In this Chapter, the Buncher and SFOFO cooling channel are introduced and described. Performance, systematic errors and tolerances are discussed. Design of the LH$_2$ absorbers is included here, in Section~\ref{BandC:absorber}. Design of the rf components and superconducting solenoid magnets are discussed in Chapters~\ref{RF:chapter} and \ref{SCSOLE}, respectively. 

The designs presented here for the bunching and cooling channels employ a
variety of magnetic-focusing lattices.  In these lattices, the solenoidal
magnetic field is periodically reversed in order to modulate the beta
function, producing periodic minima and maxima of beta, typically with
local secondary minima and maxima located in between (see Fig.~\ref{fig:beta_z}).  To be specific in our descriptions, we here define a ``cell" to be that
portion of apparatus extending from one beta minimum to the next (for
example, from one liquid-hydrogen absorber to the next in the SFOFO
cooling lattice described below).  Note that one cell of such a lattice
thus corresponds to a half-period of the magnetic field.
\section[Matching from Induction Linac to Buncher]{Matching Section from the Induction Linac to the Buncher.}       
\label{bunchcool}
After the energy spread of the beam has been reduced in the induction linacs,
the muons are distributed continuously over a distance of about 100 m. It is then
necessary to form the muons into a train of bunches prior to cooling and
subsequent acceleration. First, an 11-m-long magnetic lattice section (four 2.75 m cells) is used to
gently transform the beam from the approximately uniform  solenoidal field used
in the induction linacs to the so-called ``super-FOFO," or SFOFO, lattice used
in the remainder of the front end. This is followed by the 55-m-long rf buncher
itself, which consists of rf cavity sections interspersed with drift
regions. These two functions are performed sequentially for
design simplicity. There is a significant advantage in using the same lattice
in the buncher section as in the cooling region to follow, since it avoids
adding another complicated 6-dimensional matching section.
% Rick's version of this paragraph, shared for the buncer and cooler. 
%
%The buncher section was simulated independently using the Monte Carlo tracking
%codes ICOOL \cite{ICOOL} and Geant4 \cite{geant4}. As we will show later,
% results from these codes agree within acceptable systematic errors.

Two distinct computer codes were used to simulate this buncher and the
%D.M.K March 14 2001 : insert a space in all the \cite
cooling channel: ICOOL~\cite{ICOOL} and Geant4~\cite{G4}. There is no
shared code between the two programming environments: Fortran for ICOOL and C++ for Geant4.
The Geant4 and ICOOL implementations were based solely on the parameters listed below. 
%D.M.K March 14 2001 : The minor clarification cast doubt.
%After optimization of the matching sections and minor clarifications, good
%agreement was obtained, as shown in the performance section.
After optimization, good agreement between these two codes was obtained, as
shown in the performance section. Thus, we have high confidence that the simulated cooling performance is realistic.
%P.L, March 14-15 2001 
%Comment this out, as it is not true: ICOOL
%calculations have been done on misalignment, and are rpesented. 
% They might not be as exact as Geatn4 because of the modeling of the 
% field, but that's a quibble.
%  Based on this
% successful comparison, the more demanding magnet alignment tolerance study was
% done in the generic 3D code, i.e. Geant4.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
    
\subsection{The Transverse Matching Section}
%D.M.K March 14 2001
%The goal of the matching section is to smoothly transform the muon beam from
The purpose of the transverse matching section is to transform the muon beam smoothly from
the approximately uniform 1.25~T focusing field in the induction linac to the 2~T alternating polarity
SFOFO lattice. The 4\% rms momentum spread entering the matching section is
relatively small, so chromatic corrections are less critical than in the
minicooling field reversal. Table~\ref{BandC:tbmatch} gives coil dimensions and
current densities for the solenoid magnets used in the matching simulations.

\begin{table}[tbh]
\centering
\caption{Matching  section magnets.}
\label{BandC:tbmatch}
\begin{tabular}{|ccccc|}
\hline
$z$ & $dz$ & $r$ & $dr$ & $j$ \\
   (m) &    (m)  &   (m)   &   (m)   &  (A/mm$^2$) \\ 
\hline
  0.358 &  1.375 &  0.300 &  0.100 &     -9.99 \\
  1.733 &  0.330 &  0.300 &  0.110 &    -15.57 \\
  2.446 &  0.187 &  0.330 &  0.330 &    -33.40 \\
  2.963 &  0.187 &  0.330 &  0.330 &     35.19 \\
  4.008 &  0.330 &  0.770 &  0.110 &     67.41 \\
  5.146 &  0.187 &  0.330 &  0.330 &     43.75 \\
  5.663 &  0.187 &  0.330 &  0.330 &    -43.75 \\
  6.708 &  0.330 &  0.770 &  0.110 &    -66.12 \\
  7.896 &  0.187 &  0.330 &  0.330 &    -43.75 \\
  8.413 &  0.187 &  0.330 &  0.330 &     43.75 \\
  9.458 &  0.330 &  0.770 &  0.110 &     66.12 \\
 10.646 &  0.187 &  0.330 &  0.330 &     43.75 \\
\hline
\end{tabular}
\end{table}
%D.M.K March 14 2001 :
% Figure 19, top, has slightly different numbers than in the table 
% above (bar round-off, 37 vs 35.19 37 vs 67.. 
% Panel top, middle bottom are also misalgined.
\begin{figure}
\begin{center}
%\input{jcg-match.fig}
\includegraphics*[bb=147 153 502 694,width=3in,clip]{../template/report/ps-and-eps/jcg-match.ps}
%\input{ffDummy.fig}
\caption[Magnet configuration and axial field and beta function] {Magnet configuration, axial magnetic field and beta function of the matching section to the SFOFO lattice.} 
\label{fig:match}
\end{center}
\end{figure}

The magnet configuration at the beginning of the section, the axial magnetic
field on-axis, and the beta functions for three momenta are shown in
Fig.~\ref{fig:match}.
The magnetic lattice goes from a series of constant radius solenoids to an 
SFOFO cell structure consisting of small radius coils at each end of a cell
and a large radius coil in the middle. 
%D.M.K March 14 2001 : non-standard compare to what ? This whole paragraph
% refers to the transition, so it is confusing. P.L> decide to comment it out.
%Four magnets in the transition region have non-standard current densities. 
The axial magnetic field in a cell peaks symmetrically near the two ends and
has a smaller secondary peak in the middle. The beta functions across the match
are similar for the three momenta shown, which vary in momentum steps of
7.5\% from 185 to 215~MeV/$c.$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%D.M.K March 14 2001 : unnecessary empty space.
% P.L. does not care, up to the last editor to decide.
%\clearpage
\section{Buncher Section}
\label{buncher}    
The design principles for the lattice and details concerning the rf  and
other technical components for the buncher section will be described later.
Only the beam dynamics and performance will be described here. 

The buncher
magnetic lattice is identical to that used in the first cooling section. It
contains rf cavities in selected lattice cells and no absorbers. The main rf frequency
was chosen to be 201.25~MHz in the front end, so that the beam would fit radially
inside the cavity aperture. Power sources and other technical
components are available at this frequency. The 201.25~MHz cavities are placed at
the high-beta locations in the lattice, just as in the cooling section.
Harmonic cavities running at 402.5~MHz are placed at minimum-beta locations, corresponding to  where hydrogen absorbers are placed in the cooling section.

The buncher comprises 20~lattice cells, each 2.75~m long. Maximum bunching
efficiency is obtained by breaking the region into three rf stages, 
separated by drift regions. The locations and lengths of the buncher components
are given in Table~\ref{BandC:tbbunchz}.
%\vskip.2in

\begin{table}[tbh]
\centering
\caption[RF buncher component locations]{RF buncher component locations used in the simulations.}
\label{BandC:tbbunchz}
\begin{tabular}{|l|cccc|}
\hline
    & Length &  Frequency & Phase & Gradient \\
    & (m) & (MHz) & (deg.) & (MV/m) \\
\hline
Harmonic rf & 0.186 & 402.5 & 180 & 6.4\\
Space & 0.443 &&&\\
rf  & 4 $\times$ 0.373 & 201.25 & 0 & 6.4 \\
Space &0.443 &&&\\
Harmonic rf & 0.186 & 402.5 & 180 & 6.4\\
\hline
Drift 1& 10 $\times$  2.75 &&&\\
\hline
Harmonic rf & 0.186 & 402.5 & 180 & 6\\
Space & 0.443 &&&\\
rf  & 4 $\times$ 0.373 & 201.25 & 0 & 6 \\
Space & 0.443 &&&\\
Harmonic rf &2 $\times$ 0.186 & 402.5 & 180 & 6\\
Space & 0.443 &&&\\
rf  & 4 $\times$ 0.373 & 201.25 & 0 & 6 \\
Space & 0.443 &&&\\
Harmonic rf & 0.186 & 402.5 & 180 & 6\\
\hline
Drift 2& 3 $\times$  2.75&&&\\
\hline
Space & 0.629 &&&\\
rf  & 4 $\times$ 0.373 & 201.25 & 12 & 8 \\
Space & 0.629 &&&\\
Space & 0.629 &&&\\
rf  & 4 $\times$ 0.373 & 201.25 & 12 & 8 \\
Space & 0.629 &&&\\
\hline
Drift 3& 2 $\times$  2.75&&& \\
\hline
\end{tabular}
\end{table}

%\vskip.5in
Second harmonic (402.5~MHz) cavities are used at the entrance and exit of the
first and second stages to linearize the shape of the rf pulse. All
cavities are assumed to have thin Be windows at each end. They are modeled in
the simulation codes as perfect TM$_{010}$ pillboxes. The window radii and
thicknesses are given in Table~\ref{RF:tb18.D2}.
%D.M.K March 14 2001 :
% Dan writes: 
% Don't the multicell cavities have windows between cells also?
% Note that here the term used is "cavities", obviously refering to multicell 
% cavities.  Elsewhere these are consistently called "linacs" I prefer to use
% the term cavities everywhere.  "Linac" was the ? (can't read, presumably 
% co-notation) of a considerably longer structure than we are talking about,
% although strictly speaking it is not incorrect? Why don't we take a vote?
% (But only people whose sense of English style we respect should be counted)
%P.L, March 15 2001 
% O.K., Dan, but this is not about English litterature, nor democracy among the
% enlighted. Linac is perfectly clear. They are short, so what. 
% In the sentence above, Rick used "cell" and "cavity" as synonyms. Which is 
% fine. No change necessary here.    
%
The electric field gradient in the buncher ranges from ~6 to 8~MV/m. 
A long drift is provided after the first stage to allow the particles to
begin overlapping in space.

Figs.~\ref{BandC:buncher1} and ~\ref{BandC:buncher2} show the momentum-time distributions at
the start, and after each of the three buncher stages. Distributions are also
shown at the ends of the first and second cooling stages. In the last three
distributions,  ellipses are drawn indicating the approximate acceptance of the
cooling channel.
%
%P.L, March 14-15 2001
% This figure was still unreadable on my printer!  Panel overlaps
% seemingly random alignment of these panels. Heading of figure unnecessary.
\begin{figure}
\begin{center}
\includegraphics*[bb=135 350 510 528,clip]{../template/report/ps-and-eps/jcg-buncher1.ps}
\includegraphics*[bb=206 350 428 517,clip]{../template/report/ps-and-eps/jcg-buncher2.ps}
\includegraphics*[bb=180 360 406 516,clip]{../template/report/ps-and-eps/jcg-buncher3.ps}
\caption{Momentum-time distributions through the buncher.} 
\label{BandC:buncher1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
%%\includegraphics*[bb=135 350 510 528]{jcg-buncher1.ps}
%%\includegraphics*[bb=206 350 428 517]{jcg-buncher2.ps}
%%\includegraphics*[bb=180 360 406 516]{jcg-buncher3.ps}
\includegraphics*[bb=206 359 432 515,clip]{../template/report/ps-and-eps/jcg-buncher4.ps}
\includegraphics*[bb=179 355 395 514,clip]{../template/report/ps-and-eps/jcg-buncher5.ps}
\includegraphics*[bb=178 357 405 511,clip]{../template/report/ps-and-eps/jcg-buncher6.ps}
\caption{Momentum-time distributions through the buncher (continued).} 
\label{BandC:buncher2}
\end{center}
\end{figure}
%\afterpage{\clearpage}

It can be seen that, at the end of the buncher, most, but not all, particles are
within the approximately elliptical bucket. About 25\% are outside the bucket
and are lost relatively rapidly, and another 25\% 
%P.L, March 14-15 2001 Let us be more specific and definitive here.. 
%are lost more slowly
are lost in the cooling channel as the 
longitudinal emittance rises due to straggling and the negative slope of the
$dE/dx$ curve with energy.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\subsection{Longitudinal-Transverse Correlation}
%D.M.K March 14 2001 :
% What does "front-end latice" refers to ?
% suggest " Starting in the induction Linacs "
% P.L. answers: I suspect the correlation starts developing in the 
% decay channel, because the high transverse amplitude particle are already 
% delayed compare to the one going straight down the channel.
%  so, "front-end lattice", albeit a bit vague, is correct.
% "Starting in the induction Linacs" is definitly incorrect.
A significant coupling develops in these pre-cooling stages of the Neutrino Factory (including the
induction linac)  between a particle's longitudinal and transverse motions.
This occurs because particles with different transverse
displacements, or angular divergences, take different amounts of time to move
axially down the solenoidal lattice. They thus arrive at the cavities at
different points in the rf cycle, or at a different time with
respect to the induction linac pulse, thereby obtaining different accelerations and velocities.
%
% The fact we have also a different longitudinal focusing is of course related,
% but not essential to explaining the correlation, to first order. Also, 
% this second order effect is hard to quantify, and is not apparent in the 
% figures listed below. Hence, I would leave this out. 
% A substantial fraction of the correlation is due to the induction Linac, 
% not the bunching.
% The voltage gain or loss is too weak in the bunching section to explain
% why a muon could gain ~30 MeV and still remain in the bucket.  
The resulting  correlation, shown in Fig.~\ref{fig:corr}, 
 can be expressed as
\begin{equation}
p = p_o + C A^2,
\end{equation}
where $C$ is the correlation coefficient and the transverse amplitude is
defined as
\begin{equation}
A^2 = \frac{r^2}{\beta^2_{\perp}} + \theta^2.
\label{COOL:eq2}
\end{equation}
%P.L, March 14-15 2001 
% move this in the paragraph down, and be a bit more explicit
%Figure~\ref{fig:corr} shows 
%that there is little correlation between momentum
%and angular momentum after the induction linacs, indicating that the field
%reversal is correctly located.
%D.M.K March 14 2001 :
% Where is the angular momentum measured?  
% P.L. : Good point. I presumed this is measured at the entrance of the 
% buncher. Juan, Rick, could you ask ob ? 
\begin{figure}
\begin{center}
%\input{jcg-corrs3-new.fig}
\includegraphics*[bb=150 280 412 638,clip]{../template/report/ps-and-eps/jcg-corrs3-new.ps}
\caption[Correlations ]{(top) Correlation between momentum and angular momentum; (bottom) correlation between longitudinal momentum and transverse amplitude (see Eq.~\ref{COOL:eq2}), after the induction linac (IL2).}
\label{fig:corr}
\end{center}
\end{figure}
This quantity is evaluated at a waist in the transverse plane.

The magnitude of the momentum-amplitude correlation coefficient is seen
 from Fig.~\ref{fig:corr} to be 0.7~GeV/$c$.
This is a higher value than the 0.45~GeV/$c$ that would be obtained
without the minicooling. Ideally, the correlation should be such that forward
velocity in the following lattice is independent of transverse amplitude. A
value of approximately 1.1~GeV/$c$ would be required for this. 

Figure~\ref{fig:corr} shows 
%P.L, March 14-15 2001
% More explicit..Also, these two distinct coorelation are now listed in 
% order of appearance in the plots.
also 
that there is little correlation between momentum
and angular momentum after the induction linacs, indicating that the field
reversal in the minicooling is correctly located
with respect to the induction linacs.  

\section{Ionization Cooling Channel }

        The rms transverse emittance of the muon beam emerging from the induction
linac must be reduced to $\approx 2$ mm$\cdot$rad (normalized) in order to 
fit into the downstream accelerators and be contained in the storage ring. 
Ionization cooling is currently our only feasible option~\cite{FS1}. The
cooling channel described below, as well as the one described in the appendix, are based on extensive theoretical studies and computer simulations
performed in the same context as our previous studies~\cite{MC1, FS1, GPenn1,
JM1}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\subsection{Principle of Ionization Cooling}

        In ionization cooling, the beam loses both transverse and longitudinal
momentum by ionization energy loss while passing through an absorber.  The
longitudinal momentum is then restored to the beam in rf accelerating cavities.
This sequence, repeated many times, results in a reduction of the angular spread and thereby reduces the transverse emittance. 

Ionization cooling is limited by multiple Coulomb scattering (MCS) in the
absorbers. To minimize the MCS effect on cooling channel performance, we must have rather strong focusing at the
absorber, with $\beta_{\perp, min} \approx$ 0.4 to 0.2 m at a momentum of 200~MeV/$c$.  Strong solenoidal fields are used for this purpose. Weak focusing, 
\textit{i.e.}, too large $\beta_{\perp}$ at the absorbers, leads to
excessive  emittance growth due to MCS. Too strong focusing is hard to
achieve for such large aperture beam transport, but can also be detrimental to
the 6D beam dynamics. As the angles, or beam divergence, get too large, the
longitudinal velocity decreases too much, leading to the wrong
longitudinal-transverse correlation factor and thereby resulting in
unacceptable growth of the longitudinal emittance. Choosing the  right range of
$\beta_{\perp, min}$ with respect to the operating momentum is a key to a successful
design~\cite{FS1, JM1}. 

The approximate equation for transverse cooling in a step $ds$ along the
particle's orbit  is~\cite{MC1}
%D.M.K March 14 2001 :
% Dan suggests to cite here Neuffer + Fernow + Gallardo
% Ankenbrandt et al does not derive this formulas
% P.L. So what? This is lost in history, let it be an academical point.
\begin{equation}
\frac{d\epsilon_N}{ds} = -\frac{1}{\beta^2}\frac{dE_{\mu}}{ds}\ \frac{\epsilon_N}{E_{\mu}} +
 \frac{\beta_{\perp} (0.014 \textrm{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 absorber material.  The betatron function is determined by the strengths of the elements
in the focusing lattice. Together with the beam emittance, the beta 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} = \frac{\beta_{\perp} (14 \textrm{MeV})^2}{ 2 \beta m_{\mu} \frac{dE_{\mu}}{ds} L_R }.
\label{equi}
\end{equation}
%D.M.K March 14 2001 :
% Dan did the arithmetic and he got 290 for beta = 1. 
% Presumably, Rick did it at our operating momentum. This explains the 
% difference ==> P.L. specifies the velocity beta factor).
For a relativistic ($\beta \approx 0.87$) muon in liquid hydrogen with a
beta function $\beta_{\perp} = 8$~cm, which corresponds roughly to
confinement in a 15~T solenoidal field, the minimum achievable emittance is
about 340~mm$\cdot$mrad.

The equation for energy spread  is 
%D.M.K March 14 2001 : Missing reference here 
\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
%P.L, March 14-15 2001, let us be unequivocally affirmative here.
%will require simulation studies and hardware development for its
%D.M.K March 14 2001 : points this out
requires simulation studies and hardware development for its
optimization and application. 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, if the beam intensity is sufficiently high.

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
practical limits on magnetic field strengths, this gives an estimate of
the minimum achievable emittance for a given configuration.
Next several tracking codes were written, or modified, to study
the cooling process in detail. These codes 
use Monte Carlo techniques to track particles one at a time through the
cooling system. The codes attempt to include all relevant physical
%D.M.K March 14 2001 :to some degree is vague here. Strike
%processes to some degree, (e.g. energy loss, straggling, multiple scattering)
processes (\textit{e.g.}, energy loss, straggling, multiple scattering),
%P.L, March 14-15 2001, The r.f. fields are also correct.
%and use Maxwellian models of the focusing fields.
and use physically correct electromagnetic fields.
%D.M.K March 14 2001 : Maxwellian means what? 
% P.L. answers In the right circle, people know about Maxwell equation 
% Mawellian means that "obey maxwells equations" Good enough!  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\subsection{Concept of the Tapered SFOFO Cooling Channel}
%D.M.K March 14 2001 :
% Dan made quite a few good suggestion to improve this text. 
% However, I choose to ignore some for sake of brevity. 
% For instance, "Larmor center" and "alternating solenoid" 
% are still undefined.. 
        For optimal performance, the solenoidal field should not be kept constant during the entire
cooling process. In a cooling channel with a constant solenoidal field, 
the transverse momentum of each particle will decrease, while
the position of the Larmor center will not, causing the net angular momentum of the beam
to grow.  To avoid this, we flip the field while maintaining good focusing throughout
the beam transport and low $\beta_{\perp}$ at the absorbers.  One of the simplest
solutions (the FOFO lattice), is to vary the field  sinusoidally.  The
transverse motion in such a lattice can be characterized by its betatron
resonances, near which the motion is unstable.  The stable operating region
is between the  low momentum ($2 \pi$) and high momentum ($\pi$) phase advance per half-period of the lattice. (Note that a half-period of the lattice is one ``cell" in our notation.)
%D.M.K March 14 2001: Dan objects the order in which the 2 pi and pi are 
% introduced.  I did this because the "low" momentum comes first.       
\begin{figure*}[t]
\centering
%\input{jcg-ff.fig}
\includegraphics[bb=169 293 401 466,clip]{../template/report/ps-and-eps/jcg-ff.ps}
\caption[The longitudinal component of the magnetic field ]{The longitudinal component of the on axis magnetic field, $B_z$, for a typical SFOFO lattice.}
\label{ShortBz}
\end{figure*}
%\clearpage
        The SFOFO lattice~\cite{sessler}  is based on the use of 
alternating solenoids, but is a bit more complicated. We add a second harmonic
to the simple sinusoidal field, producing the axial field shown in
Fig.~\ref{ShortBz}. As in the FOFO case, the axial field vanishes at
the $\beta_{\perp,min}$ position, located at the center of the absorber.  This
is accomplished by using two short \textsl{focusing} coils running in opposite polarity.
However, unlike the FOFO case, the field decreases and flattens at
$\beta_{\perp,max}$, due to a \textsl{coupling} coil located midway between the focusing coils, around the rf cavity. 
The transverse beam dynamics is strongly influenced by the solenoidal field
profile on-axis and by the desired range of momentum acceptance. 

\begin{figure}
\begin{center}
%\input{ffDummy.fig}
\includegraphics[bb=85 374 334 745,clip]{../template/report/ps-and-eps/jcg-51n.ps}
%P.L, March 14-15 2001. We need to be a bit more explicit here. 
%\caption{(a) Beta functions in the (1,3) cooling lattice section; (b) phase advance as function of momentum spread.}  
\caption[Beta function and phase advance in the (1,3) lattice]{(top) Beta functions in the (1,3) cooling lattice section, at small
transverse amplitude, for 7 different momenta, spanning the entire operating
range from $155$ to $245$~MeV/$c$ above the $2\pi$ and below the $\pi$ resonance. (bottom) $\beta_{\textrm{min}}, \beta_{\textrm{max}}$ and  phase advance as a function of relative momentum. The lower curve corresponds to $\beta_{\textrm{min}}$ and the second curve from the bottom to the phase advance. The black crosses show the $\beta_{\textrm{max}}$ function.}
% P.L, March 14-15 2001. I am not sure I got this right !..
% Note also that this figure has red and green dots of the same size, 
% which is not very distinctive for color blind printer or people.. 
\label{fig:beta_z}
\end{center}
\end{figure}

This SFOFO lattice has several
advantages over the FOFO: 
\begin{itemize}
\item  The betatron resonances are usually a nuisance, since they inevitably
restrict the region of stable motion. However, in this case they  give us a
strong, approximately constant, focusing result (\textit{i.e.}, flat $\beta_{\perp}$) across the relevant momentum range, as we operate between the $2
\pi$ and $\pi$ resonances. This is illustrated in Figs.~\ref{fig:beta_z} and ~\ref{BetaPP}. Within this (albeit limited) momentum range
%P.L, March 14-15 2001 
the transverse motion is stable.
%D.M.K March 14 2001: a "period" is a length", period length is redundant.
% O.K...
\item For a given $\beta_{\perp,min}$, the SFOFO period is longer than the
corresponding FOFO period, allowing longer absorbers per
lattice cell, thereby reducing the relative amount of multiple scattering
%P.L, March 14-15 2001 in instead of from ? Can't say which is better english.
% from 
in  the absorber windows. The longer period also allows more room for all other components. 
\item The focusing coils can be located just around the absorbers, adjacent
to the rf cavity. Since the absorber has a much smaller outer diameter
than does the rf cavity, this arrangement allows the
diameter of these high-field magnets to be reduced considerably, with concomitant cost savings. 
\end{itemize}
\afterpage{\clearpage 
\begin{figure*}[t]
\begin{center}
\includegraphics[width=3in,clip]{../template/report/ps-and-eps/jcg-betas.ps}
%\input{jcg-betas.fig}
\caption[The $\beta_{\perp}$ function \textit{vs.} momentum]{The $\beta_{\perp}$ function versus momentum for the five SFOFO lattices described below.  }
\label{BetaPP}
\end{center}
\end{figure*}
}

        For a given lattice period, one can adjust independently the  location
of the two betatron resonances, or, equivalently, the nominal operating
momentum and the $\beta_{\perp,min}$ at that momentum. By adjusting these two
parameters, we can keep the $\beta_{\perp}$ symmetric about the required
nominal momentum, and independently reduce the central $\beta_{\perp}$ value.
However, this is true over only a limited momentum range. As we decrease the
coupling field and increase the focusing field, the momentum acceptance will
shrink as the $\pi$ and $2 \pi$ resonances move closer to the nominal momentum.
At this point, we are forced to change the lattice period. 

        This brings us to the second improvement over the FOFO channel used in
the previous feasibility study: $\beta_{\perp,min}$ can be ``tapered" along the
cooling channel.  One can slowly increase the focusing strength at a fixed
operating momentum, while keeping a reasonable momentum acceptance. Were we to use a fixed $\beta_{\perp,min}$, as the cooling progresses, the rms angle would
decrease.  The cooling rate would then also decrease as the heating term
due to multiple scattering becomes relatively more important.  By slowly
increasing the focusing strength (decreasing  $\beta_{\perp,min}$), we can maintain large rms angles at the absorbers ($\sigma_{x'} = \sigma_{y'}
\approx 0.1$~rad), thereby keeping the relative effect of multiple scattering to a minimum.

%P.L, March 14-15 2001 This figure says the same thing as figure 4 
% In case we need to make the document shorter, one of these two figures 
% can go.  This one below is slightly easier to understand, but we 
% need to label the 5 lines clearly in the caption, and make 6 lines, please,
% or explain there are only 5 lines while we talk about 6 lattices..
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\subsection{Description of the SFOFO Cooling Channel}

        In this subsection, we describe the cooling channel from the
viewpoint of the simulation effort. Engineering details will be given later.  

\subsubsection{Lattices}
%D.M.K March 14 2001
% Dan wrote around this section "ackward" and "confusing"
% He suggests to define more clealy the geometrical lattice cell
% P.L. Adds a definition, and removes the "length" as in period length.
%
The channel operates at a nominal momentum of 200 MeV/$c$. There are six
sections with steadily decreasing $\beta_{\perp,min}$. In the first three
lattices, labeled (1,i), i=1,3, the lattice cell length is 2.75~m, and in the other three lattices, (2,i), i=1,3, it is 1.65~m. A  cell of the
cooling lattice comprises one absorber, one linac and three coils.   The matching sections between these
sections also consist of cooling cells, which differ from  the regular cooling
cells only by the current circulating in the coils, with one  exception: A
different coil length must be used in the matching section between the (1,3)
and (2,1) lattices, where the cell length decreases from 2.75~m to 1.65~m. The
lengths of these lattice sections are specified in Table~\ref{sectionLength}. 
Coil dimensions and current densities are specified in
Table~\ref{coolCoils}. In the simulations, it is assumed that the current density is uniform
across the thickness of the coil.

\begin{table}[htb]
%\centering
\begin{center}
\caption[Length of the sections and integrated length]{Lengths of the sections and integrated length from the start of the 
cooling channel.}
\begin{tabular}{|lcc|}
\hline
Section & Length& Total length \\ 
 & (m) & (m)\\\hline
Cool (1,1) & 4 $\times$ 2.75 = 11&11 \\
Match (1,1-2) & 2 $\times$ 2.75 = 5.5&16.5 \\
Cool (1,2) & 4   $\times$ 2.75  = 11& 27.5  \\
Match (1,2-3) & 2   $\times$ 2.75  = 5.5& 33  \\
Cool (1,3) & 4   $\times$ 2.75  = 11 & 44  \\
Match (1,3) - (2,1) & 4.4 & 48.4  \\
Cool (2,1) & 12 $\times$ 1.65 = 19.8 & 68.2  \\
Match (2,1-2) & 2 $\times$ 1.65 = 3.3 & 71.5  \\
Cool (2,2) & 8  $\times$ 1.65  = 13.2 & 84.7  \\
Match (2,2-3) & 2  $\times$ 1.65  = 3.3 & 88  \\
Cool (2,3) & 12   $\times$ 1.65 = 19.8 & 107.8  \\
\hline
\end{tabular}
\label{sectionLength}
\end{center}
\end{table}

\begin{table}[htb]
\centering
\begin{center}
\caption[Geometry and current densities for the solenoids]{Geometry and current densities for the solenoids used in the simulations. The j(1,n) coil types refer to the 2.75-m-long cell, and the j(2,n) coils to the 1.65-m-long cell.
The position refers to the upstream edge of the coil and  starts from the
beginning of a cell. The radius refers to the inner radius of the
coil. The current indices refer to the
nomenclature used in the previous table.}
\label{coolCoils}
\begin{tabular}{|lccccccc|}
\hline
Type & Position & Length & 
Radius & Thickness  & j(1,1)& j(1,2)  &j(1,3)\\
& (m) & (m) & (m) & (m) &(A/mm$^2$)  &(A/mm$^2$)  &(A/mm$^2$) \\
\hline
Focusing &   0.175 &  0.167 &  0.330 &  0.175 &   75.20 & 84.17& 91.46 \\
Coupling &  1.210 &  0.330 &  0.770 &  0.080 &    98.25 & 92.42& 84.75\\
Focusing &  2.408 &  0.167 &  0.330 &  0.175 &    75.20 & 84.17& 91.46\\
\hline
  &  &  &  &  & j(2,1) & j(2,2) & j(2,3) \\
\hline
Focusing &   0.066 &  0.145 &  0.198 &  0.330  & 68.87 &  75.13&  83.48 \\
Coupling &   0.627 &  0.396 &  0.792 &  0.099  & 95.65 &  88.00&  76.52 \\
Focusing &   1.439 &  0.145 &  0.198 &  0.330  & 68.87 &  75.13&  83.48 \\
\hline
\end{tabular}
\end{center}
\end{table}
%D.M.K March 14 2001
% typos and removal of extra words (lattice cell -> cell) 
The design of the matching sections between regular sections of the
same cell length goes as follows.  In all cases, a matching section is
inserted that consists of two lattice cells, the first as in the previous cells,
the second as in the following cells, except that the currents in the central
pair of focus coils are set to an average of the currents in the previous and
following focusing coils. For instance, Table~\ref{matchCoil1-2} describes the
match between the (1,1) and (1,2) lattices. 

\begin{table}[!htb]
\begin{center}
\caption[Geometry and current density in the first matching]{Geometry and current densities for the solenoids in the first matching
section. Coil locations are given with respect to the start of the channel.
 The coil dimensions are specified in Table~\ref{coolCoils}. }
\begin{tabular}{|lcc|}
\hline
Type  & Location    & j(1,i) \\
   & (m) & (A/mm$^2$)\\ 
 \hline
\multicolumn{3}{|c|}{last (1,1)}\\
\hline
Focusing & 11.175  &  75.20  \\
Coupling & 12.210  &  98.25  \\
Focusing & 13.408  &  75.20  \\
\hline
\multicolumn{3}{|c|}{match}\\
\hline
Focusing & 13.925  & -75.20  \\
Coupling & 14.960  & -98.25  \\
Focusing & 16.158  & -80.07  \\
Focusing & 16.675  &  80.07  \\
Coupling & 17.710  &  92.42  \\
Focusing & 18.908  &  84.17  \\
\hline
\multicolumn{3}{|c|}{first (1,2)}\\
\hline
Focusing & 19.425  & -84.17  \\
Coupling & 20.460  & -92.42  \\
Focusing & 21.658  & -84.17  \\
\hline
\end{tabular}
\label{matchCoil1-2}
\end{center}
\end{table}

        The match where the cell length changes from 2.75~m down to 1.65~m
requires further attention. Although the proposed solution is not a perfect
match, its mechanical simplicity and relatively short length may actually
outweigh the benefit we might get with a slow, adiabatic match from one cell length
to the other. Note that the absorber in the matching  cell is removed,
allowing us to run the upstream and downstream rf cavity phases closer to the bunching
condition and giving us a slight local increase of the rf bucket size, 
as  well as ease of mechanical assembly. Coils and currents are listed in
Table~\ref{matchCoilP}.  The magnetic field on axis for the entire cooling
channel is  shown in Fig.~\ref{BzAllCool}. 

\begin{table}[!htb]
\centering
\begin{center}
\caption[Geometry and $j~(A/mm^2)$ in the matching (1,3) and (2,1)]{Geometry and current densities for the solenoids in the matching
section between the (1,3) and (2,1) lattices.  Coil locations are given with
respect to the start of the channel.}
\label{matchCoilP}
\begin{tabular}{|lccccc|}
\hline
Type   & Location & Length & Radius & Thickness  & j  \\
 & (m)& (m)& (m)& (m)& (A/mm$^2$)\\
% \hline
 \hline
\multicolumn{6}{|c|}{last (1,3)}\\
Focusing &  41.425 &  0.167 &  0.330 &  0.175 &  91.46 \\
Coupling &  42.460 &  0.330 &  0.770 &  0.080 &  84.75 \\
Focusing &  43.658 &  0.167 &  0.330 &  0.175 &  91.46 \\
\hline
\multicolumn{6}{|c|}{match}\\
Focusing &  44.175 &  0.167 &  0.330 &  0.175 & -91.46 \\
Coupling &  45.210 &  0.330 &  0.770 &  0.080 & -84.75 \\
Focusing &  46.393 &  0.198 &  0.330 &  0.175 & -95.24 \\
Focusing &  46.816 &  0.145 &  0.198 &  0.330 &  56.39 \\
Coupling &  47.377 &  0.396 &  0.792 &  0.099 &  95.65 \\
Focusing &  48.189 &  0.145 &  0.198 &  0.330 &  68.87 \\
\hline
\multicolumn{6}{|c|}{first (2,1)}\\
Focusing & 48.466  &  0.145 &  0.198 &  0.330 & -68.87 \\
Coupling & 49.027  &  0.396 &  0.792 &  0.099 & -95.65 \\
Focusing & 49.839  &  0.145 &  0.198 &  0.330 & -68.87 \\
 \hline
%\hline
\end{tabular}
\end{center}
\end{table}
\afterpage{\clearpage}
\begin{figure}[htb!]
\begin{center}
%\leavevmode
\includegraphics*[width=140mm]{../template/report/ps-and-eps/BzAllCool.eps}
%\end{center}
%\begin{center}
%\leavevmode
\includegraphics*[width=140mm]{../template/report/ps-and-eps/BzCoolMatch.eps}
\caption[The longitudinal field on axis for the entire SFOFO channel]{$B_z$ on axis for the entire SFOFO cooling channel (top) and 
for the matching section between the (1,3) and (2,1) lattices (bottom).}
\label{BzAllCool}
\end{center}
\end{figure}
\afterpage{\clearpage}

\subsubsection{Cooling rf}
        The lengths of the rf cavities are constrained by the lattice design, as the
focusing coils have a bore smaller than the rf cavities, and by the rf cell length, which must be optimized to give the high $R_S$ required to
reach  high gradient (see Chapter~\ref{RF:chapter}).  In the simulations, cavities are always placed in the middle of the
lattice cell.  Each rf cell can be phased individually. In order to
improve the shunt impedance of the cavity, the iris of the cell is closed with a foil. 
%D.M.K March 14 2001
% Improvement in the englease used here. 
% Dan also objects to the expression "increase with radius". 
% but has currently no better suggestion. 
        Our baseline design calls for a thin, pre-stressed beryllium window with
thickness that increases with radius. This arrangement is described in Chapter~\ref{RF:chapter}. Radius-dependent foil thickness is used because power dissipated in the foil goes like the
fourth power of the  radius (for small radius). Therefore, we benefit from more thickness
at higher radius to remove the heat. In addition, particles at large radius tend to
have large transverse amplitude and are ``warmer" than the central core. Thus,
a bit more multiple scattering can be tolerated at large radius. Windows at the end of a cavity dissipate half as much power as windows at
the boundary between two adjacent rf cells. These end windows can
be made thinner than those in the center of the cavity.   The cavity parameters used in the simulations are listed in Table~\ref{Coolrf}. The rf window
parameters are in Table~\ref{RF:tb18.D2}. 
  
  Closing the cavity iris with thin aluminum tubes arranged in a Cartesian grid can also be
considered, as briefly discussed in Chapter~\ref{APP-NCRF:grid}.

\begin{table}[!htb]
\centering
\begin{center}
\caption[Geometry and rf parameters for the linac]{Geometry and rf parameters for the cavities in the cooling channel used in the simulation study. }
\begin{tabular}{|ccccc|}
\hline
Lattice type   & No. of rf cells & Cell length& Peak field & Phase   \\ 
& & (m) & (MV/m) & (deg.)\\
%\hline
\hline
(1,i), i=1,3 & 4 &  0.466 &  15.48 &  40  \\
\hline
match & 4  &  0.466 &  15.48 &  18.8  \\
(1,3)-(2,1) & 2  &  0.559 &  16.72 &  18.8  \\
\hline
(2,i), i=1,3 & 2 & 0.559 &  16.72 &  40  \\
% \hline
\hline
\end{tabular}
\label{Coolrf}
\end{center}
\end{table}

\subsubsection{Absorbers}
%D.M.K March 14 2001
% LH2 : Dan prefers LH_2, but we are unsured what's best. Leave it as is
% for now. 
The absorber material is liquid hydrogen (LH$_2$). The LH$_2$
vessels are equipped with thin aluminum windows; their thicknesses are 360 (220)~$\mu$m , with radii of 18 (11)~cm, for the (1,i) and (2,i) lattices, respectively.   

The density of LH$_2$ is
approximately 0.071~g/cm$^3$. The energy loss, as given by the Bethe-Bloch
formula~\cite{PDG}  with a mean excitation energy of 21.9~eV, is 4.6~MeV$\cdot$cm$^2$/g. The absorber length is 35~cm for the (1,i), i=1,3 lattices
and 21~cm for the  (2,i) lattices, respectively. The muons lose 
$\approx 12$~MeV per lattice cell for the (1,i) lattices and $\approx 7$~MeV
for the (2,i) lattices, including the energy loss in the absorber windows. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\subsection{Performance}
% D.L.K, P.L. March 19 2001 : language, grammar improvement.
\begin{figure}[hbt!]
\begin{center}
%\input{jcg-63a2b.fig}
\includegraphics*[width=3.5in,clip]{../template/report/ps-and-eps/jcg-63a2b.ps}
%\input{ffDummy.fig}
\caption[Beta function in the buncher and
cooling section]{Beta function in the buncher and
cooling section; rms and maximum beam radius; rms divergence.
These results were obtained with ICOOL. }
\label{fig:rtb}
\end{center}
\end{figure}
\afterpage{\clearpage}
Fig.~\ref{fig:rtb} shows the beta functions, which step down with each
new section of the cooling lattice; also shown are the beam radius and beam divergence. The beam divergence at the absorbers is
kept approximately constant in order to minimize the effects of multiple
scattering. The $\beta_{\perp, min}$ function, derived from the beam second-order moments at the absorber centers,  is shown in
Fig.~\ref{BetaPerpAll}.


\begin{figure}[!hbt]
\begin{center}
\includegraphics*[width=125mm]{../template/report/ps-and-eps/betaPerp207.eps}
\caption[The averaged $\beta_{\perp}$ function \textit{vs.} length]{The $\beta_{\perp}$ function for the entire SFOFO cooling channel, averaged over the relevant momentum bite
and measured from the second-order moments of the beam itself, as the cooling
progresses.  
The five arrows indicate the beginning of the (1,2), (1,3), and (2,i), i=1,3
 lattice sections.  (Geant4 result.)}
\label{BetaPerpAll}
\end{center}
\end{figure}
   
The transverse and longitudinal emittances through the cooling system are shown
in Fig.~\ref{fig:emtl} and Fig.~\ref{EmittCool}. They were obtained using the ICOOL simulation code
and  the code ECALC9~\cite{ecalc9}, respectively. Emittances are computed in
ECALC9 using diagonalized covariance matrices. 
%P.L, March 14-15 2001 
%The emittance
These normalized emittance  
values are
corrected for correlations among the variables, including the strong
momentum-transverse-amplitude correlation.  

\begin{figure}[!htb]
\begin{center}
\includegraphics*[width=2.5in,angle=90]{../template/report/ps-and-eps/emit-trans-vs-s.ps}
\includegraphics*[width=2.5in,angle=90]{../template/report/ps-and-eps/emit-long-vs-s.ps}
%\input{jcg-emits.fig}
%\input{ffDummy.fig}
\caption[Transverse and longitudinal emittances in the cooling channel]{Transverse (top) and longitudinal (bottom) emittances in the cooling section, obtained with the ICOOL code. The initial and final  values are indicated. }
\label{fig:emtl}
\end{center}
\end{figure}

\begin{figure}[!hbt]
\begin{center}
%\leavevmode
\includegraphics*[width=100mm]{../template/report/ps-and-eps/Emitt2D-211.eps}
\caption[The longitudinal and transverse emittances]{The longitudinal and transverse emittances, obtained with the 
Geant4 simulation code. Notice that the length of the last lattice (2,3), has been extended by $\approx 20$~m to investigate the ultimate performance of the cooling channel.}
\label{EmittCool}
\end{center}
\end{figure}
 
The transverse emittance cools from 12 to $\approx 2$~mm$\cdot$rad. The longitudinal
emittance shows an initial rise, and then, as particles outside the  rf bucket are
lost, an approach to an asymptotic value set by the bucket size. This longitudinal emittance should naturally rise due to straggling and the
negative slope of the energy loss curve with energy. However, since the rf
bucket is already full, instead of an emittance growth we have a steady loss of
particles (\textit{i.e}, ``longitudinal scraping"), as seen in the top curve of Fig.~\ref{fig:mup}.

%P.L, March 14-15 2001 
% This figure, I bet is not in eps format. We need to work on all 
% .fig to make truly encapsulated, or correct their Bounding Boxes.
\begin{figure}[!hbt]
\begin{center}
\includegraphics*[width=4.5in]{../template/report/ps-and-eps/jcg-63ec.ps}
%\input{ffDummy.fig}
\caption[Particle transmission: in the buncher and cooling sections]{Particle transmission: number of muons per incident proton on target in the buncher and cooling sections. Top curve
is overall transmission; lower two curves are for 150~mm longitudinal acceptance with two different transverse acceptance cuts: (middle) 15~mm$\cdot$rad transverse acceptance; (bottom) 9.35~mm$\cdot$rad transverse
acceptance. This result was obtained with ICOOL.}
\label{fig:mup}
\end{center}
\end{figure}

Despite this overall loss, the number of particles within the accelerator
acceptance increases. The lower two curves in Fig.~\ref{fig:mup} give the
number of particles within the baseline longitudinal and transverse
acceptance cuts. The middle curve gives the values used in this Study (FS2). The lowest curve, shown for comparison, gives the values for the acceptances used in Feasibility Study-I (FS1)~\cite{FS1}. 
 These acceptance cuts are based on the 6D normalized beam emittances derived from  the moments of the simulated beam distribution and the estimated transverse and longitudinal beta functions:  

%P.L, March 14-15 2001 
% Should not we make the relativistice beta gamma factor explicit here in these
% formulas? 
% it did confused me
\begin{itemize}
%\item Longitudinal (FS1 \& FS2): $(dz^2)/\beta_s~+~(dp/p)^2~\beta_s~<~150$~~(mm)
%\item Transverse (FS2): $(x^2+y^2)/\beta_\perp~+~(x'^2+y'^2)\beta_\perp~<~15~$~~(mm rad)
%\item Transverse (FS1): $(x^2+y^2)/\beta_\perp~+~(x'^2+y'^2)\beta_\perp~<~9.75~$~~(mm rad)
\item Longitudinal (FS1 \& FS2): 
$(dz^2)/\beta_s~+~(dp/p)^2~\beta_s~ (\beta\gamma) ~ <~150$~~mm
\item Transverse (FS2): 
$(x^2+y^2)/\beta_{\perp}~+~(x'^2+y'^2)\beta_{\perp}~(\beta\gamma)~<~15~$~mm$\cdot$rad
\item Transverse (FS1): 
$(x^2+y^2)/\beta_{\perp}~+~(x'^2+y'^2)\beta_{\perp}~(\beta\gamma)~<~9.35~$~mm$\cdot$rad
\end{itemize}
where $\beta_s$ is the synchrotron beta function ($\beta_s=\sigma_{z}/\sigma_{dp/p}$),
and $\beta_\perp$ is the transverse $\beta$ function. Transverse and longitudinal
emittances obtained with Geant4 are shown in Fig.~\ref{EmittCool}.
At equilibrium, a transverse emittance of 2.2 mm$\cdot$rad is reached,
consistent with the ICOOL result.  

It is seen that the gain in muons  due to cooling within the accelerator
acceptance is a factor of $\approx 3$ (or $\approx 4$ if the Study-I acceptances were
used). Similar performance is obtained with the Geant4 code, as shown in 
Fig.~\ref{YieldCool}. If the particle loss from longitudinal emittance growth
could be eliminated, as might be the case if emittance exchange were used,
then these gains might double.

\begin{figure}
\begin{center}
%\leavevmode
%P.L, March 14-15 2001 
%\includegraphics*[width=150mm]{MuOPInAcc-211.eps}
\includegraphics*[width=100mm]{../template/report/ps-and-eps/MuOPInAcc-211.eps}
\caption[$\mu/p$ yield ratio for the two transverse emittance 
cuts]{The muon-to-proton yield ratio for the two transverse emittance 
cuts, clearly showing that the channel cools, \textit{i.e.}, the density in the center of
the phase space region increases. Since the relevant yield $\mu/p_{15}$ no longer
 increases for $z \leq 110$~m, the channel length was set to 108~m.
 This is a Geant4 result. }
\label{YieldCool}
\end{center}
\end{figure}

%P.L, March 14-15 2001 
% why forcing this table at the bottom of a page, below in fact a foortnote? 
%\begin{table}[tbh]
\begin{table}
\centering
\caption{Beam characteristics summary.}
\label{BandC:tbbeamsumm}
\begin{tabular}{|lccccc|}
\hline
Location     & $\sigma_x$ & $\sigma_{x'}$ & $\sigma_p$ &  $\sigma_t$ & 
$\langle p \rangle $  \\
 (end of )    & (cm) & (mrad) & (MeV/c) & (ns) & (MeV/c) \\
\hline
Induction linac & 8.6 &  95  & 118 &  & 237 \\
Matching section & 5.8 &  114  & 115 &  & 247 \\
Buncher & 5.1 &  104  & 101 & 0.84 & 238 \\
2.75 m cooler & 3.0 &  89  & 64 & 0.55 & 219 \\
1.65 m cooler & 1.6 &  94  & 28 & 0.51 & 207 \\
\hline
\end{tabular}
\end{table} 

The beam characteristics in the buncher and cooling sections are summarized in
Table~\ref{BandC:tbbeamsumm}. This table lists the properties of all the muons in the beam that survive to a given location.
The beam is cylindrically symmetric in this lattice, so the $x$ and $y$ properties
are similar. 
%P.L, March 14-15 2001 
%We see that the size steadily decreases as we proceed down the channel.
We see that the beam size steadily decreases as we proceed down the channel.
The angular divergence is kept approximately constant, maximizing cooling
efficiency. The momentum spread of the entire beam is still large after the
induction linac, but this includes very low and very high energy muons that do not get transmitted through the subsequent SFOFO lattice. For example, the range of momenta accepted in the acceleration linac is 150--300~MeV/$c.$ The rms momentum spread for muons that lie inside this momentum range varies from 
16~MeV/$c$ after the third induction linac to 21~MeV/$c$ after the 1.65~m cooling lattice.

 The decrease in energy
spread shown in Table~\ref{BandC:tbbeamsumm} is due to particle losses, since there is no longitudinal cooling or
emittance exchange. Likewise, the average momentum of the beam
decreases until it matches the acceptance of the SFOFO lattice. The time spread
refers to a single bunch in the bunch train.

The longitudinal emittance remains more or less stable, at around 30~mm.  This
is somewhat deceptive. The anti-damping slope of the energy
loss curve, straggling, and imperfections in the longitudinal-to-transverse
correlation\footnote{See Fig.~\ref{fig:corr} in the previous section.} cause particles to fall out of the
rf bucket and be scraped away due to the strong betatron resonances.
In fact, the buncher delivers a full rf bucket to the cooling section and
the longitudinal emittance cannot  grow any larger. This scraping occurs on the
combined time scales of the synchrotron period, about 20~m, and the  growth
time of the betatron resonance instability.
  

%*** Performance of the cooling channel in absence of multiple scattering. 
%Rick F. proposed a table, Paul L. would simply give the figure of merit in 
%terms of $\mu/P_{15}$ with multiple scattering off. I would write: 

The performance of the cooling channel is influenced by both multiple  scattering
and the limited momentum acceptance. 
% JCG: Need further study, it seems it is not that simple! 
%While the latter is difficult to estimate
%using computer simulations, it is straightforward to estimate the former 
%contribution by simply turning off multiple scattering in the code.
%P.L, March 14-15 2001 , may be better english:
% If so,  the relative 
%$\mu/P_{15}$ and $\mu/P_{9.75}$ yields would increase by 11 \% and 19 \%,
%respectively. 
 Without multiple scattering,  the   $\mu/p_{15}$ and $\mu/p_{9.35}$ yields would increase by approximately 20\% and 40\%, respectively. 
%%%%JCJCJCJCJCJC
\subsection{Tolerances \& Systematics}
\label{tolerances}

        The performance of the cooling channel has been evaluated based on computer 
simulations using two distinct codes. However, some parameters or assumptions in
the calculations are common in the two simulations. Since no such channel has been built yet, it is
fair to question whether the estimation of the cooling performance is robust
against small changes in these parameter values.  In addition, we need to consider the
tolerances on the mechanical alignment in such a long beam transport
system. 
%P.L. March 20 2001, let us skip over this question, since we do not provide 
% a clear answer at this point int time.
%  Do we need corrector coils, and if so, where do they need to be
% placed? 
        
\subsubsection{Sensitivity to multiple scattering model}        
        
        ICOOL treats multiple scattering by using a straightforward Moliere model, imported from the Geant3
package.  Geant4 uses an improved version of the Moliere model, but has a
tunable parameter. We have studied the sensitivity of the rms value of the
scattering angle to this parameter, in relation to the known uncertainties in
the measured values for these rms scattering angles for low-$Z$ materials. 
The sensitivity of the $\mu/{\rm P}_{15}$ yield in the relevant range of this
tunable parameter has been measured. The systematic error due to this uncertainty
is approximately 10\%.   

\subsubsection[Control of the energy loss in LH2 ]{Control of the energy loss in LH$_2$ and energy gain in the linac}
        
Because of the relatively narrow momentum acceptance  of the channel compared with 
the beam momentum spread,  the
energy loss and  the energy gain must be known in the channel to
better than $\approx 0.25 \%.$  This
tolerance can be achieved in the rf cavities, where the peak
voltage and accelerating voltage can be controlled to better than a few tenths of
a percent. 

%Lowering the LH2 density uniformly by up to 5\% does not effect the performance of the channel appreciably. 
% 
% The fluctuation in the LH2 density must be controlled. In order to get a
% feel for this effect, the LH2 density was randomly set in each absorber, 
% based on a Gaussian model, with a relative $\sigma$ of 10\% and 20\%.
% The relative $\mu/P_{15}$ yield decreased by 5\% and 10\%, respectively. 
% However, we think that we will be able to keep the density uniform to better
% than a few percent, based on the experience of the Bates group~\cite{Bates}.         
Nonuniform heat deposition within the absorbers may cause density
variations in the vicinity of the core of the beam. These could result in
reduced beam cooling as well as a net acceleration of the beam through the
cooling channel, since the reduced energy loss would then be
overcompensated by the rf accelerating gradient.  While the absorber R\&D
program has not yet reached the point where such variations may be
predicted in detail, we believe that they will be small in view
of the success of the SAMPLE collaboration at Bates Laboratory in maintaining
constant target density within tenths of a percent with 500~W of beam
heating~\cite{Bates}.

We have also explored, by simulation, the effects on muon cooling
performance of reduced absorber density.  As a first approximation the
absorber density has been reduced uniformly throughout all absorbers by 1,
3, 5, 10, and 20\%.  For density decreases up to 5\%, the cooling
performance is unaffected within the few-percent level of simulation
statistics.

 The cooling channel will require about 72 12-MW klystrons. It is likely that
 one will fail occasionally. If so, emptying an LH$_2$
 vessel and rephasing the downstream rf cavities will keep the beam on the nominal
 momentum. As an example, we have simulated the loss of rf power in a (1,1) or a (1,3) cooling cell. We find that emptying the absorber vessel and rephasing the remaining rf cavities results in a performance degradation of about 5\% (relative), allowing us to keep the cooling channel running productively.      
        
\subsubsection{ Magnet alignment}

The design of the cooling channel was optimized using ideal magnetic
fields from cylindrical current sheets. 
In an actual magnetic channel, imperfections that occur in the fabrication and assembly of the solenoids result in magnetic fields that deviate from the ideal used in the simulations by some small error field $\delta \vec{B}(x,y,z).$
A state-of-the-art magnet construction results in field errors $\frac{\delta B}{B}\le 0.1\%.$ 
These field errors produce effects, in general detrimental, that tend to increase with the length of the channel. If left uncorrected, these errors lead to mismatching and betatron oscillations, which in turn result in degradation of the cooling performance of the channel and to a decrease of the channel transmission. 

We have considered the following analytical treatment of the detrimental effects of magnet alignment errors. As the muon beam propagates along the periodic channel with a
 prescribed beta function, it encounters a series of errors of various origins,
 which are assumed to be described by a stochastic function $\delta
 \vec{B(s)}$ (we neglect the transverse coordinate dependence). The
 muons experience a series of random forces or ``kicks," which result in
 a random walk of the centroid of the beam. Statistically, the rms
 magnitude of the transverse deviation $\sqrt{<(\delta x(s))^2>}$ is a function
 of the length of the channel, $s.$ In principle, it should be possible to develop a correction algorithm such that
strategically located correction coils bring the centroid back to the
ideal trajectory, thereby minimizing the deterioration of the cooling process.

A first look at the effects of errors and the sensitivity of the present design of
the cooling channel to them has been carried out in references~\cite{ref2} and
~\cite{ref3}. Studies of the error fields due to misalignment of individual
coils and current sheets are found in references~\cite{ref4a},\cite{ref4b}. 

There are several sources of magnet imperfections that may contribute to the overall deviation from the ideal fields of the channel:
\begin{enumerate}
\item Geometric (\textsl{macroscopic}) survey errors:
%\begin{enumerate}
\begin{description}
\item [a)] transverse misalignment of solenoids, characterized by a vector $\vec{d}=\vec{d}\left(\cos{\theta},\sin{\theta},0\right)$ of magnitude $d$ and direction $\theta.$ In the simulations the values of $d$ are chosen from a Gaussian stochastic function.
\item [b)] transverse tilt of the solenoid, characterized by two angles: $\theta$ direction with respect to the $x$-axis, and the tilt, by the magnitude $\psi$, with respect to the $z$-axis. 
%\end{enumerate}
\end{description}

The Cartesian coordinates $\vec{r}=(x,y,s)$ transform  as
\begin{equation}
\vec{r'}=\vec{r}-\vec{d}
\end{equation}
 for a translation in the transverse plane and
\begin{equation}
x_i'=\mathcal{M}_{ij}x_j
\end{equation}
for a tilted magnet. The magnetic fields are calculated as
\begin{equation}
\vec{B}(x,y,s)=\vec{B'}(x',y,s') \qquad B_i(\vec{r})=\mathcal{M}_{ij}^{-1}B'_j(\mathcal{M}\vec{r})
\end{equation}  
for a translation and tilt respectively.
The transformation matrix is
\begin{equation}
\mathcal{M}= \left[\begin{array}{ccc}
 \cos{\psi}+\cos^2{\theta}(1-\cos{\psi})&\sin{\theta}\cos{\theta}(1-\cos{\psi})&\sin{\theta}\sin{\psi}\\
\sin{\theta}\cos{\theta}(1-\cos{\psi})&\cos{\psi}+\sin^2{\theta}(1-\cos{\psi})&\cos{\theta}\sin{\psi}\\
\sin{\theta}\sin{\psi}&-\cos{\theta}\sin{\psi}&\cos{\psi}
\end{array}
\right]
\end{equation}

\item Power supply fluctuations resulting in current fluctuations

\item Geometric conductor positioning, which leads to random \textsl {microscopic} field errors.
\end{enumerate}
Here we only consider the first type, \textit{i.e.}, geometric \textsl{macroscopic} field errors introduced by mispositioning of entire magnet cryostats.   

We have performed two studies with ICOOL~\cite{ICOOL}. The buncher and SFOFO cooling channel have been simulated with independent Gaussian random tilt angles and translation of the coils roughly every 5~m. The results are shown in Figs.~\ref{tilt_fig} and ~\ref{trans_fig}.

\begin{figure}
\begin{center}
\includegraphics[width=2.5in,angle=90]{../template/report/ps-and-eps/tilt-error.eps}
\caption{Transmission of the front end for different \textsl{rms} tilt angles.}
\label{tilt_fig}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=2.5in,angle=90]{../template/report/ps-and-eps/trans-error.eps}
\caption[Transmission of the front end for rms translation ]{Transmission of the front end for different rms translation errors $d$.}
\label{trans_fig}
\end{center}
\end{figure}

An independent analysis of alignment tolerance issues (microscopic field errors) has been done 
using the Geant4 package. The simulations of the buncher and
cooling channels are run in the following modes:
\begin{enumerate}
\item Random polar tilts. A Gaussian model was used to generate the tilts,
polar angles $\psi$, for each coil. Since large transverse displacements of coils are expected to be relatively easy to find and correct, we have truncated the distribution at $2 \sigma_{\psi}$. The
azimuthal $\theta$  angles were chosen randomly, between zero and $2\pi$.  The $\mu/p_{15}$ yield was measured for tens of  such simulated
channel assemblies.  The histogram in Fig.~\ref{CoolTiltHisto} shows 
that a $\sigma_{\psi}$ of 0.5 mrad gives no statistically significant
degradation of the channel performance. However $\sigma_{\psi} \approx 2.5
$ mrad would be unacceptable.
\item Random transverse displacements. Since the coils are about 15~cm long, 
a tilt of 0.5~mrad gives a lateral displacement at one end of about 75~$\mu$m.
Evidently, the coil could also shift laterally by about the same amount. 
We verified that a 2$\sigma$ truncated displacement of 100~$\mu$m has no
significant impact on the channel performance.     
\end{enumerate}
        Since the typical tolerance on accelerator magnet alignment is about
100--$\mu$m, we believe that such a channel can be assembled to the
required accuracy. 
%P.L, March 14-15 2001
% We will comment this sentence for now. This is awaiting the 
% recent "systematic misalignment" runs, where we fix $\theta = 0.$ 
% 
% Corrector coils seems to be unnecessary at this point.
 
\begin{figure*}
\begin{center}
\leavevmode
%\includegraphics*[width=150mm]{resTol212Y15p0End.eps}
\includegraphics*[width=100mm]{../template/report/ps-and-eps/TolSumAllV2.ps}
%P.L, March 14-15 2001
% Remove the 
%\caption{A histogram of the performance of 35 SFOFO cooling channels build with
%a tolerance of 0.5 mrad tilts. 
%Note that muon decays have been turned off, which explains why the nominal
%performance is 20.7 \%. I'll correct this by rescaling the performance in each
%run. (PAUL, DOES THIS STILL APPLY?)}
\caption[A histogram of the performance of 35 SFOFO channels ]{A histogram of the performance of 35 SFOFO cooling channels built with
 tolerances of 0.5 and 2.5~mrad tilts and small translations. 
\label{CoolTiltHisto}}
\end{center}
\end{figure*}
                
\subsubsection{Space charge}    
%D.N. March 20, confirm by P. L.
% Juan, you meant time
%The nominal number of muons per bunch is $N_{\mu}\approx 4/80\times 10^{12}$,
The nominal number of muons per bunch is $N_{\mu}\approx 5\times 10^{10}$,
which corresponds to $Q\approx 8$~nC. An estimate of the deleterious effect of space charge on
the beam dynamics can be found by calculating the self-electric field of a
Gaussian distribution of charge represented by the Basetti-Erskine-Kheifets
formula~\cite{BEK}
\begin{equation}
\Phi (r,z,s)=\frac{2Q}{\epsilon_o \sqrt{\pi}}\int_0^{\infty}\,dt 
{\frac{e^{-{\frac{r^2}{2{\sigma_{\bot}}^2+t}} }} {(2{\sigma_{\bot}}^2+t) }}
{\frac{e^{-{\frac{z^2}{2{\sigma_{3  }}^2+t}} }} {\sqrt{ (2{\sigma_{3}}^2+t)}}}.
\label{eq11}
%P.L, March 14-15 2001 
% relabeling this equation, clashes with the one used in the cooling intro. 
\end{equation}
The variable $z$ is defined as $z=s-c\beta t$, with $s$ the longitudinal coordinate, assuming that the centroid of the bunch is at $s=0$ at time $t=0.$ The argument $s$ in $\Phi (r,z,s)$ is there to indicate that the rms transverse size $\sigma_{\bot}$ and longitudinal size $\sigma_z$ of the beam are functions of $s.$
 This is important because the beta function varies from moderate to small values at the absorbers.

From Eq.~\ref{eq11} and the corresponding expression for the vector potential $A_s(r,z,s)=\beta\Phi (r,z,s)$ we can calculate the electric field components $E_r(r,s,t)$ and $E_s(r,s,t)$~\cite{jcg1}.
ICOOL contains this formulation and systematic studies have been carried out. The results are shown in Fig.~\ref{sp1}, where it can be seen that the number of muons per proton $\mu/p$ at the end of the cooling channel is rather insensitive to the number of muons in the bunch up to values $N_{\mu}^{critical}\approx 1\times 10^{12}$, some 20 times our intensity.

\begin{figure}[!tbh]
\begin{center}
\includegraphics*[width=10cm,angle=90]{../template/report/ps-and-eps/space_charge.ps}
\caption{$\mu/p$ \textit{vs.} $N_{\mu}$ in a bunch, assuming a Gaussian self-field. }
\label{sp1}
\end{center}
\end{figure} 
%\afterpage{\clearpage}

This approach is approximate and leaves aside potentially important phenomena:
first, the effects of induced charge in the walls of the beam pipe and in the
metallic (Be) rf windows; second, the short-range wake potential created
by the $\beta <1$ muon beam inside the cavities.
The effect of the walls of a cylindrical beam pipe on a bunch of charged
particles was also considered and has been computed with ICOOL with no noticeable effects. 
%Results are shown in Fig.~\ref{sp2}. We
%can see that for the nominal charge, no noticeable effects are detected.
%\begin{figure}[!tbh]
%\begin{center}
%\includegraphics*[bb=24 449 376 707,clip]{wall_sp.ps}
%\caption{$/mu$/p \textit{vs.} $N_{/mu}$ in a bunch including the wall effects}
%\label{sp2}
%\end{center}
%\end{figure} 
%\afterpage{\clearpage}
We note here that the presence of Be windows should mitigate any space charge
effects. However, it is rather difficult to calculate this with
precision.
% Recently, in Ref.~\cite{haipeng} it has been shown that
%MAFIA~\cite{mafia} reproduces the theoretical short range wake
%potentials extremely well for the case of a single particle with
%$\beta <1$ in a pill-box cavity. This gives us confidence that MAFIA
%solutions of the wave equation in the presence of boundary conditions
%and subluminal particles are correct. Using these fields in ICOOL or
%GEANT runs will allow a much better evaluation of the space charge
%problem. In any case, wake fields tend to decrease the effective
%transverse field seen by the bunch and consequently to reduce the
%space charge effects.

%Next we give some simple arguments to validate the above assertion. The self field $E_r$ causes %the beam to expand radially and the corresponding magnetic field focuses the beam. The radial %component of the Lorentz force is
%\begin{equation}
%r'' \approx {qE_r\over m c^2 \gamma\beta^2}(1-\beta^2)
%\end{equation}

%A long cylinder of radius $a$ and uniform charge density $\rho_o$ creates a field 
%\begin{equation}
%E_r^o \approx {\rho_o \over 2 \epsilon_o}r  \qquad (r<a)
%\end{equation}
 
%Define $\Delta ={<E_r>\over E_r^o}$ with $<E_r>=\int E_r(a,s) ds$ the average of the transverse %field at r=a. We can write
%\begin{equation}
%r'' \approx {qE_r^o\over m c^2 \gamma\beta^2}(\Delta-\beta^2)
%\end{equation}

%This equation shows that if $\Delta -\beta^2 <<1$ the radial electric field can be reduced by %introducing charges in the system, such that their $E_r$ opposes $E_r^o$ and furthermore %produces no magnetic field. Indeed, that is the case because of the induced charges in the %surface of the metallic windows, which are created to insure that there is no electric field in %the bulk of the Be window (perfect conductor approximation)~\cite{jcg2}  
\subsection{Liquid Hydrogen Absorbers}
\label{BandC:absorber}
\subsubsection{Power handling}

We estimate the maximum power dissipation per absorber to be about 300\,W,
dominated by the ionization energy loss of the muons (See Table~\ref{CPR:tb3}, which shows the absorber lengths, radii and the number of absorbers of each type).  The main technical
challenge in the absorber design is to prevent boiling of the hydrogen near the
beam axis, where the power density is greatest.  This requires that the
hydrogen flow have a significant component transverse to the beam.  We are
investigating two ways to achieve this: ``flow-through'', a design in which the absorber connects
to an external heat-exchange and temperature-control loop, and ``convection'', a design in
which the absorber vessel is itself the heat exchanger, and heat transfer
within the absorber is accomplished dominantly by convection.

The flow-through design resembles previous high-power liquid-hydrogen
targets~\cite{Mark,Bates}, which have been operated successfully at power dissipations as high as 500\,W~\cite{Bates} and have been proposed for operation at even higher dissipations~\cite{SLAC158,JLAB}.  In this approach the hydrogen is pumped
around a loop that includes the absorber vessel, as well as a heat exchanger and
a heater. 
%(see Fig.~\ref{fig:flow-through}). 
In the heat exchanger, which runs at a constant power
level, the hydrogen is cooled by counterflowing cold helium gas. The heater is
used in feedback to regulate the hydrogen temperature and compensate for
changes in beam intensity.  

Given the small emittance of conventional particle beams, liquid-hydrogen
targets tend to be narrow transverse to the beam, leading to designs in which
the natural direction of hydrogen flow is parallel to the beam. 
%(see {\em e.g.}\ Fig.~\ref{fig:flow-through}). 
To avoid boiling the liquid in the high-intensity
beam core, various design strategies are then necessary to ensure transverse
flow of the liquid~\cite{Bates,E158-milestone}. In contrast, in  our flow-through design the hydrogen
enters the absorber vessel from below and exits at the top, ensuring automatically that the
flow is transverse to the beam. The flow pattern is controlled by means of
nozzles, which must be configured so as to avoid dead regions or eddies and
to ensure adequate flushing of the windows. 

In the convection design (Fig.~\ref{fig:convection}), the interior wall of the
vessel is equipped with cooling tubes through which cold helium gas circulates.
A heater located at the bottom of the vessel is used to compensate for changes
in beam intensity. The design of the convection-cooled absorber is being guided
by two-dimensional fluid-flow calculations. The flow-through
approach is less amenable to calculation, but will be tested on the bench to
verify the efficacy of the nozzle design, first in a room-temperature model and
later at cryogenic temperature.  Prototype construction and testing programs
for both designs are now under way and will lead to high-power beam tests.
%P.L, March 14-15 2001
% Dan Kaplan suggest to resize this figure. 
% 
%
\begin{figure}[!htb]
\begin{center}
\includegraphics*[width=3.25in]{../template/report/ps-and-eps/convection.eps} 
\caption{Schematic of convection design.}
\label{fig:convection}
\end{center}  
\end{figure}

%\begin{figure}[tb]
%\centerline{\epsfxsize=3.25in\epsffile{figure1b.eps}}
%\centerline{\epsfxsize=3.25in\epsffile{convection.eps}}
%\caption{Schematic of convection design.\label{fig:convection}}
%\end{figure}

\subsubsection{Window design}

To minimize heating of the beam due to multiple scattering, the absorbers must
be equipped with thin, low-$Z$ windows.  Yet, the windows must be strong enough
to withstand the pressure of the liquid hydrogen.  We have devised a window
design that satisfies these requirements and also allows quite thin absorbers
to be built.  While a hemispherical window shape minimizes the window thickness
for a given strength, the desire to build absorbers that are thinner relative
to their diameter than a sphere leads to the ``torispherical" shape.  In the
version  specified by the American Society of Mechanical Engineers
(ASME)~\cite{ASME}, the torispherical head for pressure vessels is composed of
a central portion having a radius of curvature (the ``crown radius") equal to
the diameter of the cylindrical portion of the vessel, joined to the
cylindrical portion by a section of a toroidal surface with a radius of
curvature 6\% of the crown radius (see Fig.\,\ref{fig:torisphere}).

\begin{figure}[!htb]
\begin{center}
%\centerline{\rotatebox{270}{\epsfysize=3.5in\epsffile{torisphere.eps}}}
%P.L, March 14-15 2001
% Prof. Dan Kaplan, from the Illinois Center for Accelerator Reasearch,
% wants this way, and he might be right! 
\includegraphics*[height=3in,angle=270]{../template/report/ps-and-eps/torisphere.eps} 
%\centerline{\rotatebox{270}{\epsfysize=3.5in\epsffile{torisphere.eps}}} 
\caption[Schematic of ASME torispherical head on cylindrical vessel]{Schematic of ASME torispherical head on cylindrical vessel of diameter
$D$: solid curve shows upper half section, with dashed lines and curves 
indicating the spherical and toric surfaces from which it is 
composed.\label{fig:torisphere}}
\end{center}
\end{figure}

ASME specifies the minimum acceptable thickness of the torispherical head as
\begin{equation}
t=\frac{0.885PD}{SE-0.1P}\,,
\end{equation}
where $P$ is the differential pressure across the window, $D$ the vessel
diameter, $S$ the maximum allowable stress, and $E$ the weld efficiency.
Although previous high-power liquid-hydrogen targets have operated at
2 atm~\cite{Mark,Bates}, to keep the windows as thin as possible we have
designed for 1.2 atm. For $S$, we follow ASME recommendations and use the
smaller of 1/4 of the ultimate strength $S_u$ or 2/3 of the yield strength
$S_y$ (in practice, for aluminum alloys it is the ultimate strength that
matters). We will machine the window with an integral flange out of 
a single disk of material (Fig.\,\ref{fig:window1}), with the flange fastened 
to the assembly by bolts (Fig.\,\ref{fig:absorber}). Thus, there are no welds 
and we take $E=1$. For 1.2-atm operation, and given the ASME specification 
for 6061-T6 aluminum alloy,
$S_u=289\,$MPa, we obtain $t=530\,\mu$m for the ``Lattice 1" absorbers
($D=0.36\,$m) and $t=330\,\mu$m for the ``Lattice 2" absorbers ($D=0.22\,$m).
If necessary, the windows can be made thinner than this by tapering their thickness
as described below. In addition, less easily machinable, but stronger, aluminum
alloys (such as 2090-T81) may allow further reduction in thickness.

\begin{figure}[!tbp]
\begin{center}
%P.L, March 14-15 2001
% Resizing.. 
%\centerline{\epsfxsize=6.5in\epsffile{LAT1ABSORBER_WINDOWst2.eps}}
\includegraphics*[width=6.5in]{../template/report/ps-and-eps/LAT1ABSORBER_WINDOWst2.eps} 
\caption[Window design for the SFOFO lattice 1]{Window design for the SFOFO Lattice 1 absorbers.\label{fig:window1}}
\end{center}
\end{figure}

\begin{figure}[!tbp]
\begin{center}
%P.L, March 14-15 2001
%\centerline{\epsfxsize=8.5in\epsffile{LAT2ABSORBER_WINDOWst2.eps}}
\includegraphics*[width=5.5in]{../template/report/ps-and-eps/ABSORBER2.eps} 
\caption[Absorber assembly for SFOFO lattice 2]{Absorber assembly for SFOFO lattice 2 (flow-through design shown).
\label{fig:absorber}}
\end{center}
\end{figure}

In addition to eliminating the weld, machining the window out of a single disk
allows detailed control of the window shape and thickness profile. We have used
the ANSYS finite-element-analysis program to optimize the window shape and
profile so as to minimize the window's thickness in its central portion, where
most of the muons traverse it. The resulting shape and thickness profile are
shown in Fig.~\ref{fig:window1}.
%P.L, March 14-15 2001.  Adding a sentence to resolve the conflict 
% of 530 vs 360 microns
Therefore we have used in the simulation the smaller thicknesses of 360 $\mu$m and
220 $\mu$m for the (1,i) and (2,i) cooling lattices, respectively.
%\subsubsection{Engineering \textsl{Change Orders}} 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\afterpage{\clearpage}
\subsection{Diagnostics and Instrumentation Issues in the Cooling Channel}
There are a number of unique instrumentation problems involved in
optimizing and monitoring the performance of the cooling 
line~\cite{JimN}.  The beams will be large and intense, and a variety of 
precise measurements will be required that are both novel and
difficult.

There will be significant
backgrounds in all detectors, due either to other particles from the target 
coming down the line with the 
muons, or to x-rays and dark currents generated by the rf cavities.  
% D. M. K., March 20 : This sentence is a bit out of context.
We must consider the angular 
momentum of the beam, perhaps for the first time with any high energy
physics beam.  The beams will be intense enough so that thermal heating of 
the detectors is significant.  The environment will have high
magnetic fields, a large range of temperatures, and high-power rf cavities.  In
addition, under normal circumstances the access will be very limited,
since the rf cavities and liquid hydrogen absorbers will occupy most 
of the available space.
%P.L, March 14-15 2001.  I don't quite understand this sentence 
% See e-mail send to Jim Norrem, March 16, 3:15 P.m.
  Standard loss monitors will not be useful for the low
energy muons because the range of such particles is so short (6 
cm in Cu) and they produce no secondaries. An R\&D program is under way to look at the sensitivity and usefulness 
of different
diagnostic techniques and evaluate them in the environment of rf
backgrounds and high magnetic fields.    

On the other hand, there are a number of reasons why the tune-up and operation of the cooling channel could be fairly straightforward.  The cooling
channel will have been very thoroughly simulated by the time of
initial construction.  In addition, there are a relatively small number
of variables that control the behavior of the beam, such as currents in 
solenoids, rf parameters and liquid-hydrogen-absorber parameters, 
and these can be measured with high precision.   While 
the change in transverse beam emittance, $\epsilon_{\perp}$,
between individual cells may be difficult to measure, $\Delta 
\epsilon_{\perp}/\epsilon_{\perp} < 0.01$, the overall
size and profile of a beam with $\sim 10^{12}$ particles per macro-bunch is a 
comparatively straightforward measurement.
%  The high power muon beam 
%considered here makes measurements much easier than for the muon cooling 
%demonstration experiment, which might operate with intensities of 
%only $10^{6} - 10^{7}$ muons per bunch.

\subsubsection{Measurement precision}
The sensitivity of the system to alignment errors was described in Section~\ref{tolerances}.
Related issues involve sensitivity to various other effects: 
transverse and longitudinal mismatches 
between the cooling line and the bunching section, arc-down and 
temporary loss of an rf cavity, boiling or loss of hydrogen 
in the absorber section, inadvertent introduction of a collimator
or thick diagnostic, and mismatches at the ends of the cooling line.   
These mismatches can be either first-order (beam centroid position, 
energy, or
angle errors), or second-order (discontinuities in Twiss parameters).
Mismatches will slow down the cooling process and could  
significantly affect beam losses.   

A mismatch due to problems with the rf or absorbers would change both 
the mean beam momentum and the measured $\beta$ function downstream.
An example is shown in Fig.~\ref{BandC:fgpbeta}, where the beta functions are plotted
through regions where the momentum has changed, corresponding to an empty
hydrogen cell or a single rf cavity that is turned off, giving the 
scale of the effects that might be produced.  The changes in $\beta$ 
functions are a few percent at some positions, while at other  
points the $\beta $ functions are essentially
unchanged.  Thus, it is necessary to have
beam profile measurements done at a number of positions.
\begin{figure}[!htb]
\begin{center} 
%\includegraphics[width=3in]{norem-pbeta.eps}  
\includegraphics[width=3in]{../template/report/ps-and-eps/norem-graph.eps}  
\caption[The scale of discontinuities in $ \beta $ function]{The scale of discontinuities in $ \beta $ function when rf or absorbers are
perturbed, corresponding to empty liquid-hydrogen absorber (dashed line)
 or shorted rf cavity (dotted line).}
\label{BandC:fgpbeta} 
\end{center}
\end{figure}
Mismatches in the beam optics would persist 
until chromatic effects caused decoherence of the betatron motion 
(and perhaps subsequent recoherence due to synchrotron motion). 
The ultimate emittance growth, due to filamentation,
would be of the same order of magnitude as the change in $\beta$ 
functions.
If a change in momentum persisted through the end of the cooling line, 
it could be detected in dispersive areas of the beam transfer lines, but 
the synchrotron motion could cancel the energy fluctuation.  
Thus, it is desirable to diagnose the beam using the transverse optics.   

An R\&D program is under way to look at the sensitivity and usefulness 
of different diagnostic techniques and evaluate them in the environment 
of rf backgrounds and high magnetic fields.  Modeling is also required
to understand mismatches and realistic errors better.


\subsubsection{Angular momentum}

Angular momentum plays an important role in solenoidal ionization 
cooling channels, unlike the quadrupole channels commonly used elsewhere
in high energy physics. Due to the
rotational symmetry of the solenoid focusing field, the canonical
angular momentum is a conserved quantity when the cooling
material is absent. When absorbers are present, however, the beam
angular momentum can change, thereby yielding a residual angular
momentum at the end of a cooling channel. Non-zero beam angular
momentum creates coupling between the two transverse degrees of
freedom and causes problems for matching the beam into a
downstream quadrupole channel. Furthermore, the angular momentum
intrinsically couples with the beam emittance in the cooling
dynamics. For optimum cooling in a periodic channel, the change in the net angular momentum should be
zero. This requirement becomes the following condition~\cite{kimwang}
\begin{equation}
 \int_0^\lambda \eta(s) \beta_{\perp}(s) B(s) ds \sim 0
\label{eq:angmom}
\end{equation}
where $\lambda$ is one period, $B(s)$ is the on-axis magnetic
field, $\beta_{\perp}(s)$ is the envelope function, and $\eta(s)$ is the
ionization energy loss rate.

Measurement of beam angular momentum is a new subject. Beam
profile monitors, which measure beam distributions in $x$ and $y$, can
measure the beam emittance, but do not provide sufficient
information to directly determine the angular momentum in a
solenoid cooling channel. We are investigating whether it is possible to 
indirectly measure the angular momentum by comparing the measured emittance-damping rates. Clearly, such measurements will be difficult.
Novel ideas are being explored to directly
measure the correlation terms $\langle x p_y\rangle$~\cite{blondel}. 

Since the energy loss rate, absorber thickness, and magnetic field
are known, or can be measured with an accuracy much better than one
percent, and an envelope function measurement is likely to be accurate to a few
percent, it should be possible to determine whether Eq.~\ref{eq:angmom} 
is approximately satisfied. For machine operation 
purposes, beam angular momentum measurements are helpful, but not absolutely
necessary.
% For cooling demonstration experiments, however, good
%angular momentum measurements can provide further evidence for the
%cooling behavior. The ability to make small adjustment in the amount of 
%material in absorbers might be useful for correcting any residual angular momentum
%at the end of a cooling channel.

\subsubsection{Backgrounds}

There are a number of backgrounds that will impact 
beam diagnostics.  These backgrounds come primarily from three sources:
1) backgrounds produced by the incident protons at the target; 2) 
backgrounds produced in the cooling line, primarily from rf
cavities; and 3) backgrounds from decay electrons.  We assume that the
decay electrons can be simulated and subtracted from all measurements.

%\subsubsection{Beam Backgrounds}
\paragraph{Beam backgrounds}

The solenoidal beam transport channel for muons will also transport
all other particles whose transverse momentum is less than $\sim$100
MeV/$c$.  Thus, hadronic showers from 3 GeV protons, for example, can be 
expected in the liquid hydrogen absorbers well down the cooling line.
A large number of species are produced: $K$,
 $n$, $\gamma$, $e^{\pm}$, D, He$^{3}$, He$^{4}$, and other 
nuclear fragments should be seen in detectable quantities.
Most low-energy backgrounds from the target will be stopped in the
first minicooling absorber, but the high energy particles that are left 
will be comparatively unaffected by the rf in the line.  Timing
should be very effective in identifying and discriminating against 
these particles.   

%\subsubsection{rf Backgrounds}
\paragraph{RF backgrounds}

X-ray production from rf cavities results from dark-current electrons hitting 
the body of the cavity and nearby solid components.  The bremsstrahlung 
x-ray flux produced is then scattered and attenuated by both the 
production material and by transmission through any external structure.  
While the dark-current electron orbits will be influenced by magnetic
fields along the beam axis, the x-rays, particularly those around 1-2~MeV, 
will scatter and diffuse freely up and down the cooling line.  
The dark-current electrons from single-cell cavities, and the x-rays 
they produce, are expected to be in the 1--10~MeV range.  The 
electron and x-ray fluxes produced in this environment have  
recently been measured using a 1.3~GHz high-gradient cavity~\cite{xray}.  
Other experiments are planned using an 805~MHz cavity.
Data taken from a variety of pulsed cavities suggest that rf
 breakdown limits the total dark current, and thus the x-ray flux that
can be produced, somewhat independent of frequency.

Fluxes of dark-current electrons and 
x-rays measured in the 1.3-GHz cavity were on the order of $10^{10}$--$10^{11}$ electrons per rf pulse, or $10^{7}$ -- $10^{8}$ 
electrons per bucket, a more relevant parameter.  The number of 
bremsstrahlung photons is of the order of $n_{e}r/L_{R}$ where $n_e$ is the number of dark-current electrons, $r$ is their range, usually a few mm, and $L_{R}$ is 
the radiation length for the material, {\it e.g.}, 1.44~cm for Cu.  Because the energies 
are low, complex showers do not develop.  The photons, however, do
diffuse through the system. 

There are several ways of altering the flux of dark-current electrons 
and the x-ray flux seen by the beam diagnostics: 
\begin{itemize}
\item the geometry of the system 
can be changed so that dark-current electrons do not appear near the beam 
axis (the double-flip cooling channel option described in Section~~\ref{Chap:App-c} does this, because the rf cavities are inside of a uniform solenoidal field and, consequently, it is less likely that dark-current electrons will be guided into the diagnostic device)
\item rf electric field, $E_{rf}$, can be reduced since $I_{dark} \propto E_{rf}^{10}$ 
\item cavity can be coated with a material having low secondary electron yield. 
\end{itemize}
All of these methods are to some extent applicable to the cooling line.

%so it is difficult at this point to make precise estimates of the radiation
%flux expected in the beam diagnostics.  

\subsubsection{Access in the cooling line}
\label{BandC-SEC:access}

It is desirable to be able to insert, 
park, and possibly remove any instrumentation in the line without 
disrupting the beam vacuum.  There are a number of options for insertion 
of diagnostics into the cooling line.  

Analogous to FODO accelerator structures, it is desirable to 
insert diagnostics at every major focusing element.  While there is 
limited space in the cooling line, instrumentation 
modules compatible with the 3.7 cm expansion section that is 
a part of every cell can be used. Figure~\ref{fig:instr} shows a possible location for inserting diagnostics in the cooling cell.

\begin{figure}[!hbt]
\begin{center}
\includegraphics*[width=4cm]{../template/report/ps-and-eps/Black-norem.eps} 
%\psfig{file=Black.eps,scale=0.4}  
\caption[Possible location for diagnostic instrumentation ]{Possible location for diagnostic instrumentation in a standard cooling lattice cell. The arrows show how a diagnostic module could be removed from its parked position and inserted into the beam temporarily for measurements.}
\label{fig:instr}
\end{center}  
\end{figure}

Special SFOFO sections, without hydrogen absorbers and with only enough
rf voltage to contain the bunch, could also be used for specific 
Cherenkov counters or other devices that require more access than would 
be available in a standard cell. We have not made use of this concept as part of this Study, but it is compatible with the cooling channel.
% The expense of these special sections 
%implies that maximum use should be made of the available space in 
%expansion sections.

\subsubsection{Making and using pencil beams}

We anticipate that pencil beams will be very useful in assessing the 
alignment of all cooling, accelerator, and storage ring components. 
Since the range of the roughly 200 MeV/$c$ muons in the cooling line is only about 
6 cm in Cu, and scattering angles are large, collimation works very well.  
Thus a variety of collimators could be used at locations in the bunching 
and cooling lines to produce low transverse emittance beams, on or off
axis.  

Producing a 6D ``pencil'' may be even more useful, since 
this could be used to track the longitudinal evolution of the 
bunch through all of the acceleration and storage ring optics.  In principle,
the 6D bunch can be produced from a 4D bunch by reducing the rf
voltage in the cooling line, so that only the center of the 
longitudinal phase space is transmitted.  While the 
intensities will be reduced, muon beams in the range of $10^{10}$ to
$10^{11}$ particles should be useful and quite easily measured.

\subsubsection{Instrumentation options}

Although high-energy particles, dark-current electrons and x-rays will
be present, the signal from the primary muon beam should dominate these backgrounds.  
Nonetheless, we will look first at instrumentation options that 
offer very fast time response; these will be relatively less 
sensitive to backgrounds.  From this perspective, segmented secondary
emission monitors (SSEMs) and Faraday cups offer a significant 
advantage.  These devices have been shown to have a useful resolution 
time of the order of 150 -- 200~ps.  Since this time response is 
significantly less than the bucket length, these devices would be 
able to provide some information on the bunch shape and thus on the 
longitudinal emittance of the beam.   Faraday cups could be used 
essentially interchangeably with the SSEMs. However, they would provide
about two orders of magnitude more dynamic range, and the possibility of 
some particle discrimination using range.  

These devices can be used both to measure the beam parameters, and
to provide consistency checks for evaluating measurements by
other methods. One possibility being considered is a curtain of
scintillating fibers, with an actuator to slide the fibers into place
when beam diagnostics are needed and a safe parked area out of the way
during normal operating conditions. Likewise, a telescoping 2D
pixel-array SEM is envisioned that can be inserted to diagnose the
center or edge of the beam region.

It is possible that detectable scintillation may occur when the muon
beam passes through the liquid-hydrogen absorbers. If this is verified
in cosmic-ray tests, it would provide a diagnostic that is always
present and does not add any additional material in the beam. The
scintillation should provide a signal with good time resolution and 2D
position information.

Semiconductor arrays of polycrystalline CdTe
would be able to provide high-precision $x$ and $y$ beam profiles
in a single measurement~\cite{placidi}. The detector consists of an
array of charge-sensitive elements, each providing a signal with 
amplitude proportional to the intercepted muon beam density. 
Finally, bolometers are being developed that can measure $xy$ 
profiles from the heating in the metallic windows of the liquid-hydrogen 
absorbers.  We are also looking at high-rate ion chambers,
which would be generally useful for a wide range of measurements.
%Transition radiation, which would detect dark current electrons much 
%more efficiently than muons, may not be useful.                      
Tests of how these various devices perform in the high x-ray-flux environment of a cooling channel will be done as part of the R\&D program.
                     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{Engineering Considerations}
%P.L, March 14-15 2001, contrib. from Ed Black and P.L. 
\afterpage{\clearpage 
\begin{figure}[htb!]
\begin{center}
\includegraphics*[width=6in,clip]{../template/report/ps-and-eps/SFOFOLATTICE1rev7a.eps}
%P.L, March 14-15 2001
%\caption{Engineering detail of the 2.75 m cooling lattice cell.}
\caption[Engineering rendering of the 2.75 m cooling lattice cell]{Engineering rendering of the 2.75 m cooling lattice cell.}
\label{fig:latt275}
\end{center}
\end{figure}
}
%
The fully-integrated layout of the cooling cells for the (1,i) 
lattices and the (2,i) lattices are shown in Figs.~\ref{fig:latt275} and
~\ref{fig:latt165}. 
%respectively. In this case, a cell is defined as an absorber, a linac (2 or 4 rf cells), 3 solenoidal magnets and all the related
%interfaces and hardware. 
%\afterpage{\clearpage}

\begin{figure}[htb!]%
\begin{center}%
\includegraphics*[width=6in,clip]{../template/report/ps-and-eps/SFOFOLATTICE2rev7a.eps}
%P.L, March 14-15 2001
%\caption{Engineering detail of the 1.65 m cooling lattice cell.}
\caption[Engineering rendering  of the 1.65 m cooling lattice cell]{Engineering rendering of the 1.65 m cooling lattice cell.}%
\label{fig:latt165}%
\end{center}%
\end{figure}%

This design takes into account the constraints from
the solenoid magnet subsystem (see Figs.~\ref{fig:mag275}, \ref{fig:mag165} and
\ref{fig:mag165cs})  described  in Chapter~\ref{SCSOLE}, 
the rf cavities discussed in Chapter~\ref{RF:chapter}, and the absorber detailed in Section~\ref{BandC:absorber}.  The basic
dimensions of the cooling cells are taken from the conceptual design presented
above. However, the exact placement of various components may depart slightly
from the layout used in the simulation package. For instance,  the rf cavities are
slightly off-center longitudinally with respect to the coils (by 3.7~cm) to facilitate servicing; this will not affect beam-cooling performance. In addition, space for diagnostics is potentially
available at that location, as detailed in Sec.~\ref{BandC-SEC:access}.
\afterpage{\clearpage   
\begin{figure}[!hbt]
\begin{center}
%\includegraphics*[width=150mm]{cool_fig_1.eps}
\includegraphics[width=3.5in]{../template/report/ps-and-eps/cool_fig_1.eps}
\caption[Magnet cross section for the 2.75 m long cooling cell ]{Magnet cross section for the 2.75 m long cooling cell. 
The coils labeled ``A" is the focusing coil and ``B" is the coupling 
coil.}
\label{fig:mag275}
\end{center}  
\end{figure}

\begin{figure}[!hbt]
\begin{center}
\includegraphics*[width=3.5inc]{../template/report/ps-and-eps/cool_fig_2.eps} 
%\includegraphics*[bb=0 0 393 304]{cool_fig_2.eps}
%\special{eps:cool_fig_2.eps x=5in y=4in}
\caption[Magnet cross-section for the 1.65~m cooling cell]{Magnet cross section for the 1.65-m-long cooling cell. The coils labeled ``A" is the focusing coil and ``B" is the coupling 
coil.}
\label{fig:mag165}
\end{center}  
\end{figure}
}
 Note also that some rf feeds need to be tilted due to the space
constraint imposed by the coupling coil cryostat. The piping and auxiliary hardware for
the absorber shown in the figures correspond to the through-flow absorber design; they would,
 however, be very similar to those used in the convection absorber design.         

       Deviation from the simulation design driven by engineering or cost considerations will most likely occur. Indeed, some items have been discovered in the course of this study and are listed below:
\begin{itemize}
\item  Coil configuration: The coil positions, sizes, and current densities
 used in  the simulations (see Tables~\ref{coolCoils}--\ref{matchCoilP})
differ somewhat from those listed in Chapter~\ref{SCSOLE}, which have been engineered to reduce cost and to
allow more room, both for the absorbers and for insertion of the rf feeds.
  We have verified that the revised fields
on-axis are the same as for the original configuration, and that the
slight discrepancies in the off-axis fields do not affect the cooling
performance.
% (IDEALLY SHOULD BACK THIS UP WITH A TABLE OR GRAPH OF COOLING
%PERFORMANCE - PAUL, CAN YOU PROVIDE THIS, OR DO JUAN AND MIKE THINK IT IS
%BETTER TO SAVE THE SPACE?) (see Table/Fig. 5.XX).

\item Winding structural support: To handle the large forces exerted on the
windings, the focusing coil (``A coil") in the (2,i) cooling lattices has been split into two windings  with a
structural stainless steel plate inserted in between (see Fig~\ref{fig:mag165}.)  However, 
the simulations have been made assuming a current density uniformly
distributed over the entire radius of the coil.  We
can easily readjust the current densities of these two windings to obtain
the required magnetic field profile.
% (BOTH ON AND OFF AXIS?).

\item Absorber details: Our simulation studies have assumed
cylindrically-shaped liquid-hydrogen absorbers with flat windows of
constant thickness. However, in practice the shape will be as indicated in
Fig.\ref{fig:absorber}, with approximately spherical ends, and with tapered windows as
shown in Fig.~\ref{fig:window1}.  A closer approximation has recently been implemented
in Geant4, with the absorber ends represented as spherical caps and the
correct window thickness profile, with no adverse effects on cooling
performance.

\item rf-window design: In the rf cavities used in the simulation, the
beryllium windows are stepped in thickness, with their thickness suddenly
increasing at the radius $r_1$ (see Chapter~\ref{RF:chapter}).  In practice, the mass
distribution near $r_1$ may well be smoother (for instance, if the change in
thickness is brought about by means of chemical etching). We have
verified that, as expected, a smooth transition does not affect the
cooling performance.
\end{itemize}

\section{Conclusion: Performance of the Entire Front End}
\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=3in,angle=90]{../template/report/ps-and-eps/emit-trans-total-vs-s.ps}
%\includegraphics*[bb=23 452 380 701,width=3in,clip]{jcg-allemit.ps}
%\input{ffDummy.fig}
\caption[Transverse emittance as a function of distance]{Transverse emittance as a function of distance along the front end.}
\label{fig:emall}
\end{center}
\end{figure}
%\afterpage{\clearpage}

The transverse emittance along the entire front end is plotted in
Fig.~\ref{fig:emall}. The emittance is seen to be reduced in the minicooling at 150~m (from 17 to 13~mm$\cdot$rad), remains approximately constant through the induction linacs, althought large-amplitude particles are lost, and falls quickly in the final cooling section (from 12 to 2.8~mm$\cdot$rad).

The muon-to-proton ratio along the full system is given in
Fig.~\ref{fig:mu63}.  Particle losses prior to the buncher come
primarily from the loss of very high and very low momenta (30\%), 
plus losses from muon decay ($\approx 20$\%). The losses in the
cooling section come  from bunching inefficiency ($\approx 25$\%) and
 from loss of particles from the rf bucket as the
longitudinal emittance grows in this section ($\approx 25$\%).

As shown in the left panel of Fig.~\ref{fig:initke}, we are able to collect the muon phase space over a substancial kinetic energy range, \textit{i.e.}, approximately 50--250~MeV. The peak collection efficiency occurs for kinetic energy around 100~MeV, where about 35\% of the pions  yield accepted muons. Note the falling efficiency for pions with kinetic energy above 300~MeV.
\begin{figure}[!bth]
\begin{center}
%\input{jcg-63a2.fig}
\includegraphics*[width=3in,angle=90]{../template/report/ps-and-eps/trans-total-vs-s.ps}
%\includegraphics*[bb=95 300 430 550,width=3in,clip]{jcg-63a2.ps}
%\vskip2cm
\caption[Muons per incident proton as a function of distance]{Muons per incident proton as a function of distance along the front end. }
\label{fig:mu63}
\end{center}
\end{figure}
%\afterpage{\clearpage}

\begin{figure}[!htb]
\begin{center}
\includegraphics*[bb=0 520 441 710,width=4in,clip]{../template/report/ps-and-eps/jcg-endeff1.ps}
%\input{ffDummy.fig}
\caption[Kinetic energy distribution of initial pions]{Left plot: Kinetic energy distribution of initial pions (upper curve) and of pions that decay into muons that exit the cooling channel (lower curve). Right plot: efficiency distribution for an initial pion to produce an accepted muon by the channel.}
\label{fig:initke}
\end{center}
\end{figure}
The SFOFO cooling channel increases the number of muons per proton in
the 15~mm$\cdot$rad transverse acceptance cut by a factor of 3.5, or  a factor of 5.75 in the 9.35~mm$\cdot$rad acceptance. This is the Geant4 estimate; ICOOL numbers, listed above, are slightly lower,
consistent with systematic errors in the calculations.  The factor of 5.75 should be compared with the corresponding
enhancement achieved in Feasibility Study I, which  was approximately 2 to
2.5~\cite{FS1}. The better performance of the front-end is due primarily to our progress in understanding phase rotation and to the advantages of the SFOFO cooling lattice compared with the FOFO lattice used in Study-I.  

Engineering and
mechanical tolerances have been studied. We now know that such a channel can
be assembled and aligned with known techniques.   

\section{Matching Cooling to Acceleration }
\label{BandC:bob}
\subsection{Method}

The matching from the end of the cooling section to the start of the acceleration  is achieved by adiabatically changing the lattice parameters
from the short (1.65~m) low beta (0.18~m) SFOFO lattice (2,3) at the end
of the cooling section  to a longer (2.75~m) and higher beta (0.6~m) FOFO
lattice. The transition is made in 10 cells.

Initially there are seven 1.65-m cells, all identical in dimensions,
but with currents modified to gradually raise the central value of
the beta function, and change the form of the beta function
\textit{vs.} momentum from the SFOFO shape (Fig.~\ref{ShortBz}) to a
monotonic slope. This is followed by 3 cells with all longitudinal
dimensions scaled up to further increase the beta function. The beta
functions at the ends of the various matching cells are shown as a
function of momentum in Fig.~\ref{Mcool-acc:fg1}.

Finally, after a lone coupling coil, followed by a reversed-direction 
coil to bring the axial field down to zero, the beta function is allowed to 
rise toward the value needed for the optics of the superconducting linac. Coil dimensions and axial fields are shown in  Fig.~\ref{Mcool-acc:fg2}. 

\subsection{RF Cavities}

We introduce rf to keep the bunch together and to raise the average momentum. All currents are scaled with increasing momentum. The
rf fields (on-crest) average 8~MV/m over 19~m. Average energy gain is
3~MV/m. Each cavity consists of two rf cells, except that there are
three cells in cavity 10 and a single-cell cavity beyond the final
focus coils, for a total of 22 cells. Each rf cell is identical to
those in the 1.65-m cooling lattice. The total acceleration required
(on crest) is 152~MeV, corresponding to 6.9~MeV per cell.  The cavity
phase is set to give an actual mean acceleration of 2.6~MeV per
cavity.

%\subsubsection{Betas at cell ends}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=3.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg1.ps}
\caption[Beta function at the end of matching]{Beta functions at the end of various cells of the matching section.}
\label{Mcool-acc:fg1}
\end{center}
\end{figure}

\subsection{Superconducting Coils}
The coils have the same radial dimensions as those in the final cooling lattices, but different currents and, in the final cells, different lengths. Parameters of the matching coils are summarized in Table~\ref{Mcool-acc:tb1}. The current densities in some coils are very low and the dimensions of the coils could be reduced, provided the total currents and current centers are maintained.

\begin{table}
\begin{center}
\caption[Coils parameters of the matching cells]{Coil parameters for the cooling channel to acceleration section match.}
\label{Mcool-acc:tb1}
\begin{tabular}{|ccccc|}
\hline
  Start &  $dl$  &  $r$  &  $dr$  &   $J$ \\ 
   (m)  &    (m)  &  (m)   &   (m)  &   (A/mm$^2$) \\ 
\hline
  0.066 &  0.145 &  0.198 &  0.330  &   83.76 \\
  0.627 &  0.396 &  0.792 &  0.099  &   80.10 \\
  1.439 &  0.145 &  0.198 &  0.330  &   81.31 \\
  1.716 &  0.145 &  0.198 &  0.330 &  -81.66 \\
  2.277 &  0.396 &  0.792 &  0.099  &  -88.55 \\
  3.089 &  0.145 &  0.198 &  0.330  &  -78.99 \\
  3.366 &  0.145 &  0.198 &  0.330  &   79.32 \\
  3.927 &  0.396 &  0.792 &  0.099  &   95.67 \\
  4.739 &  0.145 &  0.198 &  0.330  &   77.58 \\
  5.016 &  0.145 &  0.198 &  0.330  &  -77.89 \\
  5.577 &  0.396 &  0.792 &  0.099  & -103.02 \\
  6.389 &  0.145 &  0.198 &  0.330  &  -75.99 \\
  6.666 &  0.145 &  0.198 &  0.330  &   76.29 \\
  7.227 &  0.396 &  0.792 &  0.099  &  111.46 \\
  8.039 &  0.145 &  0.198 &  0.330  &   74.23 \\
  8.316 &  0.145 &  0.198 &  0.330  &  -74.52 \\
  8.877 &  0.396 &  0.792 &  0.099  & -122.79 \\
  9.689 &  0.145 &  0.198 &  0.330  &  -72.31 \\
  9.966 &  0.145 &  0.198 &  0.330  &   72.58 \\
 10.527 &  0.396 &  0.792 &  0.099  &  130.03 \\
 11.339 &  0.145 &  0.198 &  0.330  &   73.92 \\
 11.616 &  0.145 &  0.198 &  0.330  &  -74.18 \\
 12.246 &  0.462 &  0.792 &  0.099  & -106.28 \\
 13.193 &  0.169 &  0.198 &  0.330  &  -54.23 \\
 13.517 &  0.169 &  0.198 &  0.330  &   54.42 \\
 14.240 &  0.528 &  0.792 &  0.099  &   89.45 \\
 15.322 &  0.194 &  0.198 &  0.330  &   41.57 \\
 15.692 &  0.194 &  0.198 &  0.330  &  -41.71 \\
 16.509 &  0.594 &  0.792 &  0.099  &  -76.98 \\
 17.726 &  0.218 &  0.198 &  0.330  &  -32.94 \\
 18.142 &  0.218 &  0.198 &  0.330  &   29.74 \\
 19.052 &  0.660 &  0.792 &  0.099 &   47.20 \\
 21.378 &  0.660 &  0.792 &  0.099  &   -1.42 \\
\hline
 \end{tabular}
\end{center}
\end{table}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=3.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg2.ps}
\includegraphics[width=3.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg3.ps}
\caption[Coil dimensions and axial field]{Coil dimensions and axial fields for the cooling-channel to acceleration-section matching section.}
\label{Mcool-acc:fg2}
\end{center}
\end{figure}

\subsection{ICOOL Simulation}
Simulation of this matching section uses a file of tracks from the end of the  baseline cooling simulation. It is important to point out that: i) No rf windows have yet been included in the simulations; ii) the window apertures have been set to 21~cm; and, iii) no optimization of the design has been attempted.
We also note that the acceleration section was designed and without the present matching. Thus, the small amount of energy gain in the matching line will lead to the elimination of a portion ($\approx 39$~m) of the initial preacceleration linac, described in Chapter~\ref{ACCE:Chapter}. (This change was accounted for properly in the cost estimate)

Figures~\ref{Mcool-acc:fg3}--\ref{Mcool-acc:fg6} show the behavior of
a number of important quantities as a function of distance along the
cooling-channel-to-acceleration-section matching line. The minimum of
the $\beta$ function shown in Fig.~\ref{Mcool-acc:fg3}, varies from 18~cm
at the beginning to $\approx 1$~m at the end of the section. Figure~\ref{Mcool-acc:fg5} shows there is a net acceleration from 210 to 270 MeV/$c$, and also that the total muon loss due to the matching is only 1.25\%. The loss inside the 
accelerator acceptance, shown in Fig.~\ref{Mcool-acc:fg6}, is
2.4\%. The selected $\mu/p$ ratios fluctuate because ECALC9 
does not correct for angular momentum when it applies the transverse 
acceptance cuts. The values are correct only at locations where 
$B_z$ is zero; these correspond to the maxima of $\mu/p$ in Fig.~\ref{Mcool-acc:fg6} (bottom). There is 
negligible growth in the normalized transverse and longitudinal 
emittances.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg4.ps}
\quad
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg5.ps}
\caption[Beta function \textit{vs.} z]{$\beta_{\perp}$ function \textit{vs.} distance along the cooling-channel-to-acceleration-section matching (left); muon momentum distribution (right).}
\label{Mcool-acc:fg3}
\end{center}
\end{figure}

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg6.ps}
\quad
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg7.ps}
\caption[rm radius \textit{vs.} z]{Rms and maximum radius \textit{vs.} $z$ (left); rms fractional momentum spread \textit{vs.} $z$ (right).}
\label{Mcool-acc:fg4}
\end{center}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg8.ps}
\quad
\includegraphics[width=2.5in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg9.ps}
\caption[Losses in the cooling to acceleration matching section]{Loss in \% (left); momentum \textit{vs.} $z$ (right), along the cooling-channel-to-acceleration-section matching line.}
\label{Mcool-acc:fg5}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\afterpage{\clearpage}
%\subsection{Plots from ECALC9}
\begin{figure}[![hbt]
\begin{center}
\includegraphics[width=3in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg10.ps}
\includegraphics[width=3in]{../template/report/ps-and-eps/jcg-match-cool-acc-fg11.ps}
\includegraphics[width=0.75\linewidth]{../template/report/ps-and-eps/jcg-match-cool-acc-fg12.ps}
%\includegraphics{jcg-match-cool-acc-fg13.fig}
\caption[Emittances \textit{vs.} z]{$\epsilon_{T}$ (top-left),$~\epsilon_{L}$ (top-right), and $\mu/p$ (bottom) \textit{vs.} $z.$ The three curves in the bottom figure are: (top) total number of $\mu/p;$ (middle) within the $\epsilon_{T}$=0.015~$m\cdot$rad and $\epsilon_{L}$=0.15~m cuts; (lower) within $\epsilon_{T}$=0.0095~$mm\cdot$rad and $\epsilon_{L}$=0.15~m cuts. }
\label{Mcool-acc:fg6}
\end{center}
\end{figure}

%\subsection{Conclusion}
%\begin{itemize}
%\item A 21 m long match has been designed
%\item The total losses are only 1.25\%
%\item Loss of mu/p with cuts is only 2.4 \%
%\item the initial beta is .18 m; the final is $\approx$ 1 m.
%\item The beam is accelerated from 212 MeV/c to 271 MeV/c
%\item The rms mom drops from 11\% to 10\%
%\item The rms bunch length changes from
%\end{itemize}
  

\begin{thebibliography}{99}
\bibitem{ICOOL}
R.~Fernow, \textsl{ICOOL, A Simulation Code for Ionization Cooling of Muons
Beams}, Proceedings of the 1999 Particle Accelerator Conference, A. Lucio and W.
MacKay, eds. (IEEE, Piscataway, NJ, 1999) p. 3020.

\bibitem{G4} The Geant4 Tool Kit is available at 
{\em http://wwwinfo.cern.ch/asd/geant4/geant4.html}

\bibitem{FS1}
\textsl{A Neutrino Source Based on an Intense Muon Storage Ring}, ed. N.~Holtkamp and
D.~Finley,  available at
http://www.fnal.gov/projects/muon\_collider/nu-factory/nu-factory.html 

\bibitem{MC1}
C.M.~Ankenbrandt~{\em et al.} Phys.\ Rev.\ ST\ Accel.\ Beams 2, 081001 (1999)

\bibitem{GPenn1}
G.~Penn {\em et al.}, Phys. Rev. Lett. 85 (4) pp. 764-767. July, (2000).

\bibitem{JM1}
J.~Monroe {\em et al.}, \textsl{Design and Simulation of Muon Ionization Cooling
Channels for the Fermilab Neutrino Factory Feasibility Study},
Phys.\ Rev.\ ST\ Accel.\ Beams 4, 041301 (2001). 

\bibitem{sessler} First suggested by A.~Sessler, LBNL.

\bibitem{PDG} The Particle Data Group, {\em Review of Particle Physics}, 
Eur.\ Phys.\ J. C {\bf 15} (2000) 163.

\bibitem{ecalc9} This code was written by G.~Penn, LBNL.

\bibitem{ref2}R.~Fernow, \textsl{Effect of a dipole error on the performance of the alternating solenoid lattice}, {http://www-mucool.fnal.gov/mcnotes/muc0035.pdf}

\bibitem{ref3}P.~Lebrun, \textsl{Sensitivity of the 15 T. Alternate Solenoid Cooling Channel to Dipole Errors caused by misalignment of the Solenoids}  \\{http://www-mucool.fnal.gov/mcnotes/muc0073.ps}

\bibitem{ref4a}J.~Gallardo, \textsl{Errors in a continuous solenoid },\\  {http://www-mucool.fnal.gov/mcnotes/muc0041.ps}; \textsl{Errors in a discrete solenoid} \\ {http://www-mucool.fnal.gov/mcnotes/muc0037.ps} 

\bibitem{ref4b}J.~Gallardo, \textsl{Note on Vector potential for a Solenoid and Errors}, unpublished (1999). 

\bibitem{BEK}M.~Bassetti and G.A.~Erskine, \textsl{Closed Expression for the Electric Field of a Two-Dimensional Gaussian Charge}, CERN Report \textbf{CERN-ISR-TH/80-06}, 1980; S. Kheifets, \textsl{Potential of a Three-Dimensional Gaussian Bunch}, DESY Report \textbf{PETRA} Note 119 (1976).

\bibitem{jcg1}F.~Zimmermann and T.~Raubenheimer, \textsl{Longitudinal space charge in final-focus systems}, Nucl.\ Instrum.\ Meth., 1997; J.~Gallardo, \textsl{Space Charge in Gaussian Beams: A Collection of formulas}, \\{http://www-mucool.fnal.gov/mcnotes/muc0038.ps}

\bibitem{haipeng}H.~Wang, \textsl{Analytical approach to the short-range wakefields},\\{http://www-mucool.fnal.gov/mcnotes/muc0117.pdf}

\bibitem{mafia} MAFIA is available from CST GmbH, Buedinger Strasse 2a, 64289
Darmstadt, Germany (www.cst.de).

\bibitem{jcg2}J.~Gallardo, \textsl{Implementing space-charge effects in cooling simulations}, BNL Report: Muon Tech. Notes Mu-019, 1998.

\bibitem{Mark}
J.W.~Mark, SLAC-PUB-3169 (1984) and references therein.

\bibitem{Bates}
E.J.~Beise {\it et al.}, Nucl.\ Instrum.\ Meth.\ {\bf A378} (1996) 383.

\bibitem{SLAC158}
R.W.~Carr {\it et al.}, SLAC-Proposal-E-158, July 1997.

\bibitem{JLAB} 
D.J.~Margaziotis, in {\sl Proc.\ CEBAF Summer 1992
Workshop}, F.~Gross and R.~Holt, eds., AIP Conf.\ Proc.\ {\bf 269},
American Institute of Physics, New York (1993), p.~531.

\bibitem{E158-milestone}
R.W.~Carr {\it et al.}, ``E158 Liquid Hydrogen Target 
Milestone Report,"
April 21, 1999, http://www.slac.stanford.edu/exp/e158/documents/target.ps.gz.

\bibitem{ASME}
\textsl{ASME Boiler and Pressure Vessel Code}, ANSI/ASME BPV-VIII-1 (American
Society of Mechanical Engineers, New York, 1980), part UG-32.

\bibitem{JimN} J.~Norem, \textsl{Instrumentation issues for the Neutrino
Source}  \\{http://www-mucool.fnal.gov/mcnotes/muc0085.ps}

\bibitem{kimwang} Kwang-Je~Kim and Chun-xi~Wang, Phys. Rev. Lett. \textbf{85} (760) 2000

\bibitem{blondel} A.~Blondel is working to develop a system based on a ring imaging Cerenkov technique.

\bibitem{xray}J.~Norem, A.~Moretti and M.~Popovic, \textsl{The Radiation Environment in and near High Gradient RF Cavities}, \\{http://www-mucool.fnal.gov/mcnotes/muc0178.pdf}

\bibitem{placidi} M.~Placidi {\it et al.}, \textsl{A CdTe Detector for Muon
Transverse Profile Measurements}, CERN-NUFACT-Note 068,  
contributed to the MuCool Instrumentation Workshop, IIT, Chicago, 
Nov. 10-11 (2000).

%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

 
%\bibitem{ref0a}B.~Palmer, \textsl{Neutrino Factory Draft Parameters} \\{http://www-mucool%.fnal.gov/mcnotes/muc0046.ps}

%\bibitem{ref1e}Jocelyn Monroe, Paul Lebrun, Panagiotis Spentzouris, \textsl{ DPGean%t and Icool Code Comparison} \\ {http://www-mucool.fnal.gov/mcnotes/muc0072.ps}

%\bibitem{SM} Y.~Semertzdis and W.~Morse, Unpublished Note BNL 2000


\end{thebibliography}