## 5 Effect of Temperature on Theoretical Spectra of Complex Dielectric Permittivity
\begin{array}{l}\left[\xi^{(\omega)}\right]_{\text{M}}^{/}\approx\varepsilon_ {\infty}\left(\left(\pi\omega T_{\text{M}}\right)^{2}\sqrt{1+\omega^{2}T_{\text {M}}^{2}}(\cos(2\Delta_{1})+\text{ch}(2\Delta_{2}))\times\frac{T_{\text{D}}} {T_{\text{M}}}+\right.\\ \left.+4(\Delta_{1}\sin(2\Delta_{1})+\Delta_{2}\text{sh}(2\Delta_{2}))+4\omega T _{\text{M}}(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2\Delta_{2})) \times\right.\\ \left.\times\left(\left(\pi\omega T_{\text{M}}\right)^{2}\sqrt{1+\omega^{2}T_{ \text{M}}^{2}}(\cos(2\Delta_{1})+\text{ch}(2\Delta_{2}))\times\frac{T_{\text {D}}}{T_{\text{M}}}+8\omega T_{\text{M}}\times\right.\\ \left.\times(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2\Delta_{2}))+4( \text{ch}(2\Delta_{2})-\cos(2\Delta_{1})))^{-1}\right.\end{array}  \tag{64a}\] \[ \begin{array}{l}\left[\xi^{(\omega)}\right]_{\text{M}}^{/}\approx \varepsilon_{\infty}\Big{(}\left(\pi\omega T_{\text{M}}\right)^{2}\sqrt{1+ \omega^{2}T_{\text{M}}^{2}}(\cos(2\Delta_{1})+\text{ch}(2\Delta_{2}))\times \frac{T_{\text{D}}}{T_{\text{M}}}+\right.\\ \left.+4(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2\Delta_{2}))-4\omega T _{\text{M}}(\Delta_{1}\sin(2\Delta_{1})+\Delta_{2}\text{sh}(2\Delta_{2})) \times\right.\\ \left.\times\left(\left(\pi\omega T_{\text{M}}\right)^{2}\sqrt{1+\omega^{2}T_{ \text{M}}^{2}}(\cos(2\Delta_{1})+\text{ch}(2\Delta_{2}))\times\frac{T_{\text {D}}}{T_{\text{M}}}+8\omega T_{\text{M}}\times\right.\\ \left.\times(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2\Delta_{2}))+4( \text{ch}(2\Delta_{2})-\cos(2\Delta_{1})))^{-1}\right.\end{array} \] (64b) Here \[\left.\Delta_{1,2}=\frac{\pi}{\sqrt{2}}\sqrt{\frac{T_{\text{D}}}{T_{\text{M} }}\Big{[}\sqrt{1+\omega^{2}T_{\text{M}}^{2}+1}\Big{]}}.\] The tangent of the dielectric loss angle at the fundamental frequency of the field \[\left[\text{tg}\delta^{(\omega)}\Big{(}\frac{T_{\text{D}}}{T_{\text{M}}}; \omega T_{\text{M}}\Big{)}\right]_{\text{M}}=\frac{\left[\xi^{(\omega)}\right]_ {\text{M}}^{/}}{\left[\xi^{(\omega)}\right]_{\text{M}}^{/}}\] (65) \[ \begin{array}{l}\left.=\left\{\frac{T_{\text{D}}(\omega T_{\text{M}})^ {2}\pi^{2}}{T_{\text{M}}}\sqrt{1+\left(T_{\text{M}}\right)^{2}}(\cos(2\Delta_{ 1})+\text{ch}(2\Delta_{2}))+4(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2 \Delta_{2}))-\right.\\ \left.-4\omega T_{\text{M}}(\Delta_{1}\sin(2\Delta_{1})+\Delta_{2}\text{sh}(2 \Delta_{2}))\right\}\times\left\{\frac{T_{\text{D}}(\omega T_{\text{M}})^{2} \pi^{2}}{T_{\text{M}}}\sqrt{1+\left(\omega T_{\text{M}}\right)^{2}}(\cos(2 \Delta_{1})+\text{ch}(2\Delta_{2}))+\right.\\ \left.+4(\Delta_{1}\sin(2\Delta_{1})+\Delta_{2}\text{sh}(2\Delta_{2}))+4\omega T _{\text{M}}(\Delta_{2}\sin(2\Delta_{1})-\Delta_{1}\text{sh}(2\Delta_{2})) \right\}^{-1}\end{array} \]  
Figures 1 and 2 show the dependencies graphs \(\left[\text{tg}\delta^{(\omega)}\Big{(}\frac{T_{\text{D}}}{T_{\text{M}}}; \omega T_{\text{M}}\Big{)}\right]_{\text{M}}=\frac{\left[\xi^{(\omega)}\right]_ {\text{M}}^{/}}{\left[\xi^{(\omega)}\right]_{\text{M}}^{/}}\), \(\varepsilon^{/}\approx\left[\xi^{(\omega)}\Big{(}\frac{T_{\text{D}}}{T_{\text{M} }};\omega T_{\text{M}}\Big{)}\right]_{\text{M}}^{/}\) for various values of the parameter \(\alpha_{1}=\frac{T_{\text{D}}}{T_{\text{M}}}\), calculated using a computer program, according to formulas (64a), (64b), (65). With this representation, relaxation processes that have the same values \(\alpha_{1}\), are depicted on the graph \(\text{tg}\delta(\omega T_{\text{M}})\) as one curve. In accordance with Figure 1, an increase in the parameter \(\alpha_{1}\) leads to an increase in the maximum and to a shift in its position to the area of small values, while at \(\alpha_{1}>1000\) the maximum position changes slightly with increasing \(\alpha_{1}\) and remains approximately identically and equal to \(\alpha_{2}\approx 0,1\). With small values of the parameter \(\alpha_{1}\) the maximum position \(\text{tg}\delta(\omega T_{\text{M}})\) determined by the criterion \(\alpha_{1}\alpha_{2}\approx 1\) or \(\alpha_{2}\approx 1\).