Theory of light absorption and non-radiative transitions in F-centres

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 assumaptions: namely, (a) that the lattice can be approximately treated as a dielectric continuum; (b) that in obtaining the vibrational wave fmnetions for the lattice, the effect of the F-centre can be considered as that of a static charge distribution. Under 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 com pare very favourably with the experimental curves for all tomperatures. 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-contres. 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 fuctions 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 01 eV. The temperature dependence of such transitions is complicated, but, for a wide range of temperatures, resombles e-W!kT. Tentative estimates show that the result is consistent with the observed steep drop of the photo-conductive current with temperature. It has long been recognized that the considerable widths of the characteristic absorption curves of F-centres (Pohl 1937) are caused by the coupling of the electronic motion in the F-centre to the ionic lattice. Owing to the coupling, while an F-centre electron is making a transition, a number of lattice vibration quanta can be created or annihilated. Thus for the same electronic transition a certain latitude in the absorption frequency results, corresponding to a greater or lesser number of vibrational quanta being annihilated or created during the absorption act. Although the basic mechanism underlying the absorption is clear, no quantitative theory of the absorption curves has hitherto been given. The usual interpretation of the optical properties of F-centres is based upon a schematic application of the Franck-Condon principle, whereby a single parameter is used to specify schematically the lattice configuration (von Hipple 1936; Seitz 1939; Mott & Gurney I948). Apart from the natural limitations of the procedure, the principle is often indifferently interpreted so that important features could well be missed (no account, for instance, is usually taken of the detailed nature of the vibrational wave functions, which are, however, well known to be important for an understanding of the intensity distributions of the absorption spectra of molecules (cf. Finkelnburg 1938)). In a recent