# Optics studies of a Muon Accumulator Ring based on FFA cells

O.R. Blanco-García<EMAIL_ADDRESS>INFN-LNF, Via E. Fermi 40, 00044
Frascati, Rome, Italy  A. Ciarma  INFN-LNF, Via E. Fermi 40, 00044 Frascati,
Rome, Italy

###### Abstract

The production of an intense, high energy and low emittance muon beam is
interesting for a possible muon collider. The Low EMittance Muon Accelerator
(LEMMA) team at the Istituto Nazionale di Fisica Nucleare (INFN), in Italy, is
studying the production of a 22.5 GeV low emittance muon beam from a high
energy positron beam at twice the muon energy impinging on a fixed thin
target.

This muon source has two main advantages: the muon beam emittance is small
thanks to the kinematics of the $e^{+}e^{-}$ annihilation process of a low
emittance positron beam, and thanks to the asymmetric collision muons are
produced with a relativistic gamma factor of 200, extending their lifetime to
0.46 ms. The disadvantage of this scheme is the low production efficiency,
which is in the order of $10^{-7}$ muon pairs per impinging positron in a
target of 1% of a radiation length, therefore, requiring a high positron rate.

The LEMMA scheme proposes to perform the muon accumulation from multiple
($10^{2}$ to $10^{3}$) positron bunches to increase the population of a single
muon bunch that is recirculated through the target using two small accumulator
rings, one per muon species. The three beams ($\mu^{+}$ and $\mu^{-}$ at 22.5
GeV and $e^{+}$ at twice the muon energy) share the same phase space at the
target on every positron bunch interaction, producing new muons inside the
preserved beam emittance.

We study the requirements and optics design of the accumulator to recirculate
the muons over the target using a Fixed Field Alternating Gradient (FFA) arc.

As a result, we achieve a compact 230 m long accumulator with two Interaction
Points, energy acceptance of $\pm 5$%, low twiss beta function at the target
$\beta^{*}_{\mu}=20$ cm, and a drift space 2$L^{*}$ of 20 cm enough to
accommodate 1% of a radiation length $X_{0}$ for several material options.
These optics parameters are obtained with magnets similar to those foreseen
for new colliders like FCC or CLIC, and could be extended further with new
magnet designs.

Simulations of the muon beam accumulation and target interaction show that the
muon beam population increases by a factor 100 in the first few hundred turns,
but, emittance grows due to multiple scattering with the target leading to
particle losses. The final muon population and emittance is limited by the
dynamic aperture of the machine.

Although the FFA and Interaction Region designs are promising, it also points
to some limitations. First, $\epsilon_{N}=5$ $\pi$ $\mu$m (or $200\times 25$
$\pi$ nm) is the minimum normalized emittance obtained with the current value
of $\beta^{*}_{\mu}$, and the production of lower $\beta^{*}_{\mu}$ values
will require quadrupole gradients above 500 T/m. Second, the achieved low
$\beta^{*}_{\mu}$ is not enough to mitigate the effect of multiple scattering
with the target over a thousand turns. Third, the current Interaction Region
for a low $\beta^{*}_{\mu}$ already reduces the energy acceptance of the
machine to only $\pm 5$%. Fourth, the FFA cells are designed to correct low
values of chromaticity, the arcs will need to be adapted consequently in
length or strength when trying to push down the $\beta^{*}_{\mu}$ and/or
increase the energy acceptance.

Further efforts could be directed to design a second order apochromatic
Interaction Region to increase the energy acceptance from $\pm 5$% up to
$\pm$10%, which would allow to accept a larger muon beam energy spread and
thus increase the production efficiency. In addition, studies on magnets with
good field regions larger than $\pm$2 cm will allow to increase dispersion and
reduce the strength of sextupole magnets used for chromatic correction leading
to larger dynamic aperture. Furthermore, quadrupole gradients beyond 500 T/m
able to work in an Interaction Region that includes a target would allow to
produce a smaller $\beta^{*}_{\mu}$ leading to a smaller muon beam emittance
and mitigating the effect of multiple scattering over more than a few hundred
turns.

The current muon accumulation results will serve as input for beam combination
studies.

##  I Introduction

One of the many challenges to aboard in accelerator physics is the feasibility
of a muon collider updateofeuropeanstrategy . Crucial parameters for this
machine are the fast production and acceleration of a high quality muon beam
measured in terms of high population, large lifetime and small emittance.

The Muon Accelerator Program (MAP) MAP  has already performed the initial
studies for a possible muon collider reaching high luminosity in the TeV scale
where positive and negative muon beams are produced as secondary particles,
from the collision of a proton beam on a target. Due to the kinematics of this
process the muon beams are produced with a very large transverse (4D) and
longitudinal (+2D) emittance that is cooled down in later stages. Progress has
been achieved in a small scale 4D emittance cooling test by the Muon
Ionization Cooling Experiment (MICE) group cooling , while 6D-cooling remains
yet untested.

The LEMMA (Low Emittance Muon Accelerator) NIM ; Antonelli:IPAC16-TUPMY001 ;
Boscolo:IPAC17-WEOBA3 ; eplusringopt ; MaricaIPAC2019  team is studying the
production of muon beams at about 22.5 GeV, with a bunch population of
$10^{9}$ particles and extremely low normalized emittance of
$\epsilon_{N}=0.040~{}\mu$m from the collision of a low emittance positron
beam and a fixed thin target.

We note that a target is considered thin if the thickness of the material
traversed by the positron beam is only a few percent of radiation length
$X_{0}$. In the case LEMMA concept the target material is Beryllium (Be) and
the target thickness is 3 mm, equivalent to 0.88% of $X_{0}$.

The LEMMA scheme has two main advantages: no emittance cooling is required and
muon lifetime is extended to 0.4 ms (more than 200 times the lifetime at rest
or equivalently 120 km of run) before the muons decay. However, the
combination of high bunch population and low emittance is challenging because
the accumulation process requires the recirculation of the beam over the
target more than a thousand times (Fig. 1), leading to energy loss and
emittance growth from multiple scattering with the target.

Figure 1: Schematic representation of the LEMMA accumulation scheme. Two muon
bunches have been produced by the interaction of the first positron bunch with
the target $T$. Muons are recirculated and synchronized with the arrival of a
second positron bunch to increase the muon bunch population with minimal
emittance growth. The cycle repeats over a thousand positron bunches.

In the LEMMA scheme, three beams ($\mu^{+}$ and $\mu^{-}$ at 22.5 GeV and
$e^{+}$ at twice the energy) occupy the same phase space at the target
location in order to create new muons without emittance growth, see Fig. 2.

Figure 2: (TOP) The three beams $e^{+}$ in red, $\mu^{+}$ in pink and
$\mu^{-}$ in violet superimpose one another at the target. (BOTTOM) Phase
space representation at the thin target location. Beams are distributed over
the same phase space, represented in colored circles. New muons are created
inside the same phase space on every positron bunch passage and muon bunches
recirculation.

In this article we concentrate in the possibility to create an accumulator
design fulfilling the LEMMA specifications addressed theoretically in muacc .
We list those requirements here:

  * •

The accumulator ring should have a very short length in order to allow a large
number of accumulation cycles.

  * •

The energy acceptance should be close to $\pm$20% due to the muon pair
production kinematic for 45 GeV positron beam. This is the minimum energy
acceptance to get about $0.7\times 10^{-7}$ muon pairs per impinging positron
in a Beryllium target 3 mm thick.

  * •

The twiss $\beta^{*}_{\mu}$ at the beam–target Interaction Point (IP) should
be at most 1 cm over $\pm$20% energy spread to create a low muon beam
emittance with large divergence, comparable to the contribution to divergence
from multiple scattering with the target over a thousand turns, mitigating the
emittance growth.

  * •

The momentum compaction factor should be small over $\pm$20% to preserve the
bunch length in the order of millimeters given by the positron beam.

In addition, we would like to remark two points:

  * •

We include in the requirements a radiofrequency (RF) cavity to recover the
beam energy loss in the target that becomes significative over a thousand
turns, equivalent to traverse several radiation lengths of material.

  * •

At the end of the accumulation, the beam should be extracted. Therefore, we
need to consider and extraction region.

A strong focusing accumulator optics and target material study
PhysRevAccelBeams.23.051001  have shown promising results on fulfilling these
requirements.

In this article, we would like to extend the possibilities to the study of an
alternative accumulator ring based in Fixed Field Alternating Gradient (FFA)
Cells because several publications show the possibility to design a small
machine length and large energy acceptance machida2012 ; sheehy2016 ;
lagrange2016 ; garland2015 . Section II states the considerations that we have
taken into account at the beginning of the accumulator study. Section III
shows the accumulator sections. Section IV shows the FFA cell for high energy
acceptance. Section V shows the modification of the FFA cell to accommodate
the insertions. Section VI shows the insertion optics for an RF and kicker.
Section VII shows the studies of beam separation. Section VIII shows the
Interaction Region design. Section IX shows the accumulator. Section X shows
the beamsize and tune of the machine. Section XI shows the muon accumulation
results for different beam energies and target materials. Section XII shows
further design improvements that have been foreseen during the design stage.
Finally, in Section XIII we conclude on the accumulation studies.

##  II Considerations before the design

There are some arbitrary assumptions we made in order to select among many
possibilities the most promising optics designs with the current magnet
technology before any accumulation study. We consider a good field region of
an arc magnet in the order of $\pm$1 to $\pm$2 cm with a maximum peak magnetic
field of 20 T. This should be a perfectly safe value for a good field region
of any magnet because for a beam with $\pm$20% energy spread and a horizontal
dispersion $\eta_{x}$=0.1 m, the displacement of the beam to first order
approximation corresponds to 2 cm ($0.1~{}\text{m}\times 0.2$). Therefore, we
have decided to pursue small dispersion along the arcs.

Sextupoles will need to be stronger in order to compensate the small
horizontal dispersion for a given value of chromaticity per cell. We prefer to
keep the dispersion small and assume that we have no limits on the achievable
sextupole strength. It also implies that second order corrections are very
sensitive and could limit the dynamic aperture of our design, but it did not
seem to be a problem due to the small muon beam emittance.

As we will see, dynamic aperture puts a limit on the accumulation achievements
due to emittance growth from multiple scattering with the target.

With respect to the optics design and as a starting point to understand the
limitations of the FFA cell, we tried a second order model in MAD-X MADX . We
recognize that it is not the best option for this design because FFA ring
studies typically require a step-by-step integrator over well defined magnetic
models that lead to better quantitative analysis of the particle trajectories
with large difference in energy. However, we consider MAD-X and the MAD-X PTC
PTC  libraries for the studies with $\pm$20% of energy spread.

Keil in emma  presents two possibilities on how to proceed:

  * •

do a Taylor expansion of the magnetic field to second order involving dipole,
quadrupole and sextupole components, or

  * •

displace the quad elements to feed down the dipole component and then add
sextupoles.

We decided to start the design following the first option, i.e. do a Taylor
expansion of the magnetic field that leads to superimposed independent dipole,
quadrupole and sextupole components that in principle could be realized with a
canted cosine theta type of magnet cantedcosmagnet ; blondel ; caspi ;
goodzeit .

##  III Accumulator sections

We divide the optics design in several minor sections in order to
systematically approach the previously listed requirements. The initial
subdivision includes:

  * •

A high momentum acceptance arc cell for the arcs

  * •

A zero dispersion cell connecting the arcs with straight sections

  * •

An interaction region common to all three beams and the target

  * •

A radio frequency cavity region

  * •

An extraction region

In the following we explore different kinds of possible cells and subsections
of the accumulator checking if they succeed in achieving the accumulator
requirements.

##  IV Large energy acceptance FFA cell

The design shown in bogomyagkov , a new concept for synchrotron line sources
at around 2 to 3 GeV, is composed by three canted-cosine magnets with dipole,
quadrupole and sextupole components superimposed to achieve a compact cell
optimized for $\pm 40$% momentum acceptance.

We have partially reproduced their results and continued to adjust the lattice
to get the minimum possible circumference for a 22.5 GeV beam.

The cell is shown in Fig. 3. It is composed by a Focusing-Defocusing-Focusing
sequence where we have dipole B components superimposed (BF-BD-BF).
Chromaticity and dispersion were minimized together via multi-parameter match
of phase advance in both planes varying the magnets lengths, dipole and
quadrupole field components while checking the momentum acceptance from
particle tracking in MAD-X PTC over $\pm$20% energy offset. Chromaticity was
corrected in a second step varying the sextupole components.

Figure 3: High energy acceptance FFA cell: (LEFT) Linear Optics. (CENTER)
Second order optics functions $Wx$, $Wy$ and $DDX$ (or $DX^{\prime}$ in the
plot) as defined in MAD-X. (RIGHT) Longitudinal phase space resulting from
particle tracking of a 98 m ring over one thousand turns starting with energy
offset of $\pm$20% in 1% steps.

The separation between BF and BD magnets is 6 cm, while the distance between
BFs in adjacent cells is 20 cm. The magnets length, and the dipole, quadrupole
and sextupole components are respectively:

  * •

BF : 42 cm, 6.165 T, 240.2 T/m, 2575 T/m2

  * •

BD : 80 cm, 5.316 T, -182.7 T/m,-2931 T/m2

The total length of the accumulator is 98 m consisting of 50 cells, 1.96 m
long each. The maximum magnetic field at 2.5 cm moving radially outwards the
reference orbit is just below 13 T, calculated as

$B=B_{d}+B_{q}x+\frac{1}{2}B_{s}x^{2},$ (1)

where $B_{d}$ is the dipole field, $B_{q}$ is the quadrupole field, $B_{s}$ is
the sextupole field, $B$ is the total sum and $x$ is the transverse horizontal
displacement from the magnet center.

The natural chromaticity is $-14$ and $-17$ units in the horizontal and
vertical planes respectively, which implies a very small chromaticity per cell
of the order of -0.25 to -0.35. Sextupoles are tuned to correct the
chromaticity on both planes.

The final momentum compaction factor is $5\times 10^{-3}$, which is a quite
large value for the needs of the muon accumulator ring. Using a cavity at 61
MHz and 400 MV, we get the required energy acceptance over more than $\pm
20$%, but, with a large bunch length of the order of one meter (see Fig. 3 and
Table 4).

We conclude from this cell design that the minimum length of an accumulator
with $\pm 20$% energy acceptance is a little less that 100 m for magnets at
about 13 T. The drawback is the large bunch length due to the momentum
compaction factor to first order.

The next step is the reduction of the momentum compaction factor.

###  IV.1 FFA with reduced momentum compaction factor

The integral expression along the reference orbit to calculate the momentum
compaction factor to first order is wiedermann

$\alpha_{c1}=\frac{1}{C}\int_{c}\frac{\eta_{x}(s)}{\rho(s)}ds,$ (2)

where $C$ is the accelerator circumference and $\rho$ is the local curvature
of the reference trajectory $s$. Using the identity $\rho\theta=L$ valid for
sector bend magnets, the integral in Eq. (2) can be approximated for a fast
calculation to the summation

$\alpha_{c1}\approx\sum_{i=1}^{N}\eta_{xi}\theta_{i}$ (3)

over the $N$ cell elements, being $\eta_{xi}$ the horizontal dispersion and
$\theta_{i}$ the bending angle at the $i$-th element. Three possibilities to
cancel $\alpha_{c1}$ arise:

  * •

Cancel the horizontal dispersion, $\eta_{x}=0$. Which is possible in some
particular types of cells called vertical FFA (vFFA) brooks , and it is not
addressed here.

  * •

Produce positive and negative values of $\eta_{xi}$, cancelling out the
summation over the cell. This is a possible approach but produces large
chromaticity due to the strong focusing to control the dispersion function. It
has not been considered in this design because it has already been explored in
PhysRevAccelBeams.23.051001 .

  * •

Negative bends, i.e. negative $\theta_{i}$. The disadvange is an increased
length of the lattice, but, given the relatively small ring found before, we
explored this idea.

Figure 4 shows the result of the inclusion of anti bends in the design. We
increased the magnetic field of the main dipoles to keep the length to 98 m.
The cell superimposed magnetic components are:

  * •

BF : 42 cm, -2.770 T, 240.2 T/m, 3.416 kT/m2

  * •

BD : 80 cm, 14.697 T, -182.7 T/m, -5.519 kT/m2

Figure 4: Reduced momentum compaction factor FFA cell: (LEFT) Linear Optics.
(CENTER) Second order optics functions $Wx$, $Wy$, $DDX$ (or $DX^{\prime}$ in
the plot) and $DPX$ as defined in MAD-X. (RIGHT) Longitudinal phase space
resulting from particle tracking of a 98 m long ring over one thousand turns
starting with energy offset of $\pm$10% in 1% steps.

We minimized the dispersion function by putting the largest dipole field in
the location of the vertically focusing magnet BD because $\eta_{x}$ follows
the horizontal beta function $\beta_{x}$, which has a minimum at the
defocusing quadrupole.

Particle tracking shows a momentum acceptance of more than $\pm$10%, and bunch
length of 150 mm, from a reduction of $\alpha_{c1}$ to $1.56\times 10^{-3}$
(see Fig. 4 and Table 4).

While any arbitrarily small value of $\alpha_{c1}$ can be obtained, the second
order momentum compaction factor $\alpha_{c2}$ is not longer negligible and
reduces the energy acceptance (See Fig. 5). We thus focus on the cancellation
or reduction of $\alpha_{c2}$ to further increase the particle energy
deviation $\delta$ acceptance.

Figure 5: Longitudinal phase space at the end of one turn particle tracking
for particles with energy offset between $\pm$20% the nominal energy. The
second order momentum compaction factor $\alpha_{c2}$ produces a non-linear
longitudinal offset incompatible with energy gain in the RF section, limiting
the energy acceptance.

We rewrite the momentum compaction factor as an expanded polynomial on energy
as in martin2011 ,

$\displaystyle\alpha_{c}$ $\displaystyle=\frac{\Delta C/C}{\delta}$ (4)
$\displaystyle\alpha_{c}(\delta)$
$\displaystyle=\alpha_{c1}+\alpha_{c2}\delta+\alpha_{c3}\delta^{2}+O(\delta^{3}).$
(5)

and we rewrite the integral

$\alpha_{c2}=\frac{1}{C}\int_{c}\left(\frac{\eta_{x}^{\prime
2}}{2}+\frac{1}{2\rho}\frac{\partial\eta_{x}}{\partial\delta}\right)ds,$ (6)

valid for relativistic beams, where $\eta_{x}^{\prime}$ is the derivative of
dispersion with respect to $s$ and $1/2\cdot\partial\eta_{x}/\partial\delta$
is the second term in the expansion of position with respect to $\delta$. It
can be rewritten using the MAD-X notation as the summation

$\displaystyle\alpha_{c2}\approx$
$\displaystyle\frac{1}{C}\left(\sum_{i=1}^{N}\frac{DPX^{2}_{i}}{2}+\frac{DDX_{i}}{\rho_{i}}\right)\Delta
s_{i}$ (7) $\displaystyle=$
$\displaystyle\frac{1}{C}\sum_{i=1}^{N}\frac{DPX^{2}_{i}}{2}\Delta
s_{i}+\frac{1}{C}\sum_{i=1}^{N}\frac{DDX_{i}}{\rho_{i}}\Delta s_{i}$ (8)
$\displaystyle=$
$\displaystyle\frac{DPX^{2}_{rms}}{2}+\frac{1}{C}\sum_{i=1}^{N}DDX_{i}\theta_{i},$
(9)

which can be evaluated over a single cell for a fast computation.

We note that $DPX^{2}_{rms}$ is always positive, therefore, the only way
remaining to cancel the second order momentum compaction factor is to produce
negative $DDX_{i}\theta_{i}$ in the cell.

The strongest dipole magnet BD creates the largest positive angle deflection
$\theta\approx 0.16$ rad. In a first attempt we produce negative $DDX$ at the
center of the cell by rematching the sextupole strength. This proved to be
effective but chromaticity was not longer cancelled, leaving the cell with
zero momentum acceptance.

As a second alternative, we could vary independently the three sextupoles in
the cell in order to cancel the horizontal and vertical chromaticities,
$dq_{x}$ and $dq_{y}$ respectively, and $\alpha_{c2}$. However, all trials
were unsuccessful because the three parameters are corrected at the expense of
ten times stronger sextupoles that effectively kill the dynamic aperture.

A third approach was to create sextupole families combining sextupoles in two
adjacent cells robin1993 , however, it also resulted unsuccessful because the
phase advance per cell ($\mu_{x}=0.1,\mu_{y}=0.3$) leaves a non orthogonal
configuration. We remind that phase advance was optimized until we got a large
energy acceptance but not orthogonality.

Taking into account the phase advance and following the ideas in fartoukh , we
increased the number of sextupole families to 5 achieving a phase advance per
super-cell of $\Delta\mu_{x}=0.5,\Delta\mu_{y}=1.5$. This allowed us to cancel
out the geometrical errors introduced by the strong sextupoles while giving
the lattice more flexibility to produce the negative DDX required to cancel
$\alpha_{c2}$. However, several matching attempts showed no improvement in the
dynamic aperture due to a large negative DDX peak produced at few sextupoles
per super cell.

After these attempts we consider that a reduction of $\alpha_{c2}$ will
require to minimize DPX. This is equivalent to increasing $\alpha_{c1}$
because the dispersion function remains almost constant.

In spite of the limitations in energy acceptance produced by the second order
momentum compaction factor, we have shown that the minimum bunch length
achievable with a simple FFA cell is in the order of 10 cm. With some
optimization efforts using stronger gradients and dipole fields adding up to
20 T, we can minimize $DPX^{2}_{rms}$ reducing the bunch length to about 2 cm,
and further work could be done to continue exploring other cell
configurations.

##  V Cell for insertions

The target location, the RF cavities and the beam extraction require dedicated
insertions in the accumulator ring. These locations should have zero
dispersion and therefore a matching section joining the insertions with the
arc cell design is needed.

As a first step to include these regions we split the ring circumference in
four equal arcs to preserve the symmetry (see Fig. 6), each starting and
ending with zero dispersion. Those four points will be occupied by two
Interaction Points (IPs) diametrically opposite, and two sections for the RF
cavities and the beam extraction, located 90∘ away from the IPs. The survey
layout is preserved as long as the RF cavity section and the beam extraction
section are of the same length.

Figure 6: Muon Accumulator ring sections. Only one ring is schematically
drawn, the other ring is a mirror reflection.

This simplified layout can be further optimized to have one only IP and a
common region for the RF cavities and extraction kickers. However, the main
goal is to study the overall parameters of a lattice design with FFA cells and
further optimization can be left for a second step to reduce the total
accumulator length in some few tens of meters.

As an open point we could still consider that the chromatic correction of two
IPs is different to one IP. In this respect we remark that the interaction
region should be designed to contribute little to chromaticity because the FFA
cell energy acceptance is deteriorated when sextupole components are modified
to correct not only the self magnet chromaticity but the total ring
chromaticity. As a way to qualitatively measure this effect we will later
refer to the natural chromaticity introduced by the arcs, and the single
insertions.

Several iterations were required in order to completely close the survey with
a total angle of $2\pi$ in the horizontal plane while minimizing the ring
circumference, dipole field and peak dispersion. The result is shown in Fig.
7.

Figure 7: (TOP) FFA cell modified to get zero dispersion on one side. (BOTTOM)
Longitudinal phase space resulting from particle tracking over a 98 m long
ring over one thousand turns starting with energy offset of $\pm$9% in steps
of 1%.

The dispersion function at the end of the arc changed and sextupole components
were rematched to achieve the chromatic correction. The effect is a reduction
of the momentum acceptance from $\pm 10$% to $\pm$9% which is not
significative.

We list the magnet components obtained in this section for reference and
include the parameter results in Table 4. However, they have been further
modified when cancelling the additional chromaticity coming from the
insertions.

  * •

B1 : 42 cm, 0.0 T, 240.2 T/m, 8.5 kT/m2

  * •

B2 : 80 cm, 1.2 T, -182.7 T/m, -13.4 kT/m2

  * •

B3 : 42 cm, 4.1 T, 240.2 T/m, 0.0 kT/m2

##  VI Extraction Kickers and RF sections

The muon beams lose few MeV when traversing a thin target
PhysRevAccelBeams.23.051001 . Although the exact amount depends on the target
length and the effect is negligible for a fraction of a radiation length of
material, the effect over hundreds or thousands of passages is relevant. This
energy needs to be recovered during each accumulation cycle and therefore an
RF cavity has been added to the design.

At the end of the accumulation cycle, the muon bunch should be extracted to
continue with the next stages of the muon acceleration chain (not yet fully
defined but a temptative proposal is shown in blancomucoll ).

There are no particular restriction about the requirements for these sections,
therefore we use twice the same optics design (one for the RF cavity and
another for an extraction kicker section) where the main criteria has been to
create a few meters long drift region for the RF cavities with large $\beta$
functions for the extraction kicker, and low chromaticity to only slightly
increase the arc sextupoles in charge of chromatic correction.

Considering the RF cavities, the optimization process started with a 15 m long
apochromatic lattice blancoprab2020  that was modified to create a 4 m long
drift, enough to allocate few RF cavities providing a total voltage in the
order of a hundred MV.

For the extraction region the beta functions were inflated to a few meters in
order to increase the effectiveness of a kick $\Delta x^{\prime}$, as in the
expression

$\Delta x\propto\Delta
x^{\prime}\sqrt{\beta_{k}\beta_{ext}}\sin\Delta\phi_{x,ext}$ (10)

where $\Delta x$ is the displacement produced by the linear propagation of a
kick at the kicker location and in a lattice with twiss optics functions
$\beta_{k}$, $\beta_{ext}$ at the kicker and extraction points respectively,
and a phase advance $\phi_{k,ext}$ between them.

In order to reduce the off-energy beta beating we opt for an apochromatic
line. Magnet gradients are kept in the order of 100 to 200 T/m, comparable to
that of the Future Circular Collider (FCC) fcchh .

The result of the optimization is a 13 m long region with a 4 m long drift and
a $\beta$ function just below 10 m, shown in Fig. 8. It is for sure a
preliminary result, because it does not match the best phase advance to
extract the beam. However, it was included to take into account the possible
increase of the accumulator ring length.

Figure 8: A 13 m long optics design for an apochromatic drift space 4 m long,
with (TOP) twiss $\beta$ functions of few meters using FCC–like quadrupoles
with gradients in the order of 100 to 200 T/m. (BOTTOM) The chromatic
functions $W_{x}$ and $W_{y}$ are very small at the entry and exit point of
the section. Two of this optics are used in the accumulator ring, one for the
RF cavities and one for the extraction kicker.

##  VII Beam separation–combination

The region containing the target is common to three beams : $\mu^{+}\mu^{-}$
at about 22.5 GeV and $e^{+}$ at double the muon energy. In the LEMMA scheme
(see Fig. 1) the interaction of the positron beam produces muon pairs that are
transported and recirculated to pass through the target a few hundreds times.
The muon beams therefore need to be separated and directed into the two muon
rings, while a line/ring will transport the remaining positrons to the next
stage in the positron chain.

When recirculated, the beams need to be recombined in order to increase the
muon bunch population while mitigating the emittance growth.

There are some aspects to take into account when considering the recombination
of the muon beam:

  * •

The multiparticle interaction of different charges at different energies. We
will not pursue any study about this subject on this article.

  * •

A second aspect is the effect of multiple scattering in the target on the
final muon beam emittance. In the article PhysRevAccelBeams.23.051001 , a
$\beta^{*}_{\mu}$ from 1 m to 10 cm was explored for a monochromatic beam,
but, proved not enough to mitigate the impact of a thousand turns through the
target. We have achieved a $\beta^{*}_{\mu}=20$ cm over $\pm$5% energy spread
using quadrupole magnets with gradient of 500 T/m, and theoretically explored
stronger focusing in order to determine the limits of such mitigation (see
Section VIII for the Interaction Region design and Section XI for the
accumulation results).

  * •

The separation and combination of the beams will produce synchrotron radiation
coming from the positron beam at high energy. We will take this into account
when considering the strength of the dipole fields.

The beam separation and combination scheme has evolved with time. Initially,
in collaboration with Susanna Guiducci from INFN and Simone Liuzzo from ESRF,
a very strong magnet was foreseen to separate the beams in short space. Figure
9 shows the schematic diagram of three beams being combined, passed through
the target and separated by 11 T 30 cm long dipole magnets. The expected
energy loss of the muon beam is negligible for a single pass, while, the
positron beam loses 0.2 GeV which is a considerable large amount of energy.
The $e^{+}$ beam radiation could be high, therefore, they suggested to think
in a lower B-Field.

Figure 9: Beam separation with 11 T dipole magnets. The positron beam loses
0.2 GeV per passage, a value that could be too high. Image courtesy of Simone
Liuzzo from ESRF.

###  VII.1 Considerations due to radiation

The positron beam energy loss expected to come as synchrotron radiation when
positrons traverse strong dipole magnets forces the separation and combination
design to consider normal conducting magnets on the order of 2 T.

We continued the design with a 1.5 T 3.3 m long dipole magnet leading to a 1.5
T and 1.2 m long septum to leave the positron beam escape the interaction
region while the muon beams are separated into assumed 16 T dipole magnets of
independent accumulator rings, see Fig. 10. The aperture has been
schematically drawn as black lines at $\pm$5 cm around the $e^{+}$ and
$\mu^{+}\mu^{-}$ trajectories.

From particle tracking in MAD-X PTC it was clear that the aperture will limit
the energy acceptance. A muon beam with large energy spread would occupy the
whole aperture when reaching the septum, so only $\pm$5% energy spread lines
have been tracked around the nominal energy line. We also verified that
positrons with energy as low as 36 GeV or -20% energy offset pass through the
separation region. At the entrance in the accumulator, beam pipes are
separated by about 10 cm, which should be a safe value to consider separated
magnets.

Figure 10: Beam separation with magnets below 2 T. A dipole field separates
the three beams while a septum directs the muon beams into the rings letting
the positron beam pass through. The positron beam loses few tens of MeVs while
the photon critical energy corresponds to strong X-rays at 2 MeV. The
separation region is less than 10 m long in total. The 16 T magnets correspond
to the beginning of the accumulator ring.

The positron beam loses 20 MeV in the 3.3 m long 1.5 T magnet, which is small
and in the same order of magnitude of energy loss due to the interaction with
a thin target. Photons are emitted from the interaction of the beam and the
dipole field with a critical energy of 2 MeV, which is a value comparable with
the FCC–ee Interaction Region expectations.

Given the acceptable critical energy of the irradiated photons and the small
energy loss of the positron beam, the design of the Interaction Region will
include magnets on the other of 1.5 to 2 T. The optics will be matched so that
the beam behavior does not perturb the arc.

###  VII.2 Exploring the idea of a separation aided by a sextupole field

As an alternative to the single dipole, we have considered the usage of a
sextupole magnet. Figure 11 shows the schematic diagram of a beam separation
assisted by a large aperture sextupole.

Figure 11: Beams separation aided by a large aperture sextupole magnet. The
three beam coming out the target are initially separated by a low dipole
field. The positron beam will continue to cross a sextupole magnet near the
axis which leaves it almost unperturbed, while, the two muon beams are kicked
away from the sextupole axis in opposite directions increasing the beam
separation before the septum. Figure 12: Beam separation design using a
combination of dipole and sextupole magnets below 2 T peak field in order to
reduce the amount of synchrotron radiation coming from the high energy
positron beam. Three beams with energy offset of $\pm$20% have been tracked to
show the energy acceptance of the separation region. The total length is less
than 30 m long.

Three beams exit the target and enter a relatively low dipole field of 0.5 T
and 1.2 m in length, after which the three beams are separated. In order to
give an extra separation kick to the muon beams while reducing the radiation
of the positron beam, a sextupole is located among the septum and the dipole.
The sextupole will kick the two muon beams in opposite directions because the
kick sign will depend on the muon charge irrespective of the crossing side, as
shown by the square term in the equations

$\displaystyle x^{\prime}$ $\displaystyle=K_{2}L(x^{2}+y^{2})$ (11)
$\displaystyle=\frac{1}{B_{\rho}}\left(\frac{\partial^{2}B_{y}}{\partial
x^{2}}\right)Lx^{2};\text{\;\;(assuming }y=0)$ (12)
$\displaystyle=\frac{q}{P}\left(\frac{\partial^{2}B_{y}}{\partial
x^{2}}\right)Lx^{2}$ (13)

where $x^{\prime}$ is the angle change in the trajectory of the particle due
to a normal focusing sextupole, $B_{\rho}$ is the particle magnetic rigidity,
$q$ is the charge of the particle, $x$ and $y$ are the transverse particle
coordinates when crossing a thin sextupole, $L$ is the length of the
sextupole, and

$\frac{\partial^{2}B_{y}}{\partial x^{2}}=B_{\rho}K_{2}.$ (14)

We have considered a gradient of 800 T/m2, a diameter of aperture equal to 76
mm, a length of 1.4 m, a magnetic peak field of 1.6 T and a good field region
of 10 mm (as the one in the FCC–ee ARC). This gives a $k_{2}=10.8$/m3 and a
$k_{2}L=15.1$/m2. Using this parameters we can estimate an additional kick of
1.5 mrad, which is small, but could lead to a reduction in the level of
radiation.

The design was continued tracking particles in pyzugoubi pyzgoubi , a python
python  interface to the tracking code Zgoubi zgoubi , allowing to estimate a
20 m long separation region taking into account the tracks of all beams with
an energy spread of $\pm$20%, see Fig. 12. It consist in a composition of
dipoles and sextupoles with reduced peak magnetic field that effectively
separates the three beams.

The total length required to separate the beams is slightly less than 30 m
long, and therefore is too long for our design. Section VIII is dedicated to
the Interaction Region design and we integrate the results of these studies
with the muon beam optics.

##  VIII Interaction Region

The interaction region of the muon accumulator ring in the LEMMA scheme is one
of the most critical parts because it is common to the three particles species
at two different energy levels ($\mu^{+}$ and $\mu^{-}$ at about 22.5 GeV and
$e^{+}$ at double the muon energy), and must mitigate the effect of multiple
scattering of beams with the target located at the Interaction Point (IP).

Multiple scattering with the target will increase the beam divergence per
passage by an amount $\Delta\sigma^{\prime}\propto\sqrt{X_{0}}/E$, where $E$
is the particle energy and $X_{0}$ is the thickness of the material in
radiation length units. We expect that the divergence growth due multiple
scattering will be more significant for the muon beam because the muon beam
has only half of the positron beam energy, and in addition the muon beams pass
a large number of times through the target (equivalent to a large $X_{0}$).

As a way to mitigate the effect one could try to focus the beams so that the
divergence is much larger than the contribution from multiple scattering over
some number of passages $N$. The contribution from multiple scattering is
uncorrelated, thus, we require a beam divergence

$\sigma^{\prime}\geq\sqrt{N}\Delta\sigma^{\prime}$ (15)

with $N\approx 100$ for the positron beam, $N\approx 10^{3}$ for the muon
beam, in the initial LEMMA proposal. This forces the interaction region design
to have a very small $\beta^{*}$ at the IP, able to produce a focal point with
large divergence from a low emittance beam $\epsilon$, according to
$\sigma^{\prime}=\sqrt{\epsilon/\beta}$.

In eplusringopt  it was found that a $\beta^{*}_{e^{+}}=50$ cm was enough to
keep low the positron beam emittance over more than 50 turns through out a
thin target of 0.89%$X_{0}$, i.e. almost half a radiation length of material
when adding all passages. In the case of the muon beam we estimated that a
$\beta^{*}_{\mu}=1$ cm will be required to avoid the emittance growth over a
thousand turns through out the material.

Due to the large muon beam energy spread coming from the kinematics of the
collision, we note that the Interaction Region should not only have a small
$\beta^{*}$ at the nominal energy, but also over the whole muon beam energy
range.

We have explored the magnet requirements to get a $\beta^{*}_{\mu}=1$ cm as
shown in Fig. 13. The design consists in two triplets, the first one reduces
the beta functions to 1 cm in both planes while the second triplet matches the
$\beta$ functions to some arbitrary value in the order of a meter and produces
the opposite chromaticity to cancel the chromatic functions at the end point.

Figure 13: Theoretical evaluation of an Interaction Region design to produce
(TOP) a $\beta^{*}_{\mu}=1$ cm over $\pm 2$% energy spread, for
$L^{*}=1~{}$cm. The design consists in two triplets, the nearest to the IP
focalize the beam, while the second produces opposite chromaticity and brings
the $\beta$ functions to some meters (possible values of a zero dispersion
cell to be connected with). (BOTTOM) Apochromaticity of the design achieved by
cancelling the $W_{x}$ and $W_{y}$ chromatic functions at the IP and at the
end points.

The space between the IP and the nearest quadrupoles is $L^{*}=1$ cm. This
means that only very thin targets could be considered.

On top of the difficulty to put a target in 2 cm ($2L^{*}$) of space, the
quadrupole magnets in the inner triplet are expected to have a gradient of 20
kT/m, i.e. be at 20 T peak with 1 mm of aperture radius. This is a gradient
that has not been yet achieved with any type of magnet technology. Quadrupole
gradients in the order of 200 to 400 T/m are foreseen in the FCC design. The
CLIC QD0 magnet pushes the gradient to a bit more than 500 T/m Modena:1427609
; testQD0 . We require at least a factor ten higher gradients.

The main reason to use such an extreme gradient is shown in Fig. 14. The
optics twiss $\beta$ and chromatic $W$ functions around the IP grow fast near
the IP due to the low $\beta^{*}$, and as a result, the dynamic aperture of
the IR is reduced because of the off-momentum beta–beating. In order to give
an order of magnitude of the $\beta$ variation with energy, we rewrite here
the expression of the chromatic functions

$W_{X,Y}=\sqrt{\left(\frac{\partial\alpha_{x,y}}{\partial
p_{t}}-\frac{\alpha_{x,y}}{\beta_{x,y}}\frac{\partial\beta_{x,y}}{\partial
p_{t}}\right)^{2}+\left(\frac{1}{\beta_{x,y}}\frac{\partial\beta_{x,y}}{\partial
p_{t}}\right)^{2}}.$ (16)

Assuming that $\alpha$ and its derivative with respect to momentum $p_{t}$
cancel at some location in the triplet, we can write

$\beta_{x,y}W_{x,y}\Delta p_{t}\approx\Delta\beta_{x,y},$ (17)

which in our inner triplet evaluates to $W_{x,y}=40$, $\Delta p_{t}\approx
0.2$ and $\beta_{x,y}=5$ m, giving $\Delta\beta_{x,y}=40$ m, which is a very
large $\beta$ excursion for a small aperture triplet, effectively limiting the
energy acceptance.

Figure 14: Zoom into the inner triplet (TOP) $\beta$ functions and (BOTTOM)
chromatic functions for a $\beta^{*}_{\mu}=1$ cm and $L^{*}=1$ cm

In spite of the problems with the magnets apertures, one could be interested
in the theoretical model behavior at different energies. Figure 15 shows the
twiss calculation for several energy offsets in steps of 1% the nominal energy
where we attach one interaction region with a mirrored interaction region to
get the twiss results.

The design is stable in the energy range of $\pm$2% the muon beam energy. This
value however is very small for the requirements of accumulation, as it would
only accept muons from a positron beam energy just above the muon pair
production threshold reducing the production efficiency to almost $10^{-8}$
muon pairs per positron, see Table 2.

Figure 15: Twiss functions $\beta^{*}$ and $\alpha^{*}$ at the IP of a
theoretical interaction region with a triplet at 20 kT/m, and $L^{*}=1$ cm for
a range of energy spread. The twiss computation succeeds in the energy range
of $\pm 2$% the muon beam energy.

From all previous aspects, we do not pursue a design achieving a
$\beta^{*}_{\mu}=1$ cm any further.

In order to reduce the $\beta^{*}$ over a large energy range we have started
from an apochromatic transport line able to focus the positron and muon beams
blancoprab2020 . The transport line was designed to have a
$\beta^{*}_{\mu}=20$ cm over $\pm$5% of energy spread, and a
$\beta^{*}_{e+}=50$ cm at twice the muon beam energy.

Figure 16 shows the result of the antisymmetric design after rematching the
transport line to the $\beta$ functions at zero dispersion cell entry side. We
see a lattice where the beta functions in horizontal and vertical plane start
with a value of 20 cm and they propagate over more than 15 m to get a beta
value close to 2.2 m.

The optimization also minimized the chromatic functions $Wx$ and $Wy$ which
will allow for a smaller off-energy beta-beating in the arc and therefore a
better chromatic correction. The chromaticity added by this section is less
than 1.5 units per plane which could be corrected by sextupoles in a quarter
of the arc running at an additional 50% strength (1.5/12 cells is
50%$\times$0.25, the arc cell chromaticity).

Figure 16: Half of the Interaction Region design to produce (TOP) a
$\beta^{*}_{\mu}=20$ cm over $\pm 5$% energy spread, for $L^{*}=10~{}$cm. The
design consists in two triplets: the nearest to the IP, at $s=0$, focalizes
the beam using a gradient of about 500 T/m, while the second produces opposite
chromaticity and brings the $\beta$ functions to some meters to match the zero
dispersion cell with magnets at 100 to 200 T/m. (BOTTOM) The low chromaticity
of the design is achieved by partially cancelling the $W_{x}$ and $W_{y}$
chromatic functions at the IP and at the end points. The chromaticity
introduced by the line is below 1.5 units on each plane.

###  VIII.1 Including the beam separation into the Interaction Region Design

Profiting from the long drifts to reduce the beta from 2 m in the arc to 20 cm
at the Interaction Region, we have included three _vertical dipoles_ with
reduced magnetic fields in the range of 1 to 2 T, mitigating the positron
synchrotron radiation, as explained in Section VII. The combination of the
three vertical dipoles produces an early and smooth separation/combination of
the three beams while cancelling the vertical dispersion $\eta_{y}$ and its
derivative with respect to $s$, $\eta^{\prime}_{y}$.

In order to keep the symmetry of the vertical dipole configuration at the
combination (i.e. before the IP) and the separation (i.e. after the IP), the
apochromatic Interaction Region has been rematched for a symmetric reflection
instead of the asymmetric solution previously described (in the asymmetric
solution the mirrored magnets polarity is inversed, while this is not the case
in the symmetric solution). The optimization process matched the
$\beta^{*}_{\mu}$, the $\beta$ at the arc cell entrance and the chromaticity
leaving a small beta-beat. It seems possible to improve the matching if
additional quadrupoles are included in the simplex minimization, but as a
first approximation it doesn’t represent a large problem on the design. Figure
17 shows the result of the modifications in the initial concept of the
interaction region.

Figure 17: Half of the Interaction Region design with IP at $s=0$ m and the
beginning of the zero dispersion arc cell at $s=16$ m. Three vertical dipoles,
with a field lower than 2 T, separate the beams while cancelling $\eta_{y}$
and $\eta^{\prime}_{y}$. The first triplet achieves a $\beta^{*}_{\mu}=20$ cm
over $\pm 5$% energy spread for $L^{*}=10$ cm using 500 T/m gradients, while
the second triplet matches the beta functions with the arc. All quads length,
strength and in-between drifts have been matched to minimize chromaticity as
in the symmetric apochromatic design.

The beams reach the arcs completely separated as a consequence of having low
magnetic field vertical dipoles in the IR. This scheme also allows to consider
that both rings will bend in the same direction with one on top of the other,
therefore, minimize the footprint of the machine, see Fig. 18, and allowing to
use a second Interaction Region (IR*) opposite to the location of the first
one, that could be also considered a muon source in the case of a second
positron source and target.

Figure 18: Rings with vertical separation can be one on top of the other. It
also allows to use a second target $T^{*}$ given a second positron beam.

The length of the machine would benefit from having multiple IPs as it will
reduce the length-over-IP-number ratio, leading to an improved muon
accumulation. In the case of this particular design we have 231 m/2 IPs for
magnets at about 14 T, but the combination arc–IR could be repeated as many
times as required because it reduces the peak magnet dipole field as the
circumference grows.

##  IX the accumulator

The muon accumulator ring design, as described before, corresponds to the
connection of the interaction region(s), the arcs, the RF cavity section and
the extraction section. We use all previous optics designs and rematch some
elements to properly adapt the twiss optics functions and correct the total
chromaticity.

Figure 30 shows the twiss functions over the entire length of the accumulator
ring while Table 5 shows the main parameters obtained with this optics. Table
4 shows only the most relevant parameters in comparison to previous design
stages. Figure 19 shows half of the accumulator ring.

Figure 19: Half of the muon accumulator ring (IP, arc and one straight section
used for RF cavities).

In Fig. 20 we show the arc cell after rematching dipole, quadrupole and
sextupole fields. For a 1 cm good field region the magnets are below 15 T,
while the individual components are

  * •

BF : 42 cm, -3.1 T, 238 T/m, 6.0 kT/m2

  * •

BD : 80 cm, 12.0 T, -183 T/m, -10.3 kT/m2

This cell sequence is composed by three BF–BD–BF magnets, where the distances
between BD and BF magnets is 6 cm while the distance between BF magnets in
adjacent cells is 20 cm.

Figure 20: Arc cell resulting from the rematch of the accumulator ring with
insertions and chromatic correction.

The zero dispersion cell is composed by the magnets:

  * •

B1 : 42 cm, 0.0 T, 238 T/m, 9.6 kT/m2

  * •

B2 : 80 cm, 1.2 T, -183 T/m, -12.9 kT/m2

  * •

B3 : 42 cm, 4.1 T, 238 T/m, 0.0 kT/m2

###  IX.1 Survey

Figure 31 shows the survey of the accumulator ring. The arcs are entirely
separated for muon species while the interaction region is common to all three
beams. The positron beam enters on one side of the IP, crosses the IP and
passes in-between the two muon arcs.

##  X Beam size and tune footprint

Figure 21 shows the expected beam size of a muon beam with a transverse
emittance of 20 $\pi$ nm rad and an energy spread of 6%, calculated as
$\sigma=\sqrt{\epsilon\beta+\eta^{2}\delta^{2}}$. One sigma of the beam relies
well inside the good field region of $\pm$2 cm along the arcs, and it is much
smaller than the 4 mm aperture at the inner triplet in the Interaction Region
with magnets of 500 T/m. However, further studies are needed to determine if
the stay clear criteria of 10$\sigma$ is required.

Figure 21: Beam size along the accumulator ring calculated as
$\sigma=\sqrt{\epsilon\beta+\eta^{2}\delta^{2}}$ for 6% energy spread and 20
$\pi$ nm rad transverse emittance. Units in mm. The black line represents the
horizontal plane while the red line represents the vertical. The beamsize at
the target location is 60 $\mu$m in both planes.

Figure 22 shows the tune footprint and the tune as a function of the energy
for the accumulator. The particular tune chosen up to know is completely
arbitrary, it must be moved to a region more distant from third and fourth
order resonances, which we believe could be achieved by a rematch of the cell
or increasing the number of cells in the arcs. However, the point of this plot
is to show the behavior of the chromatic correction on the entire energy
range.

Figure 22: Accumulator Ring tune (TOP) as a function of energy offset,
(BOTTOM) footprint due to energy offsets.

Figure 23 shows the variation of the twiss functions $\beta^{*}_{\mu}$ and
$\alpha^{*}_{\mu}$ at the interaction point with the target. It shows nearly
constant values along the expected energy range of $\pm$6%. Further studies
could be addressed to optimize these parameters, but the results from this
first design are satisfactory.

Figure 23: Accumulator Ring twiss $\beta^{*}_{\mu}$ and $\alpha^{*}_{\mu}$ vs
energy offset ranging $\pm$6% of the nominal energy. The optics effectively
focuses the beam over the expected energy range.

##  XI MUON ACCUMULATION STUDIES

The muon transverse geometrical emittance, $\epsilon_{\mu}$, for a thin target
is

$\epsilon_{\mu}=\sigma_{e^{+}}\cdot\sigma^{\prime}_{\mu}(E_{e^{+}}),$ (18)

i.e. the product of the positron beam size $\sigma_{e^{+}}$ when crossing the
target and the divergence of the produced muon beam $\sigma^{\prime}_{\mu}$
from the kinematics of the target collision with a positron beam at some
energy $E_{e^{+}}$. One could approximate the divergence to be 0.1 to 0.5 mrad
for a positron beam at 43.72 to 45 GeV, respectively. Table 2 contains the
main parameters of the muon production obtained from simulations with MUFASA
ciarma , bench-marked with Geant4 geant .

First, we calculate the muon beam size and emittance that matches the optics,
and then will use those parameters to fix the requirements on the positron
beam, as in blancoprab2020 .

Given an energy acceptance of at least $\pm$5% the muon energy and the results
listed in the Table 2, we can expect a beam divergence close to 0.3 mrad and
energy spread close to $\pm$5% from the kinematics of collision with an
$e^{+}$ beam at $E_{e^{+}}=$43.8 GeV. The muon beam size at the target that
matches the optics can be calculated as,

$\sigma_{\mu}=\beta_{\mu}^{*}\sigma^{\prime}_{\mu}=0.2\text{~{}m}\cdot
0.3\text{~{}mrad}=60~{}\mu\text{m}.$ (19)

Now, we can calculate the geometrical transverse emittance of the produced
muon beam as

$\epsilon_{\mu}=\sigma_{\mu}\sigma^{\prime}_{\mu}=60~{}\mu\text{m}\cdot
0.3\text{~{}mrad}=18~{}\pi~{}\text{nm~{}rad}.$ (20)

We assume that the positron beam at $E_{e^{+}}=43.8$ GeV, has the same beam
size given by Eq. (18), $\sigma_{e^{+}}=60~{}\mu$m. This value is above the
minimum used in alesini2019positron  for simulations of power deposition in
the target by a positron bunch population of $3\times 10^{11}$ particles.

As a free parameter to match the optics $\beta^{*}_{e^{+}}$ we have the
positron beam emittance. Thus,

$\epsilon_{e^{+}}=\frac{\sigma_{e^{+}}^{2}}{\beta^{*}_{e^{+}}}=\frac{(60~{}\mu\text{m})^{2}}{0.5~{}\text{m}}=7.2~{}\pi~{}\text{nm~{}rad},$
(21)

which is similar to the 6 $\pi$ nm rad studied for the positron ring in
Boscolo:IPAC17-WEOBA3 .

Table 1 summarizes the list of beam parameters obtained, where we also include
the positron ring energy spread given in Boscolo:IPAC17-WEOBA3 .

Table 1: Positron and muon beam parameters matching the low beta $\beta^{*}_{\mu}=20$ cm of the interaction region design. Particle | Energy | $\epsilon$ | $\sigma$ | $\sigma^{\prime}$ | $\delta E$
---|---|---|---|---|---
| (GeV) | ($\pi$ nm rad) | ($\mu$m) | (mrad) | (%) | type
$e^{+}$ | 43.8 |  7 | 60 | 0.12 |  $\pm$0.1 | (Gaus)
$\mu$ | 21.9 | 18 | 60 | 0.30 |  $\pm$5 | (Flat)
Table 2: Muon production efficiency $eff$, divergence $\sigma^{\prime}_{\mu}$ and energy spread $\delta E_{\mu}$ as a function of the positron beam energy $E_{e^{+}}$. The muon energy and divergence have a flat distribution. Efficiency has been estimated for 1% of a radiation length of Beryllium (3.5 mm of material). The muon beam divergence and energy spread are given by the collision kinematics. $E_{e^{+}}$ | $eff$ | $\sigma^{\prime}_{\mu}$ | $E_{\mu}$ | $\delta E_{\mu}$ | $E_{\mu}(1\pm\delta E_{\mu})$
---|---|---|---|---|---
(GeV) | ($10^{-8}\mu/e^{+}$) | (mradrms) | (mradmax) | (GeV) | (%rms) | (%max) | Min/Max (GeV)
43.72 |  2.3 | 0.07 | 0.16 | 21.860 |  1.40 |  2.9 | 21.20/22.50
43.80 |  3.0 | 0.14 | 0.26 | 21.900 |  2.86 |  4.9 | 20.82/22.98
44.00 |  4.6 | 0.23 | 0.42 | 22.000 |  4.81 |  8.3 | 20.17/23.83
44.50 |  7.1 | 0.37 | 0.67 | 22.250 |  7.85 | 13.6 | 19.44/25.56
45.00 |  8.7 | 0.47 | 0.83 | 22.500 |  9.90 | 17.0 | 18.67/26.33
46.00 | 10.9 | 0.62 | 1.11 | 23.000 | 12.91 | 22.6 | 17.80/28.19

The muon longitudinal emittance comes from the product of the positron bunch
length and the energy spread produced by the kinematics of the production. We
assume here the previously studied parameters of a 6 or 27 km long positron
ring in Refs. eplusringopt ; simone  with a bunch length of 3 mm, giving a
longitudinal emittance at production in the order of $6.8$ $\pi$ mm GeV
(3$\times$0.10$\times$45/2 $\pi$ mm GeV) and could be as low as 0.7 $\pi$ mm
GeV (3$\times$0.01$\times$43.72/2 $\pi$ mm GeV). However, we remark that it is
much smaller than the bunch length obtained from the optics studies, therefore
the longitudinal emittance is determined by the accumulator.

With respect to the muon beam energy, the amount of synchrotron radiation
along the accumulator is negligible while the energy loss from interacting
with the target is small for a thin target and only becomes significant when
passing several radiation lengths of material. The energy spread of the
outgoing muon beam ranges from 1 to 10% of half the positron beam energy in
the range of 43.72 to 45 GeV, respectively (See Table 2).

With respect to the muon life time at 22 GeV, we will be able to perform at
most 520 turns around the accumulator. We will not go into details of muon
decay and energy deposition in this article.

The muon pair production efficiency $eff$ has been defined in NIM  as the
ratio of muon pairs ($\mu^{\pm}$, or simply $\mu$) per positron impinging on a
fixed target. See Table 2.

###  XI.1 Beam target interaction simulation

In order to accelerate the calculation time, a dedicated fast Monte Carlo
called MUFASA, written in CERN-ROOT cernroot , has been benched-marked with
Geant4 and MDISim Burkhardt:IPAC15-TUPTY031 , and has proved to be very useful
and flexible to simulate the muon production and to study the dynamics of the
stored muon beam which interacts at each turn with the target. The simulation
starts by generating a positron beam population that interacts with the
material of the target. Then, the outgoing positrons and produced muons are
tracked through the lattice using MAD-X PTC and the output distribution is
saved to calculate positron and muon beam populations, emittances, beams
divergence and beam size. We repeat the previous process over many turns until
the muon population does not increase significantly.

###  XI.2 Simulation results

We will concentrate in a positron beam at 43.8 GeV interacting with 1%$X_{0}$
of Beryllium material, giving a production efficiency $eff=3\times 10^{-8}$
$\mu$/$e^{+}$.

Figure 24 shows the muon accumulation over several hundreds of turns in the
accumulator ring. A positron bunch population of $5\times 10^{11}$ produces
$1.5\times 10^{4}$ muon pairs per interaction. The first hundred turns
increase the population as expected to about $1.5\times 10^{6}$, but after
that the growth slows down because particles undergo multiple scattering
interacting with the target leaving the dynamic aperture of the machine. After
400 turns the accumulation saturates, as new muon only replace the amount
leaving the stable phase space.

Figure 24: Muon accumulation for a positron beam at 43.8 GeV impinging on
0.01$X_{0}$ of Beryllium as target material. The population increases as
expected in the first hundreds of turns, after that the muon beam population
grows slowly because of particle losses produced by the multiple scattering
with the target, exceeding the dynamic aperture.

Figure 28 show the emittance growth as a function of the accumulator turns, as
well as the beam dimensions in the horizontal plane. The emittance grows
slowly and proportionally to the number of turns around the accumulator ring
due to multiple scattering with the target. After a few hundred turns the
emittance growth is limited by the dynamic aperture of the machine. The final
normalized transverse emittance is approximately $10$ $\pi$ $\mu$m
(200$\times$50 $\pi$ nm).

Figure 29 shows the bunch length, energy spread and longitudinal emittance
during the accumulation period. The final emittance is approximately 10 $\pi$
GeV mm, given by the product of a 2 cm bunch length and 0.5 GeV of rms energy
spread.

We have also tested the theoretical mitigation of multiple scattering with an
ultra low $\beta^{*}_{\mu}=1$ cm. Figure 25 shows the beam emittance for a
$\beta^{*}_{\mu}=1$ cm, with the design shown in Section VIII. The emittance
growth is much smaller due to the ultra-low $\beta^{*}_{\mu}$ and it always
remains smaller than the dynamic aperture of the machine.

Figure 25: Muon beam emittance as a function of the accumulator turns for a
$\beta^{*}_{\mu}=1$ cm. The emittance growth is reduced and the beam remains
always inside the dynamic aperture of the lattice.

###  XI.3 Materials other than Beryllium

One possible way to increase the muon production efficiency is to consider
other materials for the target. The transverse and longitudinal emittance
depend on the fraction of radiation length of material interacting with the
muon beam. Therefore, using the same 1%$X_{0}$ units for other materials, we
can have the same beam dynamics while changing only the muon production
efficiency. Table 3 shows a small list of possible targets with different
lengths and efficiencies that have been foreseen.

In particular, we highlight the possibility to use 1%$X_{0}$ of liquid
Hydrogen ($H_{2}$) which will increase the muon population by a factor 2 with
respect to Be. The target dimensions fit the 20 cm ($2L^{*}$) available space
among quadrupoles close to the interaction point. A possible small mismatch of
the optics due to the target length seems negligible because the beta function
grows just to $25$ cm at the entry point of the first quadrupole magnet.

Despite of the advantages of liquid Hydrogen regarding the muon production
efficiency, there are materials that could become more practical solutions as
they withstand better the power deposition or have dimensions that better suit
the short space for beam–target interaction. Therefore, in Table 3 we include
also Carbon, a carbon based composite C/C A412 maciariello , liquid Lithium
(LLi), and a mix of 10% liquid Lithium and 90% Diamond Powder (LLi-D)
PhysRevAccelBeams.23.051001 . We note that the emittance obtained with these
materials is equal to the case of Beryllium when using the same amount of
radiation lengths of material.

Table 3: Muon production efficiency from a positron beam at 45 GeV interacting with Beryllium, liquid Hydrogen (LH2), liquid Lithium (LLi), a composition of 10% liquid Lithium and 90% Diamond powder (LLi-D), Carbon and a Carbon composite (C/C A412). Density and length for 1%X0 are included. Material | Density | 1%$X_{0}$ Length | $eff$
---|---|---|---
| (g cm-3) | (mm) | ($10^{-8}\mu/e^{+}$)
LH2 | 0.071 | 88.8 | 18.9
LLi | 0.534 | 15.5 | 10.9
Be | 1.848 |  3.5 |  8.7
LLi-D | 3.221 |  1.3 |  6.3
C | 2.267 |  1.8 |  6.2
C/C A412 | 1.7 |  2.5 |  6.2

Figure 26 show the result of accumulation with other materials, in particular,
we remark the results of using _film_ targets, as described in
PhysRevAccelBeams.23.051001 , that were created to reduce the impact of
multiple scattering. Figure. 27 shows the effect of increasing the beam energy
spread above the design specification of 43.8 GeV.

Figure 26: Muon beam accumulation with 1% of a radiation length for different
target materials interacting with a positron beam at 44 GeV with $5\times
10^{11}$ $e^{+}$ per bunch.

Figure 27: (TOP) Muon beam population as a function of the number of turns for
various positron beam energies on 1%$X_{0}$ of Be. (BOTTOM) Transverse
emittance of the accumulated beam.

##  XII Issues and Further design improvements

The biggest limit for accumulation in the design is the $\pm$5% energy
acceptance of the Interaction Region. It was already known from the beginning
that this magnet configuration would have such performance because it
corresponds to the symmetric solution of a first order apochromatic design.
However, it seems possible to increase the momentum acceptance by designing a
second order apochromatic Interaction Region. This kind of design will require
the integration of MAD-X PTC transport maps with custom scripts to calculate
the beam optics at several orders in the polynomial expansion. A similar
approach is followed in Mapclass2 mapclass2  to calculate the final beamsize.
While, a recent effort to do the symbolic polynomial calculation of the twiss
beta functions dependence on energy is available in ob1julia .

The momentum compaction factor is in the order of $10^{-4}$ bringing the bunch
length to about 2 cm. it will be beneficial to consider a reduction to
$10^{-5}$ or even $10^{-6}$ as the energy acceptance grows. Further
optimizations considering magnets below 20 T could reduce the bunch length.
However, a more interesting idea is the exploration of a cell with more than
three magnets and/or a vertical FFA brooks  lattice that provides a zero
momentum compaction factor at least to first order approximation.

On other aspects, we remark that the principle to use multiple IPs could be
beneficial as it reduces the requirements of the dipole fields to close the
ring, while the distance between IPs could, in principle, remain constant. Of
course, this would mean several positron sources available, one per IP.

The good field region of a magnet has been assumed to be at least $\pm 2$ cm.
While this is a very common value for the FCC magnets, it would be beneficial
to consider magnets with good field regions larger by a factor 10, i.e. $\pm
20$ cm. This will allow to increase the dispersion in the arcs, reducing the
sextupole strength and increasing the dynamic aperture.

The focal point has a $\beta^{*}_{\mu}=20$ cm over $\pm 5$% using quadrupole
gradients of about 500 T/m. The beam emittance growth due to multiple
scattering would benefit from a $\beta^{*}_{\mu}$ reduction, therefore an
exploration of designs with higher gradients could be interesting. Previous
work for CLIC points to gradients close to 1 kT/m or even larger silverstrov ;
modena .

With a toy design achieving $\beta^{*}_{\mu}=1$ cm over $\pm 2$% energy
spread, we have estimated that we will need gradients at least a factor 10
larger, i.e. in the order of 10 to 20 kT/m. This values are available with
plasma lenses, e.g. the E-150 experiment plasma3 ; plasma ; plasma2  at the
SLAC Final Focus Test Beam Facility (FFTB), reports gradients in the order of
$10^{6}$ T/m for electron a positron beams of $1.5\times 10^{10}$ particles
per pulse, energy of 28.5 GeV, with a horizontal and vertical beamsize in the
order of 10 $\mu$m, with a bunch length of $600$ $\mu$m and normalized
emittance of about 50 $\mu$m crossing a gas jet of $N_{2}$ or $H_{2}$. These
positron and electron beam parameters are similar to the current muon beam
parameters studied for LEMMA at the beginning of the accumulation cycle.

##  XIII Conclusions

The design of the two identical muon accumulator rings with FFA technology for
the LEMMA project has been presented.

This 231 m long optics design for the accumulator is composed by high energy
acceptance arcs based on Fixed Field Alternating Gradient (FFA) cells, two
Interaction Regions (IRs) achieving a low $\beta^{*}_{\mu}$ of 20 cm over $\pm
5$% energy spread, an extraction and a RF section with low contribution to
chromaticity.

Due to the energy acceptance and the kinematics of muon production, this
design is apt to accumulate muons at 21.9 GeV.

The Interaction Region has been conceived to combine and then separate three
beams ($\mu^{+}$ and $\mu^{-}$ at 21.9 GeV, and $e^{+}$ at twice the muon beam
energy) reducing the amount of synchrotron radiation coming from the positron
beam, and focalizing the muon beams at the Interaction Point (IP) where
$\beta^{*}_{\mu}=20$ cm. The IP marks the location for a thin target of 1% of
a radiation length of material. The distance between the nearests quads around
the IP is 20 cm (2$L^{*}$).

The arc magnets superimpose dipole, quadrupole and sextupole components that,
in principle, could be realized with a canted cosine type of magnet.

The highest quadrupole gradient of 500 T/m, like in the CLIC Final Focus, is
used in the IR to focalize the beams. All other magnets in the design have
smaller gradients in the range of 100 to 250 T/m with a good field region of
$\pm 1$ to $\pm 2$ cm and a peak magnetic field below 15 T, similar to the FCC
magnet parameters.

The RF and the extraction section have been designed to minimize the
contribution to the ring chromaticity while giving enough space for a kicker
or radiofrequency cavities of few hundred MV.

Due to the complexity of the requirements to get a high quality muon beam,
Table 4 summarizes the FFA design exploration and intermediate results on the
most relevant optics parameters. It reflects the reduction of the energy
acceptance due to the requirement of low $\beta^{*}_{\mu}$ at the target
location. For completeness, Table 5 gives a longer list of parameters of the
accumulator design.

Simulations of the beam–target interaction with MUFASA show that 100 bunches
of $5\times 10^{11}$ $e^{+}$ each at 43.8 GeV impinging on 3.5 mm of Beryllium
produce a final population of about $10^{6}$ muons at 21.9 GeV with a
normalized transverse emittance of 10 $\pi$ $\mu$m and longitudinal emittance
of 10 $\pi$ GeV mm.

The number of accumulation cycles is limited because multiple scattering with
the target increases the muon beams emittance leading to particles losses when
they populate the dynamic aperture of the machine. The initial normalized
transverse beam emittance is 5 $\pi$ $\mu$m and grows due to multiple
scattering with the target in few hundreds of turns to the final 10 $\pi$
$\mu$m. In order to further mitigate this effect we will need to produce a
smaller $\beta^{*}_{\mu}$ using quadrupole gradients above 500 T/m.

The muon population is small because the $\pm 5$% energy acceptance of the IR
limits the muon production efficiency (in the order of $10^{-8}$ muon pairs
per positron), therefore, the accumulation studies point to the need of a
redesign of the Interaction Region for a larger energy acceptance. One of the
possibilities is to create a second order apochromatic IR that could accept an
energy spread of $\pm 10$%.

Alternatively, one could consider different materials for the target. We show
the results of accumulation for few interesting ones, from where we would like
to highlight a _film_ of liquid Hydrogen (LH2) that allows to increase the
production efficiency and reduces the multiple scattering with the target
reaching a population of $10^{7}$ muons in a thousand turns.

The arc magnets could also be reoptimized for larger dynamic aperture given
that good field regions of more than $\pm 2$ cm in superconducting canted
cosine magnets are possible.

In addition, the accumulation opens up the possibility to quantitatively study
the multiparticle interaction of three beams at two different energies, one is
the high intensity low emittance positron beam and the other two are a growing
population of muons.

##  XIV Acknowledgments

This work has been financially supported by the Istituto Nazionale di Fisica
Nucleare (INFN), Italy, Commissione Scientifica Nazionale 5, Ricerca
Tecnologica – Bando n. 20069.

The authors would like to thank Manuela Boscolo, Mario Antonelli, Susanna
Guiducci, Alessandro Variola, Marica Biagini and Francesco Collamati from
INFN, and Pantaleo Raimondi and Simone Liuzzo from ESRF for useful discussions
on the muon beam production.

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Figure 28: (TOP) Transverse muon beam emittance as a function of the
accumulator turns for a $\beta^{*}_{\mu}=20$ cm. The emittance grows in the
first hundreds of turns due to multiple scattering, after that, accumulation
saturates due to the limiting dynamic aperture of the machine. (CENTER)
Horizontal beam size (BOTTOM) Horizontal divergence.

Figure 29: Muon beam (TOP) longitudinal emittance, (CENTER) bunch length and
(BOTTOM) energy spread.

Table 4: Lattice design summary. The _FFA cell_ and _Low $\alpha_{c}$_ designs refer to rings with high energy acceptance and without IP. The _Zero $\eta_{x}$_ design refers to the ring with 4 point where the dispersion is cancelled, it is a first approach to the inclusion of the insertions in the ring. The _+IR_ and _1+1* IPs_ designs refer to the inclusion of all ring sections considering 1 or 2 IPs. Parameter | Unit | Requirement | FFA cell | Low $\alpha_{C}$ | +zero $\eta_{x}$ | +IR | 1+1* IPs
---|---|---|---|---|---|---|---
Energy Acceptance | % |  $\pm$20 | $>\pm 20$ | $\pm 10$ | $\pm 9$ | $\pm 5$ | $\pm 5$
IPs |  | 1 | – | – | 1 | 1 | 2
Circumference/IP | m | 60 | 98 | 98 | 98 | 230 | 230/2
$\beta^{*}_{\mu}$ | m | 0.01 | – | 2.2 | 2.2 | 0.2 | 0.2
Bunch length | mm | 3 | $\sim 10^{3}$ |  $\sim$150 | 100 | 20 | 20
Table 5: Accumulator Ring Parameters. Parameter | Unit | Requirement | FFA design
---|---|---|---
Energy | GeV | 22.5 | 21.9
Relativistic Gamma Factor | – | 212.95 | 207.272
Length | m | 60 | 230
Revolution Frequency | MHz | 5 | 1.30275
Revolution Time |  $\mu$s | 0.2 | 0.7676
Energy Loss per Turn | MeV | – |  $3\times 10^{-6}$ (S.R.), $\sim 10$ (thin target)
Energy Acceptance | % |  $\pm$20 |  $\pm$5
Number of Bunches | – | 1 | 1
Bunch Population | – | $10^{9}$ |  $2\times 10^{6}$ (Be), $10^{7}$ (LH2 film, $E_{e^{+}}=44$ GeV)
Normalized Transverse Emittance |  $\pi$ $\mu$m | 0.04 | 5 (at production), 10 (end of accumulation)
Geometrical Transverse Emittance |  $\pi$ nm | 0.2 | 25 (at production), 50 (end of accumulation)
Longitudinal Emittance |  $\pi$ GeV mm | – | 10 (end of accumulation)
Number of IPs | – | 1 | 1+1*
Cycles of accumulation | – | 1000 |  $<400$ (Be), $<1000$ (LH2 film)
Nat. Chrom. x/y | – | – | -26.8 / -29.4
Qx/Qy/Qs | – | – | 25.6306 / 10.1525 / 0.0137
$\beta^{*}_{\mu}$ at the IP (target location) | cm | 1 | 20
Distance from IP to first magnet, $L^{*}$ | cm | – | 10
$\alpha_{C}$ | – | very small | $3\times 10^{-4}$
Bunch length | mm | 3 | 20
Straight Sections | – | – | 4 (2 IPs, RF, extraction)
Arc Magnets Peak Field | T | – | 15
Type of magnets | – | – | Canted Cosine Theta
Highest Dipole Component | T | – | 12
Highest Quadrupole Component | T/m | – | 240
Highest Sextupole Component | T/m2 | – | 11000
Arc Magnet Good Field Region | cm | – |  $\pm$2
IR Magnets Inner Triplet Peak Field | T | – | 2
Type of magnets | – | – | Dedicated function
Highest Quadrupole Gradient | T/m | – | 500
Aperture Radius | mm | – | 4
IR Magnets Second Triplet Peak Field | T | – | 5
Type of magnets | – | – | Dedicated function/superconductor
Highest Quadrupole Gradient | T/m | – | 200
Insertion Magnets Peak Field | T | – | 5
Type of magnets | – | – | Dedicated function/superconductor
Highest Quadrupole Gradient | T/m | – | 200
RF Voltage | MV | – | 150
RF Harmonic | – | – | 600
RF Frequency | MHz | – | 782
RF Drift Space | m | – | 4
Extraction Kicker Drift Space | m | – | 4

Figure 30: Accumulator Ring with arcs based on large energy acceptance FFA
cells. The first and second IPs are at $s=0$ m and $s=115$ m, respectively,
where the twiss beta functions are minimized to $\beta^{*}_{\mu}=20$ cm over
$\pm$5% energy spread for an $L^{*}=10$ cm. The RF section is located at
$s=58$ m while the extraction section is located at $s=173$ m. Four FFA arcs
join the insertions using combined function magnets at 15 T peak, foreseen to
be of canted cosine type.

Figure 31: Accumulator ring survey, units in meters. (TOP) Top view, IP is at
coordinate (0,0) (CENTER) Three dimensional representation of the Survey map
of the two rings obtained from MAD-X and plotted with Gnuplot [51]. (BOTTOM)
Side view of the positive and negative muon rings separation.