Full-pattern fitting

Figure 1. Results of full pattern fitting (after [53]) of the PXD pattern of an acid-treated phase A sample, based on an orthorhombic unit cell (refined lattice constants a = 7.940(3)A, b = 10.338(4)A, c = 11.591(5)A). The low angle peaks are somewhat asymmetric and are hence not fitted well by the assumed symmetric pseudo-Voigt peak shape.

BIS 93] BISH D.L., POST J.E., Quantitative mineralogical analysis using the Rietveld full pattern fitting method ,Hra. Miner., vol. 78, p. 932-940, 1993. 

Fig. 4.25 A typical calibration run of a mixture of two polymorphs using the Rietveld analysis. The calibration sample was prepared using 5 per cent of Form B in a mixture of Forms A and B. The upper trace shows the laboratory data for this sample. The next two rows indicate the positions expected for the diffraction peaks of Forms B and A. The bottom trace shows the rms deviation resulting from the refinement of the combination of the full patterns for the two forms against the measured pattern. The best fit is obtained for a value of 4.6 per cent Form B. (From Newman et al. 1999, with permission.)...

To account for this, a full-profile fit of the XRD pattern was made for three different crystallographic models (1) pure B structure (2) pure B2 structure (3) cubic structure with the fee sublattice for the lead atoms and two different positions for the sulfur atoms. The last model allows sulfur atoms to occupy not only octahedral interstitials in the fee sublattice as it is common for the B structure, but also the occupation of tetrahedral interstitials, which is common for the 53 structure. The best fit to experimental XRD pattern in the case of the third model is shown in Fig. 1. 

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. 

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. 

Full profile refinement is computationally intense and employs the nonlinear least squares method (section 6.6), which requires a reasonable initial approximation of many fi ee variables. These usually include peak shape parameters, unit cell dimensions and coordinates of all atoms in the model of the crystal structure. Other unknowns (e.g. constant background, scale factor, overall atomic displacement parameter, etc.) may be simply guessed at the beginning and then effectively refined, as the least squares fit converges to a global minimum. When either Le Bail s or Pawley s techniques were employed to perform a full pattern decomposition prior to Rietveld refinement, it only makes sense to use suitably determined relevant parameters (background, peak shape, zero shift or sample displacement, and unit cell dimensions) as the initial approximation. 

As a starting model, we will use the coordinates of atoms taken from Table 6.49 together with the unit cell dimensions and all profile parameters determined from Le Bail s full pattern decomposition. They are found in the input file, Ch7Exl2a.inp for LHPM-Rietica. The powder diffraction data are located in the file Ch7Exl2 MoKa.dat on the CD. As already established in section 6.18.3 (see Figure 6.43 in Chapter 6), this model of the crystal structure is also feasible but far from complete because the observed and calculated intensities do not match well. Thus, the refinement strategy will be similar to the previous example. The least squares fit here may... 

Full profile fitting can be useful for various tasks including unit cell refinement involving peak overlap, space group assignment, extracting intensities prior to structure solution, and pre-structure refinement fitting of the powder pattern. The two main methods are listed below. 

European Powder Diffraction EPDIC 8, Materials Science Forum 443 4 123 126, 2004 and WinPLOTR A Windows tool for powder diffraction pattern analysis, T. Roisnel and J. Rodriguez Carvajal, EPDIC 7 European Powder Diffraction, Pts 1 and 2 Materials Science Forum 378 3 118 123, Part 1 2 2001 GENEFP GENEFP a full profile fitting program for X ray powder patterns... 

Independent of the lattice symmetry, a linear dependence of the lattice parameters (determined by the least-squares fit to the interplanar spacing of selected reflections in the XRD pattern [45] or by the more accurate full-profile fitting analysis [46]) on the Ti content has been found (Fig. 5). The equation relating the unit cell volume to the Ti content (Table 4) is particularly usefiil for determining the real framework composition directly from XRD analysis by comparing this with the Ti content resulting from elemental analysis, the amount of possible extra-framework Ti species can be estimated [46]. 

Figure 3 Calculated X-ray diffuse scattering patterns from (a) a full molecular dynamics trajectory of orthorhombic hen egg white lysozyme and (b) a trajectory obtained by fitting to the full trajectory rigid-body side chains and segments of the backbone. A full description is given in Ref. 13.

Hot water basins are used to distribute water in crossflow towers. Here, water is pumped to an open pan over the wet deck fill. The bottom of the pan has holes through which water is distributed. Manufacturers will fit specially shaped plastic drip orifices into the holes to give the water an umbrella shape for more uniform distribution. Different size orifices are used for different flow rates. Ideally, the basin will be almost full at maximum flow. This way, sufficient depth is retained for good water distribution as turn down occurs. The turn down ratio can be extended by the addition of hot water basin weirs- a pattern of baffles perhaps 2... 

The benefit of such LFERs is that they establish patterns of regular behavior, isolating apparent simplicity and defining normal or expected reactivity. Against such patterns it becomes possible to detect widely deviant or unexpected behavior. As we saw in Chapter 7, we cannot expect great generality from the extrathermodynamic approach, so it may be necessary to define numerous model processes so as to fit a full range of situations. 

Although the specific patterns of various drugs may not fit Ainslie s model in a natural way, it performs well in explaining the general phenomenon of preference reversal. It is not clear, however, that it offers a full account of this phenomenon. Consider an example told by a former addict and cited by Lewis Yablonski (1%5). The subject s wife was about to have their first child, and he was to be present during delivery. As his... 

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