• TABLE OF CONTENTS
HIDE
 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














Title: Improvements in the relaxation theory of spectral line broadening in plasmas
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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
 Subjects
Subject: Plasma spectroscopy   ( lcsh )
Relaxation (Nuclear physics)   ( lcsh )
Physics and Astronomy thesis Ph. D
Dissertations, Academic -- Physics and Astronomy -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 161-163.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Table of Contents
    Title Page
        Page i
        Page i-a
    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
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        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
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        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
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        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 No-Quenching 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 Lyman-alpha 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 Ion-Field Dependent Atomic Density Matrix 72
(V.4) The Full Coulomb Interaction . . 79
(V.5) Ion-Field 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 Lyman-alpha 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 Lyman-alpha profile . . . 68

4 Field dependent part of the function J(CWie )
for the Lyman-beta profile a . . a 70

5 Lyman-alpha profile using a field dependent
atomic density matrix . . . . . 75

6 Lyman-alpha 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 Lyman-alpha profile using the full Coulomb
interaction. Solid line does not use a strong
collision cut-off. Dashed line uses a strong
collision cut-off. . . .. 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 Lyman-alpha 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 Lyman-alpha profiles with and
without lower state interactions . . . 106

16 A comparison of a Lyman-alpha profile computed
using the effective distribution function to the
unified theory and the experimental data . . 110

17 Static wing results for the Lyman-alpha profile
compared with the experimental data . . 113

18 The Lyman-alpha 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 ion-field 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 cut-off 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 Lyman-alpha 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 Lyman-alpha 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 Lyman-alpha 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 charged-particle fields on

the radiation, collisions between the radiator and various
neutral constituents in the plasma, and Doppler effects.

In this chapter, the Lyman-alpha 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 auto-correlation function, <(j),3o4,5

Ic") = 4ra-e 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 Lyman-alpha 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 ion-field, 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 ion-field 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
ion-ion interaction. Thus the appropriate ion microfield is
the low-frequency 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 electron-electron interaction is not fully
shielded. To explore the effects of shielding in the elec-
tron microfield, the static wing profile calculated using a
low-frequency electron microfield is compared to that obtained
using the high-frequency 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 cut-off or strong collision
cut-off. For the Lyman-alpha 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 cut-off in the second order term for frequencies
greater than the half width. However, in order to obtain
realistic results a strong collision cut-off 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 10-6 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 radiator-radiator

collisions. The broadening due to these collisions will be
neglected compared to the effect of the electron-radiator

collisions which interact through long range potentials.














(1.6) THE NO-QUENCHING 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
Lyman-alpha 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) ion-field
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 ion-field 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 ion-field.
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> di-V+-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

y4-cF 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, "a-A',^ (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 electron-proton 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 cut-off. The value of the cut-off obtained
in this manner is indicated in Figure 1. It is larger than
cut-offs 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
mg-I s



.0UO

Smg-
* 0




rzl




a-




C54










































































4 nm 0
S14 1-4


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 ion-field is retained in H'.
Including the ion-field 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
integro-differential 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
non-diagonal 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 non-diagonal 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 (l-P) to equation (2.16) and
then eliminate the operator D 2( ) from the resulting
equations. In this manner one obtains an integro-differential
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) (I-P)
-- -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 Lippmann-Schwinger
equation33

AT(w) L+ L-4 (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 ion-field 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(I-P)L) <=- >)L,
(2.39)

(2.40)











-I -
E (- L-P)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 ion-field 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 ion-field are neglected and the no-

quenching approximation made for electrons, additional
summations in equation (2.51) can be performed. Under the
no-quenching 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 Lyman-alpha 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 ion-field 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 ion-field are











on
not included. Note that ka contains the ion-field, whereas
the corresponding expression in reference 9 does not contain
the ion-field.

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

Lyman-alpha 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
radiator-electron 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 no-quenching approximation is made and
the ion-field 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 multi-dimensional 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

where the meaning of the various terms in this equation and







40
in subsequent equations in this section are explained in
Appendix C. After transforming to relative coordinates and
performing all possible integral analytically, one obtains





44 '- (-I-J J I --"1-))






(2.59)

The electron correlations provide a convergence factor
for the q integration as they do in the first term, n u 1,
of the expansion (see Section 11.6). However, the q' and
q" integration do not have suitable convergence factors for
small frequency separations. The results of performing the
q" integration will be similar to the results obtained in
the n = 1 term for i j (see equation C.23, Appendix C).
This factor diverges in the line center. Thus, the q' and
q" integration yield results which diverge in the line
center. Hence, the higher order terms are appreciable in
the line center.
The above arguments hold whether the full Coulomb
interaction or the dipole approximation is used. However,
there are differences between the two treatments which are
considered next.
In the Lyman-alpha transition, four matrix elements









appearing in the first order term (n = 1) for 9 ((a) are
found to account for most of the broadening. When these
elements are calculated using the full Coulomb interaction
they are larger than the corresponding terms in the dipole
approximation. The ratio of the dipole approximation to
the full Coulomb interaction treatment for one of the most
significant matrix elements is shown in Figure 2. The ratio

is always greater than one for the various strong collision
cut-offs considered. Therefore, the perturbation expansion

for N/(W.) using the full Coulomb interaction will break
down somewhat sooner as A) approaches zero than it will
when the dipole approximation is employed. This behavior
may partially explain the differences between the two
treatments in the line center.























o o
0


v I-
0

U 41



Ul 1



4J 1

x
01
o oi
U PP






41
0
x

-I Hl



0
14 8

0
*0
-I

4). 41




Ul
N

P1

b4





























































































0 0
N H














CHAPTER III

QUALITATIVE ANALYSIS OF THE LINE SHAPE EXPRESSION

(III.1) INTRODUCTION

The mathematical aspects of the line shape problem
necessary to interpret the theoretical results are analyzed

in this chapter. Expressions for the line shape are derived

in which the. effects of various approximations can be easily

related to the computed profiles. In certain cases,

functional aspects of the mathematical expressions are

examined rather than the complete expressions used in the

computations.











(111.2) THE OPERATOR Ca- (T(C )>

To compute a line profile the expression
T ; (<. ((a)- )

must be evaluated. This implies that the matrix elements
of the operator C) (T(W)> must be calculated. Due to
the Liouville operators occurring in (T(e )> these matrix
elements will be high order tensors requiring, in general,
four or more indices. The matrix elements of the Liouville
operator, L0, and of the effective interaction operator, / (a))
are given in equations (2.48) and (2.55) respectively. When
these are used in the expression for (w (T(c )> ), one
obtains the followings

(,- (TA))) = 0I( I)2 ^d I

-2 (- -, -F3l




[L db () -i ('- ]+(b* )/g ]


(3.1)








46
where C1 and C2 are constants. As indicated in Appendix C,
the function 4 (-&) depends on the ion-field shifted frequency.
The first bracketed term in the above expression repre-
sents the frequency separation from the line center.
Mathematically it can be interpreted as the independent vari-
able.
The second term contains the broadening effects of the
static ions. is restricted to lower states and a
is restricted to the upper states. Hence, lover state
interactions with the ion-fields are included. The lower

state ion-field broadening is identically zero for the Lyman-
alpha line in hydrogen. However, it may become important
in the Balmer beta transition having as much as a 10 percent
effect in certain cases.31
The last two terms in equation (3.1) represent the effects
of electron broadening. In the relaxation theory these terms
are not symmetric in (4 4a) and are solely responsible for
a red asymmetry in the line center that changes to blue in
the far wings of the line. The red to blue cross over is
quite apparent for the Lyman-alpha transition in ionized
helium.36,37 It also appears in the Lyman-alpha profile for
hydrogen but only in the far wings of the line, A ) 8A.
As discussed in Section (II.6), lower state perturbations
due to electron interactions are contained in the last term
in brackets in equation (3.1).
In reference 9 it is shown that for the Lyman-alpha
transition only the y = \r u 1 terms contribute to the line







47
profile. In the no-quenching approximation the states A44
are restricted to initial states. Thus, for the Lyman-alpha
transition only two indices are required to specify the
elements in the tetradic 0 (T(OW)> That is ( W -
(T(m))> ) has a convenient matrix representation. Further-
more, since a hydrogenic state having principal quantum
number n, has n2 degenerate levels, the matrix representation
will be n2 x n2; it will also have a block diagonal form as
will be shown in the next section.














(111.3) THE PARABOLIC REPRESENTATION

The Schroedinger equation without an external field is
separable in the parabolic coordinate system. The resulting
wave functions are given by38

[ ---1 1/





(3.2)

where M. ,,44 are the parabolic quantum numbers associated
with the atomic state, ,( I The other symbols are defined
in Appendix D. It should be emphasized that the parabolic
wave functions given above are not "Stark" wave functions
since they are not solutions to the Schroedinger equation
for a system consisting of a hydrogen atom in an external
field.
The parabolic representation is used for mathematical
convenience. For a given principal quantum number, the a
component of the dipole matrix element is diagonal in this
representation. Hence, the atomic density matrix occurring
in the line profile expression is diagonally the second term
in brackets in equation (3.1) is also diagonal, and many of
the other matrices reduce to a block diagonal form.. These









simplifications are particularly convenient for computational
purposes.
In the full Coulomb interaction case, the term .i y^1'
occurring in () * of the form

AN4AN ,u A"


where m is the magnetic quantum number. Thus, q " A'
is diagonal in m when the full Coulomb interaction is used.
In the dipole approximation the third term in the
expression for W (T(a~ )>, given in equation (3.1) becomes







With the results of Appendix F, the dot product R ".'
is found to contain delta function products of the form


S AO A44M A41 4
MIq. Al" I M #41 4










Thus in both the dipole interaction and the full Coulomb
interaction, the matrix ( 6& (T(&))) has the form











A I A 3 4 -



ao o o o a
1 0 0 0 0 0



W( < T >) 3 0 0 33 3+ o


4 o o 43 4+ 0

3r o o o 5'


where.the parabolic states are labeled as folloess

11) a 1100)
12) B 120-1>
13? a 1200)
147 5 1210>
15 a 1201 ) .

Blocks which arise from the same magnitude of m but
different signs are mathematically equivalent and can be
shown to give identical contributions to the line profiles
thus, the 22 block contributes the same as the 55 block.
This follows from the form of the wave function given in
equation (3.2)s the wave function depends on the sign of m
only through the term exp(imo) which does not affect the
magnitude or sign of any of the matrix elements (see
Appendices D and F).
Following a similar procedure for the Balmer beta
transition, it is found that the orders of the matrices
encountered range from 1 to 4.













(III.4) THE LYMAN-ALPHA TRANSITION

From the previous section, it is clear that the symmetry
of the atomic states in the parabolic representation greatly
simplifies the computational effort. However, there still
remains the problem of reducing the line shape expression
given in equation (2.25) to quadratures. This section
outlines the procedure for reducing the Lyman-alpha line
shape expression to a convenient calculable form.
The matrix elements of the operator (W (T(W)) ) are

((o-


+ .*AG4Sm W
T A T '/




(3.3)
where
2/3
= 2.614 M,
S= ion microfield
VW electron collision term.
AdU

In the dipole approximation W4MM is given by

w R-" R-
WA 'm e ,, (3.4)

where A,-", contains the ion-field. Using the results
derived in Appendix F we find that the non zero matrix








elements of \ are
Aw A






= :7 (A40)


VV3


-w


(3.5)


-here A 41 =- -- = d6,I is the unshifted frequency
separation. The corresponding elements of ( T (T(wi)>) are

(<- JAT all


((o- )
33


* AJa
AT
V


ST


C- (W3 \33


31-

(<- )
(- (T(w44


(I<- J-5

( )
435


tc /W
.34


a w


++ -T
B


EtICv W
4-4


= (<- ()))
-3-+


(3.6)







where -Z33 = 3ao0 44 -3a0, and the definition of c
follows from equation (3.3),

,- )
W 3&0T ) a


All other matrix elements are zero or do not enter into the
computations because of the restriction that at have matrix
elements only between the initial and final states.
The resolvent operator, K(4), appearing in the line
shape expression is the inverse of 6; For block
diagonal matrices one has


/(33 3+\ /
\f3 ++) = \+ ++)




where D is the determinant of the 2 x 2
The determinant of the 2 x 2 block
33(o
I ecw
\~S


) (4-31
=~ '= *"t /
tr


matrix.
reduces to


which can be rewritten in the following convenient forms


f^y f-3AT AIt A B
( Tr-- (3.7)
:








where A and B are complex coefficientsthat depend on the
matrix elements of W. A and B are small compared to

AA 4/./AT except in the line center (see Chapter V for
actual numerical comparisons). Hence, the determinant
provides a resonant denominator which has a minimum at ion-
fields given by

A^^

(3.8)

The function J(c E) used to compute the line profile (see
equation 2.15) will be peaked at values of the field given
in the above equation. J(GI E) can be interpreted as a
line profile from a radiating atom that is placed in a
constant ion-field and perturbed by the electrons. This
interpretation is useful since considerable physical insight
into the effects of electron broadening can be gained by
studying the function J(ou,, ) rather than the full line
profile.
For transitions more complicated than the Lyman-alpha
profile, there will be additional resonant denominators
corresponding to the various Stark shifted levels. For
example, in the Balmer beta line there are three different
z matrix elements resulting in three distinct resonant
denominators.
To compute a line shape, the imaginary part of the
following trace is required

Te ).









The matrix a occurring in the above expression is restricted
to have elements only between initial and final states. The
matrix elements of the x, y, and z components are computed
in Appendix F.
In the no-quenching approximation, the atomic density
matrix p can be written as,



I


where a <- R ^ )).
Using the above results the trace over atomic states
can be performed. The results areas


t(w Ca- s-r rA#J T 4 h.n




(3.9)

The leading term in brackets does not have a strong ion-
field dependence in the dipole approximation and is field
independent in the full Coulomb interaction treatment. In
the dipole approximation, this term can be interpreted as
representing the effects of electron broadening on the initial
levels, 13> and 14) In the dipole approximation the
K22 term depends on the dipole matrix elements having
different magnetic (m) quantum number as can be seen by
examining the \/22 matrix element given in equation (3.4).









However, the full Coulomb interaction does not mix states
having different m values as a consequence, the leading
term is field independent.
In both the dipole approximation and the full Coulomb
interaction case, the K22 contribution is significant only
in the line center. For a temperature of 20,000K and an

electron density of 3 x 1017/cm3 the K22 contribution for
the dipole approximation is significant from 0 1 A from
the line center. Whereas, for the full Coulomb interaction
treatment the range of importance is from 0 .8 A. The
net effect of this slightly different behavior is to produce
a shoulder in the line shape when the full Coulomb inter-
action is used. It is interesting to note that a similar
behavior is encountered in the dipole approximation at
higher temperatures in hydrogen and in the Lyman-alpha
transitions in ionized helium.
By taking the W/--0 limit in equation (2.54) a static
wing formula can be obtained. This is the subject of the
next chapter.














CHAPTER IV
ASYMPTOTIC WING FORMULA

(IV.1) INTRODUCTION

The purpose of the present chapter is to derive an
expression that accurately describes the wings of the profile.

In Section (IV.2) a heuristic derivation of a static ving
formula is presented. The resulting formula is shown to

be the leading term in a rigorous perturbation expansion

presented in Section (IV.3). Higher order terms in this

expansion are shown to include the effects of electron

dynamics.













(IV.2) HEURISTIC DERIVATION OF A STATIC WING FORMULA

In the derivation considered in this section, it is
assumed that the only broadening mechanisms are static fields.
Under this assumption a wing formula can be obtained from
the line profile expression (equation 3.9) presented in
Chapter III by the following procedures 1. Replace all
static ion-fields, E. by the total static field,
ET ( 7" E + f ). 2. Take the limit as the electron
dynamic terms go to zero. In this limit the line profile
becomes

I ( W = PUrQ r9 + f UT)P(+
STATIC
A*w

(4.1)

where PT( 1T ) is the microfield probability function for
the combined electron-and ion-fields, defined as ( (~ '
Since 3( ) 4+(- ) (see Section III.4) and

PT ( T. ) is symmetric4 in T the above expression is
equivalent to


SThrIe.


P (i-) P (c i + +K
4'.T
A AW


(4.2)







59
The atomic density matrix element ('3 evaluated at
ET ,< #.e is equal to 1/e where e is the natural loga-
rithm base. Thus



(4.3)

Determining the microfield distribution function for
the total field from first principals is non trivial. As a
first approximation a convolution of the electron microfield
with an ion microfield is used. The procedure for obtaining
the total microfield in terms of a convolution follows.
The microfield is defined as


q(Et) = &W.^(r -4))

(4.4)

By writing the density matrix in factored form, one can show
that the above reduces to3



= ^VTn, eS (it I. )}\ (t'e-^


q E 4.5)

where the properties of the Dirac delta distribution have
been used to obtain the last expression which is a convolution
of microfields.







60
It should be emphasized that the above expression for
the total microfield is an approximations in order to
obtain the factored form for the density matrix, the

electron-ion interaction was taken into account by an
effective potential which depends only on the ion coordi-
nates (see Section II.1 and Section 1.4). Hence, the

electron-ion interaction is contained only in the ion
microfield. It is expected that this treatment of the

electron-ion interaction should be good since the mobility
of the electrons is much larger than the ion mobility
resulting in a shielding of the ion-ion interaction. To

examine the effects of shielding on the electronselectron

interaction, line profiles obtained from convolutions of
various combinations of low-and high-frequency microfields

are compared in Chapter V.














(IV.3) THEORETICAL DERIVATION OF A WING FORMULA

In the line wings the time of interest is small; hence,
a perturbation expansion in which the time is an expansion
parameter may prove to be useful in the line wings. Such
an expansion is derived in this section and carried to
second order. The procedure for generating higher order
terms follows from a straightforward extension of the
method used to calculate the second order term.
The Hamiltonian appearing in the function, ( X), can
be written as

^ H. + MI

where

H.; s ^+ VyV V-
(4.6a)


PH /- 4 9 + Vi (4.6b)

Using the above definitions the time development
operator T(t-) appearing in ( ) can be written as





where exponents contain g third and h er order terms ha(4.7)

where exponents containing third and higher order terms have








been neglected i.e.,



.,e,(-+x ,CH"',4, HJ,..] ) 1, for n > 2.

Similarly



X ^ ('1xH,/jf
Substituting the above results in to f( C) yields

Ct> = T, (d.4 ,(* CH, fa, ,1/A l ..( ^ I.A)









(4.8)

It is convenient to regroup the above equation into
the following form

X(c) = T, (I. I'e l frni)/d + /iA )P l/,A)





(4.9)
where again terms such as {f ) have been set equal to one.







63
The following operator expansion40 is used to further
reduce (X-):

(^A)8 <4N-A)= B C[A,B] + [7C A,[8]+---
(4.10)

where A and B are operators defined as

8= (#<*/< /Jr) p4 a14a-i'x#/A)

A, ,- [I,,/')/.j. iH

(4.11)

and then applying the above to equation (4.9) for (t)yields








[H., ., ,a-f'6/) 1 r' /&)x-h )I, ))






(4.12)

Clearly the above expression will break down for large
times. Hence, as smaller frequency separations are approached,
higher order terms will become increasingly important. It
is expected that electron dynamics will certainly be









significant for frequencies less than the electron plasma
frequency; thus, for the region of the profile corresponding
to bAr << terms linear in C and perhaps even higher order
terms will be required to adequately represent the intensity.
Based on comparisons with experimental data presented in
Chapter V, this behavior indeed appears to be verified.














CHAPTER V
ANALYSIS OF THE THEORETICAL RESULTS

(V.1) INTRODUCTION

An analysis of the theoretical computations is
presented in this chapter. The theoretical results are

compared with the experimental data and with other

theoretical computations.













(V.2) THE FUNCTION J(W ,E )

The line profile involves an integration over F of
the product of P( ) times J(ko, E ). The region of the
profile in which various field dependent effects are
important can be determined by an analysis of J( ,C ).
J(o,g ) consists of a field independent term plus a
field dependent terms

J(to ) J(o) + (6 e). (5.1)

In the case of Lyman-alpha radiation, a graph of (>,o )
vs. E will have one sharp peak due to the resonance
denominator discussed in Section 111.4 (see Figure 3).
A slightly more complicated result is found when
S(WE ) vs. E is plotted for the Lyman-beta transition in
hydrogen. In this case, the ion-field removes the degeneracy
in the upper level, splitting it into three distinct levels,
two of which are shifted. Hence, the graph of the field
dependent part of J((u,E ) will have two peaks. d (u )
for the Lyman-beta transition is plotted in Figure 4. For
E = 0 both peaks coincide and occur at E = 0. As the
frequency increases two peaks appear.
A few general characteristics of (a,W ) and some of
the resulting implications are listed below.





























Figure 3. Field dependent part of the function J(W, )
for the Lyman-alpha profile.
















I





















































Ion-rield) >
































Figure 4. Field dependent part of the function J((,C )

for the Lyman-beta profile.










104


-~ o.oA


a 0.0 ~



16 Functioi
Sfor LynI




















101









100
0 8 16 24
Ion-field, E


32 40 48









1) For large frequency separations from the line
center, most of the contribution to the field integration
appearing in the line shape expression arises from large
values of the field.
2) The relative maximum of W(eJ, ) increases as the
frequency increases while the half-width decreases. This
suggests that replacing the peaks in the function (w F )
by delta function distributions should be accurate in the
line wings. This limiting behavior was used in the heuristic
derivation of the static wing formula discussed in Section
(V.21
3) The maximum value of h (, ) decreases rapidly as
the frequency increases, but the rate of decrease diminishes
for larger values of the frequency.
4) For every value of 6) ( (,~ ) will approach a
fairly constant value for field values greater than those
for which (&, f ) peaks.
From the behavior of ((u, ( ) it is easy to determine
the range in frequency for which various field dependent
effects are important. For example, a field dependent effect
that is directly proportional to the strength of the field
will be most important in the line wings. One such effect
which exhibits this dependency arises from the field depen-
dence in the atomic Boltzmann factors. This is the subject
of the next section.













(V.3) THE ION-FIELD DEPENDENT ATOMIC DENSITY MATRIX

The Stark shift in energy levels caused by the ion-field
is frequently neglected in the atomic density matrix occur-
ring in the line shape expression. Neglecting the field
assumes that the density of states is equal for all atomic
levels corresponding to a given principal quantum number.
The ion-field removes the degeneracy of the excited states
via the Stark effect yielding 2n 1 distinct energy levels
or Stark levels where n is the principal quantum number.32
The field dependent atomic density matrix weights Stark states
that are lower in energy more heavily than states of higher
energy. Since transitions from lower energy initial states
give rise to the red wing of the line profile, inclusion of
the ion-field in the atomic density matrix will cause a net
red asymmetry. To determine whether this effect is uniform
over the entire profile one can analyze the function w (w,E ).
In the previous section, curves that demonstrate the behavior
of -(w ) were presented. For the Lyman-alpha transition
the peak in (4, ) ) occurs at a field value given approxi-
mately by

e ~(5.2)

Thus, for large frequency separations, (0 (, ) peaks at
large values of the ion microfield. The exponent appearing









in the Boltzmann factors in the atomic density matrix is
directly proportional to the ion-field and hence, the larger
the field, the greater the effect in the density matrix.
Thus, the effect of including the field in the atomic
density matrix is relatively more important in the wings of
the line than it is in the line center. Profiles calculated
with and without an ion-field dependent atomic density matrix
are given in Figures 5 and 6 respectively. From these curves
it is demonstrated that including the ion-field can produce
a noticeable effect for wavelengths greater than 4 A.
The dependence of the atomic density matrix on Lampera-
ture and on the magnitude of the dipole matrix element should

also be considered. The ion-field dependent factor appearing
in the exponent of the atomic density matrix is proportional
to e Rz /RT where T is the temperature and Rz is the z
component of the atomic dipole operator. Thus, at low
temperatures, the importance of the Boltzmann factor
increases. Also, for higher series transitions Rz is larger
and the effect may become large. Since radiation from higher
series transitions will be of longer wavelength, the corres-
ponding frequency separations will be smaller. For small
values of the frequency separation the field integration
over (&>, ~ ) contributes significantly only for small values
of the field. Thus, for higher series transitions, the

product (Rz is not as large as one might otherwise expect.
For the Lyman-alpha transition in ionized helium, the z
component of the atomic dipole matrix element is smaller by a
































Figure 5. Lyman-alpha profile using a field dependent

atomic density matrix.














Lyman-alpha line
no = 3.6 x 101/om3
T 20,400 K

Pa- Pa<)


Red wing


Blue wing


6 8

AA


10
in Angatriam


I i -


0 4
































Figure 6. Lyman-alpha profile neglecting the field

dependence in the atomic density matrix.












Lyman-alpha line
no = 3.6 x 1017/O3
T 20,4000K
ea.- .(o) .i.


wing


Blue wing


) I I I


0 % 4 6


8 10 12 1

A t in Angstr6ms


102













101


100







78
factor of two compared to the matrix elements in hydrogen.

Also, the helium profiles presented in this dissertation

correspond to temperatures that are about twice as large as

those considered in the hydrogen line calculations. Thus,

the exponent appearing in the atomic density matrix is

decreased by a factor of approximately 4, and the effect

of the ion-field in the density matrix can be neglected.














(V.4) THE FULL COULOMB INTERACTION

One of the approximations used in most treatments of
the line broadening problem is the assumption that the

interaction between the perturbing electrons and the
radiating atoms is a dipole interaction. Such an approxi-
mation provides an accurate description of the interaction

for large distances between the atom and perturbing electron.
To achieve 10 percent agreement between the exact interaction
and the dipole approximation for all orientations of an
electron-proton dipole, the distance between the dipole and
the point at which the potential is measured must be greater
than 10 times the length of the dipole. For excited states

resulting in Lyman-alpha radiation, the average distance
between the nucleus and atomic electron is 4 Bohr radii.
Hence, to achieve at least 10 percent agreement between the
dipole approximation and exact interaction, the distance

between the perturbing electron and radiating atom should be
greater than 40 Bohr radii. The strong collision cut-offs
used in various theories1,291621 for the Lyman-alpha
transition are less than 15 Bohr radii which is in a region

where the dipole approximation is not highly accurate.
These conditions provided the motivation for developing a
line broadening theory which uses the full Coulomb potential

to describe the atom-perturbing electron interaction.









However, it should be emphasized that even in the full
Coulomb treatment, a strong collision cut-off should be
used to approximate symmetry effects usually neglected in
the electron density matrix (see Section II.2).
One effect of including the full Coulomb interaction
in this second order theory is to produce a shoulder at
about 0.5 1 in the Lyman-alpha profile. A shoulder is also
obtained in the dipole approximation. Although, it is not
as pronounced as in the full Coulomb treatment. In the
parabolic representation the dipole matrix elements for the
initial states have non zero x and y components between
states having different magnetic quantum numbers (see
Appendix F). However, matrix elements of the full Coulomb
interaction are diagonal with respect to magnetic quantum
number (see Appendix D). Also, the magnitude of the matrix
elements of W are different for the two approximations. The
matrix elements which appear in the function j(W) in the
full Coulomb interaction treatment are smaller than the
corresponding elements in the dipole approximation and
are primarily responsible for the more pronounced shoulder.
The numerical values for the W matrix for the two cases are
compared in Figure 7 for various strong collision cut-offs.
The computed line shape results for the full Coulomb treat-
ment are presented in Figure 8.
With either the full Coulomb interaction or dipole

approximation, the line profile can be written as

I(a) = .(L) + (p(E. Q.(4,E)i (5.3)

























oo
0 0
0 N

(% O








0
o
'u
6

M
M O








LZ
q4 0



P4*
U

o
U
0
* .6


NM









82

























oo
'-I




















SI I I I
fnfl l
IN




m u
E 1S I 0
- O S O -0
4" 1 W



0 a
N N C
4 J .4
04 0 Q
* U 4
.41 U








se ***
































Figure 8. Lyman-alpha profile using the full Coulomb

interaction. Solid line does not use a strong

collision cut-off. Dashed line uses a strong
collision cut-off.









103





Lyman-alpha profile using
the full Coulomb potential.


T 20,4000

n, 3.6 x 10l17/e3


00
102 qmax i6.


SAz 208 x A









101












10o I I I I I I
0.00 .01 .02 .03 .04 .05 .06
A *6/ 1 T









It is of interest to analyze and compare the term containing

the field integration and j (w) separately. Curves for

I()), j(&o), and P( ) f(oe )JF are given in Figure 9
for the full Coulomb interaction and in Figure 10 for the
dipole approximation. As is demonstrated in these figures,

the term j(( ), which is equal to four times the K22 matrix

element, is important only in the line center. Its region

of importance is slightly greater in the dipole approximation

than in the full Coulomb interaction case. This difference

is primarily responsible for the shoulder obtained from the

full Coulomb computations and illustrated in Figure 8.

A treatment of the line shape problem that uses a full
multi-pole expansion for the electron-atom interaction to

all orders in the coupling constant has been carried out by
M. E. Bacon, K. Y. Shen, and J. Cooper,41,42 In the case of

the Lyman-alpha transition, all multi-poles higher than

quadrupolar terms vanish due to the orthonormality of the
atomic states.

The major features of the calculation by Bacon et al.
can be summarized as follows

1. The S matrix is summed to all orders in the coupling
constant.

2. Time ordering is included in the S matrix.

3. The perturbing electron trajectories are represented
by classical paths.

4. Minimum and maximum impact parameter cut-offs are
used to account for strong collisions and electron correlations






























Figure 9. Functions I(C)), J( ), (WE) for the full
Coulomb interaction case.















I(d), j(W), P(O ) (O,)-
for the Lyman-alpha profile

using the full Coulomb

\ interaction.



102 '













101
4I











100 1








.0001 .001 01

A &,AI
4





100
.000 .00 .0






























Figure 10. Functions I(o), j((0), d(~a, E) for the
dipole approximation.








103



for the Lyman-alpha profile in
the dipole approximation





2
102




",,( IE) e)






100 0
0', \
\ \















.0001 0001 \01
\






A T
100 a
.0001 ,001 .0 1









respectively.

5. A minimum velocity cut-off is employed.
A comparison of the present theoretical results with
those of Bacon et al. are presented in Figure 11 for a
temperature of 20,0000 K and an electron density of 1.0 x

1017/cm3. Only the center and near wing of the profile are

plotted. The agreement is fairly good for wavelengths

greater than 0.4 A. However, the profiles differ consider-

ably for AX < 0.4 A.

The full Coulomb treatment presented in this disser-
tation uses a strong collision cut-off. However, reasonable

variations in the value of the strong collision cut-off does
not improve the agreement between the two theories near the

line center. Also, if electron correlations are accounted
for by a cut-off rather than by the use of the electron pair

distribution function, the agreement is not improved.

In the quantum mechanical treatment presented here, the
atom-perturbing electron potential is transformed into a

momentum representation in which convergence at zero momentum

is assured by other momentum dependent factors in the inte-

grand for 60C not equal to zero. For zero frequency separa-
tion the manner in r.hich electron correlations are treated

assures convergence in the line center (see Sections 11.6 and
II.7 for further discussion). The major convergence factor,

J(q), appears explicitly in the derivation after the time
integration appearing in % ()o) is performed. In second
order theories of spectral line broadening in plasmas time

































Figure 11. A comparison of the Lyman-alpha profile using

the full Coulomb interaction to computations
41,42
performed by Bacon et al.




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