Chapter 4 Waves in Plasmas 4.1 Representation of Waves 4.2 Group velocity 4.3 Plasma Oscillations...
-
Upload
maude-barton -
Category
Documents
-
view
237 -
download
3
Transcript of Chapter 4 Waves in Plasmas 4.1 Representation of Waves 4.2 Group velocity 4.3 Plasma Oscillations...
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.