Deformed relativistic Hartree-Bogoliubov theory in continuum with a point-coupling functional. II. Examples of odd Nd isotopes

Background: One fascinating frontier in nuclear physics is the study of exotic nuclei. The deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), which simultaneously includes the nuclear superfluidity, deformation, and continuum effects, can provide proper descriptions for both stable and exotic nuclei. In a previous work [Zhang et al., Phys. Rev. C 102, 024314 (2020)], the DRHBc theory based on the point-coupling density functionals was developed and the DRHBc calculation, previously accessible only for light nuclei, was extended for all even-even nuclei in the nuclear chart. The ground-state properties for the even-even nuclei with $8\le Z\le 120$ from the DRHBc calculations have been summarized [Zhang et al., At. Data Nucl. Data Tables 144, 101488 (2022)].

Purpose: The aim of this paper is to extend the point-coupling DRHBc theory to odd-$A$ and odd-odd nuclei and examine its applicability by taking odd-$A$ Nd isotopes as examples.

Method: In the DRHBc theory, the densities and potentials with axial deformation are expanded in terms of Legendre polynomials, and the relativistic Hartree-Bogoliubov equations are solved in a Dirac Woods-Saxon basis to include the continuum effects. For an odd-$A$ or odd-odd nucleus, the blocking effect of the unpaired nucleon(s) is taken into account with the equal filling approximation. To determine its ground state, an automatic blocking procedure is adopted, in which the orbital with the lowest quasiparticle energy is blocked during the iteration. This procedure is justified by comparing with the results from the orbital-fixed blocking calculations, in which the blocked orbital near the Fermi surface is fixed during the iteration. The ground states for both light and heavy nuclei can be provided by the automatic blocking procedure as the orbital-fixed blocking procedure, but with considerably reduced computational cost.

Results: The numerical details for even-even nuclei, including the convergence on the energy cutoff, angular momentum cutoff, and Legendre expansion, are found to be valid for odd-$A$ and odd-odd nuclei as well. The ground-state properties of odd-$A$ Nd isotopes are calculated with the density functional PC-PK1. The calculated physical observables, such as binding energies, two-neutron and one-neutron separation energies, and charge radii, are in good agreement with the available experimental data for the whole Nd isotopic chain.

Conclusions: The point-coupling DRHBc theory is extended to odd-$A$ and odd-odd nuclei by including the blocking effect. Taking Nd isotopes including both even-even and odd-$A$ ones as examples, the calculated ground-state properties with PC-PK1 are in good agreement with the available experimental data. This paper paves the way to construct the DRHBc mass table including all even-even, odd-$A$, and odd-odd nuclei in the nuclear chart.

Figure 1

Potential energy curves of ${}^{22}\mathrm{Mg}$ (a), ${}^{23}\mathrm{Mg}$ (b), and ${}^{22}\mathrm{Al}$ (c) in constrained DRHBc calculations. In (b) and (c), the results from the orbital-fixed and automatic blocking procedures are shown with dashed lines and solid lines, respectively. The unconstrained minima are also shown with filled circles.

Figure 2

Single neutron (a), (b) and proton (c), (d) orbitals near the Fermi energy in the canonical basis vs the occupation probability $v^2$ for both the prolate (a), (c) and oblate (b), (d) minima of ${}^{22}\mathrm{Mg}$ in the DRHBc calculations. The quantum numbers $m^{\pi }$ of the three orbitals nearest to the Fermi energy are given, and labeled 1, 2, and 3, where the subscript $n$ refers to neutron, and $p$ refers to proton. The Fermi energies are shown with the dashed lines.

Figure 3

Potential energy curves of ${}^{175}\mathrm{Nd}$ (a) and ${}^{174}\mathrm{Pm}$ (b) in constrained DRHBc calculations. The results of both the orbital-fixed and automatic blocking procedures are shown with dashed lines and solid lines, respectively. The unconstrained minima are also shown with filled circles.

Figure 4

Total energy $E_{\mathrm{tot}}$ and quadrupole deformation parameter ${\beta }_2$ as functions of the energy cutoff $E_{\mathrm{cut}}$ (a), angular momentum cutoff $J_{max}$ (b), and Legendre expansion truncation ${\lambda }_{max}$ (c) for ${}^{301}\mathrm{Th}$ from the DRHBc calculations. The converged values of the total energy are shifted to zero. In panel (c), the result from constrained DRHBc calculations at the quadrupole deformation ${\beta }_2^{\mathrm{cst}}=0.6$ is also shown to confirm the convergence of ${\lambda }_{max}$ at a large deformation. Here the pairing correlation is neglected.

Figure 5

Binding energy per nucleon of Nd isotopes from the DRHBc calculations as a function of the neutron number. The results in the RCHB mass table [27] and the experimental data from Ref. [7] are shown for comparison.

Figure 6

The difference between the experimental binding energy [7] and the DRHBc calculations for Nd isotopes vs the neutron number. The results of the DRHBc calculations including the rotational correction energy $E_{\mathrm{rot}}$ and the RCHB mass table [27] are also shown for comparison.

Figure 7

Two-neutron (a) and one-neutron (b) separation energies as functions of the neutron number for Nd isotopes in the DRHBc calculations. The RCHB results [27] and the available data [7] are also shown for comparison. The inset gives the detailed comparison between the calculated results and the data.

Figure 8

Neutron (a) and proton (b) Fermi energies for Nd isotopes in the DRHBc calculations vs the neutron number. The results from the RCHB mass table [27] are shown for comparison.

Figure 9

Quadrupole deformation as a function of the neutron number in the DRHBc calculations for Nd isotopes. The available data [65], only the absolute values, for even-even nuclei are shown for comparison.

Figure 10

Evolution of the potential energy curves of even-even isotopes ${}^{126,134,142,\cdots ,198}\mathrm{Nd}$ (red solid lines) and odd-$A$ isotopes ${}^{127,135,143,\cdots ,199}\mathrm{Nd}$ (blue dashed lines) from constrained DRHBc calculations. For clarity reasons, the curves have been scaled to the energy of their ground states and have been shifted upward by 1 MeV per increasing one neutron. The evolution of the ground-state deformation is shown with the red and blue circles for even-even and odd-$A$ isotopes, respectively.

Figure 11

(a) Charge radius as a function of the neutron number in the DRHBc calculations for Nd isotopes. The results in the RCHB mass table [27] and available data from Ref. [64] are shown for comparison. (b) Rms neutron radius, proton radius, and matter radius as functions of the neutron number in the DRHBc calculations for Nd isotopes. The empirical matter radii $r_0{A}^{1/3}$, in which $r_0=0.948$ fm determined by ${}^{142}\mathrm{Nd}$, are also shown to guide the eye.

Figure 12

(a) Angle-averaged neutron density distribution, i.e., the spherical component, (b) the neutron density distribution along the symmetry axis $z$ ($\theta =0^{\circ }$), and (c) the neutron density distribution perpendicular to the symmetry axis with $r_{\perp }=\sqrt{x^2+y^2}$ ($\theta ={90}^{\circ }$), for selected even-even isotopes ${}^{124,134,\cdots ,194}\mathrm{Nd}$ (solid lines) and odd-$A$ isotopes ${}^{125,135,\cdots ,195}\mathrm{Nd}$ (dashed lines) in the DRHBc calculations.

Figure 13

Neutron density distributions for ${}^{174}\mathrm{Nd}$ ($x<0$) and ${}^{175}\mathrm{Nd}$ ($x>0$) with the $z$ axis as the symmetry axis. A dashed circle is drawn to guide the eye.