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.
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, 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 and which is appropriate for treating spectroscopic data. The two radii are related by
The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909, 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.
Later studies found an empirical relation between the charge radius and the mass number, A, for heavier nuclei (A > 20):
- R ≈ r0A⅓
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.
Modern direct measurements are based on the scattering of electrons by nuclei. 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. 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.
The 2010 CODATA recommended values are:
- proton: Rp = 0.8775(51)*10-15 m
- deuteron: Rd = 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, 0.84087(39) fm: the reason for this discrepancy is not clear.
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Other relevant material
Charge radii of the nuclides: www-nds.iaea.org/publications/indc/indc-hun-0033.pdf