# An Analysis of Muon Flux from Angle Variation of the QuarkNet Cosmic Ray
Detector

Ricco C. Venterea  Department of Astronomy, Cornell University, Ithaca, NY,
14853 USA<EMAIL_ADDRESS>Urbas Ekka  School of Mathematics, University of
Minnesota, Twin Cities, MN, 55414 USA

###### Abstract

We present one of the first cosmic ray muon flux-angle variation experiments
on the QuarkNet Cosmic Ray Detector (QNCRD). We first describe QNCRD and its
calibration. The main focus is then quantifying muon flux decrease as a
function of angle from the zenith. The angle of counters of QNCRD were
incremented 15° on average every $3.1$ days over the range of 0° to 90° for a
period of approximately one month. Results showed that as the angle of the
detector increased from the zenith, muon flux decreased, which agrees with
previous studies. An estimate for the flux based on the model
$I(\theta)=I_{0}cos(\theta)^{n}$ had an exponent value of $n=1.39\pm 0.01$ for
$\theta\leq 75\degree$, an underestimate of values in other literature. These
findings provided a reasonable, although not entirely accurate, estimate for
the value of $n$ considering the duration of the study and sensitivity of the
instrument. Our results constrain the accuracy of QNCRD and provide a source
for future long-term experiments. This study also demonstrates the feasibility
of conducting science experiments in high school classrooms, increasing
science accessibility.

##  I Introduction

QuarkNet is part of a National Science Foundation funded effort to increase
science accessibility across high school classrooms in the United States. This
effort also includes training for high school science teachers and
students.Bardeen  _et al._ (2018) As part of this network, QuarkNet Detectors
are located throughout the world in high school classrooms.Arce-Larreta  _et
al._ (2022) These detectors have been used to create and study particle
physics experiments in classrooms, which range from the impact of solar
eclipses on cosmic ray muon fluxDallal  _et al._ (2022) to determining average
zenith muon flux rate.Shaffer (2010)

Muons are a byproduct of cosmic rays, a stream of particles constantly
entering Earth’s atmosphere. Cosmic rays are composed of highly energetic
particles, mostly consisting of hydrogen and helium nuclei. High-energy cosmic
rays originate from neutron stars, while low-energy rays originate from the
Sun.Gaisser  _et al._ (2016) The QuarkNet Detector measuresDallal  _et al._
(2022) muon flux momenta greater than $2$ GeV.

When cosmic rays enter Earth’s atmosphere, they collide with air molecules,
creating a cascading effect known as an air shower.Gaisser  _et al._ (2016)
After this collision, pions are produced, some of which decay into muons.
Other pions continue into earth’s atmosphere and interact with air molecules,
creating more air showers.Rao and Sreekantan (1998) Muons are similar to
electrons, with a negative charge and about 200 times as massive than the
electron.Tanabashi  _et al._ (2018)

These muons can be measured using detectors on earth. As the muons pass
through a scintillation counter, they interact with electrons, which release
photons. These photons are reflected inside the counter until they reach a
detector, where they are transformed into electric signals.

It is currently known that muon flux decreases as the angle of a muon detector
increases from the zenith. Previous muon flux studies have used more precise
detection methods at various latitudes, longitudes, and altitudes. Shukla and
SankrithShukla and Sankrith (2018) describe the theoretical and experimental
flux values for a muon detector located at sea level based on the
$\cos^{2}{\theta}$ model. They also implement their own best-fit model,
$I(\theta)=I_{0}D(\theta)^{-(n-1)}$, with $n=3.09\pm 0.03$. We also implement
this model. SchwerdtSchwerdt (2018) presents a model based on
$I(\theta)=a\cos^{2}{(b\theta+c)}+d$. Pethuraj et al.Pethuraj  _et al._ (2017)
model muonic flux as a function of angle and arrive at an exponential value of
$n=2.00\pm 0.04$ for their model $I(\theta)=I_{0}\cos^{n}{\theta}$, located
$\approx 160$ m above sea level. The cos-squared model is the main comparison
of this study, but we also present values for the Schwerdt,Schwerdt (2018)
Shukla,Shukla and Sankrith (2018) and Pethuraj et al.Pethuraj  _et al._ (2017)
models. We find the SchwerdtSchwerdt (2018) model provides the best fit for
our data compared to the aforementioned models.

While not a flux experiment like the one presented in this paper, Shaffer
presents average muon flux rate results using the QuarkNet Detector. Shaffer
used QNCRD at an angle of $0\degree$ from the zenith to measure muon flux near
Topeka, Kansas and found a flux rate of $1200$ to $1500$ events per meter
squared per minute per steradian, values significantly lower than those found
in this study. Shaffer’s plateau values were different than those used in this
study and collected data for several weeks, while this study was one month of
data. Shaffer presents a novel solution to measuring steradians using the
QuarkNet Detector, a conversion we use in this study. Shaffer’s total detector
distance between top and bottom counter was $40$ cm total, while the maximum
spacing for our counters was $13$ cm. The coincidence rate used in the
plateauing process in this study is approximately equal to the rate used by
Shaffer.Shaffer (2010) Coincidence rate is the number of counters needed to
qualify muon signals as a detection (see Figure 1).

Figure 1: Illustration of two-, three-, and four-fold coincidence, where the
muon is only detected if it passes through all counters in each coincidence
configuration.Fermilab (2012)

In Section II, we discuss features of the QuarkNet Detector and provide a
description of the calibration process of QNCRD. The experiment is described
in Section III. Results from this flux experiment can be found in Section IV,
followed by a data analysis in Section V. We follow the analysis with a
discussion in Section VI. We conclude with relevant findings and further
improvements in Section VII.

##  II Quarknet Detector

The detector used in this study is located at Irondale Senior High School,
with coordinates $45.0900\degree N,93.2072\degree W$ at an altitude of $276$
meters.

QNCRD consists of a data acquisition (DAQ) board, four scintillation counters,
Equip software, photomultiplier tubes (PMTs), Global Positioning System (GPS)
receiver, a power supply and a power distribution unit (PDU). Each plastic
counter has dimensions $25.4$ cm x $30.5$ cm x $1.27$ cm and a "cookie"
attached to one corner.QUA (2012) This cookie is the interface between the
counter and the photomultiplier tube. Each counter is wrapped in reflective
shielding to retain any signal from muon interactions. Counters will also be
referred to as channels throughout this paper, being labeled as channel $0$,
channel $1$, channel $2$, and channel $3$.

Each counter has a photomultiplier tube (PMT) that collects electric signals
as muons pass through the shielding.Lofgren (2001) The PMTs are SensTech Model
P30CW5 photodetector packages.QUA (2012) These photodetectors are connected to
a power supplyShaffer (2010) and controlled by a PDU, with voltages in the
range $0.30$ V to $5.0$ V.

The data acquisition board is the circuitry necessary for collecting electric
signals via the PMTs. This data board also generates the output data to be
uploaded to the Equip software.QUA (2012) There is a 1.25 ns resolution on the
data board, used for separating muon events and unrelated and unwanted ion
events. Shaffer (2010)

The GPS antenna is placed outside the school building at all times, while the
GPS box is located inside, next to the data acquisition board. The GPS
provides the location of the instrument as well as times of the events, which
is accurateShaffer (2010) to $24$ ns.

The Equip software records the channels in use, as well as temperature,
location, and flux. This also includes the coincidence rate. The software
provides an interface to the Cosmic Ray e-Lab, a website where data from
QuarkNet Cosmic Ray Detectors across the world are uploaded.Fermilab (2012)
Equip was installed and ran on a Windows XP operating system.

###  II.1 Plateau process

To ensure reliability of data, performance studies are conducted to measure
the time over threshold (ToT) of the PMT to a muon event.Fermilab (2001a) The
ToT is defined as the amount of time an event is above a predetermined
threshold level.Fermilab (2001b) Without this process, the flux may be an
under- or overestimate of the true value of muons passing through the
counters. Such inaccuracies would be due to a high or low voltage value of the
PMTs. Voltage values are adjusted through the power supply.

The plateau process involves setting the power supply to $0.3$ V, the lowest
voltage setting. In order to plateau one counter, another counter has to be
used as a reference. Counters 0 and 1 were stacked, channel 0 serving as a
reference. The threshold level of the three detectors was set to 300 mV by
typing TL 4 300 into the Equip software. Channel 1 was activated and read a
one-fold coincidence. One-fold coincidence means that a muon needs to travel
through one detector to be counted as a detection (Figure 1). Waiting for $10$
seconds, the voltage was increased until the digital counter on the DAQ board
was between $400$ to $600$ counts. Once the counter was within this range, the
voltage was gradually increased until the coincidence counts levelled off.

This process was repeated for counters 2 and 3, channel 0 serving as the
reference channel.pas (2009) Results for channel 3 are shown in Figure 2.
After this plateau process, data were collected over the next two weeks for
eight hours each day. This data was collected to ensure the counters were
calibrated correctly. The counters were stacked 1, 0, 2, 3, from bottom to
top, with a 0 degree angle from the horizon. Three-fold coincidence was used
for counters 0, 2, and 3, as counter 1 was determined inoperable due to
improper wrapping of the reflective shielding in that counter.

Figure 2: Plateauing coincidence rate in channel 3 is indicated by the green
triangles. The coincidence of channel 3 plateaued within a specific voltage
range. This plateauing is from the second round of calibration.

During the initial plateau process, results from counter $2$ started to cause
concern. The channel rate was displaying a peak pattern at the same time of
day. This prompted a re-calibration of QNCRD as the preliminary voltage was
set too low. The plateau process was therefore repeated, with new data
collected over two weeks. The voltages after the second plateau process are
shown in Table 1, which were used for the remainder of this study. The voltage
values were estimated from the plateau graphs where the coincidence values
were just beginning to plateau (Figure 2).

Table 1: Voltage values determined from plateau process. See Figure 2 for the voltage estimate in counter 3. Counter | Voltage (V)
---|---
3 | 0.770
2 | 0.800
0 | 0.709

##  III Experiment

Data was collected almost every day between the end of October 2019 to the
beginning of December 2019. The initial angle of the detectors was $0$ degrees
from the horizon (i.e., parallel to the horizon). The detector angle was
incremented by $15$ degrees approximately every $3.1$ days, with the study
ending with the panel surfaces perpendicular to the horizon, defined to be
$90$ degrees. Data was uploaded from the QuarkNet Detector to the Cosmic Ray
e-Lab website, where raw flux data was extracted.

Figure 3: Schematic of the detector configuration is highlighted in Figure 3$a$. Starting from the top, the counters are labeled Counter $3$, Counter $2$, Counter $0$, and Counter $1$. The individual counters are shown in Figure 3$b$ and are represented as black rectangles. Figure 3$c$ represents how the QuarkNet Detector angle was varied throughout the duration of this experiment. Note that only the top three counters were used in this study, as the fourth detector was deemed inoperable. Diagrams are not to scale. Table 2: Relative vertical spacing of counters used in this study, with negative values being measured below the origin set at the GPS box. Counter | Vertical Spacing (m)
---|---
3 | -1.5
2 | -1.565
0 | -1.63

The QuarkNet Cosmic Ray Muon Detector was set up using three counters (see
Figure 3) and three-fold coincidence. The configuration of the counters
remained stacked 1, 0, 2, 3, from bottom to top. The relative vertical spacing
measured from the QNCRD GPS box is provided in Table 2 and illustrated in
Figure 3.

##  IV Flux Results

The flux data were collected at 15° increments for a minimum of 1.5 days using
the voltage values in Table 1. The detector’s final angle ended at 90° (a
vertical orientation relative to the horizon). Data were not collected on
Wednesdays because computers in the school were automatically turned off,
preventing any data collection during this time period.

Since the QuarkNet Detector does not have the capability of measuring flux per
steradian, we use Equation 1 as described by Shaffer to convert our data to
make its analysis easier:Shaffer (2010)

$\tan{\theta}=\frac{w}{d}$ (1)

where $w$ is the width of the detector and $d$ is the total distance between
the top and bottom channel. Utilizing the fact that the angular measurement of
$32.77$ degrees from the normal equals one steradian, the width of the
QuarkNet Detector counter being $0.26$ m, and the distance between the top and
bottom counter of the detector being $0.13$ m (see Table 2), we found the
needed adjustment of data to be:McGraw-Hill (1997)

$\arctan{(\frac{w}{d})}\cdot\frac{1sr}{32.77\degree}=\arctan{(\frac{0.26m}{0.13m})}\cdot\frac{1sr}{32.77\degree}=1.936sr$

We now converted flux results from the Cosmic Ray e-Lab into units of
$events/m^{2}/min/sr$. A combination of all the muon flux measurements at
increasing detector angles is shown in Figure 4. This experiment continues to
verify the general trend that muon flux decreases as the angle of the detector
increases from the zenith.

Figure 4: Decrease of flux starting at 0° ending at 90°. These results are
from channel $0$ for detector $6709$.

While collecting data, we found discrepancies in data at $30$° and $75$°. We
attributed this to an improper alignment of the counters and a computer
malfunction. We resolved these issues by conducting the experiment again over
$1.5$ days for each affected angle.

##  V Analysis

A statistical analysis was performed on the flux data, which included removing
outliers and then comparing data to the $\cos^{2}{\theta}$ function, as this
is widely believed to be the most accurate description of muonic flux as a
function of angle.Shukla and Sankrith (2018), Pethuraj  _et al._ (2017),
Shaffer (2010), Bektasoglu and Arslan (2013), Shteinbuk (2011) We also fitted
flux data to models given by Shukla and Sankrith,Schwerdt (2018)
Schwerdt,Shukla and Sankrith (2018) and Pethuraj et al.Pethuraj  _et al._
(2017) We find Schwerdt’sSchwerdt (2018) model best represents the data
presented in this paper, which is supported by a reduced chi-squared test with
a value of $\chi{{}^{2}}_{\nu}=0.467$.

We first cut outliers in the flux data, using the $25$th and $75$th percentile
values. These outliers were based on the interquartile range of the data. Any
values lying below the $25$th percentile minus $1.5$ times the interquartile
range were cut and values above the $75$th percentile plus $1.5$ times the
interquartile range were cut. This removes the $5$ outliers as seen in Figure
4.

Due to construction occurring during measurements, it is possible that
electrical interference may have affected the results causing the outliers in
our data. Another cause of outliers could have been the relative age of the
detector, and according to Bae and Chatzidakis, detections of high zenith
angles ($\theta>60$°) saw high levels of uncertainty.Bae and Chatzidakis
(2022) However, we find outliers for $\theta<60$°.

Once we cleaned the data of outliers, we were able to generate a comparison
based on the $cos^{2}(\theta)$ model. Converting the flux in units of
steradians also helps validate our findings, as the data can now be easily
compared to other flux studies. This comparison is seen in Figure 5.

Grieder and Pethuraj et al. state that the intensity of muons follows the
empirical model:Grieder (2001), Pethuraj  _et al._ (2017)

$I(\theta)=I_{0}\cos^{n}{\theta}$ (2)

where $I_{0}$ is the vertical intensity and $\theta$ is the angle from the
zenith. However, this equation is used to approximate intensitiesGrieder
(2001) only for $\theta\leq 75\degree$. Using the curve_fit function from the
scipy.optimize package and the experimentally determined Pethuraj et
al.Pethuraj  _et al._ (2017) model, we fit our data for $\theta\leq 75$°. We
calculated an exponential value of $n=1.39\pm 0.00657$, with $r^{2}=0.973$.

We also try modeling to the Shukla and Sankrith function, given by:Shukla and
Sankrith (2018)

$I(\theta)=I_{0}D(\theta)^{-(n-1)}$ (3)

with

$D(\theta)=\sqrt{(\frac{R^{2}}{d^{2}}\cos^{2}{\theta}+2\frac{R}{d}+1)}-\frac{R}{d}\cos{\theta}$
(4)

where Shukla and SankrithShukla and Sankrith (2018) fit the ratio $R/d=174.0$.
Using this model, we find $n=2.41\pm 0.00645$ with $r^{2}=0.974$.

Finally, we compare to the Schwerdt model:Schwerdt (2018)

$I(\theta)=a\cos^{2}{(b\theta+c)}+d$ (5)

with $a$ representing a vertical stretch, $b$ a horizontal stretch, $c$ a
horizontal stretch, and $d$ a vertical shift. This model is more mathematical
in nature, representing the most general form of the cos-squared
function.Schwerdt (2018) We find this model best represents the data,
especially towards increasing values of $\theta$ and has $r^{2}=0.997$. This
function is approximated by

$I(\theta)=2241.21\cos^{2}{(1.047\theta+0.0678)}+665.13$ (6)

There are several differing values for $n$, all looking at flux data with
$\theta\leq 75\degree$. GriederGrieder (2001) states that the average value of
$n=1.85\pm 0.10$. Useche and AvilaParra and Bernal (2019) state the
experimental value for $n=1.96\pm 0.22$. Other resultsBektasoglu and Arslan
(2013),Parra and Bernal (2019) have estimated the value of $n$ to be
$n=1.95\pm 0.08$ and $n=2.11\pm 0.03$. See Figure 5 for a comparison of models
mentioned in this paper with flux data collected in this study. See also Table
3 for a comparison of exponential values from other studies.

Figure 5: Comparison of muon flux models as a function of angle with data collected in this study. Note that we only fit for $\theta\leq 75\degree$. Observe how well the SchwerdtSchwerdt (2018) model agrees with the data. The Shukla and SankrithShukla and Sankrith (2018) model and Pethuraj et al.Pethuraj  _et al._ (2017) model agree so well they overlap. Authors | Mag. Lat. (°N) | Alt. (m) |  _n_ value
---|---|---|---
Crookes and Rastin | $53$ | $40$ | $2.16\pm 0.01$
Greisen | $54$ | $259$ | $2.1$
Judge and Nash | $53$ | $0$ | $1.96\pm 0.22$
Karmakar et al | $16$ | $122$ | $2.2$
S.Pal | $10.61$ | $0$ | $2.15\pm 0.01$
S. Pethuraj et al. | $1.44$ | $160$ | $2.00\pm 0.04$
_This study_ | $45$ | $276$ | $1.39\pm 0.01$
Table 3: Comparison of muon flux data to previous studies.Pethuraj  _et al._
(2017) Note that $0$ m altitude corresponds to sea level.

We also perform a chi-squared test on the four models discussed in this study.
We find that the reduced chi-squared $\chi^{2}_{\nu}$ value for the
SchwerdtSchwerdt (2018) model best represents the data, which is also
supported by a coefficient of determination of $r^{2}=0.997$. Figure 5
displays how well this model follows the data. We summarize our statistical
results in Table 4.

Model | Equation | $\chi^{2}_{\nu}$ | $r^{2}$
---|---|---|---
Pethuraj et al. | $I_{0}\cos^{n}{\theta}$ | $5.70$ | $0.973$
Shukla | $I_{0}D(\theta)^{-(n-1)}$ | $5.35$ | $0.974$
Schwerdt | $a\cos^{2}{(b\theta+c)}+d$ | $0.467$ |  $0.99$7
cos-squared | $I_{0}\cos^{2}{\theta}$ | $22.7$ | $0.916$
Table 4: Summary of statistical tests performed on the four models presented
in this study for $\theta\leq 75\degree$.

##  VI Discussion

In general, we saw the flux decrease as the angle increased, which agrees with
measurements made by SchwerdtSchwerdt (2018) and Useche and Avila.Parra and
Bernal (2019) The reason for this flux decrease is that as the angle
increases, the cosmic ray muons are not able to penetrate the counters at
extreme angles. The majority of muons are “raining" down on the channels at
0°; the intensity of muons entering at 90° would be significantly less.

The improved model to estimate the cosmic ray muon flux for all detection
conditions is especially significant for high zenith angles ($\theta>60$°)
because the cosine-squared model is limited in use for low zenith angles due
to large uncertainties and assumes a flat earth model.Bae and Chatzidakis
(2022), Yáñez and Aguilar-Arevalo (2021)

While the reduced chi-squared test favored the Schwerdt model, a value of
$\chi^{2}_{\nu}\approx 1$ suggests the model and measurements follow error
variance. For the Schwerdt model, we found the reduced chi-squared to be
slightly less than $1$, which may suggest an improper error fit for this
model.Schwerdt (2018)

Our results highlight a significant underestimate of previous flux studies,
with most agreeing with $n=2$ for the exponential value. However, given the
duration of this study, as well as the equipment used, this presents a
precise, although not entirely accurate, flux study. Indeed, if the duration
of the study were longer, more flux data could be obtained, presenting more
values to use in best fit models. Additionally, only three counters were in
full operation. Using a fourth counter would change the flux data being
collected, since cosmic ray muons would now have to traverse four counters in
order to be classified as a detection (Figure 1). However, this would add an
additional distance of approximately $10$ cm to the top and bottom detector
distance, which may change the flux data. In this case, it would be expected
for the cosmic ray muon flux to decrease since a fourth counter would raise
the requirement of being considered a measurement. The location of the
detector may also explain why we found flux values to be below the expected
value of $n=2$. Construction was ongoing throughout the school day (8:35 AM
CST to 3:15 PM CST). Any electrical interference may have affected these
results. Our results in general suggest an under performance of the QuarkNet
Detector.

##  VII Conclusion

We conducted a short-term cosmic ray muon flux experiment to test the cos-
squared model using the QuarkNet Cosmic Ray Detector and found the muon flux
to decrease as the detector angle from the zenith increased. This agrees with
previous experiments that varied the angle of cosmic ray detectors. For
relatively low angles ($\theta<45\degree$), our results roughly correspond to
the $\cos^{2}{\theta}$ model, with several areas for improvement. We did find
discrepancies in flux data at $30\degree$ and $75\degree$, which we attributed
to experimental issues. Resolving such issues would most likely improve our
current flux model. We also found the SchwerdtSchwerdt (2018) model to best
represent the data, especially for data at high angles. The vertical shift in
this model accurately accounts for muon flux at larger detector angles,
whereas the simple cos-squared model would yield a null flux result.

Our findings are important for several reasons. Most notably, this is one of
the first type of experiments performed on the QuarkNet Cosmic Ray Muon
Detector to analyze the relation between muon flux and angle. This paper
serves as a baseline for future studies that can improve upon our current
value of $n=1.39$ for QNCRD. These results also present an accurate
representation of other flux experiments as detailed here, legitimizing the
QuarkNet Detector as a tool for scientific research and study.

While our study was short compared to previous studies, and involved equipment
with much less sensitivity than other detectors, these results are important
for improving the accuracy of the detector. High schoolers, as well as
particle physicists, may use this paper to guide their own studies similar to
this.

This experiment not only provides a constraint on the accuracy of detecting
muonic flux at varying angles for QNCRD, but also provides a framework for
future long-term studies to be implemented resembling this experiment
description. We have several suggestions that could improve the accuracy of
this detector. More data should be collected over a longer period of time for
several reasons. Ensuring that there is no seasonal, diurnal, or other
temporal variations could not have been completely verified with this short of
study. Collecting over a longer period of time would increase the data set,
improving flux collection at various angles. We suggest collecting data over
at least six months.

Using a fourth counter should improve the results. Having a four-fold
coincidence would greatly increase the accuracy of the data and may bring the
flux results down for high angles.

The effect of latitude on cosmic ray muon flux can be better studied with this
detector. Given that QNCRD is located across the globe, this paper shows that
a worldwide study may be performed to better understand how flux changes with
latitude.Arce-Larreta  _et al._ (2022) A similar study may be performed to
study variations in altitude.

###### Acknowledgements.

We would like to thank funding and support from Fermilab, QuarkNet, the
National Science Foundation, the Irondale High School administration, and Mark
Adams of Fermilab. RCV would like to thank Shane Wood for taking the time to
meet with him after school and directing this study, as well as providing
comments for this paper. From his class, RCV learned the fundamentals of
cosmic rays, but with his guidance, he has been able to explore the subatomic
world in much more depth. RCV would also like to thank Dr. Mary Sande and
Logan Doroff for allowing this study to take place in their high school
classroom. Finally, RCV would like to thank Rodney Venterea for supporting his
scientific endeavors and providing feedback for this paper.

## Data Availability

The data used in this study are publicly available here:
https://github.com/ricco-hub/cosmic_rays/tree/master/Angles.

## References

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