Semi-empirical correlation function for one and two-ionic component plasmas

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Semi-empirical correlation function for one and two-ionic component plasmas

B. Held, P. Pignolet

To cite this version:

B. Held, P. Pignolet. Semi-empirical correlation function for one and two-ionic component plasmas.

Journal de Physique, 1986, 47 (3), pp.437-446. �10.1051/jphys:01986004703043700�. �jpa-00210223�

437

Semi-empirical correlation function for one and two-ionic

component plasmas

B. Held and P. Pignolet

Laboratoire d’Electronique, ^{Avenue de} l’Université, Université de Pau, ⁶⁴⁰⁰⁰ Pau, ^France (Reçu ^le 12 juillet 1985, revise le 28 octobre 1985, accepte ^{le 18 novembre} 1985)

Résumé.

Dans cet article une expression générale semi-empirique ^{de la fonction de} corrélation g(y) ^est proposée

dans le cas des plasmas ^{à une} composante ainsi que des plasmas ^{à deux} composantes ioniques. ^Les résultats, ^en bon accord ^avec les calculs de Monte Carlo, ^{sont déduits d’une} expression analytique. ^Cette formulation permet d’entreprendre ^{des calculs} pour des mélanges ^{d’ions avec des} temps ^{de calcul non} prohibitifs.

Abstract

This paper proposes ^a general semi-empirical expression for the correlation function g(y) ^{in the one-} component plasma ^{and the two-ionic} component plasma ^{cases. The results, in} good agreement with Monte Carlo simulations, ^are deduced from an analytic expression. ^This formulation permits extensive calculations for ionic mixtures without prohibitive calculation times.

J. Physique ⁴⁷ (1986) ^{437-446 MARS} 1986,

Classification Physics ^Abstracts

52.25

1. Introduction.

During the last few _years, the Stark broadening diagnostic ^of stripped ions immersed in ionic _per- turbers have been used with success to determine the density-temperature conditions encountered in

plasmas of inertial-confinement fusion produced by

laser [1-4].

With the development ^{of laser} compression experi-

ments on exploding pusher targets ^comes the interest-

ing problem ^{of ionic} mixtures in high density-high temperature conditions.

For the diagnostics, ^{it is} necessary ^to be sure of the

experimental ^and theoretical profiles, ^{the most accu-}

rate results being given by the best fit.

In practice, the Stark broadening calculations

require knowledge of the ionic microfield. The theo- retical profile ^{is then} dependent ^{on the} microfield,

and this has widely ^motivated many works during ^the

last few _years [5] (and references given there).

The production of ion mixtures at high density

and high temperature, by ^laser compression experi-

ments, introduces the difficult problem of correlated

multicomponent plasmas.

To take these effects into account in the micro- field theory, ^{we have to} consider the inter-ionic correlations in dense, high-temperature plasmas.

The pair correlation function can be obtained in the weak coupling ^limit by ^{two theories}

^the Debye-

Hiickel [6, 7] ^{and the} Abe-Mayer ^cluster expan-

sion [8 ^to 12]

while in the strong coupling ^limit,

other methods must be used : Monte Carlo [13 ^to 17]

or hypemetted ^chain equations [18 ^to 24] for instance.

Because, ^{it is} necessary ^to know the ionic micro-

field, for the Stark broadening application, ^the pair

correlation function for ions will be at the center of

our preoccupation [25 ^to 31].

In recent works, particular attention has been

paid ^{to one or two-ionic} component plasmas ^{in view}

of applications ^to plasmas ^of inertial-confinement fusion produced by ^laser [32 ^to 36].

Unfortunately, ^all these theories and computer simulations require lengthy numerical calculations.

That is why this paper proposes ^a semi-empirical approach. ^{The ionic} pair correlation function will be derived from physical considerations and numerical results; ^the general ^{law must} be valid in two limits : weak coupling ^and strong coupling. Finally, ^{in this} general form, ^{the ionic} correlation function must agree with the empirical approach encountered in

some particular ^cases [23, 26, 33, 35, 38].

In section 2, the general correlation expression ^is given ^{for a} one-component plasma.

Section 3 is a generalization of section 2. The semi-

empirical correlation function is obtained under a

general ^form allowing applications ^{for a} plasma parameter going ^{from zero to one hundred.}

The results are presented in section 4 and compared

with other theories and computer simulations.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703043700

Finally, ^{section 5} gives general conclusions and the field of application for these results.

2. One-component plasma.

2 .1 CORRELATION ^FUNCTION - We consider a plasma

with ions of charge Ze, density ni and equilibrium temperature ^{T. The ions are assumed to be} classical,

and a uniform and rigid background neutralizes the ionic charges.

For a one-ionic-component plasma, ^the plasma parameter ^{r can be written} (Appendix A)

where F, ^{is the} electronic plasma ^parameter.

It is then useful to define the reduced length :

r; being ^the inter-ionic length.

For single-charged ions, the correlation function

can be expressed by [33, 35]

where H(y) ^{is the} screening potential.

The empirical ^form

verifies the conditions [25, 27, 35]

and respects ^the asymptotic ^{limit of} g(y)

with

I ^--

The unknown F(y) ^{function must be an} oscillating

function of y ^for large ^{values off} satisfying ^{the condi-}

tions

2.2 PHYSICAL AND NUMERICAL CONSTRAINTS. - The purpose of this section is to introduce ^a realistic and physical expression ^for H(o) ^and F(y) giving

accurate results for g(y).

Firstly, ^{it is} interesting ^{to notice} [35] ^{that a cha-}

racteristic reduced length d ^{can be} defined with the condition

Furthermore, ^a constraint at the origin ^is given

with the exact result [29, 33]

Starting ^{from the} one-component plasma screening length d ^{in the} hypemetted chain scheme computed by Tanaka and Ichimaru [35], ^a good approximation

for d as a function of r can be given by (Appendix B)

with the limits

General expressions have also been given ^for H(o) ^{in the} low-density ^{and the} high-density

limits [25, 27, 35]. ^An approach ^of H(o) ^{valid for} small, large ^and intermediate values of T can be obtained (Appendix B) ^for single-charged ^{ions :}

with the limits

An analysis ^{of the results} computed by ^{Monte Carlo}

simulations [ 13 ^to 17] ^{shows a} periodicity ^{d for} g(y)

from the first maximum when r > 3 (the ^{first maxi-}

mum corresponding to y

d).

Because of this periodicity, ^the correlation function

g(y) ^is unity ^for several values of _{y ;} consequently,

the unknown function F(y) ^{is zero for these} particular

values of y : F(y) is therefore the oscillating ^{term for} H( y) ^{with a} periodicity ^{d An} approximation ^of F(y)

can be given by ^the empirical expression proposed by ^Hubbard [23, 26, 37] for y > 1.8 :

These different constraints permit ^{one to assume}

that F(y) ^can be written in the form

where u is a useful variable for the periodic ^functions

For increasing ^values of y, ^{the first term vanishes}

439

and the developments ^of H( y) ^and g(y) give, ^{in first} approximation, expression (17).

The coefficients A, B, ^{and C are evaluated} by ^a simple ^{criterion at three} particular points, ^these

results being ^confirmed by ^{an accurate numerical}

method (Appendix C).

It is of interest to notice that, ^{in the weak} coupling

limit (F, 1), equation (15) approaches ^the Debye-

Hfckel value for an electron-ion plasma ^{with Z}

^1.

Under these conditions the electron shielding ^cannot

be ignored, ^and these calculations must be convenient-

ly ^modified by multiplying ^the pure Coulomb potential by ^an electronic screening function in the expression

of g(y) [35].

2. 3 SEMI-EMPIRICAL EXPRESSION.

To sum up, ^we propose ^a fairly simple ^{and accurate} expression ^for

the correlation function g(y) ^{in the case of} single- charged ^{ions :}

with

and

with

d, H(0), A, B, ^C being ^defined by :

3. Two-component plasma.

We now consider a plasma with ions of charges Z. e

and Zb e, ^densities na and _nb, and equilibrium ^ionic temperature T;. ^{The ions are assumed to be} classical,

and a uniform and rigid background neutralizes the ionic charge.

For a two-ionic component plasma, ^the plasma parameter ^{T is} given by (Appendix ^A)

with

where RT ^{is the} temperature ^ratio

and p ^the proportion ^{of ions « b »}

C. ^and Cb being the concentrations.

Noticing ^{that a} distinction must be made between interaction ^{« a - a »} and ^{« a -} b » (the ^central charge being Z. e), ^{we assume that the} correlation function is

given by

with

and

The screening potentials ^{at the} origin H_(0) ^and Hab(o) ^are given by (Appendix B)

with b1(i,j), b2(i, j), b3(i, j) ^and b4(i,j) being ^{defined in} Appendix ^B (B. 11) ^to (B. 14) ^and (B. 20) ^to (B. 23).

The generalization ^{of the} correlation function is obtained assuming that the characteristic reduced

length d(r) ^{and the} coefficients A(r), B(r), ^and C(F)

are functions dependent ^{on the} generalized plasma parameter ^{r for a two-ionic} component plasma.

Expressions (18) ^to (22) ^are unchanged ^{with this} assumption, the results presented in section 4 being ⁱⁿ good agreement with those obtained by ^{Monte Carlo}

simulation [38].

4. Applications.

Numerical results for the correlation function g(y) ^are presented ^{in tables I and II in the case} of one-compo- nent plasmas ^{for 0.05 F , 100. In} figures ^{1 and} 2,

these results ^are compared with Monte Carlo calcula- tions [ 13 ^to 17].

For 0.05 , F 3, ^the precision varies from a few per ^{cent to} better than ^one _per cent. For increasing

values off, ^the uncertainties can be about 10 % ^{in the}

first dip.

To sum up, the semi-empirical expression (3) ^used

for g(y) ^{is a} good approximation ^{for 0 F , 100}

and all values of y.

In figure ^3, the results presented ^{are in} good agree- ment with Monte Carlo simulations [38] ^{for a one-}

ionic component plasma (charge Za

9).

Finally, ^{because of the Stark} broadening diagnostics

of highly stripped heavy ions immersed in dense and hot plasmas ^of inertial-confined fusion produced by

laser, figure ⁴ (gaa(y) ^and gab(y)) ^{is devoted to the case}

of an ionic mixture with the conditions

wherep ^{is the} proportion of ions b » and v the correla- tion parameter (Appendix A).

5. Conclusion.

In this _paper, general semi-empirical expressions ^for

the correlation function g(y) ^have been introduced with a view toward applications ^to microfield calcula- tions for the Stark broadening diagnostics ^of highly stripped ^ions.

For the inertial-confinement fusion applications, ^the

plasma parameter ^is generally less than twenty

441

Table II.

Correlation function g( y) for ^a one-component plasma (Z

1).

Fig. ^1.

Correlation function g(y) ^{for a} one-component

plasma (Z

1) compared ^{with HNC} [19] ^{when r}

⁵

and 10.

Fig. ^2.

Correlation function g(y) ^{for a} one-component plasma (Z

1) compared with Monte Carlo [14, 15]

when r

20, 50 and 100.

Fig ^3.

Correlation function g(y) ^{for a} one-component plasma (Z

9) compared with Monte Carlo [38] ^when

r

0.52, 2.08 and 8.33.

(because ^{of the current} range of variation for the _para- meters Z., Zb, p, v), ^{and the results are} given ^with good precision.

Finally, ^this analytic formulation should permit

extensive application ^to ionic mixtures with _very short calculation time compared ^{to Monte Carlo simu-}

lations.

Appendix ^A.

PARAMETERS. - We consider a two-ionic component

plasma ^{with a} neutralizing background

For a one-electronic component plasma, ^the plasma parameter F. ^{is defined} by

where _re is the inter-electronic length

Introducing ^the electronic Debye length

the electronic plasma parameter ^becomes

This result can be generalized ^{to a two-ionic} compo- nent system, writing the two-ionic plasma parameter

where _ni is the ionic density

and Â.Di ^{the ionic} Debye length [5]

with

RT being ^the temperature ^ratio

p the proportion ^{of ions « b »}

C., ^and Cb being ^the concentration of ions.

The two-ionic plasma parameter ^{r can then be} written

If Ce ^{is the} concentration of electrons, ^{we have}

with the conditions of neutrality

and it is _easy to obtain

Using equations (A. 7), (A. 12), ^and (A. 15), ^{the two-}

ionic plasma parameter ^{can be written}

For a one-ionic component plasma ^with tempe-

rature equilibrium conditions (Te

Ti), ^the plasma parameter ^becomes

Finally, ^the electronic plasma ^parameter Te ^{can be} expressed using ^the correlation parameter ^{v :}

with

443

and F. is connected to the plasma parameter Ae [5]

by ^the equation

Appendix ^B.

EXPRESSIONS FOR d AND H(o).

^The characteristic reduced length d ^{can be} expressed ^{in a} general ^form

as a function of r, using ^the numerical results given by Tanaka and Ichimaru [35] ^{and the limits}

In particular, writing d ^{in the} general ^form

it is _easy to see that the asymptotic ^{limits corres-} pond ^to

For two particular ^{values of r} (10-2 ^and 1), ^the

curve d(r) [35] gives ^two supplementary ^relation- ships

and

Four coefficients are obtained by resolving ^this system :

General expressions ^{for the} screening potential H(y) ^{at the} origin have also been proposed ⁱⁿ papers

relating ^{to electron} screening ^and thermonuclear reactions [25, 27].

These results, ^{obtained in two} limits, ^{can be} expressed ^as

noticing that, ^{for a two-ionic} component plasma, ^a

distinction must be made between interactions « a-a »

and ^« a-b », assuming ^{that « a » is a central} ion, ^{« a »}

or « b » being ^the perturbers.

Under these conditions

in the approximation ^{of a classical system, i} and j being ^{the notations for ions « a » or « b ».}

Assuming ^that HiJ(0) ^{can be} generalized by [35]

the asymptotic ^limits give

Resolving ^this system, ^{we find}

For a one ionic component plasma (Z.

Zb

Z)

in equilibrium conditions (RT

1), ^these general expressions ^{are reduced to}

and the numerical results are in good agreement

with those of Tanaka and Ichimaru [35] for all values

of r.

Appendix ^C

DETERMINATION OF A, B, ^C.

^The determination of A, ^{Band C can be made} using ^{Monte Carlo nume-}

rical results [13 ^to 17] ^for one-component plasmas.

The choice of three particular points (i.e., ^{the first}

maximum and the first two values of y corresponding

to the condition g(y)

^{1 when r} > 3) ^{is the most} simple ^{way. This} approach gives ^{a three} equation system. ^A comparison between these results and the accurate numerical method presented in the second part ^{of this} Appendix (Figs. ^{5 to} 7) ^{shows the existence}

of a general ^{law for} A(r), B(r), ^and C(r).

Fig. ^4.

Correlation function gaa(y) ^and gab(y) ^{for a two-}

ionic component plasma (Z.

9, Zb

1, p

0.5, v = 0.8).

Fig. ^5.

Parameter A as a function of r. The solid line represents ^the analytical ^{fit for} A(r). ^The height ^of squared

dots corresponds ^{to the} approximation ^{bare of numerical}

values.

Fig. ^6.

Parameter B as a function of r. The solid line represents ^the analytical ^{fit for} B(r). ^The height ^of squared

dots corresponds ^{to the} approximation ^bare of numerical values.

Fig. ^7.

Parameter C as a function of r. The solid line represents ^the analytical ^{fit for} B(r). ^The height ^of squared

dots corresponds ^{to the} approximation bare of numerical values.

The numerical method used for the calculation of the coefficients A, B, ^{and C is} based on both, ^an iterative method for numerical solution of non-

linear equations ^{and on a least} square method.

It follows from equation (18) ^that the function

F(y) depends explicitly ^{on the} coefficients A, B, ^and

C. Thus we can write

445

Let E(A, B, C, Yi) ^{be a} discrete function defined by

i indicates the i-th value of the reduce length y, and

f ^{is the} corresponding ^value of the function F(y)

deduced from numerical data of the correlation

function gi (i

1,

M) :

where T and H(O) ^{have been} previously ^{defined in}

section 2, equations (4) ^and (5).

For each r, ^{estimated values of} the three coeffi- cients can be obtained by resolution of the following equation :

At the first order of the Taylor expansion, ^{the last} equation ^becomes

In a simple form, ^{this can} be written as follows :

where Xj ^is associated, for j ^{= 1, 2, and 3, to the coeffi-}

cients A, B, ^and C, ^{and i} corresponds ^{to the values}

of the derivatives DEij(Xt, X 2, X 3 ) ^{or the function} Ei(X(10), X2(0), x1°») ^for yi.

Thus an iterative solution can be obtained ^as follows [39] :

with the iteration XiN + 1)

x!N) ⁺ AXj ^{and the}

set of initial values X 1 (0), X 2 (0) and X 3 (0) j *

However, ^{for each} step ^of the iterative approxima-

tion equation (C. 3) ^must be verified for each _y;, and equation (C. 6) ^{becomes an} overdetermined system ^{of linear} equations. A classical least _square method is required ^to get ^an optimal ^solution.

The solution is given by ^the following ^linear system :

Thus the coefficients A, B, ^{and C or} X 1 (N), X 2 (N),

and X3(N) ^{can be} computed ^{for each r with a} relatively good approximation ^{of the order of} 10-2, ^{and the}

iterative method _converges with a small number of iterations (about ^{5 to 15} iterations).

Finally, ^the semi-empirical analytic dependence

of the coefficients A, B, ^{and C on r is obtained} by

using ^a classical least _square fit method.

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