# 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).

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.

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Homework Helper
Gold Member
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
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!

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