Analytical continuation from bound to resonant states in the Dirac equation with quadrupole-deformed potentials

Background: Resonances with pronounced single-particle characteristics are crucial for quantitative descriptions of exotic nuclei near and beyond the drip lines, and often impact halo formation and nucleon decay processes. Since the majority of nuclei are deformed, the interplay between deformation and orbital structure near threshold can lead to improved descriptions of exotic nuclei.

Purpose: Develop a method to study single-particle resonant orbital structure in the Dirac equation with a quadrupole-deformed Woods-Saxon potential. Determine the structure evolution of bound and resonant levels with deformation in this scheme, and examine the impact on halo formation in loosely bound systems, with a focus on the recent halo candidate nucleus ${}^{37}\mathrm{Mg}$.

Method: Analytical continuation of the coupling constant (ACCC) method is developed on the basis of the Dirac equation with a deformed Woods-Saxon potential. The scalar and vector terms in the deformed potential are determined by the energies of the valence neutron and nearby orbitals, which are extracted from a self-consistent relativistic Hartree-Bogoliubov (RHB) calculation with the PC-PK1 density functional.

Results: We compare the energies and widths of resonant orbitals in the recent halo nucleus candidate ${}^{37}\mathrm{Mg}$ using the ACCC method based on the Dirac coupled-channel equations with those determined from the scattering phase shift (SPS) method. It is found that the results from the two methods agree well for narrow resonances, whereas the SPS method fails for broad resonances. Nilsson levels for bound and resonant orbitals from the ACCC method are calculated over a wide range of deformations and show some decisive hints of halo formation in ${}^{37}\mathrm{Mg}$.

Conclusions: In our ACCC model for deformed potentials in the coupled-channel Dirac equations, the crossing of the configuration $1/2\left[321\right]$ and $5/2\left[312\right]$ orbitals at a deformation of approximately 0.5 enhances the probability to occupy the $1/2\left[321\right]$ orbital coming from $2p_{3/2}$ thereby explaining the recent observation of a $\mathit{p}$-wave one-neutron halo configuration in ${}^{37}\mathrm{Mg}$. The resonant $1/2\left[301\right]$ configuration plays a crucial role in halo formation in the magnesium isotopes beyond $A=40$ for a wide range of deformations larger than 0.2.

Figure 1

Energy $E$ and width $\mathrm{\Gamma }$ of the neutron resonant state $3/2\left[301\right]$ as functions of the coupling constant $\lambda$ in the quadrupole-deformed Woods-Saxon potential with $\beta =0.47$. Filled circles and crosses are the solutions of the Dirac equation, and the former is used as input in the ACCC method. Solid curves are the outcome of the Padé approximant with $L=N=4$ for energy (black) and width (red).

Figure 2

Energies and widths of the neutron resonant states for the quadrupole-deformed Woods-Saxon potential with $\beta =0.47$. Solid circles represent the results of the ACCC method, while open circles denote the results of SPS method.

Figure 3

Energies and widths of the neutron resonant state $7/2\left[413\right]$ as a function of increasing length of the ${\lambda }_i$ values interval $[{\lambda }_a,{\lambda }_b]$ for Padé approximant orders $L=N=3$ (black solid squares lines), 4 (red solid circles lines), and 5 (blue solid triangles lines). These are carried out for a quadrupole-deformed Woods-Saxon potential with $\beta =0.47$.

Figure 4

Single-neutron levels as a function of $\beta$ in a quadrupole-deformed Woods-Saxon potential whose parameters are determined by the self-consistent RHB calculation for ${}^{37}\mathrm{Mg}$. All the levels are labeled by the asymptotic Nilsson quantum numbers $\mathrm{\Omega }\left[N{n}_z\mathrm{\Lambda }\right]$.