Lattice sum for a hexagonal close-packed structure and its dependence on the c/a ratio of the hexagonal cell parameters.

Abstract

We continue the work by Lennard-Jones and Ingham, and later by Kane and Goeppert-Mayer, and present a general lattice sum formula for the hexagonal close packed (hcp) structure with different c/a ratios for the two lattice parameters a and c of the hexagonal unit cell. The lattice sum is expressed in terms of fast converging series of Bessel functions. This allows us to analytically examine the behavior of a Lennard-Jones potential as a function of the c/a ratio. In contrast to the hard-sphere model, where we have the ideal ratio of c/a=sqrt[8/3] with 12 kissing spheres around a central atom, we observe the occurrence of a slight symmetry-breaking effect and the appearance of a second metastable minimum for the (12,6) Lennard-Jones potential around the ratio c/a=2/3. We also show that the analytical continuation of the (n,m) Lennard-Jones potential to the domain n,m<3 such as the Kratzer potential (n=2,m=1) gives unphysical results.