Systematic study of even-even nuclei with Hartree-Fock+BCS method using Skyrme SIII force

Naoki Tajima, Satoshi Takahara, and Naoki Onishi,
Nuclear Theory Group, Institute of Physics, University of Tokyo
Komaba 3-8-1, Meguro-ku, Tokyo, 153, Japan
email: tajima@?????.ac.jp, staka@?????.ac.jp
Last modification: 1996/3/26,4/23 Missing links removed (y/m/d): 2001/6/26

Abstract

We have applied the Hartree-Fock+BCS method with Skyrme SIII force formulated in a three-dimensional Cartesian-mesh representation to even-even nuclei with 2 <= Z <= 114. While the details of the method of the calculations will be published elsewhere, we present and discuss in this report the results concerning the atomic masses, the quadrupole (m=0, 2) and hexadecapole (m=0, 2, 4) deformations, the skin thicknesses, and the halo radii. We also discuss the energy difference between oblate and prolate solutions and the shape difference between protons and neutrons.

Introduction

Among theoretical attempts which aim at treating all the nuclides in a single framework, the simplest category seems to be the mass formula. Hence, let us focus the discussion in this section on the nuclear masses, although the purpose of our calculation is not restricted to the masses. The most familiar mass formula may be that of Bethe and Weizsaecker [We35,BB36], which expresses the nuclear masses as a function of the number of neutrons (N) and that of protons (Z) as,
E(N,Z) = aVA + aS A2/3 + aI(N-Z)2 A-1 + aC Z2 A-1/3,
where A=N+Z. By changing the values of four coefficients, the r.m.s. error from experimental data [AW93] of even-even nuclei can be decreased down to 3.5 MeV. The form of this formula originates in the liquid drop picture of the atomic nucleus together with a parabolic approximation to the dependence on the total isospin. The first step to decrease the error is to take into account the shell effect (footnote).

Taking into account the shell effect, the TUYY mass formula [TUY88], for example, achieved an r.m.s. mass error of as small as 538 keV (footnote). The number of fitting parameters are 6 for the gross part (corresponding to the parameters of the Bethe-Weizsaecker formula) while that for the shell part is as many as 269: There is one parameter for each value of Z in an interval 1 < = Z < = 112 and one for each value of N in 1 < = N < = 157. It is not at all our intention to question the quality of the predictions of the TUYY mass formula in particular, and indeed its stabilty for extrapolations has been already demonstrated by the developpers of the foumla, but generally speaking, less number of parameters are preferable for the reliability of the extrapolation to nuclei not synthesized yet. (footnote) One usually switches to less phenomenological models in order to reduce the number of parameters.

The most elaborate one among such models seems to be the finite-range droplet model with a microscopic shell correction (FRDM), whose latest update was done by Moeller et al. [MNM95] Another extensive calculation was carried out by Aboussir et al. [APD92] in the extended Thomas-Fermi plus Strutinsky integral method (ETFSI).

The former as well as the latter methods can be regarded as approximations to the Hartree-Fock (HF) equation. The straight-forward solutions of the equation including deformation require long computation time for global calculations even with present computers. Such global results are not yet available to the public as far as we know. This is the reason why we have embarked on an extensive HF+BCS calculation.

In this paper, we utilize a widely used force of the Skyrme SIII to examine what predictions it makes for a large number of nuclei, rather than aiming at the determination of the parameters of the force through calculations requiring by two orders of magnitude as long computation time as the calculations for this paper.

Set up of the calculation

For the mean-field part of the interaction, we adopt the Skyrme SIII force. The Skyrme force is an effective interaction widely used in mean-field calculations. It is essentially a zero-range force but modified with the lowest order momentum dependences to emulate the finite-range effects, a density dependence to reproduce the saturation, and a spin-orbit coupling term. The SIII [BFG75] is one of the many parameter sets proposed for the force and features good single-particle spectra and accurate N-Z dependence of the binding energy. Click here (html file of 2KB) for further information.

For the pairing channel of the intercation, we used the seniority force Click here (html file of 2KB) for details.

The advantage to express single-particle wavefunctions on a three-dimensional Cartesian mesh is that it is suitable to describe nucleon skins, halos, and exotic shapes as well as properties of ordinary stable nuclei. Click here (html file of 6KB) for details.

The calculation has been done for N ranging from outside the proton drip line to beyond the experimental frontier in the neutron-rich side (footnote). We obtained spatially localized solutions for 1029 nuclei, together with the second minima for 758 nuclei. We imposed the D2h symmetry on the solutions, which means that only even electric multipole moments do not vanish. Click here (html file of 2KB) for an explanation of the method to search for the solutions. The other details of the method of the calculations are described in Refs. [TOT94,TTO96a].

Results of the calculation

The results of our HF+BCS calculations with Skyrme SIII force are available as data tables. The data are given not only for the ground states but also for the lowest exited local minima of the HF+BCS equation. The latter represent shape isomers.
  1. Energy-related quantities, i.e., atomic masses, nuclear binding energies, Fermi levels, and pairing gaps.
    Table for the ground states (text file of 73KB)
    Table for the lowest shape isomers (text file of 54KB)
  2. R.m.s. radii and quadrupole moments (for protons and for neutrons).
    Table for the ground states (text file of 64KB)
    Table for the lowest shape isomers (text file of 48KB)
  3. R.m.s. radius and quadrupole moment (for mass distribution).
    Mass/proton/neutron deformation parameters delta defined as 3 < Q > / 4 < r2 > .
    Table for the ground states (text file of 58KB)
    Table for the lowest shape isomers (text file of 43KB)
  4. Hexadecapole moments (for protons and for neutrons).
    Table for the ground states (text file of 76KB)
    Table for the lowest shape isomers (text file of 56KB)
  5. Shape (deformation) parameters, i.e., R0, a20, a22, a40, a42, a44, and rho0.
    The definition of the shape parameters of the HF+BCS solutions (html file of 3KB)
    Table of the mass deformations for the ground states (text file of 73KB)
    Table of the protons' deformations for the ground states (text file of 73KB)
    Table of the neutrons' deformations for the ground states (text file of 73KB)
    Table of the mass deformations for the lowest shape isomers (text file of 54KB)
    Table of the protons' deformations for the lowest shape isomers (text file of 54KB)
    Table of the neutrons' deformations for the lowest shape isomers (text file of 54KB)
  6. Skin thickness.
    The definition of the skin thickness (html file of 1KB)
    Table for the ground states (text file of 71KB)
    Table for the lowest shape isomers (text file of 52KB)
  7. Halo radius.
    The definition of the halo radius (html file of 1KB)
    Table for the ground states (text file of 25KB)
    Table for the lowest shape isomers (text file of 18KB)
  8. Expectation values of partial Hamiltonians, i.e., kinetic energy, spin-orbit energy, pairing energy, and Coulomb energy (direct part).
    Table for the ground states (text file of 73KB)
    Table for the lowest shape isomers (text file of 54KB)

Discussion of the results

  1. nuclear mass

    We compare the atomic masses obtained from our calculations with those in the latest experimental mass table [AW93] and with other mass models. Click here (html file of 8KB) for details.

  2. quadrupole moments

  3. hexadecapole moments

    We discuss the systematic trends of the axial hexadecapole deformation parameter a40 of the ground-state solutions in the N-Z plane. Click here (html file of 2KB) for details.

  4. axial aymmetry of the shape

    Non-axial deformation parameters a22, a42, and a44 have turned out very small (|a22|, |a,42| < 10-4, |a44| ~ 10-3). Click here (html file of 2KB) for details.

  5. shape difference between protons and neutrons

    One of the advantages of mean-field methods over shell-correction schemes is that the protons and the neutrons do not have to possess the same radius and deformation. It is also an advantage of the Cartesian-mesh representation over the expansion in harmonic-oscillator bases. In order to make the best use of this advantage, we have calculated the liquid-drop shape parameters separately for protons and neutrons for 1029 ground and 758 first-excited solutions. The conclusion of this investigation is that the largest differences occur in the lightest nuclei. In general, however, the shapes of proton and neutron density distributions are not remarkably different from each other. It should be reminded here that our calculation does not reach the neutron drip line except in light mass region.

  6. energy difference betweem oblate and prolate solutions

    From an investigation of the systematics of the excitation energy between the oblate and the prolate solutions, we found a clear difference between below and above the N=50 shell magic. Click here (html file of 2KB) for details.

  7. skins and halos

    As discussed in the introduction, the greatest advantage of the mesh representation for this paper is that it is fit to describe skins and halos. We show that the skin grows monotonously and regularly as nucleons are added to the nucleus. On the other hand, the halo grows very slowly except near the drip lines, where it changes the behavior completely and expands very rapidly. We also show that the neutron skin tends to make the density distribution more spherical.

Concluding remarks

References

Click here (html file of 4KB) for the list of the references cited in this paper.
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Copyright (c) 1996, Naoki Tajima, Satoshi Takahara, and Naoki Onishi