Crystal powder statistics. II. Line profiles in diffraction spectra of identical crystals and of Gaussian samples. Crystal size distributions

Abstract

The average interference function (, 7h,t(AS)) of a powder sample containing perfect crystals at a reciprocal distance AS from the peak is evaluated both for the case of identical parallelepiped crystals and for a Gaussian sample [probability of thickness d along a given crystal direction = C~ exp (-C2d2)]. In the latter case (, 7hkt(AS) ) decreases as I/AS 2 for large AS, by analogy with the Bernoullian model IAilegra, Bassi & Meille (1978). Acta Cryst. A34, 652-655] although with a smaller amplitude, for a fixed integrated intensity and half-peak width. It is shown that the Gaussian interference function, or line profile, cannot be given by any real sample, at least if its crystals neither contain holes nor have concave surfaces. Number and weight probability distributions are calculated both for the Bernoullian and for the Gaussian crystal-size statistics. As expected from the calculated line profiles, the Bernoullian statistics correspond to a larger weight percentage of crystals smaller than the average.

Preparation and characterization of Mg–Al hydrotalcite-like compounds containing cerium

“…However, a small shoulder at 2900-3000 cm −1 suggests the presence of a second type of -OH stretching vibration (possibly due to hydrogen bonding with carbonate in the interlayer spacing). A shoulder at 1650 cm −1 may be ascribed to the bending mode of interlayer water molecules [18], which decreases with decrease in the Al 3+ /Ce 3+ ratio (CMA-1-CMA-4). An intense absorption band at 1402 cm −1 is attributed to the absorption corresponding to the υ 3 mode of interlayer carbonate with a shoulder at 1490 cm −1 that splits upon lowering in symmetry with decrease in Al 3+ /Ce 3+ ratio.…”

KI7: a program written in vectorial Fortran for the simulation of powder diffraction patterns

“…Allegra, Bassi & Meille (1978) and Allegra & Ronca (1978 calculated the intensity profile of a powder of parallelepipedal crystallites characterized by different kinds of size distribution: the spherical surface was replaced, in the neighbourhood of a reciprocal-space node, by the plane tangent to it and the integral on the right-hand side of (1), with the simplified definition of the surface element, was calculated by analytical procedures. Similarly, Warren (1941) and Wilson (1949) succeeded in calculating the intensity profiles of monodimensionally disordered structures by the tangent-cylinder approximation.…”

X-ray Rietveld refinement using Debye–Scherrer geometry

“…The values of X 2 obtained, together with the number of data points (n), number of degrees of freedom (f) and the probability (p) that the Z 2 variate, for f degrees of freedom, exceeds the observed value are shown below. Allegra, Bassi & Meille, 1978;Allegra & Ronca, 1978;Langford & Wilson, 1978). Instead we merely conclude that for a well crystallized sample showing no appreciable particlesize or strain broadening and with an experiment of the present (rather low) resolution and statistical accuracy, the assumption of a basic Gaussian peak is statistically adequate.…”