Nuclear size effect

There are two sources of radiative corrections to the leading nuclear size effect, namely, the diagrams with one-loop radiative insertions in the electron line in Fig. 6.5, and the diagrams with one-loop polarization insertions in one of the external Coulomb lines in Fig. 6.6. 

Fig. 6.5. Electron-line radiative correction to the nuclear size effect. Bold dot corresponds to proton form factor slope...

Fig. 7.15. Electron polarization corrections to the leading nuclear size effect...

Table 1. Some contributions to b in units of 10-9 [9] due to vacuum polarization (VP), self-energy (SE) and finite nuclear size NS). The nuclear size effects were studied there for the main isotope of each element...

Accurate calculations for the Lamb shift and hfs of hydrogen-like atoms are limited by their nuclear structure and higher-order QED corrections. In the case of low-Z Lamb shift, the finite-nuclear-size effects can be taken into account easily if we know the nuclear charge radius. 

Table 2 also shows the QED contributions to helium-like resonance lines in vanadium as determined by Drake. The QED contributions are also expressed as a proportion of the relevant transition in ppm. The level at which our measurements test these contributions is between 5.7% and 8%. The theoretical QED contributions include mass polarization and nuclear size effects but these contribute less than 1% to the total. If the QED contributions to the 21 states are assumed to be correct, then the Is QED contribution is measured to 6%. 

A measurement of the 1S-2S frequency alone can therefore give only a modest further improvement of the Rydberg constant before its interpretation is limited by other uncertainties. At one time it has been suggested to determine the proton-electron mass ratio from the 1S-2S isotope shift. However, after the most recent measurements by VAN DYCK et al. [37], the uncertainty of the isotope shift is now dominated by nuclear size effects. 

Contrary to the mass of the nucleus, its size influences the binding energy considerably in heavy ions (Fig. 10). In studying nuclear size effects nowadays always a spherically symmetric charge distribution of the nucleus is assumed which allows a separation of the Dirac equation and corresponding wave function into an angular part and a radial part similar to the point nucleus case. The radial Dirac equation then reads [45]... 

The large influence of the nuclear size on the binding energy arises the question whether the radiative corrections of first order in alpha are also influenced significantly by nuclear size effects. The extension of the nucleus influences all wave... 

The first two terms in Eq. (2) are dominant and stem from Dirac theory, the third term comes from bound-state QED. Both contributions are indicated in Figure 7.6, and it can be seen that at high Z the QED term is of the same order of magnitude as the nuclear size effects. From Eq. (1) it is clear that for Z larger or equal to a, g is undefined. Already when Z is no longer valid, non-perturbative calcu-... 

Above we have proposed to measure the the small difference frequency df = f(1S-2S) - 3 f(2S-nS). This frequency depends critically on the Lamb shifts of the participating levels, and can provide a stringent test of QED. For n=100, the theoretical uncertainty of df is dominated by the nuclear size effect ( 70 kHz) and by approximations in the computation of electron structure corrections and uncalculated higher order QED corrections (65 kHz). The contribution of the Rydberg constant (1 kHz), the electron mass (0.05 kHz), and the fine structure constant (4 kHz) are negligible by comparison. The QED computations can be improved, and if theory is correct, a precision measurement of df can provide accurate new values for the charge radii of the proton and deuteron. 

Measurements of the 1S-2S hydrogen-deuterium isotope shift have previously been suggested as a means to measure the proton/electron mass ratio.However, after recent improved measurements of this ratio,26 the uncertainty of the isotope shift is no longer dominated by the electron mass (35 kHz), but instead by nuclear size effects (180 kHz). 

IPs of internal conversion electrons (Is and 2 s) of Cn, FI, and elements 116 (livermorium, Lv) and 118 were predicted to an accuracy of a few 10 eV using DHF theory and taking into account QED and nuclear-size effects [44]. The transition energies for different ionization states of Mt were calculated using the same approach and compared with recent experiments on the a-decay of Rg [45]. 

A further recent contribution to the theory of the Lamb shift is an argument by Borie that finite nuclear size effects should be included directly in the calculation of the lowest order level shift of 5 Si states, gq. CD contains an overall multiplying factor of 4Trenuclear charge density. Th numerical factor in CD corresponds to choosing p(r) = ZedCr) for a point nucleus. Boris s correction results from using instead a finite nuclear distribution with the result... 

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