J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Electron Acceleration by Oscillating Electric Field
Akira TSUSHIMA and Osamu ISHIHARA
Faculty of Engineering, Yokohama National University, Yokohama 249-8501, Japan(Received: 6 September 2008 / Accepted: 14 January 2009)

A unique experimental system with collisionless electrons has been developed to investigate waveparticle interaction and/or particle acceleration in turbulence. In the system, electrons are injected froma tip of electron source along a magnetic ﬁeld in a region between two electrostatic potentials. Some ofthe electrons trapped in the potentials are accelerated and other electrons are decelerated by oscillating
electric ﬁeld localized at the electrostatic potential with larger magnitude. The accelerated electrons
were observed as a current into a disk electrode placed outside the steady electrostatic potential withlower magnitude. The results show that the escaping electrons’ current was the superposition of (1)current enhanced at some discrete frequencies of the oscillating electric ﬁeld and (2) current continuously
increased with increasing the frequency of the oscillating electric ﬁeld.
Keywords: electron acceleration, stochastic acceleration, wave-particle interaction, bounce motion
The problem of the resonance was removed by terminating the transmission line in the characteristic
impedance of 50 Ω in the present experiment.

1. Introduction
Cosmic ray particles with extremely high energy
range between 108 eV and 1021 eV have been observed[1] and a power law distribution in the medium energy
range has been explained by a diﬀusive shock acceleration model at supernova. The model is based on
Fermi acceleration, the original of which was the acceleration by particle scattering against moving magnetic
inhomogeneities [2].
In laboratory plasmas, Fermi acceleration hasbeen investigated in relation to wave-particle interactions or interactions between electric ﬁeld and electrons from the wave viewpoint [3, 4, 5] and from theparticle viewpoint [6, 7, 8]. Subsequently, the experiment of electron heating has been performed in anelectron trap, where oscillating electric ﬁelds interacted with bouncing electrons [9]. Similar work wasreported in the past in an after glow plasma as a modelexperiment of the transit time heating based on Fermi
acceleration [10]. In these experiments, electrons werethermalized due to frequent electron collisions.
In the present experiment, electrons were injectedalong magnetic ﬁeld from a tip of electron source andtrapped between two electrostatic potentials [11, 12],which were apart by a distance of L � 30 cm.
Since the trapped electron density was approximately
1 × 107 cm−3 under a back ground pressure of p �
10−4 Pa, the electron was able to move back and forthapproximately 104 times between the two electrostatic
potentials and the collisionless condition was satisﬁed[12]. Then, by applying an oscillating electric ﬁeld to
one of the electrostatic potentials, the electron acceleration related to the resonance of the electron bounce
motion was observed and that related to stochastic
nature was also observed. These results are diﬀerent
from the previous results [11, 12] due to the resonanceof the transmission line for the oscillating electric ﬁeld.

2. Experimental Setup
Electrons were injected in a trap consisting of
cylindrical electrodes with an inner diameter of 20 mmalong a magnetic ﬁeld with an intensity of B = 440
G by a tip of heated LaB6 (DENKA Electron Emitter, M3 IS) placed almost on the axis of the central
electrode as shown in Fig. 1 (a). The trap was set
in a vacuum vessel (base pressure: p � 3 × 10−5 Pa)
and the magnetic ﬁeld was carefully adjusted to be directed along the axis of the central electrode and theplug electrodes (A) and (B), by two sets of correction
coils, which produced magnetic ﬁelds with the directions perpendicular to the axial direction, i.e., the vertical and horizontal directions. In the result, the accuracy of the direction of the magnetic ﬁeld was � 10−4 .
In this ﬁgure, bias potentials VA and VB to the plug
electrodes (A) and (B), respectively, generate the twoelectrostatic potentials and the electrons from the electron emitter with the acceleration potential of VE aretrapped between two electrostatic potentials. Since
the axial length of the plug electrodes is much larger
than the inner diameter, the maximum absolute values
on the electrostatic potentials (A) and (B) are almost
the values of |VA | and |VB |, respectively. In fact, thepotential proﬁle on the axis such as shown in Fig. 1(b) is formed for the typical values of VA = −80 V andVB = −60 V. The dotted line represented in Fig. 1(b) corresponds to the value of VE = −30 V and its intersections with the solid curve, which represents theaxial potential proﬁle, indicate the reﬂection pointsof an electron from the electron emitter. Then, anoscillating electric ﬁeld was applied by introducing aRF signal with a frequency fA up to 50 MHz to theplug electrode (A). The transmission line of the RF

author’s e-mail: tsushima@ynu.ac.jp

65©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

A. Tsushima and O. Ishihara, Electron Acceleration by Oscillating Electric Field

trons from its initial kinetic energy is easily evaluatedby
ΔK ≥ −e(VB − VE ) .

(a)
Central Electrode
Plug Electrode (A)Plug Electrode (B)Electron SourceCollector

3. Results

BICCG
VRF

RG

Figure 2(a) shows the current IC , due to the escaping electrons through the electrostatic potential(B), as a function of time t when the frequency fAof the oscillating electric ﬁeld was swept between 1
and 50 MHz, which is represented by a sweep voltage
Vsweep as a function of time t in Fig. 2(b) under thecondition of VA = −80 V, VB = −60 V, VE = −30 V,and ṼRF � 7.5 V. Figure 3 shows the relationshipbetween the current IC and the frequency fA , which
is obtained from Figs. 2(a) and (b). In Fig. 3, fourpeaks of IC can be easily found and their frequencies
are approximately 39.5, 42.6, 44.9, and 47.2 MHz. Besides the peak with the lowest frequency of 39.5 MHz,the diﬀerence between the two adjacent frequencies is∼ 2.2 MHz, which agrees with the bounce frequencyof an electron with a kinetic energy of ∼ 4.8 eV.
Since the density of the electron cloud between thetwo electrostatic potentials were estimated to be approximately 1 × 107 cm−3 and the space potential on
the axis of the central electrode to be approximately−25 V, the value of ∼ 4.8 eV may be considered as
the initial kinetic energy K0 of the electrons from theelectron emitter: K0 ∼ 4.8 eV. In fact, the diﬀerenceof the adjacent peak frequencies was inversely proportional to the axial length L of the two electrostatic
potentials. In that case, the gain of the kinetic energyof the escaping electron can be estimated asVCRCRAVEVH

VBVA(b)Vaxis (V)0-50

E=-eVE-100-300 -200 -1000100z (mm)200300Fig. 1 (a) Schematic of experimental setup and (b) axial
potential proﬁle, where VA = −80 V, VB = −60 V,
VE = −30 V, VH = 1.8 V, VC = 1.5 V, CG =0.1 μF, RG = 50 Ω, RA = 1.6 kΩ, RC = 10 kΩ,and B = 440 G in the typical conditiion.

signal was terminated in the characteristic impedanceof RG = 50 Ω and passed through the capacitance of
CG = 0.1 μF to the plug electrode (A) so that theproblem of the resonance of the transmission line wasremoved. In the experiment, the amplitude of the RF
signal, ṼRF , was varied up to 10 V. The amplitude ofthe potential oscillation at the reﬂection point of theelectron with an energy of E can be estimated byṼreﬂect � ṼRFA=ΔK� 6.3 .
K0

If we see Fig. 3 carefully, then, we ﬁnd that the baseof IC continuously increases with increasing fA forfA � 40 MHz. So, we may say that the current IC is
composed of (1) current enhanced at some discrete frequencies of the oscillating electric ﬁeld and (2) current
continuously increased with increasing the frequencyof the oscillating electric ﬁeld.
By the variation of ṼRF , the relationship betweenIC and fA varies, as shown in Fig. 4, where the upper,middle, and lower curves represent the cases of ṼRF =10, 7.5, and 5 V, respectively, under the condition ofVA = −80 V, VB = −60 V, and VE = −30 V. Thisﬁgure shows that the magnitude of IC increases withincreasing ṼRF ; and the discretely enhanced current ofthe type (1) is clearer for the case of the smaller ṼRF
and the continuously increasing current of the type (2)is clearer for the case of the larger ṼRF .
The relationships between IC and fA for variousvalues of VB are also shown in Fig. 5 under the condition of VA = −80 V, VE = −30 V, and ṼRF = 7.5 V.
Here, the upper, middle, and lower curves represent

E,(−eVA )

since the oscillating potential proﬁle at the electrode(A) always keeps a similar shape, where e is the elementary charge. For example, Ṽreﬂect � 3.75 V for
an electron with the injection energy of E = −eVE =30 eV when ṼRF = 10 V and VA = −80 V.
In the system, some of the electrons between thetwo electrostatic potentials (A) and (B) are accelerated or decelerated when the electrons are reﬂected
at the electrostatic potential (A) with the oscillating
electric ﬁeld. The electrons reached suﬃciently highenergy by successive acceleration can escape from theelectrostatic potential (B), the magnitude of which isless than that of the electrostatic potential (A), i.e.,|VB | < |VA |. Then, the escaping electrons are detected
as a current of IC by the collector placed farther rightof the plug electrode (B), as shown in Fig. 1 (a), andthe increase of the kinetic energy of the escaping elec-66

A. Tsushima and O. Ishihara, Electron Acceleration by Oscillating Electric Field

101

0.40

1000.20

10-100204060

IC (µA)

IC (µA)(a) 0.6080t (s)Vsweep (V)

(b)

10-2
10-3~
VRF=7.5 V~
VRF=5 V

310-430240

150fA (MHz)0020406080
Fig. 4 Relationships between IC and fA for ṼRF = 10, 7.5,
and 5 V, which are represented by upper, middle,
and lower curves, respectively, under the conditionof VA = −80 V, VB = −60 V, and VE = −30 V.

t (s)Fig. 2 (a) Current IC as a fuction of time t and (b) sweep
voltage Vsweep , corresponding to fA , as a functionof time t under the condition of VA = −80 V, VB =−60 V, VE = −30 V, and ṼRF � 7.5 V.

101
100

IC (µA)

0.6
0.5

IC (µA)~
VRF=10 V

VB=-40 V

10-1
VB=-60 V

10-2

0.4
0.3
0.2
0.1
010-3

VB=-50 V

10-4
2030

4050fA (MHz)1020304050fA (MHz)Fig. 3 Relationship between current IC and frequency fA ,which is obtained from Figs. 2(a) and (b).

Fig. 5 Relationships between IC and fA for VB = −40,−50, and −60 V, which are represented by upper,middle, and lower curves, respectively, under thecondition of VA = −80 V, VE = −30 V, and ṼRF =7.5 V.

the cases of VB = −40, −50, and −60 V, respectively.
From the ﬁgure, we can see that the magnitude of IC
decreases with increasing |VB |; and the discretely enhanced current of the type (1) is clearer for the case ofthe larger |VB | and the continuous increasing currentof the type (2) is clearer for the case of the smaller|VB |.
Figure 6 shows the relationship betweem IC andfA under the condition of VA = −80 V, VB = −40 V,
VE = −30 V, and ṼRF = 10 V. From the discussionof Figs. 4 and 5, it is considered that the continuous increasing current of the type (2) appears clearin Fig. 6. In fact, the base of IC becomes large andthe ratio of the magnitude of the peak and that of

the base for fA � 40 MHz is � 1.3, which is smaller
than the ratio of � 2.5 in the case of VA = −80 V,
VB = −60 V, VE = −30 V, and ṼRF = 7.5 V (Fig. 3).
In the frequency range of fA < 40 MHz of Fig. 6,
some peaks of IC appear and the ratios of the magnitude of the peaks and that of the base are relatively
large (> 2), although the rate of the electron acceleration is very small and IC was not observed in thecase of VA = −80 V, VB = −60 V, VE = −30 V,and ṼRF = 7.5 V (Fig. 3). Thus, with the fact thatthe diﬀerence of the frequencies of the adjacent peaks
was inversely proportional to L, we see that the electron acceleration rate increased around the frequencies
with the peaked IC due to the resonance of the elec-67

A. Tsushima and O. Ishihara, Electron Acceleration by Oscillating Electric Field[3] A. Y. Wong, N. D’Angelo, and R. W. Motley, Phys.
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IC (µA)

32100

1020304050fA (MHz)Fig. 6 Relationship between IC and fA under the condition of VA = −80 V, VB = −40 V, VE = −30 V,and ṼRF = 10 V.

tron bounce motion; and we observed as the discretely
enhanced current of the type (1). On the other hand,the continuous increasing current of the type (2), orthe base of IC increased with increasing the frequency
of fA , would be related to the stochastic nature of theelectron acceleration.

4. Summary
We have investigated the electron acceleration under the collisionless condition by the oscillating electric ﬁeld in the system, where electrons were injectedfrom a tip of electron source along the magnetic ﬁeld
between the two electrostatic potentials. The oscillating electric ﬁeld was localized at the electrostatic
potential with larger magnitude and accelerated someof the electrons trapped between the two electrostatic
potentials. Then, the suﬃciently accelerated electrons
were observed as the current into the disk electrodeplaced outside the steady electrostatic potential withlower magnitude. The results show that the escaping electron current was the superposition of (1) thecurrent enhanced at some discrete frequencies of theoscillating electric ﬁeld and (2) the continuously increasing current with increasing the frequency of theoscillating electric ﬁeld. It can be considered that thediscretely enhanced current of the type (1) was related
to the electron bounce motion and the continuously increasing current of the type (2) was to the electron’s
stochastic nature.
The experiments with the oscillating electric ﬁeld
with diﬀerent two or three frequencies have been performed to investigate particle acceleration in turbulence.
[1] J. W. Cronin, Rev. Mod. Phys. 71, S165 (1999).
[2] E. Fermi, Phys. Rev. 75, 1169 (1949).

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