Nanoparticles with cubic symmetry: classification of polyhedral shapes

Structural studies of polyhedral bodies can help to analyze geometric details of observed crystalline nanoparticles (NP) where we consider compact polyhedra of cubic point symmetry as simple models. Their surfaces are described by facets with normal vectors along selected Cartesian directions (a, b, c) together with their symmetry equivalents forming a direction family {abc}. Here we focus on polyhedra with facets of families {100}, {110}, and {111}, suggested for metal and oxide NPs with cubic lattices. Resulting generic polyhedra, cubic, rhombohedral, octahedral, and tetrahexahedral, have been observed as NP shapes by electron microscopy. They can serve for a complete description of non-generic polyhedra as intersections of corresponding generic species, not studied by experiment so far. Their structural properties are shown to be fully determined by only three parameters, facet distancesR₁₀₀,R₁₁₀, andR₁₁₁of the three facet types. This provides a novel phase diagram to systematically classify all corresponding polyhedra. Their structural properties, such as shape, size, and facet geometry, are discussed in analytical and numerical detail with visualization of typical examples. The results may be used for respective NP simulations but also as a repository stimulating the structural interpretation of new NP shapes to be observed by experiment.