ISSN 1063-780X, Plasma Physics Reports, 2016, Vol. 42, No. 9, pp. 870–875. © Pleiades Publishing, Ltd., 2016. Original Russian Text © V.E. Grishkov, S.A. Uryupin, 2016, published in Fizika Plazmy, 2016, Vol. 42, No. 9, pp. 853–858.

NONLINEAR PHENOMENA

Nonlinear Currents Generated in Plasma by a Radiation Pulse with a Frequency Exceeding the Electron Plasma Frequency V. E. Grishkov and S. A. Uryupin* Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119991 Russia National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia *e-mail: [email protected] Received December 29, 2015; in final form, March 14, 2016

Abstract—It is shown that the nonlinear currents generated in plasma by a radiation pulse with a frequency exceeding the electron plasma frequency change substantially due to a reduction in the effective electron–ion collision frequency. DOI: 10.1134/S1063780X16090038

1. INTRODUCTION The nonlinear currents arising in plasma under the action of a high-frequency field are usually described using the kinetic equation with the Landau collision integral (see, e.g., [1–6]). The use of this collision integral is justified when the field frequency ω is less than or comparable with the electron Langmuir frequency ω p . However, when ω > ω p , the electron motion in a high-frequency field should be described using the collision integral derived by V.P. Silin [7, 8] when developing the kinetic theory of fast-varying processes. As was shown in [7, 8] (see also [9]), the use of a modified collision integral leads to a change in the effective electron–ion collision frequency. If the field frequency is much higher than ω p but lower than v T / rmin , where v T is the electron thermal velocity and rmin is the minimum impact parameter, the effective collision frequency is proportional to Λ(ω) = ln(v T / ω rmin ), rather than to Λ = ln(v T / ω p rmin ) = Λ(ω) + ln(ω/ ω p ) > Λ(ω) . Such a change in the Coulomb logarithm takes place when the amplitude of electron velocity oscillations in a high-frequency field is much smaller than the electron thermal velocity. In other words, the effective collision frequency decreases in proportion to the ratio Λ(ω) / Λ . In calculations with a logarithmic accuracy, the correct result for the conductivity in a field with a frequency ω > ω p is also obtained if one uses the Landau collision integral in which Λ is replaced with Λ(ω). In this model approach, the description of the electron response to the action of the high-frequency field is substantially simpler. It is this approach that is used below to calculate the nonlinear currents generated in plasma by a high-frequency radiation pulse. In

this case, the fast-varying electron motion is considered with the electron–ion collision term proportional to Λ(ω) and the slowly varying electron motion due to the nonlinear action of the field, with a collision integral in which Λ is independent of ω . The main objective of this work is to find low-frequency nonlinear currents arising in plasma under the action of a highfrequency radiation pulse with a carrier frequency exceeding the electron plasma frequency. Such currents form the basis of the theory of generation of a quasi-stationary magnetic field [3, 5], the theory of generation of low-frequency electromagnetic radiation [10, 11], and the theory of excitation of Langmuir waves in plasma [12, 13]. Since the magnitude of the generated electromagnetic fields depends substantially on the magnitude of nonlinear currents, these currents should be calculated with a sufficient accuracy. The corresponding theory of calculation of low-frequency nonlinear currents is set forth below. It is shown that the most substantial changes in the nonlinear currents occur at times exceeding the electron mean free time. At these times, the drag current decreases by a factor of Λ(ω)/ Λ . For Λ(ω)/ Λ @ 1, the vortex current increases more than tenfold. The nonlinear current, which is proportional to the gradient of the isotropic correction to the Maxwellian distribution function that arises due to electron–ion collisions, decreases by a factor of Λ(ω)/ Λ . Finally, if Λ(ω)/ Λ  0 . 3 , then not only the magnitude but also the direction of the current proportional to the gradient of the energy density of the high-frequency field change.

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NONLINEAR CURRENTS GENERATED IN PLASMA

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2. BASIC EQUATIONS

where St(δ f ) is the electron–ion collision integral,

Let us consider the action on plasma of a pulsed field of the form

St(δ f ) = 1 ν(v ) ∂ (v 2δ αβ − v αv β ) ∂ δ f , 2 ∂v α ∂v β

1 E (r , t − z / c )e −iωt +ikz + c.c. L ⊥ m 2 −i ω t + ikz ≡ 1 E Le + c.c., 2

(1)

c 1 − ω2p / ω2 . Here, E L = ( E L,0,0); ω is the carrier frequency; k is the wavenumber; ω p = (4πne 2 / m)1/ 2 ; n is the unperturbed electron density; and e and m are the charge and mass of an electron, respectively. We assume that the amplitude of the pulse envelope varies weakly at a distance of ∼1/k and over a time of ∼1/ω, i.e.,

kL @ 1,

(2)

where L = c mt p and t p is the characteristic pulse duration. As the pulse propagates in the plasma, the amplitude of the field gradually decreases due to small dissipation. Below, we will restrict our analysis to the times shorter than 2ω2 / ω2pν ei (where

ν ei = (4/3) 2πe 4 Z Λ(ω)nm −2v T−3 is the effective electron–ion collision frequency and Z is the ion charge number), when the field dissipation can be neglected. The dispersion spread of the pulse can also be neglected if the pulse propagates over a distance shorter than kL2ω2 / ω2p @ L . To describe electron kinetics, we will use the equation for the slowly varying (over a time of 1/ ω ) correction δ f to the stationary uniform Maxwellian electron distribution function f m = n(2π) −3 / 2v T−3exp(−v 2 /2v T2 ) (see, e.g., [6]),

(

)

∂ + v ⋅ ∂ δ f − St(δ f ) = − e E ⋅ ∂ f 0 m ∂t ∂r ∂v m 2 ⎛ ⎞ ∂ fm ∂ 2 ⎟ ∂f ⎜ + 1 ⎜⎜⎜ ∂ v E ⎟⎟⎟ ⋅ m + 1 + v ⋅ ∂ V ij ⎠ 4 ⎝ ∂r ∂v 8 ∂v i ∂v j ∂t ∂r

(

(

)

The frequency ν(v ) differs from ν(v ) in that it includes Λ(ω) instead of Λ . It should be noted that, for an ultrashort pulse (ω pt p < 1), Λ should be replaced with ln(v T t p / rmin ) . On the right-hand side of Eq. (3), E 0 is the plasma electric field, varying weakly over a time of ∼1/ ω. In a homogeneous plasma, this field arises under the action of the electromagnetic pulse. In this case, the electric field E 0 is proportional to the intensity of incident radiation. In turn, the field E 0 induces the conduction current. This current was described, e.g., in [6] and is not discussed below, although Eq. (3) and its solution (10) involve the field E 0 . In formula (3), k = (0, 0, k ) ,

v E = eE L / mω,

2

2

k vT (6) ν ei , ∂ ln E L @ . ∂t ω Since Eq. (3) was derived under the assumption that the high-frequency field causes small deviations of the electron distribution function from Maxwellian, most electrons should satisfy the condition (7)

In other words, the theory of generation of low-frequency currents, which is set forth below, is applicable when the electromagnetic radiation is not too intense, i.e., when inequality (7) holds. (3)

When solving Eq. (3), we will assume that the term v ⋅ ∂δ f / ∂ r on the left-hand side of Eq. (3) is small and take into account that the tensors Ti , Tij , and Tijs , which have the form

Tij = v iv j − 1 v 2δ ij , 3 2 1 = v iv jv s − v (δ ijv s + δ isv j + δ sjv i ), 5 Ti = v i ,

∂f m 2 ν(v ) − k ⋅ v vE , v ∂v 2ω '

Tijs No. 9

(5)

When deriving Eq. (3), it was assumed that the charge number of plasma ions is high, Z @ 1. Equation (3) disregards the influence of electron–electron collisions on the distribution function, which is justified at times shorter than the reciprocal of the electron–elec−1 tron collision frequency, ν ee = Z / ν . In addition, in Eq. (3) the terms proportional to k 2 are omitted, which is justified if

2

⎡ ⎡ ∂f ⎤ ∂f ⎤ ' × ∂ ⎢k ⋅ v m ⎥ + V ij ∂ ⎢k ⋅ v ν(v ) m ⎥ ∂v i ⎣ ω ∂v j ⎦ ∂v i ⎣ ω ∂v j ⎦

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V ij = v Eiv *Ej + v *Eiv Ej .

v E max(ν ei t p , 1) ! v T2 .

)

⎡' ∂f ⎤ + 1 V ij ∂ ⎢ν(v ) m ⎥ + 1 ∂ V ij (r, t ) 4 ∂v i ⎣ ∂v j ⎦ 4 ∂t

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(4)

'

propagating at the speed c m = c 2k / ω = ∂ω/ ∂ k =

ω t p @ 1,

4 ν(v ) = 4π Ze2 n3 Λ. mv

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(8)

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are eigenvalues of the electron–ion collision operator,

St(Ti ) = −ν(v )Ti , St(Tij ) = − 3ν(v )Tij , St(Tijs ) = − 6ν(v )Tijs .

(9)

⎛ ' ∂f ∂f ⎞ ν(v ) × ⎜ 4 ν(v )v 1/ 4 ∂ ⎡v −1/ 4 m ⎤ − 4 ∂ ⎡v 4 m ⎤ ⎟ ⎢⎣ ⎥⎦ v ∂ v ⎢⎣ ∂ v ⎥⎦ ∂ v ∂ v ⎝ ⎠ t

k × v E (t ' ) − s dt 'exp [− 6ν(v )(t − t ' )]V ij (t ' ) ω

∫

2

As a result, up to terms linear with respect to E 0 and k and small gradient terms, we find from Eq. (3) that

| v (t ) |2 ∂ ⎡ 2 ∂ f m ⎤ δ f (v, r, t ) = 1 E 2 v ∂ v ⎢⎣ ∂ v ⎥⎦ 12 v t 2 v (t ' ) ∂ ⎡ ' 2 ∂f m ⎤ 1 dt ' E 2 + ⎢ν(v )v ⎥ ∂v ⎣ ∂v ⎦ 6 v t0

∫

t0

⎛ ⎡' ⎤⎞ ν(v ) ∂ f m ⎥ ⎟ ν(v ) ⎜ 3 ∂ ⎡ 1 ∂ f m ⎤ ∂ 1 ⎢ × Tijs − , v ⎜⎜ 2 ∂ v ⎢⎣v ∂ v ⎥⎦ ν(v ) ∂ v ⎢ v ∂ v ⎥ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ ⎝ where t 0 is the time at which the field is switched on at the given spatial point. If Λ(ω) and Λ are close to one another (i.e., if ω & ω p ), then the difference between

∂f + 1 V ij (t )Tij 1 ∂ ⎡ 1 m ⎤ ⎢ v ∂ v ⎣v ∂ v ⎥⎦ 8

'

the frequencies ν(v ) and ν(v ) may be neglected. In this case, formula (10) transforms into expression (40) from [6].

t

− 1 dt 'exp [− 3ν(v )(t − t ' )]V ij (t ' ) 4

∫

t0

3. NONLINEAR CURRENT DENSITY

⎛ ⎡' ⎤⎞ ν(v ) ∂ f m ⎥ ⎟ ν(v ) ⎜ 3 ∂ ⎡ 1 ∂ f m ⎤ × Tij − 1 ∂ ⎢ v ⎜⎜ 2 ∂ v ⎢⎣v ∂ v ⎥⎦ ν(v ) ∂ v ⎢ v ∂ v ⎥ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ ⎝

Expression (10) enables one to find the density of the nonlinear current generated by a high-frequency radiation pulse of form (1). Since the envelope of the pulse depends on the argument t − z / c m , it is convenient to pass to the variable τ' = t ' − z / c m when deriving the formula for the current density. When describing the action of a pulse of form (1), integration with respect to t ' in formula (10) is performed from t 0 + z / c m to t ; therefore, the limits of integration with respect to τ' are τ 0 = t 0 and τ = t − z / c m, where t 0 is the time at which the field is switched on at the point z = 0. It should be noted that, in [6], the text in two places following Eq. (43) and preceding Eq. (44) contains a mistake: t 0 should be replaced with t 0 + z / c m . With allowance for the aforesaid, the nonlinear current density is expressed as

t

+ dt 'exp [−ν(v )(t − t ' )]

∫

t0

⎧ ∂f ∂f ⎫ 2 × ⎪⎨− e E 0(t ' ) m + 1 v ⋅ ∂ v E (t ' ) 1 m ⎪⎬ v ∂ v ⎭⎪ ∂v 4 ∂r ⎩⎪ m t

2 − 1 dt ' v ⋅ ∂ v E (t ' ) {1 − exp [−ν(v )(t − t ' )]} 6 ∂r

∫

t0

∂f ⎤ ⎡' × 1 2 ∂ ⎢ν(v )v 2 m ⎥ + 1 dt ' ∂ v ⎦ 20 ν(v )v ∂ v ⎣ t

∫

t0

× {exp [−ν(v )(t − t ' )] − exp [− 3ν(v )(t − t ' )]}

(10)

τ

× v i ∂ ⎡V ij (t ' ) − 2 δ ij v E (t ' ) ⎦ 3 ∂r j ⎣

ji (r⊥ , τ) = e d v v i δ f (v, r⊥ , τ) = 4 π e d τ' 3

t ⎛ ⎡' ⎤⎞ ν(v ) ∂ f m ⎥ ⎟ 1 ∂f m ⎤ ⎡ 3 ∂ ∂ ⎜ 1 1 ⎢ ×v − + dt ' ⎜⎜ 2 ∂ v ⎢⎣v ∂ v ⎥⎦ ν(v ) ∂ v ⎢ v ∂ v ⎥ ⎟⎟ 12 t0 ⎢⎣ ⎥⎦ ⎠ ⎝ × {exp [− 3ν(v )(t − t ' )] − exp [− 6ν(v )(t − t ' )]}

⎧ ⎪ ∂f × dvv 4 ⎨exp [−ν(v )(τ − τ' )] 1 m ∂ v ∂ v ∂ ri 4 ⎪⎩ 0 2 × v E (r⊥ , τ' ) − (1 − exp [−ν(v )(τ − τ' )]) 1 1 2 6 ν(v )v ⎡' ∂f ⎤ 2 × ∂ ⎢ν(v )v 2 m ⎥ ∂ v E (r⊥ , τ' ) ∂v ⎣ ∂ v ⎦ ∂ ri

∫

2⎤

∫

⎛ ⎡' ⎤⎞ ν(v ) ∂ f m ⎥ ⎟ ∂f × Tijs ∂ V ij (t ' ) 1 ⎜ 3 ∂ ⎡ 1 m ⎤ − 1 ∂ ⎢ ∂ rs v ⎜⎜ 2 ∂ v ⎣⎢v ∂ v ⎦⎥ ν(v ) ∂ v ⎢ v ∂ v ⎥ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ ⎝ ∂f k 2 + 1 k ⋅ v 14 ∂ ⎡v 4 m ⎤ v E (t ) + 1 s V ij (t ) 10 ω v ∂ v ⎢⎣ ∂ v ⎥⎦ 4ω t

∂f × Tijs 1 ∂ ⎡ 1 m ⎤ + 1 k ⋅ v dt 'exp [−ν(v )(t − t ' )] ⎢ v ∂ v ⎣v ∂ v ⎥⎦ 10 ω

∫

t0

∫

t0

∞

∫

+ ( exp [−ν(v )(τ − τ ')] − exp [− 3ν(v )(τ − τ' )]) (11) ⎛ ⎡' ⎤⎞ ν(v ) ∂ f m ⎥ ⎟ ∂f × v ⎜ 3 ∂ ⎡1 m ⎤ − 1 ∂ ⎢ 20 ⎜⎜ 2 ∂ v ⎢⎣v ∂ v ⎥⎦ ν(v ) ∂ v ⎢ v ∂ v ⎥ ⎟⎟ ⎢⎣ ⎥⎦ ⎠ ⎝ PLASMA PHYSICS REPORTS

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NONLINEAR CURRENTS GENERATED IN PLASMA 2 × ∂ ⎡V ij (r⊥ , τ' ) − 2 δ ij v E (r⊥ , τ' ) ⎤ ⎦ 3 ∂r j ⎣

k ⎛ ' 1/ 4 ∂ ⎡ − 1/ 4 ∂ f m ⎤ + exp [−ν(v )(τ − τ' )] 1 i ⎜ 4 ν(v )v v 10 ω ⎝ ∂ v ⎢⎣ ∂ v ⎥⎦ ⎫ ν(v ) ∂ ⎡ 4 ∂ f m ⎤ ⎞ 2⎪ v − 4 ⎟ v E (r⊥ , τ' ) ⎬ . v ∂ v ⎢⎣ ∂ v ⎥⎦ ⎠ ⎪⎭

Let us first consider expression (11) at times τ − τ 0 exceeding the mean free time of thermal electrons, when ν(τ − τ 0 ) @ 1, where ν = ν(v T ). If the pulse duration t p significantly exceeds 1 / ν , then, for large times, we find from Eq. (11) that

873

increase results from the reduction in the gradient of the tensor correction for the distribution function due to inverse bremsstrahlung. The increase is especially '

large (up to 12-fold) when ν ! ν . The increase in this part of the nonlinear current may also be inferred when the form of the field is not detailed, but ω @ ω p . In other words, in the expressions for the vortex nonlinear current, the factor 16/15 (see [1–6]) should be replaced with the expression in the parentheses in front of ∂ V ij / ∂ r j . It should be noted that the increase in the nonlinear vortex current intensifies the generation of the quasi-stationary magnetic field (see [5] for details). Fourth, the factor in the parentheses in front of ∂ v E (r⊥ , τ) / ∂ ri (the third term in expression (12)) changes. The form of this term resembles the Miller force. However, there is a qualitative difference. In ' particular, if ν = ν , this contribution to the current is directed along the energy density gradient of the high2

'

τ

ν 2 ji (r⊥ , τ) = − 5 en d τ' ∂ v E (r⊥ , τ' ) ∂ ri 6ν

∫

τ0

⎛ ⎞ ' + ⎜ 64 − ν 176 ⎟ ⎜ 5 ν 15 ⎟ ⎝ ⎠ ⎛ ⎞ ' + ⎜ 2272 ν − 248 ⎟ ⎜ 45 ν 15 ⎟ ⎝ ⎠

1 en ∂ V (r , τ) ij ⊥ 2π ν ∂ r j

(12)

1 en ∂ v (r , τ) 2 E ⊥ 2π ν ∂ ri

' k 2 + 17 ν en i v E (r⊥ , τ) , 10 ν ω

Z @ ν(τ − τ 0 ) @ 1,

ν t p @ 1.

In this expression, the inequality ν(τ − τ 0 ) ! Z appeared because the electron–electron collision integral in Eq. (3) is omitted. First of all, it should be '

noted that, if ν = ν , then expression (12) passes to ' expression (44) from [6]. At ν ≠ ν , there are some distinctions. First, the drag current described by the last term in formula (12) is smaller than that presented in ' [6] by a factor of ν / ν . A similar reduction in the drag current also occurs when the explicit form of the field is not detailed. In other words, the corresponding '

expression in [4] should be reduced by a factor of ν / ν . The reduction in the density of the drag current is caused by the corresponding reduction in the momentum transferred to electrons as they collide with ions. Second, the first term in Eq. (12), which describes the contribution to the nonlinear current from the gradient of the correction for the isotropic part of the distribution function due to the absorption of high-frequency electromagnetic field, is also reduced by a fac'

tor of ν / ν . This reduction is caused by the less efficient absorption of the field with the frequency ω > ω p . Third, the factor in front of the term proportional to the derivative of the tensor V ij increases. This PLASMA PHYSICS REPORTS

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'

frequency field. At ν ≥ 0 . 3ν, the direction changes due to nonuniform electron heating caused by inverse bremsstrahlung of the high-frequency field. To conclude the discussion of expression (12), it should be noted that the first term, which contains the gradient of the field energy, changes over a time exceeding the reciprocal frequency of electron–electron collisions due to their influence on the form of the small correction to the electron distribution function [4]. Without specifying the form of the high-frequency field and for '

ν = ν , the nonlinear current arising at such times due to the nonuniformity of the isotropic correction to the distribution function was described in [4]. When '

ν < ν , i.e., at ω > ω p , the corresponding result of [4] '

should be reduced by a factor of ν / ν . When deriving expression (12), it was assumed that the characteristic time of variation in the pulsed field is greater than the electron mean free time. If the pulse is short and ν t p ! 1, then, at ν(τ − τ 0 ) @ 1, expression (12) is reduced to the approximate expression '

τ∼t p

ν ji (r⊥ , τ) = − 5 en 6ν

∫

τ0

2 d τ' ∂ v E (r⊥ , τ' ) , ∂ ri

Z @ ν(τ − τ 0 ) @ 1,

(13)

ν t p ! 1,

which describes the nonlinear current after the action of the pulse terminates. Expression (13) differs from the expression obtained in [6] in the presence of the '

additional factor ν / ν .

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GRISHKOV, URYUPIN

If the time τ − τ 0 is shorter than the electron mean free time, we obtain from Eq. (11) τ

2 ji (r⊥ , τ) = − 1 ene d τ' ∂ v E (r⊥ , τ' ) 4 ∂ ri

∫

τ0

τ

+

1

30 2π

2 eneν d τ'(τ − τ' ) ∂ v(r⊥ , τ' ) ∂ ri

∫

τ0

τ ⎛ ' ⎞ ν 1 1 1 ⎜ ⎟ − − eneν d τ'(τ − τ' ) ∂ V ij (r⊥ , τ' ) (14) ∂r j ⎜ 3 ν 5 ⎟ 2π τ0 ⎝ ⎠

∫

τ ⎛ ' ⎞ ki 2 ν 1 1 1 ⎜ ⎟ + + eneν d τ' v E (r⊥ , τ' ) , ω ⎜ 3 ν 5 ⎟ 2π τ0 ⎝ ⎠ ν(τ − τ 0 ) ! 1 .

∫

Earlier, a similar expression for the nonlinear current at such times was derived in [6] without allowance for '

the difference between the frequencies ν and ν . The first term in expression (14) coincides with the expression earlier obtained in [6], while the second is smaller by a factor of 4. It should be noted that the first term in formula (14) arises due to the ponderomotive action of the high-frequency field and agrees with the results of [14–16]. The third term in formula (14) differs from that obtained in [6]: instead of the factor 2/15, it contains an expression in parentheses, which vanishes '

'

at ν = 0 . 6ν and changes the sign at ν < 0 . 6ν . The difference of the fourth term from that in [6] is less dramatic: instead of the factor 8/15, there is an expres'

sion of the form (ν /3ν + 1/5). In this case, the maxi' mum quantitative difference is reached at ν ! ν and corresponds to the reduction in the contribution to the current by a factor of 8/3. It should be noted that, at short times, when ν(τ − τ 0 ) ! 1, the reduction in the influence of collisions on the action of the field with the frequency ω > ω p manifests itself only in the relatively small terms of expression (14) corresponding to the drag current and the current proportional to ∂ V ij / ∂ r j . For short pulses, ν t p ! 1, the term containing ∂ V ij / ∂ r j is smaller than the first term of expression (14) by a factor of [ν min(t p , τ − τ 0 )]−1 @ 1, whereas for a pulse with a duration longer or comparable with the reciprocal of the electron–ion collision frequency, it is smaller by a factor of [ν(τ − τ 0 )]−1 @ 1. The last term in expression (14) is smaller than the first term by a factor of (ν t p ) −1 @ 1.

4. CONCLUSIONS The new regularities in the generation of nonlinear currents occur due to the difference between the effective electron–ion collision frequencies when considering fast- and slow-varying electrons motions. In terms of the above model description of electron collisions, this difference is determined by the difference between the logarithms Λ and Λ(ω). Let us present a few examples when this difference can be relatively large. If the duration of the incident pulse is not anomalously short, i.e., if ω pt p @ 1, then Λ is estimated by the well-known expression

(

)

⎛ v κT ⎞ T [eV] Λ = ln ⎜⎜ T 2 ⎟⎟ = 3 ln 10 ⎝ ω p Ze ⎠ 2 (15) −3 ⎛ n [cm ]⎞ 1 Z − ln ⎜ − ln + 5 . 55, 2 ⎝ 1017 ⎟⎠ 2 where κ is the Boltzmann constant and T is the electron temperature. In turn, for Λ(ω), we have

()

(

)

⎛ ⎞ T [eV] Λ(ω) = Λ − ln ⎜ ω ⎟ = 3 ln 10 ⎝ωp ⎠ 2 −1 ⎛ ω [s ]⎞ − ln ⎜ − ln Z + 1 . 53 . 15 ⎟ 2 ⎝ 10 ⎠

()

(16)

From these relationships, for ω = 1 . 8 × 1015 s–1, T = 100 eV, Z = 4 , and n = 1017 cm–3, we obtain ' Λ  8 . 3 and Λ(ω)  3 . 7 , i.e., ν / ν  0 . 45 . Strictly speaking, for estimates with Z = 4 , the range of applicability of some relationships (see Eqs. (12), (13)) proves to be relatively narrow. However, to demon'

strate the difference between the frequencies ν and ν , this choice is not critical. The matter is that, under the ' discussed conditions, the ratio ν / ν depends weakly ' on Z . For example, for Z = 10, we have ν / ν  0 . 40 , which is close to the value of 0 . 45 obtained for Z = 4 . As the plasma density decreases, the difference ' between ν and ν increases. In particular, for n = 13 –3 10 cm and the same values of T = 100 eV, Z = 4 , and ω = 1 . 8 × 1015 s–1, we find from Eqs. (15) and (16) that the value of Λ(ω) is the same and Λ  13 , i.e., we ' have ν / ν  0 . 28 . The ratio Λ(ω)/ Λ also decreases with increasing frequency. However, in the framework of our consideration, the frequency ω should be much lower than

( )(

)

v T κT T [eV] (17) = 2 . 3 × 1015 s −1 2 . 2 Z 10 Ze Hence, for T = 100 eV and Z = 4 , we have the restriction ω  3 . 6 × 1016 s–1. According to relationships (15) and (16) and the above estimates, the difference PLASMA PHYSICS REPORTS

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between Λ(ω) and Λ (or ν and ν ), which underlies our description of nonlinear currents, should play an important role in the interaction of high-frequency radiation with an underdense plasma. In contrast, if the radiation frequency is lower than or close to the electron Langmuir frequency, i.e., the plasma density is higher than or close to the critical one, then we have ' Λ(ω) ∼ Λ and ν ∼ ν . In this case, the generation of nonlinear currents can be described by the theory developed in [2–6]. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research and the Presidium of the Russian Academy of Sciences. REFERENCES 1. V. I. Perel’ and Ya. M. Pinskii, Sov. Phys. JETP 27, 1014 (1968). 2. I. B. Bernstein, C. E. Max, and J. J. Thomson, Phys. Fluids 21, 905 (1978). 3. I. P. Shkarofsky, Phys. Fluids 23, 52 (1980). 4. K. N. Ovchinnikov, V. P. Silin, and S. A. Uryupin, Sov. Phys. Lebedev Inst. Rep., No. 2, 19 (1992).

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Translated by E. Chernokozhin