A systematic approach to the derivation of standard orientation-location parts of symmetry-operation symbols

Abstract

Automatically generated orientation-location parts, or coordinate triplets describing the geometric elements, differ frequently from the corresponding parts of the symmetry-operation symbols listed in International Tables for Crystallography [(1983), Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel]. An effective algorithm enabling the derivation of standard orientation-location parts from any symmetry matrix is described and illustrated. The algorithm is based on a new concept alternative to the 'invariant points of reduced operation'. First, the geometric element that corresponds to a given symmetry operation is oriented and located in a nearly convention free manner. The application of the direction indices [uvw] or Miller indices (hkl) gives a unique orientation provided the convention about the positive direction is defined. The location is fixed by the specification of a unique point on the geometric element, i.e. the point closest to the origin. Next, both results are converted into the standard orientation-location form. The standardization step can be incorporated into other existing methods of derivation of the symmetry-operation symbols. A number of standardization examples are given.

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.…”

“…For example, from the two equivalent descriptions of the same crystallographic direction [1 ] and [ 1] the latter is positively directed according to the rule applied in [13] and the opposite selection is compatible with geometric descriptions in ITA83. A systematic analysis of 'standard' symbols allowed to specify in [18] six conventions C1 -C6, which together with the algebraic procedures lead to unique symbols. Convention C2, in the detailed form is given in Table 3.…”

“…Even though the recipes are clear, the geometric meaning of symmetry matrices needs some additional conventions [18] to obtain a unique form, not necessarily presented in algorithms. Difficulties are connected with the positive direction of an axis line or plane traces specifications, or the selection of a point which fixes a geometric element in the space.…”

“…Procedures for doing this are well documented in the literature [8,9,[16][17][18]. Algorithms for the determination of (i) the type of rotation, (ii) the rotation angle φ given as n = 360/ φ, (iii) the sense of rotation, (iv) the orthogonal decomposition of the translation vector of the space group operator into its intrinsic and location-dependent components, (v) the orientation and the location of a geometric element, (vi) the inversion point on a geometric element in case of a rotoinversion symmetry operation, are given in the above references.…”

“…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.…”