Electricmicrofielddistribution Inneutral-Ionplasmas

ELECTRICMICROFIELDDISTRIBUTION INNEUTRAL-IONPLASMAS

pA7

Thouria CHOHRA and Mohammed Tayeb MEFTAH

LaboratoiredeDéveloppementdesEnergiesNouvelleset RenouvelablesdanslesZonesArideset Sahariennes(LENREZA),FacultédesScienceset TechnologiesetdesSciencesdelaMatière, UniversitéKasdiMerbah–Ouargla,30000Ouargla,Algeria

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ABSTRACT: The knowledge of the electricmicrofield distribution in multicomponent plasmas isa necessary conditiontothe solution ofseveral problems. Inparticular, the calculation ofthe spectral line shapes for anion, taken as radiator in a plasma consisting of neutrals and ions is one of these problems requiring such a distribution.In thiswork, we areinterestedintheelectric microfielddistributioninatwo-component plasma.To reachthisgoal,we useda usefulmethodbasedon”clusterexpansion”, widelyknowninstatisticalmechanics. Hereweonlyusethefirstterm oftheBaranger-Mozer formalism (theindependent particleapproximation).The system wedealwithconsistsofionsandneutralsimmersedinauniform neutralizingbackground. Thetotal systemisassumedtobeinthermalequilibrium andneutralatallpoints.Themaininteractions usedareion-ion and ion-neutralinteractions.

KEYWORDS: electricmicrofielddistribution, multicomponentplasma, clusterexpansion

1. Introduction

The knowledge of theprobabilitydistribution function for electric field in a multicomponentionized plasmasisaprerequisitetothesolutionof anumberof problems,in particularthatofthecalculationofthebroadeningofspectrallinesinplasmas[1,6].In relationto this problem, various theories of the electricmicrofielddistributions have been formulated.Theprimaryaim oftheseeffortshasbeentoincludeion-ioncorrelationswith various orders and thus to improve the original work done by Holtsmark [5].

2. Formalism

WeconsidertheelectricmicrofielddistributionW(E)[1],definedastheprobability

densityoffindingafield Eequaltoεatthecharge

Z1e,locatedat

r1,intwo-component

ioniccoldplasmas(TCICP)whereionsofspeciesσ=a,bcarryacharge

Zσeandneutralsof

species

σ=c,d.Here,eisthemagnitudeoftheelementarychargeandallthe

Zσ ’sare

positive.As usual,weassumethattheelectronscreening isdescribedbyDebye-Hückel’s formula. This can be justifiedonly for plasma inwhichtheelectron-electronand electron-ion couplingsarebothweakandtheplasmamaybedescribedbyclassicalmechanics.The system,whichalsoincludesauniformneutralizingbackground,isassumedtobedescribedby classical equilibriumstatistical mechanics with temperature Tand number densitiesnσ ,

N

nσ = σ Ω

and

N=∑Nσ

σ

=Na +Nb+Nc +Nd

ne =Zana +Zbnb

We introduce the composition parameter,

p= Nb ,

Na +Nb

p'=Nb

N

where

Nσis the number ofparticles of speciesσ=a,b,c,dand Ωisthe total.

The quantityλDis the electron Debye screeninglength[2]

2 KBT

λD = 2

4πnee

The dimensionless classical plasma parameter thus reads

⎛

Λ=⎜1+∑

1/2

nσ Z2⎞

e2

=0.33v3

⎜ σ=a;bne ⎟

KBTλD

r0

ne (cm )

= =0.0898

1/6 −3

v λ T1/2(K)

Theelectroncomponentwith 0sothat(4/15)(2π)3/2nr3 =1.TheHoltsmarkunitoffield

strength thus becomes

With the reduced unitβ=E/E0.

E0(KV/cm)= 2

0

The microfield distribution will be discussed under the usual isotropic form(u=kE0)

H(β)= 2β∫uF(u)sin(βu)du

(1)

in terms of its Fourier transform

π 0

F(u).

ThemathematicalquantityofinterestisobviouslyF(u).ItistheFouriertransformofthe probabilityW(E)for finding an electric field,

i n

E=E +E

i

(2)

Theelectricfieldatchargedpoint(ions)

n

E are given by,

E andtheelectricfieldatneutraledpoint(neutral)

i Nσ

E =−∑∑zσef

σ =a,bi=1

i Nσ

(r −r )r1 −ri

r1−ri

( )r1 −ri

(3)

E =−∑∑ασz1ehr1−ri

= =

r1−r

Where

σ c,d i 1

1⎡ r ⎤

i

⎛ r ⎞

f(r)=

2 ⎢1+

⎥exp⎜− ⎟

r ⎣ λD⎦

1⎡ r ⎛

⎝ λD ⎠

r ⎞ ⎤ ⎛ r ⎞

(4)

h(r)=

⎢1+

r ⎢⎣ λD

+⎜1+

⎝

⎟

λD ⎠

⎥exp⎜−2

⎥⎦ ⎝

⎟

λD ⎠

andασ

ispolarizabilitycoefficientoftheneutralofspeciesσ (α≈R3, 3

istherayonof

the neutral). One then gets

F(k)=∫exp(ik.E)W(E)dE

=∫exp(ik.E)P(r1,r2,...,rN)dr1dr2...drN

(5)

where

P(r1,r2,...,rN)is thejointprobability for finding Nparticles located atr1,r2,...,rN.

Upon introducing theauxiliary quantitiesϕ through

exp⎛

. a⎞

1 ⎡exp⎛

. a⎞ 1⎤ a

⎜ikEi ⎟= +

⎜ikEi ⎟−

=1+ϕi

⎝

exp⎛

⎠

. b⎞

⎝

1 ⎡exp⎛

⎠

. b⎞ 1⎤ b

⎜ikEj⎟= +

⎜ikEj⎟−

=1+ϕj

⎝

exp⎛

⎠

. c⎞

⎝

1 ⎡exp⎛

⎠

. c⎞ 1⎤ c

(6)

⎜ikEk⎟= +

⎜ikEk⎟−

=1+ϕk

⎝

exp⎛

⎠

. d⎞

⎝

1 ⎡exp⎛

⎠

. d⎞ 1⎤ d

Then

F(k)becomes

⎜ikEl ⎟= +

⎝ ⎠

⎜ikEl ⎟−

⎝ ⎠

=1+ϕl

F(k)=1+∑P(ri)ϕi dri +∑

∫P(rj)ϕjdrj +∑1

∫P(rk)ϕkdrk +∑1

'''

∫P(rl)ϕl drl

+∑2∫P(ri,ri' )ϕi ϕi'dridri'+∑2

∫P(rj,rj' )ϕjϕj'drjdrj'+∑2

∫P(rk,rk' )ϕkϕk'drkdrk'+(8)

a a ' b b

'' c c

∑2 ∫P(rl,rl' )ϕl ϕl dridri'+∑1∑1

∫P(ri,rj)ϕi ϕjdridrj +...

''' d d

'

' a b

Where

∑(∑

' )denotesasumonionsa(b),while

∑'(∑

'' )isasumonneutralsc(d)and

1 1 1 1

∑(∑' )

isthesumonaa(bb)pairs,andsoon.Acrucialstepinthisformalismisthe

introduction of the cluster expansions(σ,σ' =a,b,c,d)

Ω P (ri,...,ri

)=∏g1 (ri)+∑

g2 (ri,ri' )∏g1 (ri'')+...

M σ M σ σ σ

M 2

i i

ΩMPσσ'(ri,...,riM,rj,...,rjM)=

gσ(ri)

gσ'(rj)+

gσσ'(ri,rj)

gσ(ri' )

gσ'(rj' )+...

M ∏1 ∏1

i j

∑2 2

∏1 ∏1

i' j'

WhereMrefers to particleslocated atri,...,riM.

Formostcasesofpracticalinterest[2],weshallrestrictourselvestoweaklycouplessystems

(Λ≤ 1). Eq.(25) may then stop at the order Λwith

F(u)≈exp[n ha(u)+nhb(u)+nhc(u)+n

hd(u)]

(9)

And

a 1 b 1

c 1 d 1

h1 (u)=∫g1 (r1)ϕ1 dr1

σ=a,b,c,d

(10)

σ σ σ

Where

r1 denoteslocationofparticle

σ=a,b,c,dandga,

gb,

c and

gdarethe pairs

correlations functions. Making use of spherical harmonics expansion

ϕi =∑i[4π(2l+1)]

[jl(Zi

)−δl0]Yl0(θi,ωi)

σ=a,b,c,d

(11)

σ l 1/2 σ

Where

jl(Z)

l

is a spherical Bessel function, the

h1’s are expressed as

(Zi

=kEi ,

Xi =ri/λD)

nσh1

=−u

φ1 (a)

σ 3/2 σ

φσ a

= 15 nσ 1

−j Zσ

gσ X

X2dX

(12)

1 ( )

2(2π)1/2 n

3 ∫[1

0

0( 1

)]1 (

1) 1 1

Wheretheargument

a=u1/2visnottobeconfusedwiththeupperindexlabelingtheheavy

ion component. The central quantity F(u) is then well approximated by

F(u)≈exp[−u3/2(φa(a)+φb(a)+φc(a)+φd(a))]

(13)

1 1 1 1

Itcanbecomputedforanymixturethoughtheφ’sandtakingintoaccountionsandneutrals

screened by electrons with (σ=a,b,c,d)

σ Z

1 2 1

1

]exp(−X1)

σ=a,b

(14)

2 3

σ = 2αZ1av [1+X

1

+[1+X1

]2]exp(−2X)

σ=c,d

References

[1] B. Held and C. Deutsch, Phys. Rev. A 24, 540 (1981)

[2] B. Held, C. Deutsch, and M. M. Combert, Phys. Rev. A29, 880 (1984) [3] M. Baranger and B. Mozer, Phys .Rev.115, 521(1959)

[4] M. Baranger and B. Mozer, Phys .Rev. 118, 626 (1960) [5] J. Holtsmark, Ann. Physik58, 577 (1919)

[6] C. F. Hooper, Phys. Rev. A 149, 77 (1966)