Powder diffraction: Least-squares and beyond

This paper addresses some of the underlying statistical assumptions and issues in the collection and refinement of powder diffraction data. While standard data collection and Rietveld analysis have been extremely successful in providing structural information on a vast range of materials, there is often uncertainty about the true accuracy of the derived structural parameters. In this paper, we discuss a number of topics concerning data collection and the statistics of data analysis. We present a simple new function, the cumulative chi-squared distribution, for assessing regions of misfit in a diffraction pattern and introduce a matrix which relates the impact of individual points in a powder diffraction pattern with improvements in the estimated standard deviation of refined parameters. From an experimental viewpoint, we emphasise the importance of not over-counting at low-angles and the routine use of a variable counting scheme for data collection. Data analysis issues are discussed within the framework of maximum likelihood, which incorporates the current least-squares strategies but also enables the impact of systematic uncertainties in both observed and calculated data to be reduced.