Measurement of the Fermi Coupling Constant Using
Mass and Lifetime of µ Leptons
Maria Baryakhtar ∗
Steven Schowalter
†
Maximilian Swiatlowski ‡
March 10, 2009
Abstract
We present our measurements of the lifetime and mass of cosmic ray muons. The
detection of muons is performed at sea level using a simple system of three scintillators and photomultiplier tubes. We find the average lifetime of the muon to be
τµ = 2.12 ± 0.16 µs, which is in good agreement with the accepted value of 2.20 µs [2].
The muon mass is measured to be mµ = 120 ± 20 MeV/c2 , which includes the accepted value of 105.66 MeV/c2 [2]. These measurements are used to calculate the Fermi
coupling constant GF . Our value of GF is determined to be 0.19 ± 0.08 MeV fm3 ,
which is in correspondence with 0.27 MeV fm3 , the experimentally established value [2].
This experiment allows us to use a low energy setup to successfully study the weak
interaction.
∗
Produced Introduction and Background (Sections 1, 2), Appendix B, and Abstract
Produced Results and Discussion (Section 5) and Appendix A
‡
Produced Instrumentation and Procedure (Sections 3, 4), Appendix C, and Layout
†
Contents
1 Introduction
2 Background
2.1 Muon Decay . . . . . . . . . . . . .
2.1.1 Decay in Matter . . . . . . .
2.2 Effects of Relativistic Time Dilation
2.3 Muon Mass . . . . . . . . . . . . .
2.3.1 Energy Calibration . . . . .
2.4 Weak Force Coupling Constant . .
3
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3
4
4
5
5
6
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3 Instrumentation
3.1 Scintillators . . . . . . . . . . . . . .
3.1.1 Photomultiplier False Flashes
3.1.2 Efficiency Optimization . . . .
3.2 Logic . . . . . . . . . . . . . . . . . .
3.3 Autocorrelator . . . . . . . . . . . . .
3.4 Measurements . . . . . . . . . . . . .
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7
8
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10
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11
4 Procedure
4.1 Muon Lifetime Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Muon Mass Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
12
5 Results and Discussion
5.1 Determination of Muon Lifetime . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Determination of Muon Mass . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Determination of the Fermi Coupling Constant . . . . . . . . . . . . . . . . .
13
13
14
16
A Muon Time Error Analysis
17
B Muon Mass: Calibration and Error Analysis
B.1 Minimum Ionization Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Mass Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
20
C Muon and Electron Time of Flight
21
References
22
2
1
Introduction
The muon is a fundamental particle produced in the upper atmosphere as a secondary product of cosmic ray collisions with atmospheric molecules. It decays via the weak interaction
into an electron and two neutrinos, with a mean decay lifetime of 2.20 µs, longer than every
known particle other than the neutron [2]. With muons comprising most of the cosmic ray
flux at sea level, the muon is a good candidate for the study of the weak force [8, p. 8].
Our experiment consists of two main components: the muon lifetime measurement and the
muon mass measurement. In Section 2, Background, we introduce the theoretical basis for
muon creation and decay as well as the motivation for lifetime and mass measurements and
GF calculations. We describe the experimental setup for muon detection which consists of
a system of three scintillators and photomultiplier tubes (PMTs) in Instrumentation (Section 3). Using this system, the cosmic ray muons passing through the scintillators and their
decay products can be detected, along with the energy of these particles (as described in Section 4, Procedure). In Section 5, Results and Discussion, the muon lifetime and mass results
are presented with the relevant statistical analysis of data and compared to previous experimentally established values. Finally, we use the muon mass and lifetime values to calculate
the Fermi coupling constant GF that describes the strength of weak interactions.
2
Background
The muon is a lepton, originally discovered by Carl Anderson in 1936 [3]. Muons (µ− ) and
antimuons (µ+ ) are the most numerous charged particles at sea level [2]. Most of them
are decay products of pions and kaons, mesons produced at a height of about 15 km via
the interaction of cosmic ray particles with the Earth’s upper atmosphere [1]. The muon is
produced in weak decays as shown in Figure 1, and at sea level forms 80% of the total cosmic
ray flux [8, p. 8]. This very high flux rate makes the muon an ideal particle for studying
weak interactions.
µ−
d
W−
π−
νµ
u
Figure 1: The weak decay of π − which produces µ− : π − → µ− νµ
3
2.1
Muon Decay
In free space, negatively charged muons decay weakly into an electron, muon neutrino, and
electron antineutrino [5] (Figure 2):
µ− → e− νµ νe
(1)
with a corresponding antimatter process:
µ+ → e+ νµ νe
(2)
νµ
µ−
e−
W−
νe
Figure 2: Weak decay of µ− .
The decay of the muon is described by the exponential function:
N(t) = N0 e−Γµ t
(3)
where Γµ is the decay rate. In our first experiment (as described in Section 4.1), we seek to
measure the characteristic lifetime of the decay, τµ = 1/Γµ .
2.1.1
Decay in Matter
In matter, another decay channel is possible for µ− via nucleus capture:
µ− p → n νµ
(4)
Due to the relatively faster process of decay via proton capture, the mean lifetime of µ− in
matter is shortened relative to that in free space and depends on the material. Since the
positive µ+ is repelled by the nucleus, the µ+ lifetime is unaltered in matter.
4
The likelihood of the µ− capture is proportional to Z 4 , where Z is the atomic number of the
material; for light elements, the effect of this process is minimal [8, p.172]. For carbon, for
instance, Z = 6 and the mean lifetime of the muon is theoretically predicted to be between
1.5 and 1.9µs [8, p. 170].
In addition, the products of the decay described in Equation (4) are a neutron and a neutrino.
Because neutrons carry no charge, the efficiency of the scintillator in detecting neutrons is
lower than that of electrons and positrons from the decays in Equations (1) and (2). Thus,
we expect the effect of muon capture on the measured lifetime to be small, although it may
slightly decrease the value from the experimentally established 2.20 µs in free space [2].
2.2
Effects of Relativistic Time Dilation
Even with velocities within a percent of the speed of light, the travel time of the muon from
the point of creation in the atmosphere takes approximately 50µs - over 20 decay lifetimes to reach the ground for a muon emitted in the downward direction. According to Newtonian
physics, the flux would be reduced by a factor of over 1010 over the time of flight, and muons
would be undetectable at sea level. However, the flux of muons at sea level, where the lab is
located, remains large at 10−2 cm−2 s−1 sr−1 : reduced by a factor of just 5 from the peak flux
at 15 km [8].
This effect is due to time dilation predicted by the theory of special relativity. While in the
frame of the laboratory the time of flight of the muons is 50µs, the
q muon itself experiences a
v2
proper time reduced by a factor of γ: γtµ = tlab , where γ = 1/ 1 − cµ2 . Since the particles
are traveling close to the speed of light, the relativistic correction becomes non-negligible.
With muon velocities ranging from .994c to .998c, the proper time experienced by the muon
is between 3.2 and 5.5 µs, less than 2 lifetimes on average, consistent with detected muon
flux values.
The time in flight is still on the same order as, and even greater than, the lifetime of muon
decay that our experiment seeks to measure. Nevertheless, the time the muons experience in
the atmosphere prior to stopping in the detector has no effect on the decay rate measurement.
While we do not sample the quickest decaying electrons, this amounts to simply cutting
off the lowest end of the exponential: the actual decay parameter Γµ is not changed by
eliminating this data.
2.3
Muon Mass
Another experimental setup (Section 4.2) can be used to make a measurement of the muon
mass.
In order to measure muon mass, we consider the products of µ− decay: an electron and two
neutrinos (Equation (1)); the antimatter decay analysis is identical. For a muon which stops
and decays in the scintillator, the center of mass frame is the same as the lab frame. To a
5
good approximation we can assume that the rest energy of the muon is fully converted to
the kinetic energy of the e− and neutrinos, as the electron mass is only 0.5% of the muon
mass and the neutrinos are essentially massless. Then, measuring the energy distribution of
the emitted electrons will provide information regarding the initial muon mass. Specifically,
due to conservation of momentum, the magnitude of electron momentum, pe , must equal the
sum of the neutrino momenta. We can see this from the following argument. Fixing pe along
the x-axis, the only possible scenario of the decay is pictured in Figure 3, where 0 ≤ θ < π/2
and is measured from the negative x-axis.
νe
e
θ
νµ
Figure 3: Decay products of the muon. Due to conservation of energy and momentum, momenta
of the two neutrinos must be at equal angles, θ and −θ, from the direction of electron momentum.
The configuration with θ = 0 maximizes the electron momentum (and therefore energy).
The total momentum of the electron is maximized when the neutrino momenta have no y
component, that is θ = 0 (an identical argument eliminates any possible z component as
well). In this case, pe = pmax
= Σpν by conservation of momentum. Again neglecting electron
e
mass, we have Ee = pe c, so the energy of the electron is maximized when its momentum is
maximized. Thus, the maximum kinetic energy of the electron is half the rest energy of the
muon, as claimed.
1
1
Eemax = Eµ = mµ c2
2
2
(5)
By measuring the energy spectrum of emitted electrons, which is a β decay spectrum with a
cutoff at 12 mµ c2 , we can find the maximum electron energy and thereby determine the muon
mass.
2.3.1
Energy Calibration
Our instruments enable us to measure the maximum voltage of a pulse from an electron
that registers on the PMT (the method is described in more detail in Section 4.2). In order
to convert the pulse height distribution to an energy distribution, we have to calibrate the
pulse height voltage in terms of energy lost by the particle in the detector.
6
To do so, we find the muon pulse height voltage which corresponds to the minimum stopi, approximately 2 MeV g−1 cm2 ), as a function of incoming
ping power of the muon (h− dE
dx
momentum. This voltage is given by the mode in the peak height distribution of muons
which pass through all three scintillators. Since the through going muons have a relatively
random distribution of momenta, the most frequent rate of energy loss will be near a local
extremum in the stopping power vs. momentum function, which in this case is a minimum
(see Appendix B for more details).
Assuming the scintillator light output varies linearly with the amount of energy deposited,
and measuring the scintillator density and thickness, we find a conversion ratio between
pulse voltage and electron energy.
2.4
Weak Force Coupling Constant
The decay rate of the muon Γµ is proportional to the square of the amplitude of the decay
diagram (Figure 2), which depends on the product of the couplings
√ at each vertex. In this
case, the coupling at each of the two vertices is proportional to GF , the Fermi coupling
constant, so we have
Γµ ∝ G2F
(6)
A more involved calculation [7, p. 310-314] gives that the lifetime of the muon is
τµ =
192π 3 ~7
G2F m5µ c4
(7)
where c is the speed of light, ~ is Planck’s constant, and mµ is the rest mass of the muon.
Once we establish the values of τµ and mµ , we can find the Fermi coupling constant GF ,
which describes the strength of the weak interaction1 .
The weak decay of the muon is the clearest of all weak interaction phenomena in both its
experimental and theoretical aspects. Further considering the easy availability of muons at
sea level, the muon decay is an effective means of studying the nature of the weak force, and
specifically finding the Fermi coupling constant GF .
3
Instrumentation
In order to detect muon and electron events, signals from scintillators are correlated and
matched against an expected pattern. The next section (4) will discuss how these are used
1
Although GF is not equivalent to the weak coupling constant, gw , they are related by the equation
√ 2
gw
2
(~c)3
GF ≡
8
M W c2
where MW is the mass of the W bosons which mediate the weak interaction. Thus GF is sufficient and is
commonly used to describe weak interaction formulas [7, p. 313]
7
to generate lifetimes and energies; here, we focus on describing the apparatus and problems
associated with it.
3.1
Scintillators
The experimental setup consists of three scintillators stacked vertically, presenting a 5250± 30
cm2 area to a downward traveling muon. The three scintillators and corresponding photomultiplier tubes will be referred to as ‘top’, ‘middle’, and ‘bottom’ throughout according
to their physical placement. Each plastic scintillator consists of polystyrene (C8 H9 , with
density 1.08 ± .09g/cm3 ) doped with a phosphorous material, p-terphenyl. When ionizing
radiation passes through the material a light pulse is emitted. The light flash is detected by
the photomultiplier tubes placed at the ends of the scintillator. The photomultiplier uses a
high voltage (on the order of 1000 V) to convert the light pulses into a cascade of electrons
more or less linearly (i.e., the amplitude of the electrical signal from the photomultiplier is
linear with the light output, and thus with the energy of the ionizing radiation).
The signals from the photomultipliers are then fed into a bank of discriminators, each of
which outputs a NIM voltage pulse of tunable width after detecting a voltage pulse above
a given threshold. These signals are input into the logic which identifies types of events
(Section 3.2). In the muon lifetime measurements, only the discriminator outputs are used
as signals. For the muon mass measurement, the discriminated signals are still used for event
detection, but the raw output of the photomultiplier is viewed directly so as to determine
the true pulse amplitude, and thus the energy of the detected particle.
3.1.1
Photomultiplier False Flashes
The photomultiplier tubes used in the experiment have the property of producing false
signals, which occur with some regularity directly after a real signal is detected. The number
of these signals, as detailed in the autocorrelator section of Appendix A, follows a more or
less exponential decay (that is, there are exponentially fewer false pulses at longer times
than shorter times). However, the decay lifetime is on the order of magnitude of the muon
lifetime, so false signals may be mistaken for events and affect the lifetime measurement. For
instance, a false signal mistaken for a decay product will artificially shorten the measured
lifetime, and the true signal will not be detected. This will skew the overall data in the
direction of shorter mean lifetimes.
Different techniques, described below (3.1.2) and in the Procedure section (4.1), were used
to lower the effect of these false signals; however, they remain a significant pollutant of data
at the low end of the timescale, and must be dealt with statistically (see Appendix A).
8
3.1.2
Efficiency Optimization
Many different factors come into consideration for choosing the detector settings. The priority is maximizing the efficiency of the detectors so as to maximize the count of detected
muons; this can be achieved by increasing PMT voltages to increase the magnitude of output signals and decreasing threshold levels to allow more signals past the discriminator. The
other main concern is noise (particularly false flashes), which increases at higher voltages and
lower thresholds; this concern keeps voltages lower and thresholds higher. An optimization
procedure determines the optimal settings to maximize efficiency.
The optimization technique relies on the very high number of muons that go straight through
all three detectors. As most muons are very high energy when they reach the surface of the
earth, nearly the entire flux from the atmosphere passes through all the detectors without
stopping2 . In addition, the flux of muons per solid angle from a direction at angle θ to the
vertical is proportional to cos3 (θ) [6], so the chance of a muon coming in at a steep enough
angle to graze the top detector and not the bottom ones is very small. Thus, nearly 100%
of events that are detected by two of the three scintillators should be detected by the third
one as well. Employing this fact, and taking the ratio of the number of events detected by
all three detectors as compared to the number detected by just two of the detectors, we can
get an approximation of the efficiency of the excluded detector, as any event detected by
just the two should have been detected by all three. The ratio should be effectively 1 for a
perfectly efficient detector, as losses from solid angles and muons which stop in the detectors
are incredibly small as compared to the total muon flux. We chose the most efficient detector
to be in the middle, as most of the signals require at least one middle input.
Optimal voltage values are chosen by plotting efficiency against voltage; this produces a
function that increases dramatically at low voltages but soon plateaus. A value in the
middle of this plateau is chosen to guarantee the high rate of efficiency, but not so high as
to increase noise. Optimal values for discriminator threshold levels were chosen in a similar
fashion; a plot of threshold versus efficiency showed a plateau as discriminator levels are
lowered. Once again a middle value on the plateau is chosen, in an attempt to balance
guaranteeing high efficiency with noise concerns.
Initial data runs were taken with very high efficiency: 99% efficiency for the middle detector,
83% for the top, and 90% for the bottom. Voltages for the photomultiplier tubes were set
at 1160 V, 1190 V, and 1190 V respectively. All the thresholds were set to 70 mV.
After the discovery of noise problems detailed above and in Appendix A, the efficiency of
the detectors was lowered considerably, to 91%, 70%, and 78%. Voltages were changed to
1140 V, 1160 V, and 1130 V, with thresholds raised uniformly to 100 mV.
2
The measured values are 3150 counts per minute passing through all three scintillators, and 3 counts
per minute stopping in the middle scintillator. Thus 99.9% of muons pass through without stopping.
9
3.2
Logic
In order to register muon events, signals are fed out of the discriminator into a series of logic
banks. For example, to register a muon passing all the way through (used in optimizations
and muon mass measurements), the equation T ∧ M ∧ B is used: that is, all three signals
are anded together. Any time that all three detectors flash simultaneously, it is clear that a
muon has passed through all of them. It is unlikely that two separate muon events will be
anded; about 50 muons pass through the scintillators in one second, so we expect real muon
events every 0.02 s. Pulses out of the discriminator, on the other hand, are set to 100 ns for
the top and bottom detectors and 50 ns for the middle (set differently in order to guarantee
overlap); thus, since pulses are five orders of magnitude quicker than the rate of incoming
muons, we expect negligible overlap errors. Furthermore, time of flight values for muons and
electrons traveling between the scintillators are negligible; see Appendix C for details.
The muon lifetime measurement requires a signal that indicates a muon has stopped in the
middle scintillator; we call this event START, as this would start the lifetime count. The
signal is simply given by START = T ∧ M ∧ B̄; that is, a signal is detected on the top
and middle detectors but not the bottom. This either means that the muon has stopped
in the middle detector, or gone past the bottom detector at an angle; because of the close
distances between the scintillators (on the order of cm), this latter case is unlikely, and we
can interpret the event as a stopped muon.
Similarly, the muon lifetime measurement also requires a stop signal. The stop would occur
after the decay has happened; that is, an electron has been detected. After the muon
decays, the electron can move off in any direction (since the muon was at rest, and the
neutrinos take care of momentum conservation), but generally speaking we can expect the
electron to be detected not just at the middle detector, but either the top or bottom one
as well (since most electrons will have some vertical component in their momentum, and
given the large area of the scintillators, a small amount of vertical momentum will result
in the electron striking the detector). Thus, a simple STOP event would be defined as:
STOP = (T ∧ M ∧ B̄) ∨ (T̄ ∧ M ∧ B); that is, a stop event is either a bottom going or top
going electron.
However, these signals allow for START and STOP events to happen simultaneously: note
that STOP is actually defined by a START event or’ed with a different event. Moreover,
consider the scenario in which a muon stops in the middle detector, but the resulting electron
actually comes out the side; instead of admitting that there was no STOP detected, the next
START event would be recorded as a STOP. Thus, we should redefine STOP to occur only
after a valid START, within some acceptable timeframe. The logic diagram in Figure 4
illustrates the solution: STOP as it is currently defined is anded with a delayed gate signal
(that is, a delayed signal set constantly high, set to expire after a given amount of time) that
is triggered by a START event. The delay to the start of the gate is set to 100 ns, twice
as long as the 50 ns pulses outputted by the logic units, guaranteeing that simultaneous
events will not trigger a measurement; the gate length is set to 20µs, 10 times the expected
lifetime: waiting any longer is likely waiting for an electron which went out in an undetected
fashion.
10
Channel 1:
START
T
AND
M
Delay (100 ns)
!B
Gate (20 mu s)
AND
!T
M
AND
AND
OR
Channel 2:
STOP
AND
B
Figure 4: Complete logic diagram for muon lifetime measurement
3.3
Autocorrelator
In order to better study the effects of the photomultiplier false flashes, we used a Langley
Ford Instruments Model 1096 Correlator to track the autocorrelation of the signal coming
from each photomultiplier. The machine monitors an input line for a signal, and records
the correlation values of the signal with itself, offset by times from 0 to 80 bin widths.
We set it to record bins of 0.1 µs, up to a total of 8µs after the starting signal. Because
the rate of muons coming into a scintillator is significantly smaller than this timescale, any
autocorrelated signals can be assumed to be false flashes (indeed, if there were extra muons
coming in during this window, we would expect an even distribution over all times, but we
did not see this).
3.4
Measurements
To take measurements, a LabView program records data from a 100 MHz digital oscilloscope
over a GPIB connection. For the lifetime experiment, the oscilloscope is set to trigger on
a stop event, and the program measures the difference in time between the beginnings of
the start and stop pulses, on channel 1 and 2 of the oscilloscope respectively. The range of
lifetime values measurable by the oscilloscope was 0 to 18µs. For the mass experiment, the
scope is set to trigger on channel 2 (a logic pulse indicating the required condition) and to
monitor the height of the muon or electron pulses on channel 1 (the actual output from the
middle photomultiplier).
4
4.1
Procedure
Muon Lifetime Measurement
After calibration and logic setup, the muon lifetime measurement consists of monitoring the
START and STOP signals on an oscilloscope and recording the difference in time between
11
the start of their pulses. After binning the results, an exponential decay is observed, and if
instrumentation was perfect, this would be all that was necessary.
However, false flashes from the photomultiplier caused us to revise our collection procedures.
First, the autocorrelation data presented in Appendix A provided convincing evidence of data
corruption in the first 2 or 3 µs, but lowering threshold levels and voltages gave some relief
to these issues. A second solution was to eliminate the T ∧ M ∧ B̄ term before the or gate in
the STOP signal. As the muons come in from the top, the rate of false flashes coming from
the top should be greater than the rate of false flashes coming from the bottom; eliminating
the top going electron stop signal halved the rate of data acquisition, but did have a small
impact on eliminating the noise problem.
After data was collected, the curve was fitted to an exponential term with a constant background, as in equation (9). The lifetime is given by the fit parameter τµ . The bulk of
the noise problem was addressed by the statistical analysis described in Sections 5.1 and
Appendix A.
4.2
Muon Mass Measurement
To measure the mass of the muon, we measured the cutoff in the energy spectrum of outgoing
decay electrons, as described in Section 2.3. This first required calibration of voltage levels to
energy; to do this, we measured the voltage spectrum of muons passing completely through.
The peak of this curve would correspond to the minimum ionization energy, as discussed in
Appendix B.1.
The calibration was performed by setting a start signal to START = T ∧ M ∧ B; that is, a
muon completely passing through the detectors. With this input triggering the oscilloscope
on channel 2, the signal from the middle detector was fed directly into channel 1 by using a
linear fanout to split the signal between the logic and the oscilloscope. When triggered, the
LabView program performing data collection would record the minimum value present on
channel 1, which corresponds to the energy deposited by the muon as it passed through. The
density of the scintillator was also required to complete the calibration; this was recorded by
using a scale to find the mass of a scintillator and a meter stick to find the dimensions.
The electron energy spectrum (really, the voltage spectrum before being calibrated) was
measured by using the STOP signal discussed in Sections 3.2 and 4.1 (i.e., only a bottom
going electron was counted as a valid stop event, in order to minimize noise) as a trigger on
channel 2. The signal from the middle detector is once again split using the linear fanout
and the minimum recorded on channel 1.
With the calibration calculated and the cutoff determined by the statistical analysis in Section 5.2, the mass of the muon is determined to be twice the value of the cutoff energy.
12
5
5.1
Results and Discussion
Determination of Muon Lifetime
By constructing the logic pathway discussed in Section 3.2 and Section 4.1, we can determine when a muon comes to rest in the middle scintillator and correspondingly produce a
START signal. Likewise, we can determine when the stopped muon decays by observing the
production of an electron with downward velocity, producing a STOP signal. By observing
the duration between START and STOP signals, we are able to measure the time it takes for
a stopped muon to decay. Essentially this time is the lifetime of an individual muon.
This time data was measured and recorded over roughly a six day period during which
nearly 15000 events were recorded. To analyze this data, we binned the time data to create a
histogram shown in Figure 5 which measures number of events versus lifetime. The binwidths
were chosen to be 200 ns. A good binwidth for data is given by the Freedman-Diaconis’
choice [9],
h = 2(IQR)N −1/3
(8)
where h is the number of bins, IQR is the interquartile range, and N is the number of data
points in the set. According to this formula, a good binwidth for our data is 227 ns. Because
there does not exist a formula for optimal binwidth (the goodness of a specific binwidth is
dependent upon the distribution of the data), the comparable binwidth of 200 ns is a valid,
much more convenient choice.
600
Data
Fit
Background Noise Level
Number of Muons (arb)
500
400
300
[
Tμ = 2.12 ± .06 μs
200
100
0
[
5
10
15
Time (μs)
Figure 5: A histogram of muon lifetimes. The lifetimes follow exponential decay √
to some back-
ground noise level. The data was fit using 2-parameter nonlinear regression using N weighting
and excluding the first 2.0 µs. The fit has an r 2 value of 0.996 and produces a τµ of 2.12 ± 0.06 µs.
As a decay process with some time time constant, τµ , we would expect the data to be
13
modeled by the exponential decay of Equation (3). However because there is background
noise recorded by our scintillator the data is modeled by
N(t) = N0 e−t/τµ + b
(9)
where b is the constant background noise level measured by the scintillator.
To determine the muon mean lifetime, τµ , we then fit the data to Equation (9). Fitting
involved three major processes: the determination of the background noise level, b, determination of the fitting range, and finally the determination of the muon mean lifetime through
using a 2-parameter nonlinear regression test based on Equation (9).
To determine the noise background, a linear regression test was used. We created an algorithm which iteratively tested increasingly large portions of the tail end data from Figure 5
until we detected a slight correlation (p-value ≤ 0.1) between the points. We took this portion of noncorrelated tail end data to be the result of background noise measured by the
scintillator. Accordingly, the mean value of this subset was taken to be the background noise
level, b. From our data we determined the background noise level to be 10 events.
Determining the range over which we would fit the data to our model was a crucial step in
determining the mean lifetime of muons. We determined that the chance of data corruption
increased for smaller time values due to uncontrollable systematic limitations (See Appendix
A). Likewise, we found that the goodness of our fit worsened as we excluded an increasing
amount of front end data points (most likely due to the exclusion emphasizing the background
noise dominating the tail end data). Resultantly, we determined that there existed an optimal
range over which to fit. This range included data with time values t ≥ tc , where tc is the
cutoff time. All data points before and including tc were excluded from the fitting process.
To determine tc , we created an algorithm which iteratively fit the data with increasing tc
using a 2-parameter nonlinear regression test. For each fit, the τµ and the associated r 2
value were computed for tc ≤ 5 µs. These results are shown in Figure 6. We wanted tc to
be a value which corresponded to an r 2 ≥ 0.995 and a region of minimally fluctuating τµ .
Observing the Figure 6, we decided that 0.4 µs≤ tc ≤ 2.0 µs was an adequate range for tc .
Because data for smaller time values were more likely to be corrupted, we ultimately chose
our cutoff time to be 2.0 µs.
Excluding all points prior to and including tc , we were able to determine τµ to be 2.12 ±
0.16 µs. This value deviates from the accepted value, 2.20 µs, by 3.8%. However our 7.6%
error includes the accepted value.
5.2
Determination of Muon Mass
By measuring the heights of pulses produced by PMT flashes, we are able to ultimately
determine the muon mass. As discussed in Section 2.3, we can determine this mass by
observing the electron cutoff energy, Eemax . In order to measure electron energy in general we
first calibrated the PMT pulse height to the energy deposited in the scintillator as discussed in
14
0.998
2.4
0.996
0.994
r
2
2.2
0.992
τμ (μs)
0.990
2.0
0.988
r2 value
0.986
excluded from final fit
tc
0.984
0
1
1.8
τμ (μs)
2
3
4
5
Excluded Time Data (μs)
Figure 6: This plot shows the effect of excluding data up to some cutoff time tc from the fitting
process. We chose tc to be 2.0 µs because it produced a value for τµ that was in a range of relative
stability and because it produced a fit with an r 2 value > 0.995 as seen above.
Section 2.3.1. This calibration was done by measuring the pulse heights of muon which passed
through all three scintillator panels. More than 50, 000 events were observed in roughly four
hours of data acquisition. The distribution of binned pulse heights is shown in Figure 7(a).
As previously stated, the maximum, or the mode, of this distribution corresponds to a
minimum in the Bethe-Block equation (see Appendix B.1 for more information). From this
we determined that a pulse height Vµ = 9.8 ± 0.5 × 10−2 V corresponds to a dE
= 1.85 ± 0.10
dx
−1
2
MeV g cm . Knowing both the density and the thickness of the scintillator panels to be
2.5±0.2 and 1.08±0.09 g cm−3 respectively, we can determine the actual scale between pulse
height and deposited energy. Taking this scale to be approximately linear in this regime, we
found that 1 V produced by the PMTs as measured by an oscilloscope maps to an energy
of 51 ± 7 MeV.
800
mode
Vµ
= 9.8±.5
Number of Electrons (arb)
Number of Muons (arb)
1500
-2
x 10 V
1000
500
0
Decay Model
600
Vemax = 1.18±.10 V
400
200
0
0.0
0.2
0.4
0.6
0.8
1.0
Pulse Height (V)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Pulse Height (V)
(a) Histogram of muon pulse heights
(b) Histogram of electron pulse heights
Figure 7: These raw pulse heights produced by the PMTs correspond to muon and electron energies
deposited in the scintillators. Data from (a) is used to calibrate a scale from pulse height (measured
in V) to energy (measured in MeV). Data from (b) was used to determine the electron cutoff energy.
Together information from both plots were used to calculate the muon mass.
15
By observing the pulse heights produced by PMT flashes from the middle scintillator that
correspond to STOP pulses, we then measured the energy deposited by electrons produced
from muon decay. This data was recorded for roughly five days and included nearly 7, 000
events. The distribution of binned electron pulse heights, Ve , is shown in Figure 7(b). As
noted in Section 2.3, the shape of this energy distribution is determined by the kinematics
of the muon decay. Most importantly, the energy cutoff (the point at which the distribution
decays to zero) occurs at roughly one-half the mass of the decayed muon.
To determine the energy cutoff of the distribution in Figure 7(b), we used a fitting algorithm.
First we assumed that after some pulse height, the distribution could be approximately
modeled by an exponential decay. To determine this value we did an r 2 analysis of multiple
fits to using 2-parameter nonlinear regression in which we excluded an increasingly large
range of √
low pulse heights points beginning at Ve = 0. In these fits we weighted each data
point by Ne , where Ne is the number of electrons in some pulse height bin. We determined
that the smallest exclusion range to produce a fit with r 2 ≥ 0.99 was the exclusion range 0
V ≤ Ve ≤ 0.1 V. We were then able to compute the model for the decay distributions decay
as Ve approached the cutoff energy. This model approximated the number of electrons one
would expect to see (relative to the number of events observed, which was nearly 7,000) for
some Ve .
The model for the distribution’s decay was then used to determine Eemax by finding the scope
voltage at which the interval Ne ±δNe (1−σ error bars) was completely below 1. Because Ne
is a discrete value being modeled by exponential decay, we claim that an Ne ≤ 1 essentially
corresponds to no events. Therefore we take this Ve , found to be 1.18 ± 0.10 V, to be Vemax .
We take all subsequent nonzero values of Ne to be background noise produced by PMT
false flashes. Converting from Vemax , we find that Eemax = 60.8 ± 11.1 MeV. Based on the
kinematics of the muon decay we determined that this cutoff energy implies a muon mass
of mµ = 120 ± 20 MeV/c2 . This value deviates from the accepted value, 105.66 MeV/c2 , by
12%. However our 17% error includes the accepted value.
5.3
Determination of the Fermi Coupling Constant
Having determined the parameters for both the muon mean lifetime, τµ , and the muon mass,
mµ we can calculate the value for GF using Equation (7). Ultimately, from our results we
calculate that GF = 0.19 ± .08 MeV fm3 . This value deviates from the accepted value, 0.27
MeV fm3 [2], by 42%. Our 43% error just includes this accepted value.
Our calculation of GF is by no means a precision measurement. Although certain aspects of
this experiment have relatively low error (< 8%), such as the measurement of τµ , ultimately
the overwhelming error associated with the energy calibration prevented a more precise
measurement of GF .
16
Appendices
A
Muon Time Error Analysis
The main source of statistical error for τµ was due to the fitting process. Using 2-parameter
nonlinear regression, the data in Figure 5 is approximated by [4]
yi = f (β, x′i ) + ǫi
(10)
where yi is the ith expected value, β is a parameter vector, x′i is the ith row of predictors,
and ǫi is the associated random error. The likelihood, L, is given by [4]
1
exp(−
L(β, σ ) =
(2πσ 2 )n/2
2
Pn
i=1 [yi
− f (β, x′i )]2
)
2σ 2
(11)
where n is the number of data points and σ 2 is the variance. L is maximized where the sum
of squared errors, S(β), given by [4]
S(β) =
n
X
i=1
[yi − f (β, x′i)],
(12)
p
is minimized. Furthermore, each data point was weighted by Nµ , where Nµ was the number
of muons with some measured lifetime. The optimal choice for parameters with weightings
yields a statistical error of 2.8%.
An additional source of statistical error is the jitter in the electronics. However because the
number of events is so large (N = 14, 632), the contribution of the total jitter to the total
error is on the order of single nanoseconds. This is more precise than our actual measurement
of τµ and is thus disregarded.
Systematic error was also added to the total error. PMT false flashes served as a main source
of systematic error. As discussed in Section 3.3, autocorrelated events were measured by an
autocorrelator. These false flashes were found to die off approximately exponentially on the
order of τµ as seen in Figure 8. Because of this, we expect the data shown in Figure 5 to
be slightly corrupted and generally skewed in a specific direction for small t. This distortion
will affect the calculated value for τµ . To account for this skew we attributed 5% systematic
error to τµ .
Another source of systematic error in the time measurement was due to the proton capture
of muons given by Equation (4). As discussed in Section 2.1.1, this decay has a mean lifetime
on the order, but slightly less, than the muon decay we are concerned with. As a result we
would expect our data, shown in Figure 5, to be modeled by the summation of two linearly
independent exponentials above some noise floor given by
17
3000
3000
high efficiency
lower efficiency
2000
1500
1000
500
0
0
high efficiency
lower efficiency
2500
Correlation Counts
Correlation Counts
2500
2000
1500
1000
500
1
2
Time(µs)
3
0
0
4
1
(a) Bottom detector
2
3
Time(µs)
4
5
(b) Middle detector
1400
high efficiency
lower efficiency
Correlation Counts
1200
1000
800
600
400
200
0
0
1
2
3
Time(µs)
4
5
(c) Top detector
Figure 8: These plots show the existence of autocorrelated events, or false flashes, produced by
the PMTs. These events occur most frequently immediately after a true scintillator flash and
decrease as t increases. The overall amount of autocorrelated events was decreased by increasing
the discriminator threshold and increasing voltages, as described in Section 3.1.2.
N(t) = N0 (re−t/τµ + (1 − r)e−t/τµp )
(13)
where r is some relative weighting between the two decays which is determined by the
material in which the decay occurs. As stated in Section 2.1.1, the expected mean lifetime
for the proton capture of a muon would lie between 1.5 µs and 1.9 µs. Fitting to Equation (13)
gives a greater value for τµ . However, the proton capture exponential has almost completely
decayed by the time our valid data starts, due to the autocorrelation errors; therefore, a
good fit is difficult to produce as there are few data points that correspond to it. Because of
these errors, the relative uncertainty of the r, and the dependency of τµp on the material, a
better fit (relative to the fit produced using the procedure described in Section 5.1) was not
able to be produced. As a result, we determined that attributing 5% systematic error to τµ
was sufficient to account for any effect that proton capture had on the data.
18
Another possible source of systematic error would be due to systematic shifts in pulse lengths
or pulse delays. Consider the logic diagram in Figure 4. The START event is created by
2 ands, so there are two rise times (on the order of 4 ns each) before the START signal is
actually created. The STOP event on the other hand requires 2 ands, an or, and another
and; a total of 4 rise times is added onto the STOP event. Thus, there is a constant offset to
the data of 4 − 2 = 2 rise times for all muon events (and only 1 rise time when we eliminate
the top going stop event, and eliminate an or gate because of that). This systematic error
causes lifetime values to be shifted in a certain direction. However because the data assumes
an exponential decay shown in Figure 5, this time offset would not affect the overall mean
lifetime, τµ , but would only affect the leading coefficient. Therefore we are not concerned
with this type of systematic error.
Ultimately, our total statistical error is due to the error in our fit which is 2.8%. Our total
systematic error comes from both the PMT false flashes and the effect of proton capture.
These errors added in quadrature are calculated to be 7%. With this error included, we
calculate the muon mean lifetime to be 2.12 ± .06 (statistical) ±0.15 (systematic) µs. Added
in quadrature τµ = 2.12 ± 0.16µs.
B
Muon Mass: Calibration and Error Analysis
In order to calibrate the maximum electron pulse height in terms of energy and thus calculate
the muon mass, many measurements had to be made on the experimental system, a large
portion of which were imprecise due to equipment limitations.
To find the conversion ratio between pulse height in volts and energy deposited in MeV, we
used the relation
VtoMeV =
dE
−
dx
min
·h·ρ·
1
µ
Vpeak
(14)
i|
is the minimum stopping power of the muon, which is the minimum rate
where h− dE
dx min
of energy loss per density of the material and propagation depth; ρ is the density and h is
µ
the height of the scintillator, and Vpeak
is the most frequent peak voltage of through going
muons, corresponding to the minimum stopping power.
B.1
Minimum Ionization Energy
For heavy (µ and heavier) charged particles passing through matter, the stopping power
depends on the incident momentum of the particle. The relationship is given by the BetheBlock equation [10]:
dE
1 2me c2 β 2 γ 2 Tmax
δ(βγ)
2Z 1
2
−
= Kz
ln
−β −
dx
A β2 2
I
2
19
(15)
q
v2
where β = v/c and γ = 1/ 1 − cµ2 are parameters of the incoming particle, and the other
parameters are constants or properties of the material.
The rates of mean energy loss of muons for a range of incident momenta are shown in
Figure 9 [10]. The minimum stopping powers vary with material; in our experiment, the
scintillator consisted of polystyrene (C8 H9 ). The minimum ionization energy was determined
by weighing the C and H values, 1.75 ± 0.1 MeV g−1 cm2 and 4.0 ± 0.1 MeV g−1 cm2 respectively, according to the mass ratio of the two elements in the compound to give a the energy
deposited by a minimum-ionizing muon to be h− dE
i|
= 1.85 ± 0.1 MeV g−1 cm2 in the
dx min
scintillator material.
Figure 9: Mean energy loss in various materials [10]. The weighed value for polystyrene (C8 H9 )
was calculated to be 1.85 ± 0.1 MeV g−1 cm2 .
B.2
Mass Error Analysis
The density of the scintillator material was found to be 1.08 ± 0.09 g/cm3 . The height was
2.5 ± 0.2 cm, the large errors due to the tape around the scintillator making it difficult to
measure the dimensions and density.
The mode of the distribution of muons passing through was 9.8 ± 0.5 × 10−2 V (Section 5.2).
Combined with the value of Emin from Section B.1, this gives the conversion ratio to be
VtoMeV = 51 ± 7 MeV/V
where the errors are independent and thus added in quadrature.
20
(16)
e
The maximum detected electron pulse peak voltage was determined to be Vmax
= 1.2 ± 0.1
V (Section 5.2), resulting in a maximum electron kinetic energy of 61 ± 10. MeV. Thus, the
mass of the muon was established to be 120 ± 20 MeV/c2 .
C
Muon and Electron Time of Flight
Both the muon and electron time of flight between detectors has no impact on the experiment,
but for different reasons.
There is no minimum for the energy of the outgoing electron in the muon decay. Consider
the decay as drawn in Figure 10: momentum conservation gives θ ≤ π/2, but momentum
conservation also dictates that as θ approaches π/2, the outgoing electron’s momentum (and
therefore energy) approaches 0. Thus, the actual time of flight for electrons between detectors
is very widely spread. However, the middle discriminator outputs only a 50 ns pulse; to be
registered as a proper stop signal, the electron must hit the other detector within this window
(which is reduced by the comparison time of the coincidence unit). Thus, the highest possible
travel time for the electron is on the order of 50 ns, adding a possible error factor of up to
0.05 µs to each measurement. Given that the relevant timescales are on the order of µs, the
time of flight becomes completely negligible.
νe
θ
e
νµ
Figure 10: Decay products of the muon, with high θ and low Ee
However, these considerations do not save the muon lifetime calculation; consider the scenario
where a muon moves arbitrarily slowly between the middle and bottom detectors (slow
enough to escape the 50ns middle pulse). Then the coincidence unit would detect the event
as T ∧ M ∧ B̄, even though it was really a T ∧ M ∧ B, and therefore a false start signal is
created.
But from the discussion in Appendix B, it is clear that an energy on the order of 10 MeV
is deposited by a muon as it passes through the detector. The percentage of muons that
have just over 10 MeV in kinetic energy and deposit only 10 MeV is vanishingly small, as
it corresponds to a very small range of velocities and only the minimum deposited energy.
21
Thus, muons leaving a detector must have energy greater than 206 MeV, the rest energy
of the muon. Even for kinetic energies as low as 1 MeV (compared to an average kinetic
energy of about 2000 MeV for cosmic ray muons), corresponding to total energy of 207 MeV,
the travel time between scintillators is short enough. We can see this because E = γmc2 , 1
MeV kinetic energy corresponds to γ = 1.0045, and β = 0.10: i.e., even the muons on the
slow end of the spectrum have velocities about 1/10th the speed of light. As the distance
between detectors is on the order of 2 cm, the time of flight is approximately 700 ps, which
is completely negligible. Even muons with less energy would still move between detectors
quickly enough to not affect our experiment. Very few will actually move slowly enough to
outlast the 50 ns pulse; these will contribute to false starts and noise, but not significantly
so.
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[5] Nalini Easwar and Douglas A. MacIntire. Study of the effect of relativistic time dilation
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22