Cluster Approximation for Ferromagnets with First- and Second-Neighbor Exchange, with Application to the Europium Chalcogenides

The two-particle cluster approximation of Strieb, Callen, and Horwitz is extended to ferromagnets and paramagnets with nearest-and next-nearest-neighbor exchange, with particular reference to EuO, EuS, EuSe, and EuTe. Curie temperatures, magnetization curves for general values of the applied field, susceptibility, spin-correlation functions, energy, and specific-heat curves are calculated. Using the values of the exchange constants of EuS obtained from low-temperature spin-wave analysis (and no adjustable constants) we find excellent agreement with the observed Curie temperature, very close agreement with the observed "paramagnetic Curie temperature," and very good agreement with the $\lambda$-like discontinuity in the specific heat near the Curie temperature. The magnetization data for arbitrary applied fields and the susceptibility data in the paramagnetic region also can be fit closely with these values of the exchange constants. To explore the sensitivity of the theory to the exchange constants, the results of the cluster approximation are given for various ratios of the first- and second-neighbor exchange constants, including the special case of vanishing second-neighbor exchange (in this latter case the approximation generalizes the constant-coupling approximation to arbitrary spin). We find that all the presently measured magnetic properties are rather insensitive to the particular choice of exchange constants, provided that these are chosen to be consistent with the observed Curie temperature; measurement of the spin-correlation functions, however, would provide a sensitive criterion for the choice of exchange constants.