Abstract

Researchers and other QuarkNet student groups have conducted studies observing that the velocities of muon particles from cosmic ray showers reach almost the speed of light, 3*10E8 m/s. This experiment was conducted to determine if calculated velocities for the particle differed depending on the environment of the setup. Four stacked scintillator panels were used to detect incoming muons. The distance between these panels and the coincidence settings varied depending on the specific setup. Two experiments were conducted indoors: the first with a total distance of 1.4 meters between the top and the bottom panels, the second with a total distance of 2.93 meters. These conditions were repeated outdoors near the entrance to the building with only minute differences. Due to technical restraints, only three panels collected data outside.  Logic statements were implemented into excel spreadsheets to aid in the effort to analyze thousands of lines of data. It was observed that the second method, with a greater distance between panels, provided more accurate results similar to other sources. The calculated velocities using both methods were greater for the outdoor conditions than the indoor conditions. However, these differences were not significant. It was concluded that the infrastructure did not affect the measurable velocity of the muons. 

Introduction

            High energy protons in cosmic rays collide with other particles in Earth’s atmosphere to produce showers of other constituents. Many of these particles are short lived and decay into others. These interactions require a great level of energy, so many particles are stopped in the atmosphere. Muons, which are relatively easy to detect on the Earth’s surface, are produced as charged pions decay. According to the photo electric effect, muons emit photons as they pass through scintillator panels. These signals are amplified by photo multiplier tubes which create computer readable data.

            Before postulating our idea, we conducted several other muon studies such as flux and lifetime analysis. As we researched previous QuarkNet experiments, we found that many groups had attempted to calculate the velocity of muons. However, no group had conducted a velocity study both indoors and outdoors. We wondered whether the thick infrastructure of the building significantly affected the velocity of the particles.

            To collect the data, the panels were stacked on top of each other during the experiment. Therefore, the distance between the detectors was very important. The scintillation method of the detectors is not perfect; the panels are accurate to about 1.25 nanoseconds. In this time period, a particle traveling close to the speed of light would travel tens of centimeters. The panels would have to be positioned directly over each other with space between them. Fortunately, a previous group had built a scintillator panel holding device about 1.5 meters in height. This structure was utilized both indoors and outdoors to hold the panels. This method was taken to a new level as the distance was increased approximately two-fold. The structure was placed upon a vintage wooden staircase which increased the difference in height to approximately 3 meters.

            While researching old experiments, we found that previous students had calculated the velocity of muons within our building to be only about 0.5c (Muon Speed Study). We developed our procedures based on their methods. However, we adjusted our experiment according to our conditions and improved a few areas of concern. These previous students hypothesized that the velocity was so low due to the thick structure of the building. We set out to determine whether muon velocity really is affected by the building. We hypothesized that the calculated muon velocities outdoors would be greater than the calculated velocities indoors. 

Procedures

Experimental Setup:

The detectors were always stacked in order, with detector 0 at the top and 3 at the bottom. The distances between the detectors were measured for each setup and recorded, as these values are required for the data analysis. The coincidence level was always set to the maximum number of detectors. Since only three panels were available outdoors, the coincidence level was changed from four to three. Within the PuTTY software, the timing gate for the detectors was changed from 24 nanoseconds to 48 nanoseconds, depending on the distance between the panels. This insured that any detections that occurred at less than 20% of the speed of light would not be recorded.

The first trial involved the stacking structure indoors. The distance between the top and bottom panels was 1.39 meters. The panels were evenly displaced in the structure. The second trial involved placing the previous structure onto a wooden staircase. The new total distance was 2.93 meters. These experiments were repeated almost identically outdoors, but only with three panels.

We ran each setup for as much time as was available in our hectic schedule. The outdoor trials only lasted a few hours due to this time constraint whereas the indoor trials could be run overnight. The first indoor trial lasted almost 48 hours while the second, with greater distances between the panels, lasted 20 hours. The first outdoor trial lasted approximately 3 hours while the second, with greater distances, lasted 6 hours.

Observing Data:

After data collection ended for each trial, the data was uploaded into the shower study method on the cosmic ray e-lab. Under the file directory, the eventsCandidates file was downloaded and imported into excel. This file provided a readout of the order of detections for an event and the times at which each panel was triggered. Next, logic statements were implemented to check that the detectors were in order when they were triggered. The difference in trigger times between panels as well as the velocity between the first and final panels was calculated. The times were multiplied by 86400, since the times are given as fractions of a day. Because every event did not provide every measured value, the columns were filtered in excel to organize the data. The differences in trigger times were found so that a velocity could be calculated by obtaining the slope on a graph with time as the x axis and distance as the y axis.

Data Analysis:

Two methods were used to measure the velocity of the muons for each trial. First, the velocities between the first and last panels were calculated. These velocities were graphed on a histogram to easily compare the results. The second method involved the distance between the detectors and the difference in trigger times. The differences in trigger times were measured between detectors 0 and 3, 0 and 1, and 0 and 2 when the experiment was conducted indoors. Similarly, the differences in trigger times were measured between detectors 0 and 3, 0 and 2, and 2 and 3 when the experiment was conducted outdoors. The medians were found and plotted on a graph and a regression was used to find the slope of the points. This slope was another estimation of the velocity of the muons for that trial. 

Results

Data Analysis:

Figures 1-4 show the individual distributions of calculated velocities for each trial. All of the graphs are right skewed and contain velocities above the speed of light, which is impossible. This is due to the timing errors of the scintillator panels. Nonetheless, it is clear that the distributions for both outdoor trials are concentrated closer to the value of c than their respective indoor trials. Even though the first outdoor trial histogram is concentrated relatively far away from the speed of light, the values seem to be greater than in the histogram for the first indoor trial.

Using the first method to calculate velocity, the first indoor trial resulted in a mean velocity of 0.56c, while its counterpart, the first outdoor trial, resulted in a mean velocity of 0.61c. The second indoor trial, with a distance of almost twice as large, resulted in a mean velocity of 0.85c, while its counterpart, the second outdoor trial, resulted in a mean velocity of 0.86c. For both counter distances, the velocity increased when outdoors.

Figure 5-8 show the plots for each trial with the distances between the detectors and the median trigger times. The medians were used rather than the means as they are resistant measures of center.

Using the second method to calculate velocity, the first indoor trial resulted in a slope velocity of 0.26c, while its counterpart, the first outdoor trial, resulted in a slope velocity of 0.51c. The second indoor trial, with a distance of almost twice as large, resulted in a slope velocity of 1.05c, while its counterpart, the second outdoor trial, resulted in a slope velocity of 0.24c

For the first trial with a distance of 1.39 meters, the velocity increased with this method by almost twice when outdoors. However, for the second method with a larger distance, the velocity decreased by 1/4^th. These results suggest that this method, using slope to calculate velocity, might be slightly less reliable than the first method, finding the average velocity between the first and last panels.

Since the trials with larger displacement using the first method to calculate velocity provided values almost 90% of the speed of light (0.85c and 0.86c), a statistical test was conducted to provide evidence leading to a decision concerning the original hypothesis.

The calculated velocities from the indoor trial with a displacement of 2.93 meters and the outdoor trial with a displacement of 2.93 meters can be treated as two independent populations.  Therefore, a two sample t-test can be used to compare these populations. Several assumptions must be met to conduct this test. Since both samples have more than 30 trials each, their sampling distributions are normally distributed according to the Central Limit Theorem. The true population means of muon velocities and the standard deviations are not known. However, a simple random sample was not conducted due to the conditions of the experiment. Therefore, the results of the test must be analyzed somewhat cautiously.

The Null hypothesis states that the true mean velocity for muons inside of the building is equal to the true mean velocity for muons outside of the building. The alternative hypothesis states that the true mean velocity for muons inside of the building is less than the true mean velocity for muons outside of the building.

Ho: mu_indoors = mu_outdoors

Ha: mu_indoors < mu_outdoors

Figure 9 shows the results of the test as well as some other relevant statistics. However, because the calculated P-value of 0.436 is greater than the alpha level of 0.05, we fail to reject the Null hypothesis. There is no convincing evidence that the true mean velocity of the muons outdoors is greater than the true mean velocity of the muons indoors. If we assume the Null hypothesis true, there is a 43.6% chance of obtaining results as extreme as these. Figure 10 shows the resulting t distribution. 

Discussions & Conclusions

            According to the data, the velocities for both distances of displacement increased when the panels were used outdoors. For the first trials with a distance of 1.39 meters between the first and last panels, the mean velocity between these panels increased by 0.25c when conducted outdoors. However, the value was still only about 0.5c, which means these results are not very reliable. For the second trials with a distance of 2.93 meters between the first and last panels, the mean velocity between these panels increased by only 0.01c when conducted outdoors. This small difference and the results of the statistical test lead us to reject our original hypothesis. The calculated muon velocities outdoors were not significantly greater than those indoors.

 Discussion:

            The accuracy of this experiment could have been affected by the distance between the stacked detectors. Because the panels are accurate to about 1.25 nanoseconds, the distance between the detectors must be large enough that the time difference of the traveling muons can be measured. Equipment restraints only allowed the top and bottom detectors to be placed a maximum distance of 2.93 meters apart. If the detectors were placed a greater distance from one another, this experiment could have produced more accurate time differences of the muons, resulting in muon velocity measurements containing greater accuracy.

            The number of detectors used in this experiment could have also had an effect on this experiment’s accuracy. Four detectors were used in the indoor trials, whereas equipment restraints only allowed the usage of three detectors outdoors. Therefore, a four-fold coincidence study could only be conducted on the indoor trials; on the outdoor trials, three-fold coincidence studies were executed. Because a four-fold coincidence study provides the greatest data accuracy, the trials performed outside could be less accurate than those completed inside. The trials outdoor lasted much shorter than the trials indoor. Further experiments should divide time equally between all trials and attempt to subsidize at least 8 hours for each.

            The amount of data that was discarded could also be a source of error. Data that contained muon detections out of order was removed from measurement because it did not apply to the conditions of this experiment. Furthermore, this removed data could have resulted in a decrease in accuracy of this experiment’s results. Only a few differences of timing triggers were used for the regression method to calculate velocity. It might be advantageous to use every combination of differences for each trial and include them in the distance vs time graph for more accurate results. 

Bibliography

Harris, David, and Adrian Lorenzana. "Muon Speed Study." Cosmic Ray E-Lab. N.p., 8 July 2011. Web. 26 June 2014.