Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, ch. 1.3, p. 24   | 1 | 2 |

Section 1.3.1. Introduction

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

1.3.1. Introduction

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Symmetry relations using crystallographic group–subgroup relations have proved to be a valuable tool in crystal chemistry and crystal physics. Some important applications include:

  • (1) Structural relations between crystal-structure types can be worked out in a clear and concise manner by setting up family trees of group–subgroup relations (Bärnighausen, 1980[link]; Baur, 1994[link]; Baur & McLarnan, 1982[link]; Bock & Müller, 2002a[link],b[link]; Chapuis, 1992[link]; Meyer, 1981[link]; Müller, 1993[link], 2002[link]; Pöttgen & Hoffmann, 2001[link]).

  • (2) Elucidation of problems concerning twinned crystals and antiphase domains (cf. Section 1.2.7[link] , p. 18; Bärnighausen, 1980[link]; van Tendeloo & Amelinckx, 1974[link]; Wondratschek & Jeitschko, 1976[link]).

  • (3) Changes of structures and physical properties taking place during phase transitions: applications of Landau theory (Aroyo & Perez-Mato, 1998[link]; Birman, 1966a[link],b[link]; Cracknell, 1975[link]; Izyumov & Syromyatnikov, 1990[link]; Landau & Lifshitz, 1980[link]; Salje, 1990[link]; Stokes & Hatch, 1988[link]; Tolédano & Tolédano, 1987[link]).

  • (4) Prediction of crystal-structure types and calculation of the numbers of possible structure types (McLarnan, 1981a[link],b[link],c[link]; Müller, 1978[link], 1980[link], 1981[link], 1986[link], 1992[link], 1998[link], 2003[link]).

All of these applications require consideration of the relations between the atomic sites in a space group and in the corresponding subgroups.


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Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782.
Birman, J. L. (1966b). Simplified theory of symmetry change in second–order phase transitions: application to V3Si. Phys. Rev. Lett. 17, 1216–1219.
Bock, O. & Müller, U. (2002a). Symmetrieverwandtschaften bei Varianten des Perowskit-Typs. Acta Cryst. B58, 594–606.
Bock, O. & Müller, U. (2002b). Symmetrieverwandtschaften bei Varianten des ReO3-Typs. Z. Anorg. Allg. Chem. 628, 987–992.
Chapuis, G. C. (1992). Symmetry relationships between crystal structures and their practical applications. Modern perspectives in inorganic chemistry, edited by E. Parthé, pp. 1–16. Dordrecht: Kluwer Academic Publishers.
Cracknell, A. P. (1975). Group theory in solid state physics. New York: Taylor and Francis Ltd. and Pergamon.
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Müller, U. (2002). Kristallpackungen mit linear koordinierten Atomen. Z. Anorg. Allg. Chem. 628, 1269–1278.
Müller, U. (2003). Die Zahl der Substitutions- und Leerstellenvarianten des NaCl-Typs bei verdoppelter Elementarzelle (a, b, 2c). Z. Anorg. Allg. Chem. 629, 487–492.
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Wondratschek, H. & Jeitschko, W. (1976). Twin domains and antiphase domains. Acta Cryst. A32, 664–666.

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