Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals

A method for estimating the microscopic probability rate of phonon-assisted energy transfer between rare-earth $(3+)$ ions in solids was developed based on Dexter's theory for phonon-assisted energy transfer. The proposed method in this paper enables one to calculate the overlap integral from fundamental cross-section spectra of nonresonant energy transfer involving a multiphonon generation in both donor and acceptor sites. A translation of the donor emission spectrum towards the acceptor absorption scaled with the $N$-phonon emission probability represents the $m$-phonon emission band which performs the energy transfer to the acceptor. A nonvanishing overlap integral ${10}^{\ensuremath{-}2}--{10}^{\ensuremath{-}3}$ times smaller than for the resonant case is found. A multiphonon generation probability assisting the energy transfer was considered due to a combined mechanism of creation and annihilation of phonons. The participation of each phonon in the process was determined. This method was used to investigate the ${\mathrm{Tm}}^{3+}{(}^{3}{F}_{4})\ensuremath{\rightarrow}{\mathrm{Ho}}^{3+}{(}^{5}{I}_{7}),{\mathrm{Er}}^{3+}{(}^{4}{I}_{13/2})\ensuremath{\rightarrow}[{\mathrm{Ho}}^{3+}{(}^{5}{I}_{7}),{\mathrm{Tm}}^{3+}{(}^{3}{F}_{4})]$ direct energy transfers in ${\mathrm{LiYF}}_{4}$ crystals, as well the back transfers.