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. 57-58
Section 1.7.2. Databases and retrieval tools^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain, and ^{b}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
The databases form the core of the Bilbao Crystallographic Server and the information stored in them is used by all computer programs available on the server. The following description is restricted to the databases related to the symmetry relations between space groups; these are the databases that include space-group data from IT A and subgroup data from IT A1.
The programs and databases of the Bilbao Crystallographic Server use specific settings of space groups (hereafter referred to as standard or default settings) that coincide with the conventional space-group descriptions found in IT A. For space groups with more than one description in IT A, the following settings are chosen as standard: unique axis b setting, cell choice 1 for monoclinic groups; hexagonal axes setting for rhombohedral groups; and origin choice 2 (origin at ) for the centrosymmetric groups listed with respect to two origins in IT A.
The space-group database includes the following symmetry information:
The data from the databases can be accessed using the simple retrieval tools, which use as input the number of the space group (IT A numbers). It is also possible to select the group from a table of IT A numbers and Hermann–Mauguin symbols. The output of the program GENPOS contains a list of the generators or the general positions and provides the possibility to obtain the same data in different settings either by specifying the transformation matrix to the new basis or selecting one of the 530 settings listed in Table 4.3.2.1 of IT A. A list of the Wyckoff positions for a given space group in different settings can be obtained using the program WYCKPOS. The Wyckoff-position representatives for the nonstandard settings of the space groups are specified by the transformed coordinates of the representatives of the corresponding default settings. The assignments of the Wyckoff positions to Wyckoff sets are retrieved by the program WYCKSETS. This program also lists a set of coset representatives of the decompositions of the normalizers with respect to the space groups and the transformation of the Wyckoff positions under the action of these coset representatives. The programs NORMALIZER and HKLCOND give access to the data for normalizers and reflection conditions.
All maximal non-isomorphic subgroups and maximal isomorphic subgroups of indices 2, 3 and 4 of each space group can be retrieved from the database using the program MAXSUB. Each subgroup is specified by its IT A number, the index in the group and the transformation matrix–column pair (P, p) that relates the bases of and :
The column p = (p_{1}, p_{2}, p_{3}) of coordinates of the origin of is referred to the coordinate system of .
It is important to note that, in contrast to the data listed in IT A1, the matrix–column pairs (P, p) used by the programs of the server transform the standard basis of to the standard basis of (see Section 2.1.2.5 for the special rules for the settings of the subgroups used in IT A1). The different maximal subgroups are distributed in classes of conjugate subgroups. For certain applications it is necessary to represent the subgroups as subsets of the elements of . This is achieved by an option in MAXSUB which transforms the general-position representatives of by the corresponding matrix–column pair (P, p)^{−1} to the coordinate system of . In addition, one can obtain the splittings of all Wyckoff positions of to those of .
Maximal subgroups of index higher than 4 have indices p, p^{2} or p^{3}, where p is a prime. They are isomorphic subgroups and are infinite in number. In IT A1, the isomorphic subgroups are listed as members of series under the heading `Series of maximal isomorphic subgroups'. In addition, the isomorphic subgroups of indices 2, 3 and 4 are listed individually. The program SERIES provides access to the database of maximal isomorphic subgroups on the Bilbao Crystallographic Server. Apart from the parametric IT A1 descriptions of the series, its output provides the individual listings of all maximal isomorphic subgroups of indices as high as 27 for all space groups, except for the cubic ones where the maximum index is 125. The format and content of the subgroup data are similar to those of the MAXSUB access tool. In addition, there is a special tool (under `define a maximal index' on the SERIES web form) that permits the online generation of maximal isomorphic subgroups of any index up to 131 for all space groups. [Note that these data are only generated online and do not form part of the (static) database of isomorphic subgroups.]