Prolate-shape dominance in atomic nuclei within the deformed relativistic Hartree-Bogoliubov theory in continuum

The dominance of prolate over oblate ground-state deformations has been a well-known empirical fact. With the state-of-the-art deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) and taking Te, Xe, and Ba isotopes with neutron number $82⩽N⩽126$ as examples, a self-consistent and microscopic study of the prolate-shape dominance is given. According to the calculated potential-energy curves, most of these isotopes have both prolate and oblate minima and prefer prolate in the ground state, while the small amount of oblate ground-state shapes mainly appear after the major shell is half filled. It is also found that the prolate-shape dominance enhances with the proton number increasing from 52 to 56. The dominance and its enhancement can be well understood by the microscopic canonical single-particle energies obtained in the DRHBc theory, in good agreement with the schematic interpretation based on the Nilsson diagram. Finally, pairing correlations are found to bring more energy displacements to oblate minima on average but do not play a decisive role in prolate-shape dominance.

Figure 1

Quadrupole deformations as functions of the neutron number with $82⩽N⩽126$ for Te (red square), Xe (blue circle), and Ba (green diamond) isotopic chains in the DRHBc calculations with density functional PC-PK1.

Figure 2

Evolution of the potential-energy curves of Te, Xe, and Ba isotopes with even $N=82,90,\cdots ,122$ (red solid lines) and odd $N=83,91,\cdots ,123$ (blue dashed lines). For clarity, in each panel, the PEC of the lightest isotope (${}^{134}\mathrm{Te}, {}^{136}\mathrm{Xe}$, and ${}^{138}\mathrm{Ba}$) is renormalized to its ground state, and other PECs are shifted upward by 0.5 MeV per increasing one neutron. The evolution of the ground-state deformation is shown with the black squares.

Figure 3

Quadrupole deformation parameters ${\beta }_2$ of the prolate (blue squares) and oblate (green triangles) minima, as well as the energy difference (red circles) between them $E_{\text{diff}}=E_{\text{min}}\left(\text{oblate}\right)-E_{\text{min}}\left(\text{prolate}\right)$, as functions of the neutron number for Te, Xe, and Ba isotopic chains in the DRHBc calculations with PC-PK1.

Figure 4

(a) Single-neutron energies in the canonical basis of ${}^{154}\mathrm{Te}$ as functions of the deformation parameter ${\beta }_2$ obtained in the DRHBc calculations. The Fermi energy as a function of quadrupole deformation is also displayed by a dashed line. The deformation parameters of the prolate and oblate minima are marked by the vertical dashed lines. The neutron number obtained by filling all the lower levels is shown with a circle at several energy gaps. The spherical subshells $l_j$ are denoted and the $pf$ shell includes $p_{3/2}, f_{5/2}$, and $p_{1/2}$. For clarity, the orbitals stemming from the spherical subshell $l_{j=l+1/2}$ are shown in blue and those from the subshell $l_{j=l-1/2}$ are in red. (b) The same as panel (a) but for single-proton energies.

Figure 5

The impacts of pairing correlations on the prolate-oblate shape competition for Te isotopic chain obtained in the DRHBc calculations with PC-PK1. (a) Quadrupole deformation parameters of the prolate and oblate minima as functions of the neutron number with (solid symbols) or without (open symbols) pairing correlations. (b) Energy difference between the prolate local minima with and without pairing correlations (red circles), and that between the oblate ones (blue squares). (c) Energy difference between the prolate and oblate minima with (black squares) or without (olive circles) pairing correlations.