International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.6, p. 852

Section 3.6.1. Introduction

D. B. Litvina*

aDepartment 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: u3c@psu.edu

3.6.1. Introduction

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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[link]; 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[link]; Shubnikov, 1945[link]; Shubnikov et al., 1964[link]). 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[link],b[link]), 394 two-colour rod-group types (Shubnikov, 1959a[link],b[link]; Neronova & Belov, 1961a[link],b[link]; Galyarski & Zamorzaev, 1965a[link],b[link]) and 528 two-colour layer-group types (Neronova & Belov, 1961a[link],b[link]; Palistrant & Zamorzaev, 1964a[link],b[link]). Of the two-colour space groups, there are seven two-colour one-dimensional space-group types (Neronova & Belov, 1961a[link],b[link]), 80 two-colour two-dimensional space-group types (Heesch, 1929[link]; Cochran, 1952[link]) and 1651 two-colour three-dimensional space-group types (Zamorzaev, 1953[link], 1957a[link],b[link]; Belov et al., 1957[link]). See also Zamorzaev (1976[link]), Shubnikov & Koptsik (1974[link]), Koptsik (1966[link], 1967[link]), and Zamorzaev & Palistrant (1980[link]). [Extensive listings of references on colour symmetry, magnetic symmetry and related topics can be found in the books by Shubnikov et al. (1964[link]), Shubnikov & Koptsik (1974[link]), and Opechowski (1986[link]).]

The so-called magnetic groups, groups to describe the sym­metry of spin arrangements, were introduced by Landau & Lifschitz (1951[link], 1957[link]) 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[link]), 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[link]), 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[link],b[link]; Opechowski & Guccione, 1965[link]; Opechowski, 1986[link]) and the Belov–Neronova–Smirnova notation (BNS notation) (Belov et al., 1957[link]) are compared. The maximal subgroups of index [\leq] 4 of the magnetic subperiodic groups and magnetic space groups are also given.

References

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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.
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