11

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).