\section{Proton Driver}
\label{APP:Proton}
\subsection{Increasing Power to 2 or 4~MW}
%original Palmer
With an increased superconducting linac energy, the proton intensity could be increased by a factor of two (from $1\times 10^{14}$ to $2\times 10^{14}$.) A further increase of a factor of two in average proton power could be achieved by adding a 24~GeV storage ring and operating the AGS at 5~Hz. 
%These options are discussed in Section~\ref{APP:Proton}.
%\input{appendix-bc}
An upgrade to 4 MW beam power is possible by upgrading the AGS
repetition rate to 5~Hz and increasing the linac energy to 1.5~GeV,
which allows for doubling the number of protons per pulse to $2\times
10^{14}.$ To achieve the required bunch length compression a separate
compressor ring would be needed. This ring will
\begin{itemize}
\item operate below transition
\item have a small slippage factor, that is, it will be quasi-isochronous 
\item have low dispersion
\item have an  acceptance to emittance ratio $>8$ (to be compatible with the tight beam loss limit)
\item have a chromaticity correction system
\end{itemize}
Table~\ref{Proton:tb5} summarizes the key parameters of the compressor ring. As discussed in Section~\ref{improvement}, the performance penalty of operating with longer bunches is not severe, so the cost-benefit evaluation of the compressor ring must be considered carefully.
\begin{table}[!hbt]
\begin{center}
\caption{Compressor ring parameters.}
\label{Proton:tb5}
\begin{tabular}{|lc|}
\hline
Circumference (m)
&200 \\
Bending field (T)
&4.15 \\
Kinetic energy (GeV)
&24\\
Transition gamma&38.4\\
$\eta$&
0.00074\\
Betatron tune, ${x/y}$&14.8/9.2 \\
Maximum beta function, ${x/y}$ (m)&12.9/19.8\\
Dispersion function (m)&0.12 \\
Chamber radius (mm)
&25 \\
Maximum beam radius, ${x/y}$ (mm)&7.0/8.6 \\
Acceptance, ${x/y}$ (m)&48.5/31.6 \\
Beam emittance, ${x/y}$ (m) &3.8/3.8\\
Accep./emit. ratio, ${x/y}$&12.8/8.3\\
Natural chromaticity, ${x/y}$
&$-2.5/-1.7$ \\
\hline
\end{tabular}
\end{center}
\end{table}

In operation, an unmatched bunch is injected from the AGS into the compressor ring. It is extracted immediately after a bunch rotation (bunch rotation takes a quarter of a synchrotron period, {\it i.e.}, 3~ms, or 4500~turns).  Because of the very small slippage, a low rf voltage is required (see Table~\ref{Proton:tb6}.)

\begin{table}[!hbt]
\begin{center}
\caption{rf parameters of compressor ring.}
\label{Proton:tb6}
\begin{tabular}{|lc|}
\hline
RF frequency (MHz)
&5.94 \\
Harmonic number&4\\
$V_{rf}$ (kV)&200\\
Bucket height, in $\delta p/p$ &0.042 \\
Bucket area (eVs)&222 \\
Bunch area (eVs)&10\\
$f_s$, center (Hz)&91.5\\
$f_s$, edge (Hz)&82.6\\
\hline
\end{tabular}
\end{center}
\end{table}

The longitudinal parameters of the ring are summarized in Table~\ref{Proton:tb7}
\begin{table}[!htb]
\begin{center}
\caption{Longitudinal parameters of compressor ring.}
\label{Proton:tb7}
\begin{tabular}{|lcc|}
\hline
&Injection&Extraction\\
\hline
No. particles per bunch ($10^{14}$)
&0.17&0.17\\
RMS bunch length (m/ns)
&5/17&0.9/3\\
Peak current (A)
&65&363\\
Momentum spread (\%)
&0.4&2.24\\
Longitudinal emittance (eVs)
&10.5&10.5\\
Broadband impedance ($j\Omega$)
&5&5\\ 
Space-charge impedance ($j\Omega$)
&1.66&1.66 \\
Keil-Schnell threshold ($j\textrm{M}\Omega/\textrm{m}$)
&3.75&25.5\\
Effective rf voltage (kV)
&200&248\\
\hline
\end{tabular}
\end{center}
\end{table}

Clearly, the longitudinal microwave instability threshold will be low at the injection energy, because of the small slippage factor and the low $\delta p/p.$ To reduce the impedance, the vacuum chamber will have smooth tapered transitions. However, we do not plan to shield the bellows to avoid possible problems with arcing. Despite this we expect to achieve a broad impedance of $5~\Omega,$ which is acceptable.
 
We see from Table~\ref{Proton:tb7} that the combination of the broadband and the space-charge impedance is $3.34~\Omega,$ slightly lower than the Keil-Schnell (KS) threshold. Since the beam is below transition, beam instability is not expected. The overall inductive impedance below transition has a focusing effect, which increases the effective rf voltage in the bunch rotation.

\begin{table}[!bth]
\begin{center}
\caption{Transverse parameters.}
\label{Proton:tb8}
\begin{tabular}{|lcc|}
\hline
&Injection&Extraction\\
\hline
Broadband impedance ($j\textrm{M}\Omega/\textrm{m}$)
&0.51
&0.51\\
BB imp. induced tune shift
&0.0003
&0.0017 \\
Space-charge induced tune spread
&0.003
&0.016\\
Chromatic tune spread 
&0.22
&1.32 \\
Chromatic frequency (GHz)
&59.4
&59.4\\
\hline
\end{tabular}
\end{center}
\end{table}
In Table~\ref{Proton:tb8} we summarize the transverse parameters of the 
compressor ring. We find that the transverse impedance is low, as expected 
for a small ring $(Z_T\propto R).$ 
 Compared with the AGS, the compressor ring is transversely more stable 
(this is just opposite to the situation in case of longitudinal instability). 
The space-charge incoherent tune spread is small and is helped by the strong 
focusing optics. If the chromaticity were not corrected, the chromatic tune 
spread would be large. This is due to the small slippage factor, the high revolution frequency, and the high tune. For these reasons, we will control the normalized chromaticity to about 1\%.
\begin{figure}[!htb]
\begin{center}
%\input{dept.fig}
\includegraphics*[bb=133 318 419 544,clip]{../template/report/ps-and-eps/jcg-dept.ps} 
\caption{Efficiency \textit{vs.} proton bunch length.}
\label{muvst}
\end{center}
\end{figure}

The compressor ring design requires very low rf voltage; also, the
potential-well effect facilitates the short bunch production. The
required impedance is reasonable to achieve, and the
acceptance/emittance ratio of 8~units is much larger than that for existing
and proposed high intensity proton accelerators. In conjunction with
the large momentum aperture it is reasonable to expect that beam losses can be controlled. Chromaticity control at the compressor ring is not easy, however, and needs further studies.
%\afterpage{\clearpage} 
\subsection{Proton Bunch Length and a Buncher Ring}
\label{improvement}
\begin{table}[htb!]
\begin{center}
\caption{Efficiency \textit{vs.} proton bunch length.}
\vspace{2.5mm}
\begin{tabular}{|c|cc|}
\hline
rms bunch length&  ~~~~$\mu$/p~~~~ &relative \\
  (ns)       &  &\\
\hline
1 & 0.204 &1.02\\
3 & 0.20  &1.0\\
6&  0.167 &0.835\\ 
9&0.144&0.72\\
\hline
\end{tabular}
\label{muvsttab}
\end{center}
\end{table}

The minimum proton driver bunch length achievable is set by the longitudinal
emittance of the bunches and by the momentum acceptance of the AGS.
 For the baseline 1~MW case, we expect to achieve the specified
rms bunch length of 3~ns without a bunch compressor
ring.

However, if the proton bunch intensity is increased by a factor of two to reach 2 or 4~MW, as
discussed in Section~\ref{APP:Proton}, then the bunch length would be expected to increase, 
and the specified 3~ns rms bunch length could not be achieved without
increasing the momentum spread above the AGS acceptance. 

The consequence of such an increase in bunch length was simulated,
without reoptimization. (It is not expected that any reoptimization
will markedly improve the result.) The final muon per proton ratios obtained are given
in Table~\ref{muvsttab} and Fig.~\ref{muvst}. Note that the cooling
system used in this early study had larger apertures, and thus higher
performance, than the final design, but the sensitivity to bunch
length is expected to be the same. It is seen that there is relatively
little gain for pulse lengths less than 3~ns (the baseline value). For a 6~ns bunch the efficiency has dropped 16.5\%, and for
9~ns, the efficiency has dropped by 28\%.

%To avoid such losses of performance, one can employ a bunch
%compressor ring. The ring would have fixed field 
%superconducting magnets and would be much smaller in diameter,
%but have larger momentum acceptance, than the AGS. The smaller diameter
%would reduce space-charge effects in the bunched beam, and the larger
%acceptance would allow the short bunches.Properties of such a ring are described in Section~\ref{APP:Proton}.
%%%%%%%%%%%%%%%%%%%%
\section{Target}
%\subsection{ Moving metal band }
\subsection[Rotating Inconel Band Option]{Rotating Inconel Band Option}
\label{APP-band}

If unforeseen difficulties make a liquid metal target undesirable, then
a moving metal band target is a possible alternative. The
performance would be little different from the metal jet. The scheme
is discussed in Section~\ref{APP-band}. 
%A ``granular'' target has also been suggested recently as an alternative approach, but this has not been examined during Study-II.~\cite{nufact01}
%\subsection[Rotating Inconel Band Option]{Rotating Inconel Band Option}
%\label{APP-band}
\input{appendix-bb1}
\subsection{Carbon}
As demonstrated in Study-I~\cite{ref1}, a radiation cooled graphite target could be used up to 1.5~MW power level.  It appears to be a
relatively conservative solution (at 1~MW) but would sacrifice a
factor of 2 in performance and require relatively frequent
replacement. It is unclear if it could be used at 4~MW.
\subsection{Granular}
A ``granular'' target has also been suggested recently as an alternative approach, but this has not been examined during Study-II.~\cite{nufact01}.
%%%%%%%%%%%%%%%%%%%%
\section{Phase Rotation}  
\subsection{Correlation Matching}
Within the cooling lattice, particles with high transverse amplitude
travel on longer orbits than those on the axis and thus, for a given
momentum, move more slowly in the forward direction. 
In such a lattice, with active rf, the average forward
velocity is controlled by the phase velocity of the rf and constrained
to a fixed value. As a result, the stable momentum of a particle is
dependent on its amplitude
\begin{equation}
 \frac{dz}{dt}~-~\beta c~~\propto ~ A^2,
\end{equation}
where the approximately conserved particle amplitude is given by,
\begin{equation}
A~=~ {x^2+y^2 \over \sqrt{\beta}}~+~\sqrt{\beta}~(x'^2+y'^2).
\end{equation}

Such a correlation is also generated naturally in the phase rotation
process but, since the phase rotation is done in a different lattice
from the cooling, the magnitude of the correlation is not the same. As
a result, there is, in the present design, a mismatch in correlation
at the entry to the cooling channel. Study is needed to see if
performance could be improved by better matching to the optimal correlations, 
possibly
by raising the solenoid fields used in the transport and phase
rotation channels.
\subsection{Polarization}
A system of double phase rotation has been studied~\cite{pjk} that
generated a strong correlation between the muon polarization and final
time after the phase rotation. This correlation, though a little
diluted, is maintained through to the storage ring and results in
correlations between neutrino type and time of detection. The physics
need for such correlation has not been well established, and the
system requires high gradient (4~MV/m) low frequency (30~MHz) rf close
(3--6~m) to the target. The viability of rf in such high-radiation environment  has been questioned, but tests at CERN~\cite{radiation} suggest it may be feasible if it is needed.
\subsection{Bunched Beam Phase Rotation}
Cost savings may be possible by performing the phase
rotation with rf after bunching of the beam~\cite{bbpr1}. As in induction linac phase rotation, the bunch is
first allowed to drift to increase the bunch length and establish a
correlation between time and energy, but in this case the bunching is
done before the energy is corrected. The rf that performs this
bunching is acting on a beam with strong time-momentum correlation;
\textit{i.e.} a beam whose time spread is still increasing with drift distance,
and whose sub-bunches, as they are formed, have spacings that are also
increasing. This requires that the rf wavelengths used to create and
hold the bunches also rise with drift distance. After the bunches
have been formed, suitable modifications to the rf frequencies and
phases can be employed to accelerate the later bunches and decelerate
the early ones, thus ending up with a train of bunches at a single
energy, as in the conventional case.

The need to have cavities operating at many different frequencies is
certainly a complication. But since the cost of conventional rf
acceleration is likely to be less than that for induction acceleration, the
cost of the system is expected to be less. Whether it is
as efficient is less clear. For instance, non-distorting phase
rotation does not seem possible with the rf-based scheme. But this scheme has 
the interesting feature of working on muons of both signs---the bunches of the opposite
sign automatically form between the others. If both signs were
subsequently accelerated through the linacs and RLA (injected in the
opposite direction), and injected into the storage ring (also in the
opposite direction), then a factor of two in efficiency could be
achieved. This factor of two might compensate for any lower efficiency
in the phase rotation of muons of one sign. 

This solution is far from worked out, but seems worth evaluating. Injection 
into the ring must be such that timing can be used
by the detector to separate the neutrinos from the two different muon
trains.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%