Free-Energy Shift of Conduction Electrons Due to the s-d Exchange Interaction

The free-energy shift of the conduction electrons due to the exchange interaction with a localized impurity spin is calculated by perturbation expansion up to the fourth order of the exchange interaction. It is shown that the usual procedure to obtain the free-energy shift at finite temperature by first calculating the energy shift at absolute zero and then replacing the step function by the fermi distribution function involves a serious error. By a careful analysis of this point, we find a logarithmic term in the fourth order. The contribution to the entropy of this term is which is negative for both signs of J. This term represents a decrease of the engropy of the conduction electrons and may be interpreted in terms of the correlated motion of the spin polarization coupled to the localized spin. This result seems to indicate that the binding energy of the ground state in the case of negative exchange interaction is obtained by calculating the free energy at finite temperature (to infinite order) and then extrapolating the result to zero temperature.