Analytical expression for Diffraction Line Profile for polydispersive powders. New methods for Grain Size Distribution determination."

Roman Pielaszek, Witek Lojkowski

High Pressure Research Center, Polish Academy of Sciences

An analytical expression for the diffraction line profile for polydispersive powders (particularly, nanopowders) with Gamma Grain Size Distribution is derived. The expression consists of elementary functions only and can readily replace standard functions (like Gaussian, Lorentzian or Pearson) for diffraction peak fitting purposes. This allows for direct Grain Size Distribution determination using standard fitting software.

Well established Scherrer method allows for determination of the average grain size <R> of a crystalline powder by measurement of Full Width at Half Maximum (FWHM) of the diffraction peak profile. Basing on the expression derived, we propose an enhancement of this classical method. Measurement of two widths of the same peak, allows for two parameters to be distinguished: the average grain size <R> and dispersion of sizes (sigma). These parameters are sufficient to draw Grain Size Distribution (GSD) curve, that is much more informative than a single size parameter <R>.

We propose to measure widths at 1/5 and 4/5 of the peak maximum (FW1/5M and FW4/5M, respectively). A simple algebraic formula that converts measured FW1/5M and FW4/5M values into <R> and (sigma) is presented. The FW1/4/5M method proposed in this paper is especially sensitive in case of a broad diffraction maxima, i.e. for nano-sized polycrystals.