Whole powder pattern modelling

Further aspects, pros and cons of WPPF, are discussed in Chapter 5. Here it is important to underline the fact that the validity of profile fitting is limited by the basic assumption of using an a priori selected profile function without any sound hypothesis that the specific functional form is appropriate to the case of study. The consequence of this arbitrary assumption can be quite different. For example, in most practical cases, profile fitting can provide reliable values of peak position and area, whereas the effects on the profile parameters are less known and rarely considered. The arbitrary choice of a profile function tends to introduce systematic errors in the width and shape parameters, which invariably introduce a bias in a following LPA, whose consequences can hardly be evaluated. It is therefore a natural tendency, for complex problems and to obtain more reliable results, to remove the a priori selected profile functions - leading to the following section dedicated to Whole Powder Pattern Modelling methods. 

To avoid using arbitrary a priori selected) profile functions, whose parameters are not directly and uni vocally related to physically observable quantities, a different approach can be followed line profiles can be described directly in terms of physical models of the micro structure and lattice defects present in the studied material. This is the foundation of the WPPM method in this respect, modelling is opposed to fitting as the former involves the use of physical information at all stages of the analysis, whereas the latter uses an a priori selected (though flexible) form for the PD profiles. 

A brief overview of the physical basis of WPPM is given here, together with some examples of applications to real cases of study. Interested readers can find further details in the cited literature. The basic expression of WPPM considers the FT of a hkl peak profile  

The sum extends to all equivalent hid) planes composing a hkl family (that for a cubic system means all permutations of h,k,l, sign included), w ki and d ki are, respectively, weight and shift from the Bragg condition (caused by lattice defects, e.g. by faulting) for the (hkl) component. In this way the possibility that lattice defects might produce different effects on different hkl) is explicitly considered. 

The core of Equation (25) is the FT, Chki L), which, owing to Equation (13), can be written as the product of the FTs of all contributing extrinsic and intrinsic profile components  

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SCA 04] SCARDI P., LEONI M., Whole powder pattern modelling theory and applications , in MITTEMEIJER E.J., SCARDI P., Diffraction analysis of the mlcrostructure of materials, Springer Series in Materials Science, vol. 68, p. 51-91, 2004. 

Scardi. P. Leoni. M. Whole powder pattern modelling. Acta Crystallogr., A 2002. 58, 190-200. 

Leoni, M., Confente, T. Scardi, P. (2006). PM2K a flexible program implementing Whole Powder Pattern Modelling Z. Kristallogr. SuppL, 23, 249-254. 

Search, P. (2008). Recent advancements in whole powder pattern modeling Z. Kristal-logr. Suppl, 27, 101-111... 

WPPF Whole powder pattern fitting without reference to a structural model Application to X ray powder diffractometer data, H. Toraya, J. Appl. Crystallogr., 1986, 19, 440 447... 

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. ... 

A structural model is required, and the parameters in the model are adjusted programmatically, by computer, to give the best least squares fit of the whole calculated powder diffraction pattern to the whole observed pattern. Many things besides crystal structure can be and often need be considered in the calculations, e.g., instrumental profiles, preferred orientation, and contributions from background and from other phases. It is possible to refine simultaneously the structures, or at least amounts present, of two or more phases present (7J. 

Despite such restrictions, real space crystallographic methods based on genetic algorithms, Monte-Carlo methods, or simulated annealing techniques have proved to be powerful means for structure solutions from X-ray powder patterns. Provided with the unit cell, the composition and configuration of the asymmetric unit, and sufficiently texture-free diffraction data, refinable structure models can be obtained within minutes on a personal computer, even for molecules with multiple internal degrees of freedom. The resulting structure models are then refined by Rietveld techniques, which use the whole profile of the X-ray diffraction pattern for refinement . ... 

The idea of the Rietveld method is to do a step-by-step refinement of the whole powder diffraction pattern. One can constmct an initial stmctural model from a partial knowledge of the phases detected in a pattern and, gradually and progressively, improve the model in order to compare it to the experimental diffraction pattern. One of the main advantages of the method is, in principle, that it can deal with heavily overlapped peak patterns. There is no need to separate peaks, since the scattering contributions from the various phases are all summed up to calculate the intensities at each diffractogram step, not attributing a particular peak to a specific phase or a crystal plane. ... 

Detailed theoretical studies, based on EPR experimental data, on both trapped electron centres and on electron transfer toward adsorbed molecules on MgO have also appeared. In the latter paper, beside further investigation of the Fs +/N2 system, a detailed analysis of the Fs +/02" complex was reported with particular emphasis on the whole hyperfine structure of adsorbed O2 . The Fermi contact term was evaluated as = —20.3 G and the resulting dipolar tensor was = — 56 G, Byy = -1-27.5 G, B = +28.6 G. An experimental paper appeared some months later reporting a fully resolved hyperfine structure of adsorbed 2. The powder spectrum contained a large number of lines and was further comphcated by the presence of several off-axis extra features. The simulation of such a complex pattern of fines gave Uiso = 4.8 G and a dipolar tensor (J) with a value remarkably close to the theoretically calculated value. This vahdates the quahty of the model but also the capabihty of modern theoretical approaches in forecasting EPR hyperfine parameters. 

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