Determination of nanoparticle sizes by X-ray diffraction

Different procedures for analysis of particle sizes by the X-ray diffraction method are compared by the example of nanoparticles of nickel and iron(3+) oxide (Fe₂O₃). A modified Warren-Averbach method is proposed for the analysis of the X-ray diffraction line profile based on the approximation by the Voigt function, which yields stable solutions, and the efficiency of the method is shown. The analysis within the frame-work of the Warren-Averbach method makes it possible to restore the distribution function of nanoparticles (crystallites) over true diameters, which satisfactorily correlates with electron microscopy data. The applicability of the Warren-Averbach method to the estimation of crystallite sizes by the analysis of a single diffraction line is substantiated. The range of the applicability of the Scherrer, Williamson-Hall, Warren-Averbach, and modified Warren-Averbach methods to the substructure analysis by the X-ray diffraction is determined as depending on the method of nanostructure formation.