Muon Lifetime Lab Report
This is a lab report for a muon lifetime laboratory experiment carried out in advanced undergraduate physics lab....
Muon Lifetime Evan Rule April 21, 2013
Abstract In order to investigate the nature of muon particles, we carry out an experiment to measure the charge-averaged muon lifetime. Using a muon decay detector, we record the time interval from when the muon enters the detector to the time when it stops and decays via the weak force. After accounting for the background noise of the detector and averaging over measurements made with various detector thresholds, we determine the charge averaged mean muon lifetime to be τ = 1.92 ± 0.09 µs.
The first detection of muon particles was made in 1936 by Carl D. Anderson and Seth Neddermeyer, both of Caltech. Anderson and Neddermeyer had been studying cosmic radiation when they observed a particle whose behavior in a magnetic field was different from any known particle. Based on the curvature of the particle’s path, it was hypothesized that the mass of particle (eventually named the muon) was larger than that of the electron, but smaller than that of the proton. Because of the relatively large energies (≈ 105.7 MeV) necessary for the production of muons, ordinary radioactive decay, fission and fusion events are not energetic enough to produce these particles. Therefore, the primary sources of naturally occurring muons on Earth are cosmic rays. Collisions of cosmic rays with the nuclei of air molecules produce a shower of particles (see Fig. 1) which undergo further electromagnetic and nuclear interactions. In particular, some charged pions decay via the weak force into a muon plus a neutrino or antineutrino. These muons travel relatively long distances, gradually losing kinetic energy and eventually decaying via the weak force into an electron plus a neutrino or antineutrino. By observing the decay times of these muon particles, we can make a determination of the charge averaged muon lifetime.
Experiment and Data
In order to make a measurement of the charge averaged muon lifetime, we employ a muon decay detector consisting of a plastic scintillator in the shape of a right circular cylinder (see §2.1 for more details on the apparatus). Since negative muons can interact with matter via µ− + p → n + νµ ,
our determined value for the lifetime of a muon will be charge-averaged. This alternative decay pathway causes the effective lifetime of negative muons to be less than the lifetime of positive muons.
Figure 1: An overview of the process of muon production from cosmic ray interactions. We are interested in the muon particles that result from the decay of charged pions via the weak force. (Image taken from Coan and Ye)
Figure 2: A detailed view of the muon decay detector. Muons entering the scintillator trigger bursts of light that are detected by the photomultiplier tube. A second burst of light is produced when the muons decay into electrons. (Image taken from Coan and Ye)
The muon decay detector consists of a plastic scintillator and a photomultiplier tube (see Fig. 2). When muon particles enter the scintillator and pass through the plastic solvent, kinetic energy from the muon is transferred to fluor molecules, exciting their electrons and leading to the radiation of photons in the near-UV. The photomultiplier tube detects this burst of photons and triggers a timing clock. As the muons dissipate kinetic energy, they eventually reach a state where they stop and decay via the weak force into an electron plus a neutrino or an antineutrino. Since the electron mass is much smaller than the muon mass, the created electron tends to be very energetic and triggers scintillator light as it moves through the chamber. These bursts of light are also recorded by the photomultipler tube and indicate to the timing clock that the muon has decayed. This time interval between the initial and final bursts of light can be used to determine the lifetime of the muon particles. Recall that negative muons can also interact directly with protons in the scintillator material. The muon decay detector is connected to a computer which allows us to automate data acquisition. 2
The execution of the experiment is fairly straightforward and amounts to allowing the detector to observe the decay of muon particles in the scintillator. Before beginning data acquisition, though, we first use the pulser to send photon signals to the detector at regular intervals and ensure that the detector is not biased towards certain decay times. Using the included software, we set the detector to record time intervals between successive bursts of photons in the scintillator. We allow for several days for the detector to record a sufficient number of decay events. Data and Error Estimation When our data collection is complete, we have measurements of the decay times of both positive and negative muons in the scintillator chamber. The format of our data is simply the time between successive photon events recorded by the photomultiplier tube, or in the case that no secondary photon event was recorded, the number of times the detector timed-out. From these data, it is straightforward to calculate the charge-averaged mean lifetime of muons. We model the decay of muons as an exponential relation of the form −dN = λe−λt dt, N0
where N is the population of muons at time t, N0 is the initial population, and λ is a constant characteristic decay rate of muons. By determining the best fit value for λ, we also determine τ=
1 , λ
the characteristic lifetime of muon particles. However, there are several corrections which we must account for in our analysis. Since our detector has no knowledge of the nature of the events which trigger photon bursts, we must be careful to exclude background events which are different than the decay of a stopped muon in the scintillator. For example, it is possible that 2 muons might enter the scintillator and trigger successive events which might result in the detector recording a single time interval which is not the decay time of a single muon. We can account for these extraneous detections by restricting the time interval in which we look for successive photon detections and by estimating the background level by looking at large times in the decay time history.
Before we can accurately model our data, we must account for the background noise of the detector. We first disregard all muon detector time-outs, i.e. all times greater than 40,000 ns. As recommended in the lab manual, we also discard any decay times less than 80 ns, as the detector is not reliable below this limit. By collecting data for several days with the scintillator threshold set to zero, we are able to model the background as a function of the form 2
B(t) = Ae−((x−C)/D) + Ex + F,
which is simply the sum of a Gaussian and a linear function. The results of this background fitting are shown in Fig. 3. In order to keep our results consistent across different trials, we normalize our histograms by dividing the number of counts in each bin by the total number of counts for that trial. We employ a bin size of 200 ns for all trials. The results of our data collection for
”low” scintillator threshold are shown in Fig. 4. Once we have found an accurate model for the background, we then fit our data to a function of the form N (t) = C1 e−λt + C2 B(t),
where λ is the characteristic decay coefficient, from which we can obtain the mean muon lifetime, τ , via the relation 1 (6) τ= . λ Our fitting algorithm calculates the absolute error on each of the fitting parameters, from which we can directly obtain the uncertainty in τ . The results of our data fitting are shown in Table 4, where we also present measurements of the mean muon lifetime obtained without background corrections. The purpose of including these results is to gauge the effectiveness of our background correction. From the table, we see that our results (with background correction) are consistent across all 3 threshold levels. Therefore, we take the mean of these values to be the most likely value for the charge averaged mean muon lifetime, τ = 1.92 ± 0.09 µs,
where the error has been obtained by adding the relevant errors in quadrature. According to the literature, the standard value for the charge averaged mean muon lifetime is τ = 2.19703 ± 0.00004 µs. Our value is inconsistent with the accepted value at the 3-σ level (99.7%). However, because we calculate the charge averaged mean muon lifetime, we can expect our determined value of τ to be lower than the accepted free space value. In fact, we can take the mean lifetime of muons in carbon, τc = 2.043 ± 0.003 µs as an absolute lower limit of “realistic” muon lifetimes, since this decay is representative of the alternate decay pathway of negative muons. Though our result is consistent with this lower limit, it is extremely unlikely that the majority of muon decays occurred via this alternate pathway, and so we should examine our experiment for sources of systematic error.
From table 4, we see that our determined value for the mean muon lifetime when using the background correction is consistent across the 3 threshold levels, within error. When we compare these values without the background correction, however, we see that our measurements for thresholds 5 and 7 are consistent within error, but our value for the lowest threshold, 3, is significantly larger. This result can be explained by the increased background levels that are present with a lower threshold. Since the peak in the background occurs in the region of long decay times (& 10 µs), this increased background will bias our measured value toward a falsely high value for τ . Since applying the background correction makes our calculated values consistent across all thresholds, we can conclude that our data fitting routine does indeed account for background noise. Though, the fact that our determined values for τ are lower than the accepted value could be evidence that were are over-compensating for background noise. We can also make a check of systematic error by splitting our data sets in half chronologically and seeing if the mean muon lifetime changes over time. Focusing on the data obtained for threshold 5, we see that the mean muon lifetime during the first and second halves of the trial are τ = 1.73±0.07 µs and τ = 2.03 ± 0.07 µs, respectively. This discrepancy indicates that the detector has some type of time dependent bias. One possible cause of this error is that the background itself might be time dependent. We check this in the same manner as above, by splitting the background data set chronologically and fitting each half separately. In doing so, we see that all of the background 4
Figure 3: Histogram plot of the frequency of observed muon decay times, obtained with the detector threshold set to zero. 74,802 background events were recorded over a period of approximately 2 days. This data can be used to model the inherent background noise of the detector, which can then be corrected for when we model our actual data sets. We model the background as the sum of a Gaussian and a linear function.
Figure 4: Histogram plot of the frequency of observed muon decay times, obtained with the detector threshold set to 3. 9,924 decay events were recorded over a period of approximately 5 days. The trend line shows the best fit background-corrected exponential function. For this trial, we determine the charge averaged mean muon lifetime to be τ = 1.93 ± 0.05 µs.
fitting parameters agree within error. However, we also note that the center of the Gaussian has moved slightly to the right, towards the larger decay times. This could explain why our calculated mean lifetime increases with time, though given the error on our background fitting, we cannot say with certainty whether this is indeed a physical effect. Nonetheless, it would be interesting to explore this further using a more robust model of the background.
Threshold 3 5 7 3 5 7
Background correction? Yes Yes Yes No No No
τ (µs) 1.93 ± 0.05 1.92 ± 0.06 1.90 ± 0.05 2.45 ± 0.06 2.03 ± 0.03 2.04 ± 0.03
Number of decays 9,924 4,234 4,534 9,924 4,234 4,534
Table 1: Measurements of charge averaged mean muon lifetime, τ , obtained for 3 different detector threshold settings. Here, we present results obtained both with and without the background correction.
We carry out an experiment to measure the charge averaged mean muon lifetime using a scintillator and photomultiplier tube. By modeling the background noise of the detector, we are able to account for extraneous data points in our model fitting. We model the decay of muon particles as an exponential function and determine the charge averaged mean muon lifetime to be τ = 1.92 ± 0.09 µ s, a value which is significantly lower than the accepted value. We examine our data for systematic trends and determine that our calculated muon lifetimes show a strong time dependence, which may or may not be the result of a time dependent background.
References  Coan, T.E. and Ye, J., Muon Physics.