[mathjax]

$$F\left(R_{\infty}\right)=a e^{\lambda_{2} t}+c=ae^{-\frac{p_{e}}{p_{1}+p_{e}+s \exp (\frac{-E}{k T})+p_t} (s \exp (\frac{-E}{k T})+p_t)t}+c$$

$$W=W_{R}+W_{N R}$$

$$W=\frac{1}{\tau(T)}=\frac{1}{\tau_{R}}+\frac{1}{\tau_{N R}}$$

$$E_{\mathrm{nr}}=0.249 \,\mathrm{eV}$$

$$A_{\mathrm{nr}}=2.55 \times 10^{11} \,\mathrm{sec}^{-1}$$

$$k_{\mathrm{rad}}^{-1}=1.39 \,\mu \mathrm{sec}$$

$$\tau=\left[k_{\mathrm{rad}}+A_{\mathrm{nr}} \exp \left(\frac{-E_{\mathrm{nr}}}{k T}\right)\right]^{-1}$$

$$\lambda_{2} \approx-\frac{p_{m}}{p_{1}+p_{e}+p_{m}} p_{e}=-\frac{p_{e}}{p_{1}+p_{e} +s \exp (\frac{-E}{k T})} s \exp (\frac{-E}{k T})$$

$$b = -\frac{p_{e}}{p_{1}+p_{e}+p_{m}} s$$

$$F\left(R_{\infty}\right)=a e^{\lambda_{2} t}+c=ae^{b\text{exp}^{(\frac { -E }{ kT}) }t}+c$$

$$F\left(R_{\infty}\right)=a e^{\lambda_{2} t}+c$$

$$\lambda_{2} \approx b \exp \left(\frac{-E}{k T}\right)$$

$$\lambda_{2} \approx-\frac{p_{e}}{p_{1}+p_{e}+s \exp \left(\frac{-E}{k T}\right)} s \exp \left(\frac{-E}{k T}\right)$$

$$F\left(R_{\infty}\right)=a e^{\lambda_{2} t}+c=a e^{-\frac{p_{e}}{p_{1}+p_{e}+s \exp \left(\frac{-E}{k T}\right)} s \exp \left(\frac{-E}{k T}\right) t}+c$$

$$\lambda_{2} \approx-\frac{p_{m}}{p_{1}+p_{e}+p_{m}} p_{e}=-\frac{p_{e}}{p_{1}+p_{e}+p_{m}} s\text{exp}(\frac {-E }{ kT } )\approx b\text{exp}(\frac {-E }{ kT } )$$

$$-\lambda^{2}-\left(p_{1}+p_{e}+p_{m}\right) \lambda=p_{m} p_{e}$$

$$\lambda = \frac { -(p_{1}+p_{e}+p_{m})\pm \sqrt{(p_{1}+p_{e}+p_{m})^2-4p_{m}p_{e}} }{ 2 }$$

$$\lambda = \frac { -(p_{1}+p_{e}+p_{m})\pm \sqrt{(p_{1}+p_{e}+p_{m})^2-4p_{m}p_{e}} }{ 2 }$$

$$\frac{-(p_{1}+p_{e}+p_{m}) \pm \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}{2}\times\frac{-(p_{1}+p_{e}+p_{m}) \mp \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}{-(p_{1}+p_{e}+p_{m}) \mp \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}$$

$$\frac{k}{s}=\frac{\left(1-R_{\infty}\right)^{2}}{2 R_{\infty}}$$

$$F\left(R_{\infty}\right)\propto (M-m)$$

$$F\left(R_{\infty}\right)=a(-c_{1} e^{\lambda_{1} t}-c_{2} e^{\lambda_{2} t})+b$$

$$M-m= -c_{1} e^{\lambda_{1} t}-c_{2} e^{\lambda_{2} t}$$

$$F\left(R_{\infty}\right)=\frac{\left(1-R_{\infty}\right)^{2}}{2 R_{\infty}}$$

$$R(t)=a c_{2} e^{Ma}e^{\lambda_{2} t}+Q=Pe^{\lambda_{2} t}+Q$$

$$e^{(M-m) a}\approx e^{Ma}$$

$$R^{\prime}(t)=R^{\prime}(m)m^{\prime}(t)=e^{(M-m) a}ac_{2}\lambda_{2}e^{\lambda_{2} t}=ac_{2}e^{(M-m) a}(\lambda_{2}e^{\lambda_{2} t})$$

$$R(M-m)=1-e^{(M-m) a}$$

$$m\approx c_{2}\left(e^{\lambda_{2} t}-1\right)$$

$$I(T)=\frac { \omega_{1 \rightarrow 2}^{\text{rad}}I_0 }{ \omega_{1 \rightarrow 2}^{\text{rad}} +\omega_{1 \rightarrow 2}^{\mathrm{n-rad}}(T)}$$

$$I_{\text{quenching}}(T)=\frac { \omega_{1 \rightarrow 2}^{\mathrm{n-rad}}(T) I_0}{ \omega_{1 \rightarrow 2}^{\text{rad}} +\omega_{1 \rightarrow 2}^{\mathrm{n-rad}}(T)}$$

$$m=\frac { p_m }{ p_1+p_m} MI_0t\approx \frac { p_m }{ p_1} I_0t$$

$$R(m)=1-e^{(M-m)a}\approx1-e^{(M-\frac { p_m }{ p_1} I_{0}t)a}=1-e^{(M-\frac { p_m }{ p_1} D)a}=R(D)$$

$$R^{\prime}(D)=(1-R(D))\frac { p_m }{ p_1}a\propto e^{-\frac {\Delta E }{ kT } }$$

$$\begin{array}{l} \frac { dm }{ dt } =m_{e}p_{m} \\ \frac { dm_{e} }{ dt } =-m_{e}(p_{m} +p_1) +(M-m_{e}-m)p_e\end{array}$$

$$\frac{d}{d t}\left[\begin{array}{c} m \\ m_{e } \end{array}\right]=\left[\begin{array}{cc} 0 & p_{m} \\ -p_{e} & -\left(p_{m}+p_{1}+p_{e}\right) \end{array}\right]\left[\begin{array}{c} m \\ m_{e} \end{array}\right]+\left[\begin{array}{c} 0 \\ Mp_{e}\end{array}\right]$$

$$\left[\begin{array}{c} m \\ m_{e} \end{array}\right]=c_{1} e^{\lambda_{1} t} \mathbf{v}_{\mathbf{1}}+c_{2} e^{\lambda_{2} t} \mathbf{v}_{\mathbf{2}}+ \mathbf{v}_{\mathbf{3}}$$

$$\begin{array}{l} \lambda_{1}+\lambda_{2}=-\left(p_{m}+p_{1}+p_{e}\right) \\ \lambda_{1} \lambda_{2}=p_{m} p_{e} \end{array}$$

$$p_{1}\gg p_{m}$$

$$p_{1}\gg p_{e}$$

$$\mathbf{v}_{\mathbf{3}}=\left[\begin{array}{cc} M \\ 0 \end{array}\right]$$

$$c_{1}+c_{2}+M=0$$

$$\mathbf{v}_{\mathbf{1}}=\left[\begin{array}{cc} 1 \\ \frac { \lambda_{1} }{ p_{m} } \end{array}\right] \quad \mathbf{v}_{\mathbf{2}}=\left[\begin{array}{cc} 1 \\ \frac { \lambda_{2} }{ p_{m} } \end{array}\right]$$

$$c_{1}\frac { \lambda_{1} }{ p_{m}}+c_{2}\frac { \lambda_{2} }{ p_{m}}=0$$

$$\left| \lambda_1 \right|\gg \left| \lambda_2 \right| \quad \lambda_1 <0,\lambda_2<0$$

$$c_1>0,c_2<0,\left| c_2 \right|\gg \left| c_1 \right|$$

$$-c_2\approx M$$

$$\left[\begin{array}{c} m \\ m_{e} \end{array}\right]=c_{1} e^{\lambda_{1} t} \mathbf{v}_{\mathbf{1}}+c_{2} e^{\lambda_{2} t} \mathbf{v}_{\mathbf{2}}+\mathbf{v}_{\mathbf{3}}\approx c_{2} e^{\lambda_{2} t} \mathbf{v}_{\mathbf{2}}+\mathbf{v}_{\mathbf{3}}$$

$$m\approx c_{2} e^{\lambda_{2} t}+M\approx c_{2} e^{\lambda_{2} t}-c_{2}=c_{2}(e^{\lambda_{2} t}-1)$$

$$-\lambda_{2}^{2}-(p_1+p_e+p_m)\lambda_{2}=p_mp_e\approx -(p_1+p_e+p_m)\lambda_{2}\rightarrow \lambda_{2}\approx -\frac { p_m }{ p_1+p_e+p_m }p_e$$

$$e^{\lambda_{2} t}-1\approx\lambda_{2} t\quad \rightarrow m \approx c_{2}\left(e^{\lambda_{2} t}-1\right)\approx c_2\lambda_{2} t$$

$$m \approx c_{2} \lambda_{2} t\approx -\frac{c_{2}p_{e}}{p_{1}+p_{e}+p_{m}} p_{m}$$

$$\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$

$$\frac{-(p_{1}+p_{e}+p_{m}) \pm \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}{2}\times\frac{-(p_{1}+p_{e}+p_{m}) \mp \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}{-(p_{1}+p_{e}+p_{m}) \mp \sqrt{(p_{1}+p_{e}+p_{m})^{2}-4 p_{m} p_{e}}}$$

$$\lambda_{2}=\frac { 2 p_{m} p_{e} }{ -\left(p_{1}+p_{e}+p_{m}\right) - \sqrt{\left(p_{1}+p_{e}+p_{m}\right)^{2}-4 p_{m} p_{e}} } \approx \frac { p_{m} p_{e} }{ -(p_{1}+p_{e}+p_{m}) }$$

$$\lambda_{1}=\frac {2 p_{m} p_{e} }{ -\left(p_{1}+p_{e}+p_{m}\right) + \sqrt{\left(p_{1}+p_{e}+p_{m}\right)^{2}-4 p_{m} p_{e}} }$$