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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

REFERENCES

(1)Andrews, G. E. The theory of partitions. Encyclopedia of mathematics and its applications, vol. 2 (Reading, Mass., Addison-Wesley, 1976).Google Scholar
(2)Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. and Zassenhatts, H.Crystallographic groups of four-dimensional space (New York, J. Wiley, 1978).Google Scholar
(3)Bülow, R., Neubüser, J. and Wondratschek, H.On crystallography in higher dimensions I, II, III. Acta Cryst. 27 A (1971), 517–535.CrossRefGoogle Scholar
(4)Burckhardt, J. J.Die Bewegungsgruppen der Kristallographie. (Basel, Birkhäuser Verlag, 1947).CrossRefGoogle Scholar
(5)Curtis, C. and Reiner, I.Representation theory of finite groups and associative algebras (New York, Interscience, 1962).Google Scholar
(6)Plesken, W. and Pohst, M.On maximal finite irreducible subgroups of GL(n, Z). I: The five and seven dimensional cases. II: The six dimensional case. Math. Comp. 31 (138), 1977, 536–551 and 552–573.Google Scholar
(7)Schwarzenberger, R. L. E.Classification of crystal lattices. Proc. Cambridge Philos. Soc. 72 (1972), 325–349.CrossRefGoogle Scholar
(8)Schwarzenberger, R. L. E.Crystallography in spaces of arbitrary dimension. Proc. Cambridge Philos. Soc. 76 (1974), 23–32.CrossRefGoogle Scholar
(9)Schwarzenberger, R. L. E.Graphical representation of n-dimensional space groups. University of Warwick Preprint, 1978.Google Scholar