Bayesian inference of nanoparticle-broadened X-ray line profiles

Abstract

The analysis of x-ray line profile broadening can be considered as solving a series of inverse problems. There are usually two steps-removing the instrumental contribution (deconvolution), and determining the broadening contribution in terms of crystallite size and microstrain. Here we are concerned with quantifying only the size broadening, in terms of the shape and size distributions of the crystallites. We present a method that removes the instrumental broadening and determines the particle size distribution in a single step. The general theoretical framework developed makes it possible to determine the crystallite shape and average dimensions, and to fully quantify these results by also assigning uncertainties to them.In general, there are two approaches that can be adopted. The first assumes functional forms for the size distribution and shape of the crystallites, and applies least squares fitting to determine the parameters defining the size distribution [1,2]. For pragmatic reasons, this approach is often used to ensure numerical stability; however, it is based on an explicit assumption for the crystallite size distribution and does not take into account the non-uniqueness of the solution.The second approach takes into account the nonuniqueness of the problem of determining the size distribution P(D) from the experimental data, by assigning a probability to the solutions and enabling an average solution to be determined from the set of solutions; moreover, it also allows any a priori information and assumptions to be included and tested. This approach is

Crystal structure and microstructure of synthetic hexagonal magnesium–cobalt cordierite solid solutions (Mg2−2xCo2xAl4Si5O18)

“…Similar models analysed with different details can be found in works by Wilson (1962Wilson ( , 1963Wilson ( , 1970, Warren (1969), , Armstrong et al (2004) and Langford et al (1993). Overviews of the models and methods used in this branch of crystallography can be found in the survey by Langford & Louë r (1996), in books edited by Snyder et al (1999), Mittemeijer & Scardi (2004) and by Dinnebier & Billinge (2008), and also in articles by Louë r (2003) and Leoni et al (2004).…”

Kinematical diffraction by distorted crystals – dislocation X-ray line broadening

“…the analysis of finite size induced diffraction-line broadening in terms of the crystallite size distribution [e.g. 30,[67][68][69][70][71]. A crystallite size standard reference material is developed at NIST [72] which should be helpful in the characterization of size distributions using line profile analysis.…”

Characterization of Atomic Layer Deposition using X-Ray Reflectometry

“…Bayesian methods have analyzed data from a variety of scattering experiments, including X-ray diffraction and neutron diffraction and reflectivity (e.g., [6,7,8,9]. Drawing on these results, we outline how Bayesian reasoning can be applied to XRR.…”