Quadrupole collectivity and shell closure in neutron-rich nuclei near $N=126$

We present a comprehensive study on the low-lying states of neutron-rich Er, Yb, Hf, and W isotopes across the $N=126$ shell with a multireference covariant density functional theory. Beyond-mean-field effects from shape mixing and symmetry restoration on the observables that are relevant for understanding quadrupole collectivity and underlying shell structure are investigated. The general features of low-lying states in closed-shell nuclei are retained in these four isotopes around $N=126$, even though the shell gap is overall quenched by about 30% with the beyond-mean-field effects. These effects are consistent with the previous generator-coordinate calculations based on Gogny forces, but much smaller than that predicted by the collective Hamiltonian calculation. It implies that the beyond-mean-field effects on the $r$-process abundances before the third peak at $A\approx 195$ might be more moderate than that reported by Arcones and Bertsch [Phys. Rev. Lett. 108, 151101 (2012)].

Figure 1

Energies of mean-field states (MF, left), particle-number projected states ($N\&Z$, middle), and those with additional projection onto angular momentum ($J=0$, right) for (a)–(c) ${}^{190-206}\mathrm{Er}$, (d)–(f) ${}^{192-208}\mathrm{Yb}$, (g)–(i) ${}^{194-210}\mathrm{Hf}$, and (j)–(l) ${}^{196-212}\mathrm{W}$ isotopes as a function of the intrinsic mass quadrupole deformation. The energies of ${}^{190-206}\mathrm{Er}, {}^{192-208}\mathrm{Yb}, {}^{194-210}\mathrm{Hf}$, and ${}^{196-212}\mathrm{W}$ isotopes are shifted by $-4,-3,-2,$ and $-2$ MeV between two neighboring isotopes, respectively. The global energy minima are indicated by black dots.

Figure 2

Quadrupole deformation parameter $\beta$ of ground state obtained by MF, PNP ($N\&Z$), and PNAMP ($J=0$) calculations using PC-PK1 force, as a function of neutron number for ${}^{190-206}\mathrm{Er}, {}^{192-208}\mathrm{Yb}, {}^{194-210}\mathrm{Hf}$, and ${}^{196-212}\mathrm{W}$ isotopes.

Figure 3

Averaged deformation parameter ${\overline{\beta }}_{J\alpha }$ for the first two $J^{\pi }=0^{+}$ and $2^{+}$ states in ${}^{190-206}\mathrm{Er}, {}^{192-208}\mathrm{Yb}, {}^{194-210}\mathrm{Hf}$, and ${}^{196-212}\mathrm{W}$ isotopes as a function of neutron number, respectively.

Figure 4

Collective wave functions [cf. Eq. (15)] of the $0_1^{+}, 0_2^{+}, 2_1^{+}$, and $2_2^{+}$ states in ${}^{190-206}\mathrm{Er}$.

Figure 5

(a) Excitation energy $2_1^{+}$ state, (b) electric quadrupole transition strength $B(E2;0_1^{+}\to 2_1^{+})$, (c) the ratio of excitation energies $R_{42}=E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$, (d) spectroscopic quadrupole moment $Q^{\mathrm{spec}}\left(2_1^{+}\right)$, and (e) neutron-proton decoupling factor $\eta$ for Er, Yb, Hf, and W isotopes as a function of neutron number from the MR-CDFT calculations using PC-PK1. See text for details.

Figure 6

Two-neutron separation energies [(a), (b)] and their differentials [(c), (d)] in Er isotopes as a function of neutron number. The upper panels [(a), (c)] show the results from macroscopic-microscopic models and nonrelativistic energy density functionals calculations, while the lower panels [(b), (d)] are from the relativistic calculations. The discrepancy between model predictions and data is given in the insets. The data are taken from Ref. [60]. See text for details.

Figure 7

$\mathrm{\Delta }{S}_{2N}$ obtained from different calculations as a function of proton number.

Figure 8

(a) Energy surface of mean-field states (MF), particle-number projected states ($N\&Z$), and those with an additional projection onto angular momentum $J=0$ for ${}^{194}\mathrm{Er}$ by the relativistic PC-PK1. The global energy minimum of each energy surface is marked with square, triangle, and diamond symbols, respectively. The ground state by the GCM calculation is indicated with a red dot. (b) Energy gained from particle-number projection (dashed green line) and that from angular-momentum projection (solid blue line) as functions of the quadrupole deformation $\beta$.

Figure 9

Correlation energies $E_{\mathrm{Corr}}^{\mathrm{Dyn}}, E_{\mathrm{Corr}}^{\mathrm{PNP}}, E_{\mathrm{Corr}}^{\mathrm{AMP}}$, and $E_{\mathrm{Corr}}^{\mathrm{GCM}}$ for ${}^{190-206}\mathrm{Er}, {}^{192-208}\mathrm{Yb}, {}^{194-210}\mathrm{Hf}$, and ${}^{196-212}\mathrm{W}$ isotopes as a function of neutron number. The results $E_{\mathrm{Corr}}^{\mathrm{Crank}}$ calculated using the cranking prescription (18) are also given for comparison.

Figure 10

Two-neutron separation energy from the MR-CDFT calculations with different approximations, including the pure mean-field calculation (MF), with particle-number projection only (PNP), and projections onto both particle number and angular momentum $J=0$ [PNAMP($J=0$)], as well as the quantum-number projected GCM calculation (PNAMP+GCM), for ${}^{190-206}\mathrm{Er}, {}^{192-208}\mathrm{Yb}, {}^{194-210}\mathrm{Hf}$, and ${}^{196-212}\mathrm{W}$ isotopes as a function of neutron number.

Figure 11

Two-neutron separation energies $S_{2n}$ (a) and their differentials $\mathrm{\Delta }{S}_{2n}$ (b) in Er isotopes as functions of neutron number from the MR-CDFT calculations with different approximations using PC-PK1.

Figure 12

Absolute value of the differential $\mathrm{\Delta }{S}_{2n}$ at neutron number $N=126$ from different calculations as a function of proton number. The results by the Gogny force D1M are taken from Ref. [21].

Figure 13

Two-neutron separation energies (upper) and their differentials (lower) at neutron number $N=126$ in Er, Yb, Hf, and W isotopes predicted by different models.