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:

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”

7. Particle drift
Add an external electric field E┴B: Define the drift velocity: Then and u obeys the gyro-equation: Moving gyration centre!

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)

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:

11. Polarization drift
B constant in space and time, E constant in space, varies in time. Insert into Leads to: Polarization drift:

12. Polarization drift
This drift causes charge separation, “polarizes” the plasma: Insert into Maxwell: cA is Alfvén speed (Hannes Alfvén, ) Anisotropic dielectric constant!

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):

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:

15. Magnetic moment
Gyrating particle in magnetic field excites opposing magnetic field: plasma is a diamagnetic medium Associated magnetic moment: μ approx. constant:

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

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

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, (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.