Introduction and definitions

Koch, E.; Fischer, W.; Müller, U.

doi:10.1107/97809553602060000533

International
Tables for
Crystallography
Volume A
Space-group symmetry
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International Tables for Crystallography (2006). Vol. A. ch. 15.1, p. 878
https://doi.org/10.1107/97809553602060000533

Chapter 15.1. Introduction and definitions

E. Koch,^a ^* W. Fischer^a and U. Müller^b

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany, and ^bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

In Chapter 15.1, the mathematical concept of Euclidean and affine normalizers of space groups is introduced. Some crystallographic problems are mentioned for which the solution of the problem is simplified by the use of normalizers.

Keywords: normalizers; space groups; Euclidean normalizers; affine normalizers; Cheshire groups.

15.1.1. Introduction

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The mathematical concept of normalizers forms the common basis for the solution of several crystallographic problems:

It is generally known, for instance, that the coordinate description of a crystal structure trivially depends on the coordinate system used for the description, i.e. on the setting of the space group and the site symmetry of the origin. It is less well known, however, that for most crystal structures there exist several different but equivalent coordinate descriptions, even if the space-group setting and the site symmetry of the origin are unchanged. The number of such descriptions varies between 1 and 24 and depends only on the type of the Euclidean normalizer of the corresponding space group. In principle, none of these descriptions stands out against the others.

In crystal-structure determination with direct methods, the phases of some suitably chosen structure factors have to be restricted to certain values or to certain ranges in order to specify the origin and the enantiomorph. The information necessary for a correct selection of such phases and for their appropriate restrictions follows directly from the Euclidean normalizer of the space group. Similar examples are the positioning of the first atom(s) within an asymmetric unit when using trial-and-error or Patterson methods, the choice of a basis system for indexing the reflections of a diffraction pattern or the indexing of the first morphological face(s) of a crystal.

For the following problems, normalizers also play an important role: They supply information on the interchangeability of Wyckoff positions and their assignment to Wyckoff sets (cf. Section 8.3.2 and Chapter 14.1 ), needed e.g. for the definition of lattice complexes. They are important for the comparison of crystal structures, for their assignment to structure types and for the choice of a standard description for each crystal structure (Parthé & Gelato, 1984 , 1985 ). They allow the derivation of `privileged origins' for each space group (Burzlaff & Zimmermann, 1980 ) and facilitate the complete deduction of subgroups and supergroups of a crystallographic group. They enable an easy classification of magnetic (black–white or Shubnikov) space groups and of colour space groups. They may also be used to reduce the parameter range in the study of geometrical properties of point configurations, e.g. their inherent symmetry or their sphere packings and Dirichlet partitions (cf. e.g. Koch, 1984 ).

In the past, most of these problems have been treated by crystallographers without the aid of normalizers, but the use of normalizers simplifies the solution of all these problems and clarifies the common background (for references, see Fischer & Koch, 1983 ).

15.1.2. Definitions

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Any pair, consisting of a group $[{\cal G}]$ and one of its supergroups $[{\cal S}]$ , is uniquely related to a third intermediate group $[{\cal N}\!_{{\cal S}}({\cal G})]$ , called the normalizer of $[{\cal G}]$ with respect to $[{\cal S}]$ . $[{\cal N}\!_{{\cal S}}({\cal G})]$ is defined as the set of all elements $[{\bf S} \in {\cal S}]$ that map $[{\cal G}]$ onto itself by conjugation (cf. Section 8.3.6 ). $[{\cal N}\!_{{\cal S}}({\cal G}) := \{ {\bf S} \in {\cal S} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\}.]$ The normalizer $[{\cal N}\!_{{\cal S}}({\cal G})]$ may coincide either with $[{\cal G}]$ or with $[{\cal S}]$ or it may be a proper intermediate group. In any case, $[{\cal G}]$ is a normal subgroup of its normalizer.

For most crystallographic problems, two kinds of normalizers are of special interest: (i) the normalizer of a space group (plane group) $[{\cal G}]$ with respect to the group $[{\cal E}]$ of all Euclidean mappings (motions, isometries) in $[E^{3}\; (E^{2})]$ , called the Euclidean normalizer of $[{\cal G}]$ $[{\cal N}\!_{{\cal E}}({\cal G}) := \{{\bf S} \in {\cal E} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\};]$ (ii) the normalizer of a space group (plane group) $[{\cal G}]$ with respect to the group $[{\cal A}]$ of all affine mappings in $[E^{3}\; (E^{2})]$ , called the affine normalizer of $[{\cal G}]$ $[{\cal N}\!_{{\cal A}}({\cal G}) := \{ {\bf S} \in {\cal A} |\ {\bf S}^{-1} {\cal G}\ {\bf S} = {\cal G}\}.]$

The Euclidean normalizers of the space groups were first derived by Hirshfeld (1968) under the name Cheshire groups. They have been tabulated in more detail by Gubler (1982a ,b ) and Fischer & Koch (1983). The Euclidean normalizers of triclinic and monoclinic space groups with specialized metric have been determined by Koch & Müller (1990). The affine normalizers of the space groups have been listed by Burzlaff & Zimmermann (1980), Billiet et al. (1982) and Gubler (1982a ,b ). They have also been used for the derivation of Wyckoff sets and the definition of lattice complexes by Koch & Fischer (1975), even though there the automorphism groups of the space groups were tabulated instead of their affine normalizers.

References

Billiet, Y., Burzlaff, H. & Zimmermann, H. (1982). Comment on the paper of H. Burzlaff and H. Zimmermann. `On the choice of origin in the description of space groups'. Z. Kristallogr. 160, 155–157.Google Scholar

Burzlaff, H. & Zimmermann, H. (1980). On the choice of origin in the description of space groups. Z. Kristallogr. 153, 151–179.Google Scholar

Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.Google Scholar

Gubler, M. (1982a). Über die Symmetrien der Symmetriegruppen: Automorphismengruppen, Normalisatorgruppen und charakteristische Untergruppen von Symmetriegruppen, insbesondere der kristallographischen Punkt- und Raumgruppen. Dissertation, University of Zürich, Switzerland.Google Scholar

Gubler, M. (1982b). Normalizer groups and automorphism groups of symmetry groups. Z. Kristallogr. 158, 1–26.Google Scholar

Hirshfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.Google Scholar

Koch, E. (1984). A geometrical classification of cubic point configurations. Z. Kristallogr. 166, 23–52.Google Scholar

Koch, E. & Fischer, W. (1975). Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. Acta Cryst. A31, 88–95.Google Scholar

Koch, E. & Müller, U. (1990). Euklidische Normalisatoren für trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. Acta Cryst. A46, 826–831.Google Scholar

Parthé, E. & Gelato, L. M. (1984). The standardization of inorganic crystal-structure data. Acta Cryst. A40, 169–183.Google Scholar

Parthé, E. & Gelato, L. M. (1985). The `best' unit cell for monoclinic structures consistent with b axis unique and cell cell choice 1 of International Tables for Crystallography (1983). Acta Cryst. A41, 142–151.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 15.1, p. 878
https://doi.org/10.1107/97809553602060000533