Le Bail method

After the unit cell has been determined, the next stage is to prepare the intensity data for space group determination and structure solution using an appropriate profile fitting technique such as the Pawley method [30] or the Le Bail method [31]. The aim of this stage of the structure determination process is to fit the complete experimental powder XRD profile by refinement of variables that describe ... 

There are three common methods used for full-profile fitting, namely, Le Bail method,Pawley method, and... 

In Direct methods, the intensities are extracted from an indexed powder diffraction pattern by profile fitting procedures such as the Le Bail method and Pawley method. Then the integrated intensities obtained are corrected for Torentz polarization and normalized. These corrected intensity values are then subjected to routine Direct-method procedures. The structural model obtained is completed using difference Fourier maps. This method works successfully when a sufficient number of intensities could be extracted from the powder diffraction pattern. 

The Bragg residual, Rb (this figure of merit is quite important in Rietveld refinement but has little to no use during full pattern decomposition because only observed Bragg intensities are meaningful in both Pawley and Le Bail methods) ... 

The Le Bail Method. To be able to estimate R factors related to integrated intensities, Rietveld " stated a fair approximation to the observed integrated intensity can be made by separating the peaks according to the calculated values of the integrated intensities, i.e. ... 

The integrated intensities are extracted from the powder pattern via the Le Bail method. ... 

The result of the application of the DDM decomposition formula Equation (14) is similar to that of Rietveld s approximation for /obs> except that it is DDM-oriented and thus does not require a background definition. In the DDM program the formula is also used for the Bragg R factor calculation. For single non-overlapped peaks or a set of peaks with the same position, the DDM decomposition directly gives the best estimate for /obs- For partly overlapped peaks it should be iterated to arrive at an optimized set of /obs similarly to the Le Bail method. The starting intensities f can either be calculated from an existing structure model or set arbitrarily when the model is absent. Notably, for fully overlapped peaks the DDM decomposition... 

The final step before structure determination by traditional methods is extraction of integrated intensities from the powder diffraction pattern (see below). Intensities can nearly always be determined only after deconvolution of partially overlapping Bragg reflections. This can be done for individual peaks or for groups of peaks, but usually full decomposition methods are used. There are basically two techniques the iterative Le Bail method [6] and the constrained linear least-squares approach developed by Pawley [7]. 

The Le Bail method is a clever extension of the original work of Rietveld. The key difference between the two methods is that the Le Bail method uses iteratively the values of intensities obtained from the previous least squares cycle, and assumes unit values for all the peaks to initiate the refinement. Rietveld s method, on the other hand, uses the intensities calculated based on an initial model for the structure. 

Both the Pawley and Le Bail methods of intensity extraction or pattern decomposition have their own benefits and shortcomings. 

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