Symmetry operations for displacively modulated structures

Displacive modulation is defined as periodic distortion with an incommensurable k vector. If the phase of the distortion is t, a symmetry operation can be expressed as a normal space-group operation combined with a sign reversal and/or a shift in the variable t. Earlier results obtained by de Wolff [Acta Cryst. (1974), A30, 777-785] are reformulated in terms of these operations. The periodic functions defining the displacements of two symmetry-related atoms are shown to be related by a simple equation. Applications to published structures are given. The validity of this equation depends on a symmetry-adapted choice of the vector k. It is shown by a two-dimensional example that there are cases for which that choice of k requires the introduction of 'improper symmetry translations' with an extra t shift of ½ or ½. The ensuing Bravais lattice types are similar to those used for the description of magnetic symmetry.