## 2 Materials and Methods (Theoretical Bases and Methods)
The effect of proton-proton interactions (due to the low equilibrium concentration of protons \(n_{0}\approx 10^{16}\div 10^{18}\mathrm{{}_{M}}^{-3}\)) and proton-phonon interactions (due to the low oscillation frequencies of anions in comparison with protons \(\frac{\
u_{onium}}{\
u_{0,proton}}\approx 10^{-4}\div 10^{-3}\)) is not taken into account, and the motion of protons is considered against the background of a static anionic subsystem [1, 7]. Let us preliminarily note the fundamental formulas of the quasi-classical theory of dielectric relaxation in HBC [1, 2, 6, 7]. The energy spectrum \(E\) of the main charge-carrier (protons) in HBC in the quasi-classical model is assumed to be quasi-continuous [1, 2, 7]. The one-dimensional potential barrier for the proton in HBC is assumed to be parabolic \(\mathrm{U}(\mathrm{x})=\mathrm{U}_{0}\left(1-\frac{4\mathrm{x}^{2}}{8\delta_{ 0}^{2}}\right)\), where \(\delta_{0}\) denotes the potential barrier width and \(\mathrm{U}_{0}\) denotes the activation energy. Calculated by the WKB method, the quantum transparency of a parabolic potential barrier (1) unperturbed by an external field is \(D^{(0)}(E)=\exp\left(-\frac{\pi\delta_{0}\sqrt{m}(\mathrm{U}_{0}-E)}{\mathrm{ h}_{\sqrt{2}}\mathrm{U}_{0}}\right)\). Here, \(\mathrm{m}\) and \(E\) denote the mass and energy of the relaxer (proton), respectively. The transparency of the unperturbed parabolic potential barrier, averaged using the quantum canonical Gibbs distribution, is the continuous function of the crystal temperature [1, 2, 6, 7]  
\[D_{quant,tunn}^{d,-classic;(0)}(\mathrm{T})=\langle D^{(0)}(E)\rangle=\frac{1} {\mathrm{k}_{\mathrm{B}}\mathrm{T}}\int_{0}^{\mathrm{U}_{0}}\exp\left(-\frac{ E}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\right)\times D^{(0)}(E)\mathrm{d}E=\frac{\exp(- \Lambda)-\exp(-\mathrm{X})}{1-\frac{\Lambda}{\mathrm{X}}}\]  
Here, the dimensionless parameters \(\mathrm{X}=\frac{\mathrm{U}_{0}}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\), \(\Lambda=\frac{\pi\delta_{0}\sqrt{m}\mathrm{U}_{0}}{\mathrm{h}\sqrt{2}}\) are calculated as functions of the molecular parameters \(\delta_{0}\), \(\mathrm{U}_{0}\). From the condition \(\mathrm{X}=\Lambda\), we calculate, for the parabolic potential barrier (2), the critical temperature \(\mathrm{T}_{\sigma}=\frac{2}{\pi\mathrm{k}_{\mathrm{B}}\mathrm{g}}\sqrt{\frac{ \mathrm{h}^{2}\mathrm{U}_{0}}{2m\delta_{0}^{2}}}\), below which \(\mathrm{T}<\mathrm{T}_{\sigma}\), the dominant contribution to dielectric relaxation is made by the proton-tunneling inside and between ions of the anionic sublattice (quantum relaxation). At the temperatures \(\mathrm{T}>\mathrm{T}_{\sigma}\) relaxation is determined by thermally activated proton transitions (thermal activation) [2, 3]. Let us proceed to the study of additional properties to the temperature dependences indicated above \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\). The expression \(D_{quant,tum}^{q-classic;(0)}(\mathrm{T})\) can be used to calculate, and statistically average using the energy levels of the quasi-continuous spectrum, the probability of the protons quantum tunneling (quantum transfer) in the experimental temperature range in the vicinity of the maximum thermally stimulated depolarization currents' (TSDCs) density spectrum \(\mathrm{P}_{tunn}(\mathrm{T})=\frac{\exp(-\Lambda)-\exp(-\Lambda)}{1-\frac{ \Lambda}{\Lambda}}\). It is not difficult to see the identity \(\frac{\Lambda}{\Lambda}=\frac{\mathrm{T}}{\mathrm{T}_{\sigma}}\). Thermal activation probability was calculated with the help of formula \(\mathrm{P}_{therm}(\mathrm{T})=\exp\left(-\frac{\mathrm{U}_{0}}{\mathrm{K}_{ \mathrm{B}}\mathrm{T}}\right)\)[1,2,6,7]. According to the experimental data, the width of the potential barrier in HBC varies within \(\delta_{0}\approx 0.085\div 0.1\) nm [1]. The experimental activation energy of protons in HBC varies from 0.01 to 0.7 eV [1]. In order to extend the theoretical range of activation energy variation, we accept this parameter within \(\mathrm{U}_{0}=(0.01\div 1)\) eV.