Self-consistent single-nucleon potential at positive energy produced by semi-realistic interaction and its examination via nucleon-nucleus elastic scattering

Based on the variational principle, we discuss self-consistent single-particle (s.p.) potentials at positive energies, which correspond to the real part of the optical potential as the single folding potential (SFP). The nuclear-matter s.p. potential produced by the semi-realistic nucleonic interaction M3Y-P6, which has links to the bare nucleonic interaction, resembles those extracted from the empirical optical potential. Applying M3Y-P6 both to the self-consistent mean-field calculations for the target nucleus and to the SFP for the scattered nucleon, we find that the differential cross sections of the nucleon-nucleus elastic scattering are reproduced almost comparably to the empirical potentials up to $80\mathrm{MeV}$ incident energy. The results demonstrate that the s.p. potential compatible with available experimental data can be derived from a single energy-independent effective interaction in this wide energy range.

Figure 1

Energy dependence of $U_0$ and $U_1$ for homogeneous nuclear matter [see Eq. (7)] at $\rho =0.16{\mathrm{fm}}^{-3}$. The s.p. potentials given by the SCMF interactions, Skyrme-SLy4 (brown dot-dashed lines), Gogny-D1S (blue long-dashed lines), and M3Y-P6 (red solid lines), are compared with the empirical potentials, KD evaluated at $A=208$ (green dotted lines) and at $A=400$ (green short-dashed lines) and CH89 (black short-dashed lines).

Figure 2

$d{\sigma }_{\mathrm{el}}/d\mathrm{\Omega }$ of $p\text{−}A$ scattering: (a) $p\text{−}{}^{16}\mathrm{O}$, (b) $p\text{−}{}^{40}\mathrm{Ca}$, (c) $p\text{−}{}^{90}\mathrm{Zr}$, and (d) $p\text{−}{}^{208}\mathrm{Pb}$. Results of ${\mathcal{V}}^{\mathrm{SFP}}$ with M3Y-P6 plus ${\mathcal{W}}^{\mathrm{emp}}$ are depicted by red solid lines and compared with experimental data (black circles) taken from the database [61], originally reported in Refs. [19, 62, 63, 64, 65, 66, 67, 68]. For comparison, results of ${\mathcal{V}}^{\mathrm{SFP}}$ with Gogny-D1S (blue long-dashed line), and those of ${\mathcal{V}}^{\mathrm{emp}}$ of Ref. [20] (green short-dashed lines) are also displayed. Depending on ${\epsilon }_p$, the cross sections are scaled by the coefficients in the parentheses.

Figure 3

${\sigma }_{\mathrm{reac}}$ of $p\text{−}{}^{40}\mathrm{Ca}, p\text{−}{}^{90}\mathrm{Zr}$, and $p\text{−}{}^{208}\mathrm{Pb}$ scattering. Results of ${\mathcal{V}}^{\mathrm{SFP}}$ with M3Y-P6 (D1S) plus ${\mathcal{W}}^{\mathrm{emp}}$ are plotted by crosses connected with red solid lines (pluses with blue long-dashed lines), and those of ${\mathcal{V}}^{\mathrm{emp}}$ are displayed by green short-dashed lines. Black circles, squares, and triangles are experimental data [61, 69, 70, 71].

Figure 4

$d{\sigma }_{\mathrm{el}}/d\mathrm{\Omega }$ of $n\text{−}A$ scattering: $n\text{−}{}^{16}\mathrm{O}, n\text{−}{}^{40}\mathrm{Ca}$, and $n\text{−}{}^{90}\mathrm{Zr}$. Results of ${\mathcal{V}}^{\mathrm{SFP}}$ with M3Y-P6 and D1S and those of ${\mathcal{V}}^{\mathrm{emp}}$ are presented, in comparison with experimental data [61, 72, 73, 74]. See Fig. 2 for conventions.