\section{Introduction}
The Neutrino Factory~\cite{BC-mag4},~\cite{BC-mag2},~\cite{BC-mag3}, beyond approximately 18~m from the target,
requires three solenoid-based magnetic channels. In total, more than
530~m of solenoids with different magnetic strengths and bore sizes
are used in order to prepare the muon beam for injection into the muon
acceleration system. This is followed by the pre-acceleration linac with several different low-stray-field solenoids.
\section{Decay and Phase Rotation Channel Solenoids}
\label{DandPR:solenoid}
The decay and phase rotation region includes the muon decay channel,  
an induction linac (IL1), the mini-cooler and two additional induction linacs (IL2, IL3). This region extends from $z=18$~m (from the target) to $z=356$~m; within it, there are four types of solenoids.   
\begin{itemize}
\item From $z$ = 18 m to $z$ = 36 m, the decay section has a warm
bore diameter of 600~mm.  Around this warm bore is a water-cooled copper
shield that is 100 mm thick.
The solenoid cryostat warm bore is 800 mm.  The 18 m of decay solenoid
is divided into six cryostats, each 2.9-m long.  This same type of magnet is used for
the 9-m long mini-cooling
sections on either side of the field-flip solenoid.  As a result, 
there are twelve magnets of
this type.   
\item The IL1 solenoids, which extend from $z$ =
36 m to $z$ = 146 m, have a beam
aperture of 600 mm diameter.  Around the bore is a 10-mm-thick water-cooled
copper radiation shield.  The
warm bore of this magnet cryostat is 620~mm in diameter.  There are 110
magnets of this type.
\item The IL2 and IL3 solenoids, and the drift between them, extends
from $z$ = 166 m to $z$ = 356 m.
 These solenoids do not require a radiation shield and have a cryostat warm bore
diameter of 600 mm.  There are
190 magnets of this type.  
\item The field-flip solenoid
between the two mini-cooling sections
is 2.0-m long with a warm bore diameter of 400 mm.   There is only one
such magnet.
\end{itemize}
Table~\ref{fofoparams} shows the design parameters for all of these magnets; the last magnet, the 2-m long field-flip
solenoid is not included. Figure~\ref{CandPR:fg1} shows a cross section of the induction cell and
mini-cooling solenoids.
%{\flushleft \bf Baseline design of a beam window unit}
\begin{figure}
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/Phase_Rot1.eps}
\caption[Induction cell and mini-cooling solenoid]{Cross section of the induction cell and
mini-cooling solenoids.}
\label{CandPR:fg1}
\end{center}
\end{figure}
 
The basic requirements for the phase rotation solenoids, from  $z$ =
36 m through $z$ = 146 m and
from $z$ = 166 m through $z$ = 356 m, are as follows:  The magnetic induction
in the phase-rotation and
mini-cooling channel has been set to 1.25 T and the beam pipe
diameter is 600 mm.
The periodicity of the varying magnetic field on axis of the
phase rotation channel has been set to 0.5~m, to avoid potential particle losses due to resonances in the channel. This constraint 
requires the coils in the cell  to be of equal length with equal
length gaps between them.  A
1.0~m cell has two equal length coils and two equal
length spaces between coils,
yielding a period length for the magnetic field of 0.5~m.  The radial
thickness of the solenoid cryostat is  minimized to permit the induction linac structure to be
brought as close as possible to the axis of the
machine.  The distance of the induction cell from the magnetic field axis
is also influenced by the magnetic flux leakage through the gaps 
between the superconducting
coils. Therefore, the space between the
induction linac cells must be minimized; this means that the space used
for the cold mass support system,
the electrical leads, and the cryogenic feed system must be kept to a minimum.
 In addition, steering dipoles
are mounted on the inside of the solenoid coils.  The
pair of dipoles is 1.0~mm thick
and they can correct alignment errors up to 5 mrad.

Figure~\ref{CandPR:fg2} shows a cross section of a typical
superconducting solenoid designed
to generate an average induction of 1.25~T on the axis of the
phase rotation induction linac.  The inner bore
radius of the solenoid cryostat is 300~mm.  This allows a 200 MeV/$c$ muon beam
with a nominal diameter of
600~mm to pass through the solenoid without loss (except from muon decay).
The distance from the end
of the superconducting coil to the outside end of the cryostat is reduced
to 20~mm.  (If an additional
support clip were needed at the end of the coil, the coils can shortened to
accommodate the clip in the space
shown.)  The coils in the solenoid shown in Fig.~\ref{CandPR:fg2} have a length of 360~mm.  The gap between the coils is
140~mm and the space between a coil in one magnet and the coil in the next
magnet is also 140~mm.
\begin{figure}
\begin{center}
\includegraphics*[width=4.5in]{../template/report/ps-and-eps/Phase_Rot2.eps}
\caption[Cross section of the induction linac superconducting coil]{A cross 
section of the induction linac superconducting coil and cryostat.}
\label{CandPR:fg2}
\end{center}
\end{figure}

The conductor for the coils shown in Fig.~\ref{CandPR:fg2} is a standard MRI magnet
conductor that is 1 part
Nb-Ti and 4 parts RRR = 70~Cu.  This conductor has fifty-five $85\mu$m
filaments with a twist
pitch of 12.7~mm.  The bare matrix dimensions of the conductor are 0.955 mm
by 1.65 mm.  The
conductor insulation is 0.025-mm thick.  The coils are designed to be 6
layer, each one 9.6-mm thick,
including 2 mm of ground-plane insulation.  At an average design induction
of 1.25~T on axis, the coil
design current is 393~A.  The peak induction in the coil winding is
1.6~T, which gives a
coil operating temperature margin of over 2.5 K.

The coils can be wound and cast on a form that is removed
after the coil is cured.  After
curing, the coils are removed from the mold and machined at the ends and on
the outer radial surface.  After
the coil is machined, it can be shrunk fit into a 6061 Al support
structure that has been machined
so that the coils closely fit within it. The 6061 Al support
structure on the outside of the coils serves
the following functions:  1) it limits the coil strain by carrying some of
the magnet hoop forces; and 2) it
serves as a shorted secondary to protect the magnet during a quench.  A
single magnet is entirely 
self-protecting through quench-back from the support structure.  

The longitudinal space at the center of the magnet, 85~mm, is available for leads,
cryogenic services and cold mass
supports.  The cold mass of the phase-rotation solenoids
(including the 40 K shield and lower
lead assembly) is about 210~kg.  The largest forces that
will be seen between the cold mass
and room temperature will be forces due to shipping and forces introduced
due to unbalanced magnetic
fields.  The magnet cold-mass supports are designed for a force of 20,000~N
in any direction.
A pair of 60 mm diameter oriented carbon fiber tubes (with a
wall thickness of 3 mm) will be
used to carry forces from the cold mass to room temperature.

Since there is a solenoid magnet every meter down the phase rotation
channel and the drift spaces
between the phase rotation linac sections, leads must be brought out for
each of these magnets.  All of the
magnets in 25-m long sections are hooked in series and powered from a
common power supply.  Interconnects
between the solenoids use conventional copper cable.  A long
string of magnets can be run from a single
power supply in this case because the magnet coils are closely coupled inductively to
each other and to the support
structure.  

Quench-back is the primary mode of quench protection for the
string of magnets. We use
quench-back to protect a string of
these magnets as well.  When a quench is detected in one magnet, the current
in the string is discharged
through a varistor resistor, causing all coils to go normal through
quench-back from the support structure.    
Quench-back eliminates the forces between solenoids that would result when only one goes normal.  Each
1~m magnet section has
its own set of leads to room temperature.  The leads between 4 K and
50 K are made from high-temperature
superconductor (HTS).  The leads from room temperature to the
top of the HTS leads at 50 K
are gas cooled.  Gas from the refrigerator that is used to cool the
magnet shields and cold mass
support intercepts can also be used to cool the gas-cooled leads.  This gas must be returned to the refrigerator
compressor intake at room temperature. See Fig.~\ref{CandPR:fg3} for a schematic
representation of the cold mass
support system, the helium supply system, and the current leads.   The
cross section shown in Fig.~\ref{CandPR:fg3} is taken
at the center of the magnet along the magnet axis.
\begin{figure}
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/Phase_Rot3.eps}
\caption{Induction cell solenoid cold mass support system and leads.}
\label{CandPR:fg3}
\end{center}
\end{figure}

The solenoids for the decay channel and the mini-cooler section use basically the same magnet design as
those in the induction linac cells.  The primary difference is
the inside diameter of the
superconducting coil (858~mm bore versus 648~mm for the induction cell coils). Another difference is that three
coil modules share a single cryostat vacuum vessel.  Each module has its
own cold-mass support system,
but the three modules are hooked together using superconducting bus bars
cooled with two-phase helium.
There is a single set of external leads powering the modules in the magnet cryostat.
In the magnets that are next to
the flip region, individual power supplies are used to power the coils
to shape the magnetic field within the
flip region.

The solenoids for the flip region of the channel are the same as those used for the
decay channel, except that the warm
bore diameter of the magnet sections is set at 800 mm.  This diameter
should provide enough space for
the 1.75~m hydrogen absorbers that have window diameters of
600 mm.  It allows for a
50 to 70 mm space on the outside of the absorber for cooling of the
hydrogen within the absorber.  The
hydrogen absorbers will use helium coming from the refrigerator at 16 K.
 About 5,500~W of refrigeration at 16 K is needed to cool the
absorbers.  This is equivalent to
1,600~W of cooling at 4.4 K.  The cryogenic services to the hydrogen
absorber go through the 100~mm space between the magnet cryostats.  Additional room for services 
for the hydrogen absorber could be made available by
going through the magnet cryostat between magnet coil modules.

Figure~\ref{CandPR:fg4} shows a schematic representation of the solenoids in the
decay region.  All the solenoids in the decay
channel will be powered from the same
power supply. Field correction dipoles are mounted on the inside of the 
solenoid coil.
These coils are 1 mm thick and
can correct magnet alignment errors up to 5 mrad.
\begin{figure}
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/Phase_Rot4.eps}
\caption{A cross section of the solenoids in the pion decay channel.}
\label{CandPR:fg4}
\end{center}
\end{figure}


As noted earlier (see Section~\ref{DandPR:mini}), the decay region and the first induction cell 
solenoids are subject to
additional heat loads caused by
radiation emanating from the target.  Depending on the location, the
radiation heat loading is estimated to vary
between 2 and $20\mu$~W/g of cold mass.  For the well-shielded
decay solenoids, the maximum
heat leak per module is about 2.4 W.  However, the first magnet modules of IL1
may have
heating rates as high as 4.5 W per magnet module.  The additional heat load goes
down over an order of
magnitude farther down the channel.  Spent particles from the target
that remain beyond IL1
will be absorbed by the first hydrogen absorber of the
minicooling.  Half of
the radiation heat from the target is deposited in the magnet coils.  The
number of coil layers was increased to six in order
to maximize the magnet temperature margin where the magnet is subjected to
radiation heating.
The induction cell solenoids and the decay channel solenoids
are cooled by
conduction from the 6061 Al support structure.  The Al support
structure itself will be cooled by
two-phase helium flowing in attached tubes.
Two-phase helium cooling is
commonly used to cool large detector magnets; its advantages are: 
\begin{itemize}
\item{} there is very little helium inventory within the magnet
\item{} the two-phase helium tubes have a
high pressure rating;  this means that the magnet cryostat itself need not be 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 helium
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{itemize}

About twenty to twenty-five magnets are cooled in series
from the two-phase helium
refrigerator and control cryostat,  requiring a
mass-flow rate through
the flow circuit of about 2.5~g/s.  The two-phase
helium tube is attached to the
superconducting coil support structure, the base of the HTS leads and the
attachment points of the cold
mass supports.  The static heat load into a typical 1~m magnet cryostat at 
4.4 K and 40 K is summarized in Table~\ref{params_II}.

\begin{table} 
\renewcommand{\arraystretch}{0.9}
\begin{center}
\caption[Sources of heat in a 1~m long induction cell magnet]{Sources of heat at 4.4 K and 40 K in a 1-m induction cell magnet.}
\label{params_II}
\begin{tabular}{|l|c|c|}  
\hline 
Source of Heat &   4.4 K load  &  40 K load   \\
 &   (W)  &  (W)   \\
\hline
Heat flow down the cold mass supports & 0.12 & 1.9 \\
Thermal radiation through the multi-layer insulation & 0.05 & 2.0 \\
Heat flow down the helium bayonet joints & 0.03 & 1.3 \\
Heat flow down the cold mass supports & 0.12 & 1.9 \\
Heat flow down instrumentation wires  & 0.02 & 0.1 \\
Heat flow down the 400 A magnet current leads & 0.25 & --- \\
Heating due to ionizing radiation from the target & 0.0--4.5 & 0--0.5 \\
\hline
Total heat load per meter & 0.47--5.0  & 5.3--5.8 \\
\hline
\end{tabular}
\end{center}
\end{table}

%{\flushleft \bf Materials for the window tests}
The heat that is added to the two-phase helium flow stream at 4.4 K in each
meter of solenoid varies
from 0.45~W to 5.0~W, depending on the heat input due to
ionizing radiation from the
target.  The peak temperature at the inside of the superconducting
coil when the ionizing radiation heat
load is highest (a maximum value of 4.5 W in the first solenoid of IL1)
will be less than 5.5 K.  Except for heat from ionizing radiation from the
target, the heat load at 4.4 K is
dominated by the heat leak down the HTS current leads.
The heat load into the shield circuit stream is expected to vary from 5.3 W
to 5.8 W, again depending on the
heat input due to the ionizing radiation from the target.  The
shield gas comes from the refrigerator
at a temperature between 30 and 35 K.  (In the region of the mini-cooler,
the shield cooling gas comes from
the refrigerator and will enter at a temperature of 16 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 400 A gas-cooled leads require
need 0.05~g/s.
The shield gas stream picks up heat from the cold mass supports and from 
thermal radiation on the shield,
helium bayonet joints, and the instrumentation wires.  In most of the
induction cells, the expected heat load
into this stream is about 5.3 W/m.  In the first cells of IL1,
the shield
circuit may pick up as much as 5.8 W. (The extra 0.5 W is due to ionizing
radiation from the target heating
the 40 K shields.)   The helium stream temperature entering the shield
circuit from the refrigerator increases
from 22 to 24 K as it flows to the base of the gas-cooled leads.  The gas
used to cool the shields and the
cold mass support intercepts is also used to cool the gas-cooled leads
between about 60 K and room
temperature.  The HTS leads are
designed to operate with their top
end temperature below 70 K.  The gas exiting the room temperature end of
the gas-cooled leads returns
warm to the refrigerator compressor suction.  Fig.~\ref{CandPR:fg5} shows the
proposed two-phase helium
cooling system for a typical 1-m long phase-rotation
solenoid.   The refrigerator and
cryogenic distribution system for the decay solenoids, induction cell
solenoids, and the mini-cooling
section solenoids are described in Chapter~\ref{CHAP:cryo}.

\begin{figure}
\begin{center}
\includegraphics*[width=5in]{../template/report/ps-and-eps/Phase_Rot5.eps}
\caption{Cryogenic cooling system for a typical induction cell.}
\label{CandPR:fg5}
\end{center}
\end{figure}


A string of twenty-five induction cell solenoids has a self-inductance of
68.5 to 72.5~H, depending on
which induction linac they are in.  A string of six 3-m solenoids
for the decay channel and the
minicooler will have a self-inductance of 85.8~H.  The magnets in any of
the strings can be charged to full
field in less than 1,800~s, by a power supply that delivers 500~A at
voltages up to 50~V.  The power
supply controllers regulate on the voltage across the magnet during
magnet charging.  When the
magnets are charged, the current must be kept constant.
Because the magnets are closely coupled, a quench in one magnet of the
string will trigger quenches in
all of the magnets in that string.  A simple resistor across the magnet
string leads can protect the entire
string during a magnet quench.  A typical magnet quench will raise the
temperature of the magnet cold
mass to about 45 K.  The helium refrigerator is sized to allow the
magnet to be cooled back down to
4.4 K in less than two hours.  To meet this requirement,
the helium
refrigeration system must deliver helium at 10 to 15 K to the magnets at
rate of 4.5 g/s.
\subsection{Stray Fields}
The ferrite cores will carry much of the return flux; what they do not, will be 
carried in the air between the ferrite cores and the superconducting coils. 
The induction field at $R=1.5$~m is below $\approx$ 0.05~T and at $R = 2$~m 
it will be $\approx 0.025$~T.

