Intensity of N-beam X-ray diffraction: kinematical theory for a small crystal

For a small and ideally imperfect crystal, a method for calculating n-beam X-ray diffraction intensities has been developed on the basis of macroscopic intensity exchanges among the beams. This kinematical formulation results in a set of simultaneous equations that can be solved by numerical calculation. To validate the macroscopic formulae, the Darwin intensity transfer equations, which describe microscopic interactions by both diffraction and absorption, are integrated on a spherical crystal. With the hypotheses that one beam contributes to other beams as proportional to its observed intensity, the macroscopic and microscopic formulations are proved to be equivalent; quantitative evaluation of the n-beam effect thereby becomes available for practical experiments using a specimen with finite cross section of X-ray absorption. Examples of the psi-scan simulation on Si 111 and 222 are presented to characterize the present method, demonstrating the reasonable behaviour of the observed diffraction intensity while the linear absorption coefficient and the specimen size are varied.