Natural Radioactivity Homeworknow

It’s time to return to our ongoing exploration of the structure of the Standard Model. Our primary tools are Feynman diagrams, which we introduced in previous posts (part 1, part 2). By now we’ve already familiarized ourselves with quantum electrodynamics (QED): the theory of electrons, positrons, and photons. Now we’re going to start adding on pieces to build up the Standard Model. We’ll start with the muon, portrayed below by Los Angeles artist Julie Peasley. (These handmade plushes can be found at her website, The Particle Zoo.)

We’re all familiar with the electron. Allow me to introduce its heavier cousin, the muon (μ). Where did this muon come from? Or, as Nobel Prize winner I. I. Rabi once asked, “Who ordered that?” (This is still an unanswered question!) Besides its mass, the muon has the same fundamental properties as the electron: it has the same charge, feels the same forces, and—like the electron—has an anti-particle partner.

Feynman rules for QED+μ

This makes it really easy to extend our Feynman rules. We’ll call our theory “QED+μ,” quantum electrodynamics with an additional particle. We just have to write the rules for two copies of QED:

Let’s recall how to interpret this. The three lines tell us that we have three kinds or particles in the theory: electrons (e), muons (μ), and photons (γ). Recall that the matter particles, the ones whose lines have an arrow, also have antiparticles. We indicate antiparticles by arrows pointing in the wrong direction when we read the diagrams from left-to-right. The vertex rules tell us that we have two kinds of interactions: a photon can either interact with two electrons or two muons.

It’s important to note that we cannot have photon couplings that mix electrons and muons. In terms of conservation laws, we say that electron and muon number are each conserved. For example, in the theory we’ve developed so far, you cannot have a muon decay into an electron and a photon. (We’ll introduce these sorts of interactions next time when we discuss electroweak theory.)

Exercise: Is the following diagram allowed in QED + μ?

Answer: Yes! But doesn’t this violate conservation of electron and muon number? You start out with two e‘s on the left and end up with two μ’s. Hint: what are the arrows telling you?

Once you’ve convinced yourself that the above diagram doesn’t violate electron or muon conservation, let me remark that this is an easy way to produce muons at low energy electron colliders. You just smash an electron against a positron and sometimes you’ll end up with a muon-antimuon pair which you can detect experimentally.

Exercise: when we previously did electron-positron to electron-positron scattering, we had to include two diagrams. Why is there only one diagram for eμ to eμ? Hint: draw the two diagrams for ee to ee and check if the Feynman rules still allow both diagrams if we convert the final states to muons.

Detecting muons, some collider physics

If you think about this a little, you might wonder: if electrons and muons are so similar, how can experimentalists distinguish between them at a collider? Seth and Mike might scold me for skipping over some information about the interaction of charged particles through matter, but one simple way to distinguish muons from electrons is to measure their energy and momenta. We know that (away from a potential) a particle’s energy is the sum of its kinetic energy plus it’s mass energy added in quadrature E2=m2c4+p2c2 (this is the “real” version of E=mc2). Since muons are heavier than electrons, we can just check the mass of the particle by plugging in the measured energy and momentum.

Actually, this is an oversimplified picture. In order not to annoy the other US/LHC bloggers, I’d better provide a slightly less oversimplified “cartoon.” Electrons are light, so let’s imagine that they’re ping pong balls. On the other hand, muons are heavy, so let’s imagine them as bowling balls. As you probably know, the LHC detectors are big and full of stuff… by that I mean atoms, which in turn are made up of a nucleus and a cloud of electrons. We can thus imagine a sea of ping-pong balls (think of a Chuck-E-Cheese ball pit). When electrons hit this ball pit, they end up distributing all of their energy into the other balls. This happens in the electromagnetic calorimeter, or ECAL. “Calor” is Latin for heat, so you can guess that the ECAL is really just a big fancy thermometer that measures the energy that the electron dissipates. Muons on the other hand, are bowling balls that are so massive that they just barrel straight through the ball pit to get to the other side. Here’s a very scientific illustration:

I hope we don’t get any comments saying, “oh man, muons are jerks.” In fact, they’re quite the opposite: muons are the only Standard Model particles that make it all the way to the outside of the detector, making it easy for us to identify them. In fact, the big distinctive toroidal magnets on the ATLAS detector below are there to bend the path of muons to help the outermost detector determine the muon momentum by measuring the curvature of their trail.

Exercise: [for those who want to do some actual calculations, requires a high school physics background] Convince yourself that this heuristic picture is correct by calculating the final momenta of a ball colliding elastically with (a) a ball of the same mass and (b) a ball of much lighter mass.

ATLAS toroidal magnets. Image from the Interactions.org Image Bank

Neat things that muons can do

Let me make a few more semi-historical remarks: our QED+μ model is just a theoretical toy. Historically, scientists knew immediately that something was weird about the muon: unlike electrons, it decayed into other particles and seemed to interact with mesons in unusual ways. In fact, for a while people thought that muons were a kind of meson. These differences ended up being a harbinger of something more interesting: the weak force.

Exercise: convince yourself that our Feynman rules for QED+μ do not allow muon decay, i.e. μ turning into non-μ stuff.

Muons are generated in the sky when cosmic rays hit atoms of the upper atmosphere. These rain down onto the Earth and force us to put our dark matter experiments deep underground to avoid their ‘noise.’ What’s really neat, however, is that the fact that muons make it to the surface of the Earth is a rousing experimental check of relativity. We know that muons at rest decay in microseconds. In this time, it seems like there’s no way for them to traverse the kilometers (about 4 km) between the Earth and its upper atmosphere; even if they were traveling at the speed of light! (c ~ 3.  108 m/s). What’s happening is the phenomenon of time dilation!

Introducing the tau (via the Socratic method)

Exercise: the Standard Model actually has another cousin of the electron, the tau (τ), leading to three charged leptons in total. Write down the Feynman rules for the theory QED+μ+τ, i.e. the theory of electrons, muons, and taus interacting via photons. Make sure that electron, muon, and tau number are all conserved. Draw the diagram for tau production in an electron-positron collider.

Exercise: Above we argued that muons are special because they barrel right through our detectors like bowling balls through an array of ping pong balls. Taus are even heavier, shouldn’t they also make it to the outside of the detector?

Answer: This was a bit of a trick question. The logic is correct that sufficiently energetic taus should make it all the way to the outside of the detector in our QED+μ+τ theory. However, this is not the full story for electrons, muons, and taus (collectively known as leptons) in the Standard Model. Like muons, taus are unstable and will decay. In fact, they decay much more quickly than muons because they have more mass and can decay into stuff (they have more “phase space”). While muons are like bowling balls barreling through the detector, taus are more like grenades that burst into hadronic “shrapnel” inside the calorimeters. They are usually very difficult to reconstruct from the data.

A preview of things to come:

Now we’re very familiar with putting together multiple copies of QED. For now, there are only three copies we have to worry about. It is an open question why this is the case. The existence of at least three copies, however, turns out to be significant for the imbalance of matter and anti-matter in the universe. In the next post we’ll introduce the weak force and really see what we can do with these leptons.

I’m currently in the middle of my “Advancement to Candidacy” exam, so my posts might be a little more delayed than usual this month. By the end of it, however, I hope to be blogging as an official PhD candidate. 🙂

Erratum: virtual particles

I wanted to correct a misleading statement I made in my previous QED post: I discussed the visualization of virtual particles as balls that two kids toss back and forth while standing on frictionless ice. Conservation of momentum causes the two kids to slide apart as they throw and catch the ball, generating what we observe macroscopically as a repulsive force. We mentioned that it’s more difficult to see how this could give rise to an attractive force. I suggested that this is a phenomenon coming from the accumulated effect of many quantum exchanges. While this is true, there is a simpler way to understand this: pretend the ball has negative momentum! Since the particle is virtual, it is inherently quantum mechanical and needn’t have ‘on-shell’ (physical) momentum. Thus one could imagine tossing the ball with negative momentum, causing one to be deflected in the same direction as the ball was tossed. Similarly, catching the ball with negative momentum would push one in the direction that the ball came from.

Does it make sense classically? No! But that’s okay because they’re virtual particles.

That’s all for now, folks!
–Flip, on behalf of the US/LHC blog.

Tags: Feynman rules, muon, QED
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Natural Radioactivity



Radioactivity surrounds us

Radioactivity is an integral part of our environment. All living beings have been exposed to a constant flux of natural radiation on the surface of our friendly planet: to no negative effect.

The natural radioactivity of rocks:
We find the majority of naturally-occurring radioactive elements in rocks. This section of the nucleus graph shows the successive radioactive disintegrations that lead from uranium-238 to radon. The ‘descendants’ of uranium-238, present in trace amounts in these minerals, emit alpha and beta rays which remain trapped inside the rocks. Gamma rays, on the other hand, are able to escape. The sixth descendant of uranium – radon, as shown above – is a gas. Large amounts of it are therefore able to escape from the ground, thereby releasing radiation into the atmosphere.
IN2P3

Natural radioactivity is still the principal source of radiation exposure. In France, the exposure dose is 2.4 millisieverts per person per year, as opposed to 1mSv from medical examinations. This is, of course, an average, and location and lifestyle play equal roles in determining the level of exposure. Where one travels, where one lives and even whether or not one uses air conditioning are all important factors.

We are constantly being bombarded by particles of cosmic radiations : several hundred go through our bodies every second. Rocks like granite, which have become symbols of permanence and durability, contain light traces of radioactive uranium. Sitting on or walking near a block of granite exposes you to previously unencountered sources of radioactivity.

Even the food we eat or the air we breathe contains radioactive elements – either formed thanks to the intervention of cosmic rays, or as old as the solar system itself. There is absolutely no way to escape from it: even we are radioactive! Eight thousand atoms of potassium 40 or carbon 14 disintegrate in our bodies every second.

The role of radon
Continuation of the uranium decay chain. Radon, which, being a gas, escapes from its underground prison, decays comparatively quickly into lead-210: a radioactive substance which survives for decades. The chain finally ends with the stable isotope lead-206. If the radon source is deep underground, then radon may not have the time to reach the surface; in which case, the radioactivity stays in the soil. If the radon does manage to break through, however, some of its decay products can be harmful to human lungs.
IN2P3
The most important natural source of radioactivity is a rare gas known as radon. One of the products of uranium decay, radon is an ‘inert gas’ that can participate in no chemical reaction. This would seem to make it completely harmless, were it not for the fact that radon own radioactive decay produces gases poisonous to humans. The nature of the soil you live on, the construction tools used to build your house and the quality of its air conditioning are all crucial factors in determining radon exposure.

It is important to stress, at this point, that 2.4 mSv is a low dose. In certain parts of India, China and Brazil the natural exposure level rises to 10 or even 20 mSv a year. The fact that humans have survived these comparatively overexposed conditions indicates that all doses of the order of millisieverts are negligible and very likely harmless.

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Artificial Radioactivity

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