INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY Volume A SPACE-GROUP SYMMETRY Edited by THEO HAHN Contributing authors H. Arnold: Institut fu¨r Kristallographie, Rheinisch- Th. Hahn: Institut fu¨r Kristallographie, Rheinisch- Westfa¨lische Technische Hochschule, D-52056 Aachen, Westfa¨lische Technische Hochschule, D-52056 Aachen, Germany (present address: Am Beulardstein 22, D-52072 Germany. [1, 2, 10] Aachen, Germany). [2, 5, 11] H. Klapper: Institut fu¨r Kristallographie, Rheinisch- M. I. Aroyo : Faculty of Physics, University of Sofia, bulv. J. Westfa¨lische Technische Hochschule, D-52056 Aachen, Boucher 5, 1164 Sofia, Bulgaria (present address: Germany (present address: Mineralogisch-Petrologisches Departamento de Fisica de la Materia Condensada, Institut, Universita¨t Bonn, D-53115 Bonn, Germany). Facultad de Ciencias, Universidad del Pais Vasco, [10] Apartado 644, 48080 Bilbao, Spain). [Computer production of space-group tables] E. Koch: Institut fu¨r Mineralogie, Petrologie und E. F. Bertaut†: Laboratoire de Cristallographie, CNRS, Kristallographie, Philipps-Universita¨t, D-35032 Grenoble, France. [4, 13] Marburg, Germany. [11, 14, 15] Y. Billiet:De´partement de Chimie, Faculte´ des Sciences et P. B. Konstantinov: Institute for Nuclear Research and Techniques, Universite´ de Bretagne Occidentale, Brest, Nuclear Energy, 72 Tzarigradsko Chaussee, BG-1784 France (present address: 8 place de Jonquilles, F-29860 Sofia, Bulgaria. [Computer production of space-group Bourg-Blanc, France). [13] tables] M. J. Buerger †: Department of Earth and Planetary G. A. Langlet†: De´partement de Physico-Chimie, CEA, Sciences, Massachusetts Institute of Technology, CEN Saclay, Gif sur Yvette, France. [2] Cambridge, MA, USA. [2, 3] A. Looijenga-Vos H. Burzlaff: Universita¨t Erlangen–Nu¨rnberg, Robert- : Laboratorium voor Chemische Fysica, Koch-Strasse 4a, D-91080 Uttenreuth, Germany. [9.1, Rijksuniversiteit Groningen, The Netherlands (present 12] address: Roland Holstlaan 908, 2624 JK Delft, The Netherlands). [2, 3] J. D. H. Donnay†: Department of Geological Sciences, McGill University, Montreal, Quebec, Canada. [2] U. MuÈ ller: Fachbereich Chemie, Philipps-Universita¨t, W. Fischer: Institut fu¨r Mineralogie, Petrologie und D-35032 Marburg, Germany. [15.1, 15.2] Kristallographie, Philipps-Universita¨t, D-35032 P. M. de Wolff†: Laboratorium voor Technische Marburg, Germany. [2, 11, 14, 15] Natuurkunde, Technische Hogeschool, Delft, The D. S. Fokkema: Rekencentrum der Rijksuniversiteit, Netherlands. [2, 9.2] Groningen, The Netherlands. [Computer production of H. Wondratschek space-group tables] : Institut fu¨r Kristallographie, Universita¨t, D-76128 Karlsruhe, Germany. [2, 8] B. Gruber: Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, H. Zimmermann: Institut fu¨r Angewandte Physik, Lehrstuhl Malostranske´na´m. 25, CZ-11800 Prague 1, Czech fu¨r Kristallographie und Strukturphysik, Universita¨t Republic (present address: Socharˇska´ 14, CZ-17000 Erlangen–Nu¨rnberg, Bismarckstrasse 10, D-91054 Prague 7, Czech Republic). [9.3] Erlangen, Germany. [9.1, 12] † Deceased. v Contents PAGE Foreword to the First Edition /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xiii Foreword to the Second, Revised Edition /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xiii Foreword to the Third, Revised Edition /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xiv Foreword to the Fourth, Revised Edition /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xiv Foreword to the Fifth, Revised Edition /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xv Preface /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. xvi Computer production of Volume A /D. S. Fokkema, M. I. Aroyo and P. B. Konstantinov) .. .. .. .. .. .. .. .. .. .. .. .. .. xviii PART 1. SYMBOLS AND TERMS USED IN THIS VOLUME .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 1.1. Printed symbols for crystallographic items /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.1. Vectors, coefficients and coordinates .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.2. Directions and planes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.3. Reciprocal space .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.4. Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1.5. Spaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.1.6. Motions and matrices .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.1.7. Groups .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 1.2. Printed symbols for conventional centring types /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 1.2.1. Printed symbols for the conventional centring types of one-, two- and three-dimensional cells .. .. .. .. .. .. .. .. 4 1.2.2. Notes on centred cells .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 1.3. Printed symbols for symmetry elements /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.3.1. Printed symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 1.3.2. Notes on symmetry elements and symmetry operations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6 1.4. Graphical symbols for symmetry elements in one, two and three dimensions /Th. Hahn) .. .. .. .. .. .. .. .. .. .. .. 7 1.4.1. Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7 1.4.2. Symmetry planes parallel to the plane of projection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7 À À 1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes 43m and m3m only) .. .. .. .. 8 1.4.4. Notes on graphical symbols of symmetry planes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1.4.5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure .. .. .. .. .. .. .. 9 1.4.6. Symmetry axes parallel to the plane of projection .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.4.7. Symmetry axes inclined to the plane of projection (in cubic space groups only) .. .. .. .. .. .. .. .. .. .. .. .. 10 References .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 PART 2. GUIDE TO THE USE OF THE SPACE-GROUP TABLES .. .. .. .. .. .. .. .. .. .. .. .. .. .. 13 2.1. Classification and coordinate systems of space groups /Th. Hahn and A. Looijenga-Vos) .. .. .. .. .. .. .. .. .. .. .. 14 2.1.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 2.1.2. Space-group classification .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 2.1.3. Conventional coordinate systems and cells .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 2.2. Contents and arrangement of the tables /Th. Hahn and A. Looijenga-Vos) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 17 2.2.1. General layout .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 17 2.2.2. Space groups with more than one description .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 17 vii CONTENTS 2.2.3. Headline .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 17 2.2.4. International (Hermann–Mauguin) symbols for plane groups and space groups (cf. Chapter 12.2) .. .. .. .. .. .. 18 2.2.5. Patterson symmetry .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19 2.2.6. Space-group diagrams .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 20 2.2.7. Origin .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 24 2.2.8. Asymmetric unit .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 2.2.9. Symmetry operations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 26 2.2.10. Generators .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27 2.2.11. Positions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27 2.2.12. Oriented site-symmetry symbols .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28 2.2.13. Reflection conditions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 29 2.2.14. Symmetry of special projections .. .. .. .. .. .. .