Fall 2020 Measurement of Muon Time of Flight

Measurement of Muon Time of Flight

Leonardo Clarke and Avery Wold

Abstract

We measured the average speed of charged cosmic ray muons produced in high energy upper-atmosphere interactions to be (2.6394 + 0.1154)e+08 meters per second. This was done using a system of five vertically-aligned scintillators coupled with photomultiplier tubes. By limiting detector area, measurements were constrained to near-normal particle incidence. Requiring that the particle pass through all detectors reduced the probability of accidental detections. Feeding signals from this setup into a time-to-amplitude converter allowed us to determine the time Δt associated with a variable distance target detector position, Δd. Accumulating multiple Δt ∝ Δd relationships allowed us to find our measurement.

Introduction

High energy cosmic rays, namely protons and ionized nuclei, are regularly incident on the upper atmosphere of the Earth. Interactions of these cosmic rays with air molecules create short-lived particle showers, one component of which are charged muons. These muons are unstable and undergo spontaneous decay, with a rest frame lifetime of ~2.2 microseconds [1][2]. They are highly relativistic, so an Earthbound observer sees these lifetimes as much longer.

For this project, we measured the speed of incident cosmic ray muons in the aforementioned rest frame of the Earth. By measuring the time taken for incident muons to travel between a set of five photosensitive detectors, we were able to calculate the resulting speeds. We limited our analysis to those muons descending at near-vertical angles in order to minimize uncertainty in our measurement due to differing muon flight paths.

The detection of cosmic ray muons has important applications in the fields of geology and archaeology. Muons interact relatively little with ordinary matter, thus they are able to penetrate deeply into large structures before spontaneously decaying. Muon tomography, which deals with the mapping of hollow structures, takes advantage of this fact to analyze structures such as ancient pyramids and active volcanoes [3][4].

Figure 2: Left, a picture of the apparatus is shown. Right, a diagram of the panels and preprocessing electronics is shown. The AND gate symbols indicate the coincidence checks. The TAC, as shown on the far right, measures the time of flight for a given height of panel 3.

Figure 3: A simple timing diagram for a single detection. The “Stop Signal” is the signal from panel 3, which has been delayed by a fixed amount of time, as the dashed portion of the signal is where it would have been without it, whereas the “Start” is from the total coincidence of the other panels. The difference in time measurement between the start and stop signal changes with the height of panel 3.

Data Analysis

As exemplified by the plot in figure 4, the MCA histograms at each detector position are single-peaked distributions spanning hundreds of bins. It is useful to think of the bin number as a proxy for muon flight time, making the x-axis in figure 4 in units of time rather than bins. (We know this conversion from our calibration process). At each position ∆d, we began the analysis by fitting the MCA histogram data to Gaussian distribution functions. It is worth noting that we did not expect our MCA distributions to be Gaussian (see Theory section). Rather, we used these distribution profiles simply as a means of obtaining a mean bin value for all of our MCA distributions.

Figure 4: Figure from Origin 2017 displaying the MCA output from the detector at position 4. The black points represent the counts in each bin on the MCA. The error on each of the counts goes as the square root of the number of counts in the corresponding bin. The red line is the fitted Gaussian profile for this data set. Since bins containing zero counts have a corresponding error of zero, we have masked the data to restrict our analysis to the range of bins with non-zero counts.

Results & Conclusions

Figure 5: A plot from Origin 2017 showing the final measurements of Δd vs. Δt for each of the five detector positions chosen during this experiment. The black points are our calculated data points for each detector position. The red line is our linear fit to these data.

From the linear fit in figure 5, we have a slope of (−3.7887±0.01657)e-09 s/m, a reduced χ 2 value of 1.306, and a p-value of 0.2716. We convert the slope to our final value for the speed via the equation:

We therefore obtained a final speed value of (2.6394 ± 0.1154)e+08 m/s, or 88% the speed of light. This corresponds to an energy of 222.8168 ± 21.9329MeV. Our values do not agree with the literature very well, as their cited value for the average speed for charged cosmic ray muons is 2.98e + 08m/s or 99.4% the speed of light [2] , however it is within 3σ of the literature value.

It is unlikely that the cosmic ray muons sampled in this experiment had energies significantly lower than those of typical muons. Systematic measurement errors could again have come from sources such as a possible drifting of the TAC over the course of the experiment. Since calibration is a relatively quick process, an improvement for future experiments in this area would be to calibrate the TAC before gathering data at each height ∆d. This would ensure that any long-term calibration changes are accounted for.

Retrospectively, there are a number of things that could improve the quality and possible accuracy of the data. If one was able to do this experiment with more data points in the final fit, essentially more unique heights, the fit on the final data could be taken as more trustworthy, given the proper goodness of fit values. The discrepancy which the current experimenters found when compared to the aforementioned literature value could have come from a number of issues with the pre-processing electronics used. For example, the TAC could have an unknown pink noise drift associated with it, which the current experimenters did not factor into their expectations and analysis. If the fixed delay applied to panel 3 somehow scaled with the overall delay time, this could appear as a lower particle velocity. Given this, the experimenters would advocate for an investigative run of the apparatus, with data taken in one-day increments, and focusing on the extremes (ie: the highest and lowest heights tested). Doing this multiple times, with different methods of delaying the panel 3 signal could uncover whether or not the discrepancy appears from it. The location of the apparatus is also interesting, as it is stored in a basement of a four or five-story building constructed of fairly dense material. This would undoubtedly impact the penetrating energies of the muons, and as such testing in multiple different locations, or even without any obstruction to the open sky, could be informative regarding the discrepancy.

References

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[2] Pablo Solis Lulu Liu. The speed and lifetime of cosmic ray muons. 2007.

[3] G. Bonomi, P. Checchia, M. D’Errico, D. Pagano, and G. Saracino. Applications of cosmic-ray muons.

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https://doi.org/10.1016/j.ppnp.2020.103768. URL http://www.

sciencedirect.com/science/article/pii/S0146641020300156.

[4] Jacques Marteau, D Gibert, N Lesparre, F Nicollin, P Noli, and F Giacoppo. Muons tomography applied to

geosciences and volcanology. Nuclear Instruments and Methods in Physics Research Section A:

Accelerators, Spectrometers, Detectors and Associated Equipment, 695:23–28, 2012.

[5] J. Beringer, J. F. Arguin, R. M. Barnett, et al. Review of particle physics. Phys. Rev. D, 86:010001, Jul

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