Lifetime of Muons Generated by Cosmic Rays — Week 1

2.1 Prelab

After completing the prelab, you will be able to:

1. Explain how cosmic-ray muons are produced in the atmosphere and why they reach sea level.
2. Calculate the energy loss and range of muons in scintillator material using Figures Figure 1 and Figure 2.
3. Describe how a plastic scintillator converts particle energy deposition into light, and how a photomultiplier tube amplifies that light into an electrical signal.
4. Explain the role of each module in the NIM electronics chain (discriminator, delay, TAC, MCA) and why each is needed.
5. Predict the approximate rate at which muons will stop and decay in the scintillator.
6. Predict the properties (rate, polarity, duration, amplitude) of PMT output pulses from physical reasoning.

2.2 In-Lab

After completing the in-lab work, you will be able to:

1. Safely turn on and operate a photomultiplier tube following the PMT safety checklist.
2. Characterize PMT output pulses (rate, polarity, duration, amplitude) and compare to your prelab predictions.
3. Set up a discriminator threshold and measure how the trigger rate depends on threshold level.
4. Measure the cable delay that will be subtracted during analysis.
5. Configure the TAC time range appropriate for muon lifetime measurement.
6. Perform a timing calibration using a delay generator and fit the data with Python.
7. Start a long-duration data collection run in MAESTRO.

5.1 Muons and Cosmic Rays

Muons are fundamental particles, similar to electrons but approximately 200 times heavier (\(m_\mu c^2 = 105\) MeV). They are unstable, decaying with a mean lifetime of approximately 2.197 \(\mu\)s in vacuum to an electron and a pair of neutrinos:

Primary cosmic rays (mostly high-energy protons from outside the solar system) collide with nuclei in the upper atmosphere, producing showers of secondary particles including pions, which rapidly decay to muons. Many of these relativistic muons survive to reach sea level.

Key numbers:

• Sea-level muon flux: ~1 per cm\(^2\) per minute (approximately 2× higher in Boulder due to elevation)
• Mean muon energy at surface: ~4 GeV
• Energy spectrum is roughly flat from 0 to 4 GeV

The relativistic energy-momentum relation is:

where particle physicists use units of GeV for energy, GeV/\(c\) for momentum, and GeV/\(c^2\) for mass.

5.2 Energy Loss and Range

As muons travel through material, they lose energy via collisions with atomic electrons (ionization) and elastic scattering from nuclei. The rate of energy loss per unit distance (\(-dE/dx\)) depends on the particle’s momentum and the material properties.

The integral of the energy loss gives the range: the total distance a particle with a given initial momentum will travel before stopping.

5.3 Scintillator and Photomultiplier Tube

Our detector is a large block of plastic scintillator (which you can treat as carbon with density 1.05 g/cm\(^3\)). When a charged particle passes through, it excites electrons in the plastic; these excited atoms emit visible light as they de-excite. The scintillation response is fast—the light pulse rises and decays in a few nanoseconds for plastic scintillator. The amount of light is proportional to the energy deposited.

A photomultiplier tube (PMT) is attached to the scintillator to detect this light. Photons strike a photocathode, ejecting electrons via the photoelectric effect. These electrons are accelerated through a series of dynodes (typically 10–12), where each electron impact liberates several additional electrons. The overall gain is typically \(10^6\)–\(10^7\), depending on the applied voltage. This enormous amplification allows the PMT to detect single scintillation events as measurable current pulses.

Why PMTs are fragile: The dynode surfaces are made of materials optimized for electron emission (e.g., cesium-coated GaAs). When HV is applied and the PMT is exposed to bright light, massive photocurrents flow through the dynode chain, permanently damaging these surfaces. Even brief exposure can degrade the tube.

5.4 NIM Electronics: Signal Processing Chain

The detector system uses Nuclear Instrumentation Module (NIM) standard electronics to process PMT signals. Before you touch any hardware, understand the purpose of each module:

Overview of the measurement:

A cosmic-ray muon enters the scintillator and produces a light pulse (the “start” signal). If the muon stops inside the scintillator, it will eventually decay, and the decay electron produces a second light pulse (the “stop” signal). The time between these two pulses is the quantity we want to measure—it follows an exponential distribution with a time constant equal to the muon lifetime.

The electronics chain (see Figure 3) processes these signals as follows:

1. Discriminator: Converts analog PMT pulses into digital logic pulses whenever the signal crosses a threshold. This rejects noise and produces clean timing signals. The discriminator has two outputs: one goes directly to the TAC “stop” input, and the other is delayed before reaching the TAC “start” input.

2. Delay cable: A long cable (~20 feet) delays the “start” signal. This ensures that the first muon pulse (start) arrives at the TAC after being delayed, while the decay electron pulse (stop) arrives promptly. The cable delay will be subtracted during analysis.

3. Time-to-Amplitude Converter (TAC): Takes a start pulse and a stop pulse and outputs a voltage pulse whose amplitude is proportional to the time difference. If no stop arrives within the TAC’s time range, no output is produced and the TAC resets. The Ortec 437 TAC has selectable time ranges via a knob and multiplier switch.

4. Multi-Channel Analyzer (MCA): Sorts the TAC output pulses by amplitude into a histogram of channels. Each channel corresponds to a narrow range of time differences. The result is a spectrum of decay times.

NIM vs. TTL logic: NIM modules use current-based logic signals (true = −16 mA or −0.8 V into 50 Ω; false = 0 A). TTL uses voltage-based logic (true = +2.2 to +5 V; false = 0 to +0.8 V). Do not mix them. Most modules in this lab use NIM logic—check module manuals if unsure.

5.5 Prelab Question 1

Question 1: Using Figures Figure 1 and Figure 2:

(a) Given that the average cosmic-ray muon at the Earth’s surface has a kinetic energy of 4 GeV, how much energy do you expect an average muon to lose while traveling through 10 cm of plastic scintillator (density 1.05 g/cm\(^3\))?

(b) Based on this, would you expect most muons to stop inside the detector or pass through?

(c) Using Figure 2, below what momentum (in GeV/\(c\)) would a muon stop in 10 cm of scintillator? What kinetic energy does this correspond to? Hints: What R/M value does 10 cm correspond to on the y-axis when you convert the range to g/cm\(^2\)? Recall that \(m_\mu c^2 = 0.105\) GeV.

(d) Given what you know about the energy distribution of muons at the Earth’s surface (roughly uniform from 0–4 GeV), approximately what fraction of muons would stop in the detector? At a rate of ~2 muons/cm\(^2\)/min for our elevation, and given the detector area, estimate the rate of stopping muons. If you want to collect ~5,000 decay events, how long should you take data?

(e) Prediction for Week 2: Will the lifetime you measure in plastic scintillator be longer than, shorter than, or equal to the vacuum value of 2.197 \(\mu\)s? Record your prediction and reasoning—you will revisit this.

5.6 Prelab Question 2: PMT Pulse Predictions

Question 2: Before coming to lab, predict the following properties of the PMT output pulses. Record your reasoning—you will compare to your observations during the PMT turn-on procedure.

(a) How frequently will you see PMT pulses? (Use your stopping rate estimate from Q1, but also consider: will all pulses come from stopping muons, or are there other sources?)

(b) What polarity will the pulses be? (The PMT produces current pulses from electron multiplication. Think about the direction of conventional current flow and how the scope termination converts current to voltage.)

(c) Approximately how long will each pulse last? (Use the scintillator response time from the background section as a starting point. How might the cable length and scope input impedance affect the observed pulse width?)

(d) Estimate the order of magnitude of the pulse amplitude. This is a Fermi estimation—chain together what you can: a muon deposits energy in the scintillator, some fraction becomes scintillation photons, some fraction of those reach the photocathode, each photoelectron is multiplied by the PMT gain (~\(10^6\)), and the resulting charge pulse is spread over the pulse duration into a 1 MΩ load. You are not expected to get a precise answer; the goal is to reason about whether pulses will be millivolts, volts, or somewhere in between.

6.1 Step 1: PMT Turn-On and Pulse Characterization

Follow the PMT Safety Checklist in Section 4 to bring up the high voltage. Take your time—there is no need to rush this step. As part of the safety procedure, you will be monitoring the oscilloscope for signs of light leaks.

Once the PMT is at operating voltage and you have confirmed there are no light leaks, characterize the pulses you see on the scope. Sketch a typical pulse in your notebook, noting the polarity, amplitude, width, and approximate rate.

For each property you predicted in Prelab Question 2 (rate, polarity, duration, amplitude), state whether your prediction was confirmed or contradicted by your observations. For any discrepancies, explain what assumption in your reasoning was responsible.

6.2 Step 2: Discriminator Setup

Connect the PMT output to the Ortec 9302 Amplifier-Discriminator input (located on the back). The module provides an amplified analog output and two discriminator (logic) outputs.

6.3 Question 3: Discriminator Behavior

Before adjusting the discriminator:

• What do you expect will happen to the trigger rate as you raise the discriminator threshold? (Think about the distribution of pulse heights.)
• At what threshold level do you expect to see a significant drop in rate?

Now measure:

(a) Observe both the amplified signal and a discriminator output simultaneously on the scope. Sketch what you see.

(b) Adjust the discriminator threshold (screw adjust on the front panel). Use the visual scaler module to measure the discriminator output rate as a function of threshold setting. Record at least 5 data points.

(c) Set the threshold so the discriminator rate is approximately 50 Hz or less. Record your chosen threshold setting.

Compare: Does the rate-vs-threshold behavior match your prediction?

6.4 Step 3: Cable Delay Measurement

The signal path uses a long cable (~20 feet) to delay the “start” signal relative to the “stop” signal. You need to measure this delay precisely because it will be subtracted from all your timing data.

6.5 Question 4: Cable Delay

Using the two discriminator outputs, measure the time shift produced by the long delay cable.

(a) Connect one discriminator output directly to one scope channel, and route the other through the long cable to the second scope channel. Trigger on the direct signal.

(b) Measure the delay. Is it long enough to ensure that the “start” (delayed muon pulse) arrives after the “stop” (prompt decay electron pulse)?

(c) Record the delay in nanoseconds. This is a small correction you will apply to the decay times in Week 2.

6.6 Step 4: TAC Configuration

Connect the discriminator outputs to the Ortec 437 Time-to-Amplitude Converter: the delayed output to “Start” and the prompt output to “Stop.”

6.7 Question 5: TAC Time Range

Before selecting settings:

• Given that the muon lifetime is approximately 2 \(\mu\)s, what TAC time range should you select to capture several lifetimes of data? (The TAC range is set by a knob and a multiplier switch with options of 0.1, 1, or 10 \(\mu\)s.)
• Why would it be a bad idea to set the range too short? Too long?

Now test:

(a) Use the discriminator outputs (or small negative pulses from a function generator) to provide both start and stop signals to the TAC. Vary the time between start and stop.

(b) Observe the TAC output on the scope. Verify that as the time delay increases, the output pulse amplitude increases proportionally.

(c) Record your chosen TAC settings (range knob position and multiplier).

6.8 Step 5: MCA Setup

Connect the TAC output (use the 1 V full-scale output) to the ORTEC Easy-MCA input. The MCA connects to the computer via USB and is controlled by the MAESTRO-32 software.

In MAESTRO:

• Go to Display → Detector to select the detector.
• Acquire → Clear to reset the spectrum.
• Acquire → Start to begin collecting.

Verify that you see counts appearing in the MCA spectrum when you send test pulses. The horizontal axis represents channel number (proportional to time delay), and the vertical axis represents the number of events in each channel.

7.1 Question 6: Calibration Fit

(a) Record your calibration data: delay time (\(\mu\)s) vs. MCA channel number.

(b) Using Python, fit a line to the calibration data:

import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

# Your calibration data
delay_us = np.array([...])     # delay times in microseconds
channel = np.array([...])      # corresponding MCA channel numbers

# Linear model
def linear(x, slope, intercept):
    return slope * x + intercept

# Fit
popt, pcov = curve_fit(linear, channel, delay_us)
perr = np.sqrt(np.diag(pcov))

print(f"Slope:     {popt[0]:.6f} +/- {perr[0]:.6f} us/channel")
print(f"Intercept: {popt[1]:.4f} +/- {perr[1]:.4f} us")

# Plot
ch_fit = np.linspace(channel.min(), channel.max(), 100)
plt.plot(channel, delay_us, 'o', label='Calibration data')
plt.plot(ch_fit, linear(ch_fit, *popt), 'r-', label='Linear fit')
plt.xlabel('MCA Channel')
plt.ylabel('Time Delay (μs)')
plt.title('Timing Calibration')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

(c) Does the relationship look linear? Are there any systematic deviations? Calculate the residuals and check.

(d) Record your calibration parameters (slope and intercept). You will use these values in Week 2 to convert your muon data from channels to times.