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\begin{document}
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}
  \begin{tabular}{c}
  Two Novel Cases of Plasma Characterization \\
  Through Spectroscopic Modeling\\
  \end{tabular}}}
%  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\LARGE\bfseries\color{yellow} Laser-ablated Li-Ag Plasma Plume}}
  \vspace{1.5cm}
% \centerline{\psframebox[fillstyle=solid,fillcolor=Beige, framearc=.1]{\bfseries 
\begin{center}
    {\large{Manolo E. Sherrill}}\\
    \vspace{0.50cm}
    {\small\textit{Theoretical Division, Los Alamos National Laboratory}}\\
    \vspace{2.00cm}
    March 28, 2005 \\
\end{center}
\end{slide}

% ******************************************************************************
% OVERVIEW SLIDE
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Two Cases of Spectroscopic Modeling}}
\begin{multicols}{2}
\textbf{Plasmas Formed During Laser Ablation}
  \begin{itemize}
     \item Most common laser produced plasma
     \item I$_{laser}=1\times10^{7}$ -- $1\times10^{10}$ W/cm$^2$
     \item T$_{e}$ = 1-3 eV
     \item Stoichiometrically complex targets
     \item Time-dep. atomic kinetics 
     \item Opacity and lineshape
     \item {\color{red}Uniqueness/Topology/Search Tech.}
  \end{itemize}
  \vspace{1cm}
\textbf{Intense Ultrashort Laser Plasmas}
  \begin{itemize}
     \item Non-Maxwellian FEEDF 
     \item Time-dep. atomic and e$^{-}$ kinetics
     \item I$_{laser}=1\times10^{15}$ -- $1\times10^{21}$ W/cm$^2$; $\tau=20-300 fs$
     \item T$_{e}$= 1000-1500 eV (Final T.E.)
     \item Ar cluster targets
     \item {\color{red}Atomic kinetics and plasma physics?}
  \end{itemize}
\end{multicols}
\end{slide}

% ******************************************************************************
%TITLE SLIDE ABLATION
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}
  \begin{tabular}{c}
  Spectroscopic Modeling of a Collisionally Confined \\
  Laser-ablated Plasma Plume
  \end{tabular}}}
%  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\LARGE\bfseries\color{yellow} Laser-ablated Li-Ag Plasma Plume}}
  \vspace{1.5cm}
% \centerline{\psframebox[fillstyle=solid,fillcolor=Beige, framearc=.1]{\bfseries 
\begin{center}
    M. E. Sherrill and R. C. Mancini\\
    {\small\textit{Department of Physics, University of Nevada, Reno, Nevada}} \\
    J. Bailey, A. Filuk, B. Clark, and P. Lake \\
    {\small\textit{Sandia National Laboratories, Albuquerque, New Mexico}} \\
    J. Abdallah, Jr. \\
    {\small\textit{Theoretical Division, Los Alamos National Laboratory}} \\
\end{center}
\end{slide}

% ************************************************************************
%EXPERIMENTAL TARGET
\begin{slide}
{ \renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{afigs/ExpTarget.eps}
\end{center}
\caption[Target Design]{\label{fig:ExpTarget} Target Design. }
\end{figure}
}
\end{slide}

%EXPERIMENTAL CONFIGURATION
\begin{slide}
{ \renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{afigs/ExpConf.eps}
\end{center}
\caption[Experimental Configuration]{\label{fig:ExpConf} Experimental Configuration. }
\end{figure}
}
\end{slide} 

%EXP. CCD IMAGES
\begin{slide}
{ \renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[scale=0.55,clip=true,trim=0in 0.3in 0in 1.9in]{afigs/ccdimages2.eps}
\end{center}
\caption[CCD Images]{\label{fig:ExpCCDimg}  CCD Images. }
\end{figure}
}
\end{slide}

%EARLY IN TIME- LINE INDENT.
\begin{slide}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.9, clip=true]{afigs/EarlyLineout.eps}
\end{center} \caption[Early Lineout]{\label{fig:EarlyLineout} Early in time and close to the target surface, this lineout (x=28.8 $\mu$m, t=+20 nsec) displays the high density characteristics of Stark broadened line shapes.} \end{figure}
}
\end{slide}

%PT CONTRIB
\begin{slide}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.40,clip=true]{afigs/Pt_contrib.eps}
\end{center}
\caption[Pt contribution in spectral data.]{\label{fig:Pt_contrib} The superposition of two lineouts shows the location of platinum lines in the x=28.8 $\mu$m, t=+20 nsec spectrum. The spectrum of a pure platinum target was recorded during the Li-Ag trails at t=+50 nsec, x=56.7 $\mu$m from the surface with a 10 ns gate time.}
\end{figure}
}
\end{slide}

\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Spectroscopic Model}}
\textbf{Best Fit - Synthetic and Experimental Spectra}
  \begin{itemize}
     \item Improving atomic structure calculations for high Z neutrals: energy levels and wavefunctions -- cross sections and rates
     \item Reducing the number of calculated cross sections Ag$^{+1}$, {\color{red}Ag$^{+2}$}, Ag$^{+3}$. - multi-representational model
     \item Multi-element collisional radiative model - common free e$-$ pool
     \item Local plasma effects on the radiator - detailed line shapes
     \item Non-local effects on the spectra - radiation transport accurate RTE solution
     \item Plasma gradients -- self-reversal feature
  \end{itemize}
\end{slide}

%AGI CATS vs EXP.
\begin{slide}
\begin{figure}
\begin{center}
%\includegraphics[angle=0,scale=0.38,clip=true]{afigs/AgI_TvsExp.eps}
\includegraphics[angle=0,scale=0.38,clip=true]{afigs/AgExpTh.eps}
\caption[]{\label{fig:Catsvsexp} Comparison between experimentally obtained energy level structure (left) and that generated with Cowan's Atomic Structure code (CATS) (right) for neutral Ag.}
\end{center}
\end{figure}
\end{slide}

%RCE-ACE CROSS SECTION COMP.
\begin{slide}
{\renewcommand{\baselinestretch}{1}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.38]{afigs/Ag5s5pCSrce.eps}
\end{center}
\caption[Comparison of electron impact excitation cross sections generated with and without RCE modifications of the wavefunctions.]{\label{fig:AgCSrce} Comparison of electron impact excitation cross sections generated with and with
RCE modifications of the wavefunctions. The two figures correspond to
an allowed $\bf{A}$ (left) and forbidden $\bf{F}$ (right) transition.}
\end{figure}
}
\end{slide}

%RCE-ACE RATES
\begin{slide}
\begin{figure}
\begin{center}
%\includegraphics[angle=0,scale=0.45,trim=0in 2.5in 0in 0in]{Adata/ACEnRCE.eps}
\includegraphics[angle=0,scale=0.45,clip=true]{afigs/Adata/ACEnRCE.eps}
\end{center}
\caption[Electron impact excitation rate for silver generated with and without wavefunction modifications by the RCE procedure.]{\label{fig:ACEnRCE} A comparison of an electron impact excitation rate for silver generated with and without wavefunction modifications by the RCE procedure.}
\end{figure}
\end{slide}

%POPULATION HISTOGRAM
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.40,clip=true]{afigs/Kin/TempEffPop.eps}
\end{center}
\caption[Effects of Temperature on Level Populations.]{\label{fig:AtomKinetic} The effect of temperature on fractional level populations for three ionization stages in Li and five in Ag atoms, for an electron density of 1x10$^{17}$ cm$^{-3}$.}
\end{figure}
\end{slide}

%Ag GROTRIAN DIAGRAM
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90.0,scale=0.45,clip=true]{afigs/appendix1/AgI_grotExp.epsi}
\end{center}
\caption[Ag I Grotrian diagram]{\label{fig:grotAg} Ag I Grotrian diagram. }
\end{figure}
\end{slide} 

%Ag LINESHAPES
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.45,clip=true]{afigs/Results/Ag5d5p_prof.eps}
\end{center}
\caption[{\color{red}RCE corrected} neutral silver $5d-5p$ line profiles]{\label{fig:Ag5d5p.prof} Neutral silver line profiles for various plasma electron densities with the two dominant $5d-5p$ fine structure transitions.}
\end{figure}
\end{slide}

%Li LINESHAPES
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.78,clip=true]{afigs/Results/Li3d2p.prof.eps}
\end{center}
\caption[Neutral lithium $3d-2p$ line profiles]{\label{fig:Li3d2p.prof}
Neutral lithium line profiles at various plasma electron densities for the
dominant $3d-2p$ transition. {\color{red} Note the asymmetry at the peak.}}
\end{figure} 
\end{slide}

%EFFECTS O.D. ON SPEC.
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.50,clip=true]{afigs/Results/constNe.ThinNThick.eps}
\end{center}
\caption[Superposition of optically thin and thick spectra for constant N$_{e}$ series.]{\label{fig:constNe.ThinNThick}Superposition of optically thin and thick spectra for constant N$_{e}$ series.}
\end{figure}
\end{slide}

%%OPTICAL DEPTH
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=0,scale=0.50,clip=true]{afigs/Results/opd_SpVsT_N.eps}
%\end{center}
%\caption[Optical depths for constant N$_{e}$ series.]{\label{fig:opdconstNe}The corresponding optical depths for constant N$_{e}$ series.}
%\end{figure}
%\end{slide}

%SPECTRA 
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.43,clip=true ]{afigs/Comp/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}
\end{slide}

%SPECTRA
\begin{slide}
\begin{figure} 
\begin{center}
\includegraphics[angle=0,scale=0.43,clip=true]{afigs/Comp/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}
\end{slide}

%SPECTRA
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.43,clip=true]{afigs/Comp/SpZone/spec21_0.6z.eps}
\end{center}
\caption[Exp. 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}
\end{slide}

%OPACITY
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.50,clip=true ]{afigs/Comp/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}
\end{slide}
%ZONES
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.40,clip=true ]{afigs/Comp/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$_{a}$ profile results for t= +30 nsec synthetic spectra.}
\end{figure}
\end{slide}

%INSIGHT SLIDE
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Summary \& Insight}}
\textbf{Concerning spectra whose features have a larger variation in optical depth}
  \begin{itemize}
     \item REVERSE PROBLEM - Large optical depths reduces the sensitivity of the spectra and leads to ambiguity in plasma characterization.
     \item We see multiple acceptable solution domains.
     \item Solution space contains many plateaus - standard search techniques begin to fail.
     \item What does the best fit mean? What do the other plateaus mean?
     \item Brute force searches to establish this pattern.
     \item New search approach - mesh refinement: coarse to fine.
  \end{itemize}
\end{slide}


% *****************************************************************************
%DOMAIN SPACE
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.42,clip=true]{afigs/Domain.eps}
\end{center}
\end{figure}
\end{slide}

% *****************************************************************************
%%EMIS and OPAC
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=0,scale=0.42,clip=true]{afigs/Comp/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}
%\end{slide}
%
%%EMIS and OPAC
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=0,scale=0.42,clip=true ]{afigs/Comp/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}
%\end{slide}

%TITLE SIDE BOLZMANN SOLVER
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}
  \begin{tabular}{c}
  Coupled electron and atomic kinetics through the solution \\
  of the Boltzmann equation for generating \\
  time-dependent X-ray spectra \\
  \end{tabular}}}
%  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\LARGE\bfseries\color{yellow} Laser-ablated Li-Ag Plasma Plume}}
  \vspace{1.5cm}
% \centerline{\psframebox[fillstyle=solid,fillcolor=Beige, framearc=.1]{\bfseries 
\begin{center}
    M. E. Sherrill, G. Csanak, J. Abdallah, Jr.\\
    {\small\textit{Theoretical Division, Los Alamos National Laboratory}}\\
    E. S. Dodd \\
    {\small\textit{Applied Physics Division, Los Alamos National Laboratory}}\\
    Y. Fukuda, Y. Akahane, M.Aoyama, N. Inoue, H. Ueda, K. Yamakawa\\
    {\small\textit{Advanced Photon Research Center, Japan Atomic Energy Research Institute Kyoto, Japan}}\\
    A.Ya. Faenov, A. I. Magunov, T.A. Pikuz, I. Yu. Skobelev\\
    {\small\textit{Multicharged Ions Spectra Data Center of VNIIFTRI, Mendeleevo, Russia}}
\end{center}
\end{slide}
%INTRODUCTION
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Introduction}}
\textbf{Problem Type} \newline
    Spectral modeling of highly NON-EQUILIBRIUM PLASMAS (N$_{a}$=1.0$\times$10$^{20}$ cm$^{-3}$). \newline
    Time-dependent systems where the electron energy distribution function (EEDF) is NON-MAXWELLIAN. \newline
\textbf{Implementation} \newline
       \begin{itemize}
         \item \textbf{Coupled electron and atomic kinetics}  \newline
             Stepping: EK solution is propagated to time $t+1$ with level populations from the AK at $t$. Then the AK are propagated to time $t+1$ with the EEDF  from $t+1$. The $t+1$ level populations from the AK model are provided to the EK.
         \item \textbf{Simultaneous?} \newline
            Too slow!
      \end{itemize}
\end{slide}
%INTRODUCTION CONTINUE
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Introduction}}
\begin{itemize}
    \item \textbf{Common Examples} \newline
       Electron beam driven X-ray lamps for lithography \newline
       Intense short pulse laser driven plasmas
\end{itemize}

\textbf{Electron Kinetics}
\begin{itemize}
    \item \textbf{Boltzmann Transport Equation}
     \begin{equation} \label{eq:BTE}
          \frac{\partial f(\vec{r},\vec{p},t)}{\partial t} + \frac{\vec{p}}{m}\bullet \nabla_{r} f(\vec{r},\vec{p},t) + F \bullet \nabla_{p} f(\vec{r},\vec{p},t) = \left( \frac{\partial f(\vec{r},\vec{p},t)}{\partial t} \right)_{coll}
     \end{equation}
     \item \textbf{0-D Boltzmann Equation} - (Non-relativistic)
     \begin{equation} \label{eq:BE}
          \frac{\partial f(E,t)}{\partial t} = \left( \frac{\partial f(E,t)}{\partial t} \right)_{coll}
     \end{equation}
\end{itemize}
\end{slide}
%BOLTZMANN FORMALISM I
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Formalism}}
   Following J.Bretagne et al\footnote{J. Phys. D, {\bf14} pp. 1225-39} we write the Boltzmann equation for the
        free-electron distribution function in the following form,

        \begin{equation} \label{eq:gen}
           \frac{\partial}{\partial t} f(E,t) = 
           \left(\frac{\partial f}{\partial t}\right)_{el({\color{red}e-e})} +
           \left(\frac{\partial f}{\partial t}\right)_{in} +
            A(t)S(E_{p},E)
        \end{equation}

where

        \begin{equation}\label{eq:inelastic}
           \left(\frac{\partial f}{\partial t}\right)_{in} = 
           K_{ion} + {\color{red}K_{3BR}} + K_{exc} + K_{de-exc} \ .
        \end{equation}

\end{slide}
%BOLTZMANN FORMALISM II
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Formalism}}
\textbf{Ionization Contribution}
        \begin{eqnarray}\label{eq:kioncomp}
           K_{ion}(E)=\sum_{iljm}K_{iljm}^{ion}(E)
        \end{eqnarray}
        where $K_{iljm}^{ion}(E)$ is the contribution to $K_{ion}(E)$
        from the
        \begin{eqnarray}\label{eq:coll}
           e^{-}+(il) \longrightarrow e^{-}+e^{-}+(jm)
        \end{eqnarray}
        ionization process where $(il)$ indices refer to the initial ion
        and its quantum state and $(jm)$ to the final ion and its
        quantum state, respectively.  $K_{iljm}^{ion}(E)$
        can be given in the form,
        \begin{align}
           \begin{split}\label{eq:ionrate}
               K_{iljm}^{ion}(E) &= N_{il}\bigg[\int_{E+E_{0}}^{2E+E_{0}}v'\sigma_{iljm}
               (E',E'\!-\!E_{0}\!-\!E)f(E',t)\,dE'\;+ \\
               &\quad+\int_{2E+E_{0}}^{E_{max}}v' \sigma_{iljm}(E',E)f(E',t)\,dE'\;+\\  
               &\quad-vf(E,t)\int_{0}^{(E-E_{0})/2}\sigma_{iljm}(E,E'')\,dE''\bigg] 
           \end{split}
        \end{align}

\end{slide}
%BOLTZMANN FORMALISM III
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Formalism}}
\textbf{Ionization Contribution:}
        The total ionization cross section for a process
        will be denoted by $Q_{iljm}(E)$ (in units of $cm^{2}$) and
        thus it can be obtained from $\sigma_{iljm}(E,E'')$
        via the formula,
        \begin{equation}\label{eq:Q}
           Q_{iljm}(E)=\int_{0}^{(E-E_{0})/2}\sigma_{iljm}(E,E'')\,dE''
        \end{equation}
        In the present work we assumed that
        $\sigma_{iljm}(E',E)$ can be written in the following form,
        \begin{equation}\label{eq:sigQomeg}
           \sigma_{iljm}(E',E_{1})=Q_{iljm}(E')\,\Omega(E',E_{1})
        \end{equation}
        where $\Omega(E,E_{1})$ is described as follows (Astrophys. J, 381 pp 597-600 1991)
        \begin{align}
          \label{eq:2Dcross}
          \begin{split}
             \Omega(E,E_{1})&=\frac{1}{(E-E_{0})(E^{2}+aE_{0}^2)}\\
             &\quad\times\left[2(a+1)E_{0}^{2}+
             \frac{160(E+E_{0})\left(E_{1}-\frac{1}{2}(E-E_{0})^4\right)}{(E-E_{0})^3}\right]
           \end{split}
        \end{align}
\end{slide}
\begin{slide}
        and is normalized, that
        \begin{equation}\label{eq:norm}
           \int_{0}^{(E-E_{0})/2}\Omega(E,E_{1})\,dE_{1}=1
        \end{equation}

\textbf{Numerical Solution of the Boltzmann Equation}

\textbf{Bin representation} \newline
    The EEDF can be represented a series of constant values.
    \begin{equation}\label{eq:fedf} 
       f(E,t)= f(E_{u}) \text{  for  } E_{u}-w_{u}/2\le E \le E_{u} + w_{u}/2
    \end{equation}
    where $E_{u}$ is the midpoint of the u$^{th}$ bin, $w_{u}$
    is the width of the u$^{th}$ bin and $N$ is the total number of bins.

\textbf{Final Form of the Boltzmann Equation}
    \begin{equation}
       \frac{d}{d t}f(E_{u},t) = \sum_{v=1}^{N} M_{u,v}(t) f(E_{v},t)
    \end{equation}
        where the $M_{u,v}(t)$ matrix contains all the collisional contributions.
\end{slide}
%EXPERIMENT DISCRIPTION
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.55, clip=true]{bfigs/ClustExp.ps}
\end{center}
\caption[EXP]{\label{fig:na}Laser: 200mJ, I=1.0e19 W/cm$^{2}$, 30 fs, Ti:sapphire laser. Prepulse: 1 ns duration 10$^{5}$ contrast ratio. Target: $1\mu$m cluster target. FSSR:focusing spectrometers with spatial resolution, $\lambda / \delta \lambda = $1667 . Spectrum: 600 shots per spectrum.}
\end{figure}
\end{slide}

%INIT the boltz solver
\begin{slide}
\centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow} Modeling the Experiment}}
\begin{itemize}
   \item \textbf{Laser-Target Interaction\footnote{JEPT Letters, {\bf{78}} pp. 115-118}} \newline
       Target ~100K clusters in beam volume \newline
       Beam more intense in the center, less on perimeter \newline
   \item \textbf{Laser-Cluster Interaction} \newline
       1 - 10 billion atoms \newline
       ~1 $\mu$m cluster vs 10 $\mu$m main pulse \newline
       {\large{ 3D - PROBLEM}} \newline
\end{itemize}
\end{slide}

%%Time Evol. Diagram
%\begin{slide}
%  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Plasma Evolution}}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=0,scale=0.40, clip=true]{bfigs/Model3.eps}
%\end{center}
%%\caption[Model]{\label{fig:model}}
%\end{figure}
%\end{slide}
%
%%PIC Na II
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=90,scale=0.45, clip=true]{bfigs/pic2/ni_xt_2.ps}
%\end{center}
%\caption[Image4]{\label{fig:na2}N$_{a}$, 30 fs laser approaching from the left.}
%\end{figure}
%\end{slide}
%
%%PIC Ne II
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=90,scale=0.45, clip=true]{bfigs/pic2/ne_xt_2.ps}
%\end{center}
%\caption[Image5]{\label{fig:ne2}N$_{e}$, 30 fs laser approaching from the left.} 
%\end{figure}
%\end{slide}
%
%%PIC Ey II
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=90,scale=0.45,clip=true]{bfigs/pic2/ey_xt_2.ps}
%\end{center}
%\caption[Image6]{\label{fig:ey2} 30 fs laser approaching from the left.}
%\end{figure}
%\end{slide}

%Time Evol. Diagram
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Plasma Evolution}}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.40, clip=true]{bfigs/Model3.eps}
\end{center}
%\caption[Model]{\label{fig:model}}
\end{figure}
\end{slide}

%PIC EEDF BEGINNING 
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/eedf.ps}
\end{center}
\caption[Image7]{\label{fig:eedf} PIC produced EEDF at the end of the laser pulse. }
\end{figure}
\end{slide}

%SURVEY
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/survey.ps}
\end{center}
\end{figure}
\end{slide}

%ADK 30fs
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/adk.ps}
\end{center}
\end{figure}
\end{slide}

%SPECTRAL EVOLUTION
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Plasma Evolution}}
\begin{itemize}
   \item CATS -   Cowan's Atomic Structure Code
   \item ACE  -   Collisional Excitation
   \item GIPPER - Collisional Ionization
   \item 3000 fine structure levels, Ne-like - H-like ionization stages.
   \item Configurations were truncated to principle number n=3.
   \item All possible n=1,n=2 X-ray transitions within the n=3 manifold.

\end{itemize}
\end{slide}

%SPECTRAL EVOLUTION
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/specevol.ps}
\end{center}
\end{figure}
\end{slide}

%THEORY vs EXP He alpha
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/healpha.comp.ps}
\end{center}
\end{figure}
\end{slide}

%%COOL SPECTRUM
%\begin{slide}
%\begin{figure}
%\begin{center}
%\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/coolspec.ps}
%\end{center}
%\end{figure}
%\end{slide}

%%THEORY vs EXP He beta 
%\begin{slide} 
%\begin{figure}
%\begin{center}
%\includegraphics[angle=-90,scale=0.4,clip=true]{bfigs/hebeta.comp.ps}
%\end{center}
%\end{figure}
%\end{slide}

%EVOLUTION EEDF
\begin{slide}
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=0.4,clip=true]{bfigs/evoeedf.eps}
\end{center}
\caption[Image7]{\label{fig:evoeedf} The time history of the EEDF, for times after the end of the laser pulse, computed by the Boltzmann solver for a constant atomic number density (N$_{a}$=$6.0\times10^{20}$ cm$^{-3}$). The system
evolves into a Maxwellian (1197 eV) in approximately t=12.0 ps.}
\end{figure}
\end{slide}

% CONCLUSION 
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Conclusion/Summary}}
  \begin{itemize}
     \item Used 1-D particle-in-cell code to provide the initial EEDF of the Boltzmann solver.
     \item EEDF after 3 ps was far from equilibrated (13 ps)
     \item 50keV - 60keV of the EEDF is required to obtain a consistent spectra.
     \item From this model we predicted 6.0x10$^{20}$ cm$^{-3}$ represented an average density for the highly ionized contributions to the He$_{\alpha}$ spectrum.
  \end{itemize}
\textbf{Future Work}
\begin{itemize}
     \item Complete 3-body recombination
     \item Include autoionziation and dielectronic recombination
     \item Consider a non-constant volume approximation.
     \item Pursue comparisons between collisional PIC and Boltzmann Solver solutions.
\end{itemize}
\end{slide}

%INSIGHT SLIDE
\begin{slide}
  \centerline{\psframebox[fillstyle=solid,fillcolor=blue,framearc=.1,framesep=2mm]{\large\bfseries\color{yellow}Insight}}
\textbf{Plasma and Atomic Physics}
  \begin{itemize}
     \item Detailed description of the state of the plasma - The importance of plasma physics?
     \item Electron and Ion interaction?
     \item Emission spectra is of the order of the material expansion time - Coulomb explosion?
     \item Higher Z material and relativistic cross section - Plasma physicist assumption?
  \end{itemize}
\end{slide}


\end{document}

