## 3 Results
### Dielectric Loss Tangent  
Numerical studies of the theoretical spectra of the complex dielectric constant of a crystal polarized at the fundamental frequency of an alternating field \((\omega)\) should be based on calculations of the dielectric loss tangent using the quasi-classical formula (117) in combination with quasi-classical relaxation parameters (119a) and (119b) in analytical representations (120), (121), and (122). Substituting (117) into formula \(\text{tg}^{(\omega)}(\omega;\text{T})=\frac{\text{Im}[\hat{\varepsilon}( \omega;\text{T})]}{\text{Re}[\hat{\varepsilon}(\omega;\text{T})]}\) produces  
\[\text{tg}^{(\omega)}(\omega;\text{T})=\frac{\Gamma_{2}^{(\omega)}(\text{T})} {1-\Gamma_{1}^{(\omega)}(\text{T})}. \tag{150}\]  
Given representation (120), we obtain  
\[\text{tg}^{(\omega;\text{T})}(\alpha_{1},\alpha_{2})=\frac{\frac{4\alpha_{1} \alpha_{2}}{\pi^{2}}\times\Gamma_{12}^{(\omega)}(\alpha_{1},\alpha_{2})}{1- \frac{4\alpha_{1}}{\pi^{2}}\left(\Gamma_{11}^{(\omega)}(\alpha_{1},\alpha_{2}) +\alpha_{1}\Gamma_{12}^{(\omega)}(\alpha_{1},\alpha_{2})\right)} \tag{151}\]  
Expression (151) is a function of dimensionless relaxation variables \(\alpha_{1}\) and \(\alpha_{2}\), where \(\alpha_{1}(\text{T})=\frac{T_{\text{D}}(\text{T})}{T_{\text{M}}(\text{T})}\) is calculated based on (132), \(\alpha_{1}(\text{T})=\frac{\text{T}_{\text{M}}}{\pi^{2}}\times\frac{W^{(1)}( \text{T})}{W^{(0)}(\text{T})}\), in complex with (133) and (134); moreover, \(\alpha_{2}(\omega;\text{T})=\omega T_{m}(\text{T})\) becomes \(\alpha_{2}(\omega;\text{T})=\omega T_{\text{D}}(\text{T})\) in the field of diffusion relaxation (\(\text{T}<\text{T}_{cr,relax}\) ; \(T_{\text{n,D}}=\frac{T_{\text{D}}}{\text{n}^{2}}<T_{\text{M}}\) \(\text{T}_{\text{M}}\) \(\text{T}_{\text{D}}<\text{n}^{2}\)) or \(\alpha_{2}(\omega;\text{T})=\omega T_{\text{M}}(\text{T})\) in the Maxwell relaxation region (\(\text{T}>\text{T}_{cr,relax}\) ; \(T_{\text{M}}<\text{T}_{n,\text{D}}=\frac{T_{\text{D}}}{\text{n}^{2}};\text{n}^{2 }\frac{T_{\text{M}}}{T_{\text{D}}}<1\)). loss angle \(\text{tg}\delta_{M}^{(\omega)}(\omega;\text{T})=\frac{1}{\omega\text{T}_{M}}\), which is in accordance with the expression for Maxwell relaxation time \(T_{M}=\frac{\varepsilon_{0}\varepsilon_{\infty}}{\mu_{\text{mod}}^{(1)} \cdot\text{q}\cdot\text{q}_{0}}\). Here, it is linear via the field component of ion mobility coefficient \(\mu_{\text{mob}}^{(1)}=\frac{\text{q}\text{q}^{2}W(1)}{\text{k}\text{g}\text{T}}\) and takes the form \(\text{tg}\delta_{M}^{(\omega)}(\omega;\text{T})=\frac{\text{q}^{2}n_{0}\text{ }^{2}W(1)}{\text{m}\varepsilon_{0}\varepsilon_{\infty}\text{k}\text{g}\text{T}}\), providing a solution to an expression known in the physics of dielectrics as the tangent of the angle of losses of conductivity with respect to \(\text{tpob}\delta_{contact}^{(\omega)}(\omega;\text{T})=\frac{\sigma_{contact} \text{(T)}}{\omega\text{c}\cdot\text{q}\cdot\text{c}_{0}\varepsilon_{\infty}}\). In this case, the conductivity coefficient for ions is \(\sigma_{contact}\text{(T)}=\frac{\text{q}^{2}n_{0}\text{ }^{2}W(1)}{\text{k}\text{g}\text{T}}\). This result confirms, at a theoretical level, that within the temperature region, the value of critical temperature \(\text{T}_{cr,relax}\) is substantially higher, which separates the temperature regions (zones) with respect to diffusion \(\text{T}<\text{T}_{cr,relax}\) and Maxwell (\(\text{T}>\text{T}_{cr,relax}\)) relaxation. In ionic crystals, the dielectric relaxation mechanism is reduced into ion conduction processes. Quasi-classical formulas are generalized at the fundamental frequency \((\omega)\) of the alternating field for the real and imaginary components of the complex permittivity of the crystal (117) in the Maxwell relaxation region (\(\text{T}>\text{T}_{cr,relax}\) ; \(n^{2}\frac{T_{M}}{T_{D}}<1\)).