Análise da equação de Scherrer pela teoria dinâmica da difração de raios X aplicada a distribuições de tamanho de cristalitos

Abstract

The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to lar e crystallitev sizes because its derivation is based on the kinematical theory of X-ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X-ray diffraction. Due to appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X-ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6 and CeO2. The full width at half-maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coeficients below 2117.3 cm -1 the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 μm. Next, a study was carried out taking into account crystallite size distributions. The diffraction profiles were calculated by dynamic theory considering narrow and wide distributions (Gaussian and Lognormal) of crystallite size to crystals of LaB6. It was shown that the larger the value of the standard deviation, ie, the wider the distribution function, the greater the error in the crystallite size value obtained by the Scherrer equation in these profiles. It has also been shown that for any of the centered distributions in any region of size and for any standard deviation value used in this work, the integrated width (FWHM int ) of the diffraction peaks provides better results for the crystallite size in comparison to the peak width (FWHM).