Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate
The effects of neutral and dust particles for the stable or monotonically changing potential formation in an expanding magnetic field toward the divertor plate was investigated in a quasi-neutral plasma by using one-dimensional kinetic analysis. Unless there are plasma ion sources, such as ionizatio...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate / Yu. Tomita, Yu.I. Chutov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 39-41. — Бібліогр.: 5 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-802562025-02-09T13:43:49Z Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate Tomita, Yu. Chutov, Yu.I. Magnetic confinement The effects of neutral and dust particles for the stable or monotonically changing potential formation in an expanding magnetic field toward the divertor plate was investigated in a quasi-neutral plasma by using one-dimensional kinetic analysis. Unless there are plasma ion sources, such as ionization of neutral particles, the required ion flow velocity for injected ions to form the stable potential has to be larger than an ion sound velocity, i.e. the generalized Bohm’s criterion. It was clarified that the plasma ion sources mitigate this requirement. On the other hand, since the dust particles decrease the plasma ion density due to absorption to them, the required ion flow velocity for injected ions increases. The numerical values of these effects are presented. The decreasing magnetic field in a plate direction also increases this required velocity. This collaborated research was supported by the Heiwa-Nakajima Foundation. 2002 Article Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate / Yu. Tomita, Yu.I. Chutov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 39-41. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.55Hc; 52.55Rk https://nasplib.isofts.kiev.ua/handle/123456789/80256 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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English |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Tomita, Yu. Chutov, Yu.I. Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate Вопросы атомной науки и техники |
| description |
The effects of neutral and dust particles for the stable or monotonically changing potential formation in an expanding magnetic field toward the divertor plate was investigated in a quasi-neutral plasma by using one-dimensional kinetic analysis. Unless there are plasma ion sources, such as ionization of neutral particles, the required ion flow velocity for injected ions to form the stable potential has to be larger than an ion sound velocity, i.e. the generalized Bohm’s criterion. It was clarified that the plasma ion sources mitigate this requirement. On the other hand, since the dust particles decrease the plasma ion density due to absorption to them, the required ion flow velocity for injected ions increases. The numerical values of these effects are presented. The decreasing magnetic field in a plate direction also increases this required velocity. |
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Article |
| author |
Tomita, Yu. Chutov, Yu.I. |
| author_facet |
Tomita, Yu. Chutov, Yu.I. |
| author_sort |
Tomita, Yu. |
| title |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
| title_short |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
| title_full |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
| title_fullStr |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
| title_full_unstemmed |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
| title_sort |
stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2002 |
| topic_facet |
Magnetic confinement |
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https://nasplib.isofts.kiev.ua/handle/123456789/80256 |
| citation_txt |
Stable potential formation with neutral and dust particles in expanding magnetic field near divertor plate / Yu. Tomita, Yu.I. Chutov // Вопросы атомной науки и техники. — 2002. — № 4. — С. 39-41. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT tomitayu stablepotentialformationwithneutralanddustparticlesinexpandingmagneticfieldneardivertorplate AT chutovyui stablepotentialformationwithneutralanddustparticlesinexpandingmagneticfieldneardivertorplate |
| first_indexed |
2025-11-26T11:36:38Z |
| last_indexed |
2025-11-26T11:36:38Z |
| _version_ |
1849852703484674048 |
| fulltext |
divertor
plate
plasma flow
from SOL
L
z
quasi-neutral plasma
with neutral and dust particles
B(z)
φ (z)0
STABLE POTENTIAL FORMATION WITH NEUTRAL AND DUST
PARTICLES IN EXPANDING MAGNETIC FIELD NEAR DIVERTOR
PLATE
Yukihiro Tomita
National Institute for Fusion Science, Toki, Gifu 509-5292 Japan
Yuriy I. Chutov
Taras Shevchenko Kiev University, Volodymyrs'ka 64, 252017 Kiev-17, Ukraine
The effects of neutral and dust particles for the stable or monotonically changing potential formation in an expanding
magnetic field toward the divertor plate was investigated in a quasi-neutral plasma by using one-dimensional kinetic
analysis. Unless there are plasma ion sources, such as ionization of neutral particles, the required ion flow velocity for
injected ions to form the stable potential has to be larger than an ion sound velocity, i.e. the generalized Bohm’s
criterion. It was clarified that the plasma ion sources mitigate this requirement. On the other hand, since the dust
particles decrease the plasma ion density due to absorption to them, the required ion flow velocity for injected ions
increases. The numerical values of these effects are presented. The decreasing magnetic field in a plate direction also
increases this required velocity.
PACS: 52.55Hc; 52.55Rk
I. INTRODUCTION
The electrostatic potential can strongly influence the
behavior of plasma particles near a divertor region. In
fusion devices, the magnetic field is expanding in the
plate direction; i.e. the magnitude of the magnetic field is
decreasing to the plate. In this configuration, ions are
accelerated to the plate due to the gradient of the
magnetic field, so-called the mirror force. The neutral
particles play an important role to remove the heat flow
from the core plasma to the divertor plate. The dust
particles have also same effects as neutrals. In LHD
(Large Helical Device) the collection of dust particles
was reported [1], where the dust particles have the size of
about 10 µm with the surface mass density of around 100
mg/m2. In this study, the effects of neutral and dust
particles on formation of stable electrostatic potential in
nonuniform magnetic field are discussed.
II. ION VELOCITY DISTRIBUTION AND
DENSITY
The model geometry for a one-dimensional analysis
is shown in Fig 1, where the divertor plate is located at
the position of z = L. The magnitude of magnetic field B
(z), which is directed toward the z direction, decreases to
the plate ( d B / d z < 0) so slowly that the magnetic
moment of ions µ is conserved during their motion. The
neutral and dust particles are uniformly distributed in
front of the divertor plate. The plasma flow from the
point z = 0 is taken account, which corresponds the
plasma flow from the SOL (Scrape-Off Layer) region to
the divertor region. The electrons are assumed to satisfy
the Boltzmann relation with the uniform electron
temperature. In this analysis we investigate the potential
formation φ (z) with the magnetic field monotonically
decreasing in the divertor plate direction.
The ion velocity distribution function is expressed
from Boltzmann equation in ( , , )z ε µ space, where ε is
the total energy of an ion
2 / 2 ( ) ( )i z iB z zqM µ φυ + +
which is one of the constants of motion of ions:
( , , )
( , , ) ( , , ) ( , , )i
z i d
zf
z z zS Sz
ε µ
ε µ ε µ ε µυ
∂
= −
∂
Fig.1. Geometrical model for a one-dimensional
analysis.
where the charge and the mass of plasma ion are denoted
by qi and Mi , respectively. The particle velocity toward
the plate zυ should be expressed by the variables
( , , )z ε µ : 2 2[ ( ) ( ) ] / iz iB z zq Mε µ φυ = − − . In this
study, the plasma ion source Si due to electron impact
ionization of neutral atoms and the plasma ion sink Sd due
to absorption to dust particles are considered as well as
the plasma flow from the SOL region.
The ion source Si is introduced as [2,3]:
2
2
2
( , , ) ( ) ( , , )
4
exp[ ( , , ) / 2 ]
i
i e s i z
s
i s
Mz z zS n n
T
zM T
ε µ ε µσ υ υ
π
ε µυ
= < >
× −
where ns is the uniform density of the neutral atoms,
iσ υ< > is the ionization rate by electron impact and Ts is
the temperature of source ions, which is assumed spatially
uniform. In this distribution function, the ion velocity υ
Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 39-41 39
and zυ should be expressed by the function of (z, ε, µ).
Taking into account the turn of generated ions with
negative velocity in z-direction ( 0zυ < ) in the decreasing
potential and magnetic field, the density of the generated
source ions is obtained in case of a weak nonuniformity
of magnetic field and potential as following:
0
0
( ) exp[ ( ) / ]
2
exp[( / / ) ( )]
i
si s s e i i
s
L
s ei
Mz zqn n n T
T
dz e zq T T
π φσ υ
φ
= −< >
′ ′× +∫
where ne0 is the electron density at the position of φ = 0.
The injected plasma ion distribution function from
the SOL region is determined by the absorption to dust
particles. Since the dust particles is at rest due to their
heavy masses, the plasma ion absorption rate is expressed
by
( , , ) ( ) ( , , )d d d iz zfS nε µ υ υ ε µσ=
From this expression the injected ion density is obtained
as follows.
0
2
0 (0)
0
( , )2 ( )( )
( , , )
exp{ [ ( , , )] ( , , ) / ( , , ) }
i
i n
zBi
z
d d z
fB zz d dn zM
dz z z zn
µ
ε µπ µ ε
ε µυ
υ ε µ υ ε µ ε µσ υ
∞ ∞
= ∫ ∫
′ ′ ′ ′× − ∫
These expressions of plasma ion densities are used to
study the stable electrostatic potential formation.
III. POTENTIAL FORMATION
The potential formation in the quasi-neutral plasma
immersed in the expanding magnetic field is investigated
by using the charge neutrality condition with negatively
charged ( Zd ) dust particle density nd:
d i ie d i s inn n n nZ Z Z+ = + and the equality of the
density gradients for uniformly distributed dust particles:
( )i s ine
i
d dn nd n Zd z d z d z
= +
The density gradient of electron is obtained from its
Boltzmann distribution:
( )e
e
e
d e dn znd z d zT
φ=
The density gradient of plasma ion source generated by
the electron impact ionization is expressed from Eq.(3).
( )i s i
i s
s
d qn d znd z d zT
φ= −
From considering the spatial dependence of the injected
ion density Eq.(5), we can easily obtain its density
gradient.
2 2
2
ln 1 ln[ /
2
/ ]
in
z z
i
d d z dz zz
i
d n d B d B
d z d z d z
q d
n nd zM
υ υ
φ υσ υυ
⊥
−
= + < >
+ − < >< >
where the quantity z< ⋅⋅ ⋅ > denotes the average over the
velocity distribution function of the injected ions. The
equality condition (6) of the quasi-neutral plasma gives
the relation between spatial dependences of the potential,
the magnetic field and the ion velocity.
2 2
2
1 ln[ (1 /
2
/ ]
i ii s ini
z ze se e
i
d d zz zz
i
qn nZ Ze d d d B
d z d z d zn nT T
q d
nd zM
φ φ
υ υ
φ υσ υυ
⊥
−
= − + + < >
+ − < >< >
First we investigate the effects of neutral particles
without dust particles ( 0dn = ) on stable potential
formation. The spatial variation of potential and magnetic
field is expressed from Eq.(10) and 0dn = .
2 2
2
1 ln(1 )/
2
(1 ) / (1 )
e
z z
i e i e
i s i sz zi s
d BT
e d
Z T Z T
M T
υ υ φ
υ δ δ
⊥
−
+ < >
= − + + −< >
Here i sδ is defined by the ratio of the density of the
generated ions to the electron density : /ii s i s en nZδ ≡ .
In our case of decreasing potential and magnetic field,
LHS of Eq. (11) is positive. Hence in order to be formed
the monotonically decreasing potential in the decreasing
magnetic field in the plate direction, the ion flow velocity
have to be satisfied the following condition:
1 22 / (1 ) / (1 )i e
s i s i sz z
s
Z Tc
T
δ δυ
−− ≥ − +< >
where sc ( /i e iZ T M= ) is the ion sound speed.
Without plasma source inside quasi-neutral plasma (
0i sδ = ), this relation leads the generalized Bohm's
criterion [4,5]:
12 1z zυ
−− ≥< >
The effect of ion source is shown in Fig. 2 for the case of
1, 10 eVi eZ T= = , and 0.03 eVsT = , which corresponds
to the room temperature of 20 C° . This result shows ion
source in a divertor region mitigates the generalized
Bohm's criterion (for example: the 1 % of source ions
decreases this limit to 0.48).
It is clearly understand the existence of dust particles
increases the limit of ion flow velocity because they
decrease the density of plasma ions due to absorption.
The requirement for the ion flow velocity is obtained
from Eq.(10).
40
1 22
1 /
/
1 /
/
dis d e
sz z i e e
i s d d z z
s
n nZ
c Z T Tn ed d zT
δ
υ
υδ σ υ φ
−−
− +
≥< >
+ + < >
Fig. 2 Limit value of 1 22 / sz z cυ
−−< > as a function of
concentration of plasma ion generated by ionization,
where 1, 10 eVi eZ T= = and 0.03 eVsT = .
The third terms of the fraction of RHS indicate the effects
of dust particles, where note that the potential gradient is
negative ( / 0d d zφ < ). The effect of dust particles on
stable potential formation is shown in Fig. 3, where
1, 10 eV, 0.03 eV,i e sZ T T= = = 30.01, 10di s Zδ = =
and the parameter dν is defined by the ratio of the
characteristic length of potential decrease to the mean-
free path of the absorption to dust particles:
/ / /ed e d z z e d d zn Tυ φν σ υ≡ < > .
Fig. 3. Effects of dust particles ( Zd = 103 ) on the
limit value of 1 22 / sz z cυ
−−< > for the same parameters
as Fig. 2 and / / /ed e d z z e d d zn Tυ φν σ υ≡ < > .
In this case the limiting factor for injected ion flow
velocity becomes 0.58, which is compared to the value of
0.48 without dust particles.
IV. CONCLUDING REMARKS
The effects of neutral and dust particles on
monotonically changing electrostatic potential were
investigated in quasi-neutral plasma with nonuniform
magnetic field by using of one-dimensional kinetic
analysis. It is clarified that plasma ion source, such as
ionization of neutrals, mitigates the condition for the
input flow velocity. For example, in case of Zi = 1, Te =
10 eV , and Ts = 0.03 eV and the density ratio of the
generated ions to the electron density of 0.01 without
dust particles, the injected ion flow velocity have to be
larger than 0.48 cs. Please note that without neutrals this
factor is 1.0, i.e. generalized Bohm’s criterion. Plasma
ion sink, such as absorption to dust particles, increases
this required ion flow velocity. In the same values of
above plasma parameters and the density of dust particles
with Zd = 103 to the electron density of 10-4, this limit
factor increases from 0.48 to 0.58. The expanding
magnetic field in the direction of the plate increases the
required flow velocity.
There are few future issues left: 1) Two-dimensional
kinetic analysis has to be carried out in order to take into
account of ion motion in divergence free magnetic field.
2) Other processes, such as Coulomb collisions, charge
exchange collisions, should be included in this analysis.
ACKNOWLEDGMENT
This collaborated research was supported by the Heiwa-
Nakajima Foundation.
REFERENCES
[1] A. Sagara, S. Masuzaki, et al., 15th International
Conference on Plasma Surface Interactions in Controlled
Fusion Devices, May 27 - 31, 2002, Gifu, Japan, Program
and Book of Abstracts.- p. R-01 and P. Sharpe et al., ibid.
p. P3-46.
[2] G.A. Emmert, R.M. Wieland, A.T. Mense, and J.N.
Davidson, Phys. Fluids, 23 (1980) 803.
[3] R.C. Bissell and C. Johnson, and P.C. Stangeby, Phys.
Fluids, B1 (1989) 1133.
[4] D. Bohm, The Characteristics of Electrical
Discharges in Magnetic Fields, edited by A. Guthrie and
R.K. Wakerling ( McGraw-Hill, New York, 1949),
Chap.4, p.77.
[5] E.R. Harrison and W.B. Thompson, Proc. Phys. Soc.
London, 74 (1959) 145.
41
0
0.2
0.4
0.6
0.8
1
0 0.002 0.004 0.006 0.008 0.01
( 1
-
δ i
s )
/
( 1
+
Z
i T
e
δ
i
s /
T s
)
δ i s ( = Z i n i s / ne )
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0 0.2 0.4 0.6 0.8 1
1
- δ
i
s +
Z
d
n d
/ n
e
Z
i T
e δ
i
s /
T
s -
ν
d
n d
/ n
e
( nd / ne ) / 10 -4
ν d = 10 4
103
National Institute for Fusion Science, Toki, Gifu 509-5292 Japan
Taras Shevchenko Kiev University, Volodymyrs'ka 64, 252017 Kiev-17, Ukraine
III. POTENTIAL FORMATION
ACKNOWLEDGMENT
REFERENCES
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