## 2 Materials and Methods (Theoretical Bases and Methods)
According to the results of numerical calculations, the value of the polarizing field strength has practically no effect on the values of quantum transparency \(D_{quant,tunnstationary}^{\text{\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]7-\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]8{\dagger}\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]9classic}}\)\((\text{T}_{\sigma,1})\approx 0.1555\). Further, at the activation energy of \(\text{U}_{02}=~{}0.03\) eV, the critical temperature is \(\text{T}_{\sigma,2}=\frac{\text{h}\sqrt{2}\text{U}_{02}}{\pi\text{k}\text{k} \delta_{0}\sqrt{\text{m}}}\approx 68.55\) K, and the corresponding stationary quantum transparency is \(D_{quant,tunnstationary}^{\text{\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]0-\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]1{\dagger}\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]2classic}}(\text{T}_{\sigma,3}) \approx 0.008\). Further, at the activation energy of \(\text{U}_{04}=0.07\) eV, we obtain, respectively, \(\text{T}_{\sigma A}\approx 104.7119\) K, \(D_{quant,tunnstationary}^{\text{\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]3-\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]4{\dagger}\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]5classic}}(\text{T}_{\sigma,5}) \approx~{}0.00087\). The obtained numerical results indicate that the theoretical values of the stationary quantum transparency \(D_{quant,tunnstationary}^{\text{\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]6-$classic}}\)(T) are shifted towards higher critical temperatures \(\text{T}_{\sigma}\) with a decrease in the amplitude values of the quantum transparency and an increase in the energy of proton activation. Against the background of small activation energies (0.01.0.03.05, 0.07 eV), critical temperatures (39.57 K, 68.55 K, 88.49 K, 104.71 K) are distributed in the T \(<\) 105 K region with sufficiently high amplitudes of quantum transparency (0.1555, 0.031, 0.008, 0.0035) for such heavy (in comparison with electrons) charge carriers as protons. With an activation energy of 0.1 eV, the critical temperature shifts to 125.15 K, and the amplitude of the quantum transparency decreases to 0.00087. This indicates the significant contribution of quantum tunneling to the relaxation motion of protons in HBC in the temperature region T = 1-100 K. At temperatures T \(>\) 100, quantum tunneling continues to affect the relaxation of protons in HBC to a certain extent. The phenomenological model of thermally stimulated depolarization currents proposed by [PERSON] [23; 32], involves a study of the mechanism of thermal destruction of the polarized state established during polarization (in a uniform stationary electric field of strength \(\text{E}_{\text{pol},0}\) at polarization temperature \(\text{T}_{\text{pol}}\)) in the absence of an external electric field.