Condensed Matter


Algorithm for Conversion Between Geometric Algebra Versor Notation and Conventional Crystallographic Symmetry-Operation Symbols

Authors: Eckhard Hitzer, Christian Perwass

This paper establishes an algorithm for the conversion of conformal geometric algebra (GA) [3, 4] versor symbols of space group symmetry-operations [6–8, 10] to standard symmetry-operation symbols of crystallography [5]. The algorithm is written in the mathematical language of geometric algebra [2–4], but it takes up basic algorithmic ideas from [1]. The geometric algebra treatment simplifies the algorithm, due to the seamless use of the geometric product for operations like intersection, projection, rejection; and the compact conformal versor notation for all symmetry operations and for geometric elements like lines and planes. The transformations between the set of three geometric symmetry vectors a,b,c, used for generating multivector versors, and the set of three conventional crystal cell vectors a,b,c of [5] have already been fully specified in [8] complete with origin shift vectors. In order to apply the algorithm described in the present work, all locations, axis vectors and trace vectors must be computed and oriented with respect to the conventional crystall cell, i.e. its origin and its three cell vectors.

Comments: 14 Pages. 6 tables. Preprint (2009).

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[v1] 2013-06-19 03:58:59

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