Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential

The nucleon mean-field potential has been thoroughly investigated in an extended Hartree-Fock (HF) calculation of nuclear matter (NM) using the CDM3Y3 and CDM3Y6 density dependent versions of the M3Y interaction. The single-particle (SP) energies of nucleons in NM are determined according to the Hugenholtz–Van Hove theorem, which gives rise naturally to a rearrangement term (RT) of the SP potential at the Fermi momentum. Using the RT obtained exactly at the different NM densities and neutron-proton asymmetries, a consistent method is suggested to take into account effectively the momentum dependence of the RT of the SP potential within the standard HF scheme. To obtain a realistic momentum dependence of the nucleon optical potential (OP), the high-momentum part of the SP potential was accurately readjusted to reproduce the observed energy dependence of the nucleon OP over a wide range of energies. The impact of the RT and momentum dependence of the SP potential on the density dependence of the nuclear symmetry energy and nucleon effective mass has been studied in detail. The high-momentum tail of the SP potential was found to have a sizable effect on the slope of the symmetry energy and the neutron-proton effective mass splitting at supranuclear densities of the NM. Based on a local density approximation, the folding model of the nucleon OP of finite nuclei has been extended to take into account consistently the RT and momentum dependence of the nucleon OP in the same mean-field manner, and successfully applied to study the elastic neutron scattering on the lead target at the energies around the Fermi energy.

Figure 1

Total NM energy per particle $E/A$ at the different NP asymmetries $\delta$ given by the HF calculation (1) and (2), using the CDM3Y3 (lower panel) and CDM3Y6 (upper panel) interactions with their IV density dependence anew determined in the present work. The solid circles are the saturation densities of the NM at the different NP asymmetries.

Figure 2

Energy dependence of the nucleon OP in the symmetric NM (evaluated at the saturation density ${\rho }_0$ with and without the RT using the CDM3Y6 interaction) in comparison with the empirical data taken from Refs. [37] (circles), [38] (squares), and [39] (triangles). The momentum dependent factor $g\left(k\right)$ has been iteratively adjusted to the best agreement of the calculated nucleon OP (24) with the empirical data (solid line).

Figure 3

Momentum dependent scaling factor $g\left(k\right)$ obtained with both the CDM3Y3 and CDM3Y6 interactions from the best (HF+RT) fit of the nucleon OP (24) to the empirical energy dependence of the nucleon OP [39]. The points are the numerical results that are well reproduced by a cubic polynomial (solid line).

Figure 4

Density dependence of the IV part of the neutron OP in the pure neutron matter at the energy $E=1$ MeV, evaluated with and without the RT. The parameters (13) of the IV density dependence $F_1\left(\rho \right)$ of the CDM3Y6 interaction have been iteratively adjusted to give the best agreement of the HF+RT result with that of the BHF calculation by the JLM group (solid circles) [31, 32].

Figure 5

Momentum dependence of the total SP potential in the symmetric NM at the saturation density ${\rho }_0$, with the explicit contributions from the RT and HF parts. The upper panel shows the results of the extended HF calculation (23), and the lower panel shows the total SP potential (24), with the high-momentum part corrected by the $g\left(k\right)$ function determined from the observed energy dependence of the nucleon OP.

Figure 6

The same as Fig. 5 but for the total SP potential in the pure neutron matter.

Figure 7

Contributions of the RT and HF parts to the total neutron and proton SP potentials evaluated at the different NP asymmetries $\delta$ and nucleon momenta $k=k_F^{\left(\tau \right)}$, using the CDM3Y6 interaction. The results obtained at the saturation density $\rho ={\rho }_0$ are shown in the upper panel, and those obtained at the density $\rho =2{\rho }_0$ are shown in the lower panel.

Figure 8

Nuclear symmetry energy $S\left(\rho \right)$ at the NP asymmetries $\delta =0.6$ and 1 obtained with the CDM3Y6 interaction, using the parabolic approximation (28) and general definition (29). The shaded (magenta) region marks the empirical boundaries implied by the analysis of the isospin diffusion data and the double ratio of the neutron and proton spectra observed in the HI collisions [43, 45]. The square is the empirical $S$ value implied by the structure study of the GDR [46], and the triangle is that established from the analysis of the terrestrial nuclear physics experiments and astrophysical observations [44]. The circles and crosses are results of the ab initio calculations by Akmal et al. [47] and Gandolfi et al. [48], respectively.

Figure 9

Nuclear symmetry energy $S\left(\rho \right)$ given by the HF calculation using the new parameters of the IV density dependence $F_1\left(\rho \right)$ of the CDM3Y6 (upper panel) and CDM3Y3 (lower panel) interactions. These same HF results obtained with the old parameters of $F_1\left(\rho \right)$ determined in Ref. [20] are shown as the dashed lines. Other symbols are the same as in Fig. 8

Figure 10

Nuclear symmetry energy $S\left(\rho \right)$ obtained at the NP asymmetry $\delta <1$ from the (HF+RT) SP energies with the CDM3Y6 (upper panel) and CDM3Y3 (lower panel) interactions, using the parabolic approximation (31). The solid (dashed) curves show the results obtained with (without) the modification of the high-momentum part of the SP potential by the $g\left(k\right)$ function determined from the observed energy dependence of the nucleon OP. Other symbols are the same as in Fig. 8

Figure 11

Density dependence of the neutron effective mass (32) at the NP asymmetry $\delta =0,0.3$, and 0.6, obtained from the SP potential given by the CDM3Y6 interaction. The HF and HF+RT results are shown in the upper and lower panels, respectively. The solid circles are the effective-mass values determined at the saturation density ${\rho }_0$.

Figure 12

The same as Fig. 11 but for the proton effective mass.

Figure 13

The same (HF+RT) results for the neutron effective mass as in the lower panel of Fig. 11, obtained with (lower panel) or without (upper panel) the modification of the high-momentum part of the SP potential by the $g\left(k\right)$ function implied by the observed energy dependence of the nucleon OP.

Figure 14

The same as Fig. 13 but for the proton effective mass.

Figure 15

Neutron and proton effective masses (upper panel) and their splitting (lower panel) obtained at $\rho ={\rho }_0$ and the different NP asymmetries $\delta$. Both the HF and HF+RT results show a well defined linear $\delta$ dependence of the NP effective mass splitting (33).

Figure 16

$\delta$ dependence of the neutron-proton effective mass splitting obtained at $\rho =2{\rho }_0$. The HF+RT results were obtained with (solid line) or without (dashed line) the modification of the high-momentum part of the SP potential by the $g\left(k\right)$ function determined from the observed energy dependence of the nucleon OP.

Figure 17

Real $n{+}^{208}\mathrm{Pb}$ optical potential at the neutron incident energies of 30.4 and 40 MeV predicted by the single-folding calculation (37) and (38) using the CDM3Y6 interaction, with and without the contributions of the RT and the scaling $g(k(E,R))$ function. CH89 is the phenomenological OP taken from the global systematics by Varner et al. [38].

Figure 18

Elastic $n{+}^{208}\mathrm{Pb}$ scattering cross sections at the neutron incident energies of 30.4 and 40 MeV obtained with the real OP's shown in Fig. 17, in comparison with the data measured by DeVito et al. [29]. The imaginary and spin-orbital parts of the OP taken from the global systematics CH89 [38] were used in the OM calculation.

Figure 19

Local momentum $k(E,R)$ of the incident neutron (upper panel) and the scaling function $g(k(E,R))$ (lower panel) determined self-consistently from the HF+RT real folded $n{+}^{208}\mathrm{Pb}$ potential obtained with the CDM3Y6 interaction at the neutron energy $E=30.4$ MeV.