Probing spin and pseudospin symmetries in deformed nuclei by the Green's function method

(Pseudo)spin symmetries play vital roles in nuclear physics and have been studied extensively in spherical nuclei. In this work, possible spin and pseudospin symmetries in deformed nuclei are examined by solving a coupled-channel Dirac equation with quadruple deformation. The Green's function method is taken which provides a novel way to exactly determine the single-particle levels and properly describe the spacial density distributions. Taking axially-deformed nucleus $^{154}$Dy as an example, the spin doublets with a combination of Nilsson levels $\Lambda\pm 1/2[\mathcal{N},n_z, \Lambda]$ and pseudospin doublets with a combination of $\widetilde{\Lambda}\pm 1/2[\widetilde{\mathcal{N},n_z, \Lambda}]$ are determined. Different behaviors are displayed for the spin and pseudospin doublets. For the spin partners, those with smaller angular momentum $l$ and the third component $\Lambda$ owns better symmetry such as the $1p$ doublet while good pseudospin symmetry appears in partners locating close to the continuum threshold. By examining the single-particle Nilsson levels $\Omega[\mathcal{N},n_z,\Lambda]$ and the energy splittings between the partners, the conservation and breaking of SS and PSS are examined at different deformations. In the prolate side, the Nilsson levels for the spin and pseudospin doublets are almost parallel and the energy splittings are stable against varying deformations. By examining the density distributions, great similarities have been observed in the upper components for the spin doublets while great similarities in the lower component for the pseudospin doublets. Besides, these similarities maintain well at different deformations.