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

0.15 MB·

by  A. V. Latyshev

The occurrence of transverse and longitudinal electric currents in
the classical plasma under the action of N transverse
electromagnetic waves
A. V. Latyshev1 and V. I. Askerova2
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Faculty of Physics and Mathematics,

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a Moscow State Regional University, 105005,
J

1 Moscow, Radio str., 10-A
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p
Abstract
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m
Classical plasma with arbitrary degree of degeneration of electronic gas is consi-
s
a dered. In plasma N (N > 2) collinear electromagnatic waves are propagated. It
l
p
is required to find the response of plasma to these waves. Distribution function
.
s
c in square-law approximation on quantities of two small parameters from Vlasov
i
s equation is received. The formula for electric current calculation is deduced. It is
y
h demonstrated that the nonlinearity account leads to occurrence of the longitudinal
p
electric current directed along a wave vector. This longitudinal current is orthogonal
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to the known transversal current received at the linear analysis. The case of small
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v values of wave number is considered.
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Key words: Vlasov equation, classical plasma, transversal and longitudinal and
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0 transversal electric currents, nonlinear analysis.
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0 PACS numbers: 05.20.Dd Kinetic theory, 52.25.Dg Plasma kinetic equations.
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0
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1 Introduction
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:
v
i
X
In the present work formulas are deduced for electric current calculation
r
a
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
1
[email protected]
[email protected]
2
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.Wewill notice,that inwork[2]it isunderlinedexistenceof 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
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strengths
E = E ei(kjr−ωjt), H = H ei(kjr−ωjt),
j 0j j 0j
where j = 1,2,··· ,N.
Let us assume that directions of propagation of waves are collinear, that
is k k k k ··· k k .
1 2 N
We will consider a case, when the directions electric (and magnetic) fields
of waves are collinear E k E k ··· k E , (H k H k ··· k H ). Correspon-
1 2 N 1 2 N
ding electricandmagneticfields are connectedwithvectorpotentialsequalities
1∂A iω
j j
E = − = A , H = rot A , j = 1,2,··· ,N.
j j j j
c ∂t c
We take the Vlasov equation describing behavior of classical collisionless
plasma
N N
∂f ∂f 1 ∂f
+ v + e E + v, H = 0. (1.1)
j j
∂t ∂r c ∂p
" #!
j=1 j=1
X X
In the equation (1.1) f is cumulative distribution function of electrons
of plasma, E ,H (j = 1,2,··· ,N) are components of an electromagnetic
j j
field,c is the velocity of light, p = mv is momentum of electrons, v is the
electrons velocity, f(0) = f (r,v) (eq ≡ equilibrium ) is local equilibrium
eq
distribution of Fermi—Dirac
1
f (r,v) = ,
eq E − µ(r)
1 + exp
k T
B
(cid:18) (cid:19)
or, in dimensionless form,
1
f (r,v) = = f (r,P),
eq eq
1 + exp(P2 − α(r))
E = mv2/2 is the electron energy, µ is the chemical potential of electronic
gas, k is the Boltzmann constant, T is the plasma temperature, P = P/p is
B T
dimensionless momentum of the electrons, p = mv , v is the heat electron
T T T
velocity, v = 2k T/m,α = µ/(k T) is the chemical potential, k T =
T B B B
E = mv2/2 is the heat kinetic electron energy.
T T p
Lower local equilibrium distribution of Fermi—Dirac is required to us,
E −1
− µ
f (v) = 1 + exp = 1 + exp P2 − α −1 = f (P).
0 0
k T
B
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:2) (cid:0) (cid:1)(cid:3)
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It is necessary to specify, that vector potential of an electromagnetic field
A r k
( ,t) is orthogonal to a wave vector , i.е.
j j
k A r
· ( ,t) = 0, j = 1,2,··· ,N.
j j
k
It means that the wave vector is orthogonal to electric and magnetic fields
j
k E r k H r
· ( ,t) = · ( ,t) = 0, j = 1,2,··· ,N.
j j j j
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.
k E
= k (1,0,0), = E (x,t)(0,1,0).
j j j j
Therefore
1∂A iω ck
j j j
E = − = A , H = E · (0,0,1), j = 1,2,··· ,N.
j j j j
c ∂t c ω
Let us find a vector product from the equation (1.1)
ck
j
[v,H ] = E (v ,−v ,0),
j j y x
ω
j
then
N N
ck
j
v, H = E (v ,−v ,0).
j j y x
ω
" # j
j=1 j=1
X X
We find Lorentz force by means of a vector product
1 ∂f
e E + [v,H ] =
j j
c ∂p
(cid:18) (cid:19)
e ∂f ∂f
= E k v + (ω − k v ) , (j = 1,2,··· ,N).
j j y j j x
ω ∂p ∂p
j x y
(cid:20) (cid:21)
Let us notice that
∂f
0
[v,H ] = 0,
j
∂p
as
∂f
0
∼ v.
∂p
Now the equation (1.1) is somewhat simplified:
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N
∂f ∂f E ∂f ∂f
j
+ v + e k v + (ω − k v ) = 0. (1.2)
x j y j j x
∂t ∂x ω ∂p ∂p
j x y
j=1 (cid:20) (cid:21)
X
We will search the solution of equation (1.2) in the form
f = f (P) + f + f . (1.3)
0 1 2
Here
N
f = E ϕ + E ϕ + ··· + E ϕ = E ϕ , (1.4)
1 1 1 2 2 N N j j
j=1
X
where
E ∼ ei(kjx−ωt),
j
and
N N
f = E2ψ + E E ξ , (1.5)
2 j j b s b,s
j=1 b,s=1
X X
b<s
where
E2 ∼ e2i(kjx−ωjt),
j
E E ∼ ei[(kb+ks)x−(ωb+ωs)t].
b s
3 The solution of Vlasov equation in first
approximation
In this equation exist 2N parameters of dimension of length λ = v /ω
j T j
(v is the heat electron velocity) and l = 1/k . We shall believe, that on
T j j
lengths λ ,so and on lengths l energy variable of electrons under acting
j j
correspond electric field A is much less than heat energy of electrons k T
j B
(k is Boltzmannconstant, T is temperature of plasma ), i.e.we shall consider
B
small parameters
|eA |v
j T
α = (j = 1,2,··· ,N)
j
ck T
B
and
|eA |ω
j j
β = (j = 1,2,··· ,N).
j
k k Tc
j B
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If to use communication of vector potentials electromagnetic fields with
strengths of corresponding electricfields, injectedsmallparametersare expressed
following equalities
|eE |v
j T
α = (j = 1,2,··· ,N)
j
ω k T
j B
and
|eE |
j
β = (j = 1,2,··· ,N).
j
k k T
j B
We will work with a method consecutive approximations, considering, that
α ≪ 1 (j = 1,2,··· ,N)
j
and
β ≪ 1 (j = 1,2,··· ,N).
j
The equation (1.2) by means of (1.3) is equivalent to the following equations
N
∂f ∂f E ∂f ∂f
1 1 j 0 0
+ v = −e k v + (ω − k v ) (2.1)
x j y j j x
∂t ∂x ω ∂p ∂p
j x y
j=1 (cid:20) (cid:21)
X
and
N
∂f ∂f E ∂f ∂f
2 2 j 1 1
+ v = −e k v + (ω − k v ) . (2.2)
x j y j j x
∂t ∂x ω ∂p ∂p
j x y
j=1 (cid:20) (cid:21)
X
In the first approximation we search the solution of Vlasov equation in
the form
f = f(1) = f (P) + f ,
0 1
where f is the linear combination of vector potentials.
1
We have the following from the equation (2.1)
[E (iω + ik v )ϕ + E (iω + ik v )ϕ + ··· + E (iω + ik v )ϕ ] =
1 1 1 x 1 2 2 2 x 2 N N N x N
N
E ∂f ∂f
j 0 0
= −e k v + (ω − k v ) . (2.3)
j y j j x
ω ∂p ∂p
j x y
j=1 (cid:20) (cid:21)
X
ω k
j j
Let us enter the dimensionless parameters Ω = , q = , where q is
j j j
k v k
T T T
mv
the dimensionless wave number, k = T is the heat wave number, Ω is
T ℏ j
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dimensionless oscillation frequency of vector potential electromagnetic field
E .
j
In the equation (2.3) we will pass to the dimensionless parameters. We
obtain the equation
[E (q P − Ω )ϕ + E (q P − Ω )ϕ + ··· + E (q P − Ω )ϕ ] =
1 1 x 1 1 2 2 x 2 2 N N x N N
N
E ∂f ∂f
j 0 0
= −e q P + (Ω − q P ) . (2.4)
j y j j x
ω ∂P ∂P
j x y
j=1 (cid:20) (cid:21)
X
∂f ∂f
0 0
Let us notice that ∼ P , ∼ P .
x y
∂P ∂P
x y
Then
∂f ∂f ∂f
0 0 0
q P + (Ω − q P ) = Ω .
j y j j x j
∂P ∂P ∂P
x y y
(cid:20) (cid:21)
Now the equation (2.4) is somewhat simplified
[E (q P − Ω )ϕ + E (q P − Ω )ϕ + ··· + E (q P − Ω )ϕ ] =
1 1 x 1 1 2 2 x 2 2 N N x N N
N
e ∂f
0
= − E . (2.5)
j
k p v ∂P
T T T y
j=1
X
From the equation (2.5) we find
ie ∂f /∂P
0 y
ϕ = · , j = 1,2,··· ,N. (2.6)
j
k p v q P − Ω
T T T j x j
Now from the equation (2.6) we find
N
e E
j
f = · . (2.7)
1
k p v q P − Ω
T T T j x j
j=1
X
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 is defined by equality (1.5). We substitute
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(1.5) in the left-hand member of equation (2.2). We receive the following
equation
N
E2(−2iω + 2ik v )ψ +
j j j x j
j=1
X
N
+ E E (−i(ω + ω ) + i(k + k )v )ξ =
b s b s b s x b,s
b,s=1
X
b<s
N
E ∂f ∂f
j 0 0
= −e k v + (ω − k v ) . (3.1)
j y j j x
ω ∂p ∂p
j x y
j=1 (cid:20) (cid:21)
X
Let us pass in this equation to the dimensionless parameters and we will
enter the following designations
q + q Ω + Ω
b s b s
q = , Ω = .
bs bs
2 2
We receive the equation
N N
E2(q P − Ω )ψ + E E (qP − Ω)ξ =
j j x j j b s x b,s
j=1 b,s=1
X X
b<s
e2 N E2 ∂ ∂f /∂P ∂2f
j 0 y 0
= − q P − +
j y
2k2p2v2 Ω ∂P q P − Ω ∂P2
T T T (j=1 j (cid:20) x (cid:18) j x j(cid:19) y (cid:21)
X
N
E E ∂ ∂f /∂P Ω − q P ∂2f
b s 0 y b b x 0
+ q P + +
b y
Ω ∂P q P − Ω q P − Ω ∂P2
b,s=1 b (cid:20) x (cid:18) s x s(cid:19) s x s y (cid:21)
X
b<s
N
E E ∂ ∂f /∂P Ω − q P ∂2f
s b 0 y s s x 0
+ q P + .
s y
Ω ∂P q P − Ω q P − Ω ∂P2
bX,s=1 s (cid:20) x (cid:18) b x b(cid:19) b x b y (cid:21)
b<s
We find from this equation 


e2 Ξ (P)
j
ψ = − , (j = 1,2,··· ,N) (3.2)
j
2k2p2v2Ω q P − Ω
T T T j j x j
and
e2 1 Ξ (P) 1 Ξ (P)
bs sb
ξ = − + + , (3.3)
b,s
2k2p2v2 Ω qP − Ω Ω qP − Ω
T T T (cid:20) b x s x (cid:21)
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where
b < s, j = 1,2,··· ,N.
and
∂ ∂f /∂P ∂2f
0 y 0
Ξ (P) = q P − ,
j j y
∂P q P − Ω ∂P2
x (cid:18) j x j(cid:19) y
∂ ∂f /∂P Ω − q P ∂2f
0 y b b x 0
Ξ (P) = q P + ,
bs b y
∂P q P − Ω q P − Ω ∂P2
x (cid:18) s x s(cid:19) s x s y
∂ ∂f /∂P Ω − q P ∂2f
0 y s s x 0
Ξ (P) = q P + ,
sb s y
∂P q P − Ω q P − Ω ∂P2
x (cid:18) b x b(cid:19) b x b y
where
b < s, b,s = 1,2,··· ,N.
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
2d3p 2p2v
j = e vf = T T fPd3P. (4.1)
ℏ 3 ℏ 3
(2π ) (2π )
Z Z
The vector of a current density has two nonzero components j = (j ,j ,0),
x y
where j is density of transversal current, j is density of longitudinal current.
x y
Let us calculatedensity of transversal current. It is defined by the following
expression
2d3P 2d3p 2p2v
j = e v f = e v f = T T f P d3P. (4.2)
y y ℏ 3 y 1 ℏ 3 ℏ 3 1 y
(2π ) (2π ) (2π )
Z Z Z
Transversal current is directed along an electromagnetic field. Its density is
defined accordingto (4.2) onlyfirstapproximationof a cumulativedistribution
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
N
2ie2p2 E ∂f
j = T j 0P d3P. (4.3)
y y
(2πℏ)3k q P − Ω ∂P
T Z j=1 j x j y
X
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We simplify a formula (4.3)
∞
N
2ie2p2 E
j = T j ln(1 + eα−Px2)dP . (4.4)
y x
(2πℏ)3k q P − Ω
T Z j=1 j x j
−∞ X
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
N N N
j = j + j + j , (5.1)
x a bs sb
a=1 b,s=1 b,s=1
X X X
b<s b<s
where
e3p E2 Ξ (P)P d3P
T a a x
j = , (a = 1,2,··· ,N), (5.2)
a (2π~)3k2v Ω q P − Ω
T T a Z a x a
and
e3p E E Ξ P
j = T b s bs P d3P, (5.3)
bs (2π~)3k2v Ω qP − Ω x
T T Z b x
e3p E E Ξ P
j = T b s sb P d3P. (5.4)
sb (2π~)3k2v Ω qP − Ω x
T T Z s x
Here
q + q Ω + Ω
b s b s
q = , Ω = , b < s, b,s = 1,2,··· ,N.
bs bs
2 2
In these expressions one-dimensional internal integral on P is equal to
y
zero and internal integral for P are calculated piecemeal. Therefore, the
x
previous equalities becomes simpler for components of longitudinal current.
Then, internal integral we will integrate on a variable of P . Further we will
y
calculate internal integrals in plane (P ,P ) in polar coordinates. Equalities
y z
(5.2) – (5.4) come down to one-dimensional integral.
πe3p E2q ∞ ln 1 + eα−Px2 dPx
T a a
j = , (a = 1,2,··· ,N).
a (2π~)3k2v (cid:16)(q P − Ω(cid:17))3
T T Z a x a
−∞