$\beta$ decay and evolution of low-lying structure in Ge and As nuclei

A simultaneous calculation for the shape evolution and the related spectroscopic properties of the low-lying states, and the $\beta$-decay properties in the even- and odd-mass Ge and As nuclei in the mass $A\approx 70–80$ region, within the framework of the nuclear density functional theory and the particle-core coupling scheme, is presented. The constrained self-consistent mean-field calculations using a universal energy density functional (EDF) and a pairing interaction determines the interacting-boson Hamiltonian for the even-even core nuclei, and the essential ingredients of the particle-boson interactions for the odd-nucleon systems, and of the Gamow-Teller and Fermi transition operators. A rapid structural evolution from $\gamma$-soft oblate to prolate shapes, as well as the spherical-oblate shape coexistence around the neutron subshell closure $N=40$, is suggested to occur in the even-even Ge nuclei. The predicted low-energy spectra, electromagnetic transition rates, and $\beta$-decay $logft$ values are in a reasonable agreement with experiment. The predicted $logft$ values reflect the structures of the wave functions for the initial and final nuclei of $\beta$ decay, which are, to a large extent, determined by the microscopic input provided by the underlying EDF calculation.

Figure 1

The adopted spherical single-particle energies ${\epsilon }_{j_{\rho }}$ (a), (b), quasiparticle energies ${\tilde{\epsilon }}_{j_{\rho }}$ (c), (d), and occupation probabilities $v_{j_{\rho }}^2$ (e), (f) for the $2p_{1/2}, 2p_{3/2}$, and $1f_{5/2}$ orbitals for (left column) the odd neutron in the odd-$A$ Ge and even-$A$ As, and (right column) the odd proton in the odd-$A$ and even-$A$ As nuclei, calculated by the spherical RHB method.

Figure 2

SCMF and IBM $(\beta ,\gamma )$-deformation energy surfaces for the even-even ${}^{66–78}\mathrm{Ge}$ nuclei. The energy difference between neighboring contours is 200 keV. The global minimum is identified by the solid circle.

Figure 3

Comparison of theoretical and experimental [67] excitation energies of the (a) $2_1^{+}$, (b) $4_1^{+}$, (c) $0_2^{+}$, and (d) $2_2^{+}$ states of the even-even ${}^{66–78}\mathrm{Ge}$ nuclei.

Figure 4

Calculated and observed [67] $B\left(E2\right)$ values for the transitions (a) $2_1^{+}\to 0_1^{+}$, (b) $4_1^{+}\to 2_1^{+}$, (c) $0_2^{+}\to 2_1^{+}$, and (d) $2_2^{+}\to 2_1^{+}$.

Figure 5

Comparison of energy surfaces for ${}^{72}\mathrm{Ge}$ calculated with the SCMF method based on the DD-PC1 functional and mapped IBM-2 that includes configuration mixing (CM).

Figure 6

Comparison of energy spectra of ${}^{72}\mathrm{Ge}$ resulting from the mapped IBM-2 with and without the configuration mixing (CM) with the experimental spectra.

Figure 7

Comparison of the calculated and experimental [67] low-lying negative-parity excitation spectra of the odd-$N$ Ge nuclei.

Figure 8

Same as Fig. 7, but for the odd-$Z$ As nuclei.

Figure 9

Calculated $B\left(E2\right)$ and $B\left(M1\right)$ transition rates (in W.u.) between low-lying negative-parity states of odd-$N$ Ge isotopes. The available experimental data, taken from Ref. [67], are also shown.

Figure 10

Same as Fig. 9, but for the odd-$Z$ As nuclei.

Figure 11

Fractions of the $2p_{1/2}, 2p_{3/2}$, and $1f_{5/2}$ single-particle configurations in the IBFM-2 wave functions for the ${1/2}_1^{-}, {3/2}_1^{-}$, and ${5/2}_1^{-}$ states of the odd-$N$ Ge and odd-$Z$ As nuclei.

Figure 12

Same as Fig. 7, but for the low-energy positive-parity states of odd-odd As nuclei.

Figure 13

Calculated $logft$ values for the electron-capture decays from the odd-$A$ As to Ge nuclei, (a) $5/2_1^{-}\to 5/2_1^{-}$ and (b) $3/2_1^{-}\to 1/2_1^{-}$, and the ${\beta }^{-}$ decays from the odd-$A$ Ge to As nuclei, (c) $1/2_1^{-}\to 3/2_1^{-}$ and (d) $1/2_1^{-}\to 1/2_1^{-}$. The experimental data, available from Ref. [67], are also shown.

Figure 14

Same as Fig. 13, but for the ${\beta }^{+}$/EC decay $1_1^{+}\to 0_1^{+}$ from the even-$A$ As to Ge nuclei (a), and the ${\beta }^{-}$ decay $0_1^{+}\to 1_1^{+}$ from the even-$A$ Ge to As nuclei (b).