Chapter

Chapter 15.3 Examples of the use of normalizers

Space‐group symmetry

First Online Edition (2006)

Part 15. Normalizers of space groups and their use in crystallography

  1. E. Koch,
  2. W. Fischer

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000535

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Koch, E. and Fischer, W. 2006. Examples of the use of normalizers. International Tables for Crystallography. A:15:15.3:900–903.

Author Information

  1. Institut für Mineralogie, Petrologie und Kristallographie, Philipps‐Universität, D‐35032 Marburg, Germany

Publication History

  1. Published Online: 1 JAN 2006

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Abstract

In Chapter 15.3, examples for the use of Euclidean and affine normalizers for crystallographic purposes are given: (i) The derivation of Euclidean‐ and affine‐equivalent point configurations and Wyckoff positions constitutes the basis for the definition of Wyckoff sets. The derivation of all different but Euclidean‐equivalent coordinate descriptions of a certain crystal structure is described provided that the description of its space group (basis vectors and origin) remains unchanged. (ii) Each transition from one coordinate description of a crystal structure to another equivalent one necessarily causes changes in the corresponding list of structure factors: either the phases of the reflections or the phases and the indices are changed. As a consequence, the Euclidean normalizers of the space groups lead to a simple derivation of phase restrictions for use in direct methods to ‘fix the origin and the enantiomorph’. (iii) Different subgroups (or supergroups) of a given space group that play an analogous role with respect to this space group may be identified with the aid of the Euclidean or affine normalizers. (iv) The ranges of the metrical and coordinate parameters that have to be considered for geometrical studies of point configurations can be reduced with the aid of the Euclidean and affine normalizers of space groups.

Keywords:

  • normalizers;
  • point configurations;
  • Wyckoff positions;
  • equivalent crystal structures;
  • structure factors;
  • supergroups;
  • subgroups;
  • Euclidean normalizers;
  • affine normalizers