\section{Modeling the Low Density Expansion and \\ Time-Dependent Effects.}
Before the data from the Sandia experiment became available, 
extensive work was performed to
accurately model the 1-D hydrodynamic expansion of ablation plumes with
the hydrodynamic code HYAT \cite{Mancini_hydro88}.  Due to the
small amount of laser energy dispensed to the target, a careful accounting
of the laser energy coupling to the target had to be performed. 
This involved the inclusion
of a realistic equation-of-state (EoS)\cite{sesame}, and the tracking 
of phase transitions.  Early work focused
on carbon, where it was found that an expanding fluid element during ablation,
could undergo a transition across all thermal phase boundaries, and not simply 
sublimation as was customarily assumed in gas dynamic simulations. Furthermore, these 
hydrodynamic simulations went beyond the typical gas dynamic models used in modeling these 
systems \cite{Russo97}.  This work supported the need for the inclusion 
of realistic EoS for modeling laser ablation hydrodynamics.

A custom LiAg equation-of-state was generously provided by J.D. Johnson
of T-1 from Los Alamos National Laboratory to model the Sandia experiment. 
Though the expectation was that the LiAg 43-57\% binary alloy EoS was of good quality
due to the existence of a single solid phase from tabulated experimental
data, an accurate EoS
for the thin layer of lithium-oxide that it is expected to form on the target 
surface when the target was  brought up to atmosphere was not found.  
An effort to measure the reflected laser radiation 
from the target led to results varying between 20-50\% at the laser angle
of incidence used in the experiment. This made difficult to perform  
hydrodynamic simulations of the Sandia experiment. Yet, these simulations 
did permit to extract qualitative features of this low fluence laser 
ablation, such as the density and temperature time histories and the rate
of fluid expansion. This qualitative description became important in
describing the rarefied expansion of the plasma.

From numerous runs of the hydrodynamic code, the density time
history was characterized for a fluid element just leaving the surface
at +20 NSF after the laser pulse by an initial exponential followed by
a slower linear decrease. This corresponded to the continuing rapid
expansion from 5x10$^{17}$ to 1x10$^{15}$ cm$^{-3}$, followed by an
almost non-expanding fluid element with a nearly constant transverse velocity.
Temperature time histories for fluid elements just leaving the surface
were found to fall linearly from 1 eV to
0.4 eV in time.  Hydro-simulations predicted the transverse motion of
the fluid elements between 10 - 15 $\mu$m/nsec depending on the degree
of reflectivity.  This corresponded well to measurements taken from the
motion of bulk emission as seen in the Sandia CCD images (\~17 $\mu$m/nsec).
Though no guarantee exists that the brightest emissions  of the two
consecutive temporal images corresponded to the same fluid element,
the value is consistent with velocity measurements
found in the literature (\cite{RussoVel}) for this regime of laser
irradiation.

In particular, the time scale change of the two decade drop in 
atom number density due to the initial hydrodynamic expansion 
corresponds to the nanosecond time scale of the collisional processes
at these low densities.  This condition requires time-dependent
atomic kinetics instead of steady state atomic kinetics that were 
employed to model 
level populations of the high density plasma early in time and close
to the surface. Therefore, a time-dependent version of the atomic 
kinetics model was created (see Chapter 4). 
It was driven by density and temperature time-histories that 
were patterned after the time evolution of fluid
elements observed in 1D hydrodynamic simulations of the experiment.

Figure~\ref{fig:LiTotfrac} shows the time evolution of neutral Li 
total fractional population (i.e. ground plus excited states) 
over a time interval of 80 nsec (t = +20 -- 100 nsec ). 
This figure also displays the results of
the steady-state calculation driven by the same set of instantaneous
temperature and density points. The results in Figure~\ref{fig:LiTotfrac}
clearly show that time-dependent effects on the atomic kinetics are
indeed important, and that the time-dependent fractional population always lags
behind the steady-state calculation. Initially, the sudden drop in density
effectively drives ionization due to the weakening of three-body
(collisional) recombination rates. Subsequently, the temperature drop
dominates, and the system recombines. In the time-dependent calculation
this recombination proceeds at a much slower rate as compared to the
steady-state case that instantaneously adjusts to the local temperature
and density conditions. The cascade of population fed into high n levels
by 3-body recombination and driven downward toward the ground state by
collisional de-excitation  runs into a "bottle-neck" at the 1s$^{2}$3d
level which cannot depopulate fast enough as it would in the steady-state
case. As a result, the $\bf{b}$ line intensity (1s$^{2}$3d - 1s$^{2}$2p)
tends to overtake that of the $\bf{a}$ line (1s$^{2}$2p - 1s$^{2}$2s)
which is not seen in the steady state case (see Figure~\ref{fig:line_ratio}).  
This "bottle-neck" results 
in a decreased efficiency of collisional de-excitation to remove population
from the 1s$^{2}$3d level due to the larger energy difference between
1s$^{2}$3d and the 1s$^{2}$2p levels compared to energy differences between
high-n levels.  After an extensive
parameter study with temperatures and densities relevant to this 
region of the plasma plume, we found that a steady-state system would
not produce a condition where line {\bf b} was greater than {\bf a }.
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.65,clip=true ]{chap07/TD/LiTotfrac.eps}
\end{center}
\caption[Time-dependent and steady-state evolution of the total fractional population of Li atoms.]{\label{fig:LiTotfrac} A comparison of the evolution of time-dependent and steady-state atomic kinetic models for the same time history. The total fractional population of Li atoms lags in the time-dependent as compared to the steady-state results.} 
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.65,clip=true ]{chap07/TD/line_ratio.eps}
\end{center}
\caption[Time-dependent and steady-state effects on the ratio of line intensities of the Li I: $1s^{2}2p-1s^{2}2s$ to Li I:$1s^{2}3d-1s^{2}2p$.]{\label{fig:line_ratio} A comparison between time-dependent and steady-state evolution of the ratio of the lithium $1s^{2}2p-1s^{2}2s to 1s^{2}3d-1s^{2}2p$ line intensities. Cascading population from high n levels driven downward toward the ground state by collisional de-excitation  runs into a "bottle-neck" at the 1s$^{2}$3d level which cannot depopulate in the time-dependent case as fast as in the steady-state case.}
\end{figure}
}

Indeed, this time-dependent effect is observed in the data 
(see Figures~\ref{fig:tbs19_3},~\ref{fig:tbs18_8}).  First, all the lines show 
narrow and very similar line profiles.  This is indicative of the low density plasma 
in which plasma broadening effects are no longer important and line shapes are essentially
determined by the instrumental function.  Second, the Li {\bf b} line clearly overtakes the Li {\bf a}
line as times progresses.  We note that these experimental lineouts were extracted at
50 and 75 nsec after the end of the laser pulse and at distances 170 and 450 $\mu$m from the target's
surface, respectively. An expansion velocity of 10 - 20 $\mu$m/nsec was obtained from the
experimental CCD images. Thus, the experimental spectra shown in figures \ref{fig:tbs19_3},\ref{fig:tbs18_8} can be taken as produced by the same fluid element as it moves away from the target surface.  
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/TD/tbs19_3.eps}
\end{center}
\caption[Experimental lineout at x=170.0 $\mu$m and t= +50 nsec.]{\label{fig:tbs19_3}Experimental lineout at x=170.0 $\mu$m from the target surface, and t=+50 nsec after the end of the laser pulse.}
\end{figure}
}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.5,clip=true ]{chap07/TD/tbs18_8.eps}
\end{center}
\caption[Experimental lineout at x=450.0 $\mu$m and t=+75 nsec.]{\label{fig:tbs18_8}Experimental lineout at x=450.0 $\mu$m from the target surface, and t=+75 nsec after the end of the laser pulse.} 
\end{figure}
}