  We study packings of hard spheres on lattices. The partition function, and
therefore the pressure, may be written solely in terms of the accessible free
volume, i.e. the volume of space that a sphere can explore without touching
another sphere. We compute these free volumes using a leaky cell model, in
which the accessible space accounts for the possibility that spheres may escape
from the local cage of lattice neighbors. We describe how elementary geometry
may be used to calculate the free volume exactly for this leaky cell model in
two- and three-dimensional lattice packings and compare the results to the
well-known Carnahan-Starling and Percus-Yevick liquid models. We provide
formulas for the free volumes of various lattices and use the common tangent
construction to identify several phase transitions between them in the leaky
cell regime, indicating the possibility of coexistence in crystalline
materials.
