\section{Drift Sections}
\label{DandPR:drift}
%\centerline{Harold  Kirk}

%\centerline{\it Brookhaven National Laboratory}
%\subsection{Introduction}

A principal strategy for the drift sections of the capture/decay
channel is to avoid the $\pi$-resonances that will be present due to the
necessary periodic structure of the solenoidal magnetic field (resulting
from gaps between the superconducting coils).  Examples of these resonances 
are located at the minimum (odd-$\pi$) and maximum (even-$\pi$) points
shown in Fig.~\ref{CPR:fig401}.  For drift sections, these n$\pi$-resonance 
points are approximated by
\begin{equation}
p = \lambda  \frac{Bc}{2\pi n}
\end{equation}
where $p$ is in units of eV/$c,$ $B$ is the average solenoidal field in Tesla, 
$c$ is m/s and the period, $\lambda,$ is in meters.

\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in,angle=-90]{../template/report/ps-and-eps/resonances.ps}}
\caption{Location of n$\pi$-resonances in a periodic solenoidal field.}
\label{CPR:fig401}
\end{center}
\end{figure}

Particle losses in a 3~T periodic solenoidal system are confirmed in Fig.~\ref{CPR:fig411}
where the spectrum of surviving particles after a 50~m drift is compared with the
spectrum of the source particles.  The particle tracking for this example is done
with ICOOL.  Note the appearance of particle losses in the momentum band of 150--200 MeV/$c$ which agrees well with the predictions seen in Fig.~\ref{CPR:fig401}.
Hence, if we wish to avoid particle
losses in the momentum region of 100 to 300~MeV/$c$
then a channel based on a 1.5~T solenoidal field and a 1~m period should be suitable.  

\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=4.5in]{../template/report/ps-and-eps/harold-3T_1m.eps}}
\caption{Particle losses after a 50~m drift in a 3~T, 1~m periodic solenoidal field}
\label{CPR:fig411}
\end{center}
\end{figure}

We choose as the baseline for our decay channel the parameters
$B=1.25$~T and a period of 0.5~m.  We extend this periodicity throughout the capture
channel to include also the induction linac section, so that only the minicool section, with its single-flip solenoidal field does not exhibit this 0.5~m periodicity.  Using
ICOOL, we have compared the results of transporting the MARS-generated particles at the
target through the exit of the third induction linac for both the case of 0.5~m periodicity
and an artificial constant 1.25~T solenoidal field throughout the channel (excluding the
minicool segment). We find that the total throughput of muons at the exit of the
third induction linac is the same for both cases.

%\end{document}