Nuclear Fermi Distribution

For spherically symmetric nuclear charge distribution (Gaussian, Fermi, or point nucleus), the electric field at a point r outside the nucleus can be evaluated from Gauss law as... 

A somewhat more sophisticated approach to the problem of defining the nuclear size and density is to assume the nuclear density distribution, p(r), assumes the form of a Fermi distribution, that is,... 

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 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)...

One conceptually simple approach which has been used to represent temperature effects in metallic clusters is the random matrix model, developed by Akulin et al. [700]. The principles of the random matrix model, developed in the context of nuclear physics by Wigner and others, were outlined in chapter 10. The essential idea is to treat the cluster as a disordered piece of a solid. In the first approximation, the cluster is regarded as a Fermi gas of electrons, moving in an effective, spherically symmetric short range well. Without deformations, one-electron states then obey a Fermi distribution. As the temperature is raised, various scattering processes and perturbations arise, all of which lead to a random coupling between the states of the unperturbed system. One can... 

The shape of an arbitrary nuclear distribution can often be adequately described by the moments (r ") of the distribution. For the Fermi distribution above, these moments are given, to a good approximation [40], by the relations... 

Generalizations of the Fermi model to describe deformed nuclei have been applied e.g. in the study of energies for highly charged uranium [44,45]. The nuclear radius parameter c in the Fermi distribution (1) is then replaced by... 

TABLE I. Nuclear properties for the systems studied, together with a comparison of the hyperfine structure parameters obtained for a homogeneous distribution and a Fermi distribution, respectively. In addition, the x-parameters in Eqs. (4) and... 

We only briefly mention that a similar modification, i.e., a change from a Dirac delta distribution to an extended distribution, would be required for the spin-dependent electron-nucleus contact term, known as Fermi contact term, if the usual point-like nuclear magnetization distribution (the pointlike nuclear magnetic dipole approximation) is replaced by an extended nuclear magnetization distribution. 

As mentioned above, the function Pmic(r) in the PNC Hamiltonian is a weighted nuclear density function, with the weighting emphasizing the neutron density. Since there are no experimental values for the neutron density of Cs, we use instead an experimental proton density function. This proton density is taken to be a two-parameter Fermi distribution [45]... 

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... 

The effect of the Breit interaction on the wave function can be conveniently studied by comparing radial moments (r) of shells calculated with Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians. The effect is not very large as can be seen for Li- and Be-like ions in Figure 9.4. From the plot we note that the effect of the Breit interaction on the radial functions is small, but increases linearly with the nuclear charge number Z. Moreover, the four different models to describe the positive nuclear charge distribution (point-like, exponential, Gaussian shaped and Fermi) can hardly be distinguished. 

Both of these distributions present some problems for implementation in quantum chemical programs. The uniform sphere potential can be represented as the sum of the point nuclear potential and the difference between the point and uniform sphere nuclear potentials inside the nuclear radius. Then one has to perform integrals over a small finite region of space, which may have to be done numerically. A method for their evaluation has been presented by Matsuoka (1987). The Fermi distribution is difficult to represent in closed form and therefore the integrals would have to be performed numerically. An attractive alternative is to choose a Gaussian distribution,... 

The energies and are associated with the initial and final vibronic states, respectively ( (r R)i/ Fermi distribution which ensures that the initial (final) electronic state is occupied (free) at a given temperature. Note that here, in contrast with the traditional Born-Huang expansion, eqn (5), the vibrational and electronic indices are separated, since the vibronic states within a band are treated as separable. In the spirit of first-order perturbation theory, the non-adiabatic contribution of each nuclear degree of freedom is treated as an independent relaxation channel... 

Here /, is the 13C nuclear spin, S is the unpaired electronic spin, and A j- is the Fermi contact hyperfine coupling tensor. This coupling is identical for all 13C nuclei as long as the C60 ion is spherical, but becomes different for different nuclei after the Jahn-Teller distortion leading to an inhomogeneous frequency distribution. The homogeneous width of the 13C NMR lines is, on the other hand, mainly determined by the electron-nuclear dipolar interaction... 

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