\subsection{Muon front-end}
\label{Sect:FrontEnd}

The Neutrino Factory muon front-end consists of a pion decay channel
and longitudinal drift, followed by an adiabatic buncher,
phase-rotation system, and ionisation-cooling channel.

The present design is based on the lattice presented in the Neutrino
Factory Study 2A report~\cite{Albright:2004iw} and subsequently
developed in the ISS \cite{Apollonio:2009} with several modifications:
the taper from the target solenoid has been adjusted; the
solenoid-field strength in the drift, buncher, and phase rotation
sections has been reduced from 1.75 T to 1.5 T; the whole system has
been shortened; and the thickness of the lithium hydride absorbers in the
cooling section has been increased. 
These changes result in the same muon-capture performance
in a shorter bunch train, reducing requirements on some systems
downstream of the muon front-end.

\subsubsection{Decay and longitudinal drift}

Downstream of the target solenoid, the magnetic field is adiabatically
reduced from 20 T to 1.5 T over a distance of 15~m.
Over the same distance, the beam pipe radius increases from 0.075~m to
0.3~m.  
This arrangement captures within the 1.5~T decay channel a
secondary-pion beam with a large energy spread. 

The initial proton bunch is relatively short (between 1~ns and 3~ns rms,
see section \ref{Sect:ProtonDriver}) resulting in a short pion
bunch. 
As the secondary pions travel from the target they drift
longitudinally, following $ct = s/\beta_z + ct_0$, where $s$ is
distance along the transport line and $\beta_z = v_z/c$ is the
relativistic longitudinal velocity. Hence, downstream of the target,
the pions and their daughter muons develop a position-energy
correlation in this RF-free decay channel.  In the present baseline,
the longitudinal drift length $L_D$ = 57.7 m, and at the end of the
decay channel there are about 0.4 muons of each sign per incident
8~GeV proton.

\subsubsection{Buncher}

The drift channel is followed by a buncher section that uses RF
cavities to form the muon beam into a train of bunches and a
phase-energy rotating section that decelerates the leading high-energy
bunches and accelerates the late low energy bunches, so that each
bunch has the same mean energy.  
The baseline design delivers a bunch train
that is less than 80~m long. This is an improvement over the version
of the design developed for the ISS~\cite{Apollonio:2009} which
delivered a 120 m long bunch train containing the same number of
muons. 

A shorter bunch train makes some downstream systems easier
to design. 
For example, one of the constraints on the minimum length of the decay
rings is the total length of the bunch train. 
By making the bunch train shorter, it may be possible to make the
decay rings shorter. 
Also, the FFAG ring has a rather demanding kicker system, mostly
driven by the total circumference of the ring but also influenced by
the length of the bunch train.
A shorter bunch train makes these kickers slightly easier to
construct.

To determine the required buncher parameters, we consider reference
particles (0, $N_B$) at $p_0 = 233$ MeV/c and $p_{N_B} = 154$ MeV/c,
with the intent of capturing muons from an initial kinetic energy
range of 50 to 400 MeV.  The RF cavity frequency, $f_{RF}$, and phase
are set to place these particles at the centre of bunches while the RF
voltage increases along the channel.
These conditions can be
maintained if the RF wavelength, $\lambda_{RF}$, increases along the
buncher, following:
\begin{equation}
  N_B \lambda_{RF}(s) = N_B \frac{c}{f_{RF}(s)} =
  s\left(\frac{1}{\beta_{N_B}}-\frac{1}{\beta_0}\right) \, ;
\end{equation}
where $s$ is the total distance from the target, $\beta_0$ and
$\beta_{N_B}$ are the velocities of the reference particles, and $N_B$
is an integer. For the present design, $N_B$ is chosen to be 10, and
the buncher length is 31.5~m. 
With these parameters, the RF cavities
decrease in frequency from 320~MHz ($\lambda_{RF} = 0.94$ m) to
230~MHz ($\lambda_{RF} = 1.3$~m) over the length of the buncher. 

The initial geometry for the placement of the RF cavities uses
$0.4-0.5$~m long cavities placed within 0.75~m long cells.  
The 1.5~T solenoid focusing of the decay region is continued through
the buncher and the rotator section which follows.  
The RF gradient is increased from cell to
cell along the buncher, and the beam is captured into a string of
bunches, each of which is centred about a test particle position,
with energies determined by the spacing from the initial test particle
such that the $i^{th}$ reference particle has velocity:
\begin{equation}
  1/\beta_i = 1/\beta_0 + \frac{i}{N_B}
  \left(\frac{1}{\beta_{N_B}} - \frac{1}{\beta_0}\right).
\end{equation}
In the initial design, the cavity gradients, $V_{RF}$, follow a linear
increase along the buncher:
\begin{equation}
\label{eq:linear_ramp}
  V_{RF}(z) \approx 9 \frac{z}{L_B} MV/m \, ;
\end{equation}
where $z$ is distance along the buncher and $L_B$ is the length of
the buncher.  The gradient at the end of
the buncher is 9 MV/m.  This gradual increase of the bunching voltage
enables a somewhat adiabatic capture of the muons into separated
bunches, which minimises phase-space dilution.

\begin{table*}
  \caption{Summary of front-end RF requirements. The total installed RF voltage
  is 1184~MV.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
        & {\bf Length} & {\bf Number}      & {\bf Frequencies}     & {\bf Number}         & {\bf Peak gradient} & {\bf Peak power}    \\
        & [m]          & {\bf of cavities} & [MHz]                 & {\bf of frequencies} &  [MV/m]             & {\bf requirements}  \\
\hline
Buncher & 33   & 37           & 319.6 to 233.6  & 13           & 4 to 7.5    & 1--3.5 MW/freq. \\
Rotator & 42   & 56           & 230.2 to 202.3  & 15           & 12          & 2.5 MW/cavity \\
Cooler  & 75   & 100          & 201.25          & 1            & 15          & 4 MW/cavity \\
\hline
Total   & 150 & 193          & 319.6 to 201.25 & 29           &              & 562 MW \\
\hline
\end{tabular}
\label{tab:acc:fe:rf_settings}
\end{center}
\end{table*}

In the practical implementation of the buncher concept, this linear
ramp of cavity frequency is approximated by a sequence of
RF cavities that decrease in frequency along the 33 m beam transport
allotted to the buncher. The number of different RF frequencies is limited to a
more manageable 13 (1--4 RF cavities per frequency).  The linear ramp in
gradient described by equation (\ref{eq:linear_ramp}) is approximated
by the placement and gradient of the cavities in the buncher.  
Table \ref{tab:acc:fe:rf_settings} shows a summary of the RF cavities
that are needed in the buncher, rotator, and cooling sections. 

\subsubsection{Rotator}

In the rotator section, the RF bunch-spacing between the reference
particles is shifted away from the integer, $N_B$, by an increment,
$\delta N_B$, and phased so that the high-energy reference particle is
stationary and the low-energy one is uniformly accelerated to arrive
at the same energy as the first reference particle at the end of the
rotator.  For the baseline, $\delta N_B =$ 0.05 and the bunch spacing
between the reference particles is $N_B + \delta N_B = 10.05$.  This
is accomplished using an RF gradient of 12 MV/m in 0.5 m long RF
cavities within 0.75 m long cells.  The RF frequency decreases from
230.2 MHz to 202.3 MHz along the length of the 42 m long rotator
region. 
A schematic of a rotator cell is shown in figure
\ref{fig:acc:fe:rotator_schematic}. 
\begin{figure}
  \begin{center}
    \includegraphics[width=0.9\textwidth]%
      {02-AccWG/02-D-FrontEnd/Rotator_Lattice.eps}
  \end{center}
  \caption{Schematic radial cross section of a rotator cell. }
  \label{fig:acc:fe:rotator_schematic}
\end{figure}

The RF frequency is set by requiring that the reference-particles
trajectories be spaced in $ct$ by $(N_B + \delta N_B)$ wavelengths. In
practical implementation, a continuous change in frequency from cavity
to cavity is replaced by grouping adjacent sets of cavities into
the same RF frequency. The 42~m long RF rotator, then contains 56
RF cavities grouped into 15 frequencies.

Within the rotator, as the reference particles are accelerated to the
central energy (at $p =$ 233 MeV/c) at the end of the channel, the
beam bunches formed before and after the central bunch are decelerated
and accelerated respectively, obtaining at the end of the rotator a
string of bunches of equal energy for both muon species.
At the end of the rotator the RF frequency matches into the RF
frequency of the ionisation cooling channel (201.25 MHz).  The average
momentum at the rotator is 230 MeV/c. The performance of the bunching
and phase rotation channel, along with the subsequent cooling channel,
is displayed in figure \ref{fig:acc:fe:front_end_performance}, which
shows, as a function of the distance down the channel, the number of
muons within a reference acceptance. 
The phase rotation increases the ``accepted'' muons by a factor of
four.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.63\textwidth]%
      {02-AccWG/02-D-FrontEnd/eps_transverse-ai.eps}
    \includegraphics[width=0.63\textwidth]%
      {02-AccWG/02-D-FrontEnd/muons_good-ai.eps}
  \end{center}
\caption{
  Performance of the bunching and cooling channel as a function
  of distance along the channel, as simulated using the ICOOL
  code~\cite{Fernow:2005cq} and the G4beamline
  code~\cite{Roberts:2008zzc}. (top) The evolution of the rms
  transverse emittance (computed over all bunches).
  (bottom) The evolution of the number of
  muons within a reference acceptance (muons within 201.25 MHz RF
  bunches with momentum in the range 100--300 MeV/c, transverse
  amplitude squared less than 0.03\,m and longitudinal amplitude
  squared less than 0.15\,m). The cooling section starts at $s = 155$
  m, where the rms transverse emittance is 0.018 m and 0.08 $\mu$ per
  proton are in the reference acceptance. The capture performance is
  shown for a cooling channel extending to $s = 270$~m although in
  this design the cooling channel extends only to 230~m.  
  Acceptance is maximal at 0.20~$\mu$ per initial 8~GeV proton at 
  $s = 240$~m (85~m of cooling) and the RMS transverse emittance is
  0.007~m.
  At $s = 230$~m (75~m of cooling) the number of $\mu$ per proton
  is 0.19 and the transverse emittance is 0.0075~m.
  }
  \label{fig:acc:fe:front_end_performance}
\end{figure}

A critical feature of the muon production, collection, bunching, and
phase rotation system is that it produces bunches of both signs
($\mu^+$ and $\mu^-$) at roughly equal intensities. 
This occurs because the focusing systems are solenoids which focus
both signs, and the RF systems have stable acceleration for both
signs, separated by a phase difference of $\pi$. 
The distribution of muons in longitudinal phase
space for particles of both signs at the end of the rotator is shown
in figure \ref{fig:acc:fe:neg_and_pos_figure}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.9\textwidth]%
      {02-AccWG/02-D-FrontEnd/t_+-_at_buncher_end.eps}
  \end{center}
  \caption{
    Distribution of particles in longitudinal phase-space at the
    phase rotation end. $\mu+$ are shown in red and $\mu-$ are shown in
    blue.
  }
  \label{fig:acc:fe:neg_and_pos_figure}
\end{figure}

\subsubsection{Cooling channel}

The baseline cooling-channel design consists of a sequence of
identical 1.5 m long cells (figure
\ref{fig:acc:fe:cooling_schematic}).  
Each cell contains two 0.5 m-long RF cavities, with 1.1 cm thick LiH
discs at the ends of each cavity (4 per cell) and a 0.25 m spacing
between cavities. 
The LiH discs provide the energy-loss material for
ionisation cooling.  
The cells contain two solenoid coils with opposite polarity.
The coils produce an approximately sinusoidal variation of the
magnetic field in the channel with a peak value on-axis of 2.8 T,
providing transverse focusing with $\beta_\perp = 0.8$~m.  
The currents in the first two
cells are perturbed from the reference values to provide matching from
the constant-field solenoid in the buncher and rotator sections.  The
total length of the cooling section is 75~m (50 cells).  Based on the
simulation results shown in figure \ref{fig:acc:fe:front_end_performance}, the
cooling channel is expected to reduce the rms transverse normalised
emittance from $\epsilon_{N} = 0.018$~m to 
$\epsilon_{N} = 0.0075$~m.  
The rms longitudinal emittance is $\epsilon_{L}= 0.07$~m/bunch.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.9\textwidth]%
      {02-AccWG/02-D-FrontEnd/Cooling_Lattice.eps}
  \end{center}
  \caption{Schematic radial cross section of a cooling cell.}
  \label{fig:acc:fe:cooling_schematic}
\end{figure}

The effect of the cooling can be measured by counting the number of
simulated particles that fall within a reference acceptance that
approximates the expected acceptance of the downstream accelerator.

The squared amplitude $A_\perp^2$ is given by:
\begin{equation}
  A_\perp^2 = p_z/m \big[ \beta_\perp(x^{\prime2}+y^{\prime2})+
  \gamma_\perp(x^2+y^2)+2\alpha_\perp(xx^\prime+yy^\prime)+
  2(\beta_\perp\kappa - \mathcal{L})(xy^\prime-yx^\prime) \big] \, ;
\end{equation}
where $\beta_\perp$, $\alpha_\perp$, $\gamma_\perp$ are solenoidal
equivalents of the Twiss parameters, $\kappa$ is the solenoidal
focusing strength, and $\mathcal{L}$ is the dimensionless kinetic
angular momentum~\cite{Fernow:2003zz,Penn:2000mt}.

For longitudinal motion, the variables $t_c = ct$ (phase lag in periods
within a bunch multiplied by RF wavelength) and $\Delta E$ (energy
difference from centroid) are used rather than $(z, z^\prime)$. The
longitudinal squared amplitude is given by:
\begin{equation}
  A_L^2=\frac{c}{m_\mu} \left[ \frac{t_c^2}{\delta} +
    \delta \left( \Delta E - \frac{\alpha_L t_c}{\delta}\right)^2
    \right] \, ;
\end{equation}	
where $\delta$ is defined by:
\begin{equation}
  \delta= \frac{c \left<t_c^2\right>}{m_\mu\epsilon_L} \, ;
\end{equation}
$\epsilon_L$ is a normalised longitudinal emittance:
\begin{equation}
  \epsilon_L= \frac{c}{m_\mu}\sqrt{\left<t_c^2\right>\left<\Delta
    E^2\right>-\left<t_c\Delta E\right>^2} \, ;
\end{equation}
and $\alpha_L$ is a correlation factor:
\begin{equation}
  \alpha_L= \frac{c}{m_\mu \epsilon_L}\left<t_c\Delta E\right> \, .
\end{equation}
Following criteria developed using the ECALC9F program (distributed with
ICOOL), a particle is
considered to be within the acceptance of the machine if the
transverse amplitude squared $A_\perp^2$ is less than 0.03~m and the
longitudinal amplitude squared is less than 0.15~m. Note that the
transverse and longitudinal notations are not the same and
transverse-longitudinal amplitude correlations are not included.
This is a crude first approximation to the muon accelerator
acceptance, but is used in the present tables for consistent
comparisons of simulations.

Using the output from our re-optimised buncher and rotator, we have
tracked particles through the cooling channel, and obtain, within
the reference acceptances, 0.19~$\mu$ per 8~GeV incident proton. 
The acceptance criteria remove larger amplitude particles from the
distribution and the rms emittance of the accepted beam is therefore
much less than that of the entire beam.  The rms transverse emittance, 
$\epsilon_\perp$, of the accepted beam is 0.004 m and the rms
longitudinal emittance is 0.036 m.

At the end of the cooling channel, there are interlaced trains of
positive and negative muon bunches.  The trains of usable muon
bunches are 80~m long (50 bunches), with 70$\%$ of the muons in the
leading 20 bunches (30 m).  The bunch length is 0.16~m in $ct$ for
each bunch, with a mean momentum of 230\,MeV/c and an rms width 
$\delta p$ of $28$~MeV/c.  For the accepted beam, the rms bunch width
is 3.8~cm and the rms transverse momentum is 10~MeV/c.  

\subsubsection{Simulation codes}

Two independently-developed codes have been used for tracking
simulations of the muon front-end by the Monte Carlo method: ICOOL
version 3.20 \cite{Fernow:2005cq}; and G4beamline version 2.06
\cite{Roberts:2008zzc}.

ICOOL is under active development at the Brookhaven National
Laboratory. It is a 3-dimensional tracking program that was
originally written to study ionisation cooling of muon beams. The
program simulates particle-by-particle propagation through materials
and electromagnetic fields. The physics model is most accurate for
muons in the kinetic energy range of 50--1\,000~MeV, but tracking of
electrons, pions, kaons, and protons is also possible. ICOOL includes
a number of custom models for particle decay, delta-ray production,
multiple Coulomb scattering, ionisation energy loss, and energy
straggling.

G4beamline is a particle tracking and simulation program
under active development by Muons, Inc. Physics processes are modelled
using the Geant4 toolkit with the QGSP\_BERT physics package~\cite{Agostinelli:2002hh}
and it is specifically designed to simulate beam lines, and other
systems using single-particle tracking, with ease.

Both codes use semi-analytic procedures to compute electromagnetic
fields. 
Solenoid fields are generated as a sum of elliptic integrals
calculated using the solenoid-coil geometry~\cite{Fernow:2003}. 
RF cavities are modelled using a Bessel function radially and a
sinusoid in $t$ for the ideal field produced by a cylindrical pillbox
cavity. 

Good agreement is shown in the muon yield and the yield of other
particle species from the two codes
(see Figs.~\ref{fig:acc:fe:front_end_performance} and
\ref{fig:acc:fe:power_deposition}). Note that different versions of
ICOOL have been shown to disagree at the level of a few percent
\cite{Rogers:2010}.

Analysis of results is performed using ECALC9F version 2.07 in
addition to custom scripts written in Python and MATLAB. The beam has
been generated using MARS 15.07. We expect a significant systematic
error on the overall rate owing to uncertainty in the models used to
generate the input beam.

\subsubsection{RF Requirements and Design}

The RF cavities in this design are all normal conducting cavities having
29 frequencies in the range 201.25 MHz to 320 MHz. The cavities are 50 cm
long with peak field gradients in the range 4 to 15 MV/m, with the
highest voltage required for the 201 MHz cavities. The power
consumption of these cavities has been estimated semi-analytically
using standard formulae, and the results are listed in Tables
\ref{tab:acc:fe:buncher_rf_settings} and
\ref{tab:acc:fe:rot_cool_rf_settings},  together with the RF cavities
required. The position and phase of every cavity is listed in Table
\ref{tab:acc:fe:rot_rf_detailed_list}. 
\begin{table*}
\caption{Front-end RF requirements for the buncher system.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
{\bf Frequency}   &{\bf Voltage (per}      &{\bf Number of}  &{\bf Length} &{\bf Gradient} & {\bf Peak RF Power} \\
\,[MHz]\,         &{\bf frequency)} [MV]   &{\bf cavities}   &[m]          &[MV/m]         & {\bf (per frequency)} [MW] \\ %if I just use square brackets in first cell I get an error message...
\hline
319.63   &1.37   &1   &0.4 m   &4   &0.2 \\
305.56   &3.92   &2   &0.4   &5   &0.6 \\
293.93   &3.34   &2   &0.45   &4   &0.5 \\
285.46   &4.8   &2   &0.45   &5.5   &1 \\
278.59   &5.72   &2   &0.45   &6.4   &1.25 \\
272.05   &6.66   &3   &0.45   &5   &1.5 \\
265.8   &7.57   &3   &0.45   &5.7   &1.5 \\
259.83   &8.48   &3   &0.45   &6.5   &2 \\
254.13   &9.41   &3   &0.45   &7   &2.3 \\
248.67   &10.33   &4   &0.45   &5.7   &2.3 \\
243.44   &11.23   &4   &0.45   &6.5   &2.5 \\
238.42   &12.16   &4   &0.45   &7   &3 \\
233.61   &13.11   &4   &0.45   &7.5   &3.5 \\
Total   &98.1   &37   &   &   &22 \\
\hline
\end{tabular}
\label{tab:acc:fe:buncher_rf_settings}
\end{center}
\end{table*}
\begin{table*}
  \caption{Front-end RF requirements for the rotator and cooler systems.
    $f_{\text{RF}}$ is the RF frequency, $V_{\text{tot}}$ is the total voltage
    required at that frequency, $n_{\text{cav}}$ is the number of cavities
    at that frequency, $E_{\text{peak}}$ is the peak gradient in each
    cavity,
    $P_{\text{avg}}$ is the steady state RF power required per cavity,
    and $P_{\text{peak}}$ is the peak RF power required per cavity.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$f_{\text{RF}}$ [MHz]&$V_{\text{tot}}$ [MV]&$n_{\text{cav}}$&$E_{\text{peak}}$&
$P_{\text{peak}}$ [MW]\\
\hline
Rotator  &   &   &   &   \\
230.19   &18   &3   &12   &2.3 \\
226.13   &18   &3   &12   &2.3 \\
222.59   &18   &3   &12   &2.3 \\
219.48   &18   &3   &12   &2.4 \\
216.76   &18   &3   &12   &2.4 \\
214.37   &18   &3   &12   &2.4 \\
212.48   &18   &3   &12   &2.4 \\
210.46   &18   &3   &12   &2.5 \\
208.64   &24   &4   &12   &2.5 \\
206.9   &24   &4   &12   &2.5 \\
205.49   &24   &4   &12   &2.5 \\
204.25   &30   &5   &12   &2.6 \\
203.26   &30   &5   &12   &2.6 \\
202.63   &30   &5   &12   &2.6 \\
202.33   &30   &5   &12   &2.6 \\
Total   &336   &56   &   &140 \\
\hline
Cooler   &   &   &   & \\
201.25   &750   &100   &15   &4 \\
\hline
\end{tabular}
\label{tab:acc:fe:rot_cool_rf_settings}
\end{center}
\end{table*}
\begin{table*}
\caption{Full list of RF cavities. Cavities are grouped by
  frequency. Position is the position of the upstream edge of the
  first cavity in the group. $z$-Separation is the distance between the
  centres of each cavity in the group. 0$^\circ$ is bunching mode
  while 5$^\circ$ and 35$^\circ$ are partially accelerating modes.} 
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\bf $z$ Position}  &{\bf Phase}      &{\bf Peak gradient}  & {\bf Frequency} & {\bf Length} & {\bf Number} & {\bf $z$-Separation} \\
\,[m]\,             &[deg.]        &[MV/m]               & [MHz]           & [m]          &              & [m]          \\ %if I just use square brackets in first cell I get an error message...
\hline
81.48 & 0.0 & 3.42 & 319.63 & 0.40 & 1 &  0.0 \\    
85.03 & 0.0 & 4.894 & 305.56 & 0.40 & 2 & 0.40 \\
88.405 & 0.0 & 4.17 & 293.93 & 0.40 & 2 & 0.40 \\
91.01 & 0.0 & 5.34 & 285.46 & 0.45 & 2 & 0.45 \\
93.26 & 0.0 & 6.36 & 278.59 & 0.45 & 2 & 0.45 \\
95.285 & 0.0 & 4.94 & 272.05 & 0.45 & 3 & 0.45 \\
97.535 & 0.0 & 5.61 & 265.8 & 0.45 & 3 & 0.45 \\ 
99.785 & 0.0 & 6.3 & 259.83 & 0.45 & 3 & 0.45 \\
102.03 & 0.0 & 6.97 & 254.13 & 0.45 & 3 & 0.45 \\
104.28 & 0.0 & 7.65 & 248.67 & 0.45 & 3 & 0.45 \\
106.53 & 0.0 & 8.31 & 243.44 & 0.45 & 3 & 0.45 \\   
108.78 & 0.0 & 9.01 & 238.42 & 0.45 & 3 & 0.45 \\   
111.03 & 0.0 & 9.71 & 233.61 & 0.45 & 3 & 0.45 \\   
112.98 & 5.0 & 12.0 & 230.19 & 0.5 & 3 & 0.75 \\   
115.23 & 5.0 & 12.0 & 226.13 & 0.5 & 3 & 0.75 \\   
117.48 & 5.0 & 12.0 & 222.59 & 0.5 & 3 & 0.75 \\   
119.73 & 5.0 & 12.0 & 219.48 & 0.5 & 3 & 0.75 \\   
121.98 & 5.0 & 12.0 & 216.76 & 0.5 & 3 & 0.75 \\   
124.23 & 5.0 & 12.0 & 214.37 & 0.5 & 3 & 0.75 \\   
126.48 & 5.0 & 12.0 & 212.28 & 0.5 & 3 & 0.75 \\   
128.73 & 5.0 & 12.0 & 210.46 & 0.5 & 3 & 0.75 \\   
130.98 & 5.0 & 12.0 & 208.64 & 0.5 & 4 & 0.75 \\   
133.98 & 5.0 & 12.0 & 206.9 & 0.5 & 4 & 0.75 \\    
136.98 & 5.0 & 12.0 & 205.49 & 0.5 & 4 & 0.75 \\   
139.98 & 5.0 & 12.0 & 204.25 & 0.5 & 5 & 0.75 \\   
143.73 & 5.0 & 12.0 & 203.26 & 0.5 & 5 & 0.75 \\   
147.48 & 5.0 & 12.0 & 202.63 & 0.5 & 5 & 0.75 \\   
151.23 & 5.0 & 12.0 & 202.33 & 0.5 & 6 & 0.75 \\   
155.1 & 35.0 & 15.0 & 201.25 & 0.5 & 100 & 0.75 \\    
\hline
\end{tabular}
\label{tab:acc:fe:rot_rf_detailed_list}
\end{center}
\end{table*}

Several RF cavities have been constructed to support the muon
accelerator design effort~\cite{Huang:2009zzn}. A 43~cm long, 201~MHz
RF cavity has been constructed and operated at peak gradients up
to 21~MV/m and 10 more RF cavities are under construction as part of
the international Muon Ionisation Cooling Experiment
(MICE)~\cite{Alekou:2010zz}. 
Additionally, several 805~MHz cavities have been constructed and
operated at gradients up to 40~MV/m. 
Design and construction of cavities with intermediate
frequencies is not expected to present any additional difficulties.

\subsubsection{Effect of magnetic field on RF gradient}

There is some empirical evidence that suggests that magnetic fields
overlapping RF cavities, as present in the muon front-end,
may induce breakdown in the cavities~\cite{Moretti:2005zz,Palmer:2009zza}.
The performance of the muon front-end using a
reduced field has been explored using ICOOL. In
figure \ref{fig:acc:fe:reduced_rf}, the muon transmission is shown as
a function of fractional change in RF gradient in the buncher,
rotator, and cooling channel.
The simulations indicate that around the nominal gradient, muon
transmission is rather insensitive to the peak gradients. 
If the achievable RF gradient falls dramatically below the nominal
value, there is a significant effect on muon transmission. 
In order to mitigate this technical risk, several alternative
lattices have been developed and are described briefly in Appendix
\ref{sec:acc:fe:FrontEnd_Alternatives}.  
\begin{figure}
  \begin{center}
    \includegraphics[width=0.9\textwidth]%
      {02-AccWG/02-D-FrontEnd/effect_of_reduced_V_false_0.eps}
  \end{center}
  \caption{
    Fractional change of particles within the nominal acceptance
    as a function of the peak field gradient, simulated in
    ICOOL. $V_{\text{ref}}$ is the baseline gradient; all cavities were scaled
    by the constant $V/V_{\text{ref}}$. 
  }
  \label{fig:acc:fe:reduced_rf}
\end{figure}

\subsubsection{Magnet requirements and design}

Pion capture in the target system is achieved using a series of
solenoids in which the field tapers from 20~T to 1.5~T over 15~m (see
section \ref{Sect:Trgt}).
Here we describe solenoids in the drift, buncher, and rotator that
produce a constant magnetic field of 1.5~T and solenoids in the cooler
that produce the alternating solenoid field configuration. 
These coils are summarised in table \ref{tab:acc:fe:magnet_settings}.
\begin{table*}
\caption{Summary of front-end magnet requirements.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
                  & {\bf Length} & {\bf Inner radius} & {\bf Radial thickness} & {\bf Current density} & {\bf Number} \\
                  & [m]    & [m]          & [m]              & [A/mm$^2$]      &  \\
\hline
Initial transport & 0.5    & 0.68         &  0.04     & 47.5            & 180 \\
Cooling channel   & 0.15   & 0.35         &  0.15     & $\pm$107            & 100 \\
\hline
\end{tabular}
\label{tab:acc:fe:magnet_settings}
\end{center}
\end{table*}

The 1.5~T solenoids must accommodate the beam pipe, with a 30 cm
radius. Within the buncher and rotator, they must also accommodate RF
cavities with radii of 60 cm.  This can be achieved using coils with
an inner radius of 68 cm and a conductor radial thickness of 4 cm, so
that the cavities fit entirely within the coils. A coil length of 50 cm
spaced at 75 cm intervals leaves a gap of 25 cm between coils,
matching the periodicity of the cooling channel and enabling access
for room temperature services such as vacuum and RF power feeds.  The
required current for these coils is 47.5 A/mm$^2$ to give a total
current of 0.95 MA-turns. The coils are therefore large enough to
accommodate the beam pipe, RF and diagnostics, and added shielding. A
smaller radius could be used in the first 60 m, which has no RF. The
135 m transport requires 180 such magnets.

The cooling system requires strong alternating-sign coils that are
placed between RF cavities, fitting within the 25 cm inter-cavity
spaces (figure \ref{fig:acc:fe:cooling_schematic}).  The coils are 15 cm long
with inner radius 35 cm, radial thickness 15 cm and current density of
$\pm107\text{ A/mm}^2$ to give a total current of 2.4~MA-turns.  The coil
currents alternate in direction from coil to coil.  These coils
produce an on-axis solenoid field that varies from $+$2.8~T to
$-$2.8~T over a 1.5~m period, following an approximately sinusoidal 
dependence.  Maximum fields in the cooling cell volume are 5~T near
the coil surfaces. 100 such coils are needed in a 75~m cooling system.

\subsubsection{Beam Losses}

There are significant particle losses along the beam line and these
will result in a large energy deposition in superconducting magnets and
other equipment. 
Two main risks have been identified: energy deposition by all
particles may cause superconducting equipment to quench; and energy
deposition by hadrons and other particles may activate equipment
preventing hands-on maintenance. 

In figure \ref{fig:acc:fe:power_deposition}, the power deposited by
transmission losses per unit length from various particle species is
shown as a function of distance along the channel. 
Note that energy deposition in RF
windows and absorbers is not included in this calculation. 
It is expected that this equipment will absorb several kilowatts of
beam power from each particle species.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.82\textwidth]%
      {02-AccWG/02-D-FrontEnd/loss_alt_for_idr-ai.eps}
    \includegraphics[width=0.78\textwidth]%
      {02-AccWG/02-D-FrontEnd/all_power-ai.eps}
  \end{center}
  \caption{
    Power deposited by transmission losses of various particle
    species in the surrounding equipment simulated in (top) ICOOL and
    (bottom) G4Beamline.
  }
  \label{fig:acc:fe:power_deposition}
\end{figure}

In currently operating accelerators, uncontrolled hadronic losses must
be less than $\approx$1~W/m to enable ``hands-on'' maintenance without
additional time, distance, or shielding constraints.
Magnets are expected to
quench with beam losses above a few tens of W/cm$^3$. 
Several schemes are envisaged to control the beam losses and reduce
them below these values. 

Three devices are under study for reducing the transmission losses in
the front-end:
\begin{itemize}
\item 
  Low momentum protons may be removed using a proton
  absorber. This device takes advantage of the different stopping
  distance of protons compared with other particles in material;
\item 
  Particles with a high momentum, outside of the acceptance of the
  front-end, may be removed using a pair of chicanes. 
  Dispersion is induced in the beam by means of bending
  magnets or tilted solenoids
  in a  chicane arrangement and high-momentum particles are
  passed onto a beam dump. In order to retain both muon species, one
  chicane is required for each sign; and
\item 
  Particles with transverse amplitude outside of the acceptance of
  the front-end may be removed using transverse collimators.
\end{itemize}

\subsubsection{Summary}
The Neutrino Factory muon front-end captures a substantial proportion
of the muons produced by the Neutrino Factory target. Longitudinal
capture is achieved using a buncher and energy-time phase-rotation
system while transverse capture is achieved using a high field
solenoid adiabatically tapered to 1.5 T and enhanced by ionisation
cooling. 
Technical risks to the muon front-end are presented by the requirement
for high peak RF fields in the presence of intense magnetic fields and
irradiation of the accelerator hardware due to uncontrolled particle
losses.
Strategies have been outlined by which these risks can be mitigated.
Overall, the muon front-end increases the capture rate of muons in the
nominal accelerator acceptance by a factor 10.
