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 LOWFREQUENCY 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(eE )> ....................................62
6. The Classical Limit. ..? .....................................69
7. Comparisons of the Methods..................................71
8. Summary............. .... .. .................. ........... 75
III QUANTUM CORRECTIONS TO THE LOWFREQUENCY
COMPONENT MICROFIELD DISTRIBUTION................................76
1. LowFrequency 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 TwoBody 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 virialDebye 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 ionelectron interactions and
their effect approximated by a shielding of the ionion 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 lineshape
theories. The point in question is the ad hoc introduction of the
screened ion fields by the usual lineshape 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 DebyeHickel 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
hypernettedchain 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 virialDebye 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 highfrequency
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 longranged and a shortranged
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
longranged central and noncentral interactions we treat in a Debye
chain expansion,11 since these interactions of infinite range display
collective effects. The shortranged central and the screened inter
9
actions from the Debyechain 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)= 271e 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) = fef 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 fMayer functions in which
every particle in n is independently connected to every other particle
in n. The fMayer 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 fMayer function is
represented by a heavysolid 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 shortrange contribu
tions. The longrange central and all of the noncentral coulomb
interactions we treat in a DebyechainII expansion. After the long
range contributions are renormalized, all the remaining shortrange
"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
41
S)
0 ^
0 0
3) )
U0 >
^ c
Sa
Cd
N
o 4 r
.a 0
4 u
o (
0a
,C 0
( >
CO
14 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 fMayer
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 ffunction, Xfunction,
and [Bu]n/n! functions represented by heavysolid lines, lightsolid
lines, and ndashed 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 Debyechain expansion on the longrange
central and noncentral interactions we expand the noncentral fMayer
functions in powers of (Bv). With the decomposition of the ffunctions
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
r4
a)
44
CN Q
a)
Io
(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
44
hI
44
aC)
a)
bO
UI
01
141
'41
rr)
h
F
+
I
6
1 i
III
If
$
0O
?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 11 Cn Ca. Si Om
SI U
* b c C 4 0 *
I0 0 0
0l 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 44 CU
9 0 r 4J 0
4i bo 0 0 Cu > (
O ( CU U Cu 4J C14
U x Q) 0U
0 < 0 '
H[a 4 C I 
*4
*
1
cZI
+
$
/
/
/
@1
/
\
\
+ _
\ /
\ /
d
et
\
'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 () = expu (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.(1.1.251.1.32) are identical to those in I
for T(A) as given in Eqs.(2535) of I.
1.2 Corrections
In Section I.1 we expressed the microfield distribution function in
terms of a cluster expansion where the longrange interactions are
treated in a Debyechain expansion and the shortrange interactions in a
virial expansion. The two are not independent but involve a hybrid
virial Debyechain cluster expansion with the longrange collective
effects of the Debyechains modifying the shortrange 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
SBu 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 (
1a
2
a iax
3x
(1.2.3)
(1.2.4)
 e x)
2
a
q(x) 
1a2
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
1a
1
G(x) = 
1a
1 (eaax 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()
SBus(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( 2a ) E1((2a)ax)}]
a a 2a(1.2.19)
(1.2.19)
2
S )(eaxE(3ax) e axIn3E1(ax)}
a 1
4
p(x) = x [ (
 eaaxE1((2+a)ax)
+ eaaxln ( 2+ ) E((2a)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 aplateau, 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 Debyechain or by a virialexpansion. 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 aplateau. For a =
2.45 the aplateau 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
longrange collective effects of the Debyechains modify the shortrange
virial expansion. Hence, the range of validity of this formalism, when
applied to systems where particles interact through longrange
potentials, exceeds that of theories that use either expansion
separately. The hybrid virial/Debye expansion formalism can easily be
extended to the lowfrequency 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 shortranged and a weak longrange
contribution. The longrange weakly coupled part requires a Debyechain
expansion since these infinite range interactions give rise to collect
ive effects. The strongly coupled part given by the xbonds and the
effective interactions resulting from the summation of Debyechains we
treated in a virial expansion. However, instead we may identify a
cluster by the number of Xbonds 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. 1720 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: Xbonds 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 Xbonds. 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 Xbonds 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 Xbonds and lowest order in complexity in the Debye
chains we can extend the validity of our results to values of a > 2.0.
Iw
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0.
CHAPTER II
LOWFREQUENCY 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
2225
astronomical plasmas.2225 In the past a number of fruitful results
have been obtained inthis area with the application of new methods
developed in manybody physics and nonequilibrium statistical mechanics.
Among these methods are propagatoroperator 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) ionelectron interactions may be neglected and
their effect partially accounted for by shielding the ionion inter
action in an approximate fashion; and (3) electronradiator interactions
may be treated by perturbation theory. The first two assumptions allow
the introduction of the lowfrequency3 microfield distribution which
yields the Stark broadening due to the average static electronshielded
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 lowfrequency 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 lineshape theories to treat the ions through lowfrequency
instead of highfrequency 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 lowfrequency 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
lineshape 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 staticion 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 lineshape 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 secondorder theories which
assume that the perturberradiator 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 highZ radiators is in
the near wings, well inside the electron plasma frequency. Another
reason for only discussing secondorder theories is that the diffi
culties concerning the introduction of lowfrequency microfields into
the lineshape formalism are present in the secondorder theories.
Therefore, consideration of strong collisions, or socalled 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(ww. )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 radiatorplasma 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 eiHt (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 centerofmass 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 centerofmass
motion) (2.1.6b)
H = Hamiltonian for the plasma plus monopole term from
the radiatorplasma interaction (2.1.6c)
VI = radiatorperturber interaction (excluding monopole
term already in H ) (2.1.6d)
We have included the monopole part of the radiatorplasma 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 radiatorplasma 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
centerofmass variables of the radiator, such a convenient separation
would not be possible.
We now introduce two approximations present in most line broadening
calculations. The radiatorperturber 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<(miL) > d (2.1.15b)
P
where the transform has been performed and Im stands for the imaginary
part.
At this point the quasistatic ion approximation is introduced in
conjunction with some approximation to account partially for the screen
ing effect of electronion 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 electronelectron and ion
ion interactions, respectively; e and Ds the monopole radiatorelectron
e i
and ion interactions, respectively. The superscript s indicates that
we consider the electronion interaction, Vei, to produce some shielding
effect on the ions. Henceforth, the explicit electronion interaction
is dropped from the Hamiltonian. The shielding is frequently taken to
37
be the DebyeHickel result. A similar identification follows for the
electron and shieldedionelectric fields, E and The subject of
electron and shieldedionelectric fields, E and E i. The subject of
e i
the electron screening of the ionion 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 lowfrequency ion microfield by inserting into the
.+ +s
trace of Eq.(2.1.18a) a delta function 6(EE ) 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 lowfrequency ion microfield distribution is
defined as
s+ < +s)>
Qi () = (g(eE )>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,
<{wi[LR+Le+X(LRe+LRi) ]1 e= {wi[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 LippmannSchwinger expansion
<{wi[L + Le + X(LR + LRi)]}> =
+ i e R(e Ri e Re >e
+ ie
(2.1.26a)
0()= [1i(L + L)]1
R (W) = [wi(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
<{wi[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) = [wiLR]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);
{wi[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) = {wi[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 radiatorperturber 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
S1 Oe > 1 +
J(',0) = Im < d*{wiL (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
highZ hydrogenic radiators immersed in hot dense plasmas encountered in
a number of pelletimplosion 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) = 71 Im<[i1L]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 electronion interactions or assuming
static ions. Our development will closely follow that of Dufty.38
Make the following definition
<[oiL]1 <{wi[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)
<{mi[LR + LI + H (a)l >
= f dE <6(~EE i){ i[LR++ L+ H (+)]}>
= de <6(EE E ){i[LR+ L I() + H())]}l>
= f de <6(eE E+)> p{mi[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 e4 )> 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
radiatorperturber 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(mLR) 1L > p} d>R, (2.2.8b)
and
+ + +
Q (E) = <6(EE 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 ionelectron 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 nparticle 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 fMayer bonds. The expansion is based on the
definition of the ffunctions,
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 radiatorion, radiatorelectron, ionion,
ionelectron,and electronelectron. 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
,
44
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
51
C)
41
r'
CN U
11
00
C 0
o
NCCO
55
V
+
7.
4
!
6
H 0
a)
41 E
CO
0
co
Q a

, a
4 a)
a) a)
41
a) r
a
U)
U, C
S41
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 ii 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. 4144 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 ionion interactions is due to the elections and it is given by the
DebyeHUckel 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 shieldedion 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 ionion 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
= i1 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 techniques4548 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 BakerCampbellHausdorf 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) = Im / de Qi.() 
( _L1 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 electronion 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 theion creation and annihilation
operators for the spatial point, R Now the Niparticle 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 manyparticle Green's function in Eq.(2.5.5) we may
use FeynmanDyson 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 socalled
ring diagrams,5154 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 DebyeH*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 ionion and ionradiator 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
IH
co
N
r4
0
0
P4
w
41
~44
0
0)
)
4I4
(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(RIR ;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 electronelectron interaction line.
Since the ionion and ionradiator 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 corrections1720,56,57 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 ionion interaction where the screening, due to
the electrons, is treated in a Debyechain 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 ionelectron correlations through a screening of the ionion
interaction, we write
I(w) = Tl m f de Qs()
i R
1 1+
+ Le [wi(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 lineshape 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 DebyeHickel 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 electronion 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
+ {liLR() + < LRe[wi(LR + Le)I LRe>e' (2.7.4)
and for the denominator in Eq.(2.7.3), .***},
{3 iLR(c) + "Lii(iLR) LR>}
+ {wiLR(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 ionelectron interactions. Hence,
+E +
E goes over to E when electronion 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 DebyeHiickel
screening.1720 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 (0iL ) 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 (wiL L11
Tre e f eLRe(miLR) 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 lowfrequency microfields in the
lineshape 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 ionelectron correlations are neglected.
However, this approximation 'overcounts' the electrons. That is, if we
neglect ionelectron 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 lineshape
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 lowfrequency
microfields.
CHAPTER III
QUANTUM CORRECTIONS TO THE LOWFREQUENCY
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.26 In
these theories, electric fields of two types are considered: a high
frequency and a lowfrequency 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 lowfrequency 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 iontoelectron mass ratio. In this
chapter we investigate such a situation and treat quantum mechanically
the electron screening of classical ions.
III.1 LowFrequency 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 expIBVN + i(xe) kVoVN .
(3.1.3)
In Eq. (3.1.3) r. represents the position of the jth 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
DebyeHUckel 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 DebyeHickel 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.(A2), and
the generalized dielectric function,52
e (q,v) = 1v (q)w(q, e ), (3.2.1)
e n 1ee 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 electronelectron 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 ionion
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 longrange nature of the
Coulomb interaction. Also there are shortdistance 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 (rirj ) (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 (rirj).
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 qdependent 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 lowfrequency 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 lowfrequency
microfield calculation which treats the electrons in a DebyeHiickel
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 laserproduced 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 lowfrequency 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 Lymana and LymanB 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
44 G) 
II II
~l 0Cc0 0
W~
Lr4 D U
0A c 9
Qa x a.
II I II I
VJ 0 0
sII
0 II II II
0.)
5i 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 twobody function. This
twobody 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
socalled coupling parameter. This integration may be performed in the
hypernettedchain6265 (HNC) approximation to the twobody 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/2ZN1. (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 TwoBody 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 + ie1 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 Jdr2..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 eiqr 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 twobody function. This form has the advantage that
knowledge of a twobody function gives T(k) exactly, in contrast to
Refs. 3 and 5 which require knowledge of manybody 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
