Multiple diffraction of X-rays and the phase problem. Computational procedures and comparison with experiment

The general theory of n-beam X-ray diffraction (n > 2) has been developed in the framework of classical dynamical theory and applied to the Bragg case. It is shown that the crystal wave-vectors are the eigen-values of a 4n × 4n dispersion matrix. The boundary conditions are applied to a parallel-sided crystal slab and show that for an infinite thickness only 2n wave fields survive. The Umweganregung plot of Ge(222) with Cu Kα radiation has been considered in detail. The integrated intensity of an Umweganregung peak is defined here as a double integral with respect to θ (angle of incidence) and ϕ (azimuthal angle). The 222-113 and 222-{\bar 11}3 absolute integrated intensities were measured on a dislocation-free Ge crystal. Excellent agreement is obtained between experimental and calculated values. The ratio between the two integrated intensities (of the order of 7) did not change appreciably for a Ge mosaic crystal, although both reflections exhibited increases with respect to the perfect-crystal values. Since the two Umweganregung peaks considered in this experiment involve crystallographically equivalent reflections with different phases, it is suggested that the present technique can in principle be used for phase determination in crystal structure analysis.