International Tables for Crystallography

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Crystallographic symmetry
H. Wondratschek and M. I. Aroyo. International Tables for Crystallography (2016). Vol. A, ch. 1.2, pp. 12-21  [ doi:10.1107/97809553602060000920 ]

Abstract

Matrix formalism provides efficient instruments for the analytical description of crystallographic symmetry. After a brief introduction to the types of crystallographic symmetry operations, the application of the matrix formalism in the description of crystallographic symmetry operations is dealt with, including matrix–column presentations of symmetry operations and their presentation by augmented matrices. The chapter continues with the analysis of important aspects of the relation between symmetry operations and their matrix–column presentations, namely (i) the geometric interpretation of a given matrix–column pair and (ii) the determination of a matrix–column pair if the geometric meaning of the operation is known. The last section provides a detailed discussion of the key concepts of a symmetry element and its constituents, a geometric element and an element set. The matrices of the rotation parts of all space-group symmetry operations referred to conventional coordinate systems are listed.


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About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The multi-volume series comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.