International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 8687

Coordinate transformations are essential in the study of structural relationships between crystal structures. Consider as an example two phases A (basic or parent structure) and B (derivative structure) of the same compound. Let the space group of B be a proper subgroup of the space group of A, . The relationship between the two structures is characterized by a global distortion that, in general, can be decomposed into a homogeneous strain describing the distortion of the lattice of B relative to that of A and an atomic displacement field representing the displacements of the atoms of B from their positions in A. In order to facilitate the comparison of the two structures, first the coordinate system of structure A is transformed by an appropriate transformation to that of structure B. This new description of A will be called the reference description of structure A relative to structure B. Now, the metric tensors G_{A} of the reference description of A and G_{B} are of the same type and are distinguished only by the values of their parameters. The adaptation of structure A to structure B can be performed in two further steps. In the first step the parameter values of G_{A} are adapted to those of G_{B} by an affine transformation which determines the metric deformation (spontaneous strain) of structure B relative to structure A. The result is a hypothetical structure which still differs from structure B by atomic displacements. In the second step these displacements are balanced out by shifting the individual atoms to those of structure B. In other words, if represents the basis of the parent phase, then its image under the transformation = should be similar to the basis of the derivative phase . The difference between and determines the metric deformation (spontaneous strain) accompanying the transition between the two phases. Similarly, the differences between the images of the atomic positions X of the basic structure under the transformation and the atomic positions of the derivative structure give the atomic displacements that occur during the phase transition.
As an example we will consider the structural phase transition of GeTe, which is of displacive type, i.e. the phase transition is accomplished through small atomic displacements. The roomtemperature ferroelectric phase belongs to the rhombohedral space group R3m (160). At about 720 K a structural phase transition takes place to a highsymmetry paraelectric cubic phase of the NaCl type. The following descriptions of the two phases of GeTe are taken from the ICSD:
The relation between the basis of the Fcentred cubic lattice and the basis of the reference description can be obtained by inspection. The axis of the reference hexagonal basis must be one of the cubic threefold axes, say [111]. The axes and must be lattice vectors of the Fcentred lattice, perpendicular to the rhombohedral axis. They must have equal length, form an angle of 120°, and together with define a righthanded basis. For example, the vectors , fulfil these conditions.
The transformation matrix P between the bases and can also be derived from the data listed in Table 1.5.1.1 in two steps:
Combining equations (1.5.2.18) and (1.5.2.19) gives the orientational relationship between the Fcentred cubic cell and the rhombohedrally centred hexagonal cell = , whereFormally, the lattice parameters of the reference unit cell can be extracted from the metric tensor obtained from the metric tensor transformed by P, cf. equation (1.5.2.4):which gives = 4.249 Å and = 10.408 Å. The comparison of these values with the experimentally determined lattice parameters of the lowsymmetry phase [a_{hex} = 4.164 (2) Å, c_{hex} = 10.69 (4) Å (Chattopadhyay et al., 1987)] determines the lattice deformation accompanying the displacive phase transition, which basically consists of expanding the cubic unit cell along the [111] direction. (In fact, the elongation along [111] is accompanied by a contraction in the ab plane that leads to an overall volume reduction of about 1.3%.)
Owing to the polar character of R3m, the symmetry conditions following from the group–subgroup relation [cf. equation (1.5.2.11)] are not sufficient to determine the origin shift of the transformation between the high and the lowsymmetry space groups. The origin shift of in this specific case is chosen in such a way that the relative displacements of Ge and Te are equal in size but in opposite direction along [111].
The inverse transformation matrix–column pair = is necessary for the calculation of the atomic coordinates of the reference description . Given the matrix P, its inverse can be calculated either directly (i.e. applying the algebraic procedure for inversion of a matrix) or using the inverse matrices and listed in Table 1.5.1.1:(Note the change in the order of multiplication of the matrices and in Q.) The corresponding origin shift q is given byThe atomic positions of the reference description becomeThe coordinates of the representative Ge atom occupying position 4a 0, 0, 0 in are transformed to , while those of Te are transformed from 4b in to . The comparison of these values with the experimentally determined atomic coordinates of Ge 0, 0, 0.2376 and Te 0, 0, 0.7624 reveals the corresponding atomic displacements associated with the displacive phase transition. The lowsymmetry phase is a result of relative atomic displacements of the Ge and Te atoms along the polar (rhombohedral) [111] direction, giving rise to nonzero polarization along the same direction, i.e. the phase transition is a paraelectrictoferroelectric one.
References
Chattopadhyay, T. K., Boucherle, J. X. & von Schnering, H. G. (1987). Neutron diffraction study on the structural phase transition in GeTe. J. Phys. C, 20, 1431–1440.Wiedemeier, H. & Siemers, P. A. (1989). The thermal expansion of GeS and GeTe. J. Less Common Met. 146, 279–298.