CSU-Pueblo
Time Dilation Effect
of Special Relativity
Objective: Measure the stopping rate of muons, as a function of depth in the
atmosphere to demonstration the time dilation effect of special relativity
MJolley
1/1/2013
Jolley
Muon
Background
The origins of Cosmic rays are still a mystery to
scientists. In one theory Cosmic rays originate in
The muon was so unexpected
that, regarding its discovery,
Nobel laureate Isidor Isaac
Rabi famously quipped, “Who
ordered that?”
explosions that take place in outer space. They are
mostly protons accelerated in the remnants of these explosions to very high
velocities. This can result in energies of 1020
eV. (Cosmic ray 2011) Some of these
cosmic rays reach the Earth and cause
showers in the atmosphere. During cosmic
ray showers the high energy particles ionize
the atmosphere and collide with
molecules like nitrogen and oxygen.
(Figure 1) The particles are transformed
into smaller, lighter particles that swiftly
decay via the weak force, whose strength
Figure 1
can be described by the Fermi coupling constant (Gf), into the high energy
muon that we are able to detect at the Earth’s surface. There are many ways to
detect these muons.
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Muon
Discovered in 1936 by Physicist Carl David Anderson (Encyclopedia
Britanica Online 2013), the muon was originally thought to be a meson, which
led to it being named the mu meson or muon. However, the muon was found
to interact weakly with the neutron and proton, and thus classified as a lepton.
The muon is similar to the electron in characteristic behavior but has about 200
times the mass, giving it the nickname the ‘heavy electron’, and a typical
energy of about 20 GeV. (University of Rochester n.d.) This large mass and
relatively small force holding it together causes the muon to decay, usually, into
an electron or positron, and two neutrinos
(𝜇± →e± +2 ν). (Figure 2) These muons live
for only about two millionths of a second
before they decay. When a muon enters
a scintillator it loses an amount of energy,
Figure 2
around 50 MeV, passing through the wall. While inside, depending on the
remaining energy, the muon can ‘come to rest’ and emit the (𝜇± →e± +2 ν) with
these particles flying off with the muons remaining kinetic energy.
Goals
Muon decays can be detected within the scintillator tube and analyzed
by the Muon Physics software. Lifetime measurements involve muons that enter
the tube, slow, stop, and decay. These electrons will have energy of ≈ 160 MeV.
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Muon
The experimental mean lifetime of the muon is determined in the muon.exe
program that was provided with the detector, by the ‘least-squares fit’ to
histograms in the form of:
𝑡
𝑁(𝑡) = 𝑁0 𝑒 −𝜏 ,
where
and
𝑁(𝑡)𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑤𝑖𝑡ℎ 𝑙𝑖𝑓𝑒𝑡𝑖𝑚𝑒 𝑡
𝑁0 𝑖𝑠 𝑎 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑎𝑡𝑜𝑛 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑢𝑜𝑛𝑠 𝑜𝑡 𝑡𝑖𝑚𝑒 𝑡 = 0
This equation gives τ, the measured raw lifetime, which should be approximately
2.19703±0.11114 µs. With τ, the Gf constant can be calculated using the
equation:
𝐺𝑓 =
ℏ 192𝜋 3
𝜏𝜇 𝑚5
where
ħ (𝑃𝑙𝑎𝑛𝑐𝑘 ′ 𝑠 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡) = 6.58 × 10−25 𝐺𝑒𝑉 ∗ 𝑠, 𝑚 (𝑀𝑢𝑜𝑛 𝑀𝑎𝑠𝑠) = 105
𝑀𝑒𝑉
𝑐2
.
This can be compared to the accepted value of 1.17 × 10−5 𝐺𝑒𝑉 −2 (The NIST
Reference on Consants, Units, and Uncertainty 2011), to show the strength of the
weak force.
The time dilation effect of the muons can be observed by first extracting
data from the Muon Physics program at two different elevations for the number
of observed muon decays over a given time interval. A measurement for
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Muon
stopping rate is taken at a first altitude and a respective approximated rate is
determined for a second rate. The respective rate is calculated both with time
dilation effects and without. A second measurement is then taken at the
second altitude and the results are compared. The measurements at the
second altitude must take into account both the muon energy loss in transit from
the higher altitude to the lower, as well as variations in the shape of the muon
energy spectrum, which peaks around p=500MeV/c.
The energy loss can be accounted for by making a simple estimate for
the energy loss over the change in altitude given by the equation:
∆𝐸 = 𝐶𝑜 ∗ ∆𝐻 ∗ 𝜌𝑎𝑖𝑟(𝑎𝑣𝑒)
With 𝐶𝑜 = 2 𝑀𝑒𝑉⁄ 𝑔 ,
𝑐𝑚2
𝛥𝐻 = |ℎ2 − ℎ1|,
−ℎ
ℎ
1
And 𝜌𝑎𝑖𝑟(𝑎𝑣𝑒) = ∆𝐻 ∫ℎ 2 𝜌0 × 𝑒 8.4 𝑚
1
𝑔
With 𝜌0 = 1.28 × 10−3 𝑐𝑚3 .
The transit time from the higher altitude to the lower altitude in the lab
reference frame is denoted 𝑡 and is simply determined by:
𝑡=
∆𝐻
𝑐
.
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Muon
Using this t, the predicted stopping rate for another elevation can be calculated
using:
−𝑡
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 𝑅0 × 𝑒 𝜏
where R0 is a correction factor to be calculated later.
Using the corresponding Lorentz factors:
𝛾2 = 1.5,
assuming the muon stopped, on average, halfway into the scintillator,
𝛾1 =
𝐸1
𝑚𝑐 2
the gamma factor right before it enters the scintillator
with
𝐸1 = 𝐸2 + ∆𝐸,
𝐸2 = 160𝑀𝑒𝑉
the time for the path in the muon’s reference frame t’:
𝑡′ = 𝜌
𝑚𝑐
𝑎𝑖𝑟(𝑎𝑣𝑒)
𝐶𝑜
𝛾2
𝑑𝛾
1
√𝛾2 −1
∫𝛾
.
Using t’ in the form of:
𝑡 ′ = (𝑠𝑜𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟) ∗ 𝜏.
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Muon
Compute the theoretical stopping rate in Pueblo taking relativity into
consideration:
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 𝑅0 × 𝑒 (
−𝑡′
)
𝜏
.
In order to correct for the variations in the shape of the muon energy
spectrum in the region from 160 MeV- 800 MeV, a double ratio of the two
stopping rates is determined. First, ignoring the variance in the shape of the
spectrum find:
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒) = 𝑒
−𝑡′
𝜏
This is the calculated stopping rate ratio at the elevation of Pueblo. Then find the
raw stopping rate:
𝑅𝑟𝑎𝑤 =
𝑆𝑡𝑜𝑝𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑃𝑢𝑒𝑏𝑙𝑜
𝑆𝑡𝑜𝑝𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑀𝑜𝑛𝑎𝑟𝑐ℎ
This is the ratio of the two measured stopping rates at the two elevations.
Finally, take the double ratio:
𝑅𝑜 =
𝑅𝑟𝑎𝑤
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒)
This gives the muon spectrum correction factor, Ro. This factor is then
multiplied by the Ratios of the predictions:
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Muon
𝑅𝑃𝑁𝑇𝐷 = 𝑅𝑜 ∗ 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛)
𝑅𝑃𝑇𝐷 = 𝑅𝑜 ∗ 𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛)
And difference is calculated.
Equipment
The main parts of the equipment are the plastic scintillator, photomultiplier
tube, and the signal amplifier. The scintillator, photomultiplier tube, and a high
voltage supply are housed in an anodized aluminum cylinder. The high voltage
supply has an external control, with all other high voltage components
contained within the cylinder. The scintillator and photomultiplier tube are fed
to the signal amplifier. The scintillator is an organic transparent mixture of a
plastic solvent with an aromatic ring structure and one or more fluors. A particle
such a muon, carrying a charge, will lose a certain amount of kinetic energy by
either ionization or atomic excitation of the solvent. This kinetic energy causes
the electrons to excite in the fluor molecules. When this electron relaxes a
radiated blue light is emitted. This activates a timing device and readout the
time measurement when another signal is detected within 20 𝜇s. This second
signal, within this time frame, can be thought of as a muon decay event. The
Electron emitted when the muon decays excites the fluor molecules similarly.
These measurements are then sent to the muon software.
Procedure
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Muon
The data collection procedure begins with installing the software and
ensuring the wiring is properly connected. The detector is set up to distinguish
between muons which enter the detector one after another and muons which
actually decays within the detector by adjusting the duration of times between
the first pulse detected in the photomultiplier tube and a successive pulse in the
tube. Another way to ensure proper lifetime measurements is to take data over
considerable time periods.
After the software and hardware is running properly the data is collected
into a .data file which can be opened in a text reading program. At the end of
each run, a screen shot is taken and saved as a .jpg file. For each run, the .data
file is uploaded into Excel. In Excel the total time of each run is calculated. The
program time stamps the runs in UNIX time and this number car be put into an
online UNIX to month-day-year time. Excel also goes through the data and does
a sum of the values that are less than 4000 in the first row of the data file. The
values in these cells represent the time between successive signals in
nanoseconds, and indicate a decay event and therefore represent a muon
lifetime. These values are then uploaded into a Matlab file and calculations for
the Fermi Coupling constant and the influx of detected muons at the current
elevation are performed.
A predicted stopping rate is calculated for the higher stopping rate
without considering time dilation. The correction factor is used to scale the
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Muon
ratios of stopping rates and is compared to the actual stopping rate at the
higher altitude.
Data and Calculations
The manual recommends that the High Voltage (HV) adjustment on the
top of the scintillator be set to between -1100 and -1200 volts (Ye 2005). The HV
was set to 9 giving a voltage reading of -11.54 which is 1/100 time the output in
the photomultiplier tube or a reading of -1154 Volts within the photomultiplier.
The time between two pulses of light can be adjusted using the Time Adj. knob
on the photomultiplier. The output of the photomultiplier tube is connected
from the detector to the electronics box and the discriminator is set so the
output is between 180 and 220 MeV. The threshold voltage was measured to be
206 MeV.
The muon.exe program was then run for an extended period of time in
two location. The first being Pueblo, Co., having an elevation of 1420 m above
sea level (Data 2003-2012). The data was collected and stored. The detector
was then taken to an elevation of 3290 m above sea level at the base of
Monarch Mountain. (Stats and Hours n.d.) This gave a difference in elevation
(Δh) of 1870 m. Data from the Pueblo run was selected to have a collection
time comparable to that of the Monarch run.
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Muon
The 𝜏 was taken from the upper right hand side of the screen after each
run was saved.
The average of these measured lifetimes was then used in the calculation of the
Fermi Coupling Constant using the mass of the muon to be 106
𝑀𝑒𝑉
𝑐2
(al. 2012)
and ħ to be 6.58 × 10−25 𝐺𝑒𝑉 ∗ 𝑠 (The NIST Reference on Consants, Units, and
Uncertainty 2011). The value obtained for the Fermi Coupling Constant with the
Pueblo data and the Monarch data are, 1.18 × 10−5 𝐺𝑒𝑉 −2 and 1.19 × 10− 𝐺𝑒𝑉 −2
respectively. With an accepted value of 1.7 × 10−5 𝐺𝑒𝑉 −2 these measurements
give a percent error of 0.855% and 1.17% respectively.
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Muon
The stopping rate for each elevation was also determined with a MATLAB
program. The data from the muon apparatus is loaded into an Excel file. In the
Excel spreadsheet the total time duration of each run is calculated by taking the
last number stamped in the UNIX time code and subtracting the first UNIX time
code. These numbers are stored in a separate sheet in the file. Cells where
decay events are indicated by a number under 40000 and indicate the time
between successive light pulses. Numbers above 40000 indicate a time interval
where no decay events took place. The total number of muon decay events is
determined by totaling the cells with values under 40000. This number is also
stored in the separate sheet.
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Muon
The total time duration of the data collection in Monarch was 19 hours 27
minutes and 24 seconds. It was determined that the 21st set of data taken at the
pueblo elevation had a reasonably close collection time to that of the Monarch
run with a total of 22 hours 10 minutes and 39 seconds. The stopping rates for
each of the elevations was shown to be 0.0284
𝑚𝑢𝑜𝑛𝑠
𝑠𝑒𝑐
and 0.0644
𝑚𝑢𝑜𝑛𝑠
𝑠𝑒𝑐
,
respectively.
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Muon
The transit time in the observer’s reference frame, ignoring relativity is
determined to be 6.28 µsec or 𝑡 = 2.85 𝜏. Using this number the non-corrected
stopping rate ratio for Monarch is:
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑐𝑒) = 𝑅0 × 0.057
Where R0 is the correction factor.
In order to accurately predict the stopping rate ratio a few corrections
must be calculated. First, as a muon travels through the air it inevitably loses
energy. This energy loss was estimated using the air density averaged over the
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Muon
𝑔
change in elevation and resulted in 𝜌𝑎𝑖𝑟 = 972 𝑚3. Using this result ∆𝐸 =
364 𝑀𝑒𝑉 . Adding ΔE to 𝐸2 = 160𝑀𝑒𝑉 , 𝐸1 = 524 𝑀𝑒𝑉 was determined. This value
of E1 was then divided by the muons mass times c2 to determine the muon’s
gamma factor right before it enters the photomultiplier tube, 𝛾1 = 4.9.
This gamma factor is used in the integral to determine the transit time in the
particle’s rest frame, 𝑡 ′ = 2.38 𝜇𝑠𝑒𝑐. The predicted ratio in pueblo becomes
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 𝑅0 × 0.339
and the ratio which still ignores the shape of the muon energy spectrum
becomes
𝑅(𝑖𝑔𝑛𝑜𝑟𝑒 𝑠ℎ𝑎𝑝𝑒) = 0.339
The second correction that must be done was to correct for the variations in the
shape of the muon energy spectrum. First the raw stopping rates at the two
0.0284
elevations were calculated giving 𝑅𝑟𝑎𝑤 = 0.0644 = 0.441. Then a double ratio was
0.441
calculated , 𝑅𝑜 = 0.339 = 1.30.
Finally, the double ratio is used in the two predicted equations,
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑛𝑜 𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 1.30 × 0.056 = 0.0749 and
𝑅(𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)𝑃𝑢𝑒𝑏𝑙𝑜(𝑡𝑖𝑚𝑒 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛) = 1.30 × 0.339 = 0.441. The prediction which does
not take time dilation into account is off by a factor of 5.89 when compared to
the actual raw data..
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Muon
Results and Conclusion
The main goal of the experiment was to show, through measurement and
comparison, that muons travelling at very high speeds, those near the speed of
light, exhibited the time dilation effect of Einstein’s Special Relativity. This was
done by measuring the decay of muons in the photomultiplier tube and
determining the mean lifetime of these muons. This lifetime along with the time
dilated lifetime was put into ratios and compared. The comparison leads to the
conclusion that the actual measured ratio of muon decays is much more like
the calculation of the time dilated prediction rather than the non-time dilated
prediction. This shows that hypothesis that there are no time dilation effects
happening must be rejected and that the measured data is more consistent
and more supportive of the relativistic hypothesis. The Fermi Coupling Constant
calculation was fairly close to the accepted value.
References
al., J. Beringer et. 2012. "The Review of Particle Physics." Journal of Physics (URL: http://pdg.lbl.gov).
2011. Cosmic ray. Sept. Accessed Nov 2012.
http://en.wikipedia.org/wiki/Cosmic_ray#Research_and_experiments.
Data, City. 2003-2012. Pueblo Colorado. Accessed Jan 2013. http://www.city-data.com/city/PuebloColorado.html.
Encyclopedia Britanica Online. 2013. s.v."Carl David Anderson". Accessed April 2013.
http://www.britannica.com/EBchecked/topic/23589/Carl-David-Anderson.
n.d. Stats and Hours. Accessed Jan 2013. http://www.skimonarch.com/index.php/generalinfo/stats-ahours.
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The NIST Reference on Consants, Units, and Uncertainty. 2011. Fundamental Physical Constants. June 2.
Accessed Jan 2013. http://physics.nist.gov/cgi-bin/cuu/Value?bgspu.
University of Rochester. n.d. Particle at the University of Rochester. Accessed Nov 2012.
http://www.pas.rochester.edu/~pavone/particle-www/particle_physics.html.
Ye, T.E. Coan J. 2005. Muon Physics.
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