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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^-2--10^-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 Tm^3+ (^3F₄) Ho^3+ (^5I₇), Er^3+ (^4I₁₃/₂) Ho^3+ (^5I₇), Tm^3+ (^3F₄) direct energy transfers in LiYF₄ crystals, as well the back transfers.
Tarelho et al. (Mon,) studied this question.