Title: The calculation of electric microfield distributions
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Title: The calculation of electric microfield distributions
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Language: English
Creator: Iglesias, Carlos A ( Carlos Alberto ), 1951-
Copyright Date: 1981
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THE CALCULATION OF ELECTRIC
MICROFIELD DISTRIBUTIONS


BY

CARLOS A. IGLESIAS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1981

















This work is dedicated to the author's parents, Carlos A. Iglesias,

Sr., and Lilliam V. Iglesias.

















ACKNOWLEDGMENTS

I would like to thank Professor C.F. Hooper, Jr. for suggesting

this problem and for his guidance and encouragement during the course of

this work. A special thanks is due to Dr. J.W. Dufty for many valuable

discussions. Thanks are also due to Dr. Robert L. Coldwell and Lawrence

A. Woltz for providing guidance in the numerical work as well as for

lending me several excellent computer codes.

The diligence and care with which Ms. Viva Benton typed the

manuscript is very gratefully acknowledged.

Finally, a special debt of gratitude is owed to my parents for the

special understanding they have shown during the long years of this

work.


iii


















TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .....................................................ii

ABSTRACT.... ......... ............. ................. .................... ..vi

CHAPTER

I CLUSTER EXPANSION FOR THE MICROFIELD DISTRIBUTION IN A PLASMA.....1

1. Formalism..... ..... ........ .. ............................. 2
2. Corrections...,.............................................19
3. Results............................ ................. 25
4. Conclusion............................................. 26

II LOW-FREQUENCY MICROFIELD........................................33

1. Line Shape Function......... ................................35
2. Method I: Classicl Plasmas..................................47
3. Evaluation of Q (e) ........................................ 50
4. Method II...... .....................................60
5. Evaluation of <6(e-E )> ....................................62
6. The Classical Limit. ..? .....................................69
7. Comparisons of the Methods..................................71
8. Summary............. .... .. .................. ........... 75

III QUANTUM CORRECTIONS TO THE LOW-FREQUENCY
COMPONENT MICROFIELD DISTRIBUTION................................76

1. Low-Frequency Microfield Distribution........................77
2. Effective Quantum Interaction...............................78
3. Numerical Results and Analysis...............................82

IV INTEGRAL EQUATION METHOD .........................................89

1. Connection with the Chemical Potential.......................89
2. The Two-Body Function g(r;) ................................91
3. The Holtsmark Limit.........................................93
4. HNC Approximation..........................................94
5. Alternate Approximation Schemes.............................100











APPENDICES

A DERIVATION OF EFFECTIVE INTERACTIONS ............................103

B EVALUATION OF T (Z)/T (0) ....................................105
o o

C CALCULATION OF y1 ..............................................107

D CALCULATION OF THE FUNCTIONS t(x) AND p(x).......................108

E PROJECTION OPERATOR TECHNIQUES.................................110

REFERENCES ..................... ............ .........................114

BIOGRAPHICAL SKETCH.................................................118

















Abstract of Thesis Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


THE CALCULATION OF ELECTRIC
MICROFIELD DISTRIBUTIONS

by

Carlos A. Iglesias

August, 1981

Chairman: Charles F. Hooper, Jr.
Major Department: Physics

This work is a study of electric microfield distribution functions

in a plasma. The first work on this problem was done by Holtsmark, who

assumed all particles to be statistically independent. Since then,

various attempts have been made to include correlations between the

particles. This work employs a virial-Debye chain cluster expansion to

reinterpret a collective coordinate approach to the microfield

distributions developed by Hooper.

The electric microfield distributions have been used in the

calculations of the broadening of spectral lines emitted by atoms or

ions in a plasma. The application of these distributions to the line

shape problem is usually accompanied with two assumptions: (1) neglect

of ion motion, and (2) neglect of the ion-electron interactions and

their effect approximated by a shielding of the ion-ion interactions.

Here, the microfield distribution formalism for the line shape function

is retained but without assumptions (1) and (2). With this approach it










is now possible to investigate an aspect of the line broadening problem

which in the past has been frequently neglected in the usual line-shape

theories. The point in question is the ad hoc introduction of the

screened ion fields by the usual line-shape theories which treats

incorrectly the electron width and shift operator. The systematic

introduction of the shielded ion microfield clearly shows that the form

of the electron width and shift operator is more complicated than in the

usual theories.

Because of assumption (2), the ion electric microfield distribution

used in the line shapes have a shielded ion field. The electron

shielding of the ions is usually taken to be the Debye-Hickel result

which applies for a classical picture of the electrons. A study of

quantum corrections to the electron screening of the ions in the

microfield distributions including numerical results is presented.

An integral equation method for evaluating the microfield

distributions is proposed. The method is shown to simplify in the

hypernetted-chain approximation.

















CHAPTER I
CLUSTER EXPANSION FOR THE MICROFIELD
DISTRIBUTION IN A PLASMA

The electric microfield distribution function was first calculated

by Holtsmark,1 who solved it by neglecting the correlations between the

various charged particles producing the electric field. Since then,

various attempts have been made to include these correlations.2 The

purpose of this chapter is to reinterpret a collective coordinate

approach to the microfield distribution developed by Hooper.5 We will

show that the method employed in Ref.5, hereafter referred to as I, is

equivalent to a combined virial-Debye expansion similar to that

developed by Mayer.9

The system that we deal with consists of N charge particles

immersed in a uniform neutralizing background. In addition, when

treating the problem of the electric field distribution at a charged

point, a "zeroth" particle must be included. The N+1 particles interact

through the Coulomb potential. The total system is assumed to be in

thermal equilibrium and macroscopically neutral.

The reinterpretation will start by expressing the high-frequency

component electric microfield distribution in a cluster expansion

similar to the classical expansions of Ursell and Mayer.0 Then, a

split of the central interactions into a long-ranged and a short-ranged

contribution is introduced as in I. A central interaction involves the

zeroth particle and one of the N charged particles; a noncentral

interaction involves any pair from the N charged particles. The










long-ranged central and noncentral interactions we treat in a Debye-

chain expansion,11 since these interactions of infinite range display

collective effects. The short-ranged central and the screened inter-
9
actions from the Debye-chain sums we treat in a virial expansion. We

stress that although similar in technique, the cluster expansion pre-

sented here is different from previous developmentss.2,35 Corrections

to I which result from retaining additional correlations are presented.



I.1 Formalism
+ +
Define Q(e) as the probability of finding an electric field e, at a

singly charged point located at r due to N charged particles moving in

a uniform neutralizing background and contained in a volume n. Then, if

Z represents the configurational partition function of the N+1 particle

system, we may write


+ -1 + + + -BV 4 +
Q(E) = Z f'** dr drl *.drN e 6(E E) (1.1.1)


+ .th -1
where r. represents the coordinate of the j particle, =(kT) V the
I +
potential energy of the system, and E is the electric field at r due to

the N charged particles in a given coordinate configuration.

The potential energy of the system, V, is expressed as



V = e/r.. + V (1.1.2)
0=i

where VB represents the contributions to the potential energy due to the

neutralizing background.










An expression for V in terms of a Fourier expansion results in


S2kO
V 7Te 2
kO


++
-ikr 2
e i/k2


(1.1.3)


O=i

where the exclusion of the k=0 term in Eq.(1.1.3) accounts for the

neutralizing background.2

Assuming that our system is isotropic we may rewrite2 Eq.(1.1.1) as


P(s)= 27-1e fc dXXT(k4sin(ek)


(1.1.4)


where P(s) is related to Q(-) by the relation,
where P( ) is related to Q(e) by the relation,


Z2
41rQ(E)) ds = P(E)dE;


(1.1.5)


T(k) is defined by,


T(X) F Z(k)/Z,


-I- -V(2l)
Z(X) =- f-e-f dro 0 .dr0N e


V(X) I"N vij() M
o=i

++

iN 0 X 0 )Vij)
o=ij 0


(1.1.6)


(1.1.7)


(1.1.8)


where v = e2/r and V is the gradient with respect to the zeroth

particle.

The function Z(A) has the form of a configurational partition

function with the "potential energy" of the system V(X), defined in

Eq.(1.1.8). We proceed to calculate numerator and denominator of










Eq.(1.1.6) using the classical cluster expansions for configurational

partition functions which were developed by Ursell and Mayer.10

Again noting that the numerator Z(A) has the form of a

configurational partition fu-.tion, we write it in the form of a

"Helmholtz free energy," F(':


z(M) = e


(1.1.9)


F(A) given by,


-F(P) = Q2 A(P,Z).


(1.1.10)


Now the quantity A


can be expressed in terms of a cluster expansion,10


n n
A(p,a) = ZN f..f R(n,.) iT dr..
n=2 j


(1.1.11)


Here, R(n,X) is the sum of all products of f-Mayer functions in which

every particle in n is independently connected to every other particle

in n. The f-Mayer functions are defined as


-fv. .(::
f j(2) = (e 1j 1)


(1.1.12)


where


n = set of n = n + n particles
o i



n! = n !n !
o i

n n
n o i .-1
p =p p ; p = p N/I
o o











n
+
f.***f dr. = integration over set
j J
of n particles (1.1.13)



no and ni are the number of zeroth particles, here equal to 0 or 1, and

N charged particles in the cluster R(n,k), respectively.

The clusters in the expansion of A(p,k) are of two types:

(1) Clusters which do not contain the zeroth particle, no=0.

(2) Clusters which do contain the zeroth particle, no=l.

If A1(p) and A2(p,,) denote the contributions from all type (1) and (2)

clusters, respectively, then



Z(a) = exp{Al1(p) + QA2(p,.)}. (1.1.14)



A similar procedure can be applied to Z = Z(9=0), with the result



Z(a=0) = exp{Al1(p) + QA2(p,0)}. (1.1.15)



The term A (p) is independent of ; in fact, exp(2Al(p)) is the

configurational partition function for the plasma without the zeroth

particle.

Substituting Eq.(1.1.14) and Eq.(1.1.15) in Eq.(1.1.6) allows us to

write



T(A) = exp{n(A2(p,,) A2(p,0)} (1.1.16)



That is, all clusters not involving the zeroth particle cancel exactly

in Eq.(1.1.16).










A graphical representationI0 of some of the terms appearing in A2

is shown in Fig. (1.1); a black vertex represents one of the N charged

particles, a white vertex the zeroth particle. Each f-Mayer function is

represented by a heavy-solid line connecting two vertices.

As in I, we conjecture that quantitative features of the microfield

distribution will be more sensitive to central than to noncentral

interactions. A central interaction involves the zeroth particle and

one of the N charged particles; a noncentral interaction involves any

pair of N charged particles. Based on this conjecture, the details of

the central interactions are treated with greater care. Therefore, we

split the central interactions into long- and short-range contribu-

tions. The long-range central and all of the noncentral coulomb

interactions we treat in a Debye-chainII expansion. After the long-

range contributions are renormalized, all the remaining short-range

"interactions" are treated by means of a virial expansion. From an

examination of the formalism it is clear that the two expansions are not

independent, but involve a hybrid (virial/Debye) cluster expansion. It

must be emphasized that the conjecture discussed above is based on a

plausibility argument, which is justified by results.5 To carry out

this procedure, we first set



v = u + w (1.1.17)
oj oJ oj
where

w = (e /r)e o (1.1.18)
oj


In Eq.(1.1.18) a is an arbitrary, real, positive parameter which will be

independently determined, and X is the Debye length,

































0
4-1


S)
0 ^





0 0
3) )

U0 >

^ c






Sa
Cd





N
o 4 r
.a 0



4- u
o (

0a



,C 0
( >
CO
1-4 4.)







.t 0


1H 3
U n


C
i-'-

u
0
0
C


















-H
0


-'
1--



o



td
C













SI-
0

m1

Ci,
4(U 4-










0-


+~


NcC)


+


AI









2 -1/2
X = (47e2pB) -2 (1.1.19)



Substitute Eq.(1.1.17) into the expression for the f-Mayer

functions, Eq.(1.1.12). This yields a result similar to one used by

Mayer but with the difference that in the present paper only central

interactions are split



(e -1) i,j # 0

f j(a) =
fijM

xj() [+ 1 + xj()] [- u (A)]n/n! =0 (1.1.20)
n=l


where
-Bw ()
Xij() = (e o3 1). (1.1.21)



With the aid of Eq.(1.1.20) we may further separate the products

in R(p,k) into sums of products involving the f-function, X-function,

and [-Bu]n/n! functions represented by heavy-solid lines, light-solid

lines, and n-dashed lines, respectively. Clearly, there can be at most

one f or X bond directly connecting two vertices. The result of

splitting the central interactions is shown graphically for some 2 and 3

particle clusters in Fig.(1.2a). The two (-Bu)-bonds with the triple

dot in between represent the sum of graphs with all possible number

of (-Bu)-bonds as shown in Fig. (1.2b).

In order to perform a Debye-chain expansion on the long-range

central and noncentral interactions we expand the noncentral f-Mayer

functions in powers of (-Bv). With the decomposition of the f-functions

into powers of (-$v) functions we can sum simple chains of










(-Bu)- and (-Bv)-bonds as shown in Fig. (1.3). Two types of chains are

possible: the first has the zeroth particle and one perturbing ion for

endpoints, while the second has two ions for endpoints. It is

understood that the two vertices at the endpoints of Fig. (1.3) are in

general part of a more complicated graph. Hence, we are summing all

graphs which are the same except for the one sum of interactions

displayed. The intermediate ions in Fig. (1.3) do not interact with any

particles except as explicitly shown in the figure. The final form for

A2 is an infinite series of integrals involving products of the

functions, X, us, and vS; us and vs are defined graphically in Fig.

(1.3) and evaluated in Appendix A. There is the restriction that no

simple chains in the effective interactions us and vs appear in A2

because such a chain is in effect a simple chain in u and v interactions

which have been already included in the summations. The new cluster

expansion for A2 is given by


0 n
2 A2(p,a) = n T (A) + 1 2 h (2), (1.1.22)
n=1


where n T (A) is the contribution from the ring graphs presented graph-

ically in Fig.(1.4a), and hn is the set of all n+1 particle clusters,

excluding ring graphs, involving products of the functions X, us, and vs

as described above and presented graphically in Fig.(1.4b) and (1.4c)

for n=l and 2.

In Fig.(1.4) we have separated the graphs for n=l and 2 into the

subsets (bl,b2) and (cl,c2). The separation is employed since it can be

shown that only the graphs shown in (bl), (cl) are included in I.

There, only the first term in the Gram Charlierl2 expansion series for












































r-4



a)



4-4

CN Q
a)





I-o
(w a)



Q4 I


4- -4



Q) 0~
col
P.
-14
u cn











V) 0
-4 '-




4.1 CU




cY
F, -



(-'





U2 0


0









CU


co
a)














-4J
-4
















rl
0











-H




co






LW 10
(12



0) a)

O u,



CU a)
U, i-
c -






CUl
4. .








++


\* \ /






+-t
+ t _
- -F t- A
(.\ I



-I-^





\*

oi
a ~---+-
































































*1








co


a1


4-4
hI












4-4
aC)
a)

bO
UI
01
14-1
'4-1





rr)
















-h-





-F


+

I

6
1 i





III
If


$-



0-----O
?-1


/


-'





II




In
U)































bo
SC C 4






co o p e
- J 0 0


w >
o 0 4j w
3 0 4 I 0 .



4* 0- 0 > 0 a




S14a C C n
CM Cu C 3 (6
CMJ 1-1 Cn Ca. S-i Om










SI U
* b c C 4 0 *



I0 0 0
0-l P1 0
0 4 H 0 0 0
0 0 C I
4' $ 0 4 O O 0




0 0O u I
4i .0 0 I





0 U 0 H U O
4* 1 0 0 -

o 1o 0 x 0
-1 4 0 04-






0 U3 -i co 4 Ci co

00 cM 0 3 4-
( 0.. 0 i B


o cu f 0 bo .0
u0 M Cu 4 4-4 CU



9- 0 r- 4J 0
4-i bo 0 0 Cu > (

O ( CU U Cu 4-J C14
U x Q) 0U

0 < 0 '
H[-a 4 C I -





*-4

*-

1-
cZI























+
$-





/
/
/
@1
/
\
\


+ _

\ /
\ /



d
e--t--


\
'I


(N~yirF

+/


t


a



U

2Jh~LIL


CN
+




Cl
+


-o


CS.
'Il


ii I
.~nn,










the Jacobian of the collective coordinate transformation is retained.

We will show that the graphs (b2) and (c2) are neglected by such an

approximation.

Splitting up the potential appearing in Z(Z=0) in the same manner

as previously described in treating Z(Z), we are able to carry out a

similar expansion program with the result


o n
A A2(p,0) = an To(0) + P- h(O). (1.1.23)
n=l


The graphs representing the terms in Eq.(1.1.23) are topologically

equivalent to those in Eq.(1.1.22) but with X set equal to zero.

Combining Eqs.(1.1.16), (1.1.22). and (1.1.23) gives the following

result for T(k):


n
T(k) = [T (k)/T (0)] exp { i p- [h (P) h (0)]} (1.1.24)
o o n= n n n



Now consider the individual terms appearing in Eq.(1.1.24). In

Appendix B we show that the first term, T (A)/T (o), can be written as
o o



T (k)/T (0) = exp{-yL2}. (1.1.25)
0 0



In Eq.(1.1.25),



3 2 2 2
L = Zl y = aa /4(a + 1) a = r /X, and s = e/r ; (1.1.26)


r is the ion sphere radius defined by the expression







4wr 3
r p = (1.1.27)



Next, we consider the factors resulting from terms in the series

exponent. For n=l, and considering only terms shown graphically in

Fig.(1.4bl), we write

(1) (1) 1)- h ()
II (Z) p{ h1 (1)


= p d ri0 {XI(LO1)Q X(0)Q1(0)}



=3 fdx x2 eF(x)(sin[LG(x) ) s(x) sin[Lq(x)] _
o LG(x) Lq(x)
(1.1.28)



where the angular integration have been done. The functions in the

second equality are defined by



Q1 () = exp-u (rl0,)} (1.1.29)

4->
usr,) = [1 u(r0). (1.1.30)



The functions in the third equality are defined in the next section,

Eqs.(1.2.4), (1.2.5), (1.2.7) and (1.2.8).

For the second term in the series, n=2, we use the graphs in Fig.

(1.4cl) to write


(1) p (1) (1)
2 () [h (9) h (0)]

2
= f dr10dr20 [Xl (X2 ()Q (2)Q2(") -








s
-Bv
12
XI(0)x2(0)QI(0)Q2(0)](e 1). (1.1.31)



Thus, the contributions to T(J) from Figs.(1.4a), (1.4bl), and (1.4cl)

are given by



(1) 2 (1) (1)
T (1() = exp[-yL2 + I () + 1 ()]. (1.1.32)
1 2


The results in Eqs.(1.1.25-1.1.32) are identical to those in I

for T(A) as given in Eqs.(25-35) of I.



1.2 Corrections

In Section I.1 we expressed the microfield distribution function in

terms of a cluster expansion where the long-range interactions are

treated in a Debye-chain expansion and the short-range interactions in a

virial expansion. The two are not independent but involve a hybrid

virial Debye-chain cluster expansion with the long-range collective

effects of the Debye-chains modifying the short-range virial expansion.

As mentioned earlier, the graphs in Figs.(1.4b2 and 1.4c2) are not

included in the results of Eq.(1.1.32). The neglected terms can be

interpreted as correlations between the collective coordinates

introduced in I. In this section we will evaluate these contributions

to T(A) for n=1 and 2 in Eq.(1.1.24).

The corrections to Ii (), shown graphically in Fig.(1.4b2), is

given by









I () = p[h () h (0)]
1 1 1


S-Bu dr ( e )
=p f dlI {[e


+ +B( )
+ Bu (rI,)


-Bus(r )
- 1] [e


-2 ( ())2
2(Wu(rl)]


+ uS (rl) 1]}.


(1.2.1)


Performing the angular integration we get,



(2) 3 dx2 es(x)( sinlLq(x)] L2q2 (x)
0 oLq(x) 6
2)() = 3 ;o dx x2eS(X)"( -Lq(x) 6 \}

(1.2.2)



The functions that appear in the integrand are defined as follows:


x = r/r


2
s(x) = a (
1-a


2
a i-ax
3x


(1.2.3)


(1.2.4)


- e x)


2
a
q(x) -
1-a2


Combining Eqs.(1.1.28) and (1.2.2) we find that


S3*d 2 F(x)r sin[LG(x)]
I1(A) = 3dxx{e ( LG(x) -


2 2
+ L q (x)
6


2
l2 1 (x a 2 -ax
F(x) 1 ( )(-a
1-a


1
G(x) = -
1-a


1 (e-aax -ax) +a aax 2 -ax
Se -e x + ae -ae
x


1 -aax -ax) a -ax
{- (e -e J e
xx
x


-aax)I
-e .


(1.2.5)


with


(1.2.6)


- e -ax


(1.2.7)


(1.2.8)


S(Bu (rl' ))2









Before evaluating the contributions from the graphs in Fig.(1.4c)

we note that the sum of three particle clusters is a small correction5

to T(k). Hence, we only consider graphs with the lowest nonvanishing

number of (-Bv )-bonds connecting particles 1 and 2. With this

simplification the graphs in Fig.(1.4c) will be of two types: graphs
with one (-Ovs)-bond, and graphs with two (-Bvs)-bonds.
The contribution to T(A) from graphs with one (-Ovs)-bond is given

by 2 s s
(1) -Bp + + s -Bu (r ,Z) -Bu (r ,.)
I ( -)= fdr dr v (r ){x ( )X (Z)e 1 e 2
2 2 1 2 12 1 2

-Bus (r) -Bus (r2)
X1(0)X2(0) e e }, (1.2.9)


(2,1) 2 s
m21 drldr2v (rl)
2 2 1r2v 12

s + S +
-u (r ) -u (r 2)
[{(e 1 +us (rl,)-I)(e 2 + rus (r2,)-1)

-Bus (r) -Bu (r2)
(e + BuS(rl) 1)(e +Bu (r2)-1)}


-Bu (rl s + -Bu (r 2)
+ 2{(e +Bus(rl,) l)e X2()


S-Bus(rl ) -Bus (r2
(e + Bus(rl) 1) e X2(0)}] (1.2.10)


The quantity I1 (() corresponds to the graphs in Fig.(1.4cl)
(2,1)
and 1 2' (A) to the graphs in the first bracket of Fig.(1.4c2).

The integrands in Eqs.(1.2.9) and (1.2.10) are a product of
functions of (rl,X) and (r2,9) with the exception of the r12 coupling
term in vS(r12). In order to uncouple the rl,r2 dependence, we expand

vs(r12) in spherical harmonics.13









vs(r12) = I (2k+1)vk(rl,r2)Pk(cos912)
k=0



2 1x
vk(rlr2) Kk+l/2(axl)Ik+/2(ax2)/(xlx2) 1/2


(1.2.11)


(1.2.12)


and x =r /ro, x1>x2, and k=0,1,2,.... This method allows Eqs.(1.2.9)

and (1.2.10) to be reduced to a tractable double integral where the

angular integration are readily performed to yield


I()() + I2')() = 3a2 (-1)k(2k+l) f dxx2x2
Sk=0 .


Ik+ 1/(ax2)


xx l3/2
x 2 dx1x xKk k


(1) e
(k (x1x2) = e
i (


+1/2(axl)(iik x1,x2) + ik


) (x2) -Bw(x)
e {[e jk(LG(xl))-jk(Lq(x))]


-Bs(x2)
x [e jk(LG(x2))


- jk(Lq(x2))] 6k,oX(0)X2(0)1


'(2 1) s(x2)
i2) (x1,x2) = [e (2e
k l,2


-Bw(x2)


s(x1)
Jk(Lq(Xl))-"ko]


- 6k,o(l+s(x2))][e


-6k,o (e


s(x2) -w(x 2)
+ s(xl)e [2e (jo(LG(x2))-l) (jo(Lq(x2)) 1)1]}


where


(1.2.13)


(1.2.14)


(2'1 x1,x2)}


s(x I


- jk(Lq(x2 )))


(x1)


-1)[es(x2 (2e -l1)-(l+s(x2))]









L s(x2) -w(x2 )
Sk 1 {q(xl)e [2e j1(LG(x2)) l(Lq(x2))
3 k, 1 2 1 2


S( )1
+ q9x 2)[e ii (Lq(x 1 Z(x 1 )/311


(1.2.15)


The functions I and K refer to modified Bessel functions of the first

and third kind, respectively, while jk specifies a spherical Bessel

function of order k.14

The second bracket in Fig.(1.4c2) shows the graphs with a minimum

of two (-Bv )-bonds. Their approximate contribution to T(A) is given

by
(2,2)( = 22 + + s )2
12 2 f drldr2 vS(r12


2
x [ -- {u (rl,.)u (r2,z) u (rl)u (r2)}


(1.2.16)


The first term in brackets in Eq.(1.2.15) may be evaluated by

introducing the Fourier transforms as shown in Appendix C. Then,


2 4
p 2 + s 2suSs+ s
4 fdr1 dr2drv (rl2) 2 u (rl ,)u (r2,) us(rl)u (r2)}



= -Y1L2


24 2
-a a (2+l)n r 3 (a -1)(2a+l) (1
S(a 3(a+2) } +. (1.2.17)
1 2) 3 2+a 3(a+2)
12(a -1)


The term in the second bracket in Eq.(1.2.16) may be reduced to a one
dimensional integral by first integrating over r
dimensional integral by first integrating over rl,


_Iu (riOe X2 (P.)-usi)- u r 1 )e X2(0)11.O)/I








S +
p2 3 + -Bu (r 2,)
2 f drdrl2 v (rl2) {us(ll ,)e


X2(0) = ( +) dx2{[t(x2) + iLp(x2)cos2]


s(x2) -Bw(x2) iLG(x2)cos62
x e (e e


s(x2)
- t(x2)e X2(0)} = 3 fo dxx


- es(x)( sin[Lq(x)] )
Lq(x)


iLq(x2 )cos 2
- e )


2 F(x) sinlLG(x) -
{t(x)[e FX( LG(x) 1


- Lp(x)(eF()j (LG(x)) es (Lq(x))
1, (x)Jx) l(Lq(x)))j


(1.2.18)


The functions t(x) and p(x) are evaluated in Appendix D,


5
t(x)


2
-2 )[eaxE1(3ax) + eax{in3 E1(ax)}
a -1


aax -aax
e-- E ((2+a)ax) e {n( 2-a ) E1((2-a)ax)}]
a a 2-a(1.2.19)
(1.2.19)


2
S-- )(eaxE(3ax) e -axIn3-E1(ax)}
a -1


4
p(x) = x [ (


- eaaxE1((2+a)ax)


+ e-aaxln ( 2+ ) E((2-a)ax)})


3t(x)
2
a


(1.2.20)


- us(rl)e


-Bus (r2)










Eqs.(1.2.19) and (1.2.20) are only valid for values of a less than 2.

The function El is the exponential integral


-z
E (y) f dz e for y > 0. (1.2.21)
1 y z


Combining Eqs.(1.2.9), (1.2.10) and (1.2.16) we have



S(1) (2,1) (2,2) (1.2.22)
12() = I (2) + I2' ( + I 2() (1.2.22)



Thus,

T(A) = exp{-y'L2 + 1(k) + 12(0)} (1.2.23)



with Y'=+Y1 and Il(2) is given by Eq.(1.2.7) and 12(a) by Eq.(1.2.20).

The result in Eq.(1.2.22) is used in Eq.(1.1.4) to calculate P(c) at a

singly charged point in the next section.



1.3 Results

The first step in using and evaluating the present theory is to

determine the parameter a. In principle the expression for T(A) in

Eq.(1.1.24) is independent of the choice of a. However, in a practical

calculation the infinite series appearing in the exponent of Eq.(1.1.24)

is terminated and T(Z) is then no longer independent of the value

of a. The procedure for selecting a is discussed in detail in I.

Briefly, it involves finding a distinct and extended range of a values

over which the T(A) curve, hence the P(e) curve, remains stationary. In

the stability region, or a-plateau, the second term in the series is

small and the series rapidly converges.










A justification for the a selection procedure follows. For a given

a, a particular choice of a determines how much of the central

interaction is treated by a Debye-chain or by a virial-expansion. The

validity and speed of convergence of such expansions depend on the

detailed nature of the interactions treated. A best choice of a will be

that which splits the central interactions so as to optimize both

expansions, thus giving a rapid convergence of Eq.(1.1.24). Then, a

small variation of a about the best value should not significantly

affect the results provided sufficient terms are retained in the

infinite series.

Using this procedure we calculate P(e) curves and compare them with

the results of I. Figures (1.5) and (1.6) show P(e) curves for a = 1.73

and a = 2.45, respectively. From the plots it can be seen that the

corrections slightly lower and shift the peaks of the P(E) curves to

higher e values. Although not shown here, the contribution from these

corrections become smaller with decreasing values of a. In Figure (1.5)

we compare our curves with Monte Carlo results15 and in Figure (1.6)

with Molecular Dynamic results.16 The agreement for a = 1.73 is quite

good. In this case, there exists a well defined a-plateau. For a =

2.45 the a-plateau reduces to the point where the existence of a plateau

is in question; further, our results are not in agreement with the

computer experiments. The lack of agreement, together with our earlier

discussion of the a selection procedure would indicate the need to

retain more terms in the series of Eq.(1.1.24) for a > 2.0.



1.4 Conclusion

We have shown that the previously developed collective coordinate

approach to microfield distributions is equivalent to a hybrid










virial/Debye expansion. The two expansions are not independent; the

long-range collective effects of the Debye-chains modify the short-range

virial expansion. Hence, the range of validity of this formalism, when

applied to systems where particles interact through long-range

potentials, exceeds that of theories that use either expansion

separately. The hybrid virial/Debye expansion formalism can easily be

extended to the low-frequency microfield distributions in a plasma

containing multiply charged ions.7

Numerical calculations of P(e) curves including some corrections

neglected in I are presented in graphical form. These corrections cor-

respond to correlations between the collective coordinates. Their con-

tribution can be shown to be equivalent to keeping the next term in the

Gram Charlier expansion of the Jacobian of the transformation to

collective coordinates. Even though the effect of the corrections is

small, for a = 2.45 they improve agreement with the results of computer

experiment.

In Eq.(1.1.22) we have ordered the terms in the sum by number of

particles in a cluster. We now propose a different ordering based on

the splitting of the central interactions. This split separates the

central interaction into a strong short-ranged and a weak long-range

contribution. The long-range weakly coupled part requires a Debye-chain

expansion since these infinite range interactions give rise to collect-

ive effects. The strongly coupled part given by the x-bonds and the

effective interactions resulting from the summation of Debye-chains we

treated in a virial expansion. However, instead we may identify a

cluster by the number of X-bonds it contains, and by the complexity of

the weakly coupled contributions. For example, the lowest order terms










in complexity are the ring graphs,11 and the simple chains in Fig.

(1.3). Systematic correction procedures in Refs. 17-20 discuss at

length the summation of graphs of higher order in complexity.

Therefore, we propose to order the terms in Eq.(1.1.24) by the two

parameters: X-bonds and complexity of the weakly coupled interactions.

In the preceding discussion we have implicitly assumed that the

noncentral interactions constitute a weakly coupled system. This, of

course, is only true for value of a << 2.0. However, our original

conjecture is that the microfield distribution is not very sensitive to

the noncentral interactions. In this sense the noncentral interactions

can be weakly coupled to the zeroth particle.

From Fig. (1.4) we see that in this new ordering scheme the graphs

in (b2) and (c2) are of zeroth and first order in the strong coupling

parameter, the number of X-bonds. These graphs are higher order in

complexity of the weakly coupled contributions than the graphs in (a),

(bl), and (cl). We see that the graphs in (b2) and (c2) of zeroth and

first order in X-bonds are to be grouped with the ring graphs in (a) and

the graphs in (bl), respectively.

From our results in Sect. 1.3, we believe that by including terms

containing three X-bonds and lowest order in complexity in the Debye-

chains we can extend the validity of our results to values of a > 2.0.


































Iw

1 Q) U
0 >





4-4






0 t-4 .
CN4 0 wn





w -


0)

L4- 0 0/
0 41i r
0 to
0 U)
U) Q)


-H 4-4


04- 0>
o4
~0










Lr) U)
o c 1-
U) l Q) 4






o H




Ue
-4t 0


















0
J

C
L -LL
~0
2 Z
0 z



II


~o d c'j
0 U 0 0 C
a:

































4-4
.1-i 0
4-4



( a)




4.-4 U)

0
0 0



,4-
E-4
Cd










o 44~
44 0
U)P1












41-



U4 0 0
-Sw









0 0
0 t 0




4-4
~ al) t
ro a









Cc p (1


0 > r

Q p
-3 U ,-







'010
'-4C



-'-4










0




z


0 / (
>-V




/ O
4,,


IE

O --
oc





0



IN









o (D




! I I o
/o















- d d d d
-W 6 L6 6 d
0.

















CHAPTER II
LOW-FREQUENCY MICROFIELD

The spectral lines of radiating atoms or ions in a plasma are

broadened due to the perturbation of the radiator by both the ions and

the electrons of the plasma21 (assuming for simplicity a two component

plasma). Studies of the plasma broadened spectral lines have been quite

successful in determining temperatures and densities for laboratory and

22-25
astronomical plasmas.22-25 In the past a number of fruitful results

have been obtained in-this area with the application of new methods

developed in many-body physics and nonequilibrium statistical mechanics.

Among these methods are propagator-operator equations,26 Green's func-

tion techniques,27 cluster expansions,28 and formal kinetic theory of

29
time correlation functions.29 However, the problem of quantitatively

predicting the line shape is difficult in general and is often simpli-

fied by reducing the calculation to one involving independent treatments

of ion, electron, and radiator subsystems. Most past work has

accomplished this under the following assumptions: (1) ions may be

considered as static; (2) ion-electron interactions may be neglected and

their effect partially accounted for by shielding the ion-ion inter-

action in an approximate fashion; and (3) electron-radiator interactions

may be treated by perturbation theory. The first two assumptions allow

the introduction of the low-frequency3 microfield distribution which

yields the Stark broadening due to the average static electron-shielded

ion field. This is not a weak interaction effect and may not be










obtained by finite perturbation theory. The third assumption allows the

electron "width and shift" operator26 to be treated by perturbation

theory. The ions and electrons are thus treated very differently in

their interactions with the radiator, since the strong static effect and

weak dynamic effect are two different approximations. In order for this

description to be useful, the part of the line being described must be

such that the radiation occurs in a time short compared to the time

required for an ion to move significantly. On the other hand, the time

of radiation must be long compared to corresponding times for electron

motion since in finite perturbation theory the electron static effects

are not treated properly. Fortunately, due to the large ratio of ion to

electron mass, these conditions are often met over an interesting

portion of the line.

The low-frequency microfield distributions are calculated by con-

sidering a gas of ions interacting through an effective screened

potential. The ion electric fields are also assumed to be screened.

The shielding in this calculation is a way to include electron screening

effects, since the ion electric fields vary slowly overtimes on the

order of the electron relaxation times. It is necessary for the conven-

tional line-shape theories to treat the ions through low-frequency

instead of high-frequency microfields, which contain no electron

screening, in order to obtain agreement with experiments.

The purpose of this chapter is to show that the ad hoc introduction

of the low-frequency microfields by the conventional theories leads to

an incorrect expression of the electron width and shift operator. We

demonstrate this point by two similar, but different approaches to the

line-shape problem. The first approach takes advantage of the cluster










expansions developed in Chapter I in order to systematically obtain the

electron screening of the ions in the microfield distribution. The

second approach introduces an arbitrary static-ion field which must be

defined at some point in the development. The ambiguity in the choice of

the ion field is troublesome, but it allows us a freedom which is

helpful in studying the dependence of the electron width and shift

operator on the choice of microfield distribution. Both methods

introduce the microfield distribution into the line-shape function

without simultaneously introducing assumptions (1) and (2) above. It is

then possible to neglect ion motion effects at the end of the

development and compare with the conventional theory results.

We will restrict our discussion to second-order theories which

assume that the perturber-radiator interactions are weak and treat the

width and shift operator to second order in these interactions. The

second order theories are useful in laser produced plasma experiments25

since the observed part of the line profiles from high-Z radiators is in

the near wings, well inside the electron plasma frequency. Another

reason for only discussing second-order theories is that the diffi-

culties concerning the introduction of low-frequency microfields into

the line-shape formalism are present in the second-order theories.

Therefore, consideration of strong collisions, or so-called unified

theories, is not necessary for the purpose in mind here.



11.1 Line Shape Function

The radiation spectrum of a quantum system is determined

experimentally by measurement of the power radiated per unit time per

unit frequency interval, averaged over the polarization and the









direction of radiation.30,31 Since this quantity has been derived many

times in a variety of ways, its derivation will not be included here.

The power radiated when a particle makes a spontaneous dipole transition

from one quantum state to another is,31



P(w) = (4w4/3c3) 1 I26(w-w. )W (2.1.1)
i,ff


In the above, represents the matrix elements for the radiator.

The frequency, hw f, is the difference between the initial and final

energies, Ei and Ef, of the entire radiator-plasma system; and Wi is the

probability of occurence of the initial state in an ensemble. Eq.(2.1.1)

may be manipulated to read as



P(M) = (4w2/3c3)I(w) (2.1.2)



where I(M) is the line shape function defined by



I(W) = -1Re fo dt eiWtTr {p pd.d(t)} (2.1.3)
o Rp Rp


with (in units such that h=l)



a(t) = eiHt A e-iHt (2.1.4)



PRp = i>W

Here, PRp is the equilibrium density matrix, H the total Hamiltonian,

and TrRp the trace over states of the radiator and plasma system. It










has been assumed from the outset that the momentum transferred to the

radiator during the time of radiation is negligible, and therefore, that

Doppler broadening due to the motion of the radiator is independent of

the Stark broadening caused by the plasma.32 It can be shown, under

such conditions that the final line shape may be obtained by convoluting

I(w) with the Doppler profile.33 In this situation, the operators

appearing in Eq.(2.1.3) are dependent only upon the plasma degrees of

freedom and the internal degrees of freedom of the radiator; there is no

dependence on the center-of-mass variables of the radiator.

In a coordinate system with the origin at the radiator nucleus, the

Hamiltonian may be written as



H = H + H + VI (2.1.6a)
R p I
where

HR = Hamiltonian for the free radiator (no center-of-mass

motion) (2.1.6b)

H = Hamiltonian for the plasma plus monopole term from

the radiator-plasma interaction (2.1.6c)



VI = radiator-perturber interaction (excluding monopole

term already in H ) (2.1.6d)



We have included the monopole part of the radiator-plasma interaction in

H since in this frame it depends only on the plasma coordinates. The

advantage in rearranging the Hamiltonian in this way is that the polar-

ization of the plasma by the monopole part of the radiator-plasma inter-

action may be accounted for, while keeping the interaction VI short










ranged. It should be noted that if Eq.(2.1.3) were to depend on the

center-of-mass variables of the radiator, such a convenient separation

would not be possible.

We now introduce two approximations present in most line broadening

calculations. The radiator-perturber interaction VI is replaced by a

dipole interaction




VI VRe + VRi (2.1.7a)

with
++
VRe = dE (2.1.7b)
e


++
VRi = d.Ei (2.1.7c)



+ +
and Ee and Ei are the electron and ion electric fields, respectively.

This replacement of the actual charge configuration by a dipole remains

valid only if the contribution from interactions closer than a distance

r where the dipole approximation fails, are neglible. These condi-

tions are satisfied for plasmas with densities such that the average

interparticle distance is much greater than the size of the radiator.34

In the second approximation, we assume that the operator pRp can be

factored in the manner




PRp = PRPp. (2.1.8)



This approximation neglects initial correlations of radiator and plasma

due to VI only, not to the monopole contribution included in H .









The factorization of Rp is a procedure which need not appear in

the formal development of modern line shape broadening theories based on

kinetic theory.29,35'36 However, the factorization in these theories is

sometimes made for computational convenience.35

The factorization in Eq.(2.1.8) allows us to express I(w) in the

form



1(W) = 1 Re f dt e R. (2.1.9)
0 R



Here, Re stands for the real part and the brackets < >R indicate an

average over only the radiator subsystem. The operator D(t) is the time

developed radiator dipole operator, averaged over perturbers


+ +
D(t) = (2.1.10a)



<(...)> = Tr p (***) (2.1.10b)

The most general approaches to the theory of spectral line shapes

have made use of either Green's functon27 or Liouville operator26

techniques. For the purpose here, the latter seems to be the most eco-

nomical and transparent way to proceed.

The Liouville operator is defined by its action on an arbritary

quantum mechanical operator.



Ly = i[H,y]; (2.1.11)



that is, the Liouville operator gives the commutator with the Hamilton-

ian of the quantity on which it operates. Since






+
idd +
dt = [d,H],
dt



we have formally


+ Lt +
d(t) = e d.


(2.1.12)








(2.1.13)


From the Hamiltonian given in Eq.(2.1.6), we have for the Liouville

operator


L = LO + LI,


(2.1.14a)


where


(2.1.14b)


LO = LR + Lp



LRy = ilHR,y],



Lpy = i[Hp,y],



LIY = i[VI,y].


(2.1.14c)


(2.1.14d)


(2.1.14e)


The equation for the line shape can now be written in the form


I(w) = R,


(2.1.15a)


with










D(w) = -1 Re f dt et
o p


= -_ Im<(m-iL) > d (2.1.15b)
P
where the transform has been performed and Im stands for the imaginary

part.

At this point the quasi-static ion approximation is introduced in

conjunction with some approximation to account partially for the screen-

ing effect of electron-ion interactions.31'32 We first approximate the

Hamiltonian for the system,



H HR + He + Hi + d (E + E ), (2.1.16a)



where



H = T + V + D (2.1.16b)
e e ee e



H = T + V + D (2.1.16c)
i i ii i*



In Eq.(2.1.16), Te and Ti are the kinetic energy of the electrons and

ions, respectively. The Vee and Vii are the electron-electron and ion-

ion interactions, respectively; e and Ds the monopole radiator-electron
e i
and -ion interactions, respectively. The superscript s indicates that

we consider the electron-ion interaction, Vei, to produce some shielding

effect on the ions. Henceforth, the explicit electron-ion interaction

is dropped from the Hamiltonian. The shielding is frequently taken to
37
be the Debye-Hickel result. A similar identification follows for the
electron- and shielded-ion-electric fields, E and The subject of
electron- and shielded-ion-electric fields, E and E i. The subject of
e i










the electron screening of the ion-ion interaction will be treated in

greater detail later in this chapter and in the next.

Next, we assume that because of their large mass, the ion distribu-

tion is static. That is, it does not change significantly during the

time of radiation,


s
dHS
i dt [H 1H. 0.
dt 1


As a consequence of Eqs.(2.1.16) and (2.1.17) we arrive at the results


where


D(t) = Trie fiP exp[(LR + L + L i + LRe)t}),







Pe = exp{-BHe}/Ze and p = exp{-_H /Zs,



Z = Tr exp{-BH } and Z = Tr. exp{-SHs},
e e e i di iy


s + +s
Lji y = i[d*E ,y].


(2.1.18a)







(2.1.18b)


(2.1.18c)



(2.1.18d)


We can introduce the low-frequency ion microfield by inserting into the
.+ +s
trace of Eq.(2.1.18a) a delta function 6(E-E ) along with an integral
i
over the variable e. Then, because of the vanishing of the commutator
+s +
in Eq. (2.1.17), we may everywhere replace Ei by e with the result


D(t) = J de Q (e) .
i e e


(2.1.17)


(2.1.19)









Here the brackets < >e indicate an average over the electron sub-

system, LR(e) is the Liouville operator for the radiator in the external

field e, and the static low-frequency ion microfield distribution is

defined as


s+ < +s)>
Qi () = (g(e-E )>i
1 1


(2.1.20a)


where


<(..*)>s = Tr.ps(..).
i 1 1


(2.1.20b)


The result in Eq.(2.1.19) when substituted into the line shape

expression, Eq.(2.1.19), provides the starting point for most line shape

calculations,


1(w) f d'E: Qs(s)J (EW),


(2.1.21a)


+ -1 + + -1+
J (E,w) = -i Im .
e R e Re R,e

(2.1.21b)



The expression for J (E,w) in Eq.(2.1.21) can be put in the form
e


+ -1 + -1 +
Je(E,W) = -f ImR,


(2.1.22)


where LR (,w), which can be interpreted as a frequency dependent,

effective radiator Liouville operator, is amenable to perturbation

theory.31









In this work we are interested in a second order theory; that is,

we want an expression for LR(E,w) which is second order in the radiator-

perturber interaction, VI. The procedure we follow is the same as that

employed by Dufty.38 First, we introduce the coupling constant X,


L + XL (2.1.23)


which serves as our expansion parameter and will be set equal to one at

the end of the calculation. Now, we make the definition,


<{w-i[LR+Le+X(LRe+LRi) ]-1 e= {w-i[LR+Le+XLR+H(X,w)}-1,
(2.1.24)


where we have temporarily supressed the field dependence e. This

expression formally defines the operator H(X,w), a function of radiator

coordinates only, but implicitly contains broadening effects due to the

electrons. We now assume the operator H(X,w) is analytic in X, namely,



H(X,w) = H()() + XH(1 ) + X2H(2)() + *** (2.1.25)



The next step involves expanding both sides of Eq.(2.1.24) and

equating terms with like powers of X to identify terms in the pertur-

bation expansion of H(X,w). The left side of Eq.(2.1.24) may be

expanded in a Lippmann-Schwinger expansion



<{w-i[L + Le + X(LR + LRi)]}-> =
+ i e R(e Ri e Re >e


+ ie










(2.1.26a)


0()= [1-i(L + L)]1
R (W) = [w-i(LR + L )]
e


(2.1.26b)


If we use the identity


Le y(Re)>e = 0
e e


(2.1.27)


where y(Re) is an arbitrary function of the radiator and electron

coordinates, Eq.(2.1.26) reduces to


<{w-i[L R+Le +(LRe +LRi)11'>e


+ ()
+ RR(w)


0 0 2 0 0 0
+ iXRR(w)LT>eRR(W) XR lw)(LIRe (w)LI>Ri(w)


+ 0(X3),


(2.1.28a)


RR(W) = [w-iLR]1.


(2.1.28b)


The right hand side of Eq.(2.1.24) may be expanded in powers of X

using the operator identity,


-1
3A- -1 aA -1
= -A A- .


(2.1.29)


Then,


with


+ O(X 3);









{w-i[LR+XLR+H(X,w) -1 = SR(w)+iXS R(w)[L +H (w) ]S (w)


- SR(O)[LRi + H(1) ()S ()[LR + H (1)(w)IS (w)
Ri R Ri R


-i S O()H(2) S(m)O()} + 0(X3),



SR(w) = {w-i[LR + HO)(0 )]}-l.


(2.1.30a)


(2.1.30b)


Now by comparing Eqs.(2.1.28) and (2.1.30) we may identify H()(w)

appearing in Eq.(2.1.25):

H (0)() = 0 (2.1.31a)


H(1 ) = Re e


H(2) (w = i{e eR(W)e}


(2.1.31b)


(2.1.31c)


Before proceeding, we state that as a consequence of making the

dipole approximation for the radiator-perturber interaction VI, the

average, e, in Eq.(2.1.31) vanishes:


= 0.
Re e


(2.1.32)


Retaining the lowest nonvanishing contribution, H(X,w) is given by


H()) E H(2 () i = R R(w)LRe>e.


(2.1.33)


The final result for the line shape function then becomes


with








I(M) = j de Q'(c)J (,o), (2.1.34a)



where


S-1 Oe > -1 +
J(',0) = Im < d*{w-iL (c) + (L Re ()LRe j d>R
Je(CW) Ii. R(+) + (2.1.34b)



The essential feature of Eq.(2.1.34) is that the problem has been

separated into two independent calculations. The broadening due to the
s + (r
ions is contained in Qi(e), while Je (E,) contains the broadening due to

the electrons. Since the range of the approximations used to obtain

Eq.(2.1.34) are thoroughly discussed in the literature30,31; no further

comments about them will be made here. Eq.(2.1.34) has been used by

Hooper et al.39,40 to calculate Stark broadened Lyman profiles from

high-Z hydrogenic radiators immersed in hot dense plasmas encountered in

a number of pellet-implosion experiments.25



11.2 Method I: Classical Plasmas

We return to the equation for the line shape function, Eq.(2.1.15),



I(w) = R (2.2.la)



b(a) = -7-1 Im<[i-1L]-l> (2.2.1b)
P


We wish to express the line shape in the form,









I() = f dE Q(e)J(6,w), (2.2.2)


without assumptions (1) and (2).

To obtain the form of Eq.(2.2.2) for the line shape we follow

Dufty.38 There a method for introducing static microfield distributions

was developed without neglecting electron-ion interactions or assuming

static ions. Our development will closely follow that of Dufty.38

Make the following definition


<[o-iL]-1 <{w-i[LR + L + HI(W)]}-> (2.2.3)
P P1 1


This expression formally defines the operator H I(): H (w) is a function

of radiator coordinates only, but contains implicit broadening effects

due to both ions and electrons. The Liouville operator LI is defined in

Eq.(2.1.14e). With the definition in Eq.(2.2.3), it is now possible to

introduce a microfield function.

Consider the right hand side of Eq.(2.2.3)


<{m-i[LR + LI + H (a)l ->



= f dE <6(~-E-E i){ -i[LR++ L+ H (+)]}->



= de <6(E-E -E -){-i[LR+ L I() + H())]}-l>


= f de <6(eE- E+)> p{m-i[LR+ L () + H(m)]} -. (2.2.4)


Here, L ) is obtained by replacing E+ +. The results in+
Here, L (s) is obtained by replacing Ee+E by e. The results in








Eq.(2.2.4) follow from the fact that the ion microfield commutes with

everything in LR and Hi(w). Substitution of Eq.(2.2.4) into Eq.(2.2.1)

yields



I(M) = d< (<(tW e-4 )> JI (, ) (2.2.5a)
e i p I


J (,0) = --1 Im R. (2.2.5b)



The expressions in Eq.(2.2.5) are formally similar to those in

Eqs.(2.1.21) and (2.1.22) by construction. However, Eq.(2.2.5) follows

from Eq.(2.2.1) without approximation. We remark that the ensemble

average in the ion microfield distribution is over the entire plasma.

In order to determine H i(), we use the same method discussed in

obtaining a perturbation expansion for H(X,w). Then, to second order in

radiator-perturber interaction,



H(uW) H(2) () = il ,

(2.2.6)



where use has been made of the fact that



= 0. (2.2.7)
I p


With Eq.(2.2.6) we may write


I(o) = f dt Q (+) J ((,w),


(2.2.8a)










where
+1+ + -1
J (E,w) = -1 Im


-1 -1 +
<1(m-LR) 1L > p} d>R, (2.2.8b)

and


+ + +
Q (E) = <6(E-E -E )> (2.2.8c)
I e i p


The result in Eq.(2.2.8) is formally similar to the previous result in

Eq.(2.1.34). However, ion motion as well as electron motion is account-

ed for in the term,



  • p (2.2.9)



    and the ensemble averages are now over the entire plasma with no assump-

    tions about the ion-electron interactions. It will appear that the

    subtracted term in Eq.(2.2.6) removes the static part accounted for in

    the microfield function from H (m).



    11.3 Evaluation of Q (')

    The result in Eq.(2.2.8) are equally applicable to a degenerate or

    classical plasma. In this section we assume a classical picture for the

    plasma, and find that the evaluation of the microfield distribution

    function is simplified.

    As in Chapter I, we write for an isotropic plasma


    (2.3.la)


    P I () = (2c/T) f' 0 X R. in(E:X)TTe










    where

    P (e)de = 4ne2Q (c)dE (2.3.1b)



    and

    T I() = Z (2)/Z (2.3.2)
    I P P


    Z () = f di0dl. .d dN exp{-8V(2)}. (2.3.3)
    i e


    We have adopted the capitalization convention where, for example, Rj
    +
    denotes ion coordinates including the zeroth particle and rj denotes

    electron coordinates. V(Z) is the "potential" of the system


    1 + +
    V(A) = {1 i(OXe)- .V }V (2.3.4)



    where xe is the electric charge of the zeroth particle and V the sum of

    Coulomb interactions between pairs of particles,



    V e e+ + V + +
    l=i ee (r -r ) + =Ji [IR -R I)
    1=i
    N N
    + vie( R r (2.3.5)
    I=0 j=1


    The formalism developed in Chapter I is easily extended to the two

    component plasma. The evaluation of Z (9) in Eq.(2.3.3) leads to

    results similar to those in Eqs.(1.1.12) through (1.1.16) with one modi-

    fication: the clusters may now contain electrons. This is easily

    accomplished by rewriting Eq.(1.1.13),

    J = set of J = J + J + J particles (2.3.6a)
    o e i












    J! = J J J (2.3.6b)
    o e 1

    J Jo Je Ji
    n = no ne n (2.3.6c)
    o e i

    where no, ne, ni, and Jo, Je, Ji are the density and number of zeroth

    particle, ions, and electrons in an n-particle cluster, respectively.

    As in Chapter I, the clusters not containing the zeroth particle

    exactly cancel in the expression for T (A),


    TI(2) = exp{[A2(p,.) A2(p,O)]}. (2.3.7)


    The graphical representation for A2 in Eq.(2.3.7) is topologically

    similar to the graphs representing A2 in Eq.(1.1.16). The difference is

    that we replace the black vertices with black circles for the ions and

    black boxes for electrons. Some examples of the graphs are given in

    Fig.(2.1).

    The next step requires some motivation. We wish to express the

    line shape function in a form similar to that in Eq.(2.1.34). That is,

    we wish to treat most of the ion broadening through an electron screened

    ion microfield distribution without ignoring ion motion. Hence we will

    shield the ions in QI () and at the same time remove the explicit elec-

    tron broadening from QI (), transferring it into J (e,w), where it may be

    treated by perturbation theory.

    First, we observe that it is possible to separate the clusters

    appearing in A2 into two subclasses:

    (2a) clusters containing no ions except the zeroth particle.

    (2b) clusters containing the zeroth particle plus at least one

    other ion.










    The sum of clusters in subclass (2a) gives the electron microfield

    distribution if there were no ions. For the subclass (2b) clusters we

    use a seminodal expansion which is similar to the nodal expansions of

    Abe41, Meeron42, Friedman43, and Buckholtz.44 Each subclass of (2b)

    graph is to be decomposed into a collection of graphs, each having

    potential- instead of f-Mayer bonds. The expansion is based on the

    definition of the f-functions,



    f = {exp(-Bv) 1} = I {-8v/j! (2.3.8)
    j=0


    A graphical representation of the decomposition is given in Fig.(2.2a)

    where n dashed lines connecting two vertices represent the factor

    (-8v)J/j! In Fig.(2.2a) the two vertices are arbritary; each may

    represent a radiator, ion, or electron.

    The formidable number of clusters generated in the decomposition

    above, allows for a simplification which involves summing all simple

    electron chains.11 There are five cases which are shown schematically

    in Fig.(2.2b). The wiggly lines represent the screened interactions

    between the two vertices in the figure. The possibilities for the pairs

    of vertices in Fig.(2.2b) are radiator-ion, radiator-electron, ion-ion,

    ion-electron,and electron-electron. It is understood that the two

    vertices at the endpoints of Fig.(2.2b) are part of a more complicated

    graph. We are summing all terms for which the graphs are the same

    except for the one sum of interactions represented. The intermediate

    electrons in Fig.(2.2b) do not interact with any particles except as

    explicitly shown in the figure.






























    0H



    CC

    o
    -H-


    0 i





    d W


    ecd
    ,


    4-4
    5r- C!

    o 54 p
    H U











    0 0 a)
    CO




    44






    Q)

    co
    s 0 0
    S' (d
    5 S 5


    4J









    0)
    CO
    5-1





















    C)
    4-1
    r-'-


















    CN U
    1-1
    00




    C 0




    o
    NCCO





    55












    V

    +
    -7.



    4-





    !
    6

































    H 0
    a)



    4-1 E-
    CO






    0

    co
    Q a


    -
    , a


    4 a)
    a) a)
    41



    a) r
    a


    U)
    U, C






    S4-1



    U cd
    '-l

    C ca


    a) -t
    n <


    t
    '-4
    0 0


    LO








    oo
    4C)








    4.
    Co


    0 0 '$

    a m
    o 0






    10 b




    o o


    0 0
    oo







    o 1 *o u
    u) > Q
    a) a) .





    10 1

    -4 I )

    a i-i a







    ) a 1
    .3 I -4 <





























    +


    l'i i








    'I


    + 1-



    I I










    The screened interactions can be evaluated by introducing the

    Fourier transforms as in Appendix A. The effect of the screening is to

    introduce the dielectric function,



    e (q) = 1 + Pevee(q) (2.3.9a)



    v (q) = 4e2/q2. (2.3.9b)



    Hence we find that



    -Bvs (q) = -Ov (q)/e (q); o ,a = ie. (2.3.10)
    a102 a1 2 e l 2


    We have produced an expansion where at least three interaction

    bonds are connected to every electron vertex. Such vertices were

    defined in Ref. 41-44 as a node, hence the name nodal expansion has been

    adopted. In our case, an ion need only be twice connected. Therefore,

    we have developed a nodal expansion for only one species, or a seminodal

    expansion.44

    If we neglect all subclass (2b) graphs containing one or more

    electron nodes, we find that the remaining graphs contain only ions

    interacting through a screened Coulomb potential. The screening of the

    bare ion-ion interactions is due to the elections and it is given by the

    Debye-HUckel result.

    With this approximation, the subclass of (2b) graphs can be inter-

    preted as a shielded ion microfield distribution. Furthermore, the

    electron microfield distribution and the shielded-ion microfield distri-

    bution are statistically independent,








    + + + + +
    <6( E E E )> = Tr p 6(c E E )
    e 1 p p e i


    s+ + +s
    = Tr pS Tr p 6( E s)
    ii ee e i


    = f dE'{Tr .p(s' P)}{Tr p -( )} (2.3.11)
    Ii I ee e


    where p and ps are defined in Eq.(2.1.18) with VS. the sum of the Debye

    screened ion-ion interactions.

    Substituting Eq.(2.3.11) into Eq.(2.2.8) yields

    -1 + + +
    I(o) = -i7 Im f dede' Q.(s') <6(+ C' E )>
    a e e

    x + + H2 -1 +
    x R


    = -i-1 Im f de Q'(E)
    i R Re I R,e
    (2.3.12)


    We can further manipulate Eq.(2.3.12) by introducing the projection

    operator Pe defined as


    P y = Tr p y. (2.3.13)
    e ee

    Then, using Zwanzig's projection operator techniques45-48 described in

    Appendix E, we get for Eq.(2.3.12), in a second order theory,



    I(W) = -7r1 Im f de Qs(s)


    + -1 > R' (2.3.14)
    Re R Re e R










    where we have used the following properties of the projection operator


    P L y(R) = 0,
    e Re


    P2 =
    e e


    P{LR() + H2)(w)y(R) = {LR( ) + H w2) ()}y(R).


    (2.3.15a)



    (2.3.15b)


    (2.3.15c)


    Here y(R) is an arbitrary function of the radiator coordinates.

    The expression for the line shape given in Eq.(2.3.14) is of the

    form in Eq.(2.2.2), but is only applicable to classical plasmas. In

    addition, in getting to Eq.(2.3.14) we approximated QI(e). In Section

    11.7 we show that this approximation leads to complications.



    11.4 Method II

    In Section 11.3 the classical treatment of the plasma considerably

    simplified the evaluation of QI (). In particular, the identity



    exp{-BH } exp{it-(e + t )} = exp{-$H + it.(e + A )} (2.4.1)
    p e i p e i


    holds true only in the classical limit. For the quantum case, Eq.(2.4.1)

    must be replaced by the Baker-Campbell-Hausdorf formula.49 In order to

    avoid this complication, we make the definition,



    <(w IL)-l> = <( { i[L + L E+ H (w)}-> (2.4.2)
    p R Ri p










    This expression formally defines the operator HE(w), a function of

    radiator coordinates only, which contains broadening effects due to both

    ions and electrons. The Liouville operator LRi is defined as


    C + +E
    LRi y = i[d E, y] (2.4.3)


    +E
    where the ion field E. is arbitrary except that it only depends on ion
    1
    +3
    coordinates. Clearly, HE(w) will depend on the choice of Ei. With the

    definition in Eq.(2.4.2), it is now possible to introduce a microfield

    distribution function and write for Eq.(2.1.15),



    I(w) = f dE Q (E)J (E,w) (2.4.4a)



    (,w) = -Ti Im (2.4.4b)


    + += +E
    Qi () = Tr p 6(c E). (2.4.4c)
    1 pp i


    By construction the result in Eq.(2.4.4) is formally similar to

    those in Eqs. (2.1.34) and (2.3.14), but Eq.(2.4.4) follows from

    Eq.(2.1.15) without approximation.

    In order to determine HE(w), we use the same method previously used

    to obtain H (w). Then, to second order in VI,



    H () H(2)() = i{ .
    p Ri R R(2.4.5)
    (2.4.5)


    Use has been made of the fact that










    = = = 0. (2.4.6)
    Ri p Ri p Re p



    With Eq.(2.4.5), we may write for I(w),



    I(w) = -I-m / de Qi.() -

    ( _L-1 -1
    R. (2.4.7)



    Ion motion, as well as electron motion, is accounted for in the term,


    -1
    p, (2.4.8)



    and the ensemble averages are over the entire plasma with no approxima-

    tion on the electron-ion interaction. Note that the subtracted term in
    (2)
    Eq.(2.4.5) removes from H2) (w) the static part accounted for in the

    microfield function.

    Although Eq.(2.4.8) is formally similar to Eq.(2.3.14), there are

    two important differences. The first is the arbitrary ion field E..
    1
    The second is the difference in the subtracted terms that appear in

    H (2 ) and H(2) (). In order to further discuss the differences, we
    I E
    +>E
    must first select E .


    + +
    11.5 Evaluation of <6(s E )>

    It remains to determine the ion field E.. In principle, I(w) is
    1

    not dependent on a particular choice of E.. In practice, however, I(w)

    is calculated in some approximation scheme and it is no longer

    independent on the choice of E A first and simplest selection for E










    might be the electric field due to a collection of point charges.
    +E
    However, since E is a static field it is expected that the fast moving

    electrons screen the ion static field. Therefore, a 'best' choice for

    E could be an electron shielded static ion field. This, of course, is

    an ad hoc method of selection. Below we propose a plausibility argument
    4-,Z
    for selecting the field E..
    1

    We start with the formal expression for the microfield distribution

    function
    + + +4
    Q(s) = <6( E)> (2.5.1)
    P

    2
    which for an isotropic system2, can be written,


    P (E) = (2e/)7 fo di sin(e.)T () (2.5.2a)


    T (+) = (2.5.2b)


    In Eq. (2.5.2) the ensemble average is over the entire plasma. But

    because the electric field E. depends only on ion coordinates it is
    1

    possible to formally perform the trace over the electrons:

    + +E + +E
    it*E. i*E
    = Tr p e
    P p

    + +E
    iP.E.
    = Tr.{Tre p pie

    + +Ef
    i *
    = Tr p e

    + +,

    = =







    64

    E E
    which defines the brackets ( > and p, an effective ion density

    operator.

    In order to make further progress with the operator p we use the

    second quantized representation of the equilibrium reduced density

    operators,50


    + + + +

    S-


    4 + ((R) w... ^ )>p(R )...KR.)> (2.5.4)
    1 N. N. 1 p
    1 1


    The operators, + (I ) and *(k), are the-ion creation and annihilation

    operators for the spatial point, R Now the Ni-particle Green's

    function is defined by



    G(1,...,N ;1',...,N!) = -
    i 1 T ii p
    (2.5.5)



    where T orders the operators according to their value T, with the

    smallest at the right. The operator T also carries the signature

    (-1)P, where P is the number of permutations of fermion operators needed

    to restore the original ordering. Therefore, Eq.(2.5.4) may be

    expressed as
    + + + +



    + +
    = -G(1,...,N;1 +,...,N ) +
    Ni N -1
    T -1 =
    Ni N -2
    +(2.5.6)
    T = T (2.5.6)
    2 1










    The plus signs used as superscripts are intended to serve as reminders

    of the particular infinitesimal T ordering required to reproduce the

    orders of the factors in Eq.(2.5.4).

    To evaluate the many-particle Green's function in Eq.(2.5.5) we may

    use Feynman-Dyson perturbation theory.51 Each term in the perturbation

    expansion may be represented51 by a Feynman diagram. This perturbation

    theory is an expansion in the bare interaction. However, due to the

    long range of the Coulomb interaction the expansion diverges term by

    term5154 and it is necessary to sum a selected class of diagrams, whose

    sum yields a finite contribution. After examining the perturbation

    expansion, the selected class of diagrams to be summed are the so-called

    ring diagrams,51-54 which are the most divergent set of diagrams. This

    frequently used approximation in the theory of Coulomb systems is equi-

    valent to the random phase approximation (RPA) of Bohm and Pines.55

    In what follows, we will treat the electrons in the RPA. This

    approximation is most simply treated by introducing the effective two-

    body interaction defined in Fig.(2.3). This approximation to the

    effective interaction reduces in the classical limit to the Debye-H*ckel

    result.52,54 It is understood that the Green's functions at the

    endpoints of Fig.(2.3) are part of a more complicated diagram. We are

    summing all electron terms for which the diagrams are the same except

    for the sum represented. The intermediate electrons in Fig.(2.3) do not

    interact with any other particles except as explicitly shown in the

    figure. The resummation formally eliminates all the electrons from the

    diagrams and replaces the bare Coulomb ion-ion and ion-radiator inter-

    actions with effective potentials. Note that in this method the elec-

    trons do not shield the ion field E while in Method I the electrons do

    screen the bare ion field.































    0



    0

    I-H
    co


    N


    r-4



    0



    0



    P-4


    w
    41








    ~4-4
    0





    0)




    )


    4I-4
    (U
    ro 4-
    Cw
    co
    P.





    67

















    $ *1
    I I



    Il +




    l + Il

    (11 11









    The effective interactions may be expressed in terms of the

    52 53
    electron proper polarization part,5253



    U = u ( -R ;T) (2.5.7a)
    T OI

    S(RI-R ;T) = [(2r) 3BI fdq expIiq.(R -RJ)-ivT}u) (q;v)
    n=0
    (2.5.7b)



    v = 27n/B (2.5.7c)
    n
    where

    u (q;v) = v (+){1 v (q)(q ,v )-1. (2.5.7d)
    IJ n J ee e n



    In Eq.(2.5.7) To is the random phase approximation to the electron
    e
    proper polarization part, T where is defined553 as the sum of all
    e e
    electron polarization parts that can not be separated into disconnected

    parts by cutting a single electron-electron interaction line.

    Since the ion-ion and ion-radiator monopolee part) interactions

    have been screened by the electrons, it seems plausible to select I as

    follows,



    S= UE. (2.5.8)
    1 OT



    This choice of E is certainly not unique and by no means has it been
    i
    derived rigorously. Nevertheless, it seems plausible3 that if the ion-

    radiator monopole term is screened by the electrons, then the electric

    field at the radiator due to the ions should also be screened. Again we
    +emphasize that E
    emphasize that E is arbitrary and the choice given in Eq.(2.5.8)










    involves some hindsight. This particular choice for will allow us to
    1
    consider systematic corrections to the "conventional" line shape

    theories discussed in Sect. II.1. Finally we remark that the effective

    potential has been evaluated in the ring approximation, but systematic

    corrections are possible by including corrections17-20,56,57 to io in
    e
    Eq.(2.5.8d).



    11.6 The Classical Limit

    Consideration of the classical limit of an interacting quantum

    mechanical system by looking at a Green's function formulation, in that

    limit, has been discussed by Smith.58 In Ref. 58, Smith starts with the

    usual Feynman rules52 for evaluating diagrams and derives a set of rules

    which apply in the classical limit. Application of these rules to the

    diagrammatic expansion of the Green's function leads to the equation of

    state in the cluster form.

    In the grand canonical ensemble, thermodynamic quantities59 are

    functions of the volume Q, the temperature T, and the chemical potential

    p. For noninteracting bosons or fermions there are two length para-

    meters, 1/3 and the thermal wavelength, A which for a particle of

    mass m is defined as,



    A = [2nB/m] 1/2. (2.6.1)
    0o


    Bu
    In the classical limit we let the fugacity, z = e go to zero while

    the quotient (z /A ) remains finite and is in fact equal to the particle
    a a
    density.58 For an interacting system the range of the potential, 6 is

    another length parameter and the classical limit is defined by z +0









    while (z 6 /A ) remains finite for nonzero 6 Terms of higher order
    ao a o
    in (z 6 /A ) will then be dropped. As a result, there are two basic
    ao a
    simplifications over the quantum case. One is the restriction of

    diagrams that can contribute, and the other is the elimination of some

    terms in the free particle Green's function. The simplifications,

    summarized by Smith58 in the forms of rules which apply in the classical

    limit, are stated here without proof:

    (1) The only diagrams which contribute are those for which there is

    some T ordering, such that, of all the free Green's functions in a loop,

    exactly one of them has T
    two fixed points may have either zero or one propagator with T<0,

    depending on whether the two fixed points are forward or backward in

    tau, with respect to each other. As a result of this rule, no inter-

    action line can have both ends on the same loop.

    (2) In any loop, all propagators with T>0 may be replaced by -6(x)
    3
    and the one propagator with T<0 by z /A the (+) for fermions and the

    (-) for bosons. This implies that all space coordinates of a given loop

    are the same.

    Since in a coordinate representation only the diagonal elements are

    required for the reduced operator p in Eq.(2.5.5), the diagrams assoc-

    iated with the reduced distribution functions only involve closed loops

    of the radiator, ions, and electrons. In particular, the diagonal

    elements will have chains of propagators running from a fixed point, say
    + + +
    (x,T), back to the fixed point (x,T ). The propagator returning to
    + + 3
    (x,T ), thus, contributes a factor + (z /A ). This chain behaves now

    exactly as a loop. Rules (1) and (2) may now be used to evaluate p, in

    the classical limit with the result,









    S= exp{-B(Ti + Vi )}/Tr exp{-B(Ti + Vi )}, (2.6.2)



    where V.. is a screened ion-ion interaction where the screening, due to

    the electrons, is treated in a Debye-chain expansion. Therefore, in the
    E + s +
    classical limit Q (e) reduces to Q (e). However, in this limit Eq.

    (2.4.7) does not reduce to Eq.(2.3.14). The difference appears in the

    subtracted terms:


    E )-1 E
    LRi(W iLR) LRi> in Eq. (2.4.7), (2.6.3a)



    as compared to



    (L (w iLo)-1L >


    appearing in Eq. (2.3.14).



    11.7 Comparison of the Methods

    It is convenient to rewrite the results from the previous sections

    in order to compare them. First, for the conventional line shape form-

    alism described in Section 11.1, which assumes static ions and accounts

    for ion-electron correlations through a screening of the ion-ion

    interaction, we write



    I(w) = -T-l m f de Qs() i R


    -1 -1+
    + Le [w-i(L + L )] L e> d> R (2.7.1)
    Re R e LRe e R.271









    The result of Method I, which takes advantage of a cluster expansion

    formulation of the microfield distribution in order to screen the ions,

    is given by


    -1 + s + + -1
    I(w) = -i- Im j df Qs(E)



    [p- e]} >R (2.7.2)



    The second approach, Method II, introduces an arbitrary ion field. This

    field must then be chosen, and we have given plausibity arguments for

    our choice of Ei. With this method, the resulting line-shape function

    is


    1 + Z+ + + -1
    I(w) = -r- Im f de Q(E)

    i R- 2 -1 +
    + p}-' d>R. (2.7.3)



    Since Eq.(2.7.2) applies only to classical plasmas, we temporarily re-

    strict our discussion to such plasmas. The effective interactions in

    Qi(E) of Eq.(2.7.1) are usually taken to be the Debye-Hickel result,

    therefore, all three microfield distribution functions in Eqs.(2.7.1)

    through (2.7.3) are equal in the classical limit. As mentioned before,

    there are differences in the denominators appearing in the curly

    brackets of these equations. Obviously, Eq.(2.7.1) is not expected to

    agree with the other two equations since it contains assumptions (1) and

    (2); the neglect of ion motion and electron-ion correlations. At this

    time, it is instructive to make these two approximations in Eqs.(2.7.2)









    and (2.7.3). Then, for the denominator in Eq.(2.7.2), }-, we get



    {-iLR(C) + [p


    + {l-iLR() + < LRe[w-i(LR + Le)I LRe>e' (2.7.4)



    and for the denominator in Eq.(2.7.3), .***}-,



    {3 iLR(c) + "Lii(--iLR) LR>}-



    + {w-iLR(C) + e} (2.7.5)



    Notice that in Eq. (2.7.3) the ion field E is defined by Eq. (2.5.8),

    and is therefore dependent on the ion-electron interactions. Hence,
    -+E +
    E goes over to E when electron-ion interactions are neglected.

    Clearly, both methods reduce to Eq.(2.7.1) with these two approxima-

    tions.

    Let us relax the approximations immediately preceding Eq.(2.7.4)

    and examine Eq.(2.7.2) in more detail. As indicated in Section 11.3, an

    infinite number of graphs were neglected in getting to Eq.(2.3.11). In

    particular, all subclass (2b) graphs containing one or more electron

    nodes were neglected. Some neglected graphs contain the corrections to

    the simple chains of electrons: corrections to the Debye-Hiickel

    screening.17-20 But more than corrections to the electron screening of

    the ions is neglected. For example, the ion screening of the electrons

    is not included. But in the left hand side of Eq.(2.7.4) the terms in
    -1
    brackets are subtracting from (L (0-iL ) Lp > the broadening already
    I o Ip










    included in the microfield. Therefore, for a consistent treatment of

    the line shape, we require that effects neglected in the microfield

    distribution, Qi (), must also be neglected in the left hand side of

    Eq.(2.7.4). As an example, consider two terms from Eq.(2.7.4)



    e. (2.7.6)



    If we define


    eff
    pe Tr.Pp (2.7.7)



    then we may write for Eq.(2.7.6)


    eff I RwL -1 Tr1P IL (w-iL L1-1
    Tre e f eLRe(m-iLR)- LRe R Tr eeLe L LRe 0 =0

    (2.7.8)


    eff
    where we have neglected the ions screening in p For the remaining

    terms in Eq.(2.7.4) we should also be careful and neglect similar con-

    tributions. Certainly, this leads to a complicated evaluation of

    Eq.(2.7.2).

    A similar complication does not occur in Eq.(2.7.3). The term
    -1
    containing LRi in Eq.(2.7.5) exactly subtracts from
    +
    the broadening effects in Qi () (in a second order theory). For con-

    sistency all that is required is that the average over the plasma be

    evaluated identically in both terms. Also, the range of validity of

    Eq.(2.7.3) includes both quantum and classical plasmas. The difficulty

    with this approach is in justifying the selection of E..
    i2










    11.8 Summary

    The two methods developed in the previous sections demonstrate the

    problems involved with introducing low-frequency microfields in the

    line-shape function. Even though the formal expressions for the line-

    shape function derived from Methods I and II are not identical, it is

    clear that both methods modify the expression for the electron width and

    shift operator appearing in the conventional theories. In fact, the

    different forms of the width and shift operators resulting from the two

    approaches, emphasizes that the formal expression for the width and

    shift operator is very much dependent on the choice of microfield

    distribution.

    We have demonstrated that the modifications to the electron width

    and shift operator vanish if ion-electron correlations are neglected.

    However, this approximation 'overcounts' the electrons. That is, if we

    neglect ion-electron correlations in the width and shift operator and

    not in the microfield, then the implicit electron broadening in the low-

    frequency microfield is also included, to second order, in the width and

    shift operator. This, of course, is not a consistent approximation. In

    this dissertation we will not study these corrections to the electron

    broadening any further. Our intention is to simply point out the

    difficulties involved with an ad hoc treatment of the line-shape

    function.

    In the next chapter we assume that the complications discussed

    above will be resolved by future research. Having taking this point of

    view, we will proceed to investigate quantum mechanical corrections to

    the electron screening of the ions in the calculation of low-frequency

    microfields.
















    CHAPTER III
    QUANTUM CORRECTIONS TO THE LOW-FREQUENCY
    COMPONENT MICROFIELD DISTRIBUTION

    The spectral line shapes for an atom or an ion radiating in a

    plasma are determined21 by the interactions of the radiator with all of

    the components of the plasma. In relation to this problem, various

    theories of ion microfield distributions have been formulated.2-6 In

    these theories, electric fields of two types are considered: a high-

    frequency and a low-frequency component. The distribution of the high-

    frequency component is calculated3,5 by considering a gas of ions inter-

    acting through a Coulomb potential and immersed in a uniform neutraliz-

    ing background. The distribution of the low-frequency component is

    determined4'6 by considering a gas of ions interacting through an

    effective, screened potential. The shielding of the ions in the latter

    case is a way to include electron screening effects, since the ion

    electric fields vary slowly over times on the order of the electron

    relaxation times. The effective potential is chosen to be the Debye-

    HIckel result,37 which applies to classical electrons and long times.

    It is possible that the ions may be treated reasonably well with a

    classical picture, even when the density and temperature conditions

    require a quantum mechanical formulation of an electron gas. This dual

    behavior is due to the large ion-to-electron mass ratio. In this

    chapter we investigate such a situation and treat quantum mechanically

    the electron screening of classical ions.








    III.1 Low-Frequency Microfield Distribution

    The line shape function given by Eq.(2.7.1) involves the evaluation

    of the shielded, static ion distribution function



    QS() = TrlPi6(,-E) (3.1.1)



    The microfield distribution in Eq.(3.1.1) can be evaluated with the

    method developed in Chapter 1 with only simple modifications necessary

    in order to account for the effective potential.

    As in Chapter 1, the problem is to determine the electric

    microfield distribution function P(F) at a point with charge Xe, which

    is found from an evaluation of Eq.(1.1.4),



    P(E) = (2c/I) fo dX k sin(se)T(A), (3.1.2)



    where T() is given by


    1 + + + -1 ++
    T() = ZN N *** drodrl..*drN expI-BVN + i(xe)- kVoVN .

    (3.1.3)



    In Eq. (3.1.3) r. represents the position of the j-th particle,

    B=(kT)- and ZN is the configurational partition function. The

    potential energy of the system, V is expressed as,



    V = vs. (3.1.4)
    0=i

    The two particle effective interaction, vs, is usually taken to be the
    iis








    Debye-HUckel result4,6



    vs.(r) = Z.Z.e2 exp{-r/A }/r, (3.1.5a)
    1j i 3 e


    where

    Z.= X i= (3.1.5b)
    i Z iO0



    Here, Ze is the charge of one ion and e is the electron Debye length

    defined by,



    X = [4e2 P 1/2 (3.1.6)
    e e


    where pe is the electron density. By requiring the total system to be

    neutral, the ion and electron densities are related by the expression,



    Pi = ZPe



    In what follows we assume the electrons to be quantum mechanical

    while still retaining a classical picture for the ions. With this

    assumption, the Debye-Hickel result of Eq.(3.1.5) is no longer valid.



    III.2 Effective Quantum Interaction

    In order to evaluate the effective quantum interaction, it is

    convenient to introduce the Fourier transform defined in Eq.(A-2), and

    the generalized dielectric function,52



    e (q,v) = 1-v (q)w(q, e ), (3.2.1)
    e n 1-ee q)e q'n)







    52
    where ir is the proper polarization part52 of the one component electron
    e
    gas, and vee is the Fourier transform of the electron-electron Coulomb

    interaction:


    2 2
    v (q) = 4re /q
    ee


    (3.2.2)


    In terms of the dielectric function the shielded interaction may be

    expressed as,52
    expressed as,


    uj(q,v) = vij(q)(qe(,),
    ij n ij e n


    (3.2.3)


    where v.i(q) is the Fourier transform of the bare Coulomb ion-ion

    interaction,


    2 2
    v.i.(q) = 4rZ.Zj.e /q


    (3.2.4)


    In Section 11.5, we approximated the electron screening by the ring

    approximation,


    * + -2(2)-3
    S(q,v ) (q,v ) = -2(2) f dp
    e n e n


    0 + 0o
    n (p+q) n (p)
    ++ +
    hv +in E(p+q) + E(p)
    n (3.2.5)


    where n is a small positive parameter. Let e and me represent the

    chemical potential and mass of the electron, respectively, then



    no(p) = exp{-B[E(p) p1]} 1, (3.2.6a)
    P p e










    E() = h2 p2/2m (3.2.6b)
    e



    At this point we digress and examine the divergence difficulties

    that one finds in a simple perturbation expansion of the free energy of

    a system of classical point charges immersed in a uniform neutralizing

    background. The divergences are due to the long-range nature of the

    Coulomb interaction. Also there are short-distance divergences, again

    due to the Coulomb interaction which has an infinity at x=0. The

    divergences were removed by nodal expansions developed independently by

    Abe41, Meeron42, and Friedman.43 In these developments the divergences

    are systematically removed in two steps: (1) chains of Coulomb

    interactions are summed to give screened interactions, and (2) the

    resulting ladder diagrams with screened interactions are summed. Step

    (1) introduces the Debye screening length, and step (2) gives Be2 as the

    distance of closest approach.

    We return to Eq.(2.5.7) and the retardation effects in U We
    T
    point out that the contributions from the retardation effects, U (vO),

    play no role in cutting off57 the long range divergences. Also note

    that the electron proper polarization part introduced in Eq.(3.2.1) and

    evaluated in the ring approximation is given by,54


    2 2
    o (q,v) = (z /A ) fh dr exp{ h q T(-r) + 27nT
    e n e e o 2m 8h h
    e

    22 244
    S(ze/A)1 + h q 2 h 4 + ...; for n#O (3.2.7)
    4e m n 2r m n
    e e

    Since the primary contribution in the integral in Eq.(2.5.7b) is for

    q
    we neglect all but the first term in Eq.(3.2.7) for 1/q. >> Ae










    The accuracy of this assumption has not been determined for

    1/q > A however, the approximation U (v=0) appears to contain60 all
    2 e T
    the essential effects of correlations due to both the long range part of

    the Coulomb interaction and exchange.

    With this simplification, the classical limit for the ions is

    easily taken using the rules in Section 11.6. We find that the

    resulting expression for the microfield distribution Qi () is similar to

    that in Eq.(1.1.1),


    E + 1 + +
    Qi() = Z1 ... dro...drN exp{-BV }6(-E ). (3.2.8)



    However, the results in Eq.(3.2.8) involve the effective interaction


    E E
    V = u (ri-rj ) (3.2.9a)
    0=i

    u (q) = v (q){1 v (q)r (q,v=0)}- (3.2.9b)
    ij = ij ee e

    E C
    o O + +
    where u i(q) is the Fourier transform of u (ri-rj).

    The shielded interaction in Eq.(3.2.9) may be rewritten in the form



    u (q) = 4ZZ e21 + 2(q)q2}- (3.2.10)



    where X(q) can be interpreted as a q-dependent screening length defined

    by


    2 o -1)]/2
    X(q) = [-41e n (q,0)]1 (3.2.11)
    e










    The numerical evaluation of ir(q,0) is discussed in detail by C.

    Gouedard and C. Deutsch61 for an electron gas at any degeneracy. The q-

    dependence of X(q) is very weak for values of the product qX(q) less

    than 1. Therefore, it is possible to make the approximation



    u o(q) 4irZ Z.e2 X2{l+(Xoq)2}-1 (3.2.12a)



    X = A(q=0) (3.2.12b)
    o


    For values of the density and temperature considered here, the error

    introduced by the approximation in Eq.(3.2.12) is less than 1%.

    The inverse Fourier transform of Eq.(3.2.12) is easily performed



    u O(r) = ZZje2 exp{-r/Xo}/r. (3.2.13)



    The functional form of Eq.(3.2.13) is similar to Eq.(3.1.5) although

    there is a difference in the screening length. It can be shown52'54

    that in the high temperature limit X goes to X
    o e
    In the next section we calculate the low-frequency microfield for a

    gas of ions but the ions interact through the effective interaction

    given in Eq.(3.2.13).



    111.3 Numerical Results and Analysis

    The formalism developed in Ref. 6 is now easily extended to the

    case of ions interacting through V o as given in Eq.(3.2.9). The

    results are compared in Figs.(3.1) and (3.2) with a low-frequency

    microfield calculation which treats the electrons in a Debye-Hiickel










    theory. The plasma conditions in the figures are for values of the

    electron plasma parameter, r = 0.125 and 0.214, where

    e

    F = e 2/r (3.3.1)
    e o



    and ro is the average electron interparticle distance which, for an

    electron density p e is defined by



    4rp r /3 = 1. (3.3.2)
    e o



    The degeneracy parameter, y, is defined as the ratio of the noninter-

    acting electron gas Fermi temperature to the temperature of the system,



    y = TF/T. (3.3.3)



    From the two figures we see that the quantum corrections to the

    Debye screening shift the peak of the P(e) curves to a higher value

    of e. This behavior may be anticipated. In Eq.(3.2.13) o takes into

    account the fact that the electrons at the bottom of the Fermi momentum

    distribution are 'frozen' and cannot fully participate in the screening.

    Therefore, X is larger than Xe for a given electron temperature and

    density so that the electron screening is reduced and the ion electric

    fields are, on the average, larger.

    For the cases presented here, the quantum corrections are small.

    In fact, only for yPl are the effects of the corrections noticeable. It

    follows that the quantum corrections to the electron screening of the

    ions are negligible in recent laser-produced plasma experiments,2 since










    in these experiments y<0.05. The results in Figs.(3.1) and (3.2) also

    indicate that even in future experiments, where denser plasmas are

    expected, the quantum corrections to the Debye screening will still be

    small.

    Finally, we note that the low-frequency microfields are used in

    calculating line profiles which in turn are used as a diagnostic tool to

    measure electron densities and temperature. It is important to

    investigate how the quantum corrections in the microfield affects the

    line profiles. We have done this for the Lyman-a and Lyman-B lines of

    Ar+17 and C1+16. We were limited to electron densities of order 1025

    electrons/cm3 because for plasmas more dense than that, the structure of

    the profile is completely washed out. For the two cases we did

    calculate, the effect of the quantum corrections in the screening was

    small, less than 0.1%.














































    -4
    0


    0 0 C



    r4r
    o c










    LINX
    o P 0 0 .









    V) r

    4-4 G) -
    II II




    ~l 0Cc0 0
    W~























    Lr4 D U
    0A c 9


    Qa x a.



    II I II I


    VJ 0 0


    s-II

    0 II II II



    0.)
    5-i Il ..

    c 0 O*

    Li



















































    C; co
    IL6



































    Fig. 3.2 P(e) curves for r = 0.214. c is in units of e .
    e o
    24 3
    For y = 0.67; p 2 x 10 part/cm A /X = 1.13.
    25 /cm, 1.41.
    For y = 1.23; p = 5 x 10 part/cm A = 1.41.












    2.4


    P(E)


    1.8






    1.2






    0.6


    0.2 0.4 0.6
















    CHAPTER IV
    INTEGRAL EQUATION METHOD

    In this chapter we propose an integral equation approach to

    evaluate the microfield distribution function define in Eq.(1.1.4).

    This involves expressing T(A) in terms of a two-body function. This

    two-body function is formally identical to the radial distribution

    function (RDF), therefore, it is amenable to integral equation

    techniques.

    The resulting expression for T(X) involves an integration over a

    so-called coupling parameter. This integration may be performed in the

    hypernetted-chain62-65 (HNC) approximation to the two-body function,

    simplifying the expression for T(A). For simplicity we treat the same

    system as in Chapter I.



    IV.1 Connection with the Chemical Potential

    It was first noted by Morita8 that the virial expansion for T(A) is

    formally similar to that of the excess chemical potential, 6p, which is

    defined by



    exp(BS) = ZN/2ZN-1. (4.1.1)



    A comparison of Eqs.(1.1.6) and (4.1.1) shows that the two quantities

    are indeed similar.









    It is possible to write 6p in terms of the RDF.59 Introduce the

    coupling parameter C, which varies from 0 to 1 and which has the effect

    of replacing the interaction of some particle, say 1, with the jth

    particle of the system by Cv(rij). In terms of this coupling parameter,

    then, the potential energy for the system is written in the form


    N
    V = C(r i) + v(rij). (4.1.2)
    j=2 2i

    Clearly,



    ZN(=I1) = ZN (4.1.3a)

    and

    ZN(C=0) = Q ZN-_. (4.1.3b)



    Eq.(4.1.3) can be used in Eq.(4.1.1) with the result59



    6p = p f dC f dr v(r)g(r;;) (4.1.4)



    where g(r;1) is the RDF for particle 1 an any other particle.

    The RDF can be evaluated with integral equation techniques. In the

    framework of the HNC approximation, it has been shown65 that the

    integration over the parameter can be done.

    We have intentionally omitted details in this section since the

    steps are discussed elsewhere.59,65 In addition, almost the identical

    steps will be shown in the next sections for the T(A) case.










    IV.2 The Two-Body Function g(r;G)

    Because of the similarity between T(1) and 6p, it is possible to

    write an expression like Eq.(4.1.4) for T(). To do this we introduce
    +
    the parameter t which is defined as the magnitude of the vector ,



    5 = |t|. (4.2.1)



    We may rewrite the 'potential' V(Z) defined in Eq.(1.1.8) as



    V() = V + ie-1 E, (4.2.2a)



    where


    N
    + N + +
    E = E (r. ) (4.2.2b)
    j=1 o


    and is the unit vector in the direction of . The definition of

    ZN(S) follows from Eq.(4.2.2),



    ZN(c) = J dr0drl.**drN exp{-8V(t)}. (4.2.3)



    If we let C vary from 0 to in Eqs.(4.2.2) and (4.2.3), then we can

    take the field in and out of the expressions by varying C. This is

    useful since now we may write for T(A),



    AnT(Z) = An[ZN(SC=)/ZN (=0)]


    S an z ( )
    = dc
    o08






    92
    N
    S- ++V
    = fd d dr**dr (I itE(r. )/le}e- /Z N
    Nj=1 jo


    = p dC f dr (r)g(r;S) (4.2.4)
    o


    where



    ,(r) = it*.(r)/e (4.2.5)



    and

    2 -BV( ) (4.2.6)
    g(r;c) = f2 J-dr2-..drN e /ZN(). (4.2.6)



    The function g(r;) reduces to the usual RDF for C=0. Hence, for

    arbritary values of C, g(r;S) is simply the 'RDF' for the zeroth

    particle and any other particle where the system interacts through the

    'potential' V(O).

    In Eq.(4.2.4) the effect of the neutralizing background has not

    been included. The background term is given by



    f dr *(r) = *(q=0) (4.2.7)



    where *P(q) is defined as the Fourier transform


    (q f dr e
    j(q) = I dr e-iqr it*E(r)


    = .qv(q)/e


    (4.2.8)










    and

    v(q) = 4e2/q2. (4.2.9)



    With the background term included in Eq.(4.2.4), we may write



    dnT() = p / d f dr *(r)h(r;C) (4.2.10)



    where h(r;C) is the total correlation function and is defined by


    + +
    h(r;C) = g(r;) 1 (4.2.11)



    Eq.(4.2.10) is the main result of this chapter; we have expressed T(Z)

    in terms of a two-body function. This form has the advantage that

    knowledge of a two-body function gives T(k) exactly, in contrast to

    Refs. 3 and 5 which require knowledge of many-body functions. Of

    course, the price to be paid for this advantage is the integration over

    the parameter C which requires g(r;r) for all values of r between 0

    and ..



    IV.3 The Holtsmark Limit

    It is instructive to examine Eq.(4.2.10) in the Holtsmark or high

    temperature limit where the total correlation function is given by,



    lim h(r;) = e(r) 1; (4.3.1)
    T+e o


    this expression can be substituted into Eq.(4.2.10) and we find




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