Spherical Hartree-Fock Research Articles

DWIA calculations of proton momentum distributions in 12C and 16O(e, e'p) reactions

The DWIA analysis of the quasi-free (e, e'p) process on 12C and 16O has been performed within the framework of the single-particle shell model. The spherical Hartree-Fock wave functions for the effective Skyrme forces are used in describing the ground-state properties of these nuclei. The competition between the structure effects and the final proton-nucleus interaction in forming the momentum distributions of outgoing protons is studied in a wide range of recoil momenta (up to 3 fm −1). The distorting optical potential has been calculated on the basis of the Watson multiple scattering theory. The results of the analysis are compared with the new data obtained at Saclay.

Mutual support of magicities and residual effective interactions near 208Pb

We summarize experimental evidence in the lead region on the increased stability associated with neutron magic number when the proton number is magic, and vice versa. The effect is interpreted in the framework of the nuclear shell model with empirical effective interactions. Its relation to spherical Hartree-Fock calculations is pointed out and used to test Skyrme-type forces. None of the considered Skyrme interactions reproduce the effect.

EffectiveM3operator and relevant transitions inAl24,Na24,Cl34,K38, andCl38

Theoretical calculations are presented for the magnetic octupole electromagnetic matrix elements in light nuclei. Shell-model wave functions are used to calculate the $M3$ matrix elements for the cases $^{24}\mathrm{Al}$ ${1}^{+}$ \ensuremath{\rightarrow}${4}^{+}$, $^{24}\mathrm{Na}$ ${1}^{+}$ \ensuremath{\rightarrow}${4}^{+}$, $^{34}\mathrm{Cl}$ ${3}^{+}$ \ensuremath{\rightarrow}${0}^{+}$, $^{38}\mathrm{K}$ ${0}^{+}$ \ensuremath{\rightarrow}${3}^{+}$, and $^{38}\mathrm{Cl}$ ${5}^{\ensuremath{-}}$ \ensuremath{\rightarrow}${2}^{\ensuremath{-}}$. The radial matrix elements are calculated with harmonic-oscillator and spherical Hartree-Fock potentials. The comparison of the calculated and experimental matrix elements is expressed in terms of effective spin $g$ factors. Theoretical calculations for the core-polarization corrections are presented and relations between the $E2$ core-polarization charge and the $M3$ effective spin $g$ factors are derived.NUCLEAR STRUCTURE $^{24}\mathrm{Na}$, $^{24}\mathrm{Al}$, $^{34}\mathrm{Cl}$, $^{38}\mathrm{K}$, $^{38}\mathrm{Cl}$; calculations of $M3$ decay strengths; extraction of effective $M3$ operator; full ${d}_{\frac{5}{2}}\ensuremath{-}{s}_{\frac{1}{2}}\ensuremath{-}{d}_{\frac{3}{2}}$ shell-model calculations with Chung-Wildenthal Hamiltonians.

A model study of Hartree-Fock and Linear Response in coordinate space

A fast procedure for spherical Hartree-Fock is obtained by coordinate space representation and a modification of gradient iteration. Along similar lines, the corresponding Linear Response equations are derived and solved, in order to achieve a fully consistent treatment. The Linear Response equations are applied to a change in particle numbers, i.e. to the description of isotopic differences. In a model study we look for their physical and numerical properties, i.e. linearity of the response, numerical stability and consistency requirements for the Hartree-Fock basis.

Hartree-Fock Charge Distributions for 1p-Shell Nuclei

Nuclear charge distributions based on spherical Hartree-Fock calculations are employed to study the elastic electron scattering on some (spherical as well as deformed) 1p-shell nuclei. Extension to the deformed nuclei is made using the quadrupole-scattering theory. The nuclei studied are (He'), Li6 , C 12 , N 14 and 0. Fair agreement with the experimental data is obtained for not-too-large momentum transfers. § l. Introduction The nuclear charge distributions based on Hartree-Fock or projected Hartree­ Fock calculations!) are some of the non-phenomenological charge distributions em­ ployed to analyse the elastic electron scattering data on light nuclei. Charge densities based on Brueckner-Hartree-Fock calculations have also been tested suc­ cessfully in this respect.') Recently a model-independent method of determination of charge radius of a spin-zero light nucleus has been proposed.3l The method is based on the hypothesis of analyticity of the form factor in the complex q 2 plane. In this paper we have employed charge distributions based on spherical Hartree-Fock (HF) calculations4l along with the quadrupole-scattering theory to study the elastic electron scattering on some (spherical as well as deformed) 1P-shell nuclei. These HF wave functions have as their basis the Yale effective interaction derived by Shakin et al., 5l using a unitary-model-operator approach and they take into account short-range central nucleon-nucleon correlations. The cor­ rections due to the Coulomb energy and the centre-of-mass motion of the nucleus are incorporated in these calculations in a self-consistent manner. The resulting HF single particle energy levels are in fair agreement with the experimental data. The agreement of the calculated binding energies and the rms radii, however, is not as good, because the second order terms in the K-matrix are not included self-consistently.

The determination of the nuclear charge distribution of 12C from elastic electron scattering

Using an iterative procedure previously applied to 208Pb, the static nuclear charge distribution for 12C and the error and error correlations in coordinate space implied by the experimental statistical errors are determined from experimental electron scattering data at three energies. Pseudodata generated from a known charge distribution and randomized according to the experimental errors are investigated extensively to demonstrate the validity of the method. Deformed and spherical Hartree-Fock calculations with phenomenological and realistic forces are compared with experiment, and the effect of dispersion corrections on the analysis is shown to be small. Various corrections to the experimental data are discussed, and the analysis is compared with other current approaches.

Quasi-free electron scattering from 12C using Hartree-Fock wave functions

The energy spectrum and the angular correlation distributions for the reaction 12C(e, e'p) 11B are calculated within the framework of DWBA using spherical Hartree-Fock wave functions. A comparison with experiment and some of the earlier calculations is made. It is observed that a large spatial extent of the wave functions significantly improves the agreement.

Bubbles and the odd-state force

We show that the formation of bubbles in spherical Hartree-Fock calculations on 36A depends sensitively on the odd-state effective interaction, being strongly favoured, like alpha clustering, when this is repulsive

Shape-excited states of28Si

The Hartree-Fock (HF) minima for the nucleus28Si were obtained for the prolate, oblate and spherical shapes using the interaction obtained by Preedom and Wildenthal. The interaction gives rise to large energy separation between the prolate and the oblate shapes. The spherical solution is just 2 MeV above the lowest HF (oblate) minimum. The spectrum projected from the oblate HF state is in good agreement with the experimental spectrum. The transition probabilities for the different energy levels also agree reasonably well. The configuration mixing calculations performed on the basis of states projected from the three shapes indicate that there is no significant mixing of different projected states. The second 0 2 + state, thus obtained, corresponds to the third 0 3 + state in the experimental spectrum and stems dominantly from the spherical HF state. It is seen that the structure of the energy levels of28Si, especially the second 0 2 + level is very sensitive to the two body interaction. The results are compared with those obtained using the renormalised interaction of Kuo.

On Spherical Hartree–Fock Calculations with Sussex Interaction

Spherical Hartree–Fock calculations for 4He, 12C, and 16O nuclei have been carried out with Sussex relative matrix elements of Elliott et al. It is observed that the calculated binding energies for all these nuclei are considerably smaller than the experimental values. However, the Sussex matrix elements give better results for the binding energies of these nuclei compared with the truncated results of other realistic interactions in most of the cases. The rest of the calculated properties are in fair agreement with experiments.

A Hartree-Fock calculation for the stability of super-heavy nuclei

Spherical Hartree-Fock calculations have been carried out in the super-heavy region with the Skyrme interaction. Shell effects are discussed and stability against alpha and beta-decay are investigated. A strong shell effect is found to occur at N = 228.

Investigations of parity deformation in spherical nuclei

The possibility of parity deformation for spherical structures is investigated from various viewpoints. We use several criteria of stability for the spherical Hartree-Fock solution (Thouless condition, critical 0 − states in different approximations) as well as different types of two-body forces (a realistic effective interaction and the OPEP) and single-particle spectra (HF, oscillator, Woods-Saxon and experimental) in order to obtain a complete survey. The result is that parity deformation can be excluded in practically all cases. In addition, the positions of the two critical 0 − states of 16O are in good numerical agreement with experiments if realistic forces and experimental single-particle energies in connection with the random phase or Tamm-Dancoff approximation are used.

Further studies of the Hartree-Fock approximation in finite nuclei

New developments in spherical Hartree-Fock techniques make possible very fast calculations of heavy nuclei like 208Pb and the superheavy elements. With these methods, studies can be made using harmonic oscillator wave functions with a large number of radial nodes and including the physically important relative l-values. Some sample Hartree-Fock results are presented for a velocity-dependent effective interaction recently developed. Single-particle energies and wave functions, rms charge and mass radii, total binding energies, densities and overlaps of single-particle wave functions with pure harmonic oscillator wave functions have been calculated for the nuclei 16O, 40Ca, 48Ca, 56Ni, 88Sr, 90Zr and 208Pb. Studies of the 208Pb densities and overlaps strongly suggest that it is necessary to use a dimensionality somewhat greater than pure oscillator functions in calculating heavy nuclei. The origin of spin-orbit splitting in light and heavy nuclei is also discussed. In heavy nuclei, it is shown that the incomplete filling of a j = l± 1 2 doublet has an important effect on the splittings of the single-particle levels.

Spin-orbit splitting in the conventional spherical Hartree-Fock theory

The single-particle spin-orbit splitting is calculated in the conventional spherical reaction-matrix Hartree-Fock theory using empirical nuclear sizes and harmonic oscillator wave functions. The contribution from spin-saturated shells and subshells is found to be only about half to two-thirds of the experimental value for a range of nuclei. The contribution from the incomplete (i.e. spin unsaturated) subshells is found to be large but has the wrong sign and a somewhat different radial distribution. The general situation of the spin-orbit splitting is discussed in terms of the known properties of the nucleon-nucleon interaction.