% v1 of Cohen, Landen, MacFarlane - Hohlraum Radiation Modeling
% v1 on 4 Jan 04 ... completed 26 Jan 04 and sent to JJM
% v2 on 10 feb 04, after a few general comments from Joe
% v3 on 7 may 04, incorporating Nino's comments, of 18Feb04
% v4 on 17 may 04, incorporating Joe's comments on prev. version
%        as of this date, Joe's still got a couple of more comments still coming
%        OK, have more comments (18may04); 20may04 putting final touches on, 
%        before sending manuscript to Nino for further comments
%        --> sent to Nino on 21May04
% v5 on 1 jul 04, incorporating Nino's last comments (completed 5 July 2004)
% v6 on 3 sep 04, after Nino's msg about pulse shapes, and Joe finding 
%        conversion error in T_hohl calc
% v7 on 13 oct 04, incorporating capsule comparisons (and after 
%        determing that the pulseshapes don't exist)
% v8 on 1 nov 04, after getting next set of comments from Nino and Joe and 
%        incorporating LEH shield simulations - bulk of changes actually implemented
%        on 20 dec 04
%

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\begin{document}


\title{Numerical Modeling of Hohlraum Radiation Conditions: Spatial
and Spectral Variations Due to Sample Position, Beam Pointing, and
Hohlraum Geometry}

\author{David H. Cohen}
\affiliation{Department of Physics and Astronomy, Swarthmore College,
Swarthmore, Pennsylvania 19081}

\author{Otto L. Landen} \affiliation{Lawrence Livermore National
Laboratory, Livermore, California 94550}

\author{Joseph J. MacFarlane} \affiliation{Prism Computational
Sciences, 455 Science Dr., Suite 140, Madison, Wisconsin 53711}


\maketitle

\begin{abstract}

  View-factor simulations are presented of the spatially varying
  radiation conditions inside double-ended gold hohlraums and
  single-ended gold hohlraums (``halfraums'') used in inertial
  confinement fusion (ICF) and high energy density (HED) physics
  experiments [J. Lindl, Phys. Plasmas {\bf 11}, 339 (2004); M. D.
  Rosen, Phys. Plasmas {\bf 3}, 1803 (1996)].  It is shown that in
  many circumstances, the common assumption that the hohlraum
  ``drive'' can be characterized by a single temperature is too
  simplistic.  Specifically, the radiation conditions seen by an
  experimental package can differ significantly from the wall
  reemission measured through diagnostic holes or laser entrance holes
  (LEHs) by absolutely calibrated detectors.  Furthermore, even in
  situations where the radiation temperature is roughly the same for
  diagnostics and experimental packages, or for packages at different
  locations, the spectral energy distributions can vary significantly,
  due to the differing fractions of reemitting wall, laser hot spots,
  and LEHs seen from different locations.  We find that the spatial
  variation of temperature, and especially the differences between
  what diagnostics looking in the LEH measure vs.\ the radiation
  temperature on wall-mounted experimental packages, is generally
  greater for double-ended hohlraums than it is for halfraums.
  View-factor simulations can also be used to explore experimental
  variables (halfraum length and geometry, sample position, and beam
  pointing) that can be adjusted in order to, for example, maximize
  the radiation flux onto a sample, or other package.  In this vein,
  simulations of hohlraums and halfraums with LEH shields are also
  presented.

\end{abstract}


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\section{Introduction\label{sec:intro}}

Optimizing the time- and wavelength-dependent hohlraum radiation drive
onto a fuel capsule is a key component in achieving ignition in
inertial confinement fusion (ICF) experiments \cite{lindl04,rosen96}.
Although much experimental and theoretical effort has been expended in
understanding the x-ray drive characteristics of hohlraums and
optimizing the drive symmetry onto the capsule, there have been few
studies of the spatial variation of the radiation field conditions
within hohlraums, and especially within halfraums.  The x-ray spectrum
incident on a surface in a hohlraum or halfraum, whether part of the
wall, a fuel capsule, or some other object within the cylinder, will
vary with location and orientation of the surface according to the
relative view factors of wall reemission, laser hot spots, and cold
laser entrance holes (LEHs) and diagnostic holes.  Detailed
view-factor modeling can play an important role in answering questions
about this variation, and can be used to interpret diagnostics and
plan experiments.

In this paper we will present view-factor models of hohlraums and
halfraums, investigating the spatial variations of the radiation field
(both overall intensity and spectral energy distribution), and the
effects of halfraum size and geometry and of beam pointing.  One
conclusion from this modeling is that care must be taken in inferring
the drive onto an experimental package from a measurement of wall
reemission from a particular direction when using an absolutely
calibrated detector, such as DANTE \cite{kornblum86}.  A more general
conclusion is that view-factor simulations are a valuable tool for
optimizing the performance of hohlraum experiments and in interpreting
diagnostic measurements.

By their nature, view-factor simulations do not account directly for
hydrodynamics, laser-plasma interactions, or detailed atomic physics.
View-factor calculations are based on power balance between energy
source terms and radiation losses (absorption) and reemission
(reflection).  On the other hand, view-factor simulations provide a
reasonable representation of the radiation field distribution
throughout a 3-D grid of surfaces, and do so at a small fraction of
the CPU costs of multi-dimensional radiation-hydrodynamics codes.
View-factor codes can be used effectively in estimating hohlraum
radiation temperatures and predicting the radiation symmetry on ICF
capsules. The simulations we present here are relevant to all but the
latest times of laser hohlraum experiments when on-axis stagnation of
gold plasma contributes significantly to the radiation properties of a
hohlraum and the associated interpretation of diagnostics.

We will critically examine the standard analytic treatment of hohlraum
energy balance, in which the radiation properties of a hohlraum are
described by a single ``hohlraum radiation temperature.'' And although
the emission from each computational surface element in our
view-factor simulations is taken to be Planckian, the flux incident on
any given surface in a simulation (whether wall, target, or
diagnostic) can be distinctly non-Planckian.  We will show examples
where deviations from a blackbody spectrum are non-trivial.  We begin
by benchmarking DANTE measurements of a hohlraum experiment
\cite{decker97} on OMEGA \cite{boehly97}.  We then show that
experimental packages can be subject to radiation conditions that are
quite different than those seen by DANTE, even when that diagnostic is
used on an optimal LEH-viewing line of sight.

The focus of this study will be on effectively empty hohlraums and
halfraums.  However, we do investigate how the radiation conditions
change with the presence of a capsule in Sec.\ \ref{sec:capsule}. In
Sec.\ \ref{sec:halfraum} we explore the fundamental differences
between hohlraums and halfraums, in terms of both DANTE measurements
and the radiation onto an experimental package.  (Note that we use the
terms ``sample'' and ``package'' interchangeably in this paper.)
Finally, we show how variations in the beam pointing and LEH size
(Sec.\ \ref{sec:pointing}) and geometry (length, presence of disks or
foils near the LEH) of a halfraum, and internal LEH shields in a
hohlraum, (Sec.\ \ref{sec:geometry}) affect the interpretation of
diagnostics and how they can be optimized to produce the maximum
possible radiation temperature onto an experimental package.


\section{Spatial Dependence of the Radiation Drive in a Hohlraum\label{sec:hohlraum}}

We first present the results of simulations of a set of OMEGA
experiments reported on by Decker {\it et al.}  \cite{decker97}. In
these well-characterized experiments, the hohlraum temperature was
inferred by measuring the absolute flux emitted from a portion of the
hohlraum wall.  For these experiments, ten 42 degree OMEGA beams (Cone
2) and twenty 59 degree OMEGA beams (Cone 3), having a total energy of
500 J each and with 1 ns square profiles, illuminated a standard (2300
\micron\ length by 1600 \micron\ diameter) gold hohlraum, with
three-quarter (or 1200 \micron\ diameter) LEHs.  Three-quarter LEHs
are used in all the models we present in this paper, unless otherwise
noted. The beam pointing in these experiments and in our modeling was
such that all 15 beams on each side of the hohlraum made a single ring
of hot spots, centered 480 \micron\ from the LEH plane.  The beams
were all focused at the pointing spot, where they crossed the long
hohlraum axis (at the LEH plane for the Cone 3 beams and 400 \micron\
outside of the hohlraum for the Cone 2 beams).

One purpose of the Decker {\it et al.}  experimental campaign was to
show that the absolutely calibrated x-ray detector, DANTE, gives a
better sense of the hohlraum radiation conditions seen by a capsule
when it views the wall reemission through the LEH, rather than through
a diagnostic hole at the midplane.  The hohlraums in these experiments
were on the P5-P8 axis of the OMEGA target chamber, so that the DANTE
viewing angle was 37.4 degrees with respect to the hohlraum axis (see
Fig.\ \ref{fig:hohlraum_target} for a model of the hohlraum target,
including the DANTE view of this configuration).


% FIG 1: \label{fig:hohlraum_target}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.38]{figs/hohlraum_decker_1}
\includegraphics[scale=0.38]{figs/hohlraum_decker_dante}
\includegraphics[scale=0.38]{figs/hohlraum_decker_dante_3}
\includegraphics[scale=1.0]{figs/colorbar}
\end{center}
\caption{Hohlraum images generated with the VISRAD view-factor code,
  relevant to the experiments discussed in reference \cite{decker97}.
  The top panel shows the OMEGA beam pointing into the hohlraum
  cylinder seen side-on.  Note that the Cone 2 beams on each side are
  pulled back so that the beams from both cones make a single ring on
  each side of the hohlraum.  The middle panel shows the same target
  model, but from the position of the DANTE diagnostic.  The lower
  panel shows the DANTE view again, but with the beams hidden, and
  with the wall temperature at $t = 1.0$ ns displayed as a color map
  (the dynamic range in this, and all other, temperature color maps
  shown in the paper is 140 eV to 220 eV).  Note the ring of laser hot
  spots on each side of the hohlraum.  Note also in all of these
  images how structures in the model seen from the back, or outside,
  are rendered as transparent mesh to allow for an unobstructed view
  of the interior of the hohlraum.  This convention will be used
  throughout the paper. \label{fig:hohlraum_target} }
\end{figure}


We performed a series of simulations of these hohlraum experiments on
OMEGA using the VISRAD 3-D view-factor code (v3.1) \cite{jjm03}.
VISRAD computes the spatially-dependent radiation flux about a 3-D
grid of surface elements using a steady-state power balance model and
material-dependent reflection fractions (albedos).  Each surface
element is treated as a spatially thin, optically thick Lambertian
source with a Planckian frequency dependence.  Thus, the
(non-Planckian) spectrum incident on a given surface element is
composed of contributions from multiple Planckian sources. These
contributions to the radiation flux incident on each surface element
are summed for each surface element over all other source elements,
accounting for the solid angle of the source as seen from the sample,
as well as the incident angle of the source radiation onto the sample.
See Fig.\ \ref{fig:viewfactor_sketch} for a sketch showing these
geometrical considerations.  Note that the Lambertian flux on the
sample is proportional to $\cos\theta\cos\phi$.

% FIG 2: \label{fig:viewfactor_sketch}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{figs/viewfactor_sketch.eps}
\end{center}
\caption{Schematic of the view-factor calculation for an arbitrary
geometry.  The flux from any source surface element onto any other
sample surface element is proportional to the cosine of the angle
between the line of centers of the two elements and the surface normal
of the source element (because the source element is assumed to be a
Lambertian emitter) and also to the cosine of the line of centers and
the normal of the sample element (accounting for the projected cross
section of the sample as seen by the source). The line of centers is
indicated by the dashed line while the two surface normals are
indicated by arrows.
\label{fig:viewfactor_sketch} }
\end{figure}


Laser beam energy deposition is computed using realistic space- and
time-profiles for the beams (including the $f/6.7$ beam effective
focal ratio of OMEGA), in conjunction with 3-D ray-trace algorithms
for determining beam-target intersections. The 3-D ray-trace
algorithm, in which each laser beam is sub-divided into a large number
of ``beamlets,'' is used to determine which surface elements are hit
by a given portion of a laser beam.  While VISRAD also computes the
specular reflection of laser light off surfaces (glint), laser
reflectivities for all surfaces were assumed to be zero for the
simulations described here.  The target components in the calculation
are modeled as a discretized grid of surface elements.  The
time-dependent albedo and x-ray conversion efficiency (XCE) are input
parameters.  In the simulations discussed below, the albedo is based
on 1-D radiation-hydrodynamics simulations of a gold foil exposed to a
radiation drive consistent with that in the OMEGA experiments.

We note that the view-factor modeling ignores hydrodynamic motion of
the hohlraum walls, as well as non-equilibrium effects, internal
energy stored in the walls by the radiative heating of the object
elements, and detailed opacities and emissivities for the hohlraum.
This modeling also neglects the effect of temperature gradients in the
hohlraum walls, which can lead to a viewing-angle dependence of
emission temperature.  We stress that view-factor codes play a
complementary role to atomic and hydrodynamics codes.  Our goal here
is not to calculate wall motion nor the detailed atomic physics and
associated line spectra.  However, the view-factor modeling accounts
for the spatial variation of the radiation conditions, and to some
extent, the generally non-Planckian spectra in hohlraum environments
(via the summation over numerous blackbody surface elements of
different temperatures).  The time variation of the hohlraum radiation
properties in a VISRAD simulation are computed by making a series of
steady-state calculations, each using appropriate beam powers, x-ray
conversion efficiencies, and albedos.

To model the OMEGA experiments described above, we calculated the
radiation onto a surface at the position of the DANTE diagnostic every
100 ps, using the beam and hohlraum properties described at the
beginning of this section.  We assumed perfect square pulses with
exactly $1.0$ ns duration and $500$ J total energy per beam.  We
incorporated a simple model of the laser x-ray conversion efficiency,
with a linear ramp up to a value of 0.55 at 200 ps, and a constant
value thereafter.  Here, the x-ray conversion efficiency refers to the
fraction of incident laser power that is converted to x-ray radiation.
The remainder of this energy is converted into kinetic or internal
energy, or can be lost to the system via laser scattering.  In our
view-factor calculations, the partitioning of this non-radiative
energy is not modeled.

For the gold albedo, we use the results of a 1-D, time-dependent
hydrodynamics simulation of gold reemission of x-rays.  The albedo
value peaks at 0.73 at 1 ns, for a constant power drive reaching
approximately 190 eV.  In Fig.\ \ref{fig:decker} we show the assumed
XCE and calculated albedo along with the modeled DANTE temperatures
and the DANTE data obtained by Decker {\it et al.}
\cite{decker97}. Note that our modeling reproduces the observed DANTE
data at all times well within the 6\% errorbars on the data.  These
simulations do not include a capsule, in order to facilitate
comparisons to an equivalent halfraum.  In the next section we show
that the presence of a capsule simply lowers the calculated radiation
temperature and does so quite uniformly, so that there is effectively
a degeneracy between the presence of a capsule and an increase in the
XCE.


% FIG 3: \label{fig:decker}
%

\begin{figure}
\begin{center}
%\includegraphics[scale=0.2]{figs/decker_xce}
%\includegraphics[scale=0.2]{figs/decker_albedo}
\includegraphics[scale=0.45]{figs/decker_albedo_and_xce_v3}
\includegraphics[scale=0.45]{figs/decker_data_v3}
\end{center}
\caption{The top panel shows the assumed x-ray conversion efficiency
(dashed line) and calculated albedo (solid line), used as inputs to
the view-factor simulations, the results of which are shown in the
lower panel.  In the lower panel, the filled squares with error bars
are the DANTE temperature measurements from ref.\ \cite{decker97}
while the open squares are the simulated DANTE temperatures from the
view-factor calculations.  The circles are the simulated radiation
temperatures at the midplane wall of the hohlraum. \label{fig:decker}}
\end{figure}


In Fig.\ \ref{fig:decker} we also show the time-dependent radiation
temperature on the hohlraum wall at the midplane.  It is significantly
($\sim 15$ eV) lower than the DANTE temperature.  This is due to the
less favorable view factor of laser hot spots from the midplane wall
compared to DANTE, but mainly due to the contribution from the cold
LEHs.  In various hohlraum experiments, some type of sample, or other
package, is placed at the hohlraum midplane, to expose it to the
radiation drive.  Historically, hohlraum radiation conditions have
also been diagnosed from midplane wall reemission flux \cite{suter96,
  stygar01, olson03}, which is equal to the midplane wall incident
radiation multiplied by the albedo.  Clearly, hohlraum radiation
diagnostic results will vary depending on the location of the wall
reemission they sample.  Thus, while DANTE measurements through the
LEH provide valuable data on the hohlraum radiation characteristics,
they do not provide a direct measure of the radiation field seen by an
experimental package.  The differences between the wall temperatures
seen by DANTE and the radiation drive temperatures seen by a package
can be $\sim 7 - 8$\%.  This means that the radiation flux ($F_{rad} =
\int^{\infty}_0 F_{\nu} d{\nu} = {\sigma}T_R^4$; note that ``radiation
temperature,'' $T_R$, is defined by this equation), and therefore the
energy absorbed by the experimental package, can be different by $\sim
30 - 35$\% compared to what would be inferred using the DANTE
measurement directly.

It is useful to compare these detailed calculations to the simpler,
and more traditional, analytic power balance treatment.  Using
${\eta}P_L = P_{rad} = ((1 - \alpha)A_{wall} + A_{LEH}) {\sigma}T_R^4$
(see, for example, eqn.\ 1 in ref.\ \cite{suter96}) and a value of
$\eta = 0.55$ for the XCE, we find at $t=1.0$ ns when the albedo is
$\alpha = 0.73$, a value of $T_R = 192$ eV for the ``hohlraum
radiation temperature.''  This, as expected, is somewhat less than the
DANTE temperature, both in our modeling (202 eV) and in the
experiments (201 eV), since DANTE, unlike any point inside the
hohlraum, does not see any of the cold LEH regions.  It is very close
to the radiation temperature on the sample (191 eV, for a wall-facing
sample at the center of the hohlraum, 190 eV for an LEH-facing sample,
and 188 eV for a wall-mounted sample), as expected. Note that
increasing the XCE to $\eta = 0.69$ gives an analytic hohlraum
temperature that agrees with the DANTE temperature at $t = 1.0$ ns.
This higher value of the XCE is in better agreement with the values
that are usually assumed \cite{suter96}.

It is interesting also to compare the spectral energy distribution of
the radiation incident on the midplane to that measured by DANTE.  In
Fig.\ \ref{fig:hohlraum_spectra} we show the simulated DANTE spectrum
at $t=1.0$ ns along with that incident on the midplane hohlraum wall.
For reference, we also show the equivalent blackbody spectra (the
Planckian spectra having the same integrated power, or radiation
temperature, as the calculated spectra).  Note that both spectra are
harder than the equivalent blackbody spectra, and that this effect is
somewhat stronger for the midplane, where the significant view factor
of cold LEHs replacing part of the wall leads to a deficit of
low-energy photons.


% FIG 4: \label{fig:hohlraum_spectra}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figs/hohlraum_spectra_log_lin_jul04}
\end{center}
\caption{The simulated DANTE spectrum (solid black) along with the
equivalent blackbody spectrum (dashed black) for $t=1.0$ ns in the
VISRAD hohlraum simulation and the simulated spectrum incident on the
hohlraum wall at the midplane (solid gray) along with its equivalent
blackbody spectrum (dashed gray) from the same simulation time.  Note
that the radiation temperatures are 202 eV for DANTE and 188 eV for
the midplane wall. \label{fig:hohlraum_spectra}}
\end{figure}

Of course, the differences between the DANTE (through the LEH) and
midplane radiation conditions will depend on beam pointing.  In
general, the farther in the pointing, the stronger the radiation will
be at the hohlraum midplane. This is due both to the hot spots being
closer to the midplane and less radiation escaping out the LEHs. The
situation for the DANTE looking in the LEH is more complicated, and
depends on the relative fraction of the sky occupied by laser hot
spots, as seen from DANTE's position. To investigate this, we
performed two additional simulations, identical to the one presented
above, except for the beam pointing.  In the first variation, the ten
Cone 2 beams are pointed 400 \micron\ farther into the hohlraum,
giving a mean laser spot position 620 \micron\ from the LEH plane (we
refer to this pointing as ``nominal'' throughout this paper).  Like
the Cone 3 beams, they are pointed at the center of the LEH, which
creates a second ring of five hot spots on either side of the
hohlraum, closer to the midplane than the single ring in the initial
simulations, which have a mean spot position 480 \micron\ from the LEH
plane.  In the second variation, all 30 beams are pointed an
additional 200 \micron\ farther into the hohlraum, giving a mean spot
position of 820 \micron\ from the LEH plane.

% FIG 5: \label{fig:hohlraum_pointing}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.38]{figs/hohlraum_D_p2_dante}
\includegraphics[scale=0.38]{figs/hohlraum_D_p3_dante}
\includegraphics[scale=0.40]{figs/hohlraum_pointing_v2}
\end{center}
\caption{The DANTE views of the hohlraum in the two cases with
different beam pointings (top two panels).  As in the
bottom panel of Fig.\ \ref{fig:hohlraum_target}, we show a color map
of emission temperature at $t=1.0$ ns, and hide the beams for clarity.
The color scale spans 140 eV to 220 eV, and as is true for all the
color figures throughout the paper, it is identical to the map shown
at the bottom of Fig.\ \ref{fig:hohlraum_target}.  The lower panel
shows the trends of DANTE temperature (squares) and midplane
temperature (circles) as the beam pointing is changed. The filled
symbols represent simulation time $t=1.0$ ns (XCE $= 0.55$ and albedo $=
0.73$) and the open symbols represent simulation time $t=100$ ps (XCE
$= 0.28$ and albedo $= 0.18$).  The original model, used to reproduce
the experiments reported on in \cite{decker97}, has a mean laser spot
position 480 \micron\ from the LEH plane.  The first variation (620
\micron) is shown in the top panel and the second variation (820
\micron) is shown in the middle panel. We note that in this last case,
the Cone 2 beams from either side of the hohlraum hit the wall almost
exactly at the midplane, creating a single, combined ring of hot
spots.
\label{fig:hohlraum_pointing}}
\end{figure}

In Fig.\ \ref{fig:hohlraum_pointing} we show the results of this
series of simulations with varying beam pointings. The radiation drive
temperature onto the midplane wall does, in fact, increase as the beam
pointing moves farther in the hohlraum toward the midplane.  The DANTE
temperature increases almost as much, but for a different reason, as
the pointing moves inward and hence out of the partial occlusion of
the lip. As in the original simulations, these hohlraums' LEHs have
75\% the hohlraum diameter, and thus have annular lips.  The lip
certainly can affect the DANTE view of hot spots and wall reemission,
which we explore in the context of halfraums in Sec.\
\ref{sec:pointing}.  In closing, we note that the trends shown here
are very similar when we look at earlier times, where both the albedo
and the XCE are lower than at 1 ns.  The only notable difference is a
greater similarity between the DANTE and sample temperatures for the
deepest pointing at early times, as can be seen in the lower panel of
Fig.\ \ref{fig:hohlraum_pointing}.


\section{Effects of a Capsule\label{sec:capsule}}

A spherical fuel capsule located at the center of the hohlraum acts as
another sink of photons (due to the relatively low albedo of the
plastic capsule) and will lower the hohlraum temperature accordingly.
To explore the nature and magnitude of this effect, we repeated the
analytic hohlraum radiation temperature calculation with a $520$
\micron\ diameter spherical capsule in the hohlraum, and then we
repeated two of the view-factor simulations discussed above, but with
a capsule.  We assumed a capsule albedo of $\alpha = 0.3$ independent
of time, and also assumed that the capsule size remained constant over
the $1.0$ ns duration of the simulation.

The hohlraum temperature calculation discussed in the previous section
was modified to include the area of the capsule: ${\eta}P_L = P_{rad}
= ((1 - \alpha)A_{wall} + A_{LEH} + (1 - \alpha_{cap})A_{cap})
{\sigma}T_R^4$.  To calculate the hohlraum radiation temperature at $t
= 1.0$ ns, we used $\eta = 0.55$ for the XCE and $\alpha = 0.73$, as
we did previously, while the capsule albedo was assigned a value of
$\alpha_{cap} = 0.3$.  For these values, we find that the capsule's
presence lowers the hohlraum radiation temperature by 4 eV to $T_R =
188$ eV at $t = 1.0$ ns. 


% FIG 6: \label{fig:capsule_sims}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.25]{figs/DeckerBenchmark_D_empty_dante.eps}
\includegraphics[scale=0.25]{figs/DeckerBenchmark_D_capsule_dante.eps}
\includegraphics[scale=0.25]{figs/DeckerBenchmark_D_p2_empty_dante.eps}
\includegraphics[scale=0.25]{figs/DeckerBenchmark_D_p2_capsule_dante.eps}
\end{center}
\caption{VISRAD simulations of hohlraums without (left) and with
(right) fuel capsules.  The capsules are centered in the hohlraums and
have a diameter of $520$ \micron\ and an albedo of $\alpha = 0.3$.
The top two panels show the ``single ring'' pointing while the bottom
two panels show the ``nominal'' pointing in which two rings are formed
by pointing both the Cone 2 and Cone 3 beams at the LEH center.  The
color maps in all four cases are the standard emission temperature map
with the minimum set at $T = 140$ eV and the maximum at $T = 220$ eV.
\label{fig:capsule_sims}}
\end{figure}


% FIG 7: \label{fig:capsule_temps}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figs/capsule_vs_empty_decker_label.eps}
\includegraphics[scale=0.45]{figs/capsule_vs_empty_nominal_label.eps}
\end{center}
\caption{Calculated radiation temperature vs.\ time for the VISRAD
simulations shown in Fig.\ \ref{fig:capsule_sims}.  The top panel is
for the ``single ring'' pointing while the bottom panel is for the
``nominal'' pointing.  In each panel, we show the DANTE temperature
for the empty (i.e. no capsule) hohlraum as a solid line and for the
hohlraum containing a capsule as a dashed line.  The radiation
temperature seen by a wall-mounted sample at the midplane is denoted
by a dotted line for the empty hohlraum and by a dash-dot line for the
hohlraum with a capsule.
\label{fig:capsule_temps}}
\end{figure}


We repeated two of the numerical view-factor simulations: one with the
``single ring'' pointing used in the Decker {\it et al.} experiments
(corresponding to Fig.\ \ref{fig:hohlraum_target}) and the other with
the ``nominal'' pointing in which all the beams are pointed at the
middle of the LEH (corresponding to the upper right-hand panel of
Fig.\ \ref{fig:hohlraum_pointing}). In Fig.\ \ref{fig:capsule_sims} we
show the results of these two new simulations, which do, indeed, show
lower radiation temperatures than the corresponding simulations
without capsules.

In Fig.\ \ref{fig:capsule_temps} we compare the time-dependent
radiation temperature from each simulation.  At $t=1.0$ ns, the DANTE
temperature is 5 eV lower with the capsule in the case of the
``nominal'' pointing (202 eV vs.\ 206 eV) and 6 eV lower for the
``single ring'' pointing (196 eV vs.\ 202 eV), in good agreement with
the analytic power balance calculation.  The sample temperature drop
is 7 eV in the case of the ``nominal'' pointing (186 eV vs.\ 193 eV)
at $t = 1.0$ ns and 6 eV lower in the case of the ``single ring''
pointing (182 eV vs.\ 188 eV).  The slightly larger temperature
decreases due to the presence of the capsule in the case of the
wall-mounted samples arises from the large view factor of the capsule
as seen by the sample, which is mounted at the midplane wall, and thus
is quite close to the capsule.  In the case of the nominal pointing in
which the capsule's presence has a bigger effect on the sample
temperature, the capsule's blocking of some of the laser hot spots as
seen from the sample's position plays a role too.

We can also see in Fig.\ \ref{fig:capsule_temps} that the effect of
the capsule on the calculated radiation temperatures is less extreme
at earlier times, when the capsule albedo and the hohlraum wall albedo
have more similar values.  But other than these small differences in
the DANTE vs.\ sample radiation temperature and in the time-dependence
of the effect of the capsule, the capsule's effect is a uniform
lowering of the radiation temperature in the hohlraum that corresponds
to roughly 2.5 percent, or roughly 10 percent in radiation flux.  By
increasing the converted laser power by a corresponding 10 percent,
the results of the simulations without the capsule are reproduced with
the capsule.  This increase in power corresponds to increasing the XCE
from 0.55 to 0.60. Thus the results of the view-factor simulations
discussed in the remainder of this paper are applicable to hohlraums
with capsules and slightly higher XCEs, although in detail, the
temporal, spatial, and spectral dependencies will be affected by a
capsule.  If a high degree of accuracy is required in analyzing these
variations, then detailed simulations should be performed.


\section{Evolution to a Halfraum\label{sec:halfraum}}

Increasingly, indirect drive and related experiments are performed in
halfraums \cite{back00,cohen04}, which are shorter cylinders with only
one LEH, sometimes referred to as half hohlraums, or single-ended
hohlraums.  Experimental packages in halfraums are often mounted on
the end of the cylinder, opposite the LEH.  We might expect to see
similar effects to those we demonstrated in the previous sections:
spatial dependence of the drive properties within a halfraum (both in
terms of overall power and in terms of the spectral energy
distribution) and, specifically, differences between DANTE
measurements and the radiation drive incident upon an experimental
package.


% FIG 8: \label{fig:sample_orientation}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figs/sample_orientation_v3}
\end{center}
\caption{The trend of radiation temperature as a function of sample
orientation for a planar sample located at the center of a hohlraum.
The angle plotted along the x-axis is the angle between the sample
normal and the hohlraum axis, so that 0 degrees is LEH-facing, while
90 degrees is wall-facing. The filled symbols are the results from
$t=1.0$ ns, while the open symbols are from $t=100$ ps.  For
comparison, the radiation temperatures at these two times for a sample
on the wall of the hohlraum at the midplane (discussed in Sec.\
\ref{sec:hohlraum}) are 188 eV and 129 eV (denoted by Xs); nearly
identical to the centrally located, wall-facing (90 degree) results
shown here.  Finally, we note that the DANTE temperatures for these
two times are 202 eV and 143 eV, respectively.
\label{fig:sample_orientation}}
\end{figure}


Because a halfraum is essentially just half of a hohlraum, one expects
its properties to not differ appreciably from those of a hohlraum.
There are only half as many beams in a halfraum, but the wall area and
the LEH area are also about half of that in a hohlraum.  One
difference between a hohlraum and a halfraum, for experiments with
packages that are planar samples, is that a sample located at the
midplane of a hohlraum is typically mounted on the wall, or barrel, of
the hohlraum, facing the opposite wall.  In a halfraum, a planar
package is typically on the back end of the halfraum, facing the LEH.
So there is a difference in the position and orientation of the
sample, which will affect the relative view factors of hot spots and
LEH, as compared to the case of a planar sample mounted at the
midplane of a hohlraum.  In order to investigate the effect of sample
position and orientation, we first repeated our initial hohlraum
simulations (without a capsule and with the simple ``single ring''
beam pointing such that both Cone 2 and Cone 3 beams make a single
ring of hot spots 480 \micron\ from the LEH plane), but we located the
planar sample in the {\it middle} of the hohlraum, suspended where a
capsule would be.  We performed four such simulations, varying only
the sample orientation from wall facing to LEH facing.  The results of
these four simulations are shown in Fig.\
\ref{fig:sample_orientation}.


% FIG 9: \label{fig:sample_orientation_spectra}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figs/sample_orientation_spectra_log-linear}
\end{center}
\caption{Comparison of the spectrum incident on a sample at the center
of a hohlraum (at $t=1.0$ ns) when it is oriented toward the LEH (dashed line, 0
degree case in the previous figure) vs.\ the spectrum when the sample
is oriented toward the hohlraum wall (solid line, 90 degree case in
the previous figure). The LEH-facing sample has a significantly harder
spectrum, though the radiation temperature onto each is nearly
identical (190 eV vs.\ 191 eV).
\label{fig:sample_orientation_spectra}}
\end{figure}


We find that the radiation temperature onto a planar sample at the
center of the hohlraum is almost completely independent of sample
orientation at $t=1.0$ ns, when the albedo is high ($\alpha = 0.73$).
It is also nearly identical to the radiation temperature on a
wall-mounted planar sample at the midplane.  This result is relevant
for experiments that, for example, investigate shock timing on
wall-mounted packages and use the results to infer the drive onto a
capsule.

The variation among these five cases (four at the center of the
hohlraum and one on the wall) is only 3 eV at $t=1.0$ ns, with no
monotonic trend with orientation.  Indeed, this result simply reflects
the fact that cylindrical hohlraums, and their associated laser
heating schemes, have been designed to generate a nearly uniform
radiation drive onto a fuel capsule at their centers. The view factors
of hot spots and cold LEHs change in concert with each other as the
sample orientation changes. However, although the radiation
temperature is nearly independent of sample orientation, the spectral
energy distribution is not.  In Fig.\
\ref{fig:sample_orientation_spectra} we compare the spectrum incident
upon a sample facing the LEH with that incident on the same sample
facing the hohlraum wall.  The radiation temperatures in these two
cases are nearly identical (190 eV vs.\ 191 eV), but the LEH-facing
sample has a significantly harder spectrum than the wall-facing sample
($>25\%$ more flux at 2 keV; and if the gold M-band were explicitly
taken into account by the modeling, this difference could be larger).
This is because the LEH-facing sample has more high-energy radiation
incident upon it due to its larger hot-spot view factor and less
lower-energy wall radiation due to its larger LEH view factor.


% FIG 10: \label{fig:hohlraum_halfraum}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{figs/A_dante_new}
\includegraphics[scale=0.45]{figs/A_mod3_dante_new}
\end{center}
\caption{The DANTE view at $t=1.0$ ns of our original hohlraum
simulation (top) and the same view of a simulation that differs only
in having a gold disk dividing the hohlraum in half, effectively
turning it into a halfraum (bottom).  The temperature color scales are
identical in the two figures.  Note that from this viewing angle, some
of the laser hot spots on the {\it far} side of the hohlraum, caused
by beams entering the hohlraum through the far LEH, are visible, which
is, of course, not the case with the dividing disk present.
\label{fig:hohlraum_halfraum}}
\end{figure}


We also note, referring to Fig.\ \ref{fig:sample_orientation}, that
there is a somewhat larger dependence of radiation temperature on
sample orientation at early times, when the albedo is lower ($\alpha =
0.18$ at $t=100$ ps) than at late times.  The radiation temperature is
6 eV higher for the LEH-facing sample than for the wall-facing sample
at $t=100$ ps.  This is due to the fact that the wall-facing sample's
view is dominated by a significantly cooler wall in a low-albedo
situation.

Finally, it has been noticed that the DANTE temperature more closely
tracks the sample temperature in a halfraum configuration than in a
similar hohlraum configuration \cite{cohen04}.  Based on the above
analysis, we see that this is {\it not} due to the difference in the
sample position or orientation as one goes from a hohlraum to a
halfraum.  The sample radiation temperature does not change
significantly as the sample is moved from the midplane hohlraum wall
to the center of the hohlraum and turned to face the LEH.  In order to
ascertain what accounts for the better agreement between the DANTE
temperature and the sample temperature in the halfraum (recall that
this difference is about 15 eV in a hohlraum), we constructed a model
of a halfraum by simply taking our hohlraum model having the sample in
the center of the volume and facing the LEH, and inserting a gold disk
at the midplane, to effectively divide the hohlraum in half, giving
the resulting halfraum a length of 1150 \micron.  The DANTE views from
these two simulations are shown in Fig.\ \ref{fig:hohlraum_halfraum}.

In the hohlraum case, the DANTE temperature is 202 eV and the sample
temperature is 190 eV (difference of 28\% in flux).  In the halfraum
case, the DANTE temperature is 193 eV and the sample temperature is
186 eV (difference of 16\% in flux).  
% The expected hohlraum radiation
%temperature from the power balance equation actually falls by 17 eV in
%the transition from hohlraum to halfraum, because the dividing wall
%represents a significant increase in the wall area.  The sample 
%temperature falls significantly less than this because it is mounted 
%on the dividing wall, and so does not see this wall at all, viewing 
%instead proportionally more solid angle of  hot spots than does the 
%average location in the halfraum 
All temperatures are quoted for simulation time $t=1.0$ ns.  So, the
presence of the disk that divides the hohlraum in half affects the
DANTE-measured drive by about twice as much as it affects the sample
drive.  By inspecting Fig.\ \ref{fig:hohlraum_halfraum} it is clear
that the proportionally larger drop in DANTE temperature in the
halfraum case is due to the fact that in a hohlraum, DANTE sees some
of the laser hot spot emission from the {\it far} side of the
hohlraum, which is caused by the beams entering through the far LEH.
In the halfraum, DANTE sees instead wall reemission from the far end
of the halfraum (or, equivalently, the disk at the midplane of the
modified hohlraum in the case we have presented here).  In situations
where low-angle beams, which cross the hohlraum midplane, are used,
one sees similar trends, with the low-angle beams hitting the back
wall rather than the side wall of the halfraum. Such considerations
are relevant to NIF \cite{paisner94} configurations, where there are a
significant number of mid-plane-crossing low-angle beams.



\section{Beam Pointing Within a Halfraum\label{sec:pointing}}

One straightforward way to try to control the drive properties in a
halfraum is to adjust the beam pointing.  Here we explore the
dependence of the drive onto a sample mounted on the back wall of a
halfraum as the beam pointing varies.  We also monitor the DANTE
temperature as a function of beam pointing, from the usual LEH view
with a 37.4 degree angle to the halfraum axis. To simplify the
situation, we revert to the pointing used in the Decker {\it et al.}
\cite{decker97} experiments (the ``single ring'' pointing in which all
15 beams make a single ring of hot spots) and our initial modeling in
Sec.\ \ref{sec:hohlraum}.  The other halfraum properties are the same
as those we have used for the previous modeling: variable XCE and
albedo as described earlier, a halfraum length of 1150 \micron\ and a
diameter of 1600 \micron, and an LEH diameter of 1200 \micron. All 15
beams are taken to have perfect square profiles over 1 ns and total
energies of 500 J per beam.  In all the simulations presented in this
section, the beams are focused at the point at which they cross the
halfraum axis.  We make an exception for several beams in the
simulation with the deepest beam pointing, where we had to pull back
the focus position slightly in order to keep the beams from clipping
the LEH lip.

We present four simulations, depicted in Fig.\
\ref{fig:halfraum_pointing}, in which the beam pointing varies by 150
\micron\ for each simulation.  The second simulation (from the top),
with the beam pointing at the LEH plane, corresponds to the default
pointing used in the previous sections, with a hot spot distance of
480 \micron\ from the LEH plane.


% FIG 11: \label{fig:halfraum_pointing}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/F0_dante}
\includegraphics[scale=0.25]{figs/F0_sample}
\includegraphics[scale=0.25]{figs/F1_dante}
\includegraphics[scale=0.25]{figs/F1_sample}
\includegraphics[scale=0.25]{figs/F2_dante}
\includegraphics[scale=0.25]{figs/F2_sample}
\includegraphics[scale=0.25]{figs/F3_dante}
\includegraphics[scale=0.25]{figs/F3_sample}
\end{center}
\caption{Series of four halfraum simulations in which the beam
pointing was varied.  The pairs of images show the DANTE view on the
left and the view from behind the sample on the far wall, or end cap,
of the halfraum on the right; both at $t=1.0$ ns.  The top row has the
beams pointed closest to the LEH, with the pointing moving in by 150
\micron\ at each simulation, moving down the figure.  This series of
simulations has the beams pointed so that the hot spots make a single
ring, and the halfraum has a three-quarter LEH.
\label{fig:halfraum_pointing}}
\end{figure}


% FIG 12: \label{fig:halfraum_pointing_temps}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_pointing_v2}
\end{center}
\caption{The radiation temperature on a sample mounted on the end of a
halfraum (circles) and measured by DANTE (squares) for the four
different beam pointings shown in Fig.\ \ref{fig:halfraum_pointing}
(three-quarter LEH, hot spots in single ring) at two different
simulation times: $t=1.0$ ns (filled symbols) and $t=100$ ps (open
symbols).
\label{fig:halfraum_pointing_temps}}
\end{figure}


In Fig.\ \ref{fig:halfraum_pointing_temps} we show the trend of
radiation temperature on the back-wall sample along with the DANTE
temperature as a function of beam pointing. They are similar to what
is seen in hohlraums (see Fig.\ \ref{fig:hohlraum_pointing}) where
deeper pointings generate higher temperatures by decreasing LEH losses
and positioning hot spots more favorably. The flattening out of the
sample temperature trend with the deepest pointing is due to the
higher obliquity of the hot spots as seen by the sample. We note that,
as detailed in the previous section, the DANTE temperature exceeds the
radiation temperature onto the sample by between 5 and 10 eV at late
times when the halfraum albedo is high.  But at earlier times, when
the albedo is much lower, the two temperatures are more similar. When
the albedo is low, the distinction between weak wall reemission and
cold LEH is minimal relative to the hot spots, so the sample seeing
wall plus LEH and DANTE seeing solely wall give similar radiation
temperatures.


% FIG 13: \label{fig:halfraum_pointing_nolip}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/F0_nolip_dante}
\includegraphics[scale=0.25]{figs/F0_nolip_sample}
\includegraphics[scale=0.25]{figs/F1_nolip_dante}
\includegraphics[scale=0.25]{figs/F1_nolip_sample}
\includegraphics[scale=0.25]{figs/F2_nolip_dante}
\includegraphics[scale=0.25]{figs/F2_nolip_sample}
\includegraphics[scale=0.25]{figs/F3_nolip_dante}
\includegraphics[scale=0.25]{figs/F3_nolip_sample}
\end{center}
\caption{Series of four halfraum simulations in which the beam
pointing was varied. These are identical to the series shown in Fig.\
\ref{fig:halfraum_pointing}, except that the LEH is larger in these
targets, with a diameter equal to the diameter of the halfraum itself
({\it i.e.} no lip on the LEH).  The pairs of images show the DANTE view on
the left and the view from behind the sample on the far wall, or end
cap, of the halfraum on the right; both at $t=1.0$ ns.  The top row
has the beams pointed closest to the LEH, with the pointing moving in
by 150 \micron\ at each simulation, moving down the figure.
\label{fig:halfraum_pointing_nolip}}
\end{figure}


It is clear from these beam pointing simulations, and especially from
inspecting the left-hand column of Figs.\ \ref{fig:halfraum_pointing},
that the LEH lip can play an important role, as it affects the DANTE
view factors in addition to the effect it has on keeping reemitted
radiation from escaping out the LEH.  To investigate this
quantitatively, we have performed another series of four view-factor
simulations.  These are analogous to the previous set discussed in
this section (see Fig.\ \ref{fig:halfraum_pointing}), where all 15
beams are pointed to form a single ring of hot spots, and in each
successive simulation, all the beams are moved 150 \micron\ further
into the halfraum. The only difference between these new simulations
and the original ones is that in the new ones, there is no LEH lip.
That is, the LEH diameter is 100\% of the halfraum diameter.


% FIG 14: \label{fig:halfraum_pointing_temps_nolip}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_pointing_nolip_v2}
\end{center}
\caption{The radiation temperature on a sample mounted on the end of a
halfraum (circles) and measured by DANTE (squares) for the four
different beam pointings shown in Fig.\
\ref{fig:halfraum_pointing_nolip} at two different simulation times:
$t=1.0$ ns (filled symbols) and $t=100$ ps (open symbols).  In these
simulations there is no LEH lip ({\it i.e.}\ 100\% LEH) and the beams
are pointed to form a single ring of hot spots.
\label{fig:halfraum_pointing_temps_nolip}}
\end{figure}

The results of this series of simulations are shown in Fig.\
\ref{fig:halfraum_pointing_nolip}, where the emission temperature
color maps of the targets are displayed, and in Fig.\
\ref{fig:halfraum_pointing_temps_nolip}, where the trends of sample
and DANTE radiation temperature are shown.  The removal of the LEH lip
has several effects. The DANTE temperatures are lower, and somewhat
less dependent on the beam pointing.  This appears to be because the
portion of the halfraum wall nearest the LEH is much colder in these
simulations than in those with an LEH lip. In the simulations with the
lip ({\it i.e.} three-quarter LEH), the DANTE temperature increases
more dramatically as the pointing becomes deeper because the hot spots
are moving into the DANTE field of view and out of the shadow of the
lip. Here, the hot spots are already in the DANTE field of view even
for shallow pointings.  As the beams are pointed further in, the DANTE
view factor of hot spots increases, but this effect is canceled by the
larger size of the cool wall region near the LEH. 

The sample radiation temperature is somewhat lower (by as much as 10
eV) in these simulations without the lip compared to those with the
LEH lip, due to increased losses through the LEH.  The trend of
increased sample temperature with deeper pointing is apparent in these
simulations without the LEH lip, as it was in the simulations with the
lip.  The trend is somewhat stronger in the simulations without the
lip, likely because the increased LEH losses are stronger for the
shallower pointing cases.  In summary, the sample temperature drops
more than the DANTE temperature due to the absence of the LEH lip
because the sample sees a bigger cold LEH but DANTE sees more of the
hot spots than in the case with the LEH lip, which partially
compensates for the increased LEH losses.


% HALFRAUM WITH MORE REALISTIC BEAM POINTING 


% FIG 15: \label{fig:halfraum_pointing2}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/F0b_dante}
\includegraphics[scale=0.25]{figs/F0b_sample}
\includegraphics[scale=0.25]{figs/F1b_dante}
\includegraphics[scale=0.25]{figs/F1b_sample}
\includegraphics[scale=0.25]{figs/F2b_dante}
\includegraphics[scale=0.25]{figs/F2b_sample}
\includegraphics[scale=0.25]{figs/F3b_dante}
\includegraphics[scale=0.25]{figs/F3b_sample}
\end{center}
\caption{Final series of four halfraum simulations, in which the beam
pointing was varied. The beam pointing here gives two separate rings
for Cones 2 and 3, respectively, and we revert to the standard
three-quarter LEH.  The beams are moved inward by 150 \micron\ each
step down the figure, with the second row representing the nominal
pointing (all beams pointed at the center of the LEH). The left hand
column shows the DANTE view, the right hand column shows the sample
view.  The colors represent emission temperatures at $t=1.0$ ns.
\label{fig:halfraum_pointing2}}
\end{figure}

We performed a final series of simulations with varying beam pointing,
but this time with a more natural pointing configuration, in which the
aim point of both Cones 2 and 3 are the same, causing two separate
rings of hot spots on the walls of the halfraum. We also revert back
to the standard, three-quarter LEH for this series.  The nominal
pointing in this case has all 15 beams pointed (and focused) at the
LEH center.  This makes a ring of hot spots (from Cone 3) at 480
\micron\ from the LEH plane and another ring (from Cone 2) at 890
\micron\ from the LEH plane (see the second row of Fig.\
\ref{fig:halfraum_pointing2}).  As in the previous two sets of
simulations, we vary the pointing by moving all the beams inward by
150 \micron\ and then by 300 \micron, and also calculate a case in
which all the beams are pulled out 150 \micron\ from this nominal
pointing.


% FIG 16: \label{fig:halfraum_pointing_temps2}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_pointing_2_v2}
\end{center}
\caption{Temperature as a function of pointing for the third set of
simulations described in this section (see Fig.\
\ref{fig:halfraum_pointing2}), with the two beam cones making two
different rings of hot spots, and using the three-quarter LEH.  The
solid symbols are from a simulation time of $t=1.0$ ns while the open
symbols are from $t=100$ ps.  The squares are DANTE temperatures and
the circles are sample radiation temperatures.
\label{fig:halfraum_pointing_temps2}}
\end{figure}


The results of this series of simulations are summarized in Figs.\
\ref{fig:halfraum_pointing2} and
\ref{fig:halfraum_pointing_temps2}. The general trends shown
previously are also seen in this series of calculations.  The drive
temperature onto the sample is relatively independent of pointing,
except for the most extreme cases, in which it is a little
cooler. This is because the effect of the obliquity of the hot spot
view is even more extreme, with the Cone 2 beams pointed further into
the halfraum.  The DANTE temperatures are also quite independent of
beam pointing, and modestly higher than the sample radiation
temperatures (more so at the later times, when the wall albedo is
higher).



\section{Optimizing Halfraum Geometry\label{sec:geometry}}

The results from the previous section can be used to maximize the
radiation drive onto a sample mounted on the back wall of a halfraum,
as well as to relate radiation diagnostics from DANTE to the sample
drive properties.  In this section, we investigate the dependence of
drive properties on halfraum length and also on the presence of
internal LEH shields as well as a foil just outside the LEH.  For
simplicity, we keep the halfraum diameter (1600 \micron) and LEH
diameter (1200 \micron) the same for these simulations and also do not
vary the beam properties. In this first set of simulations, the beam
pointing is always at the LEH center (with the beams all focused at
this point as well).  Thus, as the halfraum length changes, the
distance of the hot spots from the sample also changes.


% FIG 17: \label{fig:halfraum_length}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/G0b_dante}
\includegraphics[scale=0.25]{figs/G0b_sample}
\includegraphics[scale=0.25]{figs/G1b_dante}
\includegraphics[scale=0.25]{figs/G1b_sample}
\includegraphics[scale=0.25]{figs/G2b_dante}
\includegraphics[scale=0.25]{figs/G2b_sample}
\includegraphics[scale=0.25]{figs/G3b_dante}
\includegraphics[scale=0.25]{figs/G3b_sample}
\end{center}
\caption{Series of four halfraum simulations in which the length of
the halfraum was varied.  From top to bottom, the halfraum lengths are
1000, 1150, 1300, and 1450 \micron.  The beam pointing and focus was
at the LEH center in all cases.  The left hand column shows the DANTE
view and the right hand column the sample view.  The colors represent
emission temperatures taken from $t=1.0$ ns in each simulation.
\label{fig:halfraum_length}}
\end{figure}


% FIG 18: \label{fig:halfraum_length_temps}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_length_v2}
\end{center}
\caption{Temperature as a function of halfraum length for the four
simulations shown in Fig.\ \ref{fig:halfraum_length}, with all
simulations having identical beam pointings with respect to the
LEH. The solid symbols are from a simulation time of $t=1.0$ ns while
the open symbols are from $t=100$ ps.  The squares are DANTE
temperatures and the circles are sample radiation temperatures.
\label{fig:halfraum_length_temps}}
\end{figure}

In Fig.\ \ref{fig:halfraum_length} we show a series of four
simulations in which the halfraum length is varied from 1000 \micron\
to 1450 \micron\ in steps of 150 \micron.  In Fig.\
\ref{fig:halfraum_length_temps} we plot DANTE and sample temperatures
at two different simulation times as a function of halfraum
length. These temperatures are relatively independent of length, with
only a slight decrease in radiation temperature for the longer
halfraums. The tendency toward lower temperatures due to the greater
wall area in the longer halfraums must be offset by fewer radiation
losses out the LEH, and for the sample, the smaller LEH solid angle as
the halfraum lengthens also tends to offset the increased wall
losses. The drop in the sample radiation temperature for the shortest
halfraums is due to the obliquity of the hot spots as seen from the
sample, especially the Cone 2 spots.


% FIG 19: \label{fig:halfraum_length_varpointing}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/G0x_dante}
\includegraphics[scale=0.25]{figs/G0x_sample}
\includegraphics[scale=0.25]{figs/G1b_dante}
\includegraphics[scale=0.25]{figs/G1b_sample}
\includegraphics[scale=0.25]{figs/G2x_dante}
\includegraphics[scale=0.25]{figs/G2x_sample}
\includegraphics[scale=0.25]{figs/G3x_dante}
\includegraphics[scale=0.25]{figs/G3x_sample}
\end{center}
\caption{Series of four halfraum simulations in which the length of
the halfraum was varied.  From top to bottom, the halfraum lengths are
1000, 1150, 1300, and 1450 \micron.  The beam pointing and focus was
varied along with halfraum length, so the distance from the hot spots
to the sample was the same for all four simulations.  The case shown
in the second row (for a halfraum of standard length, $l = 1150$
\micron), is identical to that shown in the previous set of
simulations (second row of Fig.\ \ref{fig:halfraum_length}), but the
beam pointing is 150 \micron\ outside of the LEH for the shortest
halfraum, shown in the top row, and 150 \micron\ and 300 \micron\
inside the LEH for the bottom two cases, respectively.  The left hand
column shows the DANTE view and the right hand column the sample view.
The colors represent emission temperatures taken from $t=1.0$ ns in
each simulation.
\label{fig:halfraum_length_varpointing}}
\end{figure}


% FIG 20: \label{fig:halfraum_length_varpointing_temps}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_length_varpointing}
\end{center}
\caption{Temperature as a function of halfraum length, where the beam
pointing is changed along with halfraum length, such that the hot
spots' distance from the sample mounted on the back wall is always the
same (mean hot spot distance of 530 \micron\ from the back wall,
measured along the barrel). The solid symbols are from a simulation
time of $t=1.0$ ns while the open symbols are from $t=100$ ps.  The
squares are DANTE temperatures and the circles are sample radiation
temperatures.
\label{fig:halfraum_length_varpointing_temps}}
\end{figure}


To investigate whether the slight decrease in the radiation
temperature of the back wall mounted sample is primarily due to its
distance from the hot spots, we repeated the previous series of
experiments, but with the beam pointing adjusted in each case to keep
the hot spots' distance from the sample constant as the halfraum
length was adjusted. For this series of calculations, we kept the
nominal (LEH centered) pointing for the standard halfraum length of
1150 \micron.  However, for each of the other three halfraum lengths,
we moved all 15 beams either in or out according to the halfraum
length so that the pointings, and thus the hot spots' positions, were
always the same distance from the sample.  Thus, for the 1000 \micron\
long halfraum, the beam pointing was 150 \micron\ outside of the LEH,
at the halfraum axis.  For the 1300 \micron\ halfraum, the pointing
was 150 \micron\ inside the LEH, and for the 1450 \micron\ halfraum,
it was 300 \micron\ inside. For all but the longest halfraum, all the
beams were focused at the pointing position.  For the longest
halfraum, we had to focus the beams closer to the LEH plane, to
prevent beam clipping on the LEH lip.

The results of this series of simulations are shown in Figs.\
\ref{fig:halfraum_length_varpointing} and
\ref{fig:halfraum_length_varpointing_temps}.  The primary result is
that the sample temperature is almost totally independent of halfraum
length (at both 100 ps and 1 ns).  This is, of course, counter to the
expectations of standard hohlraum temperature power balance analysis,
which would predict lower temperatures as the halfraum was lengthened
(as is seen in the first set of simulations discussed in this
section). Clearly, the fact that the sample's view factor of hot spots
is the same in each of these four cases (because the pointing is
constant relative to the sample itself) is much more important than
the addition of extra wall area as the halfraum is lengthened.
Furthermore, the LEH subtends a smaller solid angle as seen from the
location of the hot spots in the longer halfraums, so LEH losses are
minimized, even as wall losses increase. The DANTE temperature
decreases somewhat with increasing halfraum length because the wall in
the DANTE field of view includes contributions from regions farther
from the hot spots when the halfraum is longer, and these wall regions
are colder, as there is a negative axial temperature gradient
associated with the hot spots.


In order to maximize the radiation drive onto a sample, or generally
in a hohlraum or halfraum, extra walls or barriers or other complex
geometries can be employed.  Boosts of the drive onto a capsule have
been demonstrated via the use of walls on the interior of hohlraums
that block the capsule's view of the LEH \cite{amendt96}.  

We performed a simulation to investigate the effects of such ``LEH
shields.''  Again using a standard size ($2300 \times 1600$ \micron)
hohlraum with three-quarter LEHs, we put two 600 \micron\ diameter
gold disks centered on the hohlraum axis 600 \micron\ from the
hohlraum midplane.  We used the nominal beam pointing, with all 15
beams on each side of the hohlraum pointed and focused at the center
of the LEH.  As expected, and shown experimentally, these LEH shields
boost the temperature at the midplane of the hohlraum by blocking much
of the area of the LEHs as seen by a sample and also by the hot spots,
decreasing the effective view factor of cold LEH and preventing much
of the radiation losses out the LEH.  The usual emission temperature
color maps from two different views are shown for this simulation in
Fig.\ \ref{fig:LEH_shields}.

% FIG 21: \label{fig:LEH_shields}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/D_p2_shields_dante}
\includegraphics[scale=0.25]{figs/D_p2_shields_midplane}
\includegraphics[scale=0.27]{figs/D_p2_noshields_dante_new}
\includegraphics[scale=0.27]{figs/D_p2_noshields_midplane_new}
\end{center}
\caption{Emission temperature color maps at, $t=1$ ns, of the hohlraum simulation with the LEH shields (top row) compared to an identical simulation without shields (bottom row; same simulation as that shown in the top panel of Fig.\ \ref{fig:hohlraum_pointing}). The left column shows the usual DANTE view, while the right column shows a view from near the hohlraum midplane, looking toward one of the LEHs. 
  \label{fig:LEH_shields}}
\end{figure}

We monitored the radiation temperature in this simulation on a sample
at the center of the hohlraum (and facing one of the LEHs) as well as
on a midplane wall-mounted sample. The radiation temperature on these
two samples never varied by more than 2 eV (135 eV vs.\ 135 eV at
$t=100$ ps and 197 eV vs. 195 eV at $t=1$ ns). In the simulation
without the shields, the sample temperatures are nearly identical at
early times (134 eV and 135 eV at $t=100$ ps for the hohlraum-center
and wall-mounted samples, respectively) but are somewhat lower at
later times (190 eV and 193 eV at $t=1$ ns).  The lack of effect of
the LEH shields on the sample temperature at early times is due to the
fact that the shields themselves are not yet hot enough to emit
significant radiation; they are nearly as cold as the LEHs that they
block.  But at late times, the shields boost the temperature on a
sample at the center of the hohlraum by 7 eV, corresponding to 16 \% in
flux). The DANTE temperature is significantly lower in the simulation
with the shields (197 eV at $t=1$ ns vs.\ 207 eV at the same
simulation time in the case without the shields).  As can clearly be
seen in the simulation (Fig.\ \ref{fig:LEH_shields}), the lower DANTE
temperature is due the fact that the back of the shield is in the
DANTE view, blocking this diagnostic's view of some hot spots and wall
reemission. We note that, at least in this particular simulation we
have discussed, the radiation temperature on a sample at the hohlraum
center and the DANTE temperature are identical.

A strategy similar to the use of LEH shields might involve putting
metal foils {\it outside} the LEH to absorb radiation lost out the LEH
and reemit it back into the hohlraum or halfraum.  In Fig.\
\ref{fig:halfraum_foil} we show an example of this scheme, in which a
circular foil with a radius of 350 \micron\ is hung 500 \micron\
outside the LEH.  One potential advantage of this scheme is that a
foil on the exterior can be irradiated with unused beams from the
other (non-LEH-facing) side of the target chamber to provide an
additional source of x-rays to heat the halfraum.


% FIG 22: \label{fig:halfraum_foil}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.25]{figs/H4b_general}
\includegraphics[scale=0.25]{figs/H4b_sample}
\includegraphics[scale=0.25]{figs/H5b_general}
\includegraphics[scale=0.25]{figs/H5b_sample}
\end{center}
\caption{Two halfraum simulations with a metal foil just outside the
LEH.  In the simulation shown in the top two panels, there are no
beams onto the foil.  The foil simply acts to absorb and reemit
radiation that exits through the LEH.  In the bottom two panels, there
are ten beams onto the foil, which significantly increases the
radiation flux inside the halfraum.  All the snapshots show emission
temperatures at $t=1.0$ ns.
\label{fig:halfraum_foil}}
\end{figure}

We performed two simulations of the halfraum with the foil in the
configuration described above, and using the nominal pointing (all 15
beams pointed at the center of the LEH plane) and halfraum size
($l=1150$ \micron).  In one, we do not irradiate the foil at all, and
in the other, we irradiate the foil with all ten Cone 3 beams from the
other side of the halfraum, using the same power profile as the
halfraum beams (1 ns square pulses with 500 J per beam).  It is easily
seen from the color map in Fig.\ \ref{fig:halfraum_foil} that this
additional source of radiation makes the entire halfraum hotter.  In
Fig.\ \ref{fig:halfraum_spectra_foil} we compare the spectra incident
on the sample (mounted as usual on the center of the back wall of the
halfraum) from the two cases with the foil (irradiated and not) to the
standard case without the foil.  It can be seen from this figure that
while simply adding the foil makes very little difference (sample
radiation temperature of 187 eV vs.\ 185.5 eV, or 3\% in flux),
irradiating the foil makes a large difference, raising the radiation
temperature on the sample to 207 eV (representing a 55\% increase in
total flux), and roughly doubling the radiation flux at 2 keV.

% FIG 23: \label{fig:halfraum_spectra_foil}
%

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{figs/halfraum_spectra_foil_log-linear}
\end{center}
\caption{Comparison of spectra incident on the center of the sample in
our standard ($l=1150$ \micron) halfraum at $t=1.0$ ns, with the
standard beam pointing.  The dotted line represents the simulation
with no foil, the dashed line (nearly coincident with the dotted line)
represents the simulation with a foil outside the LEH, and the solid
line is from the simulation with the foil heated by ten beams.
\label{fig:halfraum_spectra_foil}}
\end{figure}


\section{Conclusions\label{sec:conclusions}}

We have presented results from numerical view-factor simulations
performed to investigate the variation of radiation conditions as a
function of spatial, geometrical, and beam pointing conditions for
hohlraums and halfraums at OMEGA.  In addition, we have presented
results showing sometimes significant differences in the hohlraum wall
temperatures viewed by DANTE and the radiation drive temperatures seen
by experimental packages attached to the hohlraum or halfraum
walls. Because the radiative flux, and thus the heating, scales as the
fourth power of the temperature, even modest differences in wall
temperatures can be significant.  View-factor simulations, such as
those presented here, provide a means of simulating hohlraum radiation
characteristics and interpreting wall emission measurements (e.g.,
DANTE), and can be of value in designing and interpreting experiments
at OMEGA and future experiments at NIF.

We find, specifically, from several series of simulations, that the
radiation drive onto a sample can differ substantially from that
measured by an absolutely calibrated x-ray detector, like DANTE, even
when the diagnostic line of sight is through an LEH.  This is
especially true in hohlraums, as compared to halfraums, and at late
times (when wall albedos are high).  In the standard hohlraum
simulations we carried out, the radiation temperature on a sample at
the hohlraum midplane is roughly 15 eV lower than the DANTE
temperature.  In standard halfraum configurations, there is good
agreement between DANTE temperatures and radiation temperatures onto a
sample mounted at the center of the back wall (roughly a 5 eV
discrepancy at 200 eV).  This better agreement is primarily due to the
fact that in a hohlraum, the DANTE temperature is boosted with respect
to a halfraum because DANTE sees some of the hot spots on the far side
of the hohlraum, from beams entering from the opposite side. We also
find that midplane radiation temperatures in hohlraums are very
similar to radiation temperatures onto a sample suspended at the
center of a hohlraum, and further, that the orientation of such a
sample has very little effect on the radiation temperature.

It was also shown that even when radiation temperatures between two
different samples, or between a sample and DANTE, are very similar,
the respective spectral energy distributions can differ significantly.
The primary trend we found is that incident spectra are harder than
the equivalent Planckian spectra having the same radiation
temperature.  Another, milder, trend is that the spectrum onto a
sample tends to be harder than that seen by DANTE.

Variations in beam pointing and halfraum length were found to have
little effect, generally, on either the sample radiation temperature
or the DANTE temperature, except in extreme cases.  The mean laser
spot position can be varied anywhere from roughly 400 \micron\ to 800
\micron\ from the LEH plane in a standard halfraum without changing
either the DANTE temperature or the drive temperature onto a sample
more than a few eV.  And a 1600 \micron\ diameter halfraum will
provide maximal drive temperatures with nominal beam pointing when the
halfraum has a length anywhere between 1100 and 1400 \micron.
However, the size of the LEH can have a significant effect on both
DANTE and sample temperatures. Furthermore, the presence of a capsule
in a hohlraum will lower both the DANTE temperature and the sample
temperature in a predictable way due to increased losses into the
low-albedo capsule.  Finally, the drive onto a sample can be increased
significantly by the inclusion of LEH shields (which also have the
effect of lowering the DANTE temperature, bringing it into much closer
agreement with the sample temperature) or by irradiating a foil placed
just outside the LEH of a halfraum.


\begin{acknowledgments}

We are grateful for support through grant DE-FG03-OOSF22023 (from the
 DOE/NLUF program) to Prism Computational Sciences, and Research
 Corporation grant CC5489 to Swarthmore College.  We also thank Rick
 Olson for useful discussions and Dave Conners for performing some
 initial calculations.

\end{acknowledgments}


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