S13MeasuringMuons

Measuring the Velocity of Cosmogenic Muons

Matthew Epland and Blake Antos

School of Physics and Astronomy

University of Minnesota - Twin Cities

Minneapolis, MN

May 15, 2013

Abstract

Using coincident scintillator panels and photomultiplier tubes the mean velocity of cosmogenic muons was measured to be <v_μ> = (0.926 ± 0.018) c which is 3.5σ below the accepted value of 0.99 c. The data collected in this experiment showed an unusual dependence on the detector energy threshold which somewhat reduces our confidence in this result and helps explain the discrepancy from the accepted value. The mean velocity measurement was made by measuring the mean time of flight of muons through the detector apparatus in many different configurations and comparing the times to the mean path length of muons through the detector generated from Monte Carlo simulations.

Introduction

Cosmic rays are streams of highly energetic particles including protons, electrons, and nuclei originating primarily from supernovae. These particles, upon interaction with our atmosphere, produce secondary particles via scattering which cascade down towards the Earth, see below.

Cosmic Ray Shower

One common particle in these shower events is the muon, a negatively charged lepton with a mass about 200 times that of an electron. The Earth is constantly bombarded with these cosmogenic muons travelling at nearly the speed of light, c. The muons’ velocity at the surface of the earth can be found by measuring the time required for the muons to travel a known distance.

In this experiment, we used a set of three scintillating detectors at known distances from each other to measure the average velocity of cosmogenic muons. Prior experiments using similar methods have measured v_μ to be (0.993 ± 0.002) c [3]. We aimed to verify this result and improve upon its precision by angling the center detector to narrow the timing distribution peaks by selecting certain muons over the others. This measurement is challenging to make as v_μ is very close to the speed of light, thus requiring timings to be made with nanosecond precision.

Background Theory

Cosmogenic muons travel through earth’s atmosphere at ultra-relativistic speeds, causing them to experience time dilation and Lorentz spatial contraction. As the average lifetime of a muon in its own rest frame is long for a particle, τ = 2.2 μs, but still short on an absolute scale. Ignoring relativistic effects would mean that most muons could only travel 660 meters, even at velocities near the speed of light. We therefore must account for relativistic effects as cosmogenic muons are generated primarily at an altitude of approximately 15 km and yet they are readily observable at the surface of the earth. The lifetime of the muon in the lab frame is τ/γ, where 1/γ ≈ 38 for a typical 4 GeV cosmogenic muon. Thus, with a time dilated lifetime of τ/γ = 83.6μs, cosmogenic muons can travel approximately 25 km which explains their presence at the earth’s surface.

Cosmogenic muons have a distribution of energies due to quantum statistics from the decay of the primary particles in the cosmic rays, and thus a distribution of velocities. This induces some uncertainty in this experiment’s velocity measurement, but by acquiring a substantial number of measurements, we can decrease this uncertainty to obtain a precise measurement of the mean velocity. The angular distribution of cosmogenic muons goes as as cos²(θ) [1], in terms of flux per unit solid angle, which can reduce the precision of our velocity measurement because of varying path lenghts through our detectors.

Experimental Setup

Detector Assembly

Cosmogenic muons were detected with a set of three plastic scintillator panels attached to Hamamatsu R1306-14 2” photomultiplier tubes (PMTs). The PMTs were powered by a high voltage DC power supply providing the specified individual voltages, which were all around 1.3 kV. The first two panels, A and B, were used to make a time of flight measurement, Δt, while the third panel, C, provided coincidence background rejection. The B panel was moved to various heights, h, and angles, α, in order to produce many different distributions of Δt that could be analyzed to determine the muons mean velocity, <v_μ>. The panels were aligned in the x and y directions to within 1 cm using a plumb bob and h was measured with an uncertainty of ± 0.2 cm.

Experimental Apparatus

Data Acquisition System

The analog PMT signals from each panel were fed into one channel of a LeCroy 821 quad discriminator. All three discriminator channels were set to the same threshold, which was initially set to its minimum at -31 mV but was later raised to -82 mV. When a PMT signal exceeded the set threshold, the discriminator would trigger, producing a standard NIM logic output pulse. The discriminator output of panels A and C were sent to a CAEN N455 quad coincidence unit, essentially an AND gate, after passing the A signal through a delay cable, D_A. This was done so that, for an A and C coincident event, the signal from A arrived 5 to 10ns after the C signal. In this way C was effectively an enable signal, allowing the coincident unit to always trigger on the A signal.

Typical Event Timing Diagram

The output of the coincidence unit was used to start our timing unit, an Ortec 566 time to amplitude converter, TAC. The discriminator signal from the B panel, delayed by a significant amount of time, D_B, was used as the stop signal for the TAC. The delay D_B was selected so that the stop signal arrived about 25ns after the start signal for midrange heights. This put all of the time intervals for all the tested heights around the middle of the TACs range. In this arrangement the TAC measured the time of flight of the muon between panels A and B, Δt, plus a constant offset that could be latter removed in the analysis process. If no stop signal was received after 50ns the TAC would reset automatically, effectively providing B coincidence rejection in addition to the C coincidence rejection done by the coincidence unit. Finally, the output voltages from the TAC were passed to a 2048 channel Ortec multichannel Analyzer, MCA, which saved the spectrum of Δt measurements on a PC.

An A, B and C coincident event rate on the order of 0.16 Hz, one every 6 seconds, was observed with the -31 mV discriminator threshold while the final threshold of -82 mV reduced the event rate to the order of 0.0047 Hz, one every 3.5 minutes.

Monte Carlo Simulations

A Monte Carlo simulation of the experimental apparatus was written in C++. The simulation generated random events in the A panel by choosing x, y and φ coordinates from a uniform distribution, setting z to be in the center of the A panel, and selecting θ from the appropriately transformed cos²(θ) distribution described in the theory section. The simulation would then check that generated muon intersected the B and C panels. If it did the path length between the middle of the A and B panels, L, would be calculated. Additionally the time difference between the A and B panels, Δt_MC, was calculated by adding the shortest time for light created at the intersection of the muon and the B panel to reach the B panel PMT to the time of flight between panels and subtracting the same shortest light propagation time in the A panel. The index of refraction for common plastic scintillators, n = 1.58, was used to find the speed of light in the scintillators while the muons velocity was set to the speed of light. By simulating 10⁵ events a mean path length value, <h>, could be calculated for each setting of h and α. The simulated Δt_MC peaks mirror those seen in the actual experiment but were considerably narrower due to the Monte Carlo not including the inherent jitter of the scintillator panels and PMT, see below. Additionally a fit of the Monte Carlo peak centers versus <h> returned a velocity of <v_μ>_MC = (1.007 ± 0.003) c. Together these two checks confirmed that the Monte Carlo simulation was working as intended and its results could be trusted.

Monte Carlo vs Experimental for h = 111.3 cm: Monte Carlo <Δt>_MC distribution (solid blue) versus experimental distribution (open circles) for α = 0, h = 111.3 cm. The experimental data was collected over 71.85 hours while the Monte Carlo simulation consisted of 10⁵ detected events, rebinned here in 0.01 ns bins. The Monte Carlo data has been shifted over to the right by 27 ns so the centers of both peaks are aligned. This shift is valid as it is accounting for cable delays and other constant delays in the data acquisition equipment that are not included in the Monte Carlo <Δt>_MC distribution.

Analysis and Results

Flat Panel Analysis

Twelve runs were done at eight evenly spaced heights with the B panel flat, α = 0. Four of the heights were done twice to verify that the data points were consistent. Most runs were done over 34 hours but three runs, at the bottom, middle and top heights, were counted for 70 hours.

For each run the data from the MCA was a histogram of counts versus Δt. The peak of each run was fit to a Lorentzian in Origin, see below. The center of the peak, x_c, was the only fit parameter of real interest as it represents the mean time of flight for the muons between panel A and B, <Δt>.

Lorentzian fit of data from the h = 111.3 cm, α = 0, 71.85 hour run: This h was in the center of the apparatus and was a fairly typical peak for this experiment. The data seen here has been rebinned 10:1. The peak center, x_c, was the main result of the fit as it and its uncertainty are used further on in the analysis as the mean time of flight, <Δt>, for this height.

Monte Carlo simulations were performed for each height resulting in mean path length values, <h>. By performing a reduced χ² linear fit on <Δt> vs <h> for all the runs the mean velocity of the muons was found, from the inverse of the fit slope, to be <v_μ> = (0.926 ± 0.018) c, see below. The cable delays are a constant offset for all the runs and therefore do not figure into the slope, but instead are factors in the y-intercept.

Plot of the mean time of flight, <Δt>, versus the mean path length, <h>, for all the flat panel runs: The fitted slope was 0.03604 ± 0.00071 cm/ns, which upon inversion gives a mean velocity of <v_μ> = (0.926 ± 0.018) c. The reduced χ² value was 2.697. The <Δt> values were determined from the individual peaks Lorentzian fits while the <h> values were calculated in the Monte Carlo simulation. The four lowest heights corresponding to the 4 largest values of <h> were measured twice to verify that the data points were reproducible.

Angled Panel Analysis

Two additional runs were performed at mid-range heights while varying α. For these runs α was set to 19.23° and 40.32° respectively while h was kept at approximately the same value; due to the position of the fulcrum on the B panel h varied slightly with α. Monte Carlo simulations for the two angled runs confirmed the hypothesis that increasing α will help narrow the Δt peaks, see below.

Monte Carlo Δt_MC peaks for α = 0, 19.23°, and 40.32°: Each peak consists of 10⁵ simulated events, rebinned to 0.1 ns for clarity. As α is increased muons that would have previously hit B begin to miss while those that do hit have different path lengths between A and B as well as different path lengths for light in the B scintillator.

However the actual data did not show the same sharpening effect as the simulation; all of the peaks had base widths of approximately 20 ns, see below.

Experimental Δt peaks for 19.23° (Left), and 40.32° (Right): Both peaks have the same base width, 20 ns, and general shape as the peak at a similar height but with α = 0°, in contrast to the Monte Carlo simulations predictions.

Threshold Dependence

Immediately after the initial assembly of the experimental apparatus and data acquisition system, a preliminary week long run through all the heights was performed. The <Δt> versus h data was clearly non-linear due to a pronounced plateau region in the middle heights and the fitted slope gave a worrying velocity of 1.6 c, see below. At this point the data acquisition system was rechecked and improved slightly before running a quicker test run through every other height which produced the same curve again. Upon learning that this bad data was reproducible the data acquisition system was completely rebuilt, care was taken to only use one cable to connect any two components, the apparatus was realigned, possible sources of ionizing backgrounds were reduced by turning off nearby florescent lights and raising the C panel off the ground by a few inches. Still the plateau persisted, so as a measure of last resort it was decided to increase all of the discriminator thresholds to -82 mV, more than twice its previous value. -82 mV was chosen specifically because the count rate still seemed decent at that value. This seemed to solve the problem as all subsequent data was linear with a reasonable value for the velocity, including the final data set.

<Δt> versus h with a discriminator threshold of -31 mV and the mid-heights plateau: Note now the error bars are much smaller than in the final data set due to the higher event rate. The <Δt> scale is also different because there were different delay cables in this initial data acquisition system setup and the TAC range was set to 100 ns. The fitted slope gives <v_μ> = (1.63 ± 0.01) c which is clearly too high, as is the reduced χ² at 458.8.

Looking back it would have been wise to calibrate each PMT individually at the beginning of the experiment by setting the PMT’s voltage to the provided specification and adjusting the discriminators threshold upwards until the count rate dropped from a high initial value to a steady plateau. The count rate would drop then level in this way as random low energy events were steady rejected leaving behind only the muon events. In effect this is what was accomplished, albeit in a cruder less efficient manner, by raising all three of the discriminators thresholds to -82 mV, which was a fairly high value. However even if the PMTs were effectively triggering at different threshold energies when all the discrimination thresholds were set to -31 mV this would not explain the observed plateau because all three discriminators have to trigger within 50 ns of each other for the TAC to record a data point, so the effective threshold simply then becomes the minimum threshold of all the panels. Indeed past groups [3] were successful when using -30 mV as their threshold. Furthermore this effective threshold was still producing good <Δt> peaks which indicates that random noise from non-muon low energy ionizing events was not a problem. Even if random noise from other events was a factor it would not explain the plateau because of the strange height dependence at midrange values of h. In the end, while usable data was produced at a high threshold of -82 mV this problem is still an unresolved matter of concern that lowers the confidence we have in our results and should be investigated further in future work.

Effects of Misalignment

In order to investigate the effects of misaligned panels additional Monte Carlo simulations with α = 0 and h = 140.3 were performed with panels B and C shifted in the x and y directions by 3 cm. Monte Carlo runs were done with both panels shifted individually and with them shifted together in the same and opposite directions. On average the 3 cm xy shifts moved <Δt>_MC from its correct value by 0.17 ns which was the same mean deviation observed between the repeated data points. As the xy alignment of the panels was known within 1 cm, 3 times as small as the simulated misalignments, it follows that the effect of any misalignment was much smaller than the statistical uncertainty and can safely be ignored as a source of systematic error. A Monte Caro run with the correct xy alignment but with h increased by 0.5 cm showed no significant change in <Δt>_MC. As h was measured to within ± 0.2 cm this also rules out the measurement of h as a systematic error.

Conclusions

Overall the experiment was successful in measuring the mean velocity of cosmogenic muons but the issues experienced with the discriminator threshold somewhat reduce the confidence we have in our result of <v_μ> = (0.926 ± 0.018) c, which is 3.5σ from the accepted value of 0.99 c [1]. The attempt to narrow the time of flight peaks by tilting the B panel was shown to be a valid approach in the Monte Carlo simulations, but unfortunately no improvement was observed experimentally as jitter in the scintillator panels and photomultiplier tubes turned out to be a limiting factor in the peaks width. As the peak’s center is the main item of interest future experiments can reduce the uncertainty on its location by simply taking longer data runs. In this experiment 24 hour runs with a few 70 hour runs were used as we were limited by the time constraints of the semester coupled with the time needed to debug the data acquisition system and by our desire to run at all 8 possible heights. At our final average event rate of 1 event every 3.5 minutes, 72 hour or week long runs at every height would significantly reduce the uncertainty of the peaks positions leading to a more precise measurement of <v_μ>. Provided that the data acquisition system is working properly, increasing the counting times is the best avenue of improvement as we are currently far from being limited by the uncertainty in the mean path length of the muon.

The last result obtained using the same experimental apparatus, Remmen and McCreary [3], cited <v_μ> to be (0.993 ± 0.002) c, a 2 parts in 1000 measurement. This is much closer to the accepted value with an uncertainty that is ~10 times smaller than what we were able to achieve. Remmen and McCreary’s analysis used multiple peak fits for each height, a primary peak and a secondary peak, which were then subtracted to find <Δt>. The analysis of our experiment used a single peak fit, which along with our lower count rate brought on by the high threshold needed to produce good data, contributed to the larger uncertainty. The 2 parts in 100 uncertainty of our experiment is a more realistic expectation of what future groups can achieve using these methods. Certainly there is room for improvement, taking more data and reducing the total solid angle of the apparatus would do much to increase this experiment’s precision. Unfortunately both of these modifications would make it more difficult to complete the experiment in the time constraints of one semesters work. If these improvements were to be implemented in future work it would be wise to calculate the inherent uncertainty in <v_μ> measurements that arises from the distribution of the velocities themselves, before time is wasted taking extra data unnecessarily.

Acknowledgments

This work was performed in the Methods of Experimental Physics II Laboratory under the supervision of Derek Lee and Kurt Wick along with Professors Pryke and Pawloski. Many thanks for all their assistance in completing this project.

References

[1] J. Beringer et al., "Review of Particle Physics," Physical Review D: Particles, Fields, Gravitation, and Cosmology, vol. 86, no. 1, pp. 305-307, 2012.

[2] J. Ziegler, "Terrestrial Cosmic Ray Intensities," IBM J. Res. Dev, vol. 40, p. 19, 1996.

[3] G. Remmen and E. McCeary, "Measurement of the Speed and Energy Distribution of Cosmic Ray Muons," Journal of Undergraduate Research in Physics, 2012.