Spectroscopic diagnostics of laser produced plasmas

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Spectroscopic diagnostics of laser produced plasmas
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viii, 128 leaves : ill. ; 28 cm.
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Delamater, Norman David, 1950-
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Laser plasmas   ( lcsh )
X-ray spectroscopy   ( lcsh )
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Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1984.
Bibliography:
Includes bibliographical references (leaves 125-127).
Statement of Responsibility:
by Norman David Delamater.
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Typescript.
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Vita.

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Full Text















SPECTROSCOPIC DIAGNOSTICS OF LASER PRODUCED PLASMAS


BY



NORMAN DAVID DELAMATER






















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



UNIVERSITY OF FLORIDA


1984


















ACKNOWLEDGMENTS


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. L.A. Woltz and R.W. Lee for

many helpful discussions, and Dr. R.L. Coldwell for providing valuable

assistance in the numerical work and for providing the curve fitting

code, RLCFIT.

I would also like to thank Mr. W. Richardson for preparing the

figures and Mrs. S. Hill for typing the final manuscript.

Finally, I would like to thank my wife, Yvonne, for her patience

and understanding throughout my graduate school career.


ii


















TABLE OF CONTENTS



PAGE

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

LIST OF FIGURES. ....................................................v

ABSTRACT...........................................................vii

CHAPTER

I. EXPERIMENT................................................

II. PLASMA MODELS .............................................6

Local Thermodynamic Equilibrium....................9
Non-LTE Steady State...............................10
Atomic Processes.................................12
Model Calculations...............................16
Line Ratios........................................23
Satellite Line Models.............................. 25
Satellite Line Calculations.........................39
Higher Order Satellites..........................41
Temperature Diagnostics................ ...........50

III. SPECTRAL LINESHAPES IN DENSE PLASMAS....................54

Introduction...................................... 54
Stark Broadening ..................................56
Radiation Transport ................................58
Non-LTE Opacity Model for Two-Level Atom...........65
Line of Best Fit Method...........................69

IV. RESULTS...............................................74

Results of Line of Best Fit Method.................74
Satellite Line Results............................. 86
Temperature Diagnostics..........................111

V. CONCLUDING REMARKS .................................... 115











APPENDIX

RATE COEFFICIENTS AND ATOMIC DATA .......................118

Energies .......................................... 118
Autolonization and Radiative Decay Rates.......... 119
Collisional Rates..................................121
Radiative Recombination.........................123

REFERENCES....................................................... 125

BIOGRAPHICAL SKETCH .. ........................... .................128


















































iv


















LIST OF FIGURES


FIGURE PAGE

1 Ionization fraction calculations
(a) Ionization fractions for Ar plasma in LTE............18
(b) Ionization fractions for Ar plasma in
coronal equilibrium.............................20
(c) Ionization fractions for Ar plasma in
collisional-radiative steady-state...............22

2 Intensity ratio versus temperature for Ar
(Is3p-ls2) line to LyB...................................27

3 Intensity ratio versus electron density for
Ar intercombination to resonance line................29

4 Schematic energy level diagram showing bound
and autoionizing levels.................................31

5 Schematic energy level diagrams showing the
six helium-like and lithium-like autoionizing
levels included in the satellite line model.............35

6 Calculated satellite emission for Ar Ly-a
helium-like satellites.................................. 43

7 Calculated satellite emission for lithium-like
satellites of Ar (1s2p-ls ) resonances lines............45

8 Density dependent helium-like satellite
line ratio, R, vs N ....................................47

9 Density dependent lithium-like satellite
line ratio, R, vs N .................................... 49

10 Intensity ratio versus temperature for Ar ID
satellite to Ly-a......................................53

11 Non-LTE uniform slab opacity broadening of
Ar+7 Ly-a .......... .................................... 64

12 Non-LTE two-level atom opacity broadening of
Ar+17 Ly-a.......................................... .68

13 Idealized line of best fit.................................72











14 Densitometer trace of spectrum from shot 7497..............77

15 Lyman series lineshape fits to spectrum
of shot 7499........... ...............................79

16 Line of best fit results...................................83

17 Line shape fit of satellites and opacity
broadened Ly-a line of shot 7499.......................92

18 Deconvolution of satellite emission
using RLCFIT for Ly-a satellite of
shots 7497 and 7496....................................96

19 Satellite line ratio, R1 determined from
RLCFIT, versus standard peak width....................100

20 Measured helium-like satellite line ratios for
several shots plotted on theoretical
calculation for density determination..................102

21 Measured lithium-like satellite line ratios for
several shorts plotted on theoretical calculation
for density determination........................... 104

22 Deconvolution of satellite emission
using RLCFIT for lithium-like satellites
of shots 7499 and 7560................................ 108














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


SPECTROSCOPIC DIAGNOSTICS OF LASER PRODUCED PLASMAS


Norman David Delamater


December 1984


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



A detailed phenomenological study of the spectroscopic analysis of

laser produced plasmas is presented in this dissertation. High

resolution X-ray spectroscopic data have been obtained in pellet

implosion experiments using the 24 beam Omega laser system at the

University of Rochester Laboratory for Laser Energetics. Line

broadening theory was applied to diagnose plasma core temperatures and

densities from observed spectral line shapes; detailed atomic physics

models were constructed to diagnose core parameters from sensitive line

intensity ratios. Previous experiments have been designed to minimize

the effects of plasma opacity, which tends to distort both the lineshape

and line ratio diagnostics. This study attempts to systematically

observe the effects of opacity on the diagnostics by a self consistent

analysis of X-ray spectra obtained from implosions of argon and neon

filled glass microballoons differing in diameter, wall thickness, and

Ar/Ne fill ratio. The subsequent analysis then includes opacity effects

through the use of a radiation transport model for the spectral lines;











atomic physics models are studied and the final results systematically

compared with the experimental results.

Typically, the X-ray spectra from dense, high temperature plasmas

show prominent satellite lines arising from doubly-excited autolonizing

states which lie near the strong resonance lines of hydrogenic and

helium-like ions. These satellites are well resolved spectroscopically

and are seen to vary in relative intensity over the range of densities

achieved in these experiments. A model is presented which has been used

to calculate the density dependence for several important satellite line

ratios and hence to diagnose electron densities in these experiments.

For the first time, a detailed comparison is presented between these

satellite line diagnostics and a line broadening analysis including the

effects of opacity.

The results of this analysis have relevance for implosive and non-

implosive plasma diagnostics. For inertial confinement fusion plasmas,

the spectroscopic information represents the critical feedback

validating hydrodynamic and atomic physics models, as well as atomic

data. Current models of pellet implosions which couple hydrodynamics

and radiation transport are highly uncertain; the constraints placed on

such models by the present results are discussed.


















viii

















CHAPTER I
EXPERIMENT



Glass microballoons of various sizes were filled with mixtures of

argon and neon and imploded using the 24-beam Omega Nd-glass laser

system at the University of Rochester Laboratory for Laser Energetics.

Experimentally observed X-ray spectra and pinhole images were analyzed

to determine plasma core parameters and to study the effects of opacity

on spectroscopic diagnostics. The proportion of argon in the fill was

varied from 5% to 100% with the intention of systematically observing

the effects of opacity on radiation transport of the principal series

lines of hydrogenic and helium-like argon. Electron temperatures and

densities, and ion densities were determined by both detailed fits of

theoretical lineshapes to experimental line profiles, and by the

measurement of various line intensity ratios which are shown to be

temperature or density sensitive. The prominent helium-like and

lithium-like satellite lines situated respectively on the red wing of

the hydrogenic and helium-like resonance lines were observed as

additional plasma diagnostics; the results were compared with the

previous methods.

The shots analyzed provided high resolution time-integrated

spectral data in the region 3 4 keV region covering the range of

hydrogenic and helium-like argon emission. Laser pulses of 1.06

microns, about 100 picoseconds in duration, and with an energy of

approximately 600 joules per pulse were used to symmetrically irradiate

1











the glass targets. Pellets of sizes 100pm (diameter) x 1pm (wall

thickness), 100pm x 3pm, 100pm x 5pm, and 150pm x 2pm were filled to 10

atmospheres total pressure with argon/neon fill ratios of 5/95, 25/75,

50/50, and 100/0 in order to systematically control the opacity in the

argon lines. Table 1 provides a list of shot parameters for those shots

providing useful data which were analyzed. A fairly complete variation

of fill ratio has been obtained for the 100pm x 1pm, and the

100pm x 3pm targets, while less complete data were obtained for the

other targets due to various experimental problems.

A flat X-ray crystal spectrometer was used to record the time-

integrated plasma emission. The instrument resolution is determined

using Bragg's law,



AE
E AG cot 0B; 1-1



if the angular spread AO is determined primarily from the finite source

size, then



AE AX
E =-cot 0, 1-2



where 0B is the Bragg angle at energy E, AX is the linear source size,

and L is the source to film distance. Using the source sizes determined

with the X-ray microscope images, and the known crystal spacing and

experimental geometry, the instrument resolution is determined to be

approximately 3.5 eV. In the analysis, the theoretical lineshapes were

convolved with a gaussian shape of full width at half maximum equal to

3.5 eV to approximately account for instrumental broadening. This was










not a major effect on the lineshapes studied since in this experiment

most of them were already considerably larger than this instrumental

width.

The spectra were recorded on Kodak 2497 film, and the film was

microdensitometered at the Lawrence Livermore National Laboratory. The

digitized neutral film density versus film distance was converted to

relative intensity versus energy in order to accurately compare the data

with theoretical lineshapes. The energy scale was established using

both argon and nearby impurity lines from the glass shell constituents

whose energies have been previously calculated and reported in the

literature.1,2 _A linear least squares fit was performed to determine

the dispersion (eV/mm) in each spectral region. It was assumed that at

the densities attained in these experiments line shifts due to plasma

effects were negligible so that the line energies calculated for the

isolated ions would be appropriate. The Kodak film used has been

absolutely calibrated and the results reported in the literature.3,4 It

has been shown that for sufficiently small film density, the film

density is linearly proportional to exposure (intensity x time). As the

film densities in this experiment do fall within the linear range and

only relative intensities are used in the analysis, the conversion from

film density to radiation intensity is linear and absolute intensities

are not calculated.

Experimentally determined lineshapes and line intensity ratios must

be compared with theoretical calculations in order to produce reliable

plasma diagnostics. In Chapter II, several detailed plasma models are

presented which were used to characterize the laser produced plasmas in

this experiment. The models provide steady state ionic population











densities; in turn these then determine a number of line intensity

ratios which depend on electron density, temperature, or both. Included

in these plasma models are an analysis of autoionizing level populations

which give rise to satellite line emission. Chapter III considers in

detail the shapes of spectral lines emitted from dense, laser produced

plasmas. Line broadening theory is used to calculate the intrinsic

shapes for the principal series lines from hydrogenic and helium-like

argon. These shapes must be modified in order to account for the

radiation's escape from the plasma. An approximate solution to the

problem of line transfer is presented. A method is presented which

systematically compares theoretical lineshapes with all observable lines

in a spectral series in order to determine electron and ion densities

which are consistent with the opacity model.

Chapter IV presents the principal results of this analysis. Line

broadening theory and line intensity ratios are used to predict plasma

core parameters for a number of plasmas covering a range of densities,

temperatures, and opacities. Chapter V discusses the significance of

these results with respect to current models of pellet implosions.





























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CHAPTER II
PLASMA MODELS



During the last decade, high power laser systems have been used to

drive pellet implosions in an attempt to produce plasma conditions

sufficient to generate significant nuclear fusion reactions in the

component hydrogen (D-T) fuel. Seed gases such as neon or argon have

been used to provide diagnostic information of plasma core parameters

and can modify the implosion dynamics by providing a mechanism for

radiative cooling. An analysis of the line spectra emitted from these

plasmas generally requires some hydrodynamic model for the pellet

implosion and a detailed atomic physics model possibly including

radiation transport to predict spectral emission.

Hydrodynamic simulation of spherically symmetric pellet implosions

can be used to produce detailed predictions of the plasma development

during the implosion and predict total ion densities, ion temperatures

and electron temperatures as a function of space and time. Currently

the detailed atomic physics models are computed in a post-process

procedure using the hydro code results as input data for the overall

ionization balance calculation. This procedure has been used5'6 to

predict the overall spectral emission with some success for certain well

characterized implosions. Major uncertainties exist in the models of

laser energy deposition in the target, electron transport through the

plasma, generation of non-thermal high energy electrons, and plasma










instabilities. When these effects become important, code predictions

can become inconsistent with the observations.

The hydro code simulations have shown that laser driven implosions

of microballoon targets result in "exploding pusher" or "ablatively

driven" implosion (or an intermediate type). The exploding pusher

target generally has a thin (4 1 pm) glass shell which is subjected to a

short (100 ps) intense laser pulse. The absorption of laser energy

results in the production of hot electrons which instantaneously explode

the shell, sending half the material outward and half the material

inward thus compressing the gas fill. Due to the effects of the hot

electrons and shock waves the gas fill can become preheated thus

reducing the maximum compression. Prior to the disassembly phase, a

stagnation phase is achieved in which the gas fill temperature is

relatively high (several keV) and density low (< liquid DT) compared to

the ablatively driven implosion. The ablative implosion is more

desirable since shock and hot electron preheating are minimized and a

slower compression is driven by the thermal conduction front, with the

achievement of higher compression and lower temperature at stagnation.

An ablative implosion is partially achieved by increasing shell

thickness such that the hot electrons cannot explode the shell. Also

reducing the laser intensity or operating wavelength has the effect of

decreasing the hot electron production and creating a more ablative

implosion.7 In the experiments to be described in this dissertation it

is expected that implosions will be obtained which are of the "exploding

pusher" type, however the thick shell targets should have an ablative

character and be of an intermediate type.











An important prediction emerging from hydro code simulations is

that spherically symmetric implosions reach a stagnation phase and the

expected range of temperature and density is such that line emission

from hydrogenic and helium-like argon ions should be prominent during

this phase. This implies that a steady-state uniform plasma model can

be used to predict the spectroscopic emission. Such models have been

previously applied successfully to predict the behavior of neon-filled

microballoon,89 and comparisons to a more detailed model show only

small differences in the diagnosed parameters.6,10,

While a steady-state uniform plasma model may be adequate when

time-integrated spectra is to be described, the more detailed

simulations will be required when time-resolved spectra is analyzed.

Equipment now becoming available is able to provide time resolution

of ~ 10 picoseconds compared to implosion times of ~ nanoseconds. These

new, time-resolved observations should provide critical tests of the

hydrodynamic codes and their inherent uncertainties.

In the experiments described here, since time-resolved spectroscopy

is not being used, a steady state atomic physics model should be

adequate to characterize the conditions in the plasma during peak

emission. In particular, the spectral line emission will be analyzed to

determine an electron temperature, Te, electron density, Ne, and ion

density, Ni, characteristic of the plasma at peak emission. This

analysis considers both lineshape and line ratio diagnostics. While

lineshape diagnostics are primarily dependent on electron density, line

ratio diagnostics depend on both Ne and Te through the ionic level

populations, the ratio of two spectral lines being given by












I N hv A
ij = i hij ij
I N hv A 2-1
mn m mn mn


where N indicates upper state level populations, hv is energy of the

transitions, and A's are Einstein coefficients for the transitions. The

level populations must be calculated from an atomic physics model.



Local Thermodynamic Equilibrium



The simplest model to assume is that of local thermodynamic

equilibrium, or LTE. Particle collisions are dominant, rates of any

collisional processes are exactly balanced by their inverse process, and

ionic populations are functions of the local values of electron

temperature and density. In fact all local ionic population densities

are given by the Saha-Boltzmann equation



NN = (6.04 x 1023) NZ z+ T3/2 kT2-2
eg g gz eV



which is valid for any two successive ionization stages where the g's

are the statistical weights and Iz is the ionization potential of stage

Z.

It is known that LTE is not a good approximation for all the ionic

levels,6,9,10 but that it may be valid for some of the upper ones. An

approximate criterion for a given level of principal quantum number n to

be in LTE with the ground state of the next highest ionization stage

is12












18 Z kT 1/2 -3
N > 7 x 108 7 ( / cm3 2-3
e 17/2 2 E
n Z EH


which is obtained by comparing the total radiative decay rates from

level n to the collisional deexcitation rate into level n. The

condition for the collision rate to be dominant gives the above

relation. The assumption of LTE for some of the levels greatly

simplifies the calculation of total level populations with a more

detailed model.



S.Non-LTE Steady-State Models ..



A more general approach to determine ionic level populations in a

plasma is to assume that overall plasma conditions are changing slowly

compared to collisional and radiative timescales so that a steady-state

condition is reached. Then the rate equations governing the population

in each level of the model can be solved for steady-state populations,

assuming all relevant collisional and radiative rate coefficients are

known. This model, also called a collisional-radiative model, contains

both the LTE model and the corona (low-density) model as limiting cases.

Consider the total populations Nj and Nj+1 in two successive stages

of ionization, j and j+1; then the solution of the rate equations gives

for the population ratio,


N S.
_= _j 2-4
N. a +D + N 2
j j+1 j+1 e j+1


where S. is the total electron collisional ionization rate from
J











j + j+l, aj+ is the total radiative recombination rate into j, Dj+1 is

the total dielectronic recombination rate into j, j+. is the total

three-body recombination rate into j.

The three body recombination process is just the inverse of

collisional ionization and is related by detailed balance as



*
N.S N 8.
S Nj+1 e j+1


where the indicates LTE population. It can easily be seen from the

last two equations that in the limit where



N Bj >> a. + D
e j+1 j+l j+1


that the population ratio becomes equivalent to that given by the Saha-

Boltzmann equation, the LTE result. Also, in the opposite limit where


N < e j+1 j+1 j+1


the conditions of coronal equilibrium are attained, where the ratio is

independent of electron density, but still dependent on temperature.

A useful plasma model for spectroscopic analysis must contain a

number of levels within each ionization stage. In practice, levels with

principal quantum n < 6 usually need only be considered since the rest

may be grouped with the next ionization stage and assumed to be in LTE

with the ground state using the criterion of equation (2-3).

An effective cutoff of principal quantum number is obtained by the

lowering of the ionization potential of an ion in a plasma, which leaves











as unbound those states with n > nmax. This effect is actually a

reduction of binding energy of an ionic electron due to the presence of

nearby ions and electrons in the plasma. It has been calculated in

13
several limiting cases using Debye-Huckel and Thomas-Fermi theory3,

though the results are uncertain in application to laser produced

plasmas in current and future experiments.

The rate equation for the ith level will have the form,


dN.
dt = -Ni (depopulation rates) 2-6

+ E N. (population rate from j)
J


The system of equations for all the levels is solved in the steady-

state, i.e. the time derivative is set equal to zero. Given the

electron temperature and density, the relevant rates in equation (2-6)

can be determined and the system of equations solved for the

populations.



Atomic Processes



The conditions generated in many laser-driven implosions are such

that many important hydrogenic and helium-like spectral features of the

gas fill are easily observed. Thus appropriate atomic physics models

generally include explicitly the bare, hydrogenic, helium-like and

lithium-like ionization stages, and group the remainder in one composite

level, whose population is usually negligible. This fact, and the

criterion of equation (2-3), is useful in keeping the total number of











levels in the model small and the solution to the equation (2-6)

tractable.

The following processes must be included in the collisional-

radiative model (for ionization species, Z, and ionic level i):

a) electron impact excitation (de-excitation)




e + N +e + N i


b) electron impact ionization (3-body recombination)




e + N + N + e + e
z,i + z+1



c) photolonization radiativee recombination)




N + hv + N + e
z + z+1


d) spontaneous emission


N + N + hv
z,j z,i


e) Dielectronic recombination (autoionization)


A**
N +e +N
z+1 + z


doubly excited


**
N +N + hv
z z,i










Appendix A gives additional details on the specific rates which

have been used in the calculations reported in this dissertation.

Hydrogenic approximations are used for photoionization rates, new

calculations of collisional excitation and ionization rates are

available in the literature;14,15,16 total dielectronic recombination

rates were calculated using the method discussed in the next section

with autoionization rates available in the literature.1

An additional process which must be included in an optically thick

plasma is photoexcitation; this is generally most important for

resonance lines. The photoexcitation rate between levels with n=l and 2

is given by



hv
R= B NJ f L(v) dv dS 2-7
4x 12 1 (v,Q)


where N1 is the ground state density and B12 is the Einstein coefficient

and L(v) is the intrinsic lineshape.

The fact that the photoexcitation rate depends on the radiation

field greatly complicates the analysis since the radiation field depends

on level populations through the transfer equation, which in the two-

level atom approximation is,


hv
SVI = [N2A21 (N1B12 N2B21)I] L(v) 2-8



In principle, it is necessary to couple the transfer equation to

the level rate equations to obtain a consistent solution for populations

and the final emergent spectral radiation. In practice it is possible

instead to modify the rate equations using the escape factor










17,18
method.17,18 In this approach, terms in the rate equations proportional

to spontaneous radiative decay in the optically thick transition are

multiplied by an escape factor, g, where g ranges from 0 to 1. The

escape factor is defined to approximately include the effects of

photoexcitation in the rate equations. A rate equation for level i can

be written


dN
dNi hv
= -NA + N.B. l L dvd 2-9
dt -N1j + N B f I(v,) L(v) dv dQ 2-9
+ other terms



since the Einstein coefficients are related we can group two of the

terms in equation (2-9) and obtain


dN.
dt = -NiA g (TO) + other terms 2-10


where g(T0), the escape factor, is given by


Ni mi 2
g(TO) = c I L dv 2-11
0) i N. m. 3 (v) (v)
J 3 2hv0


With the assumption of a uniform plasma source and a lineshape

determined by the Doppler effect, Holsteinl7 obtained the result



1
g (TO) = 2-12
T0 /Til0

where TO is the optical depth at line center. In this approximation,

the escape factor must also be included in the calculation of the

emitted intensity (equation (2-11)).













Model Calculations



The methods of the previous sections have been applied to calculate

ionic populations for steady-state argon plasmas under conditions

expected in the experiments considered in this dissertation. Figure 1

gives some results for total hydrogenic and helium-like populations as a

function of temperature under conditions of LTE, coronal, and

collisional radiative equilibrium with the electron density fixed at

1 x 1023 cm3. Coronal equilibrium rates were obtained from Shull and

von Steenberg19 and collisional-radiative calculations were done at

Lawrence Livermore National Laboratory using the code RATION.20 The

code RATION was modified to include opacity effects of the resonance

lines approximately in the rate equations with an escape factor. These

results are included in Figure 1. It can be noted that the inclusion of

opacity effects in the collisional-radiative model tends to drive the

populations away from the coronal result and towards LTE.

More complete, time dependent models are currently being

developed5'6'11 which couple hydrodynamic results with the rate

equations and transfer equation. Specifically, the hydro code produces

total ionic density and temperature as a function of time and space for

a particular implosion. This information can then be used as input to

the rate equation and transfer equation which can then be iteratively

solved to yield the final emitted, time integrated spectral

intensities. Currently these approaches have not directly coupled the

radiative transport with the hydrodynamics but use LTE opacities during

the hydrodynamics calculations and assume that implosion dynamics are























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not sensitive to non-LTE effects. While this assumption may be adequate

for D-T plasmas with small argon impurities, it is apparently not true

in the present case where the targets are argon/neon filled

microballoons. The results using the hydrodynamic code LILAC5 and a

post-processing atomic rate equation package were inadequate to model

these experiments. In general, the LILAC code did not treat the

radiation effects in these optically thick plasmas correctly and

produced results grossly inconsistent with the observations.



Line Ratios



Line intensity ratios can in general be dependent on electron

density and temperature. Suitably chosen line ratios which are

primarily dependent on either density or temperature can also provide

useful diagnostics.

To form a temperature sensitive line ratio, we consider two

transitions to the same ground state; for example Ly-B and Ly-y lines:



31 N3 31 h31
-- 2-13
I N A hv
41 4 41 h41


If the coronal approximation is valid, collisional excitation from

the ground state is balanced by radiative decay for each transition and

we can write



I31 C13 hv31
=_ 2-14
41 14 h41











From the form of the collision rate, C, given in Appendix A it is

seen that the above ratio is dependent on temperature and independent of

electron density. This is also the case in LTE [equation (2-1)].

To obtain a density dependence, we can consider two lines, one of

which has a long lifetime (low A value) such that the process of

collisonal de-excitation can become important. For two optically thin

lines, the rate equations for the level populations are




NAi. = N N C
i e g gi 2-15
NkAkg + Nk NeCk = N N C
kkg kek-g e g g+k



where the C's are collisional excitation and de-excitation-rates. It is

evident that the intensity ratio Iig/Ikg is a function of electron

density, and possibly temperature. An example of this situation occurs

in the helium-like argon resonance line of this experiment. The

resonance line transition is



is2p 1p + ls2 S



and can be compared with the intercombination line



ls2p 3P + ls2 lS



where the intercombination (ic) line has a low transition probability

(dipole forbidden) and fulfills the conditions of equation (2-15). As

the resonance line is affected by photoexcitation, it is also convenient











to consider the more nearly optically thin line ratio I(ls3p-ls )/I(ic)

which is also density dependent, and slightly dependent on

temperature. In addition, it is useful to compare lines from adjacent

ionic species and use a collisional-radiative model in the analysis.

For example, the ratio I(LyB)/I(HeB) compares a prominent hydrogenic

line with a permanent helium-like line, neither of which is greatly

affected by opacity. Figures 2 and 3 give some results of line ratio

calculations using a collisional-radiative model. These will be

referred to in chapter IV when comparing to the experimental data.



Satellite Line Models



Satellite lines arising from doubly-excited autoionizing levels are

prominent in high temperature plasmas. These lines are usually

optically thin and are useful as additional diagnostics of temperature

and density. Their presence also indicates the importance of an

additional atomic process, dielectronic recombination, which must be

included in plasma models. This process was first recognized in an

astrophysical context21'22 and the importance for laser produced plasmas

was soon recognized.23'24

The process of satellite line formation can be explained by

reference to Figure 4, which gives schematically some ionic levels for

three different ionization stages. Dielectronic recombination is a two

step process: first, a doubly excited atomic state is formed by direct

electron capture from a ground state or excited state of the next


ionization state, i.e. for Argon,
































1)

S






0)



Q)
ucm









-H 4
caa


ca



aow
U) 0

ca l








0W
.4-J






C >


4J1
co





(0 -4
'410
5-ur
Cd



C(











0)
4JCO










M--4

0)
-4
0 41


co
$4
41



01


Q))


















































N
(
0


II
w
nz


-0
d Bol


0--
o >
0Oa





























0








o4





a
c12






.r4
"-0





c to
Kd
ai

















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4J





w







4-1
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ca


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ci~
ca










w


















o
0r
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4

co
S:
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"0




































0)
Z


























a)
41
u

bO




Q ) r4




-4W 0
U,;




-' 4-1
4JO d





bo z
.r4








N O




0 )0
4 =-4.
Qa
a-L
bo r





C l,















U) H4
-4 a)
cef
N 41





















a) -
o













U, C l,
ca
EP









-4



cu 9
> U
0 4
ca Q)
p Q)
ho U





cu -
>14J$




ril























P4
cn CM
C4
Cml Cl CN






i lii


-4
CCl

l -4


CN CN CC

a. P
I t
C 1I I

------Z5


p4
w) o.
I) C


I "


0)
-4 *-
+ '-4
'-I


P4 U)
C4 CM
-z C. Ul
cl CN CM
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a)

'-1
- I
+ *H


4U


'-4

Cl


I
I



al
* i1

I
1




I





CO r






+-
+ CJ


h~l
1W7


p-1
CM

c N






I











+17 +16
N(Ar ) + e + N(Ar ,)
gs + 2Z21V


The doubly excited 2121' state may autoionize and return to the

initial state, or emit a photon and form a bound state,


+16 +16
N(Ar 2 ) + N(Ar ) + hv
2Z2z 1s2z


The net effect of these two steps is thus a recombination process

leading to a change in ionic species from Ar+17 to Ar+16

In a steady-state low density approximation the rate equation for

the doubly excited level population Ni is given by balancing electron

capture with radiative decay and autoionization,



Ni ( + E Aik) = DiNeNH 2-16
k


where .i is the autoionization rate, Di is the electron capture rate, NH

is the initial ground state level population, and Aik the Einstein

transition probability for any possible radiative decay. Autoionization

rates have been calculated from atomic structure codes for a number of

helium-like and lithium-like autoionizing levels (see Appendix).1

Electron capture rates can then be calculated from the principle of

detailed balance. Using equation (2-1) and assuming capture from a

hydrogenic ground state into a helium-like doubly excited level,


gi/gH -3/2
Di 21 V i. exp(-Ei/kT) 2-17
6.04 x 10










where Ei represents the energy of the doubly excited level above the

ionization limit (see Figure 4). Using the previous two equations it

can then be seen that the intensity of a satellite line from level i is

proportional to


TN
I. e exp (--E/kT) 2-18
13 F + E Aik
k


Due to the exponential in the above equation, kT must be large compared

to Ei for satellite emission to be prominent. The above equation also

predicts that the ratio of two satellite lines, in the coronal

approximation, is



i = ij exp(-Eim/kT) 2-19
mn mn


where


g 1 A, hvi
g i Aij ij
ij ri + E Aik
k


and Eim = Ei Em. In this approximation the relative satellite

intensities are independent of electron density.

The satellite line ratio in the high density, collision dominant

limit can easily be written down, since the populations are in the ratio

of the statistical weights of the levels



1i = -ij exp(-Em /kT) 2-20
I r im
mn mn


where













rj = g Aij hvij



It was first noticed by Seely24 that a particular group of helium-

like satellite lines (from the 2p2 3P level) was more intense than the

coronal model prediction. The coronal model is also not valid for

lithium-like satellite lines of the helium-like resonance line.

A more general collisional-radiative model for the autoionizing

level populations will yield satellite line intensity ratios as a

function of density and yield the previous results in the low and high

density limits. The model will include the following important

collisional processes (see Figure 4):

i) electron induced collisional excitation and de-excitation among

the 2222' (or ls2Z2Z') doubly excited levels.

ii) electron induced collisional ionization from the 221V' levels

to 21 (or 1s22) state of the next ionization stage

iii) electron induced inner shell excitation from bound helium-like

(lithium-like) levels.

A method similar to that of Jacobs and Davis25 was used to set up

and solve a system of rate equations for the helium-like and lithium-

like level populations giving rise to the prominent argon satellite

lines seen in this experiment. All the previously mentioned collisional

and radiative processes were included in the model in order to predict

overall satellite line emission and to compare with the experiment.

The doubly excited levels included in the analysis for both helium-

like and lithium-like satellite lines are shown in Figure 5. In either

case, six levels were included in the system of rate equations, which



















2p2

2p2


2s2p p





2s2
S


2p2


2s2p


FIGURE 5
Energy Level Diagram

(a) The six helium-like autolonizing levels
included in the satellite line calculations
are shown on a relative energy level diagram.















Is2p2 2
S


Is2p2
2D


1s2p2
P



ls2s( P)2p

2p
ls2s('P) 2p
"P


1s2s2


FIGURE 5 (continued)
(b) The six lithium-like autoionizing levels
included in the satellite line calculations are
shown on a relative energy level diagram.











were then solved for doubly excited level populations in terms of the

hydrogenic (helium-like) ground state population. It was assumed that

the population within each group could be distributed among the

sublevels of angular momentum j according to the statistical weights,

(2j+l). Atomic data for the particular argon doubly excited levels and

satellite lines were obtained from the literature, and the specific

forms for the collision rates are given in Appendix A.

The steady state rate equation for the ith doubly-excited level can

be written in a more detailed form than that given in equation (2-6):



E Q N D N N + TB N N
i (i,i') = ig g e+ ,b b e 2-21

+ Z C N. N
ex(i,j) j e


where D(i,g) is the electron capture rate from the ground state of the

higher ionization stage, TBi,b is the three body recombination rate from

the n=2 bound state of the higher ionization stage, Cex(ij) is the

inner shell collisional excitation rate from lower bound levels. The

rate matrix Q(i,i') has diagonal elements which represent the total

depopulation rate of level i due to autoionization, radiative decay, and

collisionally excited transitions, i.e.



Qi = E NC + Ti + E A + N CI 2-22
(i,i) e (i+i') if e (i+b)
i'fi f


where CI is the inverse process of TB, and ri is the inverse of D. It

is the collisionally induced mixing transitions among the doubly

excited levels which is particularly responsible for driving the

populations away from coronal equilibrium and toward LTE values.











The off-diagonal terms of Q(i,i') represent collisional excitation

or de-excitation transitions from other doubly excited levels, i', into

the level i,



Q(ii = -N C 2-23
(11') e (i'+i)


The solution for level population Ni of the system of rate equations

defined in equation (2-21) is given by




N = E Q1 [D N N + TB N N ]
i (i,i') [D(i',g) g e (i',b) b eN2
ij 2-24
-1
+ E EQ1 .C N N
i j (ii') ex(i',j) j e



which depends on the ground state, N and first excited state, Nb, of

one ionization stage and the excited bound states, N., of the next lower

ionization stage. The relative populations, Nb/Ng and Nj/Ng, were

calculated using less detailed models of the type previously described

and also using several experimentally obtained line ratios. Thus the

level populations can be written in terms of N ground state

populations. The relative intensities of several satellite lines will

then be independent of N .

It should be noted that this solution is guaranteed to reproduce

both coronal and LTE populations in the appropriate density limits.

Also the total dielectronic recombination rate may be defined as

E Ai. N,
ij 2-25
aD N N 2-25
e g











where Ni is given in equation (2-24). We can then write



-1
a= A Q D 2-26
D ,i',j ij (,) ( g) 2-26


This process is an important recombination mechanism for the total

ionization balance in a high temperature plasma.



Satellite Line Calculations



The methods of the previous section were used to calculate the

total relative satellite line intensities for argon plasmas over a

density range covering the transition from corona to LTE conditions.

Both the helium-like 222.' satellites of the Lyman alpha resonance line

and the lithium-like ls2k22' satellites of the helium-like resonance

line were included in the calculations. Equation (2-1) is used to

calculate the relative line intensities using the level populations from

equation (2-24).

The overall satellite emission is composed of a blend of many

transitions from the group of six doubly excited levels. Using the

calculated populations and the atomic data for each transition the total

relative satellite emission was obtained by summing the intensities over

all frequencies of satellite line emission for all lines and normalizing

to the intensity of the strongest satellite line. Each individual

satellite component was assumed to have, for simplicity, a Lorentzian

shape whose width was determined primarily from source broadening, which

was shown in Chapter I to be approximately 3.5eV. The Lorentzian shape


has the form













r
L 2 2 2-27
(v-v0) + (r/4-i)


where r is the full width at half the maximum intensity. Voigt profiles

were also assumed for satellite line shapes but these in general did not

prove to fit the data as adequately as did the Lorentzian shapes.

Figures 6 and 7 present the overall blend of satellite emission for

the helium-like and lithium-like satellites in argon plasmas and show

both low density (corona) to high density (LTE) emission. It can

readily be seen that the emission illustrated in these two figures

changes dramatically; this fact will be used as a density diagnostic for

experiments analyzed in this dissertation. Each of these figures is

labeled giving the component transitions of the blend in spectroscopic

notation. The usual spectroscopic notation in LS coupling is used

throughout, i.e. (2S+1)Lj.

The increased emission with increasing density, shown in the

helium-like satellites of Figure 6, comes from the 2p2 3P levels which

have low autoionization rates (see Appendix for selection rules). These

levels become populated by collisional transitions from other doubly

excited levels, a process which becomes dominant as density increases.

The calculations show that this effect will be noticeable precisely when

the condition expected in conditions of this experiment are realized.

The satellites lines of Figure 6 can just be resolved into four

major component blends. The resolvable ratio of these turns out to be a

good measure of the density effect just described. Thus the methods

described in the previous section were applied to calculate the ratio,

R,












23 3 3 3
R = I(2p P ls2p P) + I(2s2p P ls2s S) -
S1 1 2-28
I(2p2 D s2p P)


Figure 8 shows the smooth transition of this ratio from the coronal to

LTE limits. Also shown are the effects of inner shell excitation which

are predominant at low density and the effects of collisional

ionization, important at very high density.

The density effect apparent in Figure 7 for the lithium-like

satellites can also be measured by a resolvable line ratio. Using the

shorthand notation introduced by Gabriel,22 the density-dependent line

ratio is .



R = I(abcd) + I(qr) 2-29
I(jkl)


where a,b,c,d refer to transitions of the type 1s2p2 2P 1s2 2p 2P; q,r

refer to transitions of the type ls2s(IP)2p 2P Is2 2s 2S; and j,k,l

refer to transitions of the type 1s2p2 2D 1s22p 2P. This line

intensity ratio was calculated with the satellite line model described

in the previous section and the results are shown in Figure 9.

The results given in Figures 8 and 9 will be used in Chapter IV

where the respective line ratios are measured from the data and compared

to the calculations for a density diagnostic.



Higher Order Satellites



The analysis of the previous section considered only n=2 doubly-

excited levels. Higher order satellites have previously been observed




























oO

4J
Sn

o c


a 41


41






-4



0 0
0

'TJ










4 o











Q)
0
o*

+

".s

T-
^^

it
-s-s



a,
^^
^^
0,


-4c
N a



I |


4 IX
m
FCM C
0. 0
CM C


-4
CM
N C
m T
I I

E M

-N4
CM CM


,-, -, c- -N I-,
4 CM M Ir in
'^^ ^' *^' \^ _







43












Q- 0

Is

- I I























un JOl!qj) !su


Li2?
0)


to C



























m
C:

10


cO a)
5 "

1>- I
a)
Q)
k lu

co $4 Q)-
*' Q) QO C )
a), a

CI

CO 0C4 o

cl C I Ca

SHN C 1
cU~(N (N
-4 41 e m 0
a) -r(q o

(U i) hO 1 u) ^
-) cu to
4 I CM I)
I(: (oN 4.J C
S4J 04 4 0l
r~ 0 (N eq WN r-i .
() a) A ,-4 4) c
a) a) m C14 C 41 41
(N 0004 1 cc 0
(N ca. C-4- to
M -4 04N C m C04 10
-', +N 24 w ( M

aU c CU C M M I
u (N -4 '. o 0


El co cn U
41 rz C- 04
0 (. a. )



0^ 0 Q) ca 0.
U,0 e- wi i -
-o p s
4-J Ul, 0 4J

Q* ) co -i ) 3 o
CJ C~~ml, a










O j ^ / ^ r4 U ,1
0) n
CO 4J


0 -4 n 0u

Q)





E-a


























CM M CU CU



.O Io lo




I I .:
vl~


ro..."**.*.: C..--:

S ....-.
***...
.--
N


(sl!un AjDJ!qjDo) 41!sualul


CJ


-M:













ro



co


LLI












r4
ro0




O



























! o
0






0,

00
0




*1-4
a0
co





4-
c





0


Ci)
Sg



0 4-
















co
0i






<"











1
P.p
U
Jd
(ne




































m
ae,
-H Q)




10)

(N


00



cUO-


V4-a-

U



u ~r
0Q)
-H4-4



CO~x 0)
4JI 4-4

CO







41 (1)
0
-,H 41





r.4 CO
Q)O
cu U








-14-4~
4-J
-4 0

0
4-d
COO








cn~
E)U





4u-i

0
0)4


































Z

0











which arise from n=3 and higher levels. In Figure 4 some of the n=3

levels are shown, namely 2.3S' and ls2Z3Z' in the helium-like and

lithium-like stages, respectively. These levels have several channels

for radiative decay. For example, the 223V' level may follow



21m3' + 1s3V' + hv



or



203V' + 1s29 + hv



to form a satellite of Lyman a or Lyman B, respectively. Generally, the

higher order satellites of the resonance line will not be resolved and

hence will tend to broaden the resonance line.

The populations of these higher order doubly excited levels can

usually be taken to be close to the LTE limit, as seen from the

criterion of equation (2-3). Then the ratio of two satellite line

intensities arising from these levels is given by equation (2-20), the

high density limit. As will be shown in Chapter IV, these high order

satellites are seen in the experimental spectra.



Temperature Diagnostics



It has already been shown in equation (2-18) that the presence of

satellite line emission already implies a high temperature. The

intensity ratio of a satellite to resonance line provides a useful

additional temperature diagnostic. Consider an optically thin resonance











line whose upper state population is determined by collisional

excitation from the ground state. For a satellite line in the coronal

approximation, it can be shown that


I hv A D
sat s s s
-= -2-30
I C F + E A. -30
Res 12 s + Ais
i

where C12 is the collisional excitation rate for the resonance line

upper state; D and F are the electron capture rate and autoionization

rate of the satellite level. It is evident that this line ratio is a

function of temperature and atomic constants, and is not dependent on

electron density. This ratio was calculated for the strong satellite

transition,



21 1
2p2 D + ls2p P



and compared to the Ly-a resonance line and is shown in Figure 10.

Although the resonance line is not optically thin, this ratio is still a

useful temperature diagnostic (see Chapter IV).











































FIGURE 10

Intensity ratio versus temperature for the Ar
2p D ls2p P satellite to the Ly-a resonance
line of Ar the model described
by equation (2-30) was used.









'D Satellite to Lya Resonance


500 700 900 1100 1300 1500
T(eV)

















CHAPTER III
SPECTRAL LINESHAPES IN DENSE PLASMAS



Introduction



Theoretical shapes of spectral lines emitted from dense plasmas are

determined by the convolution of several processes. These include

natural broadening of the line due to the finite lifetime of the atomic

states, Stark broadening due to the collisional processes between the

radiating ion and electrons and other ions, and Doppler broadening due

to thermal motion. Stark broadening theories yield lineshapes for

transitions in ions whose energy levels are perturbed by the electric

fields due to dense plasma constituents. Under conditions generated in

current laser-produced implosion, Stark widths tend to dominate the

other causes of line broadening. The fact that Stark lineshapes are

especially sensitive to electron density and relatively insensitive to

temperature means that observed spectral lineshapes are good measures of

achieved densities in these experiments.

While observed lineshapes are primarily due to the Stark effect,

several other causes of line broadening are relevant in these plasmas.

Doppler broadening at these high temperatures must be convolved with the

Stark shape, though it is usually a small effect. More significant is

the additional opacity broadening of lines which occur when radiation

passes through the plasma. If more absorbing ions are present, or if

the plasma covers a larger region of space, there is a greater chance of

54











photon absorption and subsequent re-emission within the plasma with a

slight change in frequency. The net effect on the escaping line profile

is described through the equation of transfer, the result is generally

termed "opacity" broadening. This effect is most important for the

resonance line transitions which have the largest photon reabsorption

probabilities. Finally, all observations are affected by instrument and

source broadening, as described in Chapter I. This is generally a small

effect, basically smoothing the profile near line center.

In order to accurately compare theoretical lineshapes with

observations it is necessary to include all the above processes in the

theoretical calculations. The model theoretical lineshapes were

produced by including the broadening processes in the following order:

Stark + Doppler + Opacity + Instrument. This chapter discusses how

theoretical lineshapes are generated and applied in the analysis of the

data presented in the next chapter.

In addition, a systematic method is presented to analyze several

spectral lines in the same series while self-consistently including

opacity effects. This method attempts to formalize a procedure which

have been presented, in part, elsewhere.26 Previous theoretical

developments have generally assumed that profiles of lines originating

from high n-states are optically thin and yield a better estimate of

final core density than do those originating from lower n-states.

However, optically thick resonance line is analyzed to obtain the ground

state ion density. Here, we demonstrate how all the observed lines in a

given series can be used to obtain the "best" consistent electron and

ion densities.











Stark Broadening



A number of Stark and Doppler broadened lineshapes have been

calculated for the principal series transitions of Ar+17 and Ar+16 in

plasmas consisting of various argon/neon mixtures. The principal series

transitions considered are



Ar+ Is nk n < 6

Ar+16 s2 1s2. n < 5


The calculations covered the electron density range



22 24 -3
5 x 10 < N < 1 x 10 cm
e



and the temperature range



800eV ( T < 1200eV
e



The lineshape formalism of Tighe and Hooper27 was used with

extension to helium-like lineshapes using the method of Joyce, Woltz and

Hooper.28 Fine structure splitting of the upper state level, an

important source of asymmetry in the lineshape of the first few series

members, was included in these calculations; it was found to be

especially important for Lyman a and B lines.29

Lineshapes were calculated using the static ion approximation.

That is, it was assumed that except for times long compared to an

electron collision time the ions could be treated as creating a static

electric field at the radiating ion. In this approximation the










intensity profile is given by




0
I() = f P(e) J, de 3-1



where P(e) is the ion microfield probability distribution27 and J(w,c)

is the electron broadened line profile for a radiating ion in an ion

microfield e. The total profile is then given by an average over all

possible ion microfields. The dipole approximation is used for the

radiator-perturber interaction and the electron broadened line profile

is calculated with a second order perturbation theory in the no-

quenching approximation.30 Although most of these approximations can be

removed, they should be valid over the range of densities considered

here.29,30

The Doppler effect is included through a convolution with the Stark

profile



I )= f I(w') $ (w-w') dw' ; 3-2
S Stark Doppler


and the Doppler profile is given by the gaussian form,


1 (e-"0)
S= -- exp [- 02 3-3
Doppler 8v B6


The parameter B is the Doppler half-width given by


S2kT 1/2
-( LI) WO 3-4
me
-3 -1/2 3/2 Ryd
= 10 T Z Ryd
eV










In this treatment, it is assumed that electron collisions have no

effect on the radiating ion's motion during a radiative lifetime. This

is generally the case since the electron mass is much less than the
31
radiator's mass.3

Under the conditions produced in the current experiments, the

Doppler effect generally produces a small change about the center of the

overall Stark dominated profile. The conditions under which Doppler

broadening would predominate can be estimated using equation (3-4) and

the Stark broadening results. The condition for the Doppler effect to

dominate assuming temperatures typical for the implosions analyzed here

and assuming that we have a hydrogenic ion of charge Z is32



-3 a 3 21/4 -3
N < 5 x 10 n 3-5
e a


where a is the fine structure constant and ao the Bohr radius (cm) for

transitions from principal quantum number, n. For the Ly-a line of

Ar+17 and T ~ 1 keV this condition gives



21 -3
N < 7 x 10 cm
e


which is not satisfied in these experiments.



Radiation Transport



The passage of radiation through an absorbing medium may affect the

lineshapes finally observed, depending upon whether the material is

optically thick or thin to the radiation field I(v). In the optically











thin case, where the mean free path for photons to be absorbed is larger

than the size of the absorbing medium, the radiation field is relatively

unaffected by the medium. In the optically thick case, the observed

radiation has been subject to a number of absorption and re-emission

processes in the medium. The equation of transfer is used to describe

these processes; it may be written as,



1 ai
+ n vI = n XI, 3-6
c at



where n is the total emissivity for the medium (energy emitted per unit

volume per unit time per unit frequency per unit solid angle), X is the

opacity or absorption coefficient (units of cm-), and the photon mean

free path is 1/X. For the cases of planar and spherical geometry the

directional derivative may be written


-- I
n VI = p planar 3

a1 I 2) 1I
= + -(i-11 -L spherical
r r a



here, y=cos9 and 8 is the angle between the ray and local normal. The

time dependent term in equation (3-6) can be neglected in this

analysis. This is true because the overall plasma conditions do not

change significantly during a photon travel time through the medium (< 1

picosecond) in these experiments.

In the planar case, the transfer equation may be written in the

form



I 3-8
--= I S 3-8













where the following definitions are used



S = n/x 3-9
dT = -X dz


and S is called the source function, T is called the optical depth.

The transfer equation in the form of equation (3-8) can be formally

solved by using an integrating factor, exp(-T/U). The normally emergent

intensity from a planar medium is given by


T
I = f S(t) e dt 3-10
0


where the variable t represents the variable of optical depth and T is

the total optical depth of the medium. This result says that the

normally emergent intensity is given by the Laplace transform of the

source function.

While the source function is not in general known as a function of

optical depth, the above equation may be solved readily for a constant

source function, S, yielding



Iv = S[(1-exp(-Tv)] 3-11



In general the source function is not constant but depends implicitly on

the unknown radiation field. However in LTE, the source function is

simply given by the Planck function B ,



B 2hv (hv/kT -13-12
e -1) 3-12
v 2
c












It is useful to rewrite the transfer equation in microscopic form

for a resonance line. If we consider only a two-level atom, the atomic

processes to be considered are spontaneous emission, stimulated

emission, and photoexcitation. Then we obtain


aI + [n2 A21 (n B2 n2 21)I] L 3-13



which has the form of equation (3-6) where the right hand side

represents the difference of emission and absorption processes for the

resonance line transition of intrinsic lineshape L(v). The absorption

coefficient can be rewritten using the definition of the Einstein Bij

coefficient. We obtain,


= (n1 B12 2 B21) L() 4v
3-14
2 g n
re f n 1 ) 2
mc 12 1 g2 n1 (v)


For a uniform slab of thickness, z, the total opacity is obtained

using equation (3-9)



T = X z
2 3-15
=e 2 f1 z L = 1.64 x 106 f nz Lv)
mc 12 1 (v) 12 1 (v)



if the slab thickness is given in (cm), the intrinsic line profile has

units (eV)-1 and the density of absorbing ions has units (gm/cm3). The

constant is calculated with the argon ion mass. This form for the










optical depth assumes that the stimulated emission term of equation (3-

14) could be neglected. This is the case since gln2<
pumping mechanisms become important.

The expression for the intensity emitted from a uniform slab

[equation (3-11)] can now be written in terms of optical depth at line

center, TO, as


-TOL
I = S [1 exp( -0 ) 3-16
(0)


which is true since the ratio Tv/L(v) = T0/L(o) for frequencies within a

given lineshape, L(v). jIn this equation L(o) refers to profile value at

line center, and L(v) is the profile value at frequency v. Note that in

the optically thin limit, since the exponential argument is small, we

obtain,



I ) L(v)



and the emitted intensity is simply a reflection of the intrinsic

lineshape, L(v). It will be seen that the frequency variation of the

source function is not significant over the lineshape. In the optically

thick case, when T>1, we obtain



(v) (v)


which indicates a constant emission for frequencies in the line which

are optically thick. This gives rise to an overall profile which is

flat topped for optically thick frequencies near line center. Further










out in the wing of the line the decrease in the profile intensity leads

to an optically thin condition. The total lineshape obtained from

equation (3-16) will have a broader profile than the optically thin

intrinsic case and possibly with a flat top at line center. The effect

of the opacity model of equation (3-16) on a theoretical Ly-a profile is

shown in Figure 11 using several values of T The theoretical profile

includes the fine structure splitting as is apparent from the double

peaked profile in the result for lowest optical depth.

This model of opacity broadening was used to obtain the ratio

TO/L(o) from a series of fits to the experimental data for the

hydrogenic and helium-like argon lines. The validity of this model for.

opacity broadening will in fact be tested in this analysis since the

extent to which it can be successfully applied depends in part upon the

achievement of nearly uniform conditions during the time of maximum

emission of the spectral lines. Severe temperature and density

gradients occurring during the implosion will therefore be expected to

affect the results obtained in the next chapter.33

The solution to the transfer equation for the case of a uniformly

emitting and absorbing sphere differs from the planar solution given in

equation (3-11) by the inclusion of a cos 0 factor in the exponential,

where 0 is the angle between the outgoing ray and the sphere radius.

The length in the definition of optical depth [equation (3-15)] becomes

the sphere diameter and the total emitted flux is obtained by

integrating the intensity over the sphere,



I= f/2 S (1-e-v Cs) 27r cos 0 sinO dO 3-17
0 v













/ Key T.



S--- 6(
liii 115
I!I
/I 6(



I! I 12

'I I


I// \
I/



-1.5 -1.0 -0.5 0 0.5 1.0 1.
AE(Ryd)



FIGURE 11
Opacity broadening of Ar+17 Ly-a resonance line is shown
for several values of optical depth, T using the opacity
model of equation (3-16). The theoretical Ly-a profile is
for an electron density of 2x10 cm- temperature of
100 eV and includes the effects of fine structure.


)
0
9
D


0











It can be shown by the evaluation of the flux26 that the intensity

emitted from a slab of length z yields equivalent overall profiles to

the flux emitted from a sphere of diameter 1.5z. Thus the slab size

used in equation (3-15) was chosen to equal the measured core diameters

multiplied by the 1.5 x correction factor to approximate for total

emission from the spherical core.



Non-LTE Opacity Model For Two-Level Atoms



The model given in the previous section is inherently non-LTE,

since the source function is not required to be the LTE Planck function

and level populations are not specified. However, since the source

function depends on the unknown level populations, it is possible to

explicitly include the equations of statistical equilibrium for the

level populations into the definition of the source function to obtain a

solution for the emitted intensity which is also consistent with

statistical equilibrium equations. In the two level atom approximations

this solution can be obtained by following a straight forward numerical

procedure. While the method is inherently non-LTE it includes the

additional constraint of the statistical equilibrium equation explicitly

in the source function; simultaneously solving the transfer and rate

equations.

The source function defined in equation (3-9) can be written in the

form for the two-level atom,












"2 A21
a A12 + n2 B21
3-18
2hv3 n 12 1-1
c2 n2g1


where the final form uses the Einstein relations. In this

approximation, population mechanisms from higher lying levels and the

continuum is neglected. The ratio of level populations n1/n2 is

obtained from the statistical equilibrium equations for the two levels,



S.2n A21 + n2 B21 f L J2 dv + n2 C21

3-19

= n B12 f L Jv dv + nI C12



where L is the Stark broadened lineshape and J is the mean intensity

of the radiation field,

J = f I dp





Combining equations (3-18) and (3-19) we can obtain34 an expression for

the source function,



S = (1-E) f LV J dv + e B 3-20



where the coefficient e is given by

hv /kT
E = C2 [C21 + A21 hv /kT 1 3-21
e o 1













and e represents the probability per scattering that a photon will be

lost from the line by collisional de-excitation and thus thermalized.

Note that in the LTE limit where C21>>A21, the source function becomes

the Planck function as E becomes equal to one. The scattering term in

equation (3-20) represents photon frequency diffusion, i.e. photons

which are absorbed at frequency v can be reemitted at frequency v' and

eventually escape after multiple scatterings.

The above form for the source function can be combined with the

transfer equation (3-8); and the resulting equation can be solved by the

method of discrete ordinates for both the intensity and source function

as functions of optical depth. Then equation (3-10) can be used to find

the emitted intensity in the spectral line. The procedure described by

Avrett and Hummer35 has been used to describe the non-LTE radiative

transfer of spectral lines using Stark broadened lineshapes. The

results in this case are similar to that of the reference, which used

Doppler and Voigt lineshapes.

The results were obtained for plane-parallel geometry with the

approximation of a homogeneous, isothermal plasma and constant Planck

function. The opacity broadened Lyman alpha lineshape is shown in

Figure 12 using the two opacity broadening methods described. It is

seen that this non-LTE method shows a self reversal near line center due

to the non-coherent scattering term in the source function. This effect

is similar though not equivalent to that of a cooler absorbing region.

However, the overall shape yields a similar fit in the line wings to

that calculated from equation (3-16) for the same optical depth T.

Since the wings of the lineshape are most important in obtaining










































-1.5 -1.0 -0.5 0 0.5 1.0 1.5
A(w) Ryd


FIGURE 12

Opac y broadening of Lya with electron density of
5x10 cm and optical depth at line center, T ,
equal to 50. The solid line shows the uniform s2ab
results and the dashed line the non-LTE two level atom result.











lineshape fits to experiment; and since the theoretical is most

uncertain near line center, it is concluded that the method of equation

(3-16) can be used. It is further noted that self-reversals of the

optically thick Ly-a lines are not seen in the experiment (except as can

be explained by fine-structure corrections).



Line of Best Fit Method



A systematic approach was used to determine electron and absorbing

ion densities self-consistently when analyzing all lines in a given

spectral series. In particular, the principal series lines of Ar+17 and

Ar+16 were analyzed by fitting theoretical lineshapes to the

experimental data. Theoretical profiles used in the fitting included

the effect of Stark, Doppler, opacity, and instrument broadening as

previously described. The fitting procedure employed is similar to that

introduced by Kilkenny and Lee26 and can be called the "line of best

fit" method. This method recognizes that for a given spectral line there

are many values of ion and electron densities which can produce fits of

equivalent "quality". Recall from equations (3-15) and (3-16) that a

larger ion ground-state density yields a broader lineshape and that

Stark broadening theory predicts broader profiles for plasmas of

increasing electron densities. Then as a first step in our procedure a

line is assumed to be optically thin and fit with a Stark profile which

has no opacity correction. This optically thin profile provides an

upper limit to the inferred electron density. Next, a series of

additional fits for the same line is made, where the electron density is

progressively lower and the opacity correction (ion density) is










progressively larger. This series of fits can then be plotted in the

electron density-ion density plane to map out a line of best fit (see

Figure 13). Similar lines are constructed for each observable spectral

series member; for example Figure 13 shows Lyman a, 8, y, and 6 lines of

best fit for an idealized case. If it is conjectured that all of these

lines are emitted under the same plasma conditions (so that each is

characteristic of the same time history) then their point of

intersection should unambiguously identify the values of electron

density and the Ar+17 ground state density which are consistent for all

these lines. In practice, each line of best fit has an inherent

uncertainty depending on the quality of the fit; then the lines define a

region of consistency from which ion and electron densities can be

estimated. Previous studies have observed a marked disagreement of the

Lyman-a line of best fit with the other Lyman lines, and have relied

upon higher series members for approximately optically thin electron

density determinations. It is clear from the figure that such an

approach will tend to produce an overestimate of density, except for the

highest series members which are truly optically thin but probably not

intense enough to be observable.

An additional constraint may be placed on the electron ion density

plot with an ionization limit line. For example, all ionization

equilibrium models indicate that the maximum density in the hydrogenic

ionization state is approximately 50%, then




1
nH (< total
n
1 e
2 Z

























CO
0




cor



r-- rn C4
)0 )





>
S4J bt 4








E!!
Q0) 0H



-4



0 0)
-41




COd


0d




)0)

4J -4


0)
EO

0- 0)


4- ( c





w~0 0

G) 4- p


.a0 414


0 00

Q) 41 -4
6Uo




















C4 0l
ca w

ca




















Ed
-4-4


C C)0


u-4






















C
0


N4-~
EEc


C





4
0
00


'-
z:


0


"O

O30
x







73




The above inequality defines an excluded region in the electron density-

ion density plot. For any given value of electron density, there is a

certain maximum amount of opacity broadening which can be obtained.

This excluded region is indicated on Figure 13. A more complete

collisional-radiative model also defines an excluded region similar to

the above condition.

















CHAPTER IV
RESULTS



The methods described in the previous two chapters were used to

analyze the experimental spectra obtained from the pellet implosions of

the targets listed in Table 1. The spectra covered the 3-4 keV energy

range which included the principal series lines (Is-na or is2-Isnk for
+17 +16
n<6) of Ar and Ar A region of the spectrum which includes the

hydrogenic and helium-like Lyman alpha transitions is shown in Figure

14, a densitometer trace from shot 7497, which imploded a 150Um x 2pm

Ar/Ne filled target. Note that the 3.5eV instrumental resolution allows

separation of the fine structure components in Ar+17 Ly-a, and detailed

structure is also visible among the satellite line components.

Lineshape fits have been made for all the hydrogenic and helium-

like lines observed following the "line of best fit" method. Intensily

ratios of lines were also measured, including satellite lines, which are

sensitive to temperature or density. Results from the several different

diagnostics were then compared for consistency.



Results of Line of Best Fit Method



A number of lineshape fits were carried out for each observable

series line so that a line of best fit could be constructed on a N -Nion

plot. Each theoretical profile was fitted to the data using a least

squares procedure to determine relative intensity factors necessary to

74











obtain a high quality fit over the entire line profile. The underlying

continuum was fit both by the least squares method and by manual

estimation. It was found that the most satisfactory results were

obtained by manually inserting the background coefficients. Quality of

a fit was defined as a normalized mean square deviation, i.e.


y )2
(Yex th
Q = 2 4-1
th

where Q<0.05 represents a good quality fit. Although no specific

provision was made to ensure that area normalization was maintained, the

results of our fitting procedure were in good agreement with that

constraint. The fits were complicated in some cases by the overlapping

of neighboring lines (e.g., Ca or K impurities from the glass shell or

lines of helium-like Ar+16). The effects of overlapping lines were

treated by adjusting relative intensities, again according to a least

squares procedure. In Figures 15 (a-d) some Ar+17 Lyman series line

fits for shot 7499 are shown. These line fits are representative of

those used in the construction of a line of best fit for each spectral

series line. Recall that these lines of best fit are generated by a

sequence of line fits in which the opacity and electron density are

systematically varied.

Image analysis of X-ray microscopic data determined the approximate

diameters of the imploded cores to be 40-45 microns. These core sizes

were then included in the expression for opacity [equation (3-15)] to

calculate the density of Ar+17 ions in the ground state where the

ratio T /L(o) was determined from the best fit for a given electron

density. Electron temperatures of approximately 1 keV were deduced from





































Q)
.C
41












0+
-H




oo





44
0







-1
blo W(




(014








t~o
s,









41J

Q)

4J
Q)

0
41
*-H
C)r







77

























C

E-
t)
O
a-
















,AI!suaci Wl!-



























ca


U,

U ,
co




41




Im.
0 3


02






041i












4J.4
Q2)




'4
MaJ
0







40)




V4-4
-0
-4



CO) cu

-4-4
Q)




0)



0)u
o4







4-f 4J

-14
4-i
44
0)H

~ca


al p


cuu
4J
4

41t







rX4

4.)









































0
0 C -


ci o 2 o-


I-
F 0 -
0 0
2k 0*o u










the ratio of the Lyman alpha satellite to resonance line (see Chapter

II).

The lines of best fit for shot 7497, which employed a 150pm x 2pm,

50/50 Ar/Ne target, are shown in Figure 16(a). It is observed that a

23 -3
density inference of ne = 1.5 x 1023 cm can be made on the basis of a

consistent best-fit analysis for the Lyman-a, B, y, and 6 lines;

similarly a consistent density of Ar+17 ions in the ground state was

found to be ~ 3 x 121 cm3. It is estimated that a 20% uncertainty in

the theoretical line profiles results in a corresponding uncertainty in

the lines of best fit (LBF). Errors in emission region size will

directly affect the ion density determination but the electron density

and the general trends indicated are not sensitive to this dimension.

It was assumed that all Lyman series lines were emitted from the same

emission region, so all the LBF'S would be affected equally by errors in

size.

The ionization-limit curves shown in Figure 16(a) correspond to 30%

and 50% of the total argon ions in the ground state of Ar+17. An upper

bound on the Ar+17 ionic ground state density appears to be 50%,

regardless of the ionization model. Hence the 50% line defined in the

Ne-Ni plane represents a limit: all physically realizable results are

excluded from the region to the right of that curve. The LBF analysis

of shot 7497 implies an Ar+17 ground state density which includes

roughly 30% 50% of the total argon ions.

The lines of best fit for shot 7496, which employed a lOOm x 3im

100% Ar filled target, are shown in Figure 16(b). It is observed that

the Ly-a, -B, and -Y LBF are consistent with Ne = 2.5 x 1023 cm-3 and Ni










= 2.1 x 1021 cm-3. As in the previous case (shot 7497), less than 50%

of the total argon ions are in the Ar+17 ground state.

In the implosions of these thick shelled target, the mean free path

of hot electrons is reduced, resulting in less heating of the core and a

slower more ablative implosion, generally reaching higher densities.8

The hydrogenic x-ray spectra from the implosions of 150pm x 2pm and the

100 x 3pm targets allowed a self consistent electron and ion density

determination from the analysis of all observed Lyman series members.

In fact, this is the first time that analysis of the optically thick

Lyman alpha line (TO = 100) has been consistent with that for the other

series members. We attribute this improvement, at least in part, to the

inclusion of fine structure corrections and to an extended treatment of

the line wings. In these more ablative implosions the conditions of

line formation appear to be equivalent, or nearly so, for all the Lyman

series members; time integration effects of the spectral data produces

no serious inconsistencies. Hence, in the "right" system an optically

thick line can be used as a density diagnostic.

Next we consider two examples of experiments which employed

100pm x 1pm targets. Figures 16 (c) and (d) illustrate lines of best

fit for shots corresponding to Ar/Ne mixtures of 100/0 and 25/75,

respectively. In these shots the highest series lines, 6 and E, are

observed to correspond to average densities that are best determined to

be < 3 x 1022 cm3. This density is significantly lower than that

inferred from the Lyman-a, -0, and -y lines which imply N > 5 x

1022 cm3, assuming consistency with the 50% ionization limit. For

lower electron densities fits for these lines would require ion ground

state densities in the forbidden region, e.g. to the right of the 50%


























04 C



0


S4-i




oo
Od d







0 C4 -H
n -, c (I (a









o o

o4-J r-4 4J
Q)d a -U4

4- '4 p
41




0 4

















-44




-0 C4
10)




(I, O S i
0 a)







(4 2 U) Z
>1 0


L

























.0


ca0lxoO


0tzOIoN


czOI eN


CZ01 x eN










line. In any event the 6 and e lines are much too narrow to agree with

the density obtained from the 0 and y lines (also see Figure 15).

The results of the line fitting procedure when applied to each of

the 100 um x lum targets did not lead to single valued estimates of the

electron and ion densities. In these implosions the thin glass shell is

exploded by hot electrons which deposit their energy throughout the

shell.8 The results suggest that different Lyman series lines are most

intense at different time intervals, and hence their profiles are

representative of different temperature-density regimes. Thinner shells

tend to increase preheat of the core, so we may be observing the effects

of altered level populations due to fast electron preheat in the

100pm x lIm cases. That is, the most intense emission from the higher

lying Lyman series members may occur earlier in the thin shell targets

and at higher temperatures and lower densities. High resolution time

resolved spectroscopy is required to determine the details of these

observed differences. The electron density inferred from the

intersection of the Lyman alpha LBF and the 50% ionization limit line

was found to be about 1 x 1023 cm-3, regardless of fill ratio.

Similarly, density inferences from the other Lyman lines showed no

sensitivity to fill ratio but were also inconsistent with the Lyman-a

result. This suggests that opacity may not be as important in the

diagnosis of these shots as are the effects of implosion dynamics.

Our analysis has shown that spectroscopic plasma diagnostics using

Stark and opacity broadening of the Lyman series lines from the

compressed gas fill produces consistent results with the thick-shell

(> 2 um thickness) targets, and that inconsistencies occur with the

thin-shell targets. It is suggested that these inconsistencies are due











to the effects of implosion dynamics which result in the lines from

high-lying n-states implying significantly lower densities than do

those from the lower n-states. The time integrated spectra is

consistent with a conjecture that during these implosions the line

emission occurs in more than a single burst, each corresponding to

different temperature density regimes. Thus, although it had been

conjectured that lines from the highest observable series members, being

the most optically thin, would provide the most accurate determination

of core density, this study has been shown that lines from the lower

lying n states (e.g. Ly-a, -8, -y ) are probably more correct in

determining final core densities.

Our analysis of shots with similar target size but varying fill

ratio showed similar spectral characteristics. While it was shown that

opacity effects must be included for all lines, the variation of fill

ratio over the range of fill pressures considered did not produce

significant effects on the validity of the overall diagnostics. Opacity

may not be as important in the diagnosis of these shots, however, as are

the effects of implosion dynamics. Future high resolution time resolved

spectroscopy should provide further insights into the implosion

dynamics.

The results of the opacity broadening analysis of hydrogenic series

lines are summarized in Table 2. The results for opacity broadening

analysis of the principal series lines of Ar+16 are less complete due to

the fact that the potassium and calcium lines from the glass shell

obscured both the helium-like 0 and y lines in several shots. The

helium-like 6 line was also blended with the hydrogenic Ar+17 Lyman 8

line. An attempt was made, however, to apply the opacity broadening











method to all usable data and the results are summarized in Table 3. In

most cases, only two helium-like lines (a and B) could be analyzed

except for shot 7497 which also included the y line. Though the data is

incomplete, the indications are that the helium-like emission occurs at

electron densities almost equivalent to that of the hydrogenic emission,

while the helium-like ground state densities seem to be somewhat lower

than the hydrogenic ground state density. For temperatures above 1 keV,

this observation tends to agree with results based on the optically-

thick, collisional-radiative steady-state model discussed in Chapter II.



Satellite Line Results



Also in Chapter II, there was a discussion of the dependence of

satellite line emission on electron temperature and density in the

plasma. In particular, for the argon lithium-like and helium-like

satellites, it was shown that relative satellite line ratios are density

dependent (Figures 8, 9) and the satellite-to-resonance line ratios are

temperature dependent. These features were used to analyze the

satellite emission for the shots given in Table 1 in order to diagnose

electron temperature and density. The satellite line diagnostics were

then compared to those from the line broadening analysis for each shot.

Two complementary methods were used in the satellite line

analysis: (a) fitting the data to theoretical calculations of total

satellite emission, and (b) de-convolution of the data to extract

measurements of several satellite line ratios which are density

dependent. Both methods utilize the same theoretical calculations of

the satellite emission and produce similar results. The first method










87















d 000 0 O O
-H -H -+ -H +- -H
Q) m m m c 'T- --T
t -W I C4 CM C cM CM CM
o 000 0 0 0
Su -i -i .-H -H
S CM4 -- -M

0 z
o
W -

ca
a)* *
w ON 0
dCO CO O 0 r
-i N Ci 1 00 00 T
cHJ N c: C'J e, -H
S0 C0 0 0 cH -.
p w v C1 C9 '--4 CDl

.rq 1-4 $ v v V X- 1-4 1-4 -4

CO 0 0 cw c






U W

S a) 0 0 0 C0 O
w C C-) mn C') c CM c- 1 C 0
w 0 C0 CS, cq 4H -H 4 4 -H C
Sm0 m .n +
41 "0 0 C O CM






0 0
1*40 N c
ca m




0 4-
QrmC


4C -W N 00 cn 0 y m00 Cy) ID -
w t I I co XXX

4 0 0 0 00 00 0
U 0C 0 0 0 0 0 0
* p.1 CM LA U4 i*. '. a' LA CM
CM4 caC 0 0 0 LA -3 ONC 0 0
-4 i- 4 4 -4

w0






0 C-










88













S4 n an b f
qd N- -4 -i 0 0 0 0
a q mc -H +IN + C cN C C 4
i- I M CV 0 en -H -H 4- -H
SbO e 5 C -i 4 C ci m cn mc
J *J U 0 0 X o0 cJ CJ CD
c( \ 0 0 0 0





"M -H X XH -H
) Li^ i cJ V in X X X X





SW C) C : C




wNN i
0



01 000 00 0







* m r tr N < 4
B~ -^y $H +1 H 0 +H +1
Sl 0 0 0 0 0 0
-- 0NI N CN X CaJ CN








m0 000 C 0 0r
S+ X X X V M






Cl) en
.4 l -4. C



S C D f 0


0 0J -.4
0

U w C) r 0 0
N 00 -4 C., CN4 00 C) %. tn
u 4J 0 -

-4 n- %0 a 0 0N
0n 0 ) X X
C., Ca m m


0 s

03 rP



"J 0 0 0 0 0 0 0 0
0. __ 0 0 0 0 0 0 0 0





030



o" N. o- r N. o iO r r-.

o i()
3.
~ 31 CO










utilizes a least-squares fitting procedure over the entire range of

satellite emission, while the deconvolution method considers line ratios

over a part of the satellite emission. The deconvolution method,

however, is based in a more formal numerical procedure and produces more

accurate error estimates.

Several assumptions have been made concerning satellite line shapes

and widths. Individual satellite lines were assumed to be optically

thin with Voigt or Lorentz profiles. The effects of opacity can

generally be ignored due to the low populations of the autoionizing

levels. However, under conditions of strong satellite emission, opacity

effects may be evident in the strongest lines and this can affect line

ratio diagnostics.36 Calculations of Stark broadened lineshapes for

lines from autoionizing levels have not yet been made, but some previous

work37 has shown that the isolated satellite line width can be estimated

by adding the partial widths arising from autoionization, spontaneous

emission, and collisions. For this experiment it appears that

individual satellite lines have shapes determined primarily from

instrumental and source broadening, which when convolved with the

natural and collisional shape produces a Voigt profile (or Lorentzian

for simplicity) of full width at half-maximum of approximately 5ev. Due

to the uncertainties in shape and width of satellite components, the

sensitivity of the results to shape and width was tested. In general,

the line ratio measurements were not found to be sensitive to shape and

width when the width was allowed to vary within a narrow range of the

expected width (as determined from isolated, unblended satellite lines).










The method of satellite lineshape fitting used the same least

squares fitting procedure described previously, where the overall

satellite line shape was multiplied by a scaling factor determined by a

least squares fit to the data. A lineshape fit using this method is

shown in Figure 17 for shot 7499. Although all lineshape details are

not matched, a trend that is evident from this and other fits is that

the peak satellite emission shifts toward the resonance line as density

increased. This is evident in the results of theoretical calculations

shown in Figures 8 and 9. Due to the poorer qualities of fit obtained

in attempting to match the entire satellite lineshape to the data, this

method has somewhat greater estimated errors than the line ratio method.

The relative intensities of the satellite components necessary to

determine the ratios indicated in Figures 9 and 10 have been measured by

an effective deconvolution process using an iterative peak finding code,

RLCFIT.38 This code fits the background in terms of a Fourier series

and determines individual peaks by minimizing a weighted sum of the

difference between the data and background with respect to peak heights,

positions, and widths. Standard peak shapes employed were either

Lorentzian or Voigt. This method serves to enhance the data, but is

different from a similar procedure discussed recently by Seely.39

This method must be used with caution due to the non-uniqueness of

the curve fitting problem. It is therefore necessary to introduce as

constraint as much physical information about the spectra as possible.

Thus, standard lineshapes were chosen along with widths which were

allowed to vary by several electron volts. Though the code could be run

with a variable width option, a fixed width was chosen where the

standard peaks of fixed width were made to fit the data by varying the



























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