Time Dilation Effect of Special Relativity



CSU-PuebloTime Dilation Effect of Special RelativityObjective: Measure the stopping rate of?muons, as a function of depth in the atmosphere to demonstration the?time dilation effect?of special relativityMJolley1/1/2013BackgroundThe muon was so unexpected that, regarding its discovery, Nobel laureate Isidor Isaac Rabi famously quipped, “Who ordered that?”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. CITATION Cos11 \l 1033 (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) Figure 1leftcenter 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 CITATION Enc \l 1033 (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. CITATION Uni12 \l 1033 (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 (?± →e± +2 ν). (Figure 2) Figure 2rightcenter 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 (?± →e± +2 ν) with these particles flying off with the muons remaining kinetic energy.GoalsMuon 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:Nt=N0e-tτ , where Ntis the distribution of particles with lifetime tand 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 ?s. With τ, the Gf constant can be calculated using the equation: Gf=?τμ192π3m5where ? Planck's Constant=6.58×10-25GeV*s, m Muon Mass=105 MeVc2.This can be compared to the accepted value of 1.17×10-5 GeV-2 CITATION The11 \l 1033 (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 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*ρairaveWith Co=2MeVgcm2, ΔH=|h2-h1|, And ρairave=1?Hh1h2ρ0×e-h8.4 mWith ρ0=1.28×10-3gcm3.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-tτwhere 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 scintillatorwithE1=E2+?E, E2=160MeVthe time for the path in the muon’s reference frame t’:t'=mcρair(ave) Co γ1γ2dγγ2-1 .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).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'τThis is the calculated stopping rate ratio at the elevation of Pueblo. Then find the raw stopping rate:Rraw=Stopping rate PuebloStopping rate MonarchThis is the ratio of the two measured stopping rates at the two elevations. Finally, take the double ratio: Ro=RrawR(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(time dilation)And difference is calculated. 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 ?s. 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.ProcedureThe 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 ratios of stopping rates and is compared to the actual stopping rate at the higher altitude. Data and CalculationsThe manual recommends that the High Voltage (HV) adjustment on the top of the scintillator be set to between -1100 and -1200 volts CITATION TEC05 \l 1033 (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 CITATION Cit12 \l 1033 (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. CITATION Sta13 \l 1033 (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. 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 MeVc2 CITATION alJ12 \l 1033 (al. 2012) and ? to be 6.58×10-25 GeV*s CITATION The11 \l 1033 (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.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. 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 0.0644muonssec , respectively.The transit time in the observer’s reference frame, ignoring relativity is determined to be 6.28 ?sec or t=2.85 τ. Using this number the non-corrected stopping rate ratio for Monarch is:R(predicted)Pueblo(no time dependence)=R0×0.057Where 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 change in elevation and resulted in ρair=972gm3. 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 becomesR(predicted)Pueblo(time dilation)=R0×0.339and the ratio which still ignores the shape of the muon energy spectrum becomes R(ignore shape)=0.339The 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.. 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.References BIBLIOGRAPHY al., J. Beringer et. 2012. "The Review of Particle Physics." Journal of Physics (URL: ).2011. Cosmic ray. Sept. Accessed Nov 2012. , City. 2003-2012. Pueblo Colorado. Accessed Jan 2013. Britanica Online. 2013. s.v."Carl David Anderson". Accessed April 2013. . Stats and Hours. Accessed Jan 2013. NIST Reference on Consants, Units, and Uncertainty. 2011. Fundamental Physical Constants. June 2. Accessed Jan 2013. of Rochester. n.d. Particle at the University of Rochester. Accessed Nov 2012. , T.E. Coan J. 2005. Muon Physics. ................
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