Estimating grain-size distributions in nanocrystalline materials from X-ray diffraction profile analysis

It is well known that the Fourier analysis of X-ray diffraction peak profiles (as implemented by Warren and Averbach) can accurately determine the area-weighted average grain size of a fine-grained sample. Less well known is the fact that this method simultaneously yields a volume-weighted average grain size. Under certain circumstances, knowledge of these two weighted average grain sizes is sufficient to permit reliable estimation of the grain-size distribution, even when the distribution cannot be calculated directly from the Fourier coefficients, as is usually the case. We demonstrate this for a nanocrystalline Pd sample prepared by inert-gas condensation; average grain sizes and the grain-size distribution are estimated by X-ray diffraction profile analysis and compared with the same quantities measured directly by transmission electron microscopy (TEM). Very good agreement between the resulting average grain sizes is achieved only after compensating for the fact that diffraction profile analysis directly yields an average unit-cell column length rather than an average grain diameter. The agreement between the grain-size distributions determined by profile analysis and TEM was good for crystallite sizes larger than the area-weighted average grain size but deteriorated for smaller sizes due to the volume dependence of the diffraction peak intensity.