\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.