 Title Page 
 Acknowledgement 
 Table of Contents 
 List of Figures 
 Abstract 
 Causes of spectral line broadening... 
 The relaxation theory 
 Qualitative analysis of the line... 
 Asymptotic wing formula 
 Analysis of the theoretical... 
 Appendices 
 Bibliography 
 Biographical sketch 

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Title: 
Improvements in the relaxation theory of spectral line broadening in plasmas 

Physical Description: 
viii, 164 leaves. : illus. ; 28 cm. 

Language: 
English 

Creator: 
Whalen, Joseph Edward, 1944 

Publication Date: 
1972 

Copyright Date: 
1972 
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Subject: 
Plasma spectroscopy ( lcsh ) Relaxation (Nuclear physics) ( lcsh ) Physics and Astronomy thesis Ph. D Dissertations, Academic  Physics and Astronomy  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
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Thesis: 
Thesis  University of Florida. 

Bibliography: 
Bibliography: leaves 161163. 

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Typescript. 

General Note: 
Vita. 
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UF00098201 

Volume ID: 
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University of Florida 

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Table of Contents 
Title Page
Page i
Page ia
Acknowledgement
Page ii
Table of Contents
Page iii
Page iv
List of Figures
Page v
Page vi
Abstract
Page vii
Page viii
Causes of spectral line broadening in plasmas
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
Page 8
Page 9
Page 10
The relaxation theory
Page 11
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Qualitative analysis of the line shape expression
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Asymptotic wing formula
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Page 64
Analysis of the theoretical results
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Page 86
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Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
Page 114
Page 115
Page 116
Page 117
Page 118
Page 119
Page 120
Page 121
Appendices
Page 122
Page 123
Page 124
Page 125
Page 126
Page 127
Page 128
Page 129
Page 130
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Page 158
Page 159
Page 160
Bibliography
Page 161
Page 162
Page 163
Biographical sketch
Page 164
Page 165
Page 166
Page 167

Full Text 
Improvements in the Relaxation Theory
of Spectral Line Broadening in Plasmas
By
JOSEPH EDWARD WHALEN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972
I
PLEASE NOTE:
Some pages may have
indistinct print.
Filmed as received.
University Microfilms, A Xerox Education Company
ACKNOWLEDGMENTS
The author wishes to thank Dr. Charles F. Hooper, Jr.
for his guidance during this effort. He also wishes to
thank Dr. Larry Roszman, Dr. John O'Brien, Dr. Tony Barker,
and Dr. James Dufty for the help they have given him through
many discussions and conversations. Also, he wishes to
thank Judith Lipofsky for her programming assistance and
Sally Kirk who typed the final manuscript.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . ii
LIST OF FIGURES . . . v
ABSTRACT . . . . . . vii
CHAPTER I CAUSES OF SPECTRAL LINE BROADENING IN
PLASMAS . . . . 1
(I.1) Introduction . . . . 1
(I.2) Time of Interest . . . . 2
(1.3) Doppler Broadening .. . . 4
(1.4) Perturbing Ions and Electrons . 6
(I.5) Interactions Between the Radiators . 9
(I.6) The NoQuenching Approximation . 10
CHAPTER II THE RELAXATION THEORY . . . 11
(11.1) Introduction . . . 11
(11.2) The Density Matrix . . . 14
(11.3) The Time Development Operator, T(t) . 20
(II.4) The Average Over the Ion Microfield 21
(11.5) Application of the Zvanzig Projection
Operator Technique . . . 22
(11.6) Matrix Elements of the Operator
(11.7) Higher Order Terms in 9( () . . 37
CHAPTER III QUALITATIVE ANALYSIS OF THE LINE
SHAPE EXPRESSION . . . 44
(III.1) Introduction . 44
(III.2) The Operator Co (T(.)') . . 45
(111.3) The Parabolic Representation . . 48
(III.4) The Lymanalpha Transition . . 51
CHAPTER IV ASYMPTOTIC WING FORMULA . . 57
(IV.1) Introduction . . 57
(IV.2) Heuristic Derivation of a Static wing
Formula . . . . 58
(IV.3) Theortical Derivation of a wing Formula 61
CHAPTER V ANALYSIS OF THE THEORETICAL RESULTS 65
.ii
Page
(V.1) Introduction. . . . . 65
(V.2) The Function J(J,~E) . . . 66
(V.3) The IonField Dependent Atomic Density Matrix 72
(V.4) The Full Coulomb Interaction . . 79
(V.5) IonField Effects in the Electron Collision
Operator. . . . 102
(V.6) Effects of Lower State Interactions . 104
(V.7) The Effective Distribution Function Results 107
(V.8) Static wing Results . . . 111
(V.9) The Lymanalpha Profile for Ionized Helium 118
(V.10) Comments . . . . 119
APPENDICES 122
APPENDIX A The Effective Potential V . . 123
APPENDIX B Properties of the Operators L0 and P 126
APPENDIX C A Computational Form for <(W(4) ) '* 131
(C.1) The Full Coulomb Treatment . 131
(C.2) The Dipole Approximation . .. 144
APPENDIX D Matrix Elements of exp(iJ4*r)1 . 148
APPENDIX E Useful Matrix Relationships . 155
APPENDIX F Dipole Matrix Elements in the Parabolic
Representation . . . 157
BIBLIOGRAPHY. . . . . . 161
BIOGRAPHICAL SKETCH . . . . 164
LIST OF FIGURES
Figure Page
1 Effective probability function for an electron
a distance x from the nucleus of the hydrogen
atom a 4 a a a a a a 19
2 Ratio of the matrix element W computed using
the full Coulomb interaction Co the same term
using the dipole approximation . * a 43
3 Field dependent part of the function J(W, g)
for the Lymanalpha profile . . . 68
4 Field dependent part of the function J(CWie )
for the Lymanbeta profile a . . a 70
5 Lymanalpha profile using a field dependent
atomic density matrix . . . . . 75
6 Lymanalpha profile neglecting the field
dependence in the atomic density matrix . . 77
7 A comparison of the matrix element W33 for
various approximations. T = 20,400 K,
ne = 3.6 x 1017/cm . . . . 82
8 Lymanalpha profile using the full Coulomb
interaction. Solid line does not use a strong
collision cutoff. Dashed line uses a strong
collision cutoff. . . .. 84
9 Functions I (W), j(W), ((w>,Q.) for the full
Coulomb interaction case. . . . 87
10 Functions I((o), j(W)), &(c, E) for the dipole
approximation * * . * * 89
11 A comparison of the Lymanalpha profile using
the full Coulomb interaction to computations
performed by Bacon et al.4142. * 92
12 A comparison of red to blue ratios for various
approximations . . . . 95
Figure
Page
13 A comparison of the present computation using the
full Coulomb interaction with the complete multi
pole treatment by Bacon . . 98
14 A comparison of the present computation with the
unified theory results and with the experimental
data *. a . . . 100
15 A comparison of Lymanalpha profiles with and
without lower state interactions . . . 106
16 A comparison of a Lymanalpha profile computed
using the effective distribution function to the
unified theory and the experimental data . . 110
17 Static wing results for the Lymanalpha profile
compared with the experimental data . . 113
18 The Lymanalpha profile in ionized helium. The
full Coulomb results are compared with the dipole
treatment * * . . * 115
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
IMPROVEMENTS IN THE RELAXATION THEORY
OF SPECTRAL LINE BROADENING IN PLASMAS
By
Joseph Edward Whalen
June, 1972
Chairman Dr. Charles F. Hooper, Jr.
Major Departments Physics and Astronomy
Several improvements in previous treatments of the
relaxation theory of spectral line broadening in a plasma
are presented in this dissertations
1. To second order in the coupling constant, the theory
is extended to include ionfield effects in the atomic
Boltzmann factors and the electron collision operators.
2. The theory is extended to include full Coulomb
interaction between the radiating atoms and the perturbing
electrons.
3. The effect of using a strong collision cutoff on
the theoretical computations is examined.
4. The effects of electron symmetry are introduced
through an effective Boltzmann distribution.
5. A static ving formula is developed.
Computations are presented for the Lymanalpha transition
in hydrogen and ionized helium. For frequency separations
from the line center greater than 10 times the ion plasma
frequency, the calculated profiles for the Lymanalpha line
agree quite well with the experimental data. However, there
are noticeable differences in the theoretical results in the
line center; these discrepancies are examined.
From the behavior of the Lymanalpha profile, when the
full Coulomb interaction is used, it is concluded that
higher order terms in the coupling constant are required to
properly describe the intensity in the line center. An
explicit expression is derived for a typical higher order
term in the perturbation expansion and an upper bound is
obtained for this term. It is shown that the perturbation
expansion does indeed break down for small frequency
separations corresponding to the line center.
The calculated results using the static wing formula
are compared with the experimental data for frequency
separations greater than the electron plasma frequency.
This theory provides an inexpensive means for computing wing
profiles which may be useful to experimentalists and astro
physicists.
viii
CHAPTER I
CAUSES OF SPECTRAL LINE BROADENING IN PLASMAS
(I.1) INTRODUCTION
The discrete spectrum of radiation emitted from
isolated atoms and molecules is broadened considerably when
the radiators are placed in a plasma. The resulting
intensity distribution as a function of frequency is called
a line shape.1 The major causes of line broadening are
discussed in Chapter I. The detailed behavior of the line
shape under different approximations is investigated in
subsequent chapters.
The broadening mechanisms can be grouped into three
major categories the effects of chargedparticle fields on
the radiation, collisions between the radiator and various
neutral constituents in the plasma, and Doppler effects.
In this chapter, the Lymanalpha line of hydrogen will be
considered in order to illustrate the region of the line
shape where the different broadening effects are important;
unless otherwise stated, all numerical results pertain to
a hydrogen plasma at a temperature of 20,400 degrees Kelvin
and an electron density of 3.6 x 017/cm3. However, most of
the analysis will hold for temperatures and densities which
may vary by factors as large as 4 for the temperatures and 10
for the density; exceptions will be noted.
1
(1.2) TIME OF INTEREST
To examine the effects of the various broadening mech
anisms on the line shape, it is useful to define a time of
interest.2 The line shape can be expressed as a Fourier
transform of an autocorrelation function, <(j),3o4,5
Ic") = 4rae o t) Ac4t;
where P ( ) 1T <. (*t)> andAW4 is the frequency
separation from the line center. For times such that A.Zl,
contributions to the integral are small due to oscillations
of the exponential function. Also, for cases investigated
in this dissertation, it( E) can be shown to be small for
large times.5 Thus, contributions to the integral are
significant only for times less than l/AW Thus, t 1/a)
defines the time of interest.
Experimental profiles for the Lymanalpha line for
wavelengths less than the half width, &,\ a .15 X, are
not well defined.6'7 Using this value of aX\ the
corresponding time of interest can be calculated,
I4
tHw ^1 o.SX0o .4/.
Changes in physical quantities such as the ionfield, which
are appreciable only for times greater than g.J, have small
influence on the profile for A>P~A Hence, they will
3
be neglected in this dissertations for example, the static
ion approximation will be employed.
(1.3) DOPPLER BROADENING
Doppler broadening is caused by motion of the radiator
during the time of interest.1 Neglecting the effects of
collisions and the ionfield on the motion of the radiator,
the Doppler profile can be written as,
I&'I~P'~~, (1.1)
where M is the mass of the radiator, c the velocity of light,
JAthe Boltzmann constant, and ,D the Debye length. For
cases considered in this dissertation, the Doppler broaden
ing will be assumed statistically independent from other
broadening effects. Under this assumption the line profile
can be represented as a convolution8 of the Doppler profile
and the profile resulting from all other broadening mechanisms,
Co
I ~(a~x)= (L(~)1(a)~AA')4(4A) (1.2)
where I(AX ) is the profile arising from all other effects.
For long times of interest, which correspond to the
line center, the emotional effects of the radiator will
become more pronounced. Thus in most cases, Doppler effects
are significant only in the line center. This fact is
illustrated in reference 9.
5
The Doppler profile given in equation (1.1) also
neglects the effects of electron collisions on the motion
of the radiator.10,11 This is a valid approximation since
the momentum transferred to the radiator during an electron
collision is small due to the large difference in the
electron and atom masses.
In this dissertation, attention will be focused on
theoretical treatments of I(.AN ).
(1.4) PERTURBING IONS AND ELECTRONS
Almost all treatments of line broadening to date use
the static ion approximation. 112,13 In this approximation
the broadening effects of ion motion or ion dynamics are
neglected. In reference 9 it is shown that the relative
motion of the ions is small for times corresponding to
wavelengths greater than the plasma frequency for the ions.
For the cases investigated in this dissertation, the plasma
frequency for the ions is smaller than the half width.
Hence, the effects of ion dynamics can be neglected for
wavelengths greater than Ad J .
The criterion used in determining the region of the line
shape in which electron motion is important is the same as
that used when ion dynamics are considered. The wavelength
corresponding to the electron plasma frequency is 2 1. Thus
electron dynamics will have relatively less importance for
wavelengths greater than 2 X. As will be shown in Chapters
IV and V, a complete static treatment for both the electrons
and ions yields a theoretical profile that is in substantial
agreement with the experimental data in the line wings.
The effects of electron shielding on the ion microfield
distribution function must also be considered. A comparison
of the relative motion of the electrons to that of the ions
during the time of interest yields
(6Ai44r/Ayu) vI4
Since electron motion during the time of interest is large,
the electrons may be considered to effectively shield the
ionion interaction. Thus the appropriate ion microfield is
the lowfrequency microfield.14
The same argument can not be applied to a treatment of
the electron microfield. Because of the low mobility of
the ions, the electronelectron interaction is not fully
shielded. To explore the effects of shielding in the elec
tron microfield, the static wing profile calculated using a
lowfrequency electron microfield is compared to that obtained
using the highfrequency electron microfield.15 These two
cases should provide bounds for the eftct of ion shielding
on the electron microfield.
In the line center, 0.0 5.0 1, for the temperature
and density under investigation, the interaction between the
radiating atom and the perturbing electrons is treated as a
collision process (i.e., no static electron field broadening
is included). One approximation often used in describing
these collisions is to assume that the electrons travel in a
classical path.116 In many of these treatments logarith
mic divergences are encountered when the dipole approximation
is used, and only terms to second order in the coupling
constant are retained.17'18 These divergences are removed
by using minimum and maximum impact parameters.19'20'21,22
The maximum impact parameter is assumed to be the
Debye length or a constant close to unity times the Debye
length. Several different criteria are used to determine
the minimum impact parameter cutoff or strong collision
cutoff. For the Lymanalpha transition, the minimum
impact parameter is often in the region in which the dipole
approximation for the interaction between the radiator
and perturbing electrons is inadequate. By using the quantum
mechanical treatment developed in the relaxation theory,
employing either the dipole approximation or the full Coulomb
interaction, there is no mathematical need for a strong
collision cutoff in the second order term for frequencies
greater than the half width. However, in order to obtain
realistic results a strong collision cutoff is required.23
(1.5) INTERACTIONS BETWEEN THE RADIATORS
Collisions between the neutral radiators are neglected
in most plasma line broadening treatments. To examine the
validity of this approximation, the nature of the radiator
radiator interaction must be examined. One can compute the
average density of neutral atoms from a knowledge of the
percent ionization. The percent ionization can be determined
from recent computations by Barker.24 For an electron
density of 1.0 x 1018/cm3 and a temperature of 20,000 degrees
Kelvin, the percent ionization is approximately 75 percent
corresponding to an average distance between radiators of
approximately 1.0 x 106 cm. During the time of interest
the ratio of the average motion of the radiator to the
average separation is less than 1. Also, since the radiator
radiator interaction potential is short range, typically a
1/r6 dependency, the effective mean free path will be much
larger than the distance between radiators. Thus, during
the time of interest there will be few radiatorradiator
collisions. The broadening due to these collisions will be
neglected compared to the effect of the electronradiator
collisions which interact through long range potentials.
(1.6) THE NOQUENCHING APPROXIMATION
In the context of line broadening theories the no
quenching approximation neglects radiationless transitions
between the initial atomic states and all other atomic
states having a different principal quantum number. This
is a good approximation between states which are well
separated in energy such as the lower transitions in the
first few series in hydrogen and helium. However, quenching
can become appreciable for higher members in a series for
which the transition probability to states of different
principal quantum numbers is large. Including quenching
effects in the lower member transitions is computationally
feasible; unfortunately, this is not the case for higher
transitions because integration over the inverse of large
matrices are required. Quenching has been examined for the
Lymanalpha transition by using a matrix perturbation
expansion approach. It is shown that the effect is small
(see Chapter V). The approach used could be extended to
the higher series members. However, at this time it is
felt that such an extension would be premature since other
approximations have a more significant effect on the line
profile and should be removed first.
CHAPTER II
THE RELAXATION THEORY
(I1.1) INTRODUCTION
The relaxation theory9 of spectral line broadening
in plasmas is reformulated in this chapter to include the
following (1) the full Coulomb interaction between
radiating atoms and perturbing electrons, (2) ionfield
splitting of initial atomic levels, both in the Boltzmann
factors and in the electron collision operators.
In the relaxation theory the profile, I(&J), of a
broadened spectral line is given by the Fourier transform
of9,12,26
(2.1a)
Hence,
where Tr represents a trace over all states of the plasma
system which consists of radiators, electrons, and ions.
T(t) is the time development operator and ( is the density
matrix of the system. This dissertation is concerned with
atomic dipole radiation from partially ionized plasmasl
hence, d is restricted to be the atomic dipole operator of
the radiator.
Under the conditions outlined in Chapter I, the radia
tors can be assumed isolated from one another. Also all the
radiators will have the same spectrum of states and are
independent of one another as shown in Section (I.5). Hence,
to obtain a useful line profile expression, it is sufficient
to consider the ensemble average over all states accessible
to a single radiator. The corresponding Hamiltonian for a
radiator immersed in a gas of charged particles is
H= H. + I{ <. ( t Vh' L, isVa (2.2)
where subscripts a, e, i represent atoms, electrons, and ions,
respectively. V represents the total potential energy opera
tor, and K the total kinetic energy operator. He is the
Hamiltonian for the electrons defined by
Previous treatments of the relaxation theory used an
unperturbed atomic Hamiltonian of the form Ka + Va. In this
dissertation, the effect of the ionfield on the atomic
states is included and, hence, the unperturbed atomic
Hamiltonian in the following derivation is taken to be
Ho, = K_+ e f4< '?
where e is the electron charge, R the position operator for
the atomic electron, and Ei is the static ionfield.
To at least qualitatively account for electron shielding
(see Chapter I, Section 1.4) the operator, Vii + Vei' is
13
replaced by an effective potential, Veff, which depends only
on ion coordinates. Debye Hiickel theory27 for ions in a
neutralizing electron background is used to obtain Veff*
(11.2) THE DENSITY MATRIX
The canonical density matrix for a system whose
Hamiltonian is given by equation (2.2), with Vii + Vei
replaced by Veff, is28
&= AT) (2.3)
where is Boltzmann's constant, T is the temperature and
c is a normalization constant obtained by requiring that
T,(f) = i.
Making the static ion approximation is equivalent to
assuming that the following commutator is negligibly small
[ +V i H] 0. (2.4)
Since K + Vff commutes with itself and commutator rela
tions are distributive, equation (2.4) implies that
[Ho* uy> diV+M = 0. (2.5)
Using equation (2.5), we may write the density matrix
in the following convenient form
St: )(/(A+ T) ,I)((i t V)/JT)(2.6)
The operators Vae and H will not in general commute because
o
Vae contains the atomic electron position operator and Ho
contains the atomic electron momentum operator. However, the
exponential factor containing Vae may be expressed as
y4cF Md+#t ~ L)/j) = .O(^^(lt !tt (' .W/)(
where Vae is an effective potential which is defined to
depend only on the electron coordinates an average over
atomic electron coordinates is performed to obtain Vae. It
is assumed that the term (Vae ae) is small compared to
both Vae and AT. Hence, we neglect it in a first approxi
mation allowing equation (2.7) to be factored,
( ( HI, + H V y T )= f,,( )o /=Tk )
( , (2.8)
The entire density matrix can now be written as a product
of the density matrices for the three subsystems ... atoms,
electrons, and ions
P = C P e,.
where
=(H/(2.9a)
S .,' (2.9b)
rf, "aA',^ (2.9c)
C = normalization constant.
We next consider further approximations to It
is often stated or assumed that weak electron collisions
account for the major contribution to the electron broaden
ing. However, it is not usually pointed out that a primary
reason why a perturbing electron is relatively unlikely to
get close enough to a radiator to cause a strong collision
is due to electron symmetry effects. For the purpose of
estimating symmetry effects between the atomic electron
and perturbing electrons, an atomic model is adopted in
which the atomic electron is superimposed on the nucleus.
with this rough approximation, Va can be written as a sum
of the classical Coulomb interaction, Vel, plus a term V
arising from symmetry effects. Since the classical poten
tial produced by the above model is zero, V reduces to V .
ame
The expression for the density matrix P is factored
yielding
A a(2.10)
where the commutators between He and V. are neglected. Again,
in the spirit of the previous approximations, it is assumed
that this commutator will be small compared to AT. For
temperatures under consideration in this dissertation, the
form for C given in equation (2.10) should be adequate
when estimating the qualitative effects of electron symmetry.
The potential, Vs, is determined from the radial dis
tribution functions for electronproton and an electron
electron system, as computed by Barker.24,29 The details for
17
deriving the radial distribution functions and the corre
sponding effective potentials are discussed in Appendix A.
A plot of exp(Vs/.,T) versus the distance between the
perturbing electron and radiator is given in Figure 1. The
rapid decrease in magnitude for small distances implies a
decreased probability for strong collisions. The resulting
effect on the line profile can be represented by an effective
strong collision cutoff. The value of the cutoff obtained
in this manner is indicated in Figure 1. It is larger than
cutoffs used in other treatments of line broadening. Thus,
for this model electron symmetry effects tend to decrease
the broadening caused by the electron collisions.
r
(44
u
0
.41
44
H
0
H
41
S
I..
0
4) r
44
44
mgI s
.0UO
Smg
* 0
rzl
a
C54
4 nm 0
S14 14
co(x)
(x)d
* N
(II.3) THE TIME DEVELOPMENT OPERATOR, T(t)
The time development operator appearing in (Z) is
defined in terms of the Hamiltonian given in equation (2.2)
with Vei + Vii replaced by Veff.
= iu o+ )t/l .,,(,,. ,) t1 )
(2.11)
where the commutator relation given in equation (2.5) has
been used. Using the commutator relations given in equa
tions (2.4) and (2.5) one can show that
'The) 1 f(t/z 4t AH (2.12)
where
H,= H + H 4 + k (2.13)
T (T) i ( T ( ) is evaluated in Section (11.5) using
the Zwanzig projection operator technique.4
(II.4) THE AVERAGE OVER THE ION MICROFIELD
Using the properties of the Dirac delta function, we
can write ( t) as30
= k?T ( .4^^)^^ ^^^^
(2.14)
where QE) = 7T.(f. 4,i) is the ion microfield
distribution function.31 Using the above form for S (X)
in the expression for the line profile given in equation
(2.1b), one can show that912
^^T<^)^. (2.15)
It should be emphasized that here, unlike previous treatments
of the relaxation theory, the ionfield is retained in H'.
Including the ionfield introduces field shifted frequencies
in the effective Liouville operator occurring in ZT( 0, ).
(11.5) APPLICATION F7 THE ZWANSIG PROJECTION OPERATOR TECHNIQUE
We must now reduce the line profile expression to a
tractable form. A useful technique to effect this reduction
employs the projection operator procedure developed by
Zwanzigl4 this procedure involves defining a projection
operator, P, that projects out the relevant part of an
operator, D(XZ), that satisfies an equation of the form
y L ) (2.16)
where L is the Liouville operator. The projection operator
allows one to remove the irrelevant part of D( :) at an early
stage in the derivation, thus often simplifying the analysis.
For example, if D1 is the relevant part of D and D2 is the
irrelevant part, one has
(t) = P (t) (2.17)
n )= (l pO ) (2.18)
Applying the projection operator procedure results in an
integrodifferential equation for D ).
It is often unclear exactly what part of an operator is
relevant particularly during the earlier steps of a deriva
tion. A simple example of a relevant part of an operator
might be the diagonal elements of a matrix operator. A
projection operator which selects only the diagonal elements
of a matrix operator would be appropriate when the non
diagonal matrix elements are negligibly small or when the
nondiagonal elements in no way affect the final result. An
example of the latter situation is when an average of an
operator, A, is taken in the following manners
(A) T= (A, .
The nondiagonal elements of (eA) in no way effect the
average value of the operator A.
In the case of line broadening, one is concerned with
the behavior of the radiator, either an ion or an atom in
the cases considered in this dissertation. Hence, the
atomic subsystem can be considered the relevant subsystem.
One performs an average over the broadening effects of the
ions and electrons. The ion broadening effects are taken
into account via the microfield average. A suitable proce
dure for simplifying the electron average involves a projec
tion operator of the following form9
P= r T (2.19)
where f is defined in equation (2.10). It should be noted
2
that P is indempotent, i.e., p2 P.
The projection operator technique is now applied to the
expression appearing in equation (2.12). we define as D(t )
D()= Tit) fIlt)
(2.20)
vhich is the Heisenberg representation for p 42
The procedure for finding an expression for D1( l ) is
to apply the operator P and (lP) to equation (2.16) and
then eliminate the operator D 2( ) from the resulting
equations. In this manner one obtains an integrodifferential
equation
i4*A = NL[.(t) I 0)L/
a (2.21)
For most problems of interest the operator (2(0)
occurring in equation (2.21) is zero. This is shown for
9
the projection operator P defined in equation (2.19)
6 ()= 0 P)(o) (IP)
 h
(2.22)
= 0.
Thus equation (2.21) becomes
= PL[2.t(*> 4@ 4 i^ ( I< L)
(2*23)
The following integral must be determined to evaluate
the line shape expression
25
a )
(2.24)
where
o
An equation for ,(10) can be obtained by taking the
Fourier transform of equation (2.23). Using the procedure
outlined in reference 9 one can show that
(2.25)
( .....) is defined by (A) a TI, (A f ) where A is an
arbitrary operator. T(4)) satisfies the LippmannSchwinger
equation33
AT(w) L+ L4 (I)..)L P)L,
(2.26)
Substituting the above result into the expression for
the line profile given in equation (2.15) yields
(2.27)
where the resolvent operator K((g)) has been introduced
3.
t(v ) = (W ). (2.28)
The Liouville operator, L, encountered in the above
derivation is defined in terms of the Hamiltonian, H*,
specified in equation (2.13) and the identity operator It
L = 1'40 (2.29)
Stands for a direct product. L can be rewritten as a
sum of two operators L0 and L1 defined by,
LC t O ^ 3:9(14 + L. t (2.30)
S (2.31)
Unlike the derivation in reference 9, the ionfield depen
dence is retained in Lg. If equation (2.26) for T(a)) and
the above definitions for L1 and LO are used, (T(W))
can be reduced to
*
I
+ L L] < 1 > (2.32)
The operator Lo can be written in terms of the Hamil
tonian for the atom and electron subsystems,
Lb = Lo t (2.33)
where
(2.34)
and
LO (2.35)
With the assumption that the density matrix, t is
diagonal in the electron states, it follows from symmetry
that and
becomes
T( > ( (2.36)
where
!
The matrix elements of
(I1.6).
Five identities derived in Appendix B that are
required to further reduce V (j) are given below
(2.38)
< O(IP)L) <= >)L,
(2.39)
(2.40)
I 
E ( LP)C*E L, (2.41b)
where A is an arbitrary operator.
Using the above identities, 6 (W) can be reduced to
an expansion in powers of the operator Li9,12
AK l I# L (2.42)
Introducing the unperturbed resolvent operator, (KO%)),
defined by
k(f (t) ( C*f^ d)
equation (2.42) becomes
> 1 (2.43)
The first term in the above expansion becomes
lJ;^ ) =
= (Lh(()L,> L, to)< L,
= < L, lt L,>) (2.44)
In this dissertation the above term is the highest order
term used in the numerical results presented. However, the
order of magnitude and the qualitative behavior of higher
29
order terms are examined in Section (11.7). It is shown
that the higher order terms may become important in the
line center.
(11.6) MATRIX ELEMENTS OF THE OPERATOR (T(C))>
To obtain matrix elements of the tetradic (T()> ,
the matrix elements of La and W(&)) are required. Let the
indices ,A represent atomic states and a % represent
electron states. The matrix elements of given in
equation (2.45) are obtained by performing an electron
average of equation (B.9)
4V Y.A A (
<0 = Z L F
A 4'r (B.9)
AV .'v A vp ji, 'P (p (2.45)
The Boltzmann factor f defined in equation (B.4) is
assumed to be diagonal and is normalized to 1. Hence,
equation (2.45) becomes
(2.46)
The matrix elements of (9 ((W) > can be written as
31
L
V* f (2.47)
Matrix elements of the operator K0( O() appear in equation
(2.47). The operator K0((W) contains L O hence, to derive
an expression for the matrix elements of K0(t)) appearing
in equation (2.47), the matrix elements of L0 are required.
Combining equations (2.46) and (B.10) one has
,, ,. , ,
A tV^ .'4 4'r S
A L v ,.'r', <' m'.W
(2.48)
To simplify the matrix elements of KOg()), KO(.J) is
expanded in powers of LO/1 and then matrix elements are
taken of the individual terms in the expansion. In order
to carry out this procedure, the matrix elements of Ln are
required. One obtains,
M(L = ) ... .. 7_ .LO
Ataflj4IVp ALd t ^^
yl; v4^f'
'LII6
X 1. SMp1 ,
where the results given in equation (2.48) have been used.
Resumming the above results leads to
IVvw)
Aj.(vp, IA~eV
A.vp, MM^.'V'P' <'4* OP
(2.49)
where the superscript m signifies that transitions between
initial and final states due to the ionfield are included
in A. The above results are inserted into the expression
for ~f~ (L) given in equation (2.47). Using the proper
ties of the delta functions, we can perform two of the
summations. The results are,
"JP L, d,, o'r , wt
;K73A L .rP
vV *< < \V 0
 <"'i, ,v I,', >&,,, ,I >) 4 ,. (2.50)
When the above expression is multiplied out, four terms
are obtained. Many of the summations occurring in these
terms can be performed using the properties of the Delta
function. The final result after carrying out these sum
mations is
A ,,
(2.51)
If the transitions between initial and final atomic
states due to the ionfield are neglected and the no
quenching approximation made for electrons, additional
summations in equation (2.51) can be performed. Under the
noquenching approximation (see Section 1.4), Vae i
zero unless 4 and A&' have the same principal quantum
number. The effect of neglecting these terms is small for
initial and final states that are well separated in energy
this is the case for the Lymanalpha line. However, since
the line profile is given in terms of an integral over all
ion fields, one always encounters fields large enough to
cause transitions between initial and final states. However,
the behavior of
that the integrand is small for such large fields and,
except for the far wings of the line, can be neglected i.e.,
for iAAr > 15A. For the line profile computations
presented in this dissertation, the transitions between
initial and final states arising from the ion fields will
be neglected.
It simplifies the derivation of W (Co) if the parabolic
representation for the atomic states is used. In the para
bolic representation the z component of the atomic matrix
element R is diagonal with respect to the principal quantum
number. Using this representation and choosing a coordinate
system for the ionfield integration such that fL lies in
J.*
the R direction, the operator L0 reduced to
4b ,v , = <** eE.( a ) S 9 6,
A4 rPPJ AAV A 44' Vr ,., (4"9 fir
(2.52)
When this expression is used in the expansion for KO(6U) one
can show that
k.W) I  I k"
(2.53)
When the above form for KO(4d) is used in the equation for
; 9k (tAJ), the matrix elements reduce to an expression
similar to those in equation (3.119) in reference 9s
35
not included. Note that k contains the,
AIr
1 f its IVt 1>
(2.54)
where the nm superscript indicates that the transition
between initial and final states due to the ionfield are
on
not included. Note that ka contains the ionfield, whereas
the corresponding expression in reference 9 does not contain
the ionfield.
Computational forms for equation (2.54) are derived in
Appendix C. If Vae is taken to be the full Coulomb inter
action and if symmetry effects in P are neglected, one
obtains
00
5 a) (%)Ao
(h+ G
3 i(2.55)
36
where g4, = A' is defined in Appendix D, and D
is the Debye length. The above expression is the result
of evaluating the first term in equation (2.54). If lower
state interactions are neglected, the other terms do not
enter into the computations. In the dipole approximation
the lower state interaction is identically zero for the
Lymanalpha transition. However, when the full Coulomb
interaction is used, the lower state interaction is not
zero. Computations for the Coulomb case were performed in
which the lower state interaction was included; the results
show that these lower state interactions produce less than
a 5 percent effect on the line profile in the temperature
density cases examined. However, these interactions may
have a larger effect for the higher series transitions.34
From the above analytic form it appears that when the
integration variable approaches zero, convergence problems
arise in the line center. This would in fact be the case
if electron correlations were neglected in the electron
distribution function. However, the second term in brackets
contains electron correlations and provides convergence in
the line center. When electron correlations are neglected,
the resulting divergence in 9' (W.) produces a dip in the
center of the computed profile935
center of the computed profile.
(H.7) HIGHER ORDER TERMS IN *(O)
Most of the interactions between the radiator and
perturbing electrons are weak because the probability is
small that electrons get close enough to the radiator to
cause a strong collision (see Section 11.2). Since 9 ((O)
is expanded in terms of the coupling constant, it is
expected that the perturbation expansion for 5'(<)) will
converge rapidly for weak interactions or weak coupling.
However, in what follows, it is shown that higher order terms
may become appreciable for small frequencies the pertur
bation expansion breaks down as AW) 0 (~*oo).
The computation of higher order terms is not tractable.
Hence, only an approximate analysis of the behavior of a
higher order term in the expansion for i(s) is considered.
The procedure followed is a generalization of the method
used in obtaining an expression for the first term (see
Appendix C).
The n = 2 terms in the expansion vanish because of the
symmetry of the electron distribution function and the
radiatorelectron interaction.9 Thus, the next contributing
term is the n 3 term. There are (n + 1)2 16 such terms.
A typical term has the form
g,, 5 < I^
indices in the above term.
above expression the noquenching approximation is made and
the ionfield splitting of the atomic levels is neglected.
Under these conditions the operator k0 can be written as
ab
o (2.57)
Inserting these results into equation (2.56) yields
*(l 2 Ir .5
4,, 4 4 r"
ZA.( Mi
<.ej VIA (r i4">~ "(6 l4X( ~)) W.,
AA
,a Vali (2.58)
The terms )JL,' are the change in energy of the
perturbing electron during a collision. At the line center
WWI is small because changes in the energy of the atomic
electron are small and there are no other significant
mechanisms in this theoretical treatment which could cause
a large change in the perturbing electrons' energy. Also,
the terms Wa 3,( are small in the line center. Thus,
at the line center the exponentials do not oscillate rapidly
as they do in the wings of the line. Hence, it is expected
that higher order terms may become large in the line center
but not in the line wings since the oscillating nature of
the exponentials assures that the above term is small in
the line wings.
If the procedure used in Appendix C is generalized, an
integral expression for the n = 3 term in W (W)) can be
obtained. The following multidimensional integral is
encountered;
x here +h m+ ow ig o r' io/a. er s A) n ( h'i equ*% t o and
4p(4,+ / r r'*i)u ) 4) ( E Pr" J7j
X V. v., i.) v., V. x 4, in i >
wherI P't In< lI tevro

