Nucleus finite, effects

It is much more difficult to take into account the influence of finite dimensions and form of the nucleus (volume effect) on the atomic energy levels, because we do not know exactly the nuclear volume, or its form, or the character of the distribution of the charge in it. Therefore, in such cases one sometimes finds it by subtracting its part (22.35) from the experimentally measured total isotopic shift. Further on, having the value of the shift caused by the volume effect, we may extract information on the structure and properties of the nucleus itself. For the approximate determination of the isotope shift, connected with the differences dro of the nuclear radii of two isotopes, the following formula may be used [15] ... 

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. 

Equation (5.1) describes the size effects associated with the presence of a finite nucleus and reflects the balance between bulk and surface contributions. The limiting case L 00 represents the ideal formation of a film under complete wetting. The convention is that the system evolves towards negative values of G. When G > 0 the interface term dominates over the bulk term (low L values) while the opposite situation is found for G < 0 (higher L values). Underthe condition 9G/9L = 0, Gc and the critical length of nucleation, are given by the expressions ... 

One can consider also higher order radiative corrections to the finite size effect. Such contribution originating from very small distances and enhanced by the large logarithm ln(l/(mro)) (ro here is the radius of the nucleus) was obtained in [50, 51, 55]. Its relative magnitude with respect to the leading finite size contribution is a Za) and it is universal for S and P state. 

If the charged particle is a nucleus with no electron shells around it, the molecular stopping power depends on the velocity of the particle in the following way. At relativistic velocities v = c, the dependence of Se on v is determined by the logarithmic term in the brackets, because at v- c the factor preceding the brackets tends to some finite limit. So the molecular stopping power experiences the so-called relativistic rise. As for Se, its relativistic rise in real dense media is slowed down by the density effect (see Section V.B.2). 

All the symbols have their usual meanings. In the non-recoil limit, the motion of the nucleus is neglected and its finite mass enters only as a reduced mass of the electron. The additional terms arising from the dynamical effects of the nucleus, namely the recoil corrections and radiative-recoil corrections, have been omitted from equation 1 and will not be considered here. For more detailed discussions of the theory, see the review by Sapirstein and Yennie [3] and more recently [4,5,6], The expansion in (Za) is now carried out by expressing F and H as power series in (Za) and ln(Za) 2, as shown below in equations 2 and 3, where a is the ratio of the electron mass to its reduced mass. 

The present calculation takes into account all these effects for Coulomb-Coulomb and Coulomb-Breit interaction and omits effects l)-3) for Breit-Breit interaction. On the other hand we, in contrast to [2], carry out all the calculations in the point nucleus approximation. The corrections to the finite size of nucleus for the bound energy as well as the corrections to the finite size of nucleus for electron interaction are considered separately (see Table 1). 

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... 

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...

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