Geometrical parameterization of the crystal chemistry of P63/m apatite. II. Precision, accuracy and numerical stability of the crystal-chemical Rietveld refinement

A script developed for crystal-chemical Rietveld refinement of P6₃/m apatite with TOPAS is implemented in parallel with standard structure refinement. Least-squares standard uncertainty (s.u.) values for directly extracted crystal-chemical parameters are nearly an order of magnitude lower than those obtained indirectly by analysis of atom coordinates derived by standard Rietveld refinement. This amazing finding originates partly in the reduction of the number of refinement parameters from 21 to 17 and partly in the fact that cell data now derive from crystal-chemical parameters instead of vice versa. Great precision and accuracy otherwise funneled into unit-cell parameters is then more distributed among mostly crystal-chemical distance parameters. The least-squares s.u. values are supported by analysis of numerous refinements of the same experimental data with added artificial intensity noise. Structural parameters from single-crystal results agree better with those extracted by crystal-chemical refinement. On the basis of singular value decomposition analyses performed using the program SVDdiagnostic [Mercier et al. (2006). J. Appl. Cryst. 39, 458-465], crystal-chemical and standard Rietveld refinements are shown to have similar numerical stability. Crystal-chemical parameters extracted by direct Rietveld refinement, therefore, are more precise than, more accurate than and numerically as reliable as those derived from analysis of regular crystallographic refinement of the same data.