WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.1 A Measurement of the Cosmic Ray Muon Flux Through Large-Area Scintillators T. R. Buresh, M. J. Madsen, Z. J. Rohrbach, and J. M. Soller Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: December 15, 2010) Understanding the characteristics of the cosmic ray muon flux is vital for its use in security and archaeological applications. Anomalies in the muon flux as detected by a large-area scintillator detection array are key to these applications. As a step toward developing a detector array, we utilized the neutron detector bars from the MoNA LISA project to measure the angular dependence of the cosmic ray muon flux. Our data are consistent with the accepted cosine-squared angular dependence.
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.2 Cosmic Ray muons are created as a byproduct of cosmic ray decay after the rays enter the earth s atmosphere. Muons are elementary particles that belong to the lepton family. There are six leptons: electron, muon, tau, and an associated neutrino for each of these particles [?? ]. The muons travel towards the earth at relativistic speeds, so they can be detected on Earth s surface despite their short lifetime. The trajectories of muons are deflected by the presence of high-density materials such as lead. Being able to detect these trajectories has potential applications in the field of national defense by giving investigators the ability to detect heavy lead shielding inside trucks and other vehicles [? ] and in the field of archeology by giving explorers the ability to detect buried chambers [? ]. By knowing the angular distribution of muon flux, detectors are used to display anomalies in the muon flux through objects and provide an outline of an object with minimum disturbance to its surroundings. Measurements of muon lifetime, and the angular distribution of muons have been studied by Lawrence Berkeley Labs as aspects of larger projects as well as countless undergraduate labs [??? ]. To detect the presence of cosmic ray muons, we used 2-meter bars of scintillating plastic attached to photomultiplier tubes for detection and compilation of the muon data. Scintillator bars are good detectors for muons because when a muon enters through the scintillator it excites many atoms making a flash of light which bounces in the transparent scintillator until it reaches a photomultiplier tube (PMT) at its end [? ]. Muons that come in from the upper atmosphere reach the earth at an intensity distribution dependent on the zenith angle φ. This intensity dependence has been observed to be I 0 (φ) = cos 2 φ, (1) although it should theoretically be cos φ [?? ]. We used our setup to detect this zenith angle dependence. Information about the trajectories of the muons can be inferred by measuring the time difference between light pulses detected by two PMTs. As shown in Fig.??, the pulse of light created by scintillating plastic at an event is measured by the PMTs, which output an electrical response pulse. To measure the time difference between the two events measured by two PMTs, this pulse is sent to a Canberra Constant Fraction Discriminator (CFD), model 2126, which converts the response pulse from the PMTs into a square pulse located at the peak of the pulse. From the CFD, the difference in time between the square pulses
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.3 is converted by the Canberra Time to Analog Converter (TAC), model 2145, so the pulse from the PMT into a potential difference pulse proportional to the time between events. This pulse is sent from the TAC to the Spectech Multi-Channel Analyzer (MCA), UCS-20, where the differing voltages assigned by the TAC are placed in bins that count the events cumulatively. HV PMT CFD Start TAC Stop ΔV Δt MCA Counts Δt +1 count Bins PMT CFD ΔV Bin # HV Δt Δt FIG. 1. A schematic diagram of the evolution of the signal for each event, as it travels from the Photomultiplier Tube to the computer for analysis. As muons enter into the bars, the light pulse emitted through the plastic travels at a speed dependent upon the density of the plastic used to create the bars. Thus, before measuring the angular dependence of the muon flux, we need to know the index of refraction of our scintillating material. For a bar of length l with a PMT on either end, as shown in Fig.??, if a muon enters at a distance y µ from the end of the bar we will call End 1, then the amount of time delay before the event registers on the TAC, based upon the velocity v that the light travels in the bar, will be t 1 = l y µ v + t c1, (2) where t c1 is the time of the delay cable associated with the PMT on End 1. When the light propagates to the opposite end of the bar, End 2, then the time delay before the event
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.4 l n t c1 t c2 End 2 Plastic Scintillator End 1 y µ μ To MCA FIG. 2. The setup for the measurement of the index of refraction for the scintillator bar. When a muon enters the bar, a pulse of light is sent in both directions down the bar to the PMTs, whose time difference is measured by the MCA. registers on the TAC will be t 2 = y µ v + t c2, (3) where t c2 is the time of the delay cable associated with the PMT on End 2. The time delay associated with each PMT due to the cables was critical in insuring the firing of the PMTs occurred of the same order for every event recorded. We needed the order of the PMTs measurements to remain the same with respect to each other, so our time difference would always be positive, and not switch between positive and negative. The time difference registered on the TAC is t = t 2 t 1 = 2y µ l v + t c, (4) where t c = t c2 t c1. The maximum time difference t max will occur when the muon enters the bar at one end, where y µ = 0. t max = l v + t c. (5) Likewise, the minimum time difference t min will occur when the muon enters closest to the opposite end, where y µ = l. Therefore, the range of possible t for this setup is t min = l v + t c. (6) t max t min = 2l v. (7)
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.5 We also know that the speed of light within the bar is reliant upon the index of refraction of the scintillating plastic and the speed of light by v = c n, (8) where n is an effective index of refraction for our particular setup. Thus, we get t max t min = 2ln c n = c( t max t min ) 2l The setup for the experiment, shown in Fig.??, used a singular scintillator bar, with a PMT connected on each end. The pulses received by the PMTs were then sent to the MCA through the procedure shown above in Fig.??. We powered the PMTs using a Canberra High Voltage Power Supply set at 1500V for End 1 and 1520V for End 2, as given by the MoNA LISA group to equalize the amplifying effects of the PMTs on the pulse in the case of a muon event. The CFDs were set to a threshold of 1V, which acted as a gate, so that only muon interactions with the bars would trigger the equipment. The TAC was set to a time range of 50 ns full scale maximum between the start and stop triggers to record an event, to insure each recorded event was a muon passing through both bars. We set the End 2 PMT as the trigger for the TAC to record data, and the End 1 PMT to trigger the stop of a legitimate event, to insure we recorded legitimate cases of the same muon passing through both bars. The distribution of the number of events that were assigned to bins by the spectrometer, shown in Fig.??, is square, because the number of muons that pass through the bar should be evenly distributed across the entire length of the bar. We converted the bin numbers assigned by the spectrometer to specific times by sending a pulse of known delay through the setup, and seeing which bin the spectrometer assigned the event to [? ]. By using this conversion, we plotted the distribution of muons passing through the bar against the time separation between the PMTs in Fig.??. A critical aspect of the distribution of events is that it is not completely square. The PMTs have a Gaussian response function, so the edges of the graph are not clearly defined. By convolving a Gaussian function with the expected square function, we fit to the data to obtain a clear value for the time difference between the PMT responses. We found the value of t through this fit, then used our function relating the index of refraction to the (9) (10)
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.6 Counts 800 600 400 200 Δt min Δt max 0 0 10 20 30 40 Time ( ns) FIG. 3. The distribution of the counts against the time that each event occurs takes on the square shape that we expect. We found the time difference represented at each end by fitting the shape with an error function fit, where t is the difference in the time parameters the error function approximations. For the given data, the value of the minimum time difference is t min = 11.835 ± 0.029s and the value of the maximum time difference is t max = 37.468 ± 0.041s, where each value has an associated uncertainty of a 95% gaussian confidence interval. time difference between the PMTs response to find a value of n, which we found to be n = 1.7122 ± 0.0035(95% CI). Now, we look at the angular dependence of the muon flux. As a muon enters the top scintillator at an angle φ a distance y µ1 from the end of the bar as shown in Fig.??, it will cause a flash in the scintillator that will propagate toward a PMT at the end of the bar at speed c/n. Thus, it will take a time t 1 = ny µ2 /c + t c1 to reach the PMT and travel through the cable for to the MCA for t c1. The muon will continue to pass through the top scintillator and then propagate through the air at a speed c before entering the bottom scintillator at a distance y µ2 from the end of the bar and creating a second flash that propagates toward the bottom PMT at the same speed c/n. This flash will reach the bottom PMT and the signal will reach the MCA at t 2 = h/c cos φ + ny µ2 /c + t c2 after the first flash. Thus, the
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.7 φ l y μ1 Plastic Scintillator Cable Delay t c1 h h/cos φ Cable Delay To MCA t c2 Plastic Scintillator y μ2 μ FIG. 4. The setup for investigating the azimuthal angular dependence. Two scintillator bars are positioned directly on top of each other with vertical separation h. A muon coming in at an angle φ will enter the top scintillator, causing a flash that will be detected by the PMT, and then pass through the bottom scintillator, causing a second flash. Once the PMTs register the data, this signal will be passed through the timing equipment as explained above. time difference between the two flashes is given by t(φ) = t 2 t 1 = t c + 1 ( ) h c cos φ + n(y µ2 y µ1 ), (11) = t c + h (1 n sin φ). c cos φ (12) As mentioned above, we expect that the muon intensity varies by azimuthal angle φ as given in Eq. (??). However, we have to take into account the fact that not all muons that hit the top bar at some angle φ also hit the bottom bar. This idea is illustrated in Figure??, where we see that only a fraction w/l of the muons coming in and angle φ that hit the top bar also hit the bottom bar. By trigonometry, we see that w/l = 1 h l tan φ. (13) Also, we know that light propagation in the scintillating material attenuates at e αy for attenuation constant α. This means that probability that a flash in one bar makes it to the
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.8 φ φ φ h w s l First trajectory that hits both bars Last trajectory that hits both bars FIG. 5. Not all muons that pass through the top bar will make it to the bottom bar. Only a fraction w/l of the muons coming in at an angle φ will also hit the bottom bar. PMT is relative the the flash in the other bar making it to the PMT is given by e α yµ = e αh tan φ. (14) Therefore, we should need to modify our observed intensity I (which is proportional to probability) to say that I(φ) = (w/l)e α yµ I 0 (φ), (15) = (1 hl ) tan φ e αh tan φ cos 2 φ. (16) In our experiment, l = 2.035 ± 0.002m. Figure?? is a plot of our collected data along with our model, a parametric plot of ( t(φ), I(φ)) as given in Eqs. (??) and (??), where we have used the n found in part I, for bar separations h =.71 ±.01 m, h = 1.002 ±.002 m, and h = 2.60 ±.05 m. It is clear that this model does not agree with our data generally. However, our model is in good agreement for the angular region π/40 < φ < π/40, as seen in Fig.??. Therefore, we can measure the angular dependence of the muon flux at an azimuthal angle θ perpendicular to the bars, as shown in Fig.?? by getting data within the
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.9 Normalized Counts 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 t ns FIG. 6. A plot of our data with the parametric plot ( t(φ), I(φ)) of our model. The blue data corresponds to h =.71 ±.01 m, the green data to h = 1.002 ±.002 m, and the red data to h = 2.60 ±.05 m. It is clear that our model is in disagreement with our data, particularly for the smaller h. However, the model is in fairly good agreement for the range π/40 < φ < π/40 (indicated by the brown highlight) in all cases. φ-region in which our model is applicable for different azimuthal θ, this region is shown in Fig.??. For each azimuthal angle θ, we get the data shown in Fig.??. Bins 245 to 265 correspond to the two ends t(±π/40), sum up all the counts in the bins in this range. This gives us the data shown in Fig.??. All points are in agreement with the cosine squared model except for the 60 data point. We believe that this anomaly occurred because at this angle, we unwittingly oriented the apparatus toward a neutron howitzer, thereby substantially increasing the rate of flashes in the scintillators. We ve excluded the 60 point out of the fit for this reason. We have experimental confirmation of the cosine squared angular dependence of the muon flux. Future work can focus on improving the timing model so as to obtain better agreement with the data in Fig.??. Also, the basic method put forward in this paper can be extended
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.10 θ h Plastic Scintillator Plastic Scintillator FIG. 7. A side view of the setup shown in Fig.?? used to change the azimuthal angle θ at which the device was oriented. so as to be able to detect the muon shadow caused by lead between two scintillator arrays. [1] L. W. Alvarez. Nobel Prize Lecture. Recent developments in particle physics. (1968). [2] J. Bechhoefer, Scott Wilson. Am. J. P. 70, 4 (2002). [3] Ante Benic, Mateja Boskovic, Marin Vuksic, Branko Durdevic. Studying cosmic muons using scintillation detectors. [4] Evan Berkowitz. The Speed and Mean Lifetime of Cosmic Ray Muons. MIT Department of Physics (2006). [5] K. Borozdin et al. Los Alamos National Lab. Information Extraction from Muon Radiography Data. CITSA. (2004). [6] J. Danneskiold. Los Alamos muon detector could thwart nuclear smugglers Los Alamos National Lab. http : //www.lanl.gov/news/index.php?fuseaction = home.story&story id = 2324. [7] James Doig. The Speed and Decay of Cosmic Ray Muons. MIT (2001). [8] Lulu Liu, Pablo Solis. The Speed and Lifetime of Cosmic Ray Muons. MIT (2007).
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.11 θ=15 θ=30 θ=0 θ=45 θ=60 FIG. 8. A view of the region used to measure the angular dependence of the muon flux showing θ and φ. [9] L.W. Lupinski, R. Paudel, and M.J. Madsen. Measuring the Muon Lifetime. Wabash Journal of Physics. PHY381, (2009). [10] Muons. The Cosmic Connection. Lawrence Berkeley National Lab. http : //cosmic.lbl.gov/skliewer/cosmic Rays/Muons.htm [11] Bruno Rossi, Norris Nereson. Experimental Determination of the Disintegration Curve of Mesotrons. Physical Review. 62, 11 (1942). [12] Thornton, Stephen T. ; Rex, Andrew, Modern Physics for Scientists and Engineers Third Edition, Thomson Books, 2006. [13] Brad Vest, Jacob Gastilow. Measuring the Speed of a Muon. Wabash College Journal of Physics. 2010.
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.12 0.008 θ =0 counts second 0.006 0.004 θ =30 θ =15 θ =60 0.002 θ =45 0.000 0 100 200 300 400 500 600 bins FIG. 9. Our flux data for varying rig angles θ. As expected, the peaks of all three angular data distributions correspond to the same bin range. The decrease in count intensity per unit time, with the normalized data, agrees with the cos 2 θ intensity distribution expected.
WJP, PHY381 (2010) Wabash Journal of Physics v2.4, p.13 counts s in peak region 0.15 0.10 0.05 0.00 0 10 20 30 40 50 60 70 FIG. 10. As the angle increases from the vertical, the intensity of the muons passing through both bars decreases according to a cos 2 θ model. The fit is (.204±.014) cos 2 (θ) (.045±.011 counts/ns). The data points from the five angular distributions of 0, 15, 30, 45, and 60 degrees are shown. The data point from 60 degrees is not included in the fit due to an extremely large variation from the model. We believe that this anomaly is due to the fact that at this orientation there was an artificial neutron source in the detection area. This variation is explained by extra muons from radioactive material stored in the next room.