International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A, ch. 3.6, p. 852
Section 3.6.1. Introduction^{a}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA |
The magnetic subperiodic groups in the title refer to generalizations of the crystallographic subperiodic groups, i.e. frieze groups (two-dimensional groups with one-dimensional translations), crystallographic rod groups (three-dimensional groups with one-dimensional translations) and layer groups (three-dimensional groups with two-dimensional translations). There are seven frieze-group types, 75 rod-group types and 80 layer-group types, see International Tables for Crystallography, Volume E, Subperiodic Groups (2010; abbreviated as IT E). The magnetic space groups refer to generalizations of the one-, two- and three-dimensional crystallographic space groups, n-dimensional groups with n-dimensional translations. There are two one-dimensional space-group types, 17 two-dimensional space-group types and 230 three-dimensional space-group types, see Part 2 of the present volume (IT A).
Generalizations of the crystallographic groups began with the introduction of an operation of `change in colour' and the `two-colour' (black and white, antisymmetry) crystallographic point groups (Heesch, 1930; Shubnikov, 1945; Shubnikov et al., 1964). Subperiodic groups and space groups were also extended into two-colour groups. Two-colour subperiodic groups consist of 31 two-colour frieze-group types (Belov, 1956a,b), 394 two-colour rod-group types (Shubnikov, 1959a,b; Neronova & Belov, 1961a,b; Galyarski & Zamorzaev, 1965a,b) and 528 two-colour layer-group types (Neronova & Belov, 1961a,b; Palistrant & Zamorzaev, 1964a,b). Of the two-colour space groups, there are seven two-colour one-dimensional space-group types (Neronova & Belov, 1961a,b), 80 two-colour two-dimensional space-group types (Heesch, 1929; Cochran, 1952) and 1651 two-colour three-dimensional space-group types (Zamorzaev, 1953, 1957a,b; Belov et al., 1957). See also Zamorzaev (1976), Shubnikov & Koptsik (1974), Koptsik (1966, 1967), and Zamorzaev & Palistrant (1980). [Extensive listings of references on colour symmetry, magnetic symmetry and related topics can be found in the books by Shubnikov et al. (1964), Shubnikov & Koptsik (1974), and Opechowski (1986).]
The so-called magnetic groups, groups to describe the symmetry of spin arrangements, were introduced by Landau & Lifschitz (1951, 1957) by re-interpreting the operation of `change in colour' in two-colour crystallographic groups as `time inversion'. This chapter introduces the structure, properties and symbols of magnetic subperiodic groups and magnetic space groups as given in the extensive tables by Litvin (2013), which are an extension of the classic tables of properties of the two- and three-dimensional subperiodic groups found in IT E and the one-, two- and three-dimensional space groups found in the present volume. A survey of magnetic group types is also presented in Litvin (2013), listing the elements of one representative group in each reduced superfamily of the two- and three-dimensional magnetic subperiodic groups and one-, two- and three-dimensional magnetic space groups. Two notations for magnetic groups, the Opechowski–Guccione notation (OG notation) (Guccione, 1963a,b; Opechowski & Guccione, 1965; Opechowski, 1986) and the Belov–Neronova–Smirnova notation (BNS notation) (Belov et al., 1957) are compared. The maximal subgroups of index 4 of the magnetic subperiodic groups and magnetic space groups are also given.
References
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, 2nd ed., edited by V. Kopský & D. B. Litvin. Chichester: Wiley. [First edition 2002.]Belov, N. V. (1956a). The one-dimensional infinite crystallographic groups. Kristallografiya, 1, 474–476.
Belov, N. V. (1956b). The one-dimensional infinite crystallographic groups. Sov. Phys. Crystallogr. 1, 372–374.
Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957). 1651 Shubnikov groups. Sov. Phys. Crystallogr. 1, 487–488. See also (1955) Trudy Inst. Krist. Acad. SSSR, 11, 33–67 (in Russian), English translation in Shubnikov, A. V, Belov, N. V. & others (1964). Colored Symmetry. London: Pergamon Press.
Cochran, W. (1952). The symmetry of real periodic two-dimensional functions. Acta Cryst. 5, 630–633.
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Palistrant, A. F. & Zamorzaev, A. M. (1964b). Groups of symmetry and different types of antisymmetry of borders and ribbons. Sov. Phys. Crystallogr. 9, 123–128.
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