A Semiclassical Hartree-fock Method

We present a semiclassical approach, the partial fi resummation method, to calculate nuclear ground state properties in a selfconsistent way starting from an effective nucleon-nucleon interaction. This method is shown to be easily generalized tu describe excited nuclear systems. Selfconsistent semiclassical densities can be further used as an optimal starting point for the calculation of nuclear deformation energy surfaces like fission barriers. There has been a considerable progress over the last ten years to describe static and low energy dyriamical nuclear properties in the Hartree-Fock (HF) framework using density-dependent effective interactions of the Skyrme or Gaussian type /1/. These calculations which are relatively simple and economic for spherïcal shapes become quickly quite involved when deformed shapes or excited systems are considered. A link between this microscopicmethodand nuclear mass formulae of average nuclear properties like the liquid drop or droplet mode1 is given by the Strutinsky theorem, which corresponds to an expansion of the HF energy E(p) around its average (semiclassical) part E(P). The shell correction energy 6 E , though being small (G~/~(p)sl%) is responsible for essential nuclear properties like ground state deformations or the existence of a double hump fission barrier. The treatment of shell effects in perturbation, as done in the Strutinsky method, has been shown to reproduce the HF energy within less thanl MeV under the condition that the average density matrix is determined in a selfconsistent way / 2 / . II THE RESUMMATION METHOD In the one-body approximation of the nuclear N-body problem semiclassical quantities can most easily be obtained using the single-particle Bloch density CJ = e-% / 3 / . The latter can be used to calculate the density matrix ph (r,r4) through an inverse Laplace transform /4,5/ Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984624 C6-206 JOURNAL DE PHYSIQUE Knowing the exact Bloch density of a problem, the exact density matrix is obtained. This is however generally not the case and approximations have to be made. The simplest semiclassical approach is the Thomas-Fermi method yielding a density distribution p (2) only defined inside the classically allowed region r ç rcl, where the classical turning point relis determined through the Fermi energy h by V b ) = h . cl Calculating the Wigner transform of the Bloch density, a systematic expansion in powers of fi is obtained, called the Wigner-Kirkwood expansion, with semiclassical corrections to the Thomas Fermi approximation ~lthoucjh a very rapidly converging expansion for the energy is obtaind / 6 / , densities are confined to the classically allowed region and even diverge at the classical turning point /7/. To cure the turning point problem two methods have been proposed. One of them, the extended Thomas-Fermi (ETF) approach expresses the kinetic and spin-orbit densities as functionals £0 the local density p (?) which allows for selfconsistent calculations with density dependent effective forces in the variational sense. In the other method al1 up to first (or second) derivatives of the central potential are consistently resummed in equ. (2) to al1 orders in fi /3,10/. Resumming Fig. 1 Illustration of the local approximation of a potential (Woods-Saxon type) in the resummation method by a linear (dashed line) or a harmonic potential (dashdotted line). 3 This linearized form corresponds to a local approximation of V(r) by a constant dope potential. Resumming up to second derivatives /IO/ leads to a local approximation by a harmonic potential as shown in figure 1. Performing the Laplace inversions (eq. (1)) by the saddle point method, which can be shown to be a semiclassical expansion /Il/, density distributions are obtained which reproduce the quantum mechanical distributions on the average, washing out al1 quantum oscillations. What is espec~ally gratifying is to notice that the surface of the systern is very well reproduced, as shown in figure 2. This feature is of crucial importance if we aim at using these semiclassical densities in an iterative HF-like cycle with density-dependent effective interactions like those of the Skyrme type /12-14/. Such a selfconsistent (semiclassical) program is indeed possible and yields nuclear ground state energies and density distributions in good agreement with the results of the corresponding HF-cycle /15/. As an example we show in figure 3 the selfconsistent proton and neutron densityand potential distributions calculated with the. Skyrme SI11 force /13/ for the nucleus 208~b. The ground state energies and rms Fig. 2 Comparison of the exact, quantum-mechanical density distribution (full line) of 92 particles in a spherical Woods-Saxon potential (Vo = 44 MeV, Ro = 1.27 ~ l / 3 fm, a = 0.67 fm) with the semiclassical densities/ll/ obtained with the saddle-point method from the harmonised approximation /IO/ (dashdotted line) and the Bhaduri approach /3/ (dashed line) to the resummed Bloch densities. Fig. 3 Comparison of the selfconsistent semiclassical densityand potentialdistributions (full line) for protons and neutrons with the corresponding HF-results as obtained for the nucleus 2 0 8 ~ b with the Skyrme SI11 force. radii for a number of closed shell nuclei are shown in table 1 as obtained with the modified SkM force of ref. /14/. For the cornparison of binding energies, a Strutinsky shell correction energy 6E has of course to be taken into account. The agreement with the corresponding HF results seems quite satisfactory. Starting £rom the selfconsistent semiclassical calculations, quantum oscillations can be taken into account in a perturbative way as done for the energy by the Strutinsky method. This presents the semiclassical method like an interesting alternative for full HF calculations.