Theory of the Urbach Rule

Abstract

The two-vibrational-mode model proposed by Toyozawa and Mahr to account for the Urbach rule is considered both in the semiclassical approximation and in the full quantum theory. The semiclassical line shape accounts for both the exponential Urbach tail and the central Gaussian region of the absorption band. Quantum modifications make the semiclassical line shape incorrect for an excited state stable to odd-parity lattice distortions. The quantum line shape for an unstable excited state is shown to be qualitatively similar to the semiclassical line shape. In particular the two line shapes agree at high temperatures, but differ quantitatively at low temperatures. It is suggested that the hypotehsis of an unstable excited state is consistent with recent experimental results on intrinsic luminescence in alkali halides.