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\begin{document}
\centerline{RF in the Targetry Experiment}
\vskip 1.0cm
%\small
Once the pions have been collected at the target, it is necessary to
shape the collected pion/muon beam into a longitudinal phase space suitable
for eventual acceleration in the following cooling section. The goal
is to capture and maintain the longitudinal phase space such that the
dE/E spread is reduced to near 10\% and the pulse is contained
within a pulse length of 4$\tau <$ 6 m. Since we are interested in
capturing low-energy pions (Kinetic Energy 50-400 MeV) , these pions will have a
large initial velocity spread ( $\beta$ between 0.7 and 0.98). It is therefore
important to begin the phase rotation section as soon a practical
with an accelerating gradient as high as possible. The wavelength
of the rf cavities are constrained by the requirment that $\lambda_{rf}$/2 be greater
than the bunch length of the collected pions/muons. This bunch length in turn
is determined by the initial bunch length of the proton beam impinging on the
target and the drift distance to the initial rf cavity. For an initial
proton bunch length $\tau_{rms}$ of 1 ns and a drift distance of 3 m this
total bunch length will be 4 ns (1.2 m). This constrains the
wavelength of the initial rf cavity to be greater than $\approx$ 2.4 m (frequency
$<$ 125 MHz. As the beam moves down the phase rotation channel, it will continue
to spread in length and therefore the rf wavelength of subsequent rf cells will
need to increase and a complete phase rotation scenario will require many rf cells
of different frequencies.
A further constraint is placed on the frequencies of
the rf cells if we add the requirement that the same phase rotation channel must
be capable of handling both positive and negative bunches in separate beam
spills. This constraint can be simply satisfied by requiring the set of cavities
to be odd multiples of a fundamental harmonic, e.g. (20, 60, 100 , 140 MHz, etc. or
10, 30, 50, 70, 90, 110 MHz, etc.). We chose for convenience the series based on
the fundamental harmonic of 10 MHz. In such a scenario, the beginning rf frequency
could be 110 MHz descending to a final rf frequency of 10 MHz. Parameters for
a possible complete scenario is given in the following table:
\begin{center}
\begin{tabular}{lccc}
\multicolumn{4}{c}{Low-energy Collection Linac Parameters} \\
\noalign{\vspace{2pt}}
\hline\hline
\noalign{\vspace{2pt}}
& & & \\
RF frequency [MHz] & 90 & 50 & 30 \\
Cavity Length [cm] & 120 & 120 & 120 \\
Full Gap length [cm] & 36 & 36 & 36 \\
Cavity Radius [cm] & 90 & 206 & 126 \\
Beam Pipe Aperture [cm] & 30 & 30 & 30 \\
Q/1000 (from SFISH) & 53.4 & 71.1 & 16.8 \\
Ave Gradient [MV/m] & 4.2 & 4.0 & 2.1 \\
RF Peak Power [MW] & 1.8 & 1.2 & 4.8 \\
Ave Power (15Hz) [KW] & 2.4 & 7.8 & 12.6 \\
Stored Energy [J] & 166 & 260 & 418 \\
Linac Segment [m] & 6 & 18 & 18 \\
Total Power (15Hz) [KW] & 12 & 118 & 190 \\
& & & \\
\hline\hline
\end{tabular}
\end{center}
For the $\mu\mu$ collection system to be viable, we must satisfy ourselves that
the rf cells will work at the levels required of them and in particular
that the initial rf cells will perform as expected in the high-radiation environment
immediately following the target. Since experience operating an rf cell in a
high-radiation environment generated by a 10$^{14}$ ppp beam is limited, we propose
to establish a proof-of-principle demonstration of this issue by constructing
and operating at high gradients an rf cavity with a frequency suitable for
a muon collider collection system.
A key issue in establishing an rf test is obtaining the rf power required for
such a test. A search of available rf in the upper frequency range of our
requirements has led us to 70 MHz rf power which is now available a LBL as the
result of retirement of the HILAC facility. We therefore focus our efforts on
the design and utilization of system using this frequency. As a base line we
consider an rf accelerating cell with the following parameters:
\begin{center}
\begin{tabular}{lc}
\multicolumn{2}{c}{70 MHz rf Cavity Parameters} \\
\noalign{\vspace{2pt}}
\hline\hline
\noalign{\vspace{2pt}}
& \\
RF frequency [MHz] & 70 \\
Cavity Length [cm] & 120 \\
Full Gap length [cm] & 50 \\
Cavity Radius [cm] & 125 \\
Beam Pipe Full Aperture [cm] & 60 \\
Q/1000 (from SFISH) & 63.1 \\
Ave Gradient [MV/m] & 5.0 \\
RF Peak Power [MW] & 2.4 \\
Stored Energy [J] & 330 \\
& \\
\hline\hline
\end{tabular}
\end{center}
This cavity is shown in figure \ref{70mhz_cavity}.
\begin{center}
\begin{figure}[!hbt]
{\centering
\epsfig{file=rfcell.eps,width=4.5cm}
\caption{Superfish solution for a 70 MHz rf cavity.}
\label{70mhz_cavity}
}
\end{figure}
\end{center}
The above parameters are valid for operating the rf cavity at a level
of 2 Kilpatricks (corresponding to peak wall electric gradients of
20 MV/m). For testing purposes, it would be of interest to run the
cavity at the highest possible peak power levels in order to obtain
the maximum accelerating gradient at the front end. The r\&d program
would entail constructing a 70 MHz rf cavity, powering it to maximum
power levels without beam/target interactions and then determine
the maximum power levels achievable with the high radiation environment
present with beam/target interactions.
A further requirement of the phase rotation channel is to envelope the
entire rf channel within a solenoidal field of 1.25 T. It is desirable
to have this field be as uniform as possible in order to avoid particle
losses through resonant effects. These effects are pronounced if, for
example, one places solenoid coils between the rf cells thereby giving
the longitudinal structure of the solenoidal field an oscillating
structure with amplitude variations (see figure \ref{coil_scenarios}). We avoid
this problem by placing the entire rf channel within the coils of
the solenoids with the penalty of increasing the warm bore aperture
of the solenoids.
\begin{center}
\begin{figure}[!hbt]
{\centering
\epsfig{file=coils.eps,height=8cm}
\caption{Different locations of solenoid coils for a phase rotation system.}
\label{coil_placement}
}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[!hbt]
{\centering
\epsfig{file=profiles.eps,height=8cm}
\caption{Magnetic Induction along the beam axis for three different
coil placement scenarios.}
\label{coil_scenarios}
}
\end{figure}
\end{center}
We consider two variations of solenoids with the 240 cm warmbore
aperture required to fully envelope the 70 MHz rf cell. A superconducting
solenoid with such an aperture capable of generating a 1.25 T field
can be constructed at a cost of \$1.2 M.
\end{document}