Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.6, pp. 296-298

Section Assembling the equations into a peak and modelling the data

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Assembling the equations into a peak and modelling the data

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As previously mentioned, the broadening contributions briefly illustrated in the previous sections are employed to generate the powder peak profile for reflections from the set of planes {hkl} using equations (3.6.11)[link] and (3.6.12)[link] and where[\eqalignno {&I_{hkl}(s) \cr&\quad= k(d^*)\textstyle\int\limits_{-\infty}^\infty C(L)\exp (2\pi iLs)\, {\rm d}L \cr &\quad= k(s)\textstyle \int\limits_{-\infty}^\infty T_{\rm pV}^{\rm IP}(L)A_{hkl}^S(L)[A_{hkl}^D(L)\cos (2\pi Ls) + iB_{hkl}^D(L)\sin (2\pi Ls)] &\cr &\quad\quad\times \ldots \times [A_{hkl}^F(L)\cos (2\pi Ls) + iB_{hkl}^F(L)\sin (2\pi Ls)]\,{\rm d}L. &\cr &&(3.6.52)}]Equation (3.6.52)[link] represents an asymmetrical peak profile (the asymmetry is given by the sine terms). Fast Fourier transform and space remapping (usually s to 2θ) are then employed to generate the peaks in the measurement space; intensities are multiplied by the Lorentz and, if needed, polarization terms, and a background is added to the whole pattern. Other aberrations (affecting the position, the intensity or the shape of the peak) can be included as needed.

The various reflections are then positioned on the basis of the (reference) Bragg angle 2θB calculated from the unit-cell parameters, and a background (for example, a Chebychev polynomial) is suitably added. For completeness, thermal diffuse scattering should be included, as it can contribute to the broadening near the peak tails (see, for example, Beyerlein et al., 2012[link]). In addition, small-angle scattering can be considered to improve the WPPM result and to account for the observed increase in the background at low angle (Scardi et al., 2011[link]). The final equation is thus similar to that of the Rietveld (1969[link]) or the Pawley (1981[link]) methods,[I(2\theta) = {\rm SAXS} + {\rm TDS} + {\rm bkg} + k(2\theta)LP\textstyle\sum\limits_{hkl} I_{\{hkl \}}(x), \eqno (3.6.53)]the main difference being in the focus of the analysis and in the way that the profiles are generated.

The model parameters are then refined using a nonlinear least-squares routine (e.g. based on the Marquardt algorithm or suitable modifications, as proposed, for example, by Coelho, 2005[link]) to directly match the synthesized pattern to the experimental data. The usual weight, related to Poisson counting statistics, is employed.

As the shape of each peak is bound to the underlying physical models, the number of parameters to be refined is usually quite limited. Compared with the four parameters per peak (intensity, width, shape and position) necessary for a Scherrer-type analysis, in WPPM we refine, for example, two parameters for a domain-size distribution, three parameters for dislocations (ρ, [R_e'] and ϕ), two parameters for faulting (α and β), at most six lattice parameters, a few background parameters (e.g. four parameters) and one parameter (intensity) per peak. In addition, we can also refine some further specimen-related parameters such as a misalignment error. No atomic coordinates are involved.

A flexible software package implementing WPPM (PM2K: Leoni et al., 2006[link]) is available from the author on request ( the software includes all of the broadening models illustrated here. The user can work with any type and any simultaneous set of diffraction data (X-ray, neutrons or electrons) and build their own model with no a priori restriction on the quantity and type of parameters, the number of phases, the models or the relationships between the parameters. The WPPM method has also been implemented in the TOPAS refinement software [version 5 (Coelho, 2009[link]; Bruker, 2009[link])] using the flexible macro language provided.1 Alternative approaches

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Convolutional multiple whole profile fitting (CMWP; Ribárik et al., 2004[link]) and extended convolutional multiple whole profile fitting (eCMWP; Balogh et al., 2006[link]) have been developed to solve the same problem.

CMWP, introduced as a convolutive version of multiple whole profile fitting (MWP; Ungár et al., 2001[link]; Ribárik, 2008[link]), is very similar to the WPPM. The notable differences are:

  • (i) The instrumental profile is employed directly without interpolation and the profile of the instrumental peak closest to the peak under analysis is used. The instrumental profile imposes conditions on the range of L used in profile modelling.

  • (ii) In CMWP a subset of data points is used for speed.

  • (iii) The background is given by the user (as a spline or Legendre polynomial).

  • (iv) All points of a given (generated) peak are weighted by the same value related to the maximum intensity. However, the correct weighting scheme with individual weights for data points is available as an option.

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[link]), (e)CMWP introduces an interesting model for faults based on a parameterization of the profiles simulated with the DIFFaX software (Treacy et al., 1991[link]). 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[link]; for some applications, see, for example, Cervellino et al., 2003[link]; Cozzoli et al., 2006[link]) and the total scattering (TS) approach [also known as pair distribution function (PDF) analysis; Egami & Billinge, 2003[link]; Billinge, 2008[link]; see also Chapter 5.7[link] ]. 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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