O limite de aplicação da equação de Scherrer

Abstract

The Scherrer equation is a widely used tool in the characterization of polycrystalline samples, relating the width of the diffraction peaks with the size of the crystallites. This equation is based on the kinematical theory of X-ray diffraction, that gives reasonable results only for small crystals or crystals with high density of defects. For thick crystals, the dynamical theory of X-ray diffraction is more appropriate. Since there exists a limit for the size of the crystallites for which the kinematical theory can be applied, there is also an analogous limit in the size of the crystallites for the application of the Scherrer equation, that was determined for three crystals, Si, CeO2 and LaB6 comparing the results of the Scherrer equation with the computational simulations using X-ray dynamical theory. It has been suggested that this limit depends on the linear absorption coefficient (µ0) and the Bragg angle (θB). However, because those results were restricted for specific structures, they are not general. In this work, we develop a systematic study of the influence of several parameters in the Scherrer limit. First in similar crystals, but with µ0 ranging from 595,9 to 2726,0 cm-1, we show that there exist other parameters influencing this limit. After, in prototype structures, in which the structural parameters could be altered independently, we show how the Scherrer limit varies with the structure factor of the reflections H, -H and 0; FH, F-H and F0, respectively; the unit cell volume, V; the wavelength, λ; µ0; and the Bragg angle, θB. Finally, we confirm those results for a general structure. We show that the Scherrer limit is determined by the reduction in the penetration depth of the X-rays. In the case of µ0 = 0, this reduction is caused by the primary extinction, quantified by the extinction length (Λ0); the Scherrer limit is equal to 0.119Λ0. On the other hand, for Λ0 sufficiently large, the Scherrer limit is directly proportional to µ0= sinθB. When both effects act at the same time, the absorption decreases the Scherrer limit for large Λ0, where this reduction is proportional to Λ0. Also, we have defined a Scherrer limit based on the full width at half maximum and in this case, we showed that it is proportional to the Darwin width and not to Λ0. Finally, we developed a corrected form of the Scherrer equation, that gives the same results of the dynamical theory for µ0 = 0.