Mean-field description of a dilute <f>s</f>–<f>d</f> system

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the <f>s</f>–<f>d</f> Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a mean field and electrons in a field of barriers and wells localized at the impurity sites, are almost equal if the impurity concentration is sufficiently small or <f>s</f>–<f>d</f> coupling sufficiently weak. [Copyright &y& Elsevier]