The nature of x-ray scattering from geo-nanoparticles: Practical considerations of the use of the Debye equation and the pair distribution function for structure analysis

Interpretation of elastic scattering from nanocrystals is critical to building and testing models of the short and medium range atomic arrangements in environmentally relevant nanoparticles. Along with information about atomic arrangements, the diffraction pattern arising from a collection of nanoparticles contains information on the grain size, shape and defect structure and cannot simply be treated as a broadened collection of Bragg peaks. The Debye equation, a sum over all pairs of atoms in the particle, calculates the scattering at each value of Q, rather than at discrete values of hkl. Especially when atomic arrangements are well known, modeling using the Debye equation allows the refinement of particle shape, allowing the investigation of dominant growth axes. We show the example of 6-line ferrihydrite, fitting the diffraction pattern using a disc of 2.3(1) nm in the ab plane and 6(2) nm along the c axis. A powerful and intuitive way to help examine the diffraction pattern is to take the Fourier transform of the normalized total scattering (Bragg and diffuse), giving the pair distribution function (PDF). The PDF represents a bond length distribution of the material weighted by the respective scattering powers of the contributing atoms. Using examples from the literature, we show that by analyzing the medium range correlations (~ 5–15 Å), the structure of a nanoparticle can be distinguished using PDFs generated from model structures.

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► Broad diffraction peaks from nanoparticles are not simply the Bragg peaks broadened.
► The Debye equation, a sum over all atoms in the sample, can be used to calculate the total scattering.
► The PDF provides a conceptually simple method of viewing the diffraction pattern.
► Short and medium range peaks in the PDF are not highly affected by particle shape.
► It is easier to compare the pattern of a nanoparticle with a reference using PDF.