## 1 Introduction
### Comparative Analysis of Various Theoretical Methods for Describing Dielectric Relaxation in HBCs
Therefore, the theoretical dependencies \(J_{\text{th}}(\text{T})\) in the region of the seventh maximum density of TSDC (T\({}_{\max}\)= 290 K-in chalcanthite and T\({}_{\max}\)= 360 K-in phlogopite [1], as shown in [125] and [133]) could not be calculated numerically. It is probable that the conduction currents not taken into account in the models from [125; 133; 135] lead to a colossal excess of \(J_{\max,\exp}\) over the values of \(J_{\max,\text{th}}\) at temperatures T \(>\) 250 K.  
Thus, the existing methods for calculating the spectra of thermostimulated currents in HBCs are characterized by a number of model inaccuracies and inconsistencies between theory and experiments, both in the field of low (T \(<\) 100 K) and in the field of high (T \(>\) 100 K) temperatures. According to the results of precision measurements of the temperature spectra of the dielectric loss tangent tg\(\delta\)(T) in Onot talc (Mg\({}_{53}\)(Si\({}_{4}\)O\({}_{10}\))(OH)\({}_{2}\)) and gypsum (CaSO\({}_{4}\cdot\)0.5H\({}_{2}\)O), at the frequency of the alternating electric field \(\
u_{1}=7\cdot 10^{6}\) Hz, four maxima were found: in talc, they were at T\({}_{\max}=160\) K, 220 K, 265 K, and 310 K (Figure 29 in [8]); in gypsum, they were at T\({}_{\max}=145\) K, 210 K, 270 K, and 320 K (Figure 28 in [8]). Measurements of tg\(\delta\)(T) were also carried out at the frequency \(\
u_{2}=12\cdot 10^{6}\) [1; 8]. As with the experimental energy of activation calculated in [1] using the equation U\({}_{0,\exp}=\frac{k_{\text{H}}T_{\max,\text{T}}T_{\max,\text{T}}}{T_{\max,\text{T}}-T_{ \max,\text{T}}}\ln\left(\frac{\omega_{2}}{\omega_{1}}\right)\), without losses of conductivity, in the field of high temperatures (the fourth maximum), the essential dispersion of values U\({}_{0,\exp}\) are found (Table 6 in [8]). No low-temperature maxima tg\(\delta\)(T) in the HBCs (T \(\approx\) 70-100 K) could be measured [8]. The theoretical values of the activation energy U\({}_{0,\text{th,1}}\) calculated using kinetic theory in the linear approximation of perturbation theory [8] fall within the confidence interval of the measured values U\({}_{0,\exp}\) (Table 1). The low-temperature branch (T \(<\) 100 K) of the tg\(\delta\)(T) spectrum could not be investigated using the methods presented in [8; 135]. Of course, the difference scheme for solving the quantum kinetic equation [1], due to the bulkiness of the numerical calculation algorithm itself, is not rational in terms of optimizing the procedure for comparing the theoretical and experimental results, but it allows us to study the parameters of the low-temperature maximum \((T_{\max};\)tg\(\delta\)(T\({}_{\max}\))) depending on the thickness of the crystalline layer, ranging from 3 nm to 30 \(\upmu\)m. The activation energies U\({}_{0,\text{th,2}}\) calculated in [7] with a thickness of d = 30 \(\upmu\)m are consistent with the values U\({}_{0,\text{th,1}}\) only in the region of the first maximum (160 K in talc and 145 K in gypsum) and significantly diverge at higher temperatures (Table 1). The section of the temperature spectrum \(\rm tg\delta_{th}(T)\) at T \(>\) 350 K determined using the methods presented in [1], as reported in [8], cannot be calculated. Thus, the existing methods for studying dielectric loss spectra in HBCs are characterized by an insufficient resolution of the experimental process (with a Q-factor meter VUP of 560 [1]) and a number of model flaws relating to the construction of theoretical graphs \(\rm tg\delta_{th}(T)\) and the calculation of the activation energy \(\rm U_{0,th}\) in the temperature ranges T \(<\) 100 K and T \(>\) 350 K.  
The methods proposed in [136; 137] for describing the tunnel relaxation of protons are estimations and do not disclose the influence of shape (e.g., rectangular [1] or parabolic [71; 72; 135]) and the parameters of the potential barrier on the characteristics (amplitude, temperature position, etc.) of the theoretical maxima of the thermostimulated current and on the spectra \(\epsilon^{/}(\omega;T)\), \(\epsilon^{/}(\omega;T)\).