Determination of the average crystallite size and the crystallite size distribution: the envelope function approach EnvACS

A procedure is presented to exactly obtain the apparent average crystallite size (ACS) of powder samples using standard in-house powder diffraction experiments without any restriction originating from the Scherrer equation. Additionally, the crystallite size distribution within the sample can be evaluated. To achieve this, powder diffractograms are background corrected and long-range radial distribution functions <i>G</i>(<i>r</i>) up to 300 nm are calculated from the diffraction data. The envelope function <i>f</i> ^env of <i>G</i>(<i>r</i>) is approximated by a procedure determining the absolute maxima of <i>G</i>(<i>r</i>) in a certain interval (<i>r</i> range). Fitting of an ACS distribution envelope function to this approximation gives the ACS and its distribution. The method is tested on diffractograms of LaB₆ standard reference materials measured with different wavelengths to demonstrate the validity of the approach and to clarify the influence of the wavelength used. The latter results in a general description of the maximum observable average crystallite size, which depends on the instrument and wavelength used. The crystallite site distribution is compared with particle size distributions based on transmission electron microscopy investigations, providing an approximation of the average number of crystallites per particle.