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\title{Magnets for a muon front-end chicane}
\author{M.R. Harold}
\address{%
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK}
\begin{abstract}
A magnet system for transfer of muons from a pion decay channel
to a linear accelerator is described. The decay channel
is followed by four sets of combined function, triplet magnets,
arranged as an achromatic chicane for 190$\pm$70\,MeV muons.
\end{abstract}
\maketitle

\section{Introduction}
Use of an achromatic bending chicane to give muon beam bunches a
longitudinal, reverse phase-slip is outlined in a companion paper~\cite{REES}.
An optimum slip occurs in a 17\,m magnet lattice if gamma transition
is 1.68 for its four sets of combined function triplets. The basic
three-dipole unit bends the beam successively by 64$^\circ$,
-27$^\circ$ and 64$^\circ$, but the second two sets provide equal and
opposite bends to the first two. Focusing is provided both by
edge-angles and by shaping the poles for an FFAG-type,
vertical, field law, $B=B_0\left(r/r_0\right)^{-n}$. Achromatic
paths result for 120 to 270\,MeV muon energies.

\section{Magnet design}
Parameters of the symmetrical B1-B2-B1 triplet units are given in
Table~\ref{tab1}.

\begin{table}[htb]
\begin{minipage}{7cm}
The 0.5\,m spacing of the magnets is comparable with their 0.3\,m vertical
gap heights and it has thus been important to determine whether
field requirements are met and if the energising coils clash. The
magnet program Opera-3d has been used for separate designs of the B1
and B2 units, with yoke length adjustments made to compensate
for the considerable excess magnetic lengths resulting from the
large apertures. 
\end{minipage}\hfill
\begin{minipage}[][][t]{8cm}
\caption{B1 and B2 magnet parameters.}
\label{tab1}
\begin{tabular}{|l|cc|}
\hline
 & B1 & B2 \\ \hline\hline
Bend angle ($^\circ$) & 64.0 & -27.0 \\
Edge angle ($^\circ$) & 21.275 & 13.5 \\
$B_0$ (Tesla) & 1.4615 & 0.8221 \\
$r_0$ (m) & 0.63026 & 1.12045 \\
$n$ value (negative) & 0.15496 & 0.34746 \\
Radial width (m) & -0.6 to 0.6 & -0.4 to 0.6 \\
Vertical gap (m) & 0.3 & 0.3 \\
\hline
\end{tabular}
\end{minipage}
\end{table}
Return yokes of both magnets have been
increased in size to carry the large fringe fields. When both B1
and B2 magnets are combined in a model of the B1-B2-B1 triplet,
there are no physical clashes found but the simulations show
the fringe fields merging.  

The particle tracking facility of Opera-3d has allowed studies of
the off-energy muon orbits. Triplet magnetic fields have been set
so that a pencil beam of 190\,MeV muons is bent symmetrically 
through the required 101$^\circ$ angle. Pencil beams of muons in the
energy range 120 to 270\,MeV should then enter the triplet at an 
angle of 50.5$^\circ$ to the $z$-axis and emerge as a dispersed,
parallel beam at -50.5$^\circ$, as shown in Figure~\ref{orbits}. Correction
for the low energy muons is obtained by increasing the inside fields 
of both the B2 and B1 units. This has led to poor high energy muon
orbits,
however, because the magnet lengths are short in comparison with
their widths and the steel in B1, in particular, is very highly
saturated.

\begin{figure}[htb]
\begin{minipage}{7.5cm}
\begin{center}
\includegraphics[bb=30 42 550 800,width=7.5cm,angle=-90,clip=true]
{[jvt.muons]magnet_orbits.ps}
\caption{\label{orbits}Off-energy orbits in a chicane triplet.}
\end{center}
\end{minipage}\hfill
\begin{minipage}{7.5cm}
\begin{center}
\includegraphics[bb=30 42 550 800,width=7.5cm,angle=-90,clip=true]
{[jvt.muons]magnets.ps}
\caption{\label{triplet}Chicane triplet arrangement.}
\end{center}
\end{minipage}
\end{figure}

It has proved necessary to change from an FFAG field law and to vary both
the pole profile and the yoke length in the beam direction in order 
to achieve parallelism of the emergent muons. This has resulted in the
required edge angles of the B1 magnet not being maintained, and the
effect on muon beam optics is currently under study. The
emergence angles of the 120 to 270\,MeV muons are within -0.5$^\circ$
and +0.6$^\circ$ of the required -50.5$^\circ$, and magnet shimming
should reduce these errors to the required precision. 

Figure~\ref{triplet} gives
an idea of the appearance of the magnets, though their dimensions
have not been optimised for minimum steel weight. The file FB1B2.OPPRE
has been used for storing magnet modelling data.

Approximate weights and power consumptions are 30 tonnes and 225\,kW
for the B1, and 25\,tonnes and 100\,kW for the B2, respectively,
for 340 tonne and 2200\,kW totals. Work continues on the magnet 
designs, together with detailed muon tracking studies \cite{REES} of
transmission efficiencies. The chicane lattice option proposed \cite{REES},
with a zero gradient in B1 and a reduced gradient in B2, is also 
under consideration.

\section*{References}
\begin{thebibliography}{99}
\bibitem{REES}G.H. Rees et al., Muon front end chicane and acceleration, 
Proceedings of NuFact02.
\end{thebibliography}

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