## 2 Materials and Methods (Theoretical Bases and Methods)
The solution of kinetic Equation (11) for the process of thermally stimulated depolarization in the absence of an external electric field, when the equation \(\frac{\text{d}\Delta n_{\text{depol}}(t)}{\text{d}t}+\frac{\Delta n_{\text{ depol}}(t)}{\text{d}t}=0\), considering the initial condition \(\Delta n_{\text{depol}}(0)=\Delta n_{\text{pol}}^{(\omega=0)}(\infty)\), takes the form \(\Delta n_{\text{depol}}(t)=\Delta n_{\text{pol}}\times\exp\left(-2\int_{0}^{t} \Omega_{\text{depol}}(\text{T})\text{d}t\right)\). The advantage of this expression for excessive proton concentration in the thermally stimulated depolarization formulated in this article is as follows. First, in [4; 8], the calculation of the function \(\Delta n_{\text{depol}}(t)\) is constructed considering the effects of quantum tunneling on the statistically averaged transparency of the parabolic potential barrier, which is reflected in the expression for \(\Omega_{\text{depol}}(\text{T})\). Previously, no studies of this kind have been conducted for the potential barrier of the parabolic form. Moreover, the parabolic potential barrier model is physically closer to the structure and properties of the real crystal lattice than the rectangular potential barrier model [4]. Secondly, unlike [8], the calculation of the excess concentration of protons \(\Delta n_{\text{pol}}=\Delta n_{\text{pol}}^{(\omega=0)}(\infty)=\frac{1}{3}n_{ 0}\times\frac{\Xi_{\text{eff}}(\text{T}_{\text{pol}})}{2\Omega_{\text{st}}( \text{T}_{\text{pol}})}\) established during polarization is carried out by us in Equation (17), by taking into account the effects of additional correction elements \(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\\(\Omega_{\text{st}},\Xi_{\text{st}}\) (18) and (19). This technique increases, at a theoretical level, the contribution to the polarization of the dielectric caused by the external field of nonlinearities. The rate of proton transition probabilities at thermally stimulated depolarization, according t0 Equation (9), at the \(\xi(t)=0,\ \eta(t)=0\), takes the form  
\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]7  
The relaxation time of the ions for the process of thermally stimulated depolarization is \(\tau_{\text{depol}}(\text{T})=\frac{1}{2\Omega_{\text{depol}}(\text{T})}\). The increase in the concentration of relaxers during thermally stimulated depolarization becomes a function of temperature. For an arbitrary nonlinear law of heating with a rate \(\text{c}(\text{T})=\frac{\text{dT}}{\text{d}t}\), we have  
\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]8  
Thermally stimulated depolarization current density \(\text{j}_{\text{depol}}(t)=-\frac{\partial P_{\text{depol}}(t)}{\partial t}\), calculated on the basis of \(P_{\text{depol}}(t)=\frac{1}{2}\text{q}\delta_{0}\times\Delta n_{\text{depol} }(t)\) according to the Bucci-Kive theory [7], in the case of heating of the crystal according to the linear law \(\text{c}=\frac{\text{dT}}{\text{d}t}=const\), takes the form  
\[\frac{\mathrm{d}\Delta n^{(\omega)}(t)}{\mathrm{dt}}+2\Omega^{(\omega)}(t) \Delta n^{(\omega)}(t)=\frac{1}{3}n_{0}\Xi^{(\omega)}(t) \tag{11}\]9  
The study of the integral \(\int_{\text{T}_{0}}^{\text{T}}\Omega_{\text{depol}}(\text{T})\text{d}\text{T}\) with the integrand \(\Omega_{\text{depol}}(\text{T})\) was carried out by numerical methods. The applied scientific novelty of the Equation (22), in the context of this problem, consists of a number of additional model transformations and refinements embedded in the structure of analytical expressions for calculating quantities \(\Delta n_{\text{depol}}\) n \(\Omega_{\text{depol}}(\text{T})\) that, as noted above, carries significant physical stress associated with the effects of nonlinear polarization effects on proton relaxation kinetics during polarization in HBC.