the frequencies around $1.3\text{\,}\mathrm{GHz}$ are well suited in terms of
achievable gradient, further studies considering bucket area, beam loading and
wakefields could force to move to a lower RF frequency. Once the suitable
frequency range has been fixed, the exact RF frequency remains to be
determined based on arguments like the matching in frequency with the muon
pre-acceleration and cooling stages.

While the focus has been on the optimization of separate beam dynamics
simulations for the transverse and the longitudinal planes, full simulations
of the beam dynamics in six dimensions (6D) have yet to be performed and it
remains to be clarified which tracking tool is suitable or could be extended
to the parameters of the muon RCS chain.

The strong transient beam loading with the high-intensity single bunches also
poses challenges to the low-level RF feedback systems. A gated 1-turn delay
feedback is being considered, but the conceptual design must be made to
estimate its benefit and RF power requirement.

The extremely fast acceleration of muons in RCSs extends the synchrotron
concept in uncharted territory in terms of parameters, which are beyond those
of present accelerators. Already the conventional design tools had to be
carefully scrutinized and extended to make sure that they remain applicable
for the muon RCSs. No existing RCS has a comparable circumference, accelerates
similarly fast and with such a large synchrotron tune. In addition the concept
of interleaving super- and normal-conducting magnets is novel.

We could use existing accelerators to test operation with parameters at least
beginning to approach the parameters required for the muon RCSs. For example,
synchrotron tunes as large as $0.1$ could be tested with ions at injection
energy in the CERN SPS, to benchmark the consequences of discretised
synchrotron motion. Further beam tests may be possible in the existing RCS at
JPARC or the $8\text{\,}\mathrm{GeV}$ Booster synchrotron at Fermilab, both
accelerating proton beams within few $10^{4}$ turns.

###  0.5.4 Collider ring

Present work on the muon collider ring design focuses on a 10 TeV center of
mass machine.

#### Collider design overview

The optics design of the collider design comprises:

  * •

The interaction regions (IR) with a straight section housing the detector and
an inner focusing triplet. This is followed by a longer (almost) straight
section possibly with weak bendings and/or quadrupoles. In the present
scenario, injection, extraction for beam dumping, RF and other systems will
use the straights available in this region.

  * •

A chromatic compensation section (CCS) correcting chromatic aberrations
generated due to the small $\beta^{*}$ required to achieve sufficient
luminosity.

  * •

A matching section (MS) connecting the CCS to the arc cells. Present efforts
aim at designing an MS with sufficient tuning margin for the betatron phase
advances to allow adjustments of the working point.

  * •

The arc will be made of negative momentum compaction cells required to tune
the momentum slip factor of the whole collider ring to very small values
required to keep the bunches short ($\approx 1.5$ mm) over about 1000 turns
contributing to luminosity.

The following effects related to muon decays have to be mitigated:

  * •

Electrons and positrons lead to showers either by synchrotron radiation
generated by them towards the outside of the ring or where they impact on the
inside of the ring. Tungsten absorbers installed inside the magnet aperture
together with residual heat load and radiation to the magnet are described in
Section 0.6.5. The cryogenic system is dimensioned to cope with the residual
heat load from muon decays and is described in Section 0.6.7.

  * •

Neutrinos leave the ring almost tangential to the design trajectory and reach
Earth’s surface close to the plane defined by the collider ring with hot spots
in the direction of straight sections. Dose levels generated by these muons
reaching Earth’s surface or rather showers created by them are described in
Section 0.6.10. Measures to ensure that dose levels remain at negligible
levels are described and comprise to limit the length of straight sections
outside the IR and mechanical system described in Section 0.6.12 to
periodically deform the machine in vertical direction.

Figure 0.5.11: Twiss functions in the collider ring from the IP through the
final focusing, the CC section up the beginning of the matching section.

#### Key challenges

A major challenge of the optics design are the chromatic aberrations caused by
the small Twiss betatron function at the interaction point
$\beta^{*}=1.5\mbox{ mm}$ and the large rms momentum spread of
$\sigma_{\delta}=10^{-3}$ together with the high beam rigidity and maximum
quadrupole gradients. Twiss parameters in the most challenging part of the
muon collider ring are shown in Fig. 0.5.11. The design of the the local CC
section to correct these aberrations and, in particular, the control of higher
order effects is underway. The small $\beta^{*}$ and the large beam rigidity
unavoidably leads to large Twiss betatron functions at location of strong
magnets around the final focusing structure and, to a lesser extent, in the CC
section. A consequence is stringent requirements on unwanted multipolar
components to be evaluated once a working lattice for a perfect machine is
available. The rms bunch length has to be kept short (rms $\approx
1.5\mbox{mm}$) for about 1000 revolutions contributing to luminosity
production. This leads to stringent requirements for the momentum compaction
factor including higher order contributions and the need to control the
(average) dependence of the path length from the betatron oscillation
amplitudes using sextupoles. It has not yet been decided whether a high
gradient high frequency RF system will be used to slightly mitigate these
constraints.

Pushing the luminosity requires to push the fields of the magnets (see Section
0.6.1) to the maximum possible with existing technologies or, if feasible, to
develop new concepts. This is necessary to reduce the machine circumference
allowing to increase the number of revolutions contributing to luminosity
production and for a compact design of the final focusing structure.

The dose levels, where neutrinos reach Earth’s surface, will be kept at
negligible levels with the help of mitigation measures as installation of the
collider ring deep underground, avoidance of straight sections in regions
outside the straights housing the experiments by adding dipolar components to
quadrupoles and higher order multipoles (becoming combined function magnets)
and wobbling, i.e. periodic deformation of the ring in the vertical direction
as described in Section 0.6.12, of the machine. Challenges are the required
high precision movement system and to ensure that adverse effects on the
machine optics and performance can be controlled.

Challenges generated by muon decay products (electrons and positrons and
synchrotron radiation generated by them) and showers generated by them and
their interception by tungsten absorbers is described in Section 0.6.5.
Challenges due to the heat load from residual shower particles not intercepted
by the absorber are described in section 0.6.7 on the cryogenic system.

The machine-detector-interface (MDI) with tungsten absorbers and nozzles to
minimize beam induced background seen by the detector is described in Section
0.3.2.

#### Work since the publication of the European LDG roadmap

Substantial progress has been made on the design of a lattice for a 10 TeV
center of mass collider [13]. Despite significant improvements of the
transverse acceptance for off-momentum particles, a lattice working for
nominal beam parameters is not yet available even for a perfect machine.
Nevertheless, relaxing slightly requirements (increasing $\beta^{*}$ and the
bunch length and decreasing the momentum spread by the same factor) would
allow to find such a lattice.

Progress of Beam Induced Backgrounds (BIB) studies and Machine Detector
Interface (MDI) design optimizations are described in Section 0.3.2. BIB
mitigation strategies elaborated with the collider lattice design are
described there.

Progress on studies of radiation at Earth’s surface caused by neutrinos
generated by muon decays comprise FLUKA studies, an improved "source term"
describing radiation from a decaying muon beam without divergence and a
procedure to take the properties of the lattice into account. A detailed
description is given in Section 0.6.10.

#### Work planned for the evaluation report

Optimization aiming at designing a lattice with sufficient transverse
acceptance over the whole relevant momentum with the nominal $\beta^{*}$ will
be continued. If no such lattice is identified, parameters may have to be
slightly relaxed (increase of $\beta^{*}$ and bunch length leading to smaller
momentum spread).

Starting with a working lattice for a perfect machine without machine
imperfections, realistic scenarios will be investigated. Imperfections to be
added comprise component misalignments, magnet strength errors (possibly
caused by limited precision of power converters) and unwanted multipolar
components of magnets. Initially, a lattice without $\beta^{*}$ squeeze will
be assumed. Afterwards, correction of imperfections based on observations with
the beam will be simulated. As a result of these simulations, specifications
for component alignment tolerances, maximum magnet strength uncertainties and
maximum unwanted multipolar components will be obtained. In case some of these
conditions, e.g., maximum unwanted higher magnetic field multipolar components
cannot be met in practice, an optimized lattice design may allow to improve
and parameters may have to be relaxed.

The impact of periodic machine deformations to mitigate radiation from
neutrinos reaching Earth’s surface described in Sections 0.6.10 and 0.6.12 on
beam dynamics will be studied. Machine deformations introduce by design
vertical dispersion inside the challenging CC section with large betatron
functions and strong sextupoles possibly leading to a substantial impact on
the dynamic apertures and machine performance.

#### Next priority studies and opportunities for additional collaboration

Present studies concentrate on a $10\text{\,}\mathrm{TeV}$ center of mass
collider and no further investigations on a lower energy machine are underway.
In case a staged implementation of muon collider facility with a lower energy
collider as first stage is envisaged, thorough studies in addition to the
existing $3\text{\,}\mathrm{TeV}$ center of mass muon collider design from
MAP[1] are necessary.

Interplay of machine nonlinearities and beam-beam effect have not yet been
looked into in detail but drive the design and alignment tolerances to ensure
that the expected performance is reached. Further studies on beam-beam effect
have to be clarified.

###  0.5.5 Collective effects

#### System overview

Collective effects are a general concern over the whole muon collider, from
the proton driver to the collider ring. Different mechanisms will dominate the
system, depending on the particle’s type, their energy, the bunch emittance
and intensity.

In the proton system, the high-intensity and medium energy of the proton
bunches will make them prone to collective instabilities from space-charge
effects and interplay with impedance, in particular in the accumulator and
compressor rings. In the muon cooling system, collective effects created by
beam-matter interaction in the ionisation cooling system could make the muon
bunches unstable or lead to an increase of the emittance. These effects will
be investigated as part of muon production and cooling described in Section
0.5.2. In the muon acceleration system, the two high-intensity, counter-
rotating muon and anti-muon bunches will undergo fast acceleration to meet the
survival rate goal. The numerous RF cavities required for acceleration will
generate strong wakefields that could degrade the bunch emittance and lead to
intensity losses. Beam-beam effects will also be present in the droplet arcs
of the recirculating linacs, at two crossing points in the rapid cycling
synchrotrons (RCS) and at the two interaction points in the collider ring. In
the latter, the beam chamber will generate resistive-wall wakefields whose
magnitude will be determined by the chamber radius and materials used.

#### Key challenges

In the first RCS, an energy gain per turn of $14.7\text{\,}\mathrm{GeV}$ is
required to reach the $90\text{\,}\mathrm{\char 37\relax}$ survival rate of
the muon beams. To obtain such large accelerating gradient, $O(700)$ TESLA
superconducting RF cavities operating at $1.3\text{\,}\mathrm{GHz}$ [14] and
with an individual accelerating gradient of
$30\text{\,}\mathrm{MV}\text{\,}{\mathrm{m}}^{-1}$ will be needed, as
described in Section 0.5.3. The cavities high-order modes (HOMs) will generate
short and long-range wakefields. Combined with the high bunch intensities of
$2.7\text{\times}{10}^{12}$, $2.4\text{\times}{10}^{12}$,
$2.2\text{\times}{10}^{12}$ and $2.0\text{\times}{10}^{12}$ muons per bunch at
injection of RCS1, RCS2, RCS3 and RCS4 respectively, the strong wakefields
could disrupt the beam motion and lead to emittance blow-up and particle
losses. Moreover the fast acceleration the two bunches will undergo in the RCS
(only 17 turns for RCS1) would require the transverse damper to have a fast
response to efficiently mitigate the coherent beam instabilities. Sextupoles
magnets could be introduced to correct and control the RCS lattice
chromaticity and mitigate potential head-tail instabilities, but this would
however require further space in the RCS lattice.

In the $3\text{\,}\mathrm{TeV}$ and $10\text{\,}\mathrm{TeV}$ collider rings
the beam chamber will generate short-range resistive-wall wakefields. To reach
the highest possible magnetic field in the dipole, the magnet aperture should
be as small as possible and must host a $4\text{\,}\mathrm{cm}$ thick tungsten
shielding to intercept muon decay products, as described in Sections 0.6.1 and
0.6.5 for the $10\text{\,}\mathrm{TeV}$ collider. Because of the absence of
synchrotron radiation damping mechanism, the transverse normalised beam
emittance of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}\mathrm{rad}$
must be preserved over the storage time to reach the target luminosity. The
inner radius of the tungsten shield and its material properties, such as the
operating temperature or the use of a low electrical resistivity coating, will
influence the resistive-wall wakefield and determine the coherent stability
threshold. At the interaction points, strong beam-beam effects will arise from
the high bunch intensity of $1.8\text{\times}{10}^{12}$ muons per bunch at
injection and the small $\beta^{*}$ target of $1.5\text{\,}\mathrm{mm}$. The
short bunch length of $\sigma_{z}=$1.5\text{\,}\mathrm{mm}$$ could also lead
to beam induced heating in some devices. However the strength of collective
effects will reduce over the beam storage time in the collider thanks to the
muon decay. With a $5\text{\,}\mathrm{TeV}$ beam, the Lorentz factor
$\gamma=47323$ and the muon lifetime in the laboratory frame $\tau$ is
$\tau=\gamma\tau_{0}=47323\cdot$2.2\text{\,}\mathrm{\SIUnitSymbolMicro
s}$=$104\text{\,}\mathrm{ms}$$, equivalent to $3121$ turns in the
$10\text{\,}\mathrm{km}$ long ring.

#### Recent achievements

A transverse impedance model for the RCS1 was created using the PyWIT
framework [15], assuming the accelerator comprises 670 superconducting TESLA
type RF cavities. The HOMs used are those of the low loss type TESLA cavities
described in Ref. [16]. The three modes with the largest shunt impedance
$R_{s}$ are reported in Table 0.5.2. Figure 0.5.12 shows the real part of the
transverse horizontal dipolar beam coupling impedance obtained with this
model. The vertical impedance is assumed to be identical to the horizontal
one.

Table 0.5.2: List of transverse HOMs from the Low Loss TESLA cavity, as described in Ref. [16]. Frequency $f_{\mathrm{res}}$ [MHz] | Shunt Impedance $R_{s}$ [M$\Omega$/m] | Q factor [$10^{4}$]
---|---|---
1927.1 | 3865.11 | 1.5
2451.07 | 6155.63 | 10.0
2457.04 | 4314.02 | 5.0
Figure 0.5.12: Real part of the horizontal dipolar impedance for the RCS1,
including the HOMs created by 670 TESLA type RF cavities. The plot is zoomed
on the $1\text{\,}\mathrm{GHz}4\text{\,}\mathrm{GHz}$ range to highlight the
HOMs.

Transverse beam dynamics simulations were performed with the PyHEADTAIL [17]
macroparticle tracking code. To handle the large acceleration gradient present
in the RCS, the RF cavities must be distributed over multiple RF stations
along the ring as detailed in Section 0.5.3. Specific modules were developed
to handle in PyHEADTAIL simulations of this particular configuration of the RF
cavities [18, 19, 20].

A first stability criterion was establish through tracking simulations to find
the maximum single high-order mode shunt impedance $R_{s}$, resonance
frequency $f_{\mathrm{res}}$ and quality factor $Q$ admissible in a cavity. To
guarantee single bunch stability with a single HOM, in single-turn wakefield
regime
$R_{s}<$100\text{\,}\mathrm{M\SIUnitSymbolOhm}\text{\,}{\mathrm{m}}^{-1}$Q/f_{\mathrm{res}}^{2}$
where $f_{\mathrm{res}}$ is expressed in $\mathrm{GHz}$ whereas in multi-turn
regime
$R_{s}<$10\text{\,}\mathrm{T\SIUnitSymbolOhm}\text{\,}{\mathrm{m}}^{-1}$$.
When all the cavity HOMs are included in the impedance model, tracking
simulations show that there is a factor of two margin on the total transverse
shunt impedance $R_{s,\mathrm{tot}}$ of the modes, assuming that all modes
shunt impedance can be summed as we remain in the single-turn wakefield
regime.

These simulations were performed assuming that the chromaticity $Q^{\prime}$
is corrected to zero (i.e. using sextupoles). Assuming the RCS lattice will be
a FODO type and an average Twiss beta function of
$\overline{\beta}_{x,y}=$50\text{\,}\mathrm{m}$$, the transverse tunes can be
approximated to $Q_{x,y}\approx C_{0}/(2\pi\overline{\beta}_{x,y})=19$ with
$C_{0}=$5990\text{\,}\mathrm{m}$$ the machine circumference. The chromaticity
$Q_{x,y}^{\prime}$ was scanned from $-19+19$. With the impedance model
pictured in Fig. 0.5.12, no transverse emittance blow-up was found after 17
turns of acceleration, both with $Q^{\prime}=-19$ and $Q^{\prime}=0$. Applying
a factor two on the impedance, a large emittance blow-up appears with
$Q^{\prime}=-19$ and $Q^{\prime}=+19$. With smaller chromaticity values such
as $Q^{\prime}=-4$ or $Q^{\prime}=+4$, the transverse blow-up is still present
but limited to a few percent. Simulations including a small initial transverse
offset of the beam were also started in order to assess the impact of
injection errors on transverse coherent stability, scanning the initial
horizontal beam offset in the $0.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$ range. A transverse damper
located in the second RF station after the injection point would be required
to damp the bunch centroid motion. For larger offset such as
$1\text{\,}\mathrm{mm}$, either the damper strength should be quite strong (in
the 4-turn range) or multiple damper units should be used.

In the $10\text{\,}\mathrm{TeV}$ collider ring, the resistive-wall impedance
from the dipole magnets vacuum chambers would be the main source of
wakefields. Impedance and beam stability simulation results performed for the
$10\text{\,}\mathrm{TeV}$ collider are described afterwards. Similar studies
were also done for the $3\text{\,}\mathrm{TeV}$ collider and can be found in
Ref. [21]. Assuming that the innermost surface seen by the beam is the
tungsten shield, impedance models were created for a range of radii between
$10\text{\,}\mathrm{mm}$ and $40\text{\,}\mathrm{mm}$. To reduce the chamber
impedance, the inner surface of the tungsten shield would be copper-coated.
The material temperatures are assumed to be $300\text{\,}\mathrm{K}$, as
described in Section 0.6.7. A scan on the copper coating thickness was also
performed, from $0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to find the minimum thickness
required to shield the tungsten material. The impedance models were generated
using the ImpedanceWake2D code [22]. The chamber and material parameters used
in simulations are reported in Table 0.5.3. Figure 0.5.13 shows the transverse
dipolar impedance for a $23\text{\,}\mathrm{mm}$ radius chamber, and for
different copper coating thicknesses.

Table 0.5.3: Chamber parameters used in simulations for the $10\text{\,}\mathrm{TeV}$ collider. Chamber length and geometry |  |  |  $10\text{\,}\mathrm{km}$, circular
---|---|---|---
Copper coating thickness |  | $\mathrm{\SIUnitSymbolMicro m}$ |  0.1, 1, 10, 100 $+$ no coating $+$ infinite
Copper resistivity at $300\text{\,}\mathrm{K}$ | $\rho_{\mathrm{Cu},300\,\mathrm{K}}$ | $\mathrm{n\SIUnitSymbolOhm}\text{\,}\mathrm{m}$ | $17.9$
Tungsten resistivity at $300\text{\,}\mathrm{K}$ | $\rho_{\mathrm{W},300\,\mathrm{K}}$ | $\mathrm{n\SIUnitSymbolOhm}\text{\,}\mathrm{m}$ | $54.4$
Figure 0.5.13: Transverse dipolar impedance for a $23\text{\,}\mathrm{mm}$
radius tungsten shield in the $10\text{\,}\mathrm{TeV}$ collider. Left plot
shows the real part, right plot the imaginary part. Different copper coating
thickness are assumed, and are represented alongside two models assuming an
infinite copper thickness (orange curve) or tungsten thickness (blue curve).

Tracking simulations using PyHEADTAIL were performed with these impedance
models [23, 20]. They allowed to find the minimum chamber radius needed to
ensure transverse coherent beam stability, as well as the copper coating
thickness that should be used to reduce impedance effects. The muon decay
effect has to be included in simulations since the bunch intensity reduction
over time will reduce the coherent effects strength. A PyHEADTAIL module was
therefore implemented to reproduce this effect in simulations.

Figure 0.5.14 summarises the results of the tracking simulations. The minimum
chamber radius required to keep the beam stable is plotted as a function of
the transverse damper strength. Each curve corresponds to a different copper
coating thickness. For coating thickness larger than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the curves are superimposed on the
pure copper case. It shows that a copper coating of at least
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is required to shield the tungsten.
Results of simulations showed that if a tungsten chamber with a
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ copper coating is used, with a
50-turn transverse damper gain, the minimum chamber radius required is
$13\text{\,}\mathrm{mm}$, smaller than the $23.5\text{\,}\mathrm{mm}$
currently assumed for radiation shielding studies.

Figure 0.5.14: Minimum chamber radius required to keep the emittance growth
below $10\text{\,}\mathrm{\char 37\relax}$ after 3000 turns, for the different
copper coating thickness considered and pure copper or pure tungsten (no
copper coating).

For both RCS and collider rings the impedance models and beam dynamics codes
are made available on a dedicated Git repositories and can be found at
https://gitlab.cern.ch/muc-impedance/muc-impedance. A web page was also
developed to visualise the developed impedance models and is available at
https://muc-impedance.docs.cern.ch/.

#### Planned work

Impedance models for the RCS2, RCS3 and RCS4 will be created, following the
machine parameters reported in Section 0.6.3. Simple models for the resistive-
wall impedance of the RCS are also currently being investigated, in particular
for the pulsed magnet chambers. Because of the fast magnet ramping, in the
order of $\dot{B}=$4200\text{\,}\mathrm{T}\text{\,}{\mathrm{s}}^{-1}$$ for the
RCS1 for example, ceramic chambers with outer RF shielding stripes such as
those used in the J-PARC RCS [24] must be considered. The tracking simulations
will be ported to the XSuite [10] framework. This will allow to perform start-
to-end tracking simulation in the different RCS as well as in the collider
ring, without and with collective effects. The study of the effect of an
initial transverse offset of the beam will be refined to better assess the
required strength and location of the transverse damper. Beam-beam effects
will also be included in simulations performed within this framework.

The tracking simulation framework for the collider ring will also be migrated
to XSuite. This will allow to investigate the coherent beam-beam effects in
$10\text{\,}\mathrm{TeV}$ collider, and their possible interplay with
impedance effects. Similarly to the RCS studies, the effect of an initial
bunch offset at injection will be studied through simulations. The impact on
coherent stability of the quasi-isochronicity of the ring caused by the very
small momentum compaction factor $\alpha_{p}$ will also be investigated, as
well as the effect of the higher-order momentum compaction factors.

Collective effects studies for the proton driver part will also be started, in
association with the Linac, Accumulator and Compressor ring developments
exposed in Section 0.5.1, as well as the collective effects during the
ionisation cooling developed in Section 0.5.2. In particular the effect of
space-charge, and its interplay with impedance, on the coherent beam
instability thresholds should be investigated.

#### Next priority studies and opportunities for additional collaboration

The transverse tracking simulations for the RCS are for now limited to a
simple linear optics model. The development of a detailed lattice model would
allow to refine the coherent beam stability simulations in the RCS and ensure
through start-to-end simulations the choice of beam parameters for the RCS
chain. In particular collective effects in the hybrid synchrotrons, which will
see large horizontal beam excursions in certain sections of the ring, could be
investigated.

In collaboration with the vacuum and the magnet design teams, the technology
options for the RCS normal and super-conducting dipole vacuum chamber should
be investigated to have a better estimate of the resistive-wall impedance in
the RCS.

In the collider ring, an RF system design should be investigated in
collaboration with the lattice design and RF design teams in order to evaluate
its impact on the coherent beam stability.

Experimental study of space-charge could be performed in machine development
sessions in the CERN PS to investigate beam dynamics effects that could arise
in the proton driver Accumulator and Compressor rings. Moreover these sessions
could also be used to study coherent beam dynamic effects close to transition
for isochronous rings.

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##  0.6 Accelerator technologies

###  0.6.1 Magnets

Prior to the IMCC, the US Muon Accelerator Program (MAP) study [1, 2] provided
the most complete concept of a MuC, including an overview of the magnet
requirements. The IMCC has hence taken MAP as a starting point, identified the
key magnet challenges and technology options, and evolved the magnet
configurations to provide a current set of main magnet performance parameters.
These parameters extend the feasible performance space by considering recent
advances in superconductor and magnet technology.

Work carried out so far to evolve the MAP configurations has a strong focus on
HTS. A driving reason for this are the higher field reaches possible with HTS.
But, equally important are considerations of efficient cryogenic operation and
helium inventory. Operation in helium gas at temperatures in the range of 10 K
to 20 K can provide a coefficient of performance up to a factor four higher
than a cryoplant producing liquid helium at 4.2 K. This improved energy
efficiency will reduce the impact of energy depositions from muon decay and
triggered cascades, which applies throughout the MuC complex, and vital to the
sustainability of the accelerator complex.

The sections below describe the magnet systems for the main parts of the MuC
complex, namely the muon beam production (target, decay and capture channel),
muon cooling, acceleration and collision. We will expand on the concept and
magnet engineering, making reference to specific technologies such as Low
Temperature Superconductors (LTS, Nb-Ti or Nb3Sn), High Temperature
Superconductors (HTS, mainly considering REBCO) as well as normal conducting
(NC) iron dominated magnets. More extensive descriptions can be found in the
publications cited. We also bring the attention to the internal review of
magnet technology options that was organized during the 2023 IMCC Annual
Meeting (https://indico.cern.ch/event/1250075), with a final report presented
at the Accelerator Design Meeting of 30 October 2023
(https://indico.cern.ch/event/1337221/).

#### Magnet system for muon beam production

##### System overview

Muons are produced from the decay of pions generated from the collision of a
short, high intensity proton pulse with a target. The target, varying in outer
diameter dimensions from 150 mm to 250 mm depending on technology, is placed
within a steady-state, high field “target solenoid”, whose purpose is to
capture and guide pions into a “decay and capture channel”, also embedded in
solenoid magnets. An approximately 18 m long magnetic field profile from the
target to the end of the capture channel enables the capture of pions and has
a characteristic shape with a peak field of 20 T on the target and an
adiabatic decay to roughly 2 T at the exit of the channel [3].

To prevent heating and radiation damage from the interaction of the multi-MW
proton beam with the target, a radiation shield is necessary for the target,
decay, and capture solenoids. This shield comprises a combination of a dense
metal such as tungsten (to intercept highly energetic photons generated by the
decay electrons) and a moderator like water (to reduce neutron fluence). The
shield thickness dictates the magnet bore and is chosen to reduce to
acceptable levels the radiation damage and heat load on the solenoid coils and
surrounding infrastructure. Calculations show that with a shield approximately
500 mm thick, radiation induced heating in the coils would fall within the
range of 5 kW, local radiation dose in the range of 80 MGy, and a yearly peak
DPA in the range of $10^{-3}$. These values, and especially the radiation
damage, are at the limit of present magnet technology, thus setting a minimum
magnet bore dimension of 1200 mm used for the design of the target, decay and
capture solenoids. A larger bore dimension implies higher stored magnetic
energy, influencing electromagnetic forces, magnet protection, and cost.

To achieve the peak 20 T field profile at the target, MAP considered a hybrid
normal conducting (NC) and superconducting (SC) target solenoid configuration,
consisting of a rad-hard resistive NC insert (approximately 5 T with a 150 mm
bore), within a large bore Low Temperature Superconducting (LTS) magnet
(approximately 15 T with a 2400 mm bore) [4, 5, 6]. While this concept remains
a possibility and can be built based on the extrapolation of known technology,
the system’s stored energy (approximately 3 GJ), coil mass (about 200 tons),
and wall-plug power consumption (around 12 MW, dominated by the resistive
insert) are substantial.

##### Key challenges

The main challenge in our proposed design (see below) is radiation damage of
the superconductor and insulation, both requiring further analysis and
measurements, and especially advances in understanding the damage mechanism in
HTS. Other challenges include magnet engineering (field performance,
mechanics, stored energy and protection) and infrastructure and operating cost
(power and cooling), including integration/maintenance in a high-radiation
environment. These challenges appear manageable, and overlap broadly with
those of magnets for high magnetic field science (e.g. user facilities based
on hybrid SC and NC solenoids and all-SC solenoids), as well as magnets for
fusion devices (e.g. the central solenoid magnets for a Tokamak) [7].

##### Recent achievements

Following recent advancements in HTS magnets for fusion [8], we have proposed
an alternative design involving an HTS cable operating at 20 K. The cable
considered is shown in Fig. 0.6.1, an internally cooled conductor inspired by
the VIPER developed at MIT [9]. The analysis performed thus far indicates the
potential to eliminate the resistive insert, thus reducing the magnet bore to
the minimum of 1200 mm, roughly half the size of the US-MAP LTS coil, while
still maintaining the desired field profile for efficient muon capture. If
compared to LTS, operating at a temperature higher than liquid helium
diminishes the need for radiation heat shielding and maintains good overall
energy efficiency. The proposed system is shown schematically in Fig. 0.6.2.
It has a stored energy of $\sim$1 GJ, a coil mass of around 100 tons, and a
wall-plug power consumption of approximately 1 MW. This represents a
significant reduction compared to the previously suggested hybrid solution
from the MAP study. Our future focus will be on identifying specific HTS
conductor and winding technologies suitable for magnets of this class. The
details of the electro-magnetic and mechanical design can be found in ref.
[10], while the cooling and quench analysis can be found in ref. [11].

Figure 0.6.1: MIT VIPER cable, image taken from ref. [9]. Figure 0.6.2:
Schematic view of the HTS target, decay and capture channel proposed (top),
with a detail of the high field solenoids around the proton target area
(bottom).

##### Planned work

After completing a first iteration on the electro-magnetic, mechanic, cooling
and quench protection designs, we plan now to iterate on the free bore to
resolve known mechanical interference with target and shield, and mitigate
radiation heat load and damage. Alternative cable designs and coil geometries
are being considered, to see whether a better cost optimum can be found. We
will then integrate the design in an engineering concept, including
considerations of manufacturing, installation and maintenance. This effort can
be completed within the planned timeline. It will then be important to plan
for technology demonstrators, from cable to coil, including testing efforts.
Given the size of this magnet, this work will require considerable material
and personnel investment.

##### Next priority studies and opportunities for additional collaboration

Besides the technology demonstrators already quoted earlier, an important
study not planned with available resources is the characterization and
understanding of radiation damage in HTS. This is a recent field of research,
data is scarce and we still lack a sound understanding of the mechanism by
which particle radiation induces degradation of the critical properties of the
superconductor. Such understanding is mandatory to extrapolate measurements to
the conditions experienced by the magnets in a MuC, and thus reliably predict
the lifetime.

#### Magnet system for muon cooling

##### System overview

Muons exiting the target, decay and capture channel have a large spatial and
energy spread, and thus must be cooled in both dimensions to reach values
suitable for a collider. This 6D “beam cooling” process occurs over a long
(almost 1 km) sequence of tightly integrated absorbers (consisting of light
elements such as hydrogen), solenoids, and RF cavities. The solenoids create a
broadly increasing oscillating magnetic field in the beam direction, that when
combined with absorbers and re-acceleration (RF cavities), cools the beam in
the six dimensions of the beam phase space (position and momentum). The final
emittance of the muon beam is inversely proportional to the strength of the
final cooling solenoids. To achieve maximum cooling efficiency and minimize
the emittance there is hence a clear interest in very high field final cooling
solenoids.

The MAP design study provided a baseline of magnets (geometry and number) over
the entire cooling chain (up to the final cooling solenoids) which could
produce the desired on-axis field profile [12]. However, it is important to
note that these solenoids were created purely to best fit a desired field
profile, rather than from a magnet engineering point of view, and therefore
present un-optimized parameters (stresses, forces, volume of conductor) which
pose significant challenges. In the MAP version, in total there are
approximately 2400 solenoids in cooling channels of approximate 1 km length
(excluding the final cooling), which are divided into a total of 826 cooling
cells. There are 12 types of cooling cells (labeled A1 to A4, B1 to B8)
composed of an absorber, one or two RF modules, and two, four or six solenoids
in split pairs (depending on cell type).

For the final cooling solenoids, the MAP scheme consisted of 17 solenoids with
a bore field up to 30 T. With this field, an emittance of about 50 µm, roughly
a factor of two greater than the transverse emittance goal (25 µm), can be
achieved [12, 13]. However, other analyses [14] show that higher fields (in
the range of 50 T) offer further gains in the final emittance and beam
brightness.

##### Key challenges

As detailed in the next section, individual solenoids and cells based on the
MAP configuration pose challenges in terms of peak fields, high forces and
stresses, and quench management. The range of specifications is however within
what can be manufactured already with present technology. Indeed, it is rather
the integration of solenoids within a cell containing RF cavities and
absorbers, as well as in a series (where the field of one cell affects
neighboring ones), which poses the primary difficulty in this aspect of the
overall MuC. Furthermore, the large number of solenoids in the 6D cooling
section will require standardization and effective industrialization.

For the final cooling, in the scope of improving upon the design by the US-MAP
study, the fundamental challenge will be developing ultra high-field solenoids
(as explained in the next section) with magnetic fields reaching beyond the
current state of the art, more than 40 T in a 50 mm bore. The number of these
solenoids, which are nearly free standing, is limited. We hence expect that in
this case it will be the peak field that will pose the main challenge.

##### Recent achievements

Starting from the MAP baseline, the first step has been to characterize the
parameters of all of the solenoids during operation, as shown in Table 0.6.1.

Table 0.6.1: Summary of main cooling solenoid characteristics for the 6D cooling cells of a muon collider, based on the US-MAP configuration. For a given cell type, $B_{\mathrm{peak}}$ is the peak on axis field, $\sigma_{\mathrm{Hoop}}$ is the maximum hoop stress seen by a solenoid, and $\sigma_{\mathrm{Radial}}$ shows the minimum/maximum radial hoop stress seen by a solenoid. Cell |  $J_{E}$ (A/mm2) |  Bpeak (T) |  EMag (MJ) |  eMag (MJ/m3) |  $\sigma_{\mathrm{Hoop}}$ (MPa) |  $\sigma_{\mathrm{Radial}}$ (MPa)
---|---|---|---|---|---|---
A1 | 63.25 | 4.1 | 5.4 | 20.5 | 34 |  $-$ 4.6/0.0
A2 | 126.6 | 9.5 | 15.4 | 76.3 | 137 |  $-$ 28.3/0.0
A3 | 165 | 9.4 | 7.2 | 72.8 | 138 |  $-$ 28.5/0.0
A4 | 195 | 11.6 | 8.4 | 91.5 | 196 |  $-$ 49.4/0.0
B1 | 69.8 | 6.9 | 44.5 | 55.9 | 95 |  $-$ 13.5/0.0
B2 | 90 | 8.4 | 24.1 | 61.8 | 114 |  $-$ 20.1/0.0
B3 | 123 | 11.2 | 29.8 | 88.1 | 174 |  $-$ 36.6/0.0
B4 | 94 | 9.2 | 24.4 | 42.4 | 231 |  $-$ 23.5/19.7
B5 | 168 | 13.9 | 12 | 86.3 | 336 |  $-$ 55.7/21.1
B6 | 185 | 14.2 | 8.2 | 68.3 | 314 |  $-$ 43.1/22.3
B7 | 198 | 14.3 | 5.7 | 59.6 | 244 |  $-$ 37.4/20.7
B8 | 220 | 15.1 | 1.4 | 20.3 | 119 |  $-$ 22.9/22.1

The solenoids from the MAP optics exhibit a diverse range of parameters, from
large-bore variants with a diameter exceeding 1.5 m and a modest on-axis field
of 2.6 T, to smaller-bore with a 90 mm diameter and a high on-axis field of
13.6 T. This diversity features substantial stored magnetic energy (up to 44
MJ in the single cell B1), notable associated stresses (hoop stress up to 300
MPa and radial tensile stress reaching 20 MPa in cells B5 and B6), and quench
management challenges (energy density approaching 100 MJ/m3 of a coil in cell
A4).

While the MAP study provides a baseline, the magnets were created from a beam
physics point of view (find solenoid geometries which best fit a desired on-
axis field) rather than from a magnet engineering point of view. This is
evident in Table 0.6.1, i.e. large volumes of conductor (and therefore larger
cost) are paired with lower current density, when larger current
densities/lower volumes can be considered. Therefore, significant work has
gone into creating an optimization program to revise the coil configurations
to address identified challenges while fitting the desired on-axis field to
within some error percentage. This is currently a work in progress, and will
incorporate advancements from beam optics studies.

For the final cooling solenoids, to improve upon the results obtained by the
US-MAP study, we are considering a solenoid design with the potential to reach
and exceed 40 T, a clear bore of 50 mm, a magnet length of 500 mm, and compact
enough in size to reduce overall footprint, mass, and cost. See Fig. 0.6.3. We
are studying in particular a non-insulated HTS winding solution, where the
cooling solenoid is built as a stack of soft-soldered pancakes. To reduce the
coil size, forces, and stored energy, and meet the field target, the operating
current density target is high, 650 A/mm2. The coil size is exceptionally
small, with a 180 mm outer diameter.

Figure 0.6.3: The current 40 T and beyond conceptual design for the final
cooling solenoids. It consists of 46 identical ’modular’ and six ’correction’
pancakes, with a bore aperture of 50 mm. Figures courtesy of B. Bordini.

Two critical parameters were identified in the conceptual design phase of a
final cooling solenoid, the stress state and the selection and control of the
transverse resistance which must be a good balance between the required ramp-
rate and quench management. The mechanics design features a maximum hoop
stress of 650 MPa, ensuring no tensile stress under any operating condition.
To achieve this, the wound and soldered pancakes undergo radial loading from a
rigid external ring, providing a radial pre-compression of 200 MPa at room
temperature. This radial pre-compression is carefully chosen to nearly offset
the outward electromagnetic stress at 40 T, a field range identified as the
stress limit for a single pancake. Beyond this point, higher fields would
necessitate segmenting the coil into concentric layers with independent
support, introducing added complexity and size.

Efforts are underway to address the transverse resistance, with ongoing work
focused on reducing values typically associated with soldering tapes. Given
that most current transfer in REBCO tapes occurs through the copper coating on
the tape sides, it has been observed that minimizing or eliminating this
copper component significantly enhances transverse resistance [15]. Our
objective is to achieve protection through a low transverse resistance
(potentially with mechanisms to actively trigger a quench), while still
enabling a full ramp in less than six hours and maintaining field stability at
a flat-top better than 10 pm/s. Further details on the concept, design
studies, and recent advancements are available in Ref. [15].

##### Planned work

For the evaluation of the MAP baseline 6D cooling solenoids, one important
aspect is ongoing—the quench protection analysis. In parallel, optimization
and revision of all magnets from an engineering perspective (stress, cost,
protection, etc.) as mentioned above is underway. Tight collaboration with
beam physicists is necessary to incorporate new magnet configurations to fit
evolving on-axis field profiles. Specific manufacturing options will be
explored, and the possibility of standardization will also be considered to
manage the substantial number of solenoids slated for construction.

The final cooling solenoid is a case by itself. Besides the design and
analysis work, which will include multi-physics evaluation of electrical,
magnetic (screening currents), mechanical and thermal (quench) performance, we
have initiated supporting experimental activity. Tests are planned on HTS
tapes and pancakes to provide an experimental basis for the electro-mechanical
performance limits (stress and strain, as well as in-field measurement of
delamination) and prove the feasibility of concepts. Stacks and windings are
being manufactured to establish mechanical properties, and eventually perform
tests in field to probe performance in the range of 20 T to 40 T.

##### Next priority studies and opportunities for additional collaboration

The cryogenic cooling system needs to be defined both in concept, as well as
in practical engineering solutions. For various reasons, such as paramagnetic
effects on helium, cooling by a bath of helium is not practical. Direct and
indirect cooling need to be considered to find the optimal baseline. A second
matter that will require attention is to explore manufacturing routes for the
large number of solenoids needed by the cooling channel. Finally, a full-size
demonstration of a 6D cooling cell integration, as well as the final cooling
solenoid, will be required to validate the designs.

#### Magnet system for muon acceleration to TeV energies

##### System overview

To extend their lifetime in the laboratory reference frame, the muons must be
accelerated rapidly to relativistic momentum. The acceleration begins with an
initial linear accelerator and sequence of re-circulating linear accelerators
followed by a sequence of rapid-cycling synchrotron (RCS) and hybrid cycling
synchrotrons (HCS) [16, 17]. The RCS employs fast-ramped normal conducting
(NC) magnets, swinging from an injection to an extraction field level. In a
HCS, static superconducting (SC) magnets establish a field offset, while NC
fast-ramping magnets are powered from peak negative to peak positive field,
enabling a full field swing. This design results in a more compact accelerator
compared to an equivalent RCS.

Design concepts of NC fast-ramped magnets with a peak operating field of 1.5 T
were developed by US-MAP [18]. SC dipoles for HCS were not studied in depth,
besides setting target value for bore field and magnet length.

In the current IMCC baseline configuration, the NC dipoles in the initial RCS
need to sweep from 0.36 T to 1.8 T within 0.35 ms (equating to a rate of 4
kT/s). In the concluding HCS, the dipoles sweep from $-$ 1.8 T to 1.8 T in
6.37 ms (equating to a rate of approximately 560 T/s). Studies are ongoing to
evaluate the possibility of realizing the final slower HCS pulsating magnets
with HTS REBCO based tapes and ferric core. The SC dipoles in the first two
HCS have a nominal field of 10 T, while in the final HCS the assumption is of
16 T field. All dipole magnets, NC and SC, have a rectangular nominal aperture
of 30 mm (vertical) $\times$ 100 mm (horizontal).

##### Key challenges

Beyond magnet engineering, a fundamental challenge in constructing an
accelerator ring of the required dimensions is the substantial stored energy,
reaching several tens of MJ. Efficiently powering at a high-pulse rate with
effective energy recovery between pulses necessitates expertise in managing
peak power in the range of tens of GW. Resonant circuits combined with energy
storage systems appear to be the only viable solution. This is described in
detail in the next section, devoted to power converters.

A second challenge will be mitigating the losses originating from the iron
hysteresis, eddy currents in the lamination and coils, as well as any other
metallic component exposed to field changes. For example, the vacuum chamber
is inserted into a gap where the $\frac{dB}{dt}$ is of the order of few
thousands of Tesla per second. Standard metallic vacuum chambers cannot be
used, as the induced losses would be not sustainable leading to severe
overheating or, in the worst case, melting of the chamber. Alternative
concepts are needed to reduce the impact of these losses.

##### Recent achievements

As noted in the previous section, the primary challenge of the RCS and HCS
magnets lies in effectively managing the multi-GW power necessary to pulse
them while maintaining precise control of the field ramp shape, homogeneity,
and maximizing energy efficiency. The power required for pulsing, given a
specified ramp time and shape dictated by beam design and RF limitations, is
directly proportional to the magnetic energy stored in the ramped magnets. A
lower bound for stored energy is the magnetic energy within the beam aperture,
nominally defined as 30 mm (gap) $\times$ 100 mm (width). To limit saturation,
which can impact losses and field quality, an upper design field limit of 1.8
T has been set for the resistive magnets. This corresponds to a magnetic
energy of 3.9 kJ/m in the beam aperture, with the energy stored in the magnet
being higher. Analysis of various resistive magnet configurations, considering
different iron cross-sections, materials, coil designs, and current densities,
indicates that the lowest magnet stored energy is in the range of 5.4 kJ/m,
representing a 1.4-fold increase compared to the magnetic energy in the beam
aperture [19].

The present strategy is to realize the dipoles as single turn magnets without
return connections and let the power converters connect the ends and starts
for the coils from subsequent magnets (see section on power converters). Two
type of conductors are analysed, either flat bars cooled at the ends or hollow
direct cooled conductors.

A secondary concern involves the magnitude of resistive, eddy current, and
hysteresis losses — power drawn from the grid and dissipated. Although a
definitive target has not been established, a suitable range is considered to
be 200 to 500 J/m per pulse. After analyzing multiple configurations, it has
been determined that the best compromise between stored energy, losses, and
field quality is achieved with "H" and "Hourglass" shaped iron cores [18].
These configurations will be used in further magnetic analysis, including 3D
and end effects. It is important to note that the design of the power
converter is intricately linked with magnet design and analysis, aiming for an
optimal cost solution (CAPEX+OPEX), while considering beam dynamics and RF.
The exploration of various powering configurations has given way to
identifying two main cost drivers: capacitor-based energy storage and the
active power converters necessary for controlling ramp linearity and
reproducibility.

Finally, the steady state SC dipole magnets of the HCS’s are only in the early
stages of development. While specification of these magnets is still an active
discussion, it is accepted that the aperture will be similar to that of the
pulsed resistive magnets, i.e. 30 mm $\times$ 100 mm rectangular,
necessitating non-conventional windings. Rejecting the cos-theta coil geometry
due to its inefficiency for a rectangular aperture, we have shifted our
conceptual focus toward flat racetrack coils. These coils are simpler and more
suitable for winding HTS tapes. It appears feasible to achieve a target
magnetic field of around 10 T while operating at temperatures significantly
above liquid helium (10 to 20 K) in case HTS is used. This approach would be
highly advantageous to help with operating margin and efficiency requirements
in HCS’s, which tend to experience higher beam losses and operational
challenges.

##### Planned work

Further detailed magnetic analysis, including 3D and end effects, of the
possible NC magnet configurations (“H” and “Hourglass” shaped iron cores) is
ongoing. This will lead to a final optimized design. Tests are planned on soft
magnetic steel in a range of frequency centered around 1 kHz, and field up to
saturation, where the experimental database is sparse. These tests will
provide input for the NC magnet design.

An initial design study of magnet configurations and parameters of the SC
dipole magnets is underway, with 2D and 3D analysis. This study combined with
consideration of more complex geometries, quench protection analysis, and
engineering (mechanics, cryogenics, etc.) will provide the basis for selecting
the most suitable configuration.

##### Next priority studies and opportunities for additional collaboration

Besides the design and analysis work, and the limited experimental activity,
also in the case of the fast pulsed accelerators the program would need a
validation of the integrated system made of magnet and power converter. Given
the modular nature of the design, one such test could be performed on a single
powering cell. The configuration for this test is not yet defined, but as a
minimum it should consist of NC dipoles and power converters, including the
energy storage units. Such a set-up would allow measurement with accurate
field tracking, including field quality, losses, as well as overall efficiency
of powering and energy storage.

#### Magnet system for muon collision

##### System overview

The final stage of the muon accelerator complex is the collider ring, where
two counter-rotating bunched beams of positive and negative muons collide. A
critical design parameter for the collider ring is to maximize collisions of
stored muon beams in their limited lifetime. This is achieved by minimizing
the collider ring circumference [20], translating into a requirement of the
highest possible field in the arc dipoles. At the same time, luminosity
depends critically on the strength and aperture of the quadrupoles in the
Interaction Region (IR). In fact, not only high gradients and aperture are
important for luminosity, but also a short IR length reduces the cumulative
neutrino flux generated by muon decays in the straight sections around the
experiments and reduces chromatic aberrations. Effective radiation shielding
is essential to protect against substantial radiation and heat loads generated
by muon decay and collisions. To give representative figures, the heat load
originating from muon decay (electrons) and synchrotron radiation in the arc,
initially at 500 W/m at the level of the beam chamber, is reduced to below 5
W/m at the level of the coil, with a radiation dose below 40 MGy, by inserting
a 40 mm thick tungsten shield. The need of thick shielding results in
exceptional demands on magnet aperture [21, 22].

Assumptions in the present study of the 10 TeV collider optics include the
generation of a steady-state magnetic field up to 16 T within a 160 mm
aperture by the main arc magnets. To minimize straight sections and address
effects from the high neutrino flux, these arc magnets are assumed to serve
combined functions (e.g., dipole/quadrupole and dipole/sextupole) [20]. The
latest optics necessitate dipole fields in the range of 10 T and gradients of
the order of 300 T/m. While these field requirements combined with aperture
constraints are part of an initial assessment, they currently exceed practical
limits and will necessitate iteration. As for the IR quadrupole magnets, the
assumption from optics studies is a peak field of 20 T, also associated with
large apertures, up to 300 mm.

It should be noted that the US-MAP study investigated a reduced collider
energy of 3 TeV, which could stand as a fast-track option leveraging Nb3Sn
technology nearing demonstration.

##### Key challenges

As detailed in the next section, the present optics demands high field and
large bore combined function arc and IR magnets. While this study is evolving
and practical limits being developed for these magnets, they must be designed
to withstand large stresses (mechanics), with large stored energy (quench),
and deal with significant energy deposition and radiation dose from muon beam
decay and interaction debris.

##### Recent achievements

To advance the study, specifying the performance limits of accelerator dipole
and quadrupole magnets is crucial. We have initiated this effort through
analytical evaluations of operating margin, peak stress, hot-spot temperature,
and magnet cost, assuming a sector coil geometry [23]. While the results are
specific to this coil, extending them to other coil geometries is possible and
would not significantly alter the main outcomes.

The analytical evaluation was utilized to create design charts with maximum
magnet aperture (A) versus bore field (B) (see Fig. 0.6.4), a format
convenient for iteration with the beam optics. These A-B charts were generated
for various superconductors and operating points, including Nb-Ti at 1.9 K,
Nb3Sn at 4.5 K, and REBCO at either 4.5 K or 20 K. For a 10 TeV collider, Nb-
Ti at 1.9 K appears sub-optimal due to a low operating margin (considering the
substantial energy deposition), cryoplant efficiency, and energy consumption
concerns. Similarly, Nb3Sn at 4.5 K falls short of the required field
performance for arc magnets, providing feasible solutions only up to 14 T,
being limited by mechanics when considering the necessary 160 mm aperture. Our
initial evaluation of REBCO indicates that also in this case the available
design space does not meet the required performance. However, for REBCO,
operating margin is not a concern, and operation in the range of 10 K to 20 K
could be considered. The main limitations for REBCO arise from the cost of the
superconductor and quench protection. Cost considerations drive the current
density in an all-HTS coil toward high values where standard detect-and-dump
protection strategies are not sufficiently fast. Therefore, alternative
protection schemes need to be devised to leverage the large current carrying
capacity and margin of present REBCO conductors. Assuming a reduction in the
cost per meter of REBCO tape by a factor of three to four, and relaxing the
need for very high values of current density, a suitable design range for the
arc magnets can be defined. This range spans from a nominal aperture of 160 mm
at a reduced bore field of 12 T to a nominal bore field of 16 T but with a
reduced aperture of 100 mm. The entire range can be achieved with REBCO at 4.5
K and 20 K, while the low field range can also be reached with Nb3Sn at 4.5 K,
providing at least two technology options.

Figure 0.6.4: Dipole limitation curves as a function of stress, protection,
and margin shown against bore diameter vs. magnetic field for different
conductor material and temperature. Left: ReBCO at different operating
temperatures (Top), assuming an aspirational cost of 2500 EUR/Kg, a peak
stress of 300 MPa, and a hot spot temperature of 200 K. For example, the red
line represents the limit assuming an operating temperature of 20 K plus 2.5 K
margin. Right: Nb3Sn at 4.5 K, assuming an aspirational cost of 700 EUR/Kg, a
peak stress of 150 MPa, and a hot spot temperature of 350 K. Plots are
advanced versions from ref. [23].

The analysis for quadrupoles, using the same method outlined above and in
[23], is presently in progress (see Fig. 0.6.5). The range of feasible designs
in this case covers gradients of 300 T/m, with an aperture of 100 mm, down to
gradients of 100 T/m, and aperture of 250 mm. This is only marginally short of
the present requirements for the IR optics, which indicates that the arc
dipoles are actually the most demanding magnets.

Figure 0.6.5: Limitation curves as a function of stress and margin shown
against bore diameter vs. magnetic field gradient of collider quadrupoles
consisting of ReBCO at different operating temperatures (left) or Nb3Sn at 4.5
K (right). This is a current work in progress, with final design targets still
being decided and protection limits created. Plots courtesy of D. Novelli.

##### Planned work

We will pursue the analysis of quadrupole performance limits (for the design
of the interaction regions) and combined function magnets, necessitated by
neutrino flux mitigation. We plan to finalize the "A-B" charts for the
Superconducting (SC) dipoles and quadrupoles in the collider, incorporating an
assessment of the feasible range of combined function gradient versus dipole
strength. The main difficulty will be to extend the analysis to combined
function magnets. We will most likely need to resort on parametric Finite
Element Analysis (FEA) for this.

Following an iteration on the beam optics, we will then advance to the
examination of engineering designs for the collider arc dipole and combined
function magnets. This study will encompass stress management strategies aimed
at enhancing robustness and compensating for inevitable stress peaking
factors. Additionally, it will be of interest to explore a fast-track line for
a reduced collider energy, specifically in the range of 3 TeV, as investigated
by US-MAP, leveraging the Nb3Sn technology nearing demonstration.

##### Next priority studies and opportunities for additional collaboration

The engineering design of the arc and IR magnets would require a considerable
amount of personnel resources, well beyond what is presently allocated in the
study. Although we will focus on the most challenging elements (arc dipoles
and IR quadrupoles), other magnets may need due attention. One striking
example is the Nb3Sn option that could be used to fast-track the construction
of a first collider at 3 TeV. This design is presently beyond the foreseen
scope.

As noted in Sec. 0.6.1, the effect of radiation on HTS can be an issue. The
radiation effects of the harsh and unique environment of a Muon Collider
cannot be reproduced and tested in an existing installation, thus an
evaluation of material response will need an understanding and scaling of
existing and future data. For this specific issue we plan to rely on advances
in material characterization for magnetically confined fusion.

###  0.6.2 Power converters for the muon acceleration to TeV energies

##### System overview

The quick acceleration times can only be achieved with considerable electric
power provided by the power converter. The fast $\frac{dB}{dt}$ also carries
huge voltages with it. In order to deal with both huge power and voltages
(0.6.2), the power converters of the RCS are based on resonant circuits and
must be divided into several series/parallel connected power cells. A power
cell is the basic power converter block. Many such power cells are connected
to power one sector of the accelerator. One sector is formed by series of
connected magnets, i.e. with the same current flowing trough. The RCS is
divided into a given number of sectors, each supplied by an independent power
converter. Inside one Sector, the power cells can be connected in Cell-Load
cascade or Unified Cell to load connection mode, as illustrated in Fig. 0.6.6.
The magnets and power converters will need to be designed such that they
possess the same number of independent circuits and series connected power
cells and magnets. Figure 0.6.6 shows the case of one sector with four
separated magnets each possessing four independent circuits and four series
connected power cells per circuit.

Figure 0.6.6: Connection of the power cells with the magnets.

The choice of the connection depends upon the voltage insulation level of the
magnets and the location of the power cells. The Cell-Load cascade connection
allows a much smaller voltage insulation to ground but only makes sense if the
power cells are located close to the magnets i.e. in the tunnel. The Unified
cell to load alternative, has higher insulation requirements but power cells
can be installed on the surface without having too much cable routing.

The power cell design is based on a discontinued resonance approach where the
energy is stored in one or more capacitor banks and then discharged on the
magnets with a combination of several semiconductor switches. Two different
topologies have been identified and are shown in Fig. 0.6.7. In both cases,
the LC resonance starts by bringing the current (Field) on the negative
polarity at the injection and continues the oscillation towards positive
values during the real particle acceleration phase. It then returns back to
zero where the resonance is actively stopped by the semiconductor switch.
Capacitors are then recharged to the initial value and the power converter
waits to be restarted for the next pulse.The concept is illustrated in Fig.
0.6.8.

Figure 0.6.7: Full wave (left) and switched (right) discontinued resonance
principles Figure 0.6.8: Discontinued resonance approach

##### Key challenges

The quick acceleration requires extremely high peak power sources that can
only be economically guaranteed by making large use of resonance circuits such
as the one proposed in the previous paragraph. On the other side, resonant
circuits are not very easily controllable and it is difficult to assure high
precision among the different sectors(see Fig. 0.6.6) and high repeatability
from pulse to pulse. In addition, LC resonance doesn’t match with linear
acceleration profiles as required to minimize the RF requested power. The
above parameters will have to be carefully analysed to avoid overrating of the
power converters.

Table 0.6.2: RCS peak power and voltages for a linear Bref ramp. | RCS1 | RCS2 | RCS3 | RCS4
---|---|---|---|---
Inj Energy [GeV] | $63\text{\,}$ | $314\text{\,}$ | $750\text{\,}$ | $1500\text{\,}$
Acc. length [km] | $5.99\text{\,}$ | $5.99\text{\,}$ | $10.7\text{\,}$ | $35.0\text{\,}$
Res. mags length [km] | $3.65\text{\,}$ | $2.54\text{\,}$ | $4.37\text{\,}$ | $20.38\text{\,}$
Binj in gap [T] | $0.36\text{\,}$ | $-1.8\text{\,}$ | $-1.8\text{\,}$ | $-1.8\text{\,}$
Bextr in gap [T] | $1.8\text{\,}$ | $1.8\text{\,}$ | $1.8\text{\,}$ | $1.8\text{\,}$
B ramp time Tramp [ms] | $0.35\text{\,}$ | $1.10\text{\,}$ | $2.37\text{\,}$ | $6.37\text{\,}$
Dipoles Pmax [GW] | $111\text{\,}$ | $54\text{\,}$ | $43\text{\,}$ | $74\text{\,}$
Dipoles Vmax [MV] | $2.4\text{\,}$ | $1.1\text{\,}$ | $0.9\text{\,}$ | $1.5\text{\,}$

##### Recent achievements

The two resonating circuits reported in Fig. 0.6.7 are a concept
generalization of two possible approaches for the power converters. The full
wave resonator is the approach by which it is possible to get close to the
desired linear acceleration profile, by adding two or more resonant circuit.
The most efficient approach is to use the first and second harmonics (Fig.
0.6.9), but other approaches are possible. The circuit is very simple and, in
theory, the cost of the power electronic switch, (two thyristors) is limited.

Figure 0.6.9: Full wave resonance with active filter contribution

However, the circuit doesn’t offer any clear possibility to make a regulation
and therefore to control the current among the different sectors of the
accelerator. To overcome this limitation, additional hardware would have to be
installed (Active filter) whose cost can be high. In addition, energy storage
capacitors oscillate with bipolar voltage thus leading to a considerable less
energy dense design (and therefore higher cost). The introduction of the
switched resonance circuit makes the circuit more complicated and costly from
the power electronics point of view, but no additional magnetic components (L2
in Fig. 0.6.7) are required and energy storage capacitors only oscillate on
positive voltage so that a more favorable design is possible for these (Fig.
0.6.10).

Figure 0.6.10: Simulation of a switched resonance case

Even more important, it is possible to act on the switching on-off time
instants to play with the equivalent output voltage. When this is done using
several tens of power cells (as in the configuration shown in Fig. 0.6.6) one
realizes that it is possible to have basic control of the currents in each
sector and possibly avoid using an active filter. The control concept would be
based on an "Iterative Learning Controller" approach, which acts on the
reference voltage of each power cell to achieve matching of the current
profile across the different sectors. The study of the control algorithm
feasibility is part of the important missing efforts listed in the next
paragraphs.

##### Planned work

Because of the strong links with the design of the resistive magnets and with
other technological areas of the accelerator, it is difficult to determine the
design of the power converters alone. Instead a combined python based design
suite is being developed. A preliminary sketch is reported in Fig. 0.6.11.

Figure 0.6.11: Python Accelerator simulation suite schematic

Different Python classes shall perform the necessary computations to design
the relative parts and compute the cost of it. An integrated optimization can
then lead to a cost effective design of the accelerator as a whole.

Another very important part in the design of the power converters is with the
study of the ILC controller to verify if such an approach would allow
controlling the current among sectors with the required accuracy.

##### Next priority studies and opportunities for additional collaboration

Throughout our analysis we have assumed many design values that have a
remarkable impact on the cost and carbon footprint of the final design. As the
application is very peculiar, our assumption cannot be justified by existing
similar applications. In particular the limits of the IGBTs could be different
from that assumed in the design. As the systems only works for few ms every
200, the DC maximum voltage and peak switched current values may be higher
than normally assumed with "standard" power converters, thus reducing the
equivalent IGBT number. Also the energy density design value of the capacitors
may be higher (or lower) than assumed. Thus, the optimisation of the system
will require to build (after the design optimization effort) a full scale
power cell and study the limits of the different components. In addition, the
system behaviour and control of several connected power cells has to be
demonstrated with a string test using a limited number of full scale power
cells.

###  0.6.3 RF

#### RF system for muon cooling

##### System overview

The RF system for the muon cooling complex consists of approximately 8000 RF
cavities operating in the frequency range starting from a few hundred of MHz.
The majority of the cavities operate at 352 and 704 MHz in the initial and
6D-cooling channels. The main peculiarity of the RF system is that the RF
cavities are interleaved with strong solenoidal magnetic fields, which enhance
the cavity breakdown rate. It has been proven to affect the high gradient
operation of the RF cavities necessary to re-accelerate muons as fast as
possible due to their short life time. This also limits the choice of the RF
cavities to normal conducting ones since SRF cavities cannot operate in
magnetic field. In order to reach high gradient on the order of a few tens of
MV/m in normal conducting cavities, high peak RF power on the order of few MWs
per one single cell cavity is needed. This results in very high peak power
requirements of a few tens of GW. On the other hand, the single (or few) bunch
operation mode results in relatively short pulse lengths of $\approx 10$ µs
and relatively low duty factor of $\approx 10^{-4}$.

##### Key challenges

Several experiments on breakdown rates in cavities under high magnetic field
have been carried out in the frame of dedicated programs (International Muon
Ionization Cooling Experiment (MICE) [27], Muon Accelerator Program at
Fermilab (MAP) [2]). Higher breakdown rates leading to lower achievable
accelerating gradients were observed in copper cavities operating at 805 MHz.
A model based on the "beamlet" principle obtained good correlation with
experiments [29]. That model predicts the appearance of breakdown when the
local temperature rise exceeds a threshold beyond, which plastic deformation
and surface damage may affect cavity behavior. Additionally, it also relates
the local temperature rise to the magnetic field. The same study predicted an
improvement for cavities made of beryllium, which is corroborated by
measurements: a maximal gradient of nearly 50 MV/m is kept in presence of 3 T,
while for copper the gradient drops from 24 MV/m at 0 T to the 13 MV/m at 3 T
[29].

Integration of the RF cavities together with high-field superconducting
solenoids and absorbers in the common vacuum vessel is another engineering
challenge. The superconducting solenoids operate at cryogenic temperature
whereas RF cavities and the associated RF network operate at room temperature.
Combining all the subsystem in a compact way is a key challenge in the
engineering design of the muon cooling channel.

##### Recent achievements

A simplification of the "beamlet" model for short pulses (< 10 µs) has been
developed, where the heat diffusion in the cavity walls can be neglected. It
showed that reducing the pulse duration should strongly increase the maximal
acceptable B field, for the same accelerating field. Moreover, materials other
than copper (beryllium, aluminum, copper alloys such as CuAg) with different
mechanical properties could give other solutions to mitigate this effect.
Lower temperatures could also mitigate this effect, as it affects the field
emission of the cavity, as well as increasing the surface hardness and the
thermal conductivity . Further, the external magnetic field may affect
elliptical cavities less than pillbox cavities.

##### Planned work

Further simulations involving thermal studies must be performed in order to
estimate the local thermal diffusion and its potential effect on the breakdown
limit in different configurations. These simulations must include multiple
physics, including RF, particle tracking, and thermodynamics. A dedicated test
stand is needed for studying these possibilities, which must include a high-
power (some MW) RF source, one or several cavities and a strong magnet (some
Tesla). See chapter 0.8.2 for proposed test stands.

On the design side, based on the input from the muon cooling beam dynamics
design, a consistent set of parameters for all of the RF cavities and their
associated RF systems will be elaborated creating the backbone of the
conceptual design of the RF system for the muon cooling complex. For certain
specific cases, starting with the cavity which will be used for the
demonstrator, more advanced engineering design will be performed including 3D
RF design and thermo-mechanical simulations, as well as integration into the
cooling cells for operation at very high gradients in strong solenoidal
magnetic fields. The impact of the beam loading on the muon energy spread will
have to be investigated and appropriate mitigation measures to be proposed.

##### Next priority studies and opportunities for additional collaboration

The conceptual design of all of the cavities for the whole moun cooling
complex including front end, rectilinear initial and 6D-cooling, bunch merge
and final cooling will not be covered due to the large number of different
cavities. Design and construction of prototype cavities at nominal RF
frequency as well as testing at high gradients in strong magnetic fields is
required as next step.

#### RF system for acceleration

##### System overview

After the cooling channel, a linear accelerator and two Recirculating Linear
Accelerators (RLAs) provide an initial acceleration up to 63 GeV. The linacs
operate at the RF frequency of 352 MHz and accelerating gradient of 25 MV/m.
Downstream from the RLAs, four Rapid Cycling Synchrotrons (RCSs) will
gradually increase the energy of the two muon bunches within a few tens of
turns each up to the collision energy of 5 TeV. During the transition from the
second RLA to the first RCS, the bunches are split and continue as counter-
rotating bunches. The design choices for the RF system will be guided by the
requirements resulting from the beam dynamics simulations and the short muon
lifetime. As a result, a high RF voltage per turn are required, supplied by
hundreds of superconducting 1.3 GHz cavities operating at accelerating
gradient of 30 MV/m per RCS. A more detailed description of the acceleration
chain can be found in section 0.5.3.

##### Key challenges

All accelerators following the first RCS will be implemented as hybrid RCSs
with both normal- and superconducting magnets. Due to the nature of this
hybrid magnet design, the orbit length and, thus, the revolution period
changes during the acceleration, leading to the necessity of fast frequency
tuning capabilities for the cavities.
The large bunch charge of up to $2.7\times 10^{12}$ muons per bunch in the RCS
chain will lead to significant transient beam loading and HOM-induced power
within the cavities. While the instantaneous requirements for cavity powering
and HOM power extraction are high, the machine’s duty cycle is low. The
cavities will, therefore, be operated in a pulsed mode. The requirements for
powering and HOM extraction further increase in the RLAs. The number of passes
is $\approx$ 5, while the beam current is higher than in the first RCS due to
the higher bunch charge and lower travel time between cavity passages.
Additionally, the impact of the beam loading will not be consistent in all
cavities in an RCS due to the differing time between the passages of the
counter-rotating $\mu^{+}$ and $\mu^{-}$ bunches. The same challenge applies
to the extraction of the HOM power and cavity powering. In the RLAs, the HOM
power and cavity powering need to be investigated in detail, as the bunches
are planned to pass in buckets directly after each other, which might
significantly impact HOM power extraction requirements. In the cavity shape
design, the cavity HOMs performance has to be carefully balanced against the
fundamental mode performance, which might lead to different designs for each
ring due to the different beam currents. In the cavity design for the low-
energy acceleration, one also has to take the particle speed into account, as
some of the accelerators operate in an energy regime where the particles are
not ultra-relativistic.

##### Recent achievements

A first approximation of the power requirements for the RCS chain has been
performed using the ILC cavities, cryomodules, and powering infrastructures
[24] as a baseline. The results of which can be found in Table 0.6.3. The
9-cell ILC cavity operates at a fundamental frequency of 1.3 GHz with an
accelerating gradient of 30 MV/m. While the ILC features the same repetition
rate (5 Hz) as the muon collider, the beam current and bunch structure differ
significantly. The requirements do not consider transient beam loading, HOM
power contributions, cryogenic loss, impact of orbit change detuning, and
counter-rotating beams.

Table 0.6.3: RF parameters for the RCS chain. The average and peak RF power includes losses from the cavity to the klystron, while the wall plug power also includes the klystron efficiency. | Unit | RCS1 | RCS2 | RCS3 | RCS4 | All
---|---|---|---|---|---|---
Synchronous phase | [°] | 135.0 | 135.0 | 135.0 | 135.0 | -
Number of bunches/species | - | 1 | 1 | 1 | 1 | -
Combined beam current ($\mu^{+}$ and $\mu^{-}$) | [mA] | 43.3 | 39.0 | 19.8 | 5.49 | -
Total RF voltage | [GV] | 20.9 | 11.2 | 16.1 | 90.0 | 138.2
Total number of cavities | - | 700 | 380 | 540 | 3000 | 4620
Total number of cryomodules | - | 78 | 42 | 60 | 334 | 514
Total RF section length | [m] | 990 | 530 | 760 | 4230 | 6510
Combined peak beam power ($\mu^{+}$ and $\mu^{-}$) | [MW] | 640 | 310 | 225 | 350 | -
External Q-factor |  [$1\times 10^{6}$] | 0.94 | 1.04 | 2.05 | 7.42 | -
Cavity detuning for beam loading comp. | [kHz] | -0.69 | -0.62 | -0.32 | -0.09 | -
Max. detuning due to orbit length change | [kHz] | 0 | 2.0 | 0.33 | 0.20 | -
Total number of klystrons | - | 88 | 42 | 30 | 47 | 207
Cavities per klystron | - | 8 | 9 | 18 | 64 | -
Beam acceleration time | [ms] | 0.34 | 1.1 | 2.37 | 6.37 | -
Cavity filling time | [ms] | 0.23 | 0.25 | 0.50 | 1.81 | -
RF pulse length | [ms] | 0.57 | 1.35 | 2.87 | 8.18 | -
RF duty factor | [%] | 0.29 | 0.68 | 1.44 | 4.09 | -
Peak cavity power | [kW] | 910 | 820 | 420 | 120 | -
Total peak RF power | [MW] | 850 | 410 | 300 | 460 | -
Average RF power | [MW] | 2.42 | 2.76 | 4.27 | 19.0 | 28.4
Average wall plug power for RF System | [MW] | 3.72 | 4.25 | 6.56 | 29.2 | 43.7
HOM power losses per cavity | [kW] | 9.75 | 12.14 | 6.74 | 2.77 | -
Total HOM power losses | [MW] | 6.83 | 4.61 | 3.64 | 8.3 | 23.38

##### Planned work

Within the work package, optimised cavity geometries for the specific cases of
the muon collider RCSs and the low-energy acceleration complex are planned to
be designed. Due to the different beam- and machine parameters in the low- and
high-energy acceleration, the impact of this optimisation will be significant.
The target parameters, such as the fundamental mode frequency or HOM stability
requirements, will be discussed with the other working groups involved. For
the final choice of the frequency, the power requirements of the acceleration
and cryomodules might play a significant role.
During the acceleration, muons will constantly decay, resulting in many seed
particles for multipacting, which might lead to breakdowns. The magnitude of
this effect needs to be studied in detail and be included in the shape
optimisation.
For the chosen cavity geometry, the development of an HOM damping scheme as
well as HOM coupler and FPCs (Fundamental Power Couplers) design will be
conducted according to the requirements of the collective effects and beam
dynamic studies. Depending on the powering and beam dynamics requirements, it
might be necessary to adjust the number of cells per cavity to stay within the
power limits of the couplers. In addition to the design of the cavities, an
uncertainty quantification is planned to be conducted in order to investigate
the impact of fabrication imperfections on the performance of the cavity. The
results of this analysis will serve as a requirement for the fabrication
precision in the cavities.
The power consumption of the system will be determined for the different
frequencies under consideration. To achieve a good compromise between power
consumption and voltage stability under the different conditions in each
accelerator, simulations will be used to determine an optimal value for the
loaded quality factor $Q_{L}$ and the cavity detuning $\Delta\omega$.

##### Next Priority studies and opportunities for additional collaboration

The work package will focus on the design of the cavities and couplers for
both the high- and low-energy acceleration. The integration of the cavities
into cryomodules is not part of the work package. Possible improvements to the
fabrication procedure could enhance the accelerating gradient for the
accelerating cavities and should be examined, as these could significantly
reduce the number of required cavities.

#### RF power sources

##### System overview

The muon cooling system will require a significant amount of RF power to
supply around 8000 cavities each with around 3 MW, summing up to around 10 GW
of peak power. The frequencies required are likely to be around 352 MHz and
704 MHz. This will be driven by klystrons at this power and frequency range.

##### Key challenges

Klystrons at 352 or 704 MHz are rather large devices which are typically
located in a surface building. Using a standard commercially available 3 MW
klystron for each cavity will require a surface building on the order of 180 m
by 30 m, as well as requiring shafts for large waveguides. A key requirement
is to reduce the size of the physical footprint. This can be achieved either
by using fewer higher power klystrons to feed multiple cavities or by making
the klystrons more compact using improved technologies. For example, the
E37503 Multibeam Klystron is a 1 GHz, 20 MW klystron has an RF circuit length
of 1.5 m, Including the collector and gun, it would have a total length in
excess of 2 m. A 352 MHz klystron would be significantly longer than this.
Another issue is the wall-plug electrical power required to drive the
klystrons. Klystrons are typically around 60% efficient at their saturated
output power, however, they are typically operated at lower power to provide a
low-level RF overhead to cope with RF transients, which further reduces the
efficiency. The duty cycle of a muon collider is likely to be around 0.015%,
and including HVDC system and modulator efficiency and RF losses the total
power is close to 2 MW of average power. This could be halved to under 1 MW if
the operating efficiency could be increased to closer to 80%. An advantage of
having a higher efficiency is also a reduction in the power handling of the
collector further reducing space and cooling requirements.

##### Recent achievements

There has been significant developments in high efficiency, compact klystrons
for LHC, FCC & CLIC at CERN and Lancaster University. The use of two stage
klystrons where the gun is operated at a lower voltage and beam power making
the bunching process occur in a much shorter distance. This in turn allows
better collection of particles into a well focused bunch improving the
efficiency, while at the same time reducing the length of the klystron. A two
stage (TS) multi-beam klystron has been designed for the CLIC drive beam that
operates at 1 GHz and an output power of 24 MW, which has an efficiency of 82%
and an RF circuit length of only 0.9 m.

##### Planned work

Specific design work on 352 MHz and 704 MHz klystrons for MuCol will start in
2024 at Lancaster University. This will most likely be a two-stage klystron to
minimise the size. The number of cavities fed by each klystron will be
optimised to minimise the size of the surface building while also increasing
the efficiency.

##### Next priority studies and opportunities for additional collaboration

A prototype two-stage klystron should be constructed and demonstrated, however
this could be done for LHC, FCC or CLIC.

###  0.6.4 Target

##### System overview

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, subsequently decaying 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 due to
its suitability. Graphite allows operation at high temperatures and exhibits
remarkable resistance to thermal shock. Its extensive usage in various
facilities attests to its reliability for $2\text{\,}\mathrm{M}\mathrm{W}$
operation. However, 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.

##### Key challenges

The MAP study has laid the groundwork for a muon collider but has left certain
crucial aspects unexplored. One significant gap is the lack of an integrated
system design and optimization for the target systems for production of a
high-charge, high-quality muon beam essential for achieving the desired
luminosity. Similarly, detailed specifications for other front-end beam-
absorbers, such as beam windows and a proton dump, were not addressed until
now within the IMCC.

The integration of a MW-class production target, radiation shielding, and
their respective cooling systems within the cryostat of the SC Solenoid poses
significant challenges. For instance, in a $2\text{\,}\mathrm{M}\mathrm{W}$
facility utilizing a graphite muon production target, around
$110\text{\,}\mathrm{k}\mathrm{W}$ of thermal power are deposited on the
target. Additionally, up to 34% of the stored beam energy is deposited onto
tungsten shielding near the target, necessitating efficient heat extraction
and isolation from the surrounding solenoid. Addressing these issues requires
dedicated vessels and the routing of multiple services. The ability to align
and perform target exchanges independently, without removing the surrounding
shielding of over 20 tonnes, needs consideration. Furthermore, residual
radiation doses on the target systems necessitate designs compatible with
remote handling.

Whether employing a carbon, HLM, or fluidized tungsten target, the high-
intensity $2\text{\,}\mathrm{n}\mathrm{s}$ pulse on target results in a very
high adiabatic temperature rise of the bulk material. This temperature rise
can lead to dynamic stress-strain responses in solid targets, potentially
surpassing material limits. HLM targets, though free of structural concerns,
may induce extreme loads on the containment vessel due to pressure waves in
the liquid, necessitating in-depth design and modelling of this complex
multiphase problem. Understanding erosion management and powder handling for a
fluidized tungsten target is imperative, particularly due to limited
operational experience in present facilities.

The substantial levels of radiation exposure, encompassing hadronic and
electromagnetic showers, as outlined in section 0.6.5, pose a significant
challenge for the target system and its adjacent components. A comprehensive
study of target and beam window materials against radiation damage is
required. High integrated fluence of protons can induce atomic changes in the
crystalline structure of the target materials, resulting in conductivity loss
and increased brittleness, potentially reducing the lifespan of the device.
For graphite, radiation annealing via operation of the target at high-
temperature, coupled with larger beam sizes on target, is considered a
mitigation strategy. Proton beam windows, separating the primary vacuum from
the target confinement atmosphere, benefit from low Z materials such as
beryllium, in order to reduce their integrated dpa (displacement per atom).

Effective cooling is essential to maintain reasonable temperature conditions
for the entire system and minimize heat dissipation towards the
superconducting magnet. Simultaneously, designing a sustainable facility
requires optimal sizing of auxiliary services.

Lastly, a major challenge involves designing a proton absorber capable of
fully absorbing the entire beam if necessary at a regular dumping power,
alongside the intricate design of a beam extraction channel.

##### Recent achievements

The culmination of current studies has resulted in a baseline target design
featuring an $80\text{\,}\mathrm{c}\mathrm{m}$-long isostatic graphite rod of
$30\text{\,}\mathrm{m}\mathrm{m}$ in diameter, enclosed within a titanium
vessel for hermetic confinement. A carbon composite frame holds the rod in the
central position of the vessel. Internally, static helium gas at
$1\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ 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
$20\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$. 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 0.6.5). A tungsten shielding surrounding the pion capture solenoid is
required. To address the dissipation of heat
($680\text{\,}\mathrm{k}\mathrm{W}$ for a $2\text{\,}\mathrm{M}\mathrm{W}$
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 pie-shaped 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 $\mathrm{5}$mm ($1\sigma$) was determined as the
optimal compromise to ensure that the graphite’s dynamic response remains
within acceptable limits. This solution aligns with the requirements of a
$2\text{\,}\mathrm{M}\mathrm{W}$ production target facility and is compatible
with a proton injection of $5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ and
$2\text{\,}\mathrm{n}\mathrm{s}$ ($1\sigma$) pulses at a frequency of
$5\text{\,}\mathrm{H}\mathrm{z}$.

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.

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 model
under consideration involves a free-flow of lead in the vertical direction
(similar to a curtain), aiming to eliminate the concerns posed by the initial
concept. Ongoing work involves further modelling and development of this
concept.

##### Planned work

The plan includes defining a layout for the proton dump extraction along with
its conceptual design. Additionally, there will be a comprehensive focus on
fully integrating the target assembly system inside the cryostat, encompassing
detailed aspects of its supporting structure. Concurrently, the design of the
proton beam window, especially its cooling system, will be developed in
conjunction with a high-level integration design study. Careful considerations
will be made regarding the installation, dismantling, and maintenance of the
frontend target systems equipment. Lastly, comprehensive physics simulations,
spanning from muon production to the cooling section, are scheduled to be
conducted.

##### Next priority studies and opportunities for additional collaboration

The ongoing conceptual studies lack vital engineering data essential for the
expected operational conditions in a muon collider. There is a need for
extensive material characterization and beam tests to complement these studies
adequately. For instance, prototyping key components and carrying out beam
tests in facilities such as HiRadMat at CERN would provide valuable feedback.
Additionally, it is imperative to include the dimensioning of auxiliary
services along with respective high-level costing.

###  0.6.5 Radiation shielding

##### System overview

Radiation to equipment poses a significant challenge for all stages of the
muon collider complex, from the front-end up to the collider ring. Dedicated
shielding configurations must be designed for the different machines in order
to dissipate the radiation-induced heat and protect equipment against long-
term radiation damage. This concerns in particular the protection of
superconducting magnets in different parts of the complex, like the target and
capture solenoids of the muon source as well as the superconducting magnets in
the RCS and collider rings. The radiation shielding must prevent beam-induced
magnet quenches, reduce the thermal load to the cryogenic system, and prevent
magnet failures due to the ionizing dose in organic materials (e.g.
insulation, spacers) and due to atomic displacements in the superconductors.
In most cases, dedicated absorbers need to be embedded inside the magnet
aperture in order to attenuate secondary radiation showers before they can
reach the coils.

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 high-
temperature superconductors (HTS), has been developed within the IMCC,
combined with a new arrangement of tungsten shielding inserts for absorbing
electromagnetic and hadronic showers.

Table 0.6.4: Muon decays in the collider ring (one bunch per beam), assuming an injection frequency of 5 Hz and 1.2$\times$107 s (=139 days) of operation per year.  | 3 TeV | 10 TeV
---|---|---
Muons per bunch |  2.2$\times$1012 |  1.8$\times$1012
Circumference | 4.5 km | 10 km
Muon decay rate per unit length |  4.9$\times$109 m-1s-1 |  1.8$\times$109 m-1s-1
Power per unit length carried by decay $e^{\pm}$ | 0.411 kW/m | 0.505 kW/m
Total decays per unit length and per year |  5.87$\times$1016 m-1 |  2.16$\times$1016 m-1

Dedicated shielding configurations are also needed in the accelerators and the
collider, protecting equipment against the radiation load due to muon decay.
The power carried by decay electrons and positrons is on average about 35% of
the energy of decaying muons (the rest is carried away by neutrinos, which are
irrelevant for the radiation load to the machine). With the presently assumed
beam parameters, this amounts to about 500 W/m dissipated in the collider
ring; the collider parameters are chosen such that the power load is about the
same at 3 TeV and 10 TeV (see Table 0.6.4). The decay electrons and positrons
can have TeV energies and emit synchrotron radiation while travelling inside
the magnets. The energy is then dissipated through electromagnetic showers in
surrounding materials. In addition, secondary hadrons can be produced in
photo-nuclear interactions, in particular neutrons, which dominate the
displacement damage in magnet coils. Shielding studies for muon colliders have
been previously carried out within MAP, indicating that a continuous liner
(few centimeters of tungsten) is needed inside magnets and magnet
interconnects. The shielding requirements for collider energies up to 10 TeV
have been studied more recently within the IMCC.

Absorbers and shielding configurations might also be needed for the protection
of other accelerator systems like RF cavities. In addition, massive absorbers
are also required for the Machine-Detector Interface (MDI), in order to reduce
the beam-induced background and radiation damage in the detector. The MDI
shielding is discussed in more detail in Sec. 0.3.2 and will not be covered
here.

##### Key challenges

Designing shielding configurations for a harsh radiation environment like in
the muon collider target complex requires a good understanding of radiation
damage limits for equipment components like superconductors, insulation
materials, etc. The shielding requirements will strongly depend on R&D efforts
towards more radiation-hard solenoids and accelerator magnets. Radiation
damage effects in superconductors, in particular in HTS, are also of critical
interest for other applications like fusion reactors. The irradiation tests
and theoretical studies carried out by the fusion community are highly
beneficial for the muon collider design study. One of the key challenges is to
derive a relation between radiation damage quantities, e.g., displacement per
atom (DPA), and relevant macroscopic material properties like the critical
temperature. This is crucial for establishing a relationship between
irradiation experiments and radiation environments in the muon collider
complex. Despite the still existing uncertainties, it is expected that the
understanding of radiation effects in HTS magnets, including the possible
benefits of annealing, will significantly improve in the coming decade.

The shielding blocks inside the superconducting solenoids around the pion/muon
production target need to be tens of centimeters thick (radially), in order to
sufficiently reduce the heat load and radiation damage in the magnets. The
shielding has to be efficiently cooled since it has to dissipate a significant
fraction (about 35%) of proton drive-beam power (2 MW). Other important
aspects are the machining, handling, assembly and support of the shielding
blocks. Optimizing the shielding configuration around the target and in the
downstream capture section is crucial, since the shielding will determine the
final aperture requirements for the solenoids. Another challenge is the
overall integration of the heavy shielding inside the cryostats. Furthermore,
the shielding needs to be thermally insulated from the cryostats, considering
the significant heat dissipation in the shielding.

One of the key challenges for the collider shielding design is the overall
optimization of geometrical aspects (beam aperture, shielding thickness, coil
aperture) and thermal aspects (shielding and coil temperature, thermal
insulation, cooling scheme). This optimization critically depends on
technology choices, e.g., low-temperature versus high-temperature
superconductors, and requires a multi-disciplinary design approach including
radiation and beam physics, magnet engineering, cryogenics and vacuum. A
careful optimization of the absorber thickness is important since the
absorbers significantly impact the aperture requirements for magnets. An
essential design parameter for the shielding is the maximum allowed heat
deposition in the cold mass of collider magnets and the resulting cooling
requirements, which will be an important cost factor for facility operation
due to the associated power consumption (see Sec. 0.6.7 for details). Besides
the conceptual shielding design, one also faces different engineering
challenges. The shielding needs to be equipped with a cooling circuit in order
to extract the heat deposited by the particle showers. In addition, the vacuum
chamber needs to support the weight of the shielding absorbers ($>$100 kg/m).
Another important aspect is the raw material and manufacturing cost of the
shielding. Considering the length of the collider (10 km for the 10 TeV
machine), the shielding cross section hence needs to be carefully optimized.

##### Recent achievements

A conceptual radiation shielding design was conceived for the target area and
the downstream capture section and chicane. 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 0.6.5 shows the annual DPA and
dose in the target solenoid for different tungsten shielding thicknesses. The
studies indicate that a thickness of about 40 cm is needed for reducing the
DPA in the coils to 10-3 DPA/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 to be about 60-70 cm. The
maximum allowed DPA in the coils per year of operation needs to be further
assessed, and will depend on the expected effect of annealing cycles. A dose
of 5 MGy/year could possibly be acceptable for insulation materials.

Table 0.6.5: Radiation load on the target solenoid in terms of the maximum
displacement per atom (DPA) and the maximum absorbed dose per year of
operation for various shielding configurations. The studies considered a
graphite target. The inner aperture of the shielding around the target vessel
was 17.8 cm. A gap of 7.5 cm was assumed between the shielding and the coils
(supports, thermal insulation).

Inner radius of the magnet coils | Shielding thickness around the target |  DPA/year [$10^{-3}$] | Dose [MGy/year]
---|---|---|---
$60\text{\,}\mathrm{c}\mathrm{m}$ |  W $31.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $1.70\pm 0.02$ | $10.0\pm 0.3$
$65\text{\,}\mathrm{c}\mathrm{m}$ |  W $36.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $0.90\pm 0.02$ | $5.6\pm 0.2$
$70\text{\,}\mathrm{c}\mathrm{m}$ |  W $41.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $0.49\pm 0.01$ | $3.1\pm 0.1$
$75\text{\,}\mathrm{c}\mathrm{m}$ |  W $46.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $0.29\pm 0.01$ | $1.9\pm 0.1$
$80\text{\,}\mathrm{c}\mathrm{m}$ |  W $51.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $0.16\pm 0.01$ | $1.0\pm 0.1$
$85\text{\,}\mathrm{c}\mathrm{m}$ |  W $56.2\text{\,}\mathrm{c}\mathrm{m}$ \+ H2O $2\text{\,}\mathrm{c}\mathrm{m}$ \+ B4C $0.5\text{\,}\mathrm{c}\mathrm{m}$ \+ W $1\text{\,}\mathrm{c}\mathrm{m}$ | $0.09\pm 0.01$ | $0.6\pm 0.1$

In order to estimate the general shielding requirements for superconducting
arc magnets in the 3 TeV and 10 TeV colliders, generic shielding studies were
performed with FLUKA. The main focus of these studies was on muon decay,
whereas other source terms like beam halo losses still have to be addressed in
the future. As in the MAP studies, tungsten was assumed as absorber material
due to its high atomic number and density. For engineering reasons, pure
tungsten may be substituted by tungsten-based alloys without significantly
affecting the shielding efficiency if the alloy has a similar material
density. The simulations indicated that the total power penetrating through
the shielding is similar for all collider energies, despite the harder decay
spectrum at 10 TeV. On the other hand, the maximum power density and dose in
the coils was found to be a factor of 1.5–2 higher at 10 TeV than at 3 TeV.

Table 0.6.6 summarizes the calculated power load and radiation damage in
collider arc magnets as a function of the radial tungsten absorber thickness
(for the 10 TeV collider option). The power penetrating the absorber, mostly
in the form of electromagnetic showers, amounts to almost 20 W/m in the case
of a 2 cm shielding, and decreases to 4 W/m in the case of a 4 cm shielding.
Most of this power is deposited in the cold bore and cold mass of the
superconducting magnets. As a design criterion, the beam-induced heat load due
to muon decay shall not exceed 5 W/m, considering the additional static heat
load in the magnet cold mass and the resulting cooling requirements and power
consumption (see Sec. 0.6.7). This suggests that the tungsten shielding
thickness needs to be at least 4 cm. With such a thickness, the cumulative
radiation damage in magnets (dose and DPA) is expected to be acceptable, even
for 10 years of operation. This shows that the heat load in the cold mass
remains the driving factor for the shielding design. Together with the
required beam aperture of 2.35 cm (see Sec. 0.5.4) and other space
requirements (supports etc.), the performed shielding studies indicate that
the inner coil aperture (radius) of the collider arc magnets needs to be about
8 cm, which is a key figure for the magnet design.

Table 0.6.6: Power load and radiation damage in collider ring arc magnets (10 TeV) as a function of the radial tungsten absorber thickness. The power penetrating the shielding does not include neutrinos, since they are not relevant for the radiation load to the machine; the percentage values are given with respect to the power carried by decay electrons and positrons. The results include the contribution of both counter-rotating beams. | 2 cm | 3 cm | 4 cm
---|---|---|---
Beam aperture (radius) | 23.5 mm | 23.5 mm | 23.5 mm
Outer shielding radius | 43.5 mm | 53.5 mm | 63.5 mm
Inner coil aperture (radius) | 59 mm | 69 mm | 79 mm
Power penetrating tungsten absorber | 19.1 W/m (3.8%) | 8.2 W/m (1.6%) | 4.1 W/m (0.8%)
Peak power density in coils |  6.5 mW/cm3 |  2.1 mW/cm3 |  0.7 mW/cm3
Peak dose in Kapton (5/10 years) | 56/112 MGy | 18/36 MGy | 7/14 MGy
Peak dose in coils (5/10 years) | 45/90 MGy | 15/30 MGy | 5/10 MGy
Peak DPA in coils (5/10 years) |  8/16$\times$10-5 DPA |  6/12$\times$10-5 DPA |  5/10$\times$10-5 DPA

##### Planned work

The shielding studies for collider magnets performed so far were mostly of
conceptual nature, with the goal to establish key parameters like the required
shielding thickness. It is foreseen to progress towards a more realistic
absorber design, by optimizing the transverse absorber cross section for
dipoles and quadrupoles. Besides the contribution of muon decay, also beam
halo losses have to be studied. In addition, it is planned to repeat the
shielding studies for a wobbled machine configuration, which is essential for
mitigating the environmental impact of neutrinos. Furthermore, material and
engineering aspects have to be studied, including the choice of absorber
material (pure tungsten versus other heavy tungsten-alloys), machining and
handling, as well as the design of supports. Another important aspect is the
definition of the operating temperature and the design of the cooling circuit,
which requires a close collaboration between various experts (cryogenics,
vacuum). So far, the studies focused mainly on the collider arcs, but they
have to be extended to the experimental insertions, in particular the final
focus regions. Besides the studies for the collider, shielding studies are
also foreseen for the accelerators.

##### Next priority studies and opportunities for additional collaboration

In several cases, a detailed engineering design of the shielding
configurations will not be possible. In addition, it is out of the scope of
the present activity to proceed with the prototyping of radiation absorbers or
mock-ups of entire shielding configurations. Furthermore, with the present
resources it is not feasible to perform dedicated irradiation tests for
probing the radiation effects in superconductors like HTS. However, possible
synergies with other activities, for example within the fusion community, can
be exploited.

###  0.6.6 Muon cooling cell

The Cooling cell is the building block of a Muon Cooling section as described
in Section 0.5.2 and is considered one of the highest priority for prototyping
efforts towards the demonstration of the feasibility of a Muon Collider. The
cooling of muons requires in fact a very compact assembly comprising room
temperature RF cavities, superconducting solenoids and either Liquid Hydrogen
or Lithium Hydride absorbers (more details in chapter 0.6.8). All those
components will require space for services, power feeds and other ancillaries
which will make the mechanical integration extremely complex. Within the
European Project MuCol it has therefore been decided to have a dedicated work
package (WP8) to produce a complete 3D model of a representative cell among
the several types that will be necessary for the cooling sections. In this
chapter we describe the strategy on how we intend to proceed in phases towards
the full integration of a cell, and on the rationale behind the choice of the
cooling cell type that will be implemented. We expect to validate the choice
at the annual meeting of the IMCC collaboration that will be held in March
2024.

##### System overview

A cooling cell, in its simplest version, is defined as the assembly of an RF
structure made of several cells, surrounded by two or more superconducting
solenoids, and one absorber at each end of the cavity (See Fig. 0.6.12 for a
conceptual drawing).

Figure 0.6.12: Terminology adopted for the different components of the Cooling
Cell

Different types of cooling cells will be necessary to cope with the muon beam
characteristics along the cooling section of the complex. A description of the
layout of a possible implementation of the 6D cooling section can be found for
instance in the cooling section and in papers from the MAP programme [12, 13].
Such layout will be further optimised as discussed in Section 6.3. We will
choose the cell that can yield

  * •

an excellent cooling performance; and

  * •

an appropriate level of technical challenge.

In order to acquire the necessary experience to properly design a cooling
cell, it was agreed that the WP8 design team would be in charge as well of the
design of the "Radio Frequency cavity in a Magnetic Field Test Facility"
(RFMFTF) which encompasses a pillbox type cavity inside a superconducting
solenoid. This is considered a somewhat simpler integration exercise yet
including most of the difficulties presented by the cooling cells.

##### Key challenges

The main challenges for the integration of a cooling cell are related to the
limited space available in order to fulfil the requirements. The design will
have to cope with the following constraints:

  * •

have a large magnetic field on axis (order of a few Tesla),

  * •

have a large electric field (25 MV/m) in the RF cavity,

  * •

decide the technology to be used to limit the breakdown rate in the RF cavity
under high magnetic field,

  * •

decide whether all solenoids are in a single cryostat or in separate
cryostats,

  * •

decide whether services and ancillaries for the RF cavity (cooling, RF power
couplers, HOM damping couplers will come radially or longitudinally ,

  * •

in case the absorber, or the mitigation strategy for high breakdown rate in
the RF cavity requires the use of Liquid or Gaseous Hydrogen, understand the
safety issues and study a proper distribution network around the cell,

  * •

provide the shortest cell to cell coupling space.

##### Recent achievements

A preliminary conceptual design of the RFMFTF has been presented and
discussed, and will be the object of further studies. The parameters of such
test facility are listed in Table 0.6.7 :

Table 0.6.7: Parameters for the RFMFTF  | Unit | Parameter |  Comment
---|---|---|---
Magnetic Field on Axis | [T] | 7 |  -
RF Frequency | [MHz] | 3000 |  -
Number of solenoids | - | 2 |  In order to allow working in phase or antiphase
Number of cryostats | - | 1 |  Both solenoids will be in the same cryostat
Position of RF power coupler | - | Longitudinal |  RF Power Coupler will come from the front as all the services

##### Planned work

In the next few months we are going to complete the design of the RFMFTF that
is planned to be mostly built by INFN and placed at INFN - LASA in Milan. An
initial funding has been granted by INFN, although a full estimate of the cost
has to be established taking into account that some parts might come from in
kind contributions (e.g. cryogenics, power converters and RF power source) of
some of the labs participating to the collaboration. On January 18 and 19,
2024, a workshop on the cooling cell technology has been organised at CERN. We
presented to the community and some experts in the field the choices taken by
the team and collected feedback, in order to prepare a proposal for the
Governing board of MuCol about the type of cell for which the detailed design
will be performed. After the endorsement of the GB, the design work will start
at full speed.

##### Next priority studies and opportunities for additional collaboration

The next, not yet planned step, is the actual construction of a cooling cell.
The cost is estimated to be between 5 and 10 MCHF for the first prototype.

###  0.6.7 Cryogenics

##### Overview

A cryogenic infrastructure that enables the operation of superconducting
magnets and radio-frequency (RF) cavities is an essential part of the Muon
Collider complex. Systems requiring cryogenics will be present at most stages
of the complex, including but not limited to:

  * •

Proton Driver: cooling of high-Q RF cavities;

  * •

Front End: cooling of the target solenoid;

  * •

Cooling Channel: cooling of the magnet array, operation of the H2 absorbers;

  * •

Accelerator and Collider rings: cooling of superconducting magnets.

With sustainability in mind, one of the main drivers for the design of the
cryogenic system will be the overall electrical power consumption. The various
challenges of each part of the complex shall be tackled while keeping the
highest possible thermodynamic efficiency (i.e. temperature levels as high as
feasible) and the complexity to a minimum.

##### Key challenges

Several key challenges have been identified at the level of the Cooling
Channel, and the Accelerator and Collider rings. For the moment, the Proton
Driver and the Front End systems have not been assessed from a Cryogenics
standpoint.

Cooling channel: the aim is to economise the overall cryogenic system
investment and operation by harmonizing the demands of the various magnets
that make up the channel. The most demanding magnets of the chain are the
final cooling solenoids, where a substantial heat load needs to be extracted
from each structure. Here, the challenge is rather on designing the local heat
extraction rather than coping with the overall generated heat load. Solutions
are strongly dependent on chosen time constant and ramp scheme for the
magnets. The heat load profile and peak heat load generated during ramping
varies significantly with these two parameters; a lower time constant reduces
the heat load during ramp, but also reduces thermal and electrical stability
during operation.

Accelerator ring: the focus for the accelerator ring is to investigate
sustainable cryogenic system possibilities in a heterogeneous magnet
environment such as the accelerator ring, which will have interleaved normal
conducting (warm) and superconducting (cold) magnets, and to design a cooling
layout to cope with such arrangement. The fast ramping magnets will most
likely generate significant ramping losses, and the sheer number of cold-to-
warm transitions calls for careful design to keep heat in-leaks manageable.
Another challenge is that of magnet cooling over long distances: with the aim
of minimizing cost and complexity, one should strive to maximize the distance
required between the accelerator’s continuous-cryostat’s cryogenic-service
feed points and re-cooling stations. Considering the rather short field-free
regions between magnets (i.e. interconnects), the internal routing of cryogens
in the cryostat while satisfying minimum magnet interconnect distances needs
careful consideration.

Collider ring: Convergence on a radial build that can be achieved from both
the magnet point of view (aperture vs. field) and cryogenic/thermal design is
crucial. It has to support appropriate mitigation strategies for the heat load
deposited on the coil, such as a heat intercept between the coil and absorber
and the shielding thickness of the absorber so that the total heat load stays
below 10 W/m. As with the accelerator ring, magnet cooling over long distances
needs to be optimised: with the aim of minimizing cost and complexity one
should strive to maximize the distance required between the collider’s
continuous-cryostat’s cryogenic-service feed points and re-cooling stations.
Optimization of very short magnet interconnects compliant with magnet movement
for neutrino flux mitigation is necessary: the internal routing of cryogens in
the continuous collider cryostat and the integration of cryogenic-service
feed-points has to be solved while satisfying minimum magnet interconnect
distances and allowing for magnet movement.

##### Recent achievements

Since the definition of the road map, the principal achievement has been the
identification of the main drivers for the cryogenic design and operation of
the Muon Collider complex. The collider ring has been the subject of a more
focused study, due to the stringent conditions regarding power deposition on
the superconducting magnets. An estimation of the heat loads has been carried
out for collider ring magnets as function of operating temperature and
absorber thickness, including static heat loads from radiation and conduction
via mechanical supports. This assessment will be completed with the presently
missing contribution from resistive splices, current leads and ramping losses.
The heat load estimation evidenced the need for a heat intercept (thermal