Enumeration of closest-packings by the space group: a simple approach

Abstract

Abstract The problem of enumerating the possible closest-packings of equal spheres having a period P for each of the possible space groups is solved by systematic exploitation of the properties of the two-color symmetry group of the cyclotomic representation of the Zhdanov symbol. No sophisticated combinatorial or group-theoretical techniques are required, and for some space groups, non-recursive, simple analytical expressions are obtained through the use of Möbius Inversion Formula. A corollary to Möbius theorem, used in the derivation, has been proved.