CRYSIZ: a program for computing crystallite size and strain from the broadening of powder diffraction lines

Abstract

The program CRYSIZ is designed to take the powder diffraction line profiles for a well-crystallized sample, called a reference pattern, and for a sample of the same substance, called a broadened pattern, to produce measures of the mean crystallite size, the distribution of crystallite sizes, and the root mean square residual microstrain in the broadened sample. The data required are the two powder patterns and a series of directives to signal the calculations and plots to be done during the execution of the program. The program loads files containing the background corrected powder diffraction intensity data for both the reference and broadened patterns. Preliminary calculations find the centroids, full width at half maximums, integral breadths, spans over sum, and second moments. Two methods of deconvoluting the profile to calculate size and strain are allowed. Either the direct or the Stokes Fourier coefficient method of deconvolution may be chosen. In the direct method the profiles are extracted by numerical fitting. This method is slower but produces unfolded profiles free of ringing and the ``hook effect``. In this case the Fourier coefficients required for Warren-Averbach analysis are produced from the deconvoluted profile. In the Stokes method the diffraction pattern of each reference and broadened profile is Fourier transformed to produce a set of Fourier coefficients. The Fourier coefficients of the broadened profiles are divided by those of the reference pattern. The resulting coefficients are the Stokes coefficients. The Stokes coefficients are smoothed by a least- squares procedure in order to remove noise and quell ringing and hooking, then used as input to a reverse Fourier transform. This transform produces an ``unfolded powder line,`` which is a best estimate of the broadened profile with the reference profile and noise removed. The deconvolution of the reference profile gives a broadened profile due only to the crystallite size and strain.