Statistical physics and magnetization

[PhoenixWright]

Homework Statement

Consider a system of three aligned spins with S=1/2. There are couplings between first neighbors. Each spin has a magnetic moment $\vec{\mu} = s \mu \vec{S}$. The system is in a field $H= H\vec{u_z}$ at thermal equilibrium. The hamiltonian is:
$H=J[S(1)S(2)+S(2)S(3)] -2\mu H[S_z(1) + S_z(2)+S_z(3)]$
So, we want to calculate:
a) Possible microscopic states and their energy.
b) Internal energy U(T,H) and entropy S(T,H), for 1) T=0, H=0; 2) H=0, J <<kT
c) Partition function in closed form
d) Magnetization M(T,H).

Homework Equations

Below:

The Attempt at a Solution

I have been searching, and I think this is a 1-dimensional problem of the Ising model, but I am not sure, and I have no idea how I should start.
I know that the microscopic states of the system are 8, because there are two spin orientations (so, 2x2x2=8). But, how can I calculate the partition function?
Actually, they ask me about the internal energy and the entropy before the partition function, but all the formulas I know are:
$U=-\frac{\partial \ln Z}{\partial{\beta}}$

$S=k_B(\ln Z + \beta U)$
so I guess I need the partition function before, right? Or is there another way to obtain the entropy and the internal energy?
Thank you.

 

[Charles Link]

To calculate the partition function, you compute the energies of each of the individual spin states, and you compute $Z=\sum\limits_{i} e^{-E_i/(k_b T)}$ There are 8 possible states.

 

[PhoenixWright]

To calculate the partition function, you compute the energies of each of the individual spin states, and you compute $Z=\sum\limits_{i} e^{-E_i/(k_b T)}$ There are 8 possible states.

Thank you!