Shape phase transitions in $\mathrm{odd}\text{−}A$ Zr isotopes

Spectroscopic properties that characterize shape-phase transitions in neutron-rich odd-$A$ Zr isotopes are investigated using the framework of nuclear density-functional theory and particle-core coupling. The interacting-boson Hamiltonian of the even-even core nuclei, and the single-particle energies and occupation probabilities of the unpaired neutron, are completely determined by deformation constrained self-consistent mean-field calculations based on the relativistic Hartree-Bogoliubov model with a choice of a universal energy density functional and pairing interaction. The triaxial $(\beta ,\gamma )$ deformation energy surfaces for even-even ${}^{94-102}\mathrm{Zr}$ indicate the occurrence of a transition from triaxial or $\gamma$ soft $({}^{94,96}\mathrm{Zr})$ to prolate $\left({}^{98}\mathrm{Zr}\right)$ and triaxial $\left({}^{100,102}\mathrm{Zr}\right)$ shapes. The corresponding low-energy excitation spectra of the odd-$A$ Zr isotopes are in very good agreement with recent experimental results. Consistent with the structural evolution of the neighboring even-even Zr nuclei, the state-dependent effective deformations and their fluctuations in the odd-$A$ isotopes indicate a pronounced discontinuity around the transitional nucleus ${}^{99}\mathrm{Zr}$.

Figure 1

SCMF $(\beta ,\gamma )$ deformation energy surfaces (in MeV) for the even-even nuclei ${}^{94-102}\mathrm{Zr}$, obtained from constrained relativistic Hartree-Bogoliubov calculations using the DD-PC1 functional [29] and a separable pairing force [30]. The total SCMF energies are plotted to 5 MeV with respect to the global minimum. The energy difference between neighboring contours is 100 keV.

Figure 2

The bosonic energy surfaces based on the IBM-2 Hamiltonian in Eq. (2), with the parameters determined by the corresponding constrained SCMF calculations.

Figure 3

Low-energy excitation spectra of the even-even isotopes ${}^{94-102}\mathrm{Zr}$, calculated with the IBM-2 Hamiltonian of Eq. (2). The corresponding data, taken from the compilation of the ENSDF database [48], are included for comparison.

Figure 4

Excitation spectra for the low-lying positive [(a) and (b)] and negative-parity [(c) and (d)] states of the odd-$A$ nuclei ${}^{95-103}\mathrm{Zr}$. The experimental levels are from Refs. [13, 14, 48, 49].

Figure 5

Comparison between theory and experiment [13, 48] for the positive (left) and negative-parity (right) excitation spectrum of ${}^{97}\mathrm{Zr}$.

Figure 6

Band structure of theoretical positive (upper panel) and negative-parity (lower panel) excitation spectra of ${}^{99}\mathrm{Zr}$ in comparison to the available data [14, 48].

Figure 7

Same as in the caption to Fig. 6, but for the nucleus ${}^{101}\mathrm{Zr}$. The data are from Ref. [48, 50].

Figure 8

Probability amplitudes of the $3s_{1/2},2d_{3/2},2d_{5/2}$, and $1g_{7/2}$ single-neutron configurations in the wave functions of the calculated positive-parity yrast states ${1/2}_1^{+}$ (a), ${3/2}_1^{+}$ (b), ${5/2}_1^{+}$ (c), and ${7/2}_1^{+}$ (d) in the odd-$A$ isotopes ${}^{95-103}\mathrm{Zr}$.

Figure 9

The effective quadrupole deformation parameters ${\beta }_{\mathrm{eff}}$ (a) and ${\gamma }_{\mathrm{eff}}$ (b), and the fluctuations ${\sigma }_{\beta }$ (c) and ${\sigma }_{\gamma }$ (d), calculated for the lowest three $0^{+}$ states of the even-even Zr nuclei.

Figure 10

Same as in the caption to Fig. 9, but for several low-spin yrast states with positive (left column) and negative (right column) parity in the odd-$A$ Zr isotopes.