Abstract
A fast and exact algorithm to calculate the powder pair distribution function (PDF) for the case of periodic structures is presented. The new algorithm calculates the PDF by a detour via reciprocal space. The calculated normalized total powder diffraction pattern is transferred into the PDF via the sine Fourier transform. The calculation of the PDF via the powder pattern avoids the conventional simplification of X-ray and electron atomic form factors. It is thus exact for these types of radiation, as is the conventional calculation for the case of neutron diffraction. The new algorithm further improves the calculation speed. Additional advantages are the improved detection of errors in the primary data, the handling of preferred orientation, the ease of treatment of magnetic scattering and a large improvement to accommodate more complex instrumental resolution functions.
Highlights
The powder pair distribution function (PDF) is commonly used to characterize the local structure of a wide range of materials like disordered crystalline matter, nanoparticles, and amorphous materials including glasses and liquids (Egami & Billinge, 2012; Young & Goodwin, 2011; Playford et al, 2014; Mancini & Malavasi, 2015)
The calculated normalized total powder diffraction pattern is transferred into the PDF via the sine Fourier transform
As a means of overcoming the huge computational effort in the case of diffuse scattering that is continuously distributed in reciprocal space, we have developed a technique to calculate the PDF via the Debye scattering equation (Debye, 1915)
Summary
The powder pair distribution function (PDF) is commonly used to characterize the local structure of a wide range of materials like disordered crystalline matter, nanoparticles, and amorphous materials including glasses and liquids (Egami & Billinge, 2012; Young & Goodwin, 2011; Playford et al, 2014; Mancini & Malavasi, 2015). The PDF is obtained from a powder diffraction experiment after suitable normalization, division by the average atomic form factor, and correction for background and further experimental aspects. The model PDF is determined from a structural model by summing all interatomic distances This algorithm is used in common analysis programs such as DISCUS (Proffen & Neder, 1997; Neder & Proffen, 2008), PDFgui (Farrow et al, 2007), RMCprofile (Tucker et al, 2007) and TOPAS (Bruker, 2015; Coelho, 2018). The PDF is calculated by a detour via a calculation of the powder pattern As this calculation correctly takes the Q dependence of the atomic form factors into account, it is an exact calculation of the powder PDF for neutron, X-ray and electron diffraction. The application of the algorithm and modifications of it with respect to arbitrarily shaped finite-sized nanoparticles will be presented in a forthcoming publication
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