International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 |
International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 52-54
Section 1.6.5. Handling cell transformations^{a}Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany |
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