Boltzmann relation reliability in optical temperature sensing based on upconversion studies of Er3+/Yb3+ co-doped PZT ceramics

The fluorescence intensity ratio (FIR) of two thermally coupled levels with temperature follows the Boltzmann equation and shows an exponential nature to the temperature that is purely dependent on the energy difference between the levels. Despite the identical energy difference between the thermally coupled levels, researchers have observed varying sensitivities for various samples. In this article, the FIR and sensitivities were calculated using the Boltzmann equation by changing various parameters such as energy difference (ΔE) and the value of the constant C. The results were compared with various reports for Er³⁺/Yb³⁺ ions. After analysis, a new polynomial fit equation was used to determine the temperature sensitivities for the Er³⁺/Yb³⁺ co-doped PbZrTiO₃ phosphor in lieu of the conventional Boltzmann equation. The polynomial fit equation eliminated the dependency of the sensitivity on the inverse of the FIR factor and a flat sensitivity curve was obtained with temperature.