A set of general guidelines for structure refinement using the Rietveld (whole-profile) method has been formulated by the International Union of Crystallography Commission on Powder Diffraction. The practical rather than the theoretical aspects of each step in a typical Rietveld refinement are discussed with a view to guiding newcomers in the field. The focus is on X-ray powder diffraction data collected on a laboratory instrument, but features specific to data from neutron (both constant-wavelength and time-of-flight) and synchrotron radiation sources are also addressed. The topics covered include (i) data collection, (ii) background contribution, (iii) peak-shape function, (iv) refinement of profile parameters, (v) Fourier analysis with powder diffraction data, (vi) refinement of structural parameters, (vii) use of geometric restraints, (viii) calculation of e.s.d.'s, (ix) interpretation of R values and (x) some common problems and possible solutions.

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Rietveld refinement guidelines

A new program has been developed for ab initio crystal structure determination from powder diffraction data (X-ray and neutron). It uses global-optimization algorithms to solve the structure by performing trials in direct space. It is a modular program, capable of using several criteria for evaluating each trial configuration (e.g. multi-pattern). It is also modular in the description of the crystal content, with the possibility of describing building blocks in the sample, such as polyhedra or molecules, and with automatic adaptive handling of special positions and sharing of identical atoms between neighbouring building blocks. It can therefore find the correct structure without any assumption about the connectivity of the building blocks and is suitable for any kind of material. Several optimization algorithms (simulated annealing, parallel tempering) are available, with the possibility of choosing the convergence criterion as a combination of available cost functions. This program is freely available for Linux and Windows platforms; it is also fully `open source', which, combined with an object-oriented design and a complete developer documentation, ensures its future evolution.

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FOX, `free objects for crystallography': a modular approach to ab initio structure determination from powder diffraction

Anisotropic line-shape broadening (peak width which is not a smooth function of d-spacing) is frequently observed in powder diffraction patterns, and can be a source of considerable difficulty for whole-pattern fitting or Rietveld analysis. A model of the multi-dimensional distribution of lattice metrics within a powder sample is developed, leading naturally to a few parameters which can be varied to achieve optimal line-shape fits. Conditions on these parameters are derived for all crystal systems, and the method is illustrated with two examples: sodium p-hydroxybenzoate and rubidium fulleride.

Phenomenological model of anisotropic peak broadening in powder diffraction

The study of preferred orientation of minerals in rocks dates back to Omalius d’Halloy (1833) who attributes a special significance to the alignment of crystals as an indicator of the formation process. Much later the influence of crystal alignment on physical properties was quantified (e.g., Weissenberg 1922, Voigt 1928, Reuss 1929). Only recently has this field emerged as a coherent part of earth science research linking such branches as mineralogy, petrology, structural geology, geodynamics and seismology. The reason for this was the emergence of quantitative methods to analyze preferred orientation, or “texture” as it was first called by Naumann (1850). These methods were largely developed in collaboration with materials science and mechanics. Quantitative measurements, detailed field studies, rigorous data analysis, theories to predict textures, and improvements in characterizing seismic anisotropy in the Earth are leading to a coherent picture that is now being refined.

Though seismologists have long accepted that there is a causal relationship between anisotropic propagation of seismic waves, the deformation field and crystal orientation, the prevailing view is still largely the mythological concept that seismic fast directions align with the flow direction. While this may be approximately the case for olivine deformed under certain conditions, it is certainly no universal law, as we will try to illustrate in this review. The “fast” direction of a crystal depends on the mineral species and its crystal structure. The alignment of crystals depends on microscopic, intra-crystalline deformation systems and the deformation history. Both relationships are complex and not intuitive, but there are well-established theories to compute single crystal physical properties as well as orientation patterns. Simulations can be compared with experimental data and then applied with some caution to the macroscopic Earth.

This review is intended to provide a brief introduction, highlighting some of the issues with …

Texture and Anisotropy

!Received 19 December 2005; accepted 27 January 2006 "The deÞnitions for important Rietveld error indices are deÞned and discussed. It is shown that whilesmaller error index values indicate a better Þt of a model to the data, wrong models with poor qualitydata may exhibit smaller values error index values than some superb models with very high qualitydata. © 2006 International Centr e for Diffraction Data. #DOI: 10.1 154/1.2179804 $

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R factors in Rietveld analysis: How good is good enough?

Introduction to the Rietveld Method 1. The early days: a retrospective view 2. Mathematical aspects of Rietveld refinement 3. The flow of radiation in a polycrystalline material 4. Data collection strategies: fitting the experiment to the need 5. Background modelling in Rietveld analysis 6. Analytical profile fitting of X-ray powder diffraction profiles in Rietveld analysis 7. Crystal imperfection broadening and peak shape in the Rietveld method 8. Bragg reflection profile shape in X-ray powder diffraction patterns 9. Restraints and constraints in Rietveld refinement 10. Rietveld refinement with time-of-flight powder diffraction data from pulsed neutron sources 11. Combined X-ray and neutron Rietveld refinement 12. Rietveld analysis programs Rietan and Premos and special applications 13. Position - constrained and unconstrained powder-pattern-decomposition methods 14. Ab initio structure solutions with powder diffraction data

R. A. Young

Papers

The Rietveld method