A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements

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.