Screening effects on ionic structure in high density plasmas

Volume 59A, number 1

PHYSICS LETTERS

1 November 1976

SCREENING EFFECTS ON IONIC STRUCTURE IN HIGH DENSITY PLASMAS U. De ANGELIS and A. FORLANI Osservatorio Astronomico di Capodimonte - Napoli, Italy Received 28 April 1976 Revised manuscript received 13 September 1976 Use of the force equation and the assumption that the probability of a three-particle configuration is not substantially altered when the bare Coulomb ionic potential is modified into a screened form, lead to the conclusion that the change in the logarithm of the pair distribution function is equal to the change in the electrostatic potential as given by the Poisson-Boltzmann equation.

The Coulomb plasma has been extensively discussed as a one-component system (OCP) treating the electrons as a uniform neutralizing background and many of it’s properties are now fairly well established either analytically or numerically [1—4]. In particular the pair distribution function g0(r) and theinstructure factorfor S0(k) the OCP are coupling available [2, 4] tabular form manyofvalues of the parameter: 2e2/a, ~3(kBT)~ F~3z where a = (3/4rrp)1/3 is the radius of the sphere contaming one ion (p being the particle density). Our aim in this note is to find the deviation of the pair correlation function from the OCP value when electron screening of the ions is taken into account: the ion-ion direct interaction is changed (in k-space) 2e2/k2 from the bare potential Coulombi~’(k) potential ~‘0(k) = 4irz to a screened = ~ 0(k)/c(k, r5) where c(k, r5) is the dielectric function of the electron gas and r5 = a/a0 (a0 is Bohr’s radius) is the density parameter usually introduced in the description of the nonuniform plasma (which is parametrized by F and r5: the OCP corresponds to r5 = 0, the plasma state to r5 ~ 1, liquid metals to the density range 1 <~rs~ 7). Then the force equation [5] ,which is exact for pair potentials, is: ~

lng(R12)=—---~-—j3ji(R12)

aR1

aR1 fP(3)(R1R2R3)

—p-- ~3p(R13)dR3 aR1

30

—

(3)

(‘g 0_(R1R2R3) ~ dR g~)(R12) ~R1MR1 3) 3.

Pj

Subtracting eq. (2) the force equation written for the OCP from we have:

a

a z~ln g(R 12) 1

=

—

I~ji(Ri2)

~ 1

3 ~

— ‘-‘

)(R1R2R3) j3i~p(R1~) dR3 2)(R g~ 1R~)~R1 —~--~

(3)

where z~u(k)= u0(k) (l/e(k) 1). Rather than solving eq. (3) as it stands we try an interpretation in terms of forces: consider a fictitious system with the ions fixed at R1 ,R2, ,R~and distributed withdensity pg0(r) and suppose to change the direct ion-ion interaction from u0(r) to u(r) keepmg the ions fixed. Then the right hand side of eq. (3) gives the difference between the average forces acting —

...

~l’ ~

‘ —

where g(r) is the pair correlation function of the screened ions and p(3) is the probability of a three-ion configuration. In the limit of small r 5(~1) the available Monte-Carlo data [6] for g(r) show that the deviation from g0(r) is quite small: thus we make the assumption that the effect of screening is even smaller 3) in eq. on the three-body configuration and replace p( 2~(R (1) by it’s OCP value p~3)= pg( 3)(R 0 1R2R3)/g~ 1R2). With this assumption the force equation reads: a a R 3R 1 lng(R12)__-~~ 1 I~’-~( 12) (2)

Volume 59A, number 1

PHYSICS LETTERS

on an ion at R1 in the two cases: but since the ions are being kept fixed this must also be equal to the difference between the electric fields at R1 in the two cases. The electric field is given by —a Ze ~(R12)/3R1, 0(r) being the electrostatic potential due to an ionic distribution pg0(r) (surrounding the ion at R1 origin) and to an electronic distribution ne(r) consistent with the assumed pair interaction, Thus in the first case ne(r) = n0 is the uniform electron background (the ions interact via a bare Coulomb potential) and in the second case “e(’~)should be consistent with the assumed dielectric function e(k, rS) of the electron gas. Then we may write ~ In g(r) = —I3Ze [0(r) 00(r)] (4) where the electrostatic potentials satisfy the corresponding Poisson-Boltzmann equations: V20(r) = —4irp(r) = —4ir[Ze~(r)+Zepg 0(r) ene(r)J (5) 2q~ (r) = —4irp V 0 (r) = —4ir [Ze~(r) + Zepg0(r) en0] (6) written for the charge distributions of the fictitious system where the s-function accounts for one ion at the origin. For charge neutrality n 0 = Zp and ‘~e(’~) = n0 + 812e(T) where ~Sfle(r)is the local non-uniformity of the electron gas responsible of the screened ionic interaction. For rS ~ I the Lindhard form of the dielectric function [7] can be assumed and March [5, 7] has shown that in this approximation &ne(r) is determined from the solution of the self-consistent Schrodinger equation (i.e., written for a potential energy eØ(r)) in the Born approximation and is given in k-space by: 2~(k)[eL(k,rS)—1J (7) 4~e~e(k)—k where ~L is Lindhard’s function. Eqs. (5) and (6) can then be solved in k-space yielding: —

—

—

1 November 1976

function and S0(k) = 1 + p7i~0(k)is the OCP structure factor. Recalling eq. (4) the final solution to our problem is then given by: g(r) =g0(r) exp[—IIF(r)]; F(k) = S0(k)iXi(k). (10) Eq. (10) is the basic result of this work and gives the ionic pair correlation function of a screened, high density plasma once the structure of the corresponding OCP is known. The effect of screening is to decrease the ionic correlations and this is correctly predicted by our first order result (see eq. 10): There is also a satisfactory quantitative agreement with the “experimental” Monte-Carlo data [6] as shown in fig. 1. In conclusion we wish to add a few remarks on the basic assumptions of this 3) note. byP~3)in eq. (1) might be The substitution of p( acceptable since it enters within an integral and hategrals have smoothing properties. Replacement of the integral eq. (3) with the set of potential eqs. (4)— correspondsthetoexact the frame(6), which makes of no3), importance form of the triplet g~ theory. In fact in eq. (4) the work of thefunction Deby-Huckel difference in the potentials of average force of the “screened” and “bare” ions is seen to be coincident with the difference in the electrostatic potentials. As the effect of screening increases with F, our assumption on three-body configurations becomes weaker and the first order result of this paper is no

I r

—

4irZ2e2

r~=1

Q5L /

+p~ 0(k))g(k)S0(k) (8)

______

=

r/a

2e2 k2 (1 + p~i ZeØ~0(k)= 4irZ 0(k))=ji0(k) S0(k) where h0(r) g0(r)

—

(9)

1 is the total OCP correlation

Fig. Pair function of a screened plasma of Fvalues 10 1.and ~scorrelation = 1. Full line: this paper, dots: Monte-Carlo =

from Hubbard and Slattery [6] , broken line: OCP data from Brush et al. [2].

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Volume 59A, number 1

PHYSICS LETTERS

1 November 1976

longer sufficient for higher values of the coupling parameter: thus we propose to use it as a starting “guess” in an iterative solution of the exact force eq. (1) with some approximation for P(3)(R1R2R3) 3)(R = pg( 1R2R3)/g(R12). We are presently investigatmg the convergence on the proposed iterative solution and the relation of this work with the perturbative approach of Lado.

References

This work has been supported by a contribution from the Consiglio Nazionale delle Ricerche (CNR) through the Naples department of the CNR National Group of Astronomy.

body problem in quantum mechanics, (Cambridge Univ. Press, 1967). [8] V. Canuto, Ap. J. 159 (1970) 641. [9] F. Lado, Phys. Rev. A 135 (1964) 1013.

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[1] H. De Witt, Phys. Rev. 140 (1956) A 466.

[2] S.G. Brush, H. Sahlln and E.J. Teller, J. Chem. Phys. 45 [3] (1966) K.C. Ng,2102. Ph.D. thesis, University of Florida (1974). [4] J.P. Hansen, Phys. Rev. A8 (1973) 3096. [5] N.H. March, Liquid metals, (Pergamon, Oxford, 1968). [6] W.B. Hubbard and W.L. Slattery, Ap. J. 168 (1971) 131. [7] N.H. March, W.H. Young and S. Sampanthar, The many