# Room-temperature emission of muonium from aerogel and zeolite targets

A. Antognini1,2 P. Crivelli1 L. Gerchow1 T. D. Hume1 K. Kirch1,2 A.
Knecht2$\dagger$<EMAIL_ADDRESS>J. Nuber1,2$\ast$<EMAIL_ADDRESS>A.
Papa2,3 N. Ritjoho1,2 M. Sakurai1 A<EMAIL_ADDRESS>D.
Taqqu1 S. M. Vogiatzi1,2 J. Zhang1 L. Ziegler2 1 Institute for Particle
Physics and Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  2 Paul
Scherrer Institute, 5232 Villigen–PSI, Switzerland  3 University of Pisa and
INFN, L. Bruno Pontecorvo, Edificio C, 56127 Pisa PI, Italy

###### Abstract

A low-emittance, high-intensity atomic beam of muonium
($\mathrm{M}=\upmu^{+}+\mathrm{e}^{-}$) using superfluid helium as muon-to-
muonium converter is being developed at the Paul Scherrer Institute (PSI).
This beam could advance laser spectroscopy of muonium and allow the first
atomic interferometry experiments for the direct observation of the M
gravitational interaction. In this paper, we describe the development of
compact detection schemes which resulted in the background-suppressed
observation of atomic muonium in vacuum, and can be adapted for cryogenic
measurements. Using these setups, we compared the emission characteristics of
various muonium production targets using low momentum
($p_{\mu}=11$-$13~{}$MeV/c) muons, and observed muonium emission from zeolite
targets into vacuum for the first time. For a specific laser-ablated aerogel
target, we determined a muon-to-vacuum-muonium conversion efficiency of
$7.23\pm 0.05\text{(stat)}^{+1.06}_{-0.76}\text{(sys)}$\text{\,}\mathrm{\char
37\relax}$$, assuming thermal emission of muonium. Moreover, we investigated
muonium-helium collisions and from it we determined an upper temperature limit
of 0.3 K for the superfluid helium converter.

††preprint: APS/123-QED

##  I Introduction

Muonium (M) is a two-body exotic atom consisting of a positive muon
($\mu^{+}$) and an electron ($e^{-}$). Due to its purely leptonic composition,
hadronic effects modify the atomic levels only as loop corrections and there
are no finite-size effects, therefore it is an ideal system to test bound-
state quantum electrodynamics (QED). Precision spectroscopy of the 1S-2S
transition of M [1, 2] and the ground-state hyperfine splitting [3] contribute
additionally to the most precise determinations of the electron-to-muon mass
ratio and the muon magnetic moment. These measurements are also sensitive to
the charge equality between muon and electron [2] and hence contribute to
testing lepton universality, a topic that moved into the spotlight after the
latest results of LHCb [4] and the Fermilab $g-2$ experiment [5]. Recently,
precision measurements lead to new results on the ground-state hyperfine
splitting in muonium measured at J-PARC [6, 7] and on the Lamb shift in
muonium measured at PSI [8]. Further spectroscopy measurements using M atoms
at improved levels of precision have been proposed [9, 10] and are currently
being carried out [11, 12]. Another test of lepton universality is the search
for the lepton-flavour violating muonium-anti-muonium oscillations [13], for
which a new measurement has recently been proposed [14]. At J-PARC, the
ionization of M atoms in vacuum is part of a proposed cooling scheme for the
$\mu^{+}$ beam [15], which could enable next-generation searches for new
physics. These experiments rely on high intensity and high quality atomic
muonium beams in vacuum, and reaching higher precisions is partially limited
by presently available vacuum M sources.

Figure 1: Top view of the experimental setup for the detection of M decay
using scintillator bars and atomic electron detection. The muons entered
through the entrance detector indicated at the top of the drawing. They formed
M atoms in the target of which a fraction were emitted into vacuum. Decays of
the M atoms in vacuum were detected by coincidence of the decay $e^{+}$ in the
scintillator bars (red arrow) and atomic $e^{-}$ after electrostatic
acceleration (blue arrow). The target was mounted on a PVC frame (grey box)
outside of the electrostatic cage to repel ionization electrons.

High yields of vacuum M are frequently produced by stopping low-momentum
$\mu^{+}$ ($p_{\mu}<28$ MeV/c) in porous materials like silica powders [16,
17, 18, 19] where they can combine with electrons. Here, the initial numbers
of stopped muons per second ($\phi_{\mu}$) are converted with typically high
efficiencies up to $61\text{\,}\mathrm{\char 37\relax}$ [20], but the flux of
M atoms that can actually reach vacuum ($\phi_{\rm M_{V}}$) is strongly
dependent on the diffusion times and the consequent decay losses due to the
short muon lifetime ($\tau_{\mu}\approx 2.2~{}\upmu$s). Hence the muon-to-
vacuum-muonium conversion efficiency $\eta_{\rm M}=\phi_{\rm
M_{V}}/\phi_{\mu}$ depends on the initial muon beam momentum that defines the
implantation depth of $\mu^{+}$. Moreover, the temperature, chemical
properties, and nanoscopic structure of the converter further impact the
diffusion times. Note, that the stopping spread of muons within a certain
material increases approximately proportional to $p_{\mu}^{3.5}$ below
$p_{\mu}\approx 30$ MeV/c beam momentum [21]. Consequently, lower beam momenta
generally enable a higher fraction of the incoming muons to stop close to the
sample’s surface and be emitted into vacuum. Recent developments for high M
yields and beam qualities utilized mesoporous silica in cryogenic environments
[22] as well as silica aerogels [23, 24]. High vacuum M rates could be
achieved by laser ablation of microscopic holes in aerogels that decreased the
M diffusion times, and reached conversion efficiencies of up to $\eta_{\rm
M}\approx$3.05(3)\text{\,}\mathrm{\char 37\relax}$$ [25, 26], using
$p_{\mu}=23~{}{\rm MeV/c}$ muons. One disadvantage of these sources is that
the emerging M beam has a wide (approximately thermal Maxwell-Boltzmann)
momentum distribution, and a large angular divergence $\propto\cos\uptheta$
measured relative to the surface normal.

Aiming for a muonium gravity experiment [27] and increased precision of M
1S-2S spectroscopy, the focus of our research started with the development of
a novel cryogenic muonium source [28]. The gravity experiment needs a compact
detection setup and reliable background-free M detection methods that can
ultimately be operated in a dilution refrigerator. As an alternative solution,
a gravity measurement at higher temperatures may be possible using novel M
emitters in combination with collimators. In order to study the
characteristics of muonium emitters and test compact detection designs that
can be adapted to a cryogenic environment, we developed various detection
schemes and carried out measurements using known and novel room temperature M
sources at the $\pi$E1 beamline of the Paul Scherrer Institute (PSI).
In Section II, a compact background-suppressed M detection scheme is described
which was used to characterize the dynamics of the emitted M atoms in vacuum.
To our knowledge, our measurement was the first observation of M emission into
vacuum from zeolite samples. The same setup was also used to quantify the
impact of helium (He) gas on the mobility of emitted M atoms, which puts
boundaries on the operational temperatures in our cryogenic experiments
(Section III). Furthermore, we carried out detailed emission studies using
positron track reconstruction along one plane with MicroMegas tracker
detectors [29], which allowed us to extract the muon-to-vacuum-muonium
conversion efficiency for one aerogel target, discussed in Section IV.

##  II Background-free muonium detection

We developed a compact detection system based on tracking positrons ($e^{+}$)
from $\mu^{+}$ decay, and a coincident detection of the low-energy atomic
electrons left behind after M decay. The sketch of the experimental setup is
shown in Figure 1. Muons of $p_{\mu}=11$-$13~{}$MeV/c momentum with a momentum
spread of about $8\text{\,}\mathrm{\char 37\relax}$ (full width at half
maximum FWHM) from the $\pi$E1 beamline of PSI were guided to the experimental
setup, where they first traversed a $55~{}\mu\text{m}$ thick scintillator foil
(entrance detector) which was read out by a set of silicon photomultipliers
(SiPMs). In this process the $\mu^{+}$ lost a large fraction of their kinetic
energy and suffered significant scattering. Hence a passive copper collimator
of 15 mm thickness with an opening of 6 mm width and 10 mm height was placed
in front of the target area. The muons passing through the collimator reached
the porous muonium conversion target with kinetic energies of few $100~{}$keV
about 10 ns after the entrance signal, where they came to rest at different
implantation depths below the surface, depending on their initial energy. Due
to the distance between the entrance counter and the target, the rate of
stopped muons on the target $\phi_{\mu}$ was not known, preventing the
extraction of absolute conversion efficiencies $\eta_{\rm M}$. The stopped
$\mu^{+}$ combined with electrons to form M atoms which could diffuse in the
porous targets. The majority of M atoms decayed while diffusing through the
sample, while a fraction reached the surface and were emitted into high
vacuum, backwards relative to the muon beam, i.e. in $-z$ direction. The
vacuum M atoms propagated with a certain velocity distribution, and passed in
front of a scintillator tracker system, which consisted of small scintillator
bars (3$\times$4$\times$25 mm3) read out by SiPMs. The scintillators were
arranged in two layers, next to the drift volume of the M atoms (C1-C3) and 25
mm further away (F1-F3), and detected the decay positrons from M (red arrow in
Figure 1). By requiring coincidences between the close and far scintillator
bars (C and F) we defined three conical acceptance regions in the drift volume
for muon decays (as indicated for C2-F2 by the transparent red cone in Figure
1). The geometric acceptance for an isotropic decay in the center of these
regions was roughly $\epsilon_{\rm geo}\approx 4\times 10^{-3}$.
The events considered as muonium emission candidates all started with an
incoming muon triggering the entrance detector, followed by a coincidence
measured in one of the scintillator pairs. The time difference between the
entrance signal and the coincidence signal in the scintillator bars (typically
up to $\sim$8\text{\,}\upmu\mathrm{s}$$) was used to determine the time it
took for the M atoms to reach the acceptance region of the scintillator pairs.
However, in addition to decays of vacuum muonium in the acceptance region, the
coincidence scintillators registered a big muon-correlated background, i.e. a
background decreasing exponentially with the muon lifetime.

Figure 2: Time distributions of $e^{+}$ signal in time coincidence normalized
by the number of hits in the entrance detector. The given time is relative to
the muon entrance. Top: Time distributions without additional coincidence with
the atomic $e^{-}$ for the coincidence C2-F2 of the PVC measurement (a) and of
the Aerogel-1 measurement (b). Here, the orange lines correspond to estimates
of the exponential background from $\mu^{+}$ decay (see text for more
details). Bottom: Time distributions including the additional coincidence to
the atomic $e^{-}$ for the coincidence C2-F2 of the PVC measurement (c) and of
the Aerogel-1 measurement (d). The orange curve in the distribution for
aerogel corresponds to the simulated time distribution for M beam following a
thermal, $\cos\uptheta$ emission at $300~{}$K.

In order to suppress this background, we required a coincidence with detection
of the left-over atomic electron ($e^{-}$, blue arrow in Figure1). A similar
approach was already used to detect the decay electron in anti-muonium in
[13]. After dissociation of the M atom, the atomic $e^{-}$ have energies of a
few eV, distributed around a mean energy of $13.5\text{\,}\mathrm{eV}$ [30].
To create a measurable signal, we accelerated them towards a plastic
scintillator plate with a surface area of
$20\times$20\text{\,}{\mathrm{mm}}^{2}$$ and a thickness of 1 mm using an
electrostatic cage. Electron hits in the scintillator plate were registered by
a Hamamatsu (R7600U) Photomultiplier tube (PMT). The electrostatic cage
consisted of wires in front of the scintillator bars and a series of
rectangularly shaped electrode loops surrounding the drift volume. The voltage
of the wires in front of the scintillator bars was set to $-8~{}$kV and the
voltage of the electrode loops decreased linearly towards ground on the side
of the PMT. The cage was designed so as to repel ionization electrons that may
have been emitted from the target with energies of several eV [31]. In the
following, the requirement of two scintillator bars triggering in coincidence
with the atomic electron detector after a signal in the entrance detector will
be referred to as the triple-coincidence requirement.
Using this setup, we investigated seven different M conversion targets. Two of
these were laser ablated silica aerogel samples obtained by courtesy of G.
Marshall (TRIUMF) and co-workers. The 7 mm thick slabs of aerogel had average
densities of 29 mg/cm3 and featured microscopic vertical holes in the front
surface. The holes were ablated with a laser in a triangular grid pattern
[25]. The holes of the Aerogel-1 sample were $4$-$5$ mm deep and had diameters
of $100$-$110~{}\upmu$m and a pitch of 150 $\upmu$m. Aerogel-2 featured holes
with diameters of $175$-$240~{}\upmu$m, a pitch of 550 $\upmu$m and an
ablation depth of $\sim 1~{}$mm. Two M emitters were zeolite samples, which
are composite materials of silicon, oxygen, hydrogen and aluminum featuring
intrinsic microporosity with pores of about $0.5$ nm size. The used zeolites
were hierarchical zeolites which had been treated with alkaline to induce
mesopores with $5$-$10~{}$nm size in the material. Zeolites are known
positronium (Ps) production targets [32, 33] and $\upmu$SR studies [34, 35]
had already found that muonium atoms form in the bulk material when irradiated
with $\mu^{+}$, but to our knowledge vacuum M emission has not been observed
yet. The two samples used in this experiment were produced by P. L. Begona and
co-workers at ETH Zurich and are referred to as Zeolite-1 (HZ40-AT2) and
Zeolite-2 (CBV712-B) in this paper. They featured different nanoscopic
structures, which were a MFI structure [36, 37] for Zeolite-1 and a FAU
structure with cubic unit cells for Zeolite-2 [37]. We tested some more exotic
samples, which were a carbon nanotube enforced silica aerogel, and two
different carbon nanotube targets (single-walled CNTs, and CNT forests). No
significant vacuum M emission was observed from any of these three samples.
The measurements further described here were conducted in two sets with
similar beam conditions at $p_{\mu^{+}}\approx 11~{}\text{MeV/c}$ momentum. In
the first set, the two zeolite samples were compared to Aerogel-1. In the
second set, Aerogel-1 and Aerogel-2 were referenced to a PVC sample, that was
assumed not to produce any vacuum M. Between the two sets the geometry was
changed: An additional detector pair (coincidence C3-F3) was added to the
setup and the target was moved closer to the scintillator bars (-$x$
direction) by 5 mm to optimize the tracking conditions.
Figure 2 shows measured time distributions of the $e^{+}$ coincidences in the
case of PVC (no M emission) and Aerogel-1, with and without the additional
coincidence constraint with the atomic electron detector. The time on the
horizontal axis refers to the time difference between the $\mu^{+}$ triggering
the entrance detector and the decay $e^{+}$ being registered in the pair of
scintillator bars. The time distributions of the $e^{+}$ coincidences feature
a dominant exponential background, decaying with the lifetime of the muon.
This background stems from decays of $\mu^{+}$ stopping in front of the
scintillator bars, and from scattered decay $e^{+}$ coming from the target.
Vacuum M atoms which were emitted from the target surface and pass by the
sensitive region of the $e^{+}$ coincidence manifest as a bump on top of the
exponential background. Using the additional coincidence with the accelerated
atomic electron (triple-coincidence requirement) we obtain muonium
distributions which are free of the muon-related background. Applying the
triple-coincidence requirement to the PVC measurement (Figure 2c), only a few
background hits remain. Using these hits we can define
$\mathcal{P}^{BG}_{e^{+}e^{-}coinc}$ as the probability that a background hit
in an $e^{+}$ coincidence is accompanied by a false muonium signal in the
triple-coincidence distribution. It can be estimated by normalizing the
background hits in the triple-coincidence distribution of PVC to the
exponential background in the corresponding $e^{+}$ distribution. The
resultant probabilities for the consecutive coincidence layers along the beam
axis are given by

$\displaystyle\mathcal{P}^{BG}_{e^{+}e^{-}coinc}=\left\\{\begin{array}[]{c}\left(3.7\pm
1.8\right)\cdot 10^{-3}~{}~{}\left(\text{C1-F1}\right)\\\ \left(6.5\pm
3.2\right)\cdot 10^{-3}~{}~{}\left(\text{C2-F2}\right)\\\ \left(10.1\pm
4.1\right)\cdot 10^{-3}~{}~{}\left(\text{C3-F3}\right).\end{array}\right.$

Figure 3: Time distributions of the triple coincidence (positron in both
scintillator bars and atomic electron detected) for C1-F1 (a-c) and C2-F2
(d-f). The measured data was obtained for Aerogel-1 of the second set (a,d),
Aerogel-2 (b,e) and Zeolite-1 (c,f). The time is given relative to an entrance
hit. The orange curve corresponds to the simulated time distribution for a M
beam following a thermal $\cos\uptheta$ emission at $300~{}$K, with re-scaled
amplitude to fit the data.

Based on Monte Carlo simulations, we suspect that the dominant source for the
remaining background in the triple-coincidence distributions are not muon
decays in the target, but muon decays occurring upstream in front of the
collimator. This is a possible explanation for the increasing trend of
$\mathcal{P}^{BG}_{e^{+}e^{-}coinc}$ with increasing distance from the target.
In Figure 2d, a few remaining entries of the exponential background can be
seen in the first bins after $t=0$. Similar background characteristics were
observed for all coincidences of the aerogel measurements.
In order to study the dynamic properties of M emission we simulated the
expected time distributions for M atoms emitted from the target. To obtain
these time distributions and detection efficiencies, Monte Carlo simulations
of the $e^{+}$ detector acceptances were combined with simulations of the
atomic electron acceleration and simulations of the M atoms emitted from the
sample. The electric field in the cage needed for the $e^{-}$ acceleration was
simulated with a finite-element solver in COMSOL
Multiphysics${}^{\text{\textregistered}}$ [38]. The trajectories of atomic
electrons were then simulated by tracking them through the fieldmap using
GEANT4 [39] and G4beamline [40]. We assumed the M atoms were emitted from the
surface as a thermal beam, with a Maxwell-Boltzmann velocity distribution at
300 K, and corresponding angular distribution of the emission proportional to
$\cos\uptheta\mathrm{d}\Omega$, where $\uptheta$ is the angle of the emission
with respect to the target surface normal. In the simulations, the diffusion
time of the M atoms inside the target was neglected ($\Delta
t_{\text{Diff}}\approx 0$) compared to their time-of-flight (TOF) between the
target surface and their decay position, in agreement with the observations.
With this assumption, the starting time of the muonium atoms coincides with
the time of the entrance signal, since the incoming muons only needed about 10
ns to travel from the entrance detector to the target sample.
The time distribution obtained from the simulations is superimposed on the
triple-coincidence distribution of Aerogel-1 using an orange curve in Figure
2d with a re-scaled amplitude to best fit the data. The model is in good
agreement with the measurement. Within the time interval
$\left[600~{}\text{ns},6500~{}\text{ns}\right]$ the reduced $\chi^{2}$ is
$\chi^{2}/n_{F}\approx 36.2/30\approx 1.21$. Here, the $\chi^{2}$ function in
the limit of low counting statistics is used following [41], which is given by

$\chi^{2}=2\sum\limits_{i}N_{i}^{sim}-N_{i}^{exp}+N_{i}^{exp}\ln{\frac{N_{i}^{exp}}{N_{i}^{sim}}},$
(2)

where $N_{i}^{sim}$ and $N_{i}^{exp}$ denote the numbers of entries in the
i-th bin of the simulated and measured distributions respectively. For other
time distributions of Aerogel-1 at similar beam conditions we found reduced
$\chi^{2}$ values between $\chi^{2}/n_{F}=0.7$ and $\chi^{2}/n_{F}=2.6$ (the
lower interval limit has been adjusted to 500 ns, 600 ns and 1000 ns for the
coincidences C1-F1, C2-F2 and C3-F3 respectively, to cut away remaining
background counts at early times).
Further time distributions for Aerogel-1, Aerogel-2 and Zeolite-1 are
displayed in Figure 3. In the distributions for Aerogel-2 the peaks are wider
and occurring at later times. This indicates that the emission characteristics
are different from those of Aerogel-1 and of the thermal emission model. Also
the distributions for Zeolite-1 cannot be explained with a simple thermal
emission and feature a faster component of muonium atoms that were not
entirely thermalized. This can be understood qualitatively by the fact that
the measured hierachical zeolites entail both micropores and mesopores. While
in pores of 5 nm diameter M is expected and known to thermalize to the sample
temperature [22], for pores of 0.5 nm one needs to consider quantum mechanical
effects since de Broglie wavelength of muonium is of the same order as the
pore sizes. Therefore the lowest energy at which M can be emitted into vacuum
is limited by the ground state energy of M in the pores. This behaviour is
well studied for positronium which being approximately 100 times lighter
already experiences this effect in mesoporous materials [42, 43]. Furthermore,
we observed that all time spectra with zeolite samples feature a prominent
prompt peak at $0<t<200~{}$ns. Note, that this peak has not been observed in
any of the time spectra with aerogel samples. The peak might imply that a
fraction of the muonium atoms are formed from muons backscattering at the
surface of the zeolite sample while picking up an electron.

Table 1:  Relative vacuum yields $\xi_{\rm M}=\eta_{\rm M}/\eta_{\rm M,0}$ for the various samples. The highest conversion efficiency $\eta_{\rm M,0}$ was achieved with the Aerogel-1 sample, which is thus used as a reference. Set A | $\xi_{\rm M}$
---|---
Aerogel-1 | small holes | $1.0$
Zeolite-1 | HZ40-AT2 | $0.62\pm 0.11$
Zeolite-2 | CBV712-B | $0.66\pm 0.12$
Set B | $\xi_{\rm M}$
Aerogel-1 | small holes | $1.0$
Aerogel-2 | large holes | $0.85\pm 0.18$

The rate observed in the triple-coincidence distributions can be used to
compare the conversion efficiencies $\eta_{\rm M}$ of the samples. As the
number of muons stopping on the sample per second $\phi_{\mu}$ is not known
precisely, we define the relative yield

$\xi_{\rm M}=\frac{\eta_{\rm M}}{\eta_{\rm M,0}},$ (3)

which relates the muon-to-vacuum-muonium conversion efficiency ($\eta_{\rm
M}$) of a given sample to that of a reference sample ($\eta_{\rm M,0}$).
Aerogel-1 featured the highest conversion efficiencies and is used as the
reference sample. In order to keep the impact of different emission
characteristics low we only consider the first coincidence layer (C1-F1) which
is closest to the target surface to obtain $\xi_{\rm M}$. We define
$R^{\text{M}}_{e^{+}e^{-}}$ as the number of detected triple coincidences
normalized to the entrance counts. It is obtained by integrating the measured
triple-coincidence distribution $H_{e^{+}e^{-}}\left(t\right)$ and subtracting
the number of remaining background counts following

$\displaystyle R^{\text{M}}_{e^{+}e^{-}}$
$\displaystyle=\int\limits_{0}^{t_{max}}H_{e^{+}e^{-}}\left(t\right)\mathrm{d}t$
(4)
$\displaystyle-\mathcal{P}^{BG}_{e^{+}e^{-}coinc}\times\int\limits_{0}^{t_{max}}\text{bg}_{\text{e}^{+}}\left(t\right)\mathrm{d}t\text{,}$

where $t_{max}=8500~{}$ns was chosen as the upper integration limit. For this,
the exponential background in the $e^{+}$ coincidence is estimated based on
the number of hits in a small interval $\left[0,t_{0}\right]$ in the beginning
of the time spectrum,

$\displaystyle\text{bg}_{\text{e}^{+}}\left(t\right)=e^{-\frac{t}{\tau_{\mu}}}\times\frac{\int\limits_{0}^{t_{0}}H_{e^{+}}\left(t^{\prime}\right)\mathrm{d}t^{\prime}}{\int\limits_{0}^{t_{0}}e^{-\frac{t^{\prime}}{\tau_{\mu}}}\mathrm{d}t^{\prime}}\text{,}$
(5)

where $H_{e^{+}}\left(t\right)$ is the measured time distribution of $e^{+}$
coincidences and it is assumed that the background is decaying with the muon
lifetime $\tau_{\mu}=2197~{}$ns, as verified with the PVC sample. For the
three rows of coincidences we set $t_{0}$ to 500 ns, 750 ns or 1000 ns
respectively, such that in the interval $\left[0,t_{0}\right]$ the
contribution of vacuum M decays is negligible. To correct for different
dynamical behavior of the samples the results for $R^{\text{M}}_{e^{+}e^{-}}$
are additionally multiplied by a lifetime factor, such that the relative yield
for any sample is calculated with

$\displaystyle\xi_{\rm M}=\frac{R^{\text{M}}_{e^{+}e^{-}}\times
e^{\frac{\bar{t}}{\tau_{\mu}}}}{R^{\text{M,0}}_{e^{+}e^{-}}\times
e^{\frac{\bar{t}_{0}}{\tau_{\mu}}}},$ (6)

where $\bar{t}$ corresponds to the mean of the time distribution measured with
the sample and $R^{\text{M,0}}_{e^{+}e^{-}}$ and $\bar{t}_{0}$ indicate the
values for Aerogel-1. The $\xi_{\rm M}$ obtained in this way are summarized in
Table 1. The given uncertainties result from counting statistics. The
$\xi_{\rm M}$ given for the zeolite samples correspond to an early
measurement, which resulted in the highest measured $\xi_{\rm M}$ for these
samples. Evaluating the yields of a later measurement, it was found that the
relative yields of the zeolite samples were subject to a degrading effect and
had dropped by up to $\sim$30\text{\,}\mathrm{\char 37\relax}$$ between the
two measurements.

Figure 4: Validation of the van der Waals elastic scattering model with
measured time distributions of Aerogel-1.
a) Time distributions for three He gas densities simulated for coincidence
C1-F1 using the scattering model (arbitrary vertical scale). b)-d) Comparison
of measured time distributions for coincidence C1-F1 with simulated
distributions for the model at number densities b) $n_{0}\approx 2\cdot
10^{10}~{}\text{cm}^{-3}$, c) $n_{1}=\left(2\pm 1\right)\cdot
10^{14}~{}\text{cm}^{-3}$ and d) $n_{2}=\left(2\pm 1\right)\cdot
10^{15}~{}\text{cm}^{-3}$. The measurement at $n_{0}$ is used to fix the
global normalization of the simulation. The orange lines correspond to the
central values for $n$ while the error bands account for the possible range
given by the uncertainties of the absolute pressure determination.

From a technical point of view, it is interesting to study the detection
efficiency of the atomic electron detector. The overall efficiency of the
detection system for the atomic electron $\epsilon_{e^{-}}$ consists of two
individual efficiencies,

$\displaystyle\epsilon_{e^{-}}=\epsilon_{\rm geo}\times\epsilon_{e^{-}\rm
det.},$ (7)

where $\epsilon_{\rm geo}$ is the geometric efficiency of an accelerated
$e^{-}$ to reach the detector in case a positron coincidence was triggered,
and $\epsilon_{e^{-}\rm det.}$ is the efficiency of the atomic electron
detector (consisting of plastic scintillator and PMT) to detect the $\sim
6$-$8~{}$ keV electron. This overall efficiency can be assessed by comparing
the number of $e^{+}$ coincidences triggered by M decays with the number of
triple coincidences triggered by the same decays. In order to estimate the
number of $e^{+}$ coincidences triggered by M decays, the $\mu^{+}$ background
$\text{bg}_{\text{e}^{+}}\left(t\right)$ needs to be subtracted from the
$e^{+}$ coincidences,

$\displaystyle
R^{\text{M}}_{e^{+}}=\int\limits_{0}^{t_{max}}H_{e^{+}}\left(t\right)\mathrm{d}t-\int\limits_{0}^{t_{max}}\text{bg}_{\text{e}^{+}}\left(t\right)\mathrm{d}t\text{.}$
(8)

The overall efficiency of the atomic electron detection system can now be
calculated for each coincidence pair via

$\displaystyle\epsilon_{e^{-}}=\frac{R^{\text{M}}_{e^{+}e^{-}}}{R^{\text{M}}_{e^{+}}}.$
(9)

Table 2: Measured and simulated efficiencies for the various rows of $e^{+}$ coincidence detectors. $\epsilon_{e^{-}}$ is the overall efficiency of the atomic electron detection system. $\epsilon_{\rm geo}$ is the efficiency of the acceleration of atomic electrons towards the detector. | measured $\epsilon_{e^{-}}$ | simulated $\epsilon_{\rm geo}$
---|---|---
C1-F1 | $0.32\pm 0.03^{\text{stat}}\pm 0.04^{\text{sys}}$ | $0.45\pm 0.04$
C2-F2 | $0.34\pm 0.03^{\text{stat}}\pm 0.01^{\text{sys}}$ | $0.50\pm 0.02$
C3-F3 | $0.33\pm 0.04^{\text{stat}}\pm 0.01^{\text{sys}}$ | $0.43\pm 0.02$

Table 2 presents the weighted means over several measurements with aerogel
targets. The systematic uncertainties emerge from the subtraction of the
exponential background and were estimated based on the amount of entries
within the interval $\left[0,t_{0}\right]$ of the triple-coincidence
distributions.
By simulating the acceleration of the atomic electrons in the electric cage we
can determine $\epsilon_{\rm geo}$ \- which takes into account the coincident
detection of a positron signal - and then extract the detection efficiency of
the atomic electron detector $\epsilon_{e^{-}\rm det.}$. The computed
geometric efficiencies $\epsilon_{\rm geo}$ for electron tracking are given in
Table 2. The given systematic uncertainties on the $\epsilon_{\rm geo}$
account for the uncertain number of positron coincidences following from early
M decays, which depends on the applied energy cuts, the specific muonium
emission model and the exact positioning of the detectors. Using the
efficiencies of each coincidence row and computing the weighted mean, we find
that the atomic electron detector detects the atomic electrons accelerated to
kinetic energies of $6-8$ keV with an efficiency of

$\displaystyle\epsilon_{e^{-}\rm det.}=\left(71\pm 5\text{(stat)}\pm
6\text{(sys)}\right)$\text{\,}\mathrm{\char 37\relax}$.$ (10)

##  III Scattering of muonium atoms in helium gas

In a future muonium gravity experiment, residual helium gas may be present as
vapor due to the use of superfluid He for M production [28]. Therefore an
estimate for the elastic scattering cross section of M atoms with the residual
He gas will be needed in order to assess the risk of M atoms to scatter during
the time of the free-fall measurement. For this estimation we used the same
detection geometry described in Section II. The measurement was performed by
separating the vacuum of the target volume from the vacuum of the beam line,
and applying He gas pressures of $p_{1}=\left(8\pm 4\right)\cdot
10^{-3}~{}$mbar, $p_{2}=\left(8\pm 4\right)\cdot 10^{-2}~{}$mbar as opposed to
the high vacuum values ($p_{0}\approx 1\cdot 10^{-6}~{}$mbar). The large
uncertainties given here are not standard deviations but systematic
uncertainties that reflect the accuracy of the absolute pressure determination
for the pressure gauges used. The Aerogel-1 target was used for M production.
Due to the He environment, the $e^{+}$ coincidences were used without the HV
cage and atomic electron detector, which could not be operated with the buffer
gas present.
We calculated cross sections for elastic M-He scattering assuming a simple
spherical van der Waals interaction potential of purely electrical origin. The
ionization energy, reduced mass and polarizability of the muonium system are
approximately the same as in the hydrogen system and therefore the M-He
potential is similar to the potential between atomic H and He, which has been
studied extensively [44, 45, 46, 47, 48]. In the center-of-mass system (CMS),
following [47], we assumed the M-He potential to have a Lennard-Jones form,

$\displaystyle
V\left(r\right)=\varepsilon\left(\left(\frac{r_{m}}{r}\right)^{12}-2\left(\frac{r_{m}}{r}\right)^{6}\right),$
(11)

with $\varepsilon=6.66\cdot 10^{-4}~{}$eV indicating a particularly shallow
well and $r_{m}=3.46~{}\text{\AA}$. Solving the radial Schrödinger equation
for this potential, we expanded the total cross section in CMS in partial
waves using the expression

$\displaystyle\sigma_{tot}=\frac{4\pi}{k}\sum\limits_{l}\sigma_{l}\left(k\right)\text{,}$
(12)

where

$\displaystyle\sigma_{l}\left(k\right)=\left(2l+1\right)\sin^{2}\delta_{l}\left(k\right)\text{.}$
(13)

Here, $k$ is the wave number in CMS and $\delta_{l}$ is the scattering phase
shift for the $l$th partial wave. Likewise, the differential cross sections
were computed and transferred to scattering rates in the lab system at the
corresponding temperature, which was done by averaging over the thermal
momentum distribution of the He atoms. Since this model considers only two-
body interactions it is valid for low gas densities at which the rate of
3-body interactions is negligible.
In the simulations we assumed thermal M emission as described in Section II.
The simulations were run with He number densities $n_{1}\approx\left(2\pm
1\right)\cdot 10^{14}~{}\text{cm}^{-3}$, $n_{2}\approx\left(2\pm 1\right)\cdot
10^{15}~{}\text{cm}^{-3}$ and $n_{0}\approx 2\cdot 10^{10}~{}\text{cm}^{-3}$,
roughly corresponding to the measured pressures at 300 K. During the vacuum
measurement, the He pressure was low enough such that the uncertainty of the
pressure measurement does not make a difference. Figure 4 shows time
distributions for the coincidence C1-F1. In Figure 4a, simulations for the
three densities are compared. For the measured time spectra in the other
panels we subtracted the exponential background which we determined as
discussed in Section II. The norm of the vacuum simulation was fitted to the
vacuum measurement to extract a global normalization factor which was then
used to normalize the simulations at number densities $n_{1}$ and $n_{2}$. The
resulting curves are shown Figure 4c)-d), where the orange error bands reflect
the range given by the uncertainties of the absolute pressure determination.
Considering the large statistical uncertainty of the measured points and the
additional uncertainty of the background subtraction the computed cross
sections reproduce the measurement with better than an order-of-magnitude
level of accuracy. For a more precise test of the model, a dedicated high-
statistics measurement at well-controlled densities would be necessary.
Using the computed cross sections we can estimate the maximally allowed
density of residual helium in a gravity measurement with muonium. To ensure
that the fraction of M atoms scattering on a length scale of about 100 mm
(corresponding to the prospected length of the muonium free fall in the
gravity experiment) is below $\sim$$5\text{\,}\mathrm{\char 37\relax}$, the
helium number density needs to be reduced to around
$10^{11}~{}\text{cm}^{-3}$. Hence, ignoring a possible 3He content,
temperatures below $\approx 0.3~{}$K are needed to ensure accordingly low He
saturated vapor pressures, as given by measurements and the Clausius–Clapeyron
equation (see, e.g., Ref. [49]).

##  IV Emission study with MicroMegas detectors

In our second experimental setup, a pair of MicroMegas (MM) tracking detectors
[29] were used to study the muonium emission characteristics of aerogel more
precisely. Previously, muonium emission measurements at TRIUMF [25, 26] used
several layers of multi-wire drift chambers to track the decay positrons from
muonium atoms. The MM modules we used had an active region of
$80\times$$80\text{\,}{\mathrm{mm}}^{2}$ and were developed at ETH Zurich for
the NA64 collaboration [50].
A sketch of the experimental setup is shown in Figure 5. The muons passed
through a collimator and triggered the entrance detector when entering the
target cell. The aerogel target surface had a distance of $\sim 28$ mm from
the entrance detector, which enabled around $90\text{\,}\mathrm{\char
37\relax}$ of the muons exiting the entrance counter to stop on the target.
The target sample was held in place by thin plastic parts from the bottom and
the sides which had a minimal cross section seen from the front, minimizing
the number of muons stopping on the holders to a negligible level. Target and
collimator were mounted inside an aluminum cell that had an internal width and
height of 70 mm and 40 mm, respectively. The windows on the top and bottom of
the cell were kapton foils of $130$ $\upmu$m thickness that were glued onto
the cell. The dimensions of the cell were chosen in a way, that the number of
$\mu^{+}$ that stopped on the walls or windows in front of the target were
negligible compared to the number of M decays in that region. Two MM modules
mounted above the target cell were used as a telescope to track $e^{+}$
produced by $\mu^{+}$ and M decays inside the target cell. A large
scintillator on top of the setup was used together with the entrance detector
to trigger the MM detectors. The MM detectors were only triggered if the top
scintillator received a hit in the time window $\left[1000,~{}2000\right]$ ns
after the entrance detector. In that way, we selected events in which muonium
atoms had enough time to form, leave the sample and travel a few millimeters
in vacuum before decaying. The data acquisition system of the MM detectors
could record signals only within a limited time window of 675 ns around the
trigger, and was limited to a trigger rate of 1 kHz. Due to these limitations
in the data readout, the triggering scheme was crucial in order to separate M
decays in front of the target from $\mu^{+}$ decays on the target surface.

Figure 5: Experimental setup of the MicroMegas measurement. The $\mu^{+}$
entered from the left and triggered the entrance detector before reaching the
target. Two MM detectors were mounted above the vacuum cell in a telescope
configuration. A kapton window of $130\text{\,}\upmu\mathrm{m}$ thickness was
glued onto the vacuum cell. A scintillation counter on top of the telescope
was used to trigger the MM modules. For more details see text. Figure 6:
Distribution of $\mu^{+}$ and M decays projected to the central plane of the
vacuum cell. $e^{+}$ tracks with angles of less than $None$ to the vertical
are considered and extrapolated towards the central plane. a) Projected 2D map
of $\mu^{+}$ and M decays. The two accumulations of hits correspond to decays
in the entrance detector and the target surface. b) Projection of all decays
within $-10\leq x\leq 10~{}$mm. The black histogram corresponds to the
measured data while the orange curve corresponds to combined simulations of
background and M emission at $300~{}$K. The dashed blue line corresponds to
the background simulation (without vacuum M signal). The blue error band
accounts for the systematic uncertainty of the background simulation, which
was estimated using the discrepancy between measured and simulated
distribution on the right side of the target peak. Figure 7: Fits of emission
models with different temperatures to the data measured with the MicroMegas
detectors. a) Models without any diffusion time in the sample, i.e. $\Delta
t_{\text{Diff}}=0$. b) Same models but with a constant diffusion time of
$\Delta t_{\text{Diff}}=200~{}$ns added to the time of travel of the M atoms.
The given $\chi^{2}$ values correspond to the fits within
$\left[-12,~{}4\right]$ mm, with $n_{\text{df}}=29$ degrees of freedom.

The MM telescope allowed to produce an image of decays projected to the
central plane of the target cell. To restrict the perspectival discrepancy
between projected and actual decay position to less than 2 mm for decays in
front of the target, we considered only tracks within 10 degrees of the
vertical. The projected hit map of decays is shown in Figure 6a) with a
logarithmic color scale. The large hit accumulation on the right corresponds
to decays of stopped $\mu^{+}$ and M on the target surface, while the left
accumulation corresponds to $\mu^{+}$ decays in the entrance detector. Left of
the target peak, the cloud of M decays in vacuum is visible.
We again used Monte Carlo simulations to study the emission characteristics of
the aerogel sample. For the background simulations we considered $\mu^{+}$
decaying at the target surface and the entrance detector as well as $\mu^{+}$
that missed the target and stopped on the walls or the windows. In the
simulations, the thickness of the PCB in the MM modules was adjusted, such
that the width of the simulated peaks matched the measurement. For unknown
reasons, the PCB in the simulation needed to be thinner by roughly a factor of
two than those actually mounted in the MM detectors, which were around $3.4$
mm thick. Figure 6b) shows a projection of the measured data onto the $z$ axis
(black binned data). The dashed blue line between the peaks shows the
distribution obtained from the normalized background simulations. The light
blue error band indicates the systematic uncertainty of the background level
in this region, which was estimated conservatively based on the discrepancies
between the background simulation and the measurement on the right side of the
target peak.
An excess of the measured data over the background simulation between the two
peaks indicates the cloud of M atoms decaying in front of the target. Using
the integral of the excess we can estimate a model-independent vacuum yield
for the emission of muonium from the aerogel sample. For this, we subtract the
integrals of the simulated background distribution $N_{\text{BG}}$ from the
experimental distribution $N_{\text{exp}}$ to obtain the number of detected
muonium decays within an interval $z\in\left[-12,~{}4\right]$ mm,

$\displaystyle N^{\text{M
decays}}_{\left[-12,~{}4\right]}=\int\limits_{-12}^{4}\left(N_{\text{exp}}\left(z\right)-N_{\text{BG}}\left(z\right)\right)~{}\mathrm{d}z\text{,}$
(14)

where the interval was chosen such that the entries within the interval are
dominated by muonium decays. Normalizing this number to the total number of
decays on the target and in the cloud we obtain the vacuum yield for Aerogel-1
emitting M into $z\in\left[-12,~{}4\right]$ mm within the time window
$t\in\left[1000,~{}2000\right]$ ns:

$\displaystyle\text{yield}_{\left[-12,4\right]}=\frac{N^{\text{M
decays}}_{\left[-12,4\right]}}{N^{\text{M
decays}}_{\left[-12,4\right]}+N_{\text{target}}}\text{,}$ (15)

where $N_{\text{target}}$ is the integral of the background distribution of
simulated muon decays on the target. With this procedure we obtain

$\displaystyle\text{yield}_{\left[-12,4\right]}=\left(4.57\pm
0.07\text{(stat)}~{}^{+0.63}_{-0.39}\text{(sys)}\right)$\text{\,}\mathrm{\char
37\relax}$\text{,}$ (16)

where the uncertainty is dominated by the systematic uncertainty which we
assigned to the background level between the two peaks (indicated by the light
blue error band in Figure 6b)).
In addition to the model-independent approach we performed simulations of the
M cloud assuming M emission at various temperatures and angular distributions.
Fitting the amplitude of the obtained distributions we can determine which
emission model fits best. The orange graph in Figure 6 corresponds to the best
fit result which yields a reduced $\chi^{2}$ of $\chi^{2}/n_{\text{df}}\approx
44/29$ in the fit range $\left[-12,~{}4\right]$ mm. For this fit, emission of
thermal M at $300$ K with a $\cos\uptheta$ distribution was assumed and the
diffusion time of the M atoms inside the sample was neglected. Additionally,
fits for emission models with other temperatures were performed. For higher
temperatures, fits could be obtained by assuming non-zero diffusion times,
which demonstrates correlation between the emission temperature and the
diffusion time. It turned out that an increase in the diffusion time of about
200 ns was needed to compensate for a temperature increase of 100 K. Figure 7
shows fitted distributions for various temperatures with an additional
constant diffusion time of $\Delta t_{\text{Diff}}=200~{}$ns. The reduced
$\chi^{2}$ values for the best fits at the corresponding temperatures are
summarized in Table 3. Above 400 K, the fit quality was found to decrease
drastically, which implies that M atoms were emitted with roughly thermal
energies and after short diffusion times of maximally a few 100 ns. In order
to determine the diffusion time in the sample and the temperature
quantitatively, it is necessary to vary the time window of the measurement.
This could not be done during this study due to the limited run time and the
constraints of the MM DAQ. Since the measurement was performed using a
hardware trigger, it was also not possible to vary the timing after the
measurement. For future measurements of this kind a variation of the time
window is certainly needed. In general, the best fits were found for models
with emission following a $\cos\uptheta$ angular distribution.
From the fit of the simulated distributions to the data, it is possible to
extract a model-dependent conversion efficiency for the sample which we obtain
as

$\displaystyle\eta_{\rm M}=\frac{N^{\text{vacuum}}_{\text{M,
0}}}{N^{\text{vacuum}}_{\text{M, 0}}+N^{\text{target}}_{\mu\text{, 0}}},$ (17)

where $N^{\text{vacuum}}_{\text{M, 0}}$ is the absolute number of M atoms
emitted into vacuum from the sample and $N^{\text{target}}_{\mu\text{, 0}}$ is
the number of stopped muons remaining on the target. Both numbers can be
extracted from the simulations normalized according to the fit result.

Table 3:  Reduced $\chi^{2}$ and model-dependent conversion efficiencies $\eta_{\rm M}$ of Aerogel-1 for M emission models with various temperatures. The conversion efficiencies were obtained by fitting the simulated M distributions (assuming thermal M emission at $T=300~{}$K with a $\cos\uptheta$ distribution) to the data measured with MicroMegas detectors. The beam momentum in the measurement was at $p\approx 12.5~{}$MeV/c. Temperature [K] | Diffusion time [ns] | $\chi^{2}~{}/~{}n_{\text{df}}$ |  $\eta_{\rm M}$ [%]
---|---|---|---
300 | 0 | $44~{}/~{}29$ | $7.23\pm 0.05\text{(stat)}^{+1.06}_{-0.76}\text{(sys)}$
400 | 200 | $49~{}/~{}29$ | $6.72\pm 0.05\text{(stat)}^{+1.06}_{-0.76}\text{(sys)}$
500 | 400 | $184~{}/~{}29$ | $6.38\pm 0.05\text{(stat)}^{+1.58}_{-1.29}\text{(sys)}$

The conversion efficiency obtained in this way depends on the applied fit
range. While the lower limit of the fit was fixed to $z=-12$ mm, which is
where the M signal becomes negligible compared to the tail of decays from the
entrance detector, the upper limit for the fit was varied to study the impact
on the conversion efficiency. Figure 8 shows the dependence of the obtained
conversion efficiencies on the upper limit of the fit for the model with
$300~{}$K and no diffusion time and the model with $400~{}$K and a constant
diffusion time of $200~{}$ns. The latter model is accompanied by slightly
lower conversion efficiencies due to the losses during the time of the
diffusion. In order to extract the conversion efficiency, the fit interval
$\left[-12,~{}4\right]$ mm was chosen as a benchmark, which contains the
region in which the M distribution is dominant. The uncertainty due to the
choice of the integration limit is taken into account as a systematic
uncertainty which is estimated with the range of conversion efficiencies
between the fit ranges $\left[-12,~{}4\right]$ mm and $\left[-12,~{}-3\right]$
mm (smaller fit ranges cut away most of the M distribution). Another
systematic uncertainty is given by the discrepancy between the background
simulation and the measurement which was discussed above and is indicated by
the blue band in Figure 6. The conversion efficiencies obtained in this way
for models with various temperatures are summarized in Table 3. The latest
publication of measurements at TRIUMF [26] reports conversion efficiencies of
about 1-$2\text{\,}\mathrm{\char 37\relax}$ for comparable laser-ablated
aerogels with similar hole arrangements. Their best conversion efficiencies,
achieved with a sample with larger holes, was reported as
$3.05(3)\text{\,}\mathrm{\char 37\relax}$. In comparison, the higher
conversion efficiencies reported here (in Table 3) benefit from the narrower
stopping distribution of muons in the sample due to the low muon momentum of
around $p\approx 12.5~{}$MeV/c (momentum spread $\sim$8\text{\,}\mathrm{\char
37\relax}$$ FWHM).

Figure 8: Dependence of model-dependent conversion efficiencies on the upper
limit of the fit range. The lower limit is fixed at $z=-12~{}$mm. The two
models with the best fits are compared. The error bars contain the statistical
uncertainty only.

##  V Summary and conclusion

We developed two compact detection systems to characterize the propagation of
M atoms in vacuum, and compared various known and novel vacuum muonium
emitters. The emission of M atoms from state-of-the-art laser-ablated aerogel
samples was found to be in agreement with roughly thermal emission and the
expected $\cos\uptheta$ angular distribution. These samples reached the
highest muon-to-vacuum muonium conversion efficiencies in our studies.
Zeolites that were previously optimized for positronium conversion were found
to emit M atoms into vacuum as well, with somewhat non-thermal energy
distributions. These samples provided a relative vacuum M yield of
$66\pm$12\text{\,}\mathrm{\char 37\relax}$$ with reference to the highest-
yield aerogel sample, but their performance was not stable over time.
For the first time we used a pair of MicroMegas detectors to study emission of
muonium into vacuum. This preliminary experiment allowed us to extract an
absolute conversion efficiency for an aerogel sample and, more in general,
served as a testing ground for future muonium experiments. Based on this
experience, the MicroMegas telescope and the developed analysis tools have
been integrated into the setup which is presently aiming at the measurement of
the 1S-2S transition of M [9] to monitor M production.

We developed detailed Monte Carlo simulations to compare the measured data
with thermal emission models. In order to be able to describe the scattering
of M atoms in residual He gas in a future cryogenic gravity measurement we
calculated elastic M-He scattering cross sections based on a simple Lennard-
Jones model. The theoretical cross sections were consistent with our
measurements of M emission, where different amounts of He gas were introduced
in the vacuum chamber. This measurement placed an upper bound of
$0.3\text{\,}\mathrm{K}$ to the maximum temperature allowable for supporting a
gravity measurement with muonium produced from a superfluid helium source.

###### Acknowledgements.

The authors would like to thank the HIPA facility at PSI for the stable beam
and technical support. Special thanks to the PSI detector group, M.
Hildebrandt, A. Stoykov and F. Barchetti, and U. Greuter from the electronics
group. We are indebted to S. Kamal, G. Marshall, T. Mibe and co-workers for
providing us the laser-ablated aerogel samples, to P. L. Begona at ETHZ for
the zeolite samples, moreover to M. de Volder from Cambridge for the CNT
samples. We would like to thank G. Janka for providing support with the
MicroMegas dectectors. The work of L. Gerchow was supported by the ERC
consolidator grant 818053-Mu-MASS.

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