\section{Chromaticity correction for the linear non-scaling FFAG}

A particle with large betatron amplitude has a different
time-of-flight because the path length is longer. 
The effect exists in any ring accelerator, but turns out to be the
crucial issue in a muon FFAG for the following reasons.
Firstly, the transverse emittance of a muon beam is huge, even after
ionisation cooling. 
Secondly, a muon FFAG accelerates the beam outside the RF bucket
with an almost isochronous lattice.  
Energy gain is sensitive to the RF phase at each turn since there are
no synchrotron oscillations which average the different energy
gains at each turn.  
As a result, the longitudinal emittance blows up significantly after
acceleration in a linear, ns-FFAG \cite{Machida:2006sj}.  
Furthermore, a scheme to use an FFAG cascade to boost the muon energy
is not possible. 

It was found that chromaticity correction mitigates the
problem \cite{Berg:2007}. 
Sextupole fields introduce amplitude-dependent revolution orbits, which
cancel the increase of time-of-flight experienced by large amplitude
oscillations. 
In a chromaticity-corrected lattice as shown in figure
\ref{fig:app:acc:ffag:1}, longitudinal-emittance blowup can be
eliminated as shown in  figure \ref{fig:app:acc:ffag:2}.
\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{06-Appendix/02-05-FFAG/fig_c1.eps}
  \caption{Tune as a function of momentum for a ns-FFAG
    without (dashed lines/open circles) and with (solid lines/filled circles)
    sextupole components to correct chromaticity.}
  \label{fig:app:acc:ffag:1}
\end{figure}
\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{06-Appendix/02-05-FFAG/fig_c2.eps}%
  \includegraphics[width=0.5\linewidth]{06-Appendix/02-05-FFAG/fig_c3.eps}
  \caption{Longitudinal phase space on successive turns with (left) and
    without (right) chromaticity correction.}
  \label{fig:app:acc:ffag:2}
\end{figure}

On the other hand, there is a price to be paid.
In the chromaticity corrected lattice, the orbit shift due to
acceleration is large, as shown in figure \ref{fig:app:acc:ffag:4}.
This directly increases the magnet aperture and the cost. 
Another side-effect is the asymmetry of the time-of-flight curve shown
in figure \ref{fig:app:acc:ffag:4}. 
This implies an increase of total RF voltage.  
Finally, the most significant impact of the chromaticity correction is
the reduction of the dynamic aperture. 
Non-linearity introduced to correct chromaticity makes the dynamic
aperture smaller than 30~mm (dynamic aperture is defined the maximum
transverse amplitude which can survive during acceleration).
As a result of these adverse effects, chromaticity corrections are
not included in the present muon FFAG lattice.
\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{06-Appendix/02-05-FFAG/fig_c4.eps}%
  \includegraphics[width=0.5\linewidth]{06-Appendix/02-05-FFAG/fig_c5.eps}
  \caption{
    Left: closed position in the long straight (LS) versus momentum
    without (dashed lines, open circles) and with (solid lines, filled
    circles) chromaticity correction.  
    Right: time of flight as a function of momentum without and with
    chromaticity correction.
  } 
  \label{fig:app:acc:ffag:4}
\end{figure}
