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Some algebraic properties of crystallographic sublattices

Some algebraic properties of crystallographic sublattices In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three‐dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three‐dimensional, is a subgroup of L such that the finite quotient group, is an Abelian group of order the index of S in L. The sublattice itself in its standard form is represented by an upper triangular matrix. The index of the sublattice is given by the determinant of this matrix. It is first noted that a sublattice described by an arbitrary basis set in L may be converted to this standard form. Next the sublattice is expressed as the intersection of a set of sublattices of individual index a power of a distinct prime, i.e. where p1, p2etc. are prime numbers and is the Euclidean factorization of n. This decomposition is important because it corresponds to the Sylow decomposition of the corresponding quotient group It is also useful to be able to carry out two commutative binary operations on sublattices of L; these are to find their common sublattice of lowest index in L, which is their intersection and their common superlattice of highest index in L, given by where <> indicates the span of the sublattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A Foundations of Crystallography Wiley

Some algebraic properties of crystallographic sublattices

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References (9)

Publisher
Wiley
Copyright
Copyright © 2006 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S0108767305038225
pmid
16489245
Publisher site
See Article on Publisher Site

Abstract

In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three‐dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three‐dimensional, is a subgroup of L such that the finite quotient group, is an Abelian group of order the index of S in L. The sublattice itself in its standard form is represented by an upper triangular matrix. The index of the sublattice is given by the determinant of this matrix. It is first noted that a sublattice described by an arbitrary basis set in L may be converted to this standard form. Next the sublattice is expressed as the intersection of a set of sublattices of individual index a power of a distinct prime, i.e. where p1, p2etc. are prime numbers and is the Euclidean factorization of n. This decomposition is important because it corresponds to the Sylow decomposition of the corresponding quotient group It is also useful to be able to carry out two commutative binary operations on sublattices of L; these are to find their common sublattice of lowest index in L, which is their intersection and their common superlattice of highest index in L, given by where <> indicates the span of the sublattices.

Journal

Acta Crystallographica Section A Foundations of Crystallography – Wiley

Published: Mar 1, 2006

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