Small-angle techniques for the asymptotic analysis of X-ray diffraction peaks
Abstract
Any wide-angle X-ray scattering (WAXS) peak, relevant to a powder sample of crystallites with negligible internal disorder, is the Fourier transform of the so-called oriented stick probability function (oSPF) of the filled part of the sample, with the stick orientated along the reflexion direction. From this observation the following consequences are obtained: the correlation function used in small-angle X-ray scattering (SAXS) is the average of the former oSPF's over all possible stick orientations; any peak profile asymptotically vanishes as Srh-2, where Sr is the (specific) area of the interphase surface presented by the sample along the reflexion direction; oscillatory deviations, behaving as Sr, || cos (hL)h-2, are present only when a subset (having area Sr, ||) of the interface, after having been translated by L along the reflexion direction, superposes on itself; the angularity of the interphase surface can be measured by a natural modification of the Porod integral relation. For samples really isotropic, the above quantities should not depend on the reflexion direction and thus they should be equal to those measured by SAXS experiments. These results are applied to three ideal isotropic powder samples made up, respectively, of monodisperse spherical, cubic and cylindrical crystallites as well as to the analysis of two WAXS peaks diffracted by two real samples of zirconia powders.