PRELIMINARY REPORT ON KEY WP4 SUBSYSTEMS

Grant Agreement No: 101094300 MuCoL A Design Study for a Muon Collider complex at 10 TeV centre of mass Horizon Europe Framework Programme DELIVERABLE REPORT PRELIMINARY REPORT ON KEY WP4 SUBSYSTEMS DELIVERABLE: D4.2 Document identifier: WP4_preliminary_report.pdf DOI: 10.5281/zenodo.17733000 Due date of milestone: 30/11/2025 (End of Month 33) Work package: WP4 Lead beneficiary: UKRI Report release date: 30/11/2025 Document version: v1 Document status: Final Abstract: Muons offer a unique opportunity to build a compact high-energy electroweak collider at the 10 TeV scale. A Muon Collider enables direct access to the underlying simplicity of the Standard Model and unparalleled reach beyond it. The muon production system, based on several novel technologies, is key to delivering high luminosity. Pions are produced by a high power proton beam incident on a graphite, tungsten or liquid lead target. These pions decay to produce muons which must be captured and then cooled by several orders of magnitude. This paper presents preliminary studies of the MuCol WP4 working group to design key parts of the muon production system. MuCol Consortium, 2025 Grant Agreement 101094300 PUBLIC PRELIMINARY REPORT ON KEY WP4 SUBSYSTEMS Deliverable D4.2 Date: 30/11/2025 MuCol Consortium, 2025 For more information on MuCol, its partners and contributors please see https://mucol.web.cern.ch/ Funded by the European Union (EU). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the EU or European Research Executive Agency (REA). Neither the EU nor the REA can be held responsible for them. Delivery Slip Name Partner Date Authored by Chris Rogers (lead author) UKRI 30/11/2025 Edited by Chris Rogers (lead author) UKRI 30/11/2025 Reviewed by Daniel Schulte Roberto Losito CERN 30/11/2025 Approved by Roberto Losito [Scientific coordinator] Steering Committee 30/11/2025 Grant Agreement 101094300 PUBLIC John Back4, Carmelo Barbagallo1, J. Scott Berg5, William Bishop4,6, Bernardo Bordini1, Patricia Borges de Sousa1, Luca Bottura1, Steven Boyd4, Xavier Buffat1, Marco Calviani1, Daniele Calzolari1,2, Silvio Candido1, Carlo Carrelli7, Chris Densham6, Paula Desir´e Valdor1, Siara Fabbri1, Jose Antonio Ferreira Somoza1, Elena Fol21 Rui Franqueira Ximenes1, Dario Augusto Giove9, Alexej Grudiev1, PaulBogdan Jurj11, Rohan Kamath11, Andrea Latina1, Anton Lechner1, Giuseppe Lerner1, Roger Lewis15, Roberto Losito1, Jerzy Mikolaj Manczak1, Tim Mulder1, Emilio Nanni18,19, David Neuffer3, Jaroslaw Pasternak11,6, Alfredo Portone20, Caroline Riggall17 Chris Rogers6, Lucio Rossi8,9, Giuseppe Scarantino9,10, Daniel Schulte1, Marco Statera9, Bernd Stechauner12,1, Diktys Stratakis3, Ben Suitters6, Rebecca Taylor1, Will Thompson15,6, Luca Tricarico7, Katsuya Yonehara3, Edgar Ernesto Vera-Cardenas16,15 Dan Wilcox6, Ruihu Zhu13,14 1CERN, the European Organisation for Nuclear Research, 2Istituto Nazionale di Fisica Nucleare Sezione di Padova, 3Fermi National Accelerator Laboratory - Fermilab, 4The University of Warwick, 5BNL, Brookhaven National Laboratory, 6Rutherford Appleton Laboratory, 7ENEA, Agenzia Nazionale per le nuove tecnologie, l’energia e lo sviluppo economico sostenibile, 8Universit`a degli Studi di Milano, 9Istituto Nazionale di Fisica Nucleare Sezione di Milano, 10 Universit`a degli Studi di Roma “La Sapienza”, 11 Imperial College London, 12 Technische Universit¨at Wien, 13 Institute of Modern Physics, Chinese Academy of Sciences, 14 University of Chinese Academy of Sciences, 15 University of Sheffield, 16 Tecnol´ogico Nacional de M´exico, 17 University of Tennessee, Knoxville, 18 Stanford University, CA, 19 SLAC National Accelerator Laboratory, 20 Fusion For Energy, 21 Formerly at (1) 3 Target & Field Taper Chicane & Proton Absorber Longitudinal Capture Charge Separation Rectilinear A Bunch Merge Rectilinear B Final Cooling Charge Merge Reacceleration To 250 MeV O(1 km) Figure 1: Schematic showing the muon production system. 1 Introduction The muon collider is a unique, compact, high-energy electroweak collider concept which will produce, cool, accelerate and collide two single-bunch muon beams of opposite charge [1]. It is a paradigm-shifting tool for particle physics representing the first collider to combine the high-energy reach of a hadron collider and the high precision of a lepton collider, yielding a physics potential significantly greater than the sum of its individual parts. One of the most important and challenging sections of the Muon Collider is the muon production and cooling system. This system will generate a muon beam of unprecedented brightness and prepare it for acceleration. The muon production and cooling system is comprised of several subsystems, as shown in figure 1. The protons provided by the proton complex intersect a target to produce pions. The target is immersed in a 20 T solenoid field, yielding a high pion flux. The field is rapidly tapered to 1.5 T. Pions and muons traverse a solenoid-focused chicane where high momentum beam impurities are removed followed by a Beryllium absorber which ranges out remnant low energy protons. The remaining beam passes through a longitudinal drift where any remaining pions decay. The beam is then captured longitudinally. A multi-frequency RF system that captures the beam into 21 bunches is required owing to the initially large longitudinal beam emittance. Another solenoid chicane system splits the beam by charge species, in preparation for the ionisation cooling system. The ionisation cooling system is comprised of a series of solenoids, which focus the beam onto energy absorbers that reduce the beam’s momentum. The momentum is restored longitudinally using RF cavities, and overall there is a reduction in the beam’s emittance. Transverse, longitudinal and 6D emittance is used to characterise the volume in transverse (x, px, y, py), longitudinal (t, E) and 6D (x, px, y, py,t, E) phase space. An initial rectilinear cooling system, Rectilinear A, reduces the beam 6D emittance sufficiently that the many initial bunches can be merged into one single bunch. The beam is then cooled further to the final emittance, using a continuation of the rectilinear cooling system, Rectilinear B, and then a final transverse cooling with a sequence of high field solenoids operated with a low momentum (non-relativistic) beam. The beam is finally re-accelerated with a pre-accelerator to semi-relativistic speeds so that it can be delivered to the acceleration system. The key components in the Muon Production system, as designed by MAP, have been identified and the design has been further optimised by the MuCol team. A significant improvement in performance has been achieved. The overall performance estimate of the ionisation cooling system is summarised in figure 2. This plot shows the beam initial emittance on the right and the reduction in both emittances via the 6D rectilinear cooling (green). Further reduction in transverse emittance comes at the cost of longitudinal emittance via final cooling (orange). A more challenging final cooling design may yield improved performance (red). The improvement in performance as compared to the MAP study, which was taken as the initial design for MuCol, may be seen. The optimisation continues and performance is still improving. The MuCol team has chosen to focus on the highest risk and highest impact areas of the design, in particular the target and cooling system. Some studies have been undertaken to consider the capture system although this is beyond the scope of the MuCol study. In this report we will review the preliminary design of the muon production system as studied in MuCol. 4 Figure 2: Simulated performance in emittance space. 5 2 Target and Chicane The production of muons in the muon collider front-end involves the collision of a proton beam with a target material. This interaction triggers deep inelastic reactions that generate kaons and pions, which subsequently decay into muons. To effectively capture these particles and maintain control over their emittance, a robust solenoidal magnetic field is crucial. This magnetic field confines the charged particles along helical trajectories within the production target and subsequent beamline. To sustain the megawatt-class production target, active cooling is essential, requiring separation from the primary vacuum through beam windows. Moving downstream through the tapering sector before reaching the muon-cooling section, provision for an extraction channel for the unspent proton beam is required to accommodate a high-power dump absorber. Initially, a graphite target is being considered as the primary option. Graphite allows operation at high temperatures and exhibits remarkable resistance to thermal shock. Its extensive usage in various facilities attests to its reliability for 2 MW operation. However, it was identified that achieving the required muon bunch intensity may necessitate increasing the primary beam power up to 4 MW. Two other distinct technological alternatives are currently under investigation. The first alternative involves a heavy-liquid metal (HLM) target made of pure lead. This option addresses concerns regarding radiation damage and could potentially eliminate the necessity for integrated active target cooling within the cryostat of the superconducting (SC) solenoid. The second alternative explores a fluidized tungsten target that circulates micro-scale spheres in a closed loop. Notable features include high thermal-shock resistance and reduced susceptibility to cavitation, corrosion, and radiation damage. Both options present a possible pathway for even higher beam power delivery. 2.1 Graphite target In this section the graphite target is described. Both the 2 MW baseline target and the 4 MW alternative graphite target are described. 2.1.1 Baseline Graphite target The culmination of current studies for a 2 MW facility has resulted in a baseline target design featuring an 80 cm-long isostatic graphite rod of 30 mm in diameter, enclosed within a titanium vessel for hermetic confinement (see figure 3). A carbon composite frame holds the rod in the central position of the vessel. Internally, static helium gas at 1 bar prevents sublimation of the graphite during expected high-temperature operation (around 2000◦C). Additionally, it facilitates heat dissipation through natural convection, as opposed to a radiation-only cooled target. While direct cooling (in contact with the rod) could lower the graphite temperature, the baseline solution was preferred due to its capacity for enhancing radiation defect annealing and reducing erosion concerns. Active cooling of the inner vessel utilizes a helicoidal helium cooling circuit operating at room temperature and 20 bar. Although this setup offers a lower heat transfer coefficient compared to a liquid coolant, it mitigates the formation of pressure waves due to energy deposition on the coolant and concerns regarding water radiolysis. Positioned within the bore of the shielding assembly, the target vessel is planned to be supported on the cryostat’s upstream side, allowing easy replacement of the target without interfering with the shielding assembly. The assembly’s protection from radiation has been developed in detail (see also Section 2.2). A tungsten shielding surrounding the pion capture solenoid is required. To address the dissipation of heat (680 kW for a 2 MW facility), the approach shifted from contemplating water cooling, which posed complexities and handling limitations, towards utilizing helium gas cooling. This transition aims to resolve technological challenges associated with corrosion, erosion, and hydrogen embrittlement on the shielding material. The tungsten shielding design has evolved to adopt pieshaped segments, perforated for improved cooling efficiency, together with the engineering design of the shielding vessel. The IMCC studies have been focused on refining the target design and optimizing the pion/muon yield towards the cooling section. After careful consideration, a beam size of 5mm (1σ) was determined as the optimal compromise to ensure that the graphite’s dynamic response remains within 6 Figure 3: Conceptual design of the 2MW graphite target. 200 300 400 500 600 700 800 900 1000 1100 1200 0 10 20 30 40 50 60 70 80 PeakPowerdeposition[W/cm] Zcoordinateinsidethetargetrod[cm] 10GeV4MW5mm,targetradius15mm 5GeV2MW5mm,targetradius15mm 10GeV4MW7.5mm,targetradius22.5mm Figure 4: Peak power deposition along the graphite target rod for several proton beam configurations. The target radius is always set to 3 times the width of the Gaussian beam profile (dimensions indicated in the legend). acceptable limits. This solution aligns with the requirements of a 2 MW production target facility and is compatible with a proton injection of 5 GeV and 2 ns (1σ) pulses at a frequency of 5 Hz. The proton beam window, currently envisioned to be made of beryllium due to its low density and reduced interaction with the primary beam, will be located outside the cryostat. This component will undergo extensive radiation damage and necessitates dedicated cooling. Initially, materials such as titanium were considered for windows but were dismissed due to an expected accumulation of 10s of displacement per atom (DPA) per year. After several iterations, the decision was made to position it outside the solenoid assembly, ensuring improved accessibility for inspections, potential replacements, and increased availability of service space. 2.1.2 Alternative Graphite target Increasing the primary beam power to 4 MW could improve the collider’s performance. Beyond 2 MW, employing a graphite target technology requires active cooling with He (as opposed to indirect cooling). Additionally, the increased thermal power imposes a revision of the cooling parameters and the target vessel design. Preliminary studies suggest a feasibility path by further increasing the beam size on target and having a 10 GeV/c proton injection (figure 4). The proposed design employs a pressurized helium flow circulating through an annular channel surrounding the graphite rod, operating at 10 bar. The forced convection regime allows a significant increase in the heat transfer coefficient compared to the natural convection baseline. The helium 7 mass flow rate is in the range of 0.2-0.36 kg/s, while the Mach number is kept below 0.3 to prevent erosion of the graphite surface. The target is enclosed by a double titanium Grade 5 containment structure composed of an inner and an outer vessel, each with a wall thickness of 1 mm. The helium flow path consists of two annular gaps. The first gap, of 5 mm, between the graphite rod and the inner vessel, and the second, also 5 mm, between the inner and outer vessels, see figure 5. The inner annulus provides direct forced convection cooling of the graphite, while the outer annulus ensures efficient heat removal from the inner vessel wall, maintaining its temperature below design limits. This doublewall configuration also enhances mechanical robustness and simplifies integration of the cooling circuit with the surrounding shielding. Option A, B Option C Graphite Target Inner Vessel Outer Vessel Figure 5: Forced helium cooling configurations for the 4 MW graphite target: Option A (σ= 5 mm) and Option B (σ= 7.5 mm), showing also the steady state power density. Thermo-mechanical simulations showed that both configurations operate below the graphite tensile yield strength, with maximum principal stresses ranging between 18-22 MPa. The steadystate maximum temperature remains below 1600 ◦C, while the graphite surface stabilizes around 1400-1500 ◦C, considerably lower than the 2400 ◦C reached in the natural convection case. The forced helium flow induces a pressure drop between 0.6-1.1 bar along the annular channels, depending on curvature and internal support geometry. Option B, with a larger beam spot, reduces the local power density but increases the heat path within the graphite. Furthermore, explicit dynamic simulation evaluated the transient mechanical response of the graphite rod under the 2 ns proton pulse. The pulse duration is much shorter than the elasticwave transit time across the target radius (τ≈5.8µs for a wave speed of ∼2600 m s−1), resulting in an impulsive thermal loading that launches elastic waves through the target volume. For the both options, the maximum principal stress reached about 25 MPa with an oscillatory dynamic component contributing of a few megapascals. These values remain below the graphite tensile limit of 30 MPa, indicating that the combined thermal and dynamic effects do not compromise the target’s structural integrity. Although, further work still needs to be performed by including the effects of the vessel supports and potential degradation of graphite thermal conductivity under irradiation. 2.2 Magnet Shielding design The proton-driven MW-class muon source is one of the areas in the muon complex where massive shielding elements are needed for equipment protection. The solenoids near the production target are exposed to the secondary radiation showers generated by inelastic collision products in the target. The front-end design elaborated within MAP considered a combination of resistive copper magnet inserts and larger-aperture solenoids made of low-temperature superconductors. The latter are more sensitive to radiation than resistive magnets and were foreseen to be shielded by a thick layer of helium gas-cooled tungsten beads. A new solenoid configuration, based entirely on hightemperature superconductors (HTS), has been developed within the IMCC, combined with a new arrangement of tungsten shielding inserts for absorbing electromagnetic and hadronic showers. 8 The target team has designed a baseline radial build for the pion production target, taking into account radiation flux into the surrounding solenoid, heat load on the graphite target itself and appropriate cooling systems for the target and shielding. The radial build is explained in Table 1. A 3D FLUKA model for the target area has been constructed and the pion and muon Component Material ri[mm] re[mm] ∆r[mm] Solenoid coils HTS 700 - - Insulation Insulation 690 700 10 Vacuum Vacuum 670 690 20 Thermal shield Copper & Water 651 670 19 Vacuum Vacuum 631 651 20 Inner supporting tube Stainless-steel 619 631 12 Vacuum Vacuum 609 619 10 Outer Target shielding Tungsten 599 609 10 Neutron absorber Boron Carbide 594 599 5 Target shielding and neutron moderator Stainless-steel 589 594 5 Water 569 589 20 Stainless-steel 564 569 5 Tungsten 179 564 385 Stainless-steel 174 179 5 Vacuum Vacuum 173 174 1 Target vessel Titanium 168 173 5 Helium 155 168 13 Titanium 150 155 5 Helium 15 150 135 Target Graphite 0 15 15 Table 1: Target System radial build for a graphite target. yield through the target system has been assessed. The FLUKA geometry implementation of the target area is shown in Fig 6. The model is simplified with respect to Table 1. The yield results from simulations are presented in Table 2, assuming the standard deviation of the transverse beam position is 5 mm, the target rod has a radius of 15 mm and that all the muon and traversing the chicane can be captured if their momentum is below 500 MeV/c0. The magnet team have identified HTS technology as having greater tolerance to radiation issues. The target team have updated the design with reduced shielding and including structural elements. Proton beam energy [GeV] Yield [10−2/ GeV/p+] 3 4 5 6 7 8 9 10 µ+2.8 2.6 2.4 2.3 2.2 2.1 1.9 1.9 µ−1.8 1.8 1.8 1.8 1.7 1.7 1.7 1.7 π+1.3 1.2 1.1 1.1 1 0.98 0.92 0.9 π−0.84 0.81 0.84 0.82 0.83 0.8 0.8 0.81 Table 2: Yield per unit energy proton beam [10−2/ GeV/p+] Dedicated shielding studies were performed with the FLUKA Monte Carlo code, in order to quantify the heat load and radiation damage in the solenoids. The shielding was assumed to be composed of helium gas-cooled tungsten segments. In order to thermalize and capture neutrons, the shielding around the target was assumed to embed a layer of water and boron carbide. This can reduce the cumulative displacement damage in the coils by about a factor of two. Table 3 shows the annual DPA and dose in the target solenoid for different tungsten shielding thicknesses. The studies were carried out for a graphite target rod, assuming a proton drive beam energy of 5 GeV and beam power of 2 MW. The results indicate that a thickness of about 40 cm is needed for reducing the DPA in the coils to 1×10−3DPA/year and the dose to about 5 MGy/year. Considering the beam aperture requirements, the target vessel size and the space required for supports and thermal insulation, the minimum coil aperture (radius) for the target solenoid is hence expected 9 The peak power deposited on the target is 3638 W cm-3 . Lower intensities are seen throughout the rest of the target system. In the return pipe, the heat deposition in the tungsten is lower than in the surrounding shielding material. This is due to the reduced density compared to solid tungsten shielding blocks. To conclude, the proposed target design and 30% density tungsten shows strong potential to reach the requirements given of a 4 MW, 10 GeV beamline. Further optimisation of yields could be equivalent to or exceed that of graphite, however, caution must be taken to ensure that downstream magnets have optimal shielding as the increased load could risk inducing a quench in an optimised setup. Although the proposed 30% density tungsten powder system yields produced approximately 20% fewer particles than the graphite arrangement, further optimisation is expected to improve performance, for example by studying the interaction pipe design, which has not yet been optimised. 2.4.2 Physics Performance Development Opportunities Future simulation work will focus on refining the geometry used within FLUKA to develop a target design that balances both manufacturing feasibility with optimised performance. The outcome aims for maximising pion and muon yield output while improving thermal efficiency within the target system. To achieve these aims, various parameters will be investigated. These include the tilt angle, length, radius, wall thickness and return pipe bend radius. The target tilt and length affect the interaction length, increasing the size of the interaction region while reducing particle backscatter. Radius and wall thickness determine the number of secondaries reabsorbed within the target, while a smaller bend radius means less material for the particles to penetrate. Simulation results from FLUKA will be cross-compared with Geant4, ensuring consistency across the software. Comparing them will enable further optimisation as confidence in the design will be validated. As designs of other components in the target system progress, for example the solenoid coils, improved limits on the endurance of materials and performance can be introduced. This will allow for an optimised approach to the assessment of thermal and radiation loads on the system. 2.4.3 Thermal Management As with any high power beam intercepting device, the target performance will be limited by the capacity of the system to extract heat without compromising target integrity. A static target must extract the entire deposited power directly from the target region. The use of a flowing target reduces the continuous heat extraction to a discrete temperature rise and multiphase fluid flow problem, which can be solved separately. Instantaneous temperature rise is determined by the peak heat deposition914 K per pulse. Assuming each pulse is exposed to fresh tungsten, this means a design for a maximum operating temperature of 1000 ◦C. Tungsten has been shown to rapidly oxidise at such high temperatures[4][5], an issue which must also be tackled in the graphite baseline. This can be managed by reducing the oxygen availability in the system. Thermomechanical testing of tungsten granules will be necessary to understand degradation. Further reduction in the peak heat deposition is desirable, for example, by adapting the beam profile on the target. Reducing target temperatures will maximise target reliability. While the longitudinal profile is fixed at near-instantaneous by the acceptance of the RF capture system, the transverse profile may be adapted to balance physics performance with thermomechanical stability. Flat-topping the beam will enable a narrower target to produce more muons, and moderate flattopping, such as in figure 15 has been shown to reduce the maximum heating by approximately 30% for the same target geometry. The effect on particle yield is yet to be studied. 2.4.4 Erosion Mitigation High-velocity tungsten particles impinging on pipe walls present a challenging erosion condition which must be studied and mitigated. Solid particle erosion is a common industrial problem and is an active area of research. To achieve an understanding of the factors at play, an erosion study has been initiated, possible due to in-kind contributions from the University of Sheffield. This will assess the effects of the novel erosion that occurs due to tungsten’s extreme density, and propose 16 Figure 15: A Standard Gaussian beam (top) results in higher peak energy deposition than a flattopped “supergaussian” (bottom) for a simplified geometry. 17 system design guidelines for the target system. This will also result in increased awareness of rates of erosion which can be applied to target lifetime prediction. An erosion test capability has been developed at the University and is planned to be exploited over the next year with in-kind contributions to the collaboration from the University. Initial results show that of the candidate materials (Inconel 718, Ti-6Al-4V and stainless steel 316), Inconel shows highest resistance to erosion. Visible surface damage of the wall was observed for more malleable wall materials. Over the next year, a more detailed study of long term erosion of these materials will be conducted, particularly considering high temperature effects. Erosion of the beam window areas is of particular concern, and care must be taken to optimise the flow here to minimise particle velocities. An investigation of inspection methods is ongoing. It is necessary to propose methods of continuous process control, such as 3D electrical capacitance tomography [6], which enable in-situ measurement of the tungsten flow in the target pipe, and remote methods of inspecting erosion damage, such as Pipe Inspection Gauges. The development of radiation-hard Pipe Inspection Gauges and associated technology offers opportunity to engage with existing industry. 2.4.5 Target System Dynamics The target system is complex and difficult to study theoretically, but readily studied experimentally. The fluid dynamics of multiphase systems require in-depth physical testing to validate simulation efforts. Two types of simulation are considered: Computational Fluid Dynamics (CFD) and Discrete Element Modelling (DEM). CFD analyses of such a system are incredibly challenging as empirical models are not validated for materials as dense as tungsten, nor the multiphase solvers in CFD software designed for high volume fractions of granular material. This results in an incredibly intensive simulation that has unclear validity. In DEM individual particle grains are tracked through the system, and their interactions with each other and the propelling gas can be studied. This is also computationally expensive, requiring large amounts of GPU processing power, but is far more capable for the system in question. This method is anticipated to be useful for the tungsten target system, but is as yet untested. Empirical data of two phase systems has been successfully correlated against non-dimensional parameters by multiple authors [7], which offers opportunity for scale modelling of a target design and estimation of the flow regime in the system based on particle and fluid flow parameters. While tungsten powders yield Archimedes and Reynolds numbers in the studied range, the validity of the correlation is untested for high density materials. This analysis technique highlights parameters which must be managed to scale the model. The flow-regime map for gas-solid systems is shown in figure 16. Change of carrier gas, while fixing all other flow parameters, may move the predicted flow regime from Plug-1 in helium to the Dune/Dilute phase boundary in air. 2.4.6 Future Work Additional funding has been secured to enable material characterisation work at the Henry Royce Institute for Advanced Materials, using specialist powder handling equipment. Six powder grades and three pipe wall materials will be tested to investigate the frictional effects on the pipe wall and the internal friction of the powders at varied consolidation conditions. These tests will be conducted using existing samples available at RAL. The University of Sheffield is conducting detailed erosion studies to compare candidate wall materials and propose mitigation strategies for potential rapid erosion. This will directly input to the design of an offline flow rig, which would be a full scale representative prototype of the target system. A concept for the future implementation of such a rig is being made, which would include a phased build that enabled testing for each key subsystem in the granular flow. To enable the plans for experiments hosted at the University of Sheffield and RAL, further materials, equipment and labour support will be sought in the medium to long-term. 18 Figure 16: Flow regime map for gas-solid systems. Dotted lines drawn for 50 µm diameter tungsten spheres driven by helium and air at 1 barg. Modified from [7]. 2.5 Liquid lead target Focused research on an HLM lead target emphasized defining a high-level concept while addressing critical obstacles. An initial concept involving flow in a pipe was discarded due to high magnetohydrodynamic (MHD) losses and concerns regarding shock waves on the retaining target pipe. The current models under consideration involve a free-flow of lead in the vertical direction (similar to a curtain), aiming to eliminate the concerns posed by the initial concept (figure 17) [8], and a liquid lead jet (figure 18). Ongoing work involves further modelling and development of this concept as an alternative target technology for a 4 MW proton-driver. Free-falling curtain concept The first configuration consists of a vertically oriented free-falling curtain of liquid lead (figure 17) serving as the proton interaction medium. The flow rate is approximately 100 kg s−1, corresponding to a curtain thickness of a few centimetres and a crosssection of about 3 cm in diameter and 50 cm in length. Liquid Pb Inlet Pressure inlet Argon @ 1bar Pressure Outlet @ 1bar free surface Pb-Ar p+ beam B0 g C-Target region 51 cm 4 cm R 6 cm 5 cm possible beam interception regions Figure 17: Schematics of the proposed concept based on liquid lead free-falling curtain [8]. The liquid lead is maintained at 400 ◦C in a vessel filled with argon at 1 bar to prevent oxidation. This arrangement naturally avoids structural damage concerns since no solid material is directly exposed to the beam, and the curtain continuously renews itself between consecutive beam pulses. 19 The FLUKA studies on the muon and pion yields as well as the radiation load to the magnets in the target area showed that a liquid lead curtain induces about 3 times higher peak atomic displacement in the magnet coils compared to a graphite rod for the same proton beam parameters, while reducing the particle yields measured at the end of the tapering region [9]. Therefore, a different liquid lead geometry concept was developed. Liquid jet concept The second configuration concept involves a horizontal liquid lead jet injected into the target region (figure 18), aligned approximately parallel to the solenoidal field lines. This arrangement minimizes MHD effects and improves flow stability. Liquid Pb @ 400 ºC p+ 10 cm Container Vessel 100 cm B0 beam interception Liquid Pb Pb Pool Figure 18: Schematics of the proposed concept based on liquid lead free-falling curtain. The jet is designed to emulate a solid cylindrical target, with an equivalent interaction length of 20 cm and a diameter between 1 cm and 3 cm. The injection velocity is around 3 m s−1, yielding Reynolds numbers on the order of 105–106and Weber numbers above 104, characteristic of fully turbulent, inertia-dominated flow. Numerical models incorporating two-phase flow and magnetic effects coupling indicate that the jet remains coherent across the target region, with breakup lengths exceeding several metres (Lbreakup ≈5.9 m ≫Ltarget). To safely dissipate the kinetic energy of the jet and prevent erosion of the containment vessel, a downstream pool of liquid lead is used, allowing the jet to decelerate. Under pulsed-beam conditions, transient simulations predict the formation of localized pressure waves and cavitation zones in the jet core. However, given the high mass flow rate and short residence time, the liquid column re-establishes itself within approximately 0.1–0.2 s, shorter than the 0.2 s interval between proton pulses at 5 Hz, ensuring full jet recovery before subsequent impacts. Because the target interaction point must be within the solenoidal magnetic field, MHD coupling effects arising from the motion of the conductive liquid lead in the external field were evaluated. The magnetic Reynolds number is defined as Rem=µ0σUL, where µ0is the magnetic permeability of free space, σthe electrical conductivity of lead, Uthe characteristic flow velocity, and Lis the characteristic length scale. For both concepts Remis Rem≪1. Consequently, magnetic induction effects within the liquid are negligible, and the external solenoidal field can be considered unaffected by the flow. The magnetic field only acts to slightly damp velocity fluctuations via Lorentz forces, without significantly modifying the global flow structure or inducing additional field distortions. The FLUKA simulations showed that, with a 4 MW proton beam, a 3 cm lead jet exhibits similar problems to the ones observed with a liquid lead curtain: radiation load to the magnets of the target area increases by a factor of 3 with respect to the graphite target and the muon and pion yield is reduced. The case of a liquid lead jet of 1 cm diameter shows a more promising outcome - the radiation load is still higher by a factor of 2, but the negative muon and pion yields are increased by roughly 20% (the positively charged yield remains unaffected). However, it is not yet clear if the proton beam parameters could match the ones used in the simulation - the 3σ width of a Gaussian beam was always adjusted to the jet diameter. Further studies are needed to understand the impact of an inclined primary beam that could potentially increase the yield and 20 Figure 19: Pressure (indicating shock-wave passage) and velocity profiles over time at selected probe locations; time origin (t = 0s) marks the onset of energy deposition after the flow reached steady state (0.8826s). open a possibility of dumping the spent protons into the liquid lead pool. Shock-wave effects on liquid lead The intense thermal deposition and the short deposition time generate a compressed, high-pressure fluid region that expands through the domain in the form of shock waves [10][11]. This phenomenon is being studied to address concerns about fluidflow disruptions for both the considered concepts of free-falling curtain and axial jet, and potential damages on supporting structures caused by shockwave reflections. The energy deposition in the fluid is modelled as a high-pressure region at the initial time. The pressure within the volume is estimated using the Mie-Gr¨uneisen equation of state [12], where ∆Eis the energy deposited in the fluid region: p=pref +γ V∆E, γ =αBT ρCv =αBS ρCp . The fluid motion is solved using the Volume of Fluid (VOF) method, assuming an inviscid three-phase mixture of liquid lead, lead vapour, and argon: ∂ ∂t(αqρq) + ∇·(αqρqv) = Γq, ∂(ρv) ∂t +∇·(ρvv) = −∇p+fv. The mass generation term Γqis defined only for the liquid lead and lead vapour phases, according to the cavitation model by Singhal [13], in which phase change is allowed below a pressure threshold. The compressibility of lead is modelled using the Tait equation of state (EOS), which is widely used in UNDerwater EXplosion (UNDEX) simulations [14]. Since the Tait equation is independent of temperature, the energy equation is not solved, and the cavitation pressure is assumed constant. Argon and lead vapour are modelled with constant physical properties, derived using the ideal gas law at reference pressure and temperature. The physical model is solved in a Finite Volume Method (FVM) commercial code for a 2D case, where the chosen equation of state (EOS) for lead is implemented. The figure 19 illustrates the boundary conditions (BCs) and simulation results for a beam interaction area with a diameter of 10mm and an overpressure of 5GPa, resulting after a constant thermal deposition of 107J/m3. This value of thermal deposition is taken as the representative volumetric energy deposited by the beam into the lead in the 2 MW configuration. The first results show the propagation of shock waves within Beam Intercepting Devices (BIDs) caused by high-power beam. The results, shown in figure 20, 21 Figure 20: BCs (left) and contour of lead volume fraction with pressure isolines indicating the shock-wave region (right). highlight that the shock wave period is on the order of 1 ×10−5s and show the magnitude of the major physical variables such as pressure and velocity. They also reveal additional physical phenomena that require further investigation, such as cavitation in Heavy Liquid Metals (HLMs). Liquid-Lead System considerations The liquid metal loop is designed to ensure the continuous circulation of lead and to restore its physical conditions to the inlet parameters required by the Beam Intercepting Devices (BIDs). From a thermo-hydraulic point of view, the use of a liquid lead target offers several advantages. In fact, liquid lead, as a Heavy Liquid Metal (HLM), has a high heat capacity, allowing it to absorb large amounts of energy with relatively small temperature variations. This is due to its high density (ρ= 10580 kg/m³), high specific heat capacity (cp= 146 J/kg·K), and high boiling point (Tb= 1747 °C) [15]. The overall operation of the loop strongly depends on the design parameters of the target, particularly the mass flow rate and the thermal power deposited by the beam into the target. Although these parameters are still being defined, it is already possible to outline the general design requirements of the system. To ensure proper thermal and dynamic control of the lead, the system includes several main components: a pumping system, a heat exchanger, and a liquid lead chemistry control system. The heat exchanger is responsible for maintaining the lead temperature around 400 −450◦C during the operation phase. This value helps prevent corrosion in the piping and avoids cooling issues in the pumps when operating at high flow rates[16]. The lead chemistry control system requires an expansion tank, necessary for oxygen control through dedicated probes, and a purification system for lead oxides and argon. In addition to these components active during normal operation, a storage tank is included for holding the lead volume during start-up and maintenance phases. From the piping system point of view, every surface in contact with the liquid lead is thermally insulated and equipped with electrical heating wires providing about 0.3 kW/m of power, in order to prevent local freezing of the liquid lead. To avoid corrosion and erosion of the steel caused by liquid lead, the loop temperature is kept below 450 ◦C and the flow velocity below 1 m/s - limiting the velocity helps to reduce pressure drops within the system [16]. 3 Cooling 3.1 Rectilinear Cooling The design of the rectilinear cooling lattice, based on the MAP design [17], has been updated and optimised by the IMCC team [18]. Composed of two rectilinear cooling channels, one before 22 (Rectilinear A) and one after (Rectilinear B) the bunch merging system, the updated lattice yields an improved performance over the MAP design. The lattice design and performance are briefly summarised in this section; the detailed study including tolerance analysis can be found in [18]. The rectilinear cooling system has a periodic lattice composed of a series of solenoid magnets which have a weak dipole field superimposed to enable emittance exchange. While the previous design used tilted solenoids to generate the dipole field, the updated IMCC design employs separate dipole magnets. This design choice facilitates the independent tuning of the dipole field although more space may be required. Each periodic unit, referred to as a cooling cell, consists of solenoids with opposite polarity for tight focusing at the absorber, dipole magnets for dispersion generation, RF cavities for beam energy loss compensation and liquid hydrogen (LH2) wedge absorbers. Figure 21: Schematic of the (left) A-type and (right) B-type cooling cell layout. Two types of cooling cells are used in the present design: A-type and B-type cells, used for the pre-merge and post-merge sections, respectively. The conceptual layout of the two types of cooling cells is shown in figure 21. The primary difference between the two types of cells lies in the cell tune (number of betatron oscillations per cell). A-type cells operate with a cell tune below 1, which allows the lattice to accept a beam with a higher beam emittance, suitable for initial cooling. The A-type cell has a further advantage that the transverse betatron function at the center of the cell is equal to that at the start and end of the cell, which enables the use of an additional wedge absorber in the middle of the cell. B-type cells operate with a cell tune between 1 and 2, in a tighter focusing regime that enables cooling to lower emittances. As a result, these cells do not accept high-emittance beams and have a smaller momentum acceptance. The RF cavities are modelled as perfect cylindrical pillbox cavities with thin Beryllium (Be) windows operating in the TM010 mode. The Be windows electromagnetically seal the cavities, ensuring that the TM010 mode is an appropriate model of the real field, and have been shown to increase the operational RF gradient in multi-Tesla magnetic fields [19]. Multiple independently phased RF cavities are used in each cell. Rectilinear A and Rectilinear B lattices are composed of 4 A-type and 10 B-type stages, respectively. Each stage is a series of identical cooling cells, and the design of the cooling cell becomes more challenging (higher fields, more compact assembly, etc.) with increasing stage number. The hardware parameters for the whole rectilinear cooling system are listed in Tables 4 and 5. The performance of the rectilinear cooling system is shown in Table 6, in terms of the beam emittance and transmission at the end of each stage. The Rectilinear A section reduces the normalised transverse and longitudinal emittances from 16.96 mm and 45.53 mm to 1.24 mm and 1.74 mm, respectively, with an overall transmission rate of 49.6% including the muon decays. The Rectilinear B section reduces the normalised transverse and longitudinal emittances from 5.13 mm and 9.99 mm to 0.14 mm and 1.56 mm, respectively, with an overall transmission rate of 28.5% including the muon decays. The final transverse emittance achieved with this design is a factor of two smaller than in the previous design [17]. This improvement is expected to significantly benefit 23 Cell Stage Pipe Max. BzInt. β⊥DxOn-Axis Wedge Length Length Radius On-Axis ByWedge Len. Angle (m) (m) (cm) (T) (Tm) (cm) (mm) (cm) (deg) A-Stage 1 1.8 104.4 28 2.5 0.102 70 -60 14.5 45 A-Stage 2 1.2 106.8 16 3.7 0.147 45 -57 10.5 60 A-Stage 3 0.8 64.8 10 5.7 0.154 30 -40 15 100 A-Stage 4 0.7 86.8 8 7.2 0.186 23 -30 6.5 70 B-Stage 1 2.3 50.6 23 3.1 0.106 35 -51.8 37 110 B-Stage 2 1.8 66.6 19 3.9 0.138 30 -52.4 28 120 B-Stage 3 1.4 84.0 12.5 5.1 0.144 20 -40.6 24 115 B-Stage 4 1.2 66.0 9.5 6.6 0.163 15 -35.1 20 110 B-Stage 5 0.8 44.0 6 9.1 0.116 10 -17.7 12.5 120 B-Stage 6 0.7 38.5 4.5 11.5 0.087 6 -10.6 11 130 B-Stage 7 0.7 28.0 3.75 13 0.088 5 -9.8 10 130 B-Stage 8 0.65 46.15 2.85 15.8 0.073 3.8 -7 7 140 B-Stage 9 0.65 33.8 2.3 16.6 0.069 3 -6.1 7.5 140 B-Stage 10 0.63 29.61 2.0 17.2 0.069 2.7 -5.7 6.8 140 Table 4: Main parameters for the rectilinear cooling cell components, including the cell geometry, solenoid fields, dipole fields, beam optics and wedge absorber geometry. Liquid hydrogen is used as the wedge absorber material for all stages. RF Frequency Num. RF RF Length Max. RF Gradient RF phase (MHz) (cm) (MV/m) (deg) A-Stage 1 352 6 19 27.4 18.5 A-Stage 2 352 4 19 26.4 23.2 A-Stage 3 704 5 9.5 31.5 23.7 A-Stage 4 704 4 9.5 31.7 25.7 B-Stage 1 352 6 25 21.2 29.9 B-Stage 2 352 5 22 21.7 27.2 B-Stage 3 352 4 19 24.9 29.8 B-Stage 4 352 3 22 24.3 31.3 B-Stage 5 704 5 9.5 22.5 24.3 B-Stage 6 704 4 9.5 28.2 22.1 B-Stage 7 704 4 9.5 28.5 18.4 B-Stage 8 704 4 9.5 27.1 14.5 B-Stage 9 704 4 9.5 29.7 11.9 B-Stage 10 704 4 9.5 24.9 12.2 Table 5: Rectilinear cooling cell RF parameters. the design of the final cooling section, which aims to achieve a normalised transverse emittance of 22.5 µm. The scenario of using coupled ‘π-mode’ RF cavities has also been investigated. In this case, adjacent cavities operate with a πphase difference, and only one power coupler and feed through is required for a group of coupled RF cavities. The impact of using π-mode RF was studied on a cooling lattice comprised of ‘B-Stage 5’-like cells. Comparisons with an identical cooling channel using independently phased RF reveal no significant differences in cooling performance. More details on this work, including a sensitivity analysis can be found in [18]. After undergoing sufficient cooling in the Rectilinear A section, the initial train of bunches are merged into a single bunch by the bunch merging system. The bunch merging scheme was designed by MAP and is described in [20]. In this scheme, 21 bunches are merged in longitudinal phase space into seven bunches, which then follow seven paths of different lengths and reach a collecting funnel at the same time to be merged transversely. 24 εTεLε6D Stage Cumulative (mm) (mm) (mm3) Transmission (%) Transmission (%) Start 16.96 45.53 13500 100 A-Stage 1 5.17 18.31 492.60 75.2 75.2 A-Stage 2 2.47 7.11 44.03 84.4 63.5 A-Stage 3 1.56 3.88 9.59 85.6 54.3 A-Stage 4 1.24 1.74 2.86 91.3 49.6 Bunch merge 5.13 9.99 262.5 78.0 38.7 B-Stage 1 2.89 9.09 76.07 85.2 33.0 B-Stage 2 1.99 6.58 26.68 89.4 29.4 B-Stage 3 1.27 4.05 6.73 87.5 25.8 B-Stage 4 0.93 3.16 2.83 89.8 23.2 B-Stage 5 0.70 2.51 1.32 89.4 20.7 B-Stage 6 0.48 2.29 0.55 88.4 18.2 B-Stage 7 0.39 2.06 0.31 92.8 17.0 B-Stage 8 0.26 1.86 0.13 87.9 14.9 B-Stage 9 0.19 1.72 0.06 85.2 12.7 B-Stage 10 0.14 1.56 0.03 87.1 11.1 Table 6: Simulated performance of the rectilinear cooling lattice in terms of emittance reduction (transverse, longitudinal and 6D) and transmission, both per stage and cumulative. The MAP target for the rectilinear cooling was ε⊥=0.3 mm and εL=1.5 mm. 3.2 Final Cooling The final cooling system reduces the transverse emittance beyond that achieved by the rectilinear cooling by passing the muon beam through very high-field solenoids. The high field presents challenges in maintaining satisfactory dispersion through the solenoid, so only transverse cooling is proposed. Stronger focusing is achieved by using a low energy muon beam. The longitudinal emittance grows significantly in the system. The challenge of the final cooling lattice is to mitigate this longitudinal growth. Overall the collider performance is improved by this further emittance reduction. Three cooling channels have been studied, assuming different initial conditions. •Final-Cooling-RMAP: Beam from MAP rectilinear cooling •Final-Cooling-RB8: Beam from IMCC rectilinear B8 •Final-Cooling-RB10: Beam from IMCC rectilinear B10 Final-Cooling-RB8 is studied as a risk mitigation in case the B9 and B10 cooling cells prove too challenging to be constructed. Final cooling lattices are made from four components: ±40 T high field solenoids, lower-field matching and transport solenoids, low-frequency RF cavities, and hydrogen absorbers. These components are detailed below. The final cooling lattice is envisioned to be less than 100 m long, however the beam undergoes very dramatic changes in this process. A schematic of the FinalCooling-RMAP lattice is in figure 22. The performance of each final cooling absorber is dependent on three key parameters: The absorber length, the energy of the beam, and the energy spread. For an absorber with defined length and material, it is possible to determine an energy that provides the best transverse cooling for a given longitudinal increase. The optimum varies as a function of energy spread, beam energy and the initial longitudinal emittance. As shown in figure 23, longitudinal emittance growth has a quadratic dependence on initial energy. At low energy, significant emittance growth arises as lower energy particles lose more energy than higher energy particles. At high energy, significant emittance growth arises due to stochastic energy straggling. The analysis is guided by the minimization of the tradeoff function ∆εL,N/∆ε⊥,Nto reach maximum luminosity, where ∆εL,N and ∆ε⊥,Nis the 25 pion yield of 7.9 ×10−4π+/ POT using the target and parabolic horn shown at the top of figure 29. The graphite rod is 90 cm long and has a 6 mm radius. The horn current was fixed at 220 kA. The pion yield was calculated for pions with momenta in the 210 – 330 MeV/c range and within a transverse acceptance of 2 mm rad in the vertical and horizontal planes. The transverse acceptance is indicated by the red ellipse on the right panel of figure 29. While the best yield achieved with the ellipsoidal horn was ∼15% lower, this geometry is of interest as it can produce pion beams with upright beam ellipses which may be easier to match and transport. More details of the initial optimisation study can be found in [36]. Figure 29: (left) Target and horn conductor r−zprofiles and (right) their x−x′phase space of the captured pions. (top) Parabolic, (bottom) ellipsoidal. Preliminary thermal simulations indicate modest power deposition in the graphite target, which can be managed using a helium-based forced convection cooling system. These results also suggest that higher-Zmaterials, such as tungsten, may be viable alternatives. Building on the initial scan, a new optimisation campaign using a Bayesian optimisation framework integrated with FLUKA is now underway [37]. This study systematically explores horn geometries, target materials, and higher horn currents to identify configurations that maximise the accepted pion yield. Pion decay channel Following capture by the magnetic horn, pions are transported through a decay channel designed to maximise muon production within the desired momentum range and emittance acceptance. The baseline optics for this section, common to both TT7 and CTF3 implementations, consist of three quadrupole triplets forming a 9.5 m-long lattice. The design prioritises large momentum acceptance (±50%) and low Twiss betatron functions to minimise beam losses during transport and decay. An estimated 40% of the 300 MeV/c pions decay within the channel. This yields a muon beam with significant transverse emittance and energy spread. BDSIM tracking studies are used to evaluate the muon phase space at the decay channel exit. A transverse acceptance of 2 mm·rad defines the envelope for muons transported into the momentum-selecting chicane, with figure 30 showing the resulting beam distribution in phase space for muons in the 190-210 MeV/c momentum range. Further lattice refinements are planned to improve the usable muon yield in coordination with the chicane design. Chicane The chicane serves a dual role: it selects muons within the 190-210 MeV/c momentum window and provides a path for the extraction of unspent primary protons to a dedicated beam dump. 32 Figure 30: (left) x−x′and (right) y−y′phase space of the muons in the momentum window at the end of the pion decay section. The red ellipse indicate a 2 mm·rad single particle emittance. Additionally, the final segment of the chicane is used to host the beam preparation system (BPS). Two chicane configurations have been developed for the TT7 and CTF3 sites. The lattice components, optical Twiss functions and dispersion for the chicane segments that exclude the BPS are shown in figure 31. The MAD-X study used the Twiss functions of the muon beam emerging from the pion decay section as input and aimed to deliver a beam with βx,y = 3 m and αx,y = 0 at the entrance to the BPS. The TT7 layout uses a three-bend geometry optimised to fit within the tight spatial constraints of the tunnel; however, the study revealed that a tunnel extension is required, and the design has been developed to minimise the extent of this modification. This layout imposes significant optical challenges, including elevated beta functions that increase losses in the apertures, incomplete dispersion suppression, and overall reduced flexibility for matching into the solenoidal beam preparation system. This could be mitigated by relaxing the spatial constraints, which would require enhanced civil engineering works to extend the tunnel. The CTF3 version adopts a two-bend layout, enabled by the non-zero angle between the incoming proton beam and the cooling channel axis. The greater available space reduces constraints on lattice compactness, allowing the optics to maintain small beam sizes, achieve complete dispersion suppression, and provide flexible matching — including the delivery of beams with beta functions βx,y <1 m at the entrance to the BPS. Beam preparation system Muons originating from a proton-driven horn-target system occupy a large volume in phase space and do not satisfy the transverse emittance (ε⊥∼2 mm) and bunch length (O(100 ps)) requirements imposed by the cooling channel beam dynamics. The beam preparation system performs transverse collimation and longitudinal phase-space rotation to transform the initial muon distribution into a beam suitable for injection into the cooling cells. The system is integrated into the final segment of the chicane, and it consists of a series of solenoids that focus the beam transversely, collimators for transverse clipping, and high-gradient RF cavities to rotate the beam in the phase-energy space. A final dipole is used for momentum collimation. Figure 32 shows a BDSIM implementation of a BPS that follows the study in [28]. The most restrictive physical aperture in the BPS is set by the iris of the RF cavities, assumed to be ∼60 - 80 mm in radius for 704 MHz cavities. To avoid excessive collimation, the solenoidal optics must focus the beam to a minimum size in the regions with RF cavities. To obtain a beam with ε⊥∼2 mm at the end of the BPS, this constraint implies a required β⊥<1 m. Preliminary optics matching studies explore solenoid configurations and field profiles that can satisfy the focusing requirements and ensure integration with the upstream section of the chicane. Figure 33 shows the field and transverse betatron function profiles for a solution that achieves β⊥<1 m in the inter-solenoid regions. BDSIM tracking studies of the BPS are currently underway. Preliminary results for the CTF3 implementation indicate that ∼2×106muons/bunch are delivered at the end of the BPS. A corresponding estimate for the TT7 configuration has not yet been completed, as the BPS optics remain under study due to the dispersive nature of the beam entering the system. However, based 33 Figure 31: Lattice components, optical Twiss functions, and dispersion for the (top) TT7 and (bottom) CTF3 chicane configurations. Figure 32: BDSIM rendering of the beam preparation system concept. on tracking losses observed in the upstream part of the chicane, the delivered intensity in TT7 is expected to be up to an order of magnitude lower. 3.3.2 Cooling cell A cooling cell was designed based on the B-Stage 5 cell described in section 3.1. The beam physics design process involved an optimisation to balance practical construction considerations against achieving higher peak magnetic fields. The cooling cell schematic, as simulated in G4Beamline, is shown in figure 34. The lattice was simulated using G4Beamline v3.08 [4]. Parameters for the cooling cell are described in Tab. 12. The transverse design has been optimised in collaboration with the magnet and cooling cell teams. As with the B-Stage 5 lattice, the optics operates between cell-tune of 1 and 2, providing a balance between dynamical acceptance and tight focusing required for good operation. Preliminary longitudinal design is in place, considering impact of different dipole polarities, absorber opening angles and thicknesses and dipole fields. The cooling cell design team (WP8) have established feasibility of 60 mm radius at the cavity 34 Figure 33: (green) Longitudinal magnetic field and (blue) transverse betatron function profiles within the BPS region. Using this configuration a shorter lattice containing only three solenoids is currently explored. iris and are developing an enhanced window model to understand the possibility of enlarging the window radius to 80 mm. Options under study include graded window thickness and curved apertures to manage Lorentz forces and thermal effects on the window. In an initial phase aluminium windows are planned, owing to the handling difficulties associated with beryllium. Further studies are underway to improve longitudinal performance, such as considering the relative benefits of dispersion and absorber wedge angle and the balance between longitudinal and transverse emittance reduction. Alignment studies have been done for the Rectilinear B5 lattice, indicating challenging requirements. A beam-based alignment and correction study, including consideration of appropriate instrumentation, is in progress. Figure 34: Cooling Cell Schematic 35 Parameter Unit Value Cooling Cell Length mm 1000 Beam Physics Momentum MeV/c 200 Twiss beta function mm 130 Dispersion in X mm -61.5 Dispersion in Y mm -19.7 Beam Pipe Radius mm 81.6 Solenoid Parameters Unit Value Tol B0 T 7 0.25 B0.5 T 0 0.02 B1 T 1 0.025 B2 T 0 0.5 Coil Geometry Parameter Unit Coil 1 Coil 2 Geometry – B5-DEMO-MAG-2.4 Inner Radius mm 285 185 Length mm 211 63.4 Radial Thickness mm 76.2 71.7 Z Centre Position mm 251.8 88.1 Pancake length mm 12 12 Spacer length mm 7.9 13.7 Number pancakes – 11 3 Current Density A/mm2403.5 632.3 RF Cavity Center-to-centre distance mm 188.6 Gradient E0 MV/m 30 Iris Radius mm 81.6 Number of RF Cells 3 Frequency GHz 0.704 Synchronous Phase degree 20 Window Thickness mm 0.1 Window Material Beryllium Wedge Material LiH Opening Angle degree 10 Thickness mm 20 Alignment Horizontal Dipole Length mm 100 Polarity + - - + Field T 0.2 Z Centre Position mm 160 Field Direction Vertical Table 12: Design parameters for the Muon Cooling Demonstrator cooling cell. 36 3.4 Software Development 3.4.1 BDSIM for 6D cooling The rectilinear and final cooling systems each comprise around 20 distinct configurations of cooling cells, differing in geometry, length, magnetic field strength, RF gradient, and frequency [21]. The design and performance evaluation of these systems relies heavily on simulation studies. At present, G4Beamline [38] is the primary tool used within the MuCol study, with RFTrack [39] also applied to final cooling studies. Employing multiple simulation frameworks is essential for cross-validation, providing robustness in design choices and mitigating the limitations of any individual code. In this context, additional tools such as BDSIM can further strengthen the reliability of cooling studies. The Beam Delivery Simulation software (BDSIM) [40], developed in Europe for modelling accelerator lattices with beam-intersecting devices, has been employed in studies of future collider facilities including FCC-hh, CLIC, and ILC. In contrast, ionisation cooling simulations have historically relied on G4Beamline, which has not been actively maintained in recent years, highlighting the need for a more flexible and regularly updated alternative. BDSIM provides such a framework: it is based on Geant4 [41], benefits from active development, and offers enhanced extensibility. Early work on extending BDSIM for muon cooling, focused on implementing an ionisation cooling–dedicated beamline element, was carried out by L. Nevay [42]. This section succinctly summarises the development work which enabled using the BDSIM code for 6D ionisation cooling simulations. A detailed report of the work can be found in [43]. A new BDSIM element, muoncooler, has been implemented to enable the simulation of ionisation cooling systems. The element currently supports the four types of components required for simulating a rectilinear cooling cell: solenoid and dipole magnets, RF cavities and wedgeor cylindrically-shaped absorbers. Hence, it can be used to define and simulate a complete 6D muon cooling lattice. The fields from each element are summed in superposition to yield a 6-vector electromagnetic field at all points in the element. This ensures that fringe effects from all magnets are accounted for across the entire lattice. A 3D rendering of an example cooling channel consisting of three cells with two absorbers (wedge and cylindrical) is shown in figure 35. Figure 35: A 3D rendering of a cooling channel simulated using BDSIM. Field models The dipole magnets used in the rectilinear cooling lattice have a large aperture relative to their on-axis length. Hence, the dipole field is fringe-field dominated – fringe fields are regions at the end of a magnet where the field transitions from a nominal value inside the magnet to zero field or to the field in a neighbouring magnet. To model such a field more realistically, we implemented a dipole field model derived from the treatment described by Muratori et al. in [44], which models the fringes as Enge functions. A simplistic hard-edge model dipole field was also implemented. Additionally, two models have been implemented to estimate the magnetic field produced by a solenoid coil. In the simplest case, the solenoid is modelled as a single cylindrical current-carrying sheet. The analytic expressions for the field components follow the treatment outlined by Derby et al. [45]. For a more realistic three-dimensional representation of a solenoid coil, multiple such 37 sheets can be layered at increasing radii to approximate a cylindrical block (annulus), thereby capturing the spatial extent and field distribution of a physical solenoid more accurately. The solenoid models were validated against G4BL field maps, and a like-for-like tracking study through a single solenoid (figure 36) shows good agreement between BDSIM and G4BL. Details of all model validations can be found in the original report [43]. Figure 36: Comparison of particle positions after tracking through a single solenoid using BDSIM and G4Beamline. Preliminary 6D cooling simulations In addition to validations of the individual components, a preliminary demonstration of a 6D cooling lattice was carried out. A 80 m-long lattice based on the Muon Cooling Demonstrator cell design presented in [46] was simulated. Figure 37 shows the transverse and longitudinal emittance evolution along the channel. A mismatch occurs in the longitudinal phase space leading to an initial emittance growth, causing losses in the beam tails. Additionally, the mismatch leads to beam momentum oscillations which translate into an oscillating behaviour of the longitudinal emittance. These effects could be mitigated by preparing an appropriately matched beam. A like-for-like tracking study comparison with G4BL will be conducted as part of this benchmarking exercise. 0 10 20 30 40 50 60 70 80 z [m] 1.4 1.6 1.8 2.0 2.2 2.4 [mm] Transverse Emittance 0 10 20 30 40 50 60 70 80 z [m] 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 L [eV ms] Longitudinal Emittance Figure 37: The evolution of the (left) transverse and (right) longitudinal emittances across a ∼ 80 m lattice. 38 3.5 Alternatives and Extensions In addition to the core concepts discussed above, MuCol has considered development and integration with neighbouring subsystems and alternative designs that may be more performant than the baseline. Studies have been undertaken to develop the longitudinal capture system which is the main subsystem between the target area and the cooling system. After the longitudinal capture system, the baseline is to separate the muon charge species and then transport beam into the rectilinear cooling channel. Charge separation of the high emittance beam is more complex than a low emittance beam. An attractive alternative is to develop a combined cooling line that can cool both µ+and µ−, which also has potential to reduce cost. At the downstream end of the cooling channel, preliminary studies have been made for a cooling ‘after-burner’ which can yield lower emittances and improve facility performance. 3.5.1 Longitudinal Capture Longitudinal capture is achieved by means of a longitudinal drift, followed by a buncher region and then a region of phase-energy rotation. In the drift, faster particles travel towards the head of the bunch and slower particles fall back towards the tail, enabling an energy-time correlation to develop. Once sufficient correlation has developed, the beam enters the buncher region. RF cavities are placed in this region such that cavities further downstream have successively higher voltages. As the beam still has a distorted energy profile, the head of the bunch is moving more rapidly than the tail; successive cavities must be lower frequency to accommodate the increasingly stretched time profile of the beam. Once the beam is bunched, the rotator region is used to rectify the energy-time correlation. The RF cavity frequency and phase is offset so that the fast head of the bunch is in a decelerating region and the slow tail of the bunch is in an accelerating region. This results in a beam that has a much reduced energy spread and is separated into well contained pulses that are synchronous with the RF cavities and ready for cooling. The entire system is contained within a focusing solenoid. Parameter Unit Value Drift length m Buncher length m 10 Rotator length m 50 Rotator phase 1 degree -5 Rotator phase 2 degree +10 Number of Bunches 10 Rotator Voltage MV/m 15 Solenoid Field T 2.0 Table 13: Sample parameters for a muon collider front end. The design was performed using a bespoke tracking calculation to project reference trajectories through the RF system. The trajectories of two particles, initially synchronous, but with momenta consistent with the high-momentum and low-momentum portions of the beam, was propagated through the system. Where an RF cavity was required, the time difference between the reference trajectories was used to estimate the correct frequency and phase so that particles received a desired energy kick. The performance of the system was then estimated by tracking particles through the cooling channel using the G4Beamline tracking simulation. Overall the longitudinal capture may be modelled by several parameters, which are listed in Table 13 together with some representative values. The parameters were selected by means of a twostage optimisation. In the first stage, the parameters were adapted to yield reference trajectories having the required momentum and final RF frequency having the required frequency, chosen to match the rectilinear cooling channel design specification. This was achieved by varying the phase offsets and length of the longitudinal drift section. Because this optimisation can be performed only by calculation of the reference trajectories described above, this first optimisation was done by a straight-forward SIMPLEX optimisation. The calculation is very quick and, if a solution is found, converges in a few seconds. Non-convergence occurs if there is insufficient voltage in the 39 Figure 38: Preliminary simulated performance of the longitudinal capture system for a number of different parameters. Performance is shown for 352 MHz RF as a function of solenoid field (left) and for 176 MHz RF as a function of the number of 250 mm rotator cells (right). rotator to bring the reference trajectories to the requisite energy or if the buncher and rotator are too long so that the frequency becomes lower than required, even for a short drift section. The performance optimisation was performed in the second stage optimisation where the other parameters were allowed to vary. The problem is non-linear, under constrained and multivariate so a machine learning algorithm was employed. Particles were considered captured in the system if they were transmitted through to the end of the cooling channel and contained within an RF bucket. RF containment was estimated by tracking particles through a toy 200 cell RF system at the downstream end of the capture. This test RF system modelled a stationary bucket. Particles were considered within the RF bucket if they did not move past an unstable fixed point over the 200 cells. Optimisation has been performed for 176 MHz RF cavities and 352 MHz RF cavities. 352 MHz corresponds to the cooling system design described above, while operation at frequencies below 176 MHz necessitate complex cavity geometries that limits achievable real-estate gradient and iris radius. Performance of the two systems is shown for all configurations that the optimisation routine modelled in figure 38. 352 MHz cavities have narrow irises compared to those at 176 MHz. As discussed above upstream in the longitudinal capture system, higher frequency cavities are required than 352 MHz to generate a beam matched to 352 MHz. These narrow irises cause considerable transverse losses that are hard to mitigate. Studies have been performed with stronger solenoids to reduce transverse loss. The results in figure 38 show around a factor 3 improvement by using higher solenoid field at 352 MHz, which corresponds to the increasing focusing strength. Good performance was observed with a lattice delivering beam at 176 MHz across a wide range of parameters without the need for an aggressive optimisation campaign. However, as described in Section 3.1 352 MHz RF has been assumed for the initial cooling stage. It may be possible to develop a dual-harmonic RF system that can split the RF bucket into sub-buckets in order to capture into the lower RF frequency while maintaining the better performance at 176 MHz. Further consideration should also be given to capture efficiency of both muon charge species; RF buckets are displaced by half an RF period between charge species which may yield a significant difference in capture efficiency. Pion production in this energy range tends to favour positive charge species owing to the positive charge of protons striking the target. 3.5.2 Dual-Sign Cooling A dual-sign cooling system enables cooling of both µ+and µ−in the same equipment. This is expected to reduce the capital and operating cost of the cooling system. Separation of the two charge species may be undertaken later in the system at lower emittance, likely reducing transmission loss. On the other hand, beam loading, which may limit the available beam current, will be stronger. 40 Figure 39: Schematic of a section of the cooling system. The beam moves from left to right, near to the axis of solenoid coils (grey), RF cavities (red) and wedges (blue). Wedges are offset by 120◦ between cells. A dual-sign cooling system, dubbed a ‘helical Focusing Focusing’ lattice (hFOFO), was previously proposed. A schematic of the proposed system is shown in 39. As in the rectilinear cooling scheme, focusing is achieved via alternating-polarity solenoids. Dispersion is generated by tilting the solenoids to introduce an effective dipole field. The dipole field is rotated by 120◦rotation about the solenoid axis relative to the previous field. The dipole is periodic with three cells of the solenoid lattice, resulting in a dispersion function with the same periodicity. Solenoid focusing is independent of the charge species but the dispersion has opposite advance for different charge species. The dispersion in a solenoid is a vector defined by x y= pdx dp pdy dp !(1) Owing to the symmetry of the lattice, the vector is offset between the two charge species but the vectors do not have opposite direction. Simultaneous cooling is achieved by offsetting the wedge so that only one sign intersects the absorber. IMCC has begun preliminary investigation of this system. Lattice files previously prepared by the US-MAP project have been prepared and preliminary simulations commenced. Investigations are ongoing to study separated function dipoles which would enable steering correction independent of the solenoid field. 3.5.3 Low energy muon cooling There has been significant interest in low energy muon cooling of positively charged muons. Low energy muon cooling takes advantage of the fact that muons travelling at velocities below about 0.01chave their energy compressed. Fast muons lose more energy than slow muons until they stop completely. These techniques are not suited to negative muons, which bind strongly to atomic nuclei and become unavailable. Concepts are under active development at PSI and JPARC. The PSI scheme uses a modified Wien filter filled with rarefied helium having a density gradient [47]. A Wien filter uses crossed electric and magnetic fields to select particles having a specific velocity. The helium gas effectively reduces the electric field gradient. By employing a density gradient, off-velocity particles are 41