Moments and Cumulants of Diffraction Profiles Broadened by Stacking Faults

Line broadening in a diffraction intensity profile of powdered crystalline materials due to stacking fault has been characterized in terms of the zeroth, first, second, third, and fourth moments and the fourth cumulant. Calculations have been derived showing that the first moment causes a shift in the peak position of the profile while the third moment affects its shape. The intensity expression has been derived on the basis of usual Cartesian coordinates and also of polar coordinates indicated by the probability of the fault and the reciprocal lattice parameter as the two axes. The expressions for the fourth cumulant have also been so derived. Here we have used three different approaches to determine methods for calculating the fourth cumulant due to stacking faults. The three forms of the equations derived here are for different coordinate systems, but will arrive at the same answers.