Refinement of incommensurate structures against diffraction data from a twinned crystal

Twinning can lead to a diffraction pattern with additional reflections that are incommensurate with the reflections of a crystal with only one orientation of the structure. The integer indexing of such a diffraction pattern involves more than three reciprocal-basis vectors. Analogously, for incommensurate crystals, the original number of (3 + d) reciprocal vectors should be extended to a larger set for a twinned incommensurate crystal. In this paper, it is shown that the diffraction symmetry for a twinned crystal can be analyzed in a way analogous to the treatment of the symmetry of an incommensurate structure. The theory is implemented in a refinement program for X-ray and neutron diffraction data and allows all intensity data from isolated and overlapping reflections to be taken into account. The method can also be applied to the refinement of ordinary crystal structures. The program has been used to determine the modulated structure of the inorganic misfit layer compound (HoS)_1.23NbS₂.