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A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements
Author(s) -
Usher Tedi-Marie,
Olds Daniel,
Liu Jue,
Page Katharine
Publication year - 2018
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273318004977
Subject(s) - ellipsoid , polyhedron , function (biology) , spheres , anisotropy , embedding , algorithm , computer science , statistical physics , mathematics , materials science , geometry , physics , artificial intelligence , optics , astronomy , evolutionary biology , biology
In the structural refinement of nanoparticles, discrete atomistic modeling can be used for small nanocrystals (< 15 nm), but becomes computationally unfeasible at larger sizes, where instead unit‐cell‐based small‐box modeling is usually employed. However, the effect of the nanocrystal's shape is often ignored or accounted for with a spherical model regardless of the actual shape due to the complexities of solving and implementing accurate shape effects. Recent advancements have provided a way to determine the shape function directly from a pair distribution function calculated from a discrete atomistic model of any given shape, including both regular polyhedra ( e.g. cubes, spheres, octahedra) and anisotropic shapes ( e.g. rods, discs, ellipsoids) [Olds et al. (2015). J. Appl. Cryst. 48 , 1651–1659], although this approach is still limited to small size regimes due to computational demands. In order to accurately account for the effects of nanoparticle size and shape in small‐box refinements, a numerical or analytical description is needed. This article presents a methodology to derive numerical approximations of nanoparticle shape functions by fitting to a training set of known shape functions; the numerical approximations can then be employed on larger sizes yielding a more accurate and physically meaningful refined nanoparticle size. The method is demonstrated on a series of simulated and real data sets, and a table of pre‐calculated shape function expressions for a selection of common shapes is provided.