Low–dimensional lattices. VII. Coordination sequences

Abstract

The coordination sequence {S(n)} of a lattice or net gives the number of nodes that are n bonds away from a given node. S(1) is the familiar coordination number. Extending the work of O'Keeffe and others, we give explicit formulae for the coordination sequences of the root lattices A_d, D_d, E₆, E₇, E₈ and their duals. Proofs are given for many of the formulae and for the fact that, in every case, S(n) is a polynomial in n, although some of the individual formulae are conjectural. In the majority of cases, the set of nodes that are at most $n$ bonds away from a given node form a polytopal cluster whose shape is the same as that of the contact polytope for the lattice. It is also shown that among all the Barlow packings in three dimensions the hexagonal close packing has the greatest coordination sequence, and the face–centred cubic lattice the smallest, as conjectured by O'Keeffe.