Structural evolution in germanium and selenium nuclei within the mapped interacting boson model based on the Gogny energy density functional

The shape transitions and shape coexistence in the Ge and Se isotopes are studied within the interacting boson model (IBM) with the microscopic input from a self-consistent mean-field calculation based on the Gogny-D1M energy density functional. The mean-field energy surface as a function of the quadrupole shape variables $\beta$ and $\gamma$, obtained from the constrained Hartree-Fock-Bogoliubov method, is mapped onto the expectation value of the IBM Hamiltonian with configuration mixing in the boson condensate state. The resultant Hamiltonian is used to compute excitation energies and electromagnetic properties of the selected nuclei ${}^{66–94}\mathrm{Ge}$ and ${}^{68–96}\mathrm{Se}$. Our calculation suggests that many nuclei exhibit $\gamma$ softness. Coexistence between prolate and oblate shapes, as well as between spherical and $\gamma$-soft shapes, is also observed. The method provides a reasonable description of the observed systematics of the excitation energy of the low-lying energy levels and transition strengths for nuclei below the neutron shell closure $N=50$, and provides predictions on the spectroscopy of neutron-rich Ge and Se isotopes with $52\le N\le 62$, where data are scarce or not available.

Figure 1

Mean-field energy surfaces for the nuclei ${}^{66–94}\mathrm{Ge}$. Results have been obtained with the Gogny-D1M EDF. The energy difference between neighboring contours is 100 keV.

Figure 2

The same as in Fig. 1, but for the nuclei ${}^{68–96}\mathrm{Se}$.

Figure 3

The same as in Fig. 1, but for the mapped IBM energy surfaces.

Figure 4

The same as in Fig. 2, but for the mapped IBM energy surfaces.

Figure 5

The IBM parameters $\epsilon ,\kappa ,\chi ,{\kappa }^{\prime },\omega$, and $\mathrm{\Delta }$ are depicted as functions of the neutron number, for the $[n=0]$ and $[n=1]$ configurations. For more details, see the main text.

Figure 6

The $2_1^{+},4_1^{+},0_2^{+}$, and $2_2^{+}$ excitation energies obtained in the diagonalization of the mapped IBM Hamiltonian are plotted as functions of the neutron number, for the Ge and Se nuclei, along with the available experimental data [58].

Figure 7

Fraction of the intruder configuration in the IBM $0_1^{+}$ and $0_2^{+}$ wave functions. For more details, see the main text.

Figure 8

The $B(E2;2_1^{+}\to 0_1^{+}),B(E2;4_1^{+}\to 2_1^{+}),B(E2;0_2^{+}\to 2_1^{+})$, and $B(E2;2_2^{+}\to 2_1^{+})$ transition probabilities obtained for Ge isotopes are plotted as functions of the neutron number. Experimental data have been taken from Ref. [58].

Figure 9

The same as in Fig. 8, but for the Se isotopes. Experimental data have been taken from Ref. [58].

Figure 10

Spectroscopic quadrupole moments $Q_{sp}$ for the $2_1^{+}$ and $2_2^{+}$ states in $e\mathrm{b}$ units. The experimental values are taken from Refs. [58, 61].

Figure 11

The ${\rho }^2(E0;0_2^{+}\to 0_1^{+})$ values, obtained within the mapped IBM framework for Ge and Se nuclei, are compared with the available experimental data taken from Ref. [59].

Figure 12

Low-energy level scheme for ${}^{70}\mathrm{Ge}$. The numbers (in blue) near the arrows stand for the $B\left(E2\right)$ transition strengths in Weisskopf units. Experimental data have been taken from Ref. [58].

Figure 13

The same as in Fig. 12, but for ${}^{72}\mathrm{Se}$.

Figure 14

The same as in Fig. 12, but for ${}^{72}\mathrm{Ge}$.

Figure 15

The same as in Fig. 12, but for ${}^{74}\mathrm{Se}$.

Figure 16

The same as in Fig. 12, but for ${}^{74}\mathrm{Ge}$.

Figure 17

The same as in Fig. 12, but for ${}^{76}\mathrm{Se}$.

Figure 18

Low-energy level schemes for the $N=60$ isotones ${}^{92}\mathrm{Ge}$ and ${}^{94}\mathrm{Se}$. Note that the theoretical $0_2^{+}$ and $2_2^{+}$ levels of ${}^{94}\mathrm{Se}$ are almost degenerate. The $B\left(E2\right)$ transition strengths relevant to these states are $B(E2;0_2^{+}\to 2_1^{+})=7.7,B(E2;2_2^{+}\to 2_1^{+})=56,B(E2;2_2^{+}\to 0_1^{+})=0.087,B(E2;2_2^{+}\to 0_2^{+})=38$, and $B(E2;4_2^{+}\to 2_2^{+})=47$ W.u.

Figure 19

The Gogny-HFB energy curves for the $N=40$ isotones ${}^{72}\mathrm{Ge}$ and ${}^{74}\mathrm{Se}$ are depicted (upper panels) as functions of the axial deformation parameter $\beta$ $\left(\gamma =0^{\circ }\right)$. In the lower panels the HFB energies are shown as functions of $\gamma$. For each $\gamma$ value the parameter $\beta$ is chosen so as to minimize the energy. Results are shown for both the Gogny-D1S and Gogny-D1M EDFs.

Figure 20

Low-energy spectra obtained for ${}^{72}\mathrm{Ge}$ and ${}^{74}\mathrm{Se}$ within the fermion-to-boson mapping procedure based on the Gogny-D1S and Gogny-D1M EDFs. The experimental spectra are also included to facilitate the comparison.