Chapter 4 Waves in Plasmas

4.1 Representation of Waves

4.2 Group velocity

4.3 Plasma Oscillations

4.4 Electron Plasma Waves

4.5 Sound Waves

4.6 Ion Waves

4.7 Validity of the Plasma Approximation

4.8 Comparison of Ion and Electron Waves

4.9 Electrostatic Electron Oscillations Perpendicular to 0B

4.10 Electrostatic Ion Waves Perpendicular to

4.11 The Lower Hybrid Frequency :

4.12 Electromagnetic Waves with

4.13 Experimental Applications of the EM waves in plasmas

4.14 Electromagnetic Waves Perpendicular to ;

4.15 Cutoffs and Resonances

4.16 Electromagnetic Waves Parallel to

4.17 Experimental Consequences of EM Waves in Magnetized

Plasma

4.18 Hydromagnetic (low-frequency ion) waves along ; T=0

0B

E//k,Bk 0

plasmacold0T;0B e0

0B

plasmacold

0Te

plasmacold0T,B e0

0B

0B

4.19 Magnetosonic (low-frequency ion) waves across ; T=0

4.20 Summary of elementary plasma waves

4.21 The CMA diagram

4.1 Representation of Waves

density

trki0

10

enn

nnn

in Cartesian coordinates.

in 1D,

The density of the sinusoidal oscillation to be measured is

zkykxkrk zyx

tkxi0 ennn

tkxcosnnR 1e

phasex

1tt 2tt

1n

coordinatein0dt

d 0

in x coordinate

4.1(2)

one moves with the phase

At an observation phase

0.,q.etkx 00

2tt 1tt

1n

phvkdt

dx

0dtdx

k

0tkxdtd

In a media, there may be many waves and each has it own phase,

that is,

4.1(3)

tkxic

tkxiitkxi eEeeEeEE

or

tkxEE

cos

sincos EiEeEE ic

real

c

cERe

EImtan

in general

tkxieEE

complex

aEcEbE

x

• Why use complex amplitude?

tkxiceEE.,q.e Where is complex function .cE

4.2 Group velocity

For baEtxkkEE

baEtxkkEE

002

001

coscos

coscos

txkb

tkxa

Beating

tkxcostxkcosE2

bcosacosE2

bsinasinbcosacosbsinasinbcosacosE

bacosbacosEEEE

0

0

0

021

kvg

phv

4.2(2)

dkd

vg < C

21 EE

4.2(3)

0 0 0 00

00 0 0

...

1, ,

2

...

k k

k k

i kx k t

di k k k x k k k t

i kx k t dk

di k k x t

i k x k t dk

E x t dkd E k e

e e

e e

Dominated by

0kkdk

dv

k

kv g

0

0ph

,

)k(E

k0k

k

)k(

4.3 Plasma Oscillations

neutral

EEEE FFFF

+ + + +

+ + + +

+ + + +

+ + + + e

shifted causes the plasma oscillating with the plasma frequency.

conditions B = 0

uniform and infinite plasma

KT = 0 (cold plasma, p = 0)

ion fixed

1 D in x

4.3(2)

so,

E0E

xEE

xx

1 D

electrostatic oscillation

functions to be found: Evn ,e,e

egs :

ei00

eee

eeee

e

nnexE

E

0vnt

n

Eenvvt

vmn

4.3(3)

Perturbation theory, linearization

0EE~EE

0vv~vv

.constnnn~nn

010

010e

i010e

1st order 2nd order

10i10

101

1e11

n~nneE~x

0v~nx

n~t

E~ev~v~t

v~m

differential eq.

zero order:

0

4.3(4)

tkxi11

tkxi11

tkxi11

eEE~

enn~

evv~

11

11

v~ikv~x

v~iv~t

110

101

11

enEik

0vnikni

eEvim

algebraic eqs.

0

E

v

n

ik0e

0ikni

emi0

1

1

1

0

0

0

4.3(4)

21

m

en

m

en

0ikimiikne

0

20

p2p

0

202

002

plasma frequency.

0pp n9f

2

GHz30fcm1microwave

s/cm103c,TelsaGHz

28f2

10ce

ce

.propagatenotdoesnoscillatioplasmathe0dkd

v

valueanybecank

v

g

ph

p

srad

4.4 Electron Plasma Waves

Conditions:

.fixedion,plasmainfiniteanduniform,0KT

xinD1,0B

e

xn~x

KT3

nKT3p

1e

eee

10e n~nn

3N

N2

1N

in

orderst1

1e1010 n~x

KT3E~env~t

mn

0)v~n(x

n~t 101

)n~(eEx 110

1e

1010

n)ik(KT3

Eenvn)i(m

0vn)ik(n)i( 101

110 enE)ik(

4.4(2)

0

0

00e

ik0e

0ikni

enmni)ik(KT3

1

1

1

E

v

n

0

0

)km

KT3

m

en( 2e

0

202

2th

22p

2 vk23

e2th KTmv

21

relationdispersion

0)ik)(i(n)mi()nik(en)ik)(nik)(ik(kT3 0002

000e

4.4(3)

ph

2th2

thgv

v

23

vk

23

dkd

v

kvph

2thv)kdk2(

23

0d2

4.5 Sound waves In ordinary air,

the Navier-Stokes e.q.

the equation of continuity

0)v(t

ppv)v(

tv

(two eqs for two variable: and ) v

For a stationary equilibrium with and , the first order parts of the above eqs in Fourier transformed form

are

)p( 00 0v

)]trk(i[e

0vkii

k//vkip

vi

101

110

010

1

2

01

00

010 v

kp

vkk

pv

gs0

0ph vc)

mKT

()p

(k

v 21

21

4.5(2)

Sound speed in a neutral gas

The waves are pressure waves propagating by collision.

4.6 Ion waves

In plasma, there is no ordinary sound waves because of the absence of collisions.

The perturbation on ions can propagate through electric field.

ion fluid equation :

.]constantn,0v,0E,0B[ 0i0

nKTen

pEenv)v(t

vMn

ii

iii

0 1 0 1 11 i i i ist order i M n v en ik KT ikn )k//v( 1i

4.6(2) the balance of forces on electron requires

)KTe

1(nennne

100e

eKT1e

orderst1 ]sectionnextsee[nnKTe

nn 11ie

101e

ion equation of continuity

orderst1 1i01 vkinni )vn,unknowns3,eqs3( 1,1i,1

si

iieph v)

M

KTKT(

kv 2

1

gv

21

i

esie M

KTv,TTif

ion acoustic wave

isothermal1,efor,3,D1for ei

1i0

ie1i0ikvn

)kKTkKT(vnM

4.7 Validity of the plasma approximation

The approximation was used while is finite.

The error is going to be evaluated.

ei nn E

)nn(ekE 1e1i12

010

0nKTe

ne

11e

1i1i v0nk

n

1iii101i0i knKTkenvnM

a

c

b

d

1,1i,1e,1i vnn

b a 1ie0

202

10 en)KT

enk(

2D

2th

2pe

e

e

e0

20

e0

202 1

vKT

m

m

en

KT

enk

2D

2k1

1i1e nn

4.7(2)

1with c d

21

21

21

)MKT

M

kT(

)MKT

k1

1M

KT(

kv

)MKT

k1

1

en

KT

M

ne(

k

vnk

)kKT1k

eken(vnM

i

ii

i

e

i

ii2D

2i

eph

i

ii2D

220

e0

i0

02

1i0ii2D

20

01i0i

D

2D2D

2 )2

(k1

d ,

4.8 Comparison of Ion and Electron Waves

pi2s

2

i0

20

2D

2s

2D

2

2s

22D

2

2D

2

i0

20

21

21

)ckM

en(

),wavelengthshort(1kFor

kv

),wavelengthlong(1kFor

)ckk1

kM

en(

wavesion

0Ti 2pi

pi

svk

21

)mKT

(cs

21

i

iies )

M

KTKT(v

4.8(2)

th2th

22pe

2D

2

pe2D

2

2th

22pe

v23

k)vk23

(

),wavelengthshort(1kFor

),wavelengthlong(1kFor

)vk23

(waveselectron

21

21

k

pe

thv23

4.9 Electrostatic Electron Oscillation Perpendicular to 0B

(nonrelativistic)

terminology: k

k 0B

perpendicular

parallel

k

0B

k

1E mixed mode

k

k 1E

transverse, electromagnetic

longitudinal, electrostatic

11

1

BE

)0B(

)0B( 1

4.9(2)

For a longitudinal waves , the governing e.q.s for the motion of electron and the waves are

)E//k( 1

1e10

1e01e

01e11e

e

enE

0vnt

n

)BvE(et

vm

11e1e E,v,n

ares.q.eabovetheofcomponentsFourierthe

,zvyvxvvand,nn,zBB,xEE,xkkFor zyx1e11e0

4.9(3)

10

x01

ze

0xye

0yxe

enikE

0kvinni

0vmi

Bevvmi

BeveEvmi

)E,v,v,n(

unknowns4

s.q.e4

yx1

xe

00xe v

mi-

eBeBeEvmi

22c

ex

/1

mi/eEv

e

0ce m

eB

4.9(4)

E1

1

mi

env

knnE

e

ki

2

2cee

20

x0

10

2

2pe

2e0

02

2ce 1

m

en1

2UH

2c

2p

2

upper hybrid frequency

motionthermalnoif0vg

0B

electron orbit Planes of constant density

4.9(5)

1E,k

0B

z

xx k

ktan,fixedkfor

0xz

0z

Bk0kk

Bk2

0k

//,,

,,

pc

cp

h h

cp

zk0

p

c

0 zk

4.10 Electrostatic Ion Waves Perpendicular to 0B

(almost)

0T0,Ev

constantB,constantn

E//k

02

,Balmostk

i00

00

0

0ixiyi

0iy1ixi

01i11i

i

i

e

ez

BevvMi

BeveikvMi

Bveet

vM

)M

m(

2

.shieldingDebyeoutcarriesv,efor

xik,xEE,ionfor

21

z0B

frontswave

Ek ,

x

2

4.10(2)

1e1i

e

1

0

1e

ix01i

12

2ci

1i

ix

nn

KTe

n

n

vk

nn

)1(Mek

v

11i1eix ,n,n,v

2s

22ci

i

e22ci

2 vkM

KTk:sol

0Ti

the dispersion relation for electrostatic ion cyclotron waves

4.11 The Lower Hybrid Frequency: E//k,Bk 0

])M

m([ 21

i

e2

1e01e1

2

2ce

1e

ex

1i01i1

2

2ci

1i

ix

vk

nn)1(m

ekv

vk

nn)1(Mek

v

)1(m)1(M

vv

).neutralityquasi(nnonpproximatiaplasmaThe

2

2ce

e2

2ci

i

exx1

1e1i

LHceci

ceciei

222

21

)(

mMBe

4.11(2)

lower hybrid frequency

2piceci

2LH

1e1i

111

,nnofinsteadusedwas.qes'PoissonIf

)Mm

mM(Be

)M1

m1

(Be

Mm)mM(

ie

ei22

ie

22

2cii

2ceeei

2

4.12 Electromagnetic Waves with )plasmacold(0T;0B e0

11 BkEk

• Light waves in vacuum

112

11

EBc

BE

200c

1,0J,vacuumin

s.eqshomogeneou2,iablesvar2

122

12

12

12

12

11

Bck]Bk)Bk(k[c)Bk(kcB

t)]-exp[i(kxE,BwavesplaneAssuming

0 0B

1112 BEBc

1

2

21

ck

v

ck

ph

222

relationdispersion

1ph111 BvEBkE

21

0B

210E B

21

WE21

W

2

2ph

21

21

00B

E

c

v

B

EWW

dominateWB

dominateWE

4.12(2)

111

• In a plasma with 0B0

10

11

2

11

Ej

Bc

BE

4.12(3)

.termsourceawiths.eq2,variables3

10

11

2 Ej

Bc

]E)E([c)E(c 12

12

12

)]tkx(iexp[j,E,B 111

12

2

120

12

1 Ec

jc

iEk)Ek(k

for 1Ek

0

1

3

3

4.12(4)

1110

1222 Ejj

iE)kc(

111

0

11e1e

1e01

jEim

Eeen

EevmiEet

vmvenj

Self - consistent

10

20

1

20

01

222 Em

enE)

mi

en(

iE)kc(

2p222

p2 kc

.BdcnowithplasmainwavesEMforrelationdispersion

im/Eev 11e

4

4.12(5)

BE2

2

2p2

2

22ph WWc

kc

kv

)k2(c0dkd

2vc

dkd

v 2

ph

2

g c

k

p

)e

mnnor(,

m

enIf

.solutionkrealaforrequiredis

2

20

c0

20

p

p

2122p

11i

xxk

11

xk11

2122

ii

)(

ckkeeE,B

eE,B,ckikk

i

ip

cutoff condition

skin depth

2/122p

2/12p

2 i)(ck

4.13 Experimental Applications of the EM waves in plasmas

• Measurement of plasma density with the cutoff phenomenon

by applying waves of varying frequency.

Microwave measurement of plasma density by the cutoff of the transmitted signal.

plasma

• Microwave interferometer for plasma density measurement

index of refraction

4.13(2)

ck

vc

n~ph

glassin1vacuumin1plasmain1

A microwave interferometer for plasma density measurement.

plasmateemagic

klystronlegplasma

legreference

guidewave

rdetectopeoscillosco

shifterphase

attenuator

a

a

bb

).densityplasmahigherby(,aschangedbinphasethe,plasmawith.2

.phaseofout180arebandapathfromsignals,plasmawithout.1 o

4.13(3)

n

density

plasma

high

cutoff

change

plases

outputdetector

cutoff

density

plasmain

patternwave

The observed signal from interferometer (right) as plasma densityis increased, and the corresponding wave patterns in the plasma (left).

• plasma lens for EM waves

4.13(4)

A plasma lens has unusual optical properties, since the index of refraction is less than unity.

refractionofindex

A plasma confined in a long, linear, solenoid will trap the laserlight used to heat it only if the plasma has a density minimum on axis. The vacuum chamber has been omitted for clarity.

2CO

laser

1v

cn

ph

4.13(5)

• the effect of plasma on radio communications.

plasma

radioAM)a(vehiclereentry)b(

earthplasma induced by friction that causes a plasma cutoff for a communication blackout.

ionosphere

Exaggerated view of the earth’s ionosphere, illustrating the effect of plasma on radio communications.