Rietveld method structure factor

In order to obtain detailed structure, a knowledge of diffraction intensities is essential, the intensities being related to the structure factor. Computer-controlled single-crystal X-ray diffractometers with structure (software) packages have made structure elucidation a routine matter. The availability of synchrotron X-radiation of continuously variable wavelength has made X-ray diffraction a still more powerful structural tool for the study of solids. A technique of great utility to solid state chemists is the Rietveld treatment of powder X-ray diffraction profiles (Rietveld, 1969 Manohar, 1983). Automated structure packages for the determination of unknown structures by this method are now commercially available (see section 2.2.3). In Fig. 2.1, we show a typical set of profile data. 

In 1995, an elaborated method was developed for accurate structure analysis using X-ray powder diffraction data, that is, the MEM/Rietveld method [1,9]. The method enables us to construct the fine structural model up to charge density level, and is a self-consistent analysis with MEM charge density reconstruction of powder diffraction data. It also includes the Rietveld powder pattern fitting based on the model derived from the MEM charge density. To start the methods, it is necessary to have a primitive (or preliminary) structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. The flow chart of the method is shown in Fig. 2. 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

The space group was assigned to P2i, which is monoclinic for Y Cs2. The experimental data were analyzed in an iterative way of a combination of Rietveld analysis (Rietveld, 1969) and the maximum entropy method (MEM) (Bricogne, 1988 Collins, 1982). The MEM can produce an election density distribution map from a set of X-ray structure factors without using any structural model. By the MEM analysis (Kumazawa et al., 1993 Sakata and Sato, 1990), the Rj becomes as low as 1.5% for Y Cs2-... 

Early data analysis attempted to extract values of the individual structure factors from peak envelopes and then apply standard single crystal methods to obtain structural information. This approach was severely limited because the relatively broad peaks in a powder pattern resulted in substantial reflection overlap and the number of usable structure factors that could be obtained in this way was very small. Consequently, only very simple crystal structures could be examined by this method. For example, the neutron diffraction study of defects in CaF2-YF3 fluorite solid solutions used 20 reflection intensities to determine values for eight structural parameters. To overcome this limitation, H. M. Rietveld realized that a neutron powder diffraction pattern is a smooth curve that consists of Gaussian peaks on top of a smooth background... 

For a physical mixture, the powder diffraction pattern is the sum of the patterns of the individual materials. The diffraction pattern can therefore be used to identify the crystalline phases in a mixture. The concentrations of the crystalline phases can be determined by methods based on comparing the intensities of the diffraction peaks with standards (6-8). If the crystal structures of the phases are known, the concentration of each phase can be detamined by Rietveld analysis (20,21). In the Rietveld method, a theoretical diffraction pattern is computed and the difference between the theoretical and observed patterns is minimized. For quantitative analysis, some care should be taken with specimen preparation if accurate and reliable results are to be obtained. The effects of factors such as preferred orientation, texturing, and particle size broadening must be minimized. 

The strength of the Rietveld method lies in the refinement of the structure parameters of the whole powder sample not just of one single crystal. Rietveld [114] recognized that it is by no means necessary to allocate the measured intensities to individual reflections in a one-dimensional powder pattern. Instead, it is better to measure a diffraction pattern as exactly as possible and use it in its entirety as the basis for refinement of the structure parameters. Parameters such as the atom coordinates, temperature coefficients, and scaling factors are fitted to the diffraction data by a least-squares procedure. 

Among the most recent advances of the powder method is the determination of crystal structures from powder diffraction data. It is an application for which the resolution of the pattern is of prime importance. A series of successive stages are involved in the analysis, including the determination of cell dimensions and identification of the space group from systematic reflection absences, the extraction of structure factor moduli I hkl y the solution to the phase problem to elaborate a structure model and, finally, the refinement of the atomic coordinates with the Rietveld method. 

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... 

Having compiled a control file containing the structural information of the identified phases, an XRD pattern can be calculated using the Rietveld method to compare to the measured pattern. Through a least-squares fitting algorithm, selected parameters are optimised to obtain the best possible fit between the calculated pattern and the measured pattern. In QPA of cementitious systems the phase-dependent parameters that are varied should be limited to (1) the phase scale factors, (2) the unit cell parameters,... 

This method requires only a crude structural model as a starting model. In this analysis, the starting model was a homogeneous spherical shell density for the carbon cage. As for the temperature factors of all atoms, an isotropic harmonic model was used an isotropic Gaussian distribution is presumed for a La atom in the starting model. Then, the radius of the C82 sphere was refined as structural parameter in the Rietveld refinement. 

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. 

The result of the application of the DDM decomposition formula Equation (14) is similar to that of Rietveld s approximation for /obs> except that it is DDM-oriented and thus does not require a background definition. In the DDM program the formula is also used for the Bragg R factor calculation. For single non-overlapped peaks or a set of peaks with the same position, the DDM decomposition directly gives the best estimate for /obs- For partly overlapped peaks it should be iterated to arrive at an optimized set of /obs similarly to the Le Bail method. The starting intensities f can either be calculated from an existing structure model or set arbitrarily when the model is absent. Notably, for fully overlapped peaks the DDM decomposition... 

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. 

Quantitative analysis of phase in the multi-phase system has always been a problem. Although the XRD is the most studied technique in the past as quantitative analysis of phase, but because multicrystalline XRD has problems of bulk effect of different X-ray absorption factors, the peak-overlap and changes in the intensity of diffraction hkl is caused by difference of fine structure which cannot be solved by traditional method. Thus, the results are always unsatisfied and considered to be unreliable. Rietveld analysis has completely solved the problem. The basic equation is ... 

Rietveld multi-phase quantitative analysis is also called as model method, and can solve X-ray quantitative problem in the multi-phase system. This method has the following advantages It is free of internal standards it is free of the standard sample there is no need to separate peaks and finally, it overcomes the effect of the structure. With the scale factors of and S-y of reference sample a-Fe203 and experimental sample 7-Fe20s obtained by multi-phase Rietveld analysis, the relative abundance of amorphous phase in sample can be calculated by the following equation. 

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