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

International Tables for Crystallography (2011). Vol. A1, ch. 1.7, pp. 57-58

Section 1.7.2. Databases and retrieval tools

Mois I. Aroyo,a* J. Manuel Perez-Mato,a Cesar Capillasa and Hans Wondratschekb

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain, and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.7.2. Databases and retrieval tools

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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.

1.7.2.1. Space-group data

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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 [\overline{1}]) for the centrosymmetric groups listed with respect to two origins in IT A.

The space-group database includes the following symmetry information:

  • (i) The generators and the representatives of the general position of each space group specified by its IT A number and Hermann–Mauguin symbol;

  • (ii) The special Wyckoff positions including the Wyckoff letter, Wyckoff multiplicity, the site-symmetry group and the set of coset representatives, as given in IT A;

  • (iii) The reflection conditions including the general and special conditions;

  • (iv) The affine and Euclidean normalizers of the space groups (cf. IT A, Part 15[link] ). They are described by sets of additional symmetry operations that generate the normalizers successively from the space groups. The database includes the additional generators of the Euclidean normalizers for the general-cell metrics as listed in Tables 15.2.1.3[link] and 15.2.1.4[link] of IT A. These Euclidean normalizers are also affine normalizers for all cubic, hexagonal, trigonal, tetragonal and part of the orthorhombic space-group types. For the rest of the orthorhombic space groups, the type of the affine normalizer coincides with the highest-symmetry Euclidean normalizer of that space group and the corresponding additional generators form part of the database (cf. Table 15.2.1.3[link] of IT A). The affine normalizers of triclinic and monoclinic groups are not isomorphic to groups of motions and they are not included in the normalizer database of the Bilbao Crystallographic Server.

  • (v) The assignment of Wyckoff positions to Wyckoff sets as found in Table 14.2.3.2[link] of IT A.

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[link] 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.

1.7.2.2. Database on maximal subgroups

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1.7.2.2.1. Maximal subgroups of indices 2, 3 and 4 of the space groups

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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 [\cal H] is specified by its IT A number, the index in the group [\cal G] and the transformation matrix–column pair (P, p) that relates the bases of [\cal H] and [\cal G]: [({\bf a}',{\bf b}',{\bf c}')_{{\cal H}} = ({\bf a},{\bf b},{\bf c})_{{\cal G}}{\bi P} .\eqno(1.7.2.1)]

The column p = (p1, p2, p3) of coordinates of the origin [O_{{\cal H}}] of [{\cal H}] is referred to the coordinate system of [{\cal G}].

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 [({\bf a},{\bf b},{\bf c})_{{\cal G}}] of [{\cal G}] to the standard basis of [{\cal H}] (see Section 2.1.2.5[link] 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 [{\cal H}] as subsets of the elements of [{\cal G}]. This is achieved by an option in MAXSUB which transforms the general-position representatives of [{\cal H}] by the corresponding matrix–column pair (P, p)−1 to the coordinate system of [{\cal G}]. In addition, one can obtain the splittings of all Wyckoff positions of [{\cal G}] to those of [{\cal H}].

1.7.2.2.2. Maximal isomorphic subgroups

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Maximal subgroups of index higher than 4 have indices p, p2 or p3, 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.]








































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