Verifying Special Relativity as a Mechanism for Muon Penetration

Jasmine Mackenzie and Adiv Paradise

School of Physics and Astronomy University of Minnesota - Twin Cities

Minneapolis, MN

May 16, 2013

Links

Abstract

We observe the full-spectrum muon flux as a function of altitude via high-altitude weather balloon. We find that the logarithm of flux versus depth has a linear slope of (-0.00547\pm3.3*10^{-5})cm²/g with a reduced chi^2 of 0.23, which, while differing significantly from Bruno Rossi's 1940 finding [1] of approximately -0.003, is likely within the margins of error introduced by differences in experimentation between Rossi's setup and ours, and qualitatively matches the behavior observed by Rossi. We use a simulation to show that velocity-dependent time dilation as a mechanism for deeper muon penetration provides a model that is in close agreement with observations for the 300-700 g/cm² range, with a reduced <latex>\chi^2</latex> in that range of 0.20, and a reduced chi^2 of 6.58 over the entire range considered. We further explore possible causes of the low-altitude deviation between our model and the observed flux.

Introduction

One of the examples often given to illustrate the necessity of special relativity is that of the muon. Muons are leptons with negative unitary charge, much like electrons, but with much larger mass and smaller interaction cross-section.[2] Muons are unstable particles, and decay with a mean lifetime of 2.1969811*10^{-6} μs.[3] This means that even if they were travelling at the speed of light, the mean range of cosmogenic muons should be on the order of 600 meters, and almost none should be detected at low altitudes. Instead, considerable muon fluxes are observed at sea level.[4] Einstein's theory of special relativity predicts a velocity-dependent time dilation, such that 2.2 μs in the rest frame of a muon travelling at relativistic speeds might appear to be much longer in an observer's rest frame, allowing for deeper muon penetration.[5] This formulation was observed by Bruno Rossi in 1940 to agree with observations made at several different ground-based altitudes, and further confirmed the muon lifetime as had been measured in the lab.[1] This effect has since been confirmed by several others for various parts of the muon energy spectrum.[6],[2],[7],[8]. The characteristics of the muon energy spectrum have also been determined for various altitudes.[9] The logarithm of muon flux versus atmospheric depth is roughly linear, as shown in Fig. 1.[10] However, the degree to which velocity-dependent time dilation predicts the observed muon fluxes has only been qualitatively confirmed. We aim to use observations and models based on fundamental physical assumptions and initial conditions to quantify that goodness-of-fit.

Fig. 1: Data collected by Bruno Rossi.[10] Figure reproduced from Rossi's 1948 paper. Note the linear trend between the logarithm of the flux and the atmospheric depth. Reproduced from Rossi's 1948 paper.

Background Theory

Muons are subatomic particles produced through the interactions of energetic particles in the upper atmosphere.[2],[11] Peak muon production is in a layer at approximately 15 km above sea level,[3] though muon production can continue at lower altitudes as well. Muons have rest masses approximately 200 times that of an electron.[3] Muon decay is governed by a generic decay equation:

where N_0 is the initial population, t is the time elapsed, and tau is the lifetime of the particle.[1] In the case of particle detection, N and N_0 refer to counting rates, or numbers of particles per unit time. Since the muons are traveling through the upper atmosphere, it may be desirable to measure N not as a function of time, but as a function of altitude. Given a certain velocity v, each muon will descend in a certain length of time t a certain distance

, given by

, where z_0 is some initial altitude. Note that z<z_0. Solving for t, Eq. 1 now becomes

Note that <latex>v\tau</latex> is itself a distance, corresponding to the distance travelled during the muon's lifetime, L. Therefore, Eq. 2 can also be written . Left alone, this gives the classical equation for the detection rate of muons as a function of altitude with no absorbing medium.

Einstein's theory of special relativity predicts that objects travelling at high velocities will see a time dilation given by

where v is the object's velocity, c is the speed of light in a vacuum, tau_0 is the elapsed time in the object's rest frame, and tau is the elapsed time observed by an observer stationary relative to the object.[5] Muons are thought to travel at very high velocities, close to the speed of light. Therefore, the lifetime of the muon, tau, can instead be considered to be

when the muon is traveling at high velocity, where tau_0 is the lifetime of the muon at rest. tau_0 can be determined experimentally in the lab setting. Thus, the distance traveled by the muon before decay is given by

.For muons traveling at relativistic velocities, Eq. 2 becomes

By taking into account relativistic time dilation, Eq. 4 predicts the flux of muons of a specific energy at a given altitude for cases in which no energy is lost to ionization.

Earth's atmosphere, however, does absorb and slow down muons.[12]The energy lost by a charged particle travelling through a medium is given by the Bethe-Bloch equation, such that

where m_e and e are the electron rest mass and charge, c is the speed of light, N_A is Avogadro's number, Z is the atomic number of the target material, rho is its density, M_u is the molar mass constant, z is the charge of the particle in question, epsilon_0 is the vacuum permittivity, I is the mean excitation potential of the target, and <latex>\beta</latex> is the ratio of the particle's velocity to the speed of light.[13] This formula takes into account velocity-dependent time dilation. There is also a non-relativistic version, taken simply as the approximation where

. Numerical calculations as well as empirical observations show that for a charged incident particle such as a muon, equivalent masses of different materials (i.e., quantities with the same mass per square area ratios) have the same retarding effect.[4]

The mass of air equivalent to a given amount of lead or carbon is found by calculating atmospheric depth. Atmospheric depth is given as

where rho(h) is the atmospheric density, h is the altitude, X_0 is the depth at sea level, equal to 1030 g/cm², and h_0 is an atmospheric scale height equal to 8.4 km.[14] Rossi found that given masses of lead and carbon absorbers seemed to reduce muon fluxes less than equivalent masses of air. Since the only difference between the two absorbers was the time-of-flight through the material, this motivated the consideration of the decay equation, as shown in Eq. 1.[1]

Since muons are slowed down by the atmosphere, Eq. 4 must be revisited, and instead considered as

or rather, in its differential form:

This is important, as not only do muons not have a constant velocity (and therefore Lorentz factor), but they are also not of uniform energy. Muons are created with a spectrum of energies. This spectrum can be approximated as a power law of the form

where kappa is a scaling constant, and alpha is a constant that determines the shape of the spectrum.[15] alpha must be less than -1 for the spectrum to converge at high energies. The spectrum has been measured at various altitudes. At 15km, <latex>\alpha</latex> is expected to have a value of approximately -2.45.[9] Thus, in order to model the expected muon flux integrated over the entire spectrum at a given altitude, it is important to model not just the energy loss as a function of altitude and decaying velocity, but also the loss of muons due to spontaneous decay at a rate that depends on the time-of-flight in the muon's rest frame. Since muons of different energies slow down at different rates, Eq. 8 is useful for computing the differential loss of muons at each energy for each altitude. Since it is likely that an analytic solution does not exist, this can be done via numeric simulation.[16] Such a simulation would require only an initial spectrum and altitude, and could then compute a new spectrum at each altitude as a result of energy loss, and then reduce each population by an appropriate amount via Eq. 8.

Such a simulation would of course be subject to several limitations. There is a significant approximation being made in that such a simulation assumes a sudden injection of muons at an initial altitude. In reality, muon formation, while concentrated in an air layer at 15 km, also continues to happen over a wider range of altitudes.[2] Another important assumption is that the entire spectrum behaves as a power law. If part of the spectrum deviates from the power law used to parameterize the rest of the spectrum, then the observed muon flux will differ from the model. Finally, due to the nature of a weather balloon flight and federal flight regulations, a conventional balloon-borne apparatus cannot carry sufficient equipment so as to select for only muon events. There is significant risk of contamination by other ionizing particles, such as pions, kaons, and electrons. Pions and kaons contribute negligible amounts to the total flux--the peak ratio of pions to muons for example,

is 0.035.[6] However, contamination from high-energy electrons is a serious concern[8]. At peak muon flux, electrons make up 88% of detections, while muons make up approximately 12%. The abundance of muons rises to 75% by sea level.[2] Thus particle contaminations could also lead to the model deviating from observation. However, by fitting a simulation to observed fluxes, adapting the initial conditions to adapt the fit, it should be possible to quantitatively verify the validity of velocity-dependent time dilation as a mechanism for muon penetration.

Procedure

Payload

In our experiment, we used a high-altitude weather balloon to measure the integrated muon flux as a function of altitude. To accomplish this, we used a two-fold coincidence system of RM-60 Geiger-Müller tubes. In such a setup, two tubes are arranged such that their lengthwise axes coincide, and the thin mylar windows face each other, as shown in Fig. 2. Only ionizing radiation that passed through both counters triggered a coincidence event. Events were transmitted from the counters to a Field-Programmable Gate Array (FPGA) logic board, which recorded the number of coincidences for each set of coincidence counters in a one-second integration time on a flash memory chip. Each board listened to 2 coincidence sets, or 4 Geiger counters total. Our payload contained two boards, with 4 coincidence counters (8 Geiger-Müller tubes). Each board was powered by 4 AA batteries.

Fig. 2: A schematic of the apparatus used. Sets of two G-M tubes were arranged into two-fold coincidence counters, with their long axes parallel to the vertical. Coincident particles were recorded via FPGA board onto a flash memory chip. The payload contained 4 coincidence counters and 2 FPGA boards. One of the 4 sets had 1.5 cm of lead shielding to attempt to verify Rossi's disintegration observations.[1]

To attempt to verify and replicate Rossi's findings with the absorber, one set of counters was given 1.5 cm of lead shielding between the tubes, corresponding to an additional 17.01 g/cm². Compared at equivalent atmospheric depths, it was expected that this shielded set should see a higher flux, due to a shorter time-of-flight for incident muons. Each counter was 4.1 cm long, with a radius of 0.7 cm. The counters were separated by 0.3 cm. Thus, the telescope aperture area was given by 1.539*10^{-4} m², and the solid angle subtended was 0.06068 steradians for the lead-shielded set, and 0.0835 steradians for the three unshielded sets. The interior of the payload box was covered in aluminum foil, effectively creating a Faraday cage in order to shield the equipment from any potential radio interference from the balloon's transmitters. The payload itself was made of approximately one inch-thick Styrofoam, wrapped in black tape.

Federal ballooning regulations imposed a weight limit on our payload of approximately 2.7 kg. This meant there was insufficient space to include the shielding and counters necessary to be able to discriminate freely between muons and other ionizing particles. Another adverse impact during the flight involved the balloon's pendular motion. The weather balloon behaved as a complex frictionless pendulum, with a period of several seconds, corresponding to several integration times. Muon intensity has an angle dependence of

, where theta is the angle between the axis of coincidence and the vertical.[17],[18] Normal pendular motion, as well as any extreme motion as a result of turbulence, thus could have a serious effect on observed fluxes. To account for this, results were averaged over sliding 5-minute intervals, assuring a smoother curve, and reducing the angle dependence to a constant multiplier of the observed flux. Since this experiment is concerned with relative flux, not absolute flux measurements, the angle dependence can thus be ignored.

The balloon ascended at approximately 1000 feet per minute, reaching a maximum altitude of approximately 28 km. The apparatus recorded data on the ascent and the descent, meaning each coincidence counter returned two datasets of flux versus altitude, for 8 datasets total. The balloon was launched from St. Peter, Minnesota, and recovered a ways west of Rochester, Minnesota. This was a West-East flight, with very little variation in latitude, so the latitude effect was neligible [17].

#SecSim

Simulation

To construct the model, a numeric simulation based on Eq. 5 and Eq. 8 was performed. The simulation was written in IDL. An initial spectrum at 14 km was computed using Eq. 9, using 10,000 logarithmically-spaced bins, with a minimum energy at

, the muon rest mass, and a maximum energy at 100 TeV, such that the region of highest energy resolution was the lower end of the spectrum, where most muons were expected to fall. kappa was computed from the initial number of muons and alpha. The propagation of the simulation took place in a large array. For an initial altitude of 14 km with 5-meter intervals, the map was a 10,000x2,800 array, consisting of 28 million cells. The simulation was then advanced in 5-meter steps, at each step computing the energy loss for each bin, computing the resultant energy for those particles, and assigning them to the appropriate bin in the next row.Each new energy bin was then reduced by an appropriate amount according to Eq. 8. The model was then interpolated to the altitude-values used by the actual data, and a chi^2 analysis performed. This was repeated for multiple values of alpha, and thus multiple initial spectra, with the program code seeking out values of alpha that minimized chi^2<. Since the mean free path of cosmic ray primaries is approximately 275 g/cm², the model was pinned to the data at 313 g/cm², or approximately 10 km.[11]

Experimental Apparatus

Results

Fig. 3: Unshielded flux and shielded flux. Note that the shielded flux is far lower than the unshielded flux. Error bars are not shown for readability, however, the largest vertical error bar is approximately 3000 counts per second per square meter per steradian, which is a tiny fraction of the difference between the shielded and unshielded fluxes. The difference in depth introduced by the lead is 17.01 g/cm². The difference observed is far greater than the difference that would be introduced by a 17.01 g/cm² offset, whereas the disintegration hypothesis should predict a small enough difference that a 17.01 g/cm² offset would result in a higher shielded flux than unshielded flux.

Our data qualitatively matches that observed by Rossi and others,[1] with the exception of our lead-shielded counters. As seen in Fig. 3, we observe a decrease in counts under the lead shielding far beyond that expected by the additional stopping power of the 1.5 cm lead absorber. This suggests a possible instrumentation error in at least one of the G-M tubes used in the shielded coincidence counter. Thus, it is not possible with our data to verify the disintegration phenomenon observed by Rossi.[1]

We also note that the variation between datasets is significantly greater than that predicted by uncertainties calculated through Poisson counting statistics. This also suggests the presence of an uncertainty introduced by the instrumentation, perhaps variations in the sensitivies of individual detectors. To account for this, we define 20-meter bins, and average all unshielded points inside a given bin, using the standard deviation of points sampled for the bin as the uncertainty. We also consider only data collected at 10km and below, to try to avoid the complications of muon formation. When plotted on a logarithmic graph against atmospheric depth, as in Fig. 4, the binned data yields a linear trend as observed by Rossi and others.

Fig. 4: Binned data versus atmospheric depth, on a logarithmic plot. Note that half of the vertical error bars have been omitted for readability. Note as well the similarity to data observed by Rossi.[[[#RosSi48][10]]]

Analysis

The time-averaged data shown in Fig. 3 was produced by averaging 300 consecutive data points together. Since the time and therefore altitude assigned to this average depended on the times and altitudes of the component datapoints, it was considered that the uncertainty on the altitude of the averaged point could be found by propagating the original errors on the altitude measurements assigned to the datapoints through the arithmetic mean used to produce the data points. Thus:

where N is the number of points in the mean, in this case 300. <latex>\sigma_{z_i}</latex> was found by considering that observed coincidences could have occurred at any point throughout the integration time, so the uncertainty on the altitude of individual datapoints was half the difference between the point's altitude and the altitudes of neighboring points--in other words, half the integration bin width. The uncertainty on the averaged points themselves was computed in the same manner, with the exception that the uncertainty on the original points was given by Poisson counting statistics:

where f is the flux observed at that altitude.

However, since the variation between datasets taken at similar altitudes was much larger than the uncertainties computed purely from counting statistics and altitude uncertainty, the data was averaged into 20-meter bins. The bin edges were defined arbitrarily at rigid cutoff points, so any information about the uncertainty on the altitude was lost. The uncertainty on the altitude for a given bin is then half the bin width. The additional variation of the averaged datapoints beyond that expected by counting statistics is indicative of another source of error. Since it was not possible to precisely quantify what that error was and what its source was, the error on the averaged flux in a given bin was determined to be the standard deviation of the set of datapoints that were averaged together to arrive at that flux.

Fig. 5: Two different calculated initial spectra. Note that while the conditions used in the chosen model, alpha=-1.025, do converge, they do not converge to 0 on the upper end of the energy spectrum in the region below 100 TeV. The spectrum on the left corresponds to the spectra observed and parameterized by previous experiments.[9]

The data most closely fit a model produced with a muon injection at 14 km with a value for <latex>\alpha</latex> of -1.025, as seen in Fig. 5. The fit was pinned to the binned data at 10 km. Because the temperature data collected was deemed unreliable, as it indicated negative density at high altitudes, an atmospheric model for pressure and density was used in the simulation.[19] The overall reduced chi^2 of the model was 6.58, which is not particularly good--ideally the average variance away from the model is about one standard deviation. However, much of this deviation is due to the model and the data diverging at low altitudes. As seen in Fig. 6, the model predicts a far smaller flux than is observed for depths of 800 g/cm² and deeper. In the region of 300-700 g/cm², the model fits the data with a reduced chi^2 of 0.20.

Fig. 6: Binned data vs atmospheric depth. An exponential fit is shown, along with a model produced by a simulation based on initial conditions. The exponential fit is of the form and has a chi^2 of 0.22. The model has a chi^2 of 6.58 overall, but 0.20 in the region 300-700 g/cm². Note that the model diverges from the data at low altitudes. This is indicative of an incomplete model. Note as well that half of the vertical error bars have been omitted for readability.

Mathematically, the data can be fit with a simple exponential, appearing as a straight line on a logarithmic plot. This fit can be parameterized as where x is the atmospheric depth. This fit has a reduced chi^2 of 0.22 over the entire range observed. The similarity between that goodness of fit and the reduced chi^2 for the model over the 300-700 g/cm² range suggests that the form of the model, in which velocity-dependent time dilation is the primary mechanism for deeper muon penetration, is a quantitatively good fit. However, the specific model used, whether in its implementation or in its initial conditions and assumptions, must be in need of improvement.

One possibility is that the initial spectrum used differs somewhat from the actual spectrum at high altitudes. This is particularly likely because it has previously been observed that for muons at 15 km, alpha should be approximately -2.45,[9] and our model requires a value of -1.025. This means that our model decays muons more than they should be, as it requires higher-energy muons on average to produce the observed behavior than is expected from in situ observations at high altitudes. It's possible that the muon spectrum is not a pure power law for the entirety of the spectrum, and that modifications to the spectrum could yield better results. Indeed, it has been suggested that the high energy range of the spectrum in particular obeys a different distribution than the mid-ranges.[15]

Another possibility is that an accurate model must account for continuous muon formation, rather than a sharp initial injection. Primary cosmic rays can sometimes penetrate deep into the atmosphere, and that can lead to muon creation at lower altitudes.[2] Accounting for this could lead to an increase in predicted flux at low altitudes, better fitting the data.

A third possibility is that of contamination by other particles. The contribution of pions and kaons is negligible,[6] but the contribution of electrons can be significant. When muons decay, they produce electrons, so the deviation between the model and the observed data could be due to increased electron contamination at lower altitudes. However, the contamination from electrons is expected to be greatest at high altitudes, which would suggest higher counts than predicted by the model at high altitudes. Another interesting possibility raised by the possibility of electron contamination is that at least part of the large deviation between the observed shielded counts and the expected shielded counts could be due to the lead shielding stopping much of the electron contamination.[20]

Conclusions

We observe the vertical muon flux as a function of altitude using coincidence counters composed of Geiger-Müller tubes carried up to 28km by a high-altitude weather balloon. We were not able to verify the disintegration hypothesis as observed by Rossi, but our model suggests that velocity-dependent time dilation as the primary mechanism for deep muon penetration quantitatively fits reality well, with a reduced chi^2 of 0.20 for the region from 300-700 g/cm² closely matching the goodness-of-fit of a mathematical (non-physical) fit corresponding to a pure exponential, with a reduced chi^2 of 0.22. However, further work is necessary due to the deviation of the model from the observations at low altitudes. A variety of models must be investigated, that take into account things like non-uniform muon creation altitudes, a more sophisticated initial spectrum than initially assumed, and the possibility of electron contamination. It is also of course possible that some part of the simulation produced was flawed, so it is important that more models be created using the same technique to verify the results found in this paper.

Acknowledgments

The authors would like to thank Kurt Wick, the project advisor, for his valuable insight and assistance, particularly in the weeks leading up to the launch. We would also like to thank Dr. Clem Pryke for his advice regarding using a simulation to model our data. We also extend our thanks to Dr. Pawloski for his role in facilitating the course, along with Dr. Pryke, to the University of Minnesota School of Physics and Astronomy for providing funding and materials for this experiment, to Dr. James Flaten of the Aerospace department for providing the weather balloon and launch expertises and equipment, to Dr. Nathaniel Paradise for assisting in payload recovery, and to the farmer on whose land our payload landed.

References

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[19] B. Wilczyńska, D. Góra, P. Homola, J. Pȩkala, M. Risse, and Wilczyński, AstroparticlePhysics 25, 106 (2006).

[20] P. Gill, The Physical Review 71, 82 (1947).