Nuclear Charge Density Distribution Models

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] 

For the definition of the complete and incomplete gamma functions, F(a) and P a,x), see [28, Chap. 6]. Thus Rpa is the radius of a homogeneous charge density distribution yielding the same value for the Barrett moment as the charge density distribution under discussion. For Barrett equivalent radii parameters see, e.g., [35,46,47]. 

This section contains detailed information for a few spherical nuclear charge density distribution models. The models included here are either frequently used in electronic structure calculations or are of importance due to their use for representing nuclear charge density distributions as obtained from experiment. For further details on these models, and for a detailed discussion of a large number of other models for nuclear charge density distributions, see [41]. The following general symbols will be used in this section  

R a model-specific( ) characteristic radial size parameter, related to the extension of p r) and thus to the size of the nucleus  

Pq the normalization constant of p(r), in several cases identical with the coefficient of the constant term in Eq. (34)  

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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. 

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... 

Both requirements can be fulfilled quite easily in the following way. Different charge density distributions models can be standardized to a particular value of the moment function M p) for some fixed p. We follow the usual choice p = 2, which gives the rms radius, Eq. (44), to standardize our nuclear models. 

A relation of this form, not for a but for a nuclear radius parameter R, has been proved first by Elton [17, App. C] for the two-parameter Fermi-type charge density distribution model (see Sect. 4.5). 

In this model the nuclear charge density distribution is represented as a Fourier-Bessel expansion (see also Fig. 5) ... 

In this section we consider those applications of nuclear charge density distributions which are common practice in electronic structure calculations. We start with the electrostatic potentials resulting from the use of the most popular nuclear models, and continue with standard applications in electronic structure calculations. In the following we consider four popular nuclear models,... 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The expectation values from Eq. (38) can be used in various ways to characterize the charge density distribution. We only mention the following formulae, introduced by Ravenhall and Yennie [19] as general model-independent expressions for a characteristic nuclear radius parameter Rfci and a skin thickness parameter tva-... 

None of these implementations can be used to study effects due to variar tion of the finite nucleus model, due to their limitation to a single finite nucleus model. Of course, it is unlikely that such variations lead to significant changes in the chemical behaviour of atoms and molecules, e.g., reaction enthalpies, valence electronic charge density distribution etc. However, the finer details of the electron distribution in the vicinity of heavy atomic nuclei will be more sensitive to the variation of the finite nucleus model, but this is clearly a field in the area of atomic and nuclear physics. 

Therefore, one usually employs perturbation theory as discussed throughout this chapter and sticks to the point-charge model for atomic nuclei (or to some simplified spherically symmetric model density distribution) in quantum chemistry. The tiny effects of multipole moments of the nuclear charge density are then not included in the variational procedure for the determination of the electronic wave function. 

Since a purely theoretical, quantum mechanical determination of the nuclear structure, i.e., a determination of the nuclear state functions from which the charge and current density distributions could be obtained, is neither routinely feasible nor intended within an electronic structure calculation, we have to resort to model distributions. The latter may be rather simple mathematical functions, or much more sophisticated expressions deduced from a careful analysis of experimental data. 

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