Microscopic description of large-amplitude shape-mixing dynamics with inertial functions derived in local quasiparticle random-phase approximation

On the basis of the adiabatic self-consistent collective coordinate method, we develop an efficient microscopic method of deriving the five-dimensional quadrupole collective Hamiltonian and illustrate its usefulness by applying it to the oblate-prolate shape coexistence/mixing phenomena in proton-rich ${}^{68,70,72}\mathrm{Se}$. In this method, the vibrational and rotational collective masses (inertial functions) are determined by local normal modes built on constrained Hartree-Fock-Bogoliubov states. Numerical calculations are carried out using the pairing-plus-quadrupole Hamiltonian including the quadrupole-pairing interaction within the two major-shell active model spaces both for neutrons and protons. It is shown that the time-odd components of the moving mean-field significantly increase the vibrational and rotational collective masses in comparison with the Inglis-Belyaev cranking masses. Solving the collective Schrödinger equation, we evaluate excitation spectra, quadrupole transitions, and moments. The results of the numerical calculation are in excellent agreement with recent experimental data and indicate that the low-lying states of these nuclei are characterized as an intermediate situation between the oblate-prolate shape coexistence and the so-called $\gamma$ unstable situation where large-amplitude triaxial-shape fluctuations play a dominant role.

Figure 1

Collective potential $V(\beta ,\gamma )$ for ${}^{68,70,72}\mathrm{Se}$. The regions higher than 3 MeV (measured from the oblate HB minima) are drawn by the rose-brown color. 1D collective paths connecting the oblate and prolate local minima are determined by using the ASCC method and depicted with bold red lines.

Figure 2

Monopole-pairing and quadrupole-pairing gaps for neutrons of ${}^{68}\mathrm{Se}$ are plotted in the $(\beta ,\gamma )$ deformation plane. (upper left) Monopole pairing gap ${\Delta }_0^{(n)}$. (lower left) Quadrupole pairing gap ${\Delta }_{20}^{(n)}$. (lower right) Quadrupole pairing gap ${\Delta }_{22}^{(n)}$. See Ref. [46] for definitions of ${\Delta }_0^{(n)},{\Delta }_{20}^{(n)}$, and ${\Delta }_{22}^{(n)}$.

Figure 3

Frequencies squared $\omega {}^2$ of the LQRPA modes calculated for ${}^{68}\mathrm{Se}$ are plotted as functions of $\beta$ or $\gamma$. The LQRPA modes adopted for calculation of the vibrational masses are connected with solid lines. (top) Dependence on $\gamma$ at $\beta =0.3$. (middle) Dependence on $\beta$ along the $\gamma =0.5^{°}$ line. (bottom) Dependence on $\beta$ along the $\gamma =30.5^{°}$ line.

Figure 4

Dependence on $\beta$ and $\gamma$ of the vibrational part of the metric $W(\beta ,\gamma )$ calculated for ${}^{68}\mathrm{Se}$. (top) Dependence on $\gamma$ at $\beta =0.3$. (middle) Dependence on $\beta$ along the $\gamma =0.5^{°}$ line. (bottom) Dependence on $\beta$ along the $\gamma =30.5^{°}$ line. The cross symbols indicate values of the vibrational metric calculated for various choices of two LQRPA modes from among the lowest 40 LQRPA modes; the lowest mode is always chosen and the other is from the remaining 39 modes. The smallest vibrational metric is shown by solid line. For reference, the vibrational metric calculated using the IB vibrational mass is indicated by broken lines.

Figure 5

Vibrational masses $D_{\beta \beta }(\beta ,\gamma )$, $D_{\beta \gamma }(\beta ,\gamma )/\beta$, and $D_{\gamma \gamma }(\beta ,\gamma )/\beta {}^2$, in units of MeV${}^{-1}$ calculated for ${}^{68}\mathrm{Se}$.

Figure 6

Ratios of the LQRPA vibrational masses to the IB vibrational masses $D_{\beta \beta }/D_{\beta \beta }^{(\mathrm{IB})}$ and $D_{\gamma \gamma }/D_{\gamma \gamma }^{(\mathrm{IB})}$, calculated for ${}^{68}\mathrm{Se}$.

Figure 7

Rotational masses $D_k(\beta ,\gamma )$ in units of MeV${}^{-1}$, calculated for ${}^{68}\mathrm{Se}$. See Eq. (39) for the relation with the rotational moments of inertia ${\mathcal{J}}_k(\beta ,\gamma )$.

Figure 8

Ratios of the LQRPA rotational masses to the IB rotational masses, $D_k(\beta ,\gamma )/D_k^{(\mathrm{IB})}(\beta ,\gamma )$, calculated for ${}^{68}\mathrm{Se}$.

Figure 9

Comparison of physical quantities evaluated with the CHB $+$ LQRPA approximation and those with the ASCC method. Both calculations are carried out along the 1D collective path for ${}^{68}\mathrm{Se}$ and the results are plotted as a function of $\gamma (q)$. From the top to the bottom: (a) the collective potential, (b) monopole-pairing gaps, ${\Delta }_0^{(n))}$ and ${\Delta }_0^{(p)}$, for neutrons and protons, (c) frequencies squared $\omega {}^2$ of the lowest and the second-lowest modes obtained by solving the moving-frame QRPA and the LQRPA equations, and (d) vibrational masses, $D_{\beta \beta }$, $D_{\beta \gamma }/\beta$, and $D_{\gamma \gamma }/\beta {}^2$, and (e) rotational masses $D_k$. In almost all cases, the results of the two calculations are indistinguishable because they agree within the widths of the line.

Figure 10

Excitation spectra and $B(E2)$ values calculated for ${}^{68}\mathrm{Se}$ by means of the CHB $+$ LQRPA method (denoted CHB $+$ LQRPA) and experimental data [5, 6, 7]. For comparison, results calculated using the IB cranking masses (denoted CHB $+$ IB) and those obtained using the ($1+3$)D version of the ASCC method [denoted ($1+3$)D ASCC] are also shown. Only $B(E2)$’s larger than 1 Weisskopf unit [in the (1+3)D ASCC and/or the CHB $+$ LQRPA calculations] are shown in units of $e^2$fm${}^4$.

Figure 11

Vibrational wave functions squared $\beta {}^4|{\Phi }_{\mathit{Ik}}(\beta ,\gamma )|{}^2$, calculated for ${}^{68}\mathrm{Se}$.

Figure 12

Same as Fig. 10 but for ${}^{70}\mathrm{Se}$. Experimental data are taken from Refs. [8, 41].

Figure 13

Same as Fig. 11 but for ${}^{70}\mathrm{Se}$.

Figure 14

Same as Fig. 10 but for ${}^{72}\mathrm{Se}$. Experimental data are taken from Refs. [8, 42].

Figure 15

Same as Fig. 11 but for ${}^{72}\mathrm{Se}$.

Figure 16

Spectroscopic quadrupole moments for ${}^{68,70,72}\mathrm{Se}$. Values calculated with the LQRPA collective masses are shown with the triangles. For comparison, values calculated with the IB collective masses and those obtained with the ($1+3$)D version of the ASCC method are also shown with the squares and the circles, respectively. The filled symbols show the values for the yrast states, while the open symbols those for the yrare states.