Nuclear quantum shape-phase transitions in odd-mass systems

Microscopic signatures of nuclear ground-state shape-phase transitions in odd-mass Eu isotopes are explored starting from excitation spectra and collective wave functions obtained by diagonalization of a core-quasiparticle coupling Hamiltonian based on energy density functionals. As functions of the physical control parameter—the number of nucleons—theoretical low-energy spectra, two-neutron separation energies, charge isotope shifts, spectroscopic quadrupole moments, and $E2$ reduced transition matrix elements accurately reproduce available data and exhibit more-pronounced discontinuities at neutron number $N=90$ compared with the adjacent even-even Sm and Gd isotopes. The enhancement of the first-order quantum phase transition in odd-mass systems can be attributed to a shape polarization effect of the unpaired proton which, at the critical neutron number, starts predominantly coupling to Gd core nuclei that are characterized by larger quadrupole deformation and weaker proton pairing correlations compared with the corresponding Sm isotopes.

Figure 1

Self-consistent RHB triaxial quadrupole energy surfaces in the $\beta -\gamma$ plane ($0^{\circ }\le \gamma \le {60}^{\circ }$) for Sm and Gd isotopes. All energies are normalized with respect to the binding energy of the corresponding ground state. The contours join points on the surface with the same energy, and the separation between neighboring contours is 0.5 MeV.

Figure 2

Low-energy positive-parity (left panels) and negative-parity (middle panels) bands of ${}^{149,151,153,155}\mathrm{Eu}$ isotopes as functions of angular momentum, in comparison with the available data [44]. The excitation energies of the negative-parity states are shown relative to the corresponding lowest state. The ground-state bands of the adjacent even-even Sm isotopes are also shown in the panels on the right. The theoretical predictions for the odd-mass and even-even isotopes are obtained by using the microscopic CQC Hamiltonian and 5DCH, respectively, based on the PC-PK1 energy density functional and a finite-range separable pairing force. The choice of the Fermi level and coupling strength ($\lambda , \chi$) in the CQC Hamiltonian: ($-4.80$, 8.80), ($-8.60$, 13.4), ($-8.05$, 11.0), and ($-8.70$, 9.80) reproduces the positive-parity bands of ${}^{149}\mathrm{Eu}, {}^{151}\mathrm{Eu}, {}^{153}\mathrm{Eu}$, and ${}^{155}\mathrm{Eu}$, respectively. The corresponding values of ($\lambda , \chi$) for the negative-parity bands of the four nuclei are ($-5.95$, 5.00), ($-9.70$, 24.0), ($-6.92$, 20.0), and ($-8.00$, 19.6), respectively. $\lambda$ is in units of MeV and $\chi$ in $\mathrm{MeV}/b^2$.

Figure 3

(a)–(d) Evolution of the two-neutron separation energies $S_{2n}$, isotope shifts of the ground-state charge radii ${\langle r_c^2\rangle }_A-{\langle r_c^2\rangle }_{A-2}$, spectroscopic quadrupole moments $Q_s$, and reduced transition matrix elements $\langle J_f\left|\right|E2\left|\right|J_i{\rangle }^2$, as functions of the neutron number in odd-mass Eu isotopes, and (e)–(h) adjacent even-even Sm isotopes. The values calculated with the microscopic CQC Hamiltonian for Eu isotopes, and the 5DCH for Sm isotopes, are compared with available data [44]. Green triangles denote observables calculated with a CQC Hamiltonian that includes only Sm even-even cores.

Figure 4

(a) Probabilities of the dominant configurations, and (b) quasiparticle energies, for the ground states of Eu isotopes calculated with the CDFT-based CQC Hamiltonian. In panel (b) the experimental odd-even mass differences ${\mathrm{\Delta }}^{\left(5\right)}$ are calculated by using the five-point formula [47].