Relaxation Processes in Trivalent REE

The luminescence intensity of emission depends on a given level on the radiative and nonradiative probabilities. The quantum yield is expressed as follows  

These include ion-ion energy transfer, which can give rise to concentration quenching and non-exponential decay and relaxation by multiphonon emission, which is usually essential for completing the overall scheme, and can affect the quantum efficiency. For low concentration of rare earth dopant ions the principle nonradiative decay mechanism is a multiphonon emission. 

The ab initio calculation of the transition rate between two electronic states with the emission of p phonons involves a very complicated sum over phonon modes and intermediate states. Due to this complexity, these sums are extremely difficult to compute however, it is just this complexity, which permits a very simple phenomenological theory to be used. There are an extremely large number of ways in which p phonons can be emitted and the sums over phonon modes and intermediate states are essentially a statistical average of matrix elements. In the phenomenological approach it is assumed that the ratio of thep-th and (p - l)-th processes will be given by a couphng constant characteristic of the matrix in which the rare earth is situated and not depending 

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

---