J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing
toward a Wall
Azusa FUKANO and Akiyoshi HATAYAMA1)
Monozukuri Department, Tokyo Metropolitan College of Industrial Technology, Shinagawa, Tokyo 140-0011
1)
Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan(Received: 18 October 2009 / Accepted: 28 March 2010)

Electric potential near a wall in plasma with magnetic field increasing toward a wall is investigated analytically. A
magnetic field that is symmetric about an axis and increases monotonically toward the wall is considered. The potential
profile is analyzed by solving the plasma-sheath equation that gives the electric potential in the plasma region and thesheath region near the wall self-consistently. The potential in the plasma region depends on the profile of the magnetic field
and the ion temperature. As the magnetic field near the wall increases and the ratio of the ion temperature to the electron
temperature decreases, the potential drop in the plasma region decreases. The potential also depends on a ratio of theincrease rate of the magnetic field to the decrease rate of the potential toward the wall. When the increase rate of themagnetic field is larger than the decrease rate of the potential, the potential drop in the plasma region becomes large
compared with that in the magnetic field of which increase rate is smaller than the decrease rate of the potential.
Keywords: negative ion source, cusp magnetic field, cusp loss width, heat transmission coefficient, electric potential,
plasma region, sheath region, plasma-sheath equation,

magnetic field of which strength increases toward the wallsuch as the cusp magnetic in the negative ion sources has
not been clearly understood.
In this paper, we will investigate the potential profilenear the wall in plasma with the cusp magnetic field
increasing monotonically toward the wall. The
plasma-sheath equation is derived and solved
self-consistently over the region from the plasma to the wall,
where not only a magnitude of the magnetic field but also a
ratio of the increase rate of the magnetic field to thedecrease rate of the potential toward the wall is considered.

1. Introduction
Neutral beam injection (NBI) using negative ion
source is one of the most promising method of heatingplasma confined magnetically in Tokamak. Plasma in anegative ion source is confined by a cusp magnetic field inorder to reduce plasma loss on the wall. However, plasma
particles that arrive to the cusp magnetic field move along
the magnetic field and are lost through the cusp magnet
region on the wall. Therefore, in plasma confinement in theion source, it is important to investigate the plasma loss
through the cusp magnetic field, especially the width of theplasma loss region, the so-called ‘cusp loss width’. It has
been indicated that the cusp loss width of electron energy
depends on a heat transmission coefficient defined by theratio of the heat flux to the particle flux multiplied by theelectron temperature along the magnetic field [1]. In
general, a sheath potential is formed in the plasma near thewall surface. Since the heat transmission coefficient isdetermined by the sheath potential near the wall [2], thesheath potential needs to be investigated in plasma
confinement in the ion source.
Emmert et al. investigated formation of the potentialconsidering both the plasma region and the sheath region
self-consistently by using a plasma-sheath equation [3].
Sato et al. extended the method of Emmert et al. to a caseof magnetized plasma. In their analysis, the magnetic fieldof which strength decreases monotonically toward the wall
was considered [4]. However, the potential formation in
the plasma and the sheath regions for the case of the

2. ModelThe geometry of analytical model is shown in Fig. 1.
The electric potential φ(z) is assumed to be symmetric about
z=0 and decreases monotonically for z>0 and zero at z=0.
The magnetic field is assumed to be symmetric about z=0
and increases monotonically for z>0 and B0 at z=0. Plasma
is assumed to consist of electrons and positive hydrogenions.

3. Analysis of Electric Potential
Constant energy E of an ion in the z-direction isE=

1
M(" #2 + " //2 ) + q$ (z) ,2(1)where M is the ion mass, υ⊥ and υ// are the velocities
perpendicular and parallel to the magnetic field, and q is the!
charge of the ion, respectively. The magnetic moment isauthor’s e-mail: fukano@s.metro-cit.ac.jp598©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

A. Fukano and A. Hatayama, Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing toward a Wallφ, B-µ B(z)-qφ(z)

B
B0
−L

φ

0

L

z

0

−L

z

L

−µ B0

Fig.2 Dependence of-µB(z)-qφ (z) on z. Solid line is for thecase of µ(B(z)-B0)>-qφ(z) and dotted line is the case ofµ(B(z)-B0)<-qφ (z).

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

given byµ = (1/2) M" #2 B(z) ,(2)

E=µB(+- L)+qφ (+- L)
E-µB(z)-qφ (z)where B(z) is the magnetic field at the position z. The kinetic
equation in the phase space (z,E,µ) is described by!
$f (z, E, µ," )(3)"# // (z, E, µ)= S(z, E, µ) ,
$z

E=µB0

−L −zt

!

0

zt L

z −L

(i)

where σ(=±1) is the direction of the particle motion,f(z,E,µ,σ) is the distribution function, and S(z,E,µ) is the!
sourcefunction. The velocity parallel to the magnetic fieldis given by" // = [(2 / M){E # µB(z) # q$ (z)}]1/ 2 .

E-µB(z) -qφ (z)

−zt

0

zt L

z

(ii)

Fig.3 Energy spaces of the ion of (i) upwards convex and(ii) downwards convex

The distribution functions f(z,E,µ,σ) for the cases of (i)and (ii) are obtained for σ= ± 1 by integrating Eq. (3) for
particle trajectory on the boundary conditions. The sum of
the distribution functions about σ=±1 for each energy regionof the cases (i) becomes(4)

We assume a symmetry about z=0, that is, φ(−z)=φ(z),B(−z)=B(z), and S(−z,E,µ)=S(z,E,µ). Furthermore, we!
assumethat particles are not reflected at the wall, then theboundary condition of the distribution function isf(-L,E,µ,+1)=f(L,E,µ,−1)=0.
In the magnetic field of which strength increases toward
the wall, dependence of -µB(z)-qφ(z) on z is divided into twocases as shown in Fig. 2. The case (i) is µ(B(z)-B0)>-qφ(z),where -µB(z)-qφ(z) becomes upwards convex as shown by !
the solid line. The case (ii) is µ(B(z)-B0)<-qφ(z), where-µB(z)-qφ(z) becomes downwards convex as shown by thedotted line. Where monotonic decrease of the electric
potential and monotonic increase of the magnetic field areassumed.
In each case, particle motion is divided into someregions depending on it’s energy, which comes from the !
condition that υ// must be real number, that is,E− µB(z)−qφ(z)≥0 as shown in Fig. 3. For the case of
upwards convex, ion in the energy region of
E> µB(±L)+qφ (±L) can reach and pass thorough the centerof the plasma. Ion in the energy region ofEmin<E< µB(±L)+qφ(±L) cannot reach the wall and repeats
the reciprocation between the turning points z=−zt(E) andzt(E), where Emin=µB(z)+qφ (z). Ion in the energy region of
E<Emin cannot exist. On the other hand, for the case of
downwards convex, ion in the energy region of E>µB0 canreach and pass thorough the center of the plasma. Ion in theenergy region of Emin<E< µB0 cannot reach the center of theplasma and reflected at the turning point z=zt(E) for σ =−1
and z=−zt(E) for σ =1. Ion in the energy region of E<Emincannot exist.

599( L S( z$, E, µ)2&dz$, (E > µB(±L) + q' (±L)),** 0 % ( z$, E, µ)# f (z, E,µ," ) = ) zt ( E,//µ ) S(z$, E, µ)*2"dz$, (E min < E < µB(±L) + q' (±L)),&*% // ( z$, E, µ)+ 0(5)and that of the case (ii) becomes' L S( z$, E, µ)))2 & 0 % ( z$, E, µ)dz$, (E > µB0 ),(6)# f (z, E,µ," ) = ( L // S(z$, E,µ))2"dz$, (E min < E < µB0 ),)* & zt (E ,µ ) % // ( z$, E, µ)respectively, where z´ is the position of ion generation. In
the case (i), although the ions trapped between the turning
points go back and forth and lead the infinite ion density, we
assumed for simplicity that the trapped ions escape from thesystem due to collisions. Under this assumption, Eq. (5) is
derived by integrating only for one cycle of the trapped
particle orbit and the results for the case (i) are valid only
under the condition which this assumption is hold.
The ion density is obtained by integrating f(z,E,µ,σ)over the E-µ space as [5]n i (z) =!

2"B(z)f (z, E, µ,# )& % dE % dµ $ (z, E,µ) .
M2 #
//

(7)

Substituting Eq. (5) into Eq. (7), an equation that gives theion density ni for the case (i) is obtained

A. Fukano and A. Hatayama, Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing toward a Walln i (z) =4 "B(z) ' %dE) & #q$ (z)B(±L )M 2 ( B(z)#B 0 +q$ (±L )*&1{E #q$ (±L )}B (±L )
#q$ (z)
B(z)#B 0+ &#q$ (z)B(±L )+q$ (±L )B(z)#B 0#q$ (z )B (z )+q$ (z )B(z)#B 0*&

0
1{E #q$ (z )}B (±L )

1

* $ 0B s

L S( z ,, E, µ)
1
dz,
dµ
&
0
+ // (z, E, µ) + // ( z,, E, µ)dE &1{E #q$ (z)}B (z )#q$ (z)
B(z)#B 0{E #q$ (±L )}dµz t (E, µ ) S( z ,, E, µ)
1
dz,/.
&
+ // (z, E, µ) 0
+ // ( z,, E, µ) .

4 "B(z)
M2

+ &
+ &

'q( (z)B 0
B(z)'B 00+&0
q( (z)

dE &

dE &

dE &

1
S( z%, E, µ)
dµ
dz%
&
0
$ // (z, E, µ)$ // ( z%, E, µ)

1{E 'q( (z)}B (z )0

L S( z %, E, µ)
1
dµ
dz%
&
$ // (z, E, µ) zt $ // ( z%, E, µ)!

L S( z %, E, µ)
1
dµ
dz%/.
&
z
t
$ // (z, E, µ) $ // ( z%, E, µ) .

The integral regions of E and µ are divided to two cases
according to the conditions that υ// must be real number foreach case of (i) and (ii). The case (a) is the increase rate of
the magnetic field is smaller than the decrease rate of thepotential, that is, (B(z)-B0)/(B(z’)-B0)<φ(z)/φ (z’) for z’<z and(B(z)-B0)/(B(z’)-B0)>φ(z)/φ (z’) for z’>z. The case (b) is theincrease rate of the magnetic field is larger than the decrease
rate of the potential, that is, (B(z)-B0)/(B(z’)-B0)>φ(z)/φ (z’)
for z’<z and (B(z)-B0)/(B(z’)-B0)<φ(z)/φ (z’) for z’>z.
By interchanging the order of integrations of Eqs. (8) !
and (9) and taking sum of the integrations of them for eachcase of (a) and (b), respectively, the ion density can be
written as for the case (a)n i (z) =1{E %q, (z)}

0

L

dµ

%E p B 0

1

0

Ep

0

(E %E p )dµ

& E $ e% (z) ) , (12)M2# // (z, E, µ)exp'$*
4 " (kTi ) 2
kTi +
(

L

* dz)I(z, z))h(z)) ,

(13)

0

where I(z, z’) is given bycase (a):
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+
kTi (B( z#) $ B0 )1'*
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/
i
i
3
3 +
1/ 2 ..
3 -% {(B(z) $ B0 )q" ( z#) $ (B( z#) $ B0 )q" (z)}B( z#) ( 00, z# < z,)
3 $D-&'$
kTi (B( z#) $ B(z))(B( z#) $ B0 )* 0/0/
3 ,3 % q" ( z#) $ q" (z) (3exp&), z# > z ,
3' '
kTi
*

1/ 2
% %
( +%
( .
3exp& q" ( z#) $ q" (z) )erfc-& q" ( z#) $ q" (z) ) 0 , z# < z ,
kTi
kTi
3 '
* -,'
* 0/
3
1/ 2
3 % q" ( z#) $ q" (z) ( 2 % (B(z) $ B0 )q" ( z#) q" (z) (
$
)$
&
)
3exp&
3 '
kTi
kTi *
1 ' kTi (B(z#) $ B0 )*I(z,z') = &1/ 2% B( z#)q" ( z#) ( % B( z#) $ B(z) (3))$&
3 2 exp&' kTi (B( z#) $ B0 ) * ' B( z#) *
3
3
+%
%
((.
3 2 erfc-& B( z#) & (B(z) $ B0 )q" ( z#) $ q" (z) ))0 , z# > z ,kTi **0/3'-,' B( z#) $ B(z) ' kTi (B( z#) $ B0 )1
S( z#, E, µ) /,+ // (z, E, µ) + // ( z#, E, µ) .

(15)
Where D(z) is the Dawson function [5](10) !

x!

1
S( z#, E, µ) /,+ // (z, E, µ) + // ( z#, E, µ) .

(14)

1
S( z#, E, µ)+ // (z, E, µ) + // ( z#, E, µ)+ $ dz# $ B p %B 0 dE $ B pdµcase (b):&4 "B(z) ' L)) $ dz# $ %E p B 0 dE
20M (B p %B 0* $ B (z ){E %q, (z )}

1/ 2

L

1
{E 'q( (z)}B (z )
E
B0

1

0# "M &n i (z) = S0 %($ 2kTi '(9)!

%E s B 0

q, (z )

1
S( z#, E, µ)+ // (z, E, µ) + // ( z#, E, µ)where Ti is the temperatures, h(z) is the source strength, andS0 is the average source strength of the ion, respectively.
Substituting Eq. (12) into Eqs. (10) and (11) and integratingthem for µ and E, the ion density becomes!

'q( (z)B(z)'B 00L0S(z, E, µ) = S0 h(z)E
) 'q( (z )B 0
L S( z %, E, µ)
1# + & 0B (z )'B 0 dE & 0B 0 dµdz%&
0 $ ( z %, E, µ)
$
(z,
E,
µ
)
*
//
//
,
'q( (z )B 0
B (z )'B 0dµwhere Ep=qφ(z’), Bp=B(z’), Es=qφ (z) and Bs=B (z) for z’<zand Ep=qφ(z), Bp=B(z), Es=qφ (z’) and Bs=B (z’) for z’>z
according to the conditions that υ// must be real number.
As the source function S(z,E,µ), we use the expression
same as Emmert et al. [3]!

And substituting Eq. (6) into Eq. (7), that for the case (ii) is
obtainedn i (z) =(E %E s )

(11)

(8)

!

&
4 "B(z) ' L) $ 0 dz# $ %E s B 0 dE
2
M (B s %B 0+ $ dz# $ B s %B 0 dE $ B(z)1dµ+ // (z, E, µ)%S( z,, E, µ)dz, + & #q$ (z )B (±L )dE+q$ (±L )+ // ( z,, E, µ)
B (z )#B 0

z t (E, µ )* & B(z)1n i (z) =D(x) = " exp(t 2 )dt .

and for the case (b)

(16)

0

For the electron density ne, we use a Maxwell−Boltzmann
600!

A. Fukano and A. Hatayama, Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing toward a Wallcase (b):distribution for simplicityn e (z) = n 0 exp{e" (z) /kTe } ,.
44exp{Z$ (" % "#)}erfc {Z$ (" % "#)}1/ 2 ,"# < " ,44.
(+4exp{Z$ (" % "#)} % 2 / R(") %1 * q' w % Z$"#4& 0 R("#) %1 ) kTi
,
4
1/ 2
1
.
(
+
(
+1
4 q'
R("#) q' w
I(","#) = / % * w % Z$"-2 exp/
% Z$"#-2*#,3,3
0 R(" ) %1 ) kTi
4 ) kTi
4
1/ 2
5
.
.
+
.
1
4
R("#) % R(")
R("#)
R(") %1 ( q' w
4%
erfc7/% Z$"#/*
4 /0 R("#) 23
4
#
#
"
)
%
R(
"
)
"
)
%1
kT
R(
R(
) i
,
760
0
4
1/ 2 841(+1
4 % * q' w % Z$"-242 : ,"# > " ,4 ) kTi,334 :90(17)[where n0 is the density at z=0, −e is the electron charge, k is
the Boltzmann’s constant, and Te is the electron temperature.
!
Substituting Eqs. (13) and (17) into Poisson’s equation,
the plasma-sheath equation is derived as"2D$ e# (z) ' q S0 $ +M '1/ 2 L
e d 2#=exp&)*&) - 0 dz,I(z, z,)h( z,) ,kTe dz 2% kTe ( e n 0 % 2kTi (

(18)
2 1/2

where λD=(ε0kTe/n0e ) is the Debye length. The averagesource strength S0 is decided by the equilibrium of the fluxesof the plasma particles at the wall. We consider that !
jiw+jew=0, where jiw is the ion current density and jew is theelectron current density at the wall, then!

 S q L = n #% kTe &( exp#% e) w &( ,00$ 2"m '
e
$ kTe '
1/ 2

(23)
where the mirror ratio R can be expressed by the function ofη because that the electrical potential and the magnetic field
are assumed to vary monotonically and the coordinate s
corresponds to the value of η at each position. The(19)

normalized plasma-sheath equation (21) is solved
numerically by transforming it into a set of finite difference
equations and using a Newton method and a successiveover−relaxation method [6, 7]. The boundary conditions aredη/ds|s=0=0 and η(s=1)=0. We assume that the ion source isuniform, that is, h(z)=1.

where m is the electron mass and φw is the wall potential.
Substituting S0 given by Eq. (19) into Eq. (18), we obtain!

$ e# (z) ' 1 $ M Te '1/ 2$ e# '
e d 2# (z)"= exp&)* &) exp& w )kTe dz 2
kT
2L
m
T
% e (
%
% kTe (i(2
DL

+ - dz,I(z, z,)h( z,).
0!

The mirror ratio increasing monotonically toward thewall can be expressed by using the potential decreasing
monotonically. For the case (a), we assume the mirror ratio
similar to the expression used by Sato et al. [4](20)

4. Numerical Solution of the Plasma-Sheath
Equation[

1/ 2
1d 2" 1 $ M '= & # ) + 0 ds*I(","*)h( s*) , exp(,") , (21)2ds
2% m (

case (a):.
1/ 2;exp{Z$ (" % "#)}erfc {Z$ (" % "#)};
1/ 2
+ ( q'+1;
2 . R(") %1 ( q' w
% Z$"#- % * w % Z$"-2/*;+& 0 R("#) %1 ) kTi, ) kTi,3;;. R("#) ( q'
1 . R(") % R("#) 11/ 2 2+; 4 exp/2* w % Z$"#-2 + /R("#) 3;&,3 0
0 R("#) %1 ) kTi
;
1/ 2
5
5
.
1 . %R("#)
1 8
;
R("#)
Z$ (" % "#)27D7/
Z$ (" % "#)2 :I(","#) = / 4 exp/
0 R("#) % R(")
367 760 R("#) % R(")
3 :9
;
;
5
.
.
+
;
;
R("#)
R(") %1 ( q' w
; % D7/;% R("#) % R(") / R("#) %1 * kT % Z$"#)
,
7
0
i
0
6
;
;
1/ 2 88+1;1 :; ( q' w:"# < " ,; % * kT % Z$"-22; :: ,
)
,33 9
i
;
9
;
"# > " ,0exp{Z$ (" % "#)} ,

[

],

(24)

where α is a positive constant. Generally the plasma
parameters in the negative ion source are Te=1~5eV,!
Ti=0.3~0.5eV, ne=3×1011~3×1012cm-3 and spatial scales of
the three sides of the negative ion source is about several ten
cm~1m, respectively. Considering Horiike et al.’s
experimental parameters [8], that is, Te=2eV, ne=2×1012cm-3
and L=7.9cm, the Debye length is λD~7.4×10-4cm, thenλD/L~10-4. Since we use the mesh width Δs=Δz/L=2×10-2
from the limitation of the computational time, we choose
λD/L=5×10--2, which corresponds to a region near the wall(about 1.5×10--2cm from the wall). The relation between
the potential and the mirror ratio given by Eq. (24) for
various values of α is shown in Fig. 4, where the normalized
potential ψ=η−eφw/(kTe)=−eφ/(kTe) is shown instead of η. It
is shown that the increase rate of the mirror ratio is smaller
than the decrease rate of the potential. The profiles of themirror ratio and the potential for various values of α areshown in Fig. 5 and Fig. 6, respectively, where Z=1, τ=1and λD/L=5x10−2. As the value of α increases, the mirror
ratio becomes large, especially near the wall, and thepotential drop in the plasma region decreases. This may be
because that as the value of α increases, the ions arereflected by the increasing magnetic field near the wall andwhere η=η(s), η'=η(s'), A2=(λD2/L2)exp(−eφw/kTe), andI(η,η’) is given by!

1/ 2

R(") = exp #{" $ e% w /(kTe )}Here, we introduce the normalized variables:
η=(e/kTe)(φw−φ), s=z/L, τ=Te/Ti, Z=q/e, R=B/B0, where R is
the mirror ratio. Equation (20) is normalized as
A2

]

]

(22)
!

601

A. Fukano and A. Hatayama, Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing toward a Wall

Fig.4 Relation between the normalized potential ψ = −eφ/kTe andthe mirror ratio R given by Eq. (24) for various values of αFig.7 Profile of the normalized potential ψ = −eφ/kTe for various
values of the temperature rate τ =Te/Ti with λD/L=5x10-2(case (a)).

(case (a)).

On the other hand, for the case (b), we assume themirror ratio to be given by [4]R(") = exp[#{" $ e% w /(kTe )}] .

(25)

The relation between the potential and the mirror ratio given
by Eq. (25) for various values of α is shown in Fig. 8. It is!
shown that the increase rate of the mirror ratio is larger thanthe decrease rate of the potential. The profiles of the mirror
ratio and the potential for various values of α are shown in
Fig. 9 and Fig. 10, respectively, where Z=1, τ=1 and
λD/L=5x10−2. It is shown that as the value of α increases, themirror ratio becomes large, especially near the wall morethan the case (a). Although as the value of α increases thepotential drop decreases, dependence of the potential profileon the value of α is smaller than the case (a). This may be
because that a reflection effect of the potential on the ions issmaller than the case (a). In the case (a) the ions arereflected by the increasing magnetic field near the wall andreciprocated between the two turning points, at the sametime the ions are reflected by the potential near the wall. On
the other hand, in the case (b), since the decrease rate of thepotential is smaller than the increase rate of the magnetic
field toward the wall, the reflection effect on the ions by thepotential is smaller than the case (a). In a result, reflected

Fig.5 Profile of the mirror ratio R for various values of α with

λD/L=5x10-2 (case (a)).

Fig.6 Profile of the normalized potential ψ=−ef/kTe for various
values of α with λD/L=5x10-2 (case (a)).

reciprocated between the two turning points. As a result, theions in the plasma region increase and the potential drop
decrease. The profile of the potential for various values of
the temperature ratio τ=Te/Ti is shown in Fig. 7, where Z=1,
α=0.4 and λD/L=5x10-2. As the value of τ increases, thepotential drop in the plasma region decreases. It may be
because that the low energy ions are reflected by the strong
magnetic field near the wall and reciprocated between the
two turning points. As a result, the ions in the plasma region
increase and the potential drop decreases.

Fig.8 Relation between the normalized potential ψ=-eφ/kTe andthe mirror ratio R give by Eq. (25) for various values of α(case (b)).

602

A. Fukano and A. Hatayama, Electric Potential in Plasma and Sheath Regions with Magnetic Field Increasing toward a Wall

5. Conclusions
The electric potential near the wall has been
investigated by considering the magnetic field increasing
toward the wall. The profile of the potential has beenobtained by solving the plasma-sheath equation. The
potential drop in the plasma region depends on the profile ofthe magnetic field and the ion temperature. As the magnetic
field near the wall increases and the ratio of the ion
temperature to the electron temperature decreases, thepotential drop in the plasma region decreases. Furthermore,
the potential drop in the plasma region in the magnetic fieldof which increase rate is larger than the decrease rate of thepotential becomes large compared with that in the magnetic
field of which increase rate is smaller than the decrease rateof the potential. On the other hand, dependence of thepotential in the sheath region on the profile of the magnetic
field and the ion temperature is small.
As far as the results for the case (i) are concerned,
however, they are based on the relatively simple assumptionthat the trapped ions are untrapped and lost from the systemafter their one trapped orbit due to statistical process, e.g.,collision process. Realistic modeling for such a collision
process is essential and will be included in the modeling in
the future to obtain more robust results for the case (i).

Fig.9 Profile of the mirror ratio R for various values of α with
λD/L=5x10-2 (case (b)).

[1]
Fig.10 Profile of the normalized potential ψ =−eφ/kTe for various
values of α with λD/L=5x10-2 (case (b)).

[2]

[3]
[4]
[5]
[6]
[7]
[8]
Fig.11 Profile of the normalized potential ψ =−eφ/kTe for various
values of the temperature rate τ =Te/Ti with λD/L=5x10-2(case (b)).

ions decrease and the potential drop in the plasma region
increases compared with the case (a). The profiles of thepotential for various values of the temperature ratio τ=Te/Ti
is shown in Fig. 11, where Z=1, α=0.4 and λD/L=5x10-2. As
the value of τ increases, the potential drop decreases same as
the case (a), though the potential drop is a little larger thanthe case (a). This may be also because that the reflection
effect of the potential on the ions is smaller than the case (a).

603

A. Fukano, A. Hatayama and M. Ogasawara, JJAP 46, No.
4A, 1668(2004).
V. E. Golant, A. P. Zhilinsky and I. E. Sakharov:
Fundamentals of Plasma Physics. (John Wiley and Sons,
New York, 1980).
G. A. Emmert, R. M. Wieland, A. T. Mense, and J. N.
Davidson, Phys. Fluids 23,803(1980)
K. Sato, F. Miyawaki, and W. Fukui, Phys. Fluids B1,
725(1989).
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1974) p. 692.
J. H. Whealton, E. F. Jaegar, and J. C. Whitson, J. Comput.
Phys. 27, 32 (1978).
J. C. Whitson, J. Smith, and J. H. Whealton, J. Comput. Phys.
28, 408 (1978).
H. Horiike, M. Akiba, Y. Ohara, Y. Okumura and S. Tanaka,
Phys. Fluids 30, 3268(1987).