J. Plasma Fusion Res. SERIES, Vol.4 (2001) 5gt_594

Interaction of Electrons with an oscillating potential
TSUSHIMA AKiTa*, ISHIHARA osamu, TAKAHASHI HiToshi and MIYAHARA
Faculty of Engineering, yokohama National university, yokohama 240-g501,Kazuto

Japan(Received: 5 December 2000 / Accepted: 3 October 2001)

Abstract
A model experiment of Fermi acceleration was performed in a trap, where electrons moved
an oscillating potential and a fixed potential along a magnetic field. In the trap somebetweenof the electrons

were accelerated until they went over the fixed potential. We have observed such electrons atspecificfrequencies of the oscillating potential. A numerical study relates this resonance behavior

to thestochastic acceleration ofelectrons confined in the oscillating wall and the fixed wall.

Keywords:
fermi acceleration, electron acceleration, oscillating potential, resonance, fermi map

1. Introduction

function of z along the axis of the trap by a solid line.
The left potential wall was formed by the bias -50 V

Extremely high-energy particles have been known
to exist in space [1]. Fermi acceleration has been

applied to the leftmost ring electrode and was modulated

considered as one of the candidates of the acceleration
mechanism and has attracted many researchers from theby a sinusoidal signal with the amplitude of 3 V.
Thereby we call the left potential wall 'an oscillating

beginning of the plasma physics. Interactions between
particles and fields are primarily important and wereexperimentally studied in an electron trap [2], whereelectron heating was observed by oscillating an end
potential or two end potentials under the collisional
regime. In the present study, a similar electron trap withthe previous work has been used under the collisionlessregime. The most distinct observation was that theamount of accelerated electrons was enhanced at
specific frequencies of an oscillating end potential. This
resonant behavior of the electron acceleration has beenpotential'. The right potential wall was formed by thebias -10 V applied to the rightmost ring electrode. Notethat all the other electrodes were grounded and theheight of the right potential wall @s = -6 V was about
60 percent of the bias voltage to the rightmost ring
electrodes. We call this potential wall .a barrier
potential' since its height determines the energy ofescaping electrons along the magnetic field. The
escaping electrons were collected by a disk electrodeplaced outside of the rightmost electrode and weredetected as current 1. The electrons injected from theelectron source at the center (see Fig. l(a)) moved backand forth between the oscillating potential and thebarrier potential to form an electron cloud with ainvestigated further by the numerical study.

2. Experiment
2.1 Experimental Method
The electron trap consisting of seven ringcolumn shape. The electron density was about 2 x 107
cm-3 and the electron temperature about 2 eV. The
column radius of about 2 mm was almost as large aselectrodes with 30 mm in length and 30 mm in diameter,

which is schematically shown in Fig. l(a), was placed ina vacuum chamber with the base pressure of about 2 x10-3 Pa. Figure l(b) shows the porential profile as a*Corresponding

Debye length but much larger than the electron Larmor
radius of 0.12 mm for the magnetic field of 400 G. For@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Researchauthor's e-mail: tsushima@ ynu.ac jp591

Tsushima A. et aI., lnieraction of Electrons with an Oscillating Potential(a)ttffi=n!ry

rrdmeFs E68gr-r€ r.E@Et ffiffiEfil

EEffi ffil

Effi

ll +B - - -El.-

rir

l1

t

@rr@!*ffilE@@lE!@(b)

30

10Zzo
€
-10-8020 30 N50f (MHz)

400Z(mm)Fig.2 Current due to escaping electrons as a function offrequency of the oscillating potential for ps = -6 V

4080

andE=446G.

Fig.1 (a) Experimental setup and (b) axial potential

20

distribution along the axis.

<-13.5MH2+No

Oscillation

the energy of the injected electrons of 2 eY, the bounce
frequency between the oscillating potential and the-- lobarrier potential was about 3 MHz.

2.2 Experimental Results
Relation between the frequency of the oscillating
potential/and the current due to the escaping electrons 1
is shown in Fig. 2. The current 1 is shown to beenhanced at specific frequencies. This phenomenon

01020

indicates a resonance effect between the electron bounce
frequency and the frequency of the oscillating potential
for the electron acceleration. In Fig. 3, closed circles| 0"1
Fig. 3 Current due to escaping electrons as a function of
the barrier potential l@rl for I = 256 G.

depict the relation of the current 1 with the barrier
potential l@sl when there was the modulation of the left
potential wall with the frequency 13.5 MHz, while open
circles depict the relation when there was no3. Numerical Study
3.1 Mapping
To understand our experimental observation, wemodulation. The modulated frequency 13.5 MHz
corresponds to the lowest resonance frequency as
represented in Fig. 2. It is clear that the electron energy
increased up to about 17 eY from 5 eV by theapplied the standard mapping [3]uirr = lui + sin 0,

oscillating potential. The derivative d1ldlpsl is
proportional to the energy distribution function if weI(1)

and|it = 0i + 2nM luirl

assume that the electron energy increases monotonically,that is,/(E) * -d//dl6l . The presence of the regionsQ)

where u;, normalized by 2aa (a and a are the maximum
displacement and the frequency of the oscillating wall'with d1ldlonl > 0 suggests that some electrons decreaseenergy near the top of the barrier potential l0Bl .
592

Tsushima A. et aL, Interaction of Electrons with an Oscillatins potential

respectively), is a velocity of the particle before the i-th

lifetimes of the electrons are recorded each time and thecurrent l was estimated from the sum of the inverse of
the electron lifetimes. Figure 5 shows the current 1as afunction of the frequency/of the oscillating potentialcollision, Q is a i-th phase of the oscillating potential
wall, and M (= L /2na) is the normalized distance
between an oscillating potential wall and a fixed barrier.

The distance between the oscillating potential wall andthe fixed barrier Z is assumed to be much larger than theamplitude of the oscillating wall a. A phase space of thewall. The resonant features are clearly shown as
observed in the experiment, although someparticles is shown in Fig. 4 for M = 20 and uo = ll.5
which corresponds to E" (initial energy of electrons) = feV for parameters of a = 8.0 x 10-a m and a/2/t = 12.6

5 MHz appeared in the numerical analysis. The current
obtained in the range of high frequency above 35 MHz

discrepancies are present including the first peak aroundincreases with frequency, because the model is notapplicable to the frequency that the velocity of theoscillating potential wall becomes faster than electrons.

MHz. It is seen that some of the electrons areaccelerated beyond 10 eV and that the unoccupied
island develops in the range of about 6 eV to about I IeV in Fig. 4(a). Figure 4(b) shows the corresponding
energy distribution of the particles. It is seen that theSuch a fast movement of the wall invalidates themapping equations we used for our numerical analysis.
The resonant frequencies are found to be given by theparticles near the boundary of the island change energy

equation:

rapidly and the number of particles with that region, or/(E) with the correspomding energy, is small.

'l, (Zn-l) --Tl,f" +,[.,(n = 1,2,3...), (3)J=2_3.2 Numerical Analysis

wheref= t(2L),EE;Tn

We now set a limited potential at the fixed barrier
to compare with the experimental measurements. A
lifetime of an electron is defined as a time when theelectron escapes from the fixed barrier. The escaping
electron has an energy Eou, which is equal to elful. The

Jtn

and

f=/ (zD,lryE;TJ OUT(4)m.

4. Conclusion
We have carried out the model experiment of Fermi
acceleration and our numerical analysis clarified someof the features of stochastic acceleration observed in theexperiment. It is found that there was the resonancebetween accelerated electrons and the oscillatine
potential.

o)I{

0.010.005

0-EO0nO05O0OOt020

304050f (MHz)f(E)

Fig. 5 Current due to escaping electrons as a function ofFig. 4 (a) Phase space and (b) energy distribution of theparticles for fi" = 6 eV and M = 20.

frequency of the oscillating potential wall f lor E,"= 6 eV, E"* = 10 eV, a= 8.0 x 10{ m, and I = 0.1 m.
593

Tsushima A. et aI., lnteraclion of Electrons with an Oscillating Potential(1995); M. Tanaka et al.,Phys. Rev. Lett.81, 1163

Acknowledgements(l998).

This work was partly supported by a Grand-in-Aidfor Scientific Research (C) from the Ministry oftzl B.P. Cluggish, J.R. Danielson and C.F. Driscoll,
Phy. Rev. Lett.81, 353 (1998).
t3l A.J. Lichtenberg and M.A. Lieberman, Regular andEducation, Science Sports and Culture.

tll

Stochastic Motion (Springer-Verlag, New York,References
N. Hayashida et aI., Phys. Rev. Lett. 73, 3491
(1994); D.J. Bird et al., Astrophysics J. 441,144

1983).

594