\section{IONIZATION COOLING}
\label{subsec-compcool}
\subsection{Introduction}

The design of an efficient and practical cooling system is one of the major challenges for the muon
collider project.

For a high luminosity collider, the 6-D phase space volume occupied by the muon
beam  must be reduced by a factor of $10^5 - 10^6.$ Furthermore, this phase space reduction must
be done within a time that is not long compared to the muon lifetime ($\mu$ lifetime $\approx 2~\mu
s$). Cooling by synchrotron radiation, conventional stochastic
cooling and conventional electron cooling are all too slow. Optical stochastic
cooling \cite{ref21}, electron cooling in a plasma discharge \cite{ref22}, and
cooling in a crystal lattice \cite{ref23,ref23a} are being studied, but appear
technologically difficult. The new method proposed for cooling muons is ionization cooling. This
technique \cite{Budker78,ref1a,ref1,ref24} is uniquely applicable to muons because
of their minimal interaction with matter. It is a method that seems relatively straightforward in
principle, but has proven quite challenging to implement in practice

Ionization cooling involves passing the beam through some 
material in which the muons lose both transverse and longitudinal momentum 
by ionization energy loss, commonly referred to as $dE/dx.$ The longitudinal muon 
momentum is then restored by  reacceleration, leaving a net loss of
transverse momentum (transverse cooling). The process is repeated 
many times to achieve a large cooling factor. 

The energy spread can be reduced by introducing a transverse variation in the absorber density or
thickness (e.g.\ a wedge) 
at a location where there is dispersion (the transverse position is energy dependent). This method
results in a corresponding increase of transverse phase space and is thus an exchange of longitudinal
and transverse emittances. With transverse cooling, this allows cooling in all dimensions.
 
We define the root mean square rms normalized emittance as
\begin{equation}
\epsilon_{i,N}=\sqrt{\langle\delta r_i^2\rangle \langle\delta p_i^2\rangle-\langle\delta r_i\delta p_i\rangle^2}/m_{\mu}c
\end{equation}
where $r_i$ and $p_i$ are the beam canonical conjugate variables with $i=1,2,3$ denoting the x, y
and z directions, and $\langle...\rangle$ indicates statistical averaging over the particles. The operator $\delta$
denotes the deviation from the average, so that $\delta r_i=r_i-\langle r_i\rangle$ and likewise for $\delta p_i.$ 
  The appropriate figure of merit for cooling is the final value of the 6-D relativistically invariant
emittance $\epsilon_{6,N},$ which is proportional to the area in the 6-D phase space $(x,y,z,p_x,p_y,p_z)$ since,
to a fairly good approximation, it is preserved during 
acceleration and storage in the collider ring. This quantity is the square root of the determinant of
a general quadratic moment matrix containing all possible correlations. However, until the nature
and practical implications of these correlations are understood, it is more conservative to ignore the
correlations and use the following simplified expression for 6-D normalized emittance,
\begin{equation}
\epsilon_{6,N}\approx  \epsilon_{x,N}\times \epsilon_{y,N}\times \epsilon_{z,N} %<<<
%\left(\sigma_z^{rms}\sigma_E^{rms}\gamma \beta\right) >>>
\end{equation}
Theoretical studies have shown that, assuming realistic parameters for the 
cooling hardware, ionization cooling can be expected to 
reduce the phase space volume 
occupied by the initial muon beam by a factor of $10^5$ -- $10^6$. 
A complete
cooling channel would
consist of 20 -- 30 cooling stages, each stage yielding about a factor of 
two in 6-D phase space reduction.

 It is
recognized that understanding the feasibility of constructing a muon
ionization cooling channel is on the critical path to determining the
viability of the whole muon collider concept.
The muon cooling channel is the most novel part of a muon collider complex.
Steady progress has been made both in improving the design of sections of the channel
and in adding detail to the computer simulations. A vigorous experimental program is needed to
verify and benchmark the computer simulations.

The following parts of this section briefly describe the physics
underling the process of ionization cooling. We will show results of simulations for some chosen
examples, and outline a six year R\&D program to demonstrate the feasibility of using ionization
cooling techniques. 

\subsection{Cooling theory}

In ionization cooling, the beam loses both transverse and longitudinal momentum
as it passes through a material. At the same time its emittance is increased due to stochastic multiple
scattering and Landau straggling. The longitudinal
momentum can be restored by reacceleration, leaving a net loss of
transverse momentum. 

The approximate equation for transverse cooling in a step $ds$ along the particle's orbit  is \cite{ref2b,ref1a,ref1,ref2a,ref24b,ref24c}
  \begin{equation}
\frac{d\epsilon_N}{ds} = -{1\over \beta^2}\frac{dE_{\mu}}{ds}\ \frac{\epsilon_N}{E_{\mu}} +
 \frac{\beta_{\perp} (0.014 GeV)^2}{2\beta^3 E_{\mu}m_{\mu}\ L_R}, \label{eq1}
  \end{equation}
where $\beta$ is the normalized velocity, $E_{\mu}$ is the total energy,
$m_{\mu}$ is the muon mass,   $\epsilon_N$ is the normalized transverse emittance,
$\beta_{\perp}$ is the betatron
function at the absorber, $dE_{\mu}/ds$ is the energy loss per unit length, and $L_R$  is the
radiation length of the material.  The betatron function is determined by the strengths of the elements
in the focusing lattice \cite{pdgaccel}. Together with the beam emittance this function
determines the local size and divergence of the beam. Note that the energy loss $dE_{\mu}/ds$ is defined here
as a positive quantity, unlike the convention often used in particle physics. The first term in this
equation is the 
cooling term, and the second describes the heating due to multiple scattering.
The heating term is minimized if $\beta_{\perp}$ is small (strong-focusing)
and $L_R$ is large (a low-Z absorber). 

The minimum, normalized transverse emittance that can be achieved for a given absorber in a given
focusing field is reached when the cooling rate equals the heating rate in 
Eq.~\ref{eq1}
\begin{equation}\epsilon_{N,min} = {\beta_{\perp} (14 MeV)^2 \over
 2 \beta m_{\mu} {dE_{\mu} \over ds} L_R }\label{equi}
\end{equation}
For a relativistic muon in liquid hydrogen with a betatron focusing value of 8 cm, which corresponds
roughly to confinement in a 15~T solenoidal field, the minimum achievable emittance is about 340~mm-mrad.

The equation for energy spread  is \cite{ref1a,ref2a,ref2c}
 \begin{equation}
{\frac{d(\Delta E_\mu)^2}{ds}}\ =
-2\ {\frac{d\left( {\frac{dE_\mu}{ds}} \right)} {dE_\mu}}
 \langle(\Delta E_{\mu})^2 \rangle\ +
{\frac{d(\Delta E_{\mu})^2_{{\rm stragg.}}}{ds}}\label{eq2}
 \end{equation}
where the first term describes the cooling (or heating) due to energy loss, 
and the second term describes the heating due to straggling. $\Delta E_{\mu}$ is the rms spread
in the energy of the beam. 

Ionization cooling of muons seems relatively straightforward in theory, but
will require extensive simulation studies and hardware development for its
optimization. There are practical problems in designing lattices that can
transport and focus the large emittance  beam. There will also be effects
from space charge and wake fields.

We have developed a number of tools for studying the ionization cooling
process. First, the basic theory was used to identify the most promising
beam properties, material type and focusing arrangements for cooling. Given
the practical limits on magnetic field strengths, this gives an estimate of
the minimum achievable emittance for a given configuration.
 Next, the differential equations for cooling and heating described above were incorporated into a computer code. Allowance for the shifts in the
betatron phase advance due to space charge and aberrations was included.
This code was used to develop an overall cooling scenario, which broke the
cooling system into a number of stages, and determined the properties of the
beam, radio frequency (rf) cavities, and focusing lattice at each stage.

Finally, several tracking codes were either written or modified to study
the cooling process in detail. Two new codes (SIMUCOOL \cite{van}, and ICOOL \cite{ref25,ref25a})
use Monte Carlo techniques to track particles one at a time through the
cooling system. All the codes attempt to include all relevant physical
processes to some degree, (e.g. energy loss, straggling, multiple scattering)
and use Maxwellian models of the focusing fields. They do not yet  
take into account any space charge or wake field effects.  In addition, we
have also used a modified version of PARMELA \cite{parmela} for tracking, which does
include space charge effects, and a double precision version of GEANT \cite{ref40,paul}.

We have recently developed \cite{envelope} a model of beam cooling based on a second
order moment expansion. A computer code solving the equations for
transverse cooling gives results that agree with tracking codes. 
 The code is being extended to include energy spread and bends. It
is very fast and is appropriate for preliminary design and optimization of
the cooling channel. 
 All of these codes are actively being
updated and optimized for studying the cooling problem.

\subsection{Cooling system}

The cooling is obtained in a series of cooling stages. Each stage
consists of a succession of the following components:
\begin{itemize}
 \item Transverse cooling sections using materials in a strong focusing (low $\beta_\perp$)
environment alternated with linear accelerators. 
 \item Emittance exchange in lattices that generate dispersion, with absorbing  wedges to reduce
momentum spread.
\item Matching sections to optimize the transmission and cooling parameters of the following
section.
 \end{itemize}
The question of the best energy to use for cooling has been discussed in detail in reference \cite{palmer_lectures}. For the set of parameters of interest, a kinetic energy close to 100~MeV appears optimal. At higher energies, weaker focusing raises the heating term from Coulomb scattering, and more acceleration is required for a given amount of cooling. At lower energies, the beam divergence become large, and the rise of ${dE\over dx}$ with falling energy causes a greater increase in energy spread. There can be an advantage, initially, of using a somewhat higher energy to reduce the beam dimensions and bucket length; at the end, the energy can be dropped to attain the lowest transverse emittances at the expense of longitudinal emittance.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{sf_fg5.ps} 
\end{center}
\caption{Transverse ($\epsilon_{x,y}$), longitudinal emittance ($\epsilon_L$) and beam energy {\it vs.} stage number in the cooling sequence. }
\label{bnl_62740}
\end{figure} 
In the examples that follow it is seen that the simulated transverse cooling stages lower the 6-D emittance by a
factor of about 2.
Since the required total 6-D cooling is $O(10^6)$, about 20 such stages are required. 

We have performed calculations of complete cooling systems for the Higgs factory and for a high energy collider. These calculations are based on theoretical models of the expected cooling performance. They give an indication of the system dimensions, magnet strengths, rf frequencies and gradients, and beam parameters that will be required in a cooling system. The calculations suggest that the required cooling for a Higgs factory could be achieved in 25~stages, while the high energy collider would require 28~stages. Emittances and energies as a function of stage are shown in Fig.~\ref{bnl_62740}. The sequence can be considered to consist of 3~parts:
\begin{enumerate}
\item For the first 12~stages the primary effort is to cool in the longitudinal direction in order to reduce the bunch lengths and allow higher frequency rf to be employed. Some transverse cooling is needed in order to reduce the transverse dimensions of the beam and allow it to fit through the smaller irises at the higher frequencies. In this example, for the first stage, an energy of 300~MeV was used to calculate the reduction on the momentum spread and transverse beam dimensions. It is recognized that designing an emittance exchange at the beginning of the system that can simultaneously accept the large initial phase space of the beam and reduce the longitudinal emittance will be a major challenge. In later stages the energy is closer to 100~MeV. Solenoid focusing was assumed in all stages, with an initial field of the order of 1~T rising to about 3~T at the end.
\item In the second part ( in this example, stages 13 - 25) the 6-D emittance is reduced as far as possible. For the case of a low momentum spread Higgs collider, the required beam parameters are now achieved and the third part is not required. In this example, 80~MeV energy was used for all stages. Solenoid focusing was used in all but the last two stages, where lithium lenses were assumed.
\item For the higher luminosity and higher energy colliders, further reduction in transverse emittance is required, but this can be obtained without reduction of the 6-D phase space, by allowing the longitudinal phase space to grow. This exchange of emittances is, in this example, achieved by reducing the energy to near 10~MeV in two long lithium lens cooling stages. The same effect could  probably be achieved at similar energy, by using a hydrogen absorber with solenoid focusing. It might also be possible by using wedges.
\end{enumerate}
The total length of the system would be of the order of 600~m, and the total acceleration required
would be approximately
6 GeV. The fraction of muons remaining at the end of the cooling system
is estimated to be $\approx 60\%$. It must be emphasized that this sequence was derived without simulation of the individual stages. It serves however to guide the choice of stages to study in detail.

Three transverse cooling examples have been designed and simulated. The first uses 1.25~T
solenoids to cool the very large emittance beam coming from the phase rotation channel. The muon beam at the end of the decay channel is very intense,
with approximately $7.5\times 10^{12}$ muons/bunch, but with a large normalized  transverse
emittance $(\epsilon_{x,N}(\textrm{rms})\approx 15\times 10^3~\pi \textrm{mm-mrad})$ and a large normalized
longitudinal emittance $(\epsilon_{z,N}(\textrm{rms})\approx 612~\pi \textrm{mm}).$
  The second example 
would lie toward the end of a full cooling sequence and uses 15~T solenoids. The third example, using 31~T
solenoids, meets the requirements for the Higgs factory and could be the final 
cooling stage for this machine.

The baseline solution for emittance exchange involves the use of bent solenoids to generate
dispersion and wedges of hydrogen or LiH to reduce the energy spread. A simulated example is
given for exchange that would be needed after the 15~T transverse cooling case.

A lithium lens solution may prove more economical for the final stages, and might allow even lower
emittances to be obtained.
In this case, the lithium lens serves simultaneously to maintain the low
$\beta_{\perp}$, and provide $dE/dx$ for cooling. Similar lenses, with
surface fields of $10\,$T, were developed at Novosibirsk (BINP) and have been used, at low
repetition rates, as focusing elements at FNAL and CERN
\cite{reviewtev,ref26,ref26a,ref26b,ref27}. Lenses for the cooling application, which would operate
at 15~Hz, would need to employ flowing liquid lithium to provide adequate thermal cooling. Higher
surface fields would also be desirable.

Studies have simulated cooling in multiple lithium lenses, and have shown cooling through several
orders of magnitude \cite{Balbekov96}. But these studies have, so far, used
ideal matching and acceleration.
Cooling is also being studied in beam recirculators, which could lead to reduction of costs of the
cooling section \cite{balbekov,balbekov1}, but  full simulations with
all higher order effects have not yet been successfully demonstrated.

\subsection{15~T solenoid transverse cooling example}
The lattice consists of 11 identical 2~m long \textit{cells}. In each cell there is a liquid hydrogen
absorber (64~cm long, 10~cm diameter) in the 15~T solenoid focusing magnet (64~cm long, 12~cm
diameter). The direction of the fields in the magnets alternates 
from one cell to the next. 
Between the 15~T solenoids there are magnetic matching sections (1.3~m long, 32~cm inside
diameter) where the field is lowered and then reversed. Inside the matching sections are short,
805~MHz, high gradient (36~MeV/m) linacs.
\begin{figure*}[thb!]
\centerline{\epsfig{file=radii_beta_axial.eps,height=5.0in,width=5.0in}}
\caption[Cross section of one half period of an alternating solenoid cooling lattice ]{(a) Cross section of one half period of an alternating solenoid cooling lattice; (b) axial magnetic field \vs~z;
(c) $\beta_{\perp} $~function \vs~z.  }
\label{altsol}
 \end{figure*} 
Figure~\ref{altsol} shows the cross section of one cell of such a system, together with the betatron
function, and the magnetic field along the axis. For
convenience in modeling, the section shown in Fig.~\ref{altsol}(a) starts and ends symmetrically
in the middle of hydrogen absorber regions at the location of
the peaks in the axial magnetic field. In practice each cell would start at
the beginning of the hydrogen region and extend to the end of the rf module.

A GEANT simulation of muons traversing a section of the cooling channel is shown in Fig.~\ref{dpgeant}.


Additional simulations were performed \cite{angmom,aac_paper} using the program ICOOL. The only likely significant effects which are not yet included 
are space charge and wakefields. Analytic calculations for particle bunches in free space indicate that these effects should, for the later stages, be significant but not overwhelming.
\begin{figure*}[thb!]
\centerline{\epsfig{file=dpgeant_new.eps,height=3.0in,width=6.0in,clip=}}
\caption[GEANT simulation of muons traversing a section of cooling channel]{GEANT simulation
of muons traversing a section of the alternating solenoid cooling channel. The variation of the
magnetic field $B_z$ is shown for $1{1\over 2}$ cells of the figure.}
 \label{dpgeant}
 \end{figure*}   
A full simulation must
be done before we are assured that no problems exist. Particles are introduced with transverse and
longitudinal emittance (186~MeV/$c,$ 1400~$\pi$~mm-mrad transverse, and
1.1~$\pi$~mm~longitudinal), together with a number of naturally occurring correlations. Firstly,
the particles are given the angular momentum appropriate for the starting axial magnetic field.
Secondly, particles with large initial radius $r_o$ and/or divergence $\theta_o$ have longer
pathlengths in a solenoidal field and tend to spread out with time. This can be parameterized by
defining an initial transverse amplitude
\begin{equation}
A^2={r_o^2\over \beta_{\perp}^2 }+\theta_o^2.
\end{equation}
The temporal spreading can be minimized by introducing an initial correlation between $p_z$ and
$A^2$ that equalizes the forward velocity of the initial particles. This correlation causes the average
momentum of the beam to grow from the reference value of 186~MeV/$c$ to $\approx 195$~MeV/$c.$
Lastly, a distortion of the longitudinal bunch distribution can be introduced to reflect the asymmetric
nature of the ``alpha"-shaped rf bucket.

Figure~\ref{simu8}(a) shows the average momentum of the beam as a function of distance along
the channel. The momentum drops as the beam crosses the liquid hydrogen absorbers. The gradient
and phase
 of the rf cavities have been adjusted so that the reacceleration given to
the reference particle 
equals the mean energy loss. This causes the average momentum of the beam to
remain in a
narrow band around 195~MeV/$c.$ Figure~\ref{simu8}(b) shows the mechanical and canonical
angular
momenta as a function of distance along the channel. The mechanical angular
momentum shows
the rotational motion of the beam around the axial solenoidal field. It
periodically reverses sign
when the solenoids alternate direction. The canonical angular momentum is
defined such that it
removes the axial field dependence \cite{aac_paper}. Without the
absorbers, the
beam would have a constant (0) value for the canonical angular momentum.
However, the
presence of absorbers causes the canonical angular momentum to grow and would
lead to severe
emittance growth by the end of a long channel. This growth is stopped by
alternating the direction
of the solenoid field, as shown in Fig.~\ref{simu8}(b) Simulations have shown that 2~m is a
reasonable
(half)  period for the field, since the net growth in canonical angular
momentum is small. In
addition  synchrobetatron resonances are avoided since the periodicity of the field forces the average
betatron wavelength
to be 2~m, whereas the synchrotron oscillation wavelength seen in the simulations for this
arrangement is $\approx 14$~m.
\begin{figure*}[htb!]
\centerline{\epsfig{file=simu8.ps,height=4.0in,width=6.0in}}
\caption[Average momentum  \textit{vs.} z  ]{a) Average momentum  \textit{vs.} z; b) Average
angular momentum: mechanical (solid curve) and canonical (dashed curve), \textit{vs.} z.}
 \label{simu8}
 \end{figure*}

Figure~\ref{simu6}(a) shows the rms and maximum radius of any particle in the beam
distribution as a
function of distance along the channel. The rms radius shows that most of
the beam is confined
to within 2~cm of the axis. The peak rms radius decreases towards the end of
the channel as a
result of the cooling. The maximum particle radius is about 8~cm, which
determines the radius of
the windows required in the rf cavities.  
 Figure~\ref{simu6}(b) shows the rms momentum
spread corrected for
the correlation between $p_z$ and transverse amplitude imposed on the initial
particle distribution.
The momentum spread grows as a function of distance since the alternating
solenoid system only
cools the transverse emittance.
 Figure~\ref{simu6}(c) shows the rms bunch length s a
function of distance
along the channel. Again this grows with distance since this channel does
not cool longitudinally.

\begin{figure*}[bht!]
\centerline{\epsfig{file=simu6.ps,height=4.0in,width=6.0in}}
\caption[rms and maximum beam radii \vs~z ]{a) rms and maximum beam radii, b) rms corrected
momentum, c) rms bunch length; all \vs~z.}
 \label{simu6}
 \end{figure*}

Figure~\ref{coolingeg}(a) shows the decrease in transverse normalized emittance as a
function of distance along
the channel. The system provides cooling by a factor of $\approx $~2 in both the $x$ and $y$ transverse
phase
spaces. From the changing slope of the curve we note that the rate of cooling is
dropping. This sets $\approx$~22~m as
the maximum useful length for this type of system. It must be followed by a
longitudinal
emittance exchange region to reduce the momentum spread and bunch length
approximately
back to their starting values. Figure~\ref{coolingeg}(b) shows the increase in longitudinal
normalized emittance
in the channel due to the increase in momentum spread and bunch length. Finally,
Fig.~\ref{coolingeg}(c) shows the decrease in the 6-D normalized emittance as a function
of distance along the
channel. There is a net decrease in 6-D emittance by a factor of
$\approx$~2 in the channel. Table~\ref{cooltab} gives the initial and final beam parameters.
\begin{figure*}[bht!]
\centerline{\epsfig{file=simu7.ps,height=4.0in,width=6.0in}}
\caption[Emittance \vs~z  ]{Emittance \vs~z: a) transverse emittance $\epsilon_{T,N} (\pi \textrm{mm-mrad})$; b) longitudinal emittance $\epsilon_L (10^{-3}\pi \textrm{mm})$; and c) 6-D emittance $\epsilon_6 (10^{-12} \pi \textrm{m}^3)$.}
 \label{coolingeg}
 \end{figure*}

\begin{table*}[thb]
\caption{Initial and final beam parameters in a 15 T  transverse cooling section.}
\label{cooltab}
\begin{tabular}{llccc}
    & & initial & final & final/initial\\
\hline
Particles tracked    &        &  1000 & 980 & 0.98 \\
Reference momentum            &  MeV/$c$  &186 & 186 & 1.0\\  
Transverse Emittance & $\pi$ mm-mrad & 1400 & 600 & 0.43 \\
Longitudinal Emittance & $\pi$ mm    & 1.1 & 2.3 & 2.09 \\
6-D emittance~$\times 10^{-12}$& ($\pi$~m-rad)$^3$  & 2000 & 800 & 0.40 \\
rms beam radius in hydrogen  & cm            & 0.8 & 0.55 & 0.69 \\
rms beam radius in linac  & cm            & 2.0 & 1.4 & 0.70 \\
max beam radius in linac  & cm            & 7.0 & 7.0 & 1.0 \\
rms bunch length  & cm        & 1.5 & 2.2 & 1.5 \\ 
max bunch full width  & cm        & 13 & 19 & 1.5 \\ 
rms $\Delta p/p$              & \%        & 3.8  & 5.6 & 1.5 \\
\end{tabular} 
\end{table*}


 This simulation has been confirmed, with minor differences, by double precision GEANT \cite{paul} and PARMELA \cite{parmela} codes.

\subsection{31~T solenoid transverse cooling example}

As in the preceding example, the lattice consists of 11 identical 2~m long cells with the direction of the
fields in the solenoids alternating from one cell to the next. The maximum solenoidal field is higher
(31~T) than in the previous example, but the bore can be smaller ($6<r<8$~cm). Several hybrid magnets with at least this field have operated for many years \cite{new_mag}. More recent magnets are of even higher field. A hybrid solenoid with 45~T central field is under construction \cite{45T_mag}. However, all of these magnets are of small bore and generate their rated field over a length of only a few centimeters. Also, typical hybrid magnet longevity is not adequate for our purposes. 31~T magnets with the needed aperture appear practical, but capital and operating costs are high. High-temperature superconductors are capable of adequate current densities for this field. The challenge is to make them in sufficient lengths and to circumvent their present engineering limitations. If the capital cost of magnets employing high-Tc materials is not prohibitive, and solutions using lithium lenses are not chosen, then high-Tc magnets would be the preferred choice.   Between the 31~T solenoids there are 1.3~m long matching
sections with an inside diameter of 32~cm, superimposed on a 36~MeV/m reacceleration linac
operating at 805~MHz.

Table~\ref{cooltab2} gives the initial and final parameters for
 the 31~T example, together with the required emittances for a Higgs factory. In setting these
requirements a dilution of $20\%$ during acceleration  is assumed in each of the three emittances.

\begin{table*}[thb]
\caption{Initial and final beam parameters in a 31~T transverse cooling section.}
\label{cooltab2}
\begin{tabular}{llcccc}
    & & initial & final & final/initial& required\\
\hline
Particles tracked    &               & 4000&3984 &0.99&   \\
Reference momentum       &    MeV/$c$ & 186 & 186 &  \\
Transverse Emittance & $\pi$ mm-mrad & 460 & 240  & 0.52 & 240 \\
Longitudinal Emittance & $\pi$ mm    & 0.85 & 1.6 & 1.9& \\
6-D emittance~$\times 10^{-12}$ &($\pi$~m-rad)$^3$  & 150 & 95 & 0.63&98 \\
rms beam radius in hydrogen  & cm            & 0.44 & 0.33 & 0.75& \\
rms beam radius in linac  & cm            & .4 & 1.1 & 0.80& \\
max beam radius in linac  & cm            & 6.0 & 6.0 & 1.0& \\
rms bunch length  & cm        & 1.5 & 1.8 & 1.2& \\ 
max bunch full width  & cm        & 11 & 19 & 1.7& \\ 
rms $\Delta p/p $             & \%        & 3.5  & 5.0 & 1.4& \\
\end{tabular} 
\end{table*}

\subsection{Bent solenoid emittance exchange example}

We have been considering using a system that exchanges longitudinal and transverse emittance by exploiting dispersion in a large acceptance channel, with a low-Z wedge absorber in the region of dispersion.

In a bent solenoid, in the absence of any dipole field, there is a drift perpendicular to the bend plane
of the center of the Larmor circular orbit, which is proportional to the particle's momentum \cite{toroid,jim_bent}. In our example we have introduced a uniform dipole field
over the bend to cancel this drift exactly for particles with the reference momentum. Particles with
momenta differing from the reference momentum then spread out spatially, giving the required
dispersion (0.4 m). The momentum spread is reduced by introducing
liquid hydrogen wedges \cite{neuffer_wedges}. The hydrogen wedges would be
contained by thin beryllium or aluminum foils, but these were not included in this simulation. 

After one bend and one set of wedges, the beam is asymmetric in cross section. Symmetry is restored
by a following bend and wedge system rotated by 90 degrees with respect to the first.
Figure~\ref{cell_exch} shows a representation of the two bends and wedges. The total solenoid
length was 8.5 m. The beam tube outside diameter is 20~cm, and the minimum bend radii is 34~cm. 
\begin{figure*}[tbh!]
\centerline{\epsfig{file=double_bent_sol.ps,height=4.in,width=4.in}}
\vspace{0.5cm}
\caption{Representation of a bent solenoid longitudinal emittance exchange section}
 \label{cell_exch}
 \end{figure*}


Figure~\ref{simu4}(a) shows the magnetic fields (B$_z$, B$_y$, and B$_x$) as a function of the
position along the cell. The solenoid bend curvature is exactly that given by the trajectory of a
reference particle (equal in momentum to the average momenta given in Fig.~\ref{simu4}(b)) 
in the given transverse fields. The actual shape of the bend turns out to be very important.
Discontinuities in the bend radius can excite perturbations
which increase the transverse emittance.  We have shown, for example,
that the transverse emittance growth in a bent solenoid depends on
discontinuities of the bend radius as a function of distance, and
its first and second derivatives, the size and tilt of the solenoidal
coils, auxiliary fields and the 6-D phase space of the beam. Thus optimization is not straightforward.
One solution to this problem is to have long, adiabatic bends. However, this adds undesirable length
to the emittance exchange section.
We are studying options with coupling sections to tight bends roughly
half a Larmor length long,  which seems to minimize transverse
emittance growth while also minimizing the length of the section.
Due to similar problems, the length and longitudinal distribution of
the wedge material has also been found to affect emittance growth. For example, 
the growth can be minimized when the vector sum of the Larmor
phases at the absorber elements is small or zero.
\begin{figure*}[bht!]
\centerline{\epsfig{file=simu4.ps,height=3.75in,width=5.75in}}
\caption[Axial $B_z$ and dipole $B_y$, $B_x$, magnetic fields   ]{a) Axial $B_z$ and dipole
$B_y$, $B_x$, magnetic fields; b)
average momentum; both as a function of the position along the cell.}
 \label{simu4}
 \end{figure*}

The simulations were performed using the program ICOOL.  The maximum beam
radius is 10~cm. Transmission was 100\%.
Figure~\ref{simu2}(a) shows the rms longitudinal momentum spread relative to the reference
momentum as a function of the position along the cell. The fractional spread  decreases from an
initial value of approximately 5\%, to a final value of approximately
2.2\%. At the same time, since this is an emittance exchange, the transverse beam area grows, as
shown in Fig.~\ref{simu2}(b). One notes that the area increases not only in the regions of bends
(region 1 and 8), but also in the regions of wedges (2-6 and 9-11). This is probably due to failures in betatron matching that have yet to be understood.

\begin{figure*}[htb!]
\centerline{\epsfig{file=simu2.ps,height=3.75in,width=5.75in}}
\caption[rms longitudinal $\delta p_z$ with respect to the reference momentum as a function of z]{a)
rms longitudinal $\delta p_z$ with respect to the reference momentum and b) transverse beam area,
both as a function of z. }
 \label{simu2}
 \end{figure*}

Figure~\ref{xfig5} shows scatter plots of the transverse particle positions against their momenta. The
dispersion is clearly observed in Fig.~\ref{xfig5}(b) (after the first bend) and in Fig.~\ref{xfig5}(e)
(after the second). It is seen to be removed, with a corresponding decrease in momentum spread, in
Fig.~\ref{xfig5}(c) (after the first set of wedges) and Fig.~\ref{xfig5}(f) (after the second set of
wedges).
\begin{figure*}[thb!]
\centerline{\epsfig{file=simu5.ps,height=5.5in,width=5.5in}}
\caption[$y$ \vs~$p_z$ plots and $x$ \vs~$p_z$ plots before and after a wedge ]{$y$ \vs~$p_z$ plots:
a) at the start, b) after the first bend, c) after the first set of wedges. $x$ \vs~$p_z$ plots:  d) after the
first wedges, e) after the second bend, and f) at the end of the emittance exchange section, following
the second set of wedges.}
 \label{xfig5}
 \end{figure*}


Figure~\ref{xfig1} shows a scatterplot of the square of the particle radii vs.\ their longitudinal
momenta, (a) at the start, and (b) at the end of the emittance exchange section. The decrease in
momentum spread and rise in beam area are clearly evident. 

The initial and final beam parameters are given in Table~\ref{xtab}. Although this example demonstrate a factor of $\approx 3$ reduction in the longitudinal momentum spread, there is a $37\,\%$ increase in the 5-D phase space. The simulations must be extended to include rf so that the 6-D emittance can be studied and the emittance
exchange section can be optimized. 

\begin{figure*}[htb!]
\centerline{
\epsfig{file=simu1.ps,height=2.5in,width=3.5in}
}
\caption[Scatterplot of squared radii \vs~$p_z$]{Scatterplot of squared radii \vs~longitudinal
momentum: a) at the start, and b) at the end of the emittance exchange section.}
 \label{xfig1}
 \end{figure*}

\begin{table*}[bth!]
\caption{Initial and final beam parameters in a longitudinal emittance exchange section.}
\label{xtab}
\begin{tabular}{llccc}
    & & initial & final & final/initial\\
\hline
Longitudinal Momentum spread & MeV/$c$        & 9.26 & 3.35 & 0.36 \\
Ave. Momentum         & MeV/$c$        & 180 & 150 & 0.83 \\
Transverse size       & cm            & 1.33 & 2.26 & 1.70 \\
Transverse Momentum spread  & MeV/$c$         & 6.84 & 7.84 & 1.15 \\ 
Transverse Emittance & $\pi$ mm-mrad & 870 & 1694 & 1.95 \\
Emit$_{\textrm{trans}}^2$ $\times$ $\Delta$p$_{\textrm{long}}$& ($\pi$ m-mrad)$^2$ MeV/$c$ & 7.0 & 9.6 & 1.37
\\
\end{tabular} 
\end{table*}

Emittance exchange in solid LiH wedges, with ideal dispersion and matching, has also been
successfully simulated using SIMUCOOL \cite{dave}. Dispersion generation by weak focusing
spectrometers \cite{balbekov} and dipoles with solenoids \cite{wan} have also been studied.

\subsection{rf for the cooling systems}

The losses in the longitudinal momentum of the muon beam from the cooling media have to be
restored using rf acceleration sections. 
These rf structures are embedded in solenoidal fields that 
reverse direction within each section.
% and vary in strength
%from 0 to 5~T.
In the two transverse cooling examples above, the rf frequency is 805~MHz and  the peak gradient
is 36~MeV/m. The magnetic fields that extend over the cavities vary from 0 to 10~T, reversing in
the center. It should be pointed out that in the earlier stages, the bunches are longer, and lower
frequencies will be required.
%\begin{center}
\begin{figure*}[thb!]
\centering
\epsfig{file=rf_a.eps,height=3.5in,angle=-90}
\vspace{-2.5cm}
\caption[Two full cell sections plus two half cell sections   ]{Two full cell sections plus two half cell
sections
of the interleaved $\pi$/2 mode accelerating cavities. The
volumes labeled \textbf{C} are powered separately from the volumes labeled \textbf{D}.}
\label{rf_inter}
\end{figure*}
%\end{center}

In order to realize maximum accelerating gradients within the acceleration
cavities, we take advantage of the penetrating properties of a muon
beam by placing thin windows between each rf cell, thereby creating an
accelerating structure closely approximating the classic pill-box cavity. This permits operating
conditions in which the axial accelerating field is equal to the maximum wall field and gives a high
shunt impedance. 

The windows in the 15~T example are
16~cm diameter, $125~\mu$m thick Be foils. In the 31~T case, they are 10~cm diameter and $50~\mu$m thick. Two studies indicate that at nitrogen temperature the ohmic losses at the foils are negligible and the mechanical deformation is tolerable. \cite{holt_rf_heat}

\begin{figure*}[thb!]
\centering
\epsfig{file=rf_1.3m.eps,height=12cm,angle=-90}
\caption[A 1.3~m  acceleration section  ]{A 1.3~m  acceleration section with quarter section
cut away for viewing the inter-cavity windows.}
\label{rf_section}
\end{figure*}

 For these rf structures, we will use an interleaved cavity design in which
two parts are independently powered (Fig.~\ref{rf_inter}). 
The mode of the system will be referred to
as $\pi$/2 interleaved.
Each section supports a standing wave $\pi$ mode, with each acceleration
cell $\pi$/2 long, giving a good transit time factor.
To reduce the peak rf power requirements (by a factor of 2), we are considering operating the cells
at liquid nitrogen temperatures. 
%This also facilitates maximizing the shunt impedance. 
\begin{table*}[htb!]
\centering{\caption{Characteristics of the rf system.}
\begin{tabular}{lc}
%\multicolumn{2}{c}{RF Cavity Parameters} \\ 
%\noalign{\vspace{2pt}}
%\hline
%\noalign{\vspace{2pt}}
% &    \\
 RF frequency [MHz] & 805    \\
 Cavity Length [cm] & 8.1  \\
 Cavity Inner Radius [cm] & 14.6   \\
 Cavity Outer Radius [cm] & 21   \\
 $Q/1000$  & 2 $\times$ 20   \\
 Peak Axial Gradient [MV/m] & 36  \\
 Shunt Impedance  [M$\Omega$/m] & $2 \times 44$  \\
 $Zt^2$  [M$\Omega$/m] & $2 \times 36$ \\
 Fill Time (3 $\tau$)  [$\mu$s] & $2 \times 12$   \\
 RF Peak Power [MW/m] & $\frac{1}{2} \times 29$  \\
 Ave. Power (15Hz) [KW/m] & 5.3   \\
 Be window aperture [cm] & 16 (10 for 31 T case) \\
 Be window thickness [$\mu$m] & 127 (50 for 31~T case) \\
\end{tabular}
\label{rf_table}
}
\end{table*}

The characteristics of the rf systems currently being studied are summarized 
in Table~\ref{rf_table}. Figure~\ref {rf_section} shows a full 1.3~m section with interleaved
cavities.  
Each cell is 8.1~cm in length and the 1.3~m section consists of 16~cells.  

\subsection{The liquid lithium lens}
The final cooling element ultimately determines the luminosity of the
collider.  In order to obtain smaller transverse emittance as the muon
beam travels down the cooling channel, the focusing strength must increase, i.e.
the $\beta_{\perp}$'s must
decrease.  A current within a conductor produces an active
lens absorber, which can maintain the beam at small $\beta_{\perp}$
throughout
an extended absorber length, while simultaneously  attenuating
the beam momentum.  An active lens absorber, such as a
lithium lens, may prove to be the most efficient cooling element
for the final stages.

The cooling power of a Li lens is illustrated
in Figure~\ref{lilesim}, where the $x$ \vs~$p_x$ phase space distributions at
the beginning and at the end of the absorber are shown.  This example
corresponds to a 1~m long lens, with 1~cm radius, and a surface field of
 10~T.  The beam momentum entering the lens was 267~MeV/$c,$ with Gaussian
transverse spatial and momentum distributions:
$\sigma_x = \sigma_y = 2.89$~mm,
$\sigma_{p_x}=\sigma_{p_y} = 26.7$~MeV/$c,$ and a normalized emittance of
$\epsilon_{x,N} = 710$~mm-mrad.  The normalized emittance at the end of the
absorber was $\epsilon_{x,N} = 450$~mm-mrad (cooling factor
$\sim 1.57$), and the final beam momentum was 159~MeV/$c.$
The results were obtained using a detailed GEANT simulation of a single
stage. 

An alternative cooling scheme under study uses a series of Li
 lenses. The lens parameters would have to vary to match the changing beam emittance along the
section and in addition, acceleration of the beam between the lenses has to be included.

\begin{figure*}[hth!]
\epsfysize=4.5in
%\vspace{-1.0cm}
\centerline{
%\epsffile{lilenssim.eps}
\epsffile{lilens.eps}
}
\caption[$x$-$p_x$ phase space distribution at the beginning  and at the
end of the absorber ]{$x$-$p_x$ phase space distribution at the beginning (top figure) and at the
end (bottom figure) of the absorber described in the text.}
\label{lilesim}
\end{figure*}

Lithium lenses have been used with high reliability
as focusing elements at FNAL and CERN \cite{reviewtev,ref26a,ref26b}.  Although these lenses
have
many similar properties to those required for ionization cooling, there
are
some very crucial differences which will require significant advances in
 lens technology: ionization cooling requires longer lenses
($\sim$ 1 meter), higher fields ($\sim 10~T$), and higher operation
rates (15~Hz).  The last requirement calls for operating the lenses with
 lithium in the liquid phase. A liquid Li lens consists of a small diameter rod-like chamber filled
with
liquid Li through which a large current is drawn. 
  
 The azimuthal magnetic field
focuses the beam to give the minimum achievable emittance
$\epsilon_{x,N} \approx C \beta_{\perp}$ where the constant $C$ depends on 
the properties of the material, for example, $C_{Li}=79$ mm-mrad/cm.
The focusing term can be written as
$\beta_{\perp} \sim 0.08$[cm]$ \sqrt{p/J}$
with $p$ is the muon momentum in MeV/$c,$ and $J$
is the current density in MA/cm$^2$. Increasing $J$ is obviously desirable.
 Decreasing $p$ can also be useful. However, below about 250 MeV/$c$
the slope of ${dE\over dx}(E)$ tends to increase the longitudinal emittance.
The requirement for the highest current density causes 
large ohmic power deposition. The current density will be
limited by the maximum tolerable deposited energy, which
will produce instantaneous heating, expansion, and pressure effects.
Understanding these effects is part of the ongoing liquid Li lens R\&D.

The structural design of the lithium lens is determined by how the
pressure
pulse and heat deposition are handled.  We assume that the Li will be
flowing rapidly under high pressure, confined by electrical
insulators radially and by fairly thick Be windows longitudinally.
Operation at 15~Hz for long periods poses severe challenges.  Shock,
fatigue  and other failure modes are being evaluated, in addition
to studies of material compatibilities, corrosion and degradation to
insure
safe operation over long periods.  It seems that the
minimum required radius of the lens may be the most important parameter
to
determine,
since mechanical problems increase while losses decrease as a function of
radius.

Transferring the beam from one lens to another, with linacs to
reaccelerate and provide longitudinal focusing,
is also a challenging problem,  because of the multiple scattering
introduced
in the windows, straggling and the large divergence of the beams.
We are in the process of evaluating a number of designs for this
transfer channel, using detailed tracking simulations that include solenoids, quadrupoles and other
focusing elements together with Li lenses.

A group from BINP has designed, and is constructing, a 15~cm long liquid lithium lens prototype that will eventually be tested at FNAL.  It is planned to extend this R\&D program to design, construct, and test longer lenses.
 The design of two lenses, whose behavior will be tested at first on a
bench
and then with muon beams at the Ionization Cooling Demonstration Facility, will
then follow \cite{steveproposal}.



\subsection{Ionization cooling experimental R\&D}
An R\&D program has been proposed to 
design and prototype the critical sections of a muon ionization
cooling channel. The goal of this experimental R\&D program 
is to develop the muon ionization cooling hardware to the point
where a complete
ionization cooling channel can be confidently designed for the First Muon
Collider. Details can be found in the Fermilab proposal 
P904 \cite{steveproposal}. A summary of the R\&D program can be found in ref.~\cite{steve_sum}.
% Following is a brief description of the main elements of the experimental program.
\begin{figure*}[bht!]
%\noindent
\begin{minipage}{.85\linewidth} % fig  
\centering\epsfig{figure=muonexp7a_c.eps,width=0.85\linewidth}
\end{minipage}\hfill
\begin{minipage}{.85\linewidth} % fig 
\centering\epsfig{figure=muonexp7_c.eps,width=0.85\linewidth}
\end{minipage}
\caption{Schematic of the cooling test apparatus arrangement.}
\label{cooling_expt}
\end{figure*}
The proposed R\&D program consists of:
\begin{itemize}
\item Developing an appropriate rf re-acceleration structure. 
It is proposed to construct a 3-cell prototype rf cavity with 
thin beryllium windows, which will be tested at high power and within a high-field solenoid.
\item Prototyping initially a 2~m section, and eventually a 10~m section, 
of an alternating solenoid 
transverse cooling stage. It is proposed to test the performance of 
these sections in a muon beam of the appropriate momentum.
\item Prototyping an emittance exchange (wedge) section and measuring its 
performance in a muon beam of the appropriate momentum.
\item Prototyping and bench testing 
$\sim 1$~m long liquid lithium lenses, and 
developing lenses with the highest achievable surface fields, and hence 
the maximum radial focusing.
\item Prototyping a lithium lens--rf--lens system and measuring its 
performance in a muon beam of the appropriate momentum.
\item Developing, prototyping, and testing a hybrid lithium lens/wedge 
cooling system.
\end{itemize}

The measurements that are needed to demonstrate the cooling capability and 
optimize the design of the alternating solenoid, wedge, 
and lithium lens cooling stages will 
require the construction and operation of an ionization cooling test facility. 
This facility will need
\begin{enumerate} 
\item a muon beam with a central momentum that can be 
chosen in the range $100$-$300$~MeV/$c,$ 
\item an experimental area that can 
accommodate a cooling and instrumentation setup of initially $\sim 30$~m 
in length, and eventually up to $\sim 50$~m in length, and 
\item instrumentation to precisely measure the positions of the incoming 
and outgoing particles in 6-D phase space and confirm that they 
are muons.
\end{enumerate} In the initial design shown in Fig.~\ref{cooling_expt}, 
the instrumentation consists of 
identical measuring systems before and after the cooling apparatus \cite{lumcdpre}. 
Each measuring system consists of 
(a) an upstream time measuring device to determine the arrival time of the 
particles to one quarter of an rf cycle ($\sim\pm300$~ps), 
(b) an upstream momentum spectrometer in 
which the track trajectories are measured by low pressure TPC's on either 
side of a bent solenoid, 
(c) an accelerating rf cavity to change the 
particles momentum by an amount that depends on its arrival time, 
(d) a downstream momentum spectrometer, which is identical to the upstream 
spectrometer, and together with the rf cavity and the upstream spectrometer 
forms a precise time measurement system with a precision of a few ps. The 
measuring systems are 8~m long, and are contained within a high-field 
solenoidal channel 
to keep the beam particles within the acceptance of the cooling apparatus.

It is proposed to accomplish this ionization cooling R\&D program 
in a period 
of about 6~years. At the end of this time we believe that it will be 
possible to assess the feasibility and cost of constructing an ionization 
cooling channel for the First Muon Collider, and if it proves feasible, 
begin a detailed design 
of the complete cooling channel. 

%%%% TH LINES BELOW ARE GOOD FOR TWOCOLUMN FORMAT
%\begin{figure*}[thb!]
%\noindent
%\begin{minipage}{.85\linewidth} % fig 
%\begin{minipae}{2.05\linewidth} % fig 
%\centering\epsfig{figure=muonexp7a_c.eps,width=2.05\linewidth}
%\end{minipage}\hfill
%\begin{minipage}{2.05\linewidth} % fig 
%\centering\epsfig{figure=muonexp7_c.eps,width=2.05\linewidth}
%\end{minipage}
%\caption{Schematic of the cooling test apparatus arrangement.}
%\label{cooling_expt}
%\end{figure*}
