1. Characteristics of Muon Decay
Samuel H. Trubey
Senior Physics Major, East Carolina University, North Carolina 27858
(Dated: May 10, 2013)
Abstract
Since it was first discovered in 1937, the muon has made huge contributions to the world of particle
physics. The muon is still experimented with, still validating theories like particle decay and special
relativity. This experiment was conducted to measure the average lifespan using a muon detector
with a scintillation light inside to start and stop a timing function at the beginning and the end of a
muon’s decay (respectively). After the data is collected and processed by the software collection
programs, we will calculate the average lifespan by taking the reciprocal of the average decay rate.
The end result was only 10% the accepted value, so an error
must be considered occurring during the setup or calculation process.
I. INTRODUCTION
In the November issue of Letters to the Editor in 1937, J.C. Street and E.C. Stevenson, two
research physicists from the research lab at Harvard University, submitted this:
New Evidence for the Existence of a Particle of Mass Intermediate Between

2. the Proton and Electron
The three-counter telescope consisting of tubes 1, 2, and 3 and a lead filter L for
removing shower particles, selects penetrating rays directed toward the cloud
chamber C which is in a magnetic field of 3500 gauss. [1]
Street and Stevenson weighed the mass of this new particle at roughly130 times the rest mass of
an electron, which was wrong. However, in the following July edition, Seth H. Neddermeyer and
Carl D. Anderson, from California Institute of Technology, stepped forward and not only said
Street and Stevenson were wrong, but that they had the correct values and more information:
Cosmic-Ray Particles of Intermediate Mass
A positively charged particle of about 240 electron-masses and 10 MeV energy
passes through the glass walls and copper cylinder of a tube-counter and emerges
with an energy of about 0.21 MeV. The magnetic field is 7900 gauss. The residual
range of the particle after it emerges from the counter is 2.9 cm. [1]
This once unknown particle was being grouped into Yukawa’s meson theory of strong nuclear
forces because it was offering considerable support and so most physicists attributed it to Yukawa’s
theoretical prediction. Unfortunately, the start of World War II interrupted and it wasn’t until
December of 1946 that a discrepancy was found within the new attribution to the meson theory.
[1]
Three Italian physicists (Conversi, Pancini and Piccioni) managed to stop a negative meson
particle in carbon. What they found, however, was that the decay probability is relatively close to

3. the capture probability by nuclei. This new particle was about to have some serious identity issues.
[1]
The first identity crisis was caused by misidentifying the cosmic ray mesotron as the meson
hypothesized by Yukawa. It was resolved by the experimental discovery of two distinct particles
µ and π. The second identity crisis was created by the misconception that µ and τ had to be two
different particles. It was resolved by the breakthrough of parity non-conservation, which revealed
that these two were in reality simply the different decay modes of the same particle, the kaon. [1]
The quick discovery and slow purifying process of the muon theories helped open doors to
particle physics, and has been experimented on since its unearthing nearly 80 years ago. Since then,
however, the properties of the muon have very well defined.
The muon is a negatively charged elementary particle in the lepton family, sharing the same
charge and spin as an electron, but is roughly 207 times the electron’s weight. Comprising a large
portion of cosmic rays, which collide with the earth, they decay from pions but manages to make
it to the earths surface due to its weak interaction with matter. A muons has a mean life of roughly
2.2 µs before it decays into an electron/positron, neutrino, and antineutrino, which can easily be
illustrated as such:
µ!
"e!
+!µ+!e or µ+
!e+
+!µ+!µ.[2][3]
For this experiment, the mean lifetime of the muon will be quantized with the given ability to
measure the average time of decay at sea level from our own personal detector. The primary bulk
of the detector is a 15 cm wide and 12.5 cm tall cylindrical plastic scintillator beneath a black
anodized aluminum alloy tube. The scintillator is made from a transparent organic compound
mixture and as a charged particle passes through, it will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. [4] As high energy muons pass through

4. the scintillator, some of its kinetic energy is stripped and excites the electrons of the scintillator,
causing a photon emission. This emission then triggers a reactive process, initiating the detectors
system to begin timing and cease timing when the muon eventually breaks down and emits an
electron. It is that small frame of time where the lifetime of a muon is measured.
It is important to note that there is both a positive and negative muon (hence the two diagrams
above) and furthermore, the negative muon’s lifetime is slightly less due to its weaker interaction
with the protons of the scintillator; this is why we are measuring the mean lifetime. How do we
find the mean? Well, in a simpler setting, the situation could be expressed as a simple exponential
relation of radioactive decay:
N(t)= N0exp(!!t) , (1)
where N is the number of remaining muons after time t and N0 is the original amount of muons at
t=0. λ is the decay rate of the given particle, and τ is the lifetime of the particle, being the reciprocal
of the decay rate, 1/λ. [4] The lifetime, τ, of the muons is given by the computer muon detection
program after the collection phase has been completed, the ideal value being τµ = 2.19703 ±
0.00004 µsec. It if from this provided value that we can easily calculate the decay rate and time
distribution seen below. [4]
This friendly looking equation is, unfortunately, impractical in this situation because we do not
have a single cluster of surviving muons we can easily count, nor do they come in clumps in the
first place. Decay time distribution D(t), will allow us to notate the time-dependent probability that
a muon decays in the time interval between t and t + dt as a given function D(t)dt. So in the off
chance we did know the starting number of muons, the fraction −dN/N0 would show the average
decay amount in the time interval between t and t + dt. This is just a derivation from the relation
above:

5. −dN = N0 λ(−λ t)
dt (2)
−dN/ N0 = λ(−λ t)
dt. (3)
The first part of Eq. (3) has nothing else but the decay probability, which we are seeking! The
decay rate as a function of time can be expressed as the exponential function,
D(t) = λe(−λ t)
. (4)
This works because N0 is irrelevant here because D(t) would be true, regardless of the original
number of particles. This is true because the distribution of decay times, for newer muons entering
the detector, uses the same exponential power to describe the number of surviving muons.
Reminder, we express muon lifetime as
τ= 1/ λ. [4] (5)
II. METHODS
To begin to talk about methods, I must first mention the technical difficulty that my partner,
Wilson Hawkins, and I encountered during our data collection phase. Happening twice, I initiated
and ran the software to begin detecting muon decays and I allowed it to continue collecting data
for several days. When I returned five days later, however, the computer was powered off, having
most likely been rebooted at some point over the weekend. I would attempt to run the program
again, planning on returning the following day to analyze data collected over 24 hours but I
encountered a similar problem yet again. The computer was non-responsive for several minutes,

6. and once it began to function normally again, it was apparent that the default user account was
logged off and upon reentry, all software had already shutdown so all data collections were lost.
After consulting with my professor and lab advisor, a decision to abandon the procedure of running
the equipment was made and my partner and I were to just use data from previous, successful trials
to analyze.
With this being said, the data collection stage consisted of both hardware and software. A muon
detector was connected to a Teach Spin muon physics electronics box which was the subsequently
connected to a computer which piloted the Teach Spin data collecting software which controlled
and monitored the detector.
Inside the detector was a perpetual chain reaction. Within the muon detector was a scintillation
light, which is detected by a photomultiplier tube (PMT), which then outputs a signal fed into a
two-stage amplifier. The amplifier output supplies an adjustable threshold to a voltage
discriminator where a transistor-transistor logic (TTL) pulse is created. This produced TTL pulse
is for input signals that are above the threshold so the pulse can activate the timing circuit for the
field-programmable gate array (FPGA). Within a certain time interval, a second TTL output pulse
is sent to the FPGA input, stopping and resetting the timing circuit (the duration of this reset last
roughly 1.0 x10-3
seconds). It is not the pulses, but the time between the pulses that is important
for this experiment; the interval’s data is passed through a communications module where the
lifetime of the muon is calculated before reaching the computer (Figure 1). [4]
The extent of actual work done to set up and run the calculations was minimal. There were only
three parts to tend to and make sure they were already connected in the proper sequential order;
after confirming that the detector and the electronics box were connected by the power source and
signal cable, the electronics box was connected to the computer, we were ready to begin. The

7. electronics box was powered on and then the high voltage supply was set to 1200 Volts and the
discriminator was set to 200 mV while the “Time Adj” and the “HV Adj” dials on the muon
detector were set to ten. Once that was completed, we were ready for software operation.
The first thing that needed to be done was to configure the program “muon”, found in the
“muon_data” folder. Underneath the “Control” icon is the “configure” icon, resulting in a new
menu, as seen in Figure 2. It is here that port “com1”, a time scale of 20 microseconds for the decay
histogram, and bin number 60 were respectively selected to finalize configuration. Once this was
done, the final step is to click “Start” and allow the program to function and collect the detections
and decay rates. The judgment and determination of the event was performed completely by the
software, operating just as I have described earlier. But to delve in a little further now, the TTL
pulses, which start and stop the time detecting intervals, are triggered when a detected muon begins
to slow. As a muon begins to slow and decay, it will ultimately break down into three lesser
particles, an electron, a neutrino, and an antineutrino. The excited electron has enough energy to
surmount the threshold and trigger the scintillator, releasing another burst of light to generate the
closing TTL pulse. Once the program is finished, it consolidates all the data collected and a
histogram chart that displays the total events and the muon decay rate, the reciprocal will be take
to satisfy Eq. (5) and that will consequently provide us with the measured lifespan. [4]
III. RESULTS
Table 1 displays our final results, after the calculations of decay probability and half-life. First,
the half-life was determined with Eq. 5 by simply finding the reciprocal of the muon’s provided
lifetime value, τ=4.568±2.03µs.

8. Uncertainty is being taken into account with the propagation of an uncertainty with a power, we
use:
!q= n!x , (6)
q x
where n is -1. Half-life λ=0.219±0.097µs. With the half-life determined, the time distribution can
be calculated using Eq. 4. Once more, the uncertainty must be determined so we begin with the
distribution equation, D(t) = λe(−λ t)
. The uncertainty of the product of decay rate and exponential
distribution, the exponent, and the product within the exponent must be determined. For
demonstration, we will assign an arbitrary variable b to the exponent. For this calculation, we will
also assume that the elapse time, t, is a constant 72 hours (2.592×10!!µμs). The uncertainty of the
exponent can be determined as
!b =t!!", (7)
where != !" (5.676×10!"!").
!b =t!!"
!b = 2.592!1011
µs!0.097 !b =
2.514!1010
µs.
Now that the uncertainty of the exponent has been determined, Eq. 6 can be translated as the
propagation of uncertainty of a product, being:
2 2 (8)
!D =e! "$!b%' +"!"% ,
$ '
D # b & # "&

9. where D(t) = λe(−λ t) (0.219×!!.!"!×!"!").
"!b%2
"!"%2
!D
= e! $ ' +$ '
D # b & # "&
!D "2.514!1010
%2
"0.097%2
= e! $ 10 ' +$ '
D #5.676!10 & #0.219&
!D
= e! 0.392 D
!D
= 0.626e
D 1.135!1011
!D= 0.137!e
So the decay time distribution is calculated at 0.219×e5.676×1010 ±0.137×e1.135×1011.
IV. DISCUSSION
Our calculated decay rate is an order of power smaller than the accepted value of the average
muon decay rate. Since the detection and calculations were done by the scintillator and the lifetime
of the muon was determined by the Teach Spin software, respectively, there was hardly any room
for human error. It is possible, however, that the source of possible human error generated from
the calibration of the electronics in the beginning before the detection process even began.
Another possible source, which would cause our value to deviate from the accepted value, was
inclement weather. The weather during the 72-hour elapse time was plagued with clouds and rain.
This could have possibly skewed the amount of muons reached our detector, though I'm not sure

10. to what extent the weather would affect the muons travel when the muon travels at such high
speeds.
Given the chance to possibly run the program again and compare data between research groups,
it is more than likely that a more acceptable value would be derived, given the correct parameters
were met during the conveyance of the experiment.
V. CONCLUSION
Our value of the decay rate of the muons reach our altitude was significantly smaller than the
accepted value, so a catastrophic error must be assumed and the data collection process must be
reevaluated and possibly conducted again. I believe that the final result neither proves nor negates
the theoretical value, but rather shows there was an error present during the experiment that
generated the large divergence.
VI. ACKNOWLEDGEMENTS
I’d like to thank my colleagues and fellow classmates Sarim Akbar and Maneesh
Jeyakumar for sharing their data with me when I was unable to calculate the raw numbers
without the proper computer functions.
VII. REFERENCES
[1] Lee, T. D. December 01, 1994. A brief history of the muon. Hyperfine interactions 86, no. 1,
(accessed May 11, 2013).

11. [2] Collins Dictionary of Astronomy, s.v. "muon," accessed May 11, 2013,
http://www.credoreference.com/entry/collinsastron/muon
[3] The Penguin Dictionary of Physics, s.v. "muon," accessed May 11, 2013,
http://www.credoreference.com/entry/pendphys/muon
[4] Department of Physics. Muon Physics. User Manual, Southern Methodist University, Dallas,
Texas: Department of Physics.
TABLE 1. Final calculated results for decay rate and the time distribution of the decay rate of a
muon from the measured values detected during the experiment.
Collection Elapse Time, t (hours) 72
Total Muons 153428
Total events 959
Average Lifetime, τ (µseconds) 4.568±2.03
Decay Rate, λ (µseconds) 0.219±0.097
Decay Time Distribution, D(t) 0.219×e!.!"!×!"!" ±0.137×e!.!"#×!"!!

12. FIGURE 1. The schematic of the detector and electronics box setup, most of which is consisted
within the detector, itself. This displays the process of detection and information processing before
it is sent to the computer for analyzation. [4]
FIGURE 2. A sample screen capture during the calibration process before the detector is started
and the muon decays are measured. [4]

13. FIGURES
1

14. 2