Study of muonium emission from laser-ablated silica aerogel

The emission of muonium ($\mu^+e^-$) atoms into vacuum from silica aerogel with laser ablation on its surface was studied with various ablation structures at room temperature using the surface muon beams at TRIUMF and Japan Proton Accelerator Research Complex (J-PARC). Laser ablation was applied to produce holes or grooves with typical dimensions of a few hundred $\mu$m to a few mm, except for some extreme conditions. The measured emission rate tends to be higher for larger fractions of ablation opening and for shallower depths. More than a few ablation structures reach the emission rates similar to the highest achieved in the past measurements. The emission rate is found to be stable at least for a couple of days. Measurements of spin precession amplitudes for the produced muonium atoms and remaining muons in a magnetic field determine a muonium formation fraction of $(65.5 \pm 1.8)$% and indicate that the polarization of muonium atoms is preserved in vacuum. A projection of the emission rates measured at TRIUMF to the corresponding rates at J-PARC is demonstrated taking the different beam condition into account. The projected rates agree with the ones measured at J-PARC within the uncertainty of about $\pm$14%.


Introduction
Applications of intense beams of muons with energies below a few tens of keV ("keV muons") are growing in a variety of scientific fields. Low energy muons can be excellent probes to investigate the magnetic properties of materials using muon spin relaxation (µSR) techniques [1]. Applications can be extended to thin layers at a scale less than 10 nm for energies below 1 keV. Low energy muons can also be utilized in the form of muonium (µ + e − , or Mu) atoms, both as an analogue to a light isotope of hydrogen for surface chemistry [2] and as a purely leptonic bound state for precision tests of quantum electrodynamics (QED) [3][4][5]. Furthermore, an intense muon beam with extremely low energy naturally has a small phase space volume, thus it can provide a viable small emittance source for subsequent acceleration. The resulting muon beam of high intensity retains a small emittance, offering the possibility to extend these applications as well as open up new windows into particle physics with potential for revolutionary precision measurements.
Positive muon beams of a few MeV energy are typically obtained from the decay of positive pions stopped at or near the surface of a pion production target. These are referred to as surface muons; negative pions are captured by the target nuclei promptly and do not contribute. At their point of production, surface muons have a monochromatic energy (momentum) of 4.1 MeV (29.8 MeV/c) with 100% spin-polarization anti-parallel to the momentum direction due to the parity-violating π + → µ + ν µ decay. The conversion of MeV muons to a small emittance source of keV muons requires a cooling process. Because of the short muon lifetime (2.2 µs), muon cooling is challenging and various innovative methods are under development [6][7][8]. One such method is to thermalize the surface muons in the form of muonium atoms in an appropriate material, then allow them to diffuse into vacuum where they can be ionized with a laser resonant ionization technique [9,10]. The resulting positive muons are effectively cooled to an energy distribution characterized by the material temperature. Typically 50% of the muon beam's polarization is lost due to the hyperfine interaction with initially unpolarized electrons of the material during muonium formation.
The material can ideally be chosen for high muonium formation probability, low additional depolarization, and high efficiency of muonium diffusion.
This muon cooling method has already been demonstrated at a beamline [11] of the Materials and Life Science Experimental Facility (MLF) at the Japan Proton Accelerator Research Complex (J-PARC). The material used to thermalize muons is tungsten heated to 2000 K. Muonium is emitted from the surface at an energy of about 0.2 eV. It is proposed to extend the method further in the J-PARC muon g-2/EDM experiment [12], which is now under construction. The goal is to measure the muon anomalous magnetic moment and electric dipole moment with a totally new approach based on a small-emittance muon beam.
Here, silica aerogel has been chosen for the material of the muonium production target since it can produce thermalized polarized muonium atoms, has a porous structure that allows diffusion, and is compatible with the accelerator's vacuum requirements. The surface muon beam is to be thermalized at room temperature to the energy (momentum) of about 25 meV (2.3 keV/c). The resulting thermal muonium atoms in vacuum are ionized and the muons resulting from ionization are re-accelerated up to 212 MeV (300 MeV/c), achieving a transverse momentum spread of 0.04% (RMS) that corresponds to an unprecedentedly small emittance for a muon beam. Past studies demonstrated muonium production in silica aerogel and its emission into vacuum [13] and observed a significant improvement of the emission rate with laser ablation on the aerogel surface [14]. The present issues are to understand better the underlying processes leading to muonium emission with laser ablation, to find the optimal ablation structure, and to confirm the polarization of muonium following emission into vacuum.
To address these issues, various ablation structures were examined with the surface muon beams at TRIUMF and J-PARC in 2017. The TRIUMF muon beam was continuous with a momentum spread of about 2% (RMS) while the J-PARC beam was pulsed with a spread of about 4% (RMS). Most of the ablation structures were examined at TRIUMF utilizing the smaller beam momentum spread, in the presence of a magnetic field where the effects of polarization would be apparent. Some of them were also measured at J-PARC to demonstrate the projection of the measurements at TRIUMF into the expectations at J-PARC for the application to the muon g-2/EDM experiment. This report describes the results of these measurements and projections on the muonium emission from laser-ablated silica aerogel targets.

Target properties
The muonium production targets are prepared in the form of slabs 50 mm × 50 mm in area and typically about 9 mm in thickness. Surface muons collimated to a beam diameter of 10 mm irradiate the large face of the target, which is suspended in a vacuum chamber. The laser ablation is applied on the central 30 mm × 30 mm surface of the downstream large face, from which the muon beam exits the target.
Most of the targets are made of silica aerogel [15] with a density of 23.2 mg/cm 3 − 23.6 mg/cm 3 and a thickness of 8.7 mm − 8.9 mm. The silica aerogel was rendered hydrophobic by surface modification with trimethylsilyl groups (Si(CH 3 ) 3 ) during the production process.
A different type of material, referred to as poly(methylsilsesquioxane) (PMSQ) gel [16], is also used to explore possible variations of the target material. It is a novel organic-inorganic hybrid aerogel material, in which the silicon dioxide groups (SiO 2 ) are partially replaced with the methyl groups (CH 3 ). Its mechanical properties might be advantageous for the long-term mechanical stability compared to the fragile silica aerogel. The PMSQ gels were produced in two different drying processes, the super-critical drying process producing lower-density samples (52.5 mg/cm 3 ) and the evaporation process under the ambient condition producing higher-density samples (99.6 mg/cm 3 ). These samples are produced with thicknesses of 4 mm and 2 mm, respectively, to have a similar target mass with the silica aerogel targets. A target with no ablation, referred to as a"flat" target, is also prepared for each of the three materials. A silica plate with the thickness of 0.1 mm is used for calibration. Its high density suppresses muonium diffusion, so the muons decay in the silica plate without emission. Therefore, it can be used to evaluate the contribution of the muon decays mistracked into the vacuum region.
In addition, it is thin enough to determine the coordinate origin of the muonium emission and to evaluate the resolution of the track extrapolation used to infer the muonium decay position, as discussed later. Table 1 Summary of target thickness (T ), density (ρ), and the ablation structure parameters, hole diameter (φ), groove width (w), pitch (p) and depth (d), used in the measurements at TRI-UMF and J-PARC. The uncertainty of each parameter is uniform over the targets and described in the text. The corresponding information for the targets used in the previous study [14] is also listed for comparison.

Ablation
Target Laser ablation was introduced to produce a structure on the emission face that enhances muonium diffusion into the vacuum. This is based on the interpretation for the time evolution of the muonium population in vacuum obtained in the past study [13]. The time evolution is reasonably described by a Monte Carlo simulation assuming muonium diffusion inside the porous silica aerogel with a Maxwell-Boltzmann three-dimensional velocity distribution. The simulation indicates that the muonium emission is limited by its diffusion length of about 30 µm; which is much smaller than the range straggling of stopping muons in a target, about 2 mm (FWHM). This interpretation suggests a possible increase of the emission rate by having a structure with dimensions of order 0.1 mm on the emission face. Here, the muon stopping distribution is assumed to be peaked at or near the emission face by tuning the beam momentum of the incident surface muons.
The muonium emission rate of an aerogel target with holes on the emission face with the diameter of 0.27 mm, pitch of 0.3 mm and depth ranging in 4.5 mm − 5.0 mm was about 8 times higher than that of the flat target for the silica aerogel with the density of 29 mg/cm 3 [14]. The same ablation structure with larger pitches also obtained higher emission rates than the flat target did, but the increase tends to decrease for larger pitches. A trend indicating more muonium emission with a denser structure was less clear. Naive considerations can lead to the hypothesis that the muonium emission is dominated by the ablated surface area. In this case, the hole diameter and depth will also influence the emission rate and need to be surveyed, too. The structure geometry can also influence the emission rate. A hole structure largely contains the ablated surface inside the holes while grooves have an open structure in one direction. Extending the open structure in two directions leads to the continuously ablated area.
The properties of the targets studied are summarized in Tables 1. All the targets were used in the TRIUMF measurements, except the target S03 which was used in the J-PARC measurement only. The other targets of S01−S08 were used in both measurements. The target thickness is measured at an edge using a linear scale with the resolution of 0.25 mm while the weight is measured using an electronic scale with the resolution of 2 mg. Thus the target densities listed have uncertainties of ±0.7 mg/cm 3 . The ablation structure parameters are the hole diameter (φ) or groove width (w), pitch (p) and depth (d) as illustrated in Figs. 1-3. The holes are arranged two-dimensionally in an equilateral-triangular configuration at the vertices with the side length corresponding to a given pitch. The grooves run over the area in one direction and are arranged in parallel with a given pitch. The uncertainties of the structure parameters are considered to be ±20% for the diameter and width, ±5 µm for the pitch and ±0.25 mm for the depth as described in Sec. 2.2.

Laser ablation technique
The aerogel targets were ablated with ultra-short laser pulses from an amplified Ti:Sapphire femtosecond laser [17] with the specification listed in Table 2. Using a 300 mm focal length lens, the laser beam was focused on the surface of the target to a typical spot size of about 40 µm, with sufficient fluence to exceed the ablation threshold of about 2 mJ/cm 2 . Computer-controlled translation stages moved the target under the focused laser beam to produce a pre-programmed ablation structure, while an electronic shutter controlled the laser exposure time, or in effect, the number of laser pulses striking the surface. The pulse energy was adjusted using neutral density filters, and the laser spot size was increased if needed by reducing the diameter of the incident laser beam at the lens through a partially closed aperture. These controls were used to produce a series of structures: arrays of holes with various diameter, pitch, and depth; one-dimensional grooves; and an area of two-dimensional continuous ablation.

Table 2
Laser specification used for the ablation. The scan speed was set only for groove and continuous ablation. These images were acquired by the use of a laser-scanning confocal microscope [18], in which a 543 nm He-Ne laser scans across the surface of the target and the transmitted beam is recorded with a photo-multiplier tube to form the image. The image stack in the depth direction was utilized to obtain three-dimensional views of the hole structure.
The pitch of the holes is controlled within ±5 µm using computer-controlled translation stages. The hole diameter was controlled by tuning the laser pulse energy and exposure time, however, typically about ±20% variation in the hole diameter was observed. The depth of the holes was primarily controlled by the number of laser pulses irradiated onto the surface. This was achieved by an electronic shutter. For shallow holes of 1 mm depth, as few as 10 laser pulses were needed per hole. The depth of the holes varied by about ±250 µm under the same ablation conditions. Given the focal point is fixed at the target surface, the laser impact is less at a deeper position in the target, having more influence by the local sparseness and density of the target material and less control on the depth. It also results in a pointing shape at the bottom of the hole as shown in the cross-sectional view of Fig. 2. Therefore, the hole shape is considered to be conical. The hole structure with an enhanced conical shape was made with a laser spot size of 6 µm with the depth of 1.5 mm.
Adjacent structures sometimes run into each other in dense structures, but the influence of the structure collapse on the muonium emission is unknown. In a more macroscopic scale, the ablation on one side of the target induces the contraction of the ablated face, resulting in a warp. The warp is somewhat mitigated by having enough non-ablated margin around the ablated area in this study.

Fig. 2
Cross-sectional views of an ablated hole structure laid out with the front view (left), and their three-dimensional contour image (right). The images are obtained using a laser-scanning confocal microscope as described in the text.
Other structures can be processed by extending the hole ablation process. A channel was made by continuous raster of the laser beam in one direction across the target. The speed of raster, typically around 1 mm/s − 2 mm/s, determined the laser exposure and the resultant depth and width. For both wider channels and to achieve better definition of the walls separating each line, as many as 10 partially overlapping rasters were used. For example, the channels shown in Fig. 3 were produced by 10 line scans with 50 µm steps for each channel and 200 µm steps between channels. This resulted in about 500 µm-wide channels separated with about 120 µm-wide clean walls. A V-groove was made in a similar fashion as the channel structures, however with using a higher numerical aperture lens to produce smaller spot size of 6 µm. Such a spot size was also utilized for the enhanced conical holes.
A target with a continuously ablated area was produced by the line raster of the laser beam over the entire ablation region, while each line was overlapped with adjacent lines at 50% of the spot size. The pulse energy was set at very low level of about 30 µJ, and scan speed of 2 mm/s. This resulted in a rough surface covered with microstructures with a depth of about 400 µm. Figure 4 shows the surface morphology of the continuously ablated target.
The image on the right is acquired by Helium Ion microscope which shows pores and voids on the scale of 10 µm − 100 µm. The presence of sub-micron features is also evident.

Fig. 3
Microscopic pictures of the aerogel target with channel ablation (S22), obtained using a laser-scanning confocal microscope as described in the text. The 120 µm-wide walls between the channels are in focus.
The previous study reported that the difference in the target weight measured before and after the laser ablation was only 5% − 10% of the expected difference, which was based on the ablation geometry and target density [14]. The same comparisons were carried out with a silica aerogel target used in this study. The obtained results are summarized in Table 3.
The weight loss per ablated hole is found to be about 0.14 µg. This value is consistent with the expected weight loss based on the assumption of a conical shape for the holes.

Beam condition and experimental setup
In order to study the muonium emission from various structures on the target face, measurements were carried out at the M15 beamline of TRIUMF in 2017. The beamline delivers a continuous beam of surface muons. The typical muon rate at the beam counter,  A microscopic picture of the aerogel target with continuous ablation (S14) (left) and details of the surface morphology (right). The left image is obtained using a laserscanning confocal microscope as described in the text, showing the surface roughness over a large area of the target. The right image is acquired with Helium Ion microscope, showing the sub-micron features evidently. Table 3 Weight loss due to laser ablation for an aerogel target with ablated holes (S18).
The weights before and after ablation are measured with an electronic scale. The expected weight loss is evaluated assuming a conical shape with the intended diameter, pitch and depth, and has a range to account for the large variation of ±250 µm on the depth.

Item Value
Weight before ablation 525.45 mg Weight after ablation 523.09 mg Weight loss due to ablation 2.36 mg Number of ablated holes 16,680 Weight loss per ablated hole 0.14 µg Expected weight loss per ablated hole 0.12 µg − 0.21 µg setup is shown in Fig. 5. It was essentially the same as in our past studies [13,14] except for a few changes on the apparatus mentioned below.
The muonium production target was placed in the vacuum chamber operated at a vacuum level of around 10 −4 Pa − 10 −3 Pa. This target chamber was connected to the beam port at the upstream wall and to the vacuum pumps at the downstream wall, and had an aluminized Mylar window, with the diameter of 100 mm and thickness of 100 µm, at a side wall next to the target. The positrons decayed from muons, including those in muonium atoms, in the direction toward the Mylar window can enter the detection system with tracking capability outside of the target chamber. The reconstructed positron tracks were extrapolated back to the vertical plane containing the beam axis to infer the muon decay positions. The obtained time and spacial distributions of the muon decay positions were used for the analysis. Here, the detector coordinate system (x, y, z) is defined as shown in Fig. 5. The z-axis is along the beam center, x towards the positron detection system, and y vertically upward. The z-coordinate origin was the design position of the target downstream face. that has a 10 mm-diameter hole aligned to the last collimator hole was placed behind the housing end, and was used to veto events with unexpected beam trajectory. Another plastic scintillator was placed at z = −12 mm and was used to start a data taking cycle when fired. The thickness of this beam counter was increased from 300 µm to 380 µm in this study to improve the rejection power to the positrons in the beam based on intensity of the scintillation light.
A muonium production target with a typical thickness of about 9 mm, or the calibration target of 0.1 mm-thick silica plate, was set on a target holder, which aligned the target downstream face to be z = 0. The target holder was redesigned for this study to accommodate the larger target size than that in the past studies to mitigate the ablation warp as explained before. The target holder was made of poly(ether ether ketone) (PEEK) and held a target between fixed and removable covers on a base with gentle contact. The covered area is limited to the 6 mm peripheral edges of large faces and the side faces, using only the lower half of the target. The detector side was further cut down to remove materials that may deteriorate the tracking resolution due to the multiple scattering of positrons.
The positron detection system consisted of four layers of multi-wire drift chamber (MWDC) [19], a pair of plastic scintillators and a NaI calorimeter. The MWDC layers were used for positron tracking. A single layer consisted of a pair of wire planes tilted alternatively by ±45 degrees with respect to the x − y plane. Tracks reconstructed with good quality were required to pass through the aluminized Mylar window, centered at z = 15 mm, within a radius of 45 mm. The track slope in the x − z plane (dz/dx) was limited to be |dz/dx| < 0.1 to reduce the parallax between the true and inferred decay positions. A pair of scintillators was used to issue the trigger to acquire the event and to determine the muon decay times.
The requirement on the muon decay time to be later than 6 ns removed the prompt muon decays entirely. The NaI calorimeter was used to reduce the influence of multiple scattering effect by selecting only high-energy positrons (> 30 MeV). This selection also contributed to higher sensitivity to spin polarization information.
A major change was made on the magnets used for the spin precession study. In the past studies, a pair of ferrite magnets was used inside the target chamber. Since these magents were not specifically designed for the spin precession study, they were removed and a pair of coils was installed above and below the target chamber, providing a magnetic field up to about 1.8 mT transverse to the beam direction. Each coil, with the cross section of 35 mm × 25 mm, was wound on a frame that forms a rectangular shape with the inner dimensions of 462 mm and 162 mm in the longer and shorter directions, respectively. The two coils were positioned symmetrically to the detector coordinate origin with the gap of 440 mm. The field homogeneity is expected to be better than 0.3% over the longitudinal range used in the analysis (−20 mm < z < 40 mm). The vertical positions of muon decay points were required to be within ±20 mm in the past studies to kill positron background from the magnets inside the target chamber. This criterion was kept for consistency even without the magnets.

Beam-momentum tuning
The beam momentum was tuned to maximize the muonium emission from the target into vacuum. Since the stopping distribution is much broader than the muonium diffusion length, the maximum emission is effectively achieved by having the muonium population density, namely the Gaussian-like muon stopping distribution, peaked at or nearby the target emission face within the diffusion length. Half of the stopping muons exit the target at this condition. This is determined by measuring the fraction of decay positrons originating in the target over a certain range of beam momentum.
The corresponding fractions measured for the flat and an ablated aerogel targets, as well as their simulated ones, are shown in Fig. 6 as a function of the beam momentum. The simulation curves were obtained from the muon stopping in the corresponding targets using the Geant4 tool kit [20] with a Gaussian profile for the beam momentum. The best fit to the measured variation on the flat aerogel target was obtained with the momentum spread of 1.9% (RMS). The positron yield reaches the maximum at the beam momentum of around 22.5 MeV/c. It decreases toward the lower beam momentum since more incident muons stop in the beam counter placed in front of the target, and similarly toward the higher momentum since more muons penetrate the target. The beam momentum that makes the positron yield be half maximum in the higher-momentum side is the condition to maximize the muonium emission and is referred as "half-stop" momentum in this study. The reasonable agreement between the measurements and simulations supports the above understanding. The beam momentum is tuned for each target to account for the different structures on the target face before proceeding further studies.

Muonium emission rates
Incident surface muons slow down in the muonium production target, losing their kinetic energies by specific ionization energy loss and iteration of muonium formation/dissociation in a time scale of nano seconds or less [21]. Given the half-stop momentum condition, nearly half of them penetrate the target during this process and decay downstream of the target or hit the vacuum chamber wall. Thermalization of the rest of stopping muons to the temperature of the material involves elastic and inelastic collisions of both muons and muonium atoms with the lattice and atoms of the medium; some thermalized muons also end up in muonium atoms after capturing radiolysis electrons [22]. The empirical fraction of muons forming muonium atoms in the target is to be discussed in Sec  ablated target compared to the flat one as seen in the previous study [14]. The corresponding distribution of target decays only (z tgt dec ) was simulated for the flat aerogel target, discussed later, and is shown in the same figure for the case with no muonium emission. The simulated distribution in the positive z region rapidly drops to the level of an order of magnitude smaller than that for the measured data at z = 10 mm, while the measured data distribution extends further away from the target emission face because of the emitted thermal muonium atoms. Therefore, the muonium emission rate can be measured in the region of z > 10 mm with significantly suppressed background contribution and is defined as where the N vac e + and N all e + denote the number of positron tracks extrapolated to the vacuum region (10 mm < z < 40 mm) and all region (−20 mm < z < 40 mm), respectively, in the y − z plane.
The z tgt dec distribution for the flat aerogel target was produced taking the simulated decay points of stopping muons in the target, described in Sec. 3.2, and no decays in the beam counter were involved. The simulated z-position distribution was smeared by convoluting with the resolution shape taken from the corresponding z tgt dec distribution in the silica plate data, shown in Fig. 7 (right), which involves no muonium atoms decaying outside the silica plate because of the high density and is dominated by the extrapolation resolution because of the small thickness; the extrapolation resolution was evaluated to be about 2.5 mm (FWHM) from this distribution. The z tgt dec distribution in the silica plate data was modeled with two Gaussian functions for the target decays, sharing the mean, and single Gaussian functions for the beam counter decays and outlier events. All Gaussian parameters, other than the one shared, were allowed to vary in the fit. The obtained shape without the component for the beam counter decays was taken as the resolution function. The peak position and height of smeared simulation were adjusted to fit the z dec distribution in the flat aerogel data over the target decay dominant range (−8 mm < z < 8 mm). The smeared simulation of z tgt dec distribution was used to evaluate the background contributions of target decays and outlier events in vacuum. It was also used to estimate imperfect positioning of targets discussed later.

Table 4
Summary of the muonium emission rates in vacuum (R vac Mu ) and their statistical uncertainties measured over the ablation structures listed in Table 1 at the TRIUMF M15 beamline. The groove-ablated targets are labeled with "H" and "V", indicating the horizontal and vertical orientations, respectively, of groove configuration in the measurements. The emission rates measured in the previous study [14] are also listed for comparison. The measured muonium emission rates are summarized in Table 4. The background contributions from the target decays and outlier events evaluated using the smeared simulation for each target were subtracted from the observed entries in the vacuum region. Each of the smeared z tgt dec distributions for ablated targets is based on the muon stopping distribution for the flat target of corresponding target material, weighting the distribution near the surface according to the material loss expected from its ablation structure. The measured rates are in the range 5‰−28‰ and below 5‰ for the ablated and flat targets, respectively. Enhancement of muonium emission for all the ablated targets to the flat targets of corresponding materials is confirmed, including the PMSQ targets which have higher densities than those of the silica aerogel targets. Some of the targets show the emission rates close to the highest one measured in the previous study [14], which can be applicable to an early phase of the J-PARC muon g-2/EDM experiment [12]. instead of the background subtraction mentioned above. The resulting relative changes from the nominal results ranged in −6% to +1%. Adding the uncertainties on the imperfect target positioning and background modeling in quadrature, the relative systematic uncertainties on the emission rates were estimated to be −7% to +12%.
The dependence of muonium emission rate on the ablation structure parameters was studied to understand the emission mechanism. Naively the muonium emission rate is expected to depend on the surface area of ablation, i.e. the conical area for a hole, since the laser ablation gives a macroscopic and microscopic structures to enhance muonium atoms diffuse out of the target. No obvious correlation was, however, seen between the emission rates and such area.
Surveying the dependence on the other combinations of the ablation structure parameters, it is found that the opening fraction of the ablation region (F open ) shows such dependence for the silica aerogel targets with hole ablation as shown in Fig. 8. The dependence also applies to the ones used in Ref. [14]. The PMSQ-gel targets and the aerogel targets with groove ablation show deviations from the eye-guide with slightly lower emission rates. The Most of the ablated targets have the ablation depth of 2 mm or less, and no clear dependence of the muonium emission rate is seen on the ablation depth. The contribution of the ablation depth was specifically inspected by comparing the emission rates for the holeablated targets with different depths, keeping the other parameters the same. There are two sets of hole-ablated targets which meet this condition. The results indicate a slightly negative correlation as shown in Fig. 9 (left). The tendencies suggest an optimal depth to be less than 2 mm to maximize the muonium emission. A further study is necessary to understand the underlying mechanism and to find the optimal depth. Clearer rate dependence on the opening fraction can be seen in Fig. 9 (right) by selecting the data only from the hole-ablated aerogel targets with the ablation depth of 2 mm or less. A long-term use with constant performance is desired for the muonium production target, but a long-term exposure to the high-vacuum environment or muon beam irradiation may degrade the muonium emission rate due to the possible material modification or chargeup effect, respectively. The relative variation of the muonium emission rate was evaluated without the background subtraction over more than 50 hours, though including 10.5 hours of beam-off due to the accelerator maintenance, to address this issue and shown in Fig. 10.
The emission rate was found to be constant during this period. No physical damage from this extended exposure to the high vacuum environment was observed.

Muonium formation fraction
All µ + SR measurements refer to and rely upon the angular asymmetry of muon decay.
While the theoretical asymmetry averaged over all positron energies is 1/3, the asymmetry at the highest positron energy is 1 (assuming perfectly polarized muons) and the average over detector solid angle is somewhat smaller. When the positron detector rejects low energy The relative variation of the muonium emission rate in vacuum (R vac rel ) as a function of time (t). The rates are measured with the target S19 and normalized to its total emission rate listed in Table 4. The vertical error bars show the statistical uncertainties only.
The horizontal error bars indicate the corresponding data taking duration. The dashed line at R vac rel = 1 is given for an eye guide. The measurements were continued for 53.3 hours except for the 10.5 hours of accelerator maintenance at around t = 40. positrons and subtends a substantial solid angle, the experimental asymmetry is hard to predict. In any given experiment, therefore, the maximum asymmetry possible, A 0 , is determined empirically from a spectrum obtained using a sample that preserves the initial muon polarization; a pure silver or aluminum sample is generally a good choice. Then all the experimental idiosyncracies are assumed to be identical in the same apparatus under the same conditions for any other sample with the same density and geometry. We determined A 0 = 0.474 ± 0.004 in an aluminum plate with a magnetic field of ≈ 1.92 mT, in which the applied and ambient fields were combined.
When polarized positive muons form muonium from unpolarized electrons, half the initial states are in the | ⇑↑ eigenstate and half are in the | ⇑↓ superposition state (where ⇑ represents the muon spin direction and that of the electron), in which the muon polarization oscillates between +1 and −1 at the hyperfine frequency (in vacuum, ν 0 = 4463.302 MHz [23]), which is unresolved by most experiments. Thus the muonium polarization appears to be halved. Assuming that no "prompt" depolarization takes place during the stopping of the surface muons, the fraction f Mu of muons forming polarized muonium in the target can therefore be determined by comparing twice the measured muonium asymmetry A Mu with the maximum available asymmetry A 0 : Measuring the muonium asymmetry in aerogel targets using the same magnetic field (≈ 1.92 mT) requires fitting two muonium frequencies, ν 12 and ν 23 , corresponding to transitions between levels 1 & 2 or 2 & 3 of the muonium Breit-Rabi energy level diagram. In such a modest field, these two signals are of essentially the same amplitude but are split by ±∆ν HF = ν 2 Mu /ν 0 where ν Mu is their average. This splitting is manifest in the spectrum shown in Fig. 11 as a "beat" pattern with the first node at about 1.56 µs.
In the data comprising the spectrum in Fig. 11, there are about 12.47 3-ns bins per period of muonium precession, which makes display tedious but should provide a reliable determination of both muonium and diamagnetic asymmetries by a fit to the usual form of a transverse field µ + SR time histogram: where BG (a time-independent background) is undetectable, N 0 = (919.3 ± 1.2) counts per 3-ns bin is the overall normalization, τ µ = 2.19703 µs is the (fixed) muon lifetime, A Mu = 0.155 ± 0.004 is the initial muonium asymmetry, λ Mu = (0.28 ± 0.02) µs −1 is the muonium relaxation rate, ν 12 = ν Mu + ∆ν HF and ν 23 = ν Mu − ∆ν HF are the two frequencies of muonium precession split by twice ∆ν HF = (0.1682 ± 0.0027) MHz, whose average All the aerogel targets (even those composed of PMSQ instead of silica) yielded the same muonium amplitudes, within statistical uncertainties; therefore all such data were combined to make a much higher statistics µSR time spectrum from which the important results were extracted. For the sum of all data on aerogel targets in the same field, the fitted value of A Mu = 0.155 ± 0.004, combined with the separate determination of A 0 = 0.474 ± 0.004 from an aluminum plate, we can conclude that  The obtained muonium fraction is larger than the one obtained in the past study (0.52 ± 0.01) [13]. These studies used silica aerogel targets produced in slightly different synthesis conditions. The measurement methods and apparatus were also different. Although the discrepancy of the two muonium fractions is not understood well, it does not affect the muonium emission rates because they were measured independently.

Muonium polarization in vacuum
In contrast to the simple exponential for muon decay in all region, muon decay in a vacuum region is proportional to the probability that the muonium atom is in that region.
We assumed that the muonium atom is emitted with a velocity following the Maxwell-Boltzmann distribution and derived the corresponding time distribution assuming a travel distance . The Maxwell-Boltzmann velocity distribution in the z direction is where For a fixed distance , that gives a TOF distribution and since v z = /t, dv z /dt = − /t 2 , giving where C ≡ b 2 . Regions 1, 2 and 3 cover z ranges of 10 mm − 20 mm, 20 mm − 30 mm and 30 mm − 40 mm from the target, respectively, so that the actual D(t) in each region should in turn be weighted by the corresponding distribution D( ): Especially in region 1, there is also a distribution of emission angles to be considered. We were unable to convert this combination of distributions into an algebraic expression for fitting, so we did the next best thing: we simply substituted an average 2 for 2 , giving in Eq. (8).   Table 5.

Table 5
The fitted velocity spectrum parameter C and the rms travel distance rms = C/b obtained for each vacuum region of the listed z range from the fits described in the text. We see clear muonium precession signal in the spectra. In principle, we can use the precession amplitude to extract the muonium polarization in vacuum. However, we found it was difficult to extract a reliable value due to the following reasons. First, in order to show spectra with best statistics, the spectra were for the sum of measurements under ambient field of about 0.14 mT and the field axis had large tilt to the y-axis. This causes a tilt of the spin precession plane and thus the detector sensitivity to the polarization could be much reduced compared to that in Fig. 11 under the applied field of 1.92 mT. Second, a field variation due to the disturbance of experimental environment found over the measurements (in space and in time) causes significant relaxation of the precession and it is difficult to extrapolate the result back to the amplitude at time zero. Third, the shape of the TOF spectrum calculated from the Maxwell-Boltzmann distribution is only an approximation, since muonium atoms emerge at different angles; this interferes with the estimation of the initial muonium polarization. So, at the present stage of the analysis, we cannot conclude any reliable value for the muonium polarization. However, we should at least mention that there is no strong indication that the measurement is contradicting our expectation of muonium emission with 50% polarization.

Beam condition and experimental setup
The muonium emission rates in vacuum were also measured at J-PARC for five aerogel targets that include a flat and three ablated ones used in the TRIUMF measurements. The emission rates measured at TRIUMF for the four targets can be projected to the corresponding rates at J-PARC, taking into account the different experimental conditions. A reasonable projection, if available, can demonstrate the procedure to apply the knowledge on the muonium emission obtained under the experimental condition at TRIUMF to the simulation for the J-PARC muon g-2/EDM experiment.
The measurements were carried out with a pulsed surface muon beam at the D2 area of Muon Science Establishment (MUSE) [24] in The experimental setup was similar to that in the TRIUMF measurement shown in two layers of scintillation-fiber hodoscope (S1, S2), followed by a polyethylene absorber, and a 4×4 matrix of 20 mm-thick scintillators (S3). A positron track was determined from the hit positions of S1, S2 and S3, and was extrapolated to the y − z plane to infer the decay position in the same manner as in the TRIUMF measurements.

Muonium emission rates
The muonium emission rates were evaluated in a similar manner as in the TRIUMF measurements. The resolution of the track extrapolation was evaluated to be about 9.0 mm (FWHM) from the silica plate data, giving a wider distribution of the target decay background over the z-range compared to that in the TRIUMF measurements. To ensure the signal dominance in the emission rate evaluation, a smaller vacuum region of 20 mm < z < 40 mm was adopted in the J-PARC measurements, while the fiducial "all" region was kept to be the same (−20 mm < z < 40 mm). The contributions of the target decays and outlier events in vacuum were modeled using the silica plate data. The fraction of those backgrounds in the vacuum region was considered to be the same for the silica plate and aerogel targets at the half-stop condition. Therefore, the apparent emission rate of the silica plate, (4.80 ± 0.31)‰, was subtracted from those of the other targets.
The obtained muonium emission rates after the background subtraction are listed in Table 6 together with the statistical uncertainties. The dominant systematic uncertainty comes from the target position variation caused by a mechanical clearance of 0.6 mm in the target holder mounting used in the J-PARC measurements. Remounting of the target holder at the target replacement can give a possible variation of the target position within the clearance. The apparent emission rates were re-evaluated varying the vacuum region by ±0.3 mm, and the resulting relative changes from the nominal rates, −10% to +5%, were assigned as the systematic uncertainty.
The four aerogel targets with laser ablation have identical hole structure parameters except for the depth. Their emission rates are compared in Fig. 13 as a function of the hole depth. A trend of the rate reduction as the holes become deeper can be seen as in the TRIUMF measurements. Here, an error bar represents the quadratic sum of the statistical and systematic uncertainties excluding the statistical uncertainty of the silica plate data which was given in the background subtraction and common for all the rates.

Projection of the TRIUMF measurements
The corresponding TRIUMF measurements for the flat and three ablated targets are also projected in Fig. 13, taking into account the major differences between the two sets of the measurements. One of the major differences exists in the beam momentum spread.
The muonium diffusion model [13] indicates that the emission rate is proportional to the muon stopping density and, hence, is inversely proportional to the range spread in the muon stopping, which is largely correlated with the beam momentum spread. To account for the Table 6 Summary of the muonium emission rates in vacuum (R vac Mu ) and their statistical uncertainties measured at J-PARC D-line as described in the text.  are also not included. different beam momentum spreads, therefore, the rates in the TRIUMF measurements need to be scaled in the projection by their ratio. The range spread influences the slope of the positron yield variation of the target decays over the beam momentum, shown in Fig. 6. Comparing the variation slopes observed in the TRIUMF and J-PARC measurements, the momentum spread ratio was evaluated to be 0.6. The other major difference exists in the resolution of the track extrapolation. The difference leads to the smaller vacuum region in the J-PARC measurements as described earlier.
The ratio of the positron yields in the two regional definitions can be taken as the scale factor to account for the vacuum region difference. The scale factor was evaluated to be 0.36 using the TRIUMF data. The resolution difference also results in the event migration over a given region. This effect was calculated based on the observed track extrapolation resolutions using the TRIUMF data, giving the scale factor was of 1.1.
The projected rates of the TRIUMF measurements to the J-PARC condition were obtained applying the three scale factors at the combined accuracy of ±10% level. These rates agree with the J-PARC measurements within the uncertainty of about ±14%, where the projection accuracy of about ±10% and the about ±10% uncertainty of the J-PARC measurements are added in quadrature. The projected and measured rates also show a similar dependence on the holed depth. The agreement indicates that the projection works reasonably.

Expected thermal muonium rate in the J-PARC muon g-2/EDM experiment
The projection demonstrates that the beam property dominates the initial condition of the muonium diffusion and emission processes, while the processes themselves can be the same between the experimental conditions at TRIUMF and J-PARC; the other scalings account for the difference on how to measure the emission rates. This demonstration supports the application of the model of the two processes developed in the study of past measurements at TRIUMF [13] in the simulation to estimate the expected rate of thermal muonium atoms available for laser ionization in the J-PARC muon g-2/EDM experiment.
The simulation includes the surface muon transport through the beamline, the muon stopping and muonium diffusion in the aerogel target, and the muonium emission from the target followed by the travel in vacuum. The simulation incorporates the design of MUSE H-line [25] under construction, at which the J-PARC muon g-2/EDM experiment will be launched. The momentum spread of the surface muons is expected to be 5% (RMS) there.
The expected surface muon rate at H-line was estimated from the the measured rate at D-line, which shares the muon production target at the symmetric configuration with regard to the primary proton beam. The beam transport efficiencies at H-line and D-line were estimated using the G4beamline program [26], and their ratio was used in the rate estimation. The muon stopping was simulated with the Geant4 toolkit. All the stopped muons were assumed to form muonium atoms in the simulation and the formation fraction was applied in the rate estimation. The muonium diffusion inside the target was modeled with the Maxwell-Boltzmann velocity distribution at room temperature. The angular distribution of muonium emission was modeled as proportional to cos θ, where θ is the polar angle of the muonium emission direction with respect to the emission face normal. The laser ionization region was defined to be ±100 mm and ±25 mm regarding the beam axis in the horizontal and vertical directions, respectively, and 5 mm along the beam axis from the target emission face. The simulated time-evolution of the muonium travel shows the highest population density in the laser ionization region at the muonium arrival time of around 1 µs from the averaged arrival time of the surface muons over the two bunch structure. The expected rate of the muonium atoms for laser ionization is estimated to be 3.4 × 10 −3 per incident surface muon [12].

Summary
The muonium emission from the muonium production targets with various laser-ablated structures on their surfaces was studied at room temperature as the development of the thermal muon source for the muon g-2/EDM experiment at J-PARC. The study was intended to understand better the underlying processes leading to muonium emission with laser ablation, to find the optimal ablation structure, and to confirm the polarization of muonium following emission into vacuum. The muonium emission rates in vacuum and precession amplitudes were measured irradiating the surface muon beams at TRIUMF and J-PARC on silica aerogel targets with a typical density of 23.6 mg/cm 3 and PMSQ-gel targets with higher densities. Laser ablation was applied to produce holes or grooves with a diameter or width, respectively, of 0.07 mm − 0.5 mm varying the fractions of ablation opening and a depth of 0.3 mm − 5 mm, except for some extreme conditions. The measured emission rate tends to be higher for larger fractions of ablation opening, rather than the larger absolute surface area of ablation, and for shallower depths with a possible optimum at less than 2 mm. More than a few ablation structures reached the emission rates similar to the highest achieved in the past measurements, which would be applicable to an early phase of the J-PARC muon g-2/EDM experiment. The emission rate was found to be stable at least for a couple of days with no physical damage for the long-time exposure to the high-vacuum environment. The measurements of spin precession amplitudes for the produced muonium atoms and remaining muons in a magnetic field determined a muonium formation fraction of (65.5 ± 1.8)%. The measurements also indicated that the polarization of muonium atoms was preserved in vacuum. A projection of the emission rates measured at TRIUMF to the corresponding rates at J-PARC was demonstrated taking the different beam condition into account. The projected rates agree with the ones measured at J-PARC within the uncertainty of about ±14%. This projection procedure was applied to the estimation of the expected rate of thermal muonium atoms available for laser ionization in the muon g-2/EDM experiment at J-PARC.