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.