Ion-Induced Instability of Diocotron ModesIn Magnetized Electron Columns

Andrey Kabantsev

University of California at San DiegoPhysics Department

Nonneutral Plasma Physics Grouphttp://sdpha2.ucsd.edu/

UCSD Physics

, September 17, 2009

OUTLINEIntroduction to nonneutral plasmas. The physics of confinement.

Diocotron modes. Genesis, (negative) energy and stability.

Breaking the cylindrical and charge symmetries.Ion-induced instability of diocotron modes in electron plasmas.

Ways to mitigate/suppress the ion-induced instability.

A broader perspective on nonneutral plasmas. Conclusions.Final take-home message.

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2 2

1

[ ] .2

N

R j jj

eBP m r r constc

Cylindrical symmetry, single sign species => long confinement time

R

Nonneutral plasmas are confinedby static electric and magnetic fields

in a Penning-Malmberg trap

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2

12.

N

jj

r conePc

stB

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Exceptional particle confinement properties

Fast cyclotron radiation cooling + cryogenic walls

Ultracold plasma (Coulomb) crystals

2 2

1

1cmN

jj

r const

less than 1% of the particles can ever move out to the wall,while more than 99% of the particles are confined forever

(weeks in the experiments)

22

4 sec tesla 1sece B

with laser cooling of ions310 KiT

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0

2

, ,

( 1) ( ) , where 2 ( )m m m

mm R P W R e i

d t t t

m R R ec n n B

1m 2m 4m

Diocotron wavesDiocotron waves are flutelike density perturbations,

which are neutrally stable for an idealized "top-hat" profile ( ).They are negative energy modes, so the wal

( 0, 0)(

l dissipation p0

oce)

rm e

zmn

kr

ss may lead

to their (resistive) growth.

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Resistive wall destabilization of diocotron waves

W.D. White, J.D. Malmberg and C.F. Driscoll "Resistive Wall Destabilization of Diocotron Waves," Phys. Rev. Lett. 49, 1822

2 2 20

2 2 2

4 sin ( 2)1

s s

e

L RL R C

(Spatial) Landau Mode Damping

resonance condition wR(rc) = w/m,

n(r)

wR(r)

w/m w/m

rc rc

No damping for“top-hat” n(r) profile

Damping for adiffused n(r) profile

wR(r)

n(r)

Damping is the spiral wind-up (phase mixing) of the density perturbation near the critical radius rc , where the fluid rotation

rate wR(r) equals the wave phase rotation rate w/m

spatial (r, R = /mvelocity (, z) = /k

( )21 ( )

mc P m

P W

mr Rm R R

( )cn r r

UCSD PhysicsB. Cluggish and C.F. Driscoll "Transport and Damping from Rotational Pumping in Magnetized Electron Plasmas," Phys. Rev. Lett. 74, 4213 (1995)

Mode Damping from rotational pumping

2 2

2 2 2 2

11 2

D p

p W p W

RL R R R

Inevitable Wall Imperfections

Broken Cylindrical Symmetry

Drag of Rotating Plasmas (Negative Torque)on Static (or Slow Rotating) Asymmetries

Plasma Expansion and Heating

???????????????????????????

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2

1

BROKEN CYLINDRICAL SYMMETRY 2

. : N

jj

reBP conc

st

But a Faster Rotating Asymmetry Introduces the Positive Torque

Inward Particle Transport (Pinch), Accelerated Plasma Rotation(Still Leads to Plasma Heating)

Practically Infinite Confinement Time

Compression of Electron Cloud by Rotating Wall (Surko’s Group)

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Compression of Antiproton Clouds by Rotating Wall (ALPHA Collaboration)

( , )p e

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2

1

BROKEN E-CHARGE SYMMET2

RY: .N

j jj

BP q r constc

oppositely charged particles can move together to the wall still conserving P

and/or systems have the powerful constraint,

while the and/or any( , ) (

( , ) ( ,

systems do no

)

, ) t!p e I e

p e p e

From No Instabilities to Possible Diocotron Instabilities

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2Schematic of the H experimente

Pure electron plasma is contained in (up to) ten electrically isolated cylinders, with the cylinders S4 and S7 divided into up to 8 azimuthal sectors to excite, manipulate and detect various m 0 modes. Axial plasma confinement is provided by -100 V on the end cylinders. Radial confinement is provided by the axial magnetic field B. Plasma density z-integrated 2D-distribution n(r, ) is measured by instantaneous grounding the end cylinder, thereby allowing the plasma to stream onto a phosphor screen with attached CCD camera.

BRw fE

G1 L2 H3 S4 G5 H6 S7 G8 H9 G10

Plasma

-100 V -100 V

central density: n0 1.5107 cm-3

central potential: 0 – 30 Vplasma radius: Rp 1.2 cm (RW = 3.5 cm) equilibrium temperature: T 1 eV (D Rp /6)magnetic field: B ≤ 20 kGEB rotation frequency: fE(B) 0.1 MHz (2kG/B)axial bounce frequency: fb(T) 0.6 MHze–e collision frequency: ee(n,T) 160 sec-1

neutral pressure: P 10-11 Torr

2 2 28 ( . 8 )nmc B comp nT B

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22sin 2m R ee

I LeN

2 2 2

2 2 2

2

2

2 2 2

2

2 2

2******************

4 sin 2

2sin 2 ,

where

e

me e

me

me

R e

eN d d d I t r

I Id d t r d reN eN

I reN d

r dI

eNL

R

H2+electron

column

d

centerof

charge

centeroftrap

r

22 sin2 2

i e em R R

e e

n L Ln L

***********************************

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4.090

4.095

4.100

4.105

4.110

4.115

10-3

10-2

0 2 4 6 8 10 12

time [sec]

f1 [kHz]

d1 D1 /RW

1 0.1sec-1

1 1.5sec-1

f1 2.2Hz

f1(t)

f1(t), 1(t),

Ne(t)

i = i+ /eNe

f1 = f1Ni /Ne

i = i-1f1/f1

Modulated Ion Injection (1:15)

Typical Experimental Procedure:seed the mode, suppress the others, inject ions, watch the growth

0

0.5

1

1.5

2

0 5 10 15

B [kG]

- I+ = 15pA - I+ = 43pA

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1

eI e N

2 22sin 2 1.83sin 2m R e R ee e

I IL LeN eN

B-dependence of the single-pass m

10-4

10-3

10-2

0 2 4 6 8 10

WF_FITZ_46034

d1 = D / R

w

Time [sec]

1* = +0.0375s-1

What if we inject an electron beam instead?

1 = -1.48s-1

Ie ~ A

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Electron Beam Does Suppress Diocotron Waves !

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The double-well traps can be tried to confine particles with the opposite signs of electric charge.

In particular, they have been used recently at CERN to produce “anti-hydrogen” pairs.

+50 V -20 V 0 V -20 V +50 V

(z)

However, the powerful constraint is now broken.Is there a problem with the modes stability ?

( , )p e

2

1

constN

jj

r

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Schematic of the Double-Well (Nested) Trap Experiment

+V +V-V-V

H2+

e-B

EB

Le ½Lend½Lend

Le 53 cmLend 14 cmRw = 3.5 cm

In a double-well trap the bounce-averaged EB drift velocitiesof ions and electrons in diocotron perturbations nm are not equal.

This leads to charge separation in nm and instability of the modes.

,R r z

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m = 2

m = 3

The modes shown here are called the m = 2 and m = 3 diocotron modes.

The m = 1 diocotron mode is just a rigid off-axis spiraling of plasma column.

Images of Ion-Induced Instabilities in a Double-Well Trap

Experiments with partially neutralized electron plasma in the double-well trapshow that the diocotron modes do become unstable !

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áiñ

B, zd

2d sináiñ/2i

fR

fi

LbncLend

2 22sin / 2 2sin / 2m m i i m ie

I f feN

Schematic of Ion Drift Trajectories in the Electron Column Diocotron FrameSymmetry dictates that the average motion of trapped ions is well representedby the motion of an ion set initially at the center of elecron column.  Here, a single end run phase shift 2 , #se end mf of such runs for the diocotron cycle is 1 ( ),the average phase acquired during the diocotron cycle 2 ,

is an average ion lifetime inside the electron column, ( gives the averag

bnc m

i end bnc

i m i

f

f

e number of -cycles)

is the net change of mode frequency due to the partial space-charge neutraliztion.m i e m

df n n f

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10-4

10-3

10-2

0 2 4 6

Time [sec]

d D /RW

1 0.75sec-1

Exponential growth of the m = 1 diocotron mode over two decades in linear regime (d < dcr << 1)

2sin4

2

2

max

max

diid

i

m

iicr

f

r

rd

2 10-3

4 10-3

6 10-38 10-3

10-2

3 10-2

0 2 4 6 8 10 12Time [sec]

d D/Rw

Example of end /bnc dependenceuntrapped

ions trapped

by Vcolby Vcol

by H9

by H9

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9

1

1 9

1.52

1.64

end col

end

col

L VL H

VH

2 22sin / 2 2sin 2 /m m i i m end bnce

I f feN

10-3

10-2

5 6 7 8

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Time [sec]

dq

m = 1

m = 2

untrapped

trapped

untrapped

kHz 22.35 Hz10,sec4.14

kHz 033.4 Hz2.1,sec65.1

2

21

2

1

11

1

ff

ff

1 1 1

2 2 2

im m

e

f ff fn fn

Growth rate as a function of m - number

3.887

3.888

3.889

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.4 0.8 1.2Time [sec]

f1 [kHz]-1 [sec-1]

trapped

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Growth rate as a function of the space-charge neutralization factor

( )( ) ( ) im m m

e

n tt f t fn

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0

0.2

0.4

0.6

0 1 2

1 [Hz]f

1

1[sec ]

22sin 2 / 0.22m m end bnc m

m m i e

f f

f n n

1 3889 Hz

im m

e

nf fn

f

Growth rate as a function of the space-charge neutralization factor

CONCLUSIONS*** In the case of fast transiting ions the growth rate of diocotron modes ***

is relatively small and drops strongly with B

*** In the case of slow trapped ions the growth rate of diocotron modes ***

is defined by the neutralization (space-charge compensation) level solely,and thus may be very dangerous

*** There are various stabilization and damping techniques, out of which ***the most effective has to be chosen according to plasma and trap parameters

*** Rotating wall technique might be used to compensate the radial transport ***caused by the mode damping processes

UCSD Physics*In this presentation some illustrations from C.Surko (UCSD), J.Fajans (USB), NIST and ALPHA groups have been used.

2

2i e

m Re

n Ln

im m

e

nf

n

Final Take-Home MessagePure electron (or ion) plasmas are simple objectswith exceptional confinement properties.

Introduction of particles with an opposite sign ofelectric charge gives the way for diocotron modesto become unstable.

Instability of diocotron modes is well controllableif one knows what to trade in.

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