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The Commission on Powder Diffraction of the IUCr has undertaken a round robin of Rietveld refinement with the aims of: (i) evaluating a cross section of currently used software; (ii) examining the range and effect of various strategies of refinement; (iii) assessing the precision and accuracy (spread) of the derived parameters; (iv) comparing and contrasting various instruments and methods of data collection. These aims were addressed by circulating to 51 participants upon request: (i) two constant-wavelength X-ray and neutron powder diffraction patterns collected on PbSO4 for refinement; (ii) a sample of phase-pure monoclinic ZrO2 for both data collection and refinement. In the latter case, the raw data were requested to be returned for reanalysis with a `standard' version of Rietveld software and an `optimal' refinement protocol. A total of 23 respondents provided 18 X-ray and 20 neutron refinements of the PbSO4 crystal structure from the 'standard' data sets using 12 different Rietveld analysis programs. These results constitute Part I of the round robin and have been described previously [Hill (1992). J. Appl. Cryst. 25, 589–610]. The 28 contributors to the m-ZrO2 section of the survey were based in 12 countries and collected 27 X-ray and 14 neutron data sets, using 20 different X-ray and 11 different neutron powder diffraction instruments. The conventional X-ray instruments included 13 reflection (flat-plate) and eight transmission (capillary or thin-film) machines and used three different radiations (Co, Cu and Mo). Two additional flat-plate data sets were collected with synchrotron X-rays. The neutron data were collected on 12 constant-wavelength and two time-of-flight instruments, the former utilizing wavelengths between 1.0 and 1.9 Å. The data sets yielded 27 X-ray and 15 neutron refinements of the m-ZrO2 crystal structure. The conditions used for data collection varied widely for both types of radiation: wavelengths ranged from 0.7 to 1.9 Å, step widths from 0.01 to 0.12°2θ, step counting times from 0.1 to 46 s for X-rays and up to 30 min for neutrons, data-collection time from 4 min to 3 d, maximum step intensities from 350 to 99000 counts, minimum d spacings from 0.53 to 1.17 Å and numbers of unique reflections from 71 to 912 (not including the time-of-flight neutron data). Variations in resolution between instruments were especially marked in the case of the neutron data but were less pronounced for the X-ray machines; the two instruments situated at synchrotron X-ray sources displayed the narrowest peak widths. The peak-to-background ratios varied markedly; in descending order of peak-to-background ratio were single-wavelength X-rays (conventional and synchrotron sources, using incident-beam monochromators), two-wavelength X-rays in parafocusing (reflection) mode, two-wavelength X-rays in transmission mode and constant-wavelength neutrons. Refinement conditions were also markedly inconsistent, with the total number of refined parameters varying from 20 to 46. The major factors associated with lower accuracy of the derived crystal structure parameters were: (i) the use of insufficiently flexible peak-shape and/or background functions; (ii) omission of the high-angle data from the refinement, especially the data with d spacings below about 1 Å; (iii) use of an insufficiently wide range of diffraction angles on either side of the peak (i.e. peak truncation), especially for the reflection profiles with substantial Lorentzian (or Cauchy) character; (iv) poor instrumental resolution and/or a peak-to-background ratio less than about 50; (v) low pattern intensity (i.e. maximum step intensity less than about 2000 counts), especially at small d spacings; (vi) an observations-to-parameters ratio of less than about five. The X-ray- and neutron-data Zr-atom coordinates are distributed over a relatively narrow and similar range of values about the weighted mean values, viz 0.014 to 0.028 Å and 0.009 to 0.014 Å, respectively. On the other hand, while the values of the O-atom coordinates derived from the neutron data are determined with about the same accuracy as those of the Zr-atom ones, viz 0.006 to 0.017 Å about the mean, the corresponding values derived from the X-ray data are distributed over a very much wider range, viz 0.091 to 0.193 Å, no doubt due to the lower scattering power of the O atom. The atomic displacement (`thermal') parameters are reasonably determined with X-rays when flat-plate reflection-geometry instruments are used but transmission geometry produces very poor parameters ranging from large negative to large positive values; the poor quality of the latter results is due to the strong correlation between displacement and absorption effects and the generally smaller number of reflections included in the data. All but the lowest-resolution neutron data support a sensible anisotropic displacement ellipsoid for the atoms. The precision and accuracy of the population of crystal structural parameters produced from the participants' refinements were almost always substantially improved by reanalysis of the data using a `standard' program and an `optimal' refinement protocol. The mean probable errors, taken as the mean deviations of the individual estimates of the parameters from the weighted mean value, show that about two-thirds of the variation in the m-ZrO2 parameters is due to differences in the instrumental and data-collection conditions. The remaining one-third of the variation is due to differences in the software and/or the refinement strategy used. On average, the mean probable errors of the Rietveld parameters are larger than their derived estimated standard deviations by a factor of around two for coordinates, about five for the displacement parameters and around 16 for unit-cell dimensions. Of the X-ray instruments, flat-plate reflection-geometry ones provided the best crystal structure parameters for the sample of m-ZrO2 distributed in this study, but the quality was degraded when the data were cut off at d spacings larger than about 1 Å. The X-ray transmission geometries produced the poorest atomic parameters because of the generally poorer peak-to-background ratio and the limited range of data available (with resultant lower observations-to-parameters ratio). The results obtained with neutron data were of roughly equivalent quality to those obtained from X-rays in the case of the Zr atom, but neutrons were markedly superior for the determination of the O-atom coordinates and displacement parameters, as expected. The time-of-flight neutron and synchrotron X-ray results were not significantly different from those obtained in the conventional neutron and better-quality conventional X-ray analyses.

The concept of a variable-counting-time (VCT) strategy for use in Rietveld analysis of X-ray powder diffraction data was introduced by Madsen & Hill [Adv. X-ray Anal. (1992), 35, 39–47]. This strategy is based on a function that increases the counting time used at each step in the scan in a manner that is inversely proportional to the decline in reflection intensity that inevitably results from the combined effects of Lp factor, scattering factor and thermal vibration. The present work extends the VCT function to include the effects of reflection multiplicity, cylindrical-sample (capillary) absorption and monochromator polarization. The new algorithm has been incorporated into a PC computer program and applied to the collection of data from samples of LaB6, tourmaline, forsterite and boehmite. Subsequent analysis of the data using the Rietveld method has shown that VCT data can produce more accurate atomic coordinates and site occupancies, lower residual 'noise' in difference Fourier maps and more stable refinement of `light' atoms.

The Commission on Powder Diffraction of the International Union of Crystallography has undertaken an intercomparison of Rietveld refinements performed with two `standard' PbSO4 powder diffraction patterns: a conventional (two-wavelength) X-ray pattern collected on a Bragg–Brentano diffractometer with Cu Kα radiation and a constant-wavelength neutron pattern collected on the D1A diffractometer at the Institut Laue–Langevin. The aims of this project were: (i) to evaluate a cross section of currently used Rietveld refinement software; (ii) to examine the range and effect of various strategies of Rietveld refinement; (iii) to assess the precision and accuracy (spread) of the parameters derived by Rietveld analysis. 23 participants provided 18 refinements with the X-ray data and 20 refinements with the neutron data, using 11 different Rietveld-analysis programs. Analysis of the submitted results shows that refinement strategies play a large part in determining the detailed outcome of a Rietveld refinement. The wide variation in the values of the agreement indices obtained in these studies of the same data sets highlights the need for standardization both of the refinement procedures and of the type of data included in the algorithms used for assessing the fit. The major factors limiting the accuracy of the derived PbSO4 crystal structure parameters were: (i) use of insufficiently flexible peak shape and/or background functions; (ii) elimination of the high-angle diffraction data from the refinement; (iii) inclusion of an insufficiently wide range of diffraction angles on either side of the centroid of each peak during the step intensity calculation; and, additionally for X-rays, (iv) simultaneous release of the O-atom site-occupancy and displacement parameters. Rietveld analysis of the PbSO4 X-ray powder diffraction data provided atomic coordinates and isotropic displacement parameters for the Pb and S atoms that are precise (i.e. have small e.s.d.s) and are in reasonable agreement with the values derived from a single crystal study (viz the spread of coordinates is over the range 0.007–0.042 Å). On the other hand, the `light' O-atom parameters show relatively poor precision and have a disconcertingly wide spread of values about the weighted mean (viz 0.12–0.19 Å for the coordinates). Despite the much lower intrinsic resolution of the neutron data (i.e. peak widths some four times those of the X-ray data), the coordinates and anisotropic displacement parameters obtained for the Pb and O atoms are very precise and have a relatively narrow distribution about the single-crystal results, namely 0.004–0.020 Å for the coordinates. The range of coordinates determined from the neutron data for the relatively `light' S atom is correspondingly larger, namely 0.024–0.043 Å, about equivalent to that obtained from the X-ray data. In general, and as expected, the e.s.d.s from the Rietveld analyses are substantially smaller than the observed inter-refinement variation of the unit-cell dimensions, atomic coordinates and isotropic displacement parameters by factors of up to, respectively, 17, 5 and 22 for X-rays, and 25, 3 and 5 for neutrons. This investigation indicates that results of possibly high precision but low accuracy are not uncommon in Rietveld analysis. The disparity between individual refinements can be expected to increase further when, unlike here, the analyses are undertaken using data sets collected under diverse experimental conditions.

A weighted form of the Durbin–Watson d statistic [Durbin & Watson (1971). Biometrika, 58, 1–19] has been used to quantify the serial correlation between adjacent least-squares residuals in Rietveld refinements of step-scan powder diffraction data. Analyses of X-ray and neutron data from a range of materials have shown that the d statistic: provides a sensitive measure of the progress of a refinement and remains discriminating when other agreement indices fail, for example, when comparing results at different step widths; provides quantitative information about the significance of serial correlation present in the residuals; provides a convenient means of assessing the reliability of the estimates of the parameter variances; and provides a basis for the selection of values of step width and intensity corresponding to optimum and/or minimum use of experimental beam time.

Crystal structure parameters have been obtained for α-Al2O3, β-PbO2 and (Mg, Fe)2SiO4 by Rietveld analysis of Cu Kα X-ray powder diffraction data collected on a conventional diffractometer using counting times ranging from 0.01 to 5 s per step. For all but the 0.01 s data collected on (Mg, Fe)2SiO4 the structural parameters obtained at different counting times are statistically identical at the 3 σ level, and the spread in the values is essentially the same as that obtained by sample repacking at a fixed counting time of 1 s per step. The parameter e.s.d.'s and conventional agreement indices Rwp and RB decrease to values limited by residual model errors as the counting time increases, but the goodness-of-fit parameter becomes unacceptably larger than its ideal value of unity. When more than a few thousand counts are accumulated for the maximum step intensity in the diffraction profile a weighting scheme based solely on counting variance is inappropriate and the parameter e.s.d.'s are no longer a reflection of their accuracy.

Crystal-structure parameters and their associated estimated standard deviations (e.s.d.'s) have been determined for α-Al2O3 by Rietveld analysis of Cu Kα X-ray powder diffraction data collected on a standard diffractometer with profile step widths ranging from 0.01 to 0.32° 2θ. Estimates of the true precision of the parameters have been obtained by replication of the experiment five times. The degree of serial correlation between adjacent points in the profiles has been quantitatively determined by the use of the Durbin–Watson d statistic. Values of the conventional profile agreement indices, Rp, Rwp and goodness of fit, are not influenced by the scan-point density, but the Bragg R-factor decreases to a limiting value as the step width is decreased. The structural parameters obtained at all step widths are statistically indistinguishable at the 1σ level, although there is some loss of accuracy relative to single-crystal values at widths larger than 0.16°. The least-squares e.s.d.'s decrease linearly with the reciprocal square root of the number of steps in the pattern, as expected, but the e.s.d.'s of the unit-cell dimensions are the only ones that are significantly smaller than estimates of their true precision obtained from repetitions of the experiment. In the case of α-Al2O3, and possibly in general, the optimum step width for Rietveld analysis corresponds approximately to the minimum value of the full-width at half-maximum for well resolved peaks.