Measuring the momentum distribution of cosmic ray muons

Measuring the momentum distribution of cosmic ray muons

Tien Vo and Aaron Thomson

School of Physics and Astronomy, University of Minnesota - Twin Cities

May 08, 2021

Abstract

Using an Argon-filled threshold Cherenkov detector, we found that the momentum of cosmic ray muons from 1.41 to 2.41 GeV/c obeyed a power law distribution. The best-fit power was found to be −0.73 ± 0.05, in good agreements with results in the

literature. Additionally, we changed the reflective surface of the apparatus to aluminum tape and observed an improvement on the light collection efficiency. This allowed for higher statistics in shorter observation periods. We concluded that this apparatus

would be useful for acquiring more data in the integral intensity spectrum to calculate the differential intensity spectrum.

Introduction

Cosmic ray muons (CRMs) are produced from the collision of cosmic rays with the Earth's atmosphere. Accurate measurements of the integral and differential spectra of cosmic ray muons at sea level with momentum ranging from 1 to 3000 GeV/c have been measured using both scintillation counter detectors and spectrometers Rastin (1984); Rogers & Tristam (1984). These detection projects are, however, technically complicated. For teaching laboratories, it was demonstrated that a simpler detector, more suitable for a graduate laboratory course, can measure the distribution in a smaller range of momentum with excellent agreement Quintero (2010). This detector employed the Cherenkov radiation of muons passing through a medium to construct a momentum distribution. Based on Quintero's apparatus, Kelley & Stifter (2013) have designed a similar Cherenkov photon capturing telescope. However, they had fairly high uncertainty. Additionally, recent experiments that repeated their procedures with the same apparatus components might have been conducted with low light collection efficiency due to the poor inner reflective surfaces (see Fig. 1). Thus, our goal is to improve on the device and repeat this experiment.

Figure 1. The inner reflective surface of the gas tube in which Cherenkov radiation occurs. Previous MXP groups used an aluminized mylar sheet (left)

glued in unevenly on the surface of the tube, resulting in a wrinkled reflective surface. We have increased the light collection efficiency by applying aluminum tape (right) which leads to a smoother surface.

Theory

Figure 2. The geometry of Cherenkov radiation.

A muon of mass m with speed v = βc passes through a gas with refractive index n will emit Cherenkov radiation if v is larger than the speed of light vg = c/n in this gas. From Fig. 2, vg/v = cos θc where θc is the angle of emission. Thus, this imposes a limit on the momentum, called the threshold momentum, below which there is no emission since |cos θc| ≤ 1. The threshold momentum is determined by

The threshold momentum can also be expressed in terms of the properties of the Cherenkov medium. The polarizability is given by the Lorentz-Lorenz equation (Born, 1999)

where A is the molar refractivity, N is the number density, and NA is Avogadro's number. For Argon, A ~ 4.14e-6 cubic meter/mol (Kelley & Stifter, 2013). Then we can write

where T is the temperature, P is the pressure, and R is the gas constant and we have assumed ideal gas law. Now, from Frank-Tamm equation (Amsler et al., 2008), the number of emitted photons from a muon with momentum p and path length L through the gas can be calculated as

where λ1, λ2 mark the range of sensitive wavelengths of the photomultiplier tube (PMT) used to capture emission. α is the fine-structure constant and Z=-1 for a muon. Fig. 3 compares a few emission curves at different gas pressures. Note that the x intercept varies according to the values of the threshold momentum, and that the same muon will emit much less emission in a lower pressure gas than a higher one.

Figure 3. The number of Cherenkov photons emitted by a muon in terms of its momentum, assuming similar path length. The colored lines are for a few pressures where the threshold momentum are different.

Note that it is not possible to calculate the distribution of muon momentum from the number of emitted photon because we do not have a measurement of path length in the apparatus. Also, muons appear in a certain probability distribution function of momentum. So it is not a simple matter of integrating over the curves in Fig. 3. Instead, we must use a purely empirical form factor for the intensity, which is the power law distribution.

Method

Apparatus

Figure 4. A sketch diagram of the apparatus. The veto scintillator is placed at approximately the same height as the top panel in the bottom scintillator pair.

The two main apparatus components are (i) a radiation capturing gas cylinder and (ii) a coincidence detection system (see Fig. 4). First, a 141 cm tall metal cylinder with radius 4.9 cm is filled with pressurized argon gas through a regulator. The T-junction at the bottom is also cylindrical with the same radius. This light-tight cylinder, used by previous projects, was built in-house and tested to withstand a maximum pressure of ∼ 11 atm. A mirror is placed at 45◦ in the junction to direct the radiation through a quartz window into a Hammamatsu R2059 PMT. This PMT, used for capturing Cherenkov emission, is sensitive to wavelengths from 160 nm to 650 nm. To direct radiation along the inner surfaces of the telescope, ultraviolet-resistant aluminum tape is applied along the length of the cylinder. Next to the mirror at the T-junction, a focusing cone also made out of thick paper and aluminum tape directs light into the quartz window. On the other side of the window is the Cherenkov PMT used to capture the amount of emission.

Second, the detection system consists of scintillator panels aligned with the cylinder’s axis, each connected to a Hammamatsu R1306 PMT. The panels have a long rectangular area compared to the tube’s. Thus, to limit the detection angle, they are vertically stacked in pairs at 90◦ , forming a 353 cm2 cross-sectional area. A fifth scintillator is placed below the quartz window to veto muon shower events. The PMT pulses from each scintillator are converted into square signals with the discriminators, the outputs of which are fed into logical gates in the coincidence unit. This unit subsequently triggers a Tektronix DPO 2024B oscilloscope with a 200 MHz bandwidth and a sampling rate of 1 GHz, which saves analog signals from one top and one bottom PMT, the Cherenkov PMT, and the veto PMT for post-analysis.

Calibration

It is necessary to calibrate electronic devices for their optimal operating voltages. Those for all five scintillators were determined using the coincidence method. We vertically stacked four scintillators. The operating voltages for the PMTs connected to three of them were set at a recommended value, while that of the remaining PMT in the middle of the stack was varied. In an optimal range, the ratio between the coincidental counts from all four PMTs and those of the three fixed-voltage PMT plateaus after a linear gain (see Fig. 5). In general, we choose the operating voltage to be 100 V after the start of the plateau. The max rating voltage for these PMTs is 1500 V, so for those that plateau close to this limit, we set the voltage to be 1475 V. The Cherenkov PMT cannot be calibrated with this method, so we assume the manufacturer’s voltage of 2500 V.

Figure 5. Calibration curves of the five scintillators. The legends show the identifiers labeled on each scintillator panel in the MXP lab, and their chosen operating voltages.

The coincidence unit also needs calibration. A muon hitting the panel at different locations will result in jitters in the signals' start time because of the delay of electrons moving in the scintillators, together with the delay from the cables. Thus, it is advisable that the system is calibrated such that the timing information can be calculated from one scintillator signal out of the four in the detector. Here, we chose the bottom-most signal to be the fixed point. Fig. 6 shows a timing diagram which achieves this desired effect.

Figure 6. The digital (D) and analog (A) signals (from top to bottom, in order shown in Fig. 4) of the scintillator PMT signals.

Data collection

An event was registered whenever a particle traversed through the cross-sectional area of all four panels within a time window. The discriminator pulse length corresponding to each scintillator PMT was calibrated such that the trigger point of the coincidence unit was determined by the negative edge of the bottom most PMT signal. A Monte-Carlo simulation was done to determine the muon’s mean time-of-flight between the top and bottom pair to be ∼ 9 ns. The pulse lengths of the top pair of scintillator PMTs were extended to accommodate this time difference and the jittering (∼ 8 ns) due to the speed of electric signals within the long scintillator panels. The voltage signals from each event were saved 200 ns before and after the point of trigger in the highest temporal resolution of the scope. The vertical scale was 0.1 V/division with an offset such that the saturation occured at ∼ -0.9 V.

Analysis

Interpolation and Extrapolation

One should consider which saturation level is adequate for their plan of analysis. It is possible to perform an extrapolation for the maximum peak height during the time-above-saturation, as shown in Fig. 7. However, this depends on the accuracy of the extrapolation algorithm. Previous MXP groups have only counted the number of peaks below a certain (negative) voltage threshold, in which case the saturation is irrelevant. However, better data could be achieved by integrating around the Cherenkov signals in all voltage bins, instead of only counting the minimum peaks. Thus, the saturation significantly determines the goodness of the data. We recommend setting the scope at 0.2 V/division such that the saturation is at -2V. Most signals are smaller than this range, so the saturation should be minimal.

Figure 7. Typical Cherenkov signals of an event. The black dots are raw data. The red curves

are interpolated data with extrapolation over the plateaued range if the signal saturates. The

trigger is always at 200 ns.

Good-timing criteria

Among the minumim peak, the start time can be determined as the point of the half-width at half-minimum, at the negative edge of the signal. A good-timing criteria can be found from the statistics of these timing information (see Fig. 8). Good events are those correlated linearly with the bottom-most PMT signal.

Figure 8. 2D histograms of the Cherenkov start time (panel A) and the veto start time (panel B) in terms of the bottom start time.

Photon rates

The signals are then counted based on their maximum peaks (see Fig. 9). Note that we have inverted the (negatively powered PMT) signals to positive values. So minimum peaks become maximum here. The counts have been normalized by their respective observation time to become rates. Note that the effects of saturation is seen most clearly here, where there is an accumulation of voltages around ~ 1.1V. Instead, these histograms should roughly follow an exponential distribution (Quintero, 2010). However, to calculate the total photon intensity, we will sum over all voltage bins and divide by the solid angle and the area of the scintillators. The saturation, again, does not affect this type of analysis, but it would if one plans to integrate over the signal curves.

Figure 9. Histograms of a few observations using the telescope, set at different gauge pressures. The voltages are maximum peaks of many signals (similar to those in Fig. 7) seen during an observation time.

Results

The intensity is calculated from the rates in Fig. 9. We then fit it with a power law model and show the results as belowin Fig. 10. Note that it is the shape of the curve that is of important. The lower intensity that we have is due to an underestimation of the solid angle and the conversion efficiency between photon and muon intensity. Our momentum range is small enough that this conversion is approximately uniform across the domain. We compare our results with the two 84' papers in the literature, which used a different method for the intensity. Also, our results are compared with those of the 2013 group because we use the same gas. Our residuals are within one standard deviation with a reduced χ2 of 1.23, which indicates a good fit. Also, since we have much smaller error bars, our uncertainty is reduced by one order of magnitude.

Figure 10. Our fitted intensity curve (black) compared with those of the literature (green & red) and of the 2013 MXP group (blue).

Our data are also given in the table below.

Further works

We have demonstrated that it is possible to achieve better photon observation rates with this new apparatus. More work can be done to obtain more data points to better sample the power law distribution, as well as to calculate the differential intensity. Most of our background subtraction is done by the veto mechanism, but a curious result is found when we randomly observe radiation in the tube instead of when there is a coincidence. The noise level is at ~10%, which is significant enough for future groups to explore in further details.

Acknowledgements

The authors would like to thank Professor Roger Rusack and Professor Daniel Cronin-Hennessy for their guidance throughout the semester, Professor Kurt Wick for his helpful comments in code development, and Kevin Booth for his aid in the construction and calibration of our apparatus. We give our special thanks to Professor Rusack for having provided his high quality photomultiplier tube, which was a main component of our apparatus.

References

Amsler, C., et al. 2008, Physics Letters B, 667, 1 , doi: https://doi.org/10.1016/j.physletb.2008.07.018

Born, M. 1999, Principles of optics : electromagnetic theory of propagation, interference and

diffraction of light, 7th edn. (Cambridge ; New York: Cambridge University Press)

Kelley, T., & Stifter, K. 2013, Measuring the Momentum Distribution of Cosmogenic Muons,

Tech. rep., University of Minnesota - Twin Cities, Dept. of Physics & Astronomy

Quintero, E. A. 2010, Bsc thesis, Massachusetts Institute of Technology

Rastin, B. C. 1984, Journal of physics. G, Nuclear physics, 10, 1609

Rogers, I. W., & Tristam, M. 1984, Journal of physics. G, Nuclear physics, 10, 983