Nicolae C. Popa

Size-strain line-broadening analysis of the ceria round-robin sample

The results of both a line-broadening study on a ceria sample and a size–strain round robin on diffraction line-broadening methods, which was sponsored by the Commission on Powder Diffraction of the International Union of Crystallography, are presented. The sample was prepared by heating hydrated ceria at 923 K for 45 h. Another ceria sample was prepared to correct for the effects of instrumental broadening by annealing commercially obtained ceria at 1573 K for 3 h and slowly cooling it in the furnace. The diffraction measurements were carried out with two laboratory and two synchrotron X-ray sources, two constant-wavelength neutron and a time-of-flight (TOF) neutron source. Diffraction measurements were analyzed by three methods: the model assuming a lognormal size distribution of spherical crystallites, Warren–Averbach analysis and Rietveld refinement. The last two methods detected a relatively small strain in the sample, as opposed to the first method. Assuming a strain-free sample, the results from all three methods agree well. The average real crystallite size, on the assumption of a spherical crystallite shape, is 191 (5) A. The scatter of results given by different instruments is relatively small, although significantly larger than the estimated standard uncertainties. The Rietveld refinement results for this ceria sample indicate that the diffraction peaks can be successfully approximated with a pseudo-Voigt function. In a common approximation used in Rietveld refinement programs, this implies that the size-broadened profile cannot be approximated by a Lorentzian but by a full Voigt or pseudo-Voigt function. In the second part of this paper, the results of the round robin on the size–strain line-broadening analysis methods are presented, which was conducted through the participation of 18 groups from 12 countries. Participants have reported results obtained by analyzing data that were collected on the two ceria samples at seven instruments. The analysis of results received in terms of coherently diffracting, both volume-weighted and area-weighted apparent domain size are reported. Although there is a reasonable agreement, the reported results on the volume-weighted domain size show significantly higher scatter than those on the area-weighted domain size. This is most likely due to a significant number of results reporting a high value of strain. Most of those results were obtained by Rietveld refinement in which the Gaussian size parameter was not refined, thus erroneously assigning size-related broadening to other effects. A comparison of results with the average of the three-way comparative analysis from the first part shows a good agreement.

Crystallite Size and Residual Strain/Stress Modeling in Rietveld Refinement

We consider the problem of microstructure modeling in Rietveld refinement, in particular of crystallite size and residual strain/stress and propose the improvements to existing models. For the crystallite-size modeling, we consider the size-broadened profile given by the lognormal size distribution of spherical crystallites. We derive an analytical approximation in terms of Lorentz and Gauss functions for the size-broadened profile. Its advantage is that it can be analytically convolved with the strain-broadened and instrumentalbroadened profiles, thus replacing time-consuming numerical convolutions that are commonly used. The method is tested on two CeO2 powders; one shows “super-Lorentzian” profiles that we successfully model under the assumption of a broad lognormal size distribution. We show that the Voigt function, as an assumed size-broadened profile, fails for both very narrow and broad size distributions. In regard to the residual strain/stress modeling, we describe a novel approach to model diffraction line shifts caused by elastic residual or applied stresses in textured polycrystals. The model yields the complete texture-weighted strain and stress tensors as a function of crystallite orientations, the so-called weighted strain orientation distribution function. We also present the method to determine the average values of macroscopic strain and stress tensors that are frequently of interest. The model is particularly suitable for introduction in Rietveld-refinement programs, as the requirements on refinable parameters for all crystal Laue classes are outlined. The effects of sample symmetry are also included and conditions for strain invariance to both crystal and sample symmetries are discussed.

Spherical harmonics analysis based on the Reuss model in elastic macro strain and stress determination by powder diffraction

In this paper a new approach to macro strain/stress analysis by generalized spherical harmonics is presented. It consists of expanding the stress tensor weighted by texture in a series of generalized spherical harmonics with the ground state of expansion specific to the classical Reuss model of an isotropic polycrystal. Like previously reported models having a ground state of hydrostatic type [Popa & Balzar (2001). J Appl Cryst. 34, 187–195] and of Voigt type [Popa et al. (2014). J Appl Cryst. 34, 154–159], the actual model is appropriate for use with Rietveld refinement.

Dependence of the strain diffraction line broadening on (hkl) and sample direction in textured polycrystals

A new phenomenological approach describing the dependence of the strain diffraction line breadth on direction in both crystal and sample is presented. For a negligibly small dependence on the direction in the sample, these models reduce to those for anisotropic strain broadening that already exist in the literature and which are implemented in popular Rietveld codes. The new model is appropriate for implementation in the Rietveld programs able to process simultaneously diffraction patterns recorded in multiple directions in a sample.

Size-strain round robin: first results and the comparative analysis of the measurements

SIZE-STRAIN ROUND ROBIN: FIRST RESULTS AND THE COMPARATIVE ANALYSIS OF THE MEASUREMENTS D. Balzar N. Audebrand M. R. Daymond A. Fitch A. Hewat J. I. Langford A. Le Bail D. Louer O. Masson N. C. Popa P. W. Stephens 9 B. Toby University of Denver and NIST, Boulder, Colorado, U.S.A. University of Rennes, Rennes, France ISIS, Rutherford-Appleton Laboratory, Didcot, U.K. ESRF, Grenoble, France ILL, Grenoble, France University of Birmingham, Birmingham, U.K. University of Maine, Le Mans, France National Institute for Materials Physics, Bucharest, Romania NSLS, Brookhaven National Laboratory, Upton, New York, U.S.A. NCNR, NIST, Gaithersburg, Maryland, U.S.A