Temperature Dependence of Multiphonon Nonradiative Decay at an Isolated Impurity Center

The fluorescent lifetime of the ${}_{}{}^4{}_{}{}^{}T_2^{}{}_{}{}^{}$ excited state of ${\mathrm{Co}}^{2+}$ in KMg${\mathrm{F}}_3$ is measured from 1.6 to 550 °K. The radiative lifetime is calculated from the oscillator strength of the ${}_{}{}^4{}_{}{}^{}T_1^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}T_2^{}{}_{}{}^{}$ transition and the nonradiative decay rate deduced. This varies over four orders of magnitude. Its temperature dependence is in good agreement with that predicted for linear coupling to a continuum of harmonic phonons, with parameters deduced from the zero-phonon and one-phonon spectrum. To obtain agreement, the decrease of the energy gap with temperature, which is primarily due to anharmonicity, must be taken into account. The absolute value of the nonradiative decay rate is two orders of magnitude smaller than predicted by the model. It is shown that this discrepancy, and the much more dramatic discrepancy found in ruby, can reasonably be attributed to anharmonicity. The continuum model is compared with a discrete-frequency model, and it is shown that the "effective frequency" of the continuum, however, defined, is closer to its mean frequency than to its maximum.