Calculation of stress-induced polarization ofF-center luminescence for a vibronic model of the relaxed excited state

The polarization of $F$-center luminescence caused by an applied stress is calculated from a direct numerical diagonalization of the matrix of the interaction Hamiltonian that includes the stress splitting of the electronic states for the simplest vibronic model of the relaxed excited state consistent with cubic symmetry. In this model the $2s$ electronic state is coupled to $2p$ by a triply degenerate odd-parity vibrational mode of frequency $\omega$, and the $2s$ state is taken to lie below $2p$ by an energy $\left|E_{\mathrm{sp}}\right|=\Delta \hslash \omega$. The polarization is found in general to increase from its value at 0 K as the temperature rises, reach a maximum, and then fall, eventually approaching a limiting rate of decrease that is independent of $\Delta$ and of the vibronic coupling strength. For $\Delta \lesssim 2$ this variation with temperature is found to be too large to be consistent with the experimental observation by Hetrick and Compton for KCl, NaCl, RbCl, and NaF that the polarization is effectively independent of temperature below 140 K. The conclusion from our work that we must have $\Delta >2\mathrm{}$ for these materials agrees with an earlier estimate for KCl from magnetic-polarization data but is not in agreement with other published estimates based on the temperature dependence of the radiative lifetime and of the Stark polarization. The implications of these results are discussed.