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

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.

Symmetry of hR and Pseudo-hR Lattices

“…has been given elsewhere [9]. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from the analysis of a lattice metric to the analysis of symmetry matrices [6], (ii) from the geometric interpretation of isometric transformation based on invariant subspaces to the orthogonality concept [7] extended to splitting indices [8], (iii) and from predefined cell transformations to transformations derivable via geometric information [6,7]. It was shown that both corresponding arithmetic and geometric holohedries share the space distribution of symmetry elements and thus simplify the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction.…”

“…Classifications of unique lattice representatives obtained by the Niggli reduction or Delaunay reduction are commonly used techniques to assign the Bravais symmetry to a given lattice. Another approach, called the matrix method, directly derives isometric transformations from the lattices by B-matrices, which transform a lattice onto itself [1,5,6], or by the space distribution of orthogonalities [7], or by filtering predefined set V of 480 potential symmetry matrices [8,9]. The latter technique is applicable to a wide class of semi-reduced lattice descriptions, additionally forced by a geometric interpretation of symmetry operations.…”

Unique and Effective Characterization of Space Groups in Computer Applications

“…Next the new geometric symbolism will be proposed. A dual symbol of space-group operation is based on a dual symbol of point operation [23] and a point on geometric elements closest to the origin [18]. These modifications improve the informative properties and also reduce some conventions necessary for the standard symbols to be unique.…”

“…The problems (ii) and partially (iii) and (iv) disappear, if the orientation of a geometric element is presented in the form of the orthogonal lattice splitting (uvw) [hkl]. Advantages of using splitting indices for the characterization of point operations were described in [23]. The benefits for space operations should be even greater.…”

“…The analysis of symbol derivations as well as the contents of geometric features in the symbols, given in [15,18,23], suggests the possibility of defining new symbols, easier to calculate, universal and richer in geometric meaning.…”