\subsection{Linear non-scaling FFAG}

In the linac and RLAs described above, the cost of the RF systems 
dominates the cost of the beam lines that accelerate the beam.
A better balance between magnet and RF costs
would result by making more passes through the RF cavities.
Unfortunately, the switchyard precludes one from making a large
number
of passes in the RLA linac.
It would thus be advantageous to eliminate the switchyard and use a single
arc for all energies.  
This could be accomplished by using
a machine that allows a large energy range
within a single fixed-field arc, i.e.,   
a fixed field alternating gradient (FFAG) accelerator.

To keep the machine cost low and to allow the use of 201.25~MHz RF
cavities, we use a design that differs from the original type of FFAG
(referred to as a ``scaling'' FFAG).  
This new type of machine is called a
linear ``non-scaling'' FFAG.  
Its important features are that it is
isochronous near the middle of the energy range, limiting the
phase slippage and thus permitting the use of high RF frequencies, and
that its magnets are more compact than those of a comparable scaling
FFAG. 

We have optimised the design of a linear non-scaling FFAG
\cite{Berg:2004wc}, and the resulting parameters are given in tables
\ref{tab:acc:ffag:goals}--\ref{tab:acc:ffag:rf}.
The important criteria for this optimisation were that the long drift
length would be 5.0~m (to accommodate the septum, see below) and that
the time-of-flight be a half integer number of RF periods per cell.  
With the exception of the lattice cells containing the injection and
extraction insertions, the ring consists of identical cells, each
containing an FDF triplet. 
Most cells contain a superconducting cavity, the parameters of
which are given in table \ref{tab:acc:ffag:rf}.
The magnet aperture is increased by 30\% from what is required
for the beam to ensure that the beam is in a region of the magnet with
good field quality.  The magnet angle and shift are defined as follows
(see figure \ref{fig:acc:ffag:geom}):
upon entering the magnet, the coordinate axis bends toward the centre
of the ring by half the given angle.  The centre of the magnet is
then shifted away from the centre of the ring by the shift value.  Upon
exiting the magnet, the coordinate axis bends toward the centre of the ring
by the second half of the bend angle.
\begin{table}
  \caption{Design requirements for the FFAG accelerator.
    Energies are total energy. Transverse acceptance is
    defined as $a^2p/(\beta_\bot mc)$,
    where $a$ is the maximum beam radial displacement,
    $\beta_\bot$ is the Courant-Snyder beta
    function, $p$ is the total momentum, $m$ is the muon mass, and $c$ is the
    speed of light.  Longitudinal acceptance is $\Delta E\Delta t/mc$,
    where $\Delta E$ is the maximum energy half-width and $\Delta t$
    is the maximum half-width in time (assuming an upright longitudinal
    ellipse).}
  \label{tab:acc:ffag:goals}
  \begin{tabular}{|l|r|l|r|}
    \hline
    Injection energy &12.6 GeV&Transverse acceptance&30 mm\\
    Extraction energy&25.0 GeV&Longitudinal acceptance&150 mm\\
    Muons per train&$2.4\times10^{12}$&Muon trains&3\\
    Time between trains&120 $\mu$s&Repetition rate&150 Hz\\
    \hline
  \end{tabular}
\end{table}
\begin{table}
  \caption{Magnetic lattice parameters of the linear non-scaling FFAG.
    The short drift is between each D and F in the triplet, the
    long drift is between the F magnets.  The angle and shift are described
    in the text.  The maximum magnet field is at the magnet aperture.}
  \label{tab:acc:ffag:ring}
  \centering
  \begin{tabular}[c]{|l|r|}
    \hline
    Cells            &64        \\
    Long drift       &5.0 m     \\
    Short drift      &0.5 m     \\
    Circumference    &667 m     \\
    \hline
  \end{tabular}
  \hfil
  \begin{tabular}[c]{|l|r|r|}
    \hline
                        &D        &F       \\
    \hline
    Length (m)          &2.251117 &1.086572\\
    Angle (mrad)        &156.837  &-29.331 \\
    Shift (mm)          &41.003   &13.907  \\
    Field (T)           &4.20784  &-1.39381\\
    Gradient (T/m)      &-13.55592&18.04570\\
    Aperture radius (mm)&137      &163     \\
    Maximum field (T)   &6.1      &4.3     \\
    \hline
  \end{tabular}
\end{table}
\begin{table}
  \caption{RF parameters and derived performance of the linear non-scaling
    FFAG.  Cavity voltage is the maximum energy gain of a speed of light
    muon passing through the cavity~\cite{Ozaki:2001}.}
  \label{tab:acc:ffag:rf}
  \centering
  \begin{tabular}[c]{|l|r@{\qquad}|l|r|}
    \hline
    RF frequency     &201.25 MHz&Cavity voltage  &25.5 MV\\
    Aperture diameter&30 mm     &Cells per cavity&2      \\
    Input power      &1 MW      &RF pulse length &3 ms   \\
    Stored energy    &2008 J    &                &       \\
    Cavities         &48        &Total voltage   &1214 MV\\
    Turns            &11.6      &Decay loss      &6.7\%  \\
    \hline
  \end{tabular}
\end{table}
\begin{figure}
  \centering
  \includegraphics[width=0.75\linewidth]{02-AccWG/02-F-FFAG/RBend-Compact.eps}
  \caption{Geometry at a magnet.  $\theta$ is the angle given in
    table~\ref{tab:acc:ffag:ring}, and $\Delta x$ is the
    shift given in that same table.}
  \label{fig:acc:ffag:geom}
\end{figure}

\subsubsection{Injection and Extraction}

Injection and extraction occur at opposite ends of the ring. Each
system will have reflection symmetry to accommodate both positively
and negatively charged muons. There will be two septa in each system
with kicker magnets between them.  All the kickers in each system have
the same strength. The septum and kicker magnets have length 4.4~m,
i.e., there is 30~cm separation between each of these elements and the
adjacent main
magnets. Parameters for the injection/extraction magnets are given in
table~\ref{tab:acc:ffag:inj}.
\begin{table}
  \caption{Injection/Extraction system parameters.  Pattern contains a
    \texttt{+} for an outward kick, \texttt{-} for an inward kick,
    and a \texttt{O} for an empty drift. A
    septum is located at both ends of the kicker system.}
  \label{tab:acc:ffag:inj}
  \centering
  \begin{tabular}{|l|r|r|}
    \hline
    &Injection&Extraction\\
    \hline
    Kickers&2&4\\
    Pattern&\texttt{-O-}&\texttt{++OO++}\\
    Kicker field (T)&0.089&0.067\\
    Septum field (T)&0.92&1.76 \\
    Kicker/septum length (m) & 4.4 & 4.4 \\
    \hline
  \end{tabular}
\end{table}

A septum field below 2~T is chosen in order to minimise the
effect of the septum stray-field on the circulating beam.  
In order to
estimate the clearance needed so that the extracted or injected beam
avoids the magnet immediately following the septum, the dimensions of
the superconducting combined-function magnet (SCFM) in the J-PARC
neutrino beam line is used. The width of the J-PARC SCFM from the
inner radius of the coils to the outer radius of the cold mass is
19.8~cm~\cite{Ogitsu:2009}. This distance is used, together with the size
of the muon beam, in calculating the required septum bending
angle. Taken together, these constraints require the long drift
in the triplet to be at least 5~m in length.

In order to be feasible from an engineering perspective, the
kicker-magnet peak fields are constrained to be less than 0.1~T. The
separation between the kicked and circulating beams at the
entrance of the septum is 2~cm. 
Two kicker magnets are required for injection. Due
to the relatively high horizontal phase advance at this energy and the
requirement of mirror symmetry, the kickers need to be separated by a
single cell, resulting in an empty long drift at the centre of the
injection system. At extraction four kickers are required, i.e., two
pairs separated by two empty long drifts. Tracking results of the muon
beam through the injection and extraction systems are shown in
figure \ref{fig:acc:ffag:injext}. As can be seen from the figure,
injection is into the inside of the ring whereas the beam is extracted
to the outside.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.5\linewidth]{02-AccWG/02-F-FFAG/inj_colour-fix.eps}%
    \includegraphics[width=0.5\linewidth]{02-AccWG/02-F-FFAG/ext_colour-fix.eps}
  \end{center}
  \caption{Injection scheme (left) and extraction scheme (right) for
    the FFAG ring. The full-acceptance circulating beam at injection and
    extraction
    momenta are shown in black and the injected and extracted beams in
    blue. The red vertical lines represent the location of the kicker
    magnets (solid) and septum (dashed).}
  \label{fig:acc:ffag:injext}
\end{figure}
\begin{table}
  \caption{Magnet apertures (i.e., inner radius of magnet) in the
    Injection/Extraction system. In the rest of the ring, the magnet apertures
    are 16.3 cm and 13.7 cm for the F and D magnets, respectively.}
  \label{tab:acc:ffag:aper}
  \centering
  \begin{tabular}{|l|r|r|r|}
    \hline
    &Magnet&Number&Aperture (cm)\\
    \hline
    Injection&F&4&20.8\\
    &D&4&16.1\\
    Extraction&F&8&19.8\\
    &D&2&15.5 \\
    \hline
  \end{tabular}
\end{table}

The distribution of kickers over several cells means that a number of
large-aperture main magnets are required. The apertures are calculated
by finding the smallest circle that encloses the kicked and
circulating beam and adding an additional 1~cm clearance (for example
see figure \ref{fig:acc:ffag:aperture}). The number of these magnets and
their apertures are listed in table~\ref{tab:acc:ffag:aper}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.75\linewidth]{02-AccWG/02-F-FFAG/aperture_F.eps}
  \end{center}
  \caption{The circulating beam (black ellipses)
    enclosed by the normal F magnet aperture (black circle) and the
    additional aperture (red circle) required to accommodate the kicked
    beam (red ellipses, shown
    just before the septum at injection.}
  \label{fig:acc:ffag:aperture}
\end{figure}

\subsubsection{Kickers and Septa for the FFAG}

Travelling-wave kicker magnets are proposed both for
injection and extraction.  The kicker system consists of a power
supply charging pulse forming networks (PFNs), which are connected via
fast thyratron switches and coaxial cables to the kicker
magnet and terminated via a matching resistor.  A CX1925X~\cite{cx1925}
thyratron switch
and RG192 coaxial cables connected in
parallel could be used.  A current of about 30 kA is necessary to
produce the required 0.1~T magnetic field in an aperture of about
$0.3\text{ m}\times0.3\text{ m}$.
The same design is proposed for injection and
extraction, as the requirements with respect to rise/fall time are
very similar.  This dictates the need for careful impedance matching
to avoid any reflections, which is essential, especially for
the injection case.  In order to keep the rise- and fall-times low, the
kicker magnet must be sub-divided into four or five separate units.
The kicker uses a window-frame geometry, with the yoke
made of ferrite to enhance the field level, limit the
field leakage and improve the field quality.  The ferrite Ferroxcube
8C11~\cite{8C11} could be used for the kicker core.  The detailed
design of the kicker, including the matching capacitance scheme to
minimise reflections, remains
to be addressed in future studies.  A voltage of 60 kV is assumed,
which is compatible with existing fast thyratron switches and sets
the impedance for the magnet and termination resistor at 1~$\Omega$.
The thyratron CX 1925X is limited to 10 kA peak current, which
dictates the use of three 3~$\Omega$ PFNs connected in parallel for each
sub-kicker.  As the baseline muon pulse consists of three muon 
bunch-trains separated by 120~$\mu$s, a total of 9 PFNs for each
sub-kicker is needed to avoid requiring an extremely demanding power
supply to recharge the PFNs between the individual bunch trains.  The
total number of PFNs and switches per kicker unit installed in a
single drift is thus 36 (assuming that the kicker is subdivided into
four sub-kickers).  Power supplies of $\approx 2.5$~MW peak power
are needed for every kicker and although it is expected that the
majority of the power will be dumped into the termination resistor,
thermal issues for the kicker magnet remain to be studied.  
A schematic of the kicker system is shown in figure
\ref{fig:acc:ffag:kicksys} and some of its parameters are collected in
table \ref{tab:acc:ffag:kick}.
\begin{table}
  \caption{Parameters of the kicker system}
  \label{tab:acc:ffag:kick}
  \centering
  \begin{tabular}{|l|r|}
    \hline
    Kicker total aperture (h$\times$v)&0.3$\times$0.3 m\\
    Kicker length &4.4 m\\
    Rise/fall time (5-95$\%$)&1.9 $\mu$s\\
    Kicker max field& $\approx$0.1 T\\
    Kicker pulse duration at the top & 0.3 $\mu$s\\
    Charging voltage & 60 kV\\
    Peak current in the magnet& 30 kA\\
    Kicker inductance & 5.1 $\mu$H\\ 
    Kicker impedance &  1 $\Omega$\\
    Peak current at switch & 10 kA\\
    Repetition rate &     50 Hz\\
    Number of sub-kickers&  4-5\\
    Number of PFNs per micro-pulse per sub-kicker& 3\\
    Total number of PFNs &   36 (for 4 sub-kickers)\\
    Total averaged power per kicker &  $\approx$1.25 MW\\
    Total peak power per kicker &  $\approx$2.5 MW\\
    \hline
  \end{tabular}
\end{table}
\begin{figure}
  \includegraphics[width=0.8\textwidth]%
    {02-AccWG/02-F-FFAG/IDS_KickerSys.eps}
    \caption{Schematic of the FFAG injection/extraction kicker system
      operating with 3 muon bunch trains. The
    above system is needed for every
    sub-kicker unit.}
  \label{fig:acc:ffag:kicksys}
\end{figure}

Kicker magnets are used to suppress (create) the separation between
the injected (extracted) and the circulating beam, but in order
for the beam to clear
the external aperture of the main magnets,
septa are required.
The magnetic fields of septa for both injection and
extraction are shown in table~\ref{tab:acc:ffag:inj}. As their aperture
is large ($\approx$0.3$\times$0.3 m total), a rather high current-turn
product of
450 kA-turns is required.  Although the septum design could be based
on normal conducting technology, a DC solution would be rather
challenging due to problems with cooling the coils.  This cooling
problem could be reduced by pulsing the septum, but that would introduce
eddy currents and variable stray fields in close proximity to the
main superconducting magnets.  Moreover, a rather substantial power supply
would be
needed.  To avoid these challenges we propose to use superconducting
technology for the septa, both for injection and extraction.  As the
extraction septum is the more challenging, design work is focused on
this case. An identical device could also be used for injection.  By
using superconducting technology, high magnetic fields can easily be
obtained using very thin conductor (8 mm) with a modest current density of
200~A/mm$^2$.  The most challenging issue is the shielding of the
circulating beam region from stray fields.  In order to reduce the
field leakage, COMSOL 4.0 \cite{comsol} simulations have been
performed using various yoke geometries.  The current design is based
on a window frame magnet with the yoke extending all around both
extracted and circulating beam areas.  The septum itself, which is 2
cm thick, is subdivided into the conductor part and the yoke part. Two
yoke magnetic materials have been studied: a ``standard'' soft iron
and the soft magnetic cobalt-iron-alloy VACOFLUX
50~\cite{vacoflux}.  In addition, a chamfer was introduced on
the side facing the circulating beam in order to further limit the field
leakage.  Figure \ref{fig:acc:ffag:septum1} shows the simulations of the
stray fields using different yoke materials and with and without the
chamfer.  The leakage field at the boundary of the septum on the side
of the circulating beam is shown and was used to evaluate various
designs.  The 2D design of the extraction septum is shown in
figure \ref{fig:acc:ffag:septum2}.  Based on the electromagnetic
simulations, the septum field will be held to less than 2~T, which
dictates the length of the septum necessary to permit the beam to
clear the adjacent main magnets.  
This turns out to be an important constraint in the FFAG
lattice definition.  The detailed cryogenic design and quench limit of
the superconducting septum remain to be addressed. 
\begin{figure}
  \includegraphics[width=0.8\textwidth]{02-AccWG/02-F-FFAG/Septumleak.eps}
  \vspace*{8pt}
  \caption{Stray fields in the superconducting
    septum using standard soft iron (black, solid line), using
    advanced material-VACOFLUX 50 (red line, short dash) and
    introducing a chamfer (magenta, long dash). The simulations
    were performed using the COMSOL 4.0
    package.}
  \label{fig:acc:ffag:septum1}
\end{figure}
\begin{figure}
  \includegraphics[width=0.8\textwidth]{02-AccWG/02-F-FFAG/septum2.eps}
  \vspace*{8pt}
  \caption{2D section of the superconducting
    septum for a muon FFAG.} 
  \label{fig:acc:ffag:septum2}
\end{figure}

\subsubsection{Main Ring Magnets}

The preliminary design of the main superconducting combined-function magnets for the linear non-scaling FFAG 
has been performed using the CERN ROXIE code~\cite{roxie}. As the design of the focusing (F) magnet is more advanced, 
we present only this element. The design uses the well-established technology of Ni-Ti 
superconducting magnets fabricated with Rutherford cable. In order to simplify the geometry and allow for flexible 
beam optics tuning, the magnet design is based on separate coils for dipole and quadrupole field components, which are
assembled according to the conventional ``$\cos\theta$'' geometry.
The inner layer creates a dipole field using 3 conductor blocks,
and the outer layer creates a quadrupole field with 2 conductor blocks.
For the needs of the current design the dipole and quadrupole cables have been generated
in ROXIE based on the standard Large Hadron Collider (LHC) main dipole inner cable~\cite{Bruning:2004ej} at CERN. The cables consist of 28 strands 
and have a trapezoidal geometry. They would be constructed using the same filaments as the LHC magnet.
An iron yoke made of soft magnetic steel is placed beyond the quadrupole layer. The magnet is closed with a clamp in order to 
limit field leakage in the long straight section, where it could affect hardware components like superconducting RF cavities or kickers.
The geometry of coils and yoke for half of the F magnet is shown in figure~\ref{fig:ffag:magnet}.
The dipole field and gradient on axis have been reproduced according to the lattice design specifications with 
an accuracy of $\sim10^{-4}$. The vertical field component on the median plane of the F magnet is shown 
in figure~\ref{fig:ffag:magnetfield}. The field quality off-axis still needs to be improved, which will require an update 
of the magnet geometry. A quench analysis has been performed using ROXIE, and calculations of the temperature 
margins suggest stable magnet operation.
The main parameters of the focusing (F) FFAG magnet are collected in table~\ref{tab:ffag:magnet}. 
Future studies will address the field quality, the possible addition of higher multipoles (mainly sextupole) 
for chromaticity correction, and the cryogenic design.
\begin{table}
  \caption{Parameters of the main focusing FFAG magnet.}
  \label{tab:ffag:magnet}
  \centering
  \begin{tabular}{lr}
    Strand diameter & 1.065 mm\\
    Number of strands per cable (dipole and quadrupole)& 28\\
    Cable height (dipole and quadrupole)&15~mm\\
    Dipole cable inner/outer width & 1.58/1.75~mm \\
    Quadrupole cable inner/outer width & 1.83/1.98~mm \\
    Number of conductors in dipole blocks& 52, 25, 13\\
    Number of conductors in quadrupole blocks& 41, 10\\
    Inner radius of dipole blocks& 163~mm\\
    Inner radius of quadrupole blocks& 179~mm\\
    Inner radius of the yoke& 300~mm\\
    Yoke thickness & 100~mm\\
    Half of the yoke length (with clamp) & 840 mm\\
    Dipole current& 3190~A\\
    Quadrupole current& 8490~A\\
    Dipole current density& 332~A/mm$^2$\\
    Quadrupole current density& 885~A/mm$^2$\\
    Peak field in the conductors & 5.75 T\\
    Minimal temperature margin to quench at 2.2 K& 3.84 K\\
  \end{tabular}
\end{table}
\begin{figure}
  \includegraphics[width=0.95\textwidth]{02-AccWG/02-F-FFAG/FigFFAGMagnet.eps}
  \caption{Geometry of the design of half of the
    F magnet for the FFAG ring.}
  \label{fig:ffag:magnet}
\end{figure}
\begin{figure}
  \includegraphics[width=0.8\textwidth]{02-AccWG/02-F-FFAG/FigFFAGMagnetField.eps}
  \caption{Vertical component of the magnetic
    field on the median plane of the F magnet obtained with
    the ROXIE code.}
  \label{fig:ffag:magnetfield}
\end{figure}

\subsubsection{Beam loading}

Beam loading will cause the different bunches to gain different
amounts of energy. 
There are two time scales to consider: different energy gains within a
train, and different energy gains for different trains.

On each pass, a bunch train extracts 9.7~J from a cavity.  The last
bunch will therefore gain 62~keV less energy per cavity pass than the
first bunch.  Through the entire acceleration cycle, this amounts to
an energy difference from head to tail of 30~MV.  This is small compared
to the energy spread expected in the bunches.

Each proton driver cycle accelerates three bunches, which will
hit the target in relatively rapid succession.  Each proton driver bunch
creates a separate bunch train in the muon accelerator.
After a bunch train has completed its acceleration cycle, it has extracted
116~J from the cavities.  If no power is added to the cavities, then the
next bunch train will gain 360~MV less energy, which is too big a difference.
To replace the extracted
energy with an input power limitation of 1~MW per cavity requires 120~$\mu$s
between bunch trains and therefore between proton driver bunches.

\subsubsection{Risk Mitigation}

The use of a linear non-scaling FFAG is expected to be a more
cost-effective alternative to acceleration with an RLA.  
There are potential issues with this design, however.
It is known that the large transverse emittance
can lead to an effective longitudinal emittance growth in such a
design~\cite{Machida:2006sj}.
If necessary, this effect could be reduced by increasing the average RF gradient
and/or correcting the chromaticity, even partially~\cite{Berg:2007}.
The former can be accomplished by
increasing the drift length to 5.5~m and putting a second RF cavity in
each drift,
but this would increase the machine cost by around 25\%.  Chromaticity
correction is briefly discussed in Appendix~\ref{App:FFAG}.

One could eliminate the FFAG, and extend the energy range of the linac
and the RLAs so as to accelerate to 25~GeV.  While early estimates
led us to believe this is significantly more costly than accelerating
with an FFAG, it remains to be demonstrated in detail.  Furthermore,
the injection and extraction hardware is technologically challenging,
and likely very costly.
Should the FFAG not be found to give a
significant cost benefit, this alternative acceleration scenario without
an FFAG could be used.
