Physics

V. P. Silin

ON THE CONDUCTIVITY OF PLASMA IN STRONG ELECTRIC AND MAGNETIC FIELDS

(Presented by Academician I. E. Tamm, August 24, 1964)

Let us consider the question of the conductivity of a plasma situated in a strong constant and homogeneous magnetic field \(\mathbf B\), under conditions in which a strong high-frequency wave of circular polarization propagates in the plasma along the magnetic field. Then, with the aid of the kinetic equation for a plasma in a strong field (see \((^{1,2})\)), we obtain the generalized Ohm law:

\[ \frac{\partial \mathbf j_a}{\partial t} + [\mathbf \Omega_a \mathbf j_a] = \frac{e_a^2 N_a}{m_a} \left[ \mathbf E_1 \cos(\omega t+\delta) - \mathbf E_2 \sin(\omega t+\delta) \right] - \]

\[ - \mathbf E_1 \left[ s_{1a}\sin(\omega t+\delta) + s_{2a}\cos(\omega t+\delta) \right] - \]

\[ - \mathbf E_2 \left[ s_{1a}\cos(\omega t+\delta) - s_{2a}\sin(\omega t+\delta) \right], \tag{1} \]

where \(e_a\) is the charge; \(m_a\) the mass; \(N_a\) the number of particles; \(\mathbf \Omega_a=e_a\mathbf B/m_a c\) is the vector of the gyroscopic frequency; \(\mathbf j_a\) is the current density of particles of species \(a\); \(\mathbf E_1\) and \(\mathbf E_2\) are equal and mutually perpendicular components of the electric field \((E_1=E_2=E_0)\); finally,

\[ s_{1a} = \frac{e_a^3 N_a}{m_a} \sum_b \frac{8e_b^2 N_b}{E_0} \int_0^\infty d\tau \cos\frac{\omega\tau}{2} \int_{k_{\min}}^{k_{\max}} dk\,k \int_0^{\pi/2} d\theta\,\sin^2\theta \times \]

\[ \times \left\{ \frac{1}{m_a} \left[ \tau\cos^2\theta + \frac{\sin\Omega_a\tau}{\Omega_a}\sin^2\theta \right] + \frac{1}{m_b} \left[ \tau\cos^2\theta + \frac{\sin\Omega_b\tau}{\Omega_b}\sin^2\theta \right] \right\} \times \]

\[ \times \exp\left\{ -\frac{1}{2}v_{Ta}^2 k^2 \left[ \tau^2\cos^2\theta + 4\frac{\sin^2\theta}{\Omega_a^2} \sin^2\frac{\Omega_a\tau}{2} \right] - \right. \]

\[ \left. -\frac{1}{2}v_{Tb}^2 k^2 \left[ \tau^2\cos^2\theta + 4\frac{\sin^2\theta}{\Omega_b^2} \sin^2\frac{\Omega_b\tau}{2} \right] \right\} \times \]

\[ \times J_1\left( 2\frac{E_0 k}{\omega} \left[ \frac{e_a}{m_a(\omega-\Omega_a)} - \frac{e_b}{m_b(\omega-\Omega_b)} \right] \sin\theta\, \sin\frac{\omega\tau}{2} \right). \tag{2} \]

In formula (2), \(v_{Ta}=\sqrt{\varkappa T_a/m_a}\) is the thermal velocity; \(J_1\) is a Bessel function; \(k_{\min}\) is, in order of magnitude, equal to the reciprocal Debye radius of the ions, while \(k_{\max}\), as always, is determined by the inapplicability of perturbation theory or of classical mechanics, with the sole difference that as the particle energy one should use an expression that takes into account the oscillations of the particle under the action of the electric field. The corresponding expression for \(s_2\) differs from (2) by replacing \(\cos \omega\tau/2\) by sine.

In deriving equation (1) we neglected spatial dispersion, as is usually done in the theory of conductivity due to particle collisions. We shall be interested in the case of frequencies small in comparison with the electron Langmuir frequency of the plasma, and shall restrict ourselves to the case of singly ionized ions. Then \(s_2\) may be neglected in comparison with \(s_1\), and for \(s_{1e}\), corresponding to the electron current, we have

\[ s_{1e} = \frac{eN_e}{m^2 v_{Te}^2} \frac{8e_i^2N_i}{E_0} \int_{\varkappa_{\min}}^{\varkappa_{\max}} \frac{d\varkappa}{\varkappa} \int_0^{\pi/2} d\theta\,\sin^2\theta \int_{-\infty}^{0} d\xi \left[ \frac{d}{d\xi}J_1(\varkappa\xi\eta\sin\theta) \right] \times \]

\[ \times \exp\left\{ -\varkappa^2 \left[ \xi^2\left(1+\frac{m}{M}\right)\cos^2\theta + \sin^2\theta \left( \sin^2\xi + \frac{M}{m}\sin^2\frac{m}{M}\xi \right) \right] \right\}. \tag{3} \]

Here \(\chi=k\sqrt{2}v_{Te}\Omega_e^{-1}\), \(\eta=\sqrt{2}v_E/v_{Te}\), where
\[ v_E=E_0[e/m(\omega-\Omega_e)-e_i/M(\omega-\Omega_i)] \]
is the velocity of the relative motion of the electron and ion in the magnetic and electric fields. Naturally, such an expression is meaningful under conditions in which collisions are a small effect. The latter, for the plasma conductivity across a strong magnetic field, which is the only case of interest to us, is in fact the case. Finally, \(\xi_{\max}(\chi)=\chi^{-3/2}\chi_{\max}^{-1/2}\) is the maximum value up to which the integration over \(\xi\) should be carried out. This value arises from the condition that the particles leave the collision region owing to the Coulomb interaction \((^3)\), and corresponds to the restriction following from the condition of applicability of the perturbation theory underlying the original kinetic equation for charged particles in strong fields \((^1,^2)\).

In the limit of a very strong field, when \(\eta\gg 1\), from (3) we obtain

\[ s_{1e}=\frac{e^3N_e}{m^2}\, \frac{4\pi e_i^2N_i}{E_0^3}\, \frac{\operatorname{sgn}[e/m(\omega-\Omega_e)-e_i/M(\omega-\Omega_i)]} {[e/m(\omega-\Omega_e)-e_i/M(\omega-\Omega_i)]^2} \ln\frac{r_{\mathrm{экр}}}{r_{\min}}, \tag{4} \]

where \(r_{\mathrm{экр}}=1/k_{\min}\), \(r_{\min}=1/k_{\max}\). The dependence on the field that arises in formula (5) qualitatively corresponds to replacing the thermal velocity in the effective collision frequency by the oscillation velocity. Such an effect corresponds to strong nonlinearity.

One may speak of weak nonlinearity under conditions when the thermal velocity of the electrons is large in comparison with \(v_E\). However, even under such conditions the presence of an electric field leads to qualitative effects if the magnetic field is sufficiently large. Namely, below we shall turn to the case in which the gyration radii of the particles are smaller than the screening radius of the Coulomb field. Then the influence of the electric and magnetic fields manifests itself at large impact parameters and sufficiently long collision times, while at small impact parameters such an influence is insignificant. In this case, from formula (3) for an isothermal plasma we have

\[ s_{1e}=eN_e[e/m(\omega-\Omega_e)-e_i/M(\omega-\Omega_i)] \{v_{\mathrm{eff}}^{\Omega}+\delta v_\perp\}. \tag{5} \]

Here

\[ v_{\mathrm{eff}}^{(\Omega)} =\frac{4}{3}\, \frac{\sqrt{2\pi}\,e^2e_i^2N_i}{m^2v_{Te}^3} \ln\frac{v_{Te}}{\Omega_e r_{\min}}, \qquad \delta v_\perp= \frac{\sqrt{2\pi}\,e^2e_i^2N_i}{m^2v_{Te}^3}\,L_1, \tag{6} \]

\[ L_1=\frac{8}{\sqrt{\pi}} \int_{\chi_{\min}}^{1}d\chi \int_{0}^{\pi/2}d\theta\,\sin^3\theta \int_{0}^{\xi_{\max}(\chi)}d\xi\, \exp\left\{-\chi^2\left[\xi^2\left(1+\frac{m}{M}\right)\cos^2\theta +\psi(\xi)\sin^2\theta\right]\right\} J_1'(\chi\xi\eta\sin\theta), \quad \psi(\xi)=\sin^2\xi+(M/m)\sin^2(\xi m/M). \tag{7} \]

The integral (7) is determined by the contribution of collisions with impact parameters larger than the electron gyroradius \(\rho_e\). In a strong magnetic field, in such collisions the departure from the interaction region is slow, which leads, as is known \((^2,^3)\), to doubly logarithmic expressions, since the time during which the collision occurs proves to be considerably greater than the period of gyration. The presence of an electric field is one of the possible reasons for particles to leave the interaction region. In this case the time for an electron to leave the collision region is, in order of magnitude, equal to \(p/v_E\), where \(p\) is the impact parameter. This fact is reflected in formula (7) by the presence of the derivative of the Bessel function \(J_1'\).

In calculations with double logarithmic accuracy, one may use the approximate formula for integral (7)

\[ L_1=2\int_{\chi_{\min}}^{1}\frac{d\chi}{\chi} \int_{1/\chi}^{\min\{\xi_{\max}(\chi),\,1/\chi\}}\frac{d\xi}{\xi}\, e^{-\chi^2\psi(\xi)}. \tag{8} \]

A comparatively simple consideration makes it possible to obtain the following series of asymptotic formulas:

\[ L_1=\ln(r_{\mathrm{ecr}}/\rho_e)\ln(M/m) \quad (\rho_i\gg r_{\mathrm{ecr}}\gg \rho_e\gg r_{\min}M/m;\; v_{Ti}^{\,2}\gg v_E^{\,2}); \tag{9} \]

\[ L_1=\ln(M/m)\ln(mr_{\mathrm{ecr}}/Mr_{\min}) +\frac12\ln(Mr_{\min}/m\rho_e)\ln(M\rho_e/mr_{\min}) \]
\[ (\rho_i\gg r_{\mathrm{ecr}}\gg r_{\min}M/m\gg \rho_e;\quad v_{Ti}^{\,2}\gg v_E^{\,2}); \tag{10} \]

\[ L_1=\ln(r_{\mathrm{ecr}}/\rho_e)\ln\left(\sqrt{\rho_e r_{\mathrm{ecr}}}/r_{\min}\right) \]
\[ (\rho_e\ll \rho_i,\; r_{\mathrm{ecr}},\; r_{\min}M/m;\quad v_{Te}^{\,2}r_{\min}\gg v_E^{\,2}r_{\mathrm{ecr}}); \tag{11} \]

\[ L_1=\frac12[\ln(M/m)]^2 +\ln(r_{\mathrm{ecr}}/\rho_i)\ln\left(\sqrt{\rho_i r_{\mathrm{ecr}}}/r_{\min}\right) \]
\[ (r_{\mathrm{ecr}}\gg \rho_i\gg \rho_e\gg (M/m)r_{\min}\gg (v_E^{\,2}/v_{Ti}^{\,2})r_{\mathrm{ecr}}); \tag{12} \]

\[ L_1=\frac12\ln(\rho_e/r_{\min})\ln(M^2r_{\min}/m^2\rho_e) +\frac12\ln(r_{\mathrm{ecr}}/\rho_i)\ln(\rho_i r_{\mathrm{ecr}}/r_{\min}^2), \]
\[ (r_{\mathrm{ecr}}\gg \rho_i\gg (M/m)r_{\min}\gg \rho_e,\; r_{\mathrm{ecr}}v_E^{\,2}/v_{Ti}^{\,2}); \tag{13} \]

\[ L_1=\ln(r_{\mathrm{ecr}}/\rho_e)\ln(v_{Te}^{\,2}/v_E^{\,2}) \]
\[ (\rho_i,\; r_{\mathrm{ecr}}\gg \rho_e;\quad v_E^{\,2}\gg v_{Ti}^{\,2}\ \text{and}\ v_{Te}^{\,2}r_{\min}/\rho_e); \tag{14} \]

\[ L_1=2\ln(v_{Te}/v_E)\ln(v_E^{\,2}r_{\mathrm{ecr}}/v_{Te}^{\,2}r_{\min})+ \]
\[ +\frac12\ln(v_{Te}^{\,2}\rho_e/v_E^{\,2}r_{\min}) \ln(v_{Te}^{\,2}r_{\min}/v_E^{\,2}\rho_e) \]
\[ (\rho_i\gg r_{\mathrm{ecr}}\gg \rho_e\ll r_{\min}M/m;\quad v_{Te}^{\,2}\rho_e/r_{\min}\gg v_E^{\,2}\gg v_{Ti}^{\,2}) \tag{15} \]
or
\[ r_{\mathrm{ecr}}\gg \rho_i\gg \rho_e\ \text{and}\ r_{\min}M/m;\quad v_{Ti}^{\,2}\gg v_E^{\,2}\gg v_{Te}^{\,2}r_{\min}/r_{\mathrm{ecr}}); \]

\[ L_1=\frac12[\ln(M/m)]^2 +\ln(v_{Te}^{\,2}/v_E^{\,2})\ln(v_E^{\,2}r_{\mathrm{ecr}}/v_{Te}^{\,2}r_{\min})+ \]
\[ +\frac12\ln(v_{Te}^{\,2}r_{\min}/v_E^{\,2}\rho_i) \ln(v_{Te}^{\,2}\rho_i/v_E^{\,2}r_{\min}) \]
\[ (r_{\mathrm{ecr}}\gg (v_{Te}^{\,2}/v_E^{\,2})r_{\min}\gg \rho_i\gg \rho_e\gg (M/m)r_{\min}); \tag{16} \]

\[ L_1=\frac12[\ln(M/m)]^2 +\ln(v_{Te}^{\,2}/v_E^{\,2})\ln(r_{\mathrm{ecr}}/\rho_i) \]
\[ (r_{\mathrm{ecr}}\gg \rho_i\gg \rho_e\gg r_{\min}M/m;\quad v_{Ti}^{\,2}\gg v_E^{\,2}\gg v_{Te}^{\,2}r_{\min}/\rho_i); \tag{17} \]

\[ L_1=2\ln(v_{Te}/v_E)\ln(v_E^{\,2}r_{\mathrm{ecr}}/v_{Te}^{\,2}r_{\min})+ \]
\[ +\frac12\ln(v_{Te}^{\,2}r_{\min}/v_E^{\,2}\rho_e) \ln(v_{Te}^{\,2}\rho_i/v_E^{\,2}r_{\min}) \]
\[ (r_{\mathrm{ecr}}\gg (v_{Te}^{\,2}/v_E^{\,2})r_{\min}\gg \rho_i\gg (M/m)r_{\min}\gg \rho_e); \tag{18} \]

\[ L_1=\ln(r_{\mathrm{ecr}}/\rho_i)\ln(v_{Te}^{\,2}/v_E^{\,2}) +\frac12\ln(\rho_e/r_{\min})\ln(M^2r_{\min}/m^2\rho_e) \]
\[ (r_{\mathrm{ecr}}\gg \rho_i\gg (v_{Te}^{\,2}/v_E^{\,2})r_{\min},\; (M/m)r_{\min}\gg \rho_e). \tag{19} \]

Here \(\rho_i=v_{Ti}/\Omega_i\) is the radius of the ion gyroscopic rotation. Formulas (9)—(13) do not depend on the electric field and correspond to those obtained in work \((^3)\), with the difference that here the limits of applicability, restricted by the magnitude of the electric-field strength, have been determined. On the contrary, expressions (14)—(19) depend substantially on the electric-field strength. Consequently, in the limit of weak nonlinearity the collision frequency is substantially nonlinear.

A high-frequency field can lead to an effective decrease in the interaction in the case when its period proves to be smaller than the characteristic interaction time (see \(^{4,1,2}\)). Allowance for such an effect for \(\Omega_e \gg \omega\) is manifested in the fact that, when integrating over \(\xi\) in integral (8), as the upper limit one should take the minimum of the three expressions \(\xi_{\max}(\chi)\), \(1/\chi\eta\), \(\Omega_e/\omega\). Below we give a number of formulas characterizing \(L_1\), when the influence of the electric field also appears.*

\[ \begin{gathered} L_1(\omega)=\frac12\ln(\rho_e/r_{\min})\ln(M^2r_{\min}/m^2\rho_e) +\ln(v_{Te}^2/v_E^2)\ln(v_E/\omega\rho_i)+\\ +\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) \\ (r_{\mathrm{scr}}\gg v_E/\omega\gg \rho_i\gg r_{\min}v_{Te}^2/v_E^2\gg r_{\min}M/m\gg \rho_e); \end{gathered} \tag{20} \]

\[ \begin{gathered} L_1(\omega)=\frac12\ln(\rho_e/r_{\min})\ln(M^2r_{\min}/m^2\rho_e)+\\ +\frac12\ln(v_{Te}^2r_{\min}/v_E^2\rho_i)\ln(\rho_i v_{Te}^2/r_{\min}v_E^2)+\\ +\ln(v_{Te}^2/v_E^2)\ln(v_E^3/v_{Te}^2\omega r_{\min}) +\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) \\ (r_{\mathrm{scr}}\gg v_E/\omega\gg (v_{Te}^2/v_E^2)r_{\min}\gg \rho_i\gg (M/m)r_{\min}\gg \rho_e); \end{gathered} \tag{21} \]

\[ \begin{gathered} L_1(\omega)=\frac12[\ln(M/m)]^2 +\frac12\ln(v_{Te}^2r_{\min}/v_E^2\rho_i)\ln(v_{Te}^2\rho_i/v_E^2r_{\min})+\\ +\ln(v_{Te}^2/v_E^2)\ln(v_E^3/\omega r_{\min}v_{Te}^2) +\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) \\ (r_{\mathrm{scr}}\gg v_E/\omega\gg (v_{Te}^2/v_E^2)r_{\min}\gg \rho_i\gg \rho_e\gg (M/m)r_{\min}); \end{gathered} \tag{22} \]

\[ \begin{gathered} L_1(\omega)+\frac12[\ln(M/m)]^2+\ln(v_{Te}^2/v_E^2)\ln(v_E/\omega\rho_i)+\\ +\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) \\ (r_{\mathrm{scr}}\gg v_E/\omega\gg \rho_i\gg \rho_e\gg r_{\min}M/m;\quad v_{Ti}^2\gg v_E^2\gg v_{Te}^2r_{\min}/\rho_i); \end{gathered} \tag{23} \]

\[ \begin{gathered} L_1(\omega)=\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) +\ln(v_{Te}^2/v_E^2)\ln(v_E^3/\omega r_{\min}v_{Te}^2)+\\ +\frac12\ln(v_{Te}^2r_{\min}/v_E^2\rho_e)\ln(v_{Te}^2\rho_e/v_E^2r_{\min}), \\ (\rho_i\gg r_{\mathrm{scr}}\gg v_E/\omega\gg (M/m)r_{\min}\gg \rho_e; \\ (\Omega_e/\omega)v_E^2\gg v_{Te}^2r_{\min}/\rho_e\gg v_E^2\gg v_{Ti}^2,\ \text{or} \\ r_{\mathrm{scr}}\ \text{and}\ r_{\min}M/m\gg \rho_i\gg \rho_e;\quad r_{\mathrm{scr}}\gg v_E/\omega\gg (v_{Te}^2/v_E^2)r_{\min}; \\ v_{Ti}^2\gg v_E^2\gg (r_{\min}/r_{\mathrm{scr}})v_{Te}^2); \end{gathered} \tag{24} \]

\[ L_1(\omega)=\ln(v_{Te}^2/v_E^2)\ln(v_E/\omega\rho_e) +\ln(\omega r_{\mathrm{scr}}/v_E)\ln(v_{Te}^2/\omega v_E r_{\mathrm{scr}}) \tag{25} \]

\[ (\rho_i,\ r_{\mathrm{scr}}\gg \rho_e\gg r_{\min}M/m;\quad v_E^2\gg v_{Ti}^2\ \text{and}\ v_{Te}^2r_{\min}/\rho_e;\quad r_{\mathrm{scr}}\gg v_E/\omega\gg \rho_e); \]

\[ \begin{gathered} L_1(\omega)=\ln(v_{Te}/v_E)\ln(v_E\Omega^2/v_{Te}\omega^2) \\ (r_{\mathrm{scr}}\gg v_{Te}/\omega,\ v_E^2\gg v_{Ti}^2\ \text{and}\ v_{Te}^2r_{\min}/\rho_e); \end{gathered} \tag{26} \]

\[ \begin{gathered} L_1(\omega)=[\ln(v_{Te}/v_E)]^2 +2\ln(v_{Te}/v_E)\ln(v_E^3/v_{Te}^2r_{\min}\omega)+\\ +\frac12\ln(v_{Te}^2r_{\min}/v_E^2\rho_e)\ln(v_{Te}^2\rho_e/v_E^2r_{\min}), \\ (r_{\mathrm{scr}}\gg v_{Te}/\omega\gg (v_{Te}^3/v_E^3)r_{\min};\quad v_{Ti}^2\ll v_E^2\ll v_{Te}^2r_{\min}/\rho_e). \end{gathered} \tag{27} \]

Thus it has been shown that in strong electric and magnetic fields the conductivity depends nonlinearly on the electric-field strength. In conditions of strong nonlinearity, the reason for the appearance of the corresponding dependence is that the field strength determines the velocity of motion of the colliding particles. In conditions of weak nonlinearity, the dependence on the electric field arises when such a field takes the colliding particles out of the collision region.

P. N. Lebedev Physical Institute
Academy of Sciences of the USSR

Received
7 VIII 1964

CITED LITERATURE

1. V. P. Silin, ZhETF, 38, 1771 (1960).
2. V. P. Silin, ZhETF, 41, 861 (1961).
3. Yu. M. Aliev, A. R. Shister, ZhETF, 45, 1499 (1963).
4. V. L. Ginzburg, Propagation of Electromagnetic Waves in Plasma, Moscow, 1960.
5. A. R. Shister, Thermophysics of High Temperatures, No. 3 (1965).

* Without allowance for the influence of the electric field considered by us, the corresponding results are contained in Shister’s work (\(^{5}\)).