1. Plasma parameters interpreted using Maxwell's eqns and plasma Eqns
Physics PLASMA PHYSICS, Marty Goldman
Notes #1 Feb. 2

2. Linear waves and other truly plasma phenomena
We will use two-theory approach to plasma physics: Maxwell eqns and Plasma eqns
Maxwell's eqns for E and B in terms of j and 
Plasma eqns leading to j and  in terms of E and B 
Self-consistent
Linear waves and other truly plasma phenomena
Newton's eqns for rsj(t), s = e,i OR
Kinetic eqns for fs(r,v,t) 
Strong-coupling: Hierarchy OR
Weakly collisional: Boltzmann eqn OR
Collisionless Boltzmann eqn,/PIC simulation, OR
2-fluid eqns for ns(r,t), us(r,t)
• ps(r,t) = 0: Collisionless cold fluids
• ps(r,t) =ns(r,t)T0 Warm fluids
• Pressure tensor in collisinoless fluids
• Weakly collisional fluid transport eqns.
1-fluid eqns : MHD, eMHD
Can be derived from previous with
approxi­mations. Less physics.

3. Transverse and longitudinal parts of a vector field refer to polarization of wave

4. Spatial Fourier transforms (infinite homogeneous plasma)
Notation and conventions

5. Maxwell's eqns and wave eqn for E in terms of plasma current, J

6. Electron Plasma Frequency
MAXWELL EQNS: Transverse wave eqn. in vacuuo gives light wave freq. ω=±ck, phase velocity, vφ = ω/k = c, index of refraction, n = ck/ω = 1. Light in a plasma has different vφ and n than light in glass, because polarization currents, JT are different.
PLASMA EQNS: Light waves in cold fluid plasma eqns: JT = ωe2 ET, ω = ±√(ωe2 +c2k2), vφ = ω/k. Index of refraction, n = ck/ω= √(1-ωe2 /ω2) <1, vφ > c.
Light with ω > ωe propagates into an unmagnetized plasma.
Light with ω < ωe is reflected. Gives color to metals. Compare to light in glass: Index of refraction, n >1, vφ< c
SKIN DEPTH PARAMETERS: Electron skin depth = de = c/ωe. ω/ωe = √(1+de2k2). Light doesn't penetrate plasma ifω/ωe < 1, or de2k2 < 0.
Ion skin depth = di = c/ωi

7. EM wave in a plasma: the plasma frequency

8. EM wave in a plasma, cont'd

9. Metals: no transmission of incident light when ω<ωe
Metals: no transmission of incident light when ω<ωe. Only reflection at those frequencies
White light spectrum

10. Debye screening based on plasma density hydrostatic equilibrium

11. Electron and ion thermal velocities
Theorist's definition of thermal velocity, vs, s = (e,i)
Relation to Debye length and plasma frequency

12. Weakly-coupled (collisionless) plasmas

13. Cumulative small-deflection collision-frequency ∝ inverse number electrons in Debye sphere
Very crude approach to scattering theory:
1. Calculate small momentum-transfer to one electron colliding at large impact-parameter with (approximately stationary) ion. Use straight-line trajectory approximation.
2. Calculate cumulative-small-momentum-transfer/sec to one electron colliding with many ions at a range of large impact parameters.
3. Calculate collision-time, τei, for cumulative small- momentum-transfers to add up to large momentum transfer
4. Cumulative collision-frequency, ωei = τei-1 is larger than collision frequency associated with single large-deflection small-impact parameter event

14. Momentum transfer due to single large impact-parameter small-deflection electron-ion collision
Approximations
Neglect deviation of electron trajectory from straight line: z = v0t
Allow small perpendicular velocity, Δv⊥, to develop due to F⊥ = perpendicular component of Coulomb force between electron and ion
Validity requires Δv⊥<< v0
approximate
electron trajectory
ion, +e
imact parameter, b
true electron
trajectory
electron
F =-eq/R2
F⊥ = Fsinθ z→ θ
z = 0 v0
Δv⊥ R

15. Collision time,τei, for multiple large b (small Δv⊥) encounters to result in cumulative large-angle electron deflection x y b
b+db

16. Collision frequency,νei, for multiple large b (small Δv⊥) en-counters to produce cumulative large-angle electron deflection x y
bmin λe

17. Cyclotron frequency and Larmor-radius of charged particle in uniform B-field
Eqn of motion of particle of charge qs = se (= ±e, for s = i, e) in uniform B-field, B pointing in z-direction
Dot v into both sides ⇒ v2 = const.
Magnetic field can do no work on particles.
Can’t change kinetic energy, but can change momentum.
Decompose particle velocity, v, into components parallel and perpendicular to B:
v = v|| + v
|| and ⊥ motions independent
Constant velocities: v||, v2, v = |v|
⊥ motion circular around B.
rL and v rotate at cyclotron frequency, s
Ions clockwise, electrons counterclockwise (looking down on B)
Combined motion is spiraling || or anti-|| to B
Lorenz force provides centripetal force
z, B0 x y
z, B0 x y
rLi
rLe ve vi
electrons
ions

18. Magnetic parameters for plasma in external magnetic field
Cyclotron motion parameters
Ωs, Rs = vs⊥/Ωs (Sometines ρs is used instead of Rs)
Ratio of electron pressure to magnetic pressure: Plasma beta
β = nT(8π/B2)
Measures role of magnetic field in particle dynamics
(Ion) Alfven velocity, vAi
vAi/c = Ωi/ωi = B/√(4πnimic2), ( vAi/c)2 = magnetic pressure over nimic2
vAi = velocity of MHD waves called Alfven waves
Electron Alfven velocity, vAe
vAe/c = Ωe/ωe = B/√(4πnemec2), ( vAe/c)2 = magnetic pressure over nemec2
vAe useful velocity scaling in theory of whistler waves. (eMHD)
Ωe/ωe useful in determining importance of magnetic field in waves
Electron and ion skin depths, ds = c/ωs (useful in magnetic reconnection)
Together with vAs and Ωs forms a trio of parameters: Ωsds = vAs

19. Calculating Magnitudes of Plasma Parameters (from NRL Formulary)

20. NRL Plasma Parameters (2)

21. NRL Plasma Parameters (3)

22. Questions to ask when reading a plasma physics paper or textbook which addresses collective processes
Which Maxwell eqns are relevant?
Are electric fields longitudinal or transverse?
Is calculation self-consistent?
Does it use both Maxwell eqns & Plasma eqns?
Time-dependent (dynamical) or time-independent (stationary)?
Which plasma model is employed?
MHD, two-fluid theory, or kinetic theory?
Note- kinetic theory is the analytic theory corresponding to Particle-In-Cell simulations.
Is the plasma taken to be weakly collisional or completely collisionless?
Are nonlinear eqns linearized in the calculation?
What is the small parameter in perturbation theory?
What is the zero order state? Is it stationary? Is it thermal equilibrium?