{ \renewcommand{\baselinestretch}{1}
\chapter{Theory-Experiment Comparison and Data Analysis}
}
{\renewcommand{\baselinestretch}{1}
\section{High Density Spectral Comparison with \\ Experiment}
}
In this section, four experimental lineouts are compared with
synthetic spectra.  These four lineouts constitute the earliest spectral 
data with distinct line emission, and in turn, depict the highest density 
and temperature plasma conditions of the series of experimental lineouts.
Specifically, this data corresponds to the ablated plasma at t=20 and 30
nsecs and at distances from the target surface of 28.8 and 86.4 $\mu$m.
This data was specifically selected for the comparison with our steady state
spectral model.  Its high density and correspondingly 
strong collisional rates have been assumed to be in a steady state regime.
Issues associated with time-dependent atomic kinetics are discussed in 
the next section.

\bigskip
{\renewcommand{\baselinestretch}{1}
\subsection{Self-reversal and the Justification of Plasma \\ Non-uniformity.}
}
Up until this point the synthetic spectral model has been employed under
the uniform slab approximation.  This simplest geometry is not burdened 
by spatial variations in temperature and density and provides a
cleaner environment for the interpretation and understanding of the 
synthetic spectral results. Unfortunately, as noted in Chapter 2, the
self-reversal feature in the Li I: 3d-2p line supports the existence of
a non-uniform plasma. Though a self-reversal feature may exist in a
plasma of uniform density and temperature due to a non-uniformity in
level populations arising from a position dependent radiation field,
the plasma referred to in the four lineouts has been shown to be insensitive 
to the effects of the radiation field on the populations.  The effect of
the photon field on the populations can be approximated as an effective 
reduction of the spontaneous radiative decay rate (SRD).  This approximation 
is the basis for the use of escape factors \cite{Grim_B97ch7} to include 
the effects of radiation in atomic kinetic models. A test conducted with
the atomic kinetic model for the relevant densities and temperatures predicted
for the four early in time lineouts showed no variation in level populations
when the SRD rate was eliminated.

\bigskip
{\renewcommand{\baselinestretch}{1}
\subsection{Temperature and Density Profiles Generation.}
}
To simulate a non-uniform plasma, a series of temperature and density
profiles that define the search space must be generated.  The process
begins by selecting the only values available to every spatial zone for
T$_{e}$ and N$_{a}$, and then assumes that the profiles are symmetric
with respect to the center and that the two center zones contain the
maximum values.  A further restriction on the profiles was imposed: a zone
may take on values equal to or less than its neighbor who resides
in the direction toward the center of the profile. Or in other words,
profiles are either uniform or approximate a concave down function.
Following these restrictions, an exhaustive generation of the profiles
is performed. The requirement of spatial zones of uniform width was a
further restriction placed on the model.

\bigskip
{\renewcommand{\baselinestretch}{1}
\subsection{Trial Description}
}
Each synthetic spectrum generated was compared with an experimental lineout 
to determine the best least-squares fit.  Genetic algorithms (GA) have
been proven to be successful in the analysis of X-ray data \cite{Golovkin02}.
An exhaustive search of parameter 
space was implemented after the use of a GA produced mixed 
results.  Three trials were performed for the set of four experimental 
lineouts - they included a uniform, four- and six- spatial zone simulations.
Due to the extensive increase in parameter space with the inclusion of 
more zones, an alternative computer program was produced for speed. This
new program took advantage of the predefined temperature and density values
found in the profiles.  An exhaustive database of single zone opacity and 
emissivity curves was produced by the original parallel program for all 
permutations of T$_{e}$ and N$_{a}$ values. The new program only consisted of  
the parallel queue (PQ) and the radiation transport solver (RT) (see Chapter 5
for program details) and a small number of additional codes for manipulating 
the database.  The new program produced a 15 times speed up over the original 
synthetic spectra solver. This modification exemplifies the importance of 
the coding architecture developed in Chapter 5. 

The following are the three trial series for the comparison with
t= +20 nsec and t= +30 nsec both at x= 28.8 $\mu$m lineouts.  Results for the
x= 86.4 $\mu$m lineouts follow in a similar manner and can be found in 
Appendix C.

In the comparison shown in Figure 7.1, both the t=20 nsec lineouts proved to be the most
difficult to model. The existence of contamination along the wings
of the main spectral features, in conjunction with an asymmetry on
the high energy side of the Li I: 3d-2p line feature, hindered the
comparison.  Interestingly, the single zone calculation obtains the main
characteristics of the spectrum. Due to the uniform radiation transport
calculation, the Li I:3d-2p self-reversal cannot be approximated nor
can the slight asymmetry about the self-reversal (see Figure~\ref{fig:22_0.1}).
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/SpZone/spec22_0.1z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=20 nsec with corresponding uniform synthetic spectra.]{\label{fig:22_0.1}Experimental lineout at x=28.8$\mu$m and t=2
0 nsec with predicted T$_{e}$=1.5 eV and N$_{a}$=4.0x10$^{17}$ cm$^{-3}$ uniform
 synthetic spectra.  }
\end{figure}
}

Even with the difficulties mentioned above, the four-zone calculation
shows improvement, as shown in Figure~\ref{fig:22_0.4}, consistent with an increase
in the numerical fitness.  The self-reversal and the slight asymmetry
have now become apparent in the synthetic spectrum with the inclusion of
the gradient.  Line profiles for the most part compare relatively well
for the broadest portions of the spectral lines. Platinum contamination 
in the silver $\bf{c}$ and lithium $\bf{b}$ lines may partially explain 
the discrepancy between the spectra.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/SpZone/spec22_0.4z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=20 nsec with corresponding 4-zone synthetic spectra.]{\label{fig:22_0.4}Experimental lineout at x=28.8$\mu$m and t=20 nsec with predicted T$_{e}$=[1.1, 1.9, 1.9, 1.1] eV and N$_{a}$=[2.5, 4.5, 4.5
, 2.5] x10$^{17}$ cm$^{-3}$ 4-zone synthetic spectra.}
\end{figure}
}

For the 6-zone calculation, shown in Figure~\ref{fig:22_0.6}, the improvement in the 
numerical fitness unfortunately does not follow an obvious improvement in the spectral
comparison.  This lack of real improvement may be due to the limits of
the use of a low number of uniform zones in the RT calculation. 
Unfortunately a large
number of zones increases the search time exponentially: The addition
of one more zone on either side of the profile increases the calculation
time from a few days to a month.  Another possible remedy to improve
the comparison in the heights would
be to employ a non-uniform grid for the inclusion of narrower
zones at the perimeter, to contribute low density spectral emission 
without including their 
characteristic high opacity.  Unfortunately, a scheme has not been devised
to confidently search through such a large domain that a variable width spatial
grid would entail. 
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/SpZone/spec22_0.6z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=20 nsec with corresponding 6-zone synthetic spectra.]{\label{fig:22_0.6}Experimental lineout at x=28.8$\mu$m and t=20 nsec with predicted T$_{e}$=[1.2, 1.2, 2.0, 2.0, 1.2, 1.2] eV and N$_{a}$=[2.0, 3.5, 6.0, 6.0, 3.5, 2.0] x10$^{17}$ cm$^{-3}$ 6-zone synthetic spectra. }
\end{figure}
}

As mentioned above, part of the difficulty in the comparison arises
from an asymmetry in the high energy side of the LiI: 3d-2p line ({\bf d}). The
existence of this feature has several possible explanations.  As 
mentioned in Chapter 2, oxidation of the target, in particular with lithium
atoms after fabrication was concluded to be the origin of Li$_{2}$O and LiOH
molecules found in the target.  The strong 
$2s^{2}2p^{3}3p$ $^{5}$P - $2s^{2}2p^{3}4d$ $^{5}$D  and 
$2s^{2}2p^{3}3p$ $^{3}$P - $2s^{2}2p^{3}6p$ $^{3}$S transitions of 
neutral oxygen are found in this location, as well as other strong transitions
on the high energy side of the Li ($\bf{a}$) line where contamination is 
found.

Another possible explanation is the effects of Li$^{+0}$-Li$^{+0}$, 
Li$^{+0}$-Ag$^{+0}$ and/or Li$^{+0}$-molecule interactions for the 
broadening of the spectral line.  Estimates based on reference \cite{Grim_B97ch4}, 
for resonance (line broadening between levels of the same species connected
by dipole a transition) and Van der Waals (more general atom-atom 
interaction) broadening were found to be small in comparison to Stark 
broadening. It should be kept in mind that limitations to these 
approximations exist and in particular for Li$^{+0}$-Ag$^{+0}$ 
broadening these estimates may not be appropriate.  

Due to the possibility of contamination and the plasma's high opacity,
developing and performing more sophisticated profile calculations with 
the hopes of explaining the asymmetry would be impractical.

The comparison with the synthetic spectra for the  t=+30 nsec, x=28.8 $\mu$m 
appears much better than the t=+20 nsec data. This is also substantiated by  
an order of magnitude increase in fitness.  The asymmetry no longer dominates the
high energy side of the 3d-2p line, and the existence of contaminating features
appears to be weaker about the wings of the main features.
The uniform calculation, Figure~\ref{fig:21_0.1}, captures most of
the main features; however, the LiI:3d-2p line width is underestimated.
As a confirmation of the quality of the comparison, the LiI: 4s-2p and
4d-2p lines found in the $\bf{f}$ complex of lines compares well with the
experimental data though they have not been included in the least-squares
minimization procedure.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/SpZone/spec21_0.1z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=30 nsec with corresponding uniform synthetic spectra.]{\label{fig:21_0.1}Experimental lineout at x=28.8$\mu$m and t=30 nsec with predicted T$_{e}$=0.8 eV and N$_{a}$=1.0x10$^{17}$ cm$^{-3}$ uniform synthetic spectra.} 
\end{figure}
}

In the 4-zone calculation, Figure \ref{fig:21_0.4}, 
the self-reversal on line $\bf{b}$ is clearly displayed, and  
also a much better agreement in the width of the Li I: 3d-2p line is observed.
For the 6-zone case, Figure \ref{fig:21_0.6}, the comparison is quite favorable.  
Again platinum lines in $\bf{c}$ and $\bf{a}$ contribute
to the disparity.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true]{chap07/SpZone/spec21_0.4z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=30 nsec with corresponding 4-zone synthetic spectra.]{\label{fig:21_0.4}Experimental lineout at x=28.8$\mu$m and t=30 nsec with predicted T$_{e}$=[0.7, 1.8, 1.8, 0.7] eV and N$_{a}$=[1.0, 2.0, 2.0, 1.0] x10$^{17}$ cm$^{-3}$ 4-zone synthetic spectra.}
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true]{chap07/SpZone/spec21_0.6z.eps}
\end{center}
\caption[Experimental lineout at x=28.8$\mu$m and t=30 nsec with corresponding 6-zone synthetic spectra.]{\label{fig:21_0.6}Experimental lineout at x=28.8$\mu$m and t=30 nsec with predicted T$_{e}$=[0.7, 0.8, 2.3, 2.3, 0.8, 0.7] eV and N$_{a}$=[1.0, 1.0, 2.5, 2.5, 1.0, 1.0] x10$^{17}$ cm$^{-3}$ 6-zone synthetic spectra.}
\end{figure}
}

Figure~\ref{fig:opdep.6.all} describes the optical depth of 6-zone
synthetic spectra. Notice that the optical depth is greater for the t=+30 nsec
series.  This is due to the existence of very sharp line profiles
in the LiI: 2s-2p and 3d-2p lines that are associated with the 
low electron density. Also, the lower temperatures found in the t=+30 nsec
spectra that place more
population in the lower levels of the line transitions. Notice in this figure
that the silver lines approach optical depths as high as $\tau$=10, while
the Li I:2s-2p has $\tau$=400. Interestingly, the uniform and 4-zone optical
depths are similar to the 6-zone case; see Appendix C.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/OpdZone/optdep.6z.eps}
\end{center}
\caption[Optical depth for all four 6-zone synthetic spectra.]{\label{fig:opdep.6.all}Optical depths for all four 6-zone synthetic spectra.}
\end{figure}
}

Figure~\ref{fig:prof22} and \ref{fig:prof21} describe the 1, 4 and 6 zone
resulting profiles for the t=+20 and +30 nsec calculations respectively.
The t= +30 nsec results, show distinct differences in boundary and
core values, where the single zone solution consistently approximates
the boundary values for both T$_{e}$ and N$_{a}$. These two boundary
zones predominantly contribute the non-uniform opacity needed to reduce
Li I $\bf{a}$ and $\bf{b}$ lines with respect to the Ag I $\bf{c}$ and
$\bf{d}$ lines, while the higher density core provides the higher electron
density needed to match the widths of the spectral lines.  For the
t=+20 nsec results, the single zone solution appears to approximate the
average of the entire profile. This implies that all zones are strongly
participating in the opacity and in the line broadening, thus spectral
line widths are not primarily defined by any particular region of the
plasma, unlike the previous results.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.55,clip=true ]{chap07/SpZone/chap06.22.eps}
\end{center}
\caption[Uniform, 4 and 6 zone profile results for t= +20 nsec synthetic spectra.]{\label{fig:prof22}Uniform, 4 and 6 zone T$_{e}$ and N$_{a}$ profiles results for t= +20 nsec synthetic spectra.}
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.55,clip=true ]{chap07/SpZone/chap06.21.eps}
\end{center}
\caption[Uniform, 4 and 6 zone profile results for t= +30 nsec synthetic spectra.]{\label{fig:prof21}Uniform, 4 and 6 zone T$_{e}$ and N$_{a}$ profiles results for t= +30 nsec synthetic spectra.}
\end{figure}
}

To obtain an understanding of the sensitivity of the synthetic spectra to
the temperature and density profiles, Figure~\ref{fig:Bestfits} contains
the best, next best, 40th and 98th best solutions for the t= +20 and
+30 nsec, x = 28.8 $\mu$m synthetic spectral results. For the t= +20
nsec solution, the boundaries of the plasma show very little variation,
unlike the core spatial zones.  As mentioned before, the boundary zones
are predominantly defining the opacity need for the spectra, while the
core adds the widths of the lines.  For the t= +30 nsec spectra, though,
the core temperature shows a similar variation, however, the density
shows none.  In this plasma, the average ionization stage varies, from
1 to 1.5 through the total range of temperatures.  Therefore, the only
means to significantly raise the electron number density, and thereby
the line profiles, is through a increase in atom number density.  Due to
the broad lines found in the t= +20 nsec lineouts, the core elements must
maintain a high atom number density to obtain the proper line profiles. It
should also be pointed out that the fitness varies slowly with the change
in T$_{e}$ and N$_{a}$ profiles. This is undoubtedly an artifact of the
high opacity lessening the sensitivity of the spectra. The variation in
regards to the spectra, is seen in Figure~\ref{fig:bestspecomp}.

{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true]{chap07/SpZone/Bestfits21.0_22.0.eps}
\end{center}
\caption[6-zone temperature and density intermediate results for t= +20, +30 nsec at x = 28.8 $\mu$m synthetic spectra.]{\label{fig:Bestfits}6-zone temperature and density intermediate results associated with the  1st, 2nd, 40th and 98th best results for t= +20, +30 nsec at x = 28.8 $\mu$m synthetic spectra.}
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/SpZone/bestspecomp.eps}
\end{center}
\caption[6-zone spectral 1st, 2nd, 40th and 98th best results for t= +20, +30 nsec at x = 28.8 $\mu$m synthetic spectra.]{\label{fig:bestspecomp}6-zone  spectral  1st, 2nd, 40th and 98th best results for t= +20, +30 nsec at x = 28.8 $\mu$m synthetic spectra with corresponding experimental lineouts. The individual curves are indistinguishable.}
\end{figure}
}

\bigskip
\subsection{Lithium Lines Behaviour Under High Opacity.}
We conclude this section by describing the difference in behavior of
the Li I:3d-2p and 2p-2s lines. One critical question arises about the
behavior of these lines:  If the Li I: 2s-2p line has the largest optical
depth, then why is a large self-reversal not present in the line, such as is
found in the Li I: 3d-2p line?  The answer lies in the emissivity and opacity
functions. From Figure~\ref{fig:EmOp21} and Figure~\ref{fig:EmOp21Z}
we see that the Li I: 3d-2p $\bf{b}$ line is composed of contributions
from different spatial regions of the plasma: the wings of the line originate 
from the high density center region of the plasma, while the core of the line 
originates from the low density spatial region of the plume. This configuration
is optimal for a self-reversal in particular for a plasma large spatial  
extent.  Or in other words, we may consider the core of the line approaching
the Planckian function at a different temperature than that of the wings. On the 
other hand, the Li I: 2s-2p line is not dominated  
by contributions from different plasma regions: its 
line profile is dominated by emission and absorption from the boundaries
of the plasma.
Therefore, this line behaves as if it were produced from a uniform
plasma simply undergoing self-absorption without significant self-reversal.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true]{chap07/EmOp21/EmisOpac21_1.6z.eps}
\end{center}
\caption[ 6-zone emissivity and opacity  from t= +30 nsec x= 86.4 $\mu$m 
synthetic spectra results from plasma core and boundary zones.]{\label{fig:EmOp21}6-zone emissivity (upper curves) and opacity (lower curves) from t= +30 nsec x= 86.4 $\mu$m synthetic spectra results from plasma core and boundary zones.}
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/EmOp21/EmisOpac21_1.6z.Zm.eps}
\end{center}
\caption[6-zone emissivity and opacity  from t= +30 nsec x= 86.4 $\mu$m 
synthetic spectra results from plasma core and boundary zones.]{\label{fig:EmOp21Z}6-zone emissivity and opacity  from t= +30 nsec x= 86.4 $\mu$m synthetic spectra results from plasma core and boundary zones.}
\end{figure}
}

