Temperature dependence of magnetic excitations in singlet-ground-state systems. II. Excited-state spin waves near the Curie temperature inPr3Tl

A theory has been developed of the temperature dependence of the spin waves in a system involving transitions between all levels of the ground multiplet of an ion. The excitations out of both ground and excited states are described by the product of annihilation and creation operators for the single-ion states of the crystalline field and molecular field. The spin-waves are obtained with the random-phase approximation applied to the interlevel transition operators rather than to $S_z$ as in conventional spin-wave theory. Numerical results have been obtained for ${\mathrm{Pr}}_3$Tl that agree with the neutron measurements of Birgeneau and coworkers. There is little variation with temperature through the phase transition of the scattering from modes having wave vectors greater than $(0.25, 0.25, 0)\frac{2\pi }{a}$. This is partly because the measurements average over two groups of spin waves that interact, the spin waves out of the ground state and the excited-state spin waves corresponding to the ${\Gamma }_4\to {\Gamma }_3$ transition. The ${\Gamma }_1-{\Gamma }_4$ zone-center modes are strongly temperature dependent but do not go soft. They decrease in frequency as $T$ rises from zero to $T_C$ by factors of ∼4 and ∼6 for $S_{+}$ and $S_z$ modes, respectively. Very-low-frequency transverse ($S_{+}$) modes (∼0.02 THz) appear at elevated temperatures and do go soft as $T\to T_C$. These modes correspond to transitions between the states of the weakly exchange-split ${\Gamma }_4$ triplet.