International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.6, pp. 297-298
Section 3.6.2.7.1. Alternative approaches^{a}Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy |
Convolutional multiple whole profile fitting (CMWP; Ribárik et al., 2004) and extended convolutional multiple whole profile fitting (eCMWP; Balogh et al., 2006) have been developed to solve the same problem.
CMWP, introduced as a convolutive version of multiple whole profile fitting (MWP; Ungár et al., 2001; Ribárik, 2008), is very similar to the WPPM. The notable differences are:
The authors also suggest using the MWP procedure in other cases, for example separately measured profiles or single crystals. The MWP procedure works in Fourier space so there is no direct possibility of checking the agreement between the model and data.
In the extended version (Balogh et al., 2006), (e)CMWP introduces an interesting model for faults based on a parameterization of the profiles simulated with the DIFFaX software (Treacy et al., 1991). This simulates the diffraction pattern of a faulted structure within the tangent cylinder approximation. The proposed parameterization allows more complex faulting models, at the expense of the calculation and parameterization of the profiles for any new, intermediate or mixed case or for any peak lying outside the parameterized range. The application of (e)CMWP is limited to cubic, hexagonal and orthorhombic powders with spherical or ellipsoidal domain shapes and assumes the presence of dislocations and faults.
It is worth mentioning two more alternative approaches: the Debye scattering equation (Debye, 1915; for some applications, see, for example, Cervellino et al., 2003; Cozzoli et al., 2006) and the total scattering (TS) approach [also known as pair distribution function (PDF) analysis; Egami & Billinge, 2003; Billinge, 2008; see also Chapter 5.7 ]. Both techniques work in real space: the first creates the pattern directly from atomic positions and the other extracts the real-space information (the PDF) from the diffraction data. As the information content does not change on moving from reciprocal (measured) to direct space, provided that similar hypotheses are employed (similar microstructure and, if necessary, structure), real-space and reciprocal-space methods should give similar results. Of course there are differences, related to the way that the data are handled. For instance, it is easier to visualize the anisotropic effects in reciprocal space, as the information is contained in the peak broadening (different peaks show different breadths): in the PDF this information is sparser as it should be reflected in a variation in the correlation lengths, but also (highly integrated) in a variation of the decay of the curve. Conversely, information on the atomic arrangement (i.e. on possible defects) appears more clearly in the PDF, where variations in the distances and in coordination are well localized, contrary to the diffraction pattern, where the information is contained in the (weak) diffuse signal, in the broadening and in the peak position.
Direct-space and reciprocal-space methods thus each have their own advantages and disadvantages. Combined or comparative modelling, whenever possible, is therefore always the best solution: we can match the flexibility and immediacy of a real-space approach with the possibility of working directly with the measured data as typical in reciprocal-space methods.
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