Nature of $\gamma$ deformation in Ge and Se nuclei and the triaxial projected shell model description

Recent experimental data have demonstrated that ${}^{76}$Ge may be a rare example of a nucleus exhibiting rigid $\gamma$ deformation in the low-spin regime. In the present work, the experimental analysis is supported by microscopic calculations using the multi-quasiparticle triaxial projected shell model (TPSM) approach. It is shown that to best describe the data of both yrast and $\gamma$-vibrational bands in ${}^{76}$Ge, a rigid-triaxial deformation parameter $\gamma \approx {30}^{\circ }$ is required. TPSM calculations are discussed in conjunction with the experimental observations and also with the published results from the spherical shell model. The occurrence of a $\gamma \gamma$ band in ${}^{76}$Ge is predicted with the bandhead at an excitation energy of $\sim$2.5 MeV. We have also performed TPSM study for the neighboring Ge and Se isotopes and the distinct $\gamma$-soft feature in these nuclei is shown to result from configuration mixing of the ground-state with multi-quasiparticle states.

Figure 1

Theoretical band diagram for ${}^{76}$Ge. The labels $(K,\#)$ characterize the states, with $K$ denoting the $K$ quantum number and $\#$ the number of quasiparticles. For example, (0,0), (2,0), and (4,0) correspond to the $K=0$ ground, $K=2$ $\gamma$, and $K=4$ $\gamma$$\gamma$ band, respectively, projected from the 0-qp state. (1,$2n$), (3,$2n$), (1,$2p$), (3,$2p$), (2,4), and (4,4) correspond, respectively, to the projected two-neutron-aligned state, two-proton-aligned state, two-neutron–plus–two-proton aligned state, with different $K$ quantum numbers.

Figure 2

Comparison of the calculated band energies with available experimental data for ${}^{76}$Ge. Data are taken from Ref. [10].

Figure 3

Comparison of the TPSM calculation with experimental data [10] for ${}^{76}$Ge. Results of the spherical shell model (SSM) calculations are also shown. (a) Staggering parameter $S\left(I\right)$ for the $\gamma$ band, and (b) $B\left(E2\right)$ values for the yrast band. The $B\left(E2\right)$ data are taken from Ref. [28].

Figure 4

Probabilities of the projected configurations in the yrast-, first-, and second-excited bands.

Figure 5

Comparison of the TPSM calculation with experimental data [10] for ${}^{76}$Ge for the relation between spin $I$ and transition energy $E_{\gamma }$. Results of the spherical shell model (SSM) calculations [12] are also shown.

Figure 6

(a) Calculated $B\left(E2\right)$ values and (b) transition energies $E_{\gamma }$ for the yrast band of ${}^{76}$Ge with varying ${\varepsilon }^{\prime }$.

Figure 7

Comparison of the calculated band energies with available experimental data for ${}^{70,72,74,78,80}$Ge. Data are taken from Refs. [32, 33, 34, 35, 36].

Figure 8

Comparison of the calculated band energies with available experimental data for ${}^{76–82}$Se. Data are taken from Refs. [35, 36, 37, 38].

Figure 9

Comparison of calculated staggering parameter $S\left(I\right)$ for the $\gamma$ band before and after configuration mixing for ${}^{70–80}$Ge. $S\left(I\right)$ parameters before mixing are divided by a factor of three so that they fit in the figure.

Figure 10

Comparison of calculated staggering parameter $S\left(I\right)$ for the $\gamma$ band before and after configuration mixing for ${}^{76–82}$Se. $S\left(I\right)$ parameters before mixing are divided by a factor of three so that they fit in the figure.

Figure 11

Comparison of calculated staggering parameter $S\left(I\right)$ for the $\gamma$ band (results after configuration mixing) with different triaxial deformation parameters ${\varepsilon }^{\prime }$ for ${}^{76}$Ge and ${}^{78}$Se.

Figure 12

Comparison of the TPSM calculation of $B\left(E2\right)$ values for the yrast band with experimental data [32, 33, 34, 35, 36, 39, 40, 41, 42] for ${}^{70-80}$Ge. Results of the spherical shell model (SSM) calculations [12] are also shown for the available nuclei.

Figure 13

Comparison of the TPSM calculation of $B\left(E2\right)$ values for the yrast band with experimental data [28, 35, 36, 37, 38, 39, 40, 41, 42, 43] for ${}^{76–82}$Se. Results of the spherical shell model (SSM) calculations [12] are also shown for the available nuclei.

Figure 14

Comparison of the TPSM calculation with experimental data [32, 33, 34, 35, 36] for ${}^{70–80}$Ge for the relation between spin $I$ and transition energy $E_{\gamma }$. Results of the spherical shell model (SSM) calculations [12] are also shown for the available nuclei.

Figure 15

Comparison of the TPSM calculation with experimental data [35, 36, 37, 38] for ${}^{76–82}$Se for the relation between spin $I$ and transition energy $E_{\gamma }$. Results of the spherical shell model (SSM) calculations [12] are also shown for the available nuclei.