We have investigated the theory of the temperature dependence of nonradiative transitions of an optical ion in a solid from one electronic state to another using the proper adiabatic theory. The Hamiltonian, based on the nonadiabaticity operator, has been used to calculate the temperature dependence of nonradiative transition rates. The Hamiltonian, H_na, consists of two terms, as shown in Equation 1 below.

The first term is usually used to explain nonradiative transitions of an optical ion in solids. We have shown that the first term alone leads to a zero of the transition rates at s = z for low temperature where s represents the Huang-Rhys parameter, and z relates to the energy gap between the excited state and ground state in terms of vibrational quantum of energy, hν. This zero of the transition rate cannot be removed unless the second term is included in the calculation of the transition rates on the basis of the Fermi Golden rule and the Boltzmann distribution of occupation of the vibrational states in thermal equilibrium at temperature, T, in the excited state.

Additionally, we define three functions, U₁, U₂, and U₃ to describe the thermal dependence of the transition rate, corresponding to the two terms of H_na and a cross term that appears when applying the Fermi Golden rule to calculate the transition rates. A linear combination of these three terms leads to a physically meaningful description of the temperature-dependent, nonradiative transition rate between two electronic states. These functions are used to express the temperature dependence of the quantum efficiency of a phosphor. A comparison of results from simulations and experiments will be presented.