\section{Buncher and Cooling Channel Solenoids}
%\textcolor{red}{need to set the citations}
\subsection{Solenoid Layout and Parameters}
We proceed downstream starting at the end of the induction linacs~\cite{BC-mag1}.  
The matching section
between the last induction linac and the beginning of the bunching section
 consists of four 2.75~m long cells.  
 The focusing solenoids in this matching section  must be designed to
withstand longitudinal forces of
up to 60 metric tons that are imparted on them by the matching solenoids 
located upstream, at the end of the last induction linac.    The
%P.L, March 14-15 2001 typos, and those particularly bore won't be that warm, 
% since they host the Lh2 vessel (.O.k. 16.6 k vs 4.4 K or 2.5 K..) 
%warm bore aperture of the of the 
bore aperture of the focusing coils for a 2.75-m-ong cooling cell must be
about 650 mm in order to accommodate a liquid-ydrogen absorber (se
Fig.~\ref{fig:mag275}).
%P.L, March 14-15 2001 same as? 
% It is a bit coincidentla that the bore diameter for the cooling cells
% and the 400 MHz are about the same. 
% The
%warm bore aperture for the beam bunching cell flux
%reversal coils must be the same in order to accommodate a 402.5 MHz {\it rf} 
%cavity. 
The warm bore aperture for the focusing coils in the bunching section must
also  be $\approx$ 650 mm, in order to accommodate the 402.5 MHz rf cavitiies. 

 Room temperature service ports to the 402.5 MHz rf cavity can go out
between two focusing coils running in opposite polarity ({\it i.e.}, in 
the flux-reversal region), through the magnet cryostat servicing these two 
coils.
%P.L, March 14-15 2001 no point stating flux reversal twice in the same
% sentence..
%flux reversal coils.
 Table~\ref{BandC:tbsolencell} shows the number of cells of each type, the
minimum aperture requirements
for the magnets and the maximum coil current densities for the coils in
each cell type.  Included in Table~\ref{BandC:tbsolencell}
is the magnetic field 9.9~m from the beam axis.  Because the bunching
and cooling cell solenoids are
constantly changing polarity, there is almost no stray field from these
solenoids at a radius $R=10$~m.

\begin{table}[tbh]
\centering
\caption[Basic parameters for bunching and cooling cell]{Basic parameters for the bunching and cooling cells.}
\label{BandC:tbsolencell}
\begin{tabular}{|lcc|}
\hline
           Parameter &  2.75 m cell &   1.65 m cell \\
\hline
Number of cells of this type &  37 &    37 \\
Cell length (mm) &      2750 &  1650 \\
Maximum space for the rf cavity & 1966 &  1108 \\
Number of 201.25 MHz rf cavities per cell &       4 &     2 \\
Number of 402.5 MHz rf cavities per bunching cell &       1 &     NA \\
Focusing magnet cryostat length (mm) &  784 &   542 \\
Focusing magnet cryostat length (mm) &  283 &   209 \\
Aperture for the focusing magnet (mm) & 650 &   370 \\
Aperture for the coupling magnet (mm) & 1390 &  1334 \\
Maximum focusing coil current density (A mm$^{-2}$) &   128.04 &        99.65 \\
Maximum coupling coil current density (A mm$^{-2}$) &   99.24 & 109.45 \\
Maximum cell stored energy (MJ) &       13.2 &  17.6 \\
Maximum longitudinal warm to cold force (MN) &  0.74 &  1.20 \\
Number of longitudinal supports per coil &      4 &     6 to 8 \\
Peak induction 9.9 m from the cell axis (T) & $1.18\times 10^{-5}$ & $2.62\times 10^{-5}$ \\
\hline
\end{tabular}
\end{table}

Magnet parameters and a magnet cross section for the 2.75-m-long bunching
and cooling cell magnets are shown in Table~\ref{BandC:tbsolen275} and
Fig.~\ref{fig:mag275}. 
% We said this already.
% Note: the solenoids in the 2.75-meter long cells are
%the same for both bunching and cooling cells. 
 Magnet parameters and a magnet
cross section for the 1.65-m-long cooling cell magnets are shown in
Table~\ref{BandC:tbsolen165} and Fig.~\ref{fig:mag165}.    

\begin{figure}
\begin{center}
%\includegraphics*[width=150mm]{cool_fig3.eps} 
\includegraphics[bb=0 0 410 417]{../template/report/ps-and-eps/cool_fig_3.eps}
%\special{eps:cool_fig_3.eps x=5in y=4in}
\caption[Cross section of the 1.65 m cell focusing magnets ]{Cross section of the 1.65 m cell focusing magnets, perpendicular to
the beam.}
\label{fig:mag165cs}
\end{center}  
\end{figure}
 
\begin{table}[tbh]
\centering
\caption[Solenoid parameters for the 2.75~m bunching and cooling cell]{Solenoid parameters for the 2.75-m-long bunching and cooling cell.}
\label{BandC:tbsolen275}
\begin{tabular}{|lcc|}
\hline
 &      Focusing  &     Coupling \\
\hline
\multicolumn{3}{|c|}{Mechanical Parameters}\\
\hline
Magnet cryostat length (mm)&    784&    283 \\
Magnet cryostat bore diameter (mm)&     650&    1390 \\
SC coil length (mm)&   167&    162 \\
Inner radius of the coil (mm)&  355&    729 \\
SC coil thickness (mm)&        125&    162 \\
Distance between coils in $z$ direction (mm)&     350&    NA \\
Inner support structure thickness (mm)& 15&     0 \\
Outer support structure thickness (mm)& 20&     25 \\
Number of turns per magnet&     2304&   1472 \\
Magnet cold mass (kg)&  1430&   1245 \\
Magnet overall mass (kg)&       1870&   1570 \\
\hline
\multicolumn{3}{|c|}{Electrical Parameters and Magnetic Forces}\\
\hline
Maximum magnet design current (A)&      2320.2& 1779.9 \\
Peak induction in the windings (T)&     7.5&    6.5 \\
Magnet stored energy at design current (MJ)&    $\approx$7.9 &  $\approx$7.7 \\
Magnet self inductance per cell (H)& $\approx$2.9 & $\approx$4.9 \\
Superconductor matrix $J (A \textrm{mm}^{-2}$)&  155&    119 \\
E J$^2$ limit per magnet cell $(J A^2$ m$^{-4}$)&  $1.89\times 10^{23}$& $1.09\times 10^{23}$ \\
Force pushing the focusing coils apart (metric tons)&   329&    NA \\
% It was written "on a the Coil" Is it on one of the coil (any of them) 
% in the system, or on the "A" coils 
% I presumed the former... 
Peak fault force on a coil (metric tons)&       75.3&   75.3 \\
\hline
\end{tabular}
\end{table}

\begin{table}[tbh]
\centering
\caption{Solenoid parameters for the 1.65-m-long cooling cell.}
\label{BandC:tbsolen165}
\begin{tabular}{|lcc|}
\hline
 &      Focusing &      Coupling \\
\hline
\multicolumn{3}{|c|}{Mechanical Parameters}\\
\hline
Magnet cryostat length (mm)&    542&    209 \\
Magnet cryostat warm bore diameter (mm)&        380&    1334 \\
SC coil length (mm)&   145&    109 \\
Inner radius of inner coil&     210&    687 \\
SC coil thickness (mm)&        138&    326 \\
Distance between coils in $z$ direction (mm)&     132&    NA \\
Inner support structure thickness (mm)& 20&     0 \\
Center support structure thickness (mm)&        30&     NA \\ 
Outer support structure thickness (mm)& 40&     25 \\
Number of turns per magnet&     4480&   1974 \\
Magnet cold mass (kg)&  1995&   1750 \\
Magnet overall mass (kg)&       2430&   2290 \\
\hline
\multicolumn{3}{|c|}{ Electrical Parameters and Magnetic Forces}\\
\hline
Maximum magnet design current (A)&      1780.5& 1896.7 \\
Peak induction in the windings (T)&     8.4&    6.5 \\
Magnet stored energy at design current (MJ)& $\approx$10.7& $\approx$11.0 \\
Magnet self-inductance per cell (H)& $\approx$6.8 & $\approx$6.1 \\
Superconductor matrix J (A mm$^{-2}$)&  119&    126 \\
E J$^2$ limit per magnet cell (J A$^2$ m$^{-4}$)&  $1.51\times 10^{23}$& $1.74\times 10^{23}$ \\
Force pushing the focusing coils apart (metric tons)&   1950&  NA \\
Peak fault force on a coil (metric tons)&   122&    122 \\
\hline
\end{tabular}
\end{table}

Figures~\ref{fig:mag275} and \ref{fig:mag165} show a cross section of the
bunching and cooling cell solenoids.  The plane for the cross sections is taken
through the warm to cold supports that carry axial forces. These cross sections also show the magnet cryostats, the
coils, the coil support structure, the 30 K shields, and the vacuum vessel
around the rf cavities.  The cryostat vacuum systems are separated from
the vacuum around the rf cavities and the beam vacuum. The penetration of the hydrogen absorber
plumbing through the space between the focusing coils is not shown in
Figs.~\ref{fig:mag275} and \ref{fig:mag165}.

  Figure~\ref{fig:mag165cs} shows a cross section through the center of the
1.65-m-long cell focusing coil pair (``A'').  Note the location of the longitudinal cold
mass supports and the cold mass supports that carry forces in both directions
perpendicular to the solenoid axis.  This figure illustrates how magnet electrical leads, and helium refrigeration can be brought into the cryostat. 
Figure~\ref{fig:mag165cs} is a typical cross section that can be applied to all
of the bunching and cooling cell solenoids.

Figures ~\ref{fig:mag275} and \ref{fig:mag165} show the location of the
hydrogen absorbers within the bore of the focusing coil pair.  The hydrogen
absorber will share the same cryostat with the focusing coils.  The hydrogen
absorber and these magnets will have a common vacuum. The hydrogen absorber
will be supported from the coil package by a low thermal conductivity
support system made from a titanium tube.  Figure~\ref{fig:mag165cs}
illustrates schematically that connections to the hydrogen absorber can be made
between the focusing coils through the support structure that carries the large magnetic forces generated by these coils.
%P.L, March 14-15 2001
%carries the magnetic large forces generated by the two A coils that operate..
% No point repeating again that they run in opposite polarity 
% (See below)


\subsection{Forces}

Forces in the longitudinal direction are a serious issue for the bunching and
cooling solenoids.  The focusing coils, running in opposite polarity,  generate
large forces (up to 1950 metric tons) pushing them apart.  These forces must be
carried by a 4.4 K metallic structure between the two coils. The magnitude of
the forces pushing these coils apart depends on the spacing between the coils,
the average coil diameter and the current carried in each coil.  The inter-coil
forces are carried by either aluminum or stainless steel shells on the inside
and the outside of the coils.  The forces are transmitted to the coil end
plates, which are put in bending.  Large stresses are developed at the point
where the end plates meet the shells inside and outside the coils.  Since the
force between the focusing coils in the 1.65-m-long cooling cells is so
large,  these  coils must be divided in the
radial direction in order to reduce the bending stress in the end plates.  The
large stress in the end plates of these focusing coils in the 1.65-m-long
cooling cell dictate that the end plates and shells must be
made from 316 stainless steel.

 If the currents in all of the focusing coils and all of the coupling coils
were the same from cooling cell to cooling cell, there would be no net
longitudinal force on any of the coils.  However, the currents in the cooling
cell coils vary as one goes down the channel.  This
generates a longitudinal force in various magnet coils.  The largest
longitudinal forces will be generated at the ends of the string or when one
coil quenches and adjacent coils do not quench.  One can attach all of the
coils together with cold members, but further examination suggests that this
approach would make it difficult to assemble and
disassemble the muon cooling system.  As a result, every magnet is assumed to
have cold to warm longitudinal supports.  The cold-to-warm supports in the
magnets in the 2.75-m-long cells are designed to carry 80 metric tons (the
maximum force during a magnet fault).  These forces can be carried by four
oriented fiverglass epoxy cylindrical supports that are 50~mm in diameter with
a 4-mm-thick wall. Oriented fiberglass rods can carry stresses up to 600 MPa
in either tension or compression.

The 1.65-m-long cell magnets have longitudinal cold-to-warm supports that
are designed to carry 120 metric tons.  Figure~\ref{fig:mag165cs} shows the
location of eight of these supports on the 1.65-m-long cell focusin magnet.  A
six-support longitudinal support system would also be practical. The support
shown for the focusing coil in Fig.~\ref{fig:mag165} is designed to operate
in both tension and compression. Further engineering can define an optimum cold
mass support system for these magnets.  Compared with other heat loads into the
magnets, the longitudinal cold mass supports represent about one quarter of the
total heat leak into the magnet cryostat.

\subsection{Conductor}

The magnet conductor that is assumed for all of the coupling solenoids is a
conductor that is 7 parts copper and 1 part niobium-titanium.  This conductor
consists of strands with a copper-to-superconductor ratio of 1:1.3. The
twist pitch in the superconductor is about 10 mm.  The strands of this
conductor are attached to a pure copper matrix.  
%P.L. March 19 2001: based on the comment below and a subsequent 
% e-mail from Mike Green,  the new proposed text is just after all 
% commented lines: Old text read :
%"The overall dimensions for the
%finished conductor is 3 mm by 5 mm. The proposed conductor will carry 5100 A at
%5 T and 4.2 K.  At 7.5 T the proposed conductor will carry about 2500 A at 4.4
%K.  This conductor could be used in the 2.75-meter cell A coils, but the margin
%is rather tight. "
%P.L, March 14-15 2001 paragraph here

%" The problem occurs in the 1.65-meter long cell A
%magnet where the peak field at the
%high field point in the magnet is 8.4 T.  This coil must be operated at
%reduced temperature (say 2.5 K) when
%the proposed conductor is used." 
%P.L, March 14-15 2001
% Not sure at all we can split the A coils: this will reduce the field 
% gradient and therefore reduce the amount of transverse focusing.
% Mike G., I am sure you are well aware of this, therefore 
% you leaves it at that.  
% Therefore, in order to push the critical current  densitiy down, you 
% increase the amount of super-conductor and the reduce the temprature.
% and cools down the coil further to avoid quenching.
% You wrote:
% A re-optimization of the short cooling
%cell that moves the A coils further
%apart may be a better solution to the high field problem in the short cell
%A coils.  It is proposed that the A
%coils in the both types of cells use a conductor with a 4 to 1 copper to
%superconductor ratio.
% I understood, and write:
%" A re-optimization of the short cooling cell such that the A coils are pushed
% further apart would a solution to this critically high field problem. 
%However, this would reduce the gradient and therefore would affect the 
%cooling performance. We therefore propose to use a conductor with a 4 to 1
%copper to superconductor ratio, or conversely, coold down to 2.5 K."
%% But, but, is one or the other we have to do, or both? 
%  
%P.L, March 19 2001 New paragraph here, from Mike Green, 
% style modification from P.L., March 20
  The overall dimensions for the finished conductor for all of the bunching and
cooling solenoids are 3 mm by 5 mm. This conductor will carry 5100~A at
5 T and 4.2 K.  At 7.5~T, the proposed conductor will carry about 2500 A at 4.4
K.  The same  conductor could be used in the 2.75-m cell focusing coils but the
margin is rather tight. However, it could not be used in the 1.65-m long cell
focusing magnet, where the peak magnetic field reaches is 8.4 T within the
coil.  Therefore, this coil must be operated at reduced temperature (say 2.5
K).  To allow for greater temperature margin, all the focusing coils in 
both lattices will use a conductor with a 4:1 copper-to-superconductor
ratio. The focusing coils in the 1.65-m long cell will be cooled to 2.5 K.


The conductor will have a varnish insulation that is 0.05 mm thick. 
The layer-to-layer fiberglass epoxy insulation is 0.4~mm thick. 
The ground plane insulation around the coils is 1.6~mm thick. 
This permits the superconducting coils to be discharged with a voltage across
the leads of up to 1200~V.  Each focusing coil set and each coupling coil
is powered separately.  A quench-protection voltage of 1200~V is
adequate to protect any of the coils in the cooling cells.  Because the
conductor current density is high, the focusing coils in the 2.75-m-long
cells have the smallest safety margin when it comes to quench protection. 
Re-optimization of these coils can improve their quench protection.

The conductor currents and current densities are given for the focusing and
coupling coils in Tables.~\ref{BandC:tbsolen275} and \ref{BandC:tbsolen165}. Listed also
are the peak values that would occur in the cells operating at the highest
current. 
 The estimated stored energy occurs at the peak design current in the
coils.  
% Dan K and P.L. don't understand this sentence. Either it says the 
% that the highest the current is, the highest the stored energy is
% Quoted: 
In general, when the current density is high in the focusing coil, the current
density in the coupling coil is low.  The stored energy for the cooling cells changes
very little along the cooling channel.  The cell stored energy
shown in Table~\ref{BandC:tbsolencell} is the average stored energy for that type
of cell.  Table~\ref{BandC:tbsolenJI} shows the average coil current density and
coil current for the focusing and coupling coils in the various regions of the bunching and
cooling channel.

\begin{table}[htb!]
\centering
\caption[Coil average $j$ and $I$ for bunching and cooling channels]{Coil average $j$ and $I$ for various sections of the bunching
and cooling channel. The matching sections between lattices have been taken into
account.}
\label{BandC:tbsolenJI}
\begin{tabular}{|lcccc|}
\hline
        Section&      focusing  $j$ & focusing $I$  & coupling $j$ &  coupling $I$  \\
 & (A~mm$^{-2}$) & (A) & (A~mm$^{-2}$) & (A) \\
\hline
%P.L, March 14-15 2001, and Rick Fernow's count 
%       Bunching Cells& 22&     106.34& 1927.0& 99.24&  1779.9 \\
        Bunching cells&      105.28& 1907.7& 98.83&  1762.0 \\
%       Cooling 1-1&        106.34& 1927.0& 99.24&  1779.9 \\
        Cooling (1,1)&          105.28& 1907.7& 98.83&  1762.0 \\
        Cooling (1,2)&         117.84& 2135.3& 92.42&  1657.5 \\
        Cooling (1,3)&         128.04& 2320.2& 85.25&  1519.9 \\
        Cooling (2,1)&        82.34&  1471.1& 105.53& 1899.7 \\
        Cooling (2,2)&        89.83&  1604.9& 95.99& 1727.9 \\
%       Cooling 2-3&        99.65&  1780.5& 93.47&  1619.8 \\
        Cooling (2,3)&       99.81&  1783.2& 84.42&  1519.7 \\
\hline
\end{tabular}
\end{table}
%P. L. March 20 2001
% No quantitative information is given here, is we drop the figure. 
% Propose to comment out.. 
%Figure~\ref{fig:magmatch} is a schematic representation of the matching
%section of a 2.75-meter long
%cell to a 1.65-meter long cell.  The forces between the coils in the focusing
%magnet are quite large.  It is assumed
%that the structure around the focusing coils is stainless steel.  The focusing coil set
%shown in Figure~\ref{fig:magmatch} is the only unique
%magnet is the muon cooling channel.  
%P.L, March 14-15 2001, per consistency with Rick Fernow's count 
%There are 39 A and B coils that make
There are 41 pairs of focusing coils and coupling coils that make
up the 2.75-m-long matching, bunching and cooling cells.  
%P.L, March 14-15 2001, per consistency with Rick Fernow's count 
%There are 37 A and B coils that make up the
Likewise, there are 37 sets of coils that make up the
1.65-m-long cooling cells.
%P.L, March 14-15 2001, but, but, why mentioning this count again. 
%This is not quite necessary.. s 

% Drop this figure..
%\begin{figure}
%\begin{center}
%\includegraphics*[width=150mm]{cool_fig4.eps} 
%\includegraphics[bb=0 0 446 472]{cool_fig_4.eps}
%\special{eps:cool_fig_4.eps x=5in y=4in}
%\caption{cross section of the Matching Region between 2.75 m Cells and 1.65 m Cells.}
%\label{fig:magmatch}
%\end{center}  
%\end{figure}

The last three meters of the induction
linac channel must have thicker coils with a separate power supply on each
coil.  The 1.25~T solenoids at
the end of the induction cells must have separate longitudinal warm-to-cold
supports to carry forces (up to
60 metric tons) generated by the magnets in the first cells of the bunching
section.  

The end of the short cell cooling section must be
matched to the accelerator section downstream (see Sec.~\ref{BandC:bob}). This matching section
consists of seven standard short cooling cells with varying currents
in the coil and no hydrogen absorbers. The last three cells in this
section are longer than the standard 1.65-m cooling cell, but the coupling
 coils can be made identical to the standard coupling coils. The three focusing  coils
in the last three cells are special coils with larger spacing between
the flux reversal coils. The final two coils have the same diameter as
the short-cell coupling coils, but they are longer and powered
differently. The last two coils are considered to be part of the
solenoids in the superconductiong linac section.

\subsection{Refrigeration}
Refrigeration to the muon cooling magnets and hydrogen absorbers is
supplied at 16 K and 4.4 K.
The 4.4 K refrigeration is used to cool the superconducting coils except
for the focusing coils   in the 1.65-m-long
cell, which are cooled to 2.5 K.  The 2.5 K cooling requires an additional
heat exchanger and a vacuum
pump to produce nearly 0.3 W of cooling at 2.5 K.  Most of the heat into
the 1.65~m cell focusing coil
package is intercepted at 4.4 K.  The hydrogen absorbers are cooled from
the same refrigerator as the
solenoid magnets.  Refrigeration for the hydrogen absorbers is drawn off at
16 K.  The 16 K helium used
to cool the liquid hydrogen returns to the helium cold box at 19 K.  The
absorbers in the 2.75-m-long
cell contain 35.6~liters of liquid hydrogen.  The 1.65-m-long cell
absorbers contain about 8~liters of liquid
hydrogen. 
%P.L, March 14-15 2001
% M.G., version 2.
  The estimated heat load to the absorbers is between 120 and 130 W. 
   Table~\ref{BandC:tbsolenheat} shows the
refrigeration requirements for the 2.75-m-long cells and the 1.65~m
long cells with hydrogen
absorbers.  The equivalent 4.4 K refrigeration reflects the Carnot ratios
from 4.4 K to 16 K and the
refrigeration lost when helium returns to the compressor by bypassing the
refrigerator heat exchangers.
%P.L, March 14-15 2001
% delete the number of cells, rephrase.., correct number (M.G., v2.) 
%The equivalent 4.4 K refrigeration for each of the 22 bunching cells is
%21.1 W per cell.  About 10.4 W of
The equivalent 4.4 K refrigeration for the bunching cells is
13.3 W per cell.  About 10.5 W of
equivalent 4.4 K refrigeration are used to cool two pairs of 2000-A
gas-cooled leads from 300~K to 40~K.

\begin{table}[!tbh]
\begin{center}
\caption[Sources of heat]{Sources of heat at 2.5~K, 4.4~K, abd 16--40~K in the bunching and
cooling cell magnets. $^a$ ``A'' denotes the focusing coil; $^b$ ``B'' denotes the coupling coil}
\label{BandC:tbsolenheat}
\begin{tabular}{|lllll|}
\hline
%Source of Heat&\hspace{.0in}2.75 m Cell(W)& &\hspace{.0in}1.65 m Cell(W)& \\
Source of heat&\multicolumn{2}{c}{2.75~m Cell(W)} &\multicolumn{2}{c|}{1.65~m Cell(W)} \\
        & Coil A$^a$& Coil B$^b$& Coil A&       Coil B \\
\hline
\multicolumn{5}{|c|}{Magnet heat loads at 4.4 K} \\
\hline
        Vertical cold mass supports&    0.24&   0.24&   0.40&   0.24 \\
        Longitudinal cold mass supports& 0.36&   0.36&   0.74&   0.54 \\
        Thermal radiation through MLI&  0.16&   0.14&   0.01&   0.19 \\
        Bayonet joints and piping&      0.03&   0.03&   0.03&   0.03 \\
        Instrumentation wires&  0.02&   0.02&   0.02&   0.02 \\
        HTS current leads&      0.60&   0.60&   0.60&   0.60 \\
\hline
        Total 4.4 K heat load per coil& 1.41&   1.39&   1.80&   1.62 \\
\hline
\multicolumn{5}{|c|}{Magnet Heat Loads at 2.5 K} \\
\hline
        Vertical cold mass supports&    ---&    ---&    0.05&   --- \\
        Longitudinal cold mass supports&        ---&    ---&    0.10&   --- \\
        Thermal radiation through MLI&  ---&    ---&    0.11&   --- \\
        Bayonet joints and piping&      ---&    ---&    0.01&   --- \\
        Instrumentation wires&  ---&    ---&    0.00&   --- \\
        HTS current leads&      ---&    ---&    0.02&   --- \\
\hline
        Total 2.5 K heat load per coil& 0.0&    0.0&    0.29&   0.0 \\
\hline
\multicolumn{5}{|c|}{Magnet shield and intercept heat loads at 16 to 40 K} \\
\hline
        Vertical cold mass supports&    3.8&    3.8&    3.8&    3.8 \\
        Longitudinal cold mass supports&        7.2&    7.2&    10.8&   10.8 \\
        Thermal radiation through MLI&  2.7&    2.9&    1.9&    3.2 \\
        Bayonet joints and piping&      1.3&    1.3&    1.3&    1.3 \\
        Instrumentation wires&  0.1&    0.1&    0.1&    0.1 \\
        Gas cooled current leads&       ---&    ---&    ---&    --- \\
\hline
        Total 16 to 40 K heat load per coil&    15.1&   15.3&   17.9&   19.2 \\
\hline
\multicolumn{5}{|c|}{Hydrogen Absorber (16 K Cooling)}\\
\hline
        Cold mass supports&     1.5&    ---&    1.0&    --- \\
        Thermal radiation through MLI&  0.3&    ---&    0.2&    --- \\
        Bayonet joints and piping&      1.3&    ---&    1.3&    --- \\
        Instrumentation wires&  0.1&    ---&    0.1&    --- \\
        Thermal radiation to windows ($\epsilon$ = 0.2)& 18.4& ---&      6.9&    --- \\
%P.L, March 14-15 2001 M.G. correction, V2.
%       Beam Absorption Heating&        77.0&   ---&    81.0&   --- \\
        Beam absorption heating&        275&    ---&    110.0&  --- \\
        Circulation heater& $\approx$30 &       --- & $\approx$30 & ---  \\
%P.L, March 14-15 2001 M.G. correction, V2.
%       Total 16 K Heat Load per Coil&  128.6&  0.0&    121.5&  0.0 \\
%       Equivalent 4.4 K Refrigeration per Cell&        \hspace{.75in}54.6&&\hspace{.75in}57.6&\\
\hline
        Total 16 K heat load per coil&  326.6&  0.0&    149.5&  0.0 \\
\hline
        Equivalent 4.4 K refrigeration per cell&\multicolumn{2}{c}{100.7}&\multicolumn{2}{c|}{54.1}\\
\hline
\end{tabular}
\end{center}
\end{table}

Figure~\ref{fig:magcryo} shows a schematic representation of the refrigeration for a pair
of focusing coils with a hydrogen
absorber.  Two-phase helium at 4.4 K is used to cool the superconducting
coils.  If nineteen magnets are
cooled from a single flow circuit, the mass flow of two-phase helium should
be 8 to 10~g/s.
The flow circuit can have up to 20 magnet coils in series before the helium
is returned to the control
cryostat.  The shields, intercepts, current leads, and hydrogen absorbers
are cooled by helium that comes
from the refrigerator at 16 K.  The helium used to cool the shields and the
leads is returned to the
refrigerator compressor warm.  The rest of the 16 K helium returns to the
refrigerator at 19 K.


\begin{figure}
\begin{center}
\includegraphics*[width=150mm]{../template/report/ps-and-eps/cool_fig_6.eps} 
%\includegraphics[bb=0 0 465 373]{cool_fig_6.eps}
%\special{eps:cool_fig_6.eps x=5in y=4in}
\caption[Cryogenic cooling system for a cooling (``A'') coil]{Cryogenic cooling system within a typical cooling focusing coil cryostat.}
\label{fig:magcryo}
\end{center}  
\end{figure}

The helium used to cool the magnet shield intercepts heat from the cold-mass 
support, the bayonet
tubes, the instrumentation wires, and radiation heating through the
multi-layer insulation before it is used to
cool the gas-cooled current leads for the magnets. For the flow
circuit shown in Fig.~\ref{fig:magcryo}, the flow of
helium gas in the shield cooling circuit is dictated by the needs of the
gas-cooled current leads.  For the
current leads in the cooling and bunching magnets, this flow varies from
0.15 to 0.23~g/s.
Depending on the needs of the current leads, the temperature rise in the
shield-gas flow circuit will vary
from 14 K to 23 K.  
%P.L, March 14-15 2001.  I don't understand this. If this is clear to 
% Rick or Juan, so be it. Otherwise, we need to explain what we mean 
% by re-optimizing ( does it change the length of the coils to provide more 
% room for the leads, or change the amp-turns, or what?) 
% Or may be, dare I ask, shouldn't  we drop these statements? 
If we optimize the magnets, the lead current
might be as low as 1200 A.  With
1200 A current leads, the temperature at the top of the high T$_c$
superconducting leads would be about 50~K.

%P.L, March 14-15 2001. Paragraph, help, I am getting dizzy..
oth the focusing and the coupling magnet shields will be cooled using the
same 16 K source of gas from
the helium refrigerator, but this is not optimum from the standpoint of
overall refrigeration system
efficiency.  When the helium refrigerator cools both the hydrogen absorber
and the magnets, there will be
enough excess refrigeration capacity available to cool down the magnet
coils in a reasonable time.

The flow in the 16 K circuit to the hydrogen absorber is dictated by the
heat load in the absorber.
Without a muon beam, the heat load could be as low as 22 W.  With beam
heating and the circulation
%P.L, March 14-15 2001 changed number
%heater operating the heat load into the absorbers can approach 130 W.  The
heater operating, the heat load into the absorbers can approach 320 W.  The
temperature rise in the absorber
cooling circuit should be limited to about 2 K.  As a result, the
helium flow circuit used to cool the
%P.L, March 14-15 2001 changed number
%hydrogen absorbers should be designed to provide 12.5 grams per second of
hydrogen absorbers should be designed to provide 31~g/s of
16 K helium.  
% Added test, see Mike Green V2. Minor editing on my part.
The heat load in the hydrogen decreases along the cooling channel.
 At the end of the channel, the heat is load is expected to be as low as 
130W\footnote{This is due to beam losses}. 
Thus, the 16k helium flow rate for the 1.65-m-long cells should 
be set to about 15~g/s.  In all cases the helium will
be returned to the refrigerator cold box at around 19 K (including heating
in the return transfer line).
\subsection{Quench Protection}
The bunching section has twenty focusing magnets and twenty-one coupling magnets that have the same current in the
coils.  The number of cooling section cells where magnets carry the same
current is up to thirteen. Each magnet in the bunching and cooling sections has its own
leads.  The magnets can be
powered individually or in strings of magnets that carry the same current.
Powering magnets as a string of
magnets requires a more complicated quench-protection system that uses
diodes and resistors to cause the
string current to bypass the quenching magnet.  For sake of simplicity, each magnet has its
own power supply and quench protection system.  A 2500~A power supply for
charging and discharging a
single magnet coil (either a focusing coil or a coupling coil) should be capable of developing $\pm 7$~V.  The magnet
quench protection consists of a dump resistor across the magnet leads.
When a quench is detected, a fast
switch disconnects the power supply from the magnet.  In all cases, the
power supply control system
should permit control of the current and the voltage across the coils
as the magnet is charged and
discharged.  The power supply is not required to operate at both positive
and negative currents.  A
controller is used to control the charging and discharging voltages across
each coil and regulate the current
once the coil has reached its set current.

\subsection{Alignment}

The coupling coils can be aligned so that the solenoid axis is correct to 0.3~mrad.  The magnetic center of
the coupling coil can be maintained to about 0.3~mm.  The alignment of the focusing coils
can probably be maintained to
about 0.5 or 0.6~mrad.
%P.L, March 14-15 2001
% Still pending simulation studies to see if this statement 
% is justified. Should it be "in" or "out" radially? 
% I believe we could need a small dipole ( < 50 Gauss), but 
% it's coils can be located outside the cryostat, if space 
% is tight inside. It must cover a region where the longitudinal field
% does not flip sign. 
% Mike wrote:
%  Correction dipoles can be installed in the
% bore of the A coils that will permit
% the apparent solenoid axis to be corrected by $\pm $ 1.5 m-radians.
% P.L. would write:
  Correction dipoles could be installed if necessary, correcting the 
  apparent solenoid axis of the coils by  $\pm $ 1.5~mrad.
\subsection{Magnetic Field Outside the Solenoids}
% email from M. Green
The net magnetic moment of the cooling channel is essentially zero; consequently the field falls off quite rapidly away from the magnetic axis. Considering the long cell in isolation, the induction field at different distances from the solenoidal channel is given in Table~\ref{BC:tb-stray}
\begin{table}[!htb]
\begin{center}
\caption{Stray field at various distances from the axis of a long cooling cell.}
\label{BC:tb-stray}
\begin{tabular}{|cc|}
\hline
R & B\\
 (m) & (T)\\
\hline
$23\times 10^{-2}$ & 1.5\\
$11\times 10^{-2}$ & 2.0\\
$18\times 10^{-3}$ & 4.0\\
$56\times 10^{-5}$ & 6.0\\
$18\times 10^{-6}$ & 10.0\\
\hline
\end{tabular}
\end{center}
\end{table}
