# Statistical mechanics: Particles with spin

1. Mar 5, 2012

### SoggyBottoms

1. The problem statement, all variables and given/known data

We have N particles, each of which can either be spin-up ($s_i = 1$) or spin-down ($s_i = -1$) with $i = 1, 2, 3....N$. The particles are in fixed position, don't interact and because they are in a magnetic field with strength B, the energy of the system is given by:

$$E(s_1, ...., s_n) = -mB \sum_{i=1}^{N} s_i$$

with m > 0 the magnetic moment of the particles. The temperature is T.

a) Calculate the canonic partition function for N = 1 and the chance that this particle is in spin-up state $P_+$.

b) For any N, calculate the number of microstates $\Omega(N)$, the Helmholtz free energy F(N,T) and the average energy per particle U(N, T)/N

3. The attempt at a solution

a) $$Z_1 = e^{-\beta m B} + e^{\beta m B} = 2 \cosh{\beta m B}$$
$$P_+ = \frac{e^{-\beta m B}}{2 \cosh{\beta m B}}$$

b) The number of possible microstates is $\Omega(N) = 2^N$, correct?

I know that $U = -\frac{\partial \ln Z}{\partial \beta}$, but I'm not sure how to calculate Z here.

Last edited: Mar 6, 2012
2. Mar 6, 2012

### Liquidxlax

leave Z as the summation Z = Ʃ e-Eiβ where β = 1/KBT

so ∂ln(Z)/dβ = (1/Z)(∂Z/∂β) = [-EiƩe-Eiβ]/Z

i think b) is supposed to be (Z1)N sorry yeah your b) is right