The proposed R\&D program into targetry issues for a muon collider consists of
eight parts:
\begin{enumerate}
\item
Initial studies of liquid (and solid) target materials 
with a proton beam at the AGS.  
\item
Studies of a liquid-metal jet entering a 20-T magnet
at the National High Magnetic Field Laboratory (NHMFL) in Florida.  
\item
Studies of a full-scale liquid-metal jet in a beam of $10^{14}$ protons per
pulse, but without magnetic field.   
\item
Studies of a liquid-metal jet + proton beam + 20-T pulsed solenoid magnet.  
\item
Studies of a 70-MHz rf cavity downstream of the target in the proton 
beam, but without a magnet around the cavity.
\item    
Continuation of topic 5 with the addition of a 1.25-T, 1.25-m-radius solenoid
surrounding the rf cavity.
\item
Characterization of the pion yield downstream of the target + rf cavity.
\item
Simulation of the performance of liquid-metal targets: thermal shock, 
eddy currents.  Validation of the simulation by exploding-wire studies.
\end{enumerate}

The key requirements on the AGS for this program have been presented in the
Executive Summary.

\subsection{Initial Studies of Targets in a Proton Beam}

The first set of studies concerns the viability of various target options
with respect to their interaction with the proton beam.  The key issues are the
effects of thermal shock (sec.~2.1.2), and (for solid targets) the
effects of radiation damage.  We propose to study thermal shock in four target 
geometries:
\begin{enumerate}
\item
{\bf A free liquid-metal jet.}  The jet may be dispersed by the beam, possibly
violently.  The 20-T magnetic field should damp the dispersal, but also
distorts the jet on entry into the field.
\item
{\bf Liquid-metal in a pipe.}  There is a high probability of
damage to the pipe by the beam-induced thermal-shock.  
\item
{\bf A cylindrical solid target.}  The cylindrical geometry may enhance 
beam-induced thermal-shock damage along the axis of the target.
\item
{\bf A rectangular solid target.}  For the energy deposition required at
a muon collider source, thermal-shock damage is not expected.  If confirmed,
the primary concern will then be the long-term effect of radiation damage 
on the mechanical integrity of a moving band target.
\end{enumerate}

Details of the test targets will be presented after a discussion of the
beam.

\subsubsection{The Proton Testbeam}
 
The testbeam should approach as closely as possible the parameters of the 
primary proton beam for a muon collider.  The key parameters are 16-24 GeV
energy, $10^{14}$ protons per pulse, rms radius of 4~mm and pulse length of
2 ns.

The candidate proton testbeam is the FEB U-line at the AGS \cite{FEB,Tanaka95},
shown in
Fig.~\ref{uline4}.  The location of the U-line in the AGS-RHIC complex can
be seen in Fig.~\ref{agscomplex}.
  This beam typically is operated at 24~GeV, extracting
a single bunch per AGS cycle with up to $10^{13}$ protons in that
bunch.  The beam emittance is about 100$\pi$~mm-mrad, and the pulse length
typically is $\sigma_t = 32$~ns.  The pulse rate can be as high as 0.8~Hz.

The AGS beam parameters are well matched to the needs of the proposed studies,
and with some upgrades the AGS would be an excellent muon-collider 
source \cite{status,Roser96}; see also Appendix B.

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=6.5in,clip=]{uline4.eps} 
\end{center}
\caption{Plan view of FEB U-line at the BNL AGS.  The residual radiation levels
at various point along the tunnel are indicated.}
 \label{uline4}
 \end{figure}

All of the beam tests presented in this proposal can be performed during
parasitic use of the beam at a pulse rate of less than one every two minutes.
The initial studies (secs.~3.1 and 3.3) are in effect a series of single-pulse
experiments, while later studies (secs.~3.4-7) will be limited by the
repetition rate of the pulsed 20-T magnet.

The time scale for thermal shock is set by the target radius divided by the 
speed of sound, say (1~cm)/(3,000~m/s) = 3~$\mu$s.  Any pulse length 
less than about 1~$\mu$s will cause the same thermal shock in the targets
of concern.
Thus the AGS-bunch length of 32~ns (rms) is adequate for the thermal-shock
studies.  For the rf cavity (secs.~3.5-6) a shorter pulse is desired.   
A recent study \cite{agsbunch} indicates that the bunch length in the AGS can 
be reduced to $\sigma_t = 2$~ns if the beam is operated near 7~GeV.  That
condition should be adequate for the rf-cavity studies.

The remaining beam issue is whether a proton bunch in the FEB U-line can
cause thermal shock at the level to be encountered at a muon-collider source.
According to the model presented in sec.~2.1.2, the pressure in the target
material resulting from the thermal shock is proportional to the peak density 
of energy deposited (J/g), rather than, say, the total energy deposited.
Hence, by variable focusing of the beam, a scan can be made through the range of
thermal shock relevant to a muon-collider source.

In particular, the muon collider design \cite{status} presently calls for a 
pulse of $10^{14}$ protons in a spot size of 4-mm radius (rms).  The model 
then indicates that a pulse of $N$ protons would cause a similar
thermal shock if its radius were $4\sqrt{N/10^{14}}$~mm.  Anticipating that
pulses of $N = 10^{13}$ will be available, we
set a requirement that the U-line beam be focusable to a radius of 1~mm (rms).

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=5in,clip=]{tsoupas.eps} 
\end{center}
\caption{TRANSPORT calculation for the U-line with a focus at the neutrino
blockhouse using presently connected quads.  A transverse (95\%) emittance
of 20$\pi$~mm-mrad was assumed.}
 \label{tsoupas}
 \end{figure}

A preliminary study \cite{Tsoupas} indicates that a U-line beam with
(95\%) emittance of $20 \pi$~mm-mrad can be focused with existing quads to 
$\sigma_r = 1$~mm at the neutrino blockhouse.  Figure~\ref{tsoupas} shows
results of the TRANSPORT calculation.  For the larger emittance of
$100\pi$~mm-mrad that holds for bunches of $10^{13}$ protons, the spot size
is expected to be about $\sqrt{5}$ times larger, namely 2.4~mm (rms).

Studies are underway to determine what configuration of magnets can
produce the desired spot size of 1~mm (rms).

Of course, it is desirable that studies be made with a beam of
$10^{14}$ protons per pulse to explore possible effects that scale with the
total energy deposition in the target.  This option will be pursued in sec.~3.3.

\subsubsection{Experimental Configuration}

We propose to pursue the initial targetry studies in the neutrino blockhouse of
the FEB U-line (Fig.~\ref{uline6}), despite the relatively high radiation 
level there (Fig.~\ref{uline4}), to be able to share diagnostic facilities with 
the AGS Spallation Target Experiment (ASTE), E-938, which is studying related
issues in a mercury target. 

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=6.5in,clip=]{uline6.eps} 
\end{center}
\caption{The neutrino blockhouse of the FEB U-line, where tests of a
mercury target for neutron spallation have been performed recently.}
 \label{uline6}
 \end{figure}

The initial targetry studies are quite straightforward.  
Each of a set  of candidate targets will be exposed to a series of beam pulses 
with ever smaller radius over the range of 5~mm (rms) down to 1~mm (and even
smaller if possible).  
Since our initial
goal is understanding of possible damage of the targets by a single beam pulse,
each pulse is analyzed separately.
The primary diagnostics are visual,
and the secondary diagnostic is a measurement of the time-dependent mechanical 
strain induced in various parts of the target (discussed in sec.~3.1.3).
From these, we should learn whether there are 
important damage thresholds near the nominal targetry parameters of a 
muon-collider source.

The beam will be operated at a very low repetition rate, so the targets are
not cooled, other than by radiation (and conduction through the supports).
The solid targets are mounted on simple supports, while the liquid targets
reside in a containment vessel with walls of $1/4''$-thick aluminum.  
The beam passes through the walls of the containment vessel.  If desired,
beam ports of a different material could be added.

The targets are located on a stand
that can be positioned remotely in (and completely removed from) the beam.

\begin{figure}[thb!]
\begin{center} 
    \includegraphics*[ width=6in, clip]{liboxtop.eps}  
\end{center}
\vspace{-0.4in}
\begin{center} 
    \includegraphics*[ width=6in, clip]{liboxend.eps}  
\end{center}
\caption{Top and beam views of the setup to test simple liquid targets.}
 \label{testbox}
 \end{figure}

The layout for three initial tests of liquid targets is shown in 
Fig.~\ref{testbox}.
As discussed in sec.~2.3.3, we plan to use eutectic Ga/Sn as the initial 
liquid metal.  It is a liquid at room temperature.
As shown from right to left, these tests involved targets in the following
forms:
\begin{enumerate}
\item
A liquid metal in an open stainless-steel trough of 1-cm diameter.  The
liquid may be blown out of the trough by the beam.  The trough is contained
in a stainless-steel box with a $1/4''$-inch-thick lucite window on one
face to permit a camera to view the event.  We have verified that the Ga/Sn
alloy does not wet lucite or stainless steel (at room temperature).
\item
A liquid metal in a $1/2''$-diameter stainless steel pipe bent in a U-shape.
If the liquid is ejected from the tube by the beam, the dispersed liquid will
be seen through the lucite window of the stainless-steel chamber above the 
pipe.  It is also very possible that the pipe will be cracked by the beam,
in which case the liquid would leak into a lower stainless-steel chamber that
surrounds the pipe.
\item
A vertical liquid jet of 1-cm diameter.  The jet is part of a small closed-loop
system with a mechanical pump, all of which is inside the aluminum containment
vessel.  The vertical jet is created in a $2''$-diameter lucite housing.
The beam passes through the walls of this housing.
\end{enumerate}
The aluminum containment vessel also houses a video-rate (30 frames/s) CCD 
camera and (strobe) light for
optical viewing of the target tests.  The images from the CCD camera will be
captured by a PC-based frame grabber.

We also are considering use of a higher-speed camera. 
If the target liquid were dispersed with a velocity of 10~m/s, a rate
of 1000 frames/s would barely resolve the history of the dispersal over a 1-2~cm
path as available in the various target housings.  However, we would not
install this (expensive) instrumentation inside the 
aluminum containment vessel until we have direct experience that the
dispersal of the liquid is not too violent.

The solid-target tests will involve two configurations, cylinder and slab, of 
at least two alloys: pure nickel \cite{ODay} and a copper-nickel alloy 
\cite{cunitarget}.  We expect no single-pulse damage to metal slab targets,
but possible damage along the axis of cylindrical targets due to the
reflected, converging pressure wave (sec.~2.1.2).

After completion of the initial set of tests described above, we will then
study a a liquid-metal jet that propagates against the beam.

\subsubsection{Liquid-Metal Jet Collinear with the Beam}

As the next step towards a realistic liquid-metal target for a muon-collider
source, we will study a 3-mm-diameter liquid-metal jet that collides head-on
with the beam.

The diameter of 3 mm is set by the availability of a commercial solenoid valve
(Skinner 71215SN2MF00) with opening/closing times of 8~ms against a pressure of
30 atm.  A block diagram of the jet is shown in Fig.~\ref{hgjetdesign}, and
is based on a mercury jet demonstrated at CERN \cite{Johnson}.

The small diameter of this initial jet further emphasizes the need for a
proton beam of rms radius 1~mm or less.

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=4.5in,clip] {hgjet_design2.eps} 
%\includegraphics[height=4.5in,clip, angle=-90]{jetdesignc.eps} 
\end{center}
\caption{Block diagram of a mercury jet.}
 \label{hgjetdesign}
 \end{figure}

As mentioned before, in the initial tests we will use the nontoxic,
room-temperature-liquid alloy of gallium and tin.
This alloy has a viscosity close to that of water, so flows 
easily.  A jet is readily produced, as shown in Fig.~\ref{gasnjet}.

\begin{figure}[thb!]
\begin{center}
\includegraphics*[ width=5in, clip]{gasnjet.eps}  
\end{center}
\caption{A room-temperature jet of liquid Ga/Sn created by a syringe.}
 \label{gasnjet}
 \end{figure}

The free liquid jet will be contained within a vessel similar to that shown
in Fig.~\ref{testbox}.  Its interaction with the proton beam will be
viewed by a camera, and the shock (if any) to the vessel will be diagnosed
with the strain gauges described in the next section.

\subsubsection{Measurement of Strain in the Test Targets}

A quantitative measure of the effect of the beam-induced pressure wave on a
target is the time-dependence of the mechanical strain in the target just 
after a beam pulse.  AGS experiment E-938 is studying this issue for a
12-cm-diameter, 1-m-long mercury target, and has found that fiberoptic
strain sensors have the greatest immunity to electromagnetic noise caused
the the passage of the beam pulse \cite{Earl98}.  We therefore propose to
use this technology in the proposed studies.  The E-938 collaboration has
kindly agreed that we may share parts of the needed readout electronics.

The strain measurement is based on detection of stress-induced variations in 
the gap distance between two parts of a fiberoptic cable, shown in
Figs.~\ref{efig2}, \ref{efig6} and \ref{eppi}.  
The ends of two fiber segments are
separated by about 100~$\mu$m and free to move within a 1-cm-long glass tube.
Relative motion between the two fibers results in a change in the interference
pattern of light reflected off the two end faces.  If the fibers are
glued to the two ends of the glass tube, as shown in Fig.~\ref{efig6}, and
those ends are also glued to the material under stress, then the gap distance
equals the strain over the 1-cm length of the glass tube.

\begin{figure}[thb!]
\begin{center}
    \includegraphics*[ width=5in, clip]{earlfig2.eps}  
\end{center}
\caption{Components of a fiberoptic strain sensor.}
 \label{efig2}
 \end{figure}

\begin{figure}[thb!]
\begin{center}
\includegraphics*[ width=5in, clip]{earlfig6.eps}  
\end{center}
\caption{Two-point attachment of the strain sensor.}
 \label{efig6}
 \end{figure}

\begin{figure}[thb!]
\begin{center}
    \includegraphics*[ width=4in, clip]
    {earlfig1.eps}
%    {eppi.eps}  
\end{center}
\caption{View of a fiberoptic strain sensor.}
 \label{eppi}
 \end{figure}

The interferometric readout system \cite{FST} is capable of resolving gap
variations of about 10-20~nm, corresponding to strains of 1-$2 \times 10^{-6}$
over the 1-cm-long glass tube, \ie, 1-2 microstrain.   
From eq.~(\ref{eq456}) of  sec.~2.1.2, we see that the strain expected in
material directly exposed to the beam is
\begin{equation}
{\Delta l \over l} \approx \Delta U {\alpha \over C}, 
\label{eq457}
\end{equation}
for an energy deposition density of $\Delta U$ in a material with thermal
expansion coefficient $\alpha$ and heat capacity $C$.  With
typical values of $\Delta U = 30$ J/g, $\alpha = 2 \times 10^{-5}$ and
$C = 0.3$ J/g-$^\circ$C (Table~\ref{elements}), we expect a strain of
$2 \times 10^{-3}$.  Hence, the fiberoptic strain sensors have a resolution
about 1/1000 of the peak stress, and so can yield useful information even
when applied at positions where the stress is well below the peak amount.

Furthermore, the frequency response of the readout system exceeds 1~MHz,
and so can resolve the detailed history of the initial pressure wave and
subsequent reflections in targets of 1-cm transverse scale.

Figure \ref{efig22} illustrates data collected with this sensor system
in experiment E-938.

\begin{figure}[thb!]
\begin{center}
\includegraphics*[ width=6in, clip]{earlfig22.eps}  
\end{center}
\caption{Arrangement of the fiberoptic strain sensors on the E-938 mercury
target tank, and representative strain measurements from a beam pulse in
the U-line.}
 \label{efig22}
 \end{figure}

In our tests, the strain sensors will be applied directly to the solid targets
and to the various housings of the liquid targets.

\subsection{Liquid-Metal Jet with a 20-T Magnet}

A second key issue for a liquid-metal-jet target is whether it can move though
a 20-T magnetic field without significant distortion.  To test this,
% without first constructing a 20-T magnet, 
we propose to bring a simple liquid-metal
jet to the National High Magnetic Field Laboratory (NHMFL) \cite{NHMFL} and 
test it at Cell 4, a 20-T, 200-mm bore resistive magnet \cite{fsucell4}.

The Cell 4 magnet has a vertical bore, as shown in Fig.~\ref{fsumagnet}.  It is
available in 8-hour shifts, following approval of a proposal submitted to the
NHMFL.  We have visited the site (Tallahassee, FL) and found it well suited to
our needs.

\begin{figure}[thb!]
\begin{center}
\includegraphics*[ width=5in, clip]{fsucell4.eps}  
\end{center}
\caption{Vertical cross section through the 20-T, 190-mm bore resistive
magnet at Cell 4 of the National High Magnetic Field Laboratory.}
 \label{fsumagnet}
 \end{figure}

The baseline design for the muon-collider source (sec.~1.2.2)
calls for a target of 1-cm radius, about 30~cm long, at an angle of about 
150~mrad to the magnet axis.  The magnetic field on the target is 20 T, and
the inner diameter of the magnet coil is 24 cm.   A horizontal liquid-jet 
target should have a 
velocity of at least 10~m/s to be little affected by gravity (sec.~2.4).

We plan to perform the initial magnet tests with the 3-mm-diameter Ga/Sn
jet described in sec.~3.1.3 rather than a jet of the nominal 2-cm diameter.
%It is desirable to perform the initial magnet tests with a jet of diameter
%smaller than 2 cm; considerably less liquid metal is required, and
%the commercial valve (Skinner 71215SN2MF00) that we plan to use
%to form the jet has a diameter of only 3 mm.  
Therefore, we will need to
scale other parameters of the test to study the various eddy-current effects
as the jet enters the magnet.

We recall the key results from sec.~2.5.  The liquid metal has density $\rho$
and conductivity $\sigma$.  The jet has radius $r$ and initial velocity $v$.
The solenoid magnet has peak field $B_0$ and coil inner diameter is $D$.
Then the radial pinch perturbs the jet radius by an amount that scales as
\begin{equation}
{\Delta r \over r} \propto {\sigma  B_0^2 D \over \rho v},
\qquad \mbox{(radial pinch)},
\label{eq3.1}
\end {equation}
according to eq.~(\ref{eq4d}).  The jet loses axial velocity as it enters the 
magnet according to eq.~(\ref{eq6b}).  Since the change of axial velocity varies
with radius, initially transverse planes in the jet shear into paraboloids.
The maximum axial shear scales as
\begin{equation}
{\Delta z \over r} \propto {\sigma r B_0^2  \over \rho v}, 
\qquad \mbox{(axial shear)},
\label{eq3.3}
\end {equation}
according to eq.~(\ref{eq6c}).  Finally, for a jet at angle $\theta$ to the 
axis of the magnet, there is an additional shear (which we call angle shear)
 between opposite sides of the jet which scales as
\begin{equation}
{\Delta z \over r} \propto {\sigma  \theta  B_0^2 D  \over \rho v},
\qquad \mbox{(angle shear)},
\label{eq3.4}
\end {equation}
on multiplying eq.~(\ref{eq278}) for the rotation angle by the radius $r$.
Note that the radial pinch (\ref{eq3.1}) and the angle shear 
(\ref{eq3.4})
have the same dependence on the dimensional parameters of the system, but
the axial shear (\ref{eq3.3}) has a different dependence.  Hence, the
latter effect must be studied separately from the other two.

We desire to observe the magnitudes of the dimensionless perturbations 
(\ref{eq3.1}-\ref{eq3.4}) in
our studies of liquid jets entering a magnet.

We propose to use the eutectic Ga/Sn alloy for these tests, but
have in mind possible eventual use of mercury or a lead alloy.  Gallium has
a conductivity about 3 times larger than the heavier metals, and is half as
dense (see Table~\ref{elements}).  
Hence, the ratio $\sigma/\rho$ is about 6 times larger for gallium.
If we use a gallium jet of 3-mm diameter then $\sigma r/\rho$ is roughly
equal to that for a heavy metal target of 
2-cm diameter.  

This suggests that studies of the radial pinch and angle
shear should be performed with the ratio $B_0^2D/v$ at a value 6 times the
nominal for a muon-collider source, while studies of the axial shear should be 
performed with the ratio $B_0^2/v$ roughly at the nominal value.

Since the Cell 4 magnet at the NHMFL has the same field $B_0$ and very nearly
the same diameter $D$ as the nominal muon-collider parameters, studies of
the radial pinch and angle shear can be done with jet velocities 1/6 nominal,
while studies of the axial shear should be done at the nominal jet
velocity of 10-20~m/s.

The jet (sec.~3.1.3)
will be propelled downwards into the vertical bore of the Cell 4 magnet.
A thin-walled stainless-steel vessel will contain the jet.  The primary
diagnostic of the jet's motion will be a Hadland Photonics IMACON 790
framing streak camera available at the NHMFL.


\subsection{Liquid-Metal Jet with a Proton Beam}

The initial tests of a liquid-jet target in a proton beam (sec.~3.1)
involve beam intensities and jet radii that are both scaled down relative to
their nominal values, so as to keep the peak energy deposition density at
the nominal value.  It is still desirable to test a full-scale target in a
beam of full intensity, to test for possible effects that scale with total
energy deposition.

\subsubsection{Full-Scale Liquid-Metal Jet}

A full-scale liquid target will have a radius of 1 cm and a length of about 
30~cm.  The mass of metal in a single pulse of the jet approaches 1~kg.
If the jet velocity were as high as 30~m/s, the kinetic energy of the metal
would be 1~kJ.  At the nominal pulse rate of 15~Hz the mechanical power in
the jet would be 15~kW, which power would be dissipated as heat in the
vessel that ``catches'' the jet.

The proposed tests of a full-scale jet can be performed at a very low pulse
rate -- a few pulses/hour -- so heating of the apparatus by the jet will not 
be an issue.

However, production of even a single 2-cm-diameter jet pulse involves
considerable 
technical challenge.  The critical item is the fast valve whose opening and
closing defines the pulse length.  For a velocity of 30~m/s, a jet of 30-cm
length lasts 10 msec.  The opening and closing times of the valve must be
short compared to this; say, of order 1~ms.  But to produce that velocity, 
the metal must be pressurized to at least 50~atm.\ according to 
eq.~(\ref{eq9c}).  The valve must then be able to exert a force of order
1000~N (assuming the plunger has area 2~cm$^2$ against the pressure).

At present we have not located a commercial valve that meets such 
specifications, although the search has just begun.  A custom valve may need
to be commissioned.

For the full-scale test, we might compromise on a jet with a considerably
longer pulse length.  (Indeed, the option of a continually flowing jet is
not excluded for the muon-collider source).  Such a jet would, of course,
require a significantly larger inventory of liquid metal during the tests.

\subsubsection{Fast Extracted Beam with $10^{14}$ Protons}

As discussed in Appendix B, the AGS is now being upgraded to accelerate
1-$2 \times 10^{14}$ protons/cycle (from its present record of $6 \times 
10^{13}$).  The protons are stored in 6 bunches in
the AGS ring.  At present, only one of these bunches can be extracted into
the FEB, due to limitations of the G10 extraction kicker.

A new pulse-forming network for the kicker has been designed to render it
capable of extracting all six bunches, but the design has not been 
implemented as yet.

%xxx

\subsection{Liquid-Metal Jet with a Proton Beam in a 20-T Magnet}

The final phase of studies to establish the basic functionality of a 
liquid-metal target for a muon-collider source involves the addition of
a 20-T magnetic field around the target to capture all particle produced with
transverse momenta less than 225~Mev/$c$.

The magnetic field will perturb the motion of the jet, as discussed in 
sec.~2.5 and investigated for a scaled-down jet as in sec.~3.2.  However,
the field will damp the hydrodynamic transients caused by the beam-induced
pressure wave (sec.~2.5.2).  The complexity of these effects is such that
detailed assessment of their magnitude requires direct measurement.

The 20-T magnet need not be continuous duty.  The proposed study can be
made with a pulsed magnet that cycles a few times an hour.

\subsubsection{The 20-T Pulsed Magnet}

A pulsed magnet that can deliver a field on the target in excess of 18 T for
600~ms is sketched in Fig.~\ref{pulsed_coil}.  It is comprised of 9 tons of
circular
copper coils that are arranged in two groups: outer and inner.  A 4-MW power
supply first energizes the outer coils to a field of 10 T.  Then the
coils are switched so as to transfer some of energy stored in the outer
coils quickly into the
inner coils.  After a designated ``flattop'' length, the energy of the
both coil groups is switched into an external load.  
The time dependences of the fields and other parameters of the
system are shown in Fig.~\ref{weggel2}.

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=6.5in,clip=]{pulsed_coil.eps} 
\end{center}
\caption{Section through an arrangement of pulsed-coils that can deliver
a peak field of 20~T around the target for about 2/3 s.  The proton beam
is at an angle of 150~mrad to the axis of the magnetic field.  The beam
dump is incorporated into the magnet structure.}
 \label{pulsed_coil}
 \end{figure}

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=5in,clip=]{con_in90.eps} 
\end{center}
\caption{Time evolution of various parameters of the 20-T pulsed magnet.}
 \label{weggel2}
 \end{figure}

The coils are to be operated at liquid nitrogen temperature to reduce the
resistivity of the copper.  The temperature rise during one pulse of the
magnet is about $40^\circ$C, and the subsequent recooling time is about
10~min.  The temperature rise is higher for longer ``flattop'' lengths,
as is also the variation of the field strength during the ``flattop''.
See Fig.~\ref{weggel7}.

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=5in,clip=]{con_ends.eps} 
\end{center}
\caption{The length of the ``flattop" of the pulse and the temperature
rise $\Delta T$ in the 20-T pulsed magnet as a function of the field variation
during the ``flattop''.}
 \label{weggel7}
 \end{figure}

The present 4-MW power supply for the MPS magnet would be suitable for
energizing the above magnet.  It would be relocated to an enclosure next to
the U-line.

\subsubsection{Site of the Later Phases of the Program}

The studies proposed in secs.~3.4-7 involve larger physical facilities in the
U-line than those of secs.~3.1-3.  If an iron return yoke is added to the
1.25-T solenoid magnet of sec.~3.6, the setup will be larger in diameter than
the present U-line tunnel.  Also, installation and commissioning of these
facilities will require considerable time spent in the tunnel.  As shown
in Fig.~\ref{uline4}, the residual radiation level is quite high in the
neutrino blockhouse.

Hence, we propose that the later phases of the program be sited downstream
of the neutrino blockhouse.  A suitable location would be near Gate \#5 of
the U-line, as shown in Fig.~\ref{uline5}.

\begin{figure}[thb!]
\begin{center}
\includegraphics[width=6.25in,clip=]{uline5.eps} 
\end{center}
\caption{Possible arrangement of the targetry experiment near Gate \#5 in
the FEB U-line.  Details of the experiment as shown in Fig.~\ref{tgtexpt3}.}
 \label{uline5}
 \end{figure}

An important concern is that our apparatus does not become excessively
radioactive due to other use of the U-line.  As shown in 
Figs.~\ref{pulsed_coil}, \ref{uline5} and \ref{tgtexpt3}, the proton beam
dump is incorporated into the structure of the 1.25-20-T magnet system.

If the U-line is to continue to be used for other studies in which large
amounts of beam are delivered, our apparatus should be located downstream of
a removable beam dump.

\subsection{RF Cavity near the Target}

For the \mumu collection system discussed in sec.~2.6 to be viable, 
the rf cells must operate at the levels required of them.  In particular,
the first cells should be operated at the highest possible 
peak-power levels order to obtain
the maximum accelerating gradient at the front end.  
Also, 
the initial rf cells must perform satisfactorily in the high-radiation 
environment immediately following the target.  Since experience operating an 
rf cell in a high-radiation environment generated by a beam of 10$^{14}$ 
protons/pulse is limited, we propose
to establish a proof-of-principle demonstration of this issue by constructing
and operating at high gradients an rf cavity with a frequency suitable for 
a muon-collider collection system.

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics[width=5cm,clip]{rfcell.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{70mhz_cavity}
SUPERFISH solution for a 70-MHz rf cavity, a quarter section of 
which is shown.
}}
\end{center}
\end{figure}

A key issue is the rf power required for
such a test.  A search of available rf sources in the upper frequency range of 
our
requirements has led us to a 70-MHz rf power source which is now available at 
LBL as the
result of retirement of the HILAC facility.  We therefore focus our efforts on
the design and utilization of a system using this frequency.  As a baseline, we
consider an rf accelerating cell (Fig.~\ref{70mhz_cavity})
 with parameters given in 
Table~\ref{cavityparm} for operation of the rf cavity at a level
of 2 kilpatricks (corresponding to peak electric gradients of
20 MV/m on the cavity wall).  

\begin{table}[htb] % h = here, t = top, b = bottom, p = on a new page
\begin{center}
\parbox{5.5in}  % replace 5.5in by \hsize if want full-width caption
{\caption[ dummy]
{\label{cavityparm} 70-MHz rf-cavity parameters.
}}
\vskip6pt
\begin{tabular}{lc}
\hline\hline
RF Frequency (MHz) & 70   \\
Cavity Length (cm) & 120   \\
Full Gap Length  (cm) & 50   \\
Cavity Radius (cm) & 125   \\
Beam Pipe Full Aperture (cm) & 60  \\
$Q/1000$ (from SFISH) & 63.1 \\
Av.\ Gradient (MV/m) & 5.0   \\
RF Peak Power (MW) & 2.4   \\
Stored Energy  (J) & 330   \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

The R\&D program
would entail constructing a 70-MHz rf cavity,  first powering it to maximum
power levels without beam/target interactions, and then determining
the maximum power levels achievable with the high-radiation environment
present with beam/target interactions.

Possible breakdown of the rf cavity is associated with discharges induced
by field-emission electrons, as well as secondary emission
of electrons from the cavity walls during that passage of particles
produced in the target.  A study of the latter effect is the main goal of
phases 5 and 6 of the present proposal.

Beam-induced breakdown of the cavity is sensitive to the time of
emission of the electron in the rf cycle.  
A realistic test of cavity
performance at high beam intensities requires the time structure of the beam
to match that expected at a muon-collider source.  There, the beam pulse
should be 2 ns or less, to maximize the number of pions that are accelerated
in a single rf cycle.  Hence, we desire beam pulses in the U-line that have
similar duration.

It may not be possible to obtain such narrow beam pulse widths at 24~GeV,
but pulses with $\sigma_t = 2$~ns have been demonstrated at the AGS at 7~GeV 
\cite{agsbunch}.  At least, some portion of the rf-cavity tests should be made 
with this beam.

The rf breakdown problems will no doubt be sensitive to the total number of
protons in the beam.  Hence, we desire to perform tests with $10^{14}$
extracted protons.  At the AGS, this will be possible only for 6-bunch
extraction (sec.~3.4) over a total period of 1~$\mu$s.  While this may not
be fully equivalent to $10^{14}$ protons in a single rf bucket, tests
should be made at the highest intensities available in the U-line.

\subsection{RF Cavity Inside a 1.25-T Magnet near the Target}

A further requirement of the muon-collider phase-rotation channel 
(sec.~1.2.4) is to envelope the
entire rf channel within a solenoidal field of 1.25 T.  It is desirable
to have this field be as uniform as possible to avoid particle
losses through resonant effects.  These effects are pronounced if, for
example, one places solenoid coils between the rf cells 
(Fig.~\ref{coil_placement}a), thereby giving
the longitudinal structure of the solenoidal field an oscillating
structure with amplitude variations (Fig.~\ref{coil_scenarios}).  We avoid
this problem by placing the entire rf channel within the coils of
the solenoids, with the penalty of increasing the warm-bore aperture
of the solenoids (Fig.~\ref{coil_placement}b).

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics[height=10cm,clip]{coils.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{coil_placement}
Different locations of solenoid coils for a phase-rotation system.
}}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[height=6cm,clip]{profiles.ps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{coil_scenarios} Magnetic induction along the beam axis for three 
different coil-placement scenarios.
}}
\end{center}
\end{figure}

The beam requirements for the study of the rf cavity inside a magnetic field
are similar to those of the study without field (sec.~3.5).

We consider two variations of solenoids with the 240-cm warm-bore
aperture required to fully envelope the 70-MHz rf cell:
a superconducting magnet and a resistive magnet.

%\subsubsection{1.25-T Superconducting Magnet}

 A superconducting
solenoid with such an aperture and capable of generating a 1.25-T field
can be constructed at a cost of \$1.2~M.  A preliminary design for this
is underway at LBL.

%\subsubsection{1.25-T Resistive  Magnet}

 A resistive solenoid would be
very similar to that now used in BNL experiment E-787, and is sketched in
Fig.~\ref{tgtexpt3}.   The performance of such a magnet, with a 20-ton copper
coil and 140-ton iron return-flux yoke, is shown in Fig.~\ref{weggel3}.
The power required would be 1.2~MW.

\begin{figure}[hbt]
{\centering
\epsfig{file=fe_al4rf.eps,width=5in}
\caption{Performance of a 1.25-T resistive magnet.}
\label{weggel3}
}
\end{figure}

Similar performance could be obtained from a design with an 11-ton
aluminum coil and a 170-ton iron yoke.

\subsection{Characterization of the Pion Yield from the Target}

%The preceding sections of this document have discussed a program of targetry
%and collection technology development. 

We propose as a final phase in the R\&D program to measure the pion
yield produced and collected by the prototype system. The number of captured
pion/muons per
incident proton is a critical parameter for the success of a muon collider.
A measurement of this
quantity would confirm and extend previous measurements of the 
pion-production yield
and on the simulation codes used for designing the collection system.
%Another goal of the
%experiment is to calibrate high-current diagnostic devices that will be
%required for monitoring
%operation of the target and capture system in normal operation.
%{\sl These devices are, however, not discussed further....}

\subsubsection{Overview}

The quantity we wish to measure is $dN_\pi (B) /dP rdr d\phi$,  where 
$N_\pi(B)$ is the number of identified
pions (or muons) for a peak field $B$ in the collection solenoid, $dP$ is the
momentum bin, and $rdrd\phi$
is the cross-sectional area of the bin at some plane following the  collection
system. The
asymmetry of the production target with respect to the axis of the solenoid
causes the distribution
to be non-uniform in azimuthal angle $\phi$. 

As a primary requirement, we seek a momentum resolution of $\sigma_P/P \leq
0.05$ over the entire momentum interval $100 < P_\pi < 500$ MeV/$c$.
Another important constraint on the detector design is the need to
distinguish pions and muons from other particles (electrons, protons, kaons),
although we need not distinguish between pions and muons.

%An important criterion for the experiment is the capability of
%identifying pions and
%muons from the large flux of particles coming from the target over the momentum
%range from 100 to 500 MeV/$c$.

A schematic drawing of the proposed experiment is shown in Fig.~\ref{tgtexpt3}. 


\begin{figure}[thb!]
\begin{center}
\includegraphics[width=6.5in,clip=]{tgtexpt5.eps} 
\end{center}
\caption{Plan view of the full configuration of the targetry experiment.
}
 \label{tgtexpt3}
 \end{figure}

In this study, as for the rf-cavity studies, it would be sufficient to use a 
solid target.  However, the 20-T capture field is an essential ingredient.

\subsubsection{AGS Beam Requirements}

Unlike the previous parts of the proposal, the characterization of the pion
yield  will require slow extracted beam.  We would need only about 
$10^6$ protons per pulse (so as to have only about 10 interactions during the 
8-$\mu$s drift time of the TPC's in the
detector). For
compatibility with the 20-T pulsed solenoid the protons should be uniformly 
distributed over a $\approx 600$-ms spill. The repetition rate required is 
only 10/hour. The spot size on the target should be $\approx 0.5$~cm$^2$
($\sigma_r \approx 4$~mm).

Slow extraction has not been available up to now in the U-line.
The main obstacle  is the H10 extraction septum. Presently, it is pulsed to 
    prevent it from burning up. To operate it continuously would require a new 
    magnet and power supply (expensive).   It may be possible to build a new 
    PFN to run the magnet with a pulse that has a 600-ms-long flat top at a 
    low duty cycle. 

However, operation of the U-line with only $10^6$ protons will require
new beam diagnostics suitable for such low intensities.
%These are intensities more      similar to our secondary beam lines, where 
%we use beam counters. So, 
It may be possible to run the U-line as a 24-GeV secondary beam line with 
the H10 septum coil as the target.  In this case, the beam tune for the 
    line would be different from that during the regular FEB operation.  
Beam monitors would be needed to measure the position and angle of the 
protons incident on the test
    target, as well as a \v Cerenkov counter to establish that the beam 
    particle is a proton.

\subsubsection{Momentum Spectrometer}

Since the target and rf cavity are enclosed in a solenoidal field,
it is natural to use a bent
solenoid to provide the dispersion needed in the spectrometer. 
Particle-position measurements
will be made using two Time Projection Chambers (TPC's) that surround
a bend in the solenoid channel. This spectrometer
arrangement is
similar to one proposed for the Ionization Cooling Test Facility (MUCOOL)
experiment \cite{coolrnd,mumu97-8} at FNAL.

\paragraph{General Features of a Bent Solenoid.}

In any solenoid, individual
charged particles undergo
helical trajectories around a ``guide" trajectory. In a bent solenoid, the
guide trajectory is
deflected perpendicular to the plane of the bend of the solenoid, taken to be
the horizontal ($x$-$z$) plane. 
This deflection is known
as "curvature drift", and the amount of deflection is proportional to the
particle's momentum. 

If a vertical dipole field is superimposed over the curved part of the 
solenoid, one
reference momentum $P_0$ will pass through with its guide trajectory
undeflected \cite{mumu97-8}. Particles with momenta $P < P_0$ will
have their guide trajectory deflected, say, upwards, while
those with $P > P_0$ will have their guide trajectory deflected 
downwards.
We take the reference momentum for the spectrometer to be $P_0 = 300$~MeV/$c$, 
since that is the center of the momentum range of interest. 
The guide dipole field that leaves the reference momentum undeflected in a
solenoid whose bend has radius of curvature $R_{\rm bend}$ is 
\cite{mumu97-8}
\begin{equation}
B_G = {P_0 \over eR_{\rm bend}}.
\label{req2}
\end{equation}
For example, with $P_0 = 300$ MeV/$c$ and $R_{\rm bend} = 1.4$ m, we find
that $B_G = 0.714$~T.  

The vertical deflection of the guide
trajectory of an off-momentum particle is given by 
\begin{equation}
y_G = {P_0 \over eB_S} {\Delta P \over P_0} \theta_{\rm bend},
\label{req1}
\end{equation}
where $B_S$ is the solenoid field strength, and $\theta_{\rm bend}$ is the bend 
angle (in the plane of the bent solenoid).
It is noteworthy that the deflection is greater for weaker fields. However,
the momentum resolution obtainable in a spectrometer based on a bent solenoid 
varies as $1/(B_S \theta_{\rm bend})$, so that a stronger field gives better
resolution (eq.~(\ref{req4}) below) as is to be expected \cite{mumu97-8}.

While the beam is contained within a circular aperture, defined as $R_S$
prior to the bend, the effect of the deflection is to deform the beam
into an ellipse with semimajor axis $R_S + y_G$ after the bend.  The
downstream solenoid and detector must be large enough to accommodate this.
Hence, we desire to minimize both $R_S$ and $y_G$, which implies
large $B_S$, but small $\theta_{\rm bend}$.

Individual particles undergo Larmor oscillations around the guide
trajectories with radius of curvature given by
\begin{equation}
R_{\rm Larm} = {P_\perp \over eB_S}.
\label{req3}
\end{equation}
Since the
product  $B_S R_{\rm Larm}^2 \propto P_\perp R_{\rm Larm}$  is
adiabatically invariant, and we must have $R_S \geq R_{\rm Larm}$, we obtain
the relation
\begin{equation}
B_S R_S^2 = \mbox{constant}.
\label{req9}
\end{equation}

The weight of the coil of the bent solenoid varies roughly as $R_S$, while
the power consumed (by a resistive coil) varies as roughly as $B_S^2 R_S$.
Then according to eq.~(\ref{req9}), the power consumption will vary roughly as
$B_S^{3/2}$.  In general, this favors a lower-field, more massive coil.
Figure~\ref{weggel5} shows the power consumption calculated for various
coil options that produce a 2-T field; a 9-ton copper coil would consume 
1.42 MW.

\begin{figure}[hbt]
{\centering
\epsfig{file=bentcoil.eps,width=6in}
\caption{Power consumption of various coil options for a 2-T, 15$^\circ$ bent
solenoid. }
\label{weggel5}
}
\end{figure}

In the present case, particle trajectories evolve adiabatically from the
capture solenoid, whose field is $B_{S,0} = 20$~T, to the spectrometer 
solenoid.  Then, eq.~(\ref{req9}) also tells us that
\begin{equation}
R_S(B_S) = R_{S,0} \sqrt{B_{S,0} \over B_S}.
\label{req10}
\end{equation}
The aperture of the capture solenoid is $R_{S,0} = 7.5$ cm, corresponding to
good capture efficiency up to transverse momentum $P_{\perp,0} = 225$ MeV/$c$.

Thus, if we choose $B_S = 2$~T for the spectrometer solenoid we will have
$R_S = 24$~cm, compared to $R_S = 30$~cm in the 1.25-T magnet surrounding the
rf cavity.

From the adiabatic invariance 
we also expect the maximum transverse momentum of particles transmitted to 
a 2-T spectrometer to be 70~MeV/$c$ and the maximum Larmor radius 
to be 11.7~cm. 
%These Larmor orbits are contained inside the
%radial acceptances considered previously. 

The bend angle should be large to improve the momentum resolution, but
small to minimize the vertical dispersion of the beam.  We choose
$\theta_{\rm bend} = 15^\circ$ as a reasonable compromise.  Then
for the maximum momentum deviation of 67\% from the reference momentum, the
 guide
trajectory is deflected vertically by 9~cm. Thus, after the bend the radius 
$R_S$ of a circular
aperture would need to  be 24 + 9 = 33~cm for full acceptance. 
However, since the dispersion occurs only
in the vertical direction, we are also examining the possibility of 
constructing a racetrack-shaped
solenoid coil downstream of the bend with elliptical aperture of 
48~cm $\times$ 66~cm.

The parameters of the bent-solenoid spectrometer magnet are summarized in
Table~\ref{magtable}.

\begin{table}[htbp] % h = here, t = top, b = bottom, p = on a new page
\begin{center}
\parbox{5.5in}  % replace 5.5in by \hsize if want full-width caption
{\caption[ Short caption for the List of Tables. ]
{\label{magtable} Parameters of the analysis spectrometer magnet system.
The length $L_S$ includes both the bent and straight sections of the solenoid.
}}
\vskip6pt
\begin{tabular}{ll}
\hline\hline
$B_S$ & 2 T \\
$R_{\rm bend}$ & 1.4 m \\
$\theta_{\rm bend}$ & 15$^\circ$ \\
$R_{S,\rm max}$ & 33 cm \\
$L_S$ & 1.4 m \\
$B_G$ & 0.714 T \\
$\Delta y_{G,\rm max}$ & 9 cm \\
$R_{\rm Larm,max}$ & 11.7 cm \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Momentum Resolution of a Bent-Solenoid Spectrometer.}

A bent-solenoid spectrometer has a momentum resolution given by \cite{mumu97-8}
\begin{equation}
{\sigma_{P} \over P} \approx {P \sigma_x \over \theta_{\rm bend}
e B_s L^{5/2} } \sqrt{720 \over n},
\label{req4}
\end{equation}
when limited by position measurement accuracy $\sigma_x$.  In this relation,
$L$  is the length over which
the position measurements are made (active length of the TPC), and  $n$ is the
number of measurements per unit length. 

There is also a limit on resolution due to
multiple scattering given by \cite{mumu97-8}
\begin{equation}
{\sigma_{P} \over P} = {13.6\ \mbox{MeV}/c \over P\beta \theta_{\rm bend}} 
\sqrt{N_X},
\label{req5}
\end{equation}
where $N_X$ is the number of radiation lengths of material encountered during
the measurement.

While one can reduce multiple scattering by operating the tracking chamber
at low pressure, this would limit the ability of the chamber to provide
particle ID via $dE/dx$ measurements.  So we consider options to operate
the chamber near atmospheric pressure, with acceptable levels of multiple 
scattering and high ionization density.  

Table~\ref{gasparm} lists relevant parameters for candidate gas mixtures
and for the component simple gases.  TPC's often have been operated with
an AR/CH$_4$ (90/10) mixture because of its low saturation-drift voltage
and reasonably good ionization density.

\begin{table}[htbp] % h = here, t = top, b = bottom, p = on a new page
\begin{center}
\parbox{5.5in}  % replace 5.5in by \hsize if want full-width caption
{\caption[ Short caption for the List of Tables. ]
{\label{gasparm} Parameters for candidate chamber gases and their components,
at STP and for minimum ionizing particles.
}}
\vskip6pt
\begin{tabular}{lcccccc}
\hline\hline
Parameter & He &  Ar & CH$_4$ & iC$_4$H$_{10}$ & Ar/CH$_4$ & 
He/iC$_4$H$_{10}$ \\
 & & & & & (90/10) & (75/25 ) \\ 
\hline
Atomic number & 2 & 18 & 10 & 34  & & \\
Primary clusters/cm (STP) & 4 & 25 & 25 & 84 & 25 & 25 \\
Radiation lengths/m & 0.0002 & 0.0091 & 0.0015 & 0.0059 & 0.0083 & 0.0016 \\ 
Saturation $E$ field (kV/cm) & -- & -- & 0.9 & 1.5 & 0.2 & $\approx 1.5$ \\
Voltage for 50 cm (kV) & -- & -- & 45 & 75 & 10 & 75 \\
Saturation drift velocity (cm/$\mu$s) & -- & -- & 10 & 5 & 6
& 3 \\
Drift time over 50 cm ($\mu$sec) & -- & -- & 5 & 10 & 8 & 17 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

Table~\ref{res} includes the momentum resolution limits from 
eqs.~(\ref{req4}-\ref{req5}) evaluated at the limits of the momentum range of 
interest in a chamber.  Gas mixtures have been chosen for which
the number $n$ of samples/m is about 2,400.  The chamber length is
$L = 50$~cm, and we assume the position resolution from a single cluster is
$\sigma_x = 300\ \mu$m.  


\begin{table}[htbp] % h = here, t = top, b = bottom, p = on a new page
\begin{center}
\parbox{5.5in}  % replace 5.5in by \hsize if want full-width caption
{\caption[ Short caption for the List of Tables. ]
{\label{res} Limits on momentum resolution in the proposed spectrometer
due to position resolution of the TPC's and to multiple scattering.}}
\vskip6pt
\begin{tabular}{ccccc}
\hline\hline
$P$ (MeV/$c$) & $\beta_\pi$ & Position & Scattering 
& Scattering  \\
& & ($n = 2400$/m) & Ar/CH$_4$ & iC$_4$H$_{10}$ \\
& & & (90/10) & (225 Torr)  \\
\hline
100 & 0.58 & 0.0006 & 0.086 & 0.038 \\
500 & 0.96 & 0.003  & 0.017 & 0.008 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

We see that the Ar/CH$_4$ (90/10) gas mixture causes too much multiple 
scattering to obtain 5\% momentum resolution at 100-MeV/$c$ momentum.
This suggests the use of an atmospheric-pressure
 helium/isobutane gas mixture, as has become popular at $B$-factories
\cite{babar}.
The price for this would be a rather high operating voltage for the TPC.
Instead, we propose to use pure iC$_4$H$_{10}$ at 225~Torr,
which should permit twice as good momentum
resolution as the Ar/CH$_4$ (90/10) mixture and the same $dE/dx$ resolution, 
and requires operating the chamber at only 2.5 times higher voltage, \ie,
25~kV for a 50-cm-long chamber.

The TPC parameters are summarized in Table~\ref{tpcparm}.
For comparison, the TPC's under
development for MUCOOL \cite{coolrnd,mumu97-8}
 have a position accuracy of $\sigma_x = 200\ \mu$m, 
measurement length $L = 43$~cm, $n =$
33 measurements per meter, and 0.0002 radiation lengths of material. 

\begin{table}[htbp] % h = here, t = top, b = bottom, p = on a new page
\begin{center}
\parbox{5.5in}  % replace 5.5in by \hsize if want full-width caption
{\caption[ Short caption for the List of Tables. ]
{\label{tpcparm} Parameters of the TPC's. 
}}
\vskip6pt
\begin{tabular}{ll}
\hline\hline
Radius (TPC$_1$) & 24 cm \\
Area (TPC$_2$) & $48 \times 66\,\pi/4$ cm$^2$ \\
$L$ & 50 cm \\
$\sigma_x$ & 300 $\mu$m \\
Gas & iC$_4$H$_{10}$ @ 225 Torr \\
Voltage & 25 kV \\
Charge samples ($n$) & 2400/m \\
$N_X$ & 0.0016 rad.\ lengths \\
Drift time & 8 $\mu$s \\
Readout time & 30 ms \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\subsubsection{Particle Identification}

Particle identification is an essential component of this experiment.
Besides the pions and muons
the outgoing flux will contain a substantial background of electrons and
protons. In addition, we
expect kaon production (some of which is potentially useful as a source of
muons) at approximately 10\% of the pion rate.
However, it is not necessary for our purposes to distinguish pions from muons;
we ultimately desire to know the rate of captured muons, but
essentially all pion decays result in muons that remain captured in
the solenoid channel.

We plan to base the particle identification on a combination of 
$dE/dx$ measurements from the TPC's and a threshold \v Cerenkov counter.
A time-of-flight
system was considered, but it is difficult to construct a ``start'' counter
suitable for 100-MeV/$c$ pions (32 keV) with the required precision.

Identification of most particles will be provided by readout of the
ionization density in the TPC.
Each track will yield $\approx 2,400$ primary ionizations over the total of
one meter of path length in the TPC's, corresponding to a $dE/dx$ resolution of
5-6\% ($\sigma$).  Figure~\ref{E910dedxslow} shows data collected with a TPC
during BNL experiment 
E-910 
for which a similar ionization density held \cite{exp910} (see also Appendix A).
Kaons and protons (and light nuclei)  will be
distinguished from pions and muons over the entire interval 100-500 MeV/$c$,
and electrons are distinguished for momenta greater than about 250 MeV/$c$.

\begin{figure}[ht]
  \begin{center}
    \includegraphics[width=3in,height=4in]{E910dedxslow.eps}
    \caption{Ionization energy loss observed in the E-910 TPC for low-momentum 
               tracks. Note the overlap
             of the (nearly horizontal) electron band with other species.}
    \label{E910dedxslow}
  \end{center}
\end{figure}

Electrons will also be identified by a positive signal in an aerogel threshold
\v Cerenkov counter.  The $\beta$ of a 500-MeV/$c$ muon is 0.98, so the
radiator should have index about 1.01, as is now achieved in
high-quality aerogels \cite{Enomoto,Adachi,Nozaki,Iijima96,Iijima97,Hazumi,%
Suda,Khan,Marlow}.  
The aerogel would be subdivided into $5 \times 5 \times 15$ cm$^3$ cells, each
 viewed by a $3''$-diameter Hamamatsu fine-mesh photomultiplier tube.  A
total of about 100 cells will cover the 48~cm $\times$ 66~cm beam ellipse
at the downstream end of the bent-solenoid channel.

A charged particle other than an electron has about 5\% probability of
producing a knock-on electron that yields a signal in the aerogel detector.
In view of the expected large number of low-energy protons in the spectrometer,
the segmentation of the aerogel detector is justified.

In the system as described there would remain about a 5\% loss of pions and
muons below 250 MeV/$c$, due to mistagging from knock-ons in the aerogel.
This inefficiency could be reduced to 0.2\% at the expense of a second
layer of aerogel detectors.


\subsubsection{Expected Rates and Running Time}

We expect about 1 collected pion per interacting proton in the momentum
range of interest \cite{status}.

%{\sl The following is Rick's calculation. I have replaced it by a shorter
%one.
%The target length is 1.5 interaction lengths, so 78\% of the incident protons
%should interact. The
%TPC live-time efficiency is $10^{-3}$. In addition we include an experiment
%efficiency factor of 50\% to
%account for set up and less than ideal running conditions. Thus the expected
%rate of collected
%pions per hour in the experiment is
%($10^6$ inc.\ $p$/pulse) (0.78) (10 pulse/hr) (1 $\pi$/int.\ $p$) ($10^{-3}$) 
%(0.5) = 3900 $\pi$/hr. }

If the readout time is 30~ms per trigger, and the beam spill is 600~ms long,
then we will record about 20 triggers/pulse.  With 6 pulses/hour (limited by
the time for the pulsed 20-T solenoid to recool), we will obtain about 120
triggers/hour.

The TPC readout is to be based on the STAR TPC electronics \cite{Klein}, 
which incorporates analog storage of 512 (or 1024)
time samples in switched-capacitor arrays.  While there
are about 1,200 primary ionizations per track in the TPC, it is not necessary
to sample each ionization separately.  It should suffice to allot about
50 samples per track, clock at 6 MHz, for a total of 85 $\mu$s (if we use
512 samples, or 170~$\mu$s for 1024 samples).  If the beam
protons are separated by 10 or more time samples, \ie, by 1.6~$\mu$s or more,
we should be able to identify different protons easily.  Equivalently,
we can process 50 (or 100) proton interactions per trigger with little time 
overlap
among samples from different interactions.  With this procedure, we can
measure about 6,000 pions/hour with production kinematics of interest.

Although there is only about one pion per proton interaction of interest,
there will be about 15 tracks per interaction, mostly from slow protons
ejected from the target nuclei, in the TPC.  See Fig.~\ref{E910chmult} of
Appendix A.  Thus a trigger that contained 50 proton interactions would
consist of about 750 tracks in the TPC.

The collected pions of interest must be binned over the three phase variables 
$P$, $r$ and $\phi$.  In addition, the
field strength of the collection solenoid falls from 20~T to 18~T during the 
600-ms beam spill.
Thus the collected events must also be binned in magnetic field. If we
require a total of 6 bins per variable, there are a total of $1,296$ bins.
Then each 4-d bin receive only $\approx 15$ events per hour. If we
require $\approx 1,000$ events per bin
to achieve 3\% statistical accuracy, then we require $\approx 220$ hours of 
beam operation. It should be
emphasized, however, that during most of this time we do not need beam
delivered to the
experiment, so efficient operation in conjunction with other users may be
possible.

\subsection{Simulations of the Beam-Jet-Magnet Interaction}

The R\&D program presented in secs.~3.1-7 consists of laboratory studies,
only some of which have parameters that match the baseline design for a
muon-collider source in detail.  Also, the baseline design may
well evolve as the measurements are made and difficulties are understood.
Hence, extrapolations will be needed from the observed data to other physical 
situations.  Analytic scaling laws provide some guidance for this, but we
also desire the additional insights as may be had from numerical simulation.

Most simulations of targetry issues have emphasized thermal-induced stress
in solid targets, for example \cite{Stefanski,Tang}, with some recent work
on mercury targets for neutron spallation sources
\cite{Taleya,Taleyc,Bauer98}.  Such studies have often used the commercial
code ANSYS \cite{ANSYS}, typically modelling the target as a solid.  
The effect of a strong magnetic field on the target has not been considered
to date.

ANSYS
now combines a fluid dynamics package, FLOTRAN, and an eddy-current package,
Emag, together with its Mechanical and Thermal packages in an overall
architecture, Multiphysics.  We have begun, and propose to continue, use of
the ANSYS/Multiphysics package to model the beam-jet-magnet interaction.
An early result is shown in Fig.~\ref{ball_ansys}.

\begin{figure}[htp]  % h = here, t = top, b = bottom, p = new page
\begin{center}
  \includegraphics*[ width=4in, clip]{ball_ansys.eps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{ball_ansys} ANSYS/Emag simulation eddy currents induced in a 
conducting sphere moving through the fringe field of a solenoid magnet.
}}
\end{center}
\end{figure}

However, it is clear that the complexity of the beam-jet-magnet interaction
is beyond that studied in most ANSYS simulations, and considerable effort will
be required to obtain useful results.  We therefore also propose to
utilize research codes developed for simulation of thermal hydraulics.

\subsubsection{The HEIGHTS Simulation Package}

The HEIGHTS package has been developed at Argonne National Laboratory to study 
{\bf H}igh {\bf E}nergy {\bf I}nteractions with {\bf G}eneral 
{\bf H}eterogeneous {\bf T}arget {\bf S}ystems.  This is a 2-dimensional
particle-in-cell  (2-D PIC) code consisting of nine modules as sketched in
Fig.~\ref{hassaneinfig1}.

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
  \includegraphics*[ width=5in, clip]{hassanein1.eps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{hassaneinfig1} Block diagram of the HEIGHTS simulation package.
}}
\end{center}
\end{figure}

The HEIGHTS package will be used to study the motion of a liquid-metal jet in
 a strong and inhomogeneous magnetic field, including the hydrodynamic 
instability of the  liquid jet, thermal stresses, and the shockwave effects 
resulting from the sudden deposition of the proton energy in the liquid target.
Examples of past studies using this code include 
\cite{Hassanein1,Hassanein2,Hassanein3,Hassanein4}.

Details of the proton cascade in the target will be simulated with the
MARS Monte Carlo code, and provided as input to the HEIGHTS simulation.

\paragraph{Jet Heating and Expansion.}

We propose to perform
detailed simulation of the dynamics of a cylindrical column of radius $r_0$  of
 the liquid metal either with a free surface or confined by a solid cylinder 
(pipe).  The transport equations of continuity, motion, and heat 
balance  are to be solved in a strong magnetic field, using the 
particle-in-cell (PIC) method  in cylindrical coordinates $(r,z)$ assuming 
symmetry in azimuthal angle $\phi$.
  The problem of stability as a function of angle $\phi$ will be solved 
separately. 
 
	Because the deposited energy $Q_{\rm beam}$ depends on $r$ and $z$ it is
 necessary to calculate the motion of the medium in both coordinates.
The existence of a free surface requires the use of a Lagrangian description for
 the numerical mesh of the
 target. However, to avoid the problem of large distortion of the hydrodynamic 
cells it is necessary to use a mixed Eulerian-Lagrangian scheme. An adequate
 description can be
 achieved using the 2-D PIC method recently implemented in the
 A*THERMAL-S code, the structure of which is sketched in 
Fig.~\ref{hassaneinfig2}.

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
  \includegraphics*[ width=5in, clip]{hassanein2.eps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{hassaneinfig2} Block diagram of the A*THERMAL-S simulation package.
}}
\end{center}
\end{figure}

The results from the computer simulation will show whether a pressure wave
 is generated inside the liquid jet and, in addition, study the consequences of
 such shockwave on jet behavior and stability.  The magnitude of the pressure
 wave and its propagation/reflection will determine the severity of jet breakup
 and distortion.

In the case of a strong pressure wave that is generated inside the jet
 and, as a result, the jet is broken into energetic droplets flying inside the 
magnet, the SPLASH code (Fig.~\ref{hassaneinfig3})
can then be used to study the droplet impacts on the
 chamber wall.  The current SPLASH code models the hydrodynamic stability and 
splashing effects of a free surface of a liquid-metal layer subject to various
 forces acting on this liquid layer.  The code can also model macroscopic
 erosion of a solid target from brittle destruction due to thermal stresses.  We
 can easily implement models to study wall impact and erosion from the impinging
 of the energetic liquid-droplets.

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
  \includegraphics*[ width=5in, clip]{hassanein3.eps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{hassaneinfig3} Block diagram of the SPLASH simulation package.
}}
\end{center}
\end{figure}

\paragraph{Liquid-Jet Dynamics in a Strong Inhomogeneous Magnetic Field.}

The liquid-metal conductivity $\sigma$ is not small; therefore, the 
magnetic-field diffusion time $\tau = \mu_0 \sigma r_0^2$, eq.~(\ref{eq204}),
approaches the flight time of the jet.  Then most of the external magnetic
field penetrates into the liquid jet, and the resulting Lorentz force
deflects the trajectory of the jet.  To calculate the trajectory and 
 to choose better conditions for the energy deposition and pion production,
 detailed magnetohydrodynamic (MHD) 
analysis is necessary of the jet dynamics in an inhomogeneous 
magnetic field (near both ends of the coils). This problem can also be studied 
using the A*THERMAL-S code; including all components of the electromagnetic
 forces, however, may require some modifications of the code. 

\paragraph{Eddy-Current Effects.}

As the liquid metal propagates through the inhomogeneous magnetic field,
eddy currents are induced in the metal.  Then, because of the existence of a 
radial component of magnetic field, the jet velocity field will be distorted
(sec.~2.5.1).  This may
alter the shape of the jet from a circular form to an elliptical
 one.  Decreasing the value of the magnetic field from coil inner surface to
the $z$ axis  may stabilize this instability.  The analysis of this problem can 
be made analytically using the conventional MHD stability theory. 

\paragraph{Capillary Instability.}

Another problem is how to inject a free-surface liquid jet into an
inhomogeneous magnetic field.  During jet injection the
 hydrodynamic instability due to capillary forces evolves with characteristic
rate $\Gamma = \sqrt{2 T/\rho r_0^3}$, where $T$ is the surface tension and
$\rho$ is the density of the liquid.  The time of this instability, 
$\tau_T \approx 10/\Gamma$, is a few ms, which is comparable to or less than 
the flight time of the jet.  Therefore the
 development of this instability could result in the dividing of the liquid 
jet into small droplets with size about $0.1r_0$.  As discussed in sec.~2.5.3,
this instability is expected to be damped by the strong magnetic field of
the capture solenoid.
This  problem can also be modeled analytically using the conventional MHD 
theory and 
experimental data of time of droplet formation and droplets size.

%hhh

\subsubsection{Validation of the Simulations via Exploding-Wire Studies}

In addition to such validation of the simulations as will be possible from
the studies of secs.~3.1-4, we propose to study 
the effects of rapid energy deposition in a cylinder of liquid metal
using exploding-wire technology.  Such experiments
 could potentially be much less expensive than beam-on-target tests, and
offer greater ease in varying parameters of the system.  We 
propose to evaluate the extent to which exploding wires in liquid metals could
 provide useful, target-relevant information and set up a simple test stand to
 obtain some first results.  Our computer modeling capability will be used to 
guide the testing, interpret the results, and evaluate the relevancy to actual
 beam-on-target tests.  Initial testing would be done in a static or slowly
flowing  liquid metal and without a magnetic field.  Follow-on testing would 
include a magnetic field and target motion.  

	A schematic illustration of the suggested facility is given in 
Fig.~\ref{hassaneinfig4}.
  The liquid can be pumped from the chamber bottom to chamber top and falls 
freely in the form of a jet.  The (insulated) wire is placed in the center of 
the falling liquid jet and is heated by the electric current from the discharge
of a  condensor bank during a time $\tau_{\rm bank} \approx 1\ \mu$s.  
Facilities  with similar parameters  exist in many institutions, in particular
 at the TRINITI (Troitsk, Russia; contactperson V.~Belan) or at the 
Kurchatov Institute (Russia). 
Studies of an exploding mercury jet have been reported in
\cite{Criss95,Ansley1,Ansley2}.

\begin{figure}[ht]  % h = here, t = top, b = bottom, p = new page
\begin{center}
  \includegraphics*[ width=5in, clip]{hassanein4.eps} 
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{hassaneinfig4} An exploding-wire facility to study
rapid energy deposition in liquids. 
}}
\end{center}
\end{figure}

\paragraph{System Requirements.}

To model and simulate the processes of rapid heating of a liquid target 
and possible resulting instabilities, one needs experiments with intense 
heating of the central zone of a cylindrical liquid jet of radius $r_0$
that meet the following conditions:
\begin{enumerate}
\item
The heat deposited, $Q_0$, should be comparable to that in the muon-collider
target, roughly 30~J/g (of target material, not of wire).
\item 
The heating time $\tau_Q$ should be less than the hydrodynamic time $\tau_r =
r_0/v$, where $v$ is the speed of sound in the liquid.
\item
The length $L$ of the liquid column should  be much larger than the column 
radius $r_0$, to exclude  three-dimensional effects.
\item
The liquid should be one of the candidates (Ga/Sn, Hg, Bi/Pb, \etc) for use
as the target at a muon-collider source.
\end{enumerate}

	The best plan is to use the nominal geometrical parameters of the
muon-collider target, \ie, $r_0 = 1$~cm, $L \approx 30$~ cm.  Then for a 
candidate liquid of density $\rho = 10$~g/cm$^2$, the 
deposited energy should be $Q_0 \approx 30$~kJ.  
Deposition of energy $Q_0$ via the exploding-wire technique requires that the 
energy of the  capacitor bank, $Q_{\rm bank}$ be of order $3Q_0 \approx 
100$~kJ.  The hydrodynamic time is  $\tau_r \approx 3\ \mu$s, so the
bank discharge time $\tau_{\rm bank}$ should be 1~$\mu$s or less.

Let us estimate the electrical parameters of the exploding-wire system.
Suppose, for example, that the wire has radius $r_w = 1$ mm and is made of
copper.  Then its resistance is $R = \rho_w L / \pi r_w^2 = 1.7 \times
10^{-3}\ \Omega$ for $L = 30$~cm, using resistivity 
$\rho_w = 1.7\ \mu\Omega$-cm.  The capacitance of the bank must then be
$C = \tau_{\rm bank}/R \approx 600\ \mu$F.  The stored energy of the bank
is $U = CV^2/2 \approx 3Q_0 \approx 10^5$~J, so the operating voltage is
$V \approx 20$~kV.  The peak discharge current is $I = V/R \approx 10$~MA.
For a copper wire of radius 0.5~mm, the bank capacitance would be 150~$\mu$F,
the voltage 40~kV and the peak current 5~MA.

The mass of the 1-mm-radius copper wire would be only about 8~g, so the energy 
density would be about 3,500 J/g, which will vaporize much of the wire.

Experiments are proposed be performed in three stages: 

\paragraph{First Stage.}

	In the first stage, an exploding-wire facility with the desired
electrical parameters will be established, but for simplicity the wire will not
be immersed in a liquid metal.
Rather, a dielectric liquid such as a purified water or organic compound such 
as castor oil would be used. 
The physical processes of the shockwave formation 
and its influence on the  stability of a vertical liquid jet would be studied.
As well as establishing the experimental technique, the data will permit
first elaboration and refinement of the physical models and computer codes. 

\paragraph{Second Stage.}

	In the second stage, experiments with candidate
liquid metals will be performed.  For this, the wire would be insulated 
from the conducting liquid by a
film of a material with good dielectric properties such as, for
 example, Mylar with a thickness of a few $\mu$m.
The shockwave crosses this film in a few ns, and 
existence of this thin film would not significantly influence the physical 
phenomena under study.

\paragraph{Third Stage.}

	In this stage experiments from the second stage would be repeated 
in a strong 
magnetic field to study the effect of such field on jet stability during the 
heating of the jet.  
%However, the generated pressure due to the shockwave 
%significantly exceeds that of the magnetic pressure at least for the smaller 
%beam sizes.  

\paragraph{Diagnostics.}

	Diagnostics for the exploding-wire studies that exist at the 
Russian 
facilities include measurements for the electric parameters, pressure sensors to
 measure the shockwave parameters, and high-speed streak and frame cameras to
 monitor the liquid behavior during and after the wire explosion.  The pressure 
sensors can be placed in the liquid jet at different distances without 
disturbing the flowing jet.  Such diagnostics have been extensively used during 
high-energy deposition on target materials to simulate plasma disruption in 
future fusion reactors.