Higher order microfield effects on spectral line broadening in dense plasmas


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Higher order microfield effects on spectral line broadening in dense plasmas
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vii, 154 leaves : ill. ; 29 cm.
Kilcrease, David Parker, 1952-
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Thesis (Ph. D.)--University of Florida, 1991.
Includes bibliographical references (leaves 149-153).
Statement of Responsibility:
by David Parker Kilcrease.
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Full Text







All we know of the truth is that the absolute
truth, such as it is, is beyond our reach.

Nicholas of Cusa, 1401-1464 A.D.


I am extremely grateful to all of the people who have helped me in my

scientific career; there are too many to enumerate them all. I am also most

grateful to the taxpayers for making it possible for me to participate in the

great enterprise of scientific research.



ACKNOWLEDGEMENTS ........................................... iii

ABSTRACT ......................................................... vi


I. INTRODUCTION ................................................1

1.1 The Big Picture ................................................1

1.2 Focus of This Work ........................................... 3

1.3 Outline ................................................ ...... 5

LINE BROADENING PROBLEM............................... 8

2.1 Dense Plasma Fundamentals...................................9

2.2 Historical Background........................................ 17

2.3 Theoretical Formulation....................................... 18

III. HIGHER ORDER FIELD EFFECTS .......................... 39

3.1 The lon-Quadrupole Effect ...................................39

3.2 Atomic Data by Perturbation Theory Solution ...............54

3.3 Atomic Data by Numerical Solution ..........................70

3.4 Electron Delocalization and Field Ionization................. 78

IV. RESULTS AND DISCUSSION .................................89

4.1 The Lyman a Line...........................................91

4.2 The Lyman P Line...................... ........... ....... 92

4.3 Conclusions ........................... .................. 99


A. SOME EXPRESSIONS IN USEFUL UNITS ...................123


C. PARABOLIC COORDINATES ...............................130


E. FINE STRUCTURE CORRECTIONS .........................138

F. COMPUTER CODE DOCUMENTATION.....................144

REFERENCES ..................................................... 149

BIOGRAPHICAL SKETCH ...................................... 154

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



David Parker Kilcrease

May 1991
Chairman: Charles F. Hooper, Jr.
Major Department: Physics

Radiating atoms in dense plasmas are highly affected by their local plasma

environment. We examine the effect of this environment on hydrogenic radi-

ators by using an atomic wave function basis set that is itself a function of

the plasma electric microfield for the calculation of spectral line profiles. Our

theoretical development includes the effect of static ion perturbers up to the

quadrupole term in the multiple expansion of the radiator-perturbing ion in-

teraction, as well as dynamic electron broadening up to second order in the

radiator-perturbing electron interaction. Effects due to the presence of the uni-

form ion-electric field are treated exactly within this framework, and the field

gradient is then treated as a perturbation by inclusion of the ion-quadrupole

term. We employ a basis set for the representation of the upper state of the

transition that includes states of principal quantum number n as well as n+1

to allow for the field mixing of these two manifolds. The ion-quadrupole effect

is treated in a new way that includes ion correlations by using a renormalized


independent-particle model for the perturbing ions. We also present a pre-

liminary assessment of the importance of field ionization on the spectral line

shapes. Our results are compared with previously reported calculations using

field independent matrix elements, neglect of possible field ionization and the

independent-particle model of the ion-quadrupole effect. Hence, we are able to

identify the most important effects that arise from the use of the field depen-

dent basis set. These comparisons also allow us to examine the importance of

ion correlations when determining the ion-quadrupole effect.


In this dissertation we, that is you and I, will be examining the theory of

spectral line radiation from highly ionized atoms in dense plasmas. Before we

go into this in detail it will be helpful and illuminating to stand back and look

at the relevance of this from a wider perspective; to take a look at the big


1.1 The Big Picture

In general, a plasma can be defined as a statistical system containing mo-

bile charges.1 We will be concerned with a subcategory of this definition and

will limit ourselves to discussing electrically neutral plasmas associated with

hot highly ionized atoms. This is no great limitation since over 99 percent

of the known universe is covered in this category.2 With the aid of sophisti-

cated astronomical detection equipment, most of what we see of the universe

is electromagnetic radiation emanating from a variety of objects ranging from

ordinary stars to extremely luminescent quasistellar objects located at the edge

of the known universe. The matter in all of these entities exists almost entirely

in the plasma state; plasmas whose constituents range from ordinary ionized

atoms to exotic matter-antimatter electron pairs. The radiation reaching the

earth from these distant bodies runs the gamut from radio waves, through the

visible spectrum and into the realm of X-rays and 7-rays. It is easy to see that

an understanding of the behavior of these plasmas can and will give us great

insight into the structure and workings of the universe in which we live.


Of more mundane concerns are the plasmas encountered here on earth.

These are evident in such natural phenomena as lightning and fire. Man-

made plasmas are also numerous in example, ranging from ordinary fluorescent

lighting to terrifying large scale thermonuclear explosions that result from the

fusion of hydrogen isotopes. Since the early 1950s, one of the primary goals of

plasma physics has been to bring under control these tremendous quantities of

energy liberated by thermonuclear fusion. When this goal is achieved, we will

have succeeded in obtaining access to a nearly unlimited supply of relatively

clean and safe energy that will last into the foreseeable future.3

Due to the extremes of physical conditions under which many plasmas ex-

ist and due to their often ephemeral nature, making direct measurements of

the composition, density, temperature and other physical parameters that are

necessary for a clear understanding of their inner workings is often impossi-

ble. What is needed is an indirect nonintrusive probe of the plasma interior.

The radiation emitted or absorbed by these plasmas is the ideal answer to this

problem. Isolated, partially ionized atoms can emit or absorb radiation that

results from atomic electrons passing from one energy level to another. This

radiation is emitted or absorbed in discrete amounts leading to distinct and

identifiable sharp spectral lines. These lines have a slight width, called the

natural width, due to quantum fluctuations in the atom's own electromagnetic

field. As long as the atoms remain isolated, the spectral lines remain rela-

tively sharp. When linked to their plasma environment, however, the situation

changes radically. The amounts of energy emitted or absorbed by the atoms

are now affected by the plasma, often revealing pertinent information about

this environment. Thus we may explore the atom's surroundings indirectly

through changes in the profile of its spectral lines. This technique is employed

in many areas of science to reveal information about the internal structure

of matter in a noninvasive way by examining emitted or absorbed radiation.

Some examples of this application are nuclear magnetic resonance (NMR) and

M6ssbauer spectroscopy.4 In this dissertation we will be concerned with spec-

tral lines of emitted radiation from plasmas although the formalism applies

equally well to spectral lines originating from the absorption of radiation. The

line shapes will reveal information about the density and temperature in the

vicinity of the radiating atoms, regardless of whether they are located in a

simple laboratory experiment or in a distant star.

1.2 Focus of This Work

The main focus of this work will be on the shapes of line spectra from

dense, high temperature plasmas consisting of atoms of moderate atomic num-

ber, Z. These terms "dense" and "high" are quite relative and historically have

meant the densest and hottest plasmas obtainable in laboratory experiments

for a particular element at that particular point in time. As an example, in

the mid 1960's these terms referred to electron number densities, ne, of around

1017 cm-3 and temperatures corresponding to a few eV for elements such as

hydrogen and helium.5 Today, high density refers to ne = 1024 to 1025 cm-3

and temperatures corresponding to 600 to 1000 eV for elements such as argon

and krypton with Z = 18 and 36 respectively.6,7,8,9,10 One method of creat-

ing these dense plasma conditions in the laboratory is by laser driven inertial

confinement. Current experiments using this method have created conditions

of many times solid density for periods of a few hundred picoseconds.9 The

mechanism of compression employs high intensity laser light to symmetrically

irradiate spherical targets containing the material under investigation.11 When


this energy is rapidly deposited in the outer layers of the target (mostly by in-

verse Bremsstrahlung), these layers are quickly vaporized and blown outwards

away from the target. This propels the remainder of the target material in-

ward. This is in close analogy to the way a rocket works by combusting fuel in

an engine and expelling it in one direction in order to be propelled in the other

direction. If our laser irradiation is sufficiently symmetrical, the target will

be reduced in size many times as it is compressed, and resulting shock waves

will heat the core to a high temperature. This compression and heating can

produce the plasmas of high density and temperature that we wish to examine

in this dissertation.

Approximate descriptions of spectral line shapes emitted from dense, hot

plasmas are usually derived from theories that use an atomic physics descrip-

tion of atoms that are isolated from their environment. In reality, the atomic

wave functions for the radiating atoms can be strongly perturbed by surround-

ing charged particles, and in extreme cases it is not realistic to think of the

atoms as separate entities at all but rather as clusters of ions surrounded by

clouds of electrons. Radiation from atoms in this highly interacting environ-

ment will run the gamut from continuum to discrete. Therefore, in order to

construct a tractable method for the calculation of spectral line shapes, we

must often, somewhat arbitrarily, distinguish between the radiator and its per-

turbers. However, the more influence of the surrounding particles that we can

include in the description of our radiating atoms at the outset of our develop-

ment, the better off we will be when considering further perturbations. Our

goal in this work will be to include some of the higher order effects of the

plasma microfield on atomic spectral line emission. To this end, we will use

for our zero-order atomic wave functions, numerical solutions to Schr6dinger's

equation for an atom in a uniform electric field. These field dependent solutions

are supplemented by solutions obtained from a systematic perturbation theory

treatment for small field values. The zero order atomic Hamiltonian will thus

contain the perturbing ion-dipole interaction and higher order corrections will

describe nonuniformities in the plasma electric field. We will include the first

of these higher order corrections, the ion-quadrupole term, to account for the

effects of the field gradient.12'13,14,15 Griem16 has pointed out as early as 1954

the possible importance of this field dependence, as well as the ion quadrupole

effect, as a source of spectral line asymmetry.

The plasma electric microfield can produce additional effects on the radi-

ating atoms beyond those given by the simple field dependence of their atomic

wave functions and energy levels. Atomic bound state energies which were dis-

crete in the absence of the plasma electric field now become resonance states

with finite widths. The formerly bound electrons are now free to quantum

mechanically tunnel through the potential barrier created by the electric field.

Although atomic electrons that lie deep in the radiator potential well are little

affected by the plasma electric field, high lying electron levels are perturbed

to the point where they are, for all practical purposes, no longer bound to the

atom. This field ionization phenomenon could have important consequences

for the relative populations of the radiator excited states.

1.3 Outline

Chapter 2 focuses on the development of a general plasma line broaden-

ing theory and the accompanying approximations. We discuss fundamental

length and time scales for the interactions of the two main components of the

plasma with the radiator: the highly charged ions and the unbound plasma


electrons. We also discuss under what conditions and where in the line profile

the phenomenon of perturber motion will be important (both ion and elec-

tron). Next follows a discussion of the strength and nature of the various

interactions between the plasma particles. From this consideration, we decide

which interactions are important and which can be ignored.

A brief account of the historical development of the main ideas of spectral

line broadening theory is presented. We find the historical roots for the ideas

of Stark, Doppler and electron broadening in the late 1800's and early 1900's.

Next follows a detailed derivation of a calculable form of the line shape

function. We first present a general formalism appropriate for multi-electron

radiators that includes field dependent wave functions, an exact treatment of

the perturbing ion radiator-dipole interaction, field dependent NLTE level pop-

ulations and field dependent electron broadening. This general formalism is

then restricted to the case of hydrogenic radiators with no lower state broad-

ening. We also discuss a method for approximately including field induced

resonance widths in the line shape. Chapter 2 ends with a brief note about

other line broadening phenomenon that may need to be included before com-

parisons with experiment can be carried out.

Chapter 3 begins with the development of a theory for the approximate

evaluation of the ion-quadrupole effect. We present approximations for the

probability function for the field gradient term as well as an approximation for

the average field gradient constrained to have a given field value. These ap-

proximations include ion-ion correlations and represent an improvement over

previous theories. We then compare our calculation of this field gradient with

previous calculations and simple approximations. Next, we discuss the methods


of calculation of field dependent wave functions and matrix elements by pertur-

bation theory and by direct numerical solution of Schr6dinger's equation. The

numerical solution technique is also used to calculate resonance widths that

give an approximate measure of the lifetimes of the field dependent resonance

states produced by the presence of the plasma microfield. These lifetimes are

then used to study the effect of the resulting possible level depletion on the

line shape. We discuss a simple population kinetics model for this purpose.

In chapter 4, we present some theoretical spectral line shapes resulting

from our work. We calculate line shapes for the La and Lp lines of hydrogenic

argon at kT = 800 eV and for an electron density range of 1 x 1024 to 1 x

1025 cm-3. We then examine the importance of the various effects studied in

this dissertation on the spectral line shape, width and asymmetry. We close

with some conclusions and suggestions for further work.

Appendices discuss several useful topics and give some details of calcula-

tions presented in the main text. We have a discussion of physical units, some

details of the model used to calculate the average field gradient required for the

treatment of the ion-quadrupole effect, a discussion of parabolic coordinates

and the details of the perturbation theory calculation of the field dependent

fine structure corrections to the radiator Hamiltonian.


We will now develop a theory17,18 for the description of spectral line shapes

in dense plasmas. The physical conditions we will be interested in considering

will be pertinent to inertial confinement fusion plasmas. These plasmas will

be at, or near solid density and at relatively hot temperatures (i.e. hot enough

that the plasma is not quantum mechanically degenerate). We will also be

interested in systems where the plasma ions have much greater mass than

the electrons. This difference will allow us to consider different domains: one

defined by the fast moving or dynamic electrons and one appropriate for the

description of slow moving or quasi-static ions. This will allow us to proceed

with the development of a simplified formalism.

The physical systems and conditions that we will examine in this disser-

tation are restricted to argon radiators immersed in a pure argon plasma at

a temperature corresponding to 800 eV and an electron number density rang-

ing from 1024 cm-3 to 1025cm-3. These densities are high enough to bring

out the higher order field effects but not high enough for the plasma to be in

the degenerate electron regime. In local thermodynamic equilibrium (LTE),

the line shape is not very sensitive to small variations in temperature. Con-

sequently, we will not examine the temperature dependence of the line shape.

These physical conditions are also relevant to current experiments9 that employ

argon radiators.


2.1 Dense Plasma Fundamentals

In dense plasmas there are several fundamental length and time scales that

are important for understanding the interactions among the plasma particles.

Since our plasmas are dynamic systems, the particles are in constant motion

and travel, on the average, a characteristic distance in a given relevant time

interval. This fact allows us to establish some relations between length and

time scales. The use of these scales will allow us to classify the various plasma

particles into categories that are governed by interactions of differing strengths.

This will be important when deciding what approximations are suitable and

consistent in our description of the plasma. Without approximations we would

be hopelessly lost in a morass of ~ 1023 coupled equations of motion. A fate,

preferably, to be avoided.

We will now discuss some relevant length scales1,19,2 (see Appendix A).

The ion-sphere radius ro is an estimate of the average distance between two

ions of species j. It is given by

ro, = ( ) 1, (2.1)

where nj is the number density of species j.

Another important quantity is the Debye length AD. It is an indication

of the distance from an ion beyond which screening of its charge becomes

significant. It is given by

( kT 1/2 (2.2)
AD, = (4Z2e2n (2.2)

where k is Boltzmann's constant, T is the temperature and Ze is the charge of

species j that forms the screening cloud. For more than one component, the


resultant screening length is AX2 = Ej AX2 where the sum is over the different


The thermal de Broglie wavelength A gives an estimate of the quantum

mechanical wave nature of the plasma particles. It is given by

A 27rh2 1/2 (2.3)
A = mkT '

where m is the particle mass. Except for the factor of 27 this is just the

quantum mechanical wave length of plasma particles possessing average kinetic

energy kT. It gives us an estimate of the spatial extent of the particle wave


The radial dimension of an atomic ion can be estimated by considering the

radius of the Bohr orbit for the most probable quantum state, and writing the

radius as,

rn = aon2/Z ; (2.4)

n is the principal quantum number for the atomic ion, ao is the radius of the

first Bohr orbit for hydrogen and Z is the nuclear charge.

In order to relate time scales to characteristic lengths we need an estimate

of the velocity for particles of species j. Using the magnitude of the most

probable velocity for classical motion, we have

Vmp,j = (2kT/mj)1/2 (2.5)

We have now defined the basic quantities which will allow us to make

estimates of several important time scales of interest. Additionally, the length

scales will be used in determining appropriate scale factors for perturbation


We will now look at the time scales relevant to motion of the plasma parti-

cles, specifically the perturbing electrons and ions. It will be shown in section

2.3 that the line shape function I(Aw) can be represented by the transform of

a dipole autocorrelation function C(t) such that

I(Aw) = f exp(iAw t)C(t)dt (2.6)

where Aw is the separation from line center and C(t) is a decreasing function

of t. When t > 1/Aw, the exponential begins to oscillate rapidly causing the

contribution to the integral from the integrand in that region to be small. So,

for a given value of Aw, the integral is determined, for the most part, by C(t)

such that 0 < t < 1/Aw. Hence, we can define the time interval of interest

for the line shape at the distance from line center Aw as r = 1/Aw. If the

duration of a perturber's collision with the radiator is significantly greater than

the time of interest 7 corresponding to the part of the spectral line Aw, we may

regard the perturber as being stationary during the time of interest and treat

its perturbation as static. This is known as the quasi-static approximation.

If the duration of collision does not meet this criterion, we must regard its

perturbation as dynamic and explicitly take its time dependence into account.

For a particular separation from line center Aw, the effect of perturbing

ion motion will be negligibly small if17

F(t) << Ai(F)|, (2.7)

where F(t) is the perturbing ion electric field strength at the radiator, F(t) is its

time derivative and Awi, (F) represents the separation from line center, Aw,

due to the static linear Stark effect. Note that Awi,f(F) = wi,f(F) wi,f(0),

where i and f refer to the initial and final state of the transition, respectively,

and wi,f is the transition energy wi,f = wi-wf. The linear Stark shifted energy

for level i is wi(O) + 3niqihF/me. We use the parabolic representation (see

Appendix C) and take F = Zie/r2 where r is the distance from the radiator

to the perturbing ion, and obtain on substitution into Eq. (2.7)

Aw = 3>n-nq (2.8)
2Zmer2 r
where ni,f and qg,f are the parabolic quantum numbers for the initial and final

states and v is the perturbing ion velocity. Eliminating r from both sides of

the inequality gives

Aw > 7. (2.9)
3hZi(niqi nfqf)
We can use the most probable velocity as an estimate for v to obtain

Aw > Z Awi (2.10)
phZi(niqi nfqf)
where we have used the perturbing ion-radiator reduced mass p in the expres-

sion for v because we are only interested in the relative motion. Here, we have

defined Awi as the characteristic shift from line center due to ion motion. The

criterion for the quasistatic ion approximation to be valid is thus: Aw > Awi.

For the Lyman a line in pure argon plasma with kT = 800 eV, qi = 1 and

qf = 0 we have hAwi = 0.044 eV. For higher series members hAwi will be less.

We conclude that estimated shifts due to ion dynamics will be much smaller

than the estimated shifts due to the static-ion Stark effect. The latter are on

the order of tens to hundreds of electron volts for the physical conditions we

are interested in in this work. Consequently, we can confidently use the qua-

sistatic approximation for the perturbing ions. For low mass perturbing ions

the situation is quite different. In the case of proton perturbers hAwi = 16 eV

and ion motion will become a significant component of the line width.

The story for the perturbing electrons is, however, another matter. Due

to their much lighter mass, the electrons will be traveling much faster when

they have the same kinetic energy; their dynamic influence on the radiator

will need to be considered. If we treat the plasma ions as static, the plasma

electrons will form shielding charge clouds around them, screening the ionic

charge at a distance of roughly the Debye length.20 This is given by AD,e =

(kT/4re2ne)1/2. If we regard the time of interest for the interaction of a

perturbing electron with the screened radiator as roughly the time it takes to

cross the Debye sphere radius AD,e, we have

-1 _Ve ,mp,e
Te -= ,Ve De (2.11)
e D,e AD,e
where we have estimated Ve by its most probable value. This gives us

e-1 = V/2p,e (2.12)

where Wp,e is the electron plasma frequency given by

S 47rnee2) 1/2 (2.13)
p,e = me (2.13)

The portion of the line profile that corresponds to electron dynamics is

then given by

Aw < Wp,e ,



with a quasistatic electron approximation being applicable for Aw > Wp,e. For

our physical conditions wp,e is in the neighborhood of 10-150 eV. Examination

of the statics and dynamics of the perturbing ions and electrons allows us to

conclude that the quasistatic approximation for ions is good throughout the line

profile except at the very center for Aw less than 1 eV while electrons behave

dynamically inside wp,e. This will allow us to separate the problem of plasma

perturbations of the radiator into two regimes coupled only by the static plasma

microfield: that of purely static screened ions and purely dynamic electrons.

This will greatly simplify the line shape theory that we wish to formulate.

For our plasma particles to behave non-degenerately we require that the

interparticle spacing be much greater than the quantum mechanical wavelength

of the moving particles. For the plasma electrons this condition is

Ae < ro,e (2.15)

This requirement, except for a slight difference in the numerical constant, is

the same as EF/kT < 1, where the electron Fermi energy, EF, is given by

EF = (37r2ne)2/3 (2.16)
For our plasma at kTe = 800 eV and ne = 1 x 1025 cm-3 the above ratio is

0.21. However, for a density of ne = 1 x 1026 cm-3 the ratio becomes 0.98

and it is no longer a reasonable approximation to treat the plasma electrons

non-degenerately. We will take an electron density upper bound of ne = 1 x

1025 cm-3 to avoid the problem of electron degeneracy in this work. The

much heavier plasma ions, on the other hand, will behave classically under any

conditions we will be considering here.


To judge the importance of correlations between particles in our system we

can look at the Coulomb coupling constant P. This is the ratio of a particle's

potential energy of interaction with a neighboring particle to its kinetic energy.

If r > 1 then interactions with other particles will dominate ideal gas behavior.

If F < 1 then the kinetic energy will be most important in describing the

particle behavior.

First we look at the coupling between electrons. For the potential energy we

use the potential produced by an electron at the average interparticle spacing

ro,e. For the kinetic energy we use kT to give

Fe,e = (2.17)
For our highest density, ne = 1 x 1025 cm-3, this gives Fe,e = 0.063. We con-

clude that the plasma electrons are weakly coupled to each other and can be

treated, in a first approximation, as an ideal Boltzmann electron gas. Correla-

tions between electrons will be treated in an approximate way.21

For the ions we use ro,i and obtain

r,= T (2.18)
r,,ikT '
but since ro,i = Zl/3ro,e we have ri,i = Z5/3Fe,e. For our fully stripped argon

plasma, Z=18 and for our highest density we have Pi,i = 7.8. Clearly, then, we

must not ignore ion-ion correlations. In particular the calculation of the ion

microfield will require a careful treatment of these correlations.

For ion-electron correlations we take as the interparticle spacing ro =

(ro,e + ro,i)/2 to give

Fe,i = oT (2.19)


Noting that ro = (Z1/3 + l)ro,e/2, we can write

Fe,i = Fe,e (2.20)
(Z/^3 + 1)
For ne = 1 x 1025 cm-3 this gives Fe,i = 0.62. So ion-electron coupling is of

order one and, although not as strong as ion-ion correlations, will need to be

taken into account. This will be accomplished by a mean field approximation

using screened ion potentials.

In order to examine the effect of ion radiator interactions it will be use-

ful to make a multiple expansion of the radiator-perturbing ion interaction

potential. To do this we define an expansion parameter 6, where

6 = ra (2.21)
Here, ra = (n|lr I|n) ~ aon2/Z is the most probable orbital radius for a

radiator in the state with principal quantum number n and nuclear charge Z.


/4 1/3aon2n 1/3
6= (471/3 an2n3 (2.22)
3\3/ Z4/3
This is 6 = 8.531 x 10-9n2nel/3/Z4/3 with ne given in cm-3. At ne = 1 x

1025 cm-3 for n = 2 we have 6 = 0.16 and for n = 3 we have 6 = 0.35. The

dipole term of the multiple expansion is second order in 6 and the quadrupole

term is third order in 6. For n = 3 this gives 62 = 0.12 and 63 = 0.043. So our

expansion is quite legitimate.


2.2 Historical Background

We will now take a brief look at the historical development of spectral line

shape theory with the goal of identifying the origin of the main ideas of line


In 1889, Lord Rayleigh23 correctly explained the broadening of spectral

lines due to the Doppler effect. He used the not completely accepted idea at

that time, of a collection of randomly moving atomic radiators whose velocity

was governed by Maxwell's distribution. Interest was such that by 1895 A. A.

Michelson24 was able to summarize the possible mechanisms of line broadening

as he saw them. Among the proposed mechanisms were the following:

1) The change in the emitted radiator frequency due to neighboring molecules.

In those days the term "molecule" was used to refer to any atomic sized


2) Doppler broadening due to radiator motion;

3) Broadening of radiator emission lines due to radiation interrupting colli-

sions. Michelson developed a theory for this effect and showed that the

resulting line shape was that of a Lorentzian. These early theories consid-

ered the radiators to be classical oscillators. We will see that collisional

broadening is often proportional to the density, and hence, to the pressure,

assuming the temperature is held constant. This is where the terminology

pressure broadening originated;

4) The natural line width due to radiation damping which was viewed as the

leaking away of the oscillator's energy to the environment.

In 1901 C. Godfrey25 combined Doppler and collisional broadening in a

unified description. At that time these two effects were viewed as the most sig-

nificant causes of observed line broadening. With the advent of the quantum

theory the time was ripe for a reworking of the old classical line broadening

theory. In 1924, W. Lenz26 applied some elements of the new quantum theory

to collisional broadening but still considered the radiator as a classical oscil-

lator. Returning to the problem in 193327, Lenz incorporated line shifts and

asymmetries into his collisional line broadening theory by using finite collision

times. Previously all radiation interrupting collisions had been assumed to

occur instantaneously.

Michelson's point about the radiation from atoms being affected by atoms

in their vicinity was taken up by Holtsmark28,29 in 1919 and developed into

the theory of statistical Stark broadening. Holtsmark considered the effect

of a particular configuration of static perturbers surrounding the radiator on

the shift of the radiator's characteristic frequency. He then averaged over the

possible configurations to produce a broadened line profile. If the perturbers

are electric monopoles this theory is referred to as Stark broadening theory after

an idea originally suggested by J. Stark30 in 1914. Holtsmark also developed

similar theories for perturbing dipoles and quadrupoles.

This brings us to a point in the historical development of line broadening

theory where we see the emergence of the primary theoretical points of view

as to the physical causes of line broadening in plasmas. These are Doppler

broadening, collisional broadening and statistical Stark broadening. Many fur-

ther developments and refinements17,18 have occurred since these "early days"

of line broadening physics but we will not follow the historical development

further. We will jump to the point of present day developments.


2.3 Theoretical Formulation

In order to examine the line radiation from a plasma, we must look at the

power emitted by dipole radiation from an emitter immersed in the plasma.

At the outset we assume that the plasma consists of ionic emitters surrounded

by a plasma consisting of perturbing ions and electrons. This will allow us

to model the plasma line radiation by examining the radiation from one of

the emitters and averaging over all possible configurations of the surrounding

plasma. This is equivalent to looking at a large number of plasma radiators in

differing environmental conditions.

The resulting power radiated by dipole radiation from an emitter immersed

in a plasma, as a function of the frequency, is given by31,32

P(w) = c3 I(w) (2.23)

where I(w), the line shape function, is defined by

I(w) b 6(w wab) (blde-ifja) 2Pa (2.24)
Here, a and b refer to the initial and final states of the total radiator-plasma

system, respectively. The energy difference between these initial and final

states can be written as wab = (Ea Eb)/h. The dipole moment of the radiator

is given by d, and the population of the initial states is given by Pa which is

an eigenvalue of the density matrix p, where p = Ei pi ii) (i. The radiator

center-of-mass coordinate is given by R and the wave vector of the emitted

radiation is given by k where k = w/c. The eigenvectors ii) are eigenfunctions

of the system Hamiltonian H. Free-free and free-bound transitions will not be

examined in this dissertation so we consider only dipole radiation from atomic


Since a similar formalism can be used to describe absorption line profiles,
we will concern ourselves only with the line shape function I(w). Our system
will consist of plasma electrons, ions and the radiator which will always be
referred to in this section by the subscripts "e", "i", and "r", respectively. For

simplicity, we also set h = 1 for the remainder of this section.

We can express the Dirac delta function as an integral representation that
will allow us to incorporate the time development of the system into the dipole
moment operators (i.e. the Heisenberg representation). If we work with this
representation, we have the advantage that all of our approximations must

be made explicitly on the system Hamiltonian which will be contained in the
expression for I(w). This gives

(w) = L dei(W-(W-Wb))t (blde-'k2la) Pa
a,b (2.25)
1 roo
= Re dt eiw(t)
7r 0

0(t) = S (al eikRdlb) (bi eiwtde-ikRe- pa Ia)
a,b (2.26)
= (al d(0) b) (b d-k(t)p la)
Here we have used the fact that p Ia) = Pa la) and e-iHt Ia) e-iwat la) and the

definition of the time development of the dipole and radiator position operator,

d-k(t) = eiHtd(0) e-i'Re-iHt (2.27)

It follows that dk = dei~'. We will consider our sum over states a and b to

now be a sum over a complete set of states and replace the sum with a trace.

We are free to do this at the outset and only look at the elements of the sum

that correspond to the spectral transitions we are interested in. Now we have

(t) = Trr,e,i [4k(O) k(t)p (2.28)
= (k(0) d-k(t)
and thus

1 -. -
I(w) = -Re dteit(dk(0) dk(t)) (2.29)
7r Jo0
To address the problem of radiator motion, we will assume that the radiator

velocity' is statistically independent of perturbing ion and electron effects as

well as of the internal state of the radiator. Consequently, we can factor the

radiator translational motion from the density matrix. This gives

P = PtrPr,i,e (2.30)

Here, ptr is the density matrix for the radiator translational motion. Assuming

ideal gas behavior for the radiators, we have

Smr 3/2 _mr,2
Ptr = ( e 2T (2.31)

where mr is the radiator mass and v is its velocity. We may now separate out

the radiator's translational motion to give

I(w) = -Re o dt eit I dptreiE )) ( d(t)) (2.32)

If we assume that the radiator's motion is constant during the time of interest,

then we can write


Ri(t) = R- + Vt

This gives

I(w) = -Re dt edt d!ptre-~7kt( d (t)) (2.34)

The integral over v may be performed to give

0 1Reodt it- ( 2kt2
I(w) = Re d eit e ~- (t)). (2.35)
7r Jo
It can be shown with the aid of the Fourier convolution theorem33 that this

expression is equivalent to

I(w) = J 'd ID( w ') Is(w') (2.36)
where ID(W w') is the Doppler distribution given by

ID(w w') exp (2.37)


2a2 = 2 (2.38)
C2 \ mr /
Here, mr is the mass of the radiator and c is the speed of light. The line shape

function for the static radiator is given by

1s( 0
I(w) = -Re dt eiwt. d(t)) (2.39)

This result is the usual method for treating the Doppler effect but we see

that it follows, with suitable approximations, directly from the expression for

the power emitted by a radiator. The interaction of radiator motion with

plasma perturbations can be treated32,35 as an extension of the above devel-

opment by not making the approximation: R(t) = R + vt. We will employ

the Doppler convolution given by Eq. (2.36) to account for radiator motion

and continue with the development of IS(w). We will drop the subscript from

Is(w) for convenience in what follows. This gives

1 (00
I(w) = 1Re dt ei (d(0) d(t)) (2.40)
7r 10
We see that the line shape problem is reduced to the calculation of the

system's dipole-dipole autocorrelation function. This is the starting point for

some numerical simulation approaches.36 We, however, will continue with the

formal development of the theory along the general lines of the work of Smith

and Hooper.37

We will see below that it is possible to approximately factor the density

matrix into product form so that

P = PiPePr (2.41)

Here the individual density matrices are for the three subsystems: the plasma

ions, the plasma electrons and the radiator. We will examine the LTE case

where the density matrix becomes the simple Boltzmann factor p = e-H /Tr e-fH

Consider our system Hamiltonian

H = H + HO + HO + Vi,r + Vi,e + Vr,e. (2.42)

Here the HO's refer to the kinetic and potential energies of the respective

subsystems so that Hi = Ti + Vi and H0 = Te + Ve. The Vj,'s refer to the

potential energy of interaction between the subsystems j and k. We make the

quasistatic approximation for the ions so we assume that they do not move

appreciatively during the atomic radiation time. The use of the quasistatic

approximation allows the interaction of perturbing ions with the radiator to

be statistically screened by the rapidly moving plasma electrons. Consequently,

we will replace Vi,e +Vi with a screened ionic potential and as an approximation


V' Vi + Vi,e. (2.43)

This gives us the effective ionic Hamiltonian HP' = Ti + V'. What precise

form this screened potential should take has been the subject of considerable

work.38,39,40 Here, we will use the lowest order screening approximation for

V': the Debye-Hiickel potential.20 In order to make the density matrix factor-

ization, we will neglect Vr,e, but only in the density matrix. This amounts to

neglecting radiator-perturbing electron correlations at the initial time. This is

a reasonable approximation as long as kT > Vr,e. Now we have

H w H?' + HO + HO + Vi,r. (2.44)

The ion-radiator interaction, Vi,r, will be expressed as a multiple expansion

which we will truncate. For the density matrix we will keep only terms through

the dipole. The monopole term has only ion coordinates and is incorporated

into HP' to give H!. The dipole term contains both ion and radiator coor-

dinates but we will use a mathematical device to eliminate those of the ions

and combine it with HO to give HrI(i. Here f is the plasma microfield to be

discussed below. If we assume that the remaining effective Hamiltonians for

the three subsystems commute amongst themselves the density matrix is now

factorable with the Hamiltonian H w H, + Hr(e + Hf. We will see what these

terms are in detail shortly.


Turning from the approximate form for the Hamiltonian in the density

matrix, we consider the terms in the Hamiltonian of the dipole moment time

development operator. We make a multiple expansion41 of the radiator-ion


1 .. i .Ei(0)
Vr, = x(0) d. E() Q. +.... (2.45)
The electric fields and potentials refer to those produced by the plasma ions

at the radiator which we take to be at the origin. The effective charge of the

radiator is given by X = Z a where Z is the radiator charge and a is the

number of bound radiator electrons. The radiator dipole moment operator

is given by d and the components of the radiator quadrupole moment tensor

are given by Qij. One effect of the quadrupole term is the production of an

asymmetry in the line by intensification of the blue portion of the line relative

to the red.12 The octupole term is fourth order in the expansion parameter 6

compared to the third order of the quadrupole term. For the special case of

hydrogen-like ions, Joyce42 has shown that the octupole term, to first order in

perturbation theory, produces symmetric shifts of the hydrogenic energy levels

and thus will not contribute significantly to the asymmetry of the spectral

line. Since we are using a basis set for this calculation that is made up of

eigenfunctions of the radiator Hamiltonian that includes the field dependent

dipole term, we take the quadrupole term as the first order correction to this

Hamiltonian, and neglect all higher order terms.

We will symbolically represent all field gradient terms 9Ei(0)/zxj by the

generic term Env = QpEV. Whenever we use this term we will mean all rele-

vant partial derivatives. Our device for removing the ion coordinates from the

interaction potential Vi,r involves the introduction of a delta function in the

manner shown below. This will alow us to separate the ion coordinates from
the radiator-perturbing electron subsystem and proceed with the development
as shown below. Introducing the delta functions into Eq. (2.28) gives

(t) = dJ de, Trr,e,i [pib(- E)6(epv EP,)O() d(t)pre] (2.46)

Here, the integration variable epv is used in the same way as Etv above.

The time dependence of the radiator dipole operator is given in terms of the
total system Hamiltonian H. The ion coordinates in H that are not removed

by the delta function will commute with d(0) in the static ion approximation
and cancel.43,44 This leaves the Hamiltonian He,r that has only radiator and
perturbing electron coordinates. This is illustrated by

d(t) = eiHe d(O)e-iHer, = eiLt(0) (2.47)


Her = H0 + Ve,r + Hr(e Qi ,j (2.48)
ij J
and L is the Liouville superoperator expressed in terms of a commutator acting

on an arbitrary operator O as

LO =[Her, O] (2.49)

We have now successfully separated the system ion coordinates from the
radiator and perturbing electron subsystems. This is expressed by

I(w) = d deyW(e e^)J(, (2.50)

where W(f, eiV) is the ion probability function, and

J(w, ew,) = -Re dt eiwt ((0) d(t))e,r
7r JO

= -RIm Tre,r [d- Re,r()(pe,rd]
We have defined the resolvent operator as

Re,r(w) -i I dt ewt-iLt = [w L]- (2.52)

Re and Im refer to the real and imaginary parts respectively. The ion proba-
bility function W(f, e1v) is given by

W(, ejv) = Tri [,PI(- E)6(el Epv)]
( ( E)b(eyv E}))i.
The field E and field gradient Eu, are sums of contributions from all of the
individual system ions. We will return to the evaluation of W(C, eCv) in chapter

3. Recalling that pe,r = Pepr, we can perform the trace over plasma electron
coordinates to give

J(w, ) = -Im Trr [d- (Rer(w))e(Prd) (2.54)

We see that the effects due to the perturbing electrons have been isolated in

Rr(w) = (Rer(w))e. Note that L = L(,7 te).

Next, we proceed with the evaluation of the operator Rr(w). We now
introduce a coupling constant A as an expansion parameter. Write

L = Lo + ALI, (2.55)

where L corresponds to the Hamiltonian:

He,r = H + AVi = H, + A (Ve,r Qi (2.56)

The Hr, corresponds to Lo and Ve,r + V9 corresponds to LI. Here, Qij is
a function of the radiator's electron coordinates (for single or multielectron
radiators), Ve,r is the radiator and plasma-electron interaction, Vi is the ion-
quadrupole interaction, and Hr = Hr(,)+H0 is the combined non-interacting
radiator plasma-electron Hamiltonian containing the ion-dipole term. We will
now implicitly define an electron broadening operator M(w), by setting

Rr(w) = [w Lr(4 ev) M(w)]-1 (2.57)

where Lr(E, eyv) is the Liouville operator corresponds to the Hamiltonian:

Hr(F, epv) = Hr(,) Qijiej (2.58)
Here, Hr(E, is the Hamiltonian of the field-free radiator with the ion-dipole
term, and we have used the quadrupole approximation for the remainder of
the interaction of the plasma ions with the radiator. Specifically,

Hr() = H0 d1. (2.59)

Note that Hr(i) = Hr(fE0).
So, using Eqs. (2.52),(2.55) and (2.57), we can write,

[w- Lr(f ec) M(w)] = ([w Lo ALI-)e (2.60)

Next, we expand M(w) in powers of the coupling constant, expand both sides
of Eq. (2.60), equate coefficients of equal powers of A, and thus identify the
terms of the M(w) expansion.

The right-hand side of Eq. (2.60) can be expanded in a Lippmann-Schwinger

expansion and the left-hand side can be expanded in a Taylor series in A. First,

we will expand the right-hand side. For simplicity in notation we will denote

the full resolvent operator by Re,r(w) R, and define a zero order resolvent

operator as

R [w Lo]1 (2.61)

We can construct an iterative equation for R by noting that

R R = RORO R-1] R (2.62)

or explicitly,

1 1 1 1
S- [w L (w L)] (2.63)
-L w Lo w L w L
This can be written equivalently as,

1 1 1 1
= + L LI (2.64)
L w w-L w-L
or simply as just

R = Ro + RoLIR (2.65)

This is the Dyson equation for this system in Liouville space. This implicit

equation for R can be iterated to give a series expansion for R in powers of A:

R = A"Ro(LIRo)n (2.66)
To second order in A we have

R = Ro + ARoLIRo + A2RoLiRoLRo (2.67)

In order to examine what (. *)e does to this, look at

Rr(w) = (R)e (2.68)

To second order in the interaction Liouville operator this is

Rr(w) = (Ro)e + A(ROLIRo)e + A2(RLLIRoLJRo)e (2.69)

Here Ro is a function of Lr and Le, the zero-order-radiator and perturbing-
electron Liouville operators with Lr corresponding to Hr(e) and Le to H,0.

Tighe44 has shown that when L0 operates on objects of the form B = peb,

where b operates on radiator coordinates only, then

(Ln)e = L (2.70)

On expanding Ro this leads to

(Ro)e = [w Lr]-1 RO(w) (2.71)

For combinations of Ro with a general operator in Liouville space Tighe44

has shown that

(Ro0)e = RO()e, (2.72)


(Ro)e = (C)R .


These identities allow us to simplify the expression for the resolvent. We obtain

(R)e = R' + ARO(LI),eR0 + X2RO(LIRoL)eR0 (2.74)

For the left-hand side of Eq. (2.60) we postulated an equivalent form
for the resolvent that contains all of the perturbing electron statistics in an

implicitly defined operator M(A, w) which is analytic in the coupling constant
A. Restating this we have

(R)e = [w Lr(E e y) M(A, )] -1 (2.75)

where Lr is defined above. Expanding M(A, w) in powers of A we have

M(A,w) = M(0)(w) + AM(1)(w) + AM(2)(w) + (2.76)

To expand the right-hand side of Eq. (2.75) about A = 0 we will need the
following easily proved operator identity for some operator A in Liouville space:

BA-1 dA
= -A-l A- (2.77)

Expanding A-1(A) about A = 0 we have

A-(A) = A-(0) + A -() 2 2A1 (0) + O(3) (2.78)
OA 2 + )2

In this case we have

A(A) = [w Lr(, 0) ALr,i(euv) M(A,w)] (2.79)

with Lri(ep) corresponding to VQ. Using the identity of Eq. (2.77), along

OA( = Lr,i(e,) M(1)(w) (2.80)

a2 = -2M(2)(w) (2.81)

A-(0) = w- Lr(, 0) M()(w) (2.82)

we can evaluate the expansion of A-1 about A = 0. We obtain

(R)e = A- = A-(0) + AA-1(0) [Lr,i(ev) + M()(w)] A-(0)
A2A-1(0) [Lr,i(ep) + M(l)(w)] A-(0) [Lr,i(e ) + M(1)(w) A-1(0)
A-1()M(2)(w)A-1(0)} + O(A3) .
Comparison of the two expansions for (R)e, term by term in powers of A up
to second order, allows us to make the following identifications for the terms
in the expansion of M(A, w):
A(0) term

M()(w) = 0, (2.84)

A(1) term
M )(w) = (Lr,e)e (2.85)

A(2) term

M(2)() = (LIRoLI) (Li)eRO(LI)e .


This last expression for M(2)(w) can be further simplified if we expand LI.
Recall that LI = Le,r + Li,r with Le,r corresponding to Ve,r and Li,r to V.
This gives

M(2)(w) = (Lr,eRoLr,e)e (Lr,e)eRO(Lr,e)e
+ Lr,i((Ro R))Lr,e)e + (Lr,e(R0 R ))eLr,i -
With the use of Eqs. (2.72) and (2.73) the last two terms cancel to give

M(2)(w) = (Lr,eRoLr,e)e (Lr,e)eRO(Lr,e)e (2.88)

If the dipole approximation is made for Ve,r, we will find for the corre-
sponding Liouville operator, (Le,r)e = 0. This is easy to see by noting that
in order to evaluate this expression we need to calculate (Ee)e which must
average to zero since there is no preferred direction in the plasma. This holds
true for the case of hydrogenic or multielectron radiators. The first surviving
term in the M(w) expansion will then be to second order. So to lowest order
in A,

M(w) = M(2)(w)= (Le,rR(w)Le,r)e (2.89)

where Ro(w) = [w Lo]-1. This gives us

M(w) = -i dteit (Le,re-iLotLe,r)e (2.90)

This operator is tetradic37 because of the use of the Liouville operators and,
when evaluated as a matrix element, will give45

M(w)ab,ab, = Ib [6 a daa" daa',G(Awal.b)
+ baa' E db'b" db"biG(- ab") (2.91)
aa, dbb{G(Aab,) + G(-AWab)1

where the first term represents electron broadening of the upper state, the

second term represents electron broadening of the lower state and the third

term represents interference effects.

The many-body electron broadening function G(Awab) is given by

G(Awab) = dteiAwbt( e Ee(t))e (2.92)

with Ee(t) = eiHOt e(O)e-iHt and Awab = w (wa wb). Thus the elec-

tron perturber effects are given in terms of an autocorrelation function for the

plasma electric field at the radiator due to the plasma electrons. We have

calculated only the imaginary part of M(w) which corresponds to line width

operator. The real part corresponds to the line shift operator and yields only

a small shift44 which is ignored here.

In our approach to this calculation we approximate the effects of corre-

lations among the perturbing electrons by the use of G(Aw) = G(wp) for

|Aw| < wp, where wp is the electron plasma frequency. This procedure has

been shown to be a good approximation by a detailed treatment of dynamic

electron correlations.21 This handling of electron correlations allows us to use

single particle Coulomb waves in taking the trace over perturbing electron co-

ordinates. The Coulomb waves are single particle solutions to Schr6dinger's

equation with a central 1/r potential with effective radiator charge X, where

X = Z aON; aoN being the number of bound radiator electrons. G(Aw) is

then evaluated using the method of Tighe and Hooper.46 Other theories47 also

exist for the evaluation of G(Aw). It is instructive to point out, however, that

we can relate this electric field autocorrelation function to the charge density

autocorrelation function48 which gives us,

(jEe e(t))e_/dJ3x f3 'f
(e e(t))e = dx d 2X (p(x)p(, t))e (2.93)

This illustrates the relationship of the plasma charge-density fluctuations to the
spectral line-broadening problem. Note that this charge density correlation

function is not translationally invariant for ionic radiators as is sometimes

assumed.49 Translational invariance would imply a uniform perturbing electron

gas; for highly charged radiators, this assumption would ignore aspects of the

radiator-perturbing electron correlations.

Altogether, our approach leads to a line shape function of the following

I(w) = de depvW(-, e)J(w, ) ,v)


J(w, C, efp) = ImTrr ( [w Lr(, e-7 ) M(w)] prd} (2.94)

or in matrix element form

1 -1 (.
J(w, E ey) = Im padb'a' 'dab [w- Lr( Ep) M(W)] ab,ab (2.95)
where a and b refer to initial and final radiator states respectively.

The above equations for the line intensity I(w) are valid for the general

case of multielectron radiators with combined upper and lower state electron

broadening effects and Stark shifts. In this work we will be considering hy-

drogenic ion resonance lines which have the ground state as the lower state.

For hydrogenic ions the ground state is not subject to electron broadening and

only very weakly affected by the plasma electric microfield. Therefore we will

ignore lower state broadening and interference effects. This will leave only the

upper state term of the three terms in the matrix element expression for the

electron broadening operator M(w) given by Eq. (2.91). This gives

M(W)aa, = -- aa,, daalG(Awa,,f) (2.96)
where subscript f refers to the single lower radiator state.

Since lower state ion produced Stark shifts are unimportant here, the ra-

diator Liouville operator Lr(E, epv) in Eq. (2.95) can be evaluated to give

1 -1
J(w, E, e,) = --Im padaf dfa, [ (Hr(e ~e) f) M(w)] ,
where wy is the lower state energy of the radiator. We note that since there is

only one lower state the sum over b is absent.

For our case, the dipole and quadrupole operators become single electron

quantities with

d= -er
Qi,j = -e(3xiXj r2 i,j)

Here r and xi refer to radiator electron coordinates. The corresponding mul-

tielectron quantities are just sums of terms similar to these for each radiator


Any intrinsic line width, such as the resonance width induced by the pres-

ence of an electric field (see Chapter 3), can be approximately included in the

line profile formalism by replacing the affected level energy E by a complex

resonance energy50 E. For level j this gives

Ej Ej i (2.99)

where Ej is the energy at the center of the resonance and the presence of

irj/2 assumes a Lorentzian shape for the resonance. Fr is the full width at

half maximum of resonance j. We have chosen the minus sign in Eq. (2.99)

to be consistent with the fact that the electron broadening operator M(w) is

proportional to -i. A choice of Ej iY would correspond to a proportionality

factor of +i. The use of this complex energy is only an approximation since we

are attempting to describe the time dependent behavior of a resonant state in

a stationary state formalism. We use this procedure to approximately include

the field induced resonance width for the upper states of the Lyman series in

hydrogenic argon.

When considering the comparison of theoretical line profiles to experimen-

tal results, it is important to remember the factor w4 in the relation for the

total power, P(w) = Const. w41(w). If the radiation energy is high enough and

the line shape is broad enough, this factor will not be approximately constant

across the line shape and I(w) must be redefined to include it. Otherwise an

additional source of line asymmetry will be neglected. For experimental com-

parison, it is also important to include instrumental broadening5 as well as

opacity effects51 if significant.

In the proceeding chapter we have reviewed some fundamentals of dense

plasmas and formulated a theory of spectral line shapes. We will now examine

some of the processes and effects that can make important contributions to

these line shapes in the high density regime.

3.1 The lon-Quadrupole Effect
Previously we introduced the ion coordinate joint probability function

W(, e/,V) to describe the joint probability distribution for the microfield f to-

gether with the field gradients ev. Recall that for notational convenience, ep,
represents all of the field gradient terms. Writing out this probability function

explicitly we have

W(, epv) =( 6(E- E)6(ez Ey)6(eyy Eyy)6(ezz Ezz)

x 6(exy Ezy)6(eyz Eyz)S(ezx Ezz) )i (3.1)

=- e- #b(- E)6(e Exx)... d3Nr

where < ... >i is the ensemble average over ion coordinates, Zc is the configu-

rational partition function for the ion subsystem, N is the number of ions, and

V is the potential energy of the ion subsystem in the presence of the charged

radiator. The quantities E and Eij are the many particle electric field and field

gradients at the radiator and are functions of the perturbing ion coordinates.

Using an integral representation of the field gradient delta function,

6(e Exz) = d az e-io,"(e"-E, ) (3.2)


W(f, =)- (2r) Zc d'a e-i(O'e'+--') d3Nre b (- )ei(-,,E+-)

(2ir)6Zc d 3r 6(E-- E)eV v d6e-i(UzE:+'-) (eie)
= Q(2iJI d6ae-'i(Ox,, e +oU acyu+w )(eie)e
(21r)6 J
= (1(P(l'|,E) ,

0 = ozxxEx + ayyEyy + azzEzz + aZyExy + ayzEyz + azxEz (3.4)

We have defined the conditional averaged quantity

(eiE) f d3Nr e-3V(g- .)eie
(e d3N, (3.5)
f- d3Nre-(Vt6(-- E)
This is the ensemble average of eie where all included ion configurations are
constrained to have the microfield value f. It is normalized to give one if 0 = 0.
By defining the average in this way we have retained the benefits of using the
ordinary plasma microfield function Q(e) to describe the uniform field at the
radiator and we are now free to approximate the higher order field gradients if
we choose. The function P(E|e t) is the conditional probability for ey given
g. We take the field to be in the z direction for simplicity and without loss of

Direct evaluation of this constrained average is quite difficult if we wish to
retain all of the ion interactions. Consequently, we will make a simple approx-
imation to this function in order to proceed. We will expand the exponential
in a cumulant expansion52 by taking

(eie), = exp En (3.6)

C1 = (0)c

C2 = ((O (),)2), (3.7)

C3 = ((6 (e),)3) .
Higher order terms are more complicated. In terms of the expansion parameter
6, which we introduced in the multiple expansion (see Eq. (2.22)), we have
C1 ~ 63 and C2 ~ 66. We see this by noting that 0 goes as aEi/Oxj which goes

as 1/r3 and 1/r is of order 6. For our highest density of ne = 1 x 1025 cm-3,
this gives C1 ~ 0.043 and C2 ~ 0.0018 for the n = 3 level. The C2 term, which
is actually the variance of the quantity Theta, is therefore in the neighborhood
of 4% of the lower order term C1. Here, we will keep only the C1 term in our
approximation. This gives us

(eie) z ei(e) (3.8)

Inserting this into Eq. (3.3) gives

W(f (2_)6 Q( / df d e-i(Ezz +-)ei(e)

= Q(Ce6(ez (Ezx)c)6(eyy (Eyy),)... (3.9)

= Q(e1E6(ep (ESv)c) .

This gives us for the approximation of the conditional probability:

P( epv) 6(epv (Epv),) (3.10)

We have developed a systematic expansion for the conditional probability func-
tion and have examined the magnitude of the expansion terms. We can identify
the first term of this expansion as the approximation used by Joyce.12 Return-
ing to the expression for the spectral line intensity, Eq. (2.50) now gives

I(w) = df de, Q(e~P(f e,,)J(w, f," ev) (3.11)

The microfield function Q(g) does not depend on the direction of but
only on its magnitude, e. We can therefore preform the angular integration
and define a new microfield distribution function as

P(e) = 47re2Q(e (3.12)

Now, the intensity becomes

I(w) = J de de, P(e)P(epv)J(w, e, e). (3.13)

Using our approximation for P(e lepv) we can carry out the epv integration to

I(w) = de P(e)J(w, e, (Eg,)) (3.14)

The field gradient terms in J(w, e, e~,) have now been replaced with their
constrained averages that depend only on e so we can write J(w, e, (Epv)E) =
J'(w, e), where the prime indicates the replacement of the field gradient by
its constrained average. For the remainder of this work, we will drop the

prime with the understanding that we actually mean J'(w, e). Our use of the

approximation for P(ejEpv) has lead to a significant simplification of the line

shape formula that still retains the approximate effect of the field gradients.

In order to proceed we will need to evaluate this constrained average of

the field gradient. From the ensemble of all possible ion configurations in the

presence of the radiator we choose only those configurations that give a certain

field magnitude e at the origin. If we average a quantity F over this subset

of configurations, we have constructed the constrained field average that we

introduced above. We denote this average by

(F) = 6(-E)) (3.15)
Q() (3.15)

where Q(e) is the plasma microfield function and (* *) is the complete ion

subsystem ensemble average. This average, as we have said, is such that for

F = 1, (F), = 1. We will show that it is possible to relate this constrained

average to a functional derivative of the plasma microfield function. This will

allow us to evaluate the constrained average in terms of available simplified

models for the microfield function Q(E).

For our purposes we will take F to be an additive function of the ion


F = f(r) = d f(f)n(f) (3.16)


n(rf-) = r(f-i). (3.17)

Note that f drn(r) = Ni, the total number of ions in the system. So we can
now write

(F)e = 1 df df(-)(n(fr)6(- E))
= Jdff(Pr)nig(;E ,
where we have defined

nig(r; (E (3.19)

On expanding the delta function we have

nig( j = (2 ) e-i"(n(reiE) (3.20)

We would like to relate the average on the right-hand side of Eq. (3.20) to a
functional derivative of the plasma microfield function. To do this we introduce
G(A), the generating function of the microfield distribution, where

G(A) = In (X) = In(eiE) (3.21)

If we define a function 4, such that

= i E(r) (3.22)

then G(X) becomes a functional of or

G(A) = G[] (3.23)

Now we can cast G(A) in the form of a functional derivative, where

G(A) = dl OG)
aJo 3(1
/ Of b~6G[4b ]
= f ddl drO 6G[ (3.24)

= dl driil*- E() E G[
0 (il. E(-)
where 1= l\ and A is the unit vector in the direction of A.
Alternately, from the definition of G(X) in Eq. (3.21), we have

Sa In W(1
G(X) f dl1
(i -eig)
= dl (3.25)
0 (eif)
dl (eiff
Note that we can write,

i E d = I di. E(F b (6- r-)
i=1 (3.26)
= dr-i .* (r-n().
Using Eqs. (3.25) with (3.26) gives

( n( -eil-"E)
In Q(A) = dl dFl I d i- ( (3.27)
o' (- {eil-E)
Comparison of Eqs. (3.24) and (3.27) gives us the connection between the
constrained average considered here, (F)E, and a functional derivative of the
microfield generating functional G[O]:

(n(r-)eil E) 6G[i]
= (3.28)
(eil-E) 6(il E,(r))
which on substitution into Eq. (3.20) above, gives

1g/ dX ei= e- At() (G (3.29)
nig( Q (2) 3 E(i Er-))

Substitution into Eq. (3.18) above gives us the relationship we want:

(F) = df(r) e-iX.Q(A) G[] (3.30)

This explicitly shows that we can relate the constrained average to a functional
derivative of a microfield generating function G(A) = In Q(A). We are free to
use whatever functional model we choose for Q(X).
As an aid in evaluating these constrained averages we define the transform
of g(r'; ( by

1 6G[#]
F; 6A l (3.31)
ni b(iA E(F))
From Eqs. (3.20) and (3.28) we see that the transforms are related by

g(1; 0 = e-ilX (X~)(A ; X). (3.32)

Thus a knowledge of W(Fr; X) obtained from Q(X) will lead to g(r; e) through
the transform and then to (F)e through Eq. (3.18).
To evaluate the ion-quadrupole term recall that the perturbed radiator
Hamiltonian is

Hr(E, (EV)c) = Hr(e) E Qij(Eij) (3.33)
where the ion-quadrupole term is just Vr,i. To evaluate the (Eij), terms note
that when the angular integration are performed, the o, or azimuthal, integrals
lead explicitly to42

(Exy)E = (Eyz)e = (Ezx)c = 0 (3.34)

Together with the property cjEi = OiEj, this leads to the simplification

SQi,j(Eij) = c Qii(Eii) (3.35)
i,j i
With the field in the z direction there is no difference between the x and

y directions so (Exz), = (Eyy),. Additionally, with the tracelessness of the

quadrupole tensor41 we have

SQi,j(Eij), = Qz ((Ezz) (EZx),)

(E) 1 (3.36)

If the ion interaction potentials are screened we have V E 5 0 due to the

continuous charge distribution. However, the form of the multiple expansion

we have employed made use of V E = 0 at the origin which is not strictly true

for our screened ion potentials. We reconcile these two facts by noting that if

we restrict the screening charge density of the perturbing ions to lie outside

the Bohr orbit of the radiator's bound electron, we will have V E = 0 at the

origin. Tests of the computation of the field gradient term show that its value

upon taking V E = 0 is approximately 20% less than that obtained on taking

V E 5 0 at a field value corresponding to the maximum of the microfield

probability function at ne = 1 x 1025 cm-3 and kT = 800 eV. The difference

is less for lower densities. In light of the uncertain nature of the actual form

of the ion potential, this does not seem unreasonable. Consequently, we will

take V E = 0 in the evaluation of the field gradient term in agreement with

the form of our multiple expansion. This amounts to requiring that we have

no penetration of the radiator by the perturbing ions or their screening charge

density. For the ion-quadrupole term this gives us

Vr,i= --1Qzz(Ezz), (3.37)

with the radiator Hamiltonian given by Hr(e) + Vr,i. To evaluate Vr,i we must

calculate the constrained average term (Ezz)E, and to do this we must pick a

model for the microfield function Q(e).

In this dissertation we choose the adjustable parameter exponential approx-

imation (APEX)53,54,55,56 for the microfield. It has the advantage of incorpo-

rating correlations between the perturbing ions while retaining the functional

simplicity of the independent particle (IP) model42 of the microfield. Joyce42'12

used the IP model to evaluate the ion-quadrupole effect which we will see leads

to an overestimate of its magnitude due to the neglect of the inter-ion correla-

tions. We will compare these two models along with a nearest neighbor (NN)

approximation for the field gradient.

The APEX model for the microfield is based on the independent particle

model but with an effective inverse screening length, a. The field is effectively

renormalized in this way by requiring that two conditions54 be met: the so

called second moment condition and the local field condition (see Appendix

B). The transformed conditional distribution function W(rf; A) for the APEX

microfield distribution is given by (see Appendix B)

(; A) = g(r)ei *(r) (3.38)

where g(r) is the equilibrium radial distribution function and E*(r-) is the

APEX renormalized field. This field is given by

=*(r) = E(r(1 + ar)e-ar, (3.39)

where a is determined by the second moment of Q(E), and E(f) is the screened
field produced by the plasma ions. We calculate the radial distribution function

g(r) by integral equation methods due to Rogers.39 Lado and Dufty57 have
shown that this method of calculating g(r'; e) by using the APEX microfield
underestimates the inter-ion correlations in some cases. In particular, for the
case of a hydrogen plasma with Iii = 10 at a field value of e/eo = 0.4 (eo =
e/r2,i), the APEX calculation underestimates the maximum value of g(r'; ) by
a factor of two. We retain the APEX model, however, because of its simplicity
and significant improvement over the IP model of Joyce12,42 which ignores
ion-ion correlations entirely.

With the APEX model for g(F; e) obtained from Eqs. (3.32) and (3.38) we
can evaluate the constrained average by using Eq. (3.18) to obtain

(F), = J dFf(P)g(r)Q( *(r-F)), (3.40)

where it is understood that the microfield distributions, Q(e), are in the APEX

For the ion-quadrupole effect, the constrained average we need to evaluate
is (Ezz),. In particular, we have

F = Ezz = zz(ri). (3.41)

In general, the one perturber derivative term is given by

EU(r) = Ze 1 + r) ( 3, (3.42)
r3 A r

so for our case we have

ezz(r) = Ze ~ + 1 3 cos2 ) .(3.43)

The microfield function in the APEX approximation is54

QAPEX() = i dkk sin(ke)en hl(k) (3.44)


hl(k) = 47 drr2g(r) (j0 (kE*(r)) 1), (3.45)
nj E*(r)
where jo(x) = sin(x)/x is the zero order spherical Bessel function of the first
kind. From now on when we speak of Q(e) we will implicitly assume the APEX
approximation and suppress the subscript. Using this approximation we can
evaluate Eq. (3.40) for the case of F = Ezz to obtain

Zeni dkk2enooh(k) OO -r/A
(Ezz) e d= kk2e () drr2g(r) 0 (1-r/A)I(r). (3.46)

The angular integral IQ has been evaluated by Joyce42 and is given by

(r) d(1 3cos2O)j (k ?- *(r) (3.47)
= -8rj2(ke)j2 (kE*(r)) ,
with dQ = dpd(cos 9).
To evaluate (Ezz)E numerically, we cast its variables in unitless form by

S= E/Eo

S= Eok
a = ro,i/A

x = r/ro,i

Eo = (3.49)
We obtain

(Ez = -6 d2 en hl( )j2(k ) fo dxg(x) -(1 + ax)j2 (k*())
foi JO" dkken'hi(k) sin(ki)


hl(k) = 00 dxx2g(x) ) J( *(x)) 1) (3.51)
ni ( E*(x) (
In this form, the equation for (Ezz)c can be directly evaluated numerically.

Following Joyce42 we implement this function in the line shape calculation by

the use of a Pad6 approximate. For a particular case, (Ezz)e is calculated for

a mesh of i values. The Pad6 function is then fit to the calculated values by

minimizing the squared difference between the two while varying the coeffi-

cients of the Pade function. The resulting approximation then yields (Ezz)f

for any value of the scaled field up to i = 20 with an accuracy of about 5%

compared with the values calculated from Eq. (3.50). This is quite sufficient

for calculating most line shapes. If more accurate values of the field gradient

are required, the directly calculated values can be used.


We would like to examine the behavior of the field gradient term (Ezz)f in

the limit of large and small values of e. For e --, 0 it is easy to show that the

field dependence follows the limit of j2(ke) to lowest order in e. This will give

lim(Ezz)= = -C2 (3.52)

where C is a constant that depends on ne, T and Zi. Thus, the zero field limit is

zero. This is easy to see physically. If e = 0 there will be no preferred direction

about the radiator and the fluctuations in the field gradient Ezz will sum to

zero upon averaging over the ensemble of ion configurations. The e -- oo limit

is somewhat more complicated so we will examine the behavior of a simple

nearest neighbor model for (Ezz)E. The large field behavior of the microfield

at a charged radiator is governed by a single nearest neighbor ion.58,59 For the

strongest fields, the nearest neighbor will be significantly inside the screening

length AD,e so screening will be negligible. To approximate (Ezz)c let the field

gradient at the radiator be that produced by a single ion located at the distance

from the radiator sufficient to give a field value at the radiator of e. This gives

us the nearest neighbor approximation of

(Ezz}),N = -) /2 (3.53)

for large e.

In Fig. (3.1) we compare three different models for (Ezz)e. For purposes of

comparison, we take (e/ro,i3) = 1. We look at the APEX model which includes

ion-ion correlations, the independent particle model of Joyce42 which ignores

ion-ion correlations and the nearest neighbor model for large fields as discussed

above. We see that the APEX model gives a field gradient whose magnitude is

consistently smaller than that of the other two. This is to be expected since the

ion-ion correlations tend to cause the plasma ions to repel each other and thus

to be less likely to converge on the radiator and create a large gradient. The

ion-ion correlations will, in general, lessen plasma ion microfields by causing

a rough ordering of plasma ions. The nearest neighbor model produces the

largest gradient due to its assumption of all charge located on the z axis. It

also includes no correlation effects. The IP model of Joyce falls in between

because it includes radiator-ion correlations but no ion-ion correlations.

In Fig. (3.2) we examine the behavior of the APEX model for the field gra-

dient at several values of the electron number density. For ne = 1 x 1024 cm-3

the average Ezz gradient's magnitude is less than that of the nearest neighbor

model as was pointed out above. As the density increases the APEX field

gradient approaches that of the nearest neighbor model for all relevant val-

ues of the field (in this case we refer to a range of field values of e/eo = 0 to

5). For a particular density, we can see that the gradient also approaches the

nearest neighbor value as the field increases. This behavior can be explained

by recalling that the presence of large fields at the radiator is generally due to

the location of a nearest neighbor perturbing ion close to the radiator. The

large field behavior at constant density should be due to a nearest neighbor

ion and ion-ion correlations play little role in the interaction of two nearest

neighbors at these densities. The probability is small for having a positively

charged perturbing ion close to the positively charged radiator. It is even less

probable to have two ions close to a positively charged radiator; and we need

two for ion-ion correlations to be important. The approach of (Ezz), toward

the nearest neighbor value as density increases should be due, simply, to the

ions being forced closer to the radiator. With decreased interparticle spacing

the field gradient naturally increases.


3.2 Atomic Data by Perturbation Theory Solution

To evaluate the radiator subsystem matrix elements necessary for the cal-

culation of the spectral line shape I(w), we are free to use any complete set of

basis states that spans the Hilbert space of the system. A true representation

of an operator in this basis will thus consist of a matrix whose dimension is the

same as the number of basis vectors necessary to span the particular Hilbert

space. For our case this dimension is a nondenumerable infinity. However, a

reasonable representation of the operator can be obtained by using a truncated

set of basis vectors. For weak fields, a reasonable representation of the Hamil-

tonian for a radiator in the state with principal quantum number n can be

obtained by using eigenvectors of the field-free radiator Hamiltonian for states

n and n + 1 as basis vectors.60 For stronger fields, the need to include other

states, including positive energy states, will be necessary. In fact, the reso-

nance nature of the strong-field states can not be described without including

positive energy states to account for the unbound character of field dependent

solutions that allow for the possibility of tunneling. In order to include these

effects we will employ an alternative method using a basis set that consists of

the eigenvectors of a radiator Hamiltonian that includes the ion-dipole term

from the perturbing-ion multiple expansion. This Hamiltonian is

Hr() = H0 d- t. (3.54)

Hr(c) is diagonal in this field dependent basis set. By using this Hamiltonian,

we will automatically incorporate the characteristics of the resonate nature

of the states into our basis. Specifically, we will use Hr(e) as our zero order

radiator Hamiltonian. The first order correction to this will be the next term

in the perturbing-ion multiple expansion, the ion-quadrupole term Vr,i.

To calculate our field dependent basis we solve Schr6dinger's equation in
the position representation with the Hamiltonian Hr(e). The equation is

Hr(e)(r-) = E(r (3.55)

The natural coordinates to use for this Hamiltonian are parabolic coordinates.61,62
The wave equation will then be separable. Assuming the plasma microfield to
be in the z direction, we write Eq. (3.55) in atomic units (see Appendix A)
and scaled to Z = 1 as

V2 + + 2E 2Fz (r) = 0, (3.56)

where F is the magnitude of the plasma ion microfield. In parabolic coordinates

(see Appendix C), Schr6dinger's equation becomes

4 9 f89O 4 8 (89 1 92
+r 0 \" U + 9++rr 0 ^ +a
S+ 7 a 9 C + 777 a7 C (W2 (3.57)
+ (+4 +2E-F( 1-7) 0=0.

We exploit the separability of the equation by looking for a solution of the

(V) = f(A0)g()--e- (3.58)
We define new scaled variables and transform the two resulting separated equa-
tions. We take

x -/n

y /n (3.59)

A (1/4)n3F ,
and obtain the system of equations


{d d m2 2}
+ -Ex AX2 + A f(x) = 0 ,
dz dz Tx 4x 2

2 Ey + Ay2 +B g(y) = 0
T- ~ -JU



Field-Free Solution

For the field-free case we take A = 0 and obtain the solutions (in Z = 1

atomic units)

Ao = nl +
B0 = n2 + -+




n=nl + n2 +m + l .

Here nl and n2 are the integer valued parabolic quantum numbers.61 The

field-free eigenfunctions are

fo(x) = u(ni,m, x)


go(y) = u(n2, m, y).

u(n1,m,) = e-VXm -/2 -2m()
[(nl + m)!]3/2 n"



Lm (x) is an associated Laguerre polynomial given by63


Wdy dY




L(x) = [(n( + m)!]2 (-x)P
j- (m + p)! (nl p)! p!
For our case m is restricted to integers for which m > 0. Other definitions of

Lm (x) are quite common.33

Field Dependent Solution

In order to solve Eqs. (3.60) and (3.61) for A 0 0 we will expand the

unknowns f(x), g(y), A, B and E in powers of A. The zero order terms will

be given by the A = 0 results above. This perturbation expansion technique

closely follows the work of Hoe et al.6465 The expansions are


A = ApAP
B = Z BpAP (3.68)
f(x) = AP ah u(n1 + h,m,z)
p>O h>-ni
g(y) = AP bp u(n2 + h,m,y).
p>0 h>-n2
The sums for index h are over all possible values of nI and n2. Upon evaluation

of the coefficients ah and b^, however, we will find that all terms in the sum

are zero except for max(-nl, -2p) < h < 2p. This is also true for the sum

involving n2. We immediately see that because of the condition A + B = n,

we have

A0 + B0 = n (3.69)


Bp = -Ap for p > 0. (3.70)

The other coefficients in the expansion can be obtained by solving for the p'th
coefficients in terms of the p- 1 and lower coefficients. Thus, a knowledge of the
zero order terms will allow us to construct all higher order terms iteratively. To
accomplish this we substitute the expansions for f(x) and g(y) into Eqs. (3.60)
and (3.61), multiply by u(x) and u(y) and integrate over x and y, respectively.
For the x coordinate we obtain

Sdx u(n + k, m, x) {Ho(x) + A + x(1 + 2n2E) Ax2 f(x) = 0 (3.71)


d d m2 x
Ho(x) = -xz 4 4 (3.72)
dx dx 42 4
Following Hoe et al64,65 we call Ho(x) the field-free effective Hamiltonian for
the x coordinate. It gives

Ho(x)u(n1 + h, m, x) = -Aou(n1 + h, m, x) (3.73)


A0 = n1 + h + 2 (3.74)
Carrying out the integration gives

2p a
AP y 2 ap dxu(n + k,m,x)Ho(x)u(n1 +h,m,x)
p>O maz(-nl,-2p) (3.75)

+ A6k,h + (1/4)(1 + 2n2E) (x)k,h A (x2)k,h} 0


(X")k,h j dxxnu(nl + k,m,x)u(ni + h,m,x) (3.76)


Sdxu(ni + k,m, x)u(ni + h,m, ) = 6k,h. (3.77)

Upon expanding E and A in powers of A, we see that the p'th coefficient is

given by

p-i 2p p-i
ka = Ap-i a x)k,ha -1 + (x)k,h E vp-
i=0 h>max(-nl,-2p) i=0

If the maximum value of the sum index is less than the starting value of the

index, then the sum is defined to be zero. We will return to the special case of

k = 0 below.

Likewise, we obtain for the y coordinate

p-1 2p p-1
kbk Bp6 + ()k,hh-1 (Y)k,h E p-ibh
i=0 h>maz(-n2,-2p) i=O

For p = 0 we must have

ah = bh = 0,h. (3.80)

To obtain the energy coefficients up we take the k = 0 case of Eqs. (3.78)

and (3.79). We obtain two equations for Ap. Equating these and solving for

Vp gives

i = 1

Z [(1O + 0,h]
The sum over h is indexed from max(-nl, -2p) or max(-n2, -2p) to 2p as


We still need to evaluate the coefficients ak and bp for k = 0. To accomplish
this note that

Sdxf2(x) = 1 (3.82)


0 dyg2(y) = 1. (3.83)

Evaluating these by using the series expansions for f(x) and g(y) and exam-

ining the coefficients of a particular term of order AP will give

2p p-1
a -- a ah h (3.84)
h>maz(-ni,-2p) i=1

2p p-1
b = bb, _, (3.85)
h>maz(-n2,-2p) i=1
We now have the coefficients for calculating the energy E and the components

of the wavefunctions f(x) and g(y). To evaluate these we need the integrals

(x)k,h and (z2)k,h as well as those for (Y)k,h and (y2)k,h. These can be straight-
forwardly evaluated to give

(x)k,h = (2nl + 2k + m + 1)6k,h

/(nl + k + l)(nI + k + m + 1)6k+1,h (3.86)

v/(nl + k)(ni + k + m)k-,h,

(x2)k,h = {6(n1 + k)(nl + k + m + 1) + (m + 1)(m + 2)} Sk,h
2(ni + k + 1)(nj + k + m + 1)(2ni + 2k + m + 2)6k+l,h

2/(ni + k)(n1 + k + m)(2n, + 2k + m)6k-_,h

+ V/(nl + k + 2)(ni + k + m + 1)(ni + k + 1)(ni + k + m + 2)k+2,h

+ /(nl + k)(ni + k + m 1)(ni + k + m)(ni + k 1)bk-2,h
The relations for (Y)k,h and (y2)k,h are obtained from these formulae by taking
x -+ y and ni -- n2.

In order to compute matrix elements from the field dependent wave func-
tions given in Eq. (3.58), we also need to determine their normalization. We
take the normalization constant to be

N(ni,n2, m, A) =- dxdy (x + y)[f(x)g(y)]2 (3.88)

Expanding N(nl, n2, m, A) in powers of A we have

N(n, n2, m, A) = E NpAP (3.89)
To evaluate the coefficients Np, we expand the functions f(x) and g(y) in
powers of A, matching terms on the right- and left-hand sides to obtain

Np = E E [a ()h,k + bhb-i(Y)h,k] (3.90)
i=O h,k
The second sum is indexed from max(-nl, -2p) or max(-n2,-2p) to 2p as

appropriate. Note that

N = n (3.91)

Dipole Matrix Elements

Now that we have developed the formulae for the perturbation solution of

the Schridinger equation for a radiator in the presence of a uniform electric

field, we can go on to calculate the dipole and quadrupole matrix elements

needed for the calculation of the spectral line shape I(w). Since we know

the wave functions only in terms of their perturbation expansions, it will be

necessary to develop perturbation expansions for the matrix elements as well.

We will follow the calculation of the atomic matrix element of the coor-

dinate z as an example. The matrix elements for x and y, as well as the

quadrupole matrix elements are calculated in the same way and only the re-

sults are presented here. Further details concerning the technique can be found

in the references of Hoe et al.64,65

We wish to calculate the atomic matrix element

( /(n1, n', m', A')|z(n1, n2, m, A)) .

The normalized wave function is given by

0 = 0(nl,n2, m,A) = 2 jr(n, A (gnmA() em (3.92)
V2vrN(n1, n2, m, A)

so the matrix element of the z coordinate of the radiator's bound electron is

1 roo oo 2r
I i dx 7)* (n, n,2, m, m'')(nl 2, m,)
(ol zd1) ( 27 A)
=-j d dT dp (2 ^2)4,*(nl ,, n', m ', X)(ni, n2, m, X)
_'n dr dJ77 ( 2 _- ,72)
8 0/0


As a notation

x fn,,'A,(l/n')fnlmA(l/n)gn's,:,(i/n')g,,,mA( /n)

a =s iN(n'1nilm'')N(nln2mA) .

nal simplification, take

ZninM ( |z) .

In order to evaluate b, consider

AP = N(n', n'2, m', ')N(n,n2, m,A)


= N1 "n'3)
p'>0 p>0
= C EQ QpF '+P

E QiQp-iFP
p>0 i=0


We have defined

Qp -







WVp ZQ Qp-i .


This gives us an expansion for N in powers of the field F. Next, we use this

to obtain the expansion of $. Since 4 = vl/, we have


If we expand 4 and AN in powers of A and multiply it out, we obtain

P iFi F = F NpFP
p>0 i=O p>O

A/= pZip-i
= Eip-i + 240P*

Note that p = 0 gives

0 = v/= No

Solving for 4p gives

for p 1 ,

Ap E&1 ^p-i
pD P=1 i P-i
P- 2n t
or in terms of the more basic quantities

p p-1
P (n13)(n3)-N'iNpi E ip-
i=0 i=1






P = n2n2
1"n n

for p > 1. For the p = 0 case, we have

0 = t22 (3.105)
n/ In/ I
To proceed with the evaluation of Zn2m return to Eq. (3.93). We can
separate the functions of the two coordinates to obtain

Z 2' = F(2)G() F(0)G(2)} (3.106)


F(a) d (ifnimA,(/nflm)fnImA( /n) (3.107)


G(a) j d .Trgnm'A.,(7/n')gng2mn(l/n). (3.108)

The orthogonality condition will not apply since we are dealing with integrals
of the form fo" d( f'((/n')f((/n) and in general n' 5 n.

To evaluate Eqs. (3.107) and (3.108), we expand their left-hand sides in
powers of F to obtain

F(a) = F"a) FP
p>G. (3.109)
G(o) = G( ) FP
Expanding the right-hand sides of the integrals and rearranging as a series in
increasing powers of F allows us to equate terms of like powers in F from both
sides to obtain

F( ) E 1 '3'n"( -' -.
Fa) h ni3in3(p-i) a i ak
h,k i=0

) ) 3i3(p-i) bbk
=4P E k,h n
h,k i=0
where the sums over h and k are as before, and where

,(a =h d "u(ni + k, m, /n) u(n'l + h, m', /n') ,

au(2 + km, ) u( +) h, /)
kh =- dr q"u(n2 + k, m, n/n) u(n' + h, m', r)/n) .




Ih can be evaluated from the definition of u(nI, m, ?'/n) to obtain

I(l) ( 2nn' ~)+1
kh = +'n)

J(nl + k)!(n1 + k + m)!(n' + h)('1 + h + m)!

2n m' nx+k nI+h
q=O s=O


q --2n


(1 + -T- +q+s)!
(m + q)!(n1 + k q)!q!(m' + s)!(n' + h s)!s!

For J( just take n1 -+ n2 in Eq. (3.112).
To evaluate 'nn2 we use the expansions for F(a) and G(a). We obtain

T n' nm'
nin2lm -I-



( +2n' '
n + n/


Tln:m = T TpFP, (3.114)

Tp 1F, (2)G(. o 2'.) (3.115)
We can also introduce the expansion

Zn n2m = Z F Z P. (3.116)
Rewriting Eq. (3.113) as

Tfim' _y- 4Zfi 1 -2
Tn1n2m n2m (3.117)

and using the expansions for each side gives

Tp= ZiP-i. (3.118)
Solving this for Zp gives

Zp= T- p Zi0p-i (3.119)

If p = 0 we take the sum to be zero. This expression gives the terms in the

expansion for (O'l z \1i) given by Eq. (3.116). We have therefore successfully

calculated the perturbation expansion for the atomic matrix element of the

radiator's z coordinate.

We have followed the calculation of the z matrix element in detail in order

to demonstrate the perturbation theory method. For the remaining matrix

elements we present only the final results. The matrix elements for x and y

are calculated by the same procedure except for the angular integrals which
we evaluate explicitly. For x = v/7 cos V we have

f27 dqp cos ve-i(m'-m)p = 2 r
0 2 [ + '(3.120)
2x (A)
2 m,m'
For y = V/ sin W we have

2 dp sin e-i(m'-m)" = [m+1,m -l,m]
S2i m, m l (3.121)
27A (-)
i2 m,m *
The factor of 27r will cancel with a similar factor from the wave function nor-

The x matrix element is given by

Xnm = XpF, (3.122)

Xp = Rp i- Xip- (3.123)

S- a (3/2)G(1/2) + 1/2)G(3/2) (3.124)
The y matrix element is given by

Y-m = -i Yp FP, (3.125)

Yp i (3.126)


Kp m,m 1 +/2)G (3.127)
P p-i p-

Quadrupole Matrix Elements
The last atomic matrix element we need is the quadrupole moment tensor
component Qzz. Its operator in terms of radiator electron coordinates is given
by (taking e = 1)

zz = (3z2 r2)
1 2 (3.128)
= 2 2+ 2 4r).
The matrix elements of Qzz are then given by

(Qzzn = EQpFP, (3.129)

O S, EYip QOQ p-i
S-= (3.130)


Sp = m,m (F )G(O) + F(O)G 3F)G(1) 3F(I)G-)i
The calculation of Qzz goes beyond the work of Hoe et al64,65 but the method
is the same and is as outlined above in the calculation of the z matrix element.

For the perturbation theory calculations in this work, we will go to sixth

order, taking p = 6. This will be more than sufficient for our purposes.

3.3 Atomic Data by Numerical Solution

Next, we give a qualitative discussion of the numerical solution of Schr6dinger's

equation for a hydrogenic radiator in a uniform electric field. We have already

developed the perturbation solution useful for relatively small field values. Di-

rect numerical solution is applicable, in principal, for fields of any magnitude,

though, in practice, it turns out to be inconvenient for very small field val-

ues because it is difficult to numerically handle the resultant extremely sharp

resonances. We will discuss this point further below. Consequently, the pertur-

bation and numerical solution techniques are complementary and additionally

should serve as consistency checks since they should produce matching results

for a suitable intermediate range of fields. For large field values, the resonance

nature of the radiator states becomes an essential part of their description and

the perturbation theory used here is no longer useful for finding the field depen-

dent eigenvalues and eigenfunctions of Schrodinger's equation. We also discuss

the calculation of the width of these resonances and the numerical evaluation

of the necessary atomic matrix elements.

The presence of the uniform field at the radiator changes the very nature

of bound atomic states by turning each discrete energy level into a shape reso-

nance; in other words, there will be a solution for a particular set of quantum

numbers that includes a continuum of energy values.66,67 For a field strength

given by A, the states can be characterized uniquely by the quantum numbers

ml, A and the energy E. The quantum numbers mi and A remain discrete

but E now has a continuous spectrum. For weak field values, the density of

states is greatest for a radiator electron with an energy that closely corre-

sponds to the discrete states given by the perturbation theory solution to the

problem. As the field value increases, however, the density of states broad-

ens and the probability of finding the electron with an energy more widely

spaced from the resonance center increases. This is a result of the field in-

duced broadening of the resonance. The width, F, of the resonance is also

roughly inversely proportional to the lifetime of the electron inside the radia-

tor. This is most easily seen from the time-energy uncertainty principle but

also follows from a detailed WKB treatment of resonance decay.68 In fact, the

WKB treatment68,69 of the field ionization problem leads to a probability of

finding the quasibound electron inside the potential barrier given by the time

dependence factor e-t. Thus, when t = 1/P the probability has decreased

from one at t = 0 to 1/e = 0.37. The presence of the field lowers the Coulomb

potential along the upfield direction thus producing a potential well defined by

a finite-size potential barrier. The electron is then able to tunnel out of the

radiator. Indeed, in the presence of the field, there are now no truly bound

states in the sense of the electron being permanently associated with a partic-

ular atom baring radiative or collisional ionization. However, if the lifetime of

the electron inside the radiator potential barrier is long compared to the time

of interest for the radiator, it is effectively bound to the radiator. For even

larger field values, the plasma electric field potential will exceed the attractive

central Coulomb potential; the energy level will be above the top of the poten-

tial barrier. In this case, if the energy level is close to the top of the barrier,

the electron can still spend a significant amount of time in the vicinity of the

radiator nucleus and will retain some of the character of a bound state. As the

field is increased further this bound character will gradually be reduced. This

phenomenon results in the smooth broadening of the resonances as the field

increases until they overlap. At that point the resonances have merged into a

relatively smooth background continuum.

We need to numerically solve the system of coupled second order ordinary

differential equations (ODE's) given by Eqs. (3.60) and (3.61). In order to em-

ploy the solutions of these equations for the calculation of the matrix elements

and line shapes, and since the eigenvalues are not discrete, we will use the

Breit-Wigner62 Lorentzian parameterization of resonances and take the eigen-

functions and eigenvalues at the center of each resonance as a representative

value over the entire resonance. As long as the resonance is fairly narrow and

remains distinct, this is a reasonable approximation. It breaks down for strong

fields where the resonance becomes strongly asymmetric. Fig. (3.3) gives an

example of this phenomenon. The reason for this asymmetry lies in the shape

of the potential barrier; as the energy level goes higher it sees a thiner potential

barrier. The thiner the potential barrier, the shorter the lifetime of the state

and the greater the uncertainty in its energy. Consequently, the broader width

on the high energy side of the resonance is due to the greater uncertainty in

the energy value at that point. These resonance asymmetries have been inves-

tigated experimentally by Harmin.70 We do not need to describe this behavior

in detail since its contribution to the line shape is greatly attenuated due to

the low probability of occurrence of the relevant high field values as reflected

in a small value of the microfield probability function.

We employ the numerical solutions of Eqs. (3.60) and (3.61) by R. Mancini

and C. Hooper71 who follow the method of E. Luc-Koenig and A. Bachelier.66,67

We give here a general qualitative procedural description of the solution tech-

nique (see the above references for further details). The solutions for the equa-

tion in the scaled variable x are effectively bound and, therefore for large x

are exponentially damped. The number of nodes in the wavefunction's x com-

ponent is given by the parabolic quantum number nl. The solutions for the

equation in the scaled variable y are effectively unbound; that is, the solutions

for large y outside the potential well are oscillatory in nature. On the other

hand, for x and y -- 0 we have f(0) = g(0) = 0.

To solve the the two equations, Eq. (3.60) and Eq. (3.61) consistently,

first, we solve the effectively bound equation in x for a particular case given

by specific values of F, E, nl and ml. This will give a value for the constant

A by imposing the exponentially damped behavior of F(x) for large x (this is

an eigenvalue problem for A). We use this to determine the constant B and

proceed with the solution of the effectively unbound equation in y. We change

Eqs. (3.60) and (3.61) into a form more convenient for numerical solution by

eliminating the first order derivative with the transformations

'(xZ) = V f() (3.132)


g(y) = vg(y) (3.133)

Our two equations will then be transform to

d2 ( + Tz() =0 (3.134)


d2 g(y)
dy2 + TyG(y) = 0 ,(3.135)


A n2 m2 -1
T = -+ E Ax (3.136)
x 2 4x2

B n2 m2 -1
Ty = E-2 +Ay. (3.137)
y 2 4y2
The motivation for making this transformation is that now we have to deal

with second order ODE's with missing first derivatives which can be efficiently

integrated using the Numerov algorithm.

To solve Eq. (3.134) for an energy value E, a scaled field value A, a value

for the z component of the orbital angular momentum ml and a value for

the parabolic quantum number nl, we pick a trial value for A. We solve the

differential equation starting at x = 0 using a power series expansion; this

is used to initialize an outward numerical integration using a fourth order

Numerov algorithm.66,72 This numerical integration is continued up to the

outer turning point of the effective potential barrier. (If the energy level is

above the top of the potential barrier so that there is no outward turning

point, we use the maximum of the potential barrier.) For suitable large x

values, an exponentially damped asymptotic solution is used to begin an inward

numerical integration, again using the Numerov method. The two solutions

overlap at the outer turning point of the potential barrier (or the potential

barrier maximum if the energy level is above it). At the meeting point, we

match F(x) by adjusting a multiplicative constant, and from the matching

condition for d.(x)/dx we compute a correction to our initial guess for A.

This defines an iterative procedure which is continued until the initial guess

for A agrees within a given tolerance with the value found from matching the


To solve Eq. (3.135), the unbound equation in the scaled variable y, we

note that once we have the value of the constant A we have B = n A. This

gives us E, A, nl, ml, A and B; therefore, it is not necessary to make any initial

guesses. We start the outward integration as we have done for the variable x

using a power series solution, and then switch to the numerical integration

using the Numerov method as before. Since the constant B has already been

determined, we continue the integration outward to large y beyond the effective

potential barrier where we fit a large-y analytic oscillatory asymptotic solution

to the numerical results. This will give the value of the amplitude of g(y) for

large y. By studying the distribution of amplitudes with respect to the energy,

the width and center-location of the resonance can be estimated.

Since the values of the energy can have a continuous spectrum, we have

the normalization condition per unit energy given by

d dpj df d77 ( + r) m*,A,E Cm,A,,E = mmIA,A,6(E E') .

it can be shown62 this is equivalent to the condition Iout = 1/2r, where Iout

is the outward flux in the r7 direction at large 77. The outward flux lout can

be given in terms of the outgoing wave ,out where we have decomposed the

stationary wavefunctions into outgoing and incoming components in the r di-



The density of states can be obtained by studying the behavior of the

electron probability density near the nucleus for the component of the wave-

function associated with a constant value of the outward flux. As stated before,

we can approximate the density of states for a particular resonance using the

Breit-Wigner parameterization.62 We obtain

D ( (E) = m (Er) E ) (3.139)
n 1 n i (E Er)2 + 'ir2

where Er corresponds to the energy value at the maximum of the density of

states of the resonance, and F is the full resonance width at half maximum.

This is a reasonable parameterization as long as the resonance remains distinct.

The average lifetime rres, of the resonance can be associated with the resonance

width through65

Tres ; h/F (3.140)

This is just a manifestation of the time-energy uncertainty principal.

Once we have numerically evaluated F(x) and g(y), we are free to calculate

the matrix elements needed for the line shape. By straight forward numerical

integration we evaluate matrix elements for x, z and Qzz. We cut-off the in-

tegration over the scaled variable y at the outer turning point of the effective

potential barrier for energy values below the maximum of the potential bar-

rier. (For energies greater then the maximum of the potential barrier, we stop

the integration at the value of y corresponding to the maximum of the poten-

tial barrier.) Due to their resonance nature, the wavefunctions extend overall

space but we are only interested in the portion located near the radiator. The

numerical values of the matrix elements are not significantly sensitive to small

changes in the location of this integration cut-off. Since we now have properly

normalized wavefunctions, we can evaluate the field dependent matrix element
of a real operator 0 between two resonance states denoted by subscripts "i"
and "j" by integrating over the energy range of each resonance. The square of
the matrix element for an operator 0 can be written as,

IOi,j(F)I = dEi dEj I ( EF) ((Ei, F)1)I2 (3.141)

For narrow resonances we can approximate this results by

O i ( ) 2 (2)2 (i (Er,i,F) O1 j (Er,j,F)) (3.142)

where we have used the value of the eigenfunctions at resonance center. This
is a reasonable approximation because the main field dependence of the wave
functions, for narrow resonances, is contained in the normalization factor which
is a function of the square root of the density of states. Our evaluation of the
square of the matrix element allows us to integrate over the density of states
to obtain the 7r7i/2 factors. Since these factors are always positive and real,
the sign (and any factors of i) of Oij is reliably given by the evaluation of

(4i(Ei,F) 0 ICj(Ej,F)) at Ei = Er,i. Fig. 3.4 gives an example of the
calculation of atomic matrix elements by both perturbation theory and direct
numerical solution of Schr6dinger's equation. We see that the two solutions
join smoothly for intermediate field values and diverge for larger field values
as the accuracy of the perturbation theory solution breaks down.
As the field F goes to zero, Eq. (3.141) will reduce to the usual no field
limit where

O,j(0) = ( 0 ) (3.143)


i = u(ni, ml, )u(n2, m, ) (3.144)

which, as expected, is just the solution to Schr6dinger's equation for the field

free case. Here, the state "i" is specified by E, nl, ml and F. Since for

a solution with a discrete values of E there is a unique value of the principal

quantum number n, we can obtain n2 through the relation n = nl+n2+(ml+1

and our limiting state can be characterized by ni, n2 and ml as expected.

3.4 Electron Delocalization and Field Ionization

We have been using, as a model for the radiator-plasma interaction, the

picture of a radiator in the presence of a uniform plasma microfield produced

by the averaging of the fields from all the plasma particles. We have added

the ion-quadrupole correction to account for a relatively small field gradient at

the radiator. This picture is only a model, of course. As the plasma density

increases, it becomes unreasonable to ignore the presence of highly charged

nuclei close to the radiator. This fact creates a qualitatively different picture.

In the uniform field case, a quasibound electron will tunnel out of the radiator

and continue to accelerate toward the field source at infinity. In reality, large

fields are produced primarily by nearby charges. The quasibound electron

will then orbit about the ensuing multicenter ionic potential in a molecular

orbital74 instead of tunneling into the interparticle space; the electrons will

become delocalized. A complication to this molecular orbital picture is the

large number of free plasma electrons in the internuclear volume. It is unrea-

sonable to assume that the quasibound electrons will not strongly interact with

these free electrons as they resonate between various nuclei. Perhaps a more

accurate picture for these conditions would be the self consistent treatment of


both bound and free electrons in the presence of the multicenter potentials.

Some work has been done along these lines from an analytic point of view by

Rogers,75 and from a numerical simulation point of view by Younger et al.76

Additionally, a model consisting of a cluster of nuclei with bound electrons has

been considered by Collins and Merts77 but for Te = 0. Much work remains

to be done before these models are capable of accurately treating spectral line

transitions for plasmas of high density and temperature.

In addition to the question of whether field induced ionization or resonance

exchange of electrons between nuclei is a better description of what strong

fields do to quasibound electrons, there is the question of the level broadening

produced by these two mechanisms. The field ionization picture contains level

broadening due to the resonance character of the energy levels in the presence of

the nearly uniform electric field. This is most easily seen from the uncertainty

principle where the width is related to the resonance lifetime by AE hi/rres.

In the molecular orbital picture, an electron bound to two nuclei will still have

a discrete energy. However, as the number of nuclei in the molecule or cluster

increases, splitting is induced. If each level is m-fold degenerate and there

are N nuclei in the cluster, the splitting will be on the order of mN-fold.77

This large degree of splitting will be qualitatively similar to the intrinsic level

broadening produced in the field ionization case. We see, therefore, that the

level broadening from the two different models should produce similar results

under similar conditions, at least as far as the level width is concerned.

At this point, it appears that the molecular orbital picture may be the

most capable of describing our dense plasma since it is a truly many-body

theory. However, there is the complicating factor of the intercollision of free

and bound electrons. As the bound electrons exchange between the nuclei,


they will be subject to collisions with the free plasma electrons. If they are

then thermalized into the plasma, the situation will be more like the field

ionization case. A realistic picture probably lies somewhere between the two

over simplified models.

In the molecular orbital model, radiator levels are not depopulated. On the

other hand, in the field ionization model, all levels will be depopulated on time

scales relevant to radiative decay, for sufficiently strong fields. If the actual

phenomenon lies somewhere between these two pictures, then the resultant

effect will range somewhere between no change in the line shape due to no

level depopulation, to a substantial decrease in line radiation for large fields

due to level depletion from field ionization. The levels will not be completely

depopulated, of course, due to collisional and radiative repopulation. What

effect could these two models have on the spectral line shape?

In order to estimate the importance of this possible ionization effect on

level populations, we will examine the limiting case of field ionization due to

tunneling with no reverse process to repopulate the radiator level except the

usual collisional and radiative processes. We will construct a simple kinetic

population model to describe the influence of field induced ionization on the

relative population of the energy levels of hydrogen-like argon. We will present

only the extreme case of no back tunneling in order to ascertain the maximum

possible population depletion. As outlined above, the real situation could

lead to results ranging somewhere from no tunneling depletion at all (the pure

molecular orbital picture) to maximum depletion due to one way tunneling (the

field ionization picture). We will not look for departures from absolute LTE

populations but only relative departures from LTE among the upper states


involved in the spectral transitions; relative population differences are all we

will be able to observe from experimental line spectra.

We will examine the relative distribution of populations of the n = 3 and

4 levels of hydrogenic argon. The model includes connections between the

hydrogenic excited state levels to the ground state and the fully stripped ion

while omitting the n = 2 levels which are little affected by field ionization. At

the high densities we are concerned with in this work, collisional processes will

dominate the other population altering processes and relative LTE populations

will be very probable if we do not include the field ionization. We consider only

these few levels because we are mainly interested in the question of whether the

field ionization effect can depopulate the levels enough to produce a significant

change in the line shape.

We will incorporate into the kinetic model collisional excitation, collisional

de-excitation, collisional ionization, three body recombination, spontaneous

emission, radiative recombination and field ionization. We can easily formulate

a rate equation for the population Ni of a level denoted by the subscript "i".

The number of excited quasibound levels is denoted by N, the fully stripped ion

by the subscript "N+1" and the ground state level by "O". Our rate equation


dNi i-1 N+1
d~ = Nj neCi + Ej Nj neC d
j=0 j=i+l
N+1 i-1
E NineCj NineCf (3.145)
j=i+l j=0
N+1 i-1
+ E NjA,i E NiAi Niri
j=i+1 j=0

for i = 1 to N + 1. Note that since the fully stripped ion has no electron to

escape, FN+1 = 0. We will take the populations to be relative to the ground

state population, No. This will give Nj -* Nj/No and No -- 1 for j = 1 to

N +1. Since we are only interested in the steady state solution to these kinetic

equations, we examine the case with

d = 0, (3.146)
for all i. Next, we define the rate coefficients in Eq. (3.145):

The collisional excitation rate is given by Cfj, where78

Cf = 3.75 x 10-5 T-1/21 (i Ij) 2 e-AEi,/kT cm3 sec-1 (3.147)

Here, T is in degrees K, AEi,j is the energy difference between levels i and j,

and (il F ij) is the atomic matrix element of r in atomic units.

The collisional de-excitation rate is given by Cdj. This can be related to
the collisional excitation rate by the principle of detail balance.79 This gives

Cd. = Cf e-A j/kT (3.148)

for states of equal statistical weight.

The collisional ionization rate is given by COf e where80

N+1 = 3.46 x 105 1306) T-/2 e- {1 xi eIEl(xi)} cm3 sec-1

Here, T is in degrees K, Ii is the ionization potential of level i in eV and

xi = li/kT ,



loo e-xt
Ei(x) = dt (3.151)

is the first order exponential integral.

The three body recombination rate is given by CfN+,i. This can be related

to the collisional ionization rate by the principle of detail balance.

The spontaneous emission rate is given by the Einstein coefficient Aij,


Aij = 2.142 x 1010(AEi,j)3 (i Ij) 12 sec-1 (3.152)

Here, AEi, and (ij r' jj) are in atomic units.

The radiative recombination rate is given by AN+1,i, where81

AN+l,i = 5.197 x 10-14 neZx3/2 eZEl(xi) sec-1 (3.153)

Here, Z is the radiator charge, ne is the plasma electron number density given

in cm-3 and zx is given by Eq. (3.150).

For the steady state case and because we have taken No = 1, we can write

all of the N + 1 rate equations in the form

ao + alN1 + a2N2 + .' + aNNN + aN+1NN+1 = 0, (3.154)

where the coefficients ai do not depend on the level populations. These equa-

tions may be cast in matrix form to obtain

C=RN .



Here, C is a column vector of dimension N + 1 containing the a0 rate coeffi-

cients, N is a column vector of dimension N+1 containing the level populations

Ni and R is a N + 1 x N + 1 matrix containing the remaining rate constants.

This equation is easily solved to give the N + 1 relative populations Ni. We


N = R-C (3.156)

This effectively solves the relative populations problem for the N quasi-

bound excited state levels plus the the fully stripped ion. For the total number

of quasibound levels, we use the n = 3 and n = 4 levels of a hydrogen-like

ion. This gives a total of N = 32 + 42 = 25 levels. The effect of the lack of

inclusion of the n = 2 levels was examined by comparison with a more detailed

kinetic population model.82 This showed no significant change in the relative

populations upon inclusion of these levels.



-0.8 / -
/ /

/ _-
)- / -
S-0.6 /
: /

-0.2 -
0 -/ /

0 0.5 1 1.5 2

FIGURE (3.1) Field gradient term (Ezz)c as a function of the field for the
APEX (- ), IP ( ...... ) and NN (- -) models at ne = 1 x 1024 cm-3 and
kT = 800 eV. The field gradient is in units of e/r3,i. The three models are
discussed at the end of section 3.1.








-.** /

0 1 2 3 4 5 6


FIGURE (3.2) Approach of the constrained average field gradient term
(Ezz)e to the nearest neighbor limit for increasing density. (- -) refers to
ne = 1 x 1024 cm-3, (- -) refers to ne = 5 x 1024 cm-3, (......) refers to
ne = 1 x 1025 cm-3 and (- ) refers to the nearest neighbor model discussed
in the text. All temperatures are kT = 800 eV. The constrained average is
evaluated using the APEX model given by Eq. (3.50).




rzl -3.75

-4.00- ""


3 4 5 6 7

FIGURE (3.3) Example of the asymmetry of a resonate state discussed
in section 3.3. This is the resonance with quantum numbers n,nl,n2,m =
4,0,2,1 or n, q, m = 4, -2, 1. The resonance center is given by (- ) and the
width at half maximum is given by (- -). The field strength F is in units
of a.u./1 x 10-4 and the energy E is in units of a.u./1 x 10-2. a.u. denotes
atomic units (see Appendix A).

150 ,

125 -

o 75-

C -
S50 -


0 JI l I"I I I I "I II I I
0 0.0005 0.001 0.0015 0.002 0.0025
F (a.u.)

FIGURE (3.4) Comparison of numerical and perturbation theory results
for the calculation of a quadrupole atomic matrix element. The example given
here is a diagonal matrix element with nI = 2, n2 = 0 and m = 0. The
numerical solution as discussed in section 3.3 is given by (-) and the sixth
order perturbation theory solution calculated from Eq. (3.129) is given by
(......). All quantities are expressed in atomic units scaled for radiator charge,


In the previous chapters we have formulated a general theory of plasma

spectral line broadening and discussed many of the approximations used to

arrive at a calculable result. The theory that we use to make these calculations

has been generalized beyond previous formulations in that it includes higher-

order microfield effects on: atomic matrix elements, radiator energy levels,

state lifetimes, and level populations. We will now present and discuss the

results of these calculations.

The presence of these higher order field effects can be expected to lead to

several discernable changes in the spectral line shapes of radiators in dense

plasmas. For the physical conditions we are examining the electron number

density, ne, varies from 1 x 1024 to 1 x 1025 cm-3 and at a temperature, kT,

of 800 eV. In this range, the lowest order corrections of the ion-quadrupole

and the quadratic Stark effects will generally give rise to a blue asymmetry of

the spectral line shape. This means that the intensity of the high energy (or

blue) side of the spectral line will be enhanced over that of the low energy (or

red) side of the line. This comes about by the preferential shifting to lower

energy of components comprising the manifold of energy levels associated with

a principal quantum number n. If all of the components were shifted by an

equal amount, there would be no discernable change in the spectral line shape;

only an overall line shift. The n = 1 ground state for these transitions in

highly ionized hydrogenic ions is little affected by the field because of the

much stronger binding potential for the n = 1 state. Additionally, the ground



state experiences no linear Stark effect. Energy levels that have been Stark

shifted to the low energy side of the unperturbed upper state of the transition

correspond to quasibound electrons that are found to have the maximum value

of their probability amplitude on the upfield side of the radiator potential

well. This corresponds to the side of the origin where the potential barrier has

a maximum; the other side, the downfield side, corresponds to the potential

barrier increasing without bound. The upfield electrons are more easily affected

by the field and hence, their energy levels are shifted more. This preferential

shifting produces a spreading out of intensity on the low energy side of the line,

and thus, an increase in the peak height of the high energy side of the line.

The addition of the ion-quadrupole effect enhances this trend but also generally

produces a bunching, or lessening of the splitting amongst the energy levels

on the blue side. This further adds to the blue asymmetry of the line. The

consequences of the field dependence in the wave functions and their resulting

matrix elements are harder to characterize due to less systematic results on the

line shape. Therefore we study this effect only in combination with the other

field effects. In this dissertation, we assume that the areas of all line shapes

are normalized to one.

The possible field ionization depletion of the radiator upper level popula-

tions will lead to a lessening of intensity in the line wings. Again, the red wing

should be more strongly affected due the lower potential barrier height on the

upfield side of the potential well. For sufficiently strong fields, The Stark effect

will cause the energy levels from states with adjacent principal quantum num-

bers to overlap. Inclusion of this phenomenon is important for the accurate

representation of line merging.


4.1 The Lyman a Line
The La calculation contains the effect of the microfield on the atomic

physics through field dependent matrix elements and energy levels. This gives

rise to a field dependent fine structure correction to the radiator Hamiltonian

(see Appendix E). The field dependence appears in the dipole matrix elements,

and because we include the ion-quadrupole effect, also in the quadrupole matrix

elements. We do not study the effect of possible field ionization on the levels of

the La transition because, for this case, the resonance width r < 0.1 eV. Since

for this transition the resonance nature of the states is not important, perturba-

tion theory is adequate for the calculation of the atomic physics. At the lower

end of the density-temperature range we are interested in (ne = 1 x 1024 cm-3,

kT = 800eV), the Doppler effect is an important source of broadening so

its inclusion is essential. For the densities and temperatures we are dealing

with, the n = 2 and 3 levels are well separated in energy and do not overlap

for any relevant field strengths. Consequently, we will not need to consider

this overlap for the La line calculation. In Fig. (4.1) we have the La line at

n, = 1 x 1024 cm-3 and kT = 800 eV. The higher order field effects produce

only a slight blue asymmetry in the line which attenuates the red peak by

less than 5% in magnitude. The fine structure splitting is readily apparent

with Doppler broadening being responsible for more than half the width of

the peaks. We estimate the Doppler width from34 hwD hwi,jkT/mrc2,

where hwij is the transition energy and mr is the radiator mass. In this case

hWD ~ 3 eV. As the density goes up the Stark broadening will increase and the

Doppler broadening will become less important as its fraction of the total width

decreases. In Fig. (4.2) the density has increased to ne = 5 x 1024 cm-3; the

Stark broadening has increased and begun to obscure the fine structure split-

ting. However, the higher order field effects are still of only minor importance,

accounting for changes in peak heights of less than 4%. For ne = 1 x 1025 cm-3

there is still no significant attenuation of the red peak due to the higher or-

der effects. We conclude that, for the density range of this work, the plasma

microfield is not strong enough for these higher order effects to become impor-

tant. As the average plasma microfield strength continues to increase due to

increasing density, it will eventually become of the same order as the central

Coulomb field. At that point the higher order effects will be much more likely

to be manifest. If we go much beyond 1 x 1025 cm-3, however, the electron

degeneracy will have to be incorporated into the theory as we pointed out in

section 2.1.

4.2 The Lyman 0 Line
The L# line has its excited state electron in the n = 3 state. This makes

it much more susceptible to the effect of the plasma microfield than in the

La case. To investigate the higher order effects on this line, we will include

the ion-quadrupole effect, field dependent atomic physics, the presence of a

slight resonance width, F, for each energy level, and the overlap of the levels

of principal quantum number n = 3 and 4.

This last process allows for atomic matrix elements connecting the two

principal quantum number manifolds. It also contributes more terms to the

sum over intermediate states that is performed when evaluating the electron

broadening operator M(Aw). In the field-free atomic physics calculation, this

is also possible but the energy levels are widely separated at their unperturbed

energy values. Consequently, their contribution to the sum over intermedi-

ate states will be minimal. The decreasing value for large Aw of the electron

broadening many-body function G(Aw) will attenuate each added term if the

energy separation between levels is greater than the plasma frequency Wp,e.

In the field dependent atomic physics picture, the zero-order energy levels are

perturbed by the Stark effect and can overlap. When this occurs, the G(Aw)

function takes its maximum value for each intermediate state and the total

electron broadening term will be larger. In general, the red and blue levels

of each principal quantum number manifold interact most strongly with each

other, while the red levels of one principal quantum number and the blue lev-

els corresponding to another interact hardly at all.61 This quasi-selection rule

causes the electron broadening of the n = 3 level to be most influenced by the

red levels of the n = 4 level. This mechanism contributes a further blue asym-

metry to the Lp line by way of a broader red wing caused by this additional

electron broadening. These electron broadening effects are illustrated in Fig.

(4.4) where we show the Lp line at ne = 1 x 1025 cm-3 and kT = 800 eV for

the case of full field dependent atomic physics plus the ion-quadrupole effect.

This is compared to the same line but with all connections to the n = 4 levels

excluded from the electron broadening of the n = 3 to 1 transition. This is

accomplished by limiting the sum over intermediate states in the evaluation

of M(Aw) to dipole matrix elements connecting n = 3 levels only with other

n = 3 levels. The sums for the n = 4 to 1 transition, however, contain all n = 3

and 4 levels. The n = 3 to 1 line with the normal sum over all n = 3 and 4

intermediate states is approximately 10% broader at half maximum intensity

than the line with the restriction on the electron broadening sum.