Background: Neutron-deficient krypton isotopes are of particular interest due to the coexistence of oblate and prolate shapes in low-lying states and the transition of the ground state from one dominate shape to another as a function of neutron number. Moreover, the onset of large E2 transition strength around 76 Kr indicates the erosion of the N = 40 subshell gap. Purpose: A detailed interpretation of these phenomena in neutron-deficient Kr isotopes requires the use of a method going beyond a mean-field approach that permits the determination of spectra and transition probabilities. The aim of this work is to provide a systematic calculation of low-lying states in the even-even 68-86 Kr isotopes and to understand the shape coexistence phenomenon and the onset of large collectivity around N = 40 from beyond relativistic mean-field studies. Method: The starting point of our method is a set of relativistic mean-field plus BCS wave functions generated with a constraint on triaxial deformations (β, γ ). The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions. To examine the role of triaxiality, a configuration mixing of both particle-number-and angular-momentum-projected axially deformed states is also carried out within the exact generator coordinate method based on the same energy density functional. Results: The energy surfaces, the excitation energies of 0 + 2 , 2 + 1 , and 2 + 2 states, as well as the E0 and E2 transition strengths are compared with the results of similar 5DCH calculations but with parameters determined by the nonrelativistic mean-field wave functions, as well as with the available data. The results show a picture of oblate-triaxial-prolate shape transition. Coexistence of low-lying excited 0 + states is found to be a common feature in the neutron-deficient Kr isotopes. The underlying mechanism responsible for the shape coexistence is discussed. Conclusions:The main features of the low-spin spectra and the systematics of excitation energies and transition strengths in the neutron-deficient Kr isotopes are reproduced very well. The effects of dynamic correlations and triaxiality turn out to have important influences on the balance between the competing oblate and prolate states. An exact treatment of configuration mixing of particle-number-and angular-momentum-projected triaxial states is highly demanded to pin down these effects.

The rapid structural change in low-lying collective excitation states of neutron-rich Sr and Zr isotopes is studied by solving a five-dimensional collective Hamiltonian with parameters determined by both relativistic mean-field and non-relativistic Skyrme-Hartree-Fock calculations using the PC-PK1 and SLy4 forces respectively. Pair correlations are treated in BCS method with either a separable pairing force or a density-dependent zero-range force. The isotope shifts, excitation energies, electric monopole and quadrupole transition strengths are calculated and compared with corresponding experimental data. The calculated results with both the PC-PK1 and SLy4 forces exhibit a picture of spherical-oblate-prolate shape transition in neutron-rich Sr and Zr isotopes. Compared with the experimental data, the PC-PK1 (or SLy4) force predicts a more moderate (or dramatic) change in most of the collective properties around N = 60. The underlying microscopic mechanism responsible for the rapid transition is discussed.

The shape evolution and shape coexistence phenomena in neutron-rich nuclei at N ≈ 60, including Kr, Sr, Zr, and Mo isotopes, are studied in the covariant density functional theory (DFT) with the new parameter set PC-PK1. Pairing correlations are treated using the BCS approximation with a separable pairing force. Sharp rising in the charge radii of Sr and Zr isotopes at N = 60 is observed and shown to be related to the rapid changing in nuclear shapes. The shape evolution is moderate in neighboring Kr and Mo isotopes. Similar as the results of previous Hartree-Fock-Bogogliubov (HFB) calculations with the Gogny force, triaxiality is observed in Mo isotopes and shown to be essential to reproduce quantitatively the corresponding charge radii. In addition, the coexistence of prolate and oblate shapes is found in both 98 Sr and 100 Zr. The observed oblate and prolate minima are related to the low single-particle energy level density around the Fermi surfaces of neutron and proton respectively. Furthermore, the 5-dimensional (5D) col-

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The low-lying collective states in Sn isotopes are studied by a five-dimensional collective Hamiltonian with parameters determined from the triaxial relativistic mean-field calculations using the PC-PK1 energy density functional. The systematics for both the excitation energies of 2 + 1 states and B(E2; 0 + 1 → 2 + 1 ) values are reproduced rather well, in particular, the enhanced E2 transitions in the neutron-deficient Sn isotopes with N < 66. We show that the gradual degeneracy of neutron levels 1g 7/2 and 2d 5/2 around the Fermi surface leads to the increase of level density and consequently the enhanced paring correlations from N = 66 to 58. It provokes a large quadrupole shape fluctuation around the spherical shape, and leads to an enhanced collectivity in the isotopes around N = 58.