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\begin{document}

\title{Scaling FFAG accelerator for muon acceleration}

\classification{29.20.dk}
\keywords      {Scaling FFAG; transport line; insertion; dispersion suppressor; PRISM.}

\author{JB. Lagrange}{
  address={Graduate School of Engineering, Kyoto University Katsura, Nishikyo-ku\\
Kyoto, 615-8530,
Japan}
}

\author{T. Planche}{
   address={Graduate School of Engineering, Kyoto University Katsura, Nishikyo-ku\\
Kyoto, 615-8530,
Japan}
}
\author{Y. Mori}{
  address={Kyoto University Research Reactor Institute, Department of Nuclear Science and Engineering, 2, Asashiro-Nishi, Kumatori-cho,
Sennan-gun, Osaka, 590-0494, Japan}
}


\begin{abstract}
Recent developments in scaling fixed field alternating gradient (FFAG) accelerators have opened new ways for lattice design, with straight sections, and insertions like dispersion suppressors. Such principles and matching issues are detailed in this paper. An application of these new concepts is presented to overcome problems in the PRISM project.
\end{abstract}

\maketitle

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\section{Introduction}

Due to their large emittance, muon beams need to be manipulated by systems with large longitudinal and transverse acceptances. The scaling type of Fixed Field Alternating Gradient (FFAG) accelerator has several good features to deal with these large 6D emittances beams. Since the chromaticity is null, particles can be kept far away from harmful resonances. Moreover, since the first order dependence of the time of flight on the transverse amplitude is proportional to the chromaticity \cite{berg}, scaling FFAGs show little longitudinal amplitude growth for large transverse amplitudes. The use of this type of ring has already been proposed for muon acceleration \cite{kuno,chepsba}.

To go further, recent developments in scaling FFAGs have opened new ways for lattice design~\cite{pac09, ffag09}. Indeed, it is possible to guide particles with no overall bend in scaling FFAGs. Different types of cells can be combined in the same lattice. It offers possibilities in terms of shape for rings and transport lines, and also leads to the creation of insertions, such as dispersion suppressors.
These new concepts can be applied to overcome problems like in the PRISM project.

\section{Advanced scaling FFAG}

\subsection{Straight sections and circular sections}
The classical case of scaling FFAG is a circular section where the general form of the vertical component of the magnetic field $B$ in the mid-plane is \cite{Symon,Kolomenskii}
\begin{equation}  \label{lawbend}
B=B_0 \left(\frac{r}{r_0}\right)^k \mathcal{F}(\theta),
\end{equation}

with $r$ and $\theta$ the polar coordinates, $B_0$ the magnetic field at the radius $r_0$,  $\mathcal{F}$ an arbitrary periodic function, and $k$ the geometrical field index.
The periodic dispersion function $D$ for a given momentum $p_0$ is then defined as
\begin{equation}
D_{circ.}(p_0) = p_0\left( \frac{\partial r}{\partial p}\right)_{p_0} = \frac{r}{k+1}.
\end{equation}

But scaling FFAG sections can also be imagined with no overall bend. The general form of the vertical component of the magnetic field $B$ in the mid-plane becomes \cite{pac09, ffag09}
\begin{eqnarray} \label{B}
B=B_0e^{m(X-X_{0})}\mathcal{F}(s),
\end{eqnarray}
where $X$ is the cartesian abscissa, $s$ the curvilinear abscissa,  $B_0$ the magnetic field at the abscissa $X_0$, $\mathcal{F}$ an arbitrary periodic function and $m$ the normalized field gradient.
The periodic dispersion function $D$ for a given momentum $p_0$ is then defined as
\begin{equation}
D_{str.}(p_0) = p_0 \left( \frac{\partial X}{\partial p}\right)_{p_0} = \frac{1}{m}.
\end{equation}

FFAG straight sections could be used with circular FFAG sections, but since field laws in each section are different, it will occur a discontinuity of reference trajectories at the border of these two sections. In consequence,  to combine a straight section and a circular section, after matching a special momentum $p_0$,  dispersion can be matched with

\begin{equation} 
m=\frac{k+1}{R_0},
\end{equation} 
with $R_0$ the radius of the closed orbit for the momentum $p_0$ at the border of the cell in the circular section.
But this matching is done only to the first order in $\frac{R-R_0}{R_0}$, with $R$ the radius of the closed orbit for a momentum $p$ at the border of the cell in the circular section. Higher orders effects create a reference trajectory mismatch for momenta other than $p_0$ \cite{pac09}. By a proper choice of $p_0$, the mismatch can be minimized for the considered momentum range. 

An example has been computed by simulating the insertion of straight sections in the 150~MeV FFAG ring of KURRI, for kinetic energies between 20~MeV and 150~MeV. The mismatch $x$ as a function of kinetic energy is presented in Fig.~\ref{matching}. This maximum mismatch is around one~cm. for this case. 

%  FIGURE MATCHING
\begin{figure}[h]
    \centering
 \includegraphics[width=50mm]{matching.eps}
 \caption{Mismatch of reference trajectories between circular cells and straight cells in 150~MeV FFAG ring example. \label{matching}}
\end{figure}

In the same way, it is possible to match two circular FFAG sections with different radii $r_1$ and $r_2$. The dispersion matching condition is kept by adjusting the geometrical field index:
\begin{equation} 
\frac{k_1+1}{r_1}=\frac{k_2+1}{r_2}.
\end{equation} 
Once the reference trajectories are matched, the periodic beta-functions of the cells have also to be matched to limit the amplitude of the betatron oscillations. If a correct matching is not achievable, then an insertion with a phase advance multiple of 180~deg. can be done for one of the two different types of cells. This insertion becomes thus transparent, limiting the betatron oscillations.
%
%Dispersion Suppressor
%
\subsection{Dispersion Suppressor}
In FFAGs, dispersion suppressors can be useful at the end of a transport line, or to reduce excursion where rf cavities are set in FFAG rings. The effect of a dispersion suppressor would be to decrease the excursion, i.e. to bring closer  the reference trajectories around a matched one. This excursion reduction can even be a complete suppression (see Fig.~\ref{disup}).

A principle of a dispersion suppressor in scaling FFAGs is presented in Fig.~\ref{disup}.  The components of this scheme are three types of scaling FFAG cells, straight or circular. 
The area $1$ contains FFAG cells with a dispersion $D_1$ at the border, the area $2$, constituting the dispersion suppressor itself, contains FFAG cells with a dispersion $D_2$ at the borders, and the area $3$ contains FFAG cells with a dispersion $D_3$ at the border. The conditions to have a dispersion suppressor are a phase advance of 180~deg. for the cells of the area $2$ and the dispersion $D_2$ has to verify
\begin{equation} 
D_2 = \frac{D_1 + D_3}{2}.
\end{equation}
This principle is based on the linear theory, so is valid as long as the effect of non-linearities is negligible.
%  FIGURE DISUP
\begin{figure}[h]
    \centering
 \includegraphics[width=60mm]{disup.eps}
 \caption{Principle of a dispersion suppressor with scaling FFAG cells. The upper scheme shows the case of a complete suppression of the dispersion, the lower one the case of a partial suppression of the dispersion.   \label{disup}}
\end{figure}



\subsection{Experiment}
In order to confirm and study these new concepts, an experiment is planned at Kyoto University Research Reactor Institute. The H- Linac injector of the FFAG complex in the institute will deliver 3.5~MeV and 7~MeV beams in the FFAG beam line. This beam line (see Fig.~\ref{experiment}) is a straight FFAG section to confirm the exponential field law, and with a 180~deg. phase advance to study the dispersion suppressor principle.
%  FIGURE EXPERIMENT
\begin{figure}[!h]
    \centering
 \includegraphics[width=60mm]{experiment.eps}
 \caption{Scheme of the FFAG beam line for the experiment at KURRI.  \label{experiment}}
\end{figure}

%
% PRISM
%
\section{Application for muons: PRISM}

The PRISM (Phase Rotated Intense Slow Muon beam) project aims to realize a  low-energy muon beam with a high-intensity, narrow energy spread and high purity. For this purpose, a scaling FFAG ring has been proposed~\cite{prism}. Requirements for the FFAG ring include a large transverse and longitudinal acceptance. The original design of the FFAG ring for PRISM is based on 10 identical DFD triplets. If this design fulfills the requirements of acceptance, the excursion is very large and the injection and extraction still remains difficult. To solve this problem, we consider the use of straight cells in the lattice and a new design is proposed (see Fig.~\ref{traj-prism}). Parameters are summarized in Tab.~\ref{tablePRISM}.

%  FIGURE TRAJ
\begin{figure}[!h]
    \centering
 \includegraphics[width=60mm]{traj.eps}
 \caption{Closed orbits of 55~MeV/c, 68~MeV/c and 82~MeV/c  muons $\mu^-$ in the PRISM lattice with straight sections.  \label{traj-prism}}
\end{figure}

% TABLE PARAMETERS PRISM

\begin{table}
\begin{tabular}{lcc}
\hline
  & \tablehead{1}{r}{b}{Cicular\\section}
  & \tablehead{1}{r}{b}{Straight\\section}\\
\hline
Type					& FDF			& FDF \\
 k-value or m-value             & 2.55       & 1.3\,m$^{-1}$         \\
         Radius/Length              & 2.7\,m &  1.8\,m        \\  
          Horizontal phase advance & 60~deg. & 27\,deg. \\
Vertical phase advance & 90~deg.		& 97\,deg. \\
           Number of cells	& 2					& 3	\\
\hline
\end{tabular}
\caption{Parameters of a proposal for a new PRISM lattice}
\label{tablePRISM}
\end{table}


Particle tracking is done using our code in soft edge fields with linear fringe field falloffs. Components of the field off the mid-plane are obtained from a first order Taylor expansion, satisfying the Maxwell equations. 

The original PRISM design has a very large dispersion function ($\sim$ 1.2~m) that makes difficult the injection and the extraction. The new proposal starts then from a smaller one ($\sim$ 0.8~m). After minimizing the mismatch of the beta-functions, the bending part of the ring is made transparent to limit the effect of the remaining mismatch on the amplitude of the betatron oscillations. The resulting beta-functions for a momentum of 68~MeV/c are presented in Fig.~\ref{beta-prism}. The working point is chosen in the tune diagram so that it is far from the structural normal resonances. The present working point has a tune of 2.9 in horizontal and 6.3 in vertical.

%  FIGURE BETA PRISM
\begin{figure}[!h]
    \centering
 \includegraphics[width=50mm]{beta-prism.eps}
 \caption{Horizontal (plain red) and vertical (dotted purple) beta-functions for half of the ring of the PRISM lattice.  \label{beta-prism}}
\end{figure}

The transverse acceptance in both planes is studied by tracking over 30 turns a particle with a displacement off the closed orbit and a small deviation in the other transverse direction (1~mm). Collimators ($\pm$ 1~m in horizontal, $\pm$ 30~cm in vertical) are used to identify the lost particles. The regions drawn by the particle with the largest initial stable amplitude in the horizontal and vertical phase spaces are presented in Fig.~\ref{acceptance-prism}. Horizontal ($\sim 24000~\pi$.mm.mrad) and vertical ($\sim 6000~\pi$.mm.mrad) acceptances are then measured by the area of the biggest ellipse included in this region.

%  FIGURE ACCEPTANCEX PRISM
\begin{figure}[!h]
    \centering
 \includegraphics[width=60mm]{acceptance-prism.eps}
 \caption{Horizontal (left) and vertical (right) phase space. Two particles with an initial displacement of 15~cm and 29~cm (left) and 3.5~cm and 7~cm (right) are tracked in the lattice described in Tab. 1 over 30 turns. The dotted ellipses are the ones used to measure the acceptance in the middle of the straight section.  \label{acceptance-prism}}
\end{figure}

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\bibitem{kuno}
Y. Kuno et al.,``A feasibility study of neutrino factory in Japan", KEK report (2001).

\bibitem{chepsba}
T. Planche,``Scaling FFAG rings for rapid acceleration of muon beams", Nucl. Instr. and Meth. A 622 1 pp. 21 -- 27 (2010).

\bibitem{pac09}
JB. Lagrange et al, ``Straight section in scaling FFAG accelerator'', PAC'09, Vancouver, FR5PFP002 (2009).

\bibitem{ffag09}
JB. Lagrange et al, ``Zero-chromatic FFAG straight section'', FFAG'09, Fermilab (2009).

\bibitem{Symon} 
K. R. Symon \it{et al}., {\it Phys. Rev.} {\bf 103}, 1840-1842 (1956).

\bibitem{Kolomenskii} 
A. A. Kolomensky and A. N. Lebedev, {\it Theory of Cyclic Accelerators}, (North-Holland, Amsterdam, 1966).

\bibitem{prism}
``The PRISM Project - A Muon Source of the World-Highest Brightness by Phase Rotation 
{\it LOI for Nuclear and Particle Physics Experiments at the J-PARC} 217 (2003).

\end{thebibliography}
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\bibitem{Brown2000}
M.~P. Brown,  and K.~Austin, \emph{The New Physique}, Publisher Name,
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\bibitem{BrownAustin:2000}
M.~P. Brown,  and K.~Austin, \emph{Appl. Phys. Letters} \textbf{85},
  2503--2504 (2000).

\bibitem{Wang}
R.~Wang, ``Title of Chapter,'' in \emph{Classic Physiques}, edited by
  R.~B. Hamil, Publisher Name, Publisher City, 2000, pp. 212--213.

\bibitem{SJ:1999}
C.~D.~Smith and E.~F.~Jones,  ``Load-Cycling in Cubic Press,'' in
  \emph{Shock Compression of Condensed Matter-1999}, edited by M.~D.~F. et~al.,
  AIP Conference Proceedings 505, American Institute of Physics, New York,
  1999, pp. 651--654.

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