# Driving Mechanisms of Ion-Acoustic Activity in an m = 0 Helicon Plasma

K. P. Shamrai, V. F. Virko and V. M. Slobodyan

Institute for Nuclear Research NAS of Ukraine, 47 Prospect Nauki, 03680 Kiev, Ukraine

Abstract. Theoretical analysis of two mechanisms driving ion-acoustic activity in helicon plasmas excited by azimuthally symmetric (m = 0) antennas is presented. Electron drift current driven kinetic instability and hydrodynamic parametric instability driven by the field of the helicon pump wave and the concomitant quasi-electrostatic wave excited via linear mode conversion are examined.

Keywords: Helicon plasmas, electron drift current driven ion acoustic instability, parametric instability.

PACS: 52.35.Hr, 52.40.Fd, 52.50.Dg.

1. INTRODUCTION

Excitation of ion-acoustic (IA) turbulence in helicon plasmas was first predicted theoretically to arise from kinetic parametric instability [1]. Afterwards, many experiments have revealed that IA activity in the megahertz range of frequencies is inherent to helicon plasmas independently of design and operating conditions [2-6]. This activity is diverse and is observed as continuous broadband spectrum of noise oscillations [2,4,6], or as spiky spectrum [5,7], or as continuous spectrum with one [3,5] or many [7] spikes. In addition to kinetic parametric instability, other driving mechanisms were also considered, such as parametric decay instability [3,6] and oscillating two-stream instability [4]. Experimental evidences in support of parametric instabilities are the following: the IA waves and the HF sideband waves satisfy matching conditions on frequencies and wave numbers [3,6,7], and the IA waves demonstrate excitation threshold on the rf power [4]. It was also hypothesized that one more source for excitation of the IA waves can arise from kinetic instability driven by electron drift current streaming across the magnetic field [5].

We analyzed theoretically two probable driving mechanisms of IA activity in helicon plasmas excited by azimuthally symmetric (m = 0) antennas, in order to interpret experiments performed on two different sources in our laboratory [7]. Kinetic instability driven by various electron drifts, such as diamagnetic, Hall, and magnetic-gradient drifts, is considered in detail since in experiments the location and propagation directions of IA waves correlate well with the location and directions of the drift currents. Hydrodynamic parametric instability excited by a combined field of the helicon wave and quasi-electrostatic wave arising due to linear mode conversion is also considered.

2. EXCITATION OF IA WAVES BY ELECTRON DRIFT CURRENTS

Our experiments [7] have shown that spectra of ion-acoustic waves consist normally of continuous, noise component and, in some operating regimes, of spiky component. Continuous components have maximums in the frequency range above 200 kHz, which relates to the waves with quite short lengths, < 1.5 cm, much shorter than the discharge chamber size. For this reason, we considered the model developed for planar geometry [8] and modified it to account properly for particle collisions. Assuming that electrons drift normally to the magnetic field with the velocity u, the dispersion relation for low-frequency electrostatic waves is represented as the sum of partial dielectric permittivities of electrons and ions (α = e, i):

Σ=+αααtαpα0)/()/(12PQkvω,

where

Here, vtα = (Tα/mα)1/2, ωсα = eB0/mαc, and ρα = vtα/ωсα are the thermal velocities, gyro-frequencies, and gyro-radii of particles; k = , k2/122)(⊥+kkzz = cosθ, and k⊥ = sinθ are the total, longitudinal, and perpendicular wave numbers; θ is the wave propagation angle relative to the magnetic field; and I0,1 are the modified Bessel functions. Hereafter, the electron drift velocity will be measured in units of the ion-acoustic velocity, vS = (Te/mi)1/2, and cosθ in units of μ = (me/mi)1/2. As the ion temperature is not too small as compared with the electron one, the frequency of ion-acoustic wave, with no electron drift, takes the form . We assumed the following standard parameters relating to experimental conditions: B2/1)/31(eiSSTTkv+=ω0 = 70 G, n0 = 4×1011 cm−3, pAr = 5 mTorr, Te = 4 eV, and Ti = 0.2 eV.

Frequencies and growth rates of unstable waves computed for the standard parameters are shown in Fig. 1 as functions of the total wave number. As seen from Fig. 1(a), the growth rate first increases with the wave number, comes to the maximum at kρe ≈ 0.5, and then gradually falls and becomes negative at kρe ≈ 14 (not shown in Fig. 1) due to stabilizing effect of Landau damping on ions. The frequency of oscillations considerably exceeds the ion-acoustic frequency at kρe < 0.3, and approaches ωS at larger k [Fig. 1(b)].

Fig. 1. Frequencies and growth rates of unstable waves, at standard conditions, u/vS = 5, and cosθ = 3μ.

Fig. 2. Effect of electron and ion collisions on dispersion of unstable waves. The same parameters as in Fig. 1.

Fig. 3. Dispersion of unstable waves at various electron drift velocities. Standard parameters and cosθ = 3μ.

Fig. 4. Dispersion of unstable waves at various propagation angles. Standard parameters and u/vS = 5.

Figure 2 demonstrates the effect of collisions on the dispersion of unstable oscillations. As seen, ion collisions are negligible while electron collisions lower the growth rate, especially around the maximum [Fig. 2(a)], and result in frequency enhancement at lower k [Fig. 2(b)].

Dispersion of unstable oscillations at various values of the electron drift velocity is shown in Fig. 3. Increasing u gives rise to the increase of the growth rate, especially in the range of maximum, and to the increase of the wave frequency in the range of lower k. The growth rate and frequency also grow with increasing the angle of wave propagation relative to the magnetic field (Fig. 4).

The effect of magnetic field strength on the dispersion of unstable oscillations is demonstrated in Fig. 5. As seen,

the growth rate rises with the field, and location of the maximum of growth rate moves towards shorter wavelengths, approximately as eπρλ4max≈ [Fig. 5(a)]. However, the growth rate is found to be less than the frequency at any magnetic fields and wave numbers [Fig. 5(b)]; i.e., the instability is weak. The maximum value of the growth rate is quite small at lower magnetic fields, B0 < 50 G, and grows linearly with B0 at higher fields, as seen from Fig. 6.

Fig. 5. Dispersion of unstable waves at various magnetic fields. Standard parameters, u/vS = 5, and cosθ = 3μ..

Fig. 6. Maximum growth rate as function of the magnetic field. The same parameters as for Fig. 5.

Fig. 7. (a) The amplitude of rf electron excursion within the volume of helicon plasma source [7], at antenna current of 5 A, and (b) the frequency and growth rate of parametrically unstable oscillations as functions of the wave number.

3. PARAMETRIC EXCITATION OF THE LF OSCILLATIONS

Ion-acoustic waves can also arise from parametric instability driven by the rf oscillations of electrons relative to practically immobile ions in the field of helicon and electrostatic waves excited at the driving frequency. Figure 7(a) shows the amplitude of electron drift oscillations across the magnetic field, ζE, computed for the helicon source [7], at standard parameters. As seen, ζE has maximum under the antenna and falls when moving away. At z = 18 cm, i.e., in the range of experimental measurements [7], ζE ∼ 0.1 cm. Characteristic scale of nonuniformity of the rf electron oscillations is of the order of plasma radius, i.e., much larger than ion-acoustic wavelengths. This enables to assume the rf pump in the dipole approximation, as uniform. We used the model [9,10] modified for due respect of particle collisions, and the computer code developed in [4]. The dispersion of LF oscillations in the presence of pumping field is shown in Fig. 7(b). As seen, oscillations with wave numbers kζE ∼ 1 are unstable, with large growth rates much exceeding the growth rates of the instability driven by the stationary electron drift. Frequencies of oscillations are considerably higher than the ion-acoustic frequency owing to strong coupling to the rf pump.

5. CONCLUSIONS

Analysis shows that stationary electron drift current can efficiently drive the ion-acoustic waves over a broad range of wavelengths excepting the longest waves. Growth rates of this instability are quite high, in particular, exceed the inverse lifetime of ions in experimental devices. Frequencies of short unstable waves are close to the ion acoustic frequency while frequencies of long waves, in the 100-kHz range, can be considerably higher than ωS. Parametric instability can give rise to much higher growth rates, but applies to quite short waves only. This instability is found to results in strongly increased wave frequencies, as compared with the ion acoustic frequencies; however, this effect was not detected in experiments.

ACKNOWLEDGMENTS

This work was supported by the Science and Technology Center in Ukraine under contract No. 3068.

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