Diffraction of x rays at the far tails of the Bragg peaks

Specular reflection intensity modulated by grazing-incidence diffraction in a wide angular range

Grazing-incidence X-ray diffraction (GID) is a well known technique for the characterization of crystal surfaces. A theoretical study has been performed of the sensitivity of GID to the structure of a crystal surface and distorted nanometre-thin surface layers. To simulate GID from crystals that have a complex subsurface structure, a matrix formalism of the dynamical diffraction theory has been applied. It has been found that the azimuthal rocking curves of a crystal that has a distorted subsurface, measured over a wide angular range, show asymmetric thickness oscillations with two distinguishable sets of frequencies: one corresponding to the diffraction in the single-crystal subsurface layer and the second corresponding to the diffraction in the single-crystal substrate. Therefore, azimuthal rocking curves allow characterization of the subsurface structure of a single crystal. Furthermore, thickness oscillations induced by evanescent diffraction modulate the specular reflection intensity, showing high-intensity modulations. This will potentially allow implementation of subsurface crystal characterization using, for instance, a laboratory-scale X-ray diffractometer.

The Scherrer equation and the dynamical theory of X-ray diffraction

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 large crystallite 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. Because of the 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, LaB6and 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 coefficients below 2117.3 cm−1the 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.