Nuclear charge density

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

The variable p (r) denotes the nuclear charge density at a point r with coordinates r = xi,X2,x ), and V r) is the Coulomb potential set up at that point by all other charges (the Coulomb constant k = l/(47t o) is dropped in this description). The integration variable in (4.1) is the volume element dr = Ax dr2dx3. The origin of the coordinate system is chosen to coincide with the center of the nuclear charge. A more convenient expression can be obtained by expanding V r) at f = (0,0,0) in a Taylor series, that is,... 

Carrying out the Fourier transformation, one obtains the coordinate space representation for the third Zemach moment, in terms of the weighted convolution of two nuclear charge densities p r) [18]... 

The electrostatic energy of interaction Ee between the nuclear charge density pn ijk) at point (ijk) and the surrounding electric charge is given by 28,49)... 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

The existence of a nuclear quadrupole moment (eQ) is due to a distribution of nuclear charge density (p) with less than cubic symmetry. The nuclear charge distribution can be described by an electric quadrupole moment tensor, the components of which are related to a scalar quantity eQ, which is defined17 by... 

Nuclear charge density distributions in quantum chemistry... 

The present chapter deals with the representation of extended nuclei, and the use of suitable models for extended nuclei in theoretical approaches to the problem of electronic structure, with emphasis on the nuclear charge density distribution. We begin with a brief general description of nuclear... 

The state-dependent nuclear charge density distribution, p r), can then be obtained from the particle density distributions through convolution with the charge density distributions of the single nucleons, Pp(r) and Pn(r) respectively ... 

The charge density distributions of a single neutron, p (r), and of all neutrons (the second integral in Elq. (3)) both integrate to zero. Usually, the contribution of the neutrons is omitted in Eq. (3). In any case, normalization of this nuclear charge density distribution correctly yields the total nuclear charge Q = J d r p r) = Ze. 

With given nuclear charge density distribution p(r) and nuclear current density distribution j r), we can now obtain all quantities required to describe the resulting electric and magnetic fields from basic relations of the theory of electromagnetism [11]. 

The electric field, generated by the nuclear charge density distribution, is obtainable from the electrostatic potential or from the charge density distribution as... 

NUCLEAR CHARGE DENSITY DISTRIBUTIONS THEIR POTENTIAL AND OTHER PROPERTIES... 

The main subject of this section is the nuclear charge density distribution p r) and its relation to the resulting potential energy function V r). 

Spherical nuclear charge density distributions In the following we restrict ourselves to the case of spherical charge density distributions. In consequence, this restriction also gives spherical potentials, so that Eqs. (4) and (16) reduce to... 

Similar and additional other geometrical quantities for nuclear charge density distributions were defined by Myers [20], see also [21] and the comprehensive discussion in [22] which extends this subject to the case of non-spherical shapes. The most complete use of expectation values () (including the extension to arbitrary real powers of r) is made by the moment function M(p),... 

Another tjrpe of expectation value, which has also some importance for the characterization of nuclear charge density distributions, is the Barrett moment [24],... 

A reliable comparison of different charge density distributions, and therefore a careful analysis of differences in various ph3reical effects due to a change in the nuclear charge density, is only possible if two requirements are met. Firstly, the charge density distributions must be standardized in some way, so that they become comparable in a well defined sense. Secondly, a relation to a particular nuclide or to a full sequence of nuclides must be established. 

In those cases where particular selected nuclides (with their proton numbers Z and neutron numbers N) are to be modelled, their corresponding experimental rms radii a(Z, N) can be imposed on every suitable nuclear charge density distribution model (for experimental values of rms radii see, e.g., [7,35]). If, on the other hand, one is interested in studying trends depending on the nuclear mass number A or on the atomic number Z, an expression for the rms radius a as a function of these numbers is required. 

Nuclear charge density distribution models have also been standardized with respect to another model, in particular to the homogeneous model (see Sect. 4.3) due to the simplicity of this model. Thus, an equivalent homogeneous radius can be associated with every expectation value for a power of the radius, as introduced by Ford and Wills [45]... 

---