deposited energy. The properties of the polyvinyltoluene-based scintillator used in the muon lifetime instrument are summarized in table 1 To measure the muons lifetime, we are interested in only those muons that enter, slow, top and then decay inside the plastic scintillator. Figure 2 summarizes this process. Such muons have a total energy of only about 160 Me v 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. (See the electronics section below for more detail. A stopped muon, after a bit, decays into an electron, a neutrino and an anti-neutrino. See the next section for an important qualification of this statement. Since the electron mass is so much smaller that the muon mass,mu/me-210, the electron tends to be very energetic and to produce scintillator light essentially all along its pathlength. 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 PMT Scintillator Figure 2. Schematic showing the generation of the two light pulses(short arrows)used in determining the muon lifetime. One light pulse is from the slowing muon(dotted line) and the other is from its decay into an electron or positron(wavey line). Table 1. General Scintillator Properties Mass density 1.032g/cm3 Refractive index Base material Polyvinyltoluene Rise time 0.9ns Fall time 2.4 ns Wavelength of 423nm maximum emission Muon physics
6 Muon Physics deposited energy. The properties of the polyvinyltoluene-based scintillator used in the muon lifetime instrument are summarized in table 1. To measure the muon's lifetime, we are interested in only those muons that enter, slow, stop and then decay inside the plastic scintillator. Figure 2 summarizes this process. 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. (See the electronics section below for more detail.) A stopped muon, after a bit, decays into an electron, a neutrino and an anti-neutrino. (See the next section for an important qualification of this statement.) 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 pathlength. 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. Figure 2. Schematic showing the generation of the two light pulses (short arrows) used in determining the muon lifetime. One light pulse is from the slowing muon (dotted line) and the other is from its decay into an electron or positron (wavey line). Table 1. General Scintillator Properties Mass density 1.032 g/cm3 Refractive index 1.58 Base material Polyvinyltoluene Rise time 0.9 ns Fall time 2.4 ns Wavelength of Maximum Emission 423 nm µ e PMT Scintillator
Interaction ofFs with matter The muons whose lifetime we measure necessarily interact with matter. Negative muons that stop in the scintillator can bind to the scintillators carbon and hydrogen nuclei in much the same way as electrons do. Since the muon is not an electron, the Pauli exclusion principle does not prevent it from occupying an atomic orbital already filled with electrons. Such bound negative muons can then interact with protons →n+v before they spontaneously decay. Since there are now two ways for a negative muon to disappear, the effective lifetime of negative muons in matter is somewhat less than the lifetime of positively charged muons, which do not have this second interaction mechanism. Experimental evidence for this effect is shown in figure 3 where disintegration"curves for positive and negative muons in aluminum are shown. ( See Rossi, 1952) The abscissa is the time interval t between the arrival of a muon in aluminum target and its decay. The ordinate, plotted logarithmically, is the number of muons greater than the corresponding abscissa. These curves have the same meaning as curves representing the survival population of radioactive substances. The slope of the curve is a measure of the effective lifetime of the decaying substance. The muon lifetime ye measure with this instrument is an average over both charge species so the mean lifetime of the detected muons will be somewhat less than the free space value t=219703±0.00004usec The probability for nuclear absorption of a stopped negative muon by one of the scintillator nuclei is proportional to Z, where Z is the atomic number of the nucleus [Rossi, 1952]. A stopped muon captured in an atomic orbital will make transitions down to the k-shell on a time scale short compared to its time for spontaneous decay WHeeler]. Its Bohr radius is roughly 200 times smaller than that for an electron due to its much larger mass, increasing its probability for being found in the nucleus. From our knowledge of hydrogenic wavefunctions, the probability density for the bound muon to be found inside the nucleus is proportional to Z. Once inside the nucleus, a muons probability for encountering a proton is proportional to the number of protons there and so scales like Z. The net effect is for the overall absorption probability to scale like Z4 Again, this effect is relevant only for negatively charged muons Muon physics
7 Muon Physics Interaction of µ−’s with matter The muons whose lifetime we measure necessarily interact with matter. Negative muons that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in much the same way as electrons do. Since the muon is not an electron, the Pauli exclusion principle does not prevent it from occupying an atomic orbital already filled with electrons. Such bound negative muons can then interact with protons µ − + p → n + νµ before they spontaneously decay. Since there are now two ways for a negative muon to disappear, the effective lifetime of negative muons in matter is somewhat less than the lifetime of positively charged muons, which do not have this second interaction mechanism. Experimental evidence for this effect is shown in figure 3 where “disintegration” curves for positive and negative muons in aluminum are shown. (See Rossi, 1952) The abscissa is the time interval t between the arrival of a muon in the aluminum target and its decay. The ordinate, plotted logarithmically, is the number of muons greater than the corresponding abscissa. These curves have the same meaning as curves representing the survival population of radioactive substances. The slope of the curve is a measure of the effective lifetime of the decaying substance. The muon lifetime we measure with this instrument is an average over both charge species so the mean lifetime of the detected muons will be somewhat less than the free space value τµ = 2.19703 ± 0.00004 µsec. The probability for nuclear absorption of a stopped negative muon by one of the scintillator nuclei is proportional to Z4 , where Z is the atomic number of the nucleus [Rossi, 1952]. A stopped muon captured in an atomic orbital will make transitions down to the K-shell on a time scale short compared to its time for spontaneous decay [Wheeler]. Its Bohr radius is roughly 200 times smaller than that for an electron due to its much larger mass, increasing its probability for being found in the nucleus. From our knowledge of hydrogenic wavefunctions, the probability density for the bound muon to be found inside the nucleus is proportional to Z3 . Once inside the nucleus, a muon’s probability for encountering a proton is proportional to the number of protons there and so scales like Z. The net effect is for the overall absorption probability to scale like Z4 . Again, this effect is relevant only for negatively charged muons
1000 microsec Figure 3. Disintegration curves for positive and negative muons in aluminum. The ordinates at t=0 can be used to determine the relative numbers of negative and positive muons that have undergone spontaneous decay. The slopes can be used to determine the decay time of each charge species. (From Rossi, p168) Muon physics
8 Muon Physics Figure 3. Disintegration curves for positive and negative muons in aluminum. The ordinates at t = 0 can be used to determine the relative numbers of negative and positive muons that have undergone spontaneous decay. The slopes can be used to determine the decay time of each charge species. (From Rossi, p168.)
u/H Charge Ratio at Ground Level Our measurement of the muon lifetime in plastic scintillator is an average over both negatively and positively charged muons. We have already seen that us have a lifetime somewhat smaller than positively charged muons because of weak interactions betweer negative muons and protons in the scintillator nuclei. This interaction probability is proportional to Z, where Z is the atomic number of the nuclei, so the lifetime of negative muons in scintillator and carbon should be very nearly equal. This latter lifetime tc is measured to be t=2043+0.003 usec. [Reiter, 19601 It is easy to determine the expected average lifetime t_obs of positive and negative muons in plastic scintillator. Let n be the decay rate per negative muon in plastic scintillator and let n be the corresponding quantity for positively charged muons. If we then letn and n represent the number of negative and positive muons incident on the scintillator per unit time, respectively, the average observed decay rate n and its corresponding lifetime t_obs are given by N+at+N-A Tks=(1+p)(+# (1+p)+m where p≡NN,τ≡(λ) is the lifetime of negative muons in scintillator and t'=() is the corresponding quantity for positive muons Due to the Z effect, t =to for plastic scintillator, and we can set t equal to the free space lifetime value tu since positive muons are not captured by the scintillator nuclei Setting p=l allows us to estimate the average muon lifetime we expect to observe in the scintillator We can measure p for the momentum range of muons that stop in the scintillator by rearranging the above equation Muon physics
9 Muon Physics µ+/µ− Charge Ratio at Ground Level Our measurement of the muon lifetime in plastic scintillator is an average over both negatively and positively charged muons. We have already seen that µ − ’s have a lifetime somewhat smaller than positively charged muons because of weak interactions between negative muons and protons in the scintillator nuclei. This interaction probability is proportional to Z4 , where Z is the atomic number of the nuclei, so the lifetime of negative muons in scintillator and carbon should be very nearly equal. This latter lifetime τc is measured to be τc = 2.043 ± 0.003 µsec. [Reiter, 1960] It is easy to determine the expected average lifetime τ_obs of positive and negative muons in plastic scintillator. Let λ − be the decay rate per negative muon in plastic scintillator and let λ + be the corresponding quantity for positively charged muons. If we then let N− and N+ represent the number of negative and positive muons incident on the scintillator per unit time, respectively, the average observed decay rate λ and its corresponding lifetime τ_obs are given by where ρ ≡ N+ /N− , τ − ≡ (λ − ) −1 is the lifetime of negative muons in scintillator and τ + ≡ (λ + ) −1 is the corresponding quantity for positive muons. Due to the Z4 effect, τ − = τc for plastic scintillator, and we can set τ + equal to the free space lifetime value τµ since positive muons are not captured by the scintillator nuclei. Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the scintillator. We can measure ρ for the momentum range of muons that stop in the scintillator by rearranging the above equation:
Background readout electronics. These particles can be either charged, like electrons or muons, artts The detector responds to any particle that produces enough scintillation light to trigger neutral, like photons, that produce charged particles when they interact inside the scintillator. Now, the detector has no knowledge of whether a penetrating particle stops or not inside the scintillator and so has no way of distinguishing between light produced by muons that stop and decay inside the detector, from light produced by a pair of through-going muons that occur one right after the other. This important source of background events can be dealt with in two ways. First, we can restrict the time interval during which we look for the two successive flashes of scintillator light characteristic of muon decay events. Secondly, we can estimate the background level by looking at large times in the decay time histogram where we expect few events from genuine muon decay Fermi Coupling Constant GE Muons decay via the weak force and the fermi coupling constant GF is a measure of the strength of the weak force. To a good approximation, the relationship between the muon lifetime t and gr is particularly simple 192兀3h Ge mc where m is the mass of the muon and the other symbols have their standard meanings Measuring t with this instrument and then taking m from, say, the Particle Data group (http://www.pdg.ibl.gov)producesavalueforGf. Muon physics
10 Muon Physics Backgrounds The detector responds to any particle that produces enough scintillation light to trigger its readout electronics. These particles can be either charged, like electrons or muons, or neutral, like photons, that produce charged particles when they interact inside the scintillator. Now, the detector has no knowledge of whether a penetrating particle stops or not inside the scintillator and so has no way of distinguishing between light produced by muons that stop and decay inside the detector, from light produced by a pair of through-going muons that occur one right after the other. This important source of background events can be dealt with in two ways. First, we can restrict the time interval during which we look for the two successive flashes of scintillator light characteristic of muon decay events. Secondly, we can estimate the background level by looking at large times in the decay time histogram where we expect few events from genuine muon decay. Fermi Coupling Constant GF Muons decay via the weak force and the Fermi coupling constant GF is a measure of the strength of the weak force. To a good approximation, the relationship between the muon lifetime τ and GF is particularly simple: where m is the mass of the muon and the other symbols have their standard meanings. Measuring τ with this instrument and then taking m from, say, the Particle Data Group (http://www.pdg.lbl.gov) produces a value for GF. 192π 3 7 GF 2 m 5 c 4 τ =