Structure of krypton isotopes calculated with symmetry-conserving configuration-mixing methods

Shape transitions and shape coexistence in the ${}^{70–98}\mathrm{Kr}$ region are studied in a unified view with state-of-the-art beyond-self-consistent mean-field methods based on the Gogny D1S interaction. Beyond-mean-field effects are taken into account through the exact angular-momentum and particle-number restoration and the possibility of axial and nonaxial shape mixing. The results of the low-lying properties of these isotopes are in good agreement with the experimental data when the triaxial degree of freedom is included. Shape transitions from axial-oblate (${}^{70–72}\mathrm{Kr}$) to triaxial-prolate (${}^{74–78}\mathrm{Kr}$) and from spherical-triaxial (${}^{86–92}\mathrm{Kr}$) to axial-oblate (${}^{94–98}\mathrm{Kr}$) ground states are obtained. Additionally, low-lying $0^{+}$ excited states and quasi-$\gamma$ bands are found showing the richness of the collective structure in this region.

Figure 1

Particle number projected potential-energy surface in the $(\beta ,\gamma )$ plane for ${}^{68–98}\mathrm{Kr}$. Contour lines are separated by 1 MeV (solid lines) and by 0.25 MeV (dashed lines) and the energies are normalized to the minimum of each surface.

Figure 2

Neutron single-particle energies as a function of the axial quadrupole deformation parameter calculated for ${}^{96}\mathrm{Kr}$ with the Gogny D1S interaction. Neutron numbers in the gaps and level crossings indicate the minima in the axial potential-energy surfaces found for each nuclei (${}^{68–98}\mathrm{Kr}$). Continuous (dashed) lines correspond to positive (negative) levels and the color code represents the value of $j_z$: 1/2, 3/2, 5/2, 7/2, 9/2, and 11/2 match with black, red, green, cyan, purple, and light blue, respectively.

Figure 3

Collective wave functions for the ground state ($0_1^{+}$) and $2_1^{+}$, $0_2^{+}$, and $2_2^{+}$ excited states calculated with the SCCM method for ${}^{70–98}\mathrm{Kr}$ isotopes (from left to right and from top to bottom). One-dimensional plots represent axial calculations while “pie-like” plots represent full triaxial results (color scale: red and blue mean large and small height, respectively).

Figure 4

Excitation energies along the Krypton isotopic chain ${}^{70–98}\mathrm{Kr}$ for (a) $2_1^{+}$, (b) $4_1^{+}$, (c) $0_2^{+}$, and (d) $2_2^{+}$ states. Black boxes, blue diamonds, and red bullets represent the experimental values (taken from Ref. [13]), and the results of SCCM axial and SCCM triaxial calculations, respectively.

Figure 5

(a) Electric-quadrupole $\left(E2\right)$ reduced transition probabilities between $2_1^{+}$ and $0_1^{+}$ states. (b) Spectroscopic electric-quadrupole moments and (c) gyromagnetic factors for $2_1^{+}$ states. (d) Electric-monopole $E0$ transition strength between $0_2^{+}$ and $0_1^{+}$ states. Black boxes and red bullets represent the experimental values (taken from Refs. [5, 8, 12, 13, 14, 20, 44, 45, 46, 47, 48]) and the results of SCCM triaxial calculations, respectively. Blue dashed line in panel (c) is the $Z/A$ line provided by the collective rotor model [33].

Figure 6

Excitation energies for ${}^{70–76}\mathrm{Kr}$ isotopes. Experimental values are taken from Ref. [13].

Figure 7

Same as Fig. 6 but for ${}^{78–84}\mathrm{Kr}$ isotopes.

Figure 8

Same as Fig. 6 but for ${}^{86–92}\mathrm{Kr}$ isotopes.

Figure 9

Same as Fig. 6 but for ${}^{94–98}\mathrm{Kr}$ isotopes.