Efficient solution of particle shape functions for the analysis of powder total scattering data

Abstract

Structural characterization of powder samples via total scattering methods, in either real or reciprocal space, must take into account the effect of particle shape. Here, the shape contribution of a set of ideally isolated particles to the small-angle scattering (SAS) component of the intensity profile is modelled using the shape function [Svergun & Koch (2003). Rep. Prog. Phys. 66, 1735–1782]. The shape function is obtained by orientational averaging of common volume functions (CVFs) for a discrete set of directions. The effects of particle size and size dispersity are accounted for via scaling of the CVFs and their convolution with the underlying probability distribution. The method is applied to shapes with CVFs expressed analytically or by using discrete tables. The accurate calculation of SAS particle shape contributions up to large momentum transfer demonstrates the reliability and flexibility of modelling shape functions from sets of CVFs. The algorithm presented here is computationally efficient and can be directly incorporated into existing routines for analysis of powder total scattering data.

A simple formula for determining nanoparticle size distribution by combining small-angle X-ray scattering and diffraction results

“…In polydisperse systems of spherical NPs, L xrd ¼ 6:96=Áq stands for the XRD diameter value obtained from the diffraction peak width Áq (in reciprocal-space units), and L saxs ¼ 7:26=Áq stands for the SAXS diameter value obtained by measuring the full width at half-maximum (FWHM) of the scattering peak around q ¼ 0. For any other NP shape, numerical methods based on the Debye scattering equation can be used to compute exact values of the s and d constants (Farrow & Billinge, 2009;Cervellino et al, 2015;Scardi & Gelisio, 2016;Leonardi et al, 2022). In non-spherical NPs, d can vary from one diffraction peak to another as in the case of cubic NPs where s ¼ 5:64 and the d value falls in the range from 5.2 to 5.6, depending on the chosen diffraction peak and crystallographic orientation of the NP facets.…”

“…In the wideangle X-ray scattering (WAXS) region, what parameter of the particle size distribution (PSD) is determined by the diffraction peak width? Although there are general approaches describing how the diffraction peak line profiles are affected by size distribution (Scardi & Leoni, 2001;Leoni & Scardi, 2004;Cervellino et al, 2005;Leonardi et al, 2022), an explicit and direct answer to the above-mentioned question has been obscured by the mathematical formalism of more than 100 years of X-ray crystallography, as well summarized in Chapter 5.1 of the most recent volume of International Tables for Crystallography, Volume H (Leoni, 2019). The answer is the X-ray diffraction (XRD) peak width is defined by the median value L xrd of the fourth moment integral of the PSD.…”

Implications of size dispersion on X-ray scattering of crystalline nanoparticles: CeO2 as a case study

“…At wide angles, line-broadening analysis of diffraction peaks stands as the general approach to address size distribution. There is a very long discussion that can be traced back to the early decades of the 20th century about the actual role of size distribution in the line profile of the diffraction peaks (Jones, 1938;Bertaut, 1950;Le Bail & Loue ¨r, 1978;Langford & Loue ¨r, 1982;Langford et al, 2000;Leonardi et al, 2022). As particle dispersivity in powder samples has been seen more as a consequence of their preparation history than a physical property, most traditional methods of lineprofile analysis are primarily concerned with effects arising from crystalline imperfections and instrumentation (Cheary & Coelho, 1992;Kril & Birringer, 1998;Snyder et al, 1999;Coelho, 2000).…”