muCool: muon cooling for high-brightness μ+ beams

A number of experiments with muons are limited by the poor phase
space quality of the muon beams currently available. The muCool project
aims at developing a phase-space cooling method to transform a surface
\mu^+μ+
beam with 4 MeV energy and 1 cm size into a slow muon beam with eV
energy and 1 mm size. In this process the phase space is reduced by a
factor of 10^{9}-10^{10}109−1010
with efficiencies of 2\cdot 10^{-5}-2\cdot 10^{-4}2⋅10−5−2⋅10−4.
The beam is then re-accelerated to keV-MeV energies. Such a beam opens
up new avenues for research in fundamental particle physics with muons
and muonium atoms as well as in the field of
\muμSR
spectroscopy.

Scheme of the proposed muon compression beam line. Muons from the secondary µ + beam enter the transverse compression stage, where they are first stopped in the helium gas and then compressed in transverse (y) direction by using the combination of a vertical temperature gradient and the electric and magnetic fields. After that, they enter the longitudinal compression stage, where they are compressed in the longitudinal (z) direction and then extracted into the vacuum.
cyclotron frequency ! = eB/m. However, the presence of the He gas leads to µ + -He collisions that modify the muon motion. The deviation from theÊ⇥B direction (averaged over many collisions) will be proportional to the collision frequency ⌫c between muons and He atoms, as described by the following equation [14]: where ✓ is angle of the muon drift velocity relative to theÊ ⇥B direction. Thus, we can manipulate the muon drift direction by changing the collision frequency ⌫c.
The collision frequency can be made position dependent by having di↵erent gas densities in di↵erent regions of our setup. In the transverse compression target this is achieved by keeping the upper wall of the target at 12 K and the lower at 4 K which creates a temperature gradient and therefore also density gradient in the y-direction [15].
In the middle of the target (at y = 0, see Fig. 2 (left)), the gas density is chosen such that ⌫c ! = 1. According to the Eq. (1), at this condition the muons drift at 45 angle with respect toÊ ⇥B direction, which in our case corresponds to the +x direction (see gray trajectory in Fig. 2 (left)).
In the top part of the target, the gas density is lower, which means that ⌫c ! < 1 and muons move essentially in theÊ ⇥B direction. With our field configuration, this corresponds to muons moving y-direction while drifting in the +x direction (red trajectory in Fig. 2 (left)).
In the lower part of the cell, at larger gas densities, ⌫c ! > 1, the muons drift mostly in electric field direction, i.e in +y and +x directions (blue trajectory in Fig. 2 (left)). The result is transverse (in y-direction) compression of the muon beam. Figure 30.1: Schematic diagram of the muCool device. A surface muon beam is stopped in a cryogenic He gas target with a vertical temperature gradient inside a 5 T field. The extent of the stopped muons is reduced first in the transverse ( y), then in the longitudinal (z) direction using a complex arrangement of E-field and gas density gradient. The compressed muon beam is then extracted through an orifice into vacuum and re-accelerated along the z-axis.
The drift velocity of the µ + in a gas with E-and B-fields is given by In this equation µ is the muon mobility, ω = eB/m the cyclotron frequency of the muon, ν the The muCool setup is conceived as a sequence of stages having various density and electric 48 field conditions. In the first stage, which is at cryogenic temperatures, the muon beam is 49 stopped and compressed in y-direction (transverse compression). In the second stage, which 50 is at room temperature, the muon beam is compressed in z-direction (longitudinal direction).

51
In the third stage, the muons are extracted from the gas target into vacuum, re-accelerated in 52 −z-direction, and extracted from the B-field.

53
The 4 -MeV µ + beam with σ x, y ≈ 10 mm is degraded in a moderator and then stopped

Demonstration of transverse and longitudinal compression
The µ + drifting in x-direction then enter into the second stage, which is at room tempera-69 ture and has a field E = (0, E y , ±E z ), with E y = 2E z = 0.1 kV/cm, with a strong z-component 70 pointing towards z = 0. Because ν is small at room temperature, the µ + motion in this stage is 71 dominated by the third term of (30.1) resulting in a fast reduction of the longitudinal extent.

72
During this fast compression, the E y -component (seeÊ ×B term in (30.1)) drifts the µ + in 73 x-direction towards the extraction stage. From there, the compressed beam can be extracted 74 though a small orifice into vacuum, and moved quickly into a region of low gas pressure where 75 re-acceleration can occur. Finally the beam needs to be extracted from the solenoid through 76 an iron grid that terminates the magnetic field lines.   The GEANT4 simulation of the muon trajectories under such conditions is shown in Fig. 2 (right). Muons start at around x = 15 mm with about 10 mm spread in the y-direction and drift in +x-direction while simultaneously compressing in the y-direction. At x = 20 mm, the muon spread in y-direction is reduced to about 1 mm.

2nd stage: longitudinal compression
After the transverse compression stage, muons enter the second compression stage, which is at room temperature. The electric field now has a component parallel to the magnetic field and points towards the center of the target, which causes a muon drift into the center of the target, giving rise to the longitudinal (in z-direction) compression of the muon beam (see Fig. 3).
Additionally, there is a component of the electric field perpendicular to the magnetic field, in +y direction. Therefore, muons also drift inÊ ⇥B direction, which in this case points in +x direction, towards the final compression stage and extraction into the vacuum. (Right) Projections of the muon trajectories in the yx-plane for the big aperture and "pure drift" density conditions, i. e., without density gradient (3.5 mbar and 5 K on average). In this case, the muons also drift in the +x-direction, but without compressing in the y-direction. According to the simulation, the drift velocity is about 2 mm/µs. This value can be increased in the final setup by increasing the strength of the electric field in the y-direction.

conclusions
The longitudinal compression stage of the muCool device under development at PSI has been demonstrated. An elongated muon swarm of 200 mm length has been compressed to below 2 mm length within 2 µs. Good agreement between the simulation and the measurement has been observed.
Furthermore, the ability to drift the µ + beam in E ⇥ B-direction towards the prospective position of the extraction hole has been demonstrated by performing a measurement with the electric field having also a component perpendicular to the magnetic field.
In both cases, slightly better agreement between simulations and measurements is achieved by including small additional effective losses in the simula-   the muCool baseline efficiency using the commissioned mixed-compression target as a reference point for the compression towards the orifice. We thus assume here a target having an active region of 50 mm length operated at 10 mbar pressure with a 6-20 K temperature gradient. The stopping probability of 0.6% has been simulated assuming a surface muon beam with 10% (FWHM) momentum bite. A 3% (FWHM) momentum bite would increase the stopping probability to 1.6%. All the other entries have only been estimated and depend strongly on the upcoming R&D results.  The muCool beam can also greatly improve µSR investigations of sub-mm samples. Be-191 cause the pile-up effects in the typically 10 µs-long observation time window become increas-192 ingly unsustainable for rates exceeding 5 · 10 4 s −1 , the full HiMB-muCool potential could be 193 exploited by switching the keV-energy sub-mm beam between several µSR instruments oper-194 ating simultaneously.

195
Muon to vacuum-muonium conversion is very efficient for keV-energy muons [15]. Hence, 196 the sub-mm muCool beam at keV-energy could be converted into a high-brightness muonium 197 source. This novel muonium source could be exploited to improve on the precision of muonium 198 spectroscopy by orders of magnitude (e.g. the 1S-2S with a relative accuracy of 10 −12 [16]), 199 and could be used to study the influence of gravity on the muonium to investigate the grav-