Strength of nuclear shell effects at $N=126$ in the r-process region

We have investigated nuclear-shell effects across the magic number $N=126$ in the region of the r-process path. Microscopic calculations have been performed using the relativistic Hartree-Bogoliubov approach within the framework of the relativistic mean-field (RMF) theory for isotopic chains of rare-earth nuclei in the r-process region. The Lagrangian model NL-SV1 with the inclusion of the vector self-coupling of ω meson has been employed. The RMF results show that the shell effects at $N=126$ remain strong and exhibit only a slight reduction in the strength in going from the r-process path to the neutron drip line. This is in striking contrast to a systematic weakening of the shell effects at $N=82$ in the r-process region predicted earlier in the similar approach. In comparison the shell effects with microscopic-macroscopic mass formulas show a near constancy of shell gaps leading to strong shell effects in the region of r-process path to the drip line. A recent analysis of solar-system r-process abundances in a prompt supernova explosion model using various mass formulas, including the recently introduced mass tables based on Hartree-Fock-Bogoliubov method shows that although mass formulas with weak shell effects at $N=126$ give rise to a spread and an overproduction of nuclides near the third abundance peak at $A~190$, mass tables with droplet models showing stronger shell effects are able to reproduce the abundance features near the third peak appropriately. In comparison, several analyses of the second r-process peak at $A~130$ have required weakened (quenched) shell effects at $N=82$. Our predictions in the RMF theory with NL-SV1, which exhibit weaker shell effects at $N=82$ and correspondingly stronger shell effects at $N=126$ in the r-process region, support the conjecture that a different nature of the shell effects at the magic numbers may be at play in r-process nucleosynthesis of heavy nuclei.

Figure 1

The two-neutron separation energies $S_{2n}$ for Pb isotopes as obtained from (a) the forces with the Lagrangian model of the nonlinear scalar self-coupling of σ meson, (b) from the forces that include the nonlinear quartic coupling of ω meson, and (c) from the mass formulas FRDM [7], ETF-SI [9], and the recently obtained results from mass tables HFB-2 [39]. The experimental data are shown by the solid circles.

Figure 2

The two-neutron separation energies $S_{2n}$ for the isotopic chains from Hf ($Z=72$) to Xe ($Z=54$) in going toward the r-process nuclei and in approaching the neutron drip line, obtained from the forces with the Lagrangian model of the nonlinear scalar self-coupling of σ meson (a) NL-SH [30] and (b) NL3 [37]. The $S_{2n}$ values from the forces that include the nonlinear quartic coupling of ω meson [31, 33] are also shown in (c) with NL-SV2 and in (d) with NL-SV1.

Figure 3

The two-neutron separation energies $S_{2n}$ for the isotopic chains from Hf ($Z=72$) to Xe ($Z=54$) in going toward the r-process nuclei and in approaching the drip line, as predicted by the mass models (a) FRDM [7] and (b) ETF-SI [9]. The $S_{2n}$ values from the mass formula ETF-SI (Q) [12] that includes a quenching of the shell effects superimposed on ETF-SI are shown in (c). The results from the mass formula HFB-2 [39] are shown in (d).

Figure 4

The $S_{2n}$ values from NL-SV1 are compared to those from (a) NL-SH and (b) NL3. The shell gap region is bounded by the two vertical lines at $N=126$ and $N=128$.

Figure 5

The $S_{2n}$ values from NL-SV1 are compared to those from the mass formula (a) ETF-SI and with the quenched mass formula (b) ETF-SI (Q). The shell gap region is bounded by the two vertical lines at $N=126$ and $N=128$.

Figure 6

The $S_{2n}$ values from the FRDM are compared to those (a) from the force with strong shell effects NL-SH and (b) the mass formula with strong shell effects ETF-SI.

Figure 7

The shell gaps ${\Delta }_S$ [Eq. (12)] at $N=126$ as obtained from the RMF forces (a) NL-SH, NL-SV1, and NL-SV2 and compared with those from ETF-SI (Q). (b) The shell gaps from mass formulas ETF-SI, FRDM, and HFB-2 are shown. A comparison is also made with the ETF-SI (Q).

Figure 8

The $S_{2n}$ values from NL-SV1 are shown for nuclei across the shell closure (a) $N=82$ and (b) $N=126$ in the region of the r-process path. The evolution of the shell gaps at $N=126$ is compared to that at $N=82$.