Demonstration of Muon-Beam Transverse Phase-Space Compression
A. Antognini N. J. Ayres I. Belosevic V. Bondar A. Eggenberger M. Hildebrandt R. Iwai D. M. Kaplan K. S. Khaw K. Kirch A. Knecht A. Papa C. Petitjean T. J. Phillips F. M. Piegsa N. Ritjoho A. Stoykov D. Taqqu G. Wichmann
physics.acc-ph

	
	
	
	
	
	
	
	
























[]aldo@phys.ethz.ch
Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland
Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


[]ivanabe@phys.ethz.ch
Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland





Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland

 
 Illinois Institute of Technology, Chicago, IL 60616 USA



Present address: Tsung-Dao Lee Institute, and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland
Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland
Dipartimento di Fisica, Università di Pisa, and INFN sez. Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Illinois Institute of Technology, Chicago, IL 60616 USA


Present address: Laboratory for High Energy Physics, Albert Einstein Center for Fundamental Physics, University of Bern, CH-3012 Bern, Switzerland
Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland


Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland


Present address: Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
Institute for Particle Physics and Astrophysics, ETH Zürich, 8093 Zürich, Switzerland

muCool collaboration





We demonstrate efficient transverse compression of a 12.5 MeV/c muon beam stopped in a helium gas target featuring a vertical density gradient and crossed electric and magnetic fields.

The muon stop distribution extending vertically over 14 mm was reduced to a 0.25 mm size (RMS) within 3.5 .


The simulation including cross sections for low-energy μ^+-He elastic and  charge exchange (μ^+↔ muonium) collisions describes the measurements well. By combining the transverse compression stage with a previously demonstrated longitudinal compression stage, we can improve the phase space density of a μ^+ beam by a factor of 10^10 with 10^-3 efficiency.







Demonstration of Muon-Beam Transverse Phase-Space Compression
    G. Wichmann
    December 30, 2023
=============================================================





Next generation precision experiments with muons and muonium atoms <cit.>, such as muon g-2 and EDM measurements <cit.>, muonium spectroscopy <cit.>, and muonium gravity measurements <cit.>, require high-intensity muon beams at low energy with small transverse size and energy spread. No muon beams currently available fulfill these requirements. To improve the quality of the muon beam, phase space cooling techniques are needed. Conventional methods, such as stochastic cooling <cit.> and electron cooling <cit.>, are not applicable due to the short muon lifetime of 2.2  (at rest). Alternative beam cooling techniques based on muon energy moderation in materials have been developed <cit.>, however, they suffer from low cooling efficiencies (<10^-4).

At the Paul Scherrer Institute, we are developing a novel device that reduces the full (transverse and longitudinal) phase space of a surface <cit.> (28 MeV/c momentum) μ^+ beam by 10 orders of magnitude with 10^-3 efficiency <cit.>. 
This so-called muCool device is placed inside a 5  magnetic field, pointing in the +z-direction (Fig. <ref>).
First, a surface muon beam propagating in the -z-direction is stopped in a few mbar of cryogenic helium gas (at 10 K on average), reducing the muon energy from MeV to the eV range. 

As a result of the slowing down, the momentum spreads in all directions σ_p_x,p_y,p_z are reduced from 𝒪(MeV/c) to about 20 /c.
The 6D phase space is subsequently further reduced by compressing the spatial extent σ_x,y,z (RMS) while keeping the same momentum spread.

This is achieved by guiding the stopped muons into a sub-mm spot using a combination of strong electric and magnetic fields and gas density gradients in three stages (see Fig. <ref>).





In the first stage (transverse compression), the electric field is perpendicular to the magnetic field and at 45 with respect to the x-axis: E⃗ = (E_x, E_y, 0), with E_x=E_y≈ 1 /. In vacuum, such fields would prompt the stopped muons to drift in the -direction, performing cycloidal motion at the cyclotron frequency ω=eB/m_μ, where m_μ is the muon mass.

However, in the muCool device, the muons collide with He gas atoms with an average frequency ν_c, which depends on the gas density, elastic μ^+-He cross section, and muon energy.
These collisions lead to muon energy loss and direction change, modifying the muon motion compared to that in vacuum, so that muons drift at an angle θ relative to the -direction <cit.>:





    tanθ= ν_c/ω.


This equation can be understood as follows: for ν_c>ω, muons collide with He atoms at the beginning of the cycloidal motion, before their trajectory is significantly bent by the magnetic field, resulting in a large θ (blue trajectory in Fig. <ref>). For ν_c < ω, muons travel further in the -direction between collisions (similar to vacuum), resulting in a smaller θ (green trajectory in Fig. <ref>).





By changing the collision frequency we can thus manipulate the muon drift direction.
In the transverse compression stage, we change the collision frequency by modifying the gas density: the top of the apparatus is kept at higher temperature than the bottom (19 K versus  6 K), creating a temperature gradient in the vertical (y-) direction. The pressure is chosen so that ν_c/ω=1 at y=0. Muons at different y-positions (at fixed x-position) experience different densities, resulting in different drift directions: a muon stopped at the bottom of the target (higher density) experiences more collisions (ν_c/ω>1) and will thus drift predominately in the Ê-direction (large θ, upwards), while a muon stopped in the top part (lower density) collides less frequently with He atoms (ν_c/ω<1), resulting in drift predominately in the -direction (small θ, downwards). As a result, muons stopped at different y-positions converge towards y=0, while drifting in the +x-direction, ultimately reaching a size σ_y≈ 0.25 mm. Throughout this drift, the collisional energy loss is balanced by the acceleration in the electric field, so that ∼ 20 /c momentum spreads σ_p_x,p_y,p_z are maintained; combined with the spatial compression in the y-direction we thus obtain transverse (σ_y ·σ_p_y) phase-space compression.

Subsequently, the muons enter the second stage, at room temperature, where E⃗=(0,E_y, ± E_z), with E_y=2E_z = 0.1 kV/cm.
The E_z-component drifts the muons towards z=0, thus reducing their σ_z extent to sub-mm (longitudinal compression). Simultaneously, at this low density, the E_y-component drifts the muons in the (+x) direction, towards the extraction stage. From there, the sub-mm muon beam can be extracted though a small orifice into vacuum, re-accelerated along the z-direction with pulsed electric fields to keV energies, and extracted from the magnetic field. 

Efficient muon-beam longitudinal compression, including an -drift, has already been demonstrated <cit.>.


In this Letter we demonstrate the muon transverse compression stage of the muCool device.
For this demonstration, about 2·10^4 μ^+/s at 12.5 MeV/c were injected into the 25-cm-long target placed inside a 5-tesla solenoid. 
Before entering the target, the muons traversed a 55--thick entrance detector, several thin foils, and a copper aperture, defining the beam injection position. 
 
The transverse target gas volume was enclosed by a Kapton foil, folded and glued around two triangular PVC end-caps. 

The large thermal conductivity of single crystal sapphire plates glued to the top and bottom target walls assured homogeneous wall temperatures. The required He gas temperature gradient from 6 to 19 K was produced by heating the top sapphire and thermally connecting the bottom sapphire to a cold finger <cit.>. 

The Kapton foil enclosing the gas volume was lined with electrodes extending along the z-direction. The 45 electric field was defined by applying appropriate voltages to several of these electrodes (see Fig. <ref>), which were connected to other electrodes via voltage dividers.

Several plastic scintillator bars (A1...A3) were placed around the target to monitor the muon movement by detecting the μ^+-decay positrons. The scintillators' position resolution was improved by embedding them in collimators, mechanically decoupled from the target.

An additional downstream scintillator system (back detector) was used for aligning the magnetic field and the beam on the target axis (the muon beam follows the magnetic field lines). 
Wavelength-shifting fibers, glued inside scintillator grooves, transported the scintillation light from cryogenic to room temperature, where it was read out using silicon photomultipliers.





The probability of detecting the decay positrons versus the muon decay position is shown in Fig. <ref> for detectors A1 and A2L.
By recording time spectra for each detector (with t=0 given by the muon entrance detector), we can indirectly observe the muon drift. To demonstrate transverse compression we need to show that the muon beam vertical size decreases during the drift.
This is achieved by comparing the time spectra obtained when the temperature gradient (6–19 K at 8.6 mbar) is applied, to the time spectra with negligible temperature gradient (4–6 K at 3.5 mbar): in the first case both drift and compression are expected; in the second basically only drift (“pure drift”). 
We further increased the contrast between “compression” and “pure-drift” measurements by injecting the muons through either “top-hole” or “bottom-hole” apertures at y=±4.5 mm (see Fig. <ref>). If we were to inject the muons through a single large aperture, most muons would stop close to y=0, and would thus drift straight towards the target tip for both “compression” and “pure drift.” Contrarily, with either of the two smaller apertures, we target a narrow gas region, with distinct density conditions for “compression” and “pure drift.”



Geant4 <cit.> simulations of the muon trajectories for various experimental configurations are shown in Fig. <ref>. The simulation included the most relevant low-energy μ^+-He interactions: elastic collisions and charge exchange. The needed cross sections were appropriately scaled from the proton-He cross sections <cit.>. Simulations show that for the “compression” case, muons injected through both apertures reach the tip of the target efficiently, while in the “pure drift” case, most of the muons stop in the target walls before reaching the tip.

The measured A1 and A2L time spectra under “compression” conditions are presented in Fig. <ref> (red dots), for both injection types. The time spectra were corrected for muon decay by multiplying the counts by exp(t/2.2 μ s). The A1 counts increase with time, for both injection positions, indicating muons were gradually moving towards the tip of the target, , towards the A1 acceptance region (see Fig. <ref>). The A2L counts first increase, then decrease with time, suggesting muons first entered, then exited the detector acceptance region. After a certain time, the number of counts stays constant in both detectors, implying that the muons reached the target walls. For “bottom-hole” injection, this happens up to 1500 ns later compared to “top-hole” injection. Indeed, muons injected through the “bottom hole” travel through a denser gas than for “top-hole” injection, leading to a slower drift. All these features are consistent with the simulations of Fig. <ref> (left). 


The “pure-drift” time spectra for “top-hole” injection (black points in Fig. <ref>,  top row) differ significantly from the “compression” time spectra (red points). The A1 counts remain low at all times, suggesting muons never reached the tip of the target. The A2L counts stay constant after reaching the maximum, implying muons did not “fly by” the detector, but they stopped in the target walls within the acceptance region of the detector. This is consistent with the simulations of Fig. <ref> (right).
However, for “bottom-hole” injection, the measured “pure-drift” time spectra (black points in Fig. <ref>, bottom row) are almost identical to the “compression” time spectra (red points). This is mainly because of the worse detector resolution in the lower target region, which is further away from the detectors.



To better compare the measurements with the Geant4 simulations, the muon trajectories of Fig. <ref> were folded with the detector acceptance of Fig. <ref> to produce the corresponding time spectra.

These time spectra (dashed curves in Fig. <ref>) were fitted to the measurements using two fit parameters per time spectrum: a normalization, accounting for uncertainties in the stopping probability and detection efficiencies (detectors were not calibrated), and a flat background, accounting for beam-related stops in the walls.
Additionally, the detector positions had to be varied in the simulations because of their position uncertainty (about 1 mm). The obtained chi-squareds are relatively large, which points to systematic discrepancies between the simulations and the measurements. 

We investigated the sensitivity of the simulated time spectra to other experimental uncertainties: initial beam characteristics, foil thicknesses,  electric and magnetic field maps, He density distribution, and misalignment (tilt) between the target and the magnetic field axes.





These sensitivity studies show that the alignment uncertainty, which translates into an uncertainty in muon stopping position, is by far the dominant one: even a 0.5 mm muon-beam displacement significantly affects the time spectra. However, such precise beam alignment is not possible with our back detector (consisting of 4× 4× 5 mm^3 scintillators). 







By simulating the time spectra for various tilts and fitting them to the data, we found that the best fit (for “compression” and “pure drift” simultaneously) is obtained by rotating the target axis from the (0,0,1) direction to (0.007,0.018,0.9998) for “top-hole,” and to (0.019, 0.005, 0.9998) for “bottom-hole” injection. These tilts shift the center of the muon stopping distribution in the xy-plane by up to ∼ 3.5 . Note that the two different tilts are justified because the beam injection position was changed by adjusting the solenoid placement. 

Simulations including such tilts (solid curves in Fig. <ref>) agree with measurements significantly better. Additional smaller improvements could be achieved by simultaneously fine-tuning all other experimental parameters (within their experimental uncertainty), together with tilt and detector positions.
However, little further insight would be gained from such a time-consuming optimization, since the simulations presented here already reproduce well the main features of the measurements. Hence, the measurements demonstrate the muon-beam transverse compression.




















Next, we investigate the muon drift versus its energy by varying the electric field strength.
The drift angle θ is proportional to the μ^+-He collision frequency given by


    ν_c=Nσ_MT(E_CM)|v⃗_r|,


where N is the helium number density, σ_MT(E_CM)
the μ^+-He momentum transfer cross section, E_CM the center-of-mass energy, and v⃗_r the μ^+-He relative velocity.








For muon energies ≲ 1 eV, σ_MT scales as 1/√(E_CM)∝ 1/|v⃗_r| <cit.>, so that the collision frequency is independent of the muon energy. 

Above 1 eV, the product σ_MT|v⃗_r| decreases with energy (see Fig. <ref>), so that ν_c and θ decrease with increasing muon energy.
Hence, the muon drift direction approaches the -direction with increasing electric field (increasing energy), as shown in Fig. <ref>.



The dependence of the muon motion on the electric field strength was measured for various experimental configurations. Figure <ref> presents only the measurements for “top-hole” and “compression” configuration, which illustrate all relevant features: with decreasing HV, the A1 spectrum maxima decrease, indicating increasingly fewer muons reach the target tip, while the A2L spectra maxima shift towards later times, implying slower drift.

This is consistent with the simulated trajectories of Fig. <ref>.


The simulated time spectra for each HV and for each detector were fitted independently to the measurements with two free parameters: normalization and flat background.
A simultaneous fit for all HV values was not possible, as the normalization giving the best fit depended systematically on the HV. This might be caused by the lack of a precise definition of the electric field at the target tip, which would affect most significantly the measurements with strong electric fields, where muons reach the tip of the target.
A mismatch of the muon stopping distribution between simulations and experiment would also contribute to this systematic effect.


Still, the simulation correctly reproduces the muon drift velocity, which is given by the energy-dependent μ^+-He elastic cross section. Indeed, the observed maxima of the time spectra, occurring when muons arrive in the detector acceptance region, are consistent with simulations.

To summarize, in this Letter we demonstrate muon-beam phase-space compression in the y-direction with the muCool device. One critical aspect of this demonstration was distinguishing between simple muon drift versus drift with simultaneous reduction of the beam vertical size. Such distinction was accomplished by performing measurements with and without a vertical temperature gradient, which is needed for transverse compression, and by injecting the muon beam into different vertical target regions (up to 14  mm apart in the y-direction as shown in Fig. <ref>, left). 

The muon motion was simulated using the Geant4 toolkit including custom low-energy (<1 keV) muon processes <cit.>. Very good agreement with the measurements was achieved after accounting for small target and detector misalignments. Our modeling of low-energy μ^+-He collisions was further validated by studying the dependence of the muon drift on the applied electric field.


Simulations show that after the transverse compression stage the muon beam has σ_y=0.25 mm, 5 eV energy, and σ_p_x,p_y,p_z≈ 20 keV/c.
These beam characteristics depend only on the electric field strength and gas density (gradient) regardless of the initial beam properties.
When applying this transverse stage to a typical standard high intensity muon beam (e.g. πE5 <cit.> and μE4 <cit.> surface beams at PSI) with  σ_x,y≈ 10 mm  and σ_ p_x,p_y≈2 MeV/c at 28 MeV/c momentum (4 MeV energy), we obtain a σ_y·σ_p_y phase-space compression by a factor of ∼ 4· 10^3 with an overall efficiency of 32%. This efficiency is mainly given by muon decay, as the compression takes on average 2.3 , and includes 10% losses from muonium formation. We assumed here 100% stopping probability in He gas (achievable by slightly elongating the target in the z-direction).




While σ_p_x and σ_p_z are reduced by the dissipative slowing-down process in the transverse stage, the extension in the z-direction σ_z is reduced only in the following longitudinal stage, while σ_x is reduced by a pulsed re-acceleration process after muon extraction <cit.>.
Therefore, a total phase-space compression of 10^10 with 10^-3 efficiency is attainable by connecting the transverse compression stage demonstrated in this Letter to the previously demonstrated longitudinal compression stage <cit.>, followed by vacuum-extraction and re-acceleration <cit.>.












The experimental work was performed at the π E 1 beamline at the PSI proton accelerator HIPA. We thank the machine and beam line groups for providing excellent conditions. Simulations were performed on the Euler cluster managed by the HPC team at ETH Zürich.
We gratefully acknowledge the outstanding support received from the workshops and support groups at ETH Zurich and PSI. Furthermore, we thank F. Kottmann, M. Horisberger, U. Greuter, R. Scheuermann, T. Prokscha, D. Reggiani, K. Deiters, T. Rauber, and F. Barchetti for their help. This work was supported by the SNF grants No. 200020_159754 and 200020_172639.








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