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Title:
Stark broadening in laser-produced plasmas full Coulomb calculation
Creator:
Woltz, Lawrence A., 1949- ( Dissertant )
Hooper, Charles F. ( Thesis advisor )
Dufty, James W. ( Reviewer )
Bailey, Thomas L. ( Reviewer )
Thomas, Billy S. ( Reviewer )
Chen, Kwan Y. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
1982
Copyright Date:
1982
Language:
English
Physical Description:
v, 75 leaves : ill. ; 28 cm.
Subjects
Subjects / Keywords:
Approximation ( jstor ) Asymmetry ( jstor ) Electron scattering ( jstor ) Electrons ( jstor ) Ions ( jstor ) Line spectra ( jstor ) Plasmas ( jstor ) Power lines ( jstor ) Quantum numbers ( jstor ) Radiators ( jstor ) Coulomb excitation ( lcsh ) Dissertations, Academic -- Physics -- UF Physics thesis Ph. D Plasma radiation ( lcsh ) Plasma spectroscopy ( lcsh ) Stark effect ( lcsh )
Genre:
bibliography ( marcgt ) non-fiction ( marcgt )
Notes
Abstract:
This work is a study of the Stark broadening of spectral lines
emitted by highly charged ions in a hot, dense plasma. The line
broadening calculations of Tighe and Hooper, in which the dipole
approximation was used for the interaction between the radiating ions
and perturbing electrons, are extended by retaining the full Coulomb
radiator-perturbing electron interaction . Electron broadening is
treated to second order in the radiator-perturbing electron interaction;
ion broadening is treated by a static ion microfield probability
distribution . Perturbing electrons are treated quantum mechanically
through Coulomb wavefunctions to account for the charged radiator
.
Line profiles from this calculation are slightly broader than those
calculated using the dipole approximation . The near agreement of these
results is partly fortuitous in that the dipole approximation
overestimates the dipole part of the Coulomb interaction for perturbers
close to the radiator, partially compensating for the neglected multipoles of the interaction . This full Coulomb calculation of the
dynamic line shift due to perturbing electrons results in a
significantly smaller shift than that obtained when the dipole
approximation for the radiator-perturbing electron interaction is
used . For an Argon Lyman B line calculated for the plasma conditions of
10 electrons/cm and 800 eV, this shift is about 0.03 Ryd toward lower
energies, and it causes no noticeable asymmetry in the line. The effect
of the dynamic shift on the line is considerably smaller than that of
the static (plasma polarization) shift calculated by Skupsky, which
causes an asymmetry in the p line, with the red wing having the greater
intensity. For an Argon Lyman |3 line, the significance of ion-radiator
and electron-radiator perturbation matrix elements between the initial
radiator states, principal quantum number = 3, and the states of the
nearest adjacent energy level, principal quantum number = 4, is
examined . The electron-radiator matrix elements are found to give a
small additional broadening of the line; the ion-radiator matrix
elements are found to cause a significant nonlinear Stark shift of the
radiator energy levels. This shift causes the line to be asymmetric,
with the blue peak having greater intensity than the red peak and the
red wing having greater intensity than the blue wing .
Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Includes bibliographic references (leaves 72-74).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Lawrence A. Woltz.
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Full Text
STARK BROADENING IN LASER-PRODUCED PLASMAS:

FULL COULOMB CALCULATION

BY

LAWRENCE A. WOLTZ

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

AUGUST 1982

ACKNOWLEDGMENTS

I would like to thank Dr. C. F. Hooper, Jr., for suggesting this

problem and for his guidance during the course of this work.

I would also like to thank Dr. J. W. Dufty, Dr. C. A. Iglesias, and

R. F. Joyce for many helpful discussions, Drs. R. J. Tighe and

R. L. Coldwell for computer codes which have been of great use, and

Ms. Viva Benton for her excellent work in typing this manuscript.

Finally, I would like to thank my wife Carol and my parents,

Dr. Shreve S. Woltz and Mrs. Theresa B. Woltz, for their continued

support and encouragement during the course of this work.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ......................................................ii

ABSTRACT ....................................... ..............,.. .......iv

CHAPTER

I PLASMA SPECTRAL LINE BROADENING ..................................1

1.1 Introduction ................. .. ..........................1

1.2 The Line Shape ...........................................2

I .3 Time Scales ................................................ .4

I .4 The Ion Microfield .........................................5

I .5 Kinetic Theory .............................................. 9

I .6 Second Order Theory ......................................14

II THE FULL COULOMB RADIATOR-PERTURBER INTERACTION .................17

1.1 The Line Width and Shift Operator .........................17

11.2 Angular Momentum Sums .....................................22

11.3 Electron Correlations ....................................24

II .4 Interaction Matrix Elements Between States

of Different Principal Quantum Number .....................26

III RESULTS ..................................................... .....30

III .1 Symmetric Line Profiles ...................................30

III .2 Shifts and Asymmetries .............. ...................... 40

IV CONCLUSION....................................................56

APPENDICES

A MATRIX ELEMENTS OF M(w) .........................................59

B EVALUATION OF G(kl,k2) ..........................................63

C A COMPUTATIONAL FORM FOR A 1 2 3(x) ............................69

D NUMERICAL METHODS ............................................... 71

REFERENCES ...... .................. .................................. 72

BIOGRAPHICAL SKETCH ...................................................75

iii

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

STARK BROADENING IN LASER-PRODUCED PLASMAS:

FULL COULOMB CALCULATION

by

Lawrence A. Woltz

August, 1982

Chairman: Charles F. Hooper, Jr.

Major Department: Physics

This work is a study of the Stark broadening of spectral lines

emitted by highly charged ions in a hot, dense plasma. The line

broadening calculations of Tighe and Hooper, in which the dipole

approximation was used for the interaction between the radiating ions

and perturbing electrons, are extended by retaining the full Coulomb

radiator-perturbing electron interaction. Electron broadening is

treated to second order in the radiator-perturbing electron interaction;

ion broadening is treated by a static ion microfield probability

distribution. Perturbing electrons are treated quantum mechanically

through Coulomb wavefunctions to account for the charged radiator.

Line profiles from this calculation are slightly broader than those

calculated using the dipole approximation. The near agreement of these

results is partly fortuitous in that the dipole approximation

overestimates the dipole part of the Coulomb interaction for perturbers

close to the radiator, partially compensating for the neglected

iv

multipoles of the interaction. This full Coulomb calculation of the

dynamic line shift due to perturbing electrons results in a

significantly smaller shift than that obtained when the dipole

approximation for the radiator-perturbing electron interaction is

used. For an Argon Lyman p line calculated for the plasma conditions of

1024 electrons/cm3 and 800 eV, this shift is about 0.03 Ryd toward lower

energies, and it causes no noticeable asymmetry in the line. The effect

of the dynamic shift on the line is considerably smaller than that of

the static (plasma polarization) shift calculated by Skupsky, which

causes an asymmetry in the p line, with the red wing having the greater

intensity. For an Argon Lyman P line, the significance of ion-radiator

and electron-radiator perturbation matrix elements between the initial

radiator states, principal quantum number = 3, and the states of the

nearest adjacent energy level, principal quantum number = 4, is

examined. The electron-radiator matrix elements are found to give a

small additional broadening of the line; the ion-radiator matrix

elements are found to cause a significant nonlinear Stark shift of the

radiator energy levels. This shift causes the line to be asymmetric,

with the blue peak having greater intensity than the red peak and the

red wing having greater intensity than the blue wing.

CHAPTER I

PLASMA SPECTRAL LINE BROADENING

I.1 Introduction

Plasma broadened spectral lines have been used for many years in

determining the densities and temperatures of laboratory and astro-

physical plasmas (Griem 1964, 1974; Baranger 1962; Cooper 1966; Smith,

Cooper, and Vidal 1969, 1970). Recently, plasma broadened X-ray spectra

from highly ionized high-Z elements (e.g. neon or argon) have been used

to determine the densities of laser inertial confinement plasmas

(Yaakobi et al. 1977, 1979, 1980; Kilkenny et al. 1980; Apruzese et al.

1981).

The purpose of this work is to extend theoretical line shape calcu-

lations by Tighe (1977) and Tighe and Hooper (1976, 1978), which were

used in the studies of Yaakobi et al. The work of Tighe and Hooper is

an adaptation of the relaxation theory of line broadening (Smith 1966;

Smith and Hooper 1967; O'Brien 1970; O'Brien and Hooper 1974) to hydro-

genic radiators of arbitrary charge, appropriate for the high-Z

radiators of laser implosion experiments. In the Tighe-Hooper calcula-

tions the dipole approximation is made for the radiator-perturbing

electron interaction, an approximation which is of questionable validity

at the high densities obtained in some recent experiments. In this line

shape calculation we will not make the dipole approximation but will

retain the full Coulomb radiator-perturbing electron interaction.

2

In this chapter we outline a line broadening theory which is

obtained by use of the kinetic theory of time correlation functions

(Hussey 1974; Hussey et al. 1975), then apply several approximations to

obtain a theory in which the line width and shift operator is expanded

to second order in the radiator-perturbing electron potential. In

Chapter II we develop a computational form for the full Coulomb second

order theory. In Chapter III we present Lyman series line shapes

calculated numerically from the formalism of Chapter II. We compare

profiles calculated from this full Coulomb theory with

the results of Tighe and Hooper, and also compare this calculation

to a recent impact theory of line broadening (Griem et al. 1979).

Calculated line shapes which include dynamic shift and inelastic effects

are also presented.

I.2 The Line Shape

The power spectrum emitted by one type of ion in a plasma can be

written in terms of the ensemble-averaged radiation emitted by one ion

of that type (the radiator) as (Griem 1974)

P(W) = 4-

12p 6(w-_ ) (1.2.1)
3c3 ab a ab
where pa is the probability that the radiator-plasma system is in the
state ja>, d is the dipole moment of the radiator, and uab = (Ea-Eb)/I.
The line shape function I(w) is defined in terms of the power spectrum
by the equation
S4
P(w) = I(),
3c
where
I(u) |2 a6(u-ab). (1.2.2)
ab
We can express the Dirac delta function in I(w) as an integral to obtain
I(w) =- f- I12p ei(W-ab)tdt. (1.2.3)
ab
Here, the integrand for negative values of t is equal to the complex
conjugate of the integrand for positive values of t, so we can rewrite
this equation in terms of an integral from zero to infinity:
1(w) = Re o I 2 p e(-ab)tdt (1.2.4)
n 0 ab
or
I(W) = Re f eimt* dt
7t 0 ab
(1.2.5)
where H is the Hamiltonian for the radiator-plasma system and p is the
canonical density operator, p = e-pH/Tr e This can be written as
1 it -iHt/h iHt/t
I(W) = Re Tr f dt eit dp e-i e (1.2.6)
It 0
where Tr represents a trace over states of the radiator-plasma system.
In Eq. (1.2.1) we have neglected Doppler broadening of the line
shape due to the motion of the radiator. This will be included
approximately at the end of the calculation by convoluting the line
shape with a Doppler profile based on a Maxwell velocity distribution.
This is an approximate treatment of Doppler broadening based on the
4
assumption that the change of momentum of the radiator is negligible
during the time of radiation (Smith et al. 1971a, 1971b). The Doppler
profile is given by
-(o-u )2 2
ID() =- e (1.2.7)
where
2 2kT 2
Y =
Mc2 0
Me
M is the mass of the radiator, and o0 is the frequency of the
unperturbed transition.
1.3 Time Scales
The line shape is given by the Laplace transform of the radiator
dipole autocorrelation function (Eq. 1.2.6). Since the transform has
the property AuAt < 1, where Au is the frequency separation from the
line center, the time dependence of a perturbation of the radiator which
changes significantly in time T will be evidenced by the the part of the
line for which A < l/T. For Au >> I/T, the perturber can be considered
as static. This part of the line corresponds to radiation from initial
radiator states having average lifetimes much shorter than T. In the
plasma line broadening problem there are two characteristic perturbation
times, T 1/Wpi for perturbations due to ions and e ~ 1/pe for
1 p1 e pe
perturbations due to electrons, where pe and w are the electron and
pe pl
ion plasma frequencies.
For the plasma conditions which we will examine, the half-widths of
the first few Lyman lines fall within the region Am < w p, so we must
treat the perturbation due to the electrons as a dynamic process. In
5
the case of the ions, the region Ac < corresponds to only a small
part of the line center. We will therefore ignore ion dynamic effects
and make the approximation that the perturbing ions and radiator are
static and that the ions perturb the radiator through a static electric
field. This approximation, which is used in many line broadening
calculations, is known as the Static Ion Approximation. Lyman line
shapes which include ion motion effects have been calculated for
hydrogen plasmas (Greene 1979; Seidel 1980). There, the inclusion of
ion motion was found to cause additional broadening near the line
center. The peak of the Lyman alpha line was significantly lowered, the
peaks of the Lyman beta line were lowered slightly and the central dip
was raised.
1.4 The Ion Microfield
In this section we apply two approximations which will permit us to
write the line shape expression in the form
I(w) = f- P(E)J(W,E)
where P(E) is the low frequency ion microfield probability (Baranger and
Mozer 1959; Mozer and Baranger 1960; Hooper 1968; O'Brien and Hooper
1974; Tighe and Hooper 1977) and J(w,E) is the line shape due to the
electron broadening of radiators in the presence of the ion
microfield E. The Hamiltonian for the radiator and plasma is
H H + Hi + H +V +r Vr + Vie
r i e ri re ie
(1.4.2)
6
where Hr, Hi, and He are the Hamiltonians for the radiator, ion, and
electron systems, Vri, Vre, and Vie are the potential energies between
the radiator and ions, radiator and electrons, and ions and electrons.
The effect of the interaction Vie is to correlate the ions and electrons
such that the average electron density is greatest near the ions.
Therefore, the field at the radiator due to the electrons has a low
frequency component which tends to partially cancel the ion field there.
If the average electron spacing is much less than the Debye length, i.e.
-1/3 < kT 1/2 (1.
ne << ) I-- ) (1.4.3)
4xm e
e
where ne is the electron density, and quantum mechanical effects are not
significant (Iglesias 1982), we can treat this correlation effect to a
good approximation by considering the low frequency microfield at the
radiator to be due to Debye screened ion fields (Baranger 1962).
Similarly, we can consider the ion-ion interaction to be screened by the
electrons. With the effects of Vie approximated by screening, we
eliminate it from the Hamiltonian to obtain
s +
H = (Ki + Vi + ri) + (Hr + He + Vr + e .Xr)
= HI + H' (1.4.4)
where H' is the Hamiltonian for the ions and H' is the Hamiltonian for
the radiator and electrons. Here, Ki is the ion kinetic energy, V s is
the shielded ion-ion potential energy, He is the Hamiltonian for an
electron gas, (ri is the monopole part of the shielded ion-radiator
7
potential, e. is the shielded ion field,
1
Zex. x. -x./
i = j- (I + D (1.4.5)
j x3 D
and x and x. are the positions of the radiator electron and the j'th
r i
ion with respect to the radiator nucleus, which we choose as the origin
of our coordinate system. We will neglect terms of higher order than
dipole in the radiator-ion interaction. This approximation is much
better for the ions than for the electrons in the case of high Z per-
turbers since the ion density is 1/Zion times the electron density and
since the large ion-radiator repulsion tends to keep the ions and
radiator apart. However, for deuterium-tritium plasmas with a small
amount of high-Z impurity, terms of higher order than dipole in Vri may
have a significant effect on the line shape.
We also use the static ion approximation. We assume that the ions
do not move significantly during the lifetime of the initial radiator
state, so we make the approximation
aH'
i -= [HI,H] = 0. (1.4.6)
Then the time evolution and density operators can be factored,
iHt/fi iH't/f5 iH!t/t
e = e e (1.4.7)
and
p = PiPre (1.4.8)
where
P -H' -H/ e
Pi = e /Tri e 1 (1.4.9)
8
and
Pre = e H'/Tre eH' (1.4.10)
re re
The line shape then is given by
I id taT -iH't/ft iH't/t
I(w) = Re Tr Pi f dt e d.Tre pe e d e
(1 .4.11)
Next we multiply this equation by 8(E-E ) and integrate over e to obtain
I(w) = f Q(s)J(w,s)de
where
Q(E) = Tripi6(E-i) (1.4.12)
is the probability of finding the ion microfield E at the radiator,
J(,e) = 1Re Tr f dt et .p e-iH'(+)t/ eiH'(E )t/i
J(,) Re rre o re
and
H'(c) = H + H + V + e* .
r e er r
Since we consider an isotropic plasma, we can define the probability of
finding the ion microfield of magnitude E at the radiator by
P(s) = 4ns2Q(G) Then
I(w) = fo P(e)J(w,e)de (1.4.13)
and
J(w,t) = Re Tre d eit p e-iH'(E)t/ i eH'(E)tfi
S re 0 .4.14re
(1 .4.14)
where
H'(E) = H + H + V + eez .
r e er r
= H(r) + H +V .
e er
For convenience, we have chosen the z-axis to be in the direction of the
field E.
I.5 Kinetic Theory
Here we follow the work of Hussey (1974) and Hussey et al. (1975)
to obtain an expression for J(w,e) in terms of a width and shift
operator, M(w). The advantage of this form is that results obtained by
approximate treatments of M(() are much better than those obtained when
similar approximations are applied directly to J(w,e).
We define the Liouville operator L in terms of a commutator of the
Hamiltonian H'(s),
Lf = [H',f]/t (1.5.1)
and write J(w,e) as
J(w,e) = Re Tr J ew d.p e-iLt (1.5.2)
1 re o re
We then define the radiator and s-perturber functions
s P N! -iLt
n F(r,...,s;t) = Tr p e d (1.5.3)
(N-s)l s+1 ...N re
10
where Trs+1...N is a trace over perturbing electrons s+1 through N and n
is the electron density. We can then write J(t,) in terms of the s=0
function F(r;t) or its Laplace transform F(r;w) as
J(w,E) = i Re Tr f dt ei"t d.F(r;t)
i r o
= Re Tr a.F(r;w) (1.5.4)
i r
If we take a partial time derivative of Eq. (1.5.3) we obtain
-t + iL(r,l,...,s;t)]F(r,l ... ,s;t) =
in Trs+1[L (r,s+l) + L2(i,s+l)]F(r,l,...,s+1;t) ,
i=l
(1.5.5)
where
L(r,l,...,s;t)f = [H(r) + He(,...,s) + Ver(l,...s),f] ,
L1(r,l)f= [Vl(r,1),f]/n ,
L2(1,2)f = [V2(1,2),f]/i ,
V1(r,1) is the interaction between the radiator and perturber 1, and
V2(1,2) is the interaction between perturbers 1 and 2. Equation (1.5.5)
relates the s-perturber function F(r,l,...,s;t) to the s+1-perturber
function F(r,l,...,s+l;t), so we have an infinite hierarchy of equa-
tions, the BBGKY hierarchy (Bogoliubov 1962). We will formally close
11
the hierarchy to obtain an equation for F(r;w), and therefore J(w,E), in
terms of a width and shift operator, M(w), to be defined later.
We define the radiator and s-perturber reduced distribution
functions,
s N'
n f(r,l,...,s) = N-)! Tr .. ,
(N-s). s+1 ...N re
and the time evolution operator
U(r,l,...,s;t) = N! Tr1 e f(r,1,...,s)
n(N-s)! +1..N re
(1 .5.7)
such that
F(r,1, ...,s;t) = U(r,1, ...,s;t)F(r,1, ...,s;t=0)
= U(r,1,...,s;t)f(r,1,...,s)J.
(1.5.8)
The closure relation is obtained by eliminating I from the s=0 and
arbitrary-s members of the Laplace transform of Eq. (1.5.8),
4 1 1+
F(r,l, ...,s;w) U(r,1, ...,s;w)f(r,1 ....,s)f(r) U(r;w) F(r;
(1 .5.9)
where
N! -1 -1
U(r,, ...s;w) = s Trs+ ...N rei(w-L) f(r,, .. .,s).
n (N-s)! 1
(1 .5.10)
(1 .5.6)
This equation for s=1 is substituted for F(r,l;w) in the Laplace
transformed first equation of the hierarchy to obtain
[-iw + iL(r)]+(r;w) f(r)l =
-inTrILI(r,l)[U(r,1;c)f(r,l)f(r) -1U(r;w)-]F(r;t).
(1.5.11)
Then we separate the term in brackets on the right side of Eq. (1.5.11)
into frequency dependent and frequency independent parts by defining
U(r,...s;)f(r,...s)f()-U(r;) -1
K(r,l,...,s;w) + f(r,1,...,s)f(r)-1
(1.5.12)
and solve for F(r;w) to obtain
(r;w) = i[w L(r) B M(w)]-1 f(r)d ,
B = nTrlL (r,)f(r,)f(r)- ,
M(u) = nTrlL (r,l)K(r,1;) .
(1.5.13)
(1.5.14)
(1.5.15)
Similarly, we can use the second equation of the hierarchy (Eq. 1.5.5)
to obtain for K(r,1;w) (Hussey 1974; Hussey et al. 1975):
where
13
K(r,1;u) = [w-L(r,1) V(r,1;w)]-lf(r,1);((r,1)f(r)-1 (1.5.16)
where
V(r,l;w) = -nf(r,l)f(r)-1 Tr2L1(r,2)P12
+ nTr2[L1(r,2) + L2(1,2)]G(r,1,2;w)G(r,l;w)-1
(r,l) = L1(r,l) + nf(r,l)- Tr2[f(r,1,2)
- f(r,l)f(r) f(r,2)]L (r,2) .
(1.5.17)
The operator P12
use Eq. (1.5.16)
acts to the right to change the argument 1 to 2. We
in Eq. (1.5.15) to obtain
M(M) = nTr1lL(r,l)[w-L(r,l) V(r,1;))]f(r,l)s (r,1)f(r)-1
(1.5.18)
With Eqs. (1.5.4) and (1.5.13) we can now write J(w,E) as
J(w,E) = Im Tr d.-[w-L(r)-B-I(w)]-1 f(r)d
TI r
(1.5.19)
Here, the effects of the plasma electrons on the line shape are
contained in B and M(w). The operator B is a mean field or Hartree-
Fock type term which represents the shift of radiator energy levels due
to polarization of the plasma by the radiator. The real and imaginary
14
parts of the operator M(w) represent the dynamic shift and width,
respectively, due to finite time collisional effects. The operator
g (r,l) is the interaction of the radiator and perturber 1 statically
screened by the other perturbing electrons.
1.6 Second Order Theory
In this section we restrict the width and shift operator, M(w), to
second order in the radiator-perturbing electron interaction, Vl(r,l).
We also make use of other approximations which are common to many line
shape calculations: the neglect of interactions between perturbing
electrons (The correlation effects thus neglected will be reintroduced
in an approximate manner later in the calculation), the No-Quenching
Approximation (NQA), and the No Lower State Broadening Approximation
(NLBA).
The neglect of perturbing electron interactions permits us to write
the electron Hamiltonian as a sum of single-particle Hamiltonians,
H = H(i) (1.6.1)
i
Since the radiator is at the origin of our coordinate system, the long
range monopole part of the radiator-perturber interaction, -(Z-l)e2/xl,
is a function of only the perturber coordinate x1. Therefore we include
it in the one-electron Hamiltonian, H(1), by redefining H(1) and V1(r,1)
as
2
H(1) =_ (Z-1)e (1.6.2)
2m x
and
2 2
V1(r,l) = (1.6.3)
I- l I x
r1
15
We can now expand M(w) in this short ranged V (r,l) while retaining to
all orders the effect of the radiator monopole on the perturbing
electrons.
For the plasma conditions and Lyman lines which we will consider, a
large part of each line satisfies Au < w where Ah is the frequency
~ pe
separation from line center and wpe is the electron plasma frequency.
The electron broadening of this part of the line is primarily due to
weak electron collisions, so we can well approximate M(u) by retaining
only second order terms in an expansion in Vl(r,l) (Smith, Cooper, and
Vidal 1969). With these approximations we have
M(w) = nTr lL(r,l)[w L(r) L(1)]-1iL(r,1)f(1) (1.6.4)
where
f(r) = e-PH(r)/Trre-1H(r) (1.6.5)
and
f(l) = e-H /Tre-P (1.6.6)
n 1
In Appendix A we discuss the NQA and NLBA, and show that as a conse-
quence of these approximations, M(u) has the matrix form (Tighe 1977)
M(u)i, =
in Tr fdt eiAutv e-iH(1)t/t iH(1)t/f(
h i"
(1.6.7)
16
where Aw = w-(Ei.,-E )/Ti and the subscript i (f) represents initial
(final) states for the particular Lyman line to be calculated. We will
partially include the effects of quenching later in this work.
This form of M(w) with the dipole approximation for V1(r,l) was
used by O'Brien (1970) and Tighe (1977) in calculating Lyman line shapes
for charged radiators. Here, we will retain the full Coulomb
interaction, V,(r,l), in calculating Lyman series lines.
CHAPTER II
THE FULL COULOMB RADIATOR-PERTURBER INTERACTION
II.1 The Line Width and Shift Operator
The purpose of this chapter is to develop a computational form for
M(w) in which we make no approximations for the radiator-perturbing
electron interaction, Eq. (1.6.3). We will calculate matrix elements
of M(w) in the spherical representation, InAm>, where n is the principal
quantum number of the initial level of a Lyman transition. The perturb-
ing electron Haniltonian, H(1), is that of a particle in an attractive
Coulomb potential, so we will use the Coulomb wavefunctions to evaluate
the trace in M(u) (O'Brien 1970). The eigenfunctions of H(1) are (Alder
et al. 1956)
47t i"(-,k) k
= 3/2 ei Y*m(k)Y m(x)(kx) F (Tkx)
km (2x)3/2 m m
(2.1.1)
where
o(A,k) = argr(A + 1 + iT),
-(Z-1)
S k
F (n,kx) is the Coulomb wavef.action (Abramowitz and Stegun 1972):
-rIT
F (',p) = 2e 2 (l(2+2) p e-i I 1(+1+in,21+2,2ip)
(2.1.2)
18
and F1 is the confluent hypergeometric function. The energy
eigenvalues of H(1) are given by
H(1) J> d 2k2 k (2.1.3)
2m
With these functions we write the radiator matrix elements
of M(w) (Eq. 1.6.7) as
ineT iAt
M(u) nt 2 = h- eT f dt e
n 2n mnA2m2 h2 0
x I df dk d
3m 3
x <2n|3m31 e-iH(1)t/h l(r,1)eiH(1)t/he-PH(1) Ini2m2>, (2.1.4)
where ne is the perturbing electron density and T is the thermal
wavelength,
S232 )I/2
XT m- .---
The ket kni.m> is the product of perturber and radiator kets,
l>ln|nm>. From Eq. (2.1.1) we see that the k-angle dependence of the
perturber wavefunction can be written explicitly as
= y* (k) (2.1.5)
1=0 m=-L
We use this in the expression for M(>), obtaining four 1,m-sums over
perturber angular moment. The k,- and k2-angle integrals each are
integrals of two spherical harmonics. These can be done to obtain
Kronecker deltas which reduce the four 1,m-sums to two, which we label
S4,m4 and 6,m6. We then have
M() n dt eAt f dkl fl dk
nilml ,n)222 2 2 o o i 2
ih 2 22
if, 2 2 2 k2
i- (k1 k2)t 22m
x e e
Gn1ml,nA2m2(kl'k2)
where
3 22
GnAIm,n2m2(k ,k2) = neXTk 2 m Am Im
3m3 4m4 6m6
x
x
In Appendix B we reduce G(k,,k2) to the form
3 4
4neXe n-1
G na (kl'k2) 2 62
nA1m1,n.2m2 1 2 72 111 2 mI m2 i3 0
where
(2.1.6)
(2.1 .7)
2n-2
5=0 1 4'6=0
S(213+1)(24 +1)(26 +1) 1,1 A3 A52 (A4 15 A6)2
( 5 0 0 0 0 0 0
x [f dx F (T lklx)F (n 2k2x)AI 5(x)]2 ,
0 16 13 5 I 13S
(2.1 .8)
S5 6 0
A (x) = dxx R (A )( + x ) Rn (r)
Sn2.1.9)
x > (2.1.9)
20
Rnl(xr) is the hydrogenic radial wavefunction of the radiator, and
x< (x>) is the lesser (greater) of xr and x, the radial coordinates of
the radiator electron and perturbing electron. The 3 sum is over
initial radiator angular moment, the 15 sum is over multipoles of the
Coulomb interaction, and the 14 and 16 sums are over perturber angular
moment. The symbol
1 23)
m1 m2 m3
is the Wigner 3-j symbol (Edmonds 1957). In Appendix C we express
A 213(x) as a sum which is convenient for numerical evaluation.
The time integration in M(u) (Eq. 2.1.6) can be done by adding the
convergence factor iE (e>0) to Au, then taking the limit as e+0:
it 22
1 (k 2k )t
lirm f dt ei(Aw+ie)t e2m 1 2
lim f dt e1 6 e =
Ef0
e+O 0
i
= lim 2 2
e+0 A + -+ (k k ) + iE
2m 1 2
= P 6( IT w + -( (k k)), (2.1.10)
A + -m (k2 k2)
where P stands for the Cauchy principal part. This separates M(i) into
real and imaginary parts, 2
PH2k2
2m
S f dk k e 2m G(k,k2)
M R(A) 2 po dk o 0 2 + 2 2, (2.1.11)
fi A+2+-(k -k)
2m 1 2
and 2k2 2 2mA 1/2
a_ r_ 2m G(k1,(k1 + 2)
M (Au) = dk1 e 2 2mA 1/2- Ao>0;
I (k + )
1 t (2.1.12)
21
M (Am) = e-PBtIA MI (JAw), Am<0.
(2.1.13)
The function M (At) represents a dynamic shift of the spectral line
R
due to the interaction of the radiator with the perturbing electrons of
the plasma, M (Aw) represents the width of the line due to the
perturbing electrons. We use the fact that (O'Brien and Hooper 1974)
2k2
P o dk2 e 2n
2 2mAw 1/2
G(k ,(k1 + -t
S 2 2 = 0, A>0,
A + T- (kI k2
+ 1 2
to write MR(At) in the form
MR(AO) =
SG(k ) G i( 2 2ma 1/2
-2 fdklfdk2e e 2 2 2), >0.
6 Aw + (k k )
2m 1 2(2.1.14)
This integrand is not divergent so the principal part notation is not
necessary. Similarly, for A&<0,
MR (A) fdklodk2
2k2
[e 2m G(k ,k2)
x -----
i2k2
-e 2m a t GI) k 2m)I )1/2,k2
-e- 2m^ k2+) 1 2 k
2 T 2 2
At + (k k2)
+2m 1 2(
2.1.15)
22
Equations (2.1.12)-(2.1.15) for MR(Am) and M (Aa) are evaluated
numerically and the results are used in Eqs. (1.5.19) and (1.4.13) to
generate line shapes. We will present the results of these calculations
in Chapter III.
I .2 Angular Momentum Sums
The function G(k1,k2), which appears in Eqs. (2.1.12)-(2.1.15) for
M(w), contains infinite sums over the perturber angular moment, 14 and
16. Increasing I values in these sums correspond to increasing separ-
ation of the radiator and perturber, so for 14 and 26 greater than some
value, say V', we can use the dipole approximation for the radiator-
perturber interaction with negligible error. The part of the sum in
which the dipole approximation is used can then be summed exactly.
We separate G(kl,k2) into two parts,
G(kl,k2) = G(1)(klk2) + G(2)(k,k), (2.2.1)
where G(1) is to be calculated numerically from Eq. (2.1.8) with the
upper bound A' on the 24 and 26 sums, and G(2) is to be calculated from
a dipole approximation of Eq. (2.1.8) for the remainder of the 24 and
I6 sums. The dipole approximation for Eq. (2.1.8) is obtained by re-
stricting the 25 sum to the dipole term, A5 = 1, and replacing Eq.
(2.1.9) for A 1 3(x) by
d 1 3
A 1(x) = f dx R il(Xr)Rn (Xr)
I 2 o rr n2 Ir n r
3na
o3a 2 1/2 2 ( 1 +)1/2,
= 22[(n -1) 6, 3+I (n- ( 113 -I6 .
2Zx2 1 P3+1 1 1 3
(2.2.2)
23
We obtain this last expression from A 31(x) by replacing x /x with
213 5
x /x which is consistent with the assumption that the perturber is far
from the radiator. Since 15=1 in G(2), we can use the equation (Edmonds
1957)
(1 +1 1)= (-),-1 i(+1) l/2 (2.2.3)
0 0 0 ~(2+3)(2Y+1)
to evaluate the 3-j symbols in G(2); then we sum over 13 to obtain
G(2) (k k
nlImI 'n-2m2 'k2
3 4
4n X e 3na
2 T o)2 61 26mm2(n 2_ 1- -)f ,(k ,k2) (2.2.4)
3n2 2Z 11l 1 \^ m<2 1 12
where
fl,(k1,k2) = ({[fodxFI(11,klx)F(n2,k2x) -x ]2
2Z='+1 x
+ [f dxF (n1,klx)F-12,k2x) 1- ]2} (2.2.5)
x
The I-sum in this equation can be done analytically (Biedenharn 1956)
with the result
2 2
2k 1 o0 2 1 2 2' 2
I1T iT 2
-22'kk2 1 +-4 ( + 2 [o dx x)l( F,( lklx)F ,(T2.k2x)]
x [fo dx x1 F- 1(1 ,klx)F 1( 2,k2x)] (2.2.6)
24
If we set I' equal to one in G(2), we obtain the dipole form of
M(u) used by O'Brien (1970) and Tighe (1977). O'Brien showed that
fl(kl,k2) is related to the free-free Gaunt factor, g(kl,k2), by
fl(kl'k2) klk2g(k21,k2 .
Karzas and Latter (1961) and O'Brien (1970,1971) have calculated the
free-free Gaunt factor numerically. They reduce the integrals in
g(kl,k2) to sums which are convenient for computer evaluation. The work
of O'Brien, which was for singly ionized helium, was extended by Tighe
to radiators of arbitrary charge. We extend their work to evaluate
f ,(kl,k2) for V'>1. We use this in Eq. (2.2.4) to obtain G(2); then we
calculate M(w) from G = G(1) + G(2) according to Eqs. (2.1.12)-(2.1.15).
We calculate M(u) for increasing values of I' to determine when the
dipole approximation in G(2) gives negligible error, i.e. when M(w)
becomes independent of A'.
II.3 Electron Correlations
In calculating Eqs. (2.1.12) (2.1.15) for M (Au) and MR(AW), we
used the approximation that the perturbing electrons do not interact
with each other, thereby removing effects due to electron-electron
correlations. Here we reintroduce in an approximate manner the effect
of these correlations on the line shape.
These correlations have been shown to produce a screening of the
radiator-perturbing electron interaction (Hussey et al. 1977). This
screening is most significant for the part of the line shape
corresponding to times which are long compared to electron relaxation
25
times, that is, for frequencies inside the electron plasma frequency.
Smith (1968) and Hussey et al. (1977) show that the inclusion of corre-
lations has little effect on rM(Au) for |AI[ > )pe and that for
IA)w < pe the result for M (Aiw) including correlations is nearly equal
to the uncorrelated result evaluated at the plasma frequency, M'(upe)
Hence, to approximate the effects of correlations on M (Am), we set
MI(A') = Mi (pe) for Awl < pe
I I pe pe
To approximately include electron correlation effects in MR(Ad) we
replace the Coulomb interaction V1(r,l) in G(kl,k2) (Eq. 2.1.7) by the
-xl/7D
Debye-screened interaction V1(r,1)e ; then use the screened
G(k1,k2) in calculating Eqs. (2.1.14) and (2.1.15) for MR(6A) This
approximate treatment of correlations follows from a calculation by
Dufty and Boercker (1976), based on a ring approximation treatment of
electron-electron interactions (Brout and Caruthers 1963), in which they
obtain a fully screened form for M(w),
M(w) = nTrSl(r,1;Aw)f(rl)[Ad L(1) -S(rl)] x (rl)f(r)1,
(2.3.1)
where X(r,l;Au) is a dynamically screened radiator-perturbing electron
interaction. Then they proceed to show that for Ad << w and in the
pe
non-degenerate limit, the radiator-perturber interactions can be written
in the Debye-screened form
2 -Ix-x /D e2 -x/xD
V (r,l) = e e e (2.3.2)
s +x
Ix1-x 1
For the plasma conditions which we will consider, kD is significantly
larger than the radiator size so we approximate Eq. (2.3.2) by
26
-xl/ AD
V (r,l) = V (r,1) e (2.3.3)
Then, using the approximations of Section 1.6 (except for the neglect of
electron correlations), we can write Eq. (2.3.1) as
M(w) = nTrl V(r,l)[Aw L(1)]- V (r,l)f(1) (2.3.4)
This is the same as Eq. (1.6.7) except for the screened interactions.
Following the steps specified in Section II.1 we can obtain from Eq.
(2.3.4) the Equations (2.1.14) and (2.1.15) for MR(A)), except that
G(kl,k2) will contain the screened interaction, Eq. (2.3.3).
II.4 Interaction Matrix Elements Between States of
Different Principal Quantum Number
So far we have assumed that matrix elements of the radiator-
perturber interactions between radiator states having different
principal quantum numbers are negligible and have set them equal to zero
(see Appendix A). With these approximations and the No Lower State
Broadening Approximation, the calculation of a Lyman line shape involves
matrices which have rows and columns labeled by the initial radiator
states of the line. By using these approximations we have neglected
quenching, or radiationless transitions between radiator states of
different principal quantum number caused by the perturbing electrons.
We have also neglected Stark shifts of higher then first order in the
ion microfield and the changed oscillator strengths due to mixing of
radiator states of different principal quantum number by the ion
microfield.
27
For a given line we expect that the most significant of the
neglected matrix elements are those between the initial states of a
given spectral line (principal quantum number = n) and the states of the
nearest adjacent energy level (principal quantum number = n+1). Here we
will include these matrix elements in the calculation of the n+1 Lyman
line shape. The matrices involved in this calculation will have rows
and columns labeled by radiator states of principal quantum number n and
n+1. We write Eq. (1.5.19) for J(w,e) in the matrix form
J(c,E) = Im I dA1 [W (H(r)/-te) M(l)] fi'fi', (2.4.1)
ii'
where H(r) is the Hamiltonian for the radiator in the presence of the
ion microfield e, hwl is the energy of the radiator ground state, and i,
i' represent radiator states having principal quantum numbers n or n+l.
We will neglect the line shift, so in Eq. (2.4.1) we have set B=0, and
we will set lR(w) = 0.
Following the derivation of Eqs. (2.1.6) and (2.1.8) from Eq.
(1.6.7) we can write M() ., as
ind1 t
i1
M() n n -2 i f" dte 3 dkl dk2
"1 1mln2'" 2m2 2 n3 3 0 2
2k2
Mi( 2 2 'r k1
(k-k )t -
x e2 2 e 2m G I ln2 2m2n (k1,k2) (2.4.2)
where Au 1 = (-(Wn -W ) = m-n31 in3 is the unperturbed energy of the
states with principal quantum number n3. Here, the principal quantum
numbers nl, n2, and n3 can have the values n and n+l. The term G(k1,k2)
is given by
34
4neT e 2n
Gnl lml 'n22m2 ,n3(k I k2) 2 1 '6m1m2 m
n15-0 l4',6=0
x (213+1)(2 +1)(216+1) 11 3 5)2 14 5 6 2
(2Y5+1) 0 0 0 0 0 0
x f F (n1,klx)F6 (2,k2x)A nn3 3 5(x)dx
4 1 4 6 2 2 nn3135
x f ( ,klx)F 6(2,k2x)An2n3 1 3 (x)dx (2.4.3)
The term A(x) is given by
5 6
A (x) = dx R (xr I )R n (x) (2.4.4)
nIn3 1 3 5 0 rr n 11 5+1 x n 33 r
x>
We use Eq. (2.1.10) to do the time integration in Eq. (2.4.2) for M(w),
obtaining for the imaginary part of M(w):
P2k2
M( ) ,n22m2 = dk f dk e 2m
n I n2m n 0 1 2 0 0 2
n3%3
x G n (k1,k2)6(A (k2 k2 )) (2.4.5)
n1 1 1,n 2m2 ,n313 1 2 n31 2m 2 1
We rewrite A 1 as An = Au n where Au is the frequency
n31 n31 "3n
separation from the unperturbed line center, Aw = w-u1 Then we do one
of the k-integrations to obtain
M (A)nl1 11mn22m2
p22 G k ,(k 2mA) 1/2J
mn-1 e2m nI lm1,n2 l2m2k,nl33 (
,3 dk1 e 2m( 1/2
3=1 (k + -
2
n -- (k + -(w'-Aw))
+ I fodk2e
13=0
((k2 2 (-))1/2,k
x 2 2
(k2+ -; (W'-2.))
Ml(AW)nIm I ,2m2
n1 1'n2 2 22 2
Bn2k2
Sn-1 -hi1A |1 ,m 2m 1/2,k2)
[ n- fdk2e 2 G ((k2 I)12,k2)
3 3=0 n1 Im1,n2 22'3 3
+ (second term of Eq. (2.4.6))],
(2.4.7)
where u' = n+l ,n Here we have broken the n3-sum into two parts: the
first term in the brackets of Eqs. (2.4.6) and (2.4.7) is for n3=n; the
second is for n3=n+l. We evaluate Eqs. (2.4.6) and (2.4.7) numerically;
then use these results in Eqs. (2.4.1) and (1.4.13) to obtain line
shapes. Results of this calculation are presented in Section III.2.
Aw>O; (2.4.6)
CHAPTER III
RESULTS
III.1 Symmetric Line Profiles
Here we compare the results generated using this full Coulomb
version of the relaxation theory with those results from the relaxation
theory in which the dipole approximation is made for the radiator-
perturbing electron interaction (Tighe 1977; Tighe and Hooper 1976,
1978). We will also examine the relative importance of various multi-
poles of the Coulomb interaction in M (Am) by restricting the 15-sum in
Eq. (2.1.8) to the multipoles of interest. In this section we will
neglect the line shift, which is given by the terms B and MR(Ai), and
consider only the broadening effects of the electrons, given by M1(AW)
(Eqs. 2.1.12 and 2.1.13). We will examine line shifts and asymmetries
in section 111.2.
Comparison of Coulomb and Dipole Results
In Figures 1 and 2 we compare Lyman a and P line profiles calcu-
lated from this full Coulomb formalism with profiles in which the dipole
approximation was used. The lines appearing in these figures were
calculated for a plasma of Ar+17 ions and electrons at a density of 1024
electrons per cubic centimeter and a temperature of 800 eV, conditions
which approximate the plasmas of some recent laser implosion experiments
(Yaakobi et al. 1980). For the Lyman a lines presented here, the
neglected fine structure splitting of about 0.35 Ryd. would have a
Figure 1 Comparison of an Argon Lyman a line profile calculated from
the full Coulomb interaction with the corresponding profile
calculated from the dipole approximation for that
interaction.
Ly-a,Ar7
T = 800.0 eV
ne= .Ox1024 cc
- Full Coulomb Calculotion
--- Dipole Approximation
1.0
0.2
3
c-0.6 -
o
-j
1.4 -
-2.2-
0
0.4 0.8 1.2
A w (RYD)
B4
o -
0 1O
- 4-4
0
) 00
44
Ct 00
0 E C
QU (U
-4 4 4J
00
r-4 rI
.0
0 ri
0 a
-4 44
Clo
.14
-4 4
o c
o0 0
o c4
CM
T-4
C
.0 0
Ec
--.2 E
3 0
00 )
o 7<
<0 O
0 0 tn.
0- 0-
0 x
-j C:
(m) I
35
noticeable effect on the lines (Lee 1979), but this should be negligible
in the case of the P lines. Figure 1 shows a slightly broader central
peak for the a line calculated from the Coulomb interaction than for
the a line calculated from the dipole approximation. In Figure 2
the P line from the Coulomb calculation has a slightly less pronounced
central depression than does the corresponding dipole line. The Coulomb
and dipole calculations show little difference, but this agreement is
partly fortuitous. Although the dipole approximation of the Coulomb
interaction neglects broadening due to the other multipoles of the
interaction, it overestimates the dipole ( 5=1) part of the Coulomb
interaction for perturbers which are less than a few times the average
radiator diameter from the radiator. This nearly compensates for the
multipoles neglected in that approximation. Figures 3 and 4 show that
Lyman a and 3 lines calculated from the dipole approximation are
significantly broader than those calculated from the dipole part of the
Coulomb interaction.
Although we have removed the long range monopole from Vl(r,l) (Eq.
1.6.2), the multiple expansion of V1(r,l) still contains a monopole
contribution from the part of the perturber wavefunction which
corresponds to penetration of the radiator, as can be seen from Eq.
(C2). If we compare a and P lines calculated from the full Coulomb
interaction with lines calculated using the monopole plus dipole part of
the Coulomb interaction, the lines from the full Coulomb calculation are
only slightly broader, indicating that for these conditions the most
significant broadening of these lines is due to the monopole and dipole
parts of the interaction.
Figure 3 Comparison of an Argon Lyman a line profile calculated from
the dipole part of the full Coulomb interaction with the
corresponding profile calculated from the dipole
approximation for the full Coulomb interaction.
Ly-a, Ar+17
T= 800.0 eV
ne= 1.0 x 102/cc
- Dipole Approximation
--- Dipole Part of
Coulomb Interaction
A w (RYD)
0.2
-0.6
3
0
_J
-1.4
-2.2
QO a
0
,0 0
p8 41
7:) 0
M0 3
L) 0 0
au z
0 )) c )V 4
ho o
'4- 0 0 $1
w0 o4
0i- c 2
C. '0 0
020
CO U
-H r)
U 4
0 .0 a a
be 44 u,-.
4o
0 00
S4 0
CO c
3 4 4 C) o
-3
Oa
$l
-o
o _
o a
EE
o 0
L.C
000
E/
C-) /
0 0
L) 0>-
Scl -
L. > NI
Q x I /
0)< // :
9/
I- a, /
0 0
0/
-II
N d 0
(m)I Eo1
Comparison with Impact Theory
In Figures 5 and 6 we compare line shapes calculated from this full
Coulomb relaxation theory to line shapes calculated from a full Coulomb
impact theory (Griem et al. 1979; Kepple 1980). This impact calculation
approximately accounts for the screening of electron fields and the
finite duration of perturbing electron collisions through an appropriate
choice of cutoffs of the electron impact parameter. Griem et al. also
include, by an approximate method based on partial wave scattering cross
sections, the quantum mechanical effects of strong electron-radiator
collisions. Figures 5 and 6 show that our a and p lines are slightly
broader than those of Griem et al. In Figure 7 we compare our result
for a P line at a density of 8.2 x 1023 electrons/cm3 and a temperature
of 800 eV to the results of Griem et al. at 1024 electrons/cm3 and 800
eV. We obtain close agreement on the line wings, the part of the line
which we believe to be the most reliable for experimental density
determination. Hence, in this range of plasma conditions our
calculation would indicate a plasma density about 18% lower than would
the calculation of Griem et al.
III.2 Shifts and Asymmetries
So far we have neglected sources of asymmetry in the line
profiles. In this section we will examine two significant sources of
line asymmetry: the line shift due to perturbing electrons and radiator-
perturber interaction matrix elements between states of different
principal quantum number.
Figure 5 Comparison of an Argon Lyman a line profile calculated from
this full Coulomb formalism with an a line profile calculated
from the full Coulomb impact formalism of Griem, Blaha, and
Kepple.
Ly-a ,Ar17
T=800.OeV
ne = 1. x 024/cc
Full Coulomb Calculation
--- Kepple
-0.6-
3
00
o
_1
-1.4
-2.2
0 0.4
Aw (RYD)
0 4
p C z
4Ur
3 0-
U r- *
U 0 i4
cm
A m
4-
CO ri
ElU
444 0
O 'H
0 00
a aM
E.
0
U- a
1-0 4- 0
o ao
r-
0 4Ll En
S4- 1 -
*r4
F:.
c
o
0
U
0
0 "
-0.
Soo
0 --
C 0-
II
S0C
0 ^ |
_- s
c'J
(m)j 6oi
0
O
0
3
*1
a o .0 m
0 0
a) 0
1 '-4r U
C)M n
-o 4 4 cli
-t &4 0 40
0 C-
.--( t 1
Z 003
-4 1- b'-4
C C444 (L 3
0) 0 4)
w 00- 0o
40 0 0
m o
4440 'H C
m04 'H
40 (a m >
ca d C) c
'H 0 C) -M 0
w -4 4) CD
d a-4' a 00
4 Q)
o o o
co o
44 (4
r'
t)4
Prdm
d30 e1
tOU~rl
T -rm
[Ku
05o0 C.h'
Line Shifts
Our objective here is to examine the line asymmetry which is caused
by the static and dynamic shifts, B and MR(Aw), and to determine the
significance of the dynamic shift in comparison to the static shift. We
obtain MR(Aw) by the numerical evaluation of Eqs. (2.14) and (2.15).
Several calculations of static shifts due to mean field effects of the
plasma on the radiator have been done (Skupsky 1980; Davis and Blaha
1981; and references cited therein). Although some uncertainties remain
in these calculations, for our purposes we will use the results of
Skupsky (1980, 1982) for the static shift. These results were obtained
by the numerical solution of the Schrodinger equation for a hydrogenic
ion in a self consistent potential determined by the nonlinear Poisson
equation for electrons and neighboring ions.
The Lyman 0 lines in Figure 8 include the static shift (solid line)
and the static plus the dynamic shift (dashed line). The static shift
causes an asymmetry, the lower energy (red) wing having the greater
intensity. The dynamic shift causes a slight red shift of the line but
has no significant effect on its shape. This full Coulomb result for
the dynamic shift is significantly smaller than that obtained in the
dipole approximation (Woltz et al. 1982).
Interaction Matrix Elements
In Figure 9 we compare Lyman 0 lines calculated with (solid line)
and without (dashed line) radiator-perturbing ion and radiator-
perturbing electron interaction matrix elements between states of
principal quantum number 3 and 4 (see Section 11.4). The inclusion of
these matrix elements causes the line to be asymmetric, with the blue
*o
1-4
u W
4 14 0
0
4 m-
4-4 -4 C
C4 (- 0
mm a
0 0C
c a
imc
00
(1m
0r- 0 0)
34 0 )
Ha
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C..
I f
I
I
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-Atd
Ia
/
C
Afl
-J
O~(1 05( OhI O~ cr, 0L Oct
(~I9~NO]
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F1C 80
'U 4-4 4.1 0
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SIIC 0 44
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01. T 40 C -
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CO 0 $ O U
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4-1 w- 4J
M O
0 p *-
1C I 40
H0 (I
c o r
0 a)
c 0I
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j) &Cl
0 E 4
c"-4 44 0 >
H rt .0
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0H 0 0 0
.00 COa
to 14
( 4- 4H 0 34
4J $ 4 0
b0 HJ 0 0.
H*
>4
+ cu0
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.--I
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0 00
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3
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54
peak more intense than the red peak and the red wing more intense than
the blue wing. To distinguish between ion and electron effects, we set
the ion perturbation matrix elements equal to zero while retaining the
electron perturbation matrix elements, obtaining the solid line in
Figure 10. This line is symmetric, and it is slightly broader due to
quenching than the dashed line, in which quenching has been neglected.
The asymmetry of the p line in Figure 9 is primarily due to quadratic
Stark shifting of radiator energy levels by the ion microfield. To show
this, we have included the quadratic Stark effect in an otherwise
symmetric Lyman P line calculation by adding the quadratic shift (Bethe
and Salpeter 1977),
E= 2()4 [17n2 3q2 9m2 + 19]
16 o Z
r
to the diagonal elements of the matrix
<3qm IA MI(AUw)3q'm'>
(see Eq. 1.5.19). Here the kets |nqm> represent states of the radiator
calculated in parabolic coordinates, and the quantum number q is equal
to the difference of the quantum numbers n1 and n2 used by Bethe and
Salpeter. This calculation gives a Lyman p line which has an asymmetry
nearly identical to that of the P line in Figure 9, with the only
noticeable difference being that this calculation gives a slightly
larger peak asymmetry. The calculation of Figure 9 does not give the
exact quadratic Stark effect since it does not include radiator-ion
perturbation matrix elements between radiator states of all principal
55
quantum numbers, but the near agreement of these two calculations shows
that the matrix elements between states of n=3 and 4 are responsible for
a major part of the quadratic Stark shift. The asymmetry of Figure 9 is
similar to the asymmetry obtained by Griem (1954) from a Hydrogen Balmer
B line calculation which included the quadratic Stark effect.
CHAPTER IV
CONCLUSION
In this work we have extended the line shape formalism of Tighe
(1977) and Tighe and Hooper (1976, 1978) by eliminating the dipole
approximation which they used for the radiator-perturbing electron
interaction, retaining instead the full Coulomb radiator-perturbing
electron interaction. We calculate the line width and shift operator,
M(w), to second order in this interaction. A second order calculation
of M(w) will give good results for the part of the line profiles in
which we are interested, i.e. separations from line center less than or
on the order of the electron plasma frequency, since the electron
broadening of this part of the line is primarily due to weak electron-
radiator collisions. Our full Coulomb calculation gives Lyman a and (
lines which are only slightly broader than those calculated from the
dipole approximation, but the near agreement of the two calculations is
partly fortuitous In the dipole approximation, broadening due to the
other multipoles of the interaction is neglected, but this is partially
compensated by the fact that the dipole approximation overestimates the
dipole part of the Coulomb interaction for perturbers which are less
than a few times the average radiator diameter from the radiator.
The Lyman a and p line profiles which we have obtained from this
full Coulomb calculation are somewhat broader than corresponding line
profiles calculated by Griem et al. (1979) from a classical path impact
theory with quantum mechanical corrections for strong collisions between
56
57
the radiator and perturbing electrons. For plasma conditions on the
order of 1024 electrons/cm3 and 800 eV, density diagnostics based on our
Lyman p line would indicate a plasma density about 18% lower than would
those based on the P line of Griem et al.
We have calculated the dynamic line shift due to perturbing
electrons, obtaining a small red shift of about 0.03 Ryd. for a Lyman p
line calculated for the plasma conditions of 1024 electrons/cm3 and 800
eV. This shift is significantly smaller than that obtained in a similar
calculation in which the dipole approximation for the radiator-
perturbing electron interaction was used. The effect of the dynamic
shift calculated here is considerably smaller than that of the static
shift (Skupsky 1980, 1982), which causes an asymmetry in the p line,
with the red wing having the greater intensity. This asymmetry results
from different static shifts of the various angular momentum states
which contribute to the line.
We have included in the calculation of a Lyman p line the ion-
radiator and electron-radiator perturbation matrix elements between the
initial states of the radiator, principal quantum number = 3, and the
nearest adjacent energy level, principal quantum number = 4. The
inclusion of these matrix elements gives an asymmetric line, with the
blue peak more intense than the red peak and the red wing more intense
than the blue wing. The asymmetry is due to the matrix elements of the
ion-radiator perturbation, which cause nonlinear Stark shifts of the
radiator energy levels. The electron-radiator perturbation causes a
slight additional broadening of the line due to quenching.
Sholin (1969) has shown that when the quadratic Stark effect causes
a significant line asymmetry, the quadrupole interaction between the
58
radiator and the ions will also have a significant effect on the line
profile. Calculations of Balmer and Paschen p lines which include the
quadrupole interaction (Pittman and Kelleher 1981) give asymmetric line
profiles with the blue peak being more intense than the red peak. The
quadrupole interaction should cause a similar asymmetry in the Lyman P
line, although the magnitude of the asymmetry for the spectral lines and
plasma conditions of interest here remains to be determined. This and
the quadratic Stark effect may be significant causes of increased
intensity of the blue peak of Argon Lyman P, an asymmetry which has been
inferred from some recent laser implosion experiments (Yaakobi 1982).
APPENDIX A
MATRIX ELEMENTS OF M(w)
Here we will use the No-Quenching Approximation (NQA) and the No
Lower State Broadening Approximation (NLBA) to write M(w) (Eq. 1.6.4) in
the matrix form given by Eq. (1.6.7).
We see from Eq. (1.5.19) that since M(w) appears in the inverse
operator [w-L(r)-B-M(w)]- which can be considered as a power series
expansion, it acts on
[L(r)+B+M(w)]kf(r)a, k>O .
Here, the operator f(r) weights initial radiator states and the
operator d determines transition probabilities between initial and
final radiator states; so the only matrix elements of f(r)d which
contribute significantly to a given line are of the type [f(r)] if,
where i and f represent initial and final radiator states for that line
(Griem 1974) We can set all other matrix elements of f(r)d equal to
zero. Let us define the operator
(k) = [L(r)+B+M(w)] f(r)d. (Al)
All matrix elements of D(0) which are not of the form D(0) are set equal
if
to zero We will show that as a consequence of this matrix structure
of (0), the matrix elements of M(u)D) when restricted by the NQA and
60
the NLBA, can be written as (Smith 1966)
[M()D (0) = i M(u)ii, Di (A2)
where M(M)ii, is given by Eq. (1.6.7). Next we will show that (k) has
+(O) 6(k)
the same matrix structure as (0, so Eq. (A2) holds for all D and
the matrix form of M(W) (Eq. 1.6.7) can be used to calculate
[w-L(r)-B-M()]-1.
If we let [ and v represent arbitrary radiator states, the matrix
elements of M(w)D(, where M(w) is given by Eq. (1.6.4), can be written
as
+(0) in__ = int
[( 0)D in Trlf dt eiWt
vv f 2 (01
S-i(E +H(1))t/t i ( ) ei(Ef+H(1))t/
x Vif(1)Dif f
V ei(Ei+H(1))t/ f(1)(0) (V e V +H(1))t/-i
+ 1 ie-i(E i+H(l))t/f (0) i(EV+H(l))t/1
e p Vif(1)Dif e V
if
if eiVVV
+ifv' e f()Dif Vfve v] (A3)
where we have written the term [w-L(r)-L(1)] in M(o) as a time trans-
form and we have replaced the Liouville operators by the corresponding
commutators. Here, V represents matrix elements of V1(r,l).
Now we use the NQA and the NLBA to simplify this expression. In
the NQA, we assume that the perturbing electrons do not cause the
61
radiator to make nonradiative transitions between states of different
principal quantum number; that is, the matrix elements V are nonzero
only if the states p and V' have the same principal quantum number.
This restricts the p's to be initial states and the v's to be final
states in Eq. (A3). We will examine the significance of quenching in
Section 2.4. In the NLBA, we assume that there is no broadening of the
final radiator state by the perturbing electrons, so we set the matrix
elements Vff, equal to zero. This is a good approximation for Lyman
lines since the final (ground) state and the plasma have no dipole
interaction, which is the most significant multiple in broadening.
With these approximations, the second, third, and fourth terms in brack-
ets in Eq. (A3) are equal to zero, and the first term can be written as
[M(m)(0) ]if
in ivwt -iH( 1)t/t (0) iH(1)t/(
-2 Trl dtI e V e Vi fo)D e (A4)
fi i'i
where
Au = w-(Ei-E f)/fi
If the final states are degenerate, Eq. (A4) can be written as the
matrix product
S= M(w)ii f(O)
[M(.)< () if M()iiD i'
1
where
62
M(W) in =Tl dt eiAMtV e-iH(1)t/t eiH(1)t/f).
ii' 2 o dtii" i"i'
(A5)
We must have degenerate final states (or, as in the case of Lyman lines,
one final state) to go from Eq. (A4) to Eq. (A5) since the Ar in
M(u)ii, contains Ef.
We see from Eq. (A4) that the only nonzero matrix elements of
M())D() are [M(L)6(0)]if. This is a consequence of the NQA, the NLBA,
(O) +(0)
and the fact that the only nonzero matrix elements of D( are Df.
Similarly, it can be shown that the only nonzero matrix elements of
[L(r) + B]D(0) are {[L(r) + B](0)}if, so D() only has nonzero ele-
ments D By induction, the only nonzero elements of D for all
if
( k) (k) H0)
k > 0 are if ; so Eq. (A4) holds for b) as well as B Hence, we
can use Eq. (A5) to calculate M(w) in [w-L(r)-B-M( w)]-1
can use Eq. (AS) to calculate M(u) in [w-L(r)-B-M(w)]
APPENDIX B
EVALUATION OF G(k1,k2)
In this appendix we reduce the function Gn mn2m2(ki,k2)
(Eq. 2.1.7) to a form which is tractable for numerical calculation. We
have
G (k k) n322
n m,n12m(kl k2) = n e 2ik2 2
3m2213 4m4 6m6
x
(Bl)
The expression for Vl(r,l), Eq. (1.6.3), can be expanded in spherical
harmonics (Jackson 1975) to obtain
2 2
e e
V (r,l) = e e
1 ++
=Ane2 (22 1) <1 5 x (x)Y (xr) (B2)
1=0 m=- (2+I) ( 5+1 m m
x>
where x and x are the positions of the perturbing electron and the
radiator electron, and x< (x>) refers to the lesser (greater) of x and
xr. With this expansion and the expression for the perturber
wavefunctions, Eq. (2.1.1), the first matrix element of Vl(r,l) in Eq.
(Bl) can be written as
=
64
2 6-4 i(a(C6,k2) o(4,k)) 1
8e i e C (225+1)
5m5 5+1)
x(f dx Y~m (x)Y 5m5(x)Y m6(x))(fdxY11m (xY (r X) 3m3 r))
4 5'5 16m6 rIi rY5M5rZ3'3 r)
x (fi dx x2(klx)-(k2x)- F (l,klx)F%6(2,k2x)A 13 (x))
4 6 3 55,
1 35 1 dxxRr (x)) 25 3(X
x>
and R n(xr) is the hydrogenic wavefunction of the radiator. In
C we obtain a form of Eq. (B4) convenient for computation. The
of three spherical harmonics is given by (Edmonds 1957)
(B3)
(B4)
Appendix
integral
f dx Y 1ml(x)Y 2m2(x)Y 3m3(x)
22 33 A
(21 +1)(22+1)(213+1) 1/2
471
1 2 3)(1 2 3
Sm2 m 3 0 0 0
1 m2 3)
mI 12 m3
is the Wigner 3-j symbol. We use this equation and the fact that
Y*(x) = (-I)m Y W(x)
to obtain
2e2 16 i(o(6 ,k2)-o(q4,k1))
i6
where
65
ml-+n+m1
x I (-) 4) (21+1)(223+1)(22 +1)(2 +1)1/2
x 1 23 5 4 5 6 1 3 5 4 -5 6
-m m3 m -m4 m4 6 0 0 0 0 0 0
xx d F 4(C1 ,k1x)F 6(2,k2x)A l 5(x). (B6)
Similarly, we obtain for the second matrix element of Vl(r,l) appearing
in Eq. (BI):
2e2 4-6 i(a(4 kI)-(6 ,k2))
=k i4 e
m+m+m
X X (-1) 3 6 [(213+1)(212+1)(226+1)(224+1)]112
7m7
x (3 42 6 7 3 2 7(6 7 4)
-m 3 m2 m7 -6-m7 m4 0 0 0 0 0 0
x o dx F 6(T ,kx)F (Tl,k x)A 2(x). (B7)
We use Eqs. (B6) and (B7) to write Eq. (BI) as
Gn1mI,nl2m2 (k1'k2)
3 4
4n e Te
4eT 1 [(21 +1)(212+1)]1/2(213+1)(24+1)(216+1)
S 3 4 6 5 17
x (fo dx F C4 (,klx)F 6 (2,k2x)A 1 3 5(x))
x (o dx F% (2,k2x)Fq (l ,klx)A32 7(x))
o 16 2'2 14 X3 2 17
66
x (1 23 5 4 15 26 2 13 7 4 6 7
m 1 (- rn 4 +5M
m3m4m6 m5m
S 2 23 5 4 16 6 3 X 2 6 7 4
-m m -m4-5 m6 3 m2 7 '6 7 m4
The m-sums in
properties of
this equation can be done with the aid of the following
the 3-j symbols (Edmonds 1957):
(1) 21' 2 and 13 must satisfy the triangle inequality,
IA1-21I C 13 < (21+22), and ml+m2+m3 must equal zero in order for
the 3-j symbol
S2 3)
mI m2 '3
to be nonzero.
(2) The completeness relation:
m 1 2 m3 1 2 5i3 3 -1 3
m2 m2 m3 mi m2 ,)= 3 636m3A( 2' 3)
where A( 1, 2,3) is one if 21, 2, 3 satisfy the triangle
inequality and zero if they do not.
(3) An odd permutation of the columns of a 3-j symbol is equivalent
to mult+pliation by (- 2+
to multiplication by (-1) 1
67
21 2 1+2+3 2 2 1
1 2 3) = (-1) 1 2 3 2 1 3
m m2 m3 m2 mI m3
(4) Changing the sign of all the m's is equivalent to multiplication
1 +A2+3
by (-1) 1 2
1 2 3 -1) 2 2 3( 2 3
mi m2 m3 -m- 2-m3
A special case of this rule is that if ml=m2=m3=0, 21+ 2+ 3 must
be even or the 3-j sumbol will be equal to zero.
First we do the m4 and m6 sums,
(-)m+m6( 4 5 6)( 6 7 4) = (-) 5 (4 6 5) 4 6 7)
4m6 -m4-5 m6 -6-7 m4 m 46 m4-6 5 4-m6 7
= (-1) (25 +1)6 5 76 A(4' 65) (B9)
Next we use the Kronecker delta in m5 and -m7 to do the sum over m7,
then the m3 and m5 sums are
S(-)m3+m5 1 3 5)(2 3 5)
m3m5 -mn m3 m5 m2-n3-m5
3553 5 33 5
S(-1) (-) 1 +3 +5 ( 1 3 )(2 3 5
m3+m5 ml-3-m5 m2-m3-m5
S(-1) 1 3 5(2 +1)-161 26mm2 A(1 3,5) (B10)
Using these results, we write Eq. (B8) as
4n 3 4
G nMnY (k kI ) eT 6 6
Gn1 m,n12m2( 2 2 2 12 6mm2
n-l 2n-2 (213+1)(2a4+1)(2i6+1) (1 3 -5 4 -5 6)2
x I I ___(_1)--____ 4 5 6)2
x3=0 15=0 1 ,46=0 (215+1) 0 0 0 0 0 0
x [fo dx (n1,klx)F% (2,k2x)A1 5(x)]2 (B11)
This is the desired result, Eq. (2.1.8). We use this equation (with Eq.
(C.8) for A(x)) to calculate G(kl,k2) numerically, then the results for
G(k1,k2) are used in Eqs. (2.1.12)-(2.1.15) to obtain M(u).
APPENDIX C
A COMPUTATIONAL FORM FOR Ai 2 3(x)
Here we reduce the term A 2 (x) to a form which is practical for
computer evaluation. We start with Eq. (2.1.9),
23 6
A 3(x) f dxx R2 (x )( R x), (Cl)
1 2 3 o rr nil r +1 x n2(r
x>
where x< (x>) is the lesser (greater) of x and xr, which are the radial
coordinates of the perturber and radiator, respectively. The function
Rn (xr) is the hydrogenic radial wavefunction for a radiator with
nuclear charge Z (Messiah 1961),
S(n--1)1 1/2 -xr/2 21+1
Rn (xr)= { t(n-- e xr rn--1 (xr), (C2)
r 2n[(n+)1 r n
na
where x and xr have been scaled in terms of and L is the Laguerre
polynomial,
21+1 n-I-1 k [(n+l)!]3xk
L (+ ) = 1 (-1)x (C3)
Ln-I-i k =) (n-1-1-k)!(22+l+k)!k! (3
If we substitute Eqs. (C2) and (C3) into Eq. (CI) we obtain
A 1 2 3(x)
23 6
n-Al-1 n-i -1 kl+k2+2 + 2+2 -x x 23,0
1i 2 k k f dxrx e r 3' (C4)
k =0 k=0 k2 o r 12 3+1 x
1 2 x
69
where
Cklk2
(-1) (C5)
kl+k2 [(n- 1-1) (n-12-1)1(n+1 )!(n+l2
) 2n(n-Al-kl-l)! (n-12-k2-l)!(2 1+1+k1 )!(212+i+k2)!klik2 .
We see from Eq. (2.1.8) that A 1 3(x) will be multiplied by a 3-j
symbol containing 11, 2, and ,3' so 11, 2, and 13 must satisfy the
triangle inequality, I1 2 1 C 3 1 ( 1+A2) Also, kI and k2 are
greater than or equal to zero. Therefore the exponent of xr in Eq. (C4)
is always positive, and we can use the definite integrals
x -xr -x m (6)
x dr xm e = m!(l e- J m>o (C6)
o r r =0
J=0
and
m -xr m
fJ dx x e r= m! ex 1 2 m>0 (C7)
x r r j= 1
to write Eq. (C4) as
n-l -1 n-2 -1
A (x) 1 12 c k k
1~ 3 kl=0 k2=0 1k2
( 1- 3,0) (k ~+k2 + 1+2 +3+2) x
x (kl+k2+1+ 2+3+2)!(1 e- j=0 J
+ x (kl+k2 +1 2- 3+l1)! ex iI
Sj=0
6 1ex 2 2
Y 3,0 (kl+k+2 + 12+2) j
x (kl+k2+1+2+2)! e-x 1 2 1 2 (C8)
j=0
This equation is used in the numerical calculation of A 1 2 3(x) .
1 23
APPENDIX D
NUMERICAL METHODS
To obtain numerical results for the line profiles, we calculate
M (Aw) and MR(Aw) numerically, then use these results in a line shape
program by Tighe (1977). This program calculates J(w,s) by inverting
the matrix
[Ai (H(r)/f l ) B M(w)]
(Eq. 1.5.19), then does the ion microfield integration (Eq. 1.4.13) to
obtain I(w).
We calculate M (Aw) from Eqs. (2.1.12) and (2.1.13), using Eq.
(2.1.8) for G(k1,k2) The Coulomb wavefunctions, which appear in Eq.
(2.1.8), are calculated by a continued fraction expansion and recursion
relations (Barnett et al. 1974). For small values of the argument kx
the Coulomb wavefunctions are calculated from a series expansion
(Abramowitz and Stegun 1972). We then use a trapezoidal rule
integration to do the x-integral in G(kl,k2) The 3-j symbols in
G(kl,k2) are calculated by a Fortran function given by Vidal et al.
(1970). We make the change of variables pf 2k /2m = y to put the k-
integral of Eq. (2.1.12) into a form which can be integrated by Gauss-
Laguerre quadrature.
We calculate MR(Aw) from Eqs. (2.1.14) and (2.1.15). The k-
integral is done by Gauss-Legendre quadrature for a mesh of k2 values,
then a Simpson's rule integration is used to do the k2-integral.
71
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BIOGRAPHICAL SKETCH
Lawrence A. Woltz was born on August 29, 1949, in New Brunswick,
New Jersey. He graduated from Southeast High School, Bradenton,
Florida, in June 1967. He attended Manatee Junior College, then the
University of South Florida, where he received the degree of Bachelor of
Arts in physics in June 1971. He was a graduate student at the
University of South Florida from September 1972 until August 1974. From
September 1974 until the present he has been a graduate student in the
Department of Physics at the University of Florida.
Lawrence A. Woltz is married Carol Jean (Wideman) Woltz.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degr of
Doctor of Philosophy.
arles F. Hoper' Jr., irma
Professor of Physics I
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
J66es WY Duftyt/
Professor of Phy scs
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Thomas L. Bailey
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Billy S. Thomas
Associate Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Kwan Y. Chen
Professor of Astronomy
This dissertation was submitted to the Graduate Faculty of the
Department of Physics in the College of Liberal Arts and Sciences and to
the Graduate Council, and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August, 1982
Dean for Graduate Studies
and Research