Finite nuclear size

The finite size of the nuclear charge distribution modifies the nuclear potential near the nucleus. If one assumes a spherically symmetric nuclear charge distribution, Pnuc( )i the corresponding nuclear potential is 

Finite nuclear size corrections to energies of n = 1 and n R = 1.04 fm) and hydrogen-like uranium (R = 7.25 fm). 

The root-mean-square radius Rrma for this uniform distribution is /inns = y/S/5R. A fit to nuclear radii obtained in electron-nucleus scattering experiments and muonic x-ray measurements led to the empirical formula [27] 

In Table 1, we give the corrections to n = 1 and n = 2 states of hydrogen and hydrogenlike uranium. The correction are seen to be most important for nsi/2 and npi/2 states they grow approximately as R Z for these states and fall off roughly as n.  

The behavior of wave functions near the nucleus, which is influenced by details of the nuclear charge distribution, is important in calculations of hyperfine constants and amplitudes of parity nonconserving transitions. The basic orbitals in such calculations are obtained from self-consistent field calculations in which Pnnc T) is assumed to be a Fermi (or Woods-Saxon) distribution 

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As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

The term AEcorr contains relativistic, relativistic recoil, Lamb shift, and finite nuclear size... 

The contributions that have been considered in order to obtain precise theoretical expressions for hydrogenic energy levels are as follows the Dirac eigenvalue with reduced mass, relativistic recoil, nuclear polarization, self energy, vacuum polarization, two-photon corrections, three-photon corrections, finite nuclear size, nuclear size correction to self energy and vacuum polarization, radiative-recoil corrections, and nucleus self energy. 

At present, contributions from two-photon corrections and finite nuclear size introduce the largest uncertainty in the theoretical expressions for energy levels. Corrections from two virtual photons, of order a2, have been calculated as a power series in Za ... 

The spectrum of hydrogen and one-electron ions provides a direct test of bound-state quantum electrodynamics. Except for finite nuclear-size and mass (recoil)... 

Table 4. Lamb shift contribution for the ground state of 238U91+ ion (in eV). Here Ro denotes the nuclear radius, M is nuclear mass and ao is the Bohr radius. The finite nuclear-size correction is calculated for a Fermi distribution with (r2)1 /2 = 5.860 0.002 fm. The corrections VPVP (f) and S(VP)E are known only in Uehling approximation. The inaccuracies assigned to these rather small corrections are estimated as the average of the inaccuracies of the Uehling approximation deduced from exact results for the corrections VPVP (e) and SEVP (g),(h),(i)...

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...

In particular, for the finite-nuclear-size correction one obtains... 

Table 4. Finite-nuclear-size contribution Eq. (12) against [9]. The uncertainty of the analytic result was discussed in Ref. [4]...

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. 

The V" includes the effect of the finite nuclear size, while some finer effect, like QED, can be added to the hDCn perturbatively. The DCB Hamiltonian in this form contains all effects through the second order in a, the fine-structure constant. 

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