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A study of stark broadening of high-Z hydrogenic ion lines in dense hot plasmas

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A study of stark broadening of high-Z hydrogenic ion lines in dense hot plasmas
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Tighe, Richard Joseph, 1946-
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vi, 250 leaves : ill. ; 28 cm.

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Coordinate systems ( jstor )
Distribution functions ( jstor )
Electric fields ( jstor )
Electrons ( jstor )
Ion density ( jstor )
Ions ( jstor )
Plasmas ( jstor )
Radiators ( jstor )
Dissertations, Academic -- Physics -- UF
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Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 247-249).
Additional Physical Form:
Also available online.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Richard Joseph Tighe.

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A STUDY OF STARK BROADENING OF HIGH-Z HYDROGENIC ION
LINES IN DENSE HOT PLASMAS













By
RICHARD JOSEPH TIGHE





















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA
1977















ACKNOWLEDGMENTS


I would like to gratefully acknowledge the support of an NSF

Traineeship during the first three years of my graduate study.

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

this problem and for his guidance and encouragement during the course

of this work. Also I would like to thank Drs. J. W. Dufty, T. W.

Hussey, and F. E. Riewe for many helpful discussions. A special

thanks is due Dr. Robert L. Coldwell for providing guidance in the

numerical work as well as for lending me several excellent computer

codes.

I would like to thank Mrs. Yvonne Dixon for typing the final

manuscript, and Mr. Woody Richardson for preparing the figures.

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

for the special understanding they have shown during the long years

of this work.




















ii
rv
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS ................................................. ii

ABSTRACT .......................................................... v

SECTION I. LINE SHAPE FORMALISM ................................ 1

Introduction ................................................. 1
Causes of Line Broadening .................................. 2
The Stark Effect for Hydrogenic Ions ....................... 5
A Model for the Plasma ............... ...................... 7
The Line Shape Expression .................................. 9
Time Scales and the Line Broadening Problem ................ 11
Factorization of the Initial Density Operator .............. 13
The Line Shape in the Quasi-static Ion Approximation ....... 16
The Liouville Representation .................................... 19
A Perturbation Expansion for Rr(w) ......................... 20
A Computational Form for H()) .............................. 23
The Line Shape Formula .......... ........................... 24

SECTION II. ELECTRIC MICROFIELD PROBABILITY DISTRIBUTION
FUNCTION ......................................... 28

Introduction ................................................ 28
The Formal Calculation of T(2) ............................. 30
Introduction of Collective Coordinates ..................... 33
The Collective Coordinate Calculation ...................... 36
Asymptotic Microfield Distribution Function ................ 51

SECTION III. DISCUSSION OF THE RESULTS ......................... 54

Introduction ............................................... 54
Electric Microfield Distribution Functions ................ 55
Stark Broadened Line Profiles .............................. 85
Validity Criteria for this Theory .......................... 116

SECTION IV. CONCLUDING REMARKS ................................. 119

APPENDICES .......... ............................................. 121

A. THE INTERACTION V r .................................... 122
er


iii








page

B. THE ALGEBRA OF TETRADIC OPERATORS ....................... 124

C. QUANTUM MECHANICAL PERTURBER AVERAGES .................. 130

D. THE PARABOLIC REPRESENTATION ........................... 140

E. CALCULATION OF RADIATOR DIPOLE MATRIX ELEMENTS ......... 142

F. THE MANY-PARTICLE FUNCTION F(A) ...................... 146

G. A COMPUTATIONAL FORM FOR THE ATOMIC FACTOR ............. 150

H. NUMERICAL PROCEDURES ................................... 154

I. TABLES OF ELECTRIC MICROFIELD DISTRIBUTION FUNCTIONS ... 167

J. TABLES OF STARK BROADENED LINE PROFILES ................ 220

LIST OF REFERENCES ............................................... 247

BIOGRAPHICAL SKETCH ................................................ 250



































iv








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


A STUDY OF STARK BROADENING OF HIGH-Z HYDROGENIC ION LINES
IN DENSE HOT PLASMAS

By

Richard Joseph Tighe

June 1977

Chairman: Dr. C. F. Hooper, Jr.
Major Department: Physics

Stark broadened x-ray line profiles from highly ionized hydrogenic

ions, radiating while immersed in a hot dense plasma, are studied. The

broadening effects produced by the ions present in the plasma are

treated through the use of static electric microfield distribution

functions. The microfield distribution functions employed have the

following properties: (i) the radiating hydrogenic ion may have any

net charge; (ii) the perturbing ions and electrons in the plasma may

have different kinetic temperatures; (iii) two species of ions, of

charge zl and z2, may be present in any given ratio.

Electron broadening effects are treated by the Second-Order

Relaxation Theory as developed by O'Brien and Hooper. Effects upon the

electron perturbers due to the fact that the radiator is highly charged

are included through the use of Coulomb wave functions for the

perturbing electrons. The present work makes extensions needed for the

calculation of line profiles from higher Lyman series members. A cutoff

procedure is employed to simulate the effects due to correlations among

the electron perturbers.




v








Results are presented in graphical and tabular form for both the

electric microfield distribution functions and for the broadened x-ray

line profiles. The figures showing the microfield distribution

functions demonstrate the following effects: (i) when the value of

the radiator charge is increased, the microfield peaks are raised

and shifted to lower field values; (ii) an increase of the parameter

TR, the ratio of electron kinetic temperature to ion kinetic temperature,

causes a similar shift to lower field values; (iii) at low a values,

the microfields show a sensitivity to small concentrations of high-z

ion perturbers.

The behavior noted for the microfield distribution functions carries

over to produce similar effects in the line profiles. Although Doppler

broadening generally dominates the center of the line profiles, the

wings of the profile demonstrate clearly the behavior noted above for

the microfield distribution functions. In addition, the structure near

the line center (i.e. the shoulder in Lyman-a and the dip in Lyman-8)

shows sufficient sensitivity to the above effects so that it might be

valuable as a plasma diagnostic tool.

Directions for future research are discussed.



















vi















SECTION I


LINE SHAPE FORMALISM


Introduction


Studies of plasma-broadened spectral lines have been quite success-

ful in determining temperatures and densities for laboratory and astro-

nomical plasmas.1'2'3'4 Current investigations involving imploding

plasmas indicate that the temperature and charged-particle densities for

these plasmas will be much higher than for those discussed in the above
5,6,7,8
references.5' 8 The purpose of the present work is to extend line-

broadening theories developed previously so that they may be used to

develop an effective temperature-density diagnostic for the parameter

ranges expected in plasmas produced in the early laser-implosion

experiments.

We consider the broadening of x-ray spectral line profiles emitted

by highly-ionized hydrogenic ions immersed in hot dense plasmas. In

Section I, we outline the development of a line-shape formalism which extends
9
the work of O'Brien and Hooper to the calculation of higher Lyman series

members. Again following O'Brien and Hooper,0 we present in Section II the

calculation of electric microfield distribution functions which allow

for different ion and electron kinetic temperatures as well as for

multiply-charged ion perturbers in varying density ratios. Calculated

results for microfield distribution functions and for broadened line


1





2

profiles are presented graphically in Section III; the various approxi-

mations are also discussed. Section IV contains a summary, together

with concluding remarks.


Causes of Line Broadening


The most important mechanisms which cause broadening of spectral

line profiles are: (1) Natural broadening; (2) Doppler broadening;

and (3) Pressure broadening.

(1) In an atomic radiative process, the interaction between the

excited atom and its own radiation field gives rise to an uncertainty

in the atomic levels. This uncertainty in the excited-state energy levels

means that isolated atoms will emit lines having a finite width (referred

to as the natural width). For hydrogenic ions of nuclear charge Z, the

natural width is of order,11

2 2
S a (aZ) Z Ryd (I-1)
Nat

where a is the fine structure constant. Since (aZ)2 2 Ryd gives the

order of magnitude of the fine structure splitting for a hydrogenic ion,

this equation indicates that rNat is a factor a smaller than the fine
Nat
structure splitting. For the cases considered here, pressure broadening

effects are at least of the order of the fine structure splitting. This

means that natural broadening is at least a factor of a smaller than

pressure broadening in our cases. Consequently, we may safely neglect

natural broadening in the present work.

(2) The thermal-motion of the radiator produces another type of

broadening called Doppler broadening. This effect is due to the Doppler

shift of the radiation emitted by a moving source. If the velocity

distribution of the radiator is assumed to be Maxwellian, the resulting






3

Doppler line profile is given by



exp (W-Wo) } (1-2)
Doppler () B / 2- -

where,

2 2kT 2
S (1-3)
Mc2 o
Me2

and T is the kinetic temperature of the radiator. Here, the temperature

T is the ion kinetic temperature since the radiator is assumed to be in

equilibrium with the perturbing ions. The mass of the radiator is M;

mo is the frequency of the unperturbed transition. The Doppler half-width

is of order

2kT.
Doppler Mc2 o (1-4)



l10-3 8 Z3/2 Ryd ,(1-5)
1

where 6. is kT, in keV. From this estimate, it is easily possible to
1 1
obtain Doppler widths of the order of electron volts. In some cases,

Doppler broadening will dominate all other broadening mechanisms.

It may be shown (see Equation 8.8 of Reference (12)) that, for the

case where perturber collisions have no influence on the radiator's

trajectory (that is, the momentum transfer due to the collisions during

the time of radiation is negligible) the final Doppler-corrected line

profile is given by


I() = Stark W) Doppler -') d (1-6)


This correction is always valid when the condition, perturber mass <<

radiator mass (corresponding to electron perturbers), is met. The Quasi-

static Approximation for the ions, introduced below, allows us also to





4

neglect the influence of ion collisions on the radiator's trajectory.

Hence, the range of validity of the above decoupling of the two broaden-

ing mechanisms is the same as that of the Quasi-static Approximation for

the ions (which is well met in the present work see below).

(3) Pressure broadening effects are due to interactions between the

radiator and the particles surrounding it. These interactions remove

degeneracies and shift the radiator energy levels. The origin of the

term "pressure broadening" lies in the fact that this type of broadening

is sensitive to the density (or pressure) of the particles making up the

surrounding gas.

For highly ionized gases, Stark broadening is frequently the most

important pressure broadening mechanism. However, we may encounter

situations where Doppler and Stark broadening are equally important or

perhaps where Doppler broadening dominates.

In Stark broadening, the radiator energy levels are shifted by the

Stark effect due to the electric fields of the surrounding ions and

electrons. The radiator-perturber interaction in the dipole approximation

is given by,


V = eR'. (1-7)
Int. 1

where et is the dipole moment of the radiating ion (see also Appendix A),
4->
and 6. is the electric field at the site of the radiator due to the i-th

perturber. Stark broadening in highly-ionized gases is the primary focus

of this work.

Another type of pressure broadening that we must mention here is

Resonance broadening. This type of broadening occurs when the excited

radiator is perturbed by ground state atoms or ions which are identical

to the radiator. Also called "self-broadening," this mechanism is primarily






5

important when there is no significant ionization present.

A multipole expansion of the interaction between two neutral

hydrogen atoms gives as its lowest order nonvanishing contribution the

(quadrupole) resonance interaction [see Equation (4-98) in Reference

(1)]. When the radiators are hydrogenic ions, however, the quadrupole

term is no longer the lowest order nonvanishing contribution. The

dominant contribution to broadening between ions is produced by the Stark

effect (the dipole term) given in Equation (1-7) above. In the present

work we consider the broadening of hydrogenic-ion lines and encounter

the situation where radiator and perturber may be identical. Since

Resonance broadening results from terms higher than dipole in the multiple

expansion of the radiator-perturber interaction, we neglect it in the

course of this work. In this case, the neglect of higher terms is an

approximation whose validity improves as the value of the radiator charge

increases. Correlation effects due to the monopole interaction of two

highly charged positive ions produce an effective repulsion. This

repulsion reduces the probability that higher multipole terms will make

a contribution (i.e. the repulsion makes close collisions less likely).

In summary then, although the radiator and perturbing ions may be identical,

we here neglect the Resonance broadening and consider only the Stark

broadening interactions.


The Stark Effect for Hydrogenic Ions


In a hydrogenic ion, the interaction between the nucleus and the

single bound electron is a pure Coulomb interaction. This fact has

several important consequences. The most obvious consequence is that the

Coulomb problem can be solved exactly: we know the eigenfunctions and





6

the eigenvalues for the atomic problem. This solution displays the

well-known -degeneracy (accidental degeneracy) of the discrete eigen-

value spectrum for the pure Coulomb system. Thus we must use first-

order degenerate perturbation theory to calculate the energy level

shifts due to the linear Stark effect in the hydrogenic case. When this

calculation is carried out, we obtain the following result for the shift

of the level specified by n,q,m (for a discussion of the parabolic

representation, see Appendix D):


3 aoe
AE = nqe, (1-8)
2 Z

where a is the Bohr radius; Z is the nuclear charge; n,q are parabolic
o
quantum numbers, and E is the magnitude of the electric field.

In the general case of a non-Coulomb interaction, the -degeneracy

is absent. This means that in nonhydrogenic systems there are no level

shifts in first order due to the linear Stark effect. For these systems

the perturbation calculation must be carried to higher order with the

result that for nonhydrogenic systems, the level shifts due to the Stark

effect are smaller than in the case of hydrogenic systems. Thus hydrogenic

systems are more sensitive to pressure broadening effects through the

Stark effect than are nonhydrogenic systems. For this reason, hydrogenic

radiators are the optimum choice as emitters of broadened profiles for use

in plasma temperature-density diagnostics.

At this point, we must pause to introduce a validity criterion for

the use of the linear Stark effect in the present work. In the above

discussion we made no mention of the fine structure contribution to the

level shifts in hydrogenic ions. An assumption made throughout the present

work is that the fine structure splitting of the energy levels is negligible

compared to the Stark shifts. When this assumption breaks down, the






7

problem of computing the level shifts becomes very complex and requires

13
a numerical solution. The numerical calculation yields shifts, which

are not linear in the electric field (except, of course, in the large

field limit). By equating the Holtsmark shift with the fine structure

splitting of the upper state of the transition, we may determine a

validity criterion for the use of the linear Stark effect in line broad-
14
ening calculations. If


16 15/2 1)3/2
n > 2.4 x 101( ) (1 1)3, (1-9)
e n n

where n is the electron number density; Z is the nuclear charge of the

radiating ion; and n is the principal quantum number of the upper state,

then the linear Stark effect is appropriate for our calculations.

As we will discuss in a later section, several cases considered in

this work approach the limit of validity given above.


A Model for the Plasma


A plasma is an electrically neutral gas in which the temperature is

high enough so that some degree of ionization is present. For the purpose

of this work, a plasma is defined to be a gas which is ionized to the

extent that Stark broadening is the most important pressure broadening

mechanism.

Although normal laboratory plasmas at temperatures -10 K may have a

significant number of neutral atoms present, the work to be reported

here is aimed at treating dense, hot plasmas of laser-fusion experiments

in which case all neutrals will be strongly ionized. Therefore, our

plasma is composed of electrons and ions of various ionization stages.

The radiation we choose to study is the Lyman series of lines from the

hydrogenic ionization stages of various heavy atoms. The specific atoms






8

are selected on the basis of the predicted population densities of their

hydrogenic ionization stages.

In the formal development which follows, we employ a model in

which the plasma contains only one radiating hydrogenic ion. This model

assumes that the individual radiative processes which add together to

produce the line profile occur independently of one another. For the

present work, this is a reasonable assumption since radiative processes

in plasmas will, in general, add incoherently.15

In the above model, it may appear that we are neglecting all the

contributions from processes where two radiators interact with each

other. However, the dominant contribution from such processes, namely

the Stark broadening interaction, is contained implicitly in the treat-

ment of ion broadening through use of electric microfield probability

distribution functions (see the discussion of this procedure below).

This statement is based on the fact that in the microfield calculation,

the ion perturbers are treated as classical point particles with no

attention given to quantum problems such as particle identity or internal

structure. This means that even though a perturber may be identical to

the radiator, it produces Stark broadening effects which are independent

of its internal energy state.

An additional feature of the model is that we fix the radiator at

the origin of the coordinate system. This allows us to concentrate on

Stark broadening and add Doppler broadening at a later step. In summary,

our model consists of the single hydrogenic radiator fixed at the origin,

surrounded by a gas of perturbing electrons and ions. When this model is

employed in an ensemble average to compute a line profile, the result will

represent a profile emitted by an actual system containing many essentially

independent radiators.







The Line Shape Expression


The total power emitted by a quantum system in a spontaneous

electric dipole transition is given by16

S = ab < b I exj a> 12 ; (-10)
El 3c3


Wab is given by the Einstein formula,


ab = (Ea Eb)/ (I-11)

and c is the speed of light. Since we are considering an emission process

only, the initial state of the system has energy E and the final state
a
has energy Eb, where Ea > Eb. The sum over j includes all the components

of the dipole moment of the system.

To make the connection between Equation (1-10) and the spectrum

emitted by a plasma, we must carry out an ensemble average of this equation.

This means we must average over the initial states a by including a weight

factor pa. In addition, we must sum over final states b. Then the power

spectrum emitted by a radiating system becomes

4w
P (a) = a 6(w- -W ) 2 p (1-12)
3c a,b,j

The delta function ensures that the transition conserves energy. We now

define the line shape function I(w):


I(w) = 6(w wab) 12 pa (1-13)
a,b,j

Then P(w) (4W
Then P() = (4 ab /3c ) I() where I(w) has the property of being

applicable to absorption as well as emission processes. Now consider the

Fourier transform of I(M),





10

u(t) = dwe- I( m) ,
co

or


(t) = e abt I 2 p (1-14)
a,b,. a

From this we can see that 0(t) = [$(-t)]*. Consequently I(w) may be

expressed by

-0
I(w) = --1 Re dt eit $(t) (1-15)
o

Also 0(t) may be written in the form


0(t) = e-iabt - p (1-16)
a,b
4-
where d is the electric dipole moment of the total system. We define p
a
to be an operator that acts only on the initial states a so that we may

move it inside the matrix element:


Q(t) = e-iabt . (1-17)
a,b

Inserting the definition of ab we obtain
-b


0(t) = e-i(Ea-Eb)tt - (I-18)
a,b

Since the E's are the eigenvalues of the Hamiltonian for the system we

may also move the exponentials inside the matrix element:

S- iHt/ it/T
c(t) = . (1-19)
a,b
4- 4

S - (1-20)
a,b

where the time-development operator T(t) is defined by

T(t) = exp {-iHt/} (1-21)






11

The sum over the states a and b is just a trace operation:

-t
0(t) = Tr {d.T(t)pd T (t)} ; (I-22)

D(t) is thus obtained by performing a trace over the states of the total

quantum system. We may observe that 0(t) is the autocorrelation function

for the electric dipole moment of the system.


Time Scales and the Line Broadening Problem


We demonstrated above that I(w) can be written in terms of the

following transform:


Iw) = -T-1 Re dt eiwt D(t) (1-23)

o

Since 0(t) is an autocorrelation function, we expect it to be a smoothly

decaying function for large t. This means that for t >TR (where TR is

some critical time for the broadened emission process), the exponential

will oscillate so rapidly that the contributions to the transform will

vanish. This is a statement of a familiar property of Fourier transforms,

namely

Am R < 27 (1-24)
R-

Most of the contribution to I(w) will come for times t less than the

critical time TR. This suggests that an appropriate characteristic time

for the broadened emission process will be defined as

S 2r
TR A (1-25)
AStark

where Stark is the width of the resulting (experimental or calculated)
Stark
Stark line shape.

Processes within the system having characteristic times greater than

R will take place so slowly that they may be regarded as static.





12

Approximations of this type are useful so long as their region of

validity covers most of the line profile. This will nearly always be

the case for the motion of the ion perturbers. A characteristic time

for the ions is given by



_D 1
T = -- (1-26)
Ions v W
av p (Ions)

where XD is the Debye length for the plasma; av is the average thermal
DJ av

velocity of the ion perturbers. Essentially, TIons gives the duration of

a radiator-ion encounter; p (Ions) is the plasma frequency for the ions.

In most cases of interest, the ion plasma frequency p (Ions) is much

less than the width of the line profile:

AS T > TR (1-27)

p (Ions) Stark; Ions R (1-27)


In the formal development, we introduce an approximation in which ion

motion is neglected and ion broadening effects are treated by averaging

over static configurations of the ion perturbers. In this approximation

(known as the Quasi-static Ion Approximation), the ions provide a static

electric field which splits out the atomic levels of the hydrogenic

radiator. The average over the ions is carried out in a final step of

the line profile calculation by an integration over an electric microfield

probability distribution function.

There are, however, ions in the plasma whose velocities are greater

than av Characteristic times for these (dynamic) ions are less than TR
av R
and the effects of their motion may not be negligible. Equations (1-24)

and (1-26) allow us to make an estimate of the limits of validity for the

Quasi-static Ion Approximation:


AW To < 21
Ions

or,






13

Am < w (1-28)
p (Ions)

By this we mean that for frequency separations Aw (measured from the

unperturbed transition frequency) less than w broadening effects due

to dynamic ions will become important. For all the cases considered in

this study, the ion plasma frequency is so small compared to the width of

the profile that ion dynamics are important only over a small frequency

range at the very center of the profile. Therefore, ion dynamic effects

are neglected throughout the course of this work.

When we apply the same analysis to the perturbing electrons, we

observe some striking differences. First, the electron plasma frequency

is ~42 times greater than the plasma frequency for protons. In the case

we study here, the electron plasma frequency falls out in the wings of

the line profiles. Electron dynamics then are important over most of the

line profile and static approximations are of interest only in the far

wings. This means that the time dependence of the electric field due to

the perturbing electrons requires quite a different treatment from the

one employed when dealing with the ions. In the present work, electron

broadening is treated by a second-order time-dependent perturbation

calculation.

Thus a hierarchy of time scales is present for the line broadening

problem:


Electrons R Ions (-29)

This hierarchy gives justification for the two differing theoretical

approaches to the broadening produced by the electron and ion perturbers.


Factorization of the Initial Density Operator


The initial density operator p is chosen to be the canonical





14

Boltzmann operator given by


p = e-H / Tr {e-H} (1-30)

where B = (kT) and H is the Hamiltonian operator for the total system.



H = H0 + H0 + H0 + V + V. + V .(1-31)
r i e er ir ei

Here, H is the unperturbed Hamiltonian for the radiator, H. is the kinetic
r 1

energy of the ions plus their (ion-ion) interaction energy, and He is
e

similarly defined for the electrons. The V's are the respective inter-

action energies.

We now introduce an approximation present in most line broadening

calculations. We assume that the operator p has only diagonal matrix

elements between the initial states of the transition. Furthermore, we

assume that p may be factored in the following manner:


P = rPiPe (1-32)

In order to define the factors on the right side of Equation (1-32)

we present the following argument. We first regroup the terms in Equation

(1-31). After performing this regrouping, we may write H in the following

form


H = H + H. + H + V (1-33)
r i e er

where


H = H + V 1) (1-34)
r r ir

and


H. = H (0) + V (1-35)
1i ir ei

In this expression, VO) is the monopole contribution to V namely
ir ir'

V() = X e (1-36)
ir xp r
P





15

In this equation, X(=z-l) is the net charge of the radiator and z is
p

the charge of the ion perturber. It is important to note that in this

form this term is independent of radiator coordinates.

The term V() gives the dipole contribution to V. and is combined
ir ir
with H since it contains the radiator position operator:
r


V = eR.E. (1-37)
ir i

Terms in the multipole expansion of V. that are higher than dipole are

not considered since the present work will employ the dipole approximation

to the radiator-perturber interactions.

If we now neglect the Ver term in Equation (1-33) we may obtain the

approximate factorization of Equation (1-32). The following definitions

are possible:


Pr = e-r / Trr {e-r} (1-38)



Pi = e i / Tr {e-i} (1-39)

where the prime here indicates that we consider the term V ei to produce a

Debye shielding effect in H.. Therefore, we drop Vei and now employ only

shielded interactions in order to leave H' independent of electron
17
coordinates.1

0 0
p = e e / Tr {e- e} (1-40)

Neglect of the term Ver implies that while the electrons produce

broadening effects, they do not alter the initial distribution of radiator

states. The factorization of p is a procedure which need not appear in

the formal development of modern line broadening theories based on kinetic

theory.8 These theories, however, sometimes make a factorization for

reasons of computational convenience.





16

The Line Shape in the Quasi-static Ion Approximation


Combining Equations (1-22), (1-23), and (1-32) we obtain for the

line profile



0
-1 I R.\dteiwt t +
I(K) = Re dte Tr {d T(t)prPiPe dT (t)} (I-41)


The trace is to be evaluated using states of the entire quantum system

of radiator, electrons, and ions. The most convenient set of states for

this calculation is a set of product states. Each product state will

consist of a one-particle state for the radiator and a many-particle state

for the gas of electrons and ions. We now define this product state:


a> = Pi> x la> (1-42)

where p represents the one-particle radiator state and a represents a

many-particle state for the electrons and ions.

Now we introduce the familiar restriction that d is the dipole
1
moment operator for the radiator only. Furthermore, we restrict d to

have nonzero matrix elements only between a specific set of upper and

lower states; hence we will consider only line radiation of a specific

transition. In our case, we consider only the Lyman series of transitions

for hydrogenic ion radiators.

Now the Quasi-static Ion Approximation is introduced as follows. We

first consider the commutator of H' with H as given in the expression

d
ih H = [H', H] (1-43)
dt i 1
-1 18
Smith has shown that this commutator is proportional to m. Since the
1
-1
thermal velocity of the ions also scales as m. it is evident that the
1

limit of infinitely massive ions corresponds to the case where the ions

are static. This same limit also implies that the commutator above vanishes:






17

[H', H] = (1-44)

Several results follow immediately from this equation. The first

consequence of Equation (1-44) is that we may write


T(t) = T. (t) T (t) ; T+(t)= T+ (t) T (t) (1-45)
1 er er i

where

-itH'/T -itHer/H

T (t) = ei i ; Te(t) = e er (1-46)

and


H = H + H0 + V (1-47)
er r e er

Another consequence of Equation (1-44) is that pi commutes with pr and

T (t). Using these results we may simplify Equation (1-41), with this
er

result for the line profile:


I(w) = T1 Re dteist Tr {p d T (t)p p d T (t) (1-48)
J i er re er
o

Note that T. operators have been commuted and canceled. Next, we insert
1
->- ->-
a delta function 6(c c.) into the trace of Equation (1-48), along with
1

an integration over the variable e. This step is valid so long as the

delta function (which contains the ion coordinates) commutes with all the

other operators inside the trace. The vanishing of the commutator in

Equation (1-44) ensures the validity of this step. After this insertion,

we are free to reverse the order of the trace and integration operations:


I(M) = de T1 Re dte Tr {p.6( E.)
0 0

.:d T (t)p d T (t)}. (1-49)
er re er

The effect of inserting the delta function here is that we may replace
+ 4-
E. as it appears in T (t) and p by the integration variable e. After

making this replacement we are free to perform a partial trace over the





18
ion coordinates in Equation (1-49). The line profile in the Quasi-static

Ion Approximation is then given by


I(w) = de Q(C) J(W,C) (1-50)

where

Q(E) = Tr {pi56( Ei) } (1-51)

defines the electric microfield probability distribution function for the
-> ->
static ions: Q(E) gives the probability of finding an electric field E at
->
the site of the radiator. Now J(w,c) is defined in terms of a trace over

radiator-electron product states.

-* ->
-1 iwt +
J(w,E) = 1 Re dt e Tr {d-T (t)p pd T (t)} (1-52)
er er re er
0
4-
The operators T (t) and p are now functions of H (E) where

er r r--

r r


and C is the electric field due to the static ions. Here J(u),C) gives
4-
the electron broadened profile emitted by an ion in an external field C.

Ion broadening effects are included when the ion microfield integration

of Equation (1-50) is performed.

Before continuing, let us summarize the approximations we have made

thus far:

i) The dipole approximation to the radiator-perturber interactions


[Equation (1-7)];

ii) The factorization of the density operator [Equation (1-32)];


iii) The Quasi-static Ion Approximation [Equation (1-50)].


These approximations have been standard line broadening approximations,

the validity of which will not be tested in this work. It should be

pointed out, however, that the first two restrictions may be, in principle,






19

removed. (See References 4, 17, 19, and 20.) Our previous discussion

of ion dynamics and the validity range of the Quasi-static Ion Approxi-

mation [see Equation (1-28) and the subsequent paragraph] indicated that

for the cases considered here, the Quasi-static Ion Approximation may be

used with considerable confidence.


The Liouville Representation

In this section we introduce a notation which formally simplifies
4->
the calculation of J(wm,). Consider an arbitrary operator f whose time

dependence is generated in the following manner:

^ ^ + -iHt/l ^ iHt/h
f(t) = T(t)f(O)T (t) = e f(0) e (1-54)

If we take the time derivative of f(t), we obtain,


ih f(t) = [H,f(t)] = Lf(t) .(1-55)
dt

This equation defines the Liouville operator L and its operation on an

arbitrary operator. We may solve this equation formally for f(t):


-iLt/ ^
f(t) = e f(0) (1-56)

The Liouville representation as introduced in Equations (1-55) and (1-56)

(see also Appendix B) is essentially the "doubled atom" representation

of Baranger.2 The advantage of this notation is that it gives a "short-

hand" with which to carry out formal manipulations. That is, we may

formally perform the integration of Equation (1-52) with the following

result for J(w,e):

-1
J(W,c) = -7i Im Tre {d-K ()p p d} (1-57)
er er re

where Ke (t) is defined by
er


K (w) = -i dt e e- Ltif -L/B (1-58)
er
0





20

The operation of L is defined by

A A
Lf = [Hr, f] (1-59)


where Her is defined in Equation (1-47). We now formally carry out a

partial trace over the states of the perturbing electrons.


J(w,E) = -rIm Tr {r R (o)p 1} (1-60)


R (w) = Tr {Ker ()p } = (1-61)


The last equation defines our use of the bracket notation to indicate

an average over the perturbing electrons.

The operator Rr () is called the effective-radiator-resolvent.

Although this operator is a function of radiator coordinates only, it

includes broadening effects due to the dynamic electron perturbers.

Equations (1-58) and (1-61) give the formal definition of Rr(w). The

goal of sections to follow is the development of a useful method for

calculating Rr(w).


A Perturbation Expansion for R (w)



In this section we describe a method for approximating Rr(w), the

effective radiator resolvent operator. The expression we are interested

in is given by

-1
R (w) = = <{w-L/H} > (1-62)


where L is given by


L = LO + L+ (1-63)

This corresponds to





21

0 0
He = H + XVI = H H + XeR.- + Ve (1-64)
er 0 I r e er

Here X is a coupling constant introduced for convenience (later we will

let it equal unity). The procedure we now follow is the same as that

employed by Dufty in Reference (21). The first step in this procedure is

to make the following definition:


= {w-L /I AL. / H(a)} (1-65)
er r Ir

This expression now formally defines the operator H(w) : H(w) is a function

of radiator coordinates only, but contains broadening effects due to the

perturbing electrons. We now assume that the operator H(U) has an

expansion in powers of the coupling constant, namely,


H(w) = H(0)() + H((i)) + X2 H(2() + ... (1-66)

The next step involves expanding both sides of Equation (1-65) in

increasing powers of A and equating like powers to identify terms in the

perturbation expansion of H(w). The left side of Equation (1-65) may be

expanded in a Lippmann-Schwinger expansion by employing Equations (1-62)

and (1-63):


= + A-1



+X2 -2 + ... .(1-67)


Several identities given in Appendix B help to simplify this result.


= R (w) + X-i R ()R (w)
er r r I r

+X2-2 R (w)< L R (w)L >R() + (1-68)
r where
where





22

R O() = {(-L /H}- ; R O(i) = {m-L /Fi)1 (1-69)
r r 0 0

The right hand side of Equation (I-65) may be expanded in a Taylor series

in A with the aid of the following operator identity. For an operator A,

d -1 -1 dA -1
dA A =-A A (1-70)
dA dA

The expansion of the right hand side of Equation (1-65) is given by


{w-L /H AL. /H H(M)}- = {_-L0 / H (i )}1
r ir r


+ X{w-LO/n! H(o) )-1 [Lir/ + H(1) (() ] {-L /0 H(O) ()}-


+ A2{ I-L ( (0w)) }- [Li/ + H(1)(a) ]{Mw-L0/H H(0) ( -
r ir r


x[Lir/+ H(1)()]{- / H(O)()}-1


+ 12{ -L / H(O)()}"1 H(2)(){a-LO /h H ((0 )}" + ... (1-71)
r r

Now by comparing Equations (I-68 and (1-71) we may identify the terms H(i)

(0) appearing in Equation (1-66):


H () = 0 ; (1-72)


hH (w) = ;(1-73)
er


2H(2)w() = R(w) (1-74)
er 0 er er r er

Before proceeding, we state here (and prove in Appendix C) that, as

a consequence of making the dipole approximation for the radiator-electron

interation Ve (see Appendix A), the indicated average in Equation (1-73)

vanishes:


= 0 (1-75)
er





23

This means that if we retain the lowest order nonvanishing contribution,

H(D) is given by

-2
H(M) = 2 (1-76)

with the final result for R (w):

0 -2 -1
R (w) = {w-L /I L /E-l -2 } ; (1-77)
r r ir er 0 er

R r() expressed in this form is referred to as a second-order resolvent.


A Computational Form for H(w)


In this section we wish to develop a form for the second-order result

H(2) (t) which is convenient for computation. That is,.we wish to develop

a matrix representation of the following expression,


2 (2)
A2 H (2)() f = (1-78)
er 0 er

where f is an arbitrary radiator operator. We may insert the integral

definition of R o() and obtain
00

2 H) (w) f = -i dt e (1-79)
er er
0


=-i dt et {
Ser er er er


-iH t/t ^ tiH t/h -iH tth ^ iH tth
- +}
er er er er (1-80)

We concentrate on the first term in this expression. Denote this term by

W(w)f; then,



0
e nw ()f =te -i dt e J er er

We now take matrix elements of this expression between free radiator





24
21
eigenstates. Continuing in this manner, we may extract a matrix

representation of the tetradic operator (see Appendix B) W(w). The
9,22,23
result of this calculation is9,22,23


) = -i dt e iAP t Vu"1'(t)> (1-82)
2W(w) iav;1' v v-v6 UiM 0


where

0 0
-iH 0t/i iH t/T
AWP = W-Wv and V ,(t) = e V e e (1-83)


The calculation of the electron-averaged quantities appearing in Equation

(I-82) is discussed and carried out in Appendix C. The result for the

full expression in Equation (1-80) is given by


2 (2) =
HH (M) = 6 R r(A) )


-6 I ,I R V r(-Aw ,,)
1114 \) \) V )j '"


+ R {(-A ,)II' -w F(Aw,)} (1-84)

--
In this equation, R is the dipole moment of the bound radiator electron.

The complex function F(Aw) contains many-particle effects due to

the perturbing electrons. Its definition and the details of its calculation

are given in Appendix F. The calculation of the radiator matrix elements

appearing here is given in Appendix E.


The Line Shape Formula


We now incorporate the results of our calculations into the line

shape expression. Recall that the line profile is given by


I(w) = d P(E) J(W,c) (1-85)
0





25

where

2
P(e) = 4T2 Q() (1-86)

and we have made use of the fact that Q(e) is an isotropic function of E.

-1 +
J(W,E) = -T Im Tr {dR r(w)pr d} (1-87)


The effective radiative resolvent operator R (w) contains averaged
r
electron broadening effects:

0 -1
R (w) = {w-L /R Lir/h H(w)} (1-88)


To second order in the radiator-electron interaction, H(w) is given by

Equation (1-84).

We now insert free radiator eigenstates in order to evaluate Equation

(1-87).
-1
J(w,c) = -1Im {z E [R (w)] ; . (1-89)


This equation is the final form of Equation (1-12). These two equations

are still similar in that there is a weighted average over initial states

of the transition along with a sum over final states. Previously we

indicated that the weighting factor p is a diagonal operator acting upon

the initial states of the transition. Also, the matrix elements of a are

restricted to have non-vanishing results only between initial and final

states of specific transitions. These facts allow us to write for J(w,e):


J(,e) = -T-lm i i f [ [R (w)] ,f pi,
if I'f r if;i'fi


+ [R w)fi;i'f' p,]} (1-90)


At this point, we introduce the No-Quenching Approximation. Mathe-

matically, this approximation states that the matrix elements of the time





26

development operator for the system taken between initial and final states

of the transition must vanish. That is,

= 0, (1-91)

where H is the Hamiltonian for the system. This matrix element is

proportional to the probability amplitude for a process where the inter-

actions within the system cause a radiationless transition from the upper

state to the lower state. The No-Quenching Approximation, therefore, is

invalid when the broadening interactions mix the upper and lower states

of the transition. This means that broadening effects must be small

compared to the separation of adjacent radiator energy levels. This

statement must be considered either to be a weak collision assumption or

it must restrict us to calculate only isolated line profiles. We simply

state here that if we expand the factor R (O) fi;i'f' [appearing in

Equation (1-90)] according to previous definitions we would find factors

like that in Equation (1-91), but with H replaced by an effective Hamiltonian

operator containing averaged electron broadening effects. From this

procedure we see that the No-Quenching Approximation causes the second

term in Equation (1-90) to vanish.

We now point out that in this work we consider only Lyman series

transitions so that we need not sum over lower states.

-1
J(w,E) = -T Im [R (w)] pi(1-92)
ii'

where we define [R (w)],., by

[Rr(w)], = [R (m)] ; (1-93)

By defining


D (1-94)
Di' i





27

we obtain a simpler form for J(w,c):


J(w,E) = --T Im Z D. [R (w)]i.,p (1-95)
ii

where D is a scalar matrix we calculate in Appendix E. The sum over i

corresponds to ordinary matrix multiplication. The sum over i' represents

the desired thermal average.

The above restrictions simplify the definition of R (w):

z -1
[R (w)]i, = {Aw-eeR..,/h H() ),} (1-96)
r ii 1ii i


where Am = -w if indicates that we measure frequency in terms of separation

from the unperturbed frequency; Rz is the z-component of the position

operator for the single bound radiator electron.

-2 -+ -+
H() ii, = E r(A) E R,,, R.,i, (1-97)
if

This completes the discussion of the line shape formula. Actual

details of the numerical procedures involved in computing line profiles

will be discussed in later sections. In summary, let us list the major

approximations made in reaching this point:

i) The dipole approximation to the radiator-perturber interactions;

ii) The factorization of the density operator;

iii) The Quasi-static Ion Approximation;

iv) The second-order perturbation calculation for the effective

radiator resolvent operator;

v) The No-Quenching Approximation;

vi) No lower-state broadening (this is not an approximation for

the Lyman series).














SECTION II


ELECTRIC MICROFIELD PROBABILITY.DISTRIBUTION FUNCTION


Introduction


In Section I we indicated the steps involved in arriving at the

Quasi-static Ion Approximation. As a result of making this approxi-

mation, the line broadening problem is greatly simplified. That is,

the line profile may be written



I(M) = ds Q(E) J(w,,) (II-1)

Included in the function J(w,E) are the electron broadening effects,

calculated for the radiator placed in the static electric field of the

perturbing ions. This calculation is discussed in detail both in Section

I and in the Appendices. The electron-broadened profile is averaged over

all possible values of the ion field to produce the final Stark profile.

This average is performed when we integrate over the electric microfield

probability distribution function Q(e). The probability Q(E) of finding

an electric field e at the site of the radiator, was defined in Equation

(1-51). We now write Q(E) in the following form:



Q(E) = z exp{- V(r,.,r) dr (1i -2)

Here z is the configurational partition function and we average over con-

figuration space; V(rl,...,rN) is the total potential energy of the ions,
4-
including their interactions with the radiator; and c. is the field at the
28
28





29

site of the radiator due to the perturber located at r.:
1

E.=- V V(r ...,r) (11-3)
i X i 1, N

The usual minus sign is cancelled here because we express the gradient

in terms of r..
i
At this point we wish to recall some of the features of the model

that we employ when performing the microfield calculation. There is a

hydrogenic ion (radiator) of charge Xe fixed at the origin of our coordinate

system. The plasma perturbers consist of electrons along with two species

of ions of charges Z1 and Z2, respectively. Overall charge neutrality of

the plasma is expressed by the following relation


ne = Zn + Z2n (11-4)

where n is the number density of the electron perburbers and nl (n2) is the
e 1 (
number density of the charge Z (Z2) specie of ion perturbers. In addition

to providing charge neutrality, the electron perturbers are assumed to

produce a Debye shielding of the various interactions between the ions in

the plasma. That is, in the present work, we restrict our consideration

to the "low-frequency" microfield distribution function.024

We insert the integral definition for the delta function in Equation

(11-2):


Q(E) = Z (20r) 3exp{- V + izl(e- ZE )}d~dr (11-5)
-1 ZI (2)-3e -N
Q(c) = ... ( exp-BV + it'(e- CEi)}dkdrN (11-5)
i i


Since k is an arbitrary vector, Q(e) cannot depend on the direction of Z.

We may then perform the angular integration, arriving at the relation,




0
Q() = (22)w- T()hsin(ER) e dr (-6)
o

where





30

T() = Z 1 ... exp{-BIV -BiZZ'} drN (II-7)
i

Also, since the ions are distributed isotropically in the plasma we may

write,

2
P(e) = 47e Q(C)
S C
= 2Z -l T()sin(rE) a dZ (11-8)
o

The transform in Equation (11-8) is performed numerically; the

calculation of T(Z) forms the principal topic of this section.


The Formal Calculation of T()
--
The explicit form for V(rl,...,rN) is given by
N 2 2 N. 2 2
S +- 1 Zie / 2 Z e
V(r ,...,r) = E ---- e-ri/ + E e-rmn/
V(rl'",""N) i r
i
N N 2 N 2
N 2 ZZe -rjX NL XZle e-rj0/X
r. r
+ z 1 m Z -r e- 1
j m jm J rj

N 2
2 XZ2e2 _r/
+ E e rmO (11-9)
m rmO

We specify a convention where i and j are reserved for the specie of

density N1 and charge Z1. Subscript 0 refers to the radiator. We now

introduce the quantities:


W = XZle e-arj0/ ; (II-10)
j0 r jo


2
XZ2e 2
W Ze e-rmo0/
m0 r -; (II-11)
m0


O m
V = V0 + m W + E Wm (II-12)
j m






31


The W's represent a short-range part of the central interactions between

ions and the radiator. All the noncentral and long-range central inter-

actions are compressed into the function V0. The a is an effective range

parameter and will be discussed at a later point. Now T(A) may be written,

N 1 -+ +
T(k) = Z .- eV0 1 exp{- iW. + ix 1-V W } dr.
j j 30 3
N2 -
-1 I-3
x H. exp{-$.W + iX -*V W dr (II-13)
m

where


V0 = -V0 + iX-1.V V0 (II-14)


We now define the functions

-1 --
Xz (;j) = exp{-iWj0 + iX --V 0W. -1 (II-15)
z O JO
-4t
1-
Xz (;m) = exp-Bi W + ix--V W } -1 (11-16)


T(k) may be rewritten in terms of these new functions:

N N
V N1 N2
T(A) = z .. e {1 + Xz(;) dr. {1 + Xz (a;m)}drm.(II-17)
j m 2

The reason for writing T(A) in this form is that we are able to make a cluster

expansion of the products in the above equation. Performing this expansion,

we obtain





NN N
N1 N2
+ E X (k;j) X (;i) + E x (;n)xz (Z;m)
i

+E N 1+ 2 N
j X (;j) X (;m) ...] IT dr H dr (II-18)
j,m 2 3 m





32

We now define the functions
r N N
V0 1 N2
T. m() = ... e0 11 dr. 1 dr ,(11-19)
Jm i=j+l n=m+l n

and


Qjm () = T. m() / T 0() ; T0(k) = T00 () (II-20)

These definitions allow us to express T(A) in a greatly simplified form.


T() = T () z-1 [1 + N1 Q10 () xz (;l) drl



+ N2 01 Xz2(;1) dr1 + 1 N (N-1) Q20()


xXz l(;l))Xz (;2)drldr2 + 2 N2 (N2-1)JQ02() Xz (;l)xz (;2)drldr2



+ NIN2 Qll(xz (;l)Xz2 (;2) drdr2 + ...] (II-21)


where we have taken advantage of symmetry in order to perform the sums.

If we define a new function


h m() = .. gj (a) ( ;l)... (;j)x ( ;l)...X (a;m)


j -) m -
x H dr. n dr ,(II-22)
i m
i=l n=l

where the g's are defined in terms of the Q's through an Ursell expansion,1024

we may recognize that Equation (11-21) is just the low order terms in an

expansion of the following expression:

J m
-1 1 2
T(Z) = T (M) z exp{H E (! h. () )} (11-23)
jm

From Equations (11-13) and (11-14) we may identify





33
j m
n1 n
z = T (0) exp h{ (--- h. (0))} (11-24)
j m

The final result for T(Z) is given by

J m
n1 n2
T(Z) = [T0(Z)/T0(0)] exp {E E (- -- [hj () hj(0)])} (11-25)
j m j


Introduction of Collective Coordinates*


In this section we proceed with the definition of the collective

coordinates. A transformation to collective coordinates will allow us

to approximately perform the multi-dimensional integrations appearing in

the previous section. We may write the total potential energy V in terms

of its Fourier series:

7X2 1 2 2 -ik'r.. 2 2 -ik.r
V 42 [1 zle e 13 z22e e mn
v k [2 + 1
iij (kX)2 + 1 2 m#n (kX)2 + 1
-- -- -- -- -*->
2 -ik-r. 2 -ik.r 2 -ik.r
z z2e e jm Xz e e jO Xz2e e
+ + --- + ]2 .(II-26)
j.,m (k) + 1 j (kA) + 1 m (kk) + 1

The last two terms represent the central interaction terms. The prime

on the sum over k means that the k=0 term is to be excluded. This

exclusion ensures that the condition of charge neutrality for the plasma

is satisfied.

Now consider the following expression:
-4 --
-ik*r.. 1 -*
S(ke)2 = 1 [cos (k'r..) -i sin (k'ri.
k (k) + 1 (kX) + 1 1 ij

1 -
= (kX)2 + 1 cos (k.r ) (II-27)
k (k) + 1 ij


*The collective coordinate technique we present here follows closely
the development given in References 10 23 24 and 25






34

The second line results from the fact that the sine is an odd function.

A trigonometric identity now gives

-ik*r.. -+ 4 -

(k) +1 = (k 1 [cos (k-r) cos (k.rj)]


4---
1
+ E 2 [sin (k.r.) sin (k-r.)] (11-28)
k (kX) +1 1

The above equation is equivalent to
-> -- 4- -)- -- -> -4
-ik.r. cos(k.r.)cos(k.r.) sin(k.r )sin(k.r.)
2e 2 E-= 2 + 2 E 2 (11-29)
2 2 2
(kX) + 1 k >0 (k) + 1 k <0 (kA) + 1
z--z

We separate this expression into k > 0 and k < 0 contributions in the
z z

anticipation of the definition of the collective coordinates. We now

define

cos (k-r) ; k > 0
z -

S(k.r) =
4 -4 (11-30)
sin (k'r) ; k < 0
z

With this definition Equation (11-28) becomes
4- -4- ->-
-ik-r. 2S(k.r.)S(k.r.)
e j 1 1 (11-31)
2 2
k (kX) + 1 k (kX) + 1

The total interaction energy V may now be written in terms of the

new coordinates S:
N 2 2
42 1 2zle
V [ E S S
v 2 2 j
2k (k) + 1 i,j

N 22 N 2 N 2
2 2z2e 1 4Xzle 2 4XZ2e
++ S + E SS + 2 S S
o mn i a
m,n i m


2 2 2 2 2
4zlz2e 2z e 2z2e
+ E S.S N N (1-32)
m m a 1 a 2
1.m





35

where



1 1 2 2
o = N1zle2 + N2ze2e (11-33)


The last two terms in Equation (11-32) are needed to subtract out the self

energy terms (i-j and m-n) which are included in the first two terms.

Now we define the collective coordinates X:

2 2
X i ; (II-34)
,k j
Xj ,,,,


X2 = E S S (11-35)
2,k a m
m

The interaction energy V becomes


S27X 0 1 2 2
S(k)2 + 1 k 2k 2Xlk X,k
k (kX) +1

2 2
+ 2 2X e SO (Xk + X2k) 2] (11-36)
a 0 l,k 2,k

The constants in front of this expression may be reduced to the following:
2 2
2 6 z + Rz
2 =T a e 1] (II-37)
V 2 z + Rz '
1 2

where


R = and 0 = k T
ne Be


After defining

A ; Y = X + X
S(k)2 + k l,k 2,k
(kA) + 1

we obtain a more simplified form for V


V = [1 R ] {Z' A + 2 2 E' Ak 2ZAk} (11-38)
1 2 k k >O k
z-





36

The second term is summed over k- > 0 because
z -

; k > 0



z
f0 k z < 0


We have manipulated the interaction energy V into a quadratic form

involving the coordinates Yk. The goal of this procedure is to transform

the multi-dimensional integrals over spatial coordinates appearing in

Equation (11-22) into multi-dimensional integrals over the coordiates Yk'


The Collective Coordinate Calculation


We repeat the definitions of Wj0, Wm, and VO:

2
Xzle -r /
W. =- e-a ; (II-10)
j3 r.

2
Xz2e ,-ar /
S xz e e-amO ; (II-1l)
Wmo rmo
mO



V = V + E Wj + Z Wm (11-12)
j m

The adjustable paramete ia is well discussed in the literature 2425

We will return to the discussion of this parameter at a later point.

From an inspection of the above equations, however, we may infer that a

essentially measures the contribution of long-range central interactions to

the term V0. Proceeding in the same manner as in the previous section,

we write


2 -ik'r
W. =- k z2 e k (11-39)
JO v k 1 (A ) a





37
2 2 2
zI + Rz 2Xz e
= [ + Rz S. (11-40)
e 2 k >0 (kX) + a j
z-

or


SW0 = [ ] ^ E (I I^-41)
SWO e z + Rz 2 2 (11-41)
j 1 2 k >0 (kX) + a
z-

Also we have
2 2 .
z + Rz 2 2 Y
S+ E 0 = 2x e k (II-42)
jO +E WmO =e z + Rz 2 2
j m 1 2 k >0 (kX) + a
z--
The net result of these steps is that we now are able to write down

an explicit expression for the quantity V0.


V0 = V E W E W
0 j0 mO
J m

2 2
6 z + Rz 2 2
2 1 R z { Z A 2 + 2 E f WA 2 A ,
S 2 Rz] {k k + 2 k k() k k 2 }k (II-43)
2 + 2 k a k >0 k
z--

where

2
f (a) 2 2 (11-44)
(kX) + a

We now need to compute the gradient of V Recall from Equations (11-13

and (11-14) that this operation produces the electric field terms appearing

in the original definition of T(Z). Furthermore, only the central inter-

action terms contribute to the electric field at the site of the radiator.

By inspecting Equation (11-38) and (11-42) we see that central interactions

are contained only in the second term of Equation (11-43). Following

O'Brien and Hooper10 we obtain

z + Rz 22 -+
SV = e + Rz2] 2 fk(a)Ak kk(11-45)
1 2 k <0
We now have sufficient definitions to enable us to calculate T(0).
We now have sufficient definitions to enable us to calculate TO(R)/T0(0).





38
0 N N
1 \ V0 1 2 1 -
.. e H dr. n dr
T (A)/T (0) = m (11-46)

V 0(=0) N 2
... e n dr. n dr

N N

r -l1 2 -
... exp{- BiV + i(Xe) 'VOV} dr nH drm
e Jp{-__J m
=3 m
N1 N2
...\ exp{-.V} dr. j dr
JJ m



exp{- [AYk2 + 2bk()Yk k 1 dYk
= J x kk k k
(II-47)

... exp{- u E [Ak2 + 2bk(0)Yk]} Jk i dYk
k k

where
2 2
o zI + Rz2
u = 1 2] (II-48)
O. z + Rz

and

21 ; k >0

b() = f k(a)A x
-i
iei(Xe) k ; k <0




bk(=O) = x k(a)Ak x L (II-49)
0 ; k <0
z--

We have cancelled the self energy terms from the numerator and denominator

in Equation (II-47). Here J: is the Jacobian of the transformation from the
k
spatial coordinates in the collective coordinates.

Defining A = uAk and bk = ubk' we may write





39

...i exp{- [ k2 + 2bk()Yk] Jk dYk
TO(Z)/TO(0) = k k (11-50)


1 2
..I exp{- 1 E [A Yk2 + 2bk(0)Yk Jk 1 dY


In Reference 23 the Jacobian is shown to be exp{- 1/2 Yk2} plus small

correction terms. This reference gives a very thorough discussion of the

Jacobian transformation and the collective coordinate method. In the

present work Jacobian corrections are neglected. We give here a general

result for integrals of the type found in Equation (11-50):

S[bk(a)2
exp {- }
T()/T(0) = k +A (11-51)
2 (II-51)
1 [bk(0)]
exp {-2 I
k 1 + A

We at once see that the terms for k > 0 give equal contributions in the
z -
numerator and denominator. These terms thus cancel each other. Also, for

k < 0 the denominator goes to unity. We have
z

1 [b ]2
TO(a)/T(0) = exp{-2 1 + (11-52)
k <0 Ak
z


2 2 2 2 2 2
e z1 + Rz2 [f()]2 A (k)
= exp {- [ E 2 } (11-53)
a z + Rz2 k < 1 + A
z
This expression may be evaluated by converting the sum over k into an

integral. We obtain the following result:


T0(a)/T0(0) = exp {-yL2} (11-54)

where y is given by


Y = [a2 (1 + u)]-2 B (11-55)
e

and





40
5 3/2 4 2 3
B u + 2 [1 (1 + u)3/2] a + [2u + u2] a

1/2 2 2
4 (1 + u) [1 (1 + u) /2] a2 3 [u + u2]a


+ 2 [(1 + u) (1 + u)3 ] (11-56)

We have
2 2
8 z1 + Rz2 n
u= [ ; R = ; 6 = kT ; 6. = kT. ;
6 z + Rz2 n e e i i



4 3 r0 e
3 I n = 1 ; a = = L = e .
3 0 e A '0 2 0
r0

We now turn to the problem of calculating the Qj (k) functions which

are defined in Equations (II-19) and (II-20). A general form for these

functions is given in Equation (2-56) of Reference 23 :
i y2 7 b
-3-m 1 kk Ykbk
-j-m exp A I k (A
Qjm () =2 k + A k + Ak
Qjm = (II-57
-j 1 2 .. j m -
V-3- exp 1 k k ubk} dr. H dr
Sk 1 + A, k 1 + A i=l1 n=1 n

where yk is a new collective coordinate given by


k = al,k + a2,k

and
2 2 22
2ze j + 2z2e m
alk S(k'r) ; a E S(kr ) (11-58)
1,k a ; 2,k 1 n

First we consider Q 0():

-1 1 YkA ykbk
V exp {- E
2 k 1 + A k 1 + Ak (11-59)

YkAk Ykbk
10exp {- z 1 2 k k+
2 1+ Ak 1 k k k 1
v r~ ~k-k k






41
2
The term containing yk' when summed over all k, is proportional to
si 2 (4. + 2 -
sin ( 1rl) + cos (k.r ) = 1. Therefore, this term is independent of

r1 and may be cancelled between the numerator and denominator.



-1 k 1
v exp {- E }
Q10(a) = b (II-60)
V exp {- + } drI
k +k 1

Consider now the sum in the numerator.
22
2z e S2 *
S 1 S(kr) b )
k 1+ i+

2
2 2 4 (a2 1)
= Xze Z S(k.r) 2 2 2
k [(kk) + (1 + u)] [(k) + a ]


S ; k > 0
z -

X
(11-61)

i(xe) -16.*k ; k <0
1 Z

This sum is converted into an integral with r chosen as the polar axis.

We obtain

I = z S(x) + i z1 Lq(x) cos 6 (11-62)


where

o 2 2 1/2
S- e a -1 -aax -(1 + u) /2ax (1-
S(x)- x[ ] [e e (11-63)
0. 3x 2
i a -(1 + u)
2 1/2
-a -1 1 e-aax -(1 + u) ax
q(x) = [ [ (e e ) )
a -(1 + u) x

1/2
Sa (-aax 1/2 -(1 + u) ax (11-64
+ (ae (1 + u) e )] (1-64
x





42
-> -4
where x = r/r0 and e is the angle between Z and r. By examining Equation

(11-62) we see that I will vanish as x tends to infinity. The result of

this is that if we take the limit where v-o, N-*~, N/v = constant (thermo-

dynamic limit), the denominator in Equation (11-60) tends to unity. We

have



QIO() = -1 exp {zlS(x) + izlLq(x) cos 9} (11-65)

Similarly,


Q01() = V- exp {z2S(x) + iz2Lq(x) cos e (11-66)

We now may write down several more definitions in terms of these coordinates:


-1 2 1 -arjo /
-Bijo = -6. Xz1 e -. e J
3O



= -PzlW(x) (11-67)


where

9 2
e a -aax
-W(x) = -X e (11-68)
0. 3x
1

Also


-6iWm = -Bz2 W(x) (11-69)


The gradient terms are given by

S-> -- ax
i(xe) *0 W = iLzI cos e { 2 (1 + aax)} (1-70)


-1 4- -aax
i(xe) m0 = iLz2 cos 6 { 2 (1 + aax)} (11-71
x

In terms of dimensionless variables





43

3 2 dQ
n dr =- x dx (11-72)
1 z + RZ2 4TA '


3R 2 dQ
2 dr = + R x dx r-- (11-73)


By inserting the above expressions into either Equation (11-21) or

Equation (11-22) we may calculate I10(k).


10 (k) = n1 [hl0(9) h10(0)] (11-74)

After performing the angular integration and rearranging terms we obtain,


1 93 d xe 2 e zS(x) sin(LzG(x))
1Zl +Rz2 d 2d z S(x) 2{e z-1ZW(x) [- 1 -G] -i



sin(Lz q(x)
[ -I] } (11-75)





[ -i]} (II-75)
Lz2q (x)

For e01(k) the analogous procedure gives








-oax
Corresponding to these, we have the following thre-7


where


G(x) = q(x) +--2- (1 + aax) (II-77)
x

We now turn to the calculation of the second-order terms Q20 (),

Q02 (a), and Q11 (). Corresponding to these, we have the following three
cases.





44



Case (1): k = z1 E S(kr) (II-78a)
j=i

27 2




2e
Case (2): Yk = z2 E S(k-r ) (II-78b)
n=1



Case (3): yk = {zS(k'rl) + z2S(kr2)} (II-78c)

The collective coordinate evaluation of the second sum in the numerator of

Equation (11-57) proceeds exactly as in the evaluation of 10(Y). We may

immediately write down the following results.


Yk
Case (i): E + A = z {S(x ) + iLq(x ) cos 1 }
k k


+ zl {S(x2) + iLq(x2) cos e2} (II-79a)


Ykk
Case (2): = z2 {S(x) + iLq(xl) cos 61
k 12 + 1 1


+ z2 {S(x2) + iLq(x2) cos 62} (II-79b)



YkA;
Case (3): k 1+ = z1 {S(xl) + iLq(xl) cos e1}



+ z2 {S(x2) + iLq(x2) cos 2} (II-79c)
2 2
Next we need to evaluate the terms kA We consider y and
1+ A

concentrate on Case (1):

2 -> -+ + +
y 2 e z {S2 (k.rl) + 2S(k.rl)S(k.r2) + S2 (k.r2)} (II-80)





45
2
The terms in S when summed over all k, are independent of coordinates

as before. These terms, therefore, will cancel between the numerator

and denominator in Equation (II-57). The remaining contribution from
2
yk is given by

2 4e 2 2
Yk= 1 z1 S(k.rl) S(k.r2) (11-81)

2 4
1 YkA 2e2 2 S(k-rl)S(k.r2)
2kl+A a k 1+A j Ak
k k 4


2e2 2 cos[k-(rl-r2]
S- 1 u 2 (II-82)
k (kA) + (1 + u)

Equation (11-82) results from the application of a standard trigonometric
->-
identity. The sum over k is converted into an integral with r12 as the

polar axis. We obtain


i Yk 2 2 1/2
1 kk 2 e a e-(1 + u) ax
1 0 z 3x e 12 (II-83a)
2 k 1 + Ak 1 3x12
i 12

Similarly,


Y 2A 2 1/2
Case (1 kAk 2 e a e-(1 + u)/2ax12 (I
Case (2): -z 12 i (II-83b)
2 k 1 + Ai 3x2
k12


1 yk 8 2 1/2
Case (3): e a -(1 + u) ax
Case (3): -zlz 3x e 12 (II-83c)
2 k 1+ 1 3x 2

Now that we have obtained the form of the expressions needed to

evaluate the second order terms, we see that the integrals in the

denominator of Equation (11-56) once more tend to unity when the thermo-

dynamic limit is taken. Also we may identify factors appearing in the

second order result to be just the Ql0(k) and Q01() that we evaluated

earlier. We may then write





46
6 2 1/2
2 e ea -( + ) ax
Q20(;1,2) = Q (;1)Q 0(;2) exp {-z 6i 31 e 12},(II-84a)



6 2 1/2
2 ea -(1 + u)1 ax
Q02(k;1,2) = Q 01(;1)Qo(;2) exp {-z2 3x e 12} (II-84b)
1 12

6 2 1/2
Q1(;1,2) = Q10(a;1)Q01(a;2) exp {-zl 2 6 a- e-( + u) ac


We define the function I20(A) by

1 2
I20) = n [h20() h20()] (11-85)

where h20 () is given in terms of the Qj functions by

2
h20() = v dr1 dr2 {Q20(k;1,2) Q10 (;1)Q10(Z;2)} Xz (;1)Xz (Y;2),
S1 1
(11-86)


= 2(a)6 dxI dx2 Q10(;l)Q10(P;2)Xz (k;l)xz (Z;2) x

2 2 1/2
S2 e a -(1 + u) 1/2ax
x {exp [-z a e 1a2] -1} (II-87)
1 0. 3x
i 12

The other expressions for hj (Z) in second order are very similar so that

we concentrate here on h20(Y). Define now

S 2 1/2
2 e a -(1 + u) ax2 (II-88)
D1(x1) = exp {-z 32 e 12} -1 (11-88)


We are free to make the following expansion


D12(x12) = E (2k + 1) Vk(x1,X2) Pk (cos 12) (11-89)
k=0
where Pk is the Legendre polynomial and 012 is the angle between xl and
4->
and x2. Vk(xl,x2) is given by
7r

Vk(x,x2) = D2(xl2 )Pk(cos 6) sin 6 dO (11-90)
o





47

Inserting Equation (11-89) into Equation (11-87) we obtain


h20 () = (aX)6 E (2k + 1) dxl 5dx2 f(x1) f(x2) x
k=0O


x Vk(XlX2) Pk (cos 012) (II-91)

where


f(x) = exp {zlS(x) + iLzlq(x) cos 6}x


x {exp [-BzlW(x) + iLz VW(x) cos 8] -1} (11-92)

Pk (cos 12) may be written 26

k .(k-m)! m
Pk(cos 12) = E m [(km)! Pm (cos Pm (cos 2)cos[m(l1-2),
m=0
(11-93)
where

m1 ; m=0
m =O


2 ; m#O

If we insert this expression into Equation (11-91) we find that the only

non-vanishing contribution comes from the m=0 term. Thus the expression

for h20(k) may now be written


h20(C) = (aX) E (2k + 1) dx1 dx2 f(xl) f(x2)
k


x Vk(xlx2) Pk (cos 81) Pk (cos 2) (11-94)

The angular integrals may now be performed with the aid of the following

identity:


(i)k 47 jk (A) = ed eiAcos C P (os) (II-95)





48

where jk is the spherical Bessel function. After the angular integrals

are performed h20(Z) becomes,


h20(k) = (aX)6 (2k + 1)(-)k dxlx dx2 x



x F1(i;l) F1(;2) Vk (x12) (11-96)

where


F1(A;1) = eZ1S(x1) {e-zlW(xl) jk (LzlG(xl)) jk(Lz1q(xl))},(II-97)



Fl(0;l) = ezlS(xl) {e-BZW(X1) -11 6k,0 (11-98)

At this point Vk(x1,x2) remains unspecified. We introduce here a linearized

approximation to Equation (11-88):

6 2 1/2
(L) 2 e a -(1 + u) ax (-9
D (x12) = -z1 e 12 (II-99)
S12 1 1 12 3x

(L)

Now Vk (x 1,2) is given by
1 2 -(1 + u) ax12
Vk(L) (xx2) = 2 e a3 e- +xu)/ 1 Pk(cos 6)sine dO .(II-100)
i o 12
27
This integral has been evaluated by Swiatecki27 with the following result:


(L) 2 e a2 Kk + 1/2(a' x) k + 1/2(a' x2)-101)
V (x1'x2) = -z1 (II-101)
i jxx x>x2 ,


where a' = (1 + u)/2 a.

When we insert this result into Equation (11-96), we obtain a final form

of Io20().


I (L) = -z2 il 2 3a2 (2k 1) (-l)k {20} (11-102)
S20 + Rz
1 1 2 k





49

{20} = x3/2 k + /2(a' x2) eZlS(X2)[e-BzlW10(X2 jk(ZlLG(x2))


CO

-jk(zlLq(x2))] S x3/2 K + 12(a' xl) ezlS(X)
x


x[e- zl1W0(xl) jk(ZlLG(xl)) jk(z1Lq(x1))] dx dx2



3122
6k,0 5 3/2 1/2(a' X2) eZlS(X2 [e-z1 W10(2) -1]
0


x x2 Kl/2(a' xI) ezlS(X1 [e-BZlW 0(X) -1] dxldx2 .(11-103)
x2

Note that a factor of 2 has been cancelled and the integration now covers

one-half of the positive quadrant. This results from the fact that the

integrand is symmetric under the interchange xl --- x2. In a similar

manner we obtain

2 e R 2 2 k
I02(L) = -z2 [Z + Rz 3a2 (2k + 1) (-1) {02} (11-104)
i 1 2 k
00
{02} = x3/2 k + /2(a' x) ez2S(x2)[e-z2 Wl0(X2 jk(Z2LG(x2))
o
CO

-jk(z2Lq(x2))] x3/2 Kk + /2(a' xl) ez2S(x x
x2


[e-z2 W0(Xl jk(z2LG(x)) jk(z2Lq(xl))] dxldx2



-k,0 x/2 I1/2(a' x2) e2S(X2)[e 210(x2)l]
o





50


x x2 K2 (a' x) ez2S(x)[e-Bz2W10(x1)-1 dx1dx2 (II-105)
1xI K1/2( a' X1)
x2

The symmetry mentioned above is not present in 11(L).


i 1 + z2 k




o
x 3/2 'k 1 2(a' x2) eZ23(X2)[e-z2W10(X2) j (z2LG(x2))

00


-ji(z2Lq(x)] x32 Kk + 1/(a' xl) ezlS(x-)

X2


x[e ez1W10(x) jk(zlLG(x) jk(z1Lq(x1))] dxdx

00
+ x x3/2 k + /2(a' x2) ez2S(x2) [e-z2 10(x2 jk(zLG(x))
o
-jkz2L ) x/2 Kk + 1a 2

x2
r2
-jk(z2Lq(x2)] x32 31 +1/2(a' x) ezS(x1)



x[e-z1 10(x10 jk(zlLG(x1)) jk(zlLq(xl))] dxldx2
o

-6k,0 x 3/2 /2(a' x2 )z2S(x2 ) e-z2 10(x2 -
co


x S x /2 K/2(a' x ) ezlS(x ) [e-z1 10(x 1)-1] dx dx

x2





51




0
-6k,0 x/2 Kl/2(a' x2) ez2S(2[e-BZ210(x2-l]
o


x 2 x3/2 1/2(a' x4) ezlS(X [e-Bz1 W0(Xl)-l] dxdx2 (11-107)


We should mention here that the linearization introduced in Equation

(11-99) is not forced upon us. In a more general nonlinearized treatment,

the integral in Equation (11-90) can be evaluated numerically. In fact

the computer program needed for the nonlinear calculation has already

been developed. However, it requires considerably more CPU time than the

linearized version. We employ the linearized version because it is

quicker and, in our cases, yields results in agreement with the more

general nonlinear version.

Now let us define


I1(L) = I10(L) + I01(L)


12(L) = I20(L) + 2(L) + I11(L) ,


TI(L) = exp {-yL2 + 11(L)}


T2(L) = exp {-yL2 + I1(L) + 12(L)}

T1(L) is defined as the "first approximation" to T(L) while T2(L) is

called the "second approximation."


Asymptotic Microfield Distribution Function


In order to obtain a microfield distribution function, we must

perform the following numerical sine transform.





52


P(e) = 2e-Ii dL LT(L) sin (cL) .(11-108)
o

There are many techniques for computing this transform which yield

accurate results for small values of E (generally for e<10, in units

of eO.) However, as the value of e increases, there is a loss of

numerical precision which is characteristic of integral transforms of

this type. In order to extend P(e) to large values of e we must develop

an asymptotic expression. This problem is discussed by several

authors.1,10,23,28 We wish to outline the treatment of this subject

given in Reference 10 as it is particularly suited for this theory.

The model for this asymptotic expression is known as the nearest-

neighbor approximation (NNA). This approximation states that high fields

are produced by a single ion perturber during a close encounter with the

radiator. This model assumes that for sufficiently large values of s,

the probability that two or more perturbing ions contribute to the field

is essentially zero. The probability that a single ion produces an

electric field e is related to the probability of a close encounter with

the radiator in the following manner:


Pl("l)dE1 = 4Trl gl(rl) drl


3 2
S+ Rz x1 gl(Xl) dx1 (11-109)


2
P2(2)de2 = 41rr2 g2(r2) dr2


3R 2
Sz + Rz x2 2(x2) dx2 (II-110)
21 2

The g's are pair correlation functions, which we choose to be the non-

linear Debye-HUckel expressions.





53

2 6 1/2
a e -(1 + u) ax
gl(X1) = exp {-Xz1 a3 y e 1 (II-111)
3x1 1

2 6 1/2
a e -(1 + u) ax2} (11-112)
g2(x2) = exp {- 3xI e .


In terms of dimensionless variables

z1 -ax
1 =1-- (1 + ax1) e ax (1-113)
:1.
xI

z7
2 -- (1 + ax2) e a2 (11-114)
x2
X2

When these expressions are differentiated and inserted into Equations

(11-109) and (II-110) we obtain the following expressions for the

asymptotic microfield distribution functions.
6 2 1/2
4 e a -(1 +u) ax1
3x1 exp {-XZ1 p 3x e 1}
1 1 1 ]. 3x
P1(I) = [z + Rz2 -ax (11-115)
Sze 1 {2 + 2/ax1 + ax 1
6 2 1/2
4 e a { -(1 + u) ax21
3x2 exp {-Xz2 3x e 2}
R 1 2
P2(Z) -ax 2 (II-116)
2 (2) z1 + Rz2 z2 e 2 {2 + 2/ax2 + ax2}



PAsym (E) = P() + P2( 2) (11-117)

The present method of joining the computed P(e) and the asymptotic P(e),

which differs considerably from that appearing in the above reference, is

considered in detail in an appendix.















SECTION III


DISCUSSION OF THE RESULTS


Introduction


In this section we present graphical results for the microfield

distribution functions and x-ray line profiles calculated using the

theory as given in Sections I and II. Experimental observations of

Lyman-a for Ne X have been reported only recently.* However, it has

been verified that this theory duplicates the results for the micro-

fields and line profiles given in Reference 23.

First we present figures demonstrating the behavior of the micro-

field distribution functions under variations of the different plasma

parameters. We also point out and discuss trends and important

features which are exhibited by these curves. Following this, we

present Stark broadened x-ray line profiles computed using the above

microfield distributions. General features of these profiles are

also discussed.

After presenting the figures we consider the various approximations

made in the development of the line profile formalism in Section I.

These approximations determine several basic validity criteria for

the application of this theory.



*The observations to which we refer here were made in connection
with laser-imploded pellet studies at Lawrence Livermore Laboratory. 29

54





55

Electric Microfield Distribution Functions


In this section we present graphical results for the electric

microfield distribution functions computed using the theory of Section

II. The first three figures illustrate the behavior of the microfield

functions for multiply charged hydrogenic radiators perturbed by singly

charged ions. The field variable e is in units of (= e/r). a =

r / D. In Figures 1 and 2, the charge of the radiator is +9 and +17,

respectively. Figure 3 shows a comparison of the microfield functions

for different radiator charges at a given value of a.

Two effects are immediately obvious from these figures. First,

for fixed radiator charge X, when the value of a is increased, each

successive microfield curve has its peak shifted to lower values of E,

becomes narrower, and has its maximum value increased. This behavior

is analogous to that noted previously for =1. 0 24 25 Second,

Figure 3 indicates that for fixed a, as the value of X is increased,

the behavior is similar to that noted above when a is increased.

Furthermore, a comparison of Figures 1 and 2 with Figure 4 of

Reference 24 reveals that as X increases the relative sensitivity to

changes in a increases. This increased sensitivity is due to the fact
2
that in many functions defined in Section II the parameter a is

multiplied by X.

Figures 4 through 7 illustrate the behavior of the microfield

distribution functions for the a = 0.2 and a = 0.4 cases, corresponding

to those in Figures 1 and 2, when the parameter T is varied. This is
R
the ratio of the electric kinetic temperature to the ion kinetic

temperature.


T = kT /kT. = e /e. (III-1)
R e i e 1





56

Figures 8 and 9 illustrate the effects of T -variation on microfield
R
distribution functions for a = 0.2 and a = 0.4 when hydrogenic

Aluminum (X = +12) is perturbed by a plasma containing only Ak XIII

ions (z = +12).

As the value of the parameter TR is increased, in all cases the

microfield distribution function peaks shift to lower values of C. This

behavior might be anticipated from the form of the following relation:

i /2
a' = (1 + u)/2a, (II-2)

where, if R = 0.0,


u = e/ i *1 = TR (111-3)


The parameter a' appears in several functions in the definitions of both

I(L) and I (L). Thus, if we regard a' as a modified plasma parameter,

we see that as the value of TR is increased, the result is an effective

increase in the a value. A direct comparison of Figures 4 through 9

with Figures 1 and 2 indicates that when the value of TR is increased,

these T -variation curves indeed exhibit behavior similar to that noted
R
when the a value was increased. While the dependence on TR is obviously

strong in all cases considered, the analytic form of this dependence is

extremely complicated and difficult to assess.

1/2
To see more clearly the effect of the (1 + u)/2 factor, let us

consider the following:


a' = (1 + u)1/2 = (1 + u) /2r0 = ro/' (III-4)

where


4rrne -1/2
A' = [ 2- (1 + u)] 2 (III-5)
D e
e





57

The definition of u is given below Equation (11-56); we now write it in

the form

2 2 2 2
0e N1l z N22 e .nlz 1+ n2z2
u e + N e n (111-6)
i N1z1 + N2z2 i n

The second step in the above equation results from the overall charge

neutrality of the plasma. Inserting this result into Equation (111-5)

we obtain

2 2
2 nlZl + n2z2 -1/2
D = [4ie (- + ] (III-7)
e i

This equation is a generalization of Equation (19) in Reference 30.

Thus we obtain for a':

2 2
S nlz + nz2 1/2
at = r0 [4e2( n + 1 )]1/ (III-8)
e 1

In our computations, we regard T and n (the electron number density)

as fixed by the parameter a. This means that, for a given value of a,

an increase in TR is accompanied by a decrease in the ion temperature.

Equation 8 gives the explicit dependence of a' on the ion temperature.

In addition this equation suggests that the shift of the microfield

peaks might be due to increased importance of ion correlations as ion

temperature is lowered. This correlation effect is especially apparent

in Figures 7 and 9. Looking at these figures we see that, as TR takes

higher values (ion temperature decreases), the At microfield distributions

peak at smaller field values than do the Argon distributions. This is

to be expected since interparticle correlations are stronger in the

AR-Ak system than in the Ar-H system. On the other hand, as TR decreases

(the ion temperature increases), the figures indicate that the stronger





58

fields are more probable in the At plasma. This is due to weakened

correlations; the average interactions in this case will be stronger

in the AR-AZ system than in the Ar-H system.

In Figures 10 through 13, we present the results of computations

in which TR is set equal to unit and the parameter R is varied. Now R is

the ratio of the density of charge z2 perturbers to the density of

charge z perturbers: R = 0.0 (m) .corresponds to the case where the

ion perturbers are all of charge z (z 2). The function u is given by

2 2
z2 + Rz
u = Rz (T = 1) (III-9)
z + Rz R


Itvaries smoothly from a value of z1 to a value of z2 as R goes from

0.0 to m. Because of this we might expect that when the perturbing

ions have zl and z2 values nearly equal, the variation of R causes

very little change in P(E). Indeed this is the case for At microfields

when zI = +12 and z2 = +11. For this reason we omit figures showing

the R variation calculations forthe At system.

The most noticeable effect of an R variation occurs when z and

z2 are very different (Figures 10-13). Furthermore, it should be noted

that for very different z and z2 (with z2 > z1), most of the variation

of P(E) occurs between the values of R = 0.0 and R = 0.1. This behavior

is due to the fact that for a situation where zl << z2 and R is only

slightly greater than zero, most of the free electrons are already

contributed by perturbers of charge z2. This is illustrated by

considering, for example, the following case for argon (z = +1, z =

+17):


R = 0.0 n = ne n = 0.0
1 e 2
R = 0.1 nI = 0.37 n n = 0.037 n
1e 2 e
R = n = 0.0 n2 = 0.059 n (III-10)
=01 2 e


























Figure 1. Electric microfield distribution function P(E)
at a point having a charge of +9. e is in units
2
of E0 (= e/r ) and a = r /D. The ion perturbers
have a charge of +1.





















1 0) d
"O= D


90


6=X -0o


K '





























Figure 2. Electric microfield distribution function P(E)
at a point having a charge of +17. The ion
perturbers have a charge of +1.













O'17 '0 "Z g''l 0'I G'O 0









9'0
9'0 = D


"O = D-- \\ -6'0





I7+ *j
?'0= D





L t-X





























Figure 3. Electric microfield distribution function P(E),
for a = 0.6, at points corresponding to = 1, 9,
and 17. The ion perturbers have a charge of +1.













0*-2 S'z 0" zl:1 0"1 To. 0







331






I=X
-9'0

(\)d

-8'0





9-0 = D
* -------------_____- __ ___ ___9"O :__ Z "





























Figure 4. Electric microfield distribution function P(e) at
a point having a charge of +9. a = 0.2. The ion
perturbers have a charge of +1.











0,9- 0", 00"1 02 O'l 0
i I 0




0,0

e "0 = J.,,/






0', 0-j./
v-( )d

6=X


S' I I,































Figure 5. Electric microfield distribution function P(E) at
a point having a charge of +9. a = 0.4. The ion
perturbers have a charge of +1.








O 0 0g 0z"O1 0o












Z'O




6 =ZZ O'Z9*0
9'0
Z=ll
6=X
t"O=o = )d


O't? 8'0































Figure 6. Electric microfield distribution function P(c) at
a point having a charge of +17. a = 0.2. The ion
perturbers have a charge of +1.















O'g 0' 0 O' 0










0\0



I = '





W-O=


9'0
,-0"> "X / 3.
---- l ---- \ ---- I ---- l--------- i ____ __ ] __







































Figure 7. Electric microfield distribution
function P(c) at a point having a
charge of +17. a = 0.4. The ion
perturbers have a charge of +1.




72


1.4-





1.2-


P(E)
Tj =4.0

1.0- = 0.4
X= 17
ZX I?
ZK=I
R=0
0.8
TR=2.0



0.6-
TR= 1.0


TR= 0.5
0.4




R= 0.25
0.2-





0 1.0 2.0 3.0 4.0 5.0
E
































Figure 8. Electric microfield distribution function P(c) at
a point having a charge of +12. a = 0.2. Perturbing
ions have a charge of +12.










So'N^ o' o=u o'/ o
S0* 1l 0









gIO= =Iz







0I='Z 0,2 0
i(0)1


"0 D 900I



_'0*= o9"0





































Figure 9. Electric microfield distribution
function P(e) at a point having a
charge of +12. a = 0.4. Perturbing
ions have a charge of +12.




76


1.4





1.2


P(E)
TR = 4.0

1.0- a = 0.4
X= 12

Z,= 12
R=O
0.8-
TR= 2.0



0.6



TR= 1.0

0.4
iTR= 0.5




0.2-\ \ T 0.25





O 1.0 2.0 3.0 4.0 5.0
0 ^ ^ ^^ ^ -^ ^^ ^ E































Figure 10. Electric microfield distribution function P(E)
at a point having a charge of +9. a = 0.2.
Perturbing ions have charges +1 and +9.














O't O' O' I 00














0 '-0 = '
000













j'0






1= 'Z
l-Z.

6=X
z'0 = o- 9"0

_____1_____1_____1 --------- \----------------! --------0































Figure 11. Electric microfield distribution function P(c)
at a point having a charge of +9. a = 0.4.
Perturbing ions have charges +1 and +9.










o'g 09' 6'2 0"* O'l 0


















69*O































Figure 12. Electric microfield distribution function P(E)
at a point having a charge of +17. a = 0.2.
Perturbing ions have charges +1 and +9.































S .9'O
0"0










0"I x a


*'O= D
-9'0






























Figure 13. Electric microfield distribution function P(c)
at a point having a charge of +17. a = 0.4.
Perturbing ions have charges +1 and +9.









00 O' 0"7 0"p O' 0' 0
0











0 1 1 *-
I =-- (-)-
I = x 00=
i."0= D
9'0



L
I I I I 8 0





85

In the present work, we consider n to be fixed by the a value. Then

nI and n2 are determined by the requirement of overall charge neutrality

for the plasma (Equation II-4).

For lower a values, an increase in R causes the microfield peaks

to shift to higher field values. At higher a values, however, the

increase in R produces a shift to smaller field values. The behavior

at higher a values can be explained in terms of particle correlations.

That is, an increase in the value of R results in an increase in the

effective plasma parameter a'. At lower a values, correlations are

weaker and the observed shifts indicate that as R increases, the average

strength of interactions within the system increases.


Stark Broadened Line Profiles


In this section we display the broadened line profiles computed

using the electric microfield distribution functions shown in Figures

1 through 13. Figure 14 shows the relative contributions to the

Lyman-a Ne X line profile from the several broadening mechanisms that

are included in our calculations, together with the combined result.

The conditions represented in Figure 14 correspond to an a value of 0.4.

The relative importance of the Doppler effect is of interest. For the

temperatures discussed in this paper, the Doppler effect is much more

significant -- compared to the electron-broadening contribution -- than

was the case when dealing with more conventional plasmas (e.g., n =
17 -3
10 cm and T = 40 0000K). The qualitative features of the Stark

profile are also different: here the electron contribution produces

a sharp spike which sits on shoulders provided largely by the ions.

Figure 15 presents a plot for Ar XVIII that is equivalent to Figure 14






86

for Ne X. The qualitative information is the same.

Figure 16 shows for both Ne X and Ar XVIII, families of Lyman-a

profiles, each of which corresponds to the same T but different n.

It is evident that as n increases, the profile changes greatly, both

in shape and width, thereby illustrating the density sensitivity of

the line profile. The practical sensitivity that might be inferred

from comparison of experimental and theoretical line shapes depends

on at least two factors, (1) how much of the line profile can be

experimentally observed, and (2) the resolution of the x-ray spectro-

meter used.

Figure 17 shows for Ne X and Ar XVIII, families of three Lyman-a

profiles, each of which corresponds to the same density but different

T. The results here imply that the frequently mentioned insensitivity

of plasma-broadened line profiles to variations in T is significantly

reduced as X increases. The fact that Doppler broadening plays a more

significant role in broadening the argon lines than it does for neon

is due to the fact that the effect depends not only on mass, but also

on the radiator z.

Figures 18 through 20 show the behavior of Lyman-a profiles for

hydrogenic neon, aluminum, and argon, respectively, under variation of

the parameter TR. The electron density is the same in all these cases.

For the curves on the left the electron temperature is 1019.2 eV while

on the right it is 254.8 eV. In the neon and argon cases, the ion

perturbers have a charge of +1. For the aluminum case the ion perturbers

have a charge of +12. Three general features of these Lyman-a profiles

can immediately be noticed. First, since the electron broadening of the

unshifted central component is not directly affected by the ion





87

microfield distribution, most of the variation of the central component

observed in these figures is due to variations in the Doppler broadening.

Second, since the wings of the profile are determined almost entirely

by the microfield distribution, we immediately see that the variations

in the wings of the profile reflect the structure of the microfield

distribution functions discussed earlier. Finally, the line profiles

show a significant amount of structure in the region of the shoulder.

The electron broadening of the unshifted component plays an important

role in producing this structure. To see this more clearly, compare

the left hand TR = 4 curve with the right hand TR = 1 curve in each of

these figures. By doing this we isolate the electron broadening effects

since the ion kinetic temperatures (i.e. ion broadening and Doppler

broadening) will be the same for each curve. The electron temperatures

are the only difference between the two curves. Since electron

-1/2
broadening scales as T the electron broadening is greater in the

right hand curve. That is, the electrons. fill in the shoulder on the

right, whereas the shoulder of the left hand curve is quite pronounced.

In addition, there are significant variations in the structure of the

shoulder within each family of curves.

Figures 21 through 23 demonstrate the same T -variation calculations
R
for the Lyman-B profiles of neon, aluminum, and argon. Doppler broadening

effects are more important for these lines than for Lyman-a due to the

increase in the unperturbed transition frequency. Once again, Doppler

broadening is responsible for most of the variations around the central

dip, within a given family of curves. However, by isolating the

electron broadening effects as indicated above, we can see that these

effects also contribute significantly to the structure of the central

dip.





























Figure 14. Line profiles for the Lyman-a line of Ne X, illustrating
various contribution to the complete line profile: (a)
static-ion profile, (b) profile including static ions
and dynamic electrons, (c) profile including static ions,
dynamic electrons, and the Doppler effect, (d) Doppler
23 -3
profile. T = 809.1 eV and n = 1 x 10 cm A is in
Rydbergs.













00


COI









roo


01
~I





101
-gOl 0











-It-I --- 1 I I l v




























Figure 15. Line profiles for the Lyman-a line of Ar XVIII,
illustrating various contribution to the complete
line profile: (a) static-ion profile, (b) profile
including static ions and dynamic electrons, (c)
profile including static ions, dynamic electrons,
and the Doppler effect, (d) Doppler profile.
23 -3
T = 113.2 eV and n = 2 x 10 cm An is in
Rydbergs.














my
9"0 'O Z'O 00 2'O- i'0- 9"0- 9"0 'O Z'O 0"0 2.0- "b0O- 9"0-











ii -i
o
001






S01




------ ^ ---- i__---1 1

i01






























Figure 16. (a) Ne X and (b) Ar XVIII Lyman-a line profiles, each
of which corresponds to the same temperature but
different density. Am is in Rydbergs.













Olxl=U ] Ol xl=







o01










0I -
1-




AD 1'608=1 2'6101=1






























Figure 17. (a) Ne X and (b) Ar XVIII Lyman-a line profiles,
each of which corresponds to the same density but
different temperature.




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PAGE 1

A STUDY OF STARK BROADENING OF HIGH-Z HYDROGENIC ION LINES IN DENSE HOT PLASMAS By RICHARD JOSEPH TIGHE A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1977

PAGE 2

p^ o ACKNOWLEDGMENTS I would like to gratefully acknowledge the support of an NSF Traineeship during the first three years of my graduate study. I would like to thank Professor C. F. Hooper, Jr., for suggesting this problem and for his guidance and encouragement during the course of this work. Also I would like to thank Drs. J. W. Dufty, T. W. Hussey, and F. E. Riewe for many helpful discussions. A special thanks is due Dr. Robert L. Coldwell for providing guidance in the numerical work as well as for lending me several excellent computer codes. I would like to thank Mrs. Yvonne Dixon for typing the final manuscript, and Mr. Woody Richardson for preparing the figures. Finally, I would like to thank my wife Janette and my parents for the special understanding they have shown during the long years of this work. 11

PAGE 3

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ABSTRACT v SECTION I. LINE SHAPE FORMALISM 1 Introduction 1 Causes of Line Broadening 2 The Stark Effect for Hydrogenic Ions 5 A Model for the Plasma 7 The Line Shape Expression 9 Time Scales and the Line Broadening Problem n Factorization of the Initial Density Operator 13 The Line Shape in the Quasi-static Ion Approximation 16 The Liouville Representation 19 A Perturbation Expansion for R r (w) 20 A Computational Form for H(w) 23 The Line Shape Formula 24 SECTION II. ELECTRIC MICROFIELD PROBABILITY DISTRIBUTION FUNCTION 28 Introduction 28 The Formal Calculation of T(£) 30 Introduction of Collective Coordinates 33 The Collective Coordinate Calculation 36 Asymptotic Microf ield Distribution Function 51 SECTION III. DISCUSSION OF THE RESULTS 54 Introduction 54 Electric Microf ield Distribution Functions 55 Stark Broadened Line Profiles 85 Validity Criteria for this Theory 116 SECTION IV CONCLUDING REMARKS 119 APPENDICES 121 A. THE INTERACTION V 122 er 111

PAGE 4

page B. THE ALGEBRA OF TETRADIC OPERATORS 12 4 C. QUANTUM MECHANICAL PERTURB ER AVERAGES 130 D. THE PARABOLIC REPRESENTATION 140 E. CALCULATION OF RADIATOR DIPOLE MATRIX ELEMENTS i 42 F. THE MANY-PARTICLE FUNCTION r(Au)) 146 G. A COMPUTATIONAL FORM FOR THE ATOMIC FACTOR 150 H. NUMERICAL PROCEDURES 15 4 I. TABLES OF ELECTRIC MICROFIELD DISTRIBUTION FUNCTIONS ... 167 J. TABLES OF STARK BROADENED LINE PROFILES 220 LIST OF REFERENCES 247 BIOGRAPHICAL SKETCH 250 IV

PAGE 5

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A STUDY OF STARK BROADENING .OF HIGH-Z HYDROGENIC ION LINES IN DENSE HOT PLASMAS By Richard Joseph Tighe June 1977 Chairman: Dr. C. F. Hooper, Jr. Major Department: Physics Stark broadened x-ray line profiles from highly ionized hydrogenic ions, radiating while immersed in a hot dense plasma, are studied. The broadening effects produced by the ions present in the plasma are treated through the use of static electric microfield distribution functions. The microfield distribution functions employed have the following properties: (i) the radiating hydrogenic ion may have any net charge; (ii) the perturbing ions and electrons in the plasma may have different kinetic temperatures; (iii) two species of ions, of charge z and z„, may be present in any given ratio. Electron broadening effects are treated by the Second-Order Relaxation Theory as developed by O'Brien and Hooper. Effects upon the electron perturbers due to the fact that the radiator is highly charged are included through the use of Coulomb wave functions for the perturbing electrons. The present work makes extensions needed for the calculation of line profiles from higher Lyman series members. A cutoff procedure is employed to simulate the effects due to correlations among the electron perturbers. v

PAGE 6

Results are presented in graphical and tabular form for both the electric microfield distribution functions and for the broadened x-ray line profiles. The figures showing the microfield distribution functions demonstrate the following effects: (i) when the value of the radiator charge is increased, the microfield peaks are raised and shifted to lower field values; (ii) an increase of the parameter T the ratio of electron kinetic temperature to ion kinetic temperature, R causes a similar shift to lower field values; (iii) at low a. values, the microfields show a sensitivity to small concentrations of high-z ion perturbers. The behavior noted for the microfield distribution functions carries over to produce similar effects in the line profiles. Although Doppler broadening generally dominates the center of the line profiles, the wings of the profile demonstrate clearly the behavior noted above for the microfield distribution functions. In addition, the structure near the line center (i.e. the shoulder in Lyman-a and the dip in Lyman-3) shows sufficient sensitivity to the above effects so that it might be valuable as a plasma diagnostic tool. Directions for future research are discussed. vx

PAGE 7

SECTION I LINE SHAPE FORMALISM Introduction Studies of plasma-broadened spectral lines have been quite successful in determining temperatures and densities for laboratory and astronomical plasmas. Current investigations involving imploding plasmas indicate that the temperature and charged-particle densities for these plasmas will be much higher than for those discussed in the above references. The purpose of the present work is to extend linebroadening theories developed previously so that they may be used to develop an effective temperature-density diagnostic for the parameter ranges expected in plasmas produced in the early laser-implosion experiments. We consider the broadening of x-ray spectral line profiles emitted by highly-ionized hydrogenic ions immersed in hot dense plasmas. In Section I, we outline the development of a line-shape formalism which extends 9 the work of O'Brien and Hooper to the calculation of higher Lyman series members. Again following O'Brien and Hooper, we present in Section II the calculation of electric microfield distribution functions which allow for different ion and electron kinetic temperatures as well as for multiply-charged ion perturbers in varying density ratios. Calculated results for microfield distribution functions and for broadened line

PAGE 8

profiles are presented graphically in Section III; the various approximations are also discussed. Section IV contains a summary, together with concluding remarks. Causes of Line Broadening The most important mechanisms which cause broadening of spectral line profiles are: (1) Natural broadening; (2) Doppler broadening; and (3) Pressure broadening. (1) In an atomic radiative process, the interaction between the excited atom and its own radiation field gives rise to an uncertainty in the atomic levels. This uncertainty in the excited-state energy levels means that isolated atoms will emit lines having a finite width (referred to as the natural width). For hydrogenic ions of nuclear charge Z, the natural width is of order, 2 2 r a (aZ) Z Ryd (1-1) Nat 2 2 where a is the fine structure constant. Since (aZ) Z Ryd gives the order of magnitude of the fine structure splitting for a hydrogenic ion, this equation indicates that r,„ is a factor a smaller than the fine H Nat structure splitting. For the cases considered here, pressure broadening effects are at least of the order of the fine structure splitting. This means that natural broadening is at least a factor of a smaller than pressure broadening in our cases. Consequently, we may safely neglect natural broadening in the present work. (2) The thermalmotion of the radiator produces another type of broadening called Doppler broadening. This effect is due to the Doppler shift of the radiation emitted by a moving source. If the velocity distribution of the radiator is assumed to be Maxwellian, the resulting

PAGE 9

Doppler line profile is given by 2 / \ 1 r (dl-Un) Doppler (U) TT* 6XP { ^} (I 2) where, ft 2 2kT 2 ^t ^ p = co (1-3) Mc and T is the kinetic temperature of the radiator. Here, the temperature T is the ion kinetic temperature since the radiator is assumed to be in equilibrium with the perturbing ions. The mass of the radiator is M; to is the frequency of the unperturbed transition. The Doppler half-width is of order 2kT. ** v >\i i i. 1 'Tlnnnlfr n i Mc Doppler 2 % (1-4) 10 3 6.^ Z 3/2 Ryd (1-5) where 6. is kT in keV. From this estimate, it is easily possible to obtain Doppler widths of the order of electron volts. In some cases, Doppler broadening will dominate all other broadening mechanisms. It may be shown (see Equation 8.8 of Reference (12)) that, for the case where perturber collisions have no influence on the radiator's trajectory (that is, the momentum transfer due to the collisions during the time of radiation is negligible) the final Doppler-corrected line profile is given by 00 f I (to) =\ !„,. (a)') •_ (oj-co') dco' T ,. Stark Doppler (1-6) This correction is always valid when the condition, perturber mass << radiator mass (corresponding to electron perturbers) is met. The Quasistatic Approximation for the ions, introduced below, allows us also to

PAGE 10

neglect the influence of ion collisions on the radiator's trajectory. Hence, the range of validity of the above decoupling of the two broadening mechanisms is the same as that of the Quasi-static Approximation for the ions (which is well met in the present work see below) (3) Pressure broadening effects are due to interactions between the radiator and the particles surrounding it. These interactions remove degeneracies and shift the radiator energy levels. The origin of the term "pressure broadening" lies in the fact that this type of broadening is sensitive to the density (or pressure) of the particles making up the surrounding gas. For highly ionized gases, Stark broadening is frequently the most important pressure broadening mechanism. However, we may encounter situations where Doppler and Stark broadening are equally important or perhaps where Doppler broadening dominates. In Stark broadening, the radiator energy levels are shifted by the Stark effect due to the electric fields of the surrounding ions and electrons. The radiator-perturber interaction in the dipole approximation is given by, V T = eR'e. (1-7) Int. 1 where eR is the dipole moment of the radiating ion (see also Appendix A), -> and e. s the electric field at the site of the radiator due to the i-th l perturber. Stark broadening in highly-ionized gases is the primary focus of this work. Another type of pressure broadening that we must mention here is Resonance broadening. This type of broadening occurs when the excited radiator is perturbed by ground state atoms or ions which are identical to the radiator. Also called "self-broadening," this mechanism is primarily

PAGE 11

important when there is no significant ionization present. A multipole expansion of the interaction between two neutral hydrogen atoms gives as its lowest order nonvanishing contribution the (quadrupole) resonance interaction [see Equation (4-98) in Reference (1)]. When the radiators are hydrogenic ions, however, the quadrupole term is no longer the lowest order nonvanishing contribution. The dominant contribution to broadening between ions is produced by the Stark effect (the dipole term) given in Equation (1-7) above. In the present work we consider the broadening of hydrogenic-ion lines and encounter the situation where radiator and perturber may be identical. Since Resonance broadening results from terms higher than dipole in the multiple expansion of the radiator-perturber interaction, we neglect it in the course of this work. In this case, the neglect of higher terms is an approximation whose validity improves as the value of the radiator charge increases. Correlation effects due to the monopole interaction of two highly charged positive ions produce an effective repulsion. This repulsion reduces the probability that higher multipole terms will make a contribution (i.e. the repulsion makes close collisions less likely). In summary then, although the radiator and perturbing ions may be identical, we here neglect the Resonance broadening and consider only the Stark broadening interactions. The Stark Effect for Hydrogenic Ions In a hydrogenic ion, the interaction between the nucleus and the single bound electron is a pure Coulomb interaction. This fact has several important consequences. The most obvious consequence is that the Coulomb problem can be solved exactly: we know the eigenf unctions and

PAGE 12

the eigenvalues for the atomic problem. This solution displays the well-known Ji-degeneracy (accidental degeneracy) of the discrete eigenvalue spectrum for the pure Coulomb system. Thus we must use firstorder degenerate perturbation theory to calculate the energy level shifts due to the linear Stark effect in the hydrogenic case. When this calculation is carried out, we obtain the following result for the shift of the level specified by n,q,m (for a discussion of the parabolic representation, see Appendix D) : AE | -|nqe, (1-8) where a is the Bohr radius; Z is the nuclear charge; n,q are parabolic quantum numbers, and e is the magnitude of the electric field. In the general case of a non-Coulomb interaction, the ^-degeneracy is absent. This means that in nonhydrogenic systems there are no level shifts in first order due to the linear Stark effect. For these systems the perturbation calculation must be carried to higher order with the result that for nonhydrogenic systems, the level shifts due to the Stark effect are smaller than in the case of hydrogenic systems. Thus hydrogenic systems are more sensitive to pressure broadening effects through the Stark effect than are nonhydrogenic systems. For this reason, hydrogenic radiators are the optimum choice as emitters of broadened profiles for use in plasma temperature-density diagnostics. At this point, we must pause to introduce a validity criterion for the use of the linear Stark effect in the present work. In the above discussion we made no mention of the fine structure contribution to the level shifts in hydrogenic ions. An assumption made throughout the present work is that the fine structure splitting of the energy levels is negligible compared to the Stark shifts. When this assumption breaks down, the

PAGE 13

problem of computing the level shifts becomes very complex and requires 13 a numerical solution. The numerical calculation yields shifts, which are not linear in the electric field (except, of course, in the large field limit) By equating the Holtsmark shift with the fine structure splitting of the upper state of the transition, we may determine a validity criterion for the use of the linear Stark effect in line broadening calculations. If n > 2.4 x 10 16 (V 5/2 (1 1 ) 372 (1-9) e n n where n is the electron number density; Z is the nuclear charge of the radiating ion; and n is the principal quantum number of the upper state, then the linear Stark effect is appropriate for our calculations. As we will discuss in a later section, several cases considered in this work approach the limit of validity given above. A Model for the Plasma A plasma is an electrically neutral gas in which the temperature is high enough so that some degree of ionization is present. For the purpose of this work, a plasma is defined to be a gas which is ionized to the extent that Stark broadening is the most important pressure broadening mechanism. 4 Although normal laboratory plasmas at temperatures -10 K may have a significant number of neutral atoms present, the work to be reported here is aimed at treating dense, hot plasmas of laser-fusion experiments in which case all neutrals will be strongly ionized. Therefore, our plasma is composed of electrons and ions of various ionization stages. The radiation we choose to study is the Lyman series of lines from the hydrogenic ionization stages of various heavy atoms. The specific atoms

PAGE 14

8 are selected on the basis of the predicted population densities of their hydrogenic ionization stages. In the formal development which follows we employ a model in which the plasma contains only one radiating hydrogenic ion. This model assumes that the individual radiative processes which add together to produce the line profile occur independently of one another. For the present work, this is a reasonable assumption since radiative processes 15 in plasmas will, in general, add incoherently. In the above model, it may appear that we are neglecting all the contributions from processes where two radiators interact with each other. However, the dominant contribution from such processes, namely the Stark broadening interaction, is contained implicitly in the treatment of ion broadening through use of electric microfield probability distribution functions (see the discussion of this procedure below) This statement is based on the fact that in the microfield calculation, the ion perturbers are treated as classical point particles with no attention given to quantum problems such as particle identity or internal structure. This means that even though a perturber may be identical to the radiator, it produces Stark broadening effects which are independent of its internal energy state. An additional feature of the model is that we fix the radiator at the origin of the coordinate system. This allows us to concentrate on Stark broadening and add Doppler broadening at a later step. In summary, our model consists of the single hydrogenic radiator fixed at the origin, surrounded by a gas of perturbing electrons and ions. When this model is employed in an ensemble average to compute a line profile, the result will represent a profile emitted by an actual system containing many essentially independent radiators.

PAGE 15

The Line Shape Expression The total power emitted by a quantum system in a spontaneous electric dipole transition is given by 4 4to_, „ ,2 a \ I | < b | exj j a> J ; (1-10) J El 3c 3 oj is given by the Einstein formula, ab W ab = ^."V 7 "' (I_11) and c is the speed of light. Since we are considering an emission process only, the initial state of the system has energy E and the final state 3. has energy E where E > E, The sum over j includes all the components of the dipole moment of the system. To make the connection between Equation (1-10) and the spectrum emitted by a plasma, we must carry out an ensemble average of this equation. This means we must average over the initial states a by including a weight factor p „ In addition, we must sum over final states b_. Then the power 3. spectrum emitted by a radiating system becomes 4w „ P (w) = — ^I 6(u)-oj ) |r p (1-12) j ab a 3c a,b,j The delta function ensures that the transition conserves energy. We now define the line shape function I(u) : I(io) =1 6(d) oj ) | | 2 p (1-13) a,b,j Then P(w) = (4w /3c ) • I(w) where I(co) has the property of being ab applicable to absorption as well as emission processes. Now consider the Fourier transform of I(co)

PAGE 16

10 $(t) = \ dtoe "" l I (to) s or 't) ^ : $(t) = L e ab p (1-14) a,b, n a j From this we can see that $(t) = [$(-t)]*. Consequently I(to) may be expressed by oo I (to) = -tt" 1 Re \ dt e iut $(t) (1-15) o Also $(t) may be written in the form l(t) = I e'^ab* 1 - p (1-16) a,b -> where d is the electric dipole moment of the total system. We define p 9. to be an operator that acts only on the initial states a so that we may move it inside the matrix element: *(t) = I e" 10J ab t - (1-17) a,b Inserting the definition of to n we obtain ab $( t ) = I e~ i(E a" E b )t/ri - (1-18) a,b Since the E's are the eigenvalues of the Hamiltonian for the system we may also move the exponentials inside the matrix element: (t) I . (1-19) a,b •> -> -J - (1-20) a,b where the time-development operator T(t) is defined by T(t) = exp {-iHt/Ti} (1-21)

PAGE 17

11 The sum over the states a and ib Is just a trace operation: (t) = Tr {d-T(t)pd T (t)} ; (1-22) $(t) is thus obtained by performing a trace over the states of the total quantum system. We may observe that (t) is the autocorrelation function for the electric dipole moment of the system. Time Scales and the Line Broadening Problem We demonstrated above that 1(a)) can be written in terms of the following transform: I(w) = -ff" 1 Re \ dt e 1U)t *(t) (1-23) \ Since $(t) is an autocorrelation function, we expect it to be a smoothly decaying function for large t. This means that for t >x (where x is R K some critical time for the broadened emission process) the exponential will oscillate so rapidly that the contributions to the transform will vanish. This is a statement of a familiar property of Fourier transforms, namely Aid T n < 2ir (1-24) R Most of the contribution to I(w) will come for times t less than the critical time x This suggests that an appropriate characteristic time R for the broadened emission process will be defined as Tr e ^w • 25) where Ato is the width of the resulting (experimental or calculated) o £ ar k. Stark line shape. Processes within the system having characteristic times greater than t will take place so slowly that they may be regarded as static. R

PAGE 18

12 Approximations of this type are useful so long as their region of validity covers most of the line profile. This will nearly always be the case for the motion of the ion perturbers. A characteristic time for the ions is given by T_ = -~ ^ i, (1-26) Ions v to f v av p (Ions) where A^ is the Debye length for the plasma: V is the average thermal D av velocity of the ion perturbers. Essentially, T T gives the duration of r Ions b a radiator-ion encounter: lo T N is the plasma frequency for the ions. p (Ions) In most cases of interest, the ion plasma frequency tii ,_ is much p (Ions) less than the width of the line profile: (0 T K < Aw_„ T T > T n (1-27) p (Ions) Stark; Ions R In the formal development, we introduce an approximation in which ion motion is neglected and ion broadening effects are treated by averaging over static configurations of the ion perturbers. In this approximation (known as the Quasi-static Ion Approximation) the ions provide a static electric field which splits out the atomic levels of the hydrogenic radiator. The average over the ions is carried out in a final step of the line profile calculation by an integration over an electric microfield probability distribution function. There are, however, ions in the plasma whose velocities are greater than V Characteristic times for these (dynamic) ions are less than T„ av J R and the effects of their motion may not be negligible. Equations (1-24) and (1-26) allow us to make an estimate of the limits of validity for the Quasi-static Ion Approximation: Aw t t < 2tt Ions — or,

PAGE 19

13 Aw.< to ,, N (1-28) — p (Ions; By this we mean that for frequency separations Aw (measured from the unperturbed transition frequency) less than oj broadening effects due to dynamic ions will become important. For all the cases considered in this study, the ion plasma frequency is so small compared to the width of the profile that ion dynamics are important only over a small frequency range at the very center of the profile. Therefore, ion dynamic effects are neglected throughout the course of this work. When we apply the same analysis to the perturbing electrons, we observe some striking differences. First, the electron plasma frequency is -42 times greater than the plasma frequency for protons. In the case we study here, the electron plasma frequency falls out in the wings of the line profiles. Electron dynamics then are important over most of the line profile and static approximations are of interest only in the far wings. This means that the time dependence of the electric field due to the perturbing electrons requires quite a different treatment from the one employed when dealing with the ions. In the present work, electron broadening is treated by a second-order time-dependent perturbation calculation. Thus a hierarchy of time scales is present for the line broadening problem: T < T < x (I 29) Electrons R Ions This hierarchy gives justification for the two differing theoretical approaches to the broadening produced by the electron and ion perturbers Factorization of the Initial Density Operator The initial density operator p is chosen to be the canonical

PAGE 20

14 Boltzmann operator given by p = e~ BH / Tr {e _6H } (1-30) where g = (k T)~ and H is the Hamiltonian operator for the total system. B H = H + H + H + V + V. + V (1-31) r l e er lr ei Here, H is the unperturbed Hamiltonian for the radiator, H. is the kinetic energy of the' ions plus their (ion-ion) interaction energy, and H g is similarly defined for the electrons. The V's are the respective interaction energies. We now introduce an approximation present in most line broadening calculations. We assume that the operator p has only diagonal matrix elements between the initial states of the transition. Furthermore, we assume that p may be factored in the following manner: P = P P-P r i e (1-32) In order to define the factors on the right side of Equation (1-32) we present the following argument. We first regroup the terms in Equation (1-31). After performing this regrouping, we may write H in the following form „0 H = H + H. + H +V (1-33) r i e er where H = H + V a) (1-34) r r lr and H. = H + vf 0) + V (1-35) 11 lr ei In this expression, V. is the monopole contribution to V. namely V< 0) = X z SL. (1-36) lr A p r P

PAGE 21

15 In this equation, y( =z ~l) is the net charge of the radiator and z is the charge of the ion perturber. It is important to note that in this form this term is independent of radiator coordinates. The term V. gives the dipole contribution to V. and is combined ir ir with H since it contains the radiator position operator: V a) = eR-e. (1-37) ir 1 Terms in the multipole expansion of V. that are higher than dipole are not considered since the present work will employ the dipole approximation to the radiator-perturber interactions. If we now neglect the V term in Equation (1-33) we may obtain the approximate factorization of Equation (1-32) The following definitions are possible: p = e" 3H r / Tr {e~ 3H r} (1-38) r r p. = e" 3H l / Tr {e" 6H i} (1-39) where the prime here indicates that we consider the term V to produce a 1 ei Debye shielding effect in H.. Therefore, we drop V and now employ only shielded interactions in order to leave El independent of electron l J 17 coordinates. p e = e~ 6H e / Tr {e~ 6H e} (1-40) Neglect of the term V implies that while the electrons produce broadening effects, they do not alter the initial distribution of radiator states. The factorization of p is a procedure which need not appear in the formal development of modern line broadening theories based on kinetic 1 8 theory. These theories, however, sometimes make a factorization for reasons of computational convenience,

PAGE 22

16 The Line Shape in the Quasi-static Ion Approximation Combining Equations (1-22) (1-23) and (1-32) we obtain for the line profile oo I(io) = ^ _1 Re \ dte lwt Tr {d T(t)p p.p dT + (t)} (1-41) j r i e o The trace is to be evaluated using states of the entire quantum system of radiator, electrons, and ions. The most convenient set of states for this calculation is a set of product states. Each product state will consist of a one-particle state for the radiator and a many-particle state for the gas of electrons and ions. We now define this product state: |a> = | y> x |a> (1-42) where p represents the one-particle radiator state and a represents a many-particle state for the electrons and ions. 4Now we introduce the familiar restriction that d is the dipole 1 + moment operator for the radiator only. Furthermore, we restrict d to have nonzero matrix elements only between a specific set of upper and lower states; hence we will consider only line radiation of a specific transition, In our case, we consider only the Lyman series of transitions for hydrogenic ion radiators. Now the Quasi-static Ion Approximation is introduced as follows. We first consider the commutator of HT with H as given in the expression ih 4~ Ht [H:, H] (1-43) dt l l — 1 18 Smith has shown that this commutator is proportional to m. Since the thermal velocity of the ions also scales as m. it is evident that the limit of infinitely massive ions corresponds to the case where the ions are static. This same limit also implies that the commutator above vanishes:

PAGE 23

17 [H:, H] = (1-44) Several results follow immediately from this equation. The first consequence of Equation (1-44) is that we may write T(t) = T (t) T r (t) ; T + (t) = T^ (t) T i + (t) (1-45) where T.(t) = eitH i /B ; T (t) = e itH er /E (1-46) i er and H = H + H + V (1-47) er r e er Another consequence of Equation (1-44) is that p. commutes with p r and T (t). Using these results we may simplify Equation (1-41), with this er result for the line profile: CO I(u>) = u" 1 Re \ dte a)t Tr {p.d T (t)p D d T* (t)} (1-48) J l er r e er o Note that T. operators have been commuted and canceled. Next, we insert x > + a delta function 5(e e.) into the trace of Equation (1-48), along with > an integration over the variable e. This step is valid so long as the delta function (which contains the ion coordinates) commutes with all the other operators inside the trace. The vanishing of the commutator in Equation (1-44) ensures the validity of this step. After this insertion, we are free to reverse the order of the trace and integration operations: oo co \ de 7r~ Re \ o o + -y I(w) \ de TT "'" Re \ dte 1W Tr {p.6(e e.) .d T (t)p p d T + (t)}. (1-49) er r e er The effect of inserting the delta function here is that we may replace e as it appears in T (t) and p by the integration variable e. After i er r making this replacement we are free to perform a partial trace over the

PAGE 24

18 ion coordinates in Equation (1-49). The line profile in the Quasi-static Ion Approximation is then given by I(u) \ de Q(e) J(o),e) (1-50) where Q(e) Tr. {p.6(e e.) } (1-51) defines the electric microfield probability distribution function for the static ions: Q(e) gives the probability of finding an electric field £ at -> the site of the radiator. Now J(w,e) is defined in terms of a trace over radiator-electron product states. oo J(u,e) = tt" 1 Re \ dt e lt0t Tr {d-T (t)p p d T + (t)} (1-52) J er er re er -*The operators T (t) and p are now functions of H (e) where r er r r ->He H (e) = H + eR'e (1-53) r r -* + and £ is the electric field due to the static ions. Here J(w,£) gives 4the electron broadened profile emitted by an ion in an external field £. Ion broadening effects are included when the ion microfield integration of Equation (1-50) is performed. Before continuing, let us summarize the approximations we have made thus far: i) The dipole approximation to the radiator-perturber interactions [Equation (1-7)]; ii) The factorization of the density operator [Equation (1-32)]; iii) The Quasi-static Ion Approximation [Equation (1-50)]. These approximations have been standard line broadening approximations, the validity of which will not be tested in this work. It should be pointed out, however, that the first two restrictions may be, in principle,

PAGE 25

19 removed. (See References 4, 17, 19, and 20.) Our previous discussion of ion dynamics and the validity range of the Quasi-static Ion Approximation [see Equation (1-28) and the subsequent paragraph] indicated that for the cases considered here, the Quasi-static Ion Approximation may be used with considerable confidence. The Liouville Representation In this section we introduce a notation which formally simplifies the calculation of J(w,e). Consider an arbitrary operator f whose time dependence is generated in the following manner: f(t) = T(t)f(0)T + (t) e iHt/H f(0) e 1Ht/H (1-54) If we take the time derivative of f(t), we obtain, ih^ f(t) = [H,f(t)] = Lf(t) (1-55) This equation defines the Liouville operator L and its operation on an arbitrary operator. We may solve this equation formally for f(t): f( t ) = e iLt/R f(0) • (1-56) The Liouville representation as introduced in Equations (1-55) and (1-56) (see also Appendix B) is essentially the "doubled atom" representation 2 of Baranger. The advantage of this notation is that it gives a "shorthand" with which to carry out formal manipulations. That is, we may formally perform the integration of Equation (1-52) with the following result for J(oj,e): J(u,e) = -TT -1 Im Tr {d'K ( w )p p d} (1-57) er er r e where K (to) is defined by er 5iu)t -il dt e e K (co) = -i | dt e iU,L e~ iLt/E = {w-L/h} X er

PAGE 26

20 The operation of L is defined by Lf = [H f ] (1-59) er where H is defined in Equation (1-47) We now formally carry out a er partial trace over the states of the perturbing electrons. J(u),e) = -Trim Tr {t R r (ai)p r 1} (1-60) R Go) = Tr {K (ai)p } = (1-61) r e er e er The last equation defines our use of the bracket notation to indicate an average over the perturbing electrons. The operator R (to) is called the effective-radiator-resolvent. Although this operator is a function of radiator coordinates only, it includes broadening effects due to the dynamic electron perturbers. Equations (1-58) and (1-61) give the formal definition of R (oj) The goal of sections to follow is the development of a useful method for calculating R (oj) A Perturbation Expansion for R (lo) In this section we describe a method for approximating R (to) the effective radiator resolvent operator. The expression we are interested in is given by R.(u>) = = <{a)-i./H} _1 > (1-62) where L is given by L L Q + \ I (1-63) This corresponds to

PAGE 27

21 H = E n + XV = H + H + XeR-e + XV (1-64) er I r e er Here X is a coupling constant introduced for convenience (later we will let it equal unity). The procedure we now follow is the same as that employed by Duf ty in Reference (21) The first step in this procedure is to make the following definition: = {io-L/lT XL. /ft HU)}" 1 (1-65) er r lr This expression now formally defines the operator n'(to) : H(co) is a function of radiator coordinates only, but contains broadening effects due to the perturbing electrons. We now assume that the operator H(co) has an expansion in powers of the coupling constant, namely, H(u>) = H (0) (u>) + XH (i) (o)) + X 2 H (2) (a)) + ... (1-66) The next step involves expanding both sides of Equation (1-65) in increasing powers of X and equating like powers to identify terms in the perturbation expansion of tf(w) The left side of Equation (1-65) may be expanded in a Lippmann-Schwinger expansion by employing Equations (1-62) and (1-63): = + Xft" 1 +X 2 K" 2 + ... (1-67) Several identities given in Appendix B help to simplify this result. = R(oj) + Xn~ i R(u))R (a)) er r r I r +X 2 E _2 R(a))< L i R (oj)L i >R(w) + ... (1-68) where

PAGE 28

22 R(a)) {oj-L^/R}" 1 ; R ((o) = {oj-^/E}" 1 (1-69) The right hand side of Equation (1-65) may be expanded in a Taylor series in A with the aid of the following operator identity. For an operator A, d -1 -1 dA -1 „ ^A =-A -A (1-70) The expansion of the right hand side of Equation (1-65) is given by {03-L/n AL. /K H(w)}' 1 = {w-L/R f/ (0) (a)) } _1 r lr r + A{w-L/R H (0) (a,)} _1 [L. r /h + H (1) ( W )]{oD-L/h H (0) (w) r 1 + A 2 {w-L/R H^Ca))} -1 [L. r /h + H (1) (co)]{oo-L/K tf (0) (co) T 1 x[L ir /E + n' (1) ( u ) ] {u>-J, /E H (0) (u) } _1 + A 2 {w-L/R H^Wf 1 H (2) (a>){ai-L/ti H (0) ( W )} _1 + ... (1-71) Now by comparing Equations (1-68 and (1-71) we may identify the terms H (oi) appearing in Equation (1-66) : H (0) () ; (1-72) hH (1) (a) = ; (1-73) H 2 H (2) (o)) = R(u>) (1-74) Before proceeding, we state here (and prove in Appendix C) that, as a consequence of making the dipole approximation for the radiator-electron interation V (see Appendix A) the indicated average in Equation (1-73) vanishes : = (1-75) er

PAGE 29

23 This means that if we retain the lowest order nonvanishing contribution, f/(w) is given by H(u>) = R~ 2 ) L > (1-76) er er with the final result for R (w) : R (to) = {o)-L/h L. /h-H~ 2 }~ 1 ; (1-77) r r lr er er R (its) expressed in this form is referred to as a second-order resolvent, r A Computational Form for H(co) In this section we wish to develop a form for the second-order result (2) n (to) which is convenient for computation. That is, we wish to develop a matrix representation of the following expression, 2 (2) h fT ; (w) f =) L f> (1-78) er er where f is an arbitrary radiator operator. We may insert the integral definition of R_(w) and obtain CO dt e h 2 H (2) (co) f = -i \ dt e^ (1-79) er er CO -I .„ icot „ -iH„t/n TT -iH n t/n TT iH_t/ti TT dt e { er er er er b -iH_t/h t TT !H rt t/n -iH ft t/h T7 iH.t/h Tr - + } /T onS er er er er (1-80) We concentrate on the first term in this expression. Denote this term by W(w)f; then, 00 n W(oi)f = -l I dt e (1-81) J er er o We now take matrix elements of this expression between free radiator

PAGE 30

24 21 A eigenstates. Continuing m this manner, we may extract a matrix representation of the tetradic operator (see Appendix B) W(ai) The 9,22,23 result of this calculation is' H 2 W(co) = -i<5 t. f A -^" t A %"v W "V'v and >p' Ct) = e_iHet/K Vv elHet/R (I 83) The calculation of the electron-averaged quantities appearing in Equation (1-82) is discussed and carried out in Appendix C. The result for the full expression in Equation (1-80) is given by E 2 H (2) (w) =6 E R ,,-R" „ r(Aoj „ ) w „ up y u' y v -6 E f ,,-R „ r(-Ato „) py .i v'v" v v yv" + R ,-5 {r(-Aaj ,) r(Aoi )} (1-84) pp VV yv' y v In this equation, R is the dipole moment of the bound radiator electron. The complex function T(Aco) contains many-particle effects due to the perturbing electrons. Its definition and the details of its calculation are given in Appendix F. The calculation of the radiator matrix elements appearing here is given in Appendix E. The Line Shape Formula We now incorporate the results of our calculations into the line shape expression. Recall that the line profile is given by DO I(u)) = l de P(e) J(tt>,£) (1-85)

PAGE 31

25 where P(e) = 4 ire 2 Q(e) (1-86) + + and we have made use of the fact that Q(e) is an isotropic function of e. J(io,e) = -tt~ im Tr (d-R (co)p d) (1-87) r r r y The effective radiative resolvent operator R (a) contains averaged electron broadening effects: R r (u) = {u)-L/h L r /n H(oj)} -1 (1-88) To second order in the radiator-electron interaction, H(to) is given by Equation (1-84). We now insert free radiator eigenstates in order to evaluate Equation (1-87). J(oj,e) = -7T Im {Z Z . [R (ic)J . (1-89) i i f r pv;y* v 1 i*r This equation is the final form of Equation (1-12). These two equations are still similar in that there is a weighted average over initial states of the transition along with a sum over final states. Previously we indicated that the weighting factor p is a diagonal operator acting upon the initial states of the transition. Also, the matrix elements of J are restricted to have non-vanishing results only between initial and final states of specific transitions. These facts allow us to write for J(w,e): j( Ui e) = -u _1 Im J J f [ • [\(w)] if;i f P ., + • [R] fi;i f < i |d|f> P .,]} (1-90) At this point, we introduce the No-Quenching Approximation. Mathematically, this approximation states that the matrix elements of the time

PAGE 32

26 development operator for the system taken between initial and final states of the transition must vanish. That is, = 0, (1-91) where H is the Hamiltonian for the system. This matrix element is proportional to the probability amplitude for a process where the interactions within the system cause a radiationless transition from the upper state to the lower state. The No-Quenching Approximation, therefore, is invalid when the broadening interactions mix the upper and lower states of the transition. This means that broadening effects must be small compared to the separation of adjacent radiator energy levels. This statement must be considered either to be a weak collision assumption or it must restrict us to calculate only isolated line profiles. We simply state here that if we expand the factor R (a) f ..., f [appearing in Equation (1-90)] according to previous definitions we would find factors like that in Equation (1-91) but with H replaced by an effective Hamiltonian operator containing averaged electron broadening effects. From this procedure we see that the No-Quenching Approximation causes the second term in Equation (1-90) to vanish. We now point out that in this work we consider only Lyman series transitions so that we need not sum over lower states. -1 J(co,e) -n Im E • [R (w) ] p i (1-92) ii 1 where we define [R (co) ] by r ii' ry<-)] 1 ty^n^! • ci-93) By defining D., = • (1-94)

PAGE 33

27 we obtain a simpler form for J(co,c): JCu.e) -iT Im E D.,. [R Cm)].., p., (1-95) ... 1 1 r n l n where D is a scalar matrix we calculate in Appendix E. The sum over i corresponds to ordinary matrix multiplication. The sum over i' represents the desired thermal average. The above restrictions simplify the definition of R (oj) : [R (oj)].., = {Aa)-eeR Z .,/h HCu))..,}" 1 (1-96) r n n n where Aw = w-w. f indicates that we measure frequency in terms of separation from the unperturbed frequency; R is the z-component of the position operator for the single bound radiator electron. H(to).., = h~ 2 r(Aoj) E R..„ • R.„., (1-97) n .„ n l l This completes the discussion of the line shape formula. Actual details of the numerical procedures involved in computing line profiles will be discussed in later sections. In summary, let us list the major approximations made in reaching this point: i) The dipole approximation to the radiator-perturber interactions; ii) The factorization of the density operator; iii) The Quasi-static Ion Approximation; iv) The second-order perturbation calculation for the effective radiator resolvent operator; v) The No-Quenching Approximation; vi) No lower-state broadening (this is not an approximation for the Lyman series)

PAGE 34

SECTION II ELECTRIC MICROFIELD PROBABILITY DISTRIBUTION FUNCTION I I(w) = I de Q(e) J(oj,e) (II-l) Introduction In Section I we indicated the steps involved in arriving at the Quasi-static Ion Approximation. As a result of making this approximation, the line broadening problem is greatly simplified. That is, the line profile may be written Included in the function J(oi,e) are the electron broadening effects, calculated for the radiator placed in the static electric field of the perturbing ions. This calculation is discussed in detail both in Section I and in the Appendices The electron-broadened profile is averaged over all possible values of the ion field to produce the final Stark profile. This average is performed when we integrate over the electric microfield probability distribution function Q(e). The probability Q(e) of finding -4an electric field e at the site of the radiator, was defined in Equation (1-51). We now write Q(e) in the following form: Q(e) = z" 1 I. ..I expl-^VCr^...,^)} 6 ( E\ ?.)dr" N (II-2) Here z is the conf igurational partition function and we average over configuration space; V(r ...,r ) is the total potential energy of the ions, including their interactions with the radiator; and e. is the field at the -128

PAGE 35

29 -* site of the radiator due to the perturber located at r : ->*•-*• -> E. V. V(r_ ...,0 (II-3) i x 1> N The usual minus sign is cancelled here because we express the gradient ->in terms of r x At this point we wish to recall some of the features of the model that we employ when performing the microfield calculation. There is a hydrogenic ion (radiator) of charge x e fixed at the origin of our coordinate system. The plasma perturbers consist of electrons along with two species of ions of charges Z and Z„, respectively. Overall charge neutrality of the plasma is expressed by the following relation n = Zn, + Z_n„ (II-4) e 11 2 2 where n is the number density of the electron perburbers and n (n„) is the e 1 z number density of the charge Z (Z ) specie of ion perturbers. In addition to providing charge neutrality, the electron perturbers are assumed to produce a Debye shielding of the various interactions between the ions in the plasma. That is, in the present work, we restrict our consideration to the "low-frequency" microfield distribution function. We insert the integral definition for the delta function in Equation (II-2): 'H Ke) : \...j (2w) _3 exp{-B i V + i£-(eEe ) }d£dr (11-5) Since £ is an arbitrary vector, Q(e) cannot depend on the direction of £. We may then perform the angular integration, arriving at the relation, oo Q( e ) = (2TT 2 e) _J \ T(£)sin(e£) £ d£ (II-6) '' o where

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1 \...| exp{-6.\ 30 T(4) = Z J \...\ exp{-6.V -BlIA'*J dr N (II-7) 1 Also, since the ions are distributed isotropically in the plasma we may write, 2 P(e) = 4tte Q(e) CO = 2ir e \ T(£)sin(e£) £ d£ (II-8) M The transform in Equation (II-8) is performed numerically; the calculation of T(Jt) forms the principal topic of this section. The Formal Calculation of T(£) The explicit form for V(r ,r ) is given by N. „2 2 M 2 2 -*• -> lZe i 2 Ze V(r-,...,r„) = E -^— e r iJ /X + E — — e X ^ X 1 N r.. „ r i
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31 The W's represent a short-range part of the central interactions between ions and the radiator. All the noncentral and long-range central interactions are compressed into the function V The a is an effective range parameter and will be discussed at a later point. Now T(£) may be written, r c N i T(£) = Z X \...\ e V n exp{-0 1 W. o where -1 + iX ^ v n W-n } dr. 30 J N J x Ii exp{-3.W + i x Jl-V W dr (11-13) 1 mO mO m m 1 £-V „V (11-14) V = -6.V 0+ i X LVfa We now define the functions X z U;j) = exp{-6.W j0 + i X IV W jQ } -1 (H-15) -*. _.-> -> x (£;m) = exp{-3.W + i X AV W } -1 (11-16) A z 1 mO mO T(£) may be rewritten in terms of these new functions: 1 f f ^0 Nl T(£)=z \...\e U n {1 + x U;J)> dr n {1 + x z (^;m)}dr m (11-17) J J j z l J m 2 The reason for writing T(£) in this form is that we are able to make a cluster expansion of the products in the above equation. Performing this expansion, we obtain V N l N 2 T(£) z ] V--\ e [ 1 + S X z CAjj) + I X z U;m) j i m 2 J-J \ N 2 + z x z U;j) x z Ui + 2 x z U;n)x z tt;m) im Z]L z 2 j 3 m

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32 We now define the functions \ + N 2 n i=j+: dr. II dr n=m+l (11-19) and Q. (£) = T. (£) / T_U) ; T n (£) = T (£) (11-20) jm jm U UU These definitions allow us to express T(£) in a greatly simplified form. 1 [1 +N 1 JC -> T(£) = T Q (£) z [1 + N 1 IQ 10 (A) X z (*;D dr. + N 2 Ui (£)x z 2 a;1) viWyj Q 20 W XX (t;X x II dr. n dr (11-22) 1 1 m i=l n=l where the g's are defined in terms of the O's through an Ursell expansion, we may recognize that Equation (11-21) is just the low order terms in an expansion of the following expression: j m TU) = T (£) z" 1 exp{Z E (-^j-~ h (£) )} (11-23) j m From Equations (11-13) and (11-14) we may identify

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33 n i n 2 z = T.(0) exp {I E (—H^ h C0))> (11-24) U .Tim! ii J m J The final result for T(|) is given by n i n 2 T(£) [T (£)/T (0)] exp {E E (-rf~~ [h. (£) h. (0)])} (11-25) U U ."Jim! im p j m Introduction of Collective Coordi nates'" In this section we proceed with the definition of the collective coordinates. A transformation to collective coordinates will allow us to approximately perform the multi-dimensional integrations appearing in the previous section. We may write the total potential energy V in terms of its Fourier series: ->->-> -> /7tA 2 2 2 -ik-r. 2 2 -ik-r v = 4 IIA E .I y, z i e e ij z„e e mn V k I.,. ^ + — ^ i5 ^ j (kA) 2 + 1 2 Wn (k^) 2 + 1 > -> -•-> -> ->2 -ik-r. 2 -ik.r.„ 2 -ik-r z z e e jm Xz e e ^ jO Xz.ee m0 + S + E I + E -1 ] (11-26) j.,m (kX) + 1 j (kX) Z + 1 m (kX) + 1 The last two terms represent the central interaction terms. The prime on the sum over k means that the k=0 term is to be excluded. This exclusion ensures that the condition of charge neutrality for the plasma is satisfied. Now consider the following expression: -ik* r E -iL = y 1 k -* + •+ (kx)2 + i = I oDo? + i [cos (k r ij } 1 sln (k r ij ] I ( kA )2 + I cos (k.r..) (H-27) *The collective coordinate technique we present here follows closely the development given in References 10 > 23 24 and 25

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34 The second line results from the fact that the sine is an odd function. A trigonometric identity now gives -> -*ik • r -* -> -> y e ij y 1 TtTvV) -i ~ n [ cos (k-r.) cos (k-r.)] k (kX)-^ + i k (u) 2 + 1 J j' J + -s-> + E 5 [sin (k-r.) sin (k-r.)] (11-28) k (kX) +1 X J The above equation is equivalent to -ik-r.. cos(k.r .)cos(k.r .) sin(k. r .) sin(k.r .) E £ 1 J = 2 E 1 J+2 E 1 L (II-29) (kX) +1 k >0 (kX) +1 k <0 (kX) + 1 z — Z" We separate this expression into k > and k < contributions in the anticipation of the definition of the collective coordinates. We now define r -->j cos (kr) ; k > z — S(k-r) = / sin (k'r) ; k < z (11-30) With this definition Equation (11-28) becomes -> -• *->-*-> e~ lk r n 2S(k-r )S(k-r.) E __J_ = jJi 1_ ( (11-31) k (kX)" +1 k (kX) + 1 The total interaction energy V may now be written in terms of the new coordinates S: 9 N 2 2 I ., 1 zz n e V=^~ f E' \ [ E -S.S. V 2 k (kX) 2 + l i,j ** 1 ,22 N 2 H 2 2 2z e 1 4xz.e 2 4)(Z„e + E S S + E S.S n + E S S amn. aiO a m m,n i m 2 ,22 ,22 4z z e It. e 2z e + E — S.S — = — B, — N„ (11-32) a i m a 1 a 2

PAGE 41

35 where 2 2 2 2 a = N^e + N 2 z 2 e (11-33) The last two terms in Equation (11-32) are needed to subtract out the self energy terms (i-j and m-n) which are included in the first two terms. Now we define the collective coordinates X: p X i k = Z i,k X 2,k = E m 2 2' 2z e <* 3 9 2 2 2z 2 e — — S a m (11-34) (H-35) The interaction energy V becomes 9 2 v = 2itX E i k (kA) 2 + 1 2 2 tX l,k + X 2,k + 2X l,k X 2,k + 2 ^4~ S (X l,k + X 2,k> 2] (11-36) The constants in front of this expression may be reduced to the following: (H-37) 2 2 „ ,2 6 z, + Rz„ 2ttA a e 1 J?, v 2 z + Rz 1 where n 2 R = — and 9 = k T n e B e After defining \ (kA) 2 + 1 Y k X l,k + X 2,k we obtain a more simplified form for V 2 2 9 z + Rz„ „ 2 z, + Rz„ Tc k 1 2 k 2 2 2x e k >0 z— Vi Y, 2E/L} (11-38)

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36 The second term is summed over k. > because z — s o = k > z — ; k < z We have manipulated the interaction energy V into a quadratic form involving the coordinates Y The goal of this procedure is to transform the multi-dimensional integrals over spatial coordinates appearing in Equation (11-22) into multi-dimensional integrals over the coordiates Y, The Collective Coordinate Calculation We repeat the definitions of W.„, W „, and V : jO mO 2 X z i e /i W. =-^ear j0 /A ; (11-10) 3 r j0 2 XZ ? e -ar A \o "TV e m0 ; (II n) mO V-V +EW +ZW (11-12) J m The adjustable parameter 'a is well discussed in the literature. 24 25 We will return to the discussion of this parameter at a later point. From an inspection of the above equations, however, we may infer that a essentially measures the contribution of long-range central interactions to the term V_. Proceeding in the same manner as in the previous section, we write 4tta E 2 e tO ^

PAGE 43

e [e z 2 2 + Rz 2 2xz i e 1 KZ 2 k >0 (k*r + o z— S. J 37 (11-40) or E W JO Also we have 2 „ 2 z + Rz e z. + Rz„ l,k k >0 (kA) 2 + a 2 z~ lO m0 3 m 2 2 z^ + Rz^ ) [— -1 e L z + Rz J 9 2 2 2 X e k >0 (kA) 2 + a 2 (11-41) (11-42) The net result of these steps Is that we now are able to write down an explicit expression for the quantity V V V E W E W t0 mO J m 2 2 9 e Z l + Rz 2 2 f [^ThT 1 { ? \ \ + 2 / >n f k (a) \ Y k2 jV, (11-43) k >U k z— where f k (a) a 2 -l 2 2 (kA) + a (11-44) We now need to compute the gradient of V Recall from Equations (11-13 and (11-14) that this operation produces the electric field terms appearing in the original definition of T(£) Furthermore, only the central interaction terms contribute to the electric field at the site of the radiator. By inspecting Equation (11-38) and (11-42) we see that central interactions are contained only in the second term of Equation (11-43). Following O'Brien and Hooper we obtain 2 2 z + Rz v o v o e e t-TTzT 1 2 2 2x e „ z n f k (a) Vk k k <0 (11-45) We now have sufficient definitions to enable us to calculate T„(£)/T (0).

PAGE 44

T Q U)/T (0) M e II N 2 dr. n dr 1 m m 38 (11-46) H : ', :>o) i n j N + ^ -*• dr n dr 3 m J-J N_ N 1 -v 2 -y exp{-3 V + i( X e) 1*7X}H dr n dr^ 3 m J-S N. N, '1 2 -> exp{-3.V„> n dr. n dr i m m I iexp{-|uE t\\ 2 + 2b k (£)Y k ]} J k ; dY k J k k [\ *<" 1 U £ f\\ 2 + 2b k (0) \ ]} J k I d \ (11-47) where 2 2 8 z, + Rz„ e r _l 2, u = ?; [ Zl + rz 9 ] i 1 2 (11-48) and b k U) 9 2 2 f k^\ X 1 ; k >0 ie.(xe) i'k ; k<0 i z— b fc (£=0) = 2 2 2 X g f k (a) \ k >0 ; k <0 (11-49) We have cancelled the self energy terms from the numerator and denominator in Equation (11-47) Here T:' is the' Jacob iari" of the: transformation from the spatial coordinates in the collective coordinates. Defining A" = uA^ and b' = ub we may write

PAGE 45

39 T (£)/T Q (0) = — (11-50) exp{£ [A^Y k 2 + 2b^(0)Y k ]} J k n dY k 2 In Reference 23 the Jacobian is shown to be exp{1/2 Y } plus small correction terms. This reference gives a very thorough discussion of the Jacobian transformation and the collective coordinate method. In the present work Jacobian corrections are neglected. We give here a general result for integrals of the type found in Equation (11-50) : / [VU)] 2 exp {^ I -JE} T n U)/T n (0) = k X + K X K — (II-51) exp {— E ] k 1-+AJ We at once see that the terms for k >_ give equal contributions in the numerator and denominator. These terms thus cancel each other. Also, for k < the denominator goes to unity. We have j tb'U)] 2 T U)/T (0) = exp {E K } (H-52) U ^k<0 1 + A k z 2 2 2 2 2 2 0/ 7, + Rz^ [f (a)]\ Z (£-k) Z exp {-JI 1 2 ] E k 1+ > (H-53) a z x + Kz 2 k <0 L \ z This expression may be evaluated by converting the sum over k into an integral. We obtain the following result: T (£)/T Q (0) = exp {Y L 2 } (11-54) where y is given by y =f T 1 ^ U +u)]~ 2 B (11-55) e and

PAGE 46

B = a 5 u + 2 [1 (1 + u) 3/2 ] a 4 + [2u + u 2 ] a 3 4 (1 + u) [1 (1 + u) 1/2 ] a 2 3 [u + u 2 ]a 40 2 3/2 +2 [(1 + u) (1 + u) : (11-56) We have 9 z 2 + Rz 2 n E r x ^ i -n M £ 0. L Zl + Rz„ J n, i 1 2 1 ) = kT ; 9 = kT ; e e x l 4 Tr 3 "0 e 3 n r n 1 ; a = -y ; e Q = — r o L = V We now turn to the problem of calculating the Q. (A) functions which are defined in Equations (11-19) and (II-2.0) A general form for these functions is given in Equation (2-56) of Reference 23 : 2 Q. U) = ,,-j-m r 1 „ y k \ exp { 2 t TTa; k k E K K ; } k 1 + \ uj_m \...\ exp{4 Z y k A k J J k 1 + A/ (11-57 j -> m E y u k } n dr. n dr k 1 + AT i=l 1 n=l where y, is a new collective coordinate given by y k = a l,k + a 2,k and l l,k 2z e j + — — I S(k-r.) 1=1 2,k 1 n 2 2 2z e m +•-* — — Z S(k-r n ) (H-58) n=l First we consider Q (I): V exp -: • ^ E y ik y k b k Q 10 (*> 2 : 1 + A," 1 + A? k k k \ jexp {""J =T k A k + S E TTJr } dr i (11-59)

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41 2 The term containing y, when summed over all k, is proportional to Sin (kr ) + cos (k-r ) = 1. Therefore, this term is independent of r and may be cancelled between the numerator and denominator. V exp {E 3-™ > Q 10 U) = -— ; T ^ (11-60) -if ( y k b k .+ u j exp (-^rr^> dr i Consider now the sum in the numerator. 2 2 2z e "*" "* t = E 1 S(k-r) W£l 1 I ^K k< k [(kX) + (1 + u)] [(kX) + a ] 1 ; k > z — i( X e) 1 e.l-k ; k <0 (11-61) This sum is converted into an integral with r chosen as the polar axis. We obtain I = z S(x) + i z Lq(x) cos 6 (11-62) where 6 2 2 1 ,_ ,1/2 9 3x ^~2 [e e ] (11-63) i a -(1 + u) an r a -1 1 -aax -(1 + u) ax, q(x) = [— ] [— (e e ) a -(1 + u) x a -aax ,. ,1/2 -(1 + u) ax,-, TT ,. + — (ae (1 + u) e ) ] (11-64 x

PAGE 48

42 where x = r/r Q and 6 £ is the angle between £ and r. By examining Equation (11-62) we see that I will vanish as x tends to infinity. The result of this is that if we take the limit where v-*, N^>, N/u = constant (thermodynamic limit), the denominator in Equation (11-60) tends to unity. We have Q 10 U) = V exp { Z;L S(x) + i Z;L Lq(x) cos 6^} (11-65) Similarly, Q 01 U) = V exp (z 2 S(x) + iz 2 Lq(x) cos £ } (11-66) We now may write down several more definitions in terms of these coordinates: a tt o~l 2 1 -ar._/A -g.W -6 X z x e — e j0 JO = -g Zl W(x) (11-67) where e 2 n T7 / \ e a -aax -6W(x) = -x — -^ e (11-68) i Also 3 i W mO = _ez 2 W(X) ( XI 6 9) The gradient terms are given by -3-> _1 ~ aX i(xe) £ v Q W j0 iLz 1 cos 6 £ {^ — (1 + aax)} (11-70) x i "* "* -aax i( X e) --V W m0 = iLz 2 cos £ {^y(1 + aax)} (11-71 x In terms of dimensionless variables

PAGE 49

43 n, dr = I x 2 dx^ (11-72) 1 z, + Rz n 47T j 3R 2 df2 _„. n„ dr = — — — x dx 7— (11-73) 2 z, + Rz„ 4ir By inserting the above expressions into either Equation (11-21) or Equation (11-22) we may calculate I _(ft). I 1Q (£) = n [h 1() (i) h 1Q (0)] (11-74) After performing the angular integration and rearranging terms we obtain, CO 1 C -5 0/ \ o nr \ sin(Lz.G(x)) t r -\ 3 \ 2 z-,S(x) r -3z _W(x) 1 n i in U = T~T; — \dxxe1 le 1 [ — 1 — ^rr\ 1J 10 z, + Rz J Lz.G(x) sin(Lz q(x) ~ [ -l^qTxT-1]} (II ~ 75) For 1^(£) the analogous procedure gives 3r f : : 1 + Rz 2 ^ 0/ n o tt/ \ sin(Lz G(x) t c\ 3R 12 ZoS(x) -Bz W(x) 2 X 01 C£) z, + Rz \ dxx e 2 {e 2 [ LZ2G(X) U sin(Lz q(x) • 2 Lz q(x) -1]} (11-76) where -aax G(x) = q(x) + ^2~ (1 + aax) (11-77) x We now turn to the calculation of the second-order terms Q (ft) Q (ft), and Q (ft). Corresponding to these, we have the following three cases.

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44 Case (1) : y. = 2e z E S(k-r.) (II-78a) Case (2) y k = 2e 2 -> + z„ S S(k-r ) 2 ., m n=l (II-78b) Case (3) 2e { Zl S(kri ) + z 2 S(k-r 2 )} (II-78c) The collective coordinate evaluation of the second sum in the numerator of Equation (11-57) proceeds exactly as in the evaluation of I in (&). We may immediately write down the following results. Case (1): E A z {S(x ) + iLq(x ) cos 6 } k i + A k i X 1 L + z 1 -{S(x 2 ) + iLq(x 2 ) cos 6^ (II-79a) Case (2): I r + ; = z {S(x ) + iLq(x ) cos 6 } k k + z 2 (S(x 2 ) + iLq(x 2 ) cos 6 } (II-79b) Case (3): £ 1 + = z {S(x ) + lLqCXj) cos e i } k k + z 2 (S(x 2 ) + iLq(x 2 ) cos S^ 2,. (II-79c) Next we need to evaluate the terms k k. We consider y and 1 + AT concentrate on Case (1) : -V -V ---2 •* > y 2 -— z 2 {S 2 (k-r 1 ) + 2S(k-r 1 )S(k.r 2 ) + S 2 (k-r 2 )} (11-80)

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45 2 The terms in S when summed over all k, are independent of coordinates as before. These terms, therefore, will cancel between the numerator and denominator in Equation (11-57). The remaining contribution from 2 • • u y is given by y k = ^~ z 1 SCk-r^ S(k-r 2 ) (11-81) -> -> -> -> 2e 2 2 S(k-r 1 )S(k-r 2 ) Z l ? 1 + A^ K. k ->->•-* 2e 2 2 cos[k-(r 1 -r 2 ] = ~TT Z l U Z 5 • (11-82) k ( k x) z + (1 + u ) Equation (11-82) results from the application of a standard trigonometric identity. The sum over k is converted into an integral with r „ as the polar axis. We obtain l r y ik 2 9 e a 2 (1 + u) 1/2 2 1 rnj = z H3^ e 12 (II 83a > Similarly, 2 t y.A. e 2 1/2 r,o Q M\. X v k k 2 e a -(1 + u) ax,„ ,__ _„. C ase (2 ). 2 I —= -^ — e 12 (n-83b) i yfv 9 2 n j. ^/ 2 r^o~ ^-^. v k Tc e a -(1 + u) ax,„ ,__ „_ x Case (3): I 3-—, = 8 — — e 12 (II-83c) K K 1 12 Now that we have obtained the form of the expressions needed to evaluate the second order terms, we see that the integrals in the denominator of Equation (11-56) once more tend to unity when the thermodynamic limit is taken. Also we may identify factors appearing in the second order result to be just the Q-, n (&) and Q (£) that we evaluated earlier. We may then write

PAGE 52

46 an 1/9 Q 20 (A;1,2) = Q 10 a;l)Q 10 U;2) exp {-z^ -1 |__ e ~^ + u ) ax 12},(II-84a) i 12 ft 9 1/9 Q 02 (£;l,2) = Q 01 (£;1)Q 01 (£;2) exp {-z\ -~ |^— e" (1 + u) ax 12}, (II-84b) i 12 ft 2 1/9 Q n (£;l,2) Q 10 (£;1)Q 01 (£;2) exp {-z 2 ^ ^— e~ (1 + u) ax 12}.(Il-84c) i 12 We define the function I (£) by I 20 U) = \ nj [h 2Q (£) h 2Q (0)] (11-85) where h„_(£) is given in terms of the Q. functions by 20 jm 2 \^\ il h 2Q (£) .. v Y r 1 \ dr 2 ^Q 20 (£;1 2) Q 10 (A;DQ 10 (Ji;2)} x z d;Dx z (t}2), j4j ( (11-86) U 2 (aA) 6 j.i dx 2 Q 10 (£;l)Q 10 (£;2)x z U;l)x z U „ 2 _, .1/2 x {exp [-< ^| e U + u; aX 12] -1} (11-87) 19. 3x, „ 1 12 The other expressions for h. (£) in second order are very similar so that we concentrate here on h„ n (£). Define now D 12 (x 12 ) exp (-4 £ §ie" + ) 1/2 -12) -1 (11-88) 1 12 We are free to make the following expansion D 12 (x 12 ) -I (2k + 1) V k (x 1? x 2 ) P k (cos 6 12 ) (11-89) k=0 where P, is the Legendre polynomial and 8 „ is the angle between x and > and x V (x ,x ) is given by ir -v -> V k (x r x 2 ) | J D 12 (x 12 )P k (cOS 6) Sin 6 d9 • (H-90) o

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47 Inserting Equation (11-89) into Equation (11-87) we obtain 6! w h 2Q U) = (aA) E (2k + 1) \d Xl \dx 2 ffc^) f (x 2 ) x K— U X \ (x l' x 2 ) P k (cOS 9 12 } (H-91) where f(x) = exp {z 1 S(x) + iLz q(x) cos 9}x x {exp [-B Z;L W(x) + iL Zl VW(x) cos 6] -1} (11-92) P, (cos ) may be written P k (cos 6 12 ) = J q £m [{^g| P (cos 8l ) p£ (cos 2 )cos[ m ( W ] ( (11-93) where 1 ; m=0 e = m 2 ; m^O If we insert this expression into Equation (11-91) we find that the only non-vanishing contribution comes from the m=0 term. Thus the expression for h 20 (£) may now be written h 2Q (£) = (aX) 6 E (2k + 1) \ dx \dx 2 f( Xl ) f(x 2 ) \ dx l 1 dx 2 f ^ x i' x V k ( X;L ,x 2 ) P k (cos 8 ) P k (cos 9 ) (11-94) The angular integrals may now be performed with the aid of the following identity: i (i) k 4tt j k (A) = Vdn e iAc0s6 P k (cos6) (II-95)

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48 where j, is the spherical Bessel function. After the angular integrals are performed h„„(£) becomes, h 20 (£) = (aA) 6 E (2k + 1) (-l) k I d X] x 2 ldx^ \^Ay x F 1 (£;l) F 1 (£;2) V k (x^) (11-96) where 1 (£;1) = e*l v *l' (e P'i"^i' j k (Lz 1 G(x 1 )) j^Lz^x^) }, (11-97) F 1 (0;1) = e Z l S(x l } {e" 3z l W(x l ) -1} 5 k Q (11-98) At this point V (x ,x„) remains unspecified. We introduce here a linearized approximation to Equation (11-88) : Now V (x ,x ) is given by CO ,„ ,-. 9 6 2 f -(1 + u) x/ ax V k (L) (x r x 2 ) \ z\ f i\ S — ^ P k (cos e)sine d6 .(11-100) l o 12 27 This integral has been evaluated by Swiatecki with the following result: V < L >(x x ) = -z 2 ^ \ + l/2 (a V \ + l/2 (a V \ t x i'V z i e. 3 I s K } 1 J x x x 2 X l X 2 1/2 where a' = (1 + u) a. When we insert this result into Equation (11-96) we obtain a final form of I 20 U). 1 20 (L) = -z 2 ^ V-^\^f 3a 2 I (2k + 1) (-l) k {20} (11-102) x 1 2 k

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00 {20} = f x^ 2 I k + 1/2 (a' x 2 ) e Z l S(x 2 ) [e" Bz l W 10 (x 2 ) j^LGC^)) o CO 49 x. x[e Bz l W 10 (x l } J k (z l LG(x 1 )) J k (z 1 Lq(x 1 ))] chyh^ oo ~ 6 k,0 J x 2 I l/2 (a X 2 } e X 2 [e 1 10 2 1 X l /2 K l/2 (a X l } eZlS(Xl) C e BZ 1 W 10 (X 1 } -1] d Xl dx 2 .(11-103) CO J X 2 Note that a factor of 2 has been cancelled and the integration now covers one-half of the positive quadrant. This results from the fact that the integrand is symmetric under the interchange X*•* x„. In a similar manner we obtain I 02 (L) = -z 2 ~ [ I Rz ] 2 3a 2 I (2k + 1) (-l) k {02} (11-104) i 1 2 k CO {02} = f x 2 /2 I k + 1/2 (a' x 2 ) e Z 2 S(x 2 ) [e" 6z 2 W 10 (x 2 ) J k (z 2 LG(x 2 )) [ 3 / 2 v J X l \ + -i k ( Z2 Lq(x 2 ))] J X — ^ + 1/2 (a' x x ) e Z 2 S(x l^ x X 2 [e 6Z 2 W 10 (X 1 } J k (z 2 LG(x 1 )) j^z^q^) ) ] dx^ CO ,0 J xf Il; ~ 6 k,0 \ x^ /; I l/2 (.x 2 ) e*2 Sfa 2 ) [e" S *2 W 10 h 2 ) -l]

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50 x X oo \ X f 2K l/2 (a V e z 2 S(x l>t e 3Z2Wl CXl) -^ dx l dx 2 • t 11 105 ) The symmetry mentioned above is not present in I (L) a z ii (L) = z i z 2 ~f~ [ z + Rz ]2 R 3a2 z < 2k + D(-D k Ui) (11-106) i 1 2 k {11} f 4 /2 L + 1/9 (a' xj e 2 2 S(x 2 ) [eBz 2 W 10 (x 2 ) Jl (z LG(x 9 )) 5 2 "k + 1/2 W *2' LC iU J k vz 2^ x 2' 00 -J k (z 2 L q( x 2 ))] 1 xj[ /2 ^ + 1/2 (.' x x ) e 2 l S ^l> X 2 x[e 3Z 1 W 10 (X 1 } J k (z 1 LG(x 1 )) J k (z 1 Lq(x 1 ))] dx^ CO ] X 2 /2 \ + l/2 (a X 2 } eZ 2 S(X2) [e _Bz 2 W 10 (x 2 ) j^LG^)) + \ x *2 M MH ( 3/2 T f Zl S( Xl ) -J k ( Z2 Lq(x 2 ))] ^ X]L I k+1/2 (a' X] _) e 1 l' x[e 6Z 1 W 10 (X 1 } J k (z 1 LG(x 1 )) J k (z 1 Lq(x 1 ))] dx^ CO :,0 j x^ /2 1^ 6 k,0 \ 4 /2 I 1/2 (a' x 2 ) e z 2 S(x 2> [e'^W-l] oo v \ 3/2 t z 1 S(x 1 ) r -6z 1 W irv (x n ) -, j x \ x K (a 1 x ) e 1 1 [e 1 10 1-1] dx dx ? x 2

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51 00 I ~\,0 \ 4^ K l/2 (a x 2 } "2 S( ? 6 2 ) [" 8l 2 W 10 (x 2 ) -: x f 2 x^/ 2 I 1/2 (a' Xl ) • Z l 8(x l ) [.e" B *l W 10 (x l ) -l] d X] _dx 2 (11-107) We should mention here that the linearization introduced in Equation (11-99) is not forced upon us. In a more general nonlinearized treatment, the integral in Equation (11-90) can be evaluated numerically. In fact the computer program needed for the nonlinear calculation has already been developed. However, it requires considerably more CPU time than the linearized version. We employ the linearized version because it is quicker and, in our cases, yields results in agreement with the more general nonlinear version. Now let us define I^L) = I 10 (L) + I Q1 (L) I 2 (L) = I 2Q (L) + I Q2 (L) + I U (L) T X (L) = exp {Y L 2 + I 1 (L)} y T 2 (L) = exp {Y L 2 + I (L) + I 2 (L)} T (L) is defined as the "first approximation" to T(L) while T„(L) is called the "second approximation." Asymptotic Microfield Distribution Function In order to obtain a microfield distribution function, we must perform the following numerical sine transform.

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52 -S P(e) = 2ett J dL LT(L) sin (eL) (11-108) o There are many techniques for computing this transform which yield accurate results for small values of £ (generally for e<10, in units of e ) However, as the value of e increases, there is a loss of numerical precision which is characteristic of integral transforms of this type. In order to extend P(e) to large values of e we must develop an asymptotic expression. This problem is discussed by several I,10j23,28 TT authors. We wish to outline the treatment of this subject given in Reference 10 as it is particularly suited for this theory. The model for this asymptotic expression is known as the nearestneighbor approximation (NNA) This approximation states that high fields are produced by a single ion perturber during a close encounter with the radiator. This model assumes that for sufficiently large values of e, the probability that two or more perturbing ions contribute to the field is essentially zero. The probability that a single ion produces an electric field e is related to the probability of a close encounter with the radiator in the following manner: 2 P 1 (e 1 )de 1 = 4Tir 1 g^ (r^ d^ 2 x g (x ) dx (11-109) z, + p,z„ i 6 i v r 1 P 2 (e 2 )de 2 = 4Trr 2 g 2 (r 2 ) dr 2 3R 2 x -> g (x ) dx„ (11-110) z + Rz 2 2 6 2 v V 2 The g's are pair correlation functions, which we choose to be the nonlinear Debye-Huckel expressions.

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53 2 9 f 1/2 r \ r a e -\l + u; ax,, ,„,. ,,_ g 1 (x 1 ) = exp {X z 1 -^-e 1} f (11-111) 1 i 2 6 1/2 g 2 (x 2 ) exp {X z 2 2^~ g e 2} (11-112) 1 i In terms of dimensionless variables e = — (1 + axj) e dX l (11-113) X I Z 2 £ 2 = ~2 (1 + ax 2 ) e & 2 (II-114) x 2 When these expressions are differentiated and inserted into Equations (11-109) and (11-110) we obtain the following expressions for the asymptotic microfield distribution functions. / e 2 ft j. >!/2 4 r ea -(1+u; ax, 3x exp {X z — — — e 1} P ( e ) r 1 1 1 1 9 i 3x i i; E r L z + Rz 2 J — X — — (11-115) z..e 1 {2 + 2/ax 1 + ax } / 8 2 /I JL ^/ 2 •,4 ea -(1 + u) ax--, 3x 2 exp {X z 2 — 3— e 2} r 1 ? P 2 (£ 2 ) =[ Z;L + Rz 2 ] z 2 e~ aX 2 {2 + 2/ax 2 + a^} (II 116) P Asym (£) = W + P 2 (£ 2 ) (II ~ 117) The present method of joining the computed P(e) and the asymptotic P(e), which differs considerably from that appearing in the above reference, is considered in detail in an appendix.

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SECTION III DISCUSSION OF THE RESULTS Introduction In this section we present graphical results for the microfield distribution functions and x-ray line profiles calculated using the theory as given in Sections I and II. Experimental observations of Lyman-a for Ne X have been reported only recently.* However, it has been verified that this theory duplicates the results for the microfields and line profiles given in Reference 23. First we present figures demonstrating the behavior of the microfield distribution functions under variations of the different plasma parameters. We also point out and discuss trends and important features which are exhibited by these curves. Following this, we present Stark broadened x-ray line profiles computed using the above microfield distributions. General features of these profiles are also discussed. After presenting the figures we consider the various approximations made in the development of the line profile formalism in Section I. These approximations determine several basic validity criteria for the application of this theory. *The observations to which we refer here were made in connection with laserimploded pellet studies at Lawrence Livermore Laboratory. 29 54

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55 Electric Microfield Distribution Functions In this section we present graphical results for the electric microfield distribution functions computed using the theory of Section II. The first three figures illustrate the behavior of the microfield functions for multiply charged hydrogenic radiators perturbed by singly charged ions. The field variable s is in units of z (= e/r 2 ) a = r Q /A D In Figures 1 and 2, the charge of the radiator is +9 and +17, respectively. Figure 3 shows a comparison of the microfield functions for different radiator charges at a given value of a.. Two effects are immediately obvious from these figures. First, for fixed radiator charge x, when the value of a. is increased, each successive microfield curve has its peak shifted to lower values of e, becomes narrower, and has its maximum value increased. This behavior is analogous to that noted previously for x=l10 24 25 Second, Figure 3 indicates that for fixed a., as the value of x is increased, the behavior is similar to that noted above when a. is increased. Furthermore, a comparison of Figures 1 and 2 with Figure 4 of Reference 24 reveals that as x increases the relative sensitivity to changes in a increases. This increased sensitivity is due to the fact that in many functions defined in Section II the parameter a is multiplied by xFigures 4 through 7 illustrate the behavior of the microfield distribution functions for the a = 0.2 and _a = 0.4 cases, corresponding to those in Figures 1 and 2, when the parameter T is varied. This is R the ratio of the electric kinetic temperature to the ion kinetic temperature. T = kT /kT. = 9/6, (III-l) R e i ex v x *•'

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56 Figures 8 and 9 illustrate the effects of T -variation on microfield distribution functions for a = 0.2 and a = 0.4 when hydrogenic Aluminum (x *• +12) is perturbed by a plasma containing only A£ XIII ions (z = +12) P As the value of the parameter T is increased, in all cases the R microfield distribution function peaks shift to lower values of e. This behavior might be anticipated from the form of the following relation: a' = (1 + u) 1/2 a, (III-2) where, if R = 0.0, U = 6 e /e i z l = T R \ (IH-3) The parameter a*_ appears in several functions in the definitions of both I.. (L) and I (L) Thus, if we regard a_^ as a modified plasma parameter, we see that as the value of T is increased, the result is an effective R increase in the ji value. A direct comparison of Figures 4 through 9 with Figures 1 and 2 indicates that when the value of T is increased, R these T -variation curves indeed exhibit behavior similar to that noted when the _a value was increased. While the dependence on T is obviously R strong in all cases considered, the analytic form of this dependence is extremely complicated and difficult to assess. 1/2 To see more clearly the effect of the (1 + u) factor, let us consider the following: a' = (1 + u) 1/2 a = (1 + u) 1/2 r Q A D = r /A' D (III-4) where 2 x \ = RF 1 (i + ")] """ (IH-5) r 4irne ,., N1 -l/2

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57 The definition of u is given below Equation (11-56) ; we now write it in the form 9 e N 1 Z 1 + N 2 Z 2 9 e Vl + n 2 Z 2 u = t; Vl + n? z ; = e: n — • < II]: 6 > i 11 2 2 i The second step in the above equation results from the overall charge neutrality of the plasma. Inserting this result into Equation (III-5) we obtain D -[4.. 2 (f+ m A; n 4 r i/2 (m 7) e i This equation is a generalization of Equation (19) in Reference 30. Thus we obtain for a' : 2 2 r/ 2 / n n z + n z f a = r Q [4iTe (— + Q )] (III-8) e i In our computations, we regard T and n (the electron number density) as fixed by the parameter a.. This means that, for a given value of a, an increase in T is accompanied by a decrease in the ion temperature. Equation 8 gives the explicit dependence of a' on the ion temperature. In addition this equation suggests that the shift of the microfield peaks might be due to increased importance of ion correlations as ion temperature is lowered. This correlation effect is especially apparent in Figures 7 and 9. Looking at these figures we see. that, as T takes R higher values (ion temperature decreases) the A£ microfield distributions peak at smaller field values than do the Argon distributions. This is to be expected since interparticle correlations are stronger in the kl-kl system than in the Ar-H system. On the other hand, as T decreases R (the ion temperature increases), the figures indicate that the stronger

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58 fields are more probable in the A£ plasma. This is due to weakened correlations; the average interactions in this case will be stronger in the A£-A£ system than in the Ar-H system. In Figures 10 through 13, we present the results of computations in which T R is set equal to unit and the parameter R is varied. Now R is the ratio of the density of charge z perturbers to the density of charge z^ perturbers: R = 0.0 () .corresponds to the case where the ion perturbers are all of charge z^ (z ) The function u is given by 2 2 x + Rz 2 u = Z + rz 2 (T R = • (in-9) It varies smoothly from a value of z to a value of z as R goes from 0.0 to Because of this we might expect that when the perturbing ions have z 1 and z^ values nearly equal, the variation of R causes very little change in P(e) Indeed this is the case for A£ microfields when z 1 = +12 and z^ = +11. For this reason we omit figures showing the R variation calculations forthe A£ system. The most noticeable effect of an R variation occurs when z and z 2 are very different (Figures 10-13). Furthermore, it should be noted that for very different z and z 2 (with z 2 > z ), most of the variation of P(e) occurs between the values of R = 0.0 and R = 0.1. This behavior is due to the fact that for a situation where z z and R is only slightly greater than zero, most of the free electrons are already contributed by perturbers of charge z This is illustrated by considering, for example, the following case for argon (z = +1, z = +17): R = o.o n. n n n = 0.0 1 e 2 R 01 n. = 0.37 n n = 0.037 n 1 e 2 e R = OT l^ 0.0 n 2 = 0.059 n (111-10)

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Figure 7. Electric microfield distribution function P(e) at a point having a charge of +17. a = 0.4. The ion perturbers have a charge of +1.

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Figure 9. Electric microfield distribution function P(e) at a point having a charge of +12. a = 0.4. Perturbing ions have a charge of +12.

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85 In the present work, we consider n to be fixed by the a_ value. Then n and n are determined by the requirement of overall charge neutrality for the plasma (Equation II-4) For lower a_ values, an increase in R causes the microfield peaks to shift to higher field values. At higher a. values, however, the increase in R produces a shift to smaller field values. The behavior at higher a. values can be explained in terms of particle correlations. That is, an increase in the value of R results in an increase in the effective plasma parameter a/_. At lower a. values, correlations are weaker and the observed shifts indicate that as R increases, the average strength of interactions within the system increases. Stark Broadened Line Profiles In this section we display the broadened line profiles computed using the electric microfield distribution functions shown in Figures 1 through 13. Figure 14 shows the relative contributions to the Lyman-a Ne X line profile from the several broadening mechanisms that are included in our calculations, together with the combined result. The conditions represented in Figure 14 correspond to an a. value of 0.4. The relative importance of the Doppler effect is of interest. For the temperatures discussed in this paper, the Doppler effect is much more significant — compared to the electron-broadening contribution — than was the case when dealing with more conventional plasmas (e.g., n = 17 -3 10 cm and T = 40 000K) The qualitative features of the Stark profile are also different: here the electron contribution produces a sharp spike which sits on shoulders provided largely by the ions. Figure 15 presents a plot for Ar XVIII that is equivalent to Figure 14

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86 for Ne X. The qualitative information is the same. Figure 16 shows for both Ne X and Ar XVIII, families of Lyman-a profiles, each of which corresponds to the same T but different n. It is evident that as n increases, the profile changes greatly, both in shape and width, thereby illustrating the density sensitivity of the line profileThe practical sensitivity that might be inferred from comparison of experimental and theoretical line shapes depends on at least two factors, (1) how much of the line profile can be experimentally observed, and (2) the resolution of the x-ray spectrometer used. Figure 17 shows for Ne X and Ar XVIII, families of three Lyman-a profiles, each of which corresponds to the same density but different T. The results here imply that the frequently mentioned insensitivity of plasma-broadened line profiles to variations in T is significantly reduced as x increases. The fact that Doppler broadening plays a more significant role in broadening the argon lines than it does for neon is due to the fact that the effect depends not only on mass, but also on the radiator z. Figures 18 through 20 show the behavior of Lyman-a profiles for hydrogenic neon, aluminum, and argon, respectively, under variation of the parameter T The electron density is the same in all these cases. For the curves on the left the electron temperature is 1019.2 eV while on the right it is 254.8 eV. In the neon and argon cases, the ion perturbers have a charge of +1. For the aluminum case the ion perturbers have a charge of +12. Three general features of these Lyman-a profiles can immediately be noticed. First, since the electron broadening of the unshifted central component is not directly affected by the ion

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87 microf ield distribution, most of the variation of the central component observed in these figures is due to variations in the Doppler broadening. Second, since the wings of the profile are determined almost entirely by the microfield distribution, we immediately see that the variations in the wings of the profile reflect the structure of the microfield distribution functions discussed earlier. Finally, the line profiles show a significant amount of structure in the region of the shoulder. The electron broadening of the unshifted component plays an important role in producing this structure. To see this more clearly, compare the left hand T = 4 curve with the right hand T = 1 curve in each of these figures. By doing this we isolate the electron broadening effects since the ion kinetic temperatures (i.e. ion broadening and Doppler broadening) will be the same for each curve. The electron temperatures are the only difference between the two curves. Since electron -1/2 broadening scales as T the electron broadening is greater in the right hand curve. That is, the electrons fill in the shoulder on the right, whereas the shoulder of the left hand curve is quite pronounced. In addition, there are significant variations in the structure of the shoulder within each family of curves. Figures 21 through 23 demonstrate the same T -variation calculations R for the Lyman3 profiles of neon, aluminum, and argon. Doppler broadening effects are more important for these lines than for Lyman-a due to the increase in the unperturbed transition frequency. Once again, Doppler broadening is responsible for most of the variations around the central dip, within a given family of curves. However, by isolating the electron broadening effects as indicated above, we can see that these effects also contribute significantly to the structure of the central dip.

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116 The behavior of Lyman-a profiles for neon and argon under the variation of the parameter R is displayed in Figures 24 and 25. Figure 24 shows neon and argon profiles for a = 0.2. As we mentioned above, most of the variation occurs between R = 0.0 and R = 0.1. Notice that the shoulder becomes more pronounced as R increases. The reversal shown on the right side of Figure 25 seems to indicate that, at higher a values, particle correlations will lead to reduced line widths Figures 26 and 27 show Lyman-g profiles for neon and argon under the same R variations. The qualitative information is the same as for the Lyman-a profiles. Validity Criteria for this Theory We pointed out in Section I that the neglect of fine structure of the radiator energy levels leads to a criterion for the validity of the linear Stark effect in calculating the line profile. The requirement that the level shifts due to the Stark effect be greater than the fine structure splitting sets a lower limit for the electron density. That is, n > 2.4 x 10 16 (-V 572 (1 M 3/2 (IH-11) e n n u u An upper bound on the particle density may be determined by requiring that Stark shifts be smaller than the separation of radiator levels of different principal quantum number. This critical density is referred to as the Inglis-Teller limit: ^ in 23 9/2 -15/2 (111-12) n < 1.5 x 10 z n Vi-xjj-^j e u

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117 This restriction should ensure that the average, broadening interactions are weak enough that (1) they may be treated by perturbation theory, and (2) the No-Quenching Approximation will be valid. The densities employed in the present work fall into the range set by the above inequalities. However, there is an additional point which must be considered. The dipole approximation to the radiatorperturber interaction has been made. Higher multipole moments will become important when the radiator-perturber separation approaches the radiator size. To minimize possible contributions from the higher multipoles we should require r„ > r where On 2 n a„ r = -^ (111-13) n z is the radius of the radiator in the excited state of principal quantum number n In the present treatment, our "worst" case for this criterion 31 is Lyman-3 for neon with r_ ~ 3r Bacon has employed the Classical (J n Path Impact Theory to treat exactly the effects of higher multipole moments on Lyman-3 in hydrogen at a. values (~ .3) close to those in the present work. His work indicates that the inclusion of higher multipole terms (as well as time ordering effects) tends to fill in the central dip of Lyman6Thus, we might expect our widths to agree with Bacon's results but it is probable that there would be differences in the line center due to our lack of higher multipole moments in the radiatorperturber interaction. We mention in passing that it may be possible to adjust the correlation cutoff given in Equation (H-9) so that it accounts for strong collisions as well as for electron correlations (see the discussion of this cutoff given in Appendix H)

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118 Whereas when dealing with hydrogen plasmas the plasma parameter 2 T = e /rJkT is an indication of the strength of coupling, in plasmas containing multiply charged ions this definition must be replaced by 2 2 r' "W~ Ciii-14) 2 where = (1 + u) is a generalization of Equation (4) in Reference 32. In the present work, we require V < 1, in which region we expect the microfield calculation to be very reliable. This is equivalent to a' < 1.732 (111-15) Two of our cases (a = 0.4, T = 2.0 and T = 4.0 for aluminum) violate K. K this inequality and must be viewed with uncertainty. Equation (111-15) gives the explanation of why we restrict ourselves in this work to the consideration of ji values 0.2 and 0.4. In Section I we indicated that the Quasi-static Approximation for the ions is valid for Ago > oo T ,. In all of our cases, go t p(Ions) p(Ions) falls in the very center of the line profile so that except for a small frequency range around the line center, the ions are adequately treated as static. The plasma frequency for the electrons is given by w / w1 s = 2.72 x 10" 12 n 1/2 (111-16) p(Electrons) v in Rydberg units. This frequency falls off our scale in all the figures presented so that the electrons cannot be treated as static in the frequency range we display. The validity region of the second-order perturbation treatment in this work covers the center of the line profile and extends roughly out to the electron plasma frequency. The strong collisions, which contribute to the line profile for Aw > go ,„„ ., p(Electrons) are not adequately treated by finite-order pertubation calculations.

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SECTION IV CONCLUDING REMARKS The goal of this work has been to perform a preliminary study of Stark broadening in hot dense plasmas. By "preliminary" we mean that the results presented in the previous section should indicate the direction which future work on this problem should take. A basic conclusion resulting from the present work is as follows. Although Doppler broadening generally dominates in the center of the line profiles, the structure near the center (i.e. the shoulder of. Lyman-a and the dip of Lyman-3) shows significant sensitivity to densitytemperature variations. A large part of this structure is produced by the electron perturbers. For the cases where ions and electrons have different kinetic temperatures, the electric microfield distribution functions produce significant effects on the line shape. Since these effects can generally be discussed in terms of particle correlations, they appear to indicate that correlations will play an important role at higher densities. In addition, at low a values there is an indicated sensitivity to small concentrations of high-z ion perturbers One immediate goal of future research is the extension of line 25 broadening theories to higher densities, vL0 (e.g. densities expected in controlled fusion experiments, which are significantly greater 119

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120 than those employed in the present study) We can expect that correlation effects, quantum degeneracy effects, and strong-collision effects will become increasingly important at higher densities. These effects must be properly included in subsequent line-broadening theories. A recent experimental measurement of the Lyman-a profile for 33 hydrogen at moderate plasma density and temperature J produced an experimentaltheoretical discrepancy. The measured shape of the line 34 center was about 2.5 times wider than theoretically predicted... Griem has conjectured that this discrepancy results from an additional broadening mechanism: the screening of ions by electrons is dynamic rather than static and hence gives rise to density fluctuations in the screening electron density, which in turn provides additional line broadening. This possible explanation deserves further study since it could be important in analysis of laser implosion experiments through the use of x-ray line profiles. 35 Work in progress at the University of Florida is aimed at development of a line broadening theory capable of treating all of the above effects. Particular emphasis will be given to the study of line profiles emitted by radiators immersed in dense, highly ionized systems.

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APPENDICES

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APPENDIX A THE INTERACTION V er In the model for our plasma as discussed in Section I, we have a hydrogenic radiator fixed at the origin of coordinates. The ionic radiator has a nuclear charge of z. Using bare Coulomb interactions, we may write the interaction between the radiator and one perturbing electron as follows: 2 2 er R r 1 r The first term corresponds to the Coulomb interaction between the i-th perturbing electron and the single bound radiator electron (located at the position specified by R) The second term is the interaction between the radiator nucleus and the i-th electron. Let us expand the first term in the usual spherical harmonics, V er / JITTT) -TTT Y £m*^ Y £m ( ^ Z ^ > (A 2) £,m r i 1 > l 1 £=1 m r > We now introduce two approximations which allow us to simplify the above expression. The first step is to retain only the £=1 term in the sum of Equation (A-3) This procedure is referred to in the literature as the Dipole Approximation to the radiator-perturber interaction. Next, we take r = |r.|. This approximation neglects the possibility of "penetration of the radiator" by the perturbers. The validity of these two approximations in the classical path theories is assured by strong12 4 collision-cutoffs in the classical averages. Validity criteria 122

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123 pertaining to the present work are discussed in Section III. Now we may write 2 2 1.2,+ „i e e 4n-e R er r r 3 r\ 1 x 1 x 1 m=-l x 7 Y lm* ( V Y lm ( V (A 4) 2 1 / 2 Id e 4ire R = -X IITT + ^ x m=-l x ? V^t Co ) (A-5) where x = z ~l is the net charge of the radiating ion. The second term may be recognized as a vector dot product. V* = -X y^ + eg • !(?.) (A-6) where ->r ^ (? i } = e ^V • < A ?) Since the perturbing electrons remain "outside" of the radiator, they "see" a radiator of net charge XThis suggests that we may treat the first term of Equation (A-6) exactly by selecting as a basis set for the perturbers the free particle Coulomb wave functions for electrons moving in the field of an ion of charge X With this choice of perturber wave functions, we redefine V : er V 1 = eR • E.(r.) (A-8) er x x -> •+ where E.(r.) is the electric field at the site of the radiator due to a x I ->single perturbing electon located at r In this manner we have incorporated the effects of the radiator charge into the dynamical treatment of the perturbing electrons. The classical path theories account for these effects by using hyperbolic trajectories in performing the perturber averages.

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APPENDIX B THE ALGEBRA OF TETRADIC OPERATORS The operators L, L Q L^ R (oj) K er (w) that we have encountered in evaluating J(ou,e) are special operators and are called tetradics. In a matrix representation defined on the system of radiator plus electrons, they have four pairs of indices. An operator of the usual type defined on this system will have two pairs of indices. For an example consider the Hamiltonian H : H Q \ + H^ (B-l) Matrix elements of H are given by = (E r + E 6 ) 6 ,6 (B-2) 0' y a uu aa where u,y' refer to the state of the radiator and a, a' refer to the electrons. The Liouville operator is a special tetradic defined by its action on an arbitrary operator f li = [H,f] (B-3) From this definition, we can see that the operation of L on f will produce another operator with four indices. This means that products involving tetradic operators will have the following form: = v6 J, 3 [U uaill 0?;v g vt g f vBv B (B-4) = ve J, B y „J v „ B „ [Bo()] vatV B ;vev B X[l] vf3v'3';y"a"v"g" f y"a"v"g" (B_5) Let us consider the following 124

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125 = (E r + E 6 ) y a (E*, + E^,) = (AE^, + AEj,) (B-6) We have obviously chosen the y,a states to be eigenstates of H In order to introduce another notation, we now rederive the result of Equation (B-6). = S [L 1 „ , 1 |M J ya;y'a';v3v'B' vBv'B' E {6 ,6 ,-5 6 } IO '0' y v a B yv aB '0' vBv f l0 ,.. • (B-7) VKV The expression within curly brakets defines the matrix representation for the Liouville operator. This equation may be simplified. = E {(E r + E & ) 6 6 Q & ,6 fl „ fl „t(H M a yv aB y'v' a'B -
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126 a = I AE a AE^, .. (B-9) We have used the fact that Tr {p }' = 1 (B-10) e e We can recognize that Equation (B-9) may be also written in the form, yy' = , (B-ll) or, in terms of operators, = L r (B-12) Proceeding along the same line we may also show = l\ (B-13) and so on to higher orders. The goal of this procedure is to enable us to evaluate = <{w-L /R} _1 > (B-14) If we expand the right hand side of this equation we obtain a series whose leading terms contain the expressions found in Equations (B-12) and (B-13). We may perform the indicated averages and resum the expansion to obtain R(uj) = {co-L r /h} _1 (B-15) Employing the same type of expansion procedure, we may use Equations (B-7) and (B-8) to write down the matrix representation of R„(w) : .E r ae 6 r 1 V J yay 'a' ;vgv' g' ^ii 1 aa' u yv ag y'v'^a' [Rndo)],,,.,. ,_ Q „, ot ={u)-AE_, AE_ } &„J^ D S„ ,..,_, (B-16)

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127 This result allows us to prove two useful identities. As before, we set f = p g where g is an operator on the space of radiator states. We wish to calculate quantities such as = E a avev'V u'W'S" ^W^vgvV [L Wb • ;y Wg" E E E {u>-AE r AE e } 6 5 6 ,6 a vBv'6' y'W'3" yp aa yv aB u v aS X tL] vBv 6';y"a"v"3" %"a"v"&" (B_17) We may perform the sum over v3v'3' with the result = E E {tu-AE r } yy a y"aV'3" yy X [L] yay'a;y'V'v"3" V'a'V'B" i r -1 = E yy a = E a = yu • (B-18) In operator form we write this = R(o)) (B-19) Now let us consider X f y"a"v"S"

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128 = I ) f |y'a> a e ,-1 z S 2 [L] „ ,., {a)-AE r AE ,} X W 6 vV'V3" f yVW (B 20) We may perform the sum over u'W'g": = S E [L] i a ,., {w-AE r AE^ Q } _1 yy a vSv'S' yay a ^3v'B' vv 33' X f v3v'3' (B21) Remembering now that f = p g and e W3' = P e V S W (B 22) p we see that the delta function S means that AE will vanish. This pp 33 gives = E E [L] „ ,., {cj-AE r ,} _1 uy a v3v'3' Pay'a;v3v'3 vv' X f v3v'3' = E > a = w • (B-23) In operator form we write this = R(co) (B-24) It is convenient to summarize the identities which we have proven here. In terms of operators we have:

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= R ( = R (oj) = R (a)) (B-19) (B-24)

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APPENDIX C QUANTUM MECHANICAL PERTURBER AVERAGES In this appendix we carry out the electron averages of the operator expressions appearing in the line profile calculation of Section I, The basis set for the gas of electron perturbers is chosen to be a set of unsymmetrized products of single-particle functions. These singleparticle states are the positive energy solutions of the Schrodinger equation for an electron moving in the field of a hydrogen-like ion of charge x = z-1. An expansion in spherical harmonics of these functions 37,38 is given by *k. V = ~ZT3n (i)£ eiaU ki) Ov\m^V k i r) ((>1) l £m (2tt) where aU,k.) = arg r (, + 1 in) ; n = X; (C-2) i k. is a unit vector in the direction of k. ; J (k.r) is the Coulomb-Bessel function given by TT J £ (k.r) = e 2 r(S+2) (2k i r) ^ e"^ ^(i+l-inj 2£+2;2ik.r) (C-3) where F signifies the confluent hypergeometric function. We find that J (k.r) reduces to the usual spherical Bessel function j (k.r) when x 39 vanishes. In fact we have Um ik = (2ir)~ 3/Z e ik i r ; (C-4) x-o k i i i> $ dx = <5(k'-k) (C-5) rC K. An n-particle state is represented by an unsymmetrized product: 130

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131 ^ n; ($) = II J Ai = [ \dx. E ** (x.)^ (x.) e B 2m ] N ( C -9) J k. i i i The sum over k. resembles the Slater sum for an ion-electron system. The two-particle Slater sum for this system as given by Barker is 2 2 h k E 4>* (x)ijj (x)e _0E n + f , (N)-\ A— — dx E # uv xv, U)e 'H^T k k k

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132 where V is the volume of the system and A is the thermal wavelength of the electrons. Hence we now have o I i i -BH P e ~ V J e e • (C-12) The neglect of bound states implies that we neglect the probability that a perturbing electron will become bound on the radiator to form a doubly-excited state. This approximation is discussed in References 9 and 23There it is pointed out that the neglect of bound states has essentially the effect of introducing a close collision cutoff into the calculation. The next result we wish to obtain is the proof of an assertion made in Section I that the electron average of L vanishes. We will er make use of the results given above. From the definition of L we may er write f = f f (C-13) er er er KU ^ JJ which tells us that we are interested in calculating or, er = Tr {V p } (C-U) er uy' e uu H e ^ ^ ) Since is an operator on the Hilbert space of the radiator, the indices u,u' refer to radiator quantum numbers. The trace may be written [^A^e-^r fdxV (N) (x)V ,^ N > J ^ k w k ^erV' = hr r \ dk e 2m |dxV UV (2)V M ,^ W Cx) (C-15) rh<2 V er iS wri tten in terms of spherical tensor operators as follows: where the dipole approximation for the radiator-perturber interaction has

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133 been made. The sum over i includes all of the electron perturbers. If we then insert this above we obtain 3 r t 2 2 q „ 2 D (q) = E (-l) H e R VH [— -] Idk e 2m er up pu ,, [— 1 Jdk \ r (-q) dx ijj k (x) — ^ ijj (x) (C-17) r i The 2N-dimensional integral on the right side of this expression may be broken down into two factors. One factor consists of an integral over k. and x. essentially a single-particle matrix element involving the i-th perturber. The other factor may be recognized as just the right side of Equation (C-9) with an exponent of N-l instead of N. The result of this factorization is that we are able to cancel N-l of the factors inside the square brackets. 3 r k2,2 nX T 2 = e er py N S R k i where F. is the single-particle matrix element of the i-th perturber part of the interaction V er f r (q) F i q) -V \A } •
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134 X \ m] ( V Y 2 m 2 Ck 2 )(-l) m l [(2 4l + i)(2t 2 + 1)] 1/2 £ 1 1 £ 2 £ l 1 £ 2 X ( } U q m > T £ •£ • >21) 1 H 2 1' 2 In the expression, k. and k are unit vectors in the direction of k and k respectively. CO \;* 2 ] dr r \ (k i r) 7 % (k 2 r) < c 22 > o is the radial integral. Inserting the appropriate expression back into Equation (C-18) we have: 2 A 3 f R 2 k 2 q J £ i ,m i £ 2 m 2 x (i) 2 e ( V k) Y^Ck) Y £ ^ (k)(-l) l [(2Al + 1)(2, 2 + 1)] 1/2 -m -q m' l,\l We now point out that since p is independent of angles (isotropic) we are able to perform the angular-k integration. This gives delta functions in &-.J&0 and m ,m — causing the leading phase factors to cancel to unity. We obtain CO 3 2 2 2n V t \ C o „Rk = 1 E (-l) q R (q I dk k 2 e~ 3 "2m" E (-l) m q My J to o x (24 +1) l ; -m -q m'' T (C-24) ,1 1 L However, vanishes because of angular momentum rules. Therefore each term vanishes and we have the result

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135 = and = (C_25 > Before continuing, we point out two facts which caused to er vanish. First the fact that the perturber distribution is isotropic allowed us to perform the angular-k integration. This, along with the dipole approximation for the radiator-perturber interaction, caused to vanish from angular momentum considerations. We note in passing that a more careful treatment of not making the dipole approximation yields a small but nonvanishing contribution that may be related to the plasma polarization shift. We now turn to the evaluation of an expression appearing in the width and shift operator H (u)) The expression we wish to calculate is given by: I M ,1 = yy";y"y' yy y .'V t ^i [Md^.-4£.1 4<4-$ • 3 2 2 nA T f *N f "^w p H k -r*JL n 2 i 2 ^ = [~~] E \dk^ dk e P 2m e lfc 2m U 2 Va. (k. ,k,) „ im JJ 112 \iU A m (k 2' k l>y"y' (C ~ 26) where A. is the many-body matrix element of the interaction between the radiator and the i-th electron perturber. A i (k 1 ,k 2 ) = e 2 E(-l) q [ II 5 (k.^) ] F< q) (k^ kj (C-27) Here, F. is just the same matrix element defined in Equation (C-21) First consider the N terms in Equation (C-26) where i=m. Denote this

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136 contribution by I : 3 r/JL 2 V';yV N[ ~ ] e J dk i J dk 2 e ^T e 2m 2 x i (-i^+q' x q>q' -y -y [ fl 6 (k r k 2 )] 2 f[ q) (k ,k ) F^ q ) (k 2> k 1 ) (C-28) The squared delta functions allow us to perform N-l k -integrations and N-l k„-integrations with the result that N-l of the leading factors —r— are cancelled. The result is N V;m'V =nA x e E (-l) q+q q,q Kl 2 2 "* S H k 1 -.JL/-1 2 i 2 ^ ( \ -* t iv •* •* dk 2 e 3 ~2~ e xt 2^ U 2 V f| ^(k^k^FJ: q '(k^) .(C-29) The many-body problem has been compressed down to a calculation involving only single-particle functions. We now may insert the definitions of F. from the Equation (C-21) Once again we may take advantage of the isotropy of the perturber distribution. That is, we may again perform the angular-k integrations. The result is that the phase factors cancel to unity and we are able to perform several sums: 1 .... =^nA^ e 4 Z (-l) q+q 7r q,q 00 CO 1 dkl I 2 2 „ „ h kTi 2 2 m +m dk„ k, k„ e — ^ — e 2m 2 1 E £ (.-1) 2 1 2 2m „ 1* 1 2™2 o

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137 l l 1 l 2 2 l 1 l 2 l 2 1 £ 1 x [(2l + 1)(2£ 2 + 1)] ( } ( mi -q m^ ( -m 2 -q m^ T £ £ T £ £ (C-30) There is some cancellation of the phase factors because of a rule for the 1 • IT 41 3-j symbols : -m.-q + m 2 = (C-31) Thus (-l) q (-l) m l +m 2 = 1 (0-32) Now we are able to perform the sum over m ,m with the aid of an identitygiven in Edmonds : • • > ^ ( 3 1 2 2 J 3 )( J 1 3 2 J 3 ) S m m m„ m. m„ m' = (2j „ + 1) S 6 ,_ „„. m r m 2 12 3 12 3 J 3 ^ m^ (C-33) The result of performing this sum is 4neV 1 „ „ =-^ E ( q' 3it q 2 2 03 oo H k 1 .J! ,, 2 ,2, -it-r(k.-k. ) Hi \ dk, \ dk 2 e P 2m e 2m '2 V f (k ,k 2 ) ,(0-34) o o where l l 1 £ 2 2 f(k ,k ) = kV E (2£ + 1)(2£ + 1) ( } T. .. T .(C-35) i z l z £ ^ i z ^,^2 2>Jtl The sum over q' is recognized as the spherical tensor notation for the vector dot product. One of the sums in Equation (C-35) may be performed and the 3-j symbol evaluated explicitly (from Edmonds ) with the result

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138 f(k l'V k l k 2 l { ^ + l > T ; A 1 + *!,. A x > • (0-36) We now need to go back and consider the terms in Equation (C-26) where i/m. We denote this by I : i u e 4 ^ r n ^i N f d ,> f *> "4^ -itf (k 2 -k 2 ) Wmv e 2 [ ^T ] J dk lJ dk 2 e 2m e 2ra 2 1 • X n E n ,(-i) q+q, n 5 .(k -Uf!^ (I ,kj ->• •> *-$x n 6 (k -k ) F^" q ) (k„,k 1 ) (C-37) _/_ n l i m 2 1 nfm In this equation we have N-2 squared delta functions. Thus we may perform all of the k and k -integrations except those corresponding to particles i and m. V'jyV [ 1T ] J dk l \ dk 2 e 2m e 2m 2 1 x E (-l) q+q 6 (k -k.) 6. (k -kj q,q' m 1 2 i 1 2 x F^d^.kj) F^" q ^kj,^) (C-38) The remaining delta functions allow us to perform the k„-integrations Then we obtain 2 2 N(N-1) nA^ f + H k l ^ ; yv = e 2 ^ 2 J dk i e ~ 6 ^ ^> q+q x Fj'^Ck^k^ F „" q > (fc 1 k 1 ) (C-39) Comparing this result with Equation (C-18) we can identify the terms above as

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139 u 1 I h, ti = -r „ „ = (C-40) W ;y v 2 er yp er ]}"\i' v Therefore the neglect of correlations in the density operator leads to the result that the i^m contribution to I must vanish. The final result we have obtained is 2 2 4ne 4 A 3 f ^ k l = -^2R py"'Vy' \ dk l ) dk 2 e 2m H_ ,,2 2. x e XC 2m U 2 V f (k ,k ) (C-41) where f(k r k 2 ) k^k 2 2 E {(£ + 1) T £; / + x + £T &;£ 2 x } (C-42) (24) We conclude here by identifying f (k.. ,k.) as f(k l5 k 2 ) = k 2 k 2 2/3 g ff (k 1 ,k 2 ) (C-43) where g (k ,k ) is the free-free Gaunt factor. 2 9 23 42 43

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APPENDIX D THE PARABOLIC REPRESENTATION Schrodinger 's equation for hydrogen-like atoms is separable in parabolic coordinates. According to Bethe and Salpeter "This alternative is connected with the degeneracy of the eigenvalues belonging to like principal and different orbital quantum numbers. The connection between spherical coordinates and the parabolic coordinates g, n, and may be expressed by x = E,r\ cosij) E, = r + z y = 5n sine)) n = r z 1 —1 v z = (£ n) (j) = tan x r = ~ (| + n) (D-l) The details of the separation and solution of the Schrodinger equation in parabolic coordinates are given in some detail in Reference (11) The normalized eigenfunction is given by im. N l/2m m _.„ m ,_ _,. ; (?n) L ni+ m(^> L n 2+ m (erl) (D 2) where £= and the L's are the associated Laguerre polynomials. na o n = n + n + |m| +1 } (D-3) where n is the usual principal quantum number. For fixed m, n (or n ) runs from to n-|m|-l. Also m runs from to n-1 It is convenient to 140

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141 define a quantum number q: q = n l n 2 (D 4 ) The number q is called "the electric quantum number". Using these relations and the allowed values of n (and n ) and m, we can generate the quantum numbers n,q,m which specify the hydrogenic states in the parabolic representation. The Stark effect for hydrogen-like atoms is especially amenable to calculation within the parabolic representation. The interaction to consider is (see Appendix A) : V. nt = eze ( D _5) The electric field e defines the z-direction of the atomic system. A first-order perturbation treatment of this interaction yields the energy level shifts of a hydrogen-like atom for the linear Stark effect. When this calculation is carried out the level shifts are given by 3 a e AE nqm = 2 T nqE (D 6) Note that the energy levels are still degenerate with respect to the quantum number m. This degeneracy is a consequence of the rotational symmetry of the perturbed Hamiltonian about the z-axis and remains in all orders of perturbation theory.

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APPENDIX E CALCULATION OF RADIATOR DIPOLE MATRIX ELEMENTS In this appendix we wish to derive a computational form for the radiator dipole moment matrix elements in Equation (1-94) The expression we are interested in is -$ D li . (E-l) where | 1> represents the ground state of the hydrogen-like radiator. Employing the parabolic representation introduced in Appendix D for the hydrogenic states i and i' we have D = -<100|d|nq'm'> (E-2) where n is the principal quantum number for the upper state of the transition and q andm.ar^the parabolic quantum numbers. With the notation of Edmonds we may expand the vector dot product into a product of tensor operators. D I (-l) k <100|d |nq'm'> (E-3) n k k k k i = E E E (-1) k 1 m 1 £ 2 m 2 L L Ilk x <100|d |n& m > (E-4) The state | 100> is equivalent in the two representations |nqm> and |n&-m>. Now we must evaluate the matrix elements of the spherical tensor operators d between spherical states. From Edmonds, d k (f ) 1/2 er Y 1)k (fl) (E-5) 142

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143 Inserting this and performing the angular integrals of the spherical matrix elements we obtain: k D. .. = I £ X (-1) 11 „ „ 11 2 2 1 k ^ l 2 m 2 X ( 1} K2£ 1+ 1)C2£ 2+ 1)] (^ _, )( Q \ )( k m 2)( J Q 2 ) x <10 I er I n£ > (E-6) The last two factors here are radial matrix elements of er. In obtaining this equation we have made use of the following identity \ da \* ™ %m 2 <> Y 3 m 3 < D ? (2£+l)(2£ 9 +l)(2£„+l) A l 2 \ A % 2 \ = [ t-^ 1— ] 1/Z ^0 O'V m, m„ ; (E-7) 4tt 1 2 3 The 3-j symbols vanish unless the following angular momentum rules are met: (i) The triangle identity, K-JU i^i^ + ilj ; (E-8) (ii) m + m 2 + m 3 = ; (E-9) (iii) if m = m = m = then I + I + I = 2n (n = 0,1,2,...) (E-10) The important symmetry properties of the 3-j symbols are given by Edmonds :

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144 (iv) An even permutation of the columns leaves the numerical value of the 3-j symbol unchanged: h h h m j 2 h h m h h h m l m 2 m 3 m 2 m 3 m i m 3 m l m 2 (v) An odd permutation of the columns is equivalent to multiplication by (-1)^1 ^2 ^3: ( 1 2 3 ) = (-1) j l + V j 3 ( 2 1 h (E-12) m 1 m 2 m 3 m 2 ^ m 3 (vi) Also, i 1 3l 33 ) = (-l) j l +j 2 +j 3 ( X 2 3 ) (E-13) m m m -m -m^ -m Now we note that, in Equation (E-6) ,£=£,_ = 1 because of rules (i) and (iii) Also rule (ii) requires (-l) k+m l = 1 (E-14) Thus the angular momentum rules satisfied by the 3-j symbols have allowed us to simplify Equation (E-6) considerably: D..i = 1 I ii 1 2' k mm x ( 1 \ )( I X ) || 2 (E-15) -m -k k m„ 41 Some cancellation has occurred due to the following identity: ( J j ) m ( 1} i-n. (2j + ir l/2 (E 16) (J m -m Now rule (ii) allows us to perform the sums over mand m D... L k

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145 ,1 1 W 1 1 N | 1 2 x (n t -0 || (E-17) ^k -k 0^0 k -k Using rule (iv) we obtain D iit = £ (J ) 2 | | 2 (E-18) When this 3-j symbol is evaluated using the above identity we obtain D.., = Z -| | (E-19) A convenient form for is given by Vidal, Cooper, and Smith. n-1 n-1 =6 (-l) 1/Z(1 ^" n) [2£+l] 1/2 ( 2 J ) (E-20) mm' m-q m+q 2 2 m Inserting this into Equation (E-19) we obtain n-1 n-1 .. D... E { (_Dl/2(l-k-q-n) j2 2 n m,-k v m-q m+q ~2 ~2 _m X{6 km'^ 1/2(1+m '" q, n) I^C) "I, \)> -k,m 'I m-q m+q -m 1 9 x — | | ( (E-21) C + \ n ~^ n-1 n-1 n-1 = 6 (-l) 1 ~ n+m 2 1 ( 2 2 )( 2 2 ) mm m-q m+q -m m-q' m+q -m 2 2 2 2 x || 2 (E-22) Equation (E-22) is now in a form convenient for computation. The 3-j symbols are computed using a subroutine given in Reference (45).

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APPENDIX F THE MANY-PARTICLE FUNCTION r(Aoj) The definition of the many-particle function r(Aw) is obtained by combining Equation (1-76) with the result for given in Equation (C-41) Upon making this step we find: / o 2 2 2 4 ™ 4 f r r K k i -u £-(k 2 a r(4 ., _i f_J] ) dt J dk j dk ,ut.-( -^ e 2/2 k i> f 3tt 12 (F-l) Here n is the density of the electron perturbers in the plamsa; A is the thermal wavelength of the electrons; = (kT)" 1 for the electrons. The function £(1^, kp was defined in Appendix C. It is related to the free-free Gaunt factor 42 43 by the following: 23 f(k x k 2 ) = k 1 k 2 — g ff (k v k 2 ) (F-2) We insert a theta function into Equation (F-l) in order to extend the lower limit on the t-integration to -. OO 00 CO 4ne A r(Aaj) = -i [ ^L] 3tt 2 2 Ek l 00 00 CO \ dt\ dt \ dk 2 G(t)e 1Awt eit: fe (k 2 k l> • e 2m fCkp k 2 ) ( F 3 ) i ^V^ J ^ \ dk \ dk„ ^r]dz V^ e 1 iAut z+iri 2 2 B 2 2 ^ k 1 -it £(fcf kf) -f 2m 2 1 e 2m f(k r k 2 ) (F-4) 146

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147 We have inserted the integral definition of the theta function. 16 We define' n to be the positive infinitesimal. We may now perform the t-integration in Equation (F-4) OS CO co 2 2 r(A.) = l^l£l J dz ^ 5 dk i 5 ^ e "^ fCk l'V 6 {Aa, ~ Z "'2m~ (V k l )} • ( F ~ 5 ) Let us now define a new function of Aw: 2 2 4.3 f r ? k i 5 dk i 5 G(Au>) = -tt[ T ] \ d^ \ dk 2 e p 2m f(k ,k ) 3tt 2 6 {Aal to (k 2" k l )} (F_6) With this definition, Equation (F-5) becomes CO 5 r(Aco) = \ dz —VG(Aoj-z) (F-7) At this point we apply the following identity -^ -> P \ ~ 1tt5(z) (F-8) We then obtain r(Au) 00 oo _I P I dz G(Au-z) + [ dz G(Aw z)6(z) ^ (p 9) —CO —CO CO p J dz S£ + iG(AW) • (F 10 > This equation allows us to identify G(Aoi) as the imaginary part of the function r(Aio) :

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148 r im (Aa>) = G(Aoj) ; (F-ll) CO i'S r T (z) r_ (Aw) P \ dz m (F-12) Re n i z-Aw We now must develop a computational form for the function T (Aw) where 00 oo 2 2 4ne \ [ \ „ 5 dk i J r im (A„) -„ [— ji] \ d kl dk 2 2m £(k ,kj) JIT *' o •6 {Aco~ (k^-k*)} (F-13) We now insert the definition of Equation (F-2) and make the following change of variables: k i 4h h zr K 2 • < F 14) n h We obtain 4 3 4ne \_ 2 „ 3 n tk \ r Tt r me 4 r 2n r T (Aw) = -it [ =-] [— ] [— rl Iiri L 2 J \2 J L 4 J 2/3 3tt h me CO oo 2 f f _K 1 /6 R_ 2 2 \ d^ I dK 2 e "KjK g ff (K 1 ,K 2 )6{An-K 2 + Kj} (F-15) where 3 2 Afi = ~Ao) ; 6 R = -25K B T ; (F-16) me me are the frequency and temperature in Rydberg units. We consider first the case where Au)>0. We employ an identity from Messiah to simplify the delta function with the result

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149 4ne X 2 ,_3 Im l 2 Jl 2 J L 4 J 2/3 3tt H me 00 CO J dK l J dK 2 e ~ Kl/6R K i g ff ( K r K 2 > n\ r 2ne r 8Trm -|l/2 2-n \ -k /Q„ l jJ*" 0) [-y-H-j^-] 7 y 7 j d k;L e l'^K^OL^X^CF-lQ) o The one-dimensional integral in this equation may be evaluated rapidly and accurately using a Gauss-Laguerre quadrature formula. Proceeding in a similar manner and using the fact that g f (k ,k ) = g ffr (k ,k ) we obtain r im (Ato<0) = e l An l /0 R r im (Aoj>0) (F-20) After computing T (Aw) in this manner we may obtain T (Ao)) by lm Re numerically evaluating the Hilbert transform of Equation (F-12)

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APPENDIX G A COMPUTATIONAL FORM FOR THE ATOMIC FACTOR We indicated previously that the electron broadening operator for our case will factor into a frequency-dependent many-body part times an atomic factor. The many-body function r(Au) was discussed in Appendix F. As a prelude to the development of a computational for the atomic factor we now consider = „ -^mlRln^m > • (G-l) g m ''11 ll 11 This expression gives the form of the atomic factor in the spherical representation. That there is no internal sum over n here is a consequence of the no-quenching approximation. Using Equations (E-3) (E-5) and (E-7) we obtain = £ E (-l) k k £,m A/X/Aj-JO JO JO x (-l) m + m l [(21 + 1)(2£ 1 + 1)] 1/2 ( )( -m k m } I IV % 1 £' x [(22 + 1)(22' + 1)] 1/2 ( 0^-B^ -k m*1 (G-2) Rule (ii) (Appendix E) for 3-j symbols causes the phase factors to cancel to unity. If we also use rules (iii) (iv) and (vi) to rearrange this expression, we obtain: Z E (2£ + 1) k 2,111 tli £ 1 £, £' 1 2. £' 12. 1/2 ( )( )( )( X ) x [(22 + 1)(2£' + 1)] /Z S 0, -m k m 1 V -m' k m J (G-3) 150

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151 41 The following identity allows us to perform the sum over k,m 1 : ( 3 1 J 2 3 3 ( 3 l J 2 J 3 } E m m m ; V m ml. = (2j + 1) 6. ., 6 m im2 1 2 3 1 2 3 3 J 3 J 3 m 3 m 3 (G-4) After performing this sum we obtain a much simpler expression. tl£ 12 = 6 5 E || 2 (21. +1) ( ? (G-5) &* nun n J_ J_ *i Since the 3-j symbol must satisfy rules (i) and (iii) the sum over £ has only two terms. Performing this sum we obtain 2 A 1 £-1 2 = 6.., 6 { I I (2£-l) > ££ mm ' 2 ,£1 £+1.2 + | I (2£ + 3) { 6 } (G-6) The 3-j symbols here are a special case evaluated in Edmonds. After substituting in the values and performing some cancellation we have = 6 ££l 5^, { | | 2 j^+ | | 2 ^.(G-7) The radial matrix elements are given by Condon and Shortley: 47 CO s 3 a n 2 2 1/2 dr r R(n£) R(n £-1) =4 -**[n £ ] z z (G-8) where a is the Bohr radius and z is the nuclear charge. When this expression is used in Equation (G-7) we find q 2 2 = 6 flflt 6 — 2— -— {[ n 2 -£ 2 ]£ 11 ££ mm 2 2£+l 4z + [n 2 -(£+l) 2 ] (£+1)} (G-9) After collecting terms we arrive at a convenient form. Q 2 2 n 11 2 2 = 6 n „ 6 — ^— [n -£ -£-1] ££ mm I (G-10)

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152 Our intent here is to obtain a computational form for the atomic part of the electron broadening operator. Now we will show that Equation (G-10) is a valuable step in the development of a computational form for the atomic factor in the parabolic representation. We wish to compute = I -< n q m |r| nq'm'> (G-ll) q l m i Now the sum over q_m that is, > Inq.m > = E • (G-13) Vi I I 1 -< n £ m |R|n£ m > £ A £ 2 m 2 £ 3 m 3 U 2 2 x t (G-14) Here we may recognize that the two middle factors are just the spherical representation of the atomic factor. We now insert the result appearing in Equation (G-10) = I E £ 2 m 2 £ 3 m 3 >' f (G-15)

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153 Q 2 2 = y o 1 7 7 % m — [ n -1^-4-1] (G-16) 11 4z We again need to make use of Equation (E-20) We then obtain = I {6 (-1) l/2U+m-q-n) [2£ + 1] -l/2 Vl X n-1 n-1 2 2 £., 9a„n x ( 2 J ^-^-[n 2 -^-!] m-q m+q ,2 11 — n — n m 4z n-1 n-1 x { ( 1/2(1^-, *-n) + 1/2 2 2 1 m,m 1 m-,-q m+q' -m 9^n 2 1+m n .(q+q') 4z £ n-1 n-1 n-1 n-1 2 2 2 ? m-q m+q -m y v m-q' m+q' -m y [n -£ -£-1] (G-18) 2 2 2 2 This is the final form for the atomic part of the electron broadening operator.

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APPENDIX H NUMERICAL PROCEDURES Introduction The first two sections have developed the theory needed in computing Stark broadened spectral line profiles. The numerical procedure for producing these profiles involves four basic steps. This appendix will provide a discussion of the four programs needed, as well as the computational techniques involved. The first three programs carry out the production of the electric microfield probability distribution function P(e). The fourth program computes the Stark broadened profile for a given set of plasma parameters. Also, this final program generates plots of the Stark profiles and the Doppler-corrected profiles. In addition to discussing the numerical techniques employed, we discuss sources of error and the general reliability of the programs in their present form. The Alpha-Search Program In Section II we indicated that a is a variable effective range parameter. Essentially, a determines how much of the calculation will be treated by the collective coordinate method. It enters the calculation in the following manner: N l N 2 V V„ + I W.„ + I W n (H-l) i0 mO j=l J m=l where V is the total potential energy of the plasma and W.. is given by 154

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155 W. n = XZn 7~ e jO (H-2) J JO We recall that these W's are the short-range central interactions which are treated by cluster-expansion techniques. The term V defined by these two equations contains long-range central interations as well as all the noncentral interactionsThis term is treated by the collective coordinate method. By inspecting these two equations and comparing them with Equation II-9, we can see that as a approaches unity, the contribution of central interactions to V vanishes. In this limit, then, V contains contributions only from the noncentral interactions. In this sense, as we stated previously, the parameter a "measures" the relative importance of the short-range interactions and the collective coordinate contribution. In the calculation of P(e), two basic approximations have been made. First, we have terminated the cluster expansion at second order. Second, we have neglected correction terms in the Jacobian of the transformation from spatial coordinates to collective coordinates. As is well discussed in the literature, > > t ^ e a pp ro p r i a te choice of the value of a should make negligible the error in P(e) due to each of these two approximations. If this is the case, we should be able to locate a range of a values for which the final computed P(e) is stationary. Indeed, this is the case. The a-search program attempts to locate an "a-plateau" region over which T(L) is stationary. In all cases considered here, there is an obvious a-plateau region. The a-search program currently requires 256 k bytes of memory on the IBM 370-165 at the Northeast Regional Data Center at the University of Florida. The program computes T(L) in the first approximation at several L values (currently four values) for each a value. A typical

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Table H-l. The electric microfield distribution function P(e) is tabulated for different values of a, the effective range parameter, a = 0.8. The radiator is hydrogenlc argon (X= +17), perturbed by ions of charge +1.

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TABLE. H-l. 157 P(E) E a=0.95 a =1.05 a=1.15 0-1.25 a-1.35 0.1 0.235 0.237 0.240 0.239 0.232 0.2 0.758 0.763 0.759 0.767 0.751 0.3 1.223 1.227 1.232 1.230 1.215 0.4 1.430 1.431 1.431 1.429 1.422 0.5 1.387 1.384 1.382 1.380 1.381 0o6 1.196 1.193 1.189 1.188 1.194 0.7 0.959 0.955 0.952 0.952 0.959 0.8 0.734 0.731 0.729 0.729 0.736 0.9 0.546 0.544 0.543 0.544 0.549 1.0 0.400 0.399 0.398 0.399 0.404 1.1 0.291 0.290 0.291 0.292 0.295 1.2 0.212 0.212 0.212 0.213 0.215 1.3 0.154 0.155 0.155 0.156 0.158 1.4 0.113 0.113 0.114 0.114 0.116 1.5 0.084 0.084 0.084 0.085 0.086 1.6 0.062 0.062 0.063 0.063 0.064 1.7 0.047 0.047 0.047 0.048 0.048 1.8 0.035 0.036 0.036 0.036 0.037 1.9 0.027 0.027 0.028 0.028 0.028 2.0 0.021 0.021 0.021 0.021 0.022

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158 run for 20 a values requires vL.2 minutes of computer time. Since we employ the first approximation to T(L) the major part of the program is directed at computing the terms I 1Q (L) and I (L) These one-dimensional integrations are performed by subdividing the range of integration and using a 32-point Gauss-Legendre quadrature formula in each sub-interval. This procedure has been tested for numerical accuracy; results indicate that we have obtained at least six significant figures of numerical precision for I, Q (L) and Ig-rCL). Thus the numerical precision present in the a-search program is comfortably redundant. Presently, the choice of a requires a decision by the programmer. A future goal is to develop the program to allow the machine to make this choice. After an inspection of the output for T(L) it is a straightforward task to locate a range of a values for which T(L) is approximately stationary. In certain cases (especially a = .6,. 8), the a-plateau appears to be quite narrow. However, as is indicated in Table H~l, the final curve for P(e) is much more stable with respect to a variations than might be expected. In addition, the effect on P(e) due to a variations offers one type of error estimate for the final P(e) curves. From this point of view, we may expect that errors in P(e) will not exceed a few per cent. The P(e) Production Program The selected a value is fed as input into the P(e) production program. This program requires 384 k bytes of storage and may require from ^4.0 to 7.5 minutes of computer time (depending on the input parameters). Three major job steps are required, with intermediate results stored on magnetic disk storage. The first step produces I (L)

PAGE 165

159 The second step produces I„(L). The third step computes T(L) and performs the integral transform /to produce P(e) The final P(e) appears as printed and punched output as well as graphical display. The method employed to compute I (L) is the same as that discussed for the a search. The difference is that here we compute I (L) over a finer mesh of L values with consequently increased CPU time. For R = a normal CPU time for this step (for 95 L values) is approximately 50 seconds. This time is doubled for cases of R ^ 0. We shall only repeat here that the accuracy in the computation of 1, (L) probably exceeds six significant figures. The evaluation of I„(L) involves a two-dimensional integration so that this step requires significantly more computer time than the first step. The case R = requires about 2.8 minutes of CPU time. This time is to be doubled if R / 0. The two-dimensional integration is carried out using a product of two one-dimensional rules (Trapezoidal and Simpson rules) Also an algorithm developed especially for these half-quadrant 48 integrations is employed. Currently, the sum over k is terminated at k = 6. We obtain about three significant figures with k = 6 but it has been pointed out that for higher a. values (eg. a = /3) more terms may 49 be needed. The cost of evaluating these terms is considerable so that it is important to remember that I„(L) is expected to be a small correction to I (L) That is to say, in the same spirit in which we linearized the Debye-HUckel pair correlation function, we now recognize that an error in I~(L) as large as ten per cent will not cause more than a two or three per cent error in the final P(e). (Normally I„(L) is less than twenty per cent of the magnitude of I (L).) This gives the justification for truncating the sum over k at k = 6.

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160 The final step in this program reads as input the results of the previous step, computes T(L) in the second approximation, and evaluates the sine transform. This step requires one minute of CPU time. As we have indicated previously we must evaluate the following integral: CO P(e) 2e7T~ 1 \ L T(L) sin(EL) dL f (H-3) o where T(L) = exp {-yL 2 + I 1 (L) + I^L)} (H-4) The following approach was suggested by Coldwell. We approximate T(L) in the following manner: exp[f(x)] = E 9(x. x) 6(x x.) exp[a.x + b.] (H-5) l + l 11 i where f f i+l i f .x. f .X. a l-*i + l-*l "i" 11 + 1 1 + 11 • (H- X. -, X. 1 + 1 1 We have performed a piece-wise linear fit to the smoothe function in the exponent of T(L) If we insert this expression into the integral of Equation (H-3) we obtain J I = \ x exp [f(x)] sin(ex) dx o x b. 1+1 a .x x e sin(Ex) dx < (H-7) X. 1 This integral may be performed analytically with the result: a.x 2 2 b l a e i r e "i I = 2 e {— — [(a.x — -) sin ex i a + e a + e i i

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2a. (ex = -j — ) cos ex]} a + e 161 X i+1 (H-8) X l This computation is carried out using IBM's extended precision option to insure adequate numerical accuracy. As in all numerical transforms of this type, at large values of £ we encounter unphysical oscillations in the computed P(e). One interpretation of this phenomenon is that the numerical error is of the same order of magnitude as the exact value of the transform. With this in mind, we may regard the amplitude of these oscillations as an estimate of the error obtained in the numerical transform. Indeed, checks on the numerical accuracy of this technique indicate that, in the region where P(e) shows structure, we obtain about three significant figures of precision. Extension of the microfield table into the asymptotic region (large e) is carried out by the third computer program. The Asymptotic Microfield Program In the third program, the asymptotic microfield distribution function described in Section II is joined smoothly to the computed transform. Let us make the following definitions. The range of e values corresponding to the microfield peak we define to be Region I. The range of e values, e>100 (where we measure £ in units of e n the Holtsmark field strength), we designate Region III. The intermediate region (Region II) contains the unphysical oscillations. We assume that for e>100, P(e) is given exactly by the asymptotic form. The approach taken here is to use the reliable results from Regions I and III to interpolate into Region II. 5/2 If we scale the data in Regions I and III by e (the inverse of the asymptotic Holtsmark distribution) and take the logarithm of the

PAGE 168

162 result, we obtain a smooth curve. A fit to this curve is performed using a least-squares Monte Carlo Spline fitting routine. We simply invert the fit to obtain P(e) in Region II. In this manner we preserve the accuracy achieved in Region I and obtain a P(e) which makes a smooth transition to the asymptotic form. In order to make a statement about the numberical accuracy of this technique it is necessary to compare the fitted P(e) with the transform result in Region II. When these two results are plotted, several facts can be immediately noticed. First, as the two curves enter Region II, they overlie one another. Second, as e increases, the transform result begins to oscillate closely around the fitted P(e). These two facts tend to support our previous error estimate of a few per cent for P(e). In addition, the magnitude of P(e) in Region II is so small (^10~ ) that errors in this region are not likely to cause any difficulty in subsequent line profile calculations. The Stark Profile Program The fourth program generates the function J(u,e) discussed in Section I and carries out the microfield average over the static ions to produce the Stark broadened profile. In a final step, Doppler corrections are added by the convolution indicated in Equation (1-6) The final profile appears as printed and punched output as well as graphical display. In addition to the appropriate microfield table, basic input to this program consists of the electron number density, the electron kinetic temperature, the nuclear charge of the radiator, and the principal quantum number n of the upper level of the desired Lyman transition. The CPU times required are 2.1 minutes for Lyman-a and 6.8 minutes for

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163 Lymana Most of this time is consumed by the matrix inversions required to produce J(w,e). The matrices which must be inverted are of order 2 n u Preliminary computations of the electron broadening function r(Aw) for the cases studied in this work indicated that T (Aw) is smaller in Re 3 magnitude than Fj (Aw) by a factor of vl_0 Therefore the program does not compute r (Aw) and it is assumed to be negligible. This causes most of the shift and asymmetry of the resulting line profile to vanish. We mention here that it would be straightforward task to include T (Aw). Re However, by neglecting r (Aw) we achieve a savings of approximately 3 minutes of computer time. Another feature of the line profile calculation is an approximate treatment of electron correlations. In order to approximate the effect of electron correlations, we modify our ideal gas result for T T (Aw) such Im that, F Im (Aw i V = r im ( V < H ~ 9) This procedure is suggested by Figure 1 of Reference 22. If we impose the cutoff not at w but at some fraction or multiple thereof, we find that the line center will be lowered or raised. However the changes are small and quite insensitive to the cutoff frequency. In the Classical Path Impact Theory, the operator corresponding to r Re (Aw) is also neglected. Also, that theory treats electron correlations by a cutoff procedure yielding a result similar in form to that given in Equation (H-9.) These two facts appear to give the present theory the form of a quantum mechanical analog to the Classical Path Impact Theory.

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164 J(oi,e) is given by Equation (1-95): J(o),e) = -it Id I D, K p., (H-10) i 11 11 1 n where D and p are real. Now <£. A 1 + IB.., (H-ll) where from Equations (1-96) and (1-97) we have (after letting F (Aco) vanish) : A -5 {Aoj 4 -gn q.e} (H-12) ii ii z nz u i -v -> B.., = K" 2 r im (Aa>) ^ l lin '* nt (H-13) We have K. = C. + iF. ., (H-14) ii n n' v such that [A + IB] [C + iF] = JL and JL is the unit matrix. Now J(tD,e) reduces to J(oj,e) — rr I D. F.., p., (H-15) ... ii ii i n From Reference 19 we have: F.., = -(B + AB _1 A)" 1 .., (H-16) ii ii A final form for J(u),e) is given by J(a),e) = tt~ Z D. t (B + AB A)"*, p ., (H-17) ii' x This expression makes it obvious that for every value of Aid and e we must invert a matrix. The matrix inversion is carried out by an IBM-supplied subroutine MINV which has been modified to employ extended precision

PAGE 171

165 arithmetic. The atomic factor in Eauation (H-5) is evaluated in Appendix G. 3 The n distinct 3-j symbols required are computed by a subroutine given in Reference 33. T (Aid) is calculated in Appendix F. After J(u>,e) has been computed, the integration over the microfield distribution function P(e) is performed by a Trapezoidal Rule formula. In order to assess the numerical precision in the computation of J(o),e) we consider the last three equations in Section I. Starting with Equation (1-97) we note: (1) as discussed in Appendix F, T (Aw) is computed using a 40 point Gauss-Laguerre quadrature formula — numerical tests indicate that we obtain at least six figure accuracy; (2) in the computation of the atomic factor, we essentially are using single precision arithmetic (8 significant figures) to perform an integer arithmetic calculation. We obtain at least six figure accuracy in calculating H(Aa)) as given in Equation (1-97) Now consider Equation (1-96). In the present work, all numerical operations in the production of R (w) the effective radiator resolvent operator, (including the matrix inversion) are carried out using IBM's extended precision option (33 significant figures). We retain six figure accuracy. In the case of Equation (1-95), argument (2) above holds for the operator D. The result is that we obtain at least six figure accuracy in the calculation of J(oi, e) There are two possible checks on the accuracy of the microfield integration. The first check is to halve the integration interval and double the number of points. When this check was applied to a test case, we obtained ^4 place agreement between the two results. A second check involves extending the limit of integration. This check applied to a worst case (slowly decaying P(e), e.g., T = 0.25) yielded roughly 3 R

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166 place agreement. The result is that we obtain 3 significant figure accuracy for the Stark profiles. Actually the precision for the Doppler-corrected profiles is slightly better than this due to the "smoothing" effect of the Doppler convolution. Our conclusion is that the numerical error present in the final line profile is almost entirely due to the numerical error in the microfield functions. This allows us to set a rather conservative error bar on the final line profiles of a few per cent.

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APPENDIX I TABLES OF THE ELECTRIC MICROFIELD DISTRIBUTION FUNCTION P(e) In this appendix we present tables of the electric microfield distribution function P(e) computed using the numerical procedures discussed in Appendix H. The various parameters are defined as follows: A The plasma parameter (= r /A ) The value of this parameter is determined by the temperature and density of the electron perturbers. R The ratio of the density of the charge z specie of ion perturber to the density of the charge z specie. TEMP RATIO The ratio of the electron kinetic temperature to the ion kinetic temperature. CHARGE AT ORIGIN The net charge of the hydrogenic radiator (= z-1). zl,z2 The charges of the ion perturbers. 2 e is expressed in units of e (= e/r ) 167

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168 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB^RS A= 0.2000 R= 0,0 TEMP RATIQ= 1.00 CHARGE AT ORIGlN= 9*00 Z 1= 1,00 Z2= 9.00 E P(E) 0. 10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 0. 90E 00 0. 10E 01 0, I IE 01 0.1 2E 01 0. 13E 01 0. 14E 01 0.15E 01 0. 16E 01 0. 17E 01 0, 18E 01 0. 19c 01 0.20E 01 0.25E 01 0.3 0E 01 0.35E 01 0.40E 01 Q.45E 01 0.50E 01 0.29671E-01 0.60E 01 0.17142E-01 0.7QE 01 0.10789E-01 0.80E 01 0.72993E-02 0.90E 01 0.52378E-02 0.10E 02 0.39335E-02 0.12E 02 0.24105E-02 0.14E 02 0.15585E-02 0.16E 02 0.10512E-02 0.18E 02 0.73717E-03 0.20E 02 0.53550E-03 0.22E 02 0.40156E-03 0.24E 02 0.30975E-03 0.26E 02 0.24490E-03 0.28E 02 0.19777E-03 0-30E 02 0.16255E-03 0.35E 02 0.10503E-03 0.40E 02 0.71638E-04 0.45E 02 0.50974E-04 0.50E 02 0.37507E-04 0.60E 02 0.21932E-04 0.70E 02 0.13850E-04 0.80E 02 0.92562E-05 0.90E 02 0.64616E-05 0. 10E 03 0.46688E-05 0, 82795E-02 = 3 2355E-01 0, 70015E-0 1 0, 1 1 793E 00 0. 1 7206E 00 0. 2281 7E 00 0. 2 8 22 7E 00 0. 33099E 00 0< 37184E 00 0< 40329E 00 0* 42471E 00 0, 4 3629E 00 0. 43883E 00 0. 43352E 00 0, 42178E 00 0. 40507E 00 0 38481E 00 Q< 3G224E 00 0< 33844E 00 0 3 1427E 00 Or 20551E 00 0. 1 31 16E 00 = 85068E•01 Q 57464E-0 1 0. 40516E-01

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169 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB5RS A= 0,4000 R= 0.0 CHARGE AT ORIGIN= 9.00 TEMP RATIO= 1.00 Zl= 1.00 Z2= 9.00 P(E) 0.10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. HE 01 0. 12E 01 0. 13E 01 0.14E 01 0. 15E 01 0.16E 01 0. 17E 01 0. 18E 01 0, 19E 01 0.20E 01 0.25E 01 0.30E 01 0.3SE 01 Q.40E 01 0.45E 01 0.50E 01 0,60E 01 0.70E 01 0.80E 01 0.90E 01 0.10E 02 0.12E 02 0. 14E 02 0. 16E 02 0. 18£ 02 0.20E 02 0.22E 02 0.24E 02 0.2&E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. 10E 03 0. 19933E-01 0.76325E-01 0. 1S982E 00 0.25750E 00 0.35582E 00 0. .44323E 00 0.51 181E 00 0.55773E 00 0, .58074E 00 0.S8321E 00 0.56899E 00 0.54242E 00 0.50765E 00 0.46826E 00 0.42706E 00 0< .38613E 00 0, .34688E 00 0, 31 01 8E 00 0, .2 7649E 00 0. .24600E 00 = 1 3715E 00 0. .79675E-01 0, 48503E01 Q< 3 1248E-01 0< 2 1209E-0 1 0 1 4989E-01 0. 80386E02 0, 45998E-02 0, 2 794 3E-02 Oc 17931E-02 0< 12092E02 0, 62569E03 = 37178E-03 0.. 24365E03 0^ I6914E-03 Oc 12035E03 0, 87 183E-04 = 64236E-04 0 48092E04 0. 3&551E04 0„ 28174E•04 0. 15539E04 Oc 91622E05 0. 56849E05 Go 36742E•05 0. 16876E•05 0. 8 5331 E06 0, 46362E06 0, 26642E•06 0,., 160 13E06

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170 ELECTRIC MICROFIELO DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.6000 R= 0.0 CHARGE AT ORIGIN^ 9*00 TEMP RATIO= I. 00 Zl= 1.00 Z2 = 9.00 PIE) Oc 10E 00 = 20E 00 0. 30E 00 0. 40E 00 0* 5 0E 00 0. 60E 00 Oo 70E 00 = 80E 00 Oo 90E 00 Oo 10E 01 0., 1 IE 01 0, 12E 01 0 13E Oi 0* 14E 5 Oo 15E 01 Oo 16E 01 Oo I 7E Oi Oo 18E oa 0. 19E 01 0. 20E 01 Oo 25E 01 0. 30E 0! 0. 35E 01 Oo 40E 01 0. 45E 01 0. 50E oa Oo 60E 01 0. 70E 01 Oo 80E oi Oo 90E 01 0, 10E 02 Oo 12E 02 0. 14E 02 0. 16E 02 0, 18E 02 0. 20E 02 0. 22E 02 0. 24E 02 0. 26E 02 0. 28E 02 0. 30E 02 0. 35E 02 0. 40E 02 0. 45E 02 0. 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 0. 90E 02 0. 10E 03 0.5Q061E-01 0.18414E 00 0.36197E 00 0.53756E 00 0.67584E 00 0.76003E 00 0.78978E 00 0.77485E 00 0.72874E 00 0.66441E 00 0.59217E 00 0.51927E 00 0.45019E 00 0.38734E 00 0.33 169E 00 0.28332E 00 0.24180E 00 0.20646E 00 0.17652E 00 0.15124E 00 0.72915E-01 0.38276E-01 0.21727E-01 0. 1 3077E-01 0.8 1010E-02 0.51674E-02 0.22912E-02 0.1 1280E-02 0.6101 IE-03 0.35875E-03 0.22692E-03 0.10834E-03 0.6 1023E-04 0.37269E-04 0.23137E-04 0. 14510E-04 0.92342E-05 0.59900E-05 0.39741E-05 0.26941E-05 0.18622E-05 0.79448E-06 0.36830E-06 0. 18222E-06 0.95030E-07 0.29264E-07 0. 10237E-07 0.39452E-08 0.16415E-08 0.72675E-09

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171 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERIUR3ERS A= 0.8000 R= 0.0 TEMP RAT10= 1.00 CHARGE AT ORIGIN^ 9.00 Zl = 1.00 Z2= 9.00 E P(E> 0. 10E 00 0.12237E 00 0.20E 00 0.41742E 00 0.30E 00 0.73441E 00 0.40E 00 0.95661E 00 0.50E 00 0.10474E 01 0.60E 00 0.10281E 01 0.70E 00 0.93966E 00 0.80E 00 0.81936E 00 0.90E 00 0.69277E 00 O.IOE Oi 0.57426E 00 0.11E 01 0.47032E 00 0.12E 01 0.38264E 00 0. 13E 01 0.31040E 00 0. 14E 01 0.25178E 00 0. 15E 01 0.20455E 00 0.16E 01 0.lfc669E 00 0. 17E 01 0.13636E 00 0. 18E 01 0. 1 1204E 00 0. 19E 01 0.92505E-01 0.20E 01 0.76738E-01 0.25E 01 0.32482E-01 0.30E 01 0.15390E-01 0.35E 01 0.77776E-02 0.40E 01 0.41S04E-02 0.45E 01 0.23258E-02 0.50E 01 0.13616E-02 0.60E 01 0.52827E-03 0.70E 01 0.23S64E-03 0.80E 01 0.11756E-Q3 0.90E 01 0.64287E-04 O.IOE 02 0.37953E-04 0.12E 02 0.15666E-04 0.14E 02 0.73255E-05 0. 16E 02 0.35754E-05 0.18E 02 0.18041E-05 0.20E 02 0.93957E-06 0.22E 02 0.50419E-06 0.24E 02 0.27831E-06 0.2GE 02 0.15776E-06 0.28E 02 0.91682E-07 0.30E 02 0.54532E-07 0.35E 02 0.16307E-07 0.40E 02 0.54218E-08 0.45E 02 0.19609E-08 0.5QE 02 0.75974E-09 0.60E 02 0.13436E-09 0.70E 02 0.28175E-10 0.80E 02 0.67364E-11 0.90E 02 0.17885E-11 O.IOE 03 0.5 1737E-12

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172 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ZRS A= 0.2000 R= 0, CHARGE AT ORIGIN= 17.00 TEMP RATIO= 1.00 Zl= I. 00 22= 17.00 P(E) 0. 10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.60E 00 0.70E 00 O. 8 0E 00 0.90E 00 0. 10E 01 0. 1 IE 01 0. 12E 01 0.13E 01 0.14E 01 0. 15E 01 0. 16E 01 0.17E 01 0. 13E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0. 12E 02 0. 14E 02 0. 16E 02 0.18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 O.40E 02 0.45E 02 0.50E 02 0.60E 02 0. 70E 02 0.80E 02 0.90E 02 0.10E 03 0.9S228E-02 0.37158E-01 0.80215E-01 0.13465E 00 0.19562E 00 0.25808E 00 0.3 1733E 00 0.36968E 00 0.41225E 00 0.44355E 00 0.46313E 00 0.47149E 00 0.46980E 00 0.45964E 00 0.44278E 00 0.42099E 00 0.39590E 00 0.36892E 00 0.34I22E 00 0.31371E 00 0.19583E 00 0.12025E 00 0.75688E-01 0.49866E-01 0.34268E-01 0.24542E-01 0.13796E-01 0.83781E-02 0.53663E-02 0.36029E-02 0.25254E-02 0.13890E-02 0.36449E-03 0.58938E-03 0.42614E-03 0.31716E-03 0.24048E-03 0.18514E-03 0.1 4534E-03 0.1 1575E-03 0.93423E-04 0.57539E-04 0.37575E-04 0.25666E-04 0. 18169E-04 0.98870E-05 0.58438E-05 0.36723E-05 0.24195E-05 0.16551E-05

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173 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTJR3ERS A= 0.4000 R= 0.0 TEMP RATIO= I. 00 CHARGE AT ORIGIN^ 17.00 Zl = 1.00 12 = 17.00 E P(E> 0.10E 00 0.28835E-01 0.20E 00 0.10951E 00 0.30E 00 0.22627E 00 0.40E 00 0.35799E 00 0.50E 00 0.48363E 00 0.60E 00 0.58673E 00 0.70E 00 0.65769E 00 0.80E 00 0.69383E 00 0.90E 00 0.69796E 00 0. 10E 01 0.67612E 00 0.11E 01 0.63564E 00 0. 12E 01 0.58360E 00 O. 13E 01 0.52597E 00 0. 14E 01 0.46730E 00 0. 15E 01 0.4 1072E 00 0.16E 01 0.35815E 00 0. 17E 01 0.31061E 00 0. 18E 01 0.26843E 00 0. 19E 01 0.23153E 00 0.20E 01 0.19958E 00 0.25E 01 0.96666E-01 0.30E 01 0.50139E-01 0.35E 01 0.27997E-01 0.40E 01 0.16809E-01 0.45E 01 0.10658E-01 0.50E 01 0.70776E-02 0.60E 01 0.34344E-02 0.70E 01 0.18018E-02 0.80E 01 0.10092E-02 0.90E 01 0.60027E-03 0.10E 02 0.37715E-03 0.12E 02 0.1717SE-03 0.14E 02 0.91379E-04 0.16E 02 0.54402E-04 0.18E 02 0.34735E-04 0.20E 02 Q.22793E-04 0.22E 02 0.14962E-04 0.24E 02 0.10010E-04 0.26E 02 0.68513E-05 0.28E 02 0.47876E-05 0.30E 02 0.34090E-05 0.35E 02 0.15630E-05 0.40E 02 0.77485E-06 0.4SE 02 0.40847E-06 0.50E 02 0.22627E-06 0.60E 02 0.78005E-07 0.70E 02 0.30289E-07 0.80E 02 0.12870E-07 0.90E 02 0.58705E-08 0. 10E 03 0.28358E-08

PAGE 180

174 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBcRS A= 0.6000 R= 0.0 TEMP RATIQ= 1.00 CHARGE AT ORIGIN^ 17.00 Zl= I. 00 Z2= 17.00 E P(E) 0.10E 00 0.87047E-01 0.20E 00 0.31204E 00 0.30E 00 0.58882E 00 0.40E 00 0.82879E 00 0.50E 00 0.97782E 00 0.60E 00 0.10245E 01 0.70E 00 0.98700E 00 0.80E 00 0.89527E 00 0.90E 00 0.77760E 00 0. 10E 01 0.65488E 00 O.ilE 01 0.53982E 00 0. 12E 01 0.43866E 00 0.13E 01 0.35331E 00 0. 14E 01 0.28320E 00 0. 15E 01 0.22662E 00 0. 16E 01 0. 18145E 00 0 17E 01 0. 14561E 00 0. 18E 01 0.1 1725E 00 0.19E 01 0.94821E-01 0.20E 01 0.77057E-01 0.25E 01 0.29511E-01 0.30E 01 0.12954E-01 0. 35E 01 0.65264E-Q2 0.40E 01 0.37288E-02 0.45E 01 0.23715E-02 0.50E 01 0.15980E-02 0.60E 01 0.74376E-03 0.70E 01 0.35938E-03 0.80E 01 0.18001E-03 0.90E 01 0.93331E-04 0.10E 02 0.50016E-04 0.12E 02 0.15767E-04 0.14E 02 0.5S728E-05 0.1 6E 02 0.2 1825E-05 0. 18E 02 0.93600E-06 0.20E 02 0.43440E-06 0.22E 02 0.21561E-06 0.246 02 0.11311E-06 0.26E 02 0.61975E-07 0.28E 02 0.35056E-07 0.30E 02 0.20323E-07 0.35E 02 0.57179E-08 0.40E 02 0.17985E-08 0.45E 02 0-61783E-09 0.50E 02 0.22812E-09 0.60E 02 0.36929E-10 0. 70E 02 0.71466E-1 I 0.80E 02 0.15869E-H 0.90E 02 0.39320E-12 0.10E 03 0.10660E-12 0.80E 02 0.15869E-11 0.90E 02 0.39320E-12

PAGE 181

175 ELECTRIC MICROFtELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.8000 R0.0 TEMP RATION I. 00 CHARGE AT ORIGIN= 17.00 Zl = 1.00 Z2 = 17.00 E PIE) O. 10E 00 0.24017E 00 0.20E 00 0.7692SE 00 0.30E 00 0.12320E 01 C.40E 00 0.14314E 01 0.50E 00 0.13S16E 01 0.60E 00 0.1 1891E 01 0.70E 00 0.9S200E 00 0.80E 00 0.72853E 00 0.90E 00 0.54282E 00 O.IOE 01 0.39835E 00 O.llE 01 0.29057E 00 0. 12E 01 0.21 173E 00 0.13E 01 0.15468E 00 0. 14E 01 0. 1 1 377E 00 0. 15E 01 0.84086E-01 0. 16E 01 0.62798E-01 0.17E 01 0.47238E-01 0.18E 01 Q.35784E-01 O. 19E 01 0.27502E-01 0.20E 01 0.21 122E-01 0.25E 01 0.61281E-02 0.30E 01 0.23034E-02 0.35E 01 0.1 1 119E-02 0.40E 01 0.63823E-03 0.45E 01 0.37731E-03 0.50E 01 0.22775E-03 0.60E 01 0.84627E-04 0.70E 01 0.32703E-04 0.80E 01 0.13126E-04 0.90E 01 0.54647E-05 O.IOE 02 0.23568E-05 0.12E 02 0.48479E-06 0.14E 02 0.113Q6E-06 0. 16E 02 0.29582E-07 0.186 02 0.85947E-08 0.20E 02 0.27439E-08 0.22E 02 Q.95263E-09 0.24E 02 0.35593E-09 0.26E 02 0.14164E-09 0.28E 02 0.59404E-10 0.30E 02 0.25988E-10 0.35E 02 0.37381E-11 0.40E 02 Q.62702E-12 0.45E 02 0. 1 191 1E-12 0.50E 02 0.25071E-13 0.60E 02 0.14225E-14 0.70E 02 0.1Q465E-15 0.80E 02 0.94251E-17 O. 90E 02 0.99924E-18 O.IOE 03 0.12121E-I8

PAGE 182

176 ELECTRIC MICROFIELO DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB=RS A= 0.2830 R= 0.0 TEMP RATIQ= 1.00 CHARGE AT ORI GI N= 9,00 Zl~ 1.00 Z2= 9.00 E PCE) 0.10E 00 0.11729E-01 0.20E 00 0.45525E-01 0.30E 00 0.97467E-01 0.40E 00 0.16178E 00 0.50E 00 0.23179E 00 0.60E 00 0.30092E 00 0.70E 00 0.36351E 00 0.80E 00 0.41540E 00 0.9QE 00 0.45413E 00 O.IOE 01 0.47887E 00 0. I IE 01 0.49015E 00 0.12E 01 0.48943E 00 0.13E 01 0.47876E 00 0. 14E 01 0.46039E 00 0.15E 01 0.43653E 00 0. 16E 01 0.40914E 00 0.17E 01 0.37989E 00 0. 18E 01 0.35010E 00 0.19E 01 0.32075E 00 0.20E 01 0.292S5E 00 0.2SE 01 0.17886E 00 0.30E 01 0.1096SE 00 0.35E 01 0.69770E-01 0,40E 01 0.46423E-01 0.45E 01 0.32220E-01 0.50E 01 0.23202E-01 0.60E 01 0.13262E-01 0.70E 01 0.81944E-02 0.80E 01 0.53917E-02 0.90E 01 0.37480E-02 0.10E 02 0.27307E-02 0.12E 02 0.16169E-02 0. 14E 02 0.10505E-02 0. 16E 02 0.71261E-03 0.18E 02 0.50000E-03 0.20E 02 0.36201E-03 0.22E 02 0.26983E-03 0.24E 02 0.20656E-03 0.26E 02 0.16202E-03 0.28E 02 0.12991E-03 0.30E 02 0.10624E-03 0.35E 02 0.68949E-04 0.40E 02 0.48069E-04 0.45E 02 0.34714E-04 0.50E 02 0.25393E-04 0-60E 02 0.14034E-04 0.70E 02 0.80836E-05 0.80E 02 0.48420E-05 0.90E 02 0.30094E-05 O.iOE 03 0.19364E-05

PAGE 183

177 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.2630 R= 0.0 TEMP RATIQ= 1.00 CHARGE AT ORIGIN^ 17.00 Zl = 1.00 Z2 = 17.00 E P(E) 0.10E 00 0.14925E-01 0.20E 00 0.57726E-01 0.30E 00 0.12287E 00 0.4QE 00 0.20233E 00 0.50E 00 0.28700E 00 0.60E 00 0.36819E 00 0.70E 00 0.43876E 00 0.80E 00 0.49388E 00 0.90E 00 0.53115E 00 O.IOE 01 0.55038E 00 0. HE 01 0.55309E 00 0.I2E 01 0.S4186E 00 0. 13E 01 0.5 1981E 00 0.14E 01 0.49Q06E 00 0.15E 01 0.45549E 00 0. 16E 01 0.4 1850E 00 0. 17E 01 0.38099E 00 0.18E 01 0.34436E 00 0. 19E 01 0.30955E 00 0.20E 01 0.27714E 00 0.25E 01 0.15579E 00 0.30E 01 0.89203E-01 0.35E 01 0.53722E-01 0.40E 01 Q.34172E-01 0.4SE 01 0.22833E-01 O.SOE 01 0.15907E-01 0.60E 01 0.85994E-02 0.70E 01 0.50563E-02 0.80E 01 0.31854E-02 0.90E 01 0.21320E-02 O.IOE 02 0.15032E-02 0. 12E 02 0.84441E-03 0.14E 02 0.5276&E-03 0. 16E 02 0.34548E-03 0. 18E 02 0.23315E-G3 0.20E 02 0.16189E-03 0.22E 02 0. I 1S45E-03 0.24E 02 0.84402E-04 0.26E 02 0.63149E-04 0.28E 02 0.48266E-04 0.30E 02 0.37fjl8E-04 0.35E 02 0.21720E-04 0.40E 02 0.13628E-04 0.45E 02 0.90368E-05 0.50E 02 0.61619E-05 0.60E 02 0.29770E-05 0.70E 02 0.15079E-05 0.80E 02 0.79879E-06 0.90E 02 0.44141E-06 O.IOE 03 0.25383E-06

PAGE 184

178 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.1362 R= 0.0 TEMP RATIQ= I. 00 CHARGE AT ORIGIN= 17.00 Z 1= I. 00 Z 217.00 E P(EJ 0.69354E-02 0.27188E-01 0.59163E-01 0.10040E 00 0.14787E 00 0.19 826E 00 0.24831E 00 0.29512E 00 0.33632E 00 0.37022E 00 0.39582E 00 0.41281E 00 0.42143E 00 0.42236E 00 0.41660E 00 0.40530E 00 0.38966E 00 0.37084E 00 0.34990E 00 0.32776E 00 0.22042E 00 0.14173E 00 0.92188E-01 0.62Q17E-01 0.43359E-01 0.31434E-01 O.I8138E-01 0. 1 1373E-01 0.77231E-02 0.56134E-02 0.43160E-02 0.20816E-02 0.20932E-02 0. 1S587E-02 0. 1 1827E-02 0.91365E-03 0.71772E-03 0.57276E-03 0.46384E-03 0.38080E-03 0.31658E-03 0.20936E-03 0. 14639E-03 0. 10647E-03 0.79232E-04 0.45488E-04 0.27279E-04 0.1 7045E-04 0. 1 1071E-Q4 0.7455BE-05 0, 10E 00 = 20E 00 Oo 3 0E 00 Oc 4 0E 00 = 50E 00 Or 60E 00 Q 70E 00 0. 8 0E 00 0, 90E 00 0„ 10E 01 0. 1 IE 01 Q* 12E 01 Oo 13E 01 0* 14E 01 0, 15E 01 Oo 16E 01 0. 17E 01 Oo 18E 01 Oo 19E 01 Oo 20E 01 0. 25E 01 0. 30E 01 Oo 35E 01 Oo 40E 01 0. 45E 01 0, 50E 01 Oo 60E 01 0. 70E 01 Oo 80E 01 Oo 90E 01 0. 10E 02 Oo 12E 02 0. 14E 02 0, 16E 02 0. 18E 02 0. 20E 02 0. 22E 02 0. 24E 02 0. 26E 02 0. 2 8E 02 0. 30E 02 0. 35E 02 0. 40E 02 0. 45E 02 0. 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 0. 90E 02 0. 10E 03

PAGE 185

179 ELECTRIC MICROFIELO DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR35RS A= 0.1731 R= 0.0 TEMP RATIO= 1.00 CHARGE AT ORIGIN^ 9.00 Zl= I. 00 Z2= 9.00 E P(E) 0.75651E-02 0.29603E-01 0.64224E-01 0.10855E 00 0.15905E 00 0.21198E 00 0.26372E 00 0.3H15E 00 0.35184E 00 0.38420E 00 Q.40744E 00 0.42150E 00 0.42690E 00 0.42462E 00 0.4 1584E 00 0.40189E 00 0.38407E 00 0.363S6E 00 0.34144E 00 0.31856E 00 0.21207E 00 0.1366IE 00 0.89521E-01 0.60752E-01 0.42821E-01 0.31258E-Q1 0. 1 8068E-01 0.1 1586E-01 0.81328E-02 0.61632E-02 0.49736E-02 0.37012E-02 0.30 146E-02 0.25029E-02 0.20842E-02 0.1 7406E-02 0.1 4578E-02 0. 12244E-02 0. 10313E-02 0.87107E-03 0.73776E-03 0.49288E-03 0.33480E-03 0.231 14E-03 0. 16211E-03 0.83462E-04 0.45535E-04 0.26235E-04 O. 15908E-04 O. 1 01 16E-04 Or 10E 00 = 2 0E 00 0. 30E 00 = 40E 00 Q bOE 00 0. 60E 00 Oc 70E 00 3, 80E 00 0. 90E 00 Oc 10E Oi Oc HE 01 0. 12E 01 0. 13E 01 0* 14E 01 0. 15E 01 0, 16E 01 0* 17E 01 Oo 18E 01 o 19E 01 Oo 20E 01 0. 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0. 50E 01 0. 60E 01 0. 70E 01 0. 80E Oi 0. 90E 01 0. 10E 02 0. 12E 02 0. 14E 02 Q. 16E 02 0. 18E 02 20E 02 0. 22E 02 0. 24E 02 0. 26E 02 0. 28E 02 0. 30E 02 0. 35E 02 0. 40E 02 0. 45E 02 0. 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 0. 90E 02 0. 10E 03

PAGE 186

180 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0,2246 R= 0.0 CHARGE AT ORIGIN= 17,00 Zl = TEMP RATIO= 1,00 1.00 Z2 = 17.00 PIE) 0.1 OE 00 0.20E 00 0.30E 00 0.40E 00 O.SQE 00 0.60E 00 0.70E 00 0.80E 00 0.90E 00 0.10E 01 0. 1 IE 01 0.12E 01 0. 13E 01 0. 14E 01 0.15E 01 0. 16E 01 0. 17E 01 0.1 8E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0.10E 02 0.12E 02 0. 14E 02 0. 16E 02 0.1 8E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. 10E 03 0. 10832E-01 0.42164E-01 0.9Q6 96E-0 1 0.15150E 0.21874E 0.28648E 0.34940E 0.40331E 0.44S43E 0.47443E 0.49027E 0.49392E 0.48703E 0.47162E 0.44980E 0.42356E 0.39467E 0.36459E 0.33446E 0.30515E O. 1 8474E 0.1 1 103E 0.69143E00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 0.451 17E-01 0.30795E-01 0.21 865E-01 0.98212E-02 0.51315E-02 0.28833E-02 0.17337E-02 0.1 1 101E-02 0.53878E-03 0.31692E-03 0.2 1719E-03 0. 16670E-03 0. 13776E-03 0. 1 t 784E-03 0.10 137E-0 3 0.87375E-04 0.75455E-04 0.65286E-04 0.45838E-04 0.32557E-04 0.23386E-04 0. 16985E-04 0.92500E-05 0.52474E-05 O.3O941E-05 0. 18923E-05 0.1 1977E-05

PAGE 187

181 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBzRS A= 0-121* R= 0,0 TEMP RATIO= 1.00 CHARGE AT ORIGIN= 9.00 Z II. 00 Z2= 9.00 E P(E) 0. 10E 00 0. 2 0E 00 0. 30E 00 0. 4QE 00 0. 50E 00 0. 60E 00 0. 70E 00 0 80E 00 0. 90E 00 0. 10E 01 0, 1 IE 01 0. 12E 01 0. 13E OS 0. 14E 0! 0. 15E 01 Oo 16E 01 = 17E 01 Oo 18E 01 0. 19E Oi 0. 20E 01 Oo 25E 01 0. 30E 01 0. 35E 01 Q 40 E 01 0. 45E 01 0. 50E Oi = 60E 01 0. 70E 01 Oo 80E 01 0. 90E 01 Go 10E 02 Go 12E 02 14E 02 Oo 16E 02 Oo 18E 02 0. 20E 02 Oo 22E 02 0. 24E 02 0. 26E 02 G„ 28E 02 = 30E 02 Oo 35E 02 • 4Q£ 02 0, 45E 02 0, 50E 02 O* 60E 02 0. 70E 02 Oo 80E 02 0, 90E 02 Oo 10E 03 0. 61 114E-02 0, 23991E01 c< 52327E-01 0. 89088E-OX 0. 131 74E 00 = 17748E 00 0< 22353E 00 0, 26732E 00 0, .30672E 00 0 .34009E 00 0< .36642E 00 0, .38522E 00 0. 39653E 00 0.40077E 00 0.39868E 00 0. ,391 18E 00 0.37928E 00 0.36398E 00 0.34624E 00 0.32691E 00 0, 22780E 00 0.1S065E 00 0. 100 19E 00 0.68697E-01 0.48944E-01 0.36270E-01 0. 19896E-01 0.1 1997E-01 0.77649E-02 0.53547E-02 0.39054E-02 0.23861E-02 0. 16693E-02 0. 12662E-02 0. 10150E-02 0.848I8E-03 0.72B59E-03 0.63453E-03 0.55454E-03 0.48E>63E-03 0.42616E-03 0.31016E-03 0.228S6E-03 0. I 7047E-03 0.12865E-03 0.75814E-04 0.46660E-04 0.29917E-04 O. 19934E-04 0. 13769E-04

PAGE 188

182 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.2000 R= 0.0 TEMP RATIQ= 0.25 CHARGE AT ORIGIN^ 9.00 Zl= 1.00 Z2= 9.00 E P(E I 0. 10E 00 0. 20E 00 0. 30£ 00 0. 40E 00 0. 5 0E 00 0. 60E 00 0. 70E 00 0. 80E 00 0. 90E 00 0. 10E 01 0. i ie 01 0. I2E 01 0. 13E 01 0* 14E 01 0. 15E 01 Oo 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0. 20E 01 Oo 25E 01 0. 30E 01 Oo 35E 01 0. 40E 01 Oo 45E 01 0. 50E 01 0* 60E 01 Oo 70E 01 Oo 80E 01 Oo 90E 01 Oo 10E 02 0, 12E 02 Oo I4E 02 Oo 16E 02 0, 18E 02 Oo 20E 02 0 22E 02 Oo 24E 02 Oo 26E 02 Oo 28E 02 Oo 3 0E 02 Oo 35E 02 Oo 40E 02 0, 45E 02 Oc 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 Oo 90E 02 Oo 10E 03 0, 69169E02 Oc 2708SE-01 0. 5 8 83 IE-0 1 0, 99596E01 Oc 14624E 00 0, 19540E 00 G< 24381E 00 0. 2 8 863 E 00 0, 32761.E 00 0, 35922E 00 0 38266E 00 0< 39776E 00 0. 40492E 00 0< .40490E 00 0, 39874E 00 0< 38758E 00 0< 37256E 00 0, .35478E 00 Q c .33520E 00 Q< .31464E 00 0 -21571E 00 0< 14264E 00 0. • 95532E-01 0.66001E-01 0.47220E-01 0.34918E-01 0.208S7E-01 0.13455E-01 0.93047E-02 0.67955E-02 0.51675E-02 0.32821E-02 0.23002E-02 0. 17232E-02 0. 13429E-02 0. 10731E-02 0.86865E-03 0.70815E-03 0.58118E-03 0.48Q12E-03 0.39918E-03 0.25849E-03 0. 17367E-03 0.12077E-03 0.86731E-04 0.48812E-04 0.30391E-04 0.20550E-04 0. 14816E-04 0.11 180E-04

PAGE 189

183 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8ERS A= 0.2000 R= 0.0 TEMP RATI0= 0.50 CHARGE AT ORIGIN= 9.00 Zl = I. 00 Z2 = 9.00 E PIE) 0. 10E 00 0. 2 0E 00 0. 3 0E 00 0. 40E 00 0o 5 0E 00 0. 60E 00 Oo 7 0E 00 0. 80E 00 Oo 90E 00 Oo 10E 01 = 1 IE 01 0. 12E 01 0. 13E 01 Oo 14E 01 0. 15E 01 Oo 16E 01 Oc 17E 01 Oo I8E 01 0. 19E oa 0. 20E 01 Oc 25E 01 30E 01 Oo 35E 01 Oo 4 0E 01 Oo 45E 01 Oo SOE 01 0. 60E Oi Oc 70E 01 Go SOE 01 0. 90E 01 Oo 10E 02 Oo 12E 02 0. 14E 02 Oo 16E 02 0. 13E 02 0. 20E 02 Oo 22E 02 Oo 24E 02 Oo 26E 02 0. 28E 02 Oo 30 E 02 Oo 35E 02 0. 40E 02 Oo 45E 02 0., 50E 02 Oo 60E 02 Oo 70E 02 Oo 80E 02 Oo 90E 02 Oo 10E 03 Oc 73606E-02 0, 28802E-01 Oo 62484E-01 0 10561E 00 Oo 15474E 00 0. 2Q62SE 00 0, 25662E 00 Oc 30284E 00 0, 34256E 00 0, 37425E 00 0. 39714E 00 = 41 1 18E 00 0< 41689E 00 = 41516E 00 0, 40717E 00 Oc 39416E 00 0, 37737E 00 0, 35795E 00 0, 33690E 00 0< 31507E 00 0< 21252E 00 0. 138 78E 00 0, 92072E-01 0.63146E-01 0.44905E-01 0.33029E-01 0.19445E-01 0. 12496E-01 0.85725E-02 0.61816E-02 0.46384E-02 0.28790E-02 0. 19393E-02 0.13850E-02 0. 10247E-02 0.77259E-03 0.59203E-03 0.46077E-03 0.36397E-03 0.29161E-03 0.23681E-03 0. I 4863E-03 0. 10001E-03 0.71377E-04 0.53471E-04 0.33265E-04 0.22244E-04 0. 15501E-04 0. I 1 199E-04 0.83479E-05

PAGE 190

184 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A0.2000 R= 0.0 TEMP RATIO= I. 00 CHARGE AT ORIGIN9.00 Zl= 1.00 22= 9.00 E P(E) 0.82510E-02 0..32244E-01 0.69784E-01 0.11757E 00 0.17159E 00 0.22761E 00 0.28167E 00 0.33039E 00 0.37129E 00 0.40281E 00 0.42433E 00 0.43601E 00 0.43065E 00 0.43344E 00 0.42178E 00 0.40515E 00 0.38493E 00 0.36241E 00 0.33863E 00 0.31447E 00 0.20S68E 00 0.13U7E 00 0.85474E-0I 0.57804E-01 0.40638E-01 0.29597E-01 0. 17131E-01 0. 10842E-01 0.73268E-02 0.S2021E-02 0.38397E-02 0.22831E-02 0. 14898E-02 0.10314E-02 0.73817E-03 0.54258E-03 0.40877E-03 0.31501E-03 0.24782E-03 0. 19862E-03 0.16186E-03 O.i 0297E-03 0.69598E-04 0.49155E-04 0.36063E-04 0.21281E-04 0.13679E-04 0.92259E-05 0.64576E-05 0.46688E-05 0, 10E 00 0. 2 0E 00 0. 30E 00 0. 40E 00 0. 50E 00 0. 60E 00 0. 70E 00 0. 80E 00 0. 90E 00 0. 10E 01 0. 1 IE 01 0. 12E 01 0. 13E 01 0, 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0. 20E 01 0. 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0. 50E 01 0. 60E 01 0. 70E 01 0. 80E 01 Oo 90E 01 0. 10E 02 0. 12E 02 0. 14E 02 0* 16E 02 0. 18E 02 0. 20E 02 Oo 22E 02 Oo 24E 02 Oo 26E 02 Oo 28E 02 0. 30E 02 Oo 35E 02 Oo 40E 02 Oo 45E 02 0, 50E 02 = 60E 02 Oo 70E 02 Oo 80E 02 = 90E 02 Oo tOE 03

PAGE 191

185 IN ELECTRIC MICROFIELD O I S TR IBUT I ON FUNCTION A PLASMA CONTAINING MULTIPLY CHARGEO ION PERTUR8ERS A= 0.2000 R= 0.0 CHARGE AT ORIGIN= 9.00 Zl = TEMP 1.00 RATIO= Z2 = 2< 9< 00 00 PE) 0. IOE 00 0.2 0E 00 0.30E 00 0.40E 00 0.50E 00 0.6 0E 00 0.70E 00 0.80E 00 0.90E 00 0. IOE 01 0. I IE 01 0. 12E 01 0.1 3E 01 0. 14E 01 0. 15E 01 0.16E 01 0.17E 01 0. 18E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 Q.90E 01 0. IOE 02 0.1 2E 02 0.14E 02 0. 16E 02 0.1 8E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. IOE 03 0.10082E-01 0.39290E-01 0.84664E-01 0.14177E 00 0.20532E 00 0.26988E 00 0.33051E 00 0.38323E 00 0.42531E 00 0.45533E 00 0.47303E 00 0.47915E 00 0.47507E 00 0.46258E 00 0.44353E 00 0.41995E 00 0.39336E 00 0.36523E 00 0.33670E 00 0.30864E 00 0.19073E 00 0. I 1647E 00 0.73431E-01 0.48396E-01 0.33312E-01 0.23826E-01 0.13373E-01 0.82431E-02 0.54385E-02 0.37759E-02 0.27278E-02 0. 15555E-02 0.96490E-03 0.63921E-03 0.44385E-03 0.31878E-03 0.23590E-03 0.17936E-03 0.13972E-03 0.11 119E-03 0.90 137E-04 0.56766E-04 0.37731E-04 0.25798E-04 0. 18090E-04 0.95363E-05 0.54575E-05 0.33480E-05 0.21741E-05 0. 14756E-05

PAGE 192

186 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0-2000 R= 0.0 TEMP RATIO= 4.00 CHARGE AT ORIGIN^ 9.00 Zl= I. 00 Z2 = 9.00 E PtE) 0. 10E 00 0.13971E-01 0.20E 00 0.54165E-01 0.30E 00 0.11574E 00 0.40E 00 0.19157E 00 0.50E 00 0.27344E 00 0.60E 00 0.35330E 00 0.70E 00 0.42428E 00 0.80E 00 0.48142E 00 0.90E 00 0.52195E 00 0.10E 01 0.b4512E 00 0.11E 01 0.55191E 00 0.12E 01 0.54444E 00 0. 13E 01 0.52550E 00 0.14E 01 0.49808E 00 0.I5E 01 0.46500E 00 0. 16E 01 0.42875E 00 0.17E 01 0.39136E 00 0.18E 01 0.35437E 00 0.19E 01 0.31887E 00 0.20E 01 0.28S58E 00 0.25E 01 0.15956E 00 0.30E 01 0.9Q274E-01 0.35E 01 0.53715E-01 0.40E 01 0.33818E-01 0.45E 01 0.22409E-01 0.50E 01 0.15S07E-01 0.60E 01 0.82225E-02 0.70E 01 0.48242E-02 0.80E 01 0.30438E-02 0.90E 01 0.2Q269E-02 0.10E 02 0.14088E-02 0.12E 02 0.74952E-03 0. 14E 02 0.43435E-03 0. 16E 02 0.26946E-03 0.18E 02 0.17633E-03 0.20E 02 0.12004E-03 Q.22E 02 0.84494E-04 0.24E 02 0.61290E-04 0.26E 02 0.45669E-04 0.28E 02 0.3484SE-04 0.30E 02 0.27136E-04 0.35E 02 0.15551E-0* 0.40E 02 0.94481E-05 0.45E 02 0.59196E-05 0.50E 02 0.3ai36E-05 0.60E 02 0.17096E-05 0.70E 02 0.84023E-06 0.80E 02 0.44690E-06 0.90E 02 0.25392E-06 0.10E 03 0.1S214E-06

PAGE 193

187 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.2000 R= 0.10E 00 TEMP RATIO= I. 00 CHARGE AT ORIGIN^ 9.00 Zl= 1.00 Z2 = 9.00 P(E) 0.49883E-02 0.19604E-01 0.42837E-01 0.73121E-01 Q.10849E 00 0.14678E 00 0.I8580E 00 0.22351E 00 0.25820E 00 0.28853E 00 0.31359E 00 G.33290E 00 0.34636E 00 0.35420E 00 0.35687E 00 0.35499E 00 0.34924E 00 0.34036E 00 0.32903E 00 0.31590E 00 0.24019E 00 0.17200E 00 0.12137E 00 0.86255E-01 Q.62353E-01 0.46013E-01 0.26755E-01 0.16649E-01 0. 10981E-01 0.76200E-02 0.55222E-02 0.32282E-02 0.21 122E-02 0. 14922E-02 0. 1Q982E-02 0.82073E-03 0.62085E-03 0.47521E-03 0.36790E-03 0.28799E-03 0.22786E-03 O. 13 263E-03 0.81888E-04 0.53325E-04 0.36419E-04 0. 19163E-04 0. 11404E-04 0.73385E-05 0.4941 OE-05 0.34523E-05 0. 10E 00 0. 2 0E 00 0. 30E 00 0. 40E 00 0. 50E 00 0. 6 0E 00 0. 7 0E 00 0. 80E 00 0* 9 0E 00 0. 10E 01 0. HE Oil 0. 12E 01 0. I3E 01 0, 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0, 20E 01 0„ 25E 01 Oo 30E 01 Oo 3SE 03 0„ 40E 01 Oo 4SE 01 0. 50E 01 0. 60E 01 0. 70E 01 0. 80E 01 0. 90E 01 0. 10E 02 0. 12E 02 Oo 14E 02 0. 16E 02 0 18E 02 0 20E 02 0. 22E 02 0. 24E 02 Oo 26E 02 0.=28E 02 0, 30E 02 Oc 35E 02 Oo 40E 02 0, 45E 02 0 50E 02 Oc 60E 02 Oc 70E 02 0. 80E 02 0. 90E 02 0, 10E 03

PAGE 194

188 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.2000 R= 0.10E 13 TEMP RATIO= I. 00 CHARGE AT ORIGIN= 9.00 Zl = I. 00 Z2= 9.00 E P
PAGE 195

189 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3^RS A= 0.4000 R= 0.0 TEMP RATIO= 0.25 CHARGE AT ORIGIN= 9.00 Zl= 1.00 Z2 = 9.00 E PIE) 0.10E 00 0.12604E-01 0.20E 00 0.48683E-01 0.30E 00 0.10342E 00 Q.40E 00 0.16990E 00 O.SOE 00 0.24047E 00 0.60E 00 0.30798E 00 0.70E 00 0.36679E 00 0.80E 00 0.41323E 00 0.90E 00 0.44564E 00 0.10E 01 0.46406E 00 O.HE 01 0.46975E 00 0.12E 01 0.46471E 00 0.13E 01 0.45123E 00 0. 14E 01 0.43160E 00 0.15E 01 0.40786E 00 0.16E 01 0.38173E 00 0.17E 01 0.35458E 00 0.18E 01 0.32744E 00 0. 19E 01 0.30105E 00 0.20E 01 0.27S90E 00 0.25E 01 0.17495E 00 0.30E 01 0.11214E 00 0.3SE 01 0.74449E-01 0.40E 01 0.51421E-01 0.45E 01 0.36851E-01 O.SOE 01 0.27276E-01 0.60E 01 0.16221E-01 0.70E 01 0.1Q373E-G1 0.80E 01 0.70741E-02 0.90E 01 0.50692E-02 0.10E 02 0.37794E-02 O. 12E 02 0.23297E-02 0.14E 02 0. 16094E-02 0.16E 02 0.12Z09E-02 0.18E 02 0.96406E-03 0.20E 02 0.78928E-03 0.22E 02 0.64956E-03 0.24E 02 0.53544E-03 0.26E 02 0.44217E-03 0.28E 02 0.36589E-03 0.30E 02 0.30345E-03 0.35E 02 0.19222E-03 0.40E 02 0.12404E-03 0.45E 02 0.81816E-04 O.SOE 02 0.55339E-04 0.60E 02 0.276S3E-04 0.70E 02 0.1S615E-04 0.80E 02 0.97681E-05 0. 90E 02 0.661 89E-05 0.10E 03 0.47499E-05

PAGE 196

190 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.4000 R= 0,0 CHARGE AT ORlGIN= 9.00 TEMP RATIOa 0.50 Zl= 1.00 Z2 = 9.00 PIE) 0, 10E 00 0. 20E 00 = 30E 00 Oc 4 0E 00 0. 50E 00 Go 60E 00 0, 70E 00 0. 80E 00 Go 90E 00 0. 10E OS Oo HE 01 Oo 12E 01 Oo 13E 01 Go 14E 01 Oo 15E 01 Oo 16E OS 0. 17E 01 0. 18E 01 0. 19E 01 Oo 20E 01 0. 25E 01 Oo 30E 01 0. 35E 01 Go 40E 01 0. 45E 01 Oo 50E 01 Qo 60E 01 0. 70E 01 Oo 80E 01 Oo 90E 01 Oo 10E 02 Oo 12E 02 Oo 14E 02 0* 16E 02 Oo 18E 02 0, 20E 02 0. 22E 02 024E 02 0. 26E 02 0. 28E 02 Oo 30E 02 0. 35E 02 Oo 40E 02 0. 45E 02 0, 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 0, 90E 02 0. 10E 03 0.14940E-01 0.57535E-01 0.12163E 00 0. 19851E 00 0.27869E 00 0.35359E 00 0.41671E 00 0.46418E 00 0.49463E 00 0.S0872E 00 0.50849E 00 0.49668E 00 0.47620E 00 0.44982E 00 0.41988E 00 0.38831E 00 0.35653E 00 0.32557E 00 0.29612E 00 0.26B58E 00 0.16274E 00 0.10062E 00 0.64895E-01 0.43761E-01 Q.30726E-01 0.22339E-01 0. 12900E-01 0.80521E-02 0.53528E-02 0.37452E-02 0.27258E-02 0.15697E-02 0.97858E-03 0.64543E-03 0.441 91 E-03 0.31229E-03 0.22728E-03 0.16999E-03 0. 13036E-03 0. 10228E-03 0.8 1925E-04 0.50755E-04 0.34100E-04 0.24009E-04 0. 17287E-04 0.94703E-05 0.55444E-05 0.34394E-05 0.22416E-05 0.15218E-05

PAGE 197

191 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB-RS A= 0-4000 R= 0.0 TEMP RATIO= I. 00 CHARGE AT ORIGIN^ 9.00 Zl = 1.00 Z 2= 9.00 E P(E) 0.19902E-01 0.762Q7E-01 0.1596GE 00 0.25718E 00 0.35S45E 00 0.44285E 00 0.51 146E 00 0.55742E 00 0.5805QE 00 0.58305E 00 0.56889E 00 0.54238E 00 0.50766E 00 0.46831E 00 0.42714E 00 0.38623E 00 0.34699E 00 0.31029E 00 0.27660E 00 0.24610E 00 0.13716E 00 0.79413E-01 0.48590E-01 0.31365E-01 0.2121 1E-01 0. 14917E-01 0.80882E-02 0.4S052E-02 0.30726E-02 0.20688E-02 0. 14353E-02 0.72804E-03 0.39389E-03 0.22635E-03 0. 13756E-03 0.88044E-04 0.59092E-04 0.41414E-04 0.30 179E-04 0.22769E-04 O. 1771 IE-04 0, 10502E-04 0.G8653E-05 0.46536E-05 0.32051E-05 0. 15902E-05 0.83421E-06 0.46043E-06 0.26604E-06 0.16G13E-06 0, 10E 00 0. 20E 00 Oo 30E 00 0. 40E 00 0. 50E 00 Oo 6 0E 00 0. 70E 00 = 80E 00 Oo 9 0E 00 0. 10E 01 Oo HE 01 0. 12E 01 Oo 13E 01 Oo 14E 01 Oo 15E OS 0. 16E 01 0. 17E 01 Oo 18E 3 Oo 19E 01 0, 20E 01 Oo 25E 05 Do 30E 01 Oo 35E 01 Oo 40E 01 Oo 45E 01 0. 50E 01 0, 60E 01 Oo 70E 01 Oo 80E 01. Oo 90E 01 0. 10E 02 Go 12E 02 Oo 14E 02 Oo 16E 02 Co 18E 02 Oo 20E 02 Oo 22E 02 0. 24E 02 Oe 26E 02 Oo 2 8E 02 Oo 30E 02 Oo 35E 02 Oo 40E 02 Oo 45E 02 Oo 50E 02 Oo 60E 02 Oo 70E 02 0, 80E 02 Oo 90E 02 Oo 10E 03

PAGE 198

192 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0,4000 R= 0.0 CHARGE AT ORIGIN= 9.00 Zl = TEMP RATIO= 2,00 1,00 Z2= 9,00 P(E) 0. 10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.6 0E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. 1 IE 01 012E 01 0.1 3E 01 0. 14E 01 0.15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0.1 2E 02 0. 14E 02 0.16E 02 0. 18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. 10E 03 = 3 1 042E-01 Oc l 1753E 00 0, 24 169E 00 Oc 38000E 00 Oc 50954E 00 Oc 61303E 00 0. 68112E 00 0* 71211E 00 Go 70999E 00 0 68189E 00 Oc 63588E 00 Oc 57943E 00 Oc 51863E 00 0 4 5 793 E 00 Oc 40027E 00 = 3 4 735E 00 Oc 29997E 00 Oc 25829E 00 Oc 22209E 00 Oc 19093E 00 Oc 91839E-0 1 Oc 47435E-01 Oc 26485E-01 0 15835E-0 1 Oc 10016E-01 Oc 66323E-02 Oc 32Q3QE-02 Oc 17402E-02 Oc 10 238E-02 Oc 63476E03 Oc 40924E03 Oc 18562E-03 Oc 92768E-04 Oc 50000E-04 0. 28454E-04 Oc 16877E-04 Oc 10402E-04 Oc 66439E-05 Oc 43869E-05 0. 29867E-05 0. 20913E•05 0, 95390E-06 = 49026E-06 Oc 27278E-06 Oc 15806E-06 Oo 5666 7E-07 Oc 22000E-07 Oc 91 775E-08 Oc 40820E-08 Qc 19208E-08

PAGE 199

193 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBZRS A= 0.4000 R= 0.0 TEMP RATIO= 4.00 CHARGE AT ORIGIN9.00 Zl= 1.00 Z2= 9.00 E P(E) 0.10E 00 0.57717E-01 0.20E 00 0.21386E 00 0.30E 00 0.42460E 00 0.40E 00 0.63674E 00 0.50E 00 0.80586E 00 0.60E 00 0.90717E 00 0.70E 00 0.93670E 00 0.80E 00 0.90561E 00 0.90E 00 0.83223E 00 0. 10E Oi 0.73545E 00 0. 1 IE 01 0.63080E 00 0.12E 01 0.52907E 00 0.13E 01 0.43659E 00 0.14E 01 0.35623E 00 0. 15E 01 0.28856E 00 0. 16E 01 0.23281E 00 0.17E 01 0.18756E 00 0. 18E 01 0.151 18E 00 0. 19E 01 0. 12212E 00 0.20E 01 0.98964E-01 0.25E 01 0.37073E-01 0.30E 01 0.15799E-01 0.35E 01 0.75352E-02 0.40E 01 0.39393E-02 0.45E 01 0.22067E-02 0.50E 01 0.13061E-02 0.60E 01 0.50714E-03 0.70E 01 0.23346E-03 0.80E 01 0.11906E-03 0.90E 01 0.64587E-04 0.10E 02 0.36683E-04 0.12E 02 0.13053E-04 0.14E 02 0.50456E-05 0. 16E 02 0.20924E-05 0.18E 02 0.92673E-06 0.20E 02 0.43647E-06 0.22E 02 0.21765E-06 0.24E 02 0.11440E-06 0.26E 02 0.63103E-07 0.28E 02 0.36370E-07 0.30E 02 0.21807E-07 0.35E 02 0.70106E-08 0.40E 02 0.26206E-08 0.45E 02 0.10637E-08 0.50E 02 0.44795E-09 0.60E 02 0.87136E-10 0.70E 02 0.19006E-10 0.80E 02 0.45989E-11 0.90E 02 0.12215E-11 0.10E 03 0.35237E-12

PAGE 200

194 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.4000 CHARGE AT ORIGIN= R= 0.10E 9.00 Zl = 1.00 TEMP RATIO= I. 00 Z2= 9.00 P(E) 0.10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 O.90E 00 0.10E 01 0. 1 IE 01 0.12E 01 0. 13E 01 0.14E 01 0.1 5E 01 0. 16E 01 0. 17E 01 0. 18E 1 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0. 12E 02 0.14E 02 016E 02 0. 18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 O.IOE 03 0. 19877E-01 0.75868E-01 0.1581 IE 00 0.25319E 00 0.34751E 00 0.42996E 00 0.49347E 00 0.53515E 00 0.55555E 00 0.55744E 00 0.54463E 00 0.52113E 00 0.49058E 00 0.45600E 00 0.41972E 00 0.38342E 00 0.34827E 00 0.31501E 00 0.28405E 00 0.25559E 00 0.14939E 00 0.88827E-01 0.54741E-01 0.35097E-01 0.23368E-01 0. 16094E-01 0.83018E-02 0.47002E-02 0.28971E-02 0. 19222E-02 0. 13574E-02 0.78017E-03 0.50242E-03 0.33512E-03 0.22710E-03 0.15630E-03 G.10919E-03 0.77396E-04 0.55638E-04 0.40545E-04 0.29938E-04 0. 14815E-04 0.78819E-0b 0.44771E-05 0.26960E-05 0. 1 1324E-05 0.55216E-06 0.29575E-06 0. 16850E-06 0.1 0096E-06

PAGE 201

195 IN ELECTRIC MICROFIELD DISTR IBUTION FUNCT ION A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0-4000 CHARGE AT ORIGIN^ R= O-lOE 13 9.00 Z\ I. 00 TEMP RATIO= Z2= a. oo 9.00 P
PAGE 202

196 ELECTRIC MICROFIELU DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.2000 R= 0.0 TEMP RATIO= 0.25 CHARGE AT ORIGIN^ 12.00 Z 1 = 12.00 ZZ= 11.00 E PCE) 0.10E 00 0.16916E-02 0.20E 00 0.67025E-02 0.30E 00 0.14844E-01 0.40E 00 0.25816E-01 0.50E 00 0.39221E-01 0.60E 00 0.54588E-01 0.70E 00 0.7140GE-01 Q.80E 00 0.89118E-01 0.9QE 00 0.10721E 00 0.10E 01 0.12517E 00 0. 1 IE 01 0.14253E 00 0.I2E 01 0.15891E 00 0. 13E 01 0.1 7397E 00 0.14E 01 0.18747E 00 0.15E 01 0.19924E 00 0. 16E 01 0.20918E 00 0. 17E 01 0.21726E 00 0. 18E 01 0.22349E 00 0.19E 01 0.22794E 00 0.20E 01 0.23072E 00 0. 25E 01 0.22446E 00 0.30E 01 0.19812E 00 0.35E 01 0.16583E 00 0.40E 01 0.13508E 00 0.45E 01 0.10874E 00 0.50E 01 0.87299E-01 0.60E 01 0.56923E-01 0.70E 01 0.38251E-01 0.80E 01 0.26616E-01 0.90E 01 0.19149E-01 0.10E 02 0.14194E-01 0.12E 02 0.83716E-02 0.14E 02 0.53603E-02 0. 16E 02 0.36493E-02 0.18E 02 0.25885E-02 0.20E 02 0.18919E-02 0.22E 02 0.14139E-02 0.24E 02 0.10879E-02 0.26E 02 0.84935E-03 0.28E 02 0.67376E-03 0.30E 02 0.54257E-03 0.35E 02 0.33532E-03 0.40E 02 0.22345E-03 0.45E 02 0.15843E-03 0.50E 02 0.11795E-03 0.60E 02 0.71759E-04 0.70E 02 0.46224E-04 0.80E 02 0.30986E-04 0.90E 02 0.21528E-04 O. 10E 03 0.15439E-04

PAGE 203

197 ELECTRIC M1CRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0*2000 R= 0.0 TEMP RATIO= 0.50 CHARGE AT ORIGIN^ 12.00 Z 1= 12.00 Z2= 11.00 E P(E) 0.23951E-02 0.94670E-02 0.20884E-01 0.361 19E-01 Q.54492E-01 0.75213E-01 0.97433E-0 1 0.12032E 00 0.14305E 00 0.16491E 00 0.18530E 00 0.20373E 00 0.219S5E 00 0.23345E 00 0.24445E 00 0.25284E 00 0.25872E 00 0.26226E 00 0.26364E 00 0.26310E 00 0.23984E 00 0.20043E 00 0.16030E 00 0.12566E 00 0.97908E-0 1 0.76431E-01 0.47616E-01 0.30872E-01 0.20858E-01 0. 14630E-01 O. 10604E-01 0.60253E-02 0.37298E-02 0.24473E-02 0. 16820E-02 0. 12014E-02 0.88512E-03 0.66789E-03 0.51459E-03 0.40402E-03 0.32259E-03 0. 19524E-03 0.12579E-03 0.84928E-04 0.59690E-04 0.32423E-04 0. 19155E-04 0. 1 I875E-04 0.76647E-05 0.51272E-05 Oo 10E 00 = 2 0E 00 Oo 3 0E 00 Oo 40E 00 Oo 50E 00 Oo 6 0E 00 Oo 70E 00 Oc 80E 00 Go 90E 00 Oo 10E 01 = 1 IE OS Oo 12E 01 Oo 13E 01 Or, 14E 01 Oo I5E 0! Oo 2 6E 01 Oo 17E 01 = 18E 01 Oo 19E 01 Oo 20E oa Oo 25E 01 Oo 30E 01 Oo 35E 01 Oo 40E OE Oo 45E 01 Oo SOE 01 0. 60E 01 Oo 70E 01 Oo SOE 01 0. 90E 01 Oo 10E 02 Oo 12E 02 Oo 14E 02 Oo 16E 02 0. 18£ 02 Oo 20E 02 0. 22E 02 Oo 24E 02 0. 26E 02 Oo 28E 02 0. 30E 02 Oo 35E 02 Oo 40E 02 Oo 45E 02 0. SOE 02 Oo 60E 02 Oo 70E 02 Oo 80E 02 Oo 90E 02 Oo 10E 03

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198 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.2000 R0.0 CHARGE AT ORIGIN^ 12.00 TEMP Z\~ 12.00 RATIO= Z2= 1.00 1 1.00 P(EI 0. 10E 00 0.20E 00 0.30E 00 Q.40E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. HE 01 0.12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.4 0E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.8QE 01 0.9QE 01 0. 10E 02 0.12E 02 O. 14E 02 0. 16E 02 0. 18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0. 70E 02 0.80E 02 0.90E 02 0.10E 03 0.38992E-02 0. 15350E-01 0.33636E-01 0.57642E-0 1 0.85959E-01 0.H701E 00 0.14919E 00 0.18098E 00 0.21105E 00 0.23830E 00 0.26197E 00 0.281S5E 00 0.29684E 00 0.30783E 00 0.3 1 4 75E 0.3 1792E 0.31775E 00 0.31 473E 00 0.30930E 00 0.30192E 00 0.24893E 00 0.19129E 00 0.14267E 00 0.10547E 00 0.78170E-01 0.58439E-01 0.33887E-01 0.20720E-01 0. 1331 1E-01 0.89281E-02 0.62 129E-02 0.32814E-02 0. 18969E-02 O. 1 1696E-02 0.75767E-03 0.51224E-03 0.35912E-03 0.25944E-03 0. 19191E-03 0. 14481E-03 0. 1 1 130E-03 0.61897E-04 0.37462E-04 0.24145E-04 0. 16217E-04 0.77524E-05 0.39262E-05 0.20957E-05 0. I 1729E-05 0.68471E-06

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199 IN ELECTRIC MICROFIELO DISTRIBUTION FUNCTION A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A0,2000 R= 0.0 CHARGE AT ORIGIN^ 12.00 Zl= 12.00 TEMP RATIO= 2.00 Z2= 11.00 PIE) 0. 10E 00 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.6GE 00 0.70E 00 0.8QE 00 0.90E 00 0.10E 01 O. 1 IE 01 0. 12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0.16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.4 0E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0.12E 02 0. 14E 02 0.16E 02 0. 18E 02 0.2 0E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0. 70E 02 0.80E 02 0.90E 02 0.1 OE 03 0.74 139E-02 0.28983E-0I 0.62780E-01 0.10589E 0.15479E 0.20576E 0.25532E 0.30053E 0.33919E 0.36995E 0.39223E 0.40614E 0.41227E 0.41 157E 0-40516E 0.39419E 0.37977E 0.36289E 0.34441E 0.32503E 0.23018E 0. 15584E 0.10448E 0.70517E0.48299E00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 01 0.33685E-01 0.17394E-01 0.95934E-02 0.56194E-02 0.34717E-02 0.22465E-02 0. 10519E-02 0.55468E-03 0.32333E-03 Q.20481E-03 0.13858E-03 0.98458E-04 0.72203E-04 0.53740E-04 0.40232E-04 Q.30279E-04 0. 15220E-04 0.78924E-05 0.42162E-05 0.23170E-05 0.75733E-06 0.27245E-06 0.10665E-06 0.449Q3E-07 0.20103E-07

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A= 0. CHARGE 200 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 2000 R= 0.0 TEMP RATIO= 4.00 AT ORIGIN^ 12.00 Zl= 12.00 Z2= 11.00 E PIE) 0. 10E 00 0.16406E-01 0.20E 00 0.63290E-01 0.30E 00 0.13415E 00 0.40E 00 0.21 965E 0.50E 00 0.30947E 00 0.60E 00 0.39400E 00 0. 70E 00 0.46576E 00 0. 80E 00 0.52004E 00 0.90E 00 0.55498E 00 0, 10E 01 0.57104E 00 0. HE 01 0.57040E 00 0. 12E 01 0.55612E 00 0. I3E 01 0.53160E 00 0. 14E 01 0.50005E 00 0. I5E 01 0.46428E 00 0. 16E 01 0.42654E 00 0. 17E 01 0.38858E 00 0. ISE 01 0.35161E 00 0. 19E 01 0.3I646E 00 0. 20E 01 0.28364E 00 Q.2SE 01 0.15851E 00 0.30E 01 0.87402E-01 0.35E 01 0.49030E-01 0.40E 01 0.28312E-01 0.45E 01 0.16865E-01 0.50E 01 0.10371E-01 G.60E 01 0.42660E-02 0. 70E 01 0.19341E-02 0.80E 01 Q.96267E-03 0.9QE 01 0.52187E-03 0. 10E 02 0.30568E-03 0. 12E 02 0.12816E-03 0* 14E 02 0.66566E-04 0, 16E 02 0.40190E-04 0. 18E 02 0.26467E-04 0.20E 02 0.179Q3E-04 0.22E 02 0.12164E-04 0.24E 02 0.02992E-05 0. 26E 02 0.56855E-05 0.28E 02 0.39107E-05 0. 30E 02 0.27008E-05 0.35E 02 0.10892E-05 0.40E 02 0.45008E-06 0.45E 02 0.19043E-06 0.50E 02 0.82452E-07 0.60E 02 0.16526E-07 0. TOE 02 0.36065E-08 0.80E 02 0.85288E~09 0. 90E 02 0.21751E-09 0. 10E 03 0.59538E-10

PAGE 207

201 ELECTRIC MICROFIELO DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.4000 R= 0.0 TEMP RATIO= 0.25 CHARGE AT ORIGIN= 12.00 21= 12.00 Z2= 11.00 E PIE) 0.81058E-02 0.31499E-01 0.67568E-01 0.H246E 00 0.16171E 00 0.21090E 00 0.25623E 00 0.29489E 00 G.32524E 00 0.34671E 00 0.35966E 00 0.36501E 00 0.36405E 00 0.35814E 00 0.34858E 00 0.33648E 00 0.32274E 00 0.30807E 00 0.29 296E 0.27778E 00 0.20729E 00 0.15164E 00 O.lllOOE 00 0.82016E-01 0.61350E-01 0.46549E-01 0.28038E-01 0. 17738E-01 0. 1 1744E-01 0.80982E-02 0.57872E-02 0.32260E-02 0.19636E-02 0. 12822E-02 0.8871 IE-03 0.64215E-03 0.48028E-03 0.36653E-03 0.28243E-03 0.21924E-03 0. 17142E-03 0.95520E-04 0.55463E-04 0.33465E-04 0.20926E-04 0.90064E-0S 0.43257E-05 0.22680E-05 0.12699E-05 0.74424E-06 0. 10E 00 0. 20E 00 0. 3 0E 00 0. 40E 00 0. 50E 00 0. 6 0E 00 0. 70E 00 0. 80E 00 0. 90E 00 0. 10E 01 0. HE 01 0. 12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0. 20E 01 0. 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0. 50E 01 0. 60E 01 0. 70E 01 0. 80E 01 0. 9GE 01 0. 10E 02 0. 12E 02 0. 14E 02 0. 16E 02 0. 18E 02 0. 20E 02 0. 22E 02 0. 24E 02 0. 26E 02 0. 28E 02 0. 30E 02 0. 35E 02 0. 40E 02 0. 45E 02 0. 50E 02 0. 60E 02 0. 70E 02 0. 80E 02 0. 90E 02 0. 10E 03

PAGE 208

202 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION P£RTURB£R5 A= 0.4000 R= 0.0 CHARGE AT ORIGIN12.00 Z 1= 12.00 TEMP RATIQ= 0.50 Z2= 11.00 P(E> 0. 10E 00 0.20E 00 0.3 0E 00 0.40E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. HE 01 0. 12E 01 0. 13E 01 O. 14E 01 0.15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 1 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 1 0.80E 01 0.90E 01 1 E 2 0. 12E 02 0. 14E 02 0. 16E 02 0. 18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0. 70E 02 0.80E 02 0.9QE 02 0.1 OE 03 0. 14290E-01 0, S4852E-01 0, l 1534E 00 0, 18694E 00 Oc 26028E 00 0, 32730E 00 0, 38235E 00 0, 42258E 00 0, 44761E 00 Oc 45881E 00 0-= 45853E 00 0< 44942E 00 0, 43398E 00 0* 41434E 00 0. 39217E 00 0. 36873E 00 0. 34493E 00 0< 32139E 00 0, 29854E 00 = 27667E 00 = 18554E 00 0., 1 2358E 00 0< 83 192E-01 0, 57003E-01 Oc 39855E-01 Oc 28437E-0 1 0 1 5337E-01 0, 88056E-02 053475E-02 34055E-02 = 22S53E-02 Oc 10858E-02 0 57322E-03 0. 32433E-03 Oc 19510E-03 Q< 12398E-03 0. 82695E-04 0, 57527E-04 0, 4 1471E-04 = 30782E-04 = 23376E-04 Oc 1 2448E-04 0< 68373E-05 0< 38386E-05 O a 22009E-05 = 76805E-06 Oc 28866E-06 Qc l 1607E-06 0, 49599E-07 0, 22377E-07

PAGE 209

203 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.4000 R= 0.0 TEMP RATIO= I. 00 CHARGE AT ORIGIN12.00 Zl= 12.00 Z 2= 11.00 E PIE) 0.29234E-01 0.10988E 00 0.22343E 00 0.34653E 00 0.45808E 00 0.54409E 00 0.S9882E 00 0.62327E 00 0.62242E 00 0.60275E 00 0.57049E 00 0.53081E 00 0.48763E 00 0.44369E 00 0.40081E 00 0.36012E 00 0.32225E 00 0.2 8 752 E 0.25600E 00 0.22764E 00 0.12620E 00 0.71 157E-01 0.41322E-01 0.2477SE-01 0. 15339E-0 1 0.98159E-02 0.42972E-02 0.20876E-02 0.1 I 110E-02 0.64174E-03 0.39861E-03 0. 18427E-03 0.10189E-03 0.62722E-04 0.40641E-Q4 0.26794E-04 0.17770E-04 0. I 1849E-04 Q.79436E-05 0.53535E-05 0.36268E-05 0. 1401 1E-05 0.55828E-06 0.22922E-06 0.96858E-07 0. 18771E-07 0.40273E-08 0.94805E-09 0.24269E-09 0.66958E-10 0. 10E 00 0. 2 0E 00 0. 3 0E 00 0. 40E 00 0. 50E 00 0. 60E 00 0. 70E 00 0. 80E 00 0. 90E 00 0. 10E 01 0. t IE 01 0. 12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 0. 18E 01 0. 19E 01 0. 20E 01 0. 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0. 50E 01 0. 60E 01 0. 70E 01 0. 80E 01 0. 90E 01 0. 10E 02 0. 12E 02 0. 14E 02 0. 16E 02 0. 18E 02 0. 20E 02 0. 22E 02 0. 24E 02 0. 26E 02 0. 2 8E 02 0. 30E 02 0. 3SE 02 0. 40E 02 0. 45£ 02 0. 50E QZ 0. 60E 02 0. 70E 02 0. 80E 02 0. 90E 02 0. 10E 03

PAGE 210

204 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.4000 R= 0.0 TEMP RATI0= 2.00 CHARGE AT ORIGIN^ 12.00 Zl= 12.00 Z2= 11.00 E PtEi 0.10E 00 0.69327E-01 0.20E 00 0.25100E 00 0.30E 00 0.48131E 00 0.40E 00 0.69233E 00 0.50E 00 0.83912E 00 0.60E 00 0.907S4E 00 0.70E 00 0.90655E 00 O.OOE 00 0.85567E 00 0.90E 00 0.77537E 00 0.10E 01 0.68221E 00 0.1 IE 01 0.58760E 00 0.12E 01 0.49843E 00 0. 13E 01 0.41820E 00 0.14E 01 0.34823E 00 0. 15E 01 0.28847E 00 0.16E 01 0.23819E 00 0.17E 01 0.19632E 00 0. 18E 01 0.16168E 00 0.19E 01 0.13317E 00 0.20E 01 0.10978E 00 0.25E 01 0.43303E-01 0.30E 01 0.18016E-01 0.35E 01 0.78091E-02 0.40E 01 0.35226E-02 0.45E 01 0.165I8E-02 0.50E 01 0.8G419E-03 0.60E 01 0.21240E-03 0.70E 01 0.64313E-04 0.80E 01 0.22120E-04 0.90E 01 0.85640E-05 O. 10E 02 0.36980E-05 0.12E 02 0.92238E-06 0. 14E 02 0.31901E-06 0. 16E 02 0.14220E-06 0.18E 02 0.75921E-07 0.20E 02 0.45177E-07 0.22E 02 0.28438E-07 0.24E 02 0.18294E-07 0.26E 02 0.1I802E-07 0.28E 02 0.76281E-08 0.30E 02 0.49392E-08 0.35E 02 0.16735E-08 0.40E 02 0.57754E-09 0.45E 02 0.20081E-09 0.50E 02 0.70593E-10 0.60E 02 0.90113E-11 0.70E 02 0.12002E-11 0.80E 02 0.16665E-12 0.90E 02 0.24098E-13 0. 10E 03 0.36259E-14

PAGE 211

205 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0*4000 R= 0.0 TEMP RATIO= 4.00 CHARGE AT ORIGIN^ 12.00 21= 12.00 Z2= 11.00 E P(E) O.IOE 00 0.19266E 00 0.20E 00 0.64645E 00 0.30E 00 0.11024E 01 0.40E 00 0.13677E 01 0.50E 00 0.14011E 01 0.60E 00 0.12664E 01 0.70E 00 0.10489E 01 0.80E 00 0.81950E 00 0.90E 00 0.61477E 00 0. 10E 01 0.44860E 00 0. HE 01 0.32138E 00 0.12E 01 0.22758E 00 0.13E 01 0. 16007E 00 0. 14E 01 0.1 1224E 00 0.15E 01 0.78664E-01 0.16E 01 0.55207E-01 0.17E 01 0.3Q853E-01 0. 18E 01 0.27447E-01 0.19E 01 0.19474E-01 0.20E 01 0.13882E-01 0.25E 01 0.23946E-02 0.30E 01 0.42685E-03 0.35E 01 0.77142E-04 0.40E 01 0.14134E-04 0.45E 01 0.26254E-05 0.50E 01 0.49435E-06 0.60E 01 0.18255E-07 070E 01 0.71143E-09 0.80E 01 0.29250E-10 0.90E 01 0.12682E-H 0.10E 02 0.57962E-13 0.12E 02 0.14 159E-15 0.14E 02 0.42504E-18 0.16E 02 0.15629E-20 0.18E 02 0.70169E-23 0.20E 02 0.38344E-25 0.22E 02 0.25421E-27 0.24E 02 0.20383E-29 0.26E 02 0.19702E-31 0.28E 02 0.22885E-33 0.30E 02 0.31842E-35 0.35E 02 0.15687E-39 0.40E 02 0.22274E-43 0.45E 02 0.86726E-47 O.SOE 02 0.88086E-50 0.60E 02 0.13215E-54 0.70E 02 0.50470E-58 0.80E 02 0.32930E-60 0.90E 02 0.24634E-61 O.IOE 03 0.14177E-61

PAGE 212

206 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.2000 tt= 0-0 TEMP RATIO= 0.25 CHARGE AT ORIGIN= 17.00 Zl= I. 00 Z2 = 17.00 E PIE) 0.72 126E-02 0.28232E-01 Q.61279E-01 0.10364E 00 0.15200E 00 0.20281E 00 0.25265E 00 0.29856E 00 0.33821E 00 0.37007E 00 0.39333E 00 0.40789E 00 0.41422E 00 0.413I3E 00 0.40586E 00 0.39348E 00 0.37725E 00 0.3S832E 00 0.33 767E 0.31614E 00 0.2I413E 00 0.14010E 00 0.93015E-01 0.63797E-01 0.45362E-01 0.33359E-01 0.19766E-01 0.1267IE-01 0.86S60E-02 0.62389E-02 0.46972E-02 0.29157E-02 0. 19356E-02 0. 13498E-02 0.98256E-03 0.74192E-03 0.57749E-03 0.46045E-03 0.37372E-03 0.30691E-03 0.2S427E-03 0. 16473E-03 0. I I 196E-03 0. 7943 8E-04 0.5855QE-04 0.34995E-04 0.22981E-04 0.15976E-04 0. 11542E-04 0.86138E-05 0. 10E 00 = 2 0E 00 0. 3 0E 00 Oo 40E 00 0. 50E 00 0. 60E 00 0. 7 0E 00 0. 80E 00 0-c 90E 00 Oo 10E 01 Oo HE 01 Oo 12E 01 Oo i3E 01 0^ 14E 01 Oo 15E oa Oo I6E 01 Oo 17E oa Oo 18E 01 Oo 19E 01 Oo 20E 01 Oo 25E 01 Oo 30E 01 Oo 35E 01 0. 40E 01 Oo 45E 01 Oo 50E 01 Oo 60E 01 Oo 70E OJ Oo 80E oa Oo 90E 01 Oo 10E 02 0. 12E 02 Oo 14E 02 Oo 16E 02 o 18E 02 Oo 2 0E 02 Oo 22E 02 0. 24E 02 Co 26E 02 Oo 2 8E 02 Oo 30E 02 Oo 35E 02 0, 40E 02 Oo 45E 02 Oo 50E 02 Oo 60E 02 Oo 70E 02 Oo 80E 02 Oo 90E 02 Oo 10E 03

PAGE 213

207 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTJRBzRS A= 0.2000 R= 0.0 TEMP RATI0= 0.50 CHARGE AT ORIGIN= 17.00 Zl= 1.00 Z2= 17.00 E PIE) 0.79625E-02 0.31 132E-01 0.67453E-01 0. I 1380E 00 0.16637E 00 0.22114E 00 0.27429E 00 0.32256E 00 0.36347E 00 0.39545E 00 0.41780E 00 0.43 057E 00 0.43445E 00 0.43052E 00 0.42010E 00 0.40459E 00 0.38536E 00 0.36365E 00 0.34Q51E 00 0.31682E 00 0.20862E 00 0.13346E 00 0.87050E-01 0.58876E-01 Q.41384E-01 0.3O135E-01 0. 17540E-01 0.1 I 068E-01 0.74578E-02 0.53048E-02 0.39378E-02 0.23650E-Q2 0. 15255E-02 0. 10261E-02 0.71607E-03 0.51 703E-03 0.38525E-03 0.29544E-03 0.23256E-03 O. 18741E-03 0.1542QE-03 0.10191E-03 0.72354E-04 0.53095E-04 0.39725E-04 0.23457E-04 0. 14767E-04 0.98258E-05 0.68502E-05 0.49606E-05 0. I OE 00 0. 20E 00 0. 30E 00 0. 4 0E 00 0. 50E 00 0. 60E 00 Oo 70E 00 0. 80E 00 0. 90E 00 0. I0E 01 0. I IE 01 0. 12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0. 16E 01 Oo 17E 01 0. 18E 01 0. 19E 01 0. 20E 01 0. 25E 01 Oo 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0. 50E OS o 60E 01 0* 70E 01 Go 80E 01 o, 90E 01 0, 10E 02 Oo 12E 2 Oo 14E 02 Oo 16E 02 0. 18E 02 o 20E 02 Oo 22E 02 Oc 24E 02 Oo 26E 02 Oo 28E 02 Oo 30E 02 Oc 35E 02 0. 40E 02 0, 45E 02 Oc 50E 02 Oo 60E 02 Oc 70E 02 = 80E 02 Oo 90E 02 Oc 10E 03

PAGE 214

208 IN ELECTRIC MICROFIELD DISTRIBUTION FUNCTION A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0,2000 R= 0,0 CHARGE AT ORIGIN= 17,00 Zl = 1.00 TEMP RATIO= I. 00 Z2= 17,00 P(£J O.IOE 00 0.20E 00 0.30E 00 0.40E 00 O.SOE 00 0,60E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. 1 IE 01 0. 12E 01 0. 13E 01 0.14E 01 0.15E 01 0.16E 01 0,17E 01 0. 18E 01 0.19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.4QE 01 Q.45E 01 0.50E 01 0.60E 01 0. 70E 01 0.80E 01 0.90E 01 0. 10E 02 012E 02 0.14E 02 0,16E 02 0.18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0,35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0, 10E 03 0.95103E-02 0.37104E-01 0.80107E-01 0.13449E 00 0.19540E 00 0.25783E 00 0.31 71 OE 00 0.36939E 00 0.41 198E 00 0.44331E 00 0.46293E 00 0.47134E 00 0.46970E 00 0.45958E 00 0,44276E 00 0.42100E 00 0.39593E 00 0.36898E 00 0.34129E 00 0.31 378E 0.19584E 00 0.12005E 00 0.7575GE-01 0.499GSE-01 0.34330E-01 0.24541E-01 0. 13847E-01 0.85044E-02 0.55927E-02 0.38881E-02 0.28214E-02 0. 16142E-02 0,99775E-03 0.65892E-03 0.46090E-03 0.33851E-03 0.25877E-03 0,204l2E-03 0. 16469E-03 0. 13473E-03 0. I 1098E-03 0.69954E-04 Q.4553QE-04 0.3Q542E-04 0.21079E-04 0. 10854E-04 0.61269E-05 0.37370E-05 0.24275E-05 0. 16551E-05

PAGE 215

209 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.2000 R= 0.0 TEMP RATlO= 2.00 CHARGE AT ORIGIN= 17.00 Zl= 1.00 Z2= 17.00 E P(E> 0. 10E 00 0.20E 00 0.30E 00 0.4 0E 00 0.50E 00 0.60E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. 1 IE 01 0. 12E 01 0. 13E 01 0. 14E 01 0.1 5E 01 0.16E 01 0.17E 01 0.18E 01 0.19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0. 12E 02 0. 14E 02 0.16E 02 0. 18E 02 0.20E 02 0.22E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. 10E 03 Oo 12844E-01 0. 49898E-01 0. 10698E 00 0. 17787E 00 0. 25532E 00 0, 33205E 00 Go 40167E 00 = 45935E 00 0* 5 2 1 I E 00 0. 52880E 00 Oc 53988E 00 Oc 53695E 00 = 52238E 00 Oo 49883E 00 Oo 46895E 00 0, 43516E 00 Oo 39951E 00 Oo 36361E 00 Q, 32865E 00 Oo 29548E 00 0* 16699E 00 Oc 9474 1E-0 1 0. 56325E-01 Oc 35401E-0 1 0. 23423E-01 0* 16 194E-01 Oo 85 169E-02 Oo 50298E-02 Oo 32 150E-02 Oo 2 1483E-02 Oo 14675E-02 Oo 72575E-03 Oo 38614E-03 0* 21987E-03 Oc 1332SE -0 3 Oo 85558E-04 Oo 57861E-04 Oo 4 1006E-04 o. 30296E-04 Oo 2321 1E-04 Oo 18345E-04 Oo 1 1276E-04 0. 750G3E-05 Oo 51 121E-05 Oo 35383E-05 0. 1 7747E-05 Oo 94210E-06 Go 52652E-06 Oo 30817E-06 Go 18 789E-06

PAGE 216

210 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.2000 R= 0.0 TEMP RATIO= 4.00 CHARGE AT ORIGIN= 17.00 Zl = 1.00 Z2= 17.00 E P(E) 0.20351E-01 0.78416E-01 0.16586E 00 0.27070E 00 0.37959E 00 0.48011E 00 0.56257E 00 0.62100E 00 0.65323E 00 0.66039E 00 0.64593E 00 0.61459E 00 0.57148E 00 0.52136E 00 0.46828E 00 0.41534E 00 0.36476E 00 0.31791E 00 0.27555E 00 0.23793E 00 0.11279E 00 0.55990E-01 0.30066E-01 0. 17427E-01 0.10775E-01 0.70208E-02 0.33176E-02 0.17863E-02 0.10484E-02 0.64989E-03 0.4 1900E-03 0. 18960E-03 0.94366E-04 0.50675E-04 0.28808E-04 0. 17131E-04 0. 10618E-04 G.68397E-05 0.45643E-05 0.31460E-05 0.2232 8E-05 0.10556E-05 0.56051E-06 0.31872E-06 0.18675E-06 0.63498E-07 0.27253E-07 0. 1 1663E-07 0.53242E-08 0.25710E-0S 0. 10E 00 0. 20E 00 0. 30E 00 0. 4 0E 00 0. 5 0E 00 0. 60E 00 0. 70E 00 0. 80E 00 0. 90E 00 0. 10E 01 0. HE 01 0. 12E 01 0. 13E 01 0. 14E 01 0. 15E 01 0. 16E 01 0. 17E 01 Go 18E 01 0. 19E 01 0„ 20E 01 0. 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 o 4 5E 01 0. 50E 0! 0. 60E 01 0. 70E 01 0. 80E 01 0. 90E 01 0. 10E 02 0. 12E 02 0. 14E 02 0. 16E 02 0, 18E 02 0. 20E 02 Oo 22E 02 Oo 24E 02 26E 02 Go 28E 02 Oo 30E 02 0. 35E 02 Oo 40E 02 Oo 45E 02 Oo 50E 02 0. 60E 02 Oo 70E 02 Oo 80E 02 Oo 90E 02 Oo 10E 03

PAGE 217

211 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8=:RS A= 0.2000 R~ O.IOE 00 TEMP RATIO= I. 00 CHARGE AT QRIGIN= 17.00 2 1= 1. 00 12.= 17.00 E PIE) O.IOE 00 0.59614E-02 0.20E 00 0.23377E-01 0.30E 00 0.50896E-01 0.40E 00 0.86448E-01 0.50E 00 0.12748E 00 0.60E 00 0.17123E 00 0.70E 00 0.21499E 00 0.80E 00 0.25637E 00 0.90E 00 0.29341E 00 O.IOE 01 0.32473E 00 O.llE 01 0.3495QE 00 0. 12E 01 0.36741E 00 0. 13E 01 0.37862E 00 0. 14E 01 0.38360E 00 0.15E 01 0.38305E 00 0.16E 01 0.37779E 00 0.17E 01 0.36869E 00 0.18E 01 0.35658E 00 0. 19E 01 0.34225E 00 0.20E 01 0.32638E 00 0.25E 01 0.24093E 00 0.30E 01 0.16801E 00 0.35E 01 0. 1 1536E 00 0.40E 01 0.79594E-01 Q.45E 01 0.55719E-01 0.50E 01 0.39739E-01 O.&OE 01 0.21484E-01 0. 70E 01 0.12495E-01 0.80E 01 0.77148E-02 0.90E 01 0.50046E-02 O.IOE 02 0.33758E-02 0.12E 02 0.16593E-02 0.14E 02 0.37726E-03 0.16E 02 Q.49568E-03 0.18E 02 0.29791E-03 0.20E 02 0.18954E-03 0.22E 02 0.12705E-03 0.24E 02 0.89298E-04 0.26E 02 0.65495E-04 0.28E 02 0.4989QE-04 0.30E 02 0.39280E-04 0.35E 02 0.24198E-04 0.40E 02 0.16510E-04 0.45E 02 0.11676E-04 0.50E 02 0.04067E-05 0.60E 02 0.45832E-05 0.70E 02 0.26579E-05 0.80E 02 0.16291E-05 0.90E 02 0.10486E-05 O.IOE 03 0.70417E-06

PAGE 218

212 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8ERS A= 0.2000 CHARGE AT ORIGlN= R= 0.10E 13 17.00 £1= I. 00 TEMP RAT10= 1.00 Z2= 17.00 P(E) o.ioe oo 0.20E 00 0.30E 00 0.40E 00 0.50E 00 0.6 0E 00 0.70E 00 0.80E 00 0.90E 00 0. 10E 01 0. 1 IE 01 0. 12E 01 0.13E 01 Q.14E 01 0.15E 01 0. 16E 01 0. 17E 01 0.18E 01 0. 19E 01 0.20E 01 0.25E 01 0.30E 01 0.35E 01 0.40E 01 0.45E 01 0.50E 01 0.60E 01 0.70E 01 0.80E 01 0.90E 01 0. 10E 02 0.12E 02 0.14E 02 0.16E 02 0. I8E 02 0.20E 02 0.2 2E 02 0.24E 02 0.26E 02 0.28E 02 0.30E 02 0.35E 02 0.40E 02 0.45E 02 0.50E 02 0.60E 02 0.70E 02 0.80E 02 0.90E 02 0. 10E 03 0.49992E-02 0. 19634E-01 0.42857E-0t 0.73053E-01 0.10821E 00 0.14613E 00 0.18463E 00 0.22170E 00 0.25S69E 00 0.28535E 00 0.30986E 00 0.32882E 00 0.34221E 00 0.35028E 00 0.3S350E 00 0.35246E 00 0.34781E 00 0.34020E 00 0.33027E 00 0.31859E 00 0.24883E 00 0.18259E 00 0.13078E 00 0.93205E-01 0.66779E-01 0.48358E-01 0.26439E-01 0. 15274E-01 0.92784E-02 0.59034E-02 0.39190E-02 0. 19316E-02 0. 10772E-02 0.65903E-03 0.43066E-03 0.29562E-03 0.20995E-03 0. 1 5199E-03 0. 1 1 128E-03 0.82345E-04 0.61 559E-04 0.31022E-04 0. 16S16E-04 0.92304E-05 0.5381 1E-05 0.20233E-05 0.83515E-06 0.36710E-06 0.17063E-06 0.83380E-07

PAGE 219

213 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB^RS A= 0.4000 R= 0.0 TEMP RATIO= 0,25 CHARGE AT ORIGIN= 17.00 Z I1.00 Z2= 17.00 E PE> 0.10E 00 0.14433E-01 0.20E 00 0.55630E-01 0.30E 00 0.11776E 00 0.40E 00 0.19255E 00 0.50E 00 0.27092E 00 0.60E 00 0.34460E 00 0.70E 00 0.40727E 00 0.80E 00 0.45503E 00 0.90E 00 0.48638E 00 0. 10E 01 0.50180E 00 0.11E 01 0.50312E 00 0.I2E 01 0.49290E 00 0.13E 01 0.47393E 00 0. 14E 01 0.44888E 00 0.15E 01 0.42006E 00 Q.16E 01 0.38938E 00 0. 17E 01 0.35828E 00 0.18E 01 0.32781E 00 0. 19E 01 0.29868E 00 0.20E 01 0.27134E 00 0.25E 01 0.16540E 00 0.30E 01 0.10262E 00 0.35E 01 0.66329E-01 0.40E 01 0.44793E-01 0.45E 01 0.31485E-01 0.50E 01 0.22912E-01 0.60E 01 0.13232E-01 0.70E 01 0.82442E-02 0.80E 01 0.54982E-02 0.90E 01 0.38661E-02 O.IOE 02 0.28337E-02 0.12E 02 0.16795E-02 0. 14E 02 0.11015E-02 0.16E 02 0.78119E-03 0.18E 02 0.58585E-03 0.20E 02 0.45440E-03 0.22E 02 0.35717E-03 0.24E 02 0.28273E-03 0.26E 02 0.22534E-03 0.28E 02 0.18081E-03 0.30E 02 0.14603E-03 0.35E 02 0.88025E-04 0.40E 02 0.55109E-04 0.45E 02 Q.35755E-04 0.50E 02 0.23988E-04 0.60E 02 0.11831E-04 0.70E 02 0.64959E-05 0.80E 02 0.39000E-05 0.90E 02 0.25154E-05 O.IOE 03 0.17121E-05

PAGE 220

214 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0,4000 R= 0.0 CHARGE AT ORIGlN= 17.00 TEMP RATION 0.50 Zl= I. 00 Z2 = 17.00 PIE) 0-= 10E 00 0. 20E 00 Q. 30E 00 = 4 0E 00 0. 50E 00 0„ 60E 00 0. 70E 00 Q<= aoE 00 0-o 90E 00 0, 10E 01 0-=, 1 IE 01 0. 12E 01 0 13E 01 0. 14E 05 Oo 15E 01 0. 16E 01 0. 17E OS Oo 18E 01 Qo 19E 01 0, 20E 01 3, 25E 01 Qo 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 Oo 50E 01 Oo 60E 01 0. 70E oa Q„ 80E 01 0. 90E 01 0. 10E 02 0. 12E 02 0„ 14E 02 0. 16E 02 Oo 18E 02 0, 20E 02 Oo 22E 02 Oo 24E 02 Oo 26E 02 0. 28E 02 Oo 30E 02 Oo 35E 02 0., 40E 02 Go 45E 02 Oo 50E 02 Oo 60E 02 Oo 70E 02 Oo 80E 02 0. 90E 02 Oo 10E 03 0.1 8858E-01 0.72326E-01 0.15187E 00 0.24559E 00 0.34087E 00 0.42673E 00 0.49542E 00 0.54288E 00 0.56846E 00 0.57406E 00 0.56307E 00 0.53951E 00 0.50734E 00 0.47005E 00 0.43043E 00 0.39061E 00 0.35208E 00 0.31576E 00 0.2.8222E 00 0.25168E 00 0.14142E 00 0.82232E-01 0.50441E-01 0.32616E-01 0.22089E-01 0. 15555E-01 0.83782E-02 0.50774E-02 0.33806E-02 0.24219E-02 0.18286E-02 0. I 1468E-02 0.77776E-03 0.54961E-03 0.39371E-03 0.28477E-03 0.20793E-03 0.15323E-03 O. I 1395E-03 0.85495E-04 0.64705E-04 0.33447E-04 0. 181 77E-04 0. 10353E-04 0.61626E-05 0.24616E-05 0.1 1309E-05 0.58330E-06 0.32974E-06 0. 19944E-06

PAGE 221

215 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB5RS A= 0.4000 R= 0.0 TEMP RATIO= 1.00 CHARGE AT ORlGIN= 17.00 Zl= I. 00 Z 2= 17.00 E P(E) 0.10E 00 0.28832E-01 0.20E 00 0.10949E 00 0.30E 00 0.22624E 00 0.40E 00 0.35795E 00 0.50E 00 0.48359E 00 0.60E 00 0.58669E 00 0.70E 00 0.65765E 00 0.80E 00 0.69380E 00 Q.90E 00 0.69793E 00 0.10E 01 0.67609E 00 0. 1 IE 01 0.63561E 00 0. 12E 01 0.583S6E 00 0. 13E 01 0.52593E 00 0.14E 01 0.46726E 00 0.15E 01 0.41068E 00 0. 16E 01 0.35812E 00 0.17E 01 0.31058E 00 0. 18E 01 0.26840E 00 0. 19E 01 0.23150E 00 0.20E 01 0.19954E 00 0.25E 01 0.96755E-01 0.30E 01 0.50137E-01 0.35E 01 0.28042E-01 0.40E 01 0.16790E-01 0.45E 01 0.10640E-01 0.50E 01 0.70631E-02 0.60E 01 0.34681E-02 0.70E 01 0.18783E-02 0.80E 01 0.11058E-02 0.90E 01 0.69196E-03 O.IOE 02 0.44993E-03 0.12E 02 0.20215E-03 0.14E 02 0.97504E-04 0. 16c 02 0.50257E-04 0.18E 02 0.27554E-04 0.20E 02 0.15994E-04 0.22E 02 0.97847E-05 0.24E 02 0.62791E-05 0.26E 02 0.42074E-05 0.28E 02 0.29302E-05 0.30E 02 0.211UE-05 0.35E 02 0.10468E-05 0.40E 02 0.57993E-06 0.45E 02 0.33578E-06 0.50E 02 0.19872E-06 0.60E 02 0.74018E-07 0. 70E 02 0.29749E-07 0.80E 02 0.12809E-07 0.90E 02 0.S8654E-08 O.IOE 03 0.28358E-08

PAGE 222

216 ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.4000 R= 0*0 TEMP RATIO= 2.00 CHARGE AT ORIGIN= 17.00 Zl = I. 00 Z2= 17.00 E P(E) 0. 10E 00 0. 20E 00 0, 30E 00 0, 40E 00 Qo 50E 00 0. 6 0E 00 0, 70E 00 0. 80E 00 0. 90E 00 0 10E 0! I IE 01 Oo 12E 01 0. 13E 01 0. 14E 01 0, 15E 01 Go 16E 01 Oo 17E 1 0. 18E oa Oo 19E 01 Oo 20E 01 0. 25E oa 0. 30E 01 0, 35E 01 Oo 40E 01 0. 45E 01 0, 50E 01 Oo 60E 01 Oo 70E 01 0. 80E 01 0. 90E 01 Oo 10E 02 0, i2E 02 Oo 14E 02 Oo I6E 02 Oo 18E 02 o.= 20E 02 0. 22E 02 Oo 24E 02 Oo 26E 02 Oo 28E 02 Oo 30E 02 Oo 35E 02 Oo 40E 02 Oo 45E 02 Oo S0E 02 Oo 60E 02 Oo 70E 02 Oo 80E 02 Oo 90 E 02 Oo 10E 03 0 53124E-01 0. I9777E 00 Oo 39560E 00 Oo 59906E 00 Oo 76687E 00 Oo S7410E 00 Oo 91431E 00 Oo 89S44E 00 Oo 83319E 00 Oo 74495E 00 Oo 64582E 00 Oo 54687E 00 Oo 45508E 00 Oo 37402E 00 Oo 30 483E 00 Oo 24719E 00 Oo 19998E 00 0., 16174E 00 Oo 13100E 00 0. 10638E 00 Oo 40Q75E-01 Oo 171 19E-01 Oo 81968E-02 Oo 43029E-02 Oo 24286E-02 Oo 14484E-02 Oo 5 7625E03 Oo 27136E-0 3 Oo I4313E-03 0. 82936E-04 Oo 5 1839E-0 4 Oo 23630E-04 Oo 1 1766E-04 Oo 59930E-05 Oo 31 150E-05 Oo 16514E-05 Oo 89257E-06 Oo 49157E-06 Oo 27573E-06 Oo 15744E-06 Oo 91474E-07 Oo 2S310E-07 Oo 77220E-08 0. 25782E-08 0. 93491E-09 0 15276E-09 Oo 31 700E-1 Oo 786 16E-1 1 Q a 21956E-1 1 Oo 667Q0E-12

PAGE 223

217 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS A= 0.4000 R= 0.0 TEMP RATIO= 4.00 CHARGE AT ORIGIN= 17,00 2 1= 1.00 Z2= 17.00 E P(E) 0.11872E 00 0.42394E 00 0.79266E 00 0.10962E 01 0.12561E 01 0.12609E 01 0. 1 1469E 01 0.96810E 00 0.77215E 00 0.590376 00 0.43780E 00 0.31793E 00 0.22789E 00 0.16225E 00 0. 1 1533E 00 0.82156E-01 0.58834E-01 0.42446E-01 0.30898E-01 0.22717E-01 0.55399E-02 0. 14230E-02 0.39675E-03 0. 1 1972E-03 0.38986E-04 0. 13659E-04 0.20641E-05 0.40354E-06 0.99694E-07 0.304G1E-07 0.1 1 177E-07 0.24099E-08 0.32795E-09 0.37555E-09 0. 18744E-09 0.94630E-1 0.48203E-10 0.24772E-10 0. 12841E-10 0.67142E-1 1 0.3S404E-1 1 0.74133E-12 0. 16328E-12 0.3 7 76 IE13 0.91540E-14 0.61529E-15 0.48901E-16 0.45319E-17 0.48301E-18 0.58387E-19 0. 10E 00 0. 2 0E 00 0. 3 0E 00 0. 40E 00 0. 50E 00 0. 60E 00 0. 70E 00 Oo 80E 00 Oo 90E 00 o 10E 01 0. 1 IE 01 0. 12E 01 Oo 13E 01 0. I4E 0! 0. I5E OS Oo 16E Oi Oo 17E 01 Oo 18E 01 Oo 19E 01 Oo 20E Oi Oo 25E Oil 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E 01 0* 50E 01 0. 60E 01 Oo 70E 01 0, 80E 01 0. 90E 01 Oo 10E 02 Oo 12E 02 Oo 14E 02 Oo 16E 02 Oo 18E 02 Oo 20E 02 Oo 22E 02 0. 24E 02 Oo 26E 02 0, 28E 02 Oo 30E 02 Oo 35E 02 Oo 40E 02 Oo 4SE 02 Oo 50E 02 Oo 60E 02 0. 70E 02 Oo 80E 02 0* 90E 02 Oo 10E 03

PAGE 224

218 ELECTRIC MICROFIELO DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0.4000 R= 0.10E 00 TEMP RATIO= 1.00 CHARGE AT ORIGIN= 17.00 Zl = I. 00 Z2= 17.00 E PCE) 0.39815E-01 0.14844E 00 Q.29802E 00 0.45455E 00 0.58910E 00 0.68420E 00 0.73467E 00 0.74436E 00 0.72202E 00 0.67763E 00 0.62027E 00 0.55713E 00 0.49335E 00 0.43226E 00 0.37581E 00 0.32494E 00 0.27989E 00 0.24051E 00 0.20641E 00 0.17706E 00 0.83333E-01 0.41 135E-01 0.21560E-01 0. 12001E-01 0.70657E-02 0.43744E-02 0. 17910E-02 0.83950E-03 0.44128E-03 0.25679E-03 0. 16332E-03 0.81912E-04 0.50222E-04 0.33961E-Q4 0.23434E-04 O. 16321E-04 0. 1 1470E-04 0.81331E-05 0.58172E-05 0.41963E-05 0.30524E-05 0. 14269E-05 0.69995E-06 0.35925E-06 0.19238E-06 0.61962E-07 0.22861E-07 0.94457E-08 0.42729E-08 0.20688E-08 Oc 10E 00 0. 20E 00 Oo 30E 00 0. 40E 00 Qo 50E 00 = 60E 00 0. 70E 00 Oo 80E 00 0, 90£ 00 Oo 10E 01 0. HE 01 a. 12E 01 0. 13E 01 0. 14E 01 Oc 15E 01 Oo 16E 01 Oo 17E 1 0, 18E 01 Oo 19E 01 Oo 20E 01 Oo 25E 01 0. 30E 01 0. 35E 01 0. 40E 01 0. 45E oa Qo 50E 01 0, 60E 01 Oo 70E 01 0* 80E 01 Oo 90E 01 0. 10E 02 0. 12E 02 Oo 14E 02 0. 16E 02 0. 18E 02 Oo 20E 02 0. 22E 02 Oo 24E 02 0* 26E 02 Oo 28E 02 0. 30E 02 0. 35E 02 Oo 40E 02 Oo 45E 02 0. 50E 02 Oo 60E 02 Oo 70£ 02 0.-= 80E 02 Oo 90E 02 Oo 10E 03

PAGE 225

219 ELECTRIC MICROFIELD DISTRIBUTION FUNCTION IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS A= 0*4000 R = O.IOE 13 TEMP RATIO= 1.00 CHARGE AT ORIGIN= 17,00 Zl= I. 00 ZZ~ 17,00 E PIE) O.IOE 00 0.49364E-01 0.20E 00 0.18114E 00 0.30E 00 0.35497E 00 0.40E 00 0.52569E 00 0.50E 00 0.660Q4E 00 0.60E 00 0.74305E 00 0.70E 00 0.77522E 00 0.80E 00 0.76585E 00 0.9QE 00 0.72713E 00 0. 10E 01 0.67Q51E 00 0.11E 01 0,605Q3E 00 0, 12E 1 0.53735E 00 0.13E 01 0-47155E 00 0.14E 01 0.41018E 00 0. 15E 01 0.35452E 00 0. 16E 01 0.30S00E 00 0.17E 01 0.26156E 00 0. 18E 01 0.22384E 00 0. 19E 01 0. 19132E 00 0.20E 01 0.16345E 00 0,25E 01 0.74938E-01 0.30E 01 0.35532E-01 0.35E 01 0.17624E-01 0.40E 01 0.91662E-02 0,4SE 01 0.49835E-02 0.50E 01 0,28247E-02 0.60E 01 0.86779E-03 0.70E 01 0.29878E-03 0.80E 01 0.11074E-03 0.90E 01 0.44061E-04 O.IOE 02 0.18763E-04 0.12E 02 0.41243E-0S 0.14E 02 0.11493E-05 0. 16E 02 0.39676E-06 0.18E 02 0.16581E-06 0.20E 02 0.81968E-07 0.22E 02 0.46841E-07 0.24E 02 0.30236E-Q7 0.26E 02 0.21S43E-07 0.28E 02 0.16556E-07 0.30E 02 0,l341l£-07 0.3SE 02 0.85816E-08 0.40E 02 O.S1922E-08 0.45E 02 0.29108E-08 0.50E 02 0.15246E-08 0.60E 02 0.35260E-09 0. 70E 02 0.68629E-10 0.80E 02 0.12013E-10 0.90E 02 0.20205E-11 O.IOE 03 0.34802E-12

PAGE 226

APPENDIX J TABLES OF STARK BROADENED LYMAN SERIES LINE PROFILES In this appendix we present tables of the final Stark broadened line profiles, computed using the numerical procedures discussed in Appendix H. We tabulate the blue wing of each profile since the asymmetry present is negligible. The electron temperature is expressed -3 in units of electron volts and the electron density is given in cm The third line of parameters in each table heading identifies the electric microfield distribution function employed to produce the profile. (We designate the net radiator charge by XI.) The frequency separations (DELTA OMEGA) are expressed in Rydberg units. The middle column gives the unnormalized Stark profile in arbitrary units (relative intensity) The right-hand column gives the Doppler-corrected Stark profile, normalized to unit intensity. 220

PAGE 227

221 LYMAN ALPHA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 A=0.20 R= 0-0 TRATIO=0.25 XI= 9.0 Zl= 1.0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0. 169E 02 0.622E 01 0.735E-02 0. 12 BE 02 0.612E 01 0. 14 7E•0 1 0. 740E 01 0.583E 01 0.221E-0 I 0.439E 01 0.539E 01 0.2 9 4E-01 0. 283E 01 0.484E 01 0.368E-01 0. 198E 01 0.422E 1 0.4* 1E-01 0. 14 7E 01 0.359E 01 0.51 5E-0 1 0. 1 16E 01 0.299E 01 0.588E-01 0.950E 00 0.245E 01 0.66 2E•0 I 0.809E 00 0. 1 98E 01 0.735E-0 1 0.712E 00 0.160E 01 0.882E-01 0.595E 0.10 7E 01 1 3E 00 0.535E 00 0.773E 00 1 1 8E 00 0.506E 00 0.626E 00 0. 13 2E 00 0.492E 00 0.554E 00 1 4 7E 00 0.485E 00 0.520E 00 0.22 IE 00 0.448E 00 0.456E 00 0.294E 00 0.359E 00 0.369E 00 0.366E 00 0.259E 00 0.270E 00 0.441E 00 0. 181E 00 0. 189E 00 0.588E 00 0-899E-01 0.941E-01 0.735E 00 0.498E-01 0.519E-01 LYMAN ALPHA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE := 1019.20 ELECTRON DENSITY= 0.20E 24 A=0.2 R= 0.0 TRATIQ=1.00 XI = 9.0 21= 1.0 Z2= 9.0 del™ k STARK STARK* DOPPLER OMEGA PROFILE 0„0 0. 168E 02 0.971E 01 0.7 3 5E-02 0. 127E 02 0.924E 01 0. 147EI 0.7 39E 01 0.800E 01 0.22 IE•0 1 0.439E 01 0.636E 01 0.294E•01 0.284E 01 0.4 73E 01 0.368E01 0. 199E 01 0.337E 01 0.4 4 IE -0 I 0* 149E 01 0.238E 01 0*51 5E-0 1 0. 1 18E I 0. 172E 01 0.58 8EI 0.971E 00 0.130E 01 0.662E•01 0.833E 00 0.104E 01 0. 7 3 5E•0 1 0.738E 00 0.872E 00 0.882E01 0.624E 00 0.691E 00 0.103E 00 0.568E 00 0.608E 00 0. 1 1 8E 00 0.541E 00 0.567E 00 0. 132E 00 0.528E 00 0.54 7E 00 0. 14 7E 00 0.52 IE 00 0.536E 00 00 0,22 IE 00 0.469E 00 0.480E 0.294E 00 0.361E 00 0.371E 00 0.368E 00 0.251E CO 0.259E 00 0.441E 00 0. 170E 00 0.1 75E 00 0.58 8E 00 0.81 1E-01 0.836E•01 0.735E 00 0,438E-01 0.451E-01

PAGE 228

222 LYMAN ALPHA PROFILE FOR HYDRQGENIC NEON ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY= 0.20E 24 A=0„20 R* 0.0 TRATI0=4.00 XI= 9.0 21= 1.0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE OOPPLER 0.0 0. 168E 02 0.132E 02 0.735E-02 0. 127E 02 0.117E 02 0.147E-0 I 0.741E 01 0.849E 01 0.221E-01 0.443E 01 0.54 8E 1 0.294E-0 1 0.289E 01 0.350E 01 0.368E-0 I 0.205E 01 0.237E 01 0.441E-0 I 0. 15bE 01 0.173E 01 0.51 5E-0 1 0. 125E 01 0.136E 01 0.588E-0 I 0. 105E 01 0.1 12E 01 0.662E-0 1 0.924E 00 0.971E 00 0.73 5E-0 1 0.836E 00 0.870E 00 0.882E-0 I 0.734E 00 0.755E 00 1 3E 0.686E 00 0.701E 00 0.118E 0.662E 00 0.674E 00 0.132E 00 0.64 7E 00 0.657E 00 1 4 7E 0.634E 00 0.643E 00 0,22 IE 00 0.51 IE 00 0.519E 00 0.294E 00 0.34 IE 00 0.34 7E 0.368E 00 0.209E 00 0.213E 00 0.441E 00 0. 128E 00 0. 131E 00 0.58 8E 0.545E-01 0.554E-01 0.73 5E 0.275E-01 0.280E-01 LYMAN ALPHA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE-1019.20 ELECTRON OENSITY= 0.20E 24 A=0.20 Rss 0.10E 00 TRATID=1.00 XI= 9,0 Zl = 1.0 Z2 = 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0. 169E 02 0.980E 01 0.735E-02 0. 127E 02 0.9 33E 1 0, 14 7E-0 1 0.737E 01 0.807E 01 0.22 IE-01 0.436E 01 0.640E 01 0.29 4E-0 1 0. 280E 01 0.4 75E 01 0.368E-01 0. 195E 01 0.337E 01 0.44 IE-01 0. 144E 01 0.236E 01 0.51 5E-0 1 0. 1 12E 01 0, 169E 1 0.58 8E-0 1 0.912E 00 0.126E 01 0.662E-01 0. 769E 00 0.984E 00 0.735E-01 0.669E 00 0.810E 00 0.882E-0 1 0.546E 00 0.618E 00 3E 0.481E 00 0.525E 00 U 8E 0.448E 00 0.479E 00 0.132E 00 0.432E 00 0.455E 00 1 4 7E 0.424E 00 0.444E 00 0.221E 00 0.4 10E 00 0.424E 00 0.294E 00 0.356E 00 0.369E 00 0.368E 00 0.2 79E 00 0.290E 00 0.441E 00 0.208E 00 0.216E 58 8E 0. 1 I IE 00 0.1 16E 0.735E 00 0.625E-01 0.650E-01

PAGE 229

223 LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY^ Q.20E 24 A=0.20 R= 0.10E 13 TRATIO=1.00 Xl = 9.0 Zl = 10 Z2 = 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0. 169E 02 0.992E 01 0.735E-02 0. 127E 02 0.944E 01 0.14 7E•01 0.73 7E 01 0.816E 01 0.22 1E0 1 0.4 35E 01 0.64 7E 01 0.294E01 0.279E 01 0.479E 01 0.363E01 0. 193E 01 0.339E 01 0.44 IE•0 1 0. 142E 01 0.236E 01 0.51 5E01 0. 109E 01 0.1 68E 01 0.S8 8E01 0.881E 00 0. 124E 01 0.662E01 0. 734E 00 0.959E 00 0.735E01 0.630E 00 0.780E 00 0.88 2E•0 1 0.501E 00 0.579E 00 1 3E 00 0.430E 00 0.479E 00 1 1 8E 00 0.392E 00 0.427E 00 1 3 2E 00 0.372E 00 0.399E 00 1 4 7E 00 0.362E 00 0.385E 00 0.22 IE 00 0.357E 00 0.374E 00 0.294E 00 0.332E 00 0.349E 00 0.368E 00 0.282E 00 0.296E 00 0,44 IE 00 0.226E 00 0.237E 00 0.58 8E 00 0. 135E 00 0.142E 00 7 3 5£ 00 0.805E-01 0.847E-01 LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE^ 254.80 ELECTRON DENSITY= 0.20E 24 A=0.40 R* 0,0 TRATIO-0.25 X 1= 9.0 2 1= 1.0 Z2 = 9.0 DELTA STARK STARK.+ OMEGA PROFILE DOPPLER 0,0 0. 121E 02 0.838E 01 0.735E-02 0. 1 04E 02 0.806E 01 0, 14 7E-0 I 0.742E 01 0.719E 01 0.221E-01 0.505E 01 0.600E 1 0.294E-01 0.353E 01 0.4 75E 1 0.36 8E-01 0.259E 01 0.364E 01 0.441E-01 0. 199E 01 0.277E 01 0.51 SE-01 0. 159E 01 0.213E 01 0.588E01 0. 133E 01 0.168E 01 0.66 2E01 0. 1 14E 01 0.138E 01 0.735E-01 0. 100E 01 0. 1 17E 1 0.882E-0 1 0.830E 00 0.916E 00 0.10 3E 00 0.73 IE 00 0.783E 00 0. 1 I 8E 00 0.671E 00 0.705E 1 3 2E 00 0.630E 00 0.655E 00 0. 14 7E 00 0.598E 00 0.618E 00 0.221E 00 0.464E 00 0.477E 00 0.294E 00 0.331E 00 0.34 IE 0.36 6E 00 0.226E 00 0.233E 00 0.44 IE 00 0. 154E 00 0.159E 00 0.58 8E 00 0.770E-01 0.793E-01 0.735E 00 0.433E-01 0.446E-0 1

PAGE 230

224 LYMAN ALPHA PROFILE FOR HYDROGEN tC NEON ELECTRON TEMPERATURE^ 254.90 ELECTRON DENSITY: 0.20E 24 L=0.0 Ra* O.C TRATIO=l .0( ) Xl= 9 .0 Zl = 1.0 DELTA STARK STARK* OMEGA PROFILE DOPPLER 0.0 0. 121E 02 0.105E 02 0.73 5E02 0. 10SE 02 0.971E 01 0.14 7E1 0.745E 01 0.777E 1 0.22 1E•0 I 0.510E 1 0.569E 01 0.294E01 0.359E 01 0.407E 01 0.368E01 0.266E 1 0.297E 01 0.44 1Ea i 0.206E 01 0.226E 1 0.515E•01 0. 168E 01 0.180E 01 0.58 8E•01 0. 14 IE 01 0.150E 01 Q.662E•01 0. 123E 1 0.129E 01 0.7 3 5E•01 0.1 10E 01 0.1 I 4E 01 Q.882E•01 0.930E 00 0.959E 00 0. 103E 00 0.833E 00 0.854E 00 0.1 11 8E 00 0.770E 00 0.786E 00 0. 132E 00 0.723E 00 0.737E 00 0.147E 00 0.682E 00 0.694E 00 0.22 IE 00 0.486E 00 0.495E 0.294E 00 0.312E 00 0.318E 00 0.368E 00 0. 194E 00 0.1 97E 0.44 IE 00 0. 123E 00 0.12 5E 0.588E 00 0.558E-01 0.569E-0 1 0.735E 00 0.297E-01 0.302E-01 LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 R= O.C TRATIO=4.00 XI= 9 .0 Zl= 1.0 DELTA STARK STARK* OMEGA PROFILE DOPPLER 0.0 0. 123E 02 0.119E 02 0.73 5E02 0. 107E 02 0.105E 02 0.14 7E-0 1 0.769E 01 0.786E 01 0.22 IE01 0.534E 01 0.554E 01 0.294E•01 0.385E 01 0.398E 01 0.368E1 0.293E 01 0,30 IE 01 0.44 1E-01 0.234E 01 0.240E 01 0.51 SE•01 0. 196E 01 0.200E 01 0.588E•0 1 0.170E 01 0.1 73E 1 0.662E•0 I 0. 152E 01 0.155E 01 0.735E0 I 0. 139E 01 0.141E 01 0.882E•0 1 0. 121E 01 0.123E 01 0.103E 00 0. 109E 01 0.110E 01 0.11 8E 00 0.985E 00 0.994E 00 0. 13 2E 00 0. 889E 00 0.897E 00 1 4 7E 00 0.797E 00 0.804E 00 0.22 IE 00 0.4 06E 00 0.410E 00 0.29 4E 00 0. 193E 00 0.195E 00 0.368E 00 0.986E-01 0.996E-01 0.44 IE 00 0.563E-01 0.568E-01 0.588E 00 0.243E-01 0.246E-01 0.73 5E 00 0. 135E-01 0. 136E-01

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225 LYMAN ALPHA PROFILE FOR HYOROGENIC NEON ELECTRON TEMPERATURE= 254.80 ELECTRON OENSITY= 0.20E 24 A=0.4Q Rs= 0.10E 00 TRATIO-1.00 Xl = 9.0 Zl1.0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0. 121E 02 0.1 05E 2 0.735E-02 0. 104E 02 0.971E 01 1 4 7E-01 0.745E 01 0.777E 01 0.22 1E01 0.509E 01 0.569E 01 0.294E-01 0.359E 01 0.406E 01 0.368E-01 0.265E 01 0.296E 01 0.44 1E-01 0.206E 1 0.225E 01 0.51 5E-01 0. 167E 01 0.179E 1 0.588E-01 0. 140E 01 0.149E 01 0.662E•01 0. 122E 01 0.128E 01 0.73 5E-0 I 0. 108E 01 0.113E 1 0.882E-01 0.916E 00 0.94 5E 0.103E 00 0. 81 7E 00 0.838E 00 0.118E 00 0.753E 00 0.769E 00 0.13 2E 00 0.705E 00 0.71 9E 0.14 7E 00 0.664E 00 0.677E 00 0.221E 00 0.479E 00 0.488E 00 0.294E 00 0.31 7E 00 0.323E 00 0.368E 00 0.203E 00 0.207E 00 0.441E 00 0. 131E 00 0.1 34E 0.58 8E 00 0.599E-01 0.610E-01 0.73 5E 00 0.312E-01 0.318E-01 LYMAN ALPHA PROFILE FOR HYOROGENIC NEON ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 A=0.40 R0.10E 13 TRATIO=1.00 Xl= 9.0 Zl= 1.0 ZZ~ 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 1 2 1 E 02 0.105E 02 0.73 5E-0 2 0. 104E 02 0.970E 01 0.147E-01 0.744E 01 0.776E 01 0.221E-01 0.509E 01 0.569E 01 0.294E-0 I 0.358E 01 0.406E 01 0.368E-QI 0.265E 01 0.296E 1 0.44 1E-0 1 0.205E 01 0.225E 01 0.51 5E-01 0. 166E 01 0.179E 1 Q.58 8E-01 0.140E 01 0.1 49E 1 0.662E-0 1 0.121E 1 0.128E 01 0.735E-01 0. 10 8E 01 0.1 13E 1 0.882E-01 0.910E 00 0.939E 00 0.10 3E 0.809E 00 0.831E 00 0.118E 0.743E 00 0.760E 00 0.13 2E 0.694E 00 0.708E 00 1 4 7E 0.652E 00 0.665E 00 0.221E 00 0.470E 00 0.4 79E 0.294E 00 0.31 7E 00 0.323E 00 0.368E 00 0.209E 00 0.213E 00 0.441E 00 0. 138E 00 0. 141E 00 0.588E 00 0.642E-01 0.655E-01 0.735E 00 0.332E-01 0.338E-01

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226 LYMAN BETA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATUR£= 809.10 ELECTRON OENSITY= 0.10E 24 A=0-20 R= 0.0 TRATIQ=0.25 XI = 9,0 Zl= 1-0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE DQPPLER 0.0 0.243E 00 0.358E 00 0.735E-02 0.246E 00 0.360E 00 0.147E-01 0.254E 00 0.36 7E 00 0.221E-01 0.266E 00 0.377E 00 0.294E-01 0.282E 00 0.392E 00 0.368E-01 0.302E 00 0.410E 00 0.44 IE-01 0.325E 00 0.432E 00 0.51 5E1 0.350E 00 0.456E 00 0.588E-01 0.378E 00 0.484E 00 0.662E-01 0.407E 00 0.513E 00 0.7J5E-01 0.4 3 7E 00 0.544E 00 0.882E-0 1 0.501E 00 0.6UE 00 0.103E 00 0.566E 00 0.679E 00 0.1 18E 0.630E 00 0.74 7E 00 0.I32E 00 0.689E 00 0.812E 00 1 4 7E 0.743E 00 0.871E 00 0.221E 00 0.886E 00 0.104E 01 0.294E 00 0.845E 00 0.101E 01 0.368E 00 0.721E 00 0.869E 00 0.441E 00 0.589E 00 0.7 13E 00 0.5S8E 0.383E 00 0.465E 00 0.73 5E 0.252E 00 0.306E 00 LYMAN BETA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE= 809.10 ELECTRON DENSITY^ 0.10E 24 A=0.20 R= 0.0 TRATI0=1.00 XI= 9.0 Zl = 1.0 Z2 = 9.0 del™ STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.26 7E 00 0.335E 00 0.73 5E02 0.269E 00 0.338E 00 Q.147E01 0.278E 00 0.34 7E 0.221E•01 0.292E 00 0.361E 00 0.29 4E01 0.310E 00 0.381E 00 0.368E0 1 0.332E 00 0.405E 00 0.44 £•01 0.357E 00 0.433E 00 0.51 SE~ 01 0.386E 00 0.465E 00 0.598E01 0.4 I 6E 00 0.499E 00 0.662E0 1 0.448E 00 0.535E 00 0.735E01 0.482E 00 0.574E 00 0.882E01 0.552E 00 0.653E 00 1 3£ 00 0.622E 00 0.733E 00 0.1 18E 00 0.689E 00 0.810E 00 0.13 2E 00 0. 751E 00 0.881E 00 1 4 7E 00 0.805E 00 0.944E 00 0.22 IE 00 0.933E 00 O.llOE 01 0.2 94E 00 0.864E 00 0. 102E 01 0.36 8E 00 0.720E 00 0.851E 00 4 4 1 E 00 0.577E 00 0.633E 00 0.59 8E 00 0.365E 00 0.432E 00 0.735E 00 0.235E 00 0.278E 00

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227 LYMAN BETA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE= 809.10 ELECTRON OENSITY= 0.10E 24 A=0.20 R~ 0.0 TRATIO=4.00 XI = 9.0 Zl = 1.0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.350E 00 0.401E 00 0.735E-02 0.354E 00 0.406E 00 0.14 7E-0 1 0.366E 00 0.419E 00 0.22 1E-0 1 0.386E 00 0.440E 00 0.294E-01 0. 41 IE 00 0.468E 00 0.36 8E-0 I 0.442E 00 0.501E 00 0.441E-01 0.477E 00 0.540E 00 0.51 5E-Q1 0.516E 00 0.583E 00 0.588E-0 I 0.55 7E 00 0.628E 00 0.662E-01 0. 600E 00 0.676E 00 0.735E-0 1 0.644E 00 0.725E 00 0.882E-0 1 0.732E 00 0.823E 00 0.103E 00 0.81 7E 00 0.918E 00 0.1 I 8E 0.894E 00 0.100E 01 0.13 2E 0.959E 00 0.108E 01 0.14 7E 0. 101E 01 0.1 13E 01 0.221E 00 0.106E 01 0. 1 19E 01 0.294E 00 0. 895E 00 0.101E 01 0.36 8E 0.694E 00 0.781E 00 0.441E 00 0.527E 00 0.593E 00 0.588E 00 0.306E 00 0.344E 00 0.735E 00 0. I83E 00 0.206E 00 LYMAN BETA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY= 0.10E 24 A=0.20 R= O.IOE 00 TRATIQ=1.00 XI = 9.0 Zl= 1.0 Z2= 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.204E 00 0.269E 00 0.735E-02 0.206E 00 0.272E 00 0, 147E-01 0.212E 00 0.278E 00 0.22 1E-01 0.221E 00 0.289E 00 0.294E-0 1 0.234E 00 0.304E 00 G.368E-0! 0.250E 00 0.322E 00 0.44 1E-0 1 0.268E 00 0.344E 00 0.51 5E-01 0.289E 00 0.368E 00 0.588E-01 0.31 IE 00 0.394E 00 0.662E-0 1 0.335E 00 0.423E 00 C.735E-01 0.360E 00 0.453E 00 0.882E-0 I 0.413E 00 0.51 8E 0.103E 00 0.469E 00 0.585E 00 0. 1 1 8E 0.524E 00 0.652E 00 0.132E 00 0.5 78E 00 0.718E 00 1 4 7E 0.628E 00 0.779E 00 0.22 IE 0.793E 00 0.983E 00 0.29 4E 0.805E 00 0.100E 01 0.368E 0.724E 00 0.902E 00 0.441E 00 0.615E 00 0.768E 00 0.58 8E 0.421E 00 0.525E 00 0.735E 00 0.286E 00 0.357E 00

PAGE 234

228 LYMAN BETA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE= 809.10 ELECTRON DENSITY= 0.10E 24 A=0.20 H0.10E 13 TRATlO=l.00 Xl= 9.0 Zl = 1.0 Z2 = 9.0 DELTA STARK STARK*OMEGA PROFILE DOPPLER 0.0 0. 16 7E 00 0.231E 00 0.73SE-02 0. 168E 00 0.233E 00 0. 14 7E-0 1 0. 173E 00 0.238E 00 0.22 1E-01 0. 180E 00 0.247E 00 0.294E-0 I 0. 190E 00 0.259E 00 0.368E-0 1 0.202E 00 0.274E 00 0.44 1E-0 1 0.216E 00 0.291E 00 0.515E-01 0.232E 00 0.311E 00 0.588E-01 0.249E 00 0.333E 00 0.662E-Q1 0.268E 00 0.357E 00 0.735E-0 I 0.288E 00 0.382E 00 0.832E-01 0.331E 00 0.437E 00 0.10 3E 0.376E 00 0.495E 00 0. 1 18E 0.422E 00 0.554E 00 0.132E 00 0.468E 00 0.613E 0.147E 00 0.513E 00 0.671E 00 0.221E 00 0.683E 00 0.892E 00 0.29 4E 0.736E 00 0.963E 00 0.36 8E 0.701E 00 0.919E 00 0.441E 00 0.625E 00 0.820E 00 0.53 8E 0.458E 00 0.601E 00 0.735E 00 0.326E 00 0.429E 00 LYMAN BETA PROFILE FOR HYDROGENIC NEON ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0. 10E 24 A=0.40 R= 0.0 TRATIO=0.25 XI= 9.0 Z 1= 1.0 Z2 = 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.382E 00 0.471E 00 0.73SE-02 0.384E 00 0.474E 00 0. 147E-0 t 0.39 IE 00 0.482E 00 0.22 1E-0 1 0.402E 00 0.494E 00 0.294E-01 0.418E 00 0.51 IE 00 0.368E-0 1 0.438E 00 0.532E 00 0.44 1E-01 0.460E 00 0.557E 00 0.5 1 5E-0 1 0.485E 00 0.584E 00 0.58 8E-0 1 0.51 IE 00 0.614E 00 0.66 2E-01 0.539E 00 0.645E 00 0.735E-0 I 0.566E 00 0.678E 00 0.882E-01 0.626E 00 0.744E 00 0.10 3E 0.682E 00 0.809E 00 1 1 8E 0.735E 00 0.869E 00 0.13 2E 0.781E 00 0.923E 00 1 4 7E 0.819E 00 0.968E 00 0.221E 00 0.890E 00 1 5E 1 0. 294E 0.81 IE 00 0.963E 00 0.368E 00 0.679E 00 0.808E 00 0.44 IE 0.550E 00 0.656E 00 0.58 8E 0.355E 00 0.424E 00 0.73 5E 0.234E 00 0.2 79E 00

PAGE 235

229 LYMAN BETA PROFILE FOR HYDROGEN 1C NEON ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY= 0.10E 24 A=0.40 R0-0 TRATIO=1.00 XI = 9.0 Zl~ 1.0 Z2 = 9.0 DELT/> STARK STARK* OMEG/s i PROFILE DOPPLER 0.0 0.473E 00 0.545E 00 0.73 5E•02 0.475E 00 0.548E 00 0. 14 7E01 0.484E 00 0.559E 00 0.221EOi 0.499E 00 0.575E 00 0.29 4EI 0.520E 00 0.598E 00 0.368E•01 0.544E 00 0.625E 00 0.44 1E•0 I 0.572E 00 0.656E 00 0.51 5E01 0.603E 00 0.691E 00 0.S8 8E•01 0.636E 00 0.72 7E 00 0.662E0 I 0.670E 00 0.765E 00 0.735E01 0.704E 00 0.804E 00 0.882E01 0.771E 00 0.880E 00 0.10 3E 00 0.834E 00 0.951E 00 1 1 8E 00 0.889E 00 0.101E 01 0.132E 00 0.934E 00 0. 106E 01 0.14 7E 00 0.969E 00 0.1 10E 01 0.221E 00 0.981E 00 0.U2E 01 0.29*E 00 0.836E 00 0.955E 00 0.368E 00 0.663E 00 0.757E 00 0.44 IE 00 0.514E 00 0.587E 00 0.SS8E 00 0.309E 00 0.354E 00 0.73 5E 00 0. 193E 00 0.221E 00 LYMAN BETA PROFILE FOR HYOROGENIC NEON ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 2* A=0.40 R= 0.0 TRATIO=4.00 XI= 9.0 Zl = 1-0 Z2 = 9.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0. 758E 00 0.824E 00 0.735E-02 0.763E 00 0.829E 00 0. 147E-0 I 0.778E 00 0.846E 00 0.221E-01 0.803E 00 0.873E 00 0.294E-0 1 0.836E 00 0.908E 00 0.368E-0 I 0.876E 00 0.950E 00 0.44 1E-0 1 0.919E 00 0.997E 00 0.51SE-0 1 0.965E 00 0.1 OSE 01 0.58 8E-0 1 0. 101E 01 0. 1 10E 01 0.662E-01 0. 106E 01 0. 1 14E 01 0.735E-01 0. HOE 01 0.1 19E 02 0.882E-0 I 0. 1 18E 01 0. 128E 1 0.10 3E 0. 124E 01 0.134E 1 • 1 I 8E 0. 127E 01 0.138E 01 0.132E 0.129E 01 0. 140E 01 0.147E 00 0. 129E 01 0.139E 01 0.22 IE 0. 107E 01 0. 1 16E 01 0.294E 00 0.781E 00 0.84 7E 00 0.3&8E 00 0.551E 00 0.597E 00 C.441E 00 0.391E 00 0.424E 00 0.588E 00 0.208E 00 0.225E 00 0.73 5E 0. I22E 00 0.132E 00

PAGE 236

230 LYMAN BETA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE^ 202-30 ELECTRON DENSITY= O.iOE 24 A=0.40 R= O.IOE 00 TRATIO=1.00 XI= 9.0 Zl = 1-0 Z2= 9-0 DELTA STARK STARK* OMEGA PROFILE DOPPLER 0.0 0.462E 00 0.535E 00 0.735E-02 0.464E 00 0.538E 00 0. 14 7E-0 1 0.473E 00 0.548E 00 0.22 1E-0 1 0.488E 00 0.564E 00 0.294E-01 0.507E 00 0.586E 00 0.368E-0 1 0.531E 00 0.613E 00 0.44 1E-0 1 0-558E 00 0.643E 00 0.515E-01 0.588E 00 0.677E 00 0.588E-01 0.620E 00 0.712E 00 0.662E-0 1 0.652E 00 0.749E 00 0.735E-01 0.686E 00 0.787E 00 0.882E-01 0.751E 00 0.861E 00 0.10 3E 0.81 IE 00 0.930E 00 0. 1 1 8E 0. 865E 00 0.990E 00 0.1J2E 0-909E 00 0.104E 01 1 4 7E 0.943E 00 0I08E 01 0.221E 00 0963E 00 0.1 10E 01 0.294E 00 0.832E 00 0.955E 00 0.368E 00 0.668E 00 0.767E 00 0.441E 00 0.523E 00 0.600E 00 0.588E 00 0.31 8E 00 0.366E 00 0.73 5E 0.200E 00 0.230E 00 LYMAN BETA PROFILE FOR HYDROGEN IC NEON ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 A=0.40 Rss 0.10E 13 TRATI0=1.00 XI= 9.0 Zl= 1.0 Z2= 9-0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.457E 00 0.5 32E 00 0.735E-G2 0.460E 00 0.535E 00 0.147E-01 0.468E 00 0.545E 00 0.221E-01 0.483E 00 0.561E 00 0.294E-01 0.502E 00 0.583E 00 0.368E-01 0.52&E 00 0.609E 00 0.44 IE1 0.552E 00 0.639E 00 0.51 5E-0 1 0. 582E 00 0.672E 00 0.588E-0 I 0.613E 00 0.707E 00 0.66 2E-0 1 0.645E 00 0.743E 00 0.735E-0 1 0.677E 00 0.780E 00 0.88 2E-0 1 0.740E 00 0.852E 00 1 3E 0. 799E 00 0.920E 00 0.1 1 8E 0.851E 00 0.979E 00 0.13 2E 0.893E 00 0.103E 01 1 4 7E 0.926E 00 0.1 06E 01 0.221E 00 0.946E 00 0.109E 01 0.29 AC 0.B24E 00 0.950E 00 0.368E 00 0.668E oo 0.770E 00 0.441E 00 0.528E 00 0.608E 00 G.588E 00 0.325E oo 0.3 75E 00 0.735E 00 0.206E 00 0.237E 00

PAGE 237

231 LYMAN ALPHA PROFILE FOR HYDRQGENJC ALUMINUM ELECTRON TEMPERATURE^ 1019-20 ELECTRON DENSITY= 0.20E 24 A=0.20 R= 0.0 TRATI0=0.25 XI=12.0 Zl=12.0 Z2=11.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0*0 0.295E 02 0.507E 01 0.735E-02 0. 147E 02 0.502E 01 0.14 7E-0 1 0.588E 01 0.489E 01 0.221E-01 0.297E 01 0.468E 01 0.294E-01 0. 177E 01 0.439E 1 0.368E-01 0.U9E 01 0.406E 01 0.441E-01 0.862E 00 0.368E 1 0.515E-01 0.667E 00 0.329E 01 0.58 8E-0 I Q.543E 00 0.289E 01 0.662E-01 0.461E 00 0.251E 01 0.735E-0I 0.406E 00 0.215E 01 0.882E-01 0.344E 00 0.1 53E 1 1 3E 0.318E 00 0.106E 1 0I 18E 0.310E 00 0.751E 00 0.132E 00 0.31 IE 00 0.558E 00 1 4 7E 0.318E 00 0.450E 00 0.221E 00 0.348E 00 0.355E 00 0.29 4E 0.3 2 7E 00 0.336E 00 0.368E 00 0.2 75E 00 0.287E 00 0.441E 00 0.217E 00 0.229E 00 Q.S8 8E 0. 127E 00 0.136E 00 0.735E 00 0.757E-01 0.808E-01 LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM ELECTRON TEMPERATURE= 1019.20 ELECTRON OENSlTY= 0.20E 24 A =0.20 R= 0.0 TRATIO=1.00 XI=12.0 Z1=12.0 Z2=11.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.293E 02 0.882E 1 0.735E-02 0. 146E 02 0.853E 01 0.14 7E-0 1 0.587E 01 0.774E 01 0.221E-01 0.298E 01 0.660E 01 0.2 9 4E1 0. 181E 0! 0.531E 01 0.36 8E-0 1 0. 123E 01 0.405E 01 0.44 1E-0 1 0.917E 00 0.296E 01 0.51 5E-0 1 0.732E 00 0.21 IE 01 0.58 8E-0 1 0.61 8E 00 0.1 50E 1 0.66 2E-0 1 0.547E 00 0.109E 01 0.73 5E-0 I 0.502E 00 0.826E 00 0.882E-0 1 0.458E 00 0.578E 00 0.10 3E 0.44 8E 00 0.498E 00 1 1 8E 0.450E 00 0.475E 00 0.132E 00 0.458E 00 0.472E 00 1 4 7E 0.466E 00 0.475E 00 0.221E 00 0.445E 00 0.452E 00 0.294E 00 0.351E 00 0.360E 00 0.368E 00 0.253E 00 0.261E 00 0.44 IE 0. 177E 00 0.132E 00 0.588E 00 0.86 4E-01 0.892E-01 0.735E 00 0.453E-01 0.467E-0 1

PAGE 238

232 LYMAN ALPHA PROFILE FOR HYDROGEN IC ALUMINUM ELECTRON TEMPERATURE = 1019,20 ELECTRON DENSITY* 0.20E 24 A=0.2 R= 0.0 TRATIO=4.00 XI=12.0 Zl=12.0 Z2=ll.O DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.293E 02 0.14 3E 2 0.735E-02 0. 147E 02 0.129E 02 0-14 7E1 0.599E 01 0.956E 01 0.221E-01 0.313E 01 0.606E 01 0.294E-0 1 0. 199E 01 0.357E 01 0.368E-O1 0.144E 01 0.217E 1 0.44 1E-01 0. 1 16E 01 Q.149E 01 0.S15E-0 1 0. 101E 01 0.117E 01 0.588E-0 I 0.926E 00 0. 101E 01 0.662E-01 0.883E 00 0.929E 00 0.735E-0 1 0.861E 00 0.888E 00 0.882E-01 0.848E 00 0.857E 00 0.103E 00 0.844E 00 0.844E 00 0.118E 0.828E 00 0.827E 00 0.132E 00 0.797E 00 0.796E 00 1 4 7E Q.754E 00 0.754E 00 0.221E 00 0.458E 00 0.462E 00 0.294E 00 0.236E 00 0.239E 00 0.368E 00 0. 120E 00 0.1 22E 0.441E 00 0.642E-01 0.649E-01 0.58 8E 0.220E-01 0.222E-01 7 3 5E 0.960E-02 0.967E-02 LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM ELECTRON T£MPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 = 0.40 R= 0.0 TRATI0=0.2£ i XI=12 1.0 Z1=12.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.21 IE 02 0.825E 01 0.735E-02 0. 140E 02 0.801E 01 0. 14 7E-0 1 0.705E 01 0.734E 01 0.22 1E-0 1 0.392E 01 0.637E 01 0.294E-0 1 0.248E 01 0.525E 01 0.368E-0 1 0. 174E 01 0.415E 01 0.44 1E-01 0. 132E 01 0.317E 1 0.51 5EI 0. 10 7E 01 0.238E 01 0.588E-01 0.914E 00 0.179E 01 0.662E-01 0.81 IE 00 0.137E 1 0.73 5E-0 1 0.744E 00 0. i 10E 01 0.882E-0 I 0.666E 00 0.806E 00 0.10 3E 0.628E 00 0.687E 00 1 I 8E 0. 604E 00 0.632E 00 0.132E 0.58 3E 00 0.600E 00 1 4 7E 0.5&2E 00 0.574E 00 0.221E 00 0.429E 00 0.440E 00 G.294E 0.30 IE 00 0.310E 00 0.368E 00 0.207E 00 0.214E 00 0.441E 00 0. 143E 00 0.148E 00 0.588E 00 0.721E-01 0.743E-01 0.73 5E 0.397E-01 0.409E-01

PAGE 239

233 LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM ELECTRON TEMPERATURE^ 254.80 ELECTRON DENSITY= 0.20E 24 A=0.40 R= 0.0 TRATIO=1.00 XI=12.0 Z1=12.0 Z2=ll.O DELTA i STARK STARK* OMEGA PROFILE DOPPLER 0.0 0.21 IE 02 0.127E 02 0.7 3 5E02 0. 141E 02 0.116E 2 0. 14 7E1 0.722E 01 0.910E 01 0.22 IE01 0.414E 01 0.630E 01 0.29 4E01 0.273E 01 0.412E 1 0.368E01 0.201E 01 0.275E 01 0.44 1E01 0.1 62E 01 0.199E 1 0.51 5E•01 0.139E 01 0.1 58E 1 0.588E01 0. 12 5E 01 0.1 36E 01 0.662E•01 0. 1 16E 01 0.122E 01 0.735E0 1 0. 109E 01 0.1 13E 01 0.88 2E0 1 0. 101E 01 0.102E 01 0.10 3E 00 0.937E 00 0.943E 00 1 1 8E 00 0.866E 00 0.871E 00 0.132E 00 0.792E 00 0.797E 00 1 4 7E 00 0.717E 00 0.722E 00 0.221E 00 0.392E 00 0.397E 00 0.294E 00 0.204E 00 0.207E 00 0.368E 00 0. 109E 00 0.1 1 IE 00 0.44 1£ 00 Q.619E-01 0.626E-01 0.588E 00 0.240E-01 0.242E-01 0.735E 00 0. 1 15E-01 0.116E-01 LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM ELECTRON TEMPERATURE= 254.80 ELECTRON OENSITY= 0.20E 24 A=0.40 R= 0.0 TRATIO=4.00 XI=12.0 Zl=12.0 Z2=ll.O DELTA OMEGA 0.0 0.73 0. 14 0.22 0.29 0.36 0.44 0.51 0.58 0.66 0.73 0.88 0.10 0.11 0.13 0. 14 0.22 0.29 0.36 0.44 0.58 0.73 5E-0 2 7E-0 1 1E-0 4E-0 8E-0 1E-0 5E-0 1 8E-0 1 2E-0 1 5E-0 1 2E-0 1 3E 8E 2E 7E IE 4E 8E IE 8E 5E 00 00 00 00 00 00 00 00 00 STARK PROFILE 0.219E 02 0. 149E 02 0.800E 01 0.496E 01 0.358E 01 0.288E 01 0.248E 01 0.220E 01 0. 199E 01 0. 180E 01 0. 162E 01 0. 12 8E 01 0.990E 00 0.74 7E 00 0.558E 00 0.416E 00 1 1 4E 00 0. 482E-01 0.274E•01 0. 181E-01 0.986E-02 0.623E-02 STARK+ DOPPLER 0.177E 02 0.146E 02 0.926E 1 0.565E I 0.388E 01 0.301E 01 0.254E 01 0.224E 01 0.201E 01 0. 1 81E 01 0.163E 1 0. 129E 1 0. 100E 01 0.756E 00 0.565E 00 0.421E 00 0. 1 16E 00 0.485E-01 0.2 76E-01 0. 132E-0 1 0.991E-02 0.625E-02

PAGE 240

234 LYMAN BETA PROFILE FOR HYDROGEN IC ALUMINUM ELECTRON TEMP£RATURE= 809.10 ELECTRON DENSITY^ 0.10E 24 A=0.20 R= 0.0 TRATI0=0.25 XI=1£ !.0 Zl=12.0 OEL.TA STARK STARK+ MEG/0 l PROFILE DOPPLER 0.0 0. 113E 00 0.254E 00 0.735E02 0. 1 15E 00 0.256E 00 0.14 7E01 0.122E 00 0.261E 00 0.22 1E01 0.132E 00 0-269E 00 0.294E01 0. 146E 00 0.280E 00 0.368E01 0. 162E 00 0.294E 00 0.44 IE•01 0.181E 00 0.311E 00 0.51 5E 01 0.203E 00 0.331E 00 Q.588E•01 0.226E 00 0.352E 00 0.66 2E01 0.25 IE 00 0.376E 00 0.73 5E•0 1 0.278E 00 0.402E 00 0.882E01 0.336E 00 0.458E 0.103E 00 0.396E 00 0.51 9E 0.1 I 8£ 00 0.458E 00 0.581E 00 0. 13 2E 00 0.51 7E 00 0.643E 00 0.14 7E 00 0.573E 00 0.703E 00 0.221E 00 0.763E 00 0.927E 00 0.294E 00 0.792E 00 0.985E 00 0.368E 00 0.725E 00 0.920E 00 0.44 IE 00 0.626E 00 0.804E 00 0.588E 00 0.440E 00 0.570E 00 7 3 5E 00 0.3 06E 00 0.397E 00 LYMAN BETA PROFILE FOR HYOROGENIC ALUMINUM ELECTRON TEMP£RATURE= 809.10 ELECTRON DENSITY= 0. 10E 24 A=0.20 R= 0.0 TRATIO^l.OO XI=12.0 Zl=12.0 Z2=ll.O DELT/S STARK STARK* OMEG/> i PROFILE DOPPLER 0.0 0. 177E 00 0.262E 00 0.735E02 0. 181E 00 0.266E 00 0.14 7E1 0. 194E 00 0.276E 00 0.22 1E01 0.213E 00 0.294E 00 C.294E•01 0.239E 00 0.318E 0.36 8E01 0.269E 00 0.34 7E 4 4 I E•01 0.303E 00 0.382E 00 0.51 5E01 0.34 IE 00 0.421E 00 0.588E01 0.382E 00 0.463E 00 0.662E-0 1 0.425E 00 0.508E 00 0.7 3 5E•0 I 0.470E 00 0.556E 00 C.882E0 1 0.562E 00 0.653E 00 0.10 3E 00 0.652E 00 0.750E 00 1 1 8E 00 0.736E 00 0.841E 00 1 3 2E 00 0.81 IE 00 0.923E 00 1 4 7£ 00 0. 8 73E 00 0.993E 00 0.22 IE 00 0.990E 00 0.1 14E 01 0.294E 00 0.89 IE 00 0.103E 01 0.368E 0.730E 00 0.852E 00 0.441E 00 0.5 79E 00 0.677E 00 0.588E 00 0.358E 00 0.419E 0.73 5E 00 0.226E 00 0.264E 00

PAGE 241

235 LYMAN BETA PROFILE FOR HYOROGENIC ALUMINUM ELECTRON TEMPERATURE^ 809.10 ELECTRON OENSITY= 0.10E 24 A=0.2 R= 0.0 TRATIQ=4. 00 XI=12.0 Zl=12.0 Z2=11.0 DELTA STARK STARK* OMEGA PROFILE OOPPLER 0.0 0.403E 00 0.465E 00 0.735E-02 0.4 15E 00 0.476E 00 0.14 7E-0 1 0.452E 00 0.509E 00 0.22 1E-0 1 0.508E 00 0.561E 00 0.294E-Q1 0.579E 00 0.629E 00 0.368E-0 1 0.661E 00 0.710E 00 0.441E-0 1 0.751E 00 0.799E 00 0.515E-01 0.846E 00 0.893E 00 0.588E-01 0.942E 00 0.989E 00 0.662E-0 1 0. 104E 01 0.108E 01 0.735E-01 0. II 3E 01 0.1 18E 01 0.882E-0I 0.129E 01 0.134E 01 0.10 3E 0. 142E 01 0.1 47E 1 1 1 8£ 0. 150E 01 0.1 55E 01 0.132E 00 0. 153E 01 0.160E 01 0.14 7E 0. 153E 01 0.160E 1 0.221E 00 0. 122E 01 0.128E 01 0.294E 00 0.833E 00 0.878E 00 0.368E 00 0.557E 00 0.587E 00 0.441E 00 0.375E 00 0.395E 00 0588E 00 0. 179E 00 0.189E 00 0.735E 00 0.929E-01 0.980E-01 LYMAN SETA PROFILE FOR HYDROGEN IC ALUMINUM ELECTRON TEMPERATURE-= 202.30 ELECTRON DENSITY^ 0.10E 24 A=0.40 R= 0.0 TRATIO=0.2S i XI=12 !.0 Z1=12.C ) DELTA STARK 5TARK+ OMEGA PROFILE DOPPLER 0.0 0.31 9£ 00 0.436E 00 0.735E-02 0.324E 00 0.440E 00 0.14 7E-0 1 0.340E 00 0.453E 00 •22 1E-0 1 0.365E 00 0-474E 00 0.294E-0 1 0.39 7E 00 0.503E 00 0.36 8E-01 0.436E 00 0.537E 00 0.44 IE-0 1 0.479E 00 0.577E 00 0.51 5E1 0.S25E 00 0.622E 00 0.588E-01 0.573E 00 0.669E 00 0.662E-01 0.622E 00 0.717E 00 0.735E-0 I 0.671E 00 0.767E 00 0.832E-Q 1 0.765E 00 0.863E 00 CU10 3E 0. 84 9E 00 0.952E 00 1 I 8E 0.918E 00 0.103E 01 1 3 2E 0.971E 00 0. 109E 01 0.14 7E 0. I01E 01 0. 1 13E 01 0.22 IE 0.994E 00 1 1 3E 01 0.294E 00 0. 834E 00 Q-958E 00 0.368E 00 0.661E 00 0.763E 00 0.441E 00 0.516E 00 0.595E 00 0.58 8E 0.31SE 00 0.364E 00 0.73SE 0.200E 00 0.231E 00

PAGE 242

236 LYMAN BETA PROFILE FOR HYDROGEN IC ALUMINUM ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 A=0.40 R0.0 TRATI0=1.00 XI=12.0 Zl=12.0 Z2=lt.0 DELTA OMEGA STARK PROFILE STARK+ DOPPLER 0.0 0.73 0.14 0.22 0.29 0.36 0.44 0.51 0.58 0.66 0.73 0.88 0.10 0.1 1 0.13 0.14 0.22 0.29 0.36 0.44 0.58 0.73 5E-0 2 7E-0 1 IE-0 1 4E-0 1 8E-0 1 1E-0 I 5E-0 1 8E-0 I 2E-0 1 5E-0 1 2E-0 1 3E 8E 2E 7E IE 4E 8E IE 8E 5E 00 00 00 00 00 00 00 00 00 0.612E 0.623E 0.659E 0.713E 0.781E 0.859E 0.941E O. 103E 0. I 1 IE 0. 118E 0. 126E 0. 137E 0. 145E 0. 148E 0.148E 0. 146E O 1 I 2E 0.770E 0.521E 0.358E O. 180E 00 00 00 00 00 00 00 01 01 01 01 01 01 01 01 01 01 00 00 00 00 0.998E-0 1 0.669E 0.700E 0.732E 0.782E 0.846E 0.921E 0.100E 0.1 09E 0.1 17E 0. 125E 0. 132E 0. 144E 0.152E 0.1 56E 0. 156E 0.1 54E 0.U9E 0.819E 0.555E 0.381E 0.192E 0. 106E 00 00 00 00 00 00 01 01 01 01 01 01 01 01 01 01 01 00 00 00 00 00 LYMAN BETA PROFILE FOR HYDROGENIC ALUMINUM ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 A=0.40 ft0.0 DELTA OMEGA 0.0 0.735E-02 0. 14 7E-0 1 0.22 1E-01 0.29 4E-0 I 0.368E-0 1 0.44 IE-0 1 0.51 5E-0 1 0.58 8E-O1 0.662E-0 1 0.735E-0 I 0.882E-01 1 3E 0.118E 0.13 2E 1 4 7E 0.22 IE 00 0.2 9 4E 0.368E 00 0.441E 00 0.588E 00 0.73 5E TRATIO=4.00 XI=12.0 21=12.0 22=11.0 STARK+ DOPPLER 0.153E 01 0.156E 01 0.165E 01 0.177E 01 0.192E 01 0.206E 01 0.219E 01 0.230E 01 0.237E 01 0.241E 01 0.242E 01 0.236E 01 0.221E 01 0.201E 01 0.180E 01 0.159E 01 0.822E 00 0.445E 00 0.262E 00 0.168E 00 0.858E-01 0.526E-01 STARK PROFILE 0. 145E 01 0. 149E 01 0. 157E 01 0. 170E 01 0. 185E 01 0. 199E 01 0.212E 01 0.223E 01 0. 230E 01 0.234E 01 0.235E 01 0. 22 8E 01 0.213E 01 0. 194E 01 0. 173E 01 0. 15 3E 01 0. 792E 00 0.423E 00 0.252E 00 0. 162E 00 0.828E-01 0.507E-01

PAGE 243

237 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 A=0.20 R= 0.0 TRATIO=0.2S > xi=n '.0 Zl= 1.0 DELTA STARK STARK+ OMEGA PROFILE OOPPLER 0.0 0.558E 02 0.378E 01 0.735E-02 0. 109E 02 0.377E 01 0.147E-01 0.330E 01 0.373E 01 0.221E-01 0. 163E 01 0.366E 01 0.294E-0 I 0. 106E 01 0.358E 01 0.368E-01 0.820E 00 0.347E 01 0.441E-0 1 0.723E 00 0.335E 01 0.515E-01 0.693E 00 0.320E 01 0.58 8E-0 1 0.697E 00 0.305E 01 0.662E-01 0.714E 00 0.288E 01 0.735E-01 0.73SE 00 0.271E 01 0.882E-0 1 0.765E 00 0.236E 01 0.10 3E 0.782E 00 0.202E 01 11 8E 0.7 79E 00 0.170E 01 0.132E 00 0.741E 00 0.141E 01 0.147E 00 0.677E 00 0.1 I 7E 1 0.22 IE 0.364E 00 0.4 79E 0.294E 0. 183E 00 0.241E 00 0.368E 00 0.100E 00 0.130E 00 0.44 IE 00 0. 600E-01 0.738E-0 1 0.58 8E 0.266E-01 0.301E-01 0.735E 00 0. 146E-01 0.157E-01 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 A=0.20 R= 0.0 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.557E 02 0.680E 01 0.735E-02 0. 109E 02 0.671E 01 0. 14 7E-0 1 0.333E 01 0.643E 01 0.22 IE-01 0. 167E 01 0.599E 1 0.294E-0 1 O.lllE 01 0.543E 01 0.36 8E-0 1 0.889E 00 0.481E 01 0.44 1E-0 1 0.807E 00 0.416E 1 0.51SE-01 0.79 IE 00 0.352E 01 0.58 8E-0 1 0.805E 00 0.294E 01 0.662E-01 0.830E 00 0.243E 01 0.73SE-01 0.853E 00 0.201E 01 0.882E-01 0.878E 00 0.140E 01 1 3E 0.879E 00 0.105E 01 0.118E 00 0.853E 00 0.868E 00 0.132E 00 0.788E 00 0.762E 1 4 7E 0.700E 00 0.684E 00 0.221E 00 0.330E 00 0.359E 00 0.294E 00 0. 152E 00 0.169E 00 3 6 8E 0. 787E-01 0.851E-01 0.441E 00 0.453E-01 0.480E-01 0.58 8E 0. 190E-01 0. 197E-01 0.73 5E 0.999E-02 0. 102E-0 1

PAGE 244

238 LYMAN ALPHA PROFILE FOR HYOROGENIC ARGON ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 A=0.20 R= 0.0 TRATIQ=4.00 XI=17.0 Zl = 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.558E 02 0.125E 02 0.735E•02 0.110E 02 0.118E 02 Q.147E•0 1 0.346E 01 0.999E 1 0.221E•01 0. 185E 01 0.764E 01 Q.294E-01 0.1 34E 01 0.537E 01 0.368EI 0. 1 17E 01 0.359E 01 0.441E01 0. 1 13E 01 0.241E 01 0.5 15E-0 1 0. 1 16E 01 0.174E 01 0.588E-01 0. 120E 01 0.141E 01 0.662E-01 0. 123E 01 0.127E 01 0.735E-01 0. 123E 01 0.121E 01 0-882E-01 0. 1 18E 01 0. 1 14E 1 0.10 3E 00 0.1 08E 01 0.105E 01 0.11 8E 00 0.951E 00 0.933E 00 0.132E 00 0.792E 00 0.793E 00 1 4 7E 00 0.634E 00 0.652E 00 0.22 IE 00 0. 196E 00 0.206E 00 C.294E 00 0.707E-01 0.736E-0 1 C.368E 00 0.320E-01 0.328E-01 0.441E 00 0. 170E-01 0.174E-01 .58 8E 00 0.666E-02 0.673E-02 0.73 5E 00 0.346E-02 0.348E-02 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY^ 0.20E 24 A-0.20 R= 0.10E 00 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 DELTyi t STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.557E 02 0.671E 01 0.735E•02 0. 109E 02 0.661E 01 0.147E-0 I 0.328E 01 0.633E 1 0.221E01 0. 160E 01 0.589E 01 0.29 4E01 0. 102E 01 0.533E 01 0.368E0 I 0.779E 00 0.469E 1 0.44 l£0 1 0.673E 00 0.403E 01 0.51 5E-01 0. 635E 00 0.340E 01 0.58 8E0 1 0.631E 00 0.281E 01 0.662E-0 t 0.643E 00 0.230E 1 0.73 5E0 1 0.661E 00 0.187E 01 0.88 2E-0 1 0.693E 00 0.127E 01 0. 103E 00 0.719E 00 0.933E 00 0.1 18E 00 0.731E 00 0.770E 00 0.132E 00 0.712E 00 0.690E 00 1 4 7E 00 0.669E 00 0.642E 00 0.221E 00 0.4 10E 00 0.421E 00 0.29 4E 00 0.222E 00 0.236E 00 0.36 8E 00 0. 121E 00 0.1 29E 0.441E 00 0.68 7E-01 0.728E-01 0.58 8E 00 0.263E-01 0.274E-01 0.735E 00 0. 125E-01 0. 128E-0 1

PAGE 245

239 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRCN TEMP£RATURE= 1019.20 ELECTRON DENSITY= 0.20E 2* A=0.20 R= 0.10E 13 TRATI0=1.Q0 XI=17.0 Zl = 1-0 Z2=17.0 DELTA STARK STARK* OMEG/ i PROFILE DOPPLER 0.0 0.557E 02 0.669E 01 0.735E02 0. 109E 02 0.659E 01 0.14 7E1 0.326E 01 0.630E 01 0.22 te01 0. 158E 01 0„586E 01 0.29 4E•01 0.997E 00 0.530E 01 0.368E01 0.74SE 00 0.466E 1 0.44 1E01 0.632E 00 0.400E 01 0.51 SE 01 0.587E 00 0.336E 01 0.588E•01 0.576E 00 0.277E 01 0.66 2E-0 1 0.583E 00 0.225E 01 0.73 5E01 5 9 7E 00 0.182E 01 0.832E•01 0..627E 00 0.1 22E 1 0.10 3E 00 0.6S5E 00 0.885E 00 0.11 8E 00 0.675E 00 0.726E 00 0.132E 00 0.667E 00 0.652E 00 1 4 7E 00 0.63 7E 00 0.612E 00 0.221E 00 0.427E 00 0.432E 00 0.294E 00 0.248E 00 0.26 IE 0.36 8E 00 0. 141E 00 0.14 9E 0.44 IE 00 0.822E-01 0.867E-0 1 58 8E 00 0.313E-01 0.327E-01 0.735E 00 0. 144E-01 0. 148E-0 1 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 2< A=0.40 R= 0.0 TRATIO=0.25 Xl=17.0 Zl= 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 00 0.4 16E 02 0.662E 1 0.735E-02 0. 136E 02 0.653E 01 0. 147E-0 1 0.470E 01 0.627E 01 0.221E-01 0.244E 01 0.587E 01 0.294E-0 1 0. 163E 1 0.536E 01 0.368E-0 I 0. 129E 01 0.477E 01 0.44 IE-01 0. 1 14E 01 0.41 7E 1 0.51 5E-0 1 0. 108E 01 0.357E 01 0.588E-01 0. 105E 01 0.302E 01 0.662E-0 I 0. 104E 01 0.253E 01 0.73SE-01 0. 104E 01 0.212E 01 0.882E-01 0. 101E 01 0. 151E 01 0.10 3E 0.947E 00 0. 1 14E 1 1 1 8E 0.864E 00 0.924E 00 0.13 2E 0.76 7E 00 0.788E 00 0.14 7E 0.668E 00 0.688E 00 0.221E 00 0.304E 00 0.336E 00 0.294E 00 0. 145E 00 0.160E 00 0.36 8E 0.781E-01 0.839E-01 0.441E 00 0.466E-01 0.491E-0 1 0.58 8E 0.205E-01 0.212E-01 0.7 3 5E 0. 1 10E-01 0.113E-0 1

PAGE 246

240 LYMAN ALPHA PROFILE FOR HYDROGEN IC ARGON ELECTRON TEMPERATURE= 25*. 80 ELECTRON DENSITY= 0.20E 24 A=0.40 R= 0.0 TRATIO=1.00 XI=17.0 Zl = 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.416E 02 0.1 I 7E 2 0.735E-02 0. 137E 02 0.1 HE 02 0. 147E-0 1 0.488E 01 0.957E 01 0.22 1E-0 1 0.266E 01 0.751E 01 0.294E-0 1 0. 190E 01 0.548E 01 0.368E-01 0. 161E 01 0.385E 01 0.441E-01 0. 149E 01 0.272E 01 0.51 5E-01 0. 145E 01 0.204E 01 0.58 8E-0 1 0. 14 3E 01 0.1 67E 1 0.662E-01 0. 140E 01 0.147E 01 0.7 3 5E-0 1 0. 136E 01 0.136E 01 0.882E-0 1 0.124E 01 0. 121E 1 0.10 3E 0. 107E 01 0.106E 01 0. 118E 0.899E 00 0.907E 00 0.132E 00 0.732E 00 0.751E 00 1 4 7E 0.586E 00 0.610E 00 0.221E 00 0.191E 00 0.201E 00 0.294E 00 0.753E-01 0.761E-01 Q.368E 00 0.363E-01 0.372E-01 0.441E 00 0.203E-01 0.207E-01 0.588E 00 Q.851E-02 0.860E-02 0.735E 00 0.456E-02 0.459E-02 LYMAN ALPHA PROFILE FOR HYDROGEN1C ARGON ELECTRON TE MPER ATURE= 254.80 ELECTRON DENSITY^ 0.20E 24 A=0.40 R= 0.0 TRATIO=4.00 Xl=17.0 Zi= 1.0 Z2=17.0 DELTA STARK STARK* OMEGA PROFILE DOPPLER 0.0 0.422E 02 0.201E 02 0.735E-02 0. 143E 02 0.1 69E 2 0.147E-01 0.5&IE 01 0.106E 02 0,22 IE-0 1 0.354E 01 0.588E 01 0.294E-01 0.289E 01 0.366E 1 0.368E-0 I 0.263E 01 0.283E 1 0.44 IE-01 0.246E 01 0.249E 01 0.51 5E-01 0.228E 01 0.226E 01 0.58 8E-0 1 0.206E 01 0.204E 01 0.662E-01 0. 180E 01 0.180E 1 0.735E-0 1 0. 154E 01 0.1 55E 1 0.88 2E-0 1 0, 105E 01 0.1 08E 1 1 3E 0.690E 00 0.719E 00 0.118E 0.445E 00 0.46 7E 0.132E 00 0.291E 00 0.305E 00 0.147E 00 0. 195E 00 0.204E 00 0.221E 00 0.473E-01 0.480E-01 0.294E 00 0.212E-01 0.214E-01 0.368E 00 0. 127E-0 I 0.127E-0 I 0.441E 00 Q.856E-02 0.859E-02 0.588E 00 0.471E-02 0.473E-02 0.73 5E 0.299E-02 0.300E-02

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241 LYMAN ALPHA PROFILE FOR HYDROGEN IC ARGON ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 A=0.40 RO.IOE 00 TRATIO=1.00 Xl=17.0 21= 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE OOPPLER 0.0 4 1 7E 02 0.U8E 2 0.735E-02 0. 138E 02 0.112E 02 0. 147E-0 1 0.49 8E 01 0.967E 01 0.22 IE-0 1 0.273E 01 0.762E 01 0.29 4E-0 1 0.204E 01 0.559E 01 0.368E-01 0.175E 01 0.396E 1 0.44 1E-01 O. 163E 01 0.283E 01 0.515E-01 0. 157E 01 0.214E 01 0.58 8E-01 0. 153E 01 0.1 76E 1 0.662E-0 I 0.1 47E 01 0.15 4E 1 0.735E-01 0. 14IE 1 0.141E 01 0.882E-01 0. 123E 01 0-122E 01 0.I0 3E 0. 104E 01 0.104E 01 0.I18E 0.849E 00 0.867E 00 0.132E 00 0.68 0E 00 0.705E 00 1 4 7E 0.539E 00 0.564E 00 0.221E 00 0. 169E 00 0.1 79E 0.294E 00 0.632E-01 0.657E-01 0.368E 00 0.291E-01 0.29BE-01 0.441E 00 0. 158E-01 0.161E-01 0.58 8E 0.655E-02 0.662E-02 0.735E 00 0.364E-02 0.366E-02 LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE254.80 ELECTRON DENSITY* 0. 20E 24 A=0.40 R= O.IOE 13 TRATIO=1.00 XI=17.0 Zl= 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.418E 02 0.H9E 02 0.735E-02 0. 139E 02 0. 113E 02 0. 147E-01 0.505E 01 0.974E 01 0.22 1E-01 0.287E 01 0.769E 1 0.294E-01 0.213E 01 0.567E 1 0.368E-0 1 0. 185E 01 0.404E 1 0.441E-0 1 0. 172E 01 0.290E 01 0.515E-0 1 0.165E 01 0.220E 01 0.588E-01 0. 158E 01 0.1 80E 01 0.662E-01 0.151E 01 0.158E 01 0.735E-01 0. 142E 01 0.143E 01 0.882E-01 0. 122E 01 0.1 22E 1 0.10 3E O. 101E 01 0.I02E 01 1 1 8£ 0. 816E 00 0.839E 00 0.132E 00 0.648E 00 0.675E 00 0.14 7E 0.510E 00 0.536E 00 0.221E 00 0. 156E 00 0.1 64E 0.294E 00 C.558E-01 0.582E-01 0.368E 00 0.248E-01 0.255E-01 0.441E 00 0.133E-01 0. 135E-01 0.588E 00 0.552E-02 0.558E-02 0.735E 00 0.319E-02 0.321E-02

PAGE 248

242 LYMAN BETA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATUR£= 809,10 ELECTRON OENSITY= 0. 10E 24 A=0.20 Rat 0.0 TRATI0=0.2S XI=17.0 Z 1 = 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.262E 00 0. 1 10E 01 0.735E-02 0.285E 00 0.110E 01 0.14 7E-0 1 0.346E 00 0,1 I IE 01 0.22 IE-01 0.435E 00 0.1 1 IE 01 0.29 4E-O1 0.547E 00 0.112E 01 0*36 8E-0 t 0.6 7 7E 00 0. 1 14E 1 0.441E-01 0.818E 00 0.115E 01 0.515E-0 1 0.965E 00 0-1 17E 1 0.588E-01 0. 11 IE 01 0.119E 01 0.662E-01 0.125E 01 0.121E 01 0.735E-01 0, 137E 01 0.123E 1 0,882E-01 0. 158E 01 0,1 27E 01 G.I03E 00 0. 170E 01 0,131E 01 0.U8E 0,1 75E 01 0.134E 01 0.132E 00 0,174E 01 0.135E 01 0.14 7E 0, 168E 01 0.136E 01 0.221E 00 0, 116E 01 0.119E 01 0.29 4E 0-743E 00 0,868E 00 0.368E 00 0.484E 00 0,582E 00 0.441E 00 0.323E 00 0.387E 00 0.588E 00 0. 158E 00 0.184E 00 0.735E 00 0.874E-01 0.990E-0 1 LYMAN BETA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE^ 809*10 ELECTRON DENSITY= O.IOE 24 A=0.20 R= 0.0 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 DELTA OMEGA 0.0 0.735E-02 0.147E-0 1 0.221E-0 1 0.294E-01 0.368E-01 0.441E-01 0.51 5E-01 0.588E-01 0.662E-01 0.73 5E-0 1 0.882E-01 I 3E 0.11 8E 0.132E 00 I 4 7E 0.221E 00 0.294E 00 0.36 8E 0.441E 00 0.58 8E 0.73 5E STARK PROFILE STARK+ DOPPLER 0.312E 0.34 0E 0.417E 0.S28E 0.666E 0.824E 0.995E 0, 1 17E O. 134E 0,1 49E 0, 163E 0.183E 0. 193E 0. 194E 0. 188E O. 178E 1 1 3E 0.691E 0.430E 0.276E 0. 126E 00 00 00 00 00 00 00 01 01 01 01 01 01 01 01 01 01 00 00 00 00 0.834E 0.844E 0.871E 0.915E 0.974E 0.104E 0.1 12E 0.121E 0.130E 0.1 39E 0.147E 0. 161E 0,171E 0.1 76E 0.1 76E 0.1 72E 0.121E 0.750E 0.466E 0.298E 0.135E 00 00 00 00 00 01 01 01 01 01 01 01 01 01 01 01 01 00 00 00 00 0.671E-01 0.711E-01

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243 LYMAN BETA PROFILE FOR HYOROGENIC ARGON ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY^ 0. I OE 24 A=0.2 R= 0.0 TRATIO=4.00 XI=17.0 Zl = 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.499E 00 0.82 9E 0.735E-02 0.552E 00 0.859E 00 0. 147E-0 1 0.692E 00 0.946E 00 0.22 1E-0 1 0.892E 00 0.108E 01 0.29 4E-0 1 0.113E 01 0.125E 01 0.368E-0 1 0. 140E 01 0. 145E I 0.44 1E-Q 1 0. 167E 01 0.165E 01 0.51 5E-0 1 0. 192E 01 0.185E 01 0.588E-01 0.214E 01 0.203E 01 0.662E-Q1 0.232E 01 0.219E 01 0.735E-01 0.245E 01 0.231E 01 0.882E-01 0.256E 01 0.244E 1 0.103E 00 0.250E 01 0.24 3E 1 0.118E 0.233E 01 0.231E 01 0.132E 00 0.211E 01 0.212E 01 0.147E 00 0.187E 01 0.191E 01 0.221E 00 0.954E 00 0.990E 00 0.294E 00 0.499E 00 0.51 7E 0.368c 00 0.270E 00 0.2 79E 0.441E 00 0. 154E 00 0.159E 00 0.58 8E 0.621E-0 1 0.638E-01 0.73 5E 0.31SE-01 0.322E-01 LYMAN SETA PROFILE FOR HYOROGENIC ARGON ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY= 0.10E 24 A=0.20 R= 0.10E 00 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 DELTA OMEGA 0.0 0.73 0.14 0.22 0.29 0.36 0.44 0.51 0.58 0.66 0.73 0.88 0.10 0.11 0. 13 0.14 0.22 0.29 0.36 0.44 0.58 0.73 5E-0 2 7E-0 1 1E-0 1 4E-0 1 8E-0 1 1E-0 1 5E-01 8E-0 1 2E-0 1 5E-0 1 2E-0 1 3£ 8E 2E 7E IE 4E 8E IE 8E 5E 00 00 00 00 00 00 00 00 00 STARK PROFILE 0.232E 00 0.251E 00 0.303E 00 0.378E 00 0.473E 00 0.S85E 00 0.707E 00 0.836E 00 0.965E 00 0. 109E 01 0.121E 01 0.141E 01 0.1 55E 01 0.162E 01 0.164E 01 0.162E 01 0.122E 01 8 1 1 E 0.538E 00 0.362E 00 0.1 74E 0.915E-01 STARK+ DOPPLER 0.61 7E 0.624E 00 0.646E 00 0.682E 00 0.729E 00 0.788E 00 0.855E 00 0.928E 00 O.IOOE 01 0.1 08E 1 0.116E 01 0.130E 1 0.142E 01 0. 151E 01 0.155E 01 0.156E 01 0.128E 01 0.875E 00 0.582E 00 0.391E 00 0.188E 00 0.980E-01

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244 LYMAN BETA PROFILE FOR HYDROGEN IC ARGON ELECTRON TEMPERATURE^ 809-10 ELECTRON OENSITY= O.IOE 24 A=0.20 R= O.IOE 13 TRATIO=1.00 Xl=17.0 Zl= UO Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.206E 00 0.549E 00 Q.735E-02 0.223E 00 0.555E 00 0.147E-01 0.267E 00 0.575E 00 0.221E-0 1 0.332E 00 0.608E 00 0.294E-0 1 0.4 15E 00 0.651E 00 0.368E-0 1 0.S11E 00 0.705E 00 0.44 1E-01 0.6 I 9E 00 0.767E 00 0.515E-01 0. 732E 00 0.835E 00 0.588E-0 1 0.848E 00 0.906E 00 0.t>62E-01 0.961E 00 0.980E 00 0.735E-01 0. 107E 01 0.1 05E 1 0.882E-01 0.1 26E 01 0.1 19E 1 0.103E 00 0. 140E 01 0.131E 01 0.1 1 8E 0. 149E 01 0.140E 01 0.13 2E 0. 153E 01 0.146E 01 0.14 7E 0. 153E 01 0.149E 01 0.221E 00 0. 122E 01 0.128E 01 0.294E 00 0.848E 00 0.916E 00 0.368E 00 0.578E 00 0.628E 00 0.44 IE 00 0.398E 00 0.432E 00 0.58 8E 0. 19 7E 00 0.214E 00 0.735E 00 0. 105E 00 0.113E LYMAN BETA PROFILE FOR HYDROGENIC ARGON ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= O.IOE 24 A=0.40 R= 0.0 TRATIO=0.25 Xl=17.0 21= 1.0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE DOPPLER 0.0 0.487E 00 0.102E 01 0.735E-02 0.518E 00 0.103E 1 0. 147E-0 1 0.602E 00 0.106E 01 0.221E-01 0.724E 00 0. 1 10E 01 0.294E-0 I 0.871E 00 0. 115E 01 0.368E-0I 0.1 03E 01 0.12LE 01 0.44 1E-0 1 0. 120E 1 0.128E 01 0.51 5E-01 0. 136E 01 0.136E 01 0.S8 8E-01 0.151E 01 0.143E 01 0.662E-0 1 0. 164E 01 0. 151E 01 0.73SE-01 0. 175E 01 0.1 57E 1 0.882E-01 0. 189E 01 0.168E 01 0.10 3E 0. 193E 01 0.1 74E 1 1 1 8E 0. 189E 01 0.176E 01 0.132E 00 0. 180E 01 0.I73E 1 1 4 7E 0. 168E 01 0.167E 01 0.221E 00 0. 106E 01 0. 114E 01 0.294E 00 0.647E 00 0.706E 00 0.368E 00 0.4 07E 00 0.442E 00 0.441E 00 0.266E 00 0.287E 00 0.58 8E 0. 128E 00 0.1 37E 0.73 5E 0.707E-01 0.752E-01

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245 LYMAN BETA PROFILE FOR HYOROGENIC ARGON ELECTRON TEMP£RATURE= 202.30 ELECTRON DENSITY^ O.IOE 24 A=0.40 R~ 0.0 TRATIO=1.00 Xl=17.0 Zl = 1.0 Z2=17.0 DEL.TA i STARK STARK+ OMEG/ i PROFILE DOPPLER 0.0 0.732E 00 0.106E 01 0.735E•02 0. 783E 00 0.109E 01 0.14 7E•0 1 0.922E 00 0. 1 17E 1 0.22 1E01 0.1 12E 01 0.129E 01 0.294E•01 0. 1 35E 01 0.1 45E 1 0.368E01 0. 159E 01 0.162E 01 0.44 ie•0 1 0. 183E 01 0.180E 01 0.51 5E01 0.203E 01 0.196E 01 0.58 8E•01 0.221E 01 0.21 IE 01 0.662E01 0.234E 01 0.222E 01 0.735E•01 0.242E 01 0.231E 01 0.882E01 0.245E 01 0.237E 01 0. 10 3E 00 0.235E 1 0.232E 01 0. 1 1 8E 00 0.217E 01 0.21 8E 1 0. 132E 00 0. 196E 01 0.1 99E 1 1 4 7E 00 0.1 74E 01 0.179E 01 0.22 IE 00 0.909E 00 0.947E 00 0.294E 00 0.485E 00 0.506E 00 0.368E 00 0.273E 00 0.284E 00 0.44 IE 00 0. 165E 00 0.171E 00 0.588E 00 0.721E-01 0.744E-01 0.735E 00 0.387E-01 0.398E-01 LYMAN BETA PROFILE FOR HYOROGENIC ARGON ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 A=0.40 R= O.C 1 TRAT10=4.0( ) xi=i; '.0 Z\~ 1.0 DELTA STARK STARK4OMEGA PROFILE OOPPLER 0.0 0. 155E 01 0.181E 01 0.735E02 0.168E 01 0.191E 01 0. 14 7E-0 1 0.202E 01 0.21 6E 1 0.22 IE01 0.245E 01 0.250E 01 Q.29 4E0 1 0.288E 01 0.288E 01 0.368E01 0.325E 01 0.320E 01 0.44 1E01 0.351E 01 0.344E 01 0.51 5E01 0.364E 01 0.358E 01 0.58 8E-01 0.364E 01 0.359E 1 0.662E01 0.353E 01 0.352E 01 0.735E-01 0.335E 01 0.336E 01 0.882E•01 0.288E 01 0.292E 01 0. 103E 00 0.237E 01 0.243E 01 0. U8E 00 0. 193E 01 0.199E 01 0.13 2E 00 0. 157E 01 0.162E 01 1 4 7E 00 0.128E 01 0.1 32E 1 0.22 IE 00 0.486E 00 0.500E 00 0.294E 00 ,21 8E 00 0.223E 00 0.368E 00 „ 11 9E 00 0.122E 00 0.441E 00 0„754E-01 0.769E-01 0.58 8E 00 0..389E-01 0.396E-0 1 0.73 5c 00 0„ 241E-01 0.246E-01

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246 LYMAN BETA PROFILE FOR HYDROGEN IC ARGON ELECTRON TEMPER ATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 A=0.40 R= 0.10E 00 TRATIO=1.00 XI=17.0 Zl = 1*0 Z2=17.0 DELTA STARK STARK+ OMEGA PROFILE OOPPLER 0.0 0.844E 00 0.122E 01 0.735E-02 0.906E 00 0.125E 01 0. 147E-0 1 0. 107E 01 0.1 34E 1 0.22 1E-01 0.130E 01 0.147E 01 0-294E-0 1 0. 156E 01 0.1 64E 1 0.36 8E-01 0. 182E 01 0.182E 01 0.44 IE-01 0.207E 0! 0.200E 01 0.51 5E1 0.227E 01 0.216E 1 0.588E-01 0.242E 01 0.229E 01 0.662E-0 1 0.253E 01 0.239E 01 0.735E-01 0.257E 01 0.245E 01 0.88 2E-01 0.254E 01 0.247E 01 0.103E 00 0.239E 01 0.236E 01 0. 1 I 8E 0.2I7E 01 0.21 8E 1 0.132E 00 0. 193E 01 0.197E 01 0.14 7E 0.1 70E 01 0.1 75E 1 0.221E 00 0.853E 00 0.889E 00 0.294E 00 0.443E 00 0.461E 00 0.368E 00 0.245E 00 0.255E 00 0.441E 00 0. 146E 00 0. 151E 00 0.588E 00 0.62 7E-01 0.645E-0 1 0.735E 00 0.335E-01 0.344E-0 1 LYMAN BETA PROFILE FOR HYDROGEN IC ARGON ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 A=0.40 R0.10E 13 TRATIO=1.00 XI=17.0 Zl = 1.0 Z2=17.0 DELT/1 L STARK STARK* OMEGA PROFILE DOPPLER 0.0 0.925E 00 0.1 33E 1 0.73 5E02 0.994E 00 0.136E 1 0.147E01 0.1 I 8E 01 0.145E 01 0.22 IE01 0. 143E 01 0.1 59E 1 0.29 4E01 0. 171E 01 0.1 77E 1 0.368E01 0.1 98E 01 0.1 95E 1 0.44 1£0 1 0.222E 01 0.213E 01 0.51 5E0 1 0.242E 01 0.228E 1 0.58 8E01 0.256E 01 0.241E 01 0.662E01 0.264E Oi 0.249E 01 0.73 5E•01 0.266E 01 0.254E 01 0.88 2E-01 0.259E 01 0.252E 01 0.103E 00 0.240E 01 0.238E 01 0.1 18E 00 0.21 6E Oi 0.218E 1 0. 132E 00 0. 191E 01 0.195E 1 1 4 7E oo 0. 167E 01 0.172E 01 0.221E 00 8 1 7E 00 0.852E 00 0.294E 00 4 1 7E 00 0.434E 00 0.36 8E 00 0.228E 00 0.237E 00 0.441E 00 0, 134E 00 0.139E 00 0.58 8E 00 Q.572E-01 0.588E-0 1 0.735E 00 0.3 07E-01 0.314E-01

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LIST OF REFERENCES 1. H. R. Griem, Plasma Spectroscopy (McGraw-Hill Book Company, New York, 1964) 2. M. Baranger, Atomic and Molecular Processes D. R. Bates, Ed. (Academic Press, Inc., New York, 1962), Chapter 13. 3. J. Cooper, Plasma Spectroscopy, Rep. Progr. Phys _29, 35 (1966). 4. E. W. Smith, J. Cooper, C. R. Vidal, Phys. Rev. 185, 140 (1969); J. Quant. Spectr. Radiative Transfer 10, 1011 (1970). 5. F. E. Irons, J. Phys. B_6, 1562 (1973). 6. V. A. Batanov, V. A. Bogatyrev, N. K. Sukhodrev, and V. B. Fedorov, Zh. Eksp. Teor. Fiz. 64, 825 (1973) [Sov. Phys.-JETP 37, 419 (1973)] 7. B. Yaakobi and L. Goldman, Bull. Am. Phys. Soc. 20, 1302 (1975). 8. George F. Chapline, Hugh E. DeWitt, and C. F. Hooper, Jr., UCRL Report No. 76272, 1974 (unpublished). 9. John T. O'Brien and C. F. Hooper, Jr., J. Quant. Spectr. Radiative Transfer 14, 479 (1974). 10. John T. O'Brien and C. F. Hooper, Jr., Phys. Rev. A5, 867 (1972). 11. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of Oneand TwoElectron Atoms (Springer Verlag, Berlin, 1957). 12. E. W. Smith, J. Cooper, W. R. Chappell, and T. Dillon, J. Quant. Spectr. Radiative Transfer ri, 1547 (1971); see also J. Quant. Spectr. Radiative Transfer 11 1567 (1971) 13. Gerhart Luders, Ann. Phys. (Leipzig) 8, 301 (1951). 14. H. R. Griem and P. C. Kepple (unpublished). 15. H. R. Griem, Spectral Line Broadening by Plasmas (Academic, New York, 1974) 16. Leonard Schiff, Quantum Mechanics (McGraw-Hill Book Company, New York, 1968). 17. E. W. Smith, Dissertation (University of Florida, 1966). 247

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248 18. T. W. Hussey, Dissertation (University of Florida, 1974). 19. Le Quang Rang and D. Voslamber, J. Phys. B8, 331 (1975). 20. J. E. Whalen, Dissertation (University of Florida, 1972). 21. James W. Dufty, Phys. Rev. 187, 305 (1969). 22. E. W. Smith, Phys. Rev. 166, 102 (1968). 23. John T. O'Brien, Dissertation (University of Florida, 1970). 24. C. F. Hooper, Jr., Phys. Rev. 165, 215 (1968). 25. C. F. Hooper, Jr., Phys. Rev. 149, 77 (1966). 26. Phillip M. Morse and Herman Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). 27. W. J. Swiatecki, Proc Roy. Soc. (London) A205 283 (1951). 28. C. F. Hooper, Jr., Phys. Rev. 169 193 (1968). 29. J. Auerbach, Lawrence Livermore Laboratory internal memorandum LPIG-77-34, February 7, 1977 (unpublished). 30. Bernard Mozer and Michel Baranger, Phys. Rev. 118 626 (1960). 31. M. E. Bacon, J. Quant. Spectrosc. Radiative Transfer _12, 519 (1972). 32. Hugh E. DeWitt, Low-Luminosity Stars (Gordon and Breach, New York, 1969), Paper III-2. 33. K. Grutzmacher and B. Wende, Third International Conference on Spectral Line Shapes, London (September, 1976). 34. H. R. Griem, preprint (March, 1977). 35. James W. Dufty and David B. Boercker, J. Quant. Spectrosc. Radiative Transfer 16, 1065 (1976) 36. John D. Jackson, Classical Electrodynamics (John Wiley and Sons, Inc., New York, 1967) 37. Kurt Gottfried, Quantum Mechanics (W. A. Benjamin, Inc., New York, 1966). 38. K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Winther, Rev. Mod. Phys. _28, 432 (1956). 39. L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Addison-Wesley Publishing Company, Reading, Massachusetts, 1958). 40. A. A. Barker, Aust. J. of Phys. 21, 121 (1968).

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249 41. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957). 42. W. J. Karzas and R. Latter, Astrophys. J. Suppl. 55_, 167 (1961). 43. John T. O'Brien, Astrophys. J. L70, 613 (1971). 44. Albert Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1965). 45. C. R. Vidal, J. Cooper, and E. W. Smith, National Bureau of Standards Monograph 116 (Boulder, Colorado, 1970). 46. Alexander L. Fetter and John D. Walecka, Quantum Theory of ManyParticle Systems (McGraw-Hill Book Company, New York, 1971). 47. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, London, 1970). 48. F. W. Vickers, University of Florida (unpublished). 49. F. E. Riewe, University of Florida, private communication. 50. R. L. Coldwell, University of Florida, private communication. 51. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965), page 198.

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BIOGRAPHICAL SKETCH Richard Joseph Tighe was born on July 6, 1946, in Columbia, South Carolina. He graduated from A.C. Flora High School in June, 1964. In June, 1969, he received the Bachelor of Science degree with a major in Physics from the University of South Carolina. He spent the summer of 1969 working in the Stable Isotopes Separation Division of Union Carbide at Oak Ridge, Tennessee. In September, 1969, he enrolled in the Graduate School of the University of Florida. From that time until the present he has worked toward the degree of Doctor of Philosophy. Richard Joseph Tighe is married to Janette Cornish Gervin. He is a member of the American Physical Society. 250

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C. F; Hooper, //Jr. Chairman Professor of physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. E. D. Adams Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. P. W. Vickers Associate Professor of Computer and Information Sciences This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June 1977 Dean, Graduate School