Pattern-fitting structure refinement

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. 

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. 

The unit cell dimensions are important because they are used to calculate crystalline density which can then be used to determine percentage crystallinity using a simple two-phase model of crystalline and amorphous regions. X-ray-based unit cell determination is usually of considerable academic interest and tbe structures of all the common PAEK can be found in the literature [2, 3]. In fact PEEK and PEK were the first polymers to have their structures determined by pattern-fitting structure refinement (PFSR) [2, 3]. This technique produces more accurate unit cells by virtue of fitting X-ray peaks... 

Young, R. A., P. E. Mackie and R. B. Von Dreele (1977). Application of the Pattern-Fitting Structure Refinement Method for X-Ray Powder Diffractometer Patterns ./oMrwa/ of Applied Crystallography 10 262-269. 

The structure refinement program for disordered carbons, which was recently developed by Shi et al [14,15] is ideally suited to studies of the powder diffraction patterns of graphitic carbons. By performing a least squares fit between the measured diffraction pattern and a theoretical calculation, parameters of the model structure are optimized. For graphitic carbon, the structure is well described by the two-layer model which was carefully described in section 2.1.3. 

Fig. 6. The X-ray diffraction patterns and calculated best fits from the structure refinement program for the samples MCMB2300, iVICMB2600 and iVfCMB2800.

It is not always necessary that the fit be as good as that in Fig. 3 for useful results to be obtained. Fig. 4 shows the patterns for Rietveld refinement of a modified hydroxyapatite model for human tooth enamel (17) with both x-ray and neutron data. Here the difference curves show poorer fits than are shown in Fig. 3 and RWp is larger, being 26% for both the x-ray and the neutron cases. Nonetheless, the positions of the principal atoms were verified to ML 01 A, some of their site occupancies suggested the presence of expected substitutions, and information about the location of structurally incorporated water was revealed. 

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. 

The view about line profile analysis given in this chapter is pessimistic, it is the consequence of the complexity of the Bragg peak shapes as they occur from poorly-crystallized material. More optimistic is the future of the main whole powder pattern fitting applications (decomposition or Rietveld methods) that have moved beyond the initial stages, enabling structure determinations (almost routinely) and refinements (routinely) of moderately complex structures to even complex crystal structures such as proteins (sometimes). 

Those interested mostly in structure determination from powder diffraction see the texture problem differently. The presence of the preferred orientation makes a good pattern fitting difficult or even impossible and, consequently, a procedure is needed to correct for the texture effect in the Rietveld codes. For that it is not necessary to find the ODF, but to have a reliable model of the pole distribution whose parameters are refined together with the structure and other parameters. 

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. 

Simpro Simultaneous structure refinement of neutron, synchrotron and X ray powder diffraction patterns, J. K. Maichle, J. Ihringer and W. Prandl, J. Appl. Crystallogr., 1988, 21, 22 27 and A quantitative measure for the goodness of fit in profile refinements with more than 20 degrees of freedom, J. Ihringer, J. Appl. Crystallogr., 1995, 28, 618 619... 

Application of total pattern fitting to X ray powder diffraction data, H. Toraya and F. Marumo, Rep. Res. Lab. Engin. Mat., Tokyo Inst. Tech., 1980, 5, 55 64 and Crystal structure refinement of alpha 813X4 using synchrotron radiation powder diffraction data unbiased refinement strategy, H. Toraya,... 

The structural parameters are the atomic positions, Debye-Waller factors, electron structure factors, and electron diffraction parameters, which include the absorption potential, sample thickness, and crystal orientation. Not all parameters can be refined together. Diffraction patterns that are sensitive to certain parameters are collected and they are often refined independently. Figure 9 shows an example of a structure factor measurement by fitting CBED intensities recorded in the systematic orientation where one row of reflections are set at or near the Bragg conditions. Details about this method and its applications for structure factor measurement and atomic structure refinement are given in Ref. 18 and 43. 

The Rietveld method is a refinement technique in which the whole powder pattern is fitted by varying a number of instrumental and stmctural-model parameters. The successful use of the method is directly related to the quality of both the diffraction data and the structural model being refined. The Rietveld method is widely available for the structure refinement of powder data through such programs such as GSAS, FullProf, and Rietan. ... 

The description proposed in [26, 27] is the empirical consideration of alumina structures. The evalution of the profile fitting and Rietveld refinement procedure allow new approaches to previously intractable patterns of structural disorder materials. 

The experimental diffraction data were analyzed by a combined technique involving Rietveld analysis, the maximum entropy method (MEM), and MEM-based pattern fitting (MPF) [10-15]. Rietveld analysis, which is used to refine the crystal structure from the powder diffraction data by a least squares method, was carried out using the RIETAN-2000 program [27], which yields structure factors and their errors after structural refinement. It is known that MEM can be used to obtain a nuclear density distribution map based on neutron structure factors and their errors [5, 6, 8, 10-15, 26-29] any type of complicated nuclear density distribution is allowed so long as it satisfies the symmetry requirements. MEM calculations were carried out using the PRIMA program [29]. To reduce the bias imposed by the simple structural model in the Rietveld refinement, an iterative procedure known as the REMEDY cycle [29] was applied after MEM analysis (Fig. 6.3). In this procedure, structure factors... 

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