\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.