Time dilation effect A measurement of the muon stopping rate at two different altitudes can be used to the time dilation effect of spe lough the de configuration is not optimal for demonstrating time dilation, a useful measurement can still be preformed without additional scintillators or lead absorbers. Due to the finite size of the detector, only muons with a typical total energy of about 160 Mev will stop inside the plastic scintillator. The stopping rate is measured from the total number of observed muon decays recorded by the instrument in some time interval. This rate in turn is proportional to the flux of muons with total energy of about 160 Mev and this flux decreases with diminishing altitude as the muons descend and decay in the atmosphere After measuring the muon stopping rate at one altitude, predictions for the stopping rate at another altitude can be made with and without accounting for the time dilation effect of special relativity. A second measurement at the new altitude distinguishes between competing predictions muon's energy loss as it descends into the atmosphere, variations with energy n he the A comparison of the muon stopping rate at two different altitudes should account for the shape of the muon energy spectrum, and the varying zenith angles of the muons that stop the detector. Since the detector stops only low energy muons, the stopped muons detected by the low altitude detector will, at the elevation of the higher altitude detector, necessarily have greater energy. This energy difference AE(h) will clearly depend on the pathlength between the two detector positions Vertically travelling muons at the position of the higher altitude detector that are ultimately detected by the lower detector have an energy larger than those stopped and detected by the upper detector by an amount equal to AE(h). If the shape of the muon energy spectrum changes significantly with energy, then the relative muon stopping rates the two different altitudes will reflect this difference in spectrum shape at the two different energies. (This is easy to see if you suppose muons do not decay at all. This variation in the spectrum shape can be corrected for by calibrating the detector in a manner described below Like all charged particles, a muon loses energy through coulombic interactions with the matter it traverses. The average energy loss rate in matter for singly charged particles traveling close to the speed of light is approximately 2 Me vig/cm, where we measure the thickness s of the matter in units of g/cm". Here, s=px, where p is the mass density of the material through which the particle is passing, measured in g/cm, and the x is the particles pathlength, measured in cm. (This way of measuring material thickness in units of g/cm alle e effective thicknesses of two materials that might have very different mass densities. )A more accurate value for energy loss can be determined from the Bethe-Bloch equation Muon physics
11 Muon Physics Time Dilation Effect A measurement of the muon stopping rate at two different altitudes can be used to demonstrate the time dilation effect of special relativity. Although the detector configuration is not optimal for demonstrating time dilation, a useful measurement can still be preformed without additional scintillators or lead absorbers. Due to the finite size of the detector, only muons with a typical total energy of about 160 MeV will stop inside the plastic scintillator. The stopping rate is measured from the total number of observed muon decays recorded by the instrument in some time interval. This rate in turn is proportional to the flux of muons with total energy of about 160 MeV and this flux decreases with diminishing altitude as the muons descend and decay in the atmosphere. After measuring the muon stopping rate at one altitude, predictions for the stopping rate at another altitude can be made with and without accounting for the time dilation effect of special relativity. A second measurement at the new altitude distinguishes between competing predictions. A comparison of the muon stopping rate at two different altitudes should account for the muon’s energy loss as it descends into the atmosphere, variations with energy in the shape of the muon energy spectrum, and the varying zenith angles of the muons that stop in the detector. Since the detector stops only low energy muons, the stopped muons detected by the low altitude detector will, at the elevation of the higher altitude detector, necessarily have greater energy. This energy difference ∆E(h) will clearly depend on the pathlength between the two detector positions. Vertically travelling muons at the position of the higher altitude detector that are ultimately detected by the lower detector have an energy larger than those stopped and detected by the upper detector by an amount equal to ∆E(h). If the shape of the muon energy spectrum changes significantly with energy, then the relative muon stopping rates at the two different altitudes will reflect this difference in spectrum shape at the two different energies. (This is easy to see if you suppose muons do not decay at all.) This variation in the spectrum shape can be corrected for by calibrating the detector in a manner described below. Like all charged particles, a muon loses energy through coulombic interactions with the matter it traverses. The average energy loss rate in matter for singly charged particles traveling close to the speed of light is approximately 2 MeV/g/cm2 , where we measure the thickness s of the matter in units of g/cm2 . Here, s = ρx, where ρ is the mass density of the material through which the particle is passing, measured in g/cm3 , and the x is the particle’s pathlength, measured in cm. (This way of measuring material thickness in units of g/cm2 allows us to compare effective thicknesses of two materials that might have very different mass densities.) A more accurate value for energy loss can be determined from the Bethe-Bloch equation
dE 4T M 2mc26 Mev\ 1 2mca 0.3071 A B2 Here N is the number of electrons in the stopping medium per cm, e is the electronic charge, z is the atomic number of the projectile, Z and a are the atomic number and weight, respectively, of the stopping medium. The velocity of the projectile is B in units of the speed c of light and its corresponding lorentz factor is y. The symbol l denotes the mean excitation energy of the stopping medium atoms. Approximately, I=AZ, where A=13 ev. More accurate values for 1, as well as corrections to the Bethe-Bloch equation can be found in [ Leo, p26 A simple estimate of the energy lost ae by a muon as it travels a vertical distance H is AE=2 Mevig/cm*H P_air, where p_air is the density of air, possibly averaged over H using the density of air according to the " standard atmosphere. Here the atmosphere is assumed isothermal and the air pressure p at some height h above sea level parameterized by p= po exp(-h/ho), where po= 1030 g/cm is the total thickness of the atmosphere and ho=8. 4 km. The units of pressure may seem unusual to you but they are completely acceptable. From hydrostatics, you will recall that the pressure P at the base of a stationary fluid is P=pgh. Dividing both sides by g yields P/g=ph, and you will then recognize the units of the right hand side as g/em". The air density p, in familiar units of g/cm, is given by p=-dp/dh If the transit time for a particle to travel vertically from some height H down to sea level, all measured in the lab frame, is denoted by t, then the corresponding time in the particles rest frame is t' and given by t dh 6(h)(h) lere B and y have their usual relativistic meanings for the projectile and are measured in the lab frame. Since relativistic muons lose energy at essentially a constant rate when travelling through a medium of mass density p, dE/ds Co, so we have dE pCo dh, with Co=2 Me v/g/cm). Also, from the Einstein relation, E= ymc, dE= me' dx so dh=(mc" pco)dy Hence, dy pCo Here y is the muons gamma factor at height H and y2 is its gamma factor just before it enters the scintillator. We can take T2 =1.5 since we want muons that stop in the Muon physi
12 Muon Physics Here N is the number of electrons in the stopping medium per cm3 , e is the electronic charge, z is the atomic number of the projectile, Z and A are the atomic number and weight, respectively, of the stopping medium. The velocity of the projectile is β in units of the speed c of light and its corresponding Lorentz factor is γ. The symbol I denotes the mean excitation energy of the stopping medium atoms. Approximately, I=AZ, where A ≅ 13 eV. More accurate values for I, as well as corrections to the Bethe-Bloch equation, can be found in [Leo, p26]. A simple estimate of the energy lost ∆E by a muon as it travels a vertical distance H is ∆E = 2 MeV/g/cm2 * H * ρ_air, where ρ_air is the density of air, possibly averaged over H using the density of air according to the “standard atmosphere.” Here the atmosphere is assumed isothermal and the air pressure p at some height h above sea level is parameterized by p = p0 exp(-h/h0), where p0 = 1030 g/cm2 is the total thickness of the atmosphere and h0 = 8.4 km. The units of pressure may seem unusual to you but they are completely acceptable. From hydrostatics, you will recall that the pressure P at the base of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will then recognize the units of the right hand side as g/cm2 . The air density ρ, in familiar units of g/cm3 , is given by ρ = −dp/dh. If the transit time for a particle to travel vertically from some height H down to sea level, all measured in the lab frame, is denoted by t, then the corresponding time in the particle’s rest frame is t’ and given by Here β and γ have their usual relativistic meanings for the projectile and are measured in the lab frame. Since relativistic muons lose energy at essentially a constant rate when travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC0 dh, with C0 = 2 MeV/(g/cm2 ). Also, from the Einstein relation, E = γmc 2 , dE = mc2 dγ, so dh = (mc2 /ρC0) dγ. Hence, Here γ1 is the muon’s gamma factor at height H and γ2 is its gamma factor just before it enters the scintillator. We can take γ2 = 1.5 since we want muons that stop in the