\section{PROTON DRIVER}  
 The overview of the required parameters is followed by a description of 
% possible 
designs that have been studied in some detail.  The section concludes
with a discussion of the outstanding open issues.

\subsection{Specifications}

The proton driver requirements are determined by the design 
luminosity of the collider, and the efficiencies of muon 
collection, cooling, transport and acceleration.  The baseline 
specification is for a 4-MW, 16-GeV or a 7-MW, 30-GeV proton driver, with a repetition 
rate of 
15~Hz and $10^{14}$ protons per cycle in 2~bunches (for the 100-GeV machine) 
or 4~bunches (for the higher energies) of 
$5\times 10^{13}$ or $2.5\times 10^{13}$ protons, respectively.
Half the bunches are used to make $\mu^{-}$ and the rest for $\mu^{+}$ \cite{Norem97b}.  


The total beam 
power is several MW, which is larger than that of existing synchrotrons.   
However, except for bunch length, these 
parameters are similar to those of Kaon factories \cite{kaon} and
spallation neutron sources \cite{sns}. 
 As in those cases, the proton driver must have very low losses to permit 
inexpensive maintenance of components.


The rms bunch length for the protons on target has to be about 1~ns 
to: 1) reduce the initial longitudinal emittance of muons entering the 
cooling system, and 2) optimize the production of polarized muons.
Although bunches of up to $6\times 10^{13}$ protons per cycle have been 
accelerated, the required peak current  %at extraction for 1~ns and 4~bunches, 
is 2000~A, which is unprecedented.   

Since the collection of highly polarized $\mu$'s is inefficient (see section~\textbf{IV.G}), 
the proton driver should 
eventually provide an additional factor of two or more in proton intensity to 
permit the luminosity to be maintained for polarized muon beams.

\subsection{Possible options}

Accelerator designs are site, and to some extent, time dependent, and there
have been three studies at three different energies (30~GeV \cite{Snowmass3}, 
16~GeV \cite{fnal_study} and 24~GeV \cite{agsint95,Roser96,Ahrens97,Roser98}; see also \cite{Cho96}).  In general, if the final energy is higher, 
the required currents are lower, bunch manipulation and apertures are 
easier, and the final momentum spread and space-charge tune shifts are less.  
Lowering the final energy gives somewhat more $\pi$'s/Watt, a lower rf 
requirement ($V_{rf}\sim E^{2}$) and perhaps a lower facility cost.

In the low-energy muon collider, where two bunches of protons of $5\times 10^{13}$ are required on target,  
two bunches can be merged outside the driver. These two bunches would be extracted 
simultaneously from two different extraction ports, and fed by different 
transmission lines to the same target. By arranging the path lengths of the 
two lines appropriately, % the time of arrival of 
the two bunches can be exactly merged. 
%This is explicitly assumed in the FNAL study quoted below
%\cite{fnal_study}.

\subsubsection{A generic design}

A 7-MW collider-driver design based on parameters originally 
proposed in the Snowmass Feasibility study \cite{Snowmass3}
consists of a 600-MeV linac, a 
3.6-GeV booster and a 30-GeV driver.   Both linac and 
booster are based on the BNL Spallation Neutron Source design \cite{sns}, 
using a lower 
repetition rate and a lower total number of protons per pulse.  For the 4-bunch
 case ($2.5\times 10^{13}$ protons per bunch), the 
(95\%) bunch area is assumed to be 2~eV-s at injection and $<4.5$~eV-s at 
extraction.  The driver lattice is derived from the lattice of the JHF 
driver using 90$^\circ$ FODO cells with missing dipoles in every third 
FODO cell, allowing a transition energy that is higher than the maximum 
energy or, perhaps, imaginary.

\subsubsection{FNAL study}

If a muon collider is built at an existing laboratory, then possibilities 
abound for symbiotic relationships with the other facilities and programs of 
that laboratory.  For example, the proton driver for a muon collider might 
result from an upgrade of existing proton-source capabilities, and such an 
upgrade could then also enhance other future programs that use the proton 
beams.  

Fermilab has conceived such a proton-source development 
plan \cite{HolmesED}  %  The plan is modular, 
with three major components: an upgraded linac, a prebooster and a new booster, with the two boosters being rapid-cycling (15 Hz) synchrotrons.  The two 
synchrotrons operate in series; the four proton bunches for the muon collider 
are formed in the prebooster and then accelerated sequentially in the 
prebooster and the booster.  The plan could be implemented in stages, and other 
programs would benefit from each stage, but all three components are required 
to meet the luminosity goals of the muon colliders that have been considered so 
far.

\begin{table*}[tbh!] 
\centering
\caption{Baseline proton-driver parameters of the FNAL study. %\cite{HolmesED}.
\label{key}}
\begin{tabular}
[c]{llll}
              & \textbf{Linac} & \textbf{Booster} & \textbf{Driver} \\
\hline
Energy range (GeV)          & 1    & 3     & 16  \\
Rep.\ rate (Hz)            & 15    & 15     & 15 \\
RF voltage per turn (MV)   &         & 0.15  & 1.5  \\
Circumference  (m)        &         & 158    & 474 \\
Protons per bunch  ($\times 10^{13}$)   & &$2.5 $&$2.5 $ \\
Beam emittance [95\%] ($\pi$ mm-mrad )  &         & 200& 240 \\
Bunch area [95\%] (eV-s)     &         & 1.5 & $< 2.0$  \\
Incoherent tune shift @ Inj. &   & 0.39     & 0.39 \\
\end{tabular}
\end{table*}%

Table~\ref{key} presents the major parameters of the two rings.  
%The rationale for the major design choices is presented here.  
Whenever the needs of the muon collider itself allow some flexibility, 
the parameters have been chosen to optimize the resulting facility as a proton 
source for the rest of the future program at Fermilab.  For example, the 
machine circumferences and rf-harmonic numbers result in bunch trains that are 
compatible with the existing downstream proton machines.

A muon collider requires proton bunches that are both very intense and, at the 
pion-production target, very short. Strong transverse and longitudinal 
space-charge 
forces might disrupt such bunches in the synchrotrons unless measures to 
alleviate those effects are incorporated in the design.  The Laslett 
incoherent-space-charge tune shift quantifies the severity of the transverse 
effects.  A 
useful approximation for the space-charge tune shift $\Delta \nu_{sc}$ at the 
center of a round Gaussian beam is
\begin{equation}
\Delta \nu_{sc} = - {{3 r_{p} N_{\rm tot}}\over {2 \epsilon_{n}\beta 
\gamma^{2} b}}
\label{tuneshift}
\end{equation}
In this expression $r_{p} = 1.535\times10^{-18}$~m is the so-called 
electromagnetic radius 
of the proton, $N_{\rm tot}$ is the total number of protons in the ring, 
$\epsilon_{n}$ is the 95\% 
normalized transverse emittance, $\beta$ and $\gamma$ are the usual Lorentz 
kinematical 
factors, and $b \leq 1$ is the bunching factor, defined as the ratio of the 
average beam current to the peak current.  

The approximation (\ref{tuneshift}) implies that for a given total number of 
protons, here $10^{14}$, the 
factors in the denominator are the only ways to reduce the tune shift to a 
specified maximum tolerable value, taken as 0.4.  The bunching factor can 
be raised somewhat by careful tailoring of beam distributions, but here a 
typical value of 0.25 is conservatively assumed.  Achieving the desired beam 
intensity then requires a combination of high injection energy, here taken as 
1~GeV into the first ring, and large transverse normalized emittances, here 
assumed to be about $200\pi$~mm-mrad.  The corresponding required aperture is 
about 
13~cm in the first ring and about 10~cm in the second ring.  With such large 
apertures in rapid-cycling synchrotrons, careful design of the beam pipes for 
both rings is required to manage eddy-current effects.  Two approaches 
are under consideration.  One is a thin Inconel pipe with water cooling and 
eddy-current coil corrections integrated on the pipe, as in the AGS Booster.  
The other is a ceramic beam pipe with a conductor inside to carry beam-image 
currents, as in ISIS.

The Fermilab linac presently delivers a 400-MeV beam, and is capable with 
modest modifications of accelerating as many as $3\times 10^{13}$ protons per 
cycle at 
15 Hz \cite{Popovic}.  A significant upgrade is required in order to 
deliver $10^{14}$ protons at 1~GeV.  The energy can be raised by appending 
additional side-coupled modules to the downstream end of the linac.
Increasing the linac beam intensity probably means increasing both the 
beam current and the duration of the beam pulse.  Injection into the first ring 
is by charge stripping of the $H^{-}$ beam; the incoming beam will be chopped 
and injected into pre-existing buckets to achieve high capture efficiency.

The circumference of the second ring is set equal to that of the existing 
Fermilab Booster.  This choice provides several advantages.  First, the new 
booster could occupy the same tunnel as a relocated Booster; secondly, the 
beam-batch 
length from a full second ring matches that of the present Booster, which 
simplifies matching to downstream machines for other programs.  The output 
energy of 16~GeV then results from an assumed dipole packing fraction of 0.575 
and a peak dipole field of 1.3~T, which is the highest dipole field that 
is consistent with straightforward, nonsaturating design of magnets having 
thin silicon-steel laminations.  Driving such magnets into saturation would 
cause significant heating of the magnet yoke as well as potential problems with 
tracking between the dipoles and quadrupoles.  
%The energy of 16~GeV is a good 
%choice for pion production, because it is near the value where the linear rise 
%of pion/proton ratio with energy begins to saturate.

The prebooster also has 1.3-T dipole fields, and its circumference is one third 
that of the new booster; it operates at an rf harmonic number $h=4$.  The 
strategy for achieving the required short bunches at the target while 
alleviating space-charge effects in the rings is to start with four bunches 
occupying most of the circumference of the first small ring in order to keep 
the bunching factor large, and to do a bunch-shortening rotation in 
longitudinal phase space just before extraction from the second synchrotron.  
The four bunches are accelerated in the first ring to 3~GeV, then transferred 
bunch-to-bucket into the second ring with its harmonic number $h=12$.  At that 
energy, the kinematic factor in the tune-shift formula (\ref{tuneshift})
is large enough to 
compensate for the smaller bunching factor in the second ring. The transfer 
energy of 3~GeV between the two rings roughly equalizes their space-charge tune 
shifts.

Both rings employ separated-function lattices with flexible momentum compaction in order to raise their transition energies above their respective extraction 
energies.  This not only avoids having to accelerate beam through transition 
but also provides other advantages.  Intense beams are not subject to certain 
instabilities such as the negative-mass instability below transition and 
empirically seem less susceptible to other instabilities such as the microwave 
instability.  Also, the negative natural chromaticity is beneficial for 
stabilizing the beam below transition, thereby perhaps obviating the need for 
sextupole correctors, especially in the first ring.  Having transition not too 
far above extraction also provides substantial bucket area in which to 
accomplish beam-shortening rf manipulations.

Several potential sources of instabilities in the rings have been
examined \cite{Ng}.  Space charge is the main factor 
affecting the stability of the beams;  the rings appear to be 
safe from longitudinal- and transverse-microwave 
instabilities.  Of course, standard stabilizing methods such 
as active dampers are necessary to counteract some of the 
instabilities.  Flexible momentum-compaction lattices would 
be useful not only to raise the transition energy above the 
extraction energy, but also to allow fast changes in the slip 
factor to facilitate bunch-narrowing manipulations at 
extraction time.  
%Compensation of longitudinal space-charge 
%effects by means of ferrite-loaded inductive inserts would be 
%useful, especially for the first ring.

The magnet-power-supply circuit for each ring is a 15-Hz resonant system like 
that of the existing Booster, with dipoles and quadrupoles electrically in 
series.  This implies that the second ring will accelerate only one batch at a 
time from the first ring, which is all that the muon collider needs.  Adding 
about 15\% of second harmonic to the magnet ramp reduces the required peak 
accelerating voltage by about 25\%, which is probably worth doing, especially 
for the second ring with its large voltage requirement.  

One of the advantages 
of a two-ring system is that the two rings divide the work of accelerating the 
beam.  The rf system of the first ring is relatively modest because of its 
small circumference and small energy gain; that of the second ring is 
simplified because its high injection energy means a small rf-frequency swing 
\cite{GriffinQKerns}.

ESME simulations of longitudinal motion show that the rms bunch length is 2~nsec
as desired after the bunch rotation that occurs just before extraction 
from the second synchrotron.%~\cite{Kourbanis}.  
The bunch rotation creates momentum spreads of about 2\% with longitudinal 
emittances of about 2~eV-s per bunch.  Such spreads would contribute a few 
cm in quadrature to the beam size for a short period before 
extraction.  This is thought to be tolerable, given the large apertures that 
are required in any case.  High injection 
energies help to alleviate these longitudinal effects, which result from 
space-charge voltages having the same $1/\beta\gamma^{2}$ kinematic dependence 
as the transverse tune shifts.  

\subsubsection{AGS upgrade}

The third study \cite{agsint95,Roser96,Ahrens97,Roser98} is of an upgrade to the BNL AGS, which
should produce 
bunches larger than those required for the muon collider, but
at a lower repetition rate.   The AGS presently produces $6\times 10^{13}$ 
protons 
in eight bunches at 25~GeV and 0.6~Hz.  A 2.5-GeV accumulator ring in 
the AGS tunnel and AGS power-supply upgrade to 2.5-Hz operation would 
match the repetition rate to the 10-Hz  repetition rate of the 
booster. This would generate 1~MW beam power.  
With an additional upgrade of the linac energy to 600~MeV, an 
intensity of $2\times 10^{14}$ protons per pulse in four bunches of 
$5\times 10^{13}$ at 25~GeV and 2.5~Hz could be reached, raising the power to 
2~MW. The upgrades to the AGS accelerator complex are summarized in Fig.\ref{ags-rhic}. Other options are also under consideration, such as the addition of a 
second booster and 5-Hz operation, that would reach the baseline specification 
of 4~MW. 
\begin{figure*}
\begin{center}
\includegraphics[width=5.5in]{ags-rhic.eps} 
\end{center}
\caption[The AGS--RHIC accelerator complex]{The AGS--RHIC accelerator complex and a summary of possible intensity upgrades for the AGS.  }
\label{ags-rhic}
\end{figure*}

The AGS momentum acceptance of $\pm3$\% requires that the 
longitudinal phase space occupied by one bunch be less than 4.5~eV-s.  
This high bunch density in turn generates stringent demands on the 
earlier parts of the accelerator cycle.  In particular, Landau damping
from the beam momentum spread may guard against resistive wall 
instabilities during injection and longitudinal microwave 
instabilities after transition.   Beam stability can be restored with 
a more-powerful transverse-damping system and possibly a new low-impedance 
vacuum chamber.  The transverse microwave instability is 
predicted to occur after transition crossing unless damped by 
Landau damping from incoherent tune spread or possibly high-frequency 
quadrupoles. 



\subsection{Progress and open issues}

Conventional rf manipulations appear able to produce 1- to 2-ns proton 
bunches if enough rf voltage to overcome the space-charge forces is 
used, and the beam energy is far enough from transition so the final 
bunch rotation is fast. 
Both simulations and experimental work have been directed at 
demonstrating that a short pulse can be produced easily.  

An experiment at the AGS has shown that bunches with $\sigma_z=2$~ns can
be produced near transition from bunches with $\sigma_z \sim 8$~ns by
bunch rotation \cite{ref10,agsbunch}.  In this experiment, the AGS was
flattoped
near transition ($\sim 7$~GeV) while the $\gamma_t$-jump system
was used to bring the transition energy suddenly to the beam energy,
letting the bunch-energy spread expand and bunch length contract.  The
experiment also demonstrated that bunches are stable against microwave instabilities.  In addition, the data were used to measure the lowest two orders
of the momentum compaction factor.

The AGS bunch area, 1.5 eV-s, was comparable to that expected in the
proton driver, but the bunch charge (though as large as 3-$5 \times
10^{12}$ protons) was only about one tenth of that required by the muon
target.  The proton driver would use a flexible momentum compaction lattice which would allow tuning far from transition and permitting a fast
final bunch rotation \cite{Trbojevic97b}. In addition, the rf frequency would be higher than
that of the AGS so the buckets (and bunches) would initially be only
half as long.  Thus bunch rotation could be expected to be easier with
the new machine, which should compensate for the larger charge.

Simulations with the ESME code have also 
shown that 1-2~ns bunches of $5\times 10^{13}$ can be produced at 
extraction in a 16-GeV ring with the rf and emittance shown in Table~\ref{key}.

The efficiency of capturing and accelerating beam may be increased by 
compensation of the space-charge forces in the proton driver.  The use 
of tunable inductive inserts in the ring vacuum chamber may permit 
active control and compensation of the longitudinal space charge below 
transition (since the inductive impedance is of the opposite sign to 
the capacitive space charge).  Initial experiments at the 
KEK proton synchrotron and Los Alamos PSR \cite{PSR}
with short ferrite inserts appear to 
show a reduction in the synchrotron oscillation frequency shift 
caused by space charge and a decrease in the necessary rf voltage to 
maintain a given bunch intensity. Further experiments are 
needed to demonstrate this technique fully.

The high rf voltage and beam power and the relatively small
size of the machine require high-gradient, high-power rf cavities.
Fermilab, BNL and KEK are collaborating
on research and development of such type of cavities.%\cite{give_reference1}. 
This
work includes the study of magnet alloys and hybrid cavities using 
both ferrite and new magnet alloys, high-power amplifiers and beam-loading
compensation.

The employment of barrier-bucket \cite{give_reference2} rf cavities can effectively
generate and manipulate a gap in the beam and reduce the space-charge effect. A successful test of this scheme has recently been completed \cite{give_reference3}, and two $40-$kV barrier cavities have been built by BNL and KEK
and are being installed on the AGS. Another high-gradient
barrier cavity using magnet alloys is under study at Fermilab.
%\cite{give_reference3}.
