)] appears to be that of Frank (1975). Here two successive micrographs of a thin amorphous film are recorded (under identical conditions) and the superimposed pair used to obtain a coherent optical diffractogram crossed by fringes. The fringes, which result from small displacements of the micrographs, extend only to the band limit di 1 of information common to both micrographs, and cannot be extended by photographic processing, noise, or increased exposure. By plotting this band limit against defocus, it is possible to determine both and c . As an alternative, for thin crystalline samples of large-unit-cell materials, the parameters , c , and Cs can be determined by matching computed and experimental images of crystals of known structure. It is the specification of these parameters (for a given electron intensity and wavelength) that is important in describing the performance of high-resolution electron microscopes. We note that certain conditions of focus or thickness may give a spurious impression of ultra-high resolution [see equations (4.3.8.7) and (4.3.8.8)]. Within the domain of linear imaging, implying, for the most part, the validity of the WPO approximation, many forms of image processing have been employed. These have been of particular importance for crystalline and non-crystalline biological materials and include image reconstruction [see Section 2.5.4 in IT B (1992)] and the derivation of three-dimensional structures from two-dimensional projections [see Section 2.5.5 in IT B (1992)]. For reviews, see also Saxton (1980a), Frank (1980), and Schiske (1975). Several software packages now exist that are designed for image manipulation, Fourier analysis, and cross correlation; for details of these, see Saxton (1980a) and Frank (1980). The theoretical basis for the WPO approximation closely parallels that of axial holography in coherent optics, thus much of that literature can be applied to HREM image processing. Gabor's original proposal for holography was intended for electron microscopy [see Cowley (1981) for a review].

The aim of image-processing schemes is the restoration of the exit-face wavefunction, given in equation (4.3.8.13a). The reconstruction of the crystal potential 'p r from this is a separate problem, since these are only simply related under the approximation of Subsection 4.3.8.3. For a non-linear method that allows the reconstruction of the dynamical image wavefunction, based on equation (4.3.8.13b), which thus includes the effects of multiple scattering, see Saxton (1980b). The concept of holographic reconstruction was introduced by Gabor (1948, 1949) as a means of enhancing the resolution of electron microscopes. Gabor proposed that, if the information on relative phases of the image wave could be recorded by observing interference with a known reference wave, the phase modification due to the objective-lens aberrations could be removed. Of the many possible forms of electron holography (Cowley, 1994), two show particular promise of useful improvements of resolution. In what may be called in-line TEM holography, a through-focus series of bright-field images is obtained with near-coherent illumination. With reference to the relatively strong transmitted beam, the relative phase and amplitude changes due to the specimen are derived from the variations of image intensity (see Van Dyck, Op de Beeck & Coene, 1994). The tilt-series reconstruction method also shows considerable promise (Kirkland, Saxton, Chau, Tsuno & Kawasaki, 1995). In the alternative off-axis approach, the reference wave is that which passes by the specimen area in vacuum, and which is made to interfere with the wave transmitted through the specimen by use of an electrostatic biprism (MoÈllenstedt & DuÈker, 1956). The hologram consists of a modulated pattern of interference fringes. The image wavefunction amplitude and phase are deduced from the contrast and lateral displacements of the fringes (Lichte, 1991; Tonomura, 1992). The process of reconstruction from the hologram to give the image wavefunction may be performed by optical-analogue or digital methods and can include the correction of the phase function to remove the effects of lens aberrations and the attendant limitation of resolution. The point resolution of electron microscopes has recently been exceeded by this method (Orchowski, Rau & Lichte, 1995). The aim of the holographic reconstructions is the restoration of the wavefunction at the exit face of the specimen as given by equation (4.3.8.13a). The reconstruction of the crystal potential 'r from this is a separate problem, since the exit-face wavefunction and 'r are simply related only under the WPO approximations of Subsection 4.3.8.3. The possibility of deriving reconstructions from wavefunctions strongly affected by dynamical diffraction has been considered by a number of authors (for example, Van Dyck et al., 1994). The problem does not appear to be solvable in general, but for special cases, such as perfect thin single crystals in exact axial orientations, considerable progress may be possible. Since a single atom, or a column of atoms, acts as a lens with negative spherical aberration, methods for obtaining superresolution using atoms as lenses have recently been proposed (Cowley, Spence & Smirnov, 1997). 4.3.8.7. Alternative methods A number of non-conventional imaging modes have been found useful in electron microscopy for particular applications. In scanning transmission electron microscopy (STEM), powerful electron lenses are used to focus the beam from a very small bright source, formed by a field-emission gun, to form a small probe that is scanned across the specimen. Some selected part of the transmitted electron beam (part of the coherent convergent-

427


4. PRODUCTION AND PROPERTIES OF RADIATIONS beam electron diffraction pattern produced) is detected to provide the image signal that is displayed or recorded in synchronism with the incident-beam scan. The principle of reciprocity suggests that, for equivalent lenses, apertures and column geometry, the resolution and contrast of STEM and TEM images will be identical (Cowley, 1969). Practical considerations of instrumental convenience distinguish particularly useful STEM modes. Crewe & Wall (1970) showed that, if an annular detector is used to detect all electrons scattered outside the incident-beam cone, dark-field images could be obtained with high efficiency and with a resolution better than that of the bright-field mode by a factor of about 1.4. If the inner radius of the annular detector is made large (of the order of 10 1 rad for 100 kV electrons), the strong diffracted beams occurring for lower angles do not contribute to the resulting high-angle annular dark-field (HAADF) image (Howie, 1979), which is produced mainly by thermal diffuse scattering. The HAADF mode has important advantages for particular purposes because the contrast is strongly dependent on the atomic number, Z, of the atoms present but is not strongly affected by dynamical diffraction effects and so shows near-linear variation with Z and with the atom-number density in the sample. Applications have been made to the imaging of small high-Z particles in low-Z supports, such as in supported metal catalysts (Treacy & Rice, 1989) and to the high-resolution imaging of individual atomic rows in semiconductor crystals, showing the variations of composition across planar interfaces (Pennycook & Jesson, 1991). The STEM imaging modes may be readily correlated with microchemical analysis of selected specimen areas having lateral dimensions in the nanometre range, by application of the techniques of electron energy-loss spectroscopy or X-ray energy-dispersive analysis (Williams & Carter, 1996; Section 4.3.4). Also, diffraction patterns (coherent convergent-beam electron diffraction patterns) may be obtained from any chosen region having dimensions equal to those of the incident-beam diameter and as small as about 0.2 nm (Cowley, 1992). The coherent interference between diffracted beams within such a pattern may provide information on the symmetries, and, ultimately, the atomic arrangement, within the illuminated area, which may be smaller than the projection of the crystal unit cell in the beam direction. This geometry has been used to extend resolution for crystalline samples beyond even the information resolution limit, di (Nellist, McCallum & Rodenburg, 1995), and is the basis for an exact, non-perturbative inversion scheme for dynamical electron diffraction (Spence, 1998). The detection of secondary radiations (light, X-rays, lowenergy `secondary' electrons, etc.) in STEM or the detection of energy losses of the incident electrons, resulting from particular elementary excitations of the atoms in a crystal, in TEM or STEM, may be used to form images showing the distributions in a crystal structure of particular atomic species. In principle, this may be extended to the chemical identification of individual atom types in the projection of crystal structures, but only limited success has been achieved in this direction because of the relatively low level of the signals available. The formation of atomic resolution images using inner-shell excitations, for example, is complicated by the Bragg scattering of these inelastically scattered electrons (Endoh, Hashimoto & Makita, 1994; Spence & Lynch, 1982). Reflection electron microscopy (REM) has been shown to be a powerful technique for the study of the structures and defects of crystal surfaces with moderately high spatial resolution (Larsen & Dobson, 1988), especially when performed in a specially built

electron microscope having an ultra-high-vacuum specimen environment (Yagi, 1993). Images are formed by detecting strong diffracted beams in the RHEED patterns produced when kilovolt electron beams are incident on flat crystal surfaces at grazing incidence angles of a few degrees. The images suffer from severe foreshortening in the beam direction, but, in directions at right angles to the beam, resolutions approaching 0.3 nm have been achieved (Koike, Kobayashi, Ozawa & Yagi, 1989). Single-atom-high surface steps are imaged with high contrast, surface reconstructions involving only one or two monolayers are readily seen and phase transitions of surface superstructures may be followed. The study of surface structure by use of high-resolution transmission electron microscopes has also been productive in particular cases. Images showing the structures of surface layers with near-atomic resolution have been obtained by the use of `forbidden' or `termination' reflections (Cherns, 1974; Takayanagi, 1984) and by phase-contrast imaging (Moodie & Warble, 1967; Iijima, 1977). The imaging of the profiles of the edges of thin or small crystals with clear resolution of the surface atomic layers has also been effective (Marks, 1986). The introduction of the scanning tunnelling microscope (Binnig, Rohrer, Gerber & Weibel, 1983) and other scanning probe microscopies has broadened the field of high-resolution surface structure imaging considerably. 4.3.8.8. Combined use of HREM and electron diffraction For many materials of organic or biological origin, it is possible to obtain very thin crystals, only one or a few molecules thick, extending laterally over micrometre-size areas. These may give selected-area electron-diffraction patterns in electron microscopes with diffraction spots extending out to angles corresponding to d spacings as low as 0.1 nm. Because the materials are highly sensitive to electron irradiation, conventional bright-field images cannot be obtained with resolutions better than several nanometres. However, if images are obtained with very low electron doses and then a process of averaging over the content of a very large number of unit cells of the image is carried out, images showing detail down to the scale of 1 nm or less may be derived for the periodically repeated unit. From such images, it is possible to derive both the magnitudes and phases of the Fourier coefficients, the structure factors, out to some limit of d spacings, say dm . From the diffraction patterns, the magnitudes of the structure factors may be deduced, with greater accuracy, out to a much smaller limit, dd . By combination of the information from these two sources, it may be possible to obtain a greatly improved resolution for an enhanced image of the structure. This concept was first introduced by Unwin & Henderson (1975), who derived images of the purple membrane from Halobacterium halobium, with greatly improved resolution, revealing its essential molecular configuration. Recently, several methods of phase extension have been developed whereby the knowledge of the relative phases may be extended from the region of the diffraction pattern covered by the electron-microscope image transform to the outer parts. These include methods based on the use of the tangent formula or Sayre's equation (Dorset, 1994; Dorset, McCourt, Fryer, Tivol & Turner, 1994) and on the use of maximum-entropy concepts (Fryer & Gilmore, 1992). Such methods have also been applied, with considerable success, to the case of some thin inorganic crystals (Fu et al., 1994). In this case, the limitation on the resolution set by the electron-microscope images may be that due to the transfer function of the microscope, since radiationdamage effects are not so limiting. Then, the resolution achieved

428


4.3. ELECTRON DIFFRACTION by the combined application of the electron diffraction data may represent an advance beyond that of normal HREM imaging. Difficulties may well arise, however, because the theoretical

basis for the phase-extension methods is currently limited to the WPO approximation. A summary of the present situation is given in the book by Dorset (1995).

429



4.4. Neutron techniques

By I. S. Anderson, P. J. Brown, J. M. Carpenter, G. Lander, R. Pynn, J. M. Rowe, O. SchaÈrpf, V. F. Sears and B. T. M. Willis 4.4.1. Production of neutrons (By J. M. Carpenter and G. Lander) The production of neutrons of suf®cient intensity for scattering experiments is a `big-machine' operation; there is no analogue to the small laboratory X-ray unit. The most common sources of neutrons, and those responsible for the great bulk of today's successful neutron scattering programs, are the nuclear reactors. These are based on the continuous, self-sustaining ®ssion reaction. Research-reactor design emphasizes power density, that is the highest power within a small `leaky' volume, whereas power reactors generate large amounts of power over a large core volume. In research reactors, fuel rods are of highly enriched 235 U. Neutrons produced are distributed in a ®ssion spectrum centred about 1 MeV: Most of the neutrons within the reactor are moderated (i.e. slowed down) by collisions in the cooling liquid, normally D2 O or H2 O, and are absorbed in fuel to propagate the reaction. As large a fraction as possible is allowed to leak out as fast neutrons into the surrounding moderator (D2 O and Be are best) and to slow down to equilibrium with this moderator. The neutron spectrum is Maxwellian with a mean energy of 300 K 25 meV, which for neutrons corresponds Ê since to 1.8 A Ê 2 : En meV 81:8=l2 A Neutrons are extracted in beams through holes that penetrate the moderator. There are two points to remember: a neutrons are neutral so that we cannot focus the beams and b the spectrum is broad and

continuous; there is no analogy to the characteristic wavelength found with X-ray tubes, or to the high directionality of synchrotron-radiation sources. Neutron production and versatility in reactors reached a new level with the construction of the High-Flux Reactor at the French-German-English Institut Laue-Langevin (ILL) in Grenoble, France. An overview of the reactor and beam-tube assembly is shown in Fig. 4.4.1.1. To shift the spectrum in energy, both a cold source (25 l of liquid deuterium at 25 K) and a hot source (graphite at 2400 K) have been inserted into the D2 O moderator. Special beam tubes view these sources allowing a Ê to be used. Over 30 range of wavelengths from 0:3 to 17 A instruments are in operation at the ILL, which started in 1972. The second method of producing neutrons, which historically predates the discovery of ®ssion, is with charged particles ( particles, protons, etc.) striking a target nuclei. The most powerful source of neutrons of this type uses proton beams. These are accelerated in short bursts (< 1 ms) to 500±1000 MeV, and after striking the target produce an instantaneous supply of high-energy èvaporation' neutrons. These extend up in energy close to that of the incident proton beam. Shielding for spallation sources tends to be even more massive than that for reactors. The targets, usually tungsten or uranium and typically much smaller than a reactor core, are surrounded by hydrogenous moderators such as polyethylene (often at different temperatures) to produce the `slow' neutrons En < 10 eV used in scattering experiments. The moderators are very different from those of reactors; they are designed to slow down neutrons rapidly and to let them leak out, rather than to store them for a long time. If the accelerated

Fig. 4.4.1.1. A plane view of the installation at the Institut Laue±Langevin, Grenoble. Note especially the guide tubes exiting from the reactor that transport the neutron beams to a variety of instruments; these guide tubes are made of nickel-coated glass from which the neutrons are totally internally re¯ected.


4.4. NEUTRON TECHNIQUES particle pulse is short enough, the duration of the moderated neutron pulses is roughly inversely proportional to the neutron speed. These accelerator-driven pulsed sources are pulsed at frequencies of between 10 and 100 Hz. There are two fundamental differences between a reactor and a pulsed source. (1) All experiments at a pulsed source must be performed with time-of-¯ight techniques. The pulsed source produces neutrons in bursts of 1 to 50 ms duration, depending on the energy, spaced about 10 to 100 ms apart, so that the duty cycle is low but there is very high neutron intensity within each pulse. The time-of-¯ight technique makes it possible to exploit that high intensity. With the de Broglie relationship, for neutrons Ê 0:3966 t ls=L cm; l A where t is the ¯ight time in ms and L is the total ¯ight path in cm. (2) The spectral characteristics of pulsed sources are somewhat different from reactors in that they have a much larger component of higher-energy (above 100 meV) neutrons than the thermal spectrum at reactors. The exploitation of this new energy regime accompanied by the short pulse duration is one of the great opportunities presented by spallation sources. Fig. 4.4.1.2 illustrates the essential difference between experiments at a steady-state source (left panel) and a pulsed source (right panel). We con®ne the discussion here to diffraction. If the time over which useful information is gathered is equivalent to the full period of the source t (the case suggested by the lower-right ®gure), the peak ¯ux of the pulsed source is the effective parameter to compare with the ¯ux of the steady-state source. Often this is not the case, so one makes a comparison in terms of time-averaged ¯ux (centre panel). For the pulsed source, this is lowered from the peak ¯ux by the duty cycle, but with the time-of-¯ight method one uses a large interval of the spectrum (shaded area). For the steady-state source, the time-averaged ¯ux is high, but only a small wavelength slice (stippled area) is used in the experiment. It is the integrals of the

two areas which must be compared; for the pulsed sources now being designed, the integral is generally favourable compared with present-day reactors. Finally, one can see from the central panel that high-energy neutrons (100±1000 meV) are especially plentiful at the pulsed sources. These various features can be exploited in the design of different kinds of experiments at pulsed sources. 4.4.2. Beam-de®nition devices (By I. S. Anderson and O. SchaÈrpf) 4.4.2.1. Introduction Neutron scattering, when compared with X-ray scattering techniques developed on modern synchrotron sources, is ¯ux limited, but the method remains unique in the resolution and range of energy and momentum space that can be covered. Furthermore, the neutron magnetic moment allows details of microscopic magnetism to be examined, and polarized neutrons can be exploited through their interaction with both nuclear and electron spins. Owing to the low primary ¯ux of neutrons, the beam de®nition devices that play the role of de®ning the beam conditions (direction, divergence, energy, polarization, etc.) have to be highly ef®cient. Progress in the development of such devices not only results in higher-intensity beams but also allows new techniques to be implemented. The following sections give a (non-exhaustive) review of commonly used beam-de®nition devices. The reader should keep in mind the fact that neutron scattering experiments are typically carried out with large beams (1 to 50 cm2 ) and divergences between 5 and 30 mrad. 4.4.2.2. Collimators A collimator is perhaps the simplest neutron optical device and is used to de®ne the direction and divergence of a neutron beam. The most rudimentary collimator consists of two slits or pinholes

Fig. 4.4.1.2. Schematic diagram for performing diffraction experiments at steady-state and pulsed neutron sources. On the left we see the familiar monochromator crystal allowing a constant (in time) beam to fall on the sample (centre left), which then diffracts the beam through an angle 2s into the detector. The signal in the latter is also constant in time (lower left). On the right, the pulsed source allows a wide spectrum of neutrons to fall on the sample in sharp pulses separated by t (centre right). The neutrons are then diffracted by the sample through 2s and their time of arrival in the detector is analysed (lower right). The centre ®gure shows the time-averaged ¯ux at the source. At a reactor, we make use of a Ê At a pulsed source, we use a wide spectral band, here chosen from 0.4 to narrow band of neutrons (heavy shading), here chosen with l 1:5 A. Ê and each one is identi®ed by its time-of-¯ight. For the experimentalist, an important parameter is the integrated area of the two-shaded areas. 3A Here they have been made identical.

431


4. PRODUCTION AND PROPERTIES OF RADIATIONS cut into an absorbing material and placed one at the beginning and one at the end of a collimating distance L. The maximum beam divergence that is transmitted with this con®guration is max a1 a2 =L;

4:4:2:1

where a1 and a2 are the widths of the slits or pinholes. Such a device is normally used for small-angle scattering and re¯ectometry. In order to avoid parasitic scattering by re¯ection from slit edges, very thin sheets of a highly absorbing material, e.g. gadolinium foils, are used as the slit material. Sometimes wedge-formed cadmium plates are suf®cient. In cases where a very precise edge is required, cleaved single-crystalline absorbers such as gallium gadolinium garnet (GGG) can be employed. To avoid high intensity losses when the distances are large, sections of neutron guide can be introduced between the collimators, as in, for example, small-angle scattering instruments with variable collimation. In this case, for maximum intensity at a given resolution (divergence), the collimator length should be equal to the camera length, i.e. the sample±detector distance (Schmatz, Springer, Schelten & Ibel, 1974). As can be seen from (4.4.2.1), the beam divergence from a simple slit or pinhole collimator depends on the aperture size. In order to collimate (in one dimension) a beam of large cross section within a reasonable distance L, Soller collimators, composed of a number of equidistant neutron absorbing blades, are used. To avoid losses, the blades must be as thin and as ¯at as possible. If their surfaces do not re¯ect neutrons, which can be achieved by using blades with rough surfaces or materials with a negative scattering length, such as foils of hydrogen-containing polymers (Mylar is commonly used) or paper coated with neutron-absorbing paint containing boron or gadolinium (Meister & Weckerman, 1973; Carlile, Hey & Mack, 1977), the angular dependence of the transmission function is close to the ideal triangular form, and transmissions of 96% of the theoretical value can be obtained with 100 collimation. If the blades of the Soller collimator are coated with a material whose critical angle of re¯ection is equal to max =2 (for one particular wavelength), then a square angular transmission function is obtained instead of the normal triangular function, thus doubling the theoretical transmission (Meardon & Wroe, 1977). Soller collimators are often used in combination with singlecrystal monochromators to de®ne the wavelength resolution of an instrument but the Soller geometry is only useful for onedimensional collimation. For small-angle scattering applications, where two-dimensional collimation is required, a converging `pepper pot' collimator can be used (Nunes, 1974; Glinka, Rowe & LaRock, 1986). Cylindrical collimators with radial blades are sometimes used to reduce background scattering from the sample environment. This type of collimator is particularly useful with positionsensitive detectors and may be oscillated about the cylinder axis to reduce the shadowing effect of the blades (Wright, Berneron & Heathman, 1981). 4.4.2.3. Crystal monochromators Bragg re¯ection from crystals is the most widely used method for selecting a well de®ned wavelength band from a white neutron beam. In order to obtain reasonable re¯ected intensities and to match the typical neutron beam divergences, crystals that re¯ect over an angular range of 0.2 to 0.5 are typically employed. Traditionally, mosaic crystals have been used in preference to perfect crystals, although re¯ection from a mosaic crystal gives rise to an increase in beam divergence with a

concomitant broadening of the selected wavelength band. Thus, collimators are often used together with mosaic monochromators to de®ne the initial and ®nal divergences and therefore the wavelength spread. Because of the beam broadening produced by mosaic crystals, it was soon recognised that elastically deformed perfect crystals and crystals with gradients in lattice spacings would be more suitable candidates for focusing applications since the deformation can be modi®ed to optimize focusing for different experimental conditions (Maier-Leibnitz, 1969). Perfect crystals are used commonly in high-energy-resolution backscattering instruments, interferometry and Bonse±Hart cameras for ultra-small-angle scattering (Bonse & Hart, 1965). An ideal mosaic crystal is assumed to comprise an agglomerate of independently scattering domains or mosaic blocks that are more or less perfect, but small enough that primary extinction does not come into play, and the intensity re¯ected by each block may be calculated using the kinematic theory (Zachariasen, 1945; Sears, 1997). The orientation of the mosaic blocks is distributed inside a ®nite angle, called the mosaic spread, following a distribution that is normally assumed to be Gaussian. The ideal neutron mosaic monochromator is not an ideal mosaic crystal but rather a mosaic crystal that is suf®ciently thick to obtain a high re¯ectivity. As the crystal thickness increases, however, secondary extinction becomes important and must be accounted for in the calculation of the re¯ectivity. The model normally used is that developed by Bacon & Lowde (1948), which takes into account strong secondary extinction and a correction factor for primary extinction (Freund, 1985). In this case, the mosaic spread (usually de®ned by neutron scatterers as the full width at half maximum of the re¯ectivity curve) is not an intrinsic crystal property, but increases with wavelength and crystal thickness and can become quite appreciable at longer wavelengths. Ideal monochromator materials should have a large scatteringlength density, low absorption, incoherent and inelastic cross sections, and should be available as large single crystals with a suitable defect concentration. Relevant parameters for some typical neutron monochromator crystals are given in Table 4.4.2.1. In principle, higher re¯ectivities can be obtained in neutron monochromators that are designed to operate in re¯ection geometry, but, because re¯ection crystals must be very large when takeoff angles are small, transmission geometry may be used. In that case, the optimization of crystal thickness can only be achieved for a small wavelength range. Nickel has the highest scattering-length density, but, since natural nickel comprises several isotopes, the incoherent cross section is quite high. Thus, isotopic 58 Ni crystals have been grown as neutron monochromators despite their expense. Beryllium, owing to its large scattering-length density and low incoherent and absorption cross sections, is also an excellent candidate for neutron monochromators, but the mosaic structure of beryllium is dif®cult to modify, and the availability of goodquality single crystals is limited (MuÈcklich & Petzow, 1993). These limitations may be overcome in the near future, however, by building composite monochromators from thin beryllium blades that have been plastically deformed (May, Klimanek & Magerl, 1995). Pyrolytic graphite is a highly ef®cient neutron monochromator if only a medium resolution is required (the minimum mosaic spread is of the order of 0.4 ), owing to high re¯ectivities, which may exceed 90% (Shapiro & Chesser, 1972), but its use is Ê owing to the rather large d limited to wavelengths above 1.5A, spacing of the 002 re¯ection. Whenever better resolution at

432


4.4. NEUTRON TECHNIQUES Table 4.4.2.1. Some important properties of materials used for neutron monochromator crystals (in order of increasing unit-cell volume)

Material

Structure

Ratio of Square of incoherent to total scatteringLattice Coherent scattering constant(s) Unit-cell scattering length cross section at 300 K volume length density Ê ) V0 10 24 cm3 b (10 12 cm) 10 21 cm 4 inc =s a; c (A

Beryllium

h.c.p

Iron Zinc

b.c.c. h.c.p.

Pyrolytic graphite Niobum Nickel (58 Ni) Copper Aluminium Lead Silicon Germanium

layer hexag. b.c.c. f.c.c.

a : 2:2856 c : 3:5832 a : 2:8664 a : 2:6649 c : 4:9468 a : 2:461 c : 6:708 3.3006 3.5241

f.c.c. f.c.c. f.c.c. diamond diamond

3.6147 4.0495 4.9502 5.4309 5.6575

* 1 barn 10

28

4

16.2

0.779 (1)

9.25

6:5 10

23.5 30.4

0.954 (6) 0.5680 (5)

6.59 1.50

0.033 0.019

35.2

0.66484 (13)

5.71

< 2 10

35.9 43.8

0.7054 (3) 1.44 (1)

1.54 17.3

4 10 0

4.28 0.43 0.97 0.43 1.31

0.065 5:6 10 2:7 10 6:9 10 0.020

47.2 66.4 121 160 181

0.7718 (4) 0.3449 (5) 0.94003 (14) 0.41491 (10) 0.81929 (7)

4

4

3 4 3

Absorption Debye cross section abs (barns)* Atomic temperature AD2 Ê mass A (at l 1:8 A) D (K) 106 K2 0.0076 (8)

9.013

1188

12.7

2.56 (3) 1.11 (2)

55.85 65.38

411 253

9.4 4.2

0.00350 (7)

12.01

800

7.7

1.15 (5) 4.6 (3)

92.91 58.71

284 417

7.5 9.9

3.78 (2) 0.231 (3) 0.171 (2) 0.171 (3) 2.3 (2)

63.54 26.98 207.21 28.09 72.60

307 402 87 543 290

6.0 4.4 1.6 8.3 6.1

m2 .

shorter wavelengths is required, copper (220 and 200) or germanium (311 and 511) monochromators are frequently used. The advantage of copper is that the mosaic structure can be easily modi®ed by plastic deformation at high temperature. As with most face-centred cubic crystals, it is the (111) slip planes that are functional in generating the dislocation density needed for the desired mosaic spread, and, depending on the required orientation, either isotropic or anisotropic mosaics can be produced (Freund, 1976). The latter is interesting for vertical focusing applications, where a narrow vertical mosaic is required regardless of the resolution conditions. Although both germanium and silicon are attractive as monochromators, owing to the absence of second-order neutrons for odd-index re¯ections, it is dif®cult to produce a controlled uniform mosaic spread in bulk samples by plastic deformation at high temperature because of the dif®culty in introducing a spatially homogenous microstructure in large single crystals (Freund, 1975). Recently this dif®culty has been overcome by building up composite monochromators from a stack of thin wafers, as originally proposed by Maier-Leibnitz (1967; Frey, 1974). In practice, an arti®cial mosaic monochromator can be built up in two ways. In the ®rst approach, illustrated in Fig. 4.4.2.1(a), the monochromator comprises a stack of crystalline wafers, each of which has a mosaic spread close to the global value required for the entire stack. Each wafer in the stack must be plastically deformed (usually by alternated bending) to produce the correct mosaic spread. For certain crystal orientations, the plastic deformation may result in an anisotropic mosaic spread. This method has been developed in several laboratories to construct germanium monochromators (Vogt, Passell, Cheung & Axe, 1994; Schefer et al., 1996). In the second approach, shown in Fig. 4.4.2.1(b), the global re¯ectivity distribution is obtained from the contributions of several stacked thin crystalline wafers, each with a rather narrow mosaic spread compared with the composite value but slightly misoriented with respect to the other wafers in the stack. If the misorientation of each wafer can be correctly controlled, this

technique has the major advantage of producing monochromators with a highly anisotropic mosaic structure. The shape of the re¯ectivity curve can be chosen at will (Gaussian, Lorentzian, rectangular), if required. Moreover, because the initial mosaicity required is small, it is not necessary to use mosaic wafers and therefore for each wafer to undergo a long and tedious plastic deformation process. Recently, this method has been applied successfully to construct copper monochromators (Hamelin, Anderson, Berneron, Escof®er, Foltyn & Hehn, 1997), in which individual copper wafers were cut in a cylindrical form and then slid across one another to produce the required mosaic spread in the scattering plane. This technique looks very promising for the production of anisotropic mosaic monochromators. The re¯ection from a mosaic crystal is visualized in Fig. 4.4.2.2(b). An incident beam with small divergence is transformed into a broad exit beam. The range of k vectors, k, selected in this process depends on the mosaic spread, , and the incoming and outgoing beam divergences, 1 and 2 : k=k = cot ;

where is the magnitude of the crystal reciprocal-lattice vector ( 2=d) and is given by s 21 22 21 2 22 2 4:4:2:3 21 22 42 : The resolution can therefore be de®ned by collimators, and the highest resolution is obtained in backscattering, where the wavevector spread depends only on the intrinsic d=d of the crystal. In some applications, the beam broadening produced by mosaic crystals can be detrimental to the instrument performance. An interesting alternative is a gradient crystal, i.e. a single crystal with a smooth variation of the interplanar lattice spacing along a de®ned crystallographic direction. As shown in Fig. 4.4.2.2(c), the diffracted phase-space element has a different shape from that obtained from a mosaic crystal. Gradients in d spacing can be produced in various ways,

433


4:4:2:2

4. PRODUCTION AND PROPERTIES OF RADIATIONS including thermal gradients (Alefeld, 1972), vibrating crystals by piezoelectric excitation (Hock, Vogt, Kulda, Mursic, Fuess & Magerl, 1993), and mixed crystals with concentration gradients, e.g. Cu±Ge (Freund, Guinet, MareÂschal, Rustichelli & Vanoni, 1972) and Si±Ge (Maier-Leibnitz & Rustichelli, 1968; Magerl, Liss, Doll, Madar & Steichele, 1994). Both vertically and horizontally focusing assemblies of mosaic crystals are employed to make better use of the neutron ¯ux when making measurements on small samples. Vertical focusing can lead to intensity gain factors of between two and ®ve without affecting resolution (real-space focusing) (Riste, 1970; Currat, 1973). Horizontal focusing changes the k-space volume that is selected by the monochromator through the variation in Bragg angle across the monochromator surface (k-space focusing) (Scherm, Dolling, Ritter, Schedler, Teuchert & Wagner, 1977). The orientation of the diffracted k-space volume can be modi®ed by variation of the horizontal curvature, so that the resolution of the monochromator may be optimized with respect to a particular sample or experiment without loss of illumination. Monochromatic focusing can be achieved. Furthermore, asymmetrically

Fig. 4.4.2.1. Two methods by which arti®cial mosaic monochromators can be constructed: (a) out of a stack of crystalline wafers, each with a mosaicity close to the global value. The increase in divergence due to the mosaicity is the same in the horizontal (left picture) and the vertical (right picture) directions; (b) out of several stacked thin crystalline wafers each with a rather narrow mosaic but slightly misoriented in a perfectly controlled way. This allows the shape of the re¯ectivity curve to be rectangular, Gaussian, Lorentzian, etc., and highly anisotropic, i.e. vertically narrow (right picture) and horizontally broad (left ®gure).

cut crystals may be used, allowing focusing effects in real space and k space to be decoupled (Scherm & Kruger, 1994). Traditionally, focusing monochromators consist of rectangular crystal plates mounted on an assembly that allows the orientation of each crystal to be varied in a correlated manner (BuÈhrer, 1994). More recently, elastically deformed perfect crystals (in particular silicon) have been exploited as focusing elements for monochromators and analysers (Magerl & Wagner, 1994). Since thermal neutrons have velocities that are of the order of km s 1 , their wavelengths can be Doppler shifted by diffraction from moving crystals. The k-space representation of the diffraction from a crystal moving perpendicular to its lattice planes is shown in Fig. 4.4.2.3(a). This effect is most commonly used in backscattering instruments on steady-state sources to vary the energy of the incident beam. Crystal velocities of 9± 10 m s 1 are practically achievable, corresponding to energy variations of the order of 60 meV. The Doppler shift is also important in determining the resolution of the rotating-crystal time-of-¯ight (TOF) spectrom-

Fig. 4.4.2.2. Reciprocal-lattice representation of the effect of a monochromator with reciprocal-lattice vector on the reciprocalspace element of a beam with divergence . (a) For an ideal crystal with a lattice constant width ; (b) for a mosaic crystal with mosaicity , showing that a beam with small divergence, , is transformed into a broad exit beam with divergence 2 ; (c) for a gradient crystal with interplanar lattice spacing changing over , showing that the divergence is not changed in this case.

434


4.4. NEUTRON TECHNIQUES eter, ®rst conceived by Brockhouse (1958). A pulse of monochromatic neutrons is obtained when the reciprocal-lattice vector of a rotating crystal bisects the angle between two collimators. Effectively, the neutron k vector is changed in both direction and magnitude, depending on whether the crystal is moving towards or away from the neutron. For the rotating crystal, both of these situations occur simultaneously for different halves of the crystal, so that the net effect over the beam cross section is that a wider energy band is re¯ected than from the crystal at rest, and that, depending on the sense of rotation, the beam is either focused or defocused in time (Meister & Weckerman, 1972). The Bragg re¯ection of neutrons from a crystal moving parallel to its lattice planes is illustrated in Fig. 4.4.2.3(b). It can be seen that the moving crystal selects a larger k than the crystal at rest, so that the re¯ected intensity is higher. Furthermore, it is possible under certain conditions to orientate the diffracted phase-space volume orthogonal to the diffraction vector. In this way, a monochromatic divergent beam can be obtained from a collimated beam with a larger energy spread. This provides an elegant means of producing a divergent beam with a suf®ciently wide momentum spread to be scanned by the Doppler crystal of a backscattering instrument (Schelten & Alefeld, 1984). Finally, an alternative method of scanning the energy of a monochromator in backscattering is to apply a steady but uniform temperature variation. The monochromator crystal must have a reasonable thermal expansion coef®cient, and care has to be taken to ensure a uniform temperature across the crystal.

Table 4.4.2.2. Neutron scattering-length densities, Nbcoh , for some commonly used materials Material

Nb 10

58

Ni Diamond Nickel Quartz Germanium Silver Aluminium Silicon Vanadium Titanium Manganese

6

Ê 2 A

13.31 11.71 9.40 3.64 3.62 3.50 2.08 2.08 0.27 1.95 2.95

4.4.2.4. Mirror re¯ection devices The refractive index, n, for neutrons of wavelength l propagating in a nonmagnetic material of atomic density N is given by the expression n2 1

l2 Nbcoh ;

4:4:2:4

where bcoh is the mean coherent scattering length. Values of the scattering-length density Nbcoh for some common materials are listed in Table 4.4.2.2, from which it can be seen that the refractive index for most materials is slightly less than unity, so that total external re¯ection can take place. Thus, neutrons can be re¯ected from a smooth surface, but the critical angle of re¯ection, c ; given by r Nbcoh ; 4:4:2:5

c l is small, so that re¯ection can only take place at grazing Ê 1. incidence. The critical angle for nickel, for example, is 0.1 A Because of the shallowness of the critical angle, re¯ective optics are traditionally bulky, and focusing devices tend to have long focal lengths. In some cases, however, depending on the beam divergence, a long mirror can be replaced by an equivalent stack of shorter mirrors. 4.4.2.4.1. Neutron guides

Fig. 4.4.2.3. Momentum-space representation of Bragg scattering from a crystal moving (a) perpendicular and (b) parallel to the diffracting planes with a velocity vk. The vectors kL and k0L refer to the incident and re¯ected wavevectors in the laboratory frame of reference. In (a), depending on the direction of vk, the re¯ected wavevector is larger or smaller than the incident wavevector, kL. In (b), a larger incident reciprocal-space volume, vL, is selected by the moving crystal than would have been selected by the crystal at rest. The re¯ected reciprocal-space element, v0L, has a large divergence, but can be arranged to be normal to k0L, hence improving the resolution k0L.

The principle of mirror re¯ection is the basis of neutron guides, which are used to transmit neutron beams to instruments that may be situated up to 100 m away from the source (Christ & Springer, 1962; Maier-Leibnitz & Springer, 1963). A standard neutron guide is constructed from boron glass plates assembled to form a rectangular tube, the dimensions of which may be up to 200 mm high by 50 mm wide. The inner surface of the guide is Ê of either nickel, 58 Ni coated with approximately 1200 A Ê 1 ( c 0:12 A ), or a `supermirror' (described below). The guide is usually evacuated to reduce losses due to absorption and scattering of neutrons in air. Theoretically, a neutron guide that is fully illuminated by the source will transmit a beam with a square divergence of full width 2 c in both the horizontal and vertical directions, so that the transmitted solid angle is proportional to l2 . In practice, owing to imperfections in the assembly of the guide system, the divergence pro®le is closer to Gaussian than square at the end of a long guide. Since the neutrons may undergo a large number of re¯ections in the guide, it is important to achieve a high re¯ectivity. The specular re¯ectivity is determined by the surface roughness, and typically values in the range 98.5 to 99% are

435


4. PRODUCTION AND PROPERTIES OF RADIATIONS achieved. Further transmission losses occur due to imperfections in the alignment of the sections that make up the guide. The great advantage of neutron guides, in addition to the transport of neutrons to areas of low background, is that they can be multiplexed, i.e. one guide can serve many instruments. This is achieved either by de¯ecting only a part of the total cross section to a given instrument or by selecting a small wavelength range from the guide spectrum. In the latter case, the selection device (usually a crystal monochromator) must have a high transmission at other wavelengths. If the neutron guide is curved, the transmission becomes wavelength dependent, as illustrated in Fig. 4.4.2.4. In this case, one can de®ne a characteristic wavelength, l , given by the p relation 2a=, so that ss 2a 4:4:2:6 l Nbcoh (where a is the guide width and the radius of curvature), for which the theoretical transmission drops to 67%. For wavelengths less than l , neutrons can only be transmitted by `garland' re¯ections along the concave wall of the curved guide. Thus, the guide acts as a low-pass energy ®lter as longpas its length is longer than the direct line-of-sight length L1 8a. For example, a 3 cm wide nickel-coated guide whose characterÊ (radius of curvature 1300 m) must be at istic wavelength is 4 A least 18 m long to act as a ®lter. The line-of-sight length can be reduced by subdividing the guide into a number of narrower channels, each of which acts as a miniguide. The resulting device, often referred to as a neutron bender, since deviation of the beam is achieved more rapidly, is used in beam deviators (Alefeld et al., 1988) or polarizers (Hayter, Penfold & Williams, 1978). A microbender was devised by Marx (1971) in which the channels were made by evaporating alternate layers of aluminium (transmission layer) and nickel (mirror layer) onto a ¯exible smooth substrate. Tapered guides can be used to reduce the beam size in one or two dimensions (Rossbach et al., 1988), although, since mirror re¯ection obeys Liouville's theorem, focusing in real space is achieved at the expense of an increase in divergence. This fact can be used to calculate analytically the expected gain in neutron ¯ux at the end of a tapered guide (Anderson, 1988). Alternatively, focusing can be achieved in one dimension using a bender in which the individual channel lengths are adjusted to create a focus (Freund & Forsyth, 1979).

halo around the image point. Owing to its low thermal expansion coef®cient, highly polished Zerodur is often chosen as substrate. 4.4.2.4.3. Multilayers Schoenborn, Caspar & Kammerer (1974) ®rst pointed out that multibilayers, comprising alternating thin ®lms of different scattering-length densities (Nbcoh ) act like two-dimensional crystals with a d spacing given by the bilayer period. With modern deposition techniques (usually sputtering), uniform ®lms of thickness ranging from about twenty to a few hundred aÊngstroÈms can be deposited over large surface areas of the order of 1 m2 . Owing to the rather large d spacings involved, the Bragg re¯ection from multilayers is generally at grazing incidence, so that long devices are required to cover a typical beam width, or a stacked device must be used. However, with judicious choice of the scattering-length contrast, the surface and interface roughness, and the number of layers, re¯ectivities close to 100% can be reached.

4.4.2.4.2. Focusing mirrors Optical imaging of neutrons can be achieved using ellipsoidal or torroidal mirrors, but, owing to the small critical angle of re¯ection, the dimensions of the mirrors themselves and the radii of curvature must be large. For example, a 4 m long toroidal mirror has been installed at the IN15 neutron spin echo spectrometer at the Institut Laue±Langevin, Grenoble (Hayes et al., 1996), to focus neutrons with wavelengths greater than Ê The mirror has an in-plane radius of curvature of 15A. 408.75 m, and the sagittal radius is 280 mm. A coating of 65 Cu is used to obtain a high critical angle of re¯ection while maintaining a low surface roughness. Slope errors of less than 2:5 10 5 rad (r.m.s.) combined with a surface roughness of Ê allow a minimum resolvable scattering vector of less than 3A Ê 1 to be reached. about 5 10 4 A For best results, the slope errors and the surface roughness must be low, in particular in small-angle scattering applications, since diffuse scattering from surface roughness gives rise to a

Fig. 4.4.2.4. In a curved neutron guide, the transmission becomes l dependent: (a) the possible types of re¯ection (garland and zig-zag), the direct line-of-sight length, the criticalpangle , which is related to the characteristic wavelength l =Nbcoh ; (b) transmission across the exit of the guide for different wavelengths, normalized to unity at the outside edge; (c) total transmission of the guide as a function of l.

436


4.4. NEUTRON TECHNIQUES Fig. 4.4.2.5 illustrates how variation in the bilayer period can be used to produce a monochromator (the minimum l=l that can be achieved is of the order of 0.5%), a broad-band device, or a `supermirror', so called because it is composed of a particular sequence of bilayer thicknesses that in effect extends the region of total mirror re¯ection beyond the ordinary critical angle (Turchin, 1967; Mezei, 1976; Hayter & Mook, 1989). Supermirrors have been produced that extend the critical angle of nickel by a factor, m, of between three and four with re¯ectivities better than 90%. Such high re¯ectivities enable supermirror neutron guides to be constructed with ¯ux gains, compared with nickel guides, close to the theoretical value of m2 . The choice of the layer pairs depends on the application. For non-polarizing supermirrors and broad-band devices (Hghj, Anderson, Ebisawa & Takeda, 1996), the Ni/Ti pair is commonly used, either pure or with some additions to relieve strain and stabilize interfaces (Elsenhans et al., 1994) or alter the magnetism (Anderson & Hghj, 1996), owing to the high contrast in scattering density, while for narrow-band monochromators a low contrast pair such as W/Si is more suitable. 4.4.2.4.4. Capillary optics Capillary neutron optics, in which hollow glass capillaries act as waveguides, are also based on the concept of total external re¯ection of neutrons from a smooth surface. The advantage of capillaries, compared with neutron guides, is that the channel sizes are of the order of a few tens of micrometres, so that the radius of curvature can be signi®cantly decreased for a given characteristic wavelength [see equation (4.4.2.6)]. Thus, neutrons can be ef®ciently de¯ected through large angles, and the device can be more compact. Two basic types of capillary optics exist, and the choice depends on the beam characteristics required. Polycapillary ®bres are manufactured from hollow glass tubes several centimetres in diameter, which are heated, fused and drawn multiple times until bundles of thousands of micrometre-sized channels are formed having an open area of up to 70% of the cross section. Fibre outer diameters range from 300 to 600 mm and contain hundreds or thousands of individual channels with inner diameters between 3 and 50 mm. The channel cross section is usually hexagonal, though square channels have been

Fig. 4.4.2.5. Illustration of how a variation in the bilayer period can be used to produce a monochromator, a broad-band device, or a supermirror.

produced, and the inner channel wall surface roughness is Ê r.m.s., giving rise to very high typically less than 10 A re¯ectivities. The principal limitations on transmission ef®ciency are the open area, the acceptable divergence (note that the Ê 1 ) and re¯ection losses due to critical angle for glass is 1 mrad A absorption and scattering. A typical optical device will comprise hundreds or thousands of ®bres threaded through thin screens to produce the required shape. Fig. 4.4.2.6 shows typical applications of polycapillary devices. In Fig. 4.4.2.6(a), a polycapillary lens is used to refocus neutrons collected from a divergent source. The half lens depicted in Fig. 4.4.2.6(b) can be used either to produce a nearly parallel (divergence 2 c ) beam from a divergent source or (in the reverse sense) to focus a nearly parallel beam, e.g. from a neutron guide. The size of the focal point depends on the channel size, the beam divergence, and the focal length of the lens. For example, a polycapillary lens used in a prompt -activation analysis instrument at the National Institute of Standards and Technology to focus a cold neutron beam from a neutron guide results in a current density gain of 80 averaged over the focused beam size of 0.53 mm (Chen et al., 1995). Fig. 4.4.2.6(c) shows another simple application of polycapillaries as a compact beam bender. In this case, such a bender may be more compact than an equivalent multichannel guide bender, although the accepted divergence will be less. Furthermore, as with curved neutron guides, owing to the wavelength dependence of the critical angle the capillary curvature can be used to ®lter out thermal or high-energy neutrons. It should be emphasized that the applications depicted in Fig. 4.4.2.6 obey Liouville's theorem, in that the density of neutrons in phase space is not changed, but the shape of the phase-space volume is altered to meet the requirements of the experiment,

Fig. 4.4.2.6. Typical applications of polycapillary devices: (a) lens used to refocus a divergent beam; (b) half-lens to produce a nearly parallel beam or to focus a nearly parallel beam; (c) a compact bender.

437


4. PRODUCTION AND PROPERTIES OF RADIATIONS i.e. there is a simple trade off between beam dimension and divergence. The second type of capillary optic is a monolithic con®guration. The individual capillaries in monolithic optics are tapered and fused together, so that no external frame assembly is necessary (Chen-Mayer et al., 1996). Unlike the multi®bre devices, the inner diameters of the channels that make up the monolithic optics vary along the length of the component, resulting in a smaller more compact design. Further applications of capillary optics include small-angle scattering (Mildner, 1994) and lenses for high-spatial-resolution area detection. 4.4.2.5. Filters Neutron ®lters are used to remove unwanted radiation from the beam while maintaining as high a transmission as possible for the neutrons of the required energy. Two major applications can be identi®ed: removal of fast neutrons and -rays from the primary beam and reduction of higher-order contributions (l=n) in the secondary beam re¯ected from crystal monochromators. In this section, we deal with non-polarizing ®lters, i.e. those whose transmission and removal cross sections are independent of the neutron spin. Polarizing ®lters are discussed in the section concerning polarizers. Filters rely on a strong variation of the neutron cross section with energy, usually either the wavelength-dependent scattering cross section of polycrystals or a resonant absorption cross section. Following Freund (1983), the total cross section determining the attenuation of neutrons by a crystalline solid can be written as a sum of three terms, abs tds Bragg :

Pyrolytic graphite, being a layered material with good crystalline properties along the c direction but random orientation perpendicular to it, lies somewhere between a polycrystal and a single crystal as far as its attenuation cross section is concerned. The energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite ®lter is shown in Fig. 4.4.2.8, where the attenuation peaks due to the 00 re¯ections can be seen. Pyrolytic graphite serves as an ef®cient second- or third-order ®lter (Shapiro & Chesser, 1972) and can be `tuned' by slight misorientation away from the c axis. Further examples of typical ®lter materials (e.g. silicon, lead, bismuth, sapphire) can be found in the paper by Freund (1983). Resonant absorption ®lters show a large increase in their attenuation cross sections at the resonant energy and are therefore used as selective ®lters for that energy. A list of typical ®lter materials and their resonance energies is given in Table 4.4.2.3. 4.4.2.6. Polarizers Methods used to polarize a neutron beam are many and varied, and the choice of the best technique depends on the instrument and the experiment to be performed. The main parameter that has to be considered when describing the effectiveness of a given polarizer is the polarizing ef®ciency, de®ned as P N

4.4.2.6.1. Single-crystal polarizers The principle by which ferromagnetic single crystals are used to polarize and monochromate a neutron beam simultaneously is shown in Fig. 4.4.2.9. A ®eld B, applied perpendicular to the scattering vector j, saturates the atomic moments M along the ®eld direction. The cross section for Bragg re¯ection in this geometry is d= d FN j2 2FN jFM jP l FM j2 ; 4:4:2:9

Fig. 4.4.2.7. Total cross section for beryllium in the energy range where it can be used as a ®lter for neutrons with energy below 5 meV (Freund, 1983).

438


4:4:2:8

where N and N are the numbers of neutrons with spin parallel () or antiparallel ( ) to the guide ®eld in the outgoing beam. The second important factor, the transmission of the wanted spin state, depends on various factors, such as acceptance angles, re¯ection, and absorption.

4:4:2:7

Here, abs is the true absorption cross section, which, at low energy, away from resonances, is proportional to E 1=2 . The temperature-dependent thermal diffuse cross section, tds , describing the attenuation due to inelastic processes, can be split into two parts depending on the neutron energy. At low energy, E kb D , where kb is Boltzmann's constant and D is the characteristic Debye temperature, single-phonon processes dominate, giving rise to a cross section, sph , which is also proportional to E 1=2 . The single-phonon cross section is proportional to T 7=2 at low temperatures and to T at higher temperatures. At higher energies, E kb D , multiphonon and multiple-scattering processes come into play, leading to a cross section, mph , that increases with energy and temperature. The third contribution, Bragg , arises due to Bragg scattering in single- or polycrystalline material. At low energies, below the Bragg cut-off (l > 2d max ), Bragg is zero. In polycrystalline materials, the cross section rises steeply above the Bragg cutoff and oscillates with increasing energy as more re¯ections come into play. At still higher energies, Bragg decreases to zero. In single-crystalline material above the Bragg cut-off, Bragg is characterized by a discrete spectrum of peaks whose heights and widths depend on the beam collimation, energy resolution, and the perfection and orientation of the crystal. Hence a monocrystalline ®lter has to be tuned by careful orientation. The resulting attenuation cross section for beryllium is shown in Fig. 4.4.2.7. Cooled polycrystalline beryllium is frequently used as a ®lter for neutrons with energies less than 5 meV, since there is an increase of nearly two orders of magnitude in the attenuation cross section for higher energies. BeO, with a Bragg cut-off at approximately 4 meV, is also commonly used.

N =N N ;

4.4. NEUTRON TECHNIQUES Table 4.4.2.3. Characteristics of some typical elements and isotopes used as neutron ®lters

1 barn = 10

28

Element or isotope

Resonance (eV)

s (resonance) (barns)

l Ê A

s l (barns)

In Rh Hf 240 Pu Ir 229 Th Er Er Eu 231 Pa 239 Pu

1.45 1.27 1.10 1.06 0.66 0.61 0.58 0.46 0.46 0.39 0.29

30000 4500 5000 115000 4950 6200 1500 2300 10100 4900 5200

0.48 0.51 0.55 9.55 0.70 0.73 0.75 0.84 0.84 0.92 1.06

94 76 58 145 183 62.0 11.8 18.4 9.6 42.2 7.4

m2 .

where FN j P is the nuclear structure factor and FM j =2r0 M f hkl exp2hx ky lz is the magnetic structure factor, with f hkl the magnetic form factor of the magnetic atom at the position x; y; z in the unit cell. The vector P describes the polarization of the incoming neutron with respect to B; P 1 for spins and P 1 for spins and l is a unit vector in the direction of the atomic magnetic moments. Hence, for neutrons polarized parallel to B (P l 1), the diffracted intensity is proportional to FN j FM j2 , while, for neutrons polarized antiparallel to B (P l 1), the diffracted intensity is proportional to FN j FM j2 . The polarizing ef®ciency of the diffracted beam is then P 2FN jFM j=FN j2 FM j2 ;

4:4:2:10

which can be either positive or negative and has a maximum value for jFN jj jFM jj. Thus, a good single-crystal polarizer, in addition to possessing a crystallographic structure in which FN and FM are matched, must be ferromagnetic at room temperature and should contain atoms with large magnetic moments. Furthermore, large single crystals with `controllable' mosaic should be available. Finally, the structure

Fig. 4.4.2.8. Energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite ®lter. The attenuation peaks due to the 00 re¯ections can be seen.

factor for the required re¯ection should be high, while those for higher-order re¯ections should be low. None of the three naturally occurring ferromagnetic elements (iron, cobalt, nickel) makes ef®cient single-crystal polarizers. Cobalt is strongly absorbing and the nuclear scattering lengths of iron and nickel are too large to be balanced by their weak magnetic moments. An exception is 57 Fe, which has a rather low nuclear scattering length, and structure-factor matching can be achieved by mixing 57 Fe with Fe and 3% Si (Reed, Bolling & Harmon, 1973). In general, in order to facilitate structure-factor matching, alloys rather than elements are used. The characteristics of some alloys used as polarizing monochromators are presented in Table 4.4.2.4. At short wavelengths, the 200 re¯ection of Co0:92 Fe0:08 is used to give a positively polarized beam [FN j and FM j both positive], but the absorption due to cobalt is high. At longer wavelengths, the 111 re¯ection of the Heusler alloy Cu2 MnAl (Delapalme, Schweizer, Couderchon & Perrier de la Bathie, 1971; Freund, Pynn, Stirling & Zeyen, 1983) is commonly used, since it has a higher re¯ectivity and a larger d spacing than Co0:92 Fe0:08 . Since for the 111 re¯ection FN FM , the diffracted beam is negatively polarized. Unfortunately, the structure factor of the 222 re¯ection is higher than that of the 111 re¯ection, leading to signi®cant higher-order contamination of the beam. Other alloys that have been proposed as neutron polarizers are Fe3 x Mnx Si, 7 Li0:5 Fe2:5 O4 (Bednarski, Dobrzynski & Steinsvoll, 1980), Fe3 Si (Hines et al., 1976), Fe3 Al (Pickart & Nathans, 1961), and HoFe2 (Freund & Forsyth, 1979).

Fig. 4.4.2.9. Geometry of a polarizing monochromator showing the lattice planes (hkl) with jFN j jFM j, the direction of P and l, the expected spin direction and intensity.

439


4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.4.2.4. Properties of polarizing crystal monochromators (Williams, 1988)

Matched re¯ection jFN j jFM j Ê d spacing A Ê Take-off angle 2B at 1A Ê Cut-off wavelength, lmax A

Co0:92 Fe0:08

Cu2 MnAl

Fe3 Si

200 1.76 33.1 3.5

111 3.43 16.7 6.9

111 3.27 17.6 6.5

4.4.2.6.2. Polarizing mirrors For a ferromagnetic material, the neutron refractive index is given by n2 1

l2 Nbcoh p=;

4:4:2:11

where the magnetic scattering length, p, is de®ned by p 2B

Hm=h2 N:

4:4:2:12

Here, m and are the neutron mass and magnetic moment, B is the magnetic induction in an applied ®eld H, and h is Planck's constant. The and signs refer, respectively, to neutrons whose moments are aligned parallel and antiparallel to B. The refractive index depends on the orientation of the neutron spin with respect to the ®lm magnetization, thus giving rise to two critical angles of total re¯ection, and . Thus, re¯ection in an angular range between these two critical angles gives rise to polarized beams in re¯ection and in transmission. The polarization ef®ciency, P, is de®ned in terms of the re¯ectivity r and r of the two spin states, P r

r =r r :

4:4:2:13

The ®rst polarizers using this principle were simple cobalt mirrors (Hughes & Burgy, 1950), while Schaerpf (1975) used FeCo sheets to build a polarizing guide. It is more common these days to use thin ®lms of ferromagnetic material deposited onto a substrate of low surface roughness (e.g. ¯oat glass or polished silicon). In this case, the re¯ection from the substrate can be eliminated by including an antire¯ecting layer made from, for example, Gd±Ti alloys (Drabkin et al., 1976). The major limitation of these polarizers is that grazing-incidence angles must be used and the angular range of polarization is small. This limitation can be partially overcome by using multilayers, as described above, in which one of the layer materials is ferromagnetic. In this case, the refractive index of the ferromagnetic material is matched for one spin state to that of the non-magnetic material, so that re¯ection does not occur. A polarizing supermirror made in this way has an extended angular range of polarization, as indicated in Fig. 4.4.2.10. It should be noted that modern deposition techniques allow the refractive index to be adjusted readily, so that matching is easily achieved. The scattering-length densities of some commonly used layer pairs are given in Table 4.4.2.5 Polarizing multilayers are also used in monochromators and broad-band devices. Depending on the application, various layer pairs have been used: Co/Ti, Fe/Ag, Fe/Si, Fe/Ge, Fe/W, FeCoV/TiN, FeCoV/TiZr, 63 Ni0:66 54 Fe0:34 /V and the range of ®elds used to achieve saturation varies from about 100 to 500 Gs. Polarizing mirrors can be used in re¯ection or transmission with polarization ef®ciencies reaching 97%, although, owing to the low incidence angles, their use is generally restricted to Ê wavelengths above 2 A. Various devices have been constructed that use mirror polarizers, including simple re¯ecting mirrors, V -shaped

Fe:Fe

HoFe2

110 2.03 28.6 4.1

620 1.16 50.9 2.3

transmission polarizers (Majkrzak, Nunez, Copley, Ankner & Greene, 1992), cavity polarizers (Mezei, 1988), and benders (Hayter, Penfold & Williams, 1978; Schaerpf, 1989). Perhaps the best known device is the polarizing bender developed by SchaÈrpf. The device consists of 0.2 mm thick glass blades coated on both sides with a Co/Ti supermirror on top of an antire¯ecting Gd/Ti coating designed to reduce the scattering of the unwanted spin state from the substrate to a very low Q value. The device is quite compact (typically 30 cm long for a beam cross section up to 6 5 cm) and transmits over 40% of an unpolarized beam with the collimation from a nickel-coated guide for wavelengths Ê . Polarization ef®ciencies of over 96% can be above 4.5 A achieved with these benders. 4.4.2.6.3. Polarizing ®lters Polarizing ®lters operate by selectively removing one of the neutron spin states from an incident beam, allowing the other spin state to be transmitted with only moderate attenuation. The spin selection is obtained by preferential absorption or scattering, so the polarizing ef®ciency usually increases with the thickness of the ®lter, whereas the transmission decreases. A compromise must therefore be made between polarization,pP, and transmission, T . The `quality factor' often used is P T (Tasset & Resouche, 1995). The total cross sections for a generalized ®lter may be written as 0 p ;

4:4:2:14

where 0 is a spin-independent cross section and p =2 is the polarization cross section. It can be

Fig. 4.4.2.10. Measured re¯ectivity curve of an FeCoV/TiZr polarizing supermirror with an extended angular range of polarization of three times that of c (Ni) for neutrons without spin ¯ip, "", and with spin ¯ip, "#.

440


57

4.4. NEUTRON TECHNIQUES Table 4.4.2.5. Scattering-length densities for some typical materials used for polarizing multilayers Nb p Ê 2 10 6 A

Nb p Ê 2 10 6 A

Fe

13.04

3.08

Fe:Co (50:50)

10.98

0.52

Ni Fe:Co:V (49:49:2)

10.86 10.75

7.94 0.63

Fe:Co:V (50:48:2)

10.66

0.64

Fe:Ni (50:50) Co Fe:Co:V (52:38:10)

10.53 6.65 6.27

6.65 2.00 2.12

Magnetic layer

10

Nb 6 Ê 2 A

Nonmagnetic layer

3.64 3.50 3.02 2.08 2.08 0.27 1.95

Ge Ag W Si Al V Ti

0.27 1.95 0.27 1.95

V Ti V Ti

1.95 2.08 2.08

Ti Si Al

For the non-magnetic layer we have only listed the simple elements that give a close match to the Nb p value of the corresponding magnetic Ê 2 to be layer. In practice excellent matching can be achieved by using alloys (e.g. Tix Zry alloys allow Nb values between 1.95 and 3:03 10 6A selected) or reactive sputtering e:g:TiNx :

shown (Williams, 1988) that the ratio p =0 must be 0:65 to achieve jPj > 0:95 and T > 0:2: Magnetized iron was the ®rst polarizing ®lter to be used (Alvarez & Bloch, 1940). The method relies on the spindependent Bragg scattering from a magnetized polycrystalline block, for which p approaches 10 barns near the Fe cut-off at Ê (Steinberger & Wick, 1949). Thus, for wavelengths in the 4A Ê the ratio p =0 ' 0:59; resulting in a range 3.6 to 4 A, theoretical polarizing ef®ciency of 0.8 for a transmittance of 0:3. In practice, however, since iron cannot be fully saturated, depolarization occurs, and values of P ' 0:5 with T 0:25 are more typical. Resonance absorption polarization ®lters rely on the spin dependence of the absorption cross section of polarized nuclei at their nuclear resonance energy and can produce ef®cient polarization over a wide energy range. The nuclear polarization is normally achieved by cooling in a magnetic ®eld, and ®lters based on 149 Sm (Er 0:097 eV) (Freeman & Williams, 1978) and 151 Eu (Er 0:32 and 0:46 eV) have been successfully tested. The 149 Sm ®lter has a polarizing ef®ciency close to 1 within a Ê while the transmittance small wavelength range (0.85 to 1.1 A), is about 0.15. Furthermore, since the ®lter must be operated at temperatures of the order of 15 mK, it is very sensitive to heating by -rays. Broad-band polarizing ®lters, based on spin-dependent scattering or absorption, provide an interesting alternative to polarizing mirrors or monochromators, owing to the wider range of energy and scattering angle that can be accepted. The most promising such ®lter is polarized 3 He, which operates through the huge spin-dependent neutron capture cross section that is totally dominated by the resonance capture of neutrons with antiparallel spin. The polarization ef®ciency of an 3 He neutron spin ®lter of length l can be written as Pn l tanhOlPHe ;

4:4:2:15 3

where PHe is the 3 He polarization, and Ol Hel0 l is the dimensionless effective absorption coef®cient, also called the opacity (Surkau et al., 1997). For gaseous 3 He, the opacity can be written in more convenient units as

Ê O0 pbar l cm lA;

where p is the 3 He pressure (1 bar 105 Pa) and O 7:33 10 2 O0 . Similarly, the residual transmission of the spin ®lter is given by Tn l exp Ol coshOlPHe :

4:4:2:17

It can be seen that, even at low 3 He polarization, full neutron polarization can be achieved in the limit of large absorption at the cost of the transmission. 3 He can be polarized either by spin exchange with optically pumped rubidium (Bouchiat, Carver & Varnum, 1960; Chupp, Coulter, Hwang, Smith & Welsh, 1996; Wagshul & Chupp, 1994) or by pumping of metastable 3 He atoms followed by metastable exchange collisions (Colegrove, Schearer & Walters, 1963). In the former method, the 3 He gas is polarized at the required high pressure, whereas 3 He pumping takes place at a pressure of about 1 mbar, followed by a polarization conserving compression by a factor of nearly 10 000. Although the polarization time constant for Rb pumping is of the order of several hours compared with fractions of a second for 3 He pumping, the latter requires several `®lls' of the ®lter cell to achieve the required pressure. An alternative broad-band spin ®lter is the polarized proton ®lter, which utilizes the spin dependence of nuclear scattering. The spin-dependent cross section can be written as (Lushchikov, Taran & Shapiro, 1969) 2 1 2 PH 3 PH ;

4:4:2:18

where 1 , 2 ; and 3 are empirical constants. The viability of the method relies on achieving a high nuclear polarization PH . A polarization PH 0:7 gives p =0 0:56 in the coldneutron region. Proton polarizations of the order of 0.8 are required for a useful ®lter (Schaerpf & Stuesser, 1989). Polarized proton ®lters can polarize very high energy neutrons even in the eV range.

441


4:4:2:16

4. PRODUCTION AND PROPERTIES OF RADIATIONS 4.4.2.7.2. Rotation of the polarization direction

4.4.2.6.4. Zeeman polarizer The re¯ection width of perfect silicon crystals for thermal neutrons and the Zeeman splitting (E 2B) of a ®eld of about 10 kGs are comparable and therefore can be used to polarize a neutron beam. For a monochromatic beam (energy E0 ) in a strong magnetic ®eld region, the result of the Zeeman splitting will be a separation into two polarized subbeams, one polarized along B with energy E0 B, and the other polarized antiparallel to B with energy E0 B. The two polarized beams can be selected by rocking a perfect crystal in the ®eld region B (Forte & Zeyen, 1989). 4.4.2.7. Spin-orientation devices Polarization is the state of spin orientation of an assembly of particles in a target or beam. The beam polarization vector P is de®ned as the vector average of this spin state and is often described by the density matrix 12 1 P. The polarization is then de®ned as P Tr. If the polarization vector is inclined to the ®eld direction in a homogenous magnetic ®eld, B, the polarization vector will precess with the classical Larmor frequency !L j jB. This results in a precessing spin polarization. For most experiments, it is suf®cient to consider the linear polarization vector in the direction of an applied magnetic ®eld. If, however, the magnetic ®eld direction changes along the path of the neutron, it is also possible that the direction of P will change. If the frequency, , with which the magnetic ®eld changes is such that

dB=jBj= dt !L ;

4:4:2:19

then the polarization vector follows the ®eld rotation adiabatically. Alternatively, when !L , the magnetic ®eld changes so rapidly that P cannot follow, and the condition is known as non-adiabatic fast passage. All spin-orientation devices are based on these concepts. 4.4.2.7.1. Maintaining the direction of polarization A polarized beam will tend to become depolarized during passage through a region of zero ®eld, since the ®eld direction is ill de®ned over the beam cross section. Thus, in order to keep the polarization direction aligned along a de®ned quantization axis, special precautions must be taken. The simplest way of maintaining the polarization of neutrons is to use a guide ®eld to produce a well de®ned ®eld B over the whole ¯ight path of the beam. If the ®eld changes direction, it has to ful®l the adiabatic condition !L , i.e. the ®eld changes must take place over a time interval that is long compared with the Larmor period. In this case, the polarization follows the ®eld direction adiabatically with an angle of deviation 2 arctan =!L (SchaÈrpf, 1980). Alternatively, some instruments (e.g. zero-®eld spin-echo spectrometers and polarimeters) use polarized neutron beams in regions of zero ®eld. The spin orientation remains constant in a zero-®eld region, but the passage of the neutron beam into and out of the zero-®eld region must be well controlled. In order to provide a well de®ned region of transition from a guide-®eld region to a zero-®eld region, a non-adiabatic fast passage through the windings of a rectangular input solenoid can be used, either with a toroidal closure of the outside ®eld or with a -metal closure frame. The latter serves as a mirror for the coil ends, with the effect of producing the ®eld homogeneity of a long coil but avoiding the ®eld divergence at the end of the coil.

The polarization direction can be changed by the adiabatic change of the guide-®eld direction so that the direction of the polarization follows it. Such a rotation is performed by a spin turner or spin rotator (SchaÈrpf & Capellmann, 1993; Williams, 1988). Alternatively, the direction of polarization can be rotated relative to the guide ®eld by using the property of precession described above. If a polarized beam enters a region where the ®eld is inclined to the polarization axis, then the polarization vector P will precess about the new ®eld direction. The precession angle will depend on the magnitude of the ®eld and the time spent in the ®eld region. By adjustment of these two parameters together with the ®eld direction, a de®ned, though wavelength-dependent, rotation of P can be achieved. A simple device uses the non-adiabatic fast passage through the windings of two rectangular solenoids, wound orthogonally one on top of the other. In this way, the direction of the precession ®eld axis is determined by the ratio of the currents in the two coils, and the sizes of the ®elds determine the angle ' of the precession. The orientation of the polarization vector can therefore be de®ned in any direction. In order to produce a continuous rotation of the polarization, i.e. a well de®ned precession, as required in neutron spin-echo (NSE) applications, precession coils are used. In the simplest case, these are long solenoids where the change of the ®eld integral over the cross section can be corrected by Fresnel coils (Mezei, 1972). More recently, Zeyen & Rem (1996) have developed and implemented optimal ®eld-shape (OFS) coils. The ®eld in these coils follows a cosine squared shape that results from the optimization of the line integral homogeneity. The OFS coils can be wound over a very small diameter, thereby reducing stray ®elds drastically. 4.4.2.7.3. Flipping of the polarization direction The term `¯ipping' was originally applied to the situation where the beam polarization direction is reversed with respect to a guide ®eld, i.e. it describes a transition of the polarization direction from parallel to antiparallel to the guide ®eld and vice versa. A device that produces this 180 rotation is called a ¯ipper. A =2 ¯ipper, as the name suggests, produces a 90 rotation and is normally used to initiate precession by turning the polarization at 90 to the guide ®eld. The most direct wavelength-independent way of producing such a transition is again a non-adiabatic fast passage from the region of one ®eld direction to the region of the other ®eld direction. This can be realized by a current sheet like the Dabbs foil (Dabbs, Roberts & Bernstein, 1955), a Kjeller eight (Abrahams, Steinsvoll, Bongaarts & De Lange, 1962) or a cryo¯ipper (Forsyth, 1979). Alternatively, a spin ¯ip can be produced using a precession coil, as described above, in which the polarization direction makes a precession of just about a direction orthogonal to the guide ®eld direction (Mezei, 1972). Normally, two orthogonally wound coils are used, where the second, correction, coil serves to compensate the guide ®eld in the interior of the precession coil. Such a ¯ipper is wavelength dependent and can be easily tuned by varying the currents in the coils. Another group of ¯ippers uses the non-adiabatic transition through a well de®ned region of zero ®eld. Examples of this type of ¯ipper are the two-coil ¯ipper of Drabkin, Zabidarov, Kasman & Okorokov (1969) and the line-shape ¯ipper of Korneev & Kudriashov (1981).

442


4.4. NEUTRON TECHNIQUES Historically, the ®rst ¯ippers used were radio-frequency coils set in a homogeneous magnetic ®eld. These devices are wavelength dependent, but may be rendered wavelength independent by replacing the homogeneous magnetic ®eld with a gradient ®eld (Egorov, Lobashov, Nazarento, Porsev & Serebrov, 1974). In some devices, the ¯ipping action can be combined with another selection function. The wavelength-dependent magnetic wiggler ¯ipper proposed by Agamalyan, Drabkin & Sbitnev (1988) in combination with a polarizer can be used as a polarizing monochromator (Majkrzak & Shirane, 1982). Badurek & Rauch (1978) have used ¯ippers as choppers to pulse a polarized beam. In neutron resonance spin echo (NRSE) (GaÈhler & Golub, 1987), the precession coil of the conventional spin-echo con®guration is replaced by two resonance spin ¯ippers separated by a large zero-®eld region. The radio-frequency ®eld of amplitude B1 is arranged orthogonal to the DC ®eld, B0 , with a frequency ! !L , and an amplitude de®ned by the relation !1 , where is the ¯ight time in the ¯ipper coil and !1 B1 . In this con®guration, the neutron spin precesses through an angle about the resonance ®eld in each coil and leaves the coil with a phase angle '. The total phase angle after passing through both coils, ' 2!L=v, depends on the velocity v of the neutron and the separation L between the two coils. Thus, compared with conventional NSE, where the phase angle comes from the precession of the neutron spin in a strong magnetic ®eld compared with a static ¯ipper ®eld, in NRSE the neutron spin does not precess, but the ¯ipper ®eld rotates. Effectively, the NRSE phase angle ' is a factor of two larger than the NSE phase angle for the same DC ®eld B0 . Furthermore, the resolution is determined by the precision of the RF frequencies and the zero-®eld ¯ight path L rather than the homogeneity of the line integral of the ®eld in the NSE precession coil.

single slit with a series of slits either in a regular sequence (Fourier chopper) (Colwell, Miller & Whittemore, 1968; HiismaÈki, 1997) or a pseudostatistical sequence (pseudostatistical chopper) (Hossfeld, Amadori & Scherm, 1970), with duty cycles of 50 and 30%, respectively. The Fermi chopper is an alternative form of neutron chopper that simultaneously pulses and monochromates the incoming beam. It consists of a slit package, essentially a collimator, rotating about an axis that is perpendicular to the beam direction (Turchin, 1965). For optimum transmission at the required wavelength, the slits are usually curved to provide a straight collimator in the neutron frame of reference. The curvature also eliminates the `reverse burst', i.e. a pulse of neutrons that passes when the chopper has rotated by 180 . A Fermi chopper with straight slits in combination with a monochromator assembly of wide horizontal divergence can be used to time focus a polychromatic beam, thus maintaining the energy resolution while improving the intensity (Blanc, 1983). Velocity selectors are used when a continuous beam is required with coarse energy resolution. They exist in either multiple disc con®gurations or helical channels rotating about an axis parallel to the beam direction (Dash & Sommers, 1953). Modern helical channel selectors are made up of lightweight absorbing blades slotted into helical grooves on the rotation axis (Wagner, Friedrich & Wille, 1992). At higher energies where no suitable absorbing material is available, highly scattering polymers [poly(methyl methacrylate)] can be used for the blades, although in this case adequate shielding must be provided. The neutron wavelength is determined by the rotation speed, and resolutions, l=l, ranging from 5% to practically 100% (l=2 ®lter) can be achieved. The resolution is ®xed by the geometry of the device, but can be slightly improved by tilting the rotation axis or relaxed by rotating in the reverse direction for shorter wavelengths. Transmissions of up to 94% are typical.

4.4.2.8. Mechanical choppers and selectors Ê neutron Thermal neutrons have relatively low velocities (a 4 A has a reciprocal velocity of approximately 1000 ms m 1 ), so that mechanical selection devices and simple ¯ight-time measurements can be used to make accurate neutron energy determinations. Disc choppers rotating at speeds up to 20 000 revolutions per minute about an axis that is parallel to the neutron beam are used to produce a well de®ned pulse of neutrons. The discs are made from absorbing material (at least where the beam passes) and comprise one or more neutron-transparent apertures or slits. For polarized neutrons, these transparent slits should not be metallic, as the eddy currents in the metal moving in even a weak guide ®eld will strongly depolarize the beam. The pulse frequency is determined by the number of apertures and the rotation frequency, while the duty cycle is given by the ratio of open time to closed time in one rotation. Two such choppers rotating in phase can be used to monochromate and pulse a beam simultaneously (Egelstaff, Cocking & Alexander, 1961). In practice, more than two choppers are generally used to avoid frame overlap of the incident and scattered beams. The time resolution of disc choppers (and hence the energy resolution of the instrument) is determined by the beam size, the aperture size and the rotation speed. For a realistic beam size, the rotation speed limits the resolution. Therefore, in modern instruments, it is normal to replace a single chopper with two counter-rotating choppers (Hautecler et al., 1985; Copley, 1991). The low duty cycle of a simple disc chopper can be improved by replacing the

4.4.3. Resolution functions (By R. Pynn and J. M. Rowe) In a Gedanken neutron scattering experiment, neutrons of wavevector kI impinge on a sample and the wavevector, kF , of the scattered neutrons is determined. A number of different types of spectrometer are used to achieve this goal (cf. Pynn, 1984). In each case, ®nite instrumental resolution is a result of uncertainties in the de®nition of kI and kF . Propagation directions for neutrons are generally de®ned by Soller collimators for which the transmission as a function of divergence angle generally has a triangular shape. Neutron monochromatization may be achieved either by Bragg re¯ection from a (usually) mosaic crystal or by a time-of-¯ight method. In the former case, the mosaic leads to a spread of jkI j while, in the latter, pulse length and uncertainty in the lengths of ¯ight paths (including sample size and detector thickness) produce a similar effect. Calculations of instrumental resolution are generally lengthy and lack of space prohibits their detailed presentation here. In the following paragraphs, the concepts involved are indicated and references to original articles are provided. In resolution calculations for neutron spectrometers, it is usually assumed that the uncertainty of the neutron wavevector does not vary spatially across the neutron beam, although this reasoning may not apply to the case of small samples and compact spectrometers. To calculate the resolution of the spectrometer in the large-beam approximation, one writes the measured intensity I as

443


I/

R

3

d ki

R

4. PRODUCTION AND PROPERTIES OF RADIATIONS 3

d kf Pi ki Ski ! kf Pf kf ;

where Pi ki is the probability that a neutron of wavevector ki is incident on the sample, Pf kf is the probability that a neutron of wavevector kf is transmitted by the analyser system and Ski ! kf is the probability that the sample scatters a neutron from ki to kf . The ¯uctuation spectrum of the sample, Ski ! kf , does not depend separately on ki and kf but rather on the scattering vector Q and energy transfer h! de®ned by the conservation equations Q ki

kf ;

h!

h2 2 k 2m i

kf2 ;

M A

4:4:3:1

4:4:3:2

where m is the neutron mass. A number of methods of calculating the distribution functions Pi ki and Pf kf have been proposed. The method of independent distributions was used implicitly by Stedman (1968) and in more detail by Bjerrum Mùller & Nielson (MN) (Nielsen & Bjerrum Mùller, 1969; Bjerrum Mùller & Nielsen, 1970) for three-axis spectrometers. Subsequently, the method has been extended to perfect-crystal monochromators (Pynn, Fujii & Shirane, 1983) and to time-of-¯ight spectrometers (Steinsvoll, 1973; Robinson, Pynn & Eckert, 1985). The method involves separating Pi and Pf into a product of independent distribution functions each of which can be convolved separately with the ¯uctuation spectrum SQ; ! [cf. equation (4.4.3.1)]. Extremely simple results are obtained for the widths of scans through a phonon dispersion surface for spectrometers where the energy of scattered neutrons is analysed (Nielson & Bjerrum Mùller, 1969). For diffractometers, the width of a scan through a Bragg peak may also be obtained (Pynn et al., 1983), yielding a result equivalent to that given by Caglioti, Paoletti & Ricci (1960). In this case, however, the singular nature of the Bragg scattering process introduces a correlation between the distribution functions that contribute to Pi and Pf and the calculation is less transparent than it is for phonons. A somewhat different approach, which does not explicitly separate the various contributions to the resolution, was proposed by Cooper & Nathans (CN) (Cooper & Nathans, 1967, 1968; Cooper, 1968). Minor errors were corrected by several authors (Werner & Pynn, 1971; Chesser & Axe, 1973). The CN method calculates the instrumental resolution function RQ Q0 ; ! !0 as P M X X ; 4:4:3:3 RQ; ! R0 exp 12 ;

where X1 , X2 ; and X3 are the three components of Q, X4 !, and Q0 and !0 are obtained from (4.4.3.2) by replacing ki and kf by kI and kF , respectively. The matrix M is given in explicit form by several authors (Cooper & Nathans, 1967, 1968; Cooper, 1968; Werner & Pynn, 1971; Chesser & Axe, 1973) and the normalization R0 has been discussed in detail by Dorner (1972). [A refutation (Tindle, 1984) of Dorner's work is incorrect.] Equation (4.4.3.3) implies that contours of constant transmission for the spectrometer RQ; ! constant are ellipsoids in the four-dimensional Q±! space. Optimum resolution (focusing) is achieved by a scan that causes the resolution function to intersect the feature of interest in SQ; ! (e.g. Bragg peak or phonon dispersion surface) for the minimum scan interval. The optimization of scans for a diffractometer has been considered by Werner (1971). The MN and CN methods are equivalent. Using the MN formalism, it can be shown that

with

A

P j

j j ;

4:4:3:4

where the j are the components of the standard deviations of independent distributions (labelled by index j) de®ned by Bjerrum Mùller & Nielsen (1970). In the limit Q ! 0, the matrices M and A are of rank three and other methods must be used to calculate the resolution ellipsoid (Mitchell, Cowley & Higgins, 1984). Nevertheless, the MN method may be used even in this case to calculate widths of scans. To obtain the resolution function of a diffractometer (in which there is no analysis of scattered neutron energy) from the CN form for M, it is suf®cient to set to zero those contributions that arise from the mosaic of the analyser crystal. For elastic Bragg scattering, the problem is further simpli®ed because X4 [cf. equation (4.4.3.3)] is zero. The spectrometer resolution function is then an ellipsoid in Q space. For the measurement of integrated intensities (of Bragg peaks for example), the normalization R0 in (4.4.3.3) is required in order to obtain the Lorentz factor. The latter has been calculated for an arbitrary scan of a three-axis spectrometer (Pynn, 1975) and the results may be modi®ed for a diffractometer as described in the preceding paragraph.

4.4.4. Scattering lengths for neutrons (By V. F. Sears) The use of neutron diffraction for crystal-structure determinations requires a knowledge of the scattering lengths and the corresponding scattering and absorption cross sections of the elements and, in some cases, of individual isotopes. This information is needed to calculate unit-cell structure factors and to correct for effects such as absorption, self-shielding, extinction, thermal diffuse scattering, and detector ef®ciency (Bacon, 1975; Sears, 1989). Table 4.4.4.1 lists the best values of the neutron scattering lengths and cross sections that are available at the time of writing (January 1995). We begin by summarizing the basic relationships between the scattering lengths and cross sections of the elements and their isotopes that have been used in the compilation of this table. More background information can be found in, for example, the book by Sears (1989). 4.4.4.1. Scattering lengths The scattering of a neutron by a single bound nucleus is described within the Born approximation by the Fermi pseudopotential, 2h2 V r br; 4:4:4:1 m in which r is the position of the neutron relative to the nucleus, m the neutrons mass, and b the bound scattering length. The neutron has spin s and the nucleus spin I so that, if I 6 0, the Fermi pseudopotential and, hence, the bound scattering length will be spin dependent. Since s 1=2, the most general rotationally invariant expression for b is 2bi s I; b bc p II 1

4:4:4:2

in which the coef®cients bc and bi are called the bound coherent and incoherent scattering lengths. If I 0, then bi 0 by convention.

444


1

4.4. NEUTRON TECHNIQUES Table 4.4.4.1. Bound scattering lengths, b, in fm and cross sections, , in barns (1 barn = 100 fm2 ) of the elements and their isotopes Z: atomic number; A: mass number; I: spin (parity) of the nuclear ground state; c: % natural abundance (for radioisotopes, the half-life is given instead in annums); bc : bound coherent scattering length; bi : bound incoherent scattering length; c : bound coherent scattering cross section; i : 1 bound incoherent scattering cross section; s : total p bound scattering cross section; a : absorption cross section for 2200 m s neutrons (E = 1 Ê Ê 1. 25.30 meV, k = 3.494 A , l = 1.798 A); i = Element Z H

1

He

2

Li

3

Be

4

B

5

C

6

N

7

O

8

F

9

Ne

10

Na

11

Mg

12

Al

13

Si

14

P

15

S

16

A

I

c

1 2 3

1/2(+) 1(+) 1/2(+)

3

1/2(+)

0.00014

4

0(+)

99.99986

6

1(+)

7.5

7

3/2( )

92.5

9

3/2( )

99.985 0.015 (12.32a)

100

bc 3.7390(11) 3.7406(11) 6.671(4) 4.792(27) 3.26(3) 5.74(7) 1.483(2)i 3.26(3)

bi 25.274(9) 4.04(3) 1.04(17) 2.5(6) +2.568(3)i 0

0

1.34(2)

0

0.0454(3)

7.79(1)

0.12(3)

7.63(2)

0.0018(9)

7.63(2)

0.0076(8)

3.54(5)

1.70(12)

5.24(11)

0(+) 1/2( )

98.90 1.10

6.6460(12) 6.6511(16) 6.19(9)

0 0.52(9)

14 15

1(+) 1/2( )

99.63 0.37

9.36(2) 9.37(2) 6.44(3)

2.0(2) 0.02(2)

16 17 18

0(+) 5/2(+) 0(+)

99.762 0.038 0.200

5.803(4) 5.803(4) 5.78(12) 5.84(7)

0 0.18(6) 0

19

1/2(+)

5.654(10)

0.082(9)

20 21 22

0(+) 3/2(+) 0(+)

23

3/2(+)

24 25 26

0(+) 5/2(+) 0(+)

27

5/2(+)

28 29 30

0(+) 1/2(+) 0(+)

31

1/2(+)

32 33 34 36

0(+) 3/2(+) 0(+) 0(+)

4.566(6) 4.631(6) 6.66(19) 3.87(1)

70.5(3) 940.(4.)

767.(8.)

4.7(3) 1.231(3)i 1.3(2)

0.144(8)

3.0(4)

3.1(4)

5.56(7)

0.22(6)

5.78(9)

0.0055(33)

5.550(2) 5.559(3) 4.81(14)

0.001(4) 0 0.034(12)

5.551(3) 5.559(3) 4.84(14)

0.00350(7) 0.00353(7) 0.00137(4)

0 0.6(1) 0

11.01(5) 11.03(5) 5.21(5)

3835.(9.)

0.50(12) 11.51(11) 0.5(1) 11.53(11) 0.00005(10) 5.21(5)

1.90(3) 1.91(3) 0.000024(8)

4.232(6) 4.232(6) 4.20(22) 4.29(10)

0.000(8) 0 0.004(3) 0

4.232(6) 4.232(6) 4.20(22) 4.29(10)

0.00019(2) 0.00010(2) 0.236(10) 0.00016(1)

4.017(17)

0.0008(2)

4.018(14)

0.0096(5)

2.620(7) 2.695(7) 5.6(3) 1.88(1)

0.008(9) 0 0.05(2) 0

2.628(6) 2.695(7) 5.7(3) 1.88(1)

0.039(4) 0.036(4) 0.67(11) 0.046(6)

3.63(2)

3.59(3)

1.66(2)

1.62(3)

3.28(4)

0.530(5)

5.375(4) 5.66(3) 3.62(14) 4.89(15)

0 1.48(10) 0

3.631(5) 4.03(4) 1.65(13) 3.00(18)

0.08(6) 0 0.28(4) 0

3.71(4) 4.03(4) 1.93(14) 3.00(18)

0.063(3) 0.050(5) 0.19(3) 0.0382(8)

3.449(5)

0.256(10)

1.495(4)

0.0082(7)

1.503(4)

0.231(3)

4.1491(10) 4.107(6) 4.70(10) 4.58(8)

0 0.09(9) 0

2.1633(10) 2.120(6) 2.78(12) 2.64(9)

0.004(8) 0 0.001(2) 0

2.167(8) 2.120(6) 2.78(12) 2.64(9)

0.171(3) 0.177(3) 0.101(14) 0.107(2)

5.13(1)

0.2(2)

3.307(13)

0.005(10)

3.312(16)

0.172(6)

2.847(1) 2.804(2) 4.74(19) 3.48(3) 3.(1.) E

0 1.5(1.5) 0 0

1.0186(7) 0.9880(14) 2.8(2) 1.52(3) 1.1(8)

0.007(5) 0 0.3(6) 0 0

1.026(5) 0.9880(14) 3.1(6) 1.52(3) 1.1(8)

0.53(1) 0.54(4) 0.54(4) 0.227(5) 0.15(3)

445


1.34(2)

0.00747(1) 5333.(7.)

1.40(3)

12 13

95.02 0.75 4.21 0.02

1.34(2) 6.0(4)

0.78(3)

80.0

100

0.00 1.6(4)

0.619(11)

3/2( )

92.23 4.67 3.10

1.34(2) 4.42(10)

0.3326(7) 0.3326(7) 0.000519(7) 0

1.37(3) 0.97(7)

11

100

82.02(6) 82.03(6) 7.64(3) 3.03(5)

0.92(3) 0.46(2)

20.0

78.99 10.00 11.01

80.26(6) 80.27(6) 2.05(3) 0.14(4)

1.7568(10) 1.7583(10) 5.592(7) 2.89(3)

a

0.454(14) 0.51(5)

3(+)

100

s

1.89(5) 0.257(11)i 2.49(5)

10

90.51 0.27 9.22

i

1.90(2) 2.00(11) 0.261(1)i 2.22(2)

5.30(4) 0.213(2)i 0.1(3) 1.066(3)i 6.65(4)

100

c

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.4.4.1. Bound scattering lengths (cont.) Element Z Cl

17

Ar

18

K

19

Ca

20

Sc

21

Ti

22

V

23

Cr

24

Mn

25

Fe

26

Co

27

Ni

28

Cu

29

Zn

30

A

I

c

bc

35 37

3/2(+) 3/2(+)

75.77 24.23

9.5770(8) 11.65(2) 3.08(6)

6.1(4) 0.1(1)

11.526(2) 17.06(6) 1.19(5)

5.3(5) 4.7(6) 0.001(3)

16.8(5) 21.8(6) 1.19(5)

33.5(3) 44.1(4) 0.433(6)

36 38 40

0(+) 0(+) 0(+)

0.337 0.063 99.600

1.909(6) 24.90(7) 3.5(3.5) 1.830(6)

0 0 0

0.458(3) 77.9(4) 1.5(3.1) 0.421(3)

0.22(2) 0 0 0

0.683(4) 77.9(4) 1.5(3.1) 0.421(3)

0.675(9) 5.2(5) 0.8(2) 0.660(9)

39 40 41

3/2(+) 4( ) 3/2(+)

93.258 0.012 6.730

3.67(2) 3.74(2) 3.(1.) E 2.69(8)

1.5(1.5)

1.69(2) 1.76(2) 1.1(8) 0.91(5)

0.27(11) 0.25(11) 0.5(5) 0.3(6)

1.96(11) 2.01(11) 1.6(9) 1.2(6)

40 42 43 44 46 48

0(+) 0(+) 7/2( ) 0(+) 0(+) 0(+)

96.941 0.647 0.135 2.086 0.004 0.187

4.70(2) 4.80(2) 3.36(10) 1.56(9) 1.42(6) 3.6(2) 0.39(9)

0 0 0.31(4) 0 0 0

2.78(2) 2.90(2) 1.42(8) 0.5(5) E 0.25(2) 1.6(2) 0.019(9)

0.05(3) 0 0

2.83(2) 2.90(2) 1.42(8) 0.8(5) 0.25(2) 1.6(2) 0.019(9)

45

7/2( )

12.29(11)

6.0(3)

46 47 48 49 50

0(+) 5/2( ) 0(+) 7/2( ) 0(+)

8.2 7.4 73.8 5.4 5.2

3.370(13) 4.725(5) 3.53(7) 5.86(2) 0.98(5) 5.88(10)

50 51

6(+) 7/2( )

0.250 99.750

50 52 53 54

0(+) 0(+) 3/2( ) 0(+)

4.35 83.79 9.50 2.36

55

5/2( )

54 56 57 58

0(+) 0(+) 1/2( ) 0(+)

59

7/2( )

58 60 61 62 64

0(+) 0(+) 3/2( ) 0(+) 0(+)

63 65 64 66 67 68 70

100

100

bi

1.4(3)

4.5(5)

23.5(6)

a

2.1(1) 2.1(1) 35.(8.) 1.46(3) 0.43(2) 0.41(2) 0.68(7) 6.2(6) 0.88(5) 0.74(7) 1.09(14) 27.5(2)

2.63(3) 0 1.5(2) 0 3.3(3) 0

4.06(3) 2.80(6) 3.1(2) 4.32(3) 3.4(3) 4.34(15)

0.3824(12) 7.6(6) 0.402(2)

6.435(4)

0.01838(12) 7.3(1.1) 0.0203(2)

5.08(6) 0.5(5) E 5.07(6)

5.10(6) 7.8(1.0) 5.09(6)

5.08(2) 60.(40.) 4.9(1)

3.635(7) 4.50(5) 4.920(10) 4.20(3) 4.55(10)

0 0 6.87(10) 0

1.660(6) 2.54(6) 3.042(12) 2.22(3) 2.60(11)

1.83(2) 0 0 5.93(17) 0

3.49(2) 2.54(6) 3.042(12) 8.15(17) 2.60(11)

3.05(8) 15.8(2) 0.76(6) 18.1(1.5) 0.36(4)

3.750(18)

1.79(4)

1.77(2)

0.40(2)

2.17(3)

13.3(2)

11.22(5) 2.2(1) 12.42(7)

0.40(11) 0 0 0.3(3) E 0

11.62(10) 2.2(1) 12.42(7) 1.0(3) 28.(26.)

2.49(2)

6.2(2)

68.27 26.10 1.13 3.59 0.91

10.3(1) 14.4(1) 2.8(1) 7.60(6) 8.7(2) 0.37(7)

0 0 3.9(3) 0 0

3/2( ) 3/2( )

69.17 30.83

7.718(4) 6.43(15) 10.61(19)

0(+) 0(+) 5/2( ) 0(+) 0(+)

48.6 27.9 4.1 18.8 0.6

5.60(5) 5.22(4) 5.97(5) 7.56(8) 6.03(3) 6.(1.) E

28.(26.) 0.779(13)

6.43(6) 0.59(18) 1.7(2) 8.30(9) 2.2(3) 0.179(3)

2.56(3) 2.25(18) 2.59(14) 2.48(30) 1.28(5)

4.8(3)

5.6(3)

37.18(6)

13.3(3) 26.1(4) 0.99(7) 7.26(11) 9.5(4) 0.017(7)

5.2(4) 0 0 1.9(3) 0 0

18.5(3) 26.1(4) 0.99(7) 9.2(3) 9.5(4) 0.017(7)

4.49(16) 4.6(3) 2.9(2) 2.5(8) 14.5(3) 1.52(3)

0.22(2) 1.79(10)

7.485(8) 5.2(2) 14.1(5)

0.55(3) 0.006(1) 0.40(4)

8.03(3) 5.2(2) 14.5(5)

3.78(2) 4.50(2) 2.17(3)

0 0 1.50(7) 0 0

4.054(7) 3.42(5) 4.48(8) 7.18(15) 4.57(5) 4.5(1.5)

0.077(7) 0 0 0.28(3) 0 0

446


0 0 0

s

1.427(11) 2.80(6) 1.57(6) 4.32(3) 0.12(1) 4.34(15)

0 0 0.66(6) 0

100

19.0(3)

i

0 3.5(2) 0 5.1(2) 0

9.45(2) 4.2(1) 9.94(3) 2.3(1) 15.(7.)

5.8 91.7 2.2 0.3

c

4.131(10) 3.42(5) 4.48(8) 7.46(15) 4.57(5) 4.5(1.5)

1.11(2) 0.93(9) 0.62(6) 6.8(8) 1.1(1) 0.092(5)

4.4. NEUTRON TECHNIQUES Table 4.4.4.1. Bound scattering lengths (cont.) Element Z Ga

31

Ge

32

As

33

Se

34

Br

35

Kr

36

Rb

37

Sr

38

Y

39

Zr

40

Nb

41

Mo

42

Tc

43

A

I

c

bc

69 71

3/2( ) 3/2( )

60.1 39.9

7.288(2) 7.88(2) 6.40(3)

0.85(5) 0.82(4)

70 72 73 74 76

0(+) 0(+) 9/2(+) 0(+) 0(+)

20.5 27.4 7.8 36.5 7.8

8.185(20) 10.0(1) 8.51(10) 5.02(4) 7.58(10) 8.21(1.5)

0 0 3.4(3) 0 0

8.42(4) 12.6(3) 9.1(2) 3.17(5) 7.2(2) 8.(3.)

75

3/2( )

6.58(1)

0.69(5)

5.44(2)

74 76 77 78 80 82

0(+) 0(+) 1/2( ) 0(+) 0(+) 0(+)

0.9 9.0 7.6 23.5 49.6 9.4

7.970(9) 0.8(3.0) 12.2(1) 8.25(8) 8.24(9) 7.48(3) 6.34(8)

0 0 0.6(1.6) 0 0 0

79 81

3/2( ) 3/2( )

50.69 49.31

1.1(2) 0.6(1)

78 80 82 83 84 86

0(+) 0(+) 0(+) 9/2(+) 0(+) 0(+)

0.35 2.25 11.6 11.5 57.0 17.3

85 87

5/2( ) 3/2( )

72.17 27.83

7.09(2) 7.03(10) 7.23(12)

84 86 87 88

0(+) 0(+) 9/2(+) 0(+)

0.56 9.86 7.00 82.58

89

1/2( )

90 91 92 94 96

0(+) 5/2(+) 0(+) 0(+) 0(+)

93

9/2(+)

92 94 95 96 97 98 100

0(+) 0(+) 5/2(+) 0(+) 5/2(+) 0(+) 0(+)

14.84 9.25 15.92 16.68 9.55 24.13 9.63

99

9/2(+)

(2.13105 a) 6.8(3)

100

100 51.45 11.32 17.19 17.28 2.76 100

6.795(15) 6.80(7) 6.79(7) 7.81(2)

bi

0 0 0 185.(30.) 0 0

6.675(4) 7.80(4) 5.15(5)

i 0.16(3) 0.091(11) 0.084(8) 0.18(7) 0 0 1.5(3) 0 0

s 6.83(3) 7.89(4) 5.23(5)

a 2.75(3) 2.18(5) 3.61(10)

8.60(6) 12.6(3) 9.1(2) 4.7(3) 7.2(2) 8.(3.)

2.20(4) 3.0(2) 0.8(2) 15.1(4) 0.4(2) 0.16(2)

0.060(10)

5.50(2)

4.5(1)

7.98(2) 0.1(6) 18.7(3) 8.6(2) 8.5(2) 7.03(6) 5.05(13)

0.33(6) 0 0 0.05(26) 0 0 0

8.30(6) 0.1(6) 18.7(3) 8.65(16) 8.5(2) 7.03(6) 5.05(13)

11.7(2) 51.8(1.2) 85.(7.) 42.(4.) 0.43(2) 0.61(5) 0.044(3)

5.80(3) 5.81(12) 5.79(12)

0.10(9) 0.15(6) 0.05(2)

5.90(9) 5.96(13) 5.84(12)

6.9(2) 11.0(7) 2.7(2)

7.67(4)

0.01(14) 0 0 0

7.68(13)

25.(1.) 6.4(9) 11.8(5) 29.(20.)

8.2(4)

0 0

8.2(4)

0.113(15) 0.003(2)

6.2(2) 6.6(2)

6.32(4) 0.5(5) 0.5(5)

0.5(4) E E

6.8(4) 6.7(5) 7.1(5)

0.38(4) 0.48(1) 0.12(3)

7.02(2) 7.(1.) E 5.67(5) 7.40(7) 7.15(6)

0 0 6.88(13) 0

6.19(4) 6.(2.) 4.04(7) 0.5(5) 6.42(11)

0.06(11) 0 0 E 0

6.25(10) 6.(2.) 4.04(7) 7.4(5) 6.42(11)

1.28(6) 0.87(7) 1.04(7) 16.(3.) 0.058(4)

7.75(2)

1.1(3)

7.55(4)

0.15(8)

7.70(9)

1.28(2)

7.16(3) 6.4(1) 8.7(1) 7.4(2) 8.2(2) 5.5(1)

0 1.08(15) 0 0 0

6.44(5) 5.1(2) 9.5(2) 6.9(4) 8.4(4) 3.8(1)

0.02(15) 0 0.15(4) 0 0 0

6.46(14) 5.1(2) 9.7(2) 6.9(4) 8.4(4) 3.8(1)

0.185(3) 0.011(5) 1.17(10) 0.22(6) 0.0499(24) 0.0229(10)

7.054(3)

0.139(10)

6.253(5)

0.0024(3)

6.255(5)

1.15(5)

6.715(2) 6.91(8) 6.80(7) 6.91(6) 6.20(6) 7.24(8) 6.58(7) 6.73(7)

0 0 6.00(10) 0 6.59(15) 0 0

5.67(3) 6.00(14) 5.81(12) 0.5(5) 4.83(9) 0.5(5) 5.44(12) 5.69(12)

0.04(5) 0 0 E 0 E 0 0

5.71(4) 6.00(14) 5.81(12) 6.5(5) 4.83(9) 7.1(5) 5.44(12) 5.69(12)

2.48(4) 0.019(2) 0.015(2) 13.1(3) 0.5(2) 2.5(2) 0.127(6) 0.4(2)

5.8(5)

0.5(5)

E

6.3(7)

20.(1.)

8.1(2)

447


c

4. PRODUCTION AND PROPERTIES OF RADIATIONS Table 4.4.4.1. Bound scattering lengths (cont.) Element Z Ru

44

Rh

45

Pd

46

Ag

47

Cd

48

In

Sn

A

I

c

bc

96 98 99 100 101 102 104

0(+) 0(+) 5/2(+) 0(+) 5/2(+) 0(+) 0(+)

5.5 1.9 12.7 12.6 17.0 31.6 18.7

7.03(3) 0 0 6.9(1.0) 0 3.3(9) 0 0

103

1/2( )

102 104 105 106 108 110

0(+) 0(+) 5/2(+) 0(+) 0(+) 0(+)

107 109

1/2( ) 1/2( )

Sb

51

Te

52

c

i

s

a

0.4(1)

6.6(1)

2.56(13)

0 0

6.21(5) 0.28(2) 1;

EB A=1 Bx ; A exp y sinh y=y;

6:4:5:5 6:4:5:6

B 1=y

exp y= sinh y A

dA 1 : dy

6:4:4:4

with a2 2 2 . The path length of the diffracted beam through the crystal is D. The current density at the entrance surface is Pi0 : To ®nd formulae for the integrated intensity, it is necessary to express in terms of crystallographic quantities. 6.4.5. Primary extinction Zachariasen (1967) introduced the concept of using the kinematic result in the small-crystal limit for , while Sabine (1985, 1988) showed that only the Lorentzian or Fresnellian forms of the small crystal intensity distribution are appropriate for calculations of the energy ¯ow in the case of primary extinction. Thus, k

Qk T ; 1 T k2

6:4:5:1

where Qk V is the kinematic integrated intensity on the k scale (k 2 sin =l), Qk Nc lF2 = sin , and T is the volume average of the thickness of the crystal normal to the diffracting plane (Wilson, 1949). To include absorption effects, which modify the diffraction pro®le of the small crystal, it is necessary to replace T by TC, where

6.4.6. The ®nite crystal Exact application of the formulae above requires a knowledge of the shape of the crystal or mosaic block and the angular relation between the re¯ecting plane and the crystal surface. These are not usually known, but it can be assumed that the average block or crystal at each value of the scattering angle (2) has sides of equal length parallel to the incident- and diffracted-beam directions. For this crystal, T D sin ; D h Li;

6:4:6:1

and 6:4:6:2

The quantity L is set equal to ` for the mosaic block and to L for the crystal. 6.4.7. Angular variation of E Werner (1974) has given exact solutions to the transport equations in terms of tabulated functions. However, for the simple crystal described above, a suf®ciently accurate expression is E2 EL cos2 EB sin2 :

6:4:7:1

6.4.8. The value of x For the single mosaic block, application of the relationship T D sin leads to x Nc lF`2 ;

6:4:8:1

where ` is the average path length through the block. In the correlated block model, x is also a function of the tilts between blocks and the size of the crystal. It will be assumed for the discussion that follows in this section that the mosaic blocks are cubes of side `; and the distribution of tilts will be assumed to be isotropic and Gaussian, given by

610

611 s:\ITFC\chap 6.4.3d (Tables of Crystallography)

6:4:5:7

In these equations, x Qk TCD and y D.

x Nc 2 l2 F 2 h Li2 tanhL =2=L =2: Pi0 sinhaD ; a coshaD sinhaD

6:4:5:4

and

and Pf

6:4:5:3

1=8x

1=2

1

Here, Pi is the radiation current density (m s ) in the incident (i initial) beam, Pf is the current density in the diffracted f final beam. The distances ti and tf are measured along the incident and diffracted beams, respectively. The coupling constant is the cross section per unit volume for scattering into a single Bragg re¯ection, while , which is always negative, is the cross section per unit volume for removal of radiation from the beams by any process. In what follows, it will be assumed that absorption is the only signi®cant process, and is given by , where is the linear absorption coef®cient (absorption cross section per unit volume). This assumption may not be true for neutron diffraction, in which incoherent scattering may have a signi®cant role in removing radiation. In those cases, should include the incoherent scattering cross section per unit volume. The H±D equations have analytical solutions in the Laue case (2 0) and the Bragg case (2 ). The solutions at the exit surface are, respectively, Pf

x=2 x2 =4

EL exp yf1

6:4:4:1

2

6:4:5:2

To determine the extinction factor, E, the explicit expression for k [equation (6.4.5.1)] is inserted into equations (6.4.4.3) and (6.4.4.4), and integration is carried out over k. The limits of integration are 1 and 1. The notation EL and EB is used for the extinction factors at 2 0 and 2 rad, respectively. After integration and division by I kin ; it is found that

The radiation ¯ow is governed by the Hamilton±Darwin (H±D) equations (Darwin, 1922; Hamilton, 1957). These equations are @Pi Pi Pf ; @ti @Pf Pf Pi : @tf

tanhD=2 : D=2

6.4 THE FLOW OF RADIATION IN A REAL CRYSTAL exp D 1 2 EL 1 exp 2x; 6:4:9:2 ; 6:4:8:2 W p exp 2 2x 2 2 A : 6:4:9:3 EB where is the angular deviation of the block from the mean 1 Bx orientation of all blocks in the crystal, and is the standard deviation of the distribution. (The assumption of a Gaussian For a triangular function, W G1 jjG; for j j 1=G; W 0 otherwise, and the secondary-extincdistribution is not critical to the argument that follows.) Let the crystal be a cube of side L, and let be the probability tion factor becomes that a ray re¯ected by the ®rst block is re¯ected again by a subsequent block. The effective size of the crystal for Bragg exp D 1 EL 1 1 exp 2x ; 6:4:9:4 scattering of a single incident ray is then x 2x h Li ` L `; 6:4:8:3 2A EB Bx lnj1 Bxj: 6:4:9:5 Bx2 while the size of the crystal for all other attenuation processes is L, since, for them, the Bragg condition does not apply. The probability of re-scattering, , can readily be expressed in terms of crystallographic quantities. The full width at half-maximum 6.4.10. The extinction factor intensity of the Darwin re¯ection curve is given, after conversion 6.4.10.1. The correlated block model to the glancing-angle () scale, by Zachariasen (1945) as For this model of the real crystal, the variable x is given by 3l2 N F p c radians: 6:4:8:4 equation (6.4.8.6), with ` and g the re®nable variables. 2 sin 2 Extinction factors are then calculated from equations (6.4.5.3), The full width at half-maximum (FWHM) of the mosaic-block (6.4.5.4), and (6.4.5.5). For a re¯ection at a scattering angle of extinction distribution (6.4.8.2) p is derived in the usual way, and the 2 from a reasonably equiaxial crystal, the appropriate 2 factor is given by (6.4.7.1) as E2 E cos 2 E sin2 2. L B parameter g ( 1=2 ) is introduced to clear (to 1%) It is a meaningful procedure to re®ne both primary and numerical constants. Then , which is equal to the ratio of the secondary extinction in this model. The reason for the high widths, is given by correlation between ` and g that is found when other theories are applied, for example that of Becker & Coppens (1974), lies in gNc l2 F : 6:4:8:5 the structure of the quantity x. In the theory presented here, x is sin 2 proportional to F 2 for pure primary extinction and to Q2 for pure Insertion of hLi [equation (6.4.8.3)] in place of ` in equation secondary extinction. (6.4.8.1) for x leads to x N lF` gQ L `2 ; 6:4:8:6 6.4.10.2. The uncorrelated block model c

where Q

N02 l3 F 2 = sin 2: 6.4.9. Secondary extinction

A separate treatment of secondary extinction is required only in the uncorrelated block model, and the method given by Hamilton (1957) is used in this work. The coupling constant in the H±D equations is given by Q Ep W ; where Q Nc2 l3 F 2 = sin 2 for equatorial re¯ections in the neutron case, Ep is the correction for primary extinction evaluated at the angle , and W is the distribution function for the tilts between mosaic blocks. The choice of this function has a signi®cant in¯uence on the ®nal result (Sabine, 1985), and a rectangular or triangular form is suggested. In the following equations for the secondary-extinction factor, x Ep Q GD;

6:4:9:1

and A and B are given by equations (6.4.5.6) and (6.4.5.7). The average path length through the crystal for the re¯ection under consideration is D and G is the integral breadth of the angular distribution of mosaic blocks. It is important to note that A should be set equal to one if the data have been corrected for absorption, and B should be set equal to one if absorptionweighted values of D are used. If D for each re¯ection is not known, the average dimension of the crystal may be used for all re¯ections. For a rectangular function, W G; for jj 1=2G; W 0 otherwise, and the secondary-extinction factor becomes

When this model is used, two values of x are required. These are designated xp for primary extinction and xs for secondary extinction. Equation (6.4.8.1) is used to obtain a value for xp . The primary-extinction factors are then calculated from (6.4.5.3), (6.4.5.4) and (6.4.5.5), and Ep 2 is given by equation (6.4.7.1). In the second step, xs is obtained from equation (6.4.9.1), and the secondary-extinction factors are calculated from either (6.4.9.2) and (6.4.9.3) or (6.4.9.4) and (6.4.9.5). The result of these calculations is then used in equation (6.4.7.1) to give Es 2. It is emphasised that xs includes the primary-extinction factor. Finally, E2 Ep 2Es Ep 2; 2: Application of both models to the analysis of neutron diffraction data has been carried out by Kampermann, Sabine, Craven & McMullen (1995). 6.4.11. Polarization The expressions for the extinction factor have been given, by default, for the -polarization state, in which the electric ®eld vector of the incident radiation is perpendicular to the plane de®ned by the incident and diffracted beams. For this state, the polarization factor is unity. For the -polarization state, in which the electric vector lies in the diffraction plane, the factor is cos 2. The appropriate values for the extinction factors for this state are given by multiplying F by cos 2 wherever F occurs. For neutrons, which are matter waves, the polarization factor is always unity. For an unpolarized beam from an X-ray tube, the observed integrated intensity is given by I obs 12 Ikin E E cos2 2 . In the kinematic limit, E E 1, and the power to which cos 2

611


6. INTERPRETATION OF DIFFRACTED INTENSITIES is raised (the polarization index n) is 2. In the pure primaryextinction limit, E 1=Nc lF`, while E 1=Nc lF` cos 2. Hence, n 1. In the pure secondary-extinction limit, E 1=gQL; where g is the mosaic-spread parameter, while E 1=QgL cos 2: Hence, n 0. In all real cases, n will lie between 0 and 2, and its value will re¯ect departures from kinematic behaviour. 6.4.12. Anisotropy The parameters describing the microstructure of the crystal are the mosaic-block size and the angle between the mosaic blocks. These are not constrained in any way to be isotropic with respect to the crystal axes. In particular, they are not constrained by symmetry. For example, in a face-centred-cubic crystal under uniaxial stress, slip will occur on one set of f111g planes, leading to a dislocation array of non-cubic symmetry. In principle, anisotropy can be incorporated into the formal theory by allowing ` and g to depend on the Miller indices of the re¯ections. This has not been done in this work, but reference should be made to the work of Coppens & Hamilton (1970). 6.4.13. Asymptotic behaviour of the integrated intensity From the de®nition of the extinction factor, the integrated intensity from a non-absorbing crystal in which the block size is suf®ciently small, and the mosaic spread is suf®ciently large, will approach the kinematic limit. It is instructive to examine the behaviour in the limit of large block size and low mosaic spread. The volume of the mosaic block is v and the volume of the crystal is V . The number of blocks in the crystal is V =v L3 =`3 . The surface area of the block is v2=3 and of the crystal is V 2=3 . The subscripts L and B will again be used for the Laue and the Bragg case, respectively. The kinematic integrated intensity is given by I kin Q V l3 Nc2 F 2 V = sin 2:

6:4:13:1

6.4.13.1. Non-absorbing crystal, strong primary extinction (a) Laue case The limiting value of EL is 2=1=2 x IL 4=5Nc l2 FVv

1=3

1=2

. Hence,

= sin 2:

6:4:13:2

The dynamical theory has a numerical constant of 1=2 instead of 4=5. (b) Bragg case The limiting value of EB is x 1=2 . Hence, IB Nc l2 FVv

1=3

= sin 2:

6:4:13:3

This is in exact agreement with the dynamical theory (Ewald solution).

BCx 2Nc2 l2 F 2 =2 :

For BCx small, the integrated intensity, IB , is given by

6.4.13.3. The absorbing crystal Only the Bragg case for thick crystals will be considered here. The asymptotic values of A, B, and C are 1=2L , 1=L , and 2=L , respectively, so that

For BCx large, p IB 1=2 21

6:4:13:5

=2lNc F2 l2 Nc FV 2=3 = sin 2:

6:4:13:6

It can be shown that the parameter g (which has no relation to the parameter g used to describe the mosaic-block distribution) used by Zachariasen (1945) in discussing this case is equal to =2NCF. Hence, on his y scale, p IB =2 21 g2 : 6:4:13:7 The value he obtained is IB 8=31 2jgj; while Sabine & Blair (1992) found IB 8=31 2:36jgj:

6.4.14. Relationship with the dynamical theory Sabine & Blair have shown that the two classical limits for the integrated intensity in the symmetric Bragg case can be obtained from the Hamilton±Darwin equations when the dynamic refractive index of the crystal is explicitly taken into account. Their treatment is based on the following expression for k: Qk DT sin2 T K sinh2 D=2 k ; sinhD T K2 D=22 where K refers to the scattering vector within the crystal. Use of the relation K k and the replacement of the Fresnellian by a Lorentzian leads to equation (6.4.5.1) with the inclusion of C (6.4.5.2). The relationship between K and k, which is a function of the dynamic refractive index of the crystal, is derived in the original publication. Insertion of this expression into equations (6.4.4.3) and (6.4.4.4) and integration over k, since the diffracted beam is observed outside the crystal, leads to a dynamic extinction factor, which can be compared with the values determined from the equations given in Section 6.4.5. The integrations cannot be carried out analytically and require numerical calculation in each case. Olekhnovich & Olekhnovich (1978, 1980) have given limited expressions for primary extinction in the parallelepiped and the cylinder based on the equations of the dynamical theory in the non-absorbing case. Comparisons with the results of the present theory are given by Sabine (1988) and Sabine, Von Dreele & Jrgensen (1988).

6.4.15. De®nitions The quantity F used in these equations is the modulus of the structure factor per unit cell. It includes the Debye±Waller factor and the scattering length of each atom. (For X-ray diffraction, the scattering length of the electron is 2:8178 10 15 m.) l is the wavelength of the incident radiation. 2 is the angle of scattering. Nc is the number of unit cells per unit volume. The path length of the diffracted beam is D, while T is the thickness of the crystal normal to the diffracting plane. In practice, when the orientation of the crystal is unknown, D can be taken equal to ` or L; where these are average dimensions of the mosaic block or crystal.

612


Nc F=2V 2=3 :

IB Q =21

6.4.13.2. Non-absorbing crystal, strong secondary extinction For this condition, the limiting values of the integrated intensity are IL 4=5g 1 V 2=3 , and IB g 1 V 2=3 : In this limit, which was also noted by Bacon & Lowde (1948) and by Hamilton (1957), the intensity is proportional only to the mosaic spread and to the surface area of the crystal. No structural information is obtained from the experiment.

6:4:13:4

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James, R. W. (1962). The optical principles of the diffraction of X-rays, pp. 135±192. Ithaca: Cornell University Press. Karamura, T. & Fukamachi, T. (1979). Temperature dependence of X-ray re¯ection intensity from an absorbing perfect crystal near an absorption edge. Acta Cryst. A35, 831±835. Katayama, C., Sakabe, N. & Sakabe, K. (1972). A statistical evaluation of absorption. Acta Cryst. A28, 293±295. Kopfmann, G. & Huber, R. (1968). A method of absorption correction for X-ray intensity measurements. Acta Cryst. A24, 348±351. Krause, M. O. & Oliver, J. H. (1979). Natural widths of atomic K and L levels, K X-ray lines and several KLL Auger lines. J. Phys. Chem. Ref. Data, 8, 329±338. Lee, B. & Ruble, J. R. (1977a). A semi-empirical absorptioncorrection technique for symmetric crystals in single-crystal X-ray crystallography. I. Acta Cryst. A33, 629±637. Lee, B. & Ruble, J. R. (1977b). A semi-empirical absorptioncorrection technique for symmetric crystals in single-crystal X-ray crystallography. II. Acta Cryst. A33, 637±641. North, A. C. T., Phillips, D. C. & Mathews, F. S. (1968). A semi-empirical method of absorption correction. Acta Cryst. A24, 351±359. Rigoult, J. & Guidi-Morosini, C. (1980). An accurate calculation of T for spherical crystals. Acta Cryst. A36, 149±151. Rouse, K. D., Cooper, M. J., York, E. J. & Chakera, A. (1970). Absorption corrections for neutron diffraction. Acta Cryst. A26, 682±691. Santoro, A. & Wlodawer, A. (1980). Absorption corrections for Weissenberg diffractometers. Acta Cryst. A36, 442± 450. Schwager, P., Bartels, K. & Huber, R. (1973). A simple empirical absorption-correction method for X-ray intensity data ®lms. Acta Cryst. A29, 291±295. Stroud, A. H. & Secrest, D. (1966). Gaussian quadrature formulas. New Jersey: Prentice-Hall. Stuart, D. & Walker, N. (1979). An empirical method for correcting rotation-camera data for absorption and decay effects. Acta Cryst. A35, 925±933. Templeton, D. H. & Templeton, L. K. (1980). Polarized X-ray absorption and double refraction in vanadyl bisacetylacetonate. Acta Cryst. A36, 237±241. Templeton, D. H. & Templeton, L. K. (1982). X-ray dichroism and polarized anomalous scattering of the uranyl ion. Acta Cryst. A38, 62±67. Templeton, D. H. & Templeton, L. K. (1985). Tensor optical properties of the bromate ion. Acta Cryst. A41, 133±142. Tibballs, J. E. (1982). The rapid computation of mean path lengths for cylinders and spheres. Acta Cryst. A38, 161± 163. Wagenfeld, H. (1975). Theoretical computations of X-ray dispersion corrections. Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 13±24. Copenhagen: Munksgaard. Walker, N. & Stuart, D. (1983). An empirical method for correcting diffractometer data for absorption effects. Acta Cryst. A39, 158±166. Weber, K. (1969). Eine neue Absorptionsfactortafel fuÈr kugelfoÈrmige Proben. Acta Cryst. B25, 1174±1178. Zachariasen, W. H. (1968). Extinction and Borrmann effect in mosaic crystals. Acta Cryst. A24, 421±424.

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6. INTERPRETATION OF DIFFRACTED INTENSITIES 6.4 Bacon, G. E. & Lowde, R. D. (1948). Secondary extinction and neutron crystallography. Acta Cryst. 1, 303±314. Becker, P. J. & Coppens, P. (1974). Extinction within the limit of validity of the Darwin transfer equations. I. General formalisms for primary and secondary extinction and their application to spherical crystals. Acta Cryst. A30, 129±147. Coppens, P. & Hamilton, W. C. (1970). Anisotropic extinction corrections in the Zachariasen approximation. Acta Cryst. A26, 71±83. Cottrell, A. H. (1953). Dislocations and plastic ¯ow in crystals. Oxford University Press. Darwin, C. G. (1922). The re¯exion of X-rays from imperfect crystals. Philos. Mag. 43, 800±829. Hamilton, W. C. (1957). The effect of crystal shape and setting on secondary extinction. Acta Cryst. 10, 629±634. Kampermann, S. P., Sabine, T. M., Craven, B. M. & McMullan, R. K. (1995). Hexamethylenetetramine: extinction and thermal vibrations from neutron diffraction at six temperatures. Acta Cryst. A51, 489±497. Olekhnovich, N. M. & Olekhnovich, A. I. (1978). Primary extinction for ®nite crystals. Square-section parallelepiped. Acta Cryst. A34, 321±326.

Olekhnovich, N. M. & Olekhnovich, A. I. (1980). Primary extinction for ®nite crystals. Cylinder. Acta Cryst. A36, 22±27. Read, W. T. (1953). Dislocations in crystals. New York: McGraw-Hill. Sabine, T. M. (1985). Extinction in polycrystalline materials. Aust. J. Phys. 38, 507±518. Sabine, T. M. (1988). A reconciliation of extinction theories. Acta Cryst. A44, 368±373. Sabine, T. M. & Blair, D. G. (1992). The Ewald and Darwin limits obtained from the Hamilton±Darwin energy transfer equations. Acta Cryst. A48, 98±103 Sabine, T. M., Von Dreele, R. B. & Jrgensen, J.-E. (1988). Extinction in time-of-¯ight neutron powder diffractometry. Acta Cryst. A44, 374±379. Werner, S. A. (1974). Extinction in mosaic crystals. J. Appl. Phys. 45, 3246±3254. Wilkins, S. W. (1981). Dynamical X-ray diffraction from imperfect crystals in the Bragg case ± extinction and the asymmetric limits. Philos. Trans. R. Soc. London, 299, 275±317. Wilson, A. J. C. (1949). X-ray optics. London: Methuen. Zachariasen, W. H. (1945). Theory of X-ray diffraction in crystals. New York: John Wiley, Dover. Zachariasen, W. H. (1967). A general theory of X-ray diffraction in crystals. Acta Cryst. 23, 558±564.

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references


7.1. Detectors for X-rays

By Y. Amemiya, U. W. Arndt, B. Buras, J. Chikawa, L. Gerward, J. I. Langford, W. Parrish and P. M. de Wolff 7.1.1. Photographic ®lm*

7.1.2. Geiger countersy

In 1962, when Volume III of International Tables for X-ray Crystallography was published, photographic ®lm was the commonest detector for X-rays. Now it has been largely supplanted by the electronic devices described in other sections, but it is still much used in powder cameras and in preliminary investigation of specimens. X-rays and other radiations cause blackening of silver halide emulsions, and their intensity can be measured accordingly. The blackening of the ®lm is expressed in units of density:

Geiger-MuÈller counters (Geiger & MuÈller, 1928) are now obsolete for data collection, but are still used in portable monitors for X-rays. A cross section of a once-popular type is shown in Fig. 7.1.2.1a. The cathode C is a cylinder made of a metal such as chrome-iron, about 2 cm in diameter and 10 cm long. The anode A is a tungsten wire about 0.7 mm in diameter mounted coaxially with C and terminated by a bead to prevent destructive electrical discharges from its tip. About 1400 V DC is applied between C and A. X-rays enter at a low-absorption end window W, made of mica about 0.013 mm thick or other suitable material; beryllium would now be used. The gas ®lling may be argon at a pressure of about 55 cm Hg or krypton at a lower pressure. A small amount of halogen (0.4% of chlorine or bromine) helps to avoid destructive discharges. Separating the anode and the window is a dead space in which X-rays are absorbed but not detected. The quantum-counting ef®ciency varies with wavelength; for Cu K and its neighbours, it is about 50% and, for Mo K, it is about 10%. For the longer wavelengths, it is limited by absorption in the window and the dead space, so it is important to keep these as thin as practicable. For the shorter wavelengths, it is limited by the transparency of the gas in the sensitive volume.

D log10 I incident =I transmitted ;

7:1:1:1

where I refers to the intensity of the ordinary light incident on the ®lm. Measured densities must be corrected by subtracting the fog density DF measured on a non-exposed part of the ®lm. Important features of the photographic process for strongly ionizing radiations such as X-rays and electrons are: (i) For a given total exposure E the relationship between D and E is, to a close approximation, independent of the time variation of the intensity of the incident radiation. It does not matter whether the X-ray quanta arrive continuously or in short intense bursts (Mees, 1954). (ii) The density D increases linearly with E up to D ' 1, then increases more slowly. Photographic intensity measurements may be made either visually or by using a microdensitometer.

y Editorial condensation of the entry by W. Parrish in Chapter 3.1 of Volume III. Several papers on the relative advantages of various detectors are collected in Parrish (1962). Sections 7.1.2±7.1.4 have been slightly revised by J. I. Langford.

7.1.1.1. Visual estimation Visual estimation consists of comparing the spot or line to be measured with a series of exposure-calibrated marks similar in shape to the object of measurement, and preferably made with the same specimen and incident beam. Lack of complete similarity and unfavourable background usually cause the error of such measurements to be larger than the optimum contrast threshold of the eye. For a spot area of 1 mm2 , the latter amounts to roughly 1% or 0.004 density units, a difference that can in fact be detected under favourable circumstances (low density and low background). 7.1.1.2. Densitometry If the blackening is measured with a microdensitometer, an accuracy of 0.002 density units up to densities of at least 2 is easily attained. Higher precision is rarely required, as the limiting factors are graininess of the ®lm and irregularities in the emulsion and processing. The grains in processed X-ray ®lm are larger than those produced by visible light, and occur in clusters around each absorbed quantum. The resulting statistical ¯uctuations may be minimized by appropriate choice of densitometer slit dimensions and scanning speed. If the X-rays are not incident normally on double-coated ®lm, it may be necessary to make corrections for obliquity (Whittaker, 1953; Hellner, 1954). * Editorial condensation of the entry by P. M. de Wolff in Chapter 3.1 of Volume III.

Fig. 7.1.2.1. Detectors used for diffractometry: a Geiger counter, b side-window proportional counter, c end-window scintillation counter. The arrows X show direction of incident X-ray beam, W thin window, C cathode, A anode, SC scintillation crystal, PT photomultiplier tube.

618 Copyright © 2006 International Union of Crystallography 619 s:\ITFC\chap-7.1.3d (Tables of Crystallography)

7.1. DETECTORS FOR X-RAYS The tube is not uniformly sensitive across its diameter. The maximum sensitivity is con®ned to the cylindrical volume shown cross-hatched in the ®gure. The diameter of this sensitive area depends on the gas ®lling and the geometry of the tube. For maximum ef®ciency, the X-ray beam should be directed along and close to the anode, but should not strike it. Geiger counters are not critically temperature-dependent. Linearity of response is limited by the dead-time following the discharge initiated by the absorption of a quantum and the magni®cation of the few hundred ions produced to some millions by their acceleration under the electric ®eld. To produce this ampli®cation, a certain minimum threshold voltage is required. Above this minimum, there is a plateau extending for several hundred volts within which the number of quanta detected is essentially independent of the applied voltage and the size of the pulses is essentially independent of the energy of the absorbed quantum. Geiger counters are simple to use and show little deterioration even after prolonged use. However, since the pulses are all of about the same size, pulse-height discrimination cannot be used, and the long dead-time limits linearity of response unless special monitoring circuits are used (Eastabrook & Hughes, 1953). They have been almost completely superseded by other types of counter, described in Sections 7.1.3.±7.1.8. 7.1.3. Proportional counters (By W. Parrish) 7.1.3.1. The detector system The commonest types of detector for both powder and singlecrystal diffractometry are proportional counters and especially scintillation counters (Section 7.1.4). The detector system consists of the detector itself, a high-voltage power supply, a single-channel pulse-height analyser, and a scaling circuit, as shown schematically in Fig. 2.3.3.5. For position-sensitive detectors (Section 7.1.3.3) and solid-state detectors (Sections 7.1.4.2 and 7.1.5), multichannel analysers are necessary. The Xray manufacturers and a number of electronic companies provide complete detector systems, often integrated with the computer data-collection system. 7.1.3.2. Proportional counters Proportional counters are available in various sizes and gas ®llings. A typical detector is a metal cylinder about 2 cm in diameter and 8±10 cm long, with central wire anode and 0.13 mm Be side window, Fig. 7.1.2.1b. Some have an opposite exit window to transmit the unabsorbed beam and thus avoid ¯uorescence from the wall. The tube may be ®lled with Xe to atmospheric pressure for high absorption, and a small amount of quenching gas such as CO2 or CH4 is added to limit the discharge. When an X-ray quantum is absorbed, the discharge current is the sum of the Townsend avalanches of the secondary electrons and the gas ampli®cation is about 104 . A chargesensitive preampli®er is generally used. Some proportional counters are ®lled to several atmospheres pressure to increase the gas absorption. Very thin organic ®lm windows are used for very long wavelengths as in ¯uorescence spectroscopy. They may transmit moisture, and gas may migrate through them so that ¯ow counters are used to replenish the gas. This requires careful control of the pressure to avoid changes in the counting ef®ciency. 7.1.3.3. Position-sensitive detectors One variety of position-sensitive detector, in which the photon absorptions in different regions are counted separately, is a

special type of proportional counter. The following description applies primarily to one-dimensional detectors for powder diffractometry; two-dimensional (area) detectors are treated in Section 7.1.6. Position-sensitive detectors (PSD's) are being used in increasing number for various powder-diffraction studies. They have the great advantage of simultaneously recording a much larger region of the pattern than conventional counters. The difference in receiving apertures determines the gain in time. The position at which each quantum is detected is determined electronically by the system computer and stored in a multichannel analyser. There is a digital addition of each incident photon address and the angular address of the diffractometer. The PSD's are available in short straight form and as longer detectors with curvature to match the diffractometer focusing circle. The short detectors can be used in a stationary position to cover a small angular range or scanned. GoÈbel (1982) developed a high-speed method using a short (8 window) scanning PSD with 50 mm linear resolution in the diffractometer geometry shown in Fig. 2.3.1.12b. He was able to record at speeds of a hundred or more degrees a minute, and patterns with reasonably good statistical precision in several tens of degrees a minute. This is faster than conventional energy-dispersive diffraction and has the advantage of much higher resolution. The PSD should be selected to match best the diffraction geometry. The detector is sensitive across the 1±2 cm gasabsorption path. If the diffracted rays are not perpendicular to the window, the parallax causes broadening and loss of resolution. This becomes important in the focusing geometries and can be minimized if the diffractometer and specimen focusing circles are nearly coincident. A large loss of resolution would occur in the conventional geometry, Fig. 2.3.1.3, because only the central ray of a single re¯ection would be normal to the window. The problem is minimized in powder-camera geometry with a thin rod specimen, Fig. 2.3.4.1a, where the entire pattern can be recorded with a long, curved PSD (Ballon, Comparat & Pouxe, 1983); see also Shishiguchi, Minato & Hashizume (1986), Lehmann, Christensen, FjellvaÊg, Feidenhans'l & Nielsen (1987), WoÈlfel (1983), and Foster & WoÈlfel (1988). 7.1.3.4. Resolution, discrimination, ef®ciency The topics of energy resolution, pulse-height discrimination, quantum-counting ef®ciency, and linearity are common to proportional, scintillation and solid-state counters, and are treated in Subsections 7.1.4.3.±7.1.4.5. 7.1.4. Scintillation and solid-state detectors (By W. Parrish) 7.1.4.1. Scintillation counters The most frequently used detector is the scintillation counter (Parrish & Kohler, 1956). It has two elements: a ¯uorescent crystal and a photomultiplier tube, Fig. 7.1.2.1c. For X-ray diffraction, a cleaved single-crystal plate of optically clear NaI activated with about 1% Tl in solid solution is used. The crystal is hygroscopic and is hermetically sealed in a holder with thin Be entrance window and glass back to transmit the visible-light scintillations. The size and shape of the crystal can be selected, but is usually a 2 cm diameter disc or a rectangle 20 4 1 mm thick. A small thin crystal has been used to reduce the background from radioactive samples (Kohler & Parrish, 1955). A viscous mounting ¯uid with about the same refractive index as the glass is used to reduce light re¯ection and to attach it to the end of the photomultiplier tube. The crystal and

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620 s:\ITFC\chap-7.1.3d (Tables of Crystallography)

7. MEASUREMENT OF INTENSITIES photomultiplier are mounted in a light-tight cylinder surrounded by an antimagnetic foil. The high X-ray absorption of the crystal provides a high quantum-counting ef®ciency. A Cu K quantum produces about 500 visible photons of Ê in the scintillation crystal (which average wavelength 4100 A matches the maximum spectral sensitivity of the photomultiplier), but only about 25 will be effective in the photomultiplier operation. High-speed versions with special pulse-height analysers have recently become available; they are linear to about 1% at 105 counts s 1 and can be used at rates approaching 106 counts s 1 (see Rigaku Corporation, 1990). The detector system is as described in Subsection 7.1.3.1. 7.1.4.2. Solid-state detectors The following description applies primarily to the use of solidstate detectors in powder diffractometry. Further details of their operation and their use in energy-dispersive diffractometry are treated in Section 7.1.5. The most common form of solid-state detector consists of a lithium-drifted silicon crystal Si(Li) and liquid-nitrogen Dewar. A perfect single crystal is used with very thin gold ®lm on the front surface for electrical contact. The ®rst ampli®er stage is a ®eld-effect transistor (FET). The unit must be kept at liquidnitrogen temperature at all times (even when not in use) to prevent Li diffusion and to reduce the dark current when in use. The unit is large and heavy and, if not used in a stationary position, a robust detector arm is required, which is usually counter-balanced. The crystal is made with different-size sensitive areas and the resolution is somewhat dependent on the size of the area. In the detector process, the number of free charge carriers (the electron and electron±hole pairs) generated during the X-ray absorption changes the conductivity of the crystal and is proportional to the energy of the X-ray quantum. Details of the mechanism are given in several books [see, for example, Heinrich, Newbury, Myklebust & Fiori (1981) and Russ (1984)]. Intrinsic germanium detectors have higher absorption than silicon detectors, but they have lower energy resolution and there are more interferences from escape peaks. A mercuric iodide (HgI2 ) detector can be operated at room temperature and has high absorption (Nissenbaum, Levi, Burger, Schieber & Burshtein, 1984). They have poorer resolution than Si or Ge detectors but can be improved to FWHM 200 eV at 5.9 keV by cooling to 269 K (Ames, Drummond, Iwanczyk & Dabrowski, 1983). A small (about 16.5 10 cm), lightweight (3.2 kg) silicon detector with Peltier thermoelectric cooling is available (e.g. Kevex Corporation, 1990). This development has supplanted a number of the methods of collecting powder data. The elimination of the liquid-nitrogen Dewar and the compact size makes it possible to replace conventional detectors and the diffracted-beam monochromator in scanning powder diffractometry. The spectrum is displayed on a small screen and the window of the analyser can be set closely on the energy distribution obtained from a powder re¯ection to transmit, say, only Cu K. The monochromator can be eliminated for a large gain of intensity without loss of pattern resolution. The energy resolution is FWHM 195 eV at 5.9 keV. Elemental analysis can be performed by energy-dispersive ¯uorescence, and the background can be restricted to the narrow energy window selected. Bish & Chipera (1989) used it to obtain a 3±4 times increase of intensity, the same pattern resolution, and lower tails than with a graphite monochromator and scintillation counter in conventional diffractometry. The major limitation at present is

the limited input intensity that can be handled. The limiting (total) count rate is about 104 counts s 1 and the detector becomes markedly nonlinear at 2 104 counts s 1 . Internal dead-time corrections can extend the range by increasing the counting times. 7.1.4.3. Energy resolution and pulse-amplitude discrimination The pulse amplitudes are proportional to the energy e of the absorbed X-ray quantum so that electronic methods can be used to reduce the background from other wavelengths and sources. The rejection range is limited by the energy resolution of the detector. As noted above, the pulse amplitudes have distributions that vary around the average value A, Figs. 7.1.4.1a,b. The FWHM of the distribution increases linearly with increasing e (eV) and is proportional to e1=2 , i.e. it improves inversely with l1=2 . The ratio FWHM=A (expressed in %) is a measure of the energy resolution at a given wavelength; the smaller the ratio the better the resolution. For example, as e increases from 5 to 45 keV, the FWHM approximately doubles while FWHM=e decreases from 5 to 1%. The resolution of proportional counters is about 18% for Cu K and somewhat better for high-pressure gas ®llings; in scintillation counters, it is about 45%. The solid-state detectors have much better resolution. The best are about 2.4% (145 eV) at 5.9 keV (which is the energy of Mn K X-rays from a radioactive 55 Fe source used as a standard for calibration). The electronics include a high-voltage power supply to about 1200 V for scintillation counters and 2000±3000 V for proportional counters, and a single-channel pulse-amplitude discriminator. The latter contains pulse-shaping circuits and the ampli®er, and is designed to transmit pulses whose amplitudes lie within the selected range. The lower level rejects all pulses below the selected level and the upper level rejects the higher amplitudes (Figs. 7.1.4.1a; b). The range selected is called the window and determines the pulse amplitudes that will be counted by the scaling circuit. The multichannel analyser is generally used with solid-state detectors. It may have up to 8000 channels and sorts the pulses from the ampli®er into individual channels according to their amplitudes, which are proportional to the X-ray photon energies. The pattern can be stored and displayed on a CRT screen, but nowadays a personal computer with a suitable interface card is normally used in place of the analyser. Various programs are available for peak-energy identi®cation, spectral stripping, intensity determination, and similar datareduction requirements. The limiting count rate that can be handled by the electronics is determined by the total number of photons striking the detector. Pulse-pileup rejectors are used to stop counting momentarily when another pulse is too close in time to allow the original pulse to return to the baseline voltage. A live-time correction extends the counting period beyond the clock time to compensate for the time the analyser is gated off. About 50 000 counts s 1 is the maximum rate so that the individual powder re¯ections have a much smaller number of pulses. If good statistical accuracy is required, the count times are, therefore, much longer than in conventional diffractometry. For a given e, the pulse amplitudes of scintillation and proportional counters increase with increasing voltage (internal gain) and ampli®er setting (external gain). The detector must be operated in the plateau region for the wavelength used (Fig. 7.1.4.1c). The counts are measured as a function of the voltage and/or gain, and the plateau begins where there is no further signi®cant increase of intensity. In selecting the operating

620


7.1. DETECTORS FOR X-RAYS conditions, one should avoid excessively high voltages and ampli®er gains, which may cause noise pulses and unstable operation. Optimum settings can be determined by experiment and from manufacturer's instructions. The average pulse height should be set at about 20±25% of the full range of the pulseheight analyser. Lower settings move the low-energy tail into the noise, and high settings broaden the distribution and may be too wide for the window. The pulse-amplitude distribution can be measured with a narrow (1±3 V) upper level and increasing the lower level by small equal steps. When making this calibration, it is advisable to keep the incident count rate below 104 counts s 1 to avoid nonlinearity and pulse pileup. A plot of intensity versus lower-level setting shows the distribution, Fig. 7.1.4.1a. In some electronics, this can be done automatically and displayed on a screen. The window should be set symmetrically around the peak with the window decreasing the characteristic line intensity only a few per cent below that obtained with the lower-level set to remove only the circuit noise. The intensity change can be seen with a rate meter. Narrow windows cause a larger percentage loss of intensity than the decrease in background and, hence, the peak-to-background ratio is reduced. Asymmetric windows are sometimes used to decrease the ¯uorescence background.

7.1.4.4. Quantum-counting ef®ciency and linearity The quantum-counting ef®ciency E of the detector, its variation with wavelength, and electronic discrimination determine the response to the X-ray spectrum. E is determined by E fT fA ;

7:1:4:1

where fT is the fraction of the incident radiation transmitted by the window (usually 0.013 mm Be) and fA is the fraction absorbed in the detector (scintillation crystal or proportionalcounter gas). E varies with wavelength as shown in Fig. 7.1.4.1(d). The scintillation counter has a nearly uniform E approaching 100% across the spectrum and detects the shortwavelength continuous radiation with about the same ef®ciency as the spectral lines. The gas-®lled counters have a lower E for the short wavelengths and, therefore, may have a slightly lower inherent background; high-pressure gas counters have a higher and more uniform spectral ef®ciency. The effectiveness of electronic discrimination with a scintillation counter is shown in Fig. 2.3.5.3c for 50 kV Cu target radiation. The method cannot separate the K-doublet components because of their small energy difference, and has little effect on the K peak. The results are greatly enhanced by the addition of a K ®lter, which removes most of the K peak and a portion of the continuous radiation below the ®lter absorption

Fig. 7.1.4.1. Calculated pulse-amplitude distributions of Cu K and Mo K in the form of a integral curves and b differential curves. Resolution W =A for Cu K 50%. Analyser settings show window between lower level LL and upper level UL. c Plateaux of scintillation counter for various wavelengths and ®xed ampli®er gain. Curves normalized to same intensity at highest voltage. Noise curve is plotted in counts s 1 . Curves can be moved to higher or lower voltages by changing ampli®er gain. d Calculated quantum-counting ef®ciency (QCE) of scintillation counter as a function of wavelength (top curve) and its reduction when the pulse-height analyser is set for 90% Cu K. E.P. is escape peak at lower left.

621


7. MEASUREMENT OF INTENSITIES edge, Fig. 2.3.5.3b. The combination of discrimination and ®lter produces mainly the K doublet, Fig. 2.3.5.3d. Spectral analysis of the background of a non-¯uorescence powder sample using this method with 50 kV Cu radiation and a scintillation counter shows it to be 50±90% characteristic radiation. The linearity of the system is determined by the dead-time of the detector, and the resolving times of the pulse-height analyser and scaling circuit. The observed intensity nobs is related to the effective dead-time of the system eff by the relation ntrue nobs =1

eff nobs :

7:1:4:2

The value of eff can be measured with an oscilloscope, or with the multiple-foil method in which a number of equal absorption foils (e.g. Al 0.025 mm or Ni 0.018 mm for Cu K) are inserted in the beam one or two at a time. To make certain monochromatic radiation is used, a single-crystal plate such as Si(111), which has no signi®cant second order, and low X-ray tube voltage are employed. The linearity is determined from a regression calculation. A less accurate method is to plot nobs on a log scale against the number of foils on a linear scale. Recent developments in high-speed scintillation counters have extended the linearity to the 105 ±106 counts s 1 range. 7.1.4.5. Escape peaks The pulse-amplitude distribution may have two or more peaks, even when monochromatic X-rays are used (Parrish, 1966). Absorption of the incident X-rays by the counter-tube gas or scintillation crystal may cause X-ray ¯uorescence. If this is reabsorbed in the active volume of the counter only one pulse is produced of average amplitude A1 proportional to the incident X-ray quantum energy e1 (k constant) A1 ke1 :

7.1.5. Energy-dispersive detectors (By B. Buras and L. Gerward) In white-beam energy-dispersive X-ray diffraction, the spectral distribution of the diffracted beam is measured either with a semiconductor detector (low-momentum resolution) or with a scanning-crystal monochromator (high-momentum resolution) (see Subsection 2.5.1.3). Commercially available detectors are made of lithium-drifted silicon or germanium [denoted Si(Li) and Ge(Li), respectively], or high-purity germanium (HPGe). There are, however, other materials that are good candidates for making energy-dispersive detectors. The semiconductor detector can be regarded as the solid-state analogue of the ionization chamber. Charge carriers of opposite sign (electrons and holes) are produced by the X-ray photons. They drift in the applied electric ®eld of the electrodes and are converted to a voltage pulse by a charge-sensitive preampli®er. The energy required for creating an electron±hole pair is 3.9 eV in silicon and 3.0 eV in germanium. The number of electron± hole pairs is proportional to the energy of the absorbed photon (the intrinsic ef®ciency is discussed below). There is no intrinsic gain and one has to rely on external ampli®cation. The preampli®er employs an input ®eld-effect transistor (FET), cooled in an integral assembly with the detector crystal in order to reduce thermal noise. Usually the detector is operated at liquid-nitrogen temperature. However, Peltier-cooled silicon detectors are available, removing the maintenance concerns of cryostat cooling. The basic counting system consists further of an ampli®er, producing a near-Gaussian pulse shape, and a multichannel pulse-height analyser. It is common to use an

7:1:4:3

However, the gas or crystal has a low absorption coef®cient for its own ¯uorescent radiation, hence, some quanta of the latter of energy e2 may escape from the active volume of the counter, the amount depending on the geometry of the tube, gas, windows, etc. The average amplitude A2 of the escape pulses is A2 ke1

e2 :

7:1:4:4

A2 ke2 :

7:1:4:5

Thus, A1

The pulse-height analyser discriminates against pulses only on the basis of their amplitudes. When it is set to detect X-rays of energy e0 , it is also sensitive to X-rays of energy e0 e2 . For example, when using an NaI scintillation counter for Cu K, e0 8 keV, and for the escape X-rays I K, e2 28:5 keV. A pulse-height analyser set to detect X-rays of energy 8 keV is also sensitive to X-rays of energy 36.5 keV, because, from equations (7.1.4.3) and (7.1.4.4), A0 k:8 k36:5

28:5 A2 :

7:1:4:6

In Figs. 2.3.5.3c, d and 7.1.4.1d, the escape peak E.P. Ê , the wavelength of 36.5 keV X-rays. shows clearly at 0.35 A There may be a number of weak escape peaks arising from the stronger powder re¯ections. In practice, the escape peak should not be confused with a small-angle re¯ection. It can be tested by reducing the X-ray tube voltage to below the absorption-edge energy of the element in the detector from which it arises.

Fig. 7.1.5.1. Intrinsic ef®ciency of semiconductor detectors. The dimensions are selected to give typical best values of the energy resolution. a Si(Li), detector thickness 3 mm, Be-window thickness 25 mm. b HPGe, detector thickness 5 mm, Be-window thickness 50 mm.

622


7.1. DETECTORS FOR X-RAYS additional circuit to reject pileup pulses that can distort the spectrum at high count rates. The intrinsic ef®ciency, de®ned as the ratio of the number of pulses produced to the number of photons striking the detector, is close to 100% in a large energy range. Because of the penetrating power of high-energy X-rays, the ef®ciency declines at high energies. The low-energy limit is set mainly by the absorption in the beryllium entrance window of the detector. Fig. 7.1.5.1 shows the intrinsic ef®ciency for an Si(Li) detector and HPGe detector with typical crystal size and window thickness. It is seen that the useful photo-energy range is about 1±40 keV for the Si(Li) detector and 2±150 keV for the HPGe detector. Some minor complications of the HPGe detector are a dip in ef®ciency around the germanium K absorption edge at 11 keV and the presence of Ge K and K escape peaks in the measured spectrum. The energy resolution is commonly expressed as the full width at half-maximum (FWHM) of a peak in an energy spectrum. For a spectral peak with Gaussian shape, E(FWHM) corresponds to 2.355 times the root mean square of the energy spread. The energy resolution, including both the detector and the associated electronics, is given by EFWHM fe2n 2:355F"E1=2 2 g1=2 ;

7:1:5:1

where en is the electronic noise contribution, F the Fano factor (about 0.1 for both silicon and germanium), and " the energy required for creating an electron±hole pair. The energy resolution is generally speci®ed at 5.9 keV (Mn K) as a reference energy. Typical best values for a detector with 25 mm2 area are 145 eV (2.5%) for an HPGe detector and 165 eV (2.7%) for an Si(Li) detector. The resolution is degraded for larger detector areas. Count-rate limitations are particularly obvious in synchrotronradiation applications, where high photon ¯uxes are encountered (Worgen, 1982). The count rate is limited to below 105 counts s 1 , mainly by the pulse processing system. Cadmium telluride, mercury iodide and other wide-band-gap semiconductors could be good candidates for energy-dispersive room-temperature X-ray detectors. Until now, the best energy resolution of the Hg2 I spectrometer with both the detector and the preampli®er operating at room temperature is 295 eV (FWHM) for the 5.9 keV Mn K line, corresponding to a relative resolution of 5.0%. By lowering the noise level of the preampli®er FET with cryogenic techniques, a resolution of about 200 eV (3.4%) has been achieved (Warburton, Iwanczyk, Dabrowski, Hedman, Penner-Hahn, Roe & Hodgson, 1986). 7.1.6. Position-sensitive detectors (By U. W. Arndt) Most X-ray diffraction or scattering problems require the quantitative evaluation of a linear or of a two-dimensional pattern. Recent years have seen the development of many different types of linear and area detectors for X-diffraction purposes, that is, of position-sensitive detectors (PSD's) that allow the recording of the positions of the arrival of X-ray photons (Hendricks, 1976; Hendrix, 1982; Arndt, 1986). In addition, imaging detectors have found increasing use in related ®elds, such as in X-ray astronomy (Allington-Smith & Schwarz, 1984), in X-ray microscopy, in X-ray absorption spectroscopy, and in topography. Here, the emphasis is on the production of an image for direct viewing rather than on the making of quantitative intensity measurements; these applications, in general, require an ultra-high spatial resolution over a relatively small ®eld of view and the ability to cope with very low contrast

images: Imaging detectors for topography are discussed in Section 7.1.7. Lessons can also be learnt, and component parts utilized, from quantitative imaging devices developed for visible light. Progress in these ®elds has been covered in the Symposia on Photoelectronic Image Devices held every 3 years at Imperial College London (since 1960) and in the Wire Chamber Conferences (since 1978) and the London Position-Sensitive Detector Conferences (since 1987), both reported in full in Nuclear Instruments and Methods. Detectors are always one of the principal subjects considered at synchrotron-radiation conferences and workshops, the highlights usually being reported in Synchrotron Radiation News. Detectors feature prominently in the proceedings of the IEEE Symposia on Nuclear Science, which appear in the IEEE Transactions. Other recent reviews of X-ray detectors are by Fraser (1989), Stanton (1993), Stanton, Phillips, O'Mara, Naday & Westbrook (1993) and Sareen (1994). The detection of X-ray photons in the energy range of interest for diffraction studies (3 to 20 keV) always involves the interaction of the photon with an inner-shell electron and its complete absorption. The processes that are of interest for the construction of PSD's are of three kinds: (1) Photography. The characteristics of X-ray ®lm are discussed in Section 7.1.1. (2) The use of storage phosphors, such as europium-activated barium halide (BaFX:Eu2 , X Cl or Br) (Sonada, Takano, Miyahara & Kato, 1983; Miyahara, Takahashi, Amemiya, Kamiya & Satow, 1986), which are exposed like photographic ®lm and then scanned with a laser beam causing photonstimulated light emission of an intensity proportional to the original exciting X-ray intensity; this is measured with a photomultiplier. The plate is re-useable when the X-ray image has been erased. These X-ray detectors have a low background, a large dynamic range, and an adequate spatial resolution. See Section 7.1.8. (3) Processes that involve the production of electrons. These may be the result of the ionization of a gas; they may be due to the production of electron±hole pairs in a semiconductor; they may be produced in an X-ray photocathode; ®nally, a phosphor may be used to convert the X-rays into visible light that then produces photoelectrons from a conventional photocathode. At the present time, X-ray-sensitive photographic emulsions are mainly of historical interest. Storage-phosphor image plates have not only largely replaced photographic ®lm; they have also taken over many of the applications of electronic detectors. In the following, we are concerned only with detectors that depend on the production of electrons by incident X-rays. In all detectors, except in semiconductor detectors, the number of primary electrons is multiplied by gas ampli®cation, or in some device such as a microchannel plate or by some other intermediate process. In a PSD, the electron multiplication must take place with a minimum of lateral spread. Many methods are available for deriving the position of the ampli®ed electron stream or of the cloud of electron±ion pairs (avalanche) in an ionized gas and some of these are discussed below. However, almost any combination of photon detection, electron multiplication, and localization procedure can be used in the construction of PSD's (Fig. 7.1.6.1). 7.1.6.1. Choice of detector Detectors may either be true counters in which individual detected photons are counted or they may be integrating devices that generate a signal that is a function of the rate of arrival of

623


7. MEASUREMENT OF INTENSITIES Table 7.1.6.1. The importance of some detector properties for different X-ray patterns

" S 2 = 2 N;

7:1:6:1

where N is the number of quanta incident upon the detector and is the standard deviation of the analogue output signal of amplitude S. For a photon counter with an absorption ef®ciency

1 0 3 2 3 0 1

2 1 3 2 3 0 1

3 3 3 2 3 2 2

3 3 3 3 2 2 2

3 0 1 1 3 1 0

2 1 1 1 0 3 0

0 unimportant; 3 very important.

q, S qN; qN1=2 , and " q. An analogue detector with a DQE " thus behaves like a perfect counter that only detects a fraction " of the incident photons. Under favourable conditions, the DQE of analogue detectors for X-rays is in excess of 0.5, but " varies with counting rate and is lower for detectors with a very large dynamic range, as shown below. The DQE of CCD- and vidicon-based X-ray detectors has been discussed by Stanton, Phillips, Li & Kalata (1992a). 7.1.6.1.2. Linearity of response The linearity of a counter depends on the counting losses, which are due to the ®nite dead-time of the counter and its

Fig. 7.1.6.1. Possible combinations of detection processes, localization methods and read-out procedures in PSD's.

624


Orientation Laue

The detection ef®ciency of a detector is determined in the ®rst instance by the fraction of the number of incident photons transmitted by any necessary window or inactive layer, multiplied by the fraction usefully absorbed in the active region of the detector. This product, which is often called the absorption ef®ciency or the quantum ef®ciency, should be somewhere between 0.5 and 1.0 since the information loss due to incident photons not absorbed in the active region cannot be retrieved by subsequent signal ampli®cation. The useful ef®ciency is best described by the so-called detective quantum ef®ciency (DQE), " (Rose, 1946; Jones, 1958). For our purposes, this can be de®ned as

Topographic

7.1.6.1.1. Detection ef®ciency

Single crystal

Spatial resolution Lack of parallax Accuracy of intensity measurement High count rate capability Suitability for short time slices Suitability for short wavelengths Energy discrimination

Powder

Detector property

Fibre

Type of pattern

Solution

photons; this signal may then be digitized for recording purposes. The choice of a linear or of a two-dimensional, or area, PSD depends on its detection ef®ciency, linearity of response to incident X-ray ¯ux, dynamic range, and spatial resolution, its uniformity of response and spatial distortion, its energy discrimination, suitability for dynamic measurements, its stability, including resistance to radiation damage, and its size and weight. Before discussing a few of the many types of PSD individually, we shall examine these criteria in a general way. Some of them have been discussed by Gruner & Milch (1982). Table 7.1.6.1 shows the importance, on a scale of 0 to 3, of some of the factors in different ®elds of study. Other properties, such as stability and sensitivity, are equally important for all PSD's.

7.1. DETECTORS FOR X-RAYS processing circuits. The counting losses are affected by the time modulation, if any, of the source, as, for example, with storage rings (Arndt, 1978). Counting losses can affect the behaviour of detectors in two different ways. In most analogue detectors and in counters with parallel read-out, each pixel behaves as an independent detector and the counting loss at any point depends only on the local counting rate. In other devices, such as multiwire proportional chambers with delay-line read-out (see Subsection 7.1.6.2), the whole detector becomes dead after an event anywhere in the detector and what matters is the global counting rate. Fortunately, the fractional counting loss is the same for all parts of the pattern so that the relative intensities in a stationary pattern are not affected. 7.1.6.1.3. Dynamic range The lowest practically measurable intensity is determined by the inherent background or noise of the detector. Some form of discrimination against noise pulses is usually possible with a detector that counts individual photons, but not, of course, with integrating detectors. The maximum intensity at which a counter can operate is determined by the dead-time. In the case of an integrating or analogue detector with a variable gain, there is a trade-off between maximum intensity and DQE. Such a device can often be regarded as having an output signal with an amplitude S NV =M that is a noise-free representation of N, the number of photons detected in the integrating period of the device, where V, the maximum signal amplitude, is produced by M photons in this period. M can be varied by altering the gain of the detector. The noise can be regarded as a ®xed fraction 1=r of the maximum amplitude V that is added to the signal. Then the DQE will be " S 2 = 2 N 1 M 2 =r 2 N 1 :

7:1:6:2

This equation shows the importance of having as small a value of 1=r as possible; it also demonstrates that, for a given value of r, M can be increased only at the expense of a reduced DQE. This is valid for X-ray ®lm (Arndt, Gilmore & Wonacott, 1977), for television detectors (Arndt, 1984), for the integrating gas detectors discussed in Subsection 7.1.6.2, and for many semiconductor X-ray detectors. 7.1.6.1.4. Spatial resolution The spatial resolution of a PSD is determined by the number and size of resolution or picture elements (pixels) along the length or parallel to the edge of the detector. In most diffraction experiments, the size of the pattern can be scaled by altering the distance of the detector from the sample and what is important is the angular resolution of the detector when placed at a distance where it can `see' the entire pattern. We shall see below that linear PSD's can be made with up to 2000 pixels and that area detectors are mostly limited to fewer than 512 512 pixels. The sizes of pixels range from about 10 mm for semiconductor PSD's to about 1 mm for most gas-®lled detectors. The useful number of pixels of a detector is determined by its point-spread function (PSF). This is the relative response as a function of distance from the centre of a point image on the detector. PSF's are not necessarily radially symmetrical and may have to be speci®ed in at least two directions at right angles, for example along and perpendicular to the lines of a television raster scan. The width of the PSF at the 50% level determines the

amount of detail visible in a directly viewed image. The accuracy of intensity measurements may depend more critically upon the width of the PSF at a lower level, since a weak spot may be immeasurable when sitting on the `tail' of a very intense one. For various physical reasons, the PSF's of all PSD's, including X-ray ®lm, have appreciable tails. The spatial resolution of a detector is affected by parallax: when an X-ray beam is absorbed in a thick planar detector at an angle ' to the normal, the width of the resultant image is smeared out exponentially and its centroid is shifted by an amount sin '=. For 8 keV X-rays incident at 45 on a xenon-®lled counter, for example, this shift is about 4 mm for a ®lling pressure of 1 atm and 0.4 mm for a ®lling pressure of 10 atm. These ®gures illustrate the desirability of high-pressure xenon (Fig. 7.1.6.2) for gas-ionization detectors intended for wideangle diffraction patterns. 7.1.6.1.5. Uniformity of response All PSD's show long-range and pixel-to-pixel variations of response to larger or smaller extents. These can be corrected, in general by means of a look-up table, during data processing, but the measurements necessary for the calibration are often timeconsuming. The output signals of many analogue detectors contain ®xed-pattern noise that is synchronous with the read-out clock. This noise is usually removed during data processing, which in any case requires the subtraction of the background pattern. 7.1.6.1.6. Spatial distortion In most detectors, there is some spatial distortion of the image. Again, the necessary calibration procedure may be timeconsuming. Distortions cause point-to-point variations in pixel size, which produce response variations additional to those from other causes. Corrections for spatial distortion and for non-uniformity of response have been discussed by Thomas (1989, 1990) and by Stanton, Phillips, Li & Kalata (1992b). 7.1.6.1.7. Energy discrimination The amplitude of the signal due to a single photon is usually a function of the photon energy. The variance in this amplitude, or the full width at half-maximum (FWHM) of the pulse-height spectrum, for a monoenergetic input, depends on the statistics of the detection process. A sharp pulse-height-distribution (PHD) curve may permit simultaneous multi-wavelength measurements with a suitable counter, or at least afford a reduction of the background by pulse-height discrimination. In an analogue

Fig. 7.1.6.2. Absorption of 8 keV and 17 keV photons in argon and xenon as a function of pressure in atm column length in mm.

625


7. MEASUREMENT OF INTENSITIES detector, the variance in the primary amplitude affects the DQE (Arndt & Gilmore, 1979). 7.1.6.1.8. Suitability for dynamic measurements Many investigations, such as low-angle or ®bre diffraction studies on biological materials carried out with synchrotron radiation (Boulin, Dainton, Dorrington, Elsner, Gabriel, Bordas & Koch, 1982; Huxley & Faruqi, 1983), involve the time variation of a diffraction pattern. For very short time slots, only counters can be employed; the incoming pulses must then be gated and stored in an appropriate memory (Faruqi & Bond, 1982). Stores used for these experiments are described as histogramming memories; the contents of a given storage location are incremented by one whenever the corresponding address, which represents the position and the time of arrival of a photon, appears on the address bus (Hendricks, Seeger, Scheer & Suehiro, 1982; Hughes & Sumner, 1981). 7.1.6.1.9. Stability Stability of the performance of a detector is of paramount importance. Most position-sensitive detectors are used in connection with microcomputers, which make calibration and corrections for spatial distortion, non-uniformity of response, and lack of linearity relatively easy, provided that these distortions remain constant. The long-term stability of many detectors, notably of semiconductor devices, is affected by radiation damage produced by prolonged exposure to intense irradiation. 7.1.6.1.10. Size and weight The size and weight of the detector determine the ease with which the detector can be moved relative to the sample and thus the extent to which the diffractometer can be adapted to varying resolution and collimation conditions. Some detectors cannot be moved easily (Xuong, Freer, Hamlin, Nielsen & Vernon, 1978), or need very heavily engineered rotational and translational displacement devices (Phizackerley, Cork & Merritt, 1986). Others, such as spherical drift chamber multiwire proportional chambers, are designed for use at a ®xed distance from the sample and may only be swung about the latter but not translated (Kahn, Fourme, Bosshard, Caudron, Santiard & Charpak, 1982). In single-crystal diffraction patterns, the Bragg re¯ections may be visualized as diverging from the X-ray source while the background ± ¯uorescence X-rays, scatter by amorphous material on the specimen crystal and its mount ± diverges from the sample. Consequently, the highest re¯ection-to-background ratio is achieved by using a large detector at a large distance from the specimen. Of all the detectors discussed here, the image plate can most readily and economically be used to cover a large area; its present popularity is chie¯y due to this property (Sakabe, 1991). There is fairly general agreement on the speci®cation of an ideal X-ray area detector. It should be at least 250 mm in diameter, contain at least 1000 1000 pixels, have a large dynamic range and a high detective quantum ef®ciency for photon energies up to 20 keV and be capable of being read-out rapidly. Many suggestions have been made for improving the performance of existing detectors. It has become apparent, however, that the development of ideal, or even better, X-ray detectors is extremely expensive and, therefore, that their

development and installation can be undertaken only in central national or international laboratories such as storage-ring synchrotron-radiation laboratories. 7.1.6.2. Gas-®lled counters In all gas-®lled counters, whether one-, two-, or threedimensional, the initial event is the absorption of the incoming X-ray photon in a gas molecule with the emission of a photo-, or alternatively an Auger, electron. The detection ef®ciency depends on the fraction of the photons absorbed in the gas and this fraction is shown in Fig. 7.1.6.2 as a function of the product of gas pressure and column length for 8 and 17 keV photons on argon and xenon. The ionization energy of noble gases is about 30 eV so that one 8 keV photon gives rise to about 270 electron± ion pairs. With adequately high collecting ®elds, the electrons acquire suf®cient energy to produce further ionization by collision with neutral ®lling gas molecules; this process is often referred to as àvalanche production' or `gas multiplication'. The factor A by which the number of primary ion pairs is multiplied can be as great as ten to one hundred thousand. Up to a certain value of A, the total amount of ionization remains proportional to the energy of the original X-ray photon. The electrical signal generated at the anode of the counter is due very largely to the movement of the positive ions from the immediate vicinity of that electrode; at the same time, a corresponding pulse is induced on the cathode. The signal can be shaped to produce a pulse with a duration of the order of a microsecond. In single or multiwire proportional counters, the secondary ionization (avalanche production) takes place in the highest ®eld region, that is, within a distance of a few wire diameters of the anode wire or wires. The electrons are collected on the anode and the positive ions move towards the cathode, with very little spread of the ionization in a direction perpendicular to the ®eld gradient, that is, parallel to the wire direction. It is thus possible to construct position-sensitive devices based on such chambers. Proportional-counter behaviour is discussed in detail in many standard texts and review articles (Wilkinson, 1950; Price, 1964; Dyson, 1973; Rice-Evans, 1974). The gas ampli®cation does not have to take place in the same region of the detector as the original absorption. In so-called drift chambers, the primary ionizing event takes place in a low-®eld region where no avalanching takes place. The electrons drift through a grid or grids into a region where the ®eld is suf®ciently high for gas multiplication to occur. The drift ®eld can be made cylindrical in a linear counter (Pernot, Kahn, Fourme, Leboucher, Million, Santiard & Charpak, 1982), or spherical in an area detector (Charpark, 1982; Kahn, Fourme, Bosshard, Caudron, Santiard & Charpak, 1982), centred on the point from which the X-rays diverge, that is on the specimen; the electrons then drift in a radial direction without parallax being introduced (Fig. 7.1.6.3). In many experiments, use is made of the energy discrimination of the detector. The ratio of the full width at half-maximum to the position of the maximum of the pulse-height distribution is given by w 2:36F f =N;

where N is the number of primary ion pairs produced, F is the Fano factor (Fano, 1946, 1947), which takes into account the partially stoichastic character of the gas multiplication process, and f is the avalanche factor. For proportional counters ®lled with typical gas mixtures (argon methane), F 0:17 and f 0:65, so that for 8 keV photons w 13%, but, in the

626


7:1:6:3

7.1. DETECTORS FOR X-RAYS so-called Penning gas mixtures (e.g. noble gas and ethylene), f can approach zero at a certain ®eld strength. In a wire counter with its rapidly varying ®eld strength, f is small only for a gas ampli®cation of less than 50. The energy resolution for 8 keV photons could then be as low as 6%, but the pulses induced on the cathode wires of a MWPC are then too small to permit a precise localization. This problem has been overcome by using uniform-®eld avalanching in two regions in tandem, separated by a drift space (Schwarz & Mason, 1984, 1985). The energy information was derived after the ®rst low-gain gas multiplication process A 500: a proportion of the electrons from the ®rst avalanche then drifted into the second avalanche region which boosted the gas gain to more than 105 , necessary to give a high spatial resolution. In an alternative method (Charpak, 1982; Siegmund, Culhane, Mason & Sanford, 1982), the additive avalanche factor f is eliminated by deriving the energy information, not from the collected charge, but from the visible light pulse produced by the individual avalanches of each primary electron. 7.1.6.2.1. Localization of the detected photon There are several methods of deriving the position of the detected photon that are applicable to both linear and area detectors. (1) The charge produced in the avalanche can be collected on a resistive anode. In the case of linear detectors, the central wire can be given a low or a very high resistance. The latter type is most commonly made from a quartz ®bre coated with carbon. The emerging pulse is detected at both ends of the wire (Borkowski & Kopp, 1968; Gabriel & Dupont, 1972). Area detectors with a resistive disc anode must have at least three read-out electrodes (StuÈmpel, Sanford & Goddard, 1973). With low-resistance electrodes, the position of the event can be computed by analogue circuits from the relative pulse amplitudes (Fig. 7.1.6.4a); a preferred method with high-resistance anodes is to measure the rise times of the output pulses that are determined by the time constant formed by the input capacity of the pulse ampli®er at each output and the resistance of the path from the detection point to the output electrode (Fig. 7.1.6.4b). (2) The anode or cathode can be constructed in the form of two or more interleaved resistive electrodes insulated from each other. Provided that the charge distribution covers at least one

unit of the pattern, positional information can be derived by relative pulse height or by timing methods. Examples of this type of read-out are the linear backgammon ( jeu de jacquet) counter together with its two-dimensional variant (Allemand & Thomas, 1976), the wedge-and-strip anode developed by Anger and his collaborators (Anger, 1966; Martin, Jelinsky, Lampton, Malina & Anger, 1981), and its polar coordinate analogue (Knibbeler, Hellings, Maaskamp, Ottewanger & Brongersma, 1987), for two-dimensional read-out. The method seems capable of a higher spatial resolution than any other (Schwarz & Lapington, 1985). (3) The anode or cathode can be made from a number of sections connected to a tapped delay line (Fig. 7.1.6.4c). Positional information is derived from the time delay of the pulse relative to the arrival of an undelayed prompt pulse. Linear PSD's (LPSD's) with delay-line read out are usually made straight, but variants have been produced in the form of circular arcs (WoÈlfel, 1983; Ballon, Comparat & Pouxe, 1983). Area detectors of this type require two parallel planes of parallel wires with the wires in the two planes at right angles to one another placed on either side of the anode, which also consists of parallel wires. The prompt pulse in such a detector, the multiwire proportional chamber (MWPC), is usually taken from the anode (Fig. 7.1.6.5). In counters without a drift space, the electron avalanche always ends up on one anode wire, and there is then a pseudo-quantization in the position measurement made at right angles to the direction of the anode wires. In driftspace detectors with a narrow anode-wire spacing, the avalanche lands on more than one wire and some interpolation is possible. In the direction parallel to the anode wires, there is never any quantization and the resolution can be better than the cathode wire spacing: Although pulses are induced on several wires, the centroid of the delayed group of pulses can be measured with precision. Delay-line read-out LPSD's have reached the highest resolution in the hands of Radeka and his group (Smith, 1984). MWPC's of this type have been used for several years (Xuong, Freer, Hamlin, Nielsen & Vernon, 1978; Bordas, Koch, Clout, Dorrington, Boulin & Gabriel, 1980; Baru, Proviz, Savinov Sidorov, Khabakhshev, Shuvalov & Yakovlev, 1978; Anisimov, Zanevskii, Ivanov, Morchan, Peshekhonov, Chan Dyk Tkhan, Chan Khyo Dao, Cheremukhina & Chernenko, 1986). They have a relatively low maximum count rate (16 000:1), linearity better than 1% over the entire dynamic range, and extreme sensitivity. They display no lag or sensitivity to magnetic ®elds and they are not damaged by overlighting. Details of the performance of fully operational systems have been given by Krivanek, Ahn & Keeney (1987), Daberkow, Herrmann, Liu & Rau (1991), Kujawa & Krahl (1992), and Ishizuka (1993). To achieve optimum performance, a number of precautions must be taken including modest cooling of the CCD to suppress thermally generated charge carriers. Readout speeds are long compared with TV rates so that any noise associated with this process is minimized. In practice, high-precision correlated double-sampling techniques are used for analogue-to-digital conversion to realise this end. There is also the need to compensate for ®xed-pattern noise due to non-uniformities of dark current and due to the ®bre plate. For these purposes, use is made of a reference image recorded when the device is simply ¯ooded with uniform illumination. An alternative approach to reduce at least some of the ®xed-pattern noise is to use a lens rather than a ®bre-optic plate between the transmission ¯uorescent screen and the device (Fan & Ellisman, 1993). When all these precautions are taken, the resulting system has suf®cient sensitivity to detect single 100 keV electrons. From the above, it should be clear that the CCD-based system offers many advantages over the TV system if quantitative electron data (images or diffracton patterns) are required for subsequent computer analysis. However, its slower response speed means that its use is limited for alignment purposes and much in situ experimentation. It is advantageous, therefore, to have both systems available. 7.2.3.5. Imaging plates A recent advance using a medium related to ®lm involves the imaging plate (Mori, Katoh, Oikawa, Miyahara & Harada,

641


7. MEASUREMENT OF INTENSITIES 1986), which relies on the phenomenon of photostimulated luminescence. Here the active area is a coating of a photostimulable phosphor that can store energy when excited by electrons. The energy absorbed is then emitted as photons when the medium is illuminated with visible or infrared radiation and this signal is detected using a photomultiplier. The read signal is provided by a scanning He±Ne laser so that, as with the detectors in Subsections 7.2.3.3 and 7.2.3.4, information is recorded in parallel but accessed serially. Initial experiments suggest that the imaging plate offers higher sensitivity and a wider dynamic range than most ®lm, but currently suffers from inferior spatial resolution. Recent discussion of the performance and limitations of imaging plates has been supplied by Mori, Oikawa, Katoh, Miyahara & Harada (1988) and Isoda, Saitoh, Moriguchi & Kobayashi (1991). 7.2.4. Serial detectors 7.2.4.1. Faraday cage The Faraday cage is the most convenient means of determining electron-beam currents. It consists of a small electrically isolated cage with a small hole in it through which electrons enter. Provided the hole subtends a suf®ciently small solid angle and the inner surface of the cage is made of a material with a low back-scattering coef®cient, the probability that any electrons will re-emerge is negligible. An electrometer is normally used to measure the charge that has entered the cage. The main use of the Faraday cage is to calibrate other detectors when absolute electron intensities are required. It also serves a very important role when the total specimen exposure in an experiment must be kept below a critical value owing to the susceptibility of the specimen to radiation damage. As electron charge is being measured, the Faraday cage may be used with electrons of any energy. 7.2.4.2. Scintillation detectors One of the most widely used total ¯ux detectors is a scintillator, the output from which is coupled into a photomultiplier by a light-pipe. In the ®rst stage, an incident electron deposits its energy in the scintillator, producing a number of light photons with an energy de®ciency of up to 20% (Herrmann, 1984). By careful design of the light-pipe, an appreciable fraction of the photons will reach the photomultiplier, which should have a photocathode whose quantum yield peaks around the wavelength of the photons from the scintillator. Even though the quantum yield of the photocathode is likely to be 1 mm2 . Thus, photodiode detectors cannot normally be operated in an electron-counting mode and suffer from a low DQE whenever incident electron currents are small. To optimize their performance in this range, the device should be cooled (to reduce the thermally generated signal) and the electron beam should be scanned relatively slowly so that high-gain low-noise ampli®ers may be used for subsequent ampli®cation of the signal from the photodiode. Above the low-signal threshold, the output from the photodiode varies linearly with incident electron intensity and is once again in a form suitable for direct display or for being digitized and stored.

642


7.2. DETECTORS FOR ELECTRONS The main advantages of semiconductor over scintillation detection systems are their robustness, cheapness, and compactness. The latter is particularly valuable for certain applications in that it allows the detector to be sited very close to the specimen even when space is very con®ned. This occurs, for example, when the specimen is immersed in a magnetic lens and channelling patterns from back-scattered electrons are to be recorded. A further advantage arises if images are to be formed in scanning microscopes using signals from a number of closely positioned detectors whose shapes may be quite complex. Using lithographic techniques, several detectors may be fabricated on a single silicon substrate and, provided the gains of any succeeding ampli®ers are well

matched, a detection system with a well de®ned response function results. 7.2.5. Conclusions A wide variety of different means exists for detecting electrons. Many are almost perfect in that they add very little noise to that already present in the electron beam. However, no single detector meets all the requirements of different experiments and, before selecting a detector for a speci®c purpose, it is necessary to consider the relative importance of the attributes listed in Section 7.2.2. Once this is established, it should be straightforward to determine the optimum detector for the task in hand.

643



7.3. Thermal neutron detection By P. Convert and P. Chieux

7.3.1. Introduction

7.3.3. Neutron detection processes

In this chapter, we shall be concerned with the detection of neutrons having thermal and epithermal energies in the range Ê Given the cost and the rarity of the 0.0002±10 eV (20±0.1 A). neutron sources, it is clear that the recent trends in neutron diffractometry are more and more in the direction of designing new instruments around highly ef®cient and complex detection systems. These detection systems become more and more adapted to the particular requirements of the different experimental needs (counting rate, size, resolution, de®nition, shielding and background, TOF, etc.). It is therefore dif®cult to speak about neutron detectors and intensity measurements as such without reference to the complete spectrometers, and this should include the on-line computer. Most neutron detectors for research experiments have been created and developed using ®ssion reactors as neutron sources Ê [i:e: with an upper limit of usable energy of 0.5 eV (0.4 A)]. Given the relatively low intensity of reactor neutron beams, a very successful effort has been made to increase the detector ef®ciency and the detection area as much as possible. The recent construction of pulsed neutron sources extends the range of incident energy to at least 10 eV and generalizes the use of timeof-¯ight (TOF) techniques. A broad range of fully operational neutron detectors, well adapted to reactors as neutron sources, is commercially available, but this is not yet the case for pulsed sources. Probably due to the variation of intensity of the early neutron beams, it is a tradition in neutron research to monitor the incident ¯ux with a low-ef®ciency detector, which in the best case has a stability of the order of 10 3 , i:e: suf®cient for most experiments.

A detection process consists of a chain of events that begins with the neutron capture and ends with the macroscopic `visualization' of the neutron by a sensor (electronic or ®lm). The quality of a detection process will depend on the ef®ciency of the conversion steps and on the characteristics of the emission steps, which alternate in the process (see Table 7.3.3.1). We present below typical detection processes.

7.3.2. Neutron capture Neutrons' lack of charge and the fact that they are only weakly absorbed by most materials require speci®c nuclear reactions to capture them and convert them into detectable secondary particles. Table 7.3.2.1 lists the neutron-capture reactions that are commonly used in thermal neutron detection. The incoming thermal neutron brings a negligible energy to the nuclear reaction, and the secondary charged particles or ®ssion fragments are emitted in random P directions following the conservation-of-momentum law mi vi 0. The capture or absorption cross sections for a number of nuclei of interest are plotted as a function of neutron energy in Fig. 7.3.2.1. These cross sections are commonly expressed in barns (1 barn 10 28 m2 ). At low energies, they are inversely proportional to neutron velocity, except in the case of Gd, which has a nuclear resonance at 0.031 eV. The total ef®ciency " of neutron detection can be expressed by the equation " 1

exp Na t;

7.3.3.1. Detection via gas converter and gas ionization: the gas detector The neutron capture and the trajectories of the secondary charged particles as well as the speci®c gas ionization along these trajectories are presented in Figs. 7.3.3.1a and b. Since the gas ionization energy is about 30 eV per electron (42 eV for 3 He and 30 eV for CH4 ), there are about 25 000 ion pairs (e , He or e , CH 4 ) per captured neutron. Gases such as CH4 or C3 H8 are added to diminish the length of the trajectories, i:e: the wall effect [see Subsection 7.3.4.2b]. We give in Fig. 7.3.3.1c the proton range of an 3 He neutroncapture reaction in various gases (Fischer, Radeka, & Boie, 1983). A schematic drawing of a gas monodetector, which might be mounted either in axial or in radial orientation in the neutron beam, is given in Fig. 7.3.3.1d. For this type of detector, the ef®ciency as a function of the gas pressure, or gas-detector law, is written as "l 1

with P atm the detector-gas pressure at 293 K, t cm the Ê the detected neutron wavelength. gas thickness, and l A The numerical coef®cient b; obtained at 293 K from the ideal gas law, the Avogadro number NA , and the gas absorption cross Ê is section a (barns) at l0 1:8 A, b

273 NA a: 293 22 414 l0

Ê b 0:07417; for For 3 He, with a 5333 barns at l0 1:8 A, 10 Ê BF3 , with a 3837 barns at l0 1:8 A, b 0:0533. We give in Table 7.3.3.2 a few examples of gas-detector characteristics.

7:3:2:1

where N is the number of absorbing nuclei per unit volume, a is their energy-dependent absorption cross section, and t is the thickness of the absorbing material. The factor 1 exp Na t gives the neutron-capture ef®ciency, while is a factor 1 that depends on the detector geometry and materials (absorption and scattering in the front window) and on the ef®ciency of the secondary particles.

Fig. 7.3.2.1. The capture cross sections for a number of nuclei used in neutron detection. [Adapted from Convert & Forsyth (1983).]

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exp bPtl;

7.3. THERMAL NEUTRON DETECTION Table 7.3.2.1. Neutron capture reactions used in neutron detection n = neutron, p = H = proton, t = 3 H = triton, 4 H alpha, e = electron Capture reaction

Cross section Ê (barns) at 1 A

Secondary-particle energies (MeV)

3

He n ! t p

3000

t

0:20

p

0:57

6

Li n ! t

520

t

3:74

2:05

10

B n !7Li (93%) j! 7 Li 7 ! Li (7%)

157

Gd n ! Gd j!

conversion electrons

235

U n ! fission fragments

2100 74000 (nat Gd: 17000) 320

There are two modes of operation. In the case of direct collection of charges, the 25 000 electrons corresponding to one neutron capture (primary electrons) are collected by the anode in about 100±500 ns, and generate an input pulse in the charge preampli®er (see Section 7.3.4). If the electrical ®eld created by the high voltage applied to the anode exceeds a critical value, the electrons will be accelerated suf®ciently to produce a cascade of ionizing collisions with the neutral molecules they encounter, the new electrons liberated in the process being called secondary electrons. This phenomenon, gas multiplication, occurs in the vicinity of the thin wire anode, since the ®eld varies as 1=r. The avalanche stops when all the free electrons have been collected at the anode. With proper design, the number of secondary electrons is proportional to the number of primary electrons. For cylindrical geometries, the multiplication coef®cient M can be calculated (Wolf, 1974). This type of detection mode is called the proportional mode. It is very commonly used because it gives a better signal-to-noise ratio (see Section 7.3.4). A few critical remarks about gas detectors: (i) Some gases have a tendency to form negative ions by the attachment of a free electron to a neutral gas molecule, giving a loss of detector current. This effect is negligible for 3 He but it limits the use of 10 BF3 to about 2 atmospheres pressure, although traces of gases such as O2 or H2 O (e:g: detector materials and wall outgasing) are often the reason for loss by attachment. (ii) Pure 3 He and 10 BF3 gas detectors are practically insensitive to radiation. This is no longer the case when additional gases, which are necessary for 3 He, are used, although the polyatomic additives C3 H8 and CF4 are much better than the rare gases Kr, Xe, and Ar (Fischer, Radeka & Boie, 1983). (iii) For various reasons (the price of 3 He and 10 BF3 and the toxicity of BF3 ), neutron gas detectors are closed chambers, which must be leak-proof and insensitive to BF3 corrosion. The wall thickness must be adapted to the inside pressure, which sometimes implies a rather thick front aluminium window (e:g: a 10 mm window for a 16 bar 3 He gas position-sensitive detector; aluminium is chosen for its very good transmission of neutrons, about 90% for 10 mm thickness). 7.3.3.2. Detection via solid converter and gas ionization: the foil detector This mode of detection is generally used for monitors. In a typical design, a 10 B deposit of controlled thickness, for example

7

e spectrum

spectrum

0.07 to 0.182 up to 8

7

Li Li

0:83 1:01

0:48

Fission fragments up to 80

Ê is t 0:04 mm giving a capture ef®ciency of 10 3 at l 1 A, made on a thin aluminium plate (see Fig. 7.3.3.2). One of the two particles (, Li) produced in the solid by the capture reaction is absorbed by the plate; the other escapes and ionizes the gas. The electrons produced are collected by the aluminium plate, itself acting as the anode, or by a separate anode wire, allowing the use of the proportional mode. The detection ef®ciency is proportional to the deposit thickness t, but t must be kept less than the average range r of the secondary particles in the deposit (for 10 B, r 3:8 mm and rLi 1:7 mm), which limits the Ê The ef®ciency to a maximum value of 3±4% for l 1 A. fraction of the secondary particle energy that is lost in the deposit reduces the detector current, i:e: the signal-to-noise ratio, and worsens the amplitude spectrum (see Section 7.3.4). 7.3.3.3. Detection via scintillation In the detection process via scintillation (see Table 7.3.3.1), the secondary particles produced by the neutron capture ionize and excite a number of valence-band electrons of the solid scintillator to high-energy states, from which they tend to decay with the emission of a light ¯ash of photons detected by a photomultiplier [see Fig. 7.3.3.3a]. A number of conditions must be satis®ed: (i) The scintillation must be immediate after the neutroncapture triggering event. (ii) The scintillation decay time must be short. It depends on materials, and is around 50±100 ns for lithium silicate glasses. (iii) A large fraction of the energy must be converted into light (rather than heat). (iv) The material must be transparent to its own radiation. Most thermal neutron scintillation detectors are currently based on inorganic salt crystals or glasses doped with traces of an activating element (Eu, Ce, Ag, etc.) (extrinsic scintillators). (A plastic scintillator might be considered to be a solid organic solution with a neutron converter.) The use of extrinsic scintillators (Convert & Forsyth, 1983), although less ef®cient energetically, permits better decoupling of the energy of the photon-emitting transition (occurring now in the activator centres) from that of the valence-band electron excitation or ionization energy. The crystal or glass is then transparent to its own emission, and the light emitted is shifted to a wavelength better adapted to the following optical treatment.

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1:47 1:78

7. MEASUREMENT OF INTENSITIES Table 7.3.3.1. Commonly used detection processes 1st conversion (neutron captures)

2nd conversion

3rd conversion

Sensor

Capture/solid ! e n 157 Gd

Gas ionization

Scintillation

Film

Capture/gas n 3 He ! p t

Gas ionization ! e (3 He add. gas)

Electronics

Capture/solid n 10 B; 6 Li; 235 U ®ssion products

Gas ionization ! e (e.g. Ar CO2 )

Electronics

Capture/solid:

Fluorescence/solid:

LiF ZnS(Ag) n 6 Li ! t

LiF ZnS(Ag) !

LiF ZnS(Ag) n 6 Li ! t

LiF ZnS(Ag) !

Photoelectric effect ! e

Electronics

Ce3 enriched Li glass n 6 Li ! t

Ce3 enriched

Photoelectric effect ! e

Electronics

Li glass

!

Film

photon.

Table 7.3.3.2. A few examples of gas-detector characteristics Detection gas 10

Additional gas

BF3

Gas pressure (atm)

Useful detection volume (mm mm)

1

L 200, 50

Capture ef®ciency Ê l1A

Ê l2A

Axial Radial*

65.5% 23.4%

88.1% 41.3%

Mounting

3

He

5

L 100, 50

Axial Radial*

97.5% 84.4%

99.9% 97.5%

3

He

8

L 250, 10

Radial*

44.7%

69.5%

3

He (monitor)

2

100 40 40

C 3 H8

10

5

to 10

3

* Value calculated for the diameter.

In order to maximize the light collected by the photomultiplier [Fig. 7.3.3.3b], a light re¯ector is added in front of the scintillator, and a light coupler adapts the dimensions of the scintillator to that of the photomultiplier (PM). The area of the scintillator might be very large (up to 1 m2 ). The optimum thickness of a glass scintillator is about 1 to 2 mm, corresponding Ê to a neutron detection ef®ciency of 40 to 97% for l 1:8 A, 6 depending on the Li concentration (Strauss, Brenner, Chou, Schultz & Roche, 1983). In a Ce-doped Li silicate glass, the number of photons emitted per captured neutron is about 9000, giving ®nally about 1500 electrons at the photocathode in the optimum light-coupling con®guration. The number of photons emitted per captured neutron, the number of those reaching the photocathode, and the scintillator decay time are parameters that might differ by an order of magnitude, depending on the scintillator material. However, the glass scintillator remains for

the time being the best choice, since the possible gains given by other materials, in decay time or in the number of photons emitted, are always very severely offset by poor light output (e:g: the plastic scintillator). It is very important to maximize the number of photons per neutron reaching the photocathode, since this will help to discriminate between neutrons and rays [see Fig. 7.3.4.2e]. The optical coupling between the different parts of the detection system must be of very good quality. 7.3.3.4. Films Films are classed as position-sensitive detectors. Two types of neutron converter are used in neutron ®lm-detection processes. In the case of the scintillation ®lm system, a light-sensitive ®lm is pressed close to one or between two plastic 6 LiF/ZnS(Ag) scintillator screens (Thomas, 1972). In the case of the Gd-foil

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7.3. THERMAL NEUTRON DETECTION converter, the conversion electrons are emitted isotropically, with a main energy peak at 72 keV, and collected by an X-ray ®lm in close contact with the converter (Baruchel, Malgrange & Schlenker, 1983). In addition to the advantages given by the ®lm technique in itself (simplicity, low price, direct picture, etc.), neutron photographic methods give the best spatial resolution. However, the resolution is inversely related to the detector ef®ciency and thickness. A good compromise appears to be a thickness of 0.25 mm for a plastic scintillator [i:e: a capture ef®ciency of about 12% and a resolution of 0.1 mm for a one-screen converter Ê see the size of the ionization volumes in a scintillator, at l 1 A; Fig. 7.3.3.3a]. Here, however, as in light scattering, the optical density depends on the exposure time as well as on the incoming ¯ux (Schwartzschild effect), which necessitates a calibration (Hohlwein, 1983). For a natural Gd-foil converter, an optimum thickness is 0.025 mm, giving a resolution of 0.020 mm. The Gd-foil ®lm detector is one order of magnitude

Fig. 7.3.3.2. Typical design of a

Fig. 7.3.3.1. a Neutron capture by an 3 He atom and random-direction trajectory (Ox) of the secondary charged particles in the gas mixture. b Calculated speci®c ionization along the proton and triton trajectory in a 65% 3 He/35% CH4 mixture at 300 K and atmospheric pressure (Whaling, 1958). [Reproduced from Convert & Forsyth (1983).] c Range of a 0.57 MeV proton (from 3 He neutron capture) as a function of the pressure of various gases. [Reproduced from Convert & Forsyth (1983).] d Schematic drawing of a gas monodetector. The arrows represent the incoming beam.

B-foil detector.

Fig. 7.3.3.3. a Schematic representation of the neutron capture, secondary , and triton ionization volumes, and scintillator light emission in a cerium-doped lithium silicate glass. [Reproduced from Convert & Forsyth (1983).] b Schematic representation of a scintillation detector.

647


10

7. MEASUREMENT OF INTENSITIES less ef®cient than the scintillator, but, as in electron microscopy, the optical density is nearly proportional to the exposure. This explains the use of the Gd foil in neutron-diffraction topography. If we take into account the possible inhomogeneity of the converter and the dif®culties related to the ®lm (homogeneity, development, and photodensitometry), an accuracy of 5 to 10% is achievable in the intensity measurements under good conditions. Owing to the differences in the processes, neutron photographic techniques are much more ef®cient than those for X-rays. In the case of the plastic scintillator, the gain is about 103 , which compensates for the much lower neutron ¯uxes. 7.3.4. Electronic aspects of neutron detection 7.3.4.1. The electronic chain Each collected burst of electrons, corresponding to one captured neutron, will be successively ampli®ed, identi®ed (discriminated), and transformed to a well de®ned signal, by a chain of electronic devices that are represented in Fig. 7.3.4.1. In the case of the gas detector, the complete electronic chain is generally contained in a grounded metallic box acting as an electrical shield. The detector is connected to this box via a coaxial cable as short as possible to avoid noise and parasitic capacitance. A high-voltage power supply feeds the detector anode through a ®lter. The charge-sensitive preampli®er contains a ®eld-effect transistor (FET) to minimize the background noise, since, from the detector up to this stage, the electronic level is very low. At the output of the FET, the pulse corresponding to one neutron has an amplitude of about 20 mV. This pulse enters an operational ampli®er with adjustable gain G, which delivers a signal of about 2 V, the analogue signal (ANA). The electronic pulse-rise time (0.5 to a few ms) is adapted to the detector electron-collecting time, i:e: its amplitude is roughly proportional to the number of electrons collected at the anode. The last part of the electronic chain is a discriminator with an adjustable threshold, followed by a trigger delivering a calibrated signal (e:g: 5=0 V), called the logic signal (LOG), which is sent to a scaler. In the case of the scintillator, the photomultiplier ensures the conversion of light to electrons and produces a strongly ampli®ed electronic signal that is processed through a discriminator and trigger as for the gas detector.

are on the same scale (and not different by a factor of two). Fig. 7.3.4.2a shows characteristic shapes obtained when triggering the oscilloscope at, say, 0.2 V and observing a 10 BF3 gas detector either pulse by pulse or in a continuous way. This type of display allows the trained user to check the noise level, the amplitudes of the neutron pulses, the quality of the pulse shape, and the presence of any anomalous signal such as one due to ¯ashes of parasitic electronic or electric noise (e:g: 50 Hz). rays produce fewer electrons than neutrons, so that the corresponding pulses are of a lower amplitude (by a factor of 1=5 to 1=20). b Control of the distribution of the pulse amplitudes. The distribution of amplitudes of the various pulses in number of pulses per amplitude increment dN=dA is analysed and displayed using a multichannel analyser. Fig. 7.3.4.2b shows the amplitude spectrum for a 10 BF3 gas detector. The main peak at A0 corresponds to the main neutron-capture reaction (2.3 MeV, 93%). From its FWHM, we de®ne the detector electronic resolution A=A0 . The right-hand-side small peak is due to the 2.8 MeV, 7% capture reaction. The tail of amplitudes down to 4A0 =11 corresponds to neutrons captured near the detector walls, which stop some of the secondary-particle trajectories (wall effect). The existence of a deep valley [see Fig. 7.3.4.2b] where the discriminator threshold T is placed ensures that all captured neutrons, and nothing else, are counted. The width of this valley guarantees good detection stability. Three effects might reduce this valley. Worsening of the gas quality will reduce all the pulse amplitudes (neutron and gamma), an increase in electronic noise will result in a resolution loss affecting the valley on both sides, and a high level of radiation will produce a pile up of the independent pulses, increasing their amplitude. c Optimization of the threshold and high-voltage values. In order to set the threshold T at its optimum value for discrimination and stability, the total number of counts in the detector per unit time is plotted as a function of the threshold, for

7.3.4.2. Controls and adjustments of the electronics For the gas detector, there are basically three parameters to be adjusted, the gas-ampli®cation coef®cient M (a function of the detector high voltage), the electronic ampli®cation gain G, and the discriminator threshold T . Since these adjustments are interactive, the high voltage is initially set at the value given by the manufacturer. The gain G is adjusted in order not to saturate the ampli®er. When the electronic ampli®er power supply voltage is 5 V, a typical setting of the pulse maximum amplitudes is about 3 V. We present now the three types of operation necessary to adjust the electronics. a Control of the pulse shape. The analogue pulse ANA (see Fig. 7.3.4.1) is directly observed with an oscilloscope. Depending on the construction of the electronic chain, it is important to verify if the input of the oscilloscope (or of the multichannel analyser, see below) must be adapted with a 50

impedance or not. This will ensure that the amplitude of the analogue pulse and the accessible value of the voltage threshold

Fig. 7.3.4.1. Electronic chain following a an 3 He gas detector and b a scintillation detector.

648


7.3. THERMAL NEUTRON DETECTION a constant incident ¯ux [see Fig. 7.3.4.2c]. On this curve, the width of the plateau and a value of the slope about 10 4 (in relative variation of counts per mV) give an indication of the detector quality. A good compromise is to set the threshold T at the middle of the plateau. It is also necessary to verify that the detector high voltage, i:e: the gas-ampli®cation coef®cient M (see Subsection 7.3.3.1), is well adapted. With the value of the threshold T adjusted as above, the number of counts per unit time is plotted as a function of the high voltage [see Fig. 7.3.4.2d]. Typical values for the width of the plateau and its slope are 200 V and a few per cent per 100 V. If the high-voltage setting given by the manufacturer must be modi®ed (owing to the worsening of the gas or constraints from the electronic chain, etc.), the complete adjustment procedure of the G and T parameters must be repeated. The electronic adjustments and controls of types of detector other than 10 BF3 gas detectors are basically the same once the changes in the amplitude spectrum have been taken into account. We present in Fig. 7.3.4.2e the amplitude spectra for an 3 He gas detector with signi®cant wall effects, for a 10 B solid-deposit detector with very low ef®ciency, and for a scintillator. The energy of the secondary particles produced in an 3 He gas detector is 765 keV, about three times less than in 10 BF3 , reducing the signal-to-noise ratio; the relative importance of the wall effect is greater and extends to A0 =4. In the case of the 10 B deposit detector, only one of the secondary particles escapes the foil, so that we do not detect an amplitude A0 corresponding to the full capture-reaction energy, but only that corresponding

either to an average or Li trace. The quality of the valleys depends on the t=r (foil thickness=particle range) ratio in the 10 B solid (see Fig. 7.3.3.2). The ®gure corresponds to a monitor where t r. For the scintillator, the valley in the amplitude spectrum is not very good, even for good glasses and without radiation. The discrimination is therefore always much inferior to that of a gas detector. Moreover, the gain of the photomultiplier is very sensitive to the high voltage and has long-term stability problems. 7.3.5. Typical detection systems 7.3.5.1. Single detectors In order to measure the scattered intensities, the single detector is mounted in a shield equipped with a collimator between sample and detector. The collimator is adapted to the sample (5 to 200 Soller collimator for a powder diffractometer, or a hole adapted to the size of the beam diffracted by a single crystal). The sensitive area of the detector matches the size of the collimated diffracted beam. This geometry allows one to localize the scattered beam with adequate resolution and to avoid parasitic neutrons. In a powder diffraction measurement, the detector is scanned with a goniometer, each step being monitored. 7.3.5.2. Position-sensitive detectors With the advantages of speed and simultaneity of data collection over broad angular ranges, position-sensitive detectors

Fig. 7.3.4.2. a Characteristic 10 BF3 gas-detector analogue pulses seen on an oscilloscope. b 10 BF3 amplitude spectrum. c Plateau of a 10 BF3 detector as a function of the threshold voltage. d Typical plateau of a 10 BF3 detector in proportional mode as a function of the high voltage. e Amplitude spectra of various detectors.

649


7. MEASUREMENT OF INTENSITIES (PSDs) have seen considerable development (Convert & Chieux, 1986). They are based on the fundamental detection processes described in Section 7.3.3. Their dimensions, the introduction of various systems for position encoding and decoding, and the multiplication of the number of detection chains (with necessary adjustments and controls) have produced detection systems of a complexity that can no longer be grasped by a researcher only occasionally involved in neutron-scattering experiments. The following lines give only a very crude introduction to PSDs at a descriptive level. A PSD is always mounted in a shield that has an opening limited by the cone de®ned by the sample size and the detection area. Given the large opening in the shield, the PSD is very sensitive to parasitic neutrons, especially those coming from materials in the main monochromatic beam. However, if suf®cient care is taken with the sample environment and beamstop design, the signal-to-noise ratio may be as good as or even better than in a single detector. The scattering angle is determined by the angular position of the origin of the PSD, the sample-to-detector distance, and the location of the neutron capture in the PSD. In a PSD, each individual neutron-capture reaction in the continuous converter is localized via an internal encoding system followed by electronic decoding. There are three types of PSD: the gas resistive wire, the gas multielectrode, and the Anger camera (scintillation). (a) Gas PSD with resistance encoding. The simplest examples of this type of PSD are one-dimensional detectors in which the neutron-converter gas is contained in a cylindrical stainless-steel tube with a central, ®ne resistive wire anode. The amplitudes of the output pulses from charge-sensitive ampli®ers attached to each end of the anode are then compared to derive the positional coordinate of the neutron. An alternative method of deriving the positional information is RC (resistance capacitance) encoding followed by time decoding (Borkowski & Kopp, 1975). Twodimensional RC encoding PSDs have been developed. One of the solutions is to have a group of wires interconnected by a chain of resistors with ampli®ers at the resistor chain nodes (Boie, Fischer, Inagaki, Merritt, Okuno & Radeka, 1982). (b) Gas multi-electrode PSD. The simplest examples of this type of PSD are one-dimensional detectors with a small number of discrete electrodes, e:g: 64 parallel anode wires separated by 2.54 mm. Each anode wire has an ampli®er. A logic system analyses the amplitudes of the output pulses corresponding to one neutron that are collected by one, two, or three neighbouring wires, and decides on which wire the neutron was detected. Large curved one-dimensional PSDs have been built on this principle, e:g: 800 wires covering 80 . Two-dimensional PSDs based on the same principle use a matrix system in which a neutron gives rise to two sets of signals on two orthogonal systems of electrodes (Allemand, Bourdel, Roudaut, Convert, Ibel, JacobeÂ, Cotton & Farnoux, 1975). (c) Neutron Anger camera. The principle of the Anger camera is to detect an incident photon or particle with a continuous area of scintillator. The light produced is allowed to disperse before entering an array of photomultipliers (PMs) whose analogue output signals are used to derive the position of the incident neutron. In one dimension, the localization is achieved by a row of PMs (Naday & Schaefer, 1983). In two-dimensional PSDs, a close-packed array of PMs allows the location of the scintillation to be determined by calculating the X and Y centroids using a resistor-weighting scheme (Strauss, Brenner, Lynch & Morgan, 1981). Table 7.3.5.1 gives the characteristics of some PSDs, each example being a good compromise of detector characteristics

adapted to the instrument needs, powder diffraction, singlecrystal diffraction, and small-angle neutron scattering (SANS). From this table, it seems that the characteristics of the various types of PSD that have been presented are nearly equivalent. The homogeneity of neutron detection over the PSD sensitive area is better than 5% in all cases. Among those PSDs, the gas PSDs, which have been installed since the early 1970s, are the most commonly used. The de®nite advantage of the gas multielectrode PSD is to digitize the encoding, thus offering very good stability and reliability of the neutron localization. The resistance-encoding gas PSD has less complex electronics and permits a choice of pixel size (elementary detection area) and thereby de®nition. The Anger camera also offers the same ¯exibility in the choice of the pixel size and, because of the small thickness of the scintillator (1±2 mm), has very little parallax and is well adapted to TOF measurements. However, the sensitivity of the scintillators is much higher than that of the gas PSDs. PSDs permit a simultaneous measurement of intensities over relatively large areas. As compared with the scanned single detector, this opens the way to two possibilities, either shorter counting time (e:g: real-time experiments) or improved statistical precision. However, the high counting rates achieved, as well as the complexity of the detection, increase the effective dead time and the occurrence of defects. The ef®cient use of a PSD therefore requires a detailed understanding of its operation (detection process, encoding±decoding, data storage, etc.) and periodic tests and calibrations.

7.3.5.3. Banks of detectors When it is useful to have a large detection area without requiring spatial continuity of the detection, the solution is in the juxtaposition of several detectors. The selection of the type of detectors, the way of regrouping them, and the design of the collimation system depend on the measuring instrument. An appropriate geometry will optimize the signal-to-noise ratio and the instrumental resolution. Banks of single detectors, up to 64 covering up to 160 in the diffraction plane with Soller collimators in front of each detector, are used for powder diffractometers in reactors [D1A and D2B, Institut Laue±Langevin (1988)]. The relative position of the detectors plus collimators and their response to the neutron intensity have to be measured and calibrated. The bank of detectors is scanned in small steps over the interval between two successive detectors in order to obtain a complete diagram over the angular range of the bank. In a similar way, a juxtaposition of small PSDs with individual collimators is also used [D4, Institut Laue±Langevin (1988)]. In the case of spallation sources, the time-of-¯ight powder diffractometers are made of arrays of detectors at selected diffraction angles, to increase the detection area (Isis, 1992). Whenever it is possible, the time focusing geometry is used (Windsor, 1981). This implies a particular alignment of the detector array in order for it to become equivalent to one detector at one angle. On the same instrument [HRPD, SANDALS, LAD (Isis, 1992)], various types of detector are used ( 3 He gas single detectors of small diameter, Li glass or Li+ZnS scintillators). For each selected diffraction angle, the choice of detector depends on the required resolution, which is better for scintillators because of their small thickness, and on other properties of the detectors (background, stability, discrimination), which are better for 3 He detectors.

650


7.3. THERMAL NEUTRON DETECTION Table 7.3.5.1. Characteristics of some PSDs 1D = one dimensional; 2D = two dimensional; F= = ¯at; C= = 1D curved; X = single crystal. Gas Resistance encoding 1D 2D Location

F= MURR (USA) F= ORNL (USA)

Gas Multi-electrode

1D

Scintillation Anger camera

2D

1D

F= ILL (France) C= CENG (France)

F= ILL (France) C= ILL (France)

F= JuÈlich (FRG)

2D

F= ANL (USA)

Instrument reference

Powder (1)

SANS (2)

F= Liquids (3) C= Powder (4)

F= SANS (3,5) C= X (3)

Powder (6)

X (7)

Detection area (mm)

l 610 25:4

650 650

F= l 162:5 h 70 C= l 2096 or 80 h 70

F= 640 640

l 744 h 20

300 300

Cell size or pixel (mm)

C= 1300 80 or 64 4

10 10

F= 2.54 C= 2.62 or 0.1

F= 5 5 C= 2:54 5

0.7

0:3 0:3

Resolution (mm)

2.5

10 10

F= 3.2 C= 2.6

F= 5 5 C= 3 6:6

2.5

2:7 2:7

Ef®ciency (%)

Ê) 70 (1.3 A

Ê) 90 (4.75 A

Ê) F= 90 (0.7 A Ê) C= 52 (2.5 A

Ê) F= 75 (11 A Ê) C= 85 (1.5 A

Ê) 70 (1.2 A

Ê) 80 (1.8 A

MURR: Missouri University Research Reactor, Columbia, Missouri, USA. ORNL: Oak Ridge National Laboratory, Tennessee, USA. ILL: Institut Laue±Langevin, Grenoble, France. ANL: Argonne National Laboratory, Argonne, Illinois, USA. CENG: Centre d'Etudes NucleÂaires de Grenoble, Grenoble, France. References: (1) Berliner, Mildner, Sudol & Taub (1983); (2) Abele, Allin, Clay, Fowler & Kopp (1981); (3) Institut Laue±Langevin (1988); (4) Roudaut (1983); (5) Ibel (1976); (6) Schaefer, Naday & Will (1983); (7) Strauss, Brenner, Chou, Schultz & Roche (1983).

7.3.6. Characteristics of detection systems We shall present and comment on some characteristics of detectors plus electronic chains in operational conditions. (a) Intrinsic background (i:e: with the reactor or neutron source shut down). The intrinsic background level is about 6 counts h 1 for an 3 He, 3 bar (1 bar 105 Pa) detector ( 50 mm, L 100 mm) in operational conditions. Of course, it is very important to protect the low-level part of the ampli®er from the discriminator and trigger by separating these two stages very carefully, and from electric and electronic parasites by using good ground connections. Additional parasitic effects might be produced by (i) high-voltage ¯ashes in the detector or in dirty or de®cient plugs, (ii) microphony, and (iii) particle emission by the detector walls (e:g: uranium impurities in aluminium, or activation). (b) discrimination. The discrimination of an 3 He detector is said to be 10 8 . From our own experience, we can say that a pure 3 He detector is insensitive to a dose up to 10 mGy h 1 (1 rem h 1 ). However, additional gases increase the -detection ef®ciency (Fischer, Radeka & Boie, 1983). For a good scintillator, the discrimination is of the order of 10 4 (Kurz & Schelten, 1983). (c) Stability. Under good conditions, the gas-detector stability has been veri®ed to be better than 3 10 4 =day and

10 3 =month. The stability of operational scintillators is probably about 10 3 =day but it is known to drift over longer periods of time. (d) Dead time and non-linear effects due to the count rate. The gas detector has a dead time of 1±10 ms depending on the duration of the analogue pulse, which is ®xed by the detector plus adapted electronic chain. The scintillator has a dead time about 10 times shorter, i:e: 0.2 to 1 ms. A rule of thumb is to keep the total dead time to less than 10% of the counting time, which ®xes the maximum counting rate. This limits the effects of non-linearity of the dead time as a function of the counting rate. Non-linear effects in gas detectors are complex and due to (i) the distortion of the electrostatic ®eld near the anode by the space charge, (ii) the decrease of the high voltage at the anode produced at high counting rates by the increased ionic current passing through the high-impedance ®lter, and (iii) the possible shift of the zero level of the charge preampli®er. (e) TOF requirements. In a TOF experiment, it is important to know exactly the time and place of each neutron capture. The thickness of the detector must be as small as possible in relation to the total ¯ight path (scintillators are roughly 10 times thinner than gas detectors). The delay between the neutron capture and the logic pulse given by the ampli®er gives an additional error. Again, the scintillator is about 10 times faster.

651


7. MEASUREMENT OF INTENSITIES 7.3.7. Corrections to the intensity measurements depending on the detection system A good diffractometer on a reactor is supposed to give point by point for each solid-angle element d an exact image of the scattered intensity. This is never perfectly achieved in practice and necessitates some corrections. In all cases, it is very important to be aware that additional background might be created when part of the shielding or of the collimator intersect the monochromatic neutron beam. We suppose for the following discussion that this effect has been corrected or avoided. The wavelength dependence of the detector response (which is needed for inelasticity corrections or TOF measurements) is generally computed from the theoretical detector law [see equation (7.3.2.1)]. At each measuring point, the collected intensity is renormalized by the integrated incident neutron ¯ux, which is measured by the monitoring device. In the case of TOF measurements on a spallation source, the measured intensity must also be renormalized by the wavelength spectrum of the source, obtained from the measurement of an isotropic scatterer such as vanadium. 7.3.7.1. Single detector For up to 10% of dead time in the counting rate, the correction for the dead-time loss is generally considered as linear. If t is the electronic dead time for one neutron (1±10 ms for the gas detectors) and n the number of counts per second, the dead-time correction factor is 1=1 nt. 7.3.7.2. Banks of detectors In the case of a bank of detectors used for a powder diffractometer in a reactor, one has to calibrate the relative positions of the detectors and their response to the neutron intensity by scanning the detectors through a Bragg pattern. In the case of TOF measurements, the detector banks are installed at ®xed angles. For each detector, the measured intensity depends on the detector type, size, and distance to the sample. The neutron and background depends moreover on the detection angle. After background corrections, the intensities

measured by each detector bank are calibrated and matched using the overlaps between spectra. 7.3.7.3. Position-sensitive detectors (a) Calibration of the position. For multi-electrode PSDs, the relative position of the electrodes is ®xed and veri®ed at the time of construction. In the case of other PSDs with analogue encoding, an angular calibration is made periodically with the help of a Bragg pattern, a thin neutron beam, or a cadmium mask moved across a diffuse beam incident on the PSD (Berliner, Mildner, Sudol & Taub, 1983). (b) Calibration of the PSD homogeneity. The response function might be dependent on the position within the PSD and possibly on the intensities collected at other parts of the PSD. The homogeneity of response of a PSD, which is normally better than 5%, can be calibrated to a much higher accuracy, since the stability of the PSDs is generally very good (e:g: 0.1% or better for gas PSDs). In the case of a reactor, the classical method of calibration is the use of an isotropic scatterer such as vanadium. The calibration is made at angles that avoid the very small vanadium Bragg peaks (or with displacement of the PSD to several positions) and that keep a low and isotropic background. Calibration factors, sometimes called cell-ef®ciency coef®cients i , are then obtained. Considering the lack of isotropy of the vanadium pattern, this method is limited to about 1% accuracy. For small PSDs, a precision of 0.1% or better is obtainable by scanning the whole PSD with a step equal to the cell spacing through any nearly isotropic pattern. (c) Particular effects due to high intensities. The dead time of a PSD is complex. It depends on multiple parameters (the independent ampli®ers and the encoding±decoding procedure). However, if there is a unique decoding logic for the whole PSD, and if this gives the highest contribution to the dead time, the ratios of the peak intensities are then conserved. In the case of strong Bragg peaks, the parasitic effect of scattering by the PSD entrance window (e:g: 10 mm aluminium for high-pressure gas PSDs) is detectable and can be corrected after calibration (using an intense and well localized thin beam).

652



7.4. Correction of systematic errors

By N. G. Alexandropoulos, M. J. Cooper, P. Suortti and B. T. M. Willis 7.4.1. Absorption The positions and intensities of X-ray diffraction maxima are affected by absorption, the magnitude of the effect depending on the size and shape of the specimen. Positional effects are treated as they are encountered in the chapters on experimental techniques. In structure determination, the effect of absorption on intensity may sometimes be negligible, if the crystal is small enough and the radiation penetrating enough. In general, however, this is not the case, and corrections must be applied. They are simplest if the crystal is of a regular geometric shape, produced either through natural growth or through grinding or cutting. Expressions for re¯ection from and transmission through a ¯at plate are given in Table 6.3.3.1, for re¯ection from cylinders in Table 6.3.3.2, and for re¯ection from spheres in Table 6.3.3.3. The calculation for a crystal bounded by arbitrary plane faces is treated in Subsection 6.3.3.3. The values of mass absorption (attenuation) coef®cients required for the calculation of corrections are given as a function of the element and of the radiation in Table 4.2.4.3. 7.4.2. Thermal diffuse scattering (By B. T. M. Willis) 7.4.2.1. Glossary of symbols bej ej q Ej q Emeas E0 E1 F h h 2h H j k0 k kB mn m N q qm V vj vL 2 B 0 d d

1 d d

!j q

Direction cosines of ej q Polarization vector of normal mode jq Energy of mode jq Total integrated intensity measured under Bragg peak Integrated intensity from Bragg scattering Integrated intensity from one-phonon scattering Structure factor Planck's constant h divided by 2 Reciprocal-lattice vector Scattering vector Label for branch of dispersion relation Wavevector of incident radiation Wavevector of scattered radiation Boltzmann's constant Neutron mass Mass of unit cell Number of unit cells in crystal Wavevector of normal mode of vibration Radius of scanning sphere in reciprocal space Volume of unit cell Elastic wave velocity for branch j Mean velocity of elastic waves TDS correction factor Scattering angle Bragg angle

Thermal diffuse scattering (TDS) is a process in which the radiation is scattered inelastically, so that the incident X-ray photon (or neutron) exchanges one or more quanta of vibrational energy with the crystal. The vibrational quantum is known as a phonon, and the TDS can be distinguished as one-phonon (®rstorder), two-phonon (second-order), . . . scattering according to the number of phonons exchanged. The normal modes of vibration of a crystal are characterized as either acoustic modes, for which the frequency !q goes to zero as the wavevector q approaches zero, or optic modes, for which the frequency remains ®nite for all values of q [see Section 4.1.1 of IT B (1992)]. The one-phonon scattering by the acoustic modes rises to a maximum at the reciprocal-lattice points and so is not entirely subtracted with the background measured on either side of the re¯ection. This gives rise to the `TDS error' in estimating Bragg intensities. The remaining contributions to the TDS ± the two-phonon and multphonon acoustic mode scattering and all kinds of scattering by the optic modes ± are largely removed with the background. It is not easy in an X-ray experiment to separate the elastic (Bragg) and the inelastic thermal scattering by energy analysis, as the energy difference is only a few parts per million. However, this has been achieved by Dorner, Burkel, Illini & Peisl (1987) using extremely high energy resolution. The separation is also possible using MoÈssbauer spectroscopy. Fig. 7.4.2.1 shows the elastic and inelastic components from the 060 re¯ection of LiNbO3 (Krec, Steiner, Pongratz & Skalicky, 1984), measured with -radiation from a 57 Co MoÈssbauer source. The TDS makes a substantial contribution to the measured integrated intensity; in Fig. 7.4.2.1, it is 10% of the total intensity, but it can be much larger for higher-order re¯ections. On the other hand, for the extremely sharp Bragg peaks obtained with synchrotron radiation, the TDS error may be reduced to negligible proportions (Bachmann, Kohler, Schulz & Weber, 1985).

Differential cross section for Bragg scattering Differential cross section for one-phonon scattering Density of crystal Frequency of normal mode jq

Fig. 7.4.2.1. 060 re¯ection of LiNbO3 (MoÈssbauer diffraction). Inelastic (triangles), elastic (crosses), total (squares) and background (pluses) intensity (after Krec, Steiner, Pongratz & Skalicky, 1984).

653 Copyright © 2006 International Union of Crystallography 654 s:\ITFC\chap7-4.3d (Tables of Crystallography)

7. MEASUREMENT OF INTENSITIES Let Emeas represent the total integrated intensity measured in a diffraction experiment, with E0 the contribution from Bragg scattering and E1 that from (one-phonon) TDS. Then, Emeas E0 E1 E0 1 ;

7:4:2:1

where is the ratio E1 =E0 and is known as the `TDS correction factor'. can be evaluated in terms of the properties of the crystal (elastic constants, temperature) and the experimental conditions of measurement. In the following, it is implied that the intensities are measured using a single-crystal diffractometer with incident radiation of a ®xed wavelength We shall treat separately the calculation of for X-rays and for thermal neutrons.

involved are of small wavenumber, for which the dispersion relation can be written !j q vj q ; 7:4:2:5 where vj is the velocity of the elastic wave with polarization vector ej q. Substituting (7.4.2.5) into (7.4.2.4) shows that the intensity from the acoustic modes varies as 1=q2 , and so peaks strongly at the reciprocal-lattice points to give rise to the TDS error. Integrating the delta function in (7.4.2.4) gives the integrated one-phonon intensity E1

7.4.2.2. TDS correction factor for X-rays (single crystals) The differential cross section, representing the intensity per unit solid angle for Bragg scattering, is 0 2 d N 23 F h H 2h; V d

where N is the number of unit cells, each of volume V , and F h is the structure factor. H is the scattering vector, de®ned by Hk

k0 ;

with k and k0 the wavevectors of the scattered and incident beams, respectively. (The scattering is elastic, so k k0 2=l, where l is the wavelength.) 2h is the reciprocal-lattice vector and the delta function shows that the scattered intensity is restricted to the reciprocal-lattice points. The integrated Bragg intensity is given by Z Z 0 d E0 d dt d

2 Z Z 23 F h H 2h d dt; 7:4:2:2 N V where the integration is over the solid angle subtended by the detector at the crystal and over the time t spent in scanning the re¯ection. Using R H dH 1; with dH H d, equation (7.4.2.2) reduces to the familiar result (James, 1962) Nl3 F h 2 E0 ; 7:4:2:3 V !0 sin 2 where !0 is the angular velocity of the crystal and 2 the scattering angle. The differential cross section for one-phonon scattering by acoustic modes of small wavevector q is 1 2 d 23 F h V d

3 X H ej q E qH q 2h 7:4:2:4 m!2j q j j1 [see Section 4.1.1 of IT B (1992)]. Here, ej q is the polarization vector of the mode jq, where j is an index for labelling the acoustic branches of the dispersion relations, m is the mass of the unit cell and Ej q is the mode energy. The delta function in (7.4.2.4) shows that the scattering from the mode jq is con®ned to the points in reciprocal space displaced by q from the reciprocal-lattice point at q 0. The acoustic modes

with the crystal density. The sum over the wavevectors q is determined by the range of q encompassed in the intensity scan. The density of wavevectors is uniform in reciprocal space [see Section 4.1.1 of IT B (1992)], and so the sum can be replaced by an integral Z X NV dq: ! 23 q Thus, the correction factor E1 =E0 is given by Z 1 3 J q dq; 8

7:4:2:6

where J q

X H ej q2 j

!2 q

Ej q:

7:4:2:7

The integral in (7.4.2.6) is over the range of measurement, and the summation in (7.4.2.7) is over the three acoustic branches. Only long-wavelength elastic waves, with a linear dispersion relation, equation (7.4.2.5), need be considered. 7.4.2.2.1. Evaluation of Jq The frequencies !j q and polarization vectors ej q of the elastic waves in equation (7.4.2.7) can be calculated from the classical theory of Voigt (1910) [see Wooster (1962)]. If b e1 , be2 , be3 are the direction cosines of the polarization vector with respect to orthogonal axes x, y, z, then the velocity vj is determined from the elastic stiffness constants cijkl by solving the following equations of motion. be1 A11 v2j b e3 A13 0; e2 A12 b 2 be1 A12 b e2 A22 vj b e3 A23 0; be1 A13 b e2 A23 b e3 A33 v2j 0: Here, Akm is the km element of a 3 3 symmetric matrix A; if b q2 , b q3 are the direction cosines of the wavevector q with q1 , b reference to x, y, z, the km element is given in terms of the elastic stiffness constants by Akm

3 P 3 P l1 n1

ql b qn : cklmn b

The four indices klmn can be reduced to two, replacing 11 by 1, 22 by 2, 33 by 3, 23 and 32 by 4, 31 and 13 by 5, and 12 and 21 by 6. The elements of A are then given explicitly by

654


2 l3 H 2 F h 2 V !0 sin 2B XX H ej q 2 Ej q; !2q q q j

7.4. CORRECTION OF SYSTEMATIC ERRORS A11

q21 c11b

q22 c66b

and the axes of the reciprocal lattice. If S is the 3 3 matrix that transforms the scattering vector H from orthonormal axes to reciprocal-lattice axes, then

q23 c55b

q2 b q3 2c56b q3 b q1 2c16b q1 b q2 ; 2c15b

A22 c66b q21 c22b q22 c44b q23 2c24b q2 b q3 q3 b q1 2c26b q1 b q2 ; 2c46b

H Sh;

where h h; k; l. The ®nal expression for , from (7.4.2.11) and (7.4.2.12), is

A33 c55b q21 c44b q22 c33b q23 2c34b q2 b q3 q3 b q1 2c45b q1 b q2 ; 2c35b A12

q21 c16b

q22 c26b

q23 c45b

hT ST TSh:

q3 c25 c46 b q2 b

q1 c12 c66 b q2 ; c14 c56 b q3 b q1 b A13 c15b q21 c46b q22 c35b q23 c36 c45 b q3 q2 b q1 c14 c56 b q2 ; c13 c55 b q3 b q1 b A23 c56b q21 c24b q22 c34b q23 c23 c44 b q3 q2 b q1 c25 c46 b q2 : c36 c45 b q3 b q1 b The setting up of the matrix A is a fundamental ®rst step in calculating the TDS correction factor. This implies a knowledge of the elastic constants, whose number ranges from three for cubic crystals to twenty one for triclinic crystals. The measurement of elastic stiffness constants is described in Section 4.1.6 of IT B (1992). For each direction of propagation b q, there are three values of v2j j 1; 2; 3, given by the eigenvalues of A. The corresponding eigenvectors of A are the polarization vectors ej q. These polarization vectors are mutually perpendicular, but are not necessarily parallel or perpendicular to the propagation direction. The function J q in equation (7.4.2.7) is related to the inverse matrix A 1 by J q

3 X 3 kB T X A 1 mn Hm Hn ; 2 q m1 n1

7:4:2:12

T

7:4:2:8

where H1 , H2 , H3 are the x, y, z components of the scattering vector H, and classical equipartition of energy is assumed [Ej q kB T ]. Thus A 1 determines the anisotropy of the TDS in reciprocal space, arising from the anisotropic elastic properties of the crystal. Isodiffusion surfaces, giving the locus in reciprocal space for which the intensity J q is constant for elastic waves of a given wavelength, were ®rst plotted by Jahn (1942). These surfaces are not spherical even for cubic crystals (unless c11 c12 c44 ), and their shapes vary from one reciprocal-lattice point to another.

7:4:2:13

This is the basic formula for the TDS correction factor. We have assumed that the entire one-phonon TDS under the Bragg peak contributes to the measured integrated intensity, whereas some of it is removed in the background subtraction. This portion can be calculated by taking the range of integration in (7.4.2.10) as that corresponding to the region of reciprocal space covered in the background measurement. To evaluate T requires the integration of the function A 1 over the scanned region in reciprocal space (see Fig. 7.4.2.2). Both the function itself and the scanned region are anisotropic about the reciprocal-lattice point, and so the TDS correction is anisotropic too, i.e. it depends on the direction of the diffraction vector as well as on sin =l. Computer programs for calculating the anisotropic TDS correction for crystals of any symmetry have been written by Rouse & Cooper (1969), Stevens (1974), Merisalo & Kurittu (1978), Helmholdt, Braam & Vos (1983), and Sakata, Stevenson & Harada (1983). To simplify the calculation, further approximations can be made, either by removing the anisotropy associated with A 1 or that associated with the scanned region. In the ®rst case, the element Tmn is expressed as Z 1 k T

Tmn B 3 A 1 mn dq; 8 q2 where the angle brackets indicate the average value over all directions. In the second case,

7.4.2.2.2. Calculation of Inserting (7.4.2.8) into (7.4.2.6) gives the TDS correction factor as

3 P 3 P m1 n1

Tmn Hm Hn ;

7:4:2:9

where Tmn , an element of a 3 3 symmetric matrix T, is de®ned by Z A 1 mn k T Tmn B 3 dq: 7:4:2:10 q2 8 Equation (7.4.2.9) can also be written in the matrix form HT TH; T

7:4:2:11

with H H1 ; H2 ; H3 representing the transpose of H. The components of H relate to orthonormal axes, whereas it is more convenient to express them in terms of Miller indices hkl

Fig. 7.4.2.2. Diagrams in reciprocal space illustrating the volume abcd swept out for (a) an ! scan, and (b) a =2, or !=2, scan. The dimension of ab is determined by the aperture of the detector and of bc by the rocking angle of the crystal.

655


Tmn

kB T q 83 m

7. MEASUREMENT OF INTENSITIES

Z Z A

1

mn

displacement factor is increased from B to B B when the correction is made. B is given by

dS;

where qm is the radius of the sphere that replaces the anisotropic region (Fig. 7.4.2.2) actually scanned in the experiment, and dS is a surface element of this sphere. qm can be estimated by equating the volume of the sphere to the volume swept out in the scan. If both approximations are employed, the correction factor is isotropic and reduces to H 2 kB Tqm ; 32 v2L

7:4:2:14

with vL representing the mean velocity of the elastic waves, averaged over all directions of propagation and of polarization. Experimental values of have been measured for several crystals by -ray diffraction of MoÈssbauer radiation (Krec & Steiner, 1984). In general, there is good agreement between these values and those calculated by the numerical methods, which take into account anisotropy of the TDS. The correction factors calculated analytically from (7.4.2.14) are less satisfactory. The principal effect of not correcting for TDS is to underestimate the values of the atomic displacement parameters. Writing exp 1 , we see from (7.4.2.14) that the overall

B

8kB Tqm : 32 v2L

Typically, B=B is 10±20%. Smaller errors occur in other parameters, but, for accurate studies of charge densities or bonding effects, a TDS correction of all integrated intensities is advisable (Helmholdt & Vos, 1977; Stevenson & Harada, 1983). 7.4.2.3. TDS correction factor for thermal neutrons (single crystals) The neutron treatment of the correction factor lies along similar lines to that for X-rays. The principal difference arises from the different topologies of the one-phonon `scattering surfaces' for X-rays and neutrons. These surfaces represent the locus in reciprocal space of the end-points of the phonon wavevectors q (for ®xed crystal orientation and ®xed incident wavevector k0 ) when the wavevector k of the scattered radiation is allowed to vary. We shall not discuss the theory for pulsed neutrons, where the incident wavelength varies (see Popa & Willis, 1994). The scattering surfaces are determined by the conservation laws for momentum transfer, Hk and for energy transfer, h2 k 2

k0 2h q;

k02 =2mn

"h!j q;

7:4:2:15

where mn is the neutron mass and h!j q is the phonon energy. " is either 1 or 1, where " 1 corresponds to phonon emission (or phonon creation) in the crystal and a loss in energy of the neutrons after scattering, and " 1 corresponds to phonon absorption (or phonon annihilation) in the crystal and a gain in neutron energy. In the X-ray case, the phonon energy is negligible compared with the energy of the X-ray photon, so that (7.4.2.15) reduces to k k0 ; and the scattering surface is the Ewald sphere. For neutron scattering, h!j q is comparable with the energy of a thermal neutron, and so the topology of the scattering surface is more complicated. For one-phonon scattering by long-wavelength acoustic modes with q k0 , (7.4.2.15) reduces to k k0

" q;

where vL =vn is the ratio of the sound velocity in the crystal and the neutron velocity. If the Ewald sphere in the neighbourhood of a reciprocal-lattice point is replaced by its tangent plane, the scattering surface becomes a conic section with eccentricity 1= . For < 1, the conic section is a hyperboloid of two sheets with the reciprocal-lattice point P at one focus. The phonon

Fig. 7.4.2.3. Scattering surfaces for one-phonon scattering of neutrons: (a) for neutrons faster than sound ( < 1); (b) for neutrons slower than sound ( > 1). The scattering surface for X-rays is the Ewald sphere. P0 , P1 , etc. are different positions of the reciprocal-lattice point with respect to the Ewald sphere, and the scattering surfaces are numbered to correspond with the appropriate position of P.

Fig. 7.4.2.4. One-phonon scattering calculated for polycrystalline nickel of lattice constant a (after Suortti, 1980).

656


7.4. CORRECTION OF SYSTEMATIC ERRORS wavevectors on one sheet correspond to scattering with phonon emission and on the other sheet to phonon absorption. For > 1, the conic section is an ellipsoid with P at one focus. Scattering now occurs either by emission or by absorption, but not by both together (Fig. 7.4.2.3). To evaluate the TDS correction, with q restricted to lie along the scattering surfaces, separate treatments are required for faster-than-sound < 1 and for slower-than-sound > 1 neutrons. The ®nal results can be summarized as follows (Willis, 1970; Cooper, 1971): (a) For faster-than-sound neutrons, the TDS rises to a maximum, just as for X-rays, and the correction factor is given by (7.4.2.13), which applies to the X-ray case. (This is a remarkable result in view of the marked difference in the one-phonon scattering surfaces for X-rays and neutrons.) (b) For slower-than-sound neutrons, the correction factor depends on the velocity (wavelength) of the neutrons and is more dif®cult to evaluate than in (a). However, will always be less than that calculated for X-rays of the same wavelength, and under certain conditions the TDS does not rise to a maximum at all so that is then zero. The sharp distinction between cases (a) and (b) has been con®rmed experimentally using the neutron Laue technique on single-crystal silicon (Willis, Carlile & Ward, 1986). 7.4.2.4. Correction factor for powders Thermal diffuse scattering in X-ray powder-diffraction patterns produces a non-uniform background that peaks sharply at the positions of the Bragg re¯ections, as in the single-crystal case (see Fig. 7.4.2.4). For a given value of the scattering vector, the one-phonon TDS is contributed by all those wavevectors q joining the reciprocal-lattice point and any point on the surface of a sphere of radius 2 sin =l with its centre at the origin of reciprocal space. These q vectors reach the boundary of the Brillouin zone and are not restricted to those in the neighbourhood of the reciprocal-lattice point. To calculate properly, we require a knowledge, therefore, of the lattice dynamics of the crystal and not just its elastic properties. This is one reason why relatively little progress has been made in calculating the X-ray correction factor for powders.

Table 7.4.3.1. The energy transfer, in eV, in the Compton scattering process for selected X-ray energies Scattering angle '

Cr K 5411 eV

Cu K 8040 eV

Mo K 17 443 eV

Ag K 22 104 eV

0 30 60 90 120 150 180

0 8 29 57 85 105 112

0 17 63 124 185 229 245

0 79 292 575 849 1043 1113

0 127 467 915 1344 1648 1757

Data calculated from equation (7.4.3.1).

imprecise except in the situations where Compton scattering is the dominant process. For this to be the case, there must be an encounter, conserving energy and momentum, between the incoming photon and an individual target electron. This in turn will occur if the energy lost by the photon, E E1 E1 , clearly exceeds the one-electron binding energy, EB , of the target electron. Eisenberger & Platzman (1970) have shown that this binary encounter model ± alternatively known as the impulse approximation ± fails as EB =E2 . The likelihood of this failure can be predicted from the Compton shift formula, which for scattering through an angle ' can be written. E E1

E2

mc2 1

E12 1 cos ' : E1 =mc2 1 cos '

7:4:3:1

This energy transfer is given as a function of the scattering angle in Table 7.4.3.1 for a set of characteristic X-ray energies; it ranges from a few eV for Cr K X-radiation at small angles, up to 2 keV for backscattered Ag K X-radiation. Clearly, in the majority of typical experiments Compton scattering will be inhibited from all but the valence electrons. 7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section 7.4.3.2.1. Semi-classical radiation theory

7.4.3. Compton scattering (By N. G. Alexandropoulos and M. J. Cooper) 7.4.3.1. Introduction

For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (d=d inc , can be factorized as follows:

In many diffraction studies, it is necessary to correct the intensities of the Bragg peaks for a variety of inelastic scattering processes. Compton scattering is only one of the incoherent processes although the term is often used loosely to include plasmon, Raman, and resonant Raman scattering, all of which may occur in addition to the more familiar ¯uorescence radiation and thermal diffuse scattering. The various interactions are summarized schematically in Fig. 7.4.3.1, where the dominance of each interaction is characterized by the energy and momentum transfer and the relevant binding energy. With the exception of thermal diffuse scattering, which is known to peak at the reciprocal-lattice points, the incoherent background varies smoothly through reciprocal space. It can be removed with a linear interpolation under the sharp Bragg peaks and without any energy analysis. On the other hand, in noncrystalline material, the elastic scattering is also diffused throughout reciprocal space; the point-by-point correction is consequently larger and without energy analysis it cannot be made empirically; it must be calculated. These calculations are

Fig. 7.4.3.1. Schematic diagram of the inelastic scattering interactions, E E1 E2 is the energy transferred from the photon and K the momentum transfer. The valence electrons are characterized by the Fermi energy, EF , and momentum, kF (h being taken as unity). The core electrons are characterized by their binding energy EB . The dipole approximation is valid when jKja < 1, where a is the orbital radius of the scattering electron.

657


7. MEASUREMENT OF INTENSITIES Table 7.4.3.2. The incoherent scattering function for elements up to Z = 55 Ê 1 sin =l A Element

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.50

2.00

1H 2 He 3 Li 4 Be 5B 6C 7N 8O 9F 10 Ne 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 55 Cs

0.343 0.296 1.033 1.170 1.147 1.039 1.08 0.977 0.880 0.812 1.503 2.066 2.264 2.293 2.206 2.151 2.065 1.956 2.500 3.105 3.136 3.114 3.067 2.609 2.949 2.891 2.832 2.772 2.348 2.654 2.791 2.839 2.793 2.799 2.771 2.703 3.225 3.831 3.999 4.064 3.672 3.625 3.987 3.559 3.499 3.103 3.362 3.700 3.852 3.917 3.871 3.097 3.903 3.841 4.320

0.769 0.881 1.418 2.121 2.531 2.604 2.858 2.799 2.691 2.547 2.891 3.444 4.047 4.520 4.732 4.960 5.074 5.033 5.301 5.690 5.801 5.860 5.858 5.577 5.791 5.781 5.764 5.726 5.455 5.631 5.939 6.229 6.365 6.589 6.748 6.760 7.062 7.464 7.700 7.879 7.684 7.690 7.984 7.857 7.863 7.725 7.785 7.980 8.297 8.615 8.811 9.076 9.287 9.340 9.615

0.937 1.362 1.795 2.471 3.190 3.643 4.097 4.293 4.347 4.269 4.431 4.771 5.250 5.808 6.312 6.795 7.182 7.377 7.652 7.981 8.169 8.312 8.375 8.206 8.380 8.432 8.469 8.461 8.310 8.388 8.599 8.912 9.236 9.601 9.940 10.157 10.431 10.746 11.010 11.236 11.213 11.260 11.512 11.531 11.591 11.441 11.598 11.812 12.083 12.415 12.777 13.171 13.564 13.892 14.217

0.983 1.657 2.143 2.744 3.499 4.184 4.792 5.257 5.552 5.644 5.804 6.064 6.435 6.903 7.435 8.002 8.553 8.998 9.405 9.790 10.071 10.304 10.454 10.415 10.604 10.733 10.844 10.894 10.778 10.901 11.082 11.338 11.658 12.033 12.440 12.828 13.206 13.576 13.899 14.176 14.317 14.444 14.653 14.782 14.883 14.824 14.969 15.185 15.444 15.746 16.088 16.466 16.876 17.307 17.753

0.995 1.817 2.417 3.005 3.732 4.478 5.182 5.828 6.339 6.640 6.903 7.181 7.523 7.937 8.419 8.960 9.539 10.106 10.650 11.157 11.561 11.901 12.156 12.264 12.486 12.687 12.867 12.980 12.942 13.094 13.290 13.536 13.828 14.168 14.552 14.969 15.410 15.860 16.279 16.658 16.949 17.196 17.456 17.685 17.858 17.943 18.082 18.263 18.489 18.760 19.067 19.407 19.227 20.175 20.612

0.998 1.902 2.613 3.237 3.948 4.690 5.437 6.175 6.832 7.320 7.724 8.086 8.459 8.867 9.323 9.829 10.382 10.967 11.568 12.163 12.648 13.140 13.514 13.770 14.062 14.343 14.596 14.780 14.847 15.020 15.233 15.486 15.775 16.098 16.456 16.849 17.282 17.745 18.215 18.672 19.081 19.455 19.816 20.150 20.428 26.653 20.858 21.064 21.288 21.541 21.823 22.134 22.471 22.833 23.228

0.994 1.947 2.746 3.429 4.146 4.878 5.635 6.411 7.151 7.774 8.313 8.784 9.225 9.667 10.131 10.626 11.158 11.726 12.329 12.953 13.545 14.093 14.574 14.960 15.346 15.716 16.050 16.317 16.494 16.709 16.947 17.215 17.511 17.835 18.185 18.562 18.974 19.420 19.891 20.373 20.844 21.300 21.748 22.172 22.557 22.904 23.212 23.501 23.779 24.059 24.349 25.655 24.980 25.324 25.691

0.999 1.970 2.834 3.579 4.320 5.051 5.809 6.596 7.376 8.085 8.729 9.304 9.830 10.330 10.827 11.336 11.867 12.424 13.014 13.635 14.256 14.856 15.413 15.902 16.376 16.831 17.249 17.602 17.885 18.163 18.445 18.741 19.056 19.391 19.747 20.123 20.526 20.956 21.416 21.895 22.386 22.877 23.370 23.855 24.318 24.756 25.162 25.546 25.906 26.252 26.590 26.927 27.269 27.619 27.981

1.000 1.983 2.891 3.693 4.469 5.208 5.968 6.755 7.552 8.312 9.028 9.689 10.296 10.864 11.411 11.952 12.499 13.061 13.645 14.256 14.885 15.509 16.111 16.670 17.211 17.737 18.229 18.664 19.043 19.395 19.734 20.074 20.420 20.778 21.149 21.535 21.940 22.367 22.820 23.294 23.787 24.288 24.797 25.312 25.819 26.316 26.792 27.252 27.691 28.113 28.518 28.912 29.298 29.680 30.064

1.000 1.990 2.928 3.777 4.590 5.348 6.113 6.901 7.703 8.490 9.252 9.975 10.652 11.286 11.888 12.472 13.050 13.629 14.220 14.830 15.460 16.095 16.721 17.323 17.910 18.488 19.039 19.543 20.002 20.427 20.831 21.224 21.612 22.003 22.399 22.804 23.221 23.654 24.110 24.583 25.077 25.581 26.093 26.621 27.148 27.677 28.195 28.705 29.203 29.687 30.157 30.613 31.056 31.488 31.914

1.000 1.999 2.989 3.954 4.895 5.781 6.630 7.462 8.288 9.113 9.939 10.766 11.592 12.408 13.209 13.990 14.750 15.489 16.212 16.921 17.630 18.334 19.032 19.730 20.411 21.097 21.777 22.445 23.107 23.745 24.370 24.983 25.583 26.171 26.747 27.313 27.871 28.423 28.970 29.517 30.067 30.620 31.173 31.740 32.309 32.888 33.465 34.046 34.634 35.226 35.822 36.422 37.024 37.628 38.232

1.000 2.000 2.998 3.989 4.973 5.930 6.860 7.764 8.648 9.517 10.376 11.229 12.083 12.937 13.790 14.641 15.487 16.324 17.152 17.970 18.782 19.585 20.379 21.168 21.938 22.704 23.462 24.211 24.957 25.683 26.400 27.109 27.810 28.504 29.190 29.870 30.543 31.210 31.870 32.522 33.167 33.808 34.447 35.081 35.715 36.349 36.983 37.618 38.255 38.894 39.536 40.181 40.827 41.477 42.129

d d

inc

d d

0

SE1 ; E2 ; K; Z;

7:4:3:2

where (d=d 0 is the cross section characterizing the interaction, in this case it is the Thomson cross section, e2 =mc2 2 e1 e2 ; e1 and e2 being the initial and ®nal state photon

polarization vectors. The dynamics of the target are contained in the incoherent scattering factor SE1 ; E2 ; K; Z, which is usually a function of the energy transfer E E1 E2 , the momentum transfer K, and the atomic number Z. The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian

658


7.4. CORRECTION OF SYSTEMATIC ERRORS 2

e e pA A A: 7:4:3:3 2me2 me It produces photoelectric absorption through the p A term taken in ®rst order, Compton and Raman scattering through the A A term and resonant Raman scattering through the p A terms in second order. If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes 2 P P 2 S E2 =E1 h f j expiK rj j i i Ef Ei E; H

Table

Element Li B O Ne Mg Si Ar V Cr

j

f

7:4:3:4 where the Born operator is summed over the j target electrons and the matrix element is summed over all ®nal states accessible through energy conservation. In the high-energy limit of E EB , SE1 ; E2 ; K; Z ! Z but as Table 7.4.3.1 shows this condition does not hold in the X-ray regime. The evaluation of the matrix elements in equation (7.4.3.4) was simpli®ed by Waller & Hartree (1929) who (i) set E2 E1 and (ii) summed over all ®nal states irrespective of energy conservation. Closure relationships were then invoked to reduce the incoherent scattering factor to an expression in terms of form factors fjk : S

P j

1

j fj Kj2

j 6P k P j

k

j fjk Kj2 ;

7:4:3:5

where fj K h j jexpiK rj j j i and fjk h

k jexpiK

rk

rj j j i;

the latter term arising from exchange in the many-electron atom. According to Currat, DeCicco & Weiss (1971), equation (7.4.3.5) can be improved by inserting the prefactor E2 =E1 2 , where E2 is calculated from equation (7.4.3.1); the factor is an average for the factors inside the summation sign of equation (7.4.3.4) that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1979)]. The Waller± Hartree method remains the chosen basis for the most extensive compilations of incoherent scattering factors, including those tabulated here, which were calculated by Cromer & Mann (1967) and Cromer (1969) from non-relativistic Hartree±Fock self-consistent-®eld wavefunctions. Table 7.4.3.2 is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975). 7.4.3.2.2. Thomas±Fermi model This statistical model of the atomic charge density (Thomas, 1927; Fermi, 1928) considerably simpli®es the calculation of coherent and incoherent scattering factors since both can be written as universal functions of K and Z. Numerical values were ®rst calculated by Bewilogua (1931); more recent calculations have been made by Brown (1966) and Veigele (1967). The method is less accurate than Waller±Hartree theory, but it is a much simpler computation. 7.4.3.2.3. Exact calculations The matrix elements of (7.4.3.4) can be evaluated exactly for the hydrogen atom. If one-electron wavefunctions in manyelectron atoms are modelled by hydrogenic orbitals [with a

Sexact

Simp

0.879 0.879 0.878 0.875 0.863 0.851 0.843 0.663 0.568

0.878 0.878 0.877 0.875 0.863 0.850 0.826 0.716 0.636

SW

H

0.877 0.877 0.876 0.875 0.872 0.868 0.877 0.875 0.875

Sexact is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. Simp is the incoherent scattering factor calculated by taking the Compton pro®le derived in the impulse approximation and truncating it for E < EB . SW H is the Waller± Hartree incoherent scattering factor. Data taken from Bloch & Mendelsohn (1974).

suitable choice of the orbital exponent; see, for example, Slater (1937)], an analytical approach can be used, as was originally proposed by Bloch (1934). Hydrogenic calculations have been shown to predict accurate K- and L-shell photoelectric cross sections (Pratt & Tseng, 1972). The method has been applied in a limited number of cases to K-shell (Eisenberger & Platzman, 1970) and L-shell (Bloch & Mendelsohn, 1974) incoherent scattering factors, where it has served to highlight the de®ciencies of the Waller±Hartree approach. In chromium, for example, at an incident energy of 17 keV and a Bragg angle of 85 , the L-shell Waller±Hartree cross section is higher than the èxact' calculation by 50%. A comparison of Waller±Hartree and exact results for 2s electrons, taken from Bloch & Mendelsohn (1974), is given in Table 7.4.3.3 for illustration. The discrepancy is much reduced when all electrons are considered. In those instances where the exact method has been used as a yardstick, the comparison favours the `relativistic integrated impulse approximation' outlined below, rather than the Waller± Hartree method. 7.4.3.3. Relativistic treatment of incoherent scattering The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929) theory and the Dirac equation (see Jauch & Rohrlich, 1976). In second-order relativistic perturbation theory, there is no overt separation of p A and A A terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954); they are generally small at X-ray energies. They are of increasing interest in synchrotron-based experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988). Somewhat surprisingly, it is the spectral distribution, d2 = d dE2 , rather than the total intensity, d= d , which is the better understood. This is a consequence of the exploitation of the Compton scattering technique to determine electron momentum density distributions through the Doppler broadening of the scattered radiation [see Cooper (1985) and Williams (1977) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976) and Ribberfors (1975) have shown that the

659


7.4.3.3. Compton scattering of Mo K X-radiation through 170 from 2s electrons

7. MEASUREMENT OF INTENSITIES Compton pro®le ± the projection of the electron momentum density distribution onto the X-ray scattering vector ± can be isolated from the relativistic differential scattering cross section within the impulse approximation. Several experimental and theoretical investigations have been concerned with understanding the changes in the spectral distribution when electron binding energies cannot be discounted. It has been found (e.g. Pattison & Schneider, 1979; Bloch & Mendelsohn, 1974) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers E EB . This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from d d

Z1 E1 EB

d2 dE2 : d dE2

7:4:3:6

Unfortunately, this requires the Compton pro®le of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975)] for all elements. Ribberfors (1983) and Ribberfors & Berggren (1982) have shown that this calculation can be dramatically simpli®ed, without loss of accuracy, by crudely approximating the Compton line shape. Fig. 7.4.3.2 shows the incoherent scattering from aluminium, modelled in this way, and compared with experiment, Waller±Hartree theory, and an exact integral of the truncated impulse Compton pro®le. 7.4.3.4. Plasmon, Raman, and resonant Raman scattering In typical X-ray experiments, as is evident from Table 7.4.3.1, the energy transfer may be so low that Compton scattering will be inhibited from all but the most loosely bound electrons. Indeed, in the situation in metals where K, the momentum transfer, is less than kF (the Fermi momentum), Compton scattering from the conduction electrons may be restricted by exclusion because of the lack of unoccupied ®nal states [see Bushuev & Kuz'min (1977)].

Fortunately, in these uncertain circumstances, the incoherent intensities are low. In this regime, the electron gas may be excited into collective motion. For almost all solids, the plasmon excitation energy is 20±30 eV and, in the random phase approximation, the incoherent scattering factor becomes SE; K / K 2 =wp E h!p ; where !p is the plasma frequency. At slightly higher energies E EB ), Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977). The resultant spectrum is often referred to as the Compton±Raman band. In semi-classical radiation theory, Raman scattering is usually differentiated from Compton scattering by dropping the requirement for momentum conservation between the photon and the individual target electron, the recoil being absorbed by the atom. The Raman band corresponds to transitions into the lowest unoccupied levels and these can be calculated within the dipole approximation as long as jKja < 1, where K is the momentum transfer and a the orbital radius of the core electron undergoing the transition. The transition probability in equation (7.4.3.4) becomes P jh f jrj i ij2 Ef Ei E; 7:4:3:7 f

which implies that the near-edge structure is similar to the photoelectric absorption spectrum. Whereas plasmon and Raman scattering are unlikely to make dramatic contributions to the total incoherent intensity, resonant Raman scattering (RRS) may, when E1 EB . The excitation involves a virtual K-shell vacancy in the intermediate state and a vacancy in the L (or M or N) shell and an electron in the continuum in the ®nal state. It has now been observed in a variety of materials [see, for example, Sparks (1974), Eisenberger, Platzman & Winick (1976), Schaupp et al. (1984)]. It was predicted by Gavrila & Tugulea (1975) and the theory has been Ê berg & Tulkki (1985). The effect is treated comprehensively by A the exact counterpart, in the inelastic spectrum, of anomalous

Fig. 7.4.3.2. The incoherent scattering function, Sx; Z=Z, per electron for aluminium shown as a function of x sin =l. The Waller±Hartree theory ( ) is compared with the truncated impulse approximation in the tabulated Compton pro®les (Biggs, Mendelsohn & Mann, 1975) cutoff at E < EB for each electron group (Ð). The third curve (± ± ±) shows the simpli®cation introduced by Ribberfors (1983) and Ribberfors & Berggren (1982). The predictions are indistinguishable to within experimental error except at low sin =l. Reference to the measurements can be found in Ribberfors & Berggren (1982).

660


7.4. CORRECTION OF SYSTEMATIC ERRORS

Fig. 7.4.3.3. The cross section for resonant Raman scattering (RRS) and ¯uorescence (F) as a function of the ratio of the incident energy, E, and the K-binding energy, EB . The units of d=d are e2 =mc2 2 and the data are taken from Bannett & Freund (1975). For comparison, the intensity of Compton scattering (C) from copper through an angle of 30 is also shown [data taken from Hubbell et al. (1975)].

scattering in the elastic spectrum. It is important because, as the resonance condition is approached, the intensity will exceed that due to Compton scattering and therefore play havoc with any corrections to total intensities based solely on the latter. Although systematic tabulations of resonance Raman scattering do not exist, Fig. 7.4.3.3, which is based on the calculations of Bannett & Freund (1975), shows how the intensity of RRS clearly exceeds that of the Compton scattering for incident energies just below the absorption edge. However, since the problems posed by anomalous scattering and X-ray ¯uorescence are generally appreciated, the energy range 0:9 < E1 =EB < 1:1 is wisely avoided by crystallographers intent upon absolute intensity measurements. 7.4.3.5. Magnetic scattering Finally, and for completeness, it should be noted that the intensity of Compton scattering from a magnetic material with a net spin moment will, in principle, differ from that from a nonmagnetic material. For unpolarized radiation, the effects are only discernible at photon energies greatly in excess of the electron rest mass energy, mc2 511 keV, but for circularly polarized radiation effects at the 1% level can be found in Compton scattering experiments carried out at E1 ' 1=10 mc2 on ferromagnets such as iron. See Lipps & Tolhoek (1954) for a comprehensive description of polarization phenomena in magnetic scattering and Lovesey (1993) for an account of the scattering theory. 7.4.4. White radiation and other sources of background (By P. Suortti) 7.4.4.1. Introduction By de®nition, the background includes everything except the signal. In an X-ray diffraction measurement, the signal is the pattern of Bragg re¯ections. The pro®les of the re¯ections should be determined by the structure of the sample, and so the broadening due to the instrument should be considered as background. In the ideal angle-dispersive experiment, a well collimated beam of X-rays having a well de®ned energy (and a polarization, perhaps) falls on the sample, and only the radiation scattered by the sample is detected. Furthermore, the detector should be able to resolve all the components of scattering by

energy, so that each scattering process could be studied separately. It is obvious that only after this kind of analysis are the Bragg re¯ections (plus the possible disorder scattering) unequivocally separated from the background arising from other processes. In most cases, however, this analysis is not feasible, and the re¯ections are separated by using certain assumptions concerning their pro®le, and the success of this procedure depends on the peak-to-background ratio. The ideal situation described above is all too often not encountered, and experimenters are satis®ed with too low a level of resolution. The aim of the present article is to point out the sources of the unwanted and unresolved components of the registered radiation and to suggest how these may be eliminated or resolved, so that the quality of the diffraction pattern is as high as possible. The article can cover only a few of the possible experimental situations, and only the àlmost ideal' angledispersive instrument is considered. It is assumed that the beam incident on the sample is monochromatized by re¯ection from a crystal and that the scattered radiation is registered by a lownoise quantum detector, which is the standard arrangement for modern diffractometers. Filtered radiation and photographic recording are used in certain applications, but these are excluded from the following discussion. The wavelength-dispersive or Laue methods are becoming popular at the synchrotron-radiation laboratories, and a short comment on these techniques will be included. Other sections of this volume deal with the components of scattering that are present even in the ideal experiment: thermal diffuse scattering (TDS), Compton and plasmon scattering, ¯uorescence and resonant Raman scattering, multiple scattering (coherent and incoherent), and disorder scattering. The rest of the background may be termed `parasitic' scattering, and it arises from three sources: (1) impurities of the incident beam; (2) impurities of the sample; (3) surroundings of the sample. Parasitic scattering is occasionally mentioned in the literature, but it has hardly ever been the subject of a detailed study. Therefore, the present article will discuss the general principles of the minimization of the background and then illustrate these ideas with examples. Most of the discussion will be directed to the ®rst of the three sources of parasitic scattering, because the other two depend on the details of the experiment.

7.4.4.2. Incident beam and sample An ideal diffraction experiment should be viewed as an X-ray optical system where all the parts are properly matched for the desired resolution and ef®ciency. The impurities of the incident beam are the wavelengths and divergent rays that do not contribute to the signal but scatter from the sample through the various processes mentioned above. The propagation of the X-ray beam through the instrument is perhaps best illustrated by the so-called phase-space analysis. The three-dimensional version, which will be used in the following, was introduced by Matsushita & Kaminaga (1980) and was elaborated further by Matsushita & Hashizume (1983). The width, divergence and wavelength range of the beam are given as a contour diagram, which originates in the X-ray source, and is modi®ed by slits, monochromator, sample, and the detection system. The actual ®ve-dimensional diagram is usually given as three-dimensional projections on the plane of diffraction and on the plane perpendicular to it and the beam axis, and in most cases the ®rst projection is suf®cient for an adequate description of the geometry of the experiment.

661


7. MEASUREMENT OF INTENSITIES The limitations of the actual experiments are best studied through a comparison with the ideal situation. A close approximation to the ideal experimental arrangement is shown in Fig. 7.4.4.1 as a series of phase-space diagrams. The characteristic radiation from a conventional X-ray tube is almost uniformly distributed over the solid angle of 2, and the relative width of the K1 or K2 emission line is typically

l=l 5 10 4 . The acceptance and emittance windows of a ¯at perfect crystal are given in Fig. 7.4.4.1(b). The angular acceptance of the crystal (Darwin width) is typically less than 10 4 rad, and, if the width of the slit s or that of the crystal is small enough, none of the K2 distribution falls within the window. Therefore, it is suf®cient to study the size and divergence distributions of the beam in the lK1 plane only, as shown in Fig. 7.4.4.1(c). The beam transmitted by the ¯at monochromator and a slit is shown as the hatched area, and the part re¯ected by a small crystal by the cross-hatched area. The re¯ectivity curve of the crystal is probed when the crystal is rotated. In this schematic case, almost 100% of the beam contributes to the signal. The typical re¯ection pro®le shown in Fig. 7.4.4.2 reveals the details of the crystallite distribution of the sample (Suortti, 1985). The broken curve shows the calculated pro®le of the same re¯ection if the incident beam from a mosaic crystal monochromator had been used (see below). The window of acceptance of a ¯at mosaic crystal is determined by the width of the mosaic distribution, which may be 100 times larger than the Darwin width of the re¯ection in

Fig. 7.4.4.2. Re¯ection 400 of LiH measured with a parallel beam of Mo K1 radiation (solid curve). The broken curve shows the re¯ection as convoluted by a Gaussian instrumention function of 2 0:1 and 2 1 0:13 , which values are comparable with those in Fig. 7.4.4.4.

Fig. 7.4.4.1. Equatorial phase-space diagrams for a conventional X-ray source and parallel-beam geometry; x is the size and x0 dx=dz the divergence of the X-rays. (a) Radiation distributions for two wavelengths, l1 and l2 , at the source of width x, and downstream at a slit of width s1 . (b) Acceptance and emittance windows of a ¯at perfect crystal, where the phase-space volume remains constant, Awa l Ewe l, and the x0 ; l section shows the re¯ection of a polychromatic beam (Laue diffraction). (c) Distributions for one wavelength at the source, ¯at perfect-crystal monochromator, sample (marked with the broken line), and the receiving slit (RS); z is the distance from the source.

Fig. 7.4.4.3. Equatorial phase-space diagrams for two wavelengths, l1 (solid lines) and l2 (broken lines), projected on the plane l l1 . The monochromator at z 200 mm is a ¯at mosaic crystal, and a small sample is located at z 400 mm, as shown by the shaded area. The re¯ected beams at the receiving slit are shown for the ; and ; con®gurations of the monochromator and the sample.

662


7.4. CORRECTION OF SYSTEMATIC ERRORS question. This means that a convergent beam is re¯ected in the same way as from a bent perfect crystal in Johann or Johansson geometry. Usually, the window is wide enough to transmit an energy band that includes both K1 and K2 components of the incident beam. The distributions of these components are projected on the x; x0 ; l1 plane in Fig. 7.4.4.3. The sample is placed in the (para)focus of the beam, and often the divergence of the beam is much larger than the width of the rocking curve of the sample crystal. This means that at any given time the signal comes from a small part of the beam, but the whole beam contributes to the background. The pro®le of the re¯ection is a convolution of the actual rocking curve with the divergence and wavelength distributions of the beam. The calculated pro®le in Fig. 7.4.4.2 demonstrates that in a typical case the pro®le is determined by the instrument, and the peak-to-background ratio is much worse than with a perfect-crystal monochromator. An alternative arrangement, which has become quite popular in recent years, is one where the plane of diffraction at the monochromator is perpendicular to that at the sample. The beam is limited by slits only in the latter plane, and the wavelength varies in the perpendicular plane. An example of rocking curves measured by this kind of diffractometer is given in Fig. 7.4.4.4. The K1 and K2 components are seen separately plus a long tail due to continuum radiation, and the pro®le is that of the divergence of the beam. In the Laue method, a well collimated beam of white radiation is re¯ected by a stationary crystal. The wavelength band re¯ected by a perfect crystal is indicated in Fig. 7.4.4.1(b). The mosaic blocks select a band of wavelengths from the incident beam and the wavelength deviation is related to the angular deviation by l=l cot . The angular resolution is determined by the divergences of the incident beam and the spatial resolution of the detector. The detector is not energy dispersive, so that the background arises from all scattering that reaches the detector. An estimate of the background level involves integrations over the incident spectrum at a ®xed scattering angle, weighted by the cross sections of inelastic scattering and the attenuation factors. This calculation is very complicated, but at any rate the background level is far higher than that in a diffraction measurement with a monochromatic incident beam.

the sample (!=2 scan). The included TDS depends on these choices, but otherwise the amount of background is proportional to the area of the receiving slit. It is obvious from a comparison between Fig. 7.4.4.1 and Fig. 7.4.4.3 that a much smaller receiving slit is suf®cient in the parallel-beam geometry than in the conventional divergent-beam geometry. Mathieson (1985) has given a thorough analysis of various monochromator± sample±detector combinations and has suggested the use of a two-dimensional !=2 scan with a narrow receiving slit. This provides a deconvolution of the re¯ection pro®le measured with a divergent beam, but the same result with better intensity and resolution is obtained by the parallel-beam techniques. The above discussion has concentrated on improving the signal-to-background ratio by optimization of the diffraction geometry. This ratio can be improved substantially by an energydispersive detector, but, on the other hand, all detectors have some noise, which increases the background. There have been marked developments in recent years, and traditional technology has been replaced by new constructions. Much of this work has been carried out in synchrotron-radiation laboratories (for

7.4.4.3. Detecting system The detecting system is an integral part of the X-ray optics of a diffraction experiment, and it can be included in the phase-space diagrams. In single-crystal diffraction, the detecting system is usually a rectangular slit followed by a photon counter, and the slit is large enough to accept all the re¯ected beam. The slit can be stationary during the scan (! scan) or follow the rotation of

Fig. 7.4.4.4. Two re¯ections of beryllium acetate measured with Mo K. The graphite (002) monochromator re¯ects in the vertical plane, while the crystal re¯ects in the horizontal plane. The equatorial divergence of the beam is 0.8 , FWHM.

Fig. 7.4.4.5. Components of scattering at small scattering angles when the incident energy is just below the K absorption edge of the sample [upper part, (a)], and at large scattering angles when the incident energy is about twice the K-edge energy [upper part, (b)]. The abbreviations indicate resonant Raman scattering (RRS), plasmon (P) and Compton (C) scattering, coherent scattering (Coh) and sample ¯uorescence (K and K ). The lower part shows these components as convoluted by the resolution function of the detector: (a) a SSD and (b) a scintillation counter (Suortti, 1980).

663


7. MEASUREMENT OF INTENSITIES references, see Thomlinson & Williams, 1984; Brown & Lindau, 1986). A position-sensitive detector can replace the receiving slit when a reciprocal space is scanned. TV area detectors with an X-ray-to-visible light converter and two-dimensional CCD arrays have moderate resolution and ef®ciency, but they work in the current mode and do not provide pulse discrimination on the basis of the photon energy. One- and two-dimensional proportional chambers have a spatial resolution of the order of 0.1 mm, and the relative energy resolution, E=E ' 0:2, is suf®cient for rejection of some of the parasitic scattering. The NaI(Tl) scintillation counter is used most frequently as the X-ray detector in crystallography. It has 100% ef®ciency for the commonly used wavelengths, and the energy resolution is comparable to that of a proportional counter. The detector has a long life, and the level of the low-energy noise can be reduced to about 0.1 counts s 1 . The Ge and Si(Li) solid-state detectors (SSD) have an energy resolution E=E 0:01 to 0.03 for the wavelengths used in crystallography. The relative Compton shift, l=l, is Ê 0:024 A=l 1 cos 2, where 2 is the scattering angle, so that even this component can be eliminated to some extent by a SSD. These detectors have been bulky and expensive, but new

Fig. 7.4.4.6. Equatorial phase-space diagrams for powder diffraction in the Bragg±Brentano geometry. (a) The acceptance and emittance windows of a Johannson monochromator; (b) the beam in the l l1 plane: the exit beam from the Johansson monochromator is shown by the hatched area (z 100 mm), the beam on the sample by two closely spaced lines, the re¯ectivity range of powder particles in a small area of the sample by the hatched area (z 300 mm, note the change of scales), and the scan of the re¯ected beam by a slit RS by broken lines (z 400 mm, at the parafocus).

constructions that are suitable for X-ray diffraction have become available recently. The effects of the detector resolution are shown schematically in Fig. 7.4.4.5 for a scintillation counter and a SSD. Crystal monochromators placed in front of the detector eliminate all inelastic scattering but the TDS. The monochromator must be matched with the preceding X-ray optical system, the sample included, and therefore diffracted-beam monochromators are used in powder diffraction only (see Subsection 7.4.4.4). 7.4.4.4. Powder diffraction The signal-to-background ratio is much worse in powder diffraction than in single-crystal diffraction, because the background is proportional to the irradiated volume in both cases, but the powder re¯ection is distributed over a ring of which only the order of 1% is recorded. The phase-space diagrams of a typical measurement are shown in Fig. 7.4.4.6. The Johansson monochromator is matched to the incident beam to provide

Fig. 7.4.4.7. Three measurements of the 220 re¯ection of Ni at Ê scaled to the same peak value; (a) in linear scale, (b) in l 1:541 A logarithmic scale. Dotted curve: graphite (00.2) Johann monochromator, conventional 0.1 mm wide X-ray source (Suortti & Jennings, 1977); solid curve: quartz (10.1) Johansson monochromator, conventional 0.05 mm wide X-ray source; broken curve: synchrotron radiation monochromatized by a (; ) pair of Si (111) crystals, where the second crystal is sagittally bent for horizontal focusing (Suortti, Hastings & Cox, 1985). The horizontal line indicates the half-maximum value. In all cases, the effective slit width is much less than the FWHM of the re¯ection.

664


7.4. CORRECTION OF SYSTEMATIC ERRORS maximum ¯ux and good energy resolution. The Bragg±Brentano geometry is parafocusing, and, if the geometrical aberrations are ignored, the re¯ected beam is a convolution between the angular width of the monochromator focus (as seen from the sample) and the re¯ectivity curve of an average crystallite of the powder sample. The pro®le of this function is scanned by a narrow slit, as shown in the last diagram. The slit can be followed with a Johann or Johansson monochromator that has a narrow wavelength pass-band. In this case, there is no primary-beam monochromator, so that the incident beam at the sample is that given at z 100 mm. The slit RS is the `source' for the monochromator, which focuses the beam at the detector. The obvious advantages of this arrangement are counterbalanced by certain limitations such as that the effective receiving slit is determined by the re¯ectivity curve of the monochromator, and this may vary over the effective area. Examples of a powder re¯ection measured with different Ê radiation are given in Fig. 7.4.4.7. It instruments and 1.5 A should be noticed that scattering from the impurities of the sample and from the sample environment is negligible in all three cases. The width of the mosaic distribution of the 00.2 re¯ection of the pyrolytic graphite monochromator is 0.3 , Ê ) wide transmitted beam. which corresponds to a 180 eV (0.034 A This is almost 10 times the separation between K1 and K2 , and 70 times the natural width of the K1 line. The width of the focal line is about 0.2 mm, or 0.07 , and is seen as broadening of the re¯ection pro®le. The quartz (10.1) monochromator re¯ects a band that is determined by the projected width of the X-ray source. In the present case, the band is 15 eV wide, so that the monochromator can be tuned to transmit the K1 component only. The focal line is very sharp, 0.05 mm wide, and so the re¯ection is much narrower than in the preceding case. The third measurement was made with synchrotron radiation, and the receiving slit was replaced by a perfect-crystal analyser. The divergences of the incident and diffracted beams are about 0.1 mrad (less than 0.01 ) in the plane of diffraction, so that the ideal parallel-beam geometry should prevail. However, the

re¯ection is clearly broader than that measured with the conventional diffractometer. The reason is a wavelength gradient across the beam, which was monochromatized by a ¯at perfect crystal. On the other hand, the Ge (111) analyser crystal transmits elastic scattering and TDS only, and 2 away from the peak the background is 0.5% of the maximum intensity. The above considerations may seem to have little relevance to everyday crystallographic practices. Unfortunately, many standard methods yield diffraction patterns of poor quality. The quest for maximum integrated intensity has led to designs that make re¯ections broad and background high. It should be realized that not the ¯ux but the brilliance of the incident beam is important in a diffraction measurement. The other aspect is that the information should not be lost in the experiment, and a divergent wide wavelength band is quite an ignorant probe of a re¯ection from a single crystal. A situation where even small departures from the ideal diffraction geometry may cause large effects is measurement at an energy just below an absorption edge. Even a small tail of the energy band of the incident beam may excite radiation that becomes the dominant component of background. Similar effects are due to the harmonic energy bands re¯ected by most monochromators, particularly when the continuous spectrum of synchrotron radiation is used. Scattering from the surroundings of the sample can be eliminated almost totally by shielding and beam tunnels. The general idea of the construction should be that an optical element of the instrument `sees' the preceding element only. Inevitably, the detector sees some of the environment of the sample. The density of air is about 1=1000 of that of a solid sample, so that 10 mm3 of irradiated air contributes to the background as much as a spherical crystal of 0.3 mm diameter. Strong spurious peaks may arise from slit edges and entrance windows of the specimen chamber, which should never be seen by the detector. A complete measurement without the sample is always a good starting point for an experiment.

665



7.5. Statistical ¯uctuations By A. J. C. Wilson

7.5.1. Distributions of intensities of diffraction Intensities of diffraction have two distinct probability distributions: (1) the a priori probability that an arbitrarily chosen re¯ection of a particular substance will have a particular `true' intensity (R in the notation used below), and (2) the probability that a `true' R will have an observed value Ro . The distributions of the ®rst type depend on the symmetry and composition of the material, and are treated in Chapter 2.1 of Volume B of International Tables for Crystallography (Shmueli, 1993). The distributions of ¯uctuations of the second type, variations of the values Ro observed for a particular re¯ection, are treated here. The crude counts or counting rates are rarely used directly in crystallographic calculations; they are subjected to some form of data processing to provide ìntensities of re¯ection' Io in a form suitable for the determination of crystal structures, electron densities, line pro®les for the study of defects, etc. The resulting intensity for the jth re¯ection, Io; j , will be directly proportional to the corresponding Ro; j ; when no confusion can arise, one of the subscripts will be omitted. The value of the proportionality factor cj in Io; j cj Ro; j

7:5:1:1

will be different for different re¯ections, since they will occur at different Bragg angles, have different absorption corrections, etc.

(for the estimation of weights in re®nement processes, see Part 8) of the distribution function. 7.5.3. Fixed-time counting In the absence of disturbing in¯uences (mains-voltage ¯uctuations, unrecti®ed or unsmoothed high-tension supplies, `dead time' of the counter or counter circuits, etc.), the number of counts recorded during the predetermined time interval used in the ®xed-time mode will ¯uctuate in accordance with the Poisson probability distribution. If the `true' number of counts to be expected in the interval is N, the probability that the observed number will be No is given by pNo exp NN No =No !;

where all quantities appearing are necessarily non-negative. Both the mean and the variance of No are N. If the `true' number of counts to be expected when the diffractometer is set to receive a re¯ection is T , and the `true' number when it is set to receive the immediate background is B, the `true' intensity of the re¯ection is RT

Ro To


7:5:3:2

Bo ;

7:5:3:3

will also ¯uctuate, and can, occasionally, take on negative values. The treatment of `measured-as-negative' intensities is discussed in Section 7.5.6. Although the sum of two Poisson-distributed variables is also Poisson, the difference is not, and the probability of Ro given by (7.5.3.3) has been shown to be (Skellam, 1946; Wilson, 1978) pRo expf B T gT =BRo =2 IjRo j 2BT 1=2 ;

7:5:3:4

where In is the hyperbolic Bessel function of the ®rst kind. The mean and variance of Ro are T B and T B, respectively. If the times used for re¯ection and background are not equal, but are in the ratio of k : 1, the mean and variance of Ro To

kBo

7:5:3:5

2

are T kB and T k B, respectively. The distribution of Ro then involves a generalized Bessel function* depending on k: k 2BT k 1=2 ; pRo expf B T gT 2 k =BRo =2 IjR oj

7:5:3:6

for the properties of the generalized Bessel function, see Wright (1933) and Olkha & Rathie (1971). The probability distributions (7.5.3.4) and (7.5.3.6) seem to be very close to a normal distribution with the same mean and variance for Ro near R, but the probability of moderate deviations from R is less than normal and for large deviations is greater than normal. In applications, it is usual to work with intensities expressed as counting rates, rather than as numbers of counts, even when ®xed-time counting is used. The total intensity expressed as a counting rate is * The term `generalized Bessel function' seems to have no unique meaning in mathematics. The functions given that name by Paciorek & Chapuis (1994) belong to a different group.

666 Copyright © 2006 International Union of Crystallography

B;

provided that the time interval used for the re¯ection is equal to the time interval used for the background. In practice, the òbserved' values of To and Bo ¯uctuate with probabilities given by (7.5.3.1) with T or B replacing N, so that the observed value,

7.5.2. Counting modes Whatever the radiation, in both single-crystal and powder diffractometry, the integrated intensity of a re¯ection is obtained as a difference between a counting rate averaged over a volume of reciprocal space intended to include the re¯ected intensity and a counting rate averaged over a neighbouring volume of reciprocal space intended to include only background. If these intentions are not effectively realized, there will be a systematic error in the measured intensity, but in any case there will be statistical ¯uctuations in the counting rates. The two basic modes (Parrish, 1956) are ®xed-time counting and ®xed-count timing. In the ®rst, counts are accumulated for a pre-determined time interval, and the variance of the observed counting rate is proportional to the true (mean) counting rate. In the second, on the other hand, the counting is continued until a pre-determined number of counts is reached, and the variance of the observed counting rate is proportional to the square of the true counting rate. Put otherwise, the relative error in the intensities goes down inversely as the square root of the intensity for ®xed-time counting, whereas it is independent of the intensity for ®xedcount timing. Each mode has advantages, depending on the purpose of the measurements, and numerous modi®cations and compromises have been proposed in order to increase the ef®ciency of the use of the available time. References to some of the many papers are given in Section 7.5.7. In principle, probability distributions can be determined for any postulated counting mode. In practice, they become complicated for all but the simplest modes; this is true even for the single measurement of the total counting rate or the background counting rate, but is even more pronounced for the distribution of their difference (the re¯ection-only rate). For most crystallographic purposes, however, it is only necessary to know the mean (to correct for bias, if present) and the variance

7:5:3:1

7.5. STATISTICAL FLUCTUATIONS T =t;

7:5:3:7

where t is the time devoted to the measurement, and the variance of the counting rate is 2 T =t2 =t:

7:5:3:8

Similar expressions apply for the background, with B for the count, b for the time, and for the counting rate. For the re¯ection count, the corresponding expressions are T =t

B=b;

7:5:3:9

2

=t =b:

7:5:3:10

To avoid confusion, upper-case italic letters are used for numbers of counts, lower-case italic for counting times, and the corresponding lower-case Greek letters for the corresponding counting rates. In accordance with common practice, however, Ij will be used for the intensity of the jth re¯ection, the context making it clear whether I is a number of counts or a counting rate. 7.5.4. Fixed-count timing The probability of a time t being required to accumulate N counts when the true counting rate is is given by a distribution (Abramowitz & Stegun, 1964, p. 255): pt dt N

1! 1 tN

1

exp t dt:

7:5:4:1

The ratio N=t is a slightly biased estimate of the counting rate ; the unbiased estimate is N 1=t. The variance of this estimate is 2 =N 2, or, nearly enough for most purposes, N 12 =N 2t2 . The differences introduced by the corrections 1 and 2 are generally negligible, but would not be so for counts as low as those proposed by Killean (1967). If such corrections are important, it should be noticed that there is an ambiguity concerning N, depending on how the timing is triggered. It may be triggered by a count that is counted, or by a count that is not counted, or may simply be begun, independently of the incidence of a count. Equation (7.5.4.1) assumes the ®rst of these. Equation (7.5.4.1) may be inverted to give the probability distribution of the observed counting rate o instead of the probability distribution of the time t: p o d o N

1! 1 N

expf N

1= o N

1

1= o g d o =N

1:

7:5:4:2

There does not seem to be any special name for the distribution (7.5.4.2). Only its ®rst N 1 moments exist, and the integral expressing the probability distribution of the difference of the re¯ection and the background rates is intractable (Wilson, 1980). 7.5.5. Complicating phenomena

7.5.5.2. Voltage ¯uctuations Mains-voltage ¯uctuations, unless compensated, and unsmoothed high-tension supplies may affect the sensitivity of detectors and counting circuits, and in any case cause the probability distribution of the arrival of counts to be nonPoissonian. Backlash in the diffractometer drives may be even more important in altering the observed counting rates. As de Boer (1982) says, the ideal distributions represent a Utopia that experimenters can approach but never reach. He observed erratic ¯uctuations in counting rates, up to ten times as big as the expected statistical ¯uctuations. When care is taken, the instabilities observed in practice are much less than those of the extreme cases described by de Boer. Stabilizing an X-ray source and testing its stability are discussed in Subsection 2.3.5.1. 7.5.6. Treatment of measured-as-negative (and other weak) intensities It has been customary in crystallographic computations, but without theoretical justi®cation, to omit all re¯ections with intensities less than two or three times their standard uncertainties. Hirshfeld & Rabinovich (1973) asserted that the failure to use all re¯ections, even those for which the subtraction of background has resulted in a negative net intensity, at their measured values will lead to a bias in the parameters resulting from a least-squares re®nement. This is, however, inconsistent with the Gauss±Markov theorem (see Section 8.1.2), which shows that least-squares estimates are unbiased, independent of the weights used, if the observations are unbiased estimates of quantities predicted by a model. Giving some observations zero weight therefore cannot introduce bias. Provided the set of included observations is suf®cient to give a nonsingular normal equations matrix, parameter estimates will be unbiased, but inclusion of as many well determined observations as possible will yield the most precise estimates. Requiring that the net intensity be greater than 2 assures that the value of jFj will be well determined. Furthermore, Prince & Nicholson (1985) showed that, if the net intensity, I, or jFj2 is used as the observed quantity, weak re¯ections have very little leverage (see Section 8.4.4), and therefore omitting them cannot have a signi®cant effect on the precision of parameter estimates. The use of negative values of I or jFj2 is also inconsistent with Bayes's theorem, which implies that a negative value cannot be an unbiased estimate of an inherently non-negative quantity. There are statistical methods for estimating the positive value of jFj that led to a negative value of I. The best known approach is the Bayesian method of French & Wilson (1978), who observe that `Ìnstead of thanking the data for the information that certain structure factor moduli are small, we accuse them of assuming ìmpossible' negative values. What we should do is combine our knowledge of the non-negativity of the true intensities with the information concerning their magnitudes contained in the data.''

7.5.5.1. Dead time After a count is recorded, the detector and the counting circuits are `dead' for a short interval, and any ionizing event occurring during that interval is not detected. This is important if the dead time is not negligible in comparison with the reciprocal of the counting rate, and corrections have to be made; these are large for Geiger counters, and may sometimes be necessary for counters of other types. The need for the correction can be eliminated by suitable monitoring (Eastabrook & Hughes, 1953); other advantages of monitoring are described in Chapter 2.3.

7.5.7. Optimization of counting times There have been many papers on optimizing counting times for achieving different purposes, and all optimization procedures require some knowledge of the distributions of counts or counting rates; often only the mean and variance of the distribution are required. It is also necessary to know the functional relationship between the quantity of interest and the counts (counting rates, intensities) entering into its measurement. Typically, the object is to minimize the variance of some

667


7. MEASUREMENT OF INTENSITIES function of the measured intensities, say FI1 ; I2 ; . . . ; Ij ; . . .. General statistical theory gives the usual approximation X @F @F covIi ; Ij ; 7:5:7:1 2 F @Ii @Ij i; j where covIi ; Ij is the covariance of Ii and Ij if i 6 j, and is the variance of Ij , 2 Ij , if i j. There is very little correlation* between successive intensity measurements in diffractometry, so that covIi ; Ij is negligible for i 6 j. Equation (7.5.7.1) becomes !2 X @F 2 F 2 Ij : 7:5:7:2 @I j j These equations are strictly accurate only if F is a linear function of the I's, a condition satis®ed for the integrated intensity, but for few other quantities of interest. In most applications in diffractometry, however, the contribution of each Ij is suf®ciently small in comparison with the total to make the application of equations (7.5.7.1) and (7.5.7.2) plausible. Any proportionality factors cj (Section 7.5.1) can be absorbed into the functional relationship between F and the Ij 's. The object is to minimize 2 F by varying the time spent on each observation, subject to a ®xed total time P T tj : 7:5:7:3 j

It is simplest to regard the total intensity and the background intensity as separate observations, so that in (7.5.7.2) the sum is over n `background' and n `total' observations. With Ij expressed as a counting rate, its variance is Ij =tj [equation (7.5.3.8)], so that (7.5.7.2) becomes P 2 2 F Gj Ij =tj ; 7:5:7:4 j

where for brevity G has been written for j@F=@Ij. The variance of F will be a minimum if, for any small variations dtj of the counting times tj , * Exceptions to this statement may be important for line and area detectors, or if an interpolation function is used to estimate background. Wilson (1967) has discussed some features of the powder diffractometry case.

0

P j

Gj2 Ij tj 2 dtj ;

7:5:7:5

subject to the constancy of the total time T . There is thus the constraint P 7:5:7:6 0 dtj : j

These equations are consistent if for all j Gj2 Ij tj

2

k 2;

tj kGj Ij1=2 ;

7:5:7:7 7:5:7:8

where k is a constant determined by the total time T : P 7:5:7:9 T k Gj Ij1=2 : j

The minimum variance is thus achieved if each observation is given a time proportional to the square root of its intensity. A little manipulation now gives for the desired minimum variance " #2 1=2 1 X 2 min F @F=@Ij Ij : 7:5:7:10 T j The minimum variance is found to be a perfect square, and the standard uncertainty takes a simple form. Here, the optimization has been treated as a modi®cation of ®xed-time counting. However, the same ®nal expression is obtained if the optimization is treated as a modi®cation of ®xedcount timing (Wilson, Thomsen & Yap, 1965). Space does not permit detailed discussion of the numerous papers on various aspects of optimization. If the time required to move the diffractometer from one observation position to another is appreciable, the optimization problem is affected (Shoemaker & Hamilton, 1972, and references cited therein). There is some dependence on the radiation (X-ray versus neutron) (Shoemaker, 1968; Werner 1972a,b). A few other papers of historical or other interest are included in the list of references, without detailed mention in the text: Grant (1973); Killean (1972, 1973); Mack & Spielberg (1958); Mackenzie & Williams (1973); Szabo (1978); Thomsen & Yap (1968); Zevin, Umanskii, Kheiker & Panchenko (1961).

References 7.1.1 Hellner, E. (1954). IntensitaÈtsmessungen aus Aufnahmen in der Guinier-Kamera. Z. Kristallogr. 106, 122±145. International Tables for X-ray Crystallography (1962). Vol. III. Birmingham: Kynoch Press. Mees, C. E. K. (1954). The theory of the photographic process. New York: Macmillan. Whittaker, E. J. W. (1953). The Cox & Shaw factor. Acta Cryst. 6, 218. 7.1.2±7.1.4 Ames, L., Drummond, W., Iwanczyk, J. & Dabrowski, A. (1983). Energy resolution measurements of mercuric iodide detectors using a cooled FET preampli®er. Adv. X-ray Anal. 26, 325±330.

Ballon, J., Comparat, V. & Pouxe, J. (1983). The blade chamber: a solution for curved gaseous detectors. Nucl. Instrum. Methods, 217, 213±216. Bish, D. L. & Chipera, S. J. (1989). Comparison of a solid-state Si detector to a conventional scintillation detector±monochromator system in X-ray powder diffraction. Powder Diffr. 4, 137±143. Eastabrook, J. N. & Hughes, J. W. (1953). Elimination of dead-time corrections in monitored Geiger-counter X-ray measurements. J. Sci. Instrum. 30, 317±320. Foster, B. A. & WoÈlfel, E. R. (1988). Automated quantitative multiphase analysis using a focusing transmission diffractometer in conjunction with a curved position sensitive detector. Adv. X-ray Anal. 31, 325±330. Geiger, H. & MuÈller, W. (1928). Das ElektronenzaÈhlrohr. Z. Phys. 29, 839±841.

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7. MEASUREMENT OF INTENSITIES 7.4.4 Brown, G. S. & Lindau, I. (1986). Editors, Synchrotron radiation instrumentation. Proceedings of the International Conference on X-ray and VUV Synchrotron Radiation Instrumentation. Nucl. Instrum. Methods, A246, 511±595. Mathieson, A. McL. (1985). Small-crystal X-ray diffractometry with a crystal ante-monochromator. Acta Cryst. A41, 309±316. Matsushita, T. & Hashizume, H. (1983). Handbook of synchrotron radiation, Vol. I, edited by E. E. Koch, pp. 261±314. Amsterdam: North-Holland. Matsushita, T. & Kaminaga, U. (1980). A systematic method of estimating the performance of X-ray optical systems for synchrotron radiation. I. Description of various optical elements in position-angle space for ideally monochromatic X-rays. J. Appl. Cryst. 13, 465±471; II. Treatment in position±angle±wavelength space. J. Appl. Cryst. 13, 472±478. Suortti, P. (1980). Components of the total X-ray scattering. Accuracy in powder diffraction, edited by S. Block & C. R. Hubbard, pp. 1±20. Natl Bur. Stand. (US) Spec. Publ. No. 567. Suortti, P. (1985). Parallel beam geometry for single-crystal diffraction. J. Appl. Cryst. 18, 272±274. Suortti, P., Hastings, J. B. & Cox, D. E. (1985). Powder diffraction with synchrotron radiation. I. Absolute measurements. Acta Cryst. A41, 413±416. Suortti, P. & Jennings, L. D. (1977). Accuracy of structure factors from X-ray powder intensity measurements. Acta Cryst. A33, 1012±1027. Thomlinson, W. & Williams, G. P. (1984). Editors. Synchrotron radiation instrumentation 3. Proceedings of the Third National Conference on Synchrotron Radiation Instrumentation. Nucl. Instrum. Methods, 222, 215±278. 7.5 Abramowitz, M. & Stegun, I. A. (1964). Handbook of mathematical functions. National Bureau of Standards Publication AMS 55. Boer, J. L. de (1982). Statistics of recorded counts. Crystallographic statistics, edited by S. Ramaseshan, M. F. Richardson & A. J. C. Wilson, pp. 179±186. Bangalore: Indian Academy of Sciences. Eastabrook, J. N. & Hughes, J. W. (1953). Elimination of deadtime corrections in monitored Geiger-counter X-ray measurements. J. Sci. Instrum. 30, 317±320. French, S. & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Cryst. A34, 517±525. Grant, D. F. (1973). Single-crystal diffractometer data: the online control of the precision of intensity measurement. Acta Cryst. A29, 217. Hirshfeld, F. L. & Rabinovich, D. (1973). Treating weak re¯exions in least-squares calculations. Acta Cryst. A29, 510±513. Killean, R. C. G. (1967). A note on the a priori estimation of R factors for constant-count-per-re¯ection diffractometer experiments. Acta Cryst. 23, 1109±1110. Killean, R. C. G. (1972). The a priori optimization of diffractometer data to achieve the minimum average variance in the electron density. Acta Cryst. A28, 657±658.

Killean, R. C. G. (1973). Optimization of scan procedure for single-crystal X-ray diffraction intensities. Acta Cryst. A29, 216±217. Mack, M. & Spielberg, N. (1958). Statistical factors in X-ray intensity measurements. Spectrochim. Acta, 12, 169±178. Mackenzie, J. K. & Williams, E. J. (1973). The optimum distribution of counting times for determining integrated intensities with a diffractometer. Acta Cryst. A29, 201±204. Olkha, G. S. & Rathie, P. N. (1971). On a generalized Bessel function and an integral transform. Math. Nachr. 51, 231±240. Paciorek, W. A. & Chapuis, G. (1994). Generalized Bessel functions in incommensurate structure analysis. Acta Cryst. A50, 194±203. Parrish, W. (1956). X-ray intensity measurements with counter tubes. Philips Tech. Rev. 17, 206±221. Prince, E. & Nicholson, W. L. (1985). The in¯uence of individual re¯ections on the precision of parameter estimates in least squares re®nement. Structure & statistics in crystallography, edited by A. J. C. Wilson, pp. 183±195. Guilderland, NY: Adenine Press. Shmueli, U. (1993). Editor. International tables for crystallography. Vol. B. Reciprocal space. Dordrecht: Kluwer. Shoemaker, D. P. (1968). Optimization of counting times in computer-controlled X-ray and neutron single-crystal diffractometry. Acta Cryst. A24, 136±142. Shoemaker, D. P. & Hamilton, W. C. (1972). Further remarks concerning optimization of counting times in single-crystal diffractometry: rebuttal to Killean; consideration of background counting and slewing times. Acta Cryst. A28, 408±411. Skellam, J. G. (1946). The frequency distribution of the difference between two Poisson values belonging to different populations. J. R. Stat. Soc. 109, 296. SzaboÂ, P. (1978). Optimization of the measuring time in diffraction intensity measurements. Acta Cryst. A34, 551±553. Thomsen, J. S. & Yap, F. Y. (1968). Simpli®ed method of computing centroids of X-ray pro®les. Acta Cryst. A24, 702±703. Werner, S. A. (1972a). Choice of scans in X-ray diffraction. Acta Cryst. A28, 143±151. Werner, S. A. (1972b). Choice of scans in neutron diffraction. Acta Cryst. A28, 665±669. Wilson, A. J. C. (1967). Statistical variance of line-pro®le parameters. Measures of intensity, location and dispersion. Acta Cryst. 23, 888±898. Wilson, A. J. C. (1978). On the probability of measuring the intensity of a re¯ection as negative. Acta Cryst. A34, 474±475. Wilson, A. J. C. (1980). Relationship between òbserved' and `true' intensity: effect of various counting modes. Acta Cryst. A36, 929±936. Wilson, A. J. C., Thomsen, J. S. & Yap, F. Y. (1965). Minimization of the variance of parameters derived from X-ray powder diffractometer line pro®les. Appl. Phys. Lett. 7, 163±165. Wright, E. M. (1933). On the coef®cients of power series having exponential singularities. J. London Math. Soc. 8, 71±79. Zevin, L. S., Umanskii, M. M., Kheiker, D. M. & Panchenko, Yu. M. (1961). K voprosu o difraktometricheskikh priemah pretsizionnyh izmerenii elementarnyh yacheek. Kristallogra®ya, 6, 348±356.

676


references


8.1. Least squares

By E. Prince and P. T. Boggs The process of arriving at a model for a crystal structure may usefully be considered to consist of two distinct stages. The ®rst, which may be called determination, involves the use of chemical and physical intuition, direct methods, Fourier and Patterson methods, and other techniques to arrive at an approximate model for the structure that incorporates unit-cell dimensions, space group, chemical composition, and information with respect to the immediate environment of each atom. The second stage, which we shall call re®nement, involves ®nding the values of adjustable parameters in the model that give the best ®t between the predicted diffraction intensities and those observed in an experiment, in order to extract precise information about interatomic distances and bond angles, thermal motion, site occupancies, electron distribution, and so forth. Although there are several different criteria for the best ®t to data, such as maximum likelihood and maximum entropy, one of the most commonly used is the method of least squares. This chapter discusses both numerical and statistical aspects of re®nement by the method of least squares. Because both aspects make extensive use of linear algebra, we begin with a summary of de®nitions and fundamental operations in linear algebra (Stewart, 1973; Prince, 1994) and of basic de®nitions and concepts in mathematical statistics (Draper & Smith, 1981; Box, Hunter & Hunter, 1978). We then discuss the principles of linear and nonlinear least squares and conclude with an extensive discussion of numerical methods used in practical implementation of the technique.

8.1.1. De®nitions 8.1.1.1. Linear algebra A matrix is an ordered, rectangular array of numbers, real or complex. Matrices will be denoted by upper-case, bold italic letters, A. Their individual elements will be denoted by uppercase, italic letters with subscripts. Aij denotes the element in the ith row and the jth column of A. A matrix with only one row is a row vector; a matrix with only one column is a column vector. Vectors will be denoted by lower-case, bold roman letters, and their elements will be denoted by lower-case, italic letters with single subscripts. Scalar constants will usually be denoted by lower-case, Greek letters. A matrix with the same number of rows as columns is square. If Aij 0 for all i > j, A is upper triangular. If Aij 0 for all i < j, A is lower triangular. If Aij 0 for all i 6 j, A is diagonal. If Aij 0 for all i and j, A is null. A matrix, B, such that Bij Aji for all i and j is the transpose of A, and is denoted by AT . Matrices with the same dimensions may be added and subtracted: (A+B)ij Aij Bij . A matrix may be multiplied by a scalar: A Pijm Aij . Multiplication of matrices is de®ned by (AB)ij k1 Aik Bkj , where m is the number of columns of A and the number of rows of B (which must be equal). Addition and multiplication of matrices obey the associative law: A B C A B C; ABC ABC. Multiplication of matrices obeys the distributive law: AB C AB AC. Addition of matrices obeys the commutative law: A B B A, but multiplication, except in certain (important) special cases, does not: AB 6 BA. The transpose of a product is the product of the transposes of the factors in reverse order: ABT BT AT .

The trace of a square matrix is the sum of its diagonal elements. The determinant of an n n square matrix, A, denoted by jAj, is the sum of n! terms, each of which is a product of the diagonal elements of a matrix derived from A by permuting columns or rows (see Stewart, 1973). The rank of a matrix (not necessarily square) is the dimension of the largest square submatrix that can be formed from it, by selecting rows and columns, whose determinant is not equal to zero. A matrix has full column rank if its rank is equal to its number of columns. A square matrix whose diagonal elements are equal to one and whose off-diagonal elements are equal to zero is an identity matrix, denoted by I. If jAj 6 0, A is nonsingular, and there exists a matrix A 1 , the inverse of A, such that AA 1 A 1 A I. If jAj 0, A is singular, and has no inverse. The adjoint, or conjugate transpose, of A is a matrix, Ay , such that Aijy Aji , where the asterisk indicates complex conjugate. If Ay A 1 , A is unitary. If the elements of a unitary matrix are real, it is orthogonal. From this de®nition, if A is orthogonal, it follows that n P i1

for all j, and n P i1


Aij Aik 0

if j 6 k. By analogy, two column vectors, x and y, are said to be orthogonal if xT y 0. For any square matrix, A, there exists a set of vectors, xi , such that Axi li xi , where li is a scalar. The values li are the eigenvalues of A, and the vectors xi are the corresponding eigenvectors. If A Ay , A is Hermitian, and, if the elements are real, A AT , so that A is symmetric. It can be shown (see, for example, Stewart, 1973) that, if A is Hermitian, all eigenvalues are real, and there exists a unitary matrix, T, such that D T y AT is diagonal, with the elements of D equal to the eigenvalues of A, and the columns of T are the eigenvectors. An n n symmetric matrix therefore has n mutually orthogonal eigenvectors. If the product xT Ax is greater than (or equal to) zero for any non-null vector, x, A is positive (semi)de®nite. Because x may be, in particular, an eigenvector, all eigenvalues of a positive (semi)de®nite matrix are greater than (or equal to) zero. Any matrix of the form BT B is positive semi de®nite, and, if B has full column rank, A BT B is positive de®nite. If A is positive de®nite, there exists an upper triangular matrix, R, or, equivalently, a lower triangular matrix, L, with positive diagonal elements, such that RT R LLT A. R, or L, is called the Cholesky factor of A. The magnitude, length or Euclidean norm of a vector, x, denoted by kxk, is de®ned by kxk xT x1=2 . The induced matrix norm of a matrix, B, denoted kBk, is de®ned as the maximum value of kBxk=kxk xT BT Bx=xT x1=2 for kxk > 0. Because xT BT Bx will have its maximum value for a ®xed value of xT x when x is parallel to the eigenvector that corresponds to the largest eigenvalue of BT B, this de®nition implies that kBk is equal to the square root of the largest eigenvalue of BT B. The condition number of B is the square root of the ratio of the largest and smallest eigenvalues of BT B. (Other de®nitions of norms exist, with corresponding de®nitions of condition number. We shall not be concerned with any of these.)


A2ij 1

8.1. LEAST SQUARES We shall make extensive use of the so-called QR decomposition, which is de®ned as follows: For any n p n p real matrix, Z, there exists an n n orthogonal matrix, Q, such that R T Q Z ; 8:1:1:1 O where R is a p p upper triangular matrix, and O denotes an n p p null matrix. Thus, we have R ZQ ; 8:1:1:2 O which is known as the QR decomposition of Z. If we partition Q as QZ ; Q? , where QZ has dimensions n p, and Q? has dimensions n n p, (8.1.1.2) becomes Z QZ R;

8:1:1:3

which is known as the QR factorization. We shall make use of the following facts. First, R is nonsingular if and only if the columns of Z are linearly independent; second, the columns of QZ form an orthonormal basis for the range space of Z, that is, they span the same space as Z; and, third, the columns of Q? form an orthonormal basis for the null space of ZT , that is, ZT Q? O. There are two common procedures for computing the QR factorization. The ®rst makes use of Householder transformations, which are de®ned by HI

2xxT ;

T

8:1:1:4

small proportion of the total computing time on a vector oriented computer. 8.1.1.2. Statistics A probability density function, which will be abbreviated p.d.f., is a function, x, such that the probability of ®nding the random variable x in the interval a x b is given by pa x b

The factorization procedure for an n p matrix, A (Stewart, 1973; Anderson et al., 1992), takes as v in the ®rst step the ®rst column of A, and forms A1 H 1 A, which has zeros in all elements of the ®rst column below the diagonal. In the second step, v has a zero as the ®rst element and is ®lled out by those elements of the second column of A1 on or below the diagonal. A2 H 2 A1 then has zeros in all elements below the diagonal in the ®rst two columns. This process is repeated p 2 more times, after which Q H p . . . H 2 H 1 , and R Ap is upper triangular. QR factorization by Householder transformations requires for ef®ciency that the entire n p matrix be stored in memory, and requires of order np2 operations. A procedure that requires storage of only the upper triangle makes use of Givens rotations, which are 2 2 matrices of the form cos sin G : 8:1:1:6 sin cos Multiplication of a 2 m matrix, B, by G will put a zero in the B21 element if arctan B21 =B11 . The factorization of A involves reading, or computing, the rows of A one at a time. In the ®rst step, matrix B1 consists of the ®rst row of R and the current row of A, from which the ®rst element is eliminated. In the second step, B21 is the second row of R and the p 1 nonzero elements of the second row of the transformed B1 . After the ®rst p rows have been treated, each additional row of A requires 2pp 1 multiplications to ®ll it with zeros. However, because the operation is easily vectorized, the time required may be a

x 0;

x dx:

1 < x < 1;

and 1 R 1

x dx 1:

A cumulative distribution function, which will be abbreviated c.d.f., is de®ned by x

Rx 1

t dt:

The properties of x imply that 0 x 1, and x d x= dx. The expected value of a function, f x, of random variable x is de®ned by

n

1 R f xx dx: f x 1

n

If f x x , f x hx i is the nth moment of x. The ®rst moment, often denoted by , is themean of x. The second moment about the mean, x hxi2 , usually denoted by 2 , is the variance of x. The positive square root of the variance is the standard deviation. For a vector, x, of random variables, x1 , x2 ,. . ., xn , the joint probability density function, or joint p.d.f., is a function, J x, such that pa1 x1 b1 ; a2 x2 b2 ; . . . ; an xn bn

Rb1 Rb2 a1 a2

...

Rbn an

J x dx1 dx2 . . . dxn :

8:1:1:7

The marginal p.d.f. of an element (or a subset of elements), xi , is a function, M xi , such that pai xi bi

Rbi ai

M xi dxi

1 R 1

...

Rbi ai

...

R1 1

J x dx1 . . . dxi . . . dxn : 8:1:1:8

This is a p.d.f. for xi alone, irrespective of the values that may be found for any other element of x. For two random variables, x and y (either or both of which may be vectors), the conditional p.d.f. of x given y y0 is de®ned by C xjy0 cJ x; yyy0 ; where c 1=M y0 is a renormalizing factor. This is a p.d.f. for x when it is known that y y0 . If C xjy M x for all y, or, equivalently, if J x; y M xM y, the random variables x and y are said to be statistically independent.

679


a

A p.d.f. has the properties

2

where x x 1. H is symmetric, and H I, so that H is orthogonal. In three dimensions, H corresponds to a re¯ection in a mirror plane perpendicular to x, because of which Stewart (1973) has suggested the alternative term elementary re¯ector. A vector v is transformed by Hv into the vector kvke, where e represents a vector with e1 1, and ei 0 for i 6 1, if

8:1:1:5 x v kvke v kvke :

Rb

8. REFINEMENT OF STRUCTURAL PARAMETERS Moments may be de®ned for multivariate p.d.f.s in a manner analogous to the one-dimensional case. The mean is a vector de®ned by

R i xi xi x dx; where the volume of integration is the entire domain of x. The variance±covariance matrix is de®ned by

Vij xi hxi i xj hxj i R xi hxi i xj hxj i J x dx: 8:1:1:9 The diagonal elements of V are the variances of the marginal p.d.f.s of the elements of x, that is, Vii i2 : It can be shown that, if xi and xj are statistically independent, Vij 0 when i 6 j. If two vectors of random variables, x and y, are related by a linear transformation, x By, the means of their joint p.d.f.s are related by x By , and their variance±covariance matrices are related by V x BV y BT . 8.1.2. Principles of least squares The method of least squares may be formulated as follows: Given a set of n observations, yi i 1; 2; . . . ; n, that are measurements of quantities that can be described by differentiable model functions, Mi x, where x is a vector of parameters, xj j 1; 2; . . . ; p, ®nd the values of the parameters for which the sum n P S wi yi Mi x2 8:1:2:1

AT WAx AT W y

The model functions, Mi x, are, in general, nonlinear, and there are no direct ways to solve these systems of equations. Iterative methods for solving them are discussed in Section 8.1.4. Much of the analysis of results, however, is based on the assumption that linear approximations to the model functions are good approximations in the vicinity of the minimum, and we shall therefore begin with a discussion of linear least squares. To express linear relationships, it is convenient to use matrix notation. Let Mx and y be column vectors whose ith elements are Mi x and yi . Similarly, let b be a vector and A be a matrix such that a linear approximation to the ith model function can be written Mi x bi

p P j1

Aij xj :

8:1:2:3

Equations (8.1.2.3) can be written, in matrix form, Mx b Ax;

8:1:2:4

and, for this linear model, (8.1.2.1) becomes S y

b

AxT W y

b

Ax;

8:1:2:5

where W is a diagonal matrix whose diagonal elements are Wii wi . In this notation, the normal equations (8.1.2.2) can be written

b x AT WA 1 AT W y

b:

8:1:2:7 T

If Wii > 0 for all i, and A has full column rank, then A WA will be positive de®nite, and S will have a unique minimum at x b x. The matrix H AT WA 1 AT W is a p n matrix that relates the n-dimensional observation space to the p-dimensional parameter space and is known as the least-squares estimator; because each element of b x is a linear function of the observations, it is a linear estimator. [Note that, in actual practice, the matrix H is not actually evaluated, except, possibly, in very small problems. Rather, the linear system AT WAx AT W y b is solved using the methods of Section 8.1.3.] The least-squares estimator has some special properties in statistical analysis. Suppose that the elements of y are experimental observations drawn at random from populations whose means are given by the model, Mx, for some unknown x, which we wish to estimate. This may be written

y b Ax: 8:1:2:8 The expected value of the least-squares estimate is

T b x A WA 1 AT Wy b

AT WA 1 AT W y b AT WA 1 AT WAx x:

8:1:2:9

If the expected value of an estimate is equal to the variable to be estimated, the estimator is said to be unbiased. Equation (8.1.2.9) shows that the least-squares estimator is an unbiased estimator for x, independent of W , provided only that y is an unbiased estimate of Mx, the matrix AT WA is nonsingular, and the elements of W are constants independent of y and Mx. Let V x and V y be the variance±covariance matrices for the joint p.d.f.s of the elements of x and y, respectively. Then, V x HV y H T : Let G be the matrix AT V y 1 A 1 AT V 1 , so that b x Gy b is the particular least-squares estimate for which W V y 1 . Then, V x GV y G T . If V y is positive de®nite, its lower triangular Cholesky factor, L, exists, so that LLT V y . [If V is diagonal, L is also diagonal, with Lii V y 1=2 ii :] It is readily veri®ed that the matrix product H GLH GLT HV y H T GV y G T , but the diagonal elements of this product are the sums of squares of the elements of rows of H GL, and are therefore greater than or equal to zero. Therefore, the diagonal elements of V x , which are the variances of the marginal p.d.f.s of the elements of b x, are minimum when W V y 1 . Thus, the least-squares estimator is unbiased for any positivede®nite weight matrix, W , but the variances of the elements of the vector of estimated parameters are minimized if W V y 1 . [Note also that V x AT WA 1 if, and only if, W V y 1 .] For this reason, the least-squares estimator with weights proportional to the reciprocals of the variances of the observations is referred to as the best linear unbiased estimator for the parameters of a model describing those observations. (These speci®c results are included in a more general result known as the Gauss±Markov theorem.) The analysis up to this point has assumed that the model is linear, that is that the expected values of the observations can be expressed by hyi b Ax, where A is some matrix. In crystallography, of course, the model is highly nonlinear, and this assumption is not valid. The principles of linear least squares

680


8:1:2:6

and their solution is

i1

is minimum. Here, wi represents a weight assigned to the ith observation. The values of the parameters that give the minimum value of S are called estimates of the parameters, and a function of the data that locates the minimum is an estimator. A necessary condition for S to be a minimum is for the gradient to vanish, which gives a set of simultaneous equations, the normal equations, of the form n X @S @M x 2 wi yi Mi x i 0: 8:1:2:2 @xj @xj i1

b;

8.1. LEAST SQUARES can be extended to nonlinear model functions by ®rst ®nding, by numerical methods, a point in parameter space, x0 , at which the gradient vanishes and then expanding the model functions about that point in Taylor's series, retaining only the linear terms. Equation (8.1.2.4) then becomes Mx Mx0 Ax

x0 ;

8:1:2:10

where Aij @Mi x=@xj evaluated at x x0 . Because we have already found the least-squares solution, the estimate b x x0 AT WA 1 AT W y x0 Hy Mx0

Mx0 8:1:2:11

reduces to b x x0 . It is important, however, not to confuse x0 , which is a convenient origin, with b x, which is a random variable describable by a joint p.d.f. with mean x0 and a variance± covariance matrix V x HV y H T , reducing to AT V y 1 A 1 when W V y 1. This variance±covariance matrix is the one appropriate to the linear approximation given in (8.1.2.10), and it is valid (and the estimate is unbiased) only to the extent that the approximation is a good one. A useful criterion for an adequate approximation (Fedorov, 1972) is, for each j and k, 08 " #2 9 X n n <X = 2 @ M x @M x i 0 i 0 w i i wi @ i1 : i1 ; @xj @xk @xj (

n X i1

2 )!1=2 @Mi x0 wi ; @xk

p.d.f., or the posterior, of x. The relation in (8.1.2.14) and (8.1.2.15) was ®rst stated in the eighteenth century by Thomas Bayes, and it is therefore known as Bayes's theorem (Box & Tiao, 1973). Although its validity has never been in serious question, its application has divided statisticians into two vehemently disputing camps, one of which, the frequentists, considers that Bayesian methods give nonobjective results, while the other, the Bayesians, considers that only by careful construction of a `noninformative' prior can true objectivity be achieved (Berger & Wolpert, 1984). Diffraction data, in general, contain no phase information, so the likelihood function for the structure factor, F, given a value of observed intensity, will have a value signi®cantly different from zero in an annular region of the complex plane with a mean radius equal to jFj. Because this is insuf®cient information with which to determine a crystal structure, a prior p.d.f. is constructed in one (or some combination) of two ways. Either the prior knowledge that electron density is non negative is used to construct a joint p.d.f. of amplitudes and phases, given amplitudes for all re¯ections and phases for a few of them (direct methods), or chemical knowledge and intuition are used to construct a trial structure from which structure factors can be calculated, and the phase of Fcalc is assigned to Fobs . Both of these procedures can be considered to be applications of Bayes's theorem. In fact, Fcalc for a re®ned structure can be considered a Bayesian estimate of F.

8:1:2:12

where i is the estimated standard deviation or standard uncertainty (Schwarzenbach, Abrahams, Flack, Prince & Wilson, 1995) of yi . This criterion states that the curvature of Sy; x in a region whose size is of order in observation space is small; it ensures that the effect of second-derivative terms in the normal-equations matrix on the eigenvalues and eigenvectors of the matrix is negligible. [For a further discussion and some numerical tests of alternatives, see Donaldson & Schnabel (1986).] The process of re®nement can be viewed as the construction of a conditional p.d.f. of a set of model parameters, x, given a set of observations, y. An important expression for this p.d.f. is derived from two equivalent expressions for the joint p.d.f. of x and y: J x; y C xjyM y C yjxM x:

8:1:2:13

Provided M y > 0, the conditional p.d.f. we seek can be written C xjy C yjxM x=M y:

8:1:2:14

Here, the factor 1=M y is the factor that is required to normalize the p.d.f. C yjx is the conditional probability of observing a set of values of y as a function of x. When the observations have already been made, however, this can also be considered a density function for x that measures the likelihood that those particular values of y would have been observed for various values of x. It is therefore frequently written `xjy, and (8.1.2.14) becomes C xjy c`xjyM x;

8:1:2:15

where c 1=M y is the normalizing constant. M x, the marginal p.d.f. of x in the absence of any additional information, incorporates all previously available information concerning x, and is known as the prior p.d.f., or, frequently, simply as the prior of x. Similarly, C xjy is the posterior

8.1.3. Implementation of linear least squares In this section, we consider in detail numerical methods for solving linear least-squares problems, that is, the situation where (8.1.2.4) and (8.1.2.5) apply exactly. 8.1.3.1. Use of the QR factorization The linear least-squares problem can be viewed geometrically as the problem of ®nding the point in a p-dimensional subspace, de®ned as the set of points that can be reached by a linear combination of the columns of A, closest to a given point, y, in an n-dimensional observation space. Since this is equivalent to ®nding the orthogonal projection of point y into that subspace, it is not surprising that an orthogonal decomposition of A helps to solve the problem. For convenience in this discussion, let us remove the weight matrix from the problem by de®ning the standardized design matrix by Z UA;

where U is the upper triangular Cholesky factor of W . Consider the least-squares problem with the QR factorization of Z, as given in Subsection 8.1.1.1. For y0 Uy b, (8.1.2.5) becomes S y0 T

ZxT y0

Q y

Zx T

Zx QT y0

0

Zx;

8:1:3:2

which reduces to S QTZ y0

RxT QTZ y0

Rx y0T Q? QT? y0 :

8:1:3:3

The second term in (8.1.3.3) is independent of x, and is therefore the sum of squared residuals. The ®rst term vanishes if Rx QTZ y0 ;

8:1:3:4

which, because R is upper triangular, is easily solved for x. The QR decomposition of Z therefore leads naturally to the following algorithm for solving the linear least-squares problem:

681


8:1:3:1

8. REFINEMENT OF STRUCTURAL PARAMETERS (1) (2) (3) (4) (5)

compute the QR factorization of Z; compute QTZ y0 ; solve Rx QTZ y0 for x. compute the residual sum of squares by y0T y0 y0 QZ QTZ y0 : compute the variance±covariance matrix from V x R 1 R 1 T .

8.1.3.2. The normal equations Let us now consider the relationship of the QR procedure for solving the linear least-squares problem to the classical method based on the normal equations. The normal equations can be derived by differentiating (8.1.3.2) and equating the result to a null vector. This yields ZT Zx ZT y0 :

8:1:3:5

The algorithm is therefore to compute the cross-product matrix, B ZT Z, and the right-hand side, d ZT y0 , and to solve the resulting system of equations, Bx d. This is usually accomplished by computing the Cholesky decomposition of B, that is B C T C, where C is upper triangular, and then solving the two triangular systems C T v d and Cx v. Because Z QZ R, equation (8.1.3.5) becomes RT QTZ QZ Rx RT QTZ y0 ;

8:1:3:6

or RT Rx RT QTZ y0 :

8:1:3:7 T

It is clear that R is the Cholesky factor of Z Z, although it is formed in a different way. This procedure requires of order np2 =2 operations to form the product ZT Z and p3 =3 operations for the Cholesky decomposition. In some situations, the extra time to compute the QR factorization is justi®ed because of greater stability, as will be discussed below. Most other quantities of statistical interest can be computed directly from the QR factorization. 8.1.3.3. Conditioning The condition number of Z, which is de®ned (Subsection 8.1.1.1) as the square root of the ratio of the largest to the smallest eigenvalue of ZT Z, is an indicator of the effect a small change in an element of Z will have on the elements of ZT Z 1 and of b x. A large value of the condition number means that small errors in computing an element of Z, owing possibly to truncation or roundoff in the computer, can introduce large errors into the elements of the inverse matrix. Also, when the condition number is large, the standard uncertainties of some estimated parameters will be large. A large condition number, as de®ned in this way, can result from either scaling or correlation or some combination of these. To illustrate this, consider the matrices 2" 0 ZT Z 0 " and ZT Z

1

1

"

1

1

loss of precision, whereas an inverse for the second would be totally meaningless. It is good practice, therefore, to factor the design matrix, Z, into the form Z TS;

where S is a p p diagonal matrix whose elements de®ne some kind of `natural' unit appropriate to the parameter represented in each column of Z. The ideal natural unit would be the standard uncertainty of that parameter, but this is not available until after the calculation has been completed. If correlation is not too severe, suitable values for the elements of S, of the same order of magnitude as those derived from the standard uncertainty, are the column Euclidean norms, that is

S zj zTj zj 1=2 ; 8:1:3:9 where zj denotes the jth column of Z. This scaling causes all diagonal elements of ZT Z to be equal to one, and errors in the elements of Z will have roughly equal effects. Ill conditioning that results from correlation, as in the second example above, is more dif®cult to deal with. It is an indication that some linear combination of parameters, some eigenvector of the normal equations matrix, is poorly determined by the available data. Use of the QR factorization of Z to compute the Cholesky factor of ZT Z may be advantageous, in spite of the additional computation time, because better numerical stability is obtained in marginal situations. As a practical matter, however, it is important to recognize that an ill conditioned matrix is a symptom of a ¯aw in the model or in the experimental design (or both). Use can be made of the fact that, although determining the entire set of eigenvalues and eigenvectors of a large matrix is computationally an inherently dif®cult problem, a relatively simple algorithm, known as a condition estimator (Anderson et al., 1992), can produce a good approximation to the eigenvector that corresponds to the smallest eigenvalue of a nearly singular matrix. This information can be used in either or both of two ways. First, without any fundamental modi®cation to the model or the experiment, a simple, linear transformation of the parameters so that the problem eigenvector is one of the independent parameters, followed by rescaling, can resolve the numerical dif®culties in computing the estimates. A common example is the situation where a phase transition results in the doubling of a unit cell, with pairs of atoms almost but not quite related by a lattice translation. A transformation that makes the estimated parameters the sums and differences of corresponding parameters in related pairs of atoms can make a dramatic improvement in the condition number. Alternatively, the problem eigenvector can be set to some value determined from theory or from some other experiment (see Section 8.3.1), or additional data can be collected that are selected to make that combination of parameters determinate. 8.1.4. Methods for nonlinear least squares Recall (equation 8.1.2.1) that the general, nonlinear problem can be stated in the form: ®nd the minimum of

" ;

Sx

where " represents machine precision, which can be de®ned as the smallest number in machine representation that, when added to 1, gives a result different from 1. By the conventional de®nition, both of these matrices have a condition number for Z of 2 "="1=2 . Because numbers of order " can be perfectly well represented, however, the ®rst one can be inverted without

n P i1

wi yi

Mi x2 ;

8:1:4:1

where x is a vector of p parameters, and Mx represents a set of model functions that predict the values of observations, y. In this section, we discuss two useful ways of solving this problem and consider the relative merits of each. The ®rst is based on iteratively linearizing the functions Mi x and approximating

682


8:1:3:8

8.1. LEAST SQUARES (8.1.4.1) by one of the form in (8.1.2.5). The second uses an unconstrained minimization algorithm, based on Newton's method, to minimize Sx. 8.1.4.1. The Gauss±Newton algorithm Let xc be the current approximation to b x, the solution to (8.1.4.1). We construct a linear approximation to Mi x in the vicinity of x xc by expanding it in a Taylor series through the linear terms, obtaining Mi x Mi xc

p P j1

Jij x

xc j ;

8:1:4:2

where J is the Jacobian matrix, de®ned by Jij @Mi x=@xj . A straightforward procedure, known as the Gauss±Newton algorithm, may be formally stated as follows: (1) compute d as the solution of the linear system J T WJd J T W y

Mxc ;

(2) set xc xc d; (3) if not converged (see Subsection 8.1.4.4), go to (1), else stop. The convergence rate of the Gauss±Newton algorithm depends on the size of the residual, that is, on Sb x. If Sb x 0, then the convergence rate is quadratic; if it is small, then the rate is linear; but if Sb x is large, then the procedure is not locally convergent at all. Fortunately, this procedure can be altered so that it is always locally convergent, and even globally convergent, that is, convergent to a relative minimum from any starting point. There are two possibilities. First, the procedure can be modi®ed to include a line search. This is accomplished by computing the direction d as above and then choosing such that Sxc d is suf®ciently smaller than Sxc . In order to guarantee convergence, one uses the test Sxc d < Sxc

dT Jxc T y

Mxc ;

8:1:4:3

where, as actually implemented in modern codes, has values of the order of 10 4 . [In theory, a slightly stronger condition is necessary, but this is usually avoided by other means. See Dennis & Schnabel (1983) for details.] While this improves the situation dramatically, it still suffers from the de®ciency that it is very slow on some problems, and it is unde®ned if the rank of the Jacobian, Jb x, is less than p. Jx usually has rank p near the solution, but it may be rank de®cient far away. Also, it may be `close' to rank de®cient, and give numerical dif®culties (see Subsection 8.1.3.3). 8.1.4.2. Trust-region methods ± the Levenberg±Marquardt algorithm These remaining problems can be addressed by the utilization of a trust-region modi®cation to the basic Gauss±Newton algorithm. For this procedure, we compute the step, d, as the solution to the linear least-squares problem subject to the constraint that kdk c , where c is the trust-region radius. Here, the linearized model is modi®ed by admitting that the linearization is only valid within a limited region around the current approximation. It is clear that, if the trust region is suf®ciently large, this constrained step will in fact be unconstrained, and the step will be the same as the Gauss± Newton step. If the constraint is active, however, the step has the form Jxc T WJxc Id Jxc W y

Mxc ;

8:1:4:4

where is the Lagrange multiplier corresponding to the constraint (see Section 8.3.1), that is, is the value such that kdk c . Formula (8.1.4.4) is known as the Levenberg± Marquardt equation. It can be seen from this formula that the step direction is intermediate between the Gauss±Newton direction and the direction of steepest descent, for which reason it is frequently known as ``Marquardt's compromise'' (Draper & Smith, 1981). Ef®cient numerical calculation of d for a given value of is accomplished by noting that (8.1.4.4) is equivalent to the linear least-squares problem, ®nd the minimum of 2

J xc y M xc

: S p d 8:1:4:5

I O This problem can be solved by saving the QR factorization for 0 and updating it for various values of greater than 0. The actual computation of is accomplished by a modi®ed Newton method applied to the constraint equation (see Dennis & Schnabel, 1983, for details). Having calculated the constrained value of d, we set x xc d. The algorithm is completed by specifying a procedure for updating the trust-region parameter, c , after each step. This is done by comparing the actual value of Sx with the predicted value based on the linearization. If there is good agreement between these values, is increased. If there is not good agreement, is left unchanged, and if Sx > Sxc , the step is rejected, and is reduced. Global convergence can be shown under reasonable conditions, and very good computational performance has been observed in practice. 8.1.4.3. Quasi-Newton, or secant, methods The second class of methods for solving the general, nonlinear least-squares problem is based on the unconstrained minimization of the function Sx, as de®ned in (8.1.4.1). Newton's method for solving this problem is derived by constructing a quadratic approximation to S at the current trial point, xc , giving Sc d Sxc gxc T d 1=2dT Hxc d;

from which the Newton step is obtained by minimizing Sc with respect to d. Here, g represents the gradient of S and H is the Hessian matrix, the p p symmetric matrix of second partial derivatives, Hjk @2 S=@xj @xk . Thus, d is calculated by solving the linear system Hxc d

gxc :

8:1:4:7

[Note: In the literature on optimization, the notation r2 Sx is often used to denote the Hessian matrix of the function Sx. This should not be confused with the Laplacian operator.] While Newton's method is locally quadratically convergent, it suffers from well known drawbacks. First, it is not globally convergent without employing some form of line search or trust region to safeguard it. Second, it requires the computation of the Hessian of S in each iteration. The Hessian, however, has the form Hxc Jxc T WJxc Bxc ;

8:1:4:8

where Bx is given by Bxc jk

n P i1

wi yi

Mi xc @Mi2 xc =@xj @xk :

8:1:4:9

The ®rst term of the Hessian, which is dependent only on the Jacobian, is readily available, but B, even if it is available analytically, is, typically, expensive to compute. Furthermore,

683


8:1:4:6

8. REFINEMENT OF STRUCTURAL PARAMETERS in some situations, such as in the vicinity of a more symmetric model, Hxc may not be positive de®nite. In order to overcome these dif®culties, various methods have been proposed to approximate H without calculating B. Because these methods are based on approximations to Newton's method, they are often referred to as quasi-Newton methods. Two popular versions use approximations to B that are known as the Davidon± Fletcher±Powell (DFP) update and the Broyden±Fletcher± Goldfarb±Shanno (BFGS) update, with BFGS being apparently superior in practice. The basic idea of both procedures is to use values of the gradient to update an approximation to the Hessian in such a way as to preserve as much previous information as possible, while incorporating the new information obtained in the most recent step. From (8.1.4.6), the gradient of Sc d is gxc d gxc Hxc d:

8:1:4:10

If the true function is not quadratic, or if H is only approximately known, the gradient at the point xc d, where d is the solution of (8.1.4.7), will not vanish. We make use of the value of gxc d to compute a correction to H to give a Hessian that would have predicted what was actually found. That is, ®nd an update matrix, B0 , such that Hxc B0 xc d gxc d

gxc :

8:1:4:11

This is known as the secant, or quasi-Newton, relation. An algorithm based on the BFGS update goes as follows: Given xc and H c (usually a scaled, identity matrix is used as the initial approximation, in which case the ®rst step is in the steepest descent direction), (1) solve H c d gxc for the direction d; (2) compute such that Sxc d satis®es condition (8.1.4.3); (3) set xc xc d; (4) update H c by H c gxc gxc T =dT gxc qqT =qT d (see Nash & Sofer, 1995), where q gxc d gxc ; (5) if not converged, go to (1), else stop. If, in step (2), the additional condition is applied that dT gxc d 0, it becomes an exact line search. This, however, is an unnecessarily severe condition, because it can be shown that, if dT gxc d gxc > 0, the approximation to the Hessian remains positive de®nite. The two terms in this expression are values of dS= d, and the condition states that S does not decrease more rapidly as increases, which will always be true for some range of values of > 0. This algorithm is globally convergent; it will ®nd a point at which the gradient vanishes, although that point may be a false minimum. It should be noted, however, that the procedure does not produce a ®nal Hessian matrix whose inverse necessarily has any resemblance to a variance±covariance matrix. To get that it is necessary to compute Jb x and from it compute J T WJ 1 . If a scaled, identity matrix is used as the initial approximation to H, no use is made of the fact that, at least in the vicinity of the minimum, a major part of the Hessian matrix is given by Jxc T WJxc . Thus, if there can be reasonable con®dence in the general features of the model, it is useful to use Jxc T WJxc as the initial approximation to H, in which case the ®rst step is in the Gauss±Newton direction. The line-search provision, however, guarantees convergence, and, when n and p are both large, as in many crystallographic applications of least squares, a quasiNewton update gives an adequate approximation to the new Hessian with a great deal less computation. Another procedure that makes use of the fact that the linear approximation gives a major part of the Hessian constructs an approximation of the form

Hx JxT WJx Bx;

where B is generated by quasi-Newton methods. The quasiNewton condition that must be satis®ed in this case is Jx T WJx Bx dc qc ; where dc x

8:1:4:13

xc , and

qc Jx T W y

Mx

Jxc T W y

Mxc :

A formula based on the BFGS update (Nash & Sofer, 1995) is Bx Bxc

Gc dc dTc Gc =dTc Gdc qc qTc =qTc dc ;

8:1:4:14

where G c Jx T WJx Bxc . Since Jx and Mx must be computed anyway, this technique maximizes the use of information with little additional computing. The resulting approximation to the Hessian, however, may not be positive de®nite, and precautions must be taken to ensure convergence. In the vicinity of a correct solution, where the residuals are small, the addition of B is not likely to help much, and it can be dropped. Far from the solution, however, it can be very helpful. An implementation of this procedure has been described by Bunch, Gay & Welsch (1993); it appears to be at least as ef®cient as the trust region, Levenberg±Marquardt procedure, and is probably better when residuals are large. In actual practice, it is not the Hessian matrix itself that is updated, but rather its Cholesky factor (Prince, 1994). This requires approximately the same number of operations and allows the solution of the linear system for computing d in a time that increases in proportion to p2 . This strategy also allows the use of the approximate Hessian in convergence checks with no signi®cant computational overhead and no extra storage, as would be required for storing both the Hessian and its inverse. 8.1.4.4. Stopping rules An iterative algorithm must contain some criterion for termination of the iteration process. This is a delicate part of all nonlinear optimization codes, and depends strongly on the scaling of the parameters. Although exceptions exist to almost all reasonable scaling rules, a basic principle is that a unit change in any variable should have approximately the same effect on the sum of squares. Thus, as discussed in Subsection 8.1.3.3 (equation 8.1.3.9), the ideal unit for each parameter is the standard uncertainty of its estimate, which can usually be adequately approximated by the reciprocal of the column norm of J. In modern codes, the user has the option of supplying a diagonal scaling matrix whose elements are the reciprocals of some estimate of a typical `signi®cant' shift in the corresponding parameter. In principle, the following conditions should hold when the convergence of a well scaled, least-squares procedure has reached its useful limit: (1) Sxc Sb x; (2) Jxc T W y Mxc O; (3) xc b x. The actual stopping rules must be chosen relative to the algorithm used (other conditions also exist) and the particular application. It is clear, however, that, since b x is not known, the last two conditions cannot be used as stated. Also, these tests are dependent on the scaling of the problem, and the variables are not related to the sizes of the quantities involved. We present tests, therefore, that are relative error tests that take into account the scaling of the variables.

684


8:1:4:12

8.1. LEAST SQUARES The general, relative error test is stated as follows. Two scalar quantities, a and b, are said to satisfy a relative error test with respect to a tolerence T if ja

bj jaj

T:

8:1:4:15

Roughly speaking, if T is of the form 10 q then a and b agree to q digits. Obviously, there is a problem with this test if a 0 and there will be numerical dif®culties if a is close to 0. Thus, in practice, (8.1.4.15) is replaced by ja

bj jaj 1T ;

8:1:4:16

which reduces to an absolute error test as a ! 0. A careful examination may be required to set this tolerance correctly, but, typically, if one of the fast, stable algorithms is used, only a few more iterations are necessary to get six or eight digits if one or two are already known. Note also that the actual value depends on ", the relative machine precision. It is fruitless to seek more digits of accuracy than are expressed in the machine representation. A test based on condition (1) is often implemented by using the linear approximation to M or the quadratic approximation to S. Thus, using the quadratic approximation to S, we can compute the predicted reduction by pred Sxc

T

T

d J W y

Mxc :

8:1:4:17

Similarly, the actual reduction is act Sxc

Sx :

8:1:4:18

The test then becomes pred 1 Sxc T , act 1 Sxc T , and act 2pred . That is, we want both the predicted and actual reductions to be small and the actual reduction to agree reasonably well with the predicted reduction. A typical value for T should be 10 4 , although the value again depends on ", and the user is cautioned not to make this tolerance too loose. For a test on condition (2), we compute the cosine of the angle between the vector of residuals and the linear subspace spanned by the columns of J, cos fdT J T Wy

Mx g=dT J T WJdSx 1=2 : 8:1:4:19

The test is cos T , where, again, T should be 10 4 or smaller. Test 3 above is usually only present to prevent the process from continuing when almost nothing is happening. Clearly, we do not know b x, thus the test is typically that corresponding elements of x and xc satisfy (8.1.4.16), where T is chosen to be 10 q . A recommended value of q is half the number of digits carried in the computation, e.g. q 8 for standard 64-bit (double-precision or 16 digit) calculations. Sometimes, the relative error test is of the form jx j xc j j=j T , where j is the standard uncertainty computed from the inverse Hessian in the last iteration. Although this test has some statistical validity, it is quite expensive and usually not worth the work involved to compute. 8.1.4.5. Recommendations One situation in which the Gauss±Newton algorithm behaves particularly poorly is in the vicinity of a saddle point in parameter space, where the true Hessian matrix is not positive de®nite. This occurs in structure re®nement where a symmetric model is re®ned to convergence and then is replaced by a less symmetric model. The hypersurface of S will have negative curvature in a ®nite sized region of the parameter space for the

less-symmetric model, and it is essential to use a safeguarded algorithm, one that incorporates a line search or a trust region, in order to get out of that region. On the basis of this discussion, we can draw the following conclusions: (1) In cases where the ®t is poor, owing to an incomplete model or in the initial stages of re®nement, methods based on the quadratic approximation to S (quasi-Newton methods) often perform better. This is particularly important when the model is close to a more symmetric con®guration. These methods are more expensive per iteration and generally require more storage, but their greater stability in such problems usually justi®es the cost. (2) With small residual problems, where the model is complete and close to the solution, a safeguarded Gauss±Newton method is preferred. The trust-region implementation (Levenberg±Marquardt algorithm) has been very successful in practice. (3) The best advice is to pick a good implementation of either method and stay with it.

8.1.5. Numerical methods for large-scale problems Because the least-squares problems arising in crystallography are often very large, the methods we have discussed above are not always the most ef®cient. Some large problems have special structure that can be exploited to produce quite ef®cient algorithms. A particular special structure is sparsity, that is, the problems have Jacobian matrices that have a large fraction of their entries zero. Of course, not all large problems are sparse, so we shall also discuss approaches applicable to more general problems. 8.1.5.1. Methods for sparse matrices We shall ®rst discuss large, sparse, linear least-squares problems (Heath, 1984), since these techniques form the basis for nonlinear extensions. As we proceed, we shall indicate how the procedures should be modi®ed in order to handle nonlinear problems. Recall that the problem is to ®nd the minimum of the quadratic form y AxT W y Ax, where y is a vector of observations, Ax represents a vector of the values of a set of linear model functions that predict the values of y, and W is a positive-de®nite weight matrix. Again, for convenience, we make the transformation y0 Uy, where U is the upper triangular Cholesky factor of W , and Z UA, so that the quadratic form becomes y0 ZxT y0 Zx, and the minimum is the solution of the system of normal equations, ZT Zx ZT y0 . Even if Z is sparse, it is easy to see that H ZT Z need not be sparse, because if even one row of Z has all of its elements nonzero, all elements of H will be nonzero. Therefore, the direct use of the normal equations may preclude the ef®cient exploitation of sparsity. But suppose H is sparse. The next step in solving the normal equations is to compute the Cholesky decomposition of H, and it may turn out that the Cholesky factor is not sparse. For example, if H has the form 0 1 x x x x Bx x 0 0C C HB @ x 0 x 0 A; x 0 0 x where x represents a nonzero element, then the Cholesky factor, R, will not be sparse, but if

685


8. REFINEMENT OF STRUCTURAL PARAMETERS 1 0 1 x 0 0 0 0 x 0 0 x B0 x 0 xC B0 x 0 0 0C C B C HB @ 0 0 x x A; B0 0 x 0 0C C B B0 0 0 x 0C x x x x C B ZB 0 0 0 0 xC C: B then R has the form Bx 0 0 0 0C C B 0 1 Bx 0 0 0 0C x 0 0 x C B @x 0 0 0 0A B0 x 0 xC C RB x x x x x @ 0 0 x x A: 0 0 0 x The work to complete the QR decomposition is of order p2 operations, because each element below the main diagonal can be These examples show that, although the sparsity of R is eliminated by one Givens rotation with no ®ll, whereas for independent of the row ordering of Z, the column order can have 0 1 a profound effect. Procedures exist that analyse Z and select a x x x x x Bx 0 0 0 0C permutation of the columns that reduces the `®ll' in R. An B C Bx 0 0 0 0C algorithm for using the normal equations is then as follows: B C (1) determine a permutation, P (an orthogonal matrix with Bx 0 0 0 0C B C one and only one 1 in each row and column, and all other ZB x 0 0 0 0C B C elements zero), that tends to ensure a sparse Cholesky B0 x 0 0 0C B C factor; B0 0 x 0 0C B C (2) store the elements of R in a sparse format; @0 0 0 x 0A (3) compute ZT Z and ZT y0 ; 0 0 0 0 x (4) factor P T ZT ZP to get R; T T T 0 (5) solve R z P Z y ; each Givens rotation ®lls an entire row, and the QR decomposi(6) solve Re x z; tion requires of order np2 operations. (7) set b x PT e x. This algorithm is fast, and it will produce acceptable accuracy if 8.1.5.2. Conjugate-gradient methods the condition number of Z is not too large. If extension to the A numerical procedure that is applicable to large-scale nonlinear case is considered, it should be kept in mind that the ®rst two steps need only be done once, since the sparsity pattern problems that may not be sparse is called the conjugate-gradient of the Jacobian does not, in general, change from iteration to method. Conjugate-gradient methods were originally designed to solve the quadratic minimization problem, ®nd the minimum of iteration. The QR decomposition of matrices that may be kept in Sx 1=2xT Hx bT x; 8:1:5:1 memory is most often performed by the use of Householder transformations (see Subsection 8.1.1.1). For sparse matrices, or where H is a symmetric, positive-de®nite matrix. The gradient for matrices that are too large to be held in memory, this of S is technique has several drawbacks. First, the method works by gx Hx b; 8:1:5:2 inserting zeros in the columns of Z, working from left to right, but at each step it tends to ®ll in the columns to the right of the and its Hessian matrix is H. Given an initial estimate, x0 , the one currently being worked on, so that columns that are initially conjugate-gradient algorithm is sparse cease to be so. Second, each Householder transformation (1) de®ne d0 gx0 ; needs to be applied to all of the remaining columns, so that the (2) for k 0; 1; 2; :::; p 1; d0 entire matrix must be retained in memory to make ef®cient use of (a) k dTk gxk =dTk Hdk ; this procedure. (b) xk1 xk k dk ; The alternative procedure for obtaining the QR decomposition (c) k gxk1 T gxk1 =gxk T gxk ; by the use of Givens rotations overcomes these problems if the (d) dk1 gxk k dk . entire upper triangular matrix, R, can be stored in memory. This algorithm ®nds the exact solution for the quadratic Since this only requires about p2 =2 locations, it is usually function in not more than p steps. This algorithm cannot be used directly for the nonlinear case possible. Also, it may happen that R has a sparse representation, so that even fewer locations will be needed. The algorithm based because it requires H to compute k , and the goal is to solve the on Givens rotations is as follows: problem without computing the Hessian. To accomplish this, the (1) bring in the ®rst p rows; exact computation of is replaced by an actual line search, and (2) ®nd the QR decomposition of this p p matrix; the termination after at most p steps is replaced by a convergence (3) for i p 1 to n, do test. Thus, we obtain, for a given starting value x0 and a general, (a) bring in row i; nonquadratic function S: (b) eliminate row i using R and at most p Givens rotations. (1) de®ne d0 gx0 ; In order to specify how to use this algorithm to solve the linear (2) set k 0; least-squares problem, we must also state how to account for Q. (3) do until convergence We could accumulate Q or save enough information to generate (a) xk1 xk k dk , where is chosen by a line search; it later, but this usually requires excessive storage. The better (b) k gxk1 T gxk1 =gxk T gxk ; alternatives are either to apply the steps of Q to y0 as we proceed (c) dk1 gxk1 k dk ; or to simply discard the information and solve RT Rx ZT y0 . It (d) k k 1. should be noted that the order of rows can make a signi®cant Note that, as promised, H is not needed. In practice, it has difference. Suppose been observed that the line search need not be exact, but that 0

686


8.1. LEAST SQUARES (2) The number of times the problem (or similar ones) will be solved. If it is a one-shot problem (a rare occurrence), then one is usually most strongly in¯uenced by easy-touse, existing software. Exceptions, of course, exist where even a single solution of the problem requires extreme care. (3) The expense of evaluating the function. With a complicated, nonlinear function like the structure-factor formula, the computational effort to determine the values of the function and its derivatives usually greatly exceeds that required to solve the linearized problem. Therefore, a full Gauss±Newton, trust-region, or quasi-Newton method may be warranted. (4) Other structure in the problem. Rarely does a problem have a random sparsity pattern. Non-zero values usually occur in blocks or in some regular pattern for which special decomposition methods can be devised. (5) The machine on which the problem is to be solved. We have said nothing about the existing vector and parallel processors. Suf®ce it to say that the most ef®cient procedure for a serial machine may not be the right algorithm for one of these novel machines. Appropriate numerical methods for such architectures are also being actively investigated.

periodic restarts in the steepest-descent direction are often helpful. This procedure often requires more iterations and function evaluations than methods that store approximate Hessians, but the cost per iteration is small. Thus, it is often the overall least-expensive method for large problems. For the least-squares problem, recall that we are ®nding the minimum of Sx 1=2y0

ZxT y0

Zx;

8:1:5:3

for which gx ZT Zx

y0 :

8:1:5:4

By using these de®nitions in the conjugate-gradient algorithm, it is possible to formulate a speci®c algorithm for linear least squares that requires only the calculation of Z times a vector and ZT times a vector, and never requires the calculation or factorization of ZT Z. In practice, such an algorithm will, due to roundoff error, sometimes require more than p iterations to reach a solution. A detailed examination of the performance of the procedure shows, however, that fewer than p iterations will be required if the eigenvalues of ZT Z are bunched, that is, if there are sets of multiple eigenvalues. Speci®cally, if the eigenvalues are bunched into k distinct sets, then the conjugate-gradient method will converge in k iterations. Thus, signi®cant improvements can be made if the problem can be transformed to one with bunched eigenvalues. Such a transformation leads to the so-called preconditioned conjugate-gradient method. In order to analyse the situation, let C be a p p matrix that transforms the variables, such that x0 Cx:

8:1:5:5

Then, y0

Zx y0

ZC 1 x0 :

8:1:5:6 0

Therefore, C should be such that the system Cx x is easy to solve, and ZC 1 T ZC 1 has bunched eigenvalues. The ideal choice would be C R, where R is the upper triangular factor of the QR decomposition, since ZR 1 QZ . QTZ QZ I has all of its eigenvalues equal to one, and, since R is triangular, the system is easy to solve. If R were known, however, the problem would already be exactly solved, so this is not a useful alternative. Unfortunately, no universal best choice seems to exist, but one approach is to choose a sparse approximation to R by ignoring rows that cause too much ®ll in or by making R a diagonal matrix whose elements are the Euclidean norms of the columns of Z. Bear in mind that, in the nonlinear case, an expensive computation to choose C in the ®rst iteration may work very well in subsequent iterations with no further expense. One should be aware of the trade off between the extra work per iteration of the preconditioned-conjugate gradient method versus the reduction in the number of iterations. This is especially important in nonlinear problems. The solution of large, least-squares problems is currently an active area of research, and we have certainly not given an exhaustive list of methods in this chapter. The choice of method or approach for any particular problem is dependent on many conditions. Some of these are: (1) The size of the problem. Clearly, as computer memories continue to grow, the boundary between small and large problems also grows. Nevertheless, even if a problem can ®t into memory, its sparsity structure may be exploited in order to obtain a more ef®cient algorithm.

8.1.6. Orthogonal distance regression It is often useful to consider the data for a least-squares problem to be in the form ti ; yi , i 1; . . . ; n, where the ti are considered to be the independent variables and the yi the dependent variables. The implicit assumption in ordinary least squares is that the independent variables are known exactly. It sometimes occurs, however, that these independent variables also have errors associated with them that are signi®cant with respect to the errors in the observations yi . In such cases, referred to as èrrors in variables' or `measurement error models', the ordinary leastsquares methodology is not appropriate and its use may give misleading results (see Fuller, 1987). ^ i ; x to be the model functions that predict the Let us de®ne Mt yi . Observe that ordinary least squares minimizes the sum of the squares of the vertical distances from the observed points yi to ^ x. If ti has an error i , and these errors are the curve Mt; normally distributed, then the maximum-likelihood estimate of the parameters is found by minimizing the sum of the squares of the weighted orthogonal distances from the point yi to the curve ^ x. More precisely, the optimization problem to be solved is Mt; given by n P ^ i i ; x T Wy yi Mt ^ i i ; x Ti Wt i ; min yi Mt x; i1

8:1:6:1 where Wy and Wt are appropriately chosen weights. Problem (8.1.6.1) is called the orthogonal distance regression (ODR) problem. Problem (8.1.6.1) can be solved as a least-squares problem in the combined variables x; by the methods given above. This, however, is quite inef®cient, since such a procedure would not exploit the special structure of the ODR problem. Few algorithms that exploit this structure exist; one has been given by Boggs, Byrd & Schnabel (1987), and the software, called ODRPACK, is by Boggs, Byrd, Donaldson & Schnabel (1989). The algorithm is based on the trust-region (Levenberg± Marquardt) method described above, but it exploits the special structure of (8.1.6.1) so that the cost of each iteration is no more expensive than the cost of a similar iteration of the corresponding

687


8. REFINEMENT OF STRUCTURAL PARAMETERS ordinary least-squares problems. For a discussion of some of the statistical properties of the resulting estimates, including a procedure for the computation of the variance±covariance matrix, see Boggs & Rogers (1990). 8.1.7. Software for least-squares calculations Giving even general recommendations on software is a dif®cult task for several reasons. Clearly, the selection of methods discussed in earlier sections contains implicitly some recommendations for approaches. Among the reasons for avoiding speci®cs are the following: (1) Assessing differences in performance among various codes requires a detailed knowledge of the criteria the developer of a particular code used in creating it. A program written to emphasize speed on a certain class of problems on a certain machine is impossible to compare directly with a program written to be very reliable on a wide class of problems and portable over a wide range of machines. Other measures, including ease of maintenance and modi®cation and ease of use, and other design criteria, such as interactive versus batch, stand alone versus user-callable, automatic computation of related statistics versus no statistics, and so forth, make the selection of software analogous to the selection of a car. (2) Choosing software requires detailed knowledge of the needs of the user and the resources available to the user. Considerations such as problem size, machine size, machine architecture and ®nancial resources all enter into the decision of which software to obtain. (3) A software recommendation made on the basis of today's knowledge ignores the fact that algorithms continue to be invented, and old algorithm continue to be rethought in the light of new developments and new machine architectures. For example, when vector processors ®rst appeared, algorithms for sparse-matrix calculations were very poor at exploiting this capability, and it was thought that these

new machines were simply not appropriate for such calculations. Now, however, recent methods for sparse matrices have achieved a high degree of vectorization. For another example, early programs for crystallographic, full-matrix, least-squares re®nement spent a large fraction of the time building the normal-equations matrix. The matrix was then inverted using a procedure called Gaussian elimination, which does not exploit the fact that the matrix is positive de®nite. Some programs were later converted to use Cholesky decomposition, which is at least twice as fast, but many were not because the inversion process took a small fraction of the total time. Linear algebra, however, is readily adaptable to vector and parallel machines, and procedures such as QR factorization are extremely fast, while the calculation of structure factors, with its repeated evaluations of trigonometric functions, becomes the time-controlling step. The general recommendation is to analyse carefully the needs and resources in terms of these considerations, and to seek expert assistance whenever possible. As much as possible, avoid the temptation to write your own codes. Despite the fact that the quality of existing software is far from uniformly high, the bene®ts of utilizing high-quality software generally far outweigh the costs of ®nding, obtaining, and installing it. Sources of information on software have improved signi®cantly in the past several years. Nevertheless, the task of identifying software in terms of problems that can be solved; organizing, maintaining and updating such a list; and informing the user community still remains formidable. A current, problem-oriented system that includes both a problem classi®cation scheme and a network tool for obtaining documentation and source code (for software in the public domain) is the Guide to Available Mathematical Software (GAMS). This system is maintained by the National Institute of Standards and Technology (NIST) and is continually being updated as new material is received. It gives references to software in several software repositories; the URL is http:// math.nist.gov/gams.

688



8.2. Other re®nement methods By E. Prince and D. M. Collins

Chapter 8.1 discusses structure re®nement by the method of least squares, which has a long history of successful use in data ®tting and statistical analysis of results. It is an excellent technique to use in a wide range of practical problems, it is easy to implement, and it usually gives results that are straightforward and unambiguous. If a set of observations, yi , is an unbiased estimate of the values of model functions, Mi x, a properly weighted least-squares estimate is the best, linear, unbiased estimate of the parameters, x, provided the variances of the p.d.f.s of the populations from which the observations are drawn are ®nite. This assumes, however, that the model is correct and complete, an assumption whose validity may not necessarily be easily justi®ed. Furthermore, least squares tends to perform poorly when the distribution of errors in the observations has longer tails than a normal, or Gaussian, distribution. For these reasons, a number of other procedures have been developed that attempt to retain the strengths of least squares but are less sensitive to departures from the ideal conditions that have been implicitly assumed. In this chapter, we discuss several of these methods. Two of them, maximum-likelihood methods and robust/resistant methods, are closely related to least squares. A third one uses a function that is mathematically related to the entropy function of thermodynamics and statistical mechanics, and is therefore referred to as the maximum-entropy method. For a discussion of the particular application of least squares to structure re®nement with powder data that has become known as the Rietveld method (Rietveld, l969), see Chapter 8.6. 8.2.1. Maximum-likelihood methods In Chapter 8.1, structure re®nement is presented as ®nding the answer to the question, `given a set of observations drawn randomly from populations whose means are given by a model, Mx, for some set of unknown parameters, x, how can we best determine the means, variances and covariances of a joint probability density function that describes the probabilities that the true values of the elements of x lie in certain ranges?'. For a broad class of density functions for the observations, the linear estimate that is unbiased and has minimum variances for all parameters is given by the properly weighted method of least squares. The problem can also be stated in the slightly different manner, `given a model and a set of observations, what is the likelihood of observing those particular values, and for what values of the parameters of the model is that likelihood a maximum?'. This set of parameters is the maximum-likelihood estimate. Suppose the ith observation is drawn from a population whose p.d.f. is i i , where i yi Mi x=si , x is the set of `true' values of the parameters, and si is a measure of scale appropriate to that observation. If the observations are independent, their joint p.d.f. is the product of the individual, marginal p.d.f.s: J

n Q i1

i i :

8:2:1:1

The function i i can also be viewed as a conditional p.d.f. for yi given Mi x, or, equivalently, as a likelihood function for x given an observed value of yi , in which case it is written li xjyi . Because a value actually observed logically must have a ®nite, positive likelihood, the density function in (8.2.1.1) and its logarithm will be maximum for the same values of x:

lnlxjy


i1

lnli xjyi :

8:2:1:2

In the particular case where the error distribution is normal, and i , the standard uncertainty of the ith observation, is known, then 1 i i p exp 1=2fyi Mi x=i g2 ; 8:2:1:3 2i and the logarithm of the likelihood function is maximum when n P 8:2:1:4 S fy Mi x=i g2 i1

is minimum, and the maximum-likelihood estimate and the leastsquares estimate are identical. For an error distribution that is not normal, the maximumlikelihood estimate will be different from the least-squares estimate, but it will, in general, involve ®nding a set of parameters for which a sum of terms like those in (8.2.1.2) is a maximum (or the sum of the negatives of such terms is a minimum). It can thus be expressed in the general form: ®nd the minimum of the sum n P S i ; 8:2:1:5 i1

where is de®ned by x lnx, and x is the p.d.f. of the error distribution appropriate to the observations. If x x2 =2, the method is least squares. If the error distribution is the Cauchy distribution, x 1 x2 1 , x ln1 x2 , which increases much more slowly than x2 as jxj increases, causing large deviations to have much less in¯uence than they do in least squares. Although there is no need for x to be a symmetric function of x (the error distribution can be skewed), it may be assumed to have a minimum at x 0, so that dx=dx 0. A series expansion about the origin therefore begins with the quadratic term, and 1 P 2 k x x =2 1 8:2:1:6 ak x : k1

This procedure is thus equivalent to a variant of least squares in which the weights are functions of the deviation. 8.2.2. Robust/resistant methods Properly weighted least squares gives the best linear estimate for a very broad range of distributions of random errors in the data and the maximum-likelihood estimate if that error distribution is normal or Gaussian. But the best linear estimator may nevertheless not be a very good one, and the error distribution may not be well known. It is therefore important to address the question of how good an estimation procedure may be when the conditions for which it is designed may not be satis®ed. Re®nement procedures may be classi®ed according to the extent that they possess two properties known as robustness and resistance. A procedure is said to be robust if it works well for a broad range of error distributions and resistant if its results are not strongly affected by ¯uctuations in any small subset of the data. Because least squares is a linear estimator, the in¯uence of any single data point on the parameter estimates increases without limit as the difference between the observation and the model increases. It therefore works poorly if the actual error


n P

8. REFINEMENT OF STRUCTURAL PARAMETERS distribution contains large deviations with a frequency that substantially exceeds that expected from a normal distribution. Further, it has the undesirable property that it will make the ®t of a few wildly discrepant data points better by making the ®t of many points a little worse. Least squares is therefore neither robust nor resistant. Tukey (1974) has listed a number of properties a procedure should have in order to be robust and resistant. Because least squares works well when the error distribution is normal, the procedure should behave like least squares for small deviations whose distribution is similar to the normal distribution. It should de-emphasize large differences between the model and the data, and it should connect these extremes smoothly. A procedure suggested by Tukey was applied to crystal structure re®nement by Nicholson, Prince, Buchanan & Tucker (1982). It corresponds to a ®tting function [equation (8.2.1.5)] of the form jj < 1; 2 =2 1 2 4 =3 8:2:2:1 jj 1; 1=6 where i yi Mi x=si , and s is a resistant measure of scale. In order to see what is meant by a resistant measure, consider a large sample of observations, yi , with a normal distribution. The sample mean, y 1=n

n P i1

yi ;

8:2:2:2

is an unbiased estimate of the population mean. Contamination of the sample by a small number of observations containing large, systematic errors, however, would have a large effect on the estimate. The median value of yi is also an unbiased estimate of the population mean, but it is virtually unaffected by a few contaminating points. Similarly, the sample variance, s2 1=n

1

n P i1

yi

y2 ;

8:2:2:3

is an unbiased estimate of the population variance, but, again, it is strongly affected by a few discrepant points, whereas 0:7413rq 2 , where rq is the interquartile range, the difference between the ®rst and third quartile observations, is an estimate of the population variance that is almost unaffected by a small number of discrepant points. The median and the interquartile range are thus resistant quantities that can be used to estimate the mean and variance of a population distribution when the sample may contain points that do not belong to the population. A value of the scale parameter, si , for use in evaluating the quantities in (8.2.2.1), that has proved to be useful is m represents the median value of s i 9 m i , where yi Mi x=i , the median absolute deviation, or MAD. Implementation of a procedure based on the function given in (8.2.2.1) involves modi®cation of the weights used in each cycle by jj < 1; ' 1 2 2 ; 8:2:2:4 jj 1: ' 0; Because of this weight modi®cation, the procedure is sometimes referred to as ìteratively reweighted least squares'. It should be recognized, however, that the function that is minimized is more complex than a sum of squares. In a strict application of the Gauss±Newton algorithm (see Section 8.1.3) to the minimization of this function, each term in the summations to form the normal-equations matrix contains a factor !i , where ! d2 =d2 1 62 54 . This factor actually gives some data points a negative effective `weight', because the sum is actually reduced by making the ®t worse. The inverse of

this normal-equations matrix is not an estimate of the variance± covariance matrix; for that the unmodi®ed weights, equal to 1=i2 , must be used, but, because more discrepant points have been deliberately down weighted relative to the ideal weights, the variances are, in general, underestimated. A recommended procedure (Huber, 1973; Nicholson et al., 1982) is to calculate the normal-equations matrix using the unmodi®ed weights, invert that matrix, and premultiply by an estimate of the variance of the residuals (Section 8.4.1) using modi®ed weights and n p degrees of freedom. Huber showed that this estimate is biased low, and suggests multiplication by a number, c2 , greater than one, and given by c 1 ps2! =n!2 =!;

where ! is the mean value, and s2! is the variance of !i over the entire data set. The conditions under which this expression is derived are not well satis®ed in the crystallographic problem, but, if n=p is large and ! is not too much less than one, the value of c will be close to 1=!. ! plays the role of a `variance ef®ciency factor'. That is, the variances are approximately those that would be achieved with a least-squares ®t to a data set with normally distributed errors that contained n! data points. Robust/resistant methods have been discussed in detail by Huber (1981), Belsley, Kuh & Welsch (1980), and Hoaglin, Mosteller & Tukey (1983). An analysis by Wilson (1976) shows that a ®tting procedure gives unbiased estimates if n n X X @wi dyci 2 @wi dyci 2 i 2 i ; 8:2:2:6 @yci dx @yoi dx i1 i1 where yoi and yci are the observed and calculated values of yi , respectively. Least squares is the case where all terms on both sides of the equation are equal to zero; the weights are ®xed. In maximum-likelihood estimation or robust/resistant estimation, the effective weights are functions of the deviation, causing possible introduction of bias. Equation (8.2.2.6), however, suggests that the estimates will still be unbiased if the sums on both sides are zero, which will be the case if the error distribution and the weight modi®cation function are both symmetric about 0. Note that the fact that two different weighting schemes applied to the same data lead to different values for the estimate does not necessarily imply that either value is biased. As long as the observations represent unbiased estimates of the values of the model functions, any weighting scheme gives unbiased estimates of the model parameters, although some weighting schemes will cause those estimates to be more precise than others will. Bias can be introduced if a procedure systematically causes ¯uctuations in one direction to be weighted more heavily than ¯uctuations in the other. For example, in the Rietveld method (Chapter 8.6), the observations are counts of quanta, which are subject to ¯uctuation according to the Poisson distribution, where the probability of observing k counts per unit time is given by k lk exp l=k!:

8:2:2:7

The mean and the variance of this p.d.f. are both equal to l, so that the ideal weighting should have wi 1=li . However, li is not known a priori, and must be estimated. The usual procedure is to take ki as an estimate of li , but this is an unbiased estimate only asymptotically for large k (Box & Tiao, 1973), and, furthermore, causes observations that have negative, random errors to be weighted more heavily than observations that have positive ones. This correlation can be removed by using, after a preliminary cycle of re®nement, Mi b x as an estimate of li . This

690


8:2:2:5

8.2. OTHER REFINEMENT METHODS might seem to have the effect of making the weights dependent on the calculated values, so that the right-hand side of (8.2.2.6) is no longer zero, but this applies only if the weights are changed during the re®nement. There is thus no con¯ict with the result in (8.1.2.9). In practice, in any case, many other sources of uncertainty are much more important than any possible bias that could be introduced by this effect. 8.2.3. Entropy maximization 8.2.3.1. Introduction Entropy maximization, like least squares, is of interest primarily as a framework within which to ®nd or adjust parameters of a model. Rationalization of the name èntropy maximization' by analogy to thermodynamics is controversial, but there is formal proof (Shore & Johnson, 1980) supporting entropy maximization as the unique method of inference that satis®es basic consistency requirements (Livesey & Skilling, 1985). The proof consists of discovering the consequences of four consistency axioms, which may be stated informally as follows: (1) the result of the inference should be unique; (2) the result of the inference should be invariant to any transformations of coordinate system; (3) it should not matter whether independent information is accounted for independently or jointly; (4) it should not matter whether independent subsystems are treated separately in conditional problems or collected and treated jointly. The term èntropy' is used in this chapter as a name only, the name for variation functions that include the form ' ln ', where ' may represent probability or, more generally, a positive proportion. Any positive measure, either observed or derived, of the relative apportionment of a characteristic quantity among observations can serve as the proportion. The method of entropy maximization may be formulated as follows: given a set of n observations, yi , that are measurements of quantities that can be described by model functions, Mi x, where x is a vector of parameters, ®nd the prior, positive proportions, i f yi , and the values of the parameters for which the positive proportions ' f Mi x make the sum n P

S '0i

P

i1

0i

'0i ln'0i =0i ;

8:2:3:1

S

i1

'0i ln'0i =0i l1 1 x; y l2 2 x; y . . . ; 8:2:3:2

where the ls are undetermined multipliers, but we shall discuss here only applications where li 0 for all i, and an unrestrained entropy is maximized. A necessary condition for S to be a maximum is for the gradient to vanish. Using n @S X @S @'i 8:2:3:3 @xj @xj @'i i1 and

It should be noted that, although the entropy function should, in principle, have a unique stationary point corresponding to the global maximum, there are occasional circumstances, particularly with restrained problems where the undetermined multipliers are not all zero, where it may be necessary to verify that a stationary solution actually maximizes entropy. 8.2.3.2. Some examples For an example of the application of the maximum-entropy method, consider (Collins, 1984) a collection of diffraction intensities in which various subsets have been measured under different conditions, such as on different ®lms or with different crystals. All systematic corrections have been made, but it is necessary to put the different subsets onto a common scale. Assume that every subset has measurements in common with some other subset, and that no collection of subsets is isolated from the others. Let the measurement of intensity Ih in subset i be Jhi , and let the scale factor that puts intensity Ih on the scale of subset i be ki . Equation (8.2.3.1) becomes n X m X ki Ih 0 0 ; 8:2:3:6 ki Ih ln S Jhi0 h1 i1 where the term is zero if Ih does not appear in subset i. Because ki and Ih are parameters of the model, equations (8.2.3.5) become ! X m n m m X X X ki Ih 0 ki Ih 0 0 0; ki ln ki Ih kl ln Jhi0 Jhi0 i1 h1 i1 l1 8:2:3:7a and n X h1

0 n @S X @S @'k ; 0 @'i @'i @'k k1

8:2:3:4

k I 0 Ih ln i 0 h Jhi

n X m X h1 i1

ki Ih

0

n X l1

! ki Ih 0 0: Il ln Jhi0

These simplify to ln Ih Q

m P i1

ki0 lnki =Jhi

8:2:3:8a

Ih0 lnIh =Jhi ;

8:2:3:8b

and ln ki Q

n P h1

where Q

n P m P h1 i1

ki Ih 0 lnki Ih =Jhi :

8:2:3:8c

Equations (8.2.3.8) may Pn be solved iteratively, starting with the approximations ki h1 Jhi and Q 0. The standard uncertainties of scale factors and intensities are not used in the solution of equations (8.2.3.8), and must be computed separately. They may be estimated on a fractional

basis from the variances population means Jhi =Ih

of estimated for a scale factor and Jhi =ki for an intensity, respectively. The maximum-entropy scale factors and scaled intensities are relative, and either set may be multiplied by an arbitrary, positive constant without affecting the solution.

691


8:2:3:7b

P

where 'i 'j and i j , a maximum. S is called the Shannon±Jaynes entropy. For some applications (Collins, 1982), it is desirable to include in the variation function additional terms or restraints that give S the form n P

straightforward algebraic manipulation gives equations of the form ( !) n n X X @'i @'k '0 0 8:2:3:5 'i ln i0 0: @xj @xj i i1 k1

8. REFINEMENT OF STRUCTURAL PARAMETERS For another example, consider the maximum-entropy ®t of a linear function to a set of independently distributed variables. Let yi represent an observation drawn from a population with mean a0 a1 xi and ®nite variance i2 ; we wish to ®nd the maximumentropy estimate of a0 and a1 . Assume that the mismatch between the observation and the model is normally distributed, so that its probability density is the positive proportion 'i 'i 2i2 where i yi Letting A' 1 " n P 'i i =i2 i1

n P i1

exp 2i =2i2 ;

8:2:3:9

a0 a1 xi . The prior proportion is given by P

and

1=2

i '0 2i2

1=2

:

8:2:3:10

'i , equations (8.2.3.5) become !# n P 2 A' 'i 'j j =j 2i =i2 0 j1

" 'i i xi =i2

A ' 'i

n P j1

8:2:3:11a

!# 'j j xj =j2

2i =i2 0; 8:2:3:11b

which simpli®es to 0 P 1 n n P wi wi xi B i1 C a0 i1 B C n n @P A a P 1 2 wi xi wi xi i1 i1 0 ! 1 n n P P 2 2 B C w y i A' 'j j =j B i1 i i C j1 B C ! C; B BP C n n @ w y x 2 A P ' x = 2 A i i i j j j i ' j i1

j1

8:2:3:12

where wi may be interpreted as a weight and is given by wi 'i 2i =i4 . Equations (8.2.3.12) may be solved iteratively, starting with the approximations that the sums over j on the righthand side are zero and wi 1:0 for all i, that is, using the solutions to the corresponding, unweighted least-squares problem. Resetting wi after each iteration by only half the indicated amount defeats a tendency towards oscillation. Approximate standard uncertainties for the parameters, a0 and a1 , may be computed by conventional means after setting to zero the sums over j on the right-hand side of equations (8.2.3.12). (See, however, a discussion of computing variance±covariance matrices in Section 8.1.2.) Note that wi is small for both small and large values of i . Thus, in contrast to the robust/resistant methods (Section 8.2.2), which de-emphasize only the large differences, this method down-weights both the small and the large differences and adjusts the parameters on the basis of the moderate-size mismatches between model and data. The procedure used in this two-dimensional, linear model can be extended to linear models, and linear approximations to nonlinear models, in any number of dimensions using methods discussed in Chapter 8.1. The maximum-entropy method has been described (Jaynes, 1979) as being `maximally noncommittal with respect to all other matters; it is as uniform (by the criterion of the Shannon information measure) as it can be without violating the given constraint[s]'. Least squares, because it gives minimum variance estimates of the parameters of a model, and therefore of all functions of the model including the predicted values of any additional data points, might be similarly described as `maximally committal' with regard to the collection of more data. Least squares and maximum entropy can therefore be viewed as the extremes of a range of methods, classi®ed according to the degree of a priori con®dence in the correctness of the model, with the robust/resistant methods lying somewhere in between (although generally closer to least squares). Maximum-entropy methods can be used when it is desirable to avoid prejudice in favour of a model because of doubt as to the model's correctness.

692



8.3. Constraints and restraints in re®nement By E. Prince, L. W. Finger, and J. H. Konnert

In Chapter 8.1, the method of least squares is discussed as a technique for ®tting a theoretical model that contains adjustable parameters to a set of observations. The discussion is very general and contains very little mention of what sorts of quantities the observations are or what the model represents. In crystallography, the model is a crystal, which is constructed from identical unit cells that contain atoms, and which diffracts X-rays, neutrons or electrons in a manner that is characteristic of the arrangement of those atoms. The sample may be either a single crystal or a polycrystalline powder, and the observations are diffracted intensities, which may be ®tted directly, as in the Rietveld method for powders (see Chapter 8.6; also Rietveld, 1969), or converted to derived quantities such as integrated intensities, squared moduli of structure amplitudes, or the structure amplitudes themselves. The model generally contains a scale factor and may contain parameters describing other experimental effects, such as extinction. Each atom in the unit cell requires three parameters to describe its mean position and various parameters to describe random deviations from that position owing to thermal motion or disorder. Models that treat each atom independently, however, do not allow for the fact that a great deal more is known about a crystal initially than simply its chemical composition. Atoms have fairly de®nite sizes and tend to occupy sites whose surroundings conform to a rather limited set of common con®gurations. In this chapter, we discuss ways of using this additional information. First, we shall discuss the use of constraints to reduce the number of parameters that must be varied and account for relationships among parameters that are dictated by the laws of chemistry and physics. Then we shall discuss the use of restraints, which effectively add to the number of observations that must be ®tted by the model. 8.3.1. Constrained models The techniques of least squares are applicable for re®ning almost any model, but the question of the suitability of the model remains. The addition of parameters may reduce the residual disagreement, but lead to solutions that have no physical or chemical validity. Addition of constraints is one method of constricting the solutions. 8.3.1.1. Lagrange undetermined multipliers The classical technique for application of constraints is the use of Lagrange undetermined multipliers, in which the set of p parameters, xj , is augmented by p q (q < p) additional unknowns, lk , one for each constraint relationship desired. The problem may be stated in the form: ®nd the minimum of n P S wi yi Mi x2 ; 8:3:1:1a i1

subject to the condition fk x 0 k 1; 2; . . . ; p

q:

8:3:1:1b

This may be shown (Gill, Murray & Wright, 1981) to be equivalent to the problem: ®nd a point at which the gradient of S0

n P i1

wi yi

Mi x2

pPq k1

lk fk x

8:3:1:2

vanishes. Solving for the stationary point leads to a set of simultaneous equations of the form

@S 0 =@xj @S=@xj


k1

lk @ fk x=@xj 0

8:3:1:3a

and @S 0 =@lk fk x 0:

8:3:1:3b

Thus, the number of equations, and the number of unknowns, is increased from p to 2p q. In cases where the number of constraint relations is small, and where it may be dif®cult to solve the relations for some of the parameters in terms of the rest, this method yields the desired results without too much additional computation (Ralph & Finger, 1982). With the large numbers of parameters, and large numbers of constraints, that arise in many crystallographic problems, however, the use of Lagrange multipliers is computationally inef®cient and cumbersome. 8.3.1.2. Direct application of constraints In most cases encountered in crystallography, constraints may be applied directly, thus reducing rather than increasing the size of the normal-equations matrix. For each constraint introduced, one of the parameters becomes dependent on the remaining set, and the rank of the remaining system is reduced by one. For p parameters and p q constraints, the problem reduces to q parameters. If the Gauss±Newton algorithm is used (Section 8.1.4), the normal-equations matrix is AT WA, where Aij @Mi =@xj ;

8:3:1:4

and W is a weight matrix. A constrained model, Mi z, maybe constructed using relations of the form xj gj z1 ; z2 ; . . . ; zq :

8:3:1:5

Applying the chain rule for differentiation, the normal-equations matrix for the constrained model is BT WB, where p P Bik @Mi x=@zk @Mi x=@xj @xj =@zk : 8:3:1:6 j1

This may be written in matrix form B AC, where Cjk @xj =@zk de®nes a p q constraint matrix. The application of constraints involves (a) determination of the model to be used, (b) calculation of the elements of C, and (c) computation of the modi®ed normal-equations matrix. The construction of matrix C by a procedure known as the variable reduction method may be presented formally as follows: Designate by Z the matrix whose elements are Zjk @gj x=@xk ;

8:3:1:7

and partition Z in the form Z U; V , where V is composed of p q columns of Z chosen to be linearly independent, so that V is nonsingular. [V is shown as the last p q columns only for convenience. Any linearly independent set may be chosen.] The rows of Z form a basis for a p q-dimensional subspace of the p-dimensional parameter space, and we wish to construct a basis for z, a q-dimensional subspace that is orthogonal to it, so that all shifts within that subspace starting at a point where the constraints are satis®ed, a feasible point, leave the values of the constraint relations unchanged. This basis is used for the columns of C, which is given by Iq C : 8:3:1:8 V 1U


pPq

8. REFINEMENT OF STRUCTURAL PARAMETERS In this formulation, the columns of V correspond to dependent parameters that are functions of the independent parameters corresponding to the columns of U. Most existing programs provide for the calculation of the structure factor, F, and its partial derivatives with respect to a conventional set of parameters, including occupancy, position, isotropic or anisotropic atomic displacement factors, and possibly higher cumulants of an atomic density function (Prince, 1994). The constrained calculation is usually performed by evaluating selected elements, @xj =@zk . Because the constraint matrix is often extremely sparse, calculation of a limited sum involving only the nonzero elements is usually computationally superior to a full matrix multiplication. After adjustment of the z's, equations (8.3.1.5) are used to update the parameters. Using this procedure, it is not necessary to express the structure factor, or its derivatives, in terms of the re®ned parameters. This is particularly important when the constrained model involves arbitrary molecular shapes or rigid-body thermal motions. The need for constraints arises most frequently when the crystal structure contains atoms in special positions. Here, certain parameters will be constant or linearly related to others. If a parameter is constrained to be a constant, the corresponding row of C will contain zeros, and that column will be ignored. When parameters are linearly dependent on others, which may occur in trigonal, hexagonal, tetragonal and cubic space groups, the modi®cation indicated in (8.3.1.6) cannot be avoided. The constraint relationships among position parameters are trivial. Levy (1956) described an algorithm for determining the constraints that pertain to second and higher cumulants in the structure-factor formula. Table 8.3.1.1 is a summary of relations that are found for anisotropic atomic displacement factors, with a listing of the space groups in which they occur. Johnson (1970) provides a table listing the number of unique coef®cients for each possible site symmetry for tensors of various ranks, which is useful information for veri®cation of constraint relationships. Another important use of constraints applies to the occupancies of certain sites in the crystal where, for example, a molecule is disordered in two or more possible orientations or (very common in minerals) several elements are distributed among several sites. In both cases, re®nement of all of the fractional occupancies tends to be extremely ill conditioned, because of 0

cos ' cos cos ! sin ' sin ! R @ sin ' cos cos ! cos ' sin ! sin cos !

i1

bi aij pj ;

j1

8:3:1:9

where bi is the multiplicity of the ith site, aij is the fractional occupancy of the jth species in the ith site, and p is the total number of atoms of species j per unit cell. For a given crystal structure and composition, the bs and ps are known, and, furthermore, it is possible to write an additional constraint for the total occupancy of each site,

8:3:1:10

1 cos ' sin sin ' sin A: cos

8:3:1:11

The overall transformation of a vector, x0 , from the special coordinate system to the crystallographic system is given by x D 1 Rx0 t;

8:3:1:12

where t is the origin offset between the two systems and D is the upper triangular Cholesky factor (Subsection 8.1.1.1) of the metric tensor, G, which is de®ned by 0 1 aa ab ac 8:3:1:13 G @ a b b b b c A: ac bc cc Equations (8.3.1.12) are the constraint relationships, and the re®nable parameters include the adjustable parameters in the special system, the origin offset, and the three rotation angles. This set of parameters, although it is written in a very different manner, is a linear transformation of a subset of the conventional

694


aij 1:

If necessary, vacancies may be included as one of the n species present. In theory, (8.3.1.9) and (8.3.1.10) could be solved for n 1 m 1 unknown parameters, aij , with m n 1 constraint relations, but, in practice, at most one occupancy factor per site can be re®ned. When constraints are applied, the correlations between occupancies and displacement factors are greatly reduced. In the analysis of a crystal structure, it may be desirable to test various constraints on the shape or symmetry of a molecule. For example, the molecule of a particular compound may have orthorhombic symmetry in the liquid or vapour phase, but crystallize with a monoclinic or triclinic space group. Without constraints, it is impossible to determine whether the crystallization has caused changes in the molecular conformation. Residual errors in the observations will invariably lead to deviations from the original molecular geometry, but these may or may not be meaningful. With molecular-shape constraints, it is possible to constrain the geometry to any desired conformation. The ®rst step is to describe the molecule in a special, orthonormal coordinate system that has a well de®ned relationship between the coordinate axes and the symmetry elements. If this system is properly chosen, the description of the molecule is easy. The next step is to describe the transformation between this orthonormal system and the crystallographic axes. A standard, orthonormal coordinate system (Prince, 1994) can be constructed with its x axis parallel to a and its z axis parallel to c . If the special system is translated with respect to the standard system so that they share a common origin, Eulerian angles, !, , and ', may be used to de®ne a matrix that rotates the special coordinates into the standard system. Angle ! is de®ned as the clockwise rotation through which the special system must be rotated about the z axis of the standard system to bring the z axis of the special system into the x; z plane of the standard system. Similarly, angle is the clockwise angle through which the resulting, special system must be rotated about the y axis of the standard system to bring the z axes into coincidence, and, ®nally, angle ' is the clockwise angle through which the special system must be rotated about the common z axes to bring the other axes into coincidence. The overall transformation is given by

cos ' cos sin ! sin ' cos ! sin ' cos sin ! cos ' cos ! sin sin !

high correlations between occupancies and atomic displacement parameters. The overall chemistry, however, may be known from electron microprobe (Finger, 1969) or other analytic techniques to much better precision than it is possible to determine it using diffraction data alone. The constraining equations for the occupancies of n species in m sites have the form m P

n P

8.3. CONSTRAINTS AND RESTRAINTS IN REFINEMENT Table 8.3.1.1. Symmetry conditions for second-cumulant tensors If more than one condition is applicable for a space group, the site is identi®ed by its Wyckoff notation following the space-group symbol. The stated conditions are valid only for the ®rst equipoint listed for the position. For space groups with alternative choices of origin, the option with a centre of symmetry has been selected. (A) Monoclinic (1) Site symmetry m, 2, 2=m ± four independent elements (a) 12 23 0; one principal axis parallel to [010] All groups with unique axis b (b) 13 23 0; one principal axis parallel to [001] All groups with unique axis c (B) Orthorhombic (1) Site symmetry m, 2, 2=m ± four independent elements (a) 12 13 0; one principal axis parallel to [100] P222i; j; k; l, P2221 a; b, C2221 a, C222e; f , F222e; j, I222e; f , I21 21 21 a, Pmm2g; h, Pmc21 , Pma2c, Pmn21 , Cmm2e, Cmc21 , Amm2d; e, Ama2b, Fmm2c, Imm2d, Ima2b, Pmmmu; v, Pnnng; h, Pccmi; j, Pbang; h, Pmmak, Pnnad, Pmnaa; b; c; d; e; f ; h, Pbcmc, Pmmne, Cmcma; b; e; f , Cmcaa; b; d; f , Cmmmn, Cccmg, Cmmac; d; h; i; m, Cccae, Fmmmc; l; m, Fddde, Immml, Ibam f , Ibcac, Immaa; b; f ; h (b) 12 23 0; one principal axis parallel to [010] P222m; n; o; p, P2221 c; d, C2221 b, C222g; h, F222 f ; i, I222g; h, I21 21 21 b, Pmm2e; f , Cmm2d, Amm2c, Abm2c, Fmm2d, Imm2c, Pmmmw; x, Pnnni; j, Pccmk; l, Pbani; j, Pmmaa; b; c; d; g; h; i; j, Pmnag, Pccac, Pmmm f , Pbcn, Pnma, Cmcae, Cmmmo, Cccmh, Cmmae; f ; j; k; n, Ccca f , Fmmmd; k; n, Fddd f , Immmm, Ibamg, Ibcad, Immac; d; g; i (c) 13 23 0; one principal axis parallel to [001] P222q; r; s; t, P21 21 2, C222i; j; k, F222g; h, I222i; j, I21 21 21 c, Pcc2, Pma2a; b, Pnc2, Pba2, Pnn2, Cmm2c, Ccc2, Abm2a; b, Ama2a, Aba2, Fmm2b, Fdd2, Iba2, Ima2a, Pmmmy; z, Pnnnk; l, Pccma; b; c; d; m; n; o; p; q, Pbank; l, Pnnac, Pccad; e, Pbam, Pccn, Pbcmd, Pnnm, Cmcmg, Cmmme; f ; m; p; q, Cccmc; d; e; f ; i; j; k; l, Cmmal, Cccag; h, Fmmme; j; o, Fdddg, Immmn, Ibamc; d; h; i; j, Ibcae (2) Site symmetry mm2, 222, mmm ± three independent elements (a) 12 13 23 0 principal axes parallel to crystal axes All space groups (C) Tetragonal (1) Site symmetry m, 2, 2=m ± four independent elements (a) 12 13 0; one principal axis parallel to [100] P422l; m; n; o, P42 22 j; k; l; m, I422h; i, I41 22 f , I41 md, P42mi; j; k; l, P42cg; i, I42m f ; g, I42dd, P4=mcck; l, P4=nbmk; l, P4=nnci; j, P4=nmmi, P42 =mmco; p, P42 =mcml; m, P42 =nbch; i, P42 =nnmi; j, P42 =nmcg, I4=mmmn, I4=mcm j, I41 =amdc; d; f ; h, I41 =acde (b) 12 23 0; one principal axis parallel to [010] P41 22a; b, P43 22a; b, P4mme; f , P42 mc, I4mmd, P42ch; j, P4m2 j; k, I4m2i, P4=mmms; t (c) 13 23 0; one principal axis parallel to [001] P4, P42 , I4, I41 , P4, I4, P4=m, P42 =m, P4=n, P42 =n, I4=m, I41 =a, P422i, P421 2d, P42 22g; h; i, P42 21 2c; d, I422 f , I41 22c, P42 cmc, P42 nmb, P4cc, P4nc, P42 bc, I41 cd, P42mm, P42ck; l; m, P421 md, P421c, P4c2g; h; i, P4b2e; f , P4n2e; h, I4c2 f ; g, I42mh, I42dc, P4=mmm p; q, P4=mcce; i; m, P4=nncg, P4=mbmi; j, P4=mncc; f ; h, P4=ncce, P42=mmcq, P42 =mcm f ; k; n, P42 =nbc f ; g, P42 =nnmh, P42 =mbca; c; e; f ; h, P42 =mnmc; h; i, P42 =ncm f , I4=mmml, I4=mcmk, I41 =acdd (d) 11 22 , 13 23 one principal axis parallel to [110] P422 j; k, P421 2e; f , P41 22c, P41 21 2, P42 22n; o, P42 21 2e; f , P43 22c, P43 21 2, I422g; j, I41 22d, P4m2h; i, P4c2e; f , P4b2g; h, P4n2g, I4m2g; h, I4c2e; h, P4=mcc j, P4=nbme; f ; i; j; m, P4=nnch, P4=mncg, P42 =mmcn, P42 =nbc j, P42 =nnme; f ; k; l; m, P42 =mbcg, I4=mmmk, I4=mcme; i, I41 =amdg, I41 =acd f (e) 11 22 , 13 23 ; one principal axis parallel to [110] I41 22e, P4mmd, P4bm, P42 cmd, P42 nmc, I4mmc, I4cm, P42mn, P421 me, P4n2 f , I42mi, P4=mmmr, P4=mbmk, P4=nmmd; e; g; h; j, P4=ncc f , P42 =mcmo, P42 =mnm j, P42 =nmc f , P42 =ncmc; d; g; h; i, I4=mmm f ; m, I4=mcml (2) Site symmetry mm2, 222, mmm ± three independent elements (a) 12 13 23 0; principal axes parallel to crystal axes P422, P42 22a; b; c; d, I422c, P4mm, P42 mc, I4mm, I41 md, P42me; f , P42c, P4m2, I4m2, I42mc, P4=mmme; f ; i; l; m; n; o, P4=mcc, P4=nnc, P4=nmm, P42 =mmc, P42 =mcme, P42 =nbca; b, P42 =nnmc, P42 =nmc, I4=mmmc; g; i; j, I41 =amd

695


8. REFINEMENT OF STRUCTURAL PARAMETERS Table 8.3.1.1. Symmetry conditions for second-cumulant tensors (cont.) (C) Tetragonal (cont.) (b) 11 22 , 13 23 0; principal axes parallel to [110], [110] and [001] P421 2, P42 22e; f , P42 21 2, I422d, I41 22, P4bm, P42 cm, P42 nm, I4cm, P42mg; h, P421 m, P4c2, P4b2, P4n2, I4c2, I42me, P4=mmm j; k, P4=nbm, P4=mbm, P4=mnc, P4=ncc, P42 =mcma; c; g; h; i; j, P42 =nbcc, P42 =nnmd; g, P42 =mbc, P42 =mnm, P42 =ncm, I4=mmmh, I4=mcm, I41 =acd (3) Site symmetry 4, 4, 4=m, 4mm, 42m, 422, 4=mmm ± two independent elements (a) 11 22 , 12 13 23 0; uniaxial with unique axis parallel to [001] All space groups (D) Trigonal (hexagonal axes) and hexagonal (1) Site symmetry m, 2, 2=m ± four independent elements (a) 13 23 0; one principal axis parallel to [001] P6, P62 , P64 , P6, P6=m, P63 =m, P622i, P62 22e; f , P64 22e; f , P6cc, P6m2l; m, P6c2k, P62m j; k, P62ch, P6=mmm p; q, P6=mccg; i; l, P63 =mcm j, P63 =mmc j (b) 11 22 , 13 23 one principal axis parallel to [110] P3m1, R3m, P3m1i, R3mh, P6mme, P63 mc, P6m2n (c) 11 22 , 13 23 ; one principal axis parallel to [210] P312, P31 12, P32 12, P31mi; j, P31c, P622l; m, P6c2 j (d) 22 2 12 , 2 13 23 ; one principal axis parallel to [100] P321, P31 21, P32 21, R32, P3m1e; f ; g; h, P3c1, R3md; e; f ; g, R3c, P622 j; k, P61 22a, P65 22a, P62 22g; h, P64 22g; h, P63 22g, P62cg, P6=mmmo, P6=mcc j, P63 =mmcg; i; k (e) 22 2 12 , 23 0; one principal axis parallel to [210] P31m, P31m f ; g; k, P61 22b, P65 22b, P62 22i; j, P64 22i; j, P63 22h, P6mmd, P63 cm, P62mi, P6=mmmn, P6=mcck, P63 =mcm f ; i; k (2) Site symmetry mm2, 222, mmm ± three independent elements (a) 22 2 12 , 13 23 0; principal axes parallel to [100] and [001] P622, P62 22, P64 22, P6mm, P62m, P6=mmm, P6=mcc, P63 =mcm, P63 =mmc (b) 11 22 , 13 23 0; principal axes parallel to [110], [210] and [001] P6m2 (3) Site symmetry 3, 3, 3m, 32, 3m, 6, 6, 6=m, 6m2, 6mm, 622, 6=mmm ± two independent elements (a) 11 22 2 12 , 13 23 0; unique axis parallel to c All space groups (E) Cubic (1) Site symmetry m, 2, 2=m ± four independent elements (a) 12 13 0; one principal axis parallel to [100] P23, F23, I23, I21 3, Pm3, Pn3, Fm3, Fd3, Im3, Ia3, P432h, P42 32h; i; j, F432i, F41 32 f , I432g, I41 32 f , P43mh, I43m f , P43n, F43c, I43d, Pm3mk; l, Pn3ng, Pm3nk, Pn3mh, Fm3m j, Fm3ci, Fd3c f , Im3m j, Ia3d f (b) 11 22 , 13 23 ; one principal axis parallel to [110] P43mi, F43m, I43mg, Pm3mm, Pn3mk, Fm3mk, Fd3mg, Im3mk (c) 22 33 , 12 13 ; one principal axis parallel to [011] P432i; j, P42 32l, F432g; h, I432h, P41 32, I41 32g, Pn3nh, Pm3n j, Pn3m j, Fm3ch (d) 22 33 , 12 13 ; one principal axis parallel to [011] P42 32k, F41 32g, I432i, P43 32, I41 32h, Pn3mi, Fd3mh; i, Fd3cg, Im3mi, Ia3dg (2) Site symmetry mm2, 222, mmm ± three independent elements (a) 12 13 23 0; principal axes parallel to crystal axes P23, I23, Pm3, Pn3, Fm3, Im3, P42 32d, P43n, Pm3mh, Pm3n, Fm3c, Im3mg (b) 22 33 , 12 13 0; principal axes parallel to [011], [011] and [100] P42 32e; f , F432, I432, I41 32, P43m, F43m, I43m, Pm3mi; j, Pn3m, Fm3m, Fd3m, Im3mh, Ia3d (3) Site symmetry 3, 3, 3m, 32, 3m, 6, 6, 6=m, 6m2, 6mm, 622, 6=mmm ± two independent elements (a) 11 22 33 , 12 13 23 ; unique axis parallel to [111] All space groups (4) Site symmetry 4, 4, 4=m, 4mm, 42m, 422, 4=mmm ± two independent elements (a) 22 33 , 12 13 23 0; uniaxial with unique axis parallel to [100] All space groups (5) Site symmetry 23, m3, 43m, 432, m3m ± one independent element (a) 11 22 33 , 12 13 23 0; isotropic All space groups

696


8.3. CONSTRAINTS AND RESTRAINTS IN REFINEMENT crystallographic parameters, so that statistical tests based on the F ratio or Hamilton's R ratio (Section 8.4.2; Hamilton, 1964) may be used to assess signi®cance. Shape constraints differ from those owing to space group or chemical conditions in that the constraint equations (8.3.1.12) are not linear functions of the independent parameters. Thus, the elements of C are not constants and must therefore be evaluated in each iteration of the re®nement algorithm. Another area in which application of constraints is important arises whenever some portion of the structure undergoes thermal motion as a rigid entity. One means of determining rigid-motion parameters is to re®ne the conventional, anisotropic atomic displacement factors of all atoms individually and to ®t a librational model to the resulting thermal factors (Schomaker & Trueblood, 1968). A problem arises with this approach because the presence of libration implies curvilinear motion in the crystallographic system, and thus the probability density function for an atom that does not lie on the axis of libration cannot be described by a Gaussian function in a rectilinear coordinate system. For neutron diffraction, where H atoms have major scattering power, the effect may be large enough to affect convergence (Prince, Dickens & Rush, 1974). Anharmonic (third-cumulant) terms could be used, but the number of parameters increases rapidly, because there are as many as ten, independent, third cumulant tensor elements per atom. Thermal motions of rigid bodies are represented by a symmetric, translation tensor, T, a symmetric, libration tensor, L, and a nonsymmetric, screw correlation tensor, S (Cruickshank, 1961; Schomaker & Trueblood, 1968). Any sequence of rotations of a rigid body about a ®xed point is equivalent to a single, ®nite rotation about some axis passing through the ®xed point. This rotation can be represented (Prince, 1994) by an axial vector, k, where jkj is the magnitude of the rotation, and the direction cosines of the axis with respect to some system of orthogonal axes are given by i li =jkj li =l21 l22 l23 1=2 . An exact expression for the displacement, u, of a point in the rigid body, located by a vector r from the centre of mass, owing to a rotation k about an axis passing through the centre of mass is cos jkj=jkj2 k k r: 8:3:1:14

u sin jkj=jkjk r 1

For small rotations, the trigonometric functions can be replaced by power-series expansions, and, because of the extremely rapid convergence of these series, (8.3.1.14) is approximated extremely well, even for values of jkj as large as 0:5 rad, by u 1

jkj2 =6k r 1=2

jkj2 =24kk r: 8:3:1:15

By expansion of the vector products, this can be written 3 3 3 P P P Crijkl lj lk ll Arij lj Brijk lj lk ui j1

3 P m1

k1

Drijklm lj lk ll lm

l1

;

8:3:1:16

where the coef®cients, Arij , Brijk , Crijkl , and Drijklm are multiples of components of r. For example, k r1 l2 r3 l3 r2 , so that Ar11 0, Ar12 r3 , and Ar13 r2 . These coef®cients have been tabulated by Sygusch (1976), and expressed in Fortran source code by Prince (1994). If the centre of mass of the rigid body also moves, the total displacement of the point at r is v u t, where t is the displacement of the centre of mass from its equilibrium

position. A discussion of the effects of rigid-body

motion on diffraction intensities involves quantities like vi , vi vj , and so forth, the ensemble averages of these quantities over many unit cells, which may be assumed to be equal to the time averages for one unit cell over a long time. The rigid-body-motion tensors

are de®ned by Tij ti tj , Lij li lj , and Sij li tj . The distributions of ti and li can usually be assumed to be approximately Gaussian, so that fourth moments can be expressed in terms of second moments. Thus, li lj lk ll Lij Lkl Lik Ljl Lil Ljk , li lj tk tl Lij Tkl Sik Sjl Sil Sjk , and so forth. If the elements of t and k are with respect to their mean positions,

measured

ti li 0. Third moments, quantities like li lj tk , do not necessarily vanish, except when the rigid body is centrosymmetric, but their effects virtually always are small, and can be neglected. A particle that is part of a librating, rigid body undergoes a curvilinear motion that results in its having a mean position that is displaced from its equilibrium position. This causes an apparent shortening of interatomic distances, which must be corrected for if accurate values of bond lengths are to be derived. The displacement, d, from the equilibrium position to the mean position is (Prince & Finger, 1973) " 3 P 3

P di vi Brijk Ljk j1 k1

l1 m1

Drijklm Ljk Llm Ljl Lkm Ljm Lkl : 8:3:1:17

Anisotropic atomic displacement factors, ij , Bij , or Uij , are related by simple, linear transformations that are functions

of the unit-cell constants to the quantity ij vi vj vi vj . If the rigid body has a centre of symmetry, so that the elements of S vanish, this is given by 3 P 3 3 3 P P P Arik Arjl Lkl f3Arik C rjlmn ij Tij m1 n1 k1 l1 8:3:1:18 Arjk Crilmn 2Brikm Brjln g Lkl Lmn : Expressions including elements of S have been given by Sygusch (1976) and, in Fortran source code, by Prince (1994). Expressions for anisotropic atomic displacement factors in terms of T, L, and S that included only terms linear in the tensor elements were given by Schomaker & Trueblood (1968), who pointed out that the diagonal elements of S never appeared individually, but only as the differences of pairs, so that the expressions were invariant under the addition of a constant to all three elements. This `trace of S singularity' was resolved by applying the additional constraint S11 S22 S33 0. As was pointed out by Sygusch (1976), the inclusion of terms that are quadratic in the tensor elements removes this indeterminacy, but the effects of the additional terms are so small that the problem remains extremely ill conditioned. In practice, therefore, these elements should still be treated as underdetermined. Prince (1994) lists the symmetry restrictions for each type of tensor for various point groups. Although the description of thermal motion is essentially harmonic within the rigid-body system, the structure-factor formulation must include what appear to be anharmonic terms. Prince also presents computer routines that contain the relations between the elements of T, L, and S and the second- and third-cumulant tensor elements. As in

697


3 P 3 P

8. REFINEMENT OF STRUCTURAL PARAMETERS the case of shape constraints, the equations are nonlinear, and the elements of C must be re-evaluated in each iteration. 8.3.2. Stereochemically restrained least-squares re®nement The precision with which an approximately correct model can be re®ned to describe the atomic structure of a crystal depends on the ability of the model to represent the atomic distributions and on the quality of the observational data being ®tted with the model. In addition, although the structure can in principle be determined by a well chosen data set only a little larger than the number of parameters to be determined (Section 8.4.4), in practice, with a nonlinear model as complex as that for a macromolecular crystal, it is necessary for the parameters de®ning the model to be very much overdetermined by the observations. For well ordered crystals of small- and intermediate-sized molecules, it is usually possible to measure a hundred or more independent Bragg re¯ections for each atom in the asymmetric unit. When the model contains three position parameters and six atomic displacement parameters for each atom, the over-determinacy ratio is still greater than ten to one. In such instances, each model parameter can usually be quite well determined, and will provide an accurate representation of the average structure in the crystal, except in regions where ellipsoids are not adequate descriptions of the atomic distributions. This contrasts sharply with studies of biological macromolecules, in which positional disorder and thermal motion in large regions, if not the entire molecule, often limit the number of independent re¯ections in the data set to fewer than the number of parameters necessary to de®ne the distributions of individual atoms. This problem may be overcome either by reducing the number of parameters describing the model or by increasing the number of independent observations. Both approaches utilize knowledge of stereochemistry. A great deal of geometrical information with which an accurate model must be consistent is available at the onset of a re®nement. The connectivity of the atoms is generally known, either from the approximately correct Fourier maps of the electron density obtained from a trial structure determination or from sequencing studies of the molecules. Quite tight bounds are placed on local geometry by the accumulating body of information concerning bond lengths, bond angles, group planarity, and conformational preferences in torsion angles. Additional knowledge concerns van der Waals contact potential functions and hydrogen-bonding properties, and displacement factors must also be correlated in a manner consistent with the known geometry. In Section 8.3.1, we discuss the use of constraints to introduce this stereochemical knowledge. In this section, we discuss a technique that introduces the stereochemical conditions as additional observational equations (Waser, 1963). This method differs from the other in that information is introduced in the form of distributions about mean values rather than as rigidly ®xed geometries. The parameters are restrained to fall within energetically permissible bounds. 8.3.2.1. Stereochemical constraints as observational equations As described in Section 8.1.2, given a set of observations, yi , that can be described by model functions, Mi x, where x is the vector of model parameters, we seek to ®nd x for which the sum S

n P i1

wi yi

Mi x2

8:3:2:1

is minimum. For restrained re®nement, S is composed of several classes of observational equations, including, in addition to the ones for structure factors, equations for interatomic distances, planar groups and displacement factors. Structure factors yield terms in the sum of the form SF jFobs hj

8:3:2:2

The distances between bonded atoms and between next-nearestneighbour atoms may be used to require bonded distances and angles to fall within acceptable ranges. This gives terms of the form d dideal

dmodel 2 =d2 ;

8:3:2:3

where d is the standard deviation of an empirically determined distribution of values for distances of that type. Groups of atoms may be restrained to be near a common plane by terms of the form (Schomaker, Waser, Marsh & Bergman, 1959) p ml r

dl 2 =p2 ;

8:3:2:4

where ml and dl are parameters of the plane, p is again an empirically determined standard deviation, and indicates the scalar product. If a molecule undergoes thermal oscillation, the displacement parameters of individual atoms that are stereochemically related must be correlated. These parameters may be required to be consistent with the known stereochemistry by assuming a model that gives a distribution function for the interatomic distances in terms of the individual atom parameters and then restraining the variance of that distribution function to a suitably small value. The variation with time of the distances between covalently bonded atoms can be no greater than a few hundredths of an aÊngstroÈm. Therefore, the thermal displacements of bonded atoms should be very similar along the bond direction, but they may be more dissimilar perpendicular to the bond. If we make the assumption that the atom with a broader distribution in a given direction is `riding' on the atom with the narrower distribution, the variance of the interatomic distance parallel to a vector v making an angle v; j with the direction of bond j is (Konnert & Hendrickson, 1980) Vv 2v cos2 4v =2d02 sin4

6 sin2 cos2 . . . ; 8:3:2:5

where d0 is the normal distance for that type of bond, 2v u2a u2b , and u2a and u2b are the mean square displacements parallel to v of atom a and atom b, respectively. The restraint terms then have the form Vv2 =v2 . For isotropic displacement factors, these terms take the particularly simple form Ba Bb 2 =B2 , but with the disadvantage that, when isotropic displacement parameters are used, the displacements cannot be suitably restrained along the bonds and perpendicular to the bonds simultaneously. Several additional types of restraint term have proved useful in restraining the coordinates for the mean positions of atoms in macromolecules. Among these are terms representing nonbonded contacts, torsion angles, handedness around chiral centres, and noncrystallographic symmetry (Hendrickson & Konnert, 1980; Jack & Levitt, 1978; Hendrickson, 1985). Contacts between nonbonded atoms are important for determining the conformations of folded chain molecules. They may be described by a potential function that is strongly repulsive when the interatomic distance is less than some minimum value, but only weakly attractive, so that it can be neglected in practice, when the distance is greater than that value. This leads to terms of the form

698


jFcalc hj2 =h2 :

8.3. CONSTRAINTS AND RESTRAINTS IN REFINEMENT Ê in standard groups Table 8.3.2.1. Coordinates of atoms (in A) Table 8.3.2.1 (cont.) appearing in polypeptides and proteins; restraint relations may be determined from these coordinates using methods described by Side chains for amino acids Ala A Hendrickson (1985) Main chain, links and terminal groups Main N C C O

1.20134 0.00000 1.25029 2.18525

0.84658 0.00000 0.88107 0.66029

0.00000 0.00000 0.00000 0.78409

C terminal N C C O Ot

1.20006 0.00000 1.26095 2.32397 1.15186

0.84799 0.00000 0.86727 0.27288 2.04837

0.00000 0.00000 0.00000 0.29188 0.35987

N amino terminal N 1.20134 C 0.00000 C 1.25029 O 2.18525

0.84658 0.00000 0.88107 0.66029

0.00000 0.00000 0.00000 0.78409

N formyl terminal N 1.19423 C 0.00000 C 1.24896 O 2.10649 2.46193 Ot Ct 2.33913

0.82137 0.00000 0.88255 0.78632 0.77877 0.39064

0.00000 0.00000 0.00000 0.90439 0.93569 0.53355

N acetyl terminal N 1.19423 C 0.00000 C 1.24896 O 2.10649 Ot 2.46193 2.33913 Ct 1 3.44659 Ct 2

0.82137 0.00000 0.88255 0.78632 0.77877 0.39064 1.39160

0.00000 0.00000 0.00000 0.90439 0.93569 0.53355 0.63532

trans peptide link C 0.00000 C 0.57800 O 1.80400 N 0.33500 C 0.00000

0.00000 1.41700 1.60700 2.37000 3.80100

0.00000 0.00000 0.00001 0.00000 0.00000

cis peptide link C 0.00000 C 1.30900 O 2.38500 N 1.23500 C 0.00000

0.00000 0.79200 0.17600 2.11000 2.90700

0.00000 0.00000 0.00000 0.00000 0.00000

trans proline link C 0.00000 C 0.57800 O 1.80400 N 0.33500 C 0.00000 C 1.80000

0.00000 1.41700 1.60700 2.37000 3.80100 2.19600

0.00000 0.00000 0.00001 0.00000 0.00000 0.00000

cis proline link C 0.00000 C 1.30900 O 2.38500 N 1.23500 C 0.00000 C 2.45500

0.00000 0.79200 0.17600 2.11000 2.90700 2.93900

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.02022

0.92681

1.20938

Arg R C C C N" C N1 N2

0.02207 0.09067 0.79074 0.76228 1.57539 2.60422 1.38328

0.93780 0.23808 1.07410 0.46664 0.83569 1.65104 0.33329

1.20831 2.55932 3.57563 4.89930 5.89157 5.68019 7.11065

Asn N C C O1 N2

0.04600 0.15292 0.39364 0.06382

1.02794 0.42844 0.78048 1.27086

1.12104 2.50080 2.63809 3.52863

Asp D C C O1 O2

0.04600 0.15292 0.39364 0.06930

1.02794 0.42844 0.78048 1.21904

1.12104 2.50080 2.63809 3.46540

Cys C C S

0.01317 0.07941

0.95892 0.15367

1.18266 2.80168

Gln Q C C C O"1 N"2

0.01691 0.08291 0.20841 0.48899 0.00450

0.98634 0.32584 1.31760 2.49684 0.81846

1.16423 2.52866 3.65937 3.46331 4.87646

Glu E C C C O"1 O"2

0.06551 1.15947 1.40807 0.92644 2.16269

0.87677 1.71468 2.90920 3.06007 3.74330

1.25157 1.59818 0.72611 0.38343 1.27140

Gly G (no nonhydrogen atoms)

699


C

His H C C N1 C"1 N"2 C2

0.06434 0.52019 0.26457 0.46699 1.69370 1.75570

0.96857 0.29684 0.53405 1.05500 0.59727 0.25685

1.20324 2.46369 3.22184 4.19371 4.09040 3.02097

Ile I C C 1 C 2 C1

0.03196 0.83268 0.39832 0.77555

0.97649 2.22363 0.28853 3.32741

1.23019 0.92046 2.54980 2.01167

Leu L C C C1 C2

0.09835 0.96072 0.89548 0.73340

0.94411 2.02814 2.98661 2.79002

1.20341 1.32143 0.13861 2.62540

Lys K C C C C" N

0.03606 1.19773 1.05466 2.34215 2.16781

0.92129 1.81387 2.77178 3.51295 4.42240

1.21541 1.35938 2.53242 2.82637 3.98733

8. REFINEMENT OF STRUCTURAL PARAMETERS Ê torsion angles ( ), Table 8.3.2.2. Ideal values for distances (A), etc. for a glycine±alanine dipeptide with a trans peptide bond; distance type 1 is a bond, type 2 a next-nearest-neighbour distance involving a bond angle

Table 8.3.2.1 (cont.) Met M C C S C"

0.02044 1.00916 0.77961 2.08622

0.96506 2.05384 3.24454 4.42220

1.17716 1.00286 2.37236 1.97795

Phe F C C C1 C"1 C C"2 C2

0.00662 0.03254 1.15813 1.15720 0.05385 1.26137 1.23668

1.03603 0.49711 0.12084 0.38038 0.51332 0.11613 0.38351

1.11081 2.50951 3.13467 4.42732 5.11032 4.50975 3.20288

Pro P C C C

0.12372 0.89489 1.87411

0.78264 0.13845 0.86170

1.31393 2.22063 1.30572

Ser S C O

0.00255 0.19791

0.96014 0.28358

1.17670 2.40542

Thr T C O 1 C 2

0.00660 0.04119 1.12889

0.98712 0.14519 2.01366

1.23470 2.43011 1.21493

Trp W C C C1 N"l C"2 C2 C2 C3 C"3 C2

0.02501 0.03297 1.03107 0.62445 0.72100 1.57452 2.91029 3.37037 2.51952 1.17472

0.98461 0.36560 0.15011 0.62417 0.41985 0.72329 0.38415 0.23008 0.53303 0.20516

1.16268 2.51660 3.20411 4.42903 4.55667 5.60758 5.45120 4.28944 3.24549 3.37412

Tyr Y C C C1 C"l C C"2 C2 O

0.00470 0.18427 0.89731 0.72371 0.54776 1.63905 1.44975 0.76405

0.95328 0.27254 0.26132 0.85064 0.88971 0.38287 0.19374 1.40409

1.20778 2.54372 3.25049 4.50059 5.06861 4.37622 3.12415 6.31652

Val V C C 1 C 2

0.05260 0.13288 0.94265

0.99339 0.31545 2.12930

1.17429 2.52668 0.99811

n dmin

dmodel 4 =n4 ;

Interatomic Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

model 2 =t2 ;

to to to to to to to to to to to to to to to to to to to to

Planar groups 1 CTRM 2 LINK

C(2) C(1)

1

Distance 1.470 1.530 1.240 2.452 2.414 1.469 1.530 1.252 2.461 2.358 1.524 2.515 2.450 1.240 2.225 2.377 1.320 2.271 2.394 2.453

C(1) C(1) O(1) C(1) O(1) C(2) C(2) O(2) C(2) O(2) C(2) C(2) N(2) O(2)t O(2)t O(2)t C(1) O(1) C(1) C(1) C(2) C(1)

O(2) O(1)

O(2) N(2)

8:3:2:6

8:3:2:7

Central atom C(2)

Ala

N(2)

C(2)

C(2)

Possible nonbonded contacts Number 1 N(1) to 2 N(2) to 3 O(2) to 4 N(2) to to 5 O(2)t

O(1) O(2) C(2) O(2)t C(2)

Distance 3.050 3.050 3.350 3.050 3.350

Torsion angles N(1) C(1) C(1) C(1) C(1) N(2) N(2) C(2)

N(2) C(2) C(2) O(2)t

0.0 180.0 0.0 0.0

C(1) N(2) C(2) C(2)

Type 1 1 1 2 2 1 1 1 2 2 1 2 2 1 2 2 1 2 2 2

C(2) Chiral volume Ê 3) (A 2.492

where ideal and model are dihedral angles between planar groups at opposite ends of the bond. Interatomic distances are independent of the handedness of an enantiomorphous group. If rc is the position vector of a central atom and r1 , r2 , and r3 are the positions of three atoms bonded to it, such that the four atoms are not coplanar, the chiral volume is de®ned by Vc r1

rc r2

rc r3

rc ;

8:3:2:8

where indicates the vector product. The chiral volume may be either positive or negative, depending on the handedness of the group. It may be restrained by including terms of the form c Videal

Vmodel 2 =c2 :

8:3:2:9

Table 8.3.2.1 gives ideal coordinates, in an orthonormal Ê of various groups that are coordinate system measured in A,

700


N(1) C(1) C(1) N(1) C(1) N(2) C(2) C(2) N(2) C(2) C(2) C(2) C(2) C(2) O(2) C(2) N(2) N(2) N(2) C(2)

Chiral centres

which are included only when dmodel < dmin . Macromolecules usually gain ¯exibility by relatively unrestricted rotation about single bonds. There are, nevertheless, signi®cant restrictions on these torsion angles, which may, therefore, be restrained by terms of the form t ideal

distances

8.3. CONSTRAINTS AND RESTRAINTS IN REFINEMENT Table 8.3.2.3. Typical values of standard deviations for use in determining weights in restrained re®nement of protein structures (after Hendrickson, 1985) Interatomic distances Nearest neighbour (bond) Next-nearest neighbour (angle) Intraplanar distance Hydrogen bond or metal coordination Planar groups Deviation from plane Chiral centres Chiral volume Nonbonded contacts Interatomic distance Torsion angles Speci®ed (e.g. helix ' and ) Planar group Staggered Thermal parameters Anisotropic Ê Main-chain neighbour v = 0.05 A Ê Main-chain second neighbour 0.10 A Ê Side-chain neighbour 0.05 A Ê Side-chain second neighbour 0.10 A

d = 0.02 0.03 0.05 0.05

Ê A Ê A Ê A Ê A

Ê p = 0.02 A Ê3 c = 0.15 A Ê n = 0.50 A t = 15 3 15 Isotropic Ê2 B = 1.0 A Ê 1.5 A2 Ê2 1.5 A Ê2 2.0 A

commonly found in proteins. The ideal conformations of pairs of amino acid residues, from which the ideal values to be used in restraint terms of various types may be determined, are constructed by combining the coordinates of the individual groups. For example, consider a dipeptide composed of

glycine and alanine joined by a trans peptide link, giving the molecule C 2 O2 O1 k j k N1ÐC1Ð C1 ÐN2Ð C2 Ð C2 ÐOt 2: The origin is placed at each of the C positions in turn, and interatomic distances to nearest and next-nearest neighbours are computed. Planar groups and possible nonbonded contacts are identi®ed, and torsion angles and chiral volumes for chiral centres are computed. Table 8.3.2.2 is a summary of the restraint information for this simple molecule. In order to incorporate this information in the re®nement, these ideal values are combined with suitable weights. Table 8.3.2.3 gives values of the standard deviations of the various types of constraint relation that have been found (Hendrickson, 1985) to give good results in practice. Even for a small protein, the normal-equations matrix may contain several million elements. When stereochemical restraint relations are used, however, the matrix elements are not equally important, and many may be neglected. Convergence and stability properties can be preserved when only those elements that are different from zero for the stereochemical restraint information are retained. The number of these elements increases linearly with the number of atoms, and is typically less than 1% of the total in the matrix, so that sparse-matrix methods (Section 8.1.5) can be used. The method of conjugate gradients (Hestenes & Stiefel, 1952; Konnert, 1976; Rae, 1978) is particularly suitable for the ef®cient use of restrainedparameter least squares.

701



8.4. Statistical signi®cance tests By E. Prince and C. H. Spiegelman

In Chapter 8.1, we discussed the method of least squares and procedures for estimating the values of the adjustable parameters of a model that predicts the mean of a population from which experimental observations are drawn at random. Any model, however, will have some set of parameter values that gives the best least-squares ®t. We must now address the question of whether that best ®t is adequate, that is, whether it is plausible, given the precision of the data, to accept the hypothesis that the model really is a correct representation of the phenomena that have been measured in the collection of the data. In this chapter, we discuss the probability density function for the sum of squared residuals if the individual residuals are drawn from a normal distribution, the 2 distribution, and the conditions under which this p.d.f. may be assumed to approximate a practical case. Next, we discuss the F distribution, which is the distribution of the ratio of two independent, random variables, each of which has a 2 distribution, and its use in comparing the ®ts of constrained and unconstrained versions of a model. We also discuss a test that is useful for a more general comparison of models. Finally, we discuss the variation among data points of their effectiveness in improving the precision of parameter estimates and the application of this analysis to the optimum design of experiments. 8.4.1. The v2 distribution We have seen [equation (8.1.2.1)] that the least-squares estimate is derived by ®nding the minimum value of a sum of terms of the form Ri wi yi

Mi x2 ;

8:4:1:1

and, further, that the precision of the estimate is optimized if the weight, wi , is the reciprocal of the variance of the population from which the observation is drawn, wi 1=i2 . Using this relation, (8.4.1.1) can be written 2 Ri yi Mi x =i : 8:4:1:2 Each term is the square of a difference between observed and calculated values, expressed as a fraction of the standard uncertainty of the observed value. But, by de®nition,

i2 yi Mi x2 ; 8:4:1:3 where x has its unknown `correct' value, so that h Ri 1, and the expected value of the sum of n such terms is n. It can be shown (Draper & Smith, 1981) that each parameter estimated reduces this expected sum by one, so that, for p estimated parameters, n 2 P hS i n p; 8:4:1:4 yi Mi b x =i i1

where b x is the least-squares estimate. The standard uncertainty of an observation of unit weight, also referred to as the goodnessof-®t parameter, is de®ned by 2 P n 2 31=2 1=2 w y Mi b x 6 i1 i i 7 S 7 : G 6 8:4:1:5 4 5 n p n p From (8.4.1.4), it follows that hG i 1 for a correct model with weights assigned in accordance with (8.4.1.2).

A value of G that is close to one, if the weights have been assigned by wi 1=i2 , is an indicator that the model is consistent with the data. It should be noted that it is not necessarily an indicator that the model is `correct', because it does not rule out the existence of an alternative model that ®ts the data as well or better. An assessment of the adequacy of the ®t of a given model depends, however, on what is meant by `close to one', which depends in turn on the spread of a probability density function for G. We saw in Chapter 8.1 that least squares with this weighting scheme would give the best, linear, unbiased estimate of the model parameters, with no restrictions on the p.d.f.s of the populations from which the observations are drawn except for the implicit assumption that the variances of these p.d.f.s are ®nite. To construct a p.d.f. for G, however, it is necessary to make an assumption about the shapes of the p.d.f.s for the observations. The usual assumption is that these p.d.f.s can be described by the normal p.d.f., 1 x 2 N x; ; p exp : 8:4:1:6 2 2 2 The justi®cation for this assumption comes from the central-limit theorem, which states that, under rather broad conditions, the p.d.f. of the arithmetic mean of n observations drawn from a population with mean and variance 2 tends, for large n, to a normal distribution with mean and variance 2 =n. [For a discussion of the central limit theorem, see Cramer (1951).] If we make the assumption of a normal distribution of errors and make the substitution z x =, (8.4.1.6) becomes 2 1 z N z; 0; 1 p exp : 8:4:1:7 2 2 The probability that z2 will be less than 2 is equal to the probability that z will lie in the interval z , or 2


0

z2 dz2

R

z dz:

8:4:1:8

Letting t z2 =2 and substituting in (8.4.1.7), this becomes 2

1 R=2 t 2 p 0

1=2

exp t dt:

2 d 2 = d2 , so that 1=2 exp 2 =2 ; 2 22 2 0;

2 > 0; 2 0:

8:4:1:9

8:4:1:10

The joint p.d.f. of the squares of two random variables, z1 and z2 , drawn independently from the same population with a normal p.d.f. is 2 1 z1 z22 2 2 J z1 ; z2 ; 8:4:1:11 exp 2 2z1 z2 and the p.d.f. of the sum, s2 , of these two terms is the integral over the joint p.d.f. of all pairs of z21 and z22 that add up to s2 . 2 1 s 2 2 2 z1 s s exp z21 dz21 : 8:4:1:12 2 2 This integral can be evaluated by use of the gamma and beta functions. The gamma function is de®ned for positive real x by


R2

8.4. STATISTICAL SIGNIFICANCE TESTS Table 8.4.1.1. Values of 2 = for which the c.d.f. (2 ,) has the values given in the column headings, for various values of

0.5

0.9

0.95

0.99

0.995

1 2 3 4 6 8 10 15 20 25 30 40 50 60 80 100 120 140 160 200

0.4549 0.6931 0.7887 0.8392 0.8914 0.9180 0.9342 0.9559 0.9669 0.9735 0.9779 0.9834 0.9867 0.9889 0.9917 0.9933 0.9945 0.9952 0.9958 0.9967

2.7055 2.3026 2.0838 1.9449 1.7741 1.6702 1.5987 1.4871 1.4206 1.3753 1.3419 1.2951 1.2633 1.2400 1.2072 1.1850 1.1686 1.1559 1.1457 1.1301

3.8415 2.9957 2.6049 2.3719 2.0986 1.9384 1.8307 1.6664 1.5705 1.5061 1.4591 1.3940 1.3501 1.3180 1.2735 1.2434 1.2214 1.2044 1.1907 1.1700

6.6349 4.6052 3.7816 3.3192 2.8020 2.5113 2.3209 2.0385 1.8783 1.7726 1.6964 1.5923 1.5231 1.4730 1.4041 1.3581 1.3246 1.2989 1.2783 1.2472

7.8795 5.2983 4.2794 3.7151 3.0913 2.7444 2.5188 2.1868 1.9999 1.8771 1.7891 1.6692 1.5898 1.5325 1.4540 1.4017 1.3638 1.3346 1.3114 1.2763

the statistical library DATAPAC (Filliben, unpublished). Fortran code for this program appears in Prince (1994). The quantity n pG is the sum of n terms that have mean value n p=n. Because the process of determining the leastsquares ®t establishes p relations among them, however, only n p of the terms are independent. The number of degrees of freedom is therefore n p, and, if the model is correct, and the terms have been properly weighted, 2 n pG 2 has the chi-squared distribution with n p degrees of freedom. In crystallography, the number of degrees of freedom tends to be large, and the p.d.f. for G correspondingly sharp, so that even rather small deviations from G 2 1 should cause one or both of the hypotheses of a correct model and appropriate weights to be rejected. It is common practice to assume that the model is correct, and that the weights have correct relative values, that is that they have been assigned by wi k=i2 , where k is a number different from, usually greater than, one. G is then taken to be an estimate of k, and all elements of AT WA 1 (Section 8.1.2) are multiplied by G 2 to get an estimated variance±covariance matrix. The range of validity of this procedure is limited at best. It is discussed further in Chapter 8.5. 8.4.2. The F distribution

x

R1 0

tx

1

exp t dt:

8:4:1:13

Although this function is continuous for all x > 0, its value is of interest in the context of this analysis only for x equal to positive, p integral multiples of 1=2. It can be shown that 1=2 , 1 1, and x 1 x x. It follows that, for a positive p integer, n,p n n 1!, and that 3=2 =2, 5=2 3 =4, etc. The beta function is de®ned by Bx; y

R1 0

tx 1 1

ty

It

1

dt:

can be shown (Prince, 1994) that x y= x y. Making the substitution (8.4.1.12) becomes 2 1 1 s R s2 exp t1 t 1=2 dt 2 0 2 2 1 s B1=2; 1=2 exp 2 2 2 s 12 exp ; s2 0: 2

8:4:1:14 Bx; y t z21 =s2 ,

8:4:1:15

By a similar procedure, it can be shown that, if 2 is the sum of terms, z21 , z22 , . . ., z2v , where all are drawn independently from a population with the p.d.f. given in (8.4.1.10), 2 has the p.d.f. =2 1 2 2 _ 2 exp ; 2 > 0; 8:4:1:16 ; =2 2 =2 2 2 ; 0; 2 0: The parameter is known as the number of degrees of freedom, but this use of that term must not be confused with the conventional use in physics and chemistry. The p.d.f. in (8.4.1.16) is the chi-squared distribution with degrees of freedom. Table 8.4.1.1 gives the values of 2 = for which the cumulative distribution function (c.d.f.) 2 ; has various values for various choices of . This table is provided to enable veri®cation of computer codes that may be used to generate more extensive tables. It was generated using a program included in

Consider an unconstrained model with p parameters and a constrained one with q parameters, where q < p. We wish to decide whether the constrained model represents an adequate ®t to the data, or if the additional parameters in the unconstrained model provide, in some important sense, a better ®t to the data. Provided the p q additional columns of the design matrix, A, are linearly independent of the previous q columns, the sum of squared residuals must be reduced by some ®nite amount by adjusting the additional parameters, but we must decide whether this improved ®t would have occurred purely by chance, or whether it represents additional information. Let s2c and s2u be the weighted sums of squared residuals for the constrained and unconstrained models, respectively. If the constrained and unconstrained models are equally good representations of the data, and the weights have been assigned 2 values of the sums of squares are

by2 wi 1=i , the expected

sc n q and s2u n p, and, further, they should be distributed as 2 with n q and n p degrees of freedom, respectively. Also, s2c s2u p q, and s2c s2u is distributed as 2 with p q degrees of freedom. s2c and s2u are not independent, but s2c s2u is the squared magnitude of a vector in a p q-dimensional subspace that is orthogonal to the n p-dimensional space of s2u . Therefore, s2u and s2c s2u are independent, random variables, each with a 2 distribution. Let 21 s2c s2u , 22 s2u , 1 p q, and 2 n p. The ratio F 21 = 1 =22 = 2 should have a value close to one, even if the weights have relative rather than absolute values, but we need a measure of how far away from one this ratio can be before we must reject the hypothesis that the two models are equally good representations of the data. The conditional p.d.f. for F, given a value of 22 , is =2 1 = 2 22 1 F 1 =2 1 2 exp 1 = 2 22 F=2 ; C Fj2 =2 1 2 1 =2 8:4:2:1 and the marginal p.d.f. for 22 is =2 1 2 2 exp M 22 =22 2 2 2 =2

703


22 =2 :

8:4:2:2

8. REFINEMENT OF STRUCTURAL PARAMETERS Table 8.4.2.1. Values of the F ratio for which the c.d.f. (F, 1 , 2 ) has the value 0.95, for various choices of 1 and 2 1

1

2

4

8

15

4.9646 4.3512 4.1709 4.0847 4.0343 4.0012 3.9604 3.9361 3.9201 3.9042 3.8884 3.8726 3.8648 3.8570 3.8508

4.1028 3.4928 3.3158 3.2317 3.1826 3.1504 3.1108 3.0873 3.0718 3.0564 3.0411 3.0259 3.0183 3.0107 3.0047

3.4781 2.8661 2.6896 2.6060 2.5572 2.5252 2.4859 2.4626 2.4472 2.4320 2.4168 2.4017 2.3943 2.3868 2.3808

3.0717 2.4471 2.2662 2.1802 2.1299 2.0970 2.0564 2.0323 2.0164 2.0006 1.9849 1.9693 1.9616 1.9538 1.9477

2.8450 2.2033 2.0148 1.9245 1.8714 1.8364 1.7932 1.7675 1.7505 1.7335 1.7167 1.6998 1.6914 1.6831 1.6764

Table 8.4.3.1. Values of t for which the c.d.f. (t,) has the values given in the column headings, for various values of

2 10 20 30 40 50 60 80 100 120 150 200 300 400 600 1000

The marginal p.d.f. for F is obtained by integration of the joint p.d.f., R1 8:4:2:3 F C Fj22 M 22 d22 ; 0

yielding the result F; 1 ; 2

1 = 2 F 1 =2 1 =2 : 8:4:2:4 B 1 =2; 2 =2 1 1 = 2 F 1 2

This p.d.f. is known as the F distribution with 1 and 2 degrees of freedom. Table 8.4.2.1 gives the values of F for which the c.d.f. F; 1 ; 2 is equal to 0.95 for various choices of 1 and 2 . Fortran code for the program from which the table was generated appears in Prince (1994). The cumulative distribution function F; 1 ; 2 gives the probability that the F ratio will be less than some value by chance if the models are equally consistent with the data. It is therefore a necessary, but not suf®cient, condition for concluding that the unconstrained model gives a signi®cantly better ®t to the data that F; 1 ; 2 be greater than 1 , where is the desired level of signi®cance. For example, if F; 1 ; 2 0:95, the probability is only 0:05 that a value of F this large or greater would have been observed if the two models were equally good representations of the data. Hamilton (1964) observed that the F ratio could be expressed in terms of the crystallographic weighted R index, which is de®ned, for re®nement on jFj (and similarly for re®nement on jFj2 ), by Rw

P

wi jFo ji

jFc ji 2

P

wi jFo j2i

1=2

:

1;

and a c.d.f. for Rc =Ru can be readily derived from this relation. A signi®cance test based on Rc =Ru is known as Hamilton's R-ratio test; it is entirely equivalent to a test on the F ratio. 8.4.3. Comparison of different models

0.95

0.99

0.995

1 2 3 4 6 8 10 12 14 16 20 25 30 35 40 50 60 80 100 120

1.0000 0.8165 0.7649 0.7407 0.7176 0.7064 0.6998 0.6955 0.6924 0.6901 0.6870 0.6844 0.6828 0.6816 0.6807 0.6794 0.6786 0.6776 0.6770 0.6765

3.0777 1.8856 1.6377 1.5332 1.4398 1.3968 1.3722 1.3562 1.3450 1.3368 1.3253 1.3164 1.3104 1.3062 1.3031 1.2987 1.2958 1.2922 1.2901 1.2886

6.3138 2.9200 2.3534 2.1319 1.9432 1.8596 1.8125 1.7823 1.7613 1.7459 1.7247 1.7081 1.6973 1.6896 1.6839 1.6759 1.6707 1.6641 1.6602 1.6577

31.8206 6.9646 4.5407 3.7469 3.1427 2.8965 2.7638 2.6810 2.6245 2.5835 2.5280 2.4851 2.4573 2.4377 2.4233 2.4033 2.3901 2.3739 2.3642 2.3578

63.6570 9.9249 5.8409 4.6041 3.7074 3.3554 3.1693 3.0546 2.9769 2.9208 2.8453 2.7874 2.7500 2.7238 2.7045 2.6778 2.6603 2.6387 2.6259 2.6174

zi x i b l i1 ; n P xi2

8:4:3:1

i1

and it has an estimated variance n P

b 2l

i1

z2i

n

P 2 b l2 xi n

i1

1

n P i1

x 2i

:

8:4:3:2

The hypothesis that the two models give equally good ®ts to the data can be tested by considering b l to be an unconstrained, oneparameter ®t that is to be compared with a constrained, zeroparameter ®t for which l 0. A p.d.f. for making this comparison can be derived from an F distribution with 1 1 and 2 n 1.

Tests based on F or the R ratio have several limitations. One important one is that they are applicable only when the 704


0.90

n P

8:4:2:5

8:4:2:6

0.75

parameters of one model form a subset of the parameters of the other. Also, the F test makes no distinction between improvement in ®t as a result of small improvements throughout the entire data set and a large improvement in a small number of critically sensitive data points. A test that can be used for comparing arbitrary pairs of models, and that focuses attention on those data points that are most sensitive to differences in the models, was introduced by Williams & Kloot (1953; also Himmelblau, 1970; Prince, 1982). Consider a set of observations, y0i , and two models that predict values for these observations, y1i and y2i , respectively. We determine the slope of the regression line z lx, where zi y0i 1=2y1i y2i =i , and xi y1i y2i =i . Suppose model 1 is a perfect ®t to the data, which have been measured with great precision, so that y0i y1i for all i. Under these conditions, l 1=2. Similarly, if model 2 is a perfect ®t, l 1=2. Real experimental data, of course, are subject to random error, and jlj in general would be expected to be less than 1=2. A least-squares estimate of l is

Denoting by Rc and Ru the weighted R indices for the constrained and unconstrained models, respectively, F 2 = 1 Rc =Ru 2

1=2 : F; 1; p vF =21 F= 1=2

8:4:3:3

If we let jtj RF0 0

8.4. STATISTICAL SIGNIFICANCE TESTS

p F , and use F; 1; dF

t R0 t0

t; dt;

8:4:3:4

[where Aij @Mi x=@xj ] is a good one. Let R be the Cholesky factor of W , so that W RT R, and let Z RA, y0 y y0 , and b y0 b y y0 . The least-squares estimate may then be written b x x0 ZT Z 1 ZT y0

we can derive a p.d.f. for t, which is 1=2 t; p : 1=2 =21 t2 =

8:4:3:5

This p.d.f. is known as Student's t distribution with degrees of freedom. Setting t b l=b l , the c.d.f. t; can be used to test the alternative hypotheses l 0 and l 1=2. Table 8.4.3.1 gives the values of t for which the c.d.f. t; has various values for various values of . Fortran code for the program from which this table was generated appears in Prince (1994). Again, it must be understood that the results of these statistical comparisons do not imply that either model is a correct one. A statistical indication of a good ®t says only that, given the model, the experimenter should not be surprised at having observed the data values that were observed. It says nothing about whether the model is plausible in terms of compatibility with the laws of physics and chemistry. Nor does it rule out the existence of other models that describe the data as well as or better than any of the models tested. 8.4.4. In¯uence of individual data points When the method of least squares, or any variant of it, is used to re®ne a crystal structure, it is implicitly assumed that a model with adjustable parameters makes an unbiased prediction of the experimental observations for some (a priori unknown) set of values of those parameters. The existence of any re¯ection whose observed intensity is inconsistent with this assumption, that is that it differs from the predicted value by an amount that cannot be reconciled with the precision of the measurement, must cause the model to be rejected, or at least modi®ed. In making precise estimates of the values of the unknown parameters, however, different re¯ections do not all carry the same amount of information (Shoemaker, 1968; Prince & Nicholson, 1985). For an obvious example, consider a spacegroup systematic absence. Except for possible effects of multiple diffraction or twinning, any observed intensity at a position corresponding to a systematic absence is proof that the screw axis or glide plane is not present. If no intensity is observed for any such re¯ection, however, any parameter values that conform to the space group are equally acceptable. It is to be expected, on the other hand, that some intensities will be extremely sensitive to small changes in some parameter, and that careful measurement of those intensities will lead to correspondingly precise estimates of the parameter values. For the purpose of precise structure re®nement, it is useful to be able to identify the in¯uential re¯ections. Consider a vector of observations, y, and a model Mx. The elements of y de®ne an n-dimension space, and the model values, Mi x, de®ne a p-dimensional subspace within it. The leastsquares solution [equation (8.1.2.7)], T

1

T

b x A WA A Wy

y0 ;

8:4:4:1

is such that b y Mb x is the closest point to y that corresponds to some possible value of x. In (8.4.4.1), W V 1 is the inverse of the variance±covariance matrix for the joint p.d.f. of the elements of y, and y0 Mx0 is a point in the p-dimensional subspace close enough to Mb x so that the linear approximation Mx y0 Ax

x0

8:4:4:2

and b x y0 Zb

x0 ZZT Z 1 ZT y0 : T

8:4:4:4

T

Thus, the matrix P ZZ ZZ , the projection matrix, is a linear relation between the observed data values and the corresponding calculated values. (Because b y 0 Py0 , the matrix P is frequently referred to in the statistical literature as the hat matrix.) P 2 ZZT Z 1 ZT ZZT Z 1 ZT ZZT Z 1 ZT P, so that P is idempotent. P is an n n positive semide®nite matrix with rank p, and its eigenvalues are either 1 ( p times) or 0 (n p times). Its diagonal elements lie in the range 0 Pii 1, and the trace of P is p, so that the average value of Pii is p=n. Furthermore, n P 8:4:4:5 Pii Pij2 : j1

A diagonal element of P is a measure of the in¯uence that an observation has on its own calculated value. If Pii is close to one, the model is forced to ®t the ith data point, which puts a constraint on the value of the corresponding function of the parameters. A very small value of Pii , because of (8.4.4.5), implies that all elements of the row must be small, and that observation has little in¯uence on its own or any other calculated value. Because it is a measure of in¯uence on the ®t, Pii is sometimes referred to as the leverage of the ith observation. Note that, because ZT Z 1 V x , the variance± covariance matrix for the elements of b x, P is the variance± covariance matrix for b y, whose elements are functions of the elements of b x. A large value of Pii means that yi is poorly de®ned by the elements of b x, which implies in turn that some elements of b x must be precisely de®ned by a precise measurement of y0i . It is apparent that, in a real experiment, there will be appreciable variation among observations in their leverage. It can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that the observations with the greatest leverage also have the largest effect on the volume of the p-dimensional con®dence region for the parameter estimates. Because this volume is a rather gross measure, however, it is useful to have a measure of the in¯uence of individual observations on individual parameters. Let V n be the variance±covariance matrix for a re®nement including n observations, and let z be a row vector whose elements are zj @Mx=@xj = for an additional observation. V n1 , the variance±covariance matrix with the additional observation included, is, by de®nition, V n1 ZT Z zT z 1 ;

8:4:4:6

which, in the linear approximation, can be shown to be V n1 V n

V n zT zV n =1 zV n zT :

8:4:4:7

The diagonal elements of the rank one matrix D V n zT zV n =1 zV n zT are therefore the amounts that the variances of the estimates of individual parameters will be reduced by inclusion of the additional observation. This result depends on the elements of Z and z not changing signi®cantly in the (presumably small) shift from b xn to b xn1 . That this condition is satis®ed may be veri®ed by the following procedure. Find an approximation to b xn1 by a line search

705


8:4:4:3

8. REFINEMENT OF STRUCTURAL PARAMETERS h i1=2 Bij ZT Z zT z ZT Z zT z ii jj

along the line x b xn V n1 zT y0n1 , and then evaluate B, a quasi-Newton update such as the BFGS update (Subsection 8.1.4.3) at that point. If 1, and the gradient of the sum of squares vanishes, then the linear approximation is exact, and B is null. If

for all i and j, then (8.4.4.7) can be expected to be an excellent approximation for a nonlinear model.

706


8:4:4:8


8.5. Detection and treatment of systematic error By E. Prince and C. H. Spiegelman

8.5.1. Accuracy

b x G 2 AT WA 1 : V

Chapter 8.4 discusses statistical tests for goodness of ®t between experimental observations and the predictions of a model with adjustable parameters whose values have been estimated by least squares or some similar procedure. In addition to the estimates of parameter values, one can also make estimates of the uncertainties in those values, estimates that are usually expressed in terms of an estimated standard deviation or, according to recommended usage (ISO, 1993), a standard uncertainty. A standard deviation is a measure of precision, that is, a measure of the width of a con®dence interval that results from random ¯uctuations in the measurement process. What the experimenter who collected the data wants to know about, of course, is accuracy, a measure of the location of a region within which nature's `correct' value lies, as well as its width (Prince, 1994). In performing a re®nement, we have assumed implicitly that the observations have been drawn at random from a population the mean of whose p.d.f. is given by a model when all of its parameters have those unknown, correct values. If this assumption is incorrect, the expected value of the estimate may no longer be near to the correct value, and the estimate contains bias, or systematic error. An accurate measurement is one that not only is precise but also has small bias. In this chapter, we shall discuss various criteria by which the results of a re®nement may be judged in order to determine whether they are free of systematic error, and thus whether they may be considered accurate.

Frequently, however, there is some other, independent estimate of the variance of the observation, i2 , derived, for example, from counting statistics or from the observed scatter among symmetry-equivalent re¯ections. If this estimate is inconsistent with the hypothesis that all data points have been overweighted by a constant factor, then the assumption that the parameter estimates are unbiased but less precise than the original weights would indicate must be discarded. Instead, it must be assumed that the model is incorrect, or at least incomplete. A systematic error may be considered to cause the model to be incomplete, and may introduce bias into some or all of the re®ned parameters. (Note that in many standard statistical texts it is implicitly assumed, without so stating, that the data have already been scaled by a set of correct, relative weights. It is thus easy for the unwary reader to make the error of assuming that the practice of multiplying by the goodness-of-®t parameter is a well established procedure.) The use of (8.5.2.2) to compute estimated variances and standard uncertainties assumes implicitly that the effect of lack of ®t on parameter estimates is random, and applies equally to all parameters, even though different types of parameter may have very different mathematical relations in the model. With a model as complex as the crystallographic structure-factor formula, this assumption is certainly questionable. Information about the nature of the model inadequacies can be obtained by examining the residuals (Belsley, Kuh & Welsch, 1980; Belsley, 1991). The standardized residuals, Ri yi Mi b x=u0i , where b x is the least-squares estimate of the parameters, should be randomly distributed, with zero mean, not only for the data set as a whole but also for subsets of the data that are chosen in a manner that depends only on the model and not on the observed values of the data. Here, u0i is the standard uncertainty of the residual and is related to ui , the standard uncertainty of the observation, by u0i ui 1 Pii , where Pii is a diagonal element of the projection matrix (Section 8.4.4). A scatter plot, in which the residuals are plotted against some control variable, such as jFcalc j, sin =l, or one of the Miller indices, should reveal no general trends. The existence of any such trend may indicate a systematic effect that depends on the corresponding variable. The model may then be modi®ed by inclusion of a factor that is proportional to that variable, and the re®nement repeated. An examination of the shifts in the other parameters, and of the new row or column of the variance±covariance matrix, will then reveal which of the parameters in the unmodi®ed model are likely to have been biased by the systematic effect. When this procedure has been followed, it is extremely important to consider carefully the nature of the additional effect and determine whether it is plausible in terms of physics and chemistry. Another procedure for detecting systematic lack of ®t makes use of the fact that, if the model is correct, and the error distribution is approximately normal, or Gaussian, the distribution of residuals will also be approximately normal. A large sample may be checked for normality by means of a quantile± quantile, or Q±Q, plot (Abrahams & Keve, 1971; Kafadar & Spiegelman, 1986). To make such a plot, the residuals are ®rst sorted in ascending order of magnitude. If there are n points in the data set, the value of the ith sorted residual should be close to the value, xi , for which

8.5.2. Lack of ®t We saw in Section 8.4.2 that the sum of squared residuals from an ideally weighted, least-squares ®t to a correct model is a sum of terms that has expected value n p and is distributed as 2 with n p degrees of freedom. Further, the residuals have a distribution with zero mean. A value for the sum that exceeds n p by an amount that is improbably large is an indication of lack of ®t, which may be due to an incorrect model for the mean or to nonideal weighting or both. fThe sum, S, may be considered to be improbably large when the value of the 2 cumulative distribution function, 2 S; n p, is close to 1:0. A value for the sum that is substantially less than n p may also be an indication that the model contains more parameters than can be justi®ed by the data set. Note also that a reasonable value for the sum of squared residuals does not prove that the model is correct. It indicates that the model adequately describes the data, but it in no way rules out the existence of alternative models that describe the data equally well.g If the sum of squares is greater than n p, it is commonly assumed that the mean model is correct, and that the weights have appropriate relative values, although their absolute values may be too large. If w k2 =u2i , where k is some number greater than one, and ui is the standard uncertainty of the ith observation, the goodness-of-®t parameter, n 1=2 P G wi yi Mi b x2 =n p ; 8:5:2:1 i1

is taken to be an estimate of k, and all elements of the inverse of the normal-equations matrix are multiplied by k2 to obtain the estimated variance±covariance matrix


8:5:2:2

8. REFINEMENT OF STRUCTURAL PARAMETERS xi 2i

1=2n;

8:5:2:3

where x is the cumulative distribution function for the normal p.d.f. A plot of Ri against xi should be a straight line with zero intercept and unit slope. A straight line with a slope greater than one suggests that the model is satisfactory, but that the variances of the data points have been systematically underestimated. Lack of ®t is suggested if the curve has a higher slope near the ends, indicating that large residuals occur with greater frequency than would be predicted by the normal p.d.f. The sorted residuals tend to be strongly correlated. A positive displacement from a smooth curve tends to be followed by another positive displacement, and a negative one by another negative one, which gives the Q±Q plot a wavy appearance, and it may be dif®cult to decide whether it is a straight line or not. Because of this, a useful alternative to the Q±Q plot is the conditional Q±Q plot (Kafadar & Spiegelman, 1986), so called because the abscissa for plotting the ith sorted residual is the mean of a conditional p.d.f. for that residual given the observed values of all the others. To construct a conditional Q±Q plot, ®rst transform the distribution to a uniform p.d.f. by Ui Ri ; ; ;

8:5:2:4

where and are resistant estimates (Section 8.2.2) of the mean and standard deviation of the p.d.f., such as the median and 0.75 times the interquartile range, and represents the cumulative distribution function. Letting U0 0 and Un1 1, the expected value of Ui , given all the others, is hUi i Ui

1

Ui1 =2:

8:5:2:5

The ith abscissa for the Q±Q plot is then xi 1 hUi i; ; ;

8:5:2:6

1

where y; ; is a per cent point function, or p.p.f., the value of x for which x; ; y. Q±Q plots for subsets of the data can reveal, by nonzero intercepts, that those subsets are subject to a systematic bias. Because of its property of removing short-range kinks in the curve, the conditional Q±Q plot can be particularly useful in this application. The values of and used for the transformation to a uniform distribution, as in (8.5.2.4), should be those determined from the entire data set. A Q±Q plot will reveal data points that are in poor agreement with the model, but that do not belong to any easily identi®able subset. Because of the central limit theorem (Section 8.4.1), however, the least-squares method tends to force the distribution of the residuals toward a normal distribution, and the discrepant points may not be clearly evident. A robust/resistant procedure (see Section 8.2.2), because it reduces the in¯uence of strongly discrepant data points, helps to separate them from the body of the data. Therefore, if a data set contains discrepant points, a Q±Q plot of the residuals from a robust/resistant ®t will tend to have greater curvature at the extremes than one from a corresponding least-squares ®t. If the discrepant data points that are thus identi®ed have a pattern, this information may enable a systematic error to be characterized. 8.5.3. In¯uential data points Section 8.4.4 discusses the in¯uence of individual data points on the estimation of parameters and how to identify the data points that should be measured with particular care in order to make the most precise estimates of particular parameters. The same properties that cause these in¯uential data points to be most effective in reducing the uncertainty of a parameter estimate

when the model is a correct predictor for the observations also cause them to have the greatest potential for introducing bias if there is a ¯aw in the model or, correspondingly, if they are subject to systematic error. Reviews of procedures for studying the effects of in¯uential data points and outliers have been given by Beckman & Cook (1983), by Chatterjee & Hadi (1986), and by Belsley (1991). The effects of possible systematic error can be studied by identifying in¯uential data points and then observing the effects of deleting them one by one from the re®nement. The deletion of a data point should affect the standard uncertainty of an estimate, but should not cause a shift in its mean that is more than a small multiple of the resulting standard uncertainty. As in Section 8.4.4, we de®ne the design matrix, A, by Aij @Mi x=@xj ;

where Mi x is the model function for the ith data point, and x is a vector of adjustable parameters. Let R be the upper triangular Cholesky factor of the weight matrix, so that W RT R, and de®ne the weighted design matrix by Z RA and the weighted vector of observations by y0 Ry. The least-squares estimate of x is then b x ZT Z 1 ZT y0 ;

8:5:3:2

and the vector of predicted values is b y0 ZZT Z 1 ZT y0 Py0 ;

8:5:3:3

where P is the projection, or hat, matrix. A diagonal element, Pii , of P is a measure of the leverage, that is of the relative in¯uence, of the ith data point, and therefore of the sensitivity of the estimates of the elements of x to an error in the measurement of that data point. Pii lies in the range 0 Pii 1, and it has average value p=n, so that data points with values of Pii greater than 2p=n can be considered particularly in¯uential. Let H ZT Z be the normal-equations matrix, let V H 1 be the estimated variance±covariance matrix, and let q ZT y0 , so that b x V q. Let zi be the ith row of Z, and denote by Zi , H i , i V , qi , and b xi the respective matrices and vectors computed with the ith data point deleted from the data set. We wish to ®nd i 1=2 large values of jb xj b x i , so we need to compute V i j j=Vjj i and x . With a derivation similar to that for (8.4.4.7), it can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that V i V

V zTi zi V V zTi zi V V : 1 zi V zTi 1 Pii

8:5:3:4

Note that, if Pii 1, all elements of V i become in®nite, implying that H i is singular. Thus, if such a data point is deleted, the solution is no longer determinate. Now, b x i V i qi

8:5:3:5

and qi q

y0i zTi ;

8:5:3:6

so that, when V and b x have been computed once, it is a straightforward and inexpensive additional computation to determine whether any parameter has been strongly in¯uenced, and therefore potentially biased, by the inclusion of any data point in the re®nement. If there is any reason to be concerned about possible systematic error, the leverage of every data point included in the re®nement should be computed, and the effects of deletion of all of those with leverage greater than 2p=n should be observed.

708


8:5:3:1

8.5. DETECTION AND TREATMENT OF SYSTEMATIC ERROR 8.5.4. Plausibility of results A study of residuals to detect a pattern of discrepancies will reveal the presence of systematic error, or model inadequacy, only if different subsets of the data are affected differently. Some sources of bias, however, have noticeable effects throughout the data set, and missing parameters may mimic others that have been included, thus introducing bias without any apparent lack of ®t. To cite an obvious example, determination of unit-cell dimensions requires an accurate value of the wavelength of the radiation being used. If this value is incorrect, inferred values of the cell constants may be reproduced repeatedly with great precision, but all will be subject to a systematic bias that has no effect on the quality of the ®t. Even though the structure of the residuals in such a case reveals little about possible systematic error, it is still possible to detect it by critical examination of the estimated parameters. Even before any data have been collected in preparation for the determination of a crystal structure, a great deal is known about certain details. It is known that the crystal is composed of atoms of certain elements that are present in certain proportions. It is known that pairs of atoms of various elements cannot be less than a certain distance apart, and, further, that adjacent atoms tend to be separated by distances that fall within a rather narrow range. It is known that thermal vibration amplitudes are likely to be larger in directions normal to interatomic vectors than parallel to them, although, particularly in the case of hydrogen bonds, there may be an apparent amplitude parallel to a vector because of atomic disorder. Even when there is a particularly unusual

feature in a structure, most of the structure will conform to commonly observed patterns. Thus, if a re®ned crystal structure overall has reasonable features, such as interatomic distances that are appropriate to oxidation state and coordination number and displacement ellipsoids that make sense, one or two unusual features may be accepted with con®dence. On the other hand, if there is wide variation in the lengths of chemically similar bonds, or if the eigenvectors of the thermal motion tensors point in odd directions relative to the interatomic vectors, there must be a presumption that systematic errors have been compensated for by biased estimates of parameters. A particular problem arises when there is a question of the presence or absence of symmetry, such as a choice between two space groups, one of which possesses a centre of symmetry or a mirror plane, or a case where a symmetric molecule occupies a position whose environment has a less-symmetric point group. If symmetry constraints are relaxed, the model can always be re®ned to a lower sum of squared residuals. (For a discussion of numerical problems that occur in the vicinity of a symmetric con®guration, see Section 8.1.3.) The problem comes from the fact that the removal of the symmetry element almost always introduces too many additional parameters. Statistical tests are then quite likely to indicate that the lower-symmetry model gives a signi®cantly better ®t, but consideration of internal consistency and chemical or physical plausibility is likely to suggest that much systematic error has been absorbed by the additional parameters. The proper procedure is to devise a model with noncrystallographic constraints (see Section 8.3.2) that expresses what is, for chemical or physical reasons, known or probable. To do so may require great patience and perseverence.

709



8.6. The Rietveld method

By A. Albinati and B. T. M. Willis where s is a scale factor, mk is the multiplicity factor for the kth re¯ection, Lk is the Lorentz±polarization factor, Fk is the structure factor and Gik is the `peak-shape function' (PSF). The summation in (8.6.1.2) is over all nearby re¯ections, k1 to k2 , contributing to a given data point i. A fundamental problem of the Rietveld method is the formulation of a suitable peak-shape function. For X-rays, a mixture of Gaussian and Lorentzian components is sometimes used (see Subsection 8.6.2.2). For neutrons, it is easier to ®nd a suitable analytical function, and this is, perhaps, the main reason for the initial success of neutron Rietveld analysis. For a neutron diffractometer operating at a ®xed wavelength and moderate resolution, the PSF is approximately a Gaussian of the form r 2 ln 2 4 ln 2 2i 2k 2 Gik ; 8:6:1:3 exp Hk2 Hk

In the Rietveld method of analysing powder diffraction data, the crystal structure is re®ned by ®tting the entire pro®le of the diffraction pattern to a calculated pro®le. There is no intermediate step of extracting structure factors, and so patterns containing many overlapping Bragg peaks can be analysed. The method was applied originally by Rietveld (1967, 1969) to the re®nement of neutron intensities recorded at a ®xed wavelength. Subsequently, it has been used successfully for analysing powder data from all four categories of experimental technique, with neutrons or X-rays as the primary radiation and with scattered intensities measured at a ®xed wavelength (and variable scattering angle) or at a ®xed scattering angle (and variable wavelength). Powder re®nements are usually less satisfactory than those on single-crystal data, as the three-dimensional information of the reciprocal lattice is compressed into one dimension in the powder pattern. Nevertheless, a large number of successful re®nements by the Rietveld method has been reported; reviews have been given by Taylor (1985), Hewat (1986), Cheetham & Wilkinson (1992), Young (1993), Harris & Tremayne (1996), Masciocchi & Sironi (1997), Harris et al. (2001) and David et al. (2002). Here we shall discuss only the basic principles of the re®nement procedure.

where Hk is the full width at half-maximum (FWHM) of the peak, 2i is the scattering angle at the ith point, and k is the Bragg angle for re¯ection k. The angular dependence of the FWHM for a Gaussian peakshape function can be written in the form

8.6.1. Basic theory

Hk2 U tan2 k V tan k W ;

The model of the structure is re®ned by least-squares minimization of the residual

where U; V and W are half-width parameters independent of k (Caglioti, Paoletti & Ricci, 1958). To allow for intrinsic sample broadening and instrumental resolution, U; V and W are treated as adjustable variables in the least-squares re®nement. The tails of a Gaussian peak decrease rapidly with distance from the maximum, and the intensity at one-and-a-half times the FWHM from the peak is only about 0.2% of the intensity at the peak. Thus, no large error is introduced by assuming that the peak extends over a range of approximately 1:5Hk and is cut off outside this range. The FWHM for the Lorentzian (see Subsection 8.6.2.2) can be modelled by the relation

M

N P i1

wi yi obs:

2 yi calc: :

8:6:1:1

yi obs: is the intensity measured at a point i in the diffraction pattern corrected for the background intensity bi , wi is its weight, and yi calc: is the calculated intensity. If the background at each point is assumed to be zero, and if the only source of error in measuring the intensities is that from counting statistics, the weight is given by 1 wi yi obs: : The summation in (8.6.1.1) runs over all N data points. The number of data points can be arbitrarily increased by reducing the interval between adjacent steps. However, this does not necessarily imply an improvement in the standard uncertainties (s.u.'s; see Section 8.1.2) of the structural parameters (see Subsection 8.6.2.5), which are dependent on the number of linearly independent columns in the design matrix [equation (8.1.2.3)]. The number of independent observations in a powder pattern is determined by the extent of overlapping of adjacent re¯ections. An intuitive argument for estimating this number has been proposed by Altomare et al. (1995) and a more rigorous statistical estimate has been described by Sivia (2000). The strategy for choosing the number of steps and apportioning the available counting time has been discussed by McCusker et al. (1999) and references therein. The relation between counting statistics and the s.u.'s has been discussed by Baharie & Pawley (1983) and by Scott (1983). The calculated intensity is evaluated using the equation yi calc: s

k2 P kk1

2 mk Lk Fk Gik ;

8:6:1:2

Hk X tan k Y = cos k : The Lorentzian function extends over a much wider range than the Gaussian. A more ¯exible approach to this line-broadening problem is described by McCusker et al. (1999). The least-squares parameters are of two types. The ®rst contains the usual structural parameters: for example, fractional coordinates of each atom in the asymmetric unit and the corresponding isotropic or anisotropic displacement parameters. The second type represents `pro®le parameters' which are not encountered in a least-squares re®nement of singlecrystal data. These include the half-width parameters and the dimensions of the unit cell. Further parameters may be added to both groups allowing for the modelling of the background and for the asymmetry of the re¯ections. The maximum number of parameters that can be safely included in a Rietveld re®nement is largely determined by the quality of the diffraction pattern, but intrinsic line broadening will set an upper limit to this number (Hewat, 1986). The following indicators are used to estimate the agreement with the model during the course of the re®nement.


8.6. THE RIETVELD METHOD " 2 # 1 Profile R factor: P 2 2ik Gik 14 Lorentzian yi obs: yi calc: Hk Hk P : RP i " yi obs: 2 # 1 i 2 2ik Gik 14 Hk Hk Weighted pro®le R factor: " p 2 # 2 ln 2 2 ik 2P 2 31=2 1 p exp 4 ln 2 (pseudo-Voigt) wi yi obs: yi calc: Hk Hk 6 7 P " RwP 4 i 5 : # n 2ik 2 wi y2i obs: 2 n 21=n 1 1=n 14 2 1 Gik i Hk Hk n 12 Bragg R factor: P RI

k

Ik calc:

Ik obs: P Ik obs:

(Pearson VII) :

k

Expected R factor: 2

31=2 N P 5 : RE 4 P wi y2i obs: i

Ik is the integrated intensity of the kth re¯ection, N is the number of independent observations, and P is the number of re®ned parameters. The most important indicators are RwP and RE . The ratio RwP =RE is the so-called `goodness-of-®t', 2 : in a successful re®nement 2 should approach unity. The Bragg R factor is useful, since it depends on the ®t of the structural parameters and not on the pro®le parameters.

8.6.2. Problems with the Rietveld method One should be aware of certain problems that may give rise to failure in a Rietveld re®nement.

8.6.2.1. Indexing The ®rst step in re®nement is the indexing of the pattern. As the Rietveld method is often applied to the re®nement of data for which the unit-cell parameters and space group are already known, there is then little dif®culty in indexing the pattern, provided that there are a few well resolved lines. Without this knowledge, the indexing requires, as a starting point, the measurement of the d values of low-angle diffraction lines to high accuracy. According to Shirley (1980): `Powder indexing works beautifully on good data, but with poor data it usually will not work at all'. The indexing of powder patterns and associated problems are discussed by Shirley (1980), Pawley (1981), Cheetham (1993) and Werner (2002).

8.6.2.2. Peak-shape function (PSF) The appropriate function to use varies with the nature of the experimental technique. In addition to the Gaussian PSF in (8.6.1.3), functions commonly used for angle-dispersive data are (Young & Wiles, 1982):

where 2ik 2i 2k . is a parameter that de®nes the fraction of Lorentzian character in the pseudo-Voigt pro®le. n is the gamma function: when n 1; Pearson VII becomes a Lorentzian, and when n 1; it becomes a Gaussian. The tails of a Gaussian distribution fall off too rapidly to account for particle size broadening. The peak shape is then better described by a convolution of Gaussian and Lorentzian functions [i.e. Voigt function: see Ahtee, Nurmela & Suortti (1984) and David & Matthewman (1985)]. A pulsed neutron source gives an asymmetrical line shape arising from the fast rise and slow decay of the neutron pulse: this shape can be approximated by a pair of exponential functions convoluted with a Gaussian (Albinati & Willis, 1982; Von Dreele, Jorgensen & Windsor, 1982). The pattern from an X-ray powder diffractometer gives peak shapes that cannot be ®tted by a simple analytical function. Will, Parrish & Huang (1983) use the sum of several Lorentzians to express the shape of each diffraction peak, while Hepp & Baerlocher (1988) describe a numerical method of determining the PSF. Pearson VII functions have also been successfully used for X-ray data (Immirzi, 1980). A modi®ed Lorentzian function has been employed for interpreting data from a Guinier focusing camera (Malmros & Thomas, 1977). PSFs for instruments employing X-ray synchrotron radiation can be represented by a Gaussian (Parrish & Huang, 1980) or a pseudo-Voigt function (Hastings, Thomlinson & Cox, 1984). 8.6.2.3. Background The background may be determined by measuring regions of the pattern that are free from Bragg peaks. This procedure assumes that the background varies smoothly with sin =l, whereas this is not the case in the presence of disorder or thermal diffuse scattering (TDS), which rises to a maximum at the Bragg positions. An alternative approach is to include a background function in the re®nement model (Richardson, 1993). If the background is not accounted for satisfactorily, the temperature factors may be incorrect or even negative. The various procedures for estimating the background for X-ray, synchrotron, constantwavelength and TOF neutron powder patterns are reviewed by McCusker et al. (1999). In neutron diffraction, the main contribution to the background from hydrogen-containing samples is due to incoherent scattering. Deuterating the sample is essential in order to substantially reduce this background.

711


8. REFINEMENT OF STRUCTURAL PARAMETERS 8.6.2.4. Preferred orientation and texture Preferred orientation is a formidable problem which can drastically affect the measured intensities. A simple correction formula for plate-like morphology was given by Rietveld (1969). Ahtee, Unonius, Nurmela & Suortti (1989) have shown how the effects of preferred orientation can be included in the re®nement by expanding the orientation distribution in spherical harmonics. Quantitative texture analysis based on spherical harmonics has been implemented in the Rietveld re®nement code by Von Dreele (1997). A general model of the texture has also been described by Popa (1992). It may be possible to remove or reduce the effect of preferred orientation by mixing the sample with a suitable diluent. An additional problem is caused by particle size and strain broadening, which are not smooth functions of the diffraction angle. These effects can be taken into account by phenomenological models (e.g. Dinnebier et al., 1999; Pratapa, O'Connor & Hunter, 2002) or by an analytical approach such as that of Popa & Balzar (2002).

The determination of the elastic stresses and strains in polycrystals can be determined from diffraction line shifts using Rietveld re®nement (Popa & Balzar, 2001). 8.6.2.5. Statistical validity Sakata & Cooper (1979) criticized the Rietveld method on the grounds that different residuals yi obs:

related to the same Bragg peak are correlated with one another, and they asserted that this correlation leads to an uncertainty in the standard uncertainties of the structural parameters. Prince (1981) has challenged this conclusion and stated that the s.u.'s given by the Rietveld procedure are correct if the crystallographic model adequately ®ts the data. However, even if the s.u.'s are correct, they are measures of precision rather than accuracy, and attempts to assess accuracy are hampered by lack of information concerning correlations between systematic errors (Prince, 1985, 1993).

712


yi calc:


8.7. Analysis of charge and spin densities

By P. Coppens, Z. Su and P. J. Becker R 1 r n 1; 1 ds1 8.7.1. Outline of this chapter R n 1; 2; . . . ; n 1; 2; . . . ; n ds1 d2 . . . dn: 8:7:2:4 Knowledge of the electron distribution is crucial for our understanding of chemical and physical phenomena. It has An electron density r that can be represented by an been assumed in many calculations (e.g. Thomas, 1926; Fermi, antisymmetric N-electron wave function and is derivable from 1928; and others), and formally proven for non-degenerate that wave function through (8.7.2.4) is called N-representable. systems (Hohenberg & Kohn, 1964) that the electronic energy is As in any real system, the particles will undergo vibrations, so a functional of the electron density. Thus, the experimental (8.7.2.4) must be modi®ed to allow for the continuous change in measurement of electron densities is important for our under- the con®guration of the nuclei. The commonly used Born± standing of the properties of atoms, molecules and solids. One of Oppenheimer approximation assumes that the electrons rethe main methods to achieve this goal is the use of scattering arrange instantaneously in the ®eld of the oscillating nuclei, techniques, including elastic X-ray scattering, Compton scatter- which leads to the separation ing of X-rays, magnetic scattering of neutrons and X-rays, and e r; R N R; 8:7:2:5 electron diffraction. Meaningful information can only be obtained with data of the where r and R represent the electronic and nuclear coordinates, utmost accuracy, which excludes most routinely collected respectively, and e is the electronic wavefunction, which is a crystallographic data sets. The present chapter will review the function of both the electronic and nuclear coordinates. The basic concepts and expressions used in the interpretation of time-averaged, one-electron density r, which is accessible accurate data in terms of the charge and spin distributions of the experimentally through the elastic X-ray scattering experiment, electrons. The structure-factor formalism has been treated in is obtained from the static density by integration over all nuclear Chapter 1.2 of Volume B (IT B, 1992). con®gurations:

R r r; RPR dR; 8:7:2:6 8.7.2. Electron densities and the n-particle wavefunction A wavefunction 1; 2; 3; . . . ; n for a system of n electrons is a function of the 3n space and n spin coordinates of the electrons. The wavefunction must be antisymmetric with respect to the interchange of any two electrons. The most general density function is the n-particle density matrix, of dimension 4n 4n, de®ned as n

1; 2; . . . ; n; 10 ; 20 ; . . . ; n0 1; 2; . . . ; n

10 ; 20 ; . . . ; n0 : 8:7:2:1

In this expression, each index represents both the continuous space coordinates and the discontinuous spin coordinates of each of the n particles. Thus, the n-particle density matrix is a representation of the state in a 6n-dimensional coordinate space, and the n-dimensional (discontinuous) spin state. The pth reduced density matrix can be derived from (8.7.2.1) by integration over the space and spin coordinates of n p particles, p

1; 2; . . . ; p; 10 ; 20 ; . . . ; p0 Z n n 1; 2; . . . ; p; p 1; . . . ; n; 10 ; 20 ; . . . ; p0 ; p p 1; . . . ; n dp 1 . . . dn:

where PR is the normalized probability distribution function of the nuclear con®guration R. The total X-ray scattering can be derived from the two-particle matrix 1; 2; 10 ; 20 through use of the two-particle scattering operator. The total X-ray scattering includes the inelastic incoherent Compton scattering, which is related to the momentum density p. The wavefunction in momentum space, de®ned by the coordinates b Jb pj ; sj of the jth particle (here the caret indicates momentum-space coordinates), is given by the Dirac±Fourier transform of ; R bb 1; b 2; . . . ; b n 2h 3n=2 1; 2; . . . ; n ! P exp i=h pj rj dr1 dr2 . . . drn ; j

8:7:2:7 leading, in analogy to (8.7.2.2), to the one-electron density matrix R 1 b 1; b 10 n b 1; b 2; . . . ; b n b 1; b 2; . . . ; b n db 2 . . . db n; 8:7:2:8

8:7:2:2

According to a basic postulate of quantum mechanics, physical properties are represented by their expectation values hF i obtained from the corresponding operator equation,

b F jFj j ; 8:7:2:3 b is a (Hermitian) operator. As almost all operators of where F interest are one- or two-particle operators, the one- and twoparticle matrices 1 1 ; 10 and 2 1; 2 ; 10 ; 20 are of prime interest. The charge density, or one-electron density, can be obtained from 1 1; 10 by setting 10 1 and integrating over the spin coordinates, i.e.

and the momentum density p


1 b 1; b 1 ds1 :

8:7:2:9

The spin density distribution of the electrons, sr; can be obtained from 1 1; 10 by use of the operator for the z component of the spin angular momentum. If R sr; r0 2M 1 sz 1 1; 10 ds1 ; 8:7:2:10 sr sr; r; where M, the total magnetization, is the eigenvalue of the operator P Sz sz i:


R

i

8. REFINEMENT OF STRUCTURAL PARAMETERS 8.7.3. Charge densities

Table 8.7.3.1. De®nition of difference density functions

8.7.3.1. Introduction The charge density is related to the elastic X-ray scattering amplitude FS by the expression

R r FS exp 2iS r dS; 8:7:3:1

2 V

As Fh is in general complex, the Fourier transform (8.7.3.1) requires calculation of the phases from a model for the charge distribution. In the centrosymmetric case, the free-atom model is in general adequate for calculation of the signs of Fh. However, for non-centrosymmetric structures in which phases are continuously variable, it is necessary to incorporate deviations from the free-atom density in the model to obtain estimates of the experimental phases. Since the total density is dominated by the core distribution, differences between the total density r and reference densities are important. The reference densities represent hypothetical states without chemical bonding or with only partial chemical bonding. Deviations from spherical, free-atom symmetry are obtained when the reference state is the promolecule, the superposition of free-space spherical atoms centred at the nuclear positions. This difference function is referred to as the deformation density (or standard deformation density) p r;

where Fobs and Fcalc are in general complex. Several other different density functions analogous to (8.7.3.3), summarized in Table 8.7.3.1, may be de®ned. Particularly useful for the analysis of effects of chemical bonding is the fragment deformation density, in which a chemical fragment is subtracted from the total density of a molecule. The fragment density is calculated theoretically and thermally smeared before subtraction from an experimental density. `Prepared' atoms rather than spherical atoms may be used as a reference state to emphasize the electron-density shift due to covalent bond formation. 8.7.3.2. Modelling of the charge density The electron density r in the structure-factor expression R Fcalc h r exp2ih r dr 8:7:3:4a can be approximated by a sum of non-normalized density functions gi r with scattering factor fi h centred at ri ; P 8:7:3:5 r gi r r ri : i

8:7:3:4b

When gi r is the spherically averaged, free-atom density, (8.7.3.4b) represents the free-atom model. A distinction is often made between atom-centred models, in which all functions gr

A2 cos 2h r

0

) B1

B2 sin 2h r

a Residual map

A1 , B1 from observations calculated with model phases. A2 , B2 from re®nement model.

b X

A1 , B1 from observation, with model phases. A2 , B2 from high-order re®nement, free-atom model, or other reference state.

X deformation map

c X N deformation map [as b but]

A2 , B2 calculated with neutron parameters.

d X X N deformation map [as b but]

A2 , B2 calculated with parameters from joint re®nement of X-ray neutron data.

e X X; X N; X X N valence map

As b, c, d with A1 , B1 calculated with core-electron contribution only.

f Dynamic model map

A1 , B1 from model. A2 , B2 with parameters from model re®nement and free-atom functions.

g Static model map

model free atom , where model is sum of static model density functions.

atom r Pcore core r 3 Pvalence valence r:

8:7:3:6

This `kappa model' allows for charge transfer between atomic valence shells through the population parameter Pvalence , and for a change in nuclear screening with electron population, through the parameter , which represents an expansion < 1, or a contraction > 1 of the radial density distribution. The atom-centred, spherical harmonic expansion of the electronic part of the charge distribution is de®ned by atom r Pc core r Pv 3 valence r

lmax P l0

0 3 Rl 0 r

l P P m0

p

Plmp dlmp ; ';

8:7:3:7

where p when m is larger than 0, and Rl 0 r is a radial function. The real spherical harmonic functions d lmp and their Fourier transforms have been described in International Tables for Crystallography, Volume B, Chapter 1.2 (Coppens, 1992). They differ from Rthe functions ylmp by the normalization condition, de®ned as jdlmp j d 2 l0 . The real spherical harmonic functions are often referred to as multipoles, since each

714


A1

1=2 X

are centred at the nuclear positions, and models in which additional functions are centred at other locations, such as in bonds or lone-pair regions. A simple, atom-centred model with spherical functions gr is de®ned by

unit cell

i

0

8:7:3:2

where p r is the promolecule density. In analogy to (8.7.3.1), the deformation density may be obtained from

1 X r Fobs h Fcalc;free atom h exp 2ih r; V 8:7:3:3

Substitution in (8.7.3.4a) gives P fi h exp2ih ri : F h

1=2 X

with F A iB.

or, for scattering by a periodic lattice,

1X r Fh exp 2ih r: V h

r r

(

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES represents the component of the charge distribution r that gives a non-zero contribution to the integral for the electrostatic multipole moment qlmp ; R atom rr l clmp dr; 8:7:3:8 qlmp where the functions clmp are the Cartesian representations of the real spherical harmonics (Coppens, 1992). More general models include non-atom centred functions. If the wavefunction in (8.7.2.4) is an antisymmetrized product of molecular orbitals i , expressed P in terms of a linear combination of atomic orbitals ; i ci (LCAO formalism), the integration (8.7.2.4) leads to PP r P r R r R ; 8:7:3:9

with RP and R de®ning the centres of and , respectively, P i ni ci ci , and the sum is over all molecular orbitals with occupancy ni . Expression (8.7.3.9) contains products of atomic orbitals, which may have signi®cant values for orbitals centred on adjacent atoms. In the `charge-cloud' model (Hellner, 1977), these products are approximated by bond-centred, Gaussianshaped density functions. Such functions can often be projected ef®ciently into the one-centre terms of the spherical harmonic multipole model, so that large correlations occur if both spherical harmonics and bond-centred functions are adjusted independently in a least-squares re®nement. According to (8.7.2.4) and (8.7.3.9), the population of the two-centre terms is related to the one-centre occupancies. A molecular-orbital based model, which implicitly incorporates such relations, has been used to describe local bonding between transition-metal and ligand atoms (Becker & Coppens, 1985). 8.7.3.3. Physical constraints There are several physical constraints that an electron-density model must satisfy. With the exception of the electroneutrality constraint, they depend strongly on the electron density close to the nucleus, which is poorly determined by the diffraction experiment. 8.7.3.3.1. Electroneutrality constraint Since a crystal is neutral, the total electron population must equal the sum of the nuclear charges of the constituent atoms. A constraint procedure for linear least squares that does not increase the size of the least-squares matrix has been described by Hamilton (1964). If the starting point is a neutral crystal, the constraint equation becomes P Pi Si 0; 8:7:3:10 R where Si gr dr, g being a general density function, and the Pi are the shifts in the population parameters. For the multipole model, only the monopolar functions integrate to a non-zero value. For normalized monopole functions, this gives P Pi 0: 8:7:3:11 monopoles

If the shifts without constraints are given by the vector y and the constrained shifts by yc , the Hamilton constraint is expressed as yTc yT

Expression (8.7.3.12) cannot be applied if the unconstrained re®nement corresponds to a singular matrix. This would be the case if all population parameters, including those of the core functions, were to be re®ned together with the scale factor. In this case, a new set of independent parameters must be de®ned, as described in Chapter 8.1 on least-squares re®nements. Alternatively, one may set the scale factor to one and rescale the population parameters to neutrality after completion of the re®nement. This will in general give a non-integral electron population for the core functions. The proper interpretation of such a result is that a core-like function is an appropriate component of the density basis set representing the valence electrons.

yT QT QA 1 QT 1 QA 1 ;

8:7:3:12

where the superscript T indicates transposition, A is the leastsquares matrix of the products of derivatives, and Q is a row vector of the values of Si for elements representing density functions and zeros otherwise.

8.7.3.3.2. Cusp constraint The electron density at a nucleus i with nuclear charge Zi must satisfy the electron±nuclear cusp condition given by @ 2Zi 0i ri 0; 8:7:3:13 lim ri !0 @ri R where 0i ri 1=4 r d i is the spherical component of the expansion of the density around nucleus i. Only 1s-type functions have non-zero electron density at the nucleus and contribute to (8.7.3.13). For the hydrogen-like atom or ion described by a single exponent radial function Rr N exp r, (8.7.3.13) gives 2Z=a0 , where Z is the nuclear charge, and a0 is the Bohr unit. Thus, a modi®cation of for 1s functions, as implied by (8.7.3.6) and (8.7.3.7) if applied to H atoms, leads to a violation of the cusp constraint. In practice, the electron density at the nucleus is not determined by a limited resolution diffraction experiment; the single exponent function Rr is ®tted to the electron density away from, rather than at the nucleus. 8.7.3.3.3. Radial constraint Poisson's electrostatic equation gives a relation between the gradient of the electric ®eld r2 r and the electron density at r: r 2 r

8:7:3:14

As noted by Stewart (1977), this equation imposes a constraint on the radial functions Rr. For Rl r Nl r nl exp l r, the condition nl l must be obeyed for Rl r l to be ®nite at r 0, which satis®es the requirement of the non-divergence of the electric ®eld rV , its gradient r2 V , the gradient of the ®eld gradient r3 V ; etc. 8.7.3.3.4. Hellmann±Feynman constraint According to the electrostatic Hellmann±Feynman theorem, which follows from the Born±Oppenheimer approximation and the condition that the forces on the nuclei must vanish when the nuclear con®guration is in equilibrium, the nuclear repulsions are balanced by the electron±nucleus attractions (Levine, 1983). The balance of forces is often achieved by a very sharp polarization of the electron density very close to the nuclei (Hirshfeld & Rzotkiewicz, 1974), which may be represented in the X-ray model by the introduction of dipolar functions with large values of . The Hellmann±Feynman constraint offers the possibility for obtaining information on such functions even though they may contribute only marginally to the observed X-ray scattering (Hirshfeld, 1984). As the Hellmann±Feynman constraint applies to the static density, its application presumes a proper deconvolution of the thermal motion and the electron density in the scattering model.

715


4r:

8. REFINEMENT OF STRUCTURAL PARAMETERS 8.7.3.4. Electrostatic moments and the potential due to a charge distribution 8.7.3.4.1. Moments of a charge distribution* Use of the expectation value expression R b r dr; hOi O 8:7:3:15 b br r r . . . r r r r . . . r gives with the operator O 1 2 3 l 1 2 3 l for the electrostatic moments of a charge distribution r R 1 2 3 ...l rr1 r2 r3 . . . rl dr; 8:7:3:16 in which the r are the three components of the vector r (i 1; 2; 3), and the integral is over the complete volume of the distribution. For l 0, (8.7.3.16) represents the integral over the charge distribution, which is the total charge, a scalar function described as the monopole. The higher moments are, in ascending order of l, the dipole, a vector, the quadrupole, a second-rank tensor, and the octupole, a third-rank tensor. Successively higher moments are named the hexadecapole l 4, the tricontadipole l 5, and the hexacontatetrapole l 6. An alternative, traceless, de®nition is often used for moments with l 2. In the traceless de®nition, the quadrupole moment, , is given by R 12 r 3r r r 2 dr; 8:7:3:17 R where is the Kronecker delta function. The term rr 2 dr, which is subtracted from the diagonal elements of the tensor, corresponds to the spherically averaged second moment of the distribution. Expression (8.7.3.17) is a special case of the following general expression for the lth-rank traceless tensor elements. Z 1 l @l 1 l 2l1 M 1 2 ...l rr dr: l! @r1 @r2 . . . @rl r

lmp

and xy 32 xy :

8:7:3:19

Expressions for the other elements are obtained by simple permutation of the indices. For a site of point symmetry 1, the electrostatic moment 1 2 3 ...l of order l has l 1l 2=2 unique elements. In the traceless de®nition, not all elements are independent. Because the trace of the tensor has been set to zero, only 2l 1 independent components remain. For the quadrupole there are 5 independent components of the form (8.7.3.19). In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics clmp (IT B, 1992) multiplied by r l , which de®nes the spherical electrostatic moments * An excellent review of experimental results has appeared in the literature (Spackman, 1992).

8:7:3:20

zz 1=220 ; xx 1=2 322 1=220 ; yy 1=2 322 1=220 ; xz 3=221 ; yz 3=221 ; xy 3=222 :

8:7:3:21

8.7.3.4.1.1. Moments as a function of the atomic multipole expansion In the multipole model [expression (8.7.3.7)], the charge density is a sum of atom-centred density functions, and the moments of a whole distribution can be written as a sum over the atomic moments plus a contribution due to the shift to a common origin. An atomic moment is obtained by integration over the charge distributions total;i r nuclear;i e;i of atom i, R 8:7:3:22 1 2 3 ...l total;i rr1 r2 r3 . . . rl dr; where the electronic part of the atomic charge distribution is de®ned by the multipole expansion e;i r Pi;c core r P i;v 3i i;valence i r

lP max l0

0 03 i Ri;l i r

l P P m0

p

P i;lmp dlmp ; ';

8:7:3:23

where p when m > 0, and Rl 0i r is a radial function. We get for the jth moment of the valence density j 1 2 3 ...j Z lP max 0 P i;v 3i i;valence i r 03 i Ri;l i r l0 # l PP Pi;lmp d lmp ; ' r1 r2 . . . rj dr; m0 p

8:7:3:24

in which the minus sign arises because of the negative charge of the electrons. b for the moment operators. We get We will use the symbol O " # Z lmax X l P P j 03 b Oj Plmp dlmp Rl dr; i 8:7:3:25 l1

m0 p

where, as before, p : The requirement that the integrand be totally symmetric means that only the dipolar terms in the multipole expansion contribute to the dipole moment. If we use the traceless de®nition of the higher moments, or the equivalent de®nition of the moments in terms of the spherical harmonic functions, only the quadupolar terms of the multipole expansion will contribute to the quadrupole moment; more generally, in the traceless de®nition the lth-order multipoles are the sole contributors to the lth moments. In terms of the spherical moments, we get

716


rclmp r l dr:

The expressions for clmp are listed in Volume B of International Tables for Crystallography (IT B, 1992); for the l 2 moment, the clmp have the well known form 3z2 1, xz, yz, x2 y2 =2, and xy, where x, y and z are the components of a unit vector from the origin to the point being described. The spherical electrostatic moments have 2l 1 components, which equals the number of independent components in the traceless de®nition (8.7.3.18), as it should. The linear relationships are

8:7:3:18 Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer present in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from (8.7.3.18). For the quadrupole moments, we obtain with (8.7.3.17) xx 32 xx 12 xx yy zz xx 12 yy zz ;

R

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES R RP b lmp dlmp Rl dr: promolecule x2 dr spherical atom;i x2 dr lmp Plmp O 8:7:3:26 i PR Substitution with Rl f0 nl3 =nl 2!gr nl exp r and spherical atom;i x2 dr: 8:7:3:35 l b Olmp clmp r and subsequent integration over r gives i Z 1 nl l 2! 1 If Ri Xi ; Yi ; Zi is the position vector for atom i, each singlelmp Plmp y2lmp sin d d'; atom contribution can be rewritten as 0 l nl 2! Dlm M lm R i;xx;spherical atom i;spherical atom x2 dr 8:7:3:27 R where the de®nitions i;spherical atom x Xi 2 dr R Llm 1 Xi i;spherical atom 2 x Xi dr y y dlmp Llm clmp and clmp R M lm lmp M lm lmp 8:7:3:36 Xi2 i;spherical atom dr: 8:7:3:28 Since the last two integrals are proportional to the atomic dipole have been used (IT B, 1992). Since the ylmp functions are moment and its net charge, respectively, they will be zero for wavefunction normalized, we obtain in (8.7.3.35) gives, with

neutral spherical 1 2atoms.

Substitution R 2 2 2 r r r dr; x X ; and r i i i i 3 1 nl l 2! Llm

R P lmp Plmp : 8:7:3:29 2 1 2 2 l promolecule x dr 3 r spherical atom ; 8:7:3:37 0 nl 2! M lm R

Application to dipolar terms with nl 2, Llm 1= and M lm 3=41=2 gives the x component of the atomic dipole moment as Z 20 P : P11 d 11 R1 x dr 8:7:3:30 x 30 11 For the atomic quadrupole moments in the spherical de®nition, we obtain directly, using nl 2, l 2 in (8.7.3.29), p 30 L20 36 3 20 P 20 P20 ; 8:7:3:31 0 2 M 20 2 0 2 and, for the other elements, 2mp

30 L2m P 2mp 2 0 M 2m 2

6 P 2mp : 0 2

8:7:3:32

As the traceless quadrupole moments are linear combinations of the spherical quadrupole moments, the corresponding expressions follow directly from (8.7.3.31), (8.7.3.32) and (8.7.3.21). We obtain with n2 2 p 18 3 P 20 ; zz 0 2 9 p P 3 P yy ; 20 22 0 2 9 p 3P20 P22 ; xx 2 0 and xz

9 P 21 ; 0 2

8:7:3:33

and analogously for the other off-diagonal elements. 8.7.3.4.1.2. Molecular moments based on the deformation density The moments derived from the total density r and from the deformation density r are not identical. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as R xx total x2 dr R R promolecule x2 dr x2 dr: 8:7:3:34 The promolecule is the sum over spherical atom densities, or

atoms

and, by substitution in (8.7.3.34), P 2 r spherical atom ; xx tot xx 13 atoms

with xx

i

i x2 dr 2Xi i Xi2 qi ;

8:7:3:38b

in which i and qi are the atomic dipole moment and the charge on atom i, respectively. The last term in (8.7.3.38a) can be derived rapidly from analytical expressions for the atomic wavefunctions. Results for Hartree±Fock wavefunctions have been tabulated by Boyd (1977). Since the off-diagonal elements of the second-moment tensor vanish for the spherical atom, the second term in (8.7.3.38a) disappears, and the off-diagonal elements are identical for the total and deformation densities. The relation between the second moments and the traceless moments of the deformation density can be illustrated as follows. From (8.7.3.17), we may write R 32 12 r 2 dr: 8:7:3:39 Only the spherical density terms contribute to the integral on the right. Assuming for the moment that the spherical deformation is represented by the valence-shell distortion i:e. neglect of the second monopole in the aspherical atom expansion), we have, with density functions normalized to 1, for each atom spherical 3 Pvalence valence r

P0valence valence r 8:7:3:40

and R R P 3 r 2 dr i P valence;i valence;i i r i P0valence;i valence;i r r 2 dr P P valence;i =2i P0valence ri2 spherical valence shell i R2i Pvalence;i P 0valence;i ; 8:7:3:41 which, on substitution in (8.7.3.39), gives the required relation. 8.7.3.4.1.3. The effect of an origin shift on the outer moments In general, the multipole moments depend on the choice of origin. This can be seen as follows. Substitution of r0 r R in (8.7.3.16) corresponds to a shift of origin by R , or X, Y, Z in the original coordinate system. In three dimensions, we get, for the ®rst moment, the charge q,

717


P R

8:7:3:38a

8. REFINEMENT OF STRUCTURAL PARAMETERS Z X 1 q q; 8:7:3:42 l VT F h exp 2ih r dr

bl V and for the transformed ®rst and second moments VT Z 1X 0x x qX; 0y y qY; 0z z qZ;

bl exp 2ih r dr; 8:7:3:48 F h V 2 0 VT 2 R qR ; 0 8:7:3:43 where bl is the product of l coordinates according to (8.7.3.16), R R qR R : and l represents the moment of the static distribution if the Fh For the traceless quadrupole moments, the corresponding are the structure factors on an absolute scale after deconvolution equations are obtained by substitution of r0 r R and of thermal motion. Otherwise, the moment of the thermally r0 r R into (8.7.3.17), which gives averaged densityR is obtained. The integral VT bl exp 2ih r dr is de®ned as the shape 0 12 3R R R2 q transform S of the volume VT : P 3 R R R : 8:7:3:44

2 1X

l VT F hSVT bl ; h: V Similar expressions for the higher moments are reported in the For regularly shaped volumes, the integral can be evaluated literature (Buckingham, 1970). analytically. A volume of complex shape may be subdivided into We note that the ®rst non-vanishing moment is origin- integrable subvolumes such as parallelepipeds. By choosing the independent. Thus, the dipole moment of a neutral molecule, subvolumes suf®ciently small, a desired boundary surface can be but not that of an ion, is independent of origin; the quadrupole closely approximated. moment of a molecule without charge and dipole moment is not If the origin of each subvolume is located at ri , relative to a dependent on the choice of origin and so on. The molecular coordinate system origin at P, the total electronic moment electric moments are commonly reported with respect to the relative to this origin is given by centre of mass. ! X X 1X l VT ;i F hSVT bl ; h exp 2ih ri : 8.7.3.4.1.4. Total moments as a sum over the pseudoatom V h i i moments The moments of a molecule or of a molecular fragment are 8:7:3:49 obtained from the sum over the atomic moments, plus a Expressions for SVT for l 2 and a subvolume parallelepipecontribution due to the shift to a common origin for all but the dal shape are given in Table 8.7.3.2. Since the spherical order monopoles. If individual atomic coordinate systems are used, as Bessel functions jn x that appear in the expressions generally is common if chemical constraints are applied in the leastdecrease with increasing x, the moments are strongly dependent squares re®nement, they must be rotated to have a common on the low-order re¯ections in a data set. An example is the orientation. Expressions for coordinate system rotations have moment. Relative to an origin O, been given by Cromer, Larson & Stewart (1976) and by Su & shape transform for the dipole R Coppens (1994a). S b1 ; h r0 exp 2ih r0 dr: The transformation to a common coordinate origin requires Vt use of the origin-shift expressions (8.7.3.42)±(8.7.3.44), with, for an atom at ri , R ri . The ®rst three moments summed A shift of origin by ri leads to Z over the atoms i located at ri become 1X 10 b exp 2ih r

V ; h r dr F h S T VT 1 1 0 P V qi ; 8:7:3:45 qtotal P P exp 2ih r1 i r i q i ; 8:7:3:46 total i 1 VT r1 q; 0

and ;total

P i

0 0 0 0 i r i ri ri r i qi

in agreement with (8.7.3.46). 8:7:3:47

with ; x; y; z; and expressions equivalent to (8.7.3.44) for the traceless components . 8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation Expression (8.7.3.16) for the outer moment of a distribution within a volume element VT may be written as R 1 2 ...l rb

1 2 3 ...l dr; 8:7:3:16 VT

with b1 2 3 ...l r1 r2 r3 . . . rl , and integration over the volume VT . Replacement of r by the Fourier summation over the structure factors gives

8.7.3.4.2. The electrostatic potential 8.7.3.4.2.1. The electrostatic potential and its derivatives The electrostatic potential r0 due to the electronic charge distribution is given by the Coulomb equation, Z r dr; 8:7:3:50 r0 k jr r0 j where the constant k is dependent on the units selected, and will here be taken equal to 1. For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain Z X ZM r r0 dr; 8:7:3:51 RM r0 jr r0 j M

in which ZM is the charge of nucleus M located at RM .

718


8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES The electric ®eld E at a point in space is the gradient of the electrostatic potential at that point. Er

rr

i

@r @x

j

@r @y

k

@r : @z

8:7:3:52

As E is the negative gradient vector of the potential, the electric force is directed `downhill' and proportional to the slope of the potential function. The explicit expression for E is obtained by differentiation of the operator jr r0 j 1 in (8.7.3.50) towards x, y, z and subsequent addition of the vector components. For the negative slope of the potential in the x direction, one obtains Z Z total r r0 rx total r 0 0 dr r rx dr; E x r 2 0 0 jr rj jr rj jr0 rj3 8:7:3:53 which gives, after addition of the components, Z t rr0 r 0 0 dr: Er rr jr r0 j3

Table 8.7.3.2. Expressions for the shape factors S for a parallelepiped with edges x , y , and z ( from Moss & Coppens, 1981) j0 and j1 are the zero- and ®rst-order spherical Bessel functions: j0 x sin x=x, j1 x sin x=x2 cos x=x; VT is volume of integration. y^

S y^ r; h

Property

1

Charge

r

Dipole

VT j0 2hx x j0 2hy y j0 2hz z iVT j1 2h j0 2h j0 2hy y

r r

r r

8:7:3:54

Second moment off-diagonal

VT j1 2h

Second moment diagonal

VT 2

j1 2h j0 2h

The electric ®eld gradient (EFG) is the tensor product of the gradient operator r i @x@ j @y@ k @z@ and the electric ®eld vector E; rE r : E

r : r:

8:7:3:55

It follows that in a Cartesian system the EFG tensor is a symmetric tensor with elements rE

@2 : @r @r

8:7:3:56

The EFG tensor elements can be obtained by differentiation of the operator in (8.7.3.53) for E to each of the three directions . In this way, the traceless result rE r0

@E

r 0 n 1 3r r0 r jr r0 j5 o r r0 2 total r dr

@r Z

r 0

8:7:3:57

is obtained. We note that according to (8.7.3.57) the electric ®eld gradient can equally well be interpreted as the tensor of the traceless quadrupole moments of the distribution 2total r=jr r0 j5 [see equation (8.7.3.17)]. De®nition (8.7.3.56) and result (8.7.3.57) differ in that (8.7.3.56) does not correspond to a zero-trace tensor. The situation is analogous to the two de®nitions of the second moments, discussed above, and is illustrated as follows. The trace of the tensor de®ned by (8.7.3.56) is given by 2 @ @2 @2 r2 r r : 8:7:3:58 @x2 @y2 @z2 Poisson's equation relates the divergence of the gradient of the potential r to the electron density at that point: r2 r 4 e r 4e r: 8:7:3:59 Thus, the EFG as de®ned by (8.7.3.56) is not traceless, unless the electron density at r is zero. The potential and its derivatives are sometimes referred to as inner moments of the charge distribution, since the operators in (8.7.3.50), (8.7.3.52) and (8.7.3.54) contain the negative power of the position vector. In the same terminology, the electrostatic moments discussed in x8:7:3:4:1 are described as the outer moments.

j0 2h

j0 2h j0 2h

It is of interest to evaluate the electric ®eld gradient at the atomic nuclei, which for several types of nuclei can be measured accurately by nuclear quadrupole resonance and MoÈssbauer spectroscopy. The contribution of the atomic valence shell centred on the nucleus can be obtained by substitution of the multipolar expansion (8.7.3.7) in (8.7.3.57). The quadrupolar l 2 terms in the expansion contribute to the integral. For the radial function Rl fnl3 =nl 2!grnl exp r with nl 2, the following expressions are obtained: p 3P20 Qr ; rE11 3=5 P22 p rE22 3=5 P22 3P20 Qr ; p rE33 6=5 3P20 Qr ; 8:7:3:60 rE12 3=5P22 Qr ; rE13 3=5 P21 Qr ; rE23 3=5P21 Qr ; with

Qr r 3 3 d R1 Rr =r dr 0

3

0 =n2 n2 1n2 2 3

0 =120; in the case that n2 4 (Stevens, DeLucia & Coppens, 1980). The contributions of neighbouring atoms can be subdivided into point-charge, point-multipole, and penetration terms, as discussed by Epstein & Swanton (1982) and Su & Coppens (1992, 1994b), where appropriate expressions are given. Such contributions are in particular important when short interatomic distances are involved. For transition-metal atoms in coordination complexes, the contribution of neighbouring atoms is typically much smaller than the valence contribution.

719


j1 2h h

8. REFINEMENT OF STRUCTURAL PARAMETERS 8.7.3.4.2.2. Electrostatic potential outside a charge distribution Hirshfelder, Curtiss & Bird (1954) and Buckingham (1959) have given an expression for the potential at a point ri outside a charge distribution: ri

q r 1 3 2 3r r r 2 5 ri ri ri 5r r r r2 r r r ...; 5ri7 8:7:3:61

where summation over repeated indices is implied. The outer moments q, , and in (8.7.3.61) must include the nuclear contributions, but, for a point outside the distribution, the spherical neutral-atom densities and the nuclear contributions cancel, so that the potential outside the charge distribution can be calculated from the deformation density. The summation in (8.7.3.61) is slowly converging if the charge distribution is represented by a single set of moments. When dealing with experimental charge densities, a multicentre expansion is available from the analysis, and (8.7.3.61) can be replaced by a summation over the distributed moments centred at the nuclear positions, in which case ri measures the distance from a centre of the expansion to the ®eld point. The result is equivalent to more general expressions given by Su & Coppens (1992), which, for very large values of ri , reduce to the sum over atomic terms, each expressed as (8.7.3.61). The interaction between two charge distributions, A and B; is given by R EAB A rB r dr; where B includes the nuclear charge distribution.

rE Rp

M6P

2

jRMP j

M

jrp j5

jrp j2

Expression (8.7.3.49) is an example of derivation of electrostatic properties by direct Fourier summations of the structure factors. The electrostatic potential and its derivatives may be obtained in an analogous manner. In order to obtain the electrostatic properties of the total charge distribution, it is convenient to de®ne the `total' structure factor Ftotal h including the nuclear contribution, P

drM ; 8:7:3:64

in which the exclusion of M P only applies when the point P coincides with a nucleus, and therefore only occurs for the central contributions. ZM and RM are the nuclear charge and the position vector of atom M, respectively,

F h;

where FN h j Zj exp2ih Rj , the summation being over all atoms j with nuclear charge Zj , located at Rj . If h is de®ned as the Fourier transform of the direct-space potential, we have R r h exp 2ih r dh r2 r

R 42 h2 h exp 2ih r dh;

which equals 4total according to the Poisson equation (8.7.3.59). One obtains with R total r Ftotal h exp 2ih r dh; h Ftotal h=h2 ;

8:7:3:65

and, by inverse Fourier transformation of (8.7.3.65), 1 X r 8:7:3:66 Ftotal h=h2 exp 2ih r V (Bertaut, 1978; Stewart, 1979). Furthermore, the electric ®eld due to the electrons is given by R Er rr r r h exp 2ih r dh R 2i hh exp 2ih r dh: Thus, with (8.7.3.65), 2i X Er Ftotal h=h2 h exp 2ih r; V

Eh

8:7:3:67a

2i hF h: h2 total

Similarly, the h Fourier component of the electric ®eld gradient tensor with trace 4r is

720


Ftotal h FN h

which implies

jRMP j5

X Z e;M rM 3r r

8.7.3.4.3. Electrostatic functions of crystals by modi®ed Fourier summation

and

8.7.3.4.2.3. Evaluation of the electrostatic functions in direct space The electrostatic properties of a well de®ned group of atoms can be derived directly from the multipole population coef®cients. This method allows the `lifting' of a molecule out of the crystal, and therefore the examination of the electrostatic quantities at the periphery of the molecule, the region of interest for intermolecular interactions. The dif®culty related to the origin term, encountered in the reciprocal-space methods, is absent in the direct-space analysis. In order to express the functions as a sum over atomic contributions, we rewrite (8.7.3.51), (8.7.3.54) and (8.7.3.57) for the electrostatic properties at point P as a sum over atomic contributions. X Z e;M rM X ZM Rp drM ; 8:7:3:62 jrp j jRMP j M6P M XZ R X Z rp e;M rM MP MP drM ; 8:7:3:63 ERp 3 jrp j3 M6P jRMP j M X ZM 3R R

while rP and rM are, respectively, the vectors from P and from the nucleus M to a point r, such that rP r RP , and rM r RM rP RP RM rP RMP . The subscript M in the second, electronic part of the expressions refers to density functions centred on atom M. Expressions for the evaluation of (8.7.3.62)±(8.7.3.64) from the charge-density parameters of the multipole expansion have been given by Su & Coppens (1992). They employ the Fourier convolution theorem, used by Epstein & Swanton (1982) to evaluate the electric ®eld gradient at the atomic nuclei. A directspace method based on the Laplace expansion of 1=jRp rj was reported by Bentley (1981).

r : Eh 42 h : hh 4h : hFtotal h=h2 ; 8:7:3:67b

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES where i : k represents the tensor product of two vectors. This leads to the expression for the electric ®eld gradient in direct space, r : Er

4 X h : hFtotal h=h2 exp 2ih r: 8:7:3:68 V

(The elements of h : h are the products hi hj :) The components of E and the elements of the electric ®eld gradient de®ned by (8.7.3.67a) and (8.7.3.68) are with respect to the reciprocal-lattice coordinate system. Proper transforms are required to get the values in other coordinate systems. Furthermore, to get the P traceless rE tensor, the quantity 4=3e r 4=3V F h exp 2ih r must be subtracted from each of the diagonal elements rEii : The Coulombic self-electronic energy of the crystal can be obtained from Z Z e re r0 1 ECoulombic; electronic 2 dr dr0 ; jr r0 j R 12 e rr dr: R R Since rr dr hFh dh (Parseval's rule), the summation can be performed in reciprocal space, ECoulombic; electronic

1 X 2 F h=h2 ; 2V

8:7:3:69a

and, for the total Coulombic energy, ECoulombic; total

1 X 2 Ftotal h=h2 ; 2V

8:7:3:69b

where the integral has been replaced by a summation. The summations are rapidly convergent, but suffer from having a singularity at h 0 (Dahl & Avery, 1984; Becker & Coppens, 1990). The contribution from this term to the potential cannot be ignored if different structures are compared. The term at h 0 gives a constant contribution to the potential, which, however, has no effect on the energy of a neutral system. For polar crystals, an additional term occurs in (8.7.3.69a; b), which is a function of the dipole moment D of the unit cell (Becker, 1990), ECoulombic; total

1 X 2 2 2 Ftotal h=h2 D: 2V 3V

8:7:3:69c

To obtain the total energy of the static crystal, electron exchange and correlation as well as electron kinetic energy contributions must be added. 8.7.3.4.4. The total energy of a crystal as a function of the electron density

where Zi is the nuclear charge for an atom at position Ri , and Rij Rj Ri . Because of the theorem of Hohenberg & Kohn (1964), E is a unique functional of the electron density , so that T Exc must be a functional of . Approximate density functionals are discussed extensively in the literature (Dahl & Avery, 1984) and are at the centre of active research in the study of electronic structure of various materials. Given an approximate functional, one can estimate non-Coulombic contributions to the energy from the charge density r. In the simplest example, the functionals are those applicable to an electronic gas with slow spatial variations (the `nearly free electron gas'). In this approximation, the kinetic energy T is given by R T ck t d3 r; 8:7:3:72 with ck 3=1022 2=3 ; and the function t 2=3 . The exchange-correlation energy is also a functional of ; R Exc cx exc d3 r; with cx 3=43=1=3 and exc 1=3 . Any attempt to minimize the energy with respect to in this framework leads to very poor results. However, cohesive energies can be described quite well, assuming that the change in electron density due to cohesive forces is slowly varying in space. An example is the system AB, with closed-shell subsystems A and B. Let A and B be the densities for individual A and B subsystems. The interaction energy is written as E Ec Ec A Ec B R ck dr t A tA B tB R cx dr exc A exc A B exc B :

This model is known as the Gordon±Kim (1972) model and leads to a qualitatively valid description of potential energy surfaces between closed-shell subsystems. Unlike pure Coulombic models, this density functional model can lead to an equilibrium geometry. It has the advantage of depending only on the charge density . 8.7.3.5. Quantitative comparison with theory Frequently, the purpose of a charge density analysis is comparison with theory at various levels of sophistication. Though the charge density is a detailed function, the features of which can be compared at several points of interest in space, it is by no means the only level at which comparison can be made. The following sequence represents a progression of functions that are increasingly related to the experimental measurement.

One can write the total energy of a system as E ec T Exc ;

8:7:3:70

where T is the kinetic energy, Exc represents the exchange and electron correlation contributions, and Ec , the Coulomb energy, discussed in the previous section, is given by Z Z X Zi Zj X rr0 dr dr0 ; Zi Ri 12 Ec 12 0j j R r r ij i6j i 8:7:3:71

1; 2; . . . ; n !

electrostatic properties " 1 1; 1 ! r ! # r !

r ! F h ! I h

r ; 8:7:3:74

where the angle brackets refer to the thermally averaged functions. The experimental information may be reduced in the opposite sequence:

721


8:7:3:73

8. REFINEMENT OF STRUCTURAL PARAMETERS Table 8.7.3.3. The matrix M

1

relating d-orbital occupancies Pij to multipole populations Plm ( from Holladay, Leung & Coppens, 1983) Multipole populations

d-orbital populations Pz2 Pxz Pyz Px2 Pxy

y2

P00

P20

P22

P40

P42

P44

0.200 0.200 0.200 0.200 0.200

1.039 0.520 0.520 0.039 1.039

0.00 0.942 0.942 0.00 0.00

1.396 0.931 0.931 0.233 0.233

0.00 1.108 1.108 0.00 0.00

0.00 0.00 0.00 1.571 1.571

Mixing terms

Pz2 =xz Pz2 =yz Pz2 =x2 y2 Pz2 =xy Pxz=yz Pxz=x2 y2 Pxz=xy Pyz=x2 y2 Pyz=xy Px2 y2 =xy

P21

P21

P22

P22

P41

P41

P42

P42

P43

P43

P44

1.088 0.00 0.00 0.00 0.00 1.885 0.00 0.00 1.885 0.00

0.00 1.088 0.00 0.00 0.00 0.00 1.885 1.885 0.00 0.00

0.00 0.00 2.177 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 2.177 1.885 0.00 0.00 0.00 0.00 0.00

2.751 0.00 0.00 0.00 0.00 0.794 0.00 0.00 0.794 0.00

0.00 2.751 0.00 0.00 0.00 0.00 0.794 0.794 0.00 0.00

0.00 0.00 1.919 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 1.919 2.216 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 2.094 0.00 0.00 2.094 0.00

0.00 0.0 0.0 0.0 0.0 0.0 2.094 2.094 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.142

electrostatic properties " r # r

r

# r :

F h

I h

8:7:3:75 The crucial step in each sequence is the thermal averaging (top sequence) or the deconvolution of thermal motion (bottom sequence), which in principle requires detailed knowledge of both the internal (molecular) and the external (lattice) modes of the crystal. Even within the generally accepted Born±Oppenheimer approximation, this is a formidable task, which can be simpli®ed for molecular crystals by the assumption that the thermal smearing is dominated by the larger-amplitude external modes. The procedure (8.7.3.65) requires an adequate thermal-motion model in the structurefactor formalism applied to the experimental structure amplitudes. The commonly used models may include anharmonicity, as described in Volume B, Chapter 1.2 (IT B, 1992), but assume that a density function centred on an atom can be assigned the thermal motion of that atom, which may be a poor approximation for the more diffuse functions. The missing link in scheme (8.7.3.75) is the sequence 1 1; 1 r. In order to describe the wavefunction analytically, a basis set is required. The number of coef®cients in the wavefunction is minimized by the use of a minimal basis for the molecular orbitals, but calculations in general lead to poor-quality electron densities. If the additional approximation is made that the wavefunction is a single Slater determinant, the idempotency condition can be used in the derivation of the wavefunction from the electron density. A simpli®ed two-valence-electron two-orbital system has been treated in this manner (Massa, Goldberg,

Frishberg, Boehme & La Placa, 1985), and further developments may be expected. A special case occurs if the overlap between the orbitals on an atom and its neighbours is very small. In this case, a direct relation can be derived between the populations of a minimal basis set of valence orbitals and the multipole coef®cients, as described in the following sections. 8.7.3.6. Occupancies of transition-metal valence orbitals from multipole coef®cients In general, the atom-centred density model functions describe both the valence and the two-centre overlap density. In the case of transition metals, the latter is often small, so that to a good approximation the atomic density can be expressed in terms of an atomic orbital basis set di , as well as in terms of the multipolar expansion. Thus, d

i1 ji 4 P l0

Pij di dj ;

(

03 Rl 0 r

l P P m0 p

) Plmp dlmp ;

8:7:3:76

in which dlmp are the density functions. The orbital products di dj can be expressed as linear combinations of spherical harmonic functions, with coef®cients listed in Volume B, Chapter 1.2 (IT B, 1992), which leads to relations between the Pij and Plmp . In matrix notation, Plmp MPij ;

8:7:3:77

where Plmp is a vector containing the coef®cients of the 15 spherical harmonic functions with l 0, 2, or 4 that are generated by the products of d orbitals. The matrix M is also a function of the ratio of orbital and density-function normalization coef®cients, given in Volume B, Chapter 1.2 (IT B, 1992).

722


5 P 5 P

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES Table 8.7.3.4. Orbital±multipole relations for square-planar complexes (point group D4h )

P20 P21 P21 P22 P22

a1g eg b1g b2g

P00

P20

0.200 0.200 0.200 0.200 0.200

1.039 0.520 0.520 1.039 1.039

P40

P44

1.396 0.931 0.931 0.233 0.233

0.00 0.00 0.00 1.570 1.570

Table

P2

P40

1.039 0.520 0.520 1.039 1.039 0.00 0.00 0.00 0.00

1.396 0.931 0.931 0.233 0.233 0.00 0.00 0.00 0.00

for

trigonal

P43

P43

a In terms of d orbitals P20 P21 P21 P22 P22 P21=22 P21=22 P21 =22 P21 =22

8:7:3:78

(Holladay, Leung & Coppens, 1983). The matrix M 1 is given in Table 8.7.3.3. Point-groupspeci®c expressions can be derived by omission of symmetryforbidden terms. Matrices for point group 4=mmm (square 32, 3m, 3m) planar) and for trigonal point groups (3, 3, are listed in Tables 8.7.3.4 and 8.7.3.5, respectively. Point groups with and without vertical mirror planes are distinguished by the occurrence of both dlm and dlm functions in the latter case, and only dlm in the former, with m being restricted to n, the order of the rotation axis. The dln functions can be eliminated by rotation of the coordinate system around a vertical axis through an angle 0 given by 0 1=n arctanPln =Pln . 8.7.3.7. Thermal smearing of theoretical densities 8.7.3.7.1. General considerations In the Born±Oppenheimer approximation, the electrons rearrange instantaneously to the minimum-energy state for each nuclear con®guration. This approximation is generally valid, except when very low lying excited electronic states exist. The thermally smeared electron density is then given by

R r r; RP R dR;

Orbital±multipole relations complexes P00

The d-orbital occupancies can be derived from the experimental multipole populations by the inverse expression, Pij M 1 Plmp

8.7.3.5.

8:7:3:79

0.200 0.200 0.200 0.200 0.200 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 2.094 0.00 0.00 2.094

0.00 0.00 0.00 0.00 0.00 0.00 2.094 2.094 0.00

b In terms of symmetry-adapted orbitals*y P1 a1g P2 eg P3 e0g P4 eg eg0 eg e0g

0.200 0.400 0.400 0.00

1.039 1.039 0.00 2.942

1.396 0.310 1.087 2.193

0.00 1.975 1.975 1.397

* The electron density in terms of the symmetry-adapted orbitals is given by: 3d P1 a21g 12 P2 e2g e2g 12 P4 eg eg0 eg eg0 ; p p p 2=3dx2 p 1=3dxz ; eg p2=3dxy with: y2 p a1g d0z2 ; egp 0 p1=3dyz ; eg 1=3dx2 y2 2=3dxz ; and eg 0 1=3dxy 2=3dyz . y The signs given here imply a positive eg lobe in the positive xz quadrant. Care should be exercised in de®ning the coordinate system if this lobe is to point towards a ligand atom.

be made. The simplest is to assume a gradual variation of the thermal motion along the bond, which gives at a point ri on the internuclear vector of length R Uij ri Uij R R ri Uij R R ri =R : 8:7:3:82

where R represents the 3N nuclear space coordinates and PR is the probability of the con®guration R. Evaluation of (8.7.3.79) is possible if the vibrational spectrum is known, but requires a large number of quantum-mechanical calculations at points along the vibrational path. A further approximation is the convolution approximation, which assumes that the charge density near each nucleus can be convoluted with the vibrational motion of that nucleus,

PR r n r u Rn Pn u du; 8:7:3:80

This expression may be used to assign thermal parameters to a bond-centred function. 8.7.3.7.2. Reciprocal-space averaging over external vibrations

8:7:3:81

Thermal averaging of the electron density is considerably simpli®ed for modes in which adjacent atoms move in phase. In molecular crystals, such modes correspond to rigid-body vibrations and librations of the molecule as a whole. Their frequencies are low because of the weakness of intermolecular interactions. Rigid-body motions therefore tend to dominate thermal motion, in particular at temperatures for which kT k 0:7 cm 1 is large compared with the spacing of the vibrational energy levels of the external modes (internal modes are typically not excited to any extent at or below room temperature). For a translational displacement u, the dynamic density is given by R dyn r r uP u du; 8:7:3:83

where and are basis functions centred at R and R , respectively. As the motion of a point between the two vibrating atoms depends on their relative phase, further assumptions must

with r de®ned by (8.7.3.81) (Stevens, Rees & Coppens, 1977). In the harmonic approximation, Pu is a normalized three-dimensional Gaussian probability function, the exponents

n

where n stands for the density of the nth pseudo-atom. The convolution approximation thus requires decomposition of the density into atomic fragments. It is related to the thermal-motion formalisms commonly used, and requires that two-centre terms in the theoretical electron density be either projected into the atom-centred density functions, or assigned the thermal motion of a point between the two centres. In the LCAO approximation (8.7.3.9), the two-centre terms are represented by r P r

R r

R ;

723


8. REFINEMENT OF STRUCTURAL PARAMETERS of which may be obtained by rigid-body analysis of the experimental data. In general, for a translational displacement u and a librational oscillation !, P dyn r P Pu; !: 8:7:3:84 If correlation between u and ! can be ignored (neglect of the screw tensor S, Pu; ! PuP!; and both types of modes can be treated independently. For the translations P hitrans P r r Pu P P F 1 f h Ttr h ; 8:7:3:85 where F 1 is the inverse Fourier transform operator, and Ttr h is the translational temperature factor. If R is an orthogonal rotation matrix corresponding to a rotation !, we obtain for the librations P hilibr P Rrv Rr P!

P P F 1 f Rh ; 8:7:3:86 in which f h has been averaged over the distribution of orientations of h with respect to the molecule;

R f h f RhP! d!: 8:7:3:87 Evaluation of (8.7.3.85) and (8.7.3.86) is most readily performed if the basis functions have a Gaussian-type radial dependence, or are expressed as a linear combination of Gaussian radial functions. For Gaussian products of s orbitals, the molecular scattering factor of the product N exp r rA 2 2 N exp r rB , where N and N are the normalization factors of the orbitals and centred on atoms A and B, is given by 3=2 2 s;s r rB fstat h N N exp A 2 2 jhj exp2ih rc ; exp 8:7:3:88 v where the centre of density rc is de®ned by rc rA rB = . ForP thePtranslational modes, the temperature-factor exponent 22 i j Uij hi hj is simply added to the Gaussian exponent in (8.7.3.88) to give PP 2 jhj2 exp 22 Uij hi hj : i j For librations, we may write Rr r ulib T

As Rh r h R r h r Rh ulib , for a function centred at r,

P exp2h rc exp 2iRh rc P d! R exp2h rc exp 2iRh uclib P ! d!; 8:7:3:89 which shows that for ss orbital products the librational temperature factor can be factored out, or R s;s s;s exp 2iRh uclib P! d!: fstat f dyn 8:7:3:90

Expressions for P! are described elsewhere (Pawley & Willis, 1970). For general Cartesian Gaussian basis functions of the type r x

yA n z

zA p exp jr

rA j2 ; 8:7:3:91

the scattering factors are more complicated (Miller & Krauss, 1967; Stevens, Rees & Coppens, 1977), and the librational temperature factor can no longer be factored out. However, it may be shown that, to a ®rst approximation, (8.7.3.90) can again be used. This `pseudotranslation' approximation corresponds to a neglect of the change in orientation (but not of position) of the two-centre density function and is adequate for moderate vibrational amplitudes. Thermally smeared density functions are obtained from the averaged reciprocal-space function by performing the inverse Fourier transform with phase factors depending on the position coordinates of each orbital product h i

X 1X P f h exp 2ih r V h

rc ;

8:7:3:92

where the orbital product is centred at rc . If the summation is truncated at the experimental limit of sin =l, both thermal vibrations and truncation effects are properly introduced in the theoretical densities. 8.7.3.8. Uncertainties in experimental electron densities It is often important to obtain an estimate of the uncertainty in the deformation densities in Table 8.7.3.1. If it is assumed that the density of the static atoms or fragments that are subtracted out are precisely known, three sources of uncertainty affect the deformation densities: (1) the uncertainties in the experimental structure factors; (2) the uncertainties in the positional and thermal parameters of the density functions that in¯uence calc ; and (3) the uncertainty in the scale factor k. If we assume that the uncertainties in the observed structure factors are not correlated with the uncertainties in the re®ned parameters, the variance of the electron density is given by 2 2 k 2 2 0obs 2 calc 0obs k2 X calc 0 k

up ; k ; up obs up k p

8:7:3:93

where 0obs obs =k, up is a positional or thermal parameter, and the up ; k are correlation coef®cients between the scale factor and the other parameters (Rees, 1976, 1978; Stevens & Coppens, 1976). Similarly, for the covariance between the deformation densities at points A and B, covA ; B cov obs;A ; obs;B =k2 cov calc;A ; calc;B obs;A obs;B k=k2 ;

8:7:3:94

where it is implied that the second term includes the effect of the scale factor/parameter correlation. Following Rees (1976), we may derive a simpli®ed expression Since the for the covariance valid for the space group P1. structure factors are not correlated with each other,

724


xA m y

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES X microscopic magnetization densities are related by the simple @obs;A @obs;B 2 Fobs h cov obs;A ; obs;B ' expression @Fobs h @Fobs h Z 1 2 X 2 l mr dr; 8:7:4:1 ' 2 Fobs cos 2rA rB h V cell V 1=2 8:7:3:95 where V is the volume of the unit cell, mr is the sum of two cos 2rA rB h ; contributions: ms r originating from the spins of the electrons, P and indicates where the latter equality is speci®c for P 1, and mL r originating from their orbital motion. 1=2 summation over a hemisphere in reciprocal space. In general, the mr ms r mL r: 8:7:4:2 second term rapidly averages to zero as hmax increases, while the ®rst term may be replaced by its average 8.7.4.2. Magnetization densities from neutron magnetic elastic

scattering cos 2rA rB h 3sin u u cos u=u3 Cu; The scattering process is discussed in Section 6.1.3 and only 8:7:3:96 the features that are essential to the present chapter will be 2 with u 2jrA rB jhmax , or covobs;A Pobs;B2 ' 2=V Cu summarized here. P 2 2 2 For neutrons, the nuclear structure factor FN h is given by and obs ' 2=V 1=2 F0 , a relation 1=2 F0 P derived earlier by Cruickshank (1949). A discussion of the FN h bj Tj exp2ih Rj : 8:7:4:3 applicability of this expression in other centrosymmetric space j groups is given by Rees (1976). bj , Tj , Rj are the coherent scattering length, the temperature factor, and the equilibrium position of the jth atom in the unit 8.7.3.9. Uncertainties in derived functions cell. The electrostatic moments are functions of the scale factor, the Let r be the spin of the neutron (in units of h=2). There is a positional parameters x, y and z of the atoms, and their charge- dipolar interaction of the neutron spin with the electron spins and density parameters and Plmp . The standard uncertainties in the the currents associated with their motion. The magnetic structure derived moments are therefore dependent on the variances and factor can be written as the scalar product of the neutron spin and covariances of these parameters. an ìnteraction vector' Qh: If M p represents the m m variance±covariance matrix of FM h r Qh: 8:7:4:4 the parameters pj , and T is an n m matrix de®ned by @l =@pj for the lth moment with n independent elements, the variances Qh is the sum of a spin and an orbital term: Q and Q , s L and covariances of the elements of ml are obtained from respectively. If r0 re 0:54 10 12 cm, where is the 8:7:3:96a gyromagnetic factor (=1.913) of the neutron and re the classical M TM p T T : Thomson radius of the electron, the spin term is given by If a moment of an assembly of pseudo-atoms is evaluated, the ^ 8:7:4:5 Qs h r0 h^ Ms h h; elements of T include the effects of coordinate rotations required to transfer atomic moments into a common coordinate system. h^ being the unit vector along h, while Ms h is de®ned as * + A frequently occurring case of interest is the evaluation of the P magnitude of a molecular dipole moment and its standard rj exp 2i h rj ; 8:7:4:6 Ms h deviation. De®ning j y 2 lGlT ;

8:7:3:96b

where l is the dipole-moment vector and G is the direct-space metric tensor of the appropriate coordinate system. If Y is the 1 3 matrix of the derivatives @y=@i , 2 y YM Y T ;

8:7:3:96c

where M is de®ned by (8.7.3.96a). The standard uncertainty in may be obtained from y=2. Signi®cant contributions often result from uncertainties in the positional and chargedensity parameters of the H atoms. 8.7.4. Spin densities 8.7.4.1. Introduction Magnetism and magnetic ordering are among the central problems in condensed-matter research. One of the main issues in macroscopic studies of magnetism is a description of the magnetization density l as a function of temperature and applied ®eld: phase diagrams can be explained from such studies. Diffraction techniques allow determination of the same information, but at a microscopic level. Let mr be the microscopic magnetization density, a function of the position r in the unit cell (for crystalline materials). Macroscopic and

where rj is the spin of the electron at position rj , and angle brackets denote the ensemble average over the scattering sample. Ms h is the Fourier transform of the spin-magnetization density ms r, given by * + P ms r rj r rj : 8:7:4:7 j

This is the spin-density vector ®eld in units of 2B . The orbital part of Qh is given by * + X ir0 ^ h QL h pj exp 2i h rj ; 2h j

where pj is the momentum of the electrons. If the current density vector ®eld is de®ned by * + e X jr pj r rj r rj pj ; 8:7:4:9 2m j QL h can be expressed as ir0 ^ h Jh; 8:7:4:10 2h where Jh is the Fourier transform of the current density jr.

725


8:7:4:8

QL h

8. REFINEMENT OF STRUCTURAL PARAMETERS The electrodynamic properties of jr allow it to be written as the sum of a rotational and a nonrotational part: jr = = mL r;

8:7:4:11

where = is a `conduction' component and mL r is an òrbitalmagnetization' density vector ®eld. Substitution of the Fourier transform of (8.7.4.11) into (8.7.4.10) leads in analogy to (8.7.4.5) to ^ QL h r0 h^ ML h h;

8:7:4:12

where ML h is the Fourier transform of mL r. The rotational component = of jr does not contribute to the neutron scattering process. It is therefore possible to write Qh as ^ Qh r0 h^ Mh h;

8:7:4:13

Mh Ms h ML h

8:7:4:14

with being the Fourier transform of the `total' magnetization density vector ®eld, and mr ms r mL r:

8:7:4:15

As Qh is the projection of Mh onto the plane perpendicular to h, there is no magnetic scattering when M is parallel to h. It is clear from (8.7.4.13) that Mh can be de®ned to any vector ®eld Vh parallel to h, i.e. such that h Vh 0. This means that in real space mr is de®ned to any vector ®eld vr such that = vr 0. Therefore, mr is de®ned to an arbitrary gradient. As a result, magnetic neutron scattering cannot lead to a uniquely de®ned orbital magnetization density. However, the de®nition (8.7.4.7) for the spin component is unambiguous. However, the integrated magnetic moment l is determined unambiguously and must thus be identical to the magnetic moment de®ned from the principles of quantum mechanics, as discussed in x8:7:4:5:1:3. Before discussing the analysis of magnetic neutron scattering in terms of spin-density distributions, it is necessary to give a brief description of the quantum-mechanical aspects of magnetization densities.

8.7.4.3.1. Spin-only density at zero temperature Let us consider ®rst an isolated open-shell system, whose orbital momentum is quenched: it is a spin-only magnetism case. ^ s be the spin-magnetization-density operator (in units of Let m 2B ): P ^s m 8:7:4:16 r^ j r rj : j

rj and r^ j are, respectively, the position and the spin operator (in h units) of the jth electron. This de®nition is consistent with (8.7.4.7). The system is assumed to be at zero temperature, under an applied ®eld, the quantization axis being Oz. The ground state is an eigenstate of S^ 2 and S^ z , where S^ is the total spin: P 8:7:4:17 r^ j : S^ j

Let S and Ms be the eigenvalues of S^ 2 and S^ z . (Ms will in general be ®xed by Hund's rule: MS S.) n# ;

mSz r is proportional to the normalized spin density that was de®ned for a pure state in (8.7.2.10). mSz r MS sr:

If " r and # r are the charge densities of electrons of a given spin, the normalized spin density is de®ned as sr " r

# r

1 n"

n#

;

8:7:4:21

compared with the total charge density r given by r " r # r:

8:7:4:22

A strong complementarity is thus expected from joint studies of r and sr. In the particular case of an independent electron model, r

N P i1

j'i rj2

"; #;

8:7:4:23

where 'i r is an occupied orbital for a given spin state of the electron. If the ground state is described by a correlated electron model (mixture of different con®gurations), the one-particle reduced density matrix can still be analysed in terms of its eigenvectors i and eigenvalues ni (natural spin orbitals and natural occupancies), as described by the expression r

1 P i1

ni j

i rj

2

;

8:7:4:24

where h i j j i ij , since the natural spin orbitals form an orthonormal set, and ni 1

1 P i1

ni n :

8:7:4:18

^ ms r Ssr:

8:7:4:25

8:7:4:26

Equation (8.7.4.26) expresses the proportionality of the spinmagnetization density to the normalized spin density function. 8.7.4.3.2. Thermally density

averaged

spin-only

magnetization

The system is now assumed to be at a given temperature T. S remains a good quantum number, but all SMS states MS S; . . . ; S are now populated according to Boltzmann statistics. We are interested in the thermal equilibrium spinmagnetization density: ms r

S P MS S

pMS h

^ sj SMS jm

SMS i;

8:7:4:27

^s where pMS is the population of the MS state. The operator m ful®ls the requirements to satisfy the Wigner±Eckart theorem (Condon & Shortley, 1935), which states that, within the S ^ The ^ s are proportional to S. manifold, all matrix elements of m consequence of this remarkable property is that

726


8:7:4:20

As the quantization axis is arbitrary, (8.7.4.20) can be generalized to

8.7.4.3. Magnetization densities and spin densities

2MS n"

where n" and n# are the numbers of electrons with ": 12 and #: 12 spin, respectively. The spin-magnetization density is along z, and is given by * + P mSz r 8:7:4:19 r^ r rj SMS : SMS j jz

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES h

^ sj SMS jm

SMS i

h

^

SMS jSj

SMS i fS r;

8:7:4:28

where fS r is a function that depends on S, but not on MS . Comparison with (8.7.4.26) shows that fS r is the normalized spin-density function sr, which therefore is an invariant for the S manifold sr is calculated as the normalized spin density for any MS . Expression (8.7.4.27) can thus be written as ms r hSisr;

8:7:4:29

where hSi is the expected value for the total spin, at a given temperature and under a given external ®eld. As sr is normalized, the total moment of the system is lS hSi: The behaviour of hSi is governed by the usual laws of magnetism: it can be measured by macroscopic techniques. In paramagnetic species, it will vary as T 1 to a ®rst approximation; unless the system is studied at very low temperatures, the value of hSi will be very small. The dependence of hSi on temperature and orienting ®eld is crucial. Finally, (8.7.4.29) has to be averaged over vibrational modes. Except for the case where there is strong magneto-vibrational interaction, only sr is affected by thermal atomic motion. This effect can be described in terms similar to those used for the charge density (Subsection 8.7.3.7). The expression (8.7.4.29) is very important and shows that the microscopic spin-magnetization density carries two types of information: the nature of spin ordering in the system, described by hSi, and the delocalized nature of the electronic ground state, represented by sr. 8.7.4.3.3. Spin density for an assembly of localized systems A complex magnetic system can generally be described as an ensemble of well de®ned interacting open-shell subsystems (ions or radicals), where each subsystem has a spin S^ n , and Sn2 is assumed to be a good quantum number. The magnetic interaction occurs essentially through exchange mechanisms that can be described by the Heisenberg Hamiltonian: P P * Jnm S^ n S^ m B0 Sn ; 8:7:4:30 n<m

n

where Jnm is the exchange coupling between two subsystems, and B0 an applied external ®eld (magneto-crystalline anisotropic effects may have to be added). Expression (8.7.4.30) is the basis for the understanding of magnetic ordering and phase diagrams. The interactions lead to a local ®eld Bn , which is the effective orienting ®eld for the spin Sn . The expression for the spin-magnetization density is P ms r hSn isn r: 8:7:4:31 n

The relative arrangement of hSn i describes the magnetic structure; sn r is the normalized spin density of the nth subsystem. In some metallic systems, at least part of the unpaired electron system cannot be described within a localized model: a bandstructure description has to be used (Lovesey, 1984). This is the case for transition metals like Ni, where the spin-magnetization density is written as the sum of a localized part [described by (8.7.4.31)] and a delocalized part [described by (8.7.4.29)].

description of the magnetization density becomes less straightforward. The magnetic moment due to the angular momentum lj of the electron is 12 lj (in units of 2B ). As lj does not commute with the position rj , orbital magnetization density is de®ned as * + P lj r rj r rj lk : mL r 14 8:7:4:32 j

If L is the total orbital moment, P lj : L

Only open shells contribute to the orbital moment. But, in general, neither L2 nor Lz are constants of motion. There is, however, an important exception, when open-shell electrons can be described as localized around atomic centres. This is the case for most rare-earth compounds, for which the 4f electrons are too close to the nuclei to lead to a signi®cant interatomic overlap. It can also be a ®rst approximation for the d electrons in transition-metal ions. Spin-orbit coupling will be present, and thus only L2 will be a constant of motion. One may de®ne the total angular momentum JLS

8:7:4:34

and fJ 2 ; L2 ; S 2 ; Jz g become the four constants of motion. Within the J manifold of the ground state, ms r and mL r do not, in general, ful®l the conditions for the Wigner±Eckart theorem (Condon & Shortley, 1935), which leads to a very complex description of mr in practical cases. However, the Wigner±Eckart theorem can be applied to the magnetic moments themselves, leading to L 2S gJ;

8:7:4:35

with the Lande factor g1

JJ 1

LL 1 SS 1 2JJ 1

8:7:4:36

and, equivalently, S g

1J

L 2

gJ:

8:7:4:37

The in¯uence of spin±orbit coupling on the scattering will be discussed in Subsection 8.7.4.5. 8.7.4.4. Probing spin densities by neutron elastic scattering 8.7.4.4.1. Introduction The magnetic structure factor FM h [equation (8.7.4.4)] depends on the spin state of the neutron. Let k be the unit vector de®ning a quantization axis for the neutron, which can be either parallel " or antiparallel # to r. If I0 stands for the cross section where the incident neutron has the polarization and the scattered neutron the polarization 0 , one obtains the following basic expressions: I"" jFn k Qj2 I## jFn

k Qj2

8:7:4:38 2

I"# I#" jk Qj :

8.7.4.3.4. Orbital magnetization density We must now address the case where the orbital moment is not quenched. In that case, there is some spin-orbit coupling, and the

If no analysis of the spin state of the scattered beam is made, the two measurable cross sections are

727


8:7:4:33

j

8. REFINEMENT OF STRUCTURAL PARAMETERS I" I"" I"# I# I## I#" ;

R 1 4x sin2 :

8.7.4.4.2. Unpolarized neutron scattering If the incident neutron beam is not polarized, the scattering cross section is given by 8:7:4:40

Magnetic and nuclear contributions are simply additive. With x Q=FN , one obtains I jFN j2 1 jxj2 :

8:7:4:41

Owing to its de®nition, jxj can be of the order of 1 if and only if the atomic moments are ordered close to saturation (as in the ferro- or antiferromagnets). In many situations of structural and chemical interest, jxj is small. If, for example, jxj 0:05, the magnetic contribution in (8.7.4.41) is only 0.002 of the total intensity. Weak magnetic effects, such as occur for instance in paramagnets, are thus hardly detectable with unpolarized neutron scattering. However, if the magnetic structure does not have the same periodicity as the crystalline structure, magnetic components in (8.7.4.40) occur at scattering vectors for which the nuclear contribution is zero. In this case, the unpolarized technique is of unique interest. Most phase diagrams involving antiferromagnetic or helimagnetic order and modulations of such ordering are obtained by this method. 8.7.4.4.3. Polarized neutron scattering It is generally possible to polarize the incident beam by using as a monochromator a ferromagnetic alloy, for which at a given Bragg angle I# monochromator 0, because of a cancellation of nuclear and magnetic scattering components. The scatteredbeam intensity is thus I" . By using a radio-frequency (r.f.) coil tuned to the Larmor frequency of the neutron, the neutron spin can be ¯ipped into the # state for which the scattered beam intensity is I# . This allows measurement of the `¯ipping ratio' Rh: Rh

I" h : I# h

8:7:4:42

As the two measurements are made under similar conditions, most systematic effects are eliminated by this technique, which is only applicable to cases where both FN and FM occur at the same scattering vectors. This excludes any antiferromagnetic type of ordering. The experimental set-up is discussed by Forsyth (1980). 8.7.4.4.4. Polarized neutron scattering of centrosymmetric crystals If k is assumed to be in the vertical Oz direction, Mh will in most situations be aligned along Oz by an external orienting ®eld. If is the angle between M and h, and x

r0 Mh ; FN h

8:7:4:43

with FN expressed in the same units as r0 , one obtains, for centrosymmetric crystals,

8:7:4:44

8:7:4:45

For x 0:05 and =2, R now departs from 1 by as much as 20%, which proves the enormous advantage of polarized neutron scattering in the case of low magnetism. Equation (8.7.4.44) can be inverted, and x and its sign can be obtained directly from the observation. However, in order to obtain Mh, the nuclear structure factor FN h must be known, either from nuclear scattering or from a calculation. All systematic errors that affect FN h are transferred to Mh. For two reasons, it is not in general feasible to access all reciprocal-lattice vectors. First, in order to have reasonable statistical accuracy, only re¯ections for which both I" and I# are large enough are measured; i.e. re¯ections having a strong nuclear structure factor. Secondly, sin should be as close to 1 as possible, which may prevent one from accessing all directions in reciprocal space. If M is oriented along the vertical axis, the simplest experiment consists of recording re¯ections with h in the horizontal plane, which leads to a projection of mr in real space. When possible, the sample is rotated so that other planes in the reciprocal space can be recorded. Finally, if =2, I"# vanishes, and neutron spin is conserved in the experiment. 8.7.4.4.5. Polarized neutron scattering in the noncentrosymmetric case If the space group is noncentrosymmetric, both FN and M have a phase, 'N and 'M , respectively. If for simplicity one assumes =2, and, de®ning 'M 'N , R

1 jxj2 2jxj cos ; 1 jxj2 2jxj cos

8:7:4:46

which shows that jxj and cannot both be obtained from the experiment. The noncentrosymmetric case can only be solved by a careful modelling of the magnetic structure factor as described in Subsection 8.7.4.5. In practice, neither the polarization of the incident beam nor the ef®ciency of the r.f. ¯ipping coil is perfect. This leads to a modi®cation in the expression for the ¯ipping ratios [see Section 6.1.3 or Forsyth (1980)]. 8.7.4.4.6. Effect of extinction Since most measurements correspond to strong nuclear structure factors, extinction severely affects the observed data. To a ®rst approximation, one may assume that both I"" and I## will be affected by this process, though the spin-¯ip processes I"# and I#" are not. If y"" and y## are the associated extinction factors, the observed ¯ipping ratio is Robs

I"" y"" I"# ; I## y## I"#

8:7:4:47

where the expressions for y"";## are given elsewhere (Bonnet, Delapalme, Becker & Fuess, 1976). It should be emphasized that, even in the case of small magnetic structure factors, extinction remains a serious problem since, even though y"" and y## may be very close to each other,

728


1 2x sin2 x2 sin2 : 1 2x sin2 x2 sin2

If x 1,

which depend only on the polarization of the incident neutron.

I 12 I" I# jFN j2 jQj2 :

R

8:7:4:39

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES P mS r hSj i hsj r

so are I"" and I## . An incorrect treatment of extinction may entirely bias the estimate of x. 8.7.4.4.7. Error analysis

In the most general case, it is not possible to obtain x, and thus Mh directly from R. Moreover, it is unlikely that all Bragg spots within the re¯ection sphere could be measured. Modelling of Mh is thus of crucial importance. The analysis of data must proceed through a least-squares routine ®tting Rcalc to Robs , minimizing the error function X 1 Robs h Rcalc h2 ; " 8:7:4:48 2 R h observed

where Rcalc corresponds to a model and 2 R is the standard uncertainty for R. If the same counting time for I" and for I# is assumed, only the counting statistical error may be considered important in the estimate of R, as most systematic effects cancel. In the simple case where =2, and the structure is centrosymmetric, a straightforward calculation leads to 2 x 2 R R ; x2 R2 R 12

8:7:4:49

R 1 1 ; R2 I" I#

8:7:4:50

2 x 1 FN2 M 2 8 : x2 FN M2

8:7:4:51

one obtains the result

In the common case where x 1, this reduces to 2 x 1 1 1 1 8 2 2 2: 2 x M 8FN x

8:7:4:52

In addition to this estimate, care should be taken of extinction effects. The real interest is in Mh, rather than x: 2 M 2 x 2 FN 2 : M2 x FN2

8:7:4:53

If FN is obtained by a nuclear neutron scattering experiment, 2 FN a bFN2 ; where a accounts for counting statistics and b for systematic effects. The ®rst term in (8.7.4.53) is the leading one in many situations. Any systematic error in FN can have a dramatic effect on the estimate of Mh. 8.7.4.5. Modelling the spin density In this subsection, the case of spin-only magnetization is considered. The modelling of ms r is very similar to that of the charge density. 8.7.4.5.1. Atom-centred expansion We ®rst consider the case where spins are localized on atoms or ions, as it is to a ®rst approximation for compounds involving transition-metal atoms. The magnetization density is expanded as

8:7:4:54

where hSj i is the spin at site j, and hsj i the thermally averaged normalized spin density fj h, the Fourier transform of sj r, is known as the `magnetic form factor'. Thus, P 8:7:4:55 Mh hSj i fj h Tj exp 2h Rj ; j

where Tj and Rj are the Debye±Waller factor and the equilibrium position of the jth site, respectively. Most measurements are performed at temperatures low enough to ensure a fair description of Tj at the harmonic level (Coppens, 1992). Tj represents the vibrational relaxation of the open-shell electrons and may, in some situations, be different from the Debye±Waller factor of the total charge density, though at present no experimental evidence to this effect is available. 8.7.4.5.1.1. Spherical-atom model In the crudest model, sj r is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function Ur, the magnetic form factor is given by R1 f h U 2 r4r 2 dr j0 2hr h j0 i; 8:7:4:56 where j0 is the zero-order spherical Bessel function. For free atoms and ions, these form factors can be found in IT IV (1974). One of the important features of magnetic neutron scattering is the fact that, to a ®rst approximation, closed shells do not contribute to the form factor. Thus, it is a unique probe of the electronic structure of heavy elements, for which theoretical calculations even at the atomic level are questionable. Relativistic effects are important. Theoretical relativistic form factors can be used (Freeman & Desclaux, 1972; Desclaux & Freeman, 1978). It is also possible to parametrize the radial behaviour of U. A single contraction-expansion model [ re®nement, expression (8.7.3.6)] is easy to incorporate. 8.7.4.5.1.2. Crystal-®eld approximation Crystal-®eld effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state con®guration of the system. Thus, they have to be incorporated in the model. Taking the case of a transition-metal compound, and neglecting small contributions that may arise from spin polarization in the closed shells (see Subsections 8.7.4.9 and 8.7.4.10), the normalized spin density can be written by analogy with (8.7.3.76) as sr

5 P 5 P i1 ji

Dij di r dj r;

8:7:4:57

where Dij is the normalized spin population matrix. If d" and d# are the densities of a given spin, d" d# sr ; 8:7:4:58 n " n# the d-type charge density is d r d" d#

8:7:4:59

and is expanded in a similar way to sr [see (8.7.3.76)], PP Pij di dj ; 8:7:4:60 d r i

writing

729


Rj i;

0

with 2

j

ji

d

P

8. REFINEMENT OF STRUCTURAL PARAMETERS Pij

di dj

8:7:4:61

with " and #, one obtains Pij Pij" Pij# Dij Pij"

Pij# =n"

n# :

8:7:4:62

n

Similarly to d r, sr can be expanded as sr U 2 r

4 P

l P

P

l0 m0

p

Dlmp ylmp ; ';

8:7:4:63

where Ur describes the radial dependence. Spin polarization leads to a further modi®cation of this expression. Since the numbers of electrons of a given spin are different, the exchange interaction is different for the two spin states, and a spindependent effective screening occurs. This leads to P d r 3 U 2 r Plmp ylmp ; 8:7:4:64 lmp

with " or #, where two parameters are needed. The complementarity between charge and spin density in the crystal ®eld approximation is obvious. At this particular level of approximation, expansions are exact and it is possible to estimate d-orbital populations for each spin state. 8.7.4.5.1.3. Scaling of the spin density The magnetic structure factor is scaled to FN h. Whether the nuclear structure factors are calculated from re®ned structural parameters or obtained directly from a measurement, their scale factor is not rigorously ®xed. As a result, it is not possible to obtain absolute values of the effective spins hSn i from a magnetic neutron scattering experiment. It is necessary to scale them through the sum rule P hSn i l; 8:7:4:65 n

where l is the macroscopic magnetization of the sample. The practical consequences of this constraint for a re®nement are similar to the electroneutrality constraint in charge-density analysis x8:7:3:3:1. 8.7.4.5.2. General multipolar expansion In this subsection, the localized magnetism picture is assumed to be valid. However, each subunit can now be a complex ion or a radical. Covalent interactions must be incorporated. If r are atomic basis functions, the spin density sr can always be written as PP sr D r R r R ; 8:7:4:66

an expansion that is similar to (8.7.3.9) for the charge density. To a ®rst approximation, only the basis functions that are required to describe the open shell have to be incorporated, which makes the expansion simple and more ¯exible than for the charge density. As for the charge density, it is possible to project (8.7.4.66) onto the various sites of the ion or molecule by a multipolar expansion: P sr sj r Rj Pj Rj ; j

sj r

P j

3j Rlj j r

PP m

p

Djlmp ylmp :

The monopolar terms Dj00 give an estimate of the amount of spin transferred from a central metal ion to the ligands, or of the way spins are shared among the atoms in a radical. The constraint (8.7.4.65) becomes P P n hSn i Dj00 l; 8:7:4:68 where n refers to the various subunits in the unit cell. Expansion (8.7.4.67) is the key for solving noncentrosymmetric magnetic structure (Boucherle, Gillon, Maruani & Schweizer, 1982). 8.7.4.5.3. Other types of model One may wish to take advantage of the fact that, to a good approximation, only a few molecular orbitals are involved in sr. In an independent particle model, one expands the relevant orbitals in terms of atomic basis functions (LCAO): P 'i ci r R 8:7:4:69

with " or #, and the spin density is expanded according to (8.7.4.21) and (8.7.4.23). Fourier transform of two centre-term products is required. Details can be found in Forsyth (1980) and To®eld (1975). In the case of extended solids, expansion (8.7.4.69) must refer to the total crystal, and therefore incorporate translational symmetry (Brown, 1986). Finally, in the simple systems such as transition metals, like Ni, there is a d±s type of interaction, leading to some contribution to the spin density from delocalized electrons (Mook, 1966). If sl and sd are the localized and delocalized parts of the density, respectively, sr asl r 1

8:7:4:70

8.7.4.6. Orbital contribution to the magnetic scattering QL h is given by (8.7.4.10) and (8.7.4.12). Since = in (8.7.4.11) does not play any role in the scattering cross section, we can use the restriction jr = mL r;

8:7:4:71

where mL r is de®ned to an arbitrary gradient. It is possible to constrain mL r to have the form mL r r^ vr; since any radial component could be considered as the radial component of a gradient. With spherical coordinates, (8.7.4.71) becomes 1 @ rv; r @r

j which can be integrated as rv

R1 r

y^r jy^r dy;

one ®nally obtains 1 mL r r With f x de®ned by

730


asd r;

where a is the fraction of localized spins. sd r can be modelled as being either constant or a function with a very small number of Fourier adjustable coef®cients.

8:7:4:67

PRj is the vibrational p.d.f. of the jth atom, which implies use of the convolution approximation.

j

y1 Z

r^ y j^ry dy: yr

8:7:4:72

1 x2

f x

8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES P MS MS;n h exp 2ih Rn ; n tlt dt; 8:7:4:73 with

Zix 0

the expression for ML h is obtained by Fourier transformation of (8.7.4.72): R ML h 12 r jr f h r dr; which leads to * ML h 14

P j

lj f 2h rj f 2h rj lj

+ 8:7:4:74

by using the de®nition (8.7.4.9) of j. This expression clearly shows the connection between orbital magnetism and the orbital angular momentum of the electrons. It is of general validity, whatever the origin of orbital magnetism. Since f 0 1, ML 0 12 hLi;

8:7:4:75

as expected. 8.7.4.6.1. The dipolar approximation The simplest approximation involves decomposing jr into atomic contributions: P jn r Rn : 8:7:4:76 jr n

One obtains ML

P n

ML;n h exp 2ih Rn :

m

l

where 2

l x 2 x

Zx tjl t dt:

8:7:4:79

0

jl is a spherical Bessel function of order l, and Ylm are the complex spherical harmonic functions. If one considers only the spherically symmetric term in (8.7.4.78), one obtains the `dipolar approximation', which gives MDL;n

1 2 hLn ih 0n i;

8:7:4:80

with h 0n i

R1 0

4r 2 Un2 r 0 2hr dr;

and h j0n i

2 1 cos x: 8:7:4:81 x2 Un r is the radial function of the atomic electrons whose orbital momentum is unquenched. Thus, in the dipolar approximation, the atomic orbital scattering is proportional to the effective orbital angular momentum and therefore to the orbital part of the magnetic dipole moment of the atom. Within the same level of approximation, the spin structure factor is

0

4r 2 Un2 r j0 2hr dr:

Finally, the atomic contribution to the total magnetic structure factor is MDn hSn ih j0n i 12 hLn ih 0n i:

8:7:4:83

If hJn i is the total angular momentum of atom n and gn its gyromagnetic ratio, (8.7.4.35)±(8.7.4.37) lead to: 2 gn D Mn hJn i gn 1h j0n i h 0n i : 8:7:4:84a 2 Another approach, which is applicable only to the atomic case, is often used, which is based on Racah's algebra (Marshall & Lovesey, 1971). At the dipolar approximation level, it leads to a slightly different result, according to which h 0n i is replaced by h 0n i h j0n i h j2n i:

8:7:4:85

The two results are very close for small h where the dipolar approximation is correct. With (8.7.4.35)±(8.7.4.37), (8.7.4.84a) can also be written as MDn MDS;n hSn i

2 gn h i; 2gn 1 0n

8:7:4:84b

where the second term is the òrbital correction'. Its magnitude clearly depends on the difference between gn and 2, which is small in 3d elements but can become important for rare earths. 8.7.4.6.2. Beyond the dipolar approximation Expressions (8.7.4.74) and (8.7.4.78) are valid in any situation where orbital scattering occurs. They can in principle be used to estimate from the diffraction experiment the contribution of a few con®gurations that interact due to the L S operator. In delocalized situations, (8.7.4.74) is the most suitable approach, while Racah's algebra can only be applied to one-centre cases. 8.7.4.6.3. Electronic structure of rare-earth elements When covalency is small, the major aims are the determination of the ground state of the rare-earth ion, and the amount of delocalized magnetization density via the conduction electrons. The ground state j i of the ion is written as P j i aM jJMi; 8:7:4:86 which is well suited for the Johnston (1966) and Marshall± Lovesey (1971) formulation in terms of general angularmomentum algebra. A multipolar expansion of spin and orbital components of the structure factor enables a determination of the expansion coef®cient aM (Schweizer, 1980). 8.7.4.7. Properties derivable from spin densities The derivation of electrostatic properties from the charge density was treated in Subsection 8.7.3.4. Magnetostatic properties can be derived from the spin-magnetization density ms r using parallel expressions.

731


R1

M

and

0 x

MS;n h hSn ih j0n i;

8:7:4:77

ML;n h is the atomic magnetic orbital structure factor. We notice that f 2h r as de®ned in (8.7.4.73) can be expanded as PP l ^ f 2h r 4 i l 2hr Ylm ^r Ylm h; 8:7:4:78

8:7:4:82

8. REFINEMENT OF STRUCTURAL PARAMETERS 8.7.4.7.1. Vector ®elds

8.7.4.8. Comparison between theory and experiment

The vector potential ®eld is de®ned as Z ms r0 r r0 0 Ar dr : jr r0 j3

Since it is a measure of the imbalance between the densities associated with the two spin states of the electron, the spindensity function is a probe that is very sensitive to the exchange forces in the system. In an independent-particle model (Hartree± Fock approximation), the exchange mean ®eld potential involves exchange between orbitals with the same spin. Therefore, if the numbers of " and # spins are different, one expects Vexch" to be different from Vexch#. The main consequence of this is the necessity to solve two different Fock equations, one for each spin state. This is known as the spin-polarization effect: starting from a paired orbital, a slight spatial decoupling arises from this effect, and closed shells do have a participation in the spin density. It can be shown that this effect is hardly visible in the charge density, but is enhanced in the spin density. Spin densities are a very good probe for calculations involving this spin-polarization effect: The unrestricted Hartree±Fock approximation (Gillon, Becker & Ellinger, 1983). From a common spin-restricted approach, spin polarization can be accounted for by a mixture of Slater determinants (con®guration interaction), where the con®guration interaction is only among electrons with the same spin. There is also a correlation among electrons with different spins, which is more dif®cult to describe theoretically. There seems to be evidence for such effects from comparison of experimental and theoretical spin densities in radicals (Delley, Becker & Gillon, 1984), where the unrestricted Hartree±Fock approximation is not suf®cient to reproduce experimental facts. In such cases, local-spin-density functional theory has revealed itself very satisfactorily. It seems to offer the most ef®cient way to include correlation effects in spin-density functions. As noted earlier, analysis of the spin-density function depends more on modelling than that of the charge density. Therefore, in general, èxperimental' spin densities at static densities and the problem of theoretical averaging is minor here. Since spin density involves essentially outer-electron states, resolution in reciprocal space is less important, except for analysis of the polarization of the core electrons.

8:7:4:87

In the case of a crystal, it can be expanded in Fourier series: 2i X MS h h exp 2ih r; 8:7:4:88 Ar V h h2 the magnetic ®eld is simply Br = Ar 4 X h MS h h exp 2ih r: V h h2

8:7:4:89

One notices that there is no convergence problem for the h 0 term in the Br expansion. The magnetostatic energy, i.e. the amount of energy that is required to obtain the magnetization ms , is R Ems 12 ms r Br dr cell

2 X Ms h h MS h h : V h h2

8:7:4:90

It is often interesting to look at the magnetostatics of a given subunit: for instance, in the case of paramagnetic species. For example, the vector potential outside the magnetized system can be obtained in a similar way to the electrostatic potential (8.7.3.30): Z = mS r Ar0 dr: 8:7:4:91 jr r0 j If r0 r; 1=jr r0 j can be easily expanded in powers of 1=r 0 , and Ar0 can thus be obtained in powers of 1=r 0 . If ms r hSisr, Z =sr 0 dr: 8:7:4:92 Ar hSi jr r0 j 8.7.4.7.2. Moments of the magnetization density Among the various properties that are derivable from the delocalized spin density function, the dipole coupling tensor is of particular importance: Z 3 rni rn j rn2 i j Dnij Rn sr dr; 8:7:4:93 rn5 where Rn is a nuclear position and rn r Rn . This dipolar tensor is involved directly in the hyper®ne interaction between a nucleus with spin In and an electronic system with spin s, through the interaction energy P Ini Dnij Sj : 8:7:4:94 i;j

This tensor is measurable by electron spin resonance for either crystals or paramagnetic species trapped in matrices. The complementarity with scattering is thus of strong importance (Gillon, Becker & Ellinger, 1983). Computational aspects are the same as in the electric ®eld gradient calculation, r being simply replaced by sr (see Subsection 8.7.3.4).

8.7.4.9. Combined charge- and spin-density analysis Combined charge- and spin-density analysis requires performing X-ray and neutron diffraction experiments at the same temperature. Magnetic neutron experiments are often only feasible around 4 K, and such conditions are more dif®cult to achieve by X-ray diffraction. Even if the two experiments are to be performed at different temperatures, it is often dif®cult to identify compounds suitable for both experiments. Owing to the common parametrization of r and sr, a combined least-squares-re®nement procedure can be implemented, leading to a description of " r and # r, the spindependent electron densities. Covalency parameters are obtainable together with spin polarization effects in the closed shells, by allowing " and # to have different radial behaviour. Spin-polarization effects would be dif®cult to model from the spin density alone. But the arbitrariness of the modelling is strongly reduced if both and s are analysed at the same time (Becker & Coppens, 1985; Coppens, Koritszansky & Becker, 1986).

732


8.7. ANALYSIS OF CHARGE AND SPIN DENSITIES 8.7.4.10. Magnetic X-ray scattering separation between spin and orbital magnetism

It should also be pointed out that FM is in quadrature with FC . In many situations, the total X-ray intensity is therefore

8.7.4.10.1. Introduction

Ix jFC j2 jFM j2 :

In addition to the usual Thomson scattering (charge scattering), there is a magnetic contribution to the X-ray amplitude (de Bergevin & Brunel, 1981; Blume, 1985; Brunel & de Bergvin, 1981; Blume & Gibbs, 1988). In units of the chemical radius re of the electron, the total scattering amplitude is Ax F C F M ;

8:7:4:95

where FC is the charge contribution, and FM the magnetic part. Let e^ and e^ 0 be the unit vectors along the electric ®eld in the incident and diffracted direction, respectively. k and k0 denote the wavevectors for the incident and diffracted beams. With these notations, FC Fh e^ e^ 0 ;

8:7:4:96

where Fh is the usual structure factor, which was discussed in Section 8.7.3 [see also Coppens (1992)]. h! FM i fML h A MS h Bg: 8:7:4:97 mc2 ML and MS are the orbital and spin-magnetization vectors in reciprocal space, and A and B are vectors that depend in a rather complicated way on the polarization and the scattering geometry: k^ e^ k^ e^ 0 k^ 0 e^ 0 k^ 0 e^ B e^ 0 e^ k0 e0 k e k^ e^ k^ e^ 0

A 4 sin2 e^ e^ 0

k^ 0 e^ 0 k^ e^ :

Thus, under these conditions, the magnetic effect is typically 10 6 times the X-ray intensity. Magnetic contributions can be detected if magnetic and charge scattering occur at different positions (antiferromagnetic type of ordering). Furthermore, Blume (1985) has pointed out that the photon counting rate for jFM j2 at synchrotron sources is of the same order as the neutron rate at high-¯ux reactors. Finally, situations where the ìnterference' FC FM term is present in the intensity are very interesting, since the magnetic contribution becomes 10 3 times the charge scattering. The polarization dependence will now be discussed in more detail. 8.7.4.10.2. Magnetic X-ray structure factor as a function of photon polarization Some geometrical de®nitions are summarized in Fig. 8.7.4.1, where parallel k and perpendicular ? polarizations will be chosen in order to describe the electric ®eld of the incident and diffracted beams. In this two-dimensional basis, vectors A and B of (8.7.4.98) can be written as 2 2 matrices: ! ? 0 k^ k^ 0 2 A sin k k^ k^ 0 2k^ k^ 0 i!

8:7:4:98

For comparison, the magnetic neutron scattering amplitude can be written in the form 8:7:4:99 FMneutron ML h MS h C; ^ with C h^ r h. From (8.7.4.99), it is clear that spin and orbital contributions cannot be separated by neutron scattering. In contrast, the polarization dependencies of ML and MS are different in the X-ray case. Therefore, owing to the well de®ned polarization of synchrotron radiation, it is in principle possible to separate experimentally spin and orbital magnetization. However, the prefactor h!=mc2 10 2 makes the magnetic contributions weak relative to charge scattering. Moreover, FC is roughly proportional to the total number of electrons, and FM to the number of unpaired electrons. As a result, one expects jFM =FC j to be about 10 3 .

?

k

"f

(i and f refer to the incident and diffracted beams, respectively); k^ k^ 0 2k^ 0 sin2 B : 8:7:4:100 2k^ sin2 k^ k^ 0 By comparison, for the Thomson scattering, 1 0 e^ e^ 0 : 8:7:4:101 0 cos 2 The major difference with Thomson scattering is the occurrence of off-diagonal terms, which correspond to scattering processes with a change of polarization. We obtain for the structure factors Aif : h! ^ ^ 0 k k MS mc2 h! A?k 2i 2 sin2 fk^ k^ 0 ML k^ 0 MS g mc 8:7:4:102 h! 0 0 ^ ^ ^ Ak? 2i 2 sin 2fk k ML k MS g mc h! Akk F cos2 i 2 k^ k^ 0 f4 sin2 ML MS g: mc For a linear polarization, the measured intensity in the absence of diffracted-beam polarization analysis is A?? F

i

I jA?? cos Ak? sin j2 jA?k cos Akk sin j2 ; 8:7:4:103

Fig. 8.7.4.1. Some geometrical de®nitions.

where is the angle between E and e^ . In the centrosymmetric system, without anomalous scattering, no interference term occurs in (8.7.4.103). However, if anomalous scattering is present, F F 0 iF 00 , and terms involving F 00 MS or F 00 ML appear in the intensity expression. The radiation emitted in the plane of the electron or positron orbit is linearly polarized. The experimental geometry is 733


8. REFINEMENT OF STRUCTURAL PARAMETERS ^ k^ 0 is a vertical plane. Therefore, the generally such that k; E?0 A?? iAk? 8:7:4:106 polarization of the incident beam is along e^ 0. If a Ek0 Ak? iAkk : diffracted-beam analyser passes only k components of the diffracted beam, one can measure jA?k j2 , and thus eliminate In this case, `mixed-polarization' contributions are in phase with the charge scattering. For non-polarized radiation (with a rotating anode, for F, leading to a strong interference between charge and magnetic scattering. example), the intensity is The case of radiation with a general type of polarization is more dif®cult to analyse. The most elegant formulation involves 2 2 2 2 1 8:7:4:104 I 2 jAkk j jAk? j jA?k j jA?? j : Stokes vectors to represent the state of polarization of the incident and scattered radiation (see Blume & Gibbs, 1988). The radiation emitted out of the plane of the orbit contains an increasing amount of circularly polarized radiation. There also exist experimental devices that can produce circularly polarized radiation. For such incident radiation, Ek iE?

8:7:4:105

for left- or right-polarized photons. If E0 is the ®eld for the diffracted photons,

Acknowledgements Support of this work by the National Science Foundation (CHE8711736 and CHE9615586) is gratefully acknowledged.

734



8.8. Accurate structure-factor determination with electron diffraction By J. Gjùnnes

Several techniques have been developed for accurate determination of structure factors with electrons, mostly based on convergent-beam electron diffraction (CBED) and interpreted by dynamical scattering theory. Intensities are measured as one- or two-dimensional rocking curves, or as special intensity features in the CBED patterns, e.g. critical voltages. The main application of these methods so far has been to strong inner re¯ections from simple structures in relation to bonding effects, occupation numbers or ordering. Low-order re¯ections in electron diffraction are particularly sensitive to details in the distribution of valence electrons, through the difference Z f X in the expression for the atomic scattering amplitude f el 2

2 Z fX ; aH 2q2

8:8:1

where aH h2 =m0 e2 is the ®rst Bohr radius; the diffraction variable q 2 sin =l, and f X is the X-ray scattering factor for atomic number Z. A further advantage is that these methods can provide absolute measurement of structure factors, i.e. without a scaling procedure. The limitation to inner re¯ections may be overcome by measurement of intensities integrated across Kossel-line features in CBED patterns (Vincent, Bird & Steeds, 1984; Gjùnnes & Bùe, 1994; Tsuda & Tanaka, 1995; Tomokiyo & Kuroiwa, 1990). As distinct from the X-ray case where scattering from one electron is a convenient unit, commonly agreed upon for the atomic scattering factor as well as for the amplitude of the structure factor, there are several units and de®nitions possible in electron diffraction; see Cowley (1992), Spence & Zuo (1992). The scattering amplitude associated with the unit cell may have the same unit as the atomic scattering factor, i.e. length. The structure factor (or Fourier potential) is usually de®ned as the Fourier component, Ug , of the scattering potential that appears in the wave equation, and is expressed either in volts, as an SI unit, or in reciprocal-length squared. The latter unit follows from the common way of writing the SchroÈdinger equation in scattering theory, viz frr2 42 k2 42 Urg r 0;

charges, or the average extent of the electron clouds. For spherical neutral atoms, we obtain from (1) in the limit q ! 0 f el 0

where is the volume of the unit cell, m is the relativistic electron mass corresponding to the accelerating voltage and exp M the Debye±Waller factor. The translation to volts is Ê 2 . Results are usually given by Ug (volts) 0:00665Ug A quoted for the rest mass m0 , but in scattering calculations the factor m=m0 must be included. For comparison with theoretical calculations of charge distributions, the electron results are usually transformed to X-ray scattering factors. The structure factor associated with forward scattering has a special meaning, quite different from the X-ray case where F0X is a measure of the number of electrons. In electron diffraction, U0 is the mean inner potential ± a measure of the average screening of the nuclear

i


PP i6j

j j i i C i 0 C g C 0 C g exp2i

exp i u j t;

j t 8:8:5

where the Bloch-wave coef®cients C jh are eigenvectors, and the Anpassungen j are eigenvalues obtained by the diagonalization. The absorption coef®cients j for the Bloch waves j can be calculated from the imaginary potential U 0 r by a perturbation procedure or by non-Hermitian diagonalization (Bird, 1990). U 0 r describes the spatial variation within the unit cell of the diffuse scattering power, derived mainly from thermal scattering (Yoshioka & Kainuma, 1962). Calculations are usually based on an Einstein model (Radi, 1970; Bird & King, 1990). The CBED patterns are used in measurement strategies based on different beam con®gurations, Fig. 8.8.1, viz a two-beamlike intensity pro®les in systematic rows; b three- or four-beam cases in non-systematic con®gurations; c patterns in dense zones with strong many-beam dynamical interactions. The oneor two-dimensional intensity distributions are ®tted to theoretical calculations by a least-squares procedure with low-order structure factors and certain experimental parameters as free variables.


8:8:4

where the sum is over atomic electrons. For ions, this will diverge, but a limit can still be found by adding contributions from positive and negative ions, in which case the measured inner potential will depend upon the direction of the incident beam. U0 can be measured by interference experiments, e.g. by a biprism. For a review of experiments, see Spence (1993) and Saldin & Spence (1994). Experimental determinations of structure factors Ug have been based on various techniques: thickness fringes in bright-®eld or dark-®eld electron micrographs, Kikuchi patterns, and, in recent years, especially by convergent-beam diffraction. The intensity distribution within the CBED discs can be recorded photographically, with image plates, or by a CCD camera connected to a YAG screen in the microscope ± preferably with an energy ®lter, which will improve the signal-to-noise ratio and facilitate background subtraction (Burgess, Preston, Bolton, Zaluzec & Humphreys, 1994). Alternatives to the parallel recording in CBED may be to scan the pattern over the slit in an EELS system ± or in a modi®ed PEELS as described by Holmestad, Krivanek, Hùier, Marthinsen & Spence (1993). Because of dynamical interactions between beams, the scattered intensity will depend upon several structure factors. Re®nement of structure factors must therefore be based on extensive calculations. These are usually performed in the Blochwave representation, which is the most convenient theoretical basis (in contrast to higher-resolution imaging, where multislice dynamical calculations are commonly used). The diffracted intensity as a function of the thickness t and the diffraction condition, de®ned by the components kx , ky of the incident wave vector, can be expressed as P i Ig t; kx ; ky jC 0 Cg j2 exp 2i t

8:8:2

with the factor 42 introduced in order to conform with crystallographic conventions for reciprocal vectors and wave vectors. In this notation, the structure factor or Fourier potential may appear in units of (length) 2 : m 1 X el Ug f 2 exp Mgj exp2ig rj ; 8:8:3 m0 j

2 X 2 hrj i; 3aH

8. REFINEMENT OF STRUCTURAL PARAMETERS Systematic row. Measurement of s-fringe pro®les in CBED discs from strong inner re¯ections in systematic rows was tried by MacGillavry (1940) and developed into a method for structure-factor determination by Goodman & Lehmpfuhl (1967) and later authors. At present, this may be the commonest method for re®nement of low-order structure factors from CBED. A detailed account is given in the book by Spence & Zuo (1992) and in Spence (1993). Strong non-systematic interactions should be avoided. The intensity pro®les can often be approximated by the two-beam expression q Ug =k2 2 Ig sg 8:8:6 sin t s2g Ug =k2 ; 2 2 s Ug =k especially when Ug is substituted by an èffective potential' which may be de®ned by the corresponding gap at the dispersion surface, viz Ugeff k i j min k= ij , where ij is an extinction distance. The outer part of the pro®le (large sg ) depends mainly on the thickness, whereas the inner part is sensitive to the product tUg . Different perturbation expressions have been proposed for the effective potential. The Bethe potential X Uh Ug h Ugeff Ug 8:8:7 2ksh h is often used, e.g. in the early steps of the re®nement procedure (Gjùnnes, Gjùnnes, Zuo & Spence, 1988), and especially in order to treat weak beams beyond the typically 60±80 beams included in the Bloch-wave diagonalization (Zuo, 1993). Procedures and computer programs adapted to least-squares

Fig. 8.8.1. Schematic representations of four convergent-beam con®gurations used for structure-factor determination: a intensity pro®le of a low-order re¯ection, g; b non-systematic three- or four-beam con®guration with a strong coupling re¯ection, h; c symmetric many-beam con®guration in a dense zone; d integrated intensity measurement of high-order re¯ections using a wide aperture (Taftù & Metzger, 1985).

re®nement of structure factors from energy-®ltered line pro®les are described by Spence (1993), Zuo (1993) and Deininger, Necker & Mayer (1994). The re®nement will usually include experimental parameters (thickness, beam orientations) as well as elastic and absorptive parts of a few low-order structure factors for each pro®le ± but not high-order structure factors and thermal parameters, which are assumed. Low-order structure factors for a number of simple substances have been determined. Errors in the best results, referred to as X-ray structure amplitudes, are of the order of 0.1% ± which may be a tenth of the bonding effect in covalent compounds. See, for example, the recent study of the intermetallic compound TiAl and a variant doped with 5% Mn (Holmestad, Weickenmeier, Zuo, Spence & Horita, 1993), where the charge-density deformation distribution 1 X X r Fg (crystal) FgX (free atom) exp2ig r

g 8:8:8 was constructed from nine-low-order structure factors. Three± and four-beam, non-systematic cases. Several magnitudes can be extracted from Kikuchi or CBED patterns in such con®gurations, see e.g. Gjùnnes & Hùier (1971). In the nonsystematic critical-voltage method, the condition for extinction of line contrast is measured; in the IKL (intersecting Kikuchi line) method, one measures the separation between line segments of the split line appearing at the intersection with a strong Kikuchi band. The precision of these methods, originally developed for Kikuchi patterns, was increased considerably when CBED was used instead (Matsuhata, Tomokiyo, Watanabe & Eguchi, 1982; Taftù & Gjùnnes, 1985). A further improvement is expected when the intensity distribution over the whole CBED discs is recorded and ®tted to dynamical calculations. This has been explored recently by Hùier, Bakken, Marthinsen & Holmestad (1993); the addition of non-systematic re¯ections in a parallel row can be seen as an extension of the systematic row con®guration above. The experience so far is that the three- and four-beam con®gurations are very sensitive to structure-factor phases, but may not yield as accurate values for structure amplitudes as those obtained from the line pro®les in the systematic row. Zone axis CBED. Convergent-beam patterns around the axis of a dense zone contain extensive multiple-beam dynamical interaction. Bird & Saunders (1992) claim that this leads to high sensitivity as well as a high number of structure factors that can be determined from one CBED pattern. Their results from re®nement of structure amplitudes in f.c.c., diamond and sphalerite structures may con®rm this. From ®ltered intensities measured with a CCD camera, Saunders, Bird, Midgley & Vincent (1994) determined structure factors for silicon up to 331 from one pattern in the [110] zone. The point-spread function for the detector was deconvoluted from the raw data. Thickness, background level and a scaling factor were included in the re®nement from a grid of 21 21 intensities in each disc. Starting values were neutral atom scattering factors, absorptive scattering amplitude calculated from TDS and a preliminary thickness determination. 121 beams were included in the diagonalization, a further 270 beams by perturbation. Critical voltage and intersecting-Kikuchi-line (IKL) method. The above methods, based on line scans or two-dimensional intensity distributions in CBED discs, rely on extensive

736


ACCURATE STRUCTURE-FACTOR DETERMINATION WITH ELECTRON DIFFRACTION 8.8. ACCURATE STRUCTURE-FACTOR DETERMINATION WITH ELECTRON DIFFRACTION calculations of dynamical scattering at a number of incidentbeam directions. In calculations of the critical-voltage effect (Watanabe, Uyeda & Fukuhara, 1969; Gjùnnes & Hùier, 1971; Matsuhata & Steeds, 1987; Matsuhata & Gjùnnes, 1994), which corresponds to an accidental degeneracy of Bloch waves, only one beam direction is considered. The condition for the associated vanishing contrast (or contrast reversal) of a Kikuchi or Kossel line is determined from CBED patterns taken at a series of voltages and compared with the Bloch-wave calculations for the particular direction. This is a very sensitive method ± with the disadvantage that only a relation between structure factors is obtained. In the original experiment by Watanabe, Uyeda & Fukuhara (1969), the relation between the structure factors for a strong ®rstorder re¯ection and its second order was determined from measurement of the disappearance voltage for the second-order line, cf. formula (4.3.7.8). This second-order critical voltage depends on high-voltage microscopy ± and a strong ®rst-order re¯ection. A number of metals and simple alloy phases have been studied with this method in recent years by Fox and coworkers, see the review by Fox & Fisher (1988) and Fox & Tabbernor (1991). The extension to non-systematic cases in the normal accelerating-voltage range was ®rst shown by Gjùnnes & Hùier (1971). In principle, the Bloch-wave degeneracy will appear at a certain combination of excitation errors for any non-systematic three-beam con®guration in centrosymmetric crystals. The dif®culty is to ®nd conditions that can be measured with suf®cient precision. Matsuhata & Gjùnnes (1994) analysed a number of non-systematic critical voltages in symmetrical con®gurations and showed how they can be measured in the range below 2±300 kV for simple structures. From rutile-type SnO2 , Matsuhata, Gjùnnes & Taftù (1994) measured four critical voltages, which were analysed in terms of ionicity; structure factors were determined for two low-order re¯ections. The calculations are simpler and less time consuming than for the intensity pro®les, but also more dependent on known high-order structure factors and temperature factors. An alternative to the measurement of a critical voltage is to measure the position in the pattern where the degeneracy occurs ± as in the IKL method (Gjùnnes & Hùier, 1971; Taftù & Gjùnnes, 1985; Matsumura, Tomokiyo & Oki, 1989; Wang & Peng, 1994). Phases and absorption in multiple-beam cases. Centrosymmetry was assumed in the original derivations of the accidental Bloch-wave degeneracy leading to the critical-voltage effect. In non-centrosymmetrical crystals, we can ®nd degeneracies related to other symmetry elements, i.e. mirror planes or rotation axes or `pseudocritical voltages' when the deviation from centrosymmetry is small (Matsuhata & Gjùnnes, 1994). For larger deviations, the position of minimum contrast can be used to determine structure-factor phase invariants, based again on the Bethe approximation. Hùier & Marthinsen (1983) give the expression v( )2 u U U 2 u U U cos ' h g h h g h 2 jUgeff j jUg jt 1 sin '; Ug 2ksh Uh 2ksg 8:8:9 where the phase invariant ' 'h 'g h ' g . Subsequent studies, e.g. Zuo, Hùier & Spence (1989), showed this threebeam effect to be very sensitive to the phase invariant. From CBED pro®le measurements, Zuo, Spence, Downs & Mayer (1993) determined the structure-factor phase of the 00.2

re¯ection in Be with the remarkable precision of 0.1 . Bird (1990) pointed out that the treatment of the absorptive part needs special attention in non-centrosymmetrical crystals. Thickness fringes. Thickness fringes were used for structure-factor determination at an early stage, e.g. Ando, Ichimiya & Uyeda (1974). The analogous effect in the diffraction pattern is the measurement of a split re¯ection from a wedge, which is an interesting visualization of Bloch waves; see Lehmpfuhl (1974). Today these techniques may appear to have mainly historical interest, although thickness fringes can be an alternative for determination of Bloch-wave absorption parameters j and thereby the imaginary Fourier potentials. Comparison, evaluation and extension to integrated intensities. The convergent-beam electron-diffraction methods for determination of structure factors from small unit cells have been developed to high precision during the last 5±10 years. Least-squares ®t of one- and two-dimensional intensity distributions within CBED discs appears today as the most sensitive method for determination of the lowest-order components of the charge distribution in organic structures with small unit cells. The accuracy may be as good as or better than the best X-ray methods ± with the important provision that other structure parameters, high-order structure factors and Debye±Waller factors, are known to suf®cient accuracy. Measurements of special features, e.g. critical voltages, offer an important supplement with less extensive computation efforts. Applications have been mainly to simple structures with atoms in special positions and re®nement of low-order structure factors only. It should be noted that the deformation density may have signi®cant components beyond the low-order structure factors that are usually determined by CBED methods ± and in a range where f X < Z=2 and X-rays thus inherently more sensitive to charge redistribution. The extension of precise measurements to more re¯ections and to larger unit cells with position parameters is thus seen as a main challenge ± which may be attacked along different avenues. Higher-order re¯ection pro®les are narrower, less dynamic in character and not so suitable for the pro®le ®tting described above. An alternative is to measure integrated intensities across Kossel-line segments, as has been done in several beam con®gurations. Vincent et al. (1984) measured integrated intensities of HOLZ-line segments, i.e. with the central CBED disc around the zone axis. Holmestad, Weickenmeier, Zuo, Spence & Horita (1993) measured selected HOLZ re¯ections in less-dense zones for determination of Debye±Waller factors. Taftù & Metzger (1985) showed the sensitivity of high-order lines in a dense systematic row to atomic coordinates. This wide-angle CBED technique has been applied to coordinate re®nement in intermetallic compounds (Ma, Rùmming, Lebech, Gjùnnes & Taftù, 1992). Gjùnnes & Bùe (1994) measured intensities of a range of re¯ections in the 00l row from the superconductors YBa2 Cu3 O7 and a Cosubstituted variant. Since the high-order lines are narrow, it is possible to measure relative intensities from a number of re¯ections in one exposure. Assuming a two-beam-like shape, the integral may be related to the gap at the dispersion surface, according to the Blackman formula (Blackman, 1939)

737


R

Igtwo beam sg ; t dsg Ag =t

RAg 0

J0 x dx;

8:8:10

8. REFINEMENT OF STRUCTURAL PARAMETERS where Ag Ugeff t, and J0 is the Bessel function of zero order. For a small gap, the intensity is proportional to jU eff j2 . By many-beam calculations, Gjùnnes & Bùe (1994) showed the integrated intensities to be less sensitive to dynamical interactions along the row than that indicated from the Bethe potentials, and that relative intensities are fairly independent of thickness. Coordinate re®nement based on intensities from

a few high-order Kossel-line segments appear to produce accuracies roughly one order of magnitude poorer than good single-crystal X-ray determination. This may suggest that if some form of three-dimensional intensity data could be collected in electron diffraction the same level of accuracies as with X-rays may be attainable ± which, however, remains to be seen.

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references


9.1. Sphere packings and packings of ellipsoids By E. Koch and W. Fischer

9.1.1. Sphere packings and packings of circles 9.1.1.1. De®nitions For the characterization of many crystal structures, geometrical aspects have proved to be a useful tool. Among these, sphere-packing considerations stand out in particular. A sphere packing in the most general sense is an in®nite, three-periodic set of non-intersecting spheres (i.e. a set of nonintersecting spheres with space-group symmetry) with the property that any pair of spheres is connected by a chain of spheres with mutual contact. If all spheres are symmetryequivalent, the sphere packing is called homogeneous, otherwise it is called heterogeneous. A homogeneous sphere packing may be represented uniquely by the set of symmetry-equivalent points that are the centres of the spheres [point con®guration, cf. ITA (1983, Chapter 14.1)]. This point con®guration is distinguished by equal shortest distances giving rise to a connected graph. As all spheres of a homogeneous sphere packing must be equal in size, their common radius can be calculated as half this shortest distance. A heterogeneous sphere packing consists of at least two symmetry-distinct subsets of spheres, the centres of which form a respective number of point con®gurations. The radii of symmetrically distinct spheres can be either equal or different. In the ®rst case, the heterogeneous sphere packing may be represented by its set of sphere centres, quite similar to a homogeneous one. In the case of different sphere radii, however, the knowledge of at least some of the radii is additionally necessary. As there exists an in®nite number of both homogeneous and heterogeneous sphere packings, it is convenient to classify the sphere packings into types: two sphere packings belong to the same type if there exists a biunique mapping that brings the spheres of one packing onto the spheres of the other packing and that preserves all contact relations between spheres. The number of types of homogeneous sphere packings is ®nite whereas the number of types of heterogeneous sphere packings is in®nite. All de®nitions and properties mentioned so far may be transferred from sets of spheres in three-dimensional space to sets of circles in two-dimensional space, giving rise to heterogeneous and homogeneous packings of circles. A characteristic property of types of homogeneous sphere (circle) packings is the number k of contacts per sphere (circle): 3 k 12 for sphere packings and 3 k 6 for packings of circles. A sphere (circle) packing is called stable [close, cf. IT II (1972, Chapter 7.1)] if no sphere (circle) can be moved without moving neighbouring spheres (circles) at the same time. As a consequence, a stable sphere (circle) packing has at least four (three) contacts per sphere (circle), and not all these contacts must fall in one hemisphere (semicircle). The density of a homogeneous sphere (circle) packing is de®ned as the fraction of volume (area) occupied by spheres (circles). It may be calculated as 43

nr 3 V

for sphere packings, and as

nr 2 A for packings of circles. Here, r is the radius of the spheres (circles), n the number of spheres (circles) per unit cell, V the unit-cell volume, and A the unit-cell area. Geometric properties of different sphere (circle) packings of the same type may be different. Such properties are, e.g., the density and the property of being a stable packing.

9.1.1.2. Homogeneous packings of circles The homogeneous packings of circles in the plane may be classi®ed into 11 types (cf. Niggli, 1927, 1928; Haag, 1929, 1937; Sinogowitz, 1939; Fischer, 1968; Koch & Fischer, 1978). These correspond to the 11 types of planar nets with equivalent vertices derived by Shubnikov (1916). If, in addition, symmetry is used for classi®cation, the number of distinct cases becomes larger (31 cases according to Sinogowitz, 1939). Table 9.1.1.1 gives a summary of the 11 types. In column 1, the type of circle packing is designated by a modi®ed SchlaÈ¯i symbol that characterizes the polygons meeting at one vertex of a corresponding Shubnikov net. The contact number k is given in column 2. The next column displays the highest possible symmetry for each type of circle packing. The corresponding parameter values are listed in column 4. The appropriate shortest distances d between circle centres and densities are given in columns 5 and 6, respectively. With three exceptions (36 , 34 6, 46.12), all types include circle packings that are not similar in the mathematical sense and that differ, therefore, in their geometrical properties. The highest possible symmetry for a type of homogeneous circle packing corresponds necessarily to the lowest possible density of that type. Therefore, homogeneous circle packings of type 3.122 with symmetry p6mm are the least dense. The highest possible density is achieved by the circle packings with contact number 6 referring to triangular nets with hexagonal symmetry. All circle packings described in Table 9.1.1.1 are stable in the sense de®ned above. Only circle packings of types 3.122 and 482 may be unstable. 9.1.1.3. Homogeneous sphere packings The number of homogeneous sphere-packing types is not known so far. Sinogowitz (1943) systematically derived sphere packings with non-cubic symmetry from planar sets of spheres, but he did not compare sphere packings with different symmetry and classify them into types. Fischer calculated the parameter conditions for all cubic (Fischer, 1973, 1974) and all tetragonal (Fischer, 1991a,b, 1993) sphere packings. 199 types of homogeneous sphere packings with cubic symmetry and 394 types with tetragonal symmetry exist in all. 12 of these types are common to both systems. In a similar way, Zobetz (1983) calculated the sphere-packing conditions for Wyckoff position 6c. Using a different approach, Koch & Fischer (1995) R3m derived all types of homogeneous sphere packings with contact number k 3. Because of the unique correspondence of each homogeneous sphere packing to a graph, studies on threedimensional nets also give contributions to the knowledge on sphere-packing types. In particular, papers by Wells (1977, 1979, 1983), O'Keeffe (1991, 1992), O'Keeffe & Brese (1992) and Treacy, Randall, Rao, Perry & Chadi (1997) contain some information on sphere packings with k 3 and k 4.


9.1. SPHERE PACKINGS AND PACKINGS OF ELLIPSOIDS Table 9.1.1.1. Types of circle packings in the plane Type

k

36 32 434 33 42 34 6 44 3464 3636 63 482 46.12 3.122

6 5 5 5 4 4 4 3 3 3 3

Symmetry p6mm p4gm c2mm p6 p4mm p6mm p6mm p6mm p4mm p6mm p6mm

1a 4c 4d 6d 1a 6e 3c 2b 4d 12 f 6c

Parameters 0; 0 x; x 12 x; 0 x; y 0; 0 x; x 1 2;0 1 2 3;3 x; 0 x; y x; x

p x 14 3 p14 x 1 12 3; b=a 2 x 37; y 17 x 12

p 3

p x 1p12 2 p 1 1 x 16 3 3 6; y 6 p x 1 13 3

Table 9.1.1.2 shows examples for sphere packings with high contact numbers and high densities in the upper part and with small contact numbers and low densities in the lower part. Column 1 gives reference numbers to designate the types in the following. Column 2 displays the contact numbers k. The highest possible symmetry for each type is described in column 3. Coordinates and metrical parameters referring to the most regular sphere packings of each type are listed in column 4; the respective shortest distances d between sphere centres are given in column 5. For a sphere packing that can be subdivided into plane nets of spheres with mutual contact, the direction and the type of these nets are shown in column 6. Column 7 contains stacking information: the contact numbers to the nets above and below, and the number of layers per translation period in the direction perpendicular to the layers. The last column displays the density with respect to the parameters of column 4. For all cases, this value gives the minimal density for that type of sphere packing. The densest homogeneous sphere packings known so far may be derived from the densest packings of circles (36 in Table 9.1.1.1). Such sphere packings can always be subdivided into parallel plane layers of spheres with six contacts per sphere within each layer and with three contacts to each of the neighbouring layers above and below (cf. Fig. 9.1.1.1). Consequently, the contact number k becomes 12. As there exist two stacking possibilities for each layer with respect to the previous layer, in®nitely many stacking sequences can be derived in principle, but only two refer to homogeneous sphere packings. If for each layer the two neighbouring layers are stacked directly upon each other, a sphere packing of a two-layer type with hexagonal symmetry (type 1) results. It is called hexagonal closest packing (abbreviated h.c.p.). If for all layers the neighbouring layers are never stacked directly upon each other, a sphere packing of a three-layer type with cubic symmetry (type 2) is formed. It is designated cubic closest packing (c.c.p.). In spite of these terms, for a long time it was only known that the cubic closest packings are the densest ones that correspond to lattices (Minkowski, 1904). Only recently, Hsiang (1993) published a proof that there does not exist any packing of spheres of equal size with a higher density, but the completeness of this proof is still doubted (cf. e.g. Hales, 1994). Independently of the stacking sequences, closest packings of spheres contain ideal octahedral and ideal tetrahedral voids. The number of octahedra per unit cell equals the respective number of spheres, whereas the number of tetrahedral voids is twice as large. The distances p the centres and the vertices of p between these voids are 2d=2 and 6d=4, respectively. Within a cubic closest packing, faces are shared only between octahedral and

1 6

Density

a p p 1 2a 2 6 bp 1 7a 7 a p 1 1a 2 3 1 a 2p 1 3p3 a 2 p1a 12 16p3a 2 3a

0.9069 0.8418 0.8418 0.7773 0.7854 0.7290 0.6802 0.6046 0.5390 0.4860 0.3907

tetrahedral voids. Each edge is common to two octahedra and two tetrahedra. In contrast, piles of face-sharing octahedra are formed within a hexagonal closest packing, whereas the tetrahedra are arranged as pairs with one face in common. The other faces are shared between octahedra and tetrahedra. Again, each edge belongs to two octahedra and two tetrahedra. Densest layers of spheres may also be stacked such that each sphere is in contact with two spheres of the previous layers (cf. Fig. 9.1.1.2). Such a stacking results in contact number 10. Again, in®nitely many periodic stacking sequences are possible, but only four give rise to homogeneous sphere packings [types 9, 10, 11: cf. Hellner (1986); type 12: cf. O'Keeffe (1988)]. In the most symmetrical forms of these four cases, each sphere is located exactly above or below the middle of two neighbouring spheres of the adjacent layers. This kind of stacking gives rise to distorted tetrahedral voids only. The number of tetrahedra per unit cell is six times the number of spheres. Two kinds of differently distorted tetrahedra exist in the ratio 1:2. The twolayer type 9 corresponds to a tetragonal body-centred lattice with specialized axial ratio. Furthermore, densest layers of spheres may be stacked in a mixed sequence with three contacts per sphere to one neighbouring layer and two contacts to the other layer. This kind of stacking results in ®ve types of homogeneous sphere packings (3 to 7) with contact number 11.

Fig. 9.1.1.1. Two triangular nets representing two densest packed layers of spheres. The layers are stacked in such a way that each sphere is in contact with three spheres of the other layer.

747


1 6

p 3

Distance d

9. BASIC STRUCTURAL FEATURES Table 9.1.1.2. Examples for sphere packings with high contact numbers and high densities and with low contact numbers and low densities Type

k

1

12

2

12

Symmetry P63 =mmc 2c Fm3m 4a

Parameters

1 2 1 3;3;4

c=a 23

0; 0; 0

±

3

11

Cmca

8 f

0; y; z

4

11

P31 21

6c

x; y; z

5

11

Fdd2

16b x; y; z

6

11

P65 22

12c x; y; z

7

11

C2=m

4i

8

11

P42 =mnm 4 f

x; x; 0

9

10

I4=mmm

2a

0; 0; 0

3c

1 2 ; 0; 0

Distance d

p 6

x 12

p 2

1 2 ; c=a

p 2

2

11

10

Fddd

8a

0; 0; 0

12

10

Fddd

16g

1 1 8;8;z

p 6 p c=a 32 3 p p b=a 3; c=a 2 3 p p z 165 ; b=a 3; c=a 4 3

13

10

Cmcm

4c

0; y; 14

y 103 ; b=a 13 x

10

10

P62 22

c=a 13

14

10

Pnma

4c

x: 14 ; z

15

10

1 2 3;3;z

z 34

0; 0; z

z

1 2

y 34

p p 15; c=a 25 10 p 7 4 2 15 8 ; b=a 5 ; c=a 15

7 20 ; z

p

1 6; c=a 4 p 1 6; c=a 6

p 23 6 2 p 63

16

10

P63 =mmc 4 f R3m 6c

17

10

Cmcm

4c

0; y; 14

18

10

I41 =amd

8e

0; 0; z

p p c=a 1; b=a 3 2 p p p z 12 18 6; c=a 2 3 2 2

19

10

I4=m

8h

x; y; 0

x 176

1 4

c=a 20

10

R3

p 6,

p

1 2; y 177 17 p 14 8 17 17 21=2

4 17

4

Fd 3m

32e x; x; x

x 38

1 8

22

4

Im3m

48 j 0; y; z

y 47

3 28

23

4

I41 32

48i

x; y; z

x y 18

24

3

I41 32

24h

1 1 8 ; y; 4

y

y 14

p 3

p 42

p 6

21

(001) 36

3, 3 2

{111} 36

3, 3 3

{001} 44

4, 4 2

a

(001) 36

3, 2 4

a

(001) 36

3, 2 6

c

(010) 36

3, 2 8

a

(001) 36

3, 2 12

b

(001) 36

3, 2 12

c

±

±

0.7187

c

{110} 36

p 2; z 145

1 28

p 2

p 2; z 0 3 8

Density

2, 2 2

0.6981

0.7405

0.7187

6

2, 2 3

a

6

(001) 3

2, 2 4

a

(001) 36

2, 2 8

(001) 44

3, 3 2

c

(010) 44

3, 3 2

a

(001) 36

3, 1 4

a

(001) 36

3, 1 6

a

(010) 44

4, 2 4

a

(001) 44

4, 2 8

c

±

±

0.6619

(001)

3, 2 3

0.6347

a

1 3

1 7

(001) 3

p 6a

p 7a

34

748


p 2

x 37 ; y 17 ; z 0; c=a 17

18 f x; y; z

Stacking

a p 1 2a 2

p y 16 ; z 32 2 2 p p p b=a 3; c=a 23 6 3 p x 12 ; y 56 ; z 2 43 p p c=a 6 32 3 p x 16 ; y 34 2 1; z 0 p p b=a 43 2 2; c=a 13 3 p x 16 ; y 13 ; z 12 2 23 p p c=a 2 6 3 3 p p x 12 2 12 ; z 3 2 4 p p p b=a 13 3; c=a 16 6 13 3 p p cos 16 6 13 3

x; 0; z

Net

0.6981

0.6657

0.6657

346 p 2

143

p 2

12

1 4

12

p 6

p 3a

±

±

0.1235

1 7a

±

±

0.1033

±

±

0.0789

±

±

0.0555

1 2

p 2a 3 4

p 2a

9.1. SPHERE PACKINGS AND PACKINGS OF ELLIPSOIDS Two other types of homogeneous sphere packings (15 and 16) with contact number k 10 also refer to densest layers of spheres. In these cases, each sphere has three contacts to one neighbouring layer and one contact to the other layer that is stacked directly above or below the original layer. Cubic closest packings may also be regarded as built up from square layers 44 stacked in such a way that each sphere has four neighbouring spheres in the same layer and four neighbours each from the layers above and below (cf. Fig. 9.1.1.3). If square layers are stacked such that each sphere has contact to four spheres of one neighbouring layer and to two spheres of the other layer (cf. Fig. 9.1.1.4), sphere packings with contact number 10 result. In total, two types of homogeneous packings (17 and 18) with this kind of stacking exist. Sphere packings of type 9 may also be decomposed into 44 layers parallel to (101) or (011) in a ®ve-layer sequence. These nets are made up from parallel rhombi and stacked such that each sphere has contact with three other spheres from the layer above and from the layer below. If such layers are stacked in a two-layer sequence, sphere packings of type 13

Fig. 9.1.1.2. Two triangular nets representing two densest packed layers of spheres. The layers are stacked in such a way that each sphere is in contact with two spheres of the other layer.

Fig. 9.1.1.3. Two square nets representing two layers of spheres stacked in such a way that each sphere is in contact with four spheres of the other layer.

with symmetry Cmcm result (O'Keeffe, 1998). Sphere packings of type 14 are also build up from 44 layers, but here the rhombi occur in two different orientations (O'Keeffe, 1998). Sphere packings with high contact numbers may also be derived by stacking of other layers. Type 20, for example, refers to 346 layers where each sphere is in contact with three spheres of one neighbouring net and two spheres of the other one (Sowa & Koch, 1999). Such a sphere packing may alternatively be derived from the cubic closest packing by omitting systematically 1=7 of the spheres in each of the 36 nets. Sphere packings of types 8 and 19 (cf. Figs. 9.1.1.5 and 9.1.1.6) cannot be built up from plane layers of spheres in contact although their contact numbers are also high. Table 9.1.1.2 contains complete information on homogeneous sphere packings with k 10, 11, and 12 and with cubic or tetragonal symmetry. The least dense (most open) homogeneous sphere packings known so far have already been described by Heesch & Laves (1933). Sphere packings of that type (24) cannot be stable because their contact number is 3 (cf. Fig. 9.1.1.7). As discussed

Fig. 9.1.1.4. Two square nets representing two layers of spheres stacked in such a way that each sphere is in contact with two spheres of the other layer.

Fig. 9.1.1.5. Sphere packing of type 8 (Table 9.1.1.2) represented by a graph: k 11, P42 =mnm, 4 f , xx0.

749


9. BASIC STRUCTURAL FEATURES by Fischer (1976), it is very probable that no homogeneous sphere packings with lower density exist; those discussed by Melmore (1942a,b) with 0:042 and 0:045 are heterogeneous ones. Recently, Koch & Fischer (1995) proved that the Heesch±Laves packing is the least dense homogeneous sphere packing with three contacts per sphere. The least dense sphere packings with contact number 4 derived so far are described as type 23. All sphere packings of this type are similar in the geometrical sense and are not stable. In contrast, the sphere packings of type 22 are stable. Sphere packings of type 21 (Heesch & Laves, 1933), which have been supposed to be the most open stable ones (cf. Hilbert & CohnVossen, 1932, 1952), have a slightly higher density. On the basis of the material known at that time, Slack (1983) tried to develop empirical formulae for the minimal and the maximal density of circle packings and sphere packings depending on the contact number. A paper by O'Keeffe (1991) on four-connnected nets pays special attention to the densest and the least dense sphere packings with four contacts per sphere. 9.1.1.4. Applications Sphere packings have been used for the description of inorganic crystal structures in different ways and by several authors (e.g. Brunner, 1971; Figueiredo & Lima-de-Faria, 1978; Frank & Kasper, 1958; Hellner, 1965; Hellner, Koch & Reinhardt, 1981; Koch, 1984, 1985; Laves, 1930, 1932; Lima-de-Faria, 1965; Lima-de-Faria & Figueiredo, 1969a,b; Loeb, 1958; Morris & Loeb, 1960; Niggli, 1927; Smirnova, 1956a,b, 1958a,b, 1959a,b,c, 1964; Sowa, 1988, 1997). In the simplest case, the structure of an element may be described as a sphere packing if all atoms are interrelated by equal or almost equal shortest distances. This does not imply that the atoms really have to be considered as hard spheres of that size. Often such sphere packings are homogeneous ones with a high contact number k (e.g. Cu, Mg with k 12; Pa with k 10; W with k 8). Low values of k (e.g. diamond with k 4, white tin with k 6) and heterogeneous sphere packings (La with k 12) have also been observed for structures of elements.

Fig. 9.1.1.6. Sphere packing of type 19 (Table 9.1.1.2) represented by a graph: k 10, I4=m, 8h, xy0.

Crystal structures consisting of different atoms may be related to sphere packings in different ways: (1) The structure as a whole may be considered as a heterogeneous sphere packing. In that case, contacts at least between different spheres are present (e.g. CsCl, NaCl, CaF2 ). In addition, contacts between equal atoms may exist (I I contacts in CsI) or may even be necessary (CdI2 ). In general, a type of heterogeneous sphere packing is compatible with a certain range of radius ratios (cf. alkali halides). In special cases, a heterogeneous sphere packing may be derived from a homogeneous one by subgroup degradation (e.g. NaCl, CsCl, PtCu). (2) Part of the crystal structure, e.g. the anions or the more frequent kind of atoms, may be considered as a sphere packing whereas the other atoms are located within the voids of that sphere packing. For this approach, the atoms corresponding to the sphere packing need not necessarily be in contact (cf. e.g. the Cl Cl distances in NaCl and LiCl). Voids within sphere packings have been discussed in particular in connection with closest packings (e.g. Cl in NaCl, O in Li2 O, S in ZnS, O in olivine), but numerous examples for non-closest packings are known in addition (e.g. B in CaB6 with k 5; O in rutile with k 11, type 8 in Table 9.1.1.2; Si in -ThSi2 with k 3). Sphere packings and their voids form the basis for Hellner's framework concept (cf. e.g. Hellner et al., 1981). Voids may be calculated systematically as vertices of Dirichlet domains (cf. Hellner et al., 1981; Koch, 1984). The tendency to form regular voids of the appropriate size for the cations may counteract the tendency to form an ideal sphere packing of the anions. Examples are spinel and garnet (cf. Hellner, Gerlich, Koch & Fischer, 1979). (3) Frequently, the cations within a crystal structure are also distributed according to a sphere packing. This is explicable because the repulsion between the cations also favours an arrangement with equal but maximal shortest distances (cf. Brunner, 1971). In this sense, many crystal structures may be

Fig. 9.1.1.7. Least dense sphere packing known so far (type 24 of Table 9.1.1.2) represented by a graph: k 3, I41 32, 24h, 1 1 y. z coordinates given in multiples of 1=100. 8 ; y; 4

750


9.1. SPHERE PACKINGS AND PACKINGS OF ELLIPSOIDS described as sets of several sphere packings (one for each kind of atom) that are ®tted into each other (e.g. NaCl, CaF2 , CaB6 , -ThSi2 , rutile, Cu2 O, CaTiO3 ). Because of their importance for problems in digital communication (error-correcting codes) and in number theory (solving of diophantine equations), densest sphere packings in higher dimensions are of mathematical interest (cf. Conway & Sloane, 1988).

9.1.1.5. Interpenetrating sphere packings Special homogeneous or heterogeneous sets of spheres may be subdivided into a small number i of subsets such that each subset, regarded by itself, forms a sphere packing and that spheres of different subsets do not have mutual contact. Sets of spheres with these properties are called interpenetrating sphere packings. The cubic Laves phases are a well known example for heterogeneous interpenetrating sphere packings. The Mg atoms in MgCu2 [Fd 3m, 8a], for example, p correspond to a sphere packing with shortest distances d1 3a=4 and contact number k 4 whereas the copper atoms 16dprefer to another sphere packing with shortest distances d2 2a=4 and k 6. The shortest pdistances between centres of different spheres are d3 11a=8 > d1 d2 =2. The crystal structure of Cu2 O gives an example of a different kind. If one takes into account the size of the atoms, sphere contacts can only be expected between different spheres. As a consequence, the heterogeneous set of spheres disintegrates into two heterogeneous but congruent subsets with no mutual contact. In the case of homogeneous interpenetrating sphere packings, all i subsets have to be symmetry-equivalent. Then the symmetry of each subset is a subgroup of index i of the original space group. Homogeneous interpenetrating sphere packings with cubic symmetry have been derived completely by Fischer & Koch (1976). They may be classi®ed into 39 types. For 33 of the 39 types, the number i of subsets is 2; i is 3, 4, and 8 for 1, 3, and 2 types, respectively. Remarkable are those homogeneous interpenetrating sphere packings that are built up from sphere packings of type 24 (Table 9.1.1.2), i.e. that type with the least dense sphere packing. Combinations of 2, 4, or 8 such sphere packings result in altogether 8 different types of interpenetrating sphere packings (Fischer, 1976). The P atoms in the crystal structure of Th3 P4 give an example for such interpenetrating sphere packings built up from two congruent subsets (Koch, 1984). Complete results for other crystal systems are not available. With tetragonal symmetry, interpenetrating sphere packings are known, built up from 2, 3, or 5 congruent subsets (Fischer, 1970). Analogous interpenetration patterns are formed by hydrogen bonds within certain molecular structures (Ermer, 1988; Ermer & Eling, 1988). Interpenetrating sphere packings may be brought in relation to interpenetrating labyrinths as formed by periodic minimal surfaces or by periodic zero-potential surfaces without selfintersection (cf. e.g. Andersson, Hyde & von Schnering, 1984; Fischer & Koch, 1987, 1996; von Schnering & Nesper, 1987).

9.1.2. Packings of ellipses and ellipsoids The problem of deriving packings of ellipses in two-dimensional space or of ellipsoids in three-dimensional space may be regarded as a generalization of the problem of deriving circle packings and sphere packings. It is much more complicated, however, because a circle or sphere is fully determined by its centre and its radius, whereas the knowledge of the centre, the lengths of the two semiaxes, and the direction of one of them is needed to construct an ellipse. For an ellipsoid, the knowledge of its centre, the length of its three semiaxes, and the directions of two of them is necessary. Accordingly, the point con®guration corresponding to the ellipsoid centres does not de®ne the ellipsoid packing and not even its type. Nowacki (1948) derived 54 homogeneous èssentially different packings of ellipses'. In contrast to the de®nition of types of sphere (circle) packings (Section 9.1.1), Nowacki distinguished between similar packings with different plane-group symmetry, i.e. between packings that may differ in the orientation of their ellipses. Under an equivalent classi®cation, GruÈnbaum & Shephard (1987) derived 57 different cases of ellipse packings, thus correcting and completing Nowacki's list. Each of these 57 cases corresponds uniquely to one of the 11 types of circle packings if one takes into account only the contact relations between ellipses and circles. In eight cases, each ellipse has six contacts. Two of these cases can be derived from the densest packing of circles by af®ne transformations and, therefore, have the same density, namely 0:9069, irrespective of the shape of the ellipses (Matsumoto & Nowacki, 1966). Presumably for the other six cases this density can only be reached (but not exceeded) if the ellipses become circles. A corresponding proof is in progress (Matsumoto, 1968; Tanemura & Matsumoto, 1992; Matsumoto & Tanemura, 1995). Very little systematic work seems to be carried out on homogeneous or heterogeneous packings of ellipsoids. Matsumoto & Nowacki (1966) derived packings of ellipsoids with contact numbers 12 and high densities by af®ne deformation of cubic and hexagonal closest packings of spheres. They postulate (without proof) the following: Densest packings of ellipsoids have the same contact number and density as closest packings of spheres and can be derived always from closest sphere packings by af®ne transformations. If this assumption is true, densest packings of ellipsoids would necessarily consist of parallel ellipsoids only. Packings of ellipsoids seemed to be useful for the interpretation of the arrangements of organic molecules in crystals. The studies of Kitaigorodsky (1946, 1961, 1973), however, showed that molecular crystals may rather be regarded as dense packings of molecules with irregular shape. Heterogeneous packings of ellipsoids may possibly be adequate for the geometrical interpretation of some intermetallic compounds like cubic MgCu2 (cf. Subsection 9.1.1.4) or Cr3 Si. The ellipsoids enable the use of different àtomic radii' with respect to neighbouring atoms of the same kind or of different kinds. In MgCu2 , for example, the magnesium 8a and atoms have cubic site symmetry 43m Fd 3m; therefore can only be represented by spheres. The Cu atoms 16d with site symmetry :3m, however, may be represented by ¯attened ellipsoids of revolution.

751



9.2. Layer stacking

Æ urovicÆ, P. Krishna and D. Pandey By S. D 9.2.1. Layer stacking in close-packed structures (By D. Pandey and P. Krishna) The crystal structures of a large number of materials can be described in terms of stacking of layers of atoms. This chapter provides a brief account of layer stacking in materials with structures based on the geometrical principle of close packing of equal spheres. 9.2.1.1. Close packing of equal spheres 9.2.1.1.1. Close-packed layer In a close-packed layer of spheres, each sphere is in contact with six other spheres as shown in Fig. 9.2.1.1. This is the highest number of nearest neighbours for a layer of identical spheres and therefore yields the highest packing density. A single close-packed layer of spheres has two-, three- and sixfold axes of rotation normal to its plane. This is depicted in Fig. 9.2.1.2a, where the size of the spheres is reduced for clarity. There are three symmetry planes with indices 12:0, 21:0, and (11.0) de®ned with respect to the smallest two-dimensional hexagonal unit cell shown in Fig. 9.2.1.2b. The point-group symmetry of this layer is 6mm and it has a hexagonal lattice. As such, a layer with such an arrangement of spheres is often called a hexagonal close-packed layer. We shall designate the positions of spheres in the layer shown in Fig. 9.2.1.1 by the letter À'. This A layer has two types of triangular interstices, one with the apex angle up 4 and the other with the apex angle down 5. All interstices of one kind are related by the same hexagonal lattice as that for the A layer. Let the positions of layers with centres of spheres above the centres of the 4 and 5 interstices be designated as `B' and `C', respectively. In the cell of the A layer shown in Fig. 9.2.1.1 (a b diameter of the sphere and 120 ), the three positions A, B, and C on projection have coordinates (0, 0), (13, 23), and (23, 13), respectively.

identity period n of these layer stackings is determined by the number of layers after which the stacking sequence starts repeating itself. Since there are two possible positions for a new layer on the top of the preceding layer, the total number of possible layer stackings with a repeat period of n is 2n 1 . In all the close-packed layer stackings, each sphere is surrounded by 12 other spheres. However, it is touched by all p 12 spheres only if the axial ratio h=a is 2=3, where h is the separation between two close-packed layers and a is the diameter of the spheres (Verma & Krishna, 1966). Deviations from the ideal value of the axial ratio are common, especially in hexagonal metals (Cottrell, 1967). The arrangement of spheres described above provides the highest packing density of 0.7405 in the ideal case for an in®nite lattice (Azaroff, 1960). There are, however, other arrangements of a ®nite number of equal spheres that have a higher packing density (Boerdijk, 1952). 9.2.1.1.3. Notations for close-packed structures In the Ramsdell notation, close-packed structures are designated as nX, where n is the identity period and X stands for the lattice type, which, as shown later, can be hexagonal H, rhombohedral R, or in one special case cubic C (Ramsdell, 1947). In the Zhdanov notation, use is made of the stacking offset vector s and its opposite s, which cause, respectively, a

9.2.1.1.2. Close-packed structures A three-dimensional close-packed structure results from stacking the hexagonal close-packed layers in the A, B, or C position with the restriction that no two successive layers are in identical positions. Thus, any sequence of the letters A, B, and C, with no two successive letters alike, represents a possible manner of stacking the hexagonal close-packed layers. There are thus in®nite possibilities for close-packed layer stackings. The

Fig. 9.2.1.1. The close packing of equal spheres in a plane.

Fig. 9.2.1.2. a Symmetry axes of a single close-packed layer of spheres and b the minimum symmetry of a three-dimensional close packing of spheres.


9.2. LAYER STACKING 9.2.1.2.1. Voids in close packing

Table 9.2.1.1. Common close-packed metallic structures Stacking sequence AB; A . . . ABC; A . . . ABCB; A . . . ABCBCACAB; A . . .

Identity Ramsdell Zhdanov Jagodzinski Protoperiod notation notation notation type 2 3 4 9

2H 3C 4H 9R

11 1 22 21

h c hc hhc

Mg Cu La Sm

cyclic A ! B ! C ! A or anticyclic A ! C ! B ! A shift of layers in the same plane. The vector s can be either (1/3)1100, (1/3)0110, or (1/3)1010. Zhdanov (1945) suggested summing the number of consecutive offsets of each kind and designating them by numeral ®gures. Successive numbers in the Zhdanov symbol have opposite signs. The rhombohedral stackings have three identical sets of Zhdanov symbols in an identity period. It is usually suf®cient to write only one set. Yet another notation advanced, amongst others, by Jagodzinski (1949a) makes use of con®gurational symbols for each layer. A layer is designated by the symbol h or c according as its neighbouring layers are alike or different. Letter `k' in place of `c' is also used in the literature. Some of the common close-packed structures observed in metals are listed in Table 9.2.1.1 in terms of all the notations.

Three-dimensional close packings of spheres have two kinds of voids (Azaroff, 1960): (i) If the triangular interstices in a close-packed layer have spheres directly over them, the resulting voids are called tetrahedral voids because the four spheres surrounding the void are arranged at the corners of a regular tetrahedron (Figs. 9.2.1.3a,b). If R denotes the radius of the four spheres surrounding a tetrahedral void, the radius of the sphere that would just ®t into this void is given by 0.225R (Verma & Krishna, 1966). The centre of the tetrahedral void is located at a distance 3h=4 from the centre of the sphere on top of it. (ii) If the triangular interstices pointing up in one closepacked layer are covered by triangular interstices pointing down in the adjacent layer, the resulting voids are called octahedral voids (Figs. 9.2.1.3c,d) since the six spheres surrounding each such void lie at the corners of a regular octahedron. The radius of the sphere that would just ®t into an octahedral void is given by 0:414R (Verma & Krishna, 1966). The centre of this void is located half way between the two layers of spheres. While there are twice as many tetrahedral voids as the spheres in close packing, the number of octahedral voids is equal to the number of spheres (Krishna & Pandey, 1981). 9.2.1.2.2. Structures of SiC and ZnS

Frequently, the positions of one kind of atom or ion in inorganic compounds, such as SiC, ZnS, CdI2 , and GaSe, correspond approximately to those of equal spheres in a close packing, with the other atoms being distributed in the voids. All such structures will also be referred to as close-packed structures though they may not be ideally close packed. In the close-packed compounds, the size and coordination number of the smaller atom/ion may require that its close-packed neighbours in the neighbouring layers do not touch each other.

SiC has a binary tetrahedral structure in which Si and C layers are stacked alternately, each carbon layer occupying half the tetrahedral voids between successive close-packed silicon layers. One can regard the structure as consisting of two identical interpenetrating close packings, one of Si and the other of C, with the latter displaced relative to the former along the stacking axis through one fourth of the layer spacing. Since the positions of C atoms are ®xed relative to the positions of layers of Si atoms, it is customary to use the letters A, B, and C as representing Si±C double layers in the close packing. To be more exact, the three kinds of layers need to be written as A, B , and C where Roman and Greek letters denote the positions of Si and C atoms, respectively. Fig. 9.2.1.4 depicts the structure of SiC-6H, which is the most common modi®cation.

Fig. 9.2.1.3. Voids in a close packing: a tetrahedral void; b tetrahedron formed by the centres of spheres; c octahedral void; d octahedron formed by the centres of spheres.

Fig. 9.2.1.4. Tetrahedral arrangement of Si and C atoms in the SiC-6H structure.

9.2.1.2. Structure of compounds based on close-packed layer stackings

753


9. BASIC STRUCTURAL FEATURES Table 9.2.1.2. List of SiC polytypes with known structures in value (Verma & Krishna, 1966). The structure of the stackings in order of increasing periodicity (after Pandey & Krishna, 1982a) polytypic AgI is analogous to those in SiC and ZnS (Prager, 1983). Polytype 2.H 3.C 4.H 6.H 8.H 10.H 14.H 15.R 16.H1 18.H 19.H 20.H 21.H 21.H2 21.R 24.R 27.H 27.R 33.R 33.H 34.H 36.H1 36.H2 39.H 39.R 40.H 45.R 51.R1 51.R2 54.H

Structure (Zhdanov sequence) 11 1 22 33 44 3322 (22)2 33 23 (33)2 22 (22)3 33 (23)3 22 (22)3 44 333534 (33)2 63 34 35 (33)2 (23)3 2223 3332 (33)2 353334 (33)4 2332 (33)2 32(33)2 34 (33)4 3234 (33)2 32(33)3 (32)2 3334 (33)5 2332 (23)2 32 (33)2 32 (22)3 23 (33)6 323334

Polytype

Structure (Zhdanov sequence)

57.H 57.R 69.R1 69.R2 75.R2 81.H 84.R 87.R 90.R 93.R 96.R1 99.R 105.R 111.R 120.R 123.R 126.R 129.R 125.R 141.R 147.R 150.R1 150.R2 159.R 168.R 174.R 189.R 267.R 273.R 393.R

(23)9 3333 (33)2 34 (33)3 32 33322334 (32)3 (23)2 (33)5 35(33)6 34 (33)3 (32)2 (33)4 32 (23)4 3322 (33)4 34 (33)3 3434 (33)4 3222 (33)5 32 (33)5 34 (22)5 23222333 (33)6 32 (33)2 2353433223 (33)6 34 32(33)2 23(33)3 23 (33)7 32 (3332)4 32 (23)3 32(23)3 322332 (23)2 (3223)4 (33)8 32 (23)10 33 (33)6 6(33)5 4 (34)8 43 (23)17 22 (23)17 33 (33)21 32

9.2.1.2.3. Structure of CdI2 The structure of cadmium iodide consists of a close packing of the I ions with the Cd ions distributed amongst half the octahedral voids. Thus, the Cd and I layers are not stacked alternately; there is one Cd layer after every two I layers as shown in Fig. 9.2.1.5. The structure actually consists of molecular sheets (called minimal sandwiches) with a layer of Cd ions sandwiched between two close-packed layers of I ions. The bonding within the minimal sandwich is ionic in character and is much stronger than the bonding between successive sandwiches, which is of van der Waals type. The importance of polarization energy for the stability of such structures has recently been emphasized by Bertaut (1978). It is because of the weak van der Waals bonding between the successive minimal sandwiches that the material possesses the easy cleavage characteristic of a layer structure. In describing the layer stackings in the CdI2 structure, it is customary to use Roman letters to denote the I positions and Greek letters for the Cd positions. The two most common modi®cations of CdI2 are 4H and 2H with layer stackings A B CB . . . and A B A B, respectively. In addition, this material also displays a number of polytype modi®cations of large repeat periods (Trigunayat & Verma, 1976; Pandey & Krishna, 1982a). From the structure of CdI2 , it follows that the identity period of all such modi®cations must consist of an even number of I layers. The h=a ratio in all these modi®cations of CdI2 is 0.805, which is very different from the ideal value (Verma & Krishna, 1966). The structure of PbI2 , which also displays a large number of polytypes, is analogous to CdI2 with one important difference. Here, the distances between two I layers with and without an intervening Pb layer are quite different (Trigunayat & Verma, 1976).

A large number of crystallographically different modi®cations of SiC, called polytypes, has been discovered in commercial crystals grown above 2273 K (Verma & Krishna, 1966; Pandey & Krishna, 1982a). Table 9.2.1.2 lists those polytypes whose structures have been worked out. All these polytypes have Ê and c n 2:518 A, Ê where n is the number of a b 3:078 A Si±C double layers in the hexagonal cell. The 3C and 2H modi®cations, which normally result below 2273 K, are known to undergo solid-state structural transformation to 6H (Jagodzinski, 1972; Krishna & Marshall, 1971a,b) through a non-random insertion of stacking faults (Pandey, Lele & Krishna, 1980a,b,c; Kabra, Pandey & Lele, 1986). The lattice parameters and the average thickness of the Si±C double layers vary slightly with the structure, as is evident from the h=a ratios of 0.8205 (Adamsky & Merz, 1959), 0.8179, and 0.8165 (Taylor & Jones, 1960) for the 2H, 6H, and 3C structures, respectively. Even in the same structure, crystal-structure re®nement has revealed variation in the thickness of Si±C double layers depending on their environment (de Mesquita, 1967). The structure of ZnS is analogous to that of SiC. Like the latter, ZnS crystals grown from the vapour phase also display a large variety of polytype structures (Steinberger, 1983). ZnS crystals that occur as minerals usually correspond to the wurtzite =AB= . . . and the sphalerite =ABC= . . . modi®cations. The structural transformation between the 2H and 3C structures of ZnS is known to be martensitic in nature (Sebastian, Pandey & Krishna, 1982; Pandey & Lele, 1986b). The h=a ratio for ZnS-2H is 0.818, which is somewhat different from the ideal

9.2.1.2.4. Structure of GaSe The crystal structure of GaSe consists of four-layered slabs, each of which contains two close-packed layers of Ga (denoted by symbols A, B, C) and Se (denoted by symbols ; ; ) each in the sequence Se±Ga±Ga±Se (Terhell, 1983). The Se atoms sit on the corners of a trigonal prism while each Ga atom is tetrahedrally coordinated by three Se and one Ga atoms. If the Se layers are of A type, then the stacking sequence of the four

Fig. 9.2.1.5. The layer structure of CdI2 : small circles represent Cd ions and larger ones I ions (after Wells, 1945).

754


9.2. LAYER STACKING layers in the slab can be written as A A or A

A. There are thus six possible sequences for the unit slab. These unit slabs can be stacked in the manner described for equal spheres. Thus, for example, the 2H structure can have three different layer stackings: =A A B

B= . . ., =A A BB= . . . and =A A C C=. Periodicities containing up to 21 unit slabs have been reported for GaSe (see Terhell, 1983). The bonding between the layers of a slab is predominantly covalent while that between two adjacent slabs is of the van der Waals type, which imparts cleavage characteristics to this material. 9.2.1.3. Symmetry of close-packed layer stackings of equal spheres It can be seen from Fig. 9.2.1.2a that a stacking of two or more layers in the close-packed manner still possesses all three symmetry planes but the twofold axes disappear while the sixfold axes coincide with the threefold axes (Verma & Krishna, 1966). The lowest symmetry of a completely arbitrary periodic stacking sequence of close-packed layers is shown in Fig. 9.2.1.2b. Structures resulting from such stackings therefore belong to the trigonal system. Even though a pure sixfold axis of rotation is not possible, close-packed structures belonging to the hexagonal system can result by virtue of at least one of the three symmetry axes parallel to [00.1] being a 63 axis (Verma & Krishna, 1966). This is possible if the layers in the unit cell are stacked in special ways. For example, a 6H stacking sequence =ABCACB= . . . has a 63 axis through 0, 0. It follows that, for an nH structure belonging to the hexagonal system, n must be even. A packing nH=nR with n odd will therefore necessarily belong to the trigonal system and can have either a hexagonal or a rhombohedral lattice (Verma & Krishna, 1966). Other symmetries that can arise by restricting the arbitrariness of the stacking sequence in the identity period are: (i) a centre of symmetry at the centre of either the spheres or the octahedral voids; and (ii) a mirror plane perpendicular to [00.1]. Since there must be two centres of symmetry in the unit cell, the centrosymmetric arrangements may possess both centres either at sphere centres/octahedral void centres or one centre each at the centres of spheres and octahedral voids (Patterson & Kasper, 1959).

[00.1], the hexagonal unit cell will then be centred at 13 ; 23 ; 13 and These two settings are crystallographically equivalent for close packing of equal spheres. They represent twin arrangements when both occur in the same crystal. The hexagonal unit cell of an nR structure is made up of three elementary stacking sequences of n=3 layers that are related to each other either by an anticyclic shift of layers A ! C ! B ! A (obverse setting) or by a cyclic shift of layers A ! B ! C ! A (reverse setting) in the direction of z increasing (Verma & Krishna, 1966). Evidently, n must be a multiple of 3 for nR structures. In the special case of the p close packing =ABC= . . . [with the ideal axial ratio of 2=3], the primitive rhombohedral unit cell has 60 , which enhances the symmetry and enables the choice of a face-centred cubic unit cell. The relationship between the face-centred cubic and the rhombohedral unit cell is shown in Fig. 9.2.1.8. The threefold axis of the rhombohedral unit cell coincides with one of the h111i directions of the cubic unit cell. The close-packed layers are thus parallel to the f111g planes in the cubic close packing. 2 1 2 3 ; 3 ; 3.

9.2.1.5. Possible space groups It was shown by Belov (1947) that consistent combinations of the possible symmetry elements in close packing of equal spheres can give rise to eight possible space groups: P3m1, P 3m1,

9.2.1.4. Possible lattice types

Fig. 9.2.1.6. The primitive unit cell of the 2H close packing.

Close packings of equal spheres can belong to the trigonal, hexagonal, or cubic crystal systems. Structures belonging to the hexagonal system necessarily have a hexagonal lattice, i.e. a lattice in which we can choose a primitive unit cell with a b 6 c, 90 , and 120 . In the primitive unit cell of the hexagonal close-packed structure =AB= . . . shown in Fig. 9.2.1.6, there are two spheres associated with each lattice point, one at 0, 0, 0 and the other at 13, 23, 12. Structures belonging to the trigonal system can have either a hexagonal or a rhombohedral lattice. By a rhombohedral lattice is meant a lattice in which we can choose a primitive unit cell with a b c, 6 90 . Both types of lattice can be referred to either hexagonal or rhombohedral axes, the unit cell being non-primitive when a hexagonal lattice is referred to rhombohedral axes and vice versa (Buerger, 1953). In closepacked structures, it is generally convenient to refer both hexagonal and rhombohedral lattices to hexagonal axes. Fig. 9.2.1.7 shows a rhombohedral lattice in which the primitive cell is de®ned by the rhombohedral axes a1 ; a2 ; a3 ; but a nonprimitive hexagonal unit cell can be chosen by adopting the axes A1 ; A2 ; C. The latter has lattice points at 0; 0; 0; 23 ; 13 ; 13; and 1 2 2 3 ; 3 ; 3. If this rhombohedral lattice is rotated through 60 around

Fig. 9.2.1.7. A rhombohedral lattice a1 ; a2 ; a3 referred to hexagonal axes A1 ; A2 ; C (after Buerger, 1953).

755


9. BASIC STRUCTURAL FEATURES and Fm3m. The last space P6m2, P63 mc, P63 =mmc, R3m, R3m, group corresponds to the special case of cubic close packing =ABC= . . .. The tetrahedral arrangement of Si and C in SiC does or a plane of not permit either a centre of symmetry 1 symmetry m perpendicular to [00.1]. SiC structures can therefore have only four possible space groups P3m1, R3m1, P63 mc, and F 43m. CdI2 structures can have a centre of symmetry on octahedral voids, but cannot have a symmetry plane perpendicular to [00.1]. CdI2 can therefore have ®ve possible space groups: P3m1, P 3m, R3m, R3m, and P63 mc. Cubic symmetry is not possible in CdI2 on account of the presence of Cd atoms, the sequence =A BC ABC= representing a 6R structure.

9.2.1.6. Crystallographic uses of Zhdanov symbols From the Zhdanov symbols of a close-packed structure, it is possible to derive information about the symmetry and lattice type (Verma & Krishna, 1966). Let n and n be the number of positive and negative numerals in the Zhdanov sequence of a given structure. The lattice is rhombohedral if n n 1 mod 3, otherwise it is hexagonal. The sign corresponds to the reverse setting and to the obverse setting of the rhombohedral lattice. Since this criterion is suf®cient for the identi®cation of a rhombohedral structure, the practice of writing three units of identical Zhdanov symbols has been abandoned in recent years (Pandey & Krishna, 1982a). Thus the 15R polytype of SiC is written as (23) rather than 233 . As described in detail by Verma & Krishna (1966), if the Zhdanov symbol consists of an odd set of numbers repeated twice, e.g. (22), (33), (221221) etc., the structure can be shown to possess a 63 axis. For the centre of symmetry at the centre of a sphere or an octahedral void, the Zhdanov symbol will consist of a symmetrical arrangement of numbers of like signs surrounding a single even or odd Zhdanov number, respectively. Thus, the structures (2)32(4)23 and (3)32(5)23 have centres of symmetry of the two types in the numbers within parentheses. For structures with a symmetry plane perpendicular to [00.1], the Zhdanov symbols consist of a symmetrical arrangement of a set of numbers of opposite signs about the space between two succession numbers. Thus, a stacking j522j225j has mirror planes at positions indicated by the vertical lines.

Fig. 9.2.1.8. The relationship between the f.c.c. and the primitive rhombohedral unit cell of the c.c.p. structure.

The use of abridged symbols to describe crystal structures has sometimes led to confusion in deciding the crystallographic equivalence of two polytype structures. For example, the structures (13) and (31) are identical for SiC but not for CdI2 (Jain & Trigunayat, 1977a,b). 9.2.1.7. Structure determination of close-packed layer stackings 9.2.1.7.1. General considerations The different layer stackings (polytypes) of the same material have identical a and b parameters of the direct lattice. The a b reciprocal-lattice net is therefore also the same and is shown in Fig. 9.2.1.9. The reciprocal lattices of these polytypes differ only along the c axis, which is perpendicular to the layers. It is evident from Fig. 9.2.1.9 that for each reciprocal-lattice row parallel to c there are ®ve others with the same value of the radial coordinate . For example, the rows 10:l, 01:l, 11:l, 10:l, and 11:l all have ja j. Owing to symmetry considera01:l, tions, it is suf®cient to record any one of them on X-ray diffraction photographs. The reciprocal-lattice rows hk:l can be classi®ed into two categories according as h k 0 mod 3 or 1 mod 3. Since the atoms in an nH or nR structure lie on three symmetry axes A : 00:100 , B : 00:113; 13 , and C : 00:1 13;13 , the structure factor Fhkl can be split into three parts: Fhkl P Q exp2ih k=3 R exp 2ih k=3; P P Q zB exp2ilzB =n, where P zA exp2ilzA =n, P R zC exp2ilzC =n, and zA =n, zB =n, zC =n are the z coordinates of atoms at A, B, and C sites, respectively. For h k 0 mod 3, Fhkl P Q R

z0

exp2ilz=n;

which is zero except when l 0; n; 2n; . . .. Hence, the re¯ections 00:l, 11:l, 30:l, etc., for which h k 0 mod 3, will be extinguished except when l 0; n; 2n; . . .. Thus, only those hk:l reciprocal-lattice rows for which h k 6 0 mod 3 carry information about the stacking sequence and contain in general re¯ections with l 0; 1; 2; . . ., n 1, etc. It is suf®cient to record any one such row, usually the 10:l row with ja j, on an oscillation, Weissenberg, or precession photograph to obtain information about the lattice type, identity period, space group, and hence the complete structure (Verma & Krishna, 1966).

Fig. 9.2.1.9. The a ±b reciprocal-lattice net for close-packed layer stackings.

756


n P1

9.2. LAYER STACKING 9.2.1.7.2. Determination of the lattice type When the structure has a hexagonal lattice, the positions of spots are symmetrical about the zero layer line on the c-axis oscillation photograph. However, the intensities of the re¯ections on the two sides of the zero layer line are the same only if the structure possesses a 63 axis, and not for the trigonal system. An apparent mirror symmetry perpendicular to the c axis results from the combination of the 63 axis with the centre of symmetry introduced by X-ray diffraction. For a structure with a rhombohedral lattice, the positions of X-ray diffraction spots are not symmetrical about the zero layer line because the hexagonal unit cell is non-primitive causing the re¯ections hkl to be absent when h k l 6 3n n 0; 1; 2; . . .. For the 10:l row, this means that the permitted re¯ections will have l 3n 1, which implies above the zero layer line 10.1, 10.4, 10.7, etc. re¯ections and below the zero layer line 10:2, 10:8, etc. The zero layer line will therefore divide the 10:5, distance between the nearest spots on either side (namely 10.1 approximately in the ratio 1:2. This enables a quick and 10:2) identi®cation of a rhombohedral lattice. It is also possible to identify rhombohedral lattices by the appearance of an apparent `doubling' of spots along the Bernal row lines on a rotation photograph. This is because of the threefold symmetry which makes reciprocal-lattice rows such as 10:l, 11:l, and 01:l identical with each other but different from the other identical The extinction condition for the second set, 01:l, 10:l, and 11:l. 4; 7, etc., which is set requires l 3n 1, i:e: l 2; 5; 8, and 1; different from that for the ®rst set. Consequently, on the rotation photograph, reciprocal-lattice rows with ja j will have spots for l 3n 1 causing the apparent `doubling'. In crystals of layer structures, such as CdI2 , where a-axis oscillation photographs are normally taken, the identi®cation of the rhombohedral lattice is performed by checking for the noncoincidence of the diffraction spots with those for the 2H or 4H structures. In an alternative method, one compares the positions of spots in two rows of the type 10:l and 20:l. This can conveniently be done by taking a Weissenberg photograph (Chadha, 1977). 9.2.1.7.3. Determination of the identity period The number of layers, n, in the hexagonal unit cell can be found by determining the c parameter from the c-axis rotation or oscillation photographs and dividing this by the layer spacing h for that compound which can be found from re¯ections with h k 0 mod 3. The density of reciprocal-lattice points along rows parallel to c depends on the periodicity along the c axis. The larger the identity period along c, the more closely spaced are the diffraction spots along c . In situations where there are not many structural extinctions, n can be determined by counting the number of spacings after which the sequence of relative intensities begins to repeat along the 10:l row of spots on an oscillation or Weissenberg photograph (Krishna & Verma, 1963). If the structure contains a random stacking disorder of close-packed layers (stacking faults), this will effectively make the c parameter in®nite c ! 0 and lead to the production of characteristic continuous diffuse streaks along reciprocal-lattice rows parallel to c for re¯ections with h k 6 0 mod 3 (Wilson, 1942). It is therefore dif®cult to distinguish by X-ray diffraction between structures of very large unresolvable periodicities and those with random stacking faults. Lattice resolution in the electron microscope has been used in recent years to identify such structures (Dubey, Singh & Van Tendeloo, 1977). A better resolution of diffraction spots along the 10:l reciprocal-lattice row can be obtained by using the Laue method. Standard charts

for rapid identi®cation of SiC polytypes from Laue ®lms are available in the literature (Mitchell, 1953). Identity periods as large as 594 layers have been resolved by this method (Honjo, Miyake & Tomita, 1950). Synchrotron radiation has been used for taking Laue photographs of ZnS polytypes (Steinberger, Bordas & Kalman, 1977). 9.2.1.7.4. Determination of the stacking sequence of layers For an nH or 3nR polytype, the n close-packed layers in the unit cell can be stacked in 2n 1 possible ways, all of which cannot be considered for ultimate intensity calculations. A variety of considerations has therefore been used for restricting the number of trial structures. To begin with, symmetry and space-group considerations discussed in Subsection 9.2.1.4 and 9.2.1.5 can considerably reduce the number of trial structures. When the short-period structures act as `basic structures' for the generation of long-period polytypes, the number of trial structures is considerably reduced since the crystallographic unit cells of the latter will contain several units of the small-period structures with faults between or at the end of such units. The basic structure of an unknown polytype can be guessed by noting the intensities of 10:l re¯ections that are maximum near the positions corresponding to the basic structure. If the unknown polytype belongs to a well known structure series, such as (33)n 32 and (33)n 34 based on SiC-6H, empirical rules framed by Mitchell (1953) and Krishna & Verma (1962) can allow the direct identi®cation of the layer-stacking sequence without elaborate intensity calculations. It is possible to restrict the number of probable structures for an unknown polytype on the basis of the faulted matrix model of polytypism for the origin of polytype structures (for details see Pandey & Krishna, 1983). The most probable series of structures as predicted on the basis of this model for SiC contains the numbers 2, 3, 4, 5 and 6 in their Zhdanov sequence (Pandey & Krishna, 1975, 1976a). For CdI2 and PbI2 polytypes, the possible Zhdanov numbers are 1, 2 and 3 (Pandey & Krishna, 1983; Pandey, 1985). On the basis of the faulted matrix model, it is not only possible to restrict the numbers occurring in the Zhdanov sequence but also to restrict drastically the number of trial structures for a new polytype. Structure determination of ZnS polytypes is more dif®cult since they are not based on any simple polytype and any number can appear in the Zhadanov sequence. It has been observed that the birefringence of polytype structures in ZnS varies linearly with the percentage hexagonality (Brafman & Steinberger, 1966), which in turn is related to the number of reversals in the stacking sequence, i.e. the number of numbers in the Zhdanov sequence. This drastically reduces the number of trial structures for ZnS (Brafman, Alexander & Steinberger, 1967). Singh and his co-workers have successfully used lattice imaging in conjunction with X-ray diffraction for determining the structures of long-period polytypes of SiC that are not based on a simple basic structure. After recording X-ray diffraction patterns, single crystals of these polytypes were crushed to yield electron-beam-transparent ¯akes. The one- and two-dimensional lattice images were used to propose the possible structures for the polytypes. Usually this approach leads to a very few possibilities and the correct structure is easily determined by comparing the observed and calculated X-ray intensities for the proposed structures (Dubey & Singh, 1978; Rai, Singh, Dubey & Singh, 1986). Direct methods for the structure determination of polytypes from X-ray data have also been suggested by several workers (Tokonami & Hosoya, 1965; Dornberger-Schiff & Farkas-

757


9. BASIC STRUCTURAL FEATURES Jahnke, 1970; Farkas-Jahnke & Dornberger-Schiff, 1969) and have been reviewed by Farkas-Jahnke (1983). These have been used to derive the structures of ZnS, SiC, and TiS1:7 polytypes. These methods are extremely sensitive to experimental errors in the intensities.

Table 9.2.1.3. Intrinsic fault con®gurations in the 6H A0 B1 C2 A3 C4 B5 ; . . . structure Fault con®guration ABC sequence ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

9.2.1.8. Stacking faults in close-packed structures The two alternative positions for the stacking of successive close-packed layers give rise to the possibility of occurrence of faults where the stacking rule is broken without violating the law of close packing. Such faults are frequently observed in crystals of polytypic materials as well as close-packed martensites of cobalt, noble-metal-based and certain iron-based alloys (Andrade, Chandrasekaran & Delaey 1984; Kabra, Pandey & Lele, 1988a; Nishiyama, 1978; Pandey, 1988). The classical method of classifying stacking faults in 2H and 3C structures as growth and deformation types, depending on whether the fault has resulted as an accident during growth or by shear through the vector s, leads to considerable ambiguities since the same fault con®guration can result from more than one physical process. For a detailed account of the limitations of the notations based on the process of formation, the reader is referred to the articles by Pandey (1984a) and Pandey & Krishna (1982b). Frank (1951) has classi®ed stacking faults as intrinsic or extrinsic purely on geometrical considerations. In intrinsic faults, the perfect stacking sequence on each side of the fault extends right up to the contact plane of the two crystal halves while in extrinsic faults the contact plane does not belong to the stacking sequence on either side of it. In intrinsic faults, the contact plane may be an atomic or non-atomic plane whereas in extrinsic faults the contact plane is always an atomic plane. Instead of contact plane, one can use the concept of fault plane de®ned with respect to the initial stacking sequence. This system of classi®cation is preferable to that based on the process of formation. However, the terms intrinsic and extrinsic have been used in the literature in a very restricted sense by associating these with the precipitation of vacancies and interstitials, respectively (see, for example, Weertman & Weertman, 1984). While the precipitation of vacancies may lead to intrinsic fault con®guration, this is by no means the only process by which intrinsic faults can result. For example, there are geometrically 18 possible intrinsic fault con®gurations in the 6H (33) structure (Pandey & Krishna, 1975) but only two of these can result from the precipitation of vacancies. Similarly, layerdisplacement faults involved in SiC transformations are extrinsic type but do not result from the precipitation of interstitials (see Pandey, Lele & Krishna, 1980a,b,c; Kabra, Pandey & Lele, 1986). It is therefore desirable not to associate the geometrical notation of Frank with any particular process of formation. The intrinsic±extrinsic scheme of classi®cation of faults when used in conjunction with the concept of assigning subscripts to different close-packed layers (Prasad & Lele, 1971; Pandey & Krishna, 1976b) can provide a very compact and unique way of representing intrinsic fault con®gurations even in long-period structures (Pandey, 1984b). We shall brie¯y explain this notation in relation to one hexagonal 6H and one rhombohedral 9R structure. In the 6H ABCACB; . . . or hkkhkk structure, six kinds of layers that can be assigned subscripts 0, 1, 2, 3, 4, and 5 need to be distinguished (Pandey, 1984b). Choosing the 0-type layer in `h' con®guration such that the layer next to it is related through

B B B B B B B B B B B B B B B B B B

C C C C C C C C C C C C C C C C C C

A A A A A A A A A A A A A A A A A A



A0 jj C0 A B C B A C . . . A0 jj C1 A B A C B C . . . A0 jj C2 A C B A B C . . . A0 jj C3 B A C A B C . . . A0 jj C4 B A B C A C . . . A0 jj C5 B C A B A C . . . A B1 jj A0 B C A C B A . . . A B1 jj A1 B C B A C A . . . A B1 jj A2 B A C B C A . . . A B1 jj A3 C B A B C A . . . A B1 jj A4 C B C A B A . . . A B1 jj A5 C A B C B A . . . A B C2 jj B0 C A B A C B . . . A B C2 jj B1 C A C B A B . . . A B C2 jj B2 C B A C A B . . . A B C2 jj B3 A C B C A B . . . A B C2 jj B4 A C A B C B . . . A B C2 jj B5 A B C A C B . . .

I0;0 I0;1 I0;2 I0;3 I0;4 I0;5 I1;0 I1;1 I1;2 I1;3 I1;4 I1;5 I2;0 I2;1 I2;2 I2;3 I2;4 I2;5

Notes: (1) Dotted vertical lines represent the location of the fault plane with respect to the initial stacking sequence on the left-hand side. (2) I0;1 and I2;3 , I0;2 and I1;3 , I1;1 and I2;2 , and I1;4 and I2;5 are crystallographically equivalent.

the shift vector s (which causes cyclic A ! B ! C ! A shift), the perfect 6H structure can be written as h k k h k k h k k h k k . . . A0 B1 C2 A3 C4 B5 A0 B1 C2 A3 C4 B5 . . .. s s s s s s s s s s s There are six crystallographically equivalent ways of writing this structure with the ®rst layer in position A: (i) A0 B1 C2 A3 C4 B5 ; (ii) A1 B2 C3 B4 A5 C0 ; (iii) A2 B3 A4 C5 B0 C1 ; (iv) A3 C4 B5 A0 B1 C2 ; (v) A4 C5 B0 C1 A2 B3 ; and (vi) A5 C0 A1 B2 C3 B4 . Similarly, there are six ways of writing the 6H structure with the starting layer in position B or C. Since an intrinsic fault marks the beginning of a fresh 6H sequence, there can be 36 possible intrinsic fault con®gurations in the 6H ABCACB; . . . structure. All these intrinsic fault con®gurations can be described by symbols like Ir;s , where r and s stand for the subscript of the layer on the left- and right-hand sides of the fault plane while I represents intrinsic. Knowing the two symbols (r and s), one can write down the complete ABC stacking sequence. It may be noted that, of the 36 possible intrinsic fault con®gurations, only 14 are crystallographically indistinguishable (for details, see Pandey, 1984b). This notation can be used for any hexagonal polytype and requires only the identi®cation of various layer types in the structure. For rhombohedral polytypes, one must consider the layer types in both the obverse and the reverse settings. For example, six layer types need to be distinguished in the 9R hhk structure: Obverse:

758



Subscript notation

h h k h h k h h k . . . A0 B1 A2 C0 A1 C2 B0 C1 B2 . . . ; s s s s s s s s

9.2. LAYER STACKING Table 9.2.1.4. Intrinsic fault con®gurations in the 9R A0 B1 A2 C0 A1 C2 B0 C1 B2 ; . . . structure Fault con®guration ABC sequence ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...










Subscript notation

A0 jj C0 A C B C B A B A . . . A0 jj C1 B A B A C A C B . . . A0 jj C2 B C B A B A C A . . . A0 jj C0 B C A C A B A B . . . A0 jj C1 A B A B C B C A . . . A0 jj C2 A C A B A B C B . . . A B1 jj C0 A C B C B A B A . . . A B1 jj C1 B A B A C A C B . . . A B1 jj C2 B C B A B A C A . . . A B1 jj C0 B C A C A B A B . . . A B1 jj C1 A B A B C B C A . . . A B1 jj C2 A C A B A B C B . . . A B A2 jj B0 C B A B A C A C . . . A B A2 jj B1 A C A C B C B A . . . A B A2 jj B2 A B A C A C B C . . . A B A2 jj B0 A B C B C A C A . . . A B A2 jj B1 C A C A B A B C . . . A B A2 jj B2 C B C A C A B A . . .

I0;0 I0;1 I0;2 I0;0 I0;1 I0;2 I1;0 I1;1 I1;2 I1;0 I1;1 I1;2 I2;0 I2;1 I2;2 I2;0 I2;1 I2;2

Note: I0;0 and I1;1 , I0;1 and I1;2 , I0;2 and I2;1 , and I1;2 and I2;0 are crystallographically equivalent.

Reverse: h . . . A0

h k h h k h h k C1 A2 B0 A1 B2 C0 B1 C2 . . . . s s s s s s s s

In the obverse setting, we choose the origin layer (0 type) in the h con®guration such that the next layer is cyclically shifted whereas in the reverse setting the origin layer (0 type) in the h con®guration is related to the next layer through an anticyclic shift. Tables 9.2.1.3 and 9.2.1.4 list the crystallographically unique intrinsic fault con®gurations in the 6H and 9R structures. 9.2.1.8.1. Structure disordered crystals

determination

of

one-dimensionally

Statistical distribution of stacking faults in close-packed structures introduces disorder along the stacking axis of the close-packed layers. As a result, one observes on a singlecrystal diffraction pattern not only normal Bragg scattering near the nodes of the reciprocal lattice of the average structure but also continuous diffuse scattering between the nodes owing to the incomplete destructive interference of scattered rays. Just like the extra polytype re¯ections, the diffuse streaks are also con®ned to only those rows for which h k 6 0 mod 3. A complete description of the real structure of such one-dimensionally disordered polytypes requires knowledge of the average structure as well as a statistical speci®cation of the ¯uctuations due to stacking faults in the electron-density distribution of the average structure. This cannot be accomplished by the usual consideration of the normal Bragg re¯ections alone but requires a careful analysis of the diffuse intensity distribution as well (Pandey, Kabra & Lele, 1986). The ®rst step in the structure determination of onedimensionally disordered structures is the speci®cation of the geometry of stacking faults and their distribution, both of which require postulation of the physical processes responsible for their formation. An entirely random distribution of faults may result during the layer-by-layer growth of a

crystal (Wilson, 1942) or during plastic deformation (Paterson, 1952). On the other hand, when faults bring about the change in the stacking sequence of layers during solid-state transformations, their distribution is non-random (Pandey, Lele & Krishna, 1980a,b,c; Pandey & Lele, 1986a,b; Kabra, Pandey & Lele, 1986). Unlike growth faults, which are accidentally introduced in a sequential fashion from one end of the stack of layers to the other during the actual crystal growth, stacking faults involved in solid-state transformations are introduced in a random space and time sequence (Kabra, Pandey & Lele, 1988b). Since the pioneering work of Wilson (1942), several different techniques have been advanced for the calculation of intensity distributions along diffuse streaks making use of Markovian chains, random walk, stochastic matrices, and the Paterson function for random and nonrandom distributions of stacking faults on the assumption that these are introduced in a sequential fashion (Hendricks & Teller, 1942; Jagodzinski 1949a,b; Kakinoki & Komura, 1954; Johnson, 1963; Prasad & Lele, 1971; Cowley, 1976; Pandey, Lele & Krishna, 1980a,b). The limitations of these methods for situations where non-randomly distributed faults are introduced in the random space and time sequence have led to the use of Monte Carlo techniques for the numerical calculation of pair correlations whose Fourier transforms directly yield the intensity distributions (Kabra & Pandey, 1988). The correctness of the proposed model for disorder can be veri®ed by comparing the theoretically calculated intensity distributions with those experimentally observed. This step is in principle analogous to the comparison of the observed Bragg intensities with those calculated for a proposed structure in the structure determination of regularly ordered layer stackings. This comparison cannot, however, be performed in a straightforward manner for one-dimensionally disordered crystals due to special problems in the measurement of diffuse intensities using a single-crystal diffractometer, stemming from incident-beam divergence, ®nite size of the detector slit, and multiple scattering. The problems due to incident-beam divergence in the measurement of the diffuse intensity distributions were ®rst

759


9. BASIC STRUCTURAL FEATURES pointed out by Pandey & Krishna (1977) and suitable correction factors have recently been derived by Pandey, Prasad, Lele & Gauthier (1987). A satisfactory solution to the problem of structure determination of one-dimensionally disordered stackings must await proper understanding of all other factors that may in¯uence the true diffraction pro®les. 9.2.2. Layer stacking in general polytypic structures Æ urovicÆ) (By S. D

Thompson (1981) regards polytypes as àrising through different ways of stacking structurally compatible tabular units . . . [provided that this] . . . should not alter the chemistry of the crystal as a whole', Angel (1986) demands that `polytypism arises from different modes of stacking of one or more structurally compatible modules', dropping thus any chemical constraints and allowing also for rod- and block-like modules. The present of®cial de®nition (Guinier et al., 1984) reads: `Àn element or compound is polytypic if it occurs in several different structural modi®cations, each of which may be regarded as built up by stacking layers of (nearly) identical structure and composition, and if the modi®cations differ only in their stacking sequence. Polytypism is a special case of polymorphism: the two-dimensional translations within the layers are (essentially) preserved whereas the lattice spacings normal to the layers vary between polytypes and are indicative of the stacking period. No such restrictions apply to polymorphism. Comment: The above de®nition is designed to be suf®ciently general to make polytypism a useful concept. There is increasing evidence that some polytypic structures are characterized either by small deviations from stoichiometry or by small amounts of impurities. (In the case of certain minerals like clays, micas and ferrites, deviations in composition up to 0.25 atoms per formula unit are permitted within the same polytypic series: two layer structures that differ by more than this amount should not be called polytypic.) Likewise, layers in different polytypic structures may exhibit slight structural differences and may not be isomorphic in the strict crystallographic sense. The Ad-Hoc Committee is aware that the de®nition of polytypism above is probably too wide since it includes, for example, the turbostratic form of graphite as well as mixed-layer phyllosilicates. However, the sequence and stacking of layers in a polytype are always subject to wellde®ned limitations. On the other hand, a more general de®nition of polytypism that includes `rod' and `block' polytypes may become necessary in the future.''

9.2.2.1. The notion of polytypism The common property of the structures described in Section 9.2.1 was the stacking ambiguity of adjacent layer-like structural units. This has been explained by the geometrical properties of close packing of equal spheres, and the different modi®cations thus obtained have been called polytypes. This phenomenon was ®rst recognized by Baumhauer (1912, 1915) as a result of his investigations of many SiC single crystals by optical goniometry. Among these, he discovered three types and his observations were formulated in ®ve statements: (1) all three types originate simultaneously in the same melt and seemingly also under the same, or nearly the same, conditions; (2) they can be related in a simple way to the same axial ratio (each within an individual primary series); (3) any two types (I and II, II and III) have certain faces in common but, except the basal face, there is no face occurring simultaneously in all three types; (4) the crystals belonging to different, but also to all three, types often form intergrowths with parallel axes; (5) any of the three types exhibits a typical X-ray diffraction pattern and thus also an individual molecular or atomic structure. Baumhauer recognized the special role of these types within modi®cations of the same substance and called this phenomenon polytypism ± a special case of polymorphism. The later determination of the crystal structures of Baumhauer's three types indicated that his results can be interpreted by a family of structures consisting of identical layers with hexagonal symmetry and differing only in their stacking mode. The stipulation that the individual polytypes grow from the same system and under (nearly) the same conditions in¯uenced for years the investigation of polytypes because it logically led to the question of their growth mechanism. In the following years, many new polytypic substances have been found. Their crystal structures revealed that polytypism is restricted neither to close packings nor to heterodesmic `layered structures' (e.g. CdI2 or GaSe; cf. homodesmic SiC or ZnS; see xx9:2:1:2:2 to 9.2.1.2.4), and that the reasons for a stacking ambiguity lie in the crystal chemistry ± in all cases the geometric nearest-neighbour relations between adjacent layers are preserved. The preservation of the bulk chemical composition was not questioned. Some discomfort has arisen from re®nements of the structures of various phyllosilicates. Here especially the micas exhibit a large variety of isomorphous replacements and it turns out that a certain chemical composition stabilizes certain polytypes, excludes others, and that the layers constituting polytypic structures need not be of the same kind. But subsequently the opinion prevailed that the sequence of individual kinds of layers in polytypes of the same family should remain the same and that the relative positions of adjacent layers cannot be completely random (e.g. Zvyagin, 1988). The postulates declared mixedlayer and turbostratic structures as non-polytypic. All this led to certain controversies about the notion of polytypism. While

This de®nition was elaborated as a compromise between members of the IUCr Ad-Hoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures. It is a slightly modi®ed de®nition proposed by the IMA/IUCr Joint Committee on Nomenclature (Bailey et al., 1977), which was the target of Angel's (1986) objections. The of®cial de®nition has indeed its shortcomings, but not so much in its restrictiveness concerning the chemical composition and structural rigidity of layers, because this can be overcome by a proper degree of abstraction (see below). More critical is the fact that it is not `geometric' enough. It speci®es neither the `layers' (except for their two-dimensional periodicity), nor the limitations concerning their sequence and stacking mode, and it does not state the conditions under which a polytype belongs to a family. Very impressive evidence that even polytypes that are in keeping with the ®rst Baumhauer's statement may not have exactly the same composition and the structure of their constituting layers cannot be identical has been provided by studies on SiC carried out at the Leningrad Electrotechnical Institute (Sorokin, Tairov, Tsvetkov & Chernov, 1982; Tsvetkov, 1982). They indicate also that each periodic polytype is sensu stricto an individual polymorph. Therefore, it appears that the question whether some real polytypes belong to the same

760


9.2. LAYER STACKING family depends mainly on the idealization and/or abstraction level, relevant to a concrete purpose. This very idealization and/or abstraction process caused the term polytype to become also an abstract notion meaning a structural type with relevant geometrical properties,* belonging to an abstract family whose members consist of layers with identical structure and keep identical bulk composition. Such an abstract notion lies at the root of all systemization and classi®cation schemes of polytypes. A still higher degree of abstraction has been achieved by Dornberger-Schiff (1964, 1966, 1979) who abstracted from chemical composition completely and investigated the manifestation of crystallochemical reasons for polytypism in the symmetry of layers and symmetry relations between layers. Her theory of OD (order±disorder) structures is thus a theory of symmetry of polytypes, playing here a role similar to that of group theory in traditional crystallography. In the next section, a brief account of basic terms, de®nitions, and logical constructions of OD theory will be given, together with its contribution to a geometrical de®nition of polytypism.

symmetry principle: they consist of equivalent layers (i.e. layers of the same kind) and of equivalent layer pairs, and, in keeping with these stipulations, any layer can be stacked onto its predecessor in two ways. Keeping in mind that the layer pairs that are geometrically equivalent are also energetically equivalent, and neglecting in the ®rst approximation the interactions between a given layer and the next-but-one layer, we infer that all structures built according to these principles are also energetically equivalent and thus equally likely to appear. It is important to realize that the above symmetry considerations hold not only for close packing of spheres but also for any conceivable structure consisting of two-dimensionally periodic layers with symmetry P6=mmm and containing pairs of adjacent layers with symmetry P3m1. Moreover, the OD theory sets a quantitative stipulation for the relation between any two adjacent layers: they have to remain geometrically equivalent in any polytype belonging to a family. This is far more exact than the description: `the stacking of layers is such that it preserves the nearest-neighbour relationships'. 9.2.2.2.2. Polytype families and OD groupoid families

9.2.2.2. Symmetry aspects of polytypism 9.2.2.2.1. Close packing of spheres Polytypism of structures based on close packing of equal spheres (note this idealization) is explained by the fact that the spheres of any layer can be placed either in all the voids 5 of the preceding layer, or in all the voids 4 ± not in both because of steric hindrance (Section 9.2.1, Fig. 9.2.1.1). A closer look reveals that the two voids are geometrically (but not translationally) equivalent. This implies that the two possible pairs of adjacent layers, say AB and AC, are geometrically equivalent too ± this equivalence is brought about e.g. by a re¯ection in any plane perpendicular to the layers and passing through the centres of mutually contacting spheres A: such a re¯ection transforms the layer A into itself, and B into C, and vice versa. Another important point is that the symmetry proper of any layer is described by the layer group P6=mmm,y and that the relative position of any two adjacent layers is such that only some of the 24 symmetry operations of that layer group remain valid for the pair. It is easy to see that 12 out of the total of 24 transformations do not change the z coordinate of any starting point, and that these operations constitute a subgroup of the index [2]. These are the so-called operations. The remaining 12 operations change any z into z, thus turning the layer upside down; they constitute a coset. The latter are called operations. Out of the 12 operations, only 6 are valid for the layer pair. One says that only these 6 operations have a continuation in the adjacent layer. Let us denote the general multiplicity of the group of operations of a single layer by N, and that of the subgroup of these operations with a continuation in the adjacent layer by F: then the number Z of positions of the adjacent layer leading to geometrically equivalent layer pairs is given by Z N=F (Dornberger-Schiff, 1964, pp. 32 ff.); in our case, Z 12=6 2 (Fig. 9.2.2.1). This is the so-called NFZ relation, valid with only minor alterations for all categories of OD structures x9:2:2:2:7. It follows that all conceivable structures based on close packing of equal spheres are built on the same * This is an interesting example of how a development in a scienti®c discipline in¯uences semantics: e.g. when speaking of a 6H polytype of SiC, one has very often in mind a characteristic sequence of Si±C layers rather than deviations from stoichiometry, presence and distribution of foreign atoms, distortion of coordination tetrahedra, etc. y The direction in which there is no periodicity is indicated by parentheses (Dornberger-Schiff, 1959).

All polytypes of a substance built on the same structural principle are said to belong to the same family. All polytypic structures, even of different substances, built according to the same symmetry principle also belong to a family, but different from the previous one since it includes structures of various polytype families, e.g. SiC, ZnS, AgI, which differ in their composition, lattice dimensions, etc. Such a family has been called an OD groupoid family; its members differ only in the relative distribution of coincidence operations* describing the respective symmetries, irrespective of the crystallochemical content. These coincidence operations can be total or partial (local) and their set constitutes a groupoid (Dornberger-Schiff, 1964, pp. 16 ff.; Fichtner, 1965, 1977). Any polytype (abstract) belonging to such a family has its own stacking of layers, and its symmetry can be described by the appropriate individual groupoid. Strictly speaking, these groupoids are the members of an OD groupoid family. Let us recall that any space group * A coincidence operation is a space transformation (called also isometric mapping, isometry, or motion), which preserves distances between any two points of the given object.

Fig. 9.2.2.1. Symmetry interpretation of close packings of equal spheres. The layer group of a single layer, the subgroup of its operations, and the number of asymmetric units N per unit mesh of the former, are given at the top right. The operations that have a continuation for the pair of adjacent layers, the layer group of the pair, and the value of F are indicated at the bottom right.

761


9. BASIC STRUCTURAL FEATURES consists of total coincidence operations only, which therefore become the symmetry operations for the entire structure. 9.2.2.2.3. MDO polytypes Any family of polytypes theoretically contains an in®nite number of periodic (Ross, Takeda & Wones, 1966; Mogami, Nomura, Miyamoto, Takeda & Sadanaga, 1978; McLarnan, 1981a,b,c) and non-periodic structures. The periodic polytypes, in turn, can again be subdivided into two groups, the `privileged' polytypes and the remaining ones, and it depends on the approach as to how this is done. Experimentalists single out those polytypes that occur most frequently, and call them basic. Theorists try to predict basic polytypes, e.g. by means of geometrical and/or crystallochemical considerations. Such polytypes have been called simple, standard, or regular. Sometimes the agreement is very good, sometimes not. The OD theory pays special attention to those polytypes in which all layer triples, quadruples, etc., are geometrically equivalent or, at least, which contain the smallest possible number of kinds of these units. They have been called polytypes with maximum degree of order, or MDO polytypes. The general philosophy behind the MDO polytypes is simple: all interatomic bonding forces decrease rapidly with increasing distance. Therefore, the forces between atoms of adjacent layers are decisive for the build-up of a polytype. Since the pairs of adjacent layers remain geometrically equivalent in all polytypes of a given family, these polytypes are in the ®rst approximation also energetically equivalent. However, if the longer-range interactions are also considered, then it becomes evident that layer triples such as ABA and ABC in close-packed structures are, in general, energetically non-equivalent because they are also geometrically non-equivalent. Even though these forces are much weaker than those between adjacent layers, they may not be negligible and, therefore, under given crystallization conditions either one or the other kind of triples becomes energetically more favourable. It will occur again and again in the polytype thus formed, and not intermixed with the other kind. Such structures are ± as a rule ± sensitive to conditions of crystallization, and small ¯uctuations of these may reverse the energetical preferences, creating stacking faults and twinnings. This is why many polytypic substances exhibit non-periodicity. As regards the close packing of spheres, the well known cubic and hexagonal polytypes ABCABC . . . and ABAB . . ., respectively, are MDO polytypes; the ®rst contains only the triples ABC, the second only the triples ABA. Evidently, the MDO philosophy holds for a layer-by-layer rather than for a spiral growth mechanism. Since the symmetry principle of polytypic structures may differ considerably from that of close packing of equal spheres, the OD theory contains exact algorithms for the derivation of MDO polytypes in any category (DornbergerSchiff, 1982; Dornberger-Schiff & Grell, 1982a).

or b=4 relative to its predecessor, since the re¯ection across :m: transforms any given layer into itself and the adjacent layer from one possible position into the other. These two positions follow also from the NFZ relation: N 2, F 1 [the layer group of the pair of adjacent layers is P111] and thus Z 2. The layers are all equivalent and accordingly there must also be two coincidence operations transforming any layer into the adjacent one. The ®rst operation is evidently the translation, the second is the glide re¯ection. If any of these becomes total for the remaining part of the structure, we obtain a polytype with all layer triples equivalent, i.e. a MDO polytype. The polytype a (Fig. 9.2.2.2) is one of them: the translation t a0 b=4 is the total operation (ja0 j is the distance between adjacent layers). It has basis vectors a1 a0 b=4, b1 b, c1 c, space group P111, Ramsdell symbol 1A,* HaÈgg symbol j j. This polytype also has its enantiomorphous counterpart with HaÈgg symbol j j. In the other polytype b (Fig. 9.2.2.2), the glide re¯ection is the total operation. The basis vectors of the polytype are * According to Guinier et al. (1984), triclinic polytypes should be designated A (anorthic) in their Ramsdell symbols.

9.2.2.2.4. Some geometrical properties of OD structures As already pointed out, all relevant geometrical properties of a polytype family can be deduced from its symmetry principle. Let us thus consider a hypothetical simple family in which we shall disregard any concrete atomic arrangements and use geometrical ®gures with the appropriate symmetry instead. Three periodic polytypes are shown in Fig. 9.2.2.2 (left-hand side). Any member of this family consists of equivalent layers perpendicular to the plane of the drawing, with symmetry P1m1. The symmetry of layers is indicated by isosceles triangles with a mirror plane :m:. All pairs of adjacent layers are also equivalent, no matter whether a layer is shifted by b=4

Fig. 9.2.2.2. Schematic representation of three structures belonging to the OD groupoid family P1m1j1, y 0:25 (left). The layers are perpendicular to the plane of the drawing and their constituent atomic con®gurations are represented by isosceles triangles with symmetry :m:. All structures are related to a common orthogonal four-layer cell with a 4a0 . The hk0 nets in reciprocal space corresponding to these structures are shown on the right and the diffraction indices refer also to the common cell. Family diffractions common to all members ^ and the characteristic diffractions for of this family k 2k individual polytypes k 2k 1 are indicated by open and solid circles, respectively.

762


9.2. LAYER STACKING a2 2a0 , b2 b, c2 c, space group P1a1, Ramsdell symbol 2M, HaÈgg symbol j j. The equivalence of all layer triples in either of these polytypes is evident. The third polytype c (Fig. 9.2.2.2) is not a MDO polytype because it contains two kinds of layer triples, whereas it is possible to construct a polytype of this family containing only a selection of these. The polytype is again monoclinic with basis vectors a3 4a0 , b3 b, c3 c, space group P1a1, Ramsdell symbol 4M, and HaÈgg symbol j j. Evidently, the partial mirror plane is crucial for the polytypism of this family. And yet the space group of none of its periodic members can contain it ± simply because it can never become total. The space-group symbols thus leave some of the most important properties of periodic polytypes unnoticed. Moreover, the atomic coordinates of different polytypes expressed in terms of the respective lattice geometries cannot be immediately compared. And, ®nally, for non-periodic members of a family, a space-group symbol cannot be written at all. This is why the OD theory gives a special symbol indicating the symmetry proper of individual layers (l symmetry) as well as the coincidence operations transforming a layer into the adjacent one ( symmetry). The symbol of the OD groupoid family of our hypothetical example thus consists of two lines (Dornberger-Schiff, 1964, pp. 41 ff.; Fichtner, 1979a,b): P1 m f1 a2

1 1g

l symmetry symmetry;

where the unusual subscript 2 indicates that the glide re¯ection transforms the given layer into the subsequent one. It is possible to write such a symbol for any OD groupoid family for equivalent layers, and thus also for the close packing of spheres. However, keeping in mind that the number of asymmetric units here is 24 (l symmetry), one has to indicate also 24 operations, which is instructive but unwieldy. This is why Fichtner (1980) proposed simpli®ed one-line symbols, containing full l symmetry and only the rotational part of any one of the operations plus its translational components. Accordingly, the symbol of our hypothetical family reads: P1m1j1, y 0:25; for the family of close packings of equal spheres: P6=mmmj1, x 2=3, y 1=3 (the layers are in both cases translationally equivalent and the rotational part of a translation is the identity). An OD groupoid family symbol should not be confused with a polytype symbol, which gives information about the structure of Æ urovicÆ & Zvyagin, an individual polytype (Dornberger-Schiff, D 1982; Guinier et al., 1984). 9.2.2.2.5. Diffraction pattern ± structure analysis Let us now consider schematic diffraction patterns of the three structures on the right-hand side of Fig. 9.2.2.2. It can be seen that, while being in general different, they contain a common subset of diffractions with k 2k^ ± these, normalized to a constant number of layers, have the same distribution of intensities and monoclinic symmetry. This follows from the fact that they correspond to the so-called superposition structure with basis vectors A 2a0 , B b=2, C c, and space group C1m1. It is a ®ctitious structure that can be obtained from any of the structures in Fig. 9.2.2.2 as a normalized sum of the structure in its given position and in a position shifted by b=2, thus ^ xyz 12 xyz x; y 1=2; z:

Evidently, this holds for all members of the family, including the non-periodic ones. In general, the superposition structure is obtained by simultaneous realization of all Z possible positions of all OD layers in any member of the family (Dornberger-Schiff, 1964, p. 54). As a consequence, its symmetry can be obtained by completing any of the family groupoids to a group (Fichtner, 1977). This structure is by de®nition periodic and common to all members of the family. Thus, the corresponding diffractions are also always sharp, common, and characteristic for the family. They are called family diffractions. Diffractions with k 2k^ 1 are characteristic for individual members of the family. They are sharp for periodic polytypes but appear as diffuse streaks for non-periodic ones. Owing to the C centring of the superposition structure, only diffractions with h^ k^ 2n are present. It follows that 0k^ ^l diffractions are present only for k^ 2n, which, in an indexing referring to the actual b vector reads: 0kl present only for k 4n. This is an example of non-space-group absences exhibited by many polytypic structures. They can be used for the determination of the OD groupoid family (Dornberger-Schiff & Fichtner, 1972). There is no routine method for the determination of the structural principle of an OD structure. It is easiest when one has at one's disposal many different (at least two) periodic polytypes of the same family with structures solved by current methods. It is then possible to compare these structures, determine equivalent regions in them (Grell, 1984), and analyse partial symmetries. This results in an OD interpretation of the substance and a description of its polytypism. Sometimes it is possible to arrive at an OD interpretation from one periodic structure, but this necessitates experience in the recognition of the partial symmetry and prediction of potential polytypism (Merlino, Orlandi, Perchiazzi, Basso & Palenzona, 1989). The determination of the structural principle is complex if only disordered polytypes occur. Then ± as a rule ± the superposition structure is solved ®rst by current methods. The actual structure of layers and relations between them can then be determined from the intensity distribution along diffuse streaks (for more details and references see Jagodzinski, 1964; Sedlacek, Kuban & Backhaus, 1987; MuÈller & Conradi, 1986). High-resolution electron microscopy can also be successfully applied ± see Subsection 9.2.2.4. 9.2.2.2.6. The vicinity condition A polytype family contains periodic as well as non-periodic members. The latter are as important as the former, since the very fact that they can be non-periodic carries important crystallochemical information. Non-periodic polytypes do not comply with the classical de®nition of crystals, but we believe that this de®nition should be generalized to include rather than exclude non-periodic polytypes from the world of crystals (Dornberger-Schiff & Grell, 1982b). The OD theory places them, together with the periodic ones, in the hierarchy of the so-called VC structures. The reason for this is that all periodic structures, even the non-polytypic ones, can be thought of as consisting of disjunct, two-dimensionally periodic slabs, the VC layers, which are stacked together according to three rules called the vicinity condition (VC) (Dornberger-Schiff, 1964, pp. 29 ff., 1979; Dornberger-Schiff & Fichtner, 1972): VC layers are either geometrically equivalent or, if not, they are relatively few in kind; translation groups of all VC layers are either identical or they have a common subgroup;

763


9. BASIC STRUCTURAL FEATURES equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent [for a more detailed speci®cation and explanation see Dornberger-Schiff (1979)]. If the stacking of VC layers is unambiguous, traditional threedimensionally periodic structures result ( fully ordered structures). OD structures are VC structures in which the stacking of VC layers is ambiguous at every layer boundary Z > 1. The corresponding VC layers then become OD layers. OD layers are, in general, not identical with crystallochemical layers; they may contain half-atoms at their boundaries. In this context, they are analogous with unit cells in traditional crystallography, which may also contain parts of atoms at their boundaries. However, the choice of OD layers is not absolute: it depends on the polytypism, either actually observed or reasonably anticipated, on the degree of symmetry idealization, and other circumstances (Grell, 1984). 9.2.2.2.7. Categories of OD structures Any OD layer is two-dimensionally periodic. Thus, a unit mesh can be chosen according to the conventional rules for the corresponding layer group; the corresponding vectors or their linear combinations (Zvyagin & Fichtner, 1986) yield the basis vectors parallel to the layer plane and thus also their lengths as units for fractional atomic coordinates. But, in general, there is no periodicity in the direction perpendicular to the layer plane and it is thus necessary to de®ne the corresponding unit length in some other way. This depends on the symmetry principle of the family in question ± or, more narrowly, on the category to which this family belongs. OD structures can be built of equivalent layers or contain layers of several kinds. The rule of the VC implies that a projection of any OD structure ± periodic or not ± on the stacking direction is periodic. This period, called repeat unit, is the required unit length. 9.2.2.2.7.1. OD structures of equivalent layers If the OD layers are equivalent then they are either all polar or all non-polar in the stacking direction. Any two adjacent polar layers can be related either by operations only, or by operations only. For non-polar layers, the operations are both and . Accordingly, there are three categories of OD structures of equivalent layers. They are shown schematically in Fig. 9.2.2.3; the character of the corresponding l and operations is as follows (Dornberger-Schiff, 1964, pp. 24 ff.):

Fig. 9.2.2.3. Schematic examples of the three categories of OD structures consisting of equivalent layers (perpendicular to the plane of the drawing): a category I ± OD layers non-polar in the stacking direction; b category II ± polar OD layers, all with the same sense of polarity; c category III ± polar OD layers with regularly alternating sense of polarity. The position of planes is indicated.

l operations operations

category II

category III

Category II is the simplest: the OD layers are polar and all with the same sense of polarity (they are -equivalent); our hypothetical example given in x9:2:2:2:4 belongs to this category. The layers can thus exhibit only one of the 17 polar layer groups. The projection of any vector between two -equivalent points in two adjacent layers on the stacking direction (perpendicular to the layer planes) is the repeat unit and it is denoted by c0 , a0 , or b0 depending on whether the basis vectors in the layer plane are ab, bc, or ca, respectively. The choice of origin in the stacking direction is arbitrary but preferably so that the z coordinates of atoms within a layer are positive. Examples are SiC, ZnS, and AgI. OD layers in category I are non-polar and they can thus exhibit any of the 63 non-polar layer groups. Inspection of Fig. 9.2.2.3a reveals that the symmetry elements representing the l± operations (i.e. the operations turning a layer upside down) can lie only in one plane called the layer plane. Similarly, the symmetry elements representing the ± operations (i.e. the operations converting a layer into the adjacent one) also lie in one plane, located exactly halfway between two nearest layer planes. These two kinds of planes are called planes. The distance between two nearest layer planes is the repeat unit c0 . Examples are close packing of equal spheres, GaSe, -wollastonite (Yamanaka & Mori, 1981), -wollastonite (Ito, Sadanaga, TakeÂuchi & Tokonami, 1969), K3 [M(CN)6 ] (Jagner, 1985), and many others. The OD structures belonging to the above two categories contain pairs of adjacent layers, all equivalent. This does not apply for structures of category III, which consist of polar layers that are converted into their neighbours by operations. It is evident (Fig. 9.2.2.3c) that two kinds of pairs of adjacent layers are needed to build any such structure. It follows that only even-numbered layers can be mutually -equivalent and the same holds for odd-numbered layers. There are only ± planes in these structures, and again they

Fig. 9.2.2.4. Schematic examples of the four categories of OD structures consisting of more than one kind of layer (perpendicular to the plane of the drawing). Equivalent OD layers are represented by equivalent symbolic ®gures. a Category I ± three kinds of OD layers: one kind L25n is non-polar, the remaining two are polar. One and only one kind of non-polar layer is possible in this category. b Category II ± three kinds of polar OD layers; their triples are polar and retain their sense of polarity in the stacking direction. c Category III ± three kinds of polar OD layers; their triples are polar and regularly change their sense of polarity in the stacking direction. d Category IV ± three kinds of OD layers: two kinds are non-polar L14n and L34n , one kind is polar. Two and only two kinds of nonpolar layers are possible in this category. The position of planes is indicated.

764


category I and and

9.2. LAYER STACKING are of two kinds; the origin can be placed in either of them. c0 is the distance between two nearest planes of the same kind, and slabs of this thickness contain two OD layers. There are three examples for this category known to date: foshagite Æ urovicÆ, 1968), and (Gard & Taylor, 1960), -Hg3 S2 Cl2 (D 2,2-aziridinedicarboxamide (Fichtner & Grell, 1984). 9.2.2.2.7.2. OD structures with more than one kind of layer If an OD structure consists of N > 1 kinds of OD layers, then it can be shown (Dornberger-Schiff, 1964, pp. 64 ff.) that it can fall into one of four categories, according to the polarity or nonpolarity of its constituent layers and their sequence. These are shown schematically in Fig. 9.2.2.4; the character of the corresponding l and operations is

A slab of thickness c0 containing the N non-equivalent polar OD layers in the sequence as they appear in a given structure of category II represents completely its composition. In the remaining three categories, a slab with thickness c0 =2, the polar part of the structure contained between two adjacent planes, suf®ces. Such slabs are higher structural units for OD structures of more than one kind of layer and have been called OD packets. An OD packet is thus de®ned as the smallest continuous part of an OD structure that is periodic in two dimensions and which represents its composition completely Æ urovicÆ, 1974a). (D The hierarchy of VC structures is shown in Fig. 9.2.2.5.

category I

category II

category III

category IV

l operations

and (one set) (N 1 sets)

(N sets)

(N sets)

and (two sets) (N 2 sets)

operations

(one set)

none

(two sets)

none:

Here also category II is the simplest. The structures consist of N kinds of cyclically recurring polar layers whose sense of polarity remains unchanged (Fig. 9.2.2.4b). The choice of origin in the stacking direction is arbitrary; c0 is the projection on this direction of the shortest vector between two -equivalent points ± a slab of this thickness contains all N OD layers of different kinds. Examples are the structures of the serpentine±kaolin group. Structures of category III also consist of polar layers but, in contrast to category II, the N-tuples containing all N different OD layers each alternate regularly the sense of their polarity in the stacking direction. Accordingly (Fig. 9.2.2.4c), there are two kinds of ± planes and two kinds of pairs of equivalent adjacent layers in these structures. The origin can be placed in either of the two planes. c0 is the distance between the nearest two equivalent planes; a slab with this thickness contains 2 N non-equivalent OD layers. No representative of this category is known to date. The structures of category I contain one, and only one, kind of non-polar layer, the remaining N 1 kinds are polar and alternate in their sense of polarity along the stacking direction (Fig. 9.2.2.4a). Again, there are two kinds of planes here, but one is a l± plane (the layer plane of the non-polar OD layer), the other is a ± plane. These structures thus contain only one kind of pair of equivalent adjacent layers. The origin is placed in the l± plane. c0 is the distance between the nearest two equivalent planes and a slab with this thickness contains 2 N 1 nonequivalent polar OD layers plus one entire non-polar layer. Examples are the MX2 compounds (CdI2 , MoS2 , etc.) and the talc±pyrophyllite group. The structures of category IV contain two, and only two, kinds of non-polar layers. The remaining N 2 kinds are polar and alternate in their sense of polarity along the stacking direction (Fig. 9.2.2.4d). Both kinds of planes are l± planes, identical with the layer planes of the non-polar OD layers; the origin can be placed in any one of them. c0 is chosen as in categories I and III. A slab with this thickness contains 2 N 2 nonequivalent polar layers plus the two non-polar layers. Examples are micas, chlorites, vermiculites, etc. OD structures containing N > 1 kinds of layers need special symbols for their OD groupoid families (Grell & DornbergerSchiff, 1982).

9.2.2.2.8. Desymmetrization of OD structures If a fully ordered structure is re®ned, using the space group determined from the systematic absences in its diffraction pattern and then by using some of its subgroups, serious discrepancies are only rarely encountered. Space groups thus characterize the general symmetry pattern quite well, even in real crystals. However, experience with re®ned periodic polytypic structures has revealed that there are always signi®cant deviations from the OD symmetry and, moreover, even the atomic coordinates within OD layers in different polytypes of the same family may differ from one another. The OD symmetry thus appears as only an approximation to the actual symmetry pattern of polytypes. This phenomenon was called desymmetrization of OD structures Æ urovicÆ, 1974b, 1979). (D When trying to understand this phenomenon, let us recall the is an expression of the structure of rock salt. Its symmetry Fm3m energetically most favourable relative position of Na and Cl ions in this structure ± the right angles follow from the symmetry. Since the whole structure is cubic, we cannot expect that the environment of any building unit, e.g. of any octahedron NaCl6 , would exercise on it an in¯uence that would decrease its symmetry; the symmetries of these units and of the whole structure are not àntagonistic'.

Fig. 9.2.2.5. Hierarchy of VC structures indicating the position of OD structures within it.

765


9. BASIC STRUCTURAL FEATURES Not so in OD structures, where any OD layer is by de®nition situated in a disturbing environment because its symmetry does not conform to that of the entire structure. Àntagonistic' relations between these symmetries are most drastic in pure MDO structures because of the regular sequence of layers. The partial symmetry operations become irrelevant and the OD groupoid degenerates into the corresponding space group. The more disordered an OD structure is, the smaller become the disturbing effects that the environment exercises on an OD layer. These can be, at least statistically, neutralized by random positions of neighbouring layers so that partial symmetry operations can retain their relevance throughout the structure. This can be expressed in the form of a paradox: the less periodic an OD structure is, the more symmetric it appears. Despite desymmetrization, the OD theory remains a geometrical theory that can handle properly the general symmetry pattern of polytypes (which group theory cannot). It establishes a symmetry norm with which deviations observed in real polytypes can be compared. Owing to the high abstraction power of OD considerations, systematics of entire families of polytypes at various degree-of-idealization levels can be worked out, yielding thus a common point of view for their treatment. 9.2.2.2.9. Concluding remarks Although very general physical principles (OD philosophy, MDO philosophy) underlie the OD theory, it is mainly a geometrical theory, suitable for a description of the symmetry of polytypes and their families rather than for an explanation of polytypism. It thus does not compete with crystal chemistry, but cooperates with it, in analogy with traditional crystallography, where group theory does not compete with crystal chemistry. When speaking of polytypes, one should always be aware, whether one has in mind a concrete real polytype ± more or less in Baumhauer's sense ± or an abstract polytype as a structural type (Subsection 9.2.2.1). A substance can, in general, exist in the form of various polymorphs and/or polytypes of one or several families. Since polytypes of the same family differ only slightly in their crystal energy (Verma & Krishna, 1966), an entire family can be considered as an energetic analogue to one polymorph. As a rule, polytypes belonging to different families of the same substance do not co-exist. Al(OH)3 may serve as an example for two different families: the bayerite family, in which the adjacent planes of OH groups are stacked according to the principle of close packing (Zvyagin et al., 1979), and the gibbsite± nordstrandite family in which these groups coincide in the normal projection.* Another example is the phyllosilicates x9:2:2:3:1. The compound Hg3 S2 Cl2 , on the other hand, is known to yield two polymorphs and (Carlson, 1967; Frueh & Æ urovicÆ, 1968). Gray, 1968) and one OD family of structures (D As far as the de®nition of layer polytypism is concerned, OD theory can contribute speci®cations about the layers themselves and the geometrical rules for their stacking within a family (all incorporated in the vicinity condition). A possible de®nition might then read: Polytypism is a special case of polymorphism, such that the individual polymorphs (called polytypes) may be regarded as arising through different modes of stacking layer-like structural * Sandwiches with composition Al(OH)3 (similar to those in CdI2 ) are the same in both families, but their stacking mode is different. This and similar situations in other substances might have been the reason for distinguishing between `polytype diversity' and ÒD diversity' (Zvyagin, 1988).

units. The layers and their stackings are limited by the vicinity condition. All polytypes built on the same structural principle belong to a family; this depends on the degree of a structural and/or compositional idealization. Geometrical theories concerning rod and block polytypism have not yet been elaborated, the main reason is the dif®culty of formulating properly the vicinity condition (Sedlacek, Grell & Dornberger-Schiff, private communications). But such structures are known. Examples are the structures of tobermorite (Hamid, 1981) and of manganese(III) hydrogenbis(orthophosphite) dihydrate (CõÂsarÆovaÂ, NovaÂk & PetrÆÂõcÆek, 1982). Both structures can be thought of as consisting of a three-dimensionally periodic framework of certain atoms into which one-dimensionally periodic chains and aperiodic ®nite con®gurations of the remaining atoms, respectively, `®t' in two equivalent ways. 9.2.2.3. Examples of some polytypic structures The three examples below illustrate the three main methods of analysis of polytypism indicated in x9:2:2:2:5. 9.2.2.3.1. Hydrous phyllosilicates The basic concepts were introduced by Pauling (1930a,b) and con®rmed later by the determination of concrete crystal structures. A crystallochemical analysis of these became the basis for generalizations and systemizations. The aim was the understanding of geometrical reasons for the polytypism of these substances as well as the development of identi®cation routines through the derivation of basic polytypes x9:2:2:2:3. Smith & Yoder (1956) succeeded ®rst in deriving the six basic polytypes in the mica family. Since the 1950's, two main schools have developed: in the USA, represented mainly by Brindley, Bailey, and their coworkers (for details and references see Bailey, 1980, 1988a; Brindley, 1980), and in the former USSR, represented by Zvyagin and his co-workers (for details and references see Zvyagin, 1964, 1967; Zvyagin et al., 1979). Both these schools based their systemizations on idealized structural models corresponding to the ideas of Pauling, with hexagonal symmetry of tetrahedral sheets (see later). The US school uses indicative symbols (Guinier et al., 1984) for the designation of individual polytypes, and single-crystal as well as powder X-ray diffraction methods for their identi®cation, whereas the USSR school uses unitary descriptive symbols for polytypes of all mineral groups and mainly electron diffraction on oblique textures for identi®cation purposes. For the derivation of basic polytypes, both schools use crystallochemical considerations; symmetry principles are applied tacitly rather than explicitly. In contrast to crystal structures based on close packings, where all relevant details of individual (even multilayer) section, the structures polytypes can be recognized in the 1120 of hydrous phyllosilicates are rather complex. For their representation, Figueiredo (1979) used the concept of condensed models. Since 1970, the OD school has also made its contribution. In a series of articles, basic types of hydrous phyllosilicates have been interpreted as OD structures of N > 1 kinds of layers: the Æ urovicÆ, serpentine±kaolin group (Dornberger-Schiff & D Æ urovicÆ, 1980), the mica 1975a,b), Mg-vermiculite (Weiss & D Æ urovicÆ, 1982; group (Dornberger-Schiff, Backhaus & D Æ urovicÆ, 1984; D Æ urovicÆ, Weiss & Backhaus, Backhaus & D 1984; Weiss & WiewioÂra, 1986), the talc±pyrophyllite group Æ urovicÆ & Weiss, 1983; Weiss & D Æ urovicÆ, 1985a), and the (D Æ chlorite group (DurovicÆ, Dornberger-Schiff & Weiss, 1983;

766


9.2. LAYER STACKING Æ urovicÆ, 1983). The papers published before 1983 use Weiss & D the Pauling model; the later papers are based on the model of Radoslovich (1961) with trigonal symmetry of tetrahedral sheets. In all cases, MDO polytypes x9:2:2:2:3 have been derived systematically: their sets partially overlap with basic polytypes presented by the US or the USSR schools. The OD models allowed the use of unitary descriptive symbols for individual polytypes from which all the relevant symmetries can Æ urovicÆ & Dornberger-Schiff, 1981) as well as be determined (D Æ urovicÆ, of extended indicative Ramsdell symbols (Weiss & D 1985b). The results, including principles for identi®cation of Æ urovicÆ (1981). polytypes, have been summarized by D The main features of polytypes of basic types of hydrous phyllosilicates, of their diffraction patterns and principles for their identi®cation, are given in the following. 9.2.2.3.1.1. General geometry Tetrahedral and octahedral sheets are the fundamental, two-dimensionally periodic structural units, common to all hydrous phyllosilicates. Any tetrahedral sheet consists of (Si,Al,Fe3 ,Ti4 )O4 tetrahedra joined by their three basal O atoms to form a network with symmetry P31m (Fig. 9.2.2.6a). The atomic coordinates can be related either to a hexagonal axial system with a primitive unit mesh and basis vectors a1 , a2 , or to an orthohexagonal system p with a c-centred unit mesh and basis vectors a, b b 3a. Any octahedral sheet consists of M(O,OH)6 octahedra with shared edges (Fig. 9.2.2.6b), and with cations M most frequently Mg2 , Al3 , Fe2 , Fe3 , but Ê etc. There are three octahedral also Li , Mn2 rM 4 from the mean were automatically eliminated from the sample by STATS, other outliers were inspected carefully; d there were no compelling chemical reasons for further subdivision of the sample. 9.5.2.4. Statistics Where there are less than four independent observations of a given bond length, then each individual observation is given explicitly in the table. In all other cases, the following statistics were generated by the program STATS. (i) The unweighted sample mean, d, where d

n P i1

n P i1

di

9.5.3. Content and arrangement of the table The upper triangular matrix of Fig. 9.5.3.1a shows the 120 possible element-pair combinations that can be formed from the 15 elements As, B, Br, C, Cl, F, H, I, N, O, P, S, Se, Si, Te.

di =n

and di is the ith observation of the bond length in a total sample of n observations. Recent work (Taylor & Kennard, 1983, 1985, 1986) has shown that the unweighted mean is an acceptable (even preferable) alternative to the weighted mean, where the ith observation is assigned a weight equal to 1= 2 di . This is especially true (Taylor & Kennard, 1985) where structures have been pre-screened on the basis of precision. (ii) The sample median, m. This has the property that half of the observations in the sample exceed m, and half fall short of it. (iii) The sample standard deviation, denoted here as , where:

and 95% of the observations may be expected to lie within 2 of the mean value. For a skewed distribution, d and m may differ appreciably and ql and qu will be asymmetric with respect to m. When a bond-length distribution is negatively skewed as in Fig. 9.5.2.2, i.e. very short values are more common than very long values, then it may be due to thermal-motion effects; the distances used to prepare the table were not corrected for thermal libration. In a number of cases, the initial bond-length distribution was clearly bimodal, as in Fig. 9.5.2.3a. All cases of bimodality were resolved on chemical grounds before inclusion in the table, on the basis of hybridization, conformation-dependent conjugation interactions, etc. For example, the histogram of Fig. 9.5.2.3a was resolved into the two discrete unimodal distributions of Figs. 9.5.2.3b, c, which correspond to planar Nsp2 , pyramidal Nsp3 , respectively. The mean valence angle at N was used as the discriminator, with a range of 108±114 for Nsp3 and 117:5 for Nsp2 .

d2 =n

11=2 :

(iv) The lower quartile for the sample, ql . This has the property that 25% of the observations are less than ql and 75% exceed it. (v) The upper quartile for the sample, qu . This has the property that 25% of the observations exceed qu , and 75% fall short of it. (vi) The number n of observations in the sample. The statistics given in the ®nal table correspond to distributions for which the automatic 4 cut-off (see above) had been applied, and any manual removal of additional outliers (an infrequent operation) has been performed. In practice, a very small percentage of observations was excluded by these methods. The major effect of removing outliers is to improve the sample standard deviation, as shown in Fig. 9.5.2.1 in which a single observation is deleted. The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical, normal distribution: the mean d will be approximately equal to the median m; the lower and upper quartiles ql ; qu will be approximately symmetric about the median: m ql ' qu m,

Fig. 9.5.2.1. Effect of the removal of outliers (contributors that are > 4 from the mean) for the C C bond in Car CN fragments. Relevant statistics (see text) are: a before b after

m 1.444 1.444

0.012 0.008

ql 1.436 1.436

qu 1.448 1.448

n 32 31.

Fig. 9.5.2.2. Skewed distribution of B F bond lengths in BF4 ions: d 1:365, m 1:372, 0:029, ql 1:352, qu 1:390 for 84 observations. Note that d 6 m and that ql , qu are asymmetrically disposed about the mean d.

791


d 1.445 1.455

9. BASIC STRUCTURAL FEATURES Fig. 9.5.3.1a contains the number of discrete average bond lengths given in the table for each element pair. 682 average values are cited for 65 element pairs, of which 511 (75%) involve carbon. Bond-length values from individual structures are given for a further 30 element pairs indicated by an asterisk in Fig. 9.5.3.1a. Individual structures are identi®ed by their CSD reference code (e.g. BOGSUL), and short-form literature references, ordered alphabetically by reference code, are given in Appendix 2. For eight element pairs, the acceptance criterion (vi) was relaxed to include all available structures, irrespective of precision. These entries are denoted by a dagger in the table. No bonds were found for 25 element pairs within the subset of CSD used in this study. Each entry in Table 9.5.1.1 contains nine columns, of which six record the statistics of the bond-length distribution described above. The content of the remaining three columns: `Bond', `Structure', `Note', are now described.

aryl carbon in six-membered rings, which is treated separately from Csp2 throughout the table. The symbol is used to indicate a delocalized double or aromatic bond according to context. 9.5.3.2. De®nition of `Substructure' The chemical environment of each bond is normally de®ned by a linear formulation of the substructure. The target bond is set in bold type, e.g. Car CN (aryl cyanides); C CH2 O Car (primary alkyl aryl ethers); (C O)2 P( O)2 (phosphate diesters). Occasionally, the chemical name of a functional group or ring system is used to de®ne bond environment, e.g. in naphthalene, C2 C3; in imidazole, N1 C2. To avoid any possible ambiguity in these cases, we include numbered chemical diagrams in Fig. 9.5.3.2.

9.5.3.1. Ordering of entries: the `Bond' column For an element pair X Y, the primary ordering is alphabetic by element symbols according to the rows of Fig. 9.5.3.1a; i.e. X changes slowest, Y fastest. The complete sequence runs from As As to Te Te with bonds involving carbon in the natural position: As C. . .C C. . .C Te. Within a given X Y pair, a secondary ordering is based on the coordination numbers j of X and Y, and on the nature of the bond between them. The bond de®nition is of the form X j Y j, with j decreasing fastest for Y, slowest for X, and with all single bonds preceding any multiple bonds. For carbon, the formal hybridization state replaces (but is equivalent to) the coordination number and it is for this element that the ordering rules are most clearly required. The ordering of the most populous C C, C N, C O sections is illustrated in Fig. 9.5.3.1b. The 13 possible C C combinations follow the sequence Csp3 Csp3 , Csp3 Csp2 , Csp3 Car , Csp3 Csp1 , Csp2 Csp2 , Csp2 Car , Csp2 Csp1 , Car Car , Car Csp1 , Csp1 Csp1 , Csp2 Csp2 , Car Car , Csp1 Csp1 . The symbol Car represents

Fig. 9.5.2.3. Resolution of the bimodal distribution of C N bond lengths in Car N(Csp3 )2 fragments: a complete distribution; b distribution for planar N, mean valence angle at N > 117:6 ; c distribution for pyramidal N, mean valence angle at N in the range 108±114 .

Fig. 9.5.3.1. a Distribution of mean bond-length values reported in the table by element pair. An asterisk indicates a bonded pair represented by less than four contributors in the original data set. A `' indicates bonded pairs located when restrictions on R factor and reported e.s.d. limits were lifted (see text). b Distribution of mean bond-length values reported in the table for C C, C O, C N.

792


9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS A combination of chemical name and linear formulation is often employed to increase the precision of the de®nition, e.g. NH2 CO in acyclic amides; CC C(O) CO in benzoquinone. Finally, for very simple ions, the accepted conventional representation is deemed to be suf®cient, e.g. in NO3 , SO24 , etc. The chemical de®nition of substructure may be followed by brief qualifying information, concerning substitution, conformational restrictions, etc. For example: Csp3 Csp3 : in cyclobutane (any substituent); X C F3 (X C, H, N, O); Car NH Csp3 (Nsp3 : pyramidal). Where the generic symbol X is unquali®ed, it denotes any element type, including hydrogen. If the qualifying information is too extensive, then it will be given as a table footnote (see below). The `Substructure' column is designed to convey as much unambiguous information as possible within a small space. For Csp3 , we have employed the short forms C* and C# . C* indicates

Csp3 whose bonds, additional to those speci®ed in the linear formulation, are to C or H atoms only. C* OH would then represent the group of alcohols CH3 OH, C CH2 OH, C2 CH OH and C3 C OH. C* is frequently used to restrict the secondary environment of a given bond to avoid the perturbing in¯uence of, e.g., electronegative substituents. The symbol C# is merely a space-saving device to indicate any Csp3 atom and includes C* as a subset. 9.5.3.3. Use of the `Note' column The `Note' column refers to the footnotes collected in Appendix 1. These record additional information as follows: a additional details concerning the chemical de®nition of substructures, e.g. the omission of three- and four-membered rings; b statements of geometrical constraints used in obtaining the cited average, e.g. de®nition of planarity or pyramidality at

Fig. 9.5.3.2. Alphabetized index of ring systems referred to in the table; the numbering scheme used in assembling the bond-length data is given where necessary.

793


9. BASIC STRUCTURAL FEATURES N, torsional constraints in conjugated systems; c any peculiarities of a particular bond-length distribution, e.g. sample dominated by C* methyl; d references to previously published surveys of crystallographic results relevant to the substructure in question. We do not claim that these references are in any way comprehensive and we would be grateful to authors for noti®cation (to FHA) of any omissions. This will serve to improve the content of any future version of the table. 9.5.4. Discussion It should be remembered that this table has been derived from the organic section of CSD. We are aware that a number of organic bond types which occur very frequently in organometallics and metal complexes (e.g. C C in cyclopentadienyl, C P in triphenylphosphine, etc.) are either absent or poorly represented in this work. These omissions are recti®ed in Chapter 9.6. We also note that certain bond types listed here (e.g. As O, Si O, Si N, etc.) will occur with greater frequency in inorganic compounds. The interested reader is referred to the Inorganic Crystal Structure Database (Bergerhof, Hundt, Sievers & Brown, 1983) for a machine-readable compendium of more relevant structural data. The tabulation given here represents the ®rst stage in a major project designed to obtain the average geometries of function groups, rigid rings, and the low-energy conformations of ¯exible rings. Details of mean bond lengths, valence angles, and conformational preferences in a wide range of substructures will form the basis of a machine-readable `fragment library' for use in molecular modelling and other areas of research. The systematic survey will be extended to derive information about distances, angles, directionality, and environmental dependence of hydrogen bonds and non-bonded interactions. APPENDIX 1 Notes to Table 9.5.1.1 (1) Sample dominated by B CH3 . For longer bonds in Ê ]. CH3 , see LITMEB10 [B(4) CH3 1.621±1.644 A B 2 2 (2) p p bonding with Bsp and Nsp coplanar BN 0 15 predominates. See G. Schmidt, R. Boese & D. BlaÈser [Z. Naturforsch. Teil B (1982), 37, 1230±1233]. Ê and individual (3) 84 observations range from 1.38 to 1.62 A values depend on substituents on B and O. For a discussion of borinic acid adducts, see S. J. Rettig & J. Trotter [Can. J. Chem. (1982), 60, 2957±2964]. (4) See M. Kaftory (1983). [In The chemistry of functional groups. Supplement D: The chemistry of halides, pseudohalides and azides, Part 2, Chap. 24, edited by S. Patai & Z. Rappoport. New York: John Wiley.] (5) Bonds that are endocyclic or exocyclic to any three- or four-membered rings have been omitted from all averages in this section. (6) The overall average given here is for Csp3 Csp3 bonds which carry only C or H substituents. The value cited re¯ects the relative abundance of each `substitution' group. The `mean of means' for the nine subgroups is Ê 1.538 0:022 A. (7) See a F. H. Allen [Acta Cryst. (1980), B36, 81 96] and b F. H. Allen [Acta Cryst. (1981), B37, 890±900]. (8) See F. H. Allen [Acta Cryst. (1984), B40, 64±72]. (9) See F. H. Allen [Tetrahedron (1982), 38, 2843±2853]. (10) See F. H. Allen [Tetrahedron (1982), 38, 645±655].

(11) Cyclopropanones and cyclobutanones excluded. (12) See W. B. Schweizer & J. D. Dunitz [Helv. Chim. Acta (1982), 65, 1547±1554]. (13) See L. Norskov-Lauritsen, H.-B. BuÈrgi, P. Hoffmann & H. R. Schmidt [Helv. Chim. Acta (1985), 68, 76±82]. (14) See P. Chakrabarti & J. D. Dunitz [Helv. Chim. Acta (1982), 65, 1555±1562]. (15) See J. L. Hencher (1978). [In The chemistry of the CC triple bond, Chap. 2, edited by S. Patai. New York: John Wiley.] (16) Conjugated: torsion angles about central C C single bond is 0 20 cis or 180 20 trans. (17) Unconjugated: torsion angle about central C C single bond is 20±160 . (18) Other conjugative substituents excluded. (19) TCNQ is tetracyanoquinodimethane (see diagrams). (20) No difference detected between C2 C3 and C3 C4 bonds. (21) Derived from neutron diffraction results only. (22) Nsp3 : pyramidal; mean valence angle at N is in the range 108±114 . (23) Nsp2 : planar; mean valence angle at N is 117:5 . (24) Cyclic and acyclic peptides. (25) See R. H. Blessing [J. Am. Chem. Soc. (1983), 105, 2776±2783]. (26) See L. Lebioda [Acta Cryst. (1980), B36, 271±275]. (27) n 3 or 4; i.e. tri- or tetrasubstituted ureas. (28) Overall value also includes structures with mean valence angle at N in the range 115±118 . (29) See F. H. Allen & A. J. Kirby [J. Am. Chem. Soc. (1984), 106, 6197±6200]. (30) See A. J. Kirby (1983). [The anomeric effect and related stereoelectronic effects at oxygen. Berlin: Springer.] (31) See B. Fuchs, L. Schleifer & E. Tartakovsky [Nouv. J. Chim. (1984), 8, 275±278]. (32) See S. C. Nyburg & C. H. Faerman [J. Mol. Struct. (1986), 140, 347±349]. (33) Sample dominated by P CH3 and P CH2 C. (34) Sample dominated by C* methyl. (35) See A. KaÂlmaÂn, M. Czugler & G. Argay [Acta Cryst. (1981), B37, 868±877]. (36) Bimodal distribution resolved into 22 `short' bonds and 5 longer outliers. (37) All 24 observations come from BUDTEZ. (38) `Long' O H bonds in centrosymmetric O H O H-bonded dimers are excluded. (39) N N bond length also dependent on torsion angle about N N bond and on nature of substituent C atoms ± these effects are ignored here. (40) N pyramidal has average angle at N in the range 100±113.5 ; N planar has average angle 117:5 . (41) See R. R. Holmes & J. A. Deiters [J. Am. Chem. Soc. (1977), 99, 3318±3326]. (42) No detectable variation in SO bond length with type of C substituent. APPENDIX 2 Short-form references to individual CSD entries cited by reference code in Table 9.5.1.1 REFCODE ACBZPO01 ACLTEP ASAZOC BALXOB

794


Journal J. Am. Chem. Soc. J. Organomet. Chem. Dokl. Akad. Nauk SSSR J. Am. Chem. Soc.

Vol. 97 184 249 103

Page 6729 417 120 4587

Year 1975 1980 1979 1981

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS BAPPAJ BARRIV BAWFUA BAWGAH BECTAE BELNIP BEMLIO BEPZEB BETJOZ BETUTE10 BIBLAZ BICGEZ BIHXIZ BIRGUE10 BIRHAL10 BIZJAV BOGPOC BOGSUL BOJLER BOJPUL BOPFER BOPFIV BOVMEE BQUINI BTUPTE BUDTEZ BUPSIB10 BUSHAY BUTHAZ10 BUTSUE BUWZUO BZPRIB BZTPPI CAHJOK CAJMAB CANLUY CASSAQ CASTOF10 CASYOK CECHEX CECXEN CEDCUJ CEHKAB CELDOM CESSAU CETTAW CETUTE CEYLUN CIFZUM CIHRAM CILRUK CILSAR CIMHIP CINTEY CIPBUY CISMUM CISTED CIWYIQ CIYFOF

Inorg. Chem. Acta Chem. Scand. Ser. A Cryst. Struct. Commun. Cryst. Struct. Commun. J. Org. Chem. Z. Naturforsch. Teil B Chem. Ber. Cryst. Struct. Commun. J. Am. Chem. Soc. Acta Chem. Scand. Ser. A Zh. Strukt. Khim. Z. Anorg. Allg. Chem. J. Chem. Soc. Chem. Commun. Z. Naturforsch. Teil B Z. Naturforsch. Teil B J Organomet. Chem. Z. Naturforsch. Teil B Z. Naturforsch. Teil B Z. Anorg. Allg. Chem. Acta Chem. Scand. Ser. A Chem. Ber. Chem. Ber. Acta Cryst. Sect. B Acta Cryst. Sect. B Acta Chem. Scand. Ser. A Z. Naturforsch. Teil B Z. Anorg. Allg. Chem. Z. Naturforsch. Teil B Inorg. Chem. J. Chem. Soc. Chem. Commun. Acta Chem. Scand. Ser. A Z. Naturforsch. Teil B Inorg. Chem. Inorg. Chem. Chem. Z. Tetrahedron Lett. J. Struct. Chem. Acta Cryst. Sect. C J. Struct. Chem. Z. Anorg. Allg. Chem. J. Struct. Chem. J. Org. Chem. Z. Naturforsch. Teil B Acta Cryst. Sect. C Acta Cryst. Sect. C Chem. Ber. Acta Chem. Scand. Ser. A Isv. Akad. Nauk SSR Ser. Khim. Acta Chem. Scand. Ser. A Angew. Chem. Int. Ed. Engl. J. Chem. Soc. Chem. Commun. J. Chem. Soc. Chem. Commun. Acta Cryst. Sect. C Dokl. Acak. Nauk SSSR J. Struct. Chem. Z. Naturforsch. Teil B Z. Anorg. Allg. Chem. Inorg. Chem. Inorg. Chem.

20 35 10 10 46 37 115 11 104 30 22 486 38 37 238 37 37 493 36 116 116 38 35 29 38 474 38 23 37 36 17 22 107 24 2 40 2 508 2 48 39 40 40 117 29 38 23 40 274 2 39 511 23 23

3071 433 1345 1353 5048 299 1126 175 1683 719 118-4 90 982 20 1410 C1 1402 1230 53 829 146 146 1048 1930 738 454 31 692 2582 862 219 922 894 1809 169 4337 101-2 1879 107-2 61 207-3 5149 139 556 653 1089 763 2744 289 302 1023 1021 1458 615 281-4 485 95 1946 1790

1981 1981 1981 1981 1981 1982 1982 1982 1982 1976 1981 1982 1982 1983 1982 1982 1982 1982 1982 1982 1983 1983 1982 1979 1975 1983 1981 1983 1984 1983 1983 1981 1978 1983 1983 1983 1983 1984 1983 1984 1983 1983 1984 1984 1984 1984 1975 1983 1984 1984 1984 1984 1984 1984 1983 1984 1984 1984 1984

CMBIDZ CODDEE CODDII COFVOI COJCUZ COSDIX COZPIQ COZVIW CTCNSE CUCPIZ CUDLOC CUDLUI CUGBAH CXMSEO DGLYSE DMESIP01 DSEMOR10 DTHIBR10 EPHTEA ESEARS ETEARS FMESIB FPHTEL FPSULF10 HCLENE10 HMTITI HMTNTI HXPASC IBZDAC11 IFORAM IODMAM IPMUDS ISUREA10 LITMEB10 MESIAD METAMM MNPSIL MODIAZ MOPHTE MORTRS10 NAPSEZ10 NBBZAM OPIMAS OPNTEC10 PHASCL PHASOC01 PNPOSI SEBZQI SPSEBU TEACBR THINBR TMPBTI TPASSN TPASTB TPHOSI TTEBPZ ZCMXSP

Pages of the form n-m indicate page n of issue m.

795


J. Org. Chem. Z. Naturforsch. Teil B Z. Naturforsch. Teil B Z. Naturforsch. Teil B Chem. Ber. Z. Naturforsch. Teil B Chem. Ber. Z. Anorg. Allg. Chem. J. Am. Chem. Soc. J. Am. Chem. Soc. J. Cryst. Spectrosc. J. Cryst. Spectrosc. Acta Cryst. Sect. C Acta Cryst. Sect. B Acta Cryst. Sect. B Acta Cryst. Sect. C J. Chem. Soc. Dalton Trans. Inorg. Chem. Inorg. Chem. J. Chem. Soc. C J. Chem. Soc. C J. Organomet. Chem. J. Chem. Soc. Dalton Trans. J. Am. Chem. Soc. Acta Cryst. Sect. B Acta Cryst. Sect. B Z. Anorg. Allg. Chem. J. Chem. Soc. Dalton Trans. J. Chem. Soc. Dalton Trans. Monatsh. Chem. Acta Cryst. Sect. B Acta Cryst. Sect. B Acta Cryst. Sect. B J. Am. Chem. Soc. Z. Naturforsch. Teil B Acta Cryst. J. Am. Chem. Soc. J. Heterocycl. Chem. Acta Chem. Scand. Ser. A J. Chem. Soc. Dalton Trans. J. Am. Chem. Soc. Z. Naturforsch. Teil B Aust. J. Chem. J. Chem. Soc. Dalton Trans. Acta Cryst. Sect. B Aust. J. Chem. J. Am. Chem. Soc. J. Chem. Soc. Chem. Commun. Acta Chem. Scand. Ser. A Cryst. Struct. Commun. J. Am. Chem. Soc. Acta Cryst. Sect. B J. Chem. Soc. Dalton Trans. Cryst. Struct. Commun. Z. Naturforsch. Teil B Z. Naturforsch. Teil B Cryst. Struct. Commun.

44 39 39 39 117 39 117 515 102 106 15 15 41 29 31 40 10 19 197 104 38 31 409 105 33 29 28 97 35 17 91 17 34 102 32 30 37 28 90 33 3 92 31 5 34 34 6

1447 1257 1257 1027 2686 1344 2063 7 5430 7529 53 53 476 595 1785 895 628 697 2487 1511 1511 275 2306 1683 3139 1505 237 1381 854 621 3209 2128 643 6401 789 1336 4134 1217 333 628 5070 1416 2417 251 1357 15 5102 325 403 753 4002 1116 514 39 1064 256 93

1979 1984 1984 1984 1984 1984 1984 1984 1980 1984 1985 1985 1985 1973 1975 1984 1980 1971 1980 1971 1971 1980 1980 1982 1982 1975 1974 1975 1979 1974 1977 1973 1972 1975 1980 1964 1969 1980 1980 1980 1980 1977 1977 1982 1981 1975 1968 1977 1979 1974 1970 1975 1977 1976 1979 1979 1977

9. BASIC STRUCTURAL FEATURES Ê for bonds involving the elements H, B, C, N, O, F, Si, P, S, Cl, As, Se, Br, Te, and I Table 9.5.1.1. Average lengths A Bond As(3)

As(3)

Substructure X2

As

As

X2

As

B

see CUDLOC (2.065), CUDLUI (2.041)

As

Br

see CODDEE, CODDII (2.346±3.203)

d

m

ql

qu

n

Note

2.459

2.457

0.011

2.456

2.466

8

As(4)

C

X3 As CH3 X2 (C, O, S)As Csp3 As Car in Ph4 As X2 (C, O, S)As Car

1.903 1.927 1.905 1.922

1.907 1.929 1.909 1.927

0.016 0.017 0.012 0.016

1.893 1.921 1.897 1.908

1.916 1.937 1.912 1.934

12 16 108 36

As(3)

C

X2 X2

As As

Csp3 Car

1.963 1.956

1.965 1.956

0.017 0.015

1.948 1.944

1.978 1.964

6 41

As(3)

Cl

X2

As

Cl

2.268

2.256

0.039

2.247

2.281

10

As(6)

F

in AsF6

1.678

1.676

0.020

1.659

1.695

36

As(3)

I

see OPIMAS (2.579, 2.590)

As(3)

N(3)

X2

1.858

1.858

0.029

1.839

1.873

19

1.710

1.712

0.017

1.695

1.726

6

1.661

1.661

0.016

1.652

1.667

9

As

N

X2

As(4)N(2)

see TPASSN (1.837)

As(4)

O

X2 (O)As

As(3)

O

see ASAZOC, PHASOC01 (1.787±1.845)

As(4)O As(3)

P(3)

As(3)P(3)

X3

OH

AsO

see BELNIP (2.350, 2.362)

y

see BUTHAZ10 (2.124)

y

S

X2

As

S

2.275

2.266

0.032

2.247

2.298

14

As(4)S

X3

AsS

2.083

2.082

0.004

2.080

2.086

9

As(3)

As(3)

Se(2)

see COSDIX, ESEARS (2.355±2.401)

y

As(3)

Si(4)

see BICGEZ, MESIAD (2.351±2.365)

y

As(3)

Te(2)

see ETEARS (2.571, 2.576)

y

Bn

Bn

n 5±7 in boron cages

B(4)

B(4)

see CETTAW (2.041)

B(4)

B(3)

see COFVOI (1.698)

B(3)

B(3)

X2

B(6)

1.775

1.773

0.031

1.763

1.786

688

1.701

1.700

0.014

1.691

1.712

8

Br

1.967

1.971

0.014

1.954

1.979

7

y

B(4)

Br

2.017

2.008

0.031

1.990

2.044

15

y

Bn

C

n 5±7: B n 3±4: B n 4: B B n 3: B

1.716 1.597 1.606 1.643 1.556

1.717 1.599 1.607 1.643 1.552

0.020 0.022 0.012 0.006 0.015

1.707 1.585 1.596 1.641 1.546

1.728 1.611 1.615 1.645 1.566

96 29 41 16 24

Bn

Cl

B(5) B(4)

1.751 1.833

1.751 1.833

0.011 0.013

1.743 1.821

1.761 1.843

14 22

B(4)

F

B B

1.366 1.365

1.368 1.372

0.017 0.029

1.356 1.352

1.375 1.390

25 84

B

B

X2

C in cages Csp3 not cages Car Car in Ph4 B Car

Cl and B(3) Cl

Cl

F (B neutral) F in BF4

796


1

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS Table 9.5.1.1. Average lengths (cont.) Bond

Substructure

d

m

ql

qu

n

Note

B(4)

I

see TMPBTI (2.220, 2.253)

B(4)

N(3)

X3 B N(C)X in pyrazaboles

1.611 1.549

1.617 1.552

0.013 0.015

1.601 1.536

1.625 1.560

8 10

B(3)

N(3)

X2 B N C2 : all coplanar for BN > 30 see BOGSUL, BUSHAY, CILRUK (1.434±1.530) S2 B N X2

1.404

1.404

0.014

1.389

1.408

40

1.447

1.443

0.013

1.435

1.470

14

B(4)

O

B O in BO4 for neutral B O see Note 3

1.468

1.468

0.022

1.453

1.479

24

B(3)

O(2)

X2

1.367

1.367

0.024

1.349

1.382

35

Bn

P

n 4: B P n 3: see BUPSIB10 (1.892, 1.893)

1.922

1.927

0.027

1.900

1.954

10

B(4)

S

B(4) B(4)

1.930 1.896

1.927 1.896

0.009 0.004

1.925 1.893

1.934 1.899

10 6

B(3)

S

N B S2 (X )(N

1.806 1.851

1.806 1.854

0.010 0.013

1.799 1.842

1.816 1.859

28 10

B

O

X

S(3) S(2) )B

S

Br

Br

see BEPZEB, TPASTB

2.542

2.548

0.015

2.526

2.551

4

Br

C

Br Br Br Br Br

1.966 1.910 1.883 1.899 1.875

1.967 1.910 1.881 1.899 1.872

0.029 0.010 0.015 0.012 0.011

1.951 1.900 1.874 1.892 1.864

1.983 1.914 1.894 1.906 1.884

100 8 31 119 8

Br(2)

Cl

C* Csp3 (cyclopropane) Csp2 Car (mono-Br m,p-Br2 ) Car (o-Br2 )

see TEACBR (2.362±2.402)

2

3

4 4 4 4 y

Br

I

see DTHIBR10 (2.646), TPHOSI (2.695)

Br

N

see NBBZAM (1.843)

Br

O

see CIYFOF

Br

P

see CISTED (2.366)

Br

S(2)

see BEMLIO (2.206)

y

Br

S(3)

see CIWYIQ (2.435, 2.453)

y

Br

S(3)

see THINBR (2.321)

y

Br

Se

see CIFZUM (2.508, 2.619)

Br

Si

see BIZJAV (2.284)

Br

Te

In Br6 Te2 see CUGBAH (2.692±2.716) Br Te(4) see BETUTE10 (3.079, 3.015) Br Te(3) see BTUPTE (2.835)

Csp3

Csp3

1.581

C# CH2 CH3 (C# )2 CH CH3 (C# )3 C CH3 C# CH2 CH2 C# (C# )2 CH CH2 C# (C# )3 C CH2 C# (C# )2 CH CH (C# )2 (C# )3 C CH (C# )2 (C# )3 C C (C# )3 C* C* (overall)

1.513 1.524 1.534 1.524 1.531 1.538 1.542 1.556 1.588 1.530

797


1.581

1.514 1.526 1.534 1.524 1.531 1.539 1.542 1.556 1.580 1.530

0.007

0.014 0.015 0.011 0.014 0.012 0.010 0.011 0.011 0.025 0.015

1.574

1.507 1.518 1.527 1.516 1.524 1.533 1.536 1.549 1.566 1.521

1.587

1.523 1.534 1.541 1.532 1.538 1.544 1.549 1.562 1.610 1.539

4

192 226 825 2459 1217 330 321 215 21 5777

5, 6

9. BASIC STRUCTURAL FEATURES Table 9.5.1.1. Average lengths (cont.) Bond 3

Csp Csp (cont.)

Csp3

Substructure 3

Csp2

d

m

ql

qu

n

Note

in cyclopropane (any substituent) in cyclobutane (any substituent) in cyclopentane (C,H substituents) in cyclohexane (C,H substituents) cyclopropyl C* (exocyclic) cyclobutyl C* (exocyclic) cyclopentyl C* (exocyclic) cyclohexyl C* (exocyclic)

1.510 1.554 1.543 1.535 1.518 1.529 1.540 1.539

1.509 1.553 1.543 1.535 1.518 1.529 1.541 1.538

0.026 0.021 0.018 0.016 0.019 0.016 0.017 0.016

1.497 1.540 1.532 1.525 1.505 1.519 1.527 1.529

1.523 1.567 1.554 1.545 1.531 1.539 1.549 1.549

888 679 1641 2814 366 376 956 2682

7 8

in cyclobutene (any substituent) in cyclopentene (C,H substituents) in cyclohexene (C,H substituents)

1.573 1.541 1.541

1.574 1.539 1.541

0.017 0.015 0.020

1.566 1.532 1.528

1.586 1.549 1.554

25 208 586

8

in oxirane (epoxide) in aziridine in oxetane in azetidine oxiranyl C* (exocyclic) aziridinyl C* (exocyclic)

1.466 1.480 1.541 1.548 1.509 1.512

1.466 1.481 1.541 1.543 1.507 1.512

0.015 0.021 0.019 0.018 0.018 0.018

1.458 1.465 1.527 1.536 1.497 1.496

1.474 1.496 1.557 1.558 1.519 1.526

249 67 16 22 333 13

9 9

CH3 CC C# CH2 CC (C# )2 CH CC (C# )3 C CC C* CC (overall)

1.503 1.502 1.510 1.522 1.507

1.504 1.502 1.510 1.522 1.507

0.011 0.013 0.014 0.016 0.015

1.497 1.494 1.501 1.511 1.499

1.509 1.510 1.518 1.533 1.517

215 483 564 193 1456

C* in in in in in in

CC (endocylic): cyclopropene cyclobutene cyclopentene cyclohexene cyclopentadiene cyclohexa-1,3-diene

1.509 1.513 1.512 1.506 1.502 1.504

1.508 1.512 1.512 1.505 1.503 1.504

0.016 0.018 0.014 0.016 0.019 0.017

1.500 1.500 1.502 1.495 1.490 1.491

1.516 1.525 1.521 1.516 1.515 1.517

20 50 208 391 18 56

10 8

C* CC (exocyclic): cyclopropenyl C* cyclobutenyl C* cyclopentenyl C* cyclohexenyl C*

1.478 1.489 1.504 1.511

1.475 1.483 1.506 1.511

0.012 0.015 0.012 0.013

1.470 1.479 1.495 1.502

1.485 1.496 1.512 1.519

7 11 115 292

10 8

C* CHO in aldehydes (C*)2 CO in ketones in cyclobutanone in cyclopentanone acyclic and 6 rings

1.510 1.511 1.529 1.514 1.509

1.510 1.511 1.530 1.514 1.509

0.008 0.015 0.016 0.016 0.016

1.501 1.501 1.514 1.505 1.499

1.518 1.521 1.545 1.523 1.519

7 952 18 312 626

1.502 1.520 1.497 1.519 1.512 1.504

1.502 1.521 1.496 1.519 1.512 1.502

0.014 0.011 0.018 0.020 0.015 0.013

1.495 1.516 1.484 1.500 1.501 1.495

1.510 1.528 1.509 1.538 1.521 1.517

176 57 553 4 110 27

12 13 12 12

cyclopropyl (C) CO in ketones, acids, and esters C* C(O)( NH2 ) in acyclic amides C* C(O)( NHC*) in acyclic amides C* C(O)[ N(C*)2 ] in acyclic amides

1.486 1.514 1.506 1.505

1.485 1.512 1.505 1.505

0.018 0.016 0.012 0.011

1.474 1.506 1.498 1.496

1.497 1.526 1.515 1.517

105 32 78 15

7 14 14 14

CH3 Car C# CH2 Car (C# )2 C Car (C# )3 C Car C* Car (overall)

1.506 1.510 1.515 1.527 1.513

1.507 1.510 1.515 1.530 1.513

0.011 0.009 0.011 0.016 0.014

1.501 1.505 1.508 1.517 1.505

1.513 1.516 1.522 1.539 1.521

454 674 363 308 1813

1.490

1.490

0.015

1.479

1.503

90

C* C* C*

Csp3

Car

COOH in carboxylic acids COO in carboxylate anions C(O)( OC*) in acyclic esters in -lactones in -lactones in -lactones

cylopropyl (C)

Car

798


7 8

9 9

5

11

7

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS Table 9.5.1.1. Average lengths (cont.) Bond Csp

3

Csp2

Csp

Substructure 1

Csp2

d

m

ql

qu

n

Note

1.466 1.472 1.470 1.444

1.465 1.472 1.469 1.447

0.010 0.012 0.013 0.010

1.460 1.464 1.463 1.436

1.469 1.481 1.479 1.451

21 88 106 38

15 15 7(b) 7

1.455 1.478 1.460 1.443 1.432

1.455 1.476 1.460 1.445 1.433

0.011 0.012 0.015 0.013 0.012

1.447 1.470 1.450 1.431 1.424

1.463 1.479 1.470 1.454 1.441

30 8 38 29 280

16, 18 17, 18

CC CC

CC (conjugated) (unconjugated) (overall) CC CC CC (endocyclic in TCNQ)

CC

C(O)(

1.464 1.484 1.465

1.462 1.486 1.462

0.018 0.017 0.018

1.453 1.475 1.453

1.476 1.497 1.478

211 14 226

16, 18 17, 18

CC C(O) CC: in benzoquinone (C,H substituents only) in benzoquinone (any substituent) non-quinonoid

1.478 1.478 1.456

1.476 1.478 1.455

0.011 0.031 0.012

1.469 1.464 1.447

1.488 1.498 1.464

28 172 28

C C CC CC HOOC HOOC OOC

1.475 1.488 1.502 1.538 1.549 1.564

1.476 1.489 1.499 1.537 1.552 1.559

0.015 0.014 0.017 0.007 0.009 0.022

1.461 1.478 1.488 1.535 1.546 1.554

1.488 1.497 1.510 1.541 1.553 1.568

22 113 11 9 13 9

1.412 1.423 1.424 1.410 1.425 1.428 1.417

1.410 1.423 1.425 1.412 1.425 1.427 1.417

0.016 0.016 0.015 0.016 0.016 0.007 0.006

1.401 1.412 1.415 1.400 1.413 1.422 1.412

1.427 1.433 1.433 1.418 1.438 1.435 1.422

29 62 40 20 9 6 14

Car (conjugated) (unconjugated) (overall) cyclopropenyl (CC) Car Car C(O) C* Car C(O) Car Car COOH Car C(O)( OC*) Car COO Car C(O) NH2 Car CN C# (conjugated) (unconjugated) (overall) in indole (C3 C3a)

1.470 1.488 1.483 1.447 1.488 1.480 1.484 1.487 1.504 1.500 1.476 1.491 1.485 1.434

1.470 1.490 1.483 1.448 1.489 1.481 1.485 1.487 1.509 1.503 1.478 1.490 1.487 1.434

0.015 0.012 0.015 0.006 0.016 0.017 0.014 0.012 0.014 0.020 0.014 0.008 0.013 0.011

1.463 1.480 1.472 1.441 1.478 1.468 1.474 1.480 1.495 1.498 1.466 1.485 1.481 1.428

1.480 1.496 1.494 1.452 1.500 1.494 1.491 1.494 1.512 1.510 1.486 1.496 1.493 1.439

37 87 124 8 84 58 75 218 26 19 27 48 75 40

16, 18 17, 18

CC CC

1.431 1.427

1.427 1.427

0.014 0.010

1.425 1.420

1.441 1.433

11 280

7(b) 19

C* CC C# CC C* CN cyclopropyl (C) CC

CN

C*) (conjugated) (unconjugated) (overall)

COOH COOC* COO COOH COO COO

formal Csp2 Csp2 single bond in selected, non-fused heterocycles: in 1H-pyrrole (C3 C4) in furan (C3 C4) in thiophene (C3 C4) in pyrazole (C3 C4) in isoxazole (C3 C4) in furazan (C3 C4) in furoxan (C3 C4) Csp2

Csp2

Car

Csp1

CC

CC CN in TCNQ

Car

Car

in biphenyls (ortho substituent all H) ( 1 non-H ortho substituent)

1.487 1.490

1.488 1.491

0.007 0.010

1.484 1.486

1.493 1.495

30 212

Car

Csp1

Car Car

1.434 1.443

1.436 1.444

0.006 0.008

1.430 1.436

1.437 1.448

37 31

1.377

1.378

0.012

1.374

1.384

21

1.299 1.321 1.317 1.312 1.316

1.300 1.321 1.318 1.311 1.317

0.027 0.013 0.013 0.011 0.015

1.280 1.313 1.310 1.304 1.309

1.311 1.328 1.232 1.320 1.323

42 77 106 19 127

Csp1

Csp1

Csp2 Csp2

CC CN

CC

CC

C* CHCH2 (C*)2 CCH2 C* CHCH C* (cis) (trans) (overall)

799


18 19

10

16 17


Csp Csp (cont.)

Substructure 2

d

m

ql

qu

n

Note

(C*)2 CCH C* (C*)2 CC (C*)2 (C*,H)2 CC (C*,H)2 (overall)

1.326 1.331 1.322

1.328 1.330 1.323

0.011 0.009 0.014

1.319 1.326 1.315

1.334 1.334 1.331

168 89 493

5

in in in in

1.294 1.335 1.323 1.326

1.288 1.335 1.324 1.325

0.017 0.019 0.013 0.012

1.284 1.324 1.314 1.318

1.302 1.347 1.331 1.334

10 25 104 196

CCC (allenes, any substituents) CC CC (C,H substituents, conjugated) CC CC CC (C,H substituents, conjugated) CC Car (C,H substituents, conjugated) CC in cyclopenta-1,3-diene (any substituent) CC in cyclohexa-1,3-diene (any substituent)

1.307 1.330 1.345 1.339 1.341 1.332

1.307 1.330 1.345 1.340 1.341 1.332

0.005 0.014 0.012 0.011 0.017 0.013

1.303 1.322 1.337 1.334 1.328 1.323

1.310 1.338 1.350 1.346 1.356 1.341

18 76 58 124 18 56

CO (C,H substituent, conjugated) (C,H substituent, unconjugated) (C,H substituent, overall) in cyclohexa-2,5-dien-1-ones in p-benzoquinones (C*,H substituents) (any substituent) in TCNQ (endocyclic) (exocyclic) CC OH in enol tautomers

1.340 1.331 1.340 1.329 1.333 1.349 1.352 1.392 1.362

1.340 1.330 1.339 1.327 1.337 1.339 1.353 1.391 1.360

0.013 0.008 0.013 0.011 0.011 0.030 0.010 0.017 0.020

1.332 1.326 1.332 1.321 1.325 1.330 1.345 1.379 1.349

1.348 1.339 1.348 1.335 1.338 1.364 1.358 1.405 1.370

211 14 226 28 14 86 142 139 54

in heterocycles (any 1H-pyrrole (C2 furan (C2 thiophene (C2 pyrazole (C4 imidazole (C4 isoxazole (C4 indole (C2

1.375 1.341 1.362 1.369 1.360 1.341 1.364

1.377 1.342 1.359 1.372 1.361 1.336 1.363

0.018 0.021 0.025 0.019 0.014 0.012 0.012

1.361 1.329 1.346 1.362 1.352 1.331 1.355

1.388 1.351 1.377 1.383 1.367 1.355 1.371

58 125 60 20 44 9 40

in phenyl rings with C*,H substituents only: H C C H C* C C H C* C C C* C C (overall)

1.380 1.387 1.397 1.384

1.381 1.388 1.397 1.384

0.013 0.010 0.009 0.013

1.372 1.382 1.392 1.375

1.388 1.393 1.403 1.391

2191 891 182 3264

F C C F Cl C C Cl

1.372 1.388

1.374 1.389

0.011 0.014

1.366 1.380

1.380 1.398

84 152

in napthalene D2h C1 C2 (any substituent) C2 C3 C1 C8a C4a C8a

1.364 1.406 1.420 1.422

1.364 1.406 1.419 1.424

0.014 0.014 0.012 0.011

1.356 1.397 1.412 1.417

1.373 1.415 1.426 1.429

440 218 440 109

in anthracene D2h (any substituent)

1.356 1.410 1.430 1.435 1.400

1.356 1.410 1.430 1.436 1.402

0.009 0.010 0.006 0.007 0.009

1.350 1.401 1.426 1.429 1.395

1.360 1.416 1.434 1.440 1.406

56 34 56 34 68

1.379 1.380

1.381 1.380

0.012 0.015

1.371 1.371

1.387 1.389

276 537

1.373 1.379 1.373 1.383 1.379 1.405 1.387

1.375 1.380 1.372 1.385 1.377 1.405 1.389

0.012 0.011 0.019 0.019 0.010 0.024 0.018

1.368 1.371 1.362 1.372 1.370 1.388 1.379

1.380 1.388 1.382 1.394 1.388 1.420 1.400

30 30 151 151 10 60 28

cyclopropene (any substituent) cyclobutene (any substituent) cyclopentene (C,H substituents) cyclohexene (C,H substituents)

in CC

Car Car

substituent) C3, C4 C5) C3, C4 C5) C3, C4 C5) C5) C5) C5) C3)

C1 C2 C1 C4a C9

C2 C3 C9a C9a C9a

in pyridine (C,H substituent) (any substituents) in pyridinium cation: (N H; C,H substituents on C) C2 C3 (N X; C,H substituents on C) C2 C3 in pyrazine (H substituent on C) (any substituent on C) in pyrimidine (C,H substituents on C)

C3 C4 C3 C4

800


10 8

16 16 16

16, 18 17, 18

19 19

4 4

20 20

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS Table 9.5.1.1. Average lengths (cont.) Bond 1

Csp Csp

Csp3

Substructure 1

Cl

d

m

ql

qu

n

Note

X CC X C,H CC C,H in CC Csp2 ; ar in CC CC in CHC C#

1.183 1.181 1.189 1.192 1.174

1.183 1.181 1.193 1.192 1.174

0.014 0.014 0.010 0.010 0.011

1.174 1.173 1.181 1.187 1.167

1.193 1.192 1.195 1.197 1.180

119 104 38 42 42

15 15 15 15 15

Omitting 1,2-dichlorides: C CH2 Cl C2 CH Cl C3 C Cl

1.790 1.803 1.849

1.790 1.802 1.856

0.007 0.003 0.011

1.783 1.800 1.837

1.795 1.807 1.858

13 8 5

4 4 4

1.790 1.805 1.843 1.779 1.768 1.793 1.762 1.755

1.791 1.803 1.838 1.776 1.765 1.793 1.760 1.756

0.011 0.014 0.014 0.015 0.011 0.013 0.010 0.011

1.783 1.800 1.835 1.769 1.761 1.786 1.757 1.749

1.797 1.812 1.858 1.790 1.776 1.800 1.765 1.763

37 26 7 18 33 66 54 64

4 4 4 4 4 4 4

CC Cl (C,H,N,O substituents on C) CC Cl2 (C,H,N,O substituents on C) Cl CC Cl

1.734 1.720 1.713

1.729 1.716 1.711

0.019 0.013 0.011

1.719 1.708 1.705

1.748 1.729 1.720

63 20 80

4 4 4

Car Car

1.739 1.720

1.741 1.720

0.010 0.010

1.734 1.713

1.745 1.717

340 364

4 4

X CH2 Cl (X C,H,N,O) X2 CH Cl (X C,H,N,O) X3 C Cl (X C,H,N,O) X2 C Cl2 (X C,H,N,O) X C Cl3 (X C,H,N,O) Cl CH( C) CH( C) Cl Cl C( C2 ) C( C2 ) Cl cyclopropyl Cl Csp2

Car

Cl

Cl

Cl (mono-Cl m,p-Cl2 ) Cl (o-Cl2 )

Csp1

Cl

see HCLENE10 (1.634, 1.646)

Csp3

F

Omitting 1,2-di¯uorides: C CH2 F and C2 CH F C3 C F (C*,H)2 C F2 C* C F3 F C* C* F X3 C F (X C,H,N,O) X2 C F2 (X C,H,N,O) X C F3 (X C,H,N,O) F C( X)2 C( X)2 F (X C,H,N,O) F C( X)2 NO2 (X any substituent)

1.399 1.428 1.349 1.336 1.371 1.386 1.351 1.322 1.373 1.320

1.399 1.431 1.347 1.334 1.374 1.389 1.349 1.323 1.374 1.319

0.017 0.009 0.012 0.007 0.007 0.033 0.013 0.015 0.009 0.009

1.389 1.421 1.342 1.330 1.362 1.373 1.342 1.314 1.362 1.312

1.408 1.435 1.356 1.344 1.375 1.408 1.356 1.332 1.377 1.327

25 11 58 12 26 70 58 309 30 18

4 4 4 4 4 4 4 4 4

Csp2

F

CC

1.340

1.340

0.013

1.334

1.346

34

4

1.363 1.340

1.362 1.340

0.008 0.009

1.357 1.336

1.368 1.344

38 167

4 4

Car

F

Car Car

F (C,H,N,O substituents on C)

F (mono-F m,p-F2 ) F (o-F2 )

Csp3

H

C C H3 (methyl) C2 C H2 (primary) C3 C H (secondary) C2;3 C H (primary and secondary) X C H3 (methyl) X2 C H2 (primary) X3 C H (secondary) X2;3 C H (primary and secondary)

1.059 1.092 1.099 1.093 1.066 1.092 1.099 1.094

1.061 1.095 1.097 1.095 1.074 1.095 1.099 1.096

0.030 0.013 0.004 0.012 0.028 0.012 0.007 0.011

1.039 1.088 1.095 1.089 1.049 1.088 1.095 1.091

1.083 1.099 1.103 1.100 1.087 1.099 1.103 1.100

83 100 14 118 160 230 117 348

21 21 21 21 21 21 21 21

Csp2

H

C

1.077

1.079

0.012

1.074

1.085

14

21

Car

Csp3

H

H

Car

H

1.083

1.083

0.011

1.080

1.087

218

21

I

C*

I

2.162

2.159

0.015

2.149

2.179

15

4

Car

I

2.095

2.095

0.015

2.089

2.104

51

4

1.488 1.494 1.502 1.510 1.499

1.488 1.493 1.502 1.509 1.498

0.013 0.016 0.015 0.020 0.018

1.482 1.484 1.491 1.496 1.488

1.495 1.503 1.512 1.523 1.510

298 249 509 319 1370

Csp3 Car

CC

I N(4)

C* NH 3 (C*)2 NH 2 (C*)3 NH (C*)4 N C* N (overall)

801


9. BASIC STRUCTURAL FEATURES Table 9.5.1.1. Average lengths (cont.) Bond Csp

3

N(3)

Substructure

d

m

ql

qu

1.485

1.484

0.009

1.477

1.490

32

C* NH2 (Nsp3 : pyramidal) (C*)2 NH (Nsp3 : pyramidal) (C*)3 N (Nsp3 : pyramidal) C* Nsp3 (overall)

1.469 1.469 1.469 1.469

1.470 1.467 1.468 1.468

0.010 0.012 0.014 0.014

1.462 1.461 1.460 1.460

1.474 1.477 1.476 1.476

19 152 1042 1201

Csp3

1.472 1.484 1.475 1.473

1.471 1.481 1.473 1.473

0.016 0.018 0.016 0.013

1.464 1.472 1.464 1.460

1.482 1.495 1.483 1.479

134 21 66 240

1.454 1.464 1.457 1.462 1.458 1.478 1.479 1.468

1.451 1.465 1.458 1.461 1.456 1.472 1.476 1.471

0.011 0.012 0.011 0.010 0.014 0.016 0.007 0.009

1.446 1.458 1.449 1.453 1.448 1.467 1.475 1.462

1.461 1.475 1.465 1.466 1.465 1.491 1.482 1.477

78 23 20 15 15 6 15 15

1.485 1.509 1.533 1.537 1.552

1.483 1.509 1.533 1.536 1.550

0.020 0.011 0.013 0.016 0.023

1.478 1.502 1.530 1.525 1.536

1.502 1.511 1.539 1.550 1.572

8 12 17 19 32

1.493 1.465

1.493 1.468

0.020 0.011

1.477 1.461

1.506 1.472

54 75

1.336 1.339 1.355 1.416

1.344 1.340 1.358 1.418

0.017 0.016 0.014 0.018

1.317 1.327 1.341 1.397

1.348 1.351 1.363 1.432

10 17 22 18

1.325 1.334 1.346 1.385 1.331 1.347 1.334 1.352 1.333 1.334 1.347 1.363 1.346 1.376 1.389 1.396 1.409 1.321 1.328

1.323 1.333 1.342 1.388 1.331 1.344 1.334 1.353 1.334 1.334 1.345 1.359 1.343 1.377 1.383 1.396 1.406 1.320 1.325

0.009 0.011 0.011 0.019 0.011 0.014 0.006 0.010 0.013 0.008 0.010 0.014 0.023 0.012 0.017 0.010 0.020 0.008 0.015

1.318 1.326 1.339 1.374 1.326 1.335 1.330 1.344 1.326 1.329 1.341 1.354 1.328 1.369 1.376 1.389 1.391 1.314 1.317

1.331 1.343 1.356 1.396 1.337 1.359 1.339 1.356 1.340 1.339 1.354 1.370 1.361 1.383 1.404 1.403 1.419 1.327 1.333

32 78 5 23 20 15 6 15 380 48 26 40 192 64 38 46 28 39 140

1.372 1.370 1.357 1.349 1.370

1.374 1.370 1.359 1.349 1.370

0.016 0.012 0.012 0.018 0.010

1.363 1.364 1.347 1.338 1.365

1.384 1.377 1.365 1.358 1.377

58 40 20 44 44

1.376

1.377

0.011

1.369

1.384

44

1.465

1.466

0.007

1.461

1.470

23

C*

N in N-substituted pyridinium

Nsp3 in in in in

aziridine azetidine tetrahydropyrrole piperidine

Csp3 Nsp2 (N planar) in: acyclic amides C* NH CO -lactams C* N( X) CO (endo)

-lactams C* NH CO (endo) C* N( C*) CO (endo) C* N( C*) CO (exo) -lactams C* NH CO (endo) C* N( C*) CO (endo) C* N( C*) CO (exo) nitro compounds (1,2-dinitro omitted): C CH2 NO2 C2 CH NO2 C3 C NO2 C2 C (NO2 )2 1,2-dinitro: NO2 C* C* NO2 Csp3

N(2)

C# C*

Csp2

N(3)

CC CC CC

NN NC

Car

NH2 Nsp2 planar NH C# Nsp2 planar N (C# )2 Nsp2 planar Nsp3 pyramidal

Csp2 Nsp2 (N planar) in: acyclic amides NH2 CO acyclic amides C* NH CO acyclic amides (C*)2 N CO -lactams C* NH CO

-lactams C* NH CO

-lactams C* N( C*) CO -lactams C* NH CO -lactams C* N( C*) CO peptides C# N( X) C( C# )(O) ureas (NH2 )2 CO ureas (C# NH)2 CO ureas [(C# )n N]2 CO thioureas (X2 N)2 CS imides [C# C(O)]2 NH [C# C(O)]2 N C# [Csp2 C(O)]2 N C# [Csp2 C(O)]2 N Csp2 guanidinium [C (NH2 )3 ] (unsubstituted) (any substituent) in heterocyclic systems (any substituent): 1H-pyrrole (N1 C2, N1 C5) indole (N1 C2) pyrazole (N1 C5) imidazole (N1 C2) imidazole (N1 C5) Csp2 Car

N(2) N(4)

in imidazole (N3 Car

C4)

N (C,H)3

802


n

Note

22 5, 22 5, 22

23 14 13 13 13 13 14 14 14

23 23 23 22 23 14 14 14 13 13 13 14 14 24 25,26 25 25, 27


Substructure 2

d

m

ql

qu

n

Note 23 22 28 23 22 28 23 22 28

Car

N(3)

NH2 (Nsp : planar) (Nsp3 : pyramidal) (overall) Car NH C# (Nsp2 : planar) (Nsp3 : pyramidal) (overall) Car N (C# )2 (Nsp2 : planar) (Nsp3 : pyramidal) (overall) in indole (N1 C7a) Car NO2

1.355 1.394 1.375 1.353 1.419 1.380 1.371 1.426 1.390 1.372 1.468

1.360 1.396 1.377 1.353 1.423 1.364 1.370 1.425 1.385 1.372 1.469

0.020 0.011 0.025 0.007 0.017 0.032 0.016 0.011 0.030 0.007 0.014

1.340 1.385 1.363 1.347 1.412 1.353 1.363 1.421 1.366 1.367 1.460

1.372 1.403 1.394 1.359 1.432 1.412 1.382 1.431 1.420 1.376 1.476

33 25 98 16 8 31 41 22 69 40 556

Car

N(2)

Car

1.431

1.435

0.020

1.422

1.442

26

Car

NN

Csp2 N(3)

in furoxan ( N2C3)

1.316

1.316

0.009

1.311

1.324

14

Csp2 N(2)

Car CN C# (C,H)2 CN OH in oximes S CN X in pyrazole (N2C3) in imidazole (C2N3) in isoxazole (N2C3) in furazan (N2C3, C4N5) in furoxan (C4N5)

1.279 1.281 1.302 1.329 1.313 1.314 1.298 1.304

1.279 1.280 1.302 1.331 1.314 1.315 1.299 1.306

0.008 0.013 0.021 0.014 0.011 0.009 0.006 0.008

1.275 1.273 1.285 1.315 1.307 1.305 1.294 1.300

1.285 1.288 1.319 1.339 1.319 1.320 1.303 1.308

75 67 36 20 44 9 12 14

Car N(3)

C N C N C N

H (pyrimidinium) C* (pyrimidinium) O (pyrimidinium)

1.335 1.346 1.362

1.334 1.346 1.359

0.015 0.010 0.013

1.325 1.340 1.353

1.342 1.352 1.369

30 64 56

Car N(2)

C N (pyridine) C N (pyrazine) C N C (pyrimidine) N C N (pyrimidine) C N (pyrimidine) (overall)

1.337 1.336 1.339 1.333 1.336

1.338 1.335 1.338 1.335 1.337

0.012 0.022 0.015 0.013 0.014

1.300 1.319 1.333 1.326 1.331

1.344 1.347 1.342 1.337 1.339

269 120 28 28 56

in any six-membered N-containing aromatic ring: H C N C H H C N C C* C* C N C C* C N C (overall)

1.334 1.339 1.345 1.336

1.334 1.341 1.345 1.337

0.014 0.013 0.008 0.014

1.327 1.336 1.342 1.329

1.341 1.345 1.348 1.344

146 38 24 204

Csp1 N(2)

X

N C (isocyanide)

1.144

1.147

0.006

1.140

1.148

6

Csp1 N(1)

C* CN CC CN in TCNQ Car CN X S CN (S CN)

1.136 1.144 1.138 1.144 1.155

1.137 1.144 1.138 1.141 1.156

0.010 0.008 0.007 0.012 0.012

1.131 1.139 1.133 1.138 1.147

1.142 1.149 1.143 1.151 1.165

140 284 31 10 14

Csp3

in alcohols: CH3 OH C CH2 OH C2 CH OH C3 C OH C* OH (overall)

1.413 1.426 1.432 1.440 1.432

1.414 1.426 1.431 1.440 1.431

0.018 0.011 0.011 0.012 0.013

1.395 1.420 1.425 1.432 1.424

1.425 1.431 1.439 1.449 1.441

17 75 266 106 464

in dialkyl ethers: CH3 O C* C CH2 O C* C2 CH O C* C3 C O C* C* O C* (overall)

1.416 1.426 1.429 1.452 1.426

1.418 1.424 1.430 1.450 1.425

0.016 0.011 0.010 0.011 0.019

1.405 1.418 1.420 1.445 1.414

1.426 1.435 1.437 1.458 1.437

110 34 53 39 236

in aryl alkyl ethers: CH3 O Car C CH2 O Car

1.424 1.431

1.424 1.430

0.012 0.013

1.417 1.422

1.431 1.438

616 188

O(2)

803


19

29

5 29

9. BASIC STRUCTURAL FEATURES Table 9.5.1.1. Average lengths (cont.) Bond Csp2 O(2) (cont.)

Substructure

d

m

ql

qu

n

1.447 1.470 1.429

1.446 1.469 1.427

0.020 0.018 0.018

1.435 1.456 1.419

1.466 1.483 1.436

58 55 917

in alkyl esters of carboxylic acids: CH3 O C(O) C* C CH2 O C(O) C* C2 CH O C(O) C* C3 C O C(O) C* C* O C(O) C* (overall)

1.448 1.452 1.460 1.477 1.450

1.449 1.453 1.460 1.475 1.451

0.010 0.009 0.010 0.008 0.014

1.442 1.445 1.454 1.472 1.442

1.455 1.458 1.465 1.484 1.459

200 32 78 6 314

in alkyl esters of ; -unsaturated acids: C* O C(O) CC (overall)

1.453

1.452

0.013

1.444

1.459

112

in alkyl esters of benzoic acid C* O C(O) C(phenyl) (overall)

1.454

1.454

0.012

1.446

1.463

219

in ring systems: oxirane (epoxide) (any substituent) oxetane (any substituent) tetrahydrofuran (C,H substituents) tetrahydropyran (C,H substituents) -lactones: C* O C(O)

-lactones: C* O C(O) -lactones: C* O C(O)

1.446 1.463 1.442 1.441 1.492 1.464 1.461

1.446 1.460 1.441 1.442 1.494 1.464 1.464

0.014 0.015 0.017 0.015 0.010 0.012 0.017

1.438 1.451 1.430 1.431 1.481 1.455 1.452

1.456 1.474 1.451 1.451 1.501 1.473 1.473

498 16 154 22 4 110 27

1.397

1.401

0.012

1.388

1.405

18

C5 O5 C1 O1 H in pyranoses: C5 O 5 O1 axial : O5 C1 C1 O1 O1 equatorial : C5 O5 O5 C1 C1 O1 (overall): C5 O5 O5 C1 C1 O1

1.439 1.427 1.403 1.435 1.430 1.393 1.439 1.430 1.401

1.440 1.426 1.400 1.436 1.431 1.393 1.440 1.429 1.399

0.008 0.012 0.012 0.008 0.010 0.007 0.008 0.012 0.011

1.432 1.421 1.391 1.429 1.424 1.386 1.432 1.421 1.392

1.445 1.432 1.412 1.440 1.436 1.399 1.446 1.436 1.407

29 29 29 17 17 17 60 60 60

C4 O4 (overall (overall (overall

1.442 1.432 1.404

1.446 1.432 1.405

0.012 0.012 0.013

1.436 1.421 1.397

1.449 1.443 1.409

18 18 18

in pyranoses: O5 C1 O1 C* O5 C1 O1 C* O5 C1 O1 C*

1.439 1.417 1.409 1.435 1.434 1.424 1.390 1.437 1.436 1.419 1.402 1.436

1.438 1.417 1.409 1.435 1.435 1.424 1.390 1.438 1.436 1.419 1.403 1.436

0.010 0.009 0.014 0.013 0.006 0.008 0.011 0.013 0.009 0.011 0.016 0.013

1.433 1.410 1.401 1.427 1.429 1.418 1.381 1.428 1.431 1.412 1.391 1.428

1.446 1.424 1.417 1.443 1.439 1.431 1.400 1.445 1.442 1.426 1.413 1.445

67 67 67 67 39 39 39 39 126 126 126 126

C1 O1 C* in furanoses: values) C4 O4 values) O4 C1 values) C1 O1 values) O1 C*

1.443 1.421 1.410 1.439

1.445 1.418 1.409 1.437

0.013 0.012 0.014 0.014

1.429 1.413 1.401 1.429

1.453 1.431 1.420 1.449

23 23 23 23

1.416 1.465

1.416 1.461

0.017 0.014

1.405 1.454

1.428 1.475

29 33

in aryl C2 C3 C*

O

alkyl ethers (cont.) CH O Car C O Car O Car (overall)

C O systems in gem-diols, and pyranose and furanose sugars: HO C* OH

C1 O1 H in furanoses: values) C4 O4 values) O4 C1 values) C1 O1

C5 O5 C1 O1 C* C5 O1 axial : O5 C1 O1 O1 equatorial : C5 O5 C1 O1 (overall): C5 O5 C1 O1 C4 O4 (overall (overall (overall (overall

Miscellaneous: C# O SiX3 C* O SO2

C

804


Note

12, 29

9

16 12 12 30, 31

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS Table 9.5.1.1. Average lengths (cont.) Bond 2

Csp O(2) (cont.)

Car

O(2)

Csp2 O(1)

Substructure

d

m

ql

qu

n

CC OH CC O C* C* C(O) OH CC C(O) OH Car C(O) OH in esters: C* C(O) O C* CC C(O) O C* Car C(O) O C* C* C(O) O CC C* C(O) O CC C* C(O) O Car in anhydrides: OC O CO

1.333 1.354 1.308 1.293 1.305 1.336 1.332 1.337 1.362 1.407 1.360 1.386

1.331 1.353 1.311 1.295 1.311 1.337 1.331 1.335 1.359 1.405 1.359 1.386

0.017 0.016 0.019 0.019 0.020 0.014 0.011 0.013 0.018 0.017 0.011 0.011

1.324 1.341 1.298 1.279 1.291 1.328 1.324 1.329 1.351 1.394 1.355 1.379

1.342 1.363 1.320 1.307 1.317 1.346 1.339 1.344 1.374 1.420 1.367 1.393

53 40 174 22 75 551 112 219 26 26 40 70

in ring systems: furan (O1 C2, O1 C5) isoxazole (O1 C5) -lactones: C* C(O) O C*

-lactones: C* C(O) O C* -lactones: C* C(O) O C*

1.368 1.354 1.359 1.350 1.339

1.369 1.354 1.359 1.349 1.339

0.015 0.010 0.013 0.012 0.016

1.359 1.345 1.348 1.342 1.332

1.377 1.360 1.371 1.359 1.347

125 9 4 110 27

in in in in

phenols: Car OH aryl alkyl ethers: Car O C* diaryl ethers: Car O Car esters: Car O C(O) C*

1.362 1.370 1.384 1.401

1.364 1.370 1.381 1.401

0.015 0.011 0.014 0.010

1.353 1.363 1.375 1.394

1.373 1.377 1.391 1.408

511 920 132 40

in aldehydes and ketones: C* CHO (C*)2 CO (C# )2 CO in cyclobutanones in cyclopentanones in cyclohexanones CC CO (CC)2 CO Car CO (Car )2 CO CO in benzoquinones

1.192 1.210 1.198 1.208 1.211 1.222 1.233 1.221 1.230 1.222

1.912 1.210 1.198 1.208 1.211 1.222 1.229 1.218 1.226 1.220

0.005 0.008 0.007 0.007 0.009 0.010 0.010 0.014 0.015 0.013

1.188 1.206 1.194 1.203 1.207 1.216 1.226 1.212 1.220 1.211

1.197 1.215 1.204 1.212 1.216 1.229 1.242 1.229 1.238 1.231

7 474 12 155 312 225 28 85 66 86

delocalized double bonds in carboxylate anions: H C O2 (formate) C* C O2 CC C O2 Car C O2 HOOC C O2 (hydrogen oxalate) O2 C C O2 (oxalate)

1.242 1.254 1.250 1.255 1.243 1.251

1.243 1.253 1.248 1.253 1.247 1.251

0.012 0.010 0.017 0.010 0.015 0.007

1.234 1.247 1.238 1.249 1.232 1.248

1.252 1.261 1.261 1.262 1.256 1.254

24 114 52 22 26 18

in carboxylic acids (X COOH): C* C(O) OH CC C(O) OH Car C(O) OH

1.214 1.229 1.226

1.214 1.226 1.223

0.019 0.017 0.020

1.203 1.218 1.211

1.224 1.237 1.241

175 22 75

in esters: C* C(O) O C* CC C(O) O C* Car C(O) O C* C* C(O) O CC C* C(O) O Car

1.196 1.199 1.202 1.190 1.187

1.196 1.198 1.201 1.190 1.188

0.010 0.009 0.009 0.014 0.011

1.190 1.193 1.196 1.184 1.181

1.202 1.203 1.207 1.198 1.195

551 113 218 26 40

1.187

1.187

0.010

1.184

1.193

70

in -lactones: C* C(O) O C*

-lactones: C* C(O) O C* -lactones: C* C(O) O C*

1.193 1.201 1.205

1.193 1.202 1.207

0.006 0.009 0.008

1.187 1.196 1.201

1.198 1.206 1.209

4 109 27

13 12 12

in amides: NH2 C( C*)O (C* )(C*,H )N C( C*)O -lactams: C* NH CO

1.234 1.231 1.198

1.233 1.231 1.200

0.012 0.012 0.012

1.225 1.224 1.193

1.243 1.238 1.204

32 378 23

14 14 13

in enols: in enol esters: in acids:

in anhydrides: OC

O

CO

805


Note

12, 29 12 12

13 12 12 29, 32 12

5

12 12 12


Csp O(1) (cont.)

in amides (cont.)

-lactams: C*

-lactams: C* -lactams: C* -lactams: O*

Substructure

d

m

ql

qu

NH CO N( C*) CO NH CO N( C*) CO

1.235 1.225 1.240 1.233

1.235 1.226 1.241 1.233

0.008 0.011 0.003 0.007

1.232 1.217 1.237 1.229

1.240 1.233 1.243 1.239

20 15 6 15

13 13 14 14

1.256 1.241 1.230

1.256 1.237 1.230

0.007 0.011 0.007

1.249 1.235 1.224

1.261 1.245 1.234

24 13 20

25, 26 25 25, 27

1.800 1.791 1.806 1.821 1.841 1.813

1.802 1.790 1.806 1.821 1.842 1.811

0.015 0.006 0.009 0.009 0.008 0.017

1.790 1786 1.801 1.815 1.835 1.800

1.812 1.795 1.813 1.828 1.847 1.822

35 10 45 15 14 84

33

1.855

1.857

0.019

1.840

1.870

23

in ureas: (NH2 )2 CO (C# NH)2 CO [(C# )n N]2 CO Csp3

P(4)

C3 C2 C2 C2 C2 C2

P C* P(O) P(O) P(O) P(O) P(O)

Csp3

P(3)

C2

P

CH3 CH2 C CH C2 C C3 C* (overall)

C*

n

Note

Car

P(4)

C3 P Car C2 P(O) Car Ph3 PN P Ph3

1.793 1.801 1.795

1.792 1.802 1.795

0.011 0.011 0.008

1.786 1.796 1.789

1.800 1.807 1.800

276 98 197

Car

P(3)

C2 P Car (N )2 P Car (P N aromatic)

1.836 1.795

1.837 1.793

0.010 0.011

1.830 1.788

1.844 1.803

102 43

C* C* C* C*

SO2 SO2 SO2 SO2

1.786 1.779 1.745 1.758

1.782 1.778 1.744 1.736

0.018 0.020 0.009 0.018

1.774 1.764 1.738 1.746

1.797 1.790 1.754 1.773

75 94 7 17

S(O) C (C* CH3 excluded) S(O) C (overall) S X2 S X2 (C* CH3 excluded) S C2 (overall)

1.818 1.809 1.786 1.823 1.804

1.814 1.806 1.787 1.820 1.794

0.024 0.025 0.007 0.016 0.025

1.802 1.793 1.779 1.812 1.788

1.829 1.820 1.792 1.834 1.820

69 88 21 18 41

Csp3

S(4)

C (C* CH3 excluded) C (overall) O X N X2

Csp3

S(3)

C* C* CH3 C* C*

Csp3

S(2)

C* SH CH3 S C* C CH2 S C* C2 CH S C* C3 C S C* C* S C* (overall)

1.808 1.789 1.817 1.819 1.856 1.819

1.805 1.787 1.816 1.819 1.860 1.817

0.010 0.008 0.013 0.011 0.011 0.019

1.800 1.784 1.808 1.811 1.854 1.809

1.819 1.794 1.824 1.825 1.863 1.827

6 9 92 32 26 242

in in in in

1.834

1.835

0.025

1.810

1.858

4

1.827 1.823

1.826 1.821

0.018 0.014

1.811 1.812

1.837 1.832

20 24

C CH2 S S X C3 C S S X C* S S X (overall)

1.823 1.863 1.833

1.820 1.865 1.828

0.014 0.015 0.022

1.813 1.848 1.818

1.832 1.878 1.848

41 11 59

CC CC CC OC

C* CC (in tetrathiafulvalene) CC (in thiophene) C#

1.751 1.741 1.712 1.762

1.755 1.741 1.712 1.759

0.017 0.011 0.013 0.018

1.740 1.733 1.703 1.747

1.764 1.750 1.722 1.778

61 88 60 20

C O N

1.763 1.752 1.758

1.764 1.750 1.759

0.009 0.008 0.013

1.756 1.749 1.749

1.769 1.756 1.765

96 27 106

1.790 1.778

1.790 1.779

0.010 0.010

1.783 1.771

1.798 1.787

41 10

1.773 1.768

1.774 1.767

0.009 0.010

1.765 1.762

1.779 1.774

44 158

Csp2

Car

S(2)

S(4)

thiirane thietane: see ZCMXSP (1.817, 1.844) tetrahydrothiophene tetrahydrothiopyran

S S S S

Car Car Car

SO2 SO2 SO2

Car

S(3)

Car Car

S(O) S X2

Car

S(2)

Car Car

S S

X X2 C

C* Car

806


34 34

9

35


Substructure

d

m

ql

qu

1.764 1.777

1.764 1.777

0.008 0.012

1.760 1.767

1.769 1.785

48 47

1.679

1.683

0.026

1.645

1.698

10

1.630

1.630

0.014

1.619

1.641

14

(C*)2 CS: see IPMUDS (1.599) (Car )2 CS: see CELDOM (1.611) (X)2 CS (X C,N,O,S) X2 N C(S) S X (X2 N)2 CS (thioureas) N C( S)2

1.671 1.660 1.681 1.720

1.675 1.660 1.684 1.721

0.024 0.016 0.020 0.012

1.656 1.648 1.669 1.709

1.689 1.674 1.693 1.731

245 38 96 20

1.970

1.967

0.032

1.948

1.998

21

1.893

1.895

0.013

1.882

1.902

32

1.930

1.929

0.006

1.924

1.936

13

1.874

1.876

0.015

1.859

1.884

9

Car S(2) (cont.)

Car Car

Csp1

S(2)

NC

Csp1

S(1)

(NC

Csp2 S(1)

S S

Csp3

Se

C#

Csp2

Se(2)

CC

Car

Se(3)

Car (in phenothiazine) S X S

X

S)

Se Se

CC (in tetraselenafulvalene)

Se

Ph3

n

Note

Csp3

Si(5)

C#

Csp3

Si(4)

CH3 Si X3 C* Si X3 (C* CH3 excluded) C* Si X2 (overall)

1.857 1.888 1.863

1.857 1.887 1.861

0.018 0.023 0.024

1.848 1.872 1.850

1.869 1.905 1.875

552 124 681

Car

1.868

1.868

0.014

1.857

1.878

178

1.837

1.840

0.012

1.824

1.849

8

Si(4)

Car

Si

Si

X4

X3

Csp1

Si(4)

CC

Csp3

Te

C#

Te

2.158

2.159

0.030

2.128

2.177

13

Car

Te

2.116

2.115

0.020

2.104

2.130

72

Car

Te

Si

X3

Csp2 Te

see CEDCUJ (2.044)

Cl

Cl

see PHASCL (2.306, 2.227)

Cl

I

see CMBIDZ (2.563), HXPASC (2.541, 2.513), METAMM (2.552), BQUINI (2.416, 2.718)

Cl

N

see BECTAE (1.743±1.757), BOGPOC (1.705)

Cl

O(1)

in ClO4

1.414

1.419

0.026

1.403

1.431

252

Cl

P

(N )2 P Cl (N P aromatic) Cl P (overall)

1.997 2.008

1.994 2.001

0.015 0.035

1.989 1.986

2.004 2.028

46 111

Cl

S

Cl S (overall) see also longer bonds in CILSAR (2.283), BIHXIZ (2.357), CANLUY (2.749)

2.072

1.079

0.023

2.047

2.091

6

Cl

Se

See BIRGUE10, BIRHAL10, CTCNSE (2.234±2.851)

Cl

Si(4)

Cl Si X3 (monochloro) Cl2 Si X2 and Cl3 Si

2.072 2.020

2.075 2.012

0.009 0.015

2.066 2.007

2.078 2.036

5 5

Cl

Te

Cl Te in range 2.34±2.60 see also longer bonds in BARRIV, BOJPUL, CETUTE, EPHTEA, OPNTEC10 (2.73±2.94)

2.520

2.515

0.034

2.493

2.537

22

F

N(3)

F

C

1.406

1.404

0.016

1.395

1.416

9

F

P(6)

in hexa¯uorophosphate, PF6

1.579

1.587

0.025

1.563

1.598

72

P

P(3)

(N )2 P

1.495

1.497

0.016

1.481

1.510

10

N

C2 and F2

N

X

F (N P aromatic)

807


36

9. BASIC STRUCTURAL FEATURES Table 9.5.1.1. Average lengths (cont.) Bond F

S

Substructure

d

m

ql

qu

n

Note

43 observations in range 1.409±1.770 in a wide variety of environments F S(6) in F2 SO2 C2 (see FPSULF10, BETJOZ) F S(4) in F2 S(O) N (see BUDTEZ)

1.640 1.527

1.646 1.528

0.011 0.004

1.626 1.524

1.649 1.530

6 24

1.694

1.701

0.013

1.677

1.703

6

1.636

1.639

0.035

1.602

1.657

10

1.588

1.587

0.014

1.581

1.599

24

1.033

1.036

0.022

1.026

1.045

87

21

1.009

1.010

0.019

0.997

1.023

95

21 21 21 21, 38

F

Si(6)

in SiF26

F

Si(5)

F

Si

F

Si(4)

F

Si

F

Te

see CUCPIZ [F Te(6) 1.942, 1.937], FPHTEL [F Te(4) 2.006]

H

N(4)

X3

N

H

N(3)

X2

N

H

O(2)

in alcohols C* O H C# O H in acids OC O H

0.967 0.967 1.015

0.969 0.970 1.017

0.010 0.010 0.017

0.959 0.959 1.001

0.974 0.974 1.031

63 73 16

2.917

2.918

0.011

2.907

2.927

6

2.144

2.144

0.028

2.127

2.164

6

X4 X3

H H

I

I

in I3

I

N

see BZPRIB, CMBIDZ, HMTITI, HMTNTI, IFORAM, IODMAM (2.042±2.475)

I

O

X

I

P(3)

see CEHKAB (2.490±2.493)

I

S

see DTHIBR10 (2.687), ISUREA10 (2.629), BZTPPI (3.251)

I

Te(4)

I

I O (see BZPRIB, CAJMAB, IBZDAC11) for IO6 see BOVMEE (1.829±1.912)

Te N

y

X3 N0

X2 (N0 planar)

2.926

2.928

0.026

2.902

2.944

8

1.414

1.414

0.005

1.412

1.418

13

N(4)

N(3)

X3

N(3)

N(3)

(C)(C,H) Na Nb (C)(C,H) Na , Nb pyramidal Na pyramidal, Nb planar Na , Nb planar overall

1.454 1.420 1.401 1.425

1.452 1.420 1.401 1.425

0.021 0.015 0.018 0.027

1.444 1.407 1.384 1.407

1.457 1.433 1.418 1.443

44 68 40 139

N(3)

N(2)

in pyrazole (N1 N2) in pyridazinium (N1 N2)

1.366 1.350

1.366 1.349

0.019 0.010

1.350 1.345

1.375 1.361

20 7

N(2) N(2)

N N (aromatic) in pyridazine with C,H as ortho substituents with N,Cl as ortho substituents

1.304 1.368

1.300 1.373

0.019 0.011

1.287 1.362

1.326 1.375

6 9

N(2)N(2)

C# Car X

C# (cis) (trans) (overall) NN Car NNN (azides)

1.245 1.222 1.240 1.255 1.216

1.244 1.222 1.241 1.253 1.226

0.009 0.006 0.012 0.016 0.028

1.239 1.218 1.230 1.247 1.202

1.252 1.227 1.251 1.262 1.237

21 6 27 13 19

X

NNN (azides)

1.124

1.128

0.015

1.114

1.137

19

N(2)N(1)

NN

N(3)

O(2)

(C,H)2 N OH (Nsp2 : planar) C2 N O C (Nsp3 : pyramidal) C2 N O C (Nsp2 : planar) in furoxan (N2 O1)

1.396 1.463 1.397 1.438

1.394 1.465 1.394 1.436

0.012 0.012 0.011 0.009

1.390 1.457 1.388 1.430

1.401 1.468 1.409 1.447

28 22 12 14

N(3)

O(1)

(C )2 N O in pyridine N-oxides in furoxan ( N2 O6 )

1.304 1.234

1.299 1.234

0.015 0.008

1.291 1.228

1.316 1.240

11 14

808


37

5, 39 40 40 40

9.5. TYPICAL INTERATOMIC DISTANCES: ORGANIC COMPOUNDS Table 9.5.1.1. Average lengths (cont.) Bond N(2)

O(2)

N(3)O(1)

N(3)

N(3)

P(4)

P(3)

Substructure

d

m

ql

qu

in oximes: (C# )2 CN OH (H)(Csp2 ) CN OH (C# )(Csp2 ) CN OH (Csp2 )2 CN OH (C,H)2 CN OH (overall)

1.416 1.390 1.402 1.378 1.394

1.418 1.390 1.403 1.377 1.395

0.006 0.011 0.010 0.017 0.018

1.416 1.380 1.393 1.365 1.379

1.420 1.401 1.410 1.393 1.408

7 20 18 16 67

in furazan (O1 in furoxan (O1 in isoxazole (O1

1.385 1.380 1.425

1.383 1.380 1.425

0.013 0.011 0.010

1.378 1.370 1.417

1.392 1.388 1.434

12 14 9

in nitrate ions NO3 in nitro groups: C* NO2 C# NO2 Car NO2 C NO2 (overall)

1.239

1.240

0.020

1.227

1.251

105

1.212 1.210 1.217 1.218

1.214 1.210 1.218 1.219

0.012 0.011 0.011 0.013

1.206 1.203 1.211 1.210

1.221 1.218 1.215 1.226

84 251 1116 1733

NX2 Nsp2 : planar Nsp3 : pyramidal (overall) subsets of this group are: O2 P(S) NX2 C P(S) (NX2 )2 O P(S) (NX2 )2 P(O) (NX2 )3

1.652 1.683 1.662

1.651 1.683 1.662

0.024 0.005 0.029

1.634 1.680 1.639

1.670 1.686 1.682

205 6 358

1.628 1.691 1.652 1.663

1.624 1.694 1.654 1.668

0.015 0.018 0.014 0.026

1.615 1.678 1.642 1.640

1.634 1.703 1.664 1.679

9 28 28 78

NX P( X) NX P( X) NX P(S) NX P(S) in P-substituted phosphazenes: (N )2 P N (amino) (aziridinyl)

1.730 1.697

1.721 1.697

0.017 0.015

1.716 1.690

1.748 1.703

20 44

1.637 1.672

1.638 1.674

0.014 0.010

1.625 1.665

1.651 1.676

16 15

1.571 1.599

1.573 1.597

0.013 0.018

1.563 1.580

1.580 1.615

66 7

X2

N2, O1 N5) N2)

N5)

P(X)

(P2 N2 ring) (P2 N2 ring)

PN P Ph3 PN C,S

n

Note

N(2)P(4)

Ph3 Ph3

N(2) P(3)

N P aromatic in phosphazenes in P N S

1.582 1.604

1.582 1.606

0.019 0.009

1.571 1.594

1.594 1.612

126 36

1.600 1.633 1.642

1.601 1.633 1.641

0.012 0.019 0.024

1.591 1.615 1.623

1.610 1.652 1.659

14 47 38

35 35 35

N(3)

S(4)

C C C

N(3)

S(2)

C S NX2 Nsp2 : planar (for Nsp3 pyramidal see MODIAZ: 1.765) X S NX2 Nsp2 : planar

1.710

1.707

0.019

1.698

1.722

22

23

1.707

1.705

0.012

1.699

1.715

30

23

CN

X

1.656

1.663

0.027

1.632

1.677

36

N(2) S(2)

N S aromatic in P N S

1.560

1.558

0.011

1.554

1.563

37

N(2)S(2)

NS in NSN and NSS

1.541

1.546

0.022

1.521

1.558

37

1.748

1.746

0.022

1.735

1.757

170

1.714

1.719

0.014

1.702

1.727

16

N(2)

S(2)

SO2 SO2 SO2

NH2 NH C# N (C# )2

S

N(3)

Se

see COJCUZ (1.830), DSEMOR10 (1.846, 1.852), MORTRS10 (1.841)

N(2)

Se

see SEBZQI (1.805), NAPSEZ10 (1.809, 1.820)

N(2)Se

see CISMUM (1.790, 1.791)

N(3)

Si(5)

see DMESIP01, BOJLER, CASSAQ, CASYOK, CECXEN, CINTEY, CIPBUY, FMESIB, MNPSIL, PNPOSI (1.973±2.344)

N(3)

Si(4)

X3 Si NX2 (overall) subsets of this group are: X3 Si NHX

809


9. BASIC STRUCTURAL FEATURES Table 9.5.1.1. Average lengths (cont.) Bond

Substructure

N(3) Si(4) (cont.) N(2) N

Si(4)

Te

O(2)

X3 Si NX Si X3 acyclic N Si N in four-membered rings N Si N in ®ve-membered rings X3

Si

N

Si

X3

d

m

ql

qu

n

1.743 1.742 1.741

1.744 1.742 1.742

0.016 0.009 0.019

1.731 1.735 1.726

1.755 1.748 1.749

45 53 33

1.711

1.712

0.019

1.693

1.729

15

1.464 1.482 1.469

1.464 1.480 1.471

0.009 0.005 0.012

1.458 1.478 1.461

1.472 1.486 1.478

12 5 17

1.496

1.499

0.005

1.490

1.499

10

Note

see ACLTEP (2.402), BIBLAZ (1.980), CESSAU (2.023) O(2)

C*,H (OO) 70±85 (OO) approx. 180 (overall) OC O O CO see ACBZPO01 (1.446), CEYLUN (1.452), CIMHIP (1.454) Si O O Si C*

O

O

O(2)

P(5)

X P (OX)4 trigonal bipyramidal: axial equatorial square pyramidal

1.689 1.619 1.662

1.685 1.622 1.661

0.024 0.024 0.020

1.675 1.604 1.649

1.712 1.628 1.673

20 20 28

O(2)

P(4)

C O P( O)23 (H O)2 P( O)2 (C O)2 P( O)2 (C# O)3 PO (Car O)3 PO X O P(O) (C,N)2 (X O)2 P(O) (C,N)

1.621 1.560 1.608 1.558 1.587 1.590 1.571

1.622 1.561 1.607 1.554 1.588 1.585 1.572

0.007 0.009 0.013 0.011 0.014 0.016 0.013

1.615 1.555 1.599 1.550 1.572 1.577 1.563

1.628 1.566 1.615 1.564 1.599 1.601 1.579

12 16 16 30 19 33 70

O(2)

P(3)

(N )2 P

1.573

1.573

0.011

1.563

1.584

16

O

C (N P aromatic)

41

O(1)P(4)

C O P( O)23 (delocalized) (H O)2 P( O)2 (delocalized) (C O)2 P( O)2 (delocalized) (C O)3 PO C3 PO N3 PO (C)2 (N) PO (C,N)2 (O) PO (C,N)(O)2 PO

1.513 1.503 1.483 1.449 1.489 1.461 1.487 1.467 1.457

1.512 1.503 1.485 1.448 1.486 1.462 1.489 1.465 1.458

0.008 0.005 0.008 0.007 0.010 0.014 0.007 0.007 0.009

1.508 1.499 1.474 1.446 1.481 1.449 1.479 1.462 1.454

1.518 1.508 1.490 1.452 1.496 1.470 1.493 1.472 1.462

42 16 16 18 72 26 5 33 35

O(2)

C C C

1.577 1.569 1.580

1.576 1.569 1.578

0.015 0.013 0.015

1.566 1.556 1.571

1.584 1.582 1.588

41 7 27

1.436 1.428 1.430 1.423 1.472

1.437 1.428 1.430 1.423 1.473

0.010 0.010 0.009 0.008 0.013

1.431 1.422 1.425 1.418 1.463

1.442 1.434 1.435 1.428 1.481

316 326 206 82 104

42

1.497

1.498

0.013

1.489

1.505

90

5

1.663

1.658

0.023

1.650

1.665

21

S(4)

O O O

O(1)S(4)

C SO2 X SO2 C SO2 C SO2 in SO24

O(1)S(3)

C

O

Se

SO2 SO2 SO2

C CH3 Car

C NX2 N (C,H)2 O C

S(O)

C

see BAPPAJ, BIRGUE10, BIRHAL10, CXMSEO, DGLYSE, SPSEBU (1.597 for OSe to 1.974 for O Se)

O(2)

Si(5)

(X

O)3

Si

O(2)

Si(4)

X3 Si O X (overall) subsets of this group are: X3 Si O C# X3 Si O Si X3 X3 Si O O Si X3

1.631

1.630

0.022

1.617

1.646

191

1.645 1.622 1.680

1.647 1.625 1.676

0.012 0.014 0.008

1.634 1.614 1.673

1.652 1.631 1.688

29 70 10

1.927

1.927

0.020

1.908

1.942

16

2.133

2.136

0.054

2.078

2.177

12

O(2)

Te(6)

(X

O)6

Te

O(2)

Te(4)

(X

O)2

Te

(N)(C)

X2

810



Substructure

P(4)

P(4)

X3

P(4)

P(3)

see CECHEX (2.197), COZPIQ (2.249)

P(3)

P(3)

X2

P

P

P

P

X3

X2

d

m

ql

qu

n

Note

2.256

2.259

0.025

2.243

2.277

6

2.214

2.210

0.022

2.200

2.224

41

P(4)P(4)

see BUTSUE (2.054)

P(3)P(3)

see BALXOB (2.034)

P(4)S(1)

C3 PS (N,O)2 (C) PS (N,O)3 PS

1.954 1.922 1.913

1.952 1.924 1.914

0.005 0.014 0.014

1.950 1.913 1.906

1.957 1.927 1.921

13 26 50

P(4)Se(1)

X3

2.093

2.099

0.019

2.075

2.108

12

2.264

2.260

0.019

2.249

2.283

22

P(3)

Si(4)

PSe

X2 P Si X3 : 3- and 4-rings excluded (see BOPFER, BOPFIV, CASTOF10, COZVIW: 2.201±2.317)

P(4)Te(1)

see MOPHTE (2.356), TTEBPZ (2.327)

S(2)

S(2)

C

S

C (SS) 75±105 (SS) 0±20 (overall) in polysul®de chain S S S

2.031 2.070 2.048 2.051

2.029 2.068 2.045 2.050

0.015 0.022 0.026 0.022

2.021 2.057 2.028 2.037

2.038 2.077 2.068 2.065

46 28 99 126

S(2)

S(1)

X

NS

1.897

1.896

0.012

1.887

1.908

5

S (any)

2.193

2.195

0.015

2.174

2.207

9

S

2.145

2.138

0.020

2.130

2.158

19

2.405 2.682

2.406 2.686

0.022 0.035

2.383 2.673

2.424 2.694

10 28

2.340

2.340

0.024

2.315

2.361

15

S

S

S

Se(4)

see BUWZUO (2.264, 2.269)

S

Se(2)

X

Se

S(2)

Si(4)

X3

S(2)

Te

X S XS

Si

Te (any) Te (any)

Se(2)

Se(2)

X

Se(2)

Te(2)

see BAWFUA, BAWGAH (2.524±2.561)

Si(4)

Si(4)

X3

Te

Te

Se

X

Se

X

y

Si Si X3 three-membered rings excluded: see CIHRAM (2.511)

2.359

see CAHJOK (2.751, 2.704)

y See opening paragraph of Section 9.5.3. For numbered footnotes, see Appendix 1.

811


2.359

0.012

2.349

2.366

42


9.6. Typical interatomic distances: organometallic compounds and coordination complexes of the d- and f-block metals By A. G. Orpen, L. Brammer, F. H. Allen, D. G. Watson, and R. Taylor

9.6.1. Introduction The determination of molecular geometry is of vital importance to our understanding of chemical structure and bonding. The majority of experimental data have come from X-ray and neutron diffraction, microwave spectroscopy, and electron diffraction. Over the years, compilations of results from these techniques have appeared sporadically. The ®rst major compilation was Chemical Society Special Publication No. 11: Tables of Interatomic Distances and Con®guration in Molecules and Ions (Sutton, 1958). This volume summarized results obtained by diffraction and spectroscopic methods prior to 1956; a supplementary volume (Sutton, 1965) extended this coverage to 1959. Summary tables of bond lengths between carbon and other elements were also published in Volume III of International Tables for X-ray Crystallography (Kennard, 1962). Some years later, the Cambridge Crystallographic Data Centre (Allen, Bellard, Brice, Cartwright, Doubleday, Higgs, Hummelink, Hummelink-Peters, Kennard, Motherwell, Rodgers & Watson, 1979) produced an atlas-style compendium of all organic, organometallic and metal-complex crystal structures published in the period 1960±1965 (Kennard, Watson, Allen, Isaacs, Motherwell, Pettersen & Town, 1972). More recently, a survey of geometries determined by spectroscopic methods (Harmony, Laurie, Kuczkowski, Schwendemann, Ramsay, Lovas, Lafferty & Maki, 1979) has extended coverage in this area to mid-1977. A notable compendium of structural data, without geometric information, was given in Comprehensive Organometallic Chemistry (Bruce, 1981), covering all complexes with metal±carbon bonds. The BIDICS (Brown, Brown & Hawthorne, 1982) series, which ®nished in 1981, provided for some years a full coverage of metal complexes giving both bibliographic and geometric information. There have also been valuable annual summaries, without geometric information, on the structures of organometallic compounds determined by diffraction methods (Russell, 1988). The production of further comprehensive compendia of X-ray and neutron diffraction results has been precluded by the steep rise in the number of published crystal structures, as illustrated by Fig. 9.6.1.1. Print compilations have been effectively superseded by computerized databases. In particular, the

Fig. 9.6.1.1. Growth of the Cambridge Structural Database as number of entries Nent added annually. The structures containing d- or f-block metals are indicated by shading.

Cambridge Structural Database now contains bibliographic, chemical, and numerical results for some 86 000 organo-carbon crystal structures. This machine-readable ®le ful®ls the function of a comprehensive structure-by-structure compendium of molecular geometries. However, the amount of data now held in the CSD is so large that there is also a need for concise, printed tabulations of average molecular dimensions. The only tables of average geometry in general use are those contained in the Chemical Society Special Publications of 1958 and 1965 (Sutton, 1958, 1965), which list mean bond lengths for a variety of atom pairs and functional groups. Since these early tables were based on data obtained before 1960, we have used the CSD to prepare a new table of average bond lengths in organic compounds (see Chapter 9.5) and in metal complexes. The table given here (Table 9.6.3.3) speci®cally lists average lengths for metal±ligand distances, together with intra-ligand distances, involving bonds between the d- and f-block metals (Sc±Zn, Y±Cd, La±Hg, Ce±Lu, Th±U) and atoms H, B, C, N, O, F, Si, P, S Cl, As, Se, Br, Te, and I of ligands. Mean values are presented for 324 different bond types involving such metal± ligand bonds.

9.6.2. Methodology 9.6.2.1. Selection of crystallographic data All results given in Table 9.6.3.3 are based on X-ray and neutron diffraction results retrieved from the September 1985 version of the CSD. Neutron diffraction data only were used to derive mean bond lengths involving hydrogen atoms. This version of the CSD contained results for 49 854 single-crystal diffraction studies of organo-carbon compounds; 9802 of these satis®ed the acceptance criteria listed below and were used in the averaging procedures: (i) Structure contains a d- or f-block metal. (ii) Atomic coordinates for the structure have been published and are available in the CSD. (iii) Structure was determined from diffractometer data. (iv) Structure does not contain unresolved numeric data errors from the original publication (such errors are usually typographical and are normally resolved by consultation with the authors). (v) Only structures of higher precision were included on the basis that either a the crystallographic R factor was 0:07 and the reported mean estimated standard deviation (e.s.d.) of the Ê (corresponds to AS ¯ag 1, C C bond lengths was 0:030 A 2 or 3 in the CSD), or b the crystallographic R factor 0:05 and the mean e.s.d. for C C bonds was not available in the database (AS 0 in the CSD). (vi) Where the structure of a given compound had been determined more than once within the limits of (i)±(v), then only the most precise determination was used. The structures used in Table 9.6.3.3 do not include compounds whose structure precludes them from the CSD (i.e. not containing òrganic' carbon). In practice, structures including at least one C H bond are taken to contain òrganic' carbon. Thus, the entry for Cr CO distances has a contribution from [NEt4 ][Cr(-H)(CO)10 ] but not from K[Cr(-H)(CO)10 ] or [Cr(CO)6 ].


9.6. TYPICAL INTERATOMIC DISTANCES: ORGANOMETALLIC COMPOUNDS AND COMPLEXES 9.6.2.2. Program system

9.6.2.3. Classi®cation of bonds

All calculations were performed on a University of Bristol VAX 11/750 computer. Programs BIBSER, CONNSER, RETRIEVE (Allen et al., 1979) and GEOSTAT (Murray-Rust & Raftery, 1985a,b), as locally modi®ed, were used. A standalone program was written to implement the selection criteria, whilst a new program (STATS) was used for statistical calculations described below. It was also necessary to modify CONNSER to improve the precision with which it locates chemical substructures. In particular, the program was altered to permit the location of atoms with speci®ed coordination numbers. This was essential in the case of carbon so that atoms with coordination numbers 2, 3, and 4 (equivalent to formal hybridization state sp1 , sp2 , sp3 ) could be distinguished easily and reliably. Considerable care was taken to ensure that the correct molecular fragment was located by GEOSTAT in the generation of geometrical tabulations. Searches were conducted for all metals together and statistics for individual metal elements and subdivision of the entry for a given metal carried out subsequently. An important modi®cation to GEOSTAT allowed for calculation of metal-atom coordination number with due allowance for multihapto ligands and 2 ligands. Thus, 5 -C5 H5 , 6 -C6 H6 , and other 5 and 6 ligands were assigned to occupy 3 coordination sites, 3 and 4 ligands such as allyls and dienes to occupy 2 coordination sites, and 2 ligands such as alkenes 1 site, and so on. The approach taken in dealing with 2 bridging ligands was that when a metal±metal bond is bridged by one atom of a ligand [e.g. as in Cl, CO, OMe etc. as in (a), (b) below] then only the non-metal atom is counted as occupying a coordination site. For the relatively rare case of bridging polyhapto ligands (in which the bridging atoms are linked by direct bonds), the assignment follows logically, thus, 2 -2 ,2 -alkyne, see (c) below, occupies one site on each metal. Bridging ligands that do not have one atom bonded to both metals [e.g. acetate in (d) below] contribute to metal coordination numbers as do terminal ligands. In examples (a)±(d) below, the metal atoms therefore have coordination numbers as follows: (a), Rh 4; (b), Fe 6; (c), Co 4; (d), Rh 6. For cases where coordination number is very dif®cult to assign, notably where a metal atom is bonded to more than one other metal atom as in metal cluster complexes, no assignment was attempted.

The classi®cation of metal±ligand bonds in Table 9.6.3.3 is based on the ligating contacting atom. Thus, all metal±boron distances appear in sections 2.1±2.3 of Table 9.6.3.3, all metal± carbon distances in sections 3.1±3.22, and so on. Where intraligand interatomic distances (e.g. P C distances in tertiary phosphines) are given in Table 9.6.3.3, they are averaged over all metals and precede the individual metal±ligand interatomic distances for that ligand. Table 9.6.3.3 is designated: (i) to appear logical, useful, and reasonably self-explanatory to chemists, crystallographers, and others who may use it; (ii) to permit a meaningful average value to be cited for each bond length. With reference to (ii), it was considered that a sample of bond lengths could be averaged meaningfully if: a the sample was unimodally distributed; b the sample standard deviation was reasonably small, ideally Ê ; c there were no conspicuous outlying less than ca 0.04 A observations ± those that occurred at > 4 from the mean were automatically eliminated from the sample by STATS, other outliers were inspected carefully; d there were no compelling chemical reasons for further subdivision of the sample. It should be noted that Table 9.6.3.3 is not intended to be complete in covering all possible ligands. The purpose of the table is to provide information on the interatomic distances for ligands of the greatest chemical importance, notably for those that are simple and/or common. 9.6.2.4. Statistics Where there are less than four independent observations of a given bond length, then each individual observation is given explicitly in Table 9.6.3.3. In all other cases, the following statistics were generated by the program STATS. (i) The unweighted sample mean, d, where d

n P i1

di =n

and di is the ith observation of the bond length in a total sample of n observations. Recent work (Taylor & Kennard, 1983, 1985, 1986) has shown that the unweighted mean is an acceptable (even preferable) alternative to the weighted mean, where the ith observation is assigned a weight equal to 1=vardi . This is especially true where structures have been pre-screened on the basis of precision. (ii) The sample median, m. This has the property that half of the observations in the sample exceed m, and half fall short of it. (iii) The sample standard deviation, , where n 1=2 P 2 di d =n 1 : i1

The non-location of hydrogen atoms presents major dif®culties, both in the determination of coordination numbers for metal atoms, and for correct identi®cation of ligands (e.g. to distinguish methoxide from methanol). Care was therefore taken to exclude cases where any ambiguity existed [e.g. no data taken for M (OCH3 ) and M O(H)CH3 distances when both are present in a structure in which hydrogen-atom positions were not reported].

(iv) The lower quartile for the sample, ql . This has the property that 25% of the observations are less than ql and 75% exceed it. (v) The upper quartile for the sample, qu . This has the property that 25% of the observations exceed qu and 75% fall short of it. (vi) The number n of observations in the sample. The statistics given in Table 9.6.3.3 correspond to distributions for which the automatic 4 cut-off (see above) had been applied, and any manual removal of additional outliers (an infrequent operation) had been performed. In practice, a very small percentage of observations were excluded by these methods. The major effect of removing outliers is to improve

813


9. BASIC STRUCTURAL FEATURES the sample standard deviation, as shown in Fig. 9.6.2.1b in which four (out of 366) observations are deleted. The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical, normal distribution, the mean d will be approximately equal to the median m, the lower and upper quartiles ql ; qu will be approximately symmetric about the median m ql ' qu m, and 95% of the observations may be expected to lie within 2 of the mean value. For a skewed distribution, d and m may differ appreciably and ql and qu will be asymmetric with respect to m. When a bond-length distribution is negatively skewed, i.e. very short values are more common than very long values, then it may be due to thermal-motion effects; the distances used to prepare the table were not corrected for thermal libration. In a number of cases, the initial bond-length distribution was clearly not unimodal as in Fig. 9.6.2.1a. Where possible, such distributions were resolved into their unimodal components (as in Fig. 9.6.2.1c) on chemical or structural criteria. The case illustrated in Fig. 9.6.2.1, for Cu Cl bonds, is one of the most spectacular examples, owing to the dramatic consequences of oxidation state and coordination number (and Jahn±Teller effects) on the structures of copper complexes.

Table 9.6.3.1. Ligand index Contact atom

hydride tetrahydroborate (BH4 )

1.1 1.2

Boron

borohydrides boranes/carbaboranes boroles, borylenes, other heteroboracycles

2.1 2.2 2.3

Carbon

carbide (C) carbyne/alkylidyne (CR) vinylidene/alkenylidene (CCR2 ) acetylide/alkynyl (CCR) cyano (CN) isocyanides (CNR) carbon monoxide (CO) thiocarbonyl (CS) carbene/alkylidene (CR2 ) vinyl/alkenyl (CRCR2 ) aryl (C6 R5 ) acyl [C(O)R] alkyl (CR3 ) -alkenes (C2 R4 , allenes, etc.) alkynes (RCCR) 3 ligands (allyls, etc.) 4 ligands (conjugated dienes, etc.) 5 ligands (dienyls, etc.) 6 ligands (arenes, etc.) 7 ; 8 ligands carbaboranes, boroles miscellaneous (CO2 , CS2 , etc.)

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

nitride (N) nitrene/imide (NR) methyleneamido (NCR2 ) nitriles (NCR) isocyanate, isothiocyanate (NCO, NCS) dinitrogen (N2 ) diazonium (N2 R), diazoalkanes (N2 CR2 ) azide (N3 ) nitrosyl, thionitrosyl (NO, NS) amide (NR2 )

4.1 4.2 4.3 4.4 4.5

Nitrogen

Fig. 9.6.2.1. Effects of outlier removal and subdivision based on coordination number and oxidation state. Cu outliers [> 4 (sample) from mean]; c all data for which Cu is 4-coordinate, CuII . a b c

d 2.282 2.276 2.248

m 2.255 2.254 2.246

0.105 0.092 0.032

814


Ligand class identi®er

Hydrogen

9.6.3. Content and arrangement of table of interatomic distances Table 9.6.3.1 indicates how the interatomic distances covered in Table 9.6.3.3 are subdivided. Metal±ligand distances are grouped according to the ligand contact atom, which leads to ordering by atomic number of that contact atom. For a given contact atom (H, B, C, etc.), the ligands are grouped by type as listed in Table 9.6.3.1. The class of ligand is identi®ed numerically (e.g. alkoxides are class 5.3, alcohols class 5.23, ethers 5.24, etc.). Particular ligands are identi®ed by a third number (e.g. methoxide is ligand 5.3.1). Finally, alternative bonding modes for a particular ligand are denoted by a fourth number [e.g. terminal alkoxides 5.3.1.1, bridging 2 alkoxides 5.3.1.2]. In general, the bonding modes are arranged in the sequence 1 ; 2 ; . . . ; n , 2 ; 3 , etc., where n implies n atoms of the ligand are bonded to metal atoms, and m that m metal atoms are bonded to the ligand. Thus, acetates are represented by entries headed 5.5.2.1 1 , 5.5.2.2 (chelating, 2 ) and 5.5.2.3 (bridging, 2 ). For each ligand, the metal± ligand bonds then follow a sequence of ascending atomic

Ligand class

q1 2.233 2.232 2.233

qu 2.296 2.292 2.263

3.18 3.19 3.20 3.21 3.22

4.6 4.7 4.8 4.9 4.10

Cl: a all data; b all data without

N 366 362 153

9.6. TYPICAL INTERATOMIC DISTANCES: ORGANOMETALLIC COMPOUNDS AND COMPLEXES Table 9.6.3.1. Ligand index (cont.) Contact atom

Ligand class

Table 9.6.3.1. Ligand index (cont.) Ligand class identi®er

amidinate [RNC(R)NR] Schiff bases phthalocyanines, porphyrins, pyrroles pyrazolate, imidazolate and derivatives pyridine, polypryidyls (bpy, o-phen) pyrazines, pyridazines, pyrimidines other N2 ligands (NRNR2 , NNR2 , NRNR) triazenido (RNNNR) hydrazones and related species (NR2 NCR) oximes N-nitrite (NO2 ) amine (NR3 ) borazines

4.11 4.12 4.13

oxo (O) hydroxy (OH) alkoxy, aryloxy, etc. (OR) O-ketones (OCR2 ), urea carboxylates (O2 CR) oxalate (O2 CCO2 ) acetylacetonates [RC(O)CRC(O)CR] ; -diones (e.g. o-quinones) carbonate (CO23 ) N-oxides (e.g. pyridine N-oxide) nitrate (NO3 ) O-nitrite (NO2 ) dioxygen, peroxides phosphine oxides (OPR3 ) phosphate (PO34 ) other P O anions O-dialkyl sulfoxides (OSR2 ) sulfate (SO24 ) other S O anions (sulfonates, etc.) O-SO2 other oxyanions (e.g. ClO4 ) aqua alcohols (ROH) ethers (ROR0 ) miscellaneous (2 -acyl, 2 -CO2 , -NCO)

5.1 5.2 5.3 5.4 5.5 5.6 5.7

Fluorine

¯uoride (F) ¯uoroanions (BF4 , PF6 )

6.1 6.2

Silicon

miscellaneous

7.1

Phosphorus

phosphorus (P) phosphinidenes (PR) phosphides (PR2 ) oligo-phosphorus ligands (P3 , PR2 PR2 , PRPR, etc.) phosphines (PR3 ) diphosphines (e.g. diphos) phosphites [P(OR)3 ] aminophosphines, cyclotriphosphazenyl, misc. P N ligands

8.1 8.2 8.3 8.4

sul®des (S) thiolates (SR)

9.1 9.2

Nitrogen (cont.)

Oxygen

Sulfur

Contact atom

Chlorine

chloride (Cl)

10.1

Arsenic

arsines (AsR3 ) miscellaneous

11.1 11.2

Selenium

miscellaneous

12.1

Bromine

bromide (Br)

13.1

Tellurium

miscellaneous

14.1

Iodine

iodide (I)

15.1

4.15 4.16 4.17 4.18 4.19

5.20 5.21 5.22 5.23 5.24 5.25

8.5 8.6 8.7 8.8

9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17

number of the metal. For a given metal, the ®rst line of an entry in Table 9.6.3.3 gives statistics covering all appropriate occurrences of metal±ligand distances. Further lines give statistics for metal±ligand distances for subdivision based largely on chemical criteria (e.g. metal oxidation state or coordination number). Cases where one atom of a ligand bridges two or more metal atoms were included only when the metal atoms were all of the same type and, unless speci®ed, only when the metal±ligand distances were symmetrical (range Ê ). for distances 0.1 A In many instances, the number of structures having interatomic distances involving a given metal for a particular ligand is too small < 4 for statistics to be quoted. In these cases, individual structures, and the distances in them, are given. These structures are identi®ed by their CSD reference code (e.g. BOZMIN); short-form literature references, ordered alphabetically by reference code, are in Appendix 2. Each line of Table 9.6.3.3 contains nine columns of which six record the statistics of the bond-length distribution described above. The content of the remaining three columns: Bond, Substructure, and Note, are described below. 9.6.3.1. The `Bond' column This speci®es the atom pair to which the line refers. Therefore, in the case of triethylphosphine complexes (section 8.5.2), there are 18 lines, in which the bond column contains P C, followed by 17 entries for Ti P through to Au P, indicating statistics for both intraligand and metal±ligand atom pairs. 9.6.3.2. De®nition of `Substructure' This column provides details of any subdivision of particular metal±ligand bonds that has been applied. Thus, for terminal 815


9.3 9.4 9.5 9.6 9.7

4.14

5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19

Ligand class identi®er

S-thiocyanate (SCN) thioketones, thiourea (SCR2 ) thiocarboxylates (S2 CR ) thiocarbamates (S2 CNR2 ) xanthates (S2 COR ), dithiocarbonates trithiocarbonate (CS23 ), thioxanthates ; -dithiones phosphine sul®des dithiophosphinates (S2 PR2 ) polysulfur ligands (S2 , SSR, etc.) thioethers (SR2 ) S-SO2 , S-SO3 , etc. disul®des (RSSR) S-dialkyl sulfoxides (R2 SO) miscellaneous (2 -CS2 )

Sulfur (cont.)

4.20 4.21 4.22 4.23

Ligand class

9. BASIC STRUCTURAL FEATURES iron±chlorine bonds (in section 10.1.1.1), the second and third lines of the Fe Cl entry refer to complexes in which the iron atom is four-coordinate and in oxidation state II and III, respectively. In some cases, subdivision has been carried out

on the basis of ligand substituents in those cases where a well de®ned subdistribution was observed. For clarity, in a number of cases the ligand structure and numbering scheme are illustrated in Fig. 9.6.3.1. The reader will be aware that formal oxidation

Fig. 9.6.3.1. Diagrams of ligands in Table 9.6.3.3, showing table entry number and ligand atom labelling.

816


9.6. TYPICAL INTERATOMIC DISTANCES: ORGANOMETALLIC COMPOUNDS AND COMPLEXES Table 9.6.3.2. Numbers of entries in Table 9.6.3.3 Numbers of entries for which < 4 examples are known are given ®rst, followed by numbers of entries for which statistics are quoted (i.e. those with > 4 examples). Ligand atoms

Ligand class Metal *y Sc1 Ti2;3 V4;5;6 Cr7 Mn Fe Co8 Ni9;10 Cu11;12 Zn Y1 Zr13 Nb14 Mo15;16 Tc17;18 Ru Rh Pd Ag Cd19 Lax Cex Prx Ndx Smx Eux Gdx Tbx Dyx Hox Erx Tmx Ybx Lux Hf13 Ta23 W24;159 Re Os Ir Pt Au25 Hg26 Thx Ux

H

B

C

N

O

F

Si

P

S

Cl

As

Se

Br

Te

I

5

4

66

71

79

4

1

32

49

3

2

1

3

1

3

1, 0 0, 2, 0, 1, 1,

2 0 1 0 0

1, 2, 2, 1, 0, 1, 0, 2, 3, 2, 1, 1,

0 0 0 2 3 2 4 1 0 0 0 0

1, 1

1, 0

0, 1 2, 1

2, 0, 0, 1,

1 2 1 0

0, 6, 8, 9, 15, 10, 11, 7, 1, 3, 0, 6, 4, 11, 0, 12, 11, 9, 4,

1 8 8 15 12 33 27 20 11 2 1 8 6 25 1 25 25 15 2

0, 2 1, 1 0, 1 0, 1 0, 1

14, 6, 12, 14, 13, 5, 9, 7, 7,

5 5 13 13 19 28 23 35 16

10, 3, 17, 5, 19, 12, 14, 8, 3, 1, 0, 1, 2, 2, 0, 1,

1 5 26 5 9 18 13 6 14 1 1 0 2 1 1 0

2, 1 0, 1

1, 0, 0, 0, 1, 2,

0 2 2 2 1 0

1, 0

* No entries for Pm, Pa, and Ac.

1, 0, 2, 1, 0, 0, 1,

1 1 0 1 1 1 0

0, 1

0, 1, 2, 5, 9, 10, 12, 11, 4, 4, 8, 1, 1,

2 2 4 7 20 13 11 12 25 3 3 2 5

1, 1 2, 1, 1, 4, 10, 9, 9, 12, 11, 6, 10, 0, 7,

0, 4, 10, 5, 14, 14, 18, 9, 18, 14, 2, 5, 2, 7, 3, 11, 11, 6, 8, 12, 4, 4, 2, 7, 8, 3, 2, 1, 1, 0, 4,

4 15 10 19 10 12 22 18 31 10 6 7 9 28 5 8 14 6 2 10 7 4 5 5 6 6 4 0 2 1 4

3, 2 2 0, 1 0 2, 2 0 4, 2 1 4, 12 7 9 11, 12 6, 6 7 3, 4 4 13 8, 10 4, 0 0 5 11, 2 5, 8 3 5 16, 18

1, 0, 0, 1,

0 1 1 0

1, 0 0, 1

1, 0 3, 0 0, 1 0, 1 1, 1

1, 0 0, 1 1, 0

1, 0

2 4, 3 3 2, 6 5 1, 7 6 7, 4 13 7, 16 10 7, 11 8 12, 13 3 7, 12 7, 3

1, 2, 5, 1, 2, 5, 8, 2, 2,

1 0 11 2 13 13 7 3 0

4, 1, 3, 1, 4, 6, 4, 4, 6, 1, 0,

0 6 20 2 4 7 10 6 4 0 1

1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 2, 1,

2 2 1 2 2 2 2 3 2 1 2 2 2 1 2 2 2 1 3 0 0 0 0

1, 1, 0, 0, 0, 0, 1, 0,

0 0 2 2 2 2 1 1

0, 0, 0, 0, 1, 1,

1 2 1 2 1 1

1, 0 1, 0 1, 0 0, 1 0, 1 1, 0

2, 1, 2, 0, 0, 0, 0,

0 1 0 1 2 3 1

1, 0 0, 1 0, 1

0, 1 1, 0

0 0 1 1 1 1 3 1

1, 0

1, 0 1, 0

1, 1, 1, 1, 0, 1, 0, 0,

0, 2, 0, 0, 0, 1, 1,

2 0 2 2 1 0 2

1, 0

0, 2 1, 0, 1, 1, 0,

1 2 1 2 1

0, 1 0, 1

1, 0 1, 0

1, 0, 1, 1, 0,

1, 0 1, 0

0 1 0 0 1

0, 1 1, 2

y Superscripts refer to entries in Appendix 1.

state is not always well de®ned, where no assignment was possible then this is indicated by ( ) rather than the roman numeral used elsewhere. Finally, cases where the ligand oxidation state is variable are identi®ed (e.g. for O2 , o-quinones etc.) by references to the footnotes at the end of Table 9.6.3.3. 9.6.3.3. Use of the `Note' column The `Note' column refers to the footnotes collected in Appendix 1. These record additional information as follows:

0, 2, 4, 4, 4, 5, 6, 5, 3, 1, 1,

2 2 6 5 6 9 13 3 2 0 0

0, 1 2, 7, 4, 3, 5, 8, 7, 2, 0, 1,

4 5 3 3 5 11 3 6 1 1

1, 0 1, 0

2, 0 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0,

0 2 2 2 2 2 2 1 2 1 2

1, 0 0, 1 1, 0 0, 1

1, 1, 1, 1, 1,

0 0 0 0 0

1, 0

1, 0, 0, 1, 0, 0, 0, 0,

0 1 2 0 2 2 1 2

1, 0 0, 1

1, 0, 0, 0, 0, 1, 0,

1 2 1 2 2 0 2

0, 1

x See references 1, 20±23 in Appendix 1.

a notable features of the distribution of distances, e.g. likely bias due to dominance by one structure of substructure, skewness, bimodality (subdivisions of the entry usually follow, which remove these features whenever possible); b further details of the chemical substructure, such as the exclusion of structures with particular trans ligands; c details of exclusion criteria used for a given entry or group of entries, such as the constraint that the two M Cl distances, in bridging 2 Ê (section 10.1.1.2); d chloride complexes, differ by 0 and the vectors ai linearly independent over the rational numbers. In that case, the crystal does not have lattice periodicity and is said to be aperiodic. The above description can still be convenient, even in the case that the vectors ai are not independent over the rationals: one or more of them is then expressed as rational linear combinations of the others. A typical example is that of a superstructure arising from the (commensurate) modulation of a basic structure with lattice periodicity. Let us denote by M the set of all integral linear combinations of the vectors a1 ; . . . ; an . These are said to form a basis. It is a set of free Abelian generators, therefore the rank of M is n. The dimension of M is the dimension of the Euclidean space spanned by M fM g V

gik gkj ij :

M Z n:

9:8:4:6

n

The elements of Z are precisely the set of indices introduced above. Mathematically speaking, M has the structure of a (free Abelian) module. Its elements are vectors. So we call M a vector module. This nomenclature is intended as a generic characterization. When a series of structures is considered with different values of the components of the last d vectors with respect to the ®rst three, the generic values of these components are irrational, but accidentally they may become rational as well. This situation typically arises when considering crystal structures under continuous variation of parameters like temperature, pressure or chemical composition. In the case of an ordinary crystal, rank and dimension are equal, the crystal structure is periodic, and the vector module becomes a (reciprocal) lattice. Lattices and vector modules are, mathematically speaking, free Z modules. For such a module, there exists a dual one that is also free and of the same rank. In the periodic crystal case, that duality can be expressed by a scalar product, but for an aperiodic crystal this is no longer possible. It is possible to keep the metrical duality by enlarging the space and considering the vector module M as the projection of an n-dimensional (reciprocal) lattice in an n-dimensional Euclidean space Vs . M ! ;

f g Vs

and

Z n ;

9:8:4:7

with the orthogonal projection E of Vs onto V de®ned by M E :

9:8:4:8

This corresponds to attaching to the diffraction peak with indices h1 ; . . . ; hn the point of an n-dimensional reciprocal lattice having the same set of coordinates. The orthocomplement of V in Vs is called internal space and denoted by VI . The embedding is uniquely de®ned by the relations fasi g

i 1; . . . ; n;

fai g

9:8:4:9

is a basis of and a basis of M . The vectors aIi where span VI . The crystal density in V can also be embedded as s in Vs by identifying the Fourier coef®cients ^ at points of M and of having correspondingly the same components. ^ 1 ; . . . ; hn : ^ s h1 ; . . . ; hn h

We now consider crystal structures de®ned in the same threedimensional Euclidean space V with Fourier wavevectors that are

9:8:4:10

Then s is invariant with respect to translations of the lattice with basis

937


and

asi ai ; aIi ;

The two corresponding metric tensors g and g , gik ai ak

integral linear combinations of n 3 d fundamental ones a1 ; . . . ; an : n P H hi ai ; hi integers: 9:8:4:5

9. BASIC STRUCTURAL FEATURES asi ai ; aIi

9:8:4:11

dual to (9.8.4.9). In the commensurate case, this correspondence requires that the given superstructure be considered as the limit of an incommensurate crystal [for which the embedding (9.8.4.10) is a one-to-one relation]. As discussed below, point-group symmetries R of the diffraction pattern, when expressed in terms of transformation of the set of indices, de®ne n-dimensional integral matrices that can be considered as being n-dimensional orthogonal transformations Rs in Vs , leaving invariant the Euclidean metric tensors: gsik asi ask

and

gsik asi ask :

9:8:4:12

The crystal classes considered in the tables suppose the existence of main re¯ections de®ning a three-dimensional reciprocal lattice. For that case, the embedding can be specialized by making the choice asi

as3j

ai ; 0 a3j ; dj

i 1; 2; 3; j 1; 2; . . . ; d n

3;

9:8:4:13

and, correspondingly, asi ai ; aIi as3j 0; dj

i 1; 2; 3; j 1; 2; . . . ; d;

9:8:4:14

with di dk ik and ai ak ik . These are called standard lattice bases.

elements of the Laue group. On a standard lattice basis (9.8.4.13), the matrices R take the special form 0 E R R : 9:8:4:17 M R I R The transformation of main re¯ections and satellites is then given by R as in (9.8.4.15), the relation with R being (as already said)

where the tilde indicates transposition. Accordingly, on a standard basis one has E R M R : 9:8:4:18 R 0 I R The set of matrices E R for R elements of K forms a crystallographic point group in three dimensions, denoted KE , having elements R of O3, and the corresponding set of matrices I R forms one in d dimensions denoted by KI with elements RI of Od. For a modulated crystal, one can choose the ai i 1; 2; 3 of a standard basis. These span the (reciprocal) lattice of the basic structure. One can then express the additional vectors a3j (which are modulation wavevectors) in terms of the basis of the lattice of main re¯ections: a3j

9.8.4.2. Point groups 9.8.4.2.1. Laue class De®nition 1. The Laue point group PL of the diffraction pattern is the point group in three dimensions that transforms every diffraction peak into a peak of the same intensity.y

3P d j1

Rji aj ;

i 1; . . . ; 3 d:

9:8:4:15

The 3 d 3 d matrices R form a ®nite group of integral matrices K for K equal to PL or to one of its subgroups. A well known theorem in algebra states that then there is a basis in 3 d dimensions such that the matrices R on that basis are orthogonal and represent 3 d-dimensional orthogonal transformations Rs . The corresponding group is a 3 d-dimensional crystallographic group denoted by Ks . Because R is already an orthogonal transformation on V , Rs is reducible and can be expressed as a pair R; RI of orthogonal transformations, in 3 and d dimensions, respectively. The basis on which R; RI acts according to R is denoted by fai ; aIi g. It spans a lattice that is the reciprocal of the lattice with basis elements ai ; aIi . The pairs R; RI , sometimes also noted RE ; RI , leave invariant: R; RI ai ; aIi Rai ; RI aIi

3P d j1

Rji aj ; aIj ;

9:8:4:16

y See footnote on p. 913.

j 1; 2; . . . ; d:

9:8:4:19

d P j1

ji dj ;

i 1; 2; 3:

9:8:4:20

This follows directly from (9.8.4.19) and the de®nition of the reciprocal standard basis (9.8.4.13). From (9.8.4.16) and (9.8.4.17), a simple relation can be deduced between and the three constituents E R, I R, and M R of the matrix R: I R

E R

M R:

9:8:4:21

Notice that the elements of M R are integers, whereas has, in general, irrational entries. This requires that the irrational part of gives zero when inserted in the left-hand side of equation (9.8.4.21). It is therefore possible to decompose formally into parts i and r as follows. 1X i r ; with i R E R 1 ; 9:8:4:22 N R I where the sum is over all elements of the Laue group of order N. It follows from this de®nition that I R

i

E R

1

i:

9:8:4:23

E R:

9:8:4:24

This implies M R

I R

r

r

The matrix r has rational entries and is called the rational part of . The part i is called the irrational (or invariant) part. The above equations simplify for the case d 1. The elements 1i i are the three components of the wavevector q, the row matrix E R has the components of R 1 q and I R " 1

938


ji ai ;

i1

aIi

where R is the transpose of R 1 . In many cases, one can distinguish a lattice of main re¯ections, the remaining re¯ections being called satellites. The main re¯ections are generally more intense. Therefore, main re¯ections are transformed into main re¯ections by

3 P

The three components of the jth row of the d 3-dimensional matrix are just the three components of the jth modulation wavevector qj a3j with respect to the basis a1 ; a2 ; a3 . It is easy to show that the internal components aIi i 1; 2; 3 of the corresponding dual standard basis can be expressed as

Because all diffraction vectors are of the form (9.8.4.5), the action of an element R of the Laue group is given by Rai

R ~ R 1 ;

9.8. INCOMMENSURATE AND COMMENSURATE MODULATED STRUCTURES since, for d 1, q can only be transformed into q. One has the corresponding relations 1X with qi "Rq; 9:8:4:25 q qi qr ; N R and Rq "q (modulo reciprocal lattice ;

Rqi "qi : 9:8:4:26

The reciprocal-lattice vector that gives the difference between Rq and "q has as components the elements of the row matrix M R.

triples of one of the 32 (or 73) point groups, for each one of the two one-dimensional point groups and all homomorphisms from the ®rst to the second. Analogously, in 3 d dimensions, one takes one of the 32 (73) groups, one of the d-dimensional groups, and all homomorphisms from the ®rst to the second. If one takes all triples of a three-dimensional group, a d-dimensional group, and a homomorphism from the ®rst to the second, one ®nds, in general, groups that are equivalent. The equivalent ones still have to be eliminated in order to arrive at a list of non-equivalent groups. 9.8.4.3. Systems and Bravais classes 9.8.4.3.1. Holohedry

9.8.4.2.2. Geometric and arithmetic crystal classes According to the previous section, in the case of modulated structures a standard basis can be chosen (for M and correspondingly for ). According to equation (9.8.4.15), for each three-dimensional point-group operation R that leaves the diffraction pattern invariant, there is a point-group transformation RE in the external space (the physical one, so that RE R) and a point-group transformation RI in the internal space, such that the pair R; RI is a 3 d-dimensional orthogonal transformation Rs leaving a 3 d-dimensional lattice invariant. For incommensurate crystals, this internal transformation is unique and follows from the transformation by R of the modulation wavevectors [see equations (9.8.4.15) and (9.8.4.18) for the a3j basis vectors]: there is exactly one RI for each R. This is so because in the incommensurate case the correspondence between M and is uniquely ®xed by the embedding rule (9.8.4.10) (see Subsection 9.8.4.1). Because the matrices R and the corresponding transformations in the 3 d-dimensional space form a group, this implies that there is a mapping from the group KE of elements RE to the group KI of elements RI that transforms products into products, i.e. is a group homomorphism. A point group Ks of the 3 ddimensional lattice constructed for an incommensurate crystal, therefore, consists of a three-dimensional crystallographic point group KE , a d-dimensional crystallographic point group KI , and a homomorphism from KE to KI . De®nition 2. Two 3 d-dimensional point groups Ks and are geometrically equivalent if they are connected by a pair of orthogonal transformations TE ; TI in VE and VI , respectively, such that for every Rs from the ®rst group there is an element R0s of the second group such that RE TE TE R0E and RI TI TI R0I . Ks0

A point group determines a set of groups of matrices, one for each standard basis of each lattice left invariant. De®nition 3. Two groups of matrices are arithmetically equivalent if they are obtained from each other by a transformation from one standard basis to another standard basis. The arithmetic equivalence class of a 3 d-dimensional point group is fully determined by a three-dimensional point group and a standard basis for the vector module M because of relation (9.8.4.15). In three dimensions, there are 32 geometrically non-equivalent point groups and 73 arithmetically non-equivalent point groups. In one dimension, these numbers are both equal to two. Therefore, one ®nds all 3 1-dimensional point groups of incommensurately modulated structures by considering all

The Laue group of the diffraction pattern is a threedimensional point group that leaves the positions (and the intensities)y of the diffraction spots as a set invariant, thus the vector module M also. As discussed in Subsection 9.8.4.2, each of the elements of the Laue group can be combined with an orthogonal transformation in the internal space. The resulting point group in 3 d dimensions leaves the lattice invariant for which the vector module M is the projection. Conversely, if one has a point group that leaves the 3 d-dimensional lattice invariant, its three-dimensional (external) part with elements RE R leaves the vector module invariant. De®nition 4. The holohedry of the lattice is the subgroup of the direct product O3 Od, i.e. the group of all pairs of orthogonal transformations Rs R; RI that leave the lattice invariant. This choice is possible because the point groups are reducible, i.e. leave the subspaces V and VI of the direct sum space Vs invariant. In the case of an incommensurate crystal, the projection of on M is one-to-one as one can see as follows. The vector Hs

i1

hi ai ; 0

d P j1

mj qj ; dj

9:8:4:27

P P of is projected on H i hi ai j mj qj . The vectors projected on P P the null vector satisfy, therefore, the relation h a i i i j mj qj 0. For an incommensurate phase, the basis vectors are rationally independent, which means that hi 0 and mj 0 for any i and j. Consequently, precisely one vector of is projected on each given vector of M . Suppose now R 1. This transformation leaves the component of every vector belonging to in V invariant. If RI is the corresponding orthogonal transformation in VI of an element Rs of the point group, a vector with component HI is transformed into a vector with component HI0 . Since a given H is the component of only one vector of , this implies HI H0I . Consequently, RI is also the identity transformation. Therefore, for incommensurate modulated phases, there are no point-group elements with R RE 1 and RI 6 1. For commensurate crystal structures embedded in the superspace, this is different: point-group elements with internal component different from the identity associated with an external component equal to unity can occur. For modulated crystal structures, the holohedral point group can be expressed with respect to a lattice basis of standard form y See footnote on p. 913.

939


3 P

9. BASIC STRUCTURAL FEATURES (9.8.4.13). It is then faithfully represented by integral matrices that are of the form indicated in (9.8.4.17) and (9.8.4.18). 9.8.4.3.2. Crystallographic systems De®nition 5. A crystallographic system is a set of lattices having geometrically equivalent holohedral point groups. In this way, a given holohedral point group (and even each crystallographic point group) belongs to exactly one system. Two lattices belong to the same system if there are orthonormal bases in V and in VI , respectively, such that the holohedral point groups of the two lattices are represented by the same set of matrices. 9.8.4.3.3. Bravais classes De®nition 6. Two lattices belong to the same Bravais class if their holohedral point groups are arithmetically equivalent. This means that each of them admits a lattice basis of standard form such that their holohedral point group is represented by the same set of integral matrices. 9.8.4.4. Superspace groups 9.8.4.4.1. Symmetry elements The elements of a 3 d-dimensional superspace group are pairs of Euclidean transformations in 3 and d dimensions, respectively: gs fRjvg; fRI jvI g 2 E3 Ed;

9:8:4:28

i.e. are elements of the direct product of the corresponding Euclidean groups. The elements fRjvg form a three-dimensional space group, but the same does not hold for the elements fRI jvI g of Ed. This is because the internal translations vI also contain the `compensating' transformations associated with the corresponding translation v in V [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into one basis for V and one for VI . As for elements of a three-dimensional space group, the translational component s v; vI of the element gs can be decomposed into an intrinsic part os and an origin-dependent part as : t; tI vo ; voI va ; vaI ; with vo ; voI

n 1X Rm v; RmI vI ; n m1

9:8:4:29

where n denotes the order of the element R. In particular, for d 1 the intrinsic part voI of vI is equal to vI if RI " 1 and vanishes if " 1. The latter means that for d 1 there is always an origin in the internal space such that the internal shift vI can be chosen to be zero for an element with " 1. The internal part of the intrinsic translation can itself be decomposed into two parts. One part stems from the presence of a translation in the external space. The lattice of the 3 ddimensional space group has basis vectors ai ; aIi ; 0; dj ;

i 1; 2; 3;

j 1; . . . ; d:

9:8:4:30

The internal part of the ®rst three basis vectors is aIi

ai

d P j1

ji dj

9:8:4:31

accordingPto equation (9.8.4.20). The three-dimensional translation v i i ai then entails a d-dimensional translation v in VI given by 3 3 P P i ai i ai : 9:8:4:32 v i1

These are the so-called compensating translations. Hence, the internal translation vI can be decomposed as Pd

vI

v d;

9:8:4:33

where d j1 3j dj . This decomposition, however, does still depend on the origin. Consider the case d 1. Then an origin shift s in the threedimensional space changes the translation v to v 1 Rs and its internal part v q v to q v q 1 Rs. This implies that for the case that " 1 the part changes to q 1 Rs qr 1 Rs, because qi is invariant under R. Therefore, changes, in general. The internal translation

qr v;

9:8:4:34

however, is invariant under an origin shift in V. De®nition 7. Equivalent superspace groups. Two superspace groups are equivalent if they are isomorphic and have point groups that are arithmetically equivalent. Another de®nition leading to the same partition of equivalent superspace groups considers equivalency with respect to af®ne transformations among bases of standard form. This means that two equivalent superspace groups admit standard bases such that the two space groups are represented by the same set of 4 d-dimensional af®ne transformation matrices. We recall that an n-dimensional Euclidean transformation gs fRs jvs g if referred to a basis of the space can be represented isomorphically by an n 1-dimensional matrix, of the form Rs v s 9:8:4:35 Ags 0 1 with Rs an n n matrix and vs an n-dimensional column matrix, all with real entries. 9.8.4.4.2. Equivalent positions and modulation relations A 3 d-dimensional space group that leaves a function invariant maps points in 3 d-space to points where the function has the same value. The atomic positions of a modulated crystal represent such a pattern, and the superspace group leaving the crystal invariant leads to a partition into equivalent atomic positions. These relations can be formulated either in 3 d-dimensional space or, equally well, in three-dimensional space. As a simple case, we ®rst consider a crystal with a onedimensional occupation modulation: this implies d 1. Again, as in x9:8:1:3:2, we omit to indicate the basis vectors d1 and d1 and give only the corresponding components. An element of the 3 1-dimensional superspace group is a pair gs fRjvg; f"jI g

9:8:4:36

of Euclidean transformations in V and VI , respectively. This element maps a point located at rs r; t to one at Rr v, "t I . Suppose the probability for the position n rj to be occupied by an atom of species A is given by

940


i1

PA n; j; t pj q n rj t;

9:8:4:37

9.8. INCOMMENSURATE AND COMMENSURATE MODULATED STRUCTURES where pj x pj x 1. By gs , the position n rj is transformed to the equivalent position n0 rj 0 Rn Rrj v. As the crystal is left invariant by the superspace group, the occupation probability on equivalent points has to be the same: 0

0

PA n ; j ; t PA n; j; "t

I :

9:8:4:39

In terms of the modulation function pj this means pj 0 q n0 rj 0 pj q n rj

"I :

K rj 0

v;

where Rq "q K:

Analogously, for a displacive modulation, the position n rj with displacement uj to , where to q n rj , is transformed to n0 rj 0 with displacement "I :

" K rj 0

v;

9:8:4:42

9.8.4.4.3. Structure factor The scattering from a set of atoms at positions rn is described in the kinematic approximation by the structure factor: P SH fn H exp2iH rn ; 9:8:4:44 n

where fn H is the atomic scattering factor. For an incommensurate crystal phase, this structure factor SH is equal to the structure factor SHS of the crystal structure embedded in 3 d dimensions, where H is the projection of Hs on VE . This structure factor is expressed by a sum of the products of atomic scattering factors fn and phase factors exp2iHs rsn over all particles in the unit cell of the higher-dimensional lattice. For an incommensurate phase, the number of particles in such a unit cell is in®nite: for a given atom in space, the embedded positions form a dense set on lines or hypersurfaces of the higherdimensional space. Disregarding pathological cases, the sum may be replaced by an integral. Including the possibility of an occupation modulation, the structure factor becomes (up to a normalization factor) PPR SH dt fA HPAj t j

9:8:4:45

where the ®rst sum is over the different species, the second over the positions in the unit cell of the basic structure, the integral over a unit cell of the lattice spanned by d1 ; . . . ; dd in VI ; fA is the atomic scattering factor of species A, PAj t is the probability of atom j being of species A when the internal position is t. In particular, for a given atomic species, without occupational modulation and a sinusoidal one-dimensional displacive modulation

dt fj H exp2iH rj exp2imt

exp 2im'j :

9:8:4:47

9:8:4:48

For a general one-dimensional modulation with occupation modulation function pj t and displacement function uj t, the structure factor becomes SH

P R1 j

0

dt fj Hpj q rj t

j exp2iH

rj mt

exp2iH uj q rj t 'j :

9:8:4:49

Because of the periodicity of pj t and uj t, one can expand the Fourier series: pj q rj t j exp2iH uj q rj t 'j P Cj;k H exp2ikq rj t;

9:8:4:50

and consequently the structure factor becomes P SH fj H exp2iK rj Cj; m H; where H K mq: j

9:8:4:51 The diffraction from incommensurate crystal structures has been treated by de Wolff (1974), Yamamoto (1982a,b), Paciorek & Kucharczyk (1985), Petricek, Coppens & Becker (1985), PetrÏÂõcÏek & Coppens (1988), Perez-Mato et al. (1986, 1987), and Steurer (1987). 9.8.5. Generalizations 9.8.5.1. Incommensurate composite crystal structures The basic structure of a modulated crystal does not always have space-group symmetry. Consider, for example, composite crystals (also called intergrowth crystals). Disregarding modulations, one can describe these crystals as composed of a ®nite number of subsystems, each with its own space-group symmetry. The lattices of these subsystems can be mutually incommensurate. In that case, the overall symmetry is not a space group, the composite crystal is incommensurate and so also is its basic structure. The superspace approach can also be applied to such crystals. Let the subsystems be labelled by an index . For the subsystem , we denote the lattice by with basis vectors ai i 1; 2; 3, its reciprocal lattice by with basis vectors ai i 1; 2; 3, and the space group by G . The atomic positions of the basic structure are given by n rj ;

9:8:5:1

where n is a lattice vector belonging to . In the special case that the subsystems are mutually commensurate, there are three basis vectors a ; b ; c such that all vectors ai are integral linear combinations of them. In general, however, more than three basis vectors are needed, but never more than three times the number of subsystems. Suppose that the vectors ai i 1; . . . ; n

941


0

k

The expressions for d > 1 are straightforward generalizations of these.

expf2iH; HI rj uj t; tg;

j

For a diffraction vector H K mq, this reduces to P SH fj H exp2iK rj J m 2H Uj

where Rq "q K: 9:8:4:43

A

P R1

exp2iH Uj sin 2q rj t 'j :

To be invariant, the displacement function has to satisfy the relation uj 0 x Ruj "x

9:8:4:46

j

9:8:4:41

uj 0 to0 Ruj to

SH

9:8:4:40

In the same way, one derives the following property of the modulation function: pj 0 x pj "x

uj t Uj sin2q rj t 'j :

According to (9.8.4.45), the structure factor is

9:8:4:38

This implies that for the structure in the three-dimensional space one has the relation PA n0 ; j 0 ; 0 PA n; j; "I :

Pj t 1;

9. BASIC STRUCTURAL FEATURES form a basis set such that every ai can be expressed as an integral linear combination of them: n P Zik ak ; Zik integers; 9:8:5:2 ai k1

with n 3 do and do > 0. Then the vectors of the diffraction pattern of the unmodulated system are again of the form (9.8.4.5) and generate a vector module Mo of dimension three and rank 3 do , which can be considered as projection of a 3 do -dimensional lattice o . We now assume that one can choose aIi 0 for i 1; 2; 3 and we denote aI3j by dj . This corresponds to assuming the existence of a subset of Bragg re¯ections at the positions of a three-dimensional reciprocal lattice . Then there is a standard basis for the lattice o , which is the reciprocal of o , given by ai ; aIi ;

0; dj ;

i 1; 2; 3;

j 1; . . . ; do :

9:8:5:3

In order to ®nd the 3 do -dimensional periodic structure for which this composite crystal is the three-dimensional intersection, one associates with a translation t in the internal space VI three-dimensional independent shifts, one for each subsystem. These shifts of the subsystems replace the phase shifts adopted for the modulated structures: VI is now the space of the variable relative positions of the subsystems. Again, a translation in the superspace can give rise to a non-Euclidean transformation in the three-dimensional space of the crystal, because of the variation in the relative positions among subsystems. Each subsystem, however, is rigidly translated. For the basis vectors dj , the shift of the subsystem is de®ned in terms of projection operators : dj

3 P i1

Zi3j ai ;

j 1; . . . ; do :

9:8:5:4

P Then an arbitrary translation P t j tj dj in VI displaces the subsystem over a vector j tj dj . A translation a; aI d belonging to the 3 do -dimensional lattice o induces for the subsystem in ordinary space a relative translation over vector a aI d. The statement is that this translation is a vector of the lattice and leaves therefore the subsystem invariant. So the lattice translations belonging to o form a group of symmetry operations for the composite crystal as a whole. The proof is as follows. If k belongs to , the vector k; kI belongs to o . In particular, for k ai , one has, because of (9.8.5.2) and (9.8.5.4), ai

dj

Zi3j ;

j 1; . . . ; do ;

kI

j1

Zi3j dj

and therefore

kI dj Zi3j :

k a kI aI kI d k a aI d 0 (modulo 1), which implies that a aI d is an element of . In conclusion, one may state that the composite structure is the intersection with the ordinary space t 0 of a pattern having atomic position vectors given by t; t for any t of VI :

9:8:5:6

Such a pattern is invariant under the 3 do -dimensional lattice o . Again, orthogonal transformations R of O3 leaving the vector module Mo invariant can be extended to orthogonal

changing the position n rj into an equivalent one of the composite structure, not necessarily, however, within the same subsystem . Finally, the composite structure can also be modulated. For the case of a one-dimensional modulation of each subsystem , the positions are n rj uj q n rj :

9:8:5:8

Possibly the modulation vectors can also be expressed as integral linear combinations of the ai i 1; . . . ; 3 do . Then, the dimension of VI is again do . In general, however, one has to consider d do additional vectors, in order to ensure the validity of (9.8.4.5), now for n 3 d. We can then write q

3P d j1

Qj aj ;

Qj integers:

9:8:5:9

The peaks of the diffraction pattern are at positions de®ned by a vector module M , which can be considered as the projection of a 3 d-dimensional lattice , the reciprocal of which leaves invariant the pattern of the modulated atomic positions in the superspace given by fn rj

t uj q n rj

t qI t; tg;

for any t of VI

9:8:5:10

with dj 0 for j > do , where qI is the internal part of the 3 d-dimensional vector that projects on q . Their symmetry is a 3 d-dimensional superspace group Gs . The transformation induced in the modulated composite crystal by an element under gs of Gs is now readily written down. For the case d do 1 and gs fRjvg; f"jg, the position n rj is transformed into Rn rj v "R d1 ;

9:8:5:11

and the modulation uj q n rj into Ruj q n rj " d1

"qI d1 :

9.8.5.2. The incommensurate versus the commensurate case As said earlier, it sometimes makes sense also to use the description as developed for incommensurate crystal phases for a (commensurate) superstructure. In fact, very often the modulation wavevector also shows, in addition to continuously varying (incommensurate) values, several rational values at various phase transitions of a given crystal or in different compounds of a given structural family. In these cases, there is three-dimensional space-group symmetry. Generally, the space groups of the various phases are different. The description as used for incommensurate phases then gives the possibility of a more uni®ed characterization for the symmetry of related modulated crystal phases. In fact, the theory of higher-dimensional space groups for modulated structures is largely independent of the

942


9:8:5:7

This shows how the superspace-group approach can be applied to a composite (modulated) structure. Note that composite systems do not necessarily have an invariant set of (main) re¯ections at lattice positions.

Note that one has kI t k t, for any t from VI as is a linear operator. Because of the linearity, this holds for every k from as well. Since k; kI belongs to o and a; aI d to o , one has for their inner product:

n rj

gs : n rj ! Rn Rrj v R RI 1 vI ;

9:8:5:5

and do P

transformation Rs of O3 Odo allowing a Euclidean structure to be given to the superspace. One can then consider the superspace-group symmetry of the basic structure de®ned by atomic positions as in (9.8.5.6). A superspace-group element gs as in (9.8.4.28) induces (in three-dimensional space) for the subsystem the transformation

9.8. INCOMMENSURATE AND COMMENSURATE MODULATED STRUCTURES assumption of irrationality. Only some of the statements need to be adapted. The most important change is that there is no longer a one-to-one correspondence between the points of the reciprocal lattice and its projection on V de®ning the positions of the Bragg peaks. Furthermore, the projection of the lattice on the space VI forms a discrete set. The latter means the following. For an incommensurate modulation, the incommensurate structure, which is the intersection of a periodic structure with the hyperplane rI 0, is also the intersection of the same periodic structure with a hyperplane rI constant, where this constant is of the form 3 P i1

hi aIi

d P j1

mj aI3j :

9:8:5:12

Because for an incommensurate structure these vectors form a dense set in VI , the phase of the modulation function with respect to the basic structure is not determined. For a commensurate modulation, however, the points (9.8.5.12) form a discrete set, even belong to a lattice, and the phase (or the phases) of the modulation are determined within vectors of this lattice. Notice that the grid of this lattice becomes ®ner as the denominators in the rational components become larger. When Gs is a 3 d-dimensional superspace group, its elements, in general, do not leave the ordinary space V invariant. The subgroup of all elements that do leave V invariant, when restricted to V , is a group of distance-preserving transformations in three dimensions and thus a subgroup of E3, the three-dimensional Euclidean group. In general, this subgroup is not a three-dimensional space group. It is so when the modulation wavevectors all have rational components only, i.e. when is a matrix with rational entries. Because the phase of the modulation function is now determined (within a given rational number smaller than 1), the space group depends in general on this phase. As an example, consider a one-dimensional modulation of a basic structure with orthorhombic space group Pcmn. Suppose that the modulation wavevector is c . Then the mirror R mz perpendicular to the c axis is combined with RI " 1. Suppose, furthermore, that the glide re¯ection perpendicular to the a axis and the b mirror are both combined with a phase shift 1 2. In terms of the coordinates x; y; z with respect to the a; b and c axes, and internal coordinate t, the generators of the 3 1dimensional superspace group Pcmn00 ss0 act as x; y; z; t ! x k; y l; z m; t m n; k; l; m; n integers; x; y; z; t ! x k 12 ; y l; z 12 m; t

=2

9:8:5:13a

m 12 n; 9:8:5:13b

x; y; z; t ! x k; y l 12 ; z m; t

m 12 n;

t0 to . When has the rational value r=s with r and s relatively prime, the conditions for each of the generators to give an element of the three-dimensional space group are, respectively:

r=s

9:8:5:14d

even integer odd integer

otherwise

21 n

21 21 21

1121

21 1 c

21 cn

1c1

21 11 c

c21 n

c11

11

r odd, s even

1

r odd, s odd

In general, the three-dimensional space groups compatible with a given 3 d-dimensional superspace group can be determined using analogous equations. As one can see from the table above, the orthorhombic 3 d-dimensional superspace group leads in several cases to monoclinic three-dimensional space groups. The lattice of main re¯ections, however, still has orthorhombic point-group symmetry. Description in the conventional way by means of threedimensional groups then neglects some of the structural features present. Even if the orthorhombic symmetry is slightly broken, the orthorhombic basic structure is a better characterization than a monoclinic one. Note that in that case the superspace-group symmetry is also only an approximation. When the denominators of the wavevector components become small, additional symmetry operations become possible. Because the one-to-one correspondence between and M is no longer present, there may occur symmetry elements with trivial action in V but with nontrivial transformation in VI . For d 1, these possibilities have been enumerated. The corresponding Bravais classes are given in Table 9.8.3.2b. APPENDIX A Glossary of symbols

ai

943


2rm 2sn 4st;

r even, s odd

9:8:5:13d Note that these positions are referred to a split basis (i.e. of basis vectors lying either in V or in VI ) and not to a basis of the lattice . When the superstructure is the intersection of a periodic structure with the plane at t to , its three-dimensional space group follows from equation (9.8.5.13) by the requirement

9:8:5:14b 9:8:5:14c

where the last vector is the external part of the lattice vector sc; r=s r0; 1. The other space-group elements can be derived in the same way. The possible space groups are:

M

m n:

2rm 2sn r s 2rm 2sn s

a; b; and sc;

x; y; z; t

=2

9:8:5:14a

for m; n; r; s integers and t real. These conditions are never satis®ed simultaneously. It depends on the parity of both r and s which element occurs, and for the elements with " 1 it also depends on the value of the `phase' t, or more precisely on the product 4st. The translation group is determined by the ®rst condition as in (9.8.5.14a). Its generators are

9:8:5:13c ! x 12 k; y 12 l; z 12 m; t

rm sn 0

Vector module in m-dimensional reciprocal space m 1; 2; 3; normally m 3, isomorphic to Z n with n m. The dimension of M is m, its rank n. i 1; . . . ; n: Basis of a vector module M of rank n; if n 4 and q is modulation wavevector (the n 4 case is restricted in what follows to modulated crystals), the basis of M is chosen as a ; b ; c , q, with a ; b ; c a basis of the lattice of main re¯ections.

9. BASIC STRUCTURAL FEATURES

Lattice of main re¯ections, m-dimensional reciprocal lattice. a ; b ; c (Conventional) basis of for m 3. Direct m-dimensional lattice, dual to . Superspace; Euclidean space of dimension n m d; Vs Vs V VI . V Physical (or external) space of dimension m m 1; 2 or 3), also indicated by VE . VI Internal (or additional) space of dimension d. Reciprocal lattice in n-dimensional space, whose orthogonal projection on V is M . Lattice in n-dimensional superspace for which is the reciprocal one. asi Lattice basis of in Vs i 1; . . . ; n; if n 4, this basis can be chosen as fa ; 0; b ; 0, c ; 0, q; 1g and is called standard. An equivalent notation is q; 1 q; d ; for n 3 d, the general form of a standard basis is a ; 0, b ; 0, c ; 0, q1 ; d1 ; . . . ; qj ; dj ; . . . ; qd ; dd . asi i 1; . . . ; n: Lattice basis of in Vs dual to fasi g; if n 4, the standard basis is a; q a; b; q b, c; q c; 0; 1 0; d; for n 3 d, a standard basis is dual to the standard one given above. P3 qj Modulation wavevector(s) qj i1 ji ai ; P3 if n 4, q i1 i ai a b c ; ; ; ; P if n 4, q qi qr , with qi 1=N R in K "RRq, where "R RI , and NP is the order of K. n H Bragg re¯ections:PH i1 hi ai h1 ; h2 ; . . . ; hn ; 4 if n 4, H i1 hi ai ha kb lc mq h; k; l; m. Hs Embedding of H in Vs : for H h1 ; . . . ; hn Pn h a , one has correspondingly Hs H; HI i i Pi1 n i1 hi asi . PL Laue point group. Om Orthogonal group in m dimensions. R Orthogonal point-group transformation, element of Om. K Point group, crystallographic subgroup of Om. Rs Superspace point-group element: Rs RE ; RI R; RI element of Om Od with RE R external, and RI internal part of Rs , respectively; if n 4, superspace point-group element: R; "R with "R 1, also written R; ". Ks Point group, crystallographic subgroup of Om Od: KE External part of Ks , crystallographic point group, subgroup of Om with as elements the external part transformations of Ks . KI Internal part of Ks , crystallographic point group, subgroup of Od with as elements the internal part transformations of Ks . ro n; j Atomic positions in the basic structure: ro n; j n rj with n 2 . rn; j Atomic positions in the displacively modulated structure; d 1: rn; j ro n; j uj q rn; j 'j . In general, however, different phases 'j may occur for different components uj along the crystallographic axes. uj x Modulation function for displacive modulation with uj x 1 uj x. Modulation function for occupation modulation with pj x pj x 1 pj x.

g

Euclidean transformation in m dimensions; g fRjvg element of the space group G with rotational part R and translational part v. vo Intrinsic translation part (origin independent). Superspace group transformation d 1: gs gs fR; "jv; g fRjvg; f"jg fRs js g element of the superspace group Gs . In the 3 d-dimensional case: gs fR; RI jv; vI g fRjvg; fRI jvI g. I Internal shift d 1: I q v. Intrinsic internal shift d 1: qr v. R Point-group transformation R with respect to a basis of M and at the same time superspace point-group transformation Rs with respect to a corresponding basis of . R Superspace point-group transformation with respect to a lattice basis of dual to that of leading to R. The mutual relation is then R ~ R 1 with the tilde denoting transposition. E R; I R; M R: external, internal, and mixed blocks of R, respectively. E R; I R; M R: external, internal, and mixed blocks of R, respectively. SH Structure factor: PP SH fj H exp2iH rn; j: n

fj H

Atomic scattering factor for atom j.

APPENDIX B Basic de®nitions In the following, we give a short de®nition of the most important notions appearing in the theory and of the equivalence relations used in the tables. The latter are especially adapted to the case of modulated crystal phases. [i]

Vector module. A set of all integral linear combinations of a ®nite number of vectors. The dimension of the vector module is the dimension m of the space V (also indicated as VE and called external) generated by it over the real numbers. Its rank n is the minimal number of rationally independent vectors that generate the vector module. If this rank is equal to the dimension, the vector module is also a lattice. In general, a vector module of rank n and dimension m is the orthogonal projection on the m-dimensional space V of an n-dimensional lattice. We shall restrict ourselves mainly to the case m 3 and n 4, but the following de®nitions are valid for modulated phases of arbitrary dimension and rank. The dimension of the modulation d is n m. The modulation phases span a d-dimensional space VI (called internal or additional). [ii] Superspace. Vs is an n-dimensional Euclidean space that is the direct sum of an m-dimensional external space V (of the crystal) and a d-dimensional internal space VI (for the additional degrees of freedom). V is sometimes denoted by VE . [iii] Split basis. For the space Vs V VI , this is a basis with m basis vectors in V and d n m basis vectors in VI . [iv] Standard basis. For the m d-dimensional space Vs V VI , a standard basis in direct space is one having the last d basis vectors lying in VI (d dimension of VI dimension of the modulation). A standard basis in

944


j

9.8. INCOMMENSURATE AND COMMENSURATE MODULATED STRUCTURES reciprocal space (V identi®ed with V ) is one with the ®rst m basis vectors lying in V (m dimension of V ). [v] Conventional basis. For a lattice in three dimensions, it is a basis such that (i) the lattice generated by it is contained in as a sublattice and (ii) there is the standard relationship between the basis vectors (e.g. for a cubic lattice a conventional basis consists of three mutually perpendicular vectors of equal length). The lattice is obtained from the lattice spanned by the conventional basis by adding (a small number of) centring vectors. [For example, the b.c.c. lattice is obtained from the conventional cubic lattice by centring the unit cell with 12 12 12 .] The reciprocal basis for the conventional basis is a conventional basis for the reciprocal lattice . In the m d-dimensional superspace, a conventional basis for the lattice satis®es the same conditions (i) and (ii) as formulated above for the three-dimensional case. In addition, however, one requires that the basis is standard and such that the non-vanishing external components satisfy the relations of an m 3 conventional basis and that the corresponding internal components only involve the irrational components of the modulation vector(s) (for d 1 the basis is such that qr 0, thus qi q). Again a conventional basis for is dual to the same for . [vi] Holohedry. The holohedry of a vector module is the group of orthogonal transformations of the same dimension that leaves the vector module invariant. The holohedry of an m d-dimensional lattice is the subgroup of Om Od that leaves the lattice invariant. [vii] Point group. An m d-dimensional crystallograhic point group Ks KE ; KI is a subgroup of Om Od. With respect to a standard lattice basis its elements Rs R; RI are of the form 0 E R ; R M R I R where all the entries are integers and R is an element of an m-dimensional point group K, which is actually the same as KE . For an incommensurate modulated crystal, Ks and K are isomorphic groups. If d 1; I R " 1. [viii] Geometric crystal class. Two point groups Ks KE ; KI and Ks0 KE0 ; KI0 of pairs RE ; RI of orthogonal transformations [RE belongs to Om and RI to Od] are geometrically equivalent if and only if there are orthogonal transformations TE and TI of Om and Od, respectively, such that R0E TE RE TE 1 and R0I TI RI TI 1 for some group isomorphism RE ; RI ! R0E ; R0I . For d 1, that condition takes a simpler form because RI " 1.

[ix] Arithmetic crystal class. A group of integral matrices R [for R 2 K of Om] is determined on a basis fai ; i 1; . . . ; ng a ; b ; c ; q1 ; . . . ; qd of a vector module in reciprocal space by an m-dimensional point group K (here m 3). For modulated crystals, the transformations in direct space are given by matrices R transpose of R 1 which are of the form (9.8.4.17). Two groups 0 K 0 and K are arithmetically equivalent if and only if there is an m d-dimensional matrix S of the form SE 0 S S M SI with integral entries and determinant 1 such that 0 K 0 S K S 1 . Here SE is m m and SI is d d dimensional. An alternative formulation is: the matrix 0 groups K and K 0 determined as in equation (9.8.1.16) or in equation (9.8.1.21) are arithmetically equivalent if a the groups K and K 0 are geometrically equivalent m-dimensional point groups [the corresponding m ddimensional point groups Ks and Ks0 are then also geometrically equivalent]; b there are vector module bases a ; . . . ; qd and 0 a ; . . . ; q0d such that K on the ®rst basis gives the same group of matrices as K 0 on the second basis. [x] Bravais class. Two vector modules are in the same Bravais class if the groups of matrices determined by their holohedries are arithmetically equivalent. Two m ddimensional lattices are in the same Bravais class if their holohedries are arithmetically equivalent. In both cases, one can ®nd bases for the two structures such that the holohedries take the same matrix form. In the m ddimensional case, the lattice bases both have to be standard. [xi] Superspace group. An m d-dimensional superspace group is an n-dimensional space group n m d such that it has a d-dimensional lattice of internal translations. (This latter property re¯ects the periodicity of the modulation.) It is determined on a standard lattice basis by the matrices R of the point-group transformations and by the components i R i 1; . . . ; m d of the translation parts of its elements. The matrices R represent at the same time the elements R of the m-dimensional point group K and the corresponding elements Rs of the n d-dimensional point groups Ks . Two m d-dimensional superspace groups are equivalent if there is an origin and a standard lattice basis for each group such that the collection f K; s Kg is the same for both groups. [In previous formulae, s R is often simply indicated as s .]

References 9.1 Andersson, S., Hyde, S. T. & von Schnering, H. G. (1984). The intrinsic curvature of solids. Z. Kristallogr. 168, 1±17. Brunner, G. O. (1971). An unconventional view of the closest sphere packings. Acta Cryst. A27, 388±390. Conway, J. H. & Sloane, N. J. A. (1988). Sphere packings, lattices and groups. New York: Springer.

Ermer, O. (1988). Fivefold-diamond structure of adamantane-1,3, 5,7-tetracarboxylic acid. J. Am. Chem. Soc. 110, 3747±3754. Ermer, O. & Eling, A. (1988). Verzerrte Dreifach-Diamantstruktur von 3,3-Bis(carboxymethyl)glutarsaÈure (``MethantetraessigsaÈure''). Angew. Chem. 100, 856±860. Figueiredo, M. O. & Lima-de-Faria, J. (1978). Condensed models of structures based on loose packings. Z. Kristallogr. 148, 7±19.

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9. BASIC STRUCTURAL FEATURES 9.1 (cont) Fischer, W. (1968). Kreispackungsbedingungen in der Ebene. Acta Cryst. A24, 67±81. Fischer, W. (1970). Homogene Kugelpackungen und ihre Existenzbedingungen in Raumgruppen tetragonaler Symmetrie. Habilitationsschrift, Philipps-UniversitaÈt Marburg, Germany. Fischer, W. (1973). Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 138, 129±146. Fischer, W. (1974). Existenzbedingungen homogener Kugelpackungen zu Gitterkomplexen mit drei Freiheitsgraden. Z. Kristallogr. 140, 50±74. Fischer, W. (1976). Eigenschaften der Heesch±Laves-Packung und ihres Kugelpackungstyps. Z. Kristallogr. 143, 140±155. Fischer, W. (1991a). Tetragonal sphere packings. I. Lattice complexes with zero or one degree of freedom. Z. Kristallogr. 194, 67±85. Fischer, W. (1991b). Tetragonal sphere packings. II. Lattice complexes with two degrees of freedom. Z. Kristallogr. 194, 87±110. Fischer, W. (1993). Tetragonal sphere packings. III. Lattice complexes with three degrees of freedom. Z. Kristallogr. 205, 9±26. Fischer, W. & Koch, E. (1976). Durchdringungen von Kugelpackungen mit kubischer Symmetrie. Acta Cryst. A32, 225±232. Fischer, W. & Koch, E. (1987). On 3-periodic minimal surfaces. Z. Kristallogr. 179, 31±52. Fischer, W. & Koch, E. (1996). Spanning minimal surfaces. Philos. Trans. R. Soc. London Ser. A, 354, 2105±2142. Frank, F. C. & Kasper, J. S. (1958). Complex alloy structures regarded as sphere packings. I. De®nitions and basic principles. Acta Cryst. 11, 184±194. GruÈnbaum, B. & Shephard, G. C. (1987). Tilings and patterns. New York: Freeman. Haag, F. (1929). Die Kreispackungen von Niggli. Z. Kristallogr. 70, 353±366. Haag, F. (1937). Die Polygone der Ebenenteilungen. Z. Kristallogr. 96, 78±80. Hales, T. C. (1994). The status of the Kepler conjecture. Math. Intelligencer, 16, 47±58. È ber duÈnne Kugelpackungen. Heesch, H. & Laves, F. (1933). U Z. Kristallogr. 85, 443±453. Hellner, E. (1965). Descriptive symbols for crystal-structure types and homeotypes based on lattice complexes. Acta Cryst. 19, 703±712. Hellner, E. (1986). EinfuÈhrung in eine anorganische Strukturchemie. Z. Kristallogr. 175, 227±248. Hellner, E., Gerlich, R., Koch, E. & Fischer, W. (1979). The oxygen framework in garnet and its occurrence in the structures of Na3 Al2 Li3 F12 , Ca3 Al2 (OH)12 , RhBi4 and Hg3 TeO6 . Physik Daten ± Physics Data, 16(1), 1±40. Hellner, E., Koch, E. & Reinhardt, A. (1981). The homogeneous framework of the cubic crystal structures. Physik Daten ± Physics Data, 16(2), 1±67. Hilbert, D. & Cohn-Vossen, S. (1932). Anschauliche Geometrie. Berlin: Springer. Hilbert, D. & Cohn-Vossen, S. (1952). Geometry and the imagination. New York: Chelsea. Hsiang, W. Y. (1993). On the sphere packing problem and the proof of Kepler's conjecture. Int. J. Math. 4, 739±831. International Tables for Crystallography (1983). Vol. A. Dordrecht: Kluwer Academic Publishers.

International Tables for X-ray Crystallography (1972). Vol. II, 3rd ed. Birmingham: Kynoch Press. Kitaigorodsky, A. I. (1946). Arrangement of molecules in organic crystals. Thesis, Moscow, Russia. [In Russian.] Kitaigorodsky, A. I. (1961). Organic chemical crystallography. New York: Consultants Bureau. [Russian text published by Press of the Academy of Sciences of the USSR, Moscow, 1955.] Kitaigorodsky, A. I. (1973). Molecular crystals and molecules. New York/London: Academic Press. Koch, E. (1984). A geometrical classi®cation of cubic point con®gurations. Z. Kristallogr. 166, 23±52. Koch, E. (1985). The geometrical characteristics of the -ThSi2 structure type and of its parameter ®eld. Z. Kristallogr. 173, 205±224. Koch, E. & Fischer, W. (1978). Types of sphere packings for crystallographic point groups, rod groups and layer groups. Z. Kristallogr. 148, 107±152. Koch, E. & Fischer, W. (1995). Sphere packings with three contacts per sphere and the problem of the least dense sphere packing. Z. Kristallogr. 210, 407±414. Laves, F. (1930). Die BauzusammenhaÈnge innerhalb der Kristallstrukturen. Z. Kristallogr. 73, 202±265. Laves, F. (1932). Zur Klassi®kation der Silikate. Z. Kristallogr. 82, 1±14. Lima-de-Faria, J. (1965). Systematic derivation of inorganic close-packed structures: AX and AX2 compounds, sequence of equal layers. Z. Kristallogr. 122, 359±374. Lima-de-Faria, J. & Figueiredo, M. O. (1969a). A table relating simple inorganic close-packed structure types. Z. Kristallogr. 130, 41±53. Lima-de-Faria, J. & Figueiredo, M. O. (1969b). Systematic derivation of inorganic basic structure types: Xm Yn and Am Xn compounds, X and Y in cubic or hexagonal close packing, A in octahedral voids. Z. Kristallogr. 130, 54±67. Loeb, A. (1958). A binary algebra describing crystal structures with closely-packed anions. Acta Cryst. 11, 469±476. Matsumoto, T. (1968). Proof that the pgg packing of ellipses has never the maximum density. Z. Kristallogr. 126, 170±174. Matsumoto, T. & Nowacki, W. (1966). On densest packings of ellipsoids. Z. Kristallogr. 123, 401±421. Matsumoto, T. & Tanemura, M. (1995). Density of the p3 packing of ellipses. Z. Kristallogr. 210, 585±596. Melmore, S. (1942a). Open packing of spheres. Nature (London), 149, 412. Melmore, S. (1942b). Open packing of spheres. Nature (London), 149, 669. Minkowski, M. (1904). Dichteste gitterfoÈrmige Lagerung kongruenter KoÈrper. Nachr. Ges. Wiss. GoÈttingen Math. Phys. Kl. p. 311. Morris, I. L. & Loeb, A. (1960). A binary algebra describing crystal structures with closely packed anions. Part II: a common system of reference for cubic and hexagonal structures. Acta Cryst. 13, 434±443. Niggli, P. (1927). Die topologische Strukturanalyse I. Z. Kristalllogr. 65, 391±415. Niggli, P. (1928). Die topologische Strukturanalyse II. Z. Kristallogr. 68, 404±466. Nowacki, W. (1948). Symmetrie und physikalisch±chemische È ber EllipsenEigenschaften kristallisierter Verbindungen. V. U packungen in der Kristallebene. Schweiz. Mineral. Petrogr. Mitt. 28, 502±508. O'Keeffe, M. (1991). Dense and rare four-connected nets. Z. Kristallogr. 196, 21±37.

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O'Keeffe, M. (1992). Uninodal 4-connected 3D nets. II. Nets with 3-rings. Acta Cryst. A48, 670±673. O'Keeffe, M. (1998). Sphere packings and space ®lling by congruent simple polyhdra. Acta Cryst. A54, 320±329. O'Keeffe, M. & Brese, N. E. (1992). Uninodal 4-connected 3D nets. I. Nets without 3- or 4-rings. Acta Cryst. A48, 663± 669. Schnering, H. G. von & Nesper, R. (1987). Die natuÈrliche Anpassung von chemischen Strukturen an gekruÈmmte FlaÈchen. Angew. Chem. 99, 1097±1119. Shubnikov, A. V. (1916). On the structure of crystals. Izv. Akad. Nauk SSSR Ser. 6, 10, 755±799. Sinogowitz, U. (1939). Die Kreislagen und Packungen kongruenter Kreise in der Ebene. Z. Kristallogr. 100, 461± 508. Sinogowitz, U. (1943). Herleitung aller homogenen nicht kubischen Kugelpackungen. Z. Kristallogr. 105, 23±52. Slack, G. A. (1983). The most-dense and least-dense packings of circles and spheres. Z. Kristallogr. 165, 1±22. Smirnova, N. L. (1956a). Structure types with atomic close packing: possible structure types for the composition AB3 . Sov. Phys. Crystallogr. 1, 128±131. Smirnova, N. L. (1956b). Structure types with closest atomic packing: possible structure types for the AB4 composition. Sov. Phys. Crystallogr. 1, 399±401. Smirnova, N. L. (1958a). Structural types for close packing of atoms. III. Possible structures for the composition AB6 . Sov. Phys. Crystallogr. 3, 232±234. Smirnova, N. L. (1958b). On structural types with closest atomic packing. Possible structural types with the composition AB12 . Sov. Phys. Crystallogr. 3, 362±364. Smirnova, N. L. (1959a). The superstructures possible in closepacked structures. Sov. Phys. Crystallogr. 4, 10±16. Smirnova, N. L. (1959b). Possible superstructures in a simple cubic structure. Sov. Phys. Crystallogr. 4, 17±20. Smirnova, N. L. (1959c). Possible arrangement of atoms in the octahedral voids in the hexagonal close-packed structure. Sov. Phys. Crystallogr. 4, 734±737. Smirnova, N. L. (1964). Possible superstructures in the n-th layer of closest packing. B atoms have 4 or 2; or 4 or 1 nearest A atoms. Sov. Phys. Crystallogr. 9, 206±208. Sowa, H. (1988). Changes of the oxygen packing of low quartz and ReO3 -structure under high pressure. Z. Kristallogr. 184, 257±268. ! R3 phase transiSowa, H. (1997). Pressure-induced Fm3m tion in NaSbF6. Acta Cryst. B53, 25±31. Sowa, H. & Koch, E. (1999). Sphere con®gurations with 18h:m: Z. Kristallogr. Submitted. symmetry R3m Tanemura, M. & Matsumoto, T. (1992). On the density of the p31m packing of ellipses. Z. Kristallogr. 198, 89±99. Treacy, M. M. J., Randall, K. H., Rao, S., Perry, J. A. & Chadi, D. J. (1997). Enumeration of periodic tetrahedral frameworks. Z. Kristallogr. 212, 768±791. Wells, A. F. (1977). Three-dimensional nets and polyhedra. New York: John Wiley & Sons. Wells, A. F. (1979). Further studies of three-dimensional nets. ACA Monograph No. 8. Wells, A. F. (1983). Six new three-dimensional 3-connected nets 4.n2. Acta Cryst. B39, 652±654. È ber Kugellagerungen, WirkungsbereichsZobetz, E. (1983). U teilungen und Koordinationszahlen von Punktkon®gurationen mit trigonaler Symmetrie R3m. Z. Kristallogr. 163, 167± 180.

Adamsky, R. F. & Merz, K. M. (1959). Synthesis and crystallography of the wurtzite form of silicon carbide. Z. Kristallogr. 111, 350±361. Andrade, M., Chandrasekaran, M. & Delaey, L. (1984). The basal plane stacking faults in 18R martensite of copper base alloys. Acta Metall. 32, 1809±1816. Azaroff, L. V. (1960). Introduction to solids. London: McGrawHill. Belov, N. V. (1947). The structure of ionic crystals and metal phases. Moscow: Izd. Akad. Nauk SSSR. [In Russian.] Bertaut, E. F. (1978). The equivalent charge concept and its application to the electrostatic energy of charges and multipoles. J. Phys. (Paris), 39, 1331±1348. Boerdijk, A. H. (1952). Some remarks concerning close-packing of equal spheres. Philips Res. Rep. 7, 303±313. Brafman, O., Alexander, E. & Steinberger, I. T. (1967). Five new zinc sulphide polytypes: 10L(82); 14L (5423); 24L (53)3 ; 26L (17 423) and 28L (9559). Acta Cryst. 22, 347±252. Brafman, O. & Steinberger, I. T. (1966). Optical band gap and birefringence of ZnS polytypes. Phys. Rev. 143, 501±505. Buerger, M. J. (1953). X-ray crystallography. New York: John Wiley. Chadha, G. K. (1977). Identi®cation of the rhombohedral lattice in CdI2 crystals. Acta Cryst. A33, 341. Cottrell, A. (1967). An introduction to metallurgy. London: Edward Arnold. Cowley, J. M. (1976). Diffraction by crystals with planar faults. I. General theory. Acta Cryst. A32, 83±87. Dornberger-Schiff, K. & Farkas-Jahnke, M. (1970). A direct method for the determination of polytype structures. I. Theoretical basis. Acta Cryst. A26, 24±34. Dubey, M. & Singh, G. (1978). Use of lattice imaging in the electron microscope in the structure determination of the 126R polytype of SiC. Acta Cryst. A34, 116±120. Dubey, M., Singh, G. & Van Tendeloo, G. (1977). X-ray diffraction and transmission electron microscopy study of extremely large-period polytypes in SiC. Acta Cryst. A33, 276±279. Farkas-Jahnke, M. (1983). Structure determination of polytypes. Crystal growth and characterization of polytype structures, edited by P. Krishna, pp. 163±211. Oxford: Pergamon Press. Farkas-Jahnke, M. & Dornberger-Schiff, K. (1969). A direct method for the determination of polytype structures. II. Determination of a 66R structure. Acta Cryst. A25, 35±41. Frank, F. C. (1951). Crystal dislocation ± elementary concepts and de®nitions. Philos. Mag. 42, 809±819. Hendricks, S. & Teller, E. (1942). X-ray interference in partially ordered layer lattices. J. Chem. Phys. 10, 147±167. Honjo, G., Miyake, S. & Tomita, T. (1950). Silicon carbide of 594 layers. Acta Cryst. 3, 396±397. Jagodzinski, H. (1949a). Eindimensionale Fehlordnung in Kristallen und ihr Ein¯uss auf die Rontgeninterferenzen. I. Berechnung des Fehlordnungsgrades aus den Rontgenintensitaten. Acta Cryst. 2, 201±207. Jagodzinski, H. (1949b). Eindimensionale Fehlordnung in Kristallen und ihr Ein¯uss auf die Rontgeninterferenzen.II. Berechnung de fehlgeordneten dichtesten Kugelpackungen mit Wechselwirkungen der Reichweite 3. Acta Cryst. 2, 208± 214. Jagodzinski, H. (1972). Transition from cubic to hexagonal silicon carbide as a solid state reaction. Sov. Phys. Crystallogr. 16, 1081±1090.

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9. BASIC STRUCTURAL FEATURES 9.2.1 (cont.) Jain, P. C. & Trigunayat, G. C. (1977a). On centrosymmetric space groups in close-packed MX2 -type structures. Acta Cryst. A33, 255±256. Jain, P. C. & Trigunayat, G. C. (1977b). Resolution of ambiguities in Zhdanov notation: actual examples of homometric structures. Acta Cryst. A33, 257±260. Johnson, C. A. (1963). Diffraction by FCC crystals containing extrinsic stacking faults. Acta Cryst. 16, 490±497. Kabra, V. K. & Pandey, D. (1988). Long range ordered phases without short range correlations. Phys. Rev. Lett. 61, 1493±1496. Kabra, V. K., Pandey, D. & Lele, S. (1986). On a diffraction approach for the study of the mechanism of 3C to 6H transformation in SiC. J. Mater. Sci. 21, 1654± 1666. Kabra, V. K., Pandey, D. & Lele, S. (1988a). On the characterization of basal plane stacking faults in the N9R and N18R martensites. Acta Metall. 36, 725±734. Kabra, V. K., Pandey, D. & Lele, S. (1988b). On the calculation of diffracted intensities from SiC crystals undergoing 2H to 6H transformation by the layer displacement mechanism. J. Appl. Cryst. 21, 935±942. Kakinoki, J. & Komura, Y. (1954). Intensity of X-ray diffraction by a one-dimensionally disordered crystal. III. The closepacked structure. J. Phys. Soc. Jpn, 9, 177±183. Krishna, P. & Marshall, R. C. (1971a). The structure, perfection and annealing behaviour of SiC needles grown by a VLS mechanism. J. Cryst. Growth, 9, 319±325. Krishna, P. & Marshall, R. C. (1971b). Direct transformation from the 2H to 6H structure in single crystal SiC. J. Cryst. Growth, 11, 147±150. Krishna, P. & Pandey, D. (1981). Close-packed structures. Teaching Pamphlet of the International Union of Crystallography. University College Cardiff Press. Krishna, P. & Verma, A. R. (1962). An X-ray diffraction study of silicon carbide structure types [(33)n 34]3 R. Z. Kristallogr. 117, 1±15. Krishna, P. & Verma, A. R. (1963). Anomalies in silicon carbide polytypes. Proc. R. Soc. London Ser. A, 272, 490±502. Mesquita, A. H. G. de (1967). Re®nement of the crystal structure of SiC type 6H. Acta Cryst. 23, 610±617. Mitchell, R. S. (1953). Application of the Laue photograph to the study of polytypism and syntaxic coalescence in silicon carbide. Am. Mineral. 38, 60±67. Nishiyama, Z. (1978). Martensitic transformation. New York: Academic Press. Pandey, D. (1984a). Stacking faults in close-packed structures: notations and de®nitions. Deposited with the British Library Document Supply Centre as Supplementary Publication No. SUP 39176 (23 pp.). Copies may be obtained through The Managing Editor, International Union of Crystallography, 5 Abbey Square, Chester CH1 2HU, England. Pandey, D. (1984b). A geometrical notation for stacking faults in close-packed structures. Acta Cryst. B40, 567±569. Pandey, D. (1985). Origin of polytype structures in CdI2 : application of faulted matrix model revisited. J. Cryst. Growth, 71, 346±352. Pandey, D. (1988). Role of stacking faults in solid state transformations. Bull. Mater. Sci. 10, 117±132. Pandey, D., Kabra, V. K. & Lele, S. (1986). Structure determination of one-dimensionally disordered polytypes. Bull. MineÂral. 109, 49±67.

Pandey, D. & Krishna, P. (1975). On the spiral growth of polytype structures in SiC from a faulted matrix. I. Polytypes based on the 6H structure. Mater. Sci. Eng. 20, 243±249. Pandey, D. & Krishna, P. (1976a). On the spiral growth of polytype structures in SiC from a faulted matrix. II. Polytypes based on the 4H and 15R structures. Mater. Sci. Eng. 26, 53±63. Pandey, D. & Krishna, P. (1976b). X-ray diffraction from a 6H structure containing intrinsic faults. Acta Cryst. A32, 488±492. Pandey, D. & Krishna, P. (1977). X-ray diffraction study of stacking faults in single crystal of 2H SiC. J. Phys. D, 10, 2057±2068. Pandey, D. & Krishna, P. (1982a). Polytypism in close-packed structures. Current topics in materials science, Vol. IX, edited by E. Kaldis, pp. 415±491. Amsterdam: North-Holland. Pandey, D. & Krishna, P. (1982b). X-ray diffraction study of periodic and random faulting in close-packed structures. Synthesis, crystal growth and characterization of materials, edited by K. Lal, pp. 261±285. Amsterdam: North-Holland. Pandey, D. & Krishna, P. (1983). The origin of polytype structures. Crystal growth and characterization of polytype structures, edited by P. Krishna, pp. 213±257. Oxford: Pergamon Press. Pandey, D. & Lele, S. (1986a). On the study of the FCC±HCP martensitic transformation using a diffraction approach. I. FCC!HCP transformation. Acta Metall. 34, 405±413. Pandey, D. & Lele, S. (1986b). On the study of the FCC±HCP martensitic transformation using a diffraction approach. II. HCP!FCC transformation. Acta Metall. 34, 415±424. Pandey, D., Lele, S. & Krishna, P. (1980a). X-ray diffraction from one-dimensionally disordered 2H crystals undergoing solid state transformation to the 6H structure. I. The layer displacement mechanism. Proc. R. Soc. London Ser. A, 369, 435±449. Pandey, D., Lele, S. & Krishna, P. (1980b). X-ray diffraction from one-dimensionally disordered 2H crystals undergoing solid state transformation to the 6H structure. II. The deformation mechanism. Proc. R. Soc. London Ser. A, 369, 451±461. Pandey, D., Lele, S. & Krishna, P. (1980c). X-ray diffraction from one-dimensionally disordered 2H crystals undergoing solid state transformation to the 6H structure. III. Comparison with experimental observations on SiC. Proc. R. Soc. London Ser. A, 369, 463±477. Pandey, D., Prasad, L., Lele, S. & Gauthier, J. P. (1987). Measurement of the intensity of directionally diffuse streaks on a 4-circle diffractometer: divergence correction factors for bisecting setting. J. Appl. Cryst. 20, 84±89. Paterson, M. S. (1952). X-ray diffraction by face-centred cubic crystals with deformation faults. J. Appl. Phys. 23, 805±811. Patterson, A. L. & Kasper, J. S. (1959). Close-packing. International tables for X-ray crystallography, Vol. II, edited by J. S. Kasper & K. Lonsdale, pp. 342±354. Birmingham: Kynoch Press. Prager, P. R. (1983). Growth and characterization of AgI polytypes. Crystal growth and characterization of polytype structures, edited by P. Krishna, pp. 451±491. Oxford: Pergamon Press. Prasad, B. & Lele, S. (1971). X-ray diffraction from double hexagonal close-packed crystals with stacking faults. Acta Cryst. A27, 54±64. Rai, R. S., Singh, S. R., Dubey, M. & Singh, G. (1986). Lattice imaging studies on structure and disorder in SiC polytypes. Bull. MineÂral. 109, 509±527.

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REFERENCES 9.2.1 (cont.) Ramsdell, L. S. (1947). Studies on silicon carbide. Am. Mineral. 32, 64±82. Sebastian, M. T., Pandey, D. & Krishna, P. (1982). X-ray diffraction study of the 2H to 3C solid state transformation of vapour grown single crystals of ZnS. Phys. Status Solidi A, 71, 633±640. Steinberger, I. T. (1983). Polytypism in zinc sulphide. Crystal growth and characterization of polytype structures, edited by P. Krishna, pp. 7±53. Oxford: Pergamon Press. Steinberger, I. T., Bordas, J. & Kalman, Z. H. (1977). Microscopic structure studies of ZnS crystals using synchrotron radiation. Philos. Mag. 35, 1257±1267. Taylor, A. & Jones, R. M. (1960). The crystal structure and thermal expansion of cubic and hexagonal silicon carbide. Silicon carbide ± a high temperature semiconductor, edited by J. R. O'Connor & J. Smiltens, pp. 147±154. Oxford: Pergamon Press. Terhell, J. C. J. M. (1983). Polytypism in the III±VI layer compounds. Crystal growth and characterization of polytype structures, edited by P. Krishna, pp. 55±109. Oxford: Pergamon Press. Tokonami, M. & Hosoya, S. (1965). A systematic method for unravelling a periodic vector set. Acta Cryst. 18, 908±916. Trigunayat, G. C. & Verma, A. R. (1976). Polytypism and stacking faults in crystals with layer structure. Crystallography and crystal chemistry of materials with layered structures, edited by F. Levy, pp. 269±340. Dordrecht: Reidel. Verma, A. R. & Krishna, P. (1966). Polymorphism and polytypism in crystals. New York: John Wiley. Weertman, J. & Weertman, J. R. (1984). Elementary dislocation theory. New York: Macmillan. Wells, A. F. (1945). Structural inorganic chemistry. Oxford: Clarendon Press. Wilson, A. J. C. (1942). Imperfection in the structure of cobalt. II. Mathematical treatment of proposed structure. Proc. R. Soc. London Ser. A, 180, 277±285. Zhdanov, G. S. (1945). The numerical symbol of close-packing of spheres and its application in the theory of close-packings. C. R. Dokl. Acad. Sci. URSS, 48, 43. 9.2.2 Amelinckx, S. (1986). High-resolution electron microscopy in materials science. Examining the submicron world, edited by R. Feder, J. W. McGowan & M. Shinozaki, pp. 71±132. New York: Plenum. Angel, R. J. (1986). Polytypes and polytypism. Z. Kristallogr. 176, 193±204. Æ urovicÆ, S. (1984). Polytypism in micas. I. Backhaus, K.-O. & D MDO polytypes and their derivation. Clays Clay Miner. 32, 453±464. Bailey, S. W. (1980). Structures of layer silicates. Crystal structures of clay minerals and their X-ray identi®cation, edited by G. M. Brindley & G. Brown, pp. 1±123. London: Mineralogical Society. Bailey, S. W. (1988a) Editor. Hydrous phyllosilicates (Reviews in mineralogy, Vol. 19). Washington, DC: Mineralogical Society of America. Bailey, S. W. (1988b). X-ray diffraction indenti®cation of the polytypes of mica, serpentine, and chlorite. Clays Clay Miner. 36, 193±213.

Bailey, S. W., Frank-Kamenetskii, V. A., Goldsztaub, S., Kato, A., Pabst, A., Schulz, H., Taylor, H. F. W., Fleischer, M. & Wilson, A. J. C. (1977). Report of the International Mineralogical Association (IMA)±International Union of Crystallography (IUCr) Joint Committee on Nomenclature. Acta Cryst. A33, 681±684. Baronnet, A. (1975). Growth spirals and complex polytypism in micas. I. Polytypic structure generation. Acta Cryst. A31, 345±355. Baronnet, A. (1986). Growth spirals and complex polytypism in micas. II. Occurrence frequencies in synthetic species. Bull MineÂral. 109, 489±508. Baronnet, A. (1992). Polytypism and stacking disorder. Reviews in mineralogy, Vol. 27, pp. 231±288. Washington DC: Mineralogical Society of America. È ber die Kristalle des Carborundums. Z. Baumhauer, H. (1912). U Kristallogr. 50, 33±39. È ber die verschiedenen Modi®cationen Baumhauer, H. (1915). U des Carborundums und die Erscheinung der Polytypie. Z. Kristallogr. 55, 249±259. Belokoneva, E. L. & Timchenko, T. I. (1983). Polytypic relations in the structures of borates with a general formula RAl3 (BO3 )4 (R = Y, Nd, Gd). Kristallogra®ya, 28, 1118±1123. [In Russian.] Boer, J. L. de, van Smaalen, S., PetrÆÂõcÆek, V., DusÆek, M., Verheijen, M. A. & Meijer, G. (1994). Hexagonal closepacked C-60. Chem. Phys. Lett. 219, 469±472. Bontchev, R., Darriet, B., Darriet, J., Weill, F., Van Tendeloo, G. & Amelinckx, S. (1993). New cation de®cient perovskitelike oxides in the system La4 Ti3 O12 ±LaTiO3 . Eur. J. Solid State Inorg. Chem. 30, 521±537. Brindley, G. W. (1980). Order±disorder in clay mineral stuctures. Crystal structures of clay minerals and their X-ray identi®cation, edited by G. W. Brindley & G. Brown, pp. 125±195. London: Mineralogical Society. Burany, X. M. & Northwood, D. O. (1991). Polytypic structures in close-packed Zr(FeCr)2 Laves phases. J. Less-Common Met. 170, 27±35. Carlson, E. H. (1967). The growth of HgS and Hg3 S2 Cl2 single crystals by a vapour phase method. J. Cryst. Growth, 1, 271±277. Chamberland, B. L. (1983). Crystal structure of the 6H BaCrO3 polytype. J. Solid State Chem. 48, 318±322. CõÂsarÆovaÂ, I., NovaÂk, C. & PetrÆÂõcÆek, V. (1982). The structure of twinned manganese(III) hydrogenbis(orthophosphite) dihydrate. Acta Cryst. B38, 1687±1689. Darriet, B., Bovin, J.-O. & Galy, J. (1976). Un nouveau composeÂ de l'antimoine III: VOSb2 O4 . In¯uence steÂreÂochimique de la paire non lieÂ E, relations structurales, meÂcanismes de la reÂaction chimique. J. Solid State Chem. 19, 205±212. Dornberger-Schiff, K. (1959). On the nomenclature of the 80 plane groups in three dimensions. Acta Cryst. 12, 173. Dornberger-Schiff, K. (1964). GrundzuÈge einer Theorie von OD-Strukturen aus Schichten. Abh. Dtsch. Akad. Wiss. Berlin. Kl. Chem. 3. Dornberger-Schiff, K. (1966). Lehrgang uÈber OD-Strukturen. Berlin: Akademie Verlag. Dornberger-Schiff, K. (1979). OD structures ± a game and a bit more. Krist. Tech. 14, 1027±1045. Dornberger-Schiff, K. (1982). Geometrical properties of MDO polytypes and procedures for their derivation. I. General concept and applications to polytype families consisting of OD layers all of the same kind. Acta Cryst. A38, 483±491.

949


9. BASIC STRUCTURAL FEATURES 9.2.2 (cont.) Æ urovicÆ, S. (1982). Dornberger-Schiff, K., Backhaus, K.-O. & D Polytypism of micas: OD interpretation, stacking symbols, symmetry relations. Clays Clay Miner. 30, 364±374. Æ urovicÆ, S. (1975a). OD interpretaDornberger-Schiff, K. & D tion of kaolinite-type structures. I. Symmetry of kaolinite packets and their stacking possibilities. Clays Clay Miner. 23, 219±229. Æ urovicÆ, S. (1975b). OD interpretaDornberger-Schiff, K. & D tion of kaolinite-type structures. II. The regular polytypes (MDO polytypes) and their derivation. Clays Clay Miner. 23, 231±246. Æ urovicÆ, S. & Zvyagin, B. B. (1982). Dornberger-Schiff, K., D Proposal for general principles for the construction of fully descriptive polytype symbols. Cryst. Res. Technol. 17, 1449± 1457. Dornberger-Schiff, K. & Fichtner, K. (1972). On the symmetry of OD structures consisting of equivalent layers. Krist. Tech. 7, 1035±1056. Dornberger-Schiff, K. & Grell, H. (1982a). Geometrical properties of MDO polytypes and procedures for their derivation. II. OD families containing OD layers of M > 1 kinds and their MDO polytypes. Acta Cryst. A38, 491±498. Dornberger-Schiff, K. & Grell, H. (1982b). On the notions: crystal, OD crystal and MDO crystal. Kristallogra®ya, 27, 126±133. [In Russian.] Æ urovicÆ, S. (1968). The crystal structure of -Hg3 S2 Cl2 . Acta D Cryst. B24, 1661±1670. Æ urovicÆ, S. (1974a). Notion of `packets' in the theory of OD D structures of M > 1 kinds of layers. Examples: kaolinites and MoS2 . Acta Cryst. B30, 76±78. Æ urovicÆ, S. (1974b). Die Kristallstruktur des K4 [Si8 O18 ]: D Eine desymmetrisierte OD-Struktur. Acta Cryst. B30, 2214± 2217. Æ urovicÆ, S. (1979). Desymmetrization of OD structures. Krist. D Tech. 14, 1047±1053. Æ urovicÆ, S. (1981). OD-Charakter, Polytypie und Identi®kation D von Schichtsilikaten. Fortschr. Mineral. 59, 191±226. Æ urovicÆ, S. (1994). Classi®cation of phyllosilicates according to D the symmetry of their octahedral sheets. Ceramics-SilikaÂty, 38, 81±84. Æ urovicÆ, S. & Dornberger-Schiff, K. (1981). New fully D descriptive polytype symbols for the basic types of clay minerals. 8th Conference on Clay Mineralogy and Petrology, Teplice, Czechoslovakia, 1979, edited by S. Konta, pp. 19±25. Praha: Charles University. Æ urovicÆ, S., Dornberger-Schiff, K. & Weiss, Z. (1983). D Chlorite polytypism. I. OD interpretation and polytype symbolism of chlorite structures. Acta Cryst. B39, 547±552. Æ urovicÆ, S. & Weiss, Z. (1983). Polytypism of pyrophyllite and D talc. Part I. OD interpretation and MDO polytypes. SilikaÂty, 27, 1±18. Æ urovicÆ, S., Weiss, Z. & Backhaus, K.-O. (1984). Polytypism D of micas. II. Classi®cation and abundance of MDO polytypes. Clays Clay Miner. 32, 464±474. Effenberger, H. (1991). Structures of hexagonal copper(I) ferrite. Acta Cryst. C47, 2644±2646. Eggleton, R. A. & Guggenheim, S. (1994). The use of electron optical methods to determine the crystal structure of a modulated phyllosilicate: parsettensite. Am. Mineral. 79, 426±437. Evans, B. W. & Guggenheim, S. (1988). Talc, pyrophyllite, and related minerals. Reviews in mineralogy, Vol. 19, edited by S. W. Bailey, pp. 225±294. Washington, DC: Mineralogical Society of America.

Fichtner, K. (1965). Zur Existenz von Gruppoiden verschiedener Ordnungsgrade bei OD-Strukturen aus gleichartigen Schichten. Wiss. Z. Tech. Univ. Dresden, 14, 1±13. Fichtner, K. (1977). Zur Symmetriebeschreibung von ODKristallen durch Brandtsche und Ehresmannsche Gruppoide. Beitr. Algebra Geom. 6, 71±79. Fichtner, K. (1979a). On the description of symmetry of OD structures (I). OD groupoid family, parameters, stacking. Krist. Tech. 14, 1073±1078. Fichtner, K. (1979b). On the description of symmetry of OD structures (II). The parameters. Krist. Tech. 14, 1453±1461. Fichtner, K. (1980). On the description of symmetry of OD structures (III). Short symbols for OD groupoid families. Krist. Tech. 15, 295±300. Fichtner, K. & Grell, H. (1984). Polytypism, twinning and disorder in 2,2-aziridinedicarboxamide. Acta Cryst. B40, 434±436. Fichtner-Schmittler, H. (1979). On some features of X-ray powder patterns of OD structures. Krist. Tech. 14, 1079±1088. Figueiredo, M. O. D. (1979). CaracterõÂsticas de empilhamento e modelos condensados das micas e ®lossilicatos a®ns. Lisboa: Junta de Investigacoes CientõÂ®cas do Ultramar. Franzini, M. (1969). The A and B mica layers and the crystal structure of sheet silicates. Contrib. Mineral. Petrol. 21, 203±224. Frueh, A. J. & Gray, N. (1968). Con®rmation and re®nement of the structure of Hg3 S2 Cl2 . Acta Cryst. B24, 156. Gard, J. A. & Taylor, H. F. W. (1960). The crystal structure of foshagite. Acta Cryst. 13, 785±793. Grell, H. (1984). How to choose OD layers. Acta Cryst. A40, 95±99. Grell, H. & Dornberger-Schiff, K. (1982). Symbols for OD groupoid families referring to OD structures (polytypes) consisting of more than one kind of layer. Acta Cryst. A38, 49±54. Guinier, A., Bokij, G. B., Boll-Dornberger, K., Cowley, J. M., Æ urovicÆ, S., Jagodzinski, H., Krishna, P., de Wolff, P. M., D Zvyagin, B. B., Cox, D. E., Goodman, P., Hahn, Th., Kuchitsu, K. & Abrahams, S. C. (1984). Nomenclature of polytype structures. Report of the International Union of Crysallography Ad-Hoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures. Acta Cryst. A40, 399±404. Hamid, S. A. (1981). The crystal structure of the 11 AÊ natural tobermorite Ca2:25 [Si3 O7 :5 (OH)1:5 ]1H2 O. Z. Kristallogr. 154, 189±198. Heinrich, A. R., Eggleton, R. A. & Guggenheim, S. (1994). Structure and polytypism of bementite, a modulated layer silicate. Am. Mineral. 79, 91±106. Iijima, S. (1982). High-resolution electron microscopy of mcGillite. II. Polytypism and disorder. Acta Cryst. A38, 695±702. Ingrin, J. (1993). TEM imaging of polytypism in pseudowollastonite. Phys. Chem. Miner. 20, 56±62. Ito, T., Sadanaga, R., TakeÂuchi, Y. & Tokonami, M. (1969). The existence of partial mirrors in wollastonite. Proc. Jpn Acad. 45, 913-918. Jagner, S. (1985). On the origin of the order±disorder structures (polytypes) of some transition metal hexacyano complexes. Acta Chem. Scand. 139, 717±724. Jagodzinski, H. (1964). Allgemeine Gesichtspunkte fuÈr die Deutung diffuser Interferenzen von fehlgeordneten Kristallen. Advances in structure research by diffraction methods, Vol. I, edited by R. Brill, pp. 167±198. Braunschweig: Vieweg, and New York/London: Interscience.

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REFERENCES 9.2.2 (cont.) Jarchow, O. & Schmalle, H. W. (1985). Fehlordnung, Polytypie und Struktur von Primetin: 5,8-Dihydroxy-2-phenylchromen4-on. Z. Kristallogr. 173, 225±236. Kaneko, F., Sakashita, H., Kobayashi, M., Kitagawa, Y., Matsuura, U. & Suzuki, M. (1994). Double-layered polytypic structure of the E form of octadecanoic acid, C18 H36 O2 . Acta Cryst. C50, 247±250. Kowalski, M. (1985). Polytypic structures of chromium iron [(Cr,Fe)7 C3 ] carbides. J. Appl. Cryst. 18, 430±435. Kuban, R.-J. (1985). Polytypes of the system Fe1 x S. Cryst. Res. Technol. 20, 1649±1656. Kutschabsky, L., Kretschmer, R.-G., Schrauber, H., Dathe, W. & Schneider, G. (1986). Structure of the OD disordered 2-hydroxy-4-methoxy-2H-1,4-benzoxazin-3-one, C9 H9 NO4 . Cryst. Res. Technol. 21, 1521±1529. McLarnan, T. J. (1981a). Mathematic tools for counting polytypes. Z. Kristallogr. 155, 227±245. McLarnan, T. J. (1981b). The number of polytypes of sheet silicates. Z. Kristallogr. 155, 247±268. McLarnan, T. J. (1981c). The number of polytypes in close packings and related structures. Z. Kristallogr. 155, 269±291. MakovickyÂ, E., Leonardsen, E. & Moelo, Y. (1994). The crystallography of lengenbachite, a mineral with the noncommensurate layer structure. N. Jahrb. Mineral. Abh. 166, 169±191. Mellini, M., Merlino, S. & Pasero, M. (1986). X-ray and HRTEM structure analysis of orientite. Am. Mineral. 71, 176±187. Merlino, S., Orlandi, P., Perchiazzi, N., Basso, R. & Palenzona, A. (1989). Polytypism in stibivanite. Can. Mineral. 27, 625±632. Merlino, S., Pasero, M., Artioli, G. & Khomyakov, A. P. (1994). Penkvilskite, a new kind of silicate structure ± OD character, X-ray single-crystal (1M), and powder Rietveld (2O) re®nements of 2 MDO polytypes. Am. Mineral 79, 1185±1193. Merlino, S., Pasero, M. & Perchiazzi, N. (1993). Crystal structure of paralaurionite and its OD relationship with laurionite. Mineral. Mag. 57, 323±328. Merlino, S., Pasero, M. & Perchiazzi, N. (1994). Fiedlerite ± revised chemical formula (Pb3 Cl4 F(OH)H2 O), OD description and crystal-structure re®nement of the 2 MDO polytypes. Mineral Mag. 58, 69±78. Mogami, K., Nomura, K., Miyamoto, M., Takeda, H. & Sadanaga, R. (1978). On the number of distinct polytypes of mica and SiC with a prime layer-number. Can. Mineral. 16, 427±435. MuÈller, U. & Conradi, E. (1986). Fehlordnung bei Verbindungen MX3 mit Schichtenstruktur. I. Berechnung des IntensitaÈtsverlaufs auf den Streifen der diffusen RoÈntgenstreuung. Z. Kristallogr. 176, 233±261. Nikolin, B. I. (1984). Multi-layer structures and polytypism in metallic alloys. Kiev: Naukova dumka. [In Russian.] Nikolin, B. I., Babkevich, A. Yu., Izdkovskaya, T. V. & Petrova, S. N. (1993). Effect of heat-treatment on the crystalline structure of martensite in iron-doped, nickeldoped, manganese-doped and silicon-doped Co±W and Co±Mo alloys. Acta Metall. 41, 513±515. Pasero, M. & Reinecke, T. (1991). Crystal-chemistry, HRTEM analysis and polytypic behavior of ardennite. Eur. J. Mineral. 3, 819±830. Pauling, L. (1930a). Structure of micas and related minerals. Proc. Natl Acad. Sci. USA, 16, 123±129.

Pauling, L. (1930b). Structure of the chlorites. Proc. Natl Acad. Sci. USA, 16, 578±582. Phelps, A. W., Howard, W. & Smith, D. K. (1993). Space groups of the diamond polytypes. J. Mater. Res. 8, 2835±2839. Pring, A. & Graeser, S. (1994). Polytypism in baumhauerite. Am. Mineral. 79, 302±307. Radoslovich, E. W. (1961). Surface symmetry and cell dimensions of layer-lattice silicates. Nature (London), 191, 67±68. Reck, G. & Dietz, G. (1986). The order±disorder structure of carbamazepine dihydrate: 5H-dibenz[b,f]azepine-5-carboxamide dihydrate, C15 H12 N2 O2H2 O. Cryst. Res. Technol. 21, 1463±1468. Reck, G., Dietz, G., Laban, G., GuÈnther, W., Bannier, G. & HoÈhne, E. (1988). X-ray studies on piroxicam modi®cations. Pharmazie, 43, 477±481. Ross, M., Takeda, H. & Wones, D. R. (1966). Mica polytypes: systematic description and identi®cation. Science, 151, 191±193. Schwarz, W. & Blaschko, O. (1990). Polytype structures of lithium at low-temperatures. Phys. Rev. Lett. 65, 3144±3147. Sedlacek, P., Kuban, R.-J. & Backhaus, K.-O. (1987). Structure determination of polytypes. Cryst. Res. Technol. 22, 793±798 (I), 923±928 (II). Smaalen, S. van & de Boer, J. L. (1992). Structure of polytype of the inorganic mis®t-layer compound (PbS)1.18TiS2 . Phys. Rev B, 46, 2750±2757. Smith, J. V. & Yoder, H. S. (1956). Experimental and theoretical studies of the mica polymorphs. Mineral. Mag. 31, 209±235. Sorokin, N. D., Tairov, Yu. M., Tsvetkov, V. F. & Chernov, M. A. (1982). The laws governing the changes of some properties of different silicon carbide polytypes. Dokl. Akad. Nauk SSSR, 262, 1380±1383. [In Russian]. See also Kristallogra®ya, 28, 910±914. SzymanÂski, J. T. (1980). A redetermination of the structure of Sb2 VO5 , stibivanite, a new mineral. Can. Mineral. 18, 333±337. TakeÂuchi, Y., Ozawa, T. & Takahata, T. (1983). The pyrosmalite group of minerals. III. Derivation of polytypes. Can. Mineral. 21, 19±27. Taxer, K. (1992). Order±disorder and polymorphism of the compound with the composition of scholzite, CaZn2 [PO4 ]2 2H2 O. Z. Kristallogr. 198, 239±255. Thompson, J. B (1981). Polytypism in complex crystals: contrasts between mica and classical polytypes. Structure and bonding in crystals II, edited by M. O'Keefe & A. Navrotsky, pp. 168±196. New York/London/Toronto/Sydney/San Francisco: Academic Press. Tomaszewski, P. E. (1992). Polytypism of -LiNH4 SO4 crystals. Solid State Commun. 81, 333±335. Tsvetkov, V. F. (1982). Problems and prospects of growing large silicon carbide single crystals. Izv. Leningr. Elektrotekh. Inst. 302, 14±19. [In Russian.] Verma, A. J. & Krishna, P. (1966). Polymorphism and polytypism in crystals. New York: John Wiley. Æ urovicÆ, S. (1980). OD interpretation of MgWeiss, Z. & D vermiculite. Symbolism and X-ray identi®cation of its polytypes. Acta Cryst. A36, 633±640. Æ urovicÆ, S. (1983). Chlorite polytypism. II. Weiss, Z. & D Classi®cation and X-ray identi®cation of trioctahedral polytypes. Acta Cryst. B39, 552±557. Æ urovicÆ, S. (1985a). Polytypism of pyrophyllite and Weiss, Z. & D talc. Part II. Classi®cation and X-ray identi®cation of MDO polytypes. SilikaÂty, 28, 289±309.

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9. BASIC STRUCTURAL FEATURES Æ urovicÆ, S. (1985b). A uni®ed classi®cation and Weiss, Z. & D X-ray identi®cation of polytypes of 2:1 phyllosilicates. 5th Meeting of the European Clay Groups, Prague, 1983, edited by J. Konta, pp. 579±584. Praha: Charles University. Weiss, Z. & WiewioÂra, A. (1986). Polytypism in micas. III. X-ray diffraction identi®cation. Clays Clay Miner. 34, 53±68. Wennemer, M. & Thompson, A. B. (1984). Tridymite polymorphs and polytypes. Schweiz. Mineral. Petrogr. Mitt. 64, 335±353. White, T. J., Segall, R. L., Hutchison, J. L. & Barry, J. C. (1984). Polytypic behaviour of zirconolite. Proc. R. Soc. London Ser. A, 392, 343±358. Yamanaka, T. & Mori, H. (1981). The crystal structure and polytypes of -CaSiO3 (pseudowollastonite). Acta Cryst. B37, 1010±1017. Zhukhlistov, A. P., Zvyagin, B. B. & Pavlishin, V. I. (1990). The polytype 4M of the Ti-biotite displayed on an obliquetexture electron-diffraction pattern. Kristallogra®ya, 35, 406± 413. [In Russian.] Zoltai, T. & Stout, J. H. (1985). Mineralogy: concepts and principles. Minneapolis, Minnesota: Burgess. Zorkii, P. M. & Nesterova, Ya. M. (1993). Interlayered polytypism in organic crystals. Zh. Fiz. Khim. 67, 217±220. [In Russian.] Zvyagin, B. B. (1964). Electron diffraction analysis of clay minerals. Moskva: Nauka. [In Russian.] Zvyagin, B. B. (1967). Electron diffraction analysis of clay minerals. New York: Plenum. Zvyagin, B. B. (1988). Polytypism in crystal structures. Comput. Math. Appl. 16, 569±591. Zvyagin, B. B. & Fichtner, K. (1986). Geometrical conditions for the formation of polytypes with a supercell in the basis plane. Bull. MineÂral. 109, 45±47. Zvyagin, B. B., Vrublevskaya, Z. V., Zhoukhlistov, A. P., Sidorenko, O. V., Soboleva, S. V. & Fedotov, A. F. (1979). High-voltage electron diffraction in the investigation of layered minerals. Moskva: Nauka [In Russian.]

Kordes, E. (1939a). Die Ermittlung von AtomabstaÈnden aus der È ber eine einfache Beziehung Lichtbrechnung. I. Mitteilung. U zwischen Ionenrefraktion, Ionenradius und Ordnungszahl der Elemente. Z. Phys. Chem. B, 44, 249±260. Kordes, E. (1939b) Die Ermittlung von AtomabstaÈnden aus der Lichtbrechnung. II. Mitteilung. Z. Phys. Chem. B, 44, 327±343. Kordes, E. (1940). Ionenradien und periodisches System. II. Mitteilung. Berechnung der Ionenradien mit Hilfe atomphysicher GroÈssen. Z. Phys. Chem. 48, 91±107. Kordes, E. (1960). Direkte Berechnung der Ionenradien allein aus den Ionen-abstaÈnden. Naturwissenschaften, 47, 463. Pauling, L. (1947). The nature of the interatomic forces in metals. II. Atomic radii and interatomic distances in metals. J. Am. Chem. Soc. 69, 542±553. Pearson, W. B. (1979). The stability of metallic phases and structures: phases with the AlB2 and related structures. Proc. R. Soc. London Ser. A, 365, 523±535. Rodgers, J. R. & Villars, P. (1988). Computer evaluation of crystallographic data. In Proceedings of the 11th International CODATA Conference, Karlsruhe, FRG, edited by P. S. Glaeser. New York: Hemisphere Publishing Corp. Samsonov, G. V. (1968). Editor. Handbook of physicochemical properties of elements, p. 98. New York/Washington: IFI/ Plenum Data Corporation. Teatum, E. T., Gschneider, K. Jr & Waber, J. T. (1960). Compilation of calculated data useful in predicting metallurgical behaviour of the elements in binary alloy systems. USAEC Report LA±2345, 225 pp. Washington, DC: United States Atomic Energy Commission. Teatum, E. T., Gschneider, K. Jr & Waber, J. T. (1968). Compilation of calculated data useful in predicting metallurgical behaviour of the elements in binary alloy systems. USAEC Report LA±4003, 206 pp. [Supercedes Report LA2345 (1960).] Washington, DC: United States Atomic Energy Commission. Villars, P. & Calvert, L. D. (1991). Pearson's handbook of crystallographic data for intermetallic phases, 2nd ed. Materials Park, OH: ASM International. Villars, P. & Girgis, K. (1982). RegelmaÈûigkeiten in binaÈren intermetallischen Verbindungen. Z. Metallkd. 73, 455±462.

9.3

9.4

Brunner, G. O. & Schwarzenbach, D. (1971). Zur Abgrenzung der KoordinationsphaÈre und Ermittlung det Koordinationszahl in Kristallstrukturen. Z. Kristallogr. 133, 127±133. Daams, J. L. C. (1995). Atomic environments in some related intermetallic structure types. Intermetallic compounds, principles and practice, edited by J. H. Westbrook & R. L. Fleischer, Vol. 1, pp. 363±383. New York: John Wiley. Daams, J. L. C. & Villars, P. (1993). Atomic-environment classi®cation of the rhombohedral `ìntermetallic'' structure types. J. Alloys Compd. 197, 243±269. Daams, J. L. C. & Villars, P. (1994). Atomic-environment classi®cation of the hexagonal `ìntermetallic'' structure types. J. Alloys Compd. 215, 1±34. Daams, J. L. C. & Villars, P. (1997). Atomic enironment classi®cation of the tetragonal `ìntermetallic'' structure types. J. Alloys Compd. 252, 110±142. Daams, J. L. C., Villars, P. & van Vucht, J. H. N. (1991). Atlas of crystal structure types for intermetallic phases. Materials Park, OH: ASM International. Daams, J. L. C., Villars, P. & van Vucht, J. H. N. (1992). Atomic-environment classi®cation of the cubic `ìntermetallic'' structure types. J. Alloys Compd. 182, 1±33.

Bergerhoff, G. & Brown, I. D. (1987). Inorganic crystal structure database. In Crystallographic databases, edited by F. H. Allen, G. Bergerhoff & R. Sievers, pp. 77±95. Bonn/Cambridge/Chester: International Union of Crystallography. International Tables for X-ray Crystallography (1962). Vol. III, pp. 257±274. Birmingham: Kynoch Press. Sievers, R. & Hundt, R. (1987). Crystallographic information system CRYSTIN. In Crystallographic databases, edited by F. H. Allen, G. Bergerhoff & R. Sievers, pp. 210±221. Bonn/Cambridge/Chester: International Union of Crystallography.

9.2.2 (cont.)

9.5 Allen, F. H., Bellard, S., Brice, M. D., Cartwright, B. A., Doubleday, A., Higgs, H., Hummelink, T., HummelinkPeters, B. G., Kennard, O., Motherwell, W. D. S., Rodgers, J. R. & Watson, D. G. (1979). The Cambridge Crystallographic Data Centre: computer-based search, retrieval, analysis and display of information. Acta Cryst. B35, 2331±2339.

952


REFERENCES 9.5 (cont.) Bergerhoff, G., Hundt, R., Sievers, R. & Brown, I. D. (1983). The Inorganic Crystal Structure Database. J. Chem. Inf. Comput. Sci. 23, 66±70. Cambridge Crystallographic Data Centre User Manual (1978). 2nd ed. Cambridge University, England. Harmony, M. D., Laurie, V. W., Kuczkowski, R. L., Schwendemann, R. H., Ramsay, D. A., Lovas, F. J., Lafferty, W. J. & Maki, A. G. (1979). Molecular structures of gas-phase polyatomic molecules determined by spectroscopic methods. J. Phys. Chem. Ref. Data, 8, 619±721. Kennard, O. (1962). Tables of bond lengths between carbon and other elements. International tables for X-ray crystallography, Vol. III, Section 4.2, pp. 275±276. Birmingham: Kynoch Press. Kennard, O., Watson, D. G., Allen, F. H., Isaacs, N. W., Motherwell, W. D. S., Pettersen, R. C. & Town, W. G. (1972). Molecular structures and dimensions, Vol. A1. Interatomic distances 1960±65. Utrecht: Oosthoek. Sutton, L. E. (1958). Tables of interatomic distances and con®guration in molecules and ions. Spec. Publ. No. 11. London: The Chemical Society. Sutton, L. E. (1965). Tables of interatomic distances and con®guration in molecules and ions. Spec. Publ. No. 18. London: The Chemical Society. Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517±525. Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85±89. Taylor, R. & Kennard, O. (1986). Cambridge Crystallographic Data Centre. 7. Estimating average molecular dimensions from the Cambridge Structural Database. J. Chem. Inf. Comput. Sci. 26, 28±32. 9.6 Allen, F. H., Bellard, S., Brice, M. D., Cartwright, B. A., Doubleday, A., Higgs, H., Hummelink, T., HummelinkPeters, B. G., Kennard, O., Motherwell, W. D. S., Rodgers, J. R. & Watson, D. G. (1979). The Cambridge Crystallographic Data Centre: computer-based search, retrieval, analysis and display of information. Acta Cryst. B35, 2331±2339. Bergerhoff, G., Hundt, R., Sievers, R. & Brown, I. D. (1983). The Inorganic Crystal Structure Database. J. Chem. Inf. Comput. Sci. 23, 66±70. Brown, I. D., Brown, M. C. & Hawthorne, F. C. (1982). BIDICS-1981, Bond index to the determinations of inorganic crystal structures. Institute for Materials Research, Hamilton, Ontario, Canada. Bruce, M. I. (1981). Comprehensive organometallic chemistry, Vol. 9, pp. 1209±1520. London; Pergamon Press. Cambridge Crystallographic Data Centre User Manual (1978). 2nd ed. Cambridge University, England. Harmony, M. D., Laurie, R. W., Kuczkowski, R. L., Schwendemann, R. H., Ramsay, D. A., Lovas, F. J., Lafferty, W. J. & Maki, A. G. (1979). Molecular structures of gas-phase polyatomic molecules determined by spectroscopic methods. J. Phys. Chem. Ref. Data, 8, 619±721. Kennard, O. (1962). Tables of bond lengths between carbon and other elements. International tables for X-ray crystallography, Vol. III, pp. 275±276. Birmingham: Kynoch Press.

Kennard, O., Watson, D. G., Allen, F. H., Isaacs, N. W., Motherwell, W. D. S., Pettersen, R. C. & Town, W. G. (1972). Molecular structures and dimensions, Vol. A1. Interatomic distances, 1960±65. Utrecht: Oosthoek. Murray-Rust, P. & Raftery, J. (1985a). J. Mol. Graphics, 3, 50±59. Murray-Rust, P. & Raftery, J. (1985b). J. Mol. Graphics, 3, 60±68. Russell, D. R. (1988). Specialist periodical reports, organometallic chemistry, pp. 427±525. London: Royal Society of Chemistry. Sutton, L. E. (1958). Tables of interatomic distances and con®guration in molecules and ions. Spec. Publ. No. 11. London: Chemical Society. Sutton, L. E. (1965). Tables of interatomic distances and con®guration in molecules and ions. Spec. Publ. No. 18. London: Chemical Society. Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517±525. Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85±89. Taylor, R. & Kennard, O. (1986). Cambridge Crystallographic Data Centre. 7. Estimating average molecular dimensions from the Cambridge Structural Database. J. Chem. Inf. Comput. Sci. 26, 28±32.

9.7 Allen, F. H., Davies, J. E., Galloy, J. J., Johnson, O., Kennard, O., Macrae, C. F., Mitchell, E. M., Smith, J. M. & Watson, D. G. (1991). The development of versions 3 and 4 of the Cambridge Structural Database system. J. Chem. Inf. Comput. Sci. 31, 187±204. Baker, R. J. & Nelder, J. A. (1978). The GLIM System. Release 3. Oxford: Numerical Algorithms Group. Belsky, V. K., Zorkaya, O. N. & Zorky, P. M. (1995). Structural classes and space groups of organic homomolecular crystals: new statistical data. Acta Cryst. A51, 473±481. Belsky, V. K. & Zorky, P. M. (1977). Distribution of homomolecular crystals by chiral types and structural classes. Acta Cryst. A33, 1004±1006. Bertaut, E. F. (1995). International tables for crystallography, Vol. A, fourth, revised edition, Chap. 4.1. Dordrecht: Kluwer Academic Publishers. Brock, C. P. & Dunitz, J. D. (1994). Towards a grammar of crystal packing. Chem. Mater. 6, 1118±1127. Coutanceau Clarke, J. A. R. (1972). New periodic close packings of identical spheres. Nature (London), 240, 408±410. Donohue, J. (1985). Revised space-group frequencies for organic compounds. Acta Cryst. A41, 203±204. Evans, R. C. (1964). An introduction to crystal chemistry. Cambridge University Press. Filippini, G. & Gavezzotti, A. (1992). A quantitative analysis of the relative importance of symmetry operators in organic molecular crystals. Acta Cryst. B48, 230±234. Gavezzotti, A. (1991). Generation of possible crystal structures from the molecular structure for low-polarity organic compounds. J. Am. Chem. Soc. 113, 4622±4629.

953


9. BASIC STRUCTURAL FEATURES 9.7 (cont.) Gavezzotti, A. (1994). Molecular packing and correlations between molecular and crystal properties. Structure correlation , Vol. 2, edited by H.-B. BuÈrgi & J. D. Dunitz, Chap. 12, pp. 509±542. Weinheim/New York/Basel/Cambridge/Tokyo: VCH Publishers. Gibson, K. D. & Scheraga, H. A. (1995). Crystal packings without symmetry constraints. 1. Test of a new algorithm for determining crystal structures by energy minimization. J. Phys. Chem. 99, 3752±3764. Hahn, Th. (1995). Editor. International tables for crystallography , Vol. A, Space-group symmetry, fourth, revised edition. Dordrecht: Kluwer Academic Publishers. Kitaigorodskii, A. I. (1961). Organic chemical crystallography. New York: Consultants Bureau. Kitaigorodsky, A. I. (1945). The close-packing of molecules in crystals of organic compounds. J. Phys. (Moscow), 9, 351±352. Kitaigorodsky, A. I. (1973). Molecular crystals and molecules. New York: Academic Press. Kitajgorodskij, A. I. (1955). Organicheskaya Kristallokhimiya. Moscow: Academy of Science. Lidin, S., Jacob, M. & Andersson, S. (1995). A mathematical analysis of rod packings. J. Solid State Chem. 114, 36±41. Mezey, P. G. (1993). Shape in chemistry, an introduction into molecular shape and topology. New York/Weinheim/Cambridge: VCH Publishers. Mighell, A. D., Himes, V. L. & Rodgers, J. R. (1983). Spacegroup frequencies for organic compounds. Acta Cryst. A39, 737±740. Nowacki, W. (1942). Symmetrie und physikalisch-chemische Eigenschaften krystallisierter Verbindungen. I. Die Verteilung der Kristallstrukturen uÈber die 219 Raumgruppen. Helv. Chim. Acta, 25, 863±878. Nowacki, W. (1943). Symmetrie und physikalisch-chemische Eigenschaften kristallisierter Verbindungen. II. Die allgemeinen Bauprinzipien organischer Verbindungen. Helv. Chim. Acta, 26, 459±462. Padmaya, N., Ramakumar, S. & Viswamitra, M. A. (1990). Space-group frequencies of proteins and of organic compounds with more than one formula unit in the asymmetric unit. Acta Cryst. A46, 725±730. Patterson, A. L. & Kasper, J. S. (1959). Close packing. International tables for X-ray crystallography, Vol. II, Mathematical tables, pp. 342±354. Birmingham: Kynoch Press. Scaringe, R. P. (1991). A theoretical technique for layer structure prediction. Electron crystallography of organic molecules, edited by J. R. Fryer & D. L. Dorset, pp. 85±113. Dordrecht: Kluwer Academic Publishers. Smith, A. J. (1973). Periodic close packings of identical spheres. Nature (London) Phys. Sci. 246(149), 10±11. Williams, D. E. G. (1987). Close packing of spheres. J. Chem. Phys. 87, 4207±4210. Wilson, A. J. C. (1980). Testing the hypothesis `no remaining systematic error' in parameter determination. Acta Cryst. A36, 937±944. Wilson, A. J. C. (1988). Space groups rare for organic structures. I. Triclinic, monoclinic and orthorhombic crystal classes. Acta Cryst. A44, 715±724. Wilson, A. J. C. (1990). Space groups rare for organic structures. II. Analysis by arithmetic crystal class. Acta Cryst. A46, 742±754.

Wilson, A. J. C. (1991). Space groups rare for molecular organic structures: the arithmetic crystal class mmmP. Z. Kristallogr. 197, 85±88. Wilson, A. J. C. (1992). International tables for crystallography, Vol. C, Mathematical, physical and chemical tables, edited by A. J. C. Wilson, Chap. 9.7. Dordrecht: Kluwer Academic Publishers. Wilson, A. J. C. (1993a). Kitajgorodskij's categories. Acta Cryst. A49, 210±212. Wilson, A. J. C. (1993b). Kitajgorodskij and space-group popularity. Acta Chim. Acad. Sci. Hung. 130, 183±196. Wilson, A. J. C. (1993c). Symmetry of organic crystalline compounds in the works of Kitajgorodskij. Kristallogra®ya, 38, 153±163. [In Russian.] Wilson, A. J. C. (1993d). Space groups rare for organic structures. III. Symmorphism and inherent molecular symmetry. Acta Cryst. A49, 795±806. 9.8 Brouns, E., Visser, J. W. & de Wolff, P. M. (1964). An anomaly in the crystal structure of Na2 CO3 . Acta Cryst. 17, 614 Brown, H. (1969). An algorithm for the determination of space groups. Math. Comput. 23, 499±514. Brown, H., BuÈlow, H., NeubuÈser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of fourdimensional space. New York: John Wiley. Currat, R., Bernard, L. & Delamoye, P. (1986). Incommensurate phase in -ThBr4 . Incommensurate phases in dielectrics, edited by R. Blinc & A. P. Levanyuk, pp. 161±203. Amsterdam: North-Holland. Dam, B., Janner, A. & Donnay, J. D. H. (1985). Incommensurate morphology of calaverite crystals. Phys. Rev. Lett. 55, 2301±2304. Daniel, V. & Lipson, H. (1943). An X-ray study of the dissociation of an alloy of copper, iron and nickel. Proc. R. Soc. London, 181, 368±377. È ber die Verbreiterung der Debyelinien Dehlinger, U. (1927). U bei kaltbearbeiteten Metallen. Z. Kristallogr. 65, 615±631. Depmeier, W. (1986). Incommensurate phases in PAMC: where are we now? Ferroelectrics, 66, 109±123. Donnay, J. D. H. (1935). Morphologie des cristaux de calaverite. Ann. Soc. Geol. Belg. B55, 222-230. Fast, G. & Janssen, T. (1969). Non-equivalent four-dimensional generalized magnetic space±time groups. Report 6±68, KU Nijmegen, The Netherlands. Fast, G. & Janssen, T. (1971). Determination of n-dimensional space groups by means of an electronic computer. J. Comput. Phys. 7, 1±11. È ber Goldschmidt, V., Palache, Ch. & Peacock, M. (1931). U Calaverit. Neues Jahrb. Mineral. 63, 1±58. Grebille, D., Weigel, D., Veysseyre, R. & Phan, T. (1990). Crystallography, geometry and physics in higher dimensions. VII. The different types of symbols of the 371 monoincommensurate superspace groups. Acta Cryst. A46, 234±240. Heine, V. & McConnell, J. D. C. (1981). Origin of modulated incommensurate phases in insulators. Phys. Rev. Lett. 46, 1092±1095. Heine, V. & McConnell, J. D. C. (1984). The origin of incommensurate structures in insulators. J. Phys. C, 17, 1199±1220. Herpin, A., Meriel, P. & Villain, J. (1960). AntiferromagneÂtisme heÂlicoõÈdal. J. Phys. Radium, 21, 67.

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REFERENCES 9.8 (cont.) International Tables for Crystallography (1992). Vol. A, edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. International Tables for Crystallography (1993). Vol. B, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers. Janner, A. & Janssen, T. (1977). Symmetry of periodically distorted crystals. Phys. Rev. B, 15, 643±658. Janner, A., Janssen, T. & de Wolff, P. M. (1983). Bravais classes for incommensurate crystal phases. Acta Cryst. A39, 658±666; Wyckoff positions used for the classi®cation of Bravais classes of modulated crystals, 667±670; Determination of the Bravais class for a number of incommensurate crystals, 671±678. Janssen, T. (1969). Crystallographic groups in space and time. III. Four-dimensional Euclidean crystal classes. Physica (Utrecht), 42, 71±92. Janssen, T. & Janner, A. (1987). Incommensurability in crystals. Adv. Phys. 36, 519±624. Janssen, T., Janner, A. & Ascher, E. (1969). Crystallographic groups in space and time. I. General de®nitions and basic properties. Physica (Utrecht), 41, 541±565; Crystallographic groups in space and time. II. Central extensions, 42, 41±70. Janssen, T., Janner, A. & de Wolff, P. M. (1980). Symmetry of incommensurate surface layers. Proceeding of Colloquium on Group-Theoretical Methods in Physics, Zvenigorod, edited by M. Markov, pp. 155±166. Moscow: Nauka. Jellinek, F. (1972). Structural transitions of some transitionmetal chalcogenides. Proceedings 5th Materials Research Symposium, Washington. NBS Publ. No. 364, pp. 625±635. Kind, R. & Muralt, P. (1986). Unique incommensurate± commensurate phase transitions in a layer structure perovskite. Incommensurate phases in dielectrics, edited by R. Blinc & A. P. Levanyuk, pp. 301±317. Amsterdam: NorthHolland. Koehler, W. C., Cable, J. W., Wollan, E. O. & Wilkinson, M. K. (1962). Neutron diffraction study of magnetic ordering in thulium. J. Appl. Phys. 33, 1124±1125. Koptsik, V. A. (1978). The theory of symmetry of space modulated structures. Ferroelectrics, 21, 499. Korekawa, M. (1967). Theorie der Satellitenre¯exe. Habilitationsschrift der Ludwig-Maximilian UniversitaÈt MuÈnchen, Germany. Lifshitz, R. (1996). The symmetry of quasiperiodic crystals. Physica (Utrecht), A232, 633±647. McConnell, J. D. C. & Heine, V. (1984). An aid to the structural analysis of incommensurate phases. Acta Cryst. A40, 473±482. Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North-Holland. Paciorek, W. A. & Kucharczyk, D. (1985). Structure factor calculations in re®nement of a modulated crystal structure. Acta Cryst. A41, 462±466. Perez-Mato, J. M., Madariaga, G. & Tello, M. J. (1984). Superspace transformation properties of incommensurate irreducible distortions. Modulated structure materials, edited by T. Tsakalakos, pp. 151±159. Dordrecht: Nijhoff. Perez-Mato, J. M., Madariaga, G. & Tello, M. J. (1986). Diffraction symmetry of incommensurate structures. J. Phys. C, 19, 2613±2622.

Perez-Mato, J. M., Madariaga, G., ZunÄiga, F. J. & Garcia Arribas, A. (1987). On the structure and symmetry of incommensurate phases. A practical formulation. Acta Cryst. A43, 216±226. PetrÏÂõcÏek, V. & Coppens, P. (1988). Structure analysis of modulated molecular crystals. III. Scattering formalism and symmetry considerations: extension to higher-dimensional space groups. Acta Cryst. A44, 235±239. Petricek, V., Coppens, P. & Becker, P. (1985). Structure analysis of displacively modulated molecular crystals. Acta Cryst. A41, 478±483. Preston, G. D. (1938). The diffraction of X-rays by agehardening aluminium alloys. Proc. R. Soc. London, 167, 526±538. Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translation symmetry. Phys. Rev. Lett. 53, 1951±1953. Smaalen, S. van (1987). Superspace-group description of shortperiod commensurately modulated crystals. Acta Cryst. A43, 202±207. È ber das bemerkenswerthe Problem der Smith, G. F. H. (1903). U Entwickelung der Kristallformen des Calaverit. Z. Kristallogr. 37, 209±234. Steurer, W. (1987). (31)-dimensional Patterson and Fourier methods for the determination of one-dimensionally modulated structures. Acta Cryst. A43, 36±42. Tanisaki, S. (1961). Microdomain structure in paraelectric phase of NaNO2 . J. Phys. Soc. Jpn, 16, 579. Tanisaki, S. (1963). X-ray study on the ferroelectric phase transition of NaNO2 . J. Phys. Soc. Jpn, 18, 1181. Veysseyre, R. & Weigel, D. (1989). Crystallography, geometry and physics in higher dimensions. V. Polar and monoincommensurate point groups in the four-dimensional space E4 . Acta Cryst. A45, 187±193. Weigel, D., Phan, T. & Veysseyre, R. (1987). Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in fourdimensional space. Acta Cryst. A43, 294±304. Wolff, P. M. de (1974). The pseudo-symmetry of modulated crystal structures. Acta Cryst. A30, 777±785. Wolff, P. M. de (1977). Symmetry considerations for displacively modulated structures. Acta Cryst. A33, 493±497. Wolff, P. M. de (1984). Dualistic interpretation of the symmetry of incommensurate structures. Acta Cryst. A40, 34±42. Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a one-dimensional modulation. Acta Cryst. A37, 625±636. Wolff, P. M. de & Tuinstra, F. (1986). The incommensurate phase of Na2 CO3 . Incommensurate phases in dielectrics, edited by R. Blinc & A. P. Levanyuk, pp. 253±281. Amsterdam: North-Holland. Yamamoto, A. (1982a). A computer program for the re®nement of modulated structures. Report NIRIM, Ibaraki, Japan. Yamamoto, A. (1982b). Structure factor of modulated crystal structures. Acta Cryst. A38, 87±92. Yamamoto, A., Janssen, T., Janner, A. & de Wolff, P. M. (1985). A note on the superspace groups for one-dimensionally modulated structures. Acta Cryst. A41, 528±530. È ber einen Algorithmus zur Bestimmung Zassenhaus, H. (1948). U der Raumgruppen. Commun. Helv. Math. 21, 117±141.

955


references


10.1. Introduction

By D. C. Creagh and S. Martinez-Carrera WARNING. In this section, the main objectives of protection from ionizing radiation will be discussed and such information as may be necessary for the interpretation of legal documents relating to radiation protection will be given. The material contained herein is drawn from a wide variety of sources but principally from Vol. 26 of the International Commission on Radiological Protection (1977) (ICRP, 1977) and Part 4 (AS-2243/4) of the Standards Association of Australia (1979). It must be stressed that the recommendations made here have no legal force, nor indeed do either ICRP-26 or AS-2243/4. The precise legal requirements will be stipulated in government legislation and by the regulations pertaining to the laboratory in which the researcher is working. Notwithstanding the legal requirements, a moral requirement exists for the operator of a laboratory involved in research with ionizing radiation to be aware of the dangers involved, and to take such steps as are necessary to ensure that both he* and his workers are fully educated in the protective measures to be taken to preserve their own safety. In Vol. III of International Tables for X-ray Crystallography, Cook & Oosterkamp (1968) restricted their discussions, in the main, to the effects of X-rays and neutrons. However, with the increasing use of MoÈssbauer and other -ray techniques in crystallography and the development of nuclear magnetic resonance techniques involving the orientation of (radioactive) nuclei (NMRON), the scope of this chapter will be necessarily more general than that of IT III (1968). Finally, since this chapter can have only an advisory nature and the ®nal arbiter is the legislation of the state and local authority concerned, a list of countries that are known to have legislation concerning radiation protection is given in Table 10.3.1. Also shown in the table is the law under which control is effected and the authority responsible under the act for the implementation of radiation safety procedures. This list results from the return of questionnaires sent to all countries and is believed to be correct as of 1 October 1997.

Table 10.1.1. The relationship between SI and the earlier system of units Quantity

10.1.1.1. Ionizing radiation Ionizing radiation is de®ned as radiation that by its nature and energy has the capacity to interact with and remove electrons from (i.e. ionize) the atoms of substances through which the radiation passes. Suf®ciently energetic radiations may cause permanent changes in the nuclei of the atoms of the substance. Radiation may be propagated in the form of electromagnetic radiation (X-rays and -rays) or particles ( and particles, neutrons, protons, and other nuclear particles). In the list of de®nitions that follows SI units will be used. The relation between these SI units and the earlier system of units is given in Table 10.1.1. 10.1.1.2. Absorbed dose The energy per unit mass imparted to matter by ionizing radiation at the place of interest [SI unit gray (Gy)]. *In what follows, `he', `his' and similar pronouns are to be interpreted in a nongender-speci®c manner.

Absorbed dose [gray (Gy J kg 1 )]

1 J kg 1 0.01 J kg

Activity [becquerel (Bq s 1 )]

1 Bq 3.7 1010 Bq

2.7 1011 Ci 1 Ci

Dose equivalent [sievert (Sv J kg 1 )]

1 Sv 0.01 Sv

100 rem 1 rem

Exposure

1 C kg 1 2.58 10

100 rad 1 rad

1

4

C kg

1

3876 R 1R

10.1.1.3. Activity The number of nuclear transformations per unit time occurring in a radionuclide. 10.1.1.4. Adequate protection Protection against ionizing radiations such that the radiation doses received by an individual from internal or external sources, or both, are as low as reasonably achievable and do not exceed the maximum levels given in Table 10.1.2. 10.1.1.5. Background (radiation) Ionizing radiation other than that to be measured, but which contributes to the quantity being measured. 10.1.1.6. Becquerel (Bq) The SI unit of activity 1 Bq corresponds to one nuclear transformation per second. It replaces the curie (Ci).

An area where the occupational exposure of personnel to radiation or radioactive material is under the supervision of a designated radiation safety of®cer. 10.1.1.8. Dose equivalent Product of absorbed dose and quality factor (Subsection 10.1.1.24). This enables the dose received by individuals to be expressed on a scale common to all ionizing radiations. Where the term `dose' is used without quali®cation it is implied that `dose equivalent' is meant. 10.1.1.9. Exposure of X-ray or -radiation A measure of the radiation at a certain place based on its ability to produce ionization in air. [SI unit coulomb kg 1 . It replaces the roÈntgen (R).] 10.1.1.10. External radiation Ionizing radiation received by the body from sources outside the body.

958


Earlier

10.1.1.7. Designated radiation area

10.1.1. De®nitions

Copyright © 2006 International Union of Crystallography

SI

10.1. INTRODUCTION 10.1.1.11. Glove box

10.1.1.25. Radiation laboratory

A closed box having polymer gloves and viewing ports that is used to enclose completely radioactive materials whilst being manipulated.

A laboratory in which irradiating apparatus or sealed radioactive sources are used or stored. It does not contain any unsealed radioactive material.

10.1.1.12. Gray (Gy)

10.1.1.26. Radioactive contamination

The SI unit of absorbed dose. [SI unit 1 J kg 1 . It replaces the rad.] 10.1.1.13. Half life The period of time in which half the nuclei in a given sample of a particular radionuclide undergo decay.

The contamination of any material, surface or environment, or of a person by radioactive material. 10.1.1.27. Radioactive material

10.1.1.14. Internal radiation

Any substance that consists of, or contains any, radionuclide provided that the activity of such material is greater than 0.1 Bq kg 1 .

Radiation received from the body from sources within the body.

10.1.1.28. Radioisotope laboratory

10.1.1.15. Irradiating apparatus

A laboratory in which unsealed radioactive material is used or stored. It does not contain any irradiating apparatus.

Apparatus capable of producing ionizing radiation.

10.1.1.29. Radiological hazard

10.1.1.16. Leakage radiation All radiation except the useful beam coming from within a protective housing.

The potential danger to health arising from exposure to ionizing radiation.

10.1.1.17. Licensable quantity

10.1.1.30. Radiological laboratory

The amount of any radionuclide or mixture thereof that is permitted under statutory regulations.

A laboratory in which unsealed radioactive material and/or sealed radioactive material or irradiating apparatus is used or stored.

10.1.1.18. Maximum permissible concentration The concentration of a radionuclide in the air when breathed or water when ingested that would result in an individual receiving the maximum permissible dose (to the whole body or to a speci®c organ depending on the radionuclide in question). 10.1.1.19. Natural background Ionizing radiation received by the body from natural sources (cosmic radiation or naturally occurring radionuclides). 10.1.1.20. Non-stochastic effects Effects on a biological system in which the severity of the effect varies with the dose and for which a threshold is likely to occur. 10.1.1.21. Nuclide A species of atom characterized by the number of protons and neutrons in its nucleus. 10.1.1.22. Occupied area An area that may be occupied by personnel and where a radiation hazard may exist. 10.1.1.23. Protective housing

10.1.1.31. Radionuclide Species of atom that undergoes spontaneous nuclear transformation with consequent emission of corpuscular and/or electromagnetic radiations. 10.1.1.32. Radiotoxicity The toxicity attributable to ionizing radiation emitted by a radionuclide (and its decay products). It is related to both radioactivity and chemical effects. 10.1.1.33. Sealed source Any radioactive material ®rmly bonded within metals and sealed in a capsule or similar container of adequate mechanical strength so as to prevent dispersion of the active material into its surroundings under foreseeable conditions of use and wear. 10.1.1.34. Sievert (Sv) The SI unit for dose equivalent. 10.1.1.35. Stochastic effects

A housing of an X-ray tube or of a sealed source intended to reduce the leakage radiation to a speci®ed level.

Effects on a biological system in which the probability of an effect occurring rather than its severity is regarded as a function of dose without threshold.

10.1.1.24. Quality factor (QF)

10.1.1.36. Unsealed source

A non-dimensional factor used to reduce the biological effects of radiation to a common scale (see Table 10.1.3).

A source that is not a sealed source and that can produce contamination under normal conditions.

959


10. PRECAUTIONS AGAINST RADIATION INJURY Table 10.1.2. Maximum primary-dose limit per quarter [based on National Health and Medical Research Council (1977), as amended]

Table 10.1.3. Quality factors (QF) Type of radiation

Note: The annual MPD is typically twice the quarterly MPD MPD (i) (workers)

Part of body

X-rays, -rays, and electrons

MPD (ii) (public)

Gonads, bone marrow, whole body

30 mSv (3 rem)

2.5 mSv (2.5 rem)

Skin, bone, thyroid

150 mSv (15 rem)

15 mSv (1.5 rem)

Hands, forearms, feet, ankles

400 mSv (40 rem)

35 mSv (3.5 rem)

Organs (including eye lens)

80 mSv (8 rem)

7.5 mSv (0.75 rem)

Abdomen of female of reproductive age

13 mSv (1.3 rem)

1 mSv (0.1 rem)

Foetus between diagnosis of and completion of a pregnancy

10 mSv (1 rem)

1

Neutrons, protons, singly charged particles of rest mass not greater than one atomic mass unit of unknown energy

10

particles and multiply charged particules

20

1 Sv (dose in grays) QF.

(i) no practice ought to be adopted unless its introduction produces a positive net bene®t; (ii) all exposures should be kept as low as reasonably achievable under the existing economic and social circumstances; (iii) the dose equivalent to individuals should not exceed the limits indicated in Table 10.1.2. 10.1.3. Responsibilities 10.1.3.1. General

Note: The maximum primary dose limits as set here are advisory only, and ultimately one should strive to achieve an MPD limit as low as reasonably achievable (often referred to by the acronym ALARA), economic and social factors being taken into account.

10.1.1.37. Useful beam That part of the primary and secondary radiation that passes through the aperture, cone, or other device for collimating a beam of ionizing radiation.

10.1.2. Objectives of radiation protection Radiation protection is concerned with the protection of individuals, their offspring, and society as a whole, at the same time allowing for the participation in activities for which radiation exposure might take place. There are two aspects of these deleterious effects: the somatic effects which become manifest in the individuals themselves, and the hereditary effects which become manifest in their descendants. For the dose range involved in radiation protection, hereditary processes are regarded as being stochastic (thresholdless) processes. Some somatic effects are stochastic, and carcinogenesis is considered to be the chief risk at low doses and therefore a signi®cant problem in radiation protection. Non-stochastic processes are speci®c to particular tissues, e.g. damage to the cataract of the eye lens, non-malignant damage to the skin, damage to the bone marrow causing depletion of the red-cell count, and gonadal cell damage which impairs fertility. For these changes, the severity of the effect depends on the dose received and a clear threshold exists below which no detrimental effect has been found to occur. A balance has to be achieved between the risk of damage to individuals and the bene®ts to society in the use of the ionizing radiation in experiments. Bearing this in mind:

In laboratories using ionizing radiations, a clearly de®ned chain of responsibility has to be established with the employer accepting the responsibility for the provision of services and equipment for the implementation of radiation-protection procedures under whatever legal or administrative procedures are valid for the country in question. 10.1.3.2. The radiation safety of®cer The radiation safety of®cer (RSO) is responsible for the controlled areas within a given establishment. He (or she) is responsible to his employer for the implementation of a radiation-protection programme. His duties will vary according to the legislation and administrative arrangements applicable to his institution but will include, inter alia: (i) giving advice on working practices to management and employees; (ii) monitoring and surveying all controlled areas; (iii) maintaining all equipment for monitoring radiation levels, including personal radiation monitoring devices; (iv) keeping records of radiation levels in controlled areas, dosages to employees, stocks and locations of all radioactive materials and irradiating apparatus; (v) keeping in safe custody all radioactive materials; (vi) arranging the safe disposal of all radioactive waste; (vii) preparing the local rules concerning accident safety and emergencies; (viii) recording and reporting to the appropriate authorities all breaches of the radiation-protection rules. 10.1.3.3. The worker In English common law, the employer is responsible for the actions of his employees but this does not absolve personnel from a duty of care to their fellows. Ultimately, the responsibility for radiation protection lies with the worker concerned. He (or she) should: (i) ensure that he has an appropriate radiation dosimetry device and wears it;

960


QF

10.1. INTRODUCTION (ii) inform the RSO whenever he is to work with radioactive materials or irradiating devices; (iii) report to the RSO all known or suspected unsafe situations; (iv) be aware of the directionality of scattered beams, particularly in the case of X-rays scattered from extended single crystals; (v) be familiar with the relevant codes of practice as laid down in legislation and local instructions. 10.1.3.4. Primary-dose limits Two classes of people are envisaged

(i) persons exposed to ionizing radiation in the course of the pursuance of their duties, (ii) members of the general public. In Table 10.1.2, the maximum primary dose (MPD) for those in class (i) and class (ii) is tabulated. SI units are shown in bold type, and the earlier units are shown in parentheses in light type. Planned special exposures are permissible in emergency circumstances provided that in any single exposure twice the annual dose limit is not exceeded, and in a lifetime ®ve times the limit. Also, to allow for the different biological effectiveness of different types of radiation, the quality factor listed in Table 10.1.3 is applied to determine the dose.

961


REFERENCES

References Brodsky, A. (1982). Editor. Handbook of radiation measurement and protection, Vols I and II. Florida: CRC Press. Cook, J. E. & Oosterkamp, P. W. J. (1968). Protection against radiation injury. International tables for X-ray crystallography, Vol. III, Chap 6, pp. 333±338. Birmingham: Kynoch Press. International Commission on Radiological Protection (1977). Recommendations of the International Commission on Radiological Protection (adopted 17 January 1977). ICRP Publication 26. Oxford: Pergamon Press. International Tables for X-ray Crystallography (1968). Vol. III. Birmingham: Kynoch Press. National Heath and Medical Research Council (1977). Revised radiation protection standards for individuals exposed to ionizing radiation (as amended). ACT: NHMRC (Australia). Standards Association of Australia (1979). Safety in Laboratories. Part 4. Ionizing radiations. AS-2243/4. Sydney: Standards Association of Australia. Stott, A. M. B. (1983). Radiation protection. Nuclear power and technology, Vol. 3, edited by W. Marshall, pp. 50±77. Oxford: Clarendon Press. Other publications containing relevant material Information relevant to this part may be obtained from the sources listed below. Recommendations of the International X-ray and Radiation Protection Commission (1931). Br. J. Radiol. 4, 485.

International Commission on Radiological Protection, Clifton Avenue, Sutton, Surrey SM2 5PU, England. International Commission on Radiation Units and Measurements, 7910 Woodmont Avenue, Suite 1016, Washington, DC 20013, USA. International Atomic Energy Agency, Wagramerstrasse 5, PO Box 100, A-1400 Vienna, Austria. World Health Organization, CH-1211 GeneÁve 27, Switzerland. National Health and Medical Research Council of Australia, PO Box 100, Woden, ACT 2606, Australia. International Labour Organization, 4 route des Morillons, CH-1211 GeneÁve 22, Switzerland. OECD Nuclear Energy Agency, 38 Bd Suchet, F-75016 Paris, France. National Committee on Radiation Protection and Measurement, C/- National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. Her Majesty's Stationery Of®ce, PO Box 598, London SE1 9NH, England. National Radiological Protection Board, Chilton, Didcot, Oxford OX11 0RQ, England. Food and Drug Administration, 5600 Fishers Lane, Rockville, MD 20857, USA. Hospital Physicists Association, 47 Belgrave Square, London SW1X 8QX, England. National Competent Authorities Responsible for Approvals and Authorizations in Respect of the Transport of Radioactive Material (1997). List No. 28. International Atomic Energy Agency, Vienna.

967


references


10.2. Protection from ionizing radiation By D. C. Creagh and S. Martinez-Carrera

10.2.2.2. Open installations

10.2.1. General Because of the diversity of apparatus for the generation of ionizing radiations and the signi®cant differences that exist between laboratories within and between countries, it is not possible to give other than general guidelines as to the preventative measures to be taken. It will be assumed here that the most likely sources of exposure will be X-ray generators and radioisotopes used in the manufacture of specimens for MoÈssbauer and NMRON use, for example. The basis for this assumption is the belief that establishments maintaining neutron and particle accelerator sources will have local regulations more stringent than those of the country in which they exist ± certainly more stringent than those suggested in ICRP-26. They will also have a radiation protection of®cer who will discharge a list of duties similar to those stated in Subsection 10.1.3.2.

10.2.2. Sealed sources and radiation-producing apparatus The types of source and apparatus covered in this section include: (1) sealed sources, such as those used for radiography, and for X-ray scattering experiments; (2) apparatus that produces ionizing radiations, such as X-ray generators and particle accelerators; (3) apparatus that produces ionizing radiation incidentally, such as electron microscopes, cathode-ray oscilloscopes, and high-voltage electronic recti®ers. 10.2.2.1. Enclosed installations Most modern equipment is produced in such a form as to meet the prevailing radiation-protection regulations of the country in which it is sold, and care must be taken that safety circuits provided by the manufacturer are not defeated by staff members undertaking setting-up procedures. Such safety devices might cause visual or audible signals to be given and turn off power to the irradiating device. Many early X-ray generators, electron microscopes, etc. have by modern standards inadequate radiation-protection facilities. Where practicable, therefore, special enclosures should be fabricated to house the apparatus producing the ionizing radiation. These should be designed such that: (i) no person should have access to the interior during irradiation; (ii) access should be prevented during irradiation by the provision of fail-safe interlocks that turn off the irradiating source; (iii) no person should be able to remain in an enclosure during irradiation; (iv) a means of rapid exit should be available to an individual should by chance he (she) be within a enclosure when irradiation commences; (v) the source can be turned off from within the enclosure; (vi) during operation the dose equivalent at any accessible surface outside the enclosure shall not exceed 25 mSv (2.5 rem) per hour; (vii) when not in use, sealed sources should be capable of being housed, by remote control, within suitable shielding inside the enclosure; (viii) all interlocks should be fail-safe enabling isolation of the source in the event of the loss of electrical power.

An open installation because of operational requirements cannot have many of the safeguards suggested in Subsection 10.2.2.1. It is essential that extreme caution be exerted by the operators of such installations. They should bear in mind the following facts: (i) almost all radiation injuries in X-ray diffraction laboratories are to the ®ngers of the operators and occur when setting up monochromators close to the radiation source. Necrosis of the skin occurs within seconds under these circumstances. (ii) The beams scattered from single crystals are highly directional and very intense. Finding and monitoring these beams is usually dif®cult, and normal radiation monitors tend to underestimate the dose. 10.2.2.3. Sealed sources Sealed sources ought to be manipulated only by remote means such as forceps and long tongs. Shielding should be close to the source to minimize the risk of scattered radiation reaching other workers. Sealed sources should be registered by the RSO according to nature and activity. He (or she) is also responsible for their physical integrity and for regular examinations to detect corrosion or other damage. Note that high-activity neutron sources can activate their immediate housings and give rise to additional radiation hazards. 10.2.2.4. X-ray diffraction and X-ray analysis apparatus X-ray-generating devices such as sealed tubes and rotatinganode generators produce intense beams of small cross section and are capable of giving severe radiation burns within a second or so of exposure. Great care is necessary when working close to the exit port of these devices. Apertures in the housing enclosing the X-ray source should be covered by a shutter when the source is not being used. Interlocking devices should exist to prevent the emission of X-rays when: (i) the shutter is open without the analysing components and the beam stops being in place; (ii) the analysing device is not properly in its position in relation to the housing. Housings, shutters, shielded enclosures, and beam stops should be constructed such that the dose equivalent at any accessible point 0.05 m from their surface does not exceed 25 mSv for all practical operating conditions of the source. Warning lights and illuminated signs should be ®tted, interlocked such that they are lit when a shutter is open. 10.2.2.5. Particle accelerators The codi®cation of rules for the safe operation of high-energy particle accelerators is not simple because the various ionizing radiations produced by them require different protective procedures. Particle accelerators ought to be operated in an enclosure from a remote control room in which the dose equivalent rate does not exceed 25 mSv h 1 . A lower dose rate (2.5 mSv h 1 ) is required in adjacent areas used by non-radiation workers.


10.2. PROTECTION FROM IONIZING RADIATION The probability of mixed radiations (e.g. X-rays and neutrons) co-existing makes servicing hazardous and all maintenance should be performed under the supervision of the RSO. 10.2.3. Ionizing-radiation protection ± unsealed radioactive materials The most common controlled situation in which unsealed radioactive materials is used is in the construction of samples for use in MoÈssbauer experiments, NMRON experiments, and radioactive tracer experiments. Uncontrolled situations can occur, for example, whenever maintenance is being carried out on particle accelerators and neutron generators where radioactivity might be induced in the materials being handled by the particle beams. Great care should be taken to avoid the radioactive material being taken into the body by inhalation, ingestion or absorption through the skin or a wound. Factors that in¯uence the manner in which unsealed radioactive materials are handled include: its radiotoxicity, its volatility, the external radiation level, the nature of the work, and the design of the equipment and ultimately the design of the laboratory.

The decision concerning the manner of handling unsealed radioactive materials is the responsibility of the RSO and the safe implementaion is that of the worker. A great many rules exist concerning the handling of this material, but in the ®nal analysis the worker should: (i) think the problem through, rehearsing his actions where possible; (ii) use his common sense by minimizing the risk of breathing, eating or absorbing on his skin the radioactive material. In particular, eating, drinking or smoking in radioactive environments is to be avoided; (iii) exercise caution and wear the appropriate protective clothing at all times; (iv) be fully cognisant of the rules and regulations pertinent to the laboratory in which he is working. It is the duty of the RSO to ensure that this is so; (v) ensure that he carries the appropriate radiation monitor and the RSO records his levels of exposure regularly; (vi) have ready access to handbooks and textbooks on radiation safety, e.g. Brodsky (1982) and Stott (1983).

963


REFERENCES



967


references


10.3. Responsible bodies

By D. C. Creagh and S. Martinez-Carrera It has been stressed that the foregoing information is of an advisory nature only and that it has no legal force. Countries that have responded to the questionnaire on radiation safety produced by the Commission on Crystallographic Apparatus by 1 October

1997 are shown in Table 10.3.1. It is to the regulatory authorities designated in Table 10.3.1 that questions on radiation safety should be referred.

Table 10.3.1. Regulatory authorities Country Argentina

Legislation

Proclaimed

Responsible authority

Ð

Ð

Ministereo de Educacion, Conseuo Nacional de Investigacion CientõÂ®ca y TeÂcnica.

Australia

Code of Practice for Protection Against Ionizing Radiation Emitted from X-ray Analaysis Equipment

1/6/84

National Health and Medical Research Council, PO Box 100, Woden, ACT 2606. Note: Each state and territory has its own speci®c regulations.

Austria

Strahlen Schutzgesetz (Radiation Protection Law)

11/6/69

Federal Ministry for Health and Environment Protection, Stubenring 1, A-1010 Wien.

Belgium

Protection of the Population Against the Hazards of Ionizing Radiations

29/3/58

Service de Protection ControÃles Radiations Ionisantes.

Royal Decrees

12/3/94

MinisteÁre de la SanteÂ Publique, CiteÂ Administrative de l'Etat, Quartier Vesale 2/3, B-1010 Bruxelles.

Brazil

Normas BaÄsicas de RadioprotecËaÄo (Basic Norms of Radiation Protection)

19/9/73

ComissaÄo National de Energia Nuclear, Rua General Severiano, 90-Botafogo, 22.290, Rio de Janeiro.

Canada

Radiation Emitting Devices Regulations (1981)

Ð

Radiation and Protection Bureau, Health and Welfare (Canada), Ottawa, Ontario K1A 0K9.

Chile

Decree Concerning Radiological Protection (1976)

Ð

ComisioÂn Nacional de Investigacion, Cienti®ca y Tecnologica, Cassilia 308, Santiago.

China

Ð

Ð

National Environmental Protection Agency, 115 Nanxiaouie, Xizhemen, Beijing 100035.

Cyprus

Ð

Ð

Senior Medical Physicist, Nicosia General Hospital, Nicosia.

Czech Republic

Denmark

Act of the Ministry of Health of the Czech Republic

30/6/72

Czechoslovak State Norm No. 341725

24/7/68

Order Concerning X-ray Analytical Equipment

15/4/30

Egypt Finland

Ð Radiation Protection Act


State Of®ce for Nuclear Security, Sleszka 9, 12029 Praha 2.

National Institute of Radiation Hygiene, 378 Fredrikssundsvej, DK-2700 Brùnshùj, Copenhagen.

Ð

Atomic Energy Authority, 101 Kasr El-Eini Street, Cairo.

26/4/57

Finnish Centre for Radiation and Protection, PO Box 14, FIN-00881 Helsinki.

10.3. RESPONSIBLE BODIES Table 10.3.1. Regulatory authorities (cont.) Country France

Legislation

Proclaimed

Ð

Ð

Responsible authority MinisteÁre de la SanteÂ Publique, 31 rue de l'Ecluse, BP 35, 78110 Levesinet. CEA, 31±33 rue de la FeÂdeÂration, 75752 Paris CEDEX 15.

Germany

Gesetz uÈber die friedliche Verwendung der Kernenergie und den Schutz gegen ihre Gefahren (Atomgesetz)

23/12/59

Verordnung uÈber den Schutz vor SchaÈden durch Ionisierende Strahlen (Strahlenschutz Verordnung)

13/10/78

Bundesministerium fuÈr Verkehr, Referat A 13, Robert-Schumann Platz, D-53175 Bonn.

Verordnung uÈber den Schutz vor SchaÈden durch RoÈntgenstrahlen (RoÈntgen Verordnung)

8/1/87

Greece

Protection from Ionizing Radiation

18/1/74

Greek Atomic Energy Commission, 153 10 Aghia paraskevi-Attiki, PO Box 60228.

Hungary

Act 1/1980

1/6/80

National Atomic Energy Commission, 1374 Budapest Pf 365.

India

Atomic Energy Act of 1962

1962

Indian Atomic Energy Regulatory Board, 4th Floor, Vikram Sarabhai Bkavan Anushaktinagar, Bombay 400 094.

Ireland

Order No. 166, Nuclear Energy Act of 1971

1971

Sealed Sources SI No. 17

1972

Nuclear Energy Board, 20±22 Lr Hatch Street, Dublin 2.

Unsealed Sources SI No. 249

1972

Occupational Radiation Protection Regulation of 1981

1981

Employment of Women

1980

Italy

Legislative Act No. 185

13/2/64

Japan

The Atomic Energy Basic Law; Law concerning Prevention from Radiation Hazards due to Radioisotopes etc.

Korea, South

The Atomic Energy Law

New Zealand

Radiation Protectin Act

Norway Pakistan

Israel

Ð

The Licensing Division, Israeli Atomic Energy Commission, PO Box 7061, Tel Aviv 61070. Nuclear Safety and Health Protection Directorate, ENEA, Via Vitaliano, Brancati 48, 00144 Rome-EUR. Radiation Council, 2-2-1 Kasumigaseki, Chiyoda-ku, Tokyo.

11/3/58

Atomic Energy Bureau, Ministry of Science and Technology, The Second Integrated Government Building, Gwacheon, Kyunggi-D, 427 760.

1965

National Radiation Laboratory, Department of Health, PO Box 25-099, Christchurch.

Act Relating to the Use of X-rays, Radium, etc.

18/6/38

National Institute of Radiation Hygiene, PO Box 55, N-1345 ésteras.

Pakistan Nuclear Safety and Radiation Protection Ordinance No. IV of 1984

26/1/84

Pakistan Atomic Energy Commission, Directorate of Nuclear Safety and Radiation Protection, PO Box 1912, Islamabad.

965


10. PRECAUTIONS AGAINST RADIATION INJURY Table 10.3.1. Regulatory authorities (cont.) Country

Legislation

Proclaimed

Responsible authority

Ð

Ð

Insituto Peruano de Energia Nuclear, Direccion de Seguridad Nuclear y Proteccion Radiologica, Direccion Av. Canada No. 1470, Lima 41.

Normas Cerais parra ProteccaÄo das Pessons ~ contra as RadiacoÄes Ionizantes ~

25/11/61

Ð

Ð

Peru

Portugal

Russian Federation

Ministerio da Suude, Direccao-Geral da saude, Almeda D. Alfonso Henriques, 45, P-1056, Lisboa Codex. Ministry for Atomic Energy, Committee on Safety, Ecology, and Emergency Situations, ul. B. Ordynka 24/26, 101000 Moscow.

Singapore

The Radiation Protection Act 19B and Regulations 1974

1/9/74

Radiation Protection Inspectorate, Department of Scienti®c Services, Outram Road, Singapore 0316.

Slovak Republic

Act of the Ministry of Health of the Slovak Socialist Republic No. 65

12/9/72

Ministry of Health of the Slovak Republic, LimbovaÂ 2, SK-83341 Bratislava.

Czechoslovak State Norm No. 451725

24/7/68

Ley Reguladora de la Energia Nuclear

29/4/64

Reglamento Sobre Instalaciones Nucleares y Radiactivas

21/7/72

Reglamento s. Protection-Sanitaria contra Radiaciones Ionizantes

8/10/82

Spain

South Africa

Ð

Ð

Sweden

Act Concerning Protection Against Radiation

14/3/58

Switzerland

Federal Order (1980)

Turkey

United Kingdom

United States of America

Consejo de Seguridad Nuclear, Paseo de La Castellana 135, Madrid.

Deparment of Health and Welfare, Civitas Building, Cnr Andrew Struken St, Pretoria 0001 (Private Bag X63). Swedish Radiation Protection Institute, S-17116 Stockholm.

Ð

Of®ce FeÂdeÂral de la SanteÂ Publique, Division of Radiation Protection, CH-3003 Berne.

Radiological Health and Safety

1967

Turkish Atomic Energy Authority, Radiological Health and Safety Department, Karantil Sk. No. 67, Bakanliklar-Ankara.

Health and Safety at Work Act

1974

Ionizing Radiation Regulations

1/1/86

Health and Safety Executive, Rose Court, 2 Southwark Bridge, London SE1 9HS.

Atomic Energy Act of 1954

1954

Executive Order 10831

Department of Energy, Forestal Building, 1000 Independence Avenue SW, Washington, DC 20585.

966


Nuclear Regulatory Commission, Matomic Building, 1717th Street NW, Washington, DC 20555.

REFERENCES



967


references

AUTHOR INDEX

Author Index Entries refer to part, chapter or section numbers Abele, R. K., 7.3 Ê berg, T., 7.4.3 A Abragam, A., 2.6.2 Abrahams, J. M., 7.1.6 Abrahams, K., 4.4.2 Abrahams, S. C., 1.4, 4.2.2, 4.2.6, 5.3, 6.2, 8.1, 8.5, 9.2.2 Abramowitz, M., 6.1.1, 6.3, 7.5 Achiwa, N., 2.9 Ackermann, I., 4.3.7 Adams, L. H., 3.2 Adamsky, R. F., 9.2.1 Adlhart, W., 3.4 Agamalyan, M. M., 4.4.2 Agarwal, B. K., 4.2.6 Ahahama, Y., 4.2.5 Ahlrichs, R., 6.1.1 Ahmed, A., 5.2 Ahn, C. C., 4.3.4, 7.2 Ahtee, M., 2.3, 8.6 Airey, R. W., 7.1.6 Akashi, Y., 5.3 Akhiezer, A. I., 4.2.6 Akiyoshi, T., 2.9 Alani, R., 3.5 Albers, R. C., 4.2.3 Albertsson, J., 5.3 Albinati, A., 8.6 Alcock, N. W., 5.3, 6.3 Aldred, P. J. E., 4.2.6 Alefeld, B., 2.6.2, 4.4.2 Aleksandrov, K. S., 3.1 Alexander, E., 9.2.1 Alexander, H., 4.3.8 Alexander, L., 5.2, 5.3 Alexander, L. E., 2.3, 3.4, 5.1, 5.3, 6.2 Alexander, T. K., 4.4.2 Alexandropoulos, N. G., 7.4.3 Alkire, R. W., 3.4 Allemand, R., 2.4.2, 7.1.6, 7.3 Allen, F. H., 9.5, 9.6, 9.7 Allen, J. P., 3.1 Allen, S., 3.4 Allen, S. J. M., 4.2.4 Allewell, N. M., 3.4 Allin, G. W., 7.3 Allington-Smith, J. R., 7.1.6 Allinson, N. M., 2.7, 7.1.6 Allison, S. K., 2.3, 2.7, 4.2.1, 5.3 Allsopp, D. W. E., 2.7 Alp, E. E., 5.3 Alstrup, I., 4.2.3, 4.2.6 Alstrup, O., 2.5.1 Altarelli, M., 4.3.4 Alter, U., 5.3 Altmann, S. L., 6.1.1 Altomare, A., 8.6 Alvarez, L. W., 4.2.3, 4.4.2 Amadori, R., 4.4.2 Amelinckx, S., 3.5, 4.3.8, 9.2.2 Amemiya, Y., 7.1.6, 7.1.8 Ames, L., 7.1.4 Ammon, H. L., 3.1 AmoroÂs, J. L., 2.2, 5.3 AmoroÂs, M. C., 2.2, 5.3

d'Amour, H., 5.3 Anderegg, J. W., 2.6.1 Anderson, C. A. F., 2.3 Anderson, D., 2.2, 3.4 Anderson, D. W., 6.3 Anderson, E., 8.1 Anderson, I., 4.4.2 Anderson, I. S., 4.4.2 Anderson, J. E., 7.1.6 Anderson, R., 2.4.1 Anderson, W. F., 3.1 Andersson, B., 4.3.7 Andersson, S., 9.1, 9.7 Ando, M., 2.2, 2.7, 2.8, 5.3, 7.1.8 Ando, T., 4.3.3 Ando, Y., 4.3.7, 8.8 Andrade, M., 9.2.1 Andresen, A., 6.1.2 Andrew, N. L., 5.3 Andrews, S. J., 2.2, 2.3 Andrianova, M. E., 7.1.6 Andrus, J., 2.7 Angel, R. J., 9.2.2 Anger, H. O., 7.1.6 Anisimov, Yu. S., 7.1.6 Ankner, J. F., 2.9, 4.4.2 Ansara, I., 2.6.2 Anstis, G. R., 4.3.8 d'Anterroches, C., 4.3.8 Aoki, K., 3.4 Aoki, Y., 4.3.7 Appleman, D. E., 5.2 Apsimon, R. J., 7.1.6 Arai, T., 2.3 Arakali, S. V., 3.1 Arcese, A., 7.1.7 Archer, B. T., 4.3.3 Archer, J. M., 3.4 Argos, P., 3.4 Argoud, R., 3.4 Aristov, V. V., 4.2.6, 5.3 Armstrong, R. W., 2.7 Arndt, U. W., 2.2, 2.7, 3.4, 4.2.1, 4.2.2, 5.3, 6.2, 7.1.6 Arnesen, S. P., 4.3.3 Arnold, E., 3.4 Arnold, H., 3.4 Arnold, P., 2.5.2 Arrott, A., 6.2 Arrott, A. S., 2.8 Arsenin, V. Ya., 2.6.1 Artioli, G., 9.2.2 Artymiuk, P., 2.2 Arvedson, M., 4.3.3 Ê sbrink, S., 5.3 A Ascher, E., 9.8 Ascheron, C., 5.3 Ateiner, J., 2.3 Atoji, M., 6.1.1 Att®eld, J. P., 2.3 Auleytner, J., 5.3 Austerman, S. B., 2.7 Authier, A., 2.7 Autrata, R., 7.2 Averbach, B. L., 2.3 Avery, J., 6.1.1, 8.7

Avilov, A. S., 2.4.1, 4.3.5 Axe, J. D., 4.4.2, 4.4.3 Axelrod, H. J., 3.1 Axelsson, U., 4.2.2 Ayers, G. L., 2.3 Azaroff, L. V., 2.3, 4.2.3, 5.3, 6.3, 9.2.1 Azumi, I., 5.3 BaÎk-Misiuk, J., 5.3 Babkevich, A. Yu., 9.2.2 Bacchella, C. L., 2.6.2 Bach, H., 3.5 Bachmann, R., 2.3, 7.4.2 Backhaus, K.-O., 9.2.2 BacÏkovskyÂ, J., 5.3 Bacon, G. E., 2.6.2, 3.6, 4.4.2, 4.4.4, 6.1.3, 6.4 Bacon, J. R., 2.6.1 Badurek, G., 4.4.2 Baerlocher, Ch., 2.3, 8.6 Bagchi, S. N., 2.6.1 Baharie, E., 8.6 Bai, Z., 8.1 Baigarin, K. A., 4.2.1 Baik, D. H., 4.2.2 Bailey, D., 5.3 Bailey, I., 2.4.2 Bailey, R. L., 4.2.1 Bailey, S. W., 9.2.2 Baker, J. A., 5.3 Baker, J. F. C., 5.3 Baker, R. F., 4.3.4 Baker, R. J., 9.7 Baker, S. M., 2.9 Baker, T. W., 2.3, 5.3 Bakken, L. N., 4.3.7, 8.8 Balaic, D. X., 4.2.5 Baldock, P., 3.1 Ballon, J., 7.1.3, 7.1.6 Balzar, D., 8.6 Band, I. M., 4.2.4 Banerjee, D., 3.5 Bannett, Y. B., 7.4.3 Bannier, G., 9.2.2 Baptista, G. B., 4.2.4 Barber, D. J., 3.5 Barclay, A. N., 3.4 Barla, K., 5.3 Barna, S. L., 2.7 Barnea, Z., 4.2.5 Barnes, I. L., 5.3 Barnes, P., 3.4 Barns, R. L., 5.3 Baronnet, A., 9.2.2 Barraud, J., 2.3 Barreau, G. H., 4.2.2 Barrett, C. S., 2.3, 2.7, 4.3.5 Barrientos, J., 4.3.3 Barry, J. C., 9.2.2 Bartell, L. S., 4.3.3 Bartels, K., 3.4, 6.3 Bartels, K. S., 3.4 Bartels, W. J., 5.3 Barth, H., 2.7, 5.2 Bartl, H., 3.4 Bartunik, H. D., 3.4

968

969 s:\ITFC\index.3d (Authors Index)

Baru, S. E., 7.1.6 Baruchel, J., 2.8, 7.3 Basile, G., 4.2.2, 5.3 Basinski, Z. S., 4.3.6.2 Basso, R., 9.2.2 Batchelder, D. N., 5.3 Bateman, J. E., 7.1.6 Bateman, O., 3.4 Bates, D. R., 3.4 Bates, F. S., 2.6.2 Batson, P. E., 4.3.4 Battagliarin, M., 2.3 Batterman, B. W., 2.7 Baumhauer, H., 9.2.2 Bauspiess, W., 2.7 Bautz, M. W., 7.1.6 Bayvel, L. P., 2.6.1 Bearden, J. A., 2.3, 4.2.2, 5.2, 5.3 Beaumont, J. H., 2.3, 2.7, 4.2.5 Becherer, G., 2.6.1 Becker, J., 4.4.2 Becker, P., 4.2.2, 5.3, 8.7, 9.8 Becker, P. J., 6.3, 6.4, 8.7 Beckman, R. J., 8.5 Bednarski, S., 4.4.2 Bedzyk, M. J., 4.2.3 Beeman, W. W., 2.6.1, 6.3 Begg, G. S., 3.4 Begum, R., 4.2.6 Behlke, J., 2.6.1 Behrendt, D. R., 6.1.1 Bellamy, B. A., 2.3, 5.3 Bellard, S., 9.5, 9.6 Bellis, J. G., 7.1.6 Bellman, R., 6.1.1 Bellotto, M., 2.3, 5.2, 5.3 Belokoneva, E. L., 9.2.2 Belov, N. V., 1.4, 9.2.1 Belsky, V. K., 9.7 Belsley, D. A., 8.2, 8.5 Benedetti, A., 2.3 Beni, G., 4.2.3 Bennett, C. L., 3.4 Benoit, H., 2.6.2 Bentley, J., 8.7 Berendsen, H. J. C., 3.1 Berestetsky, V. B., 4.2.6 Berg, H. M., 5.3 Berg, W. F., 2.7 Bergamin, A., 4.2.2, 5.3 Berger, H., 5.3 Berger, J. O., 8.1 Berger, S. D., 4.3.4 Bergerhoff, G., 9.4, 9.5, 9.6 Bergevin, F. de, 4.2.5, 8.7 Berggren, K.-F., 7.4.3 Bergman, G., 8.3 Bergstrom, J. C., 4.2.1 Bergstrom, P. M. Jr, 4.2.6 Berk, N. F., 2.9 Berkum, J. van, 5.2 Berliner, R., 7.3 Berman, H., 3.2, 3.3 Berman, L. E., 4.2.5 Bernal, J. D., 2.2, 3.2, 4.3.5

AUTHOR INDEX Bernard, L., 9.8 Bernard, Y., 3.1 Berndtsson, A., 4.2.2 Berneron, M., 4.4.2 Bernstein, S., 4.4.2 Berry, B. S., 2.3 Bertaut, E. F., 1.4, 8.7, 9.2.1, 9.7 Bertin, E. P., 4.2.3 Berzina, T. S., 2.9 Besson, J. M., 2.5.1 Beth, H. A., 4.3.3 Bethe, H., 4.3.4 Bethe, H. A., 2.4.1, 4.3.1 Beu, K. E., 5.2, 5.3 Bevis, M., 5.3 Bewilogua, L., 7.4.3 Beyer, H., 4.2.2 Bhat, H. L., 3.4 Bhatt, V. P., 3.4 Bianconi, A., 4.2.3, 4.3.4 Bickmann, K., 5.3 Bieber, R. L., 5.3 Bienenstock, A., 4.2.1, 4.2.3, 4.2.6 Biggin, S., 5.3 Biggs, F., 4.3.3, 7.4.3 Bigler, E., 7.1.6 Bijvoet, J. M., 2.2, 4.2.6 Bilderback, D. H., 2.2, 4.2.5 Binnig, G., 4.3.8 Bird, D. M., 4.3.2, 4.3.7, 8.8 Bird, R. B., 8.7 Birks, L. S., 2.3 Birnbaum, H. R., 4.2.3 Bischof, C., 8.1 Bish, D. L., 2.3, 7.1.4 Bishop, A. C., 3.1 Bjerrum Mùller, H., 4.4.3 Black, D. R., 3.4 Black, R. E., 5.3 Blackman, M., 2.4.1, 4.3.1, 8.8 Blair, D. G., 6.4 Blake, A. J., 2.3 Blake, R. G., 4.3.7 Blakeslee, D. M., 3.1 Blanc, Y., 4.4.2 Blanton, T. N., 5.2 Blaschko, O., 9.2.2 BlaÈser, D., 3.4 Blech, I., 9.8 Bleeksma, J., 2.3 Bloch, B. J., 7.4.3 Bloch, F., 4.4.2, 7.4.3 Block, S., 2.3, 5.1, 5.3 Blow, D. M., 3.1, 3.4 Blum, M., 7.1.6 Blume, M., 4.2.6, 6.1.2, 7.4.3, 8.7 Blundell, S. A., 4.2.2 Blundell, T. L., 2.2, 3.1 Bùe, N., 4.3.7, 8.8 Boehli, T., 4.2.1 Boehme, R. F., 8.7 Boer, D. K. G. de, 2.9 Boer, J. L. de, 7.5, 9.2.2 Boerdijk, A. H., 9.2.1 Boersch, H., 4.3.4 Boese, R., 3.4 Boettinger, W. J., 2.7 Boeuf, A., 2.8

Boggs, P. T., 8.1 BoÈhlen, K. van, 3.4 Bohlin, H., 2.3 Boie, R. A., 7.3 Bojarski, Z., 2.3 Bokij, G. B., 9.2.2 Boød, T., 2.3 Boll-Dornberger, K., 9.2.2 Bolling, E. D., 4.4.2 Bolotina, N. B., 5.3 Bomchil, G., 5.3 Bond, C. C., 7.1.6 Bond, W. L., 2.7, 5.3 Bone, D. A., 7.1.6 Bonelle, J. P., 4.2.3 Bongaarts, P. J. M., 4.4.2 Bonham, R. A., 4.3.3 BoÈni, P., 4.4.2 Bonin, D., 7.1.6 Bonnet, M., 8.7 Bonnet, R., 3.4 Bonse, M., 2.6.2 Bonse, U., 2.2, 2.3, 2.6.1, 2.7, 4.1, 4.2.2, 4.2.5, 4.2.6, 4.4.2, 5.3 Bontchev, R., 9.2.2 Booker, G. R., 5.4.2 Boom, G., 5.2 Boothroyd, A. T., 2.6.2 Borchert, G. L., 4.2.2 Bordas, J., 2.5.1, 4.1, 5.2, 7.1.6, 9.2.1 Bordet, J., 2.4.2 Bordet, P., 3.1 Borg, I. Y., 2.3 Borgeaud, P., 5.3 Borkowski, C. J., 7.1.6, 7.3 BoÈrner, H. G., 4.2.2 Borovilova, N. V., 4.4.2 Borso, C. S., 7.1.6 Bosshard, R., 3.4, 7.1.6 BoÈttger, G., 4.4.2 Botton, G. A., 8.8 Boucherle, J. X., 8.7 Bouchiat, M. A., 4.4.2 Bouldin, C. E., 4.2.3 Boulin, C., 7.1.6 Bouman, J., 6.2 Bouquiere, J. P., 3.4 Bourdel, J., 7.3 Bourdillon, A. J., 2.5.1, 4.3.4, 5.2 Bourke, P., 4.2.1 Bourret, A., 4.3.8 Bovin, J.-O., 9.2.2 Bowen, D. K., 2.7, 4.1, 4.2.3, 5.3 Bowen, T. S., 4.2.1 Bowman, H. A., 5.3 Box, G. E. P., 8.1, 8.2 Boyarskaya, R. V., 4.3.5 Boyd, R. J., 8.7 Boyers, D. G., 4.2.1 Braam, A. W. M., 7.4.2 Bracewell, R., 2.6.1 BraÂdler, J., 2.7 Brady, R. L., 3.1, 3.4 Brafman, O., 9.2.1 Bragg, W. H., 2.2, 2.3, 5.3 Bragg, W. L., 2.2, 2.6.2, 5.3 Braillon, P., 3.5

Brammer, L., 9.5, 9.6 Brandenburg, K., 9.4 Breitenstein, M., 4.3.3 Brenner, R., 7.3 Brentano, J. C. M., 2.3 Brese, N. E., 9.1 Bretherton, L., 3.4 Briand, J. P., 5.2 Brice, M. D., 9.5, 9.6 Bricogne, G., 4.3.7, 7.1.6 Briggs, E. A., 4.2.4, 4.2.6, 7.4.3 Brindley, G. W., 9.2.2 Brister, K. E., 4.2.5 Britton, D., 3.1 Brock, C. P., 9.7 Brockhouse, B. N., 4.4.2 Brockway, L. O., 4.3.3 Brodsky, A., , 4.2.5 Brongersma, H. H., 7.1.6 Brooks, I., 2.2 Bross, H., 4.3.4 Brouns, E., 9.8 Brown, A. S., 4.2.5 Brown, B. R., 5.3 Brown, D., 4.2.1 Brown, D. B., 2.3 Brown, G. E., 4.2.6 Brown, G. M., 6.1.1 Brown, G. S., 4.2.3, 7.4.4 Brown, H., 1.4, 9.8 Brown, I. D., 9.4, 9.5, 9.6 Brown, L. M., 4.3.4, 4.3.8 Brown, M. C., 9.6 Brown, N. E., 3.4 Brown, P. J., 4.4.5, 6.1.2, 8.7 Brown, R. T., 4.2.4, 4.2.6, 7.4.3 Brown, W. D., 7.4.3 Brownell, S. J., 2.3, 5.2 Brownell, W. E., 2.3 Bruce, M. I., 9.6 BruÈhl, H.-G., 5.3 Brumberger, H., 2.6.1 Brunegger, A., 4.3.4 Brunel, M., 8.7 Brunner, G. O., 9.1, 9.3 Brydson, R., 4.3.4 Brysk, H., 4.2.6 Bubenzer, A., 4.3.4 Buchanan, D., 2.3 Buchanan, J., 8.2 Buckingham, A. D., 8.7 Budinger, T. F., 4.3.8 Budnick, J. L., 4.4.2 Bueche, A. M., 2.6.1 Buerger, M. J., 1.4, 2.2, 2.3, 3.4, 5.3, 6.2, 9.2.1 Buffat, P., 4.4.2 Buggy, T. W., 4.3.4, 7.2 BuÈhrer, W., 4.4.2 Bulkin, B. J., 4.1 BuÈlow, H., 9.8 BuÈlow, R., 1.4 Bunch, D. S., 8.1 Bunge, A. V., 4.3.3 Bunge, C., 4.3.3 Bunge, C. F., 4.3.3 Bunge, H.-J., 4.3.5 Bunkenburg, J., 4.2.1 Bunker, B., 4.2.3 Bunker, G., 4.2.3 Bunn, C. W., 3.1

969


Bunyan, P. J., 4.3.3 Burany, X. M., 9.2.2 Buras, B., 2.5.1, 2.5.2, 4.2.1, 4.2.6, 5.2, 5.3, 7.1.5 Burbank, R. D., 6.2 Burch, T. J., 4.4.2 Burdette, H. E., 2.7, 3.4 Burek, A. J., 4.2.1 Burge, R. E., 7.2 Burger, A., 7.1.4 Burgers, W. G., 2.2 Burgess, W. G., 8.8 Burgy, M. T., 4.4.2 Burke, B. E., 7.1.6 Burke, J., 5.3 Burkel, E., 7.4.2 Burla, M. C., 8.6 Burley, S. K., 3.1 Burns, R., 7.1.6 Burr, A. F., 2.3, 4.2.2 Burshtein, Z., 7.1.4 Bursill, L. A., 4.3.8 Buschert, R. C., 5.3 Buseck, P., 4.3.4 Buseck, P. R., 4.3.8 Bushnell-Wye, G., 2.3 Bushuev, V. A., 7.4.3 Busing, W. R., 3.4, 5.3 Butler, D. J., 7.1.6 Butler, E. P., 3.5 Butler, M., 2.3 Butler, R. D., 3.3 Buttiker, M., 2.9 Buxton, B. F., 4.3.7 Bychkova, V. E., 2.6.1 Byer, R. L., 4.2.1 Byrd, R. H., 8.1 Caballero, A., 4.2.3 Cable, J. W., 9.8 Caglioti, G., 2.3, 2.4.2, 4.4.3, 8.6 Cahn, J. W., 9.8 Cahn, R. W., 1.3 Calas, G., 4.2.3, 4.3.4 Calvert, L. D., 2.3, 9.3 Campbell, J. E., 3.4 Campos, C., 3.4 Camps, R. A., 4.3.8 Capasso, S., 3.1 Capel, M. S., 2.6.2 Capellmann, H., 4.4.2 Caplan, H. S., 4.2.1 Cardona, M., 4.2.2, 5.3 Cardoso, L. P., 3.4 Carlile, C. J., 2.4.2, 4.4.2, 7.4.2 Carlson, E. H., 9.2.2 Caroll, C. L., 5.3 Carpenter, J. M., 4.4.1 Carr, M. J., 2.4.1 Carr, P. D., 2.2, 3.4, 5.3 Carter, C. B., 4.3.8 Carter, C. W., 3.1 Carter, C. W. Jr, 3.1 Cartwright, B. A., 9.5, 9.6 Carver, T. R., 4.4.2 Cascarano, G., 8.6 Cascio, D., 3.4 Case, A. L., 2.8 Caspar, D. L. D., 4.4.2 Cassetta, A., 2.2

AUTHOR INDEX Castaing, R., 4.2.1, 4.3.4 Castelli, C. M., 2.7 Caticha-Ellis, S., 3.4 Catti, M., 1.3 Catura, R. C., 7.1.6 Cauchois, Y., 4.2.2 Caudron, B., 7.1.6 Caul®eld, P. B., 5.3 Causer, R., 2.3, 5.3 Cavagnero, G., 4.2.2, 5.3 Cembali, F., 5.3 Cernik, R., 5.2 Cernik, R. J., 2.3 CÏernohorskyÂ, M., 5.3 Cerva, H., 2.7 Ceska, T. A., 3.1 Chadha, G. K., 9.2.1 Chadi, D. J., 9.1 Chaimdi, M., 3.4 Chakera, A., 6.3 Chamberland, B. L., 9.2.2 Chambers, F. W., 5.3 Chambers, W. F., 2.4.1 Chan Dyk Tkhan, 7.1.6 Chance, B., 4.2.3 Chandler, G. S., 6.1.1 Chandrasekaran, M., 9.2.1 Chandrashekar, G. V., 3.1 Chang, S.-L., 5.3 Chan Khyo Dao, 7.1.6 Chantler, C. T., 4.2.6 Chapman, J. N., 7.2 Chapuis, G., 4.2.6, 7.5 Charpak, G., 2.2, 7.1.6 Chatterjee, S., 8.5 Chau, K., 4.3.8 Chayen, N. E., 3.1 Cheary, R. W., 5.2 Cheetham, A. K., 2.3, 8.6 Cheetham, G. M. T., 2.3, 3.1 Chen, C. H., 4.3.4 Chen, H., 4.2.3, 4.4.2 Chen, S. H., 2.6.1, 2.6.2 Chen, S.-H., 2.9 Chen-Mayer, H. H., 4.4.2 Cheng, T. Z., 4.3.8 Cheremukhina, G. A., 7.1.6 Chernenko, S. P., 7.1.6 Chernov, M. A., 9.2.2 Cherns, D., 4.3.8 Chesser, N. J., 4.4.2, 4.4.3 Cheung, S., 4.4.2 Chevallier, P., 5.2 Chidambaram, R., 6.1.1 Chieux, P., 7.3 Chikawa, J., 7.1.6, 7.1.7, 7.1.8 Chikawa, J.-I., 2.7 Chipera, S. J., 7.1.4 Chipman, D. R., 4.2.3 Chirino, A. J., 3.1 Chou, H. P., 7.3 Chowanietz, E. G., 7.1.6 Christ, J., 4.4.2 Christen, D. K., 2.6.2 Christensen, A. N., 2.3, 7.1.3 Christoph, A., 5.3 Chu, B., 7.1.6 Chung, S. J., 1.3 Chupp, T. E., 4.4.2 Chwaszczewska, J., 2.5.1 CõÂsarÏovaÂ, I., 9.2.2

Cisney, E., 2.3 Citrin, P. H., 4.1, 4.2.3 Clark, G. F., 2.7 Clark, S. M., 2.5.1, 3.4 Clay, R. E., 4.2.1 Clay, W. T., 7.3 Cleemann, J. C., 2.6.1 Clegg, W., 3.4, 5.3 Clementi, E., 4.4.5, 6.1.1, 6.1.2 Clifton, I. J., 3.4 Cline, J. P., 2.3 Clout, P. N., 7.1.6 Cochran, W., 5.3 Cockayne, D. J. H., 4.3.8 Cocking, S. J., 4.4.2 Cody, V., 3.1 Coelho, A., 5.2 Coene, W., 4.3.8 Coene, W. M. J., 4.3.8 Coffman, D., 4.3.3 Cohen, E. R., 4.2.1, 4.2.2, 4.2.3 Cohen, G. G., 2.7 Cohen, J. B., 2.3 Cohen, M. U., 5.2 Cohn-Vossen, S., 9.1 Cole, H., 2.7, 4.2.6, 5.3 Cole, W. F., 5.3 Colegrove, F. D., 4.4.2 Coleman, T. A., 7.1.6 Collett, B., 7.1.6 Colliex, C., 4.3.4 Collins, C. B., 4.2.1 Collins, D. M., 8.2 Colwell, J. F., 4.4.2 Comparat, V., 7.1.3, 7.1.6 Compton, A. H., 2.3, 2.7, 4.2.1, 5.3 Condon, E. U., 8.7 Conger, G. B., 7.1.6 Conolly, M. L., 3.4 Conradi, E., 9.2.2 Constenoble, M. L., 2.3 Conturie, Y., 4.2.1 Convert, P., 2.4.2, 7.3 Conway, J. H., 9.1 Cook, J. E., Cook, R. D., 8.5 Cookson, D. J., 4.2.6 Cooper, A. S., 5.3 Cooper, C. W., 2.6.1 Cooper, M. J., 4.2.3, 4.4.3, 6.3, 7.4.2, 7.4.3, 8.6 Copley, J. R. D., 4.4.2 Coppens, P., 2.2, 3.4, 6.3, 6.4, 8.7, 9.8 Cork, C., 2.2 Cork, C. W., 7.1.6 Cosier, J., 3.4 Cosslet, V. E., 4.2.3 Cosslett, V. E., 4.2.1 Cotton, J. P., 2.4.2, 7.3 Cottrell, A., 9.2.1 Cottrell, A. H., 6.4 Couderchon, G., 4.4.2 Coulter, K. P., 4.4.2 Coulthard, M. A., 4.3.1, 6.1.1 Coustham, J., 2.6.2 Coutanceau Clarke, J. A. R., 9.7 Cowley, J. M., 2.4.1, 4.1, 4.3.1, 4.3.2, 4.3.6.1, 4.3.7, 4.3.8, 8.8, 9.2.1, 9.2.2

Cowley, R. A., 4.4.3 Cox, A. R., 3.5 Cox, D. E., 2.3, 2.5.1, 4.2.6, 7.4.4, 8.6, 9.2.2 Cox, H. L. Jr, 4.3.3 Cox, M. J., 3.1 Coyle, B. A., 6.3 Cracknell, A. P., 6.1.1 Crain, J., 2.3 CrameÂr, H., 8.4 Craven, A. J., 4.3.4, 7.2 Craven, B. M., 6.4 Crawford, F. S., 4.2.3 Crawford, R. K., 2.9 Craxton, R. S., 4.2.1 Creagh, D. C., 4.2.3, 4.2.4, 4.2.5, 4.2.6, 10 Cressey, G., 2.3 Crewe, A. V., 4.3.4, 4.3.8 Crichton, R. R., 2.6.1, 2.6.2 Croce, P., 2.9 Cromer, D. T., 4.2.4, 4.2.6, 4.3.1, 6.1.1, 7.4.3, 8.7 Cross, J. O., 4.2.3 Crowder, C. E., 2.3, 5.2 Crowfoot, D., 3.2 Crozier, E. D., 4.2.3 Crozier, P. A., 4.3.4 Cruickshank, D. W. J., 2.2, 3.4, 5.3, 8.3, 8.7 Cudney, B., 3.1 Culhane, J. L., 7.1.6 Cullen, E. E., 4.2.4 Cullity, B. D., 2.3 Currat, R., 4.4.2, 7.4.3, 9.8 Curtis, C. F., 8.7 Cusack, S., 2.6.2 Cusatis, C., 4.2.6 Cuttitta, F., 3.2 Czerwinski, H., 7.4.3 Daams, J. L. C., 9.3 Dabbs, J. W. T., 4.4.2 Daberkow, I., 4.3.8 Daberkow, L., 7.2 Dabrowski, A., 7.1.4 Dabrowski, A. J., 7.1.5 Dahl, J. P., 8.7 Dainton, D., 7.1.6 Dalglish, R. L., 7.1.6 Dallas, W. J., 7.1.6 DalleÂ, D., 3.4 Dam, B., 9.8 Damaschun, G., 2.6.1 Damaschun, H., 2.6.1 Dana, E. S., 3.5 Daniel, V., 9.8 Daniels, J., 4.3.4 Daniels, P. J., 7.1.6 D'Antonio, P., 3.1 Danz, H., 3.4 D'Aprile, F., 3.4 Darriet, B., 9.2.2 Darriet, J., 9.2.2 Dartyge, E., 7.1.6 Darwin, C. G., 6.4 Das Gupta, P., 5.2 Dash, J. G., 4.4.2 Dathe, W., 9.2.2 D'Auria, S., 3.1 Davanloo, F., 4.2.1

970


David, W. I. F., 2.3, 2.5.2, 8.6 Davidson, E. R., 6.1.1 Davidson, J. B., 2.8 Davies, J. E., 9.7 Davies, N. C., 3.5 Davies, S. T., 2.7, 4.2.3 Davis, B. L., 2.3, 5.3 Dawson, B., 6.1.1 Day, M. W., 3.1 Deacon, A., 2.2 Debye, P., 2.3, 2.6.1, 6.2 DeCicco, P. D., 7.4.3 Deckman, H. W., 7.1.6 Degoy, S., 3.1 Dehlinger, U., 9.8 Deininger, C., 8.8 Delaey, L., 9.2.1 Delamoye, P., 9.8 De Lange, P. W., 4.4.2 Delapalme, A., 2.5.2, 4.4.2, 8.7 Delduca, A., 7.1.6 Delettrez, J., 4.2.1 Delf, B. W., 4.2.5, 5.2 Del Grande, N. K., 4.2.3, 4.2.4 Delhez, R., 2.3, 5.2 Dellby, N., 4.3.7, 4.3.8 Delley, B., 8.7 Deltour, J., 2.3 DeLucia, M. L., 8.7 De Marco, J. J., 4.2.4 Demasi, D., 3.1 Demierre, C., 2.2 Demmel, J., 8.1 Denesyuk, A. I., 2.6.1 Denley, D., 4.2.3 Denne, W. A., 3.4 Denner, W., 5.3 Dennis, J. E., 8.1 Dent Glasser, L. S., 3.4 Depmeier, W., 9.8 Dereniak, E. L., 7.1.6 Derewenda, Z., 7.1.6 Desai, C. F., 3.4 Descamps, J., 4.2.1 Desclaux, J. P., 4.2.2, 4.4.5, 6.1.2, 8.7 Deslattes, R., 5.2 Deslattes, R. D., 4.2.1, 4.2.2, 5.2, 5.3 Desseaux, J., 4.3.8 DeTitta, G. T., 3.1 Deutsch, M., 2.3, 4.2.2, 4.2.6, 5.3 Dewan, J. C., 3.4 Dexpert, H., 4.2.3 Dexter, D. L., 2.3 D'Eye, R. W. M., 3.4 Dickens, B., 8.3 Dideberg, O., 3.4 Dietrich, B., 5.3 Dietz, G., 9.2.2 Dietz, J., 7.1.6 DiGiovanni, H. J., 2.3 Dikovskaya, R. R., 5.3 Diller, T. C., 3.1 Dimitrov, D. P., 2.6.1 Dingley, D. J., 5.3 Dinnebier, R. E., 8.6 Di Nova, K., 4.2.1 Dischler, B., 4.3.4 Disko, M. M., 4.3.4

AUTHOR INDEX Divadeenam, M., 4.4.4 Dixon, N. E., 3.1 Dobrzynski, L., 4.4.2 Dobson, P. J., 4.3.8 Dodson, G. G., 3.4 Doi, K., 2.8 Dolin, R., 7.1.6 Doll, C., 4.4.2 Dollase, W. A., 2.3 Dolling, G., 4.4.2 Donaldson, J. R., 8.1 Dongarra, J., 8.1 Doniach, S., 2.6.1, 4.2.3 Donnay, G., 1.3 Donnay, J. D. H., 1.3, 1.4, 9.8 DoÈnni, A., 4.4.2 Donohue, J., 9.7 Dorenwendt, K., 4.2.2, 5.3 Dornberger-Schiff, K., 3.4, 9.2.1, 9.2.2 Dorner, B., 4.4.3, 7.4.2 Dorrington, E., 7.1.6 Dorset, D. L., 3.5, 4.3.7, 4.3.8 D'Orsi, C. J., 7.1.6 Doscher, M. S., 3.4 Doty, J. P., 7.1.6 Doubleday, A., 9.5, 9.6 Downing, K. H., 4.3.8 Downing, R. G., 4.4.2 Downs, J., 4.3.7, 8.8 Doyle, P. A., 4.2.4, 4.3.1, 4.3.2, 6.1.1 Drabkin, G. M., 4.4.2 Dragoo, A. L., 5.2 Draper, N., 8.1, 8.4 Dreier, P., 4.2.3, 4.2.6 Drenth, J., 3.1 Dressler, L., 5.3 Drits, V. A., 2.4.1, 4.3.5 Drum, C. M., 3.5 Drummond, W., 7.1.4 Duarte, P. W. E. P., 4.2.4 Dubey, M., 9.2.1 Duchenois, V., 7.1.6 Du Croz, J., 8.1 Ducruix, A., 3.1 Dudarev, S. L., 4.3.2 Dudley, M., 2.8 Duijneveldt, F. B. van, 6.1.1 Duisenberg, A. J. M., 3.4 Duke, P. J., 4.2.1 DuÈker, H., 4.3.8 Dumas, P., 3.4 Du Mond, J. W. M., 2.3, 2.7 Dunitz, J. D., 9.7 Dunn, H. M., 5.3 Dunning, T. H. Jr, 6.1.1 Dupont, Y., 7.1.6 Duppich, J., 4.4.2 Durand, D., 6.1.1 Durbin, R., 2.2 Durbin, R. M., 7.1.6 Durham, J. P., 4.3.4 Durham, P. J., 4.2.3 Ï urovicÏ, S., 9.2.2 D DuÈrr, J., 4.2.3 DusÏek, M., 9.2.2 Dvoryankina, G. G., 2.4.1 Dwiggins, C. W. Jr, 6.3 Dyson, N. A., 2.3, 4.2.1, 7.1.6

Early, J. G., 3.4 Eastabrook, J. N., 5.3, 7.1.2, 7.5 Ebeling, G., 4.2.2, 5.3 Eberhardt, W., 4.1 Ebert, M., 4.4.2 Ebisawa, T., 2.9, 4.4.2 Eckert, J., 4.4.3 Eddy, M. M., 2.3 Edington, J. W., 3.5, 5.4.1 Edwards, H. J., 2.3, 5.2 Edwards, S. L., 3.4 Edwards, T. H., 2.3 Effenberger, H., 9.2.2 Egelstaff, P. A., 4.4.2 Egerton, R. F., 4.3.4 Eggleton, R. A., 9.2.2 Egidy, T. V., 4.2.2 Egorov, A. I., 4.4.2 Eguchi, T., 4.3.7, 8.8 Ehrenberg, W., 4.2.1 Eichelle, G., 3.1 Eigner, W.-D., 2.6.1 Eikenberry, E. F., 2.7, 7.1.6 EiseleÂ, J.-L., 3.1 Eisenberg, H., 2.6.2 Eisenberger, P., 2.2, 4.1, 4.2.3, 7.4.3 Eklund, H., 3.4 El Korashy, A., 3.4 Elder, M., 3.4 Eling, A., 9.1 Ellinger, Y., 8.7 Ellis, T., 5.3 Ellisman, M. H., 7.2 Elsenhans, O., 4.4.2 Elsner, G., 7.1.6 Emberson, D. L., 7.1.6 Emmerich, C., 2.2 Endesfelder, A., 4.3.3 Endoh, H., 4.3.8 Endoh, T., 7.1.6 Eng, P. J., 4.2.5 Enge, H. A., 4.3.4 Engel, D. H., 4.2.6 Engel, P., 1.4 Engel, W., 4.3.4 Engelman, D. M., 2.6.2 Englander, M., 2.8 Engstrom, P., 4.2.5 Enzo, S., 2.3 Epstein, J., 4.3.3, 8.7 Erickson, J. W., 3.4 Ermer, O., 9.1 Ernst, R. R., 5.5 Ertl, G., 4.1 Escof®er, A., 4.4.2 Esquivel, R. O., 4.3.3 Esteva, J. M., 4.3.4 Evans, B. W., 9.2.2 Evans, E. H., 3.4 Evans, H. T., 2.2, 5.2 Evans, H. T. Jr, 5.3 Evans, J. C., 3.4 Evans, R. C., 2.3, 9.7 Evans, R. G., 3.5 Ewing, F., 3.1 Ewins, C., 3.5 Eyres, B. L., 3.5 Faber, W., 4.4.2 Fabian, D. J., 4.2.2

Fabri, R., 5.3 Fagherazzi, A., 2.3 Fagherazzi, G., 2.3 FaÊk, B., 2.8 Fan, G. Y., 4.3.7, 7.2 Fan, H. F., 4.3.8 Fang, Y., 6.2 Fankuchen, I., 2.3 Fano, U., 4.3.4, 7.1.6 Farabaugh, E. N., 3.5 Farge, Y., 4.2.1 Farkas-Jahnke, M., 9.2.1 Farnell, G. C., 7.2 Farnoux, B., 2.4.2, 7.3 Farquhar, M. C. M., 5.3 Faruqi, A. R., 7.1.6 Fast, G., 9.8 Favro, L. D., 6.1.1 Fawcett, T. G., 2.3, 5.2 Fearon, E. O., 5.3 Feder, R., 2.3 Fedorov, B. A., 2.6.1 Fedorov, V. V., 8.1, 8.4, 8.5 Fedotov, A. F., 4.3.5, 9.2.2 Feher, G., 3.1 Feidenhans'l, R., 2.3, 7.1.3 Feigin, L. A., 2.6.1, 2.9 Fejes, P. L., 4.3.4, 4.3.8 Felcher, G. P., 2.9 Feldman, C., 4.1 Feller, W., 6.1.1 Feng, H.-P., 7.1.6 Ferguson, I. F., 5.2 Fermi, E., 4.2.6, 7.4.3, 8.7 Ferraris, G., 1.3, 4.3.5 FerreÂ-D'AmareÂ, A. R., 3.1 Festenberg, C. V., 4.3.4 Fewster, P. F., 5.3 Fichtner, K., 9.2.2 Fichtner-Schmittler, H., 9.2.2 Fields, P. M., 4.3.8 Figueiredo, M. O., 9.1 Figueiredo, M. O. D., 9.2.2 Filhol, A., 3.4 Filippini, G., 9.7 Filscher, G., 5.3 Finger, L. W., 2.3, 2.5.1, 3.4, 8.3 Fink, J., 4.3.4 Fink, M., 4.3.3 Finlayson, H., 4.2.2 Finney, J. L., 3.4 Finzel, B. C., 7.1.6 Fiori, C. E., 7.1.4 Fiorito, R. B., 4.2.1 Fischer, D. G., 5.3 Fischer, D. W., 4.3.4 Fischer, J., 3.4, 7.3 Fischer, K., 2.7, 4.2.6 Fischer, K. F., 1.4 Fischer, P., 4.4.2, 5.5 Fischer, S., 4.4.2 Fischer, W., 1.4, 9.1 Fisher, R., 6.1.1 Fisher, R. G., 3.1 Fisher, R. M., 4.3.7, 8.8 FitzGerald, J. D., 4.3.8, 5.4.2 Fitzsimmons, M., 2.9 FjellvaÊg, H., 2.3, 7.1.3 Flack, H. D., 1.3, 4.2.2, 5.3, 6.3, 8.1

971


Flank, A. M., 7.1.6 Fleischer, M., 9.2.2 Flint, R. B., 7.2 Flower, H. M., 3.5 Fock, V., 4.2.6 Foit, F. F. Jr, 3.4 Foltyn, T., 4.4.2 Fomin, V. G., 5.3 Fontaine, A., 7.1.6 Fontecilla-Camps, J. C., 3.1 Ford, W. E., 3.5 Fordham, J. L. A., 7.1.6 FoÈrster, E., 4.2.2, 5.3 Forsyth, J. B., 4.4.2, 4.4.5, 7.3, 8.7 Forsyth, J. M., 4.1, 4.2.1 Forsythe, E., 3.1 Forte, M., 4.4.2 Foster, B. A., 7.1.3 Fouassier, M., 7.1.6 Fourme, R., 2.2, 3.4, 4.2.1, 7.1.6 Fourmond, M., 2.6.2 Fournet, G., 2.6.1, 2.6.2 Fournier, T., 3.1 Fowler, C. E., 7.3 Fox, A. G., 4.3.1, 4.3.2, 4.3.7, 6.1.1, 8.8 Fraaije, J. G. E. M., 3.1 Fraase Storm, G. M., 3.4 Frahm, A., 4.2.3 Frank, F. C., 9.1, 9.2.1 Frank, J., 4.3.8 Frankel, R. D., 4.1, 4.2.1 Frank-Kamenetskii, V. A., 9.2.2 Franklin, K. R., 2.3 Franzini, M., 9.2.2 Fraser, G. W., 7.1.6 Frauenfelder, H., 3.4 Freeborn, B. R., 2.6.2 Freeborn, W. P., 2.3 Freeman, A. J., 4.4.5, 6.1.2, 8.7 Freeman, F. F., 4.4.2 Freer, S. T., 7.1.6 French, S., 7.5 Freund, A., 4.2.6, 5.3 Freund, A. K., 4.2.5, 4.4.2 Freund, I., 7.4.3 Frevel, L. K., 2.3, 2.4.1 Frey, F., 3.4, 4.4.2 Freymann, D., 7.1.6 Fricke, H., 4.2.3 Friedli, H. P., 4.4.2 Friedman, H., 6.3 Friedrich, H., 4.4.2 Friedrich, W., 2.1, 2.2 Frishberg, C., 8.7 Fritsch, E., 4.3.4 Fritsch, M., 4.2.2, 5.3 Frolova, K. E., 4.3.5 Frolow, F., 3.4 Frueh, A. J., 9.2.2 Fryer, J. R., 3.5, 4.3.8 Fu, Z. Q., 4.3.8 Fuchs, H. F., 7.1.6 Fuchs, R., 4.3.4 Fuess, H., 3.4, 4.4.2, 8.7 Fuggle, J. C., 4.2.2, 4.3.4 Fujii, K., 4.2.2 Fujii, Y., 4.4.3 Fujikawa, B. K., 4.2.4, 4.2.6

AUTHOR INDEX Fujimoto, I., 2.7, 7.1.6, 7.1.7 Fujimoto, Z., 4.2.2 Fujita, T., 7.1.6 Fujiwara, K., 4.3.1 Fujiwara, M., 4.2.3 Fujiyoshi, Y., 4.3.7, 4.3.8 Fukahara, A., 5.3 Fukamachi, T., 2.5.1, 5.2, 6.3 Fukuhara, A., 4.3.7, 4.3.8, 8.8 Fukumachi, T., 4.2.5 Fukumori, T., 5.3 Fuller, W. A., 8.1 Fuoss, P. H., 4.2.3, 4.2.6 Furry, W. H., 6.1.1 Futagami, K., 5.3 Gabe, E. J., 5.3 Gabel, K., 4.2.5 Gabor, D., 4.3.8 Gabriel, A., 7.1.6 GaÈhler, R., 4.4.2 Gainsford, G. J., 2.3 Gaødecka, E., 5.3 Gale, B., 5.2 Gallo, R., 3.4 Galloy, J. J., 9.7 Galy, J., 9.2.2 Gamarnik, M. Ya., 5.3 Gamblin, S. J., 3.4 Gandol®, G., 2.3 Ganow, D., 5.3 Garavito, R. M., 3.1 Garcia Arribas, A., 9.8 GarcõÂa-Ruiz, J. M., 3.1 Gard, J. A., 5.4.1, 9.2.2 Gar®eld, B. R. C., 7.1.6 Garlick, G. F. J., 7.2 Garman, E. F., 3.4 Garrett, R., 4.2.5 Garrett, R. F., 4.2.5 Garroff, S., 2.9 Gasgnier, M., 4.3.4 Gaultier, J.-P., 4.3.5 Gauthier, J. P., 9.2.1 Gavezzotti, A., 9.7 Gavin, R. M. Jr, 4.3.3 Gavrila, M., 4.2.6, 7.4.3 Gay, D. M., 8.1 Gearhart, R. A., 4.2.1 Gedcke, D. A., 5.2 Geiger, H., 7.1.2 Geiger, J., 4.3.3, 4.3.4 Geist, V., 5.3 George, B., 2.6.2 George, J. D., 2.3, 5.3 Gerber, C., 4.3.8 Gerdau, E., 5.3 Gerken, M., 5.3 Gerlich, R., 9.1 Gernat, C., 2.6.1 Gerold, V., 2.7 Gerstenberg, H., 4.2.3 Gerstenberg, H. M., 4.2.3, 4.2.4 Gerward, L., 2.5.1, 2.5.2, 4.2.3, 4.2.4, 4.2.6, 5.2, 5.3 Ghose, S., 3.4 Giacovazzo, C., 8.6 Gibbons, P. C., 4.3.4 Gibbs, D., 2.9, 7.4.3, 8.7 Gibson, K. D., 9.7 GiegeÂ, R., 3.1

Gielen, P., 5.3 Giessen, B. C., 2.3, 2.5.1, 5.2 Giles, C., 4.2.5 Gill, P. E., 8.3 Gillham, C. J., 2.3, 5.2 Gilliland, G. L., 3.1, 7.1.6 Gillon, B., 8.7 Gilmore, C. J., 4.3.7, 4.3.8 Gilmore, D. J., 7.1.6 Girgis, K., 9.3 Gjùnnes, J., 4.3.3, 4.3.7, 5.4.2, 8.8 Gjùnnes, K., 4.3.7, 8.8 Glaeser, R. M., 4.3.7, 4.3.8 Glass, H. L., 5.3 Glatter, O., 2.6.1, 2.6.2 GlaÈttli, H., 2.6.2, 4.4.4 Glauber, R., 4.3.3 Glazer, A. M., 2.5.1, 3.4, 5.2, 5.3 Glazer, J., 4.3.7 Glinka, C. J., 4.4.2 Glover, I., 3.4 Glusker, J. P., 2.2 Gobel, H., 4.2.6 GoÈbel, H. E., 2.3, 7.1.3 Gobert, G., 4.4.2 Goddard, H. F., 7.1.6 Goddard, P. A., 2.7 Godwin, R. P., 4.2.1 Godwod, K., 5.3 Goetz, K., 4.2.2, 5.3 Golay, M. J. E., 2.3 Goldberg, M., 4.2.1, 8.7 Goldman, L. M., 4.2.1 Goldman, M., 4.4.4 Goldschmidt, V., 9.8 Goldsmith, C. C., 2.3 Goldsztaub, S., 4.2.1, 9.2.2 Golob, P., 4.3.4 Golovin, A. L., 5.3 Golub, R., 4.4.2 Gonschorek, W., 5.3 Gonzalez, A., 3.4 Goodhew, P. J., 3.5 Goodisman, J., 2.6.1 Goodman, P., 2.4.1, 4.3.6.1, 4.3.7, 8.8, 9.2.2 Goodson, J. H., 7.1.6 Goral, K., 2.6.1 Gorceix, O., 4.2.2 Gordon, G. E., 2.3, 2.5.1, 5.2 Gordon, R. G., 8.7 Gorshkov, A. I., 4.3.5 Goto, K., 2.7, 7.1.6, 7.1.7 Goto, N., 7.1.6, 7.1.7 Gottschalk, H., 4.3.8 Gotwals, J. K., 5.3 Goulon, J., 4.2.5 Graaff, R. A. G. de, 6.3 Graafsma, H., 3.4 Graeff, W., 2.7 Graeser, S., 9.2.2 Graf, W., 4.4.2 Grant, B. K., 2.3 Grant, D. F., 7.5 Grant, G. A., 3.5 Grant, I., 4.3.1, 4.3.2 Grasselli, J. G., 4.1 Gratias, D., 9.8 Graubner, H., 3.2

Gray, N., 9.2.2 Grebille, D., 9.8 Greegor, R. B., 4.3.4 Green, M., 2.3, 4.2.1 Green, R. E. Jr, 2.7, 7.1.7 Greenbaum, A., 8.1 Greenberg, B., 5.2 Greene, G. C., 4.4.2 Greene, G. L., 4.2.2 Greenhough, T. J., 2.2, 3.4 Greenwood, J. A., 6.1.1 Grell, H., 9.2.2 Grenville-Wells, H. J., 6.2 Greville, T. N. E., 2.6.1 Grey, D., 4.2.5 Griebner, U., 5.3 Grif®th, J. P., 3.4 Grigson, C. W. B., 2.4.1 Grimmer, H., 1.3, 4.4.2 Grimsditch, M. H., 5.3 Grimvall, G., 4.2.6 Grinton, G. R., 4.3.8 Gritsaenko, G. S., 4.3.5 Gronsky, R., 4.3.8 Grossi, F., 4.2.5 Grossman, T., 4.4.2 Grosso, J. S., 2.3 Grosswig, S., 5.3 Groves, G. W., 3.5 Grubel, G., 4.2.5 Gruber, E. E., 5.3 GruÈnbaum, B., 9.1 Gruner, S. M., 2.7, 7.1.6 Grunes, L. A., 4.3.4 Gschneider, K. Jr, 9.3 Guagliardi, A., 8.6 Gubbens, A. J., 4.3.7 Gudat, W., 4.3.4 Guetter, E., 7.2 Guggenheim, S., 9.2.2 Guidi-Morosini, C., 6.3 Guigay, J. P., 2.8 Guillemet, E., 7.1.6 Guinet, P., 4.4.2 Guinier, A., 2.3, 2.6.1, 2.6.2, 2.7, 4.3.5, 9.2.2 Gumbel, E. J., 6.1.1 GuÈnther, W., 9.2.2 Guo, C.-L., 4.2.1 Guo, S. Y., 3.2 Gupta, S. K., 3.5 Gurman, S. J., 4.2.3 Guttmann, P., 7.1.6 Guyot, P., 2.6.2 Gyax, F. N., 4.1 Hausermann, D., 4.2.5 Haag, F., 9.1 Haas, J., 2.6.1, 2.6.2 Habash, J., 2.2, 3.4 Hadi, A. S., 8.5 Haendler, H. M., 3.4 Haga, K., 4.2.3 Hagashi, Y., 5.3 Hagemann, H. J., 4.3.4 Hagen, W., 2.7 HaÈgg, G., 2.2 Hahn, Th., 1.3, 1.4, 5.3, 9.2.2, 9.7 Haider, M., 4.3.8 Hails, J. E., 2.2

972


Hainisch, B., 2.6.1 Hajdu, J., 3.4 Hale, K. F., 3.5 Hales, T. C., 9.1 Halfon, Y., 3.4 Hall, E. L., 5.3 Hall, M. M. Jr., 2.3 Halliwell, M. A. G., 5.3 Hamacher, E. A., 2.3 Hamelin, B., 4.4.2 Hamid, S. A., 9.2.2 Hamill, G. P., 2.3, 5.2 Hamilton, J. F., 7.2 Hamilton, L. D., 2.6.1 Hamilton, R., 2.3, 5.2 Hamilton, W., 2.9 Hamilton, W. A., 2.9 Hamilton, W. C., 2.2, 5.3, 6.4, 7.5, 8.3, 8.4, 8.7 Hamley, I. W., 2.9 Hamlin, R., 2.2, 3.4, 7.1.6 Hammarling, S., 8.1 Hanawalt, J. D., 2.3 Han¯and, M., 4.2.5 Hanic, F., 5.3 Hann, R. A., 3.5 Hanneman, R. E., 5.3 Hansen, N. K., 4.1 Hansen, P. G., 4.2.2 Hanson, H. P., 4.3.3 Hanson, I. R., 3.4 Harada, J., 4.2.5, 7.4.2 Harada, Y., 4.3.7, 7.2 Harding, M. M., 2.2, 2.3, 3.1, 3.4, 5.3 Hardman, K. D., 3.4 Harlos, K., 3.1 Harmon, H. E., 4.4.2 Harmony, M. D., 9.5, 9.6 Harper, R. G., 3.5 Harris, J. L., 4.2.1 Harris, K. D. M., 8.6 Harris, L. J., 7.1.6 Harris, N., 5.3 Harrison, D. C., 7.1.6 Harrison, S. C., 2.2, 7.1.6 Harrison, W. T. A., 2.3 Hart, M., 2.2, 2.3, 2.5.1, 2.6.1, 2.6.2, 2.7, 4.2.2, 4.2.5, 4.2.6, 4.4.2, 5.2, 5.3 Hartl, W. A. M., 4.3.4 Hartmann, H., 3.4 Hartmann, W., 2.7 Hartmann-Lotsch, I., 4.2.6 Hartree, D. R., 4.2.6, 7.4.3 Hartshorne, N. H., 3.1, 3.3 HaÈrtwig, J., 4.2.2, 5.3 Haruta, K., 2.7 Hasegawa, K., 7.1.6 Hasegawa, T., 5.3 Hashimoto, H., 2.7, 4.2.6, 4.3.8 Hashizume, H., 2.2, 2.3, 2.7, 4.2.5, 7.1.3, 7.4.4 Hasnain, S. S., 4.2.3 Hastings, J. B., 2.2, 2.3, 4.2.3, 4.2.6, 7.4.4, 8.6 Hastings, T. J., 3.4 Haszlo, S. E., 7.1.7 Hatton, P. D., 2.3 Hattori, S., 4.2.6 Haubold, H. G., 7.1.6

AUTHOR INDEX Haumann, J., 2.9 Hauptmann, H., 3.2 HaÈusermann, D., 2.5.1, 5.3 Hautecler, S., 4.4.2 Hawkes, D. J., 4.2.4 Hawthorne, F. C., 9.6 Hayakawa, K., 2.7, 6.3, 7.1.6 Hayes, C., 4.4.2 Hayter, J. B., 2.6.2, 2.9, 4.4.2 Hazen, R. M., 2.5.1, 3.4 Heal, K. M., 3.4 Heald, A., 4.2.3 Heath, M. T., 8.1 Heathman, S. P., 4.4.2 Hebert, H., 5.3 Heckingbottom, R., 5.3 Hedman, B., 7.1.5 Heesch, H., 9.1 Hehn, R., 4.4.2 Heidorn, D. B., 2.6.1 Heigl, A., 3.2 Heil, W., 4.4.2 Heine, S., 2.6.1 Heine, V., 4.3.4, 9.8 Heinrich, A. R., 9.2.2 Heinrich, K. F. J., 4.2.4, 7.1.4 Heise, H., 5.3 Heisenberg, W., 4.3.3 Hellings, G. J. A., 7.1.6 Helliwell, J. R., 2.1, 2.2, 2.3, 3.1, 3.4, 4.2.1, 4.2.3, 4.2.6, 7.1.6 Helliwell, M., 3.1 HellkoÈtter, H., 4.2.6 Hellner, E., 7.1.1, 8.7, 9.1 Helmholdt, R. B., 7.4.2 Hemley, R. J., 2.5.1 Henderson, R., 4.3.8 Henderson, R. J., 4.3.7 Hendricks, R. W., 2.6.1, 7.1.6 Hendricks, S., 9.2.1 Hendrickson, W. A., 3.2, 4.2.6, 8.3 Hendrix, J., 2.6.1, 7.1.6 Henins, A., 4.2.2, 5.2, 5.3 Henke, B. L., 4.2.4, 4.2.6 Hennig, M., 3.1 Henning, A., 4.2.6 Henriksen, K., 3.4 Henry, L., 4.3.4 Henry, N. F. M., 2.2, 3.1, 5.3 Hensler, D. H., 2.3 Heo, N. H., 3.4 Hepp, A., 2.3, 8.6 Herbstein, F. H., 5.3 Herino, R., 5.3 Herpin, A., 9.8 Herrman, K., 4.3.8 Herrmann, K.-H., 4.3.4, 7.2 Hertz, G., 4.2.3 Hestenes, M., 8.3 Heuss, K. L., 3.4 Hewat, A. W., 2.4.2, 5.5, 8.6 Hewett, C. A., 2.3 Hewitt, R., 4.2.3 Hey, P. D., 2.4.2, 4.4.2 Heynes, G. D., 7.1.7 Hezemans, A. M. F., 3.1 Hickling, N., 3.2 Hida, M., 4.2.3 Hidaka, M., 2.5.1, 5.2

Higashi, T., 2.2, 3.4 Higgins, S. A., 4.4.3 Higgs, H., 9.5, 9.6 HiismaÈki, P., 4.4.2 Hikaru, T., 4.2.3 Hilbert, D., 9.1 Hilczer, B., 5.3 Hildebrandt, G., 2.7 Hilderbrandt, R. L., 4.3.3 Hill, R. J., 2.3 Hilleke, R. O., 2.9 Hillier, J., 4.3.4 Hilton, M. R., 4.3.7 Himes, V. L., 2.4.1, 9.7 Himmelblau, D. M., 8.4 Hines, W. A., 4.4.2 Hinze, E., 5.2, 5.3 Hirabayashi, M., 4.2.3 Hirabayshi, M., 5.3 Hiraga, K., 4.3.8 Hirai, T., 2.7, 7.1.6, 7.1.7 Hirano, T., 7.1.6 Hirota, F., 4.3.3 Hirs, C. H. W., 2.2 Hirsch, P. B., 2.3, 3.5, 4.3.6.2, 4.3.8, 5.4.1 Hirshfeld, F. L., 7.5, 8.7 Hirshfelder, J. O., 8.7 Hirvonen, J.-P., 2.9 Hitchcock, A. P., 4.3.4 Hjelm, R. P., 2.6.2 Ho, A. H., 4.2.1 Ho, M. M., 4.3.7 Hoaglin, D. C., 8.2 Hobbs, L. W., 3.5 Hock, R., 4.4.2 Hodeau, J. L., 3.1 Hodges, C. H., 4.3.4 Hodgson, K. O., 2.6.1, 4.2.3, 4.2.6, 7.1.5 Hoerni, J. A., 4.3.3 Hofer, F., 4.3.4 Hofer, M., 2.6.1 Hoff, R. W., 4.2.2 Hoffmann, H., 2.6.1, 2.6.2 Hoffmann, R., 4.3.4 Hofmann, A., 4.2.1 Hofmann, D., 4.4.2 Hofmann, E. G., 2.3 Hùghùj, P., 4.4.2 Hohberger, H. I., 4.3.4 Hohenberg, P., 8.7 Hohlwein, D., 3.4, 7.3 HoÈhne, E., 9.2.2 Hùier, R., 4.3.7, 5.4.2, 8.8 Holden, N. E., 4.4.4 Holladay, A., 8.7 Holland, F. M., 3.5 Holmes, K. C., 2.6.1, 3.4 Holmestad, R., 4.3.7, 8.8 Holmshaw, R. T., 7.1.6 Holt, S. A., 4.2.5 HolyÂ, V., 2.9, 5.3 Holzapfel, W. B., 2.5.1 HoÈlzer, G., 4.2.2, 5.3 Hom, T., 5.3 Honess, A. P., 3.5 Hong, S.-H., 5.3 Honjo, G., 2.4.1, 9.2.1 Honkimaki, V., 4.2.1 HoÈnl, H., 4.2.6

d'Hooghe, P., 4.4.2 Hope, H., 3.4 Hopf, R., 7.1.6 Hoppe, W., 2.6.2 Horisberger, M., 4.4.2 Horita, Z., 4.3.7, 8.8 Horiuchi, S., 4.3.8 Hornstra, J., 3.4 Horota, F., 4.3.3 Horstmann, M., 2.4.1 HorvaÂth, J., 5.3 Hosemann, R., 2.6.1, 2.7 Hosokawa, N., 5.3 Hosoya, S., 2.5.1, 2.8, 4.2.6, 5.2, 9.2.1 Hossfeld, F., 2.6.1, 2.6.2, 4.4.2 HovmoÈller, S., 3.4, 4.3.7 Howard, A., 2.2 Howard, A. J., 7.1.6 Howard, C. J., 2.4.2 Howard, S. A., 2.3 Howard, W., 9.2.2 Howerton, R. J., 4.2.4, 4.2.6, 7.4.3 Howes, M. J., 7.1.6 Howie, A., 3.5, 4.3.6.2, 4.3.7, 4.3.8, 5.4.1 Hoya, H., 3.4 Hoylaerts, M., 2.6.1 Hsiang, W. Y., 9.1 Hsu, B. T., 3.1 Hu, H.-C., 6.2 Huang, D. X., 4.3.8 Huang, T. C., 2.3, 5.2, 8.6 Hubbard, C., 2.3, 5.2 Hubbard, C. R., 2.3, 5.1, 5.2, 5.3 Hubbard, K. M., 2.9 Hubbard, S. T., 2.6.1 Hubbell, J. H., 4.2.3, 4.2.4, 4.2.6, 7.4.3 Huber, H., 3.4 Huber, P. J., 8.2 Huber, R., 6.3 Huffman, F. N., 4.2.2 Huggins, F. G., 2.3 Hughes, D. J., 4.4.2 Hughes, G., 7.1.6 Hughes, J. W., 7.1.2, 7.5 Hughes, T. E., 5.2 Huke, K., 4.2.1 Hull, A. W., 2.3 HuÈller, A., 6.1.1 Hulme, R., 5.3 Hulubei, H., 4.2.2 Humblot, H., 4.4.2 Huml, K., 5.3 Hummelink, T., 9.5, 9.6 Hummelink-Peters, B. G., 9.5, 9.6 Humphreys, C. J., 4.3.6.2, 8.8 Hundt, R., 9.4, 9.5, 9.6 Hunter, B., 8.6 Hunter, J. S., 8.1 Hunter, W. G., 8.1 Hustache, R., 2.8 Hutchings, M. T., 4.4.6 Hutchinson, J. L., 4.3.8 Hutchison, J. L., 9.2.2 Huxham, M., 3.5 Huxley, H. E., 7.1.6 d'Huysser, A., 4.2.3

973


Huyton, A., 5.2 Huzinaga, S., 6.1.1 Hwang, S. R., 4.4.2 Hyde, S. T., 9.1 Hyman, A., 3.5 Hyogah, H., 4.2.6 I'anson, K. J., 2.6.1 Ibach, H., 4.3.4 Ibel, K., 2.4.2, 2.6.1, 2.6.2, 4.4.2, 7.3 Ibers, J. A., 4.3.1 Ichimiya, A., 4.3.7, 8.8 IevinÎsÏ, A., 5.3 Ihara, H., 4.2.3 Ihringer, J., 3.4 Iijima, S., 4.3.8, 9.2.2 Iijima, T., 4.3.3 Ikeda, S., 4.2.3 Ikhlef, A., 7.1.6 Illini, Th., 7.4.2 Imada, K., 3.4 Imai, K., 5.3 Imamov, R. M., 2.4.1, 5.3 Immirzi, A., 8.6 Imura, T., 5.3 Inagaki, Y., 7.3 Inagami, T., 3.4 Incoccia, L., 4.2.3 Indelicato, P., 4.2.2 Ingrin, J., 9.2.2 Inkinen, O., 2.3 Inokuti, M., 4.3.4 In't Veld, G. A., 7.1.6 Inzaghi, D., 5.3 Irie, K., 5.3 Isaacs, N. W., 3.4, 9.5, 9.6 Isaacson, M., 4.3.4 Isaacson, M. S., 4.3.4 Isherwood, B. J., 5.3 Ishida, K., 4.2.6 Ishigaki, A., 2.5.1 Ishiguro, T., 4.2.3 Ishikawa, T., 2.7, 4.2.5, 4.2.6 Ishimura, T., 4.2.1, 4.2.3 Ishizawa, N., 3.4 Ishizuka, K., 4.3.8, 7.2 Isoda, S., 7.2 Isokawa, K., 5.3 Isozaki, Y., 7.1.6 Israel, H., 4.2.6 Israel, H. I., 4.2.4 Ito, M., 2.7, 7.1.6, 7.1.8 Ito, T., 9.2.2 Ivanov, A. B., 7.1.6 Iwai, S., 3.4 Iwanczyk, J., 7.1.4 Iwanczyk, J. S., 7.1.5 Iwasaki, N., 4.2.6 Izdkovskaya, T. V., 9.2.2 Jack, A., 8.3 JaÈckel, K.-H., 5.3 Jackson, D. F., 4.2.4 Jacob, M., 9.7 JacobeÂ, J., 2.4.2, 7.3 Jacobs, L., 2.7, 4.2.2 Jacobs, S., 4.2.1 Jacobson, R. A., 3.4 Jacrot, B., 2.6.2 Jagner, S., 9.2.2

AUTHOR INDEX Jagodzinski, H., 2.3, 3.4, 9.2.1, 9.2.2 Jahn, H. A., 7.4.2 Jain, P. C., 9.2.1 James, R. W., 4.2.6, 5.3, 6.3, 7.4.2 James, V. J., 7.1.6 James, W. J., 5.3 Jancarik, J., 3.1 JaÈnig, G. R., 2.6.1 Janin, J., 3.1 Janner, A., 9.8 Janot, C., 2.6.2 Jansen, J., 3.5 Jansonius, J. N., 3.1 Janssen, R. W., 4.3.4 Janssen, T., 9.8 Jap, B. K., 4.3.7, 4.3.8 Jarchow, O., 9.2.2 JaÈrvinen, M., 2.3 JaÈschke, J., 5.3 Jauch, J. M., 7.4.3 Jauch, W., 2.5.2 Jaynes, E. T., 8.2 Jeffery, J. W., 2.2, 3.1, 3.4 Jeffrey-Hay, P., 6.1.1 Jeitschko, W., 2.3 Jelinsky, P., 7.1.6 Jelley, E. E., 3.3 Jellinek, F., 9.8 Jelonek, S., 8.6 Jenkins, R., 2.3, 4.2.1, 5.2 Jennings, L. D., 4.2.3, 4.2.5, 7.4.4 Jensen, L. H., 2.2, 3.1, 3.4, 5.3 Jensen, T., 2.5.1 Jephcoat, A. P., 2.5.1 Jepsen, O., 4.3.4 Jesson, D. E., 4.3.8 Jiang, S.-S., 2.7 Johann, H. H., 2.3 Johansson, T., 2.3 Johnson, A. L., 4.3.4 Johnson, A. W. S., 4.3.8, 5.4.1, 5.4.2 Johnson, C. A., 9.2.1 Johnson, C. K., 6.1.1, 8.3 Johnson, D., 4.3.4 Johnson, D. E., 4.3.4 Johnson, D. W., 4.3.4 Johnson, G. G. Jr, 2.4.1 Johnson, J. E., 3.4 Johnson, K. H., 4.3.4 Johnson, L. N., 2.2, 3.1, 3.4 Johnson, L. R., 5.3 Johnson, M. W., 2.5.2 Johnson, O., 9.7 Johnson, R. W., 8.2 Johnson, W. R., 4.2.2 Johnston, D. F., 8.7 Johnston, J., 3.2 Jones, A., 3.4 Jones, A. R., 2.6.1 Jones, E. Y., 3.4 Jones, P. M., 4.3.6.2, 5.4.2 Jones, R. C., 7.1.6 Jones, R. M., 9.2.1 Jong, W. F. de, 6.2 Jonson, B., 4.2.2 Jorde, C., 2.6.1 Jorgensen, J. D., 2.5.2, 5.5, 8.6

Jùrgensen, J.-E., 6.4 Jostsons, A., 4.3.7 Jouffrey, B., 4.3.4 Joy, D. C., 4.2.3, 4.3.4 Jucha, A., 7.1.6 Jurnak, F., 3.1 Kaat, E. de, 5.3 Kabra, V. K., 9.2.1 Kabsch, W., 3.4 Kaesberg, P., 2.6.1 Kaesberg, P. J., 2.6.1 Kafadar, K., 8.5 Kahn, R., 2.2, 3.4, 7.1.6 Kahovec, L., 2.6.1 Kainuma, Y., 8.8 Kaiser, A., 7.1.7 Kaiser, W., 4.4.2 Kajantie, K., 7.4.3 Kakinoki, J., 9.2.1 Kakudo, M., 4.3.5 Kakuta, N., 4.3.3 Kalata, K., 7.1.6 Kalenik, J., 3.1 Kalinin, Y. G., 4.2.1 Kalman, Z. H., 2.5.1, 5.3, 9.2.1 Kalnajs, J., 5.3 Kalus, J., 2.6.1, 2.6.2 Kamada, K., 2.8 Kambe, K., 4.3.4, 4.3.7, 4.3.8 Kaminaga, U., 7.4.4 Kamino, N., 4.2.3 Kamiya, K., 7.1.8 Kamiya, N., 7.1.6 Kammerer, O. F., 4.4.2 Kampermann, S. P., 6.4 Kane, P. P., 4.2.4, 4.2.6 Kaneko, F., 9.2.2 Kantor, B., 5.3 Kaplow, R., 2.3, 6.3 Kappler, E., 2.7 Karamura, T., 6.3 Karellas, A., 7.1.6 Karen, V. L., 9.7 Kariuki, B. M., 2.3, 3.1 Karle, I. L., 4.3.3 Karle, J., 4.2.6, 4.3.3 Karlsson, R., 3.1 Karnatak, R. C., 4.3.4 Karplus, M., 3.4 Kasai, N., 4.3.5 Kasman, Ya. A., 4.4.2 Kasper, J. S., 6.2, 9.1, 9.2.1, 9.7 Katayama, C., 6.3 Kato, A., 9.2.2 Kato, H., 7.1.6, 7.1.8 Kato, N., 2.7, 6.3 Katoh, H., 4.2.6 Katoh, T., 7.2 Katsube, Y., 3.4 Kaucic, V., 3.1 Kaufmann, E. N., 4.1 Kavinki, B. M., 8.6 Kawaminami, M., 5.2, 5.3 Kawamura, T., 2.7, 4.2.5, 7.1.7 Kawamura, T. W., 7.1.6 Kawasaki, M., 4.3.8 Kawasaki, T., 4.2.1 Kawata, H., 5.3 Kay, M. I., 6.1.1 Keast, D. J., 3.5

Keeney, R. B., 4.3.4, 7.2 Keijser, Th. H. de, 2.3, 5.2 Keil, P., 4.3.4 Kellar, J. N., 2.3 Keller, H. L., 5.3 Kelley, D. M., 3.4 Kelley, M. H., 4.3.3 Kelly, A., 3.5 Kelly, E. H., 5.2 Kelly, P. M., 4.3.7 Kendall, M. G., 6.1.1 Kendrick, H., 6.2 Kennard, C. H. L., 3.4 Kennard, O., 9.5, 9.6, 9.7 Kephart, J. O., 4.2.1 Kessler, E., 4.2.2 Kessler, E. G., 4.2.2, 5.2 Kessler, E. G. Jr, 4.2.2 Kessler, J., 4.3.3 Ketkar, S. N., 4.3.3 Keve, E. T., 8.5 Khabakhshev, A. G., 7.1.6 Kharitonov, Yu. I., 4.2.4 Kheiker, D. M., 5.3, 7.1.6, 7.5 Khejker, D. M., 5.2 Khomyakov, A. P., 9.2.2 Kihara, H., 4.3.7 Kihn, Y., 4.3.4 Kikuta, S., 2.7, 4.2.5 Killat, U., 4.3.4 Killean, R. C. G., 7.5 Kim, S., 3.4 Kim, S.-H., 3.1 Kim, Y. K., 4.2.2 Kim, Y. S., 8.7 Kimura, M., 4.3.3 Kimura, T., 4.2.3 Kincaid, B. M., 2.2, 4.1, 4.2.3 Kind, R., 9.8 King, H. E. Jr, 2.3 King, H. W., 2.3, 5.2 King, J. S., 6.2 King, M. V., 3.4, 6.1.1 King, Q. A., 4.3.2, 8.8 Kirfel, A., 4.2.6 Kirisits, M. J., 3.1 Kirk, D., 5.3 Kirkham, A. J., 5.3 Kirkland, A., 4.3.8 Kirkpatrick, H., 2.3 Kirkpatrick, H. B., 3.5 Kirkpatrick, P., 4.2.1 Kirste, R. G., 2.6.1, 2.6.2 Kishimoto, S., 7.1.8 Kishino, S., 2.7, 5.3 Kisker, E., 4.3.4 Kissel, L., 4.2.4, 4.2.6 Kiszenick, W., 5.3 Kitagawa, Y., 9.2.2 Kitaigorodskii, A. I., 9.7 Kitaigorodsky, A. I., 9.1, 9.7 Kitajgorodskij, A. I., 9.7 Kitano, T., 2.7 Kittner, R., 5.3 Kjeldgaard, M., 2.6.2 Klapper, H., 1.3 Kleb, R., 2.9 Klein, A. G., 4.4.4 Klein, O., 7.4.3 Klemperer, O., 4.3.4 Kliewer, K., 4.3.4

974


Klimanek, P., 4.4.2 Kloos, G., 2.3 Kloot, N. H., 8.4 Klug, A., 4.1 Klug, H. P., 2.3, 3.4, 5.1, 6.2 Knibbeler, C. L. C. M., 7.1.6 Knipping, P., 2.1, 2.2 Knoll, W., 2.6.2 Knop, W., 2.6.2 Knop, W. E., 1.4 Knox, J. R., 3.4 Ko, T.-S., 3.4 Kobayakawa, M., 4.2.1 Kobayashi, J., 5.3 Kobayashi, K., 4.2.6, 4.3.8 Kobayashi, M., 9.2.2 Kobayashi, T., 4.3.8, 7.2 Kobayashi, Y., 4.2.6 Koch, A., 7.1.6 Koch, B., 4.2.4 Koch, E., 1.1, 1.2, 1.3, 9.1 Koch, E. E., 4.2.1 Koch, M., 4.4.2 Koch, M. H. J., 2.6.1, 2.6.2 Koch, M. J. H., 7.1.6 Koehler, W. C., 9.8 Koester, L., 4.4.4 Kogan, V. A., 5.2 Kogiso, M., 4.3.7, 5.4.2 Koh, F., 6.3 Kohl, D. A., 4.3.3 Kohl, H., 4.3.2 Kohler, H., 2.3, 7.4.2 Kohler, T. R., 7.1.4 Kohn, W., 4.2.6, 8.7 Kohra, K., 2.2, 2.7, 4.2.3, 4.2.5 Koidl, P., 4.3.4 Koike, H., 4.3.8 Kolpakov, A. V., 4.1 Komarov, F. F., 4.2.5 Komem, Y., 5.3 Komoda, T., 4.3.8 Komura, Y., 9.2.1 Konaka, S., 4.3.3 Kondrashkina, E. A., 5.3 Konnert, J. H., 3.1, 8.3 Kopfmann, G., 6.3 Kopp, M., 7.1.6 Kopp, M. K., 7.3 Koptsik, V. A., 1.4, 9.8 Kordes, E., 9.3 Korekawa, M., 9.8 Koritsanszky, T., 8.7 Korneev, D. A., 4.4.2 Korolev, V. D., 4.2.1 KorytaÂr, D., 5.3 Koshiji, N., 5.3 Kossel, W., 5.3 Kostarev, A. L., 4.2.3 Kosten, K., 3.4 Kostorz, G., 2.6.2 Kostroun, V. O., 2.7 Kosugi, N., 4.3.4 Kottke, T., 3.4 Kovalchuk, M. V., 5.3 Kovev, E. K., 5.3 Kowalczyk, R., 5.3 Kowalski, M., 9.2.2 Koyama, K., 7.1.6 Kozaki, S., 2.3, 4.2.1, 4.2.6 Kraft, S., 4.2.2

AUTHOR INDEX Krahl, D., 4.3.4, 4.3.7, 7.2 Kraimer, M. R., 7.1.6 Kramers, H. A., 4.2.1 Kranold, R., 2.6.1 Kratky, C., 3.4 Kratky, O., 2.6.1, 2.6.2 Krause, M. O., 6.3 Krauss, M., 8.7 Krec, K., 7.4.2 Kreinik, S., 7.1.7 Kretschmer, R.-G., 9.2.2 Krigbaum, W. R., 2.6.1 Krijgsman, P. C. J., 2.6.2 Krinari, G., 4.3.5 Krinary, G. A., 4.3.5 Krishna, P., 9.2.1, 9.2.2 Krivanek, O. L., 4.3.4, 4.3.7, 4.3.8, 7.2, 8.8 KroÈber, R., 2.6.1 Kroeger, K. S., 3.4 KroÈger, E. Z., 4.3.4 Kroll, N. M., 4.2.2 Kronig, R. de L., 4.2.3 Kroon, J., 3.1 Kruger, E., 4.4.2 Kruit, P., 4.3.4 Krumpolc, M., 2.6.2 Kshevetsky, S. A., 5.3 Kuban, R.-J., 9.2.2 Kubena, J., 2.9, 5.3 Kucharczyk, D., 5.3, 9.8 Kuchitsu, K., 9.2.2 Kuczkowski, R. L., 9.5, 9.6 Kudo, S., 5.3 Kudriashov, V. A., 4.4.2 Kuetgens, U., 4.2.2 Kugasov, A. G., 4.4.2 KuÈgler, F. R., 2.6.1 Kuh, E., 8.2, 8.5 KuÈhl, W., 2.7 KuÈhlbrandt, W., 4.3.7 Kuhn, K., 4.2.1 Kuhs, W. F., 6.1.1 Kujawa, S., 7.2 Kulda, J., 4.4.2 Kulenkampff, H., 4.2.1 Kulipanov, G. N., 4.2.1 Kulpe, S., 3.4 Kumachov, M. A., 4.2.1 Kumada, J., 7.1.6 Kumakov, M. A., 4.2.5 Kumaraswamy, S., 3.4 Kumosinski, T. F., 2.6.1 KuÈndig, W., 4.1 Kundrot, C. E., 3.2, 3.4 Kuntz, I. D. Jr, 3.4 Kunz, A. B., 4.3.4 Kunz, C., 4.1, 4.2.1, 4.3.4 Kunze, G., 2.3 KuÈppers, H., 5.3 KuÈppers, J., 4.1 Kupriyanov, M. F., 5.2 Kurbatov, B. A., 5.3 Kurittu, J., 7.4.2 Kuriyama, M., 2.7, 7.1.7 Kuriyama, T., 7.1.7 Kuriyan, J., 3.4 Kurki-Suonio, K., 6.1.1 Kuroda, H., 4.2.3, 4.3.4 Kuroda, K., 2.8 Kuroiwa, T., 8.8

Kurz, R., 7.3 Kusev, S. V., 4.2.1 KuÈster, A., 3.4 Kutschabsky, L., 9.2.2 Kutzler, F. W., 4.2.3 Kuyatt, C. E., 4.2.2, 4.3.4 Kuz'min, R. N., 4.1, 7.4.3 Kuznetsov, P. I., 6.1.1 Ê ., 5.3 Kvick, A Kwong, P. D., 3.2 Laan, G. van der, 4.3.4 Laban, G., 9.2.2 Labergerie, D., 4.2.5 Labzowsky, L., 4.2.2 Laclare, J. L., 4.2.1 Ladd, M. F. C., 3.1 Ladell, J., 2.3, 5.2 Ladner, J. E., 3.1 Lafferty, W. J., 9.5, 9.6 Lagasse, A., 3.5 Laggner, P., 2.6.1 Lagomarsino, S., 2.8 Laguitton, D., 5.2 LaÈhteenmaÈki, I., 2.5.1 Laine, E., 2.5.1 Lairson, B. M., 2.2 Lambert, N., 2.6.1 Lamoreaux, R. D., 7.1.6 Lampton, M., 7.1.6 Landau, L., 4.3.4 Lander, G., 4.4.1 Landre, J. K., 2.7 Lang, A. R., 2.3, 2.7, 5.3 Lang, W., 4.2.2 Lange, B. A., 3.4 Lange, G., 3.4 Langer, J. A., 2.6.2 Langford, J. I., 2.3, 5.2, 6.2, 7.1.2 Langridge, R., 2.6.1 Lanz, H. Jr, 3.2 Lapington, J. S., 7.1.6 La Placa, S. J., 8.7 LaRock, J. G., 4.4.2 Larsen, E. S. Jr, 3.3 Larsen, F. K., 3.4 Larsen, P. K., 4.3.8 Larson, A. C., 3.4, 8.7 Larson, B. C., 5.3 Lartigue, C., 4.4.2 Laue, M. von, 2.1, 2.2 Lauer, R., 4.2.2, 5.3 Laugier, J., 3.4 Laurie, V. W., 9.5, 9.6 Laves, F., 9.1 Lawrance, J. L., 2.7 Lea, K., 4.2.1 Lea, K. R., 4.2.6 Leapman, R. D., 4.2.3, 4.3.4 Lebech, B., 2.5.2, 4.3.7, 8.8 Leber, M. L., 4.3.7 Leboucher, P., 7.1.6 Lebugle, A., 4.2.2 Leciejewicz, J., 2.5.2 Leduc, M., 4.4.2 Lee, B., 6.3 Lee, J. S., 4.3.3 Lee, P., 4.2.4, 4.2.6 Lee, P. A., 4.1, 4.2.3 Lefaucheux, F., 3.1

LeGalley, D. P., 2.3 Legrand, E., 4.4.2 Lehmann, A., 5.3 Lehmann, M. S., 2.3, 3.4, 7.1.3 Lehmpfuhl, G., 4.3.7, 4.3.8, 8.8 Leifer, K., 4.4.2 Leising, G., 4.3.4 Lele, S., 9.2.1 Lemonnier, M., 2.2, 7.1.6 Lengeler, B., 4.2.3, 4.2.6 Lenglet, M., 4.2.3 Leon, R., 4.2.5 Leonardsen, E., 9.2.2 Leopold, H., 2.6.1 LePage, Y., 1.3, 5.4.1 Le Peltier, F., 4.2.3 Lerche, M., 5.3 Lereboures, B., 4.2.3 Leroux, J., 4.2.4 LeszczynÂski, M., 3.4, 5.3 Leung, P. C., 8.7 Levi, A., 7.1.4 Levine, I. L., 8.7 Levine, M. R., 6.3 Levinger, J. S., 4.2.4 Levitt, M., 8.3 Levy, H. A., 3.4, 5.3, 6.1.1, 8.3 LeÂvy, P., 6.1.1 Lewis, M. H., 3.5 Lewis, O., 4.2.1 Lewis, R., 7.1.6 Lewis, S. J., 3.4 Lewit-Bentley, A., 3.1 Ley, L., 4.2.2 Li, F. H., 4.3.8 Li, J. Q., 4.3.8 Li, Q., 4.2.1 Li, Y., 7.1.6 Liang, K. S., 2.3 Liberman, D., 4.2.4, 4.2.6 Liberman, D. A., 4.2.6 Lichte, H., 4.3.8 Lider, V. V., 5.3 Lidin, S., 9.7 Liebhafsky, H. A., 4.2.4 Liesen, D., 4.2.2 Lietzke, R., 2.6.2 Lifchitz, E., 4.3.4 Lifshitz, R., 9.8 Lim, G., 2.3 Lim, G. S., 5.2, 5.3 Lima-de-Faria, J., 9.1 Liminga, R., 5.3 Lindau, I., 4.2.4, 7.4.4 Lindegaard-Andersen, A., 2.5.1 Lindemann, R., 2.3 Linderstrom-Lang, K., 3.2 Lindgren, I., 4.2.2 Lindhard, J., 4.3.4 Lindley, P., 3.4 Lindley, P. F., 3.1, 3.2, 3.4 Lindner, P., 2.6.2 Lindner, T., 4.3.4 Lindroth, E., 4.2.2 Lindstrom, R. M., 4.4.2 Lippman, R., 3.4 Lipps, F. W., 7.4.3 Lipscomb, W. N., 6.1.1 Lipson, H., 2.2, 2.3, 5.2, 5.3, 6.2, 9.8 Lischka, K., 2.9

975


Lisher, E. J., 4.4.5 Lisoivan, V. I., 5.3 Liss, K.-D., 4.4.2 Litrenta, T., 4.4.2 Liu, H., 7.1.6 Liu, J. W., 4.3.3 Liu, L., 4.3.8, 7.2 Livesey, A. K., 8.2 Livingood, J. J., 4.3.4 Lloyd, K. H., 5.3 Lobashov, V. M., 4.4.2 LoÈchner, U., 3.4 Loeb, A., 9.1 Long, D. C., 7.1.6 Long, N., 4.3.8 Lonsdale, K., 2.2, 5.3, 6.2 Lontie, R., 2.6.1 Looijenga-Vos, A., 9.8 Loopstra, B. O., 2.4.2 Lorber, B., 3.1 Lorenz, G., 3.4 Lotsch, H., 4.2.6 Lotz, B., 3.5 LoueÈr, D., 2.3, 5.2, 8.6 Lourie, B., 2.2 Lovas, F. J., 9.5, 9.6 Love, G., 4.2.1 Lovesey, S. W., 4.1, 6.1.2, 7.4.3, 8.7 Lovey, F. C., 4.3.8 Low, B. W., 3.2 Lowde, R. D., 2.5.2, 4.4.2, 6.4 Lowe, B. M., 2.3 Lowenthal, S., 7.1.6 Lowitzsch, K., 2.3 Lowrance, J. L., 7.1.6 Lucas, W., 4.2.2, 5.3 Lucht, M., 5.3 Luft, J., 3.1 Luft, J. R., 3.1 Luger, P., 3.4, 5.3 èukaszewicz, K., 5.3 Lukehart, C. M., 2.5.2 Lum, G. K., 4.2.2 Lumb, D. H., 7.1.6 Lund, L., 4.2.1 Luo, M., 3.4 Lushchikov, V. I., 4.4.2 Lutterotti, L., 5.2 Lutts, A., 5.3 Lutts, A. H., 5.3 Lutz, H. D., 2.3 Luzzati, V., 2.6.1, 2.6.2 Lynch, D. F., 4.3.6.1, 4.3.8 Lynch, F. J., 7.3 Lynch, J., 4.2.3, 4.3.8 Lynn, J. E., 4.4.4 Lytle, F. W., 4.2.3, 4.3.4 Lyutzau, V. G., 5.3 Ma, H., 4.3.4 Ma, Y., 4.2.3, 4.3.7, 8.8 Maaskamp, H. J., 7.1.6 MacGillavry, C. H., 3.1, 4.2.4, 4.3.1, 8.8 Machado, W. G., 2.7 Machin, K. J., 3.4 Machin, P. A., 3.4 Machlan, L. A., 5.3 Mack, B., 2.4.2 Mack, B. J., 4.4.2

AUTHOR INDEX Mack, M., 2.3, 5.2, 7.5 Mackay, A. L., 1.4, 3.4, 6.2 Mackay, K. J. H., 5.3 Mackenzie, J. K., 6.1.1, 7.5 Macrae, C. F., 9.7 Madar, R., 4.4.2 Madariaga, G., 9.8 Madsen, I. C., 2.3 Maeda, H., 4.2.3 Maeder, D. L., 3.1 Magerl, A., 4.4.2 Magorrian, B. G., 2.7 Maher, D., 4.2.3 Maher, D. M., 4.3.4 Mai, Z.-H., 2.7 Maier-Leibnitz, H., 4.4.2 Main, P., 5.3 Mair, S. L., 4.2.6, 6.1.1 Maistrelli, P., 5.2 Majkrzak, C. F., 2.9, 4.4.2 Makarova, I. P., 3.1 Makepeace, A. P. W., 2.7 Maki, A. G., 9.5, 9.6 Maki, M., 5.3 Makita, Y., 4.3.8 MakovickyÂ, E., 9.2.2 Makowski, I., 3.4 Malgrange, C., 2.8, 7.3 Malik, S. S., 2.9 Malina, R. F., 7.1.6 Malinowski, A., 2.6.2 Malinowski, M., 3.4, 5.3 Mallett, J. H., 4.2.3, 4.2.4 Malmros, G., 2.3, 8.6 Malzfeldt, W., 4.2.6 Mamy, J., 4.3.5 Mana, G., 4.2.2 Mann, J. B., 4.2.4, 4.3.1, 4.3.3, 6.1.1, 7.4.3 Mannami, M., 4.3.8 Manne, R., 4.2.1 Manninen, S., 7.4.3 Manoubi, T., 4.3.4 Mantler, M., 5.2 Manzke, R., 4.3.4 Mao, H. K., 2.5.1 Marchant, J. C. J., 7.2 Marcus, M., 4.2.3 Mardin, K. V., 6.1.1 Mardix, S., 2.7 MareÂschal, J., 4.4.2 Marezio, M., 3.1 Marks, L., 4.3.8 Marmeggi, J. C., 2.5.2 Marr, G. V., 4.2.1 Marsh, D. J., 3.4 Marsh, P., 5.3 Marsh, R. E., 5.3, 8.3 Marshall, R. C., 9.2.1 Marshall, W., 4.1, 8.7 Martens, G., 4.2.3 MaÊrtensson, N., 4.2.2 Marthinsen, K., 4.3.7, 8.8 Martin, C., 7.1.6 Martinez-Carrera, S., 10 Martini, M., 4.2.3 Maruani, J., 8.7 Maruyama, E., 7.1.6 Maruyama, H., 2.7, 7.1.6, 7.1.7 Maruyama, X. K., 4.2.1 Marvin, D. A., 2.6.1

Marx, D., 4.4.2 Marxreiter, J., 3.4 Marzolf, J. G., 4.2.2, 5.3 Masaki, N., 2.8 Masciocchi, N., 2.3, 8.6 Maslen, E. N., 4.2.6, 6.1.1, 6.3 Maslen, V. M., 4.3.4 Mason, B., 3.2 Mason, I. M., 7.1.6 Massa, L., 8.7 Massalski, T. B., 2.3, 4.3.5 Massey, H. S. W., 4.3.3, 4.3.4 Masuda, K., 4.2.1 Materlik, G., 2.2, 2.7, 4.2.3, 4.2.6 Mateu, L., 2.6.2 Mathews, F. S. 3.4, 6.3 Mathias, H. G., 3.4 Mathieson, A. McL., 7.4.4 Matsuhata, H., 4.3.7, 8.8 Matsui, J., 2.7, 7.1.7 Matsumoto, T., 9.1 Matsumura, S., 8.8 Matsunaga, K., 5.3 Matsushima, I., 7.1.6 Matsushita, T., 2.2, 2.7, 4.2.3, 4.2.5, 7.1.8, 7.4.4 Matsuura, U., 9.2.2 Matthewman, J. C., 8.6 Matthews, B. W., 3.1, 3.2, 3.4 Matthews, D., 2.2 Matthews, G. D., 4.2.2 Matzfeld, W., 4.2.3 Mauer, F. A., 5.2, 5.3 Maurice, J. L., 4.3.4 Mawhorter, R. J., 4.3.3 Max, N., 3.4 May, C., 4.4.2 May, R. P., 2.6.2 May, W., 2.5.1 Mayer, J., 4.3.7, 8.8 Mayer, J. W., 4.1 Mayers, D. F., 4.3.4 Mazzarella, L., 3.1 McCallum, B., 4.3.8 McClelland, J. J., 4.3.3 McConnell, C. H., 3.5 McConnell, J. D. C., 9.8 McCourt, M. P., 4.3.8 McCrory, L. R., 4.2.1 McCusker, L., 2.3, 8.6 McDermott, G., 3.1 McKeever, B., 3.4 McKenney, A., 8.1 McKie, C., 2.2 McKie, D., 2.2 McKinstry, H. A., 3.4 McLarnan, T. J., 9.2.2 McLaughlin, P. J., 3.4 McLean, A. D., 4.3.3, 6.1.1 McLean, R. S., 4.3.3 McMahon, M., 4.2.5 McMahon, M. I., 2.3, 2.5.1 McMann, R. H., 7.1.7 McMaster, R. C., 7.1.7 McMaster, W. H., 4.2.3, 4.2.4 McMeeking, R., 2.3 McMullan, D., 4.3.4 McMullan, R. K., 6.4 McMurdie, H. F., 3.4 McNeely, J. B., 5.3

McNeill, K. M., 7.1.6 McPherson, A., 3.1, 3.4 McSweeney, S., 2.2 Meardon, B. H., 4.4.2 Medarde, M., 4.4.2 Mees, C. E. K., 7.1.1 Megaw, H. D., 5.3 Mei, R., 4.2.3 Meier, B. H., 5.5 Meier, F., 2.7 Meier, J., 4.4.4 Meier, P. F., 4.1 Meieran, E. S., 2.7 Meijer, G., 9.2.2 Meisheng, H., 4.3.8 Meister, H., 4.4.2 Melgaard, D. K., 2.4.1 Melkanov, M. A., 4.3.3 Melle, W., 5.3 Mellema, J. E., 2.6.2 Mellini, M., 9.2.2 Melmore, S., 9.1 Mendelsohn, L. B., 4.3.3, 7.4.3 Mendelssohn, M. J., 5.3 Menke, H., 2.6.1 Menotti, A. H., 4.4.2 Menter, J. W., 4.3.8 Menzel, M., 7.2 Meriel, P., 2.6.2, 9.8 MeÂring, J., 4.3.5 Merisalo, M., 2.3, 6.1.1, 7.4.2 Merlini, A. E., 5.3 Merlino, S., 9.2.2 Merritt, E. A., 7.1.6 Merritt, F. C., 7.3 Merwin, H. E., 3.3 Merz, K. M., 9.2.1 Merzbacher, E., 2.9 Mesquita, A. H. G. de, 9.2.1 Messerschmidt, A., 3.4, 7.1.6 Metcalf, P., 7.1.6 Metchnik, V., 4.2.1 Metherell, A. J. F., 4.3.4 Metoki, N., 2.9 Metzger, T. H., 4.3.7, 8.8 Meuth, H., 4.2.3 Meyer, A. J., 5.3 Meyer, C. E., 4.3.7 Meyer, G., 2.4.1 Meyer, H., 4.3.3 Meyer-Ilse, W., 7.1.6 Meyrowitz, R., 3.2, 3.3 Meysner, L., 5.3 Mezei, F., 4.4.2 Mezey, P. G., 9.7 Michalowicz, A., 2.8 Midgley, H. G., 3.2 Midgley, P. A., 4.3.7, 8.8 Mighell, A. D., 2.4.1, 5.2, 9.7 Mihama, K., 2.4.1 Mikhailyuk, I. P., 5.3 Mikhalchenko, V. P., 5.3 Mikke, K., 2.5.2 Mikol, V., 3.1 Milch, J. R., 7.1.6 Mildner, D. F. R., 4.4.2, 7.3 Miles, M. J., 2.6.1 Milledge, H. J., 4.2.4, 5.3 Miller, A., 2.6.2 Miller, B., 4.3.3 Miller, B. R., 4.3.3

976


Miller, K. J., 8.7 Miller, P. H., 4.4.2 Millhouse, A. H., 7.4.3 Million, G., 7.1.6 Mills, D. L., 4.3.4 Milne, A. D., 2.7 Minakawa, N., 2.8 Minato, I., 2.3, 3.4, 7.1.3 Miner, B. A., 4.3.4 Mingos, D. M. P., 2.3 Minkowski, M., 9.1 Misemer, D. K., 4.2.3 Mises, R. von, 6.1.1 Misselwitz, R., 2.6.1 Mitchell, E. M., 9.7 Mitchell, J. P., 7.1.7 Mitchell, P. W., 4.4.3 Mitchell, R. S., 9.2.1 Mitra, G. B., 5.2 Mitsunaga, T., 4.2.3 Mittelbach, P., 2.6.1 Mittemeijer, E. J., 2.3 Miyahara, J., 7.1.6, 7.1.8, 7.2 Miyake, S., 9.2.1 Miyamoto, M., 9.2.2 Miyata, T., 3.4 Miyoshi, A., 2.7 Mizusaki, T., 4.3.7 Mizutani, I., 5.3 Mochiki, K., 7.1.6 Modrzejewski, A., 5.3 Moelo, Y., 9.2.2 Moews, P. C., 3.4 Moffat, K., 2.2, 3.4 Mogami, K., 9.2.2 Mohr, P. J., 4.2.2, 5.3 Moliterni, A. G., 8.6 MoÈllenstedt, G., 4.3.8 Monaco, H. L., 3.1 Montenegro, E. C., 4.2.4 Moodie, A. F., 4.3.1, 4.3.6.1, 4.3.8 Moody, P. C. E., 3.4 Mook, H. A., 4.4.2, 8.7 Moon, K. J., 2.7 Mooney, P. E., 4.3.7, 4.3.8 Mooney, T., 4.2.2 Mooney, T. M., 4.2.2 Moore, J. K., 7.1.7 Moore, L. J., 5.3 Moore, M., 2.7 Moore, P. B., 2.6.1, 2.6.2 Moran, M. J., 4.2.1 Moras, D., 3.4 Morawe, C., 2.9 Morchan, V. D., 7.1.6 Moreno, A., 3.1 Moret, R., 3.4 Moretto, R., 3.4 Morgan, B. L., 7.1.6 Morgan, C. B., 7.3 Morgan, D. V., 7.1.6 Morgenroth, W., 4.2.6 Mori, H., 9.2.2 Mori, N., 4.3.7, 4.3.8, 7.2 Moriguchi, S., 7.2 Morikawa, K., 4.3.7 Moring-Claesson, O., 2.6.1 Morosin, B., 3.4 Morris, I. L., 9.1 Morris, M. C., 2.3, 3.4

AUTHOR INDEX Morris, P. L., 3.5 Morris, R. E., 2.3 Morris, V. J., 2.6.1 Morris, W. G., 5.3 Morrison, G. R., 7.2 Morse, P. M., 4.3.3 Mort, K., 4.2.3 Mortensen, K., 2.6.2 Mortier, W. J., 2.3 Mory, C., 4.3.4 Moss, G., 8.7 Mosteller, M., 8.2 Motherwell, W. D. S., 9.5, 9.6 Motohashi, H., 2.8 Mott, N. F., 4.3.4 Mott, N. I., 4.3.3 Moudy, L. A., 5.3 Moulai, J., 7.1.6 Mountain, R. W., 7.1.6 Mount®eld, M. J., 3.5 Mourou, X., 4.2.1 Moy, J.-P., 7.1.6 Moyer, N. E., 5.3 MuÈcklich, F., 4.4.2 Mughabghab, S. F., 4.4.4 Mughier, J., 3.5 MuÈller, A., 4.2.1 Muller, J., 3.4 MuÈller, J. E., 4.2.3, 4.3.4 MuÈller, J. J., 2.6.1 MuÈller, K., 2.6.1 MuÈller, U., 9.2.2 MuÈller, W., 7.1.2 MuÈller-Heinzerling, T., 4.3.4 Munshi, S. K., 3.4 Muralt, P., 9.8 Muramatsu, T., 4.3.3 Murata, T., 4.2.3 Murray, W., 8.3 Murray-Rust, P., 9.6 Murshudov, G. N., 3.4 Mursic, Z., 4.4.2 Murthy, M. R. N., 3.4 Musil, F. J., 5.2, 5.3 Mustre de Leon, J., 4.2.3 Myklebust, R. L., 7.1.4 Myles, D., 3.4 Naday, I., 7.1.6, 7.3 Naday, Y., 4.3.4 Nagakura, S., 4.3.8 Nagel, D. J., 4.2.1 Naiki, T., 4.3.8 Najmudin, S., 3.4 Nakagawa, A., 7.1.8 Nakajima, K., 5.3 Nakajima, T., 5.2 Nakamura, T., 5.3 Nakamura, Y., 4.3.8 Nakano, Y., 4.2.5, 6.3 Nakayama, K., 2.7, 4.2.2 Nanni, L. F., 5.3 Nannichi, Y., 5.3 Napier, J. G., 4.3.7 Narayana, S. V. L., 3.4 Nash, S., 8.1 Nastasi, M., 2.9 Nathans, R., 4.4.2, 4.4.3, 6.1.2 Natoli, C. R., 4.2.3 Naukkarinen, K., 2.7 Nave, C., 2.6.1, 3.4

Navrotsky, A., 3.4 Nazarenko, V. A., 4.4.2 Necker, G., 8.8 Neder, R. B., 3.4 Neilsen, C., 7.1.6 Nelder, J. A., 9.7 Nellist, P., 4.3.8 Nelmes, R. J., 2.3, 2.5.1 Nelms, A. T., 4.2.6 Nelson, J. B., 5.2 Nemiroff, M., 5.3 Nesper, R., 9.1 Nesterova, Ya. M., 9.2.2 NeubuÈser, J., 1.4, 9.8 Neuling, H. W., 2.5.1 Nevot, L., 2.9 Newbury, D. E., 7.1.4 Newkirk, J. B., 2.7 Newman, B. A., 5.3 Newsam, J. M., 2.3 Newville, M., 4.2.3 Ng, E. W., 4.3.3 Nichols, M. C., 2.3 Nicholson, J. R. S., 2.7 Nicholson, R. B., 3.5, 4.3.6.2, 4.3.8, 5.4.1 Nicholson, W. L., 7.5, 8.2, 8.4, 8.5 Nicol, J. M., 2.3 Niculescu, V., 4.4.2 Nielsen, C., 2.2, 3.4 Nielsen, M., 2.3, 4.4.3, 7.1.3 Nielson, C., 7.1.6 Nielson, M., 4.4.3 Nieman, H. F., 3.4 Niemann, W., 4.2.3, 4.2.6 Nierhaus, K. H., 2.6.2 Niggli, P., 9.1 Niimura, N., 2.5.1 Niinikoski, T. O., 2.6.2 Niklewski, T., 5.3 Nikolin, B. I., 9.2.2 Ninio, J., 2.6.1 Nishina, Y., 7.4.3 Nishiyama, Z., 9.2.1 Nissenbaum, J., 7.1.4 Nittono, O., 2.7 Nixon, W. C., 4.2.1 Nomura, K., 9.2.2 Nomura, M., 4.2.3 Nonaka, Y., 7.1.6 Nonoyama, M., 5.4.2 Nordfors, B., 4.2.3 Norman, D., 4.2.3 Normand, J.-M., 6.1.1 North, A. C. T., 3.4, 6.3 Northwood, D. O., 9.2.2 Norton, T. J., 7.1.6 Nothnagel, A., 2.6.1 NovaÂk, C., 9.2.2 Nowacki, W., 9.1, 9.7 Nowotny, V., 2.6.2 Nugent, K. A., 4.2.5 Numerov, B. V., 4.3.3 Nunes, A. C., 4.4.2 Nunez, V., 2.9, 4.4.2 Nurmela, M., 2.3, 8.6 Nyholm, R., 4.2.2 Oba, K., 2.7, 7.1.6 Obaidur, R. M., 5.3

Obashi, M., 4.2.3 OberthuÈr, R. C., 2.6.1, 2.6.2 O'Connor, B., 8.6 Ogawa, T., 4.2.6 Ogilvie, R. E., 2.3, 5.3 Oguso, C., 7.1.6 Ohama, N., 5.3 O'Hara, S., 2.7 Ohkawa, T., 4.2.6 Ohlendorf, D. H., 7.1.6 Ohlidal, I., 2.9 Ohshima, K.-I., 3.4 Ohta, T., 4.2.3 Ohtaka, K., 4.2.4 Ohtsuki, Y. H., 4.2.4 Oikawa, T., 4.3.7, 4.3.8, 7.2 Okada, Y., 5.1, 5.3 Okamoto, S., 2.9 Okamura, Y., 7.1.6 Okazaki, A., 3.4, 5.2, 5.3 O'Keefe, M. A., 4.3.1, 4.3.2, 4.3.8, 6.1.1 O'Keeffe, M., 9.1 Oki, K., 8.8 O'Mara, D., 7.1.6 Okorokov, A. I., 4.4.2 Okude, S., 7.1.6 Okuno, H., 7.3 Old®eld, T. J., 3.1 Olejnik, S., 5.3 Olekhnovich, A. I., 6.4 Olekhnovich, N. M., 6.4 Oliver, J. H., 6.3 Olkha, G. S., 7.5 Olsen, A., 4.3.7, 4.3.8, 5.4.2 Olsen, J. S., 2.5.1, 5.2, 5.3 Olthoff-MuÈnter, K., 2.7 Omote, K., 2.3 Onitsuka, H., 5.3 Oosterkamp, P. W. J., Oosterkamp, W. J., 4.2.1 Op de Beeck, M., 4.3.8 Opechowski, W., 1.4, 9.8 Oppenheimer, I., 4.2.6 Orchowski, A., 4.3.8 Orlandi, P., 9.2.2 Orpen, A. G., 9.5, 9.6 Ortale, C., 7.1.6 Ortiz, C., 2.3 Oshima, K., 4.2.5 Ostapovich, M. V., 5.3 Ostrouchov, S., 8.1 Ostrowski, G., 2.9 Otten, E. W., 4.4.2 Ottewanger, H., 7.1.6 Otto, A., 4.3.4 Otto, J. W., 2.5.1 éverbù, I., 4.2.3, 4.2.4 Ovitt, T. W., 7.1.6 Oyanagi, H., 4.2.3, 4.2.5 Ozawa, S., 4.3.8 Ozawa, T., 9.2.2 Paakkari, T., 7.4.3 Pabst, A., 9.2.2 Pace, S., 5.3 Paciorek, W. A., 7.5, 9.8 Padawer, G. ,E., 5.3 Padmaya, N., 9.7 Pahl, R., 4.2.5 Palache, Ch., 9.8

977


Palenzona, A., 9.2.2 Palmer, R. A., 3.1 Palmer, S. B., 2.8 Panchenko, J. M., 5.2 Panchenko, Yu. M., 7.5 Pandey, D., 9.2.1 Pang, G., 5.3 Pangborn, W. A., 3.1 Pannhorst, V., 2.4.1 Panson, A. Y., 4.3.4 Pantos, E., 4.2.3 Paoletti, A., 2.3, 2.4.2, 4.4.3, 8.6 Paolini, F. R., 2.3 Papatzacos, P., 4.2.3 Papiz, M. Z., 2.3, 3.4 Parak, F., 3.4 Paretzkin, B., 3.4 Parfait, J., 2.6.1 Parfait, R., 2.6.2 Parker, N. W., 4.3.4 Parmon, V. S., 2.4.1 Parratt, L. G., 2.3, 4.2.2, 4.2.3 Parrish, W., 2.3, 2.5.1, 5.2, 5.3, 7.1.2, 7.1.3, 7.1.4, 7.5, 8.6 Parsons, D. F., 4.1 PartheÂ, E., 2.3 Pasemann, M., 4.3.8 Pasero, M., 9.2.2 Pashley, D. W., 3.5, 4.3.6.2, 4.3.8, 5.4.1 Passell, L., 4.4.2 Patel, N. B., 5.3 Patel, S., 3.1 Paterson, M. S., 9.2.1 Patt, B. E., 7.1.6 Patterson, A. L., 9.2.1, 9.7 Pattison, P., 7.4.3 PaÈtzold, H., 4.3.7 Paul, R. L., 4.4.2 Pauling, L., 9.2.2, 9.3 Paulus, M., 3.5 Pavlishin, V. I., 9.2.2 Pavlov, M. Yu., 2.6.2 Pawley, G. S., 2.3, 5.2, 6.1.1, 6.3, 8.6, 8.7 Peacock, M., 9.8 Pearce-Percy, H. T., 4.3.4 Pearl, L. H., 3.1 Pearlman, S., 2.3 Pearson, G. L., 5.3 Pearson, W. B., 9.3 Pease, D. M., 4.2.3 Pedersen, J. S., 2.6.2, 2.9 Peele, A. G., 4.2.5 Peerdeman, A. F., 4.2.6 Peerdeman, F., 5.2 Peierls, R. E., 4.2.6 Peiser, H. S., 2.3, 5.1 Peisert, A., 7.1.6 Peisl, J., 7.4.2 Peixoto, E. M. A., 4.3.3 Peltonen, H., 6.1.1 Pendry, J. B., 4.2.3, 4.3.4, 4.3.6.2 Penfold, J., 2.6.2, 2.9, 4.4.2 Peng, L.-M., 4.3.1, 4.3.2, 8.8 Pennartz, P. U., 3.4 Penner-Hahn, J. E., 4.2.3, 7.1.5 Penneycook, S. J., 3.5 Pennock, G. M., 3.5 Pennycook, S. J., 4.3.8

AUTHOR INDEX Perchiazzi, N., 9.2.2 Perez, J. P., 4.3.4 Perez-Mato, J. M., 9.8 Perez-Mendez, V., 2.2 Perfettii, P., 4.2.3 Perlman, M. L., 4.2.3 Pernot, P., 7.1.6 Perrier de la Bathie, R., 2.8, 4.4.2 Perrin, F., 6.1.1 Perry, J. A., 9.1 Persson, E., 4.2.6 Persson, H., 4.2.2 Perutz, M. F., 2.6.2 Peshekhonov, V. D., 7.1.6 Pesonen, A., 2.3 Pessen, H., 2.6.1 Peterson, R. C., 3.4 Petiau, J., 4.2.3, 4.3.4 Petit, M., 7.1.6 Petri, E., 4.3.4 Petricek, V., 9.2.2, 9.8 Petroff, J. F., 2.7, 2.8 Petrova, S. N., 9.2.2 Petsko, G. A., 3.4 Pettersen, R. C., 9.5, 9.6 Petzow, G., 4.4.2 Pfeiffer, H. G., 4.2.4 P®ster, J. C., 5.3 P¯uÈger, J., 4.3.4 P¯ugrath, J., 3.4 P¯ugrath, J. W., 7.1.6 Phan, T., 9.8 Phelps, A. W., 9.2.2 Philipp, H. R., 4.3.4 Philips, J. C., 4.2.6 Phillips, D. C., 2.2, 3.4, 6.3 Phillips, F. C., 3.1 Phillips, G. N. Jr, 3.4 Phillips, W. C., 2.3, 4.2.1, 7.1.6 Phizackerley, R. P., 7.1.6 Photen, M. L., 7.1.7 Piaget, C., 7.1.6 Pick, M. A., 5.3 Pickart, S. J., 4.4.2 Pickford, M. G., 3.4 Picot, D., 3.1 Picraux, S. T., 4.1 Piermarini, G. J., 5.3 Pierron, E. D., 5.3 Piestrup, M. A., 4.2.1 Pietraszko, A., 5.3 Pihl, C. F., 5.3 Pike, E. R., 2.3, 5.2 Piltz, R. O., 2.3 Pilz, I., 2.6.1 Pimentel, C. A., 5.3 Pincus, C. I., 4.2.1 Pinot, M., 2.6.2 Pinsker, Z. G., 2.4.1, 2.7, 4.3.5, 5.3 Pirenne, M. H., 4.2.4 Pirouz, P., 4.3.8 PlancËon, A., 4.3.5 Platzman, P. M., 7.4.3 Plechaty, E. F., 4.2.4 Plies, E., 4.3.4 Plotnikov, V. P., 4.3.5 Plotz, W., 2.9 Plummer, E. W., 4.1 Podlasin, S., 3.4, 5.3

Podolsky, R. J., 7.1.6 Pofahl, E., 5.3 Polack, F., 7.1.6 PolcarovaÂ, M., 5.3 Polidori, G., 8.6 Polizzi, S., 2.3 Pollehn, H. K., 7.1.6 Polyak, M. I., 5.3 Pommerrenig, D. H., 7.1.6 Pongratz, P., 7.4.2 Ponomarev, I. Yu., 4.4.2 Poole, M. W., 4.2.1 Popa, N. C., 7.4.2, 8.6 Popov, A. N., 7.1.6 PopovicÂ, S., 5.2, 5.3 Porat, Z., 3.5 Porod, G., 2.6.1, 2.6.2 Porsev, G. D., 4.4.2 Porteus, I. O., 4.2.3 Posselt, D., 2.6.2 Post, B., 5.3 Post, J. E., 2.3 Potts, H. R., 5.3 Poulos, T. L., 7.1.6 Pound, A., 3.2 Pouxe, J., 7.1.3, 7.1.6 Powell, B. M., 3.4 Powell, C. J., 4.2.2, 4.3.4 Powell, H. R., 2.3 Powers, L. S., 4.2.3 Prager, P. R., 9.2.1 Prandl, W., 3.4 Prasad, B., 9.2.1 Prasad, L., 9.2.1 Pratapa, S., 8.6 Pratt, R. H., 4.2.4, 4.2.6, 7.4.3 Press, W., 6.1.1 Preston, A. R., 8.8 Preston, G. D., 9.8 Preston, K. D., 2.3 Prewitt, C. T., 2.3, 3.4 Price, P. F., 4.2.6 Price, W. J., 7.1.6 Prince, E., 2.3, 4.2.2, 5.1, 5.3, 7.5, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 Prince, F. C. de, 5.3 Pring, A., 9.2.2 Prins, J. A., 2.6.1 Probst, R., 4.2.2, 5.3 Prout, C. K., 2.3 Proviz, G. I., 7.1.6 Provost, K., 3.1 Przybylska, M., 3.1, 3.4 Ptitsyn, O. B., 2.6.1 Pulay, P., 4.3.3 PuÈrschel, H. V., 2.6.1 Pusey, M., 3.1 Puxley, D. C., 3.4 Pynn, R., 2.9, 4.4.2, 4.4.3 Pyrros, N. P., 2.3 Quayle, J. A., 2.7 Queisser, H.-J., 2.7 Rabe, P., 2.7, 4.2.3, 4.2.6 Rabinovich, D., 2.2, 7.5 Rabukhin, V. B., 3.2 Rachinger, W. A., 2.3 Rackham, G. M., 4.3.6.2, 5.4.2 Radeka, V., 7.3 Rademacher, H.-J., 4.2.2, 5.3

Radi, G., 8.8 Radoslovich, E. W., 9.2.2 Rae, A. D., 8.3 Rae, W. N., 3.2 Raether, H., 4.3.4 Raftery, J., 9.6 Rai, R. S., 9.2.1 Raia, C. A., 3.1 Raiko, V. I., 4.2.1 Raj, K., 4.4.2 Ralph, R. L., 8.3 Ramachandran, G. N., 2.7 Ramakrishnan, V., 2.6.2 Ramakumar, S., 9.7 Ramaseshan, S., 4.2.6, 6.1.3 Ramesh, R., 4.3.7 Ramesh, T. G., 6.1.3 Ramsay, D. A., 9.5, 9.6 Ramsdell, L. S., 9.2.1 Randall, K. H., 9.1 Ranganath, G. S., 6.1.3 Rao, C. N., 4.3.4 Rao, Ch. P., 3.4 Rao, S., 9.1 Raoux, D., 7.1.6 Rask, J. H., 4.3.4 Rasmussen, B. F., 3.4 Rasmussen, S. E., 3.4 Rathie, P. N., 7.5 Rau, W., 4.3.8 Rau, W. D., 4.3.8, 7.2 Rauch, H., 2.7, 4.4.2, 4.4.4 Ravel, B., 4.2.3 Ravn, H. L., 4.2.2 Rayment, I., 3.4 Raynal, J., 4.3.3 Read, M. H., 2.3 Read, W. T., 6.4 Reck, G., 9.2.2 Redinbo, M. R., 3.1 Reed, M., 4.2.1 Reed, R. E., 4.4.2 Reed, S. J. B., 2.3, 4.2.1 Reeke, G. N. J., 5.3 Rees, A. L. G., 2.4.1 Rees, B., 8.7 Rees, D. C., 3.1 Rehr, J. J., 4.2.3 Reichard, T. E., 5.3 Reider, M., 3.4 Reifsnider, K., 2.7 Reilly, J., 3.2 Reim, G., 4.2.2, 5.3 Reimer, L., 4.3.4, 7.2 Reinecke, T., 9.2.2 Reinhardt, A., 9.1 Rem, P. C., 4.4.2 Remaut, G., 3.5 Remenyuk, P. I., 5.3 Remington, S. J., 3.1 Ren, G., 4.3.2 Renault, A., 4.3.8 Renda, G., 7.1.6 Rendle, D. F., 2.3 Rennekamp. R., 4.3.4 Renninger, M., 2.3, 2.7, 5.3 Resouche, E., 4.4.2 Reverchon, F., 3.5 Reynolds, G. T., 7.1.6 Reynolds, R. A., 3.1 Reynolds, R. C., 2.3

978


Rez, D., 4.3.1, 4.3.2 Rez, P., 4.3.1, 4.3.2, 4.3.4 Rhan, H., 5.3 Ribberfors, R., 7.4.3 Ricci, F. P., 2.3, 2.4.2, 4.4.3, 8.6 Rice, S. B., 4.3.8 Rice-Evans, P., 7.1.6 Richard, J. C., 7.1.6 Richards, F. M., 3.2, 3.4 Richardson, J. W., 8.6 Richardson, M. C., 4.2.1 Richardson, M. C. M., 4.2.5 Richter, D., 4.4.2 Ricker, G. R., 7.1.6 Ridou, C., 5.3 Rieck, C. D., 4.2.2 Riekel, C., 3.1 Ries-Kautt, M., 3.1 Rietveld, H. M., 2.3, 2.4.2, 5.2, 8.2, 8.3, 8.6 Rieubland, J. M., 2.6.2 Rieubland, M., 2.6.2 Riggs, P. J., 4.2.3 Riglet, P., 2.7 Rigoult, J., 6.3 Rijlart, A., 2.6.2 Rijllart, A., 2.6.2 Riley, D. P., 5.2 Rindby, A., 4.2.5 Ringe, D., 3.4 Ringe-Ponzi, D., 3.4 Rink, W. J., 3.4 Rinn, H. W., 2.3 Ripp, R., 3.4 Riquet, J. P., 3.4 Risler, J. L., 3.4 Riste, T., 4.4.2 Ritchie, R. H., 4.3.4 Ritland, H. N., 2.6.1 Ritsko, J. J., 4.3.4 Ritter, R., 4.4.2 Rizkallah, P. J., 3.1 Robert, M. C., 3.1 Roberts, K. J., 2.7, 3.4 Roberts, L. D., 4.4.2 Roberts, P.-H., 6.1.1 Robertson, B. E., 5.3 Robin, J., 4.2.1 Robinson, R. A., 4.4.3 Roche, C. T., 7.3 Rode, A. V., 4.2.5 Rodeau, J.-L., 3.1 Rodenburg, J., 4.3.8 Rodgers, J. R., 9.3, 9.5, 9.6, 9.7 Rodricks, B., 7.1.6 Rodrigues, A. R. D., 2.3, 2.7, 4.2.5 Rodriguez, S., 5.3 Roe, A. L., 7.1.5 Roe, S. M., 3.4 Roehrig, H., 7.1.6 Roetti, C., 4.4.5, 6.1.1, 6.1.2 Rogers, D. W., 3.4 Rogers, J. E., 8.1 Rogerson, I. F., 5.2 Rohe, D., 4.4.2 Rohrer, H., 4.3.8 Rohrlich, F., 7.4.3 Rollett, J. S., 5.3 Rùmming, C., 4.3.7, 8.8

AUTHOR INDEX Rooksby, H. P., 2.3, 5.1 Rooms, G., 4.4.2 Roos, B., 6.1.1 Roppert, J., 2.6.1 Rose, A., 7.1.6, 7.1.7 Rose, H., 4.3.4 Rosier, D. J. de, 4.1 Rosner, B., 5.3 Ross, A. W., 4.3.3 Ross, M., 9.2.2 Ross, P. A., 2.3 Rossbach, M., 4.4.2 Rossi, F. A., 3.4 Rossi, J., 7.1.7 Rossi, J. P., 7.1.7 Rossi, M., 3.1 Rossmanith, E., 5.3 Rossmann, M. G., 2.2, 3.4, 5.3 Rossouw, C. J., 4.3.4 Rotella, F. J., 2.5.2, 5.5 Rouault, M., 4.3.3 Roubeau, A., 2.6.2 Roudaut, E., 2.4.2, 7.3 Rourke, C. P., 6.3 Rouse, K. D., 6.3, 7.4.2 Rousseau, M., 5.3 Rousseaux, F., 2.2, 4.3.5 Roux, M., 4.3.3 Rowe, J. M., 4.4.2, 4.4.3 Rowlands, P. C., 5.3 Roy, S. C., 4.2.4, 4.2.6 Rozgonyi, G. A., 7.1.7 Rozhanski, V. N., 4.3.8 Rozhansky, V. H., 5.3 Rozhansky, V. N., 5.3 Rubin, H., 2.3 Rubinstein, I., 3.5 Ruble, J. R., 6.3 Ruckpaul, K., 2.6.1 Rudakov, L. I., 4.2.1 Rudman, R., 3.4 Rule, D. W., 4.2.1 Rullhausen, P., 4.2.4, 4.2.6 Rumpf, A., 2.7 Runov, V. V., 4.4.2 Ruoff, A. L., 2.5.1 Rush, J. J., 8.3 Russ, J. C., 7.1.4 Russell, D. R., 9.6 Russell, T. P., 2.9 Rustichelli, F., 2.8, 4.4.2 RuÈter, H. D., 5.3 Rzotkiewicz, S., 8.7 Sabine, T. M., 6.4 Sabino, E., 2.3, 5.2 Sadanaga, R., 9.2.2 Sadaoui, N., 3.1 Sadler, D. M., 2.6.1 Sagar, R. P., 4.3.3 Sagerman, G., 3.4 Sah, R. E., 4.3.4 Saito, M., 4.2.3 Saitoh, K., 7.2 Sakabe, K., 6.3 Sakabe, N., 2.2, 4.2.5, 6.3, 7.1.6, 7.1.8 Sakai, H., 2.7, 7.1.6, 7.1.7 Sakamaki, T., 3.4 Sakashita, H., 5.3, 9.2.2 Sakata, M., 7.4.2, 8.6

Sakurai, H., 4.2.1 Sakurai, K., 4.2.1 Salcido, M. M., 7.1.6 Saldin, D. K., 4.3.4, 8.8 Salemne, F. R., 7.1.6 Saloman, E. B., 4.2.3, 4.2.4 Salomonson, S., 4.2.2 Salva-Ghilarducci, A., 2.6.2 Samotoin, N. D., 4.3.5 Samsonov, G. V., 9.3 Sander, B., 5.3 SandstroÈm, A. E., 5.2 Sanford, P. W., 7.1.6 Sankey, O. F., 4.3.4 Sano, H., 4.2.4 Santiard, J. C., 2.2, 7.1.6 Santilli, V. J., 7.1.6 Santoro, A., 6.3 Sapirstein, J., 4.2.2 Sareen, R. A., 7.1.6 Sarikaya, M., 4.3.7 Sarma, R., 3.4 Sasaki, A., 4.2.3 Sasaki, H., 4.3.3 SasvaÂri, J., 5.3 Sato, F., 2.7, 7.1.6, 7.1.7 Sato, M., 3.4 Sato, S., 4.2.6 Sato-Sorensen, Y., 3.4 Satow, Y., 7.1.6, 7.1.8 Sauder, W. C., 4.2.2 Sauer, H., 4.3.4 Sauli, F., 2.2 Saunders, M., 4.3.7, 8.8 Sauvage, M., 2.7, 2.8 Savage, H. F. J., 3.4 Savinov, G. A., 7.1.6 Savitzky, A., 2.3 Sawada, M., 4.2.1, 4.2.3 Sawada, T., 4.3.3 Sawatsky, G. A., 4.3.4 Saxton, W., 4.3.8 Saxton, W. O., 4.3.8 Sayers, D. E., 4.2.3, 4.3.4 Sazaki, T., 4.3.4 Sbitnev, V. I., 4.4.2 Scarborough, G. A., 3.1 Scardi, P., 5.2, 8.6 Scaringe, R. P., 9.7 Schaefer, W., 7.3 Schaerpf, O., 4.4.2 SchaÈfer, L., 4.3.3 Schalt, W., 4.4.2 SchaÈrpf, O., 2.6.2, 4.4.2 Schattschneider, P., 4.3.4 Schauer, P., 7.2 Schaupp, D., 4.2.4, 4.2.6, 7.4.3 Schearer, L. D., 4.4.2 Schebetov, A. F., 4.4.2 Scheckenhofer, H., 2.9 Schedler, E., 4.4.2 Schedrin, B. M., 2.6.1 Scheer, J. W., 7.1.6 Scheerer, B., 4.3.4 Scheetz, B. E., 2.3 Schefer, J., 4.4.2 Scheinfein, M., 4.3.4 Schellenberger, U., 5.3 Schelten, J., 2.6.1, 2.6.2, 4.4.2, 7.3 Schenk, M., 5.3

Scheraga, H. A., 9.7 Scheringer, C., 6.1.1 Scherm, R., 2.8, 4.4.2 Scherm, R. H., 4.4.2 Scherrer, P., 2.3 Scherzer, O., 4.3.8 Schetelich, Ch., 5.3 Schick, B., 3.1 Schieber, M., 7.1.4 Schikora, D., 5.3 Schildkamp, W., 2.2 Schiller, C., 3.4 Schindler, D. G., 2.6.2 Schink, H.-J., 2.6.2 Schirber, J. E., 3.4 Schirmer, A., 4.4.2 Schiske, P., 4.3.8 Schlenker, M., 2.8, 7.3 Schlenoff, J. B., 3.4 Schlesier, B., 3.1 Schmalle, H. W., 9.2.2 Schmatz, W., 4.4.2 Schmetzer, K., 2.3 Schmider, H., 4.3.3 Schmidt, H., 5.3 Schmidt, K., 2.6.2 Schmidt, L., 4.2.1 Schmidt, V. V., 4.2.3 Schnabel, R. B., 8.1 Schnatterly, S. E., 4.3.4 Schneider, D. K., 2.6.2 Schneider, G., 9.2.2 Schneider, J., 3.4, 5.3 Schneider, J. R., 2.5.2, 4.1, 7.4.3 Schnering, H. G. von, 9.1 Schnopper, H. W., 4.2.3 Schoenborn, B. P., 2.6.2, 4.4.2, 6.1.3 Schomacher, V., 4.3.3 Schomaker, V., 4.3.3, 8.3 Schoonover, R. M., 5.3 Schrauber, H., 2.6.1, 9.2.2 Schreiner, W., 2.3, 5.2 Schreiner, W. N., 2.3, 5.2 Schrey, F., 5.3 Schreyer, A., 2.9 SchroÈder, B., 4.3.4 SchroÈder, W., 2.2, 2.7 Schroeder, L. W., 6.3 Schubert, P., 3.4 Schultz, A. J., 2.5.2, 7.3 Schultz, H., 2.3 Schulz, H., 3.4, 5.3, 6.1.1, 7.4.2, 9.2.2 Schulz, L. G., 2.7 Schulze, G. E. R., 3.2, 5.3 Schulze, H., 2.6.2 Schumacher, M., 4.2.4, 4.2.6, 7.4.3 Schurz, J., 2.6.1 Schuster, M., 4.2.6 Schutt, C. E., 2.2 Schwager, P., 3.4, 6.3 Schwahn, D., 2.6.2 Schwartz, L. S., 2.3 Schwartzenberger, D. R., 5.3 Schwarz, H. E., 7.1.6 Schwarz, W., 9.2.2 Schwarzenbach, D., 4.2.2, 5.3, 8.1, 9.3 Schweig, A., 4.3.3

979


Schweizer, J., 4.4.2, 8.7 Schwendemann, R. H., 9.5, 9.6 Schweppe, J., 4.2.2 Schweppe, J. E., 4.2.2 Schwinger, J., 4.2.1 Schwitz, W., 4.2.2 Schwuttke, G. H., 2.7, 5.3 Sco®eld, J., 4.2.6 Sco®eld, J. H., 4.2.3, 4.2.4 Scott, C. P., 7.2 Scott, H. G., 8.6 Scott, V. D., 4.2.1 Scuderi, J., 3.4 Sears, V. F., 2.9, 4.2.3, 4.4.2, 4.4.4 Seary, A. J., 4.2.3 Sebastian, M. T., 9.2.1 SeÂbilleau, F., 2.3 Secrest, D., 6.3 Sedlacek, P., 9.2.2 Seebold, R. E., 2.3 Seeds, W. E., 2.6.1 Seegar, P. A., 4.4.4 Seeger, A., 4.4.2 Seeger, P. A., 7.1.6 Seemann, H., 2.3 Segall, R. L., 9.2.2 SegmuÈller, A., 2.3, 5.3 Seip, H. M., 4.3.3 Seip, R., 4.3.3 Seka, W., 4.2.1 Seki, R., 4.2.2 Sekiguchi, A., 7.1.6 Sekiguchi, A. A., 7.1.6 Self, P. G., 4.3.8 Selsmark, B., 2.5.1 Semiletov, S. A., 2.4.1 Senemaud, C., 4.2.1, 4.2.2 Senzaki, K., 4.2.3 Serdyuk, I. N., 2.6.2 Serebrov, A. P., 4.4.2 Serughetti, J., 3.5 Servidori, M., 5.3 Sette, F., 4.3.4 Sevely, J., 4.3.4 Sevillano, E., 4.2.3 Seyfried, P., 4.2.2, 5.3 Seymann, E., 4.4.4 Shaham, H., 3.4 Sham, L. S., 4.2.6 Shankland, K., 4.3.7, 8.6 Shannon, C. E., 2.6.1 Shapiro, F. L., 4.4.2 Shapiro, S. M., 4.4.2 Sharov, V. A., 4.4.2 Shaw Stewart, P. D., 3.1 Shaw, A., 3.1 Shechtman, D., 9.8 Shekhtman, V. Sh., 5.3 Sheldon, J., 3.4 Sheldrick, G. M., 5.3 Shelud'ko, S. A., 5.3 Shenoy, G. K., 4.1 Shephard, G. C., 9.1 Sherman, I. S., 4.3.4, 7.1.6 Sherwood, J. N., 2.7, 2.8 Sheu, E. Y., 2.6.1, 2.6.2 Shibata, S., 4.3.3 Shidara, K., 2.7, 7.1.6, 7.1.7 Shields, W. R., 5.3 Shiles, E., 4.3.4

AUTHOR INDEX Shimakura, H., 4.2.6 Shimambukuro, R. L., 4.2.4, 4.2.6 Shindo, D., 4.3.8 Shinoda, G., 5.3 Shiraiwa, T., 4.2.3 Shiraiwa, Y., 4.2.1 Shirane, G., 4.4.2, 4.4.3, 6.1.2 Shirley, R., 8.6 Shishiguchi, S., 2.3, 7.1.3 Shmueli, U., 7.5 Shmytko, I. M., 4.2.6, 5.3 Shoemaker, D. P., 7.5, 8.4 Shoji, T., 2.3 Shore, J. E., 8.2 Shortley, G. H., 8.7 Shrier, A., 5.3 Shubnikov, A. V., 9.1 Shulakov, E. V., 4.2.6, 5.3 Shull, C. G., 2.8, 6.1.2, 6.1.3 Shulman, R. G., 4.2.3 Shulman, S., 2.6.1 Shuman, H., 4.3.4 Shuvalov, B. N., 7.1.6 Shvarts, D., 4.2.1 Shvyd'ko, Yu. V., 5.3 Sica, F., 3.1 Siddons, D. P., 2.3, 2.7, 4.2.6 Siddons, P., 4.2.6 Sidorenko, O. V., 4.3.5, 9.2.2 Sidorov, V. A., 7.1.6 Sieber, J., 8.6 Siegbahn, M., 4.2.1 Siegbahn, P., 6.1.1 Siegel, R. W., 4.1 Siegert, H., 4.2.2, 5.3 Siegmund, O. H. W., 7.1.6 Siegmund, W., 4.2.5 Siemensmeyer, K., 4.4.2 Sievers, R., 9.4, 9.5, 9.6 Silcox, J., 4.3.4 Sillers, I.-Y., 2.6.2 Sillou, D., 2.8 Silzer, R. M., 4.2.1 Simmons, R. O., 5.3 Simms, R. A., 7.1.6 Simon, J. P., 2.6.2 Simons, A. L., 4.2.3 Simpson, J. A., 4.3.4 Simpson, K., 2.6.2 Simpson, W. T., 6.1.1 Sinclair, H. B., 5.2 Sinfelt, J. H., 4.2.3 Singh, G., 9.2.1 Singh, K., 5.3 Singh, S. R., 9.2.1 Sinha, S. K., 2.9 Sinogowitz, U., 9.1 Sirianni, A. F., 2.3 Sironi, A., 8.6 Sirota, E. B., 2.9 Sirota, M. I., 2.4.1 Sivia, D. S., 8.6 Skalicky, P., 7.4.2 Skellam, J. G., 7.5 Skilling, J., 8.2 Skopik, D. M., 4.2.1 Skowronek, M., 7.1.6 Skupov, V. D., 5.3 Skuratowski, I. Ya., 4.2.1 Slack, G. A., 9.1

Slade, J. J., 5.3 Slade, J. J. Jr, 5.3 Slater, J. C., 7.4.3 Sleight, A. W., 2.3 Sleight, J., 4.2.1 Slingsby, S., 3.4 SÏljukicÂ, M., 5.3 Sloane, N. J. A., 9.1 Sluis, P. van der, 3.1 Smaalen, S. van, 9.2.2, 9.8 Smakula, A., 5.3 Smend, F., 4.2.4, 4.2.6, 7.4.3 Smirnov, V. V., 4.3.8 Smirnova, N. L., 9.1 Smith, A. J., 9.7 Smith, D., 4.3.3 Smith, D. G. W., 2.3 Smith, D. J., 3.5, 4.3.7, 4.3.8 Smith, D. K., 2.3, 9.2.2 Smith, D. T., 3.4 Smith, D. Y., 4.2.6, 4.3.4 Smith, G., 5.3, 7.1.6 Smith, G. F. H., 9.8 Smith, G. S., 2.3, 2.9 Smith, H., 8.1, 8.4 Smith, J. M., 9.7 Smith, J. V., 9.2.2 Smith, S. T., 2.3 Smith, S. Y., 4.3.4 Smith, T. B., 4.4.2 Smith, V. H., 4.3.3 Smith, V. H. Jr, 4.3.3, 6.1.1 Smits, D. W., 5.2 Snavely, M. K., 4.1 Snell, E., 2.2 Snigirev, A., 4.2.5 Snigireva, I., 4.2.5 Snyder, D., 4.2.1 Snyder, R. L., 2.3 Soares, D. A. W., 5.3 Soboleva, S. V., 4.3.5, 9.2.2 SoÈchtig, J., 4.4.2 Soejima, Y., 3.4, 5.3 Sofer, A., 8.1 Soff, G., 4.2.2 Soller, W., 2.3, 5.2 Somiya, S., 2.3 Somlyo, A. P., 4.3.4 Sommers, H. S., 4.4.2 Sonada, M., 7.1.6 Sonneveld, E. J., 2.3, 5.2 Sonoda, M., 7.1.8 Sorenson, D., 8.1 Sorenson, L. B., 4.2.3 Sorenson, L. O., 4.2.6 Soriano, T. M. B., 3.1 Sorokin, N. D., 9.2.2 Soures, J. M., 4.2.1 Sowa, H., 9.1 Spackman, M. A., 8.7 Spangfort, M. D., 3.1 Spargo, A. E. C., 4.3.7, 4.3.8 Sparks, C. J., 5.2, 7.4.3 Sparks, R. A., 2.3, 3.4 Sparrow, T. G., 4.3.4 Spear, W. E., 4.2.1 Spehr, R., 4.3.4 Speier, W., 4.3.4 Spence, A. J., 4.3.8 Spence, J. C. H., 4.3.4, 4.3.7, 4.3.8, 8.8

Spencer, R. C., 5.2 Spiegelman, C. H., 8.4, 8.5 Spielberg, N., 2.3, 5.2, 7.5 Spooner, F. J., 5.3 Springer, T., 2.6.2, 4.4.2 Sprong, G. J. M., 5.2 Squire, G. D., 3.4 Srinivasan, K., 2.5.2 SteÎpienÂ, J. A., 5.3 SteÎpienÂ-Damm, J., 5.3 SteÎpienÂ-Damm, J. A., 5.3 Stalick, J. K., 2.3, 2.4.1, 5.1, 5.2 Stalke, D., 3.4 Stanglmeier, F., 4.2.6 Stanley, H. B., 2.9 Stanton, M., 7.1.6 Stasiecki, P., 2.6.1 Statile, J. L., 7.1.7 Staub, U., 4.4.2 Stearns, D. G., 4.2.6 Stedman, R., 4.4.3 Steeds, J. W., 4.3.6.2, 4.3.7, 5.4.2, 8.8 Steeple, H., 2.3 Stegun, I. A., 6.1.1, 6.3, 7.5 Steichele, E., 2.5.2, 4.4.2 Steigemann, W., 3.4 Steinberger, I. T., 9.2.1 Steinberger, J., 4.4.2 Steiner, M., 4.4.2 Steiner, W., 7.4.2 Steinhauser, K. A., 2.9 Steinmeyer, P. A., 2.3 Steinsvoll, O., 4.4.2, 4.4.3 Stemple, N. R., 4.2.6 Stephens, M. A., 6.1.1 Stephens, P. W., 4.2.5, 8.6 Stephenson, S. T., 4.2.1 Stern, E. A., 4.2.1, 4.2.3, 4.3.4 Stern, R. A., 7.1.6 Steryl, A., 2.9 Stetsko, Yu. P., 5.3 Steurer, W., 9.8 Stevels, A. L. N., 2.7 Stevens, E. D., 7.4.2, 8.7 Stevenson, A. W., 7.4.2 Stevenson, M. L., 4.2.3 Stewart, G. W., 8.1 Stewart, R. F., 4.3.3, 6.1.1, 8.7 Stibius-Jensen, M., 4.2.3, 4.2.6 Stiefel, E., 8.3 Stipcich, S., 4.2.3 Stirling, W. G., 4.4.2 Stobbs, W. M., 4.3.8 Stock, A. M., 3.4 Stock, S. R., 4.2.3 Stohr, J., 4.2.3, 4.3.4 Stùlevik, V., 4.3.3 Storm, A. R., 4.2.3 Storm, E., 4.2.4, 4.2.6 Stott, A. M. B., Stout, C. D., 3.1 Stout, G. H., 2.2, 3.1, 3.4, 5.3 Stout, J. H., 9.2.2 Strack, R., 4.2.5 Stragier, H., 4.2.3 Stratonovich, R. L., 6.1.1 Straumanis, M., 5.3 Straumanis, M. E., 2.3, 5.3 Strauss, M. G., 4.3.4, 7.1.6, 7.3 Strauss, S., 7.4.3

980


Strinskii, A. N., 4.2.1 Strobel, P., 3.1 Stroud, A. H., 6.3 Stuart, A., 3.1, 3.3, 6.1.1 Stuart, D., 6.3 Stuart, D. I., 3.4 Stubbings, S. J., 3.4 Stuckey Kauffman, D., 5.3 Stuesser, N., 4.4.2 Stuhrmann, H., 4.2.1 Stuhrmann, H. B., 2.6.1, 2.6.2 StuÈmpel, J., 4.2.2 StuÈmpel, J. W., 7.1.6 Stura, E. A., 3.1 Sturhahn, W., 5.3 Sturkey, L., 2.4.1 Sturm, K., 4.3.4 Sturm, M., 4.2.6 Su, L. S., 4.3.3 Su, Z., 8.7 Suck, D., 3.4 Sudol, J., 7.3 Suehiro, S., 7.1.6 Sueno, S., 3.4 Suh, I.-H., 3.4 Suh, J.-M., 3.4 Sukharev, Y., 4.3.7 Suller, V. P., 4.2.1 Sullivan, J. D., 3.2 Sumner, I., 7.1.6 Suortti, P., 2.3, 4.2.1, 4.2.4, 4.2.6, 7.4.2, 7.4.4, 8.6 Surin, B. P., 3.1 Surkau, R., 4.4.2 Suski, T., 3.4, 5.3 Sussieck-Fornefeld, C., 2.3 Sussini, J., 4.2.5 Sutter, J., 5.3 Sutton, L. E., 9.5, 9.6 Suzuki, M., 9.2.2 Suzuki, S., 2.7 Suzuki, T., 4.3.8 Suzuki, Y., 7.1.6 Svensson, C., 5.3 Svensson, L. A., 3.1 Svergun, D. I., 2.6.1 Swann, P. R., 3.5, 4.3.4 Swanson, D. K., 3.4 Swanson, H. E., 5.2, 5.3 Swanton, D. J., 8.7 Swapp, S. M., 3.4 Sweet, R. M., 3.1, 7.1.6 Swoboda, M., 4.3.7 Swyt, C. R., 4.3.4 Sygusch, J., 8.3 Syromyatnikov, F. V., 3.2 SzaboÂ, P., 7.5 Szarras, S., 2.5.1 Szmid, Z., 2.5.1, 5.3 SzymanÂski, J. T., 9.2.2 Tabbernor, M. A., 4.3.1, 4.3.2, 4.3.7, 6.1.1, 8.8 Taft, E. A., 4.3.4 Taftù, J., 4.3.4, 4.3.7, 8.8 Tairov, Yu. M., 9.2.2 Takahashi, H., 4.3.7 Takahashi, K., 7.1.6, 7.1.8 Takahata, T., 9.2.2 Takama, T., 4.2.6 Takano, M., 7.1.6, 7.1.8

AUTHOR INDEX Takano, Y., 5.3 Takayanagi, K., 4.3.8 Takeda, H., 9.2.2 Takeda, T., 4.2.3, 4.4.2 TakeÂuchi, Y., 9.2.2 Tanaka, H., 7.1.8 Tanaka, M., 4.2.2, 4.3.7, 7.1.6, 8.8 Tanaka, N., 3.4 Tanaka, T. J., 4.2.4, 4.2.6 Tanemura, M., 9.1 Tanimoto, M., 7.1.6 Tanioka, K., 7.1.6 Tanisaki, S., 9.8 Tanner, B. K., 2.7, 4.1, 5.3 Tanoue, H., 4.2.3 Tao, K., 2.3 Taran, Yu. V., 4.4.2 Tardieu, A., 2.6.2 Tarling, S. E., 3.4 Tasset, F., 4.4.2 Tate, M. W., 2.7 Taub, H., 7.3 Taupin, D., 2.3 Tavard, C., 4.3.3 Taxer, K., 9.2.2 Taylor, A., 2.3, 4.2.1, 5.2, 9.2.1 Taylor, B. N., 4.2.1, 4.2.2, 4.2.3, 5.3 Taylor, H. F. W., 9.2.2 Taylor, J., 2.3, 5.2 Taylor, J. C., 8.6 Taylor, P. R., 6.1.1 Taylor, R., 9.5, 9.6 Tazzari, S., 4.2.1, 4.2.6 Tchoubar, C., 4.3.5 Tchoubar, D., 4.3.5 Teatum, E. T., 9.3 Teeter, M. M., 3.4 Teller, E., 9.2.1 Teller, R. G., 2.5.2 Tello, M. J., 9.8 Templer, R. H., 7.1.6 Templeton, D. H., 4.2.3, 4.2.6, 6.3 Templeton, L. K., 4.2.3, 4.2.6, 6.3 Tence, M., 4.3.4 Teng, T. Y., 3.4 Tennevin, J., 2.7 Teo, B. K., 4.2.3, 4.3.4 Terada, N., 4.2.3 Terasaki, D., 5.2 Terasaki, O., 2.5.1, 4.3.7, 5.2 Terhell, J. C. J. M., 9.2.1 Terminasov, Yu. S., 5.3 Termonia, Y., 2.3 Teuchert, W., 4.4.2 Thakkar, A. J., 4.3.3, 6.1.1 Thaller, C., 3.1 Thatcher, D. R., 3.1 Theisen, R., 4.2.4 Theobald-Dietrich, A., 3.1 Thiel, D. J., 4.2.5 Thierry, J. C., 3.4 Thiessen, M. J., 3.1 Thinh, T. P., 4.2.4 Thole, B. T., 4.3.4 Thomas, D. J., 7.1.6 Thomas, G., 3.5, 5.4.2, 7.1.6 Thomas, J. M., 4.3.4

Thomas, J. O., 8.6 Thomas, L. H., 4.2.6, 7.4.3, 8.7 Thomas, P., 7.3 Thomas, R. K., 2.9 Thomlinson, W., 2.3, 7.4.4, 8.6 Thompson, A. B., 9.2.2 Thompson, A. W., 2.2, 3.4 Thompson, D. J., 4.2.1 Thompson, J. B, 9.2.2 Thompson, P., 2.3 Thompson, T. E., 3.2 Thomsen, J. S., 4.2.2, 5.2, 5.3, 7.5 Thuesen, G., 4.2.3, 4.2.6 Thut, R., 4.4.2 Tiao, G. C., 8.1, 8.2 Tibballs, J. E., 6.3 Tighe, N. J., 3.5 Tikhonov, A. N., 2.6.1 Tikhonov, V. I., 6.1.1 Tilton, R. F., 3.4 Tilton, R. F. Jr, 3.4 Timasheff, S. N., 2.2, 2.6.1 Timchenko, T. I., 9.2.2 Timmers, J., 5.2 Timmins, P. A., 2.6.2 Tindle, G. L., 4.4.3 Tipson, R. S., 3.1 Tissen, J. T. W. M., 3.1 Tivol, W. F., 4.3.8 Tixier, R., 5.3 Toby, B. H., 2.3 Tode, G. E., 3.4 Toellner, T. S., 5.3 To®eld, B. C., 8.7 Tohji, K., 4.2.1 Tokonami, M., 9.2.1, 9.2.2 Tokumaru, Y., 5.1 Tokumoto, M., 4.2.3 Tolhoek, H. A., 7.4.3 Toman, K., 5.2 Tomaszewski, P. E., 5.3, 9.2.2 Tomimitsu, H., 2.8 Tomita, T., 9.2.1 Tomkeieff, M. V., 5.3 Tomlin, S. G., 4.2.1 Tomokiyo, Y., 4.3.7, 8.8 Tomonaga, N., 5.3 Tonomura, A., 4.3.8 Toorn, P. van, 7.2 Toraya, H., 2.3, 5.2 Tossel, J. A., 4.3.4 Tournarie, M., 2.3 Town, W. G., 9.5, 9.6 Toyoshima, N., 3.4 Trail, J., 4.2.1 Trammell, G. T., 6.1.2 Trautmann, N., 4.4.2 Travennier, M., 5.2 Travis, D. J., 7.1.6 Treacy, M. M. J., 4.3.8, 9.1 Trebbia, P., 4.3.4 Tremayne, M., 8.6 Treverton, J. A., 3.5 Trewhella, J., 2.6.1 Trigunayat, G. C., 5.3, 9.2.1 Tripathi, A. N., 4.3.3 Troitsky, V. I., 2.9 Trost, A., 2.3 Trueblood, K. N., 2.2, 8.3 Trzhaskovskaya, M. B., 4.2.4

Tse, T., 4.2.5 Tseng, H. K., 7.4.3 Tsernoglou, D., 3.4 Tsu, Y., 4.2.3 Tsuda, K., 4.3.7, 8.8 Tsuji, M., 4.3.8 Tsukimura, K., 3.4 Tsuno, K., 4.3.8 Tsutsumi, K., 4.2.3 Tsvetkov, V. F., 9.2.2 Tsypursky, S. I., 2.4.1 Tu, H. Y., 3.1 Tubbenhauer, G. A., 7.1.6 Tucker, P., 8.2 Tucker, T. N., 5.3 Tugulea, M. N., 7.4.3 Tuinstra, F., 3.4, 5.2, 9.8 Tukey, J. W., 8.2 Tulkki, J., 7.4.3 Tung, M., 3.1 Tuomi, T., 2.7 Turber®eld, K. C., 2.5.2 Turchin, V. F., 4.4.2 Turkenburg, J. P., 3.4 Turner, J. N., 4.3.8 Turner, P. S., 2.4.1, 4.2.4, 4.3.1, 4.3.2, 6.1.1 Tutton, A. E., 3.2 Tuzov, L. V., 5.3 Tzafaras, N., 3.4 Uchiyama, K., 7.1.6 Udagawa, Y., 4.2.1, 4.2.3 Uehling, E. A., 4.2.2 Ueno, Y., 7.1.8 Ugarte, D., 4.3.4 Ullrich, H.-J., 5.3 Ullrich, J. B., 4.4.2 Umanskii, M. M., 7.5 Umanskij, M. M., 5.2 Umansky, M. M., 5.3 Umeno, M., 5.3 Umezawa, K., 4.2.6 Unangst, D., 5.3 Uno, R., 2.5.1 Unonius, L., 2.3, 8.6 Unwin, P. N. T., 4.3.7, 4.3.8 Urbanowicz, E., 5.3 Ursell, H. D., 6.1.1 Usami, K., 7.1.6 Usha, R., 3.4 Uspeckaya, G. I., 5.3 Usuda, K., 5.3 Utlaut, M., 4.3.4 Uyeda, N., 4.3.8 Uyeda, R., 2.7, 4.3.7, 5.4.2, 8.8 Vacquier, V. D., 3.1 Vainshtein, B. K., 2.2, 2.4.1, 4.3.5 Vajda, I., 2.3 Valentine, R. C., 7.2 Valvoda, V., 4.1 Van Bommel, A. J., 4.2.6 Van Dyck, D., 4.3.8 Van Landuyt, J., 4.3.8 Van Mellaert, L., 2.7 Vanoni, F., 4.4.2 Vansteelandt, L., 4.4.2 Van Tendeloo, G., 4.3.8, 9.2.1, 9.2.2

981


Varghese, J. N., 4.2.5 Varnum, C. M., 4.4.2 Vaughan, D. J., 4.3.4 Veeraraghavan, V. G., 2.3 Veigele, W. J., 4.2.4, 4.2.6, 7.4.3 Veillard, A., 6.1.1 Venghaus, H., 4.3.4 Vercillo, R., 7.1.6 Vergamini, P. J., 3.4 Verheijen, M. A., 9.2.2 Verin, I. A., 3.1 Verma, A. J., 9.2.2 Verma, A. R., 9.2.1 Vernon, W., 2.2, 7.1.6 Vettier, C., 4.2.5 Veysseyre, R., 9.8 Via, G. H., 4.2.3 Victoreen, J. A., 4.2.4 Villain, F., 4.2.3 Villain, J., 9.8 Villars, P., 9.3 Vincent, M. G., 6.3 Vincent, R., 4.3.7, 8.8 Vincze, L., 4.2.5 Vineyard, G. H., 2.3 Visser, J. W., 2.3, 5.2, 9.8 Viswamitra, M. A., 9.7 Vittone, E., 4.2.2 Vittot, M., 7.1.6 Vogels, A. B. P., 2.3 Vogt, T., 4.4.2 Voigt, W., 7.4.2 Voigt-Martin, I. G., 4.3.7 Vollath, D., 4.2.4 Volz, K., 2.2 Von Dreele, R. B., 6.4, 8.6 Von Festenberg, C., 4.3.4 Voronin, L. A., 2.6.1 Vos, A., 7.4.2 Voss, R., 4.3.7, 4.3.8 Vossers, H., 3.4 Vriend, G., 3.4 Vrublevskaya, Z. V., 4.3.5, 9.2.2 Vucht, J. H. N. van, 9.3 Vvedensky, D. D., 4.3.4 Waarzak, I., 3.1 Waber, J. T., 4.2.4, 4.2.6, 4.3.1, 6.1.1, 9.3 WacheÂ, C., 5.3 Wachtel, E., 2.6.2 Wada, N., 4.2.3 Waddington, W. G., 4.3.8 Wade, R. H., 4.3.8 Wagenfeld, H., 4.2.6, 6.3 Wagner, C. N. J., 2.3 Wagner, R., 2.6.2 Wagner, V., 4.4.2 Wagshul, M. E., 4.4.2 Wait, E., 3.4 Wakabayashi, K., 7.1.8 Wakita, H., 4.2.3 Walder, V., 5.3 Walker, A. R., 5.4.2 Walker, G. A., 2.3 Walker, N., 6.3 Wall, J., 4.3.8 Wall, M. E., 2.7 Wallace, C. A., 2.7, 5.3 Waller, I., 4.2.6, 7.4.3

AUTHOR INDEX Walls, M. G., 4.3.4 Walter, G., 2.6.1 Walters, K., 4.4.2 Walton, D., 7.1.6 Wang, D. N., 4.3.7 Wang, J., 4.3.3 Wang, M. S., 4.2.6 Wang, S. Q., 8.8 Warble, C. E., 4.3.8 Warburton, W. K., 4.2.3, 7.1.5 Ward, K. B., 3.1 Ward, R. C., 7.4.2 Ward, R. C. C., 2.7 Ware, N. G., 2.3 Warren, B. E., 2.3, 4.2.5, 4.3.5 Warrington, D. H., 1.3 Waschkowski, W., 4.4.4 Waser, J., 6.2, 8.3 Washburn, J., 3.5 WasÂkowska, A., 5.3 Wassermann, G., 2.3 Watanabe, D., 4.3.7, 8.8 Watanabe, H., 4.3.7, 8.8 Watanabe, T., 4.2.3, 6.1.1 Watenpaugh, K. D., 3.4 Watkin, D. J., 2.3 Watson, D. G., 9.5, 9.6, 9.7 Watson, D. L., 2.7 Watson, K. J., 6.1.1 Watson, L. M., 4.2.2 Weaver, L. H., 3.1 Weaver, W., 2.6.1 Weber, H., 7.4.2 Weber, H.-P., 2.3 Weber, K., 6.3 Weber, P. C., 3.1 Weber, W., 4.2.6 Weckerman, B., 4.4.2 Weertman, J., 9.2.1 Weertman, J. R., 9.2.1 Wehenkel, C., 4.3.4 Weibel, E., 4.3.8 Weickenmeier, A., 4.3.2 Weickenmeier, A. L., 4.3.7, 8.8 Weigel, D., 9.8 Weik, H., 5.3 Weill, F., 9.2.2 Weill, G., 2.5.1 Weininger, M. S., 3.4 Weinstock, B., 4.3.3 Weisenberger, P., 4.2.3 Weisgerber, S., 2.2 Weiss, R. J., 6.3, 7.4.3 Weiss, Z., 9.2.2 Weissenberg, K., 2.2 Weissmann, S., 5.3 Weisz, O., 5.3 Welberry, T. R., 3.4 Wellenstein, H., 4.3.3 Wells, A. A., 7.1.6 Wells, A. F., 9.1, 9.2.1 Wells, M., 4.4.5 Welsch, R. E., 8.1, 8.2, 8.5 Welsh, R. C. J., 4.4.2 Weng, X. D., 4.3.4 Wenk, H. R., 4.3.8 Wennemer, M., 9.2.2 Wenskus, R., 7.4.3 Wenzl, H., 5.3 Werner, K., 4.4.2 Werner, P. E., 2.3, 8.6

Werner, S., 2.8 Werner, S. A., 4.4.3, 4.4.4, 6.2, 6.4, 7.5 Wertheim, G., 2.3 Wery, J. P., 3.4 West, D. R. F., 3.5 West, J. M., 3.5 West, K., 2.3 Westbrook, E. M., 3.2, 7.1.6 Westbrook, M. L., 7.1.6 Whaling, W., 7.3 Whatmore, R. W., 2.7 Whelan, M. J., 3.5, 4.3.2, 4.3.6.2, 4.3.8, 5.4.1 White, E. T., 3.2 White, T. J., 9.2.2 Whit®eld, H., 4.3.7 Whitney, D. R., 5.2, 5.3 Whittaker, E. J. W., 7.1.1 Whittemore, W. L., 4.4.2 Wichmann, E. H., 4.2.2 Wick, G. C., 4.4.2 Wicks, B. J., 3.5 Widom, J., 7.1.6 Wiedmann, L., 4.2.1 Wiegand, C. E., 4.2.2 Wien, W., 4.3.4 Wiesler, D. G., 2.9 WiewioÂra, A., 9.2.2 Wiewiorosky, J., 2.3 Wignall, G. D., 2.6.2 Wikinson, A. P., 8.6 Wiles, D. B., 2.3, 8.6 Wiley, D. C., 7.1.6 Wilhelm, T., 7.1.6 Wilkens, M., 5.2 Wilker, C. N., 4.3.4 Wilkins, J. W., 4.3.4 Wilkins, M. H. F., 2.6.1 Wilkins, S. W., 4.2.5, 6.1.1, 6.4 Wilkinson, A. P., 2.3 Wilkinson, C., 2.2 Wilkinson, D. H., 7.1.6 Wilkinson, M. K., 9.8 Will, G., 2.3, 5.2, 5.3, 7.3, 8.6 Wille, H.-C., 5.3 Wille, P., 4.4.2 Williams W. G., 4.4.2 Williams, B. G., 4.3.4, 7.4.3 Williams, D. B., 4.3.8 Williams, D. E. G., 9.7 Williams, E. J., 7.5, 8.4 Williams, G. P., 7.4.4 Williams, J. C., 3.5 Williams, J. M., 2.5.2 Williams, R., 3.4 Williams, W. G., 4.4.2 Williamson, G. K., 3.5 Willis, B. T. M., 2.2, 2.3, 2.5.2, 3.6, 4.4.6, 5.3, 5.5, 6.1.1, 6.1.3, 6.2, 7.4.2, 8.6, 8.7 Willoughby, A. F. W., 5.3 Willson, P. D., 2.3 Wilson, A. J. C., 1.4, 2.3, 2.4.2, 2.5.1, 3.3, 4.2.2, 4.3.5, 5.1, 5.2, 5.3, 6.3, 6.4, 7.5, 8.1, 8.2, 9.2.1, 9.2.2, 9.7 Wilson, A. R., 4.3.8 Wilson, C. G., 5.3 Wilson, E., 3.1 Wilson, H. R., 2.6.1

Wilson, K., 7.5 Wilson, R. J. F., 7.1.6 Wilson, R. R., 4.2.1 Wilson, S. A., 2.8 Winchell, P. G., 2.3 Windisch, D., 5.3 Windsor, C. G., 2.5.2, 4.1, 4.4.6, 7.3, 8.6 Winick, H., 4.2.1, 4.2.3 Winick, M., 7.4.3 Winkler, F. K., 2.2 Winslow, E. H., 4.2.4 Wippler, C., 2.6.2 Witters, R., 2.6.1 Wittmann, H. G., 3.4 Wittmann, J. C., 3.5 Wittono, G., 4.2.6 Witz, J., 2.2, 2.6.2 Wlodawer, A., 2.2, 6.3 Woicik, J. C., 4.2.3 Wokulska, K., 4.2.2, 5.3 Woøcyrz, M., 5.3 Wolf, B. de, 2.6.2 Wolf, J., 4.2.2, 5.3 Wolf, R. S., 7.3 WoÈlfel, E. R., 2.3, 5.3, 7.1.3, 7.1.6 Wolff, P. M. de, 1.4, 2.3, 7.1.1, 9.2.2, 9.8 Wollan, E. O., 9.8 Wolpert, R. L., 8.1 Wolstenholme, J. F. R., 5.2 Wonacott, A. J., 2.2, 3.4, 7.1.6 Wondratschek, H., 1.4, 9.8 Wones, D. R., 9.2.2 Wong, T. C., 4.3.3 Wong-Ng, W., 3.4, 5.2 Wood, G. J., 4.3.8 Wood, I. G., 3.4 Wood, R. A., 3.4 Woodruff, D. P., 4.2.3 Woodward, J. B., 4.2.6 Woolfson, M. M., 2.2, 5.3 Wooster, W. A., 2.2, 5.3, 7.4.2 Worcester, D. L., 2.6.1 Worgan, J. S., 7.1.5 Worlton, T. G., 2.5.2 Worthmann, W., 2.6.1, 2.6.2 Wright, A. F., 3.4, 4.4.2 Wright, D. J., 2.6.1 Wright, E. M., 7.5 Wright, M. M., 8.3 Wroblewski, T., 3.4 Wroe, H., 4.4.2 Wu, C. C., 2.7 Wu, D. Q., 7.1.6 Wu, Y., 5.2 Wu, Y.-Q., 4.2.1 Wulff, P., 3.2 Wunderlich, J. A., 3.2 Wurmbach, P., 2.6.2 Wyckoff, H. W., 2.2, 3.4 Xiao, Q. F., 4.4.2 Xie, J., 7.1.6 Xie, S.-D., 4.3.3 Xuong, Ng. H., 2.2, 3.4, 7.1.6 Yaakobi, B., 4.2.1 Yabuki, S., 2.6.2

982


Yagi, K., 4.3.8 Yakovlev, V. A., 7.1.6 Yakowitz, H., 5.3 Yamada, N., 5.3 Yamada, S., 2.9 Yamagishi, H., 4.3.7 Yamaguchi, M., 2.7, 7.1.6 Yamaguchi, T., 4.2.3 Yamamoto, A., 9.8 Yamamoto, M., 3.4 Yamanaka, T., 9.2.2 Yamashita, T., 7.1.7 Yamazaki, H., 3.4 Yan, D. H., 4.3.7 Yano, Y., 7.1.6 Yao, T., 4.2.1, 4.2.3 Yap, F. Y., 4.2.2, 5.2, 7.5 Yap, Y., 5.3 Yasuami, S., 5.3 Yates, A. C., 4.3.3 Yeates, T. O., 3.1 Yeh, J. J., 4.2.4 Yin, Y., 3.1 Yocum, C. F., 4.2.3 Yoder, H. S., 9.2.2 Yonath, A., 3.4 York, E. J., 6.3 Yoshida, N., 4.2.3 Yoshimatsu, M., 2.3, 4.2.1 Yoshimura, M., 2.3 Yoshioka, H., 8.8 Yoshioka, Y., 7.1.6 Young, A. C. M., 3.4 Young, H. D., 4.4.4 Young, R. A., 2.3, 4.2.5, 5.2, 6.3, 8.6 Yvon, K., 2.3 Zaanen, J., 4.3.4 Zabel, H., 2.9 Zabidarov, E. I., 4.4.2 Zaccai, G., 2.6.2 Zach, J., 4.3.8 Zachariasen, W. H., 4.2.6, 4.4.2, 6.3, 6.4 Zagari, A., 3.1 Zagofsky, A., 2.3 Zahorowski, W., 5.3 Zakharov, N. D., 4.3.8 Zaloga, G., 3.4 Zaluzec, N. J., 4.3.4, 8.8 Zanchi, G., 4.3.4 Zanevskii, Yu. V., 7.1.6 Zani, A., 5.3 Zarka, A., 2.8 Zassenhaus, H., 1.4, 9.8 Zeedijk, H. B., 3.5 Zeidler, T., 2.9 Zeisler, R., 4.4.2 Zeissler, C. J., 4.4.2 Zeitler, E., 3.5, 4.3.4, 7.2 Zemany, P. D., 4.2.4 Zemlin, F., 3.5 Zeppenfeld, K., 4.3.4 Zeppezauer, E. S., 3.4 Zeppezauer, M., 3.4 Zerby, C. D., 4.2.6 Zernicke, F., 2.6.1 Zevin, L. S., 5.2, 5.3, 7.5 Zeyen, C. M. E., 4.4.2 Zha, C. S., 2.5.1

AUTHOR INDEX Zhang, X.-J., 3.4 Zhang, Y., 2.3, 5.2 Zhao, Z. X., 4.3.8 Zhdanov, G. S., 4.1, 9.2.1 Zhou, X.-L., 2.9 Zhoukhlistov, A. P., 9.2.2 Zhu, J., 4.3.4 Zhukhlistov, A. P., 4.3.5, 9.2.2 Zimmermann, J., 4.2.1

Zimmermann, S., 4.3.4 Zinke, M., 2.6.1 Zipper, P., 2.6.1 Zirwer, D., 2.6.1 Zittlau, W., 4.3.3 Zobel, D., 3.4 Zobetz, E., 9.1 Zocco, T. G., 2.9 Zolensky, M. E., 2.3 Zolliker, P., 5.5

Zolotoyabko, E., 5.3 Zoltai, T., 9.2.2 Zorkaya, O. N., 9.7 Zorkii, P. M., 9.2.2 Zorky, P. M., 9.7 Zosi, G., 4.2.2, 5.3 Zou, X. D., 4.3.7 Zsoldos, EÃ., 5.3 Zubenko, V. V., 5.3 Zucchino, P., 7.1.6

983


Zucker, U. H., 6.1.1 ZunÄiga, F. J., 9.8 Zuo, J., 8.8 Zuo, J. M., 4.3.7, 4.3.8, 5.4.1, 8.8 ZuÊra, J., 5.3 Zurek, S., 3.4 Zvyagin, B. B., 3.5, 4.3.5, 9.2.2 Zwoll, K., 5.3

SUBJECT INDEX

Subject Index Abbe refractometer, 160 Abbe theory, 420 Abelian module, 937 Aberrations (see also Systematic errors) centroid displacements, 494 coefficients, 426 geometrical, 46, 83, 86, 493 in powder diffractometry, 46, 48, 50 line-profile breadths, 494 of an energy-dispersive diffractometer, 497 physical, 46, 85, 86, 493, 494 refraction, 492 transparency, 49 Absolute calibration of SANS data, 108 Absolute intensity in SANS, 108 Absolute measurements, 505 of lattice spacings, 505, 526, 529±533 Absorbed dose, definition of, 958 Absorption, 599, 609, 653 air, 73 anomalous, 416 coefficients, 213, 218 coefficients for Bloch waves, 735 coefficients for neutrons, 461 coefficients, linear, 599 coefficients, mass, 600 cross sections, macroscopic, 461 edges, 191, 202, 205, 206, 209, 599 edges, wavelengths of, 205±211 effects, 261 efficiency, 623 factor, 51 function, 261 in XED, 86 length, 188 minimization by suitable mounting of single crystals, 163 of generated X-rays in target, 191 photoelectric, 599 systematic error, 528 systematic error, elimination, 521±524, 528±529 X-ray, 599±608 Absorption corrections, 170, 600±608 neutron diffraction, 177 Accelerating voltage fluctuations, 424 of a transmission electron microscope, determination, 539 Accessible range of d's, 38 Accuracy, 490, 492, 707 factors determining, 501 Accuracy of lattice-parameter (lattice-spacing) determination, 505, 507, 526, 533±536 evaluation, 534 (methods of) increasing, 532±536 relative, 505 Acoustic modes, 653, 654 Activity, definition of, 958 Adequate protection, definition of, 958 Adhesives for mounting specimens, 163 Aggregation effects in SANS, 107 Air absorption, 73 Air and window transmission, 73 Air scattering, 74, 665 ALCHEMI (atom location by channelling enhanced microanalysis), 411 Alignment and angular calibration, 46

Aluminium dielectric coefficients, 402 effective number density, 411 film, 393 Ambiguities in modulated structure notation, 936 Amorphous material, diffraction from, 24 Analyser, 530 perfect-crystal, 665 Analysis of charge density, 713±734 Analysis of spin density, 713±734 Analytical extrapolation of lattice parameters, 493 Anatase, high-energy resolution spectra, 408 Anger camera for neutrons, 650 Angle definition, use of peak or centroid for, 63 Angle-dispersive diffractometry, 491, 495±496 Angle-reading error, 524 Angle-setting error, 524 Angles between crystal blocks, determination, 516 Angles in direct and reciprocal space, 4 Angles of reciprocal cell, determination, 517 Angular distribution of reflections in Laue diffraction, 29 Angular momentum, 727 orbital, 731 Angular setting errors (precession), 35 Angular-velocity factors, 596 Anharmonicity, 585, 722 Anisotropic mosaic crystals, 432 Anisotropic temperature factors, 697 Anisotropic thermal diffuse scattering correction, 654 Anode current/voltage relationship in electrochemical thinning, 175 Anomalous absorption, 416 Anomalous dispersion (scattering) (see also Dispersion), 21, 188, 241, 733 not anomalous, 241 Anomalous transmission, 116 Anti-equi-inclination setting, 31 Antiferromagnetic order, 728 Antiferromagnetism, 140 Antiferromagnets, 728 Antimorphism, 897 Antiscatter slits, 45 Aperiodic lattice, 921, 928, 937 Approximations Born, 591 Born±Oppenheimer, 713, 722, 723 commensurate, 909 convolution, 723 crystal-field, 729 dipolar, 731 first Born, 389 Glauber, 391 harmonic, 723 Hartree±Fock, 732 impulse, 657 kinematical, 260 LCAO, 723 Moliere high-energy, 260 no-upper-layer-line, 415 phase-grating, 260 projected charge-density, 423

984

c:\itfc\sindex.3d

Approximations quasi-Gaussian, 590 two-beam, 260 weak-phase-object, 423 Archimedes method for density measurement, 158 Area-detector diffractometry, 36, 170 Area detectors, geometric effects, 41 non-uniformity of response, 41 television, 630 Arithmetic crystal classes, 15, 897, 898, 911, 917, 939, 945 (3+1)-dimensional, 917 as classification of space groups, 15 classification by size, 20 derivation of, 15 four-dimensional, 15 notation for, 15 one-dimensional, 15, 16 three-dimensional, 15±20 two-dimensional, 15, 16 uses of, 15 Arithmetic equivalence, 911, 939 Arithmetic point groups, 914 Arithmetically equivalent point groups, 939 Arrangements giving partial reduction of systematic errors, 515, 514, 521±523, 526, 528±531 Associated Legendre polynomials, 581 Astigmatism, 421, 424 Asymmetric Bragg reflections, 526 Asymmetric (Straumanis) film mounting, 509 Asymmetry factor, 118 of peaks, 67 Asymptotic behaviour of SANS curves, 110 Atom-centred expansion, 729 Atom-centred models, 714 Atom-centred spherical harmonic approximation, 714 Atom location by channelling enhanced microanalysis (ALCHEMI), 411 Atomic beams, 189 Atomic dipole moment, 716 Atomic environment types, 776 Atomic form factor, 242 Atomic orbital basis, 722 Atomic quadrupole moment, 717 Atomic scattering factors, 188, 242, 259 analytical approximation for (tables), 578± 581 for electrons (tables), 263±281 free atoms (tables), 555±564 generalized, 565 ions (tables), 566±577 Atomic volumes, 774 Attenuation coefficients, 213, 230, 600 Auger shifts, 204, 205 Automation, computer-controlled, 63 Avalanche multiplication, 619, 634 Avalanche production, 626 Average structure, 913 Avogadro constant, determination of, 534 Axial divergence, 46, 50, 53, 494, 497 Axial-divergence error, 494, 523 correction for, 523 Axial holography, 427

SUBJECT INDEX Axial lengths, determination of, 532 Axial reflections, 517 Back reflection, 512±515 Backgammon ( jeu de jacquet) counter, 627 Background, 68, 661 in SANS, 108, 109 Background counting rates, 667 Background radiation, definition of, 958 Backlash in diffractometer drives, 47, 503, 667 Balanced filters, 74, 78, 79, 238 Bandwidth, 197 Basic polytypes, 763, 766, 767 Basic structural features, 745±944 Basic structure, 909 Basis conventional, 944 crystallographic, conventional, 3 crystallographic, non-primitive, 3 crystallographic, primitive, 2 primitive reciprocal, 2 standard, 944 vectors, 944 Bayerite family, 766 Bayes's theorem, 681 Beam centring, 45 Beam conditions, 120 Beam divergence, 45, 425, 498 Beam-splitting crystal, 531 Beam tilt (see also Misalignment), 524 Becquerel, definition of, 958 Bending magnets, 198 Bent crystals, 77 Berg±Barrett method, 114 Beryllium, cross section for neutrons, 439 Beryllium acetate, 663 Bessel function, 589, 666 spherical, 460, 581, 592 Best linear unbiased estimator, 680 Best overall fit, 493 Beta function, 703 Bethe approximation, 736 Bethe ridge, 411 Bethe theory for inelastic scattering, 406±408 Bias, 689, 707, 709 of midpoint of a chord, 520 of peak, 520 Bijvoet-pair intensity ratios, 251 Bijvoet-pair techniques, 251 Binding effects, 391 Biological macromolecules, SANS, 105 Birefringence, 153 of polytypes, 757 Black-body radiation in X-ray region, 198±199 Blackman curve, 81 Blind region, 34 Bloch standing waves, 411 Bloch-wave method, 259, 415±416, 426, 735 Block collimation, 99 Block polytypism, 760, 766 Boltzmann statistics, 726 Bond angles, 698 Bond lengths, 698, 813 Bond method, 498, 507, 508, 522±526, 529, 531, 534, 535±536 for small spherical crystals, 525 in multiple-crystal spectrometers, 529±531 systematic errors, 523±524 Bond-system diffractometers, 522, 524 Bonding electrons, distribution of, 425

Bonds, classification of, 791, 813 Bonse±Hart camera, 100 Bonse±Hart interferometer, 121 Born approximation, 591 first, 389 Born±Oppenheimer approximation, 713, 722, 723 Born series, 259 Borrmann effect, 113, 116, 600 Borrmann triangle, 116 Bound nuclear scattering lengths, 593 Boundaries, low-angle, 114 Bragg angle, 187 accuracy of, 505±506, 516 determination, 506, 519, 521 errors, 491, 494 from a diffraction profile, 519±521 from a photograph, 519 from a two-dimensional map of intensity, 522 measurement of, 505, 518 operational definitions, 491 Bragg±Brentano (Parrish) angle-dispersive diffractometers, 44, 495, 664 Bragg cut-off, 438 Bragg law, 505 Bragg optics, 432 Bragg reflection, 3, 432 magnetic, 591 Bravais classes, 910, 913, 940, 945 (2+l)-dimensional, 915 (2+2)-dimensional, 916 (3+l)-dimensional, 917±918 one-line symbols, 915, 920 two-line symbols, 915, 920 Bravais lattice, 3, 15, 913 Brazil twins, 11 Bremsstrahlung, 37, 191 for XED, 84 Brilliance, synchrotron radiation, 197 Brillouin zone, 657 Broadening function, 710 Brownian diffusion, 589 Broyden±Fletcher±Goldfarb±Shano update, 684 Cadmium iodide, 754, 756 Cadmium telluride detector, 623 Calculated powder patterns, 60 Calculation of the twin element, 14 Cambridge Structural Database, 790, 812 Camera methods for lattice-parameter determination, 491, 497 Camera radius extremely large, 510 uncertainty, elimination, 510 Camera tubes high-resolution TV, 633 lead oxide, 634 Cameras back-reflection, 71 Bonse±Hart, 100 cylindrical, 70 Debye±Scherrer, 42, 70 ellipsoidal mirror in SANS, 106 flat-film, 71 for recording lattice-parameter changes, 510 Gandolfi, 71 Guinier focusing, 44, 68, 70 Kossel, 512 Kratky, 99

985

c:\itfc\sindex.3d

Cameras mirror, 106 miscellaneous, 70 pinhole, in SANS, 106 pinhole, in SAXS, 100 powder, 69±71 small-angle, 99 systems for synchrotron radiation, 100 Capillary tubes for mounting specimens, 162 Carcinogenesis, 960 Cast films, 176 Castaing & Henry filter, 397 Categories of OD structures, 764 Cauchy curves, 67 Cauchy distribution, 689 Causality, principle of, 246 CBED (convergent-beam electron diffraction), 416, 540, 735 CBED disc, 417 Cell dimensions, incorrect assignment, 170 Cellulose film containers, 162, 163 Central-limit theorem, 702 Centre of gravity (centroid), 518 additivity, 518 variance, 518 Centred lattices, 3 Centred unit cells, 3 Centring conditions, 921 Centring lattice vectors, 3 Centring reflection conditions, 921 for (3+1)-dimensional Bravais classes, 935 Centroid of a reflection, 492 Centroid of wavelength distribution, 494 Ceramics, preparation of specimens, 171 Cerenkov radiation, 401 Cerium oxide (intensity standard), 500, 503 Channel-cut monochromators, 77, 121 Channelling, 189 Characteristic function, 90 Characteristic line spectrum, 191, 202 Characteristic radiation, efficiency of production, 192 Characteristic X-rays, excitation of, 510 Characterization of detectors, 639 Charge, 187 Charge-cloud model, 715 Charge-coupled devices, 629 Charge densities, analysis of, 713±734 Chemical analysis, 154 etchants for thin section preparation, 173 etching, 173 polishing, 174 properties, 154 thinning, 175 Chi-squared (2 ) distributions, 702, 703 Chiral volumes, 700 Chlorite group, 765, 769 Chlorite±vermiculite group, 769±770 Choice of reflections, 535 Cholesky decomposition, 681, 685 Cholesky factor, 678, 681, 694, 708 Choppers, 443 Chromatic aberration constant, 423, 424 Chromium oxide (intensity standard), 500, 503 Circle packings, 746±747, 752 Classification of bonds, 791, 813 of experimental techniques, 24 of space groups, 15 Cleavage, 153

SUBJECT INDEX Close-packed structures, 752, 761, 897 interstices in, 753 lattices possible, 755 notations for, 753±754, 756 polytypes, 754±756 space groups possible, 755 spheres, 747, 752 stacking faults in, 758±760 structure determination of, 756±758 symmetry of layers, 753 symmetry of stacking, 755 voids in, 753 Cobalt martensites, stacking faults in, 758 Coherent inelastic scattering, 177 Coherent multiple scattering, 661 Coherent (Rayleigh) scattering, 554 Coherent scattering cross sections, 594 Coherent scattering lengths, 594 Cohesive energy, 721 Coincidence operations, 761 Cold neutrons, 105 Collimation, 37 block, 99 in-plane, 522 of neutrons, 105, 431 systematic errors connected with, 523±524 Collimators misalignment (tilt), 523±524 misalignment (tilt), error, 524 Soller, 82, 443 Collinear structures, 591 Colour groups, 21 Column approximation, 414 Combined aberrations, 50 Combined methods, spectrometers for, 531 Commensurate approximation, 909 Commensurate modulated structures, 907±944 Comparison measurements of lattice parameters, 508 Compensating transformations, 940 Compensating translations, 940 Composite crystal structures, 907, 941 Composition surfaces, 10 Compton scattering, 90, 213, 242, 554, 599, 657±661, 663, 713 non-relativistic approximations, 657±659 relativistic treatment, 659±660 Compton shift formula, 657 Compton wavelength, 260 Computer-controlled automation, 63 Computer graphics for powder patterns, 69 Computer programs CRYSTIN, 778 data processing, 596 Computer simulation in estimation of error, 536 Computing methods for electron diffraction, 425 Concentration effects, 97 elimination of, 98 Condensed models, 766 Condition number, 678, 682 Conditional probability density function, 679 Conditional Q±Q plot, 708 Conditioning, 684 Cone-axis photography, 35 Confidence level, 64 Conic section, 515 Conical surface of an hkl reflection, 510 Conjugate-gradient methods, 686

Conservation laws, 657 Constrained models, 693 Constraints in refinement, 693, 693±701 Contact number, 747, 749 Continuous spectrum, 192 Contrast diffraction, 113, 735 extinction, 113 first-fringe, 116 match-point, 107 orientation, 113 variation in SANS, 107 variation in SAXS, 97 variation, inverse, 108 variation, spin, 108 Conventional basis, 3, 945 Conventional unit cell, 913 Conventional X-ray sources, 37 Convergent-beam electron diffraction (CBED), 80, 416, 417, 540, 735 Convolution, 66, 505, 518, 534 Convolution equations, 67 Convolution of rocking curves, 663 Convolution range, 66 Convolution square-root technique, 103 Coordination complexes, typical interatomic distances, 812±896 Coordination number, 774 Core-electron spectroscopy, 404 Core-loss spectroscopy, 404 Correction factor for absorption and extinction, 612 for powders, 657 Correction of systematic error, 653 Correlated and uncorrelated mosaic blocks, 610 Correlation coefficients, 724 Correlation energy, 391 Correlation function, 90 Correlation length, 93 Correlations between recorded intensities, 519 Corundum etching, 173 intensity standard, 500, 503 Coulombic self-electronic energy, 721 Counters backgammon ( jeu de jacquet), 627 gas-filled, 626 Geiger±MuÈller, 522 parallel-plate, 627 Counting losses, 625 Counting modes, 666 Counting rates, 666±668 background, 667 erratic fluctuations, 666 reflection only, 666 total, 666 Counting statistics, 64, 666±668 Critical-voltage effect, 416, 736 Critical wavelength, 196 Cross sections differential scattering, 260 dispersion corrections, 221 elastic differential scattering, 262 ionization, 407 of a rod-like particle, 93 PDDF of, 102, plasmon, 399 scattering and absorption, 439, 444 Cryoprotectants, 166,

986

c:\itfc\sindex.3d

Crystal(s) analysers, 56 datum orientation, 33 definition of, 908 displacively modulated, 909 edges, 3 ideal, 908 ideally imperfect, 113 ideally perfect, 113 intergrowth, 941 misalignment (tilt), 424 misalignment (tilt), error, 524 modulated, 908 monochromators, 76, 662 mosaicity, 170 normal, 908 orientation matrix, 33 real, 419 reflecting power, 590 rocking curves, 34, 37, 40 rocking widths, 33 selection, 148, 151 slippage within capillary, 165 systems, 6 Crystal classes arithmetic, 15, 911, 945 geometric, 15, 911, 913 Crystal-field approximation, 729 Crystal-lattice vector and crystal setting, 168 Crystal profile, 505 Crystal-size analysis, 81 Crystal structure determination by HREM, 419 images, 422 Crystal systems cubic, 9, 19 hexagonal, 7, 15, 19 monoclinic, 6, 16 oblique, 15 orthorhombic, 6, 16 rectangular, 15 rhombohedral, 8 square, 15 tetragonal, 7, 17 triclinic, 6 trigonal, 7, 18 Crystal thickness determination by electron diffraction, 416, 419 in transmission geometry, 512, 513 Crystalline solids, 259 Crystallite-size effects, 62 Crystallization, 148 Crystallographic system, 940 Cubic closest packing, 747 Cubic crystal system, 9, 19 Cubic harmonics, 585 Cumulant expansion, 588 Cumulative distribution function, 679 Current density, 725 Current ionization position-sensitive detectors, 628 Curvature, lattice, 114 Curvilinear density functions, 588±590 Cusp constraint, 715 Cyclic twins, 10 Cylinder elliptic, 92 homogeneous, 96 inhomogeneous, 96 Cylindrical camera, 70

SUBJECT INDEX Cylindrical collimators, 432 Cylindrical detector recording, 32 Cylindrical powder cameras, 70 Cylindrical powder specimens, 57 Cylindrical sample 2 scan, 57 for neutron diffraction, 177 d orbital occupancies, 722 Darwin width, 662 DAS (differential anomalous X-ray scattering) technique, 218 Data evaluation, 100 Data processing program for intensity factors, 596 single-crystal methods, 505, 517, 536 Databases inorganic structures, 778 organic structures, organometallic structures and coordination complexes, 790, 812 powder diffraction, 81 Datum orientation of the crystal, 33 DauphineÂ twins, 11 Davidon±Fletcher±Powell update, 684 de Broglie's law, 186 Dead-time, 619, 624, 666 Debye formula, 104 Debye±Scherrer camera, 70, 162 aberrations in, 498 Debye±Scherrer±Hull method, 42 Debye±Waller factor, 415, 729, 735 Deconvolution in SANS, 107, 111 techniques, 393 Defect types, electron diffraction, 424 Defects, 419 images, 426 lattice, 113 study of, 506, 531 viewed by an imaging system, 633 Deformation density, 714 Deformation map X ± N, 714 X ± X, 714 X ± (X+N), 714 Degrees of freedom, 703 Delay-line read-out, 627 DelbruÈck scattering, 242 Dense systems in SANS, 112 Densitometry, 618 Density, 154 Density functionals, 721 Density measurement Archimedes method, 158 flotation, 158 gradient tube (column), 156 immersion microbalance, 158 penetration or swelling of solid, 156 pycnometry, 158 vibrating-string method, 158 volumenometry, 158 Depth-profiling analysis, 58 Derivative lattice, 11 Designated radiation area, definition of, 958 Desymmetrization of OD structures, 765 Detection efficiency, 624 limits, 410 of systematic error, 498±499, 707±709 quantum efficiency, 639 systems, 397, 663

Detection processes (neutrons), 644±652 electronic aspects, 648±649 films, 646 gas ionization, 644±646 neutron capture, 644 scintillation, 645±646 Detection systems (neutrons), 649±651 Anger camera, 650 corrections, 652 gas position-sensitive, 650 position-sensitive, 649±651 single detectors, 649 Detective quantum efficiency (DQE), 624, 639 Detector recording cylindrical, 32 plane, 32 V-shaped, 32 Detector-response correction in SANS, 109 Detectors background from, 663 characterization, 639 energy-dispersive, 622, 663 for electrons, 639 for neutrons, 644, 649 gas-filled, 82 imaging, 623 in X-ray spectrometers, 522, 529±531 multiwire, 82 position-sensitive, 82, 87, 100, 113, 664 resolution of, 82 scintillation, 642, 664 semiconductor, 622±623, 629, 642 single-wire, 82 solid-state, 82, 664 with wide-open window, 522, 527 Detectors for X-rays, 618±638 Geiger counters, 618 photographic film, 498, 618 proportional counters, 619 scintillation counters, 619 solid-state detectors, 620 Diagram levels, 191 Diamagnetism, 154 Dielectric coefficients, 401 Dielectric description, 399 Difference densities, 714 Differential anomalous X-ray scattering (DAS) technique, 218 Differential methods, 527 Differential scattering cross section, 260 Diffraction contrast, 113, 735 coordinates, 31 geometry, practical realization, 36 grazing-incidence, 58 imaging, 124 intensities, 596 spot size and shape, 37, 39 topography, 113, 124 Diffraction absorption fine structure (DAFS), 254 Diffraction profile, 48, 528, 530 asymmetry of, 521 broadening of, 521 double-crystal, 528 (in) standardized (form), 519 location of, 518 narrow, 528, 530 parameters of, 528 shape of, 521 symmetric, 528

987

c:\itfc\sindex.3d

Diffractometers alignment, 46 area-detector, 36, 170 background scattering with, 664 Bragg±Brentano, 495 double-crystal in SANS, 106 for powder diffraction, 42 four-circle, 170, 516 gears, 503 inclination, 517 kappa, 36 neutron powder, 82, 652 neutron powder, high-resolution, 541 operation control, 64 Seemann±Bohlin, 492, 495 three-circle, 170 Diffractometry, 36 angle-dispersive, 491, 495±496 energy-dispersive, 491 Diffuse scattering, 261 Diffusion, Brownian, 589 Digital image processing, 635 Dimension of a lattice, 945 Dioctahedral sheet, 767 Dipolar approximation, 731 Dipole, 716 Dipole moment atomic, 717 molecular, 724 Dirac±Fock, 205 Direct and reciprocal lattices, 2 Direct crystallization, 174 Direct image, 115 Direct lattice, 412, 911 Direct-lattice parameters, 505 Direct method, X-ray detectors, 634 Direct structure analysis, 103 Direction angles of a crystal face, 4 Disc specimens, 171 Disc thinning method, 174 Dislocations, 114 Dispersion, 21, 75, 590, 600 Dispersion corrections, 241±258 for XED, 86 tables of, 255±257 theory of, 243 Dispersion surfaces, 416, 417, 736 Displacive modulation, 907 Distance distribution functions, 104 Divergent-beam techniques, 510±516 classification, 512 Dopant concentration, study of, 531 Dose equivalent, definition of, 958 Double-beam comparator, 531 diffractometer, 531 spectrometer, 531 technique, 531 Double-crystal diffraction profile, 528 Double-crystal diffractometer in SANS, 106 Double-crystal monochromator (at synchrotron), 39 Double-crystal spectrometers, 528±530 combined with double-beam technique, 531 with photographic recording, 510, 529 with symmetric (Bond) arrangement, 529 with white X-radiation, 529 Double-crystal topography, 117 Double-oscillation method, 168 DQE (detective quantum efficiency), 624 Drift chambers, 626

SUBJECT INDEX Drude model, 400 Du Mond diagram, 117 Duane±Hunt limit, 192 Dynamic measurements, 626 Dynamic R factor, 427 Dynamic range, 624 Dynamical diffraction, 80 Bloch-wave method, 41 calculations, 261 many-beam, 80 multislice method, 414 Dynamical wave amplitudes, 414 Eccentricity error, 524 elimination, 521±523, 529 EELS (electron energy-loss spectroscopy), 219, 391±412, 428 Effect on lattice parameters of electric field, 508 of irradiation, 525 of pressure, 508 of temperature, 507, 510, 516, 522, 524, 529 Effective misorientation, 119 Efficiency of the production of characteristic radiation, 192 Eigenvalue filtering, 510 Eigenvalues, 678 Eigenvectors, 678 Elastic constants, 654 Elastic differential scattering cross section, 262 Elastic scattering, 416 factors, 262 neutron, 727 Elastic specular neutron diffraction, 126 Elastic stiffness constants, 654 Elastic wave, velocity of, 654 Electric field gradient, 719 Electrical properties, 154 Electrochemical thinning, 175 Electromagnetic waves, 186 Electron beam, misalignment, 424 Electron binding energies, 203 Electron density, 90, 713 experimental, errors in, 724 thermally smeared, 723 Electron diffraction, 259 absorption effects, 188, 261 boundary conditions, 259 computing methods, 425 convergent-beam, 80, 735 crystal thickness, 188, 419 detectors for, 639±643 determination of crystal thickness, 416, 419 HOLZ technique, 538 interaction constant, 259 intensities, 416 Kikuchi technique, 538 lattice-parameter determination, 537 measurement of structure factors, 416 oriented texture patterns, 412±414 pattern analysis, 537 pattern indexing, 537 patterns, 80, 390 potential field, 259 preparation of specimens, 171 propagation function, 259 reciprocal-space representation, 412 relativistic values, 259

Electron diffraction scattering factors, 188, 259 selected-area, 80, 538 structure factors, 416 transmission function, 259 useful parameters as a function of accelerating voltage, 281 Electron diffractometry, 413 Electron distributions, 713 Electron energy-loss near-edge structure (ELNES), 408 Electron energy-loss spectrometry Castaing & Henry filter, 397 crystallographic information from, 397 parallel detection, 397 Wien filter, 396 Electron energy-loss spectroscopy (EELS), 219, 391±412, 428 aberrations in, 396 analysers for, 395 detection systems, 397 monochromators for, 395 non-characteristic background, 394 spectrometers for, 394±397 types of excitation in, 393 Electron holography, 426, 427 Electron inelastic scattering, 378 Electron kinetic energy, 721 Electron microscopy, 80, 419 preparation of specimens, 171 Electron multiplication in position-sensitive detectors, 622±623 in proportional counters, 623 Electron paramagnetic resonance, 190 Electron scattering amplitudes, 259 inelastic, 391 Electron spin, interaction with neutron spin, 725±726 Electron transitions, 261 Electron-transparent specimens, 171 Electron-tube device for measurement of intensities, 642 Electron wavelength of a transmission electron microscope, determination, 540 Electroneutrality constraint, 715 Electronic detectors, 639 Electronic instability, 424 Electrons properties, 187 scattering factors, 262 wavelength, 424 Electropolishing, 174 Electrostatic moments, 716, 717, 718 Electrostatic potential, 186, 718 Electrostatic properties, 721 Elimination of concentration effects, 98 Ellipse and ellipsoid packing, 751±752 Ellipsoid, 92 Ellipsoid of revolution, 94 Ellipsoidal-mirror SANS camera, 106 Elliptic cylinder, 92 ELNES (electron energy-loss near-edge structure), 408 Emission lines, 202, 203, 204, 206, 209 Emission-spectrum profile, 519 Empirical correction factor for preferred orientation, 61 Empirical metallic radii, 774 Enantiomorphous pairs of space groups, 20 Energy discrimination, 625

988

c:\itfc\sindex.3d

Energy-dispersive analysis, 428 detectors, 625, 641, 663 diffraction, 58, 619 diffractometer, aberrations of, 497 methods, in lattice-spacing determination, 496, 507 neutron diffraction, 87 techniques, 496 X-ray diffraction, 84, 619 Energy-filtered lattice images, 428 Energy-flow triangle, 115 Energy-flow vector, 119 Energy-loss spectrometer, 395 Energy of radiation, 187 Energy resolution, 396, 619, 620, 622 Enhanced symmetry, 13 Entrance slit, 45 Entropy maximization, 691 Epitaxic formation, 176 Epitaxic layers, study of, 516, 529 Epitaxy, 153 EPR (electron paramagnetic resonance), 190 Equatorial divergence, 497 Equatorial geometry, 516 Equi-inclination setting, 31 Equivalent origins, 15 Equivalent superspace groups, 940 Erratic fluctuations in counting rates, 667 Errors (see also Aberrations, Systematic errors) and aberrations in lattice-parameter measurements, 490 and uncertainties in wavelength, 493 in angle reading, 524 in angle setting, 524 in experimental electron density, 725 of the Bragg angle, 491 Escape peaks, 622 Estimated standard deviation, 707 of an observation of unit weight, 702 Estimates, 680 Etch figures, 153 Etching chemical, 173 corundum, 173 ion sources, 173 sputter, 173 Euclidean norm, 678 Eulerian angles, 694 Eulerian-cradle diffractometer, 517 Eulerian-geometry diffractometer, 517 Evaporated thin films, 173 Ewald sphere, 26, 526, 656 EXAFS (extended X-ray absorption fine structure), 24, 189, 213±220, 254, 409 Exchange-correlation energy, 721 Excitation errors, 414 Excitation of characteristic X-rays, 510 EXELFS (extended electron fine structure), 409 Exit beam, extremely parallel, 532 Expectation values, 679 Experimental techniques classification of, 24 for crystal structure analysis, 25 Exposure of radiation, definition of, 958 Extended electron fine structure (EXELFS), 409 Extended solids, 730

SUBJECT INDEX Extended X-ray absorption fine structure [(E)XAFS], 24, 189, 213±220, 254, 409 facilities for, 219 External space, 944 External standard, 499 External vibrations, 723 Extinction, 113, 599, 728 contrast, 113 correction factor for, 612 correction, neutron diffraction, 177 correction, XED, 86 distance, 736 length, 187 primary, 609, 610 secondary, 609, 611 symbol, 13 Extrapolated (midchord) peak, 518 Extrapolation in lattice-parameter determination, 505, 510, 521±522, 535 analytical, 493±494 graphical, 493 F distribution, 702 Face normals, 5 Face or cleavage plane of a crystal, 4 Factors determining accuracy, 501 Family diffractions, 765 Fano factor, 626 Fano plots, 408 Faraday cage, 642 Faster-than-sound neutrons, 657 Faults in polytypes, 758±760 Feasible point, 693 Fermi chopper, 443 Fermi level, 398 germanium, 406 heavy metals, 406 sulfur, 406 transition elements, 406 Fermi pseudopotential, 444 Ferrimagnetism, 154 Ferroelectricity, 154 Ferromagnetism, 154 Ferromagnets, 728 Fibre optics, 632 Fibre texture, 414 Fibres, diffraction from, 24 Film aluminium, 393 for neutrons, 646 germanium, 394 Film shrinkage, 498 error, elimination of, 509 Filters, 38, 76 balanced, 74, 78, 238 Castaing & Henry, 397 for common target elements, 79 for neutrons, 438 for X-rays, 236 graphite, 82 optimum-thickness, 238 polarizing, 438, 440 single, 78 thickness, 78 wavelength change by, 239 Wien, 396 with scintillation counters, 621 Fine-grained substances, oriented texture patterns, 412 First-fringe contrast, 116 First-order Laue zone (FOLZ), 417, 418

First/second derivatives, 65 Fixed-count timing, 667 Fixed-time counting, 666 Flat-cone setting, 31 Flat crystal, 77 Flat-crystal monochromator, 39 Flat-film camera for Laue patterns, 70 Flat particles, 95 cross-sectional inhomogeneity, 96 molecular weight, 93 Flat-specimen aberration, 47, 48 Flipping coil, radio-frequency, 728 Flipping ratios, 592, 728 Flotation, 158 Fluctuations in particle orientation, 61±62, 492 in recording counts, 492, 666 Fluorescence radiation, 657 Fluorescence scattering, 661 Fluorescence spectroscopy, 619 Fluorescence techniques, 218 Fluorescent screens, 640 Fluorophlogopite reflection angles, 503 Focal-line width, 48 Focusing diffractometer geometries, 43 Focusing geometry, 83 Focusing monochromator, 82 Focusing, neutron scattering, 443 Focusing powder camera, 70 Fog density, 618 Fog level, 640 Foil detector, 645 FOLZ (first-order Laue zone), 417, 418 Forbidden reflections, 527 Form factors, magnetic, 454, 591 Four-circle diffractometer, 516 Four-dimensional crystal classes, 16 Fourier imaging, n-beam, 422 Fourier integral, 89 Fourier-invariant expansions, 586 Fourier potential, 735 Fourier series, 89 Fourier transformation, 89 indirect, 111 techniques, 393 Free-electron gas Drude model, 400±401 Lorentz model, 400 Free-radical scavengers to improve crystal lifetime, 166 Free scattering length, 594 Frequency of space groups, 15 Fresnel diffraction theory, 259 Friedel-pair intensity ratios, 251 Friedel-pair techniques, 251 Friedel's law, 913 Fringe patterns, stacking-fault, 116 Fringe period, 419 Fringe visibility, 421 Full symbols for superspace groups, 921 Gallium selenide, 754 Gamma function, 702 Gamma rays, 187 Gas amplification, 626±627 Gas detector for neutrons, 644 Gas-filled counters, 82, 626 Gas multi-electrode position-sensitive detectors for neutrons, 650 Gas multiplication, 626 Gauss±Markov theorem, 680

989

c:\itfc\sindex.3d

Gauss±Newton algorithm, 683, 690, 693 Gaussian curves, 66, 711 Gaussian fits to X-ray scattering factors, 261 Gaussian radial functions, 724 Geiger counter, 618 Gelatine capsules, 163 Generalized Bessel function, 666 Generation of X-rays, 191 Generator stability, 72 Geometric crystal classes, 15, 911, 945 Geometrical aberrations, 41, 493 for XED, 86 Geometrical analysis of oriented texture patterns, 412 Geometrical instrument parameters, 44 Geometrical peak, 518 Geometry of SANS, 106 Germanium, Fermi level, 406 Germanium film, 394 Gibbs instability, 415 Gibbsite±nordstrandite family, 766 Gittergeister (lattice ghosts), 907 Givens rotations, 679 Glauber approximation, 391 Globular particles, 93 Glove box, definition of, 959 Gnomonic transformations, 29 Gold, dielectric coefficients, 401 Goodness-of-fit parameters, 702, 707 Gordon±Kim model, 721 Gradient tube (column) cavities, problem of, 156 Ficoll gradient, 157 inclusions, problem of, 156 shallow gradient, 157 Gram±Charlier series expansion, 586 Graphical extrapolation of lattice parameters, 493 Graphite dielectric functions, 403 monochromator, 37, 38, 51, 620 Gray, definition of, 959 Grazing-incidence diffraction, 58 Grigson scanning method, 81 Growth striations, study of, 530 Growth twins, 10 Guinier and Tennevin technique, 119 Guinier approximation, 92, 110 Guinier camera, 162 Guinier focusing, 70 Half-life, definition of, 959 Half-width, 506, 519, 522, 526, 528 (methods of) reducing, 526 minimum, 506 of wavelength distribution, 506 Hamilton's R-factor ratio test, 704 Hankel transform, 102 Hard-sphere interference model, 98 Hard X-rays, 187 Hardness, 153 Harmonics, cubic, 585 Hartree±Fock approximation, 732 model, 243 self-consistent field, 659 wavefunctions, 460 Hat matrix, 705 Heat capacity, 154 Heavy metals, Fermi level, 406 HEED (high-energy electron diffraction), 412

SUBJECT INDEX Helimagnetic order, 728 Hellmann±Feynman constraint, 715 Hermite polynomial tensors, 586 Hessian matrix, 684 Heterogeneous packing, 746 Hetero-octahedral sheet, 767 Hexacontatetrapole, 716 Hexadecapole, 716 Hexagonal closest packing, 747, 752 crystal system, 7, 15, 19 High-angle annular dark-field (HAADF) images, 428 High-angle Bragg reflections in latticeparameter determination, 509, 522, 529, 532 High-energy electron diffraction (HEED), 412 High-order Laue zone (HOLZ), 418, 424, 538, 539 High-pressure structural studies, 87 High-purity germanium detector, 622 High-resolution electron microscopy (HREM), 261, 419, 773 High-resolution energy-dispersive diffraction, 58 High-resolution experiments, 97 High-resolution powder diffractometers D2B at Institut Laue±Langevin, 541 HRPD at Rutherford Appleton Laboratory, 541 High-sensitivity lattice-parameter comparison, 531±532 High-tension supplies, unsmoothed, 667 Higher-dimensional crystallography, 908 Histogramming memories, 626 Hohenberg and Kohn theorem, 721 Hollow cylinders, 92 Hollow particles, 96 Holographic reconstructions, 427 Holohedry, 12, 939, 945 HOLZ (high-order Laue zone), 418, 424, 538, 539 Homogeneous cylinder, 96 Homogeneous packing, 746 Homogeneous particles, 93 Homogeneous triaxial bodies, 92 Homometric mapping, 751 Homo-octahedral sheet, 767 Horizontal divergence, error, 525 Horizontal Soller slits, 56 Householder transformations, 679, 686 HREM (high-resolution electron microscopy), 261, 419, 773 Hydrogen-atom scattering factors, 565 Hydrogen bonding, 906 Hyperbolic Bessel function, 666 Hyperfine interaction, 732 Hyper-resolution, 427 Hypothesis testing (no remaining systematic errors), 523 ICDD Powder Diffraction File, 81 Icosahedral viruses, SANS, 111 Ideal crystals, 908 Ideally imperfect crystals, 113 Ideally perfect crystals, 113 Idempotency conditions, 722 Idempotent, 705 Identity period, 752 Identity period determination, 508 accuracy of, 510

Ill-conditioned least-squares problems, 101 Image intensifiers, 122, 632, 635 Image processing, 427, 635 Imaging detectors, 623 Imaging plates, 426, 635, 641 Immersion microbalance, 158 Impulse approximation, 657 Incidence aperture, 53 Incident-beam monochromatization, 120 Incident-beam monochromator, 53 Inclination diffractometer, 517 Inclination of plane of specimen, 494 Incoherent elastic scattering cross section, 595 Incoherent multiple scattering, 108, 661 Incoherent scattering, 177, 554 Compton, 90 cross section, 594 functions (Table 7.4.3.2), 658 level, 109 Incommensurate modulated structures, 907±944 Index of refraction, 600 Indexing powder patterns, 541 Indirect method, X-ray detectors, 634 Indirect transformation method, 101 Indium antimonide, dielectric coefficients, 401 Induced matrix norm, 678 Inelastic coherent scattering, 109 Inelastic crystal excitations, 425 Inelastic scattering, 378, 416, 657 Bethe theory, 406±408 electrons, 391 neutrons, 391 Inelastic scattering factors for electrons (Table 4.3.3.2), 378±388 Inelastically scattered electrons, 81 Influential data points, 705, 708 Information resolution limit, 424 Infrared radiation, 187 Inhomogeneities in matter, 105 Inhomogeneous cylinders, 96 Inhomogeneous particles, 96 Inner moments, 718 Inner surface area, 109 Inorganic compounds silicates, 766±769 typical interatomic distances, 778±789 Inorganic Crystal Structure Database, 778 Insertion devices, 197 Instrument parameters, geometrical, 44 Instrumental broadening and aberrations, 47, 101 Integrated intensity for XED and powder samples, 85 formulae for, 600 of a reflection, 668 Integrated reflections, 114 Intensity, 519 distribution, two-dimensional map, 522 of characteristic lines, 191 of diffracted intensities, 554±595 standards, 500 statistics, 519 variation with take-off angle, 74 Intensity factors, 596 angular velocity, 596 data-processing programs for, 596 in single-crystal methods, 596±598 polarization, 596 trigonometric, 596±598

990

c:\itfc\sindex.3d

Interatomic distances, 778±896 in inorganic compounds, 778±789 in metals, 774±777 in organic compounds, 790±811 programs for calculating, 778 Interaxial angles, determination, 525 Interband transition, 401 Interference model, hard sphere, 98 Interferometers Bonse & Hart, 121 Fabry±Perot, 533 Interferometry, combined optical and X-ray, 533±534 Intergrowth crystal structures, 907 Interlaboratory comparison, 536 Internal space, 912, 937, 944 standard, 499 translation, 912 Interparticle interference, 97, 98 Interpenetrating packing, 751 Interplanar spacing determination accuracy of, 505 precision of, 505 Interquartile range, 690 Intersecting-Kikuchi-line method, 736 Interstices in close-packed structures, 753 Intramolecular multiple scattering, 392 Intrinsic background (neutrons), 651 Intrinsic component, 917 Intrinsic efficiency, 622 Intrinsic part of a space group, 940 Inverse contrast variation, 108 Inversion twins, 10, 12 Ion-beam thinning, 171±173 Ion-implanted silicon, 525 Ion sources for etching, 173 Ionic radii, 778 Ionicity, degree of, 425 Ionization cross sections, 407 Ionizing radiation definition of, 958 protection from, 962±963 Irradiated area displacement of, 526 exactly defined, 522 Irradiated specimen length, 45 Irradiation, study of effects of, 516, 525 Isometric point groups (crystal classes), 939 Isotopic replacement, triple, 111 IUPAC notation, X-ray diagram levels, 191 Jagodzinski notation, silicon carbide, 754 Johann monochromator, 664 Johansson monochromator, 664 Joint probability density function, 679 K doublet, 62, 510, 512±515 K1 and K 1 wavelengths in lattice-spacing determination, 521 K2 radiation, elimination, 510 K line in lattice-spacing determination, 507 Kaolinite, 769 Kappa diffractometer (definition), 37 Kappa model, 714 Kikuchi lines, 419 Kikuchi patterns, 735 Kikuchi techniques, 538 Kinematic image, 115 Kinematic theory, 590 Kinematical approximation, 80, 260, 262

SUBJECT INDEX Kinetic energy, 721 Kitajgorodskij's categories, 897 Knife-edge calibration, 498 Kossel camera, 512 cone, 510, 514 lines, 510±515 lines, intersections of, 512±513 method, 510±516 pattern, 512±514, 735 plane, 513 Kramer's constant, 192 Kramers±Kronig transform, 245 Kratky cameras, 99 Label triangulation, 111 Labelling, 97 isotopic, 108 Lagrange polynomials, 111 Lagrange undetermined multipliers, 693 Lambda curves (Laue), 39 Lambda symmetry (operations), 763 Lamellar particles, 97 Lamellar textures, 412 Lanthanum hexaboride, instrumental sample, 501 Large-scale problems, 685 Larmor frequency, 728 Laser, He±Ne, 533 Laser plasma X-ray sources, 189 Latex particles, 107 Lattice(s) aperiodic, 937 Bravais, 3, 15, 913 centred, 3 curvature, 114 defects, 113 derivative, 11 dimension, 937 direct, 911 direct and reciprocal, 2 for close-packed structures, 755 holohedry, 939 point, 2 rank, 937 twin, 10 vector, 2 vector centring, 3 vector, reciprocal, 2 Lattice bases, standard, 938 Lattice-fringe images, 421 Lattice ghosts (Gittergeister), 907 Lattice-parameter changes, study of, 507, 510, 522, 525, 529±530 Lattice-parameter determination, 490±541 aberrations in, 493 absolute, 505, 525, 529±532 for rectangular systems, 528 from one crystal mounting, 509, 510 from separate photographs, 509 HOLZ techniques, 538, 540 inter-laboratory comparison, 536 Kikuchi techniques, 538±540 least-squares methods, 498 local, 525, 529, 532 neutron diffraction, 541 of cubic lattice, 528 of deformed lattice, 513±515 of imperfect crystal, 522 of large flat slab, 507, 522, 524, 527 of perfect crystal, 522

Lattice-parameter determination of polycrystals (Kossel method), 515 of single crystals, 505±536 of small spherical crystals, 507, 525 of standard crystal, 507 powder diffraction, 491, 506, 509, 518, 521 precision, 505, 509, 515, 526, 530, 536 preliminary, 507 relative, 505 sensitivity of, 505, 507, 532 standards, 499 systematic errors in, 493, 498 wavelength problems, 492 Lattice-parameter determination methods camera, 497, 507 diffractometer, 495±496, 516±517 electron diffraction, 537±540 energy-dispersive, 496±497 neutron diffraction, 541 non-dispersive, 506, 509, 526, 533±534 polycrystalline X-ray, 491±504 pseudo-non-dispersive, 506, 526, 528 single-crystal X-ray, 505±536 synchrotron, 495 whole-pattern, 496 X-ray, 534 Lattice-parameter differences, determination of, 507, 522, 525, 528±531 Lattice-parameter measurements accuracy of, 490 discrepancy for silicon, 490 possible effect of filter, 239 Lattice parameters of silicon, 490, 499 of silver, 499 of tungsten, 499 Lattice-spacing comparators, 530 Laue class, 13, 938 Laue diffraction multiplicity distribution, 27 neutron single-crystal, 87 Laue geometry, 26, 38 Laue method, 663 Laue patterns, 27, 124 flat-film camera for recording, 70 Laue photography combined with powder diffraction, 506 Laue point group, 908, 913, 938 Laue sphere, 26 Layer-line screen (precession), 34 Layer-line screen (Weissenberg), 35 Layer lines, 414 Layer polytypism, 760, 766 Layer silicates, 414 Layer stacking, 752±773 in polytypes, 760±773 LCAO (linear combination of atomic orbitals) approximation, 715, 723 Lead oxide camera tubes, 634 Leakage radiation, definition of, 959 Least-dense sphere packings, 748, 749 Least-squares calculations, 678±688 estimator, 680 nonlinear, 682 software for, 688 Least-squares refinement, 504, 505, 510, 517 problems, 101 LEED (low-energy electron diffraction), 24 Legendre polynomial, 565 associated, 581 Lens configuration, 514

991

c:\itfc\sindex.3d

Lens-shaped figures, 512±514 Levenberg±Marquardt algorithm, 683 Leverage, 705, 708 Libration (rotational oscillation), 589 Libration tensor, 697 Librational model, 697 Librational temperature factor, 723, 724 Licensable quantity, definition of, 959 Likelihood, 689 Likelihood-ratio method, 523 Limited projection topographs, 116 Limiting resolution of X-ray detectors, 634 Line focus, 194 Line profile, 518 calculated by convolution, 662 Linear algebra, 678 Linear attenuation coefficient, 213 Linear combination of atomic orbitals (LCAO) approximation, 715, 723 Linear estimator, 680 Linearity, 621 Linearity of response, 619 Lithium-drifted germanium detector, 622 Lithium-drifted silicon detector, 622 Live X-ray topographs, 122 Local measurements (topography), 516, 525±527, 529 Location of diffraction profile, 518 Long-period polytypes, 757 Lorentz factor, 497, 596, 710 Lorentz model, 400 Lorentz±polarization factor, errors, 60, 523, 596 correction for, 523 Lorentzian curves, 66 Lorentzian functions, 711 Lorentzian profiles, 67, 400 Low-angle boundaries, 114 Low-angle reflections, confusion with escape peaks, 622 Low-energy electron diffraction (LEED), 24 Low-Q scattering, 105 Lower quartile, 813 Luminescence, photostimulated, 635 Macromolecules, biological, use of SANS, 105 Macroscopic absorption cross section, 461 Magnetic Bragg reflection, 591 domains, 124 form factors, 454, 592 interaction vector, 591 orbital structure factor, 731 ordering, 725 space group, 591 Magnetic properties, 154 of the neutron, 108 Magnetic scattering of neutrons, 590 of neutrons, elastic, 591 X-ray, 733 Magnetic structure factors, 591, 726, 727 unit-cell, 591 X-ray, 733 Magnetism, 725 Magnetization density, 591, 725 Magnetostatic energy, 731 Magnetostatic properties, 731 Magnets, bending, 197 Main reflections, 907

SUBJECT INDEX Mains-voltage fluctuations, 667 Many-beam dynamical diffraction, 80 Marginal probability density function, 679 Mass absorption coefficients, 213, 600 Mass attenuation coefficients, 213±214 tables of, 230±236 Mathematical interpretation in single-crystal methods, 536 Mathematical theory of powder diffractometry, 518 Matrix diagonalization, 425 Matrix formulae for two-circle diffractometer, 517 Maximum degree of order (MDO) polytypes, 762, 767, 768, 769, 770, 772 Maximum dimension of a particle, 93, 102 Maximum-entropy method, 428, 689 Maximum-likelihood estimate, 689 Maximum-likelihood methods, 691 Maximum oscillation angle, 33 Maximum primary dose (MPD), 960 MDO (maximum degree of order) polytypes, 762, 767, 768, 769, 770, 772 Mean, 679, 813 Mean-square broadening, 493 Measured-as-negative intensities, 667 Measured profile, 505 as a convolution, 503 Measurements of lattice parameters absolute, 505 relative, 505 Mechanical (deformation, glide) twins, 10 Mechanical properties, 153 Mechanical twins, 10 Median, 520 absolute deviation, 690 variance of, 520 Melt-grown crystals, 114 Melting point, 154 Membrane proteins, 24 Mercury iodide detector, 623 Mercury sulfide chloride, -Hg3 S2 Cl2 , 771, 772 Merohedral point groups, 12 Meso-octahedral sheet, 767 Metallic radii, empirical, 774 Metals preparation of specimens, 173 texture studies, 414 typical interatomic distances, 774±777 Methyl methacrylate resin containers, 162 Metric tensor, 694 Mezei flipper, 442 Mica containers, 162 Mica group, 765, 768±770 Microanalysis, 54 quantitative, 410 Microdensitometer, 618 Microdiffractometers, 491 Microdiffractometry, 53 Microfocus sources, 71 Microrefractometer, 160 Microtome, 171 Microwaves, 190 Midchord peak, 518 Midpoint of a single chord, 518 bias, 520 variance, 520 Miller formulae, 5 Miller indices, 5, 11 Mimetic twinning, 153

Mirror cameras, 106 Mirrors, 37 for neutrons, 436 reflection devices, 435 Misalignment, 506, 531, 535±536 diffraction, 424 of electron beam, 424 Misorientation functions, 414 Misorientation matrices, 33 Mixed-layer structures, 760 Model calculations in SAXS, 103 Model fitting in SANS, 111 Modelling of space-group frequencies, 897 Moderators for neutrons, 431 Modulated crystal structures, 907±944 examples, 936 types defined, 907 Modulation displacive, 907 occupation, 907, 913 relations, 941 Modulation transfer function (MTF), 634 MoireÂ topography, 121 Molecular beams, 189 Molecular biology, isotopic composition in SANS, 107 Molecular dipole moment, 725 Molecular geometry, 812 Molecular organic structures packing in, 897 space-group distribution of, 905 Molecular packing, 904 Molecular scattering factors, 390 Molecular weight, 93 Moliere high-energy approximation, 260 Moment, 679 Moments of a charge distribution, 716 Momentum density distributions, 659, 713 Momentum space, 713 Monitor methods, 72 Monitoring circuits, 619 Monochromatic radiation, ±2 scan, 55 Monochromatic still exposure, 30 Monochromator-scan method for diffraction, 85 Monochromators, 37, 43, 46, 51, 76, 99, 120, 395, 528, 662 alignment, 46 angular calibration, 46 channel-cut, 77, 121, 239 common types, 77 comparison of, 433 crystal, 76 different diffraction geometries, 43 diffracted beam, 44, 46 double-reflection, 239 focusing, 82 for neutrons, 432±435 for X-rays, 236 graphite, 38, 239, 620, 664±665 incident-beam, 53 Johann, 664 Johansson, 664 mosaic crystals, 432, 662 multi-reflection, 239 perfect-crystal, 39, 443, 663 pyrolytic graphite, 46, 72 quartz, 661 scanning-crystal, 622 single-reflection, 239 Monoclinic crystal system, 6, 16

992

c:\itfc\sindex.3d

Monodisperse systems, 91 Monopole, 716 Morphological properties, 153 Morphology, and mounting of single crystals, 164 Morse approximation, 389 Mosaic-crystal monochromator, 662 Mosaic spread, 432 Mosaicity, 443 neutron diffraction, 433 Moseley's law, 76 MoÈssbauer radiation, 656 MoÈssbauer spectroscopy, 189, 719 Mott±Bethe formula, 261 Mounting of specimens, 163 biological macromolecule single crystals, 165 polycrystalline specimens, 162 single crystals, 164 small organic and inorganic single crystals, 164 Moving splines, 111 Multichannel pulse-height analyser, 496 Multi-component complex, label triangulation, 111 Multidetector, 82 Multilayer materials, 171 Multilayer polarizers, 436 Multiple-beam methods, 531 Multiple-crystal techniques, 528±532 Multiple diffraction, 513 Multiple-diffraction methods, 526±528 with counter recording, 526±528 with photographic recording, 513 Multiple-diffraction pattern, 527 indexing, 527 Multiple-exposure techniques, 514 Multiple-order reflections (Laue), 27 Multiple scattering, 661 deconvolution techniques, 391 incoherent, 661 intramolecular, 392 neutron diffraction, 177 Poisson distribution, 393 problems associated with, 392 Multiple twins, 10 Multiplicity distribution in Laue diffraction, 27 Multiplicity factor, 596 Multipole expansions, 103, 714, 716, 730 Multipole functions, angle dependence, 583 Multipole model, 716 Multi-reflection devices, 121 Multislice method, 414±416, 425 programs for, 415 Multiwire detectors, 82 Multiwire proportional chamber (MWPC), 627 Muons, 189 Muskovite, 771 n-beam Fourier imaging, 422 n-particle density matrix, 713 n-particle wavefunction, 713 Natural background, definition of, 959 Near-edge fine structures, 408 Neighbours, nearest (direct) and next-nearest (indirect), 775 Net planes, 4 Neutron Anger camera, 650 Neutron-beam definition, 431 Neutron-capture reactions, 644±645

SUBJECT INDEX Neutron diffraction, 26 cross section (tables), 445±461 energy-dispersive, 87 Laue, 87 powder, 82 preparation of specimens, 177 scattering lengths (tables), 445±461 time-of-flight, 87 time-of-flight, powder, 88 white-beam, 87 Neutron polarization, 592 Neutron powder data, 68 Rietveld analysis of, 710±712 Neutron resonance spin echo (NRSE), 443 Neutron scattering, 591±595 elastic, 728 focusing, 443 form factors, 454±461 inelastic, in spectroscopy of solids, 391 magnetic, 591, 726 monochromators, 443 nuclear, 593 polarized, 728 resolution functions, 443 scattering factors, 454±461 Soller collimators, 443 spectrometers, 444 Neutron sources pulsed spallation, 189 reactors, 430 spallation, 87, 430 Neutron spin, 444 interaction with electron spin, 725±726 Neutrons absorption coefficients (Table 4.4.6.1), 461 cold, 105 faster-than-sound, 657 films for, 646 filters, 438 guide tubes, 432, 435 moderated, 430 monochromators, 432 polarized, 108, 592 properties, 187 reflectivity, 126, 433 reflectometry, 126 scattering-length densities, 441 slower-than-sound, 657 topography, 124 Next-nearest (indirect) neighbours, 775 NFZ relation, 761 NIST (National Institute of Standards and Technology) silicon standard, 495 NMR (nuclear magnetic resonance), 154, 190 No-upper-layer-line approximation, 415 Nodal reflections, 27 Noise reducer for image processing, 635 Non-crystalline samples, diffraction from, 24 Non-crystallographic symmetry elements, 907 Non-dispersive methods (techniques), 506, 526, 533±534 Non-linear least squares, 683 Non-periodic systems, 89 Non-stochastic effects, 959 Non-systematic interactions, 81 Non-uniformity of response, area detectors, 41 Normal attenuation, 214 Normal-beam equatorial geometry, 36 Normal-beam rotation method, 31 Normal crystal, 908 Normal equations, 680, 682

Normal modes of vibration, 653 Normal probability distribution function, 66, 702 Normalized spin density, 727 Notations for close-packed structures, 753, 758 Nuclear magnetic resonance (NMR), 154, 190 Nuclear reactors, 430 Nuclear scattering amplitude, 594 length, bound, 594 of neutrons, 593 Nuclear structure factors, 595, 725, 732 Nuclear Thomson scattering, 242 Numbers of reciprocal-lattice points within resolution sphere, 28 Numerical approximations to f(s), 261 Numerical methods, 685 Objective-lens defocus, 421 Oblate ellipsoid of revolution, 94 Oblique crystal system, 15 Oblique-texture electron difraction patterns, 412 Obliquity, 41 Occupation modulation, 907, 913 Octahedral sheet, 767 Octupole, 716 OD (order±disorder) diffraction pattern, 763 groupoid families, 761, 763, 765, 773 layers, 764 packets, 765 repeat unit, 764 structures, 761, 764 structures, desymmetrization of, 765 theory, 761 One-crystal spectrometers, 521±526 asymmetric (arrangement), 521±522 symmetric (arrangement), 521±526 One-dimensional crystal classes, 15, 16 One-line symbols for Bravais classes, 915, 920 One-particle reduced density matrix, 726 Open shell electrons, 727 Operations in polytypes, 762±763 Optic modes, 653 Optical activity, 153 antipodes, 153 diffractograms, 427 interferometry, 533±534 properties, 153, 160 reflectivity, graphite, 403 wavelength as a standard, 533 Optimization (of measurement), 517±520, 526±528, 532±534, 667 Optimum design of experiments, 74, 702 Orbital angular momentum, 731 Orbital magnetism, 731 Orbital moment, 727 Orbital momentum, 726 Orbitals, Slater-type, 584 Order±disorder (OD) diffraction pattern, 763 groupoid families, 761, 763, 765, 773 layers, 764 packets, 765 repeat unit, 764 structures, 761, 764 structures, desymmetrization of, 765 theory, 761

993

c:\itfc\sindex.3d

Organic compounds preparation of specimens for electron diffraction and electron microscopy, 176 space-group distribution of molecular, 905 typical interatomic distances, 790±811 Organometallic compounds label triangulation, 111 typical interatomic distances, 812±896 Orientated solidification, 176 Orientation contrast, 113 Orientation matrix, 516, 535, 537 determination, 517 four-circle diffractometer, 516±517 Orientation of crystals, 134 Orientation of the lattice relative to the point group, 15 Oriented texture patterns, 412 Origin of angular scale recovery of, 530 uncertainty of, 530, 536 Original profile, 505, 520 half-width of, 506, 520 Origins, equivalent, 15 Orthorhombic crystal system, 6, 16 Oscillation angle, maximum, 33 Oscillation photographs processing, 510 Outer moments, 718±720 Outlier removal, 813 Packing contact number in, 746±747, 752 density of, 746 heterogeneous, defined, 746 homogeneous, defined, 746 interpenetrating, 751 of circles, 746±747, 752 of ellipses and ellipsoids, 751±752 of layers, 752±773 of spheres, 746±751, 752, 904 stable, 747, 750 types, tables of, 747±748 voids in, 750 Pair-distance distribution function (PDDF), 90 Pair-production cross sections, 213 Parafocusing, 47 Parallax, 41, 624 Parallel-beam geometry, 54, 663 Parallel detection, 397 Parallel-plate counters, 627 Parallel recording, 639 Paramagnetism, 154 Paramagnets, 728 Parasitic scattering, 661 Partial coincidence operations, 761 Partial structure factors in SANS, 112 Partial wave phase shifts, 389 Partially stimulated reflections (partials), 34, 39 Particle(s), 186 latex, 107 mass in SANS, 110 maximum dimension, 102 parameters of, 91 shape in SANS, 110 shape in SAXS, 93 Particle orientation, fluctuations, 492 Particle size, 62, 89, 162 distribution, 111 Patterson synthesis, 21 Pauli principle, 412 Pauling model of hydrous phyllosilicates, 767

SUBJECT INDEX PCD (projected charge-density) approximation, 423 PDDF (pair-distance distribution function), 90 Peak asymmetry, 67 displacements, 518 height, 518 of a reflection, 492 satellite, 75 search, 65 shift, 518 variance, 520 Peak flux, 431 Peak-shape function, 710±711 Peak-to-background ratio, 65, 661 with scintillation counters, 622 Pearson VII function, 67, 711 PendelloÈsung, 250 Penetration depth, 58 of X-rays, methods of reducing, 525 Peptides, standard coordinates, 699 Per cent point function, 708 Perfect-crystal analysers, 665 Perfect-crystal monochromators, 444, 663 Perfect single crystals, 510, 519 Persistence length, 93 Petrographic sections, 171 Phase analysis, electron diffraction, 412 Phase diagrams, determination of, 510 Phase-grating approximation, 260 Phase identification, 42, 81 from electron-diffraction patterns, 81 Phase-space analysis, 661 Phase-space diagrams, 661 Phase transitions, study of, 509, 522, 525, 529 Phonon absorption, 656 Phonons, 261, 653 Phosphor screens, 634 Phosphoric acid as etchant, 173 Phosphors, 630 storage, 635 Photoabsorption measurements, 406 Photoconductive layer amorphous Se±As alloy, 635 lead oxide, 634 X-ray-sensing, 634 Photo-effect data, theoretical, 221 Photoelectric absorption, 599 Photographic emulsions, 640 Photographic film graininess, 640 properties of, 640 shrinkage, 498 Photographic methods electron diffraction powder pattern, 81 single-crystal, 508±516 single-crystal, classification of, 508 Photographs cone-axis, 36 setting (precession), 35 upper-layer (precession), 35 upper-layer (Weissenberg), 35 zero-layer (precession), 35 zero-layer (Weissenberg), 34 Photomultiplier tube, 619 Photon energy, 84 Photon-induced X-ray analysis, 189 Photon interaction cross sections, tables of, 223±229

Photon noise, 633 Photon scattering cross section, 213 Photons, 186 Photostimulated luminescence, 635 Phyllosilicates, 413, 766±771 Physical aberrations, 493, 493 for XED, 86 Physical constraints, 715 Physical properties, relation to crystal structure, 151±153 Picture elements (pixels), 634 Piezoelectricity, 154 Pinhole cameras in SANS, 106 in SAXS, 100 PIX (proton-induced X-ray analysis), 189 Pixels (picture elements), 634 Planck's law, 186 Plane detector recording, 32 Plane-wave topography, 121 Plasmas, 191 Plasmon(s), 261, 403 cross section, 399 dispersion, 398 energies in metals, 397 excitation energy, 660 lifetime, 399 scattering, 657, 661 Plasticity, 153 Pleochroism, 153 Point group(s), 939 arithmetic, 913 arithmetically equivalent, 939 definition of, 945 equivalence class, 939 geometrically equivalent, 939 Laue, 908, 913 merohedral, 12 orientation of the lattice relative to, 15 reducible, 940 Point lattice, 2 Point row, 3 Point-spread factor, 40 Point-spread function (PSF), 625 Poisson distribution, 393, 666, 690 difference of two, 666 sum of two, 666 Poisson statistics, 519 Poisson's electrostatic equation, 719 Polarization, 193 dependence, 732 incident neutron, 727 index, 611 neutron, 592 rotation of, 593 vector, 654 Polarization factor, 51, 596 for XED, 85 Polarized neutron scattering, 728 Polarized neutrons, 108, 592 Polarized radiation, circularly, 734 Polarizers, multilayer, 435 Polarizing filters, 438 microscope, 154 mirrors, 440 neutron guides, 431 Polishing, 174 Polycrystalline samples for neutron diffraction, 177 Polydisperse systems, 89, 99

994

c:\itfc\sindex.3d

Polymers isotopic composition in SANS, 107 preparation of specimens for electron diffraction and electron microscopy, 176 texture studies, 414 Polypeptides, restraints in refining, 699±700 Polysynthetic twins, 10 Polytypes, 754±756, 760±773 basic, 762 families, 761 faults in, 758±760 layer stacking in, 760±773 long-period, 757 maximum degree of order (MDO), 762 mixed-layer structures, 761 regular, 762 rod, 760, 766 simple, 762 standard, 762 turbostratic structures, 761 Polytypism definition, 760 layer, 766 oriented texture patterns, 412 rod, 760, 766 Position-sensitive detectors, 82, 87, 100, 113, 619, 623±633, 664 choice of detectors, 623 detection efficiency, 623±624 detector properties, 623±625 for neutrons, 649, 652 localization of detected photon, 627 photographic film, 623 size and weight, 626 software packages, 633 storage phosphors, 634, 635±638 Positron annihilation spectroscopy (PAS), 189 Positrons, 189 Posterior probability density function, 681 Potential scattering, 594 Powder cameras, 70 Powder diffraction, 42, 664 advantages of synchrotron, 54 combined with Laue photography, 506 electron techniques, 80 methods, basic, 55 neutron techniques, 82 special factors in, 596 standards, 498±499 tilted-beam techniques, 80 Powder diffraction data, Rietveld analysis of, 710±712 Powder diffractometry, mathematical theory of, 518 Powder methods compared with single-crystal methods, 506 Powder-pattern geometry, 80 Powder-pattern indexing, 541 Powder-pattern intensities, 80 Powder patterns calculation of, 60 computer graphics for, 69 Powders, correction factor for, 657 Poynting vector, 119 Precautions against radiation injury, 958±967 Precession geometry, 35 setting, 168 Precision, 490, 492, 497, 501, 707 of parameter estimates, 702

SUBJECT INDEX Precision of lattice-spacing determination, 505 and profile shape, 517±519 (methods of) increasing, 518, 533±534 relative, 505 Preferred orientation, 60, 80, 162, 712 empirical correction factor, 61 minimization of, 60 Preparation of crystals, 153 of specimens for electron diffraction and electron microscopy, 171 of specimens for neutron diffraction, 177 Pressure, effect on lattice parameters, study of, 508 Primary dose limits, 960 Primary extinction, 609, 610 Primitive crystallographic basis, 2 reciprocal basis, 2 unit cell, 2 Principle of causality, 246 Prior probability density function, 681 Probability density function, 681 Probability distribution function (p.d.f.), 707 Profile fitting, 65, 492, 710 computer procedures, 491 functions, 66, 710 in oscillation photographs, 510 Profile parameters, 710 Projected charge-density (PCD) approximation, 423 Projection matrix, 705, 706 Projection topograph, 115 Prolate ellipsoid of revolution, 94 Promolecule, 714 Promolecule density, 714 Propagation function, 415 Proportional counters, 619 Protection from ionizing radiation, 962±963 Proteins label triangulation, 111 restraints in refining, 699±701 Proton-induced X-ray analysis (PIX), 189 Pseudo-atom moments, 718 Pseudo-non-dispersive methods, 506, 526, 528±533 Pseudo-Voigt function, 67, 711 Pulse-amplitude discrimination, 73, 620 Pulse-amplitude distributions, 621 Pulse-height analyser, 622 Pulse-height discrimination, 619 Pulse-height distribution, 626 Pulsed neutron source, 711 Pulsed (spallation) neutron source, 87, 189, 430±431 Pycnometry, 158 Pyroelectricity, 154 Pyrolytic graphite, 43, 77, 438, 665 cross section for neutrons, 439 Pyrophyllite, 768 Q±Q (quantile±quantile) plot, 707, 708 QED corrections, 204, 205 QR decomposition, 679 Quadrupole, 716 Quadrupole moment, 717 Quality factor (QF), definition of, 959 Quanta, 186 Quantile±quantile plot, 707, 708 Quantitative microanalysis, 410

Quantum counting efficiency, 621 Quantum efficiency, 624 Quartz monochromator, 664 Quartz twins, 11 Quasi-Gaussian approximation, 590 Quasi-Newton methods, 683 Quasicrystals, 908 R factors, 68, 710±711 dynamical, 427 weighted, 68 Racah's algebra, 731 Radial constraint, 715 Radial distribution function, 97 Radiation damage, 166, 417, 626, 630 Radiation injury definition of terms, 958±960 possible sources, 962±963 precautions against, 958±967 regulatory authorities, 964±966 responsibilities, 960±961 Radiation protection, 957, 962 Radiation safety officer, 960 Radiations used in crystallography electromagnetic waves, 186 particles, 186, 259 Radioactive samples, 619 Radioactive sources, 189 Radio-frequency flipping coil, 728 Radionuclides, 75, 196 definition of, 959 Radiotoxicity, definition of, 959 Radius of gyration, 91 of the cross section, 92 of the thickness, 92 Radoslovich model, 769 Raman effect, 153 Raman scattering, 657, 660 resonant, 657, 660, 661 Raman spectroscopy, 189 Ramsdell notation, 752 Rank of a lattice, 937 Rate-meter measurements, 63 Ratio method for powder samples, 509 for single crystals, 509 Ratio of lattice spacing to optical wavelength, 533 Rational twin axis, 10 Rayleigh criterion, 427 Rayleigh scattering, 214, 242, 554, 599 Rayleigh scattering data, theoretical, 221 RBS (Rutherford backscattering), 189 Reactors, 430 Real crystals, 419 Real solids, 401 Real structure determination, 516, 531 errors due to, 528 Receiving slit, 45 aperture, 53 width, 48 Reciprocal cell picture of, three-dimensional, 509 picture of, two-dimensional, 509 picture of, undeformed, 509 Reciprocal lattice, 412 angles, determination, 517 geometry, 513 layer lines and crystal setting, 168 parameters, determination, 517

995

c:\itfc\sindex.3d

Reciprocal lattice point, 415 vector, 3 Recording counts, fluctuations of, 492 Recording range, 52 Rectangular crystal system, 15 Reducible point groups, 939 Reference crystal(s), 531±532 Reference sample in SANS, 109 Refinement least-squares, 503, 505, 510, 517 of structural parameters, 677 problems, least-squares, 101 Rietveld, 56, 82, 541 Rietveld, using XED, 86 Reflecting power of a crystal, 590 Reflection angles, 499 conditions, for a twinned crystal, 13 conditions, special, 921 of light, 153 Reflection electron microscopy (REM), 428 Reflection high-energy electron diffraction (RHEED), 428 Reflection-only counting rates, 666 Reflection specimen, ±2 scan, 44, 53 Reflection topographs, 113, 114 Reflection twins, 10, 12 Reflections integrated, 114 main, 907 multiple-order (Laue), 27 nodal, 29 satellite, 907 single-order (Laue), 27 Reflectivity function, 528 Refraction, 527 correction for, 492, 505, 523, 536 effects, 81 Refractive index, 81, 154, 160, 189, 599±600 immersion media for measurement of, 160 Regular polytypes, 762 Regulatory authorities, 964 Relative measurements of lattice spacing, 505 Relative molecular mass in SANS, 110 Relativistic corrections, 390 Relativistic effects, 186, 260, 262 REM (reflection electron microscopy), 428 Remanent systematic error, testing for, 498 Repeated twins, 10 Residual, 707 Residual map, 714 Resolution in XED, 85 sphere, 27 Resolution errors detector element, 106 gravity, 106 in SANS, 106 slit, 106 wavelength, 106 Resolution functions in neutron scattering, 443 Resonance scattering, 594 Resonant Raman scattering, 657, 660, 661 Restraints in refinement, 691, 693±701 RHEED (reflection high-energy electron diffraction), 428 Rho operations, 763 Rhombohedral crystal system, 8 Ribosomes, scattering curves from, 111

SUBJECT INDEX Rietveld method, 56, 82, 422, 493, 496, 541, 690, 710 background, 711 indexing, 711 peak-shape function, 710±711 preferred orientation, 712 problems with, 711 using XED, 86 Rigid-motion parameters, 697 Robust/resistant methods, 689 Rock minerals, preparation of specimens, 171 Rocking curves, 37, 39, 188, 662 double-crystal, 529 Rod-like particles, 94 molecular weight, 93 radial inhomogeneity, 96 Rod polytypes, 760, 766 Rotating-anode tubes, 71, 189, 194 Rotation diagrams, 414 Rotation geometry setting with moving-crystal methods, 168 Rotation method, 29 normal beam, 31 Rotation of polarization, 593 Rotation/oscillation geometry, 31 Rotation twins, 10, 12 Rotational oscillation (libration), 589 Rutherford backscattering (RBS), 189 Rutile, intensity standard, 503 Sample mean, 813 Sample median, 813 Sample standard deviation, 813 SANS (small-angle neutron scattering), 105, 110 Satellite peaks, 75 reflections, 907 Saticon television camera tubes, 630 SAXS (small-angle X-ray scattering), 89 Sayre's equation, 428 Scale factor, estimation of, 691 Scanning-crystal monochromator, 622 Scanning electron microscopy (SEM), 540 Scanning range, 519±520 Scanning transmission electron microscope (STEM), 427 Scanning tunnelling microscope, 428 Scattering coherent multiple, 661 Compton, 242, 554, 599, 661 DelbruÈck, 242 diffuse, 261 elastic, 416 electron, 259 fluorescence, 661 inelastic, 416, 657 low-Q, 105 magnetic, 730 magnetic X-ray, 730 multiple, 661 multiple, deconvolution techniques, 393 multiple, incoherent, 661 multiple, intramolecular, 392 multiple, neutron diffraction, 177 multiple, Poisson distribution, 393 multiple, problems associated with, 392 neutrons, magnetic, 591 neutrons, nuclear, 593 nuclear and magnetic, 435 parasitic, 661

Scattering plasmon, 657, 661 potential, 594 Raman, 657 Rayleigh, 242, 554, 599 resonance, 594 resonant Raman, 657, 660, 661 spin-flip, 591 thermal diffuse, 416, 653, 655, 657, 661 Thomson, 733 Scattering amplitudes, 389 for electrons, 263±281 nuclear, 594 Scattering cross sections coherent, 594 Compton, 213 elastic, 213 elastic differential, 262 incoherent, 594 incoherent elastic, 595 inelastic, 213 magnetic, 593 nuclear, 591 pair-production, 213 Rayleigh, 213 total, 213, 594 total (tables), 223±229 Scattering factors atomic, 554, 566 complex, 188, 262 electron, 188, 259 for electrons, molecular, 390 for electrons, partial wave (Table 4.3.3.1), 286±377 for neutral atoms, 263 free atoms, 555 generalized, 565 hydrogen-atom, 565 interpolation, 565 magnetic, 461 parameterization, 262, 461 X-ray, Gaussian fits, 261 X-ray incoherent, 389 Scattering functions, 89 incoherent (Table 7.4.3.2), 658 Scattering intensities calculation of, 104 neutron, 105 Scattering-length densities, 105 match-point, 107 Scattering lengths, 444 bound nuclear, 594 coherent, 594 density, 105 density, match-point, 107 for neutrons, 188, 444 free, 594 total, 91 Scattering surfaces, 656 Scattering vector, 3, 90 Scherzer focus, 422, 423, 424, 426 SchroÈdinger wave equation, 186, 415, 735 Scintillation detectors, 619, 642, 664 Screen menu (CRT) for diffractometeroperation control, 64 Screenless rotation technique for largemolecule data collection, 169 Screw correlation tensor, 697 Secant methods, 683 Secondary extinction, 609, 611 Section topograph, 115

996

c:\itfc\sindex.3d

Seemann±Bohlin diffractometers, 495 Seemann±Bohlin geometry, 43 Seemann±Bohlin method, 52 advantages, 53 Selected-area channelling patterns, 540 Selected-area diffraction patterns, 428 Selected-area electron diffraction, 80, 538 Selection of crystals, 151 Self-centring slit, 45 Self-consistent field (Hartree±Fock) method, 243 SEM (scanning electron microscopy), 540 Semiconductor crystals, 428 Semiconductor detectors, 629, 642 Sensitivity (of lattice-spacing determination) increasing, 505 (methods of) highest, 531 Separation plots, structural, 774 Serial recording, 639 Serpentine±kaolin group, 766±769 Setting ±2, 47 anti-equi-inclination, 31 azimuthal, 168 equi-inclination, 31 flat cone, 31 Guinier, 39 photograph (precession), 35 precession geometry, 168 rotation geometry, 168 stationary crystal, 168 Setting angles in standard diffractometers, 516 Shadowing, 189 Shannon±Jaynes entropy, 691 Shape function, 520, 523 Shape of profile affected by collimation, 520 and precision, 519±521 Shape transform, 718 Sheets dioctahedral, 767 hetero-octahedral, 767 homo-octahedral, 767 meso-octahedral, 767 octahedral, 767 tetrahedral, 768 trioctahedral, 767 Short symbols for superspace groups, 921 Siegbahn notation, 191 Sievert, definition of, 959 Sigma symmetry (operations), 763 Signal-to-noise ratio, 633, 645 Significance tests, 702 Silicates (phyllosilicates), 766±771 Silicon, lattice parameter of, 490, 495, 499 Silver, lattice parameter of, 499 Silver behenate reflection angles, 503 Simple polytypes, 762 Simulations in SAXS, 103 Simultaneous reflection, 526 Single crystal characterization, 525 Laue diffraction, neutron, 87 monochromators (at synchrotron), 39 topography, 114 XED methods, 87 Single-crystal methods compared with powder methods, 506 photographic, 508±516 with counter recording, 516±533

SUBJECT INDEX Single-crystal X-ray techniques, 26, 505±536 classification of, 25 Single filters, 78 Single-order reflections (Laue), 27 Single-particle scattering, 110 Single-wire detectors, 82 Skewness, 586 Slater determinant, 722 Slater-type orbitals, 584 Slits antiscatter, 45 design, 45 self-centring, 45 Slower-than-sound neutrons, 657 Small-angle approximation, 80 Small-angle cameras, 99 Small-angle neutron scattering (SANS), 105, 110 Small-angle X-ray scattering (SAXS), 89 Small angles of incidence, 525 Small particles essential, 56 line broadening from, 62 Small spherical crystals, lattice-parameter determination of, 507, 525 Solid-state detectors, 82, 620, 642, 664 Solid-state effects, 400 Solid-state valence-band theory, 415 Soller collimators, 82, 432, 443 Soller slits, 46, 56, 82, 494, 521±522 Solutions, diffraction from, 24 Somatic effects, 960 Sound velocity, 656 Source intensity distribution and size, 73 Sources of X-radiation, 507 Space-group frequencies statistical modelling of, 897±906 tables, 905 Space groups and arithmetic crystal class, 15±20 arranged by arithmetic crystal class, 16 classification of, 15, 897 closest-packed, 897 distribution of molecular organic structures, 897 enantiomorphous pairs, 20 for close-packed structures, 755 frequency of, 15 impossible, 897 limitingly close-packed, 897 magnetic, 591 one-line symbol, 920 permissible, 897 symmetry, 695 two-line symbol, 921 Spallation neutron sources, 87, 189, 430±431 Spark erosion, 174 Sparse matrices, 685 Sparse matrix methods, 701 Spatial distortions, 41, 625, 633 Spatial non-uniformity, 633 Special reflection conditions, 921 for (3+1)-dimensional space groups, 934 Specific heat, 154 Specific isotopic labelling, 107 Specific surface, 93 Specimen aberrations, 48 absorption, 497, 498 displacement, 494, 498, 499 displacement error, correction, 528

Specimen factors, 60 fluorescence, 43, 78 focusing circle, 44 irradiated length, 45 mounting, 162 orientation, 44 preparation, 171, 177, 503 surface displacement, 48, 499, 503 transparency, 494, 497, 499 transparency aberration, 50 Specimen-tilt and beam-tilt error correction, 524 Spectral breadth, 189 Spectral brightness, 197 Spectral profiles, 48 Spectral purity, 72 Spectrometers, 395 asymmetric, 521±522 combined (techniques), 531 double-beam, 531 double-crystal, 510, 528±530 one-crystal, 521±526 stability of, 532 symmetric, 521±526, 529±531 time-of-flight, 444 triple-axis, 444, 531±532 triple-crystal, 531±532 Spectroscopy electron energy-loss, 391 infrared, 189 Raman, 189 Sphalerite, 754 Sphere(s), 92, 94 close-packing, 746, 752, 761 hollow, 92 of reflection (Laue sphere), 26 packing, 747 Spherical aberration, 421 Spherical Bessel function, 460, 565, 592 Spherical harmonic approximation, atomcentred, 714 Spherical harmonic functions, 581, 714, 722 Spherical harmonic multipole model, 715 Spherical symmetry, 103 Spherically symmetric particles, 96 Spin flipper, 442 of neutrons, 443 polarization, 388 Spin-contrast variation, 108 Spin density analysis of, 713±734 errors, 729 Spin-flip processes, 728 Spin-flip scattering, 591 Spin-magnetization densities, 725, 727, 731 Spin-orbit coupling, 727 Spin-orientation devices, 442 Spin-polarization effect, 732 Spin structure factor, 731 Spin-turn coil (flipper), 442 Split basis, 944 Spot size and shape, 37, 39 Sputter etching, 173 Sputtered thin films, 173 Square crystal system, 15 Square-root technique, convolution, 103 Square-wave modulation transfer function, 634 Stability of spectrometers, 532 Stability of X-ray sources, 72

997

c:\itfc\sindex.3d

Stable packing, 746, 750 Stacking faults, 754, 762 fringe patterns, 116 Stacking sequence, determination of, 757 Standard basis, 944 Standard crystal, 507, 531±532 lattice-parameter determination of, 507 Standard deviation, 679 Standard lattice bases, 938 Standard polytypes, 762 Standard reference materials, 498 Standard specimens, 501 Standard uncertainty, 681, 707 Standards intensity, 500 powder-diffraction, 498±499 Static model map, 714 Stationary-crystal method, 168 Stationary-phase focus, 422 Statistical errors of lattice-parameter determination, 505, 519, 523 Statistical fluctuations, 69, 492, 666 Statistical modelling, 904 Statistical significance tests, 702±705 Statistical validity in general, 702±705 of Rietveld method, 712 Statistics, 679 of recorded counts, 519 Poisson, 519 STEM (scanning transmission electron microscope), 427 Step size and count time, 64 Stereochemical constraints, 698 Stereographic projection of Kossel pattern, 513 Stereographic transformation, 29 Stibivanite, 769±772 Still exposure, monochromatic, 31 Still photographs for initial crystal setting, 169 Stochastic effects, 959 Stopping rules, 684 Storage phosphors, 623, 635 Storage rings, 196 synchrotron-radiation sources, 199 Strain, measurement of, 510, 516, 529 Strainmeter, 510 Straumanis film mounting, 509 Stress internal, 528 study of, 510, 516, 522 Strip-chart recordings, 63 Stroboscopic X-ray topography, 120 Structural classes, 904 Structural separation plots, 774 Structure amplitude, complex, 261 Structure analysis direct, 103 electron diffraction, 413 Structure determination of close-packed stackings, 756±758 Structure factor(s), 590, 941 determination, 735 magnetic, 725, 728, 730 magnetic orbital, 731 magnetic, unit-cell, 591 magnetic X-ray, 733 measurement by electron diffraction, 416 nuclear, 595, 725, 730 partial, 112 SANS, 112

SUBJECT INDEX Structure factor(s) spin, 731 X-ray, 737 Structure imaging, electron diffraction, 424 Structure prediction, 897 Structure refinement, 426 Student's t distribution, 704 Subfamilies, 769 Sub-grains, 114 Sublimed films, 176 Sulfur, Fermi level, 406 Superficial layers (see also Epitaxic layers), study of, 525 Superlattices, determination of, 525 Supermirrors, 435 Superposition structure, 763 Superspace, 944 embedding, 908 Superspace groups, 909, 912, 916, 940, 945 (2+1)-dimensional (table), 920 (2+2)-dimensional (table), 921 (3+1)-dimensional (table), 922±934 equivalent, 940 full symbols, 921 short symbols, 921 symbols for, 921 Superstructure, 919 Surface diffraction, 24 Surface of a particle, 93 Surface plasmons, 403 Surface-roughness scattering, 108, 128 Surface structure, 428 Symmetric arrangement in single-crystal methods, 509, 521±526, 529±531 Symmetry conditions for second cumulant tensors, 695±696 elements, non-crystallographic, 907 enhanced, 13 group, 908 of Patterson synthesis, 21 spherical, 103 Symmorphism, 15, 897 Synchrotron radiation, 54, 99, 114, 119, 187, 191, 596, 623, 653, 665, 711 camera systems for, 100 determination of wavelength, 495 facilities (for EXAFS), 219 for XED, 84 sources, 38, 198, 495 special applications, 189 spectrum, 197 Synchrotron X-ray topography, 120 Systematic errors (see also Aberrations), 490, 492, 501, 653±665, 707 background, 661±665 connected with collimation, 523±524 detection and treatment, 498±499, 707±709 estimation of, 535 in counter-diffractometer methods, 518, 535 in divergent-beam methods, 515 in photographic methods, 508, 515, 535 in single-crystal spectrometers, 521, 522, 523 in the Bond method, 523±525 of wavelength determination, 535 plasmon scattering, 660 Raman scattering, 660±661 reduced experimentally, 512, 515, 521, 526, 528±530

Systematic errors (see also Aberrations) reduced by detailed analysis of Kossel patterns, 512±515 reduced by extrapolation, 505, 535 reduced by least-squares refinement, 517 remanent, 408 specimen displacement, 517, 531 testing for, 498±499 thermal diffuse scattering, 653±657 white radiation, 661±665 Systematic interactions, 81 Take-off angle, 74 Talc±pyrophyllite group, 768±770 Tangent formula, 428 Tau operations, 764 Television area detectors, 630 Television camera tubes, 632 TEM (transmission electron microscopy), 171, 428, 540 Temperature correction, 524 Temperature dependence of lattice parameters, study of, 507, 530 Temperature factor(s), 586 anisotropic, 697 generalized, 586 librational, 724 Tensors, symmetry of, 695±696 Tetragonal crystal system, 7, 17 Tetrahedral sheet, 767 Texture axis, 412 basis, 412 fibre, 413 lamellar, 412 patterns, 412 Theoretical photo-effect data, 221 Theoretical Rayleigh scattering data, 221 Thermal diffuse scattering, 415, 653, 656, 661, 711 correction, anisotropic, 655 correction factor, 654 correction factor for thermal neutrons, 656 error, 653 Thermal effects, error connected with, 515 Thermal expansion, 154 study of, 510, 516, 522, 525, 529 Thermal neutron detection, 644±652 detection process, 644±648 detection systems, 649±651 electronic aspects, 648 via gas ionization, 645 via scintillation, 645 Thermal smearing, 723 Thermodynamic properties, 154 Thickness distance distribution function of, 103 fringes, 735 of crystal (sample), 512, 513 of lamellar particles, 93 Thin films and thinning, 173 Thin sections, 171, 174 Thinning solution, 175 Thomas±Fermi method, 659 Thomas±Fermi model, 243 Thomson formula, 90 Thomson scattering, 733 by a free electron, 242 Three-axis spectrometers, 444 Three-beam fringes, 422 Three-dimensional crystal classes, 15±20

998

c:\itfc\sindex.3d

Tilt(s) of beam, 524 of crystals, 530 Tilt-series reconstruction method, 427 Time-averaged flux, 431 Time-constant errors, 492 Time-of-flight neutron diffraction, 87, 431 Time-of-flight SANS, 106 Time-of-flight spectrometers, 444 Time reversal, 591 Topography, 113±123, 516, 525±527 detectors suitable for, 634 Topotaxy, 154 Total coincidence operations, 761 Total counting rates, 666 Total external reflection, 525 Total scattering cross section, 594 Total scattering lengths, 91 Townsend avalanches, 619 Trace of S singularity, 697 Transformation(s) compensating, 940 gnomonic, 29 stereographic, 29 twins, 10 Transition elements, Fermi level, 406 Transition-radiation X-rays, 192 Translation, internal, 912 Translation tensor, 697 Translations, compensating, 940 Transmission coefficients, 601 Transmission electron microscopy (TEM), 428, 540 preparation of specimens, 171 Transmission factor for XED, 86 Transmission function, 414, 432 Transmission geometry, 512, 513, 525 crystal thickness for, 512, 513 Transmission method, advantage of, 52 Transmission specimen, ±2 scan, 49 Transmission topographs, 113, 114, 124 Transparency aberration, 49 Traverse topograph, 115 Triaxial bodies, homogeneous, 92 Triclinic crystal system, 6 Tricontadipole, 716 Trigonal crystal system, 7, 18 Trigonometric intensity factors, 596 Trimercury dichloride disulfide, 766, 771±772 Trioctahedral sheet, 767 Triple-axis spectrometers, 444, 531±532 Triple-crystal spectrometers, 531±532 Triple isotopic replacement, 111 Triple-reflection scheme, 532 Truncation level, 518 optimum, 520 Trust-region methods, 683 Tungsten lattice parameter of, 499 reflection angles, 499, 502 Turbostratic structures, 760 TV cameras, X-ray-sensitive, 633 Twin(s) axis, 10, 11 axis, rational, 10 boundary, 10 Brazil, 11 centre, 10 centred lattice, 11 components, 10

SUBJECT INDEX Twin(s) cyclic, 10 DauphineÂ, 11 element, 10, 14 growth, 10 index, 11 interface, 10 inversion, 10, 12 lattices, 10 law, 10 mechanical (deformation, glide), 10 mimetic, 153 multiple, 10 operation, 10 plane, 10, 11 polysynthetic, 10 primitive lattice, 11 quartz, 11 reflection, 10, 12 repeated, 10 rotation, 10, 12 simulated Laue class of, 13 transformation, 10 Twinned crystal, reflection conditions, 13 Twinning, 10 by merohedry, 12 by pseudomerohedry, 12 in polytypes, 762 reciprocal-space implications, 12 Two-beam approximation, 80, 260 Two-circle diffractometers, 517 matrix formulae, 517 Two-dimensional crystal classes, 15, 16 Two-line symbols for Bravais classes, 915, 920 Ultramicrotomy, 171 Ultraviolet radiation, 187, 189 Umweganregung, 527 Undulators, 197 Uniformity of response, 625 Unit cell conventional, 913 conventional or centred, 2 magnetic structure factor, 591 primitive, 2 volume, 2 Unsmoothed high-tension supplies, 667 Upper-layer photographs (precession), 35 Upper-layer photographs (Weissenberg), 35 Upper quartile, 813 V-shaped detector recording, 32 Valence map, X ± X, X ± N, and X ±(X+N), 714 Vanadium, scattering from, 594 Variable reduction method, 693 Variance, 679 of centroid, 520 of measure of location, 519 of median, 520 of peak, 520 of single midpoint of chord, 520 Variance±covariance matrices, 680, 692, 707 Variances of measured intensities (recorded counts), 519 Variations in cell parameters, 522 VC (vicinity condition), 763 VC layers, 765 VC structures, 765

Vector(s) basis, 944 energy-flow, 119 lattice, 2 module, 907, 937, 944 Poynting, 119 reciprocal-lattice, 3 scattering, 3 Velocity of sound, 656 Velocity of the elastic wave, 654 Vermiculites, 765 Vertical divergence, 82 Vertical inclination correction for, 522 of incident beam, 522 of reflected beam, 522 Vibrating-string method for density measurement, 158 Vibration, normal modes, 653 Vicinity condition (VC), 763 Viruses, SANS, 106, 111 Visual estimation, 618 Voids in close-packed structures, 753 Voigt function, 67, 711 Volume of a homogeneous particle, 92 plasmons, 398 Volumenometry, 158 Voronoi polyhedron, 774 Waller±Hartree method, 659 Wave amplitudes, dynamical, 414 Wavefunction, 186 Wavelength calibration, 55 Wavelength determination, 506, 528, 533 accuracy of, 526 errors in, 541 Wavelength filter, 528 Wavelength normalization (Laue), 39 Wavelength problems, 492 Wavelength selection, 75 easy, 55 Wavelength shifts, 197 Wavelengths

-rays, 187 determination, 533 distribution, 506 errors, 492 synchrotron radiation, 187 uncertainty, 536 X-rays, 187, 191, 200, 201, 206, 209 Wavevector, 186 Weak-peak measurement, 65 Weak-phase-object (WPO) approximation, 423, 427 Weighted R factors, 68 Weissenberg camera, setting of single crystals, 168 Weissenberg diffractometer, 517 Weissenberg geometry, 34 White-beam energy-dispersive X-ray diffraction, 622 White-beam neutron diffraction, 87, 124 White radiation, 661 in double-crystal spectrometer, 529 in lattice-parameter determination, 507, 508, 529 streaks and crystal setting, 169 topography, 119 Whole-powder-pattern fitting, 68 Wien filter, 396

999

c:\itfc\sindex.3d

Wigglers, 197 Wigner±Eckart theorem, 727 Window thinning method, 174 WPO (weak-phase-object) approximation, 423, 427 Wurtzite, 754 Wyckoff positions, 914 XAFS (extended X-ray absorption fine structure), 24, 189, 213±220, 254, 409 as a short-range-order phenomenon, 214 data analysis, 217 XANES (X-ray absorption near-edge structure), 214±220, 258, 403 XED (X-ray energy-dispersive diffraction), 84 XPS (X-ray photoemission spectroscopy), 189 X-ray absorption, 599±612 X-ray absorption coefficients, 220 absolute measurement of, 214 data analysis of EXAFS, 217±218 experimental techniques, 214 X-ray absorption near-edge structure (XANES), 214±220, 258, 401 X-ray absorption spectra, 213±241 X-ray attenuation coefficients, 220 X-ray background over a spot, 34 X-ray beam extremely parallel, 532 highly divergent, 507, 508, 510±516 in single-crystal techniques, 507 well collimated, 507, 508, 536 X-ray diffraction detectors for, 618±638 texture patterns, 412 X-ray dispersion corrrections, 241 X-ray energies, 236 X-ray energy-dispersive diffraction (XED), 84 X-ray generators, 72 X-ray imaging systems, 633 X-ray incoherent scattering factors, 389 X-ray interferometry, 201 combined with optical interferometry, 533±534 X-ray levels, 191 X-ray microanalysis, 82 X-ray microscopy, 189 X-ray optics, 37 X-ray phosphors, 631 X-ray photoemission spectroscopy (XPS), 189 X-ray powder techniques, 42±79, 492±503 aberrations in, 47±50 energy-dispersive, 58 filters, 78±79 focusing geometries, 43 history, 42±43 literature, 42±43 microdiffractometry, 53±54 monochromators in, 43, 76±78 parallel-beam geometries, 54 Seemann±Bohlin geometry, 43, 52±53 Soller slits in, 50, 56 specimen fluorescence in, 43 zero position, 46 X-ray scattering, 554±590 magnetic, 733 X-ray-sensitive TV cameras, 633 X-ray source(s), 191 conventional, 37 in the sample, 510 laser plasma, 189

SUBJECT INDEX X-ray source(s) on the sample, 510 outside the sample, 510 radioactive, 195 synchrotron, 38, 196 X-ray tube, 193 X-ray spectra, 71, 74 Bremsstrahlung, 191 X-ray spectrometers Bragg, 510 double-crystal, 528 symmetric, 510, 521, 529 triple-crystal, 530 X-ray techniques, single-crystal, 26 X-ray topography, 115, 516, 525±527

X-ray tubes, 71±74, 193 loading, 195 power dissipation in, 195 X-ray wavelengths, 187, 191, 200±212, 221 conversion factors, 191 in single-crystal methods, 506 X-rays hard, 187 properties, 187 soft, 187 special applications, 189 Z-module, 907 Zebra patterns, 119 Zeeman polarizers, 442

1000

c:\itfc\sindex.3d

Zero-angle calibration, 494 Zero-error elimination, 517, 521, 523, 529 Zero-layer photographs (precession), 35 Zero-layer photographs (Weissenberg), 34 Zero line, 415 Zero-order Laue zone (ZOLZ), 418 Zero plane, 415 Zero-point correction, 517 Zero setting, 528 Zhdanov notation, 752 Zinc oxide (intensity standard), 503 Zinc sulfide, 754 ZOLZ (zero-order Laue zone), 418 Zone axis, 3, 10 Zone equation, 4

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