The enumeration of neighbors on cubic and hexagonal-based lattices

Radii and occupation numbers have been calculated for the first 50 shells of neighbors on each atomic sublattice for the CsCl, NaCl, zincblende, wurtzite, and CaF2 binary lattices. We present the results in tabular form along with rules for extending the tables to higher shell numbers. A sublattice approach is used and tables are given for key cubic and hexagonal-based sublattices. The generality of the sublattice approach is such as to allow easy application of the tables to more complex lattice structures or to such problems as enumeration of preferred interstitial sites. A number-theoretic explanation is offered for previously observed difficulties in obtaining a simple expression for the radius of the m-th shell in cubic-based structures.