Short description: Measure of the size of atomic nuclei

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 has some similarity to that of defining a radius for the entire atom; neither have well defined boundaries. However, basic liquid drop models of the nucleus imagine a fairly uniform density of nucleons, theoretically giving a more recognizable surface to a nucleus than an atom: the latter being composed of highly diffuse electron clouds with density gradually reducing away from the centre.

For individual proton and neutron, and small nuclei, the concept of size and boundary can be less clear. A single nucleon needs to be regarded as a “color confined” bag of 3 valence quarks, binding gluons and so called “sea” quark antiquark pairs. Additionally the nucleon is surrounded by its Yukawa pion field responsible for the strong nuclear force. Indeed it could be difficult to decide even whether to include the surrounding Yukawa meson field as part of the proton or nucleon size or if that should be regarded as separate entity.

Fundamentally important are realizable experimental procedures to measure some aspect of size, whatever that may mean in the quantum realm of atoms and nuclei.

Foremost, the nucleus can be modeled as a sphere of positive charge for the interpretation of electron scattering experiments: 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 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, rd (obtained from scattering data), and the bound-state charge radius, Rd, 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

$\displaystyle{ R_{\rm d} = \sqrt{r_{\rm d}^2 + \frac{3}{4}\left(\frac{m_{\rm e}}{m_{\rm d}}\right)^2 \left(\frac{\lambda_{\rm C}}{2\pi}\right)^2}, }$

where me and md are the masses of the electron and the deuteron respectively while λC is the Compton wavelength of the electron.[5] For the proton, the two radii are the same.[5]

History

Main page: Physics:Geiger–Marsden experiment

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):

Rr0A13

where the empirical constant r0 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: Rp = 0.8751(61)×10−15 m
deuteron: Rd = 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

1. 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)fm2).
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
3. Foldy, L. L. (1958), "Neutron–Electron Interaction", Rev. Mod. Phys. 30 (2): 471–81, doi:10.1103/RevModPhys.30.471, Bibcode1958RvMP...30..471F .
4. Friar, J. L.; Martorell, J.; Sprung, D. W. L. (1997), "Nuclear sizes and the isotope shift", Phys. Rev. A 56 (6): 4579–86, doi:10.1103/PhysRevA.56.4579, Bibcode1997PhRvA..56.4579F .
5. Template:CODATA 1998
6. Geiger, H.; Marsden, E. (1909), "On a Diffuse Reflection of the α-Particles", Proceedings of the Royal Society A 82 (557): 495–500, doi:10.1098/rspa.1909.0054, Bibcode1909RSPSA..82..495G .
7. 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 .
8. Blatt, John M.; Weisskopf, Victor F. (1952), Theoretical Nuclear Physics, New York: Wiley, pp. 14–16 .
9. Sick, Ingo (2003), "On the rms-radius of the proton", Phys. Lett. B 576 (1–2): 62–67, doi:10.1016/j.physletb.2003.09.092, Bibcode2003PhLB..576...62S .
10. Sick, Ingo; Trautmann, Dirk (1998), "On the rms radius of the deuteron", Nucl. Phys. A 637 (4): 559–75, doi:10.1016/S0375-9474(98)00334-0, Bibcode1998NuPhA.637..559S .
11. Template:CODATA 2002
12. Antognini, A.; Nez, F.; Schuhmann, K.; Amaro, F. D.; Biraben, F.; Cardoso, J. M. R.; Covita, D. S.; Dax, A. et al. (2013). "Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen". Science 339 (6118): 417–420. doi:10.1126/science.1230016. PMID 23349284. Bibcode2013Sci...339..417A.

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