\subsection{Linac and RLA}

The muon acceleration process involves a complex chain of accelerators
including a (single-pass) linac, two recirculating linacs (RLAs) and an
FFAG ring~\cite{Bogacz:2008zzb}.  
This section will discuss the linac, the two RLAs and the chicanes,
shown schematically in figure \ref{fig:acc:rla:sketch}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
      {02-AccWG/02-E-LinacRLA/linac_rlas.eps}
  \end{center}
  \caption{
    Layout of the linac and recirculating linacs connected by
    chicanes. 
  }
  \label{fig:acc:rla:sketch}
\end{figure}

Acceleration starts after ionisation cooling at 230\,MeV/c and proceeds
to 12.6 GeV. 
The pre-accelerator captures a large muon phase space and accelerates
muons to relativistic energies, while adiabatically decreasing the
phase-space volume so that effective acceleration in the RLA is
possible.
The RLA further compresses and shapes the longitudinal and
transverse phase spaces, while increasing the energy. 
An appropriate choice of multi-pass linac optics based on FODO
focusing assures a large number of passes in the RLA. 
The proposed ``dog-bone'' configuration facilitates simultaneous 
acceleration of both $\mu^+$ and $\mu^-$ species through the
requirement of mirror-symmetric optics in the return ``droplet''
arcs. 

The linac consists of superconducting RF cavities and iron-shielded
solenoids grouped in cryo-modules \cite{Bogacz:2003uk},
while the recirculating linacs RLA~I and RLA~II consist of
superconducting RF cavities and quadrupoles. 
The linac is required to accelerate 0.22~GeV/c muons coming from the
muon front-end to 0.9~GeV/c and, given these relatively low energies,
solenoidal transverse focusing has been chosen so that the beam
preserves its initial horizontal-vertical phase-space coupling.
The transfer to RLA~I is performed through the double chicane,
``chicane~1'', which consists of a vertical dipole spreader (at the
beginning), horizontal bending magnets, a vertical dipole combiner (at
the end), and quadrupoles for transverse focusing
\cite{Bogacz:2009zzc}.
In this manner both positive and negative muons can be transferred,
while keeping RLA~I at the same height as the linac, a decision taken
to simplify the civil engineering.
With the beam now being relativistic, quadrupole focusing in a
FODO-lattice is preferred for chicane~1.
In addition, a number of sextupoles must be inserted at positions
where the dispersion caused by bending has a maximum. 
Entering RLA~I, the beam performs 4.5 passes at an average gain of
0.6~GeV/pass with an excursion along the return arcs after each pass.
Single-cell, superconducting cavities provide acceleration along RLA~I
while transverse focusing is achieved by quadrupoles in a FODO
arrangement along RLA~I and its arcs. 
Similarly, through another double chicane, ``chicane~2'', the 3.6-GeV
beam is transferred to RLA~II where it again performs 4.5 passes but
now at an average gain of 2~GeV/pass. 
Eventually, the beam is extracted and injected into the FFAG at
12.6~GeV.
Section \ref{sec:acc:rla:track} reports the latest results of
muon-beam tracking and the work which has been performed to allow
tracking using realistic field maps for both RF cavities and
solenoids.

\subsubsection{Overall lattice description }
\paragraph{Linear Pre-Accelerator}

A single-pass linac raises the total energy from 0.244 GeV to 0.9 GeV.
This makes the muons sufficiently relativistic to facilitate further
acceleration in the RLA. 
The initial phase-space of the beam, as delivered by the muon
front-end, is characterised by significant energy spread; the linac
has been designed so that it first confines the muon bunches in
longitudinal phase-space, then adiabatically super-imposes
acceleration over the confinement motion, and finally boosts the
confined bunches to 0.9~GeV. 
To achieve a manageable beam-size in the front-end of the linac, short
focusing cells are used for the first six cryo-modules
\cite{Bogacz:2005zr}.
The beam size is adiabatically damped with acceleration, allowing the
short cryo-modules to be replaced with intermediate length
cryo-modules, and then with 11 long cryo-modules. 
Consequently, the linac was split into three consecutive sections
(referred as: the upper, middle and lower linac sections) each section
being built of a particular type of cryo-module as shown in
figure \ref{fig:acc:rla:modules}. 
The important parameters for each section are summarised in
table \ref{tab:acc:rla:linacsumm}.
Each linac section is configured with periodic FOFO 
cells, matched at the section junctions, as illustrated in 
figure \ref{fig:acc:rla:transv} \cite{Berg:2005ut}.
Periodicity within each section is maintained by scaling the solenoid
fields in consecutive cryo-modules linearly with increasing momentum as
summarised in table \ref{tab:acc:rla:table1}. 
\begin{figure}
  \begin{center}
    \includegraphics[width=0.98\textwidth]%
      {02-AccWG/02-E-LinacRLA/modules_text.eps}
  \end{center}
  \caption{
    Layout of the three cryo-module types. 
    They share the same solenoid and cavity type throughout in this
    study.
  } 
  \label{fig:acc:rla:modules}
\end{figure}
\begin{table}
  \caption{Summary of important linac parameters.}
  \label{tab:acc:rla:linacsumm}
  \begin{tabular}{|l|r|r|r|}
    \hline
    {\bf Linac section} & {\bf upper} & {\bf middle} & {\bf lower}  \\
    \hline
    Cell length (m)     & 3           & 5            & 8            \\
    Cryo-modules         & 6           & 8            & 11           \\
    Maximum $\beta$ function (m) & 2.90 & 4.93       & 8.25         \\
    RF cells            & 6           & 16           & 44           \\
    \hline
  \end{tabular}
\end{table}
\begin{table}
  \caption{For each cryo module, the kinetic energy at the exit of the
    module, the solenoid field (for a 1~m solenoid), and the RF phase
    (zero is on-crest). Taken from the OptiM model.}
  \label{tab:acc:rla:table1}
  {\bf Short cryo-modules} \\[0.2ex]
  \begin{tabular}{|l|r|r|r|r|r|r|}
    \hline
    Kin. energy (MeV)&141.6&145.1&149.0&153.1&157.6&162.4\\
    Field (T)&$-1.04$&1.06&$-1.07$&1.09&$-1.12$&0.89\\
    RF phase (deg.)&$-73.3$&$-71.6$&$-69.8$&$-68.1$&$-66.4$&$-64.6$\\
    \hline
  \end{tabular}\\[0.2ex]
  {\bf Medium cryo-modules} \\[0.2ex]
  \begin{tabular}{|l|r|r|r|r|r|r|r|r|}
    \hline
    Kin. energy (MeV)&174.3&187.3&201.3&216.5&232.6&249.7&267.7&286.6\\
    Field (T)&$-0.99$&0.98&$-1.03$&1.08&$-1.14$&1.21&$-1.28$&1.03\\
    RF phase (deg.)&$-62.11$&$-59.13$&$-56.22$&$-53.31$&$-50.40$&$-47.52$&$-44.67$&$-41.85$\\
    \hline
  \end{tabular}\\[0.2ex]
  {\bf Long cryo-modules} \\[0.2ex]
  \begin{tabular}{|l|r|r|r|r|r|r|r|r|r|r|r|}
    \hline
%    &1&2&3&4&5&6&7&8&9&10&11\\\hline
    Kin. energy (MeV)&326.4&368.6&412.6&458.3&505.5&553.7&602.8&652.6&702.8&753.3&803.9\\
    Field (T)&$-1.13$&1.17&$-1.29$&1.41&$-1.54$&1.68&$-1.81$&1.95&$-2.09$&2.23&$-2.37$\\
    RF phase (deg.)&$-38.1$&$-33.8$&$-29.5$&$-25.5$&$-21.5$&$-17.8$&$-14.2$&$-10.8$&$-7.6$&$-4.5$&$-1.7$\\
    \hline
  \end{tabular}
\end{table}
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
      {02-AccWG/02-E-LinacRLA/transv_optics.eps}
  \end{center}
  \caption{Transverse FOFO optics of the entire linac; the upper,
    middle and lower periodic sections uniformly matched at the
    junctions.}
  \label{fig:acc:rla:transv}
\end{figure}

One of the main requirements of the single-pass pre-accelerator linac
is to compress adiabatically the longitudinal phase-space volume in the 
course of acceleration. 
The initial longitudinal acceptance of the linac (chosen to be
2.5~$\sigma$) calls for ``full bucket acceleration''; with an initial
momentum acceptance of $\Delta p/p = \pm$17\% and a bunch length of  
$\Delta\phi$= $\pm$102$^\circ$ (in RF phase). 
To perform adiabatic bunching one needs to drive rather strong
synchrotron motion along the linac. 
The profile of the RF-cavity phases is organised so that the phase of
the first cavity is shifted by 73$^\circ$ (off crest) and then the
cavity phase is gradually changed to zero by the end of the linac, see
table \ref{tab:acc:rla:table1} and figure \ref{fig:acc:rla:long}a.  
In the initial part of the linac, when the beam is still not relativistic, 
the far-off-crest acceleration induces rapid synchrotron motion (one full 
period, see figure \ref{fig:acc:rla:long}b), which allows bunch
compression in both bunch-length and momentum spread as illustrated in
figure~\ref{fig:acc:rla:long}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
      {02-AccWG/02-E-LinacRLA/long_ab1.eps}
  \end{center}
  \caption{Longitudinal matching of a single-pass linac: a) the bunch
    starts far off-crest to capture the beam at low energy and then
    moves to the crest as the energy increases and the longitudinal
    acceptance
    improves; b) the synchrotron phase advances by a full period
    from the beginning to the end of the linac; c) the rate of
    energy gain increases as the bunch moves closer to crest.
    d) Longitudinal acceptance matched
    inside the separatrix and optimised for `full bucket'
    acceleration.}
  \label{fig:acc:rla:long}
\end{figure}

To maximise the longitudinal acceptance, the initial position of the bunch
is shifted relative to the centre of the bucket, to keep the
beam boundary inside the separatrix~\cite{Bogacz:2003uk}, as
illustrated in figure~\ref{fig:acc:rla:long}d. 
The synchrotron motion  
also suppresses the sag in acceleration for the bunch head and tail. In our 
tracking simulation we have assumed a particle distribution that is Gaussian 
in 6D phase space with the tails of the distribution truncated at 2.5 $\sigma$, 
which corresponds to the beam acceptance. Despite the large initial energy 
spread, the particle tracking simulation through the linac does not predict 
any significant emittance 
growth~\cite{Bogacz:2009zzc}.
Results of the simulation are illustrated in figure
\ref{fig:acc:rla:dpp} which shows the longitudinal phase-space at the
end of the linac as simulated by ELEGANT~\cite{Borland:2000,elegant};
tracking through the individual field maps of the linac's RF cavities
and solenoids (figure \ref{fig:acc:rla:dpp}a) and by a simple matrix
based code OptiM \cite{optim} (figure \ref{fig:acc:rla:dpp}b).
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
    {02-AccWG/02-E-LinacRLA/dpp.vs.s.eps}
  \end{center}
  \caption{Longitudinal phase-space compression at the linac end as
    simulated by ELEGANT~\cite{Borland:2000,elegant} (left) and
    OptiM~\cite{optim} (right).}
  \label{fig:acc:rla:dpp}
\end{figure}

\paragraph{RLA I}

The dog-bone RLA I is designed to accelerate simultaneously the 
$\mu^+$ and $\mu^-$ beams from 0.9 GeV to 3.6 GeV. 
The injection energy into the 
RLA and the energy gain per RLA linac (0.6 GeV) were chosen so that 
a tolerable level of RF phase slippage along the linac could be 
maintained ($\sim20^\circ$ in RF phase). To suppress chromatic effects,
90$^\circ$ FODO optics was used as a building block for both the
linac and the return arcs. 
The layout and optics of the linac's periodic cell is shown in figure
\ref{fig:acc:rla:fodo}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
    {02-AccWG/02-E-LinacRLA/fodo.eps}
  \end{center}
  \caption{Periodic 90$^\circ$ FODO 6~m linac cell. A pair
    of 2-cell cavities is interleaved between quadrupoles.  The cavity
    length is 1 RF wavelength and the elements are equally spaced.}
  \label{fig:acc:rla:fodo}
\end{figure}

The focusing profile along the linac was chosen so that beams with a
large energy spread could be transported within the given
aperture. 
Since the beam is traversing the linac in both directions, a
`bisected' focusing profile \cite{Ankenbrandt:2009zza} was chosen for
the multi-pass linac.
Here, the quadrupole gradients scale up with momentum to maintain
90$^\circ$ phase advance per cell for the first half of the linac
(figure \ref{fig:acc:rla:multipass}a), and then are mirror reflected
in the second half, as illustrated in figure
\ref{fig:acc:rla:multipass}b.  
The complete focusing profile along the entire linac is summarised in
table \ref{tab:acc:rla:table2}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
    {02-AccWG/02-E-LinacRLA/multipass.eps}
  \end{center}
  \caption{FODO-based multi-pass linac optics. The quadrupole
    gradients are scaled up with momentum for the first half of the
    linac, then they are mirror reflected in the second half. The
    injection chicane magnets are located at the middle of the linac
    (marked in blue).}
  \label{fig:acc:rla:multipass}
\end{figure}
\begin{table}
  \caption{Gradient, cryo-module by cryo-module along the
    0.6 GeV linac.  The quadrupole centres start at longitudinal position
    46~cm with a spacing between quadrupole centres of 3~m.  The FODO lattice
    starts at one end of the linac with the first F quad in the table,
    and is reflection symmetric about the
    centre of the last D quad shown in the table.
    There are RF cavities
    between each of the quadrupoles, except there are no cavities between
    the central D magnet and its adjacent F quadrupoles. In the first full
    pass, each cavity causes
    the beam to gain 25~MeV, starting with a kinetic energy of
    1103.9~MeV at the beginning of that pass.}
  \label{tab:acc:rla:table2}
  \begin{tabular}{|l|r|r|r|r|r|r|r|}
    \hline
    F Gradient (kG/cm)&0.402&0.385&0.368&0.352&0.335&0.318&0.301\\
    D Gradient
    (kG/cm)&$-0.394$&$-0.377$&$-0.360$&$-0.343$&$-0.325$&$-0.308$&$-0.301$ \\
    \hline
  \end{tabular}
\end{table}%

At the ends of the RLA linacs, the beams need to be directed into the
appropriate energy-dependent (pass-dependent) droplet arc for
recirculation \cite{Bogacz:2005zr}. 
The entire droplet-arc architecture is based on 90$^\circ$
phase-advance cells with periodic beta functions, as described in
figure \ref{fig:acc:rla:inward}.  
\begin{figure}
  \begin{center}
    \includegraphics[width=0.85\textwidth]%
      {02-AccWG/02-E-LinacRLA/inward.eps}
  \end{center}
  \caption{Periodic 90$^\circ$ FODO 6~m inward-bending cell, including
    a pair of rectangular bends interleaved between quads.  All
    elements are equally spaced.}
  \label{fig:acc:rla:inward}
\end{figure}

For practical reasons, horizontal rather than vertical beam separation
has been chosen.
Rather than suppressing the horizontal dispersion created by the
spreader, it has been matched to that of the outward
60$^\circ$ arc.  This is partially
accomplished by removing one dipole (the one furthest
from the spreader) from each of
the two cells following the spreader.
To switch from outward to inward bending, three transition cells are used,
wherein the four central dipoles are removed.  The two remaining dipoles
at the ends bend the same direction as the dipoles to which they are closest.
To facilitate simultaneous acceleration of both $\mu^+$
and $\mu^-$ bunches, a mirror symmetry is imposed on the
droplet arc optics (oppositely charged bunches move in opposite
directions through the arcs)~\cite{Bogacz:2008zzb}.
This puts a constraint on the exit/entrance
Twiss functions for two consecutive linac passes, 
namely 
$\beta_{out}^n = \beta_{in}^{n+1}$ and 
$\alpha_{out}^n = -\alpha_{in}^{n+1}$, 
where $n = 0, 1, 2 ...$ is the pass index. 
Complete droplet arc optics for the lowest energy arc (1.2 GeV) are
shown in figure \ref{fig:acc:rla:compact}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
      {02-AccWG/02-E-LinacRLA/compact.eps}
  \end{center}
  \caption{Compact droplet arc optics, featuring uniform periodicity
    of beta functions and dispersion, which is suppressed geometrically
    via `missing dipoles' at the arc ends.}
  \label{fig:acc:rla:compact}
\end{figure}

The momentum compaction is relatively large (6.5 m), which guarantees
significant rotation in longitudinal phase-space as the beam
passes through the arc.
This effect, combined with off-crest acceleration in the subsequent
linac, yields further compression of the longitudinal phase-space as
the beam is accelerated \cite{Bogacz:2009zzc}.

All higher arcs are based on the same bends (1~m, 1.1~T dipoles) and
they are configured via extending Arc 1 by additional inward and
outward cells as summarised in table \ref{tab:acc:rla:table3}.  The transition
regions always to have three cells.
The quadrupole strengths in the arcs are scaled up linearly with
momentum to preserve the 90$^\circ$ FODO lattice, as summarised in
table \ref{tab:acc:rla:table3}.  Since the linac does not have a
90$^\circ$ phase advance per cell for later passes, the quadrupoles
near the spreader must be modified to achieve matching, as described
in Table~\ref{tab:acc:rla:arcmatch}.  The matching designs will
need to be slightly modified in future work to ensure that common
magnets (namely the F magnet at the end of the linac and the ends
of all the arcs) all have the same parameters.
\begin{table}
  \caption{Higher arcs optics architecture; arc circumference and 
    quadrupole strength scaled up with momentum.  $E$ is the arc energy,
    $p/p_1$ is the ratio of the momentum to the first arc momentum,
    $n_{\text{out}}$ is the number of outward bending cells in each outward
    bend (the total is twice this), $n_{\text{in}}$ is the number of inward
    bending cells, $L$ is the arc length, and $G_F$ and $G_D$ are the
    focusing and defocusing quadrupole gradients, respectively.}
  \label{tab:acc:rla:table3}
  \begin{tabular}{|r|r|r|r|r|r|r|}
    \hline
    {\bf $E$} (GeV) & {\bf $p/p_1$} & {\bf $n_{\text{out}}$} & {\bf $n_{\text{in}}$} & {\bf $L$} (m)
    & {\bf $G_F$} (kG/cm)& {\bf $G_D$} (kG/cm)\\
    \hline
    1.2&1&2&10&130&0.385&$-0.349$\\
    1.8&3/2&3&15&172&0.578&$-0.524$\\
    2.4&2&4&20&214&0.770&$-0.698$\\
    3.0&5/2&5&25&256&0.962&$-0.873$\\
    \hline
  \end{tabular}
\end{table}
\begin{table}
  \caption{Strengths of matching quadrupoles in arc~2 of RLA~I.  Values
    for other arcs have not yet been computed.}
  \label{tab:acc:rla:arcmatch}
  \begin{tabular}{|l|rrrrrr|}
    \hline
    Gradient (kG/cm)&0.5248&$-0.5260$&0.6321&$-0.6878$&0.6151&$-0.5463$\\
    \hline
  \end{tabular}
\end{table}
\paragraph{RLA II}

RLA~II is designed to accelerate the $\mu^+$ and $\mu^-$ beams from
3.6~GeV to 12.6~GeV. 
The injection energy and the energy gain per linac (2~GeV) were chosen
so that a tolerable level of RF-phase slippage along the linac could  
be maintained. 
The optics configuration is almost identical to that of RLA~I. It is  
again based on a 90$^\circ$ FODO lattice and a bisected multi-pass
linac. 
The main difference is that the fundamental cell-length was extended to
12~m (cf.\ 6~m for RLA~I).  
The layout and optics of the linac's periodic cell is shown in figure
\ref{fig:acc:rla:fodo12}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.85\textwidth]%
    {02-AccWG/02-E-LinacRLA/fodo12.eps}
  \end{center}
  \caption{Periodic 90$^\circ$ FODO linac cell (12~m), including
    two pairs of 2-cell cavities interleaved between quadrupoles.}
  \label{fig:acc:rla:fodo12}
\end{figure}

As for RLA I, a bisected focusing profile was chosen for the
multi-pass linac in RLA~II, with the quadrupole gradients
scaled up with momentum in the first half of the linac and mirror
reflected in the second half, as illustrated in
figure \ref{fig:acc:rla:bisect}. 
The complete focusing profile along the entire linac is summarised in
table \ref{tab:acc:rla:table4}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
      {02-AccWG/02-E-LinacRLA/bisect.eps}
  \end{center}
  \caption{Bisected multi-pass linac optics. The quadrupole gradients
    are scaled up with momentum for the first half of the linac and
    mirror reflected in the second half.}
  \label{fig:acc:rla:bisect}
\end{figure}
\begin{table}
  \caption{Gradient and energy profile, cryo-module by cryo-module along
    the 2 GeV linac.  As in table~\ref{tab:acc:rla:table2}, except that
    the spacing between quadrupole centres is 6~m, the first quad centre is
    at 133~cm, the energy gain between quadrupoles is 50~MeV, and
    the kinetic energy is 2894~MeV at the beginning of the first full pass.}
  \label{tab:acc:rla:table4}
  \begin{tabular}{|l|r|r|r|r|r|r|}
    \hline
    Sequence&1&2&3&4&5&6\\
    F grad. (kG/cm)&0.200&0.196&0.190&0.183&0.176&0.169\\
    D grad. (kG/cm)&$-0.201$&$-0.194$&$-0.187$&$-0.181$&$-0.174$&$-0.167$\\\hline
    Sequence       &7       &8       &9       &10      &11      & \\
    F grad. (kG/cm)&0.163   &0.156   &0.149   &0.142   &0.135   & \\
    D grad. (kG/cm)&$-0.160$&$-0.153$&$-0.146$&$-0.139$&$-0.135$& \\
    \hline
  \end{tabular}
\end{table}

All higher arcs are based on the same bends (2~m, 2.1~T dipoles) and
they are configured via extending Arc 1 by additional inward and
outward cells as summarised in table \ref{tab:acc:rla:table5}. 
Magnets to achieve these dipole fields will likely need to be superconducting.
The quadrupole strengths in the arcs are scaled up linearly with 
momentum to preserve the 90$^\circ$ FODO lattice, as summarised in
table \ref{tab:acc:rla:table5}. 
\begin{table}
  \caption{As in table~\ref{tab:acc:rla:table3}, but for arcs in RLA~II.
    $B$ is field for the dipoles.}
  \label{tab:acc:rla:table5}
  \begin{tabular}{|r|r|r|r|r|r|r|r|}
    \hline
    {\bf $E$} (GeV) & {\bf $p/p_1$} & {\bf $n_{\text{out}}$} & {\bf $n_{\text{in}}$} & {\bf $L$} (m)
    & {\bf $G_F$} (kG/cm)& {\bf $G_D$} (kG/cm)&$B$ (T)\\
    \hline
    4.6 &1  &2&10&260&1.45&$-1.42$&2.10\\
    6.6 &1.43&3&15&344&2.08&$-2.04$&2.01\\
    8.6 &1.87&4&20&428&2.71&$-2.66$&1.96\\
    10.6&2.30&5&25&512&3.34&$-3.27$&1.94\\
    \hline
  \end{tabular}
\end{table}%

\paragraph{Injection double chicane}

To transfer both $\mu^+$ and $\mu^-$ bunches from one accelerator to
the next, which is located at a different vertical elevation, we
use a compact double chicane \cite{Bogacz:2008zzb}.
A FODO lattice is used with a 90$^\circ$ phase advance horizontally
and a 120$^\circ$ phase advance vertically.
Each ``leg'' of the chicane involves four horizontal and two vertical 
bending magnets, forming a double achromat in the horizontal and
vertical planes, while preserving periodicity of the beta functions
\cite{Bogacz:2009zzc}.  The lattice for RLA~I is described in
table~\ref{tab:acc:rla:chicane} and figure~\ref{fig:acc:rla:chicane}.
The injection chicane
for RLA~II will be similar, but has yet to be completed.
The chicanes will have pairs of sextupoles to correct vertical emittance
dilution, located at the peaks of the vertical dispersion
(see figure~\ref{fig:acc:rla:chicane}).
Their integrated strengths (in the first injection
chicane) will be approximately 0.01~kG/cm.
\begin{table}
  \caption{Lattice of injection chicane into RLA~I.  The left hand table
    gives the dipole parameters, the right hand table gives the quadrupole
    parameters.  The dipoles are 60~cm long, the quadrupoles are 50~cm
    long.  The chicane is 30~m long and designed for a kinetic energy
    of 803.877\,MeV.  $s$ is the longitudinal position of the end of
    the magnet, $B$ is the
    magnetic field of the dipole (H for horizontal, V for vertical),
    $\theta$ is the bend angle of the
    dipole, $G$ is the quadrupole gradient, and $\varepsilon_{\text{in}}$
    and $\varepsilon_{\text{out}}$ are the entrance and exit angles,
    respectively.  Zero entrance and exit angles would correspond to a sector
    bend; an edge angle with the same sign as the bend angle rotates the
    edge such that a vector perpendicular to the edge and pointing out of the
    magnet rotates away from the centre of curvature (MAD convention).}
  \label{tab:acc:rla:chicane}
  \begin{tabular}[c]{|crrrrr|}
    \hline
    Type&$s$ (cm)&$B$ (kG)&$\theta$ (deg.)&$\varepsilon_{\text{in}}$ (deg.)&$\varepsilon_{\text{out}}$ (deg.)\\\hline
    H& 120&7.024468&8.01381&0&9\\
    H& 290&$-7.024468$&$-8.01381$&$-9$&0\\
    V&504&$-4.682978$&$-5.34254$&0&$-6$\\
    V&2306&4.682978&5.34254&6&0\\
    H&2520&$-7.024468$&$-8.01381$&$-9$&0\\
    H&2690&7.024468&8.01381&0&9\\
    \hline
  \end{tabular}
  \begin{tabular}[c]{|rr|}
    \hline
    $s$ (cm)&$G$ (kG/cm)\\\hline
    100&$-0.3682620$\\
    400&0.3090135\\
    700&$-0.3623229$\\
    1000&0.3121510\\
    1300&$-0.3641247$\\
    1600&0.3121510\\
    1900&$-0.3641247$\\
    2200&0.3090135\\
    2500&$-0.3623229$\\
    2800&0.3172734\\
    \hline
  \end{tabular}
\end{table}
\begin{figure}
  \includegraphics[width=0.95\linewidth]{02-AccWG/02-E-LinacRLA/ABogacz-110113-2b.eps}
  \caption{Diagram of the injection chicane lattice.}
  \label{fig:acc:rla:chicane}
\end{figure}

The chicane and RLA lattices have not yet been simulated with 
realistic field maps but only with usual beam optics codes such as
MADX \cite{mad-x} and OptiM \cite{optim}. 
As an example, the horizontal and vertical betatron functions of
chicane~1 are shown in figure \ref{fig:acc:rla:beta_chicane1}. 
\begin{figure}
  \begin{center}
    \includegraphics[width=0.95\textwidth]%
    {02-AccWG/02-E-LinacRLA/madx01_text.eps}
  \end{center}
  \caption{Horizontal and vertical betatron functions as the beam
    passes through chicane 1, calculated with MADX.}
  \label{fig:acc:rla:beta_chicane1}
\end{figure}

\paragraph{Arc crossings}

One unfortunate feature of the dogbone lattice is that the arcs will cross
each other.  To address this, the arcs will have a vertical bypass in the
region of the crossing, one arc going up and the other down, as shown
in figure~\ref{fig:acc:rla:bypass}.
\begin{figure}
  \includegraphics[width=0.95\linewidth]{02-AccWG/02-E-LinacRLA/overpass.eps}
  \caption{Diagram of the arc bypass lattice.  The vertical dipoles are
    10 cm long, centred in the gap between the adjacent dipole and quadrupole.}
  \label{fig:acc:rla:bypass}
\end{figure}

\subsubsection{Magnets}

The most important magnet types of the linac and RLA sections are:
superconducting solenoids (linac) and normal conducting dipoles,
combined dipole/sextupoles and quadrupoles (RLAs and chicanes).
A detailed numerical simulation has been performed for the
superconducting solenoids, while the design of normal conducting
magnets is in progress.  
As shown in figure \ref{fig:acc:rla:solenoid} a configuration 
of two superconducting coils with opposite currents shielded by a 5~cm
thick iron wall is good enough to prevent longitudinal field leakage
which may cause quenching of the neighbouring superconducting
cavities. 
Also, it has been shown that the coils are far from the quench limit
and may be made thinner.  
\begin{figure}
  \begin{center}
    \includegraphics[width=0.35\textwidth]%
      {02-AccWG/02-E-LinacRLA/short2.eps}\hspace{0.4cm}
    \includegraphics[width=0.6\textwidth]%
      {02-AccWG/02-E-LinacRLA/solenoid_field_dens.eps}
  \end{center}
  \caption{Layout of the shielded superconducting solenoid (left) and its 2D
    magnetic field map across the axial plane (right) for a peak
    magnetic field of 2~T.}
  \label{fig:acc:rla:solenoid}
\end{figure}

\subsubsection{RF system}

The lattice designs for the acceleration systems are based on the Study~II
RF-cavity designs \cite{Ozaki:2001}, the parameters of which are summarised
in table \ref{tab:acc:rla:cav}.
\begin{table}
  \caption{Important parameters for the Study~II superconducting RF cavity
    designs~\cite{Ozaki:2001}.  When single-cell cavities are used,
    appropriate parameters are cut in half.}
  \label{tab:acc:rla:cav}
  \begin{tabular}{|l|r|r|}
    \hline
    RF frequency (MHz)&201.25&201.25\\
    Cells per cavity&2&2\\
    Aperture diameter (mm)&300&460\\
    Energy gain/cavity (MeV)&25.5&22.5\\
    Stored energy/cavity (J)&2008&1932\\
    Input power/cavity (kW)&1016&980\\
    RF on time (ms)&3&3\\
    Loaded $Q$&$10^6$&$10^6$ \\
    \hline
  \end{tabular}
\end{table}

We have studied the optimisation of the cavity design.
The layout of the standing wave superconducting RF cell has been
optimised for a number of parameters.
The target energy gain over the whole 66 cells of the linac is 
$\sim 665$~MeV, or a little more than 10 MeV/cell as shown in figure
\ref{fig:acc:rla:cavity}.  
Both single-cell and double-cell ($\pi$-mode) cavities have been
simulated with specialised codes including Superfish
\cite{superfish,Halbach:1976cp}, CST-MWS \cite{cst-mws} and Comsol
\cite{comsol} in order to obtain the electric- and magnetic-field
maps necessary for particle tracking.  
\begin{figure}
  \begin{center}
    \begin{tabular}{l}
      length = 0.7448 m\\
      radius = 0.6854 m\\
      resonance freq. = 201.247 MHz\\
      quality fact. = 24.67$\times 10^9$\\
      transit time factor = 0.650 - 0.690\\
      peak $\hat{E}$ =  26.17 MV/m\\
      $|E|^{max}_{surf}$ = 21.70 MV/m\\
      $|H|^{max}_{surf}$ = 48.06 kA/m\\
      stored energy = 712 J
    \end{tabular}\hspace{4em}
    \raisebox{-0.5\height}{\includegraphics[width=0.3\textwidth]{02-AccWG/02-E-LinacRLA/ellipse2b.eps}}\\
    \includegraphics[width=0.55\textwidth]{02-AccWG/02-E-LinacRLA/single_cell_2.eps}%
    \includegraphics[width=0.45\textwidth]{02-AccWG/02-E-LinacRLA/cavity.eps}
  \end{center}
\caption{Main parameters of the superconducting RF cell and its electric field maps.}
\label{fig:acc:rla:cavity}
\end{figure}

The peak electric fields are 23.09~MV/m in the upper linac and
26.17~MV/m the middle and lower linacs.  
In previous work~\cite{Ozaki:2001}, the references to 15~MV/m,
17~MV/m, and 17~MV/m cavities included the transit time factor for a
$\beta=1$ particle ($26.17\text{ MV/m}\times 0.65 = 17$~MV/m, 0.65 is the
transit time factor).   
Here $0.90\leq\beta \leq 0.99$ and the cavity phases were offset to
introduce $\sim 1$ synchrotron period over the whole linac.  

\subsubsection{Beam tracking}
\label{sec:acc:rla:track}

First, using only solenoid field maps (no-acceleration), Gaussian muon
bunches with very large emittances have been tracked through many
cells of the three linac cell types in order to obtain
the transverse acceptances.  
A significant number of muons are lost in the early part of the
linac.
Later in the linac, the beam is transported without additional loss
making it possible to determine the rms transverse emittances shown in
figure \ref{fig:acc:rla:au}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.85\textwidth]{02-AccWG/02-E-LinacRLA/admittance_up_A.eps}\\    
    \begin{tabular}{|c| c | l| l| l| l| l| l| l|}
      \hline
      \multicolumn{2}{|c|}{linac}            & $\bar{\varepsilon}_{x}$ & $\sigma_{x}$ & $\sigma_{x'}$ & $\bar{\varepsilon}_{y}$ & $\sigma_{y}$ & $\sigma_{y'}$  \\
      \multicolumn{2}{|c|}{section}          &      [mm rad]           & [mm]         & [mrad]        & [mm rad]                & [mm]         & [mrad]         \\ \hline
      \textit{\textbf{\textit{upper}}} & MIN &  5.7                    & 78           & 43            & 5.7                     &  82          & 40             \\
      \textit{\textbf{\textit{middle}}}& MIN &  3.4                    & 79           & 26            &  3.3                    &  79          & 25             \\
      \textit{\textbf{\textit{lower}}} & MIN &  2.2                    & 87           & 15            &  2.1                    &  88          & 16 \\
    \hline
    \end{tabular}
  \end{center}
  \caption{Stable tracks in the upper linac 36 cells downstream
    from the injection point (top), the \textit{y}-phase space at the two
    positions indicated (middle) and figures of the normalised
    transverse emittances and standard deviations (bottom).}
  \label{fig:acc:rla:au}
\end{figure}

End-to-end, particle-tracking studies are in progress for the 
linac using the codes GPT~\cite{gpt}, G4beamline~\cite{Roberts:2008zzc},
OptiM and ELEGANT.  
As an example, figure \ref{fig:acc:rla:trup} shows the longitudinal and
transverse phase-space evolution for the upper linac while figure
\ref{fig:acc:rla:g4bl} shows the particle trajectories and
kinetic-energy increase along the entire linac. 
\begin{figure}
  \begin{center}
    \includegraphics[width=0.84\textwidth]%
      {02-AccWG/02-E-LinacRLA/entrance_up_A.eps}\\
    \begin{tabular}{|c |c | l |l |l |l |l |l |l |}
      \hline
      \multicolumn{2}{|c|}{\textit{phase}}         &$\bar{\varepsilon}_{x}$ & $\sigma_{x}$ & $\sigma_{x'}$ & $\bar{\varepsilon}_{y}$ & $\sigma_{y}$ & $\sigma_{y'}$  \\
      \multicolumn{2}{|c|}{\textit{space}}         & [mm rad]               & [mm]         & [mrad]        & [mm rad]                & [mm]         & [mrad]     \\ \hline
      \multirow{2}*{\textit{\textbf{transverse}}}  & A & 4.9                & 81           & 34            &  5.0                    & 87           & 31         \\
                                                   & B & 4.5                & 61           & 33            &  4.6                    & 62           & 36 \\ 
      \hline
      \multicolumn{2}{|c}{\textit{}}                & \multicolumn{2}{|l|}{(1/$\beta\gamma$)$\times$ $\bar{\varepsilon}_{||}$}& \multicolumn{2}{|l|}{$\sigma_{t}$} &\multicolumn{2}{|l|}{$\sigma_{E_{tot}}$}              \\
      \multicolumn{2}{|c}{\textit{}}                  & \multicolumn{2}{|l|}{[ms eV]}             & \multicolumn{2}{|l|}{[ns]}           &  \multicolumn{2}{|l|}{[MeV]}         \\ \hline
      \multirow{2}*{\textit{\textbf{longitudinal}}}& SU1& \multicolumn{2}{|l|}{2.7}                & \multicolumn{2}{|l|}{0.72}        &   \multicolumn{2}{|l|}{13}     \\ 
      & C  & \multicolumn{2}{|l|}{1.7}                & \multicolumn{2}{|l|}{0.55}        &   \multicolumn{2}{|l|}{15}  \\ \hline
    \end{tabular}
  \end{center}
  \caption{GPT particle tracks through the upper linac (top),
    the \textit{y}-phase space at its beginning/end (middle) and figures
    of the normalised emittances and standard deviations (bottom).}
  \label{fig:acc:rla:trup}
\end{figure}
\begin{figure}
  \begin{center}
    \includegraphics[width=0.87\textwidth]%
      {02-AccWG/02-E-LinacRLA/a-22.eps}\\
    \includegraphics[width=0.87\textwidth]%
      {02-AccWG/02-E-LinacRLA/a-61.eps}
  \end{center}
  \caption{
    G4beamline particle tracking showing vertical projections of the
    trajectories (top) and kinetic energy (bottom) along the entire
    linac.
  }
  \label{fig:acc:rla:g4bl}
\end{figure}

As figure \ref{fig:acc:rla:trup} shows, there is noticeable (adiabatic)
reduction of the transverse beam-size as the beam undergoes
acceleration,
although the phasing of the RF cavities is not yet fully optimised.
Future studies will focus on controlling the longitudinal-transverse
phase-space coupling with the aim of transporting the beam along the whole
linac with minimum losses.  
