Diffraction theory of quasicrystals

Abstract

Abstract The diffraction problem of quasicrystals is solved with the aid of shift operators, s q(x, y, z) (q = x, y, z). Starting from certain sets of latent lattices which have to be optimized according to chemical composition, density, and diffraction pictures, it is shown that periodic structures are determined by as many parameters as the number of independent positions. Furthermore, the Fourier coefficients (Fc's) are strictly periodic. Structure factors are given by sums of products of Bessel functions (Bfs). The amplitudes of Fc's enter the arguments of Bf's, while their phases determine phase factors. The diffraction pictures show periodic properties, as far as the orders of Bf's and Fc's are concerned. This periodicity is violated only by the arguments of Bf's. Hence direct calculation of phases of reflections with certain ambiguities is possible. Quasicrystals are characterized by incommensurate modulations with an infinite number of possible Fc's which can not be derived from diffraction patterns. This difficulty is overcome by the observation of extinction rules, limiting the number of reflections observed. The orders of Fc's, Bf's respectively, entering the sum mentioned above, are correlated by convolution properties in diffraction. Taking into account their periodicities and the existence of sets of latent lattices, regularities in the diffraction pattern result which are briefly discussed.