Crystal Lattice of Spin-1 Particles: Chemical Potential?

[akoe]

Homework Statement

A crystal lattice consists of a spin 1 particle at each lattice point. Spin 1 particles can have z-components of magnetic moment that take on the values +μ_Z, 0, and -μ_Z. In an external magnetic field B, each spin can have an energy U = -μ_ZB, so the possible energies are +μ_ZB, 0, and -μ_ZB. We place the crystal lattice so that it is in thermal equilibrium with a reservoir at temperature T as well as in an external magnetic field B. Your answers should be in terms of any of μ_Z, B, T, N, and fundamental constants.
a. Write down the single particle partition function for one of these spin 1 particles located at a lattice point.
b. Determine the probability of finding the particle in the "up" state.
c. The system consists of N such spin 1 particles. Write down the total partition function for these N spin 1 particles. Let's assume that the particles only interact with the external magnetic field and the thermal reservoir.

Homework Equations

Z = ∑e^{-[E(s)-μN(s)]/kT}

The Attempt at a Solution

I think that I could do the rest of the problem if I could figure out how to calculate the chemical potential for this particular set up, and then calculate the partition function from there. I thought about trying to calculate it by N = ∑\frac{1}{e^{(ε-μ)/kT}-1}, but I don't think that can be right for part A, because we aren't even looking at the N particles yet.

Thanks

 

[Oxvillian]

Hello akoe - these are localized particles, so you can just use Boltzmann statistics.

 

[bloby]

I think canonical formalism is more appropriate here than grand canonical. Then you don't have to bother with adjusting the chemical potential.

http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)