## 2 Materials and Methods (Theoretical Bases and Methods)
From the above comparison, it is obvious that the model of a parabolic pattern in the physical sense, in contrast to the model of a rectangular pattern, is closer to the real process of the migration of structural defects and can be used in a more rigorous calculation of the theoretical spectra of thermally stimulated depolarization currents in layered crystals. Taking the increment in the potential energy of the proton in the electric field \(\mathrm{E}(t)=\mathrm{E}_{0}\mathrm{e}^{\mathrm{i}\omega t}\) in the approximation of a weak field inhomogeneity at distances of the order of the lattice constant \(|\Delta\mathrm{U}(t)|=\frac{1}{6}\mathrm{q}\delta_{0}(\varepsilon_{\infty}+2) \mathrm{E}_{0}\mathrm{e}^{\mathrm{i}\omega t}\)[7], where \(\varepsilon_{\infty}\) is the permittivity associated with inductive polarization and ion charge, we write:  
\[W^{\mathrm{q}-classic(\pm)}(t)=\frac{\mathrm{v}_{0}}{2}\bigg{(}\exp\!\bigg{(} -\frac{\mathrm{U}_{0}\pm|\Delta\mathrm{U}(t)|}{\mathrm{k}_{\mathrm{B}}\mathrm{ T}}\bigg{)}+D_{quant,tunn}^{q-classic(\pm)}(t)\bigg{)} \tag{4}\]  
The quantum transparency of a parabolic potential barrier perturbed by an external electric field will be calculated in the approximation \(\frac{|\Delta\mathrm{U}|}{\mathrm{U}_{0}}<<1\)[7] for a proton moving with potential energy \(|\Delta\mathrm{U}(t)|\) in this direction, either along the field \(D_{quant,tunn}^{q-classic(-)}(\mathrm{U}_{0}-|\Delta\mathrm{U}(t)|;E)\) or against the field \(D_{quant,tunn}^{q-classic(+)}(\mathrm{U}_{0}+|\Delta\mathrm{U}(t)|;E)\). In [1,2], this kind of calculation was performed for a rectangular potential barrier. In this case, the choice of the parabolic form of the potential barrier allowed, at the theoretical level, for the mathematical model to be brought closer to the re-spatial structure of the crystal potential field. Then,  
\[\mathrm{D}^{(\pm)}(E)=\exp\!\bigg{(}-\frac{\pi\delta_{0}\sqrt{\mathrm{m}}( \mathrm{U}_{0}\pm|\Delta\mathrm{U}(t)|-E)}{\hbar\sqrt{2\mathrm{U}_{0}}}\bigg{)} \tag{5}\]  
Unlike earlier works [1,2,7], in this article, the statistical averaging of non-stationary quantum transparency will be carried out with a stricter consideration of the effects of the corrections \(\xi(t)=\left|\frac{\Delta\mathrm{U}(t)}{\mathrm{k}_{\mathrm{B}}\mathrm{T}} \right|\), \(\eta(t)=\Lambda\left|\frac{\Delta\mathrm{U}(t)}{\mathrm{U}_{0}}\right|\) due to the electric field. To create a parabolic potential barrier, these transformations will be completed for the first time. Then, based on  
\[D_{quant,tunn}^{q-classic(\pm)}(t)=\frac{1}{\mathrm{k}_{\mathrm{B}}\mathrm{T}} \int\limits_{2|\Delta\mathrm{U}(t)|,0}^{\mathrm{U}_{0}\pm|\Delta\mathrm{U}(t)| }\exp\!\bigg{(}-\frac{E}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\bigg{)}\times \mathrm{D}^{(\pm)}(\mathrm{U}_{0}\pm|\Delta\mathrm{U}(t)|;E)\mathrm{d}E \tag{6}\]  
and considering Equation (5), we obtain  
\[D_{quant,tunn}^{q-classic(+)}(t)=\frac{\exp(-\Lambda)\exp(\eta(t))\exp(-2\xi( t))-\exp(-X)\exp(-\xi(t))}{1-\frac{\Lambda}{X}} \tag{7}\]  
\[\mathrm{X}^{2}\left(\frac{\Lambda}{\mathrm{X}}\left(1+\frac{1}{\mathrm{X}} \right)-1\right)\exp(-\mathrm{X})=\Lambda\exp(-\Lambda),\]0  
Calculating the kinetic coefficients leads to:  
\[ \begin{array}{c}\Omega^{(\omega)}(t)=\frac{W^{q-classic(-)}(t)+W^{q-classic( +)}(t)}{2}=\\ =\frac{\mathrm{v}_{0}}{2}\bigg{(}\exp(-X)\mathrm{ch}(\xi(t))+\frac{\frac{1}{2} \exp(-\Lambda)\exp(\eta(t))(1+\exp(-2\xi(t)))-\exp(-X)\mathrm{ch}(\xi(t))}{1- \frac{\Lambda}{X}}\bigg{)}^{\prime}\end{array}  \tag{9}\]\[\Xi^{(\omega)}(t)=W^{q-classic;(-)}(t)-W^{q-classic;(+)}(t)= \tag{10}\] \[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]  
Here, \(\omega\) is the cyclic frequency of the external field. In this case, the relation \(\frac{\mathfrak{n}(t)}{\xi(t)}=\frac{\Lambda}{X}=\frac{\mathrm{T}}{\Gamma_{ \alpha}}\), which is important for the quasi-classical model, is fulfilled.