An analytical approximation for a size-broadened profile given by the lognormal and gamma distributions

The size-broadened profile given by the lognormal and gamma size distributions of spherical crystallites is considered. An analytical approximation for the size-broadened profile is derived that can be analytically convolved with the strain-broadened and instrumental-broadened profiles. The method is tested on two CeO₂ powders; one shows `super-Lorentzian' profiles that were successfully modelled under the assumption of a broad lognormal size distribution. It is shown that the Voigt function, as a common model for a size-broadened profile, fails for both very narrow and broad size distributions. It is argued that the size-broadened line profile is not very sensitive to variations in size distribution and that an apparent domain size or even column-length distribution function can correspond to significantly different size distributions.