Bayesian method for the analysis of diffraction patterns using BLAND

Abstract

Rietveld refinement of X‐ray and neutron diffraction patterns is routinely used to solve crystal and magnetic structures of organic and inorganic materials over many length scales. Despite its success over the past few decades, conventional Rietveld analysis suffers from tedious iterative methodologies, and the unfortunate consequence of many least‐squares algorithms discovering local minima that are not the most accurate solutions. Bayesian methods which allow the explicit encoding of a priori knowledge pose an attractive alternative to this approach by enhancing the ability to determine the correlations between parameters and to provide a more robust method for model selection. Global approaches also avoid the divergences and local minima often encountered by practitioners of the traditional Rietveld technique. The goal of this work is to demonstrate the effectiveness of an automated Bayesian algorithm for Rietveld refinement of neutron diffraction patterns in the solution of crystallographic and magnetic structures. A new software package, BLAND (Bayesian library for analyzing neutron diffraction data), based on the Markov–Chain Monte Carlo minimization routine, is presented. The benefits of such an approach are demonstrated through several examples and compared with traditional refinement techniques.

The benefits of a Bayesian analysis for the characterization of magnetic nanoparticles

“…The histogram of each individual parameter exhibits a narrow and symmetric peak (i.e. a normal (Gaussian) distribution) which indicates that the solution found from the Bayesian refinement is a strong minimum of the 𝜒 2 function [18] . The median values of these distributions (i.e.…”

“…Regarding the cross-correlation plots in figure 2(c), an isotropic pattern in the center of the box indicates that there is no strong correlation between the values of two parameters [18,21] . This is the case for example for the pairs (𝑀 𝑆 , 𝜎 𝑚 ) and (𝑏, 𝜎 𝑚 ); they are essentially independent of each other.…”

Transient phase fraction and dislocation density estimation from in-situ X-ray diffraction data with a low signal-to-noise ratio using a Bayesian approach to the Rietveld analysis

“…realized by means of adaptive Markov Chain Monte Carlo (MCMC) simulation algorithms [20][21][22], which are based on the Bayes theorem. It is demonstrated in [23][24][25] that this Bayesian approach can also be successfully applied to the Rietveld method. It is shown in this work that by applying the Bayes theorem on the Rietveld analysis not only lattice constants and phase fractions but also dislocation densities can be deduced from diffraction data even in case of a low signal-to-noise-ratio.…”

Optimization of reflectometry experiments using information theory

“…In the recent past, Bayesian statistical methods have found applications in reflectometry for robust global model fitting and the determination of confidence limits on model parameters (Sivia & Webster, 1998;Kirby et al, 2012;Maranville et al, 2016;Lesniewski et al, 2016). In particular, the work of Sivia and co-workers has provided a solid foundation for the ISSN 1600-5767 application of Bayesian statistics to reflectivity data, discussing aspects such as parameter estimation, model selection and experimental design.…”

“…In particular, the work of Sivia and co-workers has provided a solid foundation for the ISSN 1600-5767 application of Bayesian statistics to reflectivity data, discussing aspects such as parameter estimation, model selection and experimental design. Our work concerning experimental optimization adds to this foundation by introducing model fitting based on a Monte Carlo Markov chain (MCMC) simulation, which by design yields a sample of the posterior parameter density function (PDF) (Yustres et al, 2012;Braak & Vrugt, 2008;Lesniewski et al, 2016). A measure of the information gain from a given experiment is obtained from comparing the entropies of the posterior and prior PDFs, which represent the knowledge about the sample after and before the experiment, respectively.…”