#
A.M.E.L.I.E.
Apparatus for Muon Experimental Lifetime Investigation and Evaluation

Angelo Maggiora111INFN, Torino Section, retired researcher

###### Abstract

The muon is one of the first elementary particles discovered. It is also known
as heavy electron, and it’s the main component of cosmic rays flux at sea
level. Its flow is continuous, 24h/7d, and it is free. It is natural and does
not have any radio protection banning or limitation to its use in schools and
can be managed safely by the students. We propose to build a light, small and
didactic apparatus to measure the lifetime of the muons. It is useful tool to
introduce the modern physics, particle physics, particles instability and
decay, special relativity etc. It can be used for small didactic but complete
experiments for measurement of muon rate and lifetime, correction and
equalization of data collected etc. A useful instrument to introduce and teach
the scientific method to the students. Last but not least, do not contain any
dangerous system like high voltage or explosive gas and the cost must be
cheap.

###### Contents

  1. 1 Introduction.
  2. 2 The physics of primary cosmic rays.
    1. 2.1 Composition of cosmic rays.
    2. 2.2 Energy of cosmic rays.
  3. 3 Physics of the muons.
    1. 3.1 The decay.
    2. 3.2 The lifetime.
    3. 3.3 The Fermi coupling constant $G_{F}$.
    4. 3.4 Muons in matter.
    5. 3.5 Muons flux and intensity.
    6. 3.6 Muon production and their path.
      1. 3.6.1 Energy loss and interaction with matter.
      2. 3.6.2 Muon decay and absorption.
    7. 3.7 The muon mean free paths and special relativity.
    8. 3.8 Time arrival and Statistics of uniform random events.
  4. 4 Design and optimization of the AMELIE detector.
    1. 4.1 Experimental setup.
    2. 4.2 Scintillator detectors.
    3. 4.3 FLUKA simulations.
      1. 4.3.1 Optimization of the detector
      2. 4.3.2 Number of optical photons vs energy of the muon.
      3. 4.3.3 Uniformity of optical photons collection.
      4. 4.3.4 Muon decay.
      5. 4.3.5 Primary cosmic rays and the muon shower simulation.
    4. 4.4 Front-end electronic.
      1. 4.4.1 TDC characteristics and measure of muon decay time.
    5. 4.5 PSpice simulations.
      1. 4.5.1 Simulation of SiPM signal.
      2. 4.5.2 SiPM Amplitude/Charge measurement.
    6. 4.6 Particles Identification.
  5. 5 Description of AMELIE detector.
    1. 5.1 Front panel.
    2. 5.2 Back panel.
    3. 5.3 Cyclical recording.
    4. 5.4 Adjustment of the operating parameters.
  6. 6 Data acquisition and analysis.
    1. 6.1 Data quality check.
    2. 6.2 Data tacking for SiPM gain $G=5\times 10^{6}$.
    3. 6.3 Background & background subtraction.
      1. 6.3.1 Fake events.
      2. 6.3.2 Negative Muons absorption in nuclear matter.
    4. 6.4 Data analysis for SiPM gain $G=6\times 10^{6}$.
    5. 6.5 Data analysis for SiPM gain $G=7\times 10^{6}$.
    6. 6.6 Summary of the three rounds of data taking.
  7. 7 Conclusions.

##  1 Introduction.

Our earth is continuously bombarded by a stream of elementary particles called
cosmic rays. They come mainly from the sun but also from deep cosmic space,
supernovae, pulsar etc. The cosmic rays were discovered in 1912 by Victor Hess
by bringing an electroscope in balloon ascents, see Fig 1 left.

Figure 1: Left: Austrian physicist Victor Hess on a hydrogen balloon ascension
up to 5300m, 1912 ; Right: the cosmic rays shower

In the November, 1912 issue of the German journal Physikalische Zeitschrift,
Hess suggested:

”The results of my observation are best explained by the assumption that a
radiation of very great penetrating power enters our atmosphere from above.”

The mechanism of formation of cosmic rays arriving on earth is now well known,
after more than 100 years of research, although their origin is still the
subject of research. The composition of primary cosmic rays is: about 90%
simple protons (i.e., hydrogen nuclei); 9% alpha particles, identical to
helium nuclei; and 1% the nuclei of heavier elements. These fractions vary
highly over the energy range of cosmic rays. A very small fraction are stable
particles of antimatter, such as positrons or antiprotons [1] [2]. Further
details are shown in Chap. 2.1.

When it reach the stratosphere, the proton interacts with some nucleus of our
atmosphere and produces a shower of pions, gammas, neutrinos etc. see Fig 1.
The only particle that survives at sea level, thanks to relativistic time
dilation, is the muon.

The purpose of this paper is to describe an innovative particle detector
called A.M.E.L.I.E. (Apparatus for Muon Experimental Lifetime Investigation
and Evaluation), dedicated to measuring the average lifetime of muons that
stop within it. We will show how the design and construction of this didactic
detector, small, simple, inexpensive, allows high school students to approach
and understand how physics measurements are made.

In this way, they are taught the principles of the Galilean scientific method,
the collection of data, their possible correction, the analysis and the fit of
the data using mathematical functions and the rudiments of statistics.

The paper is organized as follows: In Chap. 2, cosmic ray physics is briefly
covered, followed by the physics of the muon in Chap. 3. AMELIE’s conceptual
design is described in Chap. 4, together with the fluka simulations of muon
decay time, the optical collection, uniformity and electronic simulations
using PSPice. This is followed by chapters on the description of data taking,
quality check, data analysis and muon physics in nuclear matter, see Chap. 6.
Finally, Chap. 7 contains the conclusions and general considerations on the
purpose, quality and educational value of the apparatus.

##  2 The physics of primary cosmic rays.

Cosmic rays are high energy charged particles, originating in outer space,
that travel at nearly the speed of light and strike the Earth from all
directions.

The term ”cosmic rays” usually refers to galactic cosmic rays, which originate
in sources outside the solar system, distributed throughout our Milky Way
galaxy.

However, this term has also come to include other classes of energetic
particles in space, including nuclei and electrons accelerated in association
with energetic events on the Sun, solar flares etc.

Since cosmic rays are charged, positively charged protons or nuclei or
negatively charged electrons, their paths through space can be deflected by
magnetic fields (except for the highest energy cosmic rays). On their journey
to Earth, the magnetic fields of the galaxy, the solar system, and the Earth
scramble their flight paths so much that we can no longer know exactly where
they came from.

###  2.1 Composition of cosmic rays.

Figure 2: Cosmic ray elemental abundances compared to abundances in present-
day solar system material. Abundances are normalised to Si=103.

Cosmic rays include essentially all of the elements in the periodic table;
about 89% of the nuclei are hydrogen (protons), 10% helium, and about 1%
heavier elements. The relative composition of primary cosmic rays is shown in
Fig. 2. A broad and exhaustive description is provided by [3].

Cosmic ray abundances are from AMS-02 (H,He) ([4], [5]), ACE/CRIS (Li-Ni)
([6], [7]), and TIGER/SuperTIGER (Cu-Zr) ([8], [9]). Solar system abundances
are from Table 6 of Ref. [10].

The common heavier elements (such as carbon, oxygen, magnesium, silicon, and
iron) are present in about the same relative abundances as in the solar
system, but there are important differences in elemental and isotopic
composition that provide information on the origin and history of galactic
cosmic rays.

Electrons constitute about 1% of galactic cosmic rays.

###  2.2 Energy of cosmic rays.

The range of energies encompassed by cosmic rays is truly enormous, starting
at about $10^{7}$ eV and reaching $10^{20}$ eV for the most energetic cosmic
ray ever detected, which can be millions of times the energies reached at the
LHC.

The energies of the order of $10^{20}$ correspond to the energy of a golf ball
falling from a table, but, in this case, this energy is concentrated in a
single particle with a diameter of less than $10^{-15}$ m.

Figure 3: Left: The cosmic ray energy spectrum clearly shows that the cosmic
ray flux drops off dramatically as we go to higher energies. The spectrum
exhibits a ‘knee’ and an ‘ankle’, both of which deviate from the standard
exponential decline (blue line) [11]. Right: Spectrum of muons at
$\theta=0^{o}$ with the exception that hollow diamond markers represent data
at $\theta=75^{o}$. Different markers represent data obtained from different
experiments. The solid line plots Eq. 15. This graph is obtained from the
particle data group [12].

By plotting this range of energies against the number of cosmic rays detected
at each energy we generate a cosmic ray spectrum which clearly shows that the
number of cosmic rays drops off dramatically as we go to higher energies, see
Fig 3.

Roughly speaking, for every 10 increase in energy beyond $10^{9}$ eV, the
number of cosmic rays per unit area falls by a factor of 1,000. However, if we
look at the spectrum more closely we can see a knee at $10^{15}$ eV and an
ankle ùat $10^{18}$ eV.

The origin of these changes in the steepness of the spectrum is still the
subject of intense study, but it is assumed that they distinguish between
populations of cosmic rays originating via different mechanisms. Current
suggestions are that the cosmic rays with energies less than about $10^{10}$
eV are primarily solar cosmic rays produced in solar flares and coronal mass
ejections, while those with energies between $10^{10}$ eV and the knee at
$10^{15}$ eV are galactic cosmic rays produced in the shocks of supernova
remnants. The origin of the cosmic rays with energies between the knee and
ankle is unclear. They are again thought to be produced within the Galaxy, but
the energies are too high for them to be accelerated by the shocks of
supernova remnants. The origin for the ultra-high energy cosmic rays (UHECRs)
below the ankle is also a mystery, but many have suggested that these may be
created outside of our Galaxy.

As previously mentioned, the primary cosmic rays interact with the nuclei of
the stratosphere, rarefied, and generate a shower of particles. At sea level,
all the particles are muons from the $\pi$ decay.

$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ (1)

$\pi^{-}\rightarrow\mu^{-}+\overline{\nu}_{\mu}$ (2)

The energy of the mouns is shown in Fig. 3, right.

##  3 Physics of the muons.

The fundamental constituent of the matter are shown Fig. 4. They can be
divided into two categories called hadrons and leptons, to which five bosons
mediating the interactions are added. The figure does not include the
“graviton” although physicists believe it should be added to existing bosons,
and a unified theory including gravity has not yet been formulated.
Gravitational waves , which are a manifestation of the existence of the
“graviton”, have only been discovered in recent years.

Figure 4: The Standard model, the basic constituents of the matter.

The hadrons are defined as the particles that interact through strong
interaction and they are constituted by quarks. Quarks are bounded up into
hadrons, forming the baryons (semi-integer spin) such as protons and neutrons
and the mesons (integer spin) such as the pions and kaons.

The leptons category include electrons, muons and their associated neutrinos.
As the electrons, the muons can carry a positive or negative electric charge
($\mu^{+}$ or $\mu^{-}$).

As mentioned earlier, muons are abundantly produced in the atmosphere by
cosmic rays and, because they are quite penetrating, they can reach the
ground, enter the houses or schools through the walls or roof of the building,
and be detected with a suitable apparatus.

Muons were first detected and investigated by Bruno Rossi in the 1930s and
1940s [13] and in 1947 by Neddermeyer and Anderson [14] in surveys on cosmic
rays.

Initially the muon particle was associated with the Yukawa particle postulate
in 1935, the Yukawa’s mesotron (later denominated as $\pi$ meson), a force
carrier of the strong interaction. But it was demonstrated in 1947 that the
muon did not interact through the strong interaction.

Consequently, the muons could not be the Yukawa $\pi$ meson. The discovery of
the Yukawa’s particle, the $\pi$ meson was made in 1947 by Lattes, Ochialini,
and Powel [15] in the cosmic rays using emulsion techniques. Around one year
later Lattes at Chacaltaya-Bolivia (5200 m above sea level) obtained the first
experimental evidence of the $\pi\rightarrow\mu$ decay [16].

The rest mass of the muon is:

$M_{\mu}=105.66\hskip 5.69054ptMeV/c^{2}$ (3)

approximately 207 times the mass of the electron. See Particle Data Group
report [3].

###  3.1 The decay.

The muon travels a relatively long distance (muons are quite penetrating and
they can reach the ground) while losing its energy and it decays by the weak
interaction into an electron plus a neutrino and an anti-neutrino

$\mu^{-}\rightarrow e^{-}+\overline{\nu}_{e}+\nu_{\mu}$ (4)
$\mu^{+}\rightarrow e^{+}+\nu_{e}+\overline{\nu}_{\mu}$ (5)

$\mu^{-}$$\nu_{\mu}$$\overline{\nu}_{e}$$e^{-}$$W^{-}$

$\mu^{+}$$\overline{\nu}_{\mu}$$\nu_{e}$$e^{+}$$W^{+}$

Figure 5: Left: Feymann diagrams of $\mu^{-}$ and, Right, $\mu^{+}$ decay

The branching ratios for these modes (called as normal modes) are near 99%.
Even so, there are some experimental evidences for other modes as the
radioactive muon decay observed specially in negative muons with energies
above 10 MeV.

$\mu^{-}\rightarrow e^{-}+\overline{\nu}_{e}\nu_{\mu}+\gamma$ (6)

with a branching ratio around 1.4%. There are also other (SUSY-GUT) decay
modes called as exotic. In this case, only the upper bound of the branching
ratios has been observed, and they are less than 10-11 at 90% confidence
level.

###  3.2 The lifetime.

Decays of unstable nuclei and particles, including muons, are characterized by
the decay probability in a unit of time, $\lambda$. The mean lifetime of the
particle, $\tau$, is defined as $\tau=1/\lambda$

Decays follow the exponential radioactive decay law:

$N(t)=N_{0}\exp^{-t/\tau}$ (7)

where $N_{0}$ is the number of unstable particles at the beginning, $t=0$, and
N(t) is the number of particles that survive until time t.

We know, see Particle Data Group report [3], that the lifetime of the muon in
vacuum and at rest is:

$\tau_{0}=2.1969811\pm 0.0000022\hskip 5.69054pt\mu s$ (8)

###  3.3 The Fermi coupling constant $G_{F}$.

The Fermi coupling constant $G_{F}$, was introduced by Fermi in order to
phenomenologically ther strength of the new “weak force”.

Any renormalizable theory becomes predictive only once it has been supplied
with a sufficient number of experimental inputs so as to fix the free
parameters that appear in its Lagrangian. In order for the subsequent
theoretical predictions to be as accurate as possible, these inputs are chosen
from the experimentally best-measured quantities available. In the case of the
Standard Model of electroweak interactions these are the electromagnetic
coupling constant, $\alpha$, the Fermi coupling constant, $G_{F}$, and the
mass of the $Z_{0}$ boson, $M_{Z}$. Their recognized best values, along with
their absolute and relative errors that they represent are:

$\alpha=1/(137.0359895\pm 0.0000061)$ (9) $G_{F}=(1.16639\pm 0.00002)\times
10^{-5}\hskip 14.22636ptGeV^{-2}$ (10) $M_{Z}=91.1867\pm 0.0021\hskip
14.22636ptGeV$ (11)

The Fermi coupling constant is obtained from the muon lifetime, $\tau_{\mu}$,
via a calculation in the Fermi model, in which the weak interactions are
represented by a contact interaction.

The Feynman amplitude $\mathcal{M}$ describes the transition amplitude of the
process. It is derived from the Feynman rules for interaction vertices and the
propagators involved. The Feynman amplitude $\mathcal{M}_{F}$ for Fermi four-
point interaction, as shown in Fig. 6, is:

$\mathcal{M}_{F}\approx\sqrt{2}G_{F}$ (12)

$\mu^{+}$$\bar{\nu}_{\mu}$$\nu_{e}$$e^{+}$$G_{F}$ Figure 6: Feynman diagram
for muon decay in the Fermi effective theory

where $G_{F}$ is the Fermi constant which was introduced by Fermi to
phenonemologically describe the strength of the weak force.

The Fermi interaction is in reality not point-like but that only appear point-
like due to the fact that the weak interation is mediated by weak bosons with
very large masses according to the Standard Model shown in Fig. 5.

Using the full Feynman amplitude, the muon lifetime can be calculated as:

$\tau^{-1}_{\mu}=G^{2}_{F}\frac{m^{5}_{\mu}}{192\pi^{3}}$ (13)

which can be used to determine the Fermi coupling constant, $G_{F}$, of the
weak interaction.

###  3.4 Muons in matter.

Figure 7:  Muon charge ratio as a function of the muon momentum from Refs.
([17], [18], [19], [20], [21]).

The muons are produced by primary cosmic rays interacting with nucleons of the
upper atmosphere, see Chap. 2. Due to the different charge of primary protons,
electrons, neutrons or heavy nuclei, the number of positive and negative
charged muons differs at sea level and the ratio is shown in Fig. 7.

###  3.5 Muons flux and intensity.

The average muon vertical flux is latitude dependent. At sea level and for
magnetic latitudes above 45°, the intensity is $1.1\times 10^{-2}$
muons/cm2/sr/sec which corresponds to a flux through a horizontal area of
$1.8\times 10^{-2}$ muons/ cm2/sec [4], with a angular distribution
approximately $cos^{2}(\theta)$ dependence, with $\theta=0$ at the zenith.
This works out to a good rule of thumb of about 1 muon/cm2/minute.

###  3.6 Muon production and their path.

As said before, the muons are produced by the interactions of protons with the
nuclei of the atoms that constitute our atmosphere, principally nitrogen,
oxygen and argon.

The high energy cross-section for inelastic proton scattering on atomic
nitrogen is $2.65\times 10^{-25}$ cm2.

The nuclear interaction length is:

$\lambda_{n}\approx 35\times A^{1/3}\>[g\;cm^{-2}]$ (14)

Averaging over the atomic constituents of air, the mean free path for
inelastic interactions is 89 g/cm2. Our atmosphere pressure at sea level, one
interaction length $\lambda$, is 1033 g/cm2. Thus, a primary proton must
traverse 11.6 interaction lengths to arrive at the Earth’s surface intact.

This means that the muons observed at ground level, the most likely height for
production will be in the neighborhood of the first interaction length, where
63.2% of interactions take place.

The vertical distribution of the atmosphere is approximately exponential with
a scale height, $h_{0}$, of 7600 m.

Thus, one interaction length depth in the atmosphere corresponds to a height,
h, determined by: $89=1033\times e^{(-h/h_{0})}$.

Solving, we find h must be 18,600 m above sea level.

####  3.6.1 Energy loss and interaction with matter.

During their time of flight, muons can lose energy through Coulomb scattering,
ionization loss, Compton scattering and Bremsstrahlung. Since muons are
massive, they mostly lose energy through ionization. On average, cosmic ray
muons lose 2 GeV throughout their course towards sea level to ionization [22].
Upon reaching ground, the energy spectrum of muons are analytically derived in
Eq. 15 and experimentally shown in Fig. 3.

$\frac{dN_{\mu}}{dE_{\mu}d\Omega}\approx
0.14E_{\mu}^{-2.7}[cm^{2}ssrGeV]^{-1}\times[\frac{1}{1+\frac{1.1E_{\mu}\cos\theta}{115GeV}}+\frac{0.54}{1+\frac{1.1E_{\mu}\cos\theta}{850GeV}}]$
(15)

The flux of cosmic ray muons depends on the incident zenith angle. In
particular, the relation is

$I(\theta hE)=I(0^{o})\cos^{n(Eh)}\theta$ (16)

where $\theta$ is the zenith angle, h is the vertical distance traveled by the
muon, E is the energy of the muon, and $n(E,h)$ is an empirically determined
constant. At sea level experiments showed that $n\approx 2$.

When a charged particle passes through matter, it can lose energy and be
deflected from its incident direction. For charged heavy particles, such as
muons, these effects are primarily due to inelastic collisions with atomic
electrons of the material. The amount of energy transferred during every
collision is small, but in dense media the interaction cross section can be
large and many collisions per path length can occur, so the cumulative effect
can lead to a substantial energy loss.

A quantum-mechanical description of how this energy loss relates to its
relevant quantities is shown mathematically by the Bethe-Bloch formula that
describes the stopping power of a material for different incident particles.
For experimental purposes this formula was slightly altered to include
realistic experimental effects. The formula is the following:

$-\frac{dE}{dx}=2\pi
N_{a}r^{2}_{e}m_{e}c^{2}\rho\frac{Z}{A}\frac{z^{2}}{\beta^{2}}[ln(\frac{2m_{e}\gamma^{2}v^{2}W_{max}}{I^{2}})-2\beta^{2}-\delta-2\frac{C}{Z}]$
(17)

with

$2\pi N_{a}r^{2}_{e}m_{e}c^{2}=0.1335MeVcm^{2}/g$ (18)

The maximum energy transfer is produced by a knock-on collision . For an
incident particle of mass M:

$W_{max}=\frac{2m_{e}c^{2}\eta^{2}}{1+2s\sqrt{1+\eta^{2}}-s^{2}}$ (19)

where $s=m_{e}/M$ , M is the mass of an incident particle, and
$\eta=\beta\gamma$. For the case of muons, since the mass of a muon
$M=m_{\mu}$ is much greater than the mass of an electron $m_{e}$ , we have:

$W_{max}\approx 2m_{e}c^{2}\eta^{2}$ (20)

The Bethe-Bloch formula is plotted in Fig. 8 as a function of momenta of
incident particles. The curve is decreasing at lower energies until a certain
point where the curve becomes relatively constant. Particles at these momenta
where the value of $dE/dx$ is minimal are call minimum ionizing particles.
Their energy deposition per unit distance traveled over a piece of material is
relatively constant. Cosmic ray muons travel at relativistic speeds at ground
level, and are typically minimum ionizing particles. A commonly used
approximation for $dE/dx$ is 2 MeV/(g/cm2).

Figure 8: Energy dependence of the energy deposition per distance traveled of
different charged particles in different media.

####  3.6.2 Muon decay and absorption.

The positive and negative muons have a different behaviour after they are
brought to rest in matter. While the positive muon has in matter only the
possibility to decay, the negative muon can replace one of the outermost
electrons in the atom, having the same charge, and thus be captured by the
host atom forming a muonic atom.

Just like electrons, negative muons fall on stable Bohr levels with radius
$r_{\mu}$

$r_{\mu}=\frac{(n\hbar)^{2}}{M_{\mu}^{red}Ze^{2}}$ (21)

where $M_{\mu}^{red}$ is the reduced mass of the muon-nucleus system and Z is
the atomic number of the host element. The mass of a muon is about 200 times
larger than that of an electron. This results in a Bohr radius 200 times
smaller than for the corresponding electron level with the same main quantum
number. Consequently, there is a non vanishing overlap of the wave function of
the muon with the wave function of the nucleus and this can lead to a capture
of the negative muon by the nucleus. For$Z\geq 40$ the Bohr orbit is already
inside the nucleus.

Due to its different flavour, there is no exclusion by the Pauli principle for
the muon and electrons and, within a short time $10^{-13}$s, it cascades to
the 1s level.

In this state, there is a considerable overlap of the muon wave fnnction with
the nucleons. The muon has now the possibility either to decay or to be
captured by the nucleons.

The capture process is described by the Feynman diagram show in Eq. (22) ad
Feymann diagram 9 .

$\mu^{-}+p\rightarrow\nu_{\mu}+n$ (22)

$u$$u$$d$$d$$u$$d$$\mu^{-}$$\nu_{\mu}$$W^{\pm}$$p$$n$

Figure 9: Feynman diagram of a negatively charged muon captured by a proton.
The proton is transformed into a neutron and a muonic-neutrino is emitted. The
W-boson is the mediator of the interaction.

The competition between these processes reduces the mean lifetime of the
observed decay of the negative muon as compared with the free decay. Only
muons not already captured in the orbit can decay. Expressed in terms of
process rates $\Lambda=1/\tau$, the total disappearance probability is written
as the sum of the capture and of the decay probabilities (either one or the
other):

$\Lambda_{total}=\Lambda_{decay}+\Lambda_{capture}$ (23)

The nuclear capture rate is determined by the overlap of the muonic orbit with
the nucleus and it follows roughly a power law of the atomic number
($\Lambda_{capture}\sim Z^{4}$), [23].

In compounds, the nuclear capture is preferred by the nucleus with the larger
nuclear capture rate, i.e. larger atomic number. For instance, in a plastic
scintillator material (polyvinyl-tolnol, 2-CH3C6H4CH=CH2 monomer) only the
carbon capture is significant.

The capture probability for this condition is very close to 1. For light
nuclei, how-ever, with charge number of $\approx 10$ or less, also negative
muons will preferentially decay in an analogue fashion to $\mu^{+}$ decay.

The decay probability for free muons at rest, $\lambda=1/\tau$, is the same
for positive and negative muons.

$\lambda^{+}=\lambda^{-}\hskip 11.38109pt$or$\hskip
11.38109pt\tau^{+}=\tau^{-}$

If negative muons are stopped in matter the measured decay probability
$\lambda^{-}_{m}$ is always larger than the decay probability $\lambda^{+}$ or
$\lambda^{-}$ for free muons. So the equation

$\lambda^{-}_{m}>\lambda^{+}=\lambda^{-}$

holds or, respectively

$\tau^{-}_{m}<\tau^{+}=\tau^{-}$

In the case of muon capture the process competes with free decay. The
$\mu-$capture shortens the effective lifetime of $\mu$. It results to:

$\frac{1}{\tau}=\frac{1}{\tau_{0}}+\frac{1}{\tau_{c}}$ (24)

where $\tau_{c}$ is the $\mu-$capture lifetime.

Because the muons produced by cosmic radiation are a mixture of positive and
negative muons, one expects the following time behavior for the number of
disintegrated muons:

$N(t)=N(\mu^{-},t_{0})e^{-(t-t_{0})/\tau_{0}}e^{-(t-t_{0})/\tau_{c}}-N(\mu^{+},t_{0})e^{-(t-t_{0})/\tau_{0}}$
(25)

because the positive muons do not undergo capture.

Z | nucleus of material | average lifetime $\tau_{\mu}[\mu s]$ | capture rate $\times 10^{3}(s^{-1})$
---|---|---|---
| $\mu^{+}$ | $2.1969811\pm 0.0000022$ | 455.16
1 | ${}^{1}H$ | $2.19490$ | 0.450
12 | ${}^{12}C$ | $2.028$ | 37.9
13 | ${}^{27}Al$ | $0.864$ | 705.
20 | Ca | $0.334$ | 2546.
82 | Pb | $0.0748$ | 12985.
Table 1: Some total capture rates and meanlife for $\mu^{-}$ in nuclei [23].
See also Fig. 10.

The capture of negative muons by nuclei is accompanied by secondary effects,
see [24]. During the electromagnetic cascade down to the 1s level, photons by
radiative transitions and Auger electrons are emitted. For heavy nuclei,
nuclear Auger effect (neutron emission from nucleons) or prompt fission may
occur also. The total capture rate is not easy to calculate as the final
nucleons is excited to an unknown energy, which is the critical parameter from
the theoretical point of view. Nuclear gamma transitions, neutron evaporation
or even charged particle and cluster emission can follow the muon capture.

Figure 10:  Mean lifetime of negative muons in different materials measured in
[25]

However, from the experimental point of view, the measurement is straight-
forward, the determination of the muon lifetime after stopping in the relevant
material is simple.

###  3.7 The muon mean free paths and special relativity.

The first approach to measure the travel time and the distance of muons
produced by primary cosmic rays is the Newtonian dynamics. We can consider the
muon lifetime at rest given by Eq. 8, the speed of secondary muons close to
the speed of light, we find the distance d that the muon can travel:
$d=c\tau=660m$, (see Eq. 28) incompatible with the distance of 10 - 20 km from
the earth’s surface where the interactions of primary cosmic rays take place.
With the Newtonian dynamics, can also calculate the probability of survival of
secondary muons:

$\displaystyle t=\frac{d}{c}$ $\displaystyle t=\frac{15\times 10^{3}}{3\times
10^{8}}=5\times 10^{-5}s$ $\displaystyle N(t)=N_{0}e^{\frac{t}{\tau}}$
$\displaystyle\frac{N}{N_{0}}=e^{-\frac{5\times 10^{-5}}{2.2\times
10^{-6}}}=1.3\times 10^{-10}$

This is clearly incompatible with reality. One has to use special relativity
[26] and Lorentz transformations in cases where the speed is close to that of
light.

To obtain the mean range of a fast muon, we must take into account the time
dilation and distinguish between apparent life and proper life. The lifetime
of a muon is equal to $\tau_{0}$ in the rest system of the muon (8). Due to
relativistic time dilation, a muon with a velocity $v$ appears to have a
lifetime

$\tau^{\prime}=\tau_{0}\gamma\hskip 14.22636pt;\hskip
14.22636pt\gamma=\frac{1}{\sqrt{1-(v/c)^{2}}}$ (26)

The mean range of a muon with velocity $v$ is therefore:

$R(v)=v\tau^{\prime}=v\tau_{0}\gamma$ (27)

or

$R(v)=\tau_{0}\frac{p}{m_{\mu}}=\frac{p}{m_{\mu}c}\tau_{0}c\hskip
5.69054pt;\hskip 5.69054pt\tau_{o}c=0.66km$ (28) $\displaystyle p=mv$
$\displaystyle m=\frac{m_{\mu}}{\sqrt{1-(v/c)^{2}}}=m_{\mu}\gamma$
$\displaystyle L=\frac{L_{0}}{\gamma}$ $E=\gamma mc^{2}$ (29)

with $L_{0}$ proper length and $L$ observed length.

If we consider an energy of 20 GeV Fig. (3) we can find $\gamma$ from equation
(29). We also need to consider the rest mass of muons which is m =
$105.7\times 10^{6}$ eV:

$\gamma=189.2$ (30)

and, then, the muon proper distance L, given by Eq. 28 is:

$x^{\prime}=\frac{20000}{189.2}=105.7\>[m]$ (31)

consequently, the percentage of muons that will reach the ground by using the
proper time $\gamma\tau$ is:

$\frac{N}{N_{0}}=e^{-\frac{5\times 10^{-5}}{189.2\times 2.2\times
10^{-6}}}=0.89$ (32)

i.e. a significant fraction of the muons produced in the stratosphere are
capable to reach the earth surface.

It is evident that the muon, produced in the stratosphere at about 10-30 km
high, reaches the earth only thanks to the dilation of its average life time
caused by special relativity.

###  3.8 Time arrival and Statistics of uniform random events.

The arrival of cosmic rays is a random process. So we expect it to follow the
Poisson distribution for rare events:

$P(n,t)=\frac{(rt)^{n}e^{-rt}}{n!}$ (33)

where r is the average event rate, namely the average number of events
percentage unit time.

Figure 11: Left: Poissonian distribution at different average values
$\lambda$; Right: time between two consecutive events for average rate 4 s-1

Then the probability of observe n events in the time interval $\Delta$t is
given by Eq. 33 and show in Fig. 11.

##  4 Design and optimization of the AMELIE detector.

The basic idea that guided the design of the AMELIE detector was to measure
the time between the arrival of a muon and its decay in a plastic scintillator
block, see Fig. 12. The start signal is given by the ionization produced by
the incoming muon, while the stop signal is given by the electron/positron
produced by its decay.

###  4.1 Experimental setup.

The experimental setup, shown in Fig. 12, consists of a cube of plastic
scintillator of dimensions $15\times 15\times 15\;cm^{3}$, coupled to a solid
state Silicon Photo Multiplier (SiPM).

Figure 12: Scheme of the detector setup. The apparatus is based on a cube of
scintillator detector of dimensions $15\times 15\times 15\;cm^{3}$. The
$6\times 6\;mm^{2}$ SiPM detector is placed on one corner cut at $45^{o}$.

The size of the scintillator is a trade-off the counting time to get reliable
results and the cost of the scintillator. Size can be increased to benefit
data acquisition time, increasing cost. Instead of the plastic scintillator, a
large volume of liquid scintillator, enclosed in an airtight light-proof bin,
can be used. The same SiPM, facing the liquid scintillator, would constitute
the photon detector.

The scintillator is a general purpose plastic scintillator like PS-923
produced by DETECT [27].

The Silicon Photo Multiplier (SiPM) is a new sensor
MICROFICHE-60035-SMUT-TR , produced by ON SEMI [28].

Its active area is $6\times 6mm^{2}$, high quantum efficiency and gain are
shown in Fig. 13. From the figures, it is clear that the quantum efficiency of
the SiPM matches perfectly with that of the scintillator.

Plastic scintillator and SiPM have been covered by a thin sheet of aluminized
mylar, covered, in turn, by a black box made from black cardstock and black
adhesive tape. The aluminized mylar sheet reflects light inwards to maximize
the number of photons hitting the SiPM in order to maximize light collection.
It is placed on a edge surface cut at 45o. This position maximizes the
uniformity of light collection throughout the volume.

The output of the small frontend SiPM card consist of 4 wires: 1) $V_{bias}$;
2) SiPM output; 3) temperature sensor; 4) ground.

The front-end electronic, described in detail in Sec. 4.4, is placed on the
top of the scintillator block. It contain a $V_{bias}$ DC-DC converter,
threshold and Sample&Hold circuits. The frontend electronic is controlled by a
microcontroller based on ESP32. The data are shown on a OLED display and
recorded on a $\mu$SD card for off-line evaluation.

Figure 13: Left: relative light output of BC-400 scintillator; Center: Quantum
efficiency of Onsemi 60035 SiPM; Right: SiPM gain versus overvoltage
$V_{bias}=B_{reakDown}+O_{verVoltage}$

Although the detector is small in size, compact and cheap, it has all the
characteristics of any modern physics experiment, such as those at CERN on the
LHC accelerator.

In particular:

  * •

uses detectors of innovative design, widely used in modern physics
experiments;

  * •

coupled with state-of-the-art sensors;

  * •

modern, high-performance front-end electronics and digitization of incoming
data;

  * •

has a microcontroller, which is a computer with which essential parts of an
experiment are carried out:

    * –

data recording on $\mu$SD, the modern version of magnetic tapes, for off-line
data analysis;

    * –

slow control, i.e., cyclic control of system operating parameters;

    * –

display and system monitor during data taking;

    * –

ability to set and change detector operating parameters.

###  4.2 Scintillator detectors.

Scintillators are charged particle detectors widely used in modern and past
physics experiments in the following detectors:

  * •

hadron and electron calorimetry;

  * •

time of flight measurements;

  * •

trigger counters;

  * •

veto counters;

  * •

particles identification by number of photons.

There are three types of organic scintillators: 1) crystalline; 2) liquid; 3)
plastic.

In Amelie, the plastic scintillator was used, but the system can be easily
adapted to a liquid scintillator.

The working principle is the same for all. Charged particle through a medium
generates a track of excited molecules. It, loses energy by ionization,
emission of delta rays etc., see Fig. 14. Some type of molecules releases a
small fraction ($\approx$3%) of energy as optical photons. The scintillation
process is particularly marked in substances containing aromatic rings
(polystyrene, polyvinyltoluene etc.)

The excitation is generated by the absorption of a photon while de-excitation
generates the emission of a longer wavelength photon. The effect of wavelength
difference between absorption and emission peaks is called Stokes’ shift
(greater Stokes’ shift is better as it minimize self-absorption). Fluors are
used as “waveshifter”.

The dE/dx converted into visible light detected by photosensors.

Figure 14: Left: Plastic scintillator; Right: excited level of organic
compounds

The most common bases used in plastic scintillators are the aromatic plastics,
polymers with aromatic rings as pendant groups along the polymer backbone,
amongst which polyvinyltoluene (PVT) and polystyrene (PS) are the most
prominent.

With a density range of a plastic scintillator from 1.03 to 1.2 gr cm3, the
photon yields are about 1 photon per 100 eV of energy deposit [4]. This means
that a muon of 50 MeV that is arrested in the block, see Chap. 4.3, emits
500000 photons in all directions.

light output | 56 % of antatacene
---|---
rise time | 0.9 ns
decay time | 3.3 ns
bulk attenuation length (BAL) | 250-450 cm
technical attenuation length (TAL) | 150-250 cm
wavelength of max emission | 418 nm
n. of H atoms per cm3 | $5.17\times 10^{22}$
n. of C atoms per cm3 | $4.69\times 10^{22}$
n. of electrons per cm3 | $3.33\times 10^{23}$
density | 1.023 g/cm3
Table 2: Physical characteristic of DETEC P923A plastic scintillator.

###  4.3 FLUKA simulations.

The FLUKA code [29] is a general purpose Monte Carlo code for the interaction
and transport of hadrons, leptons, and photons from keV (with the exception of
neutrons, tracked down to thermal energies) to cosmic ray energies in any
material. It has many applications in high energy experimental physics and
engineering, shielding, detector and telescope design, cosmic ray studies,
dosimetry, medical physics and radio-biology.

The history of FLUKA goes back to 1962-1967, when J. Ranft was working at CERN
on hadron cascades. Its development has never ceased since then and, today,
continues as part of a collaboration between CERN and INFN.

Fluka has been used extensively in Compass to evaluate the environmental
irradiance caused by the beam and secondary particles in the various
measurements made from 2000 to the present.

As part of the AMELIE project, Fluka has been used for:

  * •

optimization of the detector;

  * •

evaluation of counting rate by simulation of primary cosmic rays and muon
shower.

####  4.3.1 Optimization of the detector

In order to optimize the experimental setup and to find the optimal
configuration, the experimental setup, shown in Fig. 12, has been extensively
simulated using Fluka.

The following simulations has been done:

  * •

optimization of the collection of optical photons by the SiPM placed on one
corner of the scintillator cube;

  * •

study of the energy loss of low energetic muons on the scintillator cube;

  * •

study of the number of optical photons vs muon energy and muon stopped in the
scintillator.

The material of the scintillator is P923A, with a density of $1,032gr/cm^{3}$,
with a composition of H 0.084757, C 0.915243.

The $\mu^{-}$ source simulating cosmic muons was placed along the Z axes at Z
= -10 cm.

_Limit of Fluka simulations_

However, one must keep in mind that Fluka, like any simulation program, has
inherent limitations. The dependence of the number of photons on wavelength in
the scintillator, shown in Fig. 13 left, is not considered. It is simplified
with a mono energetic photon of 2.9 nm.

In any case, although the absolute values on the number of photons collected
by the SiPM cannot be considered reliable, we believe that the relative ratios
are sufficient for the purpose of this study.

####  4.3.2 Number of optical photons vs energy of the muon.

For this simulations, the source of $\mu^{-}$ was placed at the center of the
cube, X = 0 ; Y = 0 ; Z = -2.5cm from the top surface.

Figure 15: Left: energy loss in the scintillator cube for muons that cross
completely the cube. Below 60 MeV the muons are stopped in the cube; Center:
Bragg peak for muons stopped in the cube, 40, 50, 60 MeV, and crossing the
cube 70 MeV ray flux; Right: number of photons collected by SiPM.

In agreement with the increase of Stopping Power at low energies, shown in
Fig. 8, the energy loss of muons during the path increase at low energies.

Below an energy of 60 MeV, all $\mu^{-}$ are stopped in the scintillator block
and the Bragg peaks are clearly shown in Fig. 15, center.

Consequently, also the number of optical photons generated by the muon
increases at low energies, increase from about 150 photons/primary per muon at
the minimum of ionization (MIP) to 300-360, as shown in Fig. 15, right.

It is evident that, by suitably raising the threshold with respect to the MIP
muons, it is possible to distinguish the muons that stop in the scintillator
block from those that cross it. These are the muons of interest for the
measure of their average life.

####  4.3.3 Uniformity of optical photons collection.

For this simulations the energy of the was 50 MeV, low enough to stop the muon
in the block.

The position of the muon beam was at the center and at the four corner of the
block.

The number of photons collected by the SiPM in each position is shown in Tab.
3

X [cm] | Y [cm] | n. of SiPM optical photons
---|---|---
0 | 0 | 352.6
5 | 5 | 315.5
5 | -5 | 302.3
-5 | 5 | 300.3
-5 | -5 | 628.7
Table 3: Number of optical photons collected by SiPM vs position of muon
hitting the TOP side, the same side of SiPM

With the exception of the last position, just in front of the SiPM, the number
of photons is uniform and above 300 photons.

####  4.3.4 Muon decay.

The decay process of the muon is a three body decay shown in Eq. 4 and 5. The
energy corresponding to the mass of the muon at rest, 105.66 $MeV/c^{2}$, is
shared between the two neutrinos and the electron/positron. Of course we
cannot see the two neutrinos as they are not charged. We only see the light
emitted by the shower produced by the charged lepton. This means that the
energetic spectrum of the three particles are continuous with a maximum of 53
$MeV/c^{2}$ for the charged lepton.

Figure 16: Fluka simulations. $E_{\mu^{+}}$ = 50 MeV at x = y = 0. Left:
Energy of positrons from muons decay; Right: Muon decay time.

In Fig. 16 left, the energy of electron/positron is shown. The electron or
positron will produce a shower until the energy is completely absorbed. In
Fig. 17 the energy of the positron shower is shown.

Again, the energy released in the scintillator block is high and the SiPM
threshold can be left higher than minimum ionization particles.

The Fig. 16 right shown the muon decay time from fluka simulations. The fit of
decay time is 2.32$\mu$s

Figure 17: Left: Energy density of muons decay positrons; muons energy =
$50MeV/c^{2}$; position X = 0, Y = 0; Right: cut of the left histogram. The
peak is the Bragg peak of Fig. 15

####  4.3.5 Primary cosmic rays and the muon shower simulation.

This point is of great importance in the project. If the events rate is high,
this must be taken into account in the design of the front-end and acquisition
electronics. If the rate is too low, the measurement may not be feasible in a
reasonable time by the students.

The primary cosmic rays, the shower of secondary particles on Torino (Italy)
(35∘N , 7.7∘E) produced by them, were simulated using Fluka code.

Fluka provides a complete calculation to determine particle fluxes in the
atmosphere at ground level at the geographical location of the chosen
location. It take care of:

  * •

the spectrum and composition of primary cosmic rays at the local interstellar
medium;

  * •

the determination of changing conditions in the solar wind magnetic field and
the resulting interaction with the inward flow of galactic cosmic rays from
the local interstellar medium;

  * •

the determination of the trajectories of cosmic rays through the Earth’s
geomagnetic field;

  * •

the transport of the surviving incident cosmic rays through the Earth’s
atmosphere to various depths.

Fig. 18 shows histograms of primary cosmic and muon shower.

Figure 18: Left: Fluka simulations of primary protons; Right: Muos at ground;
The simulation is centered at the Torino (Italy) position. To be compared with
Fig. 3.

From the simulations, the average $\mu^{+}$ / $\mu^{-}$ ratio, for energy
below 70 Mev/c is:

$R_{\mu^{+}/\mu^{-}}=\frac{N_{\mu^{+}}}{N_{\mu^{-}}}=0.240=24\%$ (34)

and the percentage of $\mu^{+}+\mu^{-}$ with energy below 70 MeV/c over the
total is:

$N_{\mu}=2,2\%$ (35)

70 MeV/c is the limit beyond which muons pass through the scintillator block,
as shown by the simulations in Cap. 4.3.

Taking into account the cosmic rate at sea level is about (rule of thumb)
$r=0.01s^{-1}cm^{2}sr^{-1}$, and the AMELIE cube is $15\times 15cm^{2}$, the
total mouns rate is $2.25s^{-1}$ and the rate of low energetic muons stopped
in the cube is in the order of 2 - 3 per minute.

Altough, in any case, these numbers should be taken with a grain of salt, they
are, however, reasonably indicative of the feasibility of the measure. The
percentage depends strongly on the materials that can slow down muons that
overlie or surround the detector. The simple simulation performed involves
only air above the detector up to the stratosphere.

###  4.4 Front-end electronic.

The front-end electronics is divided into three parts 19. A small PCB that
houses the SiPM attached directly to the scintillator and a second PCB that
houses the analog circuitry, the part that transforms the analog signal
provided by the SiPM to digital, and a third PCB with the remaining
electronics, microcontroller and TDC (Time to Digital Converter).

The front-end electronic is placed on top of the scintillator cube, see Fig.
24.

Figure 19: Scheme of electronic

The microcontroller, based on ESP32 chip, will manage the full front-end
electronic. The current setup and data are shown on a e-paper display. The
working parameters can be modified by a knob and/or push-buttons. Finally, the
data are recorded on a $\mu$SD card for off-line analysis.

The optical photons from stopped muon or decay lepton shower are collected by
a SiPM. A fast current pulse is generated by each photons burst by a fast
trans-conductance amplifier. Since the trans-conductance amplifier is keep to
a reference signal Vref, the output will therefore be negative with respect to
Vref. The transconductance analog signal is converted into a logic signal by
two fast discriminators, VThHigh and VThLow.

The charge of the SiPM signal is proportional to the number of photons
collected. Consequently, the peak of the signal is also proportional to the
number of photons collected. A pick-sensing circuit, followed by sampling-and-
hold, adapts the fast SiPM signal to the microcontroller’s internal ADC.

The gain of the SiPM is controlled by Vbias, generated on-board by DAC and DC-
DC converter, under control of microcontroller. A DC-DC converter will
generate the required Vbias = 25-32 V. The DAC is controlled by the ESP32
using the I2C bus. The noise generated by DC-DC oscillator is filtered by a
onboard dual T filter. The reference voltage Vref is controlled by a stable
“voltage reference” onboard. In order to have a large dynamic range, Vref will
be 4.096 V.

From the Fluka simulations, see Sec. 4.3, the number of photons collected by
SiPM generated by stopped muons or leptons from muon decay are, at least, two
time the photons generated by a high energetic muon that cross the
scintillator block.

To take advantage of this typical feature of muon decay, two discriminators
were installed, one with a low threshold and a second with a higher threshold.

The low threshold discriminator, is suitable to detect minimum ionization
particles, and the second, with high threshold, to detect stopped muon or
decay lepton. The high threshold signal is detected by the microcontroller by
polling the input pin. It is used for time measurements using the TDC.

####  4.4.1 TDC characteristics and measure of muon decay time.

In order to measure the decay time of the muon, we plan to use the TDC7200
chip produced by Texas Instruments [30].

The chip is designed to be used in LIDAR applications or ultrasonic sensing
measurements such as water flow meter, gas flow meter, and heat flow meter.

Its main features are: 1) resolution 55 ps; 2) two ranges: 12 to 500 ns and
250 ns to 8 ms; 3) readout through a SPI bus, TDC time is stored in internal
registers.

The chip require a simple but very stable 8 MHz clock and the device has an
internal self-calibrated time base which compensates for drift over time and
temperature. Self-calibration enables time-to-digital conversion accuracy in
the order of picoseconds. This accuracy makes the TDC7200 ideal for our
application, indeed, in reality, it exceeds the requirements, and the time
resolution is downgraded to 1 ns.

The TDC has separated START and STOP input, while, in our case, we have a
single SiPM which generates both the START and STOP signals.

A little logic had to be built to separate the two signals, as shown in Fig.
19.

It is evident that the measurement of time between two signals constitutes the
distinguishing feature of this detector. Special care was taken in the testing
and calibration of the TDC by designing and constructing a signal pulser
capable of generating a double pulse in the range provided by the gate. The
circuit, shown in Fig. 20 emulates the SiPM by injecting a charge equivalent
to the transconductance amplifier.

Figure 20: Test circuit capable of generating a double pulse at the input of
the front-end board.

The TDC is designed to generate an interrupt signal to the microcontroller
only when it receives a STOP signal. This interrupt signal is not used by
microcontroller.

Only the high threshold signal is used to generate TDC START and STOP signals.

The START will also trigger a not-retriggerable flip-flop gate with a signal
width of 12 $\mu$s. This signal inhibits the generation of a second START
signal and allows the generation of a STOP signal if it arrives in the gate
time.

The TDC and charge ADC are then readout by microcontrolled and the system is
set for the next event.

###  4.5 PSpice simulations.

The front-end electronics has been extensively simulates using LTspice in
order to optimize the active and passive components. LTspice was developed by
Linear Technology, now acquired by Analog Devices [31]. It is a graphical
interface to the well known SPICE, Simulation Program with Integrated Circuit
Emphasis, a circuit simulation program developed in 1975 by Electronics
Research Laboratory California University in Berkeley.

####  4.5.1 Simulation of SiPM signal.

The current generated from 1 to 40 photo electrons in the scintillator, varies
between 13 and 512 $\mu$A.

The analog parameters of the various components are optimized. The simulated
circuit simulated with PSpice is shown on the top of Fig. 21.

The SiPM photoelectron response is simulated by a exponential current pulse,
shown Fig. 21 (A):

$I(t)=I_{pulsed}\times(1-e^{-\frac{t-T_{RiseDelay}}{\tau_{Rise}}})-I_{pulsed}\times(1-e^{-\frac{t-T_{FallDelay}}{\tau_{Fall}}})\\\
$ (36)

where:

$I_{pulsed}=12.8\mu A$ to $524.9\mu A$ i.e. 1 to 40 photoelectrons;

$T_{RiseDelay}=2ns$

$\tau_{Rise}=6ns$

$T_{FallDelay}=2ns$

$\tau_{Fall}=50ns$

The photoelectrons collected by SiPM are amplified by the optical detector.
The gain of SiPM is shown in Fig. 13. Then, the charge generated by
photoelectrons is:

$Q=N_{phot}\times G\times 6.241\times 10^{18}=\int_{0}^{\infty}I(t)\,dt\hskip
28.45274pt[C]$ (37)

In the PSpice simulations the gain of the SiPM was set to $3.5\times 10^{6}$
and the integral of the current signal show in Fig. 21 (A) is the charge for
$N_{phot}=1\;to\;40$.

The results of the simulations are shown in Fig. 21. When compared to the
measured values of Fig. 23, the simulations are quite realistic and enough for
this letter of intent.

Figure 21: PSpice simulations of front-end electronics. The curves relate to a
variable number of photoelectrons from 1 to 40. Top: Scheme of PSpice
simulation; (A): SiPM current generated from 1 to 40 photoelectrons; (B):
output of transconduttance amplifier with $V_{ref}=4V$; (C): output of fast
inverter amplifier; (D): output of peak-sensing amplifier, the peak is delayed
by 1.2 $\mu$s.

####  4.5.2 SiPM Amplitude/Charge measurement.

In addition to the TDC time measurement, we want to monitor also the amplitude
of the:

  * •

MIP muons crossing the scintillator block;

  * •

the energy released by the muons stopped in the scintillator block;

The SiPM signal is a short and unstable signal, see Fig. 21-A . The ADC of the
microcontroller, conversely, requires a stable signal throughout the
conversion time. To adapt the signal to the ADC, a pick-sensing circuit
followed by sample&hold was designed.

The peak sensing output signal is proportional to the charge collected by
SiPM. This signal is delayed by 1.2 $\mu$s from fast SiPM signal. The input
signal of Sample&Hold chip is the delayed by 12. $\mu$s and will stabilize its
output for a time required by microcontrolled ADC to digitize the fast input.
The scheme of the peak sensing amplifier, which use two Schottky diodes and
capacitors to integrate the charge, is shown in Fig. 22.

Figure 22: Scheme of peak sensing amplifier, simulated with PSpice. See Fig.
21

In Fig. 23 the oscilloscope signal for a minimum ionization muon is shown. The
detector used is a scintillator tile of dimensions 15x15x1 cm3 with the same
SiPM of this proposal [28].

On the left, the fast signal of SiPM is shown; In the center, the output of
peak sensing amplifier is shown and, on the right, is the output of the
Sample&Hold circuit is shown. The sample&hold is triggered by the High
Threshold discriminator signal delayed by 1.2 $\mu$s in order to hold the peak
of the signal. The start of ADC readout is further delayed by 7 $\mu$s by
microcontroller in order to have a stable signal from the sample&hold. The ADC
readout time is between the two vertical lines where the analog signal is
stable as shown on the oscilloscope.

Figure 23: Left: output signal of SiPM transconductance amplifier for a
minimum ionizing particle; Center: output of the peak sensing circuit; Right:
output of Sample&Hold DS1843. The two vertical lines are the start and stop of
microcontroller ADC conversion. The 300 ns Sample%Hold input signal is
generated on the top of peak sensing circuit, shown by the small peak. The
upper signal is always the output of the discriminator (NIM level).

###  4.6 Particles Identification.

It would be possible to identify the type of incident particle and/or $\mu$
charge by adding time of flight detector or magnetic fields. This is routinely
done in physics experiments at large laboratories such as CERN in Geneva. This
would have greatly complicated the detector without adding anything to the
educational purposes for which it was built. The solid angle and,
consequently, the counting rate would also be greatly reduced. Therefore, it
was decided not to identify the incident particles but to adopt some
techniques in data analysis that allow, nevertheless, to achieve the goal,
indeed, showing the data analysis approach techniques used in real
experiments.

##  5 Description of AMELIE detector.

The main aim of the project is the measurement of the muon lifetime in a
school environment with the use of simple and cheap instruments.

In other words, the idea is to bring students to the knowledge of modern
physics by allowing them to carry out real scientific experiments, collect the
data, make any corrections, analyze the results and compare them with existing
models or data.

The purpose of this apparatus is to measure the lifetime of the muon, one of
the elementary particles that reach the ground produced by cosmic rays.

The Amelie apparatus and its display is shown in Fig. 24 left. In the same
figure, the e-paper display is also shown.

Figure 24: Detector assembly and the display

The system handles four operating parameters, essential for the set-up of the
detector:

  * •

SiPM Vbias [V];

  * •

VHighThresold [mV]:

  * •

VLowThresold [mV];

  * •

Recording cycle [h].

In addition, some other parameters as atmospheric pressure, atmospheric
temperature, and of the SiPM, are only informative of the date and time,
environmental conditions. Absolute and relative rates as well as acquisition
time also belong to this category. This parameters are very important for the
monitoring of data acquisition and for data quality check.

Finally, three histograms are shown, two containing the charge signal for low
and high threshold and the last, the most important one, the time of TDC. The
y-axis of histograms is autoranging. When a value reaches the maximum value,
the range of y is doubled.

The display used is a 400 x 200 pixels e-paper. This type of display is very
visible even without back lighting, does not consume anything except during
refresh, and the data is not erased even if the power is removed. The problem
with this display is the long refresh times. Specifically, this display takes
0.4s to partially refresh the data. In this time the microcontroller is
dedicated full time to refresh and any data from the SiPM would be lost.

To avoid having a high dead time during the acquisition, refresh is done every
second during the first 10s after the power-on. Then it is done every 10m,
unless any button is operated, especially the arrow indicating machine status.
In this case, the refresh is done every second for the next 10s.

This configuration is perfectly compatible with the detector’s purpose of long
acquisition times, hours or days, without changing any parameters.

System commands and outputs are shown in Fig. 25.

Figure 25: Left: front view with commands; Right: back view with output
signals, power switch and $\mu$SD card

###  5.1 Front panel.

The front panel is shown in Fig. 25 at left.

Three switches on the right and a rotary encoder on the left are provided for
changing parameters. The encoder also functions as a switch. When pressed, the
microcontroller stores all operating parameters in the internal e-prom memory.
This memory is read when the detector is turned on, and in operating
parameters, at the time of power off, are reset. This procedure is used to
restart the system in case of power failure at the previous conditions.

Each of the three switches has three positions. One central, neutral, one
upward and one downward. How they work is shown in Tab. 4.

switch | position | function
---|---|---
left | up | START
down | STOP
central | up | RESET
down | RECORD DATA on $\mu$SD OR CLOSE THE FILE
right | up | MACHINE STATUS ARROW+
down | MACHINE STATUS ARROW-
rotary encoder | push | STORE CONFIGURATION on e-PROM
rotation | SET PARAMETER UNDER ARROW
Table 4: Operating switches

The bottom right part of the display shows the current status of the system
via icons. The icons are shown in Fig. 26

$\mu$SD plugged

$\mu$SD unplugged

$\mu$SD recording

DAQ running

DAQ stopped

Figure 26: Icons showing the status of the system configuration.

In addition, the system has a $\mu$SD memory card where data from each event
are stored. In the recording state, two files are provided on $\mu$SD. The
first one stores the data of each event above the high threshold, the second
one, in addition to this data, also the time given by the TDC in ns.

The REC button is a toggle switch. The REC state changes each time it is
pressed. Also, when the recording is turned off, the file on $\mu$SD is
closed. This action is very important in case of manual operation of the
system for proper prevention of recorded data.

Files written to $\mu$SD are automatically numbered at each run. The maximum
number of runs is 120. After that, you have to empty the $\mu$SD. The current
run number is shown in the right pane at the bottom of the display.

###  5.2 Back panel.

The rear panel is shown in Fig. 25 at right.

In the rear panel contains the power supply jack (V = 7 $\div$ 12 Vdc), the
main switch, the digital outputs of low and high threshold signals, the analog
signal of SiPM, the $\mu$SD, and the micro-USB connector of the
microcontroller. The analog and digital signals must be terminated at
$50\Omega$.

###  5.3 Cyclical recording.

The detector allows two types of recording on $\mu$SD card: 1) totally manual;
2) automatic cyclic recording.

The selection is made by setting the REC cycle parameter in the window at the
bottom right. If set to zero, the user decides whether to start data
acquisition, stop it, and record the data to $\mu$SD, using the switches on
the front panel. The process has no set time limit.

Cyclic recording on $\mu$SD is enabled by setting values $>$0 in the window
and setting START and REC with the switches. The recording cycle can range
from 1 to 100 hours. At the end of each cycle the files are closed, all
counts, graphs etc. are reset and two new files are opened with the increased
number. The ”REC cycle” box shows a bar indicating the percentage of time in
the current cycle, and the bottom box shows the number of files open in that
cycle.

The cycle time can be changed by bringing the arrow indicating ”machine
status” to the proper box using the right switch. Turning the ”rotary encoder”
clockwise or counterclockwise, sets the desired cycle time.

###  5.4 Adjustment of the operating parameters.

The operating parameters, Vbias and high or low threshold, can be changed by
bringing the arrow indicating ”machine status” to the desired box. Then, they
are increased or decreased by turning the ”rotary encoder” clockwise or
counterclockwise.

It is advisable to memorize the chosen status by pressing the ”rotary encoder”
button.

##  6 Data acquisition and analysis.

Before starting data taking, preliminary checks were made on the apparatus to
have confidence in the quality of the data.

###  6.1 Data quality check.

Using the test pulser circuit in Cap. 4.4.1 it is possible to simulate a
single or double pulse of the SiPM. The time between two pulses given by the
TDC7200 was compared with the time between two output pulses using a digital
oscilloscope. Both gave the same result within 1 ns.

By connecting the SiPM sensor to the front-end circuit, it was possible to
verify the analog output of the sensor. The AMELIE analog output signal of the
SiPM was connected to an oscilloscope that can integrate the input signal for
a long time. The output signal is shown in Fig. 27. Also visible, on the
right, some second signals that follow the first one within the gate time.

Figure 27: Output signal of the SiPM

###  6.2 Data tacking for SiPM gain $G=5\times 10^{6}$.

In order to check for systematic errors, three data sets were collected at
different SiPM gains, $5\times 10^{6}$, $6\times 10^{6}$ and $7\times 10^{6}$.
The procedure for collecting the data was the same in all cases. Data were
collected in runs of 24 hours each, using the cycling recording facility, for
a totoal of 20 runs for each SiPM gain has been recorded.

For each run, using the data recorded on $\mu$SD, several tests were carried
out for each run to check the stability of the system during the 24h of data
taking and the quality of the data collected.

Figure 28: Left: Data quality check; test of Vbias, VLowThresh and VHighThresh
stability; Right: measured time between two cosmic rays.

In particular, Vbias, VHighThresh and VLowThresh signals has been carefully
checked. The stability is optimal as shown by the graphs in Fig. 28 left.

Using the recorded data, the histogram of the charge released by each cosmic
ray was made for high and low threshold, as shown in Fig. 29.

In the right figure, you can clearly see the two components of the signal: 1)
SiPM electronic noise (in purple); 2) SiPM signal (in blue).

In the left figure, it can be seen that the electronic noise is cancelled by
raising the threshold.

Figure 29: Cosmic rays charge for a SiPM gain of $5\times 10^{6}$: Left: High
Threshold; Right: Low Threshold

It is interesting to verify experimentally the dependence of the rate of
cosmic rays on atmospheric pressure, as shown in the histogram in Fig. 30
left. The correlation between cosmic ray frequency and atmospheric pressure is
clearly visible. As atmospheric pressure increases, the frequency of cosmic
rays decreases.

It is then possible to make the histogram of the times between the arrival of
one cosmic ray and the following, Fig. 30 the left.

This measure has already been discussed theoretically in Chap. 3.8.

From this histogram one can deduce the frequency of cosmic rays as an
alternative to the simple division $R=N_{cosmic}/T_{time}$.

The frequency from the graph appears to be: $R=1/\tau$.

Figure 30: Left: Rate of cosmic rays VS. air pressure; Right: SiPM charge for
TDC events

In Fig. 30 right, the SiPM charge is shown for only those events that provided
a signal from the TDC, that is, with a double signal in the gate time.
Comparing with Fig. 29 it is evident that for these events the SiPM signal is
much larger than that of the single muons. In Chap. 4.3.4 this phenomenon was
discussed.

At the conclusion of all consistency checks, you can proceed with the analysis
of the TDC data.

Tab. 5 shows the average measured values of a some paramenters. It is
interesting to compare the rate of cosmic muons interacting with the detector,
about 4 $s^{-1}$, with those giving a TDC interrupt, START, and STOP signal in
gate time, which is about 1 every 2 minutes (30 per hour).

For each run, a histogram was made of the timing provided by the TDC between 0
and 10 $\mu$s. The mean value and standard deviation were calculated for each
bin. The result is plotted in Fig. 31.

Figure 31: TDC time

The histogram clearly shows the negative exponential trend, typical of
particle decays. The data were fit using a function shown in Eq. 7, as shown
in Chap. 3:

$N(t)=N_{0}\exp^{-t/\tau}$ (38)

from 0.6 to 10 $\mu$s. Below 500 ns, the TDC data are unreliable both because
of the overlapping widths of the START and STOP signals and because the
TDC7200 is used in ”Mode 2,” between 250 ns and 8 ms, while ”Mode 1” is usable
between 12 and 500 ns.

The decay time given by the fit is 1.406 $\mu$s, far from the official muon
decay value of 2.197 $\mu$s, as indicated in Eq. 8.

This means that some physical phenomena stated in Chap. 3.6.2 and Chap. 3.8
were not taken into account.

###  6.3 Background & background subtraction.

As discussed in previous chapters on muon physics, there may be physical
contributions that alter the lifetime measurement even significantly. In
particular:

  * •

“fake events”;

  * •

$\mu^{-}$ absorption in nuclear matter.

####  6.3.1 Fake events.

The time distribution of cosmic muons is flat. This means that the probability
of having two cosmic muons in gate time is not zero and varies with the
frequency of them.

“Fake events” are defined as events that simulate decay but due to the
subsequent arrival of two cosmic muons in the gate time = 12 $\mu$s.

Unfortunately, as shown in Chap. 3.8, the trend of these events is exponential
as the decay of muon end can be easily confused with the lifetime events.

This correction is, however, small. If we look at the events above
$3\sigma=6.6\mu$s the muon lifetime, where the probability to have a real muon
decay is only $0.3\%$, and, therefore we expect to have only “fake-events”,
from Fig. 31 left, we see that this number of events is negligible.

However, this correction was taken into account by using a fit with two time
coefficients:

$f(t)=D\times e^{-t/T}+n\times e^{-t/\tau}$ (39)

Where the coefficients D and T are taken from Fig. 30 left.

The result is shown in Fig. 32 at left. The blue line is the fake events
correction.

Figure 32: Left: TDC fit with fake events for SiPM gain $G=5\times 10^{6}$;
Right: the same bat for $G=6\times 10^{6}$.

“Fake-events” could be eliminated by surrounding the cube with scintillation
detectors in anti-coincidence. Cosmic rays, in fact, originate from outside
instead of inside the cube. This would lead to detector complication,
increased cost, and complicated operation, which cannot be justified by the
small measured contribution.

It was chosen, therefore, to use the technique of background subtraction too
save on hardware cost and complication, and to allow the students to “see”-
and understand the random background.

But, it is evident from Fig. 32, that such this small noise rate is
statistically insignificant for background considerations. Then this
contribution will be ignored in subsequent analyses.

####  6.3.2 Negative Muons absorption in nuclear matter.

The configuration of the AMELIE detector does not include the possibility of
separating $\mu^{-}$ from $\mu^{+}$, a possibility offered only by complex and
expensive experiments on accelerator machines or balloons or satellites.

As discussed in Chap. 3.6.2, in nuclear matter, the behavior of $\mu^{-}$
differs from that of $\mu^{+}$. In particular, it should be kept in mind that
$\mu^{-}$ tend to replace electrons in atomic orbitals and be absorbed by the
nucleus.

Not only that, it must be taken into account that, at the ground level, the
number of $\mu^{-}$ differs from that of $\mu^{+}$:

$R(\mu^{+}/\mu^{-})=1.30\pm 0.05$ at 0.86 GeV/c

$R(\mu^{+}/\mu^{-})=1.27\pm 0.01$ at 0.52 GeV/c

See Chap. 3.4 and Fig. 7. and also [24].

The competition between absorption and decay processes, reduce the meanlife of
the obseved decay of the negative muon as compared with the free decay.

Expressed in terms of process rates $\Lambda=1/\tau$, the total disappearance
probability is written as the sum of the capture and of the decay
probabilities:

$\Lambda_{total}=\Lambda_{capture}+\Lambda_{decay}$

$\Lambda_{decay}=Q\times 1/\tau_{\mu^{+}}$

$\Lambda_{total}^{-1}=\tau_{\mu^{-}}<\;\tau_{\mu^{+}}=2.197703\;\mu s$

Using the different lifetimes of muonic atoms compared with the lifetime of
the positive muon, it is possible to measure the ratio of positive to negative
muons in a flux, without using a magnetic spectrometer.

In general, if a device detects decay electrons coming from muons stopped in
different materials, the total decay curve will be a superposition of several
decay laws:

$\frac{dN}{dt}=N_{+}c_{0}\frac{1}{\tau_{0}}e^{-\frac{t}{\tau_{0}}}+N_{-}\sum_{j=1}^{m}c_{j}\frac{1}{\tau_{j}}e^{-\frac{t}{\tau_{j}}}$
(40)

where:

$N_{+}$ , $N_{-}$ |  \- number of $\mu^{+}$, $\mu^{-}$ which enter the detector
---|---
m |  \- number of materials in the detector
$c_{0}$ |  \- detection efficiency for $\mu^{+}$ stopped in all materials
$c_{j}$ |  \- detection efficiency for $\mu^{-}$ stopped in material
$\tau_{0}$ |  \- mean lifetime of $\mu^{+}$ (2.19703 $\mu$s)
$\tau_{j}$ |  \- mean lifetime of $\mu^{-}$ in material j

This method is also called “delayed coincidences” from historical reasons.

Taking into account the composition of the plastic scintillator, (H-11, C-10),
(Chap. 4.2), the above formula become:

$\frac{dN}{dt}=\frac{N_{0}}{R+1}\left[Rc_{0}\frac{1}{\tau_{0}}e^{-\frac{t}{\tau_{0}}}+c_{h}\frac{1}{\tau_{h}}e^{-\frac{t}{\tau_{h}}}+c_{c}\frac{1}{\tau_{c}}e^{-\frac{t}{\tau_{c}}}\right]$
(41)

where:

$N_{0}$ |  = $N_{+}+N_{-}$
---|---
$R(\mu^{+}/\mu^{-})$ |  = $N_{+}/N_{-}\approx 1.27$
$c_{h}$ |  = detection efficiency for $\mu^{+}$ stopped in hydrogen
$\tau_{h}$ |  = 2.19490 $\mu$s (mean lifetime of $\mu^{+}$ in hydrogen)
$c_{c}$ |  = detection efficiency for $\mu^{+}$ stopped in carbon
$\tau_{c}$ |  = 2.028 $\mu$s (mean lifetime of $\mu^{+}$ in carbon)

In this material, at the absorption of $\mu^{-}$ shown in Eq. 22 on hydrogen,
we must add the absorption on carbon:

$\mu^{-}+\>^{12}C\rightarrow\nu_{\mu}+\>^{13}B^{*}$ (42)

where ${}^{13}B^{*}$ has at unknown excitation energy. The “mean” excitation
energy in the nuclear muon capture is around 15 to 20 MeV. This is well above
the nucleon emission threshold in all complex nuclei. Thereby, the neutron
emission is the preferred channel, compared to other channels like: emission
of charged particles, gamma rays or delayed fission.

Using the formula [41] and keeping the value of the average $\mu^{+}$ life
fixed at the value [8], using the data acquired in the $5\times 10^{6}$ gain
runs, the average lives of $\mu^{-}$ in the scintillator can be determined,
particularly on hydrogen and carbon, as shown in Fig. 31.

Figure 33: $\mu^{-}$ and $\mu^{+}$ mean lifetime in plastic scintillator,
consisting primarily of hydrogen and carbon. 20 runs of 24h each, with SiPM
gain $5\times 10^{6}$.

Statistics is a very important parameter for improving any measurement. In
this analysis, 24h runs were made but, given the stability of the system,
taking into account that muons stopping in the cube are about one every two
minutes, much longer time runs can also be implemented.

###  6.4 Data analysis for SiPM gain $G=6\times 10^{6}$.

The same procedure used for the analysis of data acquired with a SiPM gain of
$5\times 10^{6}$, shown in the previous chapter, was used by increasing the
gain to $6\times 10^{6}$.

Fig. 34 shows the SiPM charge signal at low threshold, high threshold and for
events that gave a TDC signal.

Figure 34: Left: SiPM charge signal for low threshold; Center; SiPM charge
signal for high threshold; Right: SiPM charge signal for TDC events. SiPM gain
$G=6\times 10^{6}$

In Fig. 32 , the muon decay times for different SiPM gains are shown.

It is evident that the exponential curvature given by the $\tau$ coefficient,
does not depend on the SiPM gain. This means that the measurement is not
affected by systematic errors dependent on the operating parameters of the
system but only on the times provided by the TDC, already calibrated and
tested as described in Chap. 4.4.1.

Fig. 35 shows the average muon lifetime measured with a SiPM gain of $6\times
10^{6}$. The same figure shows the fit made with the formula [41].

The muon lifetimes in hydrogen and carbon are similar to those shown in Fig.
4.4.1 with a gain of the SiPM of $5\times 10^{6}$.

Figure 35: $\mu^{-}$ and $\mu^{+}$ mean lifetime in plastic scintillator,
consisting primarily of hydrogen and carbon. 20 runs of 24h each, with SiPM
gain $6\times 10^{6}$.

###  6.5 Data analysis for SiPM gain $G=7\times 10^{6}$.

In order to verify the stability of the system and the consistency of the
collected data, a third round of 20 runs was made with a SiPM gain of
$G=7\times 10^{6}$.

Fig. 36 shows the SiPM charge signal at low threshold, high threshold and for
events that gave a TDC signal.

Figure 36: Left: SiPM charge signal for low threshold; Center; SiPM charge
signal for high threshold; Right: SiPM charge signal for TDC events. SiPM gain
$G=7\times 10^{6}$.

Conversely, Fig. 37 shows the time measured by the TDC , raw data on the left
and with the time measurement in Carbon and hydrogen on the right.

Figure 37: Left: TDC fit with fake events for SiPM gain; Right: $\mu^{-}$ and
$\mu^{+}$ mean lifetime in plastic scintillator. SiPM gain $G=7\times 10^{6}$.

###  6.6 Summary of the three rounds of data taking.

As mentioned above, three cycles of data taking were done at three different
values of the SiPM gain $5\>,6\>,7\times 10^{6}$. For each cycle, 20 runs of
24 hours each were made.

Tab. 5 summarizes all relevant parameters of each cycle.

| SiPM gain
---|---
| $5\times 10^{6}$ | $6\times 10^{6}$ | $7\times 10^{6}$
Average n. of low thresh. ev. per run |  343350 $\pm$ 4716 |  392222 $\pm$ 5134 |  433560 $\pm$ 3580
Average n. of high thresh. ev. per run |  249759 $\pm$ 4439 |  276409 $\pm$ 2360 |  253269 $\pm$ 3121
Average n. of TDC events per run |  632 $\pm$ 31 |  990 $\pm$ 42 |  804 $\pm$ 22
Low Thresh. rate [$s^{-1}$] |  3.97 $\pm$ 0.05 |  4.49 $\pm$ 0.06 |  4.97 $\pm$ 0.03
High Thresh. rate [$s^{-1}$] |  2.79 $\pm$ 0.05 |  3.16 $\pm$ 0.03 |  2.90 $\pm$ 0.03
$\mu$ decay rate [$min^{-1}$] |  0.43 $\pm$ 0.02 |  0.68 $\pm$ 0.03 |  0.55 $\pm$ 0.02
$\Delta$t between two siccessive events [$s^{-1}$] |  0.252 $\pm$ 0.003 |  0.223 $\pm$ 0.003 |  0.201 $\pm$ 0.001
row TDC data decay constant $\tau$ [ns] |  1440.3 $\pm$ 30.3 |  1470.9 $\pm$ 48.2 |  1440.2 $\pm$ 48.0
$\mu^{-}$ lifetime in hydrogen [ns] | 1979.9 | 1589.9 | 1407.9
$\mu^{-}$ lifetime in carbon [ns] | 1706.8 | 1603.6 | 1418.4
SiPM charge median for TDC ev. [au] | 708.0 | 770.8 |
Table 5: Summary of the three data tacking rounds at different SiPM gains.

From the Tab. 5, it can be stated that, while the decay constant of the raw
data from the three rounds lies in the statistical error, the convergence fit
of negative muon lifetime in hydrogen and carbon is not stable and, then, a
much higher statistics is required.

The frequency of TDC events is affected by the environment in which the
detector is placed. For example, the frequency increases greatly if the
detector is very close to thick concrete walls that tend to slow down cosmic
muons, instead of in the center of the room.

##  7 Conclusions.

An innovative detector has been designed. Its main purpose is to measure the
average lifetime of muons and to study the absorption of negative muons in
nuclear matter. The detector is designed for high school students but can also
be used in university physics laboratories.

The idea of muons lifetime measurement with stoppend cosmic muons is not new.
Many setups of this type have been successfully implemented in the physics
laboratories of many universities around the world. They require, however,
expensive detectors, photomultipliers using high voltages of 1,000 volts or
more, modular NIM electronics, expensive and complex to handle.

AMELIE is innovative compared to the others setup because it use SiPM optical
detectors instead of the usual photomultipliers. This avoids having high
voltages generators, cables, connectors, and offers an additional safety
margin for students. In addition, use a Time to Digital Converter (TDC) that
are to state of the art of this type of converter.

AMELIE is a detector that is compact, easy to use and records data for off-
line data analysis in universal .csv format. Is a complete system, requires no
additional equipment other than common PCs with free Open Source programs like
“Libre Office” or “gnuplot”.

Taking into account the AMELIE simplicity, low cost and educational purpose of
this apparatus, the results obtained do not differ much from the values
measured with complex and expensive experimental apparatus shown in Tab. 1. It
should be kept in mind that the purpose of the apparatus is not to make
publishable measurements in peer-reviewed scientific journals, measurements
widely available in the literature, but educational and didactic.

The detector is very stable, reliable and the measurement does not appear to
have systematic errors due to the particular setting of functional parameters
or relevant background that could invalidate the measures.

In addition, the detector uses cosmic rays which are present all over the
world 24h/7d, free of charge and consistent over time.

It allows professors to introduce modern physics, the world of elementary
particles with simple, didactic but very formative measurements, data analysis
and data correction. Other accessible fields are statistics, data analysis,
corrections and counting over time.

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