Charge radius

The rms charge radius is a measure of the size of an atomic nucleus, particularly of a proton or a deuteron. It can be measured by the scattering of electrons by the nucleus and also inferred from the effects of finite nuclear size on electron energy levels as measured in atomic spectra.

Definition

The problem of defining a radius for the atomic nucleus is similar to the problem of atomic radius, in that neither atoms nor their nuclei have definite boundaries. However, the nucleus can be modelled 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.

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^[1][2] and which is appropriate for treating spectroscopic data.^[3] The two radii are related by

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

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.^[3] For the proton, the two radii are the same.^[3]

History

Main article: Geiger–Marsden experiment

The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909,^[4] 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.^[5]

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

R ≈ r₀A^⅓

where r₀ is an empirical constant of 1.2–1.5 fm. This gives a charge radius for the gold nucleus (A = 197) of about 7.5 fm.^[6]

Modern measurements

Modern direct measurements are based on the scattering of electrons by nuclei.^[7][8] 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 finite 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.^[3] 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.^[9]

The 2010 CODATA recommended values are:

proton: R_p = 0.8775(51)*10^-15 m

deuteron: R_d = 2.1424(21)*10^-15 m

Recent work on the spectrum of muonic hydrogen (an exotic atom consisting of a proton and a negative muon) indicates a significantly lower value for the proton charge radius, 0.84184(67) fm: the reason for this discrepancy is not clear.^[10]

References

1. ^ Foldy, L. L. (1958), "Neutron–Electron Interaction", Rev. Mod. Phys. 30: 471–81, Bibcode 1958RvMP...30..471F, doi:10.1103/RevModPhys.30.471 .
2. ^ Friar, J. L.; Martorell, J.; Sprung, D. W. L. (1997), "Nuclear sizes and the isotope shift", Phys. Rev. A 56: 4579–86, arXiv:nucl-th/9707016, Bibcode 1997PhRvA..56.4579F, doi:10.1103/PhysRevA.56.4579 .
3. ^ ^{a b c d} Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998". J. Phys. Chem. Ref. Data 28 (6): 1713–1852. doi:10.1103/RevModPhys.72.351. 
4. ^ Geiger, H.; Marsden, E. (1909), "On a Diffuse Reflection of the α-Particles", Proc. Roy. Soc., Ser. A 82: 495–500, Bibcode 1909RSPSA..82..495G, doi:10.1098/rspa.1909.0054 .
5. ^ Rutherford, E. (1911), "The Scattering of α and β Particles by Matter and the Structure of the Atom", Phil. Mag., Ser. 6 21: 669–88, doi:10.1080/14786440508637080 .
6. ^ Blatt, John M.; Weisskopf, Victor F. (1952), Theoretical Nuclear Physics, New York: Wiley, pp. 14–16 .
7. ^ 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 .
8. ^ 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 .
9. ^ Mohr, Peter J.; Taylor, Barry N. (2005). "CODATA recommended values of the fundamental physical constants: 2002". Rev. Mod. Phys. 77 (1): 1–107. Bibcode 2005RvMP...77....1M. doi:10.1103/RevModPhys.77.1. 
10. ^ Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D.; Biraben, François; Cardoso, João M. R.; Covita, Daniel S.; Dax, Andreas et al. (2010), "The size of the proton", Nature 466 (7303): 213–16, Bibcode 2010Natur.466..213P, doi:10.1038/nature09250, PMID 20613837