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. 290-291
Section 3.6.2.5. Whole powder pattern modelling (WPPM)^{a}Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy |
The techniques briefly illustrated in the previous section, as well as other alternatives appearing in the literature before the beginning of this century, lack full completeness for quantitative microstructure analysis. The whole powder pattern modelling method attempted to fill this gap. Starting with the same ideas as in Section 3.6.2.1 (i.e. peaks as convolution), it uses equations (3.6.5) and (3.6.6) to generate the peaks within a fully convolutional approach. The peak profiles are therefore generated from the Fourier transform of each broadening component; the resulting h(x) function accounts just for the shape of the profile, which in turn can be represented as (Scardi & Leoni, 2002)where includes all constant or known functions of (e.g. structure factor, Lorentz–polarization factor etc.), whereas is the Fourier transform of the peak profile. The term k is a function of ; it is not necessarily a function of s, as the peak is actually centred in .
Equation (3.6.11) assumes that the broadening sources act on the entire family of symmetry-equivalent reflections {hkl} and therefore that a multiplicity term [included in ] can be used: however, certain types of defects (e.g. faults) can act independently on each of the symmetry-equivalent reflections. Equation (3.6.11) then becomes more correctlywhere w_{hkl} is a weight function depending on the lattice symmetry and actual broadening source, s_{hkl} = = s − δ_{hkl} is the distance, in reciprocal space, from the centroid of the peak hkl, and δ_{hkl} is the shift from , the Bragg position in the absence of defects. The sum is over independent profile subcomponents selected on the basis of the specific defects (e.g. two for the {111} family in f.c.c. when faults are present; selection is based on the value of |L_{0}| = |h + k + l|, i.e. 3 or 1).
According to equation (3.6.5), the function is the product of the Fourier transforms of the broadening contributions. In a real material, broadening is mostly due to the specific nature of the instrument, to the finite size of the coherently diffracting domain (size effect) and to the presence of defects such as, for example, dislocations and faults (Cheary & Coelho, 1992; van Berkum, 1994; Scardi & Leoni, 2002). Taking these into account,where T^{IP}(L) and are the FTs of the instrumental and domain-size components, respectively, whereas the terms in angle brackets <> are average phase factors related to lattice distortions (ψ_{hkl}) and faulting (ϕ_{hkl}).
Equation (3.6.13) is the core of the WPPM method: as indicated by the ellipsis, any other broadening source can easily be considered by including the corresponding (complex) Fourier transform (i.e. the corresponding average phase factor) in equation (3.6.13). Expressions are known for several cases of practical interest (see, for example, Scardi & Leoni, 2002, 2004, 2005; Leoni & Scardi, 2004; Leineweber & Mittemeijer, 2004; van Berkum, 1994; Cheary & Coelho, 1992).
The approach is strictly valid when the broadening sources can be considered as diluted and independent (i.e. uncorrelated defects). If this does not apply, then cross-terms should be considered and the whole approach revised. In fact, here we assume that the structure factor can be factored and the lattice is fully periodic in three dimensions: under these conditions, structure (peak intensity) and microstructure (peak shape) can be decoupled as the peak positions can be determined in a straightforward way. Extended defects (e.g. faults) cause the appearance of diffuse effects and the displacement of the Bragg peaks: in order to calculate the diffraction pattern, the structure and the microstructure must be simultaneously known (see, for example, Drits & Tchoubar, 1990).
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