Multiphonon Orbit-Lattice Relaxation of Excited States of Rare-Earth Ions in Crystals

A systematic study has been made of multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals. By measuring both excited-state fluorescence lifetimes and multiphonon relaxation quantum efficiencies, pure multiphonon transition rates were determined for the following set of crystals: (1) La${\mathrm{Br}}_{3}$: ${\mathrm{Nd}}^{3+}$, La${\mathrm{Br}}_{3}$: ${\mathrm{Dy}}^{3+}$, and La${\mathrm{Br}}_{3}$: ${\mathrm{Ho}}^{3+}$; (2) La${\mathrm{F}}_{3}$: ${\mathrm{Er}}^{3+}$ and La${\mathrm{F}}_{3}$: ${\mathrm{Ho}}^{3+}$; (3) Sr${\mathrm{F}}_{2}$: ${\mathrm{Ho}}^{3+}$ and Sr${\mathrm{F}}_{2}$: ${\mathrm{Er}}^{3+}$; and (4) ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$: ${\mathrm{Er}}^{3+}$ and ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$: ${\mathrm{Ho}}^{3+}$. Lifetimes were measured both in the near infrared using the fluorescence phase-shift technique, and in the visible region using pulse-excitation techniques. Measurements were performed at 4.2\ifmmode^\circ\else\textdegree\fi{}K to determine spontaneous multiphonon transition rates, as well as at higher temperatures. The temperature dependences were fitted with a theoretical model involving the stimulated emission of phonons due to the thermally populated phonon modes. These results indicate that the decay involves the emission of high-energy optical phonons and occurs in approximately the lowest order permitted by energy conservation and cutoff in the phonon spectrum. The spontaneous transiton rate was found to have a systematic exponential dependence on the transition energy to the terminal level, that is, on the energy gap. This behavior is interpreted as being due to the simple convergence of the orbit-lattice perturbation expansion. With the precise features of the interacting phonon modes and atomic levels averaged out for these high-order processes, the only critical parameter within a given lattice is the energy gap, i.e., the order of the process. The dependence on the order has been extracted by normalizing the energy gap by the optical-phonon cutoff energy. The rate for a given order and the swiftness of the convergence reflect the strength of the orbit-lattice interaction. The relative behavior for a given host can be explained through its known crystalline properties, in particular the phonon frequency distribution function and the static crystal field. The observed order dependence in ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$ indicates a relative weakness in the orbit-lattice interaction for crystals with strong covalency.