Novel analytical model for the determination of grain size distributions in nanocrystalline materials with low lattice microstrains by X-ray diffractometry

We have developed a novel, analytical model for the determination of grain size distributions in nanocrystalline (nc) materials with low internal stresses by X-ray diffractometry (XRD). The model assumes explicitly that the XRD peaks are pseudo-Voigtian and that the grain size distributions are lognormal, both of which are assumptions amply supported by the experimental evidence. It was found analytically that the grain size dispersion depends on the shape of the XRD peaks only, whereas the grain size median depends on both the shape and width of the XRD peaks. In addition, the theoretical predictions resulting from the model were validated using standard XRD peaks obtained by computer simulation from first principles. Particular emphasis is given to the discussion of the validity limits of the model, and to the analysis of the influence of the characteristics of the grain size distributions on the nature of the XRD peaks. We then show how to calculate the average and apparent grain sizes from the grain size distribution determined with the model, and how this compares with the Scherrer method. Implications for the characterization of (undistorted and distorted) nc-materials are indicated, and a case study of an nc-powder of cubic ZrO2 is presented. The application of the model itself is simple, involving only the fit of a pseudo-Voigt function to a single XRD peak followed by the use of two equations. This suggests that the model may have an important role to play in the characterization of nc-materials