Shape vibration and quasiparticle excitations in the lowest $0^{+}$ excited state in erbium isotopes

The ground and first excited $0^{+}$ states of the ${}^{156–172}\text{Er}$ isotopes are analyzed in the framework of the generator coordinate method. The shape parameter $\beta$ is used to generate wave functions with different deformations which together with the two-quasiparticle states built on them provide a set of states. An angular momentum and particle number projection of the latter spawn the basis states of the generator coordinate method. With this ansatz and using the separable pairing plus quadrupole interaction we obtain a good agreement with the experimental spectra and $E2$ transition rates up to moderate spin values. The structure of the wave functions suggests that the first excited $0^{+}$ states in the soft Er isotopes are dominated by shape fluctuations, while in the well deformed Er isotopes the two-quasiparticle states are more relevant. In between, both degrees of freedom are necessary.

Figure 1

Energy curves for ${}^{156–172}\text{Er}$ calculated in the HFB approach (dashed lines), and in the PNAMP one (continuous lines). The bullet on each curve represents the experimental value of the deformation [36].

Figure 2

$0_2^{+}$ energies calculated by three different model spaces. The model spaces 1, 2, and 3 correspond to the wave functions of Eqs. (21), (22), and (8), respectively. The experimental data [37] are also shown for comparison.

Figure 3

The calculated spectra of the yrast bands and the $0_2^{+}$ bands of ${}^{156–172}\text{Er}$, compared with the experimental data. The data are taken from [37].

Figure 4

The ratio $R_{4/2}=E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$ for the erbium isotopes. The data are taken from [37].

Figure 5

Calculated $B(E2,I\to I-2)$ values for the yrast bands and the $0_2^{+}$ bands of ${}^{156–172}\text{Er}$, compared with available data. The data are taken from [37].

Figure 6

Calculated $B(E2,0_2^{+}\to 2_1^{+})$ values for ${}^{156–172}\text{Er}$, compared with experimental data in neighboring nuclei with the same neutron numbers. The data are from [37] except for ${}^{168}\text{Er}$ and ${}^{170}\text{Er}$, which are taken from Refs. [25, 41], respectively.

Figure 7

Calculated $E0$ transition probabilities for ${}^{156–172}\text{Er}$, compared with the experimental data where available. The data are taken from [43] for ${}^{164,168}\text{Er}$ and from [44] for ${}^{166}\text{Er}$.

Figure 8

Mikhailov plot for the nuclei ${}^{170}\text{Er}$ and ${}^{158}\text{Er}$ for transitions from the $0_2^{+}$ band to the ground band compared with the experimental data where available. The data are taken from [37].

Figure 9

Absolute squares of the collective wave functions calculated for the ground states and the $0_2^{+}$ states in ${}^{156–172}\text{Er}$. The thick continuous black lines represent $|p_0^{\sigma I}{\left(\beta \right)|}^2$ and the thick dashed red lines ${\sum }_{\left\{ij\right\}}{\left|p_{ij}^{\sigma I}\left(\beta \right)\right|}^2$ as a function of the deformation parameter $\beta$ for the ground states. The thin blue and magenta lines represent the corresponding amplitudes for the $0_2^{+}$ states.

Figure 10

Same as Fig. 9 but for angular momentum $6\hslash$ (top panels) and $10\hslash$ (lower panels), and two nuclei ${}^{158}\text{Er}$ (left panels) and ${}^{166}\text{Er}$ (right panels).

Figure 11

Contribution of the two-quasiparticle states to the ground states and the $0_2^{+}$ states in ${}^{156–172}\text{Er}$, measured by the summation of the absolute square of the collective amplitudes over the two-quasiparticle states and all shapes. The contributions from the most relevant two-quasiparticle configurations are also shown.