Nanoparticles with Cubic Symmetry: Classification of Polyhedral Shapes

TL;DR

This paper classifies cubic-symmetry polyhedral shapes of nanoparticles using geometric parameters, providing a comprehensive phase diagram and detailed structural analysis to aid nanoparticle characterization and simulation.

Contribution

It introduces a systematic classification of cubic-symmetry polyhedra based on three key parameters, enabling complete structural description and interpretation of nanoparticle shapes.

Findings

Polyhedra are fully characterized by three structure parameters.

A phase diagram classifies all possible cubic-symmetry polyhedra.

Structural properties are detailed with analytical and numerical methods.

Abstract

The shape of crystalline nanoparticles (NP) can often be described by polyhedra with flat facet surfaces. Thus, structural studies of polyhedral bodies can help to describe geometric details of NPs. Here we consider compact polyhedra of cubic point symmetry Oh.as simple models. Their surfaces are described by facets with normal vectors along selected directions (a, b, c) together with their symmetry equivalents forming a direction family {abc}. For given {abc} this yields generic polyhedra with up to 48 facets where we focus on polyhedra with facets of {abc} = {100}, {110}, and {111}, suggested for metal NPs with cubic lattices. The resulting generic polyhedra, cubic, rhombohedral, and octahedral, can serve for the description of non-generic polyhedra as intersections of corresponding generic species. Their structural properties are shown to be fully determined by only three structure parameters, facet distances R100, R110, and R111 of three types of facets. This provides a phase diagram to completely classify the corresponding Oh symmetry polyhedra. Structural properties of all polyhedra, such as shape, size, and facet geometries, are discussed in analytical and numerical detail with visualization of characteristic examples. The results may be used for respective nanoparticle simulations but also as a repository assisting the interpretation of structures of real compact nanoparticles observed by experiment.