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

International Tables for Crystallography (2011). Vol. A1, ch. 1.6, p. 44

Section 1.6.1. Introduction

Ulrich Müllera*

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

1.6.1. 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 (see following sections).

  • (2) Elucidation of problems concerning twinned crystals and antiphase domains (see Section 1.6.6[link]).

  • (3) Changes of structures and physical properties taking place during phase transitions; applications of Landau theory (Aizu, 1970[link]; Aroyo & Perez-Mato, 1998[link]; Birman, 1966a[link],b[link], 1978[link]; Cracknell, 1975[link]; Howard & Stokes, 2005[link]; Igartua et al., 1996[link]; Izyumov & Syromyatnikov, 1990[link]; Landau & Lifshitz, 1980[link]; Lyubarskii, 1960[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 (see Section[link]).

  • (5) Solution of the phase problem in the crystal structure analysis of proteins (Di Costanzo et al., 2003[link]).

Bärnighausen (1975[link], 1980[link]) presented a standardized procedure to set forth structural relations between crystal structures with the aid of symmetry relations between their space groups. For a review on this subject see Müller (2004[link]). Short descriptions are given by Chapuis (1992[link]) and Müller (2006[link]). The main concept is to start from a simple, highly symmetrical crystal structure and to derive more and more complicated structures by distortions and/or substitutions of atoms. Similar to the `diagrams of lattices of subgroups' used in mathematics, a tree of group–subgroup relations between the space groups involved, now called a Bärnighausen tree, serves as the main guideline. The highly symmetrical starting structure is called the aristotype after Megaw (1973[link]) or basic structure after Buerger (1947[link], 1951[link]); other terms used in the literature on phase transitions in physics are prototype or parent structure. The derived structures are the hettotypes or derivative structures. In Megaw's (1973[link]) terminology, the structures mentioned in the tree form a family of structures.

The structure type to be chosen as the aristotype depends on the specific problem and, therefore, the term aristotype cannot be defined in a strict manner. For example, a body-centred packing of spheres (space group [Im\overline 3m]) can be chosen as the aristotype for certain intermetallic structures. By symmetry reduction due to a loss of the centring, the CsCl type (space group [Pm\overline 3m]) can be derived. However, if all the structures considered are ionic, there is no point in starting from the body-centred packing of spheres and one can choose the CsCl type as the aristotype.


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