Measuring the Momentum Distribution of Cosmic Ray Muon.

Ryan Thomas and Will Fischer

University of Minnesota - Twin Cities

Methods of Experimental Physics 2014

Abstract

This experiment used a carbon-dioxide based Cherenkov detector to measure the integral flux of vertical muons in the momentum range of 3.5 to 1 GeV/c. The data was fit to an exponential curve, with a coefficient of 0.0108±0.0004 muons per second per cm^2 per sr and an exponent of -0.62±0.02.

Introduction

Cosmic ray muons are generated when cosmic rays strike the upper atmosphere. This generates a show of secondary particles, some of which decay to muons traveling at relativistic velocities. These muons were detected by the Cherenkov radiation they produced when they travel faster than the speed of light in a medium.

Theory

Charged particles generate photons when they travel through a medium with velocity such that n>c/v, where n is the index of refraction of the medium and c is the speed of light. This can be parameterized by β=v/c, so that Cherenkov radiation is generated when β≥ 1/n. The threshold momentum for a muon to generate Cherenkov radiation can then be found from the relativistic momentum equation.

The number of photons produced by a muon can be related to the velocity of the muon, the index of refraction, and the spectral range from the following equation.[1]

The index of refraction of a gas can be found from the Lorentz-Lorenz equation [2]

where A is the molar refractivity, T is the temperature in Kelvin, R is the universal gas constant, and P is the pressure. A plot of this threshold momentum versus pressure can be found below.

Apparatus

A schematic of our apparatus can be found above. It consisted of a pressure tube filled with variable pressures of CO2 gas, two sets of crossed scintillator panels to eliminate non-vertical muons, and a set of discriminators, coincidence unit, oscilloscope, and PC to read in pulses from the Cherenkov detecting photomultiplier tube. The pressure tube is rated for a max pressure of 150PSI.

Method

The scintillator panels were calibrated to detect all the muons that passed through all four panels. This then triggered the oscilloscope to read in pulse information from the Cherenkov PMT. Data was taken at a range of pressures, from 15PSI to 150PSI, which corresponds to muon momentums of between 3.5 and 1 GeV/c. The data was then analyzed to determined if the muon that triggered the scintillator panels also generated Cherenkov radiation. This was then converted into a rate of muons above the threshold that were hitting the tube, and using a Monte Carlo simulation to produce the acceptance of the tube-panel assembly, was converted into an absolute flux of muons in (cm^2 sr^2 s)^-1. Data was collected for 24 hours at each pressure.

Results

Analyzed results. Reference data was taken from [3]. The two data points on the far left of the plot were excluded from the fit due to equipment malfunctions during the data taking, and are shown only for reference. The exponential fit had a chi-squared of 5.13, which corresponds to a probability-to-exceed of 74%. Errors were found from standard Poisson counting statistics and standard error propagation.

Future Improvements

The primary area for future improvements is simply taking a greater sample of data. It may also be possible to analyze the pulse-heights from the PMT to determined roughly the momentum of the muon that caused it, but this would required calibration of the PMT to determined single-photon pulse sizes.

Conclusion

Our measurements resulted in an absolute flux that was relatively close to previous results, although the shape of the curve was considerably different. It is unknown what effect caused this, but it indicates a systematic error, especially at low pressures, that caused an underdetection of muons. Attempts to adjust the data for effects caused by differences in path lengths of non-vertical muons or inefficiencies in the detector were made, but failed to improve the results.

Acknowledgments

We would like to thank Clem Pryke for his advice during the project, Kurt Wick for his expertise with the equipment, and Roger Rusack for the loan of his PMT.

References

[1] J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).

[2] "Lorentz-Lorenz Equation." A Dictionary of Chemistry. 2008.

[3] Rastin, B.C. "An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c."Journal of Physics G: Nuclear Physics. 10. (1984): 1609-1628.

[4] Kelley, T. and Stifter, K. "Measuring the Energy Distribution of Cosmic Ray Muons Using a Cerenkov Detector." University of Minnesota. 2013.