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% v7: edited by Vernon, July 2008
% v8: further editing by David, incoporating comments from MB, Aug 2008


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\begin{document}

\title{Spectroscopic Measurements of Temperature and Plasma Impurity
  Concentration During Magnetic Reconnection at the Swarthmore
  Spheromak Experiment}

\author{Vernon H. Chaplin}

\author{Michael R. Brown}

\author{David H. Cohen}

\author{Tim Gray} \affiliation{Department of Physics and Astronomy,
  Swarthmore College, Swarthmore, Pennsylvania 19081}

\author{Chris D. Cothran} \affiliation{Naval Research Laboratory,
  Washington, DC 20375} \affiliation{Department of Physics and
  Astronomy, Swarthmore College, Swarthmore, Pennsylvania 19081}




\maketitle

\begin{abstract}
  Electron temperature measurements during counter-helicity spheromak
  merging studies at the Swarthmore Spheromak Experiment (SSX) [M. R.
  Brown, Phys. Plasmas {\bf{6}}, 1717 (1999)] are presented.  VUV
  monochromator measurements of impurity emission lines are compared
  with model spectra produced by the non-LTE excitation kinematics
  code PrismSPECT [J. J. MacFarlane {\it et al.}, in {\it Proc.\ of
    the Third Conf.\ on Inertial Fusion Science and Applications}
  (2004)] to yield the electron temperature in the plasma with 1
  ${\mu}$s time resolution. Average $T_{\rm e}$ is seen to increase
  from 20 eV to $30-35$ eV during spheromak merging.  Average C {\sc
    {iii}} ion energy, measured with a new ion Doppler spectrometer
  (IDS) [C. D.  Cothran {\emph{et al.}}, Rev. Sci. Instrum.
  {\bf{77}}, 063504 (2006)], likewise rises during magnetic
  reconnection, also peaking at 30-35 eV, though this peak is earlier
  in time than the electron temperature peak.  The VUV emission line
  measurements are also used to constrain the concentrations of
  various impurities in the SSX plasma, which are dominated by carbon,
  but include some oxygen and nitrogen. A comparison of time-dependent
  and steady state modeling results indicates that excitation and
  ionization equilibrium are achieved within several ${\mu}$s of
  spheromak formation, and therefore steady-state simulations of the
  plasma during reconnection are adequate to describe the spectral
  line formation. A burst of soft X-ray emission is seen during
  reconnection with a new four channel detector (SXR).  Evidence is
  seen in the data for spectral changes in the soft X-ray emission as
  reconnection progresses, although our single-temperature equilibrium
  spectral models are not able to provide adequate fits to all the SXR
  data.

\end{abstract}

%\keywords{spheromak, flow, heating}

\section{Introduction}

Magnetic reconnection is the process driving the dynamics in spheromak
merging and relaxation \cite{Brown99} as well as in several
astrophysical scenarios \cite{Brown06b}.  During reconnection,
magnetic energy is rapidly converted to electron and ion heat, plasma
flow, and energetic particle beams \cite{Brown02,Brown02b}. However,
the partitioning of energy among various channels is not fully
understood.  Spheromak dynamics have been studied at the Swarthmore
Spheromak Experiment (SSX) in a number of geometries
\cite{Geddes98,Cothran03a,Cothran05,Brown06a,Brown06b}.  We have
recently been studying spheromak merging in a prolate 0.4 m diameter,
0.6 m length, 3 mm wall copper flux conserver at SSX (see Fig.\
\ref{Fig: SSX schematic}). In these experiments, merging of a pair of
counter-helicity spheromaks generates magnetic reconnection dynamics
near the midplane.  

\begin{figure}[h]
\begin{center}
\includegraphics[width=3 in]{figs/ssx_schmatic_mod_eps2eps.eps}

\caption{\label{Fig: SSX schematic} Schematic of the SSX, shown in
  cross-section. The device is cylindrically symmetric about the
  horizontal axis in the figure.  Spheromak plasmas are formed in the
  guns on the east and west sides of the vacuum chamber and ejected
  into the main flux conserver.  The lines of sight of the VUV
  monochromator, the soft x-ray detector, and the ion Doppler
  spectrometer are at the midplane, aligned with the short axis of the
  chamber and perpendicular to the long axis, and here are indicated
  schematically by the white squares (see Fig.\ \ref{Fig: SXR photo}).
  Arrays of magnetic probes are shown schematically as six black lines
  in the figure. The probes are removed hen we take spectroscopic
  data.  The contours represent the magnetic field lines of an FRC,
  the plasma structure that forms when two counter-helicity spheromaks
  merge.}
\end{center}
\end{figure}

Local and global magnetic structure of SSX spheromaks has been studied
with up to 600 individual internal magnetic probes operated
simultaneously at 1.25 MHz using a multiplexer system
\cite{Landreman03}.  Line averaged electron density is monitored with
a quadrature HeNe laser interferometer \cite{Brown02}.  We can scan
density in a range $n_e = 1-10 \times 10^{14}~{\rm cm^{-3}}$, and find
a typical value of $n_e = 5 \times 10^{14}~{\rm cm^{-3}}$.  The
composition of the SSX plasma is primarily hydrogen, with trace
concentrations of carbon, oxygen, and other impurities.  The plasma is
fully ionized and fully magnetized, with typical magnetic fields of
$0.1$ T and an ion gyroradius, $\rho_{\rm i} \ll R$, where $R = 0.2$ m
is the outer flux conserving boundary of the plasma (defined by a
cylindrical copper wall).  The Lundquist number $S$, the ratio of the
resistive magnetic diffusion time $\tau_{\rm R}$ to the Alfv\'en
transit time $\tau_{\rm A}$, is large for SSX, $S \approx 1000$.
Accordingly, the global structure of SSX spheromaks is fully in the
magnetohydrodynamic (MHD) regime ($S \gg 1, \rho_{\rm i} \ll R$).
Previous studies have constrained the plasma temperatures $T_{\rm e}
\approx T_{\rm i} \le 100$ eV \cite{Brown08,Cothran03a,Cothran06}.

Because magnetic reconnection is dynamically three-dimensional, it is
often difficult to identify the reconnection site within the
experimental volume or to determine whether the reconnection process
has convected into the line of sight.  In the experiments reported
here, spectroscopic data are line-integrated over a 1 cm wide chord
through a diameter at the midplane with a time resolution of about
$1~{\rm \mu s}$.  In addition, an ensemble of 20-30 discharges are
typically averaged for each measurement.

Line averaged electron temperature and impurity concentration levels
are inferred from model fits to data from a vacuum ultraviolet (VUV)
spectrometer.  Data are analyzed with a non-LTE excitation/ionization
kinematics code. This modeling is also used to explore the extent to
which time-dependent, non-equilibrium excitation and ionization has to
be accounted for.  Mean ion energy is measured with an ion Doppler
spectrometer (IDS) \citep{Cothran06}.

In section II, details of the spectral modeling using the PrismSPECT
code are presented. In section III, determinations of impurity
concentration from the VUV spectrometer are presented.  In section IV,
we present measurements of the plasma electron temperature made using
carbon emission line intensity ratios and compare these results with C
{\sc {iii}} ion temperatures measured with IDS. In section V, we
discuss the progress of additional electron temperature measurements
utilizing model fits to data from a four-channel soft x-ray (SXR)
array. A summary is presented in section VI.


\section{Spectral Modeling}

\subsection{PrismSPECT}

We analyzed the SXR and VUV data using the PrismSPECT non-LTE
excitation/ionization code \cite{PrismSPECT,MacFarlane04}.
Simulations were run for spatially uniform plasmas with a variety of
temperatures, densities, and compositions.  Model spectra and line
strength ratios from these simulations were compared to spectroscopic
data from SSX to determine best fit values for the plasma parameters
of interest. We also used time-dependent simulations to study the
plasma's approach to ionization/excitation equilibrium after spheromak
formation.

PrismSPECT models all significant radiative and collisional (including
three-body) processes, though not charge exchange.  In its time
dependent mode, the code computes atomic level populations at each
time step by integrating the relevant rate equations forward in time,
and the calculated level populations and transition probabilities are
used to produce a model spectrum.  In the steady-state mode,
equilibrium ionization balances and level populations are calculated
by inverting a rate matrix.  In both modes emissivities are calculated
from the level populations and the emitted spectra are transported
through the plasma taking optical depth effects into account. Doppler
shifts from bulk motion are not accounted for in PrismSPECT, but all
relevant line-broadening mechanisms -- including the effects of
natural, Stark, and thermal broadening -- are accounted for
\cite{Golovkin06}. This level of detail goes significantly beyond the
coronal approximation, which we show below is necessary for accurate
modeling of some of the emission lines we measure.

%\begin{figure}[h]
%\begin{center}
%\includegraphics[width = 2.4 in]{figs/Fig2a.eps}
%\includegraphics[width = 2.4 in]{figs/Fig2b.eps}
%\includegraphics[width = 2.4 in]{figs/Fig2c.eps}

%\caption{\label{Fig: Timedep vs. Steady Spectra} Comparison of spectra
%  from a 30 eV time-dependent simulation (top two panels) with a 30 eV
%  steady-state spectrum (bottom panel).  The ion density in the time
%  dependent simulation was $8\times10^{15}$ ions cm$^{-3}$ during the
%  first 10 ${\mu}$s (a typical value for the SSX plasma in the gun),
%  decreasing linearly to reach $5\times10^{14}$ ions cm$^{-3}$ (the
%  same value used for the steady state simulation) at $t =$ 30
%  ${\mu}$s. Initial atomic level populations were calculated based on
%  LTE at 0.025 eV in the time-dependent simulation.  The
%  time-dependent spectrum at $t =$ 5 ${\mu}$s already shares many
%  qualitative features with the steady-state spectrum, and by $t =$ 32
%  ${\mu}$s the two spectra are essentially indistinguishable.  }
%\end{center}
%\end{figure}

Atomic models for PrismSPECT simulations are drawn from the ATBASE
database \cite{ATBASE}.  Many of the higher energy levels of impurity
ions in the SSX plasma produce no strong spectral features, so we were
able to make simplifications to the full ATBASE models and reduce
simulation runtimes without sacrificing accuracy.  Tests of a variety
of atomic models revealed 200 energy levels per carbon ion and 50
levels per nitrogen ion to be sufficient, while the oxygen model used
contained 50 levels for O {\sc iii} through O {\sc vii} and one level
for the other ions.  Hydrogen in SSX is almost 100\% ionized, and line
radiation following recombination of protons and electrons was found
to make only negligible contributions to SXR signals, so we used only
the ground state of neutral hydrogen and ionized hydrogen in our
atomic model for H.  For each element, adding more levels per ion
stage beyond these described above did not lead to any significant
change in the calculated spectra.


\subsection{Equilibration Time}

The spheromak formation and expansion phases of an SSX discharge are
characterized by turbulence and rapidly changing temperatures and
densities \cite{Geddes98}.  Although it is difficult to estimate the
time-varying conditions at the beginning of a discharge with a high
degree of precision, the plasma conditions should not be varying
significantly as the spheromaks approach the midplane. Here we
investigate the extent to which the plasma is in ionization and
excitation equilibrium. If it is, then the time history of the
spheromak plasma is not important, and its spectrum can be modeled
with a simple collisional-radiative equilibrium approach, in which the
density, temperature, and composition are the only important
parameters.

The experiments we report on here focus primarily on the merging of
two spheromaks rather than on the spheromak formation process itself.
The merging and associated reconnection generally occurs at about $t =
30$ to 40 ${\mu}$s and lasts for 20 to 30 ${\mu}$s, after which the
plasma achieves a quasi-steady, relaxed configuration. The VUV
monochromator and SXR view the plasma along fixed lines of sight
through the midplane (see Fig.\ \ref{Fig: SSX schematic}), so measured
signals are typically zero for approximately the first 30 ${\mu}$s of
a discharge and peak around $t = 50-60$ ${\mu}$s.  Steady-state
simulations will therefore be sufficient if they can accurately
describe the conditions in the plasma at $t > 30$ ${\mu}$s.

To test this, we calculated the evolution of the excitation and
ionization of a hydrogen plasma with trace levels of carbon for a set
of assumed density and temperature values as a function of time, based
on our estimates of the conditions before, during, and after spheromak
formation. A variety of initial conditions were tested, with electron
temperatures in the plasma gun ranging from 0.025 eV to 60 eV. We
computed ultraviolet spectra as well as quantities such as the mean
charge state at many times during these simulations, and saw that
these tended to approach a steady state after $\sim 10$ $\mu$s. We
then compared the spectra at late times, between 30 and 40 $\mu$s,
with spectra calculated assuming equilibrium conditions. Indeed, these
late-time spectra are nearly identical.

\begin{figure}[t]
\begin{center}
%\includegraphics[width = 3 in]{figs/C20eVden5e14spectrum.eps}
%\includegraphics[width = 3 in]{figs/C40eVden5e14spectrum.eps}
\includegraphics[width = 3 in]{figs/Fig3a.eps}
\includegraphics[width = 3 in]{figs/Fig3b.eps}
\caption{\label{Fig: PrismSPECT spectra} Model spectra from
  steady-state simulations with $T_{\rm e}$ = 20 eV (top) and $T_{\rm
    e}$ = 40 eV (bottom).  Both simulations assume a uniform 40 cm
  thick plasma composed of 99\% hydrogen and 1\% carbon with an ion
  density of $5\times10^{14}$ cm$^{-3}$.  The temperature dependence
  of the $I_{97.7} / I_{155}$ line ratio is evident. Note also the
  presence of the Lyman edge at 91.2 nm, indicating that radiative
  recombination of hydrogen, rather than bremsstrahlung, is the
  dominant continuum process at work.}
\end{center}
\end{figure}

We can explore this numerical result analytically. Using the
expression for the collisional excitation cross section
\cite{Osterbrock}, we find the mean excitation time in SSX with
$n_{\rm i} = 5\times10^{14}$ ions cm$^{-3}$ and $T_{\mathrm e} = 30$
eV to be:
\begin{equation}
t\sim\frac{\sqrt{3kTm_{e}^{3}}}{n{\pi}{\hbar}^{2}}\sim2\times10^{-7}\,\,\mathrm{s.}
\end{equation}
Therefore, in a period of several ${\mu}$s, each ion in SSX will be
collisionally excited many times, indicating that equilibrium will be
reached on a time scale ($<$ 10 ${\mu}$s) that is short relative to
the plasma lifetime, consistent with the results of the detailed
simulations.  Based on the observed insensitivity of late-time
simulation results to initial conditions, tested for a range of
initial temperatures and atomic level populations, we conclude that we
can safely use steady-state simulations to model the VUV and SXR
spectra.

\subsection{Line Ratio Modeling}

\begin{figure}[h]
\begin{center}
%\includegraphics[width = 3 in]{/home/vchapli1/Paperfigures/97over155vsdensity.eps}
\includegraphics[width = 3 in]{figs/Fig4.eps}
\caption{\label{Fig: 97 / 155} Simulated $I_{97.7} / I_{155}$ line
  ratio plotted as a function of temperature for three different
  plasma densities (given in cm$^{-3}$). Note the mild density dependence of this line ratio. }
\end{center}
\end{figure}


The SSX plasma, in the time leading up to reconnection, should be
close to coronal equilibrium, in which all upward atomic transitions
are assumed to be caused by collisions between ions and electrons, and
all downward transitions are assumed to occur by spontaneous emission.
Similarly, ionization is by electron impact and is assumed to be
balanced by radiative recombination.  Since collisional ionization and
radiative recombination both scale with the square of the plasma
density, the overall ionization balance in a coronal plasma is density
independent, and the ratio of the intensities of two lines from
different ionization stages of the same element can serve as an
electron temperature diagnostic.  The primary line ratio used for our
temperature determinations was C {\sc{iii}} 97.7 nm / C {\sc{iv}} 155
nm (hereafter written as $I_{97.7} / I_{155}$).  Representative
spectra, calculated for two different temperatures, are shown in Fig.\
\ref{Fig: PrismSPECT spectra}, where these two carbon lines are
prominent.

\begin{figure}[h]
\begin{center}
%\includegraphics[width = 3 in]{/home/vchapli1/Paperfigures/C_ionization_balance.eps}
\includegraphics[width = 3 in]{figs/Fig5.eps}
\caption{\label{Fig: C ionization balance} Carbon ionization balance
  vs. electron temperature for an electron density of $5\times10^{14}$
  cm$^{-3}$.}
\end{center}
\end{figure}

Although the SSX plasma is close to coronal, we ran simulations that
included the full range of atomic processes and found a noticeable
density dependence in the $I_{97.7} / I_{155}$ ratio in the relevant
density and temperature regimes, as shown in Fig.\ \ref{Fig: 97 /
  155}.  Although the deviation from coronal equilibrium is slight, C
{\sc{iii}} and C {\sc{iv}} are subdominant ion stages at T $>$ 10 eV
(see Fig.\ \ref{Fig: C ionization balance}, where we show the
temperature-dependent carbon ionization balance), so small absolute
changes in their abundances can lead to large relative changes in the
corresponding line intensity ratio.  Because of this effect, it was
necessary to obtain independent density measurements in order to use
line ratios for precise calculations of electron temperature. We use
an electron density value of $n_{\mathrm e} = 5 \times 10^{14}$
cm$^{-3}$ based on measurements with the laser interferometer.


\section{Impurity Determinations from VUV Line Measurements\label{Sec: Impurities}}

\begin{figure}[t]
\begin{center}
\includegraphics[width = 3 in]{figs/Fig6a.eps}
\includegraphics[width = 3 in]{figs/Fig6b.eps}
\caption{\label{Fig: Sample VUV data} Sample VUV monochromator data.
  Top: A measurement of the C {\sc iv} 155 nm line intensity during a
  discharge with counter-helicity spheromak merging in SSX. Bottom: A
  measurement of the C {\sc iii} 97.7 nm line intensity during a
  single spheromak discharge. }
\end{center}
\end{figure}

Individual emission line strengths are measured in SSX using a vacuum
ultraviolet (VUV) monochromator with a focal length of 0.2 m.  Photons
enter the device through a slit of adjustable width and strike a
reflective diffraction grating, which selects and refocuses a narrow
bandwidth around the desired central wavelength.  Light leaving the
diffraction grating is directed through an exit slit of adjustable
width and into an 800 V photomultiplier tube (PMT).  500 ${\mu}$m was
found to be the optimal exit slit width, corresponding to a spectral
resolution of 2 nm.  Signals from the PMT pass through a Stanford
Research Systems SR570 current amplifier and are registered at 10 ns
intervals by an oscilloscope and transferred to a computer using
LabView. Sample traces from individual discharges for two different
lines are shown in Fig.\ \ref{Fig: Sample VUV data}.


\begin{table}[h]
\caption{\label{Tab: Lines observed}Impurity emission lines measured with the VUV monochromator. }
\small
\begin{tabular*}{3.4 in}{@{\extracolsep\fill}llll} \hline\hline
  Ion & ${\lambda}$ (nm) & Transition & Typical Signal\footnote{Typical signals quoted are averaged over the 40-60 ${\mu}$s interval of each discharge.} \\ \hline
  C {\sc iii} & 97.7 & $^{1}P_{1} \rightarrow \,\,^{1}S_{0}$ & 10 ${\mu}$A\\[1ex]
  C {\sc iv} & 155 & $^{2}P_{1/2, 3/2} \rightarrow \,\,^{2}S_{1/2}$ & 1-2 mA\\[1ex]
  C {\sc v} & 227.4 & $^{3}P_{0, 1, 2} \rightarrow \,\,^{3}S_{1}$ & not detected\\[1ex]
  C {\sc iii} & 229.7 & $^{1}D_{2} \rightarrow \,\,^{1}P_{1}$ & 10 ${\mu}$A\\[1ex]
  N {\sc iv} & 76.5 & $^{1}P_{1} \rightarrow \,\,^{1}S_{0}$ & not detected\\[1ex]
  N {\sc v} & 124 & $^{2}P_{1/2, 3/2} \rightarrow \,\,^{2}S_{1/2}$ & 100 ${\mu}$A\\[1ex]
  O {\sc iv} & 55.4 & $^{2}P_{1/2, 3/2} \rightarrow \,\,^{2}P_{1/2, 3/2}$ & not detected\\[1ex]
  O {\sc v} & 63.0 & $^{1}P_{1} \rightarrow \,\,^{1}S_{0}$ & 5-10 ${\mu}$A\\[1ex]
  O {\sc vi} & 79 & $^{2}D_{3/2, 5/2} \rightarrow \,\,^{2}P_{3/2}$ & not detected\\[1ex]
  O {\sc vi} & 103.5 & $^{2}P_{1/2, 3/2} \rightarrow \,\,^{2}S_{1/2}$ & not detected\\\hline\hline
\end{tabular*}
\end{table}

PrismSPECT was used to identify the carbon, nitrogen, and oxygen lines
most likely to be strong in SSX, and we tuned the monochromator to
measure these lines.  The results are summarized in Table \ref{Tab:
  Lines observed}.  Five emission lines produced signals that could be
reliably distinguished from background noise: C {\sc{iii}} 97.7 nm, C
{\sc{iv}} 155 nm, C {\sc{iii}} 229.7 nm, N {\sc{iv}} 124 nm, and O
{\sc{v}} 63.0 nm.  The strongest of these was C {\sc{iv}} 155 nm,
which consistently produced PMT currents over 1 mA.  C {\sc{v}} is the
dominant carbon ionization stage at SSX temperatures (see Fig.
\ref{Fig: C ionization balance}); however, being helium-like, its
resonance lines arise in the K-shell, and so are outside the
wavelength range accessible to the monochromator.  Therefore it is not
necessarily surprising that no C {\sc{v}} lines were detected. The C
{\sc v} line at 227 nm is from a transition between two excited
levels, and so is not expected to be extremely strong.

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{figs/NV124overCIV155.eps}
\includegraphics[width = 3 in]{figs/Fig7a.eps}
\caption{\label{Fig: N and O to C ratios} Line ratios calculated from
  PrismSPECT simulations with 0.1\% carbon, nitrogen, and oxygen
  impurities and $n_{i} = 5 \times 10^{14}$ cm$^{-3}$.  The N {\sc v}
  124 nm line is predicted to be at least 8 times stronger than the C
  {\sc iv} 155 nm line for plausible SSX temperatures (top), and the O
  {\sc v} 63.0 nm line is predicted to be over 970 times stronger than
  the C {\sc iii} 97.7 nm line (bottom).  The carbon lines are
  significantly stronger in the data than the nitrogen and oxygen
  lines, and those measurements, combined with the modeling shown here
  allow us to place an upper limit on the concentrations of nitrogen
  and oxygen relative to carbon in the SSX plasma.}
\end{center}
\end{figure}

The comparative strengths of C, N, and O lines can yield information
about the relative fractions of these impurities present in SSX.
These relative concentrations are a vital input for the simulations of
soft x-ray spectra to be discussed in Section \ref{Sec: SXR}.  Of
course, line ratios involving C, N, and O depend on temperature, the
the main parameter we are trying to constrain.  Nevertheless, limits
can be placed on relative impurity concentrations through comparisons
of VUV line measurements with simulations.

We consider the ratios N {\sc{v}} 124 nm / C {\sc{iv}} 155 nm and O
{\sc{v}} 63.0 nm / C {\sc{iii}} 97.7 nm, in order to constrain the
concentrations of N and O relative to C. We use the different carbon
lines in order to minimize the temperature dependence of each
diagnostic line ratio. As shown in Fig.\ \ref{Fig: N and O to C
  ratios}, with equal concentrations of C, N, and O present, the
simulated $I_{124} / I_{155}$ ratio is $> 8/1$ for all plausible SSX
plasma temperatures and about 20/1 for $T_{\rm e} = 30$ eV.  Yet, the
measured intensity ratio is 1/15, implying a concentration of N that
is roughly 300 times lower than that of C.  Similar modeling for
oxygen shows that the $I_{63.0} / I_{97.7}$ ratio is $>$ 970/1 for all
plausible SSX temperatures.  However, measured strengths of the two
lines are approximately equal, implying that carbon is $\sim 1000$
times more abundant than oxygen in SSX. The low concentrations of N
and O are not surprising, as copper gasket seals are used on the SSX
device and we employ an intensive program of high temperature bakeout
and helium glow discharge conditioning before running the experiment.


%\begin{figure}[h]
%\begin{center}
%\includegraphics[width= 3 in]{figs/Fig14.eps}
%\caption{\label{Fig: C III levels} Partial C\, {\sc iii} energy level
%  diagram.  Wavelengths corresponding to selected transitions are
%  given in nm.}
%\end{center}
%\end{figure}

%Stronger than expected signals were likewise observed for the C
%{\sc{iii}} 229.7 nm line.  Simulated $I_{97.7} / I_{229.7}$ and
%$I_{155} / I_{229.7}$ ratios exceeded the measured ratios by over an
%order of magnitude.  The 229.7 nm line is produced by an electron
%transition into the upper level of the 97.7 nm transition, so the
%observed ratio $I_{97.7} / I_{229.7} \sim 1$ could only occur if a
%substantial fraction of electrons were collisionally excited out of
%the 2p level before they had time to make the transition back to the
%ground state (see Fig. \ref{Fig: C III levels}).  We have yet to
%identify a likely mechanism by which this could occur. A strong 229.7
%nm line has been independently observed in at least two other
%spheromaks \cite{Turner,Cunningham}, although no publication we know
%of addresses the issue explicitly.  In the case of SSX, we cannot rule
%out a misidentification of the line; the C {\sc{v}} 227.4 nm line is
%expected to be strong at high temperatures (see Fig. \ref{Fig:
%  PrismSPECT spectra}).


\section{Temperature Measurements}

\subsection{Electron Temperature}

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{figs/Te_VUV_merging.eps}
\includegraphics[width = 3 in]{figs/Te_VUV_single.eps}
\caption{\label{Fig: TfromVUV} Electron temperatures for counter
  helicity merging (top) and single spheromak discharges (bottom)
  derived from $I_{97.7} / I_{155}$ measurements averaged over 25
  discharges for each line.  An ion density of $5 \times 10^{14}$
  cm$^{-3}$ was assumed.  Dashed lines give the uncertainty range,
  defined as the temperatures implied by a line ratio one standard
  deviation above or below the mean.}
\end{center}
\end{figure}

During the merging of two spheromaks, reconnection converts stored
magnetic energy into thermal and kinetic energy.  Understanding the
details of this energy budget is vital for modeling coronal heating
and other astrophysical phenomena.  Electron temperatures during
counter helicity spheromak merging were calculated by comparing
measured and simulated (Fig.\ \ref{Fig: 97 / 155}) values of the
carbon line ratio $I_{97.7} / I_{155}$.  Models show that even the
strongest emission lines in SSX are optically thin, so this comparison
is quite straightforward. We determine an average electron temperature
at each 1 $\mu$s bin from the mean line ratio taken from 25
discharges. The standard deviation of the line ratio at each time, for
the 25 discharges, was used to define the uncertainty in the electron
temperature. The results of the analysis of the $I_{97.7} / I_{155}$
ratio measurements are shown in Fig.\ \ref{Fig: TfromVUV}.


The average electron temperature during counter helicity discharges
increases from 20 eV to 35 eV during the $35-70 ~{\mu}$s time period
(reconnection begins to occur by $t \sim 40$ ${\mu}$s), providing
substantial evidence for magnetic energy being converted into heat.
$T_{e}$ rises rather gradually, however.  A lesser temperature
increase ($\sim 5$ eV) is seen during single spheromak discharges, as
the plasma relaxes into a lower-energy configuration around $t = 50$
${\mu}$s.  This measurement provides something of a control,
indicating that the heating observed during reconnection is
significant.

\subsection{Ion Temperature}

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3in]{figs/Fig11_corr.eps}
\caption{\label{Fig: Ion Temperature} Average ion temperature during
  counter helicity merging calculated from IDS measurements of the C
  {\sc iii} 229.7 nm line. These temperatures represent the mean of
  ten separate discharges during which the IDS view chord was along a
  diameter through the midplane. The measurement shown is a direct
  characterization of the line width, which we attribute completely to
  thermal broadening (after eliminating profiles with clearly
  non-thermal shapes). The data prior to $\sim 30$ $\mu$s, when
  spheromak merging begins, is too noisy to be reliable. }
\end{center}
\end{figure}

Ion temperature and flow dynamics are monitored at the midplane of SSX
with an ion Doppler spectrometer (IDS).  The IDS instrument achieves
high spectral resolution (0.0075 nm per pixel; equivalent to thermal
broadening of carbon at a temperature of only 3 eV) and can measure
the width and Doppler shift of impurity emission lines with 1 ${\mu}$s
or better time resolution \cite{Cothran06}. Non-thermal line profiles
are sometimes seen, especially early in the reconnection process.  The
most obvious non-thermal profiles, which have double-peaked
morphologies suggestive of bi-directional jets, have been removed from
the datasets used in this analysis.  However, some non-thermal
contribution to the line widths may still be present, and so the
quantity we derive from the measured line widths could perhaps be
better characterized as a mean ion energy, rather than strictly a
temperature. Fig. \ref{Fig: Ion Temperature} shows the mean C {\sc
  iii} ion energy during counter helicity merging, derived from the
second moment of the C {\sc iii} 229.7 nm lineshape data, averaged
over ten discharges.  Heating to 35 or 40 eV is seen as soon as
reconnection is initiated, and subsequent cooling to roughly 20 eV is
seen to occur around 40 $\mu$s, as the merging spheromaks begin to
approach their relaxed, equilibrium configuration.


% THOUGH WE'VE ELIMINATED THE FOLLOWING DISCUSSION; WE MIGHT WANT TO 
% REVIVE SOME OF THE ERROR ANALYSIS AND PUT IT ELSEWHERE
%As mentioned above, non-thermal broadening of
%the C {\sc iii} 229.7 nm line can lead us to overestimate $T_{i}$, but
%this effect would tend to increase the discrepency between $T_{e}$ and
%$T_{i}$.  A wavelength-dependence in the signal strength of the VUV
%monochromator is certainly a possibly; between 97.7 nm and 155 nm the
%uncertainty is likely 10 $\%$ or less, corresponding to a shift in
%$T_e$ of $<$ 3 eV in the 20-40 eV range.  Likewise, a factor of four
%uncertainty in the ion density would affect the calculated $T_{e}$ by
%as much as 5 eV (see Fig. \ref{Fig: 97 / 155}).  We were unable to
%take data with the VUV monochromator and measure density with the
%interferometer simultaneously, so we cannot rule out this possibility
%with $100\%$ certainty; however, interferometer data from other
%discharges suggests that $n_{i}$ is constrained between
%$3\times10^{14}$ cm$^{-3}$ and $6\times10^{14}$ cm$^{-3}$ for all
%discharges with the standard gas valve delay.  Therefore, the net
%effect of the aforementioned sources of systematic uncertainty on our
%$T_{e}$ measurement should be no more than $\pm$ 5 eV.  Referring back
%to Figs. \ref{Fig: TfromVUV} and \ref{Fig: Ion Temperature}, $T_{e} >
%T_{i}$ in SSX remains a noteworthy result, particularly at $t > 60$
%${\mu}$s.


\section{SXR Measurements\label{Sec: SXR}}

\begin{figure}[h]
\begin{center}
\includegraphics[width= 3 in]{figs/plasmaflange_small.eps}
\caption{\label{Fig: SXR photo} Location of SXR at the midplane on the
  SSX machine (see the schematic in Fig.\ \ref{Fig: SSX schematic}).
  The four wires at the top of the image carry current from the SXR
  photodiodes.  Clockwise from upper left, the diodes are filtered by
  foils made of Al, Zr, Sn, and Ti. The port for the IDS is visible at
  the bottom center of this image. Note the hydrogen Balmer series
  recombination emission is visible through the windows in the vessel
  in this photograph. }
\end{center}
\end{figure}

The SSX soft x-ray detector (SXR) \cite{Cothran03a,
  Hansen,Hokin,Soukhanovskii}, mounted at the mid-plane (see Fig.\
\ref{Fig: SXR photo}) and sampling a chord of plasma in the
reconnection region, measures the intensity of EUV and soft X-ray
emission from the SSX plasma with a time resolution better than 10 ns.
The SXR consists of a matched set of four International Radiation
Detectors AXUV silicon p-n junction photodiodes filtered by thin films
of aluminum (100 nm thick), titanium (50 nm), tin (100 nm), and
zirconium (100 nm).  The variation in the spectral response functions
of the filters at EUV and soft x-ray energies, shown in Fig.\
\ref{Fig: SXR responsivities}, allows us to garner information not
just about the time-dependent intensity of the soft X-ray emission,
but also about broad spectral properties of the emission.  Ultimately,
we intend to use this data to supplement the VUV line ratio analysis
and independently derive the electron temperature in SSX.  Although
the analysis of the SXR data lacks the simplicity of the line ratio
measurement we make with the VUV monochrometer, it has the distinct
advantage of allowing us to determine electron temperatures for
individual discharges, since the flux through all four filters can be
measured simultaneously. In addition, the SXR is sensitive to
significantly higher energy photons than the VUV monochrometer, and
thus can potentially measure emission from hotter plasma and even
emission produced by a possible high-energy, non-Maxwellian component
of the electron population.

\begin{figure}[h]
\begin{center}
\includegraphics[width=3 in]{figs/Fig9a.eps}
\includegraphics[width=3 in]{figs/Fig9b.eps}
\caption{\label{Fig: SXR responsivities} SXR filter responsivities \cite{filter_transmissions}.
  The colored lines show the response function of the filtered diodes
  in the 10 eV to 500 eV range, and the black line shows the response
  function of the unfiltered diode.  Line emission at E $>$ 500 eV was
  negligible at all simulation temperatures. The lower panel shows the
  same responsivity curves, but zoomed in on the range below 100 eV,
  where the bulk of the plasma emission is expected. }
\end{center}
\end{figure}


\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{figs/SXR_data4.eps}
\caption{\label{Fig: SXR data} Top: SXR data from a discharge with
  counter-helicity merging, smoothed over 1 ${\mu}$s intervals. Note
  the onset of strong soft X-ray emission around 30 $\mu$s, when
  reconnection begins.  }
\end{center}
\end{figure}

In Fig.\ \ref{Fig: SXR data} we show the time-dependent soft X-ray
signals measured in the four different SXR filters in one
counter-helicity spheromak merging shot.  The X-ray emission ramps up
quite rapidly at the onset of reconnection, around 30 $\mu$s, and is
sustained for several tens of $\mu$s, just the time frame that
magnetic measurements show reconnection occurring, with the relaxation
to a merged, equilibrium configuration typically around 60 $\mu$s
corresponding to a strong decrease in the SXR signal, as we expect for
thermal emission arising from the heating associated with magnetic
reconnection.  We note that the signal in the Al bandpass is stronger
than that seen in Ti or Zr, which is not surprising given the higher
and broader spectral responsivity of the Al channel, shown in Fig.\
\ref{Fig: SXR responsivities}.  The very strong Sn channel signal is
quite unexpected, however, as the responsivity of this channel is not
the highest at any photon energy.  We verified that there is no
low-energy leak (down to 8 eV) by blocking this channel with UV fused
silica and sapphire filters.  However, there must be a problem with
this channel, as there is no physical basis for the relative signal
being this high, so we exclude the Sn channel data from the analysis
we describe below.

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{figs/SXR_data3.eps}
\includegraphics[width = 3 in]{figs/SXR_ratios.eps}
\caption{\label{Fig: SXR ratios} Top: SXR data for the Al, Zr, and Ti
  filters from a discharge with counter-helicity merging, smoothed
  over 1 ${\mu}$s intervals.  Bottom: Time evolution of filter ratios
  for the discharge.  Sn filter data is omitted in order to more
  clearly show the time evolution of the other three ratios.}
\end{center}
\end{figure}


In Fig.\ \ref{Fig: SXR ratios} we again show the SXR filter traces,
but at an expanded scale afforded by the elimination of the Sn data.
The time-dependence of each signal appears similar, but in the lower
panel, where we show the time-dependent channel intensity ratios, it
is clear that the spectral signal of the X-ray emission changes in the
interval $35~{\mu}{\rm s} < t < 60$ $\mu$s.  This is exactly the time
frame of reconnection and relaxation toward a magnetic equilibrium
that is seen in the magnetic data. The Ti/Al ratio is the most
constant of the three during this time; the changes in the spectral
energy distribution seem to be primarily due to the Zr signal.
Looking at the response curves (Fig.\ \ref{Fig: SXR responsivities}),
Zr has a significantly harder spectral response than Al and unlike Ti,
has no significant response at low energies ($< 60$ eV).  Since the
change seen in the channel intensity ratios with time (lower panel of
Fig.\ \ref{Fig: SXR ratios}) is in the sense of the Zr signal becoming
weaker faster than the signal in the other two channels, this implies
a spectral softening with time during reconnection.


\begin{figure}[h]
\begin{center}
\includegraphics[width= 3 in]{figs/SXR30eVspectrum.eps}
\caption{\label{Fig: SXR Spectrum} Model spectrum from a steady-state
  simulation with $T_{e} = 30$ eV and $n_{i} = 5\times10^{14}$
  cm$^{-3}$.  Impurity concentrations of 1\% carbon, 0.001\% oxygen,
  and 0.0033\% nitrogen were included.  The plasma emits primarily at
  $E < 150$ eV, but SXR will also measure substantial emission from
  the C {\sc v} and C {\sc vi} resonance lines between 300 and 400 eV.
  Note that the Zr channel has the lowest responsivity at these
  energies, which is counter to what is seen below 100 eV.}
\end{center}
\end{figure}


We model the soft X-ray emission from the SSX plasma using the same
type of PrismSPECT simulations we described earlier in the context of
the VUV line ratio modeling.  For the SXR analysis, we include C, N,
and O in the ratios determined from the VUV line intensity
measurements, and perform similar non-LTE, steady state simulations,
calculating the emergent spectrum as a function of temperature at high
resolution between 10 and 500 eV.  The absolute impurity
concentrations (relative to hydrogen) are not important as long as
they are above one part in $10^4$, as our simulations show that the
spectra are completely dominated by impurity line emission even at low
concentration. A representative model spectrum is shown in Fig.\
\ref{Fig: SXR Spectrum}. We multiply the temperature-dependent model
soft X-ray spectra by the filter responsivities and integrate over
photon energy to produce a model of the relative signal strength in
each of the three (Zn, Al, Ti) channels.  We then compare the measured
intensity ratios from the three channels to the temperature-dependent
model predictions.  Note that by using ratios of the intensities we
avoid errors due to our uncertain knowledge of the exact emissivity in
the volume of plasma sampled by the SXR (which is affected by the
detailed density distribution in the sampled volume, as well as the
absolute impurity concentrations).

By comparing these model channel intensity ratios to the measured
ratios in 1 $\mu$s intervals, we can, in principle, determine the time
evolution of the electron temperature during a single discharge. The
model-data comparison procedure for two time snapshots is shown in
Fig. \ref{Fig: T Fitting}. What we find, however, is that there are
times (as in the bottom panel in the figure) where no single
temperature or small range of temperatures can explain all three
intensity ratios.  Perhaps the response at high energies is sensitive
to some non-equilibrium physics that we do not include in our models.
We have modeled the inclusion of Cu and other possible impurities, but
thermal line emission from additional species does not appear to
explain the discrepancies.

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{figs/SXRFit1.eps}
\includegraphics[width = 3 in]{figs/SXRFit2.eps}
\caption{\label{Fig: T Fitting} Illustration of the process used to
  fit SXR data to model spectra and determine $T_{e}$.  Channel intensity ratios
  calculated from models are plotted as a function of temperature
  (solid lines).  Measured ratios at $t = 35$ ${\mu}$s (top
  panel) and $t = 55$ ${\mu}$s (bottom panel) during a typical single
  spheromak discharge are plotted as points at the location of the
  calculated best-fit temperature for the time step.  For illustrative
  purposes, 30\% error in the measured ratios is assumed.  At $t = 35$
  ${\mu}$s, a good fit is achieved at $T = 30$ eV.  However, at $t =
  55$ ${\mu}$s, all three measured ratios cannot match the models at
  any one temperature, and the best-fit temperature of 30 eV may not
  accurately reflect the actual $T_{e}$ in the plasma. }
\end{center}
\end{figure}



\section{Summary}

We have shown that high time resolution UV spectroscopy is a useful
tool for characterizing the physical properties of the SSX plasma
during and after magnetic reconnection.  Electron temperature can be
estimated from impurity emission line ratios, but it is important to
take into account deviations from coronal equilibrium, which requires
detailed spectral modeling.  This same modeling also shows that
ionization and excitation equilibrium is reached within less than 10
$\mu$s after the plasma is discharged from the guns, so that
steady-state simulations are sufficient for describing the plasma
emission at the time that reconnection occurs (beginning 30 or 40
$\mu$s after the plasma discharge) and, furthermore, that the exact
conditions in the plasma guns do not need to be known in order to
accurately model the plasma properties around the time of
reconnection.

Comparing model line ratios with measured UV line emission from
various impurity species allows us to place constraints on the
relative impurity concentrations in the plasma.  Carbon is the
dominant impurity in SSX, with a concentration approximately 300 times
greater than that of nitrogen and 1000 times greater than that of
oxygen.  However, even the less abundant elements make a significant
contribution to the UV and soft X-ray line emission. 

For the first time in SSX we have made precise, high time resolution
measurements of the electron temperature during counter helicity
spheromak merging.  Energy released by magnetic reconnection causes
$T_{e}$ to rise from approximately 20 eV to 35 eV on average during
the 40 $\mu$s to 70 ${\mu}$s interval of a discharge. Average ion
energies, as determined by line width measurements using an ion
Doppler spectrometer also rise to roughly 35 eV, but much more
rapidly; essentially as soon as reconnection begins. The delay in the
electron heating may be an effect of the much greater mobility of
electrons, compared to ions, and our sampling of a limited volume of
the reconnecting plasma.  In this case, the hot ions created thermally
or non-thermally during reconnection stay within the centimeter-scale
volume sampled by the IDS and VUV instruments, while the hot
electrons, with their higher velocities and larger mean free paths,
escape the region quickly.  The electrons would have to be heated
throughout the bulk of the SSX chamber before an appreciable signal in
the carbon line ratios (or the SXR, presumably) could be seen.

Utilizing the information about impurity concentrations derived from
UV line intensity measurements, we have developed a framework for
monitoring electron temperature during individual discharges using a
broadband soft x-ray detector, and the same spectral modeling tools
that we used successfully to model the UV emission lines. The soft
X-ray signal measured by the SXR increases dramatically during
reconnection, and also shows a modest change in its spectral energy
distribution. However, our steady-state non-LTE models (which assume
statistical equilibrium in the excitation/ionization state of the
plasma) cannot reliably reproduce the time evolution of the SXR
signals, indicating that additional physics -- for example,
non-Maxwellian particle distributions -- must be included in future
modeling.


\begin{acknowledgments}

  The authors gratefully acknowledge the technical assistance of S.
  Palmer and J. Haldeman at Swarthmore, and J. MacFarlane at Prism
  Computational Sciences, as well as some preliminary modeling done by
  V. Swisher at Swarthmore. This work was supported by DOE grant XXX
  and by a Eugene M. Lang summer research fellowship from the
  Provost's Office at Swarthmore College.

\end{acknowledgments}


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