Measuring the Momentum Distribution of Cosmogenic Muons

Tyler Kelley and Kelly Stifter

School of Physics and Astronomy

University of Minnesota - Twin Cities

Minneapolis, MN, 55455

May 15, 2013

Abstract

The cosmic ray muon momentum distribution from 1.5 to 3.5 GeV/c is measured using an argon gas threshold Cherenkov detector. The momentum distribution is fit to a power law distribution with exponent α, based on expectation of known empirical data. A value of -0.84±0.19 is measured for α, which is consistent to within 0.4σ with the established empirical value of -0.76±0.05.

Introduction

The main goal of this experiment is to measure the shape of the cosmic ray muon momentum distribution for momenta between 1.5 and 3.5 GeV/c by using an argon gas threshold detector. This is done by recording the number of muons detected at different momentum thresholds and plotting the total counts against the corresponding threshold momenta. The result obtained is fit to a power law with exponent α, as shown below in equation 1.

The measured value of α is compared to results of a previous experiment done by B.C. Rastin that tested the same phenomenon in a different way and measured an α value of -0.76±0.05 in the low momentum regime between 1.5 and 3.5 GeV/c [2].

Theory

Cherenkov radiation is produced when a charged particle travels faster than the speed of light in the medium through which it passes. The particle excites other atoms or molecules around it with the electromagnetic field it produces, and they quickly fall back down to the ground level by emitting Cherenkov photons. The emitted Cherenkov radiation travels at the speed of light in the medium, given by c/n, where c is the speed of light in a vacuum and n is the index of refraction. The geometry of the slower moving Cherenkov light and the superluminal muon results in the Cherenkov photons forming a cone-shaped front, as shown in figure 1.

Considering the geometry of figure 1, the exact angle, θ_c, that the photons are emitted at relative to the moving particle can be determined as a function of the index of refraction and β, the ratio of the particle’s speed to the speed of light in a vacuum, as shown in equation 2.

The minimum particle velocity that will produce Cherenkov radiation is c/n, or the speed of light in the medium. When a particle is traveling at exactly that speed, photons will be emitted at an angle of 0°, according to equation 2. The Cherenkov angle reaches a maximum as the speed of the particle approaches the speed of light in a vacuum. Since there is a threshold velocity, there is a corresponding threshold momentum that can be calculated from the equation of relativistic momentum of a particle.

To calculate the threshold momentum, the threshold condition of β=1/n must be substituted.

Since muons are the particles in question, the particle mass, m, is known and the threshold momentum is only a function of the index of refraction. The index of refraction is adjusted by varying the pressure of the medium. This relation is expressed for a dilute gas in the Lorentz-Lorenz formula, in equation 5 [4].

In equation 5, A is the molar refractivity of the medium, P is the pressure, R is the universal gas constant, and T is the temperature. Putting equations 4 and 5 together, equation 6 can be obtained.

The experimental set-up is capable of producing pressures between 2 and 10 atm, which correspond to muon threshold momenta of about 3.5 GeV/c and 1.5 GeV/c, respectively. The total range of threshold momenta that the apparatus is capable of capturing with the allowed pressures is shown below in figure 2.

In order to determine how many muons are present above any given momentum, the index of refraction is varied to adjust the muon momentum threshold by changing the pressure of the Cherenkov medium. At each pressure, a different signal strength is expected, which is determined by the number of photons produced. The number of photons produced per length can be calculated using the Frank-Tamm equation, which relates energy loss per unit length to wavelength, particle velocity, and the index of refraction. It is shown in equation 7 [adapted from 1]. In this case, λ indicates wavelength and x indicates path length of the muon. In order to determine the number of photons produced per unit length, it is necessary to integrate over the wavelength range of the PMT.

After integrating over λ, equation 7 can be expanded, as shown in equation 8.

The function quickly reaches a plateau as β goes to one, so a value based on only the index of refraction can be obtained [5]. Note that the assumed medium is a gas with an index of refraction close to one.

The indices of refraction explored in this experiment are all within .003 of each other, so the signal strength is not expected to vary widely. At any given pressure, light in the range of the PMT is collected from muons with momenta greater than or equal to the threshold value. It produces an approximately 10ns pulse in the PMT signal. The number of pulses at any given pressure is a count of muons at or above the corresponding threshold momentum.

Method

Apparatus

The main component of the experiment is a vertically supported stainless steel tube that contains the Cherenkov medium of argon gas. The tube is 122 cm long, and has an inner diameter of 9.8 cm and an outer diameter of 10.2 cm. Attached to the bottom of the tube is another stainless steel tube section, in a T shape. The T has the same diameter as the main tube, is approximately 27cm long, and adds 19cm to the height of the tube. On one end of the T, a pressure gauge, safety valve, and release valve are fastened to the flange. A fused silica window, capable of withstanding pressures of up to 10 atm, and a Hamamatsu R2059 photomultiplier tube are placed in the opposing flange. This PMT was chosen due to its sensitivity in the 160 to 650nm range. It has a peak wavelength of 420nm, and a peak quantum efficiency of about 20%. The PMT is mechanically mounted within a plastic tube that is completely closed to light, in order to protect it from any direct light and to reduce the background.

Below the T-junction, an elliptical mirror is placed at a 45° angle, in order to reflect the Cherenkov light to the PMT mounted in the flange. In order to collect as much light as possible, a cone of aluminized mylar is fashioned and placed between the mirror and the PMT, which focuses the light and directs it towards the window of the PMT. To further increase the efficiency of light transmission, the inner surface of the straight pipe is also lined with aluminized mylar. A schematic of the experimental setup is shown below in figure 3.

A combination of four scintillator paddles is used in this experiment. Each paddle independently triggers on the presence of a muon, so a coincident trigger in all four paddles indicates a coincident muon event in all four. In order to pick out the vertical muons that likely pass all the way through the tube, two of the paddles are situated above the pipe and the other two paddles are situated underneath the pipe. The two scintillators in each pair lie perpendicular to each other in order to form a 353.4 cm² cross section of scintillator that lies directly above or below the pipe. This is done in order to reduce the number of muons capable of passing through all four of the scintillator paddles without passing through the pipe, and also to reduce the variation in Cherenkov radiation intensity which depends on path length of the muon through the argon. Due to the size and geometry of available scintillating paddles, it is impossible to construct a cross section that exactly fits the area of the tube.

In previous experiments [6], the PMT was simply mounted on the bottom of the tube. Whenever a muon went straight through the tube it would produce Cherenkov radiation in the windows of the PMT and the apparatus’s flange. In order to account for this, the PMT is mounted off to the side, about 16cm away from the middle of the tube. This is about 4cm outside the cross section, so a muon cannot pass through all four paddles as well as the flange window, except in the case of a muon shower event. This effect is taken in to account during background subtraction.

Each paddle has its own PMT to convert any scintillation into an electrical signal. The signal from each scintillator paddle is then sent to a discriminator module with the minimum threshold value of 30mV to convert any pulse that is above that threshold into a digital pulse. The minimum value of 30mV is chosen because many of the signals are expected to be quite small. Those digital pulses are fed into a coincidence unit, where an output of HI represents a muon passing through all four scintillator paddles.

In order to have 4-fold coincidence events, the top pair of paddles is put in coincidence with windows of 25ns, as well as the bottom pair with the same window. The two pairs are then put in coincidence with windows of 100ns. This is to account for any variation in path length, so that as many muon events as possible are recorded. This does open the door for false events, but they are accounted for in the background subtraction. Everything is calibrated by adding delay cables to the PMT connections in order to account for the differences in cable lengths and PMT signal generation speeds.

#SecCherenkovMedium

Choice of Cherenkov Medium

Many Cherenkov detectors use air, CO₂, N₂, or water. This experiment uses argon gas due to a number of factors. The first is that it is widely available, and not very expensive. The second is that for the pressures available to us, it produces indices of refraction that allow us to study the desired range of muon momenta. The third and largest consideration is the absorption of other viable alternatives as compared to argon in the ultraviolet. The ultraviolet is of particular concern, because the number of photons produced per length goes as , as shown in equation 7.

The main viable alternative is nitrogen. Argon is the medium of choice because its mean free path at standard temperature and pressure is 7.22m, as compared to 6.76m in nitrogen [7]. Argon also absorbs less in the ultraviolet, as can be seen on the right-hand side of figure 4, below.

Data Collection

When a four-fold coincidence event is detected, an oscilloscope that is connected to the Cherenkov PMT signal triggers on an enable signal that is sent from the coincidence module. This data is then stored in a computer that interfaces with the oscilloscope through a LabView instrument driver which records the trace of the signal and appends each trace to a .dat file. The vertical scale of the oscilloscope is set such that signals of 8V or less can be measured. About 5% of signals still saturate the oscilloscope, and it is assumed that these are the shower events. The horizontal scale is set in such a way that allows the capture of all events to their full extent while still maintaining the highest resolution possible. The oscilloscope triggers on a 4-fold coincidence, but the timing is dictated by the bottom pair of paddles since that signal always comes later than that of the top. This means that to see the full extent of the signal, the initial position is shifted back almost 100ns, and the signal comes at about 40ns which, is approximately the middle of the window. The limited settings of the oscilloscope allow no smaller resolution.

Calibration

The optimal voltage for the PMTs in the scintillator paddles needs to be determined in order to obtain the maximum detection efficiency while minimizing the noise. This was done by splitting the signal from a single PMT and feeding both signals into a discriminator, one with a threshold of 30mV and the other with a threshold of 130mV. The events from the scintillator were counted on two separate channels for the two different thresholds. The ratio of high threshold to low threshold was minimized to find to operating voltages. The measured voltages were up to 100V away from the recommended operating voltages.

The optimal voltage for the Cherenkov PMT is not determined. It is a brand new tube, and the manufacturer’s specs note an operating voltage of -2500V. It is simply held at this value for the duration of the data taking process.

Analysis

For each event trace, the maximum peak height is found. Since the pulses are actually a negative signal, the most negative point is taken as the peak. The peak heights are then split in to 160 bins of width 50mV, which are then normalized to obtain the number of counts per minute per bin. This is done for all five pressures, as well as the background. The normalized background data is then subtracted off by bin for every pressure. The total contribution of the background is 1.57±0.01 counts per minute. This is less than previous experiments [6], so it can be concluded that moving the PMT off to the side reduced the background counts by about 50%.

To calculate the total number of muons per minute above the corresponding threshold momentum, the bins have to be added up. The lower peak height bins are completely dominated by background in the form of noise in the PMT signal. In order to account for this, a cut of 200mV is instituted in order to exclude the background dominated data, which was identified by the scale of the error bars. The cut is shown for the 10.21 atm run in figure 5.

After the cut, the counts are added to obtain a normalized counts value. Data for each pressure can be seen in table 1, below.

All analysis was carried out within MatLab. Due to the fact that pulse heights are not calibrated to specific momenta, the efficiency of this cut cannot be evaluated.

The number of counts did not drastically increase from previous experiments [6]. The strength of the signals did increase about ten-fold, but it is unclear whether the cause is a different Cherenkov medium, or a more sensitive PMT. Overall, no definitive conclusions can be reached about the effectiveness of argon as a Cherenkov medium as compared to nitrogen.

Results

The two constants that characterize the momentum distribution are the coefficient, and the exponent α. These values are determined to be 0.46±0.05 and -0.84±0.19, respectively, by doing a two parameter fit to minimize the χ². The χ² value is .61. This value is low because the uncertainties on each point are large. They cannot be reduced, however, because they simply come from Poisson counting statistics. Since there are only three degrees of freedom, this corresponds to a p-value of .89, which is acceptable despite the low χ². The final fit can be seen in figure 6, below.

Since this experiment takes place in the low-momentum regime, the results cannot be directly compared to the accepted α value of -2.72±0.01 since it corresponds to the primary momentum distribution which breaks down at low momenta. This region is shown highlighted in blue in figure 7 [adapted from 2].

In order to determine the α value in the range of momenta that are examined, highlighted in red in figure 7, the data shown below in table 2 is taken from the original paper and analyzed.

The errors are not stated explicitly for these momenta, so they are extrapolated from the rest of the given data. The relation at low momenta is no longer a simple power law, but can be approximated as such over a small range of momenta. The data can be described by a power law with an exponent of -0.76±0.05 and a coefficient of 0.47±0.02. These values are also found by performing a two parameter fit to minimize χ². A reduced χ² value of 1.03 shows that the data is consistent with a power law over this small range of momenta, as shown in figure 8, below.

This fit indicates that the α value that is relevant to this experiment is -0.76±0.05, which is consistent with the measured value of -0.84±0.19. Both data sets can be seen in figure 9, below.

Through the results of this experiment, a value of -0.84±0.19 is determined for the α value of the momentum distribution for cosmogenic muons in the low-momentum regime between 1.5 and 3.5GeV/c. The value for α is consistent with the empirically obtained value of -0.76±0.05, to within 0.4σ.

Improvements

Though our result agrees with the expected value, the experiment could be improved in order to obtain a lower uncertainty and a higher degree of accuracy. The largest problem is the low number of muon events detected. One way to raise the number of counts, and therefore lower the uncertainty, would be to take data for longer periods of time. Unfortunately, this is impractical because the amount of time that it would take to significantly decrease the uncertainty far exceeds the time allotted for this project. Another method would be to take more data points in order to more closely determine the shape of the distribution. This is the preferred method, because it is more feasible. To improve this method even further, a wider range of pressures could be studied, though this would require an alteration of the apparatus. The easiest change would be to adapt the tube to accommodate negative pressures, but since that would probe higher threshold momenta, the number of counts would be even less. A last method would be to further alter the apparatus, and create a larger cross-sectional area of the tube, thereby detecting more muons. Overall, there are a number of extensions that could improve the experiment, but many of them are outside the scope of the project.

Acknowledgments

Thank you to our external advisor, Professor Rusack, for all of his HEP knowledge and for the fancy PMT; our internal advisor, Professor Pawloski, for his repeated editing and advice, as well as knowledge of analysis techniques; Kurt Wick, for all of his patience, technical expertise, and equipment; and the University of Minnesota Physics Department for the use of their facilities.

References

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