Introduction to Plasma Physics and Plasma-based Acceleration

1. Introduction to Plasma
Physics and Plasma-based
Acceleration
Charged particles in external fields

2. 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

3. 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

4. Examples
Solar loop Tokamak plasma

5. 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:
Gyro-equation:
Cyclotron frequency:
Cyclotron radius:
m
ZeB


ZeB
mv


 
B
v
m
Ze
dt
v
d 





6. Helical motion
Particle “gyrates” around
field lines, while gyration
centre moves freely along
field lines
In collision-poor plasma:
ρ takes over from “mean free
path”
mfp
coll 

 



7. Particle drift
Add an external electric field E┴B:
Define the drift velocity:
Then and u obeys the gyro-
equation:
Moving gyration centre!
 
B
v
E
m
Ze
dt
v
d 






E
E v
v
u
B
B
E
v









 ;
2
0


 B
v
E E



 
B
u
m
Ze
dt
u
d 





8. Particle drift
– Particle executes gyro-
motion around magnetic field
– Particle is accelerated for
half its orbit, decelerated for
the other half
– Periodic variation in gyro-
radius
– After averaging over fast
gyro-motion, a net drift
remains: ExB-drift
– Particle not fully confined to
magnetic field line.

9. Particle drift
In general, a force F┴B leads to a drift velocity:
If F is independent of the charge, then its drift
will cause charge separation (not for ExB drift)
2
1
B
B
F
Ze
vF


 


10. Diamagnetic drift
Assume a magnetised plasma with
slowly varying density n and
temperature T.
Electron pressure: P = nkT
Force on plasma:
Diamagnetic drift speed:
 
nkT
n
P
n
F 





1
1

2
ZenB
B
P
vd

 




11. Polarization drift
B constant in space and time, E
constant in space, varies in time.
Insert into
Leads to:
Polarization drift:
2
B
B
E
u
v



 

  
B
v
E
m
Ze
dt
v
d 






B
u
m
Ze
B
B
dt
E
d
dt
u
d 







 
2
t
E
B
v
i
e
i
pe



 


.
,
1

12. Polarization drift
This drift causes charge separation, “polarizes” the
plasma:
Insert into Maxwell:
cA is Alfvén speed (Hannes Alfvén, 1908-1995)
Anisotropic dielectric constant!
i
i
e
e
pi
i
pe
e
p m
n
m
n
t
E
B
v
Ze
n
v
e
n
J 






 


;
2







0
2
2
2
2
2
||
2
0
0 ;
1
1
1 B
c
t
E
c
c
c
t
E
c
J
B A
A


















 

13. Gradient-B drift
–Assume a magnetic field with slowly-
varying (over length LB) magnitude B.
–Leads to varying gyro-radius
–Leads to drift velocity, just like ExB drift (u
denotes gyro-velocity):
 
















 u
O
u
L
u
O
B
B
u
B
B
Ze
m
v
B
B ~
2
2
2



14. Curvature drift
Assume magnetic field with gradual
curvature. Radius of curvature:
Particle moving along field line feels
centrifugal force (perp. to B):
Leads to drift velocity:
2
c
c R
R
b
b







2
2
|| c
c
c R
R
mv
F



 
b
b
v
B
B
Ze
m
vc







 2
||
2

15. Magnetic moment
Gyrating particle in magnetic field excites
opposing magnetic field: plasma is a
diamagnetic medium
Associated magnetic moment:
μ approx. constant:


 2
;
2
2
2



 
Ze
I
B
mu
I
 








 O
t
B
B
dt
d

16. Magnetic mirror
Time-independent B-field cannot
perform work on particle, so kinetic
energy is conserved:
Particle moving in direction of increasing
B feels opposing force: mirror force
B
b
dt
du
m
dt
d
mv
B
mv
mv








 





||
2
||
2
2
|| 0
;
2
1
2
1
2
1

17. 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

18. Earth’s magnetosphere
Earth’s magnetic field confines
magnetosphere plasma and deflects
harmful solar wind plasma
Image: Rice
University

19. 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)

20. 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.

22. Summary
– A magnetic field confines charged
particles: they can move along but not
across.
– External forces, or changes in the B-
field itself, can introduce cross-field drift,
thus breaking confinement.
– Confinement can imply keeping plasma
out as well as in.