Isometries

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 8 | 1 | 2 |

Section 1.2.2.5. Isometries

Hans Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.2.5. Isometries

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Isometries are special affine mappings, as in Definition 1.2.2.1.1. The matrix W of an isometry has to fulfil conditions which depend on the coordinate basis. These conditions are:

(1) A basis $[{\bf a}_1, \, {\bf a}_2, \, {\bf a}_3]$ is characterized by the scalar products $[({\bf a}_j \, {\bf a}_k)]$ of its basis vectors or by its lattice parameters $[a,\,b,\,c,\,\alpha,\, \beta]$ and $[\gamma]$ . Here $[a,\,b,\,c]$ are the lengths of the basis vectors $[{\bf a}_1,\,{\bf a}_2,\,{\bf a}_3]$ and $[\alpha,\,\beta]$ and $[\gamma]$ are the angles between $[{\bf a}_2]$ and $[{\bf a}_3,\, {\bf a}_3]$ and $[{\bf a}_1,\,{\bf a}_1]$ and $[{\bf a}_2]$ , respectively. The metric matrix M (called G in IT A, Chapter 9.1 ) is the $[(3\times3)]$ matrix which consists of the scalar products of the basis vectors: $[{\bi M} = \left (\matrix{a^{2} & a\,b\,\cos\gamma & a\,c\,\cos\beta \cr b\,a\,\cos\gamma & b^{2} & b\,c\,\cos\alpha \cr c\,a\,\cos\beta & c\,b\,\cos\alpha & c^{2}} \right). ]$ If W is the matrix part of an isometry, referred to the basis ( $[{\bf a}_1,\ {\bf a}_2,\ {\bf a}_3]$ ), then W must fulfil the condition $[{\bi W}^{\rm T}\,\,{\bi M}\,{\bi W} = {\bi M}]$ , where $[{\bi W}^{\rm T}]$ is the transpose of W.
(2) For the determinant of W, $[\det({\bi W}) = \pm{1}]$ must hold; $[\det({\bi W}) = + 1]$ for the identity, translations, rotations and screw rotations; $[\det({\bi W}) = -1]$ for inversions, reflections, glide reflections and rotoinversions.
(3) For the trace, $[\rm{tr}({\bi W}) = W_{11} + W_{22} + W_{33} = \pm(1 + 2 \cos{\varphi})]$ holds, where $[\varphi]$ is the rotation angle; the + sign applies if $[\det({\bi W}) = + 1]$ and the − sign if $[\det({\bi W}) = -1]$ .

Algorithms for the determination of the kind of isometry from a given matrix–column pair and for the determination of the matrix–column pair for a given isometry can be found in IT A, Part 11 or in Hahn & Wondratschek (1994).

References

Hahn, Th. & Wondratschek, H. (1994). Symmetry of crystals. Introduction to International Tables for Crystallography, Vol. A. Sofia: Heron Press.Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 8