# Thermal Relaxation Spectra for Evaluating Luminescence Quantum Efficiency of CASN:Eu2+
## 4 Relation Between Photothermal and Photoluminescence Excitation Spectra
\(\hbar\omega_{v}:\) phonon energy, \(E_{00}:\) energy of zero phonon line, \(E_{\text{fl}}:\) fluorescence peak energy, \(q_{\text{gr}}\), \(q_{\text{ex}}\) are the equilibrium configurations at electronic ground and excited states, respectively. \(W_{\text{LR}}\): magnitude of electron-lattice interaction defined by the energy difference between the excited-state adiabatic potential energy at \(q_{\text{gr}}\) and that at \(q_{\text{ex}}\). which is normalized by \(\beta nE_{\text{ex}}\) to give the photoluminescence excitation spectrum:  
\[PLES(E_{\text{ex}})=\frac{\eta E_{\text{fl}}}{E_{\text{ex}}}\Big{(}1-e^{-\alpha_{ \text{ex}}L}\Big{)} \tag{6}\]  
Thus, we finally obtain the relation between \(PTES(E_{\text{ex}})\) and \(PLES(E_{\text{ex}})\) as:  
\[PTES(E_{\text{ex}})=\frac{E_{\text{ex}}-\eta E_{\text{fl}}}{\eta E_{\text{fl}} }PLES(E_{\text{ex}}) \tag{7}\]  
In Figure 5, we assumed \(\eta=0.75,\ 0.8,\ 0.9,\ 1\) and \(E_{\text{fl}}=E_{00}-S\hbar\omega_{v}=1.9\)eV (the peak of the photoluminescence spectrum). The PTES agrees with the transformed PLES within \(\eta=0.8\) to \(1\), consistent with \(\eta=0.9\) independently measured with the absolute quantum efficiency measurement system. Even though agreement is fairly good, there is evident residual difference. Possible reasons for the discrepancy are as follows. Firstly, thermoluminescence from CASN:Eu\({}^{2+}\) may distort PLES and/or PTES, causing departure from ideal behavior expected in the standard energy relaxation model. CASN:Eu\({}^{2+}\) is known not only for high quantum efficiency but also for excellent thermal-quenching behavior. The mechanism of the latter nature is studied [17; 18] and it is revealed that photoexcited electrons are partially trapped at trapping sites and thermally excited to yield thermoluminescence. This may also affect apparent luminescence quantum efficiency. Secondly, it can be interpreted at face value, i.e., the luminescence quantum efficiency \(\eta\) depends on the excitation light energy as follows:  
\[PTES(E_{ex})=\frac{E_{ex}-\eta(E_{ex})E_{fl}}{\eta(E_{ex})E_{fl}}PLES(E_{ex}) \tag{8}\]  
For example, if \(\eta(E_{ex})=1-0.2\Big{(}E_{ex}-E_{fl}\Big{)}\), we obtain excellent agreement as shown in \[S=\frac{W_{LR}}{\hbar\omega_{v}},\ F_{m0}=\frac{e^{-S}S^{m}}{m!},\ \sum_{m}F_{m0}=1,\ \text{and}\ \sum_{m}mF_{m0}=S \tag{1}\]0 In fact, calculation of absorption spectrum of CASN:Eu\({}^{2+}\)[19] shows that absorption increases monotonically up to 3.6 eV, inconsistent with the PLES presently observed, where the absorption levels off around 2.5 eV and decreases above 2.6 eV. This may be indicative of the onset of a non-radiative channel by higher energy excitation, such as intersystem crossing into the radiatively inactive triplet state. Therefore, it is reasonable for the luminescence quantum efficiency tends to decrease with energy as assumed in \[S=\frac{W_{LR}}{\hbar\omega_{v}},\ F_{m0}=\frac{e^{-S}S^{m}}{m!},\ \sum_{m}F_{m0}=1,\ \text{and}\ \sum_{m}mF_{m0}=S \tag{1}\]1 The other possibility is that \(\alpha_{ex}L\) increases with photon energy to be in the absorption saturation region (\(1-e^{-\alpha_{ex}L}\approx 1\)) above 2.6 eV. It is usually difficult to judge which is the case, but the improved agreement between the PTES and the transformed PLES in Figure 7 supports the former case. In order for the comparison between PTES and PLES to be valid, it is not necessary that the PLES is proportional to the absorption spectrum since the PTES and PLES have the same (\(1-e^{-\alpha_{ex}L}\)) dependence. Thirdly, excitation and observation geometry was not the same between PTES and PLES. In order to estimate the effect of geometry, we measured PLES in the geometry similar to PTES to find that almost the same PLES as shown in Figure 5 was obtained. However, we need further to consider the effect of possible difference between the optical and thermal thickness of the sample as discussed in Reference [20]. Therefore, we derive a more rigorous formula by taking these effects into account below.