An aid to the structure analysis of incommensurate phases

Group theory is used to establish three results likely to be useful in solving the crystal structures of complicated incommensurate phases. In the first of these it is demonstrated that an incommensurate structure with paired scattering vectors ± q must contain two different component structures, one modulated with cos q.r and the other with sin q.r. The second theorem states that the two components have different but related symmetries if the average structure has at least one element in its space group which turns q into -q. In that case, each aspect of the modulation is assigned uniquely by symmetry to either the cosine or sine factor. The third result concerns the Patterson function that may be constructed from the intensity scattered by the incommensurate modulation. This is also necessarily two-dimensional, the plus difference Patterson function being the sum of the Patterson functions obtained separately for the two component structures, while the minus difference Patterson function contains cross terms between the two components. Other symmetry arguments are mentioned, including symmetry signatures in Patterson functions, and systematic equalities in satellite intensities which arise from systematic extinctions in the scattering from one component or the other.