Whole pattern fitting

Rietveld refinement [25, 26] is a method of whole pattern refinement, where a calculated diffraction pattern for a structure model is a least-squares fit to an observed diffraction pattern. Originally, it was used as a means of verifying proposed structure models. For zeolites, Rietveld refinement is still used for the same purpose and provides details of the structure including atomic positions of framework atoms and cation sittings. Data with accurate intensities and well-resolved peaks are needed for the most accurate work, and so often a synchrotron source is used for data collection since it can provide higher intensity and peak resolution than an in-house diffractometer. However, modern in-house diffractometers often provide good enough data for some refinements. 

During the last five years, a powerful new method of getting crystal structural information from powder diffraction patterns has become widely used. Known variously as the Rietveld method, profile refinement1, or, more descriptively, whole-pattern-fitting structure refinement, the method was first introduced by Rietveld (X, 2) for use with neutron powder diffraction patterns. It has now been successfully used with neutron data to determine crystal structural details of more than 200 different materials in polycrystalline powder form. Later modified to work with x-ray powder patterns (3, X) the method has now been used for the refinement of more than 30 crystal structures, in 15 space groups, from x-ray powder data. Neutron applications have been reviewed by Cheetham and Taylor (5) and those for x-ray by Young (6). 

The whole-pattern-fitting structure-refinement method, which was first introduced by Rietveld and used for neutron diffraction powder patterns, does yield from x-ray diffraction patterns correct, refined structural information for linear polymers. Remarkably precise lattice parameters are obtained incidentally in the use of the method. The method lends itself to improved estimations of the fraction of amorphous and crystalline materials, or of two polymorphic forms, present. As improved profile functions come in to use, the method promises to provide crystallite size information, almost as a spin-off benefit. 

YOUNG ET AL. Rietveld Whole-Pattern-Fitting Method... 

The whole-pattern-fitting structure-refinement method is not limited to specimens containing randomly oriented crystallites. The method does permit an orientation parameter to be refined, though it was not done in this work. Crystallites in the molded polypropylene specimen probably were oriented to some small extent nevertheless, the refinement method worked well. The sensitivity of the method to crystallite orientation and the applicability of the method to polymers with oriented crystallites such as in drawn films or fibers are yet to be determined. 

The whole-pattern-fitting structure-refinement method can give correct, refined structural information for linear polymers. Use of generalized coordinates, in which accurately known structural information can be introduced as known parameters, makes the least-squares fitting problem tractable. Excellent fits of x-ray diffraction data for isotactic polypropylene permitted the selection of the correct space group and a preferred model. 

Historically, the observation that volumes of reacting gases always simplified to ratios of small, whole numbers is called Gay-Lussac s Law of Combining Volumes. In the preceding example, the volumes of NO to 02 to N02 fit the pattern of 2 1 2. This observation further strengthened Dalton s argument for an atomic theory of matter. 

We have found that in a temperature interval above Tc and below some T 300 K the nuclear spin relaxation for a broad class of cuprates comes from two independent mechanisms relaxation on the stripe -like excitations that leads to a temperature independent contribution to 1 /63i and an universal temperature dependent term. For Lai.seSro.nCuC we obtained a correct quantitative estimate for the value of the first term. We concluded from eq.(l) and Fig.3 that "stripes always come about with external doping and may be pinned by structural defects. We argue that the whole pattern fits well the notion of the dynamical PS into coexisting metallic and IC magnetic phases. Experimentally, it seems that with the temperature decrease dynamical PS acquires the static character with the IC symmetry breaking for AF phase dictated by the competition between the lattice and the Coulomb forces. The form of coexistence of the IC magnetism with SC below Tc remains not understood as well as behaviour of stoichiometric cuprates. 

In BLH-NhaCo()2, the guest species in the galleries are highly disordered. MEM is quite effective for detailed structure analysis of such an intercalation compound. In MEM-based whole-pattern fitting (MPF), crystal structures are expressed not by structure parameters such as fractional coordinates and atomic displacement parameters but by electron densities in the unit cell. Therefore, MPF allows us to represent the disordered atomic configuration in a more appropriate way than conventional Rietveld analysis adopting a split-atom model. The... 

In the pharmaceutical community, quantitative analyses has conventionally been based on the intensity of a characteristic peak of the analyte. It is now recognized that phase quantification will be more accurate if it is based on the entire powder pattern.This forms the basis for the whole-powder-pattern analyses method developed in the last few decades. Of the available methods, the Rietveld method is deemed the most powerful since it is based on structural parameters. This is a whole-pattern fitting least-squares refinement technique that has also been extensively used for crystal structure refinement and to determine the size and strain of crystallites. 

Peak by peak analysis is widely used in X-ray diffraction and significant developments in digital calculations have made it possible to define very precisely the actual peak profiles. It sometimes happens, however, that the peaks overlap too much for it to be possible to separate their contributions. It then becomes necessary to provide additional information which makes it possible to correlate either the relative positions of the different peaks or their intensities. This is referred to as whole pattern fitting. 

Figure 4.3b. Fitting of the whole pattern obtained with a sample fired at 800°C...

Measurements of integral breadths by whole pattern fitting... 

Since the early 1980s, a new way of quantifying structural defects has appeared. It consists of applying the developments implemented in stmctural analysis to the increase in peak width [KEl 83, LUT 90, THO 87b, LEB 92, LEB 99], This method, which relies on Rietveld s stmctural analysis method, consists of a whole pattern fitting of the diffraction pattern. 

Figure 6.11. Whole pattern fitting of the diffraction pattern reported in Figure 6.9. The calculated curve is shown in grey...

In the two examples we have just described, the diffraction peaks associated with the different phases of zirconia show a significant overlap. Only whole pattern fitting can lead to reliable results. 

HON 92] HONKIMAKl V., SUORTTl P., Whole pattern fitting in energy dispersive powder diffraction , / Appl. Cryst, vol. 25, p. 97-104,1992. 

UNG 99b] UNGAR T., LEONIM., SCARDIP., The dislocation model of strain anisotropy in whole pattern fitting the case of an Li-Mn cubic spinel , J. Appl. Cryst, vol. 32, p. 290-295,1999. 

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