International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, ch. 1.7, pp. 6668
Section 1.7.4. Relations of Wyckoff positions for a group–subgroup pair of space groups^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E48080 Bilbao, Spain, and ^{b}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
Consider two group–subgrouprelated space groups . Atoms that are symmetrically equivalent under , i.e. belong to the same orbit of , may become nonequivalent under , (i.e. the orbit splits) and/or their site symmetries may be reduced. The orbit relations induced by the symmetry reduction are the same for all orbits belonging to a Wyckoff position, so one can speak of Wyckoffposition relations or splitting of Wyckoff positions. Theoretical aspects of the relations of the Wyckoff positions for a group–subgroup pair of space groups have been treated in detail by Wondratschek (1993) (see also Section 1.5.3 ). A compilation of the Wyckoffposition splittings for all space groups and all their maximal subgroups is published as Part 3 of this volume. However, for certain applications it is easier to have the appropriate computer tools for the calculations of the Wyckoffposition splittings for : for example, when is not a maximal subgroup of , or when the space groups are related by transformation matrices different from those listed in the tables of Part 3 . The program WYCKSPLIT (Kroumova, PerezMato & Aroyo, 1998) calculates the Wyckoffposition splittings for any group–subgroup pair. In addition, the program provides further information on Wyckoffposition splittings that is not listed in Part 3 , namely the relations between the representatives of the orbit of and the corresponding representatives of the suborbits of .
To simplify the notation, we assume in the following that the group , its Wyckoffposition representatives and the points of the orbits are referred to the basis of the subgroup .

The program WYCKSPLIT calculates the splitting of the Wyckoff positions for a group–subgroup pair given the corresponding transformation relating the coordinate systems of and .
Input to WYCKSPLIT:
The program needs as input the following information:
Output of WYCKSPLIT:
WYCKSPLIT can treat group or subgroup data in unconventional settings if the transformation matrices to the corresponding conventional settings are given.
Example 1.7.4.1.1
To illustrate the calculation of the Wyckoffposition splitting we consider the group–subgroup pair (No. 136) > (No. 65) of index 2, see Fig. 1.7.4.1. The relation between the conventional bases of the group and of the subgroup is retrieved by the program MAXSUB and is given by , . The general position of splits into two suborbits of the general position of : This splitting is directly related to the coset decomposition of with respect to . As coset representatives, i.e. as points which determine the splitting of the general position, one can choose and (referred to the basis of the subgroup).

Sequence of calculations of WYCKSPLIT for the splitting of the Wyckoff positions and of , No. 136, with respect to its subgroup , No. 65, of index 2. are the orbits of in the basis of . 
The splitting of any special Wyckoff position is obtained from the splitting of the general position. The consecutive steps of the splittings of the special positions and are shown in Fig. 1.7.4.1. First it is necessary to transform the representatives of to the basis of , which gives the orbits and . The substitution of the values x = 0, y = 0, z = 0 in the coordinate triplets of the decomposed general position of (cf. the corresponding output of WYCKSPLIT) gives two suborbits of multiplicity 2 for the position: and . The assignment of the suborbits to the Wyckoff positions of (cf. Table 1.7.4.1) is straightforward. Summarizing: the Wyckoff position splits into two independent positions of with no sitesymmetry reduction:

No splitting occurs for the case of the special position orbit: the result is one orbit of multiplicity 8, . The assignment of is also obvious: there are five Wyckoff positions of of multiplicity 8 but four of them are discarded as they have fixed parameters 0 or (Table 1.7.4.1). The orbit belongs to the Wyckoff position .
As expected, the sum of the sitesymmetry reduction factors equals the index of in for both cases (cf. Section 1.5.3 ). The loss of the fourfold inversion axis results in the appearance of an additional degree of freedom corresponding to the variable parameter of .
References
Kroumova, E., PerezMato, J. M. & Aroyo, M. I. (1998). WYCKSPLIT: a computer program for determination of the relations of Wyckoff positions for a group–subgroup pair. J. Appl. Cryst. 31, 646.Wondratschek, H. (1993). Splitting of Wyckoff positions (orbits). Mineral. Petrol. 48, 87–96.