Coordinate systems in crystallography

Wondratschek, H.

doi:10.1107/97809553602060000516

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

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International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 732

Section 8.3.1. Coordinate systems in crystallography

H. Wondratschek^a ^*

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

8.3.1. Coordinate systems in crystallography

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The matrices W and the columns w of crystallographic symmetry operations $[\hbox{\sf W}]$ depend on the choice of the coordinate system. A suitable choice is essential if W and w are to be obtained in a convenient form.

Example

In a space group I4mm, the matrix part of a clockwise fourfold rotation around the c axis is described by the W matrix $[4^{-}\ 00z: \pmatrix{\phantom{0\;}0 &1 &0\hfill\cr - 1 &0 &0\hfill\cr \phantom{0\;}0 &0 &1\hfill\cr}]$ if referred to the conventional crystallographic basis a, b, c. Correspondingly, the matrix $[m\ 0yz: \pmatrix{- 1 &0 &0\hfill\cr \phantom{0}0 &1 &0\hfill\cr \phantom{0}0 &0 &1\hfill\cr}]$ represents a reflection in a plane parallel to b and c. These matrices are easy to handle and their geometrical significance is evident. Referred to the primitive basis $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf c}']$ , defined by $[{\bf a}' = {1 \over 2}( - {\bf a} + {\bf b} + {\bf c})]$ , $[{\bf b}' = {1 \over 2}({\bf a} - {\bf b} + {\bf c})]$ , $[{\bf c}' = {1 \over 2}({\bf a} + {\bf b} - {\bf c})]$ , the matrices representing the same symmetry operations would be $[4^{-}: \pmatrix{1 &\phantom{0}0 &- 1\hfill\cr 1 &\phantom{0}0 &\phantom{0\;}0\hfill\cr 1 &- 1 &\phantom{0\;}0\hfill\cr}; \quad m: \pmatrix{1 &\phantom{0}0 &\phantom{0\;}0\hfill\cr 1 &\phantom{0}0 &- 1\hfill\cr 1 &- 1 &\phantom{0\;}0\hfill\cr}.]$ These matrices are more complicated to work with, and their geometrical significance is less obvious.

The conventional coordinate systems obey rules concerning the vector bases and the origins.

(i) In all cases, the conventional coordinate bases are chosen such that the matrices W only consist of the integers 0, 1 and −1, that they are reduced as much as possible, and that they are of simplest form, i.e. contain six or at least five zeros for three dimensions and two or at least one zeros for two dimensions. This fact can be expressed in geometric terms by stating that `symmetry directions (Blickrichtungen) are chosen as coordinate axes' (axes of rotation, screw rotation or rotoinversion, normals of reflection or glide planes); cf. Section 2.2.4 and Chapter 9.1 . Shortest translation vectors compatible with these conditions are chosen as basis vectors. In many cases, the conventional vector basis is not a primitive but rather a nonprimitive crystallographic basis, i.e. there are lattice vectors with fractional coefficients. The centring type of the conventional cell and thus the lattice type can be recognized from the first letter of the Hermann–Mauguin symbol.

Example

The letter P for $[E^{3}]$ (or p for $[E^{2}]$ ), taken from `primitive', indicates that a primitive basis is being used conventionally for describing the crystal structure and its symmetry operations. In this case, the vector lattice L consists of all vectors $[{\bf u} = u_{1}{\bf a}_{1} + \ldots + u_{n}{\bf a}_{n}]$ with integral coefficients $[u_{i}]$ , but contains no other vectors. If the Hermann–Mauguin symbol starts with a `C' in $[E^{3}]$ or with a `c' in $[E^{2}]$ , in addition to all such vectors u all vectors $[{\bf u} + {1 \over 2}({\bf a} + {\bf b})]$ also belong to L. The letters A, B, I, F and R are used for the conventional bases of the other types of lattices, cf. Section 1.2.1 .

In a number of cases, the symmetry of the space group determines the conventional vector basis uniquely; in other cases, metrical criteria, e.g. the length of basis vectors, may be used to define a conventional vector basis.
(ii) The choice of the conventional origin in the space-group tables of this volume has been dealt with by Burzlaff & Zimmermann (1980). In general, the origin is a point of highest site symmetry, i.e. as many symmetry operations $[\hbox{\sf W}_{j}]$ as possible leave the origin fixed, and thus have $[{\bi w}_{j} = {\bi o}]$ . Special reasons may justify exceptions from this rule, for example for space groups $[I2_{1}2_{1}2_{1} \equiv D_{2}^{9}]$ (No. 24), $[P4_{3}32 \equiv O^{6}]$ (No. 212), $[P4_{1}32 \equiv O^{7}]$ (No. 213), $[I4_{1}32 \equiv O^{8}]$ (No. 214) and $[I\bar{4}3d \equiv T_{d}^{6}]$ (No. 220); cf. Section 2.2.7 . If in a centrosymmetric space group a centre of inversion is not a point of highest site symmetry, the space group is described twice, first with the origin in a point of highest site symmetry, and second with the origin in a centre of inversion, e.g. at 222 and at $[\bar{1}]$ for space group $[Pnnn \equiv D_{2h}^{2}]$ (No. 48); cf. Section 2.2.1 .¹ For space groups with low site symmetries, the origin is chosen so as to minimize the number of nonzero coefficients of the $[w_{j}]$ , e.g. on a twofold screw axis for space group $[P2_{1} \equiv C_{2}^{2}]$ (No. 4).

A change of the coordinate system, i.e. referring the crystal pattern and its symmetry operations $[\hbox{\sf W}]$ to a new coordinate system, results in new coordinates $[\specialfonts{\bbsf x}']$ and new matrices $[\specialfonts{\bbsf{W}}']$ ; cf. Section 5.1.3 .

References

Burzlaff, H. & Zimmermann, H. (1980). On the choice of origins in the description of space groups. Z. Kristallogr. 153, 151–179.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 732