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Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application

Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties... A fixed set, that is the set of all lattice metrics corresponding to the arithmetic holohedry of a primitive lattice, is a natural tool for keeping track of the symmetry changes that may occur in a deformable lattice [Ericksen (1979). Arch. Rat. Mech. Anal.72, 1–13; Michel (1995). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek & S. Walcerz. Singapore: Academic Press; Pitteri & Zanzotto (1996). Acta Cryst. A52, 830–838; and references quoted therein]. For practical applications it is desirable to limit the infinite number of arithmetic holohedries, and simplify their classification and construction of the fixed sets. A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and geometric description were analyzed and offered as the framework for dealing with the symmetry of reduced lattices. This matrix space covers all arithmetic holohedries of primitive lattice descriptions related to the three shortest lattice translations in direct or reciprocal spaces, and corresponds to the unique list of 39 fixed points with integer coordinates in six‐dimensional space of lattice metrics. Matrices are presented by the introduced dual symbol, which sheds some light on the lattice and its symmetry‐related properties, without further digging into matrices. By the orthogonal lattice distortion the lattice group–subgroup relations are easily predicted. It was proven and exemplified that new symbols enable classification of lattice groups on an absolute basis, without metric considerations. In contrast to long established but sophisticated methods for assessing the metric symmetry of a lattice, simple filtering of the symmetry operations from the predefined set is proposed. It is concluded that the space of symmetry matrices with elements from {0, ±1} is the natural environment of lattice symmetries related to the reduced cells and that complete geometric characterization of matrices in the arithmetic holohedry provides a useful tool for solving practical lattice‐related problems, especially in the context of lattice deformation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A Foundations of Crystallography Wiley

Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application

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References (20)

Publisher
Wiley
Copyright
Copyright © 2011 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S0108767311020113
pmid
21844646
Publisher site
See Article on Publisher Site

Abstract

A fixed set, that is the set of all lattice metrics corresponding to the arithmetic holohedry of a primitive lattice, is a natural tool for keeping track of the symmetry changes that may occur in a deformable lattice [Ericksen (1979). Arch. Rat. Mech. Anal.72, 1–13; Michel (1995). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek & S. Walcerz. Singapore: Academic Press; Pitteri & Zanzotto (1996). Acta Cryst. A52, 830–838; and references quoted therein]. For practical applications it is desirable to limit the infinite number of arithmetic holohedries, and simplify their classification and construction of the fixed sets. A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and geometric description were analyzed and offered as the framework for dealing with the symmetry of reduced lattices. This matrix space covers all arithmetic holohedries of primitive lattice descriptions related to the three shortest lattice translations in direct or reciprocal spaces, and corresponds to the unique list of 39 fixed points with integer coordinates in six‐dimensional space of lattice metrics. Matrices are presented by the introduced dual symbol, which sheds some light on the lattice and its symmetry‐related properties, without further digging into matrices. By the orthogonal lattice distortion the lattice group–subgroup relations are easily predicted. It was proven and exemplified that new symbols enable classification of lattice groups on an absolute basis, without metric considerations. In contrast to long established but sophisticated methods for assessing the metric symmetry of a lattice, simple filtering of the symmetry operations from the predefined set is proposed. It is concluded that the space of symmetry matrices with elements from {0, ±1} is the natural environment of lattice symmetries related to the reduced cells and that complete geometric characterization of matrices in the arithmetic holohedry provides a useful tool for solving practical lattice‐related problems, especially in the context of lattice deformation.

Journal

Acta Crystallographica Section A Foundations of Crystallography – Wiley

Published: Jan 1, 2011

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