Theory of nonradiative quenching of hot luminescence inFcenters

A quantitative theory of nonradiative quenching of hot luminescence in $F$ centers excited by an extremely short laser pulse is given on the basis of the Franck-Condon principle in optical excitation and the momentariness of nonradiative transitions (NRT) in the cases of strong electron-(accepting mode) phonon coupling. The treatment given differs from the qualitative considerations by Dexter, Klick, and Russell and Bartram and Stoneham in two important aspects, namely, the strength of the coupling between the $1s$-like ground state and the $2p$-like excited state as well as the damping constant of the lattice vibration are taken into account. It will be shown that the coupling strength via a promoting-mode phonon is just intermediate so that its nonperturbative treatment in the region of crossing of the two potential surfaces is essentially required. For strong electron-phonon coupling, the interferences between different passages through the crossing point $X$ are found to cancel out mainly due to the initial wide distribution of the excitation spectrum. This fact is taken into account by applying the Landau-Zener formula of NRT's for mutually independent passages through $X$. Accordingly, a systematic method to determine the classical motions of the lattice is presented. It will be shown that the system may have many more chances to pass over $X$ for damping oscillation on the $2p$ potential surface than on the $1s$ potential surface as long as the damping is slow. In this way, the probability for an excited $F$ center to return to its ground state will be numerically calculated for $\frac{1}{4}<\Lambda <\frac{1}{2}$ where $\Lambda$ is the excited-state lattice relaxation energy divided by the optical-absorption energy. The result shows that the Dexter, Klick, and Russell postulate that $\Lambda <\frac{1}{4}$ might be a sufficient and necessary condition for luminescence is inadequate.