Nuclear charge distribution finite

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... 

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... 

The electrostatic interaction of s-electrons with a nucleus of finite dimensions produces shifts in the nuclear energy levels. Since nuclear charge distributions generally vary from one nuclear state to another, the magnitude of the shift will also depend on the nuclear state. It may be shown (5) that a gamma ray photon emitted in a transition from an excited state e > to the ground state g > will be shifted in energy by an amount... 

The point charge model is sufficiently accurate if one is interested in valence properties of atoms and molecules, however, more realistic finite nucleus models may be used instead. In recent years a Gaussian nuclear charge distribution (Visser et al. 1987),... 

The finite electronic charge density at the nucleus gives rise to an electrostatic interaction with the nuclear charge distribution which represents a correction of the Coulomb potential of the nucleus experienced by electrons. In an optical transition a change 8p of the average relativistic total charge density across the nucleus may occur that manifests itself in the field shift contribution... 

Figure 6.6 Comparison of ground-state energies E[glZ scaled by I7 obtained tor hydrogen-iike atoms from Schrodinger quantum mechanics (horizontal line on top at -0.5 hartree), from Dirac theory with a Couiomb potential from a point-like nucleus (dashed line) and from Dirac theory with a finite nuclear charge distribution of Gaussian form (thin black line). The highest energy of the positronic continuum states, -2meC, appears as a thick black line, which is bent because of the l/Z scaling.

Figure 6.7 This figure comprises the radial functions shown already in Figure 6.1, in Hartree atomic units, pius those obtained for Gaussian nuclear charge distribution for the one-electron atom with Z = 170. Oniy in the case of a finite nucleus, quantum mechanical states of atoms with nuciear charges Z > c are defined (c 137.037 in Hartree atomic units).

This review article presents a good overview on all finite-nuclear charge distributions which have been invented to model (mainly radial symmetric) extended nuclear charges. The number of such models is huge and this review provides a very useful classification of the many different empirical approaches. 

The isomer shift of the absorption lines in the Mossbauer spectrum, also sometimes known as the chemical shift, the chemical isomer shift or the centre shift, is a result of the electric monopole (Coulomb) interaction between the nuclear charge distribution over the finite nuclear volume and the electronic charge density over this volume. This shift arises because of the difference in the nuclear volume of the ground and excited states, and the difference between the electron densities at the Mossbauer nuclei in different materials. In a system where this electric monopole interaction is the only hyperfine interaction affecting the nuclear energy levels, the nuclear ground and excited states are unsplit, but their separation is different in the source and absorber by an amount given by the isomer shift <5. 

At this point it is useful to summarize some of the features of the solutions for point and finite nuclear charge distributions. 

At least inside the nucleus, P and Q are essentially Gaussian in shape. This means that in a method using a Gaussian basis set a nuclear charge distribution with a finite radius is preferred to a point nucleus the basis then has the right behavior at the origin, and the demands on the basis are smaller because of the cutoff in the potential (Visser et al. 1987, Ishikawa et al. 1985). 

If a nuclear charge distribution with a finite radius is to be used, the question of the functional form of the distribution must be raised. The point nuclear model was simple now we have to consider some form of the nuclear charge distribution that bears some relation to experimentally determined distributions. The Coulomb potential at a point r from a charge distribution p (r) is... 

The source of the electric field can be an externally applied field, or it can originate in the components of the nuclear potential that are not included in the internal component of the field (that is, the nuclear potential V). Such components arise from the nonspherical nature of the nucleus, the lowest-order term of which is the quadrupole moment. The implementation of a finite-nuclear model is quite straightforward we simply expand the nuclear charge distribution in a series ... 

The evaluation of the spin-orbit integrals now reduces to the same form as the one-electron spin-orbit integrals with a finite Gaussian nuclear charge distribution. 

The finite charge density of s-electrons at the nucleus gives rise to an electrostatic interaction with the nuclear charge distribution, generating a shift contribution which, in lowest... 

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 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. ... 

We have also performed the calculation of hyperfine coupling constants the electric quadrupole constant B and magnetic dipole constant A, with inclusion of nuclear finiteness and the Uehling potential for Li-like ions. Analogous calculations of the constant A for ns states of hydrogen-, lithium- and sodiumlike ions were made in refs [11, 22]. In those papers other bases were used for the relativistic orbitals, another model was adopted for the charge distribution in the nuclei, and another method of numerical calculation was used for the Uehling potential. 

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