Relaxation Process in Trivalent Rare Earths

The luminescence intensity of emission from a given level depends on the radiative and non-radiative probabilities. The quantum yields is expressed as 

Where B and a are characteristics of the host (a is negative). Experimental data have shown that the approach is very good for a large variety of hosts. In this way all multiphonon rates can be inferred from a few measured rates. 

The dominant emission process is the one which requires the least number of phonons to be emitted. The minimum number of phonon required for a transition between states separated by an energy gap AE is 

Where htWmax is the maximum energy of optical phonons. With increasing temperature the simulated emission of phonons by termal phonons increases the relaxation rate W according to 

The non-radiative relaxations in the rare earth ions are related to their excited state populations and are governed by the energy difference between the emitting level and the next lower level, separated by the number of phonons of the host. In the weak-coupling case, for the rare earth ions, the temperature dependence of the non-radiative rate is given by  

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