The **rms charge radius** is a measure of the size of an atomic nucleus, particularly the proton distribution. It can be measured by the scattering of electrons by the nucleus. Relative changes in the mean squared nuclear charge distribution can be precisely measured with atomic spectroscopy.

## Definition

The problem of defining a radius for the atomic nucleus is similar to that of defining a radius for the entire atom; 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.

This definition of charge radius can also be applied to composite hadrons such as a proton, neutron, pion, or kaon, that are made up of more than one quark. In the case of an anti-matter baryon (e.g. an anti-proton), and some particles with a net zero electric charge, the composite particle must be modeled as a sphere of negative rather than positive electric charge for the interpretation of electron scattering experiments. In these cases, the square of the charge radius of the particle is defined to be negative, with the same absolute value with units of length squared equal to the positive squared charge radius that it would have had if it was identical in all other respects but each quark in the particle had the opposite electric charge (with the charge radius itself having a value that is an imaginary number with units of length).^{[1]} It is customary when charge radius takes an imaginary numbered value to report the negative valued square of the charge radius, rather than the charge radius itself, for a particle.

The best known particle with a negative squared charge radius is the neutron. The heuristic explanation for why the squared charge radius of a neutron is negative, despite its overall neutral electric charge, is that this is the case because its negatively charged down quarks are, on average, located in the outer part of the neutron, while its positively charged up quark is, on average, located towards the center of the neutron. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. But, this is only the simplest of a variety of theoretical models, some of which are more elaborate, that are used to explain this property of a neutron.^{[2]}

For deuterons and higher nuclei, it is conventional to distinguish between the scattering charge radius, *r*_{d} (obtained from scattering data), and the bound-state charge radius, *R*_{d}, which includes the Darwin–Foldy term to account for the behaviour of the anomalous magnetic moment in an electromagnetic field^{[3]}^{[4]} and which is appropriate for treating spectroscopic data.^{[5]} The two radii are related by

where *m*_{e} and *m*_{d} are the masses of the electron and the deuteron respectively while *λ*_{C} is the Compton wavelength of the electron.^{[5]} For the proton, the two radii are the same.^{[5]}

## History

The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909,^{[6]} under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester, UK. The famous experiment involved the scattering of α-particles by gold foil, with some of the particles being scattered through angles of more than 90°, that is coming back to the same side of the foil as the α-source. Rutherford was able to put an upper limit on the radius of the gold nucleus of 34 femtometres.^{[7]}

Later studies found an empirical relation between the charge radius and the mass number, *A*, for heavier nuclei (*A* > 20):

*R*≈*r*_{0}*A*^{⅓}

where the empirical constant *r*_{0} of 1.2–1.5 fm can be interpreted as the Compton wavelength of the proton. This gives a charge radius for the gold nucleus (*A* = 197) of about 7.69 fm.^{[8]}

## Modern measurements

Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei.^{[9]}^{[10]} There is most interest in knowing the charge radii of protons and deuterons, as these can be compared with the spectrum of atomic hydrogen/deuterium: the nonzero size of the nucleus causes a shift in the electronic energy levels which shows up as a change in the frequency of the spectral lines.^{[5]} Such comparisons are a test of quantum electrodynamics (QED). Since 2002, the proton and deuteron charge radii have been independently refined parameters in the CODATA set of recommended values for physical constants, that is both scattering data and spectroscopic data are used to determine the recommended values.^{[11]}

The 2014 CODATA recommended values are:

- proton:
*R*_{p}= 0.8751(61)×10^{−15}m - deuteron:
*R*_{d}= 2.1413(25)×10^{−15}m

Recent measurement of the Lamb shift in muonic hydrogen (an exotic atom consisting of a proton and a negative muon) indicates a significantly lower value for the proton charge radius, 0.84087(39) fm: the reason for this discrepancy is not clear.^{[12]}

## References

**^**See, e.g., Abouzaid, et al., "A Measurement of the K0 Charge Radius and a CP Violating Asymmetry Together with a Search for CP Violating E1 Direct Photon Emission in the Rare Decay KL->pi+pi-e+e-", Phys. Rev. Lett. 96:101801 (2006) DOI: 10.1103/PhysRevLett.96.101801 https://arxiv.org/abs/hep-ex/0508010 (determining that the neutral kaon has a negative mean squared charge radius of -0.077 ± 0.007(stat) ± 0.011(syst)fm^{2}).**^**See, e.g., J. Byrne, "The mean square charge radius of the neutron", Neutron News Vol. 5, Issue 4, pg. 15-17 (1994) (comparing different theoretical explanations for the neutron's observed negative squared charge radius to the data) DOI:10.1080/10448639408217664 http://www.tandfonline.com/doi/abs/10.1080/10448639408217664#.U3GYaPldVUA**^**Foldy, L. L. (1958), "Neutron–Electron Interaction",*Rev. Mod. Phys.*,**30**(2): 471–81, Bibcode:1958RvMP...30..471F, doi:10.1103/RevModPhys.30.471.**^**Friar, J. L.; Martorell, J.; Sprung, D. W. L. (1997), "Nuclear sizes and the isotope shift",*Phys. Rev. A*,**56**(6): 4579–86, arXiv:nucl-th/9707016, Bibcode:1997PhRvA..56.4579F, doi:10.1103/PhysRevA.56.4579.- ^
^{a}^{b}^{c}^{d}Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998" (PDF).*Journal of Physical and Chemical Reference Data*.**28**(6): 1713–1852. Bibcode:1999JPCRD..28.1713M. doi:10.1063/1.556049. Archived from the original (PDF) on 2017-10-01. **^**Geiger, H.; Marsden, E. (1909), "On a Diffuse Reflection of the α-Particles",*Proceedings of the Royal Society A*,**82**(557): 495–500, Bibcode:1909RSPSA..82..495G, doi:10.1098/rspa.1909.0054.**^**Rutherford, E. (1911), "The Scattering of α and β Particles by Matter and the Structure of the Atom",*Phil. Mag.*, 6th Series,**21**(125): 669–88, doi:10.1080/14786440508637080.**^**Blatt, John M.; Weisskopf, Victor F. (1952),*Theoretical Nuclear Physics*, New York: Wiley, pp. 14–16.**^**Sick, Ingo (2003), "On the rms-radius of the proton",*Phys. Lett. B*,**576**(1–2): 62–67, arXiv:nucl-ex/0310008, Bibcode:2003PhLB..576...62S, doi:10.1016/j.physletb.2003.09.092.**^**Sick, Ingo; Trautmann, Dirk (1998), "On the rms radius of the deuteron",*Nucl. Phys. A*,**637**(4): 559–75, Bibcode:1998NuPhA.637..559S, doi:10.1016/S0375-9474(98)00334-0.**^**Mohr, Peter J.; Taylor, Barry N. (2005). "CODATA recommended values of the fundamental physical constants: 2002" (PDF).*Reviews of Modern Physics*.**77**(1): 1–107. Bibcode:2005RvMP...77....1M. doi:10.1103/RevModPhys.77.1. Archived from the original (PDF) on 2017-10-01.**^**Antognini, A.; Nez, F.; Schuhmann, K.; Amaro, F. D.; Biraben, F.; Cardoso, J. M. R.; Covita, D. S.; Dax, A.; Dhawan, S.; Diepold, M.; Fernandes, L. M. P.; Giesen, A.; Gouvea, A. L.; Graf, T.; Hänsch, T. W.; Indelicato, P.; Julien, L.; Kao, C. -Y.; Knowles, P.; Kottmann, F.; Le Bigot, E. -O.; Liu, Y. -W.; Lopes, J. A. M.; Ludhova, L.; Monteiro, C. M. B.; Mulhauser, F.; Nebel, T.; Rabinowitz, P.; Dos Santos, J. M. F.; Schaller, L. A. (2013). "Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen".*Science*.**339**(6118): 417–420. Bibcode:2013Sci...339..417A. doi:10.1126/science.1230016. hdl:10316/79993. PMID 23349284.