S18_Time of Flight and Energy of Cosmic Muons
Introduction
Cosmic ray muons are subatomic particles created as an indirect result of the interaction between cosmic rays and Earth's atmosphere. Muons and neutrinos make up the majority of cosmic ray particles at Earth's surface [1]. Because muons are very penetrating, they have practical application in the field of muon tomography, which is a technique that uses cosmic ray muons to generate 3D images of large volumes [2]. We measured the speed of cosmic ray muons at Earth's surface by measuring the time of flight for muons to travel a known distance. Muons were detected using scintillator panels, and time measurements were made with a Time-to-Amplitude Converter (TAC).
Theory
Muons have a half-life of approximately 2.2us, which means that, in order for an appreciable amount of muons to reach Earth's surface before decaying, they need to be moving close to the speed of light. The flux of muons at Earth's surface is given in [1] to be approximately 70 s-1m-2sr-1. This experiment only considers vertical muons that travel through our detectors, however using an approximate solid angle of 0.05 sr and approximate detector size of 0.1 m2, we still expect to detect a muon every minute or so.
We relate time of flight to velocity by the equation
This equation shows that time is linearly related to travel distance, with a slope of 1/v where v is the muon velocity.
Experimental Setup
Muons are detected using scintillator panels, which are flat boards made of a scintillating material that releases a photon when a particle passes through it. We detect this photon-- and thus a muon "hit"-- using a photomultiplier tube (PMT) connected to the scintillator panel. The PMT produces an analog signal which is digitized by connecting the output of the PMT to a discriminator. The discriminator allows us to customize the input threshold voltage and pulse width.
In total 5 panels are used, two at the top, two at the bottom, and one in the middle. In order to only measure vertical muons, we use a conicidence unit connected to the discriminator outputs. The coincidence unit is effectively an AND gate, and outputs a pulse if all the panels are triggered within a certain timeframe specified by the discriminator pulse width. This event is called "full coincidence" and occurs in the case where a muon strikes all the panels, and is thus moving vertically. There is a non-zero probability that full coincidence is detected by two separate muons triggering all the panels, however that probability is extremely low given the relatively low flux of muons and the short discriminator pulse width.
Figure 1: Diagram showing the vertical placement of the scintillator panels. Panel C is the "measurement panel".
Figure 2: Diagram showing how panels A and B and panels D and E are crossed to reduce the valid "strike zone", shown in red.
Figure 1 above shows how the panels are stacked. Figure 2 shows how the upper two and lower two panels are crossed. We crossed these panels to limit the "strike zone" of the muons. In order for full coincidence to occur, the upper two and lower two panels must all be triggered, therefore a muon has to hit this reduced strike zone otherwise it wont be measured. This ensures that detected muons are moving closer to perfectly vertical, however it decreases the number of full coincidence muons.
The panel in the middle is used as a "measurement panel" and is the panel that is moved up and down to vary the muon travel distance. This panel is connected to a delay circuit set to a relatively large delay. We then measure the time between when full coincidence is detected and this delayed measurement panel signal using the TAC. The result is that our time measurement is not the time of flight, but related to the time of flight by the equation:
where t is time of flight, t' is the time we measure with the TAC, and k is some constant.
Figure 3: Timing diagram of panel outputs and coincidence unit output. Panel C's signal is delayed and sent to the TAC stop input. The coincidence unit's output is sent to the TAC start input.
Figure 3 above shows a timing diagram of the experiment. The important part is that TOF + Mach. delay + Time measured = Constant delay. Combining that with the fact that the Mach. delay doesn't change when we change the height of the measurement panel, we get the relationship between time of flight and time measured derived above.
The time measurement data was collected using a multichannel analyzer connected to the computer. Running on the computer is a data collection program called MAESTRO. MAESTRO collects time measurement data as a histogram, where each bin corresponds to a time interval, and the bin height represents the number of muons detected with time measurements in that time interval. This histogram comprises the raw data in the experiment.
Figure 4: Full experimental setup diagram. Panel C is situated in between panels A, B, D, and E to ensure that, if full coincidence is detected, the muon also hit panel C.
Analysis
Calibration was done by using a pulse fed into the delay boxes. The pulse was split, where one fed into the start on the TAC, while the other went into the delay boxes first, shown in figure 5.
Figure 5: The setup for calibrating the bins in Maestro. Each delay box only gave a delay up to 63ns, so two delay boxes in series were used. Throughout the whole experiment two delay boxes were used, so to keep system delays consistent, two were also used here. (Original figure)
Every setting on the delay box was tested, resulting in delays ranging from 1ns to 126 ns. The peaks were then observed in Maestro, and the resulting data and its χ values are plotted in figure 6a and 6b.
Figure 6a: Each delay setting was tested between 1ns and 126ns. Two delay boxes were used, both with delays ranging from 0ns to 63ns. The resulting linear fit for the calibration using LSQFit was Bin#=13.3 + (1.0249±0.0005)*Delay(ns).
All wires that were used for calibration were used in all data collection to keep the delays due to the wires constant. The constant offset from the TAC was ignored, as only the slope, called the calibration factor, is used in the final speed calculation.
Figure 6b: The χ value for each delay setting from LSQFit. While the actual χ are all small, there was a prominent pattern in the data. This means that there was a second order effect in our data, and a linear fit might not be the best for our calibration.
Figure 6b shows that our calibration linear fit model might be flawed, but since the χ values were mostly at ±0.5 or below, the second order effects had a small effect on our results. A data set for each of the 6 heights was collected, creating a distribution for each set. Figure 6 below shows the raw histogram for the lowest height.
Figure 7: The figure created in Matlab shows the number of hits in each bin. Note that all points on the chart are at least 1 hit.
The data was then fit to a Gaussian curve to be able to find the peak bin number. The distribution was not a true Gaussian, and was more closely related to Lorentzian, but a Gaussian fitting was used for ease.
Figure 8: The standard Gaussian plus a vertical offset was used on each height to produce a plot like the one shown above. The peak from the fitting was used to find the peak bin number for that height, which was then paired with the height to determine muon velocity. (Matlab plot).
The Gaussian fit like the one in figure 8 was performed on each data set obtained, and the peak bin number vs height was plotted. The uncertainty for the peak bin number was the standard error of the Gaussian,
where σ is the standard deviation, and N is the amount of data points. The resulting plot in figure 9a shows a linear trend, which was then confirmed by the χ values in figure 9b.
Figure 9a: The resulting trendline from LSQFit for the Bin# vs height was Bin# = 48-(3.43±0.11)*Height. The constant offset did not factor in determining muon speed, since calculations were done with the slope only. The negative sign is due to our setup, primarily equation 5. The absolute value of the slope was used.
Figure 9b: There is no visible pattern in the χ values from LSQFit. The χ values are of the expected size, meaning that our trendline was a good fit for the data.
With 6 heights worth of data, the resulting absolute value of the slope of the Bin# vs height was (3.43±0.11) in units of peak bin# per meter. To obtain m/ns the calibration factor was applied to convert bin number to ns.
Using the equation above, the speed of the muons was then calculated to be
The accepted value of the speed of light, c, is 0.2998m/ns. Therefore, our measured muon speed was (0.997±0.033)c. The error was calculated using standard error propagation formulas on the speed formula, giving the result
The error came from 3 main sources, the calibration slope error, the error in the Gaussian fitting finding the peak bin number, and error in height. Error in the Gaussian fitting was encoded in the data trials slope error. The error in height was determined to be negligible, since we were able to accurately measure the heights of the panels up to 1.5mm, and heights up to 2 m were used.
Results
We determined from our data that the speed of cosmic ray muons at Earth’s surface is (2.99 ± 0.01)×108 m/s, or (0.997 ± 0.033)c. Looking at the specific contributions to the error, the data trials slope term was on the order of 1000x larger than the calibration factor term. This means that our data could still be improved by reducing the error in the Gaussian fitting. The simplest way to do that would be to continue averaging more data points. Unfortunately, even if enough data is averaged, the calibration error would still give a resolution of ±0.0005c. To achieve higher resolution than that, a different setup needs to be used, possibly a more accurate method to measure the time delays.