## 8 Conclusions (with Elements of Results Analysis)  
1. By the methods of the quasi-classical kinetic theory of the relaxation polarization and conductivity in crystals with ion-molecular type of chemical bonds (proton-relaxationpolarization and proton conduction in HBC), taking into account the quantum effects caused by tunnel transitions of the main charge carriers or relaxers (first of all, protons, in HBC) on chemical bonds, for the model of a one-dimensional potential crystalline field, the transcendental Equation (8c) was built. (8d) allows the calculation of the critical temperature \(T_{\rm cr,relax}\) separating the temperature regions (zones) of diffusion (\(\rm T<T_{\rm cr,relax};T_{\rm D}<T_{M}\)) and Maxwell (\(\rm T>T_{\rm cr,relax};T_{M}<T_{D}\)) relaxation and establishes the effects of molecular parameters of relaxers (lattice constant \(a\); potential barrier width \(\rm\delta_{0}\); activation energy (potential barrier height) \(\rm U_{0}\); equilibrium concentration of relaxers \(n_{0}\) (in the state of thermodynamic equilibrium in the absence of an external disturbing field); natural frequency of relaxer oscillations in an unperturbed potential well \(\rm\gamma_{0}\)). It has been found that the critical temperature for dielectric relaxation \(\rm T_{\rm cr,relax}\) (dependent, in addition to molecular parameters, also on the thickness of the crystal) is associated with another previously calculated critical temperature for microscopic transitions of the relaxers over the potential barrier \(\rm T_{\rm cr,move}=\frac{\hbar\sqrt{2\rm U_{0}}}{\pi\delta_{0}\sqrt{m_{\rm KB }}}\), which separates the areas of tunnel (\(\rm T<T_{\rm cr,move}\)) and thermally activated relaxation (\(\rm T>T_{\rm cr,move}\)) by ions (protons in HBC). 2. The analysis of basic positions of the physical and mathematical model of the nonlinear relaxation polarization in the hydrogen-bonded crystal (HBC) was performed [1; 2; 3; 4]. It was established that the existing methods [1; 3] are incorrect when calculating the polarization of the dielectric at the frequencies of the field, which are multiplied by fundamental frequency (\(r\omega\)), starting with the second frequency harmonic \(\rm P^{(3\omega)}(t)\). An improvement in the methodology for solving the system of equations of the basic phenomenological model (expressions (2)-(7)), in relation to the regions of abnormally high non-linearities (1-10 K, 100-1000 kV/m; 550-1500 K, 10-1000 MV/m) is required ([1], p. 43; [2], p. 74). 3. Generalized (in the k-th approximation of the perturbation theory, by parameter \(\gamma\)) analytical solutions of the nonlinear system of the Fokker-Planck and Poisson equations with blocking electrodes at the boundaries of the crystal were obtained. Recurrence expression (18) was constructed to calculate the complex amplitudes \(\rm\Re_{k}^{(r\omega)}(n,\tau)\) of the relaxation modes \(\rm\rho_{k}^{(r\omega)}(x;t)=\sum_{n=1}^{\infty}\rm\Re_{k}^{(r\omega)}(n,\tau) \cos(\frac{\tau\rm m\Sigma}{d})\) generated in the stationary polarization mode at the frequency \(r\omega\) (20). Volumetric charge density \(\rm\rho(x;t)\) was calculated with accuracy to the third frequency harmonic of the variable field (22), (23). In this case, \(\rm\rho^{(\omega)}\)\(\sim\rm E_{pol}(t)\) is the first, \(\rm\rho^{(2\omega)}\)\(\sim\rm\left(E_{pol}(t)\right)^{2}\) is the second and \(\rm\rho^{(3\omega)}\)\(\sim\rm\left(E_{pol}(t)\right)^{3}\) is the third field approximation. 4. The generalized (in case of arbitrary approximation by the frequency of variable field \(r=2\lambda+1\)) non-linear by the field \(\rm E_{pol}(t)\) expressions for complex dielectric permittivity (CDP) \(\hat{\varepsilon}^{(\Omega)}\) calculated at multiple frequencies \(\Omega=\{\omega;2\omega;3\omega;\ldots;2\lambda\omega;(2\lambda+1)\omega;\ldots\}\) were constructed in the form of decomposition into a series by even frequency harmonics of variable field \(\hat{\varepsilon}^{(\Omega_{2\lambda+1})}\)\(\sim\rm e^{2i\lambda\omega t}\). Here \(\lambda=\{0,1,2,3,\ldots\}\).