Rietveld refinement procedure

Historically (1960 s and 1970 s) mostly used on single crystals, it has since then been generalised for powders using Rietveld refinement procedures. 

B. H. O Connor and M. D. Raven, Application of the Rietveld refinement procedure in assaying powdered mixtures. Powder Dijfr., 1988, 3, 2-6. 

In the recent years, sophisticated modeling tools have become available, such as the Cerius (8), where various modules aUow the analysis of crystallization, crystal growth, and material form characterization. In brief, this technique uses a simulated annealing and a rigid-body Rietveld refinement procedure, whereby the calculated and measured XRPD patterns are compared if they agree sufficiently, the structure is deemed to be solved. Other modules offered by Cerus include ... 

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. 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

The process is cyclic the procedure stops when all anions have been positioned. For each feasible model the profile residual is calculated and those corresponding to the best agreement factors are selected for Rietveld refinement. 

The residual difference after a successful DDM refinement or/and decomposition can be considered as a scattering component of the powder pattern free of Bragg diffraction. The separation of this component would facilitate the analysis of the amorphous fraction of the sample, the radial distribution function of the non-crystalline scatterers, the thermal diffuse scattering properties and other non-Bragg features of powder patterns. The background-independent profile treatment can be especially desirable in quantitative phase analysis when amorphous admixtures must be accounted for. Further extensions of DDM may involve Bayesian probability theory, which has been utilized efficiently in background estimation procedures and Rietveld refinement in the presence of impurities.DDM will also be useful at the initial steps of powder diffraction structure determination when the structure model is absent and the background line cannot be determined correctly. The direct space search methods of structure solution, in particular, may efficiently utilize DDM. 

Powder neutron diffraction data for Bi 8 0 2 3 12 collected at Brookhaven National Laboratory s High FIux Beam Reactor. Details of the experimental procedure and Rietveld refinement have been reported previously (18). Starting parameters for Ce doped and pure Bi Mo 0. were taken from a single crystal x-ray diffraction study of Bi,.Ho 0-,. by Van den Elzen and Rieck (19a). 2 J 12... 

When a powder is examined, many diffracted beams overlap, (see Section 6.11), so that the procedure of structure determination is more difficult. In particular this makes space group determination less straightforward. Nevertheless, powder diffraction data is now used routinely to determine the structures of new materials. An important technique used to solve structures from powder diffraction data is that of Rietveld refinement. In this method, the exact shape of each diffraction line, called the profile, is calculated and matched with the experimental data. Difficulties arise not only because of overlapping reflections, but also because instrumental factors add significantly to the profile of a diffracted beam. Nevertheless, Rietveld refinement of powder diffraction patterns is routinely used to determine the structures of materials that cannot readily be prepared in a form suitable for single crystal X-ray study. 

Samples with increasing Ti content have been synthesized in EniChem laboratories following a procedure described in the original patent [1], The total insertion of Ti atoms in the MFI framework has been witnessed by comparison of the amount of Ti found in the samples by chemical analysis with the cell volumes obtained by Rietveld refinement of powder XRD data, see Ref [4] for more details. 

This clearly shows several important features of the Rletveld refinement procedure. Firstly, it involves inversion of an M M matrix. Secondly, it works only if the starting values are already close to a minimum, because of the linear approximation In the derivation of equation (2). Thirdly, whatever minimum is obtained, it is not necessarily the global minimum one is looking for but may be a local minimum, since the method is a local one. Finally, because of finite width of every reflection, the calculated intensity at each step contains contributions from several neighboring reflections. In our case up to 5 possible reflections can contribute to the Intensity calculated at a single point in the diffraction pattern. The program used for Rietveld refinement in this work is an extensively modified version of REFIN (FORTRAN V), written by A. Imralrzi (5). 

The RIR method is attractive for fast and easy quantitative analysis but it has typical accuracy around 10 % and it is not as rigorous as fuU pattern analysis such as Rietveld refinement (see below). It has to be used with care due to peak overlaps and it is limited to the quality and accuracy of the actual peak fitting procedure. 

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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