Nonlinear theory of multiphonon relaxation of excited rare-earth ions in laser crystals

A general formula for the probability of multiphonon nonradiative transitions between J multiplets of 4f(N) states of trivalent rare-earth ions in a crystal derived for a nonlinear mechanism is discussed. A technique is developed for calculating the quantities involved in this formula. Particular attention is given to calculating the spectral density J((p))(Omega), which is the Fourier transform (at the transition frequency) of the pth power of the correlation function K(t) of host ion displacements. Based on the central limit theorem from probability theory, an approximation is proposed for the spectral density J((p))(Omega). The theoretical values of nonradiative multiphonon transition rates are compared with experimental ones.