Correlations and microfields in dense plasmas

J. Quanf. Specfrosc.

Printedin Great

Radar.

7’ransfer Vol. 44, No. 1, pp. 11-17,1990

0022-4073/90 S3.00 + 0.00 Pcrgamon Pressplc

Britain

CORRELATIONS AND MICROFIELDS DENSE PLASMAS

IN

BERNARDHELD Laboratoired’Electronique,des Gaz et des Plasmas,I.U.R.S.-Universitide Pau, Avenuede l’llniversitb, 64000-PauCedex,France

Abstract-This paper proposes a general semiempirical expression for the correlation function g(y) for the two-ionic component plasma. The microfield calculation at a charged point is accomplished using an iterative method. This formulation permits extensive applications to ionic mixtures without prohibitive calculation times.

INTRODUCTION With the development of laser compression experiments on exploding pusher targets comes the interesting problem of ionic mixtures in high density-high temperature conditions.’ For the Stark broadening diagnostic of stripped ions immersed in ionic perturbers, it is necessary to be sure of the experimental and theoretical profiles. But, in plasmas under extreme or unusual conditions, the traditional theoretical approach is certainly not a realistic way for the study of spectral line formation. In general, these unusual situations are encountered in correlated multicomponent plasmas. That is why this work proposes a semiempirical approach for the ionic pair correlation function with several applications being given for the low-frequency microfield. The ionic pair correlation function is derived from physical considerations and numerical results. The general law is given in an analytic form valid in the weak coupling and strong coupling limits. The low-frequency component of the microfield at a charged point is deduced from the second moment rule, using an iterative method. FORMALISM We consider a plasma with ions of charges Z,,e and Zbe, densities n, and nb, and equilibrium temperature T. The ions are assumed to be classical and, in first approximation, the two-body potential interaction is given by the Debye-Hiickel screened potential. If N is the total number of ions and E the total microfield at the origin, it is convenient to introduce the Fourier transform of the probability distribution W(E),’ F(k) =

s

exp(ik*E)W(E)

d3E.

(1)

Using a cluster expansion method, it is possible to express F(k) by the development of integral functions of correlation functions g(r). The probability distribution W(E) is then obtained by inverting the Fourier transform, 1 W(E) =&

exp( - ik -E)F(k) d3k.

(2)

With this method, it is necessary to take care of the correlation function g(r) and the resummation over all orders for the development of F(k). These points are particularly important for plasmas under extreme or unusual conditions. That is why this paper proposes a general semiempirical expression for the correlation function and an iterative method for the calculation of the low-frequency component of the microfield at a charged point. 11

BERNARD HELD

12

Finally, for extensive application, description:*

it is useful to introduce

four parameters

for the plasma

(i) Correlation (3) where r, is the mean-electronic

length and JDc the Debye length for the electrons;

(ii) Proportion P = MC, where C,,(C,) is the concentration

(4)

+ C,),

of ions,

(iii) and (iv) Number of charges on ions a and b; Z,, 2,. Under these conditions, the two-ionic plasma parameter F can be written, J- = @?/z”‘)r,)

(5)

where Fe is the electronic plasma parameter, Fe = (2/15)(2x)“W,

(6)

Z=z,+p(Z,+Z,),

(7)

Z is the average ionic charge,

and 2 the root mean square ionic charge, 2’ = z; + p(Zi - z;>.

CORRELATION

(8)

FUNCTION

The ionic pair correlation function will be derived from physical considerations and numerical results. The semiempirical expression must be valid in the weak coupling and the strong coupling limits. The correlation function can be expressed by

[

g&> = exp -$ where H,(y)

+ Hii(y)

1,

(9)

is the screening potential and Tii = (ZiZj/2”3)r,,

_Y= (r/r/);

(10)

ri being the inter-ionic length. The empirical form

H,(Y) =

H,(o) H,.(o)

+yY

(11)

+.KY)

I/

verifies the conditions: H&J

= 0) = H,(o)

H,(y

= co)=0

(12)

lim g(y) = 1. ?‘+m

(13)

and respects the asymptotic limit:

Finally, the unknown function f(y) satisfying the conditions

must be an oscillating function of y for large values of r

f(y = 0) = 1; f(y = co) = 0.

(14)

Correlations and microfields in dense plasmas

13

An analysis of the Monte-Carlo results in the strong coupling limit shows a periodicity d for the correlation function g(Y) from the first maximum when r > 3.3 Using the empirical expression proposed by Hubbard for Y > 1.8, 1 -A’

g(y)=

ev( - B’Y) Y

it is possible to writef(Y)

COS(C’Y+ D’),

in the form f(y)

= exp(-Ay)

y

- cu cos u

[

1)

where u is a useful variable for describing the periodic function, u = (2n/d)y.

(17)

Finally, the reduced length d and the screening potential at the origin H,(o) can be expressed using the well known weak coupling limit (r 4 1),5 d = 0.95/P’*,

H,(o) = 1.73 zizjZT,)‘*;

(18)

and strong coupling limit (r % 1),6 d = 1.67,

H,(o) = (0.92 2”3[(Zi + Zj)5’3- Z;” - Zi”3] + 0.2843 z2’3 x [(Zi + Zj)“’ - zp

- zj”’ ]}r e+ 0.4600 Z-*“[(Zi + Zj)2’3- Z:‘3 - Z;‘3]. (19)

Assuming that d and H,(o) can be generalized by a, + a2P

d=

r,,

=

4

(6

IV

+

1

(20)

,

[

H,(o)

m

bz(i,

Wl”

r3,2

1 + b3(i, j)r;‘2 + b4(i, j)r,

(21)

’ ’

we find,’ using the weak and strong coupling limits: a, =0.915,

a*= 2.42,

n =0.862,

m =0.58,

and K=

15 [ 2(2a’

1 I/*

(22)

bl(i. j) = KG j),

(23) (24)

b,(i,j)

KC j) = KKY’(i, j)’

(25)

(26) with K’(i, j) = ZiZjz, K;” (i, j) = (0.9 P3[(Zi + Zj)u3 - Zj’3 - Zj”] + 0.2843 *P’[(Zi

(27) + Zj)4’3- Z!‘3 - ZT’3]}, (28)

KT (i, j) = 0.46 Z-2/3[(Zi + Zj)*” - Zf” - Z,“‘].

(29)

14

BERNARD

HELD

The last three coefficients (A, B, C) are evaluated by a simple criterion at three particular points, using Monte-Carlo numerical results for a one-component plasma: A=

1.69 - 1.4exp(-0.19~)+0.54exp(-0.32T)-0.83exp(-l.84T), I+ 0.0012 r

(30)

B = 0.4 + 0.5 exp( - 0.1062 r) - 0.694 exp( - 0.2706 r) - 0.206 exp( - 11.83 r),

C = 0.15559 exp(-0.00605

7355.5449 r) - [(14.7455)2 + yz]z.

(31) (32)

These results are generalized, assuming that the coefficients A(r), B(T), C(T) are functions dependent on the generalized plasma parameter r for a two-ionic component plasma. This analytic formulation permits extensive applications to ionic mixtures with very short calculation times compared to Monte-Carlo simulations, the results being in very good agreement with numerical simulations (Figs. 1 and 2). IONIC

The probability formalism,

distribution

MICROFIELD

H(P) of the scalar microfield /3 is deduced from the general

uF(u)sin(jIu) du,

(33)

u = kE,;

(34)

with /I = E/E,,

E, = e/r:,

where F(u) is given by2 F(u) = exp( - u3”[t+GI + high order terms]},

(35)

with lcll=#?+ti?

(36) (37)

(38)

rmO.52 V-0.2;P-0;

l-=8.33

1.4

2..2b’9

V-0.8;

p-0;

z,,‘zb‘9

t

s

1.2

-

1.0

-

0.8

-

0.6

-

%I -

This work

function g(y) for a one-component compared with Monte-Carlo when l- = 0.52.

This work

---MC

---MC

Fig. I. Correlation plasma (Z = 9)

-

0.4

-

0.2

-

Fig. 2. Correlation plasma (Z = 9)

function g(y) for a one-component compared with Monte-Carlo when r = 8.33.

Correlations and microfields in dense plasmas

15

where jO is the spherical Bessel function of order zero and

Uo=Z,$(l +x)exp(-x)

(39)

U,=Z,$(l +x)exp(-x)

(40)

with x = r/A,,

a = u’/‘V.

(41)

The asymptotic behaviour (fi % 1) of H(b) can be given with an analytic expression (42) where Z!‘2 ] Yirt,;,p*

(43)

In practice, the first order term $, is calculated classically and a corrective function is introduced to serve as a substitute for the resummation over the higher order terms. Assuming that this correction is proportional to the first order terms, the coefficient of proportionality is determined by an iterative method using the second moment rule. The general second moment rule expression is given by8

(E.

E)

=

=(v;V), (Z&92

where POis the gradient with respect to the position of the central charge (Z,e) and Y the total potential energy of the system. Using the non-linearized Poisson-Boltzmann equation, it is possible to introduce the correlation function, and the second moment rule can be written as9

+ 5.1: where UJX) is, in first approximation,

[exp($)

the screened Debye-Hiickel

uii(r) = z,z.n kgT

- l]g&)x’dx}, potential,

exp(-x)

lJe

x

(45)

’

(46)

with n

_ e

2(2n)“2 u3*

15

(47)

In practice, the microfield is calculated in four steps, assuming that, F(u) = exp[-u3/2($,

- k’tj,)],

(48)

where k’ll/, is the corrective term: -the first order terms I++,and the second moment rule (j12), are deduced from the correlation function g(y) by the semiempirical method,

BERNARDHELD

16

kb=0.304

kb =0.415 v-0.2;p-l;z,-zb-s

v-0.2;p-0.5;z,-9;Zb’l

0.6 -

This work T,,

---

NNN

0.4 G 2

---

0.2

APEX

0

Fig. 3. Microfield distributions of H(B) for a one-component plasma with v = 0.2, p = 1, Z, = Z,, = 9. Iteration converged at kh = 0.415.

Fig. 4. Microfield distributions of H(B) for a two-component plasma with v = 0.2, p = 0.5, Z, = 9, Z,, = 1. Iteration converged at kk = 0.304.

-the microfield distribution H(B) is calculated without higher order terms (k’ = 0), -verification of the condition of normalization

s0 -calculation

%0dfi=l,

(49)

of the second moment @‘>c =

3oP’MP) dp. (50) s0 After comparison with the second moment rule and an incrementation on k’, the last three steps are still performed and so on, until convergence is reached at some value k’ = k;. This iterative method gives results (Figs. 3,4) in good agreement with the more recent results given by other theories: Tighe-Hooper calculations,‘0 adjustable parameter exponential (APEX) approximation,” and the next-nearest-neighbour (NNN) approximation.‘2 Other cases were studied for different values of the parameters u, p, Z,, Z, and these results are available on request. CONCLUSION

In this paper, a semiempirical expression for the correlation function g(y) has been introduced with a view toward applications to microfield calculations by a general iterative method. These methods can be generalized for strongly coupled systems and the sensitivity of the convergence with the two-body potential interaction can be used to test other effective two-body potential functions. It is also of interest to note that the accuracy of these results could certainly be improved by introducing the electronic degeneracy in the electronic screening function and using an effective two-body potential function solution of the non-linearized Poisson-Boltzmann equation. For Stark broadening calculations in plasmas under extreme or unusual conditions, it is necessary to note carefully the mean inter-ionic length. In the strong coupling limit, it is certainly not realistic to consider separately the radiators and the perturbers. In this case, the traditional approach is certainly not a realistic way for the study of spectral line formation and thus the microfield calculation becomes an academic problem. REFERENCES 1. 2. 3. 4.

Rapport GRECO Interaction Laser Matibre, Rapport d’activitk Ecole Polytechnique, B. Held, J. Phys. 45, 1731 (1984). S. G. Brush, H. L. Sahlin, and E. Teller, Bull. Am. Phys. Sot. 10, 232 (1965). W. B. Hubbard, Astrophys. J. 146, 858 (1966).

Palaiseau

(1985).

Correlations and microfields in dense plasmas 5. 6. 7. 8. 9. 10. 11. 12.

17

S. Tanaka and S. Ichimaru, J. Phys. Sot. Japan 53, 2039 (1984). E. E. Salpeter, Ausf. J. Phys. 7, 373 (1954). B. Held and P. Pignolet, J. Phys. 47, 437 (1986). C. A. Iglesias, J. L. Lebowitz, and D. McGowan, Phys. Rev. A 28, 1667 (1983). B. Held and P. Pignolet, J. Phys. 48, 1951 (1987). R. J. Tighe and C. F. Hooper, Phys. Rev. A 17, 410 (1978). C. A. Iglesias, C. F. Hooper, and H. E. De Witt, Phys. Rev. A 28, 361 (1983). C. A. Iglesias, H. E. De Witt, J. L. Lebowitz, D. McGowan, and W. B. Hubbard Phys. Rev. A 31, 1698 (1985).