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Incommensurate Crystallography$
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Sander van Smaalen

Print publication date: 2007

Print ISBN-13: 9780198570820

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198570820.001.0001

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SYMMETRY OF MODULATED CRYSTALS

SYMMETRY OF MODULATED CRYSTALS

Chapter:
(p.48) 3 SYMMETRY OF MODULATED CRYSTALS
Source:
Incommensurate Crystallography
Author(s):

Sander Van Smaalen

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198570820.003.0003

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.

Keywords:   diffraction, incommensurately modulated crystal, modulation, point symmetry, space group, structure factor, superspace, superspace group, symmetry, symmetry restriction

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