J. Plasma Fusion Res. SERIES, yol.2 (1999\ 352_355

Electron Heating at Sub-Harmonic Electron cyclotron
Frequency in an Electron Beam-plasma System
MAEHARA Tsunehiro, AoNo Shinji, SHINKAI Megumi, ursuNoMlyA Shohei,
SUGAWA Masao and SUGAYA Reiji
Department of Physics, Faculty of science, Ehime ILniversity, Matsuyama 790-g577, Japan(Received: 9 December 1998 / Accepted:22 April 1999)

Abstract
Electron heating occurs in an electron beam-plasma system at a=31{D..by launching a space charge
wave on Vu = 1500V and 1000V, where V5 denotes the acceleration voltage of beam electrons.
Experimental results are qualitatively consistent with the nonlinear coupling coefficient of the subharmonic electron cyclotron (EC) resonance. This suggests that the electron heating is due to the subharmonic EC resonance.

Keywords:
space charge wave, sub-harmonic electron cyclotron resonance

1. lntrodustion
Ion heating at sub-harmonic ion cyclotron

a - klvtt= m a--.

2*

frequencies (a- ma.tl2) was investigated [1-6]. Here,m is an odd integer. Recently, electron heating at half
the electron cyclotron frequency by extraordinary wave,The term ft11v11 can be neglected since ft1; << ft1 in thespace charge wave, and then the resonance condition isgiven by a = ma..l2. Therefore, the resonance atwhich can be applicable for ITER, is investigatedtheoretically [7]. In these circumstances, experimental
studies for the sub-harmonic electron cyclotron (EC)frequencies is useful. However, only a few studies werecarried out [8]. In this paper, we study sub-harmonic EC
resonance by employing a space charge wave in anelectron beam-plasma system at a = 32a". (a". : lelB / m.).
Sub-harmonic heating is due to the nonlinear
Landau (cyclotron) damping resulting from the selfinteraction of waves. The resonance condition for
nonlinear Landau damping in a magnetic field is given
by c+ - ror, - (kll - k') vn = moJce, where (a4, k) and (c4,,k') are frequencies and wave vectors of two waves, andm is an integer. In the case of self-interaction, (r4, k) =(-ay, -k') = (ar, k) and then (coy,, k") = (2a, 2k). The
resonance condition is presented bya =)a." (i.e. m = 3) can be occurred.
In this paper, the first experimental results arepresented and are qualitatively compared with thenonlinear wave-particle coupling coefficient. In Section
2, we describe the experimental results. In Section 3, thekinetic wave equation and the transport equation aredescribed. In Section 4, the numerically calculated

coupling coefficients are compared with theexperimental results. Finally, we present a summary inSection 5.

2. Experimental Results
2.1 Experimental procedures
Experiments are performed in a linear device ofabout 9 cm diameter and 50cm length. A mono-energetic electron beam from electron gun is@ l 999 by The Japan Society of Plasma

Corre sponding author's e-mail : maehara@ phys. sci. ehime -u.ac.jp352

Science and Nuclear Fusion ResearchMaehara T. et al., Electron Heating at Sub-Harmonic Electron Cyclotron Frequency in an Electron Beam-Plasma System

continuously injected along a magnetic field of 8s :by applying the external RF signal (P = lW). Electron

7OG (a"./2n = l98MHz). Acceleration voltage, current
and diameter of electron beam are V6 500-1500V, 16 =temperature before applying the extemal RF signal is Z"s:

= 7eY, 4eY for V5 = 1500V and V6 = 1000V,

respectively. For Vo = 1500V, Aln" has a peak at aloJce= 1.5, and becomes small in ola".< 1.4 or ala""> 1.6.

0.5-2mA and Db =2mm (diameter of the anode hole ofthe electron gun is 2mm), respectively. Length from theFor Vs - 1000V, AT" has a smaller peak at ala." = l-5,
and AZ" remains positive at oloce= 1.4 or 1.6. This isdifferent from the former case (v6 = 1500v)' Thefractional change of electron temperature A7"/2"6 is thesame order for both cases. The fractional change ofelectron density L,n"lnao versus (o/oce are shown in fig. Igun to target is = 40cm. Langmuir probe is placed 20cm

away from the gun at the plasma center (r = Omm).

Working gas is argon, and gas pressure is 1.5mTorr. In
the beam-generated plasma, electron density is n" = 5lgto.ttr-:. The space charge waves are excited by
19"
applying an external RF signal with the frequency al2n= 250-350MHz and the power P = lW to the electrode(b). In both cases, electron density decreases withincreasing in AZ". Electron density before applying theexternal RF signal is n"s = 6, 10 x 10l0cm-3 for Vo =1500V and Vr = 1000V, respectively. Figure I (c) showsof the gun.

2.2 Electron heating
Figure I (a) shows AT" versus o)|a"" for Vo =the fractional change of energy density L(n"T")l (n"oT"o)versus (o/@ce. In spite of decrease in electron density,energy density increases at o)l@ce = 1.5 due to theincrease in electron temperature for both cases.

1500V (solid circles) and V6 = 1000V (open circles).
Here, AZ" means the change of the electron temperature

3. Kinetic Wave Equation and Transport(a)K/\*-1lsIJ

EquationzVb=rSq0tV)(re-7ev)

Experimental results show that electron heating

Vw:=try-v)
cv)

occurs at a/a".= 3/2. This suggests that the electron
heating is due to the sub-harmonic EC resonance. Here,
we describe the kinetic wave equation and the transport

P'-o. \,!1'b- ,//\(Te-4O -O--

equation for the sub-harmonic resonance.
We assume that the velocity distribution functions
for bulk and beam electrons are the Maxwellian andshifted Maxwellian, respectively. The kinetic wave
equation for the sub-harmonic resonance is presented aslTs(b)[e]:vb=1500(v)+ot = 2yo[Jo+ €g0)1dt<luol' ,oF

t 0.1E(1)\to*t* ,_ r 4,frt ,a*,o
a*=-( 2 ,) -Aa;)x

ffi'o,..'

nemevieEF,ro-T,"

1.5(t) / otce

Fig. 1 (a) Change of the electron temperature (Ai"), (b)fractional change of the electron density lL,n"ln.ol,(c) fractional change ofthe energy density (Ln.T.ln"o?:o) versus ala". lor % = 1500V (solid circles)
and Vo = 1000V (open circles), respectively. For
the case of % = 1500V, lu= 1.7m4, De = 6 x1010cm-3, 7"0 = 7eV. For the case of % = 1000V, /r =0.8mA, n =10 x 1010cm-3, T"= 4eV, where /o is thebeam current. In both cases, a..l2tr =198MHz isfixed.

A=

f dte-'2 r{ot) J,(pt) ^ s(pt) ^ -,(pt),B=

f dt"-'' ntet)J.,q(et)J.(2pt),

JJ

C= B(q -) r),
353

J-J

Maehara T' et al., Electron Heating at Sub-Harmonic Electron Cyclotron Frequency in an Electron Beam-plasma Systemdarlf reaches to zero.

D=e2p2 l-(-zp2)/2,First, we discuss the case of V5 - 1500V.
Experimental results show that electron density

E = e,A [@ - m | 2)2 - t)r I? - m / z)2 _ \-t,decreases at the same time that electron heating occurs.

This is similar to the electron heating due to the2at^'€t=l+ t=",UL -^--

explosive instability via non-linear Landau damping ofspace charge wave and Trivelpiece-Gould (TG) mode in
an electron beam-plasma system [10]. However, in thekr14"-4 --r^,I.(,1.) [ | + (o,zp((^)|,
v* -previous experiments [0], heating occurs in the lower
range (alrt4" = 1.1(2tDr- 0"") / 2krv r. ,experiments, heating occurs only around oila"" - 7.5.

where p = k1v7./o)r", (,, = G4 - rol".)lk1vn, €,a = (@r k11v11-- 1.45), and LTJT"0 has some peaks.

This is because TG mode with various frequencies canbe scattered and it heats electrons. In the presentThis is qualitatively consistent with the coupling
coefficient of the sub-harmonic resonance. Therefore,
the cause of the present electron heating is not nonlinear Landau damping of space charge wave and TGra.,)/k11v11, 2. = k2rv2r"l2a", roou is the plasma

frequency for beam electrons, vr" (yru) is the thermalvelocity of bulk (beam) electrons, vo =.[ 2 ,Vo/rb. Jn it
the Bessel function of the q-th order, I is the modified

mode, but the sub-harmonic EC resonance.

Bessel function of the r-th order, E1 is the electric fieldof the wave (@k, k), U1 is the energy density of thewave, Zo is the plasma dispersion function, and the othernotations are standard. The linear damping rate 7* is
negligible small at a1,la"" = 3/2. Transport equation
indicating time evolutions of energy and momentum
densities of bulk electrons are presented as:dAU^| t7
2€o @oaol Enl- u o .
=;;ap,,=- 2€nk,,aol,Eol,)

J,UoQ)

It is found from eq. (l) that explosive instability canoccur when a1 > 0. This is possible if a negative-energy
wave (da1,e1/dtop < 0) is scattered from bulk electronssIt

tet.

4. Discussion
We define the non-dimensional coupling coefficientas ob : laplx (n"m.v\./2). Since slow mode of the spacecharge wave is the negative-energy wave, or > 0. On
the other hand, ai < 0 for fast mode of the space charge
wave.
Fig. 2 (a) The non-dimensional coupling coefficient o6

Figure 2 shows the coefficient q) versus alo."for
Vu = 1500V and V5 = 1000V, respectively. Parametersversus rola4" lor % = 1500V. Parameters

correspond to the experimental condition (a".l2nfor numerical calculation are chosen from the=198GH2, lr= 1.7mA, D" = 6 x 1010cm3, T"=7eVl.

experimental condition. In the case of V6 = 15OOV, thecoefficients for both modes become maximum at alo)ce: 1.48. In the case of V6 - 1000V, the coefficients forboth modes become maximum at (Dl(Dce: 1.45, andthese increase again below oJl@".- 1.38 because lda*e*/(b) The non-dimensional coupling coefficient aoversus a/a.. lor % = 1000V. Parameters

correspond to the experimental condition (a.J2n=198MH2, /u = 0.8mA, n" =10 x 1010cm3, 7" = 4eV).
In both cases, we assume that electron beam iscold(i.e.4=0).
354

Frequency in an Electron Beam-Plasma System
Maehara T. et al., ElectonHeating at Sub-Harmonic Electron Cyclotron

Next, we discuss the case of Vo = 1000V'

resonance. The experimental results and the coefficientshow qualitative agreement' This suggests that theelectron heating is due to the sub-harmonic EC

Experimental results show that the electron temperature
increases and the electron density decreases by applying

the external RF signal in the range of aila".= l'4-l'6'
The coefficient oh increase with ala""beyond a/a".=
1'4 and becomes maximum at alace= l'45' Both A?"
and o6 show similar dependence in oila."> l'4'ln ala.. < 1.4, AT" and aa shows different behavior' Furtherresonance.

References

lll M. Ono et c/., Phys. Rev. Lett. 52'37 (1984).
t2l M. Ono et a/., Phys. Lett. 54, 2339 (1985)'
t3l K. Toi et al., tn Proc- of Tenth Inte. Conf' onstudies are needed in this region.

Plasma Phys. and Controlled Nuclear Fusion Res'The second harmonic component of the spacecharge waves can be coupled with electrons via theLondon. 1984 (lntetnational Atomic Energy
Agency. Vienna, 1985), Vol.l' P.523.

linear third harmonic resonance. In order to investigate
this linear effect, we apply the RF signal with thefrequency o)lo)."-- 3 and the small power P = 0.lW toelectron temperature is measured.

t4l M. Porkolab, Phys. Rev. Lett. 54' 434 (1985).
t5l R.Sugaya, Phys. Fluids 30, 1730 (1987).
t6l H. Abe et a/., Phys. Rev. Lett. 53' I153 (1984).
[7] T. Maehara and R. Sugaya, Phys. Lett. A 247, 3O9

5. Summary

t8l M. Sugawa, Phys. Rev. Lett. 61, 543 (1988); M.

In launching a space charge wave' electron heating
occurs in an electron beam-plasma system at a = )a*t9l

the electrode of the gun. However, no increase in(1998).
Sugawa, J. Phys. Soc. Jpn. 48,465 (1992).
R.Sugaya, J. Phys. Soc. Jpn.60,518 (1991)'tl0l R. Sugaya et al., J. Phys. Soc. Jpn. 64, 2018

on Vo = 1500V and 1000V. Experimental results in bothcases are compared with the numerically calculated
nonlinear coupling coefficient for the sub-harmonic EC(199s).

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