1. Electron Acoustic Waves in Pure Ion Plasmas F. Anderegg C. F
Electron Acoustic Waves in Pure Ion Plasmas F. Anderegg C.F. Driscoll, D.H.E. Dubin, T.M. O’Neil University of California San Diego
supported by NSF grant PHY

2. Overview
We observe “Electron” Acoustic Waves (EAW) in magnesium ion plasmas.
Measure wave dispersion relation.
We measure the particle distribution function f(vz , z = center) coherently with the wave
A non-resonant drive modifies the particle
distribution f(vz) so as to make the mode resonant
with the drive.

3. Electron Acoustic Wave: the mis-named wave
EAWs are a low frequency branch of standard electrostatic plasma waves.
EAWs are non-linear plasma waves that exist at moderately small amplitude.
Observed in: Laser plasmas Pure electron plasmas Pure ion plasmas

4. Other Work on Electron Acoustics Waves
Theory: neutralized plasmas Holloway and Dorning 1991
Theory and numerical: non-neutral plasmas Valentini, O’Neil, and Dubin 2006
Experiments: laser plasmas Montgomery et al 2001 Sircombe, Arber, and Dendy 2006
Experiments: pure electron plasmas Kabantsev, Driscoll 2006
Experiments: pure electron plasma mode driven by frequency chirp Fajan’s group 2003

5. Theory
Electron Acoustic Waves are plasma waves with a slow phase velocity
w ≈ 1.3 k v
This wave is nonlinear so as to flatten the particle distribution to avoid strong Landau damping.

6. Dispersion relation Infinite homogenous plasma (Dorning et al.)
Trapping “flattens” the distribution in the resonant region (BGK)
Landau damping
“Thumb diagram”

7. Dispersion Relation Infinite size plasma (homogenous) Fixed lD / rp
Langmuir wave
EAW
kz lD
w / wp
Fixed lD / rp
k = 0.25
Trapped NNP
(long column finite radial size)
kz lD
w / wp
Experiment:
fixed kz
vary T and measure f
Fixed kz
TG wave
EAW

8. Penning-Malmberg Trap

9. Density and Temperature Profile
rp ~ 0.5 cm
0.05eV < T < 5 eV
Mg+
B = 3T
n ≈ 1.5 x 107 cm-3
Lp ~ 10cm

10. Measured Wave Dispersion
Trivelpiece Gould
EAW
Rp/lD < 2

11. Received Wall Signal Trivelpiece Gould mode
The plasma response grows smoothly during the drive
10 cycles
21.5 kHz

12. Received Wall Signal Electron Acoustic Wave
During the drive the plasma response is erratic.
Plateau formation
100 cycles
10.7 kHz

13. Fit Multiple Sin-waves to Wall Signal
Electron Acoustic Wave
The fit consist of two harmonics and the fundamental sin-wave,
resulting in a precise description of the wall signal
data
fit
Wall signal [volt +70db]
Time [ms]

14. Wave-coherent distribution function
Record the Time of Arrival of the Photons
photons
Photons are accumulated in 8 separate phase-bin
Wall signal [volt +70db]
35.5
time [ms]
36.0

15. Distribution Function versus Wave Phase
Trivelpiece Gould mode
f = 21.5 kHz T = 0.77 eV
f(vz, z=0)
The coherent distribution function shows oscillations dv of the entire distribution
These measurements are done in only one position (plasma center, z~0)

16. Distribution Function versus Wave Phase
Electron Acoustic Wave
f = 10.7 kHz
T = 0.3 eV
f(vz, z=0)
The coherent distribution function shows:
- oscillating Dv plateau at vphase - dv0 wiggle at v=0 Dv
dv0
These measurements are done in only one position (plasma center, z=0)

17. Distribution Function versus Phase

18. Distribution Function versus Phase

19. Distribution Function versus Phase

20. Distribution Function versus Phase
Shows wiggle of the entire distribution
4000
Velocity [m/s]
-4000
Small amplitude 90
180
270
360
Phase [degree]
Trivelpiece Gould mode
This measurement is done in only one position (plasma center)

21. Distribution Function versus Phase
18055_18305;23 Dv
Shows:
trapped particle island of half- width v
dv0 wiggle at v=0
Velocity [m/s]
dv0
-2000 90
180
270
360
Phase [degree]
Electron Acoustic Wave
This measurement is done in only one position (plasma center)

22. Model Two independent waves Collisions remove discontinuities
18055_18305;23
2000
Two independent waves
Collisions remove discontinuities
Velocity [m/s]
-2000 90
180
270
360
Phase [degree]
Electron Acoustic Wave

23. Island Width Dv vs Particle Sloshing dv0
Trapping in each traveling wave gives Dv
The sum of the two waves gives sloshing dv0
Linear theory gives:

24. Frequency VariabilityTG
100 cycles TG
EAW
100 cycles TG
EAW
100 cycles
100 cycles
Large amplitude drives are resonant over a wide range of frequencies

25. Frequency “jump” f response f drive
100 cycles TG
EAW
f response
f drive
The plasma responds to a non-resonant drive by re-arranging f(v) such as to make the mode resonant

26. f(v) evolves to become resonant with drive!
Non-resonant drive modifies the particle distribution f(vz) to make the plasma mode resonant with the drive.

27. Particle Response Coherent with Wave
Fixed frequency drive
100 cycles at f =18kHz
The coherent response give a precise measure of the phase velocity

28. When the Frequency Changes kz does not change
T ≈ 1.65 eV
kz = p / Lp
1.4 vth < vphase< 2.1 vth
Plasma mode excited over a wide range of phase velocity:

29. Range of Mode Frequencies
Trivelpiece Gould
EAW
When the particle distribution is modified, plasma modes can be excited over a continuum range, and also past the theoretical thumb.

30. Chirped Drive The frequency is chirped down from 21kHz to10 kHz
The chirped drive produce extreme modification of f(v)
Damping rate
g/w ~ 1 x 10-5

31. Summary
Standing “Electron” Acoustic Waves (EAWs) and Trivelpiece Gould waves are excited in pure ion plasma.
Measured dispersion relation agrees with Dorning’s theory
We observe: - Particle sloshing in the trough of the wave - Non-linear wave trapping Close agreement with 2 independent waves + collisions model
Surprisingly: Non-resonant wave drive modifies the particles distribution f(v) to make the drive resonant. Effectively excites plasma mode at any frequency over a continuous range

34. Distribution Function versus Phase
Shows wiggle of the entire distribution
Velocity 90
180
270
360
Large amplitude
Phase [degree]
Trivelpiece Gould mode
This measurement is done in only one position (plasma center)

35. Typical Parameters n ≈ 1.5 x 107 cm-3 rp ~ 0.5 cm
0.05eV < T < 5 eV
Mg+
B = 3T
n ≈ 1.5 x 107 cm-3
Lp ~ 10cm
Standing wave phase velocity

36. Stability f (v) => This plasma is stable
Penrose criteria predicts instability if
satisfied
and k satisfies
k < 96 m-1
Our
= 230 m-1 is larger than the maximum
=> This plasma is stable
allowed by Penrose criteria

37. Chirped Drive The frequency is chirped down from 21kHz to10 kHz
Received signal [ Volt +70db ]
Time [ms]
The frequency is chirped down from
21kHz to10 kHz

38. Particles Coherent Response
Trivelpiece Gould mode
vph
vph
The coherent response changes sign at v = 0 (almost no particle are present at the phase velocity)

39. Particles Coherent Response
Electron Acoustic Wave
vph
vph
The coherent response changes sign at: v = 0 at the wave phase velocity

40. Distribution Function versus PhaseDv
Shows:
trapped particle island of half- width v
dv0 wiggle at v=0
Velocity [m/s]
dv0
-2000 90
180
270
360
Phase [degree]
Electron Acoustic Wave
This measurement is done in only one position (plasma center)