Acceleration of a muon beam is a challenging task because of its large
phase space and short lifetime. In the design concept
presented here, acceleration starts after ionization cooling at 210
MeV/c and proceeds to 20~GeV, where the beam is injected into a
Neutrino Factory storage ring. The key technical issues, beyond the
basic physics parameters of Table~\ref{tab:acc:parm}, are:
 
\begin{table}[!bht]
  \caption{Main parameters of the muon accelerator driver.}
  \label{tab:acc:parm}
  \begin{center}
    \begin{tabular}{|lc|}
      \hline
      Injection momentum (MeV/$c$)/Kinetic energy (MeV)&210/129.4 \\
      \hline
      Final energy (GeV)&20 \\
     \hline
      Initial normalized acceptance (mm-rad)&15\\
      \quad rms normalized emittance (mm-rad)&2.4\\
      \hline
      Initial longitudinal acceptance, $\Delta pL_b/m_\mu$ (mm)&170\\
      \quad momentum spread, $\Delta p/p$&$\pm0.21$\\
      \quad bunch length, $L_b$ (mm)&$\pm 407$\\
      \quad rms energy spread&0.084\\
      \quad rms bunch length (mm)&163\\
      \hline
      Number of bunches per pulse&67\\
      \hline
      Number of particles per bunch\textbf{/}per pulse&
      $4.4\times10^{10}$\textbf{/}$3\times10^{12}$\\
      \hline
      Bunch frequency\textbf{/}accelerating frequency (MHz)&201.25\textbf{/}201.25\\
      \hline
      Time structure of muon beam&6 pulses at 50 Hz; 2.5 Hz repetition rate\\
      \hline
      Average beam power (kW)&150 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{itemize}
\item choice of acceleration technology (superconducting versus normal conducting
cavities) and, related to it, rf frequency choice 
\item choice of
acceleration scheme 
\item capture, acceleration, transport and
preservation of a large-phase-space muon beam 
\item  accelerator performance issues, such as potential
collective effects (\textit{e.g.}, cumulative beam breakup) resulting from the
high peak current.  
\end{itemize}
To minimize muon decay, the highest possible
accelerating gradient is required; that is the main driver for the
proposed scheme. The muon accelerator driver consists of a 2.87~GeV 
linac and a four-pass recirculating linear accelerator,
as shown in Fig.~\ref{fig:acc:layout}.

\begin{figure}[tb]
  \centering
  \includegraphics{../template/report/ps-and-eps/accel-010307-f01-fix.eps}
  \caption{Layout of the muon accelerator driver.}
  \label{fig:acc:layout}
\end{figure}

Very large transverse and longitudinal accelerator acceptances drive
the design to low rf frequency. If we were to use normal-conducting rf (NCRF) cavities, the required gradients would be of the order of 15~MV/m, which, in turn, demands extremely high peak  power rf sources.  Superconducting rf (SCRF)
cavities are a much more attractive solution. RF power can then be
delivered to the cavities over an extended time, and thus the rf source
peak power can be reduced. Another important advantage of SCRF cavities
is that their design is not limited by a requirement of low shunt
impedance, and therefore their aperture can be significantly larger then would be practical for NCRF cavities. 
Taking into account the required longitudinal and transverse
acceptances, and that the beam is already bunched at 201.25 MHz at the
source (ionization cooling), a 201.25 MHz rf frequency has been
chosen for both the linear accelerator and the recirculator. This
choice provides adequate stored energy to accelerate multiple
passes of a single-pulse bunch train without the need to refill the
extracted energy between turns.

The short muon lifetime essentially excludes use of a conventional circular
accelerator, and demands either a high-gradient conventional or
recirculating linac (FFAG rings could be also considered).  While recirculation provides significant cost
savings over a single linac, it cannot be utilized at low energy for
two reasons. First, at low energy the beam is not sufficiently
relativistic and will therefore slip in phase on subsequent
passes, thus significantly reducing acceleration efficiency.
Second, there are major difficulties associated
with injection of a beam with the large emittance and energy spread
characteristic of a muon source. Beam pre-acceleration in a linear
accelerator to about 2.5 GeV makes the beam sufficiently relativistic,
and adiabatically decreases the phase-space volume, so that further
acceleration in recirculating linacs is possible.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f02.eps}
  \caption[Decay of muons in acceleration.]{Decay of muons in the course
    of acceleration.  The dotted line is the decay in the linac, and
    the solid line is the decay in the recirculator.  Vertical drops
    correspond to the beam transport in arcs.}
  \label{fig:acc:decay}
\end{figure}

Cost considerations favor multiple passes per stage, but practical
experience commissioning and operating recirculating linacs dictates
prudence.  Experience at Jefferson Lab suggests that for a given large
initial emittance and energy spread, a ratio of final-to-injected
energy below 10-to-1 is prudent, and the number of passes should be
limited to about five. We therefore propose a machine architecture
(see Fig.~\ref{fig:acc:layout}) featuring a 0.13-to-2.48 GeV
straight ``preaccelerator'' linac, followed by a  2.48-to-20 GeV four-pass
recirculating linac (RLA). Figure~\ref{fig:acc:decay} shows the loss of muons during the course
of acceleration. While the RLA gives a significant
contribution, the major loss fraction comes from the linac. Note
that the arcs (vertical drops in Fig.~\ref{fig:acc:decay} ) do not contribute much to the
decay, which justifies the choice of normal conducting bends, and
triplet focusing discussed below.

\section{Linear Accelerator}
\subsection{Matching}

It is important to provide a section that matches the beam ellipse
from what it is in the cooling section to what it is in this linac.
That matching section is described in Section~\ref{BandC:bob}.  The
lattice not only matches the beam ellipses, it accelerates by a
small amount, approximately 59~MeV.  Using the phase graph in
Fig.~\ref{fig:acc:phase}, we compute that approximately the first
38~m of the linac described in this Chapter is no longer required, that is, only the last 4 short cryomodules are needed.  In this Chapter, however, we
ignore the acceleration in the matching section, and
assume that we start accelerating at a momentum of 210~MeV/$c$.

\subsection{Linac General Parameters and Lattice Period Layout}
\label{LGPandLPL}
\begin{table}[!tb]
  \caption{Main parameters of linear accelerator.}
  \label{tab:acc:linacmain}
  \begin{center}
    \begin{tabular}{|lc|}
      \hline
      Injection momentum (MeV/$c$)/Kinetic energy (MeV)&210/129.4\\
      \hline
      Final momentum (MeV/$c$)/Kinetic energy (MeV)&2583/2480\\
      \hline
      Total linac length (m)&433\\
      \hline
      Acceptance: initial/final (no emittance dilution) (mm-rad)&7.5/0.62\\
      \hline
      Momentum spread: initial/final&$\pm0.21$/$\pm 0.075$\\
      \hline
      Total bunch length: initial/final (mm)&814/190\\
      \phantom{Total bunch length: initial/final}(deg.)&197/46 \\
      \hline
      Total installed accelerating voltage (GeV)&2.87\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[!tb]
  \caption{Parameters for the linear accelerator cryomodules.}
  \label{tab:acc:cryoparm}
  \begin{center}
    \begin{tabular}{|lccc|}
      \hline
      &Short&Intermediate&Long\\
      \hline
      Number of periods&11&16&19\\
      Total length of one period (m)&5 &8 &13 \\
      Number of cavities per period&1&2&4\\
      Number of cells per cavity&2&2&2\\
      Number of couplers per cavity&2&2&2\\
      Cavity accelerating gradients (MV/m)&15&15&17 \\
      Real-estate gradient (MV/m)&4.47 &5.59&7.79\\
      Aperture in cavities ($2a$) (mm)&460 &460&300\\
      Aperture in solenoids ($2a$) (mm)&460 &460&360\\
      Solenoid length (m)&1 &1&1.5 \\
      Solenoid maximum field (T)&2.1 &2.1 &4.2\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f03.eps}
  \caption[Layouts of cryomodules.]{Layouts of short (top), intermediate
    (middle) and long (bottom) cryomodules.  Blue lines are the SC
    walls of the cavities.  Solenoid coils are indicated in red, and
    BPMs in yellow.}
  \label{fig:acc:cryomod}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f04.eps}
  \caption[Beam envelopes along the linear accelerator.]{Beam envelopes of
    the entire beam ($2.5\sigma$) along the linear accelerator.}
  \label{fig:acc:linacenv}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f05.eps}
  \caption[Beta functions along the linear accelerator.]{Beta functions
    along the linear accelerator.  The beta functions are computed in
    the frame that rotates with angular frequency $\omega=eB_s/2pc,$
    so that the beam motion is decoupled.}
  \label{fig:acc:linacbeta}
\end{figure}

The requirement for a large acceptance of the accelerator necessitates a large aperture
and tight focusing at its upstream end. Considering the large aperture,
the tight space, the moderate beam energy and the need for strong focusing in both
planes, solenoidal focusing is superior to triplet focusing, and
has been chosen for the entire linac. To achieve a manageable beam
size in the initial portion of the linac, short focusing cells are used for the
first 11 cryomodules. As the beam size adiabatically damps during
acceleration, 
intermediate-length cryomodules can be used and finally, when the energy reaches
0.75 GeV long (standard) cryomodules are utilized for the remainder of the linac. In comparison with the
standard 13 m cryomodules, the short and intermediate-length
cryomodules have increased aperture and, consequently, reduced
accelerating gradient. The main parameters of the linac and its three cryomodule types are presented in Tables~\ref{tab:acc:linacmain} and
\ref{tab:acc:cryoparm}. Figure~\ref{fig:acc:cryomod} depicts the
layouts of short, intermediate-length and long cryomodules.
Figures~\ref{fig:acc:linacenv} and \ref{fig:acc:linacbeta} present the
beam envelope and beta-function along the linac.

The layout of cryomodules and the arrangement of SC cavities are
determined by the requirement to keep power through the fundamental coupler
at an acceptable level and to have cavities sufficiently decoupled. The
coupler power limitation (below 0.5 MW) requires 1 coupler per cell, so there is a coupler at each end of the two-cell
cavity.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f06.eps}
  \caption{Electrical circuit model for calculation of cavity coupling.}
  \label{fig:acc:circuit}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[width=\textwidth]{../template/report/ps-and-eps/accel-010307-f07.eps}
  \caption[Dependence of cavity voltage on frequency.]{Dependence of cavity
    voltage on frequency.  Solid lines: voltage for normally powered
    cavity; dashed line: voltage for improperly functioning cavity
    with corresponding power generator off.  Left figures, cavity is
    not detuned;  right figures, cavity is detuned by 5 bandwidths.
    Top figures, $\delta=0.1/Q$; bottom figures, $\delta=1/Q$.}
  \label{fig:acc:coupling}
\end{figure}

The coupling coefficient determined as $\delta=C_3/C_1$ (see
Fig.~\ref{fig:acc:circuit}), should be sufficiently small,
\begin{equation}
  \delta\leq\dfrac{1}{10Q},
\end{equation}
to have a possibility to bypass cavities not  functioning properly.
Figure~\ref{fig:acc:coupling} demonstrates the effects of cavity
coupling and detuning on the cavity voltage. For a loaded $Q$ of
$5\times10^5$, the required cavity decoupling should be below
$2\times10^{-6}$.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f08.eps}
  \caption[Attenuation of electromagnetic waves between two cavities.]
  {Attenuation of electromagnetic waves between two cavities for short
    (left) and long (right) cryomodules.  The attenuation is
    approximated by
    $\delta=\exp(-L\sqrt{(\mu_0/a)^2-(2\pi/\lambda)^2})$.}
  \label{fig:accel:atten}
\end{figure}

Such decoupling requires significant distances between nearby
cavities. For an estimate, we take the coupling between cavity
cells to be 5\%, and then, using results presented in
Figure~\ref{fig:accel:atten}, find that the distance between
cavities has to be more than 110 cm for short cryomodules and 70 cm
for long cryomodules. Taking into account that the fundamental and
HOM couplers are located in the same space, these distances were
increased to 150 and 100 cm, respectively. BPMs are located inside the
solenoids to reduce effects of EMI coming from the rf cavities.

There is an additional limitation on the layout of the linac
determined by a requirement that all cavities be treated and placed under vacuum
in a clean room and then kept under vacuum thereafter. To accomplish this, each cryomodule must have vacuum valves at both
ends, with corresponding transition modules from liquid-helium
temperature to room temperature. To achieve the maximum
real-estate accelerating gradient, the focusing solenoids are also
located inside the rf cryomodules.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f09.eps}
  \caption{Layout of short solenoid and plot of its magnetic field lines.}
  \label{fig:acc:shortsol}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010607-f10.eps}
  \caption[Dependence of the magnetic field on longitudinal coordinate.]
  {Dependence of magnetic field on longitudinal coordinate.  The solid
    line is the field on axis; the dotted line is the field at the
    cavity radius (23~cm); the dashed line is a fit to
    $B(s)=(B_0/2)\{1-\tanh[(s-L/2)/a]\}$, where $a=19.5$~cm.  Vertical
    lines show positions where the SC screen and cavity start.}
  \label{fig:acc:solmagfield}
\end{figure}

Considering the large aperture required by the beam size, the question of
focusing linearity of the solenoids must be addressed. The dependence of
focusing strength on radius can be approximated by the following
expression:
\begin{equation}\Phi\equiv\dfrac{1}{F}\approx
  \left(\dfrac{e}{2pc}\right)^2
  \left(\int B^2 ds+\dfrac{r^2}{2}\int B'^2ds\right)
  \approx L\left(\dfrac{eB_0}{2pc}\right)^2
  \left(1+\dfrac{r^2}{3aL}\right),
  \label{eq:acc:solfoc}
\end{equation}
where $L$ and $a$ are the solenoid length and radius. As can seen 
from Eq.~\ref{eq:acc:solfoc}, to reduce the nonlinearity one needs
to increase the solenoid length and aperture. Increasing length
directly decreases the real-estate gradient; increasing the aperture
requires a larger distance between the solenoid and cavity to shield
the magnetic field and, in the final analysis, also decreases real-estate
gradient. An aperture increase also makes solenoids more expensive and
less reliable. The length of the short solenoid has been chosen to be 1~m, 
as a compromise between these conflicting requirements. The length
of the long solenoids is determined by the magnetic field limitation and
is chosen to be 1.5 m. The concept of the short solenoid and plots of
magnetic lines are shown in Fig.~\ref{fig:acc:shortsol}. Design details are given in Chapter~\ref{SCSOLE}. To achieve
fast field drop between the solenoid and the cavity, the solenoid has an outer
bucking coil that cancels its magnetic flux. It also has a
SC shield at its outer surface. Together, these give a magnetic
field less than 0.1~G inside the cavity, as depicted in Fig.~\ref{fig:acc:solmagfield}.

\subsection{Longitudinal Beam Dynamics}
Because the initial bunch length and energy spread are very large, the
bunch length is more than a half rf wavelength
($\Delta\phi=\pm89^\circ$), and the momentum spread is about
$\pm21\text{\%}$.  Decreasing these values (via adiabatic damping)
to a manageable level is the most important assignment of the beam
acceleration in the linac. The final linac energy is determined
by demanding a velocity sufficiently close to the velocity of light that
there is no significant rf phase slip for higher passes in the
subsequent recirculator.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f11.eps}
  \caption{RF (left) and synchrotron (right) phases along the linac.}
  \label{fig:acc:phase}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f12.eps}
  \caption[Beam boundary inside separatrix at the beginning of the
  linac.]{Beam boundary (solid line) inside separatrix (dashed line)
    shown at the beginning of the linac.}
  \label{fig:acc:beamsep}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f13.eps}
  \caption[Beam boundary at the beginning, middle, and end of the
  linac.]{Beam boundary at the beginning (dashed line), in the middle
    (dotted line), and at the end of the linac.}
  \label{fig:acc:beamboundary}
\end{figure}

To perform adiabatic bunching, the rf phase of the cavities is shifted
by $73^\circ$ at the beginning of the linac and is gradually changed to
zero at the end of the linac, as shown in Figure~\ref{fig:acc:phase}. In the
first half of the linac, when the beam is still not sufficiently
relativistic, the offset results in synchrotron motion, which prevents a sag in acceleration for the bunch head and tail, and allows bunch
compression in both length and momentum spread to $\Delta
p/p=\pm7.5\text{\%}$ and $\Delta\phi=\pm23^\circ$. However, the rf phase
offset also reduces the effective accelerating gradient, so that a total
voltage of 2.87~GV is required to provide a beam acceleration of 2.35~GeV.
To maximize longitudinal acceptance, the initial position of the beam is shifted
relative to the center of the bucket.  Figure~\ref{fig:acc:beamsep}
depicts the position of the beam boundary inside the separatrix; 
Figure~\ref{fig:acc:beamboundary} shows how the initially elliptical
boundary of the bunch longitudinal phase space is transformed to the
end of the linac.

\subsection{Transverse Beam Dynamics and Tracking}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f14.eps}
  \caption[Dependence of the maximum beta function with the tune advance per
  cell.]{Dependence of maximum beta function on phase advance per
    cell for a beam line with solenoidal focusing and period length 6~m.}
  \label{fig:acc:bmaxtune}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f15.eps}
  \caption[Dependence of beta function maximum on
  momentum.]{Dependence of relative change of the maximum beta function
    on the relative momentum change for different values of the phase
    advance per cell.}
  \label{fig:acc:bmaxmom}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f16.eps}
  \caption[Relative emittance change.]{Relative emittance change after
    passing 50 solenoidal lenses of 1~m length.
    $\varepsilon_n=15\text{ mm-rad}$; vertical lines show betatron
    tune spread in the beam: $\Delta\nu/\nu\approx\Delta
    p/p=\pm26\text{\%}$.}
  \label{fig:acc:emitnu}
\end{figure}

Betatron phase advance per cell, $\nu$, is an important parameter
to determine the properties of the beam transport in the linac. There are a
few considerations that must be taken into account. First, large
beam emittance and limited aperture in the cavities require
minimization of the beam size for a given period length. As 
seen from Fig.~\ref{fig:acc:bmaxtune}, this requires $\nu$ close to
0.25. Second, we must minimize the dependence of the beta function
on momentum.  For the same initial conditions, the
beta function oscillates relative to its nominal value if the  momentum is
changed.  Figure~\ref{fig:acc:bmaxmom} shows the ratio of maximum
beta function achieved in the course of oscillations to the maximum 
beta function at equilibrium energy. For a momentum
spread of $\pm20\text{\%}$ a phase advance below 0.25 is clearly preferred. Third,
the solenoids are short compared with their aperture, and therefore
they have significant non-linearity in their focusing. 
From Eq.~(\ref{eq:acc:solfoc}), for $a$ = 19 cm, $r$ = 23 cm and $L$ = 1 m, there is a change of focusing strength of 9\% at the beam boundary.
Such nonlinear fields can cause strong nonlinear resonances, even for
a small number of lattice periods.  Figure~\ref{fig:acc:emitnu} shows
 how the beam emittance changes for different values of the phase advance 
per cell
after traversing  a channel with 50 solenoidal lenses. One can see the very
strong effect of the 1/4-resonance, which spreads over the 0.21--0.24 tune range. The 1/6-resonance is also visible, but does not
produce so large an effect.  In reality, this effect would be  much smaller, because of
adiabatic damping of the beam size with acceleration.  Taking everything
into account, we choose a phase advance of 0.175.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f17.eps}
  \caption[Longitudinal phase space in the linac.]{Longitudinal
    phase space at the beginning (left) and the end (right) of the
    linac. }
  \label{fig:acc:longps}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f18.eps}
  \caption[Beam intensity, momentum spread, emittance, and 
  envelope along the linac]{Beam intensity, rms momentum
    spread (top), beam emittances (normalized to the initial linac
    energy) and beam envelopes (bottom) along the linac.}
  \label{fig:acc:beamevol}
\end{figure}

The particle distribution for tracking has been chosen to be Gaussian in
6D phase space, but the tails of the distribution are truncated at
$2.5~\sigma$, corresponding to the beam acceptance presented in
Table~\ref{tab:acc:parm}.  Despite the large initial energy spread,
particle tracking through the linac does not exhibit any significant
emittance growth, with 0.2\% beam loss coming mainly from particles at
the longitudinal phase space boundary. Figure~\ref{fig:acc:longps}
presents longitudinal phase space at the beginning and the end of
the accelerator. Figure~\ref{fig:acc:beamevol} shows the beam emittance, 
beam envelope, and beam intensity along the linac. The sudden
increases and then decreases of the envelopes correspond to a particle
motion instability, with subsequent particle scraping. The decay of
muons is not taken into account in this beam intensity plot.

\subsection{Injection Chicane}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f19.eps}
  \caption{Diagram of the injection chicane.}
  \label{fig:acc:injec}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f20a.eps}
  \caption{Beta functions and dispersion in the injection chicane.}
  \label{fig:acc:injecbeta}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f20b.eps}
  \caption{Beam envelopes in the injection chicane.}
  \label{fig:acc:injecenv}
\end{figure}

An injection chicane is used to inject the beam into the RLA. A
simplified scheme is presented in
Figure~\ref{fig:acc:injec}. The chicane is built from four
dipoles with four quadrupole triplets between them to form an achromat. 
A standard three-dipole chicane cannot be
used because the chicane has to be sufficiently long to bypass
the incoming higher energy arcs. In addition, two quadrupole triplets are used immediately before the chicane in place of the solenoidal focusing in the linac. An
advantage of triplet focusing is that it easily provides the long straight sections
necessary for beam separation at injection.  Triplet focusing also
naturally matches the solenoidal focusing. The period length is 15 m
to coincide with the period length of the downstream RLA
linac. The betatron phase advance per cell is chosen to be $90^\circ$, as this is
preferable from the viewpoint of compensating chromatic effects.
Figures~\ref{fig:acc:injecbeta} and \ref{fig:acc:injecenv} depict
beta functions, dispersion, and beam envelopes in the chicane.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f21.eps}
  \caption[Variation of beam emittance and beam envelopes along the
  linac.]{Variation of beam emittance (normalized to the initial
    linac energy) and beam envelopes along the linac.}
  \label{fig:acc:emitvar}
\end{figure}

In the linac, chromatic effects are suppressed by the periodicity of
the focusing and do not require special correction. Unfortunately,
it does not quite work the same way when we introduce bends;  
a sextupole chromatic corrector is required for the horizontal plane. This is achieved by introducing sextupole components into the field of the
focusing quadrupoles of the six triplets. Four of these quadrupoles are located at the
maximum of the dispersion function for chromaticity compensation and the
other two are located in front of the chicane for compensation of the
nonlinearity introduced by the strong chromaticity-correction quadrupoles, as
shown in Fig.~\ref{fig:acc:injecbeta}. Tracking studies showed that the
nonlinearity of the sextupole fields can be cancelled for a small beam momentum 
spread, but only partial cancellation can be
achieved for a momentum spread as large as $\pm10\text{\%},$  the value at the end of the linac (see Fig.~\ref{fig:acc:longps}). Strong sextupole components, required for
good correction of second-order dispersion, cause emittance 
growth that is too large, due to inadequate cancellation of nonlinearities, and therefore a
partial compensation of second-order dispersion is preferable. In
tracking studies, values of all six sextupole components were varied
independently to minimize the overall emittance growth through the
chicane. It was found that if all sextupole components are
proportional to the corresponding quadrupole components (the preferable
technical choice), the emittance growth is close to its minimum value.
Such a choice required only one additional type of quadrupole, and
therefore it was adopted. The optimal ratio of sextupole to quadrupole
components is $S/G=0.355~ \text{m}^{-1}$, which corresponds to a 7\%
correction of quadrupole gradient at a radius of 20~cm.
Figure~\ref{fig:acc:emitvar} depicts the beam envelopes and the beam
emittances normalized to the initial linac energy,
$\varepsilon\gamma\beta/\gamma_0\beta_0$. The
horizontal emittance grows by 13\% and the vertical by 3\%, with no
losses. Maximum horizontal beam size occurs at the last chicane
triplet and is equal to $\pm19\text{ cm}$.

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f23.eps}
  \caption{Layout of the injection chicane at the separation point.}
  \label{fig:acc:chicsep}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270]{../template/report/ps-and-eps/accel-010307-f24.eps}
  \caption{Cross section of the injection chicane at the separation point.}
  \label{fig:acc:injxsecsep}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270]{../template/report/ps-and-eps/accel-010307-f25.eps}
  \caption{Cross section of the injection chicane at a focusing quadrupole.}
  \label{fig:acc:injxsecfoc}
\end{figure}

\begin{table}[!tb]
  \caption{Parameters of the injection chicane quadrupoles.}
  \label{tab:acc:injquad}
  \begin{center}
    \begin{tabular}{|l|c|c|c|c|c|}
      \hline
      &Number of&Maximum&Length&Aperture&Built-in\\
      &magnets&gradient&&&sextupole, $S/G$\\
      &&(T/m)&(m)&(m)&(m$^{-1}$)\\
      \hline
      Focusing quad&3&4&1.40&0.15&0\\
      \hline
      Defocusing quad&14&4&0.75&0.15&0\\
      \hline
      Large aperture quad&4&4&1.40&0.181&0.355\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[!tb]
  \caption{Parameters of injection dipoles.}
  \label{tab:acc:injdip}
  \begin{center}
    \begin{tabular}{|l|c|c|c|c|c|}
      \hline
      &Number of&Maximum&Length&Gap&Width\\
      &magnets&field&&&\\
      &&(T)&(m)&(m)&(m)\\
      \hline
      Short dipole&4&1.7&0.6&0.30&0.30\\
      \hline
      Long septum&1&1.7&1.2&0.30&0.38\\
      \hline
      Short septum&1&1.7&0.6&0.30&0.38\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

Figure~\ref{fig:acc:chicsep} shows the injection chicane in the vicinity of
the separation point. To minimize emittance growth, the bend angles of the
chicane dipoles are chosen to be as small as possible.  The separation
is determined by the beam sizes and the space required for the septum
magnet coil.  Figures~\ref{fig:acc:injxsecsep} and
\ref{fig:acc:injxsecfoc} show cross sections of the injector chicane at
the separation point and in the center of the focusing quad. Taking into
account the large apertures of the magnets (about 30~cm) and their
comparatively modest magnetic fields ($<2\text{ T}$), it looks
preferable to use magnets with SC coils but with the field formed by the
cold-iron core.  Such a choice allows compact
magnets and a significant reduction in required power.
Tables~\ref{tab:acc:injquad} and \ref{tab:acc:injdip} present
parameters of dipoles and quadrupoles used for the injection chicane.

\subsection{Solenoid Magnets}
The linac contains several different solenoid magnets to provide
focusing.  The end of the matching section consists of a pair of low-stray-field solenoids with adjustable currents for tuning.  The short
and intermediate modules have a 1-m focusing solenoid with a
1.25-m straight section at each end. Likewise, the long modules have a 1.5~m solenoid with a 1.25-m separation to the first rf cavity cell.

Unlike focusing quadrupoles, focusing solenoids produce a stray field
that reaches far from the magnet.  Superconducting rf cavities are
sensitive to magnetic fields even at the gauss ($10^{-4}$~T) level, so as noted in Section~\ref{LGPandLPL}\}, a
key parameter in the superconducting focusing solenoid design is
getting rid of the solenoid stray field in the rf cavities.  The
approaches one can use to eliminate the stray field in the rf cavities
are described in Chapter~\ref{SCSOLE}.

\section{Recirculating Linac (RLA)}
\subsection{Arc Optics, Spreaders, and Recombiners:  Design Choices}

For multiple practical reasons, horizontal rather than vertical beam
separation was chosen. If 
vertical separation were chosen, one of the drawbacks would be an enormous
vertical aperture of the vertical spreader/recombiner dipole. Furthermore, instead of having to suppress vertical dispersion created by the spreaders and recombiners we can use horizontal separation with no dispersion suppression; it is
matched to the horizontal dispersion of the arc. Finally, to ensure a
compact arc architecture, very short matching sections (three triplets)
were implemented in the spreaders and recombiners.

The 4-pass RLA beam transport system uses a single dipole separation
of beams at the end of each linac to allow independent recirculation
of each pass. Individual recirculation arcs are based on a periodic
triplet-focusing structure, which is a smooth continuation of the linac
focusing. Betatron phase advance per cell is chosen to be $90^{\circ}$,
which is preferable from the point of view of compensating chromatic effects.
The period length (about 11~m) is slightly shorter  than
for the linacs, to achieve the desired small value of $M_{56}$ (about
1.4~m).

Triplet focusing has a few advantages over the FODO
focusing structure. First of all, it has larger distance between
quadrupoles, which significantly simplifies spreader/recombiner design.
Spreading and recombining the beams with a FODO lattice would be
much more complicated. Furthermore, triplet focusing allows
simple and smooth beam envelope matching from linac to recirculating
arc, which is very important for beams with such a large energy
spread. Finally, triplet focusing has only half the
chromaticity of the vertical beam envelope compared with FODO
focusing, so it requires chromatic corrections in only one plane.

Another crucial beam transport issue is to maintain manageable beam
sizes in the arcs. This calls for short cells and for putting
stringent limits on dispersion and beta functions (beam envelope).
Since spreaders and recombiners were chosen in the horizontal plane,
the uniform focusing and lattice regularity is broken in that plane, 
and the horizontal beam envelope requires special attention. On the
other hand, the vertical beam size remains small due to maintaining
uniform focusing (unbroken periodic symmetry) and small beta functions
in that plane.

Furthermore, there is a need for high periodicity and a smooth
transition between different types of optics, \textit{e.g.}, linac-arc-linac, to
alleviate emittance dilution due to chromatic aberrations (second-order-dispersion).
\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010411-f33.eps}
  \caption[Geometric layout of odd numbered arcs.]{Geometric layout
    (bird's eye view) of odd numbered arcs, including
    spreader/recombiner configuration.  Crosses mark locations of
    magnets (dipoles and quadrupoles). All distances are in cm.}
  \label{fig:acc:arcgeom}
\end{figure}

Finally, the required large momentum acceptance necessitates
introduction of a chromatic correction scheme using three sextupole families for the
off-momentum orbit and path length. As in other recirculating linacs,
and unlike storage rings and synchrotrons, correction of betatron
``tunes'' is unnecessary. Figure~\ref{fig:acc:arcgeom} shows a geometric
layout of arcs 1, 3, 5 and 7.

\subsection{Longitudinal Dynamics in Recirculating Linac}

\begin{table}[!tb]
  \caption{Parameters for acceleration in the recirculator.}
  \label{tab:acc:rlaparm}
  \begin{center}
    \begin{tabular}{|lccccc|}
      \hline
      &Kinetic energy&Gang phase&Total energy&Horizontal&Vertical\\
      &&&spread, $2\Delta p/p$&acceptance&acceptance\\
      &(GeV)&(deg)&(\%)&(mm-mrad)&(mm-mrad)\\
      \hline
      Entrance&2.480&0&15.0&669&638\\
      \hline
      Arc 1&4.756&-23&11.3&384&350\\
      \hline
      Arc 2&6.884&-23&8.9&292&253\\
      \hline
      Arc 3&9.017&-23&6.7&244&202\\
      \hline
      Arc 4&11.140&-23&5.8&216&171\\
      \hline
      Arc 5&13.284&-20&5.0&198&150\\
      \hline
      Arc 6&15.462&-16&4.4&187&134\\
      \hline
      Arc 7&17.690&-5&3.4&178&122\\
      \hline
      Exit&20.000&&3.2&157&108\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f26.eps}
  \caption[Boundaries of longitudinal phase space in the
  recirculator.]{Boundaries of the beam longitudinal phase space at
    different locations in the recirculator; $M_{56}=1.4~\text{m}$.}
  \label{fig:acc:rlalong}
\end{figure}

The beam bunch length and energy spread are still too large at the RLA input,
and further compression is required in the course of
 the acceleration. To achieve this, the beam is accelerated off-crest with
non-zero $M_{56}$ (momentum compaction). This causes synchrotron
motion, which suppresses the longitudinal emittance growth related to
nonlinearity of the accelerating voltage. Without synchrotron motion, the
minimum beam energy spread would be set by the nonlinearity of the rf
voltage at a value of $(1-\cos\phi)\approx6\text{\%}$ for a bunch length of $\phi=20^\circ$. 

Synchrotron motion causes particle motion within the bunch and
equalizes the total average energy gain of particles in the tail and particles in the core. The parameters of acceleration are presented
in Table~\ref{tab:acc:rlaparm}, and corresponding boundaries of
longitudinal phase space are presented in
Figure~\ref{fig:acc:rlalong}. We chose to have approximately the same $M_{56}$ for all the arcs; the optimum value is about 1.4~m.
Optimal detuning of the rf phase from the on-crest position is different for
different arcs. Although longitudinal motion is still
quite nonlinear, it is possible to reduce the energy spread by a factor of 4.7,
to $\pm1.6\text{\%}$, with emittance dilution of about 75\%.  In
these calculations for arc optics we
assume that the longitudinal displacement is a function of momentum
only, given by the linear function, $\Delta L/L=M_{56}\Delta p/p$. The
horizontal and vertical acceptances of the arcs listed in
Table~\ref{tab:acc:rlaparm} are presented with emittance dilutions of
9\% to 4\% per arc. This is supported by preliminary
tracking results.  Final details of the beam dynamics depend on the beam
transport optics and can be determined only by tracking (discussed
below).

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f27.eps}
  \caption[Boundaries of longitudinal phase space for first and last
  bunch.]{Boundaries of the beam longitudinal phase space at the end
    of the recirculator for the first bunch (line with crosses) and
    the last bunch (solid line).  $M_{56}=1.4\text{ m};$ the energy droop
    of 0.5\% per pass corresponds to $3\times10^{12}$ muons in the
    bunch train.}
  \label{fig:acc:loading}
\end{figure}

Since the beam intensity is high, beam loading has to be taken into
account. It causes the rf voltage to droop by about 0.6\% per pass,
yielding about 2.4\% loss in acceleration for the tail bunch of the
last pass. This is comparable with the energy aperture of the later arcs and
their optics tuning must be done with energy droop taken into
account. In this scenario, the first and the last bunches see
different accelerating voltage and experience different longitudinal
dynamics. Fortunately, accelerating off-crest resolves this issue as
well.  In this case, after acceleration in the first linac, the last
bunch experiences less acceleration; but then, because of smaller
energy, the bunch comes faster through the first arc and is accelerated
with smaller rf phase, causing higher acceleration, in the next linac.
In other words, the bunch center of the last bunch experiences
synchrotron motion relative to the center of the first bunch. This 
suppresses the effect of accelerating voltage droop.
Figure~\ref{fig:acc:loading} shows the longitudinal phase space for the
first and the last bunches at the end of the accelerator. The acceleration
has been optimized so that the energy spread of both bunches is
the same. Although the beam loading significantly changes the
bunch shape, the energy droop cannot be seen.

\subsection{General Parameters and Period Layout of RLA Linac}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f28.eps}
  \caption{Layout of an RLA linac period.}
  \label{fig:acc:rlalinac}
\end{figure}

Both RLA linacs have the same period. One period comprises a
cryomodule with four SC cavities and a cryomodule with a quadrupole triplet;
the layout is presented in Figure~\ref{fig:acc:rlalinac}. The design
and parameters of the cavities are the same as for the cavities of
the long cryomodule of the preaccelerator linac (see
Table~\ref{tab:acc:cryoparm}). However, in contrast to the preaccelerator linac,
which has just one cryomodule per period, the RLA period has separate
cryomodules for cavities and for magnets. This solution is preferred due to the increased
length of the RLA period. The design and parameters of the triplets of the
first RLA linac are similar to those of the small triplets of the injection chicane,
but with higher focusing gradients ranging from 3.2 -- 6.7~T/m. 
Quadrupoles of the second RLA linac have a similar design but with 1.5 times
smaller aperture (100~mm).  Their focusing gradient ranges
from 6.2 -- 9.7~T/m.

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f29a.eps}
  \caption{Beta functions for the first pass of the first RLA linac.}
  \label{fig:acc:betafirst}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f29b.eps}
  \caption{Beta functions for the last pass of the first RLA linac.}
  \label{fig:acc:betalast}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f30a.eps}
  \caption{Beam envelopes for the first pass of the first RLA linac.}
  \label{fig:acc:env1first}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f30b.eps}
  \caption{Beam envelopes for the last pass of the first RLA linac.}
  \label{fig:acc:env1last}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f31a.eps}
  \caption{Beam envelopes for the first pass of the second RLA linac.}
  \label{fig:acc:env2first}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010307-f31b.eps}
  \caption{Beam envelopes for the last pass of the second RLA linac.}
  \label{fig:acc:env2last}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics*{../template/report/ps-and-eps/accel-010411-f32.eps}
  \caption[Tune diagram for one period of the first pass in the RLA
  linac.]{Tune diagram for one period of the first pass in the RLA
    linac.  Solid line shows tunes where $\beta_x=\beta_y$ for the
    highest energy pass.  The cross shows the chosen tune,
    $Q_x=0.273$, $Q_y=0.204$, and the circle around it corresponds to
    the tune changes corresponding to 10\% energy spread.}
  \label{fig:acc:tunediagram}
\end{figure}

\begin{table}[!tb]
  \caption{Parameters of the RLA linac periods.}
  \label{tab:acc:rlalin}
  \begin{center}
    \begin{tabular}{|l|c|c|}
      \hline
      &Linac 1&Linac 2\\
      \hline
      No. of periods&24&24\\
%      \hline
      Total length of one period (m)&15 &15 \\
%      \hline
      No. of cavities per period&4&4\\
%      \hline
      No. of cells per cavity&2&2\\
%      \hline
      No. of couplers per cavity&2&2\\
%      \hline
      Cavity accelerating gradient (MV/m)&17 &17 \\
%      \hline
      Aperture in cavities, $2a$ (mm)&300&300\\
%      \hline
      Aperture of quadrupole, $2a$ (mm)&300 &200 \\
%      \hline
      Focusing quad length (m)&1.4 &1.4 \\
%      \hline
      Defocusing quad length (m)&0.75 &0.75 \\
%      \hline
      Quadrupole gradient (T/m)&3.2--6.7 &6.2--9.7 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

Figures~\ref{fig:acc:betafirst}--\ref{fig:acc:env1last} show the
beta-functions and beam envelopes for the first and the last passes in
the first RLA linac. Beta functions of the first pass in the second
RLA linac are the same as for the first RLA linac.  The last pass beta functions
of the second RLA linac are smaller than the corresponding beta functions of
the first RLA linac because of a smaller energy difference between the last and
the first passes. Figures~\ref{fig:acc:env2first} and
\ref{fig:acc:env2last} show the beam envelopes for the first and the
last passes in the second RLA linac. The difference between the
vertical and horizontal beam sizes for the last pass is related to a
larger horizontal emittance resulting from higher horizontal emittance
growth. The focusing structure for both linacs is chosen 
to give the same betatron phase advance per cell for the first-pass
beam. The requirement to have similar horizontal and vertical
beta functions for the higher passes determines that the horizontal
and vertical phase advances are not equal.
Figure~\ref{fig:acc:tunediagram} shows the line on the tune diagram
where the horizontal and vertical beta functions are approximately
equal for the last pass of the first RLA linac.  Parameters of linac
periods are presented in Table~\ref{tab:acc:rlalin}.

Each cavity cryomodule has vacuum valves at both ends and is
delivered to the tunnel under vacuum. These valves are slow;
it is not feasible to build a sufficiently fast valve to
prevent a major vacuum failure in a chamber of so large an 
aperture. Therefore each linac is separated from the arcs with 0.5-mm
beryllium windows. This approach also resolves the question of differential
pumping between the high vacuum in the RLA linacs and the low vacuum in the arcs,
which otherwise would be a major issue with a vacuum chamber of such large
aperture. The design and size of the windows are similar to the beryllium
windows used for the ionization cooling. Altogether there are 5
windows: one in the injection chicane, and one at each end of both
RLA linacs. Multiple scattering causes the total emittance growth of
about 5\% for windows of 0.5-mm thickness. The contribution to
emittance growth from different passages through the windows is almost
even: the beam of higher energy experiences smaller scattering, but it
has a proportionately larger beta function in the RLA.

\subsection{Beam Dynamics in the RLA Linacs}

\begin{table}[!tb]
  \caption{Acceptable nonlinear fields of quadrupoles.}
  \label{tab:acc:nonlin}
  \begin{center}
    \begin{tabular}{|c|c|c|c|}
      \hline
      &4th order&6th order&8th order\\
      &(octupole)&&\\
      \hline
      $\left|F_n\right|$&$<0.015$&$<0.02$&$<0.02$\\
      \hline
    \end{tabular}
  \end{center}
\end{table}

To choose a working point we took into account the following
considerations. First, due to the symmetry of the quadrupole field, the lowest
nonlinearity of its field has sixth-order symmetry, so we would
like to avoid sixth-order resonances. Second, the beam size
should be close to its minimum for a given period length. Third, the
chromaticity of the beam envelopes should be minimized. The chosen tunes
of $Q_x=0.273$, $Q_y=0.204$ satisfy the above requirements. For higher
passes, the phase advance per cell is not constant and grows from the
beginning to the end of the linac, causing the tune to cross a few
resonances. The most sensitive is the second pass, during which tunes cross
the sixth-order resonances. Nevertheless, the tracking showed that
all higher passes are less sensitive to quadrupole nonlinearity than the
first pass. The first RLA linac is more sensitive to quadrupole
nonlinearity because it has larger beam size for all passes. Therefore, it
sets the limit for acceptable quadrupole nonlinearity. For
simulations we assumed that nonlinear terms are proportional to the
quadrupole gradients. The nonlinearity is described by the parameter
\begin{equation}
  F_n = \dfrac{1}{Ga}\dfrac{a^{n-1}B_n}{n!},
\end{equation}
which determines a relative correction of the gradient at the
reference radius $a$. Table~\ref{tab:acc:nonlin} summarizes results of simulations
performed for the first RLA linac. The values correspond to a
reference radius of 10~cm, which is close to the beam envelope in the
focusing quadrupoles. We find that the requirements for quadrupole
nonlinearity are very modest and should be easily met.  Summarizing, we can conclude that an
accuracy of the quadrupole field integral better than 1\% at the reference
aperture of 100~mm is sufficient. With this assumption, 
tracking in the linacs did not exhibit any significant emittance
growth, demonstrating that the periodicity
of motion is not badly broken in the arcs.

\begin{figure}[!tb]
  \centering\includegraphics{../template/report/ps-and-eps/accel-010307-f33.eps}
  \caption[RF phase for different passes through the RLA linacs.]{RF
    phase for different passes through the first (left) and second
    (right) RLA linacs.  The solid line is pass~1, the dotted line is
    pass~2, the dashed line is pass~3, and the dot-dashed line is
    pass~4.}
  \label{fig:acc:phaseslip}
\end{figure}

As was already mentioned, there is a significant rf phase slip for
the beam during different passes because of different particle velocities
at different energies. Figure~\ref{fig:acc:phaseslip} presents rf
phases for the beam at different passes, assuming that the cavity
phases are set so that the second-pass beam is on crest. We see
that the first-pass beam in the first RLA linac has phase variations
in the range of about $-19$ to $12^{\circ}$. This reduces the effective
accelerating gradient by 1.2\%, but it does not otherwise produce any significant
effect for higher passes.

\subsection{Odd Arcs: Proof-of-principle Lattice Design}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010411-f34.eps}
  \caption[Arc 1 optics.]{Arc 1 optics.  Beta functions and the horizontal
    dispersion for the end of linac 1 and arc 1 (matched to both
    adjacent linacs).}
  \label{fig:acc:arc1}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010411-f35.eps}
  \caption[Arc 2 optics.]{Arc 2 optics.  Beta functions and the horizontal
    dispersion for the end of linac 2 and arc 2 (matched to both
    adjacent linacs).}
  \label{fig:acc:arc2}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010411-f36.eps}
  \caption[Arc 3 optics.]{Arc 3 optics.  Beta functions and the horizontal
    dispersion for the end of linac 3 and arc 3 (matched to both
    adjacent linacs).}
  \label{fig:acc:arc3}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010411-f37.eps}
  \caption[Arc 5 optics.]{Arc 5 optics.  Beta functions and the horizontal
    dispersion for the end of linac 5 and arc 5 (matched to both
    adjacent linacs).}
  \label{fig:acc:arc5}
\end{figure}

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=\textwidth]{../template/report/ps-and-eps/accel-010411-f38.eps}
  \caption[Arc 7 optics.]{Arc 7 optics.  Beta functions and the horizontal
    dispersion for the end of linac 7 and arc 7 (matched to both
    adjacent linacs).}
  \label{fig:acc:arc7}
\end{figure}

Lattices for four odd arcs and one even arc of the RLA are illustrated in
terms of the beta functions and dispersion in
Figs.~\ref{fig:acc:arc1}--\ref{fig:acc:arc7}. Short matching
sections in spreaders and recombiners (consisting of six quadrupoles) allow
us to match all Twiss functions and to smoothly join  regions of
different optics of the adjacent linacs. 

The number of periodic cells in the arc was chosen and tuned so that
the desired value of momentum compaction factor required for optimum
longitudinal phase space compression ($M_{56}=1.4\text{ m}$) is built
into the arc optics.

Similar optics for one full turn in the RLA (linac1-Arc1-linac2-Arc2),
as illustrated in Figures~\ref{fig:acc:arc1} and \ref{fig:acc:arc2},
was put to the test for beam transport properties in Study I. A 
multi-particle simulation was carried out for a particle distribution having a realistic large momentum
spread (10\%). The particle distribution used for
tracking was Gaussian in 6D phase space with the
tails of the distribution truncated at $2.5\sigma$, 
corresponding to the beam acceptance presented in
Table~\ref{tab:acc:rlaparm}.  Despite the large initial energy spread,
particle tracking through the linac does not exhibit any significant
emittance growth. There was less than 0.5\% beam loss, coming mainly from
particles at the longitudinal phase space boundary. The simulation
also showed that the chromatic corrections via three families of
sextupoles in the spreaders/recombiners are a very effective means of
emittance dilution control.

%\end{document}