Triaxial-shape dynamics in the low-lying excited $0^{+}$ state: Role of the collective mass

Background: Nonyrast states in neutron-rich nuclei are being investigated experimentally. These states reveal various aspects and details of the nuclear structure, such as the fluctuation around the axially symmetric shape.

Purpose: The beyond-mean-field effects in neutron-rich nuclei with $N\simeq 28$ are investigated. We focus on the role of collective mass in triaxial-shape dynamics.

Method: We employ the five-dimensional quadrupole collective Hamiltonian method with the potential obtained in a constrained Hartree-Fock-Bogoliubov approach with a Skyrme energy-density functional and the collective-mass functions obtained by the cranking approximation. The method includes triaxial deformations.

Results: We find that ${}^{42}\mathrm{Mg}, {}^{40}\mathrm{Si}, {}^{44}\mathrm{S}$, and ${}^{46}\mathrm{S}$ show $\gamma$-soft: A flat behavior in the potential energy surface along the triaxial deformation. Their low-lying spectra show a strong nucleus dependence, while those obtained with a collective mass assumed as constant are similar to each other. The energy ratio $E\left(0_2^{+}\right)/E\left(2_1^{+}\right)$ and the $B\left(E2\right)$ ratio $B(E2;0_2^{+}\to 2_1^{+})/B(E2;2_1^{+}\to 0_1^{+})$ show a unique property of the $0_2^{+}$ state, while the energy and $B\left(E2\right)$ ratios in neutron-deficient $\gamma$-soft nuclei with $N=78$ do not depend on nucleus so much.

Conclusions: Low-lying spectra are determined by not only the potential energy but also the collective mass. We clarify the important role of the collective mass in low-energy dynamics in the neutron-rich $N\approx 28$ nuclei.

Figure 1

Potential energy surface in the $\beta -\gamma$ plane of Mg, Si, and S isotopes with $N=26$, 28, and 30 obtained from constrained DFT calculations with the ${\mathrm{SkM}}^{*}$ EDF.

Figure 2

Potential energy as a function of $\gamma$ at a fixed $\beta$ for the selected nuclei. The potential is shifted to $V(\gamma =0^{\circ })=0$.

Figure 3

Low-lying excitation spectra and $B\left(E2\right)$ values in units of $e^2$ ${\mathrm{fm}}^4$ of ${}^{44}\mathrm{S}$ (top) and ${}^{46}\mathrm{S}$ (middle) with the cranking mass (left) and the constant mass (right). Those of the E(5)–${\beta }^4$ and Wilets-Jean models are shown at the bottom. In the E(5)–${\beta }^4$ and Wilets-Jean models, the value of $D$ in the constant mass is determined by fitting $E\left(0_2^{+}\right)=4$ MeV and the $B\left(E2\right)$ values are normalized to $B(E2;2_1^{+}\to 0_1^{+})=50$ $e^2$ ${\mathrm{fm}}^4$.

Figure 4

Same as Fig. 1, but for the neutron-deficient $N=78{}^{134}\mathrm{Ba}, {}^{136}\mathrm{Ce}, {}^{138}\mathrm{Nd}, {}^{140}\mathrm{Sm}$, and ${}^{142}\mathrm{Gd}$ nuclides.

Figure 5

Same as Fig. 2, but for the $N=78$ nuclei.

Figure 6

$R_{0/2}$ (filled square), $R_{2/2}$ (filled circle), and $R_{4/2}$ (filled triangle) for the selected nuclei with the cranking mass (a) and constant mass (b). The ratios obtained with the E(5)–${\beta }^4$ and E(5)–${\beta }^6$ models and the Wilets-Jean model (WJ) are included in (b). Note that the $R_{0/2}$ value for the Wilets-Jean model is 3.9. (c) shows the ratios of the available experimental data from [14] for ${}^{44}\mathrm{S}$, from [20] for ${}^{40}\mathrm{Si}$, and from [44] for the $N=78$ nuclei.

Figure 7

Decomposition of the $0_2^{+}$ energy to the vibrational kinetic energy $T_{\text{vib}}$ and the potential energy $V$ divided by the total energy with the cranking mass for the selected nuclei.

Figure 8

$B\left(E2\right)$ ratio for the selected nuclei with the cranking mass (a) and constant mass (b) together with E(5)–${\beta }^4$ and E(5)–${\beta }^6$ models and the Wilets-Jean model (WJ).

Figure 9

Vibrational wave functions $|{\mathrm{\Psi }}_{\alpha 00}{(\beta ,\gamma )|}^2$ multiplied by ${\beta }^4{\left|W(\beta ,\gamma )R(\beta ,\gamma )\right|}^{1/2}$ of the $0_1^{+}$ (top) and $0_2^{+}$ (bottom) states with the cranking and constant masses for ${}^{44}\mathrm{S}$ and ${}^{46}\mathrm{S}$. The right two columns show those with the E(5)–${\beta }^4$ and Wilets-Jean models.