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 Δ both greater and less than the phonon-cutoff energy ω₀. In the former case, the coupling is short range, falling off as (ω₀2Δ)^2[(r_jj^'a)^-2], where r_jj^' 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 Δ (i.e., when λ>hνₛΔ, where νₛ is the sound velocity), the coupling due to transverse phonons falls off very slowly, ${~}{({{Δ}}{{}{{ν}}ₜ})}²{[cos({{r}_{j{j}^{{'}}}{Δ}}{{}{{ν}}ₜ)}]}{{r}_{j{j}^{{'}}}}$, where ${{ν}}ₜ$ is the transverse sound velocity, a result similar to that of McMahon and Silsbee. Finally, in the limit of vanishing ${Δ}$, our results go over smoothly to a range dependence similar to that found by Sugihara, and Aminov and Kochelaev, ${~}{1}{{R}³}$. Numerical estimates are made for the strength of the coupling coefficient.