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

International Tables for Crystallography (2006). Vol. A, ch. 5.2, p. 86

## Section 5.2.1. Transformations

H. Arnolda

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

### 5.2.1. Transformations

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Symmetry operations are transformations in which the coordinate system, i.e. the basis vectors a, b, c and the origin O, are considered to be at rest, whereas the object is mapped onto itself. This can be visualized as a motion' of an object in such a way that the object before and after the motion' cannot be distinguished.

A symmetry operation transforms every point X with the coordinates xyz to a symmetrically equivalent point with the coordinates . In matrix notation, this transformation is performed byThe matrix W is the rotation part and the column matrix w the translation part of the symmetry operation . The pair (W, w) characterizes the operation uniquely. Matrices W for point-group operations are given in Tables 11.2.2.1 and 11.2.2.2 .

Again, we can introduce the augmented matrix (cf. Chapter 8.1 ) The coordinates of the point , symmetrically equivalent to X with the coordinates xyz, are obtained by or, in short notation, A sequence of symmetry operations can be obtained as a product of matrices .

An affine transformation of the coordinate system transforms the coordinates of the starting point as well as the coordinates of a symmetrically equivalent pointThus, the affine transformation transforms also the symmetry-operation matrix and the new matrix is obtained by

#### Example

Space group (85) is listed in the space-group tables with two origins; origin choice 1 with , origin choice 2 with as point symmetry of the origin. How does the matrix of the symmetry operation 0, 0, z; 0, 0, 0 of origin choice 1 transform to the matrix of symmetry operation z; , 0 of origin choice 2?

In the space-group tables, origin choice 1, the transformed coordinates are listed. The translation part is zero, i.e. . In Table 11.2.2.1, the matrix W can be found. Thus, the matrix is obtained:

The transformation to origin choice 2 is accomplished by a shift vector p with components , 0. Since this is a pure shift, the matrices P and Q are the unit matrix I. Now the shift vector q is derived: . Thus, the matrices and are By matrix multiplication, the new matrix is obtained: If the matrix is applied to x′, y′, z′, the coordinates of the starting point in the new coordinate system, we obtain the transformed coordinates , By adding a lattice translation a, the transformed coordinates are obtained as listed in the space-group tables for origin choice 2.