## 3 Results
### Numerical Study of the Loss Tangent of Dielectric Materials
According to the data in Figure 11, the parameters of high-temperature (160 K, 220 K, 265 K, 310 K) theoretical maxima \(2^{\prime}\)-\(5^{\prime}\) calculated using a strict quantum-mechanical model (according to the expression (27) in [4]) completely match the results of the quasi-classical nonlinear model for theoretical maxima 2-5 (given in Table 2 and Figure 10). According to the results of calculating \(\mathrm{tg}\delta_{\mathrm{th}}^{(\mathrm{v})}(\mathrm{T})\) using the methods of quantum kinetic theory [4] for the theoretical maximum \(6^{\prime}\) (Figure 11), established at a temperature T = 105 K from the expression (27) in [4] with the maximum amplitude \(\mathrm{tg}\delta_{\mathrm{th}}^{(\mathrm{v})}(105)\approx 0.5\times 10^{-3}\), at absolute zero temperatures (from the expression (27) in [4]), we have \(\mathrm{tg}\delta_{\mathrm{th}}^{(\mathrm{v})}(0)\to 0.25\times 10^{-3}\). The description of the graphs is similar to Figure 10 but with elements of the quantum theory of low-temperature polarization in HBC. Similar patterns appear in a comparative analysis of the results of strict nonlinear quasi-classical and quantum-mechanical models of dielectric relaxation in a gypsum sample CaSO\({}_{4}\cdot\)0.5H\({}_{2}\)O (Figures 12 and 13). Figure 11: Temperature spectra of dielectric loss tangent in onotlalc crystals modernized by quantum kinetic theory methods [4]. Figure 12: Temperature spectra of the dielectric loss tangent in gypsum. Figure 13: Temperature spectra of dielectric loss tangent in a gypsum sample modernized by quantum kinetic theory methods [4]. The calculation results of the angle of dielectric loss tangent in the region of low temperatures in a gypsum sample using the quasi-classical nonlinear model of dielectric relaxation are presented in the form of a theoretical maximum \(6^{\prime}\) established at a temperature T = 100 K with maximum amplitude \(\text{tg}\delta_{\text{th}}^{(\
u)}(100)\approx 0.8\times 10^{-3}\). At absolute zero temperatures (from Formula (151)), we have \(\text{tg}\delta_{\text{th}}^{(\
u)}(0)\to 0.25\times 10^{-3}\). According to the results of calculating \(\text{tg}\delta_{\text{th}}^{(\
u)}(\text{T})\) using the methods of quantum kinetic theory [4] for the theoretical maximum \(6^{\prime}\), established at a temperature T = 97 K with the maximum amplitude \(\text{tg}\delta_{\text{th}}^{(\
u)}(97)\approx 0.85\times 10^{-3}\), at absolute zero temperatures (from the expression (27) in [4]), we have \(\text{tg}\delta_{\text{th}}^{(\
u)}(0)\to 0.5\times 10^{-3}\). The results obtained are consistent with the data on onot lalc. The experimental graph 1 was obtained from measurements on gypsum CaSO\({}_{4}\)-0.5H\({}_{2}\)O pre-calined at T\({}_{ac}=393\) K, at a polarization temperature T = 300 K and an alternating field frequency \(\
u=7\) MHz. Theoretical graphs (2-5) were calculated near the frequencies corresponding to maxima on the experimental graph 1, using the quasi-classical kinetic theory methods (from expression (151), with parameters (120), (121), (122) as described in this article). Additionally, theoretical graphs (2\({}^{\prime}\)-5\({}^{\prime}\)) were computed using direct quantum-mechanical calculations based on the model equations (from expression (27) in [4]), combined with the comparison function minimization method (MCF-method) [73; 171; 172]. The theoretical maximum \(6^{\prime}\) has not yet been experimentally identified. From the comparison and analysis of data from Tables 2 and 3 of this article, Tables 1, 2 from [4], and Tables 1-4 from [3], it is evident that the second and fourth maxima in Figures 10-13 are associated with the relaxation of Bjerrum ionization defects H\({}_{3}\)O\({}^{+}\) and OH\({}^{-}\). The third maximum is a dipole and is caused by rotations of crystallization water molecules. The fourth maxima is associated with the relaxation of VL and VD type complexes.