Hartree-Fock calculations for finite nuclei with equivalent nonlocal potentials

The method of constructing equivalent regular two-body potentials by a unitary transformation of the two-body Hamiltonian has been generalized to spin-parity dependent nuclear potentials containing tensor- and spin-orbit terms. Starting from the Gammel-Christian-Thaler potential, which includes tensor forces, we obtained a class of equivalent regular, but nonlocal potentials depending on a parameterλ — the range of nonlocality. — These potentials have been used in a Hartree-Fock calculation for the closed-shell nuclei He⁴, C¹², O¹⁶, Si²⁸, S³², Ca⁴⁰. The calculated binding energies show a slowλ-variation with a minimum in the region of 0.7 f. The nuclear radii decrease with increasingλ and are in general too small. The sequence of single particle levels of the nuclei with closedl- shells is in agreement with that obtained with the usual nuclear shell model potential including spin-orbit coupling.     

Hyperon properties in finite nuclei using realistic YN interactions

Single-particle energies of Λ and Σ hyperons in several nuclei are obtained from the relevant self-energies. The latter are constructed within the framework of a perturbative many-body approach employing present realistic hyperon-nucleon interactions such as the models of the Jülich and Nijmegen groups. The effects of the non-locality and energy dependence of the self-energy on the bound states are investigated. It is also shown that, although the single-particle hyperon energies are well reproduced by local Woods-Saxon hyperon-nucleus potentials, the wave functions from the non-local self-energy are far more extended. Implications of this behavior for the mesonic weak decay of Λ hypernuclei are discussed.     

Multi-configuration Hartree-Fock theory in nuclei

A variational method for the self-consistent solution of the nuclear many body problem with the inclusion of correlations is formulated. The trial function in this multiconfiguration-Hartree-Fock (MCHF) theory is a linear combination of unrestricted Slater determinants. The MCHF equations are given and a simple procedure for solving them is outlined. A great advantage of this method is that it also yields the excited states. It is shown that the trial function is stable against particle-hole excitations. Therefore the Slater determinants differ from each other at least by two particle — two hole excitations. This method is applied to the Lipkin model. In the MCHF method the difference to the exact solution is reduced by a factor three to ten compared with the corresponding value in the HF approach.     

Hartree-Fock calculations of bubble nuclei

Hartree-Fock calculations of ³⁶Ar and of ²⁰⁰Hg in spherical configurations exhibit densities which are very much depleted in the interior region. The depletion agrees well with previous theoretical predictions of bubble nuclei.     

Hartree-Fock calculations of super-heavy nuclei

Hartree-Fock calculations are performed in the region of super heavy nuclei using zero range (Skyrme) forces and the finite range force G-0; the latter predict shell closure for the (α-unstable) nucleus N = 218, Z = 126 while the former do not. Calculations performed using the density matrix expansion (DME or DMEX) approximations give results similar to the Skyrme force. The problem is traced to an inadequacy of the DME, incorrectly placing the nodeless n = 1 high ? states.     

Deformed Hartree-Fock calculation of proton-rich nuclei

We perform Hartree-Fock+BCS calculations for even-even nuclei with 2 ≤ Z ≤ 82 and N ranging from outside the proton drip line to the experimental frontier on the neutron-rich side. The ground state solutions are obtained for 737 nuclei, together with shape-coexistence solutions for 480 nuclei. Our method features the Cartesian-mesh representation of single-particle wavefunctions, which is advantageous in treating nucleon skins and exotic shapes. The results are compared with those of the finite-range droplet model of Møller et al. as well as the experimental values.     

On the accuracy of Hartree-Fock calculations in light nuclei

Kato's theory provides rigorous mathematical estimates for the accuracy of approximate solutions of the eigenvalue equation of an arbitrary Hermitean operator H, the mean-square deviation of H representing a measure for the quality of an approximate solution. Using a quadrupole-quadrupole interaction, we calculate the mean-square deviations of HF, HFP, and PHF solutions for some nuclei in the sd shell, and check the accuracy of the Hartree-Fock solutions by applying Kato's estimates.     

Generalized shells in nuclei: Hartree-Fock calculations for bubble nuclei

Hartree-Fock calculations in the bubble degree of freedom have been performed on a variety of spherical nuclei. Of particular importance are incipient bubble configurations in ³⁶Ar, ⁶⁸Se, ⁸⁴Se, ¹⁰⁰Sn, ¹¹⁶Ce, ¹³⁸Ce and ²⁰⁰Hg, each of which possesses a binding energy which is comparable to that of the normal spherical closed-shell configuration. The densities of the above nuclei display strong deviations from a uniform shape, and give rise to depletions in the nuclear interior. These nonuniformities are due both to the absence of low angular momentum states in otherwise normally occupied spherical shells, and also to strong self-consistency effects. The nonuniformities in the mass density are further enhanced for nuclei whose neutron and proton densities have depressions or peaks at approximately the same distance from the center of the nucleus. A depression of the central density is most pronounced in the nuclei ³⁶Ar, ¹³⁸Ce and ²⁰⁰Hg. Interior depletions of the density are associated with the relatively higher energies of low angular momentum single-particle levels as compared to high angular momentum single-particle levels. This effect can give rise to moderately large gaps at the Fermi surface. Finally, it is shown that in a bubble configuration, the spin-orbit splitting of low lying doublets is sometimes reversed, and that this effect is especially pronounced for levels with low angular momentum.     

The calculation of Dirac sea effects on finite solitons

The effects of polarization of the Dirac sea on finite solitons in a simple theory in which fermions interact with a single scalar field are studied. The mass shift for a given background scalar field is computed numerically and compared to approximations arising from expansions in inverse powers of the effective fermion mass and in powers of derivatives of the background scalar field. The conditions under which such approximations succeed are discussed. When such approximations work one can derive local equations of motion for the soliton fields which include the effects of polarizing the Dirac sea. These new equations are studied and energy minimization is used to explore the effects of the Dirac sea on the structure of the soliton. Calculations for a typical Friedberg-Lee soliton are presented, and it is shown that, while the approximations do not work well for fields employed to model the quark structure of nucleons, they do provide an upper bound for the mass of the soliton. A scalar field typical of those used to model ¹⁶O in quantum hadrodynamics is also studied, and it is shown that, when the effective potential is supplemented by the next term occurring in a derivative expansion, the renormalized shift in the energy of the Dirac sea is well approximated.     

Self-consistent hartree description of finite nuclei in a relativistic quantum field theory

Relativistic Hartree equations for spherical nuclei are derived from a relativistic nuclear quantum field theory using a coordinate-space Green function approach. The renormalizable field theory lagrangian includes the interaction of nucleons with σ, ω, ρ and π mesons and the photon. The Hartree equations represent the “mean-field” approximation for a finite nuclear system. Coupling constants and the σ-meson mass are determined from the properties of nuclear matter and the rms charge radius in ⁴⁰Ca, and pionic contributions are absent for static, closed-shell nuclei. Calculated charge densities, neutron densities, rms radii, and single-nucleon energy levels throughout the periodic table are compared with data and with results of non-relativistic calculations. Relativistic Hartree results agree with experiment at a level comparable to that of the most sophisticated non-relativistic calculations to date. It is shown that the Lorentz covariance of the relativistic formalism leads naturally to density-dependent interactions between nucleons. Furthermore, non-relativistic reduction reveals non-central and non-local aspects inherent in the Hartree formalism. The success of this simple relativistic Hartree approach is attributed to these features of the interaction.     

Self-consistent description of finite nuclei based on a relativistic quark model

Relativistic Hartree equations for spherical nuclei have been derived from a relativistic quark model of the structure of bound nucleons which interact through the (self-consistent) exchange of scalar (σ) and vector (ω and ϱ) mesons. The coupling constants and the mass of the σ-meson are determined from the properties of symmetric nuclear matter and the rms charge radius in ⁴⁰Ca. Calculated properties of static, closed-shell nuclei from ¹⁶O to ²⁰⁸Pb are compared with experimental data and with results of Quantum Hadrodynamics (QHD). The dependence of the results on the nucleon size and the quark mass is investigated. Several possible extensions of the model are also discussed.     

Signature effects in odd mass ytterbium nuclei in an angular momentum projected Hartree-Fock analysis

We explain the signature effects in the spectrum of odd mass Yb nuclei in a projected Hartree-Fock calculation without assuming gamma asymmetry. Rotation-alignment of nucleons in largej orbits is responsible for the signature dependence in energy and transition rates.     

Renormalization effects in a field theory of finite nuclei

In case of thep-α system, the Coulomb exchange potential of the resonating group model and of the fish bone optical model are compared with each other. It is seen that the difference between the two approximations arises mainly from the unrealistic shortdistance behaviour of the proton-protone ²/r potential, which has been assumed in both models. In case of the α-¹⁶O system, it is demonstrated that the importance of the Coulomb exchange potential can be very much reduced by an off-shell transformation.     

Spin symmetry in Dirac negative-energy spectrum in density-dependent relativistic Hartree-Fock theory

The spin symmetry in the Dirac negative-energy spectrum and its origin are investigated for the first time within the density-dependent relativistic Hartree-Fock (DDRHF) theory. Taking the nucleus ¹⁶O as an example, the spin symmetry in the negative-energy spectrum is found to be a good approximation and the dominant components of the Dirac wave functions for the spin doublets are nearly identical. In comparison with the relativistic Hartree approximation where the origin of spin symmetry lies in the equality of the scalar and vector potentials, in DDRHF the cancellation between the Hartree and Fock terms is responsible for the better spin symmetry properties and determines the subtle spin-orbit splitting. These conclusions hold even in the case when significant deviations from the G -parity values of the meson-antinucleon couplings occur.