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The Rietveld [J. Appl. Cryst. (1969), 2, 65-71] method of structure refinement from powder diffraction patterns is widely used with neutron data and increasingly so with X-ray data. A computer program for the application of the method with X-ray data, or with neutron nuclear scattering data, has been written ab initio in an effort to make it versatile, user oriented, portable, convenient, and largely self-contained yet reasonably comprehensive. It is of modular construction to facilitate user-desired modifications, one of which may be additions to the choice of four profile functions now included. The program accommodates either one or two wavelengths (e.g. c~1 and 0~ 2 or 2 and 2/2). It permits simultaneous refinement of the structures of two phases and the background. AbstractIntroduction A computer program has been written with the aim of calculating the domain of an atom in a structure. The domain may be limited by planes situated half-way between the atoms, or at a distance that takes into account the relative radii of the atoms. Data concerning this domain and the corresponding coordination polyhedron are computed and printed. Punched cards, to be used in a special plotting program, can also be produced. Three types of weighted coordination number are calculated.

Reflection profile functions used or suggested for use in Rietveld refinements with fixed-angle or fixedwavelength powder diffraction data, obtained with neutrons or with X-rays, are reviewed briefly. The fixed-wavelength X-ray case is considered in detail. In tests of fitting to individual resolved reflection profiles, at least eight different profile functions, including some representations consisting of "learned' differentiable number arrays, have been show to be satisfactory at the less than 3~0 misfit level. Several authors have made comparative studies of subsets of the six analytic functions, Gaussian (G), Lorentzian (L), modified Lorentzian (ML), intermediate Lorentzian (IL), pseudo-Voigt (p-V), and Pearson VII (PVII). In this work, the performance of all six, in Rietveld refinements with data sets from seven different specimens with differing degrees of broadening, have been intercompared on the basis of difference plots, R,,.p, and R e.The Gaussian function was consistently the worst performer, while the pseudo-Voigt and Pearson VII were the best. For this reason, for reasons of simplicity, and for convenient though crude physical interpretation of the Lorentz-fraction parameter, the pseudo-Voigt is identified as the function of choice among these six. It is emphasized that a need still exists for a better profile function soundly based on physical reality.

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The degree to which physically significant precision in crystal-structural details can be obtained with selected routine procedures has been assessed and some illustrative applications to analyses of bonding effects and of impurity substitutions have been made. Least-squares refinements with X-ray data for three single crystals of mineral hydroxyapatite and with neutron diffraction data for a fourth of the same origin yielded R = 2% in each case (_> 40 parameters adjusted and > 500 reflections). Seemingly minor extinction corrections improved the mutual agreement among separate measures of some parameters from 4a to the final la (typically <5%) found for all, even anisotropic thermal, parameters in the X-ray cases. Final results were insensitive to reasonable changes in the weighting scheme. Comparisons of X-ray and neutron results, which generally agreed within 2~r, showed systematic differences associated with the oxygen atoms bonded to phosphorus in the phosphate group. Analysis of the final R-value also suggested (1) real differences, among the crystals, smaller than the o-'s associated with individual parameters and (2) either a systematic inadequacy of the refinement model, or similar residual systematic errors (such as thermal diffuse scattering contributions), in both neutron and X-ray data. Direct refinement for the degree of fluorine substitution for OH in Cal0(PO4)6(OH)2 led to the same result, 8 at.% substitution, with both X-ray and neutron data. Analysis of the decrements found with both X-ray and neutron data in the apparent site-occupancy factors for the Ca atoms showed that a simple substitution of Mg2+ for Ca 2÷ at the same site is not in itself a sufficient substitutional model for this case.