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I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) + S_z(2)S_z(3)] - 2\mu B[S_z(1)+S_z (2) + S_z (3)]$$ where $B$ is a magnetic field in the $z$ axis direction. This hamiltonian is given in an exercise.

I have to obtain the microscopic states of the system: $2\times2\times2 =8$, because we have $2$ possible spin orientations ($-1/2$ and $1/2$).

Now, how can I calculate the partition function? Is it the sum of $e^{-\beta E}$ for all the possible values of $E$?

I am also asked to calculate the entropy of the system and the internal energy for $B=0$. Do I have to use the expressions for $U$ and $S$ given by: \begin{align*} U&=-\frac{\partial \ln Z}{\partial{\beta}}, \\ S&=k_B(\ln Z + \beta U) . \end{align*} Can I calculate $U$ and $S$ without $Z$? Because in the exercise I am asked to find $Z$ after calculating $U$ and $S$. I don´t know if it is possible.

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  • $\begingroup$ Thank you! You are rigth, my hamiltonian had a mistake. I have already edit the question. I think I have a Ising model, but with spins taking values $\pm 1/2$. $\endgroup$ – Eva Martin Dec 6 '18 at 15:20
  • $\begingroup$ Good. I will delete my original comment (and in a short while, this one too) so as to keep things tidy. $\endgroup$ – user197851 Dec 6 '18 at 15:21
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As you realize, we shouldn't give complete answers to homework-and-exercises questions, but you seem to be asking simply for confirmation that your method is basically correct.

The partition function is the sum of Boltzmann factors for all the microstates. In this case, you have already identified the $8$ microstates, so you can just calculate $Z$ by directly summing up all $8$ terms. You will see that some of them have the same energy, which makes the final expression simpler.

Although there are various alternative formulae for $U$ (as the average energy, expressed as a sum of $p_i E_i $ terms, where $p_i$ is the probability of each microstate) and $S$ (related to a sum of $p_i \ln p_i$ terms), they still involve knowledge of $Z$ (since $p_i=\exp(-\beta E_i)/Z$), and they are equivalent to the formulae you already gave. So I think there is no way around evaluating $Z$ first.

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