Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T13:20:19.586Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of crystal lattices

Published online by Cambridge University Press:  24 October 2008

R. L. E. Schwarzenberger
R. L. E. Schwarzenberger
Affiliation:
University of Warwick
Get access Rights & Permissions [Opens in a new window]

Extract

There are two natural ways in which to attempt a classification of the lattices which occur in crystallography or, more generally, of n-dimensional lattices. The first method is algebraic and proceeds by classifying the symmetry groups of the lattices. Thus an n-dimensional lattice is a discrete subgroup

of Rn and determines a symmetry group

which is a subgroup of the orthogonal group On. Two lattices T1 and T2 are said to determine the same crystal system if the symmetry groups G(T1) and G(T2) are conjugate: that is there exists a linear isomorphism ø: Rn → Rn such that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 325 - 349
Copyright
Copyright © Cambridge Philosophical Society 1972

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baily, W. L. and Borel, A.Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 84 (1966), 442–528.CrossRefGoogle Scholar
(2)Bravais, A. Mémoire sur les systémes formés par des points distribues régulierement sur un plan ou dans l'espace. J. École Polytechnique, Paris 1850; English translation, Crystallo-graphic Society of America, 1949.Google Scholar
(3)Buerger, M. J.Elementary crystallography, an introduction to the fundamental geometrical features of crystals (Wiley, New York, 1956).Google Scholar
(4)Frankenheim, M. L.System der Krystalle (Breslau, 1842).Google Scholar
(5)Lejeune-Dirichlet, P. G.Uber die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. reine angewandte Math. 40 (1850), 209–227.Google Scholar
(6)Mackay, A. L. and Pawley, G. S.Bravais lattices in four dimensional space. Acta Cryst. 16 (1963), 11–19, corrected by Belov, N. L. and Kuntsevich, T. S. Acta Cryst. 24A (1968), 42–51 and 25A (1969), Supplement p. 3.CrossRefGoogle Scholar
(7)Robertson, S. A., Carter, S. and Morton, H. R.Finite orthogonal symmetry. Topology 9 (1970), 79–95.CrossRefGoogle Scholar
(8)Spanter, E.The homology of Kuinmer manifolds. Proc. Am. Math. Soc. 7 (1956), 155–160.CrossRefGoogle Scholar
(9)Van Der Waerden, B. L.Die Reduktionstheorie der positiven quadratischen Formen. Acta Math. 96 (1956), 265–293.CrossRefGoogle Scholar