## Contents

1. Note that there is NO eating or drinking in the 111-Lab anywhere, except in rooms 282 & 286 LeConte on the bench with the BLUE stripe around it. Thank You the Staff.

The muon is an unstable particle, with a lifetime on the order of microseconds, which decays by means of the weak interaction into an electron and two neutrinos. If we start out with No muons, then the number of muons present at a time t later is N(t) = Noexp(-t/T), where T is the mean life.

In this experiment, cosmic-ray muons that enter a large tank of liquid scintillator trigger the emission of a pulse of photons. These photons are detected by a photomultiplier tube. If the muon stops inside the tank, another pulse of light is produced when the muon decays. The time difference between the pulses is a mean for measuring the lifetime. An histogram of these time differences for many decaying muons resembles an exponential distribution from which the mean lifetime can be determined.

You will get acquainted with a number of electronic devices that are quite common in laboratories. Data analysis using a computer and program writing are required. You will spend time in setting up the equipment. Once that's done and data-taking is started, you don't need to be in the lab, except for checking out how everything is working.

1. Pre-requisites: None
2. Days Alloted for the Experiment: 6
3. Consecutive days: No

All pages in this lab. Note To print Full Lab Write-up click on each link below and print separately

Reprints and other information can be found on the Physics 111 Library Site

This lab will be graded 20% on theory, 20% on technique, and 60% on analysis. For more information, see the Advanced Lab Syllabus.

# Before the Lab

Complete the following before your experiment's scheduled start date:

1. View the Muon Lifetime video.
2. Complete the MUO Pre Lab and Evaluation sheets. Print and fill it out. The Pre-Lab must be printed separately. Discuss the experiment and pre-lab questions with any faculty member or GSI and get it signed off by that faculty member or GSI. Turn in the signed pre-lab sheet with your lab report.

1. † B. Rossi, "Interpretation of Cosmic-Ray Phenomena", Rev. Mod. Phys. 20, 537 (1948).
Gives the cosmic ray flux at sea level. Figure 6 on p. 545 is important for your calculations. Searchable Page
2. † D. Griffiths, Introduction to elementary particles, 1987.
Read the introduction (pp. 1-10). The result of a tedious calculation of the muon lifetime is given in pp. 307-309.
3. R.D. Evans, The Atomic Nucleus, McGraw-Hill (1955).
4. C.S. Sutton et al., "Undergraduate Cosmic Ray Muon Decay Experiments with Computer Interfacing", Computers in Physics, Nov/Dec, 76 (1987), #QC52.C658.
5. O.C. Allkofer, Introduction to Cosmic Radiation, Verlag Karl Thiemig, Munchen (1975). Read Sections 4.1, 9.1-9.6, 9.12, and 10.1-10.4.
6. "RCA-4522 Photomultiplier Tube Specifications”, RCA Electronic Components, June, 1 (1968).

More References

You should keep a laboratory notebook. The notebook should contain a detailed record of everything that was done and how/why it was done, as well as all of the data and analysis, also with plenty of how/why entries. This will aid you when you write your report.

# Introduction

The goal for this lab is to measure the lifetime of the muon. This is a relatively simple experiment, but one that can achieve surprisingly good precision if you are careful. Calibrating the electronics properly is important and is worth doing carefully. Most of the adjustments are made in the analysis using software. Don't rush through the adjustments, but ensure you understand what you are doing and how your changes affect the data. Get the settings right before spending all day and night recording data; otherwise you only find out later that the data are useless. Once you have the data and a nice plot on the screen, you still need to work hard to make a proper analytical fit and to extract a correct value for the lifetime.

Muons are particles that have a mass over 200 times that of the electron. Muon decay is of great importance in the study of weak interactions, one of the fundamental forces in nature. The value of the muon lifetime is one of the most precisely known constants, and is currently measured with a precision of a few parts per million. In this experiment, low-energy muons are primarily created from decays of particles produced in the interactions of high-energy cosmic-rays with the earth's atmosphere. Eighty percent of the cosmic rays at sea level are muons, positively or negatively charged. They have a mean lifetime of a few microseconds.

They decay by the following processes:

$\mu^+\rightarrow e^++\nu_e+\bar{\nu}_{\mu}$ (1)

and

$\mu^-\rightarrow e^-+\bar{\nu}_e+\nu_{\mu}$ (2)

By the CPT theorem, the lifetimes of the negative and positive muons in vacuum (i.e. due to reactions (1) and (2)) are equal. However, in matter, negative muons, when brought to rest in the close neighborhood of some nucleus (Z,A), can also disappear through the competing reaction (where Z is the atomic number [the number of protons] and A the atomic mass [the number of protons plus neutrons]):

$\mu^-+(Z,A)\rightarrow(Z-1,A)+\nu_{\mu}$ (3)

The rate of the muon-capture process (3) depends strongly on the charge Z of the nucleus which captures the muon. In carbon, the rate of the muon capture (3) is about 10% the decay rate in (2).

Most of the muons that are incident on the detector pass straight through it. Only the lowest energy particles come to rest within the scintillating liquid itself.

# Apparatus

## Overview

Figure 1. Block Diagram of Muon Lifetime Experiment

The large liquid scintillation counter is located in a shaft which can be accessed from room 275 Le Conte. It is filled with 125 gallons, 500 kg (1/2 cubic meter) of mineral oil in which a scintillating substance is dissolved. The passage of fast charged particles through the liquid gives rise to short pulses of light a few nanoseconds in duration. These flashes of light called scintillations are converted into electrical pulses by photomultiplier tubes located at the top and bottom of the detector (Figure 1). Each photomultiplier views half of the tank, which is divided by an opaque baffle in the horizontal mid plane.

A photomultiplier tube (PMT) is a detector that converts single photons into large pulses of electric current by successive multiplying stages (Figure 2).

Figure 2 Photomultiplier tube

In the shaft of room 275 Le Conte Hall (Get a GSI to let you into the room ) are located:

• a high voltage supply and a high voltage divider for the photomultiplier tubes. The high voltage of the photomultiplier tube normally is left on and is NOT adjusted by the student. See the staff in the 111-LAB. These photomultiplier tubes are irreplaceable. Do not mess with them.
• a voltage meter to read the individual high voltages. Under no circumstances should the high voltage exceed 2200 V!
• one large liquid scintillator with 500 kg of doped mineral oil
• cables leading to 286 Le Conte where the rest of the apparatus is located.

At the 286 LeConte Muon Lifetime Station are located:

• SR445A 4-channel amplifier, for amplifying the signals from the top PMT (white cable) and bottom PMT (black cable)
• Tektronix TDS 360; Digital Storage Scope
• Tektronix 465 Oscilloscope (Analog)
• Miscellaneous equipment for calibration of the system and 50 ohm terminators
• Stanford Research Systems Model DG645 Digital Delay Generator
• National Instruments Digitizing Card # NI-5114

This Digitizer works together with a LabVIEW program that does two things: (1) it is an oscilloscope, (2) it is a differential detector logic system.

## How the Experiment Works

Muons are derived from the process of cosmic ray particles striking the nuclei of atoms and molecules in the atmosphere. A muon that passes through the liquid scintillator tank generates scintillation. A photomultiplier tube picks up the light pulse and sends an electrical signal through a cable to our experiment. A muon that enters, stops, and subsequently decays in the liquid scintillator gives rise to two pulses separated by a few microseconds, one when it enters the liquid, and a second one when it decays. The pulses can be viewed with an oscilloscope or digitized and stored on the computer. You are going to measure the time difference between these correlated pulses, as well as the amplitude and width of each pulse. After many such observations, you will obtain a distribution of the number of counts vs. time difference. Ideally, when these data are plotted, they will form an exponential decay curve, whose decay constant is the lifetime of the the muon.

Figure 4

When the two pulses come down the cable, they are routed to an amplifier. Then they go to the National Instruments Digitizer Card NI 5114. You configure triggering and timing properties of the digitizer in the LabVIEW programs. The LabVIEW program Muon Detection Program coupled to the digitizer card detects pulses from two photomultiplier tubes, measures their arrival times, amplitudes, and widths, and, for a subset of pulses that satisfy a preset criteria, writes the data to disk. You will read the data offline, filter it to reject noise, plot the time difference between two pulses in the same photomultiplier tube, and use the distribution to measure the muon lifetime.

## Explanation of the LabVIEW Programs

Note: Before you use these programs you should have viewed the signals with a real oscilloscope. Follow the signal paths starting at the Muon signal cables from the PMTs to the place where they connect to the Digitizer. What polarity are they, positive or negative? How wide, how high, etc.

• Note: High voltage on the PMTs are as follows, Top tube =1900Volts Negative and bottom tube = 2300 Volts Negative with the SR445A 4 each amplifier, setup top PMT 1 -amp, bottom PMT 2 amps.

This experiment uses two different LabVIEW programs: View Signals and Muon Detection Program. The purpose of View Signals is for you to choose the best input parameters by inspecting the signals and seeing how changes in the input parameters affect the signals and counting rates. ( This reads the input signals like an oscilloscope). The optimal input parameters should then be used in the program Muon Detection Program, whose purpose is to save pulses to data files, from which you calculate the muon lifetime.

A high-speed digitizer installed in the back of the computer receives two signals. The top PMT is connected to Channel 0 of the digitizer, and the bottom PMT is read out by Channel 1. The digitizer is configured and initialized when you click Start Acquisition in either of the Muon LabVIEW programs. You can set the input range for each channel separately; 2 V is about right for both channels (the offset is set to 20% of the range, so 2V corresponds to the range [-0.4V..+1.6V]). The input impedance is set to 50 ohms and the digitizer uses full bandwidth. The user control Range (microseconds) determines the minimum length of the record (counting from the trigger), i.e. the maximum difference between the trigger and the second pulse. The triggering event is set to be at 12% of the way through each record, so that the digitizer can see pulses just before the trigger. The user control Min sample rate specifies the minimum sample rate (leave it at 250 MHz). The type of triggering is set to analog edge triggering. You can select to trigger on Channel 0, Channel 1, or External (which we do not use). The trigger slope is set by the user control Pulse polarity, and the user control Trigger level sets the trigger level.

Both programs initiate a record acquisition by the digitizer and, once the voltage exceeds the trigger threshold, retrieve voltage waveform data for both channels, incrementing the Trigger Count by one. The programs remove the voltage offset by averaging the voltages of the first 200 samples of the record, all of which should occur before the triggering event, and subtracting that voltage from all the voltage waveform data. The programs then search for up to two pulses below (for negative Pulse polarity) or above (for positive Pulse polarity) the Channel 0(1) threshold, which you set. This threshold is relative to the offset, not ground. The time difference between the first and second pulses has to be less than the Range, which you set, and greater than the Pulse 2 delay, which you set. You can specify how many pulses in each channel is required for the data to be written to disk. The Output Count is incremented if exactly that number is found in each channel. Whether or not the output condition is satisfied, the process repeats with the next data record.

Muon Detection Program measures the pulse height and width in addition to time. To reduce noise, the programs pass the waveforms through a 29-tap low-pass filter. You set the cut-off frequency of the filter with the control Lower-cut off frequency. Pulse 1 threshold and Pulse 2 threshold correspond to the unfiltered signal. Unfiltered pulses that meet these thresholds will have their amplitudes reduced upon filtering.

In the program View Signals, after clicking Start Acquisition, the program continues to acquire records and search them for pulses until you click Stop Acquisition. This program graphs the Raw signals acquired by the digitizer, the Signal with offset removed and the Filtered signal -- for each channel (Channel 0 in white, Channel 1 in red).

The program Muon Detection Program works somewhat differently. When you click Start Acquisition, the program prompts you to enter two different file paths to save data: a temporary local data file and a network data file. The temporary file is supposed to be on local disk ("C:/") so that network connectivity does not affect performance. The temporary file is copied to the network storage (your "Documents/" directory) when the program terminates. After the files are open, the program begins acquiring records and searching them for pulses. When an output condition is satisfied (i.e. exactly the specified number of pulses is found in each channel, up to the maximum of 2), the program writes out a record of 12 floating-point numbers to the temporary file: time, amplitude, and width of four pulses: 1st pulse in Channel 0, 2nd pulse in Channel 0, 1st pulse in Channel 1, 2nd pulse in Channel 1. If a given pulse is not found in the record, the time, amplitude, and width are assigned zero values.

As the program runs, it updates a histogram of decay events, where the x-axis specifies the time between the two registered pulses. If everything is working correctly, this should mimic the Muon lifetime decay curve.

The program measures the amplitude and the full-width-half-max of each peak from the signal out of the filter. The filter removes a lot of the noise, so the amplitude and width information are more accurately measured with a filter. Timing information is more accurately measured before the signal is filtered.

### Program Controls

Flip polarity
In order for the program to run successfully, a positive signal with respect to ground is expected on both channels. However, the incoming signal from the experiment may be negative, which would crash the program. Enabling Flip Polarity negates the incoming signal, so that a negative signal becomes positive. If the incoming signal is already positive, Flip Polarity should be disabled.
Select Trigger
This control is identical to selecting a trigger channel on an oscilloscope. When a large enough pulse (see Trigger Level) is seen on the selected channel, the program collects data on both channels for a time duration set by Range, and then processes those signals for peaks.
Trigger Level (V)
This defines the signal amplitude which is required for the trigger to be activated.
Pulse Detection Threshold (V)
Once the trigger is activated, the program scans the collected signal for pulses. This control defines the minimum amplitude of a signal in order for it to be recognized as a pulse. Any part of the signal which is above this threshold will be recognized as a pulse, and any part which is below this threshold will not be recognized as a pulse.
Pulse 2 Delay (μs)
Once the trigger is activated, we wish to ignore any pulses that occur shortly after, since they may simply be noise, as opposed to a muon decay event. This control allows the user to define how long this ignore period should last. This control should be large enough to ignore this noise, but not so large as to ignore a significant portion of the actual muon decay events.
Range (μs)
After the trigger is activated, the program must collect the signal which is to be analyzed for some amount of time. This control allows us to set that time duration. After this time has passed, the program analyzes the collected data for pulses, and then returns to listening for the next trigger event. This control should be just large enough to encompass a large majority of decay events.
Exact Number of Pulses
Once the program has identified all of the peaks existing on the most recently received signal, it must determine whether or not these pulses represent a muon decay event. The event is registered as a true muon decay event only if the number of pulses on Channel 0 and on Channel 1 match exactly with the specified Exact Number of Pulses. There are three suggested use cases: [CH0 = 2, CH1 = 0]: This signifies a muon which entered the chamber connected to channel 0, and then decayed in that same chamber at a time at least as long as defined by Pulse 2 delay, but no longer than Range. Since there are only two pulses on this channel and no pulses on the other channel, we are convinced that another muon did not interfere with the measurement. For this setting, the user should set the trigger on Channel 0. [CH0 = 0, CH1 = 2]: Same as above, only for the chamber connected to channel 1. For this case, the user should set the trigger on Channel 1. [CH0 = 1, CH1 = 1]: This represents a muon event in which a muon is first detected in one chamber, but then its decay is detected in another chamber, and no other pulses are recorded. It is suggested that for this case, the trigger Channel is set to the Channel which is connected to the chamber which is on top, since more likely than not, we expect the muon to come from the above the surface of the Earth, and not below.
Vertical Range
This control allows the user to define the vertical range of the digitizer channel. This is analogous to setting the vertical range of a channel on a regular oscilloscope. This value should be just large enough to encompass the amplitude of any received pulse.
Low Pass Filter Cutoff
The program passes a received signal through a Low Pass filter, in order to reduce noise. This sets the cutoff for that filter.
Min Sample Rate
This parameter sets the sample rate of the digitizer.

# Procedure

## Inspecting the apparatus and the signal

1. Inspect the apparatus in 275 Le Conte, the SPS room: (Get GSI to let you into the room.)
• Check the voltage setting of the PMTs but do not change (the knobs on the power supply are accurate, not the power supply's meter).
• Notice the color of the co-axial cables going to 275 Le Conte.
• The cables that originate in 275 Le Conte should be connected to the amplifier. Check the gain setting of the amplifier. Start at x5 gain for the top PMT (white cable) and x5 (if the amplified signal is still not big enough, then amplify it by another x5 gain) for the bottom PMT (black cable).
• Using a fast digital storage oscilloscope (Tektronix TDS-360), observe the output of the amplifiers.
• The scope should be set on the fastest sweep, high sensitivity (~0.2 V/div), and high beam-intensity, and triggered internally with a threshold just short of "free run."
• The cable must be terminated at the oscilloscope with a 50Ω (what happens if it is not? Try it).
• You should see pulses at ~100-1000 Hz up to ~1.0 volt in amplitude, followed occasionally (~1 Hz rate) by smaller delayed pulses. Many of these are muon decays, others are the result of various sources of noise.
• Note the polarity, typical magnitude, and width of the pulses

## Determine optimal settings using the LabVIEW program View Signals

Program operation and description National Instruments Digitizer VI

1. Now use the LabView Program View Signals on the desktop.
• If the pulses coming into the digitzer card are negative, Flip Polarity should be enabled.
• Estimate the average height of the primary and delayed pulses.
2. Inspect the amplified pulses with ViewSignals VI. Vary the frequency cutoff and observe how the signal and noise change. Pick the frequency cutoff that optimizes signal/noise ratio. Set the thresholds so that the counting rates (both input and output) make sense. Both should be safely above the noise level.
• For the standard data taking (i.e. when you will be collecting data for measuring the muon lifetime), you will require 2 pulses in the top PMT (Channel 0) and 0 pulses in the bottom PMT (Channel 1). What does this setting mean?
3. Use a pulse generator SRS DG645 to simulate a sequence of two pulses of approximately the same size, width, and separation as the output of the detector in Room 275 Le Conte. SRS DG645 Manual. The manual may also be found in C:/Support/MUO/DG645m.pdf
• With the Program & apparatus counting muons, estimate the Trigger rate and the Output rate that are displayed on the program Front Panel. Do these rates agree with your calculations for the muon incidence and stopping rates (from the Pre-lab)? If the rates do not make sense, you might try adjusting the program thresholds (what does this do?). How might changing the threshold settings affect the accuracy of your data?

## Data Collection

1. Set up the Main program & apparatus and start taking data. Transfer your settings from the View_Signals VI to the main VI, Muon_Detection_Program.
2. Save files properly. Otherwise, the program may crash.
• When you Start Acquisition, the main program will ask you for four (2) unique file names to save data.
• The final file MUO final filtered pulse data.txt should have the path My Documents\LabView Data\. It will be saved when the program finishes or when you Stop Acquisition Early.
• The temporary file MUO temp filtered pulse data.txt should be saved to the local C: hard drive in C:\LabView Local Save\"login name" for speed. If you do not save it locally, the program may crash. If there is a program error, your data should be recoverable from this file.
• Note: These files are needed for the program to function. Keep in mind they are duplicate files, one set saved to the local hard drive and the other set saved to the network drive.

Note: You MUST remember to Start the program first, then push the Start Acquisition button. To STOP remember that you must push Stop Acquisition Early button first before you push the RED program stop button; otherwise your data will not be saved correctly.

## Calibration of Electronics

You will need to run three types of calibration:

1. Time resolution of the digitizer. You will first need to make sure that the time resolution of the digitizer is small compared to the muon lifetime. This can be done by comparing the time recorded by two PMTs for through-going muons, i.e. muons that go through both halves of the scintillator tank. They will produce two signals, one in each PMT. Set Muon Detection Program to record events with exactly 1 pulse in each channel, trigger on Channel 0 (trigger level and pulse thresholds should be set around 0.1 V; check settings with View Signals program to make sure the counting rate is reasonable). Collect sufficient amount of data, and plot the time difference between pulses in Channel 1 and Channel 0 (you may need to apply additional cuts on pulse amplitude if you set the thresholds too low). Fit the distribution to a Gaussian function. Explain why the mean of the distribution is not zero. The standard deviation of the Gaussian is the time resolution. Compare it to the muon lifetime. Do you need to apply any correction to the lifetime data ?
2. Efficiency of the digitizer. Since you will determine the muon lifetime from a histogram of counts vs time, you will need to make sure that each time interval is counted by the apparatus equally. To check this, you need to present the apparatus with a-priori uniform distribution of time differences. This is done by combining signals from two uncorrelated sources: a through-going muon, and an external pulse generator (DG645). As long as the counting rates from the two sources are similar, it's likely that the two trigger pulses will come from different sources.
• First, set up the apparatus according to the procedure: Calibration setup Procedure for Muon Lifetime Calibration.
• Now use a ZSCJ-2-1 Power Splitter to add the Delay Generator signal to the Muon pulse, and run it to Channel 0 of the digitizer. The Power Splitter just takes signals from input 1 and input 2, and outputs their sum on output S. Specifically: plug the AB output of the Delay Generator into input 1 of the power splitter, plug the amplified Muon signal into input 2 (it does not matter if you use the top or bottom chamber), and then plug Channel 0 of the digitizer into output S of the Power Splitter.
• Use the Scope Soft Front Panel to check the signal. Do you see the generator peaks? The Muon signal peaks? The polarity of both should be the same (be sure to remember if the polarity is positive or negative). If you wish to change the polarity of the Delay Generator pulses, press SHIFT and then press either 8 or 9. 8 corresponds to positive polarity, which 9 to negative.
• Run the Muon Detection Program. Remember to select the appropriate polarity. Set the trigger level to be above the amplitude of the Delay Generator peaks (0.6V works well). Set the Pulse Detection Threshold to about 0.2V. Set the Exact Number of Pulses to be 2 on Channel 0 and 0 on Channel 1.
• Check the data rate: the rate of single and double pulses should be roughly equal, and counting at a few Hz.
• Collect the data: a total of more than 100k output pulses would be more than enough.
• Save you data, and repeat the procedure for Channel 1 of the digitizer.
• Data analysis: look at the distribution of time differences dt in each channel. Apply selection on pulse heights if the thresholds are set too low (e.g. the second pulse in each channel should be from DG645, and therefore very constant in amplitude). Select the interval where the counting rate is most uniform. Determine if the data are consistent with constant counting rate, or if you need to make a correction to the real muon decay data.
1. Plot the histogram of counting rate vs dt for each channel. Remove bins that obviously over-count, if any.
2. If there is an obvious slope to the distribution, you may not have set the thresholds correctly, and the VI is triggering on two muon pulses, instead of one muon and one generated pulse. Try adjusting thresholds in the VI, after carefully inspecting the amplitudes of the muon and generator pulses. You may also be able to select the correct pairs of pulses in the RAW file. The generator pulse has a very stable amplitude, typically between 0.1-0.2 V after the digital filter. Select events in the RAW file in which one of the pulses has this specific amplitude, and the other is larger.
3. Once you have a reasonable distribution of counting rate vs dt, convert it to log scale, and fit to a straight line. Is the slope significant ? If it is, how does it affect your muon data ?
3. Calibration of the digitizer clock. Even though the VI reports the time interval dt in seconds, you need to check that the time scale is accurate. To do this, you need to send pairs of pulses from DG645 generator to the digitizer, vary the time interval between the pulses, and reconstruct this time interval with the VI.
• First, reconfigure the internal trigger of the Delay Generator to 12.5kHz (this should be half the value used in the previous part). If the value is kept at 25kHz, a third pulse may be detected (why?).
• Configure the CD output of the Delay Generator to send out a 100ns pulse some time after the AB pulse. Remember that you set the AB pulse to be sent 1μs after the internal trigger, so setting the CD pulse to begin at 3 μs will correspond to a time difference of 2 μs between the pulses. Pick the amplitude and polarity to match that of pulse AB.
• Plug output AB into input 1 of the Power Splitter, and plug output CD into input 2 of the power splitter. Plug output S into the digitizer channel to be tested.
• Run the Soft Scope Front Panel and make sure that the signal is what you expect. Note the time difference between the pulses.
• Run the Muon Detection program to collect data. Be sure to set the trigger at a low enough value for the pulses to be picked up (0.2V works well).
• You should see just one of the histograms bins being filled. You should collect data for enough time differences to be sure that the digitizer reads all time differences correctly, from 1μs to 40μs.
• Repeat this procedure for the other digitizer channel.
4. Linearity of the pulse height measurement. With the setup above, set the time difference between AB and CD to some fixed value (e.g. 5 us), and vary the amplitude of AB and CD. Collect data for each setting, plot the reconstructed pulse heights versus the amplitude set by the DG645 generator. How linear is the digitizer ? Can you also think of a way to check the linearity of the 40 dB amplifier ?

# Analysis

1. Collect data with the default setting (low thresholds, require 2 pulses in Channel 0 and 0 pulses in Channel 1). You may run overnight or over the weekend to collect a large enough dataset, 100k-1M pulses.
2. Inspect the distribution of pulse heights and pulse widths for both the first and second pulses. Can you explain the features ? Apply a cut to remove low-amplitude noise pulses. See how the distribution of time separation dt changes when you cut on pulse height.
3. Correct the dt plot for any non-uniformity measured in Step 2 of the Calibration section
4. Apply time calibration from Step 3 of the Calibration section
5. Make a semi-log plot of counting rate D vs. time separation dt. On this plot, fit the background (from about 10 to 20 μsec) to a straight line (with the slope not necessarily zero). Extrapolate the line under the muon decay region and subtract the background in each channel from D, yielding D2. THINK: should the subtraction be done on a linear or semi-log scale ? How does the choice of the background fit influence your results ?
6. Make a semi-log plot of D2 vs dt. Make sure errors are correctly computed for log(D2). Make a least-squares fit to a straight line (over a time interval where such a fit makes sense). The slope of this line is inversely proportional to the "effective" muon lifetime. Also show lines corresponding to "effective" lifetimes that are one standard deviation higher and lower than the mean lifetime. All lines should be normalized to the same total number of counts. After you make a fit, also check and see how the calculated exponential curve fits the measured curve. Change the limits of the fit by adding or deleting data points and the beginning and end of the data interval; observe how the lifetime values and the χ2 of the fit change. In other words, cut and trim your data to observe the effects on the lifetime measurement. Justify any omission of data when you quote your final answer.
7. Be sure to read Evans Chapter 15 at this point. As indicated in the lab write-up, some negatively charged muons (μ - ) undergo nuclear capture in the carbon nuclei of the mineral oil in the scintillation tank. These muons generate a start pulse when they enter the detector but do not generate a stop pulse when they are captured, or at any later time. Therefore their decay times are not detected. Although the reprints show that the entire population of (μ - 's cease to be muons with the combined decay rate of λcombined = λdecay + λcaptured, it may not be obvious that the free-decaying μ - 's will also decay with this combined rate. This is a subtle point that you should be sure to ask about if you need to.
• Note that $\lambda_{decay}\cong\frac{1}{2.2\mu s}$ for both positive and negative muons, and that $\lambda_{capture}\cong\frac{1}{26\mu s}$ is the rate of capture in the material (carbon) of the detector.
• As discussed above, there are two different populations of muons that you measure, μ - and μ + , and they decay with the different decay rates, $\lambda_{combined}^{\mu^-}$ and $\lambda_{decay}^{\mu^+}$ respectively. The histogram of decay-times that you will get from the equipment (barring background and noise) is the sum of two exponentials: $\mathrm{Histogram}(t)=A \exp(-t\lambda_{combined}^{\mu^-})+B \exp(-t\lambda_{decay}^{\mu^+})$, where A and B are constants. To analyze these data properly requires a non-linear curve fitting program. But because $\lambda_{combined}^{\mu^-}$ and $\lambda_{decay}^{\mu^+}$ are very close to each other, it is difficult to get good results. Instead, it is recommended that you fit your data to a single exponential: $\mathrm{Histogram}(t)=Ae^{-t\lambda_{effective}}$. This gives you an "effective" life-time. The use of the word "effective" is only to signify the answer the computer gave you. It is possible to fit any curve to a single exponential and get an answer; the computer just does the best it can. So when you are asked to estimate the true lifetime, expect to make some approximations and don't expect to come up with an exact answer.
• Assuming that half of the observed decays come from μ - , with apparent decay rate $\lambda_{combined}^{\mu^-}$, estimate the magnitude and sign of the difference between the true muon lifetime and the "effective" lifetime (for μ + and μ - together) measured in this experiment.
8. What is your value of the muon lifetime and its uncertainty, and how does this value compare with the published lifetime? IMPORTANT: estimate systematic uncertainties, i.e. think how the procedure, calibration, etc. may bias your results.
9. What is the value and uncertainty of the weak fine structure constant determined in this experiment? How does this measurement compare with the published value?

## Note on Analysis Tools

This analysis can be done with a variety of tools, but we would recommend one of the more common packages designed for high-volume datasets. Thus, while it's possible to do a rudimentary version of the analysis in Excel, we do not recommend it, and we do not provide pre-packaged histograms of dt for that reason. MATLAB is installed on the Advanced Lab computers, including the convenient Curve Fitting and Statistics add-on packages, as well as additional home-made fitting tools. Comprehensive tutorials for MATLAB are available at Intro to Matlab and elsewhere on the web. If you want to use MATLAB outside the 111 lab, it is available for students at a discount at the University Store. There is also a GPL (free) clone Octave, which is compatible with MATLAB at about 90% level (that is, about 90% of the commands are implemented, but the Curve Fitting package, regrettably, is not available). Octave can be downloaded for Windows, MacOS, and Linux.

The tool of choice for High Energy and Nuclear physics is ROOT. It is a sophisticated data statistical analysis package, specifically designed for analyzing large data sets, curve fitting, etc. While MATLAB is best at linear algebra and signal processing, ROOT is most natural for reading a large data set, applying cuts, and performing non-linear fits. ROOT is available for free download from CERN, and there is good documentation and tutorials on the Root's main site. We also provide a sample (bare bones) analysis examples below: Muon Analysis in ROOT

# Computer simulation of experiment and analysis

Write a program in Matlab (or other software of your choice) to simulate muon decays and accumulate their lifetime distribution. Fit the generated data using whatever means you used above to determine the muon lifetime. Software routines that simulate data are called Monte Carlo programs (see ref. 10 or 11). By fitting this computer-generated data you can directly verify that the fitting procedure reproduces the input parameters (i. e., number of events and lifetime). These Monte Carlo simulations are common in complex experiments to model the experimental system and check the analysis methods.

Computers usually generate random numbers ri uniformly distributed over the interval (0,1), that is, the successive values, called iterates, of ri are described by the probability distribution

$P_u(r)=\left \{ 1: 0. (7.1)

You wish to simulate a series of muons whose lifetimes ti follow an exponential probability distribution

$P_{\mu}(t)=\frac{1}{\tau}e^{-\frac{t}{\tau}}, t\ge 0$, (7.2)

where τ= mean muon lifetime. Prove that

ti = - τlnri (7.3)

will convert iterates from the uniform distribution to the exponential distribution $see ref. 10 or 11$

Write a program to

• generate a number of exponential iterates, ti, and
• sort them into time channels Nk, which roughly correspond to the width and range of the experiment channels. (Increment Nk by 1, if ti falls into the range of channel k).
• and store the data on the disk drive.

Fit these data with your fitting program. Investigate the dependency of the fitting program output on the number of events simulated. How many exponential iterates are needed to get good results from the fit?

A more detailed Monte Carlo analysis of your experiment is possible. For instance, generate the accidental background and study the influence of accidentals, statistics and fit -range on the results of your analysis.

Program operation and description [1]

# References

1. Particle Data Group reference page: http://pdg.lbl.gov
2. National Instruments Waveform Digitizer Manual.
3. RCA Photo Multiplier Tube Manual.
4. B. Rossi, Cosmic Rays, McGraw Hill (1964).
History and background.

Other reprints and reference materials can be found on the Physics 111 Library Site