The Muon: 1970

In 1970, Hall, Lind, and Ristenen (Univ. of Colorado at Boulder) published a paper in the American Journal of Physics (AJP, vol. 38, No. 10) on “A Simplified Muon Lifetime Experiment for the Instructional Laboratory.” Basically, it articulates precisely the experiment at the heart of a similar instrument at SMU. Muons are produced in cosmic rays raining down on the atmosphere. Some muons make it all the way to sea level. Some of those are moving slowly enough to be stopped when passing through material. If that material gives off light in response to the slowing, stopping, and then decay of the muon, it is possible to use the light to measurement the lifetime of the muon.

An excerpt from the Hall et al. paper, showing their collected data (counts vs. channel, where one channel represents about 100ns of time) and the results of a least-squares fit to the data to extract the lifetime of the muion.

Hall et al. reported on a run of their experiment of 695 hours (about 29 days!). I’ve had nothing but time on my hands, and after discovering the Hall paper when I started playing around with the SMU instrument I was inspired to repeat their experiment.

Data from the SMU muon detector.

As of today, I have 695 hours of data from the muon detector at SMU. Based on a model fitted to the data (an exponential decay function added to a flat background), I find the lifetime of the muon to be 2170 \pm 29 nanoseconds (ns). The accepted lifetime is 2196 ns. The Hall et. al result using a similar but earlier version of the experiment found 2106 \pm 58. (note: they quote the half-life, but that is easily converted to the lifetime [average life of the muon] by dividing the half-life by ln(2)).

In 1970, as now, the lifetime of the muon has not changed within the resolution of two 695h data sets, taken independently and 50 years apart. There is a wonder in the power of scientific investigation to reveal those things that are steady and constant in the cosmos.

What I learned this week

I have really thrown myself into physics, since I am stuck at home (a) because there is a pandemic and (b) because SMU won’t let me on campus until tomorrow (because I was abroad when they ended work-related international travel 2 weeks ago). This has been a grand opportunity. Here are some things I learned this week.

UPROOT and UPROOT-METHODS

UPROOT is awesome. It lets me utilize natively in Python files created in ROOT. No more do I need to have ROOT compiled in the background, along with its Python interfaces. I can just import UPROOT and load ROOT files into Python Pandas dataframes, which anyway are how I prefer analyzing data these days.

UPROOT was introduced to me by my former PhD student Matthew Feickert and my current PhD student Chris Milke. I’ve been using it for several weeks to work on a project with one of my undergraduate research students. However, for high-level physics operations, like dealing with four-vector mathematics, ROOT is hard to beat. Turns out, there is a solution.

UPROOT-methods! These are implementations of interfaces akin to C++ classes in ROOT that do cool physics things… like vector arithmetic! I just learned about this today and already did some Lorentz Transformations on particle vectors. I’m pretty happy about this.

MatPlotLib Subplot Gridding!

Sometimes you just want to layout a bunch of graphs in a single plot in a non-uniform way. Consider the following graph:

An analytic model is fitted to data from a muon detector at SMU. The top plot is the number of muon candidates vs. their measured lifetime. The bottom plot is the “model pull” – the difference in each bin of the lifetime measurement between the real counts and the fitted model prediction, divided by the standard deviation of the counts in the bin.

I need to show the fit of an analytic model (an exponential lifetime model) to the data coming from a muon detector in the basement of Fondren Science Building at SMU; below that, I need to show how well the model describes the data after optimizing the model parameterization to reproduce the data. To do this, I need a big plot at the top and a short plot at the bottom. I needed plot grid layouts!

I found this article extremely helpful: https://towardsdatascience.com/subplots-in-matplotlib-a-guide-and-tool-for-planning-your-plots-7d63fa632857?gi=b00175d89216 (it’s how I made that figure above!)

Online Teaching Tips

How do you mute all those jerks with hot mics in Zoom? WHY WON’T MY F**KING MAC LET ME SHARE MY DESKTOP?!?!?!

Check out these tweets.

Planning new experiments and particle colliders is fun

I’ve been participating in a workshop (online only) hosted by Temple University on physics and detector design ideas for the Electron-Ion Collider, a project planned for construction at Brookhaven National Accelerator Laboratory. I’m still just beginning to think about bottom quarks and how to use them to probe structure in protons and nuclei, and the discussions at this workshop have got me thinking about how this problem changes when going from the LHC to a different collider designed to probe such matters with high precision.

What have I learned? I have a lot to learn.

The Joy of the Muon

Muons are a gateway drug. They are just difficult enough to detect that they are really not obvious to humans. They are just easy enough to stop in material that, once you learn to spot them, you want to stop them and watch them do what they do. What do muons do?

They decay.

In about 2 millionths of a second, that muon you just captured is gone – evaporated into a particle spray containing an electron and two neutrinos. This fact allows us to measure the lifetime of the muon. Being captured by an atom resets their quantum clock to zero. What happens after that, and when it happens, tells us the probability that a muon, nearly at rest, will decay after a certain amount of time.

If you can capture all of this in a detector system, you can measure the lifetime of the muon.

Thanks to my colleagues, Tom Coan and Jingbo Ye, we have an awesome little muon detector in the basement of Fondren Science Building. Thanks to our awesome “Internet of Things” developer, Guillermo Vasquez, we have a Raspberry Pi computer connected to the detector that accepts data from it. I had some fun writing python code to read and save the data to disk, and then I used a Jupyter notebook to analyze it.

And here is the joy of the muon: a measurement of its lifetime from an ensemble of >1000 decayed muons and assuming an exponential decay model. The accepted value of the lifetime is 2196.9811(22) nanoseconds, where the numbers in parentheses are the uncertainty on the last two decimal places of the accepted lifetime measurement. Not bad. Not bad at all.

This image is updated every minute or so, representing an updated data sample from the detector (and an updated fit of the model to the data). The rate of “good muons” through the detector is about 3 per 15 minutes, so don’t hold your breath. Reload this page daily to see new results!

P.S. What’s the data coming in from the electronics, you ask? It’s the number of clock cycles between the flash of light that signals muon capture by an atom, and the flash of light that signals the decay of the muon (and the exiting of the electron from the medium). The clock speed is 50MHz, so a period of 20ns. Each clock cycle is thus 20ns of time.

Pi for dessert

Mmmmmmm. Pi.

Let’s end this day on a note of wonder. It’s Pi Day! (March 14, or 3-14). Pi is an irrational number… it cannot be written as the ratio of two integers. It’s a number that represents the ratio of the circumference of a circle to its own diameter.

It shows up everywhere when you try to describe nature. Here are a few of my favorite places it shows up.

Coulomb’s Law

Determined from experimental evidence by Augustin de Coulomb, this describes the degree of force that one electric charge exerts on another:

    \[ F = k \frac{q_1 q_2}{r^2} \]

Coulomb’s constant, k, is written in System International Units as k = \frac{1}{4 \pi \varepsilon_0}.

Biot-Savart Law

This is the law that describes the magnetic field generated from an electric current. In its differential form:

    \[ B = \frac{\mu_0}{4 \pi} \frac{i d\ell \sin\theta}{r^2} \]

General Relativity

The Einstein Field Equations of General Relativity can be written compactly in tensor notation as

    \[ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu} \]

The Heisenberg Uncertainty Principle

Sure you can measure the position of a subatomic particle. Sure you can measure its momentum. But you cannot make a measurement of both of them that yields unlimited precision on them at the same time. In fact, the uncertainty on them both is bounded by this relationship:

    \[ \Delta p \Delta x \ge \frac{h}{4\pi} \]

where h is Planck’s Constant, 6.626 \times 10^{-34} \mathrm{J \cdot s}.