Pseudospin symmetry in the resonant states of nuclei

Pseudospin symmetry for the resonant states in ${}^{208}\mathrm{Pb}$ is investigated by solving the Dirac equation with Woods-Saxon vector and scalar potentials in combination with an analytic continuation in the coupling-constant method, where the influences of the coupling-constant interval and the Padé approximant order are analyzed, and the stable and convergent energies and widths are obtained. Pseudospin breaking is shown in correlation with the nuclear mean field shaped by the central depth ${\Sigma }_0$, a radius (range) R, and a diffusivity a, which play an important role in the splittings of energy and width. The energy-level crossings appear in several pseudospin partners of resonant states, where the decay time is found to be different for the pseudospin doublets even when their energies are fully degenerate.

Figure 1

(Color online) The energies of resonant states ɛ as a function of λ obtained from solving the Dirac equation with bound boundary condition. The crossings of ɛ and the horizontal line are the branch points ${\lambda }_0$.

Figure 2

(Color online) The calculated energies as a function of (${\lambda }_b-{\lambda }_0$) for the resonant states shown in Fig. 1, where panels (a), (b), (c), and (d) represent, respectively, the calculations with (3,3), (4,4), (5,5), and (6,6) PAs.

Figure 3

(Color online) The same as Fig. 2, but for the calculated widths as a function of (${\lambda }_b-{\lambda }_0$).

Figure 4

The single-particle levels in ${}^{208}\mathrm{Pb}$ calculated by the ACCC method in the Dirac equation for a Woods-Saxon potential with the parameters $R=7$ fm, ${\Delta }_0=650$ MeV, ${\Sigma }_0=-66$ MeV, and $a=0.6$ fm.

Figure 5

Pseudospin energy splittings for the resonant states in ${}^{208}\mathrm{Pb}$ as a function of the diffusivity for the neutron pseudospin partners shown in Fig. 1. The filled marks represent the corresponding pseudospin partners that have been lowered to the bound states. The vertical line corresponds to the fitted Woods-Saxon parameters for ${}^{208}\mathrm{Pb}$ given in the caption of Fig. 4.

Figure 6

The same as Fig. 5, but for the width splittings of pseudospin partners in ${}^{208}\mathrm{Pb}$.

Figure 7

Pseudospin energy splittings for the resonant states in ${}^{208}\mathrm{Pb}$ as a function of the radius for the neutron pseudospin partners shown in Fig. 1. The filled marks represent the corresponding pseudospin partners that have been lowered to the bound states. The vertical line corresponds to the fitted Woods-Saxon parameters for ${}^{208}\mathrm{Pb}$ given in the caption of Fig. 4.

Figure 8

The same as Fig. 7, but for the width splittings of pseudospin partners in ${}^{208}\mathrm{Pb}$.

Figure 9

Pseudospin energy splitting for the resonant states in ${}^{208}\mathrm{Pb}$ as a function of the depth of the Σ potential for the neutron pseudospin partners shown in Fig. 1. The filled marks represent the corresponding pseudospin partners that have been lowered to the bound states. The vertical line corresponds to the fitted Woods-Saxon parameters for ${}^{208}\mathrm{Pb}$ given in the caption of Fig. 4.

Figure 10

The same as Fig. 9, but for the width splittings of pseudospin partners in ${}^{208}\mathrm{Pb}$.