Rate Equation Models for Excited-State Dynamics

Rate Equation Models for Excited-State Dynamics 

At this level of description, this ensemble model assumes homogeneous doping and excitation densities such that the entire ensemble of ions can be described as having identical environments at all times, and the entire ensemble participates equally in the photodynamics of the system. 

The same parameters are used for each of the two simulation modes, and involve the usually encountered situation of (A 2a + k2b) ki. The specific parameters used are hsted in the figure caption. Importantly, all simulations reflect low-excita-tion-densities and low-upconversion rates, where the upconversion processes contribute in only a perturbative way to the intermediate-level populations and decay rates (i.e., kiNi ENi or 2wetu )- This corresponds to a power range that is experimentally easily accessible. As shall be described in Sect. 5, the use of higher pumping powers may in some cases lead to significant and interesting deviations from the behavior described in this section. 

To determine fully the kinetic parameters governing upconversion using Eq. (10) it is imperative to know the laser-induced excitation densities, Nj and N2, in steady state, or Mi(0) and 2(0) in time-dependent studies. These parameters may be determined from careful measurement of the physical properties of the sample, the excitation configuration, and the experimentally absorbed power. These numbers are not easily reliably determined, however, and they are therefore more commonly estimated or taken as experimental unknowns in the use of these equations for simulations. The difficulty with which absolute excitation densities are determined is one of the practical limitations of this rate equation model. 

A second commonly invoked kinetic model for ETU processes is the so-called Dimer Model [22,23], in which pairs are treated as distinct isolated entities. Such scenarios are often encountered when trivalent rare-earth metal ions are substituted into divalent host lattices of the CsNiClj type. In these hosts, only small concentrations of M + ions can typically be incorporated, and it has been shown that more than 90% of all ions are introduced as (M + - vacancy - M +) pairs to satisfy charge compensation requirements [24, 25]. The ions are thus introduced as isolated pairs. In this model, three excitation populations are con- 

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