A new look at interrelation between Stokes shift and heat release parameter in vibronically assisted optical spectra of activated crystals

In this paper we address the long standing problem of the relationship between the Stokes shift E_Stokes and Pekar-Huang-Rhys (heat release) parameter S, in vibronically assisted optical spectra of activated crystals. We focus on resolving the ambiguity of the alternative definitions E_Stokes=2Sℏω and E_Stokes=(2S−1)ℏω widely accepted in textbooks and numerous articles. We propose a new framework for the analysis of the Stokes shift in the broad multiphonon optical bands based on the analysis of the spectral distributions within the linear electron-vibrational coupling model. Using the exact quantum-mechanical expressions for the moments of the Pekarian type spectral distribution we demonstrate that the Stokes shift can be described by the universal expression E_Stokes=(2S−δ(T))ℏω where the temperature dependent term δ(T) varies in the range between 1 (low temperature) and 0 (high temperature). This result is valid for a discrete Pekarian, when the vibronic sidebands are spectrally resolvable (and maximum of the band cannot be strictly defined) and for a more common case of structureless bands arising under the condition of prevailing interactions with phonons having continuous spectrum. It is emphasized that the parameter δ(T) can be determined experimentally by studying the measured spectral moments of the optical band and does not require numerical modeling, which is necessarily based on a specific (often not entirely justified) assumption regarding the shape and broadening of the individual lines of a discrete Pekarian. In a more general context we emphasize that the use of the single-mode Pekarian approach in the cases of a significant contribution of crystal vibrations is associated with a significant risk of misinterpretation of experimental data.