A one-dimensional, spin-one model of magnetism

Consider a one-dimensional macroscopic model with spin-one quantum mechanical magnetic moments located on each of N sites. The Hamiltonian is

where each σj takes on the values -1, 0, or +1, J is an exchange interaction parameter, h is a magnetic field, and D is a parameter representing a ‘crystal field'. The entire system is in contact with a thermal reservoir at temperature T.

1. Write down a formal expression for the partition function of this system. You do not have to evaluate any sums or integrals that appear in your expression for this part of the problem.

2. For general values of the parameters, derive a formal expression for the quantity

in terms of a derivative of the partition function with respect to an appropriate parameter.

3. For J = 0, but arbitrary values of h and D, calculate the partition function for this system.

4. For J = 0, but arbitrary values of h and D, calculate the free energy of this system.

5. For J = 0 and h = 0, calculate the entropy of this system.