Kinetic Study of the Linear Stage of Thermocurrent. Instability in the Framework of the Boltzmann Equation for a Model Gas. N. A. Dyatko, I. V. Kochetov, and ...
ISSN 1063780X, Plasma Physics Reports, 2011, Vol. 37, No. 6, pp. 528–534. © Pleiades Publishing, Ltd., 2011. Original Russian Text © N.A. Dyatko, I.V. Kochetov, A.P. Napartovich, 2011, published in Fizika Plazmy, 2011, Vol. 37, No. 6, pp. 567–573.
PLASMA INSTABILITIES
Kinetic Study of the Linear Stage of Thermocurrent Instability in the Framework of the Boltzmann Equation for a Model Gas N. A. Dyatko, I. V. Kochetov, and A. P. Napartovich Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow oblast, 142190 Russia Received November 11, 2010
Abstract—The linear stage of thermocurrent instability is investigated for a model gas in which the integral of inelastic collisions of electrons with gas particles has a divergent form and the frequencies of elastic and inelastic collisions are independent of the electron velocity. The proposed approach consists in the reduction of the Boltzmann equation for electrons in an inhomogeneous plasma to a set of equations for the moments of the electron velocity distribution function. The instability growth rate and the wave phase velocity as func tions of the perturbation wavenumber are calculated, the maximum growth rate and the corresponding wave number are determined, and the dependence of these quantities on the degree of plasma quasineutrality is examined. It is demonstrated that the model satisfactorily (both qualitatively and quantitatively) describes the linear stage of thermocurrent instability in helium. DOI: 10.1134/S1063780X11050047
1.INTRODUCTION The effect of thermocurrent instability can be explained qualitatively as follows. Let us consider a weakly ionized quasineutral plasma in a homogeneous electric field, assuming that the electron mobility is much higher than the ion mobility (this assumption is valid in most cases) and the typical spatial scale of inhomogeneity of the plasma parameters is much larger than the electron energy relaxation length λ u (in this case, the diffusion fluxes are negligibly small as compared to the drift flux). Then, in the first approxi mation, the relationship between the electric field strength E and the electron density ne can be written as j e (x) = eneµ e E = const,
To the best of our knowledge, such instability was considered for the first time in [1], where the set of continuity equations for electrons and ions,
(1)
where je is the current density, e is the electron charge, µ e is the electron mobility, and x is the coordinate along the electric field. Let us assume for simplicity that µ e = const . If a longitudinal inhomogeneity of the electron density (e.g., a hump in the electron den sity, see Fig. 1) arises in plasma, then, according to Eq. (1), the change in the electric field will have the opposite sign (the field will decrease) and the electron temperature will also change accordingly (decrease). In this case, the thermodiffusion electron flux caused by the gradient of the electron temperature Te, jTD ~ ∂Te (x)/ ∂x , is directed opposite to the diffusion flux j D ∼ ∂ne (x)/ ∂x . If jTD > j D , then the inhomoge neity will grow. 528
ne(x) n0 E(x)
∂ne ∂ 2 − ( Dene ) − ∂ (Wene ) = 0, ∂t ∂x 2 ∂x
(2)
∂ni ∂ + (Wi ni ) = 0 ∂t ∂x
(3)
Diffusion
∂n e ( x ) J D ∼ ∂x
∂T e ( x ) Thermodiffusion J TD ∼ ∂x
E0 Te(x) T0
x
Fig. 1. Diagram illustrating the development of ther mocurrent instability via the enhancement of the electron density perturbation due to the thermodiffusion effect.
KINETIC STUDY OF THE LINEAR STAGE
was studied in the longwavelength limit k −1 Ⰷ λ u (where k is the perturbation wavenumber) under the assumption of plasma quasineutrality. In Eqs. (2) and (3), De, We, Wi, ni, and t are the electron diffusion coefficient, electron drift velocity, ion drift velocity, ion density, and time, respectively. The electron diffu sion coefficient and the drift velocity were assumed to be known functions of the local field. It was shown that instability develops if the following condition is satis fied: ∂ ⎛ De ⎞ > 0. (4) ⎜ ⎟ ∂ E ⎝W e ⎠ It follows from the results of [1] that, under the condi tion k −1 Ⰷ λ u , the instability growth rate Γ is propor tional to the perturbation wavenumber squared, k2. The author of [1] did not take into account the renormalization of the longitudinal diffusion coeffi cient caused by the electron density inhomogeneity in plasma [2, 3]. Later, an expression for the electron flux along the electric field was derived in [4] with allow ance for the renormalization caused by the electron density inhomogeneity and spatiotemporal variations in the electric field. In the framework of the approach proposed in [4], a more general criterion for the onset of thermocurrent instability in the limit k −1 Ⰷ λ u was obtained. Calculations of the corresponding renor malization coefficients for different gases showed that thermocurrent instability can occur in helium, nitro gen, and carbon dioxide [4–6]. It was also shown that violation of plasma quasineutrality stabilizes perturba tions [4]. Coulomb collisions and superelastic colli sions with vibrationally excited molecules also have a stabilizing effect [5, 6]. Moreover, it was shown in [7] that application of a magnetic field across the electric field appreciably affects the range of E/N values (where N is the number density of gas atoms and mol ecules) at which this instability exists. In [8, 9], thermocurrent instability was studied using a set of hydrodynamic equations, including the balance equation for the mean electron energy. How ever, this approach is applicable only if the electron energy distribution function is Maxwellian. In a weakly ionized gasdischarge plasma, the electron dis tribution, as a rule, appreciably differs from Max wellian and the application of such an approach is unjustified. Thermocurrent instability cannot be comprehen sively described in the framework of transport equa tions only [1, 4], because, in the approximation of plasma quasineutrality, they result in an unbounded increase in the instability growth rate with decreasing perturbation wavelength, Γ ~ k2. Therefore, perturba tions growing at the fastest rate should be described in the framework of kinetic theory. For the first time, the kinetic approach was applied to study the linear stage of thermocurrent instability in a model gas in [10], PLASMA PHYSICS REPORTS
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where the Boltzmann equation for the electron veloc ity distribution function in an inhomogeneous plasma was reduced to an infinite set of equations for the moments of the distribution function. It was shown that the growth rate has a maximum at k −1 ~ λ u . Then, in [11], an inhomogeneous equation for perturbations of the electron distribution function in helium was solved numerically. It was found that the calculated instability growth rate for helium agreed rather well with the results obtained in [10] for a model gas. Note that, in recent papers [12, 13], an attempt was made to describe the nonlinear stage of thermocurrent instabil ity. However, these studies were performed in the framework of the transport equations, which are valid at k −1 Ⰷ λ u , while the fastest perturbations with k −1 ∼ λ u will develop first of all [10, 11]. Paper [10] is in fact a conference paper abstract in which both theoretical and experimental results were only briefly reported. In particular, the theoretical model was not described. Below, we present a detailed description of both the theoretical method applied in [10] to study the linear stage of thermocurrent insta bility and the results obtained. A novel (as compared to [10, 11]) result is calculation of the instability growth rate with allowance for charge separation. 2. MODEL Let us first consider a homogeneous timeindepen dent Boltzmann equation for the spherically symmet ric part of the electron velocity distribution function ( F00 ) in the electric field [1, 14], e 2 E 02 ∂ ⎛ v 2 ∂F00 ⎞ 0 (5) ⎜ ⎟ + C 0(F0 ) = 0, 3m 2v 2 ∂v ⎝ ν m ∂v ⎠ where v is the electron velocity, m is the electron mass, E0 is the homogeneous electric field, and ν m is the transport frequency of electron–neutral collisions. We will study a model gas in which the electron–neutral collision integral has the form
(6) C0(F00 ) = 12 ∂ ν uv 3F00 , v ∂v where ν u is the frequency of electron energy losses. The collision integral has form (6) if electrons lose their energy mainly due to elastic collisions. We assume that the frequencies ν u and ν m are independent of the electron velocity,
(
ν m(v) = const,
)
ν u(v) = const.
The distribution function tron density,
F00
(7)
is normalized to the elec
∞
∫
4π v F0 dv = ne . 0
2
0
0
(8)
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The solution to Eq. (5) is the Maxwellian distribution function ⎛ 2⎞ 2e 2 E 2 (9) F00(v) = C exp ⎜ − v 2 ⎟ , v 02 = 2 0 , 3m ν mν u ⎝ v0 ⎠ where the coefficient С is found from normalization condition (8),
()
v0 3 3/2 3 (10) Γ = C π v 0. 2 2 Passing from the velocity distribution to the electron energy distribution, we obtain 3
ne = C 4π 0
⎛ ⎞ 0 0 F0 (u) ~ exp ⎜ − u0 ⎟ , u = mv , Te = 2 ⎝ Te ⎠ 2
mv 02 2
,
(11)
where Te0 is the electron temperature. The electron drift velocity and diffusion coefficient are ∞
0 We
3 eE ∂ F0 eE 0 = μ e E 0 = − 1 4π 0 v dv = ne 3 m ν m ∂ v mν m
the instabiity criterion [1], which is applicable in this case, is always satisfied,
⎛ De0 ⎞' ⎛ eE 0 ⎞' e ⎜ 0⎟ = ⎜ ⎟ = 3mν ν > 0. ν ν 3 m W ⎝ ⎠ ⎝ e ⎠ E0 m u E0 m u
The corresponding expression for the instability growth rate in the limit k Ⰶ λ u−1 is [1] 0
Γ ≈ W i 0λ u k 2 =
(12)
and ∞
4 v0 Te 0 0 , = De = 1 4π v F0 dv = 2ν m ne 3 ν m mν m 2
∫
0
(13)
0
respectively. The upper index “0” in F00 , ne0, W e0 , and De0 refers to steadystate values in the homogeneous electric field. It should be noted that, in the case at hand (ν m(v) = const ), the electron mobility is inde pendent of the shape of the distribution function. Accordingly, all the coefficients determining the renormalization of diffusion fluxes along the electric field are zero [4]. In particular, the coefficients of lon gitudinal and transverse diffusion are equal to one another. The typical spatial relaxation length of the distribu tion function (or the electron mean free path in terms of inelastic collisions) is defined as the distance at which an electron drifting in an electric field loses the energy (acquired from the electric field) equal to its temperature, 0
λ ueE 0 = Te0 ⇒ λ u =
Te . eE 0
Γ≈
λu =
eE 0
=
2eE 0
=
=
We0 3ν u
=
Wi 0 W0 = 3 i 0 ν u ν u. λu We
(18)
To calculate the instability growth rate in the entire range of k, let us consider the timeindependent Bolt zmann equation for the electron distribution function in an inhomogeneous plasma [1, 14] (the time deriva tive can be omitted, because Γ ν u ),
( )
2 v 2 ∂ F0 − ev ∂ E ∂F0 − eE ∂ ⎛ v 3 ∂F0 ⎞ 3ν m ∂x 2 3mν m ∂x ∂v 3mv 2 ∂v ⎜⎝ ν m ∂x ⎟⎠ (19) 2 2 ⎛ v 2 ∂F0 ⎞ 1 ∂ 3 e E ∂ + 2 2 ⎜ ν uv F0 = 0, ⎟+ 3m v ∂v ⎝ ν m ∂v ⎠ v 2 ∂v
(
)
instead of continuity equation (2) for electrons. The idea of the method is to replace Eq. (19) with a set of equations for the moments of the electron distri bution function. Let us define the dimensionless nth moment as ∞
∫
∞
4π v v F0 dv M n(x) =
2n
∫
4π v 2nv 2F0 dv
2
=
0
v 02nne0
0
.
∞
(20)
∫
v 02n 4π v 2 F00dv
(14)
0
Using expressions (9), (12), and (13) for v 02 , De, and We, the expression for λu can be rewritten as
De0 We0
(17)
In the opposite limit, k λ u−1, the growth rate should vanish, because, in this case, electrons do not feel individual shortwavelength fluctuations over their path length ~ λ u . It is physically clear that the growth rate is maximum at λ u k ~ 1. If we formally substitute λ u k = 1 into expression (17), then we obtain
0
mv 02
Wi (λ uk ) 2 . λu
0
∫
Te0
(16)
De0 3ν u
It follows from this definition that M 0 ( x) =
.
(15)
In the model gas under study, thermocurrent insta bility develops for any electric field strength Е, because
ne (x) . ne0
(21)
Using expression (9) for F00 , the values of M n0 corre sponding to the unperturbed electron distribution function can be calculated, PLASMA PHYSICS REPORTS
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∫
4π v v
M n0
2n
2
F00dv
(λu/Wi0 ) × Γ = (λu/Wi0 ) × Re(iω) 0.06
( ) ()
Γ n+3 2, = 0 ∞ = 3 Γ v 02n 4π v 2F00dv 2
∫
(22) 0.05
0
0 0 M 1 = 3 , M 2 = 15 , ..., 2 4 (23) (2n + 1)!! 0 Mn = , .... n 2
M 0 = 1, 0
Then, multiplying all terms in Eq. (19) by 4πv 2n 0 v 0 ne (where n = 1, 2, 3, …) and integrating them over the velocity, we obtain the following set of equations: 2n
2 1 ∂ M n + (2n + 1) e ∂ M E ( n−1 ) 2 3ν m ∂x 2 ∂ 3mν mv 0 x ∂M n−1 + (2n − 2) eE 2 3mν mv 0 ∂x
(24) 2 2 e E M n −2 + (2n − 2)(2n − 1) 2 3m ν mv 04 ν − (2n − 2) u2 M n−1 = 0, n = 1, 2, 3, .... v0 Let us analyze this set together with continuity equa tion (2) for ions,
∂ni ∂ + (Wi ni ) = 0, Wi = μ i E, μ i = const (25) ∂t ∂x under the assumption of plasma quasineutrality, ne = ni . The standard procedure of linearization of Eqs. (24) and (25), E = E 0 + E exp(iω + ikx), 0 M n = M n + M n exp(iω + ikx),
0.01
N = 30
0
0.1
0.2
0.3
0.4
0.5
0.6 0.7 ξ = kλu
Fig. 2. Growth rate of thermocurrent instability as a func tion of ξ = λuk. The solid lines show the results obtained by solving different numbers of equations N for the moments of the electron distribution function, and the dashed line shows the result obtained in the hydrodynamic approach Eq. (17). The plasma is assumed to be quasineutral.
iω
ni n + ikμ i E 0 0i + ikμ i E 0 E = 0. 0 E0 ni ni
(27)
We divide all terms in Eqs. (26) and (27) by E / E 0 , make the substitution K k = M k / E , and pass from k E0 to the variable ξ = kλ u . Taking into account that ne = ni and M 0(x) = ne (x)/ ne0 , set of equations (26) and (27) can be rewritten as
(n − 1)(2n − 1)K n−2
0
e 2 E 02 M n −2 3m 2ν mv 04 2e 2 E 02 M n0−2 E + (2n − 2)(2n − 1) 3m 2ν mv 04 E 0 ν − (2n − 2) u2 M n−1 = 0, n = 1, 2, ..., v0 + (2n − 2)(2n − 1)
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N=4
(28)
= − (2n − 2)(2n − 1)M n0−2 − (2n + 1)iξM n0−1, n = 1, 2, 3, ...,
i ω = −i ξ
E + (2n − 2) ikeE 0 M n −1 E0 3mν mv 02
PLASMA PHYSICS REPORTS
0.02
+ ((2n + 1)iξ + (2n − 2)iξ − (2n − 2)) K n−1 − 2ξ K n
2 ikeE 0 M n−1 − k M n + (2n + 1) 3ν m 3mν mv 02
ikeE 0 M n0−1 + 1) 3mν mv 02
0.03
2
ni = ni0 + ni exp(iω + ikx) yields the following set of equations:
+ (2n
N=5
0.04
0
M −1 = 2,
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Wi λu
⎛ 1 ⎞ ⎜1 + K ⎟ . ⎝ 0⎠
(29)
In order to solve set (28), we consider N equations only, i.e., n = 1, 2, 3, …, N, ignoring the term contain ing K N in the last equation (n = N). The resulting set of N equations contain N unknowns, K 0, K 1, ..., K N −1 . Note that the first equation (n = 1) also contains K −1 , but the coefficient by this term is zero. As a result, the set of equations takes the form
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(λu/Wi0 ) × Γ = (λu/Wi0 ) × Re(iω) 0.05 0.04 0.03 0.02 0.4 0.01 0 0.6 –0.01 0.8 –0.02 1.0 –0.03 2.0 –0.04 –0.05 0 0.1 0.2 0.3 0.4
–Vph/Wi0 = (–Im(iω)/k)/Wi0 0.04 0.2
0.4 0.6
d/λu = 0 0.02
0.8 0 1.0 –0.02 2.0 –0.04 0.5
0.6 0.7 ξ = kλu
Fig. 3. Growth rate of thermocurrent instability, calculated with allowance for charge separation (Poisson’s equation) for different values of the parameter d/λu. The number of equations for the moments of the electron distribution function is N = 30.
3iξK 0 − 2ξ 2 K 1 = −3iξ,
n = 1, 3K 0 + ( 7iξ − 2) K 1 − 2ξ 2 K 2 = −6 − 15 iξ, n = 2, 2 10K 1 + (11iξ − 4) K 2 − 2ξ 2 K 3 = −30 − 105 iξ, n = 3, 4 … (30) (N − 1)(2N − 1)K N −2 + ((4N − 1)iξ − (2N − 2)) K N −1 (2N − 3)!! = − (2N − 2)(2N − 1) N −2 2 (2N − 1)!! − (2N + 1)iξ , n = N, N −1 2 ⎛ 1 ⎞ (31) ⎜1 + K ⎟ . ⎝ 0⎠ Set of equations (30) and (31) was solved as follows. For a given value of ξ, set of linear equations (30) was solved numerically by the sweep method and the cal culated value of K 0 was substituted into Eq. (31). As a result, the instability growth rate as a function of ξ, Γ(ξ) = Re(iω), was determined. 0
i ω = −i ξ
d/λu = 0 0.2
Wi λu
3. RESULTS AND DISCUSSION The solid lines in Fig. 2 show the dependences Γ(ξ) calculated using different numbers of equations in set (30). The calculations show that the solution Γ(ξ) rapidly converges with increasing number of equations N. The curve for N = 5 practically coincides with that for N = 30. Moreover, even for N = 4, the solution within the interval ξ ∈ [0, 0.6] is close to the limiting
0
0.1
0.2
0.3
0.4
0.5
0.6 0.7 ξ = kλu
Fig. 4. Phase velocity of oscillations, calculated with allowance for charge separation (Poisson’s equation) for different values of the parameter d/λu. The number of equations for the moments of the electron distribution function is N = 30.
one. The maximum growth rate is achieved for ξ0 =
(
)
λuk0 ≈ 0.38 and Γ(k0 ) ≈ 0.05 Wi 0 / λ u . As was expected (see Eq. (18)), Γ(k0 ) ν u . We also note that, in the range of ξ under study, we have K 0(ξ) ~ 1 and the absolute value of the real part of K 0 is much larger than that of the imaginary part. For comparison, the dashed line in Fig. 2 shows the dependence Γ(ξ) calculated using the hydrodynamic approach. According to Eq. (17), in this case, (λu/W i 0 )Г = ξ2. As was noted above, the qualitative criterion of the applicability of the hydrodynamic approach is the inequality ξ 1. Comparison of the growth rates calculated using the hydrodynamic and kinetic approaches shows that, for ξ = 0. 1, the differ ence is ~7%, whereas for ξ = 0.15, it reaches ~20%. Therefore, the criterion of the applicability of the hydrodynamic approach for description of ther mocurrent instability can be written as ξ ≤ 0. 1. In the framework of our approach, the influence of charge separation in plasma on the development (growth rate) of thermocurrent instability can also be taken into account. For this purpose, we replace the relationship ne = ni with Poisson’s equation,
∂E = 4π(n − n ). i e ∂x
(32)
Linearizing this equation, we obtain
ikE = 4π(ni − ne ). PLASMA PHYSICS REPORTS
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Eliminating ni from Eqs. (27) and (33), we obtain the following expression for iω : iω = −iξ
−1 0 ⎞ ⎤ W i ⎡ ⎛ iξ E 0 + K 0 ⎟ ⎥ , ⎢1 + ⎜ λ u ⎣⎢ ⎝ λ u 4π en0 ⎠ ⎦⎥
(34)
(λu/Wi0 ) × Γ 0.07 0.06
3
(a)
0.05
where n0 is the steadystate value of the electron and ion densities (n0 = ni0 = ne0 ). Using expressions for the relaxation length of the distribution function λ u (for mula (15)), Maxwellian neutralization time of the space charge τ M = 1/(4πen0μ e ), and Debye radius d =
Te /(4πe n0 ), expression (34) can be rewritten as 0
533
2 0.04 0.03 1 0.02
2
iω = −iξ
(
)
0.01
−1
⎤ Wi ⎡ ντ 1 + iξ u M + K 0 ⎥ ⎢ 3 λu ⎣ ⎦ 1 − 2 0 ⎞ ⎤ W ⎡ ⎛ ⎛ ⎞ = − iξ i ⎢1 + ⎜ iξ ⎜ d ⎟ + K 0 ⎟ ⎥ . ⎟ ⎥ λ u ⎢ ⎜⎝ ⎝ λ u ⎠ ⎠ ⎦ ⎣ 0
0
(35)
(b)
For ξ(d /λ u ) 1 (or ξν u τ M /3 1), the plasma can be considered quasineutral [15]. In this case, the term 2 iξ ( d / λ u ) in expression (35) can be ignored (we recall that K 0(ξ) ~ 1, see above). Then, expression (35) is reduced to formula (31). If this inequality is not satis fied, violation of quasineutrality results in stabilization of perturbations. For given values of d and λu, quasineutrality is violated, first of all, for shortwave length perturbations. Figures 3 and 4 show the calculated growth rate and phase velocity V ph (k) = − Im(iω(k))/ k as functions of ξ for different values of the parameter d / λ u . It can be seen from Fig. 3 that, as this parameter increases, the maximum growth rate decreases, while the position of the maximum shifts toward long wavelengths (the value of ξ0 decreases). An appreciable difference from the case of quasineutral plasma (d / λ u = 0 ) is observed already at d / λ u = 0.2 . For d / λ u ≥ 1, perturbations are damped (Γ < 0) at any wavelength. Thermocurrent instability should result in the for mation of striations with a typical scale length of ~1/k0, propagating with a velocity of V ph (k0 ) = − Im(iω(k0 )) / k0 . This results in oscillations of the electrode voltage with a characteristic frequency f ( k0 ) ~ k0V ph (k0 ). It follows from Figs. 3 and 4 that, for d /λ u = 0 , the phase velocity is V ph (k0 ) ≈ 0.3Wi 0 and the frequency is f ≈ 0.12 Wi 0 / λ u . As the ratio d / λ u increases, the values of k0, phase velocity V ph (k0 ), and frequency f decrease. To illustrate, let us estimate the values of λ u , Wi, d, and ξ0 for the conditions of [10]: a discharge in helium, the pressure P ~ 1–4 Torr, and the current density je ~ 20–100 μA. Let the gas temperature and 2
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0.6 1 0.4 2 0.2 3 0
0.1
0.2
0.3
0.4
0.5
0.6 ξ = kλu
Fig. 5. (a) Growth rate and (b) phase velocity of ther mocurrent instability as functions of ξ = k λ u for different values of the reduced field: E/N = (1) 5, (2) 6, and (3) 7 Td. The solid lines show the results obtained in this study for the model gas (N = 30, d/λu = 0), and the dashed lines show the results calculated for He in [11].
the reduced field be Tg = 300 K and E0/N = 5 Td, respectively (according to [4–6], thermocurrent insta bility in helium takes place in the range E0/N ≈ 4– 7 Td). The electron transport coefficients in helium were calculated by numerically solving the timeinde pendent homogeneous Boltzmann equation for elec trons. The method of solution and the set of cross sec tions for electron scattering from He atoms are described in [16]. At low pressures, the main ion in discharge tubes of radius ~1–2 cm is He+. The mobil ity of He+ ions in He at moderate electric fields is μi [cm–2 s–1 V–1] ≈ 10 × (760/P [Torr]) × (Tg[K]/273) [17]. The obtained estimates are given in the table. It can be seen that, at P = 1 Torr, the ratio d/λu is small in the entire range of currents under study; hence, the frequency f is practically independent of Je. For P =
534
DYATKO et al.
Typical values of λu, Wi, d, and ξ0 for a discharge in helium at Tg = 300 K
1.2 0.3
0.1
4 Torr, the ratio d/λu is small for je = 100 μA and reaches a fairly large value of 0.32 for je = 20 μA. Therefore, as the current density decreases (and, accordingly, the ratio d/λu increases), plasma quasineutrality is violated, which leads to a decrease in the oscillation frequency. For the same current, the value of f increases nearly in proportion to the pres sure, at least for je = 100 μA, when the ratio d/λu is small. In [10], oscillations of the gap voltage were observed experimentally and it was noted that the typ ical values of the oscillation frequencies and their dependence on the pressure qualitatively agreed with the calculated results. The dependences of (λu/W i 0 )Γ and Vph/W i 0 on ξ, calculated for the model gas, are universal, because they are independent of the reduced electric field. Of course, this is not the case for an actual gas. Figure 5 compares the results obtained in this study with those calculated for helium [11]. As was shown in [11], ther mocurrent instability is most pronounced (its growth rate is maximum) for E/N = 5–7 Td; therefore, Fig. 5 shows the results obtained in this range of reduced fields. In plotting the curves, the results of [11] were recalculated into the variable E/N for λu = 1.2 cm (for Р = 1 Torr and Tg = 300 K, see table). It follows from Fig. 5a that the maximum growth rate obtained in this study and its position ξ = kλu agree well with the results of calculations for helium. The phase velocities (see Fig. 5b) in the range ξ = 0.3–0.4, in which the growth rate is maximum, are also quite close to one another. Thus, our model satisfactorily (both qualitatively and qualitatively) describes the linear stage of thermocur rent instability in helium. Probably, this approach, based on the method of moments, can be also used to investigate the nonlinear stage of instability. REFERENCES
0.08 0.32
0.38 0.3
d/λu
f = k0Vph(k0), s–1
1 4
d, cm
ξ0 = λuk0
1.25
Je = 100 μA f = k0Vph(k0), s–1
Wi, 104 cm/s
2.1
d/λu
ξ0 = λuk0
Te = De/μe, eV
1.13
λu = De/We, cm
We, 106 cm/s
5
d, cm
P, Torr
E/N, Td
Je = 20 μA
1180 2900
0.045
0.04 0.16
0.38 0.35
1180 4080
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Translated by E.G. Baldina
PLASMA PHYSICS REPORTS
Vol. 37
No. 6
2011
