Design of a compact, cryogen-free superconducting solenoid for the electron lens of the Fermilab Integrable Optics Test Accelerator (IOTA)

The proposed electron lens for Fermilab's Integrable Optics Test Accelerator (IOTA) will broaden its capabilities by enabling new research in nonlinear integrable optics, space-charge compensation, proton beam cooling, and more. The electron lens is based on a 5–10 keV, 1–2 A electron beam, shaped using a 0.7 m long, 0.8 T solenoidal magnetic field. A cryogen-free superconducting solenoid has been designed to provide this solenoidal field, taking into consideration the constraints on space, utilities, and infrastructure in the IOTA experimental hall. The solenoid is made of copper stabilized niobium-titanium conductor, conduction-cooled using 4 K closed-cycle cryocoolers. This paper describes the overall design of the solenoid encompassing its mechanical construction, current leads optimization, cryogenic thermal modeling that provides estimates of cooldown time and static/dynamic heat loads, and quench analysis.


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: Acceleration cavities and superconducting magnets (high-temperature superconductor; radiation hardened magnets; normal-conducting; permanent magnet devices; wigglers and undulators); Cryogenics and thermal models; Accelerator Subsystems and Technologies; Cryocoolers

Introduction
The Integrable Optics Test Accelerator (IOTA) is a 40 m circumference storage ring at Fermilab, dedicated to beam physics, accelerator technology, and education [1]. A new electron lens is being built for IOTA. This lens will serve as a unique test bed for studying novel applications such as nonlinear integrable optics, space-charge compensation, generation of Landau damping, cooling of proton beams to reduce their emittance, and more. The electron lens will be made of a low-energy electron beam (5-10 keV, 1-2 A) generated by a thermionic gun and transported by solenoids and a toroidal section into the interaction region. The interaction region of the electron lens has a 0.8 T Main Solenoid with a warm bore enclosing a meter-long section of the main IOTA proton beam line. Figure 1 schematically shows the layout of the electron lens comprising of the Main Solenoid, an electron source and a collector, gun and collector solenoids, transport solenoids, and toroidal sections. Further details of the IOTA electron lens including development status and research plans are elaborated in refs. [1][2][3].
Electron lenses previously built for the Tevatron collider at Fermilab [4] and RHIC at Brookhaven National Laboratory [5], as well as the ones planned for LHC at CERN [6], require 5-7 T solenoidal fields. They use solenoids made of superconducting NbTi conductor cooled with liquid helium at ∼ 4.5 K. For the 0.6 T electron-lens solenoid in SIS18 at GSI, on the other hand, a resistive magnet is being considered [7]. The IOTA electron lens Main Solenoid requires a field of 0.8 T. A superconducting option implies that the solenoid will be physically smaller than those for the Tevatron, RHIC and LHC, with less stored energy, looser quench protection requirements, and less stringent cryogenic cooling demands. A design study has been carried out using copper stabilized NbTi conductor, where the coil is conduction-cooled using 4 K cryocoolers. This paper presents electromagnetic, mechanical, and cryogenic aspects of the design study. To our knowledge, this is the first instance where an electron lens employs a cryogen-free superconducting magnet.

Main Solenoid design 2.1 Requirements and constraints
The electron lens requires a solenoidal field of 0.8 T in a 'good field' region of at least 600 mm in length and 30 mm in diameter. Field homogeneity, / ( is the radial field), in the good field region should be < 2 × 10 −4 . The Main Solenoid assembly should possess a bore of diameter of at least 50 mm to accommodate the IOTA beam pipe, plus clearance for flanges, beam position monitors, feedthroughs, and other instrumentation.
Although the field and bore size requirements are modest, the existing IOTA beam line infrastructure puts space and weight constraints on the new electron lens. The Main Solenoid should preferably fit in a total longitudinal space of 1.4 m, so that the existing devices upstream and downstream (IOTA quadrupoles and related instrumentation) are not displaced (see figure 1). The lifting device in the IOTA experimental hall limits individual component weight to approximately 10 kN when lifting for assembly. We further note that although the IOTA hall does not have specific -2 -constraints on utilities such as cooling water and electrical power, efforts must be made to keep additional installations to a minimum.
The constraint on the possible use of a superconducting solenoid is that the IOTA experimental hall, although adjacent to the FAST superconducting linac, does not have readily available cryogen supply and recovery facilities. A self-contained, dry cryocooling system must therefore be used. The solenoid should cool down to its base operating temperature within 24 hours so that the electron-lens downtime between successive experimental runs is reasonable. The cryocooling scheme should also be able to cool the solenoid to its base temperature within a few hours after recovering from a quench, in order to have a manageable impact on operations.

Evaluation of design options
Based primarily on the requirements of field and good field region, we evaluated resistive and superconducting options for the Main Solenoid. The resistive design uses a 20 mm × 20 mm copper conductor with 15 mm diameter hole for cooling water. A copper stabilized NbTi conductor of size 1.25 mm × 0.8 mm is selected for the superconducting option. This cable was used on a similar magnet designed at CERN for the AEGIS experiment [8]. Table 1 lists the design input parameters, namely coil turns, coil inner and outer diameters, and the operating current required to generate 0.8 T. The design output parameters, viz. the stored energy, coil weight, electrical and cooling requirements are also compared in table 1. The superconducting solenoid outperforms the resistive solenoid in all of the criteria compared in table 1, mainly because superconductors can carry significantly higher current density (> 500 A/mm 2 for the chosen NbTi conductor) than resistive copper (typically limited to about 10 A/mm 2 , Wilson : 1983). Higher current density implies a smaller, lighter coil as well as a smaller operating current. For the present case, the superconducting coil has (a) five-fold smaller operating current, (b) three-fold smaller outer diameter (size of the coil), (c) three-fold smaller stored energy at 0.8 T, and (d) almost forty-fold smaller weight. The moderate size, weight, and stored energy in -3 -the superconducting option makes it possible to use conduction cooling based on compact, closedcycle 4 K cryocoolers. The total electrical power requirement for the superconducting solenoid is 28 kW assuming two Cryomech PT415 cryocoolers, which is smaller than the 41 kW needed for the resistive option. Cryocooler technology is mature, with demonstrated reliability in medical, industrial and research applications. The higher cost of a superconducting solution is also manageable. Low-frequency vibrations generated by the cryocoolers can be mitigated with mechanical dampers. Because of these considerations, the conduction-cooled superconducting option is clearly favored for the IOTA electron lens.

Mechanical design
The Main Solenoid uses commercial NbTi conductor manufactured by Luvata with its parameters listed in table 2. The solenoid has one main coil that generates the 0.8 T solenoidal field. Four corrector dipole coils (two short coils upstream and two longer ones downstream) are used to adjust position and angle of the electrons in the lens and to control the overlap with the circulating beam. Additional trim layers are added at the two ends of the main coil to sustain the 0.8 T axial field at the edges. The main coil and the two trim coils are electrically connected in series. Table 3 shows parameters of the main coil and trim coils as well as the solenoidal fields produced by them when operated with a current of 200 A.  Figure 2 shows a computer model of the Main Solenoid assembly. The main coil and the trim coils are first wound on the outer surface of an aluminum former, which also acts as a quench protection layer for the coils. After winding, the main and trim coils are impregnated in epoxy to prevent winding layer movement. A second cylindrical aluminum shell, called the main coil support, is added around the epoxy impregnated coils. The main coil support provides for attachment of the coil -4 -hangers. Outside of the main coil support are the corrector coil and its support made of OFHC copper. The corrector coil support has slots to reduce its weight. Aluminum strain-relief posts for each of the ten current leads of the five coils are mounted on the main coil support, as depicted in figure 2.

Main coil support
Quench protection layer  This does not satisfy the good field region requirement defined in section 2.1. Several options are being actively considered for lengthening the good field region beyond 0.6 m and reducing the field homogeneity below 0.02%. These include increasing the solenoid length, reducing the inner diameter, and inserting ferromagnetic plugs with 30 mm diameter holes at the coil edges.

Cryostat design
Due to the absence of liquid cryogen supply and recovery facilities in the IOTA experimental hall, a dry 4 K cryo-cooling scheme is proposed for the Main Solenoid. The dry, cryogen-free cooling system eliminates the piping required to circulate liquid/gaseous cryogens as well as a cryo-liquid bath that would otherwise be needed to cool the solenoid. Furthermore, the absence of cryogenic liquids and/or compressed gases eliminates the possibility of over-pressurization of the cryostat due to accidental scenarios such as a sudden loss of insulating vacuum or magnet quench. The cryogenfree cooling scheme thereby offers a compact cryostat with relaxed safety, operational procedures, and utility requirements.

Main features of the cryostat
The Main Solenoid cryostat, depicted in figure 4, has the standard three-temperature level cooling scheme enabled by two-stage 4 K cryocoolers: 1. A vacuum vessel made of stainless steel SS304 operating at room temperature and enclosing all of the cold mass. The vessel is pumped to a vacuum < 10 −6 torr so that convective heat load to the cold mass inside is restricted to the level of residual gas conduction [10]. 2. A thermal radiation shield made of aluminum alloy Al6061-T6 enclosing the Main Solenoid assembly and conduction cooled to the cryocooler stage 1 inside the vacuum vessel. The thermal shield is wrapped with multi-layer insulation to reduce the radiation heat load on cryocooler stage 1. 3. The Main Solenoid assembly conductively connected to the cryocooler stage 2 and enclosed by the thermal radiation shield.

Current leads
The five pairs of current leads are arranged inside the cryostat, as depicted in figure 5. The composite leads include phosphor bronze sections (resistive leads) between room temperature and cryocooler stage 1 followed by HTS-110 Cryosaver TM sections (BSSCO HTS leads [11]) between cryocooler stage 1 and stage 2. The colder ends of the HTS leads are connected to the coil NbTi terminations coming from the coils. During solenoid operation, the heat flow to cryocooler stage 1 through the resistive leads comprises of ambient heat conduction and Joule dissipation in the leads. The resistive leads are designed with optimal length to cross section area ratio, / CS , to minimize the total heat load to cryocooler stage 1. The optimization was done by numerically solving McFee's equation [12]:  where is operating current, and are warm end and cold end temperatures of the resistive leads, is the electrical conductivity, is the thermal conductivity, and ,min is the total heat load at , given by: (3.2) Figure 6 summarizes the results of the resistive leads optimization routine. The parameters = 200 A, = 300 K, and = 50 K result in ,min = 10 W and optimal / CS = 2.8 m −1 . The optimization used temperature dependent and for phosphor bronze taken from NIST Monograph 177 [13], assuming 5% tin content by weight in the alloy. The curve in figure 6 is the temperature -7 -profile along the length of the lead assuming 170 mm length and 61 mm 2 cross section. We note that the optimized current lead has an adiabatic boundary = 0 at , which is a necessary condition for minimizing the total heat load at through the lead [12]. Commercial off-the-shelf HTS-110 CryoSaver TM current leads [11] are deployed between cryocooler stage 1 and stage 2. The HTS lead uses a BSSCO superconducting wire encased in a fiberglass composite body for structural integrity. The lead terminations have flat nickel plated copper endcaps with bolt holes. The selected HTS lead has current rating of 250 A at 64 K, which exceeds the design current of 200 A at the expected lead operating temperature of < 50 K.

Cryocooler conduction cooling
The conduction cooling arrangement for the current leads and the solenoid assembly is depicted in figure 5. Foil-shaped straps made of copper are used for conduction cooling. An OFHC copper strap connects stage 1 of the cryocooler to the interconnect of the resistive and HTS current leads. A sapphire washer is placed between the strap and each interconnect to electrically isolate the strap from the current leads while ensuring good thermal conductance across the contact. The washers are supplied by HTS-110 and a single washer is usable with interconnects carrying up to 1 kA. A similar OFHC copper foil strap is deployed between the cryocooler stage 2 and interconnects of the HTS-NbTi leads.
The solenoid coil assembly is connected to the cryocooler stage 2 using larger foil straps made of high purity copper (residual resistivity ratio ∼500). Compared to OFHC, this copper grade possesses much higher thermal conductivity below ∼ 40 K, which helps to reduce the cooldown time of the solenoid assembly to its base temperature of 4 K. Near the end of the coil assembly, the foil strap branches out into three sections that are bolted individually to the main coil, main coil support, and corrector coil support. The other termination of the strap is bolted to the cryocooler stage 2.  The model requires temperature dependent thermal conductivities, specific heat capacities, and emissivities of the cryostat components for solving the transient heat diffusion equation. The thermal conductivities and specific heat capacities for copper, aluminum alloy, G10, Stycast 2850, and stainless steel are taken from the NIST database [14] while those for NbTi are taken from [15]. The material emissivities listed in table 5 are from ref. [16].
For bolted thermal contacts, i.e. at the two ends of the coil cooling link (see figure 5), thermal conductance is derived from the experimental data published in [17]. Figure 7a shows this data for temperatures in the range 3-300 K. Temperature dependent cooling capacity of the Cryomech PT415 cryocooler is displayed in figure 7b. The solid lines are least-squares fits to the data from ref. [18].
The thermal model uses the following assumptions: 1. Thermal shield and magnet assembly are considered as lumped masses. Spatial temperature gradients are neglected so that the cooldown time may be reasonably estimated without going into the complexity of spatial dimensions.
-9 -  2. Thermal masses of current leads are evaluated in the following way: (a) Resistive current leads at the temperature of cryocooler stage 1 (b) HTS current leads at the temperature of cryocooler stage 2. 3. Cryocooler stages 1 and 2 cool independently -the cooling capacity of one does not depend on the temperature of the other. 4. Thermal contact conductances between the following are neglected as they are not expected to significantly influence the magnet assembly cooldown rate: (a) thermal shield and cryocooler stage 1 (b) resistive current leads and cryocooler stage 1 (c) HTS current leads and cryocooler stage 1 (d) HTS current leads and cryocooler stage 2. 5. Thermal contact resistance is considered for the following contacts: (a) magnet assembly and its thermal link (b) the magnet assembly thermal link and cryocooler stage 2. 6. Thermal shield and magnet assembly are wrapped with 30 layers of multilayer insulation (MLI).
-10 - Figure 8 represents the principal heat transfer channels during the cooldown and operation of the solenoid. These channels, which include conduction and radiation heat transfer, are shown as a thermal resistance network. The tables in figure 8 define the temperature nodes of interest (where temperature is to be calculated) and the heat transfer channel corresponding to each thermal resistance. Since residual gas conduction is negligible due to high vacuum in the cryostat, the heat transfer modes are conduction through solid elements, surface to surface radiation, and heat flow across bolted contacts. The modes are mathematically defined as follows: 1. Conduction heat transfer conduction between two nodes at temperatures 1 and 2 : 2. Surface-to-surface radiation radiation : where is the Stefan-Boltzmann constant, and are the surface area and the emissivity of surface , and MLI is the number of MLI blankets between the two surfaces. 3. The rate of heat transfer across a bolted contact, contact , with thermal conductance contact : (3.5) An energy balance applied to each of the six temperature nodes defined in figure 8 leads to the system of equations: where the left-hand side represents the rate of change of internal energy at each node and the right-hand side denotes the net rate of heat flow through the node (total heat in minus total heat out). The system of equations (3.6) is a coupled system of six ordinary differential equations, where the heat transfer rates are strongly dependent on the unknown temperatures. The system is solved using the odeint library available with scientific Python.
The thermal model results are summarized in figure 9 and tables 6 and 7. With the existing configuration, the solenoid assembly takes 21 h to cool down from room temperature to 4 K. The cooldown rate is primarily governed by the total thermal conductance of the coil cooling link, which comprises two thermal contacts (link-coil and link-cryocooler) and bulk conductance of the cooling link. Figure 9a shows that the temperature drop across the link ( LD − LU ) is much greater than that across the contacts ( M − LD and LU − CC2 ), indicating that the bulk imposes the most thermal resistance in the cooling link. The bulk thermal resistance can be reduced by increasing the cross section of the coil cooling link. For instance, figure 9b shows the cooldown curves that would results from a link with twice the cross section. The solenoid assembly in this case cools down in 18 h. For even faster cooldown, the cooling link thermal resistance can be further reduced by increasing its cross section or by reducing the contact resistance by applying thermal interface materials (for example, indium foil) at the cooling link bolted contacts. In any case, cooldown times below 24 h are considered acceptable. Table 6 and table 7 list the steady-state temperatures and heat loads on the Main Solenoid assembly. Two cases are considered: when the magnet is not powered, = 0 A, and when the magnet is fully powered, = 200 A. In both the cases, the temperatures are < 4.5 K, which is suitable for solenoid operation. Furthermore, the heat loads are well within the cooling budget of the PT415 cryocoolers.

Quench analysis
A quench can result from a part of the superconducting coil warming above NbTi superconducting transition temperature. Following a quench, NbTi will turn resistive and a large portion of the circulating current will transfer to the copper stabilizer in the conductor. The current (and stored energy) will then dissipate in the coil resistance causing the coil (a) to further warm up and (b) to induce internal voltage. Determining the peak coil temperature and peak coil voltage following a quench are two goals of the present quench analysis. We note that the low operating current of 200 A -12 -2021 JINST 16 T03009   Quench parameters are calculated using OPERA 3D QUENCH code following the conservative adiabatic coil heating approach [9,19,20]. This approach assumes that immediately after a quench the current supply turns off and an external dump resistor comes in series with the coil. These processes can be activated by a quench protection system. The circulating current (and stored energy) then dissipates in the coil and the dump resistor. The present simulations use two values of the dump resistor, 0.5 Ω and 10 mΩ along with temperature dependent thermal conductivity and -14 -

JINST 16 T03009
heat capacity of the conductor. The quench is initiated in the middle of the solenoid by a short heater pulse that locally warms the coil. Figure 10 presents the time profiles of coil current, temperature, and voltage following the simulated quench. The transients in all the three parameters decay within one second of the quench initiation. With the smaller dump resistor the current decay takes longer and the coil gets to warmer temperature. However, the coil peak voltage is smaller. The choice of the dump resistor thus involves a trade-off between the peak coil temperature and peak coil voltage. Nonetheless, both upper values of peak coil voltage (320 V) and peak coil temperature (42 K) are reasonable for epoxy impregnated coils such as the one being designed for the Main Solenoid. Figure 10d depicts the 3D temperature profile in the coil with a 10 mΩ dump resistor, 2 s after quench. This is well beyond the timescale of the transients and so can be considered as a steady state. The temperature profile in the coil shows that only a small part of the coil actually warms up to 45 K, while a large section still is at the base temperature of 4.5 K. This indicates that after recovery from the quench the coil can be cooled back to its base temperature within a reasonably short time with cryocooler conduction cooling.
The present quench analysis neglects the effects of eddy currents induced in the aluminum quench protection layer of the coil assembly (see figure 2). These eddy currents are expected to further lower the peak quench parameters in the coil [21].

Summary and outlook
We presented the first results of the design of a superconducting solenoid for the IOTA electron lens at Fermilab. The design is based on commercially available NbTi cable and on 4 K pulse tube cryocoolers for conduction cooling. The solenoid assembly is compact, lightweight, and more energy efficient compared to a resistive solenoid. A detailed thermal model shows that the static and dynamic heat loads in the solenoid are well within the cooling capacity of the selected cryocoolers. The thermal model further predicts that the solenoid assembly would cool down to its operating temperature in less than a day. The quench analysis using an external dump resistor shows that the coil would be thermally stable during a quench event and be able to cool back down in a few hours after recovering from the quench.
Future work includes refining the electromagnetic design to satisfy the field quality requirements and a detailed engineering design, followed by manufacture, testing, and commissioning in the IOTA storage ring.