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The Proton Size Problem

November 3, 2016

It's easy to measure the diameter of a billiard ball to great accuracy, but not that easy to measure the diameter of a tennis ball. A tennis ball is fuzzy, so it's hard to say where a tennis ball ends and the rest of the universe begins. The same is true of the subatomic particles, which, because of their quantum nature, exist as fuzzy blobs. The electron is an interesting case. While experiments show that it is a "point particle" with no diameter, we must conclude by Heisenberg's uncertainty principle that it does occupy a nonzero volume.

Werner Heisenberg (right), alongside Wolfgang Pauli, at the 1927 Solvay Conference

Werner Heisenberg (right), alongside Wolfgang Pauli, at the 1927 Solvay Conference.

(Photo by Benjamin Couprie, Institut International de Physique de Solvay, via Wikimedia Commons.)

While protons act as if they are charged spheres, the edge of the sphere is not sharply defined. The proton charge radius is therefore taken as a root-mean-squared value. There have been many measurements of this charge radius, and the accepted value is 0.8751 femtometers (fm, 1015 meter) with an uncertainty of about 0.7%.[1]

It's not known why a proton should have a particular charge radius, let alone the particular value that we measure. Also, the
proton has a magnetic moment, acting as if it were a spinning ball of charge. Otto Stern discovered the large magnetic moment of the proton, and he was awarded the 1943 Nobel Prize in Physics for this discovery.

Today, we know that the proton is a composite particle composed of three quarks, two up quarks of electric charge +2/3 e and spin 1/2 ↑, and one down quark of electric charge -1/3 e and spin 1/2 ↓. Adding the charges and spins of these gives us the values that we observe for the proton.

Quark structure of the proton

Quark structure of the proton.

The three quarks (two up and one down) are held together by appropriately named gluons.

(Image by Arpad Horvath, modified, via Wikimedia Commons.)

As I wrote in two previous articles (Fat Protons, July 12, 2010, and Proton Size Problem, March 20, 2013), proton size measurements based on hydrogen atoms in which muon replace electrons have yielded a significantly smaller charge radius than previous measurements, giving rise to what's now termed the proton radius puzzle.[2-4]

When a hydrogen atom is constructed from a muon and a proton, the proton appears to have a slightly smaller radius than that of a normal hydrogen atom of an electron orbiting a proton. Although muons are 207 times heavier than electrons, all their other properties are identical, and theory predicts that the proton shouldn't behave differently. ArXiv lists 61 papers containing "proton radius" in their title from 2010 to the present.

Muonic hydrogen provides a better system than electonic hydrogen for measurement of the proton charge radius, since the higher mass muon is in a nearer orbit of the proton than an electron. These measurements have been based on pulsed laser spectroscopy of the Lamb shift of the muonic energy levels. The Lamb shift, named after Nobel Physics Laureate, Willis Lamb, is the small difference in energy between the 2S1/2 and 2P1/2 electron energy levels.

Muonic hydrogen

The Bohr models of hydrogen and muonic hydrogen drawn to scale.

(Illustration by the author using Inkscape.)

This is a difficult experiment, since muons have a mean lifetime of just 2.197 microseconds. In this short time span, you need to create muonic hydrogen and measure the Lamb shift. Fortunately, the experimental teams have been able to create a few hundred muons each second, and a fraction of these replaced electrons in hydrogen gas.

Earlier this year, the same research group that made the original discovery of the proton size problem conducted further experiments using muons with deuterium nuclei, which are protons combined with a neutron, and muons.[5-6] They found a proton radius several standard deviations smaller than that of normal, electron deuterium. These repeated confirmatory measurements show that, aside from some unknown experimental or theoretical error, there might be some new physics to be discovered, such as a previously unknown fundamental force acting between muons and protons, but not between electrons and protons.[5-6]

Some new physics might explain the anomalous magnetic dipole moment of the muon.[6] The lead principal investigator on these experiments, Randolf Pohl of the Max Planck Institute of Quantum Optics (Garching, Germany) is more cautious.[6] The proton size calculation requires a value of the Rydberg constant, which is supposedly known to great accuracy (10 973 731.568 508 m-1).[7] Pohl thinks that the problem might be in this constant.[6]

As typical in the sciences, works continues to unravel the true story. Another team is attempting a measurement that doesn't require the Rydberg constant, and Pohl's group plans to examine muonic helium, a muonic atom with two protons.[6]


  1. NIST, "CODATA Value: proton rms charge radius".
  2. Randolf Pohl, et al., "The size of the proton," Nature, vol. 466, no. 7303 (July 8, 2010), pp. 213-216.
  3. Aldo Antognini, et al. "Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen," Science, vol. 339 no. 6118 (January 25, 2013), pp. 417-420.
  4. Helen S. Margolis, "How Big Is the Proton?" Science, vol. 339 no. 6118 (January 25, 2013), pp. 405-406.
  5. Randolf Pohl, et al., "Laser spectroscopy of muonic deuterium, Science, vol. 353 no. 6300 (August 12, 2016), pp. 669-673, DOI: 10.1126/science.aaf2468.
  6. Natalie Wolchover, "New Measurement Deepens Proton Puzzle," Quanta Magazine, August 11, 2016.
  7. NIST, "CODATA Value: Rydberg constant".                    

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