A system contains N~ sites and N~ electrons. At a given site there is only one accessible orbital state, but that orbital state can be occupied by zero, one, or two electrons (of opposite spin). The site energy is zero if the site is either empty or singly occupied, and it is ε if the site is doubly occupied. In addition there is an externally applied magnetic field which acts only on the spin coordinates.
a) Calculate the chemical potential μ as a function of the temperature and the magnetic field.
b) Calculate the heat capacity of the system.
c) Calculate the initial magnetic susceptibility of the system (i.e., the magnetic susceptibility in small magnetic field).