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

MJolley1/1/2013

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Background

The origins of Cosmic rays are still a

mystery to scientists. In one theory Cosmic rays

originate in 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 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.

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

The muon was so unexpected that,

regarding its discovery, Nobel laureate Isidor

Isaac Rabi famously quipped,

“Who ordered that?”

Figure 1

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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, 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. 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:

N (t )=N0 e−tτ ,

Figure 2

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whereN ( t ) is t he distributionof particles wit h lifetime t

and N 0 is anormalizaton 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=ℏτ μ192 π3

m5

where

ħ (Planc k ' sConstant )=6.58×10−25GeV∗s ,m (MuonMass )=105 MeVc2

.

This can be compared to the accepted value of 1.17×10−5GeV−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 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

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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 )

WithCo=2MeVgcm2

, ΔH=¿h2−h1∨,

And ρair (ave )=1∆ H∫

h1

h2

ρ0×e−h8.4m

With ρ0=1.28×10−3 gcm3

.

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=∆ Hc .

Using this t, the predicted stopping rate for another elevation can be

calculated using:

R( predicted)Pueblo(no time dilation)=R0×e−tτ

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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=E1mc2

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 '= mcρair(ave)Co

∫γ 1

γ 2 dγ√γ 2−1

.

Using t’ in the form of:

t '=(some factor )∗τ.

Compute the theoretical stopping rate in Pueblo taking relativity into

consideration:

R( predicted)Pueblo(time dilation)=R0×e(−t 'τ ).

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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 PuebloStopping 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:

RPNTD=Ro∗R( predicted)Pueblo(no time dilation)

RPTD=Ro∗R (predicted )Pueblo(timedilation )

And difference is calculated.

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EquipmentThe 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

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

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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

ratios of stopping rates and is compared to the actual stopping rate at the

higher altitude.

Data and Calculations

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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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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 be106 MeVc2

(al.

2012) and ħ to be 6.58×10−25GeV∗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, 1.18×10−5GeV−2 and

1.19×10−¿GeV−2¿ respectively. With an accepted value of 1.7×10−5GeV−2 these

measurements give a percent error of 0.855% and 1.17% respectively.

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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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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 muonssec and

0.0644 muonssec , respectively.

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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 timedependence)=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

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over the change in elevation and resulted in ρair=972 gm3 . Using this result

∆ E=364MeV .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 '=2.38μsec. The predicted ratio in pueblo becomes

R( predicted)Pueblo (timedilation)=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 elevations were calculated giving Rraw=0.02840.0644

=0.441. Then a double ratio

was calculated , Ro=0.4410.339

=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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Results and ConclusionThe 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.

Referencesal., 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/Pueblo-Colorado.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-a-hours.

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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.