Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)566–569
000–000
J. Plasma Fusion

Particle-Trapping Effects in Electron-Beam-Plasma
TAKEDA Tsuyoshi and YAMAGIWA Keiichiro
Faculty of Science, Shizuoka University, Shizuoka, 422-8529, Japan(Received: 9 December 2003 / Accepted: 12 February 2004)

Abstract
Wave packets and velocity distribution functions of electron-beams in an electron-beam-plasma system are investigated to find out particle-trapping effects. The dependency between the packet amplitudes and the distribution
spreads, and the value of the bounce frequency indicate that the potentials of the packets trap the beam-electrons. The
trapping appears to induce the beam scattering and the deformation of the packets.
Keywords:
particle-trapping, electron-beam-plasma, wave packet, velocity distribution function of electron-beam,coaxial probe, energy analyzer

beams are observed by using a coaxial probe. An energy analyzer [8] is adopted so as to observe velocity distributionfunctions of the beams. Figure 1 illustrates the schematic ofexperimental apparatus. A synchronized system with a test
wave signal is introduced in the above observation. The signal, which is a single amplitude modulation wave, consists ofcarrier frequency of 90 MHz and half-width time of 50 ns,
and is given to a control grid of beam gun to excite an initialwave packet. The signal is also given as a trigger to digital
oscilloscopes simultaneously. Signals detected by the analyzer and the coaxial probe are amplified by a low frequency
amplifier (DC–8 MHz) and a high frequency amplifier(0.1–1300 MHz), respectively. These amplified signals arereceived on 2-channel of the oscilloscopes. The observations
are carried out at each of the axial positions of 128 points.
Typical experimental parameters are as follows: electron
plasma temperature 0.76 eV, electron plasma density ne = 1 ×1014 m-3, electron-beam velocity vb = 4.2 × 106 m/s, and electron-beam density nb /ne = 0.3, 0.6, 1.2 %.

1. Introduction
In an electron-beam-plasma system, it is well known
that wave packets developing as an initial growth process inelectron time scales obey the beam-mode properties
described by a linear equation [1]. But the packets can notcontinue to grow infinitely, and begin to shift to the processes of saturation and damping before long. Since the packet iseasy to be accompanied with nonlinear phenomena [2,3], it isnecessary to discuss modulational instability [4], particletrapping [5], and so on. Yamagiwa et al. [6] observed that awave packet evolves into a packet train as continuously emitting some wave packets in a weak electron-beam plasma,
nb /ne < 0.3 % (nb and ne are the electron-beam and the plasma
density, respectively), and discuss its phenomenon by a nonlinear theory of Yajima et al. [7]. But Akimoto et al. [5]showed by computer simulations that the trapping effects
outstand in the nonlinear regime of this system. It has beeninteresting that velocity distribution functions of electronbeams are investigated experimentally. In this study, theauthors investigate nonlinear phenomena in the case of strong
beams, nb /ne ≥ 0.3 %, and the relationships between the packets and the distribution functions so as to pick up the signs ofbeam-electron trapping.

3. Results and discussions
Figure 2 shows that a wave packet in the case of nb /ne =1.2 % evolves temporally along the beam path. Here k0z andωpet is respectively the axial space and the time. The characterization parameters of wave packets against nb /ne are given
in Table 1. Here ωpe is the electron plasma frequency, k thewave number, ω (= ωr + iωi, ωr = Re(ω), ωi = Im(ω)) the frequency, k0(≡ ωpe /vb) the defined wave number, and vφ (=ωr /k) the phase velocity. ωr is almost constant due to being
almost determined by the frequency of the test wave. kincreases and vφ decreases with nb /ne. This result means thatthe packets obey the beam-modes because the dispersibility

2. Experimental procedure
Plasma is produced in a cylinder chamber, whose sizes
are 0.26 m in diameter and 1.2 m in length, filled with argon
gas by DC-discharges between four heated filaments and its
wall. Then the plasma is confined in full-line cusps [9] produced by magnets. Electron-beams injected into the plasma
are pulsed, and travel one-dimensionally along z-axis magnetic field induced by coils. Wave packets excited by the©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: sttaked@ipc.shizuoka.ac.jp1566

Takeda T. et al., Particle-Trapping Effects in Electron-Beam-Plasma
Table 1 The characterization parameters of the packets
against the beam density nb /ne.

nb /ne (%)
0.3
0.6
1.2

k/k0
1.01
1.07
1.10

ωr /ωpe
0.9
0.9
0.9

ωi /ωpe
0.066
0.10
0.12

vφ /vb
0.89
0.84
0.82

with the maximum amplitudes at each upper side, the distributions mapped on v − z plane at each left side, and the distributions in several positions k0z at each right side. With
increasing nb /ne, the spread of these distributions becomeslarger towards the low side of the velocity space. In the caseof nb /ne = 1.2 %, the spread becomes unusual in k0z = 30 − 40
as two peaks arising. Those spreads exist around the peak
positions of the packets, which seem to be involved by thepacket amplitudes.
When a wave potential traps beam-electrons, thedependency between the potential φ and the trapping radius∆vt is described by ∆vt2 = 4eφ/m [2,5], where e/m is the specific charge of electron. Figure 4 plots on double logarithmic
chart the packet amplitude against the velocity width ∆v
between the phase velocity and the lower edge of the spread.
The liner line fitted by (∆v/vb)2 implies that such plotting is
consistent with the dependency. The upper horizon of thisfigure is the magnitude of the potential into which the trapping radius is converted, which gives φ = 0.56 V in nb /ne 1.2%. Then the bounce frequency ωt (= ek2φ/m) is 0.083 ωpe.
Since ωt is near ωi, the trapping effects can not be neglected.
These above results indicate that the packet potentials trap
the beam electrons. The distribution spreads seem to be constructed by the beam scattering due to trapping and detrapping. The local difference of the beam density generates thatof ωi and vφ, which may induce the deformation of the packet.

Fig. 1 Experimental apparatus.

depends on nb /ne. After the packet grows linearly in ωpet <90, it shifts to saturation and damping process.
Unfortunately, modulation instabilities or wave packet trains
are not observed in the case of the above beams. Instead ofthose, the asymmetry deformation appears in the packet.
The correlations between wave packets and velocity distributions fb for nb /ne = 0.3, 0.6, 1.2 % are respectively shownin Fig. 3(a), 3(b) and 3(c). These figures show the packets

4. Conclusions
The authors investigate wave packets and velocity distribution functions of electron-beams to find out particle-trapping effects in an electron-beam-plasma system. The deformation of the packets and the distribution spreads of thebeams are observed by using a coaxial probe and an energy
analyzer, respectively. From the dependency between thepacket amplitudes and the distribution spreads, and thebounce frequency estimated, it is concluded that the potentials of the packets trap the beam-electrons. The authors
guess that the beam scattering due to trapping and detrapping
induces the distribution spreads and the deformation of thepackets.

Acknowledgments
The authors would like to acknowledge useful discussions with Professor SAEKI Koichi and Assistant ProfessorAKIMOTO Kazuhiro.

Fig. 2 The temporal evolution of the packet at time intervalof ωpet = 4.5 in the case of nb /ne = 1.2 %.

2567

Takeda T. et al., Particle-Trapping Effects in Electron-Beam-Plasma

Fig. 3 The correlations between the packets and the distributions fb for nb /ne = 0.3 (a), 0.6 (b), and 1.2 % (c). The packets with themaximum amplitudes are displayed at each upper side, the distribution mappings for v − z plane at each left side,and thedistributions of several positions at each right side.

3568Takeda T. et al., Particle-Trapping Effects in Electron-Beam-Plasma

References[1] R.J. Briggs, Advances in Plasma Physics, ed. A. Simonand W.B. Thompson (Interscience, New York, 1971)
Vol. 4, p. 43.
[2] A. Hasegawa, Plasma Instabilities and Nonlinear
Effects (Springer-Verlag, Berlin Heidelberg New York,1975).
[3] M.V. Goldman, Rev. Mod. Phys. 56, 709 (1984).
[4] T. Intrator, C. Chan, N. Hershkowitz and D. Diebold,
Phys. Rev. Lett. 53, 1233 (1984).
[5] K. Akimoto, Y. Omura and H. Matsumoto, Phys.
Plasmas 3, 2559 (1996).
[6] K. Yamagiwa, T. Itoh and T. Nakayama, Invited Papers,XXIIIrd ICPIG 1997, Toulouse (EDP Sciences, 1997)
Vol. 7, p. C4-413.
[7] N. Yajima and M. Tanaka, Prog. Theor. Phys. Suppl. 94,138 (1988).
[8] R.L. Stenzel, W, Gekelman, N. Wild, J.M. Urrutia and
D. Whelan, Rev. Sci. Instrum. 54, 1302 (1983).
[9] K.N. Leung, T.K, Samec and A. Lamm, Phys. Lett. A
51, 490 (1975).

Fig. 4 The dependency between the packet amplitudes andthe distribution spreads ∆v. The upper horizon is thepotential φ. The fitting line is proportional to (∆v/vb)2.

4569