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Introduction to Plasma
Physics and Plasma-based
Acceleration
Charged particles in external
fields
Plasma confinement
Plasma cannot be confined by solid
walls. So how to confine it?
– Gravity, e.g. in stars and planets
– Magnetic fields, e.g. in tokamaks
or in the earth’s ionosphere
– Strong laser beams, e.g. in inertial
confinement fusion
– Not at all, e.g. in laser-driven
wakefields
Magnetic confinement
Magnetic Lorentz force is always
perpendicular to B and to (charged)
particle velocity:
– Motion across magnetic field lines
severely restricted
– Motion along magnetic field lines is
“free”
Closed magnetic field lines can trap
large plasma volumes
Examples
Solar loop
Tokamak plasma
Charged particle in
homogeneous B-field
Particle (ion or electron) with charge Ze,
mass m, speed v in homogeneous field
with strength B, revolves around field lines:

dv Ze  

vB
Gyro-equation:
dt m
ZeB
Cyclotron frequency:  
m
mv
Cyclotron radius:  
ZeB


Helical motion
Particle “gyrates” around
field lines, while gyration
centre moves freely along
field lines
In collision-poor plasma:
   coll
  mfp
ρ takes over from “mean
free path”
Particle drift
Add an external electric field E┴B:
dv Ze   


E  v  B
dt m
Define the drift velocity:
 

EB   
vE 
; u  v  vE
2
B
  
Then E  vE  B  0 and u

equation: du  Ze u  B
dt
m

obeys the gyro-

Moving gyration centre!
Particle drift
– Particle executes gyromotion around magnetic
field
– Particle is accelerated for
half its orbit, decelerated
for the other half
– Periodic variation in gyroradius
– After averaging over fast
gyro-motion, a net drift
remains: ExB-drift
– Particle not fully confined
to magnetic field line.
Particle drift
In general, a force F┴B leads to a drift
 
velocity:

1 FB
vF 
Ze B 2
If F is independent of the charge, then its
drift will cause charge separation (not for
ExB drift)
Diamagnetic drift
Assume a magnetised plasma with
slowly varying density n and
temperature T.
Electron pressure: P = nkT

1
1
F



P


nkT 
Force on plasma:
n
n
Diamagnetic drift speed:

P  B
vd  
ZenB 2
Polarization drift
B constant in space and time, E
constant in space,
varies
in time.
 

dv Ze   
  EB
Insert v  u  2 into dt  m E  v  B 
B
Leads to:



du dE B Ze  



uB
dt
dt B 2 m
Polarization drift:

v pe,i

1 E

 e.i B t
Polarization drift
This drift causes charge separation,
“polarizes” the plasma: 



 E
J p  ne ev pe  ni Zev pi  2  ;   ne me  ni mi
B t
Insert into Maxwell:
2
1 E|| 1  c 2  E
B
  B  0 J 0  2
 2 1  2 
; c A2 
c t c  c A  t
0 
cA is Alfvén speed (Hannes Alfvén, 1908-1995)
Anisotropic dielectric constant!
Gradient-B drift
–Assume a magnetic field with slowlyvarying (over length LB) magnitude B.
–Leads to varying gyro-radius
–Leads to drift velocity, just like ExB
drift (u denotes gyro-velocity):

 u


m B u2 B
vB 

~ O
u   Ou 
2
 LB 
Ze B
2 B


Curvature drift
Assume magnetic field with gradual
curvature. Radius of curvature:
 

2
b  b   Rc Rc
Particle moving along field line feels
centrifugal force
(perp.
to B):


Fc  mv Rc R
2
||
2
c
Leads to driftvelocity:
 


m B
2
vc 

v
|| b  b
2
Ze B

Magnetic moment
Gyrating particle in magnetic field
excites opposing magnetic field:
plasma is a diamagnetic medium
Associated magnetic moment:
mu2
   I 
; I  Ze 2
2B
2
μ approx. constant:
d
 B

 O 
dt
B t
Magnetic mirror
Time-independent B-field cannot
perform work on particle, so kinetic
energy is conserved:
1
2
1
2
du||
1
2
  mv||2  mv2  B  mv||2 ;
m
dt

  b  B
d
0
dt
Particle moving in direction of
increasing B feels opposing force:
mirror force
Magnetic mirror
–Since μ is a constant, an
increase in B causes an
increase in u┴
–Since ε is a constant, u||
must decrease
–This causes the particle
to reflect eventually
–Another method of
plasma confinement
Earth’s magnetosphere
Image: Rice
University
Earth’s magnetic field confines
magnetosphere plasma and deflects
harmful solar wind plasma
Mini-magnetosphere
Magnetosphere
keeps harmful
solar particles out.
No magnetosphere
on Mars?
No problem, bring
your own.
L. Gargaté et al., Plasma
Phys. Contr. Fusion 50,
074017 (2008)
Aurora borealis
Charged particles forced to follow
Earth’s magnetic field lines when they
get near.
They only reach the surface near the
poles.
Ionisation/excitation of atmospheric
atoms produces light effects: aurora.
Summary
– A magnetic field confines charged
particles: they can move along but
not across.
– External forces, or changes in the Bfield itself, can introduce cross-field
drift, thus breaking confinement.
– Confinement can imply keeping
plasma out as well as in.