A Microscopic−Macroscopic Analysis for Mixed Energy Transfer Schemes in Doped Amorphous Solids

Abstract

In this study we propose a methodology to treat mixed and more complex energy transfer schemes to reach the time-dependent intensities of luminescence in doped amorphous materials. We start outlining the differential equations for the time variation of microscopic probabilities of being in the relevant states and then transforming them into equations for the variation of the relevant macroscopic states populations and solve those equations. By this method, statistical approaches to the initially excited and up-converted states transient populations for up-conversion processes in the presence of cross relaxation in lanthanide-monodoped amorphous solid are calculated. The resultant formulations produce plots that show correct general tendencies and are coherent with what would be expected for systems exhibiting both mechanisms, hence they could be used in the fitting of experimental curves to calculate some important parameters. The new solution method is more convenient that the classical analysis because it permits the introduction of more realistic time dependent functions for the interacting optical centers and allows showing the time dependence of the macroscopic energy transfer rates in the equations for the dynamics of the involved populations. We apply our method to the particular case of mixing of up-conversion and cross relaxation phenomena; however, because of its general characteristics, we suggest it could be applied to other mixings or more complex schemes.