Finite Nuclear Size Effects

Finally, we note that analytic expressions are available for the radial moments r ) of hydrogen-like Dirac atoms [130]. 

Theoretical nuclear physics does not provide a unique model function for the positive charge distribution derived from quantum chromodynamics. That is why there is a certain degree of arbitrariness in the choice of such functions. In the spherically symmetric case, the radial Poisson equation 

Many model potentials (pnuc f) have been used [131] but two have become most important in electronic structure calculations. These are the homogeneous and the Gaussian charge distributions. The homogeneously or uniformly charged sphere is a simple model for the finite size of the nucleus. It is piecewise defined, because the positive charge distribution is confined in a sphere of radius R. The total nuclear charge -f-Ze is uniformly distributed over the nuclear volume 4 rR /3, 

The radius R has to be fixed empirically and may be understood as the size of the nucleus. This charge density distribution leads through Eq. (6.150) and multiplication by qe = to the homogeneous electron-nucleus potential energy operator 

outside the spherical nucleus, for r R, the ordinary Coulomb attraction governs the electron-nucleus interaction. 

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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. 

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... 

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. 

Table 5. Lamb shift contribution for the ground state of 208Pb81+ i0n (in eV). The notations are the same as in Table 4. The finite nuclear size correction is calculated for a Fermi distribution with (r2 1,/2 = 5.505 0.001 fm. The SESE (a) (irred) correction is obtained by an interpolation from the known values for Z = 70, 80,92. The inaccuracy of the Uehling approximation for VPVP (f) and S(VP)E corrections is neglected. The zero value presented for the nuclear polarization is due to the cancellation of the usual nuclear polarization [35] with the mixed nuclear polarization (NP)-vacuum polarization correction [36]. The latter effect arises when the nucleus interacts with a virtual electron-positron pair. For lead, due to the collective monopole vibrations, specific for this nucleus, mixed NP-VP effect becomes rather large. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for 208Pb, making this ion especially suitable for the most precise theoretical predictions...

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...

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. 

The theoretical energy levels are determined to high accuracy by the Dirac eigenvalue, quantum electrodynamic effects such as the self energy and vacuum polarization, finite-nuclear-size corrections, and nuclear motion effects. 

Nuclear Finite Size At low Z, the effect of the finite nuclear size on level energies is given by... 

From Uqi = (h/Sri ) this parameter is the intemuclear distance corrected for the effects of finite nuclear size. 

As a matter of fact, for hydrogenic atoms with nuclear charge Z > f37, it could even be imaginary. In practice, other effects, such as finite nuclear size, become important at that stage, and the imaginary y remains mainly a formal problem. 

Thus far in our discussion of relativistic expressions for properties we have assumed that the nuclei are represented by point charges. However, schemes for actual calculation of relativistic wave functions normally use nuclei with finite size in order to avoid problems with the weak singularity of the Dirac equation at the nucleus—and also because the nucleus really does have a finite size. The use of a point nucleus to calculate properties therefore appears somewhat inconsistent. At the very least we should know what errors we incur by using a point nucleus, and we will therefore discuss the low-order effects of finite nuclear size for electric and magnetic fields. 

We next consider the effect of finite nuclear size on the nuclear spin Hamiltonian. The electric moments were derived by considering the Coulomb interaction of the nuclear charge density, expanded in a multipole series, with the electrons. By analogy, the magnetic moments are derived by considering the Gaunt interaction of the nucleus with the electrons. It is at this point that we must consider, at least as a formal entity, the nuclear wave function, and from it obtain a nuclear spin density that interacts with the electron spin density. 

The operators of the Dirac Eq. (5) are 4 x 4 matrix operators, and the corresponding wave function is therefore a four-component (4c) vector (spinor). The V" includes the effect of the finite nuclear size, while some finer effect, like QED, can be added to the hocB perturbatively, although the self-energy QED term is more difficult to treat [36,47,48]. The DCB Hamiltonian in this form contains all effects through the second order in a, the fine-structure constant. 

Visscher L and Dyall K 1997 Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions. At. Data Nucl. Data Tables 67(2), 207-224. Autschbach J 2009 Magnitude of finite nucleus size effects in relativistic density functional computations of indirect nmr nuclear spin-spin coupling tensors. ChemPhysChem 10, 2274-2283. 

The last class of corrections contains nonelectromagnetic corrections, effects of weak and strong interactions. The largest correction induced by the strong interaction is connected with the finiteness of the nuclear size. 

Radiative Corrections to the Nuclear Finite Size Effect... 

Total radiative correction to the nuclear finite size effect has the form [20, 21]... 

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