International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.5, pp. 283-284
Section 3.5.2.2.2. Le Bail method^{a}Université du Maine, Institut des Molécules et Matériaux du Mans, UMR CNRS 6283, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France |
In order to be able to estimate R factors related to integrated intensities, Rietveld (1969) stated [see also the book The Rietveld Method edited by Young (1993)]: `a fair approximation to the observed integrated intensities can be made by separating the peaks according to the calculated values of the integrated intensities,' i.e.where w_{j,hkl} is a measure of the contribution of the Bragg peak at position 2θ_{hkl} to the diffraction profile y_{j} at position 2θ_{j} [corresponding to equation 7 in Rietveld (1969)]. is the sum of the nuclear and magnetic contributions for neutron diffraction, or is more simply for X-rays. The sum is over all y_{j}(obs) that can theoretically contribute to the integrated intensity I_{hkl}(obs). Bias is introduced here by apportioning the intensities according to the calculated intensities; this is why the observed intensities are said to be `observed', in quotation marks, in the Rietveld method. These `observed' intensities are used in the R_{B} and R_{F} calculations (residuals on intensities and structure-factor amplitudes, respectively). They are also required for Fourier-map estimations, which, as a consequence, are less reliable than those from single-crystal data.
A process using the Rietveld decomposition formula iteratively for WPPD purposes was first applied in 1988 (Le Bail et al., 1988) and much later was called the `Le Bail method' or `Le Bail fit', or `pattern matching' as well as `profile matching' in the FULLPROF Rietveld program (Rodriguez-Carvajal, 1990). In the original computer program (named ARITB), arbitrarily all equal values are first entered in the above equation, instead of using structure factors calculated from the atomic coordinates, resulting in `I_{hkl}(obs)' which are then re-entered as new values at the next iteration, while the usual profile and cell parameters (but not the scale) are refined by least squares (ARITB used profile shapes represented by Fourier series, either analytical or learned from experimental data, providing an easy way to realize convolution by broadening functions modelling size–strain sample effects, possibly anisotropic). Equipartition of exactly overlapping reflections comes from the strictly equal result from equation (3.5.1) for Bragg peaks at the same angles which would have equal starting calculated intensities. Not starting from a set of all equal values avoids equipartition for the exactly overlapping reflections but produces I_{hkl}(obs) keeping the same original ratio as the ones. It is understandable that such an iterative process requires starting cell and profile parameters as good as the Rietveld method itself. The process is easier to incorporate within an existing Rietveld code than the Pawley method, so that most Rietveld codes now include structure-factor amplitudes extraction as an option (generally multiphase), with the possibility of combining Rietveld refinement(s) together with Le Bail fit(s).
A non-exhaustive list of programs applying this method (either exclusively or added within a Rietveld code) includes MPROF (Jouanneaux et al., 1990), later renamed WinMPROF; FULLPROF (Rodriguez-Carvajal, 1990); EXTRACT (Baerlocher, 1990); EXTRA (Altomare et al., 1995); EXPO (Altomare et al., 1999), which is the integration of EXTRA and SIRPOW.92 for solution and refinement of crystal structures; and RIETAN (Izumi & Ikeda, 2000). Then followed most well known Rietveld codes (BGMN, GSAS, MAUD, TOPAS etc.) or standalone programs (AJUST by Rius et al., 1996). In the work of the Giacovazzo group, many modifications of the values for SDPD purposes were applied before or after the extraction and were integrated in EXPO2011 (Altomare et al., 2011): obtaining information about the possible presence of preferred orientation by statistical analysis of the normalized structure-factor moduli; using the positivity of the Patterson function in the decomposition process, this having been considered previously (David, 1987; Estermann & Gramlich, 1993); characterization of pseudotranslational symmetry used as prior information in the pattern-decomposition process; multiple Le Bail fits with random attribution of intensity to the overlapping reflections, instead of equipartition, followed by application of direct methods to large numbers of such data sets; use of a located structure fragment for improving the pattern-decomposition process; and use of probability (triplet-invariant distribution functions) integrated with the Le Bail algorithm. Another approach for solving the overlapping problem was proposed by using maximum-entropy coupled with likelihood evaluation (Dong & Gilmore, 1998). The list of structure solutions made from intensities extracted by using the Pawley and Le Bail methods is too long to be given here; a partial list (>1000 first cases, including those using |F_{hkl}| values extracted by other methods) can be found on the web (Le Bail, 2007). The first application of the Le Bail method was to the structure solution of LiSbWO_{6} (Le Bail et al., 1988) using the ARITB software.
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