Collective Excitations and Magnetic Ordering in Materials with Singlet Crystal-Field Ground State

The low-lying magnetic excited states ("Frenkel spin excitons") in rare-earth compounds with a singlet crystal-field ground state are studied. We consider the effect of exchange interaction between the ionic moments as it increases from zero towards the critical value necessary for magnetic ordering at zero temperature. For simplicity, we consider the two-level system where the first-excited crystal-field state is also a singlet. By employing a pseudospin formalism, we avoid many of the fundamental difficulties encountered in previous treatments which introduce fermion operators for each of the crystal-field states. Applying techniques similar to those used in treating standard spin problems such as the Heinsenberg ferromagnet, we obtain an improved collective-excitation spectrum with a gap which decreases as the exchange increases. The critical value of exchange for magnetic ordering found is substantially greater than that obtained from molecular-field theory, and also is somewhat greater than that found with the constant-coupling approximation.