100 years of Debye's scattering equation

Abstract

Debye's scattering equation (DSE) has spanned a century of scientific development, from the dawn of quantum mechanics and the investigation of the structure of atoms and molecules to the era of nanotechnology, paving the way tototal scatteringmethods. The formulation offers the most accurate representation of the intensity scattered by randomly oriented atomic aggregates, constructed by superimposing the signal from each atomic distance in the molecule. The present paper reviews some of the milestone applications, from the interpretation of the intensity curves from gases and vapours, to aggregates of increasing size and more extended order. Important developments, aimed at mitigating the prohibitive computational complexity of the DSE, and state-of-the-art methods for the characterization of static and dynamic displacements are also discussed.

Structure analysis of supported disordered molybdenum oxides using pair distribution function analysis and automated cluster modelling

“…While heptamolybdate and the two octamolybdate isomers have similar local structure motifs in the clusters, the relations between the individual cluster units in the crystal structures vary significantly. This is done by using the Debye scattering equations to calculate the PDFs (Gelisio & Scardi, 2016) from atomic coordinates applying the Diffpy-CMI program package . 5(d).…”

“…To simplify the models, we therefore attempt to treat the clusters as finite structural models, without assuming periodicity or symmetry. This is done by using the Debye scattering equations to calculate the PDFs (Gelisio & Scardi, 2016) from atomic coordinates applying the Diffpy-CMI program package . The xyz coordinates of the cluster models used can be found online (Lindahl Christiansen et al, 2019).…”

Diffraction line profiles of spherical hollow nanocrystals

“…For small crystal structures, the calculation of the intensity I ( s ), with the scattering vector s = 2 π ⋅ k , can be also performed considering direct crystalline space, using the Debye scattering equation 20 –22…”

Extending the Debye scattering equation for diffraction from a cylindrically averaged group of atoms: detecting molecular orientation at an interface

“…Gases, amorphous solids and polycrystalline powders are approximate realizations of these requirements. The Debye equation is now over 100 years old and its long history of applications in crystallography has been celebrated in recent work [see for example Gelisio & Scardi (2016), Scardi et al (2016)]. It has been useful for predicting the scattered intensity of simulated isotropic polycrystalline or amorphous structures derived from empirical interaction potentials or from density functional theory (Billinge & Levin, 2007;Cervellino et al, 2003;Leonardi et al, 2013;Leonardi & Bish, 2016).…”