Section 1.1.4. Applications of group–subgroup relations

Phase transitions . In 1937, Landau introduced the idea of the order parameter for the description of second-order phase transitions (Landau, 1937). Landau theory has turned out to be very useful in the understanding of phase transitions and related phenomena. Such a transition can only occur if there is a group–subgroup relation between the space groups of the two crystal structures. Often only the space group of one phase is known (usually the high-temperature phase) and subgroup relations help to eliminate many groups as candidates for the unknown space group of the other phase. Landau & Lifshitz (1980) examined the importance of group–subgroup relations further and formulated two theorems regarding the index of the group–subgroup pair. The significance of the subgroup data in second-order phase transitions was also pointed out by Ascher (1966, 1967), who formulated the maximal-subgroup rule: `The symmetry group of a phase that arises in a ferroelectric transition is a maximal polar subgroup of the group of the high-temperature phase.' There are analogous applications of the maximal-subgroup rule (with appropriate modifications) to other types of continuous transitions.

The group-theoretical aspects of Landau theory have been worked out in great detail with major contributions by Birman (1966a,b), Cracknell (1975), Stokes & Hatch (1988), Tolédano & Tolédano (1987) and many others. For example, Landau theory gives additional criteria based on thermodynamic arguments for second-order phase transitions. The general statements are reformulated into group-theoretical rules which permit a phase-transition analysis without the tedious algebraic treatment involving high-order polynomials. The necessity of having complete subgroup data for the space groups for the successful implementation of these rules was stated by Deonarine & Birman (1983): ` there is a need for tables yielding for each of the 230 three-dimensional space groups a complete lattice of decomposition of all its subgroups.' Domain-structure analysis (Chapter 3.4 in IT D; Janovec & Přívratská, 2003) and symmetry-mode analysis (Aroyo & Perez-Mato, 1998) are further aspects of phase-transition problems where group–subgroup relations between space groups play an essential role. Domain structures are also considered in Section 1.2.7 .

In treating successive phase transitions within Landau theory, Levanyuk & Sannikov (1971) introduced the idea of a hypothetical parent phase whose symmetry group is a supergroup of the observed (initial) space group. Moreover, the detection of pseudosymmetries is necessary for the prediction of higher-temperature phase transitions, cf. Kroumova et al. (2002) and references therein.

In a reconstructive phase transition, there is no group–subgroup relation between the symmetries of the two structures. Nevertheless, it has been pointed out that the `transition path' between the two structures may involve an intermediate unstable structure whose space group is a common subgroup of the space groups of the two phases (Sowa, 2001; Stokes & Hatch, 2002).

Overlooked symmetry . In recent times, the number of crystal-structure determinations with a wrongly assigned space group has been increasing (Baur & Kassner, 1992; Marsh et al., 2002). Frequently, space groups with too low a symmetry have been chosen. In most cases, the correct space group is a supergroup of the space group that has been assigned. A criterion for the correct assignment of the space group is given by Fischer & Koch (1983). Computer packages for treating the problem can be made more efficient if the possible supergroups are known.

Twinning can also lead to a wrong space-group assignment if it is not recognized, as a twinned crystal can feign a higher or lower symmetry. The true space group of the correct structure is usually a supergroup or subgroup of the space group that has been assumed (Nespolo & Ferraris, 2004).

Relations between crystal structures . Working out relations between different crystal structures with the aid of crystallographic group–subgroup relations was systematically developed by Bärnighausen (1980). The work became more widely known through a number of courses taught in Germany and Italy from 1976 to 1996 (in 1984 as a satellite meeting to the Congress of the International Union of Crystallography). For a script of the 1992 course, see Chapuis (1992). The basic ideas can also be found in the textbook by Müller (1993).

According to Bärnighausen, a family tree of group–subgroup relations is set up. At the top of the tree is the space group of a simple, highly symmetrical structure, called the aristotype by Megaw (1973) or the basic structure by Buerger (1947, 1951). The space groups of structures resulting from distortions or atomic substitutions (the hettotypes or derivative structures) are subgroups of the space group of the aristotype. Apart from many smaller Bärnighausen trees, some trees that interrelate large numbers of crystal structures have been published, cf. Section 1.3.1 . Such trees may even include structures as yet unknown, i.e. the symmetry relations can also serve to predict new structure types that are derived from the aristotype; in addition, the number of such structure types can be calculated for each space group of the tree (McLarnan, 1981a,b,c; Müller, 1992, 1998, 2003).

Setting up a Bärnighausen tree not only requires one to find the group–subgroup relations between the space groups involved. It also requires there to be an exact correspondence between the atomic positions of the crystal structures considered. For a given structure, each atomic position belongs to a certain Wyckoff position of the space group. Upon transition to a subgroup, the Wyckoff position will or will not split into different Wyckoff positions of the subgroup. With the growing number of applications of group–subgroup relations there had been an increasing demand for a list of the relations of the Wyckoff positions for every group–subgroup pair. These listings are accordingly presented in Part 3 of this volume.