International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.8, p. 913

Another type of modulation, the occupation modulation, can be treated in a way similar to the displacive modulation. As an example consider an alloy where the positions of the basic structure have spacegroup symmetry, but are statistically occupied by either of two types of atoms. Suppose that the position r is occupied by an atom of type A with probability and by one of type B with probability and that p is periodic. The probability of finding an A atom at site is with . In this case, the structure factor becomes where and are the atomic scattering factors. Because of the periodicity, one has Hence, where Δ(H) is the sum of δ functions over the reciprocal lattice of the basic structure: Consequently, the diffraction peaks occur at positions H given by (9.8.1.7). For a simple sinusoidal modulation [m = ±1 in (9.8.1.29)], there are only main reflections and firstorder satellites (m = ±1). One may introduce an additional coordinate t and generalize (9.8.1.27) to which has (3 + 1)dimensional spacegroup symmetry. Generalization to more complex modulation cases is then straightforward.