Particle Physics Activities for High School Physics Students

Muon Decay in the MINERvA Experiment

Margaret Menzies – Harbor City International School, Duluth, MN

Jackie Kitchenhoff - Northland High School, Remer, MN

Dr. Richard Gran - University of Minnesota Duluth, Duluth, MN

Students use actual MINERvA detector event data to find and record decay events from a particle called a muon, understand how atomic and subatomic decay processes work by seeing them happen. They calculate the average lifetime and half-life for the muon and also study conservation of energy

Conservation Laws of Collisions

(Billiard Ball Collisions on the Particle Scale)

Paul Conrow – Rochester City School District, Rochester, NY

Carol Hoffman – Hilton Central School District, Hilton, NY

Jeffrey Paradis – Rush-Henrietta Central School District, Henrietta, NY

Paul Sedita - Canandaigua City Schools, Canandaigua, NY

Dr. Kevin McFarland – University of Rochester, Rochester, NY

Students use actual MINERvA detector event data to find and record particle collision events between a neutrino and neutron (within the nucleus), understand how conservation of momentum and energy on the particle scale. Calculations made will help determine the original momentum and kinetic energy of the incoming neutrino.

Editing, Technical Work

Dr. Nathaniel Tagg– Otterbein University, Westerville, OH

Rev. July 2015

Table of Contents

You will find all of the materials listed below (and additional PowerPoint presentations) on the web page: http://neutrino-classroom.org

Particle Physics Activities for High School Physics Students

Table of Contents

Introduction

Why do you want MINERvA in your curriculum?

The essential questions you can explore with this material

Technical details on the Arachne program

Suggested Activity Sequence

Particle Pre-Assessment for Students

Activity: Radioactive Decay of M&M’s

Directions for Simple Arachne

Activity: Modeling Radioactive Decay with Dice

Activity: Finding Muon Decays with MINERvA

Identifying Decay (Michel) Electrons with Arachne

Activity: Measuring the Muon Lifetime

Activity: Energy of Decay Electrons (Understanding why the electrons in muon decay have this distribution of energy)

Quiz: ‘What is MINERvA’?

Activity: Build a Model of MINERvA

Activity: Conservation of Momentum when Neutrinos Interact

Activity: Neutrino Scattering and the Target Nucleus

Standard Model Poster Scavenger Hunt

Arachne  Scavenger Hunt

Additional background information

Merging Excel Spreadsheets from Student Groups

Teacher background Information for Introducing Histograms to Students

Glossary of Terms

Introduction

Why do you want MINERvA in your curriculum?

Explore cutting edge science in the fascinating world of particle physics.

Students use real particle physics data to explore two standard physics topics

1.      Radioactive decay: the random process and lifetime of a particle called a muon.

2.      Measuring conservation of momentum and energy when particles collide.

Insight into how practicing scientists creatively solve problems and explore the unseen world of fundamental particles

You don't have to be Einstein to get involved with particle physics.

Choose as much or as little as you like to add depth to your physics curriculum, or offer parts as an optional activity for students or an after school science club.

The essential questions you can explore with this material

1. What do your students already know about particle physics? How is the MINERvA project being used to detect particles that are difficult to observe? What are the signatures of decay and momentum events in the detector?
2. What is radioactive decay, and what is characteristic about the decay of the muon particle? How do I see that radioactive decay is a random process?
3. When subatomic particles collide, their identities often change.  Is momentum still conserved?
4. How do I visually represent particle physics data, in order to better draw conclusions? Why are histograms used to analyze data?
5. What is the mean lifetime (and half-life) for decay? What is the distribution of random decay times?
6. How can I use conservation of momentum of the final products of a neutrino collision to learn about the neutrino and the neutron that collided? Was the neutron moving inside the nucleus?
7. How do scientists deal with different interpretations of the same data? What else can be observed in MINERvA?
8. How does special relativity govern the relationships between speed, momentum and energy? What is the connection between E=mc2 and the energy spectrum of the electrons from muon decay?

Technical details on the Arachne program

Arachne is named after a character from mythology who lost a weaving contest to MINERvA. The Arachne program was created in order to visualize particles that were recorded by the MINERvA detector. It pulls calibrated data from a server at Fermilab and then draws a few pictures and graphs to summarize what happened in the detector.

Arachne requires a recent version of a web browser, such as Firefox version 4.0, Safari version 4.0, and Chrome version 5.0. Arachne relies on Java and parts of the HTML5 standard to more dynamically render the images and make them interactive. Because it does not yet support the latter, the Internet Explorer browser will not run Arachne.

The day before your class, be sure to check that the server is responding. It occasionally goes down.

Suggested Activity Sequence

There are a number of interesting things so see in the MINERvA data, but we have chosen two different threads that illustrate and reinforce ideas that are already part of your physics and chemistry curriculum.

Muon Decay is an example of radioactive decay. In chemistry or physics class, the concept seems abstract, something that happens unseen inside lumps of material like uranium or aging trees. These activities let you observe real decays and measure their decay time one-by-one, in the comfort of your own classroom.

In the MINERvA detector particles with charge produce visible tracks that can be followed, which is what this activity is all about, using real-event data showing muon decay to Michel electrons. They are named for Louis Michel (1923-1999), the French physicist who provided the first documentation of the decay sequence of muons to an electron and two neutrinos: A development not so different from Marie Curie's pioneering initial work on radioactive decay. MINERvA is capable of detecting an array of events. We will be focused on muon decay events in these activities, but students are likely to observe other events if they do any exploring of data on their own. (Arachne Scavenger Hunt) Further background on the types of events detected in MINERvA is outlined in a PowerPoint presentation on Types of Events.

Billiard Ball Collisions involving conservation of momentum and energy, is a familiar introductory physics topic. The same conservation principles and visualization applies to particle interactions, too. The simplest examples even look like the collision of two billiard balls, but also reveal something of the inner workings of the atomic nucleus.

Below, we suggest a sequence of activities for each topic. Depending on your class, schedule, or other considerations, you can shorten the activities, or in some cases assign pieces as homework instead of doing them in class.

Muon Decay Activities

Introductory Activities - (I)         Data Analysis- (DA)     Enrichment Activities - (E)

Prior Knowledge Activities:

(I) Particle Pre-Assessment for Students

(I) Particle Adventure1

(I) Standard Model Poster scavenger hunt

(DA) Histogram Activities3

Classroom Activities:

(I) Activity: Modeling radioactive decay with dice (60 minutes)

(I) Activity: Radioactive Decay of M&M’s      

(I) What is MINERvA? – PowerPoint (20 minutes)

(I) Directions for Simple Arachne (Arachne basics for muon decay) - Tutorial (20 minutes)

(I) Activity: finding muon decays with MINERvA- teacher guided (30 minutes)

(I) History of the Neutrino “Norton Nabs a Nu” – PowerPoint (30 minutes)

(E) Types of Events – PowerPoint (30 minutes)

(DA) Activity: Measuring the muon lifetime (90 minutes), data analysis, calculations and    

                         graphing data (180 minutes),

Optional/Extension Activities:

(E) Arachne Scavenger Hunt

(DA) Neutrino Birth and Death PowerPoint

(I) Activity: Build a model of MINERvA

(E) Activity: Energy of Decay electrons (Understanding why the electrons in muon decay have this distribution of energy)    

(I) Particle Events in 60 seconds

(I) See what the MINERvA detector is doing right now

(E) Neutrinos in 60 seconds

(E) Kevin McFarland’s neutrino talk for students

Billiard Ball Collisions (Conservation Laws) on the Particle Scale

Introductory Activities - (I)         Data Analysis- (DA)     Enrichment Activities - (E)

Prior Knowledge Activities:

(I) Particle Pre-Assessment for Students

(I) Particle Adventure1

(I) Standard Model Poster scavenger hunt

(DA) Histogram Activities2 

Classroom Activities:

(I) History of particle physics and MINERvA – PowerPoint (30 minutes)

(I) What is MINERvA? – PowerPoint (30 minutes)

(I) Directions to Simple Arachne – Tutorial (30 minutes)

(I) Bug and Truck Collision – PowerPoint (30 minutes)

(I) Activity: Conservation of momentum when neutrinos interact– teacher guided (60 minutes)

(I) Ten collision sampler (30 minutes)

(I) MINERvA Momentum Model – PowerPoint (30 minutes)

(I) History of the Neutrino “Norton Nabs a Nu” – PowerPoint (30 minutes)

(DA) Activity: Neutrino scattering and the target nucleus (60 minutes), Continue data analysis

Optional/Extension Activities:

(E) Arachne Scavenger Hunt

(DA) Neutrino Birth and Death PowerPoint

(I) Activity: Build a model of MINERvA

(I) See what the MINERvA detector is doing right now

(E) Neutrinos in 60 seconds

(E) Kevin McFarland’s neutrino talk for students

1 - Particle Adventure is an excellent online resource that provides a fairly quick introduction to particle physics and fundamental particles. It was produced and is maintained by the Particle Group of the Lawrence Berkeley National Laboratory. Here is the website: http://www.particleadventure.org/index.html. Once at the website, choose “The Standard Model” option. This is divided into four major parts. For a good unit introduction we suggest using “What is fundamental?”, and “What is the world made of?” “What holds it together?” follows these first two and should be considered optional as it provides background on fundamental forces and their interactions which is not critical to this unit's focus. “Particle decays and annihilations” follows this section and should be used with students to provide some overview of decay

 2 - Histogram activities are provided if you think your students could use further background in the purposes, construction and appropriate uses of histograms for scientific data.

Particle Pre-Assessment for Students

Goal

A quick assessment for teachers to use to gauge the content that students already have in the topic of modern physics.

Notes for the Teacher

Each student enters physics with a different set of knowledge.  This questionnaire is meant for the students to take so that the teacher can determine what Prior Knowledge Activities might be appropriate so that the students feel more comfortable heading into the unit.

Student Instructions and Worksheet

Particle Physics Pre-Assessment

Directions: Answer each of the following questions with honesty and thorough thought.  The answers that you give are going to help your teacher structure your particle physics unit.

1. How comfortable are you with the main principles and concepts of particle physics?

2. What, if any, value do you see in learning particle physics in high school? Explain why.

3. Where would you get information about particle physics?

4. What scenarios would you use to illustrate momentum conservation in physics class?

5. What is the connection between momentum and particle physics?

6. What are the fundamental particles of matter?

7. What do you know about quarks and/or neutrinos?

8. What is antimatter?

9. In addition to electric charge, what is conserved in particle reactions?

10. What is E = mc2 all about?

11. What does LHC stand for & why has it been in the news in the past year?

12. What are the basic principles of physics that explain the operation of particle accelerators (feel free to include diagrams)?

13. What is a histogram (include a chart with axis titles)?

14. What are histograms used for?

15. Why are histograms important to particle physics analysis?

Activity: Radioactive Decay of M&M’s

Goal

This simulation provides a simple example of the rate at which a radioactive isotope decays.

Materials

    M&M™ candy pieces

    resealable bag

    graph paper

Teacher's Notes

    Some naturally occurring isotopes of elements are not stable. They slowly decompose by discarding part of the nucleus. The isotope is said to be radioactive. This nuclear decomposition is called nuclear decay. The length of time required for half of the isotope to decay is the substance's half-life. Each radioactive isotope has its own particular half-life. However, when the amount of remaining isotope is plotted against time, the resulting curve for every radioisotope has the same general shape.

    Hint: Make sure you use candies with printing on one side (plain M&Ms™).

Answers to Extensions

    Half-life is the length of time required for one half of an isotope to decay.

    The half-life of M&Ms™ in this activity was 10 seconds.

    At the end of two half-lives, 1/4 of the original sample remained and 3/4 of the sample had decayed into a new element.

    The graph is a decreasing logarithmic curve.

    The shape of the graphs will be almost the same.

    The shape of the graphs will be almost the same.

Student Instructions and activity sheet 

Procedure

    Place 50 atoms (M&Ms™) in the bag.

    Seal the bag and gently shake for 10 seconds.

    Gently pour out candy.

    Count the number of pieces with the print side up—and record the data. These atoms have "decayed".

    Return only the pieces with the print side down to the bag. Reseal the bag.

    Consume the "decayed atoms”.

    Gently shake the sealed bag for 10 seconds.

    Continue shaking, counting, and consuming until all the atoms have decayed.

    Graph the number of undecayed atoms vs. time.

Data and Observations

Questions

    What is a half-life?

    In the experiment, what was the half-life of the M&Ms™?

    At the end of two half-lives, what fraction of the atoms had not decayed?

    Describe the shape of the curve drawn in step 9.

    Repeat the experiment three more times, starting with 30 atoms, 80 atoms, and 100 atoms of ‘candium’. Compare the resulting graphs.

    Repeat the experiment using half-lives of 5 seconds, 20 seconds, and 1 minute. Compare the resulting graphs.

Directions for Simple Arachne

(Arachne basics for muon decay)

Goal:

The information below should be reviewed by both the teacher and the students so that everyone has a basic knowledge of how the Arachne site works and relays data

http://minerva05.fnal.gov/Arachne/simple.html

DATA:

• (example) MN 1020/10 v7r5p1 – represents the run and Sub-Run number of the event
• Entry – This is 16000 ns of time in which a ‘snapshot picture of the event was captured
• Prev/Next Gate – this changes the event viewed through the detector during a different 16000 ns of time and corresponds to the Entry #
• Prev/Next Slice – this gives you a specific time of the event within the given 16000 ns gate

[Analogy:  the Run is like a roll of film…the Entry (Gate) is each individual picture on the film… the Slice contains the details within each picture]

TIME HISTOGRAM:

• Gives you the indication of time (in ns) where informative events may have occurred.  A grey highlighted cursor moves across the histogram as you change Slices.  
• The y-axis is measured Energy in MeV or in Photoelectrons (PE)

   PH HISTOGRAM:

• PH stands for Pulse Height and is indicative to the energy (y-axis) deposited in the detector as the particles move through
• The energy levels can be displayed in different units (see Options) such as PE and MeV
• A value of 100 or greater indicates a useful event

BIG XZ VIEW:

• This area gives you the ‘picture of the event’ in which we will look for the presence of a muon, proton or electron moving through the modules of the detector
• The x-axis is the numbered modules that the particle is being detected in.
• The y-axis is scintillator strip number found within the modules.  The distance between the centers of adjacent  triangles is 1.6 cm
• The magnifier can be held over the tracks to get a clearer view of the energy changes detected. Blue signifies the highest change in energy.
• ECAL and HCAL windows: Electronic Calorimeter and Hadronic Calorimeter are used as absorbers to stop particles by putting more material (lead and steel, respectively) in the particle’s way.

MAGNIFIER WINDOW:

MAGNIFIER WINDOW:

• When the cursor is held over the green path line, the path  line will become red.  Clicking the left mouse key will produce the window shown.

• This drop down menu allows you to see details about the particle.  
• Track # - each green path is assigned a number
• Time – time in ns of the event
• Minos – the momentum of the particle (most likely a muon) in the ‘front face’ of the Minos detector at Fermi lab
• If Muon /If Proton – momentum and KE data of the proton and muon are used to determine the momentum and KE of the original neutrino and neutron prior to the collision.  The proton’s path will be shorter, with more energy deposited in a short distance, and the muon’s path will reach beyond the detector, since it loses little energy as it travels.  (See 3D display).  This information is helpful when determining if the track is that of a proton or muon so that the correct data is recorded. By clicking on the Copy button, the needed information will be copied into an excel document for analysis

HIT MAPS:

• Detector views the event, as seen in the Big XZ view, but from different angles

SLICE INFO:

• Mean Time - (Average) time for the event in that slice.  Comparison of the mean time (of a muon path) in one slice to the mean time of the presence of the electron in a different slice is indicative to how long it takes the muon to decay
• No. ID hits – # of hits inside the detector
• No. OD hits - # of hits outside the detector
• Visible Energy – used to determine conservation of momentum and energy during muon decay. By measuring the visible energy during different slices, the amount of energy transferred from the muon to the electron can be determined.

3D DISPLAY:

• This area shows a moving 3-

         dimensional view of the

         particles’ paths through the

         detector

• The 2 hexagonal structures represent the front and back of the MINERvA (Main Injector Neutrino ExpeRiment v-A) detector. The particles’ paths are depicted by the lines starting with the dot (toward the right side of the diagram) and proceeding to the end of the detector (toward the left side.
• The structure to the far left is the front face of the MINOS (Main Injector Neutrino Oscillation
  Search) detector.  The MINOS detector is also involved with neutrino detection and is part of an experiment that spans from Fermi lab to Tower Soudan Underground Physics Laboratory in Soudan, Minnesota
• If a particle extends beyond the end of the MINERvA detectors and enters in MINOS, than it is most likely a muon particle.

Activity: Modeling Radioactive Decay with Dice

This is the first activity in learning about radioactive decay

Goals

1.      Start with a physical, tactile simulation of radioactive decay.

2.      See the exponential and random nature of a decay process directly.

3.      Play dice with the Universe.

Radioactive decay, as it is often taught, is abstract. We talk about piles of radioactive material, a “decaying exponential” mathematical function, and how it is dangerous, but these things are happening at an atomic level we can't actually see or experience with our hands or eyes directly. The two variations here let us do that, making clear both the intrinsically random, but at the same time the intrinsically exponential nature of the process. Then in muon decay activity two, with data from MINERvA, we will take an actual decay process which is also usually unseen, and make it visible too.

Prior Knowledge:

Examples of radioactive decay from the news or elsewhere, such as carbon-14, the M&Ms™ lab,  medical tracer isotopes, or the storage of nuclear waste.

Materials:

Dice

Paper to record the data, graph paper to make a histogram

Calculator

Notes for the teacher

What is radioactive decay, and what other things in nature share the same characteristics?  The process of radioactive decay, of atomic isotopes or fundamental particles, is intrinsic to the universe and to particle physics. The mathematically characteristic exponential decay (and the related exponential growth) is found in lots of places in nature; anywhere the rate of change of something is proportional to the amount of that something. Radioactive isotopes, bacteria populations, investments in a bank account. The other characteristic of radioactive decay is its inherent randomness, which is pretty astounding by itself, and is a core feature of quantum mechanical systems.  The random nature also leads to some interesting issues in observing and measuring the decay, problems shared with other inherently random things such as political polling and the ubiquitous ±3% error and separately with rolling dice. These activities use the dice process to give you a tactile, physical experience with what is really happening in these other processes that are nanoscopic or abstract.

There are two related pieces here. Rolling one hundred dice several times, and rolling a single die multiple times, then repeating that a hundred times. Here, we've assumed you are able to do both, possibly saving the second one as an out-of-lab homework activity. It is reasonable that, due to time or material constraints, you choose to do only the second one or only the first. If you picked this up with the MINERvA muon decay activities, the second one is the most like what the students will experience looking at muon decay.

The two ways are first summarized here, together.  The student instructions break them into two separate, with specific instructions and questions for the group doing the activities.

First way: roll one hundred dice (multiple times) to represent a sample of a hundred radioactive isotopes decaying over time. This does a good job of illustrating the idea of "half-life" and the exponential decrease in the activity of a radioactive sample over time. The question you are asking is “how many remain after some time”. Such samples might be medical isotopes, waste from your local nuclear reactor, or the uranium and thorium atoms that remain on earth long after they were formed ten billion years ago in supernovae.

Second way: roll a single die (multiple times) to represent how long a single muon lives, then do it again for the next muon, and the next, one hundred times. This is a good way to illustrate the range of lifetimes you will observe if you have a sample of unstable particles such as muons. The question you are quantifying is “how long (what range of times) does each particle live”. Such samples are common in particle and nuclear physics.

In fact, these two illustrations are just two ways of looking at the same phenomena. After doing one or both activities, it should be clear that you are observing a random process, observing many at once, or one at a time doesn't matter, and they both lead to the exponential decay that is characteristic of the random process. This will also help overcome the problem that you can't “see” radioactive isotopes or muons; it makes these real but hidden details plain. As a bonus, you will also observe how the randomness also means you will get somewhat different results when you repeat a measurement with otherwise identical initial conditions, such as you might have noticed applies to political polling, and maybe baseball, among other everyday phenomena.

Doing the second variation only: If you are not able to obtain enough dice, you can do the second activity only.  And anyway, the second variation better matches the muon decay activities involving the MINERvA data.  When you prepare the instructions for students, trim out the references to the first variation.

Doing the second variation as homework:  If you think it is likely that your students have at least one board game at home with dice, you can assign the second activity (which takes more time) as homework, and have each student bring their results in for discussion and be ready for the muon data.

Using spreadsheets or other software:  This exercise is designed to be done with paper and pencil and calculator, but for students who are prepared, interested, and want to, you can use computer software such as Excel or more advanced software like Matlab, to explore things more quantitatively.  We recommend you press on quicker to get to the muon decays in the MINERvA data, but you might know just the student who wants something more advanced.

Instructions for students

What is radioactive decay, and what other things in nature share the same characteristics?  The process of radioactive decay, of atomic isotopes or fundamental particles, is intrinsic to the universe and to particle physics. The mathematically characteristic exponential decay (and the related exponential growth) is found in lots of places in nature; anywhere the rate of change of something is proportional to the amount of that something: radioactive isotopes, bacteria populations, investments in a bank account. The other characteristic of radioactive decay is its inherent randomness, which is pretty astounding by itself, and is a core feature of quantum mechanical systems.  The random nature also leads to some interesting issues in observing and measuring the decay, problems shared with other inherently random things such as political polling and its ubiquitous ±3% error and separately with rolling dice. These activities use the dice process to give you a tactile, physical experience with what is really happening in these other processes that are nanoscopic or abstract.

First way: one hundred at once.

You need: a large supply of dice (about one hundred), a cup or bucket large enough for them, a sheet of paper to record data, and some graph paper for graphing your results.

Imagine: you have a supply of some radioactive isotope that has a 1/6 chance of decaying in the next minute. How much of that isotope remains after six minutes? How about after 20 minutes? How much time does it take for approximately half the sample to decay?

Put all the dice in your cup or jar and roll them on the table.

Separate all the dice that turned up “1”, these are the particles that decayed.

Count and record those that decayed and those that remain, measured after “one minute” passed.

Separate the decayed “ones” into a pile that you can see later, while you roll the non-ones again.

Consider: is the pile of decayed dice about as many as you expect? Was it exactly as many as you expected?

Put the others (the ones showing 2-6) back into the cup.

Repeat all these steps and record a measurement of what decayed during and what remains after two minutes. Separate this next batch of dice that turned up “1” into its own pile next to the first – you will save the progression of decays so you can see the process visually, in addition to the numbers you have recorded.

Repeat again for the third minute.

Repeat again and again until they are all decayed.

You now have a sequence of piles you can see (and numbers recorded that you can graph) that represent how many decays happened during each minute. Describe you can see both these seemingly contradictory properties: a) the number of decays is proportional to the number available to decay and b) the number that decayed is random.

Let’s represent the situation with a couple graphs. Graph the number that decayed in each minute, using a histogram or bar chart. Separately, graph the number that remain undecayed in the sample, using a histogram or bar chart.

Estimate from your graph, by counting, how much time it takes for approximately half the sample to decay. Draw a mark or an arrow on the horizontal axis of each graph indicating where this time is.

Look up the exponential decay function; if you have a graphing calculator or similar program, plot it with a constant of (0.16666666 = 1/6), in other words, plot with its negative sign: e-(1/6)x .  If you have a regular calculator, compute ten or so values and make another graph of it on your paper.   Does that function describe the data you graphed? If you have access to the right software, you might even be able to plot your data and ask the software to fit an exponential, but sketching this one by hand would be good enough. Does the inherent randomness of this process make it difficult to see the exponential nature, and if so, can you think of a change in your procedure

How do you feel about the idea that there was likely a die that remained and didn't decay for fifteen to twenty rolls? Is it possible that one could remain for a hundred or more rolls?

Second way: simulating one hundred separate muons

You need: one (or a few) dice, a bit of time, a few pieces of paper for recording your data, and a couple more for graphing.

Imagine: you are creating, and then observing an unstable particle like a muon or a neutron, and can measure how long each one of them lives, one by one. Each one has a 1/6 chance of decaying in the next one nanosecond, you will count the number of nanoseconds until each one decays. Do you expect short lifetimes to be more common, or long lifetimes, or something in the middle? What is the longest you expect to see a particle live? Can you guess what the average (mean) lifetime will be, when you are done and have analyzed all the decays?

Take your die and count the number of rolls until you roll a “1”, and record it on your piece of paper.

Repeat about 100 times, each time recording the number of rolls until you roll the number “1”. You have some quality time during this activity to consider the questions mentioned in the “imagine” paragraph above, and what you learned from activity one. (You could divide the effort among three or four people and get it done in 1/3 or 1/4 the amount of time, if you are in a hurry.)

Let’s graph the results. Look at the data, and count the number of times where the muon lasted up to one nanosecond. Record that on a graph in the 0-1 interval. Count the number of times it took two rolls, and record that in the 1-2 interval. You are making a histogram of the decay time. Continue until you have filled all the intervals; toward the end many of them will be filled with zeros, with occasional intervals where a few long-lived particles fall. 

Now compute the average life, in nanoseconds. Clearly you can average all 100 numbers you recorded by adding them and dividing by 100. You might see that you can do this more quickly from your graph than you can from the sheet with 100 numbers on it, but either way works. Draw a mark on the horizontal axis of your graph to represent the average lifetime.

Look up the exponential decay function; if you have a graphing calculator or similar program, plot it with a constant of (0.16666666 = 1/6), in other words, plot with the negative sign: e-(1/6)x . Does that function describe the data you graphed?

Does the inherent randomness of this process make it difficult to see the exponential nature, and where is it the most difficult? Why? 

How do you feel about the idea that there was likely a muon that remained and didn't decay for fifteen to twenty rolls? Is it possible that one could remain for a hundred or more rolls?

Now what are your answers to these questions: do you expect short lifetimes to be more common, or long lifetimes, or something in the middle? What is the longest you expect to see a particle live?

If you have done both exercises, your one graph from the second exercise is most like which graph from the first exercise? And it should look approximately like the other graph from the first exercise, but 6 times smaller. In these cases, they should look similar enough that you can recognize the similarity, but they certainly won't be identical.

How would you explain why they should look so similar but not identical?

What would happen if you repeated one or the other exercise again, would the results be identical? For both exercises, you had one die that lasted the longest, but might you ever expect to see a particle last ten-times that long? What would happen if you had the luxury of doing 1,000 trials, or 1,000,000 trials, or an Avogadro’s number (6.02 x 1023) number of trials?

The function that best describes these graphs is a decaying exponential Ne-t/τ, (sometimes also written Ne-λt ) where our situation is such that t is time, N is either the initial number of particles that might decay, or the initial activity which is 1/6 of the number of particles, and τ (tau) is the mean lifetime which is 6 seconds or nanoseconds depending on which activity you are doing or λ which is the probability that a particle will decay in the next bit of time. This function occurs a lot for interesting random and non-random situations, but random processes often give rise to behavior that can be described mathematically like this.

Notice, in the second activity you directly measured the value of this parameter τ tau = mean lifetime. If you recognized how the second activity is the same as the first, then you have again measured it. Go back to your first data set and calculate the mean lifetime there. Is it the same? Should it be? What is the accuracy of your measurement and the role of random fluctuations in your measurement? If you had a need to make a more accurate measurement, how would you do it?

Advanced questions:

You might notice that the spot where you marked “half-life” in the first activity is not at all in the same location as the spot you marked “mean-life” in the second activity. These are different. Think carefully about how you knew where to mark those spots and try to describe the subtle difference between them.

Political polling has been mentioned as another random process that we suppose you have encountered. Though it doesn't obviously give rise to an exponential distribution, but it clearly does give rise to fluctuations between repeated measurements of apparently the same thing, such as the public's support one or the other political candidate. If you don't expect two polling firms to get the same answer if they poll on the same night, how can you still draw conclusions from their results?

I teased that the same issues involving random processes might also apply to the game baseball. Think on that.

If you are comfortable enough with your calculus, you can derive the connection between the probability to decay and the exponential function. An internet search or textbook will get you started.

If you are comfortable enough with programming, and can find a pseudo-random number generator to use, you can explore further and faster than you can with dice. Using the same 1/6 probability, code the procedure into a program. Use a programming platform like C++, Java, Matlab, Mathematica, or possibly even a spreadsheet. When it's ready, play with the mean lifetime and compare the exponential to high statistics histograms. Then play with changing the time step or probability.

Activity: Finding Muon Decays with MINERvA

This is the suggested second activity for learning about radioactive decay.  It is suggested that this be a teacher guided activity.

Goals

         To observe and analyze specific examples of a particular radioactive decay process: muon decay.

         Practice taking data for the main activity to quantify the nature of decay times.

         Reinforce the randomness inherent in the decay process.

Prior knowledge

         Know a few fundamental particles: muons, electrons, neutrinos.

         Know the basics of radioactive decay (from the dice activity or elsewhere)

         Have an understanding of the MINERvA experiment and the structure of the detector.

Materials

         Website access to Arachne (Firefox, Chrome, Safari, not Internet Explorer see note)

         Practice run data sheet handout for ten events.

         Follow-up questions handout.

Notes for the teacher

The MINERvA detector and what it records. Depending on the order in which you have chosen to do the material in this packet, this is the first use of MINERvA data for you. In brief, MINERvA is a detector located in a neutrino beamline at Fermilab, near Chicago. As is typical of a particle detector, MINERvA only records the passage of charged particles; neutral particles (neutrinos, neutrons, and photons) are invisible until they interact with an atomic nucleus in the detector and transform into or produce new charged particles. The main view of these interactions is the XZ view, which is the top view of the activity in the detector, and you can see things travel from front to back, left to right. Many charged particles, especially muons, look like simple and long “tracks” of activity left behind as the particle passed through. Some particles, especially electrons, look like little splashes of activity that are well localized and may be made of only a few hits. We have another activity [link to it] with material that gives a broader background to the detector and what it was designed for.

Muon decay: If you did the dice activity, or other activity, you are familiar with the basics of radioactive decay. A particle or nucleus spontaneously (and randomly) changes into one or more new particles (or a new nucleus and some particles). The resulting state is energetically favorable, so there is always additional energy available to become kinetic energy. Specifically, the muon decays similar to the classic “beta decay” (similar to carbon-14 and also like neutron decay) giving an electron and two neutrinos as products. If you are familiar with the standard model of particles, muons are a heavy cousin to the electron; the technical term for this family of particles is leptons. So the muon decays into its less massive lepton cousin. The neutrinos are neutral, so they are usually unseen as they escape the point of the decay, so the signature is a muon stopping in the detector then the electron from the decay appearing. We can write this reaction in a form similar to how we right chemical reactions

μ-       →          e-           +         anti-ve           +         vμ

muon decays to electron,  anti-electron neutrino and muon neutrino

Particle interactions and conservation laws: This is a good place to look at rules for particle interactions. Notice first that the charge stays the same on both sides of the decay, so charge is conserved. Both the muon and the electron are in the lepton family, but particle physicists say they have different “flavor”, muon flavor and electron flavor. Notice then that there is still one of each flavor on the right side after the decay has happened. In fact, we have carefully written that there is an electron and an electron anti-neutrino, a particle and anti-particle pair, so the left side has no electron-ness at all while the right side the electron-ness cancels out or sums to zero, like adding a +1 and a -1 together. This chemical reaction style statement doesn't say so, but of course momentum and energy were conserved. The muons you will see came to rest with no kinetic energy and no momentum. For momentum to be conserved, neutrinos must have come out in roughly the opposite direction to the electron (but we won't see them). For energy to be conserved, the rest energy (mass) of the muon, 107 MeV, was converted according to E=mc2 into the kinetic energy of the three products, some of which was given to the electron which we measured; half or less goes to the electron.

Measuring Muon Decay- Initial Ten Practice Run Data Sheet

Other tips on this activity or questions that may come up:

Where do the muons come from, if this data is from a neutrino beam? Most of these muons were one of the neutrinos, before it interacted in the rock in front of the detector. After it changed from a muon neutrino into a muon it travelled out of the rock, through the air a bit, and into the front of the MINERvA. You can see other neutrinos interacting in the MINERvA detector itself (not in the rock) too, as you look through the data for muons and their decay.

Answers to the follow-up questions

1.      As you flip through the time-slices of Muon Decay Event One, how many have activity that is track-like? How many are small splashes of energy?

Two slices have tracks, four slices have splashes, one slice has both (though there may be different interpretations of whether these shorter ones are tracks or splashes) and two slices that have nothing.

2.      There were actually two time-slices in Event One, each with a small splash, that occurred at the end of the muon track. How did you know which one was the decay electron and which one to reject as some other unrelated thing?

The splash in slice two had energy that was too high, 152.1 MeV, which is well over the expected range of 60 MeV or less.  It is just a coincidence that something else happened in the right location.

3.      Considering all eleven links, which one has the most time between the decay electron and its parent particle? Which one has the least time?

Event five = MV/2397/16/921 has the highest at 6877 ns.  

Event seven = MV/2397/16/1453 has the lowest with 1075 ns.

4.      In addition to all the muons that stopped in the middle and decayed, what was the most interesting other thing you saw?

A variety of answers are possible, though there is a neutrino interaction in the same link as seven in MV/2397/16/1453 that we think stands out as interesting.

5.      The muon you find is always traveling left to right, then stopping somewhere in the middle. How about the electrons?

Since the muon is at rest when the decay happens, the electron can go in any direction it chooses (it chooses randomly), so it may or may not move in the same direction as the muon travelled.  In these ten decays you see all the variations.  In contrast, if the muon was moving (its not), then to conserve momentum, the decay electron would probably be moving in roughly the same direction.

6.      Describe in your own words, what is radioactive decay, and specifically how does a muon decay.

Radioactive decay happens when an unstable particle or a whole atom tries to settle into a more stable state.  It happens randomly, with both long and short times.  (In the next activity, we will measure the characteristic decay time, despite the randomness.)  A muon specifically decays to form a Michel electron and two neutrinos, of which only the electron is visible in the MINERvA data.  For sure I notice that charge is conserved. There is a negative charged thing on both sides.

[If you have covered all the conservation laws, then the answer might continue as follows.  Rules of particle decay require charge to be conserved (-1 on both sides).  Lepton family/flavor must be conserved:  there is muon-type flavor on both sides, there is no electron flavor in the initial state, but there is both electron and anti-electron (neutrino) flavor in the final state, which is like adding +1 and -1 for a net zero. ]

Identifying Decay (Michel) Electrons with Arachne

To access the Arachne program, start by clicking on the following web link.

Muon Decay Event One: http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=306&slice=-1&phCutLow=0.5

The display shows all the activity in the MINERvA detector for a single, short pulse of neutrinos.  Of the few buttons on the page, you are most concerned with the “Prev slice” and “Next slice” which steps you forward and backward in time within the 16,000 nanosecond long burst of data.  “All Hits” shows all the activity in this burst of particles, regardless of time.

Click the ‘next slice’ button to go through each slice until you find the two special slices. First, one slice has a nice long particle that came in the front (left edge) of the detector and stopped somewhere in the middle. Second, another, later slice has a small burst of energy, maybe only a few hits, that starts right where the other long particle track ended – with no more than a one segment gap. Use magnifying glass effect to help you confirm your choice. It is also helpful to point your finger(s) at the end(s) of one or more tracks to mark its specific location, to see that it matches. Also take a look at the 3D display on the bottom (after re-clicking the all-hits button), you can see a visualization of the green reconstructed tracks that are overlaid on the activity in the detector.

Shortcut: you can type the 1-9 keys to jump to slices in the event, useful for jumping around in time.

To be concrete, after clicking Next Slice, you should see this (this is only a part of the screen, not the whole screen).

As you flip to the next slice, it looks interesting, is seems to be in almost the right spot (maybe the separation is two spaces, not one or zero).  And it has only a few hits, which is right.  But the visible energy is too high!  It’s more than the muon’s rest energy could have given to the electron.  This slice tried to fool us.  Don’t worry, this rarely happens, but it is a nice lesson.

Click another two slices ahead, and here is another little burst of activity, just a few hits, located at the end of the muon track.  It has a reasonable amount of energy, 34 MeV.  Here is our match!

For this event you should find the long track in slice one and the small burst of energy in slice four. This is a good place to answer the first question on the worksheet.

Sometimes the slices are back to back, but more often there is a gap of several slices between the end of the first track and the continuation of the path by the electron. In muon decay, the muon travels, giving up the energy you see as the track, until it’s all gone and the muon stops. After a long while (hundreds or thousands of nanoseconds!) it decays to a Michel electron. This electron has some kinetic energy and it will continue on in some direction, it could be backwards, upwards, or any other direction. You might ask your teacher “where did the electron get its kinetic energy, if the muon had stopped and is at rest?” Teachers love questions like that, because then you all get to talk about Einstein's famous E=mc2.

The gap of several slices isn't important, but the gap in time is. Look in the box labeled “Slice Info” and notice the Mean Time (in nanoseconds) and the Visible Energy (in units of mega electron-volts MeV). Get the Mean Time for both slices, and calculate how many nanoseconds elapsed between the end of the muon and the appearance of the electron.

In the “Slice Info” box, also, get the Visible Energy for the electron from the decay. It is expected that the electron can carry between zero MeV and up to half the energy of the muon at rest, which is 107 MeV. The other energy is carried off, unseen, by two neutrinos. The final distinguishing feature of muon decay is this energy, which should be less than 60 MeV. If you think you have a splash that matches (using your finger on the screen) in space, but has more energy than this, it indicates something other than muon decay.

Look for the mean time value in the Slice information box for slice 1. That data is already entered in the appropriate data slot. We have also done that for slice 4. Note that for slice 4 you will also need the visible energy value. Subtract the mean time for the muon slice (slice 1) from the mean time for the electron slice (slice 4), in this example you get 1410 nanoseconds. Check that the data recorded on your data table matches what is shown on the Arachne screen for each slice.

The final thing you should get from the slice box is the Event ID which is MV/2397/16/575.  This is a unique identification number, so you can use it to make sure you know which events you are talking about and not get them mixed up with others.

Now we can get started looking at nine more pulses of the beam at Fermilab. Go one by one through the following nine links to find the muon and its decay electron. There is one pair in each of these nine.

Muon Decay Event Two:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=344&slice=-1&phCutLow=0.5

Muon Decay Event Three:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=456&slice=-1&phCutLow=0.5

Muon Decay Event Four:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=469&slice=-1&phCutLow=0.5

Muon Decay Event Five:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=496&slice=-1&phCutLow=0.5

Muon Decay Event Six:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=668&slice=-1&phCutLow=0.5

Muon Decay Event Seven:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0016_numip_v05_1007120650_RecoData_DST_v10r6.root&entry=786&slice=-1&phCutLow=0.5

Muon Decay Event Eight:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0017_numip_v05_1007120720_RecoData_DST_v10r6.root&entry=157&slice=-1&phCutLow=0.5

Muon Decay Event Nine:

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0017_numip_v05_1007120720_RecoData_DST_v10r6.root&entry=502&slice=-1&phCutLow=0.5

Muon Decay Event Ten::

http://minerva05.fnal.gov/Arachne/simple.html?filename=/minerva/data/users/minervapro/outreach/muondecayDST/v10r6_2397/97/MV_00002397_0017_numip_v05_1007120720_RecoData_DST_v10r6.root&entry=536&slice=-1&phCutLow=0.5

For each one you will start with “All Hits”. Flip through the slices to find the track followed by the decay electron. Working together, determine the time (in nanoseconds) for each slice in the pair and log them in your data sheet. Make note of places where you have difficulty or questions, and work on the follow-up questions on the worksheet. Good luck, and you and your partner are on your own.

Student Chart

Questions from activity Getting started finding muon decays

1.      As you flip through the time-slices of Muon Decay Event One, how many have activity that is track-like? How many are small splashes of energy?

2.      There were actually two time-slices in Event One, each with a small splash, which occurred at the end of the muon track. How did you know which one was the decay electron and which one to reject as some other unrelated thing?

3.      Considering all eleven links, which one has the most time between the decay electron and its parent particle? Which one has the least time?

4.      In addition to all the muons that stopped in the middle and decayed, what was the most interesting other thing you saw?

5.      The muon you find is always traveling left to right, then stopping somewhere in the middle. How about the electrons?

6.      Describe in your own words, what is radioactive decay, and specifically how does a muon decay.

Activity: Measuring the Muon Lifetime

This is the suggested third activity in the muon radioactive decay lessons.  In this exercise, students do self-guided work to measure the muon lifetime in individual events and can observe the exponential nature of the decay.

Goals:

1.      Measure a large enough sample to the exponential nature of the decay time.

2.      Measure the average muon lifetime using that sample.

3.      The random nature of the decay process leads to variation in mean lifetime for different samples.

Prior Knowledge:

1.      Making a style of graph called a histogram of the decay time data

2.      The basics of radioactive decay (from our dice activity or carbon-14 or similar.)

3.      How to find decay events using the MINERvA data and Arachne.

4.      Logarithms and natural logarithms often covered in Algebra-II or Pre-Calc course.

Materials:

1.      Website access to Arachne and this webpage with links to many events.

2.      Excel data sheet (available for download here)

3.      Calculators

4.      Graph paper for drawing a graph

5.      Page of follow-up questions

Notes for the teacher:

Students are on their own taking data. Now that they have all practiced with the same ten events, the students (in groups probably) will be given their own set of fifteen events. There is a web page with groups of links, assign one group of links to each group of students, and make sure they know which set is theirs. Once they click on each link, they should fill out the line in the spreadsheet similar to how they did in the previous activity, including the event ID, the muon and electron time, and the electron energy. There is other information available to record, but it isn't necessary.

Filling out and combining the spreadsheets: Not only will the students calculate a decay time from their fifteen events, but this activity works best if you have a means of combining all their spreadsheet information into one large spreadsheet, and thus creating a very large data set. This means that you should consider your computer lab setup. If there are enough computers for teams to sit side by side at two different computers, entering data in the spreadsheet at the same time as they view it on the next computer works well. You can also have students jot down data on a hard copy and then flip between applications to enter it as they go as well. Otherwise, determine where in the process you would like them to enter data in the spreadsheet and be sure that students are clear on where they should do that.

After students have collected their data, they will be instructed to analyze their small set of data for average lifetime and half-life. You should expect quite a bit of variation in the data obtained from this small data set. They will compare their answers with three other groups to see what kind of variation there is in the data sets. You can choose to stop students and do some analysis at this point in the process, and talk about the difficulties with small data sets and variations in results. You may want to wait until all have completed their data analysis of both their small group and whole class set, this will depend on the type of student groups you are working with, and whether they will need some additional reinforcement that they are on the right track before proceeding to the next part of the activity.

Histograms: Students will be making a couple of histogram style graphs. You might want to allow them to determine their own parameters for graphing, or for ease of class discussion choose to have them use the same scale. Intervals of 500 nanoseconds in decay time works fairly well. The graph below shows a sample of data.

Measurement uncertainty: The results that students get for average lifetimes and half-lives are likely to vary fairly dramatically. This is of course due to small sample size.  The same issue is also present in familiar contexts like political polling, for example. Comparing the whole group sample size to the small individual groupings is a good discussion, and is part of the activity.

A step further is to take a look at the average lifetime data calculated by the individual groups, and calculate how much the data vary. This will be reassuring to those students who are likely to think that a value different from the “so called” right answer is wrong. The uncertainty of decay is an important concept in the bigger scientific picture that clearly tells us that things don't always fall exactly as a single accepted value says they should have. An easy way to do this in the classroom is to set up a range line on the board (see example graph below) and have students place their groups calculated value for average lifetime on the scale and include the value. Then it is an easy job to take the average of all of the individual group results, find the range of measurement variance, the largest variance from the average value is your error range. This is a reasonable estimate of event variance. If you would like to take your class further in this discussion you can Google ‘standard deviation calculation’. You may have a student ideally suited for a little extra job along these lines.

Computing mean life, simple and not simple: It is reasonable, and mostly correct, to simply calculate the mean life by making the average of the data and be done.  There is actually a subtle problem with this activity, and you can decide whether you want your class to realize the problem and correct for it, or enjoy the new concepts they learned without worrying about the subtle thing.  You or your students may (will?) notice that there are no decays with decay time less than 500 nanoseconds. There is a technical issue with how MINERvA records data that causes it to record data very short times after the muon in a different way (that we can't measure with Arachne), or even not record it at all. Because of this, we have hidden any decays that happen sooner than 500 nanoseconds. But wait! For an exponential decay, that’s the interval that has the most decays! If they are not there, isn't that going to make our mean life too high?

Well, it won’t be so very high that the students will notice or think they are getting the “wrong” answer.  So no harm is done?  Or you can consider two ways of correcting it.  The boring but effective way, but one that is often used by practicing scientists, is to estimate what events were missing from that first interval, include that estimate into your average, and consider that the more accurate number with some uncertainty because your estimate probably wasn’t perfect.  Okay, this is totally valid, as long as you include that in your discussion of the results.

The interesting argument that leads to a correction to the average lifetime invokes an understanding of the dice activity in a very special way.  If you want your students to try it, cut and paste the following thing into the instructions and questions for your students.

[For students: if the teacher wants them to consider it.]

There is a problem here, there is data missing between 0 and 500 ns.  That’s actually the interval where we expect most of the events to be, so our average will be too high by a little bit.  But the fix is interesting.  Remember the dice activity, especially the version where you threw 100 dice at the same time, and separated out those with a “one”, and then rolled again?  Each time you rolled, it was the same random 1/6 chance, the dice don’t remember how many rolls went by without decaying.  You could have ignored that first roll, and just as well pretended you had started out with roughly 85 dice.  Or maybe you didn’t know it, but there was a prior roll you didn’t see that had 115 dice, and you walked in with only 100 left.  Your analysis of the lifetime for the dice wouldn’t have been affected by it.  

Likewise, we can analyze the average lifetime of the muons as if those that decayed in the first interval never happened, and the lifetime clock started 500 nanoseconds later, for the second roll.  If this is convincing, then mathematically we have two (equivalent) choices.  Either subtract 500 nanoseconds from every event we are putting into the mean lifetime calculation, or subtract 500 nanoseconds from the average.  Do you see that these two are equivalent?  If you do, then the latter choice is easier, and will give you a more accurate measurement.

Looking at only student small group results should result in an average lifetime for muons of 2.29 μs. Note this is converted from the nanoseconds used in the graphing activity to microseconds- just move the decimal three places to the left. The average muon half-life should be somewhere in the range of 1.56 μs.

If all of the student groups’ average time difference is calculated we get a range of times from 1.591 to 3.079 μs. The largest difference is 0.793 μs providing the error range for this set of data. The next step asks students to make the same calculations using the whole group data. The data yields 2.32 μs for the average lifetime in this scenario, and an half-life of 1.61 μs. Graphs below for time difference, show the distribution for the assigned group data at the time of writing this guide.

This trend with increased data, the exponential shape becomes more clear, and the mean lifetime measurement is more accurate, can be taken to its logical extreme, if only you have the time or some way to automate the process (or both).  Here is the equivalent histogram for a very large data set, about 10,000 events, which include exactly the same decay events you measured, but also include four months of data in spring and summer 2010.  You can use this histogram you’re your students if you want them to better see how lots of data overcomes the random fluctuations inherent in the smaller samples.

Teacher Answer Key – Note: the first 10 events are from the previous activity  The visible energy  and times listed in this table might differ a little bit from the one in Arachne because of software and calibration updates that happened after this answer key was made.  The results of you and your student’s studies will still work.