An analytical model for the determination of crystallite size and crystal lattice microstrain distributions in nanocrystalline materials from the variance of the X-ray diffraction peaks

Abstract

An analytical model for the determination of crystallite size and crystal lattice microstrain distributions in nanocrystalline (nc) materials by X-ray diffractometry (XRD) is presented. It entails generalizing the variance method to establish analytically the connection between the variance coefficients of the physically broadened XRD peaks and the characteristic parameters of explicit distributions of crystallite sizes and crystal lattice microstrains, which results in a more detailed characterization of the nc-materials. The proposed model is generic in nature and has the potential to be used under the assumption of different mathematical functions for the two distributions, which suggests that it may have an important role to play in the characterization of nc-materials. Nevertheless, the specialization to the case of nc-materials with log-normal crystallite size distribution and three typical types of lattice microstrains is used as an illustration and to formulate explicit analytical expressions of interest. Finally, the usefulness of the proposed model is demonstrated on standard XRD profiles.