The occurrence of transverse and longitudinal electric currents in the classical plasma under the action of N transverse electromagnetic waves

TL;DR

This paper investigates how multiple electromagnetic waves induce both transverse and longitudinal electric currents in classical plasma, revealing nonlinear effects that generate a longitudinal current orthogonal to the linear transverse current.

Contribution

It derives a distribution function and current formula showing nonlinear plasma response to multiple electromagnetic waves, highlighting the emergence of a longitudinal current.

Findings

Longitudinal current appears due to nonlinearity in plasma response.

Longitudinal current is orthogonal to the linear transverse current.

Results are valid for small wave numbers.

Abstract

Classical plasma with arbitrary degree of degeneration of electronic gas is considered. In plasma N (N>2) collinear electromagnatic waves are propagated. It is required to find the response of plasma to these waves. Distribution function in square-law approximation on quantities of two small parameters from Vlasov equation is received. The formula for electric current calculation is deduced. It is demonstrated that the nonlinearity account leads to occurrence of the longitudinal electric current directed along a wave vector. This longitudinal current is orthogonal to the known transversal current received at the linear analysis. The case of small values of wave number is considered.

Equations

$${\text{E}}_j={\text{E}}_{0j}{e}^{i\left({\text{k}}_j\text{r}-{\omega }_{j}t\right)},{\text{H}}_j={\text{H}}_{0j}{e}^{i\left({\text{k}}_j\text{r}-{\omega }_{j}t\right)},$$

$${\text{E}}_j=-\frac{1}{c}\frac{\partial {\text{A}}_j}{\partial t}=\frac{i{\omega }_j}{c}{\text{A}}_j,{\text{H}}_j=\mathrm{rot}{\text{A}}_j,j=1,2,\cdots ,N.$$

$$\frac{\partial f}{\partial t}+\text{v}\frac{\partial f}{\partial \text{r}}+e\left(\sum _{j=1}^N{\text{E}}_j+\frac{1}{c}\left[\text{v},\sum _{j=1}^N{\text{H}}_j\right]\right)\frac{\partial f}{\partial \text{p}}=0.$$

$$f_{eq}\left(\text{r},v\right)=\frac{1}{1+\exp \left(\frac{\mathcal{E}-\mu (\text{r})}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)},$$

$$f_{eq}\left(\text{r},v\right)=\frac{1}{1+\exp \left({\mathrm{P}}^2-\alpha (\text{r})\right)}=f_{eq}\left(\text{r},P\right),$$

$$f_0\left(\text{v}\right)={\left[1+\exp \left(\frac{\mathcal{E}-\mu }{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)\right]}^{-1}={\left[1+\exp \left({\mathrm{P}}^2-\alpha \right)\right]}^{-1}=f_0\left(P\right).$$

$${\mathbf{k}}_j\cdot {\mathbf{A}}_j(\mathbf{r},t)=0,j=1,2,\cdots ,N.$$

$${\mathbf{k}}_j\cdot {\mathbf{E}}_j(\mathbf{r},t)={\mathbf{k}}_j\cdot {\mathbf{H}}_j(\mathbf{r},t)=0,j=1,2,\cdots ,N.$$

$${\mathbf{k}}_j=k_j(1,0,0),{\mathbf{E}}_j=E_j(x,t)(0,1,0).$$

$${\text{E}}_j=-\frac{1}{c}\frac{\partial {\text{A}}_j}{\partial t}=\frac{i{\omega }_j}{c}{\text{A}}_j,{\text{H}}_j=\frac{c{k}_j}{\omega }{E}_j\cdot \left(0,0,1\right),j=1,2,\cdots ,N.$$

$$\left[\text{v},{\text{H}}_j\right]=\frac{c{k}_j}{{\omega }_j}{E}_j\left(v_y,-v_x,0\right),$$

$$\left[\text{v},\sum _{j=1}^N{\text{H}}_j\right]=\sum _{j=1}^N\frac{c{k}_j}{{\omega }_j}{E}_j\left(v_y,-v_x,0\right).$$

$$e\left({\text{E}}_j+\frac{1}{c}\left[\text{v},{\text{H}}_j\right]\right)\frac{\partial f}{\partial \text{p}}=$$

$$=\frac{e}{{\omega }_j}{E}_j\left[k_j{v}_y\frac{\partial f}{\partial p_x}+\left({\omega }_j-k_j{v}_x\right)\frac{\partial f}{\partial p_y}\right],(j=1,2,\cdots ,N).$$

$$\left[\text{v},{\text{H}}_j\right]\frac{\partial f_0}{\partial \text{p}}=0,$$

$$\frac{\partial f_0}{\partial \text{p}}\sim \text{v}.$$

$$\frac{\partial f}{\partial t}+v_x\frac{\partial f}{\partial x}+e\sum _{j=1}^N\frac{E_j}{{\omega }_j}\left[k_j{v}_y\frac{\partial f}{\partial p_x}+\left({\omega }_j-k_j{v}_x\right)\frac{\partial f}{\partial p_y}\right]=0.$$

$$f=f_0\left(P\right)+f_1+f_2.$$

$$f_1=E_1{\phi }_1+E_2{\phi }_2+\cdots +E_N{\phi }_N=\sum _{j=1}^N{E}_j{\phi }_j,$$

$$E_j\sim e^{i(k_{j}x-\omega t)},$$

$$f_2=\sum _{j=1}^N{E}_j^2{\psi }_j+\sum _{\begin{array}{c}b,s=1\\ b<s\end{array}}^N{E}_b{E}_s{\xi }_{b,s},$$

$$E_j^2\sim e^{2i(k_{j}x-{\omega }_{j}t)},$$

$$E_b{E}_s\sim e^{i[(k_b+k_s)x-({\omega }_b+{\omega }_s)t]}.$$

$${\alpha }_j=\frac{\left|e{A}_j\right|v_T}{c{k}_{B}T}(j=1,2,\cdots ,N)$$

$${\beta }_j=\frac{\left|e{A}_j\right|{\omega }_j}{k_j{k}_{B}Tc}(j=1,2,\cdots ,N).$$

$${\alpha }_j=\frac{\left|e{E}_j\right|v_T}{{\omega }_j{k}_{B}T}(j=1,2,\cdots ,N)$$

$${\beta }_j=\frac{\left|e{E}_j\right|}{k_j{k}_{B}T}(j=1,2,\cdots ,N).$$

$${\alpha }_j\ll 1(j=1,2,\cdots ,N)$$

$${\beta }_j\ll 1(j=1,2,\cdots ,N).$$

$$\frac{\partial f_1}{\partial t}+v_x\frac{\partial f_1}{\partial x}=-e\sum _{j=1}^N\frac{E_j}{{\omega }_j}\left[k_j{v}_y\frac{\partial f_0}{\partial p_x}+\left({\omega }_j-k_j{v}_x\right)\frac{\partial f_0}{\partial p_y}\right]$$

Full text

Key words: Vlasov equation, classical plasma, transversal and longitudinal and transversal electric currents, nonlinear analysis.

PACS numbers: 05.20.Dd Kinetic theory, 52.25.Dg Plasma kinetic equations.

1 Introduction

In the present work formulas are deduced for electric current calculation in classical collisionless plasma. At the solution of the kinetic Vlasov equation describing behaviour of classical degenerate plasmas, we consider as in          decomposition distribution functions, and in decomposition of quantity of the self-conjugate electromagnetic field the quantities proportional to square of intensity of an external electric field. In such nonlinear approach it appears that the electric current has two nonzero components. One component of an electric current it is directed along vector potentials of electromagnetic fields. These components of an electric current precisely same, as well as in the linear analysis. It is a "transversal"  current.

Those, in linear approach we receive known expression of a transversal electric current.

The second nonzero an electric current component has the second order of smallness concerning quantities intensity of electric fields. The second electric current component is directed along a wave vector. This current is orthogonal to the first a component. It is a "longitudinal"  current.

Occurrence of a longitudinal current comes to light the spent nonlinear analysis of interaction of electromagnetic fields with plasma.

Nonlinear effects in plasma are studied already long time [1]–[10].

In works [1] and [6] nonlinear effects in plasma are studied. In work [6] nonlinear current was used, in particular, in probability questions decay processes. We will notice, that in work [2] it is underlined existence of nonlinear current along a wave vector (see the formula (2.9) from [2]).

In experimental work [3] the contribution normal field components in a nonlinear superficial current in a signal of the second harmonic is found out. In works [4, 5] generation of a nonlinear superficial current was studied at interaction of a laser impulse with metal.

We will specify in a number of works on plasma, including to the quantum. These are works [11]–[17].

2 The Vlasov equation

Let us demonstrate, that in case of the classical plasma described by kinetic Vlasov equation, the longitudinal current is generated and we will calculate its density. It was specified in existence of this current more half a century ago [1].

Let us consider that the $N$ electromagnetic waves are propagated with strengths

[TABLE]

where $j=1,2,\cdots,N.$

Let us assume that directions of propagation of waves are collinear, that is $\textbf{k}_{1}\parallel\textbf{k}_{2}\parallel\cdots\parallel\textbf{k}_{N}.$

We will consider a case, when the directions electric (and magnetic) fields of waves are collinear $\textbf{E}_{1}\parallel\textbf{E}_{2}\parallel\cdots\parallel\textbf{E}_{N}$, $\left(\textbf{H}_{1}\parallel\textbf{H}_{2}\parallel\cdots\parallel\textbf{H}_{N}\right)$. Corresponding electric and magnetic fields are connected with vector potentials equalities

[TABLE]

We take the Vlasov equation describing behavior of classical collisionless plasma

[TABLE]

In the equation (1.1) $f$ is cumulative distribution function of electrons of plasma, $\textbf{E}_{j},\textbf{H}_{j}(j=1,2,\cdots,N)$ are components of an electromagnetic field,$c$ is the velocity of light, $\textbf{p}=m\textbf{v}$ is momentum of electrons, ${\bf v}$ is the electrons velocity, $f^{(0)}=f_{eq}(\textbf{r},v)$ (eq $\equiv$ equilibrium ) is local equilibrium distribution of Fermi—Dirac

[TABLE]

or, in dimensionless form,

[TABLE]

$\mathcal{E}=mv^{2}/2$ is the electron energy, $\mu$ is the chemical potential of electronic gas, $k_{B}$ is the Boltzmann constant, $T$ is the plasma temperature, $\textbf{P}=\textbf{P}/p_{T}$ is dimensionless momentum of the electrons, $p_{T}=mv_{T}$, $v_{T}$ is the heat electron velocity, $v_{T}=\sqrt{2k_{B}T/m},\alpha=\mu/(k_{B}T)$ is the chemical potential, $k_{B}T=\mathcal{E}_{T}=mv_{T}^{2}/2$ is the heat kinetic electron energy.

Lower local equilibrium distribution of Fermi—Dirac is required to us,

[TABLE]

It is necessary to specify, that vector potential of an electromagnetic field ${\bf A}_{j}({\bf r},t)$ is orthogonal to a wave vector ${\bf k}_{j}$, i.е.

[TABLE]

It means that the wave vector ${\bf k}_{j}$ is orthogonal to electric and magnetic fields

[TABLE]

For definiteness we will consider, that wave vectors N of fields are directed along an axis $x$ and electromagnetic fields are directed along an axis $y$, i.e.

[TABLE]

Therefore

[TABLE]

Let us find a vector product from the equation (1.1)

[TABLE]

then

[TABLE]

We find Lorentz force by means of a vector product

[TABLE]

[TABLE]

Let us notice that

[TABLE]

as

[TABLE]

Now the equation (1.1) is somewhat simplified:

[TABLE]

We will search the solution of equation (1.2) in the form

[TABLE]

Here

[TABLE]

where

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

3 The solution of Vlasov equation in first approximation

In this equation exist $2N$ parameters of dimension of length $\lambda_{j}={v_{T}}/{\omega_{j}}$ ($v_{T}$ is the heat electron velocity) and $l_{j}={1}/{k_{j}}$. We shall believe, that on lengths $\lambda_{j}$,so and on lengths $l_{j}$ energy variable of electrons under acting correspond electric field $A_{j}$ is much less than heat energy of electrons $k_{B}T$ ($k_{B}$ is Boltzmann constant, $T$ is temperature of plasma ), i.e. we shall consider small parameters

[TABLE]

and

[TABLE]

If to use communication of vector potentials electromagnetic fields with strengths of corresponding electric fields, injected small parameters are expressed following equalities

[TABLE]

and

[TABLE]

We will work with a method consecutive approximations, considering, that

[TABLE]

and

[TABLE]

The equation (1.2) by means of (1.3) is equivalent to the following equations

[TABLE]

and

[TABLE]

In the first approximation we search the solution of Vlasov equation in the form

[TABLE]

where $f_{1}$ is the linear combination of vector potentials.

We have the following from the equation (2.1)

[TABLE]

[TABLE]

Let us enter the dimensionless parameters $\Omega_{j}=\dfrac{\omega_{j}}{k_{T}v_{T}}$, $q_{j}=\dfrac{k_{j}}{k_{T}}$, where $q_{j}$ is the dimensionless wave number, $k_{T}=\dfrac{mv_{{}_{T}}}{\hslash}$ is the heat wave number, $\Omega_{j}$ is dimensionless oscillation frequency of vector potential electromagnetic field $\textbf{E}_{j}$.

In the equation (2.3) we will pass to the dimensionless parameters. We obtain the equation

[TABLE]

[TABLE]

Let us notice that $\dfrac{\partial f_{0}}{\partial P_{x}}\sim P_{x}$,   $\dfrac{\partial f_{0}}{\partial P_{y}}\sim P_{y}$.

Then

[TABLE]

Now the equation (2.4) is somewhat simplified

[TABLE]

[TABLE]

From the equation (2.5) we find

[TABLE]

Now from the equation (2.6) we find

[TABLE]

Thus first approximation is defined by equality (2.7).

4 The solution of Vlasov equation in second approximation

In the second approach we search for the decision of Vlasov equation (1.2) in the form of (1.3) in which $f_{2}$ is defined by equality (1.5). We substitute (1.5) in the left-hand member of equation (2.2). We receive the following equation

[TABLE]

[TABLE]

[TABLE]

Let us pass in this equation to the dimensionless parameters and we will enter the following designations

[TABLE]

We receive the equation

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We find from this equation

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

Thus the decision of Wigner equation is constructed and in the second approach. It is defined by equalities (1.5) and (3.2)–(3.3).

5 Density of transversal electric current

The density of electric current according his definition is equal

[TABLE]

The vector of a current density has two nonzero components $\textbf{j}=\left(j_{x},j_{y},0\right)$, where $j_{x}$ is density of transversal current, $j_{y}$ is density of longitudinal current.

Let us calculate density of transversal current. It is defined by the following expression

[TABLE]

Transversal current is directed along an electromagnetic field. Its density is defined according to (4.2) only first approximation of a cumulative distribution function. The second approximation of a cumulative distribution function does not make a contribution to a current density. Thus, in an explicit form transversal current is equal

[TABLE]

We simplify a formula (4.3)

[TABLE]

6 Density of longitudinal electric current

We will investigate longitudinal current. By means of decomposition (1.5) we will present longitudinal current in the following form

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

Here

[TABLE]

In these expressions one-dimensional internal integral on $P_{y}$ is equal to zero and internal integral for $P_{x}$ are calculated piecemeal. Therefore, the previous equalities becomes simpler for components of longitudinal current. Then, internal integral we will integrate on a variable of $P_{y}$. Further we will calculate internal integrals in plane $(P_{y},P_{z})$ in polar coordinates. Equalities (5.2) – (5.4) come down to one-dimensional integral.

[TABLE]

and

[TABLE]

[TABLE]

$\quad b<s,\qquad b,s=1,2,\cdots,N.$

Let us find numerical density the concentration of particles of plasma answering to distribution of Fermi—Dirac

[TABLE]

where

[TABLE]

We will enter plasma (Langmuir) frequency in expression before integrals

[TABLE]

and numerical density (concentration) $N_{0}$. We will express numerical density through a thermal wave number. Then

[TABLE]

Here

[TABLE]

is the dimensionless plasma (Langmuier) frequency, $\sigma_{\rm l,tr}$ is the longitudinal-transversal conductivity,

[TABLE]

Now we will write down components of longitudinal current in form

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Here

[TABLE]

In equalities (5.5) $J_{1},J_{2},\cdots,J_{N},J_{12},J_{21},\cdots,J_{bs},J_{sb}$ are the dimensionless parts of density of longitudinal current. Thus, a longitudinal part of current is equal

[TABLE]

If to enter transversal fields

[TABLE]

then equality (5.6) can be written down in an invarianty form

[TABLE]

Let us consider a case of small values of a wave number. From (5.6) follows that at small values of wave numbers for density of longitudinal current we receive

[TABLE]

Remark

At calculation of the integrals entering the dimensionless parts of density of longitudinal current it is necessary to use Landau’s rule. According to this rule, for example, for integrals $J_{a}$ we will have

[TABLE]

where $a=1,2,\cdots,N.$

7 Conclusions

We found out dependence of transversal and longitudinal current, generated in classical plasma by $N$ transverse electromagnetic waves. In this work we consider the effect of the nonlinear character of the interaction of electromagnetic fields with collisionless Maxwell classical plasma. We considered the Vlasov equation and his solution, the method of successive approximations has been found of the distributions functions. We found formulas for electric current in collisionless classical plasma.