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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 42

Section 1.4.1.1. Introduction

H. Wondratscheke

1.4.1.1. Introduction

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Space groups describe the symmetries of crystal patterns; the point group of the space group is the symmetry of the macroscopic crystal. Both kinds of symmetry are characterized by symbols of which there are different kinds. In this section the space-group numbers as well as the Schoenflies symbols and the Hermann–Mauguin symbols of the space groups and point groups will be dealt with and compared, because these are used throughout this volume. They are rather different in their aims. For the Fedorov symbols, mainly used in Russian crystallographic literature, cf. Chapter 3.3[link] . In that chapter the Hermann–Mauguin symbols and their use are also discussed in detail. For computer-adapted symbols of space groups implemented in crystallographic software, such as Hall symbols (Hall, 1981a[link],b[link]) or explicit symbols (Shmueli, 1984[link]), the reader is referred to Chapter 1.4[link] of International Tables for Crystallography, Volume B (2008)[link].

For the definition of space groups and plane groups, cf. Chapter 1.3[link] . The plane groups characterize the symmetries of two-dimensional periodic arrangements, realized in sections and projections of crystal structures or by periodic wallpapers or tilings of planes. They are described individually and in detail in Chapter 2.2[link] . Groups of one- and two-dimensional periodic arrangements embedded in two-dimensional and three-dimensional space are called subperiodic groups. They are listed in Vol. E of International Tables for Crystallography (2010[link]) (referred to as IT E) with symbols similar to the Hermann–Mauguin symbols of plane groups and space groups, and are related to these groups as their subgroups. The space groups sensu stricto are the symmetries of periodic arrangements in three-dimensional space, e.g. of normal crystals, see also Chapter 1.3[link] . They are described individually and in detail in the space-group tables of Chapter 2.3[link] . In the following, if not specified separately, both space groups and plane groups are covered by the term space group.

The description of each space group in the tables of Chapter 2.3[link] starts with two headlines in which the different symbols of the space group are listed. All these names are explained in this section with the exception of the data for Patterson symmetry (cf. Chapter 1.6[link] and Section 2.1.3.5[link] for explanations of Patterson symmetry).

References

International Tables for Crystallography (2008). Vol. B, Reciprocal Space. Edited by U. Shmueli, 3rd ed. Heidelberg: Springer.
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, edited by V. Kopský & D. B. Litvin, 2nd ed. Chichester: John Wiley. [Abbreviated as IT E.]
Hall, S. R. (1981a). Space-group notation with an explicit origin. Acta Cryst. A37, 517–525.
Hall, S. R. (1981b). Space-group notation with an explicit origin; erratum. Acta Cryst. A37, 921.
Shmueli, U. (1984). Space-group algorithms. I. The space group and its symmetry elements. Acta Cryst. A40, 559–567.








































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