Periodic lattice microscopic rate equation model for upconversion dynamics and determination of energy transfer microparameters

The upconversion luminescence (UCL) assisted by inter-ion energy transfer has received widespread attention. The inter-ion energy transfer microparameters are the fundamental physical quantities determining the UCL process. However, the experimental exploration of the microparameters has not been reported due to challenges in dynamics models. Here, we report on a universal periodic lattice microscopic rate equation model for upconversion dynamics and determination of the microparameters of energy transfer between ions. In periodic lattices, we propose a processing method for ionic interactions, improving the accuracy of the microscopic model. The random doping of luminescent ions in host lattices often leads to randomness in lattice geometric modeling. Therefore, based on the proposed evaluation criterion for the randomness of lattice modeling, we develop a multiple modeling method for periodic small-scale lattices to replace single modeling of large-scale lattices. This eliminates the influence of randomness in lattice geometric modeling on microscopic model calculations, significantly reducing the computational complexity of the microscopic model (reducing computational time by three orders of magnitude) while ensuring model accuracy. Accurate correlation is actualized from micro-ion interactions to macroluminescence. Using this microscopic model and combined with experimental data on luminescence decay curves at multiple energy levels and ion doping concentrations, 19 inter-ion energy transfer microparameters of $E{r}^{3+}$ ions in β-$\mathrm{NaY}{F}_4$:$E{r}^{3+}$ are determined.