## 1 Introduction
### Basic Principles of Quasi-Classical Model of Ion Relaxation Polarization
Reorientation of a protonated anion due to the transition of protons inside it \[\text{HSO}_{4}^{-}\rightarrow\text{SO}_{4}\text{H}^{-}.\]Transition of a proton between cells of the anionic sublattice, and the newly formed protonated anion acquires the opposite orientation, \[\text{SO}_{4}\text{H}^{-}\to\text{HSO}_{4}^{-}.\] 5. Reorientation of the second protonated anion \[\text{HSO}_{4}^{-}\to\text{SO}_{4}\text{H}^{-}.\] 6. Transition of a proton to a water molecule and the formation of a defect \(\text{H}_{3}\text{O}^{+}\): \[\text{SO}_{4}\text{H}^{-}+\text{H}_{2}\text{O}\to\text{SO}_{4}\text{H}^{-}+ \text{H}_{2}\text{O}^{+}.\]  
Therefore, in order to simplify the mathematical model of migration polarization in HBCs, we consider the transfer of only one type of charge carriers, protons, and the oppositely charged ions (anions) are considered to be slow-moving and capable of forming a homogeneous potential force background in the field in which protons move [1]. From the condition of the transitions of charge carriers (protons) between neighboring states (potential pits) equal to a number of \(\text{i}-1,\text{i},\text{i}+1\) follows a system of kinetic equations [1; 2; 3; 4]. In HBCs, as well as in other ionic dielectrics similar to them in terms of structure and lattice properties, if the condition \(\overset{\rightarrow}{\text{E}}\parallel\overset{\rightarrow}{\text{C}}\) is fulfilled, the geometrical model of the crystal potential for conduction ions is assumed to be one-dimensional. When taking into account the influences on ionic conductivity imposed by additional factors related to the interactions of conduction ions with other ionic subsystems embedded in the crystal structure of a  
Figure 1: Scheme of the \(\text{H}_{3}\text{O}^{+}\) ion movement due to the gradual transfer of the proton in a chalcanthite crystal \(\text{CuSO}_{4}\text{-}\text{5H}_{2}\text{O}\)[3]. The arrows show the movement of the proton; the numbers indicate the stages of its movement. given dielectric, the transition to a three-dimensional model of ionic relaxation polarization is natural. Thus, based on Figure 1 and continuing the methodology from [3], we adopt a model of a one-dimensional triple symmetric potential well to describe the motion of the relaxerot (proton) in hydrogen-bonded crystals (HBC). Figure 2 illustrates the unperturbed system, while Figure 3 shows the system perturbed by an electric field. In the absence of external perturbation, the transition probability rates of the proton with energy \(E\) in all directions are equal in magnitude  
\[W_{\mathrm{i-1;\lambda}}=W_{\mathrm{i+1;\lambda}}=W^{(0)},\ W_{\mathrm{i; \lambda-1}}=W_{\mathrm{i;\lambda+1}}=W^{(0)}.\]  
When an external perturbation is applied, the transition probability rates of the proton in the direction of the field are higher than against the field \(W_{\mathrm{i-1;\lambda}}^{(-)}>W_{\mathrm{i+1;\lambda}}^{(+)},\ W_{\mathrm{i; \lambda-1}}^{(+)}<W_{\mathrm{i;\lambda+1}}^{(-)}\). In the scale of a one-dimensional spatially periodic potential field U(x), in which the triple symmetric well model is embedded in the system, we obtain a picture of a multi-well symmetric crystalline potential (Figures 4 and 5), which is more closely approximated to the  
Figure 3: The model of a perturbed by an electric field \(\mathrm{\overset{\rightarrow}{E}}\) one-dimensional triple symmetric potential well for the proton in HBC. Figure 2: The model of an unperturbed one-dimensional triple symmetric potential well for the proton in HBC. real dielectric structure. In this model: d--thickness of the crystal; U\({}_{0}\)--activation energy of the proton at the bond; U\({}_{\rm max}\)'proton escape work from the crystal; N\({}_{\rm W}\)'total number of potential wells in the model. In general, the one-dimensional kinetic equation of transfer of the generalized most mobile charge carriers (ions with a certain charge sign, called conduction ions) with respect  
Figure 4: The model of an unperturbed one-dimensional multi-well symmetric crystalline potential for the proton in HBC.