Electric microfield distribution in multicomponent plasmas

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Electric microfield distribution in multicomponent plasmas

B. Held

To cite this version:

B. Held. Electric microfield distribution in multicomponent plasmas. Journal de Physique, 1984, 45

(11), pp.1731-1750. �10.1051/jphys:0198400450110173100�. �jpa-00209915�

Electric microfield distribution in multicomponent plasmas

B. Held

Laboratoire d’Electricité, ^{Avenue Louis} Sallenave, Université de Pau, 64000 Pau, France

(Reçu ^{le 14 mai} 1984, accepté ^{le 5} juillet 1984)

Résumé.

²⁰¹⁴

Un formalisme général ^est présenté pour les parties ^Haute Fréquence ^{et Basse} Fréquence ^{du micro-} champ ^dans

^un

plasma ^{à trois} composantes (une composante électronique

²⁰¹⁴

température Te, ^deux composantes ioniques

²⁰¹⁴

température Ti). Ce formalisme est présenté ^sous

^une

^forme générale permettant ^des applications

allant des plasmas faiblement corrélés

aux

plasmas ^fortement corrélés, les effets physiques ^étant introduits explici-

tement par la fonction de corrélation de paire. ^{Les résultats sont} présentés pour les deux parties ^du microchamp ^et

une attention particulière

^a

^été portée

^sur

^{les effets} rencontrés dans les plasmas de fusion inertielle produits par laser

(corrélation, proportion, température, charge). ^{Ces effets montrent une très} grande sensibilité du microchamp

Basse Fréquence vis-à-vis du paramètre plasma ^A.

Abstract.

²⁰¹⁴

A general formalism is presented ^{for the} High Frequency and the Low Frequency parts of the electric microfield in three component plasmas (one electronic

²⁰¹⁴

temperature Te, ^{two ionic}

²⁰¹⁴

temperature Ti). ^This

formalism is presented ⁱⁿ

^a

general ^form allowing ^extensive applications ^from weakly ^to strongly correlated plas-

mas, the physical ^effects being introduced explicitly by

^means

^of

^a

two-body correlation function. Results

are

pre- sented for the two parts of the electric microfield and particular attention has been paid ^to effects encountered in

plasmas of inertial-confined-fusion produced by ^laser (correlation, proportion, temperature, charge). ^These

effects show that the Low Frequency part of the electric microfield is very sensitive ^to the plasma parameter ^A.

Classification

Physics ^Abstracts

52.70

1. Introduction.

In dense and hot plasmas of inertial-confinement- fusion produced by ^{laser, the Stark} broadening ^dia- gnostics ^of stripped ions immersed in ionic perturbers

appears ^to be the best method for determining ^the density-temperature conditions [1-8].

Because, ^for the Stark broadening calculations

[9-13], ^it is necessary ^to know the electric microfield,

the _purpose of this _paper is to present ^a general ^for-

malism for multicomponent plasmas.

The problem ^of electric microfield distributions was

first solved a long ^time ago by ^Holtsmark [14-16] ⁱⁿ

the low-density, high-temperature ^limit (non ^corre-

lated system). Since then _many works have been devoted to these calculations [17-77] ^{and a} general

formalism was presented by Baranger ^{and Mozer} [27-

28] using ^the approximation ^{of a} weakly ^correlated

one component plasma, ^but Hooper et ^al. [30-32]

were the first to propose ^{an accurate} numerical code for multicomponent plasmas.

Further to these theoretical approaches, ^numerical

methods were developped ^to predict ^{the microfield}

distributions. The Monte Carlo calculations [78]

now give ^accurate results for two component plas-

mas [79-80] and numerical simulations for the ionic component ^{of the} plasma allow the inclusion of the effect of ion dynamics ^{on stark} profiles [81-90].

A generalization ^{of the} Baranger-Mozer ^formalism

is presented in this paper for multicomponent plasmas including ^strong correlation effects by ^a resummation up ^to infinity ^{of the most} important ^chain diagrams [92-93].

A comparison between the results presented ^here

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450110173100

and other works [80, 91] gives ^accurate values for a

plasma parameter ^{A less than} unity. ^This theory ^can

be extended to higher ^{values of A} by including ^into

the model new conditions on the resummation and

on the two body correlation function [91].

The electric field is separated ^{into two} parts : ^the

High Frequency part (H.F.) being ^{relative to electrons}

and the Low Frequency part (L.F.) ^{to ions.}

For the H.F. part, the model is ^an electronic _gaz with uniform neutralising background, ^{while the L.F.}

part corresponds ^{to the time} average of the electric

field, ^the average being ^{taken over times} long compared

to electronic relaxation times but short compared ^to

ionic times. Keeping in mind the Stark broadening application, the electric microfield distribution must be calculated on an emitter; this emitter can be an atom or an ion and we must establish a distinction between the Neutral Point case (N.P.) ^{and the} Charged

Point case (C.P.).

Finally, ^{the two} time-independent parts ^{of the} scalar microfields (H.F. ^and L.F.) ^{will be calculated}

in the two cases (N.P. ^and C.P.) using ^the assumption

that the medium is isotropic.

The model developed ^{in this} paper is devoted to three component plasmas (one electric and two ionic),

with the assumption ^{of local} thermodynamic equili-

brium for electrons (Te) ^{and ions} (T;), and the des-

cription ^{of the} plasma ^is given by ^five parameters : the correlation parameter ^v (measuring the relative

importance of the energy of interaction compared ^to

the thermic energy), ^the proportion coefficient _p

(between ^{ions a and} b), ^the temperature ^rate RT (between electrons and ions) and the number of

charges Za, Zb of the ionic perturbers.

In section 2, ^the general formalism is developed ^for (ç) particles, (ç) being ^{electrons or ions.}

Section 3 is devoted to applications of the formalism to the H.F. and the L.F. parts of the microfield. Defi- nitions are given for the five parameters ^and physical

effects are introduced into the theory by ^{means of the} two-body correlation function.

In section 4 are presented the results for the H.F.

and the L.F. part ^{of the} microfield, in the N.P. and C.P. cases.

Section 5 gives ^the general conclusions and supple-

ments to the main text are presented ^{in an extensive} Appendix.

2. Formalism.

We consider a plasma ^{with two} species ^of particles’

and ç. ^{We call N the total number of} particles ^with

coordinates r,, r2,

^...,

rN :

where N, ^and N4 ^are respectively ^{the number of} particles ^of types ^and ç.

For a certain spatial repartition ^of particles, ^the

total microfield at an arbitrary point ^{0 is} given by :

where EJ(E4) ^{is the} microfield produced by ^the parti-

cle ((j) ^{located at a} point Mj(rj) ^(Mk(rk))’

We wish to calculate the probability distribution

w(E), ^{but it is more} convenient to consider the Fourier transform :

If {f(rl’ r2,

^...,

rN) ^{is the} probability ^{of occurrence}

of a given configuration ^{of the N} particles, ^{we can}

write :

It is then possible ^to replace ^the exponential by ^the development :

where qJ) ^{and (p4 k are defined by :}

By ^an integration ^{over all the} possible positions, F(k) ^{becomes :}

E4 (.E4) being the ^{sum on all} particles (ç), (r) ^{the sum on all} pairs, ^etc...

If we take M particles among the N and assume that correlations introduce small corrections, ^{it is} possible

to express the probability ^function 5’(rj, r2l .... rN) by ^{a series} containing increasing orders of correlations :

where V is the total volume and gc, g2,

^...

correlations of first and second order.

The Fourier transform of probability ^{can then be written}

It is then simpler ^{to define} G(k) functions :

I(I;) ^{being the sum on all possible} combinations of p particles, Ip(I;p) ^{on all possible} distinct combina- tion of two clusters containing p particles, ^etc...

If we call n,(n,) ^{the density of particles (ç) and assume that} n, ^and nÇ ^{remain constant when} N,, N ç ^{and V}

become very large, ^{we can write :}

When N, ^and N4 ^{approach infinity, the} Gp(k) functions have a fairly simple expression :

with :

We obtain the probability distribution W(E) by inverting the Fourier transform :

with :

3. General results.

3.1 PARAMETERS. 2013 We assume that the plasma ^is composed ^{of two ionic} components (a, b) ^{and an}

electronic one. The description ^{of such a} plasma ^is possible ^{with five} parameters (v, p, RT, Za, Zb) :

-

Correlation :

where ro ^is approximately ^{the mean} inter electronic

length ^and AD. ^{the Debye length for the electrons} (appendix A).

-

Proportion :

where Ca(Cb) is the concentration of ions.

-

Temperature ^{rate :}

where Te(T;) ^is the electronic (ionic) temperature.

-

Number of charges ^{on ions a and b :}

Starting with these five parameters, ^the plasma parameter ^can easily be written (appendix A) :

with :

where AD ^{is the} Debye length ^{for the} plasma.

3.2 CORRELATION FUNCTION.

^-

If we call V,,(r) ^the

interactive potential ^between particles C and j ^{at a}

distance r, the two body correlation function is defined by :

with :

To connect the g,,(r) correlation function with

g 1 (r), g 1 (r) ^and g’4(r) probability functions, ^{we assume}

that there is an C particle, ^with Z, ^{charges, at the}

centre of the system.

The one-body probability ^functions gc(r) ^and gi(r) ^can easily be written starting ^{from the} g,,(r)

correlation functions :

- gí (r) represents the correlation between an , particle (coordinate r) ^and the C particle ^{located at the}

centre (coordinate 0) :

-

g4(r) ^is the correlation between the centre and

a ç particle (coordinate r) :

The two-body correction gj4(r) ^can be written :

and gÇ(r) is obtained explicitly by ^a linearization of the g,,(r) function. This approximation ^{is corrected} by ^a resummation over all the orders [92-93] ^and g%4(r) ^{becomes :}

with :

and the reduced length :

The general expressions ^{for the} electronic (H.F.)

and the ionic (L.F.) parts ^are easily ^{deduced :}

3. 3 GENERAL EXPRESSIONS.

^-

Before deducing gene- ral expression for the microfield from the formalism, it _{is very} useful to introduce :

-

the reduced microfield :

with :

-

a new variable for the expression of F(k) :

with :

-

a new variable for functions cp) ^{and (p4 :}

with these notations, ^the general expressions ^{for the}

microfield produced ^{at the} origine by ^{an electron or}

an ion located at Xj (Xk) ^are given by :

for an electron

If we call H(fl) ^the probability distribution of the scalar microfield. ^{we can write :}

with :

for the first and second orders of correlation.

The functions h2(u) ^are calculated expanding every factor in the integrant ⁱⁿ spherical ^harmonics (appen-

dix B); h 1 (u) ^and h2 (u) ^{can then be written :}

where the h(Xl’ ^{x2) functions are} deduced from the development ^into spherical harmonics of 0,,(x) ^(appen-

dix B).

The F(u) function becomes :

with :

and :

3.4 ONE ^AND TWO-COMPONENT SYSTEMS.

^-

The one component system ^{is deduced from the} general expres-

sions with the condition : _p

⁼

0 and it _{is easy} to show that (appendix A) :

The function F(u) ^{is then} given by :

For the electronic part (H.F.), , ^{is an electron e} with the conditions :

and for the ionic part (L.F.), , ^{is an ion a with :}

For two ionic-components systems, ^{we have to} introduce the proportion p and functions F(u), t/J l’ ql2, ^become

__ ,

which verify the one-ionic component ^limit (p

⁼

0).

3 . S PARTICULAR CASES.

^-

The microfield distribu- tion can be calculated at a charged point (C.P.) ^{or at a}

neutral point (N.P.). ^{In the first case, there is a} charged particle ^with Z, ^{charges at the centre of the system}

and in the second case, there are no charged particles

at the centre.

We now consider the N.P. case, deduced from the C.P. situation (section 3. 2) writing :

and we find :

The general results of section 3.4 are also valid in N.P. conditions with :

another particular ^{case is} also caused by high tempe-

rature-low density conditions; ^the system ^{is then} equivalent ^to independent particules without correla- tion :

The general ^{results are then reduced to :}

and

The general expressions of section 3. 3 become :

For the electronic part (H.F.) ^{we find (p}

⁼

0, nÇ

⁼

⁰⁾

the Holtsmark distribution :

with (appendix C) : ie

⁼

^{1 and for the two ionic} part (L.F.) :

with (appendix C) :

The Holtsmark distribution is also the limit of an

ionic microfield with the following conditions :

or

When there is no correlation between particules (v =o),

the functions g/§’ and g/§4 ^{are the same at a neutral or at}

a charged point and the results are independent ^of

the temperature ^rate RT.

3 . C) ASYMPTOTIC BEHAVIOUR. - We consider first a

correlated system (v # 0) ^{and we establish a distinc-}

tion between a neutral point (N.P.) ^{case and a} charged point (C.P.) ^{case to take the} asymptotic ^limit.

Let us assume that we are in the N.P. case. With an

appropriate change ^of variable, ^{we can write} (appen-

dix C) :

and

with

FE(a) being ^the development ^of F(u) ^{in the} vicinity ^of

a

⁼

0. As it is shown in appendix C, ^the functions 4/1 and t/J 2 ^are given by developments of a and if we call

t/J(a) ^{the sum on all these} developments :

F£(a) ^can be written :

For a one component system

with

and for a two-ionic component system :

In each _case, the constants A, B, ^{C have to be calcu-} lated from functions t/11 ^and ql2’

After a development ^{of the} powers of t/1(a), ^{we find :}

If we ^now consider a charged particle (Z,) ^{at the centre of the system,} it is then possible ^{to make the nearest} neighbour approximation ^to obtain the asymptotic development ^of H(fl). ^{If we assume that the} probability ^of finding microfield fl ^{at the} origin ^is equal ^{to the} probability ^of finding ^a charged particle ^{at a distance r} (appen-

dix C), ^{we can write :}

and with the notation :

we find :

with :

The total microfield becomes :

and, ^{as a first} approximation, ^{it is} possible ^to replace ^the exponential integral ^term by unity if P ^is sufficiently large.

For a one component system (p

⁼

_{0, n4}

⁼

0) ^{we find :}

-

for electrons :

-

for ions :

in the limit of large ^{values of} fl.

For a two-ionic component system :

Finally, ^we should consider a non correlated system (v

⁼

0). ^{In this case} (see ^section 3. ), F(u) is reduced to the first terms oa,, oab . ^The asymptotic distribution

H(fl) is obtained as a limit case of N.P. calculations, writing :

and

without distinction between N.P. ^or C.P. cases.

It is obvious that we will find the asymptotic

Holtsmark distribution for electrons (K

⁼

1) ^{and also}

for ions if :

or

4. Applications.

Because of the Stark broadening diagnostics ^of highly stripped heavy ^{ions immersed in} dense and hot

plasmas of inertial confinement

^-

fusion produced by ^laser, the results presented ^{below are devoted to the}

L.F. part ^{of the} microfield in the C.P. case (Figs. 1 ^to 8).

Fig. ^1.

^-

Correlation effects (v) ^at

^a

charged point (L.F.)

for

a

one component plasma.

Fig. ^2.

^-

Correlation effects (v) ^at

^a

charged point (L.F.)

for

a

two component plasma.

Correlation effects are presented ⁱⁿ figures ^{1 and} 2 ; when v increases (i.e. ^for increasing ^{values of the}

electronic density ne ^or decreasing values of the elec- tronic temperature T e), the microfield H(fl) ^{is shifted}

towards smaller values of the reduced microfield f

with a pinch ^{effect on} the distribution and an enhan- cement of the maximum. The weaker field, produced by charged particules far from the centre, ^are intro- duced with increasing ^values of the correlation v, this effect‘ being ^accentued by ^{the presence of a} charged particle ^{at the centre.}

This effect is also observed on H.F. part ^{of the} microfield in the N.P. case and in the C.P. case [51-52].

In figures ^{1 and 2} (p

⁼

1, RT

⁼

^1, Zb

⁼

1) ^{it is also} possible ^to observe the correlation effects for various values of the central charge Za. ^When Za increases, ^a repulsive ^{effect on the} perturbers Zb ^{appears and the}

microfield is slightly ^shifted towards weaker fields with a pinch ^{effect on the} distribution. When the central charge ^{and the} perturbers ^{are the same} (p

⁼

0, RT

⁼

1 ), ^{the distance} between the perturbers ^{and the}

centre increases with Z. (repulsive effect), but the field

being proportional ^to Za, the maximum of the dis- tribution shifts towards stronger fields and the peak

value of H(f3) ^decreases.

Composition ^{effects are} illustrated by figures ^{3 and 4}

when the charge ^{on the} perturbers (Zb = 1 ) ^{are less}

than the charge ^on the central ion (Za). ^In figure ³ (v

⁼

0,.Zb

⁼

^{1) a shift towards the weaker} fields with

a pinch effect is observed when p ^increases (progressive

introductions of singley charged ions into the plasma).

In the case of correlated plasma (Fig. 4), ^{the shift} towards the weaker fields with increasing ^values of p

is accentuated by the correlation effect. The correlation

being ^more important for small values of p, ^{the resul-}

tant shift is maximum for p equals ^zero and minimum for _p equal unity : ^the global composition ^effect

decreases when correlations increase (Figs. 3, 4).

Fig. ^3.

^-

Proportion ^effects (p) ^at

^a

charged point (L.F.)

for a non correlated plasma.

Fig. ^4.

^-

Proportion ^effects (p) ^at

^a

charged point (L.F.) for

a

correlated plasma.

Temperature ^{effects are} presented ⁱⁿ figures ^{5 and 6.}

The maximum of the distribution H(P) increases with

RT, this effect growing ^with Za : ^{the cloud of electrons} surrounding each ion is perturbed by ^a change ⁱⁿ RT leading ^{to a} modification in the shield effect.

In figures ^{7 and} 8, the effects of the central charge (Za) ^{are shown. In the case of a one} component plasma,

when Za increases, ^{there is a} competition ^between

the Coulomb repulsion and the enhancement of Za :

the maximum of H(fl) decreases and the distribution is shifted towards stronger fields. The ^« correlation shift » towards weaker fields increasing ^with Z., ^the global Za charge effect decreases when v increases.

Fig. ^5.

^-

Temperature ^{rate effects} (RT) ^at

^a

charged point (L.F.) ^for

^a

^one component plasma.

Fig. ^6.

^-

Temperature ^{rate effects} (RT) ^at

^a

charged point (L.F.) ^for

^a

^two component plasma.

Fig. ^7.

^-

Charge ^effects (Za) ^at

^a

charged point (L.F.) ^for

a

non correlated plasma.

5. Conclusion.

In this _paper, the H.F. part and the L.F. part ^of the microfield have been introduced as particular ^{cases of}

a general formalism valid for a three-component plasma. Physical ^{effects are} easily introduced into the

theory by ^{means of the} two-body correlation function, allowing ^{extensive studies} (such ^as diffraction and symmetry ^effects encountered in highly ^{dense and hot} plasmas [53-54]).

With the introduction of five parameters (v, p, RT, Za, Zb), ^{it is} possible ^to observe the different effects

on the L.F. part of the microfield in view of application

Fig. ^8.

^-

Charge ^effects (Za) ^at

^a

charged point (L.F.) ^for

a

correlated plasma.

to the Stark broadening. ^In practice, the results have been presented ^{for values} of the coefficient of correla- tion v encountered in plasmas ^{of current interest in}

inertial-confinement-fusion studies (i.e. ^for high ^den- sity, high temperature).

A general survey of the curves (Figs. 1 ^to 8) permits

one to notice the importance ^{of the} plasma parameter A for the interpretation ^{of the} results; however, particular ^{attention must be} paid ^to density-tempe-

rature effects because a dense plasma ^{is not} necessarily

a correlated plasma, the correlation being dependent

on the temperature (A3); ⁱⁿ consequence, the Holtsmark limit ^can be reached in high temperature conditions for the H.F. part ^{or the L.F.} part ^{of the} microfield.

With the results presented ^{in this} paper, it is possible

to deduce a general law for the normalized distribution of the microfield : ^{a shift of} H(fl) ^{towards the weaker}

fields is always accompanied by ^a pinch of the distri- bution (and ^an enhancement of the maximum to

respect ^the normalisation conditions), ^{this law} being

valid for all physical ^effects.

In summary, this shift _appears when the correlation

or the proportion (for Zb Za) increase and when the

charge Za ^decreases (for ^{a one-ionic} component

plasma); ^the proportion (p) ^and charge ^effects (Za)

are reduced by the correlation effects. The temperature effect is responsible ^{for a} modification in the electronic shield surrounding each ion and a small shift is also observed when RT ^increases (this tendency increasing

with the central charge Za).

Finally, keeping ^{in mind the} applications ^{of the}

Stark broadening, ^{it is} important ^{to note that the} proportion, temperature ^or charge ^{effects can some-}

times be interpreted ^as correlation (density-tempe- rature) effects : this is a difficulty ^{for the} diagnostic ^{if the} plasma caracteristics ^are insufficiently ^known.

Appendix ^A.

^-

Parameters.

We consider a three component plasma : ^{one electronic}

e and two ionic a and b. The description ^{of this} plasma

is possible ^{with five} parameters : ^v (correlation), p

(proportion ^{of ions} b), RT (temperature rate), Za - Zb (number ^of charges ^{on ions a and} b).

These parameters ^are fully ^defined using ^electronic density ne, electronic and ionic temperatures T,, Ti.

We have first to introduce the mean inter electronic

length ^defined by :

and the Debye length for the electrons :

kB ^being the Boltzmann constant.

The coefficient of correlation v is then given by :

and the proportion p by :

where Ca ^and Cb, the concentrations of ions a and b,

are related to Ce, the concentration of electrons by :

and neutrality :

Introducing ^the density ^of particles :

with

it is _easy to obtain the rates :

Finally ^{it is useful to introduce} RT, ^the temperature

rate (when ^{electron and ions are at two} equilibrium temperatures Te ^and Ti) :

with these definitions, ^{it is} easy ^to explicit ^the Debye length and ^{and the} plasma parameter ^A.

The Debye length ^{for a} plasma ^{with two} tempe-

ratures is given by [94] :

with

From the preceding definitions, ^the Debye length

for the ions can be written :

and the Debye length ^{for the} plasma ^{becomes :}

with

It now becomes interesting ^to express the plasma

parameters from these definitions.

The electronic plasma parameter Ae ^defined by :

becomes

and the plasma parameter ^{A :}

becomes :

Appendix ^B.

^-

Correlation function and spherical ^harmonics developments.

The two-body correction function d24(r) ^can be written from the two-body correlation function g,ç(r) :

It has been shown [92-93], that it is possible ^{to make a} resummation on all the orders of development

for a two-body correlation function :

writing

we find

with

The h2(u) ^{function can then be expressed} explicitly ^{with a} spherical ^harmonics development. Field functions

c ⁴ and correlation functions 0,,(x) ^{can be expanded :}

By integration, ^the functions fC4(X 11 ^{X2) can be found [52] :}

with

Appendix ^{C. -} Asymptotic behaviour of H (fl).

Cl GENERAL EXPRESSIONS FOR THE FIELDS.

^-

Before

establishing asymptotic behaviour of H(fl) ^{at a neutral}

or at a charged point, ^{we have to} express the variables

U j ^{and U4} of the fields.

By definitions, ^{we can write :}

For the H.F. part, Eje is the field of one electron at a distance rj :

For the L.F. part, Ej(E:) ^{is the} field-screaned by ^the

electrons

^-

of an ion a(b) ^{at a distance} rj(rk) :

and we find :

with the definitions :

The distribution of H(P) ^{will be} expressed from P,

the reduced microfield, ^defined by :

where Eo is the field of an electron at a distance ro :

ro being approximately ^{the mean} inter electronic

length (section 3 .1 ).

Eo ^{can be also} expressed explicitly by :

C2 ASYMPTOTIC FORM AT A NEUTRAL POINT (V # 0).

-

We will now examine the neutral point ^{case and it is} important ^to calculate the 04 ^and 2 functions. The first order functions can easily ^{be found} using general

results for the spherical Bessel functions j, [95-96].

Integrating t/Je by ^{parts :}

we find, ^{for the H.F.} part :

The calculation is a little more complicated ^{for the L.F.} part :

In practice, these functions are calculated for small values of the variable ^a.

After the change of variable :

and the development ofjo(Uf) :

we can write :

with

Integrating by parts, ^we find, for small values of a :

which verify ^{the non} correlated limit (v

⁼

0), given ^here by :

For the second order functions, ^{it is} possible ^{to show} [52] ^that g/gE ^is proportional ^{to the} variable a in the

vicinity ^{of zero. Under these} conditions, ^{it is} possible ^{to write :}

with

and

A, B, ^C being calculated for each particular ^{case from the} corresponding ^curves.

We have now to perform ^the calculation of the asymptotic behaviour of H(fl) ⁱⁿ the neutral point ^{case when}

the system ^is correlated (v # 0).

After an appropriate change of variable, ^{we can write:}

with

in the asymptotic fl limit, ^and

FE(a) being ^the approximation ^of F(u) in the limit of small values of a :

After a development ^of [I - 41(a)]", ^{we find :}

with

and

After integration, HN(fl) ^becomes [52] :

C3 ASYMPTOTIC FORM AT A CHARGED POINT (V ^# 0).

-

We now consider a charged particle (Z,) ^{at the}

centre of the system ^{and we make the nearest} neigh-

bour approximation.

If P(r) ^{is the} probability ^of finding ^a particle ^between

0 and r and P’(r) ^the contrary probability, ^{we can}

write :

For ^r + dr, ^{wL- have :}

and noting ^{that the} probability P(r) ^is necessarily

proportional ^{to the} length ^r, this result can be _expres-

sed :

where p ^{is the} density probability ^of finding ^a particle between r and r + dr.

In consequence, P(r) ^{becomes :}

If we assume that the probability ^of finding ^{a micro-} field P ^{at the} origine ^{0 is} equal ^{to the} probability ^of finding ^a charged particle ^{at a distance r from} 0, ^{we can} write :

We must now express the probability ^{p dr.}

The definition of the correlation function for two bodies permits ^{one to write :}

With one ( particle ^{located at the} origine ^and assuming ^the hypothesis ^of isotropy :

this integral shows that 4 Hr’ ^9,,(r) dr is the proba- bility ^of finding a j particle between r and ^r + dr.

For a system containing N, particles, ^{we find : 1}

with

The distribution H(fl) ^{then becomes :}

It now useful to introduce a new variable _p defined

by :

Using ^the definition of ro (appendix A), ^{we find :}

and

The total distribution is then given by :

The reduced microfield f3 ^{can be} expressed starting

from definitions of Ej, Ej ^{and Eo.}

For the H.F. part :

and for the L.F. part :

For large ^values of P, ^{p is} approximately ^{zero and the} exponential integral ^{term can be} replaced by unity.

Therefore, ^{it is} possible ^to write for the ions :

The asymptotic behaviour of the one component

electronic system (H.F.) ^{is then} given by :

and for a two component ^ionic system (L.F.) :

If rigorous ^{treatment is} required, ^{it is} necessary ^to compute ^the exponential integral ^term.

C4 ASYMPTOTIC FORM FOR A NON CORRELATED PLASMA

(v

⁼

0).

^-

^{Since we have no} correlation (v

⁼

0), ^we

return to the neutral point ^condition (no correlation between the centre of the system ^{and the} charged particles) ^{with :}

- rB.

Consequently, ^{we find :}

For the H.F. part :

For the L.F. part :

These results can be expressed ^{in the} general ^{form :}

and

The asymptotic ^{form for a non} correlated plasma ^is

then given by ^the asymptotic ^{form at a neutral} point

with the condition :

because in this case : tf¡(a)

⁼

^0.

The distribution H(fl) then becomes :

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