# One-dimensional Ising Model in a three spin chain

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.

• 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 $\pm 1/2$. – Eva Martin Dec 6 '18 at 15:20
• Good. I will delete my original comment (and in a short while, this one too) so as to keep things tidy. – user197851 Dec 6 '18 at 15:21

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.