Transcript Radial transport in bounded cylinders and physics of

Radial transport in bounded
cylinders and physics of
“universal” profiles
Francis F. Chen
UCLA
With Davide Curreli, Univ. of Illinois, Champaign-Urbana
Gaseous Electronics Conference, Austin TX, Monday, October 22, 2012
Why is n(r) peaked on axis?
2
P(r) (/ cm )
1.2
1.0
Parabolic Profile: R = 1.47 Ohms
0.8
Square Profile: R = 1.71 Ohms
0.6
0.4
0.2
0.0
0
1
2
r (cm)
3
n (1011 cm-3)
2
KTe (eV)
1
RF Bz field
skin depth
0
0
5
r (cm)
10
15
3
4
Take a very simple model
It has ends!
Normal sheath
e
HIGH DENSITY
+
LOWER DENSITY
e
+
SHEATH
B
½
½
 KTe  e p
 KTe 
n
e
 n
,


 2 m 
 M 
e p
1/2
 M 
 ln 

KTe
2

m


Let the ions diffuse inwards
e
+
+
SHEATH
APPARENT
ELECTRON FLOW
ION DIFFUSION
HIGH DENSITY
LOWER
DENSITY
e
B
(a)
The sheath has to adjust to neutralize tube 2
It is AS IF electrons had followed the ions!
But ACTUALLY, they have not moved across B.
1
2
Electrons fall into a Maxwellian
But they still keep their temperatures Te(r)
e
+
+
SHEATH
e
+
E
B
(b)
n  n0ee / KTe
ION DIFFUSION
HIGH DENSITY
LOWER
DENSITY
1
2
A little simple algebra
Ion equation of motion:
Mv (nv)  Mnv v  enE  Mn io v  en( v  B)  KTin  0
Ion equation of continuity:
 (nv)  nnn Pi (r )
Simplify the collision terms:
Pi (r )    v ion (r )
Pc (r )    v cx (r )   io / nn
Use the Boltzmann relation:
e / KTe
n  n0e

 n0e
E  ,   e / KTe , and cs  (KTe / M )½
3 equations for 3 unknowns: n, v, 
Eliminate n and  to get an equation for v(r):
cs2
dv
 2 2
dr cs  v
 v

v2
   nn Pi (r )  2 nn ( Pi  Pc ) 
cs
 r

Non-dimensionalize:
u  v / cs ,
du
1

dr 1  u 2
k (r )  1  Pc (r ) / Pi (r )
 u nn
2 
   Pi (1  ku ) 
 r cs

An O.D.E. for plasma profiles
We had:
Rescale r:
Finally:
du
1

dr 1  u 2
 u nn
2 
   Pi (1  ku ) 
 r cs

  (nn Pi / cs )r
du
1

d  1 u2

u
2
1  ku   


K contains the plasma information:
k (r )  1  Pc (r ) / Pi (r )
A universal profile!
Rescale  so that a  1 in each case
V / Cs
Solutions for three values of k
1.0
1.0
0.8
0.8
a
0.6
a
a
0.0
-0.2
n/n0
-0.4
0.6
-0.6
0.4
0.4
-0.8
v/cs
0.2
eV/KTe
0.2
-1.0
0.0
0.0
0.5
1.0

1.5
2.0
0.0
-1.2
0.0
0.2
0.4
0.6
0.8
1.0
r/a
This profile is independent of pressure, size, and magnetic field.
It depends on KTe, but is always peaked at the center.
Title