Dynamical Properties of Induced-Moment Systems

The low-temperature excitation spectrum for ferro- and antiferromagnets with large crystalline-field splittings is derived using the Bogoliubov transformation. Specific application is made to those systems where the single-ion ground state is a singlet and the magnetization of the coupled ions arises from the Van Vleck temperature-independent susceptibility. ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Eu}}^{3+}$ in cubic and hexagonal environments are used as examples. The specific heat, magnetization, and nuclear-nuclear coupling are computed for both the ordered and paramagnetic regimes. The results in the paramagnetic limit display a characteristic energy gap $\Delta {[1-\alpha (\frac{K}{\Delta }\left)\right]}^{\frac{1}{2}}$, where $\Delta$ is the crystalline field splitting (or, in ${\mathrm{Eu}}^{3+}$, the spin-orbit splitting) between the ground singlet and first excited state, $K$ is the exchange integral, and $\alpha$ is a constant proportional to the Van Vleck temperature-independent susceptibility. In the magnetically ordered state, the same behavior is found for longitudinal excitations in ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Eu}}^{3+}$ salts with a gap equal to ${({\alpha }^2{K}^2-{\Delta }^2)}^{\frac{1}{2}}$, while transverse excitations are found to exhibit a linear dispersion law without an apparent gap in the limit of complete isotropy. The phase transition between the paramagnetic and ordered regimes is shown to be of second order. However, the entropy of ordering is shown to be reduced from the usual value $\mathrm{NK}$ $\ln \mathrm{}(2S+1)\mathrm{}$.