\section{Minicooling Absorbers}
\label{DandPR:mini}
%\centerline{Harold  Kirk, Dan Kaplan}

%\centerline{\it Brookhaven National Laboratory, Illinois Institute of Technology}
%\subsubsection{Introduction}

The baseline design includes two ``minicooling" liquid-hydrogen absorbers, each
30~cm in radius and 1.75~m long, preceded by a thin beryllium sheet and
separated by a field flip.  ICOOL
simulations have been used to propagate MARS-generated secondary particles from
the primary target through the initial induction linac module to the minicool
absorbers.  Tables~\ref{CPR:tb11} and ~\ref{CPR:tb22} give the estimated power 
deposition in these absorbers for each important particle species generated 
at the target. For each mode the total power deposited in the absorber
is about 5 kW.  As seen in Fig.~\ref{CPR:fig30},
the power dissipation in the first absorber is peaked at the absorber's upstream end.
This energy dissipation peak is due to the arrival of low-energy protons which are generated
at the target and conducted down the capture/decay channel.  They are not removed
by the induction linac because they are out of time with the higher velocity
mesons and electrons.
A beryllium sheet placed immediately before the liquid hydrogen absorber 
is used to absorb the low-energy protons and reduce the peak energy deposition 
in the 
first several cm of liquid hydrogen. However, even without this beryllium 
absorber foil, we find the volume power density in the liquid hydrogen to be 
manageable.

\begin{table}[b]
\begin{center}
\caption[Power dissipation: positive muons]{Power dissipation in the 
$\mu^+$ collection mode.}
\label{CPR:tb11}
%\vspace{0.4 cm}
\begin{tabular}{|lccccc|}  
\hline
       &  \multicolumn {5}{c|}{ Power }   \\
       &  \multicolumn {5}{c|}{ (kW) } \\
\hline
       & $e$  &    $\mu$   &  $\pi$  & $K$  & $p$ \\
\hline
positives       & 0.42  &    2.02   &  0.14  & 0  & 0.86 \\
negatives       & 0.43  &    1.29   &  0.24  & 0  & - \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption[Power dissipation: negative muons]{Power dissipation in 
the $\mu^-$ collection mode.}
\label{CPR:tb22}
%\vspace{0.4 cm}
\begin{tabular}{|lccccc|}  
%       &   &    &    &   &  \\
 \hline
       &  \multicolumn {5}{c|}{ Power }   \\
       &  \multicolumn {5}{c|}{ (kW) }   \\
       & $e$  &    $\mu$   &  $\pi$  & $K$  & $p$ \\
\hline
positives       & 0.42  &    1.45   &  0.19  & 0  & 0.94 \\
negatives       & 0.45  &    1.90   &  0.14  & 0  & - \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in,angle=90]{../template/report/ps-and-eps/harold-dpower_dl.eps}}
\caption{Power deposition along the length of the first minicool section.}
\label{CPR:fig30}
\end{center}
\end{figure}

\subsection{Handling the Average Heat Load}

Even without detailed refrigeration studies, we can be confident
about the overall power-handling capability of these absorbers based on
experience with the Fermilab 15-foot bubble chamber. The 15-foot bubble chamber
was cooled by a 6.7\,kW refrigerator~\cite{CPR:Kilmer-private}. While considerably
larger than a minicooling absorber, the bubble chamber had substantially 
lower beam-induced
power dissipation; the large refrigeration plant was required to remove the heat 
generated by the work done on the fluid by the rapid-cycling expansion piston. 

Each minicooling absorber will have
refrigeration requirements comparable to that of the 15-foot bubble chamber.
On the other hand, Table~\ref{CPR:tb3} shows that the refrigeration needed
for the minicooling absorbers dominates that of the cooling channel itself
(see Section~\ref{BandC:absorber})   %   ??????????
and the extra complexity and reliability impact of operating two
15-foot-bubble-chamber equivalents is worth avoiding if a practical alternative
is available; these considerations motivate the minicooling alternatives
considered in Section~\ref{APP-mini-cool}.

\begin{table}
\begin{center}
\caption[Absorber parameters]{Absorber parameters.}
\label{CPR:tb3}
%\scriptsize
\begin{tabular}{|lccccc|}
\hline
Absorber & Length & Radius & Number & $P$ & $P_{tot}$ \\
 & (cm) & (cm) & & (kW) & (kW) \\
\hline
Minicool & 175 & 30 & 2 & $\approx$5 & $\approx$10 \\
SFOFO lattice 1 & 35 & 18 & 16 &  $\approx$0.3 & $\approx$4 \\
SFOFO lattice 2 & 21&  11 & 36 &  $\approx$0.1 & $\approx$3 \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Handling the Peak Power Density}

Figure~\ref{CPR:fig30} shows the power dissipation \textit{vs.}\ position 
along the absorber,
and Fig.~\ref{CPR:fig31} shows the transverse distribution.  For comparison, the
liquid-hydrogen target built for SLAC Experiment 158~\cite{CPR:SLAC158} is designed
to handle 700\,W, uniformly distributed over 1.5\,m of length but with
about 1\,mm rms\ transverse beam size.  While the power per cm at the
upstream end of the first minicooling absorber is more than 10 times that in 
SLAC E158, the power per cm$^3$ is only about 10$^{-3}$ of that in E158. We
therefore conclude that the peak power density will not pose a problem and can
be handled using one or the other of the approaches described elsewhere in this
report (see Section~\ref{BandC:absorber}).

\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in,angle=0]{../template/report/ps-and-eps/harold-dpower_dr.eps}}
\caption{Transverse power deposition in the first minicool section.}
\label{CPR:fig31}
\end{center}
\end{figure}
\subsection{Window Design}
Unlike the case for the SFOFO absorbers, in the minicooling
absorbers muon multiple scattering is dominated by the hydrogen, and
muon-cooling performance hardly depends on the details of the window design.
Furthermore, a hemispherical window shape, which minimizes the window thickness
for a given strength, is practical for absorbers such as these, whose length
far exceeds their diameter.  The American Society of Mechanical Engineers
(ASME)~\cite{CPR:ASME} specifies the minimum acceptable thickness for a
hemispherical window as
\begin{equation}
t=\frac{0.5PR}{SE-0.1P}\,,
\end{equation}
where $P$ is the differential pressure across the window, $R$ the vessel
radius, $S$ the maximum allowable stress, and $E$ the weld efficiency. 
For $S$,
we follow ASME recommendations and use the smaller of 1/4 of the ultimate
strength, $S_u$, or 2/3 of the yield strength, $S_y$; in practice, 
for aluminum
alloys, it is the ultimate strength that matters. If we machine the window with
an integral flange out of a single disk of material, as for the cooling channel  absorbers, there are no welds and we can take $E=1$. For 1.2 atm
operation, and given the ASME specification for 6061-T6 aluminum alloy,
$S_u=289\,$MPa and  we obtain $t=250\,\mu$m. While a detailed finite-element analysis
(taking into account the stresses imposed on the spherical shell by the stiff
flange) may result in a somewhat thicker window, even windows as thick as 1\,mm
have been shown by simulation to have negligible effect on muon-cooling
performance.

\section{Summary}
The guiding principle of the design for the Capture/phase rotation
section of the front end for the neutrino factory has been to
achieve good performance while utilizing components based on 
at-hand technology.  Key components include large
aperture superconducting solenoids, 260~m of
induction linac, and 3.5~m of liquid hydrogen absorbers.  
The gradients required for the induction linacs are between
1.55 and 1.0~MV/m. The muon beam delivered to the buncher has
an rms energy spread, $\frac{\delta E}{E}$ of 4.4\% and contains 0.49 $\mu$/p within
the $\pm 3.5\sigma$ boundaries of this energy spread.

%We find that all components proposed for this section
%can be built and will perform
%reliably based on current established technology.
%The guiding principle of the design for the capture/phase rotation 
%section of the front end for the Neutrino Factory has been to 
%achieve good performance while utilizing components based on  
%at-hand technology.  Key components include large 
%aperture superconducting solenoids, 260~m of 
%induction linac, and 3.5~m of liquid-hydrogen absorbers.   
%The gradients required for the induction linacs are between 
%1.55 and 1.0~MV/m. The muon beam delivered to the buncher has 
%an rms energy spread, $\frac{\delta E}{E}$ of 10\%. We 
%find that all of these components can be built and will perform 
%reliably based on current established technology. 
  %    put   it  here.

%\end{document}