International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 9.1, p. 742

Section 9.1.1. Description and transformation of bases

H. Burzlaffa and H. Zimmermannb*

a Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@knot.uni-erlangen.de

9.1.1. Description and transformation of bases

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In three dimensions, a coordinate system is defined by an origin and a basis consisting of three non-coplanar vectors. The lengths a, b, c of the basis vectors a, b, c and the intervector angles [\alpha = \angle ({\bf b},{\bf c})], [\beta = \angle ({\bf c},{\bf a})], [\gamma = \angle ({\bf a},{\bf b})] are called the metric parameters. In n dimensions, the lengths are designated [a_{i}] and the angles [\alpha_{ik}], where [1 \leq i \lt k \leq n].

Another description of the basis consists of the scalar products of all pairs of basis vectors. The set of these scalar products obeys the rules of covariant tensors of the second rank (see Section 5.1.3[link] ). The scalar products may be written in the form of a [(3\times 3)] matrix [({\bf a}_{i}\cdot {\bf a}_{k}) = (g_{ik}) = {\bf G};\quad i,k = 1, 2, 3,] which is called the matrix of the metric coefficients or the metric tensor.

The change from one basis to another is described by a transformation matrix P. The transformation of the old basis (a, b, c) to the new basis [({\bf a'},{\bf b'},{\bf c'})] is given by [({\bf a}',{\bf b}',{\bf c}') = ({\bf a},{\bf b},{\bf c})\cdot {\bi P.}] The relation [{\bf G}' = {\bi P}^{\bi t}\cdot {\bf G}\cdot {\bi P} \eqno(9.1.1.1)] holds for the metric tensors G and [{\bf G}'].








































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