A new approach to phason disorder for a decagonal quasicrystal: the moment series expansion of the tiling distribution function for AlCuRh

A method is proposed of calculating the geometric term of the structure factor for quasicrystals, which enables incorporation of the phason disorder. The scheme is based on the series expansion of the structure factor with moments of the distribution function as coefficients. A distribution function is a mathematical object that is constructed for reference vertices of the tiles in the quasilattice. It encloses the entire structural information of the underlying quasilattice, together with the inherent disorder, necessary to calculate the diffraction pattern. By tuning the value of the distribution moments through the refinement procedure, it is possible to obtain a very good agreement of this new model of the decagonal AlCuRh phase with the experimental data, reflected in the crystallographic R factor of 6.08%. The characteristic bias of the calculated diffraction peak intensities observed for the low-intensity reflections is significantly diminished, confirming its origin being, to some extent, related to phason disorder. Additionally, it is no longer necessary to use the general Debye–Waller factor for phasons, as the new formula accommodates this type of structural disorder. However, the best result was obtained for the model combining the new approach with the Gaussian corrective term.