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\title[Frictional Cooling]{Frictional Cooling}

\author{Raphael Galea$^*$ \dag \footnote[3]{Email: galea@nevis.columbia.edu}\\
{\small $^*$ in collaboration with A. Caldwell\dag\ and S. Schlenstedt\ddag}}

\address{\dag\ Nevis Laboratories, Columbia University, New York, New
York, USA}
\address{\ddag\ DESY, Zeuthen, Germany}

%\begin{abstract}
%The concept of frictional
%cooling will be discussed as a possible candidate for the cooling channel
%of a Muon Collider.
%\end{abstract}

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\begin{abstract}
To build a Muon Collider, on the order of $10^{12}$ muons
must be produced and then collected in a small phase space to achieve
interesting luminosities. The muons are produced by targeting a
bunched proton beam on a target in a region of strong magnetic field.
The pions resulting from the proton interactions are constrained to
helical trajectories by the magnetic field, and decay to muons
in some drift space. The resulting muons occupy
 very large phase space. They must then be collected and cooled
before they can be injected into an accelerator chain. Reducing this
phase space of the produced muons, by at least $10^6$, is a
critical issue in producing a high luminosity Muon Collider.
The scheme considered here is
based on frictional cooling in gases\cite{fric}. This in principle gives
the possibility to cool both $\mu^+$ and $\mu^-$ to the sufficient
emittance. 
\end{abstract}

\section{Basic Ideas}

The basic idea of frictional cooling is to bring the muons into a
kinetic energy range where the energy loss per unit distance increases
with kinetic energy. A constant accelerating force can be applied to
the muons resulting in an equilibrium energy. A sample $dE/dx$ curve
is shown in Fig. \ref{dedxoverview}, 
where it can be seen that this condition can
be met for kinetic energies below a few KeV or kinetic energies beyond
about 200 MeV. At the high energy end, the change in $dE/dx$ with
energy is only logarithmic, whereas it is approximately proportional
to the speed at low energies. Below the $dE/dx$ peak, muons are too
slow to ionize the atoms. The processes leading to energy loss:
excitation, elastic scattering on nuclei and charge exchange reactions
yield differences for $\mu^+$ and $\mu^-$, with significantly large
energy loss rates for $\mu^+$\cite{barkas}.
\begin{figure}
\begin{center}
\epsfig{file=figs/dEdx-overview.eps,height=7cm,width=7cm}
\end{center}
\caption{\label{dedxoverview} $dE/dx$ in Helium as a function of
kinetic energy, T, for $\mu^+$. The effective $dE/dx$ resulting from
an external electric field is superimposed. An equilibrium kinetic
energy near 1 KeV would result. The nominal scheme discussed for a
Neutrino Factory would cool muons near T=200 MeV.}
\end{figure}

Operating in this energy regime, an electric field can be applied
which compensates for the energy loss. Several issues become apparent:
\begin{itemize}
\item $dE/dx$ is very large in this region of kinetic energy, so we
need to work with a low average density gas in order to have a reasonable
electric field strength.  5 MV/m is used in all simulations.
\item The electric and magnetic fields must be applied perpendicular
to each other or the muons
will never get below the $dE/dx$ peak. At the higher energies, the
muons follow the magnetic field lines with slow drift and do not pick
up energy from the electric field. Once the muons are slow, the
electric force is no longer small compared to the magnetic force and
the muons will drift out of the volume at a definite Lorentz angle
given by the ratio of the magnetic and electric  fields. 
\item Muonium formation ($\mu^+ + {\rm Atom}\rightarrow {\rm Mu}$) is
significant at low $\mu^+$ energies. In fact, the muonium formation
cross section dominates over electron stripping cross section in all
gases except Helium\cite{nakai}. Data from Agnello
\etal\cite{agnello} is parameterized and included in our simulations.
\item A possibly fatal problem for $\mu^-$ is the loss resulting from
muon capture. ($\mu^- + {\rm Atom}\rightarrow \mu{\rm Atom} + e^-$)
The cross section for this process has been calculated up to 80
eV\cite{cohen}. 
Hydrogen and
Helium seem to be the best candidate media. 
\end{itemize}

\section{Simulation of Frictional Cooling Scheme}

Given the conditions outlined in the previous section, a scheme was
simulated from the production of the pion beam to the extraction of the
cooled muons from the cooling section.

The target scheme differs from that of the Neutrino Factory.
MARS simulations were performed to optimize the pion
production for pions with kinetic energies less than 120 MeV. 
The pions are extracted transverse to the target.  
The forward pion production from a proton impinging
on a target has tails to large kinetic energies which this scheme will
not cool. Moreover, there are roughly equal numbers of
$\pi^+$ and $\pi^-$ produced on either side of the target. This allows
for both signs to be cooled at the same time by developing a symmetric
channel on either side of the production target. To this end a new
symmetric magnetic capture section was designed with an initial
strength of 20 T tapering down to 5 T for the remaining drift and
cooling sections. The drift distance was also maximized to optimize
the muon yield. As a result of these investigations the following optimal
conditions were determined: a 2 GeV proton driver on a Cu or W target
followed by 28m of drift. 

At the end of the drift region, there is a correlation between the
longitudinal momentum of the muons and the arrival time. This allows
for a phase rotation, where time varying electric fields are used to
increase the number of muons at lower momenta. The muons are then
input into the cooling channel which consists of a series of cooling
cells interspersed between solenoids. Due to the repetitive nature of
the scheme a ring design was adopted for the cooling section in which
the phase rotation would be performed in the solenoidal regions
in between the cooling cells. A simple model was used for the phase
rotation portion and consisted of a flat 5 MV/m field for some time,
$t_1$, after which the field went  linearly to zero  at a time,
$t_2$. These parameters were also optimized, as well as the 
time window for the beam to be accepted 
into the ring. The cooling cells contained
Helium gas for $\mu^+$ and Hydrogen for $\mu^-$. The emittance values
were calculated to the edge of the cooling cells, only for those muons which
were cooled (see Table \ref{emittances}). 

\begin{table}
\caption{\label{emittances} Emittance Reduction factors for a Ring
Frictional Cooling Scheme.}
\begin{indented}
\item[]\begin{tabular}{@{}lllll}
\br
Target & Muon sign & Cooling Medium & Yield ($\mu$/p)&
$\epsilon_{6D}/\epsilon'_{6D}$($1\times 10^6$)\\
\mr
Cu & + & He & 0.005 & 22\\
Cu & - & He & 0.002 & 0.06\\
Cu & - & $H_2$ & 0.003 & 0.8\\
W  & + & He & 0.006 &18\\
W  & - & He & 0.003 & 0.06\\
W  & - & $H_2$ & 0.004 & 0.6\\
\br
\end{tabular}
\end{indented}
\end{table}

\section {Nevis Lab and Ongoing Improvements}

The simulations to date have not included thin windows which are
required in order to extract the cooled muons from the cooling cells.
This will be included in the next iterations.
A simpler architecture is being considered; one in which the all the
cooling cells are collected into one large cooling volume followed by
a region of phase rotation. This phase rotation will reflect those
muons which do not cool in the first pass through the cooling volume.
This linear channel is more practical to construct and avoids the
difficulties of kicker systems needed for injection and extraction
into  any ring scheme. 

A lab has been setup at Nevis in order to address some of the key
experimental issues for frictional cooling. Experiments are underway
in order to demonstrate the frictional cooling concept using protons
at the 4 MV Van der Graaff accelerator located at the Radialogical
Research Accelerator Facility located on the Nevis site. 

Frictional Cooling has the potential to cool muon beams to the
required emittance for use in a Muon Collider. 


\section*{References}
\begin{thebibliography}{99}
\bibitem{fric} M.Muhlbauer, \etal, {\it Hyperfine Interact. \bf 119}, 305
(1999).
\bibitem{barkas} W.H.Barkas, W.Birnbaum and F.M.Smith, {\it Phys. Rev.
\bf 101}, 778 (1956).
\bibitem{nakai} Y.Nakai, T.Shirai, T.Tabata and R.Ito, {\it At. Data
Nucl. Data Tables \bf 37}, 69 (1987).
\bibitem{agnello} M.Agnello, \etal, {\it Phys. Rev. Lett. \bf 74}, 371
(1995).
\bibitem{cohen} J.S.Cohen, {\it Phys. Rev. \rm A \bf 62}, 1160 (2000).
\end{thebibliography}



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