Full pattern decomposition

The fundamental parameters approach has been recently implemented in some of the current programs. It gives, to a certain extent, physical meaning to the parameters involved in pattern decomposition. This method attempts to model the contributions from various instrumental components (such as monochromators, slits) and geometry to the observed peak profile shapes. Since this is done considering the relevant physics involved from the generation, diffraction, and detection of PXRD, it leads to more physical meaning than the two methods described earlier. This method can be used not only to perform full pattern decomposition, but also effectively to do a standardless refinement of the sample effects such as crystallite size and microstrain. 

The background should never be subtracted prior to full pattern decomposition and full profile-based Rietveld refinement. In these cases it is approximated using various analjdical functions with coefficients, which are refined along with other parameters. Thus computed background is then added to the intensity calculated as a function of Bragg angle. 

The two related full pattern decomposition methods in common use today were suggested by Pawley and by Le Bail et al Pawley s approach is based on Eq. 2.48 and full pattern decomposition in the case of dual wavelength data, when Kai/Ka2 doublets are present, is performed by solving the following system of equations using a least squares minimization ... 

When peak shape functions and their parameters, including Bragg reflection positions, are known precisely and the background is modeled by a polynomial function with j coefficients, the solution of Eq. 6.6 is trivial because all equations are linear with respect to the unknowns (Bj, see Eq. 4.1, and / ). It facilitates the use of a linear least squares algorithm described in section 5.13.1. In practice, it is nearly always necessary to refine both peak shape and lattice parameters in addition to Bj and h to achieve a better precision of the resultant integrated intensities. Thus, a non-linear least squares minimization technique (see next section) is usually employed during full pattern decomposition using Eq. 6.6. 

Both the full pattern decomposition and Rietveld refinement are based on the non-linear least squares minimization of the differences between the observed and calculated profiles. Therefore, the non-linear least squares method is briefly considered here. Assume that we are looking for the best solution of a system of n simultaneous equations with m unknown parameters (n m), where each equation is a non-linear function with respect to the unknowns, Xu X2,. .., In a general form, this system of equations can be represented as... 

Since both Pawley and Le Bail full pattern decompositions are based on finding a least squares solution of Eq. 6.6, the problem may be considered solved and a pattern deconvoluted when the best possible fit between the... 

It is, therefore, of utmost importance to have certain numerical figures of merit that quantify the quality of the least squares fit and therefore, may be used to estimate the reliability of the extracted integrated intensities and observed structure factors. The following figures of merit are customarily used to characterize both the full pattern decomposition and Rietveld refinement quality. 

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) ... 

By solving crystal structures of different classes of materials," we will illustrate only a few of the possible approaches to the ab initio structure solution from powder diffraction data. Whenever possible the structure factors obtained from full pattern decompositions should be used until the coordinates of all atoms are established. In some cases it may be necessary to re-determine individual structure factors based on the nearly completed structural model, especially when locations of lightly scattering atoms are of concern after all strongly scattering species have been correctly positioned in the unit cell. This re-determination may be routinely performed during Rietveld refinement and will be briefly discussed in Chapter 7. 

Full pattern decomposition usually begins with the refinement of the background while keeping peak shape and instrumental parameters fixed at their default values, and unit cell dimensions fixed at their best-known values. We begin with only two parameters representing the background as a... 

Table 6.2. Figures of merit obtained at different stages during the full pattern decomposition of the powder diffraction pattern of LaNi4 g5Sno,i5 using Le Bail approach incorporated in...

A specific of the Le Bail s approach nearly always requires that early refinement includes the simplest suitable background function with all other relevant parameters kept fixed at their default or approximately known values. This ensures the proper determination of the individual integrated intensities, which are initially set to identical values, and the overall success of the full pattern decomposition. 

The results of the full pattern decomposition are shown in Table <5.5 as a list of peak shape and lattice parameters, and also in Table d.4 as a list of Miller indices with the corresponding individual and their standard... 

Table 6.4. The list of Bragg reflections with their corresponding observed structure factors squared determined from Le Bail s full pattern decomposition of the powder diffraction...

The progression of the Le Bail full pattern decomposition is illustrated in Table 6.29 and the results are shown in Figure 6.21. Bragg peaks were represented by the pseudo-Voigt function with Howard s asymmetry correction. 

The array of the individual structure factors determined from Le Bail s full pattern decomposition was processed using WinCSD (see footnote No. 4 on page 515) and, according to a combined figure of merit, one of the possible solutions was notably better than the others. The subsequently calculated E-map is listed in Table 6.31. 

This value is considerably lower than Rp reachable during full pattern decomposition because the extended background-only ranges are usually excluded from the semi-manual profile fitting. Furthermore, an independent treatment of positions and full widths at half maximum of Bragg peaks observed within the processed range enables a better fit between the observed and calculated intensities. 

Figure 6.38. The results of Le Bail s full pattern decomposition of the powder diffraction pattern of GdsGe. The discrepancies between the observed and calculated profiles are small and all residuals are low, indicating that the unit cell dimensions are accurately determined and that the chosen peak shape function (pseudo-Voigt) is a good choice for this experiment.

Figure 6.40. The observed (circles) and calculated (lines) intensities in a fragment of powder diffraction pattern of GdsSi4. The calculated intensity has been normalized to match the observed profile. Peak shape, background and lattice parameters employed to compute the calculated profile have been obtained by a full pattern decomposition of the observed data using Le Bail s technique, as shown in Figure 6.39. Note, that the difference plot has been compressed ten-fold for clarity.

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