Resonant phonon-assisted energy transfer in ruby from 29-cm−1-phonon dynamics

The dynamics of 29-${\mathrm{cm}}^{-1\mathrm{}}$ phonons resonantly trapped in the $\overline{E}\to 2\overline{A}$ excited-state resonance of ${\mathrm{Cr}}^{3+}$ ions in ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ is studied as a function of excited ion concentration, spot size, and ${\mathrm{Cr}}^{3+}$ concentration employing pulsed optical techniques for generating and detecting the phonons. It is shown that serious differences exist between the present pulsed experiments and the cw optical experiments described elsewhere. The observed saturation of the bottlenecked lifetime at high excited-ion concentrations and its independence of spot size is shown to be consistent with a resonant-phonon-assisted energy transfer (RPAET) among nonresonant ions, which shifts the phonons out of the resonance. Detailed computer calculations of this rate are performed which include a random distribution of ${\mathrm{Cr}}^{3+}$ pairs with an exchange which decays exponentially with separation. Several models are considered for the minimum pair separation which leaves the pairs resonant with the single ions. Calculations for both microscopic and macroscopic broadening are examined. We find that the calculated rates are consistent with the observed magnitude of the saturated lifetime and its dependence on ${\mathrm{Cr}}^{3+}$-ion concentration assuming microscopic broadening and a value of the exchange an order of magnitude smaller than previously estimated for average pair separations in 1% ruby. Implications for the effects of RPAET on the problem of energy transfer in ruby are discussed and it is shown that the absence of RPAET in fluorescence line narrowing and transient grating experiments is consistent with its dominant role in 29-${\mathrm{cm}}^{-1\mathrm{}}$-phonon dynamics assuming these smaller values for the exchange.