## 2 Materials and Methods (Theoretical Bases and Methods)  
The methods of the quasi-classical theory of dielectric relaxation [1; 4] make it possible to accept a hydrogen ion (proton) as a physical relaxer, whose motion with different values of molecular parameters (activation energy U\({}_{0}\); natural oscillation frequency \(\
u_{0}\); potential barrier width \(\delta_{0}\); lattice constant \(a\); equilibrium concentration \(n_{0}\)) in the vicinity (on the set of points of the continuum measure) of the temperatures of the corresponding monorelaxation maxima determines the features of thermally stimulated depolarization in various temperature ranges [1; 2; 4]. This approach makes it possible to classify HBC by its electrophysical properties in a wide range of field parameters (0.1-1 MV/m) and temperatures (50-550 K), as proton semiconductors and dielectrics (PSD) [1; 3; 4]. Theoretical studies carried out using the phenomenological theory of dielectric relaxation [1; 4] point to the classical mechanism of relaxation polarization caused by the thermal activation of protons in the HBC at high temperatures (100-450 K). The calculations of the parameters of relaxers successfully correspond to their measured values in the region of high-temperature maxima of thermally stimulated depolarization currents (TSDC) [1; 4]. In the low-temperature range (50-100 K), the activation energies and equilibrium proton concentrations in HBC, calculated according to the classical theory, differ markedly (by 20-50%) [1] from the values calculated from an analysis of the experimental spectra of thermally stimulated depolarization currents (TSDC) [1,4]. This indicates the need to more strictly consider the quantum effects (associated with proton-tunneling transitions) on the theoretical TSDC density spectra in the nitrogen temperature region. This approach is further justified in the region of ultra-low temperatures (1-10 K), when the dimensional effects associated with a high probability of the quantum transition of protons inside clusters of nanometer size (1-10 nm) are manifested in HBC [1]. Recent theoretical studies have shown that the tunnel transitions of protons continue to influence the dielectric relaxation in HBC in the region of sufficiently high temperatures (250-550 K) during the formation of nonlinear space-charge polarization [1,3,7,8]. The aim of this paper is to improve the existing phenomenological theory of dielectric relaxation in HBC [1,4] by extending it to the low-temperature region (50-100 K). On this basis, a mathematical substantiation of the features of low-temperature thermally stimulated depolarization was revealed in the experiment, associated with quantum proton transitions through hydrogen bonds against the background of small activation energies and the width of the potential when the statistically averaged transparency of the potential barrier for protons reaches relatively high values. The Bucci-Rive method [1,4] will be used as the experimental methodological basis of the study, and the theoretical methodology of the study will be based on solutions of the kinetic equation describing proton transport in a potential field of a crystal lattice when disturbed by an external field, with potential parabolic barriers [1,3,7]. Based on this goal, the following research objectives were formulated:  
1. Use a more stringent account of the effects that the external electric field corrections had on the statistically averaged non-stationary quantum transparency of the parabolic potential barrier for relaxers (protons) in HBC compared to previous studies [1, 2, 3, 4, 5, 6]. This kind of correction for a parabolic potential barrier was not previously calculated. 2. Write a generalized solution of the quasi-classical kinetic equation of nonlinear dielectric relaxation using a small parameter of perturbation theory in HBC [1,6]. As a geometric model of the undisturbed crystalline potential, we assumed a double symmetrical potential pit with a barrier of parabolic shape.