\section{Linear Accelerator Solenoids}
The requirement of a large acceptance for the pre-accelerator linac requires large apertures and strong focusing in both planes. Clearly, solenoids are superior to quadrupole triplets (see Chapter~\ref{ACCE:Chapter}).

The present design contains several different solenoid magnets. The matching section has a pair of low-stray-field solenoids with adjustable currents. The short and intermediate cryomodules have a 1~m solenoid and the long one has a 1.5~m solenoid. 

Solenoids produce  stray fields that have adverse effects on the superconducting rf cavities; therefore, a very important design feature of the solenoids is the need to eliminate the stray fields. The solenoids satisfy the following conditions:   
\begin{enumerate}
\item are designed to produce zero net magnetic moment.
  This means that the coil that produces the solenoidal field is
  bucked by a coil or coils that are larger in diameter.
\item The field from the bucking coils is be distributed in the
  same way as the solenoid field.  This suggests that the bucking
  solenoid be around the focusing solenoid so that the return flux
  from the focusing solenoid is returned between the focusing solenoid
  and the bucking solenoid.
\item The solenoid pair is surrounded by iron, except where the
  muon beam passes through it.
\item An iron flux shield is installed between the solenoid magnet package 
and the rf cavity cells.
\item The superconducting rf cells nearest the focusing solenoid are 
 covered with a type 2 superconducting shield.  This will
  not shield the earth's magnetic field, but it will shield the
  remaining stray flux from a nearby solenoid. A superconducting
  shield was used to shield the stray field from a superconducting
  inflector magnet that is located within the good field region (good
  to better than 1 part in a million) of the $g-2$ experiment at
  BNL.
\end{enumerate}
It is unlikely that all five steps will be needed to sufficiently reduce 
the stray
field in the rf cavities arising from the adjacent solenoid.  The linac solenoids 
are designed to have  bucking solenoid coils on the outside of the main solenoid.  The bucking coil is the same
length as the main solenoid, and its radius and current are set
so that the solenoid pair produces zero net magnetic moment.  In order
for a solenoid of average radius $R_1$ with a total current $I_1$
to have zero net magnetic moment, a bucking coil of radius $R_2$  larger than 
 R$_1,$ must be around it.  The
total current of the bucking solenoid $I_2$ can be calculated using the
expression
\begin{equation}
  I_2 = -I_1\dfrac{R_1^2}{R_2^2}.
\end{equation}
If the coils in both the outer and the inner solenoids in a system of
solenoids with zero net magnetic moment are evenly distributed, the
induction generated at the center of the nested solenoid pair will be
given by
\begin{equation}
  B_0=\dfrac{\mu_0I}{L}(n_1\cos\beta_1-n_2\cos\beta_2),
\end{equation}
where
\begin{align}
  \beta_1&=\tan^{-1}\left(\dfrac{2R_1}{L}\right)&
  \beta_2&=\tan^{-1}\left(\dfrac{2R_2}{L}\right),
\end{align}
$I$ is the current in the solenoid pair, $L$ is the length of the
nested solenoid pair, $n_1$ is the number of turns in the inner
focusing solenoid, and $n_2$ is the number of turns in the outer
bucking solenoid.  Because of the zero net magnetic moment condition,
$R_2=(n_1/n_2)^{1/2}R_1$ with $I_1=n_1I$ and $I_2=n_2I$ .

\begin{table}[!tb]
  \caption{Superconducting solenoid parameters for the linear accelerator.}
  \label{tab:acc:solparm}
  \begin{center}
    \begin{tabular}{|lccc|}
      \hline
      &Short&Intermediate&Long\\
      \hline
      \multicolumn{4}{|c|}{Mechanical Parameters}\\
      \hline
      Beam bore diameter (mm)&460&460&300\\
      Solenoid cryostat length (mm)&1260&1260&1710\\
      Solenoid cryostat outer diameter (mm)&1180&1180&1060\\
      Iron shell length (mm)&1300&1300&1750\\
      Iron shell outer diameter (mm)&1240&1240&1120\\
      Iron shell thickness (mm)&9.5&9.5&9.5\\
      Coil length for both coils (mm)&1000&1000&1500\\
      Inner coil average radius (mm)&254&254&182\\
      Inner coil thickness (mm)&10.4&10.4&31.2\\
      Number of inner coil layers&8&8&24\\
      Number of inner coil turns&4840&4840&21816\\
      Outer coil average radius (mm)&520.6&520.6&453.6\\
      Outer coil thickness (mm)&2.6&2.6&5.2\\
      Outer coil center gap (mm)&50&50&50\\
      Number of outer coil layers&2&2&4\\
      Number of outer coil turns&576&576&3512\\
      Solenoid cold mass (kg)&376&376&746\\
      Solenoid cryostat mass (kg)&166&166&238\\
      Iron shell mass (kg)&485&485&581\\
      \hline
      \multicolumn{4}{|c|}{Magnetic and Electrical Parameters}\\
      \hline
      Solenoid average magnetic induction (T)&2.1&2.1&4.2\\
      Solenoid magnetic length (m)&$\approx1.0$&$\approx1.0$&$\approx1.5$\\
      Magnet design current (A)&469.6&469.6&274.0\\
      Peak induction in the inner coil $B_p$ (T)&
      $\approx2.9$&$\approx2.9$&$\approx5.8$\\
      Magnet conductor $I_c$ at 4.4 K and $B_p$ (A)&
      $\approx1100$&$\approx1100$&$\approx590$\\
      SC current density (A mm$^{-2}$)&307&307&180\\
      Solenoid stored energy (MJ)&0.421&0.421&1.306\\
      Solenoid self inductance (H)&3.82&3.82&34.8\\
      $EJ^2$ limit (A$^2$m$^{-4}$J)&
      $3.97\times10^{22}$&$3.97\times10^{22}$&$4.23\times10^{22}$\\
      \hline
    \end{tabular}
  \end{center}
\end{table}
Table~\ref{tab:acc:solparm} presents the mechanical and electrical
parameters for the short and long module focusing solenoids. The
matching solenoids at the start of the channel are similar to the
first focusing solenoids.  All of these solenoids are designed to have
zero net magnetic moment.  The solenoids in
Table~\ref{tab:acc:solparm} are assumed to have a warm bore and a warm
iron shell around the solenoid pair.  It should be noted that the
magnet bore does not have to be warm.  A cold bore solenoid will be
somewhat smaller and the bore can be a cryopump for the beam vacuum.
The iron shield around the magnet pair does not have to be warm either, as
long as it does not carry large forces.  The inner coils and the outer
coils of the solenoid in Table~\ref{tab:acc:solparm} have an even
number of layers.  This allows the solenoid leads to be brought out
together at one end.  The outer solenoid is split with a 50~mm gap
between the two coils.  This allows the leads and helium cooling tube
for the inner solenoid to be brought out through the outer solenoid.
Electrical connections and helium into the magnet can be brought in at
the center of the solenoid, thus minimizing the stray field that might
be produced at or near the connection point.  The solenoid pair is
assumed to be supplied with current through a single set of high
temperature superconductor (HTS) and gas-cooled electrical leads.
Since the nested magnets are hooked in series, the focusing solenoids
have zero net magnetic moment at all magnet currents.

\begin{figure}[!tbp]
  \centering\includegraphics{../template/report/ps-and-eps/accel-green-010410-f1.eps}
  \caption{A cross section, parallel to the magnetic axis, of a short 1-m solenoid.}
  \label{fig:acc:magside}
\end{figure}
Figure~\ref{fig:acc:magside} shows a cross section of the short
solenoid (1.0~m long with 2.1~T in the inner bore) in a plane
that contains the magnetic axis; it also shows
the separation of the inner coil and the bucking coil, as well as the magnet cryostat, an electrical lead, and the iron shield around
the actively shielded solenoid.  The center of the cryostat has no
iron shield around it because there is very little magnetic flux
leaking outside the bucking solenoid.  Not shown in
Figure~\ref{fig:acc:magside} is iron flux shield that is about 300~mm
from the end of the magnet cryostat.  This shield further reduces the
field in the rf cavity.

\begin{figure}[!tb]
  \centering\includegraphics[scale=0.95]{../template/report/ps-and-eps/accel-green-010410-f2.eps}
  \caption[A cross section of the long 1.5~m solenoid]{A cross section, perpendicular to the magnetic axis, of the long 1.5-m solenoid.}
  \label{fig:acc:magend}
\end{figure}
Figure~\ref{fig:acc:magend} shows a cross section of the long focusing
solenoid (1.5~m long with 4.2~T in the inner bore) in a plane
 perpendicular to the solenoid axis.  Figure~\ref{fig:acc:magend} shows
the 24~layer inner coil and a 4~layer bucking coil; it also shows a cold mass support system that can
be used for both types of focusing solenoids.  The cold mass support
system carries predominantly gravitational loading during magnet
operation.  The support system is designed to carry shipping loads due
to acceleration generated by the truck.  Because the focusing
solenoids are decoupled magnetically from each other, there are no
loads imposed on the solenoid by nearby magnets.  Also shown in
Fig.~\ref{fig:acc:magend} are the magnet current leads and some of
the 4.4~K and 40~K helium plumbing for the magnet.
Figures~\ref{fig:acc:magside} and \ref{fig:acc:magend} represent
typical cross sections that can be applied to both types of focusing
solenoids.

\begin{table}[!tb]
  \caption{The sources of heat at 4.4~K and 40~K in a 1.5~m long
    focusing solenoid.}
  \label{tab:acc:magheat}
  \begin{center}
    \begin{tabular}{|l|c|c|}
      \hline
      &4.4 K load&40 K load\\
      Source of heat&(W)&(W)\\
      \hline
      Heat flow down the cold mass supports&0.12&1.9\\
      Thermal radiation through the multi-layer insulation&0.10&4.0\\
      Heat flow down the helium bayonet joints&0.03&1.3\\
      Heat flow down instrumentation wires&0.02&0.1\\
      Heat flow down the 280 A magnet current leads&0.45&---\\
      \hline
      Total heat load per magnet&0.72&7.3\\
      \hline
    \end{tabular}
  \end{center}
\end{table}
The focusing solenoids are cooled by conduction from the 6061-aluminum
support structure.  The aluminum support structure will be cooled by
two-phase helium flowing in tubes attached to it.  Two-phase helium
cooling is commonly used to cool large detector magnets.  The
advantages of two-phase tubular cooling are as follows:
\begin{enumerate} 
\item there is
very little helium inventory within the magnet 
\item the tubes carrying
the two-phase helium have a high-pressure rating.  This means that the
magnet cryostat is not a pressure vessel 
\item two-phase helium cooling
does not require a cold compressor or a helium pump to circulate the
helium through the magnet cooling system 
\item the temperature of the
helium in a two-phase cooling circuit decreases as it moves
along the flow circuit 
\item the pressure drop along a two-phase helium flow circuit is lower than for a supercritical helium forced flow
circuit
\end{enumerate}
  The static heat load into the magnet cryostat at 4.4~K and
40~K for the 1.5-m-long focusing solenoid is shown in
Table~\ref{tab:acc:magheat}.  The 4.4~K heat load into a short
solenoid is estimated to be about 0.50~W.  Most of the difference is
heat flow down the HTS leads.

All of the magnets in an acceleration section are
cooled in series from the two-phase helium refrigerator and control
cryostat.  Whether this refrigerator is the same one that cools the
superconducting RF cavities depends on the operating temperature of
the rf cavities.  Cooling for 23 or 24 magnets requires a mass flow
rate through the two-phase 4.4~K flow circuit of about 2.5~g/s.
  The two-phase helium tubes would be attached to the
inner coil support structure, the outer coil, the attachment points of
the cold-mass supports, and the base of the HTS leads.

The heat load into the shield circuit helium stream is expected to
vary from 6.1 to 7.3~W, depending on the length of the magnet.  The
shield gas comes from the refrigerator at a temperature of 30~K. This
gas enters the magnet cryostat through a single vacuum insulated tube.
The helium flow in this tube is dictated by the needs of the
gas-cooled leads between 50~K and room temperature.  The mass flow
through the shield circuit is governed by the needs of the gas-cooled
leads.  The short solenoid leads will need 0.05~g/s; the
long solenoid leads will need 0.035~g/s.  Gas exits
from the gas-cooled electrical leads at room temperature.  It returns
warm to the refrigerator compressor suction.  In the short solenoid,
the shield gas enters the gas-cooled leads at about 55~K.  In the long
solenoid, the top of the HTS leads will be about 70~K.  The same HTS
leads can be used for both magnets.

For sake of simplicity, each magnet has its own power
supply and quench protection system.  A 500-A power supply can be used
for charging and discharging a single magnet at $\pm5$~V.  The charge
time with 3~V across the short magnet is about 600~s.  The long
focusing solenoid will take about 3200~s to charge with 3~V
across 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.  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.  Both coils in
the magnet go normal through quench-back.

The focusing solenoids can be aligned so that the solenoid axis is
correctly placed to about 0.5~mrad.  The magnetic center of the
B coil can also be maintained to about 0.3~mm.  
