J. Plasma Fusion Res. SERIES, Vol.4 (2001) 387-390

Study of Electrostatic Potential in the GAMMA 10 End
Region with Variation of End Plate Resistance
TATEMATSU YOshinori*, SAITO Teruo, IMAIZUMI YusuKe, NISHIDA Keiichi, YOKOYAMA Eiji,
ISHIKAWA Masao, KAJIWARA KenI), KATANUMA Isao and YATSU Kiyoshi
Plasma Research Center, University of Tsukuba
Tsukuba, Ibaraki 305-8577, Japan])Naka Fusion Research Establishment,

Japan Atomic Energy Research InstituteNaka, Ibaraki 3II-0193, Japan(Received: 5 December 2000 / Accepted:27 August 2001)

Abstract
An experiment of variation of an end plate resistance over several orders of magnitude is carried outin GAMMA 10. The axial potential distribution in the end region is measured for a wide range of an end
plate net current. A potential model which calculates a potential distribution in front of a current carrying

wall is developed. The potential depth from a mirror throat (B = 3 T) to the end plate (B = 0.01 T) is
calculated with this model. The experimental results are well explained by this model.

Keywords:
electrostatic potential, net current, secondary electron, sheath, tandem mirror

1. Introduction

To investigate the relation between the net current
and the potential distribution, we actively broke theEnd plates have been installed in front of end wallsof the tandem mirror GAMMA l0 and connected to a
vacuum vessel with a high resistance R6,p of 275 ko-.
They are supposed to control potential formation andsuppress non-ambipolar diffusion along magnetic fields.
We have developed a potential model for an end regionof a tandem mirror on the basis of floating condition offloating condition on the end plate by varying theresistance R6p in a wide range [7]. Then we havepotential distribution in the end region was wellextended our potential model to include the effect of thenet current. This paper describes the extended modeland discusses the observed potential distribution in theend region for resistance variation.
In the present paper, the experimental result of end
plate resistance variation is described in section 2. The
potential model including the effect of the net current isreproduced by the model. However, the sheath potential

developed in section 3. Then discussion and conclusion

depth in front of the end plate was a little shallower thanfollow in sections 4 and 5.

that expected by the model. A detailed analysis of anexperiment, in which secondary electrons emitted fromthe end plates are suppressed, has been shown that a

2. Experiment of End Plate Resistancethe end plates [,2]. From comparison of the model
calculations with experimental results [3-6], the axial

Variation

current balance does not hold and a negative net current
enters the end plate even for the standard high resistanceFundamental electron cyclotron resonance heating(ECRH) is applied in an end mirror cell for plugcase [4].
@2001 by The Japan Society of Plasma

Corresponding author's e-mail: tatema@ prc.tsukuba.acjp387

Science and Nuclear Fusion ResearchTatematsu Y. et al., Study of Electrostatic Potential in the GAMMA 10 End Region with Variation of End Plate Resistancepotential of the vacuum vessel (0 V).

potential generation. The cell is located inside the mirror
throat shown in Fig. 1. The axial profile of the magnetic
field strength in the end region is also shown in Fig. l.
The magnetic field strength decreases from 3 T at themirror throat to 0.01 T on the end plate in the GAMMA

A result of potential measurement for varying end
plate resistance is shown in Fig 2. The end plate
potential @6p is deeply negative for large REp. It
gradualty increases as R6p decreases to around 1 kf). For
small Rr", it approaches to 0 V. Potentials @1a7, Qs4 and@rys behave similarly to Q", for the resistancevariation. However, they approach positive values for

10 end region. Warm electrons created by ECRH andions flow out of the mirror throat and enter the endplate. They induce a secondary electron emission fromsmall R6p.
The ratio of the net current to the electron cuffent,In",lI", on the end plate is plotted against the end plate
resistance R6p in Fig. 3. For large R6p, 1,",/1" is small but
remains finite, which indicates that the current balancedoes not hold in front of the end plate. As R6p becomesthe end plate.

A part of warm electron flux out of the mirror
throat is reflected by a potential drop in front ofthe end
plate and secondary electrons emitted from the end plate

are returned by the magnetic mirror. Ion and warm
electron current densities Ii and -1" are measured with amulti-grid type electrostatic enegry analyzer on the end
plate and the secondary electron current density Iu= TI"

small. it reaches 0.6.

is evaluated by the secondary electron emission
coefficient 7 of the stainless steel end wall [4]. The
coefficient 7 includes the effect of magnetic mirror

0.5reflection [4]. The net current density is determined as
I n", = I - I, - 1," on the end plate and I,* ) 0 for electron

0VA"A.fI lv
a Or'aj'9,-lIv

I
*A.

,@ep

- 1.5-f IowI/l^ -0.5current excess.
To measure the axial space potential distribution of
the end region, Langmuir probes have been installed
midway in the end region as shown in Fig. 1. We referto them as MT-, WI- and WE-probes, respectively. The
corresponding potentials are represented as @1a7, @s,1
and @s,6. These values are measured in reference to thevrI

ls-l 10P ttttttt',

rc2 103 104 105 106
REP (o)Fig. 2 A result of potential distribution for an experiment3.sronsJ(J2.5oa-l^FL------:--.->
warm electforsof the end olate resistance variation. Potentials
are measured in reference to that of the vacuum

secondaryelectrons<-vessel.

IY;H 1.5 mirendI0.50;

0.6-cellt00.5

wE EPMT\{I111213"o'46 0.3

14z (m)0.20.1

Fig. 1 Distribution of magnetic field strength B in theGAMMA 10 end region. lons and warm electrons

0r-flow out of the mirror throat and secondary

1o-l I

electrons are emitted from the end plate (EP).
Potential distribution is studied between themirror throat (MT) and the end plate. To measure101

REP (O)

Fig. 3 The ratio of the net current to the warm electron
one on the end plate is plotted against the end
plate resistance Bgr.

axial space potential. Langmuir probes have been
installed at the positions marked by Ml, Wl andWE,388Tatematsu Y . et al., Study of Electrostatic Potential in the GAMMA 10 End Region with Variation of End plate Resistance

3. Potential Model Including the Net
Current

temperature, and the electron mass, respectively. When( is positive, the net flow is negative and electron flux

In this section the theoretical potential model [1,2]exceeds ion flux.

is extended to include the net current. The model

An example of calculated potentials is shown inFig.4. as a function of (with T=O.5. The normalized

calculates the axial potential distribution between themirror throat and the end plate.
The potential distribution is determined as follows[2]. Velocity distribution functions of the ions and thewarm electrons are given as bi-Maxwellians at themirror throat and that ofthe secondary electron is set as
a Maxwellian at the end plate. The density and current
for each particle species are calculated from the velocity
distribution with an assumption of a collisionlesspotential V/stands for the potential drop from the mirror
throat divided by ?, as V = e(aD - apM)lkT". Figure 4
indicates the normalized potentials at the probe positions

WI and WE and the end plate EP. As the parameter (increases, the normalized end plate potential tyBpincreases, on the other hand VlwE and r4r, slightly
decrease. The potentials for ( = 0 agrees with the results

calculated with the former model under the floatine
condition [2].
Variation of the potentials with the net current isexplained as follows. When 1,,, is permitted, it does notneed to reflect warm electrons in front of the end plate,thus r1r6p increases. For larger (, I/I"becomes smaller
and ion density becomes smaller than the electron one
midway the end region, thus p1az7 and W,s go down tokeep the charge neutrality by reflecting warm electrons
toward the mirror throat and by making the electron

plasma. As boundary conditions, current balance on theend plate and charge neutrality, p = qni- e(n" + n,") = 0,

at the mirror throat are imposed, where q and e are theion and electron electric charges, respectively. An
additional condition is the continuity of charge
neutrality, 0pld7 = 0, at the mirror throat. To satisfy allthese conditions, ion should have a drift energy
corresponding to ion acceleration due to the potential
drop from the plug position to the mirror throat. The
potential distribution is determined with charge

density decrease.

neutrality except a region close to the end plate where asheath with a width of several times the Debye lensth is

4. Discussion

formed.

We compare the model calculation and theexperimental results by using the following relation
between Io^ and y,Here, we permit the net current density In", and,introduce a dimensionless parameter,(=-+^l4I^",,
V kT.
-t-^eN,n(o-y)------_--'
I"f, f mTi B2p

Inn _(1)rwhere N;s, 7", and m arc the ion density at the mirror
throat for the Maxwell distribution. the electron(--t-\'/vQ)

MTe BMr

for warm electrons with one component isotropic
temperature. Here M is the ion mass, and Br7 and 86, isthe magnetic field strength at the mirror throat and at theend plate, respectively.
The relation between In",lI" and I r/u" I is shown for-ldifferent values of 7in Fig. 5. The experimental data arealso plotted.The model values with T = 0.5 - 0.6 agreewith the data points for I ys, | = 3 or for a high end
plate resistance. This value of yis consistent with the-2

value evaluated with the stainless steel secondary
electron emission coefficient and the magnetic mirror-3

reflection [4,8].

-4

For small I tlrrrl , the model value with T = O.4

0r2345agrees with the data points. When I yrr" I is small,potential acceleration of the secondary electrons
decreases and the effect of the magnetic mirror
reflection increases, thus 7 is smaller than that for( (10-s)
Fig. 4 Calculated normalized potentials with the net
current parameter (.

I389vrrl = 3.

Tatematsu Y . et aL, Study of Electrostatic Potential in the GAMMA 10 End Region with Variation of End Plate Resistance

0.6

1.t
--J'- - -.:.:.-.{;experiment

0.5

0.5C)

0.4()C)

0.3C)

o.2-x''

0.1

experiment' \0our modelr\(-0.5
1=0.5-1040 0.5 1 1.5 2 2.5 3 3.5 4tvepl

tvnpl

Fig.6 l,",ll"is plotted as a function of lVtrl with 7= 0.5.
Dotted curve is calculated with the Ordonez's
model and the solid one is calculated with ourpotential model. Dashed curve is also the result ofour model, however the potential distribution isFig.5 t^",/1" is plotted as a function of lVnl for different
values of /. The dashed curves indicate the regionof non-monotonic solution. The experimental data
are also plotted.

not monotonic.

The value of In",lI" is determined by the flux
balance, however, the model has no solution of amonotonic potential distribution between the mirror
throat and the end plate for small I ty"rl. The dashedcurves indicate the region of non-monotonic solution in

investigated. The theoretical model including the net
current was developed and the experimental results
agreed with the model indications. However, small
potential depth is attained in the experiment, monotonicFig. 5.
Ordonez has studied a sheath potential including a
net current for a non-magnetized plasma [9]. Figure 6
plots In",lI" calculated with the Ordonez's model against
potential depth I ry6p | , which corresponds to t4" in Ref[9] for a hydrogen plasma with T/7" = 0.2 and T= O.5.

solutions with small potential depth are not obtained in
the theoretical model.
The authors wish to thank the members of theGAMMA 10 group for stimulating discussions.

The result of our model is also plotted in the samefigure. For the Ordonez's model the net current becomes

ion excess (1n", 10) with lVrrl , 2.6. In thetll

References
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(1992\.

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data than the Ordonez's model though our model has no
monotonic solution with small | 14r" | . rne reason ofthis difference is mainly whether the effect of thet2l Y. Tatematsu et al., J. Phys. Soc. Jpn. 63, 558(1994).

[3] T. Saito et al.,Phys. Plasmas 2,352 (1995).
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t6l K. Kajiwara et al., J. Phys. Soc. Jpn.70, 421(200r).

5. Conclusion[7] T. Saito et al.,Tras. Fusion Tech.39, 143 (2001).
t8l K. Kajiwara et al., in Proceedings of 1996

A negative net current flows in the end plate in theGAMMA 10 tandem mirror even in the case of a highend plate resistance. The experiment of end plate

International Conference on Plasma Physics,
edited by H. Sugai and T. Hayashi, Yol.2, l3l4

resistance variation was carried out and the relation
between the net current and potential distribution was(1997').

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390