## 5 Effect of Temperature on Theoretical Spectra of Complex Dielectric Permittivity
Considering \(\gamma\mathrm{P}_{1}^{(\omega)}(\mathrm{t})=\varepsilon_{0}\left(\hat{ \varepsilon}_{1}^{(\omega)}-\varepsilon_{\infty}\right)\cdot\mathrm{E}_{ \mathrm{pol}}(\mathrm{t})\)[1], calculating \(\hat{\alpha}_{1}^{(\omega)}=\varepsilon_{0}\varepsilon_{\infty}\left(\Gamma _{1}^{(\omega)}-\mathrm{i}\Gamma_{2}^{(\omega)}\right)\); \(\hat{\varepsilon}_{1}^{(\omega)}=\varepsilon_{\infty}\left(1+\Gamma_{1}^{( \omega)}-\mathrm{i}\Gamma_{2}^{(\omega)}\right)\), we find  
\[\left[\hat{\varepsilon}_{1}^{(\omega)}\right]=\mathrm{Re}\!\left[\hat{ \varepsilon}_{1}^{(\omega)}\right]=\varepsilon_{\infty}\!\left(1+\Gamma_{1}^{( \omega)}\right)\!,\left[\hat{\varepsilon}_{1}^{(\omega)}\right]^{//}=\mathrm{ Im}\!\left[\hat{\varepsilon}_{1}^{(\omega)}\right]=\varepsilon_{\infty}\Gamma_{2}^{( \omega)},\left(\mathrm{tg}\delta^{(\omega)}(\mathrm{T})\right)_{1}=\frac{ \left[\;\varepsilon_{1}^{(\omega)}\right]^{/}}{\left[\hat{\varepsilon}_{1}^{( \omega)}\right]}=\frac{\Gamma_{2}^{(\omega)}}{1+\Gamma_{1}^{(\omega)}}. \tag{46}\]  
In the case of \(\mathrm{T}>>\mathrm{T}_{\mathrm{cr,relax}}\), \(\mathrm{T}_{\mathrm{M}}\ll T_{\mathrm{D}}\), by substituting \(\Gamma_{1,\mathrm{M}}^{(\omega)}\approx\frac{1}{1+\omega^{2}T_{\mathrm{M}}^{2 }}\), \(\Gamma_{2,\mathrm{M}}^{(\omega)}\approx\frac{\omega\mathrm{T}_{\mathrm{M}}}{1+ \omega^{2}T_{\mathrm{M}}^{2}}\) in (46), we obtain the expressions (39) approximated in the first approximation by the parameter \(\gamma\)  
\[\left[\hat{\varepsilon}_{1}^{(\omega)}\right]_{\mathrm{M}}=\varepsilon_{ \infty}\!\left(1+\frac{1}{1+\omega^{2}T_{\mathrm{M}}^{2}}\right)\!,\left[\hat{ \varepsilon}_{1}^{(\omega)}\right]_{\mathrm{M}}^{/}=\varepsilon_{\infty}\frac {\omega\mathrm{T}_{\mathrm{M}}}{1+\omega^{2}T_{\mathrm{M}}^{2}}\!,\left(\mathrm{ tg}\delta^{(\omega)}(\mathrm{T})\right)_{1,\mathrm{M}}=\frac{\omega T_{ \mathrm{M}}}{2+\omega^{2}T_{\mathrm{M}}^{2}}.\] (46a) From this, by virtue of \(\left[\hat{\varepsilon}_{1}^{(\omega=0)}\right]_{\mathrm{M}}=\varepsilon_{ \mathrm{SM,1}}(\mathrm{T})=\varepsilon_{\infty}\!\left(1+\Gamma_{1\mathrm{M}}^ {(\omega=0)}\right)=2\varepsilon_{\infty}\!\left[\hat{\varepsilon}_{1}^{( \omega=0)}\right]_{\mathrm{M}}^{/}=0\), entering the designation \(\varepsilon_{S,\mathrm{M},1}-\varepsilon_{\infty}=\varepsilon_{\infty}\), we have \[\left[\hat{\varepsilon}_{1}^{(\omega)}\right]_{\mathrm{M}}=\varepsilon_{\infty} +\frac{\varepsilon_{\mathrm{SM,1}}-\varepsilon_{\infty}}{1+\omega^{2}T_{ \mathrm{M}}^{2}},\left[\hat{\varepsilon}_{1}^{(\omega)}\right]_{\mathrm{M}}^{ //}=\frac{(\varepsilon_{\mathrm{SM,1}}-\varepsilon_{\infty})\omega T_{ \mathrm{M}}}{1+\omega^{2}T_{\mathrm{M}}^{2}},\] (46b) \[\left(\mathrm{tg}\delta^{(\omega)}(\mathrm{T})\right)_{1,\mathrm{M}}=\frac{( \varepsilon_{\mathrm{S,M},1}-\varepsilon_{\infty})\cdot\omega T_{\mathrm{M}}}{ \varepsilon_{\mathrm{S,M},1}+\varepsilon_{\infty}\cdot\omega^{2}T_{\mathrm{M} }^{2}},\;\left[\left(\mathrm{tg}\delta\right)_{1,\mathrm{M}}\right]|_{\omega \rightarrow0}=0,\;\left[\left(\mathrm{tg}\delta\right)_{1,\mathrm{M}}\right]|_{ \omega\rightarrow\infty}=0,\] (46c) which is consistent with the results of the linear theory of dielectric losses [1,2] in the temperature region \(\mathrm{T}>>\mathrm{T}_{\mathrm{cr,relax}}\).