Phonon-Induced Ion-Ion Coupling in Paramagnetic Salts

The zero-point phonon vibrations in a paramagnetic crystal are shown to give rise to an effective ion-ion coupling. Expressions are derived for ion-energy transitions $\Delta$ both greater and less than the phonon-cutoff energy $\hslash {\omega }_0$. In the former case, the coupling is short range, falling off as ${\left(\frac{\hslash {\omega }_0}{2\Delta }\right)}^{2\left[{\left(\frac{r_{j{j}^{\prime }}}{a}\right)}^{-2\mathrm{}}\right]}$, where $r_{j{j}^{\prime }}$ is the ion-ion separation distance and $a$ the lattice constant. In the latter case, a slowly decreasing oscillatory range dependence is obtained. When the ionic separation is greater than the phonon wavelength appropriate to the transition energy $\Delta$ (i.e., when $\lambda >\frac{h{\nu }_s}{\Delta }$, where ${\nu }_s$ is the sound velocity), the coupling due to transverse phonons falls off very slowly, $\sim {\left(\frac{\Delta }{\hslash {\nu }_t}\right)}^2\frac{\left[cos\right(\frac{r_{j{j}^{\prime }}\Delta }{\hslash {\nu }_t)}]}{r_{j{j}^{\prime }}}$, where ${\nu }_t$ is the transverse sound velocity, a result similar to that of McMahon and Silsbee. Finally, in the limit of vanishing $\Delta$, our results go over smoothly to a range dependence similar to that found by Sugihara, and Aminov and Kochelaev, $\sim \frac{1}{R^3}$. Numerical estimates are made for the strength of the coupling coefficient.