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\begin{document}

\title[]{Muon Phase Rotation and Cooling: \\
Simulation Work at CERN}

\author{G.~Franchetti\dag, S.~Gilardoni\ddag, K.~Hanke\ddag, E.~B.~Holzer\ddag, A.~Lombardi\ddag, 
M.~Migliorati\S $\parallel$, F.~Tazzioli$\parallel$ and C.~Vaccarezza$\parallel$}

\address{\dag\ GSI, Darmstadt, Germany}

\address{\ddag\ CERN, Geneva, Switzerland}

\address{\S\   Univ. La Sapienza, Dip. Energetica, Rome, Italy}

\address{$\parallel$ INFN-LNF, Frascati, Italy}

\begin{abstract}
We present simulation work performed at CERN and collaborating institutes
concerning the muon front-end of a neutrino factory and a possible muon
cooling experiment. The front-end in the CERN scheme of a neutrino factory
has been revised, eliminating the 44~MHz cavities and starting
directly with 88~MHz. Two options for a muon cooling experiment have been
studied, one based on the 88~MHz CERN cooling channel, one based on the
200~MHz super FOFO channel of the US study~II design. A figure of merit
for the cooling efficiency is discussed.
\end{abstract}

%Uncomment for PACS numbers title message
\pacs{29.27.Bd, 29.27.Eg, 41.75.Lx}

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%\submitto{\JPA}

% Comment out if separate title page not required
%\maketitle

\section{Introduction}
The front-end of the CERN scheme for a neutrino factory, which was
based on using both 44~MHz and 88~MHz cavities, has now been re-designed with the
goal of eliminating the 44~MHz cavities. The main part of this paper
will be dedicated to the new front-end. Furthermore, we present
simulations of a muon cooling experiment.

\section{New 88~MHz Front-End}
The CERN reference scenario~\cite{refscen} used 44~MHz cavities to perform
the phase rotation and the first stage of cooling. The cavities were assumed
to provide an average gradient of 2~MV/m and had a 2~T solenoid housed
around the beam pipe~\cite{gerigk}. For 4~MW beam power on target, the 
reference scenario could
provide some 10$^{21}$ muons/year in the recirculator acceptance.
The engineering study for the cooling experiment revealed that a 4~T
average field was within the reach of a superconducting solenoid with the
required dimensions to fit around a 88~MHz cavity~\cite{gerigk}. This triggered a
review of the whole front-end. In particular the goal was to design a system
with the same efficiency (10$^{21}$ muons in the recirculator acceptance) but without 
using the bulky 44~MHz cavities. The bottleneck of the longitudinal
acceptance is the high rf frequency of the recirculator: the 200~MHz
recirculator of the CERN reference scenario offers an acceptance of the
order of 0.1~eV~s and as longitudinal cooling is not foreseen at this
stage, the number of particles that can be potentially accepted must already lie
within this acceptance before transverse cooling. Several
competing factors came into the trade-off between starting with either a 44~MHz or a
88~MHz system. A 44~MHz system allows for a more efficient RF gymnastics to
concentrate particles from the target into the wanted acceptance. A 44~MHz
system is also effective at a distance of 30 meters from the target where
most of the pions have decayed into muons, thus allowing operation at a
very modest gradient. On the other hand, an 88~MHz system, placed closer to
the target (about 15 meters) mitigates the semi-relativistic effects which
spoil the longitudinal emittance. 
Also, due to the higher gradient (4~MV/m), fewer 
cavities are needed. Assuming that it is possible to produce a number of 88~MHz
cavities with 40~cm aperture, a system employing only 88~MHz has been
proved to perform as well as the CERN reference scenario. The phase rotation
is performed by 8 RF cavities and the fundamental cooling cell consists of
6 cavities, half a meter of hydrogen and a half meter long matching solenoid. The
parameters are reported in Table~\ref{tab88mhz}. 
%Phase space plots at the input
%and output of the phase rotation section are 
%shown in Fig.~\ref{phrot}.

\begin{table}[h]
\begin{center}
\small
\begin{tabular}{|l|c|c|c|c|c|}
\hline
 & decay & rotation & cooling & acceleration I & acceleration II \\
\hline
length [m] & 15  & 8 & 90 & $\approx$10 & $\approx$450 \\
\hline
aperture [cm] & 40 & 40 & 40 & 30 & 20 \\
\hline 
B-field [T] & 4 & 4 & 4 & 4 & quadrupoles \\
\hline
frequency [MHz] & & 88 & 88 & 88 & 88-200 \\
\hline
gradient [MV/m] &  & 4 & 4 & 4 & 4-10 \\ 
\hline 
E$_{kin}$ [MeV] &  & 200 & 200 & 300 & 2000 \\
\hline
\end{tabular}
\caption{Parameters of the 88~MHz front-end.}
\label{tab88mhz}
\end{center}
\end{table}
\normalsize
%
%\begin{figure}[h]
%\begin{center}
%\includegraphics[width=10cm,bb = 47 457 490 580,clip]{phrot.eps}
%\caption{Longitudinal phase space at input (left) and output (right) 
%of phase rotation section.}
%\label{phrot}
%\end{center}
%\end{figure}

\section{Cooling Experiment Simulations}
We have studied in detail a Muon Cooling Experiment based on the
88~MHz CERN cooling channel. 
%The baseline scenario is a set-up
%of 8 cavities with two LH$_2$ absorbers of 47~cm length at 
%the entry and exit of the acceleration section. The emittance
%diagnostics is located upstream and downstream of the channel.
The simulations have been performed using field maps obtained
from an engineering design of the solenoids and cavities.
The detailed set-up and beam dynamics of this channel are discussed
in~\cite{nfnote108}. 
For nominal settings and
input beam parameters, the
transmission is 100\% and
the transverse emittance at the exit of the cooling channel
is reduced by about 3.7\% per plane. 
%Furthermore, we have varied
%the input beam parameters and explored the cooling efficiency
%for different regimes.
%\\
%A cooling experiment based on only 4 cavities has also been studied.
%The cooling efficiency drops down in proportion and is probably beyond
%the sensitivity of the diagnostics apparatus.
%\\
As an alternative to the experiment based on 88~MHz cavities, a set-up using
the 200~MHz cavities from the US study~II~\cite{study2} was also simulated~\cite{epac}.
This cooling experiment is based on a total of 4 pillbox cavities at 200~MHz with 
absorbers at the entry and exit as in the case of the 88~MHz set-up. 
The main difference is the arrangement of the magnetic coils: at 200~MHz, they cannot
be integrated in the cavity but must be outside the cavities at a large radius.
This drastically changes the beam dynamics of the system, as there is much stronger
inter-plane coupling than in the case of the 88~MHz system. However, the overall cooling
efficiency of both systems is comparable.
In order to diagnose the cooling efficiency in the presence of coupling,
an algorithm was developed to 
measure 4D and 6D cooling rather than the 2D projections.
The orientation of the hyper-ellipsoid is adapted to the rms muon 
distribution. 
%The condition for a particle to be inside the hyper-ellipsoid 
%is a generalization of the 2D case to k dimensions:
%\begin{equation}
%X_0^T \Sigma^{-1} X_0 < (\frac{\epsilon_{acceptance}}{\epsilon_{rms}})^{2/k}.
%\label{eq:cut}
%\end{equation}
%$\Sigma$ is the beam sigma matrix, $\epsilon$ is the kD emittance and 
%$X_0$ is the coordinate vector of the particle.
%\\
Figure~\ref{merit} shows the number of muons inside a normalized 4D 
volume of $(0.015 \; \pi 
\mbox{m}\;\mbox{rad})^2$ along the z-position of the cooling experiment.
The left plot corresponds to the 
88~MHz cooling experiment, the right plot to the 200~MHz scenario.
The upper curves are the
number of muons in a 4D ellipsoid, the lower ones in the 2D projections.
%
%In the 88~MHz case the correlations between the x and y plane stay
%below 0.08 along the channel. In the 200~MHz case the
%correlations are 0, 0.7, 0.6 and 0.14 respectively at the 4 analyzed positions.
%At the points of high correlation the method of 2D projections can not be 
%used to measure the cooling performance.
%Counting the muons in 4D (or 6D) yields a stable figure of merit in
%the presence of correlations.
%The 4D ellipsoid looks at a more central core of the
%distribution, hence the higher density of muons and a slightly larger
%factor of merit. In the 88~MHz simulation
%the number of muons increases by 9.9\% in the 4D hyper-ellipsoid
%as compared to 7.6\% in the 2D projections. 
%In the 200~MHz simulation
%the
%figure of merit calculated in the 4D ellipsoid and the 2D projections
%is 13.3\% and 9.9\% respectively. 
%This method gives a comparable figure of merit for the two experiments. 
\\
The beam dynamics simulations reported in this section were performed
using the tracking code PATH. The results of this code were cross-checked
and found to be consistent with the code GEANT~IV~\cite{simone}. 
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=14cm]{merit.eps}
\caption{Number of muons along the 88~MHz cooling experiment (left plot)
and along the 200~MHz cooling experiment (right plot). The upper curves correspond
to a 4D ellipsoid, the lower curves to a 2D ellipsoid.}
\label{merit}
\end{center}
\end{figure}

\section{Conclusions}
We have successfully designed a new front-end for the CERN
neutrino factory scheme using only 88~MHz cavities,
which shows the same performance as the previously proposed scheme
working at both 44~MHz and 88~MHz.
For the cooling experiment, we have completed a study of two options:
one based on the 88~MHz CERN cooling channel, the other one based on
the 200~MHz US study~II super FOFO channel. We find comparable cooling
efficiency for both set-ups, even though the beam dynamics (coupling) is
very different.

\section{References}

\begin{thebibliography}{9}

\bibitem{refscen}
A.~Lombardi, {\sl A 40-80 MHz System for Phase Rotation and Cooling},
CERN Nufact Note 34 (2000).

\bibitem{gerigk}
R.~Garoby, F.~Gerigk, {\sl Cavity Design for the CERN Muon Cooling
Channel}, CERN Nufact Note 87 (2001).

\bibitem{nfnote108}
M.~Aleksa, J.F.~Amand, R.~Garoby, F.~Gerigk, K.~Hanke, E.~B.~Holzer,
E.~S.~Kim, A.~Lombardi, M.~Migliorati, S.~Russenschuck, F.~Tazzioli,
C.~Vaccarezza, {\sl Beam Dynamics Study of a Muon Ionization Cooling
Experiment}, CERN Nufact Note 108 (2002).

\bibitem{study2}
S.~Ozaki, R.~Palmer, M.~Zisman, J.~Gallardo (ed.),
{\sl Feasibility Study-II of a Muon-Based Neutrino Source},
BNL 52623 (2001).

\bibitem{epac}
M.~Migliorati, L.~Palumbo, F.~Tazzioli, C.~Vaccarezza, K.~Hanke, E.~B.~Holzer,
A.~Lombardi
{\sl Beam Dynamics Simulation of a Muon Cooling Experiment with 200~MHz 
Cavities in the Framework of the CERN Cooling Study}, conf. proc. EPAC~2002,
Paris (2002)

\bibitem{simone}
S.~Giani, S.~Gilardoni, V.~M.~Grichine, 
{\sl GEANT~IV Simulation of Phase Rotation for Neutrino Factory},
CERN Nufact Note 102 (2001).

\end{thebibliography}

\end{document}

