# Crystal Lattice of Spin-1 Particles: Chemical Potential?

1. Jun 13, 2014

### akoe

1. The problem statement, all variables and given/known data
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 +μZ, 0, and -μZ. In an external magnetic field B, each spin can have an energy U = -μZB, so the possible energies are +μZB, 0, and -μZB. 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 μZ, 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.

2. Relevant equations
Z = ∑e-[E(s)-μN(s)]/kT

3. 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

2. Jun 13, 2014

### Oxvillian

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

3. Jun 14, 2014