Proton size is surprisingly hard to measure.

by Matthew Francis, DXS physics editor 

That’s about the size of it

The story is familiar: we and our surroundings are made of atoms of various sorts, combined in ways to make solids, liquids, gases, cells, books, and so forth. Atoms are made of electrons, protons, and (except for hydrogen) neutrons. Less familiar knowledge is that while electrons are fundamental, protons and neutrons are comprised of quarks.

Even that short paragraph tells us a lot about the nature of things: matter is hierarchical, with small things combining in various ways to make bigger and bigger things, up to our bodies, planet, and Universe. But how big are these things? For macroscopic objects like us, that’s easy to answer with rulers, tape measures, and the like. For the cells in our bodies, we can use microscopes. Viruses are (mostly) smaller than the wavelengths of visible light, so they’re invisible even with the most powerful ordinary microscopes; for those, we need to use electron microscopes. (That’s why, while bacteria were discovered centuries ago, viruses weren’t seen until 1931, though medical scientists inferred their existence several decades earlier.) Atoms are smaller still—much smaller. A human cell can contain 100 trillion atoms of various types, mostly hydrogen, carbon, and oxygen, with plenty of others to bulk things out.

Let’s think about some numbers for size, then: unless you’re a kid or a giant, your height is probably between 1 and 2 meters. One of my fingernails is about 1 centimeter across (1/100 of a meter). A red blood cell is a little less than 10 micrometers—10 millionths of a meter—across. A typical atom is a few hundred picometers (100 trillionths of a meter) across, or ten thousand times smaller than that blood cell. An atomic nucleus is a few quadrillionths of a meter across; the unit of measure for that is the femtometer.

The classic film “Powers of Ten” is a pretty good start for understanding how big and small things are in our Universe; if you haven’t seen it, give it a try now before we delve into the real question of the post: how big is a proton?

Measuring the really tiny things
Not even an electron microscope can help us see atoms, much less atomic nuclei or protons. The name itself tells us why: electron microscopes use electrons, which interact with the atoms in ways that change both. That’s similar to the reason we can’t use smaller wavelengths of light to see atoms or even viruses: cranking down the wavelength cranks up the energy of the light at the same time, to the point where it can damage the virus or break apart the atoms. However, that doesn’t mean we can’t use either electrons, other particles, or various forms of light to study atoms; we just have to accept that we may be changing the system we’re studying, and plan accordingly.

Protons are actually pretty easy to study as particles go: they’re stable, they’re relatively massive (10,000 times more than electrons), and they have an electric charge, so we can steer them with electric and magnetic fields. Contrast that with the Higgs boson, which is also pretty massive for a particle, but which is electrically neutral and decays so quickly into other things that we don’t actually see it in our detectors—only its decay products. So, you might think that measuring a proton’s radius is simple, but it’s not.

To see why this is, let’s step back again for a moment. When you consider the size of an atom and the size of an atomic nucleus, you see that there’s a huge discrepancy: nuclei are really tiny by comparison. You may have heard that most of your body is actually empty space, which is more or less true. Electrons are likewise miniscule (in a sense they have no size, but that’s a story for another day), so most of an atom is not occupied by any particle.

However, the size of an atom is dictated by the forces that hold it together, particularly the electromagnetic force. When you pick up a pencil, the electrons in your fingers repel the electrons in the pencil, allowing you to grip and suspend it in midair. The pencil’s atoms therefore “sense” the atoms in your hand and vice-versa, making each solid to each other. The emptiness within the atom, therefore, isn’t actually relevant in that context. (That’s the main reason I don’t like the “solar system” model of an atom, but that’s something to get into in another post.)

On the other hand, the size of an atomic nucleus depends on how you’re studying it. Nuclei involve three forces: the electromagnetic force, the weak force, and the strong force. The electromagnetic force has the greatest reach, but the other two are important if you can get close in. If you bombard a nucleus with high-energy particles, you no longer see the proton and neutrons: you see the quarks that make them up. If you bombard it with particles with too little energy, they may not get close enough for measurement purposes. All of this is the province of nuclear physics, which (despite the stereotypes) mostly isn’t about weapons or building power plants.

The simplest way to measure the size of a proton involves shooting electrons at it, and measuring the paths the electrons take as they feel the influence of the various forces. Because of those forces, in fact, the proton can’t be said to have a single size! Instead, physicists use three different size measurements, which are all pretty close to each other, but not exactly the same. The one most important to us for this post is the charge radius. Electron bombardment measurements found that to be about 0.88 femtometers.

However, electron bombardment only gets us so far; if we want better accuracy, we need another method.

Measuring the proton using weird hydrogen
Hydrogen is the most common element in the Universe, and the simplest: it consists of a proton and an electron. As a result, we know a lot about it, to an amazing degree of accuracy. However, there are still some weird things at the fringes physicists still are studying, for the sake of knowing everything there is to know about matter.

In an earlier post, I described how every element in the periodic table has its own unique spectrum. Hydrogen’s spectrum is one of the simplest, again because there aren’t that many ways things can be arranged in the atom. To calculate the basic spectrum, the size of the proton doesn’t matter. Just as the mass of the Sun is what dictates the size of Earth’s orbit, its the mass and electric charge of the proton that dictates the structure of hydrogen. Size, so long as it isn’t too big, doesn’t affect orbits. (The size of the Sun does make a difference for the amount of light we receive.)

However, because of the nature of quantum mechanics, there’s a small possibility that the electron in a hydrogen atom will overlap with the proton. Mostly that doesn’t matter—the probability is too small for us to worry about in most instances. That does mean that the size of the proton affects the spectrum of hydrogen in certain very sensitive measurements, since a larger proton means more probability of overlap.

To turn it around, sensitive measurements of the spectrum could tell us the size of the proton. In particular, there are two energy configurations in hydrogen that are very close but not quite equal. Those configurations depend on the proton’s size and magnetic properties, so a lot of work has been done over the decades to measure the transition between the configurations, something known as the Lamb shift (named for Willis Lamb, not for sheep). The Lamb shift actually led to the theory of quantum electrodynamics in the 1940s and ’50s, but we don’t need to worry about that for now.

To amplify the effect, researchers made an exotic form of hydrogen using a muon instead of an electron. Muons are the heavier, more unstable cousins of electrons (I think we all have relatives like that), so they can be substituted in for electrons in certain places. A muonic hydrogen atom is about 186 times smaller than a normal hydrogen atom, meaning the size of the proton makes a much bigger difference to the behavior of the system.

However, the muon’s instability means you have to move fast: in 2 millionths of a second, half the muons in your experiment will decay. (Personally, I find it amazing this experiment can be done at all, since the time for all of this is measured in small fractions of millionths of seconds.) Scientists in Europe sent beams of muons into hydrogen atoms, knocking the electrons out and forming muonic hydrogen.

A very small number of the muonic hydrogen atoms could be manipulated into exhibiting the Lamb shift, by using lasers. The light emitted from the Lamb shift lies in the microwave portion of the electromagnetic spectrum (though not the same part used in microwave ovens), so they used a sensitive detector to snag the small number of photons as they came out.

What they found was astonishing: the size of the proton they measured was significantly smaller than the size obtained from electron bombardment: 0.84 femtometers. That may not seem shockingly different than 0.87 femtometers, but the ranges of possible values didn’t overlap at all, so these results are in complete conflict.

Why is that? We don’t know yet. It’s possible that with such delicate measurements there could be hidden inaccuracies, just like with the “faster-than-light” neutrino story from 2011. However, it’s also possible that the muons themselves are revealing a new physical phenomenon, one we haven’t seen before. That’s an exciting possibility, unexpected from our original naive question about the size of a proton.

Why should we care?
I’m sure many of us have asked the question about how big an atom is, or how big a nucleus is, or how big a proton is. It’s a natural question, born of human curiosity. The constant probing is part of what makes us human; earlier probings into the structure of matter led to quantum mechanics, which is the foundation of nearly all our modern technology, from computer chips to fluorescent lights to lasers. The initial question—how big is a proton?—may be simple, but answering it leads us to the deepest mysteries of the Universe. I for one find that thrilling.

Reference: Aldo Antognini et al., Proton structure from the measurement of 2S-2Sp transition frequencies of muonic hydrogen. Science 339 (2013), 417. Link (for subscribers to Science): DOI 10.1126/science.1230016