Florence Tondeur

Nuclear mass formula via an approximation to the Hartree-Fock method

We present the first nuclear mass table to be based entirely on microscopic forces. The calculations are performed using the extended Thomas-Fermi plus Strutinsky integral method, a semiclassical approximation to the Hartree-Fock method that includes full Strutinsky shell corrections; BCS pairing corrections are added. The eight active parameters of the underlying Skyrme and δ-function pairing forces are fitted to all the 1492 mass data (1988 compilation) for A ≥ 36; the rms error of this fit is 0.736 MeV. Our tabulation covers the range 36 ≤ A ≤ 300 and reaches beyond the neutron- and proton-drip lines. In addition to the calculated masses, we show the calculated neutron- and proton-separation energies and beta-decay energies. We also give for each nucleus in the table the model predictions for the deformation parameters and deformation energy at equilibrium (with axial and left-right symmetry assumed) and for the charge radii.

Fission barriers of neutron-rich and superheavy nuclei calculated with the ETFSI method

Abstract Using the ETFSI (extended Thomas–Fermi plus Strutinsky integral) method, we have calculated the fission barriers of nearly 2000 exotic nuclei, including all the neutron-rich nuclei up to A=318 that are expected to be relevant to the r-process, and all superheavy nuclei in the vicinity of N=184 , with Z≤120 . Our calculations were performed with the Skyrme force SkSC4, which was determined in the ETFSI-1 mass fit. For proton-deficient nuclei in the region of N=184 we find the barriers to be much higher than previously believed, which suggests that the r-process path might continue to mass numbers well beyond 300. For the superheavy nuclei we typically find barrier heights of 6–7 MeV.

A Hartree-Fock-Bogoliubov mass formula

In order to have more reliable predictions of nuclear masses at the neutron drip line, we here go beyond the recent mass formula HFBCS-1 and present a new mass formula, HFB-1, based on the Hartree-Fock-Bogoliubov method. As with the HFBCS-1 mass formula, we use a 10-parameter Skyrme force along with a 4-parameter δ-function pairing force and a 2-parameter phenomenological Wigner term. However, with the original HFBCS-1 Skyrme force (MSk7), the rms error becomes unacceptably large and a new force fit is required. With the isoscalar and isovector effective masses constrained to be equal, the remaining 15 degrees of freedom are fitted to the masses of all the 1754 measured nuclei with A ≥ 16, N - Z > 2 given in the 1995 Audi-Wapstra compilation. The rms error with respect to the masses of all the 1888 measured nuclei with Z, N ≥ 8 is 0.764 MeV. A complete mass table, HFB-1 (available on the Web), has been constructed, giving all nuclei lying between the two drip lines over the range Z, N ≥ 8 and Z ≤ 120. A comparison between HFB-1 and HFBCS-1 mass tables shows that the HFBCS model is a very good approximation of the HFB theory, in particular for masses, the extrapolated masses never differing by more than 2 MeV below Z ≤ 110. We also find that the behaviour of shell gaps far away from the region of beta stability does not depend on whether the HFBCS or HFB methods are used, in particular, no quenching of astrophysical interest arises from replacing the BCS method by the Bogoliubov method.

Static nuclear properties and the parametrisation of Skyrme forces

Abstract We present a systematic study of the dependence of static nuclear properties on the parameters of the effective interaction used in the Hartree-Fock (HF) and extended-Thomas-Fermi (ETF) models. For this purpose, a set of trial Skyrme forces, which are constrained by a fit to nuclear radii and binding energies, is developed. This leaves six free parameters: the spin-orbit strength, the nuclear-matter compression modulus, the isoscalar and isovector contributions to the effective masses, the value of the exchange parameter x3 (governing the surface-symmetry properties) and the coefficient of the “gradient-symmetry” term |▽ρn − ▽ρp|2 in the energy-density functional. The influence of these parameters on various properties is studied: droplet-model parameters, quality of the fit to experimental masses, extrapolation of masses, fit to charge radii, charge distributions and neutron-skin thicknesses, semiclassical fission barriers, and Landau parameters. Indications are given of the directions which could be followed in order to improve the fit to experimental data. Several correlations remaining in the results suggest that a larger number of degrees of freedom obtained by additional terms could be useful.

Thomas-Fermi approach to nuclear-mass formula. (IV). The ETFSI-1 mass formula

Abstract We summarize the main features of the first nuclear-mass table to be based entirely on microscopic interactions. A semi-classical approximation to the HF-BCS method is adopted, with full Strutinsky shell corrections included. The 9 parameters of the underlying Skyrme and δ-function pairing forces are fitted to all 1492 mass data for A ⩾ 36; the r.m.s. error of this fit is 0.730 MeV. Our tabulation covers the range 36 ⩽ A ⩽ 300, goes out to the neutron-drip line and extends beyond the proton-drip line. Equilibrium deformations of all nuclei are calculated, with axial symmetry assumed. We also calculated several fission barriers using the same force with no further adjustment of parameters; a satisfactory agreement with experiment is obtained.

Thomas-Fermi approach to nuclear mass formula: (III). Force fitting and construction of mass table

Abstract The ETFSI method, developed in two earlier papers, is here used to construct a complete mass table. Since the method allows for interpolation both in the ( N , Z ) plane and with respect to deformations, without losing the characteristic shell-model fluctuations, it is some 2000 times faster than the HF-BCS method for a given force. The present table is calculated using a preliminary Skyrme-type force with δ-function pairing, fitted to a restricted data set of 491 spherical nuclei. The resulting rms error for all 1492 measured nuclei, spherical and deformed, with A ⩾ 36 is e rms = 0.868 MeV, achieved with just 9 parameters. The main experimental trends in ground-state deformations are well followed. The symmetry coefficient of nuclear matter corresponding to our force is 27.5 MeV. Ways of rapidly improving the fit are indicated.

A skyrme functional for Hartree-Fock calculations of nuclear masses and density distributions

Abstract A Skyrme functional is presented which is designed for calculations of nuclear masses and density distributions in a constrained time invariant Hartree-Fock + BCS model. With an effective mass equal to the nucleon mass, the experimental masses are reproduced for spherical nuclei with a rms deviation of 1 MeV. A good fit to experimental density distributions is obtained, including equal charge radii for 40Ca and 48Ca, and taking into account experimental data for the neutron skin thickness in 208Pb.

Nuclear-matter incompressibility from fits of generalized Skyrme force to breathing-mode energies

Abstract In an attempt to extend the range of values of K v , the incompressibility of symmetric nuclear matter, for which fits to the measured breathing-mode energies are possible, we investigate generalized Skyrme-type forces with a term that is both density- and momentum-dependent. Acceptable fits are found to be possible only for values of K v in the range 215±15 MeV. For higher values fits are impossible, while for lower values fits are achieved only by introducing an unphysical collapse of nuclear matter. Thus our generalization of the Skyrme force does not permit a significantly wider range of values of K v than that already given by force SkM ∗ . However, with a view to having a more reliable estimate of the compressional properties of the highly neutron-rich nuclear matter that comprises the core of collapsed stars, we present a new version of this latter force giving a much better fit to the masses of neutron-rich nuclei. Comparison with force SkM ∗ also shows that the value of K v extracted from the breathing-mode energies is essentially independent of the choice of effective mass. By providing a counter-example, we show that K v cannot be extracted from masses and charge distributions alone. As for the third-order coefficient K ′, we cannot be more precise than to say that it lies in the interval 700 ± 500 MeV.

Pairing with a delta interaction in the energy density nuclear mass formula

Abstract The variations of the average pairing strength in the ( N , Z ) plane are studied with a δ-interaction in the frame of the self-consistent energy density formalism. It is found that the same δ-interaction with constant strength can be used for protons and neutrons in spherical nuclei near the stability line. This interaction is used to study the extrapolation of the average pairing strength to deformed nuclei, to the superheavy region and to the regions of the drip lines. The consequences of the results for the stability of superheavy nuclei and for magic numbers far from the stability region are examined.

Thomas-fermi approach to nuclear mass formula

Abstract With a view to having a more secure basis for the nuclear mass formula than is provided by the drop(let) model, we make a preliminary study of the possibilities offered by the Skyrme-ETF method. Two ways of incorporating shell effects are considered: the “Strutinsky-integral” method of Chu et al., and the “expectation-value” method of Brack et al. Each of these methods is compared with the HF method in an attempt to see how reliably they extrapolate from the known region of the nuclear chart out to the neutron-drip line. The Strutinsky-integral method is shown to perform particularly well, and to offer a promising approach to a more reliable mass formula.