General remarks

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000542

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

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International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 54 | 1 | 2 |

Section 2.1.7.1. General remarks

Hans Wondratschek^a ^* and Mois I. Aroyo^b

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

2.1.7.1. General remarks

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The group–subgroup relations between the space groups may also be described by graphs. This way is chosen in Chapters 2.4 and 2.5 . Graphs for the group–subgroup relations between crystallographic point groups have been published, for example, in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and in IT A (2005), Fig. 10.1.4.3 . Three kinds of graphs for subgroups of space groups have been constructed and can be found in the literature:

(1) Graphs for t-subgroups, such as the graphs of Ascher (1968).
(2) Graphs for k-subgroups, such as the graphs for cubic space groups of Neubüser & Wondratschek (1966).
(3) Mixed graphs, combining t- and k-subgroups. These are used, for example, when relations between existing or suspected crystal structures are to be displayed. An example is the `family tree' of Bärnighausen (1980), Fig. 15, now called a Bärnighausen tree.

A complete collection of graphs of the first two kinds is presented in this volume: in Chapter 2.4 those displaying the translationengleiche or t-subgroup relations and in Chapter 2.5 those for the klassengleiche or k-subgroup relations. Neither type of graph is restricted to maximal subgroups but both contain t- or k-subgroups of higher indices, with the exception of isomorphic subgroups, cf. Section 2.1.7.3 below.

The group–subgroup relations are direct relations between the space groups themselves, not between their types. However, each such relation is valid for a pair of space groups, one from each of the types, and for each space group of a given type there exists a corresponding relation. In this sense, one can speak of a relation between the space-group types, keeping in mind the difference between space groups and space-group types, cf. Section 1.2.5.3 .

The space groups in the graphs are denoted by the standard HM symbols and the space-group numbers. In each graph, each space-group type is displayed at most once. Such graphs are called contracted graphs here. Without this contraction, the more complex graphs would be much too large for the page size of this volume.

The symbol of a space group $[{\cal G}]$ is connected by uninterrupted straight lines with the symbols of all maximal non-isomorphic subgroups $[{\cal H}]$ or minimal non-isomorphic supergroups $[{\cal S}]$ of $[{\cal G}]$ . In general, the maximal subgroups of $[{\cal G}]$ are drawn on a lower level than $[{\cal G}]$ ; in the same way, the minimal supergroups of $[{\cal G}]$ are mostly drawn on a higher level than $[{\cal G}]$ . For exceptions see Section 2.1.7.3. Multiple lines may occur in the graphs for t-subgroups. They are explained in Section 2.1.7.2. No indices are attached to the lines. They can be taken from the corresponding subgroup tables of Chapter 2.3 , and are also provided by the general formulae of Section 1.2.8 . For the k-subgroup graphs, they are further specified at the end of Section 2.1.7.3.

References

Ascher, E. (1968). Lattices of equi-translation subgroups of the space groups. Geneva: Battelle Institute.Google Scholar

Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Comm. Math. Chem. 9, 139–175.Google Scholar

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)Google Scholar

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.) (Abbreviated IT 35.)Google Scholar

Neubüser, J. & Wondratschek, H. (1966). Untergruppen der Raumgruppen. Krist. Tech. 1, 529–543.Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 54