A perturbation expansion for the free energy of the s-d exchange Hamiltonian

The first- and second-order terms of a perturbation expansion of free-energy density of the s-d exchange Hamiltonian with random impurity positions, in the presence and absence of an external magnetic field B, are evaluated. The reduced s-d Hamiltonian Hr(B) is chosen for the Hamiltonian of the unperturbed system. For weak fields B, the expansion takes different forms at temperatures T> Tc and T < Tc, Tc being the temperature at which Hr(B) exhibits a second-order phase transition.