Download Effect of two-temperature electrons distribution on an electrostatic

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Effect of two-temperature electrons distribution
on an electrostatic plasma sheath
J. Ou, N.Xiang, C.Y.Gan, and J.H.Yang
IPP
2013-5
Abstract
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A magnetized collisionless plasma sheath containing two-temperature electrons is
studied using a one-dimensional model in which the low-temperature electrons are
described by Maxwellian distribution (MD) and high-temperature electrons are
described by truncated Maxwellian distribution (TMD). Based on the ion wave
approach, a modified sheath criterion including effect of TMD caused by high temperature electrons energy above the sheath potential energy is established
theoretically. The model is also used to investigate numerically the sheath structure
and energy flux to the wall for plasmas parameters of an open divertor tokamak-like.
Our results show that the profiles of the sheath potential, two-temperature electrons
and ions densities, high-temperature electrons and ions velocities as well as the
energy flux to the wall depend on the high-temperature electrons concentration,
temperature and velocity distribution function associated with sheath potential. In
addition, the results obtained in the high-temperature electrons with TMD as well as
with MD sheaths are compared for the different sheath potential.
Jing Ou et. el.al, Phys. Plasmas (To be published 2013-07)
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Outline
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Motivation and background
Sheath model for two-temperature electrons plasma
Sheath criterion
Sheath structure
Summary
Effect of the Sheath on the boundary plasma of
tokamak
Bohm criterion as the boundary condition for the edge
plasma model
Fluid model (B2, UEDGE, EDGE2D):
Onion-Skin model
sheath structure determined the energy flux to the wall
sheath heat transmission coefficient
particle recycling
Sheath criterion
Bohm criterion ( without collision and magnetic field)
V  Cs  kTe / mi
Ti  Te
Modified Bohm criterion (without collision)
Vix  Cs cos 
Vix20  cos 2  
Cs  k ( Ti  Te ) / mi
isothermal flow
Ti
Te
 1
V  Cs  k (Ti  Te ) / mi
----Used in Fluid model B2, UEDGE, EDGE2D
Two-temperature electrons plasma
Edge tokamak
1) Edge heating by waves
2) Core plasmas through a stochastic magnetic field
3) ELMs
Microelectronics manufacturing industry and
material precessing
plasma with negatively ions
Two-temperature electrons with Boltzmann
distribution
Bohn criterion
Interpretation of tokamak Langmuir probes with the
presented a fast electron component
Heat transmission through plasma sheath with
energetic electrons.
The model is justified under the assumption of the two-temperature
electrons in the state of local thermal equilibrium.
Electrons with cut-off velocity distribution
Some of the tail electrons with their energy greater than the sheath
potential energy do not bounce back into the plasma and then have a
cut-off velocity distribution.

VH2
1
exp( 2 )

2VtH
f H (VH )   2 I H VtH

0

VcH  VH  
otherwise
Effect of TMD on Bohm criterion
Sheath model
 A collisionless electrostatic plasma sheath in steady state.
 plasmas consisted of two-electrons and singly charged ions
 Low-temperature electrons are majority and high-temperature
electrons are minority
 The distribution of low-temperature electrons follow the
Boltzmann relations (Maxwellian distribution MD); hightemperature electrons with truncated Maxwellian distribution
(TMD)
 B is in the x-z plane .
Sheath model –equations
Low-temperature electrons
nL  nL ,0 exp(
e
)
k BTL
High-temperature electrons
VH 
 V 
2
H
VtH
V
exp[( cH ) 2 ]
2 I H
2VtH
VtH2
 IH
{  IH 
VcH
2VtH
exp[ (
VcH
2VtH
) 2 ]}
THeff  me ( VH2  VH2 )
nH 
nH ,0
V
I H exp[( cH ) 2 ]
V
2VtH
I H ,0 exp[( cH ,0 ) 2 ]
2VtH
Sheath model –equations
Ions
nV
i ix
0
x
Vix
Vix
Vix
Vix
1 Pi
e  e

 BViy sin  
mi x mi
mi ni x
x
Viy
x

e
B (Viz cos   Vix sin  )
mi
Viz
e
  BViy cos 
x
mi
Poisson equation
 2
e


(n  n  n )
0 i L H
x 2
Sheath model –normalized equations
Normalized parameters
  x / De
  e / k BTL
N L  nL / nL ,0
N i  ni / ni ,0
  ci /  pi
  mi / me
TH*  THeff / TL
N H  nH / nH ,0
ui  Vi / Vrs
  (ni ,0 / nL,0 )1/ 2
  TH / TL
De   0 k BTL / nL ,0 e2
Normalized equations
N L  e
NH 
uiy
uiz
   cos 

uix
IH /
e
I H ,0
Ni
N

(  uiy sin  )
 2 i
 uix  Ti / TL 
uix
u

  2 ix
(  uiy sin  )

uix  Ti / TL 
uiy

  (sin  
uiz
cos  )
uix
 2
 N L  ( 2  1) N H   2 Ni
2

uH 
 uH2 
 (
e
2 I H
w  )/ 
( w   ) (

[  IH 
e
 IH

TH*   [1 
( w   ) / 
 IH
w  )/ 
e(w  )/  
]
1
e 2(w  )/  ]
2
2 I H
Sheath criterion - ion wave approach
Plasma sheath boundary
Quasi-neutral condition:
E0   / 
 0
ni ,0  nL ,0  nH ,0
0
Ion wave approach:

k
(
k BTe
k BTi 1/ 2
1

)
2 2
mi 1  k D mi
F.F.Chen, Introduction to Plasma Physics (Plenum, New York,1974)
等离子体物理学导论 4.7 等离子体近似的有效性 P58
Sheath criterion
Sagdeev potential

VS ( )   [ 2 N i  N L  ( 2  1) N H ]d
0
N i

 0

N i ,0
2
ix ,0
u
 Ti / TL
(1   sin 2  )
Integrated form of Poisson equation
1  2 1 
( )  (
2 
2 
d 2VS
d 2
 0
2
)
 VS ( )
 0
Ni N L
N
 [

 ( 2  1) H ]  0  0
 

2
N L / 
N H

 0

 0
N H ,0

 N L ,0
(1   )
Sheath criterion -(Effect of TMD)
e w / 

)
2  I h ,0  w / 
主要影响因素：
  TH / TL
  (ni ,0 / nl ,0 )1/ 2
w
高温电子温度与背景电子温度之比
背景离子浓度与背景电子浓度之比
鞘层电势
Sheath criterion -(Modified Bohm criterion)
uix ,0
Ti  2 (1   sin 2  ) 1/ 2
[ 
]
2


1
TL
1
(1   )

1）without the high temperature electron ：
 2 1
2）two-temperature electrons with Max-distribution：   0
uix 0  [
uix 0
Ti
 cos 2  ]1/2
TL
Ti  2 (1   sin 2  ) 1/ 2
[ 
]
 2 1
TL
1


Velocity of ion entering the sheath decreases with increasing sheath potential.

Influence on the Bohm criterion is small.
Numerical results
Default parameters
- similar to an open divertor tokamak edge plasma .
B  1T
  50
nL ,0  1018 m3
TL  10eV
Input parameters

 2 w
Boundary conditions
uiz ,0  uix ,0 tan 
E0  0.01  1
0  0
Ti / TL  1
Sheath structure
Electrostatic sheath and magnetic pre-sheath
Sheath structure
Effect of the high-temperature electrons temperature.
Sheath structure
Effect of the high-temperature electrons density.
Sheath structure
Effect of the divertor biasing
Sheath structure
Comparison of effect of MD with TMD
Effective temperature of the high temperature
electrons in the sheath
the effective temperature in the plasma –sheath edge increases with the
decreasing biasing.


the effective temperature at the wall is independent of the biasing.
 Vh2  1
Vh 
2

Th*  Vh2  Vh2
Th*  1 
2

 0.3633
Power flux to the wall

Power flux decreases in the case of truncated-Max for large biasing.

power flux tends equal in the two cases of the truncated-Max and Max
distribution for small biasing.
qi 
5 Ti 1 2
 uix ,0   w
2 TL 2
qL  2
 
e
2
w
qH  TH* , w
 1 
e
'
2 I H ,0
w
/
qt   2 qi  qL  ( 2  1)qH
HT-7 belt limiter
Summary & Conclusions
 A criterion for a two-temperature electrons plasma
sheath in an oblique magnetic field is established
theoretically with a fluid model.
The properties of the sheath depend on not only the
plasma balance equations but also the sheath boundary
conditions.
The change of sheath structure is mall with variation of
the biasing but profiles of the high temperature electron
density and velocity.
power flux depends on the biasing.