## 1 Introduction
### Basic Principles of Quasi-Classical Model of Ion Relaxation Polarization  
As the main (most mobile) carriers of electric charge (relaxers) in dielectrics, ions (of an arbitrary sign of charge) are accepted and capable of performing relaxation (reciprocal) movement in the potential field of a crystal lattice (or sublattice) perturbed by an external electric field. The basic equations of ion motion in the crystal, in fact, describe the diffusion of ions (cations moving in the direction of the lines of force of the external electric field and anions moving against the field) against the background of the interaction of this type of ion with a potential background generated by heavier, inactive (in comparison) ions localized in the nodes of the crystal lattice. From a quantum theory perspective of crystal lattice oscillations, the most mobile (light) ions form an ion subsystem moving in the field of the phonon subsystem formed by vibrations of inactive, massive (compared to the data) ions or ion groups (ion clusters), and the interaction effects of the ion and phonon subsystems are described using the corresponding quantum mechanical operators (similar to the electron-phonon interaction operators known in solid-state theory [4], which is referenced in describing the quantum properties of metals or semiconductors and dielectrics with high electron conductivity). For example, a model of the quantum motion of hydrogen ions (protons) in the field of the phonon subsystem formed by oscillations of ions belonging to the anion sublattice (the anion component of the phonon subsystem for protons) in an HBC is effective from the perspective of quantum kinetic theory [4]. Regarding the quantum theory of proton relaxation polarization in an HBC, one cannot but count the influence on the proton subsystem from the side of the phonon subsystem formed by vibrations of ions even heavier than anions that belong to the additional sublattice (the ion component of the phonon subsystem for protons) in the HBC. As noted in Section 2.1, in the framework of this study, the issues of the quantum kinetic theory of proton relaxation polarization are not examined, limiting the scope to quasi-classical approximations in the description of the quantum model of the proton tunneling motion in the HBC. Applied to the processes of ion relaxation polarization in dielectrics of various classes (except HBCs), at the level of the quasi-classical kinetic theory of dielectric relaxation, in view of the colossal masses of ions or ion groups in comparison with protons, there is no strict need to consider the tunnel quantum component in the kinetic coefficients (e.g., in the diffusion and mobility coefficients) for ions. However, in the mathematical description of proton relaxation in an HBC, it is necessary to consider that due to the relatively small mass of the proton compared to the masses of the anionic sublattice ions, the quantum transparency coefficient of the potential barrier for protons increases significantly (up to 0.0001-0.15), especially in the context of the small activation energies (\(\mathrm{U}_{0}\approx\) 0.01-0.1 eV) and potential barrier widths (\(\delta_{0}\approx\) 0.08-0.1 nm) characteristic of crystals with hydrogen bonds for protons on bonds in the low-temperature region (for protons relaxing in the region T = 50-100 K). In this connection, even within the framework of the methods of quasi-classical kinetic theory [125; 126; 127; 128; 129; 130; 131], a rather strict account of proton quantum tunneling is required in the recording and study of the coefficients of the quasi-classical kinetic equation of proton relaxation [122; 123; 124].