Magical Mathematical Formulas for Nanoboxes.

Hollow nanostructures are at the forefront of many scientific endeavors. These consist of nanoboxes, nanocages, nanoframes, and nanotubes. We examine the mathematics of atomic coordination in nanoboxes. Such structures consist of a hollow box with n shells and t outer layers. The magical formulas we derive depend on both n and t. We find that nanoboxes with t = 2  or  3, or walls with only a few layers generally have bulk coordinated atoms. The benefits of low-coordination in nanostructures is shown to only occur when the wall thickness is much thinner than normally synthesized. The case where t = 1 is unique, and has distinct magic formulas. Such low-coordinated nanoboxes are of interest for a myriad variety of applications, including batteries, fuel cells, plasmonic, catalytic and biomedical uses. Given these formulas, it is possible to determine the surface dispersion of the nanoboxes. We expect these formulas to be useful in understanding how the atomic coordination varies with n and t within a nanobox.