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Application of matrix decomposition algorithms for singular matrices to the Pawley method in Z‐Rietveld
Author(s) -
OishiTomiyasu R.,
Yonemura M.,
Morishima T.,
Hoshikawa A.,
Torii S.,
Ishigaki T.,
Kamiyama T.
Publication year - 2012
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889812003998
Subject(s) - rietveld refinement , matrix (chemical analysis) , algorithm , decomposition , powder diffraction , intensity (physics) , diffraction , nonlinear system , computer science , physics , mathematics , crystallography , chemistry , optics , quantum mechanics , organic chemistry , chromatography
Z‐Rietveld is a program suite for Rietveld analysis and the Pawley method; it was developed for analyses of powder diffraction data in the Materials and Life Science Facility of the Japan Proton Accelerator Research Complex. Improvements have been made to the nonlinear least‐squares algorithms of Z‐Rietveld so that it can deal with singular matrices and intensity non‐negativity constraints. Owing to these improvements, Z‐Rietveld successfully executes the Pawley method without requiring any constraints on the integrated intensities, even in the case of severely or exactly overlapping peaks. In this paper, details of these improvements are presented and their advantages discussed. A new approach to estimate the number of independent reflections contained in a powder pattern is introduced, and the concept of good reflections proposed by Sivia [ J. Appl. Cryst. (2000), 33 , 1295–1301] is shown to be explained by the presence of intensity non‐negativity constraints, not the intensity linear constraints.