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The decomposition of crystal families

Published online by Cambridge University Press:  24 October 2008

J. D. Jarratt
J. D. Jarratt
Affiliation:
University of Auckland, New Zealand
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Extract

In the course of their enumeration of all 4-dimensional space groups, Brown, Bülow, Neubüser, Wondratschek and Zassenhaus have introduced concepts appropriate to the study of n-dimensional crystallography for n ≥ 4 (see (2), (3)). One such concept is that of a crystal family. Families seem to be particularly useful as a framework within which to study higher dimensional crystallography, primarily because they determine a classification of all the standard crystallographic objects and overcome the traditional confusion over crystal systems (see (2); pp. 16–17, (9)). In this paper, techniques are developed for the determination of all rationally decomposable families in a given dimension from the indecomposable families of lower dimensions. These techniques place emphasis on three geometric invariants of families: the decomposition pattern; the canonical decomposition pattern; and the number of free parameters. This, it is felt, further reinforces their position as fundamental objects. The key result (Theorem 5.4) is:

a family uniquely determines, and is uniquely determined by, the constituent families in the canonical decomposition.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 2 , September 1980 , pp. 245 - 263
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

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