Crystal powder statistics. II. Line profiles in diffraction spectra of identical crystals and of Gaussian samples. Crystal size distributions

Abstract

The average interference function ({\cal J} hkl ( ΔS )) of a powder sample containing perfect crystals at a reciprocal distance ΔS from the peak is evaluated both for the case of identical parallelepiped crystals and for a Gaussian sample [probability of thickness d along a given crystal direction = C 1 exp (- C 2 d 2 )]. In the latter case ({\cal J} hkl ( ΔS )) decreases as 1/ ΔS 2 for large ΔS , by analogy with the Bernoullian model [Ailegra, Bassi & Meille (1978). Acta Cryst. A 34 , 652-655] although with a smaller amplitude, for a fixed integrated intensity and half-peak width. It is shown that the Gaussian interference function, or line profile, cannot be given by any real sample, at least if its crystals neither contain holes nor have concave surfaces. Number and weight probability distributions are calculated both for the Bernoullian and for the Gaussian crystal-size statistics. As expected from the calculated line profiles, the Bernoullian statistics correspond to a larger weight percentage of crystals smaller than the average.