## 5 Conclusions  
This work is devoted to improving the schemes and methods of the existing quasiclassical kinetic theory of dielectric relaxation [1; 2; 3; 4; 5] and, on this basis, numerical calculations of the parameters of relaxers (protons) in HBC over a wide temperature range (50-550 K) and fields (0.1-1 MV/m). The subject of these improvements is the development of a more stringent, in comparison with [1; 2; 3; 4; 5], theoretical method to consider the effects of corrections to the non-stationary quantum transparency of a parabolic potential barrier caused by an electric field. In previous works [1; 6], such corrections to the parabolic potential barrier were not calculated. A double-symmetrical potential pit with a parabolic potential barrier was adopted as a geometric model of the crystalline potential field. The solutions of the nonlinear quasi-classical kinetic Equation (11), in continuation [1; 6], are constructed in the most general form in Equation (16), considering the additional elements of \(\exp(-2\underline{\varepsilon}(t))\) in expressions for non-stationary quantum transparency in Equations (7) and (8). The diffusion and mobility coefficients of protons are constructed to model the parabolic double-symmetric potential well of forms that are perturbed by an external variable field in Equations (9) and (10). The Bucci-Rive phenomenological formula (Equation (22)) for thermally stimulated depolarization current density (TSDC) was first written first, considering the stationary quantum transparency of the parabolic potential proton barrier (Equation (24)). When calculating the initial polarization of the crystal, the updated stationary Equations (17)-(19) obtained in this work were used. Previously, these tasks were not solved. In this case, the choice of the parabolic form of the potential barrier allows, at the theoretical level, for the mathematical model to be brought closer to the real spatial structure of the crystalline potential field. This is evident from the comparison of the numerical values of the relaxer parameters (activation energy, equilibrium concentration) calculated from the experiment and on the basis of Equation (22), considering tunneling and excluding proton tunneling, using the example of crystals of phlogopite, onot talc, muscovite and chemically pure chalcanthite (Tables 1-4). It can be seen from Tables 1-4 that accounting for proton tunnel transitions in the Equation (22) significantly approximates the theoretical values of activation energy and equilibrium concentration of protons relaxing in low-temperature (50-100 K) and in high-temperature (350-550 K) maxima regions of TSDC density, compared to their experimental values in HBC. Thus, in this work, at a higher theoretical level, in comparison with [4], it was confirmed that the quantum effects associated with proton tunnel transitions in the HBC anion sublattice dominate the region of low-temperature (50-100 K) TSDC maxima. In continuation of this [1; 7], we found that quantum effects continue to significantly affect the dielectric relaxation kinetics in proton semiconductors and dielectrics (PSD) in the region of high-temperature (250-550 K) TSDC maxima. At the same time, in continuation of [1; 4], we carried out precision measurements of temperature spectra of TSDC density for chalcanthite crystals. The effect of alloying impurities' concentrations and crystal calcination temperatures on the properties and parameters of experimental maxima in the TSDC spectrum of chalcanthite was established. We described, in the form of chemical equations, the physical mechanism of the quantum tunnel motion of HBC protons with a complex crystal structure (Figure 5). A significant effect of proton tunneling on the amplitudes of theoretical TSDC density maxima has been established. It follows from Figures 6-9 that, taking into account quantum effects in the Bucci-Rive formula (Equation (24)), the theoretical amplitudes of TSDC can be approximated in comparison to experimental ones over almost the entire temperature range (50-550 K).