## 1 Introduction
### Comparative Analysis of Different Ion-Relaxation Polarization Models  
In HBCs, according to the results of precision measurements, the maximum density of the TSDC \(J(\mathrm{T})\) and \(\mathrm{tg}\delta(\mathrm{v},\mathrm{T})\) appear in the temperature range \(\mathrm{T}=70\)-\(450\) K at a strength of \(\mathrm{E_{0}}\approx\left(10^{5}\)-\(10^{6}\right)\)\(\frac{\mathrm{V}}{\mathrm{M}}\) and field frequency \(\mathrm{v}=\left(10^{3}\)-\(10^{7}\right)\) Hz [1] and are explained by the relaxation motion of various types of Bjerrum defects (regarding orientation, such as \(\mathrm{H_{3}O^{+}}\) and \(\mathrm{OH^{-}}\), and ionization, such as \(\mathrm{L}\) and \(\mathrm{D}\)) and water molecules (structural and adsorbed) in the electric field [1; 125; 126; 133; 134; 135]. From the point of view of the quasiclassical kinetic theory of proton relaxation polarization and conduction [1], the physical relaxer in HBCs is a proton moving in the vicinity of the temperature \(\mathrm{T_{max}}\) of each mono-relaxation maximum of the experimental spectra \(J(\mathrm{T})\) and \(\mathrm{tg}\delta(\mathrm{T})\) with different values of characteristic (geometric and molecular) parameters: activation energy, \(\mathrm{U_{0}}\); natural frequency, \(\mathrm{V_{0}}\); equilibrium concentration, \(n_{0}\); and the width of the potential barrier, \(\delta_{0}\)[71]. The potential pattern of the proton is modeled as a one-dimensional periodic potential \(\mathrm{U_{c}(x)}\) perturbed by an external electric (polarizing) field \(\mathrm{E_{pol}(t)=E_{0}\exp(i\omega t)}\) directed toward the crystal axis \(\overrightarrow{\mathrm{C}}\left\|\overrightarrow{\mathrm{E}}\left[1\right]\). The proton-proton and proton-phonon interactions in [1; 70] are not considered, and the influence of temperature on the mechanism of the relaxation movement of protons is reflected in the kinetic coefficients of \(W^{(l)}(\mathrm{U_{0}},\delta_{0},\mathrm{V_{0}};\mathrm{T})\)[72]. The mathematical model in [71; 125] is based on the system of nonlinear Fokker-Planck and Poisson equations solved by methods of perturbation theory by decomposition into power series by a small parameter \(\gamma=\frac{\mathrm{u}_{mub}^{(1)}\cdot\mathrm{a}E_{0}}{\mathrm{D}_{diff}^{(0)}}\)[1; 125], where \(\mathrm{D}_{diff}^{(0)}=a^{2}W^{(0)}\), \(\mathrm{u}_{mub}^{(1)}=\frac{\mathrm{q}a^{2}W^{(1)}}{\mathrm{k_{B}T}}\) denote diffusion and mobility coefficients, respectively; \(\mathrm{q}\) denotes pro ton charge, and \(a\) is the lattice constant [70,71,73]. The coefficients \(W^{(l)}\) for the parabolic potential barrier model are calculated in [65] and consistent with (10):  
\[W^{(0)}(\mathrm{T})=\frac{\mathrm{\
u_{0}}}{2}\Big{(}\mathrm{exp}(-\mathrm{X}( \mathrm{T}))+D^{(0)}(\mathrm{T})\Big{)},\ W^{(1)}(\mathrm{T})=\frac{\mathrm{ \
u_{0}}}{2}\Big{(}\mathrm{exp}(-\mathrm{X}(\mathrm{T}))+D^{(1)}(\mathrm{T}) \Big{)}. \tag{55}\]  
In (55), the statistically averaged transparency of the potential barrier [65] is  
\[D^{(0)}(\mathrm{T})=\frac{\mathrm{exp}\ (-\Lambda)-\mathrm{exp}\ (-\mathrm{X}( \mathrm{T}))}{1-\frac{\Lambda}{\mathrm{X}(\mathrm{T})}},\ D^{(1)}(\mathrm{T})= \frac{\frac{\Lambda}{\mathrm{X}(\mathrm{T})}\ \mathrm{exp}\ (-\Lambda)-\mathrm{exp}\ (- \mathrm{X}(\mathrm{T}))}{1-\frac{\Lambda}{\mathrm{X}(\mathrm{T})}}, \tag{56}\]  
where \(=\frac{\mathrm{U}_{0}}{\mathrm{kg}}\), \(\Lambda=\frac{\pi\delta_{0}\sqrt{\mathrm{mU}_{0}}}{\mathrm{h}\sqrt{2}}\), and \(\mathrm{m}\) is the mass of a proton. Since for HBCs, the condition \(\zeta_{0}=\frac{\mathrm{qE}_{0}a}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}<<1\) works in almost the entire experimental range of change \(\mathrm{E}_{0}\) and \(\mathrm{T}\)[1,125], taking into account \(\frac{W^{(l)}}{W^{(0)}}\)\(\leq\) 1 [135], the parameter smallness condition \(\gamma=\frac{\zeta_{0}W^{(l)}}{W^{(0)}}\) is applied for any set of parameters of relaxers \(\mathrm{U}_{0},\ n_{0},\ \ \mathrm{\
u_{0}},\ \ \mathrm{and}\ \ \delta_{0}\) involved in the establishment of volume-charging polarization [1,135].