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

International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 52-54

## Section 1.6.5. Handling cell transformations

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: mueller@chemie.uni-marburg.de

### 1.6.5. Handling cell transformations

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It is important to keep track of the coordinate transformations in a sequence of group–subgroup relations. A Bärnighausen tree can only be correct if every atomic position of every hettotype can be derived from the corresponding positions of the aristo­type. The mathematical tools are described in Sections 1.2.2.3 , 1.2.2.4 and 1.2.2.7 of this volume and in Part 5 of Volume A.

A basis transformation and an origin shift are mentioned in the middle of a group–subgroup arrow. This is a shorthand notation for the matrix–column pair (Seitz symbol; Section 1.2.2.3 ) or the 4 × 4 matrix (augmented matrix, Section 1.2.2.4 ) to be used to calculate the basis vectors, origin shift and coordinates of a maximal subgroup from those of the preceding space group. If there is a sequence of several transformations, the overall changes can be calculated by multiplication of the 4 × 4 matrices.

Let be the 4 × 4 matrices expressing the basis vector and origin changes of several consecutive cell transformations. are the corresponding inverse matrices. Let a, b, c be the starting (old) basis vectors and , , be the (new) basis vectors after the consecutive transformations. Let and be the augmented columns of the atomic coordinates before and after the transformations. Then the following relations hold: is the vector of the origin shift, i.e. its components are the coordinates of the origin of the new cell expressed in the coordinate system of the old cell; is a zero vector. Note that the inverse matrices have to be multiplied in the reverse order.

#### Example 1.6.5.1

Take a transformation from a cubic to a rhombohedral unit cell (hexagonal setting) combined with an origin shift of (in the cubic coordinate system), followed by transformation to a monoclinic cell with a second origin shift of (in the hexagonal coordinate system). From Fig. 1.6.5.1 (left) we can deducewhich in matrix notation isIn the fourth column of the 4 × 4 matrix we include the components of the origin shift, : The inverse matrix can be calculated by matrix inversion, but it is more straightforward to deduce it from Fig. 1.6.5.1; it is the matrix that converts the hexagonal basis vectors back to the cubic basis vectors:The column part (fourth column) of the inverse matrix is (cf. equation 1.2.2.6): Therefore, isSimilarly, for the second (hexagonal to monoclinic) transformation, we deduce the matrices , and the column part of , , from the right image of Fig. 1.6.5.1:For the basis vectors of the two consecutive transformations we calculate: which corresponds toand an origin shift of in the cubic coordinate system. The corresponding (cubic to monoclinic) coordinate transformations result fromwhich is the same as

 Figure 1.6.5.1 | top | pdf |Relative orientations of a cubic to a rhombohedral cell (hexagonal setting) (left) and of a rhombohedral cell (hexagonal setting) to a monoclinic cell (the monoclinic cell has an acute angle β).