A new nodal expansion for classical plasmas

ANNALS

OF PHYSICS

86, 197-232 (1974)

A New Nodal

Expansion THOMAS

L-O,

Lawrence

Licermore

for Classical

Plasmas*

J. BUCKHOLTZ

Laboratory,

Livermore,

California

94550

Received March 16, 1973

In calculating the equation of state for plasmas we find that diagrammatic expansions for the free energy become unwieldy at high density. At best, many terms must be retained in order to obtain meaningful results. We present a new expansion technique which can be applied to plasmas in the interiors of Jupiter and white dwarf stars. In such cases the older techniques are unsatisfactory because of the size of the ion coupling parameter. Our work yields expansions for which this parameter is supplanted by ion correlation functions, which can be supplied by external computations. In this paper we assume a two-species plasma of classical particles, thereby focusing on combinatorial techniques. The final result is a new nodal expansion in terms of ion correlation functions and an electron coupling parameter.

1. INTRODUCTION Diagrammatic expansions are used to calculate thermodynamic properties of gases and liquids [l]. Such an expansion consists of a power series in a coupling parameter, such as the particle density, with coefficients involving cluster integrals of functions of the interparticle potential [2]. This technique is particularly successful at low density because reasonable accuracy can be achieved with the first few terms [3]. At higher densities more terms must be computed, and the series become unwieldy or divergent. Results have been obtained by two methods: Monte Carlo simulation and selective summation of diagrams. In the former, a system of particles is simulated on a computer; for example, Brush, Sahlin, and Teller (cited as BST) [4] and Hubbard and Slattery [5] determine pressures, free energies, and pair correlation functions for nuclei in dense stellar matter. One class of summation schemes gives nodal expansions [6-81, which are useful for a Coulomb gas at moderate density [9]. Other summations yield a change of coupling constant * Work performed under the auspices of the U.S. Atomic Energy Commission and supported in part by the National Science Foundation.

197 Copyright All rights

0 1974 by Academic Press, Inc. of reproduction in any form reserved.

19s

THOMAS J. BUCKHOLTZ

and the introduction of correlation functions. Zwanzig [lo] treats the case in which the interparticle potential may be divided into dominant and subordinate parts. Bellemans and De Leener [ll] derive an electron-lattice interaction term for metallic hydrogen at zero-temperature. We are interested in material at high density, such as metallic hydrogen. In examining perturbation seriesfor the free energy one finds that the electrons are weakly coupled but the ions are strongly coupled. The strong coupling makes the seriesintractable. As in some of the above-cited work, we sum classesof diagrams and eliminate a troublesome parameter. The resulting expansion is in terms of the electron coupling parameter and ion correlation functions. The appropriate correlation functions are those for a one-speciessystem (ions in a uniform background of electrons) and are assumedknown from BST-like calculations. Our technique can be applied [12] to fully-ionized dense plasmassuch as might be found in the interior of Jupiter or white dwarf stars. The nuclei are strongly coupled and may be treated classically; the electrons are weakly coupled but must be treated as degenerate. Our expressions give results in agreement with the weakly coupled ion schemesand with the Bellemans and De Leener work for crystallized nuclei. In the intermediate regions the results are new; in particular, we compute the most significant ion-electron interaction term. The current paper focuses on the combinatorial analysis involved in resumming expansions. As such, it is convenient to assumethat both speciesmay be treated classically. In the next chapter we discussthe case of weak short-range potentials and derive expansions in terms of ion correlation functions and the electronelectron and ion-electron Mayer functions. The Coulomb case is treated in the following chapter; we replace the Mayer functions with screenedpotentials. This expansion is then recast into a nodal expansion (triply connected diagrams) similar to those of Meeron [6], Abe [7], and Friedman [S] (cited as M-A-F) for the one-component case.

2. WEAK POTENTIALS The normal two-species expansions for weak potentials involve series whose coefficients are integrals over Mayer cluster functions. The coefficients differ depending on whether the seriesis in powers of the activities or of the densities. The designation “weak potential” means that we assume the cluster integrals converge. In this section we review the usual expansion and use our summation schemeto derive two new series.Each is a seriesin an electron coupling parameter with coefficients involving integrals over the ion correlation functions and ionelectron and electron-electron Mayer functions.

NODAL

EXPANSION

FOR

CLASSICAL

199

PLASMAS

A. The Expansion in Activities 1. Mayer Cluster Theory- Notation This section establishes our notation and the basis for our work. We review previous results for two-component systems; the discussion parallels a fuller treatment of the one-species case found in Hill [2]. The canonical partition function may be written as 1 z N+N- = Il”,“+N+!

1 ATN-N-!

. exp[-PW,

s d3& ..* d3RN+ d3rl *‘- d3rN-

,..., RN+

, 5

,...,

(2.1)

rN-)l.

We have adopted the convention that + denotes ions and - electrons; N+ and N- are the total number of particles of the respective species. We also use a capitalization convention. For example, R denotes ion coordinates and r denotes electron coordinates. The thermal wavelengths (1+ are defined in terms of Plan&s constant h, Boltzman’s constant k, , the temperature T, and the masses m+ of the particles L’L = h/(2vm*kkBT)1/2.

(2.2)

/I = IlkBT.

(2.3)

/3 is the reciprocal temperature

U is the potential energy. We assume that U is the sum of potentials between pairs of particles: U(R, ,. . ., rl ,... > = u++@, > RJ + u++@, 3 R3) + a*. + u++@N+--1 , RN+) +

u+-(Rl

+

u--(5

, r,> , r2)

+ +

...

+

-a* +

+

a+-@~+ u--(rN_-l,

u+-R

, r,> , TN-) TN-).

(2.4)

The Mayer f-functions are defined by f& = epBU++ - 1.

(2.5a)

f+-(R, , r3) = e-Bu+-(Rl,r~) - 1.

(2.5b)

For example, we have We will often use abbreviated notations h, = fR fi = f&

, RJ) = f++(R, , RJ, , rd = f+-@I , rd,

fii = f(ri , rj) = f--(ri

, rj).

(2.5~)

200

THOMAS

In terms of thef-functions,

ZN+N- =

AYN’N

J. BUCKHOLTZ

we can write 1 A3NfN

+ + +-

-

-. I

j

d3R1

-.a d3RN+ d3rl -a* d3rN-

x l
(2.6)

The next step is to convert to an expansion in connected diagrams. For each pair (1, i) of nonnegative numbers of ions and electrons (except (0,O)) there is defined a cluster integral bli = J$ d j d3Rl *-a d3RI d3r, a*- d3ri S,...,,,..., . .

.

(2.7)

V is the volume of the system. S,, and S,, are defined to be unity; otherwise, S,...,,,...i is the sum over all products of f-functions satisfying the following criteria:

(i) The coordinates R, ,..., RI, rl ,..., ri must each appear in at least one f-function. (ii) For eachf-function, the arguments must be two distinct elements from the list R, ,..., RI , rl ,..., r( . (iii) Nof-function is raised to any power other than zero or one. (iv) Consider the I + i coordinates (R, ,..., rJ as vertices of a graph. In this graph, two vertices are joined by an edge if and only if the points are both arguments of onef-function. The graph thus formed must be connected. (For a discussion of graph theory and statistical mechanics, see Uhlenbeck and Ford [13].) As a result of Eq. (2.7), we can write Eq. (2.6) in the form (2.8)

where each rnli is a nonnegative integer, and the prime on the summation denotes restriction by the conditions T imli = N-

for all I,

c ImIi = N+

for all i.

symbol

(2.9)

I

The grand-canonical defined

partition

function, 2, is now defined. The activities [rt are (5 = e”*/Ah3,

(2.10)

NODAL

EXPANSION

where (Y+ and 01~are /I multiplied species. Then we have

FOR

201

PLASMAS

by the chemical potential

2 = N Fe,, fPt!fe +* , Condition

CLASSICAL

C’ n $J @It) Ii

(Vbri)mrd-

(2.9) can be removed by taking an appropriate X =

,zo

5+Kihi

of the appropriate

logarithm.

2

(2.11) Defining (2.12)



we can verify that evx = 2.

(2.13)

Thus we have log 2 = -(/3F - /lG)/V,

X = (l/V)

(2.14)

where F is the free energy of the system and G is given by W/V

= P +a+ + p-a- .

(2.15)

Here, ph are the number densities of the two species. The above completes the review of the expansion of the free energy in powers of the activities. The purpose of the remainder of Section 2.A is to sum terms of Eqs. (2.7) and (2.12) in order to obtain a new expansion for the free energy. We will remove all appearances of the ion&functions (f++) in favor of an expansion in terms of Ursell cluster functions. The Ursell functions are closely related to the various density functions of a one-species system of ions. The electron f-function (f-J and the mixed f-functions (f+J will remain in our expressions. At the end of the section we collect the results in the form of a set of rules for a diagrammatic expansion of the free energy. 2. Density Functions- Graphical Notation - Ursell Functions

We next define the density and Ursell functions for the ions. In so doing, we introduce the graphical notation used extensively in the remainder of this paper. In the present subsection we consider a system of ions only. With regard to Eqs. (2.7) and (2.12), this means that we set i = 0. The J-particle ion-density function n,(R, ,..., R,) is given [14] by ~J@I ,..., RI) = f 5,’ ITI

#II N+P’=P+

(I y J)!

s

d3RJ+I ... d3RI 81...r,

(2.16)

202

THOMAS J. BUCKHOLTZ

where E,..., is identical to S1...1, in Eq. (2.7) except for the following Criterion (iv):

change in

(iv’) The graph need not be connected; however, each maximal connected subgraph must contain at least one of the vertices R, ,..., RJ . In other words, each integrated vertex (RL , .I + 1 < L < Z) must be connected to a root vertex (RL , 1 < L < J) by a path of edges. After introducing graphical notation in the next paragraph, we will give examples of Eq. (2.16). A given term in R,..., consists of a product off-functions. Eachf-function has a pair of coordinates as arguments. For example, two terms in R,,, in the expansion of n2 are (2.17) h&3 = f++R , RJ f++R 9&I and (2.18) h&1&3 = f++@, 3 R,)f++@, , Wf++(R, 3 Rd. As indicated above, each term corresponds to a graph in which the points represent coordinates and the edges symbolize f-functions. Terms (2.17) and (2.18) are represented in Fig. 1 graphs (a) and (b), respectively. Notice that n, is a function of R, and R, ,* we have distinguished the coordinates of integration (RJ with brackets. a)

Eq. (2.17)

.(3)

x I

A ;

b) Eq. (2.18)

x(3)

A X-X I c)

2

Eq. (2.21) (3) =

2i

FIG.

x 1

i ;

(3)

(4)

1

i

x 1

x 2

1. Some graphs in the activity

+

3(‘” 1

expansion

2

of n2(RI , Rz).

A sample term in R,,,, is .fisf3~f2~ = f++@, 2 Rdft+@s

, R,)f+,(R,

3 Rd.

(2.19)

Often, we will not label integrated coordinates. A graph is said to be connected if for any two points there is a path of edges

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

203

between them. Thus, graphs (a) and (b) in Fig. 1 are connected. Notice that the ones in Fig. l(c) are not connected but do satisfy the weaker condition (iv’). There are two maximal connected subgraphs in the third term: vertices 1 and 4 connected by an edge and vertices 2 and 3 connected by the other edge. For future reference we also define double-connectivity. A graph is doubly connected if and only if, after the removal of any one point and of all edges in which it participates, the resulting graph is connected. The graph in Fig. I(a) is not doubIy connected; we can disconnect it by removing point 3. The graph in Fig. l(b) is doublyconnected. “, @,) = E+(X) 1 +

Ef

j-d3R2 (

i x

)

+ $ Sd3R*d3R3( 2i +x\x/x + p‘ ) 1 1 7 + ... FIG. 2. Graphical representation of the activity expansion for the density function n, .

FIG.

%*

3. Graphical representation of the activity expansion for the two-particle density function

204

THOMAS

J. BUCKHOLTZ

Expansions of q and n2 are given in Figs. 2 and 3. The actual expression for n2 is +A

9 R,) = f+2(1 + fiz) + 5,” j- d3& (fu +

523

+ $ j- d3R,d3R,(2f,,f,,

+

fi3f23

+

fizfis

+

fizfis

+ fiJ2-m~

+ 2fisf34 + -.) + ***.

(2.20)

The term 5,” * 1 corresponds to the void product in R,, . The graphs with combinatorial factors other than unity correspond to groups of similar terms. For example, we have used the simplification symbolized in Fig. I(c)

2f3h4

replaces fi3f2* +A&,

(2.21)

.

As represented in Fig. 3, the terms in (2.20) follow rules (i), (ii), (iii), and (iv’). The other density functions n, have expansions similar to that for n2 . It should be noted that each nI is a function of only I - 1 independent variables. It is useful to have simplified notation: n = n, = p+ = N+/V = n,(RJ,

any RI ,

(2.22)

n&G - Rd = n3(Rl , RJ.

In the next subsection we will find that the Ursell functions are of more interest to us than the density functions. The former, denoted by x1, I = 1,2,..., are defined from the density functions by the relations (2.23a)

n,@J = xdRA 4h

(2.23b)

3 RJ = X@I 7R,) + xdR,) xdh),

n3& y RZ , R3) = x3(& , KY R3> + x2R , W x1(R3) + xz@ 3 R3> xdRd + x& >W XI(&) + x1(R) xdRd x1(R3h etc.

(2.23~)

The relations can be inverted xdRJ X& x3&

(2.24a)

= ndRA

, RJ = n2(Rl , R2) - ndR3 n,(RJ,

, Rz , R3) = MG

, R2 , RJ - [n&G , R-J nd&)

+ 24@1) nd&J nl(R3h etc.

(2.24b) + cyclic permsl (2.24~)

205

NODAL EXPANSION FOR CLASSICAL PLASMAS

Again, we introduce simplified notation

x1 = xdRA any RI , - RJ = x&G , W

x&

(2.25a) (2.25b)

From Eqs. (2.23) and (2.24) it follows that

xz(R) = n%dR) - 11,

(2.26a)

where g is the pair correlation function [g(m) = 11. We define j(R) = g(R) - 1. There are diagrammatic

(2.26b)

expansions for the Ursell functions also [Is] (2.27)

XJ@, >...,RJ) = f t+%(R, ,..., RJ), I=J

where the 6,(*-a) here are generalizations

h(R, ,..., RJ) =

(I

:

J>!

of Eq. (2.7) @RJ,,

s

**-

d3&

&...I,

I 2 J.

(2.28)

The terms in S... satisfy (i), (ii), (iii), and (iv); the graphs are connected. Figure 4 gives graphical expressions for b,(R, , R,) and b,(R, , R,). b2 CR,, R2,

=

x-x I

b3($,rR2)

=

/ x-x I

2 x (3)

.(3) +.A I

2

2

.(3) + !A 1 FIG.

2

4. Terms appearing in xz , the second Ursell function.

3. The Role of the Ursell Functions We can now proceed with calculations. Starting with Eqs. (2.7) and (2.12) for the free energy, we collect similar terms and eliminate all appearances off++ functions. In their place are left various Ursell functions. In this subsection we demonstrate how this substitution is achieved. The next subsection contains rules for the new diagrammatic expansion.

THOMAS J. BUCKHOLTZ

206

As a first step, we write (2.12) as x==

f

(2.29)

x,,

k=O

where X, contains each diagram having precisely k ions appearing in f+- functions. While this segregation of terms is far from the one we ultimately achieve, it gives useful results, especially for the terms X, , k = 0, 1, and 2. We consider these cases next. The first term in (2.29) X0 , contains all terms in (2.12) having no f+- functions. Because each diagram is connected, the only terms appearing are those involving only one species X0 = -/-IF+/ V - ,8F-/ V, (2.30a) -pF+/V

= +, log Z, = :I + $!

1 d3R, ‘.’ d3RI S,...,, ,

(2.30b)

j d3rl *a*d3ri Ss,..+ .

(2.30~)

/ -fsF_/V

= + log Z- = & t

$

Z, denotes the grand canonical partition function for a one-species system containing only particles of type *. X,, gives the free energy assuming that particles interact only with like particles; it is assumed calculable by normal one-species considerations. The next term, X, , has one ion interacting with electrons in each term. Mixedspecies terms can also be represented by diagrams. Notation for two-species diagrams is given in Fig. 5. By definition, XI contains all connected diagrams having Y

(or x *I

ion coordinate

(J! )

(or

. i) I Y-X

J)

f+-,

ion-ion

X-.

(or

I X-e

j)

f,.,

ion-electron

.-.

(or

i

i)

f-.,

electron-electron

(‘L

cqq)

(or x-x

FIG.

--.

election

coordinate

(r;)

f-f”nction

(f++ .(&, , &J))

f-function

(f+_,

(RI'Lj))

f-function

5. Notation for two-species diagrams.

exactly one ion which appears in f+- functions. Some terms contributing to XI are shown in Fig. 6. We have made a classification into types, according to the topological structure of the f+- and f-- functions appearing. For the simplest class (a) we have given explicitly all of the permutations of ion coordinates in the diagrams having less than four ions. The terms in (b) have similar permutations, which are indicated by combinatorial coefficients. We write the coefficient of the

NODAL

a)

Clors

I

X’.

EXPANSION

q -

*

L!-

-

'2 I-*-.

FOR

2

2

3x'

3x

al 1 Nx-.

CLASSICAL

“\’,,x-.

'x Nx-.2 ;p. w' 3 'XQ. 'x.3 px-.

207

PLASMAS

3x-x-x-.2'

2 3

3,-?-L.

?-La-.

x-x-x-.

1 2 x-x-x-.

2' x-x-x-. 3

1

.., 3

X’ b) 1

Clorr x’

0 (’

2

2x-

\.

c) Cl.rs a -

2

3x\xR' / \.

< .-

‘.2X

'.,X,2_!

,.2x-x-?-!

('.1).(3)X

2.3x'x'<

. ..

.

I., x-!-T

d) Class

x\,/* 3x0 \.

-~-!-?'.3~)x,!,?

2.3x-x-x< ...

*%,?,! X'

I.3

4\.

0 -.

"\

'I:

('.2).(3)

*\ ('*3)~(3)x,~-4~~**

--<<

e) Other classes

a*-

(2.3).('2)

X-X-x:-:...

FIG. 6. Some diagrams contributing

to Xl .

last diagram given as 2 * 3. This distinguishes the factor 3, representing the number of possible candidates for the role of argument in the two f+- functions, from the factor 2, representing another symmetry of the diagram (interchange of the leftmost two ions). The reason for this distinction will become apparant soon. Class (c) is the simplest one having more than one permutation of electron coordinates. For class (d) we have added the right-hand factor to give the electron symmetry. The general factor for terms in XI is of the form (2.31)

(S * I) * s,

where I is the number of ions and S and s are integers. Returning to algebraic expressions, we simplify the expression for XI XI = ,&

+ $

.

g

.

j- d3R, 1.. @RI d3r,

...

d3ri

c

nfJKl-bfL1 c JK

1

;hk

(2.32a)

(2.32b) (2.32~)

208

= ,zo $ $

THOMAS

J. BUCKHOLTZ

j d3& ... d3& d3r1 -a. d3ri 1 n,(R,) nf(R1, 2 G

rJ nfjlc

. (2.32e)

jk

The manipulation is much less formidable than it may appear, but we have been cautious in presenting the steps. Equation (2.32a) is the definition of X, . The symbol CC denotes the requirement that the diagrams represented by the particular choices for the triple product must be connected. Examples of products which do appear are given in Fig. 6. Ion coordinate RL represents the one ion participating in an f+- . In (2.32b) we recognize the 1 possible values of L. In (2.32~) we do the manipulation symbolized by (2.33)

This corresponds to the recognition equally simple and important

of the factor I in Eq. (2.31). Step (2.32d) is

&I=

(IT

l)! -

Finally, in (2.32e) we have recognized that we have found n,(R,), as defined in Eq. (2.16). The factors S are precisely the ones needed. The reader may wish to check the factors by referring to Fig. 2. It should be noted that (2.32e) can be written also as

x, = c L 5-' j @R,*a. d3R1 d3rl I.i>O v I!

... d3ri c x1@,) nf(R, c

L

, rL>nfjk

.

(2.35)

jk

This can be verified using Eqs. (2.27) and (2.28) or may be seen from definition (2.23a). The important result is the discovery of the factor xl(-) (or nl(-)) in Eq. (2.35) (Eq. (2.32e)). We now consider the term xz, for which similar summations can be done. Here the functions xZ and n, will be found. The topology of the diagrams will determine which of the two is correct. The least complicated set of graphs of A’, to be summed is given in part (a i) of Fig. 7. The factors given are of the form s* (

ICI-

1)

2

1*

(2.36)

NODAL

a)

EXPANSION

Some diagramr i)

Class

FOR

CLASSICAL

in X2 3

(I * 1);:.

(2.

3) “-“;.

(I.3)x(x). x

(2.3)+.

(1 -3)

x(;)

(2 .6) x

x’ i)

209

PLASMAS

x x-x\. X.-X@

...

l3-.

Class

q I’

X-X-. (I

.I).

(2.3).2

2i-

x-. (1. b)

3).2

/-* “\I

X-e

1 x-.

(1.

3).2x<::’

*a’

Some diagrams not in X2, ii (1.1).2

X-. x-.

(2.

3).2x-;I:

(2.6).2;1:1:

FIG. 7. Graphs relevant to the discussion of X2 .

By comparison with the expansion in Fig. 3, it is found that the factors are precisely correct for the recovery of the function &r, . The complete result for the sum is x3.f = $ s d3RI d3& d3r !edR,

- R,) f+-(R, , r) f+-(RZ , r).

(2.37)

Next, consider the class X,,ii given in part (a ii) of Fig. 7. The third combinatorial factor refers to the electron symmetry s. By comparison with Eqs. (2.27) and (2.28) and with Fig. 4, we obtain the result J’w = t

y

s d3RI d3R3 d3rI d3r, +x2@, - R,) 2f(R, , rl)f(R2 , r,).

(2.38)

The immediate question is that of why X,,< involves in2 while X,,ii depends on QxZ . In Fig. 7(b) are given some diagrams that do not occur in X,,ii, because they are disconnected. It is seen that these terms would sum to

+ y I d3RI d3&d3rId3r2 &xI(Wx1(&J 2fR , r,)f(R,, r,>.

(2.39)

Recall the relation between n2 , xZ , and n, (= x1) given in Eq. (2.23b). The sum of (2.38) and (2.39) contains n2 . But, because of connectedness properties, only xZ applies. The result we have found can be generalized by the following two rules:

210

THOMAS

J. BIJCKHOLTZ

(a) A x-function is to be considered as connecting each of its arguments. (b) A class of diagrams sums in such a manner as to retain all of the terms of the appropriate case of Eq. (2.23) which leave the diagram connected. The above examples provide cases conforming to these rules. There are more cases for X, , none of which provide any new insight; the interesting cases for X, are too tedious to present herein. To complete this subsection, we give some physical motivation for the two rules. We consider the case of short-ranged potentials rapidly as

f+-CR, 4 -+ 0 f--b,

/ R - r / + co,

rapidly as 1rl - r2 j + co.

3 r2> - 0

(2.40)

Recall that n,(R, - R,) + n12

as IR,--R21-+co.

(2.41)

We see then that expressions (2.37)-(2.39) behave as (2.37) = O(l), (2.38) = O(l), (2.39) = O(V),

(2.42)

as the volume V becomes large. The last result is easily verified (2.39) = 0 I+

[s d3R d3r nlf(Rl , rd]‘! (2.43)

= 0 (+

* v2) = O(V).

Thus, we see that in order to get intensive results expression (2.39) cannot be present. 4. Rules for the Expansion in Terms of Activities

We now give the complete expansion of X (Eq. (2.12)) in terms of ,$- , the electron activity, and the functions x1 , f+- , and f-- . X can be written as

ii,@,

,...,

fi fi (f(&, L=l

I=1

I[

rJYLg

(2.44) ltfk

NODAL EXPANSION FOR CLASSICAL PLASMAS

211

EC denotes the sum over all distinct connected diagrams, as specified by Z, and the exponents pLt and qkn . Each exponent can be zero or one. For each term there is an associated graph having the I + i labeled points R, ,..., RI, r1 ,..., ri . If pLL(qkn)is one, then there is an edge in the graph connecting the points R, and r,(r, and r,). Z, represents any term in the sum in the expansion (Eq. (2.23)) of the density function n1in terms of Ursell functions. An Ursell function is considered to give rise in the graph to edgesconnecting once each pair of its arguments. For example, the term x3(R1, R, , R4) x1(R3) arising in the expansion of n4 provides three edges: R,R, , R,R, , and R,R, . Each such ??I,is legal for the sum, providing that the graph thus constructed is connected. All topologically-distinct labeled connected graphs contribute to the sum. As an example, consider the following three candidates for the (I, i) = (2,3j term: x2(Rl , kJJC@, , rJ f@h , rS)f(r2 , r3),

(2.45)

xz(RI , Rd fR

(2.46)

, rl)fR

x1@&) xl(RJfR

, rs>f(r2 , rd,

, rdf&

, r3)f(r2 , rJ.

(2.47)

The corresponding labeled graphs are given in Fig. 8. Expressions (2.45) and (2.46) are connected. The graphs, if treated as unlabeled, are the same; however, (2.45)

: 5, x -.q I 5*x-.

(2.46)

-. r2

:

4

A!, x -.L1 I 52 x -L&--i2 (2.47)

: R, x-*q

%“--(3-;

-2

FIG. 8. Graphs associated with expressions (2.45), (2.46), and (2.47).

as labeled graphs they are topologically distinct. Hence, they both contribute to the free energy in Eq. (2.44). There are twelve distinct labeled graphs corresponding to this particular unlabeled one. We say that “the weight of the unlabeled graph is twelve.” Notice that twelve is the reciprocal of the factor l/(2! * 3!) which multiplies these terms in (2.44). To conclude this example, we remark that (2.47) is not connected and, hence, does not occur in Eq. (2.44).

212

THOMAS

J. BUCKHOLTZ

Finally, notice that we can combine Eqs. (2.30) and (2.44) to obtain x = --P(F+

- C e

jh(R,

+ F-;-)/V + , go +c +,. g . j d3R, *a* d3RI d3rl .*a d3ri

,...,

RI)

[ fi

L=l

k

(f(b.,

G’L’][,i!

fil

(f(rk,

rn)pn]/.

(2.48)

Z=l n#k

B. Free Energy as a Function of Densities In the previous section we derived an expansion in powers of the electron activity [- . In the current section we do similar calculations in order to obtain an expansion in terms of the electron density, p- , which we henceforth denote by p. This new expansion is motivated by the simplifications which occur in the usual cluster expansions when one transforms a series in the activity into one in the density. Just as in the usual theory, we will find that the transformation replaces the criterion of connectedness by one of double connectedness. Thus, the number of terms to be calculated at a given order is reduced. Because the manipulations for deriving the density expansion are similar to those for the activity case, we will shorten the treatment. The well known expansion for the free energy in terms of the densities of the two species is given by Eq. (2.14) and the relation x=

1 nrpiBIli , #

(2.49)

where n is the ion density (p+), p the electron density (p-), and BIi is a cluster integral analogous to bli in Eq. (2.7). Indeed, we have [16] (2.50)

where D.,. consists of terms satisfying the same criteria (following Eq. (2.7)) as S.,. , with one exception. Criterion (iv) is replaced by that of double connectedness of the associated graph. There exist expansions for the various ion distribution functions in terms of the ion density n [17]. For example, n2 may be expressed as shown in Fig. 9. We now perform the summation of diagrams for a few sample cases. The first results are analogous to Eq. (2.30) f nIBI = -fiF+/V, I=1

i$ p”Boi = -/K/V.

(2.51)

NODAL EXPANSION

FOR CLASSICAL PLASMAS

213

FIG. 9. Expansion of n2 in terms of the density n. a) aa a-. b) Class

d)

D,,,=

X-* %

Ll

Class 0-a cl-.x

FIG. 10. Graphs relevant to some cases of the expansion in densities.

Next come the terms having i = 1 and onef+- function. Here, the simplification caused by double connectedness is apparent; there is only one graph contributing. It is shown in Fig. IO(a); the algebraic expression is

P + j- d3R, d3r, nf(R1, rJ.

(2.52)

Next, consider the i = 1 case having exactly twof+- functions. The graphs given in Fig. IO(b). Notice that the graph given in part (c) of the figure does appear; it is not doubly connected. By combining each coefficient shown in with the appropriate l/1!, we see that we would recover in2 if we also had

are not (b) the

214

THOMAS J. BIJCKHOLTZ

diagram in (c). But the diagram in (c) corresponds to the term iz9 * 1 = xl2 in the expansion in Fig. 9. The l/1! = l/2 that would appear for (c) is precisely that factor we need. The result is $I, - +x12= $x2 .

(2.53)

Thus, case(b) sumsto 1 3 Pt

j d3Rl d3& d3r x2& , RJ fR

, r1)f(R2 , rd.

(2.54)

In contrast to the above, consider the caseshown in Fig. IO(d). All of n2 occurs this time. The result is k g + j d3R, d3R, d3r, d3r, ?zz(R,, RJf(Rl

2rlMRl

, &f(R2

y r1MR2 3G. (2.55)

The resulting expansion is similar to the previous one. We must consider that a x function singly connects each pair chosen from its arguments. The resulting term must have the appropriate connectedness. In the activity expansion the connectednessis single; here it is double. Thus, the analog of Eq. (2.48) is x = --P(F+ + F-Y v + ,;,

+ +,. $ . j d3R, ... d3RI d3r, -a*d3ri

where Cdc denotes the sum over doubly connected diagrams of x, f+- , and f-lines between the points R, ,..., R, , rl ,..., ri . As an example of the relative simplicity of this expansion, notice that the terms shown in Fig. 10(e) are entirely absent. 3.

THE

COULOMB

PLASMA

Expansion (2.56) is satisfactory for casesof weak forces; however, it fails for Coulomb forces. In particular, for R - r + O&JR, r) has an essentialsingularity. Also, the slownesswith which the Mayer functions tend to zero at large spatial separation presents problems. In the usual cluster expansions one overcomes the latter difficulty by expanding the f-functions in powers of the potentials u and then collecting similar terms in order to form an expansion in screened potentials. We also must use similar techniques for the Coulomb case. In the following section we discussthe expansion in powers of the potentials and the derivation of potentials screenedby electrons. In Section 3B we find that

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

215

it is necessary to sum a class of ring-like diagrams in order to remove divergences arising from the long-range nature of the potential. This process is analogous to the summation of the ring diagrams in the ordinary theory; the results are given in Section 3C. We then give a seminodal expansion in Section 3D. In the following section we sum some other classes of chains. Section 3F presents the rules for a complete expansion of the free energy for the two-species classical Coulomb system. This result parallels the nodal expansion of M-A-F. In Section 3G we discuss the convergence of some of the terms in our expansion. A. Expansion in Potentials and Screening by Electrons Each diagram in Eq. (2.56) is to be decomposed into a collection of diagrams, each having potentials instead of f-functions. The expansion is based on Eq. (2.9, the definition off f+-CR 4 = exp[-flu+-@,

r)l -

1

= fl 4 [-flu+-CR, r>l”> f--(rl

, r-J = f

-$ [-pzf--(rl

?I=1

, r,)]“.

(3.1)

.

Figure 11 gives symbolically these decompositions. We use thin lines to indicate potentials. The factor (-/3)“/ n ! must be understood to apply to a collection of n edges between two specific vertices. f ;t* -=

=

-b* -

*

+

&P”*,)2

+

+

-

+

;(-P”**)3 =-

+ I.. +...

FIG. 11. Diagrammatic expansion of an f-function.

The expansion for X becomes

(3.2)

216

THOMAS

J. BUCKHOLTZ

The symbol Cdc refers to the double connectedness property inferred from the “parent” term in Eq. (2.56) from which a term in (3.2) arises. The varying of the nonnegative integers sLZand tk, is included in the symbol &o . Note that sLl(tkn) is zero if and only if pLl(qkn) is zero in the corresponding parent term. Having created a formidable collection of terms, we look for simplifications. The first one is the summation of linear chains. There are three cases, which are shown graphically in Fig. 12.

v __ ‘+r,’

=o-m

=O)

.-.+ 0 m

l!

i=o

b)

+

21 ._._._. 0

+ 3!.-.-.-.--‘+... 0 (1)(2)(3)3

(1x2)m

m

2

Two ions as endpoints “++(fr,,

c)

._._. 0 (1) I

Mixed

ER)

.y2

y-.4+

2 p-‘-.-y+

6 y-.-.-.4,+..’

endpoints - 0-. , *

v+-@,‘L,)-pm;-

FIG.

-

12.

+ *-.+ I

2-.-.-.+... 1

I

I

Summation of chains for electron screening.

The first case represents the screened interaction between two electrons labeled 0 and co. It is understood that these two electrons are part of a more complicated graph. We are summing all terms for which the graphs are the same except for the one chain of interactions represented. The electrons labeled (l), (2), (3),... do not interact with any particles except as explicitly shown in the figure. Let the prototype graph for the summation have j electrons, including 0 and co. The expression for the ith term in the chain will include &

pjfi (j ? i, s d3r, -** d3ri i! [-flu-&, * [-/3u--(rl ,

rz>] * a*. [-flu-.&, l

r,)].

, r,)] (3.3)

The factor pj+(/(j + i)! arises as in (3.2) from the total number of electrons in the diagram. The combinatorial factor (jy) = (j + i)!lj! i! gives the number of ways of partitioning j + i identical objects into two groups, one ofj objects, and other of i. This represents a partial accounting of the allowed permutations of electron indices. The factor i! comes from the number of permutations of the indices (I), (2),..., (i). This is the coefficient shown in Fig. 12(a). The -fiu-factors

correspond

to the potential

lines.

NODAL

EXPANSION

FOR

CLASSICAL

217

PLASMAS

The summation of the terms of the form (3.3) is best done in wave-number space. The Fourier transform of a general function h(r) is defined by

The inverted relation is

h(k) = + s d3r eik’%(r).

(3.4a)

‘k)=

(3.4b)

d3k ePik’%(k).

&&

The explicit forms of interest here are (3.5a)

uJrl

, r2) = @/I rl - r2 I, u+-(R, r) = -zeZ/i R - r 1, u--(k)

= 4ne2/Vk2,

(3.5b) (3.6a)

w+-(k)

= -4n-ze2/ Vk2.

(3.6b)

The new symbols are e for the negative of the charge of an electron, z for the charge quantum number of a nucleus, and k for a vector in wave-number space. Expression (3.3) can now be written as i

pj * pi (-

#$y’l

- exp{--i[k,

1 d3r, ... d3ri d3k, ... d3ki+l zl--(k,)

... u-Jk,,,)

* (q - rl) + a.. + ki+l . (ri -- r,)])

= $- pi . p”(--PV>i+l

&

j d3k, ... d3ki+I exp[-i(k,

. S3(kl - k,) 1.. a3(ki - k,+l) u--(k,) 1 p i . p”(-/w>“” = j’?

j d3k u%(k)

&

. y. - ki+l . r,)]

... uJki+,) exp[--ik

* (r, - r,)].

(3.7)

The diagrams of Fig. 12(a) can be summed to give the screened potential. Notice that the factor pj/j! conveniently applies to the prototype diagram, which has j electrons. Dropping this factor, we find that the sum is given by

1s

p 8.rr3

d3k exp[--ik

. (r. - r,)]

f

(3.8a)

[-/3PVu--(k)]i+1

i=O

d3k exp[--ik

* (ro - b>l

+u--(k) 1 + ~pJ,u--(k)

We recall that the factor associated with a line is -flu.

.

(3.8b)

By analogy, the above

218

THOMAS

result is -,%-thus

J. BUCKHOLTZ

; v...- has the dimensions of a potential.

The screened potential

is

U--M

I

d3kexp[--ik . (rl - cd 1 + pPVu--~k~

i

d3kexp[--ik . (b - r2>l1 + bnpe2,$,k2 (3.9)

he2/k2

=& =

j&

s d3k

(2$3

ev--ik

* 6.1 -

r2)l

1 +

1

k2h-2

3

where X- is the electron Debye screening length XI2 = 4rrpe2/3.

(3.10)

Finally, using Eq. (3.4b), we see that -PUk)

= 2

(3.11)

1 + iaAd2 .

Looking back at Eq. (3.8b) we see that the effect of the screening is to introduce the denominator 1 + /3pVuJk) into an expression, which would otherwise be the first term on the right-hand side of Fig. 12(a). Analogous results occur for o++ and v-- . We find

(-PI2 v++R 3R,) =~ (2:)”

-Pv+-CR,

9 = czLj3

s

d3k exp[--ik

* (R, - R,)] ~-~~;(k!;kp

, (3.12)

ev--ik *CR - +I1+ppvu--(k) . sd3k --PVu+-04

(3.13)

We introduce the ion Debye screening length h, AT2 = 47rn(ze)2 j3.

(3.14)

p = zn

(3.15)

The condition of charge neutrality can be used to show that h-2 = 2x+2.

(3.16)

Results analogous to Eq. (3.11) are 1 1 1 C-6)” v++(k) = - Vn -X+2k2 1 + L2k2 ’

(3.17) (3.18)

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

219

The u-potentials, Eqs. (3.11) (3.17), and (3.18), each have a l/V factor, as do Eqs. (3.6). u++ differs from v+- and a-- in that it has as its simplest term a twopotential interaction. B. The Ring-Like

Subdiagrams

We now evaluate the lowest order terms in Eq. (3.2). It is convenient again to represent terms by diagrams. We use dots for electrons, boxes for arguments of X-functions, and lines for potentials. Examples are given in Fig. 13(a). The first a)

Sample term in Eq. (3.2) q -.

2 b)

a

d)

I

a-.

Example of a rummed chain =

“=.

+

q

i) The lowest-order

The counter-term x--x

.Q.

diogrom involving q

ii)

+

+

...

two rpecier

-.

in -fiF+/V

and -BF-/V .-.

FIG. 13. Some diagrams discussed in Section 3B.

one is part of the (1, i) = (2, 1) term. The second and third are some of the (2,2) diagrams. In the first two, only xz is legal for n”, , while in the third term x1 * x1 is also legal. Part (b) of the figure gives an example of the summation of a chain. The summed result is also called a diagram. It is convenient to introduce the notion of a subdiagram. We have seen that for a given structure of boxes, dots, lines, and springs there may be more than one acceptable product of x-functions. We shall call each legal combination, along with the remainder of the graph, a subdiagram of the given structure. A subdiagram has its “x-structure” denoted by loops enclosing ion boxes whose coordinates are arguments of the same x-function. Figure 13(c) gives two examples. In the first diagram are two subdiagrams whose characteristics are given, respectively, by the expressions

x&h, Rdb++OG3W2

(3.19)

XI&) x~@z)b++R >W2.

(3.20)

and

220

THOMAS J. BUCKHOLTZ

The second diagram has only one subdiagram because the x1 * x1 combination does not give a doubly connected graph. The simplest term in expansion (3.2) involving both species is shown in Fig. 13(d i). It is f$ j d3R d3r

zezp IR--r/

=

znpe2fi 1 d3x &

= p2e2fi- 4n jamx dx

(3.21)

and is divergent at large distance. Fortunately, there are two terms which combine with this one to give zero. They are shown in Part (ii) of Fig. 13(d). Their values add to give the negative of (3.21)

.& j

d3R,

d3R

-flzze2 2 I R,

zzz

2 Ge2/3

-

+ R,

T&

j

d3rl

d3r2

, r-ii”:,

,

I

J’ d3xL). 1x1

(3.22)

This “charge neutrality cancellation” is identical to one appearing in the normal two-species Coulomb theory [18]. For our theory it is necessary to ignore the (1,l) diagram having only one u-function. Alternatively, we can consider this cancellation as a renormalization of the otherwise infinite quantities -/IF+/ V and --/IF-/V. Are there any more “long-range” divergences? In the ordinary theory it is necessary to sum a series of terms called ring diagrams in order to remove divergences appearing in each term [19]. The ring terms are shown in Fig. 14. The first .z.

x=.

.-. .:

X-. I

X=X

i

I

J’\ x6 L-l Lx/ ! ... i

FIG. 14. The ring terms in the ordinary theory.

column consists of the electron-only ring terms. They are all divergent, but sum to a finite result. In our theory they are part of F- . The ion-only ones, found in the last column, are included in F+ . The remaining terms contain both species. We discuss next the contributions of the terms in our expansion which contain the mixed-species ring diagrams.

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

221

The term “ring-like” shall apply to those subdiagrams having the topology of a circle. The ring-like subdiagrams are shown in Fig. 15. Notice that we are using

3 1 T

1 6

4 ... I fi

a..

FIG. 15. The ring-like subdiagrams. I counts the number of ions, C the number of screened interactions. The coefficient is the weight of the subdiagram divided by I!.

screened interactions. Each ring-like subdiagram includes an infinity of ring terms from the ordinary expansion. To see this, all one need do is replace each box by a cross, each loop enclosing two boxes by a potential line, and each spring by the summation it represents. Each ring-like subdiagram, other than the {I, C} = { 1, l> term, contains more than ring terms. Each mixed-species ring term is contained in a ring-like subdiagram. Before discussing the divergence of these subdiagrams it is necessary to be able to compute the various quantities in them. We will transform all of the terms to expressions in wave-number space. As in the ordinary theory, there is conservation of a wave-number at each vertex. A proof that this property applies here would merely repeat much of the work done in deriving electron screening. The result is always an integral over one-wave-number variable of a quantity having a factor for each item in the ring. Equation (3.17) indicates the factor associated with a spring. A box enclosed in own circle corresponds to a factor n. A pair of boxes enclosed in a Ioop gives a factor x3 = n2b. We do not need complete knowledge of x2(k) for this discussion because we are interested only in long-range considerations; we defer a more detailed discussion until the next section. Long-range in R corresponds to small values of k (= 1k I), so we examine the k + 0 behavior of x2(k). We will let - denote “behaves as . . . as k --f 0.” n,(R) is spherically symmetric, and nl is independent of position; thus, xZ is even in k. x2(R) approaches zero sufficiently rapidly so that xZ - a finite

222

THOMAS

.I.

BUCKHOLTZ

value [in contrast, trz(k) - const . a3(k)]. It can be shown in general that x2(k) +. 0 (see, for example, Eq. (3.37)). Thus, we assume x2(k) - const # 0. The {I, C}th ring-like subdiagram obeys the relation

-~~RLu,c~ V- .c

k2 dk [v++(k)]C

[d(k)]‘-”

-

s

k2 dk (l/k2)c.

(3.23)

It follows that only the two C = 1 subdiagrams do not have long-range divergences. We can explicitly sum the ring-like subdiagrams. The coefficient for the (1, C}th subdiagram, as shown in Fig. 15, is (1/2C)(,$). This rule fails for the (1, l} term; however, recalling that this term comes from a (-/3~)~/2! term in anf-function expansion (not from a -/3u/l term, as do all the others), we see that it has an “extra” factor of $. The result is -P’RL,v

=

7

’ Q ’ j d3kjl Ig & (I : c) [(-13)2V~++(W C~“BWl’-” n2’-’

=&j d3k gljfo&(;)(P”Vu++F n”W)” =&j d3k El&(~~2v~++)C (1+nj>” =

- ; & j d3k log 11 - [1 + d(k)1 h+2@(1; hm2k2)1.

Some of the quantities in Eq. (2.34) are important symbols. We define 4, Y, and D by

(3.24a) (3.24b) (3.24~)

(3.244

enough to be given special

444 = 1 + aW,

(3.25a)

Yor) = nPVv++@) 4(k)? D(k) = 1 - Y(k).

(3.25b) (3.2%)

Equation (3.24d) can be written as +FRL/V

= - ; &

1 d3k log / D(k)/.

(3.26)

C. Evaluation of FRL

Equation (3.24) represents the first mixed-species contribution to the free energy. In this section we examine the properties of expressions (3.24).

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

223

The first property desired is the vanishing of the term if the electrons are considered as a uniform background. This limit is defined by e ---f 0,

(3.27a)

pe = me = const,

(3.27b)

p-a,

const,

Fl =

ze = const.

(3.27~)

In (3.27a) the number of electrons is made infinite, while the charge of each becomes zero. Equation (3.27b) assures charge neutrality, while (3.27~) assures that we are not changing the ion properties. Equations (3.27) give the conditions assumed for the Monte-Carlo calculations of BST, from which we compute values of j. In this limit the only affected quantity in Eq. (3.24d) is X- , the only one depending on the electron parameters. From Eqs. (3.10) and (3.27) we see that h- becomes infinite; hence Y is zero and D unity. The result is, as expected, FRL = 0.

The other one-species limit case we have n+

(3.28)

is that of a uniform background

co,

z+O,

nz == const,

of ions. In this (3.29a) (3.29b)

g(k) ---f 0.

As a consequence of (3.29a) we find xi2 + 0.

(3.30)

Assuming nj is well behaved, we find Y(k) =

’

-&I

1 + Xm2k2h+2k2

- t- n&WI

- 0.

(3.31)

This limit also yields the result of zero FRL . We have claimed to have removed a long-range divergence by performing the ring-like sum. But, does the ring-like sum actually converge? Before answering this question, we introduce some dimensionless units. Let L be a distance characteristic of the inter-ion spacing L = (4m/3)-1/3.

(3.32)

Following BST, we introduce the dimensionless parameter r = pz2e2/L,

(3.33)

224

THOMAS

J. BUCKHOLTZ

which measures the ratio of potential energy to kinetic energy of the ions. As a result of these definitions, Eq. (3.24d) becomes -@'RL/V

= - ;&z&j-

d3Klog ( 1 - $

1 + z;a/(3r)

d(K) 1, (3.34)

where K = Lk

(3.35)

is a dimensionless wave-number. We make definitions corresponding to (3.25) in terms of dimensionless parameters 4(K) = 1 + n2(K),

(3.36a)

3r 1 'cK) = F 1 + #,(3r) D(K) = 1 -

(3.36b)

&K),

Y(K).

(3.36~)

FRL is now formulated

in terms of L, T, z, and 4. 4 may be expanding in a “series” in the parameter lY For small r, Dewitt has given the result [20] nj(K)

= 4mX+3

i

-

A 1 + W@+I-W=

fl= d +2KA,/L arctan (q)

(K~~+l-W4

+ [I + WO+/Wl + ...I + ***I,

(3.37)

where A+

=

(3.38)

(477&+3)-l

is the classical plasma parameter. Use of the equality 3r = L=/h,=

(3.39)

in (3.36a) and (3.37) yields

W) = 3ry

K2

+i

(3rrFi2)2

arctan

&

-

(3.40)

Expanding about K = 0 gives

4(K)-$ =$[l-$(I-

- &

K4 + O(K6) + i2(3ry= y)

+ O(K”)].

(3.41)

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

225

For some values of I’, results for (b can be obtained by Fourier transforming results of BST. Hubbard has done transforms of approximations of the BST pair correlation functions for various values of r between 1 and 40 [21]. Figure 16 gives copies of these results for 4. MML

I

G s

3

5

2 l-=1

0 m

K

FIG. 16. Hubbard’s values for $(K). Taken from Ref. [21].

In general, 4(K) rises as an even power of K away from the value 4(O) = 0. Thus, Y is well behaved at small K. For small r we can use (3.41) in (3.36~) to obtain

D(K) -..+1 - (1 - g,.

[I - g

(1 - J!y!,] (3.42)

-

!g

(z

+

1

-

$

p).

Whether or not r is small enough for (3.42) to apply, it is seen that (3.34) is well behaved at K M 0. We can compute results for (3.34) using Hubbard’s curves for 4(K) for r = 1,2, 3, 5. For r = 4 we used the “MML theory” (see Ref. [21]) approximation of K -: 1 &vIML(K) = 0 =l

K:P- 1.

(3.43)

We set L = 1, which fixes a unit of length, and choose z = 1. The results are shown in Table I in the middle column. The results behave in a regular manner, increasing in r away from the r = 0 value of r;X,. = 0 implied by (3.31). It is interesting to note the results if we do not carry the summation of the ring-like terms to completion. Suppose we had stopped at Eq. (3.24~) and then interchanged the summation and the integration. The result for -/3FRJ V would be (3.44)

226

THOMAS

J. BUCKHOLTZ TABLE

Ring-Like

a MML

Contributions

I

to the Free-Energy for Classical Hydrogen

r

Logarithm form

Summation form

1 2 3 4” 5

0.13 0.39 0.73 1.2 2.0

0.13 0.39 0.13 co cc

approximation

to 4.

In the third column of Table I we have tabulated results given by (3.44). There is agreement for I’ = 1,2,3. But the summation form diverges for r = 4 and 5 because Y(K) is sometimes greater than unity. Indeed, for r = 5, Y(K) is larger than one for 0 < K < 1.94. The series C (1/2C) Yc does not converge in this region. Equation (3.34), not (3.44), provides continuation of the low-r results. D. Seminodal Expansion In this section we present the diagrammatic expansion, as developed so far, for the Coulomb plasma. This involves the reformulation of Eq. (3.2) in terms of the screened potentials V-- , v++ , and v+- . The expansion is given by X = -@‘+

+ P-)/V - ,8FRL/V +

c

t~~Sd3R,..,d3HId3r~...d3ri

I.i>l U.i>#

In the first term, the tildes denote the removal discussed in Section 3B. The next term is the and is given explicitly by Eq. (3.24d). The last but with new potentials. The p, s, and t are

.

.

of the I = 2 and i = 2 terms, as sum of the ring-like subdiagrams term resembles terms in Eq. (3.2), nonnegative integers. The parent

NODAL

EXPANSION

FOR

CLASSICAL

PLASMAS

227

diagram must be doubly connected. The prime on the interior summation sign denotes the effect of using the screenedpotentials: no electron can participate in only two v-interactions. The reader should note the appearance of three potentials in Eq. (3.49, as opposed to only two in Eq. (3.2). The third potential is needed in order to sum the linear chains which start and end at ion coordinates (seeFig. 12(b)). We have not produced an expansion in which every point participates in at least three interactions. Each electron must be part of at least three v-interactions; however, an ion need only be twice-connected, counting both v-interactions and X-connections. Of course, each ion must participate in at least one z)-interaction. For the ordinary theory, M-A-F have presented a “nodal” expansion in which each particle partakes in at least three interactions. We have achieved in our work so far such a requirement for one of the two species.We, therefore, call the results presented in this section a “seminodal” expansion. In the next two sections we explore the possibility of making a complete nodal expansion. E. Summation

of Chains

ln Section 3A we summed linear chains, each having electrons as intermediate points. The computation was done in wave-number space and was facilitated by the property that one momentum “flowed” through the chain. In Fig. 17 are diagrammed some other chains having similar flow properties. In this section we shall sum the first three of these chains. The four chains of Fig. 17 differ by their endpoints. In case (A) there are two electrons. Cases(B) and (C) have one and two ion endpoints, respectively. The ion endpoints are specifically restricted so as not to participate in xz interactions for the beginning of the chain. In case (D) a xz interaction is allowed. We could sum case(D) also, but, for the nodal expansion given in the next section, the result is not needed.

D’

_FIG.

595/86/2-3

Y i

ax3-

17. Chains discussed in Section

3E.

228

THOMAS

di

Calculation

J. BUCKHOLTZ

of the double-darh 0

We abbreviate

by

0 and

8

u+p

3

= o.(l+jo++ojQ+...)+~.

bYf’

(I+fo+;iofo+*-)

*6.(l+o~+o-/of+~~)+o?-. = (o+fQ). (1++o+fOfO*~.~)

(l+of+ofOf+“~)

+(i+o~).(l+o~+o7(oj+“~) = (0:~2j0+1).(l+fO+~~+...) 1 2

+

ng - 2ji2 Y ++2

I

“26

+ “fi2”+&

I - np V”,,

I -“g

I I - “9 2 V”++

“&+ + 1 - n P2V”++

FIG.

18. Intermediate stages in the summation of chains.

The first step is the summation indicated in Fig. 18(a). By analogy with Eqs. (3.3)(3.1 l), the result is

fPZ?++ l 1 - n~Vv++

.

What good is this result? We replace each series of boxes connected by springs with a pair of boxes connected by a “single-dash” interaction in all elements to be summed in the three cases. The result for case (A) is shown in Fig. 18(b). Notice that single-dash interactions may never be attached to both sides of the same box. The next step is suggested by Fig. 18(b) and is shown in Fig. 18(c). The manipulation required is indicated symbolically in Fig. 18(d). The first line is the definition of the double-dash. The second and third lines follow because all of the factors commute. The fourth line is the wave-number space expression equivalent to the third line. By multiplying numerator and denominator by 1 - ngz Vu,, , we obtain the following expression for the double-dash interaction:

WGXWV

n6 + n26P2v++ + nB2u++)

1 - n/Y Vv++(l + ni) The denominator

(3.47)

*

is the expression D which appeared in the ring-like

sum.

NODAL

EXPANSION

= f

FIG.

.

FOR

(1

+

p +

CLASSICAL

(y)*+

229

PLASMAS

... )

19. Calculation of three chains.

The three chains (A), (B), and (C) may now be reexpressed as shown in Fig. 19. The results are w-- = L- - nfiVut-(l -+ n&/D, (3.48) (3.49) w- = P-/D, (3.50) w++ = c,,lD. In each case D enters as a screening factor. G. Nodal Expansion

Using the summations of the last section, we can now present a nodal expansion. The complete free energy is divided into four parts --BW

G is given in Eq. (2.15). F” gives the one-species contributions of an infinity by charge neutrality -pF”/V

= log 2, + log 2

= (logZ+ - x-x) = -p
(3.51)

= --B(G + F” + Fm + -F&V.

= - ; &

after the cancellation

+ n--e

+ (log-z

- v-v)

(3.52)

subdiagrams 1 d3K log 1D(K)].

The fourth piece is the sum of the nodal terms and is described below.

(3.53)

230

THOMAS

A term belongs to the nodal conditions:

J. BUCKHOLTZ

part if, diagrammatically,

it satisfies

several

(a) It consists of boxes, points, resistors, and loops only. It must have at least one box and at least one point. (b) Each box is enclosed by exactly one loop. (Of course, loops do not enclose points.) (c) A point or box may be connected to any other point or box by any number of resistors. A resistor must connect two distinct items. (d) Each point must be an endpoint of at least three resistors. (e) Each box must be an endpoint for at least one resistor. (f) Three endpoints of resistors must occur within each loop. Before giving the algebraic expression corresponding to each term, let us explain the above rules. The boxes (points) represent ion (electron) coordinates. Boxes enclosed by a loop represent the coordinates of an Ursell function. Resistors represent the screened interactions IV++ , IV+- , and IV-- (Eqs. (3.48)-(3.50)). They replace the v,, , u+- , and u-- of the seminodal expansion. Rule (d) is a carry over from the seminodal rules. Rule (e) is evident: If an ion participated in zero reactions its coordinate could be integrated over. Rule (f) represents the result of introducing the w-potentials. There are no longer any x2 functions having each coordinate participating in only one interaction. For a given term satisfying (a)-(f) there is an algebraic expression. The rules given below assume that one has transformed the w-potentials into coordinate space representations. For a graph there is (i) (ii) (iii) (iv) (v) (vi) I resistors; (vii) I resistors; (viii) 1 resistors.

a factor l/V; a factor pi/i!, where i is the number of points; a factor l/Z!, where Z is the number of ion boxes; an integral s d3R, ... d3R,, d3r, ... d3ri ; a factor xL(RJ1 ,..., RJ2) for a loop enclosing precisely boxes RJ1 ,..., R,2 ; a factor (l/Z!)(-/Jw-_)~ f or a pair of points connected by precisely a factor (l/Z!)(-pw+J a factor

(l/Z!)(p2w+.,)7

f or a point and box connected

by precisely

f or a pair of boxes connected

by precisely

In the nodal expansion we sum over all topologically distinct graphs, where each graph has all of its points and boxes labeled. A permutation of point labels and/or box labels can produce a distinct term. (Alternatively, we can use unlabeled graphs and combinatorial factors.)

NODAL EXPANSION

FOR CLASSICAL PLASMAS

231

H. The Zero of D(K) The factor D(K), defined in Eq. (3.36), appears in FRL and in the three w-potentials used in the nodal expansion. In this section we discussthe possibility for and effect of a zero in D(K) for positive K. Section 3C contains results for nuclear charge z of unity. We found that D(K) is always positive for r lessthan three. For larger values of r, D(K) can be negative over a finite nonzero interval, starting at the origin, of positive K. In general, for any z there might be values of r having negative D(K); we let r, denote the infinum of such values of r. For example, we have found that 3 < r, < 5. For r > I’, there must be a zero of D(K). BecauseD(K) appears in the ring-like sum and in the three w-potentials in the nodal expansion, the zero has two effects. For the former case, we have seenthat the logarithm form [Eq. (3.34)] provides a smooth continuation in r, while the summation form [Eq. (3.44)] diverges for r > I’, . For the w-potentials, it is at best difficult to define sensibletransforms in order to obtain configuration-space representations when r > r, . We have found [12] that the difficulties may be overcome by treating the electrons quantum mechanically. In the physically reasonable approximation that the mass of an ion is much greater than that of an electron, it can be shown that the quantum analog of Y(K) is always numerically less than the classical value. However, numerical inaccuracies in the Fourier transform 2 have precluded final resolution of the problem.

4. CONCLUDING

REMARKS

Expansions such as the ones derived herein may be used to compute equations of state. The pressure P is obtained by a partial differentiation at constant temperature ,8P = -(a/a V) ,8F.

(4.1)

Each term in an expansion gives a contribution to the pressure. An important result of our expansion is an orderly cataloging of the mixedspeciesterms. For systemswith weakly coupled electrons the normal one-species expansion may be used for p. The thermodynamics of the ions may be treated by Monte Carlo methods, such as those of BST. Such computer results often include a pair correlation function for ions in a uniform seaof electrons. This is precisely the function neededto compute FRL . In principle, the higher correlation functions could also be obtained, and one could compute higher mixed-speciesterms. Numerical calculations [12] indicate that the expansions provide a reasonable treatment of the interaction between the two fluids of ions and electrons. The

232

THOMAS 3. BUCKHOLTZ

expansions have been formulated in reasonable generality, so that systems with non-Coulombic interactions could be studied. Of course, the charge neutrality terms must be appropriately reinstated.

ACKNOWLEDGMENTS Hugh Dewitt stimulated my interest in statistical mechanics and was advisor for the thesis research which included material presented herein. Much of this work was done at the Lawrence Livermore Laboratory. The National Science Foundation, through its Graduate Fellowship program, supported my studies at the University of California, Berkeley.

REFERENCES 1. See, e.g., P. A. EGELSTAFF, “An Introduction to the Liquid State,” Academic Press, New York, 1967. 2. T. L. HILL, “Statistical Mechanics,” McGraw-Hill, New York, 1965. 3. H. E. DEWITT, in “Low-Luminosity Stars” (S. S. Kumar, Ed.), Gordon and Breach, New York, 1969. 4. S. G. BRUSH, H. L. SAHLIN, AND E. TELLER, J. Chem Phys. 45 (1966), 2102. Cited as BST. 5. W. B. HUBBARD AND W. L. SLATTERY, Astrophys. J. 168(1967), 131. 6. E. MEERON, J. Chem. Phys. 28 (1958), 630. 7. R. ABE, Progr. Theoret. Phys. (Kyoto) 22 (1959), 213. 8. H. L. FRIEDMAN, “Ionic Solution Theory Based on Cluster Expansion Methods,” Interscience, New York, 1962. Refs. [6-81 cited as M-A-F. 9. F. DEL RIO AND H. E. DEWITT, Phys. Fluids 12 (1969), 791. 10. R. W. ZWANZIG, J. Chem. Phys. 22 (1954), 1420. 11. A. BELLEMANS AND M. DELEENER, in “Advances in Chemical Physics,” Vol. VI (I. Prigogine, Ed.), Interscience, New York, 1964. 12. T. J. BUCKHOLTZ, “Statistical Mechanics of Two-Species Plasmas, with Applications to White Dwarf Stars and Metallic Hydrogen,” Ph.D. dissertation, University of California, Berkeley, 1971; UCRL-51055. 13. G. E. UHLENEIECK AM) G. W. FORD, in “Studies in Statistical Mechanics,” Vol. I (J. de Boer and G. E. Uhlenbeck, Eds.), North-Holland, Amsterdam, 1962. 14. Ref. [13, pp. 141 ff]. 15. Ref. [13, p. 1431. 16. H. E. DEWITT, Notes on the Calculation of Thermodynamic Functions of Plasmas from Quantum Statistical Mechanics, unpublished, p. 13 ff., personal communication. 17. Ref. 113, pp. 162 ff.]. 18. Ref. 116, pp. 2Off.l. 19. H. E. DEWITT, J. Nucl. Energy C 2 (1961), 27. 20. H. E. DEWITT, Phys. Rev. A 140 (1965), 466. 21. W. B. HUBBARD, Astrophys. J. 146 (1966), 858.