Cosmic Ray Muons and the Muon Lifetime Experiment CRM University of Florida | Department of Physics PHY4803L | Advanced Physics Laboratory Objective eventually collide with an air molecule and ini- tiate a hadronic shower|a cascade of particles Four scintillation paddles and coincidence (mostly pions) that may undergo further nu- techniques are used to determine the overall clear reactions. Neutral pions (π0) decay in flux and angular distribution of cosmic ray into two gamma rays with a very short life- muons. The muon lifetime is measured us- time less than 10−17 s, which in turn gener- ing rare events where, after passage of a muon ate electromagnetic showers (e+, e−, γ) that into a scintillator is detected, its decay is also are not very penetrating. Charged pions (π+, detected a short time later. The distribution π−) that do not undergo further nuclear re- of the decay times provides information about actions will decay in-flight into muons (µ+, the average muon lifetime. Statistical uncer- µ−) and neutrinos (ν , ν ): π+ ! µ+ + ν , tainties appropriate for Poisson variables are µ µ µ π− ! µ− + ν with a lifetime of 26 nanosec- employed throughout the experiment. µ onds. Both the muon and its corresponding neutrino are classified as leptons | particles References that do not participate in nuclear reactions. 1. CMS Collaboration, Measurement of the The neutrinos have an extremely tiny capture charge ratio of atmospheric muons with cross-section, and typically pass through the the CMS detector, Physics. Letters B Vol- Earth without any further interactions. ume 692, August 2010, pp. 83-104. Since most cosmic rays and the nuclei they interact with are positive, positive cos- mic muons are more abundant than negative Introduction muons. Studies of cosmic ray muons below Cosmic rays are high-energy particles|mostly a momentum of 100 GeV/c using the Com- protons and alpha particles with a small frac- pact Muon Solenoid at CERN (see reference 1) tion of heavier nuclei and other subatomic par- found the ratio of positive to negative muons ticles such as electrons, positrons and antipro- to be 1.28. tons. Their origins in supernovae, quasars, Muons were discovered in cosmic rays by and other exotic astronomical events and how C. Anderson and S.H. Neddermeyer in 1937. they acquire their sometimes colossal energy There are two kinds of muon, the negative µ− (over 1020 eV) is a topic of current research. and its antimatter partner, the positive µ+. Cosmic ray muons are created when cos- They are essentially heavy versions of the elec- mic rays enter earth's atmosphere where they tron and its antimatter partner, the positron, CRM 1 CRM 2 Advanced Physics Laboratory having the same spin and charge, but with Exercise 1 (a) Explain the difference be- 2 a mass mµ = 105:66 MeV/c approximately tween dP in Eq.1 and dPe(t) in Eq.2. (b) 207 times larger than the electron. Muons Show that the expectation value for the decay are unstable | decaying into an electron or time hti | defined as the muon lifetime τµ | is + + positron and two neutrinos: µ ! e +νe+νµ, equal to 1=Γ. (c) Show that the muon \half- − − µ ! e + νe + νµ with an average lifetime life" (the time at which half of a large sam- τµ = 2:197 µs | about 100 times longer than ple of muons will have decayed) is given by that of the charged pion. t1=2 = τµ ln 2. Because the muon undergoes a 3-body de- cay, the kinetic energy of the emitted electron The differential flux of cosmic ray muons or positron is not fixed but has a broad distri- (per unit time, per unit area, per unit solid bution of values with a maximum (endpoint angle) at the surface of the Earth is approxi- energy) of 53 MeV in the rest frame of the mately described by: muon. This kind of energy spectrum is similar dN ≈ I cosk θ (3) to nuclear beta-decay (another 3-body decay) dA dΩ dt 0 where a neutron inside a nucleus decays into a proton, an electron, and an anti-neutrino. In where θ is the polar angle with respect to ver- −2 −1 −1 fact, the neutrino's existence was first postu- tical, k ≈ 2, and I0 ≈ 100 m sr s at sea lated to explain why electrons from beta-decay level. I0 can vary by a few percent with lati- are not emitted with a fixed energy as would tude and altitude as well as with atmospheric be predicted if the neutron decayed into only temperature and pressure. There is no ex- a proton and electron. pected dependence on the azimuthal angle φ. ◦ Once created, the muon decay is a com- Eq.3 is not expected to be valid for θ > 80 pletely random event that does not depend on where the Earth's curvature becomes an im- its past history. The probability dP of de- portant consideration. cay in the next infinitesimal time interval dt Solid angle is a three-dimensional analog of is independent of how long it has lived since an included angle in a two-dimensional plane. creation and is given by: Shown in Fig.1, an arbitrary solid angle Ω can be defined by the area A it would cover on a dP = Γdt (1) sphere of radius R centered at the apex of the solid angle. where the decay rate Γ is the inverse of the A lifetime: Γ = 1/τ . Ω = (4) µ R2 This decay process implies that the proba- Solid angles are expressed in the dimension- bility of a muon decay in the interval from t to less units of steradian, abbreviated sr.1 One t+dt (given that the muon exists at t = 0) fol- steradian is the solid angle covered by an area lows the exponential probability density func- of 1 m2 on a sphere with a 1 m radius. No- tion: tice that the solid angle for covering the entire dP (t) = Γe−Γtdt (2) e sphere (area 4πR2) is 4π sr. Here, the time t represents the time for a par- 1 ticular decay to occur and will be called a de- The units of steradian should be dropped where inappropriate; for example, in A = ΩR2 (from Eq.4), cay time. In one part of this experiment, you the units on the left are those of area (m2) and on will measure a large sample of decay times and the right they are solid angle times length squared compare with this exponential distribution. (sr m2=m2). August 6, 2021 Cosmic Ray Muons and the Muon Lifetime CRM 3 Muons lose energy as they travel through the atmosphere and other materials. The mean energy loss per unit length (called the stopping power) for any charged particle traversing a block of matter is governed by the Bethe-Bloch equation: dE Z 1 = −Kz2 · (7) dx A β2 "1 2m c2β2γ2T 2 δ # ln e max − β2 − 2 I2 2 p Here β = v=c and γ = 1= 1 − β2 are the usual relativistic factors, Z and A are the atomic number and mass of the medium, z is the charge of the incident particle, Tmax is the maximum kinetic energy that may be trans- Figure 1: The solid angle Ω subtended from ferred to an electron in a collision, and K, I, the origin of a sphere of radius R by an arbi- and δ are atomic factors. trary area A on the sphere is Ω = A=R2. A scaled version of the stopping power is given as a function of momentum for muons incident on copper in Fig.3. For reasons to Figure2 shows the geometry for Eq.3. be discussed shortly, the values are in units dN=dt ≈ I cosk θ dA dΩ would be the rate at 0 of MeV cm2/g and must be multiplied by the which muons pass through an area dA coming density of copper (8.94 g/cm3) to get the stop- from a polar angle θ within the solid angle dΩ. ping power in MeV/cm. The area dA should be considered to have its The general shape of this graph is common normal along the incoming direction as shown to charged particles other than muons. At in Fig.2a and thus the area orientation would low momentum, charged particles rapidly lose vary as θ or φ varies. Experimentally, the area energy as they ionize atoms in the medium element is sometimes fixed in the horizontal and the stopping power is high. The stopping plane with the area normal oriented vertically power decreases with increasing momentum as in Fig.2b. A comparison between equal ef- and approaches a minimum as the particle mo- fective areas in the two cases is demonstrated mentum gets into the relativistic regime. It in Fig.2c with then increases only gradually from the mini- dA = dA0 cos θ (5) mum as the particle momentum continues to increase. Thus, for an area element in a horizontal Figure3 can also be used for materials plane, Eq.3 would be other than copper. The scaling principle is that the actual energy lost (not the stopping dN k+1 power) should be roughly the same for pas- 0 ≈ I0 cos θ (6) dA dΩ dt sage through different materials as long as the where 0 ≤ θ ≤ π=2, i.e., the muons only come product of the travel length and the density from the upper half plane. of the material is the same | passage through August 6, 2021 CRM 4 Advanced Physics Laboratory Figure 2: Muons arrive from all overhead directions (solid angles) and their flux is described as a number per unit time per unit area per unit solid angle.