Nucleus charge distribution

Total (Epnai) and electronic energy curve of Th 73+ with the point-like (Epn) and finite (Ejn) nuclei charge distributions (38). The total energy is the sum of the electronic energy and Z, h/R term. All energies are in Hartree. 

There are, however, differences among the expectation values of these observables which deserve a few comments. The charge distribution has the aspect of a local quantity, expressed as a scalar. We shall make use in the following of the total (electrons and nuclei) charge distribution ... 

It is well known that for heavy atoms the effect of the finite nucleus charge distribution has to be taken into account (among other effects) in order to describe the electronic structure of the system correctly (see e.g. (36,37)). As a preliminary step in the search for the effect of the finite nuclei on the properties of molecules the potential energy curve of the Th 73+ has been calculated for point-like and finite nuclei models (Table 5). For finite nuclei the Fermi charge distribution with the standard value of the skin thickness parameter was adopted (t = 2.30 fm) (38,39). 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear... 

The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

The shape of the nucleus is best described by a power series, the relevant term of which yields the nuclear quadrupole moment. In Cartesian coordinates, this is represented by a set of intricate integrals of the type J p (r)(3x,x, — 6-jr )Ax, where x, = x, y, z, and pfifi) is the nuclear charge distribution (4.12). The evaluation of Pn(r) for any real nucleus would be very challenging. 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The influence of a noncubic electronic charge distribution interacting with a Mossbauer nucleus may be exemplified by using point charges, for which the EFG is easy to calculate. A point charge <7 at a distance r = +y from a... 

The charge distribution of the nucleus was taken to be spherically symmetric and represented by a single Gaussian of the form ... 

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... 

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

Quadrupolar nuclei Those nuclei, which because of their spin quantum number (which is always >1/2), have asymmetric charge distribution and thus posses an electric quadrupole as well as a magnetic dipole. This feature of the nucleus provides an extremely efficient relaxation mechanism for the nuclei themselves and for their close neighbors. This can give rise to broader than expected signals. 

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

Fig. 1 (a) Schematic representation of the spherical and non-spherical charge distribution in a nucleus. The value of electric quadrupole moment Q for the quadrupolar nucleus depends on the isotope under consideration, (b) The quadrupolar interaction arises from the interaction of Q with surrounding electric field gradient (EFG)... 

A second relation between p(r) and 4>(r) may be obtained by noting that the sources of 0(r) are the point charge Ze of the nucleus, located at the origin and the charge distribution due to the N electrons. Treating the charge density —ep(r) of the electrons as continuous, Poisson s equation of electrostatics may be used to write... 

Asymmetry in the ligand environment, either geometric or in charge distribution (or both), affect the asymmetry parameter, tp An r = 0 value corresponds to complete axial symmetry, whereas r = 1 corresponds to pure rhombic symmetry. Electric monopole interactions between the nuclear charge distributions and the electrons at the nucleus cause a shift of the nuclear ground and excited states. These interactions are known as the isomer shift, 8. Both the Mossbauer source and the absorber (the sample of interest) experience an isomer shift, and it is customary to quote 8 relative to a standard, usually Fe metal or Na2[Fe(CN)5NO] 2H2O at... 

The sign of V(r) at any point in space depends on whether the nuclear contribution (positive) or the electronic (negative) dominates. For neutral, spherically-averaged free atoms, V(r) is positive everywhere, decreasing monotonically with radial distance from the nucleus [3, 4]. When atoms interact to form molecules, however, the concomitant polarization of their charge distributions (relatively minor but very important) results in the development of regions of negative potential. These occur primarily... 

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