International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.1, pp. 3-4
Section 1.1.3. Development of the theory of group–subgroup relations^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,^{b}Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and ^{c}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
The systematic survey of group–subgroup relations of space groups started with the fundamental publication by Carl Hermann (1929). This paper is the last in a series of four publications, dealing with
In paper (IV), Hermann introduced two distinct kinds of subgroups. The translationengleiche subgroups of a space group have retained all translations of but belong to a crystal class of lower symmetry than the crystal class of (Hermann used the term zellengleiche instead of translationengleiche, see the footnote to Definition 1.2.6.2.1 in Chapter 1.2). The klassengleiche subgroups are those which belong to the same crystal class as , but have lost translations compared with . General subgroups are those which have lost translations as well as crystal-class symmetry. Hermann proved the theorem, later called Hermann's theorem, that any general subgroup is a klassengleiche subgroup of a uniquely determined translationengleiche subgroup of . In particular, this implies that a maximal subgroup of is either a translationengleiche subgroup or a klassengleiche subgroup of .
Because of the strong relation (homomorphism) between a space group and its point group , the set of translationengleiche subgroups of is in a one-to-one correspondence with the set of subgroups of the point group . The crystallographic point groups are groups of maximal order 48 with well known group–subgroup relations and with not more than 96 subgroups. Thus, the maximal translationengleiche subgroups of any space group can be obtained easily by comparison with the subgroups of its point group . The kind of derivation of the space-group types by H. Heesch (1930) also gives access to translationengleiche subgroups. In IT 35, the types of the translationengleiche subgroups were listed for each space group [for a list of corrections to these data, see Ascher et al. (1969)]. A graph of the group–subgroup relations between the crystallographic point groups can also be found in IT 35; the corresponding graphs for the space groups were published by Ascher (1968). In these lists and graphs the subgroups are given only by their types, not individually.
The group–subgroup relations between the space groups were first applied in Vol. 1 of Strukturbericht (1931). In this volume, a crystal structure is described by the coordinates of the atoms, but the space-group symmetry is stated not only for spherical particles but also for molecules or ions with lower symmetry. Such particles may reduce the site symmetry and with it the space-group symmetry to that of a subgroup. In addition, the symmetry reduction that occurs if the particles are combined into larger structural units is stated. The listing of these detailed data was discontinued both in the later volumes of Strukturbericht and in the series Structure Reports. Meanwhile, experience had shown that there is no point in assuming a lower symmetry of the crystal structure if the geometrical arrangement of the centres of the particles does not indicate it.
With time, not only the classification of the crystal structures but also a growing number of investigations of (continuous) phase transitions increased the demand for data on subgroups of space groups. Therefore, when the Executive Committee of the International Union of Crystallography decided to publish a new series of International Tables for Crystallography, an extension of the subgroup data was planned. Stimulated and strongly supported by the mathematician J. Neubüser, the systematic derivation of the subgroups of the plane groups and the space groups began. The listing was restricted to the maximal subgroups of each space group, because any subgroup of a space group can be obtained by a chain of maximal subgroups.
The derivation by Neubüser & Wondratschek started in 1965 with the translationengleiche subgroups of the space groups, because the complete set of these (maximally 96) subgroups could be calculated by computer. All klassengleiche subgroups of indices 2, 3, 4, 6, 8 and 9 were also obtained by computer. As the index of a maximal non-isomorphic subgroup of a space group is restricted to 2, 3 or 4, all maximal non-isomorphic subgroups of all space groups were contained in the computer outputs. First results and their application to relations between crystal structures are found in Neubüser & Wondratschek (1966). In the early tables, the subgroups were only listed by their types. For International Tables, an extended list of maximal non-isomorphic subgroups was prepared. For each space group the maximal translationengleiche subgroups and those maximal klassengleiche subgroups for which the reduction of the translations could be described as `loss of centring translations' of a centred lattice are listed individually. For the other maximal klassengleiche subgroups, i.e. those for which the conventional unit cell of the subgroup is larger than that of the original space group, the description by type was retained, because the individual subgroups of this kind were not completely known in 1983. The deficiency of such a description becomes clear if one realizes that a listed subgroup type may represent 1, 2, 3, 4 or even 8 individual subgroups.
In the present Volume A1, all maximal non-isomorphic subgroups are listed individually, in Chapter 2.2 for the plane groups and in Chapters 2.3 and 3.2 for the space groups. In addition, graphs for the translationengleiche subgroups (Chapter 2.4 ) and for the klassengleiche subgroups (Chapter 2.5 ) supplement the tables. After several rounds of checking by hand and after comparison with other listings, e.g. those by H. Zimmermann (unpublished) or by Neubüser and Eick (unpublished), intensive computer checking of the hand-typed data was carried out by F. Gähler as described in Chapter 1.4 .
The mathematician G. Nebe describes general viewpoints and new results in the theory of subgroups of space groups in Chapter 1.5 .
The maximal isomorphic subgroups are a special subset of the maximal klassengleiche subgroups. Maximal isomorphic subgroups are treated separately because each space group has an infinite number of maximal isomorphic subgroups and, in contrast to non-isomorphic subgroups, there is no limit for the index of a maximal isomorphic subgroup of .
An isomorphic subgroup of a space group seems to have first been described in a crystal–chemical relation when the crystal structure of Sb_{2}ZnO_{6} (structure type of tapiolite, Ta_{2}FeO_{6}) was determined by Byström et al. (1941): `If no distinction is drawn between zinc and antimony, this structure appears as three cassiterite-like units stacked end-on-end' (Wyckoff, 1965). The space group of Sb_{2}ZnO_{6} is a maximal isomorphic subgroup of index 3 with of the space group (, No. 136) of cassiterite SnO_{2} (rutile type).
The first systematic study attempting to enumerate all isomorphic subgroups (not just maximal ones) for each space-group type was by Billiet (1973). However, the listing was incomplete and, moreover, in the case of enantiomorphic pairs of space-group types, only those with the same space-group symbol (called isosymbolic space groups) were taken into account.
Sayari (1976) derived the conventional bases for all maximal isomorphic subgroups of all plane groups. The general laws of number theory which underlie these results for plane-group types p4, p3 and p6 and space-group types derived from point groups 4, , , 3, , 6, and were published by Müller & Brelle (1995). Bertaut & Billiet (1979) suggested a new analytical approach for the derivation of all isomorphic subgroups of space and plane groups.
Because of the infinite number of maximal isomorphic subgroups, only a few representatives of lowest index are listed in IT A with their lattice relations but without origin specification, cf. IT A (2002), Section 2.2.15.2 . Part 13 of IT A (Billiet & Bertaut, 2005) is fully devoted to isomorphic subgroups, cf. also Billiet (1980) and Billiet & Sayari (1984).
In this volume, all maximal isomorphic subgroups are listed as members of infinite series, where each individual subgroup is specified by its index, its generators and the coordinates of its conventional origin as parameters.
The relations between a space group and its subgroups become more transparent if they are considered in connection with their normalizers in the affine group and the Euclidean group (Koch, 1984). Even the corresponding normalizers of Hermann's group play a role in these relations, cf. Wondratschek & Aroyo (2001).
In addition to subgroup data, supergroup data are listed in IT A. If is a maximal subgroup of , then is a minimal supergroup of . In IT A, the type of a space group is listed as a minimal non-isomorphic supergroup of if is listed as a maximal non-isomorphic subgroup of . Thus, for each space group one can find in the tables the types of those groups for which is listed as a maximal subgroup. The supergroup data of IT A1 are similarly only an inversion of the subgroup data.
References
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Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745.
Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Minéral. Cristallogr. 96, 327–334.
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