Phonon Bottleneck and Frequency Distribution in Paramagnetic Relaxation at Low Temperatures

Abstract

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 $T_{1}^{}{}_{}{}^{\ensuremath{'}}$ and the width of lattice excitation $\ensuremath{\Delta}{\ensuremath{\nu}}_{L}$ have been derived from several points of view as functions of the spin-resonance linewidth $\ensuremath{\Delta}{\ensuremath{\nu}}_{s}$, the line shape, and the intrinsic spin-lattice relaxation time ${T}_{1}$. In the presence of a phonon bottleneck $T_{1}^{}{}_{}{}^{\ensuremath{'}}>{T}_{1}$ and $\ensuremath{\Delta}{\ensuremath{\nu}}_{L}>\ensuremath{\Delta}{\ensuremath{\nu}}_{s}$. 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.