J. Plasma Fusion Res. SERIES, Vol.2 (1999) 368-3'70

Nonlinear Waves Observed in an Electron Beam-Plasma
KAWAKAMI Retsuo. MORI lchiro and MORIMOTO Toshifumir
The University of Tokushima, Minami-Josanjima, Tokushima 770-8506, JapantTakuma Radio Technical CoIIege, Kohda, Takuma, Kagawa 761-1192,

Japan(Received: 9 December 1998 / Accepted: 3 June 1999)

Abstract
Intense wave packets with hyperbolic secant forms and strongly modified wave packets with infirm
amplitudes, which are excited by interaction of an electron beam with a plasma in a magnetic mirror

field, are observed experimentally. It is found that the intense wave packet has the nature that an increase
in its amplitude decreases its width. The nature is in agreement with that obtained by solving thenonlinear Schrcidinger equation. It is also observed that density depletions appear at a certain gas
pressure, where the intense wave packets are excited.

Keywords:
electron beam-plasma, magnetic mirror field, axial plasma density, density depletion, intense wave
packet, sech form

1. lntroduction

electron beam which has an energy of 2keV, a current of20mA and a density of l.l x l0ram-3 is passed through ahole slit with a diameter of 8mm and injected

The beam-plasma interaction has been studied as amethod of heating a plasma by many authors andinteresting phenomena on waves have been found incontinuously into a chamber,0.4m in length and 0.16m
in diameter, immersed in a magnetic mirror field with amirror ratio of 1.4, the strength of which is 80 G at thecenter of the chamber. The beam. then. interacts withexperiments [-4]. However, much still remains to beinvestigated because nonlinear wave phenomena occur
actively when waves excited by the plasma instabilitygrow.
In general, it is well known that in a beam-plasma
system a plasma is formed as ionization of neutral gas

argon gas at 0.10 to 0.30mTorr in the chamber andproduces a plasma, as shown in Fig. l. The figure shows

distributions of plasma density at 0.10 to 0.30mTorr as ain high-frequency fields and the secondary ionization
produces an abrupt increase in plasma density. At gas

function of axial distance from a beam injectionentrance, for ion saturation current 1;. varies inproportion to plasma density n if temperature for plasmapressures where plasma density increases abruptly in anelectrons Z" is constant: d. -electron beam-plasma under a magnetic mirror field, we

increases monotonously with increasing gas pressure athave observed temporal evolutions of characteristic
wave packets which correspond to nonlinear waves. In
this paper our observation on the wave packets with

0. l0 to 0.21mTorr. We name the plasma state at suchinteresting natures will be discussed.

increases abruptly at 0.22 to 0.30mTorr and the plasma

by beam electrons and secondarily by plasma electrons,f2". Plu.rna density

gas pressures the first stage of the beam-plasma
discharge (BPD). The plasma density, however,state above 0.22mTon is different from the first staee. It

2. Experiment
Our experiment is characterized as follows. An@1999 by The Japan Society of Plasma

Corresponding author's e-mail: retsuo@ ee.tokushima-u.ac.jp
368

Science and Nuclear Fusion Research

Kawakami R. et al., Nonlinear Waves Observed in an Electron Beam-Plasma5s.

cAs PRESSURE 0.30 [mTorr]

zuiFttlC)

zotrtrfF(A)EU)

2UJ)Eo0.04 0.08 0.r2 0.16 0.20 0.24 0.28 0.32

0.36

AXIAL DISTANCE IMI=Fig. 1 Axial distributions of ion saturation current withgas pressures between 0.10 and 0.30mTorr.

A

12oT
11.5!sec

Fig. 2 A sample of wave packets, (A) and (B). observed

should be noted in particular that density depletions

at 0.245m under 0.22mTorr. The bottom twofigures show redrawn views of the wave packets

appear at O.22mTon. We name such a plasma state the(A) and (B), respectively.

second stage of the BPD.

In the second stage a plasma has density n n l}ta 10r5m-3 and temperature Z" = 3eV; as a result, theplasma parameter A"= n Lo3 becomes an order of lOa -105>> 1, where i,p is the Debye length
In otherwords, collective-cooperative behavior becomes active?
gin the second stage. Waves excited by interaction
between the beam and the plasma are detected by aprobe and observed directly by a digital oscilloscope.

F6ztu
Fz

3. Observation
Wave packets with intense amplitudes appear
intermittently at 0.22mTorr in the second stage. A
sample of wave packets observed at 0.245m from the350

beam injection entrance is shown in the top of Fig. 2. In

the sample the two wave packets are identified bynotations (A) and (B), respectively. The bottom twofigures are redrawn views of the wave packets (A) and(B). For each wave packet, its envelope has a wellknown shape of a hyperbolic secant, i.e., sech. A peakto-peak value of the wave packet (A) is 2.6V and its full
width at half maximum (FWHM) is 0.74psec, and forthe wave packet (B) the former is 4V and the latter is
0.53psec. It is seen that amplitude of the wave packet
increases with decreasing its FWHM. The relationshipbetween the amplitude and the width is in agreementwith a characteristic of solution lS(x, t) | , which
satisfies the nonlinear Schrcidinger (NLS) equation,

450

Fig. 3 Frequency spectrums corresponding to the wave
packets (A) and (B) in Fig. 2.

where P and Q are parameters which stand for thestrength of dispersion and nonlinearity in the NLS
equation [5]. In eq. (1) an increase in A corresponding tothe amplitude decreases FWHM ot lS(-r, r) | , for theFWHM is given by 2 sech-ru/2llf-QIPA\.
Figure 3 shows frequency spectrums of these wavepackets. The spectrums of (A) and (B) are almost thesame and each of the spectrums has a localized structurewith a FWHM of 3MHz around a carrier frequency of396MHz.
Such wave packets, however, do not appear withexpressed byls t", rtl =A sech [rOno.1,400
FREQUENCY [MHz]

(1)
369

Kawakami R. et aI., Nonlinear Waves Observed in an Electron Beam-Plasma

Eulol

(a)

0.'l(b) |l,v'FJ.LI

I= -0.1

4-n&fs.
F

6zIIJ

10s sec

F(a)EUJoltJz(b)

!7.

(L400

= -o.'t
4.2L0I

1.50

450500FREAUENcY [MHz]lrISeC

Fig. 5 Frequency spectrums corresponding to the wave
packets (a) and (b) in Fig. 4.

Fig. 4 A sample of wave packets. (a) and (b), observed

at 0.245m under 0.30mTorr. The bottom twofigures show redrawn views of the wave packets(a) and {b), respectively.

experimentally, which are excited by interaction of anelectron beam with a plasma column in a magnetic
mirror field, at the second stage of the BPD where anabrupt increase in plasma density occurs. It turns outthat the wave packets vary sensitively with gas pressure:wave packets at 0.22mTon have intense sech forms andfrequency spectrums, 3MHz in FWHM, localizedaround a carrier frequency of 396MHz, whereas wave

the further increase of gas pressure. A sample of wave
packets, (a) and (b), observed at the same position under

0.30mTon is shown in Fig. 4. Likewise, in the sample
the two wave packets are labeled (a) and (b) respectively and the bottom two figures show redrawings of
the wave packets (a) and (b). There is no appearance ofdensity depletions at this gas pressure. Amplitudes of
the wave packets are infirm compared with those atpackets at 0.30mTorr have infirm amplitudes, which arestrongly modified at rapidly decreasing parts, andfrequency spectrums with two peak values of intensity.

O.22mTon, or the amplitudes become about one tenth asmuch as those at 0.22mTon. Each envelope of the wave
packets no longer follows a sech form and is strongly
modified at a rapidly decreasing part. Frequency
spectrums of the wave packets (a) and (b) broaden incomparison with those of (A) and (B), as shown in Fig.
5, and have peak values of intensity at about 438 andIn addition, it is observed that density depletions appear
at the gas pressure where the intense wave packets areexcited.

It also turns out that the intense wave packet withthe sech form has the nature that its amplitude increases

with decreasing its FWHM; the relationship between theamplitude and the width is in agreement with thatobtained by solving the NLS equation.

442MHz. Here, a collision effect is not taken into
account for the reason that an increase in plasma density
between 0.22 and 0.30mTorr is slight. It is seen fromthese figures that the wave packets (a) and (b) are
affected by some nonlinear interaction effect, such aslll

References
W.D. Getty and L.D. Smullin, J. Appl. Phys. 34,342r (1963).
[2] T. Yamamoto, K. Ohniwa, H. Akimune and T.

the nonlinear Landau damping.

These results lead to the thing that the density
depletions appear with the intense wave packets at
0.22mTon.It is likely that pressures of the intense wave
packets cause the density depletions through theSuita, J. Phys. Soc. Ipn.22,277 (1967).

t3l K. Yatsui, Y. Yamamoto and K. Saeki, Phys. Lett.
39t^,42s (1972).

ponderomotive force as a nonlinear effect.

[4] I. Mori and K. Ohya, Phys. Rev. Lett. 59 1825

4. Gonclusion

t5l N. Yajima, M. Oikawa, J. Satsuma and C. Namba,J. Phys. Soc. Jpn. 45,643 (1978).

(1987).

We have observed characteristic wave packets
370