This second Feasibility Study, 
(``Study II"), commissioned by BNL Director John~Marburger, 
uses BNL site-specific proton driver specifications and a BNL-specific 
layout of the storage ring, in
particular, the pointing angle of the straight sections. It is a
follow-up to the FNAL
specific (``Study I") study commissioned by the Fermilab Director,  that was completed in April 2000~\cite{INTRO:ref1}
and is site specific in the same spirit, that is, 
in each study there are a few site-dependent parts; otherwise, the studies
are generic. The primary difference is that this study is aimed at a
lower muon energy (20 GeV), but higher intensity (for physics
reach). Figure~\ref{EPP:fgphysics} has been adapted from a figure in
the physics study~\cite{INTRO:ref9}. Both studies were carried out
jointly with the Neutrino Factory and Muon Collider
Collaboration~\cite{EPP:collaboration} which has over 140 members from
many institutions in the U.S. and abroad.


The design and simulated performance are summarized here; specific
details can be found in the chapters that follow.

The efficiency of producing muons at the end of the cooling channel is
$\approx$~0.17~$\mu$/p with 24~GeV protons. This higher efficiency
translates, per MW of proton beam power, into about $6\times $ that
found in Feasibility Study I~\cite{INTRO:ref1}.

The higher efficiency is achieved by: 
\begin{enumerate}
\item using a liquid mercury target
\item using three induction linacs to achieve nearly non-distorting phase 
rotation into a longer bunch train with less momentum spread 
\item tapering the focusing strength in the cooling system so that the 
angular spread of the muons being cooled is maintained at a near-constant 
value
\item increasing the transverse acceptance of the muon acceleration and 
storage ring.
\end{enumerate}
%section{ Introduction}


\begin{figure}[!bth]
\begin{center}
%\vskip2.7in
%\includegraphics[totalheight=5in]{../template/report/ps-and-eps/study2c.eps}
\includegraphics[totalheight=4in]{../template/report/ps-and-eps/physics01.ps}
%\begin{center}
\caption[Muon decays in  a straight section \textit{vs.} muon energy]{ Muon decays in  a straight section per $10^{7}\,$s \textit{vs.} muon energy, with fluxes required for different physics searches assuming a 50~kT detector. Simulated performance of the two studies is indicated.
}
\label{EPP:fgphysics}
\end{center}
\end{figure}

The components of the system are shown schematically in Fig.~\ref{sch_fig} (in the Preface).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Components}
\subsubsection{Proton Driver}
The proton driver is an upgrade  of the Brookhaven
Alternating Gradient Synchrotron (AGS) and uses most of the existing
components and facilities.  The existing booster is replaced by a 1.2~GeV 
superconducting proton linac. The AGS repetition rate is increased
from 0.5~Hz to 2.5~Hz.  The total proton charge ($10^{14}$ ppp) is
only 40\% higher than the current performance of the AGS. The six bunches are extracted
separately, spaced by 20~ms, so that the target, induction linacs
and rf systems that follow, need only be designed to deal with single
bunches at an average repetition rate of 15~Hz, instantaneous rate of 50~Hz. The average power
would be 1~MW. A possible future upgrade to $2 \times 10^{14}$~ppp and 
5~Hz could
give an average beam power of 4~MW (see, Section~\ref{APP:Proton}). In that scenario, a 1/4 circumference, 
fixed-field,
superconducting bunch compressor ring would be added to reduce the rms bunch
length, at the higher intensity, to 3~ns.

\subsubsection{Target \& Capture}
%\vskip.2in
A high {\it Z}, (mercury) jet target is chosen to give a high yield of pions per
incident proton power ($\approx~ 1.9 ~\times $ that for carbon, which was 
the choice in Study I). 
 

The jet is continuous, is 1~cm diameter, and enters the target
enclosure at a vertical angle of 100~mrad with respect to the magnetic axis. 
The proton beam intersects the
jet at an angle of 33~mrad (\textit{i.e.}, its trajectory is 67~mrad to the magnetic 
axis). The geometry is
shown in Fig.~\ref{EPP:target-fig}. It is assumed that the thermal shock
from the interacting proton bunch fully disperses the mercury. In this
case, the jet must have a velocity of 30~m/s to be replaced before
the next bunch. Perturbations to the jet by the capture magnetic field
are controlled by placing the jet nozzle inside the field, so that the
jet only sees 1~T field changes before it has passed beyond the
production region.

\begin{figure}
\begin{center}
\vskip1.1in
{\small
\hskip.3in
\input{palmer-tgt1.fig}}
\caption{Mercury jet target geometry. }
\label{EPP:target-fig}
\end{center}
\end{figure}
%\begin{figure}
%\begin{center}
%\includegraphics[width=0.75\linewidth]{KMcDonald-Target.eps}
%\caption{Mercury jet target geometry. }
%\label{EPP:target-fig}
%\end{center}
%\end{figure}



\begin{figure}
\begin{center}
{\small
\vskip.3in
\hskip.65in
\input{palmer-tgt2.fig}}
\vskip.65in
%\begin{center}
\caption[Mercury enclosure, mercury-pool beam dump]{Mercury enclosure, mercury-pool beam dump, and solenoid capture magnets.}
\label{EPP:fgcapture}
\end{center}
\end{figure}


Pions emerging from the target are captured and focused down the decay
channel by a solenoidal field that is 20 T at the target center, and
tapers down, over 18 m, to a periodic (50 cm) superconducting solenoid
channel ($<B_z>\approx 1.25$~T) that continues through the phase
rotation to the start of bunching.

Figure~\ref{EPP:fgcapture} shows a section of the 20 T hybrid magnet, the front
end of the taper, the mercury containment, and the mercury pool proton
beam dump. The 20 T solenoid, with a hollow copper conductor magnet insert and
superconducting outer coil, is not different in character from the higher
field (up to 45 T), but smaller bore, magnets at several existing
laboratories. However, the magnet insert in this design is made with hollow
copper conductor and ceramic insulation to withstand
radiation. MARS~\cite{EPP:mars} simulations of radiation levels show that,
with the shielding provided, both copper and superconducting magnets
could have a lifetime greater than 20~years, even at 4~MW.

\subsubsection{Phase Rotation}
Pions, and the muons into which they decay, are generated in the
target over a very wide range of energies, but in a short time pulse
(3~ns rms). This large energy is phase rotated 
using drifts and induction linacs into a pulse with a longer time duration
and a lower energy spread. The muons first drift to
spread out their time, the induction linacs then decelerate the early
ones and accelerate those later. Three induction linacs (with lengths
100, 80, and 80 m) are used in a system that reduces distortion in the
phase-rotated bunch, and allows all induction units to operate with
unipolar pulses~\cite{EPP:refnon}. The 1.25-T beam transport solenoids are placed inside
the induction cores to avoid saturating the core material. The induction
units are similar to those being built for DARHT\cite{daarht}.

Between the first and second induction linacs, two hydrogen absorbers
(each 1.7~m long and 30~cm radius), with a magnetic field reversal between
them, are introduced to reduce the transverse emittance (``minicooling").

%Fig.\ref{EPP:fgrotation}) shows distribution of pions immediately after the
%target and b) after the phase rotation. It is seen that there is
%relatively little distortion in the rotated distribution.
%\begin{figure}
%\begin{center}
%\input{ndrot.fig}
%\caption[Phase rotation: kinetic energy \textit{vs.} time]{Phase Rotation: The kinetic energy \textit{vs.} time, before (TOP) and after (BOTTOM) the phase rotation}
%\label{EPP:fgrotation}
%\end{center}
%\end{figure}
%\afterpage{\clearpage}
\subsubsection{Buncher}
The long bunch (400~ns) after the phase rotation is bunched at 201.25~MHz
prior to cooling and acceleration at that frequency. The bunching is
done in a lattice identical to that at the start of cooling, and is preceded 
by a matching section from the 1.25~T
solenoids into this lattice. The bunching has three stages, each
consisting of rf (with increasing acceleration) followed by drifts
with decreasing length (27.5~m, 11~m, 5.5~m). In the first two rf
sections, second harmonic rf is used together with the 201.25~MHz to
improve the capture efficiency.
\subsubsection{Cooling}
Transverse emittance cooling is achieved by lowering the beam energy
in hydrogen absorbers, interspersed with rf acceleration to keep the
average energy constant. Transverse and longitudinal momenta are
lowered in the absorbers, but only the longitudinal momentum
is restored by the rf. The emittance increase from Coulomb scattering is
minimized by maintaining the focusing strength so that the angular spread
of the beam at the absorber locations is large. This is achieved by 
keeping the focusing strength
inversely proportional to the emittance; \textit{i.e.}, increasing as the
emittance is cooled. This could be achieved by a simple
solenoid, but such a field also must be reversed periodically to avoid a
growth of angular momentum.  For
this study, a modified Focus-Focus (SFOFO)~\cite{EPP:refsfofo} lattice is
employed. The solenoidal fields in each cell alternate in sign and the field
shape is chosen to maximize the momentum acceptance ($\pm 22$\%).

Figure~\ref{palmer-coolemit} shows a simulation of cooling, the emittance 
 falls along the length of the channel.
\begin{figure}
\begin{center}
\hskip.3in\input{palmer-coolemit.fig}
%\begin{center}
\caption{Transverse emittance along the cooling channel.}
\label{palmer-coolemit}
\end{center}
\end{figure}
\subsubsection{Acceleration}
A 20~m SFOFO matching section, using normal conducting rf systems, 
matches the beam optics to the requirements of a 2.5~GeV 
superconducting rf
linac with solenoidal focusing. 
The linac is in three parts. The first part has a single 2~cavity
unit per cell. The second, as a longer period becomes possible, has
two 2~cavity units per cell. The last section, with still longer period, accommodates four 2~cavity units per cell.

This linac is followed by a single, recirculating linear accelerator
(RLA) that raises the energy from 2.5~GeV to 20~GeV, in 4~passes. This
RLA uses the same 4~cavity superconducting structures. The arcs have
an average radius of 62~m. The final arc has a dipole field
of 2~T.            
%%  check this

\subsubsection{Storage Ring}               
After acceleration in the RLA, the muons are injected into the upward 
straight of a
racetrack shaped storage ring with a circumference of $\approx 358$m.   
High field superconducting arc magnets
are used to minimize the arc length and maximize the
fraction (35\%) of muons that decay in the downward straight and
generate neutrinos headed towards the detector at the WIPP facility in
Carlsbad, 2903~km away.  All
muons are allowed to decay; the total
heating from the decay electrons is 42~kW (126~W/m). This load is too high
to be dissipated in the superconducting coils. A magnet design has
been chosen~\cite{skrinsky} that allows the majority of these electrons
to pass out between separate upper and lower cryostats, and be
dissipated in a dump at room temperature. To maintain the
vertical cryostat separation in focusing elements, skew quadrupoles
are employed in place of standard quadrupoles.

In order to maximize the average bending field, Nb$_3$Sn pancake coils
are employed.   One coil of
the bending magnet is extended and used as one half of the previous or
following skew quadrupole, (see Chapter~\ref{MSR:chap}).

Figure~\ref{EPP:fgsection} shows a cross section of the ring, which is kept
above the water table and is placed on a roughly $30$~m high berm. The 110~m 
high BNL stack is also shown for scale.

%\begin{figure}
%\begin{center}
%\vskip-.5in
%\hskip.35in\input{palmer-trans2.fig}
%\begin{center}
%\vskip-.2in
%\caption{Cross section through ring and cover. }
%\label{EPP:fgsection}
%\end{center}
%\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{../template/report/ps-and-eps/mole-hill.eps}
\caption[Top view and cross section through ring and berm]{Top view and cross section through ring and berm. The 110~m tall tower, drawn to scale, gives a sense of the height of the ring on the BNL landscape.}
\label{EPP:fgsection}
\end{center}
\end{figure}

\subsection{Performance}

Complete simulations up to the start of acceleration have been
performed using the code MARS~\cite{EPP:mars} (for pion production) followed by
ICOOL~\cite{icool} (for transport, phase rotation and cooling). These results 
have been confirmed by GEANT4~\cite{EPP:GEANT4}. They show an average of 
0.17 final muons per initial proton on the target, 
\textit{i.e.}, $0.0071\mu$/p/GeV, (considering the energy of the initial beam). 
This can be compared with a value of $0.0011\mu$/p/GeV produced in 
Study I~\cite{INTRO:ref1}. The gain $(6\times)$
comes from: 
\begin{itemize}
\item use of mercury, instead of carbon as a target (1.9
$\times$)
\item use of three, instead of only one, phase
rotation induction linacs (2 $\times$)
\item use of a more efficient, tapered cooling channel design (1.4 $\times$)
\item use of a larger accelerator acceptance (1.2 $\times$)
\end{itemize}

The muons delivered to the ring with a 1~MW (4~MW) proton driver would be:

\begin{eqnarray}
 \mu/{\rm year}&=
10^{14}(\textrm{ppp})\times 2.5~(\textrm{Hz}) \times 10^7~(\textrm{s})\times 0.17~(\mu/p)\times 0.81~(\textrm{acc. efficiency})\nonumber \\
&=3.4\times 10^{20}~(=13.6\times 10^{20})
\end{eqnarray}
and the number of muons decaying in the production straight section would be
$$1.2\times 10^{20}~~(=~4.8\times 10^{20})$$.


\subsection{Conclusions}

This Study II shows significant improvements $(6\times)$ over Study-I, 
yet there remains the possibility of further gains.  Cooling of the
longitudinal emittance~\cite{exchange} and the capture of both
signs~\cite{bbpr} appear possible and, together might improve overall
performance by a factor between 2 and 4. 

 %  better summary
 %    feasible
 %    prformance enough for good physics
 %    suitable for BNL site
 %    most comp0onents are generic  (cooling??)
 %    CAN DO GOOD PHYSICS





