International
Tables for
Crystallography
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. 297-298

Section 3.6.2.7.1. Alternative approaches

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Matteo.Leoni@unitn.it

3.6.2.7.1. Alternative approaches

| top | pdf |

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.

References

Balogh, L., Ribárik, G. & Ungár, T. (2006). Stacking faults and twin boundaries in fcc crystals determined by X-ray diffraction profile analysis. J. Appl. Phys. 100, 023512.Google Scholar
Billinge, S. J. L. (2008). Local structure from total scattering and atomic pair distribution function (PDF) analysis. In Powder Diffraction: Theory and Practice, edited by R. E. Dinnebier & S. J. L. Billinge. London: Royal Society of Chemistry.Google Scholar
Cervellino, A., Giannini, C. & Guagliardi, A. (2003). Determination of nanoparticle structure type, size and strain distribution from X-ray data for monatomic f.c.c.-derived non-crystallographic nanoclusters. J. Appl. Cryst. 36, 1148–1158.Google Scholar
Cozzoli, P. D., Snoeck, E., Garcia, M. A., Giannini, C., Guagliardi, A., Cervellino, A., Gozzo, F., Hernando, A., Achterhold, K., Ciobanu, N., Parak, F. G., Cingolani, R. & Manna, L. (2006). Colloidal synthesis and characterization of tetrapod-shaped magnetic nanocrystals. Nano Lett. 6, 1966–1972.Google Scholar
Debye, P. (1915). Zerstreuung von Röntgenstrahlen. Ann. Phys. 351, 809–823.Google Scholar
Egami, T. & Billinge, S. J. L. (2003). Underneath the Bragg Peaks. Structural Analysis of Complex Materials. Oxford: Elsevier.Google Scholar
Ribárik, G. (2008). Modeling of Diffraction Patterns Properties. PhD thesis, Eötvös University, Budapest.Google Scholar
Ribárik, G., Gubicza, J. & Ungár, T. (2004). Correlation between strength and microstructure of ball-milled Al–Mg alloys determined by X-ray diffraction. Mater. Sci. Eng. A Struct. Mater. 387–389, 343–347.Google Scholar
Treacy, M. M. J., Newsam, J. M. & Deem, M. W. (1991). A general recursion method for calculating diffracted intensities from crystals containing planar faults. Proc. R. Soc. Lond. Ser. A, 433, 499–520.Google Scholar
Ungár, T., Gubicza, J., Ribárik, G. & Borbély, A. (2001). Crystallite size distribution and dislocation structure determined by diffraction profile analysis: principles and practical application to cubic and hexagonal crystals. J. Appl. Cryst. 34, 298–310.Google Scholar








































to end of page
to top of page