Rietveld fitting

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

The structure derived from a Rietveld fit of a neutron diffraction pattern of a 6-line ferrihydrite which showed more and sharper lines (Fig. 2.9, lower) than an XRD pattern, was in agreement with the structure proposed by Drits et al. (1993) except that it was not necessary to assume the presence of hematite in order to produce a satisfactory fit (Jansen et al. 2002). The unit cell of the defect free phase had a = 0.29514(9) nm and c = 0.9414(9) nm and the average domain size derived from line broadening was 2.7(0.8) nm. Since forced hydrolysis of an Fe solution at elevated temperatures will ultimately lead to hematite, it is likely that incipient hematite formation may occur under certain synthesis conditions. Neither these studies nor Mbssbauer spectroscopy, which showed only a singular isomer shift at 4.2 K characteristic of Fe, supported the presence of " Fe (Cardile, 1988 Pankhurst Pollard, 1992). However, the presence, at the surface, of some Fe with lower (<6) coordination, perhaps as tetrahedra (Eggleton and Fitzpatrick, 1988) which may have become unsaturated on heating, has been suggested on the basis of XAFS results (Zhao et al. 1994). 

Fig. 6 Results from Rietveld refinement of the disordered crystal structure of the P polymorph of p-formyl-tranj--cinnamic acid. The disorder concerns two orientations of the formyl group as shown in (a). The crystal structure in (b) shows only the disorder component of higher occupancy. The results from Rietveld refinement shown at the bottom are for (c) an ordered model comprising only the major orientation of the formyl group, and (d) the final disordered model (right side). Apart from the description of the order/disorder of the formyl group, all other aspects of these refinement calculations were the same. A slight improvement in the quality of the Rietveld fit for the disordered model is evident...

Fig. 1 XRD patterns of the MnO nanoparticies with average diameters of (a) 6, (b) 8.5, (c) 10 and (d) 14 nm along with the Rietveld fits. Difference patterns are shown below the observed patterns.

The successful Rietveld fit of the data, in which the K ions occupy all of the available sites in a virtually unchanged Cm lattice, is a direct experimental indication of the microscopic... 

FIG. 1 X-ray diffractogram of TDAE-Cgo compound. Data were taken at wavelength 1,2995 A with a channel-cut Ge(lll) monochromator and Ge(220) analyser, The smooth curve is a Rietveld fit, as described in text. To define the baseline, we have drawn a piecewise linear curve through regions of the spectrum where there are no allowed diffraction peaks. Inset shows a diagram of the TDAE molecule. The lower panel shows residuals, to the same vertical scale. Regions around untreated f.c.c. Ceo peaks excluded from the fit are absent from the model curve and residuals. 

We have taken a diffraction pattern of a second sample at 11 K. The degree of contamination by unreacted Cso is slightly less in that sample. We achieved similar R factors in Rietveld fits. The lattice parameters were a = 15.807(1) A, = 12.785(1) A, c = 9.859(1) A, =94.02(1)°. We have also studied a sample prepared from C o dissolved in toluene, described in ref. 5, which had a smaller saturation moment of 0.11/xb per formula unit, It showed a much smaller amount of coexisting f.c.c. Cfto, but essentially the same monoclinic pattern. This shows that this is a simple two-phase system, and hence confirms the validity of ignoring the f.c.c. Qo peaks in the refinement of Fig. 1. Interestingly, the diffraction lines from this sample were not uniformly sharp in particular, lines with h and / both nonzero were so broad as to be virtually undetectable. Evidently, there were solvent molecules present, which acted both to degrade the lattice perfection and the magnetic properties. 

To eorreet for instrumental misalignment an internal standard, e.g. 10% silieon (Si) or eorundum (a-Al203) powder (1 am Fisher Comp. 122651 K) should be mixed with the sample. The peak width must be eorreeted for the instrumental broadening using a well crystalline standard, sueh as lanthanum boride LaB. As an example, a Rietveld fit of an X-ray pattern from a mixture of synthetic hematite and goethite is shown in Fig. 3-9. 

Fig. 3-9. Rietveld-fit of an X-ray diffi actogram. The sample is an artificial mixture of synthetic hematite, goethite and corundum (internal standard) in a ratio of a b c. The X-ray diffractogram (upper) is followed by the x-ray line positions of the three components (middle) and the differences between the measured and the fitted intensities (residuals). Courtesy J. Friedl and H. Stanjek).

From D2b data we obtained a structure model through the iterative generation of stacking variants delivering best possible Rietveld fits from a single structural model. The... 

A L6 A and Rietveld fit (solid line) of best fitting model the ticks mark theoretical peak positions for ice Ic (above) and Ih (below), with lattice constants calculated from those of the refined model... 

The Rietveld fittings described here are simple illustrations of the method. As we have already said, the objective in this chapter is to give a few elements on this refinement process which is described in much greater detail in other books. Readers interested in knowing more will find a complete presentation of Rietveld refinements, with every step described in detail, in [PEC 03],... 

Fig. 4.6 Selected X-ray diffraction patterns observed experimentaiiy at iow temperature for Sii36 clathrate showing the very smaii shifts in key reflection profiies even at high angles. The broad background near 20-25° 2 is due to the glass capillary used to contain the sample this was removed from the pattern during the Rietveld fitting procedure. Careful analysis of the entire data sets provide a value for the unit cell volume at each temperature with associated errors (from [66])...

The room temperature X-ray powder diffraction patterns of these compounds are shown in Fig. 2a. A structural transition from orthorhombic symmetry (space group Pnma) for samples with x O.4 to rhombohedral symmetry (space group R-3c) for 0.5perovskite phases for the x = 0.8 composition was observed. Representative Rietveld fits to the X-ray diffraction data for the samples LPS20, LNSC50 and LNSC80 are shown in Fig. 2b, 2c and 2d, respectively. 

Fig. 2. (a) X-ray diffraction patterns for the series Lm-xAxFeOs-s with x-0.2 to x-0.8, all obtained by the ceramic route. Rietveld fits to the X-ray diffraction data for samples LPS20 (b), LPSC50 (c) and LSC80 (d). 

Fig. 5. Rietveld fits to room XRD patterns for LSSB-ss and LSSB-gn samples. In each case, lattice parameters (a, b, c), unit cell volume (V) and theoretical density (p) are included.

Fig. 7. (a) X-ray powder diffraction patterns at room temperature for all samples, (b) and (c) show the Rietveld fits to the X-ray powder diffraction patterns at room temperature for... 

Fig. 7.27 Rietveld fitting curve of Fea04 reduced from ")r-Fe203 (Dot line for the observation curve, solid line for the calculated curve, below curves is the difference value, a short vertical strokes eu e positions of Bragg reflections)...

F. 6.2 a XRD patterns and the Rietveld fits of Cu2ZnSnS4 and Cu2FeSnS4 nanocrystals, b TEM and c HRTEM images for Cu2ZnSnS4 nanocrystals, d TEM and e HRTEM images for Cu2FeSnS4 nanocrystals... 

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