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

A general program for the refinement of quasicrystalline structures us ing diffraction data is presented. The program can be used for both ic osahedral and polygonal quasicrystals. The refinement process is based on the fitting of the structural model to experimental diffraction da ta and observed density and chemical composition. Superspace formalism is used for the structure description and the hypersurfaces in supers pace 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 at omic-surface contours from a spherical shape do not contain very short wave components in a significant amount. The contours of each symmetr y-independent atomic hypersurface in internal space are parametrized i n terms of linear combinations of radial functions (surface harmonic) invariant for the hypersurface point group in internal space. This all ows a continuous refinement of the structure in terms of symmetry-adap ted parameters consistent with the symmetry restrictions resulting fro m the postulated superspace symmetry. The program requires an initial very approximate guess of the structure in terms of 'spherical' hypers urfaces 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 exp ansion of the surface contours. In general, the number of surface harm onics 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. Th e physical consistency of the refined hypersurfaces with respect to th e predicted density, chemical composition and interatomic distances is controlled by including additional 'penalty functions' in the paramet er to be minimized.