Single-Particle Resonant States in Relativistic Hartree-Fock Theory: A Green's Function Approach

Abstract

Relativistic Hartree-Fock theory is combined with the Green's function method in coordinate space to study both single-particle bound and resonant states within a unified framework. Within this approach, single-particle resonance energies and widths are unambiguously extracted from the density of states, and the influence of the Coulomb exchange effects on proton resonances in $N=82$ isotones are systematically examined. It is found that the exact treatment of Coulomb exchange terms reduces proton resonance energies of approximate $0.09\sim0.21$ MeV, a significantly smaller effect than that obtained from the phenomenological treatment. Moreover, except for rather narrow resonances, the proton resonance widths are visibly reduced by the Coulomb exchange terms, also being much less pronounced than the phenomenological approach. Notably, clear shell effects are observed in the isotonic evolutions of the resonance energy reductions for specific resonances. All these highlight the necessity of a microscopical and exact treatment of the Coulomb exchange terms.

Figures (6)

• Figure 1: (Color online) Contours for the Green's function on the complex energy plane, with a rectangle measuring 0.15 MeV in width and 0.20 MeV in length, and the integration step size reading 0.01 MeV. The red crosses, blue hollow circles and thick green line represent bound states, resonant states $E - i\Gamma/2$, and the continuum states, respectively.
• Figure 2: (Color online) Density of states $n_\kappa(\varepsilon)$ (MeV$^{-1}$) as functions of the energy $\varepsilon$ (MeV) for the neutron orbits of $^{120}$Sn. The results are calculated by the RHF-GF method with PKO1 and $R_{\rm max}$=20 fm (blue solid lines), in comparison with the result for the free-particle background obtained (red dotted lines).
• Figure 3: (Color online) Energies $E$ (MeV) and half-widths $\Gamma/2$ (MeV) of the neutron resonant states and weakly bound one $3p_{1/2}$ in $^{120}$Sn. The results are obtained by the RHF theory with the Green's function (GF) method, as compared to those with the real stabilization method (RSM) YangW2024APS73.062102.
• Figure 4: (Color online) Plots (a) and (b) display the energies $E$ (MeV) and widths $\Gamma$ (MeV) of the proton resonant states for the $N=82$ isotones, respectively. The results are calculated by the RHF theory with the GF method, in which the Coulomb exchange terms are treated exactly.
• Figure 5: (Color online) (a) Coulomb exchange reductions $\Delta E$ (MeV) in resonance energies for proton resonant states of the $N=82$ isotones; (b) Interaction matrix elements $V_{ab}^{\text{Cex}}$ (keV) contributed by the Coulomb exchange terms between proton bound $1g$ states and resonant states. The results are given by the RHF-GF method with PKO1.