Section 15.1.1. Introduction

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).