Electron Acoustic Wavesin 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-0354979

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.

Electron Acoustic Wave: the mis-named wave

• EAWs are a low frequency branch of standard electrostatic plasma waves.

• Observed in: Laser plasmasPure electron plasmas Pure ion plasmas

• EAWs are non-linear plasma waves that exist at moderately small amplitude.

Other Work on Electron Acoustics Waves

• Theory: neutralized plasmas Holloway and Dorning 1991

• Theory and numerical: non-neutral plasmasValentini, O’Neil, and Dubin 2006

• Experiments: laser plasmas Montgomery et al 2001Sircombe, Arber, and Dendy 2006

• Experiments: pure electron plasmas Kabantsev, Driscoll 2006

• Experiments: pure electron plasma mode driven by frequency chirp Fajan’s group 2003

Theory

Electron Acoustic Waves are plasma waves with a slow phase velocity

This wave is nonlinear so as to flatten the particle distribution to avoid strong Landau damping.

0

0.5

1

-4 -3 -2 -1 0 1 2 3 4

vz / v

EAW

TG

≈ 1.3 k v

Dispersion relation• Infinite homogenous plasma (Dorning et al.)

0=ε(k,)=1−p2

k2 dvLandau∫k∂f0∂vkv−

0≈1−p2

k2 P dvk∂f0∂vkv−∫ −iπp2

k2∂f0∂v/k

Landau damping

0≈1−p2

k2 P dvk∂f0∂vkv−∫ “Thumb diagram”

Trapping “flattens” the distribution in the resonant region (BGK)

Dispersion RelationInfinite size plasma(homogenous)

Langmuir wave

EAW

kz D

/

p

Fixed D / rp

k = 0.25

Trapped NNP(long column finite radial size)

kz D

/

p

Experiment: fixed kz vary T and measure f

Fixed kz

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4

T [eV]

TG wave

EAW

Penning-Malmberg Trap

Density and Temperature Profile

0

5

10

15

20

-1.5 -1 -0.5 0 0.5 1 1.5x(cm)

1940 -198

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5x(cm)

1940 -198

Mg+

B = 3T

0.05eV < T < 5 eV rp ~ 0.5 cm

Lp ~ 10cmn ≈ 1.5 x 107 cm-3

0

5

10

15

20

25

30

0 0.5 1 1.5

T [eV]

Measured Wave Dispersion

Rp/D < 2

EAW

Trivelpiece Gould

Received Wall Signal

Trivelpiece Gould mode

The plasma response grows smoothly during the drive

10 cycles 21.5 kHz

Received Wall SignalElectron Acoustic Wave

100 cycles 10.7 kHz

During the drive the plasma response is erratic.

Plateau formation

Fit Multiple Sin-waves to Wall Signal

The fit consist of two harmonics and the fundamental sin-wave, resulting in a precise description of the wall signal

Electron Acoustic Wave

fitdata

Time [ms]

Wal

l sig

nal [

volt

+70

db]

Wave-coherent distribution function

Record the Time of Arrival of the Photons

Photons are accumulated in 8 separate phase-bin

time [ms]

Wal

l sig

nal [

volt

+70

db]

photons

35.5 36.0

Distribution Function versus Wave Phase

The coherent distribution function shows oscillations v of the entire distribution

These measurements are done in only one position (plasma center, z~0)

f(vz,

z=0)

f = 21.5 kHzT = 0.77 eV

0o

45o

90o

135o

180o

225o

-6000 -4000 -2000 0 2000 4000 6000

315o

ion velocity [m/s]

270o

Trivelpiece Gould mode

0o

45o

90o

135o

180o

225o

-4000 -2000 0 2000 4000ion velocity [m/s]

315o

270o

before wave

after wave

Distribution Function versus Wave Phase

The coherent distribution function shows:

- oscillating v plateau at vphase

- v0 wiggle at v=0

These measurements are done in only one position (plasma center, z=0)

f(vz,

z=0)

f = 10.7 kHzT = 0.3 eV

Electron Acoustic Wave

v

v0

T=0.3

T=0.4

Distribution Function versus Phase

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Distribution Function versus Phase

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Distribution Function versus Phase

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Distribution Function versus Phase

This measurement is done in only one position (plasma center)

Trivelpiece Gould mode

Small amplitude

Vel

ocit

y [m

/s]

-4000

Shows wiggle of the entire distribution

4000

Phase [degree]

0 90 180 270 360

Distribution Function versus Phase

Shows:- trapped particle

island of half-

width v

- v0 wiggle at v=0

This measurement is done in only one position (plasma center)

Electron Acoustic WavePhase [degree]0 90 180 270

v

v0

Vel

ocit

y [m

/s]

-2000

360

18055_18305;23

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Model

•Two independent waves

•Collisions remove discontinuities

Electron Acoustic WavePhase [degree]0 90 180 270

Vel

ocit

y [m

/s]

-2000

360

18055_18305;23

2000

Island Width v vs Particle Sloshing v0

Trapping in each traveling wave gives v

The sum of the two waves gives sloshing v0

Linear theory gives:100

1000

10 100 1000

δv0 at v=0 [m/s] (half-width)

Δv = ( 2 δv0 v

ph )1/2

0v = 2 δv0 v phase( )1/2

Frequency Variability

Large amplitude drives are resonant over a wide range of frequencies

0

200

400

10 15 20 25 30fresponse

[ ]kHz

10 mV drive

TG100 cycles

0

200

400

10 15 20 25 30fresponse

[ ]kHz

60 mV drive

TG

EAW

100 cycles

0

200

400

10 15 20 25 30

100mV drive

fresponse

[ ]kHz

TG

EAW

100 cycles

10 15 20 25 300

200

400300mV drive

fresponse

[kHz]

100 cycles

Frequency “jump”

0

200

40060mV rivε

TG

EAW

f response

f drive10 15 20 25 30

frequency [kHz]

The plasma responds to a non-resonant drive by re-arranging f(v) such as to make the mode resonant

100 cycles

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.

0

5

10

15

-6000 -3000 0 3000 6000

before wave

with wave

wf3_PhoSum_37456_37655___.txt;2

Below TG mode, 19kHz drive

relative velocity [ m/s ]

0

5

10

15

-6000 -3000 0 3000 6000

relative velocity [ m/s ]

Resonant with TG mode, 21.8kHz drive

before wave

with wave

wf3_PhoSum_37717_37916___.txt;3

Particle Response Coherent with Wave

Fixed frequency drive 100 cycles at f =18kHz

-8

-6

-4

-2

0

2

4

6

8

-3 -2 -1 0 1 2 3 4

v / vth

T = 1.75 eVv

th= 2646. m/s

WF19371-19571

vphase

vphase

The coherent response give a precise measure of the phase velocity

When the Frequency Changes kz does not change

k z = π

/ L p

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20 250

0.5

1

1.5

2

mode frequency [kHz]

rp /

D ~ 2

= 1.65 T eVT ≈ 1.65 eV

1.4 vth < vphase< 2.1 vth

Plasma mode excited over a wide range of phase velocity:

0

5

10

15

20

25

30

0 0.5 1 1.5

T [eV]

Range of Mode Frequencies

EAW

Trivelpiece Gould

When the particle distribution is modified, plasma modes can be excited over a continuum range, and also past the theoretical thumb.

Chirped Drive

The chirped drive produce extreme modification of f(v)

The frequency is chirped down from

21kHz to10 kHz

Damping rate ~ 1 x 10-5

-8000 -4000 0 4000 80000

40

80

ion velocity [m/s]

with wave

vφ 2

0

40

80

before wave

vφ1

= 1.3 T eV

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

Distribution Function versus Phase

This measurement is done in only one position (plasma center)

Shows wiggle of the entire distribution

Trivelpiece Gould mode

Vel

ocit

y

Phase [degree]0 90 180 270 360 Large amplitude

Typical Parameters

Mg+

B = 3T

0.05eV < T < 5 eV rp ~ 0.5 cm

Lp ~ 10cmn ≈ 1.5 x 107 cm-3

D

=4 π n e

2

⎛

⎝⎜

⎞

⎠⎟

1 / 2

= 0 . 2 4 c mT

e V

1 / 2

n7

1 / 2

k Tf

r= 5

B3 T

n7

[ k H z ]

Standing wave phase velocity

vp h a s e = = 2 f L p [ m s ]

1 0 k H z

⎛

⎝⎜

f ⎞

⎠⎟

k

= 2 0 0 0

vt h

=k T

m= 2 0 0 0 T

eV

1 / 2

[ m s ] ν ii ≅ 1 s −1 n7 T−3

2eV

Stability

Penrose criteria predicts instability if

-8000 -4000 0 4000 80000

40

80

120

ion velocity [m/s]

v0

f (v)

f (v0) − f (v)

v−v0( )2−∞

∞

∫ dv < 0

k < p2 f (v0)−f (v)

v−v0( )2−∞

∞∫ dvand k satisfies

satisfied

k < 96 m-1

= 230 m-1 is larger than the maximum

=> This plasma is stable

k⊥= 1rp

2

ln(rw rp)Our

allowed by Penrose criteria

Chirped Drive

The frequency is chirped down from

21kHz to10 kHz

Rec

eive

d si

gnal

[ V

olt +

70db

]

Time [ms]

-1

0

1

-6000 -3000 0 3000 6000

ion velocity [m/s]

Particles Coherent Response

The coherent response changes sign at v = 0 (almost no particle are present at the phase velocity)

vph vph

Trivelpiece Gould mode

f ~∂ f

0

v−vph

∂v

Particles Coherent Response

-20

0

20

-4000 -2000 0 2000 4000

ion velocity [m/s]

The coherent response changes sign at: v = 0 at the wave phase velocity

vph vph

Electron Acoustic Wave

f ~∂ f

0

v−vph

∂v

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Distribution Function versus Phase

Shows:- trapped particle

island of half-

width v

- v0 wiggle at v=0

This measurement is done in only one position (plasma center)

Electron Acoustic WavePhase [degree]

0 90 180 270 360

v

v0

Vel

ocit

y [m

/s]

-2000