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Physical Review Letters

Phonon-Assisted Energy Transport in Inhomogeneously Broadened Systems

Theodore D. Holstein, S. K. Lyo, R. Orbach

Published Apr 12, 1976• 111 citations
Publisher

Abstract

Recent experiments have demonstrated Anderson localization in inhomogeneously broadened paramagnetic spin systems. We examine two-phonon processes which can frequency shift the individual spin packets, and thereby allow energy transfer to take place. We make specific application to ruby, and show that the dominant phonon-induced energy-transfer rate is directly proportional to the (spin-nonflip) $\overline{E}\ensuremath{\rightarrow}2\overline{A}$ rate [varying as $\mathrm{exp}(\ensuremath{-}\frac{42}{T})$], and inversely proportional to the square of the energy mismatch of the participating spin packets.

DOI
10.1103/physrevlett.36.891

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Recent experiments have demonstrated Anderson localization in inhomogeneously broadened paramagnetic spin systems. We examine two-phonon processes which can frequency shift the individual spin packets, and thereby allow energy transfer to take place. We make specific application to ruby, and show that the dominant phonon-induced energy-transfer rate is directly proportional to the (spin-nonflip) E‾\ensuremath→2A‾\overline{E}\ensuremath{\rightarrow}2\overline{A}E\ensuremath→2A rate [varying as exp(\ensuremath−42T)\mathrm{exp}(\ensuremath{-}\frac{42}{T})exp(\ensuremath−T42​)], and inversely proportional to the square of the energy mismatch of the participating spin packets.
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 JJJ by $\ensuremath{\sim}\frac{{\ensuremath{\tau}}_{R}^{\ensuremath{-}1}\ensuremath{\hbar}}{(\frac{{r}_{0}}{\ensuremath{\lambda}})}$. Here ${\ensuremath{\tau}}_{R}$ is the radiative lifetime, r0{r}_{0}r0​ the distance between sites, and $\ensuremath{\lambda}$ the photon wavelength involved in the radiative transfer. This replacement results in a line center to wing transfer rate equal to 1.1 \ifmmode\times\else\texttimes\fi{} 102{10}^{2}102 ${e}^{\ensuremath{-}\frac{42}{T}}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for ruby (independent of concentration for a photon mean free path smaller than the sample size).
The first experimental evidence for distinct spectral and spatial energy transfer in ruby via nonradiative coupling is presented. Monochromatic laser excitation of the R1{R}_{1}R1​ line results in a rise of the inhomogeneous background on a millisecond time scale, consistent with a one-phonon---assisted process and contrary to recent theory. However, time-resolved studies of trap fluorescence suggest a rapid, coherent, spatial transfer within the R1{R}_{1}R1​ line which does not shift frequency.
The time-dependent spectral redistribution in fluorescence line narrowing in an inhomogeneously broadened system is investigated by means of simple theoretical models and Monte Carlo calculations. The two-site phonon-assisted energy transfer rate is assumed to be site-symmetric and a Lorentzian function of energy mismatch. For the Monte Carlo calculations initial excitations are made at the wing, shoulder, and center of the inhomogeneously broadened Gaussian band. For the wing excitation it is argued that, because of the above Lorentzian factor, the intensity in the initial site of excitation decays with a much smaller back-transfer rate than for the center excitation, simplifying the theoretical treatment. This effect is demonstrated by the Monte Carlo study. A refinement of this model is developed for a general situation, and its predictions are compared with the Monte Carlo results.
The critical concentration for single-ion-single-ion energy transfer in ruby is calculated using Anderson's method for transport between ions which are characterized by inhomogeneously broadened energy levels. The excitation on the chromium impurity is assumed to propagate by virtue of superexchange interactions. Using a continuum approximation, we find that the single-ion excitation is localized below concentrations of \ensuremath{\sim}0.3-0.4%, a range not inconsistent with that concentration at which a sudden change in the fluorescent lifetime of ruby occurs.