Radiationless relaxation of rare-earth ions in solids: Phenomenological effects of f→f and f→d transitions and the origin of multiphonon relaxation in first-order perturbation theory

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Abstract
In this paper, the phenomenological effects of d→f and f→f processes are examined. The relationship between radiationless multiphonon decay and excitation processes is obtained in terms of the first-order adiabatic relaxation theory. A detailed analysis is given of the experimentally observed temperature dependence of the lifetimes and intensities of Sm2+ 5D0 fluorescence in KCl and SrF2. From this analysis, rate constants for the nonradiative d?5D0 processes are evaluated. The observed characteristics of radiationless f→f and f→d relaxation processes are compared in terms of the theoretical dependence of the rate constant on the energy gap, on temperature, and on the frequencies of the phonon modes of the host crystals. The main features of the first-order relaxation theory are compared with those of an earlier high-order treatment in the crude Born–Oppenheimer approximation. The role of potential-surface displacements (signifying coupling strengths) in the first-order perturbation treatment of multiphonon relaxation is discussed. It is shown that in the zero-displacements limit, the first-order multiphonon rate constant is identically zero. In this limit, multiphonon processes can only arise from high-order terms in the perturbation expansion.