What are helicons?
description
Transcript of What are helicons?
![Page 1: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/1.jpg)
![Page 2: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/2.jpg)
What are helicons?
Helicons are partially ionized RF discharges in a magnetic field.
They are basically whistler modes confined to a cylinder.
They are much different than in free space; they have E-fields.
OLD
NEW
Long cylinder Permanent magnet
![Page 3: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/3.jpg)
Helicons pose unending problems
UCLA
• Why does the amplitude oscillate along the cylinder?
• Why is a right-helical antenna better than a left one?
• What causes the high ionization efficiency?
• Why does an endplate near the antenna increase n?
• Why is the ion temperature so high?
• Why is a half-wavelength antenna better than a full?
• Why is the density peaked at the center?
Most discharge theorists treat only collision cross sections and ion distribution functions.
![Page 4: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/4.jpg)
The Trivelpiece-Gould mode: edge ionization
UCLA
Trivelpiece-Gould mode
Helicon mode
An electron cyclotron wave near the edge deposits most of the RF energy
0
1000
2000
3000
4000
5000
0.000 0.005 0.010 0.015 0.020 0.025r (m)
P(r
)
(arb
.)
2.5E+11
4.0E+11
6.3E+11
1.0E+12
n (cm-3)
![Page 5: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/5.jpg)
Edge ionization should give a hollow profile
UCLA
B
n
r
0
1
2
3
4
5
-25 -20 -15 -10 -5 0 5 10 15 20 25r (cm)
n (1
01
1/c
m3),
KT
e (
eV)
0
2
4
6
8
10
12
14
16
18
Vs (V
)
n11KTeVsVs(Maxw)
65 Gauss
But density is almost always peaked at center, even in KTe
is peaked at the edge.
![Page 6: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/6.jpg)
Previous attempt for an ICP
UCLA
Skin depth
with FL
without FL
![Page 7: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/7.jpg)
Let’s take the simplest realistic problem
UCLA
Eliminate all unnecessary features, and not length!
L
aB
+
rLi >> a
rLe << a-
Treat a 1D problem in radius r
![Page 8: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/8.jpg)
The problem is how to treat the ends
UCLA
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
e
e
+
½ ½ 1/2
, ln2 2
pe pe e
e
eKT KT Mn n eM m KT m
The sheath drop is normally independent of density
![Page 9: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/9.jpg)
Ion diffusion upsets the balance
UCLA
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+e
e
APPARENT ELECTRON FLOW ION DIFFUSION 1
2
(a)
The short-circuit effect “moves” electrons across B.
Sheaths change to preserve neutrality.
Electrons can now follow the Boltzmann relation.
This happens in nanoseconds.
![Page 10: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/10.jpg)
Sheath drops interchange, creating Er
UCLA
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+
e
e
ION DIFFUSION+
-
E
(b)
1
2
/0
ee KTn n e
( 0)
![Page 11: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/11.jpg)
In equilibrium, n is peaked on center
UCLA
BSHEATH
+
LOWER DENSITY 2
e
+
HIGHDENSITY 1
e
+ e LOWEST DENSITY 3
ION DIFFUSION
+
-
E
Er and diffusion must be outward if axial flow is slow.n(r) is flat in the limit of all ionization at edge.
![Page 12: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/12.jpg)
Three equations in 3 unknowns: v, n, and
UCLA
( ) ( ) /c cx io nP r v r n
( ) ( ) 0io iM n Mn en Mn en KT n v v v Bv v E v
Ion equation of motion:
Ion equation of continuity: ( ) ( )n in nn P r v
( ) ( )i ionP r v r
Use the Boltzmann relation:
Simplify the collision terms:
/0 0
ee KTn n e n e
½, / , and ( / )e s ee KT c KT M E
![Page 13: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/13.jpg)
Reduce to one dimension in r
UCLA
2 2
2 2 2( ) ( )s
n i n i cs s
cdv v vn P r n P P
dr rc v c
Eliminate n and to get an equation for v(r):
/ , ( ) 1 ( ) / ( )s c iu v c k r P r P r Non-dimensionalize:
22
1(1 )
1n
is
ndu uP ku
dr r cu
This is an ordinary differential equation for all the plasma
profiles.
![Page 14: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/14.jpg)
Rescale r to see structure of the equation
UCLA
( / )n i sn P c r
( ) 1 ( ) / ( )c ik r P r P r
22
11
1
du uku
d u
22
1(1 )
1n
is
ndu uP ku
dr r cu
We had:
Rescale r:
Finally:
k contains the plasma information:
![Page 15: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/15.jpg)
Solutions for uniform pressure and KTe
UCLA
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
V /
Cs
a
a
a
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0r / a
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
n/n0
eV/KTe
v/cs
Solutions for three values of k Rescale so that a 1 in each case
This profile is independent of pressure, size, and magnetic field.This profile is independent of pressure, size, and magnetic field.
It depends on It depends on KTKTee, but is always peaked at the center., but is always peaked at the center.
![Page 16: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/16.jpg)
This profile IS modified:
UCLA
• When Te is changed or varies with r
• When nn varies with r (neutral depletion, treated later)
• When k varies with r
But the central peaking remains
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0r / a
n /
n0
110100
p (mTorr)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1r / a
n /
n0
234
KTe (eV)
![Page 17: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/17.jpg)
Ionization balance restricts KTe for real r
UCLA
1( )n i e
drnv n P T
nr dr
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5r [cm]
V /
Cs
3.00
3.49
4.00
KTe (eV)
22
1(1 )
1n
is
ndu uP ku
dr r cu
Our previous dimensional equation
Solved simultaneously
![Page 18: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/18.jpg)
Improved Te – p0 relation
UCLA
0
2
4
6
8
10
1 10 100p0 (mTorr)
KT
e (
eV)
2.5
5
10
Tube radius (cm)
Old, radially averaged data:M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed. (Wiley-Interscience, Hoboken, NJ, 2005).F. F. Chen and J.P. Chang, Principles of Plasma Processing (Kluwer/Plenum, New York, 2002),
![Page 19: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/19.jpg)
The EQM program solves simultaneously:
UCLA
22
1( ) 1 (1 / )
1n
i c is
ndu uP r u P P
dr r cu
1( )n i e
drnv n P T
nr dr
2n n iD n n nP
Ion motion
Neutral depletion
Ionization balance
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5r (cm)
n (1
012
cm-3
) &
p
/ p
0
1
5
10
p0 (mTorr)
![Page 20: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/20.jpg)
Last step: iteration with HELIC
UCLA
0
2
4
6
8
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
n (1
011
cm
-3)
0
1
2
3
4
5
6
Pr (K
W/m
2)n
Pr
13.56 MHz65G, 400W
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
KT
e (e
V)
14.4
14.6
14.8
15.0
15.2
p (mT
orr)
KTe (eV)p (mTorr)
13.56 MHz65G, 400W
Lc
a b
h
Loop antenna
Helical antenna
B0
![Page 21: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/21.jpg)
Another layer off the onion!
UCLA
![Page 22: What are helicons?](https://reader036.fdocuments.in/reader036/viewer/2022062423/56814579550346895db24af6/html5/thumbnails/22.jpg)
Title
UCLA