Nucleus 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. 

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...

This is a result of the use of a point charge as a model for the nucleus. As discussed elsewhere in this volume, there are a number of alternative models using a finite nuclear size, and the choice of model affects the behavior of the wavefunction at the nucleus. We will not go into a detailed discussion here, but just summarize the facts about the behavior of the wavefunction at small r which will be of relevance to our later discussion. Point nucleus. 

Equation 3.5 ignores distortion of y>s(0) hy the finite nuclear size which will cause an overestimation of the shift. It should also be mentioned that there is a second possible contribution to electron density at the nucleus if electronic states of the atom are occupied. However, this term will always be much less than the equivalent i-term. Furthermore, the electronic state is not usually encountered in applications of Mossbauer spectroscopy and will therefore not be considered f urther. 

As stated in Section 1, one of fhe goals of this work is to use the comparison between theory and experiment for the isotope shift to determine the nuclear charge radius for various isotopes of helium and other atoms. One of the most interesting and important examples is the charge radius of the Tialo nucleus He. For a light atom such as helium, the energy shift due to the finite nuclear size is given to an excellent approximation by... 

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. 

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... 

This distribution function is obtained straightforwardly from the known analytic expressions for the radial functions. Finally, the energy shift from first-order perturbation theory is obtained, for any finite nucleus model, in terms of a series expansion in X = 2ZR, where i is a model-specific radial nuclear size parameter, as... 

For finite nucleus models with a well-defined nuclear size parameter R, beyond which the nuclear charge density is exactly( ) zero and the nuclear potential is exactly( ) given as —Z/r (r > R), the energy shifts can be obtained directly from the matching condition for the logarithmic derivatives L (r) = / r)/P [r) of the radial functions in the inner... 

The solution to this dilemma is to recognize that the nucleus has a finite size, and that this should be accounted for. Ishikawa and coworkers showed that the use of a finite nucleus instead of a point nucleus allowed for more compact basis sets [12] and also eliminated problems with basis set balance close to the nucleus [13]. Visser et al. [14] performed a full relativistic optimization of exponents for the one-electron atoms Sn and U with and without a finite nucleus, showing that the use of a finite nuclear radius significantly decreased the maximum exponent. 

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

The first term in (4.6), Jp (r)r dr, depends only on the radial distribution of the nuclear charge. This term represents the so-called nuclear monopole moment, note that it is related to the extended finite size of the nucleus. ... 

Due to the finite size of the nuclear charge distribution, the relative distance between the nucleus and the electron is not constant but is subject to additional fluctuations with probability p r). Hence, the energy levels experience an additional shift... 

For calculations of the first order corrections for uranium ions we took into account the effect of finite size of nucleus. To perform it the Dirac equation for the states lsi/2, 2.s- /2, 2pi/2 was solved with the potential that corresponds to a Fermi distribution for the nuclear charge... 

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