FCC versus HCP via parametric density

We consider the asymptotic behavior of finite sphere packings in the face centered cubic lattice (fcc), the hexagonal closest packing (hcp) and related periodic structures called Barlow packings. We use concepts as the parametric density and the density deviation and compare packings in different periodic structures. We prove that for any structure M there is a range for the parameter ϱ such that the regular octahedron in fcc is asymptotically ϱ-denser than any polytope in M. This result has a physical and a mathematical aspect: (a) Most of the noble gases cristallize in fcc. So it is a model for this physical fact. The Lennard–Jones potential does not reflect this fact. (b) Further, it shows that large sphere packings of the critical lattice converge faster to the density δL=π/18 than other periodic sphere packings. So it is a finite contribution to the Kepler problem.