Excitations in Magnetic Systems with a Singlet Ground State. II

The pseudospin formalism developed in a previous work to study the magnetic properties of the singlet-triplet system has been extended to the present case where the crystal-field-only levels are two singlets and one doublet, again in the presence of exchange interactions. The lowest-four crystal-field levels of the ${\mathrm{Pr}}^{+}$ in ${\mathrm{Pr}}_3$Tl and the hexagonal sites of double hcp Pr are believed to have such a level structure. Using a simplified hybrid molecular-field and random-phase approximation, we obtain the excitation spectrum both in the ferromagnetic and paramagnetic regions. Of particular interest is that the present calculation does not show full soft-mode behavior at critical point, similar to the singlet-triplet case. The present model has been used to look at the experimental work on the excitation modes propagating along the hexagonal sites of double hcp Pr single crystals using Bleaney and Rainford's level schemes. Discussions and suggestions are provided.