Tables for
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2010). Vol. E, ch. 5.2, pp. 395-396   | 1 | 2 |

Section 5.2.1. Introduction

V. Kopskýa* and D. B. Litvinb

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:

5.2.1. Introduction

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The global symmetry of an ideal crystal is described by its space group [\cal G]. It is also of interest to consider symmetries of local character. The classical example is that of the site symmetries, which are the symmetries of individual points in a crystal. These are completely described and classified as a part of the standard description of space groups in International Tables for Crystallography, Volume A , Space-Group Symmetry (IT A, 2005[link]). The results of this procedure contain two types of information:

  • (i) site symmetries of individual points under the action of the group [\cal G] and

  • (ii) orbits of points under the action of the group [\cal G].

This information, apart from its use, for example, in the consideration of the splitting of atomic levels in the field of the site symmetry, provides the background for the description of crystal structure: points of the same orbit are occupied by identical atoms (ions) and the environment of these atoms (ions) is also identical. A complete description of the structure is reduced to a description of the occupation of individual Wyckoff positions.

Analogously, we may consider the symmetries of planes transecting the crystal and of straight lines penetrating the crystal, called here the sectional layer groups (symmetries) and the penetration rod groups (symmetries). Here we look again for the two types of information:

  • (i) symmetries of individual planes (straight lines) under the action of the group [\cal G] and

  • (ii) orbits of planes (straight lines) under the action of the group [\cal G].

The general law that describes the connection between local symmetries and orbits of points, planes or straight lines is expressed by a coset resolution of the space group with respect to local symmetries. The orbits of planes (straight lines) have analogous properties to orbits of points. The structure of the plane (straight line) and its environment is identical for different planes (straight lines) of the same orbit. This is useful in the consideration of layer structures, see Section[link], and of structures with pronounced rod arrangements.

Layer symmetries have also been found to be indispensable in bicrystallography, see Section[link]. This term and the term bicrystal were introduced by Pond & Bollmann (1979[link]) with reference to the study of grain boundaries [see also Pond & Vlachavas (1983[link]) and Vlachavas (1985[link])]. A bicrystal is in general an edifice where two crystals, usually of the same structure but of different orientations, meet at a common boundary – an interface. The sectional layer groups are appropriate for both the description of symmetries of such boundary planes and the description of the bicrystals.

The sectional layer groups were, however, introduced much earlier by Holser (1958a[link],b[link]) in connection with the consideration of domain walls and twin boundaries as symmetry groups of planes bisecting the crystal. The mutual orientations of the two components of a bicrystal are in general arbitrary. In the case of domain walls and twin boundaries, which can be considered as interfaces of special types of bicrystals, there are crystallographic restrictions on these orientations. The group-theoretical basis for an analysis of domain pairs is given by Janovec (1972[link]). The consideration of the structure of domain walls or twin boundaries involves the sectional layer groups (Janovec, 1981[link]; Zikmund, 1984[link]); they were examined in the particular cases of domain structure in KSCN crystals (Janovec et al., 1989[link]) and of domain walls and antiphase boundaries in calomel crystals (Janovec & Zikmund, 1993[link]), see Section[link], and recently also in fullerene C60 (Janovec & Kopský, 1997; Saint-Grégoire et al., 1997).

The first attempts to derive the sectional layer groups systematically were made by Wondratschek (1971[link]) and by using a computer program written by Guigas (1971[link]). Davies & Dirl (1993a[link]) developed a program for finding subgroups of space groups, which they later modified to find sectional layer groups and penetration rod groups as well (Davies & Dirl, 1993b[link]). The use and determination of sectional layer groups have also been discussed by Janovec et al. (1988[link]), Kopský & Litvin (1989[link]) and Fuksa et al. (1993[link]).

The penetration rod groups can be used in the consideration of linear edifices in a crystal, e.g. line dislocations or intersections of boundaries, or in crystals with pronounced rod arrangements. So far, there seems to be no interest in the penetration rod groups and there is actually no need to produce special tables for these groups. Determining penetration rod groups was found to be a complementary problem to that of determining sectional layer groups (Kopský, 1989c[link], 1990[link]).

The keyword for this part of this volume is the term scanning, introduced by Kopský (1990[link]) for the description of the spatial distribution of local symmetries. In this sense, the description of site symmetries and classification of point orbits by Wyckoff positions are a result of the scanning of the space group for the site symmetries.

The Scanning tables, Part 6[link] , give a complete set of information on the space distribution of sectional layer groups and of the penetration rod groups. They were derived using the scanning-group method and the scanning theorem, see Section[link]. The tables describe explicitly the scanning for the sectional layer groups. The spatial distribution of (scanning for) the penetration rod groups is seen directly from the scanning groups, which are given as a part of the information in the scanning tables.

The sectional layer groups and the penetration rod groups are subgroups of space groups and as such act on the three-dimensional point space. The examples of particular studies in Section 5.2.5[link] emphasize the importance of the exact location of sectional layer groups with reference to the crystal structure and hence to the crystallographic coordinate system. In the usual interpretation, Hermann–Mauguin symbols do not specify the location of the group in space. In the scanning tables, each Hermann–Mauguin symbol means a quite specific space or layer group with reference to a specified crystallographic coordinate system, see Sections[link] and[link].

The layer and rod groups can also be interpreted as factor groups of reducible space groups (Kopský, 1986[link], 1988[link], 1989a[link],b[link], 1993a[link]; Fuksa & Kopský, 1993[link]). Our choice of standard Hermann–Mauguin symbols for frieze, rod and layer groups reflects the relationship between reducible space groups and subperiodic groups as their factor groups, see Section 1.2.17[link] . In the case of the layer groups, our choice thus substantially differs from that made by Wood (1964[link]). The interpretation of subperiodic groups as factor groups of reducible space groups also has consequences in the representation theory of space and subperiodic groups. Last but not least, this relationship reveals relations between the algebraic structure of the space group of a crystal and the symmetries of planar sections or of straight lines penetrating the crystal. These relations, analogous to the relations between the point group and symmetries of Wyckoff positions, will be described elsewhere.

It should be noted finally that all the information about scanning can be and is presented in a structure-independent way in terms of the groups involved. The scanning tables therefore extend the standard description of space groups.


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