In the first approximation, the Earth’s magnet seems to be a dipole inclined to the rotation axis at an angle of 11˚ and having a strength of 0.3 G at the magnetic equator. To date, the amplitudes of more than a dozen harmonics following the dipole have been measured, which decrease according to the power law with a break at the eighth harmonic. The dipole mode accounts for about 90% of the intensity. The residual field (full minus dipole) has the form of a finite number of anomalies occupying areas with sizes from hundreds to two thousand kilometers. Chaotic fluctuations of the direction of the dipole moment with characteristic time periods of 10³ - 10⁴ years have been observed. When averaging over these fluctuations, the average Earth dipole will be oriented along the rotation axis. Consequently, rotation must have a strong influence on the evolution of the magnetic field. After a characteristic time period of about 10⁵ years reversals (inversions) of the magnetic dipole direction take place. The process is random ( [1] , pp. 266-267).

Modes those are higher than the dipole mode result from turbulent fluctuations of currents of electrically conducting liquid in the Earth’s outer core. Mathematical modeling shows that if they are considered as white noise acting on the dipole mode, then model states occur that explain the inversion of the Earth’s magnetic field [3] .

Figure 1 shows a simplified scheme of the Earth’s structure with an indication of the names of the regions and the distances from the surface to the boundaries of characteristic states of matter ( [6] , V2, p. 79).

The depth distribution of pressure, temperature and density according to the model “Earth-2” by V. N. Zharkov, V. P. Trubitsyn and P. V. Samsonenko is given in Table 1 in ( [7] , pp. 26-27). An interpolation by parabolic splines was performed on the tabular data for the temperature and the density of matter in the Earth’s interior. The range of density values (5.56 - 10.08) g/cm³ at a depth of 2920 km was replaced by the average value of 7.82 g/cm³. The depth parameter was recalculated as the distance from the center. The density parameter was normalized to the Earth mass. All parameters were converted to the international measurement system SI.

The obtained dependences of the absolute temperature $T_z\left(r\right)$ and density ${\rho }_n\left(r\right)$ of matter in the Earth’s interior, which are further used in the calculation of ionization and polarization of free electrons in the Earth’s mantle, are shown in Figure 2 .

Vertical markers indicate the radius of the inner core $R_j=1.25\times {10}^6\text{m}$ and the inner radius of the mantle $R_{mu}=4.35\times {10}^6\text{m}$. The largest radius value corresponds to the boundary of the upper mantle $R_{mo}=6.34\times {10}^6\text{m}$.

Table 2 shows the concentration of oxides of the main elements of the Earth’s mantle in mass percent (according to Ringwood A. E.) ( [8] , p. 12), their molecular weight, number of atoms in a molecule, density, and relative dielectric permittivity [9] , as well as the first ionization potential (energy) of the molecules [10] [11] and [12] . Oxides, the content of which in the Earth’s mantle is less than 1%, are excluded from the calculation. These data were used in the further calculation of the ionization and polarization of charges in the Earth’s mantle.

Temperature fluctuations of atoms in the Earth’s mantle can be considered as a phonon gas, subject to Boltzmann statistics. The number of molecules with an energy value greater than a given $w_0$, at $w_0\gg k_{b}T$, can be represented by the expression ( [13] , p. 207):

$N\left(w_0\right)=2N\sqrt{\frac{w_0}{\pi k_{b}T}}{\text{e}}^{-\frac{w_0}{k_{b}T}}$

where: N is the total number of molecules in the considered volume;

$k_b$ is the Boltzmann constant;

T is the temperature of the substance.

This expression is also valid for the density of molecules if we take the derivative with respect to volume of both parts of the equation and if the number of molecules per unit volume is statistically significant:

${\delta }_i\left(r\right)=2n_i\sqrt{\frac{w_i}{\pi k_b{T}_z\left(r\right)}}{\text{e}}^{-\frac{w_i}{k_b{T}_z\left(r\right)}}$

where: $n_i\left(r\right)=\frac{{\rho }_n\left(r\right)m_i}{AMU\cdot M_i}$ is the number of molecules of the i-th oxide per unit volume;

$m_i=\frac{p{r}_i}{\Sigma p{r}_i}$ is the fraction of the i-th oxide in the mixture;

$p{r}_i$ is the percentage content of the i-th oxide ( Table 2 );

AMU is the atomic mass unit constant;

$M_i$ is the molar mass of a molecule of the i-th oxide;

$w_i$ is the first ionization potential of the molecule;

r is the distance from the center of the Earth.

The resulting density is obtained by summing over all components of the mixture:

${\delta }_{is}\left(r\right)=\Sigma {\delta }_i\left(r\right)$ (1)

The logarithm of the relative ion density ${\delta }_{is}\left(r\right)$ in the Earth’s mantle is shown in Figure 3 . Vertical markers indicate the conventional limits of polarization $R_p=5.9\times {10}^6\text{m}$ and ionization $R_i=6.2\times {10}^6\text{m}$ obtained below.

As can be seen from the figure, the lower part of the mantle is strongly ionized. The ion density decreases exponentially and reaches a critical value in the region of the conditional ionization boundary $R_i$. The ratio of the number of ions to the number of atoms per unit volume at characteristic points was:

$k_i\left(R_{mu}\right)={\delta }_{is}\left(R_{mu}\right)/n_a\left(R_{mu}\right)=6\times {10}^{-9}$

$k_i\left(R_p\right)={\delta }_{is}\left(R_p\right)/n_a\left(R_p\right)=7\times {10}^{-13}$

$k_i\left(R_i\right)={\delta }_{is}\left(R_i\right)/n_a\left(R_i\right)=7\times {10}^{-16}$

There are at least 10⁸ neutral atoms per ion. This is also true for free electrons, whose partial pressure can be neglected compared to the partial pressure of the phonon gas. Thus, the matter of the mantle is in the state of a semiconductor, the “holes” of which are “frozen” in the basic matter, and the free electrons are in the state of an electron gas. In other words, the polarization of free electrons in the mantle matter is possible.

Under the influence of the radial component of the phonon gas pressure gradient, free electrons are displaced in the direction of the Earth’s surface (polarized), and the equilibrium state is described by the equation of the equilibrium of forces acting on an elementary volume of electron gas in an electric field created by holes:

$E_p\left(r\right)\cdot \Delta q_e+{\nabla }_{r}p\left(r\right)\cdot \Delta V$ (2)

where: $E_p\left(r\right)=\frac{q_p}{\epsilon {\epsilon }_o{S}_n\left(r\right)}$ is the strength of the electrostatic field created by the positive charge of holes;

${\epsilon }_o$ is the permittivity of vacuum;

$\epsilon$ is the relative permittivity of the mantle;

$\text{Δ}{q}_e\left(r\right)$ is a change in the charge of free electrons in a volume $\text{Δ}V$;

$\Delta V=S_n\left(r\right)\cdot \Delta r$ is the volume increment;

$S_n\left(r\right)=4\pi r^2$ is the spherical surface area;

$\text{Δ}r$ is the increment of the radius;

${\nabla }_{r}p\left(r\right)=p\left(r\right)\cdot \psi \left(r\right)$ is the pressure gradient in the radial direction;

$p\left(r\right)=n_a\left(r\right)\cdot k_b\cdot T_z\left(r\right)$ is the pressure of the phonon gas;

$n_a\left(r\right)=\Sigma n{a}_i\left(r\right)$ is the number of atoms in a unit volume;

$n{a}_i\left(r\right)=N_i\cdot m_i\left(r\right)$ is the density of the atoms of the i-th oxide;

$N_i$ is the number of atoms in an oxide molecule;

$\psi \left(r\right)=\frac{{\nabla }_r{n}_a\left(r\right)}{n_a\left(r\right)}+\frac{{\nabla }_r{T}_z\left(r\right)}{T_z\left(r\right)}$ is the gradient function (a function equal to the sum of the relative values of the radial gradients of atomic density and temperature in the mantle).

By integrating (2), we obtain the distribution of the charge of holes in the mantle. At negative values of the gradient function, it has a real value and characterizes the displacement of free electrons from the center to the Earth’s surface.

$q_o\left(r\right)=\sqrt{-2\epsilon {\epsilon }_o{\displaystyle {\int }_{R_{mu}}^r{S}_n{\left(r\right)}^{2}p\left(r\right)\psi \left(r\right)\text{d}r}}$

Integrating by parts, we obtain an expression that is more convenient for calculations:

$q_o\left(r\right)=\sqrt{2\epsilon {\epsilon }_o\left(p\left(R_{mu}\right)S_n{\left(R_{mu}\right)}^2-p\left(r\right)S_n{\left(r\right)}^2-16\pi {\displaystyle {\int }_{R_{mu}}^{r}r\cdot S_n{\left(r\right)}^{2}p\left(r\right)\text{d}r}\right)}$(3)

Expression (3) is used to calculate the magnetic moment and the integral of forces acting on free electrons beyond the polarization boundary. Below $R_{mu}$ there is a conducting liquid core, the polarization of whose charges is not considered within the framework of the proposed mechanism.

The relative dielectric permittivity of the medium for a homogeneous mixture is estimated with the Landau-Lifshitz equation ( [14] , p. 69):

$\epsilon ={\left(\frac{{\displaystyle {\sum }_i^{n}m{t}_i\cdot \sqrt[3]{{\epsilon }_i}}}{{\displaystyle {\sum }_i^{n}m{t}_i}}\right)}^3=8.4$

where: $m{t}_i=m_i/M_i$ is the fraction of molecules of the i-th oxide;

${\epsilon }_i$ is the relative dielectric permittivity of the i-th oxide;

$n=5$ is the number of oxides taken into account.

If the density of the free electrons is low, Equation (2) loses its physical meaning. Each atom must experience on average at least one collision per second with free electrons distributed in the volume:

$u_s\left(r\right)\sqrt[3]{n_a\left(r\right)}\cdot k{p}_{ea}\left(r\right)\ge 1\cdot s^{-1}$ (3.1)

where: $u_s\left(r\right)=\sqrt{\frac{8k_b{T}_z\left(r\right)}{\pi m_e}}$ Is the arithmetic mean velocity of the free electrons ( [13] , p. 207);

$m_e$ is the mass of an electron;

$k{p}_{ea}\left(r\right)={\delta }_i\left(r\right)/n_a\left(r\right)$ is the ratio of the number of ions (or free electrons) to the number of atoms in a unit volume.

Inequality (3.1) is a physical condition for the application of expression (3). Its solution gives the value of the ionization boundary $R_i$ ( Figure 3 ). The following expression is to be considered as a criterion for the polarization boundary of free electrons:

$\ln \left(N_d\left(r\right)k_p\left(r\right)\right)\le 0$ (3.2)

where: $N_d\left(r\right)=\frac{4}{3}\pi R_D{\left(r\right)}^3{\delta }_i\left(r\right)$ is the Debye number of the plasma (number of charge particles of one sign within the region bounded by the Debye length:

$R_D\left(r\right)=\sqrt{\frac{\epsilon {\epsilon }_o{k}_b{T}_z\left(r\right)}{{\delta }_i\left(r\right)q_e^2}}$;

$k_p={\delta }_p\left(r\right)/{\delta }_i\left(r\right)$ is the polarization coefficient;

${\delta }_p\left(r\right)=\frac{\text{d}{q}_p\left(r\right)}{q_e\text{d}V}$ is the number of holes per unit volume;

$\text{d}V=S_n\left(r\right)\text{d}r$ is the change in volume;

$\text{d}r$ is the change in radius;

$q_e$ is the elementary charge.

From a physical point of view, expression (3.2) means that the average number of free electrons leaving the Debye screening region as a result of polarization does not exceed one. The mantle substance remains quasi-neutral. The solution of expression (3.2) gives the magnitude of the polarization boundary as $R_p=5.64\times {10}^6\text{m}<R_i$. Its value is located in the region of pronounced ionization ( Figure 3 ).

Below the polarization boundary, intense electron-hole recombination with the release of thermal energy should occur. Most likely, this is due to a sharp decrease in the density of the phonon gas, which leads to a loss of free electron energy ( Figure 4 ). The maximum energy release corresponds to the radius $R_r=5.46\times {10}^6\text{m}<R_p$, obtained from the solution of the equation:

$\frac{\text{d}}{\text{d}r}\left(\frac{{\nabla }_r{T}_z\left(r\right)}{T_z\left(r\right)}\right)=0$

By analogy with the Maxwell distribution, we apply the Fermi-Dirac distribution to the region of electron-hole recombination, the center of which is the $R_r$, analogue of the chemical potential. The probability of polarization of a free electron will be equal to:

$p\left(r\right)={\left(1+{\text{e}}^{\frac{r-R_r}{{\Delta }_r}}\right)}^{-1}$

where: ${\Delta }_r=\frac{R_{r2}-R_{r1}}{6}=99\text{km}$ is a length constant that characterizes the decrease in the charge of holes by e times on a given segment (e is the constant of the natural logarithm);

$R_{r1}=5.1\times {10}^6\text{m}$ and $R_{r2}=5.7\times {10}^6\text{m}$ are the lower and upper limits of the region of recombination, which are obtained as a result of solving the equation: ${T^{\prime }}_z\left(r\right)=0$ ranging from $R_{mu}$ to $R_p$, where ${T^{\prime }}_z\left(r\right)={\Delta }_T\left(r\right)-median\left({\Delta }_T\left(r\right)\right)$;

${\Delta }_T\left(r\right)=\frac{{\nabla }_r{T}_z\left(r\right)}{T_z\left(r\right)}$ is a function of the temperature gradient.

This probability corresponds to the probability density of electron-hole recombination:

$f\left(r\right)=\frac{1}{4{\Delta }_r{\cosh }^2\left(\frac{R_r-r}{2{\Delta }_r}\right)}$

Figure 4 shows the functions of the temperature gradient and probability density of electron-hole recombination normalized at point $R_r$, where $k_T=\frac{f\left(R_r\right)}{T_z\left(R_r\right)}$ is the normalization coefficient.

The distribution of the positive charge of holes in the earth’s mantle can be represented by the expression:

$q_p\left(r\right)=\frac{q_o\left(r\right)}{1+{\text{e}}^{\frac{r-R_r}{{\Delta }_r}}}$ (3.3)

Figure 5 shows the distribution of the positive charge of holes in the earth’s mantle and the terms of the gradient function. The horizontal marker marks the median value of the temperature gradient function. Vertical markers mark the recombination center and polarization boundary.

The change in the force acting in the center of the recombination region on the total charge of free electrons located in the elementary volume, due to the change in charge, has the form:

$\Delta F\left(r\right)=E\left(r\right)\Delta q_{pe}$ (4)

where: $E\left(r\right)=\frac{q_o\left(r\right)+q_{pe}}{\epsilon {\epsilon }_o{S}_n\left(r\right)}$ is the electrostatic field strength;

$q_o\left(r\right)+q_{pe}$ is the charge near the polarization boundary;

$q_{pe}$ is the charge of the free electrons;

$\text{Δ}{q}_{pe}$ is the change in the charge of free electrons.

$F\left(r\right)={\displaystyle {\int }_0^{-q_o\left(r\right)}\frac{\left(q_o\left(r\right)+q_{pe}\right)d{q}_{pe}}{\epsilon {\epsilon }_o{S}_n\left(r\right)}}=-\frac{q_o{\left(r\right)}^2}{2\epsilon {\epsilon }_o{S}_n\left(r\right)}$

By integrating (4) taking into account the equality of charges of different signs, we obtain the integral of the attractive forces directed toward the center of the Earth and acting on free electrons outside the recombination region.

$F_g\left(R_{r1}\right)=-\frac{q_o{\left(R_{r1}\right)}^2}{2\epsilon {\epsilon }_o{S}_n\left(R_{r1}\right)}=-1.1\times {10}^{24}\text{N}$

The integral of the gradient pressure forces directed away from the center and acting on the free electrons distributed in the Earth’s mantle:

$F_m\left(R_{r1}\right)={\displaystyle {\int }_{R_{mu}}^{R_{r1}}{S}_n\left(r\right)p\left(r\right)\psi \left(r\right)\text{d}r}=-1.5F_g\left(R_{r1}\right)$

This value exceeds the integral of the attraction forces by 50% and confirms the possibility of keeping the charges in the polarized state.