# The mysterious size of the proton

One of the stories to come out of atomic physics in the first part of this year that grabbed my attention was the continuing saga of the puzzle of the proton size. I really find this interesting because it shows that there are still surprises waiting for us in physics, even when we’ve though that we’ve understood everything as well as possible. Read on for a description of the experiment and what’s going on.

Protons, being made of quarks rather than being fundamental point particles, have a size, and unless you get really close, it looks like a round sphere. In hydrogen, this affects the energy levels of the electron, by a very small, but detectable, amount.  If you measure the Lamb shift, the name for the effect where the lowest energy excited states of hydrogen, the 2S and 2P states, have very slightly different energies, you can, with a bunch of help from theory, work backwards to calculate the proton radius.

The Lamb shift was measured first in 1947 by Willis Lamb and Robert Retherford, and Lamb won the Nobel Prize in Physics in 1955 for (amongst other things), this discovery.

Willis Lamb with the experiment (source: AIP)

In their experiment, Lamb and Retherford produced hydrogen atoms in the 2S state by shooting a beam of electrons into a gas. The excited atoms passed through a region of adjustable magnetic field, which causes an energy difference between the 2S and 2P states (and would still even if there was a Lamb shift of zero). Microwave radiation applied while the atoms are in the magnetic field could induce transitions from the 2S state to the 2P state if they absorbed a photon of the right energy. Any atoms in the 2P state very quickly de-excite to the ground state and lose their excess energy as a photon. However, any atoms left in the 2S state would deposit their energy in a detector, which excited electrons and could be seen as a current.
The microwave frequency corresponding to the minimum current in the detector is the frequency of the photon needed to drive the transition between the two states.
Lamb and Retherford measured this frequency for different magnetic fields, and extrapolating to zero magnetic field, they found to their surprise, a non-zero difference, which we know today as the Lamb shift.

Until recently, the proton charge radius was determined from experiments measuring the Lamb shift in hydrogen, and was averaged with some complementary experiments that collide electrons and protons, and measure the angle at which the electron scatters from the proton. These two methods give results that are pretty much in agreement with each other.

Instead of using normal hydrogen, a group in 2010 conducted a new experiment where they formed a hydrogen-analogue, muonic hydrogen, which replaces hydrogen’s electron with a muon. A muon is a particle from the same family as electrons (the leptons), but with around 200 times the mass, and a very short lifetime – around 2 microseconds. They can be produced in particle accelerators, like the one in the Paul Scherrer Institute in Switzerland used in the new experiment.

There are a couple of advantages to using muonic hydrogen. First, the 2S-2P transition has a higher energy, in the infrared regime. This means that lasers are used to stimulate the transition, which can be stabilised much more easily than microwave sources.
Second, the heavier mass of the muon means that, on average, the muon spends more time closer to the proton than an electron would, and so is more sensitive to the proton’s radius.

In the experiment, a beam of muons was shot into a hydrogen gas target, with some of the muons forming muonic hydrogen in the 2S state. Then, the laser pulse was fired to try to induce transitions to the 2P state. Just like in regular hydrogen, the 2P state decays quickly to the 1S state, and emits an X-ray photon. By counting the number of X-ray photons as a function of the laser frequency, the experimenters could build up a resonance curve and find the 2S-2P energy difference.

A pretty schematic of the experiment at PSI (from the Science paper)

Up until this experiment, the value quoted for the proton charge radius was 0.8768 +/- 0.0069 fm.
That’s almost one femtometre, 10^-15 metres. That means if you take a hydrogen atom, and blow it up to the size of a football stadium (which seems to be traditional — don’t try this at home ;)), the proton is about the size of a grain of sand in the middle.

In the first results with muonic hydrogen, the charge radius they calculated was  0.84184 +/- 0.00067 fm.
If you know anything about physics, you know that if you want to compare two numbers, you need to see what the difference is compared to the uncertainty. You’ll notice that the new measurement is around 10 times more precise than the old (thanks to the advantages of using the muon), so the uncertainty in any comparison is almost entirely due to the older value, 0.0069 fm. The difference is 0.35 fm, and the uncertainty is 0.0069, just about five times smaller. Five times is a huge difference in precision measurements, and is really calling for an explanation.

Now, the same team has taken more data (using essentially the same method, but with a 2S-2P transition with a different nuclear spin — some of you might know that each of the atomic states is actually split into several sub-states due to the spin of the nucleus). They get an improved value for the charge radius of 0.84087 +/- 0.0039 fm, which agrees with their value from a few years ago, and still disagrees with the value if you use an electron.

Why the difference? Well, nobody knows. It could be that the interaction between the electron and the proton is subtly different than that between the muon and the proton, but data from other experiments don’t leave a lot of room for that to be the case. It could also be that there’s a systematic effect that noone’s though of yet.
It’s up to the experts and the future students in this field to work it out!