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4) Magnetic confinement of plasma
Due to the shielding in the plasma, there is almost no control with electric fields. A control is possible
with magnetic fields, as particles are bound to the field lines. This is called plasma confinement. It is of
fundamental interest for the magnetic confinement fusion (Tokamak, Stellarator).
Example in nature: Guidance of
High energetic particles of the
cosmic radiation by the earth magnetic
field to the poles  generation of
polar lights.
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4.1
We solely take single particle motion in the magnetic field into account. If we include collective effects,
we need magneto hydrodynamik (MHD) theory.
Lorentz forcet:
 
 
m  v  FL  q  v  B


FL  B

m  v||  0 ,
As
yields
the magnetic field has no influence on the motion parallel to the
magnetic field. The direction of the velocity changes perpendicular to the magnetic field, hence the
kinetic energy of the particles is constant (orbit motion).
v2
m   m c2 r  q  v  B  q c r  B
r

c 
q
B
m
cyclotron frequency
 ce Hz   1.76 1011  BT  ,  ci 
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(4.1)
me
 ce ,
mi
4.2
Radius on orbit:
r
m  v
qB
 Gyration of electrons  Plasma diagnostic
 Injection of an electromagnetic wave of  = ce  plasma heating
Using
1
2
m  v2  kT
(2 degrees of freedom perpendicular to B) the Lamor radius rL is
Te eV 
m  v
2mkT
6
m
rL 

 3.4 10 
qB
qB
BT 
(4.2)
Example: Te = 1 eV, B = 1 T → rL = 3.4 m
Let’s now investigate
 
 
mv  F  qv  B
Assuming an additional force F acting on the particles  no closed orbit
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4.3
For constant B und F in time and space
 guiding center-Ansatz
The center motion is calculated via
 
  
 mv v  B
m  
rc  r  rg , rg 


v B
2
q B v B q B
The velocity of the guiding center is give by

    
  
m   
1
vc  rc  r  rg  v 
v  B  v 
( F  q  v  B)  B
2
2
qB
qB
with
  
  


( v  B )  B  ( v  B ) B  B 2 v   B 2 v
 
  F B
vc  v|| 
q  B2
the following result of the motion is derived
and


mv||  F||
(4.3)
The second term is called drift velocity. This term does not accelerate and is directed prependicular to
B and F|.
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4.4
Examples:

1.)
elektric field


F  qE

E x B –drift
 

EB
vD 
B2
The drift velocity is not
dependent on the charge of the
particle.
ExB-drift in direction of the x-axis. The gyration radii are not drawn to
scale!
 

m g  B


v

D
F  mg
2.) Gravitation

q  B2

g
The particle gains energy in the direction of || . The drift is charge dependent and leads to a charge
 

 
g  B  mi

j

n
e

(
v

v
)

n
e

m

e
i
e
e
e
separation introducing the current g
e  B2  Z

For sources this drift is negligible. More important are drifts due to gradients.
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4.5
3.)
 

 B-Drift, if B  f ( x, t )
 curvature drift
 
 B  0
Due to
a gradient in B leads to a
curvature of the field lines. This results in a
2

v|| 
ˆ
F

m


r
c.
centrifugal force
Rc
 2  
m

v

rˆc  B
||
vD 

Rc
q  B2

ˆ
ˆ
rc dT
B


Rc ds
B
The centrifugal force is in direction of

 B
and hence
 2


m

v

||
vD  
 B  B
3
qB
Particle drift in a toroidal magnetic field. The drift results in a charge
separation and the resulting ExB-drift moves the plasma outward
Example: toroidal magnetic field
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4.6
Adiabatic invariance of the magnetic moment
A charged particle on a circular orbit creates a current in the magnetic field.
I q
v
2  r
q

2
magnetic dipole moment:



q2
2 
1
 m  I   dF  nF  I  F  nF  2 q c  rL  nF 
B  rL2  nF
2m
F

Assuming the Lamor radius
v m
rL  
Bq

2
 v m 
q
m  v2 W
m 
B  

 
2m  B  q 
2B
B
2
(4.4)
How does the magnetic moment change with a slow drift of the guiding centre?
Change of the magnetic flux  through the orbit area!
2
W  q 
 c and
dW W   q  rL dB
dB



 m
dt
c
c
dt
dt
Gyration period

C 
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d
dB
   q  rL2 
dt
dt
if c and rL are introduced.
4.7
On the other hand
dW d
d m
dB
 ( m B)  B
 m
dt
dt
dt
dt
d m
 0 ,  m  const .
hence:
dt
The magnetic moment is constant under a slow drift of the guiding center (slow in comparison to the
orbit motion).

Example:
adiabatic invariant
magnetic mirror
If a charged particle moves into a zone with stronger
magnetic field, the kinetic energy of the gyration
increases due to the invariance of the magnetic
moment. As the total kinetic energy in of the particle
does not change (no collisions) the kinetic energy in
Direction of the guiding center motion must be reduced.
In order to reduce v|| completly
 m ( Bmax  B1 )  W  W2   W1  W||,1
The particle starts at position 1 between the coils. At Pos. 2 is the maximum of the magnetic field Bmax.
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4.8
B2 W2 
W

 1
B1 W1
W1

B2
W
1 

B1
W1 W1
W||,1
2
 v1|| 
  cot  2
 
 v1 
B2/B1 is called mirror ratio.

v1
v1||
The loss con eis characterized by
  arcsin(
Bmin
)
Bmax
(4.5)
This equation demonstrates that a magnetic mirror is not perfect. Particles with small velocity
components perpendicular to the magnetic field are not reflected, but slowed down (as their angles
towards the axis are smaller then .
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4.9
Particle motion within the earth magnetic field. At the polar regions magnetic mirrors
exist and the curvature drift lead to an equatorial current.
Magnetic confinement is used in the field of ion source in




Electron Cyklotron Resonanz Ion Source (ECRIS) (resonance heating of plasms using rf)
Confinement of ions in an Electrone Beam Ion Source (EBIS)
Multicusp-Ion Sources
Motion of ions in an EBIS or a Penning trap
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4.10