22P161-P
Potential Distribution in Surface Produced Negative Ion Source with Magnetic
Field Increasing toward a Wall
表面生成負イオン源における壁に向かって増加する磁場中での電位分布
Azusa Fukano1), Akiyoshi Hatayama2)
深野あづさ1), 畑山明聖2)Tokyo Metropolitan College of Industrial Technology
1-10-40, Higashi-Ohi, Shinagawa-ku, Tokyo 140-0011, Japan1)
東京都立産業技術高等専門学校〒140-0011 東京都品川区東大井1-10-40
Faculty of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan2)
慶應義塾大学理工学部〒223-8522 神奈川県横浜市港北区日吉3-14-1
Electric potential near a wall for the plasma with the surface produced negative ions with magnetic field
increasing toward a wall is investigated analytically. The potential profile is derived analytically by using aplasma-sheath equation. It is shown that the negative potential peak is formed in the sheath region near theplasma grid (PG) surface for a case of the strong surface production of negative ions or low temperaturenegative ions. It is also shown that as the increase rate of the magnetic field becomes large, the negative
potential peak becomes small.

1. Introduction
In the surface produced negative ion source,
negative ions produced on a plasma grid (PG) are
accelerated by potential difference between theplasma and the PG. However, the general sheathpotential formed in the plasma accelerates thesurface produced negative ions toward an inner ofthe ion source. It has been shown that a negative
potential peak is formed near the PG surface andthe negative ions have a possibility to be returned to
the PG surface for a case without a magnetic field[1]. In the JT-60U negative ion source in JAEA
(Japan Atomic Energy Agency), permanentmagnets are embedded in an extraction grid (EXG)
in the extraction region to suppress the accelerationof the extracted electrons [2]. These magnets
produce the cusp magnetic fields increasing toward
the PG surface inside the negative ion source.
The effect of the magnetic field on the potential
distribution near the extraction region for theplasma with the surface produced negative ions will
be studied analytically, where the magnetic field
increasing toward the PG surface is considered. The
potential distribution is obtained by using aplasma-sheath equation [3] and dependence on theamount and the temperature of the surface produced
negative ions and the profile of the magnetic field isexamined.
2. Analysis of the Electric Potential
The magnetic field is assumed to beperpendicular to the wall near the PG surface andthe problem is treated as one dimensional model inz-direction. It is assumed that negative hydrogenions H− are produced only on the PG surface andthe background plasma consists of electrons andpositive hydrogen ions H+. The geometry ofanalytical model is shown in Fig. 1. The electricpotential φ(z) is assumed to be symmetric about z=0
and zero at z=0. The magnetic field is assumed tobe symmetric about z=0 and B0 at z=0.
Constant energies E and E− of a positive ion anda negative ion in the z-direction are
E=

1
M(" #2 + " //2 ) + q$ (z)
2(1)

E" =1
2
2
M" (# $"+ # //") " q% (z)2(2)!

where M and M− are the ion masses, υ⊥ and υ⊥− arethe velocities perpendicular to the magnetic field,
υ// !and υ//− are the velocities parallel to the magneticfield, and q and −q are the charges of the positiveion and the negative ion, respectively. The magnetic
moments for the positive ion and the negative ion
are given byφ
 ,BBB0-Lφ

0Lz

Fig.1 Geometry of potential and magnetic field in
the analysis model.

µ = (1/2) M" #2 B(z)(3)

2µ" = (1/2) M"# $"
B(z)

(4)

1/ 2 .
+%% "q$ ( z#) + q$ (z) ("q$ (zmin ) + q$ (z) ( 0I" (z, z#) = exp&)erfc-&)
kTi"
kTi"
'*-,'
* 0/1/2# B(z ) " B(z) &"$ min'% B(zmin ) (where !
B(z) is the magnetic field at the position z. The
kinetic equations in the phase space for the positive!
! the negative ion are described byion and$f (z, E, µ," )"# // (z, E, µ)= S(z, E, µ)$z"# //$ (z, E $ , µ$ )%f$ (z, E $ , µ$ ," )= S$ (z, E $ , µ$ )
%z

(5)
(6)

!

!

where σ (=±1) is the direction of the particle motion,f(z,E,µ,σ) and f−(z,E−,µ−,σ) are the distribution!
functions, and S(z,E,µ) and S−(z,E−,µ−) are thesource functions, respectively.
The distribution functions are obtained byintegrating Eqs. (5) and (6) for particle trajectory on
the boundary conditions. The positive ion density ni
and the negative ion density ni− are obtained byintegrating distribution functions over the E−µ andE−−µ− spaces, respectively. As the electron density
ne, we use a Maxwell−Boltzmann distribution forsimplicity. Substituting the densities ni, ni− and ne
into Poisson’s equation and using the source
functions same as Emmert et al. [3], theplasma−sheath equation is derived as$ e" ( z) 'd 2" ( z) n 0e=exp&)
2dz#0% kTe (% MTe (1/ 2% e+ w ( L
en 0"'* exp'* - dz,I ( z, z,) h ( z,)
2#0 L(1+ $ ) & meTi )& kTe ) 0!
+!

!

1/ 2% e+ w ( L
en 0" % M$Te ('* exp'
* - dz,I$ ( z, z,) h$ ( z,)
2#0 L(1+ " ) & meTi$ )& kTe ) 0(7)Where D(z) is the Dawson function.
4. ! Numerical Solution of the Plasma-Sheath
Equation
The plasma-sheath equation (7) is normalized
and solved numerically by transforming it into a setof finite difference equations. We assume themirror ratio R=B/B0 =exp[α{ η−eφw/(kTe)}1/2] which
is similar to the expression used by Sato et al. [4],where η=(e/kTe)(φw−φ) and α is a positive constant.
The potential profile for various values of α isshown in Fig. 2, where τ=Te/Ti=2, τ−=Te/Ti−=10,
β=S0–/S0=0.8, where S0 and S0− are the averagesource strengths of the positive ion and the negativeion, and λD/L=5x10−2, where λD is the Debye length,
where the normalized potential Φ =−η=(q/kTe)(φ−φw) is shown. It is shown that the sheath potential
has the negative peak near the wall for the case of
the increase rate of the magnetic field is small. The
potential profiles for various values of β and thetemperature ratio τ−=Te/Ti are also calculated.

where z´ is the position of ion generation, Ti and Ti−
are the temperatures, h(z) and h−(z) are the sourcestrengths of the positive ion and the negative ion,respectively, me and Te are the mass and thetemperature of the electron, and φw is the wall
potential and% %( +%(1/ 2 .
3exp& q" ( z#) $ q" (z) )erfc-& q" ( z#) $ q" (z) ) 0
kTi
kTi
3 '* -,'
* 0/
3
1/ 2
3
2 % (B(z) $ B0 )q" ( z#) $ (B( z#) $ B0 )q" (z) (&)
3+#
kT
(B(z)
$B)
1'*i0
3
3% q" ( z#)B( z#) ( % B(z) $ B( z#) (1/ 2 2
3 2 exp&)+&)
1
33' kTi (B( z#) $ B0 ) * ' B( z#) *I ( z,z') = &% (q" ( z#) $ q" (z))B( z#) (+ +% (q" ( z#) $ q" (z))B( z#) (1/ 2 .
3
3 2 exp& kT (B( z#) $ B(z)) )-D-&$ kT (B( z#) $ B(z)) ) 0'*, -,'* 0/ii33 +1/ 2 ..
3 -% {(B(z) $ B0 )q" ( z#) $ (B( z#) $ B0 )q" (z)}B( z#) ( 00, q" ( z#) > q" (z),)
3 $D-&'$kTi (B( z#) $ B(z))(B( z#) $ B0 )* 0/0/
3 ,3 % q" ( z#) $ q" (z) (3exp&), q" ( z#) < q" (z) ,3' 'kTi*(8)!

!

&{#q$ ( z%) + q$ (z)}B(zmin ) + {q$ ( z%) # q$ (zmin )}B(z) )"exp(+kTi# {B(zmin ) # B(z)}'*1/ 2 .
+%((9)#q$(z)+q$(z)B(z){}
minmin"erfc-&) 0-,' kTi# {B(zmin ) # B(z)} * 0/Fig.2 Profile of the normalized potential Φ for variousvalues of α with τ =2, τ−=10, β=0.8, λD/L=5x10−2.
References[1] A. Fukano and A. Hatayama, AIP Conference
Proceedings (to be published).
[2] M. Kisaki, et. al., AIP Conference Proceedings, 1097,344 (2009).
[3] G. A. Emmert, R. M. Wieland, A. T. Mense and J. N.
Davidson, Phys. Fluids, 23, 803 (1980).
[4] K. Sato, F. Miyawaki, and W. Fukui, Phys. Fluids
B1, 725(1989).