Phonon-assisted radiative transfer

The radiative-transfer rate for inhomogeneously broadened optical lines is calculated. Photon emission and absorption is supplemented by one- and two-phonon processes to achieve energy conservation. These photon-assisted radiative transfer processes are shown to be effective at low concentrations, where electrostatic and exchange processes are weak, in the intermediate or strong radiative trapping regime. Specific application is made to the case of dilute ruby, and explicit transfer rates are calculated. The coherence factors, which undermine nonradiative single-phonon-assisted energy transfer for a small energy mismatch, are not important for radiative transfer, if the photon mean free path is longer than the wavelength of the energy matching phonon. The one-phonon process is found to vary linearly with temperature, and to be independent of the energy mismatch. However, the calculated rate for the case of ruby is found to be very slow. The two-phonon-assisted radiative-transfer process rates are calculated. They dominate the one-phonon-assisted rate for all reasonable temperatures. The results of an earlier paper are shown to be related to phonon-assisted radiative transfer if one replaces the exchange coupling between transfer sites $J$ by $\sim \frac{{\tau }_R^{-1\mathrm{}}\hslash }{\left(\frac{r_0}{\lambda }\right)}$. Here ${\tau }_R$ is the radiative lifetime, $r_0$ the distance between sites, and $\lambda$ the photon wavelength involved in the radiative transfer. This replacement results in a line center to wing transfer rate equal to 1.1 × ${10}^2$ $e^{-\frac{42}{T}}$ ${\sec }^{-1\mathrm{}}$ for ruby (independent of concentration for a photon mean free path smaller than the sample size).