The nucleon-nucleus potential in the Hartree-Fock approximation with Skyrme's interaction

Abstract

We investigate the connection between the self-consistent nuclear field in the Hartree-Fock (HF) approximation with the Skyrme interaction and the real part of the phenomenological nucleon-nucleus optical potential. We show that a simple local transformation u(r) = ( m ∗(r) m ) 1 2 u L (r) of the radial wave function u( r) leads to an ordinary Schrödinger equation for u L( r). The energy-dependent potential V L( r, E) which appears in this equation is identified with the real part of the nucleon optical potential. We decompose V L( r, E) into a central part V av, a symmetry part V sym and a spin-orbit part V s.o.. The radial form factors which characterize V av, V sym and V s.o. are expressed in closed form in terms of the total nuclear density ϱ( r) and the neutron excess density Δϱ( r). Using force parameters obtained by fitting bound-state properties of closed-shell nuclei, we calculate V L numerically and compare to empirical potentials. We find that V av strongly resembles a Woods-Saxon form, with depth V 0 ≈ − 43 MeV, radius R 0 = 1.10A 1 3 + 0.75 fm and diffuseness a ≈ 0.55 fm; V av also contains an energy-dependent part of depth 0.43 E and radius R 0 E = R 0 − 0.25 fm. It is shown that V sym is proportional to Δϱ( r) times a smooth function of the density. The calculated symmetry potential changes from a surface form for medium nuclei (Ca) to a surface plus volume for heavier nuclei (Zr, Sn, Pb). The form factor for V s.o. is close to a Thomas form, but peaks at a radius inside that of the central potential.