A KINETIC THEORY OF SPECTRAL LINE BROADENING IN PLASMAS
By
THOMAS WILLIAM HUSSEY
A DISSERTATIW. PRESENTED 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
IA Introduction 1
IB Causes of Line Broadening .. 2
IC Stark Broadening in Plasmas 3
ID The Line Shape . 4
IE Model for the Plasma .. 7
II. UNIFIED THEORIES . 13
IIA Introduction 13
IIB Unified Theory of Vidal, Cooper, and Smith 15
IIC Theory of Capes and Voslamber .... 19
III. FORMAL KINETIC THEORY APPLIED TO LINE BROADENING 24
IIIA The Hierarchy .. .. 24
1III Formal Closure 27
IIIC Short Time Limit . 37
IV. WEAK COUPLING LItITTS 41
IVA Introduction .. 41
IVB Second Order Theories 47
IVC Unified Theories Expansion in e 50
IVD Unified Theory Random Phase Approximation 54
V. DISCUSSIMN OF RESULIS . 62
VA Fully Shielded Uhified Theory 62
VB 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 noquenching 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
IA 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 manybody 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 noninterfering 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 manybody 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.
IB 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 halflife 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.
IC 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 halflife 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 halflives 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 halflives. 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 halflife 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 halflife. These two limits will be discussed further in
Section IE of this dissertation.
ID 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) (ID2)
0(t)Tr {d.T(t)(pc)Tt(t)}. (ID3)
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 plasmaradiator
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) (ID4)
and can be written formally as
SiHt
T(t)=e ; T(0)=1; (ID6)
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 (II6)
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)]* (ID7)
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 (ID9)
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. (ID10)
I ea la
It is convenient now to define the Liouville operator, L, given by4
Lf= [H,fl; Lf=HaH ,f] etc. (ID11)
aI n a9
We can now define the Liouville time development operator,
eiLtf f fe t(t) (ID12)
With this definition the autocorrelation function, given by ID3,
becomes
(t)Tr{~*eiLtpa}. (ID13)
Equation ID13 is now seen to satisfy the Liouville equation, a
property that will be utilized in Chapter III.
IE Model For the Plasma
In order to evaluate equation ID2, 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 socalled noquenching 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 ID2.
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> (IE2)
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. (IE3)
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 noquenching approximation, allows us to rewrite IE3
,(t)= (IE4)
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 ID6, 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, (IE6)
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 ionion interactions that appear in p
by some effective interactions which attempt to account for electron
screening of the ions. On the other hand, the perturberatom 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 ID13,
(t)TrdeiLtP}(IE6)
can be divided into a trace over atomic coordinates and a trace over
perturber coordinates
m(t)=Tra{d*Trie{e ipe)pad}. (pI7)
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 IE7. This allows us to
write equation ID8 in the following form
I(W )= deP(c)J(We) (IE8)
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 (IE10)
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
(IE.1)
L(c)=La (e)+L e+Le (I E12)
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.1114 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
IIA 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 socalled 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 electronatom
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
19671820 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 oneelectron 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.
IIB ,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 (IIBI)
where the approximation that pae=ape has been utilized. The form
of IIB1 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 }. (IB2)
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 (lP)L (t)}
As =eL (c); L (t)=ei(La +Le^(acL (IIB)4)
Op a ea ea
T is the time ordering operator and P is the projection operator
defined by
Pf=PeTre f}. (IIB6)
In equation IIB3 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 II4 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 IIB6 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 electronatom 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 electronatom 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 dteiLet(1P)EL (j)eiLet}
exp ni ) j ea
i. t
zIT { tdt'L (j.t')} (IIB6)
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 IIB7 is all orders in the electronatom
interaction. It was shown that in this approximation
(1P)L (1)L (1). (IIB8)
e Cea
One important limitation imposed by taking only the lowest order
term in the density expansion is that electronelectron 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
quasiparticles, 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 electronelectron correlations
in their theory.
Another VCS approximation, the neglect of time ordering from
equation IIB7, has been discussed in detail by Smith, Cooper, and
Roszman,28 and it will not be further considered here. Integrating
by parts, equation IIB7 becomes
iM( )=hAitA pjdteiAoptAWop. (IIB9)
Finally VCS assume that the time dependence of R can be neglected
in the expression for L (1,t); hence
L (lt)='eie ()t()ei1e()t (IIB10)
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 IIB9 is extended to infinity, the time develop
ment operator in the interaction representation goes over to the
Smatrix 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 (IB1l)
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.
IIC 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{eiL(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  (IIC2)
3 3 3^k
where we have defined
6(a,lo .N;t) tNe"iL(E )tpp (IIC3)
and where we have explictly made the classical path approximation.
Taking partial traces of IIC2, 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 (II4)
where
(a,l"**s;t) Trs+1*..N A{a(a," *N;t)}. (IIC6)
The first two equations of this hierarchy are
( +iLa)I(a;t)=in fd(2)l(a,l;t), (IIC6)
( 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). (IIC7)
ax av
1 1
Since J(we) could be written as
J(wE)= LR Rdte Tr {C*aat)}, (IIC8)
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)) (IIG9)
where the reduced density matrix is given by30
p(l1.s)= *** d(s+1) d(N)oe.
If the closure relationship, IIC9, is substituted into the
second equation of the hierarchy, IIC7, and if only those terms
which are lowest order in an expansion in the electronelectron
coupling strength are kept, then IIC7 becomes
4oV)*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), (IIC10)
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 IIC10 is recognized as the Vlasov
operator, V(1).27 If the Laplace transforms of IIC6 and IIC10
are solved simultaneously, we find that
.(al;w)= i L a(l1(a; W); (IIC11)
wL L (1)+iv .V V(l)
a ea 1 1
V(l) is again the Vlasov operator, and Le is the electronatom
ea
interaction, statically shielded by the electronelectron pair
correlation function. If equation IIC11 is substituted into
the right hand side of the Laplace transform of IIC5 then we get
D(a,l;)= wL pd (IIC12)
a
where the memory operator, M(w), is given by
M(o)=inf d(l)Le (1) a . Ls (1). ,
ea ea (IIC13)
wLa 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 (IIC14)
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 electronatom interactions. Note, however, that
the electronelectron interaction appearing in U(t) is not shielded
while the two interactions appearing around it are. Thus it is not
possible to further simplify 11014 by performing an integration
by parts as was done in equation IIB10.
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 cutoff
procedure employed by VCS. Its main weakness, however, is the
closure hypothesis, equation IIC7. Capes and Voslamber show
that this is closely related to the impact approximation (binary
collision approximation for electronatom 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 IIC14.
CHAPTER III
FORMAL KINETIC THEORY APPLIED TO LINE BROADENING
IIIA 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 electronatom interactions
in the initial density matrix.
From equation IE9 the line shape function for electron
broadening is given by
Jd(w)= I R dte) tTrae {eiL()tp d)
11 e0 ae 0a
J(wE)= R Trae{. i P } (IIIA1)
e ae aN )
This can be written in the alternative form:
J()= 1R Tr { d D(a, **N;w)} (IIIA2)
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 IIC3;
(IIIA3)
where V is the system volume and paN is given by
pN=eH() r eBH(
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) (IIIA4)
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(ei 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
(IIIA6)
(a, Nt) eiL()t
(IIIA6)
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(alN;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) (IIIA8)
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
(IIIA7)
(IIIA10)
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
selfenergy 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.
IIIB 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 VNeiLtpaN }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
(IIIB3)
U(a[t)Tr N.VNei LtPN }p(a)
)(a)Tr*..**N N d NP (a).
(IIIB4)
The first step in effecting our closure will be to eliminate p(a);
in IIIB1 in favor of D(a;t) in equation IIIB3. Hence, assuming
that an inverse exists for U(a;t) and i(a;w), the functional
which result are
(11166)
or the Laplace transformed version,
D(a,L1***sID(a; w))=U(a,l"*s; w)U" (as w)D(a;).
(IIIB6)
U(a,lo**s;t) and U(a,1'..s;.w)'1(a;w) will in general be extremely
complicated operators. Equation IIIB6 gives us a formal method
(IIIB1)
(IIIB2)
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, IIIA10, may therefore
be written
( +iL ) D(a;t)=inTr L ea(1)U(alt)U (at)}(at) (IIIB7)
This can be cast in a different form,34
a ft
(11+iL )D(a;t)=ij dt'x(t't)D(at) (IIIMB8)
by introducing a collision operator, x(t't), which is nonlocal
in time. This is a particularly easy form to Laplace transform.
Inspection of equation III3B 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))
tD
=U(a,l..***s;t=)U a;t0)D(a;t). (IIIB9)
This shows that x(t't) has a singular contribution at t'=t.
Extracting this part explicitly from x(tt) yields
( +iLa)D(a;t)=iBD(ast)i) dt'M(t t)5(a;t') (IIIB10)
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 IIIB10 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 (1111)
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 electronatom interactions
are neglected from the density matrix. Consistent with the
separation of the singular part of the collision operator, x(t't),
from the nonsingular part, we write )(a,1*'*s;tjD(a;t)) as the sum
of its short time limit, equation IIIB9, 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)(IIIB12)
where
(a,***sst=0)=0. (IIIB13)
It can be seen that i(a,ljt) is related to the nonsingular part
of the collision operator, x(t't). Equation IIIB9, 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 IIIB9 and III812.
From the definition of U(a,l***s;t) given by IIIB2 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) .
(IIIB14)
Substituting IIIB14 into IIIB12 and taking the Laplace transform
gives
$(a,1*s(a;)=p(al***s)p" (a)D(a)1Pal*s ). (IIIB15)
The closure relationship, equation IIIB6 in conjunction with
equation IIIB15 enables us to write:
(a.l*s;W)=K(al***s;))6(a;W) (IIIB16)
K(a***s;w)=U(a***s)U (a)p(a,***s)p a). (IIIB17)
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 IIIA10,
(wL )6(a:s);=Tr {L (l)(1a.l;W)}=ip(a))
a 1 ea
and by using equations IIIB15 and IIIB16 we arrive at
(WL )D(a;w)nTr {L (1)p(a,l)p "(a)} (a;w)
Tr ( )K(a ;w)(a ))}(a;i)=ip(a) (IIIpB18)
This can be compared to equation IIIB11 to get
uL a.BM(w)
B=hTriL a(l)P (a.l)p'" (a)}
M(w )r=hT l (l 1)(a,l;)} (IIIB19)
The operator, B, as it is defined in equation IIIB19, 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 IIIA10 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 IIIB15:
(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). (IIIB20)
If these relations are substituted into equation IIIB20 the
resulting expression is
(_ +iL(l))p(a,l)pa 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)}. (IIIB21)
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) (I3)
and where P.. is the permutation operator, defined in Appendix A.
Taking the transform of equation IIIB22 we have
(WL(1))(adla)+p(al)p1 (a)nTr {Lea (l)P(a,ls g)}
ea
nTr {(L(2) +L )P(a.,2;e)}=GL (1)D(a;) (III824)
where we have used equation IIIB13. If we use the definition that
P(al..*s;w)=K(a.l***.os;)(aiw) then equation IIIB24 becomes
(wL(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) (IIIB25)
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 )} (II1Br6)
which is discussed in Appendix B. With the use of equation IIIB26
we solve equation IIIB25 formally to get a convenient form for
K(aal;w)
(IIIB27)
K(a,l;w)=  . GLea(1) .
TL V a,,e) a
The memory operator then becomes
M( )=inTrj{L (1) (1) 1 Lea(1)1.
ea L V(a,1;w)
(IIB.28)
It is now possible to analyze V(a,lw) in exactly the same manner
that we analyzed the collision operator, x(tt), that is, we
separate V(al;w) into frequency dependent and frequency independent
parts:
(IIIB29)
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 III129 can best by analyzed by continuing
to the next equation of the hierarchy.
V(a,l)=VV(a,lu.=)r+c(a.lj) .
(IIIB30)
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)} (IIIA10)
3 ea"e ee ee
From Appendix B, equations B9 and B10, 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). (IIIB31)
Then in equation B1l 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 IIIB11. 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)). (IIIB32)
Following a procedure analogous to that used to derive equation
IIIB12 we now separate this last expression into a short time
functional and a time dependent remainder term:
(IIIB33)
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 IIIB33 yields
K(a,l***s;w I K(a,l;w) )=K(a.1***s;t=OI K(a,.l ))+X(al*** s;,).
(IIIB34)
Hence, comparing this result with equation B13, 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
(IIIB35)
(IIIB36)
The similarity of the above with equations IIIB12 and IIIB17
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 IIIB 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 III35, 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
(wL(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).
LL W(a,1,22w)
(IIIB38)
V (a,1; )=inTr {(L(2)+L ) .. ...S(a,1,2)}, (IIIB39)
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 Nbody effects hidden in the operator W(al,2;wj).
IIIC 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 IIIB10 we rewrote the first equation of the
hierarchy in the form:
(IIIB10)
where B represented the singular, time independent part of the collision
operator, X(t't), and M(tt) represented its time dependent,
nonsingular part. From the Laplace transform of equation
IIIB10 we got
a dt >
( +iL )(at)=BD(a;t)i dt'M(t't)D(a;t),
at a Jo
(IIIB10)
(a;W)= uaLBM(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 IIIB11, 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
(IIIB28)
V(a,l;w)=inTr {(L (2)+L21) .i S(a,1,2)}. (IIIB39)
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 N1 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 IIIB11. If
a
wit WTa (ai;u>
ir(w)=inTr {Lea (1)
we were to neglect M(w) then all electronatom effects would
be contained in B and the time development of P d would be governed
a
by the mean field electronatom interactions. The analysis of the
separation of V(a,liw) into V(a,.li; ) and V (a,l;w) in equation
IIIB28 is very similar. If we were to neglect V (a,l;w), the
time development represented by the resolvent operator,
equation IIIB28, would be governed by the exact interaction of
the atom with one electron, L(1), plus the mean field effects
of the remaining N1 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 N1 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 Nl 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
IVA Introduction
In Section IIIC 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 noquenching 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 electronelectron
correlations, and the neglect of electronatom 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 electronatom interaction. The
electronelectron interaction is treated extensively in the plasma
theory literature.37,38 The electronatom 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 electronatom interaction
should be independent of the one we use for the electronelectron
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 electronelectron 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); (IVA1)
ae ae'ae ae ae ae
V e(j)eeVe(j); L (j)=X L (j). (IVA2)
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
electronatom 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 electronelectron coupling
constant may not be a valid expansion parameter, an expansion in
the plasma parameter is the betterchoice.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)"
xx' ; vvv=  (IVA3)
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 = (IVA4)
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 electronatom coupling parameter becomes
e2a0
S (IVA5)
ea T f
From Chapter III the line shape due to electron broadening
is
J(.E)= a R TrL W) p(a)l} (IVA6)
e a WL 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 Bi(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)  (IVA7)
wLa 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 iL BSince ^i
a a
is not necessarily small, an expansion of the form shown in
equation IVA7 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
(IIIB25)
M()=inTr {L a(l) 1 GL (1)p' (a). (IIIB28)
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 IIIB28 that causes the
difficulty. Again, we could expand the resolvent in the manner
suggested by equation IVA7, 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"
IVB Second Order Theories
The simplest class of approximations for G and M(u) in
equation IIB28 are those which retain only the lowest order
nonvanishing terms in an electronatom 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 electronatom coupling
parameter. Even here, however, there is some attitude in
approximating the electronelectron 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 IIIB8 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 IIIB23
GLea (1)pl(a)=fO (V )p (a)L (l)p1 (a) o (IVBl)
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). (IVB2)
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 D12 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). (IVB3)
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 D12 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 (IV4)
L (r ") is an electronatom interaction which is statically shielded
ea 1
by the electronelectron 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 E16
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 IVB3, 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). (Ivsa )
0 1 ea 1
However, the form of equation IVB5 is not simple to evaluate since
a modified Vlasov operator,37 equation E16, appears in the denominator.
We have shown in Appendix F that any expression having the general
form of equation IVB6 can be cast in the form of a dynamically
shielded electronatom 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) (IV6)
ea 1
where, in this case, the dynamically shielded interaction is given
exactly by equation F25:
LD = L (it
ea f2dIT d E Ie 
where (k;wLa) is a frequency dependent dielectric function
E()(~L )=[l dv .1 C(k)f(v)]. (IVB7)
Swa '1 .L 4 1
a 1 1
If the static shielding defined by equation IVB4 is substituted
for the dynamic shielding of equation IVB6, we finally arrive
at a second order theory, including electron correlations, similar
to that studied by several authors.19,20'41"43
IVC 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 IVB1 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). (IVC1)
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 B13 and IIIB39, 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). (IVC2)
~ ~ e ea G
The above will be recognized as the same result that was obtained
by VCS, whose theory was outlined in Section IIB.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 NI
electrons. As mentioned in Section IIB, 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 N1 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 D15:
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 D15 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 D15 reduces to
V(a.l;,w=)=n d(2)L21F (v )P ,V(1) (IVC3)
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) (IIIB39)
c ea ee oL 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 IVB:
L (1) f (vl)
G = 4T Lea(*)p (a)
p Wp 3LD3 ea 1
Sa=L ; r h(rI 2ea( 2
(IVB3)
(IVB4)
Substitution of equation IVC3, together with equation IVB4 into
the general expression for the memory operator, equation IIIB28,
gives,
M(w)=in d(i)L ea(i) v )p(a)1 (a) (IVC4)
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 tLiv 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.. ee*e
ea 1 E(jdtL )
2 af (v ) i
( v, o(IVC6)
a k7 1 3 vi ,L +itv1
a 1
where this last quantity has the form of the usual plasma dielectric
function, with w, replaced by wL a
It should be noted here that the mean field operator appearing
in the denominator of equation IVC4, and therefore the dynamic
shielding of equation IVC6, differs slightly from that appearing
in the denominator of equation IVB5. In that second order theory
we took the denominator to zeroth order in the plasma parameter
and derived the generalized Vlasov operator, equation E16; in this
case we have expanded in the electronelectron coupling strength.
IVD Ulified heory Random Phase Ap roximation
In Section IVA we suggested that for large regions of
temperature and density an expansion in the plasma parameter
would be preferable to an expansion in the electronelectron
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 IVA. From Appendix D, equations D,
D6, and D7, 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 D12 we can express G L".". pl(a) in
dimensionless units:
L (1) f ) (V.
G Lp 1(a)= P(a(aa,l)La(r1 )+ jr p(a)g(a,2)L (r)
W P Wee ea ea 2
f(aLf 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 electronatom coupling
strength, which means that g(al**s)=g(l**s), equation D12 becomes
L (1) f (v 1) ,
G A1 p (a)p(a)=ea ea L ( 2)L 1
(IV.D1)
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 ).
(IVD2)
This result may be substituted into equation IVD1 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 (IVD3)
P a ea ea
This is the same as the result obtained in Section IVB,
We must now evaluate V(a,cI)=V(al;w=)+ c(a,l;w) to first
We mus
order in the electronatom 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:
(IVD4)
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 21 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
(IVD5)
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 IVD5 will be recognized as the
modified Vlasov operator, which in Appendix E, equation E17, 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 D17),
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 C11 can be shown to be first order in either A or A.
ea
Since the free streaming part of the effective resolvent in equation
D17 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((). (IVD6)
1 10 1 0 1
The contributions to V\l'(a,l;w) may be determined by examining
equations D16 and D17 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
40'2 fd(2')L (r 2)P (IVD7)
A ea( 2 21
where we have demanded that the above be no more than first order
in Xea. We now substitute equation IVD2 into equation IVD7
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)fO(v ) fdr 'dv' L (r ')P (IV08)
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 )(1f (v )f (v ))
eea( 2 2 ea 2 1 2 02 01 1
+Xeaf 0(v 1 )d2 d'2Leaa 2 '(1 '2 O')(1f0('2) )21
xf (v )Jd dL f (v fdi3dt3h(i )1( f 0(v ) dfr v)).
(IVD9)
From equation IVD8 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 IVD9 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
(IVD10)
where X(P.i) represents the contributions that contain P... We now
combine equation IVD10 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 ) (IVD11)
In equation IVD11 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 IVD11:
p p
o ) ()La i( 3 +i a)f(v) 'p"(a) (IVn12)
) Z ,.s P~af f ,. (, ) (IvD12)
00 1
where we have recognized that
L)+a 1 ea r 2 h( )ea (IVD13)
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 IVE12
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 electronatom
integration. The resulting expression is IVD14
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), (IVD14)
where the dynamically shielded electronatom interaction is given
by equation F17:
LD ,d L (1)D (a.U e )e
ea ea
It should be emphasized that no approximation was made in going
from equation IVD12 to equation IVD14. 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 F20. Equations IVE14 and F17
give the expression for iM() correct to first order in ea
While equation IVD14 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 F20 and F23 indicates that, in this approx
imation, D"1(a,lt,';w) becomes
D_1(a,k ; o;)=
c(ki, L )
a
e(k,La)=[lfd ' f (v )C(k)]. (IVD15)
8 1
which gives for LD (r)
ea
LD ( r ),dkei l r e a
ea e(s;wL )
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
VA Fully Shielded Unified Theory
In Section IIIB 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 IVD, 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
(IIIB28)
(IVD12)
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 electronatom interaction appearing there:
iX2 L
p(a)f (v )Lr )p a). (IVD14)
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 IVD14.
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 IVD12 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 IVD, 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 IIC), 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 electronelectron pair correlation function.4
The fact that all interactions in equation IVA1 are the same allows
us to integrate by parts, following the procedure of VCS outlined
V 
in Section IIB. Applying this procedure yields
M(a)=in(wLa )dr dv [U(t)l] fo (v ) (La),
U(t)Te { i dtL ( (WrA2)
exp t 0 ea
where U(t) is the time development operator in the interaction
representation and T is the time ordering operator. Equation
IVA3 closely resembles the expression which VCS evaluated, except
that now the electronatom interaction appearing in the exponential
is shielded. VCS evaluated the integral in equation IVA2 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.
VB 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 IIIC 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
IIIA10, 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
IIIB12, 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 IIIA10 is
(*+iL(1))p(a,1)pol(a1 (at+ ( +iL(1))(altt)
at at
inTr {(L (2)+L21 )p(a.l2)p1(a) (a;t)
2 ea ee
21
inTr {(L (2)+L )P(a1..2;t)}. (IIIB22)
2 ea ee
We may simplify equation IIIB22 by recalling the first equation
of the hierarchy
(  +iL )6(a;t)=inTr {Lea(1)(a,1st)}. (IIIA10)
at a 1 ea
With the separated expression for 6(a,l;t) this equation becomes
D6*(ast)=iL D(a;t)inTr {L (l)p(a;t)p'1(a))6(ajt)
at a 1 ea
inTr {L (l)P(al;t)} (A1)
1 ea
which, when substituted into equation IIIB22, 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). (A2)
The two terms in brackets on the right hand side of equation A2
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) (A3)
[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). (A4)
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 A3 and A4,
=[P(a)L anTr 1 {p(al)L a (1)}]p.1(a)(a;t) (A5)
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) (A6)
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
A5 and A6 are substituted into equation A2 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). (A7)
1 ea
Equation A7 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) (A8)
with which we can rewrite equation A7 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). (A9)
It is now possible to Laplace transform equation A9 with the result
(WL(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). (A10)
1 e 6ea
From equation IIIB16 we recall that P(al1*s;)=K(al*..s; )6(a;w)
hence we are able to get an equation for K(al;,);
(uL(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). (A11)
1ea ea
Finally in this appendix we will examine the operator GL (1),
ea
defined by equation A8, noting that all of the terms on the right
hand side of that equation contain a factor of the electronatom
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 A8:
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). (A12)
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
(A13)
APPENDIX B
ANALYSIS OF V(a,l;w)
The purpose of this appendix is to validate equation IIIB26,
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}, (IIIB26)
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 IIIB6 and IIIB6 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
IIIB17;
K(a,1***sw)=U(a,l'***sGm)0 (a; w)p(a,l***s)p'l(a). (IIIB17)
From equations IIIB1 and IIIB3 we have the definitions of
O(a,l*..s;w) and U(aIw):
U(al***ss)raN.}VN P p1a) (IIIB)
U(aJ) ...r N L paN)p (a). (IIIB3)
Making use of the fact that paN and L commute, and substituting
equations IIIB1 and IIIB3 into equation IIIB17 we arrive at
K(a***se;w )=Trs***N{N aN i" (p (a)[Trl.  pN (a)] 
. .b'eeN..N Lr
p(a,1...s)p1 (a). (Bl)
Noting that
$a = li p(a)( (IIIB19)
a
i
U( = L M( (B2)
a
We can rewrite equation B1 in a more useful form:
K(ali)=iTr +1*N}p'(a)[{L BM()]p(al**s)p'l(a) (B3)
where B and M(w) are given by equations IIIB19 and IIIB28
respectively. We now make use of the identity,
i i i i
_= _ i(L.L )  (B4)
L wL wL ( a wL (
a a
in order to get
i i
K(a,l*s )=iTrs+...{p N[ i Lea(j)] } x
3 a
xp1 (a)[wLa M( )]p (a,1*"s)pl (a); (B5)
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
xx(wL ), but we cannot since L will not in general commute
(.LL a a
Sa
with p(a)TrN {PaN}. However, from the first equation of
the equilibrium hierarchy, equation A5, 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). (B6)
a 1 ea
If this is substituted into equation B6 we get
5+14** a*N wsL ea
+iTrs+1...N aN 1 (a)[nTr p(a,1)Lea(1)p'(a)+() ]. (B7)
We now consider equation IIIB28
M(w)=inTr {L a() GL ea(1)p'1 ( a),
1 ea aL 1 4(a,1;w) ea
which we will substitute into equation B7 and which is consistent
with the formal definition of V(a,l;w) in equation IIIB26. Now
we can extract a factor of L (1)p l(a) from all of the terms
ea
in equation B7 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)p1 (p (a)
ea s+l*N*{aN urL ii ea
+iTrs N N }p 1(a)[nTr {p(a,l)P }L (1)pl(a)
s+1***N aN L 2 21 ea
inTr {L (1) GL (1)p'1(a)}. .( (B9)
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) (B10)
which, when substituted into equation IIIB26, 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 (B1l)
Thus, we have justified equation IIIB26 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,;); (B12)
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 }, (B13)
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)}. (B14)
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 B9 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). (B15)
When this is substituted into equation B10 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), (B16)
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 (IIIA10)
at 3 ea ee ee
into an equation for X(al,,2;w) defined by equation III B36. As
in Appendix B we use equation IIIB12 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)]. (C1)
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)}, (IIIA10)
1 ea
which can be substituted into equation C1 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(at)inTr {(L (3)+L +Le)p(a,1.2,3) }p (a)6(a;t).
WD1 t 3 {(Lea ee ee
(c2)
76
The Laplace transform of equation C2 yields
xp(a,1,2.3)} l(a)p(al2)p'l(a)nTr {Lea (1(aD1)p(a (C3)
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 IIIB13
and IIIB16, respectively. It is now useful to take the inverse
Laplace transform of equation C3,
( (2)
( L2 )K(a,l,2;t)i (ea,l,2)YPl(a)nTr 3{Lea(3)K(a,3t)}
inTr3{(L (3)+L3+L3S2 )K(a,l.232t)}, (C4)
3 ea ee ee
where we have noted that
K(a,l,2;t=O)=p(a,l,2)p7 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 C4 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 IIIB26. We start by using equation IIIB33
K(a,1.s;t K(alt))=K(a,;K(a1. st=OjK(al;t))+X(al*..;t) (C6)
where
K(al***s;t=OIK(a.l;t))=X(a.1s...i^==) 'Cl(a, w)K(alt) (C7)
Again the correspondence between the t*O limit and the wo limit
should be noted. Equation C7 may be substituted into equation C4
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)]. (C8)
Using the inverse Laplace transform of equation A11i we now
eliminate a K(al;t) from equation C8:
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(agliaFo)K(al;t)+i (a,l.2; IF)x
xX1(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)}. (C9)
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) (C10)
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)pl(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). (C11)
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 onecomponent 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 xfx + ' (D2)
Wp LD D Wp
If equation D1 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.(D3)
8v2 2 4rr
1 12
There are two observations that are worth making about equation D3.
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 D2, 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 Dl, is first order in
the electronelectron coupling strength, e2, the right hand side
of equation D3, 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(D4)
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 electronelectron coupling strength;
thus, in general, a systematic expansion in the plasma parameter
will include different terms than a systematic expansion in
the electronelectron 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 D2:
S )i (IIIB17)
D(a; )=L BM(a) a
a
6(a; )=. p(a)dT. (D5)
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) (D6)
()L e(1) L (1) p
=d(1) G (D7)
n" d PV(aul G) P
 p
p p
where the trace operation of equations IIIB25 and IIIB26 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= .. . (D8)
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 electronatom coupling strength to be
= (D9)
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 (D11)
where f (v.) is the MaxwellBoltzman 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 D10 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 )] (D12)
and when this is substituted into equation D7, 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)2D13)
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 IIIB8 we get
L(l) L iv L (1) L
L .. , + .... = . .i 0 + L ( ). (D14)
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 IIIB30
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 (D15)
ea 21
Using equation D11, 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 (D16)
ea 1 21
V (aljw) defined by equation IIIB39,
V (aw)=in fd(2)(L ea(2)+L ee) (2), i S(a,1,2) (IIIB4)
is more difficult to scale. It can be shown that all terms in
S(a,l 2), given by equation C12, 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 IIIB39 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
(D17)
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; ) (B10)
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 B15, 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)] (E1)
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), (E2)
where we recall from Appendix D that
p(12)=f0(v )f (v )(l+h(12)). (D11)
0 2
Equation E2 is not yet in a convenient form to determine
its inverse. In order to do this ve will Fourier transform E2
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) (E3)
where the operator Pv is defined by
PVF( f)=dv f (v)F (r). (E4)
It is now possible to solve E3 for Lea()p' (a) with the result
L (t)p (a)=wl[+P nh(a) 1 1 (a)fl(v )FT [X(a,l; ==)L (1)pl(a)
ea v 0 1 ea
.=(1P inh()L pl(a)f'l(v )FT[X(a,l um=)L (l)pl(a) (E6)
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) (E6)
1+nh(ik)
with the inverse Fourier transform
nC(12)= f e ("d2i 1 nh(k) (E7)
The inverse transform of E6 can now be taken with the result:
Lea (1)p (a)=(lnC) (a)f a,(v e)(apla )L (1)p1a )
(E8)
CF(1)=nfd(2)f (v )C(12)F(2). (E9)
We are now able to rewrite equation B10 in the following form,
K(a...***s ==K(a,1; ))=3C(a,***s. s==),(l)p" (a)f (v )K(a.,l;).
(E10)
To be consistent, ((a.l"*s;w==) must be taken to the same order
in the coupling strength as3C,'(a,1;w==), hence E0O 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)). (E11)
31 0 1
Now that we have calculated K(al1,2;u=jK(als,)) to zeroth
order in the electron atom coupling strength we can now calculate
V(a,l;ow=) in the same approximation by substituting El1 into B16
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 ). (E12)
(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 )(lC)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
(E13)
g(1,2,3) can be eliminated from equation E13 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). (E14)
1 1 0 1 ee 0 1 o 3
This can now be substituted into E13 to give
nd(2)Lee (vf f (v )njd(3)g(123)f (v )(lC)f" (v )K(al;o)
=hnd(2)[L +i )1 ]f (v )f0 (v )g(12)(l1)f (v )K(a,l;w). (E15)
I'ee i1 1fo 1 0 2 0 2
Equation E15 may now be combined with E12 to yield
V(a,l.c)=inV1 *l f0 (v1) d(2)f(vq)g(12)(1C)fol(v )P (E16)
Frcm the definition of C, this can be shown to be equivalent to
V(a,l;= )=in1" f 0(v ) f ). (E17)
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). (IIIB28)
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 IIIB28 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 electronatom interaction. This
allows us to rewrite equation IIIB28 in the form:
M()=in d()LD ) n GL. )p (a (Fl)
ea WL') 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 IIIB 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).
(F2)
wLa+i~ 0' iL (l)V(a,1;) ea
a V I ea
To show that the form for M(w) indicated in equation F1 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) (F3)
where F(a,l;w) is an aribtrary function. We will now specify that
Z(a,l,2;w) be a function of rz2 and v, only so that we can
rewrite equation F2 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 ). (F4)
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 F1 in which is is fdviK(a,l;w) for which we must actually
solve. Then we rewrite equation IIIB28
SL +iV '~.L (1) (a,1;W))K(al; )
ar r~V1 ) d2K(ar v 2w)
=GL (l)p (a). (F5)
ea
We now define
o(ar v,.)= L  GL (1)p1 (a) (F6)
WL +iv *V L ()V (a, ) ea
a 1 1 ea
which allows us to rewrite equation F6,
dv K(a,r* I(k)") v1dr2 VL+ Id_x
w L +iv*V L )CV(al;W)
a 1 1 ea
^(aw2 11e) ld^K(a ,r2v2 1) jd*^KO( arp ). (F7)
For convenience in notation we will define
R(a,r v z; W))= (F8)
aL +iv *FL (1)' (a,l; )
a 1 1 ea
We now Fourier transform equation F7
fdvK(ak,tvlv;a)+ 7 dv1 dk(at,' vsa;))<(a AV sj)x
x fdv2K(ak'2v. 2o)=d O(a ,)* (F9)
This allows us to define the operator
D(at,e ;w)=6(lt )+ IT djk(a 4, 1I;)
(a'.'v ;w) (F10)
so that equation F9 may be written,
dk D(att ;w ) dv K(a v'I ;w)= fdIao(a. ,i;w). (F11)
Now we define
d (lMa j)Do(ad ^^;)( ) (F12)
with the useful result
d k(a p )=f d51 "(aAA0) d Ro(a t s) (F13)
In order to utilize equation F13 we note that equation F1
can be rewritten in the form
()=e c n sst(itute F3 ito F14 (F14)
We can now substitute F13 into F14 to get
9(w)= &dy [ AdLf ea )l(aViW) fdv o(a.k ,. s) (F15)
with the inverse Fourier transform
D +
e 9 fd 1 (ar 1 V 11
=infdr 1LD 1:GL (1)p (a): (F16)
) 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 ei Z (F17)
ea ) ea (F17)
The form of equation F15 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 electronatom
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 oL +ia *" tL +iv *v ea
a i 1 a I 1
xR(a,r ,v1 ;) (F18)
which in turn allows us to rewrite the second term on the left
hand side of equation F6:
d f R2(a ,r s ))(a 1 ;w) dv2K(a,r2.' 2;)= fd x
f fL +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 2x;) dv2K(a,r 22Vsow). (F19)
This can be Fourier transformed with the result
FT dv dR 2(a, 21))1(ar2 v1;v ) dv2K(ar2' 2;w)
SL 4v *1
a 1
 L ea(t)R(a, t sj) (a,k ,v ) dv 2K(a, +v2; W)
W'Lal"^ ea (F20)
Combining this result with equation F8, F9, and F10 suggests
that D(ak,k';w) should be divided into two parts: one which is
local in kspace 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(kk)R(ak kv;O)
a 1
x (aFlt W s). (F21)
If we assume that e(;sULa) the mean field part of V(a,l;w) is the
Vlasov operator, then e(k;uLa) is the usual frequency dependent
dielectric constant. We now have:
fd, (a t ~D 1oD)(a ) a+D 1)(a, , s)]=6(2') (F22)
or, using equation F21
5"'1(a,t,'; )e(k;o La)+ a dt I1 (a, U 3a)
x D(l)(a,k S,.; ))=(04)
(F23)
