Theta series and magic numbers for close-packed spherical clusters

In the face-centered cubic lattice each sphere is surrounded by a shell of 12 spheres, then shells of 6 and 24, the whole configuration forming a spherical cluster of 43 spheres. This paper presents a mathematical investigation into the possible numbers of spheres in such spherical clusters in the square and hexagonal lattices in two dimensions and the simple, face-centered and body-centered cubic lattices and the hexagonal close packing in three dimensions, for various choices for the center of the cluster. The theta series of these clusters are calculated and the resulting nuclearities (or magic numbers) are tabulated. Applications to cluster chemistry and physics, communication theory, and other fields are highlighted. In particular, the application of the present atom counting results to electron counting of close-packed high nuclearity metal clusters is illustrated with examples.