Crystal structure Rietveld method

Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

Application of the Rietveld Crystal Structure Refinement Method to Cellotetraose... 

The conventional crystal structure refinement method by means of the Rietveld method suffers from considerable uncertainty largely because of the unfavorable data-to-parameter ratios. The complexity of the problem generally depends on the number of atoms located in the asymmetric unit. For nonstoichiometric guest-host materials, the difficulty in structural determination of crystalline powders lies in how to determine dynamical disorder and partial... 

Applications The general applications of XRD comprise routine phase identification, quantitative analysis, compositional studies of crystalline solid compounds, texture and residual stress analysis, high-and low-temperature studies, low-angle analysis, films, etc. Single-crystal X-ray diffraction has been used for detailed structural analysis of many pure polymer additives (antioxidants, flame retardants, plasticisers, fillers, pigments and dyes, etc.) and for conformational analysis. A variety of analytical techniques are used to identify and classify different crystal polymorphs, notably XRD, microscopy, DSC, FTIR and NIRS. A comprehensive review of the analytical techniques employed for the analysis of polymorphs has been compiled [324]. The Rietveld method has been used to model a mineral-filled PPS compound [325]. 

Although simple intensity correction techniques can be used to develop very adequate XRPD methods of quantitative analysis, the introduction of more sophisticated data acquisition and handling techniques can greatly improve the quality of the developed method. For instance, improvement of the powder pattern quality through the use of the Rietveld method has been used to evaluate mixtures of two anhydrous polymorphs of carbamazepine and the dihydrate solvatomorph [43]. The method of whole pattern analysis developed by Rietveld [44] has found widespread use in crystal structure refinement and in the quantitative analysis of complex mixtures. Using this approach, the detection of analyte species was possible even when their concentration was less than 1% in the sample matrix. It was reported that good quantitation of analytes could be obtained in complex mixtures even without the requirement of calibration curves. 

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. 

Most of the unknown structures is determined from single crystal diffraction and refined from powder diffraction. Refinement is done with the Rietveld method, which is a least square fitting of the computed pattern to the measured one, while structure parameters are treated as the primary fitting parameters. This is in contrast to the procedure in pattern decomposition, which is outlined above (where not the structure parameters, but the peak intensities were the primary fitting parameters). Beside the... 

A method known as Rietveld analysis has been developed for solving crystal structures from powder diffraction data. The Rietveld method involves an interpretation of not only the line position but also of the line intensities, and because there is so much overlap of the reflections in the powder patterns, the method developed by Rietveld involves analysing the overall line profiles. Rietveld formulated a method of assigning each peak a gaussian shape and then allowing the gaussians to overlap so that an overall... 

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). 

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. 

Post, J. E. Veblen, D. R. (1990) Crystal structure determinations of synthetic sodium, magnesium, and potassium bimessite using TEM and the Rietveld method. Amer. Mineral., 75, 477-89. 

A complete understanding of the structure of the material under study or application is a sine qua non condition for the successful research or use of the material. In the case of powders, the best way to decipher the structure of new materials is the Rietveld method. This methodology was initially developed by Hugo M. Rietveld in 1969 [23] as a procedure for refining crystal structures using neutron powder diffraction data. To implement the method in practice, certain information about the estimated crystal structure of the phase or phases of interest in the diffraction profile under test is necessary. 

A perfect crystal structure model is very helpful for theoretical calculations, reaction mechanism analysis, and some physical property analysis such as conductivity, magnetic susceptibility, chemical potential, etc. Powder XRD (or neutron diffraction) Rietveld refinement is one of the most popular methods used to characterize crystal structure. 

The best structure solution (i.e. the structure with lowest Rwp in the final generation) for the a phase is shown in Fig. 5a and the best structure solution for the f5 phase is shown in Fig. 5b. For comparison, the known crystal structures [61,62] of the a and (5 phases are also shown. In each case, the structure solution obtained from the GA calculation is in excellent agreement with the known structure. In each case, the maximum distance between an atom in the structure solution and the corresponding atom in the known crystal structure is less than 0.5 A, and the structure refines readily (using the Rietveld method) to the known crystal structure. We emphasize that the L-glutamic acid molecule has a significantly different conformation in the a and f3 phases, and the GA structure solution calculations have successfully found the correct conformation for each phase. 

In cases where it is difficult to obtain large crystals, chemists may logically conclude that powder diffraction experiments offer a suitable alternative. However, powder diffraction experiments " are usually restricted to large samples of smaller unit cells than those commonly involved in a typical organometallic structure. In addition, because the nature of the diffraction data is one-dimensional (rather than three-dimensional in a single-crystal experiment), peak overlap is a serious problem. This difficulty is largely alleviated by the use of the Rietveld method, " in which the overall profile of the diffraction pattern is essentially scanned stepwise to yield hundreds of individual intensity measnrements. Fnr-thermore, deuteration is almost always necessary in a powder... 

Because of the general similarities in the diffraction patterns, and the lack of clearly resolvable distinguishing peaks, they employed the Rietveld method (Young 1993). In the Rietveld method, the entire experimental diffraction pattern for each solid phase is used as a basis for comparison. For structure determination using powder diffraction, this comparison is made with a structural model used to generate a calculated pattern. In quantitative analysis of polymorphic phases, the known crystal stmctures are used to generate the standard diffraction patterns and these are then refined against the experimental powder pattern of the mixture to obtain the relative amounts of the polymorphs. 

MacLean et al. (2000) have recently smdied the dimorphic behaviour of the pigment precursor ( latent pigment) derivative of 8-VI (R = COOr-but, R = H) (abbreviated DPP-Boc). The latency is due to the thermal decomposition reaction of both polymorphs resulting in the commercially important pigment DPP. The a form of DPP-Boc contains three half molecules in the asymmetric unit (see also Ellern et al. 1994) while the form contains one half molecule per asymmetric unit. Hence, they are easily distinguishable by solid state NMR as well as by X-ray powder diffraction. The crystal structure solution from powder data and Rietveld refinement of both polymorphs is an exemplary smdy demonstrating the potential of these methods in determining the detailed crystal structure of these compounds which are often difficult to crystallize. 

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 Rietveld method is employed both to finalize the model of the crystal structure, when necessary, e.g. to locate a few missing atoms in the unit cell by coupling it with Fourier series calculations, and to confirm the crystal structure determination by refining positional and other relevant parameters of individual atoms together with profile variables. The fully refined... 

Y° is the observed total intensity in the i point, yk,t is the calculated intensity of the A reflection in the point. The main difference between Rietveld and Le Bail decompositions is in the calculated intensities. The former technique uses intensities computed from the model of the crystal structure, while the latter approach uses intensity obtained from the previous cycle during the decomposition. Initially, all calculated intensity values in the Le Bail s method are set to arbitrary identical quantities, t)rpically unity. 

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. 

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