Muon Decay Time Distribution 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 ndt, where n 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)n dt, or dN/N(t)=-ndt some time t and no is the number of muons at t=O. The"lifetime "t of a muon / t ?p Integrating, we have N(t)=No exp(-n t), where N(t) is the number of surviving muons at reciprocal of n, t 1/. 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 By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between t and t+ dt is given by d(t)dt. If we had started with No muons hen the fraction -dN/No that would on average decay in the time interval between t and t+ dt is just given by differentiating the above relation -dn= non exp(-n t)dt dN/No=n exp(h t) dt The left-hand side of the last equation is nothing more than the decay probability we seek, so D(t)=exp(n t). This is true regardless of the starting value of No. That is, the distribution of decay times, for new muons entering our detector, is also exponential with he very same exponent used to describe the surviving population of muons. Again, what ve call the muon lifetime ist =1n Because the muon decay time is exponentially distributed, it does not matter that the whether you examine it at early times or late times, its e-folding time is the sate ln muons whose decays we detect are not born in the detector but somewhere above us the atmosphere. An exponential function always "looks the sa the sense th Detector Physics The active volume of the detector is a plastic scintillator in the shape of a right circular ylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing together one or more fluors with a solid plastic solvent that has an aromatic ring structure a charged particle passing through the scintillator will lose some of its kinetic energy by ionization and atomic excitation of the solvent molecules. Some of this deposited energ is then transferred to the fluor molecules whose electrons are then promoted to excited states. Upon radiative de-excitation, light in the blue and near-UV portion of the electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A typical photon yield for a plastic scintillator is l optical photon emitted per 100 eV of Muon physics5 Muon Physics Muon Decay Time Distribution 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)λ dt, or dN/N(t) = −λdt. Integrating, we have N(t) = N0 exp(−λ t), where N(t) is the number of surviving muons at some time t and N0 is the number of muons at t = 0. The "lifetime" τ 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. By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between t and t + dt is given by D(t)dt. If we had started with N0 muons, then the fraction −dN/N0 that would on average decay in the time interval between t and t + dt is just given by differentiating the above relation: −dN = N0λ exp(−λ t) dt −dN/ N0 = λ exp(−λ t) dt The left-hand side of the last equation is nothing more than the decay probability we seek, so D(t) = λ exp(−λ t). This is true regardless of the starting value of N0. That is, the distribution of decay times, for new muons entering our detector, is also exponential with the very same exponent used to describe the surviving population of muons. Again, what we call the muon lifetime is τ = 1/λ. 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. Detector Physics The active volume of the detector is a plastic scintillator in the shape of a right circular cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing together one or more fluors with a solid plastic solvent that has an aromatic ring structure. A charged particle passing through the scintillator will lose some of its kinetic energy by ionization and atomic excitation of the solvent molecules. Some of this deposited energy is then transferred to the fluor molecules whose electrons are then promoted to excited states. Upon radiative de-excitation, light in the blue and near-UV portion of the electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of