The aim of this project is to detect the muon decay events and to measure the muon mean lifetime.
For this purpose we will use the scintillation detector described in the following post : Scintillation Detector for Cosmic Muons.
The μ lepton (muon), the main component of the secondary cosmic rays, is an elementary particle with spin 1/2, mass equal to 105,65 MeV and mean life of 2,2µs. As mentioned earlier, the μ are produced mainly in the upper atmosphere by the decay of particle π. At the production time they have relativistic speeds and because of the phenomenon of time dilation they can reach the sea level.
Muons were the first elementary particles to be found unstable, i.e. subject to decay into other particles. At the time of Rossi’s pioneering experiments on muon decay, the only other fundamental” particles known were photons, electrons and their antiparticles (positrons), protons, neutrons, and neutrinos. Since then dozens of particles and antiparticles have been discovered, and most of them are unstable. In fact, of all the particles that have been observed as isolated entities, the only ones that live longer than muons are photons, electrons, protons, neutrons, neutrinos and their antiparticles. Even neutrons, when free, suffer beta decay with a half-life of 15 minutes.
Muons decay through the process :
The decay times for muons are easily described mathematically. Suppose at some time t we have N(t) muons. If the probability that a muon decays in some small time interval dt is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays, then the change dN in our population of muons is just dN = −N(t)λdt, or dN/N(t) = −λdt. Integrating, we have :
N(t) = N0e(-λt)
N(t) is the number of surviving muons at some time t
N0 is the number of muons at t = 0
The “lifetime” t of a muon is the reciprocal of λ, τ = 1/ λ
This simple exponential relation is typical of radioactive decay.
Now, we do not have a single clump of muons whose surviving number we can easily measure. Instead, we detect muon decays from muons that enter our detector at essentially random times, typically one at a time. It is still the case that their decay time distribution has a simple exponential form of the type described above.
Because the muon decay time is exponentially distributed, it does not matter that the muons whose decays we detect are not born in the detector but somewhere above us in the atmosphere. An exponential function always “looks the same” in the sense that whether you examine it at early times or late times, its e-folding time is the same.
Muon Decay Measure Method
So the scintillator/photomultiplier assembly, when properly configured, produces several electronic events every second, almost all of which are due to muons passing through the scintillator. But the truly exciting fraction of those events is due to the muons which arrive with much-less-than-average kinetic energy, because such muons can lose enough energy in the scintillator to come to rest inside it. In coming to rest, they will deposit the last of their kinetic energy, typically of order 50 MeV, so they will still produce a scintillator flash as they come to rest.
The muons that come to rest then live a relatively long time, on the order of order microseconds, inside the scintillator; but eventually each of them decays in an electron (positron) plus a neutrino and a antineutrino. Nearly all of the rest energy (105 MeV) of the stopped muons appears as kinetic energy of the three particles; on average, the electron (positron) gets a third of this energy, about 35 MeV. (The two neutrinos carry away the rest of the energy undetectably.) But such an electron is a charged particle, itself certain to cause ionization as it moves through the scintillator. Conveniently, the typical energy deposited in this ionization process is about the same as that deposited by a muon-in-transit, or a muon stopping, so the very same scintillator/PMT configuration is also suitable for detecting stopped muons subsequently decaying at rest.
To measure the muon’s lifetime, we are interested in only those muons that enter, slow, stop and then decay inside the plastic scintillator. Such muons have a total energy of only about 160 MeV as they enter the tube. As a muon slows to a stop, the excited scintillator emits light that is detected by a photomultiplier tube (PMT), eventually producing a logic signal that triggers a timing clock. A stopped muon, after a bit, decays into an electron, a neutrino and an anti-neutrino. Since the electron mass is so much smaller that the muon mass, mμ/me ~ 210, the electron tends to be very energetic and to produce scintillator light essentially all along its path. The neutrino and anti-neutrino also share some of the muon’s total energy but they entirely escape detection. This second burst of scintillator light is also seen by the PMT and used to trigger the timing clock. The distribution of time intervals between successive clock triggers for a set of muon decays is the physically interesting quantity used to measure the muon lifetime.
For the data acquisition of muon decay we used an oscilloscope with time of persistence set to “infinite” : in this way the pulses due to muons decay remain displayed and can be measured.
In detail the trigger threshold should be set to 200mV, in mode “peak detect” so as to acquire even the shortest pulses, the time base should be set to 400ns or to 800ns, and the graph must be moved to the left part of display so as to be able to use all the right with a width of about 10μsec.
Time measurement is made using the measurement cursors normally available on the oscilloscope.
With the detector and the process above described we have acquired 400 events.
Distributing them in bin with temporal width of 1 μsec we obtain the first chart shown below, from which we obtain with exponential fit a value of τμ =2 μsec.
Distributing them in bin with temporal width of 0,5 μsec we obtain a value of τμ =2,5 μsec.
Using the statistic method MLE (maximum Likelihood Estimation) for the exponential time constant, it can be shown the most likely value of the time constant of the exponential decay is equal to the average of the measurements.
For n = 400 (400 events have been acquired) we obtain the values :
τμ = 2,078 (measurements average)
σ = 0,011
The value that is obtained is probably slightly underestimated since the frame window taken into examination had up to 7 usec. This value is, however, in agreement with the result that you should get which stays between the theoretical value of 2,2 μsec for positive muons which is equal to the value measured in empty space and the value of 2,04 μsec for negative muons which are affected by the interactions with the nuclei of the scintillator material.
|Exponential fit with bin 1 μsec||2 μsec|
|Exponential fit with bin 0,5 μsec||2,5 μsec|
|MLE Statistic||2,078 ± 0,011 μsec|
Pdf document with the project description : RilevatoreMuoni_ENG