Muon physics T.E. Coan and. Ye 051110.0
Muon Physics T.E. Coan and J. Ye v051110.0
Contents INTRODUCTION OUR MUON SOURCE MUON DECAY TIME DISTRIBUTION DETECTOR PHYSICS… INTERACTION OF MATTER 335579 U/H CHARGE RATIO AT GROUND LEVEL BACKGROUNDS…10 FERMI COUPLING CONSTANT GF. TIME DILATION EFFECT ELECTRoNICS 15 SOFTWARE AND USER INTERFACE Control 21 Data file format Rate meter 4 Muons through detector Muon Decay Time histogram 24 THE LIFETIME FITTER MUON DECAY SIMULATION UTILITY SOFTWARE REfERENCES Instrumentation and Technique osmic Rays 27 General Muon Lifetime in Matter....... GETTING STARTED SUGGESTED STUDENT EXERCISES HOW TO GET HELP Muon physics
2 Muon Physics Contents INTRODUCTION ................................................................................................................ 3 OUR MUON SOURCE ........................................................................................................ 3 MUON DECAY TIME DISTRIBUTION ................................................................................. 5 DETECTOR PHYSICS ......................................................................................................... 5 INTERACTION OF µ − ’S WITH MATTER................................................................................ 7 µ + /µ− CHARGE RATIO AT GROUND LEVEL ....................................................................... 9 BACKGROUNDS.............................................................................................................. 10 FERMI COUPLING CONSTANT GF.................................................................................... 10 TIME DILATION EFFECT ................................................................................................. 11 ELECTRONICS................................................................................................................. 15 SOFTWARE AND USER INTERFACE ................................................................................. 19 Control ...................................................................................................................... 21 Data file format......................................................................................................... 23 Monitor ..................................................................................................................... 24 Rate Meter................................................................................................................. 24 Muons through Detector........................................................................................... 24 Muon Decay Time Histogram................................................................................... 24 THE LIFETIME FITTER .................................................................................................... 25 MUON DECAY SIMULATION........................................................................................... 25 UTILITY SOFTWARE ....................................................................................................... 25 REFERENCES .................................................................................................................. 27 Instrumentation and Technique ................................................................................ 27 Cosmic Rays.............................................................................................................. 27 General ..................................................................................................................... 28 Muon Lifetime in Matter ........................................................................................... 28 GETTING STARTED....................................................................................................... 29 SUGGESTED STUDENT EXERCISES.................................................................................. 31 HOW TO GET HELP.......................................................................................................... 33
Introduction The muon is one of nature's fundamental"building blocks of matter"and acts in many ways as if it were an unstable heavy electron, for reasons no one fully understands Discovered in 1937 by C w. Anderson and s.H. Neddermeyer when they exposed a cloud chamber to cosmic rays, its finite lifetime was first demonstrated in F Rasetti. The instrument described in this manual permits you to measure the charge averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muon as a function of height above sea-level and to demonstrate the time dilation effect of that can be used for experimental tests of standard probability distribution n numbers special relativity. The instrument also provides a source of genuinely rando Our muon source The top of earth's atmosphere is bombarded by a flux of high energy charged particles produced in other parts of the universe by mechanisms that are not yet fully understood The composition of these "primary cosmic rays" is somewhat energy dependent but a useful approximation is that 98%o of these particles are protons or heavier nuclei and 2%0 are electrons Of the protons and nuclei, about 87%o are protons, 12%o helium nuclei and the balance are still heavier nuclei that are the end products of stellar nucleosynthesis See simpson in the reference section for more details The primary cosmic rays collide with the nuclei of air molecules and produce a shower of particles that include protons, neutrons, pions(both charged and neutral), kaons, photons electrons and positrons. These secondary particles then undergo electromagnetic and nuclear interactions to produce yet additional particles in a cascade process. Figure 1 ndicates the general idea. Of particular interest is the fate of the charged pions produced in the cascade. Some of these will interact via the strong force with air molecule nuclei but others will spontaneously decay (indicated by the arrow) via the weak force into a muon plus a neutrino or antineutrino: π→μv 兀→μv The muon does not interact with matter via the strong force but only through the weak and electromagnetic forces. It travels a relatively long instance while losing its kinetic energy and decays by the weak force into an electron plus a neutrino and antineutrino We will detect the decays of some of the muons produced in the cascade (Our detection efficiency for the neutrinos and antineutrinos is utterly negligible. Muon physics
3 Muon Physics Introduction The muon is one of nature’s fundamental “building blocks of matter” and acts in many ways as if it were an unstable heavy electron, for reasons no one fully understands. Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F. Rasetti. The instrument described in this manual permits you to measure the charge averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons as a function of height above sea-level and to demonstrate the time dilation effect of special relativity. The instrument also provides a source of genuinely random numbers that can be used for experimental tests of standard probability distributions. Our Muon Source The top of earth's atmosphere is bombarded by a flux of high energy charged particles produced in other parts of the universe by mechanisms that are not yet fully understood. The composition of these "primary cosmic rays" is somewhat energy dependent but a useful approximation is that 98% of these particles are protons or heavier nuclei and 2% are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and the balance are still heavier nuclei that are the end products of stellar nucleosynthesis. See Simpson in the reference section for more details. The primary cosmic rays collide with the nuclei of air molecules and produce a shower of particles that include protons, neutrons, pions (both charged and neutral), kaons, photons, electrons and positrons. These secondary particles then undergo electromagnetic and nuclear interactions to produce yet additional particles in a cascade process. Figure 1 indicates the general idea. Of particular interest is the fate of the charged pions produced in the cascade. Some of these will interact via the strong force with air molecule nuclei but others will spontaneously decay (indicated by the arrow) via the weak force into a muon plus a neutrino or antineutrino: π + → µ + νµ π −→ µ − νµ The muon does not interact with matter via the strong force but only through the weak and electromagnetic forces. It travels a relatively long instance while losing its kinetic energy and decays by the weak force into an electron plus a neutrino and antineutrino. We will detect the decays of some of the muons produced in the cascade. (Our detection efficiency for the neutrinos and antineutrinos is utterly negligible.)
μ Figure 1. Cosmic ray cascade induced by a cosmic ray proton striking an air molecule Not all of the particles produced in the cascade in the upper atmosphere survive down to sea-level due to their interaction with atmospheric nuclei and their own spontaneous decay. The flux of sea-level muons is approximately I per minute per cm"(see http://pdq.lbl.govformoreprecisenumbers)withameankineticenergyofabout 4 Gev Careful study [pdg Ibl. gov] shows that the mean production height in the atmosphere of the muons detected at sea-level is approximately 15 km. Travelling at the speed of light, he transit time from production point to sea-level is then 50 usec. Since the lifetime of at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sea level muon flux is qualitative evidence for the time dilation effect of special relativity Muon physics
4 Muon Physics Figure 1. Cosmic ray cascade induced by a cosmic ray proton striking an air molecule nucleus. Not all of the particles produced in the cascade in the upper atmosphere survive down to sea-level due to their interaction with atmospheric nuclei and their own spontaneous decay. The flux of sea-level muons is approximately 1 per minute per cm2 (see http://pdg.lbl.gov for more precise numbers) with a mean kinetic energy of about 4 GeV. Careful study [pdg.lbl.gov] shows that the mean production height in the atmosphere of the muons detected at sea-level is approximately 15 km. Travelling at the speed of light, the transit time from production point to sea-level is then 50 µsec. Since the lifetime of at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel muon flux is qualitative evidence for the time dilation effect of special relativity. π + p µ + ν π 0 γ γ e + e e + − e − π − µ − ν n π +
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 physics
5 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
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 =
