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.

The atomic-scale response of dielectrics/ferroelectrics to electric fields is central to their functionality. Here we introduce an in situ characterization method that reveals changes in the local atomic structure in polycrystalline materials under fields. The method employs atomic pair distribution functions (PDFs), determined from X-ray total scattering that depends on orientation relative to the applied field, to probe structural changes over length scales from sub-Ångstrom to several nanometres. The PDF is sensitive to local ionic displacements and their short-range order, a key uniqueness relative to other techniques. The method is applied to representative ferroelectrics, BaTiO3 and Na½Bi½TiO3, and dielectric SrTiO3. For Na½Bi½TiO3, the results reveal an abrupt field-induced monoclinic to rhombohedral phase transition, accompanied by ordering of the local Bi displacements and reorientation of the nanoscale ferroelectric domains. For BaTiO3 and SrTiO3, the local/nanoscale structural changes observed in the PDFs are dominated by piezoelectric lattice strain and ionic polarizability, respectively.