\section{Beam Windows}
%N. Simos \& Hans Ludewig
\subsection{Upstream Proton Beam Window}
The upstream and downstream beam windows isolate the incoming proton beam
transport and pion decay channel from the mercury vapor atmosphere near the
target.

The upstream proton beam window will see the full beam before it hits the 
target.  The resulting pulsed energy deposition excites pressure waves that 
must be sustained by the window for over $10^8$ cycles per year.
Since the lifetime of the
window is expected to be limited, provisions for its periodic replacement 
are part of its design.

 The proton beam
window is a double wall structure with a gap between the
two walls that allows for active cooling. The interior face of the window will be exposed to mercury vapor, so the window
material must be mercury compatible.  Candidate window materials include
beryllium and Ti90-Al6-V4 alloy (whose short-term 
compatibility with mercury has recently been verified).

To assess the viability of candidate window materials, an ANSYS finite-element 
analysis was performed, including both the thermal aspect of the
beam/window interaction and the resulting thermal shock.  The energy deposition
in the window material was computed using the MARS code~\cite{mars14}--\cite{muc0194}. 
Figure~\ref{TgtW:fg1} shows results for a 1-mm-thick
beryllium window intercepting six pulses of $1.7 \times 10^{13}$ 24-GeV protons
with $\sigma_r=1$~mm.
%\afterpage{\clearpage}
\begin{figure}[!htb]
\begin{center}
%\includegraphics*[width=3in,angle=90]{simos_fig1.eps}
%\includegraphics*[width=3in,angle=90]{simos_fig2.eps}
\includegraphics*[width=3in,angle=-90]{../template/report/ps-and-eps/simos1.ps}
\includegraphics*[width=3in,angle=-90]{../template/report/ps-and-eps/simos2.ps}
\caption[ANSYS modeling of the proton beam window ]
{ANSYS model of a 1-mm-thick beryllium window subject to a 
train of six micro-pulses  of $1.7 \times 10^{13}$ 24-GeV protons per pulse
with $\sigma_r = 1$~mm).  Top: transient thermal response; bottom: von Mises
stress.}
\label{TgtW:fg1}
\end{center}
\end{figure}
 Figure~\ref{TgtW:fg1} (left) shows the temperature rise of one of the walls of 
 the beryllium window during a train of six micro-pulses that arrive 20~ms
apart. Bunches of these six micropulses arrive at a frequency of 2.5~Hz. 
The temperature rise per micro-pulse, at the center of the beam,
is approximately 10$^\circ$C. In steady-state conditions, coolant flowing 
between the walls, would limit the temperature in the window to 
$\approx 116^\circ$C above ambient, assuming a heat removal coefficient of
100~W/m$^2\cdot {}^\circ$C.  

Figure~\ref{TgtW:fg1} (right) shows the von Mises stress induced in the Be
window by a single micropulse. The peak stress is about 90 MPa
while the yield strength of beryllium is between 186 and 262~MPa. 
 We note that the beam spot on the window will certainly be
larger than that assumed here.
The spot size at the window is related to that at the target by
\begin{equation}
\sigma_{r,\rm window} = \sigma_{r,\rm target}
\sqrt{1+ \frac{L^2_{\rm window}}{\beta^{\star 2}}},
\label{window.1}
\end{equation}
where, $\sigma_{\rm target} = 1.5$~mm, $L_{\rm window}$ is the distance from
the window to the target, and $\beta^\star$ is the betatron parameter of the
beam focus (not yet determined).  Clearly, large $L$ and small $\beta^\star$
provide greater safety margin for the beam.   In the present design,
$L_{\rm window} \approx 3.3$~m, but parameters of the proton beam focus,
including $\beta^\star$, have not been set. In any case, we have taken a very conservative estimate of the spot size, so we have a significant safety margin.

%The ANSYS model indicates that it would be impractical to operate a double
%beam window with atmospheric pressure coolant between the window pair and
%vacuum on the other sides.  Then, a peak von Mises stress of 400 MPa would be 
%experienced
%at the edge of a 5-cm-diameter Be window, which is well beyond the yield
%stress.  A cooled double window should be operated at atmospheric pressure.


%\begin{figure}
%\begin{center}
%\includegraphics*[width=3.5in]{simos_fig4.eps}
%\caption[von Mises shock stresses]
%{von Mises shock stresses in a 1-mm-thick stainless steel beam window induced by
%a micro-pulse of 16 TP, 24 GeV, and $\sigma_r = 2$ mm.}
%\label{TgtW:fg4}
%\end{center}
%\end{figure}

%\begin{figure}
%\begin{center}
%\includegraphics*[width=3.5in]{simos_fig5.eps}
%\caption[Radial stresses in a 1-mm stainless steel window]
%{Radial stresses in a 1-mm-thick stainless steel window induced by a 
% micro-pulse of 16 TP, 24 GeV, and $\sigma_r = 2$ mm.}
%\label{TgtW:fg5}
%\end{center}
%\end{figure}
\subsection{Downstream Beam Window}
The downstream beam window is located on the magnetic axis at $z = 6$~m and 
will be approximately 36~cm in diameter.  It intercepts forward secondary 
particles, but not the unscattered proton beam.  The baseline window design is a pair of 2-mm-thick Be plates with active cooling between them.

A MARS calculation of the power deposition and radiation dose in the Be
window is shown in Fig.~\ref{Tgt:be-window-dose}.  The dose is high enough
that the Be window is not a lifetime component.  A preliminary concept for
window replacement is shown in Fig.~\ref{Tgt:fg4}.

The mechanical design of the downstream window is governed by the following:
\begin{itemize}
\item Large window diameter (36 cm)
\item Pressurized active coolant in the gap of the double wall
\item Vacuum environment on the downstream side
\end{itemize}
The principal design challenge is to maintain mechanical integrity against
the pressure differential over the large window area.
Failure due to beam-induced stress is a lesser concern for this window.

Three variations of the basic design concept are being considered, as
 shown in Fig.~\ref{TgtW:fg7}.  For a window with flat plates, as in
Fig.~\ref{TgtW:fg7}~c), the stress at the edge of the plates due to a one
atmosphere pressure differential is above the yield strength.  To relieve the
stress the windows should be curved, as in Fig.~\ref{TgtW:fg7} a) and b).
Option a) in which the two windows have equal but opposite curvature, appears
to be more favorable, with a steady-state temperature gradient of only
$30^\circ$C.  If no coolant were used, the temperature gradient would be
250$^\circ$C.
\begin{figure}
\begin{center}
\includegraphics*[width=3.5in]{../template/report/ps-and-eps/simos_fig7.eps}
\caption[Three double window designs for the downstream beam window ]
{Three double layer designs for the downstream beam window.}
\label{TgtW:fg7}
\end{center}
\end{figure}
\section{Mercury Deflectors}
Two components of the mercury handling system present unusual design
challenges in view of the disruptive effect of the proton-mercury interaction: 
i) the mercury jet nozzle and ii) the entrance baffles to the mercury pool that serves as the proton beam absorber.

\subsection{Mercury Jet Nozzle}
Pressure waves generated in the mercury jet during its interaction with
the proton beam will travel back to the nozzle, which must withstand the
pressure wave.  An ANSYS model of the effect of a pulse of $1.7 \times 10^{13}$
24-GeV protons on a 5-mm-radius mercury jet indicates a peak stress of
3800 MPa.
  The resulting pressure wave propagates to the nozzle in about
100 $\mu$s where the pressure pulse will be about 100 MPa, as shown in
Fig.~\ref{TgtW:fg11}.

\begin{figure}
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/simos_fig9.eps}
\includegraphics*[width=3in]{../template/report/ps-and-eps/simos_fig11.eps}
\caption[Pressure profile in the Hg jet ]
{ANSYS model of the pressure wave in the mercury jet induced by a pulse of 
$1.7 \times 10^{13}$ 24-GeV protons.  Left: the pressure profile just after
the proton pulse;
right: the pressure profile when the wave reaches the nozzle after 100 $\mu$s.}
\label{TgtW:fg11}
\end{center}
\end{figure}

The nozzle must be constructed of a material with yield strength well above
100 MPa to have the desired lifetime of $> 10^8$ cycles.
\subsection{Entrance Baffles to the Mercury Pool} 
Both the unscattered proton beam and the undisrupted mercury jet enter
a pool of mercury at $2.25 < z < 5$ that serves as the proton beam absorber.
Details of this concept are shown in Fig.~\ref{TgtW:fg14}.
\begin{figure}
\begin{center}
\includegraphics*[width=4in,clip]{../template/report/ps-and-eps/simos_fig14.eps}
\caption[The mercury pool/proton beam absorber ]
{Schematic of the mercury pool that serves as the proton beam absorber.}
\label{TgtW:fg14}
\end{center}
\end{figure}

The undisrupted mercury jet has mechanical power $\pi \rho r^2 v^3/2 \approx
10$ kW for $r = 5$~mm and $v = 30$~m/s.  This power will agitate the mercury
pool unless the impact of the jet is mitigated by a set of diffusers 
submerged in the pool.  The diffusers will consist of stainless steel mesh and a bed
of tungsten balls.

The unscattered part of the proton beam retains about 10\% of the initial
beam power, which is sufficient to disperse a significant volume of mercury as
it enters the pool.  A set of stainless-steel-mesh 
baffles will direct the ejected mercury
droplets back into the pool.  The design must be robust enough to survive at
least one pulse in which the mercury jet was not present and the full proton
beam entered the pool.   
\afterpage{\clearpage}