DETERMINATION OF QUASI-CRYSTALLINE STRUCTURES - A REFINEMENT PROGRAM USING SYMMETRY-ADAPTED PARAMETERS

A general program for the refinement of quasicrystalline structures using diffraction data is presented. The program can be used for both icosahedral and polygonal quasicrystals. The refinement process is based on the fitting of the structural model to experimental diffraction data and observed density and chemical composition. Superspace formalism is used for the structure description and the hypersurfaces in superspace describing the atomic positions are assumed to be parallel to the internal space. No additional a priori assumption on the form of the atomic hypersurfaces is necessary except that the deviations of the atomic-surface contours from a spherical shape do not contain very short wave components in a significant amount. The contours of each symmetry-independent atomic hypersurface in internal space are parametrized in terms of linear combinations of radial functions (surface harmonic) invariant for the hypersurface point group in internal space. This allows a continuous refinement of the structure in terms of symmetry-adapted parameters consistent with the symmetry restrictions resulting from the postulated superspace symmetry. The program requires an initial very approximate guess of the structure in terms of 'spherical' hypersurfaces of which only the symmetry centres are known with confidence. The continuous parametrization of the hypersurfaces does not a priori restrict their form, except in its degree of complexity or fine detail, which is limited by the number of terms considered in the linear expansion of the surface contours. In general, the number of surface harmonics considered should be consistent with the accuracy allowed by the experimental data set. The refinement process can be performed either by a full least-squares method or by means of a simplex algorithm. The physical consistency of the refined hypersurfaces with respect to the predicted density, chemical composition and interatomic distances is controlled by including additional 'penalty functions' in the parameter to be minimized.