Rietveld-based Methods

Whole pattern (especially Rietveld-based) methods for the determination of phase abundance have the potential to produce more accurate and precise 

While the Rietveld teehnique was initially developed for the refinement of erystal strueture, other parameters that must be refined to ensure best fit between the observed and ealeulated patterns eontain useful, non-struetural information that ean be of interest to the analyst. These inelude peak width and shape, which can be related to crystallite size and strain (see Chapter 13), and the Rietveld scale factor, which, in a multiphase mixture, relates to the amount of the phase present. 

Recalling Equation (1), the expression in the second square bracket shows that the constant C in Equation (5) is inversely proportional to the square of the unit cell volume (F ), while Hill has demonstrated that the individual reflection intensity I is proportional to the Rietveld scale factor, S  

Coupled with the knowledge that the phase density (g cm can be calculated from the mass of the unit cell contents (ZM, where Z = the number of formula units in the unit cell and M=the molecular mass of the formula unit) and the unit cell volume (V)  

In a similar manner to that used for the single peak approach, the need to measure K and measure or calculate can be eliminated by the addition of an 

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This chapter focuses on the application of quantitative phase analysis (QPA) techniques for the extraction of phase abundance from diffraction data. Rather than repeat the extensive coverage of the QPA methodology covered in other texts, the focus will be on the basis and application of the most commonly used techniques. These were identified from participant responses to the recent round robin on QPA sponsored by the International Union of Crystallography (lUCr) Commission on Powder Diffraction (CPD). By far the greatest number of participants in that study used whole pattern (Rietveld based) methods but there are still several users of traditional single peak based methods and there are still many applications for which these methods suffice. Issues in the measurement of precision and accuracy will also be discussed. 

To test the validity of Rietveld based methods for QPA, the round robin organizers undertook the analysis of all eight mixtures in the Sample 1 suite using the following refinement strategy ... 

The Te Bail method is a clever extension of the original work of Rietveld. The key difference between the two methods is that the Te Bail method uses iteratively the values of intensities obtained from the previous least squares cycle, and assumes unit values for all the peaks to initiate the refinement. Rietveld s method, on the other hand, uses the intensities calculated based on an initial model for the structure. 

Lasocha W, Schenk H (1997) A simplified, textirre-based method for intensity determination of overlapping reflectiorts in powder diffraction. J Appl Crystallogr 30 561-564 Lee Y, Cahill C, Hanson J, Parise JB, Carr S, Myrick ML, Preckwinkel UV, Phillips JC (1999) Characterization of K -ion exchange into Na-MAX using time resolved synchrotron X-ray powder diffraction and rietveld refinement. In 12th Int l Zeolite Assoc Meeting, Baltimore, Maryland, IV 2401-2408... 

These were largely due to inappropriate use of the method being applied. For the Rietveld based techniques, these errors included (i) entry of incorrect crystal structure information, including atom coordinates and thermal vibration parameters, (ii) incorrect space group notation, (iii) incorrect atom site occupation parameters, (iv) allowing structure parameters (especially thermal parameters) to refine to physically unrealistic values, and (v) not completing the refinement. 

H.M. Rietveld developed a practical method to analyze powder diffraction data. The Rietveld refinement method [6] is based on modeling a diffraction pattern according to structural information available for the phases previously identified in the pattern. This model is calculated by computer software after introduction of some crystallographic data space group symmetry, atomic species and positions within the cell, occupancies, lattice parameters, line broadening parameters, and experimental conditions. The term refinement refers to the cyclic improvement process used in the estimation of the set of parameters that can model the diffraction pattern as close as possible to the observed pattern. Usually, the Rietveld refinement is accomplished by minimizing the sum of the differences of calculated and observed intensities for each angular step of the diffraction pattern and by a... 

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. 

The x-//J-Si3N4 ratio is measured by X-ray diffraction [228-231]. In earlier investigations only some peaks where chosen [229-231]. New techniques based on the Rietveld method use the whole angle range of the XRD diffraction pattern [219,228]. This results in more accurate data even in textured samples [228], The amount of amorphous Si3N4 also can be analysed by this method using an internal standard [224]. 

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. 

In conventional Rietveld refinements, a structural model based on the arrangement of atoms in the unit cell is generally used. However, the imaging ability of MEM in the MEM/Rietveld method enables one to visualise more detailed features like the bonding electron distribution associated with hybridised orbitals [86]. was found to adopt a pseudo-monoclinic cell with... 

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... 

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. 

Weekend experiments normally take more than a day. They are conducted when precise peak shapes are desired, for example, when microstmctural properties are analyzed based on the anomalies of Bragg peak shapes, or when the crystal stmcture of the material is determined and refined by using the Rietveld method (see Chapter 7). 

Le Bail s approach is also based on Eq. 6.6 and it differs from the Pawley s method in that the individual intensities remain unaltered during each least squares cycle. They are extracted from a total observed intensity of the pattern between the least squares cycles after subtraction of the background. The extraction is performed using a decomposition approach that has been employed in the Rietveld method since its development, in which intensity observed in every point of the powder pattern is divided among different reflections proportionally to their calculated intensities ... 

Both the full pattern decomposition and Rietveld refinement are based on the non-linear least squares minimization of the differences between the observed and calculated profiles. Therefore, the non-linear least squares method is briefly considered here. Assume that we are looking for the best solution of a system of n simultaneous equations with m unknown parameters (n m), where each equation is a non-linear function with respect to the unknowns, Xu X2,. .., In a general form, this system of equations can be represented as... 

VMRIA VMRIA a program for a full profile analysis of powder multiphase, neutron diffraction time of ffight (direct and Fourier) spectra, V. B. Zlokazov, V. V. Chernyshev, J. Appl. Crystallogr., 1992, 25, 447 451 and DELPHI based visual object oriented programming for the analysis of experimental data in low energy physics, V. B. Zlokazov, Nucl. Instrum. Methods Rhys. Res. A, 2003, 502(2 3), 723 724 Rietveld refinement of TOF data... 

PRIMA) entropy method from powder diffraction data, F. Izumi and R. A. Dilanian, in Recent Research Developments in Physics, Transworld Research Network, Trivandrum, 2002, Vol. 3, Part II, pp. 699 726 and Beyond the ability of Rietveld analysis MEM based pattern fitting, F. Izumi, Solid State Ionics, 2004, 172, 1 6 ... 

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