## View Blog | Read Bio

### The size of the proton

There is a new measurement of the size of the proton and it turns out that protons are smaller than we thought they were.

At some point in your education you probably got introduced to the Bohr model of the atom.  The nucleus is made up of protons and neutrons, and electrons orbit around the nucleus.  In the Bohr model, electrons orbit the nucleus in circular orbits like the Earth orbits the Sun, but these orbits are only allowed to have some radii (which correspond to an integer number of de Broglie wave lengths).  Electrons can transition between these levels and when they do, they either absorb a photon (in the case of an electron being excited from, say, the ground state to an excited state) or emit a photon (in the case of an electron going from an excited state to a lower state.)  This is shown below:

The Bohr model isn’t exactly right – but it’s close enough to get some feel for what’s going on.  In a more precise quantum mechanical picture, the electron isn’t actually orbiting the nucleus – it’s smeared out in what we call a wave function.  The square of the wave function tells us how likely we are to find the electron in a given place.  The ground state orbital (the shape of the wave function of the electron in the atom) is spherical.  The lowest excited state has four different possible orbitals, one spherical (S) and three which are shaped like a dumbbell (P), a sort of 3D figure-8.

What you probably learned in school was that these S and P orbitals have exactly the same energy – and they almost do.  In a simple model, the nucleus is just a point particle – meaning it exists just at a single point, with no size in any dimension.  But protons aren’t point particles – they’re just very small.  In the S orbitals, the electron spends most of its time near the nucleus, but in the P orbitals, the electron spends less of its time near the nucleus.  This difference in how much time the electron spends near the nucleus leads to a very small shift in the energy of the orbitals, called the Lamb shift.  The Lamb shift is measured by measuring the photon emitted when an electron goes from the P to the S orbital in the second shell.  It depends on the mass of the electron and the size of the proton.  (Here’s the explanation of the Lamb shift on the experiment’s web site.)

In this new measurement, they looked at hydrogen with a muon (the heavier cousin of the electron) instead of an electron.  Because the muon is about two hundred times heavier than the electron, it spends more time near the nucleus than the electron, meaning it’s more sensitive to the Lamb shift than the electron.  Previously, the best measurement of the diameter of the proton was 0.877±0.007 femtometers (m) and this measurement measured it to be 0.8418±0.0007 fm.  A femtometer is 10-15 meters.  If you were a proton (you’re somewhere between 1-2m tall), this would mean traveling one millimeter would be like traveling from the Earth to the Sun (1011 m).  This measurement would be like finding out that you’re 5’5″ instead of 5’8″ by looking at how long it takes for you to walk between Milwaukee, WI and Chicago, IL (150 km) and Milwaukee, WI and Madison, WI (141 km)*.

The fact that this measurement is so far off from our expectations indicates one of the following:

• The precise calculation we’re comparing to is flawed.The proton is actually a really complicated object – perhaps we forgot an important component.
• The measurement has some flaw we haven’t figured out yet.  Maybe there was some systematic shift that wasn’t taken into account.
• Our theory is flawed.  This could indicate some physics beyond the Standard Model – exactly what we’re looking for at the LHC.

We have to seriously consider the first two options, but the third would obviously be very exciting.

So why I am writing about this here?  First, it illustrates that there are other ways of studying fundamental particle physics than by slamming things together.  Second, it’s an interesting result that may hint at exciting new physics we’re hoping to see at the LHC.  Third, it’s a great segue into my next post…

*Yes, this analogy breaks down at some point.  Don’t take it too far.