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DoesDoes Quasi Neutrality Quasi Neutrality Remain Valid in PairRemain Valid in Pair--Ion Ion
Plasmas ?Plasmas ?
HH. Saleem . Saleem Physics Research Division Physics Research Division
(PRD), PINSTECH, (PRD), PINSTECH, P.O.NiloreP.O.Nilore. . Islamabad, PakistanIslamabad, Pakistan
22
Plan of the talk Plan of the talk 1.1. Introduction Introduction 2.2. PairPair--Ion (PI) Plasmas Ion (PI) Plasmas 3.3. Criterion to Determine (PI) Criterion to Determine (PI)
PlasmasPlasmas4.4. Instability of A New Mode and Instability of A New Mode and
Nonlinear Dynamics Nonlinear Dynamics 5.5. Break Down of Quasi NeutralityBreak Down of Quasi Neutrality6.6. Conclusions Conclusions
33
1. Introduction1. Introduction
Recently the efforts have been made Recently the efforts have been made to produce pure pairto produce pure pair--ion fullerene ion fullerene plasmas in laboratories [1. Phys. Rev. plasmas in laboratories [1. Phys. Rev. LettLett.. 91 91 205005 (2003); 2. Phys. 205005 (2003); 2. Phys. Rev. Rev. LettLett. . 9595 , 175003 (2005)]., 175003 (2005)].
( )±60c
0.5 vT T e+ = =8 3~10n n cm−
+ −=
0 ~ 0.3B T
44
Three kinds of electrostatic waves, Three kinds of electrostatic waves, propagating only in the direction propagating only in the direction parallel to the external magnetic field parallel to the external magnetic field have been discussed in the pure pairhave been discussed in the pure pair--ion plasma [2].ion plasma [2].
These are the ion plasma wave These are the ion plasma wave (IPW), the ion acoustic wave (IAW) (IPW), the ion acoustic wave (IAW) and the third one has been named as and the third one has been named as the intermediate frequency wave the intermediate frequency wave (IFW) because it(IFW) because it’’s frequency lies in s frequency lies in between the frequencies of the other between the frequencies of the other two waves i.e. the IPW and the IAW.two waves i.e. the IPW and the IAW.
55
66
77
Important PointsImportant PointsA.A. It will be shown that the It will be shown that the
experimental observations itself experimental observations itself indicate the existence of indicate the existence of electrons in the system at a electrons in the system at a significant level. Therefore it significant level. Therefore it does not seem to be a pure pairdoes not seem to be a pure pair--Ion plasma. Ion plasma.
B.B. QuasiQuasi--neutrality is not a neutrality is not a reasonable approximation in reasonable approximation in such plasmas, when these are such plasmas, when these are perturbed.perturbed.
88
Definition of a PlasmaDefinition of a Plasma
A quasi neutral statistical ensemble A quasi neutral statistical ensemble of charged particles which exhibits of charged particles which exhibits collective behavior due to long range collective behavior due to long range electromagnetic forces. Neutral electromagnetic forces. Neutral atoms (molecules) may also be atoms (molecules) may also be present in the system.present in the system.Ex: HEx: H-- PlasmaPlasma
3 30 0
0 0 0
; ;i e
i e
n H cm n e cmn n n
+ − − −= =
= =
99
Let:
Electron Debye lengthL System’s dimension
Collision frequency of charged particles with neutrals
Plasma oscillation frequencyThe Plasma demands1) (i.e. em forces
dominate)2) (quasi neutrality)
3)
(statistical ensemble)
22 20 0
20 0
;eDe pe
e
T n en e m
λ ω⎛ ⎞ ⎛ ⎞∈
= =⎜ ⎟ ⎜ ⎟∈⎝ ⎠ ⎝ ⎠
Deλ
ν
peω
peν ω<<
LDe<<λ2
041 ( )3d D eN nπλ<< =
1010
Quasi Neutrality (Consider an EI-Plasma)Perturbation
1.1
Assumption :
For a linear wave
If wavelength
Then the long wavelength perturbation does not see the effects of charge separation. Therefore
1.2may be used
ϕ−∇=−∈
=∇ 2ei
0
)nn(qE.ϖ
2 2 1Deλ ∇ <<
2 2k∇ →
Deλ <<
ei nn =
E ϕ= −∇
0 0e in n=
1111
But or 1.3(1.2 together with 1.3) is the concept of quasi neutrality in plasmas.
IAW in EI- plasmas:
i)
ii) uniform plasma.
ion dynamics:
1.4
1.5
0≠ϕ 0E ≠
0 0 ˆB B z=v
0
( v . )v
( v )i i t i i
i i i
m n
en E B z p
∂ + ∇
= + × − ∇
rvv r v
.( v ) 0t i i in n∂ +∇ =v
1212
Linearization:
Perturbation 1.6
1.7
1.8
Electron dynamics :
1.9
e x p ( )zi k z tω∝ −
0 0v ( )i t iz z
i i z i
m n enT n
ϕγ
∂ = −∂− ∂
/0
0 (1 / )
ee Te
e
n n en e T
ϕ
ϕ=
≈ +
( )0em →
0 v 0t i z izn n∂ + ∂ =
1313
Using 1.10Eqs . (1.4--1.9) yield linear dispersion relation as,
1.11
where is ion sound speed
i en n=
2 2e i iz
i
T T kmγω
⎛ ⎞+= ⎜ ⎟⎝ ⎠
1/2
= e i is
i
T Tcmγ⎛ ⎞+
⎜ ⎟⎝ ⎠
1414
For
(1.11) becomes
(Non dispersive IAW) 1.12
If we have
(Dispersive IAW) 1.13
for
2 2 2s zc kω =
2 2 0D e kλ ≠
2 22
2 21s z
De
c kk
ωλ
=+
2; ei e s
i
TT T cm
<< =
2 2piω ω= 2 21 Dekλ<<
1515
Electron-Ion and Pair Plasmas: a. EI- Plasmas: Let
in general
zBB 00 =ϖ
;e im m≠ 0 0i e
i e
eB eBm m
Ω = << Ω =
pi peω ω<<
pee
pii
ccω
=λ>>ω
=λ
Di Deλ λ≠ ( )e iT T≠Q
2 2v ; ve iT e T i
e i
T Tm m
= =
1616
Slow time scales or Fast time scalesHybrid time scale
b. Pair plasmas m+ = m- , No diff in time scales. Ion acoustic wave does not exist in such plasmas.
Examples : Electron-Positron Plasmas in
i) Pulsar Magnetospheres ii) Active Galactic Nuclie (AGN) iii)Laboratory Pair-Ion plasmas:i) Recent claims that pure pair-ion
fullerene plasmas have been produced in laboratories(?)
t∂ iΩ<<ω
peet ,~ ωΩ∂peepi ,, ωΩ<ω<Ωω
1717
2. Pair2. Pair--Ion PlasmasIon Plasmas
Let constant, and consider the plasma to be homogeneous. [3.Phys.Lett.A 350,375(2006)]. For -species
2.2
≡= zBB 0
ϖϕ−∇=E
ϖ
0 ,v ( v )t Zm n n q E B pα α α α α α α∂ = + × −∇rr r
zt z z
q pv Em m n
α αα
α α α
∂∂ = −
0 0.v v 0t z zn n nα α α α α⊥ ⊥∂ + ∇ + ∂ =r v
2 2( ) v ( )t t
t
q E E zm
p z pm n m n
αα α α
α
α α α
α α α α
⊥ ⊥
⊥ ⊥
∂ + Ω = ∂ + Ω ×
Ω ∇ × ∇− − ∂
r rr r
r
α
2.1
2.3
2.4
1818
Eqs. (2.2-2.4) give,
2.5
Writing Eq. (2.5) for and then subtracting one equation from the other, we obtain
2.6
where is the ion gyro frequency.
Also and has been assumed
2 2 2 2 2 2 2 2 2
2 2 2 2 20 0
( ) v v
( ) 0
T T z
z
k k nn q n qk k
m m
α α α α α
α α α αα
α α
ω ω ω
ω ϕ ω ϕ⊥
−Ω − + Ω
− − −Ω =
i0i m/qB=Ω
qqq == −+ immm == −+
α = ±
2 2 2 2 2 2 2 2 2
0 0 2 2
0 0 2 2 2
[ ( ) v v ]( )
( )
( ) ( ) 0
i T i T i z i
i
z ii
k k n nqn n k
mqn n k
m
ω ω ω
ω ϕ
ω ϕ
+ −
+ − ⊥
+ −
−Ω − + Ω −
− +
− + −Ω =
1919
Let and be the ratio of specific heats. The Poisson equation reads
2.7
Let us assume for the time being that electrons are also present in the system and they obey the Baltzmann density distribution,
2.8
The set of equations (2.5--2.7) yields a few simple but interesting results. Let us discuss the limiting cases one by one. We observe that a new mode which may be called a finite frequency pair plasma convective cell (PPCC) can exist in such systems in the quasi neutral approximation. Let’s assume , then Eq.(2.6) with and
gives,2.9
1/2, v ( / )i Ti i i iT T T T mγ+ −= = = iγ
.q
)nn( 20 ϕ∇∈
−=− −+
0 e x p ( )e ee
en nTϕ
=
0n e≈−+≈nn
2
22 2z
ikk
ω⊥
= Ωiω << Ω
2020
Note:
In dusty plasmas a similar mode has been investigated. It may exist due to the presence of stationary dust in EI plasmas.
Dispersion relation is
2.100
0
e i zi
i e
n m kn m k
ω⊥
= Ω
2121
3. Criterion to Determine a PI Plasma3. Criterion to Determine a PI Plasma
IAW can not be observed in pure PI plasma. We
want to find out a criterion to determine .
Let , then for
Eqs.(2.6) & (2.8) yield,
00en
n+
0k,nn =≅ ⊥−+
2 2 2 2 20( ) v 3 .1s z T i z
q N c k ke
ω = +
where
And
If the frequency of
the wave may become larger than .
0 0
00
11e
n nNn+ −+ + ∈
= =− ∈
0
0 .nn−
+
∈=
2 2iω << Ω
01 N<<2 2s zc k
2222
We may have also even if The observation of IAW indicates the presences of electrons in the system. Therefore we expect
in the experiment as the plot of acoustic wave in Fig.2 of [2] shows (our Fig 1). Since holds if therefore we can determine by measuring the frequency of IAW. If and Eqs. (2.6) and (2.8) yield,
ω<iΩ izs kc Ω<
0 0en ≠( )0nkckc 0
zssszs ≠≠ωω< −Θ
01 ,N< 0n
0, 0iT k⊥= ≠ ,nn −+ ≈
4 2 2 2 20
2 2 20
( )
0 3 .2
i s
s z i
q N c ke
q N c ke
ω ω− Ω +
+ Ω =
2 2 2sc k ω<
e
2323
In the electron ion plasma case and for the ion cyclotron wave we have therefore Eq.(3.2) yields the well known dispersion relation
In the present situation, is possible along with therefore we retain the last term in Eq.(3.2). It gives for q = e,
3.3
,kkz ⊥<<
.kc 22s
2i
2 +=ω Ω,kcN z
22s0
2<ω
2 2 2 20
2 2 2 2 2 2 2 1 / 20 0
1 [( )2
( ) 4 ) ]
i s
i s s z i
N c k
N c k N c k
ω = Ω + ±
Ω + − ΩThis is the modified ion cyclotron wave dispersion relation. In the limit it reduces to the ion acoustic wave:
3.4
This is obliquely propagating IAW. In our opinion,(6.3) gives the so called IMF wave of Ref [2]. But should also be measured along with
iΩ<<ω2 2
2 02 2
0
/1 /
s z
s
q N c k eq N k e
ωρ
=+
0 1N =
01 N<<
k⊥ .eT
22
2s
si
cρ =Ω
2424
4. Instability of A New Mode and Nonlinear 4. Instability of A New Mode and Nonlinear Dynamics Dynamics
Let there be an external electric field and let the plasma flow be along y- direction such that both positive and negative ions move with the same shear velocity, and, hence the background current is zero. The steady state demands
For Eqs.(2.1) gives,
00E ϕ∇−=ϖϖ
0. . v ( ) v ( ) v ( )ie x x x+ −= =
.vBE 000 −=
t i∂ < Ω
0
1 1v ( ) ( v . )v
v v 4.1
t
E p
E z zBα α α
α
α
⊥ ⊥= × − ∂ + ∇ ×Ω
= +
v vv v vv v
v v
2525
and
4.2Note :
Then (4.2) yields
4.3
( v . )vt z zq Em
αα α
α
∂ + ∇ =v
0
20 0
0
2.( v v ) [
1 ( ) . ] ( )
p p tiB
zB
ϕ ϕ ϕ ϕ
+ −
⊥ ⊥ ⊥
∇ − = − ∂ +Ω
× ∇ + ∇ ∇ +
v v v
v vv
00
1 ( ) .
( v v ) ,z z
zt B
bz
ϕ ϕ
ϕ
⊥ ⊥
+ −
⎡ ⎤∂+ × ∇ + ∇⎢ ⎥∂⎣ ⎦
∂− = −
∂
v vv
2626
and continuity Eqs.yield the nonlinear Eq.
4.4
where a = - 2/ and b = 2q /mi
00
20
1 ( ) .
( ) ( v v ) ,z z
zt B
az
ϕ ϕ
ϕ ϕ
⊥ ⊥
⊥ + −
⎡ ⎤∂+ × ∇ + ∇⎢ ⎥∂⎣ ⎦
∂∇ + = − −
∂
r rr
)B( 0 Ω
Assuming the linear perturbation of the form , we obtain from
above Eqs.
4.5
where the superscript double prime indicates the second-order differentiation with respect to x . Let where and denote the realand imaginary parts of the frequency, respectively.
)tzkyk(exp)x( zy ω−+ιϕ
''202
20
2
20
vv
0 ,( v )
yy
y
z
y
kd kd x k
kk
ϕ ϕ ϕω
ϕω
− +−
Ω− =
−
,rω ω ιγ= + rω γ
2727
Then the instability condition turns out to be
4.6
where
'' 2 20 2
2 2 4 2 20 0 0
v 2 0 ,4
y z ik k ωω γ ω γ ω
Ω+ =
+ +
Let us consider the nonlinear stage of instability in a frame which is moving with velocity components and in the plane with . Eq. (7.3) can be written in this frame as
4.7
where Then the solution of Eq. (4.3) becomes
4.8
where F is an arbitrary function of the given argument. We choose a linear form of this function F = with an arbitrary constant.
yUzU −yz zy U/U=α
0
0
[ ( ). ]
[(v v ) ] 0,z y
z z
e B U x
B bxα⊥ ⊥
+ −
×∇ Φ − ∇
− − =
r
.0ϕ+ϕ=Φ
)xUB(FbxB)vv( y00zz −=α−− −+ Φ
)xUB( y0−Φ)xUB( y0−Φ
0 0( v )r ykω ω= −
0F 0F
2828
Thus the above equation becomes 4.9
where . Similarly, Eq. (4.4) can be written as
4.10
which yields
4.11
where is a constant. It is well know that equation (4.11) admits dipolar and tricolor vortex solutions.
0 0( )z zv v F B g x+ −− = Φ +
0( )yg b F Uα= −
20 0 0( ) ( ) 0,yBU x a F B xα⊥ ⊥∇ Φ− ×∇ ∇ Φ− =
20 0 0
00
(
) 0,
yG G B U
F B αα
⊥∇ Φ− Φ+
− =
0G
2929
5. Break down of quasi neutrality in PI 5. Break down of quasi neutrality in PI plasmas.plasmas.
5.1
Let and
5.2
0
1. ( )eE q n q n en+ + − −∇ = − −∈
v
eqq == −+ ϕ−∇=Eϖ
)nnn(ee
0
2 −−∈
=ϕ−∇ −+
)nn
nn
nn()
Te(
e0
e
e0e0e
22De
−+ −=ϕ
∇λ−
It , then is assumed and for ion continuity eq is used. then quasi neutrality yields (for ) ,
5.3
0n =−Te/e
0ee enn ϕ=+n
)n~n( ie 0k =⊥
2z
2s
2s kc=ω
3030
Since for low frequency IAW, therefore quasi neutrality is valid in general.If , then the above dispersion relation becomes,
5.4
For , and
(short wavelength limit), (5.4) becomes
5.5
Note : .
Small implies large .
1k22De <<λ
2 22
2 21s z
De
c kk
ωλ
=+
2 2 1Dekλ <
22De k1 λ<< 0k =⊥
2pi
2 ω=ω
2 02
0
eDe
e
Tn e
λ ∈=
0enDeλ
3131
If and , then for the situation seems to be very common in Pair-ion-electron (pie) plasmas.The real freq in pie plasmas becomes,
5.6
0n ≠− +−≈nn 0 en n +<<2 21 Dekλ<
rω2 2
2
v3( ) (1 )2
Tir s
s
kkω ωω
= +
where
5.7
Using kinetic model for , the dispersion relation for IAW can be written as,
5.8
2 2
0 2 21s
sDe
c kNk
ωλ
=+
0Ti≠
0)z(wi1k
11 j2Dj
2j =π+λ
∑+
3232
where
Here is called plasma dispersion function
if 5.9a
5.9b
For
For analytical analysis of IAW,let us assume,
5.10
,v
:kv2
zpj
TjDj
Tjj ω
=λω
=
j
jTj
2/1
j
2j0
pi mT
v,)m
qn4( =π
=ω
)z(w j
1)z(w j ≈ 1z j <<2
2 4
1 3w( ) (1 ) exp( )2 4j j
j jj
iz zz zzπ
= + + + −
Tevk
v <<ω
<<±
1 | | .jz<<
3333
If ,
If for , then we have for
5.11
1k22De <<λ 2 2 2 2 2
0 3vs T iN c k kω⇒ = +2 21 Dekλ<< 0 0en n +<<
r iω ω γ= −
0 2 2
2 20 0
( ) (1 )
1 3(1 )2 2
r p i
D
D e e
k
n kk n N
ω ω
λλ
+
+ +
≈ + ∈
⎡ ⎤− +⎢ ⎥
⎣ ⎦
where and . Eq. (5.11)
(may hold) 5.12
0
0
nn
+
−∈= ⇒pi rω ω<
0 2
0pi
i
n em
ω+ +=∈
in pie plasmas forLet
5.13
0 0en ≠
02 2 0
1 0 0 2( )eD e D e
Tnn n e
λ λ ++
− +
∈=∈ =
3434
For example if
Then . If is assumed
(which is equivalent to in case of ei plasmas) , then in pie
plasmas, we find and hence
19N,n9.0n 000 == +−
2 210De Deλ λ += 2 2 10De kλ + =
10k 22De =λ +
101=∈2 2 210 1Dekλ = >>
Imaginary part of the frequency can be written as,
5.14where
5.15
5.16
5.17
±± γ+γ=γ ie
e0e N γ=γ ±
i20e N γ=γ ±
1/ 2 1/ 22 2 2( ) ( ) ( )
8 (1 )e s
ei De
m c kkm k
πγλ
≈+
3535
5.18
For above relation reduce to the case of ei plasmas where [5. Akhiezeret et. al. 1975]
1/ 2 3 / 22 2 2
2
2 2
( ) ( ) ( )8 (1 )
( )exp( )
2 v
e si
i Dc
Ti
T c kkT k
kkγ
πγλ
ω
≈ ×+
−
0n 0 =−1N 0 =
In the limit we obtain
5.19
and
5.20
2 21 Dekλ<<
kck
1mm
8)k( s22
D
2
i
ee ⎥
⎦
⎤⎢⎣
⎡
λ∈−π
≈γ+
±
23/2
2 2 2
2
2 2
(1 )( ) ( )8 ( )
exp2
ei
i D
sTi
TkT k
c kk k
γ
πγλ
ω
±
+
⎡ +∈≈ ⎢⎣
⎤⎛ ⎞− ⎥⎜ ⎟⎜ ⎟⎥⎝ ⎠⎦
3636
6. Conclusion6. ConclusionI. Damping analysis:
Therefore
For the same in ei and pie plasmas, we find
i)
ii) always because
2
2 2exp 1 2 v
and 1 or 1
r
Tikω⎛ ⎞
− <<⎜ ⎟⎝ ⎠∈≤ ∈<<
ii γ≤γ ±
2 2De kλ +
ii γ<γ ±
e eγ γ± < 2(1 ) 1− ∈ <
3737
So for 6.1
II. may hold because
even for The quasi neutrality is not a good approximation in such systems.Therefore IAW can be easily excited in pair-ion plasmas comprising electrons. This can be a possible explaination for the density fluctuation associated with IAW in experiment [ 2]. Hence these plasmas do not seem to be the pure ion plasmas.
III. A new mode may be interesting in pair plasmas. It can also become unstable in certain situations.
IV. for IAW in pair-ion plasmas having electrons. Measuring and , one
can estimate .
γ<γ± 2 21 .Dekλ<<
k1 2Deλ<< 11 <<∈
.1k2De ≤λ
s rc k ω<<eT sc k
00en
n+
3838
1. W.Oohara and R. Hatakeyama, Phys. Rev. Lett. 91, 205005(2003).
2. W.Oohara and R. Hatakeyama, 95, 17 5003(2005) Phys. Scr. T116, 101(2005).
3. H. Saleem, J. Vranjes and S. Poedts, Phys. Lett. A 350, 375(2006).
4. H. Saleem, Phys. Plamas (in Press).5. A. I. Akhiezer, I. A. Akhiezer, R. V.
Polovin, A. G. Sitenko, and K. N. Stepnov, Translated by D. ter Haar, Plasma Electrodynamics, vol. 1 (Pergamon Press 1975).