%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{Introduction and Overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{figure}[t!] % fig 1
\centering
\includegraphics[width=5.0in]{../template/report/ps-and-eps/magnetplan.eps}
\caption[A conceptual illustration of the rotating band setup]{A conceptual illustration of the setup for a
pion
production target based on a rotating inconel
band.}
\label{layout}
\end{figure}

\begin{figure}[t!] % fig 1
\begin{center}
\includegraphics[height=1.7in]{../template/report/ps-and-eps/bandxsec.eps}
\hspace{0.5in}
\includegraphics[height=1.7in]{../template/report/ps-and-eps/bandplan.eps}
\caption[Passage of the proton beam through the target band]{
Passage of the proton beam through the target band
shown in
cross-sectional (left) and plan (right) views. The
horizontal
position of the beam spot in the band webbing varies
along the
interaction region due to the curvature of the band.
The plan
view shown in the right plot is anamorphic, with a 10:1
aspect
ratio.}
\label{band_and_beam}\end{center}
\end{figure}

 As a backup scenario to the baseline mercury jet
target
design, we present here a solid-target option
that is based upon an Inconel Alloy 718 target in a rotating band
geometry.
Similar conceptual designs for rotating band targets
have been presented
previously~\cite{PAC99_band_target,nufact99_band_target,RAL_band_target}
for use at both muon colliders and neutrino factories.
A more
detailed report on this particular conceptual design
can be found
in reference~\cite{bandnote}.

  A plan view of the targetry setup for the band
target option is
shown in Fig.~\ref{layout}. An Inconel target band
threads
through the solenoidal magnetic capture channel to
tangentially intercept
the proton beam. The circulating band is cooled by
passage through a
water tank located in a radiation-shielded maintenance
enclosure.

 Inconel 718 is a
niobium-modified nickel-chromium-iron superalloy that
is widely
used in nuclear reactors
and particle accelerator applications because of
its high strength, outstanding weldability, resistance
to creep-rupture due to radiation damage and
resistance to corrosion
from air and water. The Inconel target band has an
I-beam
cross section. The band dimensions and positioning
relative
to the proton beam are shown in
Fig.~\ref{band_and_beam}.
The proton pulse structure and bunch charges were
assumed to be identical
to the base-line target scenario.
Table~\ref{target_band_specs} presents the parameter
specifications
that have been assumed for the Inconel target band and
the incident
proton beam.


\begin{table}[htb!]
\begin{center}
\caption[Specifications of the Inconel target band ]{Specifications of the Inconel target band and
assumed proton
beam parameters.}
\label{target_band_specs}
\begin{tabular}{|lc|}
\hline
Target band radius, R (m)        &  2.5    \\
Band thickness, t (mm)    &  6  \\
Band webbing height, h (mm)      &  60\\
Full width of band flanges (mm) &  40  \\
Beam path length in band, L (cm)  &  30  \\
Proton interaction lengths, $\lambda$     & 1.81 \\
Density of Inconel 718, $\rho$ (${\rm g.cm^{-3}}$)  &  8.19 \\
Mass of band (kg)                 &  98.8 \\
Band rotation velocity, v (m/s)      &  1 \\ 
Proton energy (GeV)                  &  24  \\
Protons/bunch                   &  $1.7 \times
10^{13}$ \\
Bunches/fill                    &  6 \\
Time between extracted bunches  &  20 ms \\
Repetition rate for fills       & 2.5 Hz \\ 
Horizontal beam-channel angle, $\alpha$ (mrad)    &  100 \\
Beam spot size at target (horizontal), $\sigma_x$ (mm)   & 1.5 \\
Beam spot size at target (vertical), $\sigma_y$ (mm)   & 15.0\\
\hline
\end{tabular}
\end{center}
\end{table}

\subsubsection{Mechanical Design Considerations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{figure}[t!]
\begin{center}
\includegraphics[width=5.0in]{../template/report/ps-and-eps/target_cooling.eps}
\caption[A conceptual illustration of the cooling
setup]{A conceptual illustration of the cooling
setup for the
Inconel target band.}
\label{band_cooling}
\end{center}
\end{figure}

  As is evident from Fig.~\ref{layout}, threading the
target
band through the pion capture channel requires only 
 slight variations 
on the channel design assumed for the baseline
mercury jet target option.
An entry port must be incorporated into the iron plug
in the upstream
end of the capture solenoid and an exit port traverses
the
tungsten shielding and then passes between the
solenoidal magnet coil
blocks and out of the pion decay channel. The exit
port can either be
designed into the magnet cryostat or else the cryostat
can be partitioned
longitudinally into two cryostats so the band can exit
between them.
The radius of the third magnet coil block from the
upstream end of
the channel must be increased by approximately 10 cm
relative to the
baseline design in order to provide adequate space
for the band to
exit the channel. A modest reoptimization of the
magnet coil currents
in this region can restore
the baseline magnetic
field specifications.


  No detailed consideration has yet been given to the
design of the beam
dump. As is clear from Fig.~\ref{layout},
the target band exit port is far enough upstream from
the beam dump for
it to be essentially ddecopled from the beam dump
design.

  The band is guided and driven by several sets of
rollers located
around its circumference, as shown in
Fig.~\ref{layout}. A few
hundred watts~\cite{bandnote} of drive power will be
required due
to the eddy current forces from the band entering and
exiting the
20~T solenoid. Following the lead of the BNL
g--2 target
design~\cite{gminus2}, the roller assemblies will all
incorporate
self-lubricating graphalloy~\cite{graphalloy} bushings
that are
compatible with high radiation environments.

The pion production region of the target is
in an air environment, with beam window positions
shown in
Fig.~\ref{layout}.
This simplifies target maintenance and target band
replacement
by avoiding any requirement to break and re-establish
seals in
a high radiation environment.
Activated air and gases from the target
and interaction region
are continuously diluted and then vented from the
target hall into
the outside atmosphere following the procedure
adopted~\cite{gminus2}
for the BNL g--2 target.

  The heated portion of the band rotates through a 2-m-long
cooling tank~\cite{bandnote} whose conceptual design
is shown in
Fig.~\ref{band_cooling}. The band entrance and exit
ports in the
ends of the tank also serve as the water outlets. Both
the heat
transfer rates and water flow rates are
found~\cite{bandnote}
to be relatively modest and the water flows due to its
gravitational
head alone with no need for forced flow.

  The rotation of the target band has the desirable
dilution effect
that the rate of radiation damage on any particular
section
of the band material is reduced by roughly two orders
of magnitude
relative to a fixed target geometry, since the 15.7-m-band circumference 
corresponds to 95 interaction
lengths.
Hence, each target band may last for several
years~\cite{bandnote}
before requiring replacement.

  The heavily irradiated used bands will
be remotely extracted by progressively clamping
and then shearing off 1 meter lengths and dropping
them into a hot box.
After the removal of the hot box, the band maintenance
area can then
be accessed and the new band progressively installed
by welding
together, in situ,
eight 1.96-m-long chords of target band that have
been previously
cast (or otherwise prepared) into the correct I-beam
cross section
and circumferential curvature. Beam-induced stresses
on the welds
are minimized by welding on the flanges of the I-beam
rather than
on the central webbing; the flanges are not directly
exposed to
the proton beam and will also receive much smaller
energy depositions
from secondary particles than the central webbing.



\subsubsection{Simulations of Pion Yields and
Beam-Induced Stresses} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 Full MARS~\cite{MARS} tracking and showering Monte
Carlo simulations
were conducted~\cite{bandnote} for 24 GeV protons
incident on the
target, giving predictions for the pion yield and
energy deposition
densities.

 The yield per proton for pions plus kaons plus muons
at 70 cm
downstream from the central intersection of the beam
with the target
was predicted~\cite{bandnote} to be 0.715 (positive)
and 0.636 (negative)
for the momentum range 0.05$<p<$0.80~GeV/$c,$ and
0.304 (positive) and 0.288 (negative) for the kinetic
energy
range 32$<E_{kin}<$232~MeV that approximates the
capture
acceptance of the entire cooling channel. Note that
the material
in the flanges of the I-beam was not included in the
calculation;
their inclusion might result in a small change in the
predicted
yield. For comparison, the predicted yield was 18\%
higher for
the identical geometry but with the band material
artificially changed from Inconel to mercury.

  Approximately 7\% of the proton beam energy is
deposited in the
target as heat and the maximum instantaneous energy
deposition from
a single proton bunch is approximately 13 J/g, which
corresponds
to a temperature rise of approximately 29 ${}^{\circ}$C.
Detailed 3-dimensional maps of energy deposition
densities were
generated for input to dynamic target stress
calculations~\cite{bandnote}
using the commercial ANSYS finite-element analysis
code.

 For the ANSYS simulations, the target band geometry
was discretised
into a 3-dimensional mesh containing approximately 30,000 elements.
It was conservatively assumed that all of the
deposited energy
from a proton pulse is instantaneously converted into
a local
temperature rise.


\begin{figure}[t!] % fig 1
\begin{center}
\includegraphics[height=3.0in]{../template/report/ps-and-eps/band_2ns_first_seqv2.eps}
\caption[Predicted von Mises stress distribution ]{Predicted von Mises stress distribution for
the target band
model at $1~\mu s$ after the arrival of the proton
pulse. For
simplicity, the flanges of the I-beam have been
neglected in this
particular simulation run.
The maximum predicted von Mises stress at this time is
171~MPa.}
\label{band_vonMises_initial}
\end{center}
\end{figure}


\begin{figure}[t!] % fig 1
\begin{center}
\includegraphics[height=3.0in]{../template/report/ps-and-eps/band_seqv_trend_fixed.eps}
\caption[Predicted
time dependence of von Mises stresses ]{
Predicted
time dependence of von Mises stresses on the target
band. The time
origin corresponds to the arrival of the proton
pulse.}
\label{band_vonMises_relax}
\end{center}
\end{figure}


 The von Mises stress (\textit{i.e.}, the deviation from the
hydrostatic state
of stress) was found to be initially zero but to
develop and fluctuate
over time as the directional stresses relax or are
reflected from material
boundaries. Figure~\ref{band_vonMises_initial} gives a
snap-shot of the predicted von Mises stress
distribution at
$1\mu$s after the arrival of a proton pulse and
Fig.~\ref{band_vonMises_relax}
shows the time development of the predicted
stress at the position of maximum stress.
For Fig.~\ref{band_vonMises_relax}, a ``fixed edge''
constraint has been applied to the band model with a
simple rectangular
cross section that is shown in the preceding figure;
this is intended
to better approximate the stiffening from the I-beam
flanges in the
actual band without the extra computing capacity
required to simulate
the more complicated true geometry.
The predicted 190 MPa peak value, in both time and
position,
for the von Mises stress
from a single proton bunch is much less than the the
740 MPa
(or 1100 MPa) yield strength for annealed (or
precipitation
hardened) Inconel 718 and is also well below its
fatigue strength.

  The band rotation speed, 1 m/s, advances the band by
40 cm between
successive beam fills. This presents a fresh 30 cm
chord of target
band for each beam fill, but the energy depositions
from the 6 bunches
within the fills are largely superimposed. However,
the pile-up of
stresses is not considered serious since any
significant level of
von Mises stress is expected to die out well within
the 20 millisecond
time span between successive bunches (\textit{i.e.},
approximately 3000 times
the time interval plotted in
Fig.~\ref{band_vonMises_relax}),
leaving only the relatively benign
hydrostatic stresses.


\subsubsection{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%

  In summary, the Inconel rotating band target design
appears to
be a promising backup option to the baseline mercury
jet target.
The pion yield appears slightly lower than the mercury
baseline,
although this has yet to be fully optimized. The
engineering design
looks manageable and initial simulations of target
stresses
are encouraging.

