Green's function method for the spin and pseudospin symmetries in the single-particle resonant states

We investigate the spin and pseudospin symmetry in the single-particle resonant states by solving the Dirac equation containing a Woods-Saxon potential with Green's function method. Taking double-magic nucleus ${}^{208}\mathrm{Pb}$ as an example, three spin doublets $3d, 2h$, and $1j$ and three pseudospin doublets $3\tilde{p}, 1\tilde{i}$, and $1\tilde{j}$ are obtained for the single-neutron resonant states. By analyzing the energy splittings, we find that the threshold effect plays an important role in resonant pseudospin doubles. Besides, there is a reversed level structure of pseudospin doublets in the continuum. Differently, all the width splittings of either the spin doublets or the pseudospin doublets are systematically positive and the splittings are very small except $1\tilde{j}$ doublet. Further studies show that the splittings of the energies and widths for the resonant (pseudo)spin doublets are independent. Besides, the similarity properties of the wave functions of the spin and pseudospin doublets still maintain well in resonant states.

Figure 1

Density of neutron states $n_{\kappa }\left(\varepsilon \right)$ for different blocks in ${}^{208}\mathrm{Pb}$ obtained by solving a Dirac equation with Woods-Saxon mean-field potentials by Green's function method (blue solid line), in comparison with the results for free neutrons calculated with potentials $V=S=0$ (red solid line). The black-dashed line at $\varepsilon =0$ MeV indicates the position of the continuum threshold.

Figure 2

Single-particle levels and the mean-field potential $\mathrm{\Sigma }\left(r\right)$ for neutrons in ${}^{208}\mathrm{Pb}$. The neutron magic numbers 2, 8, 20, 28, 50, 82, and 126 and the candidate 184 are given.

Figure 3

Spin doublets (a) and pseudospin doublets (b) of the single-neutron spectra in ${}^{208}\mathrm{Pb}$.

Figure 4

Reduced SO splitting $\mathrm{\Delta }{E}_{\mathrm{SO}}=({\varepsilon }_{j_{<}}-{\varepsilon }_{j_{>}})/(2l+1)$ (a) and reduced PSO splitting $\mathrm{\Delta }{E}_{\mathrm{PSO}}=({\varepsilon }_{{\tilde{j}}_{<}}-{\varepsilon }_{{\tilde{j}}_{>}})/(2\tilde{l}+1)$ (b) versus their average single-particle energies $E_{\mathrm{av}}=({\varepsilon }_{j_{<}\left({\tilde{j}}_{<}\right)}+{\varepsilon }_{j_{>}\left({\tilde{j}}_{>}\right)})/2$ in single-neutron spectra of ${}^{208}\mathrm{Pb}$. For the spin doublets, $j_{<}=l-1/2$ and $j_{>}=l+1/2$, and for the pseudospin doublets, ${\tilde{j}}_{<}=\tilde{l}-1/2$ and ${\tilde{j}}_{>}=\tilde{l}+1/2$. Spin(pseudospin) doublets with the same $l(\tilde{l}$) are linked by lines.

Figure 5

The same as Fig. 4 but for the reduced SO width splitting $\mathrm{\Delta }{\mathrm{\Gamma }}_{\mathrm{SO}}=({\mathrm{\Gamma }}_{j_{<}}-{\mathrm{\Gamma }}_{j_{>}})/(2l+1)$ (a) and the reduced PSO width splitting $\mathrm{\Delta }{\mathrm{\Gamma }}_{\mathrm{PSO}}=({\mathrm{\Gamma }}_{{\tilde{j}}_{<}}-{\mathrm{\Gamma }}_{{\tilde{j}}_{>}})/(2\tilde{l}+1)$ (b).

Figure 6

Reduced energy splittings $\mathrm{\Delta }{E}_{\left(\mathrm{P}\right)\mathrm{SO}}$ versus reduced width splittings $\mathrm{\Delta }{\mathrm{\Gamma }}_{\left(\mathrm{P}\right)\mathrm{SO}}$ for the (pseudo)spin doublets in ${}^{208}\mathrm{Pb}$.

Figure 7

Distribution function ${\rho }_g\left(r\right)=\mathrm{Im}{\mathcal{G}}_{\kappa }^{\left(11\right)}(r,r;\varepsilon +i\epsilon )$ and ${\rho }_f\left(r\right)=\mathrm{Im}{\mathcal{G}}_{\kappa }^{\left(22\right)}(r,r;\varepsilon +i\epsilon )$ in the coordinate space for the single-neutron resonance spin doublets (a) $3d$, (b) $2h$, and (c) $1j$ in ${}^{208}\mathrm{Pb}$ at the resonant energy $\varepsilon ={\varepsilon }_{j_{>}\left(j_{<}\right)}$ extracted from $n\left(\varepsilon \right)$. For comparison, the functions for the spin doublets with smaller angular momenta $j_{<}$ are multiplied by a factor of 12.12, 2.50, and 0.05, respectively. With those factors, the density of states $n\left(\varepsilon \right)$ integrated by ${\rho }_g\left(r\right)+{\rho }_f\left(r\right)$ in Eq. (9) at $\varepsilon ={\varepsilon }_{j_{>}\left(j_{<}\right)}$ have the same height.

Figure 8

The same as Fig. 7 but for the pseudospin doublets (a) $3\tilde{p}$, (b) $1\tilde{i}$, and (c) $1\tilde{j}$ in ${}^{208}\mathrm{Pb}$. Besides, to depict the small components $\mathrm{Im}{\mathcal{G}}_{\kappa }^{\left(22\right)}(r,r;\varepsilon +i\epsilon )$ clearly, all of them are enlarged by 10 times.