\section{Overall Design Motivation}
The storage ring is the part of the Neutrino Factory that delivers the
neutrino beam to the detector.  As such, its effectiveness can be
defined by the ratio of the number of neutrinos aimed in
the direction of the detector to the total number of muons injected
into the storage ring.  Since we consider here sending neutrinos
to a single detector, we use the storage ring geometry
shown in Fig.~\ref{fig:sr:RingIntro}.  The straight sections are
aligned parallel to a line from the storage ring to the detector.
Thus, the muons decaying in one of the two straight sections
(the ``production'' straight)
contribute to the neutrinos headed toward the detector, while muons
decaying in the rest of the ring do not.  The
effectiveness $f_S$ of the storage ring is the ratio of the length of
the production straight section to the circumference of the storage ring:
\begin{equation}
  f_S = \dfrac{L_S}{C} = \dfrac{L_S}{2(L_S+L_A)},
  \label{eq:sr:decayfrac}
\end{equation}
where $L_S$ is the length of one straight section, and $L_A$ is the
length of $180^\circ$ of arc.
\begin{figure}[!bt]
  \centering \includegraphics[width=\textwidth]{../template/report/ps-and-eps/RingIntro.eps}
  \caption{Storage ring diagram.}
  \label{fig:sr:RingIntro}
\end{figure}

In the case of a detector at the WIPP facility, the storage ring must
be tilted $13.1^\circ$ from horizontal to have a straight section
pointing toward the detector.  It follows that there must be a
substantial elevation change from the top of the ring to the bottom.
For this study, we have constrained ourselves to keep the bottom of
the storage ring tunnel at least 10 feet above the water table.  At
the Brookhaven site, the water table averages 48 feet above sea level,
with a seasonal variation of $\pm5\text{ feet}$.  Thus, the bottom of
the storage ring tunnel will be at 63 feet above sea level.  The
highest ground elevation on the Brookhaven site where the storage ring
could be realistically placed is about 90 feet above sea level.  Thus,
there is not enough room to keep the entire storage ring underground,
and a hill is needed.  For large $L_S$, the height of the
hill increases linearly with $L_S$, and the volume of fill required
for that hill increases as $L_S^3$.  Thus, there is an economic
incentive to keep the ring circumference small.

\begin{figure}[!tb]
  \centering
  \includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/DecayFraction.eps}
  \caption[Fraction of decays in a straight \textit{vs.}\ $L_S/L_A$.]{Fraction
    of decays in a straight ($f_S$) as a function of
    the ratio of the length of a straight to the length of an arc.}
  \label{fig:sr:decayfrac}
\end{figure}
One can see from Eq.~(\ref{eq:sr:decayfrac}) that $f_S$ depends only
on the ratio $L_S/L_A$.  That dependence is plotted in
Fig.~\ref{fig:sr:decayfrac}.  One can conclude two things from Fig.~\ref{fig:sr:decayfrac}.
First, for a given $f_S$, the shorter the arc is, the shorter the straight
section can be.  Second, beyond $L_S/L_A\approx2$,
it takes a very large change in $L_S/L_A$ to increase $f_S$ by
even a small amount.

We summarize in Table~\ref{SRING:tb} the values of some parameters of the storage ring.
\begin{table}[tb]
\begin{center}
\caption{Muon storage ring parameters.}
\label{SRING:tb}
\begin{tabular}{|lc|}
\hline
Energy (GeV) & 20 \\ 
Circumference (m) & 358.18 \\ 
Normalized transverse acceptance (mm-rad) & 30 \\ 
Energy acceptance (\%) & 2.2 \\ \hline
\multicolumn{2}{|c|}{Arc} \\ \hline
Length (m) & 53.09 \\ 
No. cells per arc & 10 \\ 
Cell length (m) & 5.3 \\ 
Phase advance ($\deg $) & 60 \\ 
Dipole length (m) & 1.89 \\ 
Dipole field (T) & 6.93 \\ 
Skew quadrupole length (m) & 0.76 \\ 
Skew quadrupole gradient (T/m) & 35 \\ 
$\beta _{\text{max}}$ (m) & 8.6 \\ \hline
\multicolumn{2}{|c|}{Production Straight} \\ \hline
Length (m) & 126 \\ 
$\beta _{\text{max}}$ (m) & 200 \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Design Choices for Optimizing Arcs}

It is clear from Fig.~\ref{fig:sr:decayfrac} that it is beneficial
to minimize the arc length.
Whereas Study-I focused on a storage ring with an energy of
50~GeV, here we consider a 20 GeV storage ring,
at least partly for this reason. (there are several other
reasons, related to machine cost and the insensitivity of CP violation
physics to the beam energy, that contributed to this decision as well.)

\begin{figure}[!tb]
  \centering\includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/saggitta.eps}
  \caption[Photograph of a pancake coil.]{Photograph of a pancake coil.  This
    also illustrates a method to provide sagitta and reverse bend
    using Kevlar strings.}
  \label{fig:sr:pancake}
\end{figure}
One way to decrease the arc length of the storage ring is to use
high-field bending magnets.  $\text{Nb}_3\text{Sn}$
superconductor can achieve very high fields, but due to its brittle nature,
it is difficult to wind a $\cos\theta$ magnet
using it.  However, winding a pancake-type coil
(Fig.~\ref{fig:sr:pancake}) is not a problem with
$\text{Nb}_3\text{Sn}$.  For this reason, we have chosen to use
$\text{Nb}_3\text{Sn}$ pancake coil magnets in the arcs.

Another consideration for the arc magnet design is the decay of the
muons.  The superconducting magnet coils must be shielded from the
decay electrons, which remain primarily in the horizontal plane of the
beam.  For a $\cos\theta$ magnet, this would be accomplished by
putting a tungsten shield inside the magnet, increasing the required
magnet aperture.  The pancake coil configuration has the advantage
that it can be designed with no coil in the midplane, eliminating the
necessity for coil shielding.

\begin{figure}[!tb]
  \centering
  \includegraphics[width=\textwidth]{../template/report/ps-and-eps/ArcCell.eps}
  \caption[Demonstration of the effect of inter-magnet spacing on arc length.]
  {Demonstration of the effect of inter-magnet spacing on arc
    length.  The top drawing (a) is the arc cell for a 50 GeV lattice from
    the Fermilab design study.  Scaling that lattice to 20~GeV,
    but leaving the inter-magnet spacing fixed,
    does not reduce the cell length to 40\% of the original
    length (b).  In (c), we see that
    using combined-function instead of separated-function magnets can
    reduce the cell length substantially.  In (d),
    we show
    what happens if the inter-magnet spacing is completely
    eliminated.}
  \label{fig:sr:arccell}
\end{figure}
One of the primary obstacles to reducing the arc length is the
required spacing between magnets.  As demonstrated in
Fig.~\ref{fig:sr:arccell}, if one simply scales the magnets from the
50~GeV storage ring in Study-I by 40\%, the arc is
in fact longer than 40\% of the 50~GeV arc length, since the
inter-magnet spacing must remain roughly the same.  Indeed,
it might be necessary to increase the inter-magnet spacing due to larger beam
sizes, and thus larger magnet aperture, at the lower energy.
Some gain can be achieved by eliminating some of the gaps by using
combined-function magnets.  Ideally, it would be best to eliminate the
inter-magnet gaps altogether.  The pancake coil design achieves this
by using coil configurations where one coil (of two) continues
through each transition between magnets.

\subsection{Choice of Straight Length}

Once the length of the arc is minimized, the length of the straight
determines the number of neutrinos decaying in the direction of the
detector.
For this study, there is an added consideration: a hill
must be built to accommodate the storage ring.
Thus, we chose a goal of $f_S=0.35$.  In
the design a straight length of $L_{\textrm{S}}=126$~m was chosen along with an
arc length of $L_A=53.09\text{ m}$.  This requires a hill
with a height of 43~m and having $6.4\times10^5\text{ m}^3$ of
fill.

\subsection{Lattice Parameters}
\label{sec:sr:latparm}
\begin{table}[!tb]
  \caption{Input beam parameters.}
  \label{tab:sr:beam}
  \centering
  \begin{tabular}{|lc|}
    \hline
    Energy (GeV)&20\\
    Normalized transverse acceptance (mm-rad)&30\\
    Ratio of full width to rms width&2.4\\
    Relative momentum spread (full) (\%)&$\pm 1.9$\\
    \hline
  \end{tabular}
\end{table}
The input beam parameters are given in Table~\ref{tab:sr:beam}.  There
two primary issues related to these parameters that need to be
dealt with.  First, since the magnets operate at very high field, their
aperture should be kept small to minimize peak fields
and maintain field uniformity.  Second, to give a reasonable
uncertainty at the detector, the angular spread in the beam must
be kept well below the angular spread in the neutrinos due to decays.
Specifically, we require that $\sigma_\theta<0.1/\gamma=0.53\text{ mrad}$.

The first constraint requires that, in the arcs, the beta
functions must be kept small.  The second constraint requires that,
in the production straight, the
Twiss parameter gamma ($\gamma=(1+\alpha^2)/\beta$) be small.  Note
that, in the production straight, not only does $\beta$ need to be
large, but $\alpha$ must be small.  The above requirement on $\sigma_\theta$
means that $\gamma<0.011\text{ m}^{-1}$ in the production straight.  It
will turn out that this constraint is not met for that
lattice as is; later in this chapter we describe changes that
must be made to achieve this.  The reason for the difficulty
is that, since $\beta$ goes
from being very small in the arcs to being very large in the
production straight, there is necessarily an intermediate region which
has moderate $\beta$ and large $\alpha$.

\section{The Lattice}
\subsection{Effect of Magnet Choice on Lattice Design}
\begin{figure}[!tb]
  \centering
  \includegraphics[width=\textwidth]{../template/report/ps-and-eps/ArcMagnets.eps}
  \caption[Arc magnet layout.]{Arc magnet layout.  Above is a top view of
    the arc
    (straightened out); in the middle is a side view.
    At the bottom are two cross sections of the magnet lattice,
    at the points indicated in the side view.  Coil current directions are
    indicated, with + being out of the page.}
  \label{fig:sr:arcmags}
\end{figure}
Figure~\ref{fig:sr:arcmags} shows a diagram of the arc cell layout.
The arcs will consist of two sequences of racetrack
coils, one placed above the other, but with the magnets overlapping.
The direction of the current alternates from one magnet in the
sequence to the next.  There are two types of magnetic fields in this
lattice, illustrated in the two cross sections in
Fig.~\ref{fig:sr:arcmags}: dipole fields (the right cross section),
and \textit{skew} quadrupole fields (the left cross section).
Note that this lattice gives rise to skew
quadrupole fields rather than upright quadrupole fields.
A single coil on top covers one quadrupole and both bends in
any given cell.  That coil has no ends (and therefore no wasted space)
at the ends of the skew quadrupole field in the section that it
overlaps.  The lower coils at that same skew-quadrupole region do
in fact have ends.  But, from an efficiency standpoint, those ends
are not ``wasted'': they merely create a transition from a bending
region to a skew quadrupole region.  In essence, the
transition is really a combined-function section, rather than an empty
section.

\begin{figure}[tb]
  \centering
  \includegraphics[width=30mm]{../template/report/ps-and-eps/EigenPlanes.eps}
  \caption[Eigenplanes for the pure skew decoupled lattice]
  {Eigenplanes for the pure skew decoupled lattice used for
    this storage ring.  The eigenplanes are rotated $45^\circ$ with
    respect to the horizontal.}
  \label{fig:sr:eigen}
\end{figure}
\begin{figure}[!tb]
  \centering
  \includegraphics{../template/report/ps-and-eps/BendTilt.eps}
  \caption[Generating upright quadrupole in bends.]
  {Raising part of each coil to generate an upright quadrupole
    component in the bend.}
  \label{fig:sr:tilt}
\end{figure}
Thus, the lattice is completely skew, with the eigenplanes
shown in Fig.~\ref{fig:sr:eigen}.  However, since the dipoles are
focusing in the
horizontal plane, which is not one of the eigenplanes, they
introduce coupling between the two skew planes.  To avoid this we
create an upright quadrupole with
\begin{equation}
  \dfrac{dB_y}{dx} = -\dfrac{eB_y^2}{2p},
\end{equation}
where $p$ is the reference momentum.  Then, the focusing is the
same in the horizontal and vertical planes, and therefore
cylindrically symmetric.  Cylindrically symmetric focusing does not
produce coupling between the skew planes.  This upright
quadrupole is created by raising or lowering one side of each coil pack within
the bending region, as shown in Fig.~\ref{fig:sr:tilt}.  The amount
that the coils need to be raised or lowered is actually very small,
about 1~mm.  Generating this
amount of vertical shift in the coils will be straightforward.

\subsection{Lattice Design}

\begin{figure}[!tb]
  \centering
  \includegraphics[width=0.75\textwidth]{../template/report/ps-and-eps/ArcLayout.eps}
  \caption[Layout of the arc.]
  {Layout of the arc.  The ``E'' cells have no dipoles.  The
    linear magnets in the central six cells are identical, and are
    the same as those in the ``D'' cells.  The triangles indicate the
    placement of sextupoles.}
  \label{fig:sr:layout}
\end{figure}
Figure~\ref{fig:sr:layout} shows the layout of an arc.  The phase
advance per cell is chosen to be $60^\circ$.  This gives reasonable values
for the cell length,
the beta functions, and the dispersion.  It also gives a reduced
swing in the dispersion functions, leading to a lower vertical dispersion.

To accomplish dispersion suppression, we employ the pattern of cells
 shown in Fig.~\ref{fig:sr:layout}.  There are six central cells,
surrounded by two cells having no bend, and on the ends there are
two cells that are identical to the central cells.  For 
a $60^\circ$ phase advance per cell,
this gives
dispersion suppression and matches the dispersion in the central cells
automatically.
 Arc cells with no dipoles will contain warm quadrupoles.
This permits the use of collimation in those cells, and confines the
decay shower to a region with warm magnets.

As discussed above, the production straight has large beta functions, so as to
minimize the angular spread of the neutrinos decaying toward the
detector.  The other straight, where we inject,
has beta functions roughly twice those in the arcs.  That section
will also be used to adjust the overall ring tunes.

\begin{table}[!tb]
  \caption[Magnet parameters]
  {Magnet parameters.  Only four of the utility
    straight quadrupoles require the strength shown; others are as low
    as 6.7~T/m.  The arc dipole and SC quadrupole parameters are really
    idealized parameters for a lattice made from standard magnets;
    the actual fields are provided by overlapping coils as described
    in the text.}
  \label{tab:sr:magparm}
  \centering
  \begin{tabular}{|lccccc|}
    \hline
    &Number&Length&Field (T)/&$R_{\text{pole}}$&$B_{\text{pole}}$\\
    &&(m)&Gradient (T/m)&(mm)&(T)\\
    \hline
    Arc dipole&32&1.89&6.93&&\\
    Arc SC quadrupole&32&0.76&35&&\\
    Arc NC quadrupole&16&0.65&27.2&47&1.28\\
    Production straight quadrupole&2&1.85&5.0&175&0.87\\
    &2&5&3.6&200&0.72\\
    &2&2.25&6.4&125&0.8\\
    &2&0.76&23.2&50&1.16\\
    Utility straight quadrupole&28&1.9&11.6&70&0.81\\
    \hline
  \end{tabular}
\end{table}
Table~\ref{tab:sr:magparm} gives the parameters for the magnets used
in the storage ring.

\begin{figure}[!tb]
  \centering \includegraphics[width=\textwidth]{../template/report/ps-and-eps/ring_optics_eigen.eps}
  \caption[Beta functions for the storage ring.]{Beta functions for the
    storage ring, in the 45$^\circ$-rotated eigenplanes.}
  \label{Ring_info_mu_mf09b}
\end{figure}
\begin{figure}[!tb]
  \centering \includegraphics[width=\textwidth]{../template/report/ps-and-eps/arc_optics_10cell2s.eps}
  \caption[Beta functions and the dispersion function for arc.]
  {Beta functions and the dispersion function for an arc of the
    storage ring.  The beta functions are in the 45$^\circ$-rotated
    eigenplanes, while the dispersion functions are projected into
    the standard horizontal and vertical planes.}
  \label{Arc_info_mu_mf09b}
\end{figure}
Lattice functions for the 20~GeV muon storage ring using compact skew
combined function arc cells are shown in
Figs.~\ref{Ring_info_mu_mf09b} and \ref{Arc_info_mu_mf09b}. Here the
beta functions, ($\beta_A,\beta_B$), are given for the 45$^\circ$-rotated 
betatron eigenplanes (A,B) shown in Fig.~\ref{fig:sr:eigen}, but
the eigenplane dispersion functions ($\eta_A,\eta_B$) are projected to
dispersion in the normal horizontal-vertical coordinate system according to 
the relationships, $\eta_x =
\dfrac{\eta_A + \eta_B}{\sqrt{2}}$ and $\eta_y = \dfrac{\eta_A -
  \eta_B}{\sqrt{2}}$.

By design, the dispersions in the A and B eigenplanes are nearly
equal, so the effective vertical dispersion is much smaller than the
horizontal dispersion. With this skew lattice, the horizontal
dispersion is nearly constant across the arc, whereas the vertical
dispersion oscillates with small amplitude about zero. The arc cells
without bending match the dispersion to zero for both eigenplanes in
the straight sections.

The lattice shown in Fig.~\ref{Ring_info_mu_mf09b} has a ratio
between the lengths of arcs and straight sections such that the
geometric decay ratio, $f_s$, is just over 35\%. The central 93~m of
the production straight has a Twiss gamma function of
$0.01~\text{m}^{-1}$, which meets the requirements for the angular
divergence described in Sec.~\ref{sec:sr:latparm}.  Within that
central section, there are no quadrupoles; this appears to be the most
straightforward way to minimize $\alpha$, and therefore $\gamma$, in
that section.  Beta functions in the return straight are
intermediate in magnitude between the values in the arcs and
the production straight in order to facilitate injection.

\subsection{Chromatic Correction Sextupoles}
\label{sec:sr:chrom}

For chromatic correction, a skew sextupole has been placed at each
skew quadrupole in the central section of the arcs (see
Fig.~\ref{fig:sr:layout}).  A skew sextupole, like the dipole and skew
quadrupole magnets, requires no coil in the midplane of the magnet.
These sextupoles must be very strong, since they correct not
only the chromaticity generated in the arcs, but also the
chromaticity generated in the straights.  Since the straights are
significantly longer than the arcs, the sextupoles require high
strengths.

\begin{figure}[tb]
  \centering\includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/SexCorrect.eps}
  \caption[Diagram showing cancellation of sextupole nonlinearities.]
  {Diagram showing cancellation of the sextupole
    nonlinearities due to the choice of families and the $60^\circ$
    phase advance per cell.  The left diagram shows the first-order
    cancellation, the right diagram shows that, given the right
    relationship between the A, B, and C family strengths, there will
    be a second-order cancellation as well.  Note that putting
    sextupoles in the D or E regions would not give a similar
    cancellation.}
  \label{fig:sr:sexcancel}
\end{figure}
The sextupoles are divided into A, B, and C families, as indicated in
Fig.~\ref{fig:sr:layout}.  All sextupoles in a given family have
identical strengths (this is true separately for the sextupoles at
focusing and defocusing quadrupoles).  Due to the $60^\circ$ 
phase advance per cell, there is an automatic first-order cancellation between
the nonlinear terms from these sextupoles, as shown in
Fig.~\ref{fig:sr:sexcancel}.  The families can be chosen to
give a second-order cancellation of the nonlinearities as well, as
illustrated in the second part of Fig.~\ref{fig:sr:sexcancel}.

The required sextupole strength to achieve zero chromaticity is 
$S=B''=78~\text{T}/\text{m}^2$.  This sextupole strength could be reduced by
putting sextupoles in the remaining arc cells, but there
is no simple cancellation scheme for the nonlinearities in that case,
as illustrated in Fig.~\ref{fig:sr:sexcancel}.

With one degree of freedom in our three families needed to correct the
chromaticity, and another needed to correct the second-order
nonlinearities, there is still one remaining degree of freedom in each
plane.  We use this to minimize the
nonlinear momentum dispersion in the production straight.  Writing
the transverse momentum of the closed orbit
in the center of the production straight as a function of
$\delta=\delta p/p$) as
$p_{x0}(\delta)=D_{p1}\delta+D_{p2}\delta^2+\cdots$, the sextupoles could
be used to eliminate $D_{p2}$ (note that $D_{p1}$ is already eliminated in
our dispersion suppression scheme).

\subsection{Coil End Effects}
\label{sec:sr:end}

\begin{figure}[!tb]
  \centering
  \includegraphics{../template/report/ps-and-eps/EndSolenoid.eps}
  \caption[Arc magnet cross section.]
  {Arc magnet cross section at the skew quadrupole end of the
    upper coil pack.}
  \label{fig:sr:endsol}
\end{figure}
There are two end effects with which we will be primarily concerned.
The first is a solenoid generated by the ends of the coils.  The
reason for the solenoid field is seen in
Fig.~\ref{fig:sr:endsol}.  Where one coil ends,
there is a transverse current, which leads to a longitudinal field. (In
standard magnet designs, the symmetry of the return coils
eliminates this solenoid field.  Since our design lacks this
symmetry, there is a net solenoid field.)

It is important to note that this solenoidal field will affect the
linear lattice; it is not just a nonlinear effect.  There are two
effects: the first is a rotation, the other is a focusing effect.  The
focusing can just be taken into account in the linear lattice, but the
rotation is more difficult since it can couple the eigenplanes
together.
The coupling can be removed from the lattice, if necessary, by rotating
the skew quadrupoles by the amount that each end rotates.  One
quadrupole sign is rotated in one direction; the other is rotated in
the opposite direction.  The amount of rotation can be calculated
precisely knowing only the current in the coil and the height
of the coil above the axis.  The coils need to be displaced by only 
about 1~mm to put this rotation in, which is comparable to the coil
displacement we already must achieve to put the upright quadrupole
field into the bending sections.

\begin{figure}[!tb]
  \centering
  \includegraphics{../template/report/ps-and-eps/EndSextupole.eps}
  \caption[Cross-section as one approaches the end of the top coil.]
  {Cross-section as one approaches the end of the top coil,
    showing the coils on top moving closer to each other.  This has a
    nonzero sextupole field.}
  \label{fig:sr:endsex}
\end{figure}
The second end effect we consider is a sextupole
contribution.  This is not the sextupole-order contribution caused by
the longitudinal derivative of the dipole field.  Rather, it comes
from the fact that the coil symmetry changes when the coil starts to
cross at the end.  Consider the dipole cross section in
Fig.~\ref{fig:sr:arcmags}.  That cross section will have a nonzero
sextupole field unless the coils are placed in precisely the correct
places. (For a single wire coil, that would be along a line through
the center, $30^\circ$ from horizontal.)  The main body of the magnet
is designed with the coils placed to eliminate that sextupole
contribution.  However, when the coils are moved out of the position
that zeros the sextupole  contribution so that they can cross over, as
depicted in Fig.~\ref{fig:sr:endsex}, there will be a net sextupole
field at that point.  This sextupole field can be hundreds of times as
large as the sextupole component in the body of a realistic magnet.

\begin{figure}[!tb]
  \centering\includegraphics[angle=270,width=0.5\textwidth]{../template/report/ps-and-eps/end-profile.eps}
  \caption[Longitudinal variation in the sextupole field ]{Longitudinal variation in the sextupole field at the
    transition from the dipole to the skew quadrupole region.}
  \label{fig:sr:b2err}
\end{figure}
Uncorrected, this sextupole field would decrease the dynamic
aperture of the ring.  However, immediately adjacent to the end
that generates this sextupole is another coil end that generates
an opposite sextupole.  As shown in Fig~\ref{fig:sr:b2err}, the
integrated sextupole is zero over a very short
distance (a few tens of cm), and should not be a problem for beam dynamics.

\subsection{Correction and Tuning}

In principle, since the short coils in the skew quadrupole regions can
be powered separately from the longer coils, it is possible to perform
both dipole and quadrupole corrections by varying their current.
To create a ``pure'' correction requires more than one magnet
working in concert, since one coil does not produce a pure
dipole or a pure skew quadrupole at a given point.

Alternatively, an additional coil could be added opposite the short
coil to allow pure dipole or pure skew quadrupole corrections at a
given point.  The coil could be either superconducting or
normal conducting, since
it need not generate large fields.

To adjust the overall tune of the ring, the quadrupoles in the utility
straight will be used.

\subsection{Tracking}
Tracking studies were performed on a single arc cell using
COSY INFINITY~\cite{sr:cosy}.  A more detailed analysis is given in
\cite{sr:msucl-1197}.  The arc cell used is slightly different
from the one described earlier in this chapter.  A diagram of the cell
used in the tracking is shown in Fig.~\ref{fig:sr:trackcoil}.
\begin{figure}[!tb]
  \centering\includegraphics[width=12cm]{../template/report/ps-and-eps/bars.eps}
  \caption{Arc coil configuration used for tracking.}
  \label{fig:sr:trackcoil}
\end{figure}  Note that the
short coils with a reversed current direction are missing.
Instead of having a pure skew quadrupole section, there is a
combined-function section which is part dipole and part skew
sextupole.  The phase advance in this cell is $60^\circ$, just as in
the actual lattice.  We expect the results from tracking this cell to
be similar to what we find from tracking the actual lattice, since the
only real difference between the two is a slight redistribution of the
linear components.
\begin{table}[!hbt]
  \caption[Optical description and parameters for the arc cell]
  {Optical description and parameters for the arc cell.
    $k_1=({\partial} B_y/{\partial} x)/(B \rho)$,
    $k_2=({\partial}^2 B_y/{\partial} x^2)/(B \rho)$,
    $(B \rho)$ for a 20 GeV muon is 67.064332~Tm,
    B$_D$ is the dipole field strength, and
    B$_Q$ and B$_S$ are the quadrupole and sextupole field strength
    at the aperture, $r=6.5$~cm, divided by 7.02296~T.}
  \label{tab:sr:cell}
  \begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Section & Starting & Length & Tilt  & Deflection & $k_1$ (m$^{-2}$)   & $k_2$ (m$^{-3}$)  \\
        & position &        & angle & (B$_D$)    & (B$_Q$) & (B$_S$) \\
& (m) &(m)&$({}^{\circ})$& & &  \\
\hline
(a)     & 0.00 & 0.55 & 45 & 57.6~mrad & -0.00548  & 0 \\
        &       &       &              & (7.02296~T)     &(-0.02389~T)&  \\
\hline
(b)     & 0.55 & 1.55 & 45 & 81.8~mrad & -0.00137  & 0 \\
        &       &       &              & (3.51148~T)     &(-0.00597~T)&  \\
\cline{4-7}
(b')    &       &       &  0 &    0~mrad & -0.30269  & -0.01932 \\
        &       &       &              &                &(-1.31950~T)&(-0.002737~T)\\
\hline
(c)     & 2.10 & 0.55 & 45 & 57.6~mrad & -0.00548  & 0 \\
        &       &       &              & (7.02296~T)     &(-0.02389~T)&  \\
\hline
(d)     & 2.65 & 0.55 & 45 & 57.6~mrad & -0.00548  & 0 \\
        &       &      &              & (7.02296~T)     &(-0.02389~T)&  \\
\hline
(e)     & 3.20 & 1.55 & 45 &81.8~mrad & -0.00137  & 0 \\
        &       &       &              & (3.51148~T)     &(-0.00597~T)&  \\
\cline{4-7}
(e')    &       &       &  0 &    0~mrad &  0.30269  &  0.01317 \\
        &       &       &              &                & (1.31950~T)&( 0.001866~T)\\
\hline
(f)     & 4.75 & 0.55 & 45 & 57.6~mrad & -0.00548  & 0 \\
        &       &       &              & (7.02296~T)     &(-0.02389~T)&  \\
\hline
\end{tabular}
\end{center}
\end{table}
Table~\ref{tab:sr:cell} lists the magnet
parameters used in the COSY-INFINITY tracking.
In the combined-function areas, there are regions of overlapping
bend, upright quadrupole, skew quadrupole, and skew sextupole (these
are the regions with only a single coil, labeled as (b)/(b') and
(e)/(e') in Table~\ref{tab:sr:cell}).  To model this properly in
COSY INFINITY, these sections are divided into 10 subsections of
length $\Delta L=0.155$~m, each of
which consists of a bend/upright-quadrupole section of length $\Delta L$,
a negative drift of length $-\Delta L$,
and a skew-quadrupole/skew-sextupole section of length $\Delta L$.
The skew sextupole strength is chosen to be sufficient to precisely
cancel the chromaticity.  COSY INFINITY also uses the full kinematic
Hamiltonian (the full square root, as opposed to simply
$p_x^2/2+p_y^2/2$) in its tracking.

\begin{figure}[!tb]
  \centering
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F0xaDA.ps}
  \hspace{-1.6cm}
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F0xyDA.ps}
  \caption[Tracking particles in the arc cell.]
  {Tracking particles in the arc cell starting at $A=10,...,70$~cm,
    showing $A$-$p_A$ motion (left) and $A$-$B$ motion (right).
    The scales of the pictures cover $\pm150$~cm in the $A$ plane,
    $\pm100$~cm in the $B$ plane, and $\pm0.5$ in the $p_A$ plane.
    Recall that $A$ and $B$ are the diagonal eigenplanes.}
  \label{fig:sr:tr-thick}
\end{figure}
\begin{figure}[!tb]
  \centering
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F0xa.ps}
  \hspace{-1.6cm}
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F0xy.ps}
  \caption[Low-amplitude tracking in the arc cell.]
  {Tracking particles in the arc cell,
    starting at $x=0.5,...,3.5$~cm,
    showing $A$-$p_A$ motion (left) and $A$-$B$ motion (right).
    The scales of the pictures cover $\pm7.5$~cm in the $A$ plane,
    $\pm5$~cm in the $B$ plane, and $\pm0.025$ in the $p_A$ plane.}
  \label{fig:sr:tr-idealsm}
\end{figure}
Figure~\ref{fig:sr:tr-thick} shows tracking results for the arc cell as
described above.  The dynamic
aperture for this idealized lattice is much larger than needed.
Figure~\ref{fig:sr:tr-idealsm} demonstrates, that for amplitudes within
the magnet aperture, the lattice is extremely linear and well
decoupled.

\begin{figure}[!tb]
  \centering
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F3xa.ps}
  \hspace{-1.6cm}
  \includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLv3F3xy.ps}
  \caption[Tracking with end fields.]{Tracking with end fields.
    Initial conditions and scales as in Fig.~\ref{fig:sr:tr-idealsm}.}
  \label{fig:sr:endfield}
\end{figure}
The next step is to add end fields to these magnets.  An Enge-function
falloff model \cite{sr:prstab-3124001} was used to model these end
fields.  Fig.~\ref{fig:sr:endfield} shows the results.  Note that
there is now significant coupling between the planes.  Linear coupling
is, in principle, correctable as described earlier, but there is a
nonlinear coupling that will inevitably be there, as  is the case in an
upright lattice.  The beam still is well within the dynamic aperture.

\begin{table}[!tb]
  \caption[Multipole components in the arc cell]
  {Multipole components in the arc cell.
 The value given in the table is the maximum of the magnetic field in
Tesla for that multipole component at a radius of 6.5 cm.}
\label{tab:sr:multipole}
\begin{center}
\begin{tabular}{|cc||c||c|c|}
\hline
 & & Double Coil Region & \multicolumn{2}{|c|}{Single Coil Region (b)} \\
\cline{4-5}
 & & Normal & Normal & Skew \\
\hline
 2 & Sextupole & $-0.721874127471$ & $-0.360937063736$ &  0                \\
 3 & Octupole  &  0              &  0              &  0.100208577080   \\
 4 & Decapole  &$ -0.325677875510$ &$ -0.162838937755$ &  0                \\
 5 &Dodecapole&  0              &  0              &  0.105845309541   \\
 6 & 14-pole   &$ -0.048154527567$ &$ -0.024077263783$ &  0                \\
 7 & 16-pole   &  0              &  0              &$ -0.111799108203 $    \\
\hline
\end{tabular}
\end{center}
\end{table}
Next, multipole components in the body of the magnets are added.  We
started with multipole components from a 2D model of the region with
both coils, which was computed using Poisson
\cite{sr:poisson,sr:poisson-pac93} and Opera 2D.  Independently, we
performed a 3D field simulation using a bar magnet model, where the
field can be expressed analytically \cite{sr:AIEP108book,sr:barfield}.
Two models were used: one where the bar magnets were infinite in
extent, and a second where there were four magnets laid out as in
Fig.~\ref{fig:sr:trackcoil}.  The bar magnet models were constructed
to match as closely as possible the 2D model described above.  The
multipole fields in the real system were computed by starting with the
values from the original 2D model, and scaling them by the
ratio of the values in the second bar magnet model to those in the
first bar magnet model.  The values of these computed multipole
components are given in Table~\ref{tab:sr:multipole}.

\begin{figure}[bt!]
\centering
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0xa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0xy.ps}
\\
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3xa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3xy.ps}
\caption[Tracking with additional multipole components.]
{Tracking the arc cell with the additional multipole
components in Table~\ref{tab:sr:multipole}.
Without (top two) and with (bottom two) fringe fields.}
\label{fig:sr:tr-CLm}
\end{figure}
\begin{figure}[bt!]
\centering
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0sodxa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0sodxy.ps}
\\
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3sodxa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3sodxy.ps}
\caption[Tracking with reduced multipole components.]
{Tracking with the additional multipole components,
  with the normal sextupole, skew octupole and normal decapole
  strengths
  reduced to 10\% of their initial values.
  Without/with (top/bottom) fringe fields.}
\label{fig:tr-CLmsod}
\end{figure}
Results of tracking with these multipole components are shown in
Fig.~\ref{fig:sr:tr-CLm}.  The dynamic aperture is unacceptably
small.  However, the multipole components shown in
Table~\ref{tab:sr:multipole} are extremely large.  As described
later in Sec.~\ref{sec:sr:magdes}, we believe we can make those
multipole components much smaller.  It was found that if the
sextupole, skew octupole, and decapole components were reduced to 10\%
of the values in Table~\ref{tab:sr:multipole}, the dynamic aperture
was acceptable, even with fringe fields (see, Fig.~\ref{fig:sr:tr-CLm}).

\begin{figure}[!tb]
\centering
\includegraphics[width=8cm,angle=270]{../template/report/ps-and-eps/Bz0.ps}
\caption{Field profile of solenoidal field component from the
  bar magnet model.}
\label{fig:sr:Bz0}
\end{figure}
\begin{figure}[hbt!]
\centering
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0solxa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF0solxy.ps}
\\
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3solxa.ps}
\hspace{-1.6cm}
\includegraphics[width=6.2cm,angle=270]{../template/report/ps-and-eps/CLmF3solxy.ps}
\caption[Tracking with solenoidal fields added.]
{Adding the solenoid components
to tracking with the additional multipole components,
with the sextupole, octupole and decapole strengths
10\% of the initial.
Without/with (top/bottom) fringe fields.}
\label{fig:sr:tr-CLmsol}
\end{figure}
The bar magnet model gives a solenoidal field as shown in
Fig.~\ref{fig:sr:Bz0}.  In that model, the solenoidal field doesn't
become small anywhere.  The solenoidal field was modeled using several
short solenoids with a Gaussian-shaped profile.  There are three
effects of this solenoidal field: linear coupling, linear focusing,
and additional nonlinearities (from the longitudinal derivative of the
field).  Figure~\ref{fig:sr:tr-CLmsol} shows the results of tracking
with these solenoidal fields added.  The solenoidal fields have a
significant impact on the dynamic aperture of the machine.  Further
study is needed to examine to what extent this effect can be
mitigated.

\begin{table}[bt!]
\caption{Linear tunes in each study case for the two orthogonal planes.}
\label{tab:sr:tune}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& No fringe field effects & With fringe field effects \\ 
\hline
Initial approximation
& 0.166667 & N/A \\
& 0.166667 & \\
\hline
Thick lens model
& 0.168422 &  0.168040 \\
& 0.168422 &  0.166919 \\
\hline
With solenoids
& 0.162584 &  0.162190 \\
& 0.174157 &  0.172703 \\
\hline
\end{tabular}
\end{center}
\end{table}
Table~\ref{tab:sr:tune} indicates the strong effect of the solenoidal
fields on the tune.  In particular, the correction scheme
described in Sec.~\ref{sec:sr:chrom} for canceling geometric
nonlinearities will not work as well when the phase advance is not
exactly $60^\circ$.  The tune split comes from coupling from the
solenoids, which can be corrected as described in
Sec.~\ref{sec:sr:end}.  Thus, this decoupling, along with an
adjustment of magnet strengths to compensate for the focusing effect
from the solenoids, may restore the dynamic aperture.

\section{Magnets}

\begin{figure}[!tb]
  \centering\includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/neutrino-cross-section-D-323.eps}
  \caption{An engineering design of the magnet cross section.}
  \label{fig:sr:engrxsec}
\end{figure}
\begin{figure}[!tb]
  \centering\includegraphics[width=\textwidth]{../template/report/ps-and-eps/xupdn-a-model-view-iron5.eps}
  \caption[Three-dimensional view of the storage ring magnets.]
  {Three-dimensional view of the storage ring magnets.  Iron (blue) is
    shown for only half the length.}
  \label{fig:sr:mag3d}
\end{figure}
Figures~\ref{fig:sr:engrxsec} and \ref{fig:sr:mag3d} show diagrams of
the arc magnets to be used in the storage ring.  Each coil pack with
its associated hardware will be built in the lab, and the individual
cryostats will then be connected together appropriately in the
tunnel.

The vertical aperture of the dipole is 80~mm.  The actual vertical
separation between the upper and lower coils is 130~mm, to accommodate
space taken by the support structure, heat shield, and
cryostat.  The aperture in the horizontal plane is much larger than in
the vertical plane, since the space under the coils is also available
due to the open-midplane design (see Figs.~\ref{fig:sr:engrxsec} and
\ref{fig:sr:mag3d}).  The actual horizontal coil aperture, 240~mm,
is dictated by the minimum bend radius at the end of 120~mm.  
Of course the beam cannot take advantage of the region under the coil,
since the field
quality is poor there.  It is important to know the field in
this region, however, since it is needed for tracking of decay electrons.  The
region of good field quality can be determined from the harmonics
given in Section~\ref{sec:sr:magdes}, or by a field profile within the beam tube.

The operating field of the dipole is 6.93~T, and the quench field is
over 8~T.  This gives an operating field margin of over 15\%.  As
mentioned earlier, the
maximum field on the conductor at quench excludes the possibility of
using NbTi at 4.2~K operating temperature.  The coils, therefore, are
made of Nb$_3$Sn superconductor.  Large bend
radius in the ends allows the use of the ``react and wind'' technique
in a pancake (racetrack) coil geometry.  These pancake coils must have
a large sagitta, due to the small size of the ring. The reverse
curvature in the coil is provided by Kevlar strings as shown in
Fig.~\ref{fig:sr:pancake}.

In this design, most of the energy from the decay particles is
deposited in the warm-iron yoke. A warm-iron design allows
the heat generated by decay particles to be removed efficiently. This means that a tungsten inner liner to protect the
superconducting coils is not needed.

An engineering design of the magnet is shown in
Fig.~\ref{fig:sr:engrxsec}. The coils are located in a cryostat that is
placed inside the warm iron yoke.  Upper and lower coils will have
separate cryostats that are placed in a common vacuum vessel. A major
design consideration is to minimize the cryostat volume, in particular
the vertical separation between the upper and lower coils.

The Lorentz forces are large, due to the high field and large aperture of
this magnet.  Horizontal forces are contained in a self-supporting
collar structure. The vertical forces are transmitted to the cryostat
with a coldmass support post, which in turn is connected to the iron
and contains the force. The posts are designed to minimize the heat
leak, which in the present design is comparable to the heat load from
the decay products.

\subsection{Magnetic Design}
\label{sec:sr:magdes}

The magnet cross section is based on two double-layers of pancake
coils (total four layers) with a curvature that approximately
follows the beam trajectory. The total height of the coil pack is 40~mm
(made with four stacks of 10-mm-wide cable), and the width is 60~mm.
The cross section has spacers (equivalent to
the wedges in a cosine theta design) to optimize the field quality and to
reduce the peak field in the conductor. The peak field in the
conductor is higher when the coil and magnet are made compact, in
particular, when there is a large gap between the upper and lower coils
at the midplane of the magnet. In the present design, the peak field is
about 50\% of the central field.

\begin{table}[!tb]
  \caption[Field errors in the dipole magnet section.]{Estimated
    field errors in the pure dipole magnet section at a 20~mm
    reference radius.  $\left<b_n\right>$ and $\left<a_n\right>$ are
    the expected means of the normal and skew terms.  $d(b_n)$ and
    $d(a_n)$ are systematic uncertainties arising from design and
    manufacturing errors, and $\sigma(b_n)$ and $\sigma(a_n)$ are
    the random uncertainties in those values.  Note that $n=2$
    corresponds to the sextupole term.}
  \label{tab:sr:diperr}
  \begin{center}
    \begin{tabular}{|c|ccc|ccc|}
      \hline
      $n$&$\left<b_n\right>$&$d(b_n)$&$\sigma(b_n)$
      &$\left<a_n\right>$&$d(a_n)$&$\sigma(a_n)$\\
      \hline
      1&0&0.2&0.2&0&1&2\\
      2&$-1$&1&2&0&0.1&0.5\\
      3&0&0.1&0.1&0&0.3&1\\
      4&$-1$&1&1&0&0.05&0.2\\
      5&0&0.03&0.03&0&0.1&0.5\\
      6&$-0.3$&0.2&0.1&0&0.03&0.1\\
      7&0&0.03&0.01&0&0.03&0.1\\
      8&$-0.1$&0.1&0.02&0&0.03&0.1\\
      9&0&0.03&0.01&0&0.03&0.1\\
      10&$-0.03$&0.02&0.02&0&0.03&0.1\\
      \hline
    \end{tabular}
  \end{center}
\end{table}
\begin{table}[tb]
  \caption[Field errors in the skew quadrupole magnet
  section.]{Field errors in the skew quadrupole magnet section at a
    20~mm reference radius.  See Table~\ref{tab:sr:diperr} for
    definitions.}
  \label{tab:sr:skewerr}
  \begin{center}
    \begin{tabular}{|c|ccc|ccc|}
      \hline
      $n$&$\left<b_n\right>$&$d(b_n)$&$\sigma(b_n)$
      &$\left<a_n\right>$&$d(a_n)$&$\sigma(a_n)$\\
      \hline
      1&0&0.2&0.2&0&1&2\\
      2&$-0.5$&0.5&1&0&1&0.5\\
      3&0&0.1&0.1&2&2&1\\
      4&$-0.5$&0.5&0.5&0&0.05&0.2\\
      5&0&0.03&0.03&1&1&2\\
      6&0&0.2&0.1&0&0.03&0.1\\
      7&0&0.03&0.01&0.5&0.5&0.3\\
      8&0&0.1&0.05&0&0.03&0.1\\
      9&0&0.03&0.01&0.1&0.03&0.1\\
      10&0&0.02&0.01&0&0.03&0.1\\
      \hline
    \end{tabular}
  \end{center}
\end{table}
The expected skew and normal harmonics are given in
Tables~\ref{tab:sr:diperr} and \ref{tab:sr:skewerr}. These are not
final values, as the magnetic design is not yet optimized.
Table~\ref{tab:sr:diperr} gives the field errors in the dipole section,
where the upper and lower coils have the same polarity, and
Table~\ref{tab:sr:skewerr} gives the field errors in the skew
quadrupole section, where the upper and lower coils have opposite
polarity.  Field harmonics are expressed in terms of the normal
and skew harmonic coefficients, $b_n$ and $a_n$, defined by
\begin{equation}
  B_y+iB_x = 10^{-4}B_0\sum_{n=0}^\infty(b_n+ia_n)
  \left(\dfrac{x+iy}{R}\right)^n,
\end{equation}
where $x$ and $y$ are the horizontal and vertical coordinates, $R$ is
the reference radius, and $B_0$ is the field strength at $x=R$ and
$y=0$ in the dipole straight section.  Field harmonics are
normalized to this same $B_0$ in both sections.  The reference radius
$R$ is 20~mm.


\section{Beam Flux to Detector}

\begin{figure}[tb]
  \centering
  \includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/nurate.eps}
  \caption[Relative event rate at the detector.]
  {Relative event rate at the detector for a 30~GeV muon beam
    as a function of the rms beam angular divergence, with
    data taken from \cite{sr:geer}.  Curves
    are fit to this data with the functional form
    $A/(1+\sigma_\theta^2/\sigma^2)$, where $\sigma_\theta$ is the
    beam divergence, and $\sigma$ and $A$ are fit parameters.  This
    form is the central flux of two overlapping Gaussians, one with
    divergence $\sigma_0$ (the beam) and the other with divergence
    $\sigma$ (the decay divergence).  The fit $\sigma_0$ values are 1.64 mrad
    ($\nu_e$) and 1.41 mrad ($\nu_\mu$ and $\nu_e\to\nu_\mu$).}
  \label{fig:sr:eventrate}
\end{figure}
There are really two quantities that we must deliver to the detector:
a sufficient flux of neutrinos, and a sufficiently small uncertainty
in that flux.  It turns out that the latter may be the more difficult
challenge.  To understand this, consider Fig.~\ref{fig:sr:eventrate}.
In that figure, we show event rate as a function of beam divergence
(for a round beam), and fit a curve to that.  This is based on the
fact that, for an elliptical Gaussian beam divergence, with angular
divergences of $\sigma_p$ and $\sigma_q$ in the two directions, and a
Gaussian divergence of the neutrino decays with RMS divergence
$\sigma$, the central flux is expected to be proportional to
\begin{equation}
  \dfrac{1}{\sqrt{(1+\sigma_p^2/\sigma^2)(1+\sigma_q^2/\sigma^2)}}.
\end{equation}
While the basis for this model is not particularly accurate, it
nonetheless gives a reasonably accurate representation of the
simulated data when you fit to $\sigma$.  Based on the fit to the data
in Fig.~\ref{fig:sr:eventrate}, we take $\sigma=0.42/\gamma$.

From this curve, one can understand why a larger divergence beam
contributes more to the uncertainty: the uncertainty comes from the
uncertainty in the angular spread times the slope of that curve.  When
one has a larger divergence, the curve has a larger slope.  Another
way to view this is that if the angular spread in the beam is
much smaller than the angular divergence from the decays, the
uncertainty in that angular spread has little effect on the total
angular spread in the beam.
\begin{figure}[!tb]
  \centering
  \includegraphics[width=0.5\textwidth]{../template/report/ps-and-eps/prod09-flux.eps}
  \caption[Flux at the detector and the uncertainty in that flux.]
  {Flux at the detector, and the uncertainty in that flux, as a
    function of position in the production straight.  A relative
    uncertainty in the angular divergence of 15\% in each plane is
    assumed and, within a given plane, those uncertainties are assumed
    to be completely correlated.  The correlation assumption is
    correct for the contribution to the uncertainty from the emittance
    uncertainty, but only partially correct for the contribution from
    the uncertainty in the lattice functions (the uncertainties must
    be propagated properly).}
  \label{fig:sr:flux}
\end{figure}
Using this model, we can plot the flux and the uncertainty in that
flux relative to a beam with zero angular spread.  This is shown in
Fig.~\ref{fig:sr:flux}.  The resulting relative flux is
$0.759\pm0.014$.  Thus, despite the fact that the flux uncertainty
from the middle of the straight section is almost exactly 1\% (add the
two planes in quadrature), the ends of the production straight
contribute disproportionately to the uncertainty, despite their
relatively small contributions to the total flux.  Also, note the
reduced flux: instead of having 35\% of the muons decaying in a very
low divergence beam toward the detector, we have only 27\% of the muons
decaying from the low divergence portion of the straight section.

It is clearly important to eliminate this uncertainty from the ends of
the production straight.  These ends are a matching section from the
relatively low beta functions in the arcs to the high beta function in
the straight.  Thus, they will necessarily have a large angular
divergence.  One workaround is to point the matching sections in a
different direction from that of the detector by bending slightly less
than $180^\circ$ in the arcs, and inserting a bend just before the
``good'' section of the production straight.  One drawback to this is
that the production straight becomes shortened.
However, since the ends contribute relatively little to the total
flux, not much is lost: whereas originally the relative flux was
$0.759\pm0.014$, the relative flux would become $0.700\pm0.007$
after the workaround.

The required bending angle has been estimated to be 29~mrad.  This
calculation assumed that the ends produced a flat beam with an
angular spread in the wide dimension equal to the largest angular
spread anywhere in the end, and required that this beam contribute
less than 0.1\% to the total flux.

\section{Instrumentation}

\begin{table}[!tb]
  \caption[Required precision for muon beam measurements.]{Required
    precision for muon beam measurements \cite{sr:geer}.}
  \label{tab:sr:meas}
  \begin{center}
    \begin{tabular}{|lrrl|}
      \hline
      momentum              & $ \delta p_{\mu} $&$\leq $&$0.3\%$             \\
      momentum spread       & $ \Delta \sigma_{p}/\sigma_{p} $&$<$&$0.17 $\\
      divergence            & $\sigma_{\theta} $&$\leq $&$0.1 / \gamma $  \\
      & $\delta \sigma_{\theta}/\sigma_{\theta} $&$\leq $&$0.2  $ \\
      polarization          & $\delta P_{\mu} $&$<$&$0.01$                   \\
      direction             & $\delta \theta_{\mu}$&$<$&$0.6\ \sigma_{\theta}$ \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

The storage ring presents some new beam instrumentation problems.
There are of course the usual measurements of emittance, divergence,
closed orbit, injection, extraction, beam loss and beam energy.
However, since we are interested in the properties of the neutrino
beam produced by the muon beam, it is helpful to have additional
information.  The precision with which the parameters of the muon beam
must be known to achieve sufficient precision in the neutrino beam
flux have been determined by Geer \cite{sr:geer}, and these are listed
in Table~\ref{tab:sr:meas}.  As indicated in this table, for instance,
the polarization of the muon beam should be measured, and precision
measurements of the beam direction in the straight section would be
useful.  The majority of instrumentation for the muon beam in the
storage ring should utilize proven technology.  The primary difficulty
is that precision measurements are complicated by the
presence of decay electrons in the beam, since electrons can cause
showers.

The fraction of primary decay electrons in the beam is $L/\gamma \tau c$,
where $L$ is a path length in the storage ring, and $\gamma \tau
c \sim 125$~km is the decay length at 20 GeV.  In 
bending magnets, these electrons would be swept out within 
a few meters of their creation, so the electron contribution 
would be negligible.  Quadrupoles produce less efficient
sweeping.
The fraction of muons that will decay in the 126~m straight sections, 
however, is $\sim0.001$, and the electron/muon ratio 
at the downstream end of the straight will be $\sim0.001F_{s}$, 
where $F_{s}$ is a factor that depends on the probability of 
electrons showering and being swept from the vacuum pipe.  
These backgrounds are relatively low, but 
they may not be insignificant.  Since estimates of the electron background 
are difficult, it may be desirable to have precision 
measurements external to the ring for 
determining the neutrino beam direction, profile and divergence. 

We anticipate that the 6D ``pencil'' beams used to tune up the 
accelerator will also be useful in tuning up and operating the 
storage ring.

Semertzidis and Morse~\cite{sr:SM} have looked at using the
$g-2$ frequency of the muons to determine the muon beam energy.  They
consider measurement using the very substantial signal from synchrotron 
radiation from decay electrons.  Synchrotron radiation is nonlinear 
in energy, which amplifies the oscillations.  The beam momentum spread 
is also measurable using $g-2$, since the beam will be dephased by 
the spread in $\gamma$.

The beam size and divergence in the quadrupole-free decay straight section
of length, $2L$, can be measured by comparing beam profile
measurements at the ends, $\sigma_{e}$, and middle, $\sigma_{m}$, of
the straight section.  We assume that $\alpha=0$ in the center of the
straight section.  The beam divergence is
$\sigma_{\theta}=\sqrt{\sigma_{e}^{2}-\sigma_{m}^{2}}/L$, and the beam
emittance is $\epsilon_{\perp}=\sigma_{m}\sigma_{\theta}$.  Since
there will be no focusing in this part of the straight section, it should
be possible to perform a Monte Carlo calculation and subtract the
contributions from any decay electrons.  The beam size can be measured
using visible transition radiation from foils inserted in the beam, or
a variety of other fast detectors.

In order to separate the contributions to the neutrino flux from
decays in the upstream and downstream matching sections, bending
magnets have been introduced.  Using a near detector located a few
hundred meters from the straight section, the precise profile of the
contributions from the three sections to the downstream detectors can
be evaluated with statistics $10^5$ times larger than will be available in
the far detector.  We assume a dense, fine-grained detector consisting
of tungsten or other heavy plates interspersed with hodoscopes or
liquid-argon calorimeters.  This could be located in a 52~m deep
shaft, 200 m downstream of the decay straight, where the three
``beams'' would have a Gaussian radius of about 1~m.  Rates could be
high, on the order of 25 events/fill for a 1-m-thick detector
($\sim$0.5 kHz).  With this rate, it may be possible to measure and
subtract the background contribution of neutrinos produced in the
upstream and downstream matching sections.  Such a system needs
further analysis, and is not considered part of the baseline
design.

The polarization of the muons in the storage ring can be measured by
looking at the momentum distribution of the decay electrons moving
in the beam direction.  Roughly
$8\times10^{4}$ decays/m per turn will generate about 100 W/m of 
signal.  These electrons are produced close to the beam as part of the
fan of electrons that is swept inward by the bending magnets.
The muon beam polarization is measured from the electron decay spectrum. 
A shower calorimeter, which can absorb the shower from 
forward-going electrons close to the beam,
can be instrumented to look at the power deposition rate at the $\mu$s 
time scale.  Detection with a calorimeter should be relatively linear 
with electron energy.  The precision with which the Monte Carlo 
calculations could be done would be crucial, and this work is under way.

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\end{thebibliography}

%%%%%\end{document}
