Accounting for unknown systematic errors in Rietveld refinements: a Bayesian statistics approach

A method has been developed to address the effects of systematic errors in Rietveld refinements using powder diffraction data. Relevant errors were categorized into multiplicative, additive and peak-shape types. Corrections for these errors were incorporated into structural refinements using a Bayesian statistics approach, with the corrections themselves treated as nuisance parameters and marginalized out of the analysis. Structural parameters refined using the proposed method represent probability-weighted averages over all possible error corrections. The developed formalism has been adapted to least-squares minimization algorithms and implemented as an extension to the Rietveld software package GSAS-II. The technique was first tested using neutron and X-ray diffraction data simulated for PbSO₄ and then applied to the equivalent experimental data sets for the same compound. The results obtained using the simulated data confirmed that the proposed method yields significantly more accurate estimates of structural parameters and their uncertainties than standard refinements. The benefits were particularly significant for joint refinements using neutron and X-ray diffraction data because accounting for systematic errors enabled more adequate weighting of the individual data sets.