Phonon Bottleneck and Frequency Distribution in Paramagnetic Relaxation at Low Temperatures

The general problem of the imprisonment of resonant quanta due to reabsorption has been studied for phonons in a lattice of paramagnetic centers. Expressions for the apparent spin-lattice relaxation time $T_1^{}{}_{}{}^{\prime }$ and the width of lattice excitation $\Delta {\nu }_L$ have been derived from several points of view as functions of the spin-resonance linewidth $\Delta {\nu }_s$, the line shape, and the intrinsic spin-lattice relaxation time $T_1$. In the presence of a phonon bottleneck $T_1^{}{}_{}{}^{\prime }>T_1$ and $\Delta {\nu }_L>\Delta {\nu }_s$. Detailed agreement is found with Holstein's theory of photon trapping in atomic gases, and inconsistencies in other treatments of phonon trapping are illuminated. The present study gives a detailed physical picture of the imprisonment process. Special attention is given to the nature of the bottleneck when the phonon interruption rate becomes comparable with the spin-resonance linewidth.