# Thermal Relaxation Spectra for Evaluating Luminescence Quantum Efficiency of CASN:Eu2+
## 4 Relation Between Photothermal and Photoluminescence Excitation Spectra  
Let us formulate the relation between PTES and PLES in terms of the luminescence quantum efficiency (or quantum yield) based on radiative transition in the conventional configurational coordinate model in the harmonic approximation as shown in Figure 6[16]. We can reasonably assume a typical non-radiative relaxation process where rapid non-radiative relaxation occurs from the higher to lower excited states and from the vibrationally excited state into the bottom of the adiabatic potentials in the lowest excited state and in the ground state. It is assumed that the radiative transition occurs with the luminescence quantum efficiency (\(\leq\)1) from the bottom of the adiabatic potential of the lowest excited state into the vibrationally excited levels in the adiabatic potential of the ground state. Figure 5: (**a**) Photothermal excitation, photoluminescence excitation, and photoluminescence spectra of Eu\({}^{2+}\)-doped CaAlSiN\({}_{3}\) phosphors (CASN:Eu\({}^{2+}\)). (**b**) Transformed photoluminescence excitation spectra with various luminescence quantum yields \(\eta\) compared with the photothermal excitation spectrum of CASN:Eu\({}^{2+}\). The photothermal excitation spectrum (PTES) is compared with the transformed photoluminescence excitation spectrum (PLES) expressed by \((E-1.9\eta)\)/1.9\(\eta\) PLES(\(E\)). As usual as depicted in Figure 6, we define the magnitude of electron-lattice interaction \(W_{LR}\) and phonon energy \(\hbar\omega_{v}\) to obtain the Huang-Rhys factor \(S\) and the Franck-Condon factor \(F_{m0}\) as follows:  
\[S=\frac{W_{LR}}{\hbar\omega_{v}},\ F_{m0}=\frac{e^{-S}S^{m}}{m!},\ \sum_{m}F_{m0}=1,\ \text{and}\ \sum_{m}mF_{m0}=S \tag{1}\]  
The conversion ratio \(P\) of excitation energy to thermal energy when luminescence quantum efficiency is 100% (assuming that relaxation within the adiabatic potential occurs non-radiatively):  
\[P=\frac{E_{\text{ex}}-E_{\text{fl}}}{E_{\text{ex}}}=\frac{E_{\text{ex}}-(E_{00} -\sum_{m}m\hbar\omega_{v}F_{m0})}{E_{\text{ex}}} \tag{2}\]  
where \(E_{\text{ex}}\) is the excitation photon energy, \(E_{\text{fl}}\) is the fluorescence photon energy, and \(E_{00}\) is the zero phonon energy. The photothermal excitation signal intensity (raw signal) \(PTE(E_{\text{ex}})\) is expressed by:  
\[PTE(E_{\text{ex}})=\frac{E_{\text{ex}}-\eta(E_{00}-S\hbar\omega_{v})}{E_{\text {ex}}}\beta(E_{\text{ex}})n(E_{\text{ex}})E_{\text{ex}}\Big{(}1-e^{-\alpha_{ \text{ex}}L}\Big{)}, \tag{3}\]  
where \(\eta\) is the luminescence quantum efficiency, \(\beta(E_{\text{ex}})\) is the spectral response function of the measurement apparatus, \(n(E_{\text{ex}})\) is the photon flux density of the excitation light, and \(\alpha_{ex}\) is the absorption coefficient of the sample at \(E_{ex}\). Then, the photothermal excitation spectrum \(PTES(E_{\text{ex}})\) of the sample is obtained by normalizing \(PTE(E_{\text{ex}})\) with that of activated carbon (\(\eta=0,\ \alpha_{ex}L\gg 1\)), \(\beta nE_{\text{ex}}\), as:  
\[PTES(E_{\text{ex}})=\frac{E_{\text{ex}}-\eta(E_{00}-S\hbar\omega_{v})}{E_{\text {ex}}}\Big{(}1-e^{-\alpha_{\text{ex}}L}\Big{)}=\frac{E_{\text{ex}}-\eta E_{ \text{fl}}}{E_{\text{ex}}}\Big{(}1-e^{-\alpha_{\text{ex}}L}\Big{)}. \tag{4}\]  
This equation shows PTES is proportional to the absorption spectrum (\(\alpha_{\text{ex}}\)) if \(\eta=0\) and \(\alpha_{ex}L\ll 1\). The raw signal intensity for the photoluminescence excitation spectrum is expressed by:  
\[PLE(E_{\text{ex}})=\frac{\eta(E_{00}-S\hbar\omega_{v})}{E_{\text{ex}}}\beta(E _{\text{ex}})n(E_{\text{ex}})E_{\text{ex}}\Big{(}1-e^{-\alpha_{\text{ex}}L} \Big{)}=\frac{\eta E_{\text{fl}}}{E_{\text{ex}}}\beta nE_{\text{ex}}\Big{(}1-e ^{-\alpha_{\text{ex}}L}\Big{)}\, \tag{5}\]  
Figure 6: Radiative and non-radiative relaxation in the configurational coordinate model [16].