Theoretical Studies of Nonlinear Relaxation Electrophysical Phenomena in Dielectrics with Ionic–Molecular Chemical Bonds in a Wide Range of Fields and Temperatures

Abstract

This paper is devoted to the development of generalized (for a wide range of fields (100 kV/m–1000 MV/m) and temperatures (0–1500 K) in the radio frequency range (1 kHz–500 MHz)) methods for the theoretical investigation of the physical mechanism of nonlinear kinetic phenomena during the establishment of the relaxation polarization, due to the diffusion motion of the main charge carriers in dielectrics with ionic–molecular chemical bonds (hydrogen-bonded crystals (HBC), including layered silicates, crystalline hydrates and corundum–zirconium ceramics (CZC), etc.) in an electric field. The influence of the nonlinearities equations of the initial phenomenological model of dielectric relaxation (in HBC-proton relaxation) on the mechanism for the formation of volume–charge polarization in solid dielectrics is analyzed. The solutions for the nonlinear kinetic Fokker–Planck equation, together with the Poisson equation, for the model of blocked electrodes are built in an infinite approximation (including all orders k of smallness without dimensional parameters) of perturbation theory for an arbitrary order r of the frequency harmonic of an alternating external polarizing field. It has been established that the polarization nonlinearities in ion-molecular dielectrics, already detected at the fundamental frequency, are interpreted in the mathematical model (for the first time in this work) as interactions of the relaxation modes of the volume charge density calculated on different orders of spatial Fourier harmonics. At the fundamental frequency of the field, an analytical generalized expression is written for complex dielectric permittivity (CDP), which is expressed analytically in terms of special relaxation parameters, which are quite complex real functions in the fields of frequency and temperature. The theoretical CDP and the dielectric loss tangent spectra studied depend on the nature of the relaxation processes in the selected temperature range (Maxwell and diffusion relaxation; thermally activated and tunneling relaxation), which is relevant from the point of view of choosing exact calculation formulas when analyzing the optimal operating modes of functional elements (based on dielectrics and their composites) for circuits of instrumentation, radio engineering and power equipment in real industrial production.

1. Introduction

In the last twenty years in the industry, hydrogen-bonded crystals (HBC)-based materials (and their composites) have been widely used in insulating and cable technology, capacitor technology (electrically controlled nonlinear capacitors), physicochemical technologies (solid state electrolytes), laser technology (regulators of laser radiation parameters and electric shutters), radio electronics, radio engineering (elements of radiation generators and microwave control systems), etc. [1,2].

The materials of the class of HBC, from the point of view of mineralogy, belong to the class of layered crystals. They are divided into crystalline hydrates and layered silicates, which differ in the presence of a hydrogen sublattice in their structure and the possibility of hydrogen ions (protons) moving along hydrogen links. The displacement of protons in HBC along hydrogen bonds can be carried out both due to thermal activation (classical transitions) and tunneling (quantum transitions). Therefore, the imposition of an electric field on the crystal (directed along the crystal axis) leads to the diffusion motion of protons in the direction of field lines (proton conductivity). Thus, according to electrophysical properties, HBC are classified as proton semiconductors and dielectrics (PSCD) [1,2,3,4].

According to the results of experiments, in HBC, in a wide range of polarizing field strengths (100 kV/m–100 MV/m) and temperatures (50–550 K), volume –charge polarization is established. It possesses a relaxation nature and significantly depends on the parameters of the crystal structure, of the external electrical field (amplitude and frequency of EMF) and the selected temperature range. Polarization phenomena at low temperatures (50–100 K) appear as quantum polarization due to the tunneling motion of protons inside and between ions of the anion sublattice; at high temperatures (450–550 K), they appear as a nonlinear space charge polarization due to the mixed type of proton motion (both due to thermal activation and tunneling) [1].

The analytical study of the nonlinear relaxation polarization is an applied scientific direction relevant for electrophysics and electrical engineering. It allows, in a wide range of polarizing field intensities (100 kV/m–1000 MV/m) and temperatures (0–1500 K), the identification of the patterns of behavior of the current spectra of thermally stimulated polarization and dielectric losses in the dielectrics class of PSCD, in dielectrics and a wider range of crystals with a type of conductivity similar to HBC in terms of physical mechanism. This type of crystal has ionic–molecular chemical bonds, characterized by high (and in some cases very high) ionic conductivity (perovskites, clay minerals, corundum–zirconium ceramics (CZC) [5], etc.). It should be noted that some materials of this class belong to high-temperature superconductors. For example, high-temperature superconducting ceramics (HTSC-ceramics) and ceramics created on the basis of oxide high-temperature superconductors. In microwave components, thin films of HTSC ceramics on single-crystal substrates are used. In zirconium oxide crystals stabilized by ion implantation, the formation of nanoclusters is observed, and conductivity maxima are found due to the resonant tunneling of electrons through dielectric potential barriers with transparency depending on the cluster size, electron energy and voltage on the structure [6].

The area of abnormally high nonlinearities is of special interest. It manifests in PSCD in the range of ultra-low (helium) temperatures (1–10 K) in the region of weak fields (100 kV/m–1000 kV/m) and the range of ultra-high temperatures (550–1500 K) in the region of strong fields (10–1000 MV/m) ([1], p. 43; [2], p. 74). Conditions of this type can be defined as extreme for the operation of the heterogeneous functional elements of process circuits of electrical (instrumentation, computer, etc.) and power (insulation coatings of current removal elements of electric generators and transformers of thermal power plants) plants and systems ([3], p. 7). The practical use and scientific practical significance of the non-linear elements based on PSCD are detailed more in ([1], pp. 41–42) and pertain to space technologies, nanotechnologies and high-voltage technology [3,4].

In [1], a detailed analysis of the influence of voltage source parameters (amplitude and frequency of EMF), dielectric temperature and thickness on the properties of the complex dielectric permittivity (CDP) spectra ${\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)$ for PSCD was performed; effects of the interaction of relaxation modes at the main field frequency $\mathsf{\omega }$ [3] were detected. It can be the basis for the development of experimental methods for amplifying signals in the radio frequency range (microwave generators).

Since control measuring and power equipment during operations, in most cases, is tuned to the fundamental frequency of the alternating field $\mathsf{\omega }$, further theoretical studies of the behavior of the parameters of relaxation processes (probability rates of transitions of relaxers, relaxation times, kinetic coefficients, etc.) and the polarization itself, depending on temperature, field strength and dielectric structure parameters, on the fundamental frequency harmonic ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)$ are possible. It is important to construct algorithms for calculating higher-order polarization harmonics (starting from the third ${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(t\right)$) based on recurrent analytical expressions, which also allow the calculation of the components of the complex permittivity in an arbitrary approximation with respect to the polarizing field.

2. Mathematical and Physical Model of the Nonlinear Relaxation Polarization. Setting of the Research Objective

In [3], nonlinear solutions of the semi-classical kinetic equation in complex with the Poisson equation for the model of blocking electrodes were obtained with accuracy to the cubic term (by a small parameter $\mathsf{\gamma }$ [3], pp. 13–14). By the method of mathematical induction in the infinite approximation of the perturbation theory at the main frequency $\mathsf{\omega }$, the frequency harmonic of polarization ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)$ ([3], p. 16) was calculated and the nonlinear frequency–temperature spectra of the CDP were constructed and studied in detail ([1], pp. 43–49). At the same time, the polarization harmonics ${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(t\right)$ following ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)$ and multiplied by the frequency $3\mathsf{\omega }$ according to the methodology [1,3] can no longer be calculated.

The purpose of this paper, in the continuation of [1,2,3], is to develop generalized (for any approximation of the perturbation theory) methods for the analytical study of the influence of the nonlinearities of the kinetic equations of the original phenomenological model ([3], p. 13) on the mechanism for the formation of the volume–charge distribution in the dielectrics type of hydrogen-bonded crystals (HBC). The results of these studies can be used in the mathematical description and modeling of the ionic conductivity and relaxation polarization processes in crystals with ionic–molecular chemical bonds, a special case of which are hydrogen-bonded crystals (characterized by proton conductivity).

This paper is devoted to the development of generalized (for a wide range of field and temperature parameters) analytical methods for solving the nonlinear kinetic Fokker–Planck equation, together with the Poisson equation, in an infinite approximation (including all orders of smallness from k = 0 to $\mathrm{k}=\infty$ without dimensional parameter $\mathsf{\gamma }<1$) of perturbation theory for an arbitrary order of the frequency harmonic of an alternating polarizing field ${\mathrm{P}}^{\left(r\mathsf{\omega }\right)}\left(t\right)$. Theoretical studies of the complex dielectric permittivity (CDP) spectra of dielectrics with a compound crystalline structure (dielectrics with ionic–molecular chemical bonds and hydrogen-bonded crystals (HBC)) at the fundamental frequency harmonic of the polarization ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)$ are carried out in this work. On this basis, constructed formulas for calculating the CDP components depend on the nature of the relaxation processes in the selected temperature range (Maxwell and diffusion relaxation; thermally activated and tunneling relaxation).

The model we are developing will formally be applicable to the description of the nonlinear polarization kinetic phenomena and other solid dielectrics similar to HBC in the type of crystal lattice structure and in the mechanism of the ionic conductivity. Mainly, it could be for the case of diffusion transfer of the slightly coupled ions (with ${\mathrm{U}}_0\approx \left(0.01÷1\right)$ eV activation energies) at attachment places (equilibrium states) in the electric field. In the continuation of the papers [1,3], the calculation of the k-th component of volume charge density ${\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)$ will be carried out from a recurring expression suitable in any approximation of the perturbation theory. The results [1,3] will be considered as private cases of the generalized method.

Since for HBC the condition ${\mathsf{\varsigma }}_0=\frac{{\mathrm{qE}}_{0}a}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}<1$ works in almost the entire theoretical range of parameter change ${\mathrm{E}}_0$, T, the condition of $\mathsf{\gamma }=\frac{{\mathsf{\varsigma }}_0{W}^{\left(1\right)}}{W^{\left(0\right)}}<1$, taking into account that $\frac{W^{\left(1\right)}}{W^{\left(0\right)}}\le 1$ ([1], p. 43; [2], p. 73), is fulfilled for any set of the parameters of the relaxers ${\mathrm{U}}_0$, $n_0$, ${\mathsf{\nu }}_0$, ${\mathsf{\delta }}_0$ ([4], pp. 150–152) involved in volumetric–charge polarization. In this paper: ${\mathrm{E}}_0$—module of alternating electric field intensity; $a$—crystal lattice parameter; q-ion charge (in HBC-proton) ([1], p. 43).

The effect of the temperature on the mechanism of the relaxation movement of the protons is reflected in kinetic coefficients ${\mathrm{W}}^{\left(l\right)}\left({\mathrm{U}}_0{,\mathsf{\delta }}_0{,\mathsf{\nu }}_0;\mathrm{T}\right)$ calculated in the l-approximation of the perturbation theory, taking into account both thermally activated (classical) and tunnel (quantum) transitions of ions (in HBC-protons) ([4], p. 154)

$$W^{\left(l\right)}\left(\mathrm{T}\right)=\frac{{\mathsf{\nu }}_0}{2}\left(\exp \left(-\mathrm{X}\right)+\langle {\mathrm{D}}^{\left(l\right)}\rangle \right),\langle {\mathrm{D}}^{\left(l\right)}\rangle =\frac{{\mathsf{\Lambda }}^l\exp \left(-\mathsf{\Lambda }\right)-{\mathrm{X}}^l\exp \left(-\mathrm{X}\right)}{{\mathrm{X}}^{l-1}\left(\mathrm{X}-\mathsf{\Lambda }\right)}.$$

(1)

In (1), $\mathrm{X}=\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}T}$, $\mathsf{\Lambda }=\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}$, $\mathrm{m}$—$\mathrm{ion} \left(\mathrm{in} \mathrm{HBC}-\mathrm{proton}\right) \mathrm{mass};{\mathsf{\nu }}_0$—natural frequency of oscillations of the ion (in HBC-proton) in the potential well; ${\mathsf{\delta }}_{0, }{\mathrm{U}}_0$—width and high (activation energy) of the potential barrier for the ion (in HBC-proton) ([4], p. 153). The potential barrier in expression (1) is taken to be parabolical ([4], p. 152).

3. Studies of the Nonlinear Kinetic Equation

The phenomenological model of the ions diffusion transfer in the dielectric (protons in HBC) in the polarizing electric field is based on a system of the nonlinear equations of the Fokker–Planck and Poisson type (expressions (11), (14) of [3], p. 13)

$$\frac{\partial \mathsf{\rho }}{\partial \mathsf{\tau }}=\frac{{\partial }^2\mathsf{\rho }}{\partial {\mathsf{\xi }}^2}-\mathsf{\theta }\mathsf{\rho }-\mathsf{\gamma }\frac{\partial }{\partial \mathsf{\xi }}\left(\mathsf{\rho }\mathrm{z}\right),$$

(2)

$$\frac{\partial \mathrm{z}}{\partial \mathsf{\xi }}={\phi }_1\mathsf{\rho },$$

(3)

considering initial and boundary conditions ([3], p. 13)

$$\mathsf{\rho }\left(\mathsf{\xi },0\right)=0,$$

(4)

$$\frac{\partial \mathsf{\rho }}{\partial \mathsf{\xi }}{|}_{\mathsf{\xi }=\left\{0;\frac{\mathrm{d}}{a}\right\}}=\mathsf{\gamma }\left(n_0+\mathsf{\rho }\right)\mathrm{z}{|}_{\mathsf{\xi }=\left\{0;\frac{\mathrm{d}}{a}\right\}},$$

(5)

$${\int }_0^{\mathrm{d}/a}\mathrm{z}\left(\mathsf{\xi };\mathsf{\tau }\right)\mathrm{d}\mathsf{\xi }=\frac{\mathrm{d}}{a}\exp \left(\frac{\mathrm{i}\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right).$$

(6)

In (2)–(6), the following designations are adopted: $\mathsf{\rho }\left(\mathsf{\xi };\mathsf{\tau }\right)=n\left(\mathrm{x};\mathrm{t}\right)-n_0$ is an ion excess concentration over its steady-state concentration $n_0$; $z\left(\mathsf{\xi };\mathsf{\tau }\right)=\frac{\mathrm{E}\left(\mathrm{x};\mathrm{t}\right)}{{\mathrm{E}}_0}$, $\mathsf{\xi }=\frac{\mathrm{x}}{a}$, $\mathsf{\tau }=W^{\left(0\right)}t$, ${\phi }_1=\frac{a\mathrm{q}}{{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{E}}_0}$, $\mathsf{\theta }={\phi }_1{\mathsf{\gamma }n}_0$, $\mathsf{\gamma }=\frac{{\varsigma }_0{\mathrm{W}}^{\left(1\right)}}{{\mathrm{W}}^{\left(0\right)}}=\frac{{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}{aE}_0}{{\mathrm{D}}_{\mathrm{diff}}^{\left(0\right)}}$, where ${\mathrm{D}}_{\mathit{diff}}^{\left(0\right)}=a^2{W}^{\left(0\right)}$, ${\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}=\frac{\mathrm{q}{a}^2{W}^{\left(1\right)}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}-$ are diffusion and mobility factors (coefficients) for ions (protons in HBC) ([1], p. 44, p. 75; [4], p. 156); ${\mathsf{\epsilon }}_{\infty }$—high-frequency dielectric constant of crystal; d—dielectric thickness ([3], pp. 12–13).

We also introduce an additional dimensionless small parameter ${\Xi }_0=\frac{8{\phi }_1{n}_0{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\mathsf{\pi }}^2}$ < 1. It is important from the point of view of the further expansion of solutions to Equation (2) in series in the powers of the parameter ${\mathsf{\Lambda }}_0={\sum }_{\mathrm{s}=1}^{\infty }\frac{{\sin }^2\left(\frac{\mathsf{\pi }\mathrm{s}}{2}\right)}{{\mathrm{s}}^2\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}$ ([3], p. 16), which, in turn, determined the interaction parameter of relaxation modes, at the base frequency $\mathsf{\omega }$ [3].

The authors of [3] have built their solutions to the equation systems (2)–(6) by methods of the perturbation theory using power series ([3], p. 12)

$$\mathsf{\rho }\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{k}=1}^{+\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right), \mathrm{z}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{k}=0}^{+\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\mathrm{z}}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)$$

(7)

in the third approximation by parameter $\mathsf{\gamma }$. In [3], based on the functions ${\mathsf{\rho }}_1\left(\mathsf{\xi };\mathsf{\tau }\right)$, ${\mathsf{\rho }}_2\left(\mathsf{\xi };\mathsf{\tau }\right)$, ${\mathsf{\rho }}_3\left(\mathsf{\xi };\mathsf{\tau }\right),$ by the method of mathematical induction the recurrence formulas have been devised for calculating relaxation modes ${\mathsf{\rho }}_{\mathrm{k}}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, ${\mathsf{\rho }}_{\mathrm{k}}^{\left(2\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$—volume density of charge $\mathsf{\rho }\left(\mathsf{\xi };\mathsf{\tau }\right)$, calculated in the k-th approximation of the perturbation theory, at the frequencies $\mathsf{\omega }$, $2\mathsf{\omega }$ of the variable field, respectively, (expressions (18), (19) from [3], p. 15). On this basis, the first two frequency harmonics ${\mathsf{\rho }}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi };\mathsf{\tau }\right)$, ${\mathsf{\rho }}^{\left(2\mathsf{\omega }\right)}\left(\mathsf{\xi };\mathsf{\tau }\right)$ of the function $\mathsf{\rho }\left(\mathsf{\xi };\mathsf{\tau }\right)$ are calculated. They are expressions (20), (21) of ([3], c.16) and the frequency harmonics of polarization ([3], p. 16)

$${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)=\frac{8a\mathrm{q}{n}_0\mathsf{\gamma }}{{\mathsf{\pi }}^2\left(1-{\Xi }_0\right)}\times {\sum }_{\mathrm{n}=1}^{+\infty }\left[\frac{{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\right]\times \exp \left(\frac{\mathrm{i}\mathsf{\omega }\mathsf{\tau }}{W^{\left(0\right)}}\right); {\mathrm{P}}^{\left(2\mathsf{\omega }\right)}\left(t\right)=0.$$

(8)

In (8), ${\mathsf{\tau }}_n=\frac{{\mathsf{\tau }}_{\mathrm{n},\mathrm{D}}{\mathsf{\tau }}_{\mathrm{M}}}{{\mathsf{\tau }}_{\mathrm{n},\mathrm{D}}+{\mathsf{\tau }}_{\mathrm{M}}}-$ is the dimensionless relaxation time for n-th relaxation mode, where, according to expressions ${\mathsf{\tau }}_{\mathrm{n}}=T_{\mathrm{n}}{W}^{\left(0\right)}$, ${\mathsf{\tau }}_{\mathrm{n},\mathrm{D}}=T_{\mathrm{n},\mathrm{D}}{W}^{\left(0\right)}$, ${\mathsf{\tau }}_{\mathrm{M}}=T_{\mathrm{M}}{W}^{\left(0\right)}$, the formula is built for the dimension relaxation time $T_n=\frac{T_{\mathrm{n},\mathrm{D}}{T}_{\mathrm{M}}}{T_{\mathrm{n},\mathrm{D}}+T_{\mathrm{M}}}$ determined with the help of ${\mathrm{T}}_{\mathrm{n},\mathrm{D}}=\frac{T_{\mathrm{D}}}{{\mathrm{n}}^2}$—diffusion relaxation time for n-th mode, $T_{\mathrm{D}}=\frac{d^2}{{\mathsf{\pi }}^2{\mathrm{D}}_{\mathit{diff}}^{\left(0\right)}}$, $T_{\mathrm{M}}=\frac{{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}{\mathrm{q}{n}_0{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}}$—Maxwell relaxation time ([1], p. 44, [3], p. 15, [4], p. 60). In this case, the ratio of relaxation times allows us to construct the impotent equality of the semi-classical kinetic theory of the relaxation processes $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{{\mathrm{d}}^2\mathrm{q}{n}_0{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}}{{\mathsf{\pi }}^2{\mathrm{D}}_{\mathit{diff}}^{\left(0\right)}{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}=\frac{{\mathrm{d}}^2{n}_0{\mathrm{q}}^2}{{\mathsf{\pi }}^2{a}^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\times \frac{W^{\left(1\right)}}{W^{\left(0\right)}}$, where the critical temperature separating temperature diffusion zones and Maxwell relaxation $T_{\mathrm{cr},\mathrm{relax}}$ may be calculated. Here we use the ratio for the diffusion and mobility factors $\frac{{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}}{{\mathrm{D}}_{\mathit{diff}}^{\left(0\right)}}=\frac{\mathrm{q}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\times \frac{W^{\left(1\right)}}{W^{\left(0\right)}}$, with the expressions for the zeroth and linear approximations of the perturbation theory in terms of the small dimensionless parameter ${\varsigma }_0=\frac{{\mathrm{qE}}_{0}a}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}<1$ for the rate of the probability of the transition of the relaxer through the potential barrier. Therefore, according to (1)

$$W_{}^{\left(0\right)}\left(\mathrm{T}\right)=\frac{{\mathsf{\nu }}_0}{2}\left(\exp \left(-\mathrm{X}\right)+〈{\mathrm{D}}_{}^{\left(0\right)}〉\right); 〈{\mathrm{D}}^{\left(0\right)}〉=\frac{\exp \left(-\mathsf{\Lambda }\right)-\exp \left(-\mathrm{X}\right)}{1-\frac{\mathsf{\Lambda }}{\mathrm{X}}},$$

(8a)

$$W_{}^{\left(1\right)}\left(\mathrm{T}\right)=\frac{{\mathsf{\nu }}_0}{2}\left(\exp \left(-\mathrm{X}\right)+〈{\mathrm{D}}_{}^{\left(1\right)}〉\right); 〈{\mathrm{D}}^{\left(1\right)}〉=\frac{\frac{\mathsf{\Lambda }}{\mathrm{X}}\exp \left(-\mathsf{\Lambda }\right)-\exp \left(-\mathrm{X}\right)}{1-\frac{\mathsf{\Lambda }}{\mathrm{X}}},$$

(8b)

we can write the transcendental equations for critical temperature

$$\frac{{\pi }^2{\mathsf{\epsilon }}_0^{}{\mathsf{\epsilon }}_{\infty }^{}{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}{{\mathrm{d}}^2{n}_0{\mathrm{q}}^2}=\frac{\exp \left(-\frac{\pi {\mathsf{\delta }}_0\sqrt{\mathrm{m}{\mathrm{U}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}\right)}{\frac{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}\exp \left(-\frac{\pi {\mathsf{\delta }}_0\sqrt{\mathrm{m}{\mathrm{U}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}\right)},$$

(8c)

$$\frac{{\pi }^2{\mathsf{\epsilon }}_0^{}{\mathsf{\epsilon }}_{\infty }^{}{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}{{\mathrm{d}}^2{n}_0{\mathrm{q}}^2}=\frac{\exp \left(-\frac{\pi {\mathsf{\delta }}_0\sqrt{\mathrm{m}{\mathrm{U}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}\right)}{\exp \left(-\frac{\pi {\mathsf{\delta }}_0\sqrt{\mathrm{m}{\mathrm{U}}_0}}{\hslash \sqrt{2}}\right)-\frac{{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}}\right)}.$$

(8d)

Here, ${\mathrm{T}}_{\mathrm{cr},\mathrm{move}}=\frac{\hslash \sqrt{{2\mathrm{U}}_0}}{{\pi \mathsf{\delta }}_0\sqrt{\mathrm{m}}{\mathrm{k}}_{\mathrm{B}}}-$ is determined from equality $\frac{\mathsf{\Lambda }}{\mathrm{X}}=1$, another critical temperature separating the regions (areas) of tunneling (quantum), when $\frac{\mathsf{\Lambda }}{\mathrm{X}}<1$, $\mathrm{T}<{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}$, and thermally activated (classical), when $\frac{\mathsf{\Lambda }}{\mathrm{X}}>1$, $\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}$, transitions of relaxers through a potential barrier.

An attempt to apply a similar method to the calculation of the function $\mathsf{\rho }\left(\mathsf{\xi };\mathsf{\tau }\right)$ in the approximation of k at the frequency $3\mathsf{\omega }$ gives such bulky expressions ${\mathsf{\rho }}_4^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, ${\mathsf{\rho }}_5^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, etc., that the conclusion of the recurring formula ${\mathsf{\rho }}_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$ requires a fundamentally new, more general approach. The first and simplest member of this series is the expression ${\mathsf{\rho }}_3^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$ formulated ([3], p. 17). Let us move on to the implementation of the general analytical method. Then, substituting the series (7) into the system (2)–(6) gives

$$\frac{\partial {\mathsf{\rho }}_{\mathrm{k}}}{\partial \mathsf{\tau }}=\frac{{\partial }^2{\mathsf{\rho }}_{\mathrm{k}}}{\partial {\mathsf{\xi }}^2}-\frac{\partial }{\partial \mathsf{\xi }}\left({\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\sum }_{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}\right)-{\mathsf{\theta }\mathsf{\rho }}_{\mathrm{k}},$$

(9)

$$\frac{\partial {\mathrm{z}}_{\mathrm{k}}}{\partial \mathsf{\xi }}={\phi }_1{\mathsf{\rho }}_{\mathrm{k}},$$

(10)

$${\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };0\right)=0,$$

(11)

$$\frac{\partial {\mathsf{\rho }}_{\mathrm{k}}}{\partial \mathsf{\xi }}{|}_{\mathsf{\xi }=\left(0;\frac{\mathrm{d}}{a}\right)}=\left[n_0{z}_{\mathrm{k}-1}+{\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\sum }_{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}\right]{|}_{\mathsf{\xi }=\left(0;\frac{\mathrm{d}}{a}\right)}·$$

(12)

Here ${\mathsf{\rho }}_0\left(\mathsf{\xi };\mathsf{\tau }\right)=0$, ${\mathrm{z}}_0\left(\mathsf{\xi };\mathsf{\tau }\right)=\exp \left(\frac{\mathrm{i}\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)$ ([3], p. 13). In all subsequent approximations

$${\int }_0^{\mathrm{d}/a}{\mathrm{z}}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)\mathrm{d}\mathsf{\xi }=0, \mathrm{k}\ge 1.$$

(13)

We expand ${\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)$ into the Fourier series according to the orthogonal functions $\cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$, on the segment $0\le \mathsf{\xi }\le \frac{\mathrm{d}}{a}$:

$${\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)=\sum _{\mathrm{n}=1}^{\infty }{\Re }_{\mathrm{k}}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right).$$

(14)

The function ${\mathsf{\rho }}_{\mathrm{k},\mathrm{n}}\left(\mathsf{\xi };\mathsf{\tau }\right)={\Re }_{\mathrm{k}}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$ has the meaning of the relaxation mode of the n-th order k-th approximation of the perturbation theory, ${\Re }_{\mathrm{k}}\left(\mathrm{n},\mathsf{\tau }\right)=\frac{2a}{\mathrm{d}}{\int }_0^{\mathrm{d}/a}{\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)\cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }-$ is the complex amplitude of the relaxation mode ${\mathsf{\rho }}_{\mathrm{k}}\left(\mathsf{\xi };\mathsf{\tau }\right)$. According to

$$\frac{{\partial }^2{\mathsf{\rho }}_{\mathrm{k}}}{\partial {\mathsf{\xi }}^2}÷\frac{2a}{\mathrm{d}}{\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}{\int }}\frac{{\partial }^2{\mathsf{\rho }}_{\mathrm{k}}}{\partial {\mathsf{\xi }}^2}}\cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }=-\frac{{\pi }^2{\mathrm{n}}^2{a}^2}{{\mathrm{d}}^2}{\Re }_{\mathrm{k}}+\frac{2a}{\mathrm{d}}\left({\left.\frac{\partial {\mathsf{\rho }}_{\mathrm{k}}}{\partial \mathsf{\xi }}\right|}_{\mathsf{\xi }=\frac{\mathrm{d}}{a}}{\left(-1\right)}^{\mathrm{n}}-{\left.\frac{\partial {\mathsf{\rho }}_{\mathrm{k}}}{\partial \mathsf{\xi }}\right|}_{\mathsf{\xi }=0}\right),$$

(14a)

$$\begin{gathered} \frac{\partial }{\partial \mathsf{\xi }}\left({\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}}\right)÷\frac{2a}{\mathrm{d}}{\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}{\int }}\left(\frac{\partial }{\partial \mathsf{\xi }}\left({\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}}\right)\right)}\cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }= \\ =\frac{2a}{\mathrm{d}}\left\{{\left.\left[{\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}}\right]\right|}_{\mathsf{\xi }=\frac{\mathrm{d}}{a}}\right.{\left(-1\right)}^{\mathrm{n}}-{\left.\left[{\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}}\right]\right|}_{\mathsf{\xi }=0}+ \\ +\frac{\pi \mathrm{n}a}{\mathrm{d}}{\displaystyle \underset{0}{\overset{\raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}{\int }}\left({\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\displaystyle \sum _{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}}\right)}\sin \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }, \end{gathered}$$

(14b)

staking into account (13), rewrite (10) in the form of an operator equation

$$\begin{gathered} \frac{\partial {\Re }_k\left(\mathrm{n},\tau \right)}{\partial \mathsf{\tau }}+\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}{\Re }_{\mathrm{k}}\left(\mathrm{n},\mathsf{\tau }\right)=\frac{2a}{\mathrm{d}}\left\{n_0\left(z_{\mathrm{k}-1}{|}_{\mathsf{\xi }=\frac{\mathrm{d}}{a}}\cdot {\left(-1\right)}^{\mathrm{n}}-z_{\mathrm{k}-1}{|}_{\mathsf{\xi }=0}\right)\right.- \\ \left.-\frac{\pi \mathrm{n}a}{\mathrm{d}}{\int }_0^{\mathrm{d}/a}\left(z_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\sum }_{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}\right)\sin \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }\right\}, \end{gathered}$$

(15)

where $\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}=\frac{{\pi }^2{\mathrm{n}}^2{a}^2}{{\mathrm{d}}^2}+\mathsf{\theta }$. Integrating (15) with ${\Re }_{\mathrm{k}}\left(\mathrm{n};0\right)=0$, we get

$$\begin{gathered} {\Re }_{\mathrm{k}}\left(\mathrm{n};\mathsf{\tau }\right)= \{-\frac{2a}{\mathrm{d}}{n}_0(1-{\left(-1\right)}^{\mathrm{n}}){\int }_0^{\mathsf{\tau }}\exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{z}}_{\mathrm{k}-1}\left(0,\mathsf{\tau }\right)\mathrm{d}\mathsf{\tau }- \\ \left.\begin{array}{c}-\frac{2a}{\mathrm{d}}\cdot \frac{\pi \mathrm{n}a}{\mathrm{d}}{\int }_0^{\mathsf{\tau }}({\int }_0^{\frac{\mathrm{d}}{a}}\left({\mathrm{z}}_0{\mathsf{\rho }}_{\mathrm{k}-1}+{\sum }_{\mathrm{m}=1}^{\mathrm{k}-2}{\mathrm{z}}_{\mathrm{m}}{\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}\right)\sin \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)\mathrm{d}\mathsf{\xi }\times \\ \times \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right)){\mathrm{d}\mathsf{\tau }}^{}\end{array}\right\}\times \exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right). \end{gathered}$$

(16)

Using additional decomposition

$$\begin{gathered} {\mathsf{\rho }}_{\mathrm{k}-1}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{s}=1}^{\infty }{\Re }_{\mathrm{k}-1}\left(\mathrm{s};\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{s}a}{\mathrm{d}}\mathsf{\xi }\right), \\ {\mathrm{z}}_{\mathrm{k}-1}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{s}=1}^{\infty }{\Re }_{\mathrm{k}-1}\left(\mathrm{s};\mathsf{\tau }\right)\left[\frac{{\phi }_1}{\frac{\pi \mathrm{s}a}{\mathrm{d}}}\sin \left(\frac{\pi \mathrm{s}a}{\mathrm{d}}\mathsf{\xi }\right)-\frac{{\phi }_2}{\frac{{\pi }^2{\mathrm{s}}^2{a}^2}{{\mathrm{d}}^2}}\left(1-{\left(-1\right)}^{\mathrm{s}}\right)\right], \end{gathered}$$

$$\begin{gathered} {\mathsf{\rho }}_{\mathrm{k}-\mathrm{m}-1}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{p}=1}^{\infty }{\Re }_{\mathrm{k}-\mathrm{m}-1}\left(\mathrm{p};\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{p}a}{\mathrm{d}}\mathsf{\xi }\right), \\ {\mathrm{z}}_{\mathrm{m}}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{l=1}^{\infty }{\Re }_{\mathrm{m}}\left(l;\mathsf{\tau }\right)\left[\frac{{\phi }_1}{\frac{\pi \mathrm{la}}{\mathrm{d}}}\sin \left(\frac{\pi \mathrm{la}}{\mathrm{d}}\mathsf{\xi }\right)-\frac{{\phi }_2}{\frac{{\pi }^2{l}^2{a}^2}{{\mathrm{d}}^2}}\left(1-{\left(-1\right)}^l\right)\right] \end{gathered}$$

and denoting ${\phi }_2={\phi }_1\frac{a}{\mathrm{d}}$, we convert (16) to the recurring formula

$$\begin{gathered} {\Re }_{\mathrm{k}}\left(\mathrm{n},\mathsf{\tau }\right)=\frac{8n_0{\phi }_1}{{\pi }^2}\sum _{\mathrm{s}=1}^{\infty }\left\{\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{s}}^2}\times \left[\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}\right]\right\}-\frac{4a}{\mathrm{d}}\cdot \\ \cdot \sum _{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{n}}^2-{\mathrm{s}}^2}\times \left[\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\cdot \exp \left(\left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}+\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathsf{\tau }}^{}\right){\mathrm{d}\mathsf{\tau }}^{}\right]\right\}+ \\ +\frac{8{\phi }_1}{{\pi }^2}\sum _{\mathrm{m}=1}^{\mathrm{k}-2} \sum _{\mathrm{p}=1}^{\infty } \sum _{l=1}^{\infty }\left\{\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right){\sin }^2\left(\frac{\pi l}{2}\right)}{l^2\left({\mathrm{n}}^2-{\mathrm{p}}^2\right)}\right.\times \\ \times \left.\left[\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\times {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-\mathrm{m}-1}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}\left(l{,\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}\right]\right\}. \end{gathered}$$

(17)

Since the generation of relaxation modes with complex amplitudes ${\Re }_{\mathrm{k}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ begins with the first order of perturbation theory [1]

$${\Re }_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

from the first term of (17) it is not difficult to see

$${\Re }_2^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times \frac{8n_0{\phi }_1}{{\pi }^2}\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times {\sum }_{\mathrm{s}=1}^{\infty }\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{s}}^2\left(\frac{1}{{\mathsf{\tau }}_s}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_n}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

$${\Re }_3^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-\frac{4{\mathrm{an}}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^2\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times {\left({\sum }_{\mathrm{s}=1}^{\infty }\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{s}}^2\left(\frac{1}{{\mathsf{\tau }}_s}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\right)}^2\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

where we get the recurring formula

$${\Re }_{\mathrm{k}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-\frac{4{\mathrm{a}n}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-1}\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times {\mathsf{\Lambda }}_0^{\mathrm{k}-1}\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}}.$$

(17a)

To develop a recurring expression for the complex amplitudes ${\Re }_k^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ of higher orders (multiples of frequency $r\mathsf{\omega }$), we rewrite the expression (17) as

$$\begin{gathered} {\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=\frac{8n_0{\phi }_1}{{\pi }^2}{\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{s}}^2}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-1}^{\left(r\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)\right]\right\}- \\ -\frac{4a}{\mathrm{d}}\cdot {\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{n}}^2-{\mathrm{s}}^2}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(\left(r-1\right)\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)\right]\right\}+ \\ +\frac{8{\phi }_1}{{\pi }^2}\sum _{\mathrm{m}=1}^{\mathrm{k}-2} \sum _{\mathrm{p}=1}^{\infty } \sum _{f=1}^{\mathrm{m}} \sum _{l=1}^{\infty }\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right){\sin }^2\left(\frac{\pi l}{2}\right)}{l^2\left({\mathrm{n}}^2-{\mathrm{p}}^2\right)}\times \\ \times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\left(r-f\right)\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(f\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)\right]. \end{gathered}$$

(18)

In Equation (18) uses integral operators

$${\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-1}^{\left(r\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}^{\left(r\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}=\frac{{\Re }_{\mathrm{k}-1}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(r\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)},$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(\left(r-1\right)\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}^{\left(\left(r-1\right)\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\cdot \exp \left(\left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}+\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathsf{\tau }}^{}\right){\mathrm{d}\mathsf{\tau }=}^{} \\ =\frac{{\Re }_{\mathrm{k}-1}^{\left(\left(r-1\right)\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)\cdot \exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(r\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\left(r-f\right)\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(f\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\left(r-f\right)\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(f\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}= \\ =\frac{{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\left(r-f\right)\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right){\Re }_{\mathrm{m}}^{\left(f\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(r\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}. \end{gathered}$$

As the generation of the relaxation modes with the amplitudes ${\Re }_k^{\left(2\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ begins from the second order of the perturbation theory, according to (18)

$$\begin{gathered} {\Re }_{\mathrm{k}}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-\frac{4a}{\mathrm{d}}\times {\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{n}}^2-{\mathrm{s}}^2}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)\right]\right\}+ \\ +\frac{8{\phi }_1}{{\pi }^2}{\sum }_{\mathrm{m}=1}^{\mathrm{k}-2} {\sum }_{\mathrm{p}=1}^{\infty } {\sum }_{l=1}^{\infty } \left\{\begin{array}{c}\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right){\sin }^2\left(\frac{\pi l}{2}\right)}{l^2\left({\mathrm{n}}^2-{\mathrm{p}}^2\right)}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)\right]\\ \end{array}\right\}. \end{gathered}$$

(18a)

In (18a) uses integral operators

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\exp \left(\left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}+\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathsf{\tau }}^{}\right){\mathrm{d}\mathsf{\tau }}^{}= \\ =\frac{{\Re }_{\mathrm{k}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)\cdot \exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left({\mathrm{l},\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}= \\ =\frac{{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}. \end{gathered}$$

According to (17a)

$${\Re }_{\mathrm{k}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-2}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-2}{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

$${\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-\mathrm{m}-2}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-\mathrm{m}-2}{\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

$${\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)=-\frac{4{a\mathrm{n}}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{m}-1}\cdot {\mathsf{\Lambda }}_0^{\mathrm{m}-1}{\sin }^2\left(\frac{\pi l}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_l}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}},$$

we get the recurrent relation from (18a)

$$\begin{gathered} {\Re }_{\mathrm{k}}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=\left(\mathrm{k}-1\right){\left(\frac{4a}{\mathrm{d}}\right)}^2{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-2}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-2}\cdot {\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times \\ {\wp }_2\left(\mathrm{n}\right)\times \frac{\exp \left(2\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

(18b)

where ${\wp }_2\left(\mathrm{n}\right)={\sum }_{\mathrm{s}=1}^{\infty }\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{\left({\mathrm{n}}^2-{\mathrm{s}}^2\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}$.

As the generation of the relaxation modes with the amplitudes ${\Re }_k^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ begins from the third order of the perturbation theory, according to (18)

$$\begin{gathered} {\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=\frac{8n_0{\phi }_1}{{\pi }^2}\sum _{\mathrm{s}=1}^{\infty }\left\{\frac{{\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{s}}^2}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-1}^{\left(3\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)\right]\right\}- \\ -\frac{4a}{\mathrm{d}}\cdot \sum _{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right)}{{\mathrm{n}}^2-{\mathrm{s}}^2}\times \left[{\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)\right]\right\}+ \\ +\frac{8{\phi }_1}{{\pi }^2}\sum _{\mathrm{m}=1}^{\mathrm{k}-2} \sum _{\mathrm{p}=1}^{\infty } \sum _{l=1}^{\infty } \frac{{\mathrm{n}}^2{\cos }^2\left(\frac{\pi \mathrm{n}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right){\sin }^2\left(\frac{\pi l}{2}\right)}{l^2\left({\mathrm{n}}^2-{\mathrm{p}}^2\right)}\cdot {\mathrm{K}}^{\left(\mathsf{\tau }\right)}[{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)+ \\ +{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)]. \end{gathered}$$

(18c)

In (18c), integral operators

$${\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-1}^{\left(3\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}^{\left(3\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}=\frac{{\Re }_{\mathrm{k}-1}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)},$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left(\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}{\mathsf{\tau }}^{}\right){\Re }_{\mathrm{k}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{s},\mathsf{\tau }}^{}\right)\cdot \exp \left(\left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}+\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathsf{\tau }}^{}\right){\mathrm{d}\mathsf{\tau }}^{}= \\ =\frac{ {\Re }_{\mathrm{k}-1}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)\cdot \exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot \\ \cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}=\frac{{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right){\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

$$\begin{gathered} {\hat{\mathrm{K}}}^{\left(\mathsf{\tau }\right)}\left({\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\right)=\exp \left(-\frac{\mathsf{\tau }}{{\mathsf{\tau }}_{\mathrm{n}}}\right)\cdot \\ \cdot {\int }_0^{\mathsf{\tau }}{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left({\mathrm{p},\mathsf{\tau }}^{}\right){\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l{,\mathsf{\tau }}^{}\right)\cdot \exp \left(\frac{{\mathsf{\tau }}^{}}{{\mathsf{\tau }}_{\mathrm{n}}}\right){\mathrm{d}\mathsf{\tau }}^{}=\frac{{\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right){\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \end{gathered}$$

have been introduced.

We get the contribution to amplitude ${\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ from the consequent recursion (18c), taking into account (18b)

$${\Re }_{\mathrm{k}-1}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{s},\mathsf{\tau }\right)=\left(\mathrm{k}-2\right){\left(\frac{4a}{\mathrm{d}}\right)}^2{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-3}\cdot {\mathrm{s}}^2{\cos }^2\left(\frac{\pi \mathrm{s}}{2}\right)\times {\wp }_2\left(\mathrm{s}\right)\times \frac{\exp \left(2\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)},$$

(18d)

where

$${\wp }_2\left(\mathrm{s}\right)={\sum }_{\mathrm{m}=1}^{\infty }\frac{{\sin }^2\left(\frac{\pi \mathrm{m}}{2}\right)}{\left({\mathrm{s}}^2-{\mathrm{m}}^2\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{m}}}+i\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

Substituting of (18d) from (18c) into addend gives

$${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_2=-\left(\mathrm{k}-2\right){\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-3}\cdot {\mathrm{n}}^2\times {\wp }_3\left(\mathrm{n}\right)\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)},$$

(18e)

where the notation

$${\wp }_3\left(\mathrm{n}\right)=\sum _{\mathrm{m}=1}^{\infty }\left\{\frac{{\mathrm{m}}^2{\wp }_2\left(\mathrm{m}\right){\cos }^2\left(\frac{\pi \mathrm{m}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{\left({\mathrm{n}}^2-{\mathrm{m}}^2\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{m}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\right)}\right\}$$

is introduced. It is used in numerical calculations as

$${\wp }_3\left(\mathrm{n}\right)={\sum }_{\mathrm{m}=1}^{\infty }{\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{m}}^2{\cos }^2\left(\frac{\pi \mathrm{m}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{\left({\mathrm{n}}^2-{\mathrm{m}}^2\right)\left({\mathrm{m}}^2-{\mathrm{s}}^2\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{m}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\right)}\right\}.$$

From (18c) it directly follows that ${\wp }_3\left(\mathrm{n}\right)\ne 0$ only for the odd modes n. We consider the contribution of the amplitude ${\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ from the third component of recursion (18c). For this purpose, from (18b) we calculate

$$\begin{gathered} {\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-\mathrm{m}-2}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-\mathrm{m}-2}{\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}}, \\ {\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)=-\frac{4{an}_0}{\mathrm{d}}\times {\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{m}-1}\cdot {\mathsf{\Lambda }}_0^{\mathrm{m}-1}{\sin }^2\left(\frac{\pi l}{2}\right)\times \frac{\exp \left(\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_l}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}}, \\ {\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right)=\left(\mathrm{k}-\mathrm{m}-2\right){\left(\frac{4a}{\mathrm{d}}\right)}^2{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-\mathrm{m}-3}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-\mathrm{m}-3}\cdot {\mathrm{p}}^2{\cos }^2\left(\frac{\pi \mathrm{p}}{2}\right)\times {\wp }_2\left(\mathrm{p}\right)\times \frac{\exp \left(2\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}, \\ {\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)=\left(\mathrm{m}-1\right){\left(\frac{4a}{\mathrm{d}}\right)}^2{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{m}-2}\cdot {\mathsf{\Lambda }}_0^{\mathrm{m}-2}\cdot l^2{\cos }^2\left(\frac{\pi l}{2}\right)\times {\wp }_2\left(l\right)\times \frac{\exp \left(2\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_l}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}. \end{gathered}$$

The contribution from product of ${\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right)\cdot {\Re }_{\mathrm{m}}^{\left(\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)$ will be equal to zero, since ${\cos }^2\left(\frac{\pi \mathrm{p}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)=0$ for any p values.

Non-zero contribution gives only the product ${\Re }_{\mathrm{k}-\mathrm{m}-1}^{\left(\mathsf{\omega }\right)}\left(\mathrm{p},\mathsf{\tau }\right)\cdot {\Re }_{\mathrm{m}}^{\left(2\mathsf{\omega }\right)}\left(l,\mathsf{\tau }\right)$. After that, from (18c)

$${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_3=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-3}\cdot {\mathrm{n}}^2\times {\wp }_3\left(\mathrm{n}\right)\times \frac{\left(\mathrm{k}-3\right)\left(\mathrm{k}-2\right)}{2}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

(18f)

The total contribution to the amplitude $\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]$ from the terms ${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_2$ and ${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_3$ gives

$${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_{2,3}=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}\cdot {\mathsf{\Lambda }}_0^{\mathrm{k}-3}\cdot {\mathrm{n}}^2\times {\wp }_3\left(\mathrm{n}\right)\times \frac{\left(\mathrm{k}-1\right)\left(\mathrm{k}-2\right)}{2}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

(18g)

Expression (18g), by virtue of (18c), is nonzero only for odd modes, therefore, it is further necessary to take into account the contribution to the amplitude $\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]$ from the first term of the recurring expression (18c). Each expression (18g) of the g-th approximation, substituted in subsequent approximations in the first term (18c), provides the following contributions to the amplitude ${\Re }_k^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$:

$${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_{1,\mathrm{g}}=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}{\mathsf{\Lambda }}_0^{\mathrm{g}-3}{\mathsf{\Lambda }}_1{\mathsf{\Lambda }}_2^{\mathrm{k}-\mathrm{g}-1}\frac{\left(\mathrm{g}-1\right)\left(\mathrm{g}-2\right)}{2}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

(18h)

In (18h), the sum of the series ${\mathsf{\Lambda }}_1={\sum }_{\mathrm{p}=1}^{\infty } \left\{\frac{{\wp }_3\left(\mathrm{p}\right)\cdot {\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\right\}$, by virtue of (d1), reduces to the form

$${\mathsf{\Lambda }}_1={\sum }_{\mathrm{p}=1}^{\infty } {\sum }_{\mathrm{m}=1}^{\infty } {\sum }_{\mathrm{s}=1}^{\infty } \left\{\frac{{\mathrm{m}}^2{\cos }^2\left(\frac{\pi \mathrm{m}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)}{\left({\mathrm{p}}^2-{\mathrm{m}}^2\right)\left({\mathrm{m}}^2-{\mathrm{s}}^2\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{s}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{m}}}+\mathrm{i}\left(2\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\right)\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)\right)}\right\}.$$

In (18h), the sum of the series ${\mathsf{\Lambda }}_2={\sum }_{\mathrm{p}=1}^{\infty }\left\{\frac{{\sin }^2\left(\frac{\pi \mathrm{p}}{2}\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\right\}$ is also introduced.

The total contribution of expression (18h) to the complex amplitude ${\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ in the approximation $\mathrm{k}\ge 4$ is determined by summing the elements ${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_{1,\mathrm{g}}$ by g from g = 3 to g = k − 1: ${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_1={\sum }_{\mathrm{g}=3}^{\mathrm{k}-1}{\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_{1,\mathrm{g}}$.

Then

$${\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_1=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3}\cdot {\mathsf{\Lambda }}_1\cdot {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times {\sum }_{\mathrm{g}=3}^{\mathrm{k}-1}\frac{\left(\mathrm{g}-1\right)\left(\mathrm{g}-2\right)}{2}{\mathsf{\Lambda }}_0^{\mathrm{g}-3}\cdot {\mathsf{\Lambda }}_2^{\mathrm{k}-\mathrm{g}-1}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

(18i)

The complete expression for the complex amplitude ${\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)={\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_{2,3}++{\left[{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\right]}_1$ becomes

$$\begin{gathered} {\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\left(\frac{8n_0{\phi }_1}{{\pi }^2}\right)}^{\mathrm{k}-3} \\ \times \left\{\frac{\left(\mathrm{k}-1\right)\left(\mathrm{k}-2\right)}{2}{\mathsf{\Lambda }}_0^{\mathrm{k}-3}\cdot {\mathrm{n}}^2{\wp }_3\left(\mathrm{n}\right)+{\mathsf{\Lambda }}_1{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times {\sum }_{\mathrm{g}=3}^{\mathrm{k}-1}\frac{\left(\mathrm{g}-1\right)\left(\mathrm{g}-2\right)}{2}{\mathsf{\Lambda }}_0^{\mathrm{g}-3}\cdot {\mathsf{\Lambda }}_2^{\mathrm{k}-\mathrm{g}-1}\right\}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}. \end{gathered}$$

(18j)

Further, representing (7) in the form of $\mathsf{\rho }\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{r=1}^{\infty }{\sum }_{k=r}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\mathsf{\rho }}_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, $\mathsf{\rho }\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{r=1}^{\infty }{\mathsf{\rho }}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, we will write the expansion of the frequency harmonic number r for the function $\rho \left(\mathsf{\xi },\mathsf{\tau }\right)$ in power series by the powers of the parameter $\mathsf{\gamma }$

$${\mathsf{\rho }}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{\mathrm{k}=r}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\mathsf{\rho }}_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right).$$

(19)

The decomposition components (19) are represented by the k-th order relaxation modes of the perturbation theory generated at $r\mathsf{\omega }$ frequency, respectively

$${\mathsf{\rho }}_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi };\mathsf{\tau }\right)={\sum }_{\mathrm{n}=1}^{\infty }{\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right).$$

(20)

Expression (20), by virtue of (14), has the meaning of superposition of the relaxation modes of the n-th order ${\mathsf{\rho }}_{\mathrm{k},\mathrm{n}}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi };\mathsf{\tau }\right)={\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$ with the amplitudes calculated according to the recurring formula (18), taking into account (17a), (18b), (18j). Then, from (19), (20) we obtain

$${\mathsf{\rho }}^{\left(r\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=\mathrm{r}}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right).$$

(21)

When r = 1, r = 2, from (21) we obtain ${\mathsf{\rho }}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=1}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\Re }_{\mathrm{k}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$ and ${\mathsf{\rho }}^{\left(2\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=2}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\Re }_{\mathrm{k}}^{\left(2\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$. Taking (17a), (18b) into account, we confirm the expressions (20), (21) from [1,7].

When r = 3, according to ${\mathsf{\rho }}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)={\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=3}^{\infty }{\mathsf{\gamma }}^{\mathrm{k}}{\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cdot \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right)$ and taking account of (18j), we obtain

$$\begin{gathered} {\mathsf{\rho }}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)=-{\left(\frac{4a}{\mathrm{d}}\right)}^3{n}_0{\mathsf{\gamma }}^3\times {\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=3}^{\infty }{\left(\frac{8n_0{\phi }_1\mathsf{\gamma }}{{\pi }^2}\right)}^{\mathrm{k}-3}\left\{{\mathsf{\Lambda }}_0^{\mathrm{k}-3}\times \frac{\left(\mathrm{k}-1\right)\left(\mathrm{k}-2\right)}{2}\times {\mathrm{n}}^2{\wp }_3\left(\mathrm{n}\right)\right.+{\mathsf{\Lambda }}_1{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times \\ \left.{\sum }_{\mathrm{g}=3}^{\mathrm{k}-1}\frac{\left(\mathrm{g}-1\right)\left(\mathrm{g}-2\right)}{2}{\mathsf{\Lambda }}_0^{\mathrm{g}-3}\cdot {\mathsf{\Lambda }}_2^{\mathrm{k}-\mathrm{g}-1}\right\}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\times \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right). \end{gathered}$$

(22)

From where, after calculating the sums of the series, we get

$${\mathsf{\rho }}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)=-\frac{64a^3{n}_0{\mathsf{\gamma }}^3}{{\mathrm{d}}^3{\left(1-\frac{8n_0{\phi }_1{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\pi }^2}\right)}^3}\times {\sum }_{\mathrm{n}=1}^{\infty }\left\{{\mathrm{n}}^2{\wp }_3\left(\mathrm{n}\right)+\frac{8n_0{\phi }_1{\mathsf{\Lambda }}_1\mathsf{\gamma }}{{\pi }^2}\cdot \frac{{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{1-\frac{8n_0{\phi }_1{\mathsf{\Lambda }}_2\mathsf{\gamma }}{{\pi }^2}}\right\}\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\times \cos \left(\frac{\pi \mathrm{n}a}{\mathrm{d}}\mathsf{\xi }\right).$$

(23)

Based on the expression [1]

$${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\tau }\right)=\frac{\mathrm{q}}{\mathrm{d}}{\int }_0^{\mathrm{d}}{\mathrm{x}\mathsf{\rho }}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)\mathrm{dx},$$

(24)

taking account of (23)

$${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\tau }\right)=\frac{128a^3\mathrm{q}{n}_0{\mathsf{\gamma }}^3}{{\mathrm{d}}^2{\pi }^2{\left(1-\frac{8a\mathrm{q}{n}_0{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\pi }^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{E}}_0}\right)}^3}\times {\sum }_{\mathrm{n}=1}^{\infty }\left\{{\wp }_3\left(\mathrm{n}\right)+\frac{8a\mathrm{q}{n}_0{\mathsf{\Lambda }}_1\mathsf{\gamma }}{{\mathrm{n}}^2{\pi }^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{E}}_0\left(1-\frac{8a\mathrm{q}{n}_0{\mathsf{\Lambda }}_2\mathsf{\gamma }}{{\pi }^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{E}}_0}\right)}\right\}\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\times \frac{\exp \left(3\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\mathsf{\tau }\right)}{\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\left(3\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}.$$

(25)

Expression (25), in addition to (8), confirms that odd r relaxation modes ${\mathsf{\rho }}_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, as well as ${\mathsf{\rho }}_{\mathrm{k}}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, give nonzero contribution to polarization. Manifestation of this pattern at higher frequencies $\left(5\mathsf{\omega }\right)$,$\left(7\mathsf{\omega }\right)$,…, $\left(\left(2\mathsf{\lambda }+1\right)\mathsf{\omega }\right)$ is obvious.

4. The Complex Dielectric Permittivity

The dielectric polarization, in the general case, calculated as

$${\mathrm{P}}^{\left(\mathsf{\Omega }\right)}\left(\mathrm{t}\right)={\sum }_{\mathsf{\lambda }=0}^{\infty }{\hat{\mathsf{\alpha }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\mathsf{\lambda }+1}\right)}{\mathrm{E}}_{\mathrm{pol}}^{2\lambda +1}\left(\mathrm{t}\right).$$

(26)

In (26) ${\hat{\mathsf{\alpha }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}$ is a complex function defined on set ${\mathsf{\Omega }}_{2\lambda +1}=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega };\dots ;2\lambda \mathsf{\omega };\left(2\lambda +1\right)\mathsf{\omega }\right\}$ of parameters of relaxers and temperature ${\hat{\mathsf{\alpha }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}\left({\mathrm{U}}_0{,\mathsf{\delta }}_0{,\mathsf{\nu }}_0,n_0,a;\mathrm{T}\right)=\frac{{\mathrm{P}}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathrm{t}\right)}{{\mathrm{E}}_{\mathrm{pol}}^{2\lambda +1}\left(\mathrm{t}\right)}$, which has meaning of $2\lambda$ order component from decomposition of the polarization ${\mathrm{P}}^{\left(\mathsf{\Omega }\right)}\left(\mathrm{t}\right)$ into a series according to odd degrees of intensity of the polarizing field ${\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$. Here $\lambda =\left\{0,1,2,3,\dots \right\}$. The frequency harmonic of the polarization, of order $r=2\lambda +1$, according to (7), ${\mathrm{P}}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathrm{t}\right)=\frac{\mathrm{q}}{\mathrm{d}}{\sum }_{\lambda =0}^{\infty }{\int }_0^{\mathrm{d}}{\mathrm{x}\mathsf{\rho }}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)\mathrm{dx}$, taking into account (20), becomes

$${\mathrm{P}}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathrm{t}\right)=-\frac{2\mathrm{qd}}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{k}=2\lambda +1}^{\infty }\frac{{\mathsf{\gamma }}^{\mathrm{k}}\cdot {\Re }_{\mathrm{k}}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)}{{\mathrm{n}}^2}\times {\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right).$$

(27)

Comparing the formula ${\mathrm{P}}^{\left(\mathsf{\Omega }\right)}\left(\mathrm{t}\right)={\hat{\mathsf{\alpha }}}^{\left(\mathsf{\Omega }\right)}\cdot {\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$ ([4], p. 151) with the expression (26), let us write the complex dielectric susceptibility (CDS) in the form of

$${\hat{\mathsf{\alpha }}}^{\left(\mathsf{\Omega }\right)}={\sum }_{\lambda =0}^{\infty }{\hat{\mathsf{\alpha }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}{\mathrm{E}}_{\mathrm{pol}}^{2\lambda }\left(\mathrm{t}\right),{\hat{\mathsf{\alpha }}}^{\left(\mathsf{\Omega }\right)}={\sum }_{\lambda =0}^{\infty }{\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}.$$

(28)

In (28) ${\hat{\mathsf{\alpha }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}\cdot {\mathrm{E}}_{\mathrm{pol}}^{2\lambda }\left(\mathrm{t}\right)={\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}=\frac{{\mathrm{P}}^{\left(\left(2\lambda +1\right)\mathsf{\omega }\right)}\left(\mathrm{t}\right)}{{\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)}$.

Taking the relaxation complex polarization as ${\mathrm{P}}^{\left(\mathsf{\Omega }\right)}\left(\mathrm{t}\right)={\mathsf{\epsilon }}_0\left({\hat{{\epsilon }}}^{\left(\mathsf{\Omega }\right)}-{\mathsf{\epsilon }}_{\infty }\right){\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$ ([4], p. 151), we find the complex dielectric permittivity (CDP)

$${\hat{{\epsilon }}}^{\left(\mathsf{\Omega }\right)}={\mathsf{\epsilon }}_{\infty }+\frac{{\hat{\mathsf{\alpha }}}^{\left(\mathsf{\Omega }\right)}}{{\mathsf{\epsilon }}_0}.$$

(29)

Substitution (28) into (29) gives

$${\hat{{\epsilon }}}^{\left(\mathsf{\Omega }\right)}={\mathsf{\epsilon }}_{\infty }+{\sum }_{\lambda =0}^{\infty }{\hat{{\epsilon }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}{\mathrm{E}}_{\mathrm{pol}}^{2\lambda }\left(\mathrm{t}\right), {\hat{{\epsilon }}}^{\left(\mathsf{\Omega }\right)}={\mathsf{\epsilon }}_{\infty }+{\sum }_{\lambda =0}^{\infty }{\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}.$$

(30)

In (30) ${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}={\hat{\mathsf{\epsilon }}}_{2\lambda }^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}{\mathrm{E}}_{\mathrm{pol}}^{2\lambda }\left(\mathrm{t}\right)=\frac{{\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}}{{\mathsf{\epsilon }}_0}$; limiting the approximation of $\lambda =1$,

$${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1,{\mathsf{\Omega }}_3\right)}={\mathsf{\epsilon }}_{\infty }+\frac{1}{{\mathsf{\epsilon }}_0}\left({\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}+{\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}\right)$$

(31)

we find the CDP with accuracy to the quadratic term along the field

$${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1,{\mathsf{\Omega }}_3\right)}={\mathsf{\epsilon }}_{\infty }+{\hat{\mathsf{\epsilon }}}_0^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}+{\hat{\mathsf{\epsilon }}}_2^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}{\mathrm{E}}_{\mathrm{pol}}^2\left(\mathrm{t}\right).$$

(32)

Substituting (8) and (25) in expressions of the form

$$\begin{gathered} {\hat{\mathsf{\epsilon }}}^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}={\hat{\mathsf{\epsilon }}}_0^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}=\frac{{\hat{\mathsf{\alpha }}}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}{{\mathsf{\epsilon }}_0}=\frac{{\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{t}\right)}{{\mathsf{\epsilon }}_0{\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)}, \\ {\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}={\hat{\mathsf{\epsilon }}}_2^{{}^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}}{\mathrm{E}}_{\mathrm{pol}}^2\left(\mathrm{t}\right)=\frac{{\hat{\mathsf{\alpha }}}^{{}^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}}}{{\mathsf{\epsilon }}_0}=\frac{{\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{t}\right)}{{\mathsf{\epsilon }}_0{\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)}, \end{gathered}$$

we obtain

$${\hat{\mathsf{\epsilon }}}^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}={\hat{\mathsf{\epsilon }}}_0^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}=\frac{{\mathsf{\epsilon }}_{\infty }\left({\Gamma }_1^{\left(\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}{1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}+{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}},$$

(33)

where relaxation parameters are introduced that characterize the first frequency harmonic of the polarization (a multiple of the first power of the alternating field (8))

$${\Gamma }_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{T_{\mathrm{n}}}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{{\mathrm{n}}^2}\right)}\right], {\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{\mathsf{\omega }{T}_{\mathrm{n}}^2}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{{\mathrm{n}}^2}\right)}\right].$$

(33a)

The expression ${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1\right)}={\mathsf{\epsilon }}_{\infty }+{\hat{\mathsf{\epsilon }}}_0^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}$ formally coincides with the formulae (2) of ([1], p. 44), which indicates the validity of the “zero” field of approximation and consistent with the expressions obtained in [3] for CDP ([3], p. 17).

Also, based on (25), we obtain a multiple of the second power of the alternating field component of the complex permittivity

$${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}\right)}=\frac{{\mathsf{\epsilon }}_0^2{\mathsf{\epsilon }}_{\infty }^3{\pi }^4{\mathrm{E}}_0^2}{4{\mathrm{q}}^2{n}_0^2{\mathrm{d}}^2}\left(\frac{{\Gamma }_{1,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_{1,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}}{{\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}+{\mathrm{i}\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}}+\frac{{\Gamma }_{2,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_{2,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}}{{\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}+{\mathrm{i}\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}}\right)\cdot \exp \left(2\mathrm{i}\mathsf{\omega }t\right).$$

(34)

In (34) the following notations are adopted:

$$\begin{gathered} {\Gamma }_{1,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}={\left(\frac{8}{{\pi }^2}\right)}^3{\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{m}=1}^{\infty }{\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{m}}^2{\cos }^2\left(\frac{\pi \mathrm{m}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{\left({\mathrm{n}}^2-{\mathrm{m}}^2\right)\cdot \left({\mathrm{m}}^2-{\mathrm{s}}^2\right)\cdot \left(1+{\mathsf{\omega }}^2{T}_{\mathrm{s}}^2\right)\cdot \left(1+4{\mathsf{\omega }}^2{T}_{\mathrm{m}}^2\right)\cdot \left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right.\times \\ \times \left.\frac{T_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{n}}}{T_{\mathrm{M}}^3}\times \left(1-{\mathsf{\omega }}^2\left(2T_{\mathrm{s}}{T}_{\mathrm{m}}+6T_{\mathrm{m}}{T}_{\mathrm{n}}+3T_{\mathrm{n}}{T}_{\mathrm{s}}\right)\right)\right\}; \end{gathered}$$

(34a)

$$\begin{gathered} {\Gamma }_{1,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}={\left(\frac{8}{{\pi }^2}\right)}^3{\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{m}=1}^{\infty }{\sum }_{\mathrm{s}=1}^{\infty }\left\{\frac{{\mathrm{m}}^2{\cos }^2\left(\frac{\pi \mathrm{m}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{s}}{2}\right){\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{\left({\mathrm{n}}^2-{\mathrm{m}}^2\right)\cdot \left({\mathrm{m}}^2-{\mathrm{s}}^2\right)\cdot \left(1+{\mathsf{\omega }}^2{T}_{\mathrm{s}}^2\right)\cdot \left(1+4{\mathsf{\omega }}^2{T}_{\mathrm{m}}^2\right)\cdot \left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right.\times \left.\frac{T_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{n}}}{T_{\mathrm{M}}^3}\times \left(\mathsf{\omega }\left(T_{\mathrm{s}}+2T_{\mathrm{m}}++3T_{\mathrm{n}}\right)- \\ 6{\mathsf{\omega }}^3{T}_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{n}}\right)\right\}; \end{gathered}$$

(34b)

$${\Gamma }_{2,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}={\left(\frac{8}{{\pi }^2}\right)}^4{\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{p}=1}^{\infty }{\sum }_{\mathrm{m}=1}^{\infty }{\sum }_{\mathrm{s}=1}^{\infty }\frac{\left(1-{\mathsf{\omega }}^2\left(2T_{\mathrm{s}}{T}_{\mathrm{m}}+6T_{\mathrm{m}}{T}_{\mathrm{p}}+3T_{\mathrm{p}}{T}_{\mathrm{s}}\right)-3\mathsf{\omega }{T}_{\mathrm{n}}\left(\mathsf{\omega }\left(T_{\mathrm{s}}+2T_{\mathrm{m}}+3T_{\mathrm{p}}\right)-6{\mathsf{\omega }}^3{T}_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{p}}\right)\right)}{\left(1+{\mathsf{\omega }}^2{T}_{\mathrm{s}}^2\right)\left(1+4{\mathsf{\omega }}^2{T}_{\mathrm{m}}^2\right)\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{p}}^2\right)\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)};$$

(34c)

$${\Gamma }_{2,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}={\left(\frac{8}{{\pi }^2}\right)}^4{\sum }_{\mathrm{n}=1}^{\infty }{\sum }_{\mathrm{p}=1}^{\infty }{\sum }_{\mathrm{m}=1}^{\infty }{\sum }_{\mathrm{s}=1}^{\infty }\frac{\left(3\mathsf{\omega }{T}_{\mathrm{n}}\left(1-{\mathsf{\omega }}^2\left(2T_{\mathrm{s}}{T}_{\mathrm{m}}+6T_{\mathrm{m}}{T}_{\mathrm{p}}+3T_{\mathrm{p}}{T}_{\mathrm{s}}\right)+\mathsf{\omega }\left(T_{\mathrm{s}}+2T_{\mathrm{m}}+3T_{\mathrm{p}}\right)-6{\mathsf{\omega }}^3{T}_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{p}}\right)\right)}{\left(1+{\mathsf{\omega }}^2{T}_{\mathrm{s}}^2\right)\left(1+4{\mathsf{\omega }}^2{T}_{\mathrm{m}}^2\right)\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{p}}^2\right)\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)};$$

(34d)

$${\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}={\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}\cdot {\mathsf{\Phi }}_1^{\left(3\mathsf{\omega }\right)}-{\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}\cdot {\mathsf{\Phi }}_2^{\left(3\mathsf{\omega }\right)}, {\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}={\mathsf{\Phi }}_2^{\left(3\mathsf{\omega }\right)}\cdot {\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}+{\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}\cdot {\mathsf{\Phi }}_1^{\left(3\mathsf{\omega }\right)};$$

(34e)

$${\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}=\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)\cdot \left({\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2-3{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2\right), {\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}={\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(3{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2-{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2\right)$$

(34f)

$${\mathsf{\Phi }}_1^{\left(3\mathsf{\omega }\right)}=1-{\Gamma }_1^{\left(3\mathsf{\omega }\right)}, {\mathsf{\Phi }}_2^{\left(3\mathsf{\omega }\right)}={\Gamma }_2^{\left(3\mathsf{\omega }\right)};$$

(34g)

$${\Gamma }_1^{\left(3\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{3T_{\mathrm{n}}}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right],{\Gamma }_2^{\left(3\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{3\mathsf{\omega }{T}_{\mathrm{n}}^2}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+9{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right].$$

(34h)

Based on (32), taking (33) and (34) into account, we obtain

$${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1,{\mathsf{\Omega }}_3\right)}={\mathsf{\epsilon }}_{\infty }\left(\frac{{1-\Gamma }_1^{\left(\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2}+\frac{{\mathsf{\epsilon }}_0^2{\mathsf{\epsilon }}_{\infty }^2{\pi }^4{\mathrm{E}}_0^2}{4{\mathrm{q}}^2{n}_0^2{\mathrm{d}}^2}\left({\chi }_1^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}-\mathrm{i}{\chi }_2^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}\right)\cdot \exp \left(2\mathrm{i}\mathsf{\omega }t\right)\right).$$

(35)

The abridged notations are adopted in (35):

$$\begin{gathered} {\chi }_1^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}=\frac{{\kappa }_1^2-{\kappa }_2^2}{{\kappa }_1^2+{\kappa }_2^2},{\chi }_2^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}=\frac{2{\kappa }_1{\kappa }_2}{{\kappa }_1^2+{\kappa }_2^2}; \\ {\kappa }_1={\mathsf{\Psi }}_1{\mathsf{\Psi }}_3+{\mathsf{\Psi }}_2{\mathsf{\Psi }}_4, {\kappa }_2={\mathsf{\Psi }}_1{\mathsf{\Psi }}_4-{\mathsf{\Psi }}_2{\mathsf{\Psi }}_3; \\ {\mathsf{\Psi }}_1={\Gamma }_{1,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}+{\Gamma }_{1,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}+{\Gamma }_{2,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}+{\Gamma }_{2,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}; \\ {\mathsf{\Psi }}_2={\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}{\Gamma }_{1,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}-{\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}{\Gamma }_{1,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}+{\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}{\Gamma }_{2,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}-{\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}{\Gamma }_{2,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}; \\ {\mathsf{\Psi }}_3={\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}{\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}-{\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}, {\mathsf{\Psi }}_4={\mathsf{\Phi }}_2^{\left(\mathsf{\omega }\right)}{\mathsf{\Phi }}_1^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}+{\mathsf{\Phi }}_2^{\left(\mathsf{\omega };3\mathsf{\omega }\right)}{\mathsf{\Phi }}_1^{\left(\mathsf{\omega }\right)}. \end{gathered}$$

Separating the real and imaginary parts in (35), we get

$$\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1,{\mathsf{\Omega }}_3\right)}\right]={\mathsf{\epsilon }}_{\infty }\left(\frac{{1-\Gamma }_1^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2}+\frac{{\mathsf{\epsilon }}_0^2{\mathsf{\epsilon }}_{\infty }^2{\pi }^4{\mathrm{E}}_0^2}{4{\mathrm{q}}^2{n}_0^2{\mathrm{d}}^2}\left({\mathsf{\chi }}_1^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}\cos \left(2\mathsf{\omega }t\right)+{\mathsf{\chi }}_2^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}\sin \left(2\mathsf{\omega }t\right)\right)\right),$$

(36)

$$\mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1,{\mathsf{\Omega }}_3\right)}\right]={\mathsf{\epsilon }}_{\infty }\left(\frac{{\Gamma }_2^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2}+\frac{{\mathsf{\epsilon }}_0^2{\mathsf{\epsilon }}_{\infty }^2{\pi }^4{\mathrm{E}}_0^2}{4{\mathrm{q}}^2{n}_0^2{\mathrm{d}}^2}\left({\mathsf{\chi }}_2^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}\cos \left(2\mathsf{\omega }t\right)-{\mathsf{\chi }}_1^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}\sin \left(2\mathsf{\omega }t\right)\right)\right).$$

(37)

It is easy to see that the first terms in the right parts of (36), (37) also formally coincide with the CDP components calculated in the “zero” field approximation ([1], p. 44; [3], p. 17).

The second terms in the right parts of (36), (37), respectively, reflect the influence of the polarizing field on the CDP spectra, in the regions of both strong fields (10–1000 MV/m) and ultra-high temperatures (550–1500 K)—high-temperature volumetric-charge polarization; and weak fields (100 kV/m–1000 kV/m) and ultra-low temperatures (1–10 K)—low-temperature quantum tunnel polarization.

The functions ${\Gamma }_{1,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}$, ${\Gamma }_{1,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}$ reflect the non-linear effects of the third order on the field frequency associated with the interaction of the relaxation modes generated at frequencies $\mathsf{\omega }$, $2\mathsf{\omega }$, $3\mathsf{\omega }$ with the corresponding relaxation times $T_{\mathrm{s}}{T}_{\mathrm{m}}{T}_{\mathrm{p}}$. An additional relaxation mode generated at the frequency of the $3\mathsf{\omega }$ field with relaxation time ${\mathsf{\tau }}_{\mathrm{p}}$ appears in functions ${\Gamma }_{2,1}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}$, ${\Gamma }_{2,2}^{\left(\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right)}$. At the same time, the parameters ${\Gamma }_1^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_2^{\left(\mathsf{\omega }\right)}$ reflect the non-linear effects of the first order in frequency, with relaxation times ${\mathsf{\tau }}_{\mathrm{n}}$. Parameters ${\mathsf{\Lambda }}_1$, ${\mathsf{\Lambda }}_2$ of (18h), (23) are a measure of the effect of the non-linear effects of the third order (interaction between the relaxation modes at frequencies $\mathsf{\omega }$, $2\mathsf{\omega }$, $3\mathsf{\omega }$ and corresponding relaxation times ${\mathsf{\tau }}_{\mathrm{s}}{\mathsf{\tau }}_{\mathrm{m}}{\mathsf{\tau }}_{\mathrm{p}}$) on the polarization. So, provided that the parameters ${\Xi }_0=$$=\frac{8n_0{\phi }_1{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\pi }^2}<<1$, ${\Xi }_1=\frac{8n_0{\mathsf{\phi }}_1{\mathsf{\Lambda }}_1\mathsf{\gamma }}{{\pi }^2}<<1$, ${\Xi }_2=\frac{8n_0{\phi }_1{\mathsf{\Lambda }}_2\mathsf{\gamma }}{{\pi }^2}<<1$ are small, the expression (23) is determined by the relaxation mode ${\mathsf{\rho }}_3^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$, corresponding to the third order of the perturbation theory (k = 3). This is the simplest (by the structure of complex amplitudes ${\Re }_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$) of the set of modes ${\mathsf{\rho }}_{\mathrm{k}}^{\left(3\mathsf{\omega }\right)}\left(\mathsf{\xi },\mathsf{\tau }\right)$ generated by the frequency $\left(3\mathsf{\omega }\right)$, which is consistent with ([3], p. 17).

Numerical calculation according to the formulas (36), (37) will allow analysis in more detail of the influence of the non-linear effects of the third order on the field frequency, on polarization, depending on the values of the parameters ${\mathrm{U}}_0$, $n_0$, ${\mathsf{\nu }}_0$, ${\mathsf{\delta }}_0$ ([4], pp. 150–152) and the thickness of the dielectric.

The mathematical model developed in this paper is suitable for theoretical studies, previously experimentally installed in crystals of zirconia alumina ceramics (ZAC) (ZrO₂-Y₂O₃)-n (Al₂O₃), at the temperature of T = 1250 K and frequency of 1 kHz, abnormally high dielectric permittivity ε = 5 $\cdot$ 10⁶ [5]. The nonlinearities of the first and third order can be considered as the reason for the semi-ferroelectric effect [6] associated with the rearrangement of the oxygen subarray in the ZAC, near the phase transition point.

The areas closest to electrical and radio engineering in the practical application of this model are laser and fiber optic technology: femtosecond lasers, optical light manipulators, incident radiation direction detectors, differential phase and frequency analyzers, etc. [8,9,10,11].

5. Effect of Temperature on Theoretical Spectra of Complex Dielectric Permittivity

Further study of the analytical dependencies (36), (37) is a separate theoretical task, the solution of which, together with expressions (33a), (34a)–(34h), will allow us to establish in more detail the influence of the nonlinear effects on dielectric loss tangent spectra and thermos-stimulated polarization and depolarization currents. It depends both on the patterns as macroscopic relaxation processes occurring throughout the dielectric and related to polarizing parameters (external) electric field (EMF amplitude and frequency) and temperature and microscopic processes associated with transitions of main charge carriers (ions of both charge signs in the general case, and protons in HBC) through a potential barrier. The parameters of the potential barrier (reflecting the molecular mechanism of interaction of the conduction ions with the crystal lattice, and protons with an anionic sublattice in HBC) significantly determine the efficiency of both classical (thermally activated) and quantum tunnel transitions of microparticles (relaxers) through the potential barrier. Within the framework of the quasi-classical model, they determine the influence of classical effects on the nonlinear volume–charge polarization in the field of sufficiently high temperatures (250–550 K) and quantum effects on tunneling polarization manifested in HBC due to the quantum relaxation movement of protons in the region of low (50–100 K) and ultra-low (4–25 K) temperatures.

The analytical expressions for the relaxation coefficients (33a) calculated at the base frequency $\left(\mathsf{\omega }\right)$ of the alternating electric field in the areas of diffusion $\left(\mathrm{T}<{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}\right)$ and Maxwell $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}\right)$ relaxation, away from the critical temperature ${\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$, should differ significantly in the type of functional dependence of these parameters on temperature. In this case, we will analyze the dispersion expressions (36), (37) taking into account the temperature dependencies for the complete relaxation time $T_{\mathrm{n}}=\frac{T_{\mathrm{n},\mathrm{D}}}{\frac{T_{\mathrm{n},\mathrm{D}}}{T_{\mathrm{M}}}+1}=\frac{T_{\mathrm{D}}}{\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}+{\mathrm{n}}^2}$,$T_{\mathrm{n}}=\frac{T_{\mathrm{M}}}{1+\frac{T_{\mathrm{M}}}{T_{\mathrm{n},\mathrm{D}}}}=\frac{T_{\mathrm{M}}}{1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}}$.

We convert the “zero” component of the complex dielectric permittivity (CDP) by the polarizing field from (32), taking into account (33), into ${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_1\right)}={\mathsf{\epsilon }}_{\infty }+{\hat{\mathsf{\epsilon }}}_0^{{}^{\left({\mathsf{\Omega }}_1=\left\{\mathsf{\omega }\right\}\right)}}={\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}=\frac{{\mathsf{\epsilon }}_{\infty }}{1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}+{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}}$. From where, separating the real and imaginary components, we obtain the dispersion ratios for the CDP components calculated in the nonlinear approximation of the perturbation theory (up to infinite approximation by the small dimensionless parameter $\mathsf{\gamma }=\frac{{\varsigma }_0{W}^{\left(1\right)}}{W^{\left(0\right)}}<1$) at the main frequency $\left(\mathsf{\omega }\right)$ of the external polarizing alternating electric field, taking into account the effects manifested, within the phenomenological model of the relaxation polarization in the interaction of the first-order relaxation modes by the field frequency

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}^{}=\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]=\frac{{\mathsf{\epsilon }}_{\infty }\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}{{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2}, {\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}^{//}=\mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]=\frac{{\mathsf{\epsilon }}_{\infty }{\Gamma }_2^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)}^2}.$$

(38)

In (38), the relaxation coefficients (33a) are a converging infinite functional series, obtained as a result of solutions of the Fokker–Planck Equation (2) together with the Poisson Equation (3) in an infinite approximation of the perturbation theory by a small dimensionless parameter $\mathsf{\gamma }=\frac{{\varsigma }_0{W}^{\left(1\right)}}{W^{\left(0\right)}}<1$ according to Equation (7), at the fundamental frequency of the alternating electric field and thereby are the consequences of sufficiently stringent solutions to the generalized nonlinear kinetic Equation (2). They allow us to calculate the spatially inhomogeneous bulk density of the excess electric charge in arbitrary approximation (multiplicity r) on frequency harmonics (21). In particular, cases that are used to calculate the values of polarization of the dielectric at the first two frequencies (r= = 1, r = 3) from expressions (8) and (25) are important for comparison with the experiment. In this case, temperature dependencies of the relaxation parameters (33a) corresponding to the first frequency harmonic of polarization (8) are similar to temperature dependencies for the simplest forms of the relaxation parameters (34h), corresponding to the third frequency harmonic of polarization (25). This allows us, with a certain degree of accuracy, to judge the significant influence of the temperature range on the mechanism of the nonlinear relaxation processes when generating polarization of the dielectric in the wide range of field parameters. It is obvious that in the region of weak fields (0.1–1 MV/m) the non-linearities will appear due to low-temperature (10–50 K) polarization effects (quantum diffusion polarization), and in the region of strong fields (10–1000 MV/m) the non-linearities will appear due to high-temperature (550–1500 K) polarization effects (nonlinear bulk-charge polarization of the mixed type) [7,12,13,14,15,16].

In the field of Maxwell relaxation $\left(T_{\mathrm{M}}<<T_{\mathrm{n},\mathrm{D}};T_{\mathrm{M}}<<T_{\mathrm{D}}\right)$, when $\frac{T_{\mathrm{M}}{\mathrm{n}}^2}{T_{\mathrm{D}}}<<1$, taking $T_{\mathrm{n}}\approx \approx T_{\mathrm{M}}$, due to the identity ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2}\right]=\frac{{\pi }^2}{4}$, from (33a)

$$\begin{gathered} {\Gamma }_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{T_{\mathrm{n}}}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right]\approx \frac{1}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}, \\ {\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{8}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\frac{{\mathsf{\omega }\mathrm{T}}_{\mathrm{n}}^2}{T_{\mathrm{M}}}{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{\mathrm{n}}^2\right)}\right]\approx \frac{\mathsf{\omega }{T}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}, \end{gathered}$$

convert (38) into

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{}={\mathsf{\epsilon }}_{\infty }\frac{1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}={\mathsf{\epsilon }}_{\infty },{\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}={\mathsf{\epsilon }}_{\infty }\frac{{\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}={\mathsf{\epsilon }}_{\infty }\frac{{\mathsf{\epsilon }}_{\infty }}{\mathsf{\omega }{T}_{\mathrm{M}}}.$$

(39)

Here ${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{1}{1+{\mathsf{\omega }}^2{{\rm T}}_{{\mathrm{M}}^2}}$, ${\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{{\mathsf{\omega }\mathrm{T}}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{{\rm T}}_{{\mathrm{M}}^2}}$.

Tangent of dielectric loss angle ${\left[{\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right]}_{\mathrm{M}}=\frac{{\Gamma }_{2,M}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)}{1-{\Gamma }_{1,M}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)}$ [1], according to (39), takes asymptotic form

$${\left[{\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right]}_{\mathrm{M}}=\frac{1}{\mathsf{\omega }{T}_{\mathrm{M}}}=\frac{n_0{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}\mathrm{q}}{{\mathsf{\omega }\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}.$$

(40)

Substituting ${\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}=\frac{\mathrm{q}{a}^2{W}^{\left(1\right)}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}$ [1,2] in (40) and comparing the obtained result with the known expression for conductivity loss $\mathrm{tg}\mathsf{\delta }=\frac{\mathsf{\sigma }}{{\mathsf{\omega }\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}$ [2], we find the coefficient of ion (in HBC-proton) relaxation electrical conductivity

$${\mathsf{\sigma }}_{\mathrm{e},\mathrm{pr}}\left(\mathrm{T}\right)=\frac{n_0{\mathrm{q}}^2{a}^2{W}^{\left(1\right)}\left(\mathrm{T}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}.$$

(41)

In the field of the diffusion relaxation $\left(T_{\mathrm{D}}<T_{\mathrm{M}};T_{\mathrm{n},\mathrm{D}}<T_{\mathrm{M}}\right)$, the parameters (33a) converted into

$${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}=\frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right],{\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}=\frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right],$$

at the area $T_{\mathrm{n},\mathrm{D}}<T_{\mathrm{M}}, \mathrm{when}$ $\frac{T_{\mathrm{D}}}{{\mathrm{n}}^2{T}_{\mathrm{M}}}<<1$, are reduced to equalities

$${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2}\right],{\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right].$$

(42)

In the region of ultra-low frequencies, when $\mathsf{\omega }{{\rm T}}_{\mathrm{D}}<<1$, based on (42), taking into account the identity ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4}\right]=\frac{{\pi }^4}{48}$, passing to the approximate equations of the form ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4}\right]=\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}$, ${\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\to 0$ and using formal expressions of the form (38)

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{/}={\mathsf{\epsilon }}_{\infty }\frac{1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right)}^2}={\mathsf{\epsilon }}_{\infty },{\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{//}={\mathsf{\epsilon }}_{\infty }\frac{{\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right)}^2}$$

(43)

we obtain

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{/}\approx {\mathsf{\epsilon }}_{\infty }\frac{1}{1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}}=\frac{{\mathsf{\epsilon }}_{\infty }}{1-\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}};{\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{//}\to 0.$$

(44)

Further, in the diffusion relaxation $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}={\left(\frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{D}}}\right)}^2<<1$, when the values of the Debye shielding radius are much larger than the crystal thickness ${\mathrm{r}}_{\mathrm{D}}>>\mathrm{d}$ [1,2], taking $\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}<<1$, from (44) we obtain an expression for the real component of the CDP corresponding to the classical law for dipole polarization

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{D}}^{}\approx {\mathsf{\epsilon }}_{\infty }\left(1+\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}\right)={\mathsf{\epsilon }}_{\infty }\left(1+\frac{{\pi }^2{\mathrm{d}}^2}{{12\mathrm{r}}_{\mathrm{D}}^2}\right).$$

(45)

Another criterion of validity of formulae (8) and (25) of the nonlinear approximation of the perturbation theory investigated above is their ultimate transition to results of the linear kinetic theory of the relaxation polarization [7,12,13,14,15,16]. To this end, we simplify the expression for the polarization (8).

Taking the condition $\frac{8\phi n_0{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\pi }^2}<<1$ in (8), by virtue of $\frac{1}{1-\frac{8\phi n_0{\mathsf{\Lambda }}_0\mathsf{\gamma }}{{\pi }^2}}\to 1$ we obtain the expression ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{t}\right)\approx \frac{4a\mathrm{q}{n}_0\mathsf{\gamma }}{{\pi }^2}\times {\sum }_{\mathrm{n}=1}^{+\infty }\left[\frac{\left(1-{\left(-1\right)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left(\frac{1}{{\mathsf{\tau }}_{\mathrm{n}}}+\mathrm{i}\frac{\mathsf{\omega }}{W^{\left(0\right)}}\right)}\right]\times \exp \left(\mathrm{i}\mathsf{\omega }t\right)$ corresponding at the fundamental frequency of the field $\mathsf{\omega }$, the first approximation by parameter $\mathsf{\gamma }$ [17], where ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(\mathrm{t}\right)\to {\mathsf{\gamma }\mathrm{P}}_1^{\left(\mathsf{\omega }\right)}\left(\mathsf{\tau }\right)={\hat{\mathsf{\alpha }}}_1^{\left(\mathsf{\omega }\right)}\cdot {\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$. Considering ${\mathsf{\gamma }\mathrm{P}}_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{t}\right)={\mathsf{\epsilon }}_0\left({\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}-{\mathsf{\epsilon }}_{\infty }\right)\cdot {\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$ [1], calculating ${\hat{\mathsf{\alpha }}}_1^{\left(\mathsf{\omega }\right)}={\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }\left({\Gamma }_1^{\left(\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)$; ${\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_1^{\left(\mathsf{\omega }\right)}-{\mathrm{i}\Gamma }_2^{\left(\mathsf{\omega }\right)}\right)$, we find

$${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}^{}=\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_1^{\left(\mathsf{\omega }\right)}\right),{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}^{//}=\mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]={\mathsf{\epsilon }}_{\infty }{\Gamma }_2^{\left(\mathsf{\omega }\right)},{\left({\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right)}_1=\frac{{\left[{ \mathsf{\epsilon } ^}_1^{\left(\mathsf{\omega }\right)}\right]}^{//}}{{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}^{}}=\frac{{\Gamma }_2^{\left(\mathsf{\omega }\right)}}{1+{\Gamma }_1^{\left(\mathsf{\omega }\right)}}.$$

(46)

In the case of ${\mathrm{T}>>\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$, $T_{\mathrm{M}}\ll T_{\mathrm{D}}$, by substituting ${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{1}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}$, ${\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{\mathsf{\omega }{T}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}$ in (46), we obtain the expressions (39) approximated in the first approximation by the parameter $\gamma$

$${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{}={\mathsf{\epsilon }}_{\infty }\left(1+\frac{1}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\right),{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}={\mathsf{\epsilon }}_{\infty }\frac{{\mathsf{\omega }\mathrm{T}}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2},{\left({\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right)}_{1,\mathrm{M}}=\frac{\mathsf{\omega }{T}_{\mathrm{M}}}{2+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}.$$

(46a)

From this, by virtue of ${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{M}}^{}={\mathsf{\epsilon }}_{\mathrm{S},\mathrm{M},1}\left(\mathrm{T}\right)={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }=0\right)}\right)=2{\mathsf{\epsilon }}_{\infty }$, ${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{M}}^{//}=0$, entering the designation ${\mathsf{\epsilon }}_{S,\mathrm{M},1}-{\mathsf{\epsilon }}_{\infty }={\mathsf{\epsilon }}_{\infty }$, we have

$${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{}={\mathsf{\epsilon }}_{\infty }+\frac{{\mathsf{\epsilon }}_{\mathrm{S},\mathrm{M},1}-{\mathsf{\epsilon }}_{\infty }}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2},{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}=\frac{\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{M},1}-{\mathsf{\epsilon }}_{\infty }\right)\mathsf{\omega }{T}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2},$$

(46b)

$${\left({\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right)}_{1,\mathrm{M}}=\frac{\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{M},1}-{\mathsf{\epsilon }}_{\infty }\right)\cdot \mathsf{\omega }{T}_{\mathrm{M}}}{{\mathsf{\epsilon }}_{\mathrm{S},\mathrm{M},1}+{\mathsf{\epsilon }}_{\infty }\cdot {\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}, \left[{\left(\mathrm{tg}\mathsf{\delta }\right)}_{1,\mathrm{M}}\right]{|}_{\mathsf{\omega }\to 0}=0, \left[{\left(\mathrm{tg}\mathsf{\delta }\right)}_{1,\mathrm{M}}\right]{|}_{\mathsf{\omega }\to \infty }=0,$$

(46c)

which is consistent with the results of the linear theory of dielectric losses [1,2] in the temperature region ${\mathrm{T}>>\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$.

In the case of ${\mathrm{T}<<\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$, $T_{\mathrm{D}}<<T_{\mathrm{M}}$, by rewriting (46) while taking into account (42), we have

$${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{}={\mathsf{\epsilon }}_{\infty }\left(1+\frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2}\right]\right),{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{//}=\frac{4T_{\mathrm{D}}{\mathsf{\epsilon }}_{\infty }}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right].$$

(46d)

From here, by virtue of ${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{D}}^{}={\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}\left(\mathrm{T}\right)={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }=0\right)}\right)={\mathsf{\epsilon }}_{\infty }\left(1+\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}\right)$, ${\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{D}}^{//}=0$, ${\left[{\hat{\mathsf{\epsilon }}}_{11}^{\left(\mathsf{\omega }=0\right)}\right]}_{\mathrm{M}}^{//}=0$, entering the designation ${\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}-{\mathsf{\epsilon }}_{\infty }=\frac{{\pi }^2{T}_{\mathrm{D}}{\mathsf{\epsilon }}_{\infty }}{12T_{\mathrm{M}}}$, we have

$$\begin{gathered} {\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{}={\mathsf{\epsilon }}_{\infty }+\frac{48\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}-{\mathsf{\epsilon }}_{\infty }\right)}{{\pi }^4}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2}\right],{\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{//}=\frac{48\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}-{\mathsf{\epsilon }}_{\infty }\right)}{{\pi }^4}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right], \\ {\left({\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right)}_{1,\mathrm{D}}=\frac{\frac{48}{{\pi }^4}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}-{\mathsf{\epsilon }}_{\infty }\right)(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right]}{{\mathsf{\epsilon }}_{\infty }+\frac{48}{{\pi }^4}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{\left({\mathsf{\epsilon }}_{\mathrm{S},\mathrm{D},1}-{\mathsf{\epsilon }}_{\infty }\right)(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2}\right]}, \end{gathered}$$

(46e)

$$\left[{\left(\mathrm{tg}\mathsf{\delta }\right)}_{1,\mathrm{D}}\right]{|}_{\mathsf{\omega }\to 0}=0,\left[{\left(\mathrm{tg}\mathsf{\delta }\right)}_{1,\mathrm{D}}\right]{|}_{\mathsf{\omega }\to \infty }=0,$$

(46f)

which is also consistent [2] in the temperature region ${\mathrm{T}<<\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$.

In the theoretical studies of the solution schemes of the nonlinear kinetic Equation (2) and its annexes to the issues of studying the mechanisms of the relaxation polarization described by formulas (8), (25), (21), it is impossible not to consider the question of the generalized calculation of the relaxation parameters (33a) for the wide range of field and temperature parameters.

Introducing dimensionless variables ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}$, ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}$, we present the generalized expressions (33a) in the form of

$${\Gamma }_1^{\left(\mathsf{\omega }\right)}=\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2+{\mathsf{\alpha }}_2^2\right)}\right],{\Gamma }_2^{\left(\mathsf{\omega }\right)}=\frac{4{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2+{\mathsf{\alpha }}_2^2\right)}\right].$$

(47)

It is convenient for further analysis of the effects of various kinds of the polarization nonlinearities (interactions of the multiple relaxation modes of both the same frequency and different field frequencies; quantum microscopic effects associated with proton tunneling in HBC; the phenomena of nonlinear volume–charge polarization, etc.) and temperature per dispersion expressions (38) with numerical calculation of frequency–temperature spectra of the complex dielectric permittivity (CDP).

When calculating by formulae

$${\Gamma }_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}\left({\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)+{\mathsf{\alpha }}_1{\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right), {\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)=\frac{4{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2}{{\pi }^2}\times {\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)$$

(48)

it becomes possible, regardless of the properties of the parameters ${\Gamma }_1^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_2^{\left(\mathsf{\omega }\right)}$, in a certain temperature range (see particular cases ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$, from (42) and ${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}$), to convert the sum of infinite series to an explicit analytical form. So, calculated in the functions of temperature and frequency of an alternating polarizing field, infinite sums of converging series

$${\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2+{\mathsf{\alpha }}_2^2}\right], {\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2+{\mathsf{\alpha }}_2^2\right)}\right]$$

(49)

taking into account identities

$$\begin{gathered} S_1^{\left(\pm \right)}={\sum }_{\mathrm{n}=1}^{\infty }\frac{1}{{\mathrm{n}}^2+{\mathsf{\alpha }}_1\pm {\mathrm{i}\mathsf{\alpha }}_2}=-\frac{1}{2{\left(z_1^{\left(\pm \right)}\right)}^2}-\frac{\pi }{2z_1^{\left(\pm \right)}}\mathrm{ctg}\left(\pi z_1^{\left(\pm \right)}\right), \\ {S}_2^{\left(\pm \right)}={\sum }_{\mathrm{n}=1}^{\infty }\frac{{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^2+{\mathsf{\alpha }}_1\pm {\mathrm{i}\mathsf{\alpha }}_2}=-\frac{1}{2{\left(z_1^{\left(\pm \right)}\right)}^2}-\frac{\pi }{2z_1^{\left(\pm \right)}}\cos \mathrm{ec}\left(\pi z_1^{\left(\pm \right)}\right), \\ {z}_1^{\left(\pm \right)}=-z_2^{\left(\pm \right)}=\mathrm{Re}\left[z_1^{\left(\pm \right)}\right]\mp \mathrm{iIm}\left[z_1^{\left(\pm \right)}\right], \\ \mathrm{Re}\left[z_1^{\left(\pm \right)}\right]=\sqrt{\frac{\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}-{\mathsf{\alpha }}_1}{2}}, \mathrm{Im}\left[z_1^{\left(\pm \right)}\right]=\sqrt{\frac{\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}+{\mathsf{\alpha }}_1}{2}}, \end{gathered}$$

according to

$$\begin{gathered} S^{\left(\pm \right)}={\sum }_{\mathrm{n}=1}^{\infty }\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^2+{\mathsf{\alpha }}_1\pm {\mathrm{i}\mathsf{\alpha }}_2}=\frac{\pi }{2z_1^{\left(\pm \right)}}\left(\cos \mathrm{ec}\left(\pi z_1^{\left(\pm \right)}\right)-\mathrm{ctg}\left(\pi z_1^{\left(\pm \right)}\right)\right)=\frac{\pi }{2z_1^{\left(\pm \right)}}\mathrm{tg}\left(\frac{\pi z_1^{\left(\pm \right)}}{2}\right)=\frac{{\pi }^2}{4}\times \frac{\mathrm{tg}\left(\frac{\pi z_1^{\left(\pm \right)}}{2}\right)}{\frac{\pi z_1^{\left(\pm \right)}}{2}}, \\ {S}^{\left(\pm \right)}={\sum }_{\mathrm{n}=1}^{\infty }\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^2+{\mathsf{\alpha }}_1\pm {\mathrm{i}\mathsf{\alpha }}_2}=\frac{{\pi }^2}{4}\times \frac{\mathrm{tg}\left({\mathsf{\Delta }}_1\mp {\mathrm{i}\mathsf{\Delta }}_2\right)}{{\mathsf{\Delta }}_1\mp {\mathrm{i}\mathsf{\Delta }}_2},{ \mathsf{\Delta }}_1=\frac{\pi }{2}\sqrt{\frac{\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}-{\mathsf{\alpha }}_1}{2}}, {\mathsf{\Delta }}_2=\frac{\pi }{2}\sqrt{\frac{\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}+{\mathsf{\alpha }}_1}{2}}, \end{gathered}$$

take the form

$${\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)=\frac{1}{{2\mathsf{\alpha }}_2\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}}\times \frac{{\mathsf{\Delta }}_1\cdot \mathrm{sh}\left({2\mathsf{\Delta }}_2\right)-{\mathsf{\Delta }}_2\cdot \sin \left({2\mathsf{\Delta }}_1\right)}{{\mathrm{ch}}^2\left({\mathsf{\Delta }}_2\right){\cos }^2\left({\mathsf{\Delta }}_1\right)+{\mathrm{sh}}^2\left({\mathsf{\Delta }}_2\right){\sin }^2\left({\mathsf{\Delta }}_1\right)},$$

(50)

$${\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)=\frac{{\pi }^2}{4\left({\mathsf{\alpha }}_{1,}^2+{\mathsf{\alpha }}_2^2\right)}\times \left[1-\frac{2\left({\mathsf{\alpha }}_1\left({\mathsf{\Delta }}_1\mathrm{sh}\left({2\mathsf{\Delta }}_2\right)-{\mathsf{\Delta }}_2\sin \left({2\mathsf{\Delta }}_1\right)\right)+{\mathsf{\alpha }}_2\left({\mathsf{\Delta }}_1\cdot \sin \left({2\mathsf{\Delta }}_1\right)+{\mathsf{\Delta }}_2\mathrm{sh}\left({2\mathsf{\Delta }}_2\right)\right)\right)}{{\mathsf{\alpha }}_2{\pi }^2\left({\mathrm{ch}}^2\left({\mathsf{\Delta }}_2\right){\cos }^2\left({\mathsf{\Delta }}_1\right)+{\mathrm{sh}}^2\left({\mathsf{\Delta }}_2\right){\sin }^2\left({\mathsf{\Delta }}_1\right)\right)\sqrt{{\mathsf{\alpha }}_1^2+{\mathsf{\alpha }}_2^2}}\right].$$

(51)

Special cases have application significance. In the field of the diffusion relaxation, at temperatures far from critical ${\mathrm{T}<<\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$, in a wide range of the variable field frequencies ${ \mathsf{\omega }\mathrm{T}}_{\mathrm{D}}>0$, when the conditions ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<<1$, ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}>0$, ${\mathsf{\Delta }}_1={\mathsf{\Delta }}_2=\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}$ are met, taking the variable $\mathsf{\xi }=\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}$ and moving to (49) and, respectively, in (50), (51), in the area of ultra-low temperatures $\left({\mathsf{\alpha }}_1\to 0\right)$, to the limits of ${\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)=\underset{{\mathsf{\alpha }}_1\to 0}{\lim }\left({\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)\right)$, ${\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)=\underset{{\mathsf{\alpha }}_1\to 0}{\lim }\left({\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)\right)$, we write approximate expressions

$${\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)={\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi }\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^4+{\mathsf{\alpha }}_2^2}\right]=\frac{{\pi }^4}{{16\mathsf{\xi }}^3}\times \left[\frac{\mathrm{sh}\left(\mathsf{\xi }\right)-\sin \left(\mathsf{\xi }\right)}{{\cos }^2\left(\frac{\mathsf{\xi }}{2}\right){\mathrm{ch}}^2\left(\frac{\mathsf{\xi }}{2}\right)+{\sin }^2\left(\frac{\mathsf{\xi }}{2}\right){\mathrm{sh}}^2\left(\frac{\mathsf{\xi }}{2}\right)}\right],$$

(52)

$${\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)={\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(\mathsf{\xi }\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\alpha }}_2^2\right)}\right]=\frac{{\pi }^6}{{16\mathsf{\xi }}^4}\left(1-\frac{1}{2\mathsf{\xi }}\times \left[\frac{\mathrm{sh}\left(\mathsf{\xi }\right)+\sin \left(\mathsf{\xi }\right)}{{\cos }^2\left(\frac{\mathsf{\xi }}{2}\right){\mathrm{ch}}^2\left(\frac{\mathsf{\xi }}{2}\right)+{\sin }^2\left(\frac{\mathsf{\xi }}{2}\right){\mathrm{sh}}^2\left(\frac{\mathsf{\xi }}{2}\right)}\right]\right).$$

(53)

In the ultra-low frequency region, according to $\mathsf{\omega }{T}_{\mathrm{D}}<<1$, taking in expressions (52), (53) ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}\to 0$ and, moving to the limits at $\mathsf{\xi }=\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\to 0$, we obtain the expressions ${\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left[{\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left(0{;\mathsf{\alpha }}_2\right)\right]=\frac{{\pi }^4}{48}$, ${\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left[{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left(0{;\mathsf{\alpha }}_2\right)\right]=\frac{{\pi }^6}{480}$. They can act as a criterion for the validity of the results of previously performed calculations when moving from equalities (49) to (50), (51) and from (50), (51) to (52), (53), since the expressions for the ${\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left[{\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left(0{;\mathsf{\alpha }}_2\right)\right]$, ${\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left[{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left(0{;\mathsf{\alpha }}_2\right)\right]$ limits completely coincide with the results of the direct calculations of the sums of infinite series resulting from their (49) while meeting the formal conditions ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\to 0$, ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}\to 0$ when ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^4}\right]=\frac{{\pi }^4}{48}$, ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^6}\right]=\frac{{\pi }^6}{480}$. Moreover, it is not difficult to see that built formally, on the basis of (50), (51), for reasons of the smallness of a parameter ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<<1$ expressions (52), (53), in physical relation, fully satisfying the conditions of the diffusion relaxation $\left(T_{\mathrm{D}}<T_{\mathrm{M}};T_{\mathrm{n},\mathrm{D}}<<T_{\mathrm{M}}\right)$, coincide with the parameters ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$ from the equalities (42). Therefore, after substitution (52), (53) in expressions (43), the generalized dispersion expressions can be obtained that are suitable for calculating the components of the complex dielectric permittivity (CDP) in the field of diffusion relaxation at arbitrary frequencies of a variable field according to generalized formulae (43). Then, based on the partial spectral expressions (46), approximate formulas of the linear approximation of perturbation theory (46.4) for the CDP components in the diffusion relaxation region $\left({\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<<1\right)$, far from the critical temperature ${\mathrm{T}<<\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$ but not at absolute zero, according to (52), (53), are reduced to a transcendent form

$$\begin{gathered} \mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}_{1\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right]={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_1^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)\right)={\mathsf{\epsilon }}_{\infty }\left(1+\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}\left({\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)+{\mathsf{\alpha }}_1{\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)\right)\right)= \\ ={\mathsf{\epsilon }}_{\infty }(1+\frac{{\mathsf{\alpha }}_1}{\pi \sqrt{{2\mathsf{\alpha }}_2^3}}\left[\frac{\mathrm{sh}\left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)-\sin \left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}{{\cos }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{ch}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)+{\sin }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{sh}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}\right]+ \\ +{\left(\frac{{\mathsf{\alpha }}_1}{{\mathsf{\alpha }}_2}\right)}^2\left(1-\frac{1}{2\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}}\times \left[\frac{\mathrm{sh}\left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)+\sin \left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}{{\cos }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{ch}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)+{\sin }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{sh}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}\right]\right), \end{gathered}$$

(54)

$$\begin{gathered} \mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}_{1\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right]={\mathsf{\epsilon }}_{\infty }{\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)=\frac{4{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2}{{\pi }^2}{\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left(0{;\mathsf{\alpha }}_2\right)= \\ =\frac{{\mathsf{\epsilon }}_{\infty }{\mathsf{\alpha }}_1}{{\mathsf{\alpha }}_2}\left(1-\frac{1}{2\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}}\times \left[\frac{\mathrm{sh}\left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)+\sin \left(\pi \sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}{{\cos }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{ch}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)+{\sin }^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right){\mathrm{sh}}^2\left(\frac{\pi }{2}\sqrt{\frac{{\mathsf{\alpha }}_2}{2}}\right)}\right]\right). \end{gathered}$$

(55)

It is not difficult to see that the expressions (54), (55), in comparison with (46d), (46e), are more obvious functional dependencies of the CDP components on the frequency of the alternating field (against the background of a weak influence of temperature). They are also more convenient for numerical calculations of the theoretical frequency spectra of the CDP and analysis of the effects of field and temperature parameters on the tangent spectra of the dielectric loss angle ${\left({\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right)}_{1,\mathrm{D}}=\frac{\mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}_{1\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right]}{\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}_{1\mathrm{D}}^{\left(\mathsf{\omega }\right)}\right]}$ in the diffusion relaxation region, far from the critical temperature ${\mathrm{T}<<\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$, at the main field frequency, in the linear approximation of perturbation theory.

The study of the laws of dielectric relaxation at arbitrary temperatures (including close to critical ${\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$) when ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>0$, in the low frequency range of the variable field ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}<<1$, when in expressions (49), and accordingly in (50), (51), at the frequency of the field tending zero, the conditions ${\mathsf{\alpha }}_2\to 0$, ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>0$, ${\mathsf{\Delta }}_1=0$, ${\mathsf{\Delta }}_2=\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}$ can be assumed, is an important question for theory. From there, entering the variable $\mathsf{\zeta }=\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}$ and, going to (49) and (50), (51), to limits ${\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left({\Gamma }_{11}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)\right)$, ${\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=\underset{{\mathsf{\alpha }}_2\to 0}{\lim }\left({\Gamma }_{12}^{\left(\mathsf{\omega }\right)}\left({\mathsf{\alpha }}_1{;\mathsf{\alpha }}_2\right)\right)$ we will write approximate expressions

$${\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2}\right]=\frac{{\pi }^4}{32}\times \frac{\mathrm{th}\mathsf{\zeta }-\mathsf{\zeta }\times \left(1-{\mathrm{th}}^2\mathsf{\zeta }\right)}{{\mathsf{\zeta }}^3}$$

(56)

$${\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)={\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2{\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}^2}\right]=\frac{{\pi }^6}{128}\times \left(\frac{3\left(\mathsf{\zeta }-\mathrm{th}\mathsf{\zeta }\right)-{\mathsf{\zeta }\mathrm{th}}^2\mathsf{\zeta }}{{\mathsf{\zeta }}^5}\right).$$

(57)

In this case, the analytical expressions (56), (57) are functions only of temperature.

Also note that for the expressions (56), (57), as for the expressions (52), (53), the criteria of reliability of results obtained during the formula transformations (49)–(50), (51) and (50), (51) to the form (56), (57) are met. This is confirmed by the transitions in formulae (56), (57) to the limits ${\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_1\to 0}{\lim }\left[{\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)\right]=\frac{{\pi }^4}{48}$, ${\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left(0;0\right)=\underset{{\mathsf{\alpha }}_1\to 0}{\lim }\left[{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)\right]=\frac{{\pi }^6}{480 }$, which completely coincide with the results of direct calculations of the sums of infinite series resulting from (49) when the formal conditions ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}\to 0$, ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\to 0$ are met simultaneously, when ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^4}\right]=\frac{{\pi }^4}{48}$, ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{1-{\left(-1\right)}^{\mathrm{n}}}{{\mathrm{n}}^6}\right]==\frac{{\pi }^6}{480}$.

Note that, although the expressions (56), (57), when performing inequality ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<1$, in a certain relation satisfy the conditions of the diffusion relaxation $\left(T_{\mathrm{D}}<<T_{\mathrm{M}};T_{\mathrm{n},\mathrm{D}}<T_{\mathrm{M}}\right)$, in the general case (by virtue of ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>0$), do not coincide with the parameters ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}$ of equalities (42), because formulae (56), (57) are constructed primarily for the low frequency range of the variable field (${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}<<1$). This means that, for a mixed type of relaxation (near the critical temperature), when calculating the components of the complex dielectric permittivity (CDP), it is advisable to substitute equations (56), (57) in expressions (38). Then, based on the partial spectral expressions (46), the approximate formulas of the linear approximation of the perturbation theory for the CDP components, at ultra-low frequencies (${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}\to 0$), according to (56), (57) and (48), are reduced to the form

$$\begin{gathered} \mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]={\mathsf{\epsilon }}_{\infty }\left(1+{\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)\right)={\mathsf{\epsilon }}_{\infty }\left(1+\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}\left({\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)+{\mathsf{\alpha }}_1{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)\right)\right)= \\ ={\mathsf{\epsilon }}_{\infty }\left\{1+\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}(\frac{{\pi }^4}{32}\times \frac{\mathrm{th}\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)-\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)\times \left(1-{\mathrm{th}}^2\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)\right)}{{\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)}^3}\right.+ \\ +\frac{{\pi }^6{\mathsf{\alpha }}_1}{128}\times \left.\left(\frac{3\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}-\mathrm{th}\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)\right)-\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right){\mathrm{th}}^2\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)}{{\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)}^5}\right))\right\}, \end{gathered}$$

(58)

$$\mathrm{Im}\left[{\hat{\mathsf{\epsilon }}}_1^{\left(\mathsf{\omega }=0\right)}\right]={\mathsf{\epsilon }}_{\infty }{\Gamma }_2^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)={\mathsf{\epsilon }}_{\infty }\frac{4{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2}{{\pi }^2}{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=0.$$

(59)

The conversion of formulae (48) for the case ${\mathsf{\alpha }}_2\to 0$ makes them more convenient for comparison with the experiment. To do this, rewrite (48) using (56), (57), entering the variable $\mathsf{\zeta }=\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}$. Then

$${\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}\left({\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)+{\mathsf{\alpha }}_1{\Gamma }_{11}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)\right)=\frac{1}{2\mathsf{\zeta }}\times \left(\mathrm{th}\mathsf{\zeta }-\mathsf{\zeta }\times \left(1-{\mathrm{th}}^2\mathsf{\zeta }\right)+3\left(\mathsf{\zeta }-\mathrm{th}\mathsf{\zeta }\right)-{\mathsf{\zeta }\mathrm{th}}^2\mathsf{\zeta }\right)=1-\frac{\mathrm{th}\mathsf{\zeta }}{\mathsf{\zeta }}$$

(60a)

$${\Gamma }_2^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)={\mathsf{\epsilon }}_{\infty }\frac{4{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2}{{\pi }^2}{\Gamma }_{12}^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=\frac{{\pi }^4{\mathsf{\alpha }}_2}{8}\times \left(\frac{3\left(\zeta -\mathrm{th}\zeta \right)-\zeta {\mathrm{th}}^2\zeta }{{\zeta }^3}\right)=0.$$

(60b)

On the other hand, from (47), with ${\mathsf{\alpha }}_2\to 0$, we get the same result as in (60a)

$${\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)=\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}\right]=\frac{4{\mathsf{\alpha }}_1}{{\pi }^2}\times \frac{1}{{\mathsf{\alpha }}_1}{\sum }_{\mathrm{n}=1}^{\infty }\left((1-\left(-1{)}^{\mathrm{n}}\right)\right)\left[\frac{1}{{\mathrm{n}}^2}-\frac{1}{{\mathrm{n}}^2+{\mathsf{\alpha }}_1}\right]=1-\frac{\mathrm{th}\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)}{\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}}.$$

Here ${\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left({\mathrm{n}}^2+{\mathsf{\alpha }}_1\right)}\right]=\frac{{\pi }^2}{4}\times \frac{1}{{\mathsf{\alpha }}_1}\left(1-\frac{\mathrm{th}\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right)}{\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}}\right)$.

It is not difficult to see that the identity (60.1) makes it possible to significantly simplify temperature dependencies in generalized expressions (38)

$$\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }=0\right)}\right]={\mathsf{\epsilon }}_{\infty }\frac{1}{1-{\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left({\mathsf{\alpha }}_1;0\right)}={\mathsf{\epsilon }}_{\infty }\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\mathrm{cth}\left(\frac{\pi }{2}\sqrt{{\mathsf{\alpha }}_1}\right).$$

(61)

Expression (61) is suitable for investigating the temperature dependencies of the real component of the complex dielectric permittivity $\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }=0\right)}\right]$ over the wide temperature range at the variable field frequency of zero (stationary relaxation polarization). It results from a generalized variance expression (38) constructed at the base frequency of the variable field in an infinite approximation of the theory of perturbations over a small parameter. It contrasts the expression (57), constructed on the basis of linear expressions (46) in the first approximation of the theory of perturbations by the parameter. This expression is more limited in terms of physical completeness of coverage of the nonlinear polarization effects associated with the effects of the interactions of multiple relaxation modes of the basic field frequency on the dielectric relaxation mechanism in the low and ultra-low temperature region. Thus, the expression (61), which is generalized to the wider range of the physical nonlinearities, is more effective in detecting the effects of quantum phenomena (due to tunnel transitions of relaxers (protons) between anionic sublattice ions) on the mechanism of low-temperature migration polarization (50–100 K) in proton semiconductors and dielectrics and their nanoscale layers (1–10 nm) at ultra-low temperatures (1–10 K). Within the framework of this scientific paper, we do not investigate the temperature relationships of the statistically averaged quantum transparency coefficient of a potential barrier for various configurations of the energy spectrum (discrete degenerate or non-degenerate; quasi-continuous) tunneling protons depending on the parameters of the undisturbed potential crystal lattice field (hydrogen bond field). Detailed studies of this issue carried out in the works [1,2,4,14,15] showed that the nature and properties of the quantum transparency coefficient for protons in the hydrogen-bonded crystals (HBC) are most qualitatively disclosed for the discrete degenerate model (split into sublevels within individual energy zones, as a result of the effect of neighboring potential pits on a proton localized in the region of a given potential pit). The numerical values of the statistically averaged quantum transparency coefficient are significantly dependent on the parameters of the potential barrier (especially from its height, which makes sense of the activation energy), crystal thickness and temperature range. Thus, in [16], it was established that accounting for splits of proton energy levels in the HBC on sublevels (in an unperturbed potential field) with the proton activation energy of 0.01 eV leads to a shift in the temperature maximum of the statistically averaged transparency coefficient from 55 K with an amplitude of 0.12 (in the absence of energy spectrum degeneration) to 150 K with an amplitude of 0.14 (in the presence of energy spectrum degeneration). Ref. [14] shows the effects of the thickness of the crystal layer on quantum transparency and dielectric constant of the crystal of the HBC type at nanoscale thicknesses of the sample, when, in layers with a thickness of 10 nm, in the case of the zone structure of the energy spectrum of protons with an activation energy of the proton of 0.01 eV, there is an increase in the amplitude of the quantum transparency coefficient to 0.98–0.99 (proton superconductivity) at a maximum temperature of 10–15 K, and the amplitude of the stationary dielectric permittivity $\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }=0\right)}\right]$ calculated according to formula (61) reaches abnormally high values (2.5–3.0 million) [16]. This indicates the formation of the ferroelectric state in HBC due to quantum tunnel movement of protons in hydrogen sublattice of HBC nanofilms in the region of liquid helium temperatures.

The effects of the quantum effects on the frequency spectra of the complex dielectric permittivity should be carried out according to the generalized dispersion formulas (38), taking into account the most common expressions for relaxation coefficients in the form (50), (51). This will allow taking into account the influence of all the above-mentioned parameters of the crystal lattice structure and the parameters of the energy spectrum of protons, crystal thickness and temperature on the parameters of maxima of the CDP by frequency and amplitude. The advantages of expressions (38), compared to the particular linear equations (46) are that formulae (39) are constructed, as noted above, at the base frequency of the alternating polarizing field, everything up to infinite is taken into account, approximation, by a small dimensionless parameter of the perturbation theory when solving the Fokker–Planck Equation (2) and, as a consequence, the results of the effects of the spatially inhomogeneous polarization-induced electric field in the dielectric. This is very important in studies of the effects of the volume–charge distribution of relaxers in the mixed-type relaxation region, manifested in the region of sufficiently high temperatures (250–450 K), when, against the background of the dominant mechanism of thermally activated proton transitions [1,2,15,16], proton tunneling continues to make a significant contribution to the kinetics of nonlinear thermally activated depolarization [1,2]. As noted in [3,4], the polarization nonlinearities associated with the formation of charge–volume polarization in dielectrics with the ion-molecular type of chemical bond and manifesting in the region of ultra-high temperatures (550–1500 K) and strong electric fields (10–100 MV/m) [7,14], result from a mixed type of relaxer movements (protons) activated both by classical (due to thermal motion and the interaction of protons with the crystal lattice) transitions and quantum tunnel transitions.

Numerical studies and analysis of the properties of the frequency spectra of the complex dielectric constant, performed according to the generalized dispersion expressions (39) and (50), (51), are a separate task, the results of which will be presented in a future work.

6. Properties of Theoretical Spectra of Complex Dielectric Permittivity in the Region of Maxwellian Relaxation

From the point of view of organizing and planning experiments to study the electrical properties of circuit elements of control and measuring and radio-electronic equipment used in the electric power industry, insulating and cable technology and industrial electronics, it is relevant to study the frequency–temperature spectra of the dielectric loss tangent in the temperature range T = 150–550 K [2], when the main contribution to the ion-relaxation polarization in dielectrics is made by the Maxwellian relaxation of the main charge carriers (in ionic dielectrics-interstitial ions and in HBC-protons localized on hydrogen bonds).

In the area of Maxwellian relaxation at temperatures higher than critical $\left(T>T_{\mathrm{cr},\mathrm{relax}}\right)$, when $T_{\mathrm{M}}<T_{\mathrm{n},\mathrm{D}}$ [1], by approximating parameters (33a)

$${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}=\frac{8}{{\pi }^2}{\displaystyle \sum _{\mathrm{n}=1}^{\infty }\left[\frac{{\sin }^2\left(\frac{\pi \mathrm{n}}{2}\right)\cdot \left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}{{\mathrm{n}}^2\left({\left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{M}}{}^2\right)}\right]}, {\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}=\frac{8}{{\pi }^2}{\displaystyle \sum _{\mathrm{n}=1}^{\infty }\left[\frac{{\sin }^2{\left(\frac{\pi \mathrm{n}}{2}\right)}^{\mathrm{n}})\cdot \mathsf{\omega }{T}_{\mathrm{M}}}{{\mathrm{n}}^2\left({\left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{M}}{}^2\right)}\right]},$$

(62)

with respect to the dimensionless parameter ${\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}<1$, we bring the expressions (38)

$${\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{/}={\mathsf{\epsilon }}_{\infty }^{}\frac{1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}, {\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}={\mathsf{\epsilon }}_{\infty }^{}\frac{{\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}$$

(63)

to the form

$$\begin{gathered} {\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{/}\approx {\mathsf{\epsilon }}_{\infty }\left({\left(\pi \mathsf{\omega }{T}_{\mathrm{M}}\right)}^2\sqrt{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)\times \frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right.+ \\ +4\left({\Delta }_1\sin \left(2{\Delta }_1\right)+{\Delta }_2\mathrm{sh}\left(2{\Delta }_2\right)\right)+4\mathsf{\omega }{T}_{\mathrm{M}}\left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)\times \\ \times \left({\left(\pi \mathsf{\omega }{T}_{\mathrm{M}}\right)}^2\sqrt{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)\times \frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}+8\mathsf{\omega }{T}_{\mathrm{M}}\right.\times \\ {\left.\times \left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)+4\left(\mathrm{ch}\left(2{\Delta }_2\right)-\cos \left(2{\Delta }_1\right)\right)\right)}^{-1}, \end{gathered}$$

(64a)

$$\begin{gathered} {\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}\approx {\mathsf{\epsilon }}_{\infty }\left({\left(\pi \mathsf{\omega }{T}_{\mathrm{M}}\right)}^2\sqrt{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)\times \frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right.+ \\ +4\left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)-4\mathsf{\omega }{T}_{\mathrm{M}}\left({\Delta }_1\sin \left(2{\Delta }_1\right)+{\Delta }_2\mathrm{sh}\left(2{\Delta }_2\right)\right)\times \\ \times \left({\left(\pi \mathsf{\omega }{T}_{\mathrm{M}}\right)}^2\sqrt{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)\times \frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}+8\mathsf{\omega }{T}_{\mathrm{M}}\right.\times \\ {\left.\times \left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)+4\left(\mathrm{ch}\left(2{\Delta }_2\right)-\cos \left(2{\Delta }_1\right)\right)\right)}^{-1} \end{gathered}$$

(64b)

Here ${\Delta }_{1,2}=\frac{\pi }{\sqrt{2}}\sqrt{\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\left[\sqrt{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}\mp 1\right]}$.

The tangent of the dielectric loss angle at the fundamental frequency of the field ${\left[\mathrm{tg}{\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right]}_{\mathrm{M}}^{}=\frac{{\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}}{{\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{/}}$ is

$$\begin{gathered} {\left[\mathrm{tg}{\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right]}_{\mathrm{M}}^{}= \\ =\left\{\frac{T_{\mathrm{D}}{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2{\pi }^2}{T_{\mathrm{M}}}\sqrt{1+{\left(T_{\mathrm{M}}\right)}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)+4\left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)\right.- \\ \left.-4\mathsf{\omega }{T}_{\mathrm{M}}\left({\Delta }_1\sin \left(2{\Delta }_1\right)+{\Delta }_2\mathrm{sh}\left(2{\Delta }_2\right)\right)\right\}\times \left\{\frac{T_{\mathrm{D}}{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2{\pi }^2}{T_{\mathrm{M}}}\sqrt{1+{\left(\mathsf{\omega }{T}_{\mathsf{M}}\right)}^2}\left(\cos \left(2{\Delta }_1\right)+\mathrm{ch}\left(2{\Delta }_2\right)\right)\right.+ \\ {\left.+4\left({\Delta }_1\sin \left(2{\Delta }_1\right)+{\Delta }_2\mathrm{sh}\left(2{\Delta }_2\right)\right)+4\mathsf{\omega }{T}_{\mathrm{M}}\left({\Delta }_2\sin \left(2{\Delta }_1\right)-{\Delta }_1\mathrm{sh}\left(2{\Delta }_2\right)\right)\right\}}^{-1}. \end{gathered}$$

(65)

Figure 1 and Figure 2 show the dependencies graphs ${\left[\mathrm{tg}{\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right]}_{\mathrm{M}}^{}=\frac{{\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}}{{\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{/}}$, ${\mathsf{\epsilon }}_{}^{/}\approx {\left[{\widehat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right]}_{\mathrm{M}}^{/}$ for various values of the parameter ${\alpha }_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}$, calculated using a computer program, according to formulas (64a), (64b), (65). With this representation, relaxation processes that have the same values ${\alpha }_1$, are depicted on the graph $\mathrm{tg}\mathsf{\delta }\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)$ as one curve. In accordance with Figure 1, an increase in the parameter ${\alpha }_1$ leads to an increase in the maximum and to a shift in its position to the area of small values, while at ${\alpha }_1>1000$ the maximum position changes slightly with increasing ${\alpha }_1$ and remains approximately identically and equal to ${\alpha }_2\approx 0,1$. With small values of the parameter ${\alpha }_1$ the maximum position $\mathrm{tg}\mathsf{\delta }\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)$ determined by the criterion ${\alpha }_1{\alpha }_2\approx 1$ or ${\alpha }_2\approx 1$. The curves $\mathrm{tg}\mathsf{\delta }\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)$ in the area of large values ${\alpha }_2$ are nearly congruent.

In conclusion, it should be pointed out that the anomalously high dielectric permittivities ε = 5 million found experimentally in [5], in samples of corundum–zirconium ceramics (CZC), with an alternating field frequency of 1 kHz, at the point T = 1250 K, can be theoretically explained and further investigated at a higher analytical level using expressions (64a), (64b), (65) that indicate the correspondence of the formulated theoretical methodology to the experimental regularities that manifest themselves in high-temperature ionic superconductors near the second-order phase transition temperature (quasi-ferroelectric effect), which is topically for the design of the theoretical methods for forecasting the nonlinear electrophysical properties of hydrogen-bond ferroelectrics (KDP, DKDP) used in laser technology as regulators of electromagnetic radiation parameters and electric gates [1,2,3,4].

In this regard, promising are the designs of high-speed nonvolatile memory devices based on thin films of ferroelectrics with a rectangular hysteresis loop (RHL), characterized by anomalously long times remanent polarization relaxation, increased mechanical performance and thermal stability.

For the practical application of the calculation formulas (64a), (64b), (65), it is convenient to express them in terms of the static permittivity ${\mathsf{\epsilon }}_S^{}$ and permeability at high frequencies ${\mathsf{\epsilon }}_{\infty }^{}$. In order to obtain such phenomenological relations, we will use the expression obtained from (61) in the Maxwellian relaxation region, when ${\alpha }_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>>1$ and, at $\zeta =\frac{\pi }{2}\sqrt{{\alpha }_1^{}}>>1$, by virtue of $\zeta cth\zeta \approx \zeta$, we have ${\mathsf{\epsilon }}_S^{}\approx {\mathsf{\epsilon }}_{\infty }^{}\frac{\pi }{2}\sqrt{{\alpha }_1^{}}$, where $\frac{2{\mathsf{\epsilon }}_S}{{\mathsf{\epsilon }}_{\infty }}=\pi \sqrt{\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}}}$. Then, based on (64a), (64b), (65) we have

$$\begin{gathered} {\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right)}^{/}\approx {\mathsf{\epsilon }}_s\frac{1+\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2}{1+{\left(\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}\right)}^2{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2}, \\ {\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right)}^{//}\approx {\mathsf{\epsilon }}_s\frac{\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}{1+{\left(\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}\right)}^2{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2}, \\ \mathrm{tg}{\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\approx \frac{\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}\mathsf{\omega }{T}_{\mathrm{M}}}{1+\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}{\left(\mathsf{\omega }{T}_{\mathrm{M}}\right)}^2}. \end{gathered}$$

(66)

Additional terms

$${\left(\mathsf{\omega }{T}_M\right)}_{\left[\max ;{\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\right)}^{//}\right]}\approx \frac{2}{\pi }\sqrt{\frac{T_{\mathrm{M}}}{T_D}}=\frac{{\mathsf{\epsilon }}_{\infty }}{{\mathsf{\epsilon }}_s},{\left(\mathsf{\omega }{T}_M\right)}_{\left[\max ;{\mathrm{tg}}^{\left(\mathsf{\omega }\right)}\left[\mathsf{\delta }\right]\right]}\approx \sqrt{\frac{2}{\pi }\sqrt{\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}}}=\sqrt{\frac{{\mathsf{\epsilon }}_{\infty }}{{\mathsf{\epsilon }}_s}}.$$

(67)

In formal form, the dispersion relations (66) resemble the classical expressions for the Debye dispersion [3], but differ from them in coefficients, which leads to other expressions for determining the maxima of the functions ${\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathrm{M}}\right)\right)}^{//}$ и ${\mathrm{tg}}^{\left(\mathsf{\omega }\right)}\left[\mathsf{\delta }\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathsf{M}}\right)\right]$, as well as their values.

The Debye expressions for the complex permittivity give the maximum value of the function ${\mathrm{tg}}^{\left(\mathsf{\omega }\right)}\left[\mathsf{\delta }\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathsf{M}}\right)\right]$, reduced to $1-\frac{{\mathsf{\epsilon }}_{\infty }}{{\mathsf{\epsilon }}_s}$, compared with (66), but under the condition that is typical for dielectrics with a large depth of dispersion, when the magnitudes of the maxima will be approximately the same. The Debye relations [3] for determining the position of the maxima ${\left(\mathsf{\omega }{T}_M\right)}_{\left[\max ;{\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\right)}^{//}\right]}=\frac{{\mathsf{\epsilon }}_{\infty }}{{\mathsf{\epsilon }}_s}$ and ${\left(\mathsf{\omega }{T}_M\right)}_{\left[\max ;t{g}^{\left(\mathsf{\omega }\right)}\left[\mathsf{\delta }\right]\right]}=\sqrt{\frac{{\mathsf{\epsilon }}_{\infty }}{{\mathsf{\epsilon }}_s}}$ are several $\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}$ times higher than the corresponding values calculated by formulas (66), (67). The Debye relations [3] for determining the position of the maxima of functions ${\left({\widehat{\mathsf{\epsilon }}}_{}^{\left(\mathsf{\omega }\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathsf{M}}};\mathsf{\omega }{T}_{\mathsf{M}}\right)\right)}^{//}$, ${\mathrm{tg}}^{\left(\mathsf{\omega }\right)}\left[\mathsf{\delta }\left(\mathsf{\omega }\right)\right]$ are $\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}$ several times higher than, the corresponding values calculated by the formulas (66), (67). Thus, Debye expressions, which are used quite often to determine the parameters of defects during space charge relaxation, lead to a significant error in determining the relaxation time, increasing it by a factor of $\frac{{\mathsf{\epsilon }}_s}{{\mathsf{\epsilon }}_{\infty }}$.

7. Scientific and Practical Significance of the Study Results

Currently, quite a large amount of experimental data have been accumulated on the use of proton semiconductors and dielectrics (PSCD), mainly in the field of electrochemical technologies and physical chemistry (when developing solid-state electrolytes (perovskites, orthoperiodates and alkali metal biperiodates of alkali metals) [18,19,20,21,22,23,24,25]. At the same time, there are not many practical applications of PSCD properties [17,26,27,28,29,30,31,32] in the field of theoretical electrical engineering [33,34,35,36,37,38,39,40,41], physical electronics and microelectronics [42].

Theoretical developments aimed at applying quantum kinetic theory of proton conductivity and polarization [7,12,13,14,15,16,29,30] to questions and quantum theory of high-temperature superconductivity [43] are relevant.

Semi-empirical studies of migration of adsorbed protons on the surface of single-layer carbonaceous nanotubes [44] are not complete, due to the lack of a strict theoretical justification: (1) the relationship between the configuration of the tube surface and the dominant physical mechanism of proton transfer (tunnel or thermally activated proton transfer through a potential barrier); (2) temperature dependence of the probability of tunnel crossings of protons; (3) the forms of potential relief and activation energy for protons. The methods of quantum kinetic theory, the foundations of which are laid by the authors [1,2], will allow, using the apparatus of the density matrix [12,13,15,16,26,27], to consider in more detail the quantum mechanism of the tunnel transfer of protons in nanoscale materials with high-proton conductivity at a stricter theoretical level.

The study of the processes of charge accumulation and relaxation in nanoscale PSCD (low-temperature electret effect) in the development of hydrogen energy fuel cells [45,46,47,48,49] in the field of space technologies and for electrochemical technologies [50,51,52,53,54] is an important issue.

Theoretically discovered electrophysical properties of the nano-sized state of HBC [2,3,13,27] open the prospects for further research of tunnel diffusion polarization in proton semiconductors and dielectrics (PSCD). This could be used to develop the theoretical foundations of universal software and hardware designed to predict properties and optimize the numerical values of the characteristic parameters of structure defects in materials of the PSDC class, used as working elements of circuits of various electrical and radio electronic devices. The development of the generalized nonlinear kinetic theory of the thermally activated depolarization will allow one to adapt the structure of mathematical and computer modeling algorithms in the case of electrophysical processes in heterogeneous functional elements of devices based on HBC (MDS, MSM structures) operating in the wide range of field parameters (100 kV/m–1000 MV/m) and temperatures (0–1500 K).

Mathematical modeling of quantum proton transfer in systems from thin potential barriers with potential pits containing quantum-dimensional energy levels is relevant in the development of physical principles and schemes for the operation of resonant tunnel diodes (RTD) and quantum field effect transistors based on PSCD, for microelectronics, radio electronics and quantum electronics [55,56,57,58,59].

The results of research in the field of tunnel diffusion polarization in materials of the PSCD class will find future application in the development of schemes for numerical optimization of structure parameters and in the prediction of the properties of ferroelectrics of the HBC class (KDP, DKDP). In particular, in studies of the effects related to the influence on the non-linear optical processes of the second order (generation of the second harmonic, parametric generation and amplification of light, frequency mixing, electro-optical effect) [60,61,62,63], non-linearities of the higher order (effect of self-exposure of laser radiation) are relevant for the technique of femtosecond lasers [64,65].

The manifestation of the rectangular loop of a hysteresis (RLH) [65] with an abnormally long time of the relaxation of residual polarization (to 10 years) in a class HBC ferroelectric material (KDP [60,61,62,63,64,65], triglycine sulfate (TGS) [66], a seignette salt, etc.) allows the use of these materials in condensers of non-volatile high-speed memory devices (cells of DRAM, FeRAM memory, etc.) and electronic computers [67].

The scientific and practical significance of the theoretical methods developed in this paper consists in the development of universal algorithms for computer programs in the format of modern fast-acting software and hardware, which are designed not only for analysis but also for forecasting, with a high degree of accuracy, the results of scientific and production experiments on measuring and calculating the parameters of functional elements based on dielectrics and semiconductors about the complex structure of a crystal lattice [68,69,70,71,72,73,74,75].

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–relaxation polarization 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 $T_{\mathrm{cr},\mathrm{relax}}$ separating the temperature regions (zones) of diffusion $\left(\mathrm{T}<{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}};T_{\mathrm{D}}<T_{\mathrm{M}}\right)$ and Maxwell $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{relax}};T_{\mathrm{M}}<T_{\mathrm{D}}\right)$ relaxation and establishes the effects of molecular parameters of relaxers (lattice constant a; potential barrier width ${\mathsf{\delta }}_0$; activation energy (potential barrier height) ${\mathrm{U}}_0$; 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 ${\mathsf{\nu }}_0$). It has been found that the critical temperature for dielectric relaxation ${\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$ (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 ${\mathrm{T}}_{\mathrm{cr},\mathrm{move}}=\frac{\hslash \sqrt{{2\mathrm{U}}_0}}{{\pi \mathsf{\delta }}_0\sqrt{\mathrm{m}}{\mathrm{k}}_{\mathrm{B}}}$, which separates the areas of tunnel $\left(\mathrm{T}<{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}\right)$ and thermally activated relaxation $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}\right)$ 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 $\left(r\mathsf{\omega }\right)$, starting with the second frequency harmonic ${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(t\right)$. 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 $\mathsf{\gamma }$) 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 ${\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)$ of the relaxation modes ${\mathsf{\rho }}_{\mathrm{k}}^{{}^{\left(r\mathsf{\omega }\right)}}\left(\mathrm{x};\mathrm{t}\right)={\sum }_{\mathrm{n}=1}^{\infty }{\Re }_{\mathrm{k}}^{\left(r\mathsf{\omega }\right)}\left(\mathrm{n},\mathsf{\tau }\right)\cos \left(\frac{\pi \mathrm{nx}}{\mathrm{d}}\right)$ generated in the stationary polarization mode at the frequency $r\mathsf{\omega }$ (20). Volumetric charge density $\mathsf{\rho }\left(\mathrm{x};\mathrm{t}\right)$ was calculated with accuracy to the third frequency harmonic of the variable field (22), (23). In this case, ${\mathsf{\rho }}^{\left(\mathsf{\omega }\right)}{~\mathrm{E}}_{\mathrm{pol}}\left(t\right)$ is the first, ${\mathsf{\rho }}^{\left(2\mathsf{\omega }\right)}~{\left({\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)\right)}^2$ is the second and ${\mathsf{\rho }}^{\left(3\mathsf{\omega }\right)}~{\left({\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)\right)}^3$ 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 ${\mathrm{E}}_{\mathrm{pol}}\left(\mathrm{t}\right)$ expressions for complex dielectric permittivity (CDP) ${\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\Omega }\right)}$ calculated at multiple frequencies $\mathsf{\Omega }=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega };\dots ;2\mathsf{\lambda }\mathsf{\omega };\left(2\mathsf{\lambda }+1\right)\mathsf{\omega };\dots \right\}$ were constructed in the form of decomposition into a series by even frequency harmonics of variable field ${\hat{\mathsf{\epsilon }}}^{\left({\mathsf{\Omega }}_{2\lambda +1}\right)}{~\mathrm{e}}^{2\mathrm{i}\lambda \mathsf{\omega }t}$. Here $\lambda =\left\{0,1,2,3,\dots \right\}$. A particular case at $\lambda =0$ was examined in [1,3]. The analytical expressions for the complex dielectric permittivity (CDP) were obtained with accuracy $\lambda =1$ on the set ${\mathsf{\Omega }}_3=\left\{\mathsf{\omega };2\mathsf{\omega };3\mathsf{\omega }\right\}$ in a quadratic field approximation. They reflect the non-linear polarization effects of the third order in field frequency, which can serve as a theoretical basis for creating microwave generators by tripling the frequencies of the radio range.

5. Based on the expression for dielectric polarization at the fundamental frequency of the alternating polarizing (external) field ${\mathrm{P}}^{\left(\mathsf{\omega }\right)}\left(t\right)$ (see formula (8)) the generalized nonlinear analytic expressions are constructed (38). They allow us to examine and numerically calculate theoretical frequency and temperature spectra of the complex dielectric constant of the dielectric at the main frequency of the polarizing field, taking into account the nonlinear polarization effects caused by the effects on polarization of the spatially heterogeneous electric field induced in the crystal during volume–charge polarization (in the region of high temperatures) or due to the tunnel movement of ions (primarily protons), in low-temperature quantum polarization (in the low-temperature region) in the hydrogen-bonded crystals (HBC). Formally, according to the spectral expressions (38), the nonlinear polarization effects in the dielectric appear in the form of additional terms reflecting the interactions the of relaxation modes with different (by the Fourier order number of the harmonic of the bulk charge density) relaxation times $T_{\mathrm{n}}$ at the fundamental frequency of the external field $\left(\mathsf{\omega }\right)$. In this paper, the relaxation ${\Gamma }_1^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)$, ${\Gamma }_2^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)$ parameters are introduced for the first time. These parameters are dimensionless values written in the form of converging number series in the functions of the frequency of the alternating polarizing field and temperature (33a) and, further, converted to analytical expressions (48) calculated together with (50), (51) in the functions of dimensionless parameters ${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}$, ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}$. Further studies of the expressions (50), (51), for particular cases of diffusion relaxation over the wide range of field frequencies (${\mathsf{\alpha }}_1=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<<1$, ${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}>0$) in the form of (52), (53), and for the case of mixed type relaxation in the low frequency region of the field (${\mathsf{\alpha }}_2=\mathsf{\omega }{T}_{\mathrm{D}}<<1$) in the form of (56), (57) confirms both a high degree of mathematical accuracy in justifying the expressions (51), (52), and the physical meaning laid down in the preceding equalities (33a).

6. In parallel with the results indicated in paragraph 5 of the conclusions, based on the expression $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}={\left(\frac{\mathrm{d}}{{\mathrm{r}}_{\mathrm{D}}}\right)}^2$, we introduce the Debye screening length ${\mathrm{r}}_{\mathrm{D}}=\mathrm{d}\sqrt{\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}}$. This allows dividing the temperature regions (zones) of relaxation of the diffusion type $\left(T_{\mathrm{D}}<T_{\mathrm{M}}\right)$, when the relaxation parameters (33a) are written as

$${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}=\frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right],{\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}=\frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\left({\mathrm{n}}^2+\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right],$$

and the Maxwell type $\left(T_{\mathrm{M}}<T_{\mathrm{D}}\right)$, when the relaxation parameters (33a) are recorded differently

$${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}=\frac{4}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}{{\mathrm{n}}^2\left({\left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2\right)}\right], {\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}=\frac{4}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{T}_{\mathrm{M}}}{{\mathrm{n}}^2\left({\left(1+{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}\right)}^2+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^{\begin{array}{c}\\ 2\end{array}}\right)}\right].$$

Additional criteria of the validity of the dispersion expressions (38) and related relaxation parameters (33a) are established. So, for the case of diffusion relaxation at temperatures much lower than the critical $\left(T_{\mathrm{n},\mathrm{D}}<<T_{\mathrm{M}};\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}{<<\mathrm{n}}^2\right)$, according to the limit expressions (42), in the low frequency region of the field $\mathsf{\omega }{{\rm T}}_{\mathrm{D}}<<1$, when equals (42) are approximated to the form ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2}\right]\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^4}\right]=\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}$, ${\Gamma }_{2,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{4T_{\mathrm{D}}}{{\pi }^2{T}_{\mathrm{M}}}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{{\rm T}}_{\mathrm{D}}}{{\mathrm{n}}^2\left({\mathrm{n}}^4+{\mathsf{\omega }}^2{T}_{\mathrm{D}}^2\right)}\right]\to 0$, polarization goes into a quasi-stationary mode. According to the dispersion expressions (43), which are the same dispersion expressions in (38) written formally, for the diffusion type case $\left(T_{\mathrm{D}}<<T_{\mathrm{M}}\right)$ relaxation, the real and imaginary components of the dielectric constant are converted to a form corresponding to classical dipole polarization

$$\mathsf{\omega }{{\rm T}}_{\mathrm{D}}<<1:{\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{}\approx {\mathsf{\epsilon }}_{\infty }\frac{1}{1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}}\approx \frac{{\mathsf{\epsilon }}_{\infty }}{1-\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}}\approx {\mathsf{\epsilon }}_{\infty }\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}, {\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{//}\to 0.$$

The only clarification is that here the relaxation time ratio $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{A}{\mathrm{T}}\times \frac{W^{\left(1\right)}\left(\mathrm{T}\right)}{W^{\left(0\right)}\left(\mathrm{T}\right)}$ is a more complex function of temperature T than in the classical model. It takes into account not only thermally activated (classical) transitions of particles (relaxers) ${\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}_{\mathrm{therm}}\approx \frac{A}{\mathrm{T}}$, but also tunneling of particles (relaxers) through a potential barrier (see item 1 of the conclusions)

$$\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{A}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\times \frac{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}, \frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{A}{\mathrm{T}}\times \frac{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}{\frac{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}{T_{}}\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)},$$

which is reflected in the expressions (8a), (8b) for the transition probability rates of the relaxers $W^{\left(0\right)}\left(\mathrm{T}\right)$, $W^{\left(1\right)}\left(\mathrm{T}\right)$. The values included in the right part of the formula for the parameter $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}$ are disclosed in the item 1 of the conclusions and in Section 1 and 2 of this paper. Here, a constant $A=\frac{{\mathrm{d}}^2{n}_0{\mathrm{q}}^2}{{\pi }^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{k}}_{\mathrm{B}}}$ with a temperature dimension is adopted. In particular, in the case of the dominant contribution of thermally transitions of relaxers in the process of polarization formation $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}\right)$, it is mentioned in item 1 of the conclusions by the critical temperature ${\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}=\frac{{\mathrm{d}}^2{n}_0{\mathrm{q}}^2}{{\pi }^2{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }{\mathrm{k}}_{\mathrm{B}}}$ separating the areas of diffusion $\left(\mathrm{T}<{\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}\right)$ and Maxwell relaxation $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}\right)$.

7. It was found that for the case of Maxwell relaxation at temperatures much higher than critical $\left(T_{\mathrm{M}}<<T_{\mathrm{n},\mathrm{D}};{\mathrm{n}}^2\frac{T_{\mathrm{M}}}{T_{\mathrm{D}}}<<1\right)$, the above partial $\left(T_{\mathrm{M}}<T_{\mathrm{D}}\right)$ formulas for relaxation coefficients ${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}$, ${\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}$ take the limit form of

$${\Gamma }_{1,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{4}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2\right)}\right]=\frac{1}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}, {\Gamma }_{2,\mathrm{M}}^{\left(\mathsf{\omega }\right)}\approx \frac{4}{{\pi }^2}{\sum }_{\mathrm{n}=1}^{\infty }\left[\frac{(1-\left(-1{)}^{\mathrm{n}}\right)\cdot \mathsf{\omega }{T}_{\mathrm{M}}}{{\mathrm{n}}^2\left(1+{\mathsf{\omega }}^2{T}_{{\mathrm{M}}^2}\right)}\right]=\frac{\mathsf{\omega }{T}_{\mathrm{M}}}{1+{\mathsf{\omega }}^2{T}_{\mathrm{M}}^2}.$$

In this case, the dispersion expressions (38) are converted to the analytical form (39) corresponding to the generalized spectra of the complex dielectric constant of the crystal, calculated by the methods of the quasi-classical kinetic theory, on the fundamental frequency harmonic polarization (8) formed at temperatures much higher than the critical relaxation temperature $\left({\mathrm{T}>>\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}\right)$

$${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{}={\mathsf{\epsilon }}_{\infty }\frac{1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}={\mathsf{\epsilon }}_{\infty },{\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{M}}^{//}={\mathsf{\epsilon }}_{\infty }\frac{{\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}}{{\left(1-{\Gamma }_{1\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2+{\left({\Gamma }_{2\mathrm{M}}^{\left(\mathsf{\omega }\right)}\right)}^2}=\frac{{\mathsf{\epsilon }}_{\infty }}{\mathsf{\omega }{T}_{\mathrm{M}}}.$$

The expression constructed on the fundamental frequency harmonic of polarization (8) for the tangent of the dielectric loss angle in the temperature region $T>>T_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}$ is reduced to the form ${\left[{\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right]}_{\mathrm{M}}=\frac{{\Gamma }_{2,M}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)}{1-{\Gamma }_{1,M}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)}\approx \frac{1}{\mathsf{\omega }{T}_{\mathrm{M}}}$ corresponding to the conduction losses caused by macroscopic effects associated with the through movement of ion relaxers (ion conductivity currents) in the dielectric in the region of ultra-high temperatures (550–1550 K). This is consistent with experimental data on measuring the temperature spectra of the current density of the thermally activated depolarization (TSCD) in the hydrogen-bonded crystals in the vicinity of an ultra-high temperature maximum (450–550 K). Note that the application of the quasi-classical (corresponding to the system of nonlinear equations (2), (3)) formulas for Maxwell’s relaxation time $T_{\mathrm{M}}=\frac{{\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}{n_0{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}\mathrm{q}}$, where ${\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}=\frac{\mathrm{q}{a}^2{W}^{\left(1\right)}\left(\mathrm{T}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}-$ calculated in linear approximation by the parameter ${\mathsf{\zeta }}_0=\frac{\mathrm{q}{E}_0\mathrm{a}}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}<1$, the ion mobility coefficient [61,62] gives the expression for the tangent of the dielectric loss angle in the temperature region much higher than the critical relaxation temperature $\left({\mathrm{T}>>\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}\right)$ to the form ${\left[{\mathrm{tg}\mathsf{\delta }}^{\left(\mathsf{\omega }\right)}\left(\mathrm{T}\right)\right]}_{\mathrm{M}}=\frac{{\mathsf{\sigma }}_{\mathrm{ion}}^{\left(1\right)}\left(\mathrm{T}\right)}{{\mathsf{\omega }\mathsf{\epsilon }}_0{\mathsf{\epsilon }}_{\infty }}$ (see formula (41)). Here, the value ${\mathsf{\sigma }}_{\mathrm{ion}}^{\left(1\right)}\left(\mathrm{T}\right)={\mathrm{n}}_0{\mathsf{\mu }}_{\mathrm{mob}}^{\left(1\right)}\left(\mathrm{T}\right)\mathrm{q}$ makes sense of the conductivity coefficient for ions calculated as a function of the temperature and crystal structure parameters (see listing of these parameters in the item 6 of the conclusions)

$${\mathsf{\sigma }}_{\mathrm{ion}}^{\left(1\right)}\left(\mathrm{T}\right)=\frac{n_0{\mathrm{q}}^2{a}^2{\nu }_0}{2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\left(\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)+\frac{\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}{1-\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}}\right).$$

8. From the results noted in the conclusions in item 6, it was found that in the low-frequency range $\mathsf{\omega }{{\rm T}}_{\mathrm{D}}<<1$, during diffusion relaxation at temperatures near the critical $\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}<{\mathrm{n}}^2;\mathrm{T}\le {\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}\right)$, while maintaining the above approximation ${\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}\approx \frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}$, when $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{A}{\mathrm{T}}\times \frac{W^{\left(1\right)}\left(\mathrm{T}\right)}{W^{\left(0\right)}\left(\mathrm{T}\right)}\to 1$, the real component of the dielectric constant reaches abnormally high values of ${\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }\right)}\right]}_{\mathrm{D}}^{}\approx {\mathsf{\epsilon }}_{\infty }\frac{1}{1-{\Gamma }_{1,\mathrm{D}}^{\left(\mathsf{\omega }\right)}}\approx \frac{{\mathsf{\epsilon }}_{\infty }}{1-\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}}>>1$ and in dielectrics with the complex crystal lattice structure (mica, talc, vermiculites, ceramics, etc.) there is a quasi-ferroelectric effect established earlier experimentally for samples of corundo–zirconium ceramics [13].

In the case of the dominant contribution of thermally activated relaxer transitions $\left(\mathrm{T}>{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}\right)$, when $\frac{{\mathrm{T}}_{\mathrm{D}}}{T_{\mathrm{M}}}\to {\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}\right)}_{\mathrm{therm}}\approx \frac{{\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}}{\mathrm{T}}$, by virtue of $\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}\to 1$, the point of phase transition of the crystal to the quasi-magnetic electric state is determined from the expression ${\mathrm{T}}_{\mathrm{c};\mathrm{therm}}=\frac{{\pi }^2}{12}{\mathrm{T}}_{\mathrm{cr},\mathrm{relax};\mathrm{therm}}$.

In general, when $\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}=\frac{A}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\times \frac{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)}$, by virtue of $\frac{{\pi }^2{T}_{\mathrm{D}}}{12T_{\mathrm{M}}}\to 1$, by analogy with (8c), (8d), a transcendent equation generalized to the wide temperature range is constructed to determine the $T_c$ point of the phase transition of the crystal to the quasi-hydroelectric state $\frac{{\pi }^2}{12}\times \frac{A}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}=\frac{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\frac{{\mathrm{T}}_{\mathrm{c}}}{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{c}}}\right)}{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{c}}}\right)}$. After transformations, this expression takes the form $\frac{{\pi }^2}{12}\times \frac{A}{{\mathrm{T}}_{\mathrm{c}}}=\frac{\frac{{\mathrm{T}}_{\mathrm{cr},\mathrm{move}}}{{\mathrm{T}}_{\mathrm{c}}}\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{c}}}\right)}{\exp \left(-\frac{{\pi \mathsf{\delta }}_0\sqrt{{\mathrm{mU}}_0}}{\hslash \sqrt{2}}\right)-\exp \left(-\frac{{\mathrm{U}}_0}{{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{c}}}\right)}$. Comparison of numerical values of temperatures ${\mathrm{T}}_{\mathrm{c}}$ and ${\mathrm{T}}_{\mathrm{cr},\mathrm{relax}}$ is established from the solution of Equation (8c) or (8d).

9. The theoretical scheme has been developed for a more stringent, in comparison with item 7 of the conclusions, analysis of the behavior of the real component of the dielectric constant in the low-frequency range $\mathsf{\omega }{{\rm T}}_{\mathrm{D}}<<1$, with a mixed dielectric relaxation mechanism ($\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>0$) when the relaxation parameters (33a), (47), (48) are converted to the form (60a), (60b). This, as in the case of (42), also allows us to proceed to the study of the quasi-stationary polarization mode, according to the condition $\mathsf{\omega }{{\rm T}}_{\mathrm{D}}\to 0$, when ${\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};0\right)=1-\frac{\mathrm{th}\mathsf{\zeta }}{\mathsf{\zeta }}$, ${\Gamma }_2^{\left(\mathsf{\omega }=0\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};0\right)\to 0$. Variable $\mathsf{\zeta }=\frac{\pi }{2}\sqrt{\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}}$ has been entered. The first of the dispersion expressions (38) is brought to a stationary form, which made it possible to present the generalized analytical expression for static dielectric constant in the form convenient for comparing the results of the theory and experiment $\mathrm{Re}\left[{\hat{\mathsf{\epsilon }}}^{\left(\mathsf{\omega }=0\right)}\right]={\mathsf{\epsilon }}_{\infty }\frac{1}{1-{\Gamma }_1^{\left(\mathsf{\omega }=0\right)}\left(\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}};0\right)}={\mathsf{\epsilon }}_{\infty }\mathsf{\zeta }\mathrm{cth}\mathsf{\zeta }$. This is a relevant result for this field of research, allowing us to identify, at the theoretical level, in the wide temperature range (0–1500 K), the influence of the microscopic and macroscopic mechanisms of relaxation processes in dielectric (diffusion and Maxwell relaxation; tunnel and thermally activated relaxation; electret effect; quasi-ferroelectric effect, etc.) on the nonlinear polarization effects associated primarily with the interactions of relaxation modes of the volumetric charge at the fixed frequency of an alternating field. In this matter, we limited ourselves to the fundamental frequency harmonic of polarization in the form of (8) in the framework of this paper. The study of the non-linear relaxation processes on the frequency harmonics of the polarization of the higher orders of magnitude, at least from the third ${\mathrm{P}}^{\left(3\mathsf{\omega }\right)}\left(t\right)$ (see formula (25)), etc., is the subject of a separate research and will be carried out in subsequent works.

10. The results obtained in this paper relate primarily to the category of applied theoretical studies in the field of dielectric physics and condensed state physics. They are aimed at improving existing and developing additional schemes for the theoretical analysis of nonlinear kinetic phenomena associated with the relaxation transfer of the most mobile charge carriers (relaxers) in the crystal structure of dielectrics with an ion-molecular type of chemical bonds (ceramics, layered silicates, crystalline hydrates, perovskites, vermiculites, clay minerals, halloysite, etc.) when polarizing dielectrics in variable external fields. The results obtained in this paper described in claims one to eight of the conclusions are relevant and scientifically significant for a number of fields of modern science. In particular, we could point out their potential application in the development of theoretical ideas about quantum mechanisms for the formation of a ferroelectric state due to the tunneling of protons in the hydrogen sublattice of KDP, DKDP crystals, which are of significant practical importance for laser technology (regulators of radiation parameters), nonlinear optics and optoelectronics (electro-optical sensors, the sensors of the deformations of hard rocks and building materials, the parts of machines and mechanisms, etc.). They are also significant for information technologies (thin-film ferroelectric elements of microcircuits for fast-acting non-volatile memory devices with abnormally long residual polarization retention time (up to 10 years)). A description of the prospects for the practical application of the results obtained for various branches of modern technology and technology is made in Section 7 of this paper.