Diffraction symmetry of incommensurate structures

The symmetry of the diffusion pattern for an incommensurate displacive structure is analysed in a general context. The rotational symmetry and systematic extinctions in the diffraction diagram of this type of structure are shown to be directly based on a particular transformation property of the amplitudes of the atomic modulation waves intervening in the distortion for a subgroup of the basic structure space group. Hence a general three-dimensional framework can be established, where diffraction symmetry can straight-forwardly be used to restrict and relate the amplitudes corresponding to the atomic modulation waves of the distortion. It is shown how a full direct exploitation of diffraction symmetry can therefore be accomplished to determine the characteristics of the incommensurate distortion, without referring to the superspace group formalism. The equivalence of the present simpler approach with the latter is then demonstrated. Finally a concept of atomic scattering modulation factors is proposed to reduce the expression of the structure factor for these phases to a very simple form analogous to those used in commensurate cases.