A kinetic theory of spectral line broadening in plasmas

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A kinetic theory of spectral line broadening in plasmas
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Thesis--University of Florida.
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A KINETIC THEORY OF SPECTRAL LINE BROADENING IN PLASMAS


By

THOMAS WILLIAM HUSSEY


A DISSERTATIW. PRES-ENTED T(
THE U.iIVERSITY
IN PARTIAL FULFILLMENT OF
DEGREE OF DOCTOR


STiE GRADUATE COUNCIL OF
OF FLORIDA
THE REQUIREMENTS FCR THE
OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1974












ACIONW EDGEMENTS

The author wishes to express his gratitude to Dr. Charles F.

Hooper, Jr. for his continued guidance and encouragement during

the course of this work. He also wishes to thank Dr. James W.

Dufty for many enlightening discussions. Finally the author

expresses his appreciation to his wife, Barbara, for her patience

and understanding during the long years involved.











TABLE OF CONTENTS


Page

ACKNOWJLEIDGEIENTS . . ii

LIST OF TABIES . . v

ABSTRACT . vi

CHAPTER

I. PLASMA LINE BROADENING .. . 1

I-A Introduction . 1

I-B Causes of Line Broadening . .. 2

I-C Stark Broadening in Plasmas . 3

I-D The Line Shape . . 4

I-E Model for the Plasma . .. 7

II. UNIFIED THEORIES . . 13

II-A Introduction . 13

II-B Unified Theory of Vidal, Cooper, and Smith 15

II-C Theory of Capes and Voslamber .... 19

III. FORMAL KINETIC THEORY APPLIED TO LINE BROADENING 24

III-A The Hierarchy . .. .. 24

1III- Formal Closure . 27

III-C Short Time Limit . . 37

IV. WEAK COUPLING LItITTS . 41

IV-A Introduction .. 41

IV-B Second Order Theories . 47

IV-C Unified Theories Expansion in e 50

IV-D Unified Theory -Random Phase Approximation 54








V. DISCUSSIMN OF RESULIS . . 62

V-A Fully Shielded Uhified Theory . 62

V-B Conclusion . 64

APPENDICES

A. EQUATION FCR K(a,l;w) . 67

B. ANALYSIS OF V(a,l;w) . ... 71

C. KINETIC EQUATICN FCR X(a,,12;w) . 76

D. DII'ENSICJLESS UNITS .. .. .. 80

E. DIRECT CORRELATION FLNCTICN . 86

F. DYNAMIC SHIEDING . . 90

REFERENCES .. .. 96

BIOGRAPHICAL SKETCH . . 99











LUST OF TABLES


Table Page

1. Some Weak Coupling Limits for the Denominator of
M(W) . .* 46









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


A KINETIC THEORY OF SPECTRAL LINE BROADENING IN PLASMAS

By

Thomas William Hussey

December, 1974

Chairman: Charles F. Hooper
Major Department: Physics

In this dissertation a formal kinetic theory is used to cast

the line shape function into a form that, while similar to the

"unified" theories of Smith, Cooper, and Vidal and of Voslamber,

does not introduce some of the usual approximations. The resulting

line shape function explicitly includes the initial correlations

between the atom and perturbers, and also demonstrates the natural

separation of plasma mean field and collisional effects. The

classical path and no-quenching approximations are discussed and

ultimately employed; however, they are not required in the formal

development. The weak coupling limit is considered as a systematic

approximation to the formal results. It is shown that different

ways of applying this limit lead to different expressions for the

memory operator, some of which correspond to existing theories.

One approximation is considered which systematically incorporates

the effects of electron correlations within the framework of a

unified theory. In addition, a practical approximation suitable

for a strongly interacting plasma is discussed.











CHAPTER I
PLASMA LINE BORADENING

I-A Introduction


For a large range of temperature and density, most of the radiation

emitted by gaseous plasmas is due to atomic transitions. If individual

radiators are considered to be isolated and stationary, then the

width of spectral lines will only be due to natural line broadening.

On the other hand if the radiators move and also interact with each

other and/or the plasma, pressure and Doppler line broadening must be

taken into account. In a plasma in which a significant percentage of

the particles are ionized the dominant factor in pressure broadening

will be Stark boradening, the understanding of which requires the

study of both atomic physics and the many-body physics of plasmas.

Plasma spectroscopy has been shown to be a particularly

useful diagnostic technique for laboratory and astrophysical plasmas.1

This is due to the fact that radiation emitted by the plasma acts as

a non-interfering probe, in other words, it is not necessary to disturb

the radiating system in order to measure the spectrum of the emitted

radiation. Impetus exists, therefore, to calculate theoretically

the emitted (absorbed) line shapes. The atomic physics of light

atoms is well understood, so it is the many-body theory that presents

the chief obstacle. In Chapters I and II of this dissertation we

discuss some of the recent approaches.to this problem; in Chapters III

and IV an alternative approach based on kinetic theory is proposed

and developed.








I-B Causes of Line Broadening

If an excited atom or molecule were alone in a radiation field

which initially contains no photons5'the excited electrons could

make a transition to a state of lower energy thereby emitting

electromagnetic radiation with a frequency approximately equal to

the energy difference between the initial and final states in units

of -. In order to obtain a power spectrum for such a system, we

consider an ensemble of excited atoms or molecules and take an ensemble

average of the radiation emitted. The power spectrum thus obtained

in composed of several very narrow emission lines at the characteristic

natural frequencies of the atom or molecule in question. The theory

describing these natural lines is discussed by Heitler2 who points out

that the widths of optical lines for atoms are on the order of 10"4

Angstroms.

A calculation of a line shape that includes Doppler broadening3

requires a knowledge of the velocity distribution of the radiating

atoms. This subject along with a very thorough discussion of the

general line broadening problem is given by Griem.1

When these excited atoms or molecules are exposed to time dependent

perturbations due to their interaction with other particles of the

gas, degeneracies of their excited state energy levels may be removed

and the half-life for a given transition may be appreciably altered.4

This is called pressure broadening. In this case the power spectrum

for the radiators in question will show a redistribution of frequencies.

A typical power spectrum contains broad lines which are usually centered








near characteristic natural frequencies; hence one says that the

natural lines have been broadened and perhaps slightly shifted by

the perturbations.

If significant numbers of particles are ionized then the

strongest of the pressure broadening mechanisms will involve the

interaction of ions and electrons with the radiating particles.

Since electric fields are involved this interaction is usually

called Stark broadening. For atoms in many laboratory plasmas

this effect is quite important, typical broadened widths being on

the order of 103 times larger than the natural width. In fact,

the broadening of spectral lines in fairly dense plasmas (e.g. 1016

per cm3, 104 degrees K) is dde almost entirely to the Stark effect

which is also several orders of magnitude larger than the broadening

due to neutral atom pressure broadening mechanisms.

For typical plasma conditions dealt with in this dissertation

Doppler boradening3 will only be significant near the very line

center. However, this effect is easily included in the formalism

developed in Chapters III and IV.


I-C Stark BroadeninQ in Plasmas

There are two boradening agents responsible for the Stark effect

in ionized gases: ions and electrons. The broadening caused by each

of these is considerably different due to the difference in their

velocity distributions.4 In order to illustrate this difference,

we note that the length of time which is of importance in line broadening

is the half-life for the excited state of the atom. In a few half-

lives we may assume that an atom, originally in an excited state,








has radiated, hence any process that takes many half-lives will be

almost static from the point of view of the excited atom. It is

just this fact that enables us to distinguish electrons from ions.

The ions, being much heavier than the electrons, move more slowly

and, for most plasma problems, the distribution of the ions does

not change appreciably during a few half-lives. This is the basis

for the "quasistatic" or "statistical" approximation which was

developed by Holtsmark.5,6 The method of treating the electron

collisions represents an example of the opposite extreme, in which

a process takes a very small fraction of a half-life and may be

regarded as instantaneous. An approximation based on this limit,

the impact approximation,1,7 has been used to treat the electrons

since their great speed causes most of them to pass rapidly by the

atom producing collisions of very short duration compared to the

excited half-life. These two limits will be discussed further in

Section I-E of this dissertation.


I-D The Line Sha

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.1,4 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

4w2
P( )= C3 I(3L (I--)







where I(w) is usually referred to as the line shape. It is often
convenient to write I(w) in terms of its Fourier transform o(t),


I(W)= a r dteiwte- Yt(t) (ID-2)

0(t)-Tr {d.T(t)(pc)Tt(t)}. (I-D-3)

The damping factor, y which represents the effect of the natural

half width of the spectral line of interest was introduced in an
ad hoc manner by Smith,4 While the effect of this term is negligible
over most of the spectral line, it is a convenient quantity to introduce
since it will insure the convergence of certain indeterminate integrals.
Tr{***} denotes the trace operation taken over the plasma-radiator

system. d is the dipole operator for the radiator and T(t) is the
time development operator for the system The latter satisfies the
usual equation of motion;


ti t T(t)=H(t) (I-D-4)

and can be written formally as
SiHt
T(t)=e ; T(0)=1; (I-D-6)

where H is the Hamiltonian operator for the system, the form of which
will be specified later. The density matrix for the plasma radiator
system when it is in thermal equilibrium is given by


p=e -H/Tr {e }H

0=.l/KBT (I-I-6)


where K is Boltzmann's constant and T is the temperature.








The Fourier transform of the line shape, 4(t), can be interpreted

as the autocorrelation function for the amplitude of the wave train

of the radiation emitted when a radiator makes a spontaneous dipole

transition.7 An important property of D(t) is that


(-t)=[ (t)]* (I-D-7)

which allows us to write I(w) as a Laplace, rather than a Fourier

transform,


( e)= IR rdteite- t).t (I(t8)

This is an important difference since the initial value of any equation

of motion may be specified more easily at t=0 rather than in the

difficult limit t +-.o

The Hamiltonian operator H is taken as the sum of three terms,

H=H +H +HI (I-D-9)
a pI

where Ha is the Hamiltonian for the isolated radiator, H is the

Hamiltonian for the isolated plasma, and HI is the Hamiltonian for

the interactions between the radiator and the particles of the

plasma. H and HI can be further subdivided into the contributions
P
from electrons and ions

Hp=He +Hi +Hie

H = a+H. (I-D-10)
I ea la

It is convenient now to define the Liouville operator, L, given by4


Lf= [H,fl; L-f=HaH ,f] etc. (I-D-11)
aI n a9








We can now define the Liouville time development operator,

e-iLtf f fe t(t) (I-D-12)


With this definition the autocorrelation function, given by I-D-3,

becomes


-(t)-Tr{~*e-iLtpa}. (I-D-13)


Equation I-D-13 is now seen to satisfy the Liouville equation, a

property that will be utilized in Chapter III.


I-E Model For the Plasma


In order to evaluate equation I-D-2, additional approximations

are required. There have been a number of different theoretical

approaches to the problem of line broadening in plasmas, each of

which has employed a different set of assumptions. In this section

we will discuss a few of the approximations that are common to most

of the theoretical approaches. We will further consider what

restrictions these place on the model of the plasma.


No Quenching Aooroximation

One important approximation invoked by most theoretical

calculations of a line shape is the so-called no-quenching approxi-

mation. This has been discussed in detail by many authors1,4,7 and

we will merely outline its implications here.

Any complete set of states may be used to evaluate the trace

operation and the matrix elements contained in equation I-D-2.

However, it is usually most convenient to use the complete set









formed by the direct product of the free radiator states denoted

by Ia> and the free plasma states denoted by ia>,


jI a>= a>c> (.-l)

These states satisfy the following eigenvalue equation

H ala>=Ea a>

H ja>=Ea a>
Hp[ c> i >

(Ha+Hp )aa>=(E aE) a> (I-E-2)

where E and E are the energy eigenvalues for the free radiator
a a
and free plasma respectively. The initial and final states of the

radiator will be denoted by [i> and jf> respectively.
The autocorrelation can be rewritten by expressing the trace

as a sum over products of matrix elements,

<(t)= x
a,a'a "aa a
x. (I-E-3)

We shall be interested in the radiation that results when an atom

spontaneously decays from some excited state of principal quantum

number n to some state of lower principal quantum number n'. For

low lying levels of hydrogenic atoms it is commonly assumed that

radiationless transitions are improbable; hence, T(t) and Tt(t)

will be assumed to have matrix elements only between states

of the same principal quantum number. This restriction, which

is the no-quenching approximation, allows us to rewrite I-E-3










,(t)=- (I-E-4)


where we have used the fact that 3, a pure atomic operator, will be

diagonal between free plasma states.


Classical Path Approximation

For simple radiating systems for which the atomic physics is

well understood the real problem that must be contideredlis the

statistical average over perturber states. In general this is an

extremely difficult problem, but one which can be simplified by

employing the classical path approximation.1,7 In this approximation

the wave packets of the perturbers are assumed to be small enough

so that they do not overlap either with each other or with the

radiator. This allows us to view the perturbers as point particles

traveling in classical trajectories and interacting with classical

potentials. The effect of this approximation is to replace p.

defined by equation I-D-6, by its classical analog and correspondingly

to replace the trace over perturber states by integrals over

perturber coordinates. These simplifications are probably valid

for ions except at extreme densities and low temperatures, but

since electrons are much lighter and their wave packets much larger

they could present a problem. Several authors have shown, however,

that over much of the temperature and density ranges considered by

experiment the electrons will indeed behave like classical particles *7


Factoring The Initial Density Matrix

An additional approximation that is made throughout much of







the literature'4,7,8 is the factorization of the density matrix,

=naPiPee, (I-E-6)

where Pas Pi and pe represent the density matrices for the radiator, the

ions, and the electrons respectively. This factorization implies that

the interactions among the ions, the electrons, and the radiator have

been neglected. Static correlations between the ions and the electrons

are not ruled out entirely, however, because they can be partially

accounted for by replacing the ion-ion interactions that appear in p

by some effective interactions which attempt to account for electron

screening of the ions. On the other hand, the perturber-atom interactions

are neglected in p; the effect of this exclusion will be discussed

later. Using this factored form of the density matrix, the average

over perturber coordinates in equation I-D-13,

(t)Trd-e-iLtP}(I-E-6)

can be divided into a trace over atomic coordinates and a trace over

perturber coordinates

m(t)=Tra{d*Trie{e ipe)pad}. (pI--7)

Baranger has shown that this approximation is equivalent to the

neglect of back reaction; that is to say the trajectories of the

perturbers are assumed to be unaffected by their interactions with

the radiator.


Static Ion Annoxiamation

Ultil recently most theories of line broadening have assumed

that the ions are static: they are assumed to be so heavy that their








distribution is not appreciably altered during the radiative

lifetime of the atom. This is equivalent to assuming that the

ions are infinitely massive. Smith4 has shown that this approximation

implies that the kinetic energy part of the ion Hamiltonian will

commute with the potential energy part. The effect of this commutation

is that the free ion Hamiltonian does not appear in the Liouville

time development operator in equation I-E-7. This allows us to

write equation I-D-8 in the following form

I(W )= deP(c)J(We) (I-E8)

where

j(wE)= -R fdteitTr {d'ei iL()tp de (I.-E9)
e e aeae

and where P(E) is the static microfield distribution calculated by

Hooper.9,10 It will be noted that all of the dependence on the

ion field variable is contained in the Liouville operator, L(e).

The associated Hamiltonian is then given by

H(c)=4 +e-*R+H +H (I-E-10)
a e ae

where the ion radiator interaction is given in the dipole

approximation. is the position vector of the atomic electron

and t is the electric field strength at the radiator. For

convenience in notation we will define

Ha ()=H +ei~-
which allows us to white
which allows us to write


H(c)Ha ()+Hae +He


(I-E-.1)








L(c)=La (e)+L e+Le (I E-12)


As a result of the static ion approximation we are able to

concentrate on the electron broadening of an atom placed in an

external field; that is, we determine J(ce).

The effect of ion motion has been shown to be important only
4
in the very center of a spectral line, but there are, nevertheless,

certain physical situations where it is observable. Several authors

have discussed this problem.11-14 It will be ignored, however, through-

out the remainder of this dissertation, and the ions will henceforth

be assumed to be static.













CHAPTER II
UNIFIED THEORIES

II-A Introduction


Before the late 1950's it was thought that all Stark broadening

resulted from ions since the fast moving electrons were expected to
15 16
have no net effect. It was first shown by Kolb and Griem in

1958 and by Baranger7 that electron dynamics would be important

and result in a considerable amount of broadening. Exact

calculations of line profiles that include the effect of the electrons

are not in general possible, hence a number of theoretical approaches

have appeared, each of which uses a different set of approximations.

All line broadening theories may be divided into two broad

categories, those which are fully quantum mechanical are those

which make the classical path assumption for the perturbers. Several

fully quantum mechanical theories have been developed; however the

only successful calculations of entire line profiles have been

based on theories in which the classical path approximation has been

made. In the remainder of this chapter we will discuss only these

theories. A fully quantum mechanical treatment, valid over the

entire line shape, would be preferable, of course, but this has not

yet been possible. It should be noted that the formalism developed

in Chapter III of this dissertation is not limited to the classical

path approximation and might be used as a starting point for an all

order, fully quantum mechanical calculation.







The first line shape calculations that included electron

broadening realistically were the so-called impact theories of Kolb

and Griem, and of Baranger. While these approaches employ a factored

density matrix and make the static ion and classical path. approximations,

they also make the impact approximation and the completed collision

assumption. The completed collision assumption treats the electrons

as moving so fast that collisions can be considered instanteous.

Baranger has shown that this is equivalent to a Markov approximation

that leads to a considerable simplification. Also included in the

impact theory is the impact approximation. This assumes that

close collisions between atom and electrons will occur one at a

time and, hence, further simplifies the calculation by considering

only binary electron atom collisions.

The early impact theory calculations also took the collision

operator (Chapter III) to second order in the electron-atom

interaction. Any theory that makes this approximation is called

a second order theory. It should be pointed out that not all

second order theories are impact theories: Smith and Hooper in

196718-20 developed a fully quantum mechanical second order theory

without the completed collision or impact approximations.

It will be observed in Chapter IV that second order theories

are valid only in the line center, while one-electron theories,21

which result from expanding J(we) in powers of the inverse frequency

separation are valid only in the line wings. Recently several unified

theories have been developed that are valid both in the line wings

and in the line center.







II-B ,1ified Theory of Vidal. Cooper and Smith

The first successful unified theory calculations were carried

out by Vidal, Cooper, and Smith (VCS).22,23 Although they used the
classical path and static ion approximations together with a
factored density matrix they included all orders of the electron
atom perturbation.
The starting point for the VCS calculation is the electron
broadening line shape, given by

J(wE)= eR dtei tTrae .e iL( )tpa (IIB-I)

where the approximation that pae=ape has been utilized. The form
of II-B-1 can be simplified by separating the trace over atomic
coordinates from that over electron coordinates and by performing
the Laplace transform:

J(we)= ReTra( .Tr {' p}p }. (I-B-2)

VCS then employ a projection operator technique developed by Zwanzig24

ind by Mori2526 to obtain the result:


( w) e a arL a M()
a
where

M(w)=i f dte"AwOPt
Jo ea ea

G(t)--T {- L dt (l-P)L (t-)}

As =e-L (c); L (t)=ei(La +Le^(acL (II-B)4)
Op a ea ea







T is the time ordering operator and P is the projection operator

defined by

Pf=PeTre f}. (II-B-6)


In equation II-B-3 all of the complicated dependence on the N electrons

has been transferred into the effective atomic operator, M(w). This

is a particularly useful functional form for the line broadening

problem since M(w) can be interpreted as a frequency dependent width

and shift operator.8

Equation II--4 is still not a form which is amenable to calcu-

lation. One means of approximating this equation is to set the time

development operator in II-B-6 equal to 1. This leads to the so-

called second order theories which receive their name from the

fact that M() is second order in the electron-atom interaction.

Many line broadening calculations have been made using this

approximation: from the early calculations of the impact theory

to a fully quantum mechanical treatment by Smith and Hooper.18A '

However, this approximation breaks down in the line wings. In other

words when L and therefore. M(), is large, it is incorrect to Stop
ea
at finite order in an electron-atom perturbation expansion.

Since the unified theory is supposed to be valid in the line

wings as well as in the line center, setting the time development

operator equal to 1 is not allowable. VCS showed that to lowest

order in the density the time development operator could be

approximated in the following wayj


G(t)--T {- L dte-iLet(1-P)EL (j)eiLet}
exp ni ) j ea
i. t
zIT {- tdt'L (j.t')} (II-B-6)
j exp -i ea








with M(w) given by


M(W)=-iNfdtei'bpt ea exp 0G ea ea

The average is performed over the coordinates of particle 1 only.

It should be noted that II-B-7 is all orders in the electron-atom

interaction. It was shown that in this approximation

(1-P)L (1)-L (1). (II-B8)
e Cea

One important limitation imposed by taking only the lowest order

term in the density expansion is that electron-electron interactions

are omitted both from p and L (j,t). VCS partially correct for
ea
this by assuming that the electrons can be replaced by shielded

quasi-particles, in other words they assume that the electric

field in e *R is shielded. We will see in the next section

how Capes and Voslamber included electron-electron correlations

in their theory.

Another VCS approximation, the neglect of time ordering from

equation II-B-7, has been discussed in detail by Smith, Cooper, and

Roszman,28 and it will not be further considered here. Integrating

by parts, equation II-B-7 becomes

iM( )=hA-itA pjdteiAoptAWop. (II-B-9)


Finally VCS assume that the time dependence of R can be neglected

in the expression for L (1,t); hence


L (lt)='e-ie ()t()ei1e()t (IIB-10)
ea








Instead of actually performing a spatial integration over the

time dependent shielded electric field, e4i (1)te'(1)e,()

VCS replace it by an unshielded one and cut off the resulting

integral over the free particle trajectories at the Debye sphere.

It is interesting to note that if the upper limit of the integral

in the exponential of II-B-9 is extended to infinity, the time develop-

ment operator in the interaction representation goes over to the

S-matrix and we regain an all order impact theory. This is entirely

equivalent to the completed collision assumption. Thus the

expression for M(w) finally evaluated by VCS was

(wI)=-inAWop dtei OPtA)op (I-B-1l)

The resulting matrix was inverted to yield an expression for the

line shape.

The numerical results obtained by this formulation of the

line broadening problem has been shown to agree well with most

currently available experimental results. Nevertheless, the

approximations that have gone into this development are not entirely

transparent. The density expansion was truncated in a way that

precludes the possibility of going to higher order. In addition,

there is no way of examining the validity of the way in which

electron correlations were included.

It is also possible to formulate a systematic kinetic theory

approach to line broadening. This method, as developed by Capes

and Voslamber,27j29 will be discussed in the next section.








II-C Theory of Capes and Voslamber

A systematic treatment of the line broadening problem may

be effected by viewing the radiating atom and the N electrons as

an N+1 particle system obeying a Liouville equation. Capes and

Voslalber27,29 have developed such a theory that included electron

correlations in a more systematic fashion. This section will outline

their approach, indicating its advantages and where it differs from

the VCS theory.

Capes and Voslamber make several of the usual approximations

including the classical path approximation, the neglect of perturber-

atom interactions from the density matrix, and the static ion

approximation. They do not, however, neglect electron correlations,

and their theory offers an understanding of what effect these correlations

will have on the line shape.

As in the VCS theory, Capes and Voslamber start with the

equation

J(ae)= tRe dteitTr a{dTre{e-iL(e)t ^ (II--)
aT e jf e a

which they reduce to a simpler form by utilizing the fact that the

integrand obeys a Liouville equation:

( +iL +iEL (j))(a.l*N;t)+(Ev.f --* -^ -* -)l.(al*N;t)=0
at a ea j j m xj v
j kj ax. av
e2 r -- (II-C-2)
3 3 3^k

where we have defined

6(a,lo .N;t) tNe"iL(E )tpp (II-C-3)








and where we have explictly made the classical path approximation.
Taking partial traces of II-C-2, a BBGKY hierarchy of kinetic equations
results
a i N 9( a -
a ea
+iLa +i L e(j)) (a,l-..st)+ )E (vj -V -. -)D(a St)
a j=l j=1 m =+1 ax. 3 av

=-infd(s+l)L.a(s+l)(al ....ss+lt)-mfd(s+) a Ssi. --- D(al **st)
i= xi avi (II--4)

where

(a,l"**s;t) Trs+1*..N A{a(a," *N;t)}. (II-C6)

The first two equations of this hierarchy are

( +iLa)I(a;t)=-in fd(2)l(a,l;t), (II-C-6)

( tiL +iLe (1))4' )#(a.l;t)=-in fd(2)L (2)6(a,1,2;t)
at a ea 1 1 ea

m +
D- fd(2) -^21 --(a.lV2st). (II-C-7)
ax av
1 1
Since J(we) could be written as

J(wE)= LR Rdte Tr {C*aat)}, (II-C-8)

Capes and Voslamber then solved for I(a;t). To do so they assumed
that M(a.l,2;t) could be expressed as an approximate functional of
l(a;t) and i(adl;t). This resulted in a pair of coupled differential
equations which were solved simultaneously. Their closure
relationship based on the weak coupling approximation was

I(al,2;t)=D(a,l;t)Vp(2)+6(a,lst)Vp(l)+6(ast)V2(p(12)-p1p( 2)) (II-G-9)








where the reduced density matrix is given by30


p(l1.-s)= *** d(s+1) d(N)oe.

If the closure relationship, II-C-9, is substituted into the
second equation of the hierarchy, II-C-7, and if only those terms

which are lowest order in an expansion in the electron-electron

coupling strength are kept, then II-C-7 becomes
4o-V)*V 21 ,
( +iL +iLea(1)+5 )i(a,;t)=- -- d(2) (at)
at a ea m 3v f X
av 1ax
1 1

-info(vJ)fd(2)Le(2)D(a,2;t)-nfo(vl) d(2)Lea(2)g(12)p(2)t(ast)s


P(12)=p(1)p(2)g(12); fo(v )--(l), (II-C-10)


where the symmetry properties of the interactions have been used
to eliminate some of the resulting integrals. The first term on
the right hand side of equation II-C-10 is recognized as the Vlasov
operator, V(1).27 If the Laplace transforms of II-C-6 and II-C-10
are solved simultaneously, we find that


.(al;w)= i L a(l1-(a; W); (II-C-11)
w-L -L (1)+iv .V -V(l)
a ea 1 1

V(l) is again the Vlasov operator, and Le is the electron-atom
ea
interaction, statically shielded by the electron-electron pair
correlation function. If equation II-C-11 is substituted into
the right hand side of the Laplace transform of II-C-5 then we get


D(a,l;)= w-L pd (II-C-12)
a








where the memory operator, M(w), is given by


M(o)=-inf d(l)Le (1) ---a .-- Ls (1). ,
ea ea (II-C-13)
w-La -L (1)+iv *) -1 V(1)
a ea 1 1

Using a technique similar to the one presented in Appendix F Capes

and Voslamber showed that this equation reduces to

()=-iN d() dteiAoptLD (lt)U(t)L5 (1)
f 2f ea ea

U(t)=r T ie dt'IeaL t)j (II-C-14)


where Le (1.t) is a dynamically shielded interaction and U(t) is
ea
the interaction representation time development operator. This

result is identical to that of VCS except for the shielding which

appears in the electron-atom interactions. Note, however, that

the electron-electron interaction appearing in U(t) is not shielded

while the two interactions appearing around it are. Thus it is not

possible to further simplify 11-0-14 by performing an integration

by parts as was done in equation II-B-10.

The main strength of the approach of Capes and Voslamber is

the fact that their theory includes the effect of electron

correlations in a way which is preferable to the ad hoc cut-off

procedure employed by VCS. Its main weakness, however, is the

closure hypothesis, equation II-C-7. Capes and Voslamber show

that this is closely related to the impact approximation (binary

collision approximation for electron-atom interactions, and like

the impact approximation, there is no clear cut way of improving

upon it. In the next chapters we will close the hierarchy in a




23



way in which approximations may be more systematically made, and

we will suggest an approximation procedure which leads to a more

inclusive result than found in equation II-C-14.












CHAPTER III
FORMAL KINETIC THEORY APPLIED TO LINE BROADENING

III-A The Hierarchy

In this section we will develop a kinetic theory of line

broadening in plasmas similar to that derived by Voslanber.27,29

The formalism will be developed for an atom, perturbed by an
external field, and immersed in a one component plasma; therefore

the static ion approximation is implicit. However, two other

frequently employed approximations will be avoided: the classical

path approximation and the neglect of electron-atom interactions

in the initial density matrix.

From equation I-E-9 the line shape function for electron

broadening is given by


Jd(w)= I R dte) tTrae {-e-iL()tp d)
11 e0 ae 0a

J(wE)= R Trae{. i P } (III-A-1)
e ae aN )

This can be written in the alternative form:





J()= 1R Tr { d D(a, **N;w)} (III-A-2)
7 e ae f

where we have defined the operator t(al...*N;t) and its Laplace

transform. i(al...N;w), as in equation II-C-3;













(III-A-3)


where V is the system volume and paN is given by

pN=e-H() r e-BH(

H(e)=H (e)+EH (j)+ Z H i+EH (j).
a i
In the above equation H (j) represents the kinetic energy of

particle j, Hij represents the interaction between particles i and
ee
j, and Ha(j) represents the interaction of the atom with electron J.
The Liouville operator corresponding to H(E) is given by

L=L +EL (j)+ z L i+ EL (j) (III-A-4)
a e i
where here and henceforth the functional dependence of L And L on
a
the ion microfield, will be suppressed. We also define the reduced

functions,


1(a 1.. ; t)=VSTr s *N(e-i i\ a}

D(al*..s; )---VTrS, + {. -L pa a
s*1** 1-


p(al.*.s)--Trs. {paN }.

Hence the expression for J(We) can be written as


J(ws)= RiR dteitTr {d.D(a;t)},


J(w)) ;R Tr {d.D(a;f) }
Tre a


(III-A-6)


(a, Nt) e-iL()t


(III-A-6)







and thus the problem is to determine 5(ast) or D(a;").

We start by noting that the operator I(a,1...N;t) obeys the

Liouville equation,31


( +iL)D(al--N;t)=0,
at


from which we can now generate a hierarchy of kinetic equations

by taking partial traces;


( iL())(a 1 st)-in Tr (L (s)+Lis+1)
at i= Ts+1* *N ea ee
i=1
I5(al.*.s+l;t) (III-A-8)
S 5
LsL + E L (j)+ E L (j)+ E Li
a j= ej ea i

where the thermodynamic limit has been assumed. The first three

equations of this hierarchy are given explicitly:


( +iLa)(a;t)=-inTri{Lea(1)(al;t)}
at- ea


( +iL(l))d(a,l;t)=-inT2 {(Lee(2 e )(a,l,2;t)} ,
(at +iL(2))(a,2;t)=-in3{(Lea(3)+L+ee)a 2,3)}
a (2) )N.l,2;t)iin~ (L (3)L +L 32 )t1(a.l.2,3;t)}.
atea ee ee


The approach followed by Capes and Voslanber to solve for D(a;t) was

to use a closure relationship, expressing D(a,l,2;t) as an

approximate functional of D(a;t) and D(a,1;t). While this procedure
led to a closed set of coupled differential equations that could be

readily solved, its weakness lay in the nature of the closure

relationship. They justify their method of closure by relating it

to an expansion in the coupling parameters; but the limitations of


(III-A-7)


(III-A-10)








their technique are not clear and a method of improving upon their

results is not obvious.

The procedure followed by Vidal, Cooper, and Smith2223 was not

as inclusive as the one used by Capes and Voslamber, but it does

have one important advantage: the expansion in the density, used

in order to get an expression for the memory operator, is well

understood and can be related to a diagrammatic expansion of the

self-energy operator.

In this dissertation we combine the advantages of the

hierarchy approach of Capes and Voslamber with those of the weak

coupling limit. In the remainder of this chapter we will develop

a formally exact method to close the hierarchy.32,33 In Chapter IV

we will apply the weak coupling limit to the formally exact

expression for 5(a;t). It will be seen in Chapter IV that, depending

on how the weak coupling limit is applied, we can reproduce several

of the existing theories of electron broadening as approximations

to the exact theory. Furthermore, we then generate our own

approximation procedure which enables us to develop a more systematic

theory.



III-B Formal Closure


First Equation of the Hierarchy

Rather than immediately employ an approximate closure

relationship, we will proceed formally to obtain an exact relationship

by observing that 5(a,l,.***st) represents a linear map of an

atomic function onto a space containing functions of electron

coordinates as well as atomic coordinates;












U(a,l..*s;t)=Tr s+*N VNe-iLtpaN }p'l(a)

U_(aL_**s )N 1 N 1- p-(a).


For the specific case where the trace is taken over all N electrons
we get

D(ait)=U(ast)p(a)c


(IIIB--3)


U(a[t)Tr N.VNe-i LtPN }p-(a)

)(a)Tr*..**N N d NP (a).


(III-B-4)


The first step in effecting our closure will be to eliminate p(a);
in III-B-1 in favor of D(a;t) in equation III-B-3. Hence, assuming
that an inverse exists for U(a;t) and i(a;w), the functional
which result are


(111-6-6)


or the Laplace transformed version,


D(a,L1***sID(a; w))=U(a,l"*s; w)U" (as w)D(a;).


(III-B-6)


U(a,lo**s;t) and U(a,1'..s;.w)'1(a;w) will in general be extremely
complicated operators. Equation III-B-6 gives us a formal method


(III-B-1)


(III-B-2)


b(als;tl~(a;t))=v(arl...s t)v~l(a t)~t(a t),







of closing the hierarchy of equations at any level (any value of s).
The first equation of the hierarchy, III-A-10, may therefore
be written


(- +iL ) D(a;t)=-inTr L ea(1)U(alt)U (at)}(at) (III-B-7)

This can be cast in a different form,34

a ft
(11+iL )D(a;t)=-ij dt'x(t'-t)D(at) (IIIMB-8)

by introducing a collision operator, x(t'-t), which is non-local
in time. This is a particularly easy form to Laplace transform.
Inspection of equation III3-B- shows that as t- 0, t(a,l...s;t)
approaches a time independent functional of b(a;t)

rim 6(al*"s;t ID(a;t)=((a,1. *s;t=0 I(a;t))
t-D

=U(a,l..***s;t=)U a;t0)D(a;t). (III-B-9)

This shows that x(t'-t) has a singular contribution at t'=t.
Extracting this part explicitly from x(t-t) yields


( +iLa)D(a;t)=-iBD(ast)-i) dt'M(t -t)5(a;t') (III-B-10)

where B is time independent. The operator M(t'-t) is now non-
singular and the integral in which it appears vanishes as t- 0.
If equation III-B-10 is Laplace transformed we find that


(-La)D(a s ; )=iD( a;t=0)+M( o)(w a; a)+B( a; )

D(asa)= La Pa (11-11)
a







This result displays the same functional form obtained by VCS where

M(w) together with B plays the role of the VCS memory operator.
It can be shown that B vanishes when electron-atom interactions
are neglected from the density matrix. Consistent with the
separation of the singular part of the collision operator, x(t'-t),
from the non-singular part, we write )(a,1*'*s;tjD(a;t)) as the sum
of its short time limit, equation III-B-9, and a time dependent
remainder which vanishes at t=O:


D(a,l***s;t D(at))=U(a,l...***ss;t=0)U 1(at=0)(a;t)+; (a,1***s;t)(III-B-12)

where

(a,***sst=0)=0. (III-B-13)
It can be seen that i(a,ljt) is related to the non-singular part
of the collision operator, x(t'-t). Equation III-B-9, which is
sometimes called the short time limit, is discussed in Section C
of this chapter.
We now make a few observations about I(a,l*...s;t=0O(a;t))
and P(al.*.s;t), given by equations III-B-9 and III-8-12.
From the definition of U(a,l***s;t) given by III-B-2 we have


U(a,1...s;t=0)=p(al**s)p" (a); U(a,t=0)=1

U(a,l ***.t=O)U (a,t=0)=p ( a,l***s)p (a) .


(III-B-14)


Substituting III-B-14 into III-B-12 and taking the Laplace transform
gives

-$(a,1*s|(a;)=p(al***s)p" (a)D(a)1Pal*s- ). (III-B-15)








The closure relationship, equation III-B-6 in conjunction with
equation III-B-15 enables us to write:

(a.l-*s;W)=K(al--***s;))6(a;W) (III-B-16)

K(a***s;w)=U(a***s)U (a)-p(a,***s)p a). (III-B-17)

The functional form of the operators, B and M(wl can now be
exhibited explicitly. Considering the Laplace transform of the
first equation of the hierarchy, equation III-A-10,

(w-L )6(a:s);=Tr {L (l)(1a.l;W)}=ip(a))
a 1 ea

and by using equations III-B-15 and III-B-16 we arrive at

(W-L )D(a;w)-nTr {L (1)p(a,l)p "(a)} (a;w)

-Tr ( )K(a ;w)(a ))}(a;i)=ip(a) (IIIpB-18)

This can be compared to equation III-B-11 to get



u-L a.B-M(w)

B=hTriL a(l)P (a.l)p'" (a)}

M(w )r=hT l (l 1)(a,l;)} (III-B-19)

The operator, B, as it is defined in equation III-B-19, is a simple
funcfon of well defined operators and can be calculated. The operator
M(w), on the other hand, contains the formal operator, K(a,l;w),
which is obtained from the next equation of the hierarchy.








Second Equation of the Hierarchy


The approach used to cast the first equation of the hierarchy

in the form of a linear kinetic equation is well known. However,

it has been shown that if we close the second equation of the

hierarchy in the sane manner as the first, a useful result,
32s33
which may be easily and systematically approximated, is obtained3233

The approximation technique used will be discussed in detail in
Chapter IV.
From equation III-A-10 we have the second equation of the

hierarchy,

+iL(l))D(a,1;t)=-inTr2{(L (2)+L21)D(al,2;t)}.
at

In order to put this into a form where K(a,l;w) appears explicitly,

we first use equation III-B-15:


(al;t)=p(a,l)p"l(a)D(ast)P(a,ldt)

t(a,1,2;t)=p (a,1,2)p- (a)D(a;t)+P(a,1,2;t). (III-B-20)

If these relations are substituted into equation III-B-20 the

resulting expression is

(_ +iL(l))p(a,l)p-a 1(a)D(a;t) (3. +iL(1))P(al;t)
at at

=-inTr {(L (2)+L2 )p(a,1,2)p (a)}D(ast)


-inTr2{(L (2)+Lee )P(a,1,2;t)}. (III-B-21)


This equation is developed further in Appendix A where the

properties of the equilibrium hierarchy for an N+1 particle system







30
are used.30 The resulting kinetic equation is

( +iL(1))P(a,l;t)=-iGL (l)p .(a (ait)+ip(CaD)x
ea

xp (a)nTr ILe (1 )P(a;t)}-inTr {(L (2)+L21)-P(a1.2;t)}(III..B22)
ea 2 ea ee

where the operator GL (1) is given by
ea

GLea(l)p'l (a) (at)=[p(al)-np(a,l)p1 (a)nTr 2 {(a2)P 21

+nTr2 p(a ,12)P21 ]La ()p-(a)(a;t) (I--3)

and where P.. is the permutation operator, defined in Appendix A.
Taking the transform of equation III-B-22 we have

(W-L(1))(adla)+p(al)p-1 (a)nTr {Lea (l)P(a,ls g)}
ea

nTr {(L(2) +L )P(a.,2;e)}=GL (1)D(a;) (III-824)

where we have used equation III-B-13. If we use the definition that
P(al..*s;w)=K(a.l***.os;)(aiw) then equation III-B-24 becomes

(w-L(1))( a, I;w)+p(a,l)p"- l(a)nTr1 ({l e )K(ai;)}

21
-nTr2 {(Lea (2)+Lee) (a,1,2;-)=GLea (1) (IIIB-25)

Thus we have a kinetic equation for K(a,l;w) in terms of K(a.1,2;w )
and a source term, GL (1). We now make the formal definition
ea

V(a,1aw)K(adjw)=hTr2 21 w
V(al)Tr2 {(Lea (2)+L1ee)K(a,2; )

"p(aD1)p'(a)nTr 1 (1)K(a,;l )} (II1-Br6)
which is discussed in Appendix B. With the use of equation III-B-26







we solve equation III-B-25 formally to get a convenient form for

K(aal;w)


(III-B-27)


K(a,l;w)= -- .--- GLea(1) .
T-L -V a,,e) a

The memory operator then becomes


M( )=-inTrj{L (1) (1) 1 Lea(1)1.
ea -L -V(a,1;w)


(II-B.-28)


It is now possible to analyze V(a,l-w) in exactly the same manner

that we analyzed the collision operator, x(t-t), that is, we

separate V(al;w) into frequency dependent and frequency independent

parts:


(III-B-29)


The infinite frequency term, V(a,l;w)=-), is analyzed in Appendix B

with the result that

V(a,1;w= )K(a,1;w)=hTr {(L (2)+L )x
2 ea ee


1 ea


K(a,1...ss))=, (al-*s; )L (1).
ea


The collisional part of III1--29 can best by analyzed by continuing

to the next equation of the hierarchy.


V(a,l)=VV(a,lu.=)r+c(a.lj) .


(III-B-30)








Third Equation of the Hierarchy

We already have expressions for M(w) and J(we) in terms of

V(a,l;w=") which are formally more inclusive than any previously

derived. In order to fully understand what part of V(al;) is

included in its short time limit, V(a,l;w=v), we must understand

what is excluded from it; that is, we must look at V (a,1;w). To

do this we will require the third equation of the hierarchy,

(t -+iL(2))(a.l,2;t)-inTr3{ (L (3)+Lt +L1 )(al,2,3;t)} (III-A-10)
3 ea"e ee ee

From Appendix B, equations B-9 and B-10, we know that:

K(a,l ssaw )=e(ald*ss;o)L (1)p '(a)
ea

K(a,1**s;wa)=i(a,1.**.s;w)) (a,l;w)K(al;w). (III-B-31)

Then in equation B-1l we defined the operator V(ail;w):

V(a,lw)K(a,l;a))=nTr2 {(Lea (2)+L 21)( (adl,2; )9(al;wa)}K(alaw)

-p(al)p I(a)NTr 2L (2)P }K(a,1e). ;-'
2 ea 21
We now want to divide V(a,l;w) into frequency dependent and

independent parts, in a manner suggestive of the separation of the

collision operator in equation III-B-11. In order to do this we

consider the inverse transform of K(al.**s;w) and write it as a

functional of the inverse transform of K(a,l;w),

K(a,1*" s; t)=K(al" .s;t K(a,l;t)). (III-B-32)

Following a procedure analogous to that used to derive equation

III-B-12 we now separate this last expression into a short time

functional and a time dependent remainder term:








(III-B-33)


K(a,l".s;tl K(a,l;t))=K(a,l. s;t=i0 K(a,l;t))+X(a, l- s;t).

The Laplace transform of equation III-B-33 yields

K(a,l***s;w I K(a,l;w) )=K(a.1***s;t=OI K(a,.l ))+X(al*** s;,).


(III-B-34)


Hence, comparing this result with equation B-13, we observe that

the infinite frequency limit is related to the short time limit:


K(al1...***sjw (ala))=9(al-***s ;=) (a,l;w,=-)k(a,l;w)

+X(a.l*s ) (a


(IIIB-35)


(III-B-36)


The similarity of the above with equations III-B-12 and III-B-17
suggests that the approach used to determine the functional form
of V (al;w) from the third equation of the hierarchy will follow
that used in Section III-B to find the functional form of M(w)
from the second equation of the hierarchy.
With the exception of an intermediate step, the algebra involved
in converting the third equation of the hierarchy into one for

x(al1,2;w), defined by equation III--35, is similar to that
used to convert the second equation of the hierarchy from an
equation for d(a,1;t) to one for K(a,l;).32,33 The result of
this conversion, determined in Appendix C, is

(w-L(2))x(a.1.2;w) (a,2w~m) ) 1 l(aiw= )x

x[nTr { (L(2)+ (al,2;w )]-nTr3 (La(3)+L32+
where S(a 2) is a complicated, frequency independent source term

where S(al,2) is a complicated, frequency independent, source term.


-f(aZ***s~;W=0 ) .3 (alw***s.s;)] K(al jw).







If we further define an operator, W(al.2;w), as follows,

W(a,lP,2;W)x(a,1,2,-w)=((al,2;sa=)X 1(a.l;w=)x

TnTr {(La(2)+L2 )X(a,1.2; )}-nTr {(L (3)+L31+
2 ea ee 3 ea ee
+L32) (a,1,2,3jw) A

we can formally solve for x(aAl,2;w) and Vc(al;w) in terms of
S(a,1,2) and W(a.l,2;w):

1
x.(a,1,2;w )= 2)-- S(a.1,2).
L-L -W(a,1,22w)


(III-B-38)


V (a,1; )=-inTr {(L(2)+L ) -.. ...S(a,1,2)}, (III-B-39)
c 2 ea ee (-L a.1.2;w)

where we recognize that Vc(a,l;w) is still formally exact with all
the complicated N-body effects hidden in the operator W(al,2;wj).


III-C Short T L

In the next chapter we will consider some possible weak
coupling limits to the memory operator, RM(o). Before we do this,
however, it will be useful to consider an alternative approximation
method. In equation III-B-10 we rewrote the first equation of the
hierarchy in the form:


(III-B-10)


where B represented the singular, time independent part of the collision
operator, X(t'-t), and M(t--t) represented its time dependent,
nonsingular part. From the Laplace transform of equation

III-B-10 we got


a dt >
( -+iL )(at)=-BD(a;t)-i dt'M(t'-t)D(a;t-),
at a Jo








(III-B-10)


(a;W)= ua-L-B-M(w) P
a


B was called the short time limit of x(t'-t) and M(w) vanished in

the infinite frequency limit. Hence, the memory operator was divided

into a frequency dependent and a frequency independent part. We

will see in this section that it is useful to apply the analogous

separation to higher order equations in the hierarchy.3536

Equation III-B-11, above, is exact and has the same form as

the expressions for M(w) and c(a,l;w) which are also exact:
c


GL (1)p (a)}
ea


(III-B-28)


V(a,l;w)=-inTr {(L (2)+L21) .i S(a,1,2)}. (III-B-39)
2 ea ee -L -W(a,1,2;wu)

Thus, in formally closing the first equation of the hierarchy, we

have put all the effects of the N electrons into the atomic

operators, B and M(w). Similarly, in formally closing the second

equation of the hierarchy, we have cast the evaluation of MR()

into the form of an effective two body problem (the atom and one

electron) where the operators, V(al;twm) and Vc(al;w) contain

the effects of N-1 electrons. Thus, in both the expression for

6(av ) and that for M(w) the many body effects have been divided

into a short time (infinite frequency) limit and a frequency dependent

part that vanishes at t=O (w ). The former terms contain mean

field effects while the latter contain collisional effects.

Next we consider the physical significance of B and M(o) as

they appear in the effective atomic resolvent which governs the

time development of the operator p a, equation III-B-11. If
a


wit WTa (ai;u>


ir(w)=-inTr {Lea (1)








we were to neglect M(w) then all electron-atom effects would

be contained in B and the time development of P d would be governed
a
by the mean field electron-atom interactions. The analysis of the

separation of V(a,liw) into V(a,.li; ) and V (a,l;w) in equation

III-B-28 is very similar. If we were to neglect V (a,l;w), the

time development represented by the resolvent operator,

equation III-B-28, would be governed by the exact interaction of

the atom with one electron, L(1), plus the mean field effects

of the remaining N-1 electrons included in V(a,l;w='). Another

way of stating this is that the atom is perturbed by a single

electron moving in a static background due to the other N-1 electrons.

This discussion suggests a possible systematic approximation

method for calculating the line shape function, I(w). Keeping

only B in the expression for I(w) is the crudest approximation and

leads only to a slight shift in the location of the line center.

Making the approximation at the next level, that is, neglecting

only V (a,l;w) from the denominator of the expression for M(w)

includes the interaction of the atom with one electron exactly,

together with the mean field effects of the other N-l electrons.

It will be observed in the next chapter that most existing

theories of electron broadening can be obtained by taking some

sort of weak coupling limit of the result of this approximation

to M(w). It is possible to go further and keep only the short

time limit of W(a,l,2;w) which appears in the denominator of the

expression for Vc(a,l;w), but since retaining only V(a,l;=F-)

yields an expression for the memory operator which goes beyond




4U



most existing theories, the nature of Vc(al;iw) will not be emphasized

here.

The approximation procedure outlined in this section should

be regarded as a possible alternative to the weak coupling limit.

It is hoped that this method will serve as a starting point for a

fully quantum mechanical unified theory applicable to a high

density plasma.












CHAPTER IV
WEAK COUPLING LIMITS

IV-A Introduction


In Section III-C we discussed an exceedingly useful

procedure for obtaining approximate line shape functions by taking

the short time limits of the different collision operators. In

this chapter, however, we will consider a different approximation

procedure which, we shall, show, parallels more closely the

results of existing theories of line broadening. This technique,

called the weak coupling limit, involves a perturbation expansion

in some sort of coupling parameters; the impact approximation

mentioned in Chapter II is related to this technique. In this

chapter we identify possible expansion parameters and explore

some expansion techniques. We will find that variation in

the weak coupling methods lead to different expressions for the

memory operator, some of which we will relate to existing theories.

One expansion will be developed that leads to an expression for

the line shape that contains several of the previously developed

theories as approximations, and which can be systematically

carried further than any of them.

In Chapter III the results were exact within the limitations

of the static ion approximation; neither the classical path nor

the no-quenching approximations were made. However, the bulk of

the literature concerning line broadening has been within the

framework of these approximations, and they will also be assumed







in the remainder of this dissertation. On the other hand it

will be shown that several of the important approximations which

have been introduced in an ad hoc manner by many authors,16,17,22,23

the impact approximation, the neglect of electron-electron

correlations, and the neglect of electron-atom correlations from

the density matrix, will follow from simple expansions of the

memory operator in the various coupling parameters.

However, a difficulty that arises in the coupling constant

expansion as applied to the line broadening problem in the existence

of two intrinsically different types of interactions: the electron-

electron interaction and the electron-atom interaction. The

electron-electron interaction is treated extensively in the plasma

theory literature.37,38 The electron-atom interaction, however,

will contain atomic operators which are unrelated to the fundamental

lengths of the usual plasma problem. This suggests that the

coupling constant we use for the electron-atom interaction

should be independent of the one we use for the electron-electron

interaction. In plasma line broadening, the concept of truncating

a hierarchy of kinetic equations by expanding simultaneously in

two independent parameters was carried out first by Capes and
27
Voslamber.27 While also using a two parameter expansion, we

will employ a different electron-electron coupling parameter

and will apply the weak coupling limit in a somewhat different way.

We now discuss possible expansions and expansion parameters

further. The most straightforward method is to expand in powers

6f the coupling constants, Aee and e defined by the relations,
ee ea








V ae(j)=xae V (j); L ae(j)= L (j); (IV-A-1)
ae ae'ae ae ae ae
V e(j)eeVe(j); L (j)=X L (j). (IV-A-2)
ee eeee ee ee ee

This is the approach used by Capes and Voslanber and we will later

show that our technique for approximating the kinetic equations

gives a result very much like theirs if we go to the same order

in the two coupling constants as they did. The expansion in the

electron-atom coupling constant is a good one, as seen in

Appendix D, and we will use it in this chapter. On the other hand

it has been shown in the literature that for large ranges of

plasma temperature and density where the electron-electron coupling

constant may not be a valid expansion parameter, an expansion in

the plasma parameter is the better-choice.37

The procedure involved in expanding in the coupling

constant for an interaction is a simple one which orders contributions

to quantities being expanded in powers of that constant. The

concept involved in the plasma parameter expansion, however, is

slightly more subtle; before the coupling parameter is ever

identified, all expressions are first scaled to an appropriate

set of units. The natural units with which to scale the plasma

problem are

w (4irne2/m) I=(KT/4ne2)"

x-x' ; vv-v= -- (IV-A-3)
D p Pp
As shown in Appendix D, where the equations involved in the line

broadening problem are scaled to these lengths, the appropriate








expansion parameter is the plasma parameter:


A = (IV-A-4)


where N represents the number of particles in the Debye sphere.

It is also observed that if the kinetic equations of line broadening

are scaled, the electron-atom coupling parameter becomes

e2a0
S--- (IV-A-5)
ea T f


From Chapter III the line shape due to electron broadening

is

J(.E)= a R TrL W) p(a)l} (IV-A-6)
e a W-L aBP
a

where the factors, pa, B, and M(w), will in general contain terms

which include all orders in all of the coupling parameters.

Expanding p(a), which appears in the numerator to lowest order

in Xee and Aea is equivalent to replacing it by e H/Tr{e B},

where Ha is the unperturbed atomic Hamiltonian. The resolvent
i
operator -L B-i(m) is more difficult to approximate, however.
a
A straightforward perturbation expansion which is first order in

any of the coupling constants, yields

= 1- rl i(B+M(w)) 1) -- (IV-A-7)
w-La -BM(w) -L -L a

where (B+M(w))) represents the lowest order, nonvanishing, term

in a perturbation expansion of B+1 i(). Near the line center, where

M(w) approaches M(O), the effective atomic resolvent appearing in

the line shape function becomes i-L -B-Since ^-i
a a








is not necessarily small, an expansion of the form shown in

equation IV-A-7 will not be a good one.39,40 Thus B and M(w),

which may be viewed as "width and shift" operators for the line

shape, should be retained in the denominator; it is then possible

to expand B and M(w) themselves in the various coupling parameters.

In the remainder of this chapter we will discuss the possible

methods of expanding them.

From Chapter III we recall that the operators B and M(w)

are given by


B iTr {L (1)p(a,l)p (a)
1 ea


(III-B-25)


M()=-inTr {L a(l) 1 GL (1)p' (a). (III-B-28)
1 ea -L -V(ala) ea

The expansion of B in the various coupling parameters is a straight-

forward problem, but M(U), which contains the formally exact

operator, V(a,l;,), is difficult to deal with. It is the

denominator appearing in equation III-B-28 that causes the

difficulty. Again, we could expand the resolvent in the manner

suggested by equation IV-A-7, but this expansion would also be

invalid in the line center. Therefore, we examine the operators

in the denominator, L(l)+V(a,l;w), in the various weak coupling

limits. The results of these approximations to the denominator of

the expression for M(w) are displayed in Table 1 and are discussed

in detail by the remaining sections of this chapter.










TABLE 1. Some Weak Coupling Limits for the Denominator of M(w)




Some Weak Coupling Limits. RESULT
for the Denominator of M(w)


SXa=A =0
ea ee


second order theory
no shielding


Xea=0; A =0


A =1; A =0
ea ee

a =l; A==0 for G;
A e=l for denominator of M(w)


S=1; A =0
ea


second order theory
"random phase approximation"

unified theory of VCS


unified theory of Capes
and Voslamber


unified theory
"random phase approximation"







IV-B Second Order Theories


The simplest class of approximations for G and M(u) in

equation II-B-28 are those which retain only the lowest order

nonvanishing terms in an electron-atom coupling constant

expansion. Second order theories are so named because by taking
L(1)*V(a,;w) and G to zeroth order in xea we restrict the memory

operator, M(w), to second order in the electron-atom coupling

parameter. Even here, however, there is some attitude in

approximating the electron-electron interactions; in this section
we will consider the cases where ee=0 and A=0,

As the first approximation to both G and L(1)+V(a,l;~), we

consider A =0 and A =0. From equation III-B-8 we have
ee ea
L(1)=L -iV +L (1) which to zeroth order in both coupling constants
a 1 1 ea
becomes L (-iv The operator, GLea(1)p (a), can be simplified

if we realize that to lowest order in Xea p(a.1)=(a)P(1) and to

lowest order in Aee, p(1,2)=p (1)p(2). Hence from equation III-B-23

GLea (1)pl(a)=fO (V )p (a)L (l)p-1 (a) o (IV-B-l)
ea 0 1 ea

Since all contributions to V(a,l;o) contain at least one factor of

Lip or Ll (j), thus this operator will not contribute in this
ee ea
approximation and F(I) becomes:

M(w)=-in d(l)La (1) f (v )p(a)Lea (1)p'(a). (IV-B-2)
f (-L +iV*1 V ea

The quantum mechanical analog of this is essentially the expression

that was calculated by Smith and Hooper.18
(1)
Another result is obtained by taking G and L +V(al.;,)

to zeroth order in the plasma parameter rather than to zeroth order







in ee4143 Again, in this approximation we have L(1)=L -iv( *
=ea 1 *1
but G and V(a,l;w) will both be more complicated than they were
for the case where Xee=O. In equation D-12 of Appendix D the
operator GLe (l)p'(a) is scaled to dimensionless coordinates;

G 1p '(a)= [p(a)g(a,1) eaLea ( 2p(a)x


x(g(a,l,2)-g(a,l)ga,,.l))Lea(r2 ) ]p(a). (IV-B-3)

If we now observe that, for Xea=0 g(al,**s)=g(1..*s) and if we
recall that H(r r2), defined by G(r r )=l+h(rlr ) is the pair
distribution function which, when scaled to dimensionless coordinates,
is proportional to A, then equation D-12 may be written in the form,
L (1) f (v -)
G ~- [p-'(a)- P(a)xea La( )+ d p(a)h( r -2 XL ( e )]x

f (v -)
1(a)= -n'3 p(a) LS 1 7()
wpIeaea 1 2e
L(S 1)=L (r)+ (r 2 e)a(2 ) (2IVh4)
ea 1 ea 2 1 2 ea (IV--4)

L (r ") is an electron-atom interaction which is statically shielded
ea 1
by the electron-electron pair correlation function. In Appendix E
it is shown that K(a,l,2;w=m K(a,l;w)) to zeroth order in Xea is
given by

K(al,2s;w0= K(al;)))=f (v )f (v )[(l+P21 )(l+h(12))

+nfd(3)f (v )g(123)P1 ](1-)f1 (v )K(a,l;w).
(V2 31 0 1
Hence from equation E-16

V(a,l;w=-)=--in, *- f (v )f'~(v )
1 1 0 1 0 1







where
C=N d(2)f (v2)D(12)P2

C(12) is the direct correlation function.45 When this, along with
equation IV-B-3, is substituted into the expression for the memory
operator we find that

mi A2 L
f47 "i2A*d L a +i(V
= -'i "d jvL (r ')i(w 1+i" in ^ f (v I )dfo (v ""x
SA J 1 1 1 1 0 1 0

xP(a)f 0(v A)La (V' ) p7'(a). (Iv-sa )
0 1 ea 1

However, the form of equation IV-B-5 is not simple to evaluate since
a modified Vlasov operator,37 equation E-16, appears in the denominator.
We have shown in Appendix F that any expression having the general
form of equation IV-B-6 can be cast in the form of a dynamically
shielded electron-atom interaction; thus,

4- 'DL 0
= v I (L d .LD ,a + i4 vl f0 (v 1)p(a)x
0 A 1 1 ea 1
p p

x L (a) (IV--6)
ea 1

where, in this case, the dynamically shielded interaction is given
exactly by equation F-25:

LD = L (it
-ea f2dIT d E Ie -


where (k;w-La) is a frequency dependent dielectric function

E()(~-L )=[l dv .1 C(k)f(v)]. (IV-B-7)
Swa '1 .-L -4 1
a 1 1
If the static shielding defined by equation IV-B-4 is substituted
for the dynamic shielding of equation IV-B-6, we finally arrive








at a second order theory, including electron correlations, similar

to that studied by several authors.19,20'41"43


IV-C Unified Theories Expansion in ee

Zeroth Order in ee

Another useful result is obtained by taking the denominator

of the effective two particle resolvent operator in M(o) to first

order in Xea and to zeroth order in Xee. Again, note that it is

only the denominator of M(w) that we are expanding to first order

in Xea and not the entire function. With this expansion we

arrive at an expression for M(u) which includes all orders in the

interaction of the atom with only one electron. Such an expression

for i(w) yields a line shape function that is valid in the line

wings as well as in the line center; hence, it is called a unified

theory. It should be observed that the result obtained in this

approximation will entirely neglect electron correlations.

In equation IV-B-1 it was determined that zeroth order in

Xea and ee G is given by

GLea(1)p-(a)=f (v1 )p(a)Lea(1)p"(a). (IV-C-1)

In this approximation the operator L(1)+(a,l;j) is just as simple

to evaluate. L(1) is retained in full, while 9(a,l;w)-V(a,1;w-)+

V (a,1;w), given by equations B-13 and III-B-39, will vanish to

zeroth order in xee and ea* The resulting form for the memory

operator is

M( -in d(l)L (l) i f (v 1) (a)Lea(1)l (a). (IV-C-2)
~ ~ e ea G







The above will be recognized as the same result that was obtained

by VCS, whose theory was outlined in Section II-B.22,23 It will
be recalled that their theory used a density expansion of the
expression for the memory operator. The advantage of their result
was that it included all orders in an expansion of the interaction
of the atom with only one electron; while its primary disadvantage
was the neglect of the correlation effects of the other N-I
electrons. As mentioned in Section II-B, VCS partially compensate
for this neglect by including curoffs in the spatial integration.
In the remainder of this chapter we will show that the mean field
effects due to the N-1 additional electrons can be included

systematically via an expansion in either Xee or A.

First Order in >ee
We have just seen the expression that is obtained for the
memory operator by taking its denominator to first order in
ea and zeroth order in X ee We next approximate L(1)+V(al;o)
to first order in Xea or in Aee
To first order in X L(l) is retained in full; but V(a,ljw)
is complicated and hence must be approximated. The mean field
part of this operator, given explictly in Appendix D, equation D-15:

V(a,l;j0Foo)=fd(2)(Le(2)+Lee){p(a)f (v1)f (v ) [g(a1l,2)(l+P2)

Snfd(3)f (v )g(a.1,2,3)P -g(a,12)n d(2)f0(v )g(a,2)P21])

x{p(a)f (v ) [g(al)+nJd(2)(g(a,1.2)-g(a.l)g(a.2))P ]}r

-np(a)f (v )g(al) [d(2)L (2)P 2
0 1 ea 21








We have purposely not written this equation in terms of the
dimensionless coordinates related to the plasma parameter because
we want to expand it in Xee not A. With the requirement that
equation D-15 be taken to first order in either e or X and
ea ea
since there is an explicit factor of either Le or Le appearing
ee ea
under the integral, the remainder of the integrand must be
zeroth order in both of the expansion parameters. Ihder this

restriction g(al'**s)=l and equation D-15 reduces to

V(a.l;,w=)=n d(2)L21F (v )P ,V(1) (IV-C-3)
f ee o 1 21

where V(1) is the usual Vlasov operator and P.. is the permutation
operator defined in Appendix B. The collisional term, V (al;w),
is given by

V (al;)=-ind(2)(L (2)+L21) (2) ---(a,l,2) (III-B-39)
c ea ee o-L -W(a,1,2;o)

It can be shown that S(a,1l2), appearing in the integranc is first
order in either Aea or Xee; thus we neglect V (a.1;j). We now
have a result for the denominator of M(w) which is exactly the
same as that derived by Capes and VoSlamber. If we further
take the operator GLea(1)p'l(a) to zeroth order in A (not first
order in ee) the resulting operator is the same as that obtained
in Section IV-B:


L (1) f (vl-)
G = 4--T Lea(*)p (a)
p Wp 3LD3 ea 1

Sa=L ; r h(rI 2ea( 2


(IV-B-3)


(IV-B-4)







Substitution of equation IV-C-3, together with equation IV-B-4 into

the general expression for the memory operator, equation III-B-28,

gives,


M(w)=-in d(i)L ea(i) v )p(a)1 (a) (IV-C-4)
fL iV( ea

which is identical to the result obtained by Capes and Voslamber.
If we now apply a technique very similar to the one they used, and

which is developed in Appendix F, this equation becomes


M(w)=-in d(1)L i f (v )p(a)L pi(a) (IVC- 6)
S ea t-Liv -L (1) a 1 ea
a 1 1 ea

where the generalized dynamic shielding is given by
-+-

LD ()=Id'-Ldt L (-)D (a~'(a- )e'-ir -
ea ea

It is interesting to note that to lowest order in X the above
ea
expression reduces to

L (4.) -ig i
L ( )= d e.. e-e*e
ea 1 E(jdtL )

2 af (v ) i
( v, o(IV-C-6)
a k7 1 3 vi ,-L +it-v1
a 1

where this last quantity has the form of the usual plasma dielectric

function, with w, replaced by w-L a

It should be noted here that the mean field operator appearing

in the denominator of equation IV-C-4, and therefore the dynamic

shielding of equation IV-C-6, differs slightly from that appearing

in the denominator of equation IV-B-5. In that second order theory

we took the denominator to zeroth order in the plasma parameter







and derived the generalized Vlasov operator, equation E-16; in this

case we have expanded in the electron-electron coupling strength.


IV-D Ulified heory Random Phase Ap roximation

In Section IV-A we suggested that for large regions of

temperature and density an expansion in the plasma parameter

would be preferable to an expansion in the electron-electron

coupling strength. We will see in this section that an expansion

in this parameter, coupled with an expansion in ?ea, is a distinct

improvement over theories presented in previous.sections; the

result is an all order, or unified, theory in which the effects of

electron correlations are systematically accounted for.

Since all operators in this section will be expanded in the

plasma parameter we will express them in terms of the dimensionless

coordinatesxdefined in Section IV-A. From Appendix D, equations D-,

D-6, and D-7, we recall that


W (a a)i(a -
p p p "p

L (1)
.n.1d(l)p(a)f (v)g(a,1)a" ... p( 1a),

L (1) .G L (1)
S=.in d() ---- G _a-(a).
1p p V ( a1) "p
p p

Since the operator B will involve only a static shift, it will not

be emphasized in this section; instead we will emphasize a study

of .
rp oL (1)
From equation D-12 we can express G L".". pl(a) in







dimensionless units:


L (1) f ) (V.
G L-p 1(a)= P(a(aa,l)La(r1 )+ jr p(a)g(a,2)L (r-)
W P Wee ea ea 2

-f(a-Lf gAa jl) ea(it ,

where the spatial parts of the reduced distribution functions have
GL (1)
not yet been scaled. Since we are requiring that -a.-- p (a)
"p
be taken to lowest nonvanishing order in the electron-atom coupling
strength, which means that g(al**s)=g(l**s), equation D-12 becomes
L (1) f (v -1) ,
G -A1 p (a)p(a)=ea ea L ( 2)L -1
(IV.D-1)
It has been shown44 that reduced distribution functions, when
expressed in dimensionless units, may be easily expanded in A:

g(rI or2)=l+h(r1 ^r2)=l+Ah(r1 -72

+ -+ -+ -++
g(r or or )=1+h(r r )+h(r r )+h(r r )+h(r -r .r )
1 2 3 12 1 3 23 1 2 3

=l+Ah(if 2, -)+Ah(r -1 -)+Ah(it 2* )+A2h( 1 4 2 -).
(IV-D-2)
This result may be substituted into equation IV-D-1 with the result
L (1) f (v1-)
G (a)= P a' ea[Lea(r -)+ dr 2( r 2')Lea (r')]p' (a)
P P D eaaea ()
fo(v P X (IV-D-3)
P a ea ea

This is the same as the result obtained in Section IV-B,
We must now evaluate V(a,cI)=V(al;w-=)+ c(a,l;w) to first
We mus








order in the electron-atom coupling strength and zeroth

order in the plasma parameter. It will be convenient to separate

this approximation to V(a,l;w) into two parts:


(IV-D-4)


While V)o (a,l;w) is zeroth order in both X and A. Vi1o(a,1;w)
ea
is first order in Aea and zeroth order in A. In Appendix D

w4 have expressed the frequency independent contribution to V(al,;)

in dimensionless units:

Y~a ) Le X.1-( 2A L rf.---)p(a)f (v If (v I
Wy2 2 A e 2ea 2 0 1 0 2
P 2 12 1

x[g(a,12)(l+P )+ I fd(3')f (v ')(g(a.l,2,3)-g(a,1,2)g(a,3))P 3)]}M


x{p(a)f (v I [g(a.l)+ k Id(2)f (v ")(g(al,2)-g(al)g(a.2))P ]r1


- g(a,) 2 d 2 ( 2)P
21A J --2--1 a2
where L i(- We now identify the terms
ee r12 a
above equation that are zeroth order in both parameters.

terms having integrals containing L (j) certainly can't
ea
order, so we will consider only those terms which do not

explicitly contain this factor;

Ve(al;a)=i dr dv/(- I (v)fV2
e far r' av
[g(2Br r12 )V1
[g(2)(P21)+ d(3)fo(v )(g(123)-1)P D}f (v
lg12(lP 1 f 0 A31 01


[1+ d(2-)fo(v ')(g(12)-l)P 21".


in the

The

be zeroth


(IV-D-5)


In the last equation we have used the fact that when a +0,
ea


V(a,l;)~ 1,o(a,l 0)+V 0o(a,l1 0).
(arl w)+v








g(al1*.s)-g(1...s). Equation IV-D-5 will be recognized as the

modified Vlasov operator, which in Appendix E, equation E-17, is

shown to reduce to

0 0 1
VO 0(adlg =)=-inv vifo1(vl)d )(v

We must now look at V c(a,lw), given in dimensionless coordinates

by Appendix D (equation D-17),
V (a,lsw) A
-1-=--id d(2 )(-ILL (r )+i( -- )---) x
p A ea 2 12
(2) 1 1
i('-W ))1( 12)
wp ^
in order to determine whether it contributes, to zeroth order in

both parameters. The operator, S(a,l,2), given in Appendix C

by equation C-11 can be shown to be first order in either A or A.
ea
Since the free streaming part of the effective resolvent in equation

D-17 is zeroth order in both parameters, we see that there will be

a contribution to V (a,l;w) which is first order in Xea and

zeroth order in A, but no contribution which is zeroth order in

both parameters. Thus we have identified the contribution to

V(al;w) which is zeroth order in both parameters:

O (a .lw)-in*f (v Cfv )c((). (IV-D-6)
1 10 1 0 1
The contributions to V\l'(a,l;w) may be determined by examining

equations D-16 and D-17 more closely.

The operator V1'O(al;w) is still very complicated, but

there is a class of terms, each having a very simple functional

form, that will be seen to have a straightforward physical

interpretation. Let us now write out the contributions to V(a,1;W)






which contain integrals with a factor of L (l) in the integrands
ea
V (a,l;w==)
~ ~= d *dV- LaL (r ){f (v ')f (v2^)[g(12)(l+P
p 1 1 A ea 1 0 02 21
+ ld(3-)f (V3')(g(123)-1)P 31]f (v 1)


[l+fd(2) )(r 2)(g(12)-l)P 2 -1

4-0'2 fd(2')L (r 2)P (IV-D-7)
A ea( 2 21

where we have demanded that the above be no more than first order
in Xea. We now substitute equation IV-D-2 into equation IV-D-7
and keep terms no higher than zeroth order in A:
V (a,l1 ==) X
Va ---c=i ? eafdid2 d L ( L ) {f (v ')f (v ) [( +h( ))x
WP ea 2 2 ea 2 0 2 0 1 A 2

x(1+P )+ d(3-)f (v )(h( 4 +h( r 4 -) +h(.2 4 A)+(r( r ))
21 0 2 1 2 3 1 2 3

xP ]}(l-)f-O(v ) fdr 'dv' L (r ')P (IV-0-8)
31 0 1 A 2 2 ea 2 21

where the results of Appendix E have been utilized. Making use
of the symmetry properties of L (j), together with the definition
of C, the above equation simplifies:
V (a,;j=cO)
=1d 'dv2 'L () -)h(i r ")f (v )(1-f (v )f (v ))
eea( 2 2 ea 2 1 2 02 01 1

+Xeaf 0(v 1 )d2 -d'2-Leaa 2 '(1 '2 O')(1-f0('2) )21

xf (v -)Jd d-L f (v fdi-3dt3-h(i )1-( f 0(v ) dfr v)).
(IV-D-9)
From equation IV-D-8 we observe that all but one of the terms contain
a permutation operator either explicitly, or implicitly through the
operator C. We will separate out these terms and rewrite








equation IV-D-9 in the abbreviated form.
V 1 0 (a, 1= r )
= d 2d T" h2 ( Th(2 ')f (V2 )+X(P.i)
a ea 2 2 ea 2 1 2 0
P


(IV-D-10)


where X(P.i) represents the contributions that contain P... We now

combine equation IV-D-10 with the expressions for Vo(a,1;w),

VP ,o(adj-=b), and V"o (a,l; ):
e C

^^^ in ^.l ^v,)^Cfs v,)+X "dr 'dvL Cr N rc ^fr
e c
V(h.1 ) :-inv f0 ) -o1(V ea 2 dV2 2 2-)h( --)


S(O)=e_ 0 (al;l.=p)+9co(a 1;a)+X(Pij ) (IV-D-11)

In equation IV-D-11 we have explicitly separated out the simple

multiplicative frequency independent terms and also the term which

is zeroth order in both expansion parameters, from V(I.1:/)
p
Hence, A(w), which is first order in Xea contains integral operators,

some of which will be frequency dependent. We now explicitly write

out the memory operator, with V(al;w) given by equation IV-D-11:

p p

o ) (-)La -i(- 3 +i a)f(v) 'p"(a) (IV-n-12)
) Z ,.s P~af f ,. (, -) (Iv-D-12)
00 1


where we have recognized that


L)+a 1 ea r 2 h( )ea (IV-D-13)
ea ea L
This last expression for the memory operator is still not in

a particularly convenient form. However, there is a transformation

procedure, outlined in Appendix F, that will cast equation IV-E-12

in a form which we can more easily interpret. This technique

puts the contributions to the denominator, which are in the form








of integral operators, into a dynamically shielded electron-atom

integration. The resulting expression is IV-D-14

S, a 1 dL (LD )i +i i "- ea L -,())-1
W A ea A d eaea 1
P P
p(a)f0(v, )L a( ) (a), (IV-D-14)


where the dynamically shielded electron-atom interaction is given

by equation F-17:

LD ,d L (-1)D (a.U e )e--
ea ea

It should be emphasized that no approximation was made in going

from equation IV-D-12 to equation IV-D-14. The operator D'(aIt#t;)

includes all of the effects of the operator A(w) as well as the

effects of the modified Vlasov operator; its functional form

may be inferred from equation F-20. Equations IV-E-14 and F-17

give the expression for iM() correct to first order in ea

While equation IV-D-14 is exact to that order it is difficult to

evaluate in general. However, one simple approximation to

DS1(a,X^';I), that keeps only those contributions that are

zeroth'-order in both parameters is particularly useful.

Examination of equations F-20 and F-23 indicates that, in this approx-

imation, D"1(a,lt,';w) becomes

D_1(a,k -; o;)=
c(ki, -L )
a


e(k,-La)=[l-fd --' f (v )C(k)]. (IV-D-15)
8 1








which gives for LD (r)
ea

LD ( -r ),dke-i l r e a
ea e(s;w-L )
a

E~fCs -L ) will be recognized as a generalized dynamic shielding

function. It is interesting to further approximate dynamic

shielding by static shielding, giving a theory for which all of

the interactions are shielded statically. This will be discussed

further in the next chapter.










CHAPTER V
DISCUSSION OF RESULTS

V-A Fully Shielded Unified Theory

In Section III-B we developed an expression for the memory

operator:

M(G)=-in d(l)L (1) i GL (1p (a).
f ea (1) L) ea

In Section IV-D, LM and V(al;-) were taken to first order in

ea and all terms in 1M(w) were taken to zeroth order in the
ea
plasma parameter:


1()-i r|- d- 'L ( i(w- L A- +i --e L5 (r' )
Xea L
p A f 1 Iea 1 1 1 eaea 1
+in *, 'f (vo ,1)cf1(v I)-Z())' p(a)f (V )LSeaT ,p1(a).
1 10 1 0 1 0 1 ea


(III-B-28)










(IV-D-12)


We then proceeded to separate the operators appearing in the
denominator into two parts: those containing a factor of the

permutation operator, Pij. and those not containing it. The

terms containing P.i were combined with the leading L (l) to form
13 ea
a dynamically shielded interaction (Appendix F). The remaining
terms containing no factor of P.. were left in the denominator to

effectively shield the electron-atom interaction appearing there:
-iX2 -L



p(a)f (v )Lr )p a). (IV-D-14)
0 1 1)e

L (r )fdkdk'L (-k )D (a,^,-;w)e-*
ea 1 j ea







m.1 + +-
where the operator, D (a.kk ;1), contains all the effects

included in A(w). In this section we discuss some of the

consequences of the functional form of equation IV-D-14.

This result is interesting because it leads to a unified theory

in which all of the interactions are shielded. The unified theory

of VCS did not explicitly include shielding, while the unified

theory of Capes and Voslamber led to an expression for M(() in which

only those interactions appearing in the numerator were shielded.

The weak coupling limit used to develop equation IV-D-12 is more

systematic than that used by either VCS of Capes and Voslamber and

leads to the fully shielded result presented here.

As indicated in Section IV-D, a further approximation would

be to take the shielding function D" (ak,kii;w) to zeroth order

on both A and X This leads to a result similar to that of
ea
Capes and Voslamber (Section II-C), except for the static

shielding appearing in the denominator. As a further approximation

to this last result we replace dynamic shielding by static

shielding. This yields an expression for the memory operator

in which all of the interactions are statically shielded in the

same way:

M() in d(l)Lea(r ')i(La+iv -L a(r,))-1p(a)fo(v,)L)a "p (a),




44
Lea(r 1)=L ar n d(2)f 0(v2)h(r 1 2) ea r)


where h(r r ) is the electron-electron pair correlation function.4

The fact that all interactions in equation IV-A-1 are the same allows

us to integrate by parts, following the procedure of VCS outlined




V -


in Section II-B. Applying this procedure yields


M(a)=-in(w-La )dr dv [U(t)-l] fo (v ) (-La),

U(t)-Te {- i dtL ( (WrA-2)
exp t 0 ea

where U(t) is the time development operator in the interaction

representation and T is the time ordering operator. Equation

IV-A-3 closely resembles the expression which VCS evaluated, except

that now the electron-atom interaction appearing in the exponential

is shielded. VCS evaluated the integral in equation IV-A-2 by

neglecting time ordering, assuming that electrons followed

straight line trajectories, and cutting off the spatial

integration at the Debye sphere. The numerical results which

they obtained using their method have, in general, agreed well with

experiment in spite of the ad hoc way in which they accounted for

electron correlations. Since an integral over a statically

shielded interaction can frequently be approximated with good

accuracy, by a bare interaction and some sort of cutoff, we

can understand why the procedure employed by VCS was so successful.



V-B Conclusion

In Chapter III we developed a general expression for the

line shape function for a dipole radiator immersed in a one

component perturbing fluid. The calculation was fully quantum

mechanical and it was never necessary for us to specify the nature

of either the perturbing fluid or the radiator. A formally exact

expression for the line shape was obtained in a form suitable for







approximation. Then in Chapter IV we assumed classical perturbers

interacting with the radiator via a Coulomb potential, and we

applied the weak coupling limit. It was determined that the weak

coupling limit could be applied in a variety of ways and we

discovered that we could reproduce several of the existing theories

of line broadening, depending on how these limits were used. We

also developed a unified theory, which was fully shielded, by

expanding M(w) to zeroth order in the plasma parameter.

We should emphasize the generality of the method employed

here. While we have used some of the ideas of Gross5 in our

development of the short time limit, this work was primarily

based on a technique developed by Mazenko32*33 to study the general

problem of time correlation functions. The development of Chapter III

was similar to a study of the velocity correlation function made

by Mazenko32 except that we have two different kinds of particles:

perturbers and radiators; hence we have two different kinds of

interactions. In Chapter IV we restricted our consideration to

a plasma for which the classical path approximation was valid,

and applied a weak coupling limit based on the nature of these

interactions.

We should note, however, that it is in general, possible

to parallel more closely the work of Gross or Mazenko and bypass

the weak coupling limit. In Section III-C we discussed the use of

short time limits to formally close the BBGKY hierarchy of kinetic

equations. Hopefully it will be possible using this approximation

procedure, to evaluate a fully quantum mechanical expression for

the line shape without applying the weak coupling limit. Such

an expression would be necessary to properly treat very dense

plasmas, such as laser produced plasmas.45





































APPENDICES











APPENDIX A
EQUATION FOR K(a,1; )

In this appendix the second equation of the hierarchy, equation

III-A-10, will be put in a form that allows us to formally solve

for the operator, K(a,lw)j, and in turn, to arrive at a formal

expression for the memory operator, M(M). We have used equation
III-B-12, which separates 6(a,l;t) and 6(a,1,2;t) into their

short time functionals of 3(a;t) and remainder terms, R(a,lt) and

P(a,1,2;t), respectively. The result obtained by substituting
this separated form into III-A-10 is

(*+iL(1))p(a,1)pol(a1 (at+ ( +iL(1))(altt)
at at

-inTr {(L (2)+L21 )p(a.l2)p-1(a) (a;t)
2 ea ee
21
-inTr {(L (2)+L )P(a1..2;t)}. (III-B-22)
2 ea ee

We may simplify equation III-B-22 by recalling the first equation

of the hierarchy

( -- +iL )6(a;t)=-inTr {Lea(1)(a,1st)}. (III-A-10)
at a 1 ea

With the separated expression for 6(a,l;t) this equation becomes

D-6*(ast)=-iL D(a;t)-inTr {L (l)p(a;t)p'1(a))6(ajt)
at a 1 ea

-inTr {L (l)P(a-l;t)} (A-1)
1 ea








which, when substituted into equation III-B-22, yields


(_ '+L()P)(a.lt)inTr2{(Lea(2)+L21)P(a,1,2;t)}
2t ea ee
-ip(a.,1)p'(a)nTr L Lea(1)P(alst)=-i[L(1)p(al)

21 -1
4nTr {(L (2)+L (a,1,2)} ] (a)D(ast)


+ip(a,l)p (a) [Lap(a)+nTr {Lea(1)p(al) } ]P'(a)D(a;t). (A-2)

The two terms in brackets on the right hand side of equation A-2
can also be simplified by looking at the first two equations of the
equilibrium hierarchy:

[L p(a)+nTr {L a(1)(a,1)}]p (a)D(a;t)
a 1 ea

r***N{La }p'1(a)6(alt) (A-3)

[L(1)p(al)+nTr {( L (2)+L21)p(a,l,2) }]p (a)(a;t)
2 ea ee

-Tr2. *N {LpaN }p1(a)(a;t). (A-4)

It will be noted that Lp aN would vanish except for the fact that
it operates on a function of atomic coordinates, p1(a)(a;t).
The operators, L and paN, will, however, commute; that is,
Ip M aNL, and we rewrite equations A-3 and A-4,




=[P(a)L anTr 1 {p(al)L a (1)}]p.1(a)(a;t) (A-5)
a 1 ea]p(a)a;t)

[L(1)p(a,l)+nTr {(L (2)+L21 )p(ae2)}p1(a)6(at)
2 ea ee







=[p(al)(L +Lea (1))+nTr2 {p(a,l,2)L (2) ] p-'(a)(a;t) (A-6)

where we have also made use of the fact that terms with L (j) and
e
Lij operating on purely atomic operators, vanish. If equations
ee
A-5 and A-6 are substituted into equation A-2 we get

(. +iL(1))f(al;t)+inTr {(L (2)+L )P(al,2;t) 1

-inp(a,1) pl(a)Tr {La ()P(a.l;t) --i p(al)L (l)p'"(a)D(a;t)
1 e a e

-inTr {p(a1,2)L ea(2) }p-(a)i(a;t)+inp(al,)pil.(a)x

xTr {p(a1)L e()lip'(a,))(at). (A-7)
1 ea

Equation A-7 is useful because with it we have an equation
for P(a,l;t) and P(a1l,2;t) in terms of 6(a;t). We can define
a time independent atomic operator, GLea(l), given by

GL (1)p '(a)6(a;t)=p(a,l)L (1)p1 1(a)D(a;t)+nT 2{p(al,12)La (2)
ea ea ea

xp '(a)6(ast)-np(al)p'1(a)Tr. Ip(a,1)L () }p'1(a)D(a;t) (A-8)

with which we can rewrite equation A-7 in a more compact manner;

( +iL(1))P(al;t)+inTr {(L (2)+L2)f(a,l,2;t)}
at 2 ea ee

-inp(a,l) p (a)Tr l{L (1)P(a,l;t) =-iGL (1)p'1(a)D(a;t). (A-9)

It is now possible to Laplace transform equation A-9 with the result

(W-L(1))?(a,1; )-nTr {(L (2)+L )e(a,1,2; w)}
2 ea 9 Q











--~
+np(a.1)p (a)Tr {L a(l)P(a,l; )}=GL ()p l(a)D(a;w). (A-10)
1 e 6ea

From equation III-B-16 we recall that P(al1*s;)-=K(al*..s; )6(a;w)

hence we are able to get an equation for K(al;,);


(u-L(l))K(a,l;wa)-inTr {(L (2)+L21 )R(a,1,2;w)}
2 ea ee
-11
np(a,l)p''(a)Tr {L a()K(a,)}=GL a(1)pl(a). (A-11)
1ea ea

Finally in this appendix we will examine the operator GL (1),
ea
defined by equation A-8, noting that all of the terms on the right

hand side of that equation contain a factor of the electron-atom

interaction, L (j). If we define a permutation operator, P..i
ea x3
that will change the functional dependence of all operators

appearing to its right from functions of particle j coordinates to

functions of particle i coordinates, then we can rewrite equation A-8:


GL p (a) (agt)=p(al)Le (1)p (a) 5(a;t)


nTr2{ (a,,2)P21 L ea(p (a)D(a t)

-op(a,1)p (a)Tr (p(a,2)P2 } L(ea(1).- a(a;t). (A-12)

Thus we are able to define

Gp(a,1)+nTr2{p(a,1,2)P21 )-(a,l)p (a)nTr 2{p(a,2)P 21


(A-13)










APPENDIX B
ANALYSIS OF V(a,l;w)

The purpose of this appendix is to validate equation III-B-26,

V(a,lw)R(a,ls-)=nTr {L a(2)+Le )(a,l,2;w)

-p(a,1)p'1(a)nTr {L (l)K(a,1;w)j}, (III-B-26)
1 ea

and to study the short time (infinite frequency) limit of V(al,;w).
Our first step is to express K(a,1,2;w) as some functional of
K(a,l;c). Following Mazenko32,33 we will use exactly the same
technique for expressing K(a,l,2;w) as a functional of K(al;w)
that we used to express t(al;t) as a functional of If(a;t).
Equations III-B-6 and III-B-6 suggest that we should attempt
to express K(al**-.s;) as a linear map of a function spanning the
space of the coordinates of the atom and particle 1 onto the
space of the coordinates of the atom and s particles.
First we recall the definition of K(a,1**s;s), equation
III-B-17;

K(a,1***sw)=U(a,l'***sGm)0- (a; w)-p(a,l***s)p'l(a). (III-B-17)

From equations III-B-1 and III-B-3 we have the definitions of
O(a,l*..s;w) and U(aIw):

U(al***ss)raN.}VN P p-1a) (III-B-)

U(aJ) ...r N -L paN)p- (a). (III-B-3)

Making use of the fact that paN and L commute, and substituting
equations III-B-1 and III-B-3 into equation III-B-17 we arrive at









K(a***se;w )=Trs***N{N aN i" (p (a)[Trl. --- p-N (a)] -
. .b'eeN..N -Lr

-p(a,1...s)p1 (a). (B-l)
Noting that

$a = li p(a)( (III-B-19)
a
i

U( = -L M( (B-2)
a

We can rewrite equation B-1 in a more useful form:

K(al-i)=-iTr +1*N}p'(a)[{-L -B-M()]-p(al**s)p'l(a) (B-3)


where B and M(w) are given by equations III-B-19 and III-B-28

respectively. We now make use of the identity,

i i i i
_= -_ -i(L-.L ) --- (B-4)
--L w-L w-L -( a w-L (
a a

in order to get

i i
K(a,l-*-s )=-iTrs+...{p N[- -i Lea(j)] }- x
3 a
xp1 (a)[w-La- M( )]-p (a,1*"s)pl (a); (B-5)
21
we have observed that the result of L (j) and L operating on purely
ea ee
atomic operators is zero. It would be tempting to cancel
x--x(w-L ), but we cannot since L will not in general commute
(.L-L a a
Sa
with p(a)-TrN {PaN}. However, from the first equation of
the equilibrium hierarchy, equation A-5, we know that

pe (a)[L aB] =P (a)[LaP(a)+nTr {Lea(l)P(aDl)]p'1(a)

=p^(a)[p(a)L +nTr {p(al)L (1)}]p' (a)

=L (a)p (a)nTr P(a.D)L p( 1 a). (B-6)
a 1 ea








If this is substituted into equation B-6 we get


5+14** a*N ws-L ea

+iTrs+1...N -aN 1 (a)[nTr p(a,1)Lea(1)p'(a)+() ]. (B-7)

We now consider equation III-B-28

M(w)=-inTr {L a() GL ea(1)p'1 ( a),
1 ea a-L 1 4(a,1;w) ea

which we will substitute into equation B-7 and which is consistent
with the formal definition of V(a,l;w) in equation III-B-26. Now
we can extract a factor of L (1)p l(a) from all of the terms
ea
in equation B-7 by introducing the permutation operator defined in
Appendix A. Hence we are able to define the operator, X (a,l*'*ss;).

K(a,1. *s;aw)= ,(a,1"ss; )L (1)p (a)
ea

C(a,1"**"ss)L a(1)p-1 (p (a)
ea s+l*N*{aN ur-L ii ea

+iTrs N N }p 1(a)[nTr {p(a,l)P }L (1)p-l(a)
s+1***N aN -L 2 21 ea

-inTr {L (1) GL (1)p'1(a)}. .( (B-9)
2 ea c-(1)(a,1; ) ea

We will now assume that j(a,l;w) has an inverse and write

K(a,l ***s; R(a,l; ))= 2(al ***ss w)} '(a,l;s)K(a,l; w) (B-10)

which, when substituted into equation III-B-26, yields

V(a,1;w)K(alw)=nTr ((L (2)+L )((al; )"' (adl; W)}K(al;w)
2 ea ee

-p(a,1)p- (a)nTr {La(2)P2 }K(a,l ;)
2 ea 21

V(adlij=Tr {(L (2)+L )e (ael2; ) l(alia)}
2 ea ee








-P(al)p (a)nTr{Lea(2)P21 (B-1l)

Thus, we have justified equation III-B-26 and in doing so we have
derived an exact formal expression for V(a,1;w).
We will now separate V(a,l;w) into a frequency dependent and
independent parts
V(a,1;w)-V(al;=-")+V(a,1,;); (B-12)
21
V(a,1; I==)nTr{(L (2)+L ) X(a,1, 2; (==) ( a,1;a==)}
2 ea ee
(a,1)p (a)nTr2 {L (2)P }, (B-13)

Vc(a,a1;)=nTr2 (L (2)+L2l)fC(al,2;w)P.'l(a,l;o)

-3( a, 1,2;w~=) X")(a,1;=-m)}. (B-14)

It is possible to further analyze the expression for V(al,;w=)
by realizing that im -I = Neglecting terms that are
higher than first order in 1 from equation B-9 allows us to
write

X(a,,1e***,S;w= )L (1)p 4(a)= 6 [Tr* a Pj
ea W s+l*.4J PaN Pi
-p(a,l**..*s)p (a)Tr2 {p(a,2)P21 }]L (1)p (a). (B-15)

When this is substituted into equation B-10 we get

X(a,1.***s ) a=)."1 (a,l;w==)K(a,l;w)=[Tr+...N{PaN Pj}

-np(a,l..s)p (a)Tr2{p(a,2)P21 ][Tr2...N{paN P

-np(a,l)p 1(a)Tr2 p(a,2)P21J K(a,1;w), (B-16)

where it will be noted that the permutation operator, P.., operates
on everything that appears to its right. Hence, we are able to
write V(a,l;w-=) explicitly in terms of known functions, while





75


the formally exact collisional term, V (a1l;4), requires the

next equation of the hierarchy.











APPENDIX C
KINETIC EQUATION FOR X(a,l,2;w)

In this appendix we will transform the third equation of the

hierarchy,

( +iL(2))D(a,l,2;t)=-inTr {(L (3)+L31+L32)5(a,1i2,3;t)A (III-A-10)
at 3 ea ee ee

into an equation for X(al,,2;w) defined by equation III- B-36. As
in Appendix B we use equation III-B-12 to get

( +iL(2))p(al,2)p- (a)D(at)+( *+iL(2))(a,,2;t)
at at

=inTr (L (8)+L ee+Lee )[p(a.l,2;*3)p.1 (a)(a;t)

+P(a,1,2,3;t)]. (C-1)

From the first equation of the hierarchy one obtains

-a (a;t)=-iLa(a;t)-inTr {L (1)p(a,l)}p (a) (a;t)
at 1 ea

-inTr {L (1)e(al;t)}, (III-A-10)
1 ea
which can be substituted into equation C-1 to give

( t +iL(2))(a,l,2;t)-inTr {L (1)P(all;t)}
at I ea

+inTr {(L (3)+L3+L32 )(a,,2,3,t)-i (2)p(a,l,2)pl(a)D(a;t)
3 ea ee ee

+p(a.1,2)p-(a)La (a;t)+ip(a,1,2)p(a)nTr{Lea () p(a,)}

P '(a)D(a-t)-inTr {(L (3)+L +Le)p(a,1.2,3) }p (a)6(a;t).
WD1 t 3 {(Lea ee ee
(c-2)
76








The Laplace transform of equation C-2 yields









xp(a,1,2.3)} l(a)-p(al2)p'l(a)nTr {Lea (1(aD1)p(a (C-3)

where we have explicitly used the fact that (a,5l"s;t=O)=O and
the relation P(a,l*.es;)=--K(a,l'..s;.w)Da;u), equations III-B-13
and III-B-16, respectively. It is now useful to take the inverse
Laplace transform of equation C-3,
( (2)

( L2 )K(a,l,2;t)-i (ea,l,2)YPl(a)nTr 3{Lea(3)K(a,3-t)}

-inTr3{(L (3)+L3+L3S2 )K(a,l.232t)}, (C-4)
3 ea ee ee

where we have noted that

K(a,l,2;t=O)=-p(a,l,2)p-7 I iL +iL(2)p(a,1.2)"1(a)
a

+inTr3{(L (3)+L31+L32)p(al.3))p'(a)
e ea ee ee
-in(a,l1,2) p (a)nTr {L ()p(al)}p)ea (C-)


Equation C-4 gives a relation between K(a,1;t), K(a,l,2Jt),
and K(a,1,2,3At) which will be closed in a manner analogous to
that shown in equation III-B-26. We start by using equation III-B-33

K(a,1.s;t K(alt))=K(a,;K(a1. st=OjK(al;t))+X(al*..;t) (C-6)
where








K(al***s;t=OIK(a.l;t))=X(a.1s...i^==) 'Cl(a, w)K(alt) (C-7)

Again the correspondence between the t-*O limit and the w-o limit
should be noted. Equation C-7 may be substituted into equation C-4
with the result that

( +iL(2))X(a,1,2;w ==)3Cl-(a,1; =m)K(al;t)
at

+( +iL(2))X(a,1,2;t)'-p(aL,2)pi'(a)nTr {La(3)K(a,3;t)}
at 3 ea

=-inTr {(L (3)+L 3+LS2 3[(al)23;Fm)X (a.1; F" )x
ea ee ee

xK(a,1it)+X(a,l,2,3;t)]. (C-8)

Using the inverse Laplace transform of equation A-11i we now
eliminate a K(al;t) from equation C-8:

a( +iL(2))X(a,l,2;t)+inTr3{(L (3)+L 1+L32)X(a.l,2,3;t)}
at 3 ea ee ee

-iX(a,l,2i. ==()X'"l(al o)nTr {(Le (2)+L21)X(aAl,2jt)}

=-iL(2)kL (a,1.2 ; Fm)%" l(agliaF-o)K(al;t)+i (a,l.2; IF)x

xX-1(a,1; w=)L(1)K(al;t)+p(a.1,2)p 1l(a)nTr {La (3)K(a,2;t)}

-3(a,1,2; F=) (a, 1; 0=)p(a,l)p- l(a)nTr 3 {Lea(3)K(a,3;t)}

+inTr {(L (3)1++L32) [l(ao1,2,3;.Fw (a,1; u )K(a,1;t)}
3 ea ee ee
+i((a.,l.2; c~)inl" l(al; =F=)nTr2 (Lea(2)+L21)( al,2; Fm)

3.C' (alh=)K(a,};t)}. (C-9)

Since X(a,l..*s; w):=(a,l*s;w )K(al; ), the Laplace transform of
the abovecequation is







(arL(2)) x(a,1,2; w)+X(al,1,2j ul=)X" (a,l; uFp) x

xnTr {(L (2)+L )x(a,1"2;a )}-nTr {(L (3)+LL+L32 )
2 ea( + 3 ea ee ee

xj(al,2.3jw)}=S(a,1*2) (C-10)

where S(a,1,2) is a frequency independent, inhomogeneous source term
given by

S(al.2)K(a,l1)=-) (a,1,2)X' (al==)[L(1)(a-,l;)

-p(a.l)p-l(a)nTr {L (2)K(a,l;w)}+nTr {(L (2)+L21)x
2 ea 2 ea ee

x(a,1.2;=o)3'Cl tal;w=o)K(a,l;w)]+L(2)N (a, 1 ,2;= o)x

x"LJ (a,l =o")Kk(a,l;o)-P (a,1.2)p" (a)nTr {L (3 )Ka, 3;w)}
3 ea
nTr { (L ,(3)+L3'+L32 )((a,1,2,3;w=o))X' (al;w=o)x

xK(alw). (C-11)

This is the result needed in order to solve for V (a,~l;).
c











APPENDIX D
DIMENSICNLESS UNITS


In order to understand the justification for a plasma

parameter expansion, let us first look at the linearized Vlasov

equation for a one-component plasma:

S+V1 )f(1)(r 1Av1;t)= n -- f(vl) d(2) f(1)(2 2t)(
at I In D 0 1 2sV2


where M2 -' is the interparticle potential. In the literature
12
it has been shown that the natural units should be

o =(4mne 2/m); LD(KT/47ne2)2;


-W+W x-fx + '--- (D-2)
Wp LD D Wp

If equation D-1 is scaled with these dimensionless units the result

is

S1)ne2 l 'f(O)(v )d 2) r(0
p aV1 ar 12

x f ()(t)=f (v ) d(2-) a -f r A t.(D-3)
8v2 2 4rr
1 12
There are two observations that are worth making about equation D-3.

The first of these is the fact that whenever L21 appears in a
ee
kinetic equation which is scaled to the dimensionless units

defined by equation D-2, it will usually be accompanied by a factor
of he second observation is that, even though the right
p
hand side of the Vlasov equation, equation D-l, is first order in

the electron-electron coupling strength, e2, the right hand side








of equation D-3, when expressed in dimensionless units, is

zeroth order in this parameter. Since this term will not be small

for long range forces, this suggests that perhaps e2 is not

the natural expansion parameter. Let us examine the following

expression:

JL j=i e + =i 4* [*.J
p 1 2 2 1 1 2 2 12


= [ q* Wn+9 ]6,~ ~-a =iA r4 r v(D-4)
1 1 2 2 r12 r1 vI r2 v2r12

where we identify P= r as the plasma parameter. For typical

experimental temperatures and densities this parameter is about

0.05; this suggests that it might be a good expansion parameter.

It should be noticed that, as in the case of the linearized

Vlasov equation, the factor of Lee will often appear under a
P
three dimensional spatial integral and be multiplied by a

factor of the density. In this event, the scaling to dimensionless

units produces a term which is zeroth order in the plasma parameter

but first or higher order in the electron-electron coupling strength;

thus, in general, a systematic expansion in the plasma parameter

will include different terms than a systematic expansion in

the electron-electron coupling strength. The advantages of a

plasma parameter expansion over an expansion in the coupling

constant for systems with long range forces are discussed in
37,38
most texts on plasma physics.37

It will now be useful to rewrite some of the quantities that

appear in the line broadening formalism of Chapter III as functions

6f the dimensionless variables defined by equation D-2:









S )i (III-B-17)
D(a; -)=-L -B-M(a) a
a

6(a; )=--------. p(a)dT. (D-5)
p B l
W 7- W p
Note that both B and M(M) appear with a factor of -
p
=B Id(1)p(a,1) LD p (a) (D-6)


()L e(1) L -(1) p
=-d(1) G (D-7)
-n" d PV(au-l G) P
-- -p
p p
where the trace operation of equations III-B-25 and III-B-26 has

been replaced by integrals indicating that the classical path
approximation has been made. Again, note that every factor of Lea()


P

L (1) 2 r, 2 R2 r
a= ----.R= .. -----. (D-8)
p bnf 1 \p- | UpO L, n I- | 3
p p |
If we take R to be approximately the Bohr radius, a then we can
define the electron-atom coupling strength to be

=--- (D-9)
ea .pe

which, for typical experimental conditions, is about 0.01; hence,
we can expect it to be a good expansion parameter.
It is now useful to examine G which is given classically by

L (1) L (1) L (2) 1
G ^' -P" (a)=p(al) -- p- (a)'nd(2)p(a,1,) ..-- -' (a)

P P P
"n (1) e ) 1) 2
~p al P 1 D








The simplest way to deal with p(a,l.**s) is to define30

p(a,***s)=p(a)p(l)""*p(s)g(a,1-*s)

p(j)?fo(vj)/A (D-11)

where f (v.) is the Maxwell-Boltzman velocity distribution. We will
o 3
not scale g(a,l**s) here; but it should be noted that to zeroth

order in Xea, g(al**s)-g(l**s), where g(1...s) is the usual

spatial reduced distribution function which may easily be expanded

in the plasma parameter, A.44 Since L ea(j) is a function of

r. only, equation D-10 becomes

L _(l) fj f 0 (vi X
G p (a)= 3 p(a)[g(al)Xea eaLal)+ d 'x
W A ea eaji A f

x(g(a,1,2)-g(a,l)g(a,2))Lea (2 )] (D-12)

and when this is substituted into equation D-7, that equation

becomes, in dimensionless coordinates,

a 1 11=. re dd -rL (r ')- i f (v )p(a)x
Wp Wp 1 ea V(a.l) e Il
"P W
xp p
x[g(a,l)L r X )+ dr (g(a,1,2)-g(a,1)g(a,2))Lear( p*(a)2D-13)
ea( 1 +A J2 (2 p(a)4D1

All that remains now is to express L(1)+V(a,l;w) in dimensionless

coordinates. Using equation III-B-8 we get

L(l) L iv L (1) L
L .. -, + .... = -. .i 0 + L ( ). (D-14)
w w w w ea i
p p p p
The operator V(al; u) will be treated in two parts: the mean field

part, V(a,l;w =), and the collisional part, Vc(al;w). The
former was given by equation III-B-30

V(ali-;:=n-)-d(2)( Le(2)+Lee2)[p(a,1,2)(l+P2








4nfd(3)p(a,1.2.3)P3 -np(a1.2)p (a) fd(2)p(a,2)P ]x

x[p(a,1)4n fd(2)p(al,2)P3^pal) )pl(a) d(2)P(aA2)P21l

-p(a>)p1(a) d(l)L e()P9 (D-15)
ea 21

Using equation D-11, the above may be scaled to give

a1: = d Lea 2 0+i(2 L ( 2 1-+){p(a)f (V If (V 2 )x
pr 12 01 0
1

x[g(a,l,2)(1+P21)+ I fd(3')f (v )(g(a,1.223))g(a2)(a3))P3 ]}x

x{p(a)f 0(v ) [g(aPl)+ I fd(2 )f (v -)(g(a1.2)-g(a.l)g(a,2))P21 r

-Ag(a,1) jd(l-)L (r 1)P (D-16)
ea 1 21

V (aljw) defined by equation III-B-39,

V (aw)=-in fd(2)(L ea(2)+L ee) (2), i S(a,1,2) (III-B-4)


is more difficult to scale. It can be shown that all terms in
S(a,l 2), given by equation C-12, contain a factor of L L ,
ea ee
or (g(12)-1) which does not appear under an integral; thus this
term is always first order in either ea or A. In dimensionless
coordinates, equation III-B-39 becomes

Pi ala)=-i f d(2 )( L a *)+i 1 ,) -
W A ea 2 ar 1 12 av 1
p 1 12 1
i(- 2) W(..I2W) -lS(a,1,2);
P P
L(2) L
S- 1 -2 2 v+i r )+v ea ea1 ea ea La 2
p p 1 1 2 2
(D-17)








showing that V (a,l;w) is at least first order in a or A. Since
c ea
no real attempt is made to study V (al;w) in this dissertation,

other than to identify its order in an expansion in 1ea and A, we

will not attempt to evaluate W(a,1,2;w). It should be noted,

however, that to lowest order on both Xea and A, this operator can

be evaluated.











APPENDIX E
DIRECT CCRRELATICN FUNCTION

In Appendix B we obtained the formal result,

K(a,lo***os; )= (a,1**s; )X'l(a )K( )a,l; ) (B-10)

with the short time limit,

K(al'***"s IK(a,l a))= (a,l'** "s; (=)k "l(a,la =-)K(al;w ).

The purpose of this appendix is to examine the operator "I (a.l;w=o)
which appears in the operator V(a,l;s=)), in the limit Xea O.
In this limits C (a,1;o=), defined in equation B-15, becomes:

X(alw==)L (1)p'(a)= [p(a)f (v )L (1)p-'(a)-a d(2)p(12)
ea W 0 1 ea f

L (2)p-'(a)-nf0 (Vl)d(l)p(a)f (v )L (1)p (a)] (E-1)

where the classical limit has also been taken. By taking advantage
of the symmetry of L (l) we find,
ea

X(als;w))L (1)p' (a): p(a)[f(v )L (1)
ea A 1 ea

+nfd(2)p(a)f (v )h(12)L a(2)]p.1(a), (E-2)

where we recall from Appendix D that

p(12)=f0(v )f (v )(l+h(12)). (D-11)
0 2
Equation E-2 is not yet in a convenient form to determine
its inverse. In order to do this ve will Fourier transform E-2







FT[Xl(a13,w )La(1)pl- (a) = p (a)[fo(v ) (1+jdv2h(t)f (v2)L(ea))]pl (a)

= p(a)f (v )[1+Pnh()]La(k)p'1(a) (E-3)

where the operator Pv is defined by

PVF( f)=dv f (v)F (r). (E-4)

It is now possible to solve E-3 for Lea()p' (a) with the result

L (t)p- (a)=wl[+P nh(a) -1 -1 (a)f-l(v )FT [X(a,l; ==)L (1)p-l(a)
ea v 0 1 ea

.=(1-P inh()L p-l(a)f'l(v )FT[X(a,l um=)L (l)p-l(a) (E-6)
V 1+nh(Jt) 0 1 ea

where we have used the fact that P2=P. We now define the direct
46
correlation function46 given by

nC(lt)- nh(t)- (E-6)
1+nh(ik)

with the inverse Fourier transform

nC(12)= f e ("d2i 1 nh(k) (E-7)

The inverse transform of E-6 can now be taken with the result:

Lea (1)p (a)=(l-nC) (a)f a,(v e)(apla )L (1)p-1a )
(E-8)
CF(1)=nfd(2)f (v )C(12)F(2). (E-9)

We are now able to rewrite equation B-10 in the following form,

K(a...***s ==K(a,1; ))=3C(a,***s. s==),(l-)p" (a)f (v )K(a.,l;).
(E-10)
To be consistent, ((a.l"*s;w-==) must be taken to the same order







in the coupling strength as3C,'(a,1;w==), hence E-0O becomes

K(al.**s;(=IK(a ,l w))=f (v )f ( (v ) [(+P )(l+h(l2))*n d(3)f (v )
0 1 0 2 21 0 3

g(123)P 31](-r) v )K(a,ld)). (E-11)
31 0 1
Now that we have calculated K(al1,2;u=j-K(als,)) to zeroth
order in the electron atom coupling strength we can now calculate
V(a,l;ow=) in the same approximation by substituting E-l1 into B-16
21
and considering only the integral over L21
ee

V(a, l,-=)= ) d(2)Leef (v )f (v )[(l+P )(l+h(12))+

+nfd(3)fo(v )g(123)P ]-(16)f(v ). (E-12)
(0 3 31 0 1

The term containing g(1,2,3) may be rewritten

fd(2)ee21f (v )f (v )n d(3)g(123)f (v )(l-C)f'ol(v )K(a,3;)
J ee 0 1 0 2 0 3 0 3

lhfd(2)[njd(3)L f (v )f (v )g(123)]f (v2 )(l-)"1 (v2)K(a,2,w)
ee 0 )1f0 3 0 2 0 2
(E-13)
g(1,2,3) can be eliminated from equation E-13 by using the second
equation of the equilibrium hierarchy:

-i(v + *v )f (v )f (v )g(12)+L21f O( )f (V2)g(12)
1 1 2 2 0 1 0 2 ee 0 1 0 2

=aff (v )d(3)L3f (v )f (v )g(123).

After integrating this equation over v we get

-iv f (v )g(12)+L21'f (v )g(12)=-nfd(3)Lof (v )f (v )g(123). (E-14)
1 1 0 1 ee 0 1 o 3
This can now be substituted into E-13 to give

nd(2)Lee (vf f (v )njd(3)g(123)f (v )(l-C)f" (v )K(al;o)

=hnd(2)[-L +i )1 ]f (v )f0 (v )g(12)(l1-)f (v )K(a,l;w). (E-15)
I'ee -i-1 1fo 1 0 2 0 2








Equation E-15 may now be combined with E-12 to yield

V(a,l.c)=inV1 *l f0 (v1) d(2)f(vq)g(12)(1-C)fol(v )P (E-16)

Frcm the definition of C, this can be shown to be equivalent to

V(a,l;= )=in1" f 0(v ) f ). (E-17)

47
This is the result obtained by Zwanzig47 and others for the short

time kinetic equation correcting the Vlasov equation.












APPENDIX F
DYNAMIC SHIELDING


The general expression for the memory operator in the

classical path approximation is given by


1(w)=-infd(1)Le (1) ,- GL (1)p"1(a). (III-B-28)
J ea (L)-1)(a..;J) ea

In Chapter IV we observe that many of the existing classical

approximations to the line shape can be obtained by applying, some

weak coupling limit to this equation. Expressions derived there

for the denominator of equation III-B-28 have some terms which

contain the permutation operator, Pi. A convenient technique for

dealing with expressions of this type is to use them to form a

generalized dynamically shielded electron-atom interaction. This

allows us to rewrite equation III-B-28 in the form:

M()=in d()LD ) n GL. )p (a (F-l)
ea W-L')- -V(a,1;) I ea

where V'(a,l;w) contains all contributions to V(a,l;w) which are

not in the form of integral operators, and L (1) contains the
ea
rest.

From Section III-B we recall that the memory operator can be

written, in the classical path approximation, as


M()=nfr1 dvi L (e )K(als)
1 1 ea 1


GL (l) (a).


(F-2)


w-La+i~ 0' -iL (l)-V(a,1;) ea
a V I ea

To show that the form for M(w) indicated in equation F-1 is possible,


...... ... Jl .. i, ,,


R(K 1 f )=







we will now consider the operator V(a,lm;). We will define an
operator (a,1,2;w) such that V(a,l;() becomes



V(a,1;w)F(a,.1;w)lr '(a,l;w)f(al;)1)+fd(2) (a,l,2;a))F(a,l1;a) (F-3)

where F(a,l;w) is an aribtrary function. We will now specify that

Z(a,l,2;w) be a function of rz-2 and v, only so that we can
rewrite equation F-2 in the form

V(a,lp)F(a,1; w)=V'(a,1; a)F(a,l;w)+fdr 2(a,-r+ 2 'W)

fdv2 (a,r2,va ). (F-4)

The motivation for this restriction, which is valid for all of the
approximations to V(a,l;w) considered in Chapter IV, comes from
equation F-1 in which is is fdviK(a,l;w) for which we must actually
solve. Then we rewrite equation III-B-28

S-L +iV '~-.L (1)- (a,1;W))K(al; )

ar r~V1 ) d2K(ar v 2w)

=GL (l)p (a). (F-5)
ea
We now define

o(ar v,.)= ----L --- GL (1)p1 (a) (F-6)
W-L +iv *V -L ()-V (a, ) ea
a 1 1 ea
which allows us to rewrite equation F-6,

dv K(a,r* I(k)") v1dr2 -V--L+ Id---_x
w- L +iv*V -L )-CV(al;W)
a 1 1 ea

^(aw2 11e) ld^K(a ,r2v2 1) jd*^KO( arp ). (F-7)







For convenience in notation we will define

R(a,r v z; W))= (F-8)
a-L +iv *F-L (1)-' (a,l; )
a 1 1 ea
We now Fourier transform equation F-7

fdvK(ak,tvlv;a)+ 7 dv1 dk-(at-,' vsa;))<(a AV sj)x

x fdv2K(ak'2v-. 2o)=-d- O(a ,)* (F-9)

This allows us to define the operator

D(at,e ;w)=6(l-t )+ IT djk(a -4, 1I;)

(a'.'v ;w) (F-10)

so that equation F-9 may be written,
dk D(att ;w ) dv K(a v'I ;w)= fdIao(a. ,i;w). (F-11)

Now we define

d (lMa j)Do(ad ^^;)(- ) (F-12)

with the useful result

d k(a p )=f d5-1 "(aAA0) d Ro(a t s) (F-13)

In order to utilize equation F-13 we note that equation F-1
can be rewritten in the form

()=e- c n sst(-itute F-3 ito F-14 (F-14)

We can now substitute F-13 into F-14 to get
9(w)=- &dy [ AdLf ea- )l(aV-iW) fdv o(a.k ,. s) (F-15)







with the inverse Fourier transform



D -+
e 9 fd- 1 (ar 1 V 11

=-infdr 1LD 1----:---GL (1)p (a): (F-16)
) 1 ea 1 ,-L -L (1)-L (1)^ (also) ea

LD (r ), the generalized dynamically shielded electron atom interaction,
ea
is defined by

LD ( r-" f 1 fd IMit -e-i Z (F17)
ea ) ea (F-17)

The form of equation F-15 is particularly useful because it
incorporates all of the effects of the difficult mean field integral

operators contained in (al,2;w) into the dynamic shielding

operator, D (aAk,k; w).
The shielding supplied by G, in the case of the electron-atom

interaction appearing in the numerator, and by V'(a,lj;) in the
denominator is easy to calculate in the weak coupling limit, and
is discussed in Chapter IV. On the other hand the dynamic

shielding of LD (r ) is more difficult and requires a more
ea
complicated approximation procedure. We start by noting that
S(a., lv l;t) may be written;

F(air ,v )= i(L (1)Va,1;w))x
1 1 o-L +ia *" t-L +iv *v ea
a i 1 a I 1
xR(a,r ,v1 ;) (F-18)

which in turn allows us to rewrite the second term on the left

hand side of equation F-6:

d f R2(a ,r s ))(a 1 ;w) dv2K(a,r2.' 2;)= fd x
f f-L +i "v


d 2 Ox 2 1 ;)fd 2K(a' 2 2)-i d d -L +n'
a 1 1

a id







(Lea (1)+V'(a,1-))R(ar vcv (ar 2-x;) dv2K(a,r 22Vsow). (F-19)

This can be Fourier transformed with the result

FT dv dR 2(a, 21))1(ar2 v1;v ) dv2K(ar2' 2;w)


S-L 4v *1
a 1
--- L ea(t--)R(a, t- -sj) (a,k ,v ) dv 2K(a, +v2; W)
W'Lal"^ ea (F-20)

Combining this result with equation F-8, F-9, and F-10 suggests
that D(ak,k-';w) should be divided into two parts: one which is
local in k-space and one which is nonlocal,

D(a,;s s)=D() (a, Ot'k w) +D(l)(a,kmk*Z w)

D(O)(ak,k 6(-^) [1+i dv a (a. ;w)1=6(-k )E(k*-L )
a 1

(1)(ta(t ,-)= d^y ddk- L *---- L(ea(k-k)R(ak --kv;O)
a 1
x (aFlt W s). (F-21)

If we assume that e(;sU-La) the mean field part of V(a,l;w) is the
Vlasov operator, then e(k;u-La) is the usual frequency dependent
dielectric constant. We now have:

fd,- (a t ~D- 1oD)(a- ) a+D 1)(a, ,- s)]=6(2-') (F-22)

or, using equation F-21

5"'1(a,t,'-; )e(k;o -La)+ a dt I1 (a, U 3a)


x D(l)(a,k S,.; ))=(0--4)


(F-23)