J. Plasma Fusion Res. SERIES, Vol.4 (2001) 345-348

Particle Simulation of Potential Formation due toLocal ECR along Diverging Magnetic-Field Lines
ISHIMORI Takayuki, KANEKO Toshiro*, ISHIGURO Seijir. HATAKEYAMA Rikizo and SATO Noriyoshi
Graduate School of Engineering, Tohoku University, Sendai 980-8579, JapantNational Institute for Fusion Science. Toki 509-5292. Jaoan(Received: 5 December 2000 / Accepted: 27 August 2001)

Abstract
Plasma potential formation due to local electron cyclotron resonance (ECR) along diverging
magnetic-field lines is investigated by using a two-and-a-half dimensional particle simulation. When an
electromagnetic wave for ECR is applied to a plasma flow, a potential drop in the downstream region

from the ECR point is enhanced. In the case of the small wave amplitude, the potential structure isformed based on an ambipolar mechanism. In the case of the large wave amplitude, on the other hand,trapped electrons in the downstream region induce a further increase in the potential drop superimposedon the ambipolar potential.

Keywords:
local electron cyclotron resonance, diverging magnetic-field lines, particle simulation, ambipolar plasma
potential, trapped electrons

1. lntroduction
Plasma potential formation along diverging

strong increase in the electron pitch angle.

On the other hand we have qualitatively demonstrated the potential formation due to local ECR at amagnetic gradient region in basic experiments with a Qmachine plasma apparatus, where a large potential drop
is observed in the downstream region from the ECR
point [7]. It is also shown that the potential formation isclosely related to electron and ion energies parallel tothe magnetic field after the ECR heating and that ionsmagnetic-field lines is an important subject concerning
plasma thrust by expansion in a magnetic nozzle [], ion
impinging in reactive plasma processing [2,3], ion conic
generation in the ionosphere [4], and thermal flow to adivertor region in a fusion oriented plasma [5]. As forthe potential formation in a laboratory experiment using
an electron cyclotron resonance (ECR) discharge plasma

to the field profile of a diverging magnetic-field

are accelerated up to the value equivalent to thepotential drop. However, this experimental work isconfiguration [6]. Since an electron pitch angle is larger

limited to a phenomenological investigation and thethan ion's even at the entrance of the diverging

quantitative dependence ofpotential structure on plasma
parameters is not clarified. Concerning a particle

source, it is reported that the potential profile is similar

configuration in such a plasma, a charge separation
takes place at the magnetic gradient region, resulting in
the potential decrease toward the downstream region insimulation on this problem, the relationship between thepotential and the plasma parameters was investigated byorder to cancel this charge separation. When ECR

Hooper [8]. In this simulation, however, the electron

heating is artificially realized by changing thelocally occurs at a magnetic gradient region, the charge
separation is expected to be more enhanced because of a*

C o rre sp ondin g autho r' s e - mail : knne ko @ e c e i, t o hoku. acjp345@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchIshimori T. et al., Particle Simulation of Potential Formation due to Local ECR along Diverging Magnetic-Field Lines

perpendicular energy distribution function at the ECR

the condition of the low density plasma.

point in the absence of an electromagnetic wave

Simulation parameters used in this paper are as
follows. The ion to electron mass ratio mifm.is fixed at400. The ratio of the electron cyclotron to electron
plasma frequency is ro".s/rop"s = 5, and the ratio of theapplied electromagnetic wave to electron cyclotron
frequency is rrla1""5 = 0.9. The normalized waveamplitude Eu1= Eu/17.r/"1.p"5)) is changed from 0 to 1.2.

propagation. Thus, the correlation between the wave
heating and the potential structure has not been known.

Therefore, in order to clarify the above-mentioned
points in the fashion of quantitative analysis, weinvestigate the dependence of the potential structure on

the wave amplitude and the time evolution of spatial
potential-profile through a two-and-a-half dimensional

particle simulation which realizes local ECR byHere, 2"5 and ,1p"5 are the electron temperature and theDebye length, respectively. The time step width A/ is

externally applying an electromagnetic wave.

0.02arpls. The system sizes L, and L, are 512.1p"5 and2. Simulation ModelWe employ a two-and-a-half dimensionalparameters in the reservoir.

electrostatic particle simulation code on magnetizedplasmas. Our simulation code applies the standard
particle-in-cell (PIC) method [9], which follows particle
motions in a self-consistent electric field generated byparticles and an external stationary magnetic field. A

3. Simulation Results

l28LD.s, respectively. The subscript "S" stands for theA typical example of the plasma potential eQlT"s atllLo"s = 64 in the steady state (a6"s/ = 3000) with R--r =2.0 is presented in Fig. 2(b), where an arrow at x/),p.s 291 indicates the position of the ECR point. For Eu= 0.0(dotted line), the potential decreases slightly toward thecollector along the magnetic-field lines and its profile issimilar to that of the magnetic-field strength . For E, =simulation system with magnetic-field lines isschematically shown in Fig. l. Plasma particles are
assumed to be emitted at x = L, and y = 0.4Ly - 0.6Ly
n5 and temperatures 15. A floating collector is placed at0.2 (solid line), on the other hand, the potential
decreases strongly in the downstream region from the.r = 0 and }l = 0 - Zr. The simulation system is initially

ECR point. The electrons are accelerated by the ECR inempty and a particle injection with a constant rate starts
at t = 0 from the emitter. The boundaries at -{ = Lo ! =0, and y - L, are grounded while the collector (-r = 0)
potential is floated. These conditions correspond to theQ-machine experimental configuration. An external

the direction perpendicular to the magnetic field andaccelerated in the axial direction by -ptYxB force, while
the ions are not directly affected by the ECR. Thus, thepotential drop which decelerates electrons andaccelerates ions is expected to be formed in thestatic magnetic field B is shown in Fig. 2(a) and adownstream region under the quasi-neutrality condition

from a plasma reservoir with electron and ion densities

degree of magnetic divergence R--r is defined as thereciprocal of mirror ratio R. which is the ratio of B atx = L, to that at x = 0. An externally-applied(a)electromagnetic wave with right-hand circular polarization propagates toward the emitter and the wave
amplitude E, is assumed to be spatially constant under

Ly Collector

1(0 = oceU)m; o.sECR(b)Plasma

EmitteraoF(l)v/:1r-10^eu=ooll-5- Er, = 0.2

0128 256 384

512

x / l.Des

LxFig. 2 Spatial profiles of (a) magnetic field strength and(b) plasma potential eflT"" al ylh"s = 64 with R.-1
= 2.0 lor Ep = 0.0 (dotted lines) and 0.2 (solid lines)
at a4"sf = 3000.

Fig. 1 Schematic of simulation system and diverging
magnetic-field line346

Ishimori T. et al., Particle Simulation of Potential Formation due to Local ECR along Diverging Magnetic-Field Lines(p : magnetic moment). Here, a potential peak is also

point is considered to cause the saturation of e"11/2"5 inobserved around the ECR point. Since the only electrons

the downstream region. Furthermore, since the electrons

are accelerated by the ECR in the axial direction as
described above, the ions are left around the ECR point.
Thus, the ion rich condition is realized and the potential

are deceleratedby leQlT"sl which increases even for Er2

0.4, e"nlT.s in the downstream region is considered todecrease gradually for Eu>0.4.
In order to investigate the difference of leQlT"sl ande"1/2"5 dependences on larger Euas shown in Fig. 3 indetail, temporal evolutions of leQlT.sl, electron densityn", and ion density n1 in the downstream region (x/,1,p.5 128) with R,-r = 2.0 for Eu= 0.2 and 0.8 are presented
in Figs. 4 and5, respectively. In both the cases of Eu=
0.2 and 0.8, leQlT6l increases as time goes by in thepeak is expected to be formed there.

Since the potential drop le@/I.51, which is thepotential difference between the emitter and thedownstream region (xl\"s = 128), is considered todecelerate and reflect the electrons so as to satisfy thequasi-neutrality condition in the downstream region,
leQlT.sl is expected to be related to an average electronenergy parallel to the magnetic-field lines e"1/2"5 andresult in accelerating ions up to the same energy of theelectrons. In order to clarify the relation between them,leQlT"sl and e";1/T"5 in the steady state (r+"st = 3000 4000) are plofted as a function of E, with R^-t = 2.0 at xl
Lo"s= I28, as given in Fig. 3. leQlT"tl increases with anincrease in .6u, being followed by a gradual saturation

same way, but the time when the potential attains to themaximum and its value are different. In the case of E, =0.2, leQlT.sl starts to apparently increase at rones/ = 1200,saturates after attaining to the maximum value at ,Jbes/ =1500, and the local maximum value is observed only
once. The time of (or.st = 1200 corresponds to the time
when n. starts to increase in the downstream region as

for Eu2 0.8. On the other hand, e"1/2.5 increases in theshown in Fig. a(b). The other time of @p"5r = 1500

same way as leQlT.sl, but gradually decreases for EuZ

corresponds to the time when n; starts to increase, which

0.4. The decrease in e"1/2"5 seems to be related to themeans that the plasma flow arrives at the observation

efficiency in an energy transfer from the wave topoint (.r/.1p"5 = 128).In the case of Eu= 0.8. on the otherelectrons at the ECR point. Then, we measure average

hand, leQ/7"51increases not only at or"st = 1200, whichis the same as in the case of Ep= 0.2,but also at apest =electron energies parallel t"tr,lT.s and perpendicular€eLJTes to the magnetic-field lines at ECR point in thesteady state. As a result, it is found that the passing timeof electrons in the ECR region becomes shorter due toincreased €.1JT.s, and thus t.tJT.s does not increase

2200, attaining to the maximum value just after thesecond increase. Furthermore, the times of the first andthe second increases in leQlT"sl correspond to the timesof the first and second increases in r", respectively.
These second increases inleQlT"sl and n" are observed in
the time scale when the plasma flow arrives at theeven if Eu is increased. Therefore, the saturation of thetotal energy E",lT"s (= €"t,/T"s * t.ylT.s) at the ECR

collector, which will be discussed in Section 4 in20

201515(a) 20

troooFao
1-(l)E

10-=-10o

Er. = 0.2

T15o)q,)

(b) 0
^4
N'g3
a2

o.4

0.8't.2€1c0E,,

01000 2000 3000 4000

Fig. 3 The absolute value of potential drop le@/[51opest(closed circles) and the average electron energy
parallel to the magnetic field e",,/[. (closedtriangles) at xl),s.s = 128 as a function of E" withR^t = 2.0 in steady state.

Fig. 4 Temporal evolutions of (a) the absolute value of
potential drop le@/il"l and (b) the density n/n" at
x/h.s='128, y/1t = 64 with R^' = 2.0 lor Er= 9.2.

"347

Ishimori T. et al., Particle Simulation of Potential Formation due to Local ECR along Diverging Magnetic-Field Lines(a)magnetic mirror is enhanced with an increase in Er,
because the electrons can not pass through the mirror
field due to the strong -ttY1B force with large p in thecase of large Eu. Thus, le@/Q5l continues to increase dueto the trapped electrons even though e"1/2"5 decreases
with an increase in fl.

20Er, = 0.8oo 15F 10o50NoXa325. Conclusion
The plasma potential formation under the1configuration of diverging magnetic field is investigated00by using a two-and-a-half dimensional particle

1000 2000 3000 4000

simulation. The potential structure equivalent to theexperimental result which has been demonstrated isconfirmed to be observed. The absolute value of thepotential drop in the downstream region from the ECR
point increases depending on the wave amplitude Eu. In
the case of the small Eu, the potential drop is generated
as an ambipolar diffusion process, the value of whichcorresponds to the parallel electron energy €e1. In thecase of the large Eu, on the other hand, it is found that
leQlT"sl more increases due to the electrons trapped in
the downstream region between the magnetic mirror andthe sheath potential in front of the collector. Thus, it ispossible to generate the potential structure whichefficiently accelerates ions without the requirement forOoest

Fig. 5 Temporal evolutions of (a) the absolute value of
potential drop leplilsl and (b) the density n/n atilh.s = 128, Yl)o"s = 64 with R^'' = 2.0 for Eu = g.g.

connection with the electron dvnamics.

4. Discussion
Let us mention a mechanism of the potentialformation. In the case of the small wave amplitude (Er=0.2), leQlT.sl starts to increase as soon as the plasma

flow arrives at the ECR point (46".t = 1200), as alreadyshown in the temporal evolution of leQlT6l, n", and ny ofFig. 4. Since the large amount of electrons areaccelerated by ECR and supplied to the downstream
region at that time, n. at xlhe"s = 128 extremely

high e"11.

Acknowledgment
The computations were performed at the Computer

Center of Tohoku University and at the Advanced
Computing System for Complexity Simulation at theincreases, causing the increase in leQlT6l. When ionssubsequently arrive at the observation point (xllo"t =128), leQlT"sf saturates because the excess of thenegative charge density begins to be compensated by thepositive ions. Here, leQlT"sl is approximately equivalent
to the e"n/2"5 in the downstream region. In this sense theNational Institute for Fusion Science.

tll

potential formation is due to the ambipolar diffusion

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