Charge radius

This section needs to be updated. The reason given is: as reported in Science, the proton radius puzzle has possibly been solved. A new measurement using the Lamb shift in ordinary hydrogen agrees with that using muonic hydrogen.. Please update this article to reflect recent events or newly available information. (September 2019)

The problem of defining a radius for an atomic nucleus (proton) is similar to the problem of atomic radius, in that neither atoms nor their nuclei have definite boundaries. However, the nucleus can be modeled as a sphere of positive charge for the interpretation of electron scattering experiments: because there is no definite boundary to the nucleus, the electrons "see" a range of cross-sections, for which a mean can be taken. The qualification of "rms" (for "root mean square") arises because it is the nuclear cross-section, proportional to the square of the radius, which is determining for electron scattering.

The internationally accepted value of a proton's charge radius is 0.8768 fm (see orders of magnitude for comparison to other sizes). This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium.

However, in 2010 an international research team published a proton charge radius measurement via the Lamb shift in muonic hydrogen (an exotic atom made of a proton and a negatively charged muon). As a muon is 200 times heavier than an electron, its de Broglie wavelength is correspondingly shorter. This smaller atomic orbital is much more sensitive to the proton's charge radius, so allows more precise measurement. Their measurement of the root-mean-square charge radius of a proton is "0.84184(67) fm, which differs by 5.0 standard deviations from the CODATA value of 0.8768(69) fm". In January 2013, an updated value for the charge radius of a proton—0.84087(39) fm—was published. The precision was improved by 1.7 times, increasing the significance of the discrepancy to 7σ. The 2014 CODATA adjustment slightly reduced the recommended value for the proton radius (computed using electron measurements only) to 0.8751(61) fm, but this leaves the discrepancy at 5.6σ.

The international research team that obtained this result at the Paul Scherrer Institut in Villigen includes scientists from the Max Planck Institute of Quantum Optics, Ludwig-Maximilians-Universität, the Institut für Strahlwerkzeuge of Universität Stuttgart, and the University of Coimbra, Portugal. The team is now attempting to explain the discrepancy, and re-examining the results of both previous high-precision measurements and complex calculations involving scattering cross section. If no errors are found in the measurements or calculations, it could be necessary to re-examine the world's most precise and best-tested fundamental theory: quantum electrodynamics. The proton radius remains a puzzle as of 2017. Perhaps the discrepancy is due to new physics, or the explanation may be an ordinary physics effect that has been missed.

The radius is linked to the form factor and momentum-transfer cross section. The atomic form factor G modifies the cross section corresponding to point-like proton.

The atomic form factor is related to the wave function density of the target:

${\displaystyle G(q^2)=\int <!--\; \int \; -->e^{iqr}\mathrm{\psi <!--\; \psi \; -->}(r{)}^{2}d{r}^3}$

The form factor can be split in electric and magnetic form factors. These can be further written as linear combinations of Dirac and Pauli form factors.

Pressure inside the protonedit

Since the proton is composed of quarks confined by gluons, an equivalent pressure which acts on the quarks can be defined. This allows calculation of their distribution as a function of distance from the centre using Compton scattering of high-energy electrons (DVCS, for deeply virtual Compton scattering). The pressure is maximum at the centre, about 1035 Pa which is greater than the pressure inside a neutron star. It is positive (repulsive) to a radial distance of about 0.6 fm, negative (attractive) at greater distances, and very weak beyond about 2 fm.

Charge radius in solvated proton, hydroniumedit

The radius of hydrated proton appears in the Born equation for calculating the hydration enthalpy of hydronium.