Magnetic Excitons in Real Singlet-Ground-State Ferromagnets: Application toPr3Tl and fcc Pr

We have studied the magnetic-exciton behavior expected in cubic systems containing ${\mathrm{Pr}}^{3+}$ with a crystal-field-only ${\Gamma }_1$ singlet ground state. We use this study to discuss the experimentally observed behavior in ${\mathrm{Pr}}_3$Tl and fcc Pr. In particular, we show that crystal-field states lying higher than the ${\Gamma }_4$ first-excited triplet have an important qualitative effect on the magnetic-exciton dispersion relationship. The most important difference from the results for a singlet-triplet model is the appearance of a substantial gap at $\overrightarrow{\mathrm{q}}=0\mathrm{}$ (∼18°K in a typical realistic case) for the transverse excitons in the ferromagnetic state; while the gap for the longitudinal modes would also be much different if one neglected the higher-lying crystal-field states (∼75% greater in a typical realistic case). The necessity of including effects of crystal-field states above the ${\Gamma }_4$ triplet led us to adopt an effective boson (i.e., Bogoliubov-type) approximation valid only as the temperature approaches zero. We then use our knowledge of the random-phase-approximation (RPA) results for the singlet-singlet problem to discuss the expected temperature dependence of the excitation spectrum. The existing theory, including the effects of all crystal-field levels, is quite successful in quantitatively predicting the experimental magnetic-exciton behavior at low temperature in ${\mathrm{Pr}}_3$Tl. On the other hand, the existing theory offers no explanation for the absence of any measurable change with temperature of the measured dispersion relationship even when going to temperature well above the Curie temperature in ${\mathrm{Pr}}_3$Tl and fcc Pr. Incidental to our discussion of the magnetic-exciton behavior, we treat the macroscopic magnetiżation variation with temperature in ${\mathrm{Pr}}_3$Tl including all crystal-field levels in a molecular-field theory.