## 1 Introduction
### Comparative Analysis of Different Ion-Relaxation Polarization Models
Applying the methods of the kinetic theory of proton relaxation [1,65,70,125,126] to the calculation of the temperature spectra of the thermostimulated depolarization currents (TSDC) of chalcanthite \(\mathrm{CuSO}_{4}\cdot\mathrm{5H}_{2}\mathrm{O}\) shows good agreement with the experimental results when calculating the parameters of relaxers in the high-temperature region (\(\mathrm{T}_{\mathrm{max}}=\) 170,206,230,246 K), where the main contribution to dielectric relaxation is made by thermally activated (classical) proton transitions through the potential barrier [71]. At low temperatures (\(\mathrm{T}_{\mathrm{max}}=\) 94 K and \(\mathrm{T}_{\mathrm{max}}=\) 138 K), there is a significant discrepancy between the theoretical \(\mathrm{U}_{0,\mathrm{theory}}\) and measured \(\mathrm{U}_{0,\mathrm{exp}}\) activation energy values (see Table on page 82 in [135]). Moreover, for high-temperature maxima, the theoretical values of the current density amplitudes \(J_{\mathrm{max,theory}}\) are 2-4 orders of magnitude lower than the experimental \(J_{\mathrm{max,exp}}\), and for low-temperature maxima, \(J_{\mathrm{max,theory}}\) and \(J_{\mathrm{max,exp}}\) are consistent (see Figure 1 on page 81 [135]). Calculating the spectra \(J_{\mathrm{theory}}(\mathrm{T})\) using the density matrix apparatus [66,133] showed the correspondence of the values \(\mathrm{U}_{0,\mathrm{theory}}\) and \(\mathrm{U}_{0,\mathrm{exp}}\) (see the table on page 12 in [65]), although the discrepancies between the values \(J_{\mathrm{max,theory}}\) and \(J_{\mathrm{max,exp}}\) at high temperatures remained (Figure 1 on page 12 [65]), which the authors [65] explained to be due to the influence of conduction currents unaccounted for in the mathematical model [3,4] in the high-temperature range. A similar situation occurs when calculating the spectra of \(J_{\mathrm{theory}}(\mathrm{T})\) in phlogopite\({}^{\ast}\)KM\({}_{3}(W\mathrm{IS}_{3}\mathrm{O}_{10})(\mathrm{OH})\)\({}_{2}\)[1,135]. The dependence \(J_{\mathrm{theory}}(\mathrm{T})\) in the region of the seventh (due to the relaxation of the volumetric charge) experimental maximum density TSDC (where \(\mathrm{T}_{\mathrm{max}}=\) 290 in chalcanthite [1,125] and \(\mathrm{T}_{\mathrm{max}}=\) 360 K in phlogopite [1,135]) could not be calculated using the methods reported in [125] and [133]. This is probably due to nonlinear effects unaccounted for in [125,133] in regard to the formation of volume-charge polarization in the dielectric. The components of the complex permittivity (CDP) \(\varepsilon^{\prime}(\mathrm{\
u}\mathrm{T})\), \(\varepsilon^{\prime}/(\mathrm{\
u}\mathrm{T})\)[65,66], constructed by taking proton tunneling into account, in the field of quantum diffusion relaxation in HBCs (\(\mathrm{T}<\) 100 K) differ from the laws of classical Debye dispersion [1,135]. The theoretical assessment of the influence of third-order nonlinearities on the field in [6] is incomplete. Since the theoretical spectra \(\mathrm{tg}\delta(\mathrm{\
u}\mathrm{T})\) in [65,66,70,71] were not numerically calculated, a comparative analysis of the effectiveness of the methods used in [70,71] was carried out according to the following qualitative indicators:  
1. As the small parameter of the theory of indignations \(\mathrm{\gamma}=\zeta_{0}(\mathrm{T})\frac{W^{(1)}(\mathrm{T})}{W^{(0)}( \mathrm{T})}\)[1] is expressed through the small parameter \(\zeta_{0}=\frac{\mathrm{qE}_{0}a}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\)[1,125], polarization decomposition in a row, to within a certain degree (linear, square, etc.) of a member of \(\mathrm{\gamma}\) in accordance with [67,69,70,135], reflects the mechanism of the relaxation process in the dielectric, depending on both temperature and factors (parameters) relating to the microscopic effects of interactions of the relaxer with the crystal lattice.