International Union of Crystallography. Commission on Powder Diffraction. Rietveld refinement round robin. II. Analysis of monoclinic ZrO₂

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.