The Enumeration of Neighbors on Cubic and Hexagonal-Based Lattices

In discussing phenomena involving the interaction of ions in a crystalline lattice it is often necessary to know the radii and occupation numbers of near-lying shells of lattice sites. Such information is extremely important, for example, in the interpretation of donor-acceptor pair recombination spectra 1 ' 2 and in calculations of ion pairing 3,4 and other defect clustering phenomena. The present work was motivated by the apparent lack of any generally available tables or formulae for calculating these m-th neighbor shell parameters for common lattices. Shell radius formulae and partial tables have been published 1 ' 2 for the interpretation of pair spectra in materials with zincblende lattices but these tables are inadequate for other applications. Wood5 and FerrisPrabhu 0 have given slightly more complete treatments but do not present sufficiently general rules to allow indefinite extension of their tables.* The methods which will be described here differ from those * In fact, if the diamond lattice radius rules given bj' Ferris-Prabhu 0 were used to extend his table beyond the 25 shells which he lists, one would err in predicting the radius of the 28th shell and would have all higher shells improperly numbered. F u r t h e r errors would be made for much higher shell numbers. 355