# Partition function for hard spheres on a lattice

1. Mar 4, 2013

### Yoran91

Hi everyone,

I'm reading some lecture notes on statistical physics and thermodynamics and I'm stuck at an expression for a partition function which I really don't understand.

The chapter is on mean field theory and the discussion is about hard spheres on a lattice. The interaction of the hard sphere is $\beta U = \infty$ if $r<\sigma$ and $\beta U = 0$ if $r>\sigma$ as usual, where $\sigma$ is the diameter of the spheres.

Now it's said that a single hard sphere is treated exactly and the other spheres are located at their 'ideal' lattice positions. This supposedly leads to $Z_N=\prod_{i=1}^N V_i$ where $V_i$ is the free volume in which the center of mass of particle i can move.

I really don't see this. I'm expecting an $h^3$ or $\Lambda ^3$ to appear somewhere, obtained by integrating over the momenta of such a sphere, but it isn't there. Why is this the partition function?