mean life time of cosmic ray muon

Midshipman 1/C TAIMOOR. ALI. RAJA20 February 2012

Department of Physics, United States Naval Academy

Mean Lifetime of the Cosmic Ray Muon

USNA, Department of Physics Mean Lifetime of the Cosmic Ray Muon

Abstract:We measured the mean lifetime of muons that were produced by cosmic rays high in the atmosphere and made their way to sea level. We stopped some of the muons in a cube of plastic scintillator and used photomultiplier tubes (PMT) to detect the light that they and their decay electrons produced. We processed the PMT signals using standard NIM electronics and measured the time intervals between them using a time to amplitude converter and a multichannel analyzer. From the distribution of decay time intervals, we determined that the stopped muons had a mean lifetime of (2.174 ± 0.019) µs. This is slightly less than the accepted value for free muons, about 2.197 µs, but our smaller value may be due to the capture of negative muons in the scintillator material.

I. INTRODUCTION

Cosmic Rays:A steady rain of particles accelerated by intergalactic magnetic fields and moving with

relativistic speeds falls on the Earth at all times. We know these particles as cosmic rays [1]. The intensity of cosmic rays increases with increasing altitude, which shows that cosmic rays come from outer space. The intensity also varies with geomagnetic latitude, which shows that cosmic rays consist of charged particles that are deflected by the Earth’s magnetic field [2].

Muons:The muon is a fundamental particle with the same charge and spin as an electron, but with a

mass about 207 times greater than an electron’s. Like the electron, the muon experiences only the weak and electromagnetic interactions. For this reason, electrons and muons are both called leptons. Unlike the electron, however, the muon is unstable. So for many purposes, the muon can be considered to be a heavy, unstable electron.

Origin of Muons: The muons we studied were produced high in the earth’s atmosphere by collisions between

air molecules and the so-called primary cosmic rays, which come from outer space. The approximate composition of the primary cosmic radiation is 87% protons and 13% neutrons, the latter being bound in nuclei [3].

The charged particles that reach the Earth’s surface are not primary, but secondary; that is, they are produced by nucleon–nucleus interactions. The most abundant products of these interactions are pions, which may be either electrically neutral or charged. Neutral pions decay into gamma-ray pairs in about 10-16 seconds. Charged pions decay into muons and neutrinos in about 10-8 seconds according to these schemes:

π+ → μ+ + υμ

π- → μ- + ῡμ

Figure 1 illustrates this process of production and decay. Muons make up more than half of the cosmic radiation reaching the surface of the Earth, the other half being mainly electrons, positrons and photons [4].

Midn. T. A. Raja Page 1 of 16


Fig. 1: Cosmic ray interactions in the atmosphere. From Reference [4]

Decay of Muons:During the 1940s, experiments conducted using nuclear emulsions showed that a muon

decays into an electron with a kinetic energy that may have any value from zero to about 50 MeV. This means that there must be at least three particles among the decay products, two of which must be neutral because they produce no tracks in the emulsions. Now we know that a muon decays into an electron, and two different types of neutrinos, in the processes shown below and in Figure 2:

µ- → ℮- + υμ + ῡe

µ+ → ℮+ + υe + ῡµ

Fig. 2: Feynman diagram for muon decay. From Reference [6]



The decay of the muon is characterized by a decay constant λ and a mean lifetime τ, which are defined in terms of the relationship

dN = -λNdt = -1/τ Ndt

Here, N represents the number of muons present at some time (usually taken to be t=0) and dN represents the change in that number during a subsequent small time interval dt.

This relationship can be integrated to give the number of muons remaining as a function of time:

N(t) = N(0) e-λt = No e-t/τ

This expression also describes the expected distribution of time intervals between the arrival of a single muon in a detector and its decay into an electron.

The mean lifetime of free muons has been measured to high precision. A recent value isτ = (2.197034±0.000021) μs [7]

If negative muons are stopped in a material such as a plastic scintillator, they may become attracted to and captured by nuclei of the atoms making up that material. This process does not occur with positive muons, which are repelled by the positive nuclei. Nuclear capture results in a shorter observed lifetime for negative muons stopped in materials. For carbon, which makes up much of the plastic scintillator material, the lifetime of stopped negative muons is

τ=(2.0263 ± 0.0015 ) µs [8]

II. MATERIALS AND METHODSThe major equipment used in the experiment includes a large plastic scintillator,

photomultiplier tubes, signal-processing electronics, and a computer-interfaced multichannel analyzer. This equipment is shown in Figure 3.

Fig. 3: Apparatus used in muon lifetime measurement



Scintillator:The scintillator is the large black cube at the top of Figure 3. It converts the energy deposited by charged particles that pass through it into visible light. These flashes are picked up by the photomultiplier tubes attached to the left-hand side of the cube and converted to an electrical signal. The scintillator used in this particular experiment is a transparent plastic with special light-emitting molecules added. It is made by Rexon Components, Inc, and its dimensions are 12 x 12 x 12 inches. It and the photomultiplier tubes are enclosed in a black, opaque covering to exclude stray light.

Photomultiplier tubes:Photomultiplier tubes make use of the photoelectric effect to convert the faint light emitted by the scintillator into a more easily analyzed burst of electrons. Light from the scintillator first falls on a photocathode, which emits a small number of so-called photoelectrons. Shaped electric fields direct them through a series of dynodes, which are electrodes made of a material with good secondary electron emission. [9] Each dynode emits more electrons than it receives, creating a large shower of electrons at the output electrode. In this experiment, the photomultipliers used are Rexon model 2000P, Serial Numbers 9738 and 9739. A bias voltage of negative 2000 V is applied to each tube, and a voltage-divider circuit apples the appropriate voltage to each dynode. The output of the photomultiplier tubes is shown in Figure 4. The oscilloscope input is terminated in 50 ohms.

Fig. 4: Output of the photomultiplier tubes.



Electronic Components:The main signal processing electronic components used in this experiment are shown

in the block diagram of Figure 5. They include:- Fast Timing Amplifier (Ortec FTA 820)- Constant Fraction Discriminator (Ortec 583B)- Coincidence Circuit (Lecroy 465)- Time to Amplitude Converter (Ortec 567)- Timer and Counter (Ortec 871)- Multi-Channel Analyzer (Amptek MCA8000A)

Fig. 5: Block diagram of the electronics.

The main function of the electronics is to make sure that every signal that is counted is genuine and related to the decays of muons. The Fast Timing Amplifier (FTA) is a broadband amplifier that increases the amplitude of the pulses coming from the photomultiplier tubes. These pulses include noise as well as signals generated by charged particles passing through the scintillator. The Constant Fraction Discriminator (CFD) is used to exclude noise pulses and to generate a precise timing signal. The Coincidence Circuit (CC) produces an output pulse only when the two photomultiplier tubes detect scintillations within about 50 nanoseconds of one


FTA

FTA


another. This further reduces the effects of noise. The Time to Amplitude Converter (TAC) is used to measure the time interval between successive outputs of the Coincidence Circuit. In a sense, the TAC behaves like a stopwatch. A pulse at the Start input starts the stopwatch, and a pulse at the Stop input stops it. The output of the TAC is a positive signal whose amplitude, in the range 0 – 10 V, is proportional to the time interval between Start and Stop pulses. In this experiment, the maximum interval was set to 20 µs. The Multichannel Analyzer (MCA) sorts the TAC pulses into a spectrum with counts on the vertical axis and bin number, proportional to the amplitude of the TAC pulse, on the horizontal axis. A time calibration of the apparatus permits the horizontal axis to be labeled in terms of time intervals.

Following the signal:We measured the signals at the output of each module in the block diagram, and the results

are illustrated in the figures below. In each case, the input is a pair of fast negative pulses from the double-pulse generator, which mimic the output of the photomultiplier tubes. The time interval between the pulses is always about 2.2 µs, approximately the mean lifetime of the muon.

Figure 6: Output of the Pulse Generator.

Figure 7: Output of the FTA



Figure 8: Output of the CFD

Figure 9: Output of the Coincidence Circuit

Figure 10: Output of the TAC



III. DATA ANALYSIS

Time Calibration:We performed a time calibration by sending pairs of identical pulses with known and

variable time spacing through our circuit. The pulses were obtained from a double-pulse generator and the TAC output was sent to the MCA. We used Microsoft Excel to calculate a linear regression formula for the time calibration as shown below. The equation:

y = 0.0195x + .1635simply means that the time corresponding to channel ‘x’ is 0.1635μs + 0.0195 μs*x.See Appendix A for linear regression formulas for first two time calibrations.

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

12

14

16

18

20

f(x) = 0.0195009043585039 x + 0.163483829512503R² = 0.999996625051903

Time Calibration # 3

Channel Number

Tim

e (u

s)

Figure 11: Time calibration curve.

Muon Lifetime Measurement:After we successfully completed the time calibration, we replaced the output of the pulse

generator with the outputs of the PMTs and again connected the TAC output to the MCA. The result was a plot of the number of muon decays versus the time interval between the TAC Start and Stop pulses. The Amptek multichannel analyzer was used to convert analog signals to digital. It simply assigns the pulse to a specific channel depending on its height. The plot was displayed and analyzed on a notebook computer running the ADMCA software.

We repeated the experiment several times over periods varying from about 2 days to about a week.

The data for each run was saved in a text file, which was converted in Microsoft Excel from counts-vs-channel to a counts-vs-time using the time calibration conducted previously. Finally, the data was imported into Origin 8.5 [10] where it was plotted and fit to a decaying exponential of the form

Y = y0 + A*exp (-t/τ)The results for Run 4 are shown below.



Figure 12: Muon decay curve obtained in Run 4.

See Appendix B for data for all runs.

Table 1: Fit parameters obtained in Run 4.

The value t1 returned by the fitting program corresponds to τ, the time constant for the equation and is our value for the mean observed lifetime. In this run, we found that = (2.15 ± 0.02) µs, where the quoted uncertainty is that associated with the fitting process alone. Other uncertainties, such as those associated with the time calibration, are not included.

The value y0 represents a constant background, presumably due to so-called accidental coincidences. These can arise if two unrelated charged particles pass through the detector during the 20-µs TAC time window. The background count per channel was estimated using the following expression:

N A=(Rs∗Δt∗Rs ) T

NCH

Here, Rs = rate of start stop pulses at TAC (about 20 s-1) Δt = TAC time window (20 µs) T = duration of run (1.94 days or 1.68 105 s) NCH = Number of channels in spectrum (1024)



The estimated background count per channel was equal to 1.31, which is in close agreement with y0 value of 1.47 as shown in Table 1.

Data Summary:Our results for four different data runs are summarized in Table 2 below. The weighted

average of the measured mean lifetime is (2.174 ± 0.019) µs. The quoted uncertainty represents the average fitting error only. Possible additional uncertainties associated, for example, with the time calibrations, have not been assessed or included.

Table 2: Measured lifetimes for several different runs.

IV. CONCLUSIONWe measured the mean lifetime of muons stopped in a plastic scintillator and obtained

an average value of (2.174 ± 0.019) µs, which is close to the accepted value of 2.197 µs for the lifetime of free muons. The fact that our measured lifetime is smaller than the accepted value is probably due to the capture of negative muons by the scintillator material.

V. ACKNOWLEDGEMENTSI would like to acknowledge Midshipman 2/C R. M. Simpson for his assistance in every phase of the experiment and Professor David Correll for his guidance throughout the course of experiment.



VI. REFERENCES

[1] Rossi, Bruno Benedetto. Cosmic rays. New York: McGraw-Hill, 1964.

[2] Friedlander, Michael W. A thin cosmic rain: particles from outer space. Cambridge, MA: Harvard UP, 1989,2000.

[3] Thompson, M. G. "Energetic Muons." Cosmic rays at ground level. Comp. A. W. Wolfendale. London: Institute of Physics, 1973.

[4] "Cosmic Rays." Hyperphysics.com. 06 Nov. 2011 <http://hyperphysics.phy-astr.gsu.edu/hbase/astro/cosmic.html>.

[5] The Speed and Decay of Cosmic-Ray Muons: Experiments in Relativistic Kinematics- The Universal Speed Limit and Time Dilation. Tech. MIT Department of Physics, November 29, 2010. Print.

[6] Muon Decay. Digital image. Wikipedia, the Free Encyclopedia. Web. 07 Nov. 2011. <http://en.wikipedia.org/wiki/File:Muon_Decay.png>

[7] K. Nakamura et al. (Particle Data Group), Journal of Physics G 37, 075021 (2010).

[8] T. Suzuki et al., Physical Review C 35, 2212 (1987)

[9] Measurement of Muon Lifetime. Tech. University of Michigan, November, 2001. Print.

[10] www.originlab.com

[11] T. E. Coan and Y. Ye, “Muon Physics Manual”, www.matphys.com/muon_manual.pdf, page 9.

[12] B. Vulpescu et al., Journal of Physics G 27, 977 (2001)



APPENDIX ATIME CALIBRATIONS

a. Calibration 1: CHANNEL # TIME (us)

196.94 4

401.86 8

610.96 12

810.96 16

1002003004005006007008009000

5

10

15

20

f(x) = 0.019499781975 x + 0.149100141855R² = 0.999929319897504


Channel Number

Tim

e (u

s)

This calibration was done on 07OCT2011 and was used for Run 1 and Run 2.

b. Calibration 2: CHANNEL # TIME (us)

94.98 2197.10 4299.21 6401.88 8504.58 10607.01 12709.42 14811.97 16914.00 18

0 200 400 600 800 10000

5

10

15

20

f(x) = 0.01952412087 x + 0.15084029446R² = 0.999999438596675


Channel

Tim

e (u

s)



This calibration was done on 28NOV2011 and was used for Run 3.

c. Calibration 3: CHANNEL # TIME (us)

94.23 2197.00 4299.00 6401.55 8504.55 10607.00 12709.16 14813.23 16914.00 18

0 200 400 600 800 10000

5

10

15

20

f(x) = 0.019500904359 x + 0.16348382951R² = 0.999996625051903


Channels

Tim

e (u

s)

This calibration was done on 04DEC2011 and was used for Run 3 and Run 4.



APPENDIX BEXPONENTIAL FITS TO THE DATA FOR ALL FOUR RUNS

a. Run 1:

b. Run 2:



c. Run 3 with Time Calibration 2:

d. Run 3 with Time Calibration 3:

e. Run 4:


mean life time of cosmic ray muon

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