WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.1 Measuring Cosmic Ray Muon Decay Constant and Flux R.C. Dennis, D.T. Tran, J. Qi, and J. Brown Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: March 17, 2015) We constructed an apparatus to measure cosmic ray muon flux as well as the muon lifetime. The primary components of our apparatus include scintillating material, photomultiplier tubes, and an oscilloscope. We found the muon lifetime to be τ = 2.05 ± 0.14 µs (95% CI).
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.2 I. INTRODUCTION Muons are one of the six elementary particles belonging to the lepton family. Cosmic ray muons are created as cosmic rays impact nuclei in the earth s atmosphere about 15km above sea level. The fact that they travel at relativistic speeds makes it possible for them to be detected even at sea level due to time dilation. As muons enter a scintillating bar, there is a small chance that the muons will collide with an electron or a nucleus in the scintillating bar, causing them to come to a stop[2][3]. Once they stop traveling at relativistic speed, they decay in our time frame, and the resulting electron excites the plastic to emit light, which can be captured for measurement[4][7]. The measurement of these muon s lifetimes and their angular distribution is a common undergraduate physics lab[5][10]. By understanding the work done in large institutions[2][3][4][9] as well as the work done in classrooms, we aim to create a simplified experiment for measuring the muon decay constant and eventually the angular distribution of the flux. This angular flux should match the model given by Aguilar and Ho[1][6]. II. MODEL Muons are elementary particles similar to electrons. They carry a 1 electric charge and 1/2 spin. The muons we can observe on earth mainly come from the cosmic radiation, which carries very high energies. The muons from cosmic rays travel at relativistic speeds. This is why they can travel through the atmosphere despite the short lifetime. In order to detect muons, scintillator bars are used. The scintillator bar can interact with ionizing radiation. When muons hit the scintillator bar, they can interact with the molecules in the scintillator bar to excite them. Then, molecules in the excited state will quickly drop to their ground state and emit photons. Those photons are detectable by a photomultiplier. Through this, we can count the rate of muons hitting the scintillator bar. Using the difference in the registered times by photomultipliers on both ends of the scintillator bar, we can find the position of the muons hitting the bar[2]. When we use the scintillator bars to detect muons, there is chance that some muons do not have enough energy to go through the bar. The muons that stay in the bar will quickly decay in accordance with µ e + ν e + ν µ (1)
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.3 or µ + e + + ν e + ν µ, (2) which both emit photons that can be detected by the photomultipliers. Since the average lifetime of muons are short (about 2.19 µs in vacuum), this second light pulse s signal will quickly be caught by the photomultiplier and the oscilloscope will show two peaks for the muon decay events. Just like the decay of most unstable particles, muon decay is also an exponential process, which means the number of muons remaining after time t is N(t) = N 0 e t/τ (3) where τ is the mean lifetime of the muon. Thus, by fitting our data of decay time of each muons, we can find the mean lifetime of muons. In the end, we also want to investigate the angular distribution of flux of cosmic muons. Theoretically, the flux of cosmic ray muons depends on the incident zenith angle. This relation can be described as: I(θ, h, E) = I 0 cos n(e,h) (θ) (4) where I 0 is the intensity of muons with zenith angle 0, θ is the zenith angle, h is the vertical distance traveled by muons, and E is the energy of the muon. The power of cos θ is a function about h and E, which is determined empirically. At sea level, experiments show that n 2. So in our experiment, we will use the relation [1][6] I(θ) = I 0 cos 2 θ. (5) III. SETUP Our first objective was to measure the muon lifetime. If our apparatus accurately measures this quantity, then we can be assured that we are properly capturing and analyzing muons. Our apparatus for measuring the muon decay rate is a simplified version of FIG 4 consisting of a small piece of scintillator material attached to a photomultiplier tube base assembly. This tube is attached to high voltage and the anode output runs into an oscilloscope. It is important to note that this scintillator bar was wrapped in a layer of aluminum foil followed by a layer of electrical tape and then completely covered with a thick, dark felt sheet. This eliminates light from the room and ensures that any light in the apparatus is
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.4 due to cosmic rays. The only portion of the scintillator material that is not covered in foil is the portion connected to the PMT. The PMT and scintillator material are separated by an optical couplant with a similar index of refraction (to prevent reflection and loss of light). As explained in the model, when muons traverse this bar, photons are emitted. These photons are captured by the photomultiplier tube and then converted into electrical signals to be read by the oscilloscope. IV. DATA ANALYSIS The simplest setup for collecting data from our apparatus is to have the oscilloscope set to trigger when a single event occurs and collect the signal reading from the PMT. When a muon enters the scintillator bar, it will typically go all the way through and exit the other side cause photons to be produced. It is possible, however, for the muon to lose enough energy that it can no longer make it through the bar. In this case, the muon comes to a stop and then decays inside the bar. In doing so it will emit a light pulse. This pulse should show up alongside the data that was triggered by the muon s entrance in the bar causing two peaks to appear in our oscilloscope rather than one. In FIG 1 we can see an example of a decay event in the raw data collected by our oscilloscope. Essentially, the oscilloscope triggered because a light pulse entered the PMT and then while the oscilloscope was collecting data from this event, another light pulse entered the device. However, by setting the oscilloscope to capture four events and displaying the averaged data, we were able to capture more decay events. The reason this works stems from the fact that the oscilloscope does not capture every decay event because some events occur while it is processing data and resetting to capture another event. By calculating the averaged data, we were able to collect four events at once and when a decay event occurs along with three single events, our averaged output still shows a second peak, but the amplitude of said peak is smaller. Fortunately, what we are really interested in is the time between the muon hitting the bar and decaying and this parameter is not lost by averaging. Notice here that there are three potential problems we plan to face: There is no guarantee that this is a decay event because it is possible that the second peak came not from a decay event, but from a separate muon entering the apparatus at an opportune time. We will discuss later how to deal with this issue.
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.5 It is possible that the decay events are occurring so quickly that we cannot register them as separate events (i.e. the two peaks are so close together that we assume this to be a single peak). It is also possible that the decay event happens so long after the original event that it is not captured in the same triggering of the oscilloscope (i.e. the peaks are so far apart that we register them as two separate muon events rather than a muon and a decay event). FIG. 1. This figure shows the capture of a muon that decayed. The graph indicates that there are two peaks occurring within a short period of time. The muon first entered the scintillator, emitting a light pulse in this process, which is the first peak triggered. Then, it was trapped inside the scintillator, decayed and the plastic emitted another light pulse, which is seen as the second peak. To sift through this data, we developed a computer program that checks the number of peaks in each trigger. If there are two peaks, the data is collected; otherwise, it is ignored. Remember previously that this data is averaged over four events. This means that there
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.6 exists the potential for the same decay event to be captured twice. To remedy this, we have our program check to make sure the location of the second peak is different from the one that was captured last. It is possible (though very unlikely) that we have two separate decay events that happen at the same time in succession in which case, we would accidentally be dumping good data. This should have no discernible affect on our results. After collecting many decay events, we subtracted the relative time of arrival for the first peak from the relative time of arrival for the second peak. Naturally, this value gives the time it took for the muon to decay. Next, we created a histogram of the data ignoring the first few bins. We outlined the reason for ignoring these bins above; these are all going to be underestimates because of the unintentional dumping of very close decay events. The ordinary model for muon decay is N(t) = N 0 e t/τ where N(t) is the number of events that occur before a time t and τ is the mean lifetime. Naturally, N 0 is the total number of events. However, this is not what our histogram reflects. We have the data to analyze this model, but because of the missing events on the tails of our exponential distribution, it makes much more sense to put all of that data points within a certain range into a single bin. This will give the derivative of the above function. The good news is that the derivative of an exponential function like this will take a similar form: dn(t) dt = N 0 τ e t/τ. However, there still exists this issue of collecting extraneous data points from mistaking two muons that enter the apparatus at nearly the same time and a decay event. Fortunately, these events should occur at random separations in time, so the end result is that our histogram data is shifted up by some factor (we call this factor b 2 ). This means that our final model is of the form: dn(t) dt = Ae t/τ + b 2 (6) where b 2 is positive and real. We found b 2 to be 0 s 1 with high precision which means that the error due to the background is negligible. V. CONCLUSIONS By finding the height of the peaks (in number of events) for each bin in our histogram, we fit our model to the data and added error bars. The result of this fit and 4029 decay events is a decay constant of τ = 2.05 ± 0.14 µs (95% CI). (7)
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.7 FIG. 2. This is a log plot showing the number of events versus the lifetime of muons with error bars. We use this data to fit a decay function due to the theory (black lines). With that fitting, we determined the mean lifetime of muons. This is slightly below the accepted value for muon decay in a vacuum, τ = 2.19703 ± 0.00004 µs, because the negative muons may also decay from proton capture, an effect which will always lower the value measured in experiment. In conclusion, we managed to find the muon decay constant with acceptable precision using only basic software, a high voltage supply, a photomultiplier tube, a scintillator bar, and an oscilloscope. No additional circuit components were necessary to achieve the desired result and we have shown the capability to move on to the next portion of our experiment. VI. FUTURE WORK In the future we plan to measure the angular distribution of the muon flux using a two bar setup. We will first put the bars in an intersecting arrangement (See FIG 4). We will collect two sets of data: the data from the top bar and the data from the bottom bar. By using a simple model, we will be able to determine where on that bar the muon struck. Our
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.8 software will only collect data when all four PMTs are triggered so that we can say with confidence that the muon entered our apparatus in the given region. If we do the same for the bottom bar, we may extrapolate where in the intersecting region this muon hit. We then use this data to calibrate our device as we expect that the average position of all of these muons will be in the center of the detecting region. After calibrating the device, we plan to change the setup (See FIG 3) so that the bars are parallel. By following the same procedure, we can determine at what angle the muon enters the top bar. We expect that the angular distribution will be proportional to cos 2 (θ).[6][1]
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.9 FIG. 3. This figure shows two scintillator bars in a stacked arrangement. When a muon hits the bar, a tiny portion of its kinetic energy is converted into detectable light with a high scintillator efficiency. This process of emission of visible radiation from the substance is called fluorescence. Essentially, when the muons pass through the crystalline structure, individual molecules in the plastic are excited into electronic states, which then relax by emitting light. The detectable light then propagates down the bar in both directions where it is collected by a photomultiplier on each end. The bars are connected to the photomultipliers using an optical couplant with a similar index of refraction so that the path of the light is not disturbed. This particular arrangement is useful because the muon enters the top bar at some angle such that the position it strikes the top bar and the bottom bar are different. The emitted light pulse is captured at each of the four ends of these bars and the data collected from this can be used to determine the angle at which it struck the apparatus. By doing this for a great deal of muons, we can determine the angular dependence of the flux.
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.10 FIG. 4. This is the overall setup. Two scintillation bars are placed perpendicular and on top of each other, making a detection area. The two scintillation bars are connected to the four PMTs at the ends of each scintillation bars, which send the signals to the oscilloscope. A muon decay event should trigger the oscilloscope. [1] J. A. Aguilar, et al., Zenith distribution and flux of atmospheric muons measured with the 5-line ANTARES detector. Astroparticle Physics, Volume 34, Issue 3, p.179-184, (2010) [2] J. Brown, et al., Going Beyond the Neutron Drop-Line with MoNA, Nuclear Physics News, Volume 20, Issue 3, p.23-26 [3] T. Baumann, et al., Construction of a modular large-area neutron detector for the NSCL, Nuclear Instr. and Meth. A543, 517-527 (2005) [4] T. Baumann, et al., Fabrication of a modular neutron array: A collaborative model to
WJP, PHY381 (2015) Wabash Journal of Physics v3.3, p.11 undergraduate research, American Journal of Physics (2005) [5] R.E. Hall, D.A. Lind, R.A. Ristinen, A Simplified Muon Lifetime Experiment for the Instructional Laboratory, American Journal of Physics, Volume 38, p.1196 (1970) [6] C. Y. E. Ho. Cosmic Ray Muon Detection using NaI Detectors and Plastic Scintillators, American Institute of Physics, Volume 3, (2008) [7] G. F. Knoll, Radiation Detection and Measurement 3rd, Ann Arbor, MI: John Wiley and Sons, Inc. (2000) [8] Roger J. Lewis, Automatic measurement of the mean lifetime of the muon, American Journal of Physics, Volume 50, p.894 (1982); [9] T. Baumann, et al., MoNA The Modular Neutron Array,Nuclear Instruments and Methods in Physics Research A50533 (2003) [10] T. Coan, T. Liu, J. Ye, A compact apparatus for muon lifetime measurement and time dilation demonstration in the undergraduate laboratory, American Journal of Physics, Volume 74, p.161-164 (2006).