Theory of light absorption and non-radiative transitions in <i>F</i> -centres

TLDR

A quantitative theory based on the Franck‑Condon principle describes F‑centre absorption band shapes, assuming the lattice behaves as a dielectric continuum and treating the F‑centre as a static charge, while explicitly accounting for electron‑lattice coupling. The study aims to evaluate the probabilities of non‑radiative transitions relevant to photo‑conductivity after F‑centre light absorption. Adiabatic F‑centre electron wave functions are derived using perturbation methods. The absorption constant is expressed through Bessel functions and matches experimental data across temperatures, while non‑radiative decay to the ground state is negligible but transitions to the conduction band become significant when the energy gap is ≤0.1 eV, with a temperature dependence resembling e^{-w/kT} that explains the observed steep decline of photo‑conductive current.

Abstract

A quantitative theory for the shapes of the absorption bands of F -centres is given on the basis of the Franck-Condon principle. Underlying the treatment are two simplifying assumptions: namely, ( a ) that the lattice can be approximately treated as a dielectric continuum; ( b ) that in obtaining the vibrational wave functions for the lattice, the effect of the F -centre can be considered as that of a static charge distribution. Underlying these assumptions, it is shown that the absorption constant as a function of frequency and temperature can be expressed in terms of the Bessel functions with imaginary arguments. The theoretical curves for the absorption constant compare very favourably with the experimental curves for all temperatures. Also considered in the paper are the probabilities of non-radiative transitions, which are important in connexion with the photo-conductivity observed following light absorption by F -centres. The treatment given differs from the qualitative considerations hitherto in one important aspect, namely, the strength of the coupling between the electron and the lattice is taken into account. The adiabatic wave functions for the F -centre electron required for the discussion are obtained by perturbation methods. The probability for an excited F -centre to return to its ground state by non-radiative transitions is shown to be negligible; similar transitions to the conduction band are, however, important if the excited state is separated from the conduction band by not much more than 0·1 eV. The temperature dependence of such transitions is complicated, but, for a wide range of temperatures, resembles e - w/k T . Tentative estimates show that the result is Consistent with the observed steep drop of the photo-conductive current with temperature.