\documentclass [12pt]{iopart}
\usepackage{graphicx,subfigure}

\begin{document}
\title{Muon front-end chicane and acceleration}
\author{G.H. Rees, S.J. Brooks, M.R. Harold and C.R. Prior}
\address{%
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK}

\begin{abstract}
A muon front-end system is described which includes an initial pion 
to muon decay channel, a reverse phase slip bending chicane and a 
190-400\,MeV, 88\,MHz muon linac. A chicane is preferred to a phase
rotation
system as it leads to a more linear acceleration of the muons in a
shielded, 88\,MHz linac. The design proposed is compatible with the
linac-compressor ring option for the proton driver of the CERN 
neutrino factory scheme.
\end{abstract}    

\maketitle

\section{Introduction}
In the decay channel, the large energy spread of the pion-muon bunches 
causes a shearing in longitudinal phase space. An initial 1\,ns rms 
bunch extends by $\sim$ 70 \% in a 30 m channel for muon
energies of 190 $\pm$ 70\,MeV. The shearing may be reversed, and
the initial beam orientation in longitudinal phase space almost restored, 
by transit through a special bending chicane. The non-linearities in 
the subsequent acceleration are then reduced.

Comparisons may be made with linac phase rotation systems. The bunch
frequency of the protons on target and the emerging pions is 44\,MHz
and a harmonic of this may be chosen for both phase rotation and acceleration.
Use of a sinusoidal 88\,MHz rotation field results in an increase of
bunch area after filamentation. The effect is enhanced for a 44\,MHz
field due to its reduced focusing and the return of non-linearity
when the frequency is raised for linac acceleration. A linear phase
rotation, using a multi-harmonic system or an induction linac, is
not considered as the high bunch frequency leads to added cost and
complexity. A chicane, followed by an 88\,MHz linac, is the preferred
solution, resulting in longitudinal bunch areas of $\sim$0.5\,eV sec.

Regular solenoid focusing is used before and after the chicane and
the total system length is $\sim$130\,m, for no transverse cooling.
There is, however, an initial 20\,T solenoid in the decay channel,
followed by a few adjustable solenoids, to optimize the muon 
distribution from the chicane. Aperture radii are 0.1\,m in the 20\,T unit,
0.3\,m in the following solenoids, then 0.25 m, 0.2 m and finally
0.15 m at the end of the channel and in the muon linac.
The aperture reduction provides some energy and
betatron selection. Chicanes may take $\mu^+$ or $\mu^-$
but the design is for $\mu^+$.

The required reverse phase slip in the 17\,m chicane is obtained by
the choice of a magnet lattice with a gamma-transition of 1.68,
which is below the muon $\gamma$ range. In addition, the lattice
transverse focusing is set for approximate input and output
matching and for achromatism at the centre and end point.
\section{Chicane lattice}
The chicane lattice consists of twelve combined function magnets
arranged in four groups of three, as indicated in Figure~\ref{chicane}. 

\begin{figure}[htb]
\centering{
\includegraphics[bb=170 292 440 686,width=6cm,clip=true]%
{[jvt.muons]muon_chicane.ps}
\caption{\label{chicane}Muon chicane.}}
\end{figure}

In each group, the three magnets bend the muon beam successively either by
64$^\circ$, -27$^\circ$ and 64$^\circ$ or by equal and opposite
amounts. By this means, the first two groups introduce a total
horizontal bend of 202$^\circ$ and the next two, -202$^\circ$. The
axis of the downstream linac is then well displaced from that of
the decay channel so that it may be appropriately shielded. Strong
transverse focusing is obtained using magnets of large input and
output edge angles and small bending radii, but of low gradient,
FFAG, pole profiles. The end field extent and symmetry are
important parameters due to the small bending radii. Two magnet
types are involved, B1 units for 64$^\circ$ bends, B2 for 27$^\circ$,
providing a symmetrical B1-B2-B1 triplet.     

In the lattice design, the magnet lengths, bend angles, edge angles,
separations and pole profiles have been varied to obtain the required
features. These include a $\gamma_t$ value of 1.68 for a 17\,m total
chicane length, a horizontal betatron phase shift of 180$^\circ$ 
for the basic triplet group, and maximum lattice parameters of
$<1.8$\,m ($\alpha_p$) and $<2.0$\,m (peak $\beta$), 
1.5\,m (input $\beta$).
For these maximum values, it is assumed there is no input dispersion
and vertical betatron motion is matched, with the horizontal and
vertical input $\beta$-values assumed equal (for a round beam).

The horizontal-betatron, phase-shift choice provides both for achromatism at
the chicane centre and end points and for periodic horizontal
$\beta$-values over the four triplet groups. In addition,
symmetry for the magnet end fields results in symmetry of the triplet
$\beta$-functions, as shown in Figure~\ref{beta}, which also shows the effect
of the reverse bends on the dispersion. Only half of the chicane is
displayed, but the $\beta$-functions are repeated in the second
half, while the dispersion is inverted. Similar lattice functions
are also obtained with a different triplet, which has the advantage
of a zero gradient B1 magnet, but with the disadvantage of a narrower
pole on its inner radius.

\begin{figure}[htb]
\begin{minipage}{8cm}
\centerline{\small DISPERSION}
\includegraphics[bb=58 135 516 470,width=8cm,clip=true]
{[jvt.muons]muon_chicane_disp.ps}
\end{minipage}\hfill
\begin{minipage}{8cm}
\centerline{\small BETATRON}
\includegraphics[bb=58 135 516 470,width=8cm,clip=true]
{[jvt.muons]muon_chicane_beta.ps}\end{minipage}

\caption{\label{beta}Chicane lattice parameters.}
\end{figure}

A linear lattice code has been used for the initial design,
assuming hard edge magnets, but correcting for end fields. Effects
found due to the extent and asymmetry of the end fields have been 
confirmed by muon tracking studies using the Parmila linac
simulation code. The B1-B2-B1 triplet was then modelled using the
magnet code, Opera-3d, as reported in~\cite{HAROLD}, and the fields have been
entered in a new code, Muonplayer~\cite{BROOKS}, which includes pion decays
and tracks target pions and decay muons through both the decay
channel and the chicane. The code uses accurate expansions for the
solenoid end fields. The two sets of cell parameters are given in 
Table~\ref{parameters}.

\begin{table}[htb]
\caption{\label{parameters}Chicane cell parameters.}
\vskip 3mm
\begin{tabular}{lccc}
\hline \\
Unit & Length (m) & Angle ($^\circ$) & Gradient (m$^{-2}$) \\ \\
B1 & 0.176 & 16.0 & -0.39010 \\
B2 & 0.176 & -9.0 & -0.27678 \\
B1, B2 end & 0.330 & 0.0 & 0.0 \\
B1E edge & 0.000 & 21.275 & 0.0 \\
B2E edge & 0.000 & 13.5 & 0.0 \\
O1 & 0.4935 & 0.0 & 0.0 \\
O2 & 0.170 & 0.0 & 0.0 \\ \\
B1 & 0.176 & 16.0 & 0.0 \\
B2 & 0.176 & -9.0 & -0.13222 \\
B1, B2 end & 0.330 & 0.0 & 0.0 \\
B1E edge & 0.000 & 15.25 & 0.0 \\
B2E edge & 0.000 & 13.5 & 0.0 \\
O1 & 0.4713 & 0.0 & 0.0 \\
O2 & 0.2144 & 0.0 & 0.0
\end{tabular}
\begin{tabbing}
Cell structure: \= B2EX, \= B2EX, \= 4*B1X, \= B1EX, \= B2EX, \kill
Cell C1: \> O1,\> B1E,\> 4*B1,\> B1E,\> O1: \\
Cell C2: \> B2E,\> B2,\> B2,\> B2,\> B2E: \\ \end{tabbing}
\begin{tabbing}
Cell structure: \=  \kill
Cell structure:\> 2*(O2, C1, C2, C1, O2),2*(O2,-C1,-C2,-C1, O2)
\end{tabbing}
\hrulefill
\end{table}
Lattice studies are still continuing, in conjunction with tracking
evaluations. The points at issue are the following: motion of muons
along the inner and outer off-momentum orbits (shown in Figure~\ref{orbits}),
end field symmetry
and curtailment, the choice between the two lattice solutions 
found, and the required modifications of the FFAG pole profiles.
Adjustments for the magnet pole profiles are needed to preserve achromatism
for the far off-momentum orbits.

\begin{figure}[htb]
\begin{center}
\includegraphics[bb=14 14 728 585,height=10cm,clip=true]
{[jvt.muons]off_energy_orbits3.eps}
\caption{\label{orbits}Off-energy orbits.}
\end{center}
\end{figure}

\section{Solenoid focusing}
Regular solenoid focusing is used downstream and just upstream of the
chicane but there is an initial 20 T solenoid in the decay channel,
followed by lower field units.
These are set at the field levels
found by an optimizer in Muonplayer, which seeks a maximum
transmission of muons after the chicane. Betatron matching into and
out of the chicane is not feasible over the entire muon energy range
from 120 to 270\,MeV. Instead, the mis-matching is limited for the
full energy range, as far as is practical.

The solenoids in the 81 m muon linac have an aperture radius of 0.15 m
while those in the 30 m decay channel have 0.1 m in the 20 T unit,
0.3 m
in the following units, then 0.25 m, 0.2 m and finally 0.15 m. Some
momentum and transverse collimation is obtained due to the progressive
aperture reduction ahead of the chicane.
The adjustable solenoids
have to be set differently for an optimum transmission from the
target to the chicane output, in place of its input.
\section{Muon linac}
The 88\,MHz linac for accelerating muons from 190 to 400\,MeV uses 
the cavity design of the CERN front-end scheme. This is a single cell
unit, with a 0.15\,m bore radius, a 0.9\,m length and it includes a 0.4\,m
superconducting solenoid within the inner nose cone. There are
ninety such units in an overall 81\,m length. If cooling is added,
every fifth cavity in the cooling interval is replaced by an 
ionizing unit, but the focusing pattern is retained. The
use of 0.4\,m solenoids at intervals of 0.9\,m is also adopted in the
decay channel.

An accelerating field gradient of 4\,MV/m is proposed ($E_0LT=3.6$\,MV) 
for each cavity, but the synchronous phase varies smoothly
from -66$^\circ$ at 190\,MeV to -39$^\circ$ at 400\,MeV. On leaving
the chicane, the beam has an upright distribution in longitudinal
phase space and there is acceptable non-linearity during
acceleration for the parameters chosen. The energy at which to 
change to a higher accelerating frequency has yet to be decided.
\section{Simulation results}
Pion and muon tracking studies with the Muonplayer code were 
initially made for the decay channel. After lengthy optimizing
of the solenoids, a transmission efficiency of $\sim$7\%
$\mu^+$ per $\pi^+$ was obtained from a 10\,mm radius,
tantalum target, with normalised rms transverse emittances of
$\sim$6500\,($\pi$)\,mm\,mr for the 120 to 270\,MeV $\mu^+$.
This translates to $\sim$0.67\% $\mu^+$ per (proton GeV) for the
2.2 GeV protons on target.

Tracking of muons was then extended into the chicane and 400\,MeV
linac. At first, simulations were made with the Parmila linac code,
using hard edged models for both the chicane magnets and the linac solenoids.
The optimized figure of merit found for muon yields was $\sim$0.51\%
$\mu^+$ per (proton GeV). More accurate simulations, however, with
the Muonplayer code, using field maps from the Opera-3d magnet
models, indicate that the yields are reduced to $\sim$0.25\% $\mu^+$
per (proton GeV).

Studies have now been re-directed to the design of a racetrack
cooling ring in which the two long straight sections have solenoid
focusing and the two arcs are formed from 180\(^{0}\) chicane units,
each formed from two, B1-B2-B1 triplet sets with bending angles
(64\(^{0}\), -38\(^{0}\), 64\(^{0}\)). The muon front end has the
30 m decay channel, as before, but this is followed by a 44 MHz rf
phase rotation channel, which reduces the muon energy range to 180
\(\pm\) 35 MeV for injection into the cooling ring.

\section*{References}
\begin{thebibliography}{99}
\bibitem{HAROLD}M.R. Harold, Magnets for a muon front end chicane, 
Proceedings of this conference, NuFact02.
\bibitem{BROOKS}S.J. Brooks, Muonplayer tracking code, Rutherford 
Appleton laboratory.
\end{thebibliography}


\end{document}
