Trapping of a Resonant Phonon by a Pair of Paramagnetic Ions

The trapping of a resonant phonon by a pair of paramagnetic ions is studied in the limit of zero temperature with the aid of the Heitler damping formalism. The probability amplitudes of the states (spin 1 up, spin 2 down, no phonon), (spin 1 down, spin 2 up, no phonon), and (spin 1 down, spin 2 down, one phonon) are computed. Provided coherence is maintained between the two spins, the transfer of energy to the crystal lattice takes place in the time $T_{10}{(1-\frac{sin{k}_0{r}_{12}}{k_0{r}_{12}})}^{-1\mathrm{}}$. Here $T_{10}$ is the spin-lattice relaxation time for an isolated ion at zero degrees, $k_0$ is the wave vector of the resonant phonon, and $r_{12}$ is the distance between the spins. The relation of this result to the general problem of spin-lattice relaxation at low temperature is discussed.