Crystal powder statistics. I. Lorentzian line profiles in diffraction spectra of Bernoullian samples

A Bernoullian powder sample is defined as an ensemble of parallelepiped crystals where the probability of any layer being on the surface is independent of its size as well as of the number of its predecessors, although being different for the three types of layers parallel to the crystal faces. It is shown that the line profile of any reflection is given by the Lorentzian, or Cauchy, distribution I(e) = A/(1 + K2e2), where e is the reciprocal coordinate measured from the peak with intensity A, and 2/K is the half-peak width, provided the average size of the crystals is at least of the order of 10 unit cells along each of the three edges.