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\title{Muon collection with an AG channel}

\author{Bruno Autin  
\footnote[3]{To
whom correspondence should be addressed (bruno.autin@cern.ch)}
, Andr\'e Verdier\dag\ and Fran\c{c}ois M\'eot\ddag}

\address{\dag\ CERN, Switzerland}

\address{\ddag\  CEA, France}

\begin{abstract}
 A new pion collection system has been proposed for the CERN neutrino
factory. It consists of an AG channel which matches several
horns, with one target per horn, to an AG decay channel. The effect
of  pion decay in this channel has been  investigated by means of a
ray-tracing code which includes Monte Carlo simulation of pion decay
into muons. The AG decay channel is a little less efficient than
a solenoid decay channel but it is much more economical and robust.
\end{abstract}

%\maketitle

\section{Introduction}
 A pion collection system has been proposed for the CERN neutrino
factory~\cite{tsukuba} which features an ensemble of four horns placed
close to each other and an alternating gradient matching section which
delivers the four beams into the pion decay channel. The advantages of
such a system are detailed in~\cite{tsukuba}.
The performance of a complete system including a solenoid decay channel
were analysed in~\cite{epac02}.

In the present design, the decay channel is made from AG cells.
In this paper this
channel is compared with solenoid decay channels.

\section{Funneling section}
The basic design of~\cite{epac02} has been kept but the quadrupoles
have been lengthened to maintain the field of the pole below 0.5T,
taking into account a superimposed dipole component to steer the beam
and match the 
dispersion. The edge focusing of the quadrupoles has been included in
the matching procedure. As a result, there is only a small difference 
between the optics computed either with transfer matrices by means of
MAD~\cite{mad} or with numerical integration of trajectories by means
of ZGOUBI~\cite{zgoubi} which includes the actual variation of the
field at the end of the quadrupoles.

Two quadrupoles have been added to match the funneling section to the
AG decay channel. The optics functions are shown in figure\ref{optd}.

\section{AG decay channel}
The decay channel is made of 14 cells with a phase advance such that
the average $\beta$-functions are minimum. It has been investigated
whether tuning the gradients according to the average energy of the
beams is attractive. The different between  this tuning and an uniform
tuning is not significant.
\begin{figure}   \begin{center} \begin{turn}{-90}
\epsfig{file=
/afs/cern.ch/user/v/vrd/public/conferences/nufact/london02/optEedge.eps,
height=15cm,width=7cm}   \end{turn}
\end{center} \vskip -5mm  \caption{\label{optd}Optics functions and beam
  trajectory in the funnelling section. The beam enters the system on
  the left 0.4m above the axis, with an angle of 0.093rad downwards.}
\end{figure}

Trajectories of pions and muons with a uniforn energy distribution
between 100MeV and 400MeV are plotted in
figure~\ref{trjmupi}. It is clear that a radius of at least 20cm is
necessary to avoid pion losses.
\begin{figure}[h]   \begin{center}
\epsfig{file=
/afs/cern.ch/user/m/missy/meot/nuFact/Log/x.vs.XLab_pimu.eps,
height=8cm,width=15cm}
\end{center}   \caption{\label{trjmupi}Pions and muons trajectories
in the funnelling section. The enegy spread of the pions is uniform 
between 100MeV and 400MeV. 
The pions accupy the dark part of the plot, i.e. their amplitudes stay
below 0.2m}     \end{figure}

\section{Efficiency of the new system}
Trajectories of pions and muons have been tracked with different inner
radii of the quadrupoles, assuming that a particle hitting the
aperture is lost. The efficiency defined as the ratio between
incomming pions and outgoing muons is shown in figure~\ref{efff}
\begin{figure}[h]   \begin{center}
\epsfig{file=
/afs/cern.ch/user/m/missy/meot/nuFact/Log/partNvsEpsxzB.eps,
height=7cm,width=15cm}
\end{center}   \caption{\label{efff}Efficiency of three different
  decay channels as a function of the total beam normalised
  emittance. For an emittance of 0.01m (left point), the efficiency of
  the AG channel is 25\% while the efficiency of the solenoid channel
  is 33\%.}     \end{figure}

\section{Conclusion}
Using an AG decay channel with room temperature quadrupoles is
attractive as it is less sensitive to 
particle losses than a superconducting system. However the efficiency
of the AG channel is about 76\% of that of the solenoid channel.

An interesting outcome of this study is the fact that the dispersion
matching is extremely efficient. It opens the possibility of using
seven horns (at the expense of a very wide first quadrupole).

\section*{References}
\begin{thebibliography}{4}

\bibitem{tsukuba}B. Autin, A. Verdier, Multiple beam collection. Proceedings
 of the Neutrino Factory workshop in Tsukuba (May 2001). Also NuFact Note 100
 (October 2001) CERN.

\bibitem{epac02}F. M\'eot, B. Autin, A. Verdier, A new pion collection 
system for the CERN neutrino factory. EPAC02, Paris, June 2002. 

\bibitem{mad} H. Grote and F.C. Iselin, The MAD program (Methodical
  Accelerator Design) version 8.16, User's reference manual,
  CERN/SL/90-13(AP), (rev. 4) (March 27, 1995).

\bibitem{zgoubi}F. M\'eot and S. Val\'ero, Zgoubi usre's guide, Report
CEA/DSM/DAPNIA/SEA-97-13, Saclay, October 1997.

\end{thebibliography}

\end{document}





\epsfig{file=
/afs/cern.ch/user/m/missy/meot/nuFact/Log/x.vs.XLab_pi.eps,
height=8cm,width=15cm}

\bibitem{spectro}F.  M\'eot and N. Willis,
Ray-trace computation with Monte Carlo simulation of particle decay,
internal report CEA/LNS/GT/88-18, Saclay  (1988)

\bibitem{yelnufact02}The Neutrino Factory Steering Group, The study of
  an European Neutrino Factory Complex, CERN yellow report to be
  published (P. Gruber editor).

\bibitem{bopt}BeamOptics, A program for analytical beam optics,
  B. Autin editor, CERN 98-06, p 76-78 (6 August 1998)

\begin{itemize}
\item from the World Wide Web
\item from the World Wide Web
\end{itemize}

\begin{equation*}
\frac{1}{M_{\rm a}}\left(\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}\right)^{-1}
\end{equation*}
instead of
\begin{equation*}
\frac{1}{M_{\rm a}}\biggl/\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}.
\end{equation*}


\begin{table}
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and $\backslash${\tt lineup} to assist in aligning columns on the 
decimal point. The width of the 
table and rules is set automatically by the 
preamble.} 

\begin{indented}
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\br                              
$\0\0A$&$B$&$C$&\m$D$&\m$E$&$F$&$\0G$\cr 
\mr
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\0\039.7&\-60&0.74&$-51.9$&$-0.208$&47.2 &146\cr 
\0123.7 &\00 &0.75&$-57.2$&\m---   &---  &---\cr 
3241.56 &60  &0.60&$-48.1$&$-0.29$ &41   &\015\cr 
\br
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\end{indented}
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