## 1 Introduction
### Comparative Analysis of Various Theoretical Methods for Describing Dielectric Relaxation in HBCs  
Currently, theoretical descriptions of the kinetics of the volume-charge polarization of HBCs are made by accounting for the nonlinear effects associated with the influence of second- and third-order nonlinearities in the polarizing field on the parameters of the spectra of thermally stimulated depolarization currents (TSDCs) [1; 2; 3; 72; 75; 125; 133] and dielectric losses [1; 4; 68; 71; 120; 121]. In regions with sufficiently high temperatures (T \(>\) 250 K), these effects manifest in the form of a non-linear dependence of the amplitude of the density of the TSDCs on the modulus of the electric field strength [1], and in low-temperature regions (T \(\approx\)70-100 K), when the main contribution to relaxation is made by quantum proton transitions, this leads to deviation from the classical laws of Debye dispersion [2]. The proton relaxation models proposed in [1] are based on a mathematical apparatus applicable only to a certain experimental temperature range, and when deviating from this range, significant discrepancies arise between the theoretical and measured values of the relaxer parameters [125; 132; 133; 134]. The methods presented in [125; 133; 134] do not allow for the in-detail study of the high-temperature and dielectric loss tangent \(\mathrm{tg\delta(T)}\). The numerical calculation of the proton activation energy in the vicinity of the first two (low temperature) mono-relaxation maxima of TSDC density in chalcanthite (CuSO\({}_{4}\cdot\)5H\({}_{2}\)O (T\({}_{\max,\,1}=94\) K; T\({}_{\max,\,2}=138\) K) [1]) and phlogopite mica (KMg\({}_{53}\)(AlSi\({}_{3}\)O\({}_{10}\))(OH\({}_{2}\)) (T\({}_{\max,\,1}=100\) K; T\({}_{\max,\,2}=130\) K) [1]) using the methods presented in [125] yields a significant discrepancy between theory and experiments. So, for chalcanthite, U\({}_{0,\exp,\,1}=0.07\)\(\pm\) 0.01 eV, and U\({}_{0,\text{th, 1}}=0.13\) eV; U\({}_{0,\text{exp, 2}}=0.11\)\(\pm\) 0.01 eV, and U\({}_{0,\text{th, 2}}=0.21\) eV (corresponding to the table on page 82 in [125] and Table 1 on page 136 in [135]). For phlogopite, U\({}_{0,\exp,\,1}=0.05\)\(\pm\) 0.01 eV, and U\({}_{0,\text{th, 1}}=0.08\) eV; U\({}_{0,\text{exp, 2}}=0.17\)\(\pm\) 0.02 eV, and U\({}_{0,\text{th, 2}}=0.2\) eV (from Table 2 on page 136 in [135]). In the region where the high-temperature maxima \(J_{\exp}(\text{T})\) of chalcanthite (T\({}_{\max}=170,206,230,246\) K (Figure 1 on page 81 in [125] and Figure 3 on page 134 in [135])) and phlogopite (T\({}_{\max}=178,206,235,260\) K (Figure 4 on page 135 in [135])), the values U\({}_{0,\text{th}}\) and U\({}_{0,\exp}\) agree well; however, the amplitudes of the theoretical maxima \(J_{\max,\text{th}}\) are 2-4 orders of magnitude lower than the measured \(J_{\max,\text{exp}}\). The use of the density matrix in the WKB approximation allows us to account for the quasi-sensitivity of the proton energy spectrum [133], leading to the agreement of the values U\({}_{0,\text{th}}\) and U\({}_{0,\exp}\) at low temperatures and at high temperatures, as expected, the influence of quantum effects on the values of U\({}_{0,\text{th}}\) is insignificant (see the table on page 12 in [133] and Table 3.4 on page 140 in [135]). Moreover, the ratios of \(J_{\max,\text{th}}\) to \(J_{\max,\text{exp}}\) for all maxima are practically the same [133; 135]. The disadvantage of the mathematical model in [133] is the bulkiness of the formula for calculating \(J_{\text{th}}(\text{T})\) (see expressions (28) and (29) on pages 10 and 11 in [133]). Also, when displaying working formulas (on pages 80 and 81 in [125], and (26) on page 10 in [133]), the nonlinear effects at volume-charge polarization occurring in the area of sufficiently high temperatures (T \(>\) 250 K) cannot be studied.