Test of pseudospin symmetry in deformed nuclei

Pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian with scalar and vector mean fields equal and opposite in sign. This symmetry imposes constraints on the Dirac eigenfunctions. We examine extensively the Dirac eigenfunctions of realistic relativistic mean field calculations of deformed nuclei to determine if these eigenfunctions satisfy these pseudospin symmetry constraints.

Figure 1

The neutron spectra in MeV for the pseudospin doublets in ${}^{168}\mathrm{Er}$.

Figure 2

Wave functions in ${\left(\text{Fermi}\right)}^{-3∕2}$ as a function of $z$ and $\rho =1,3,\text{and}5\text{fm}$ for the neutron pseudospin doublet $\left[402\right]5∕2$ and $\left[404\right]7∕2(\tilde{\Lambda }=3)$ in ${}^{168}\mathrm{Er}$. In each segment, the top row shows (from left to right) the relations in (i) Eq. (12a), involving $f_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{+}$ and $f_{\tilde{\eta },\tilde{\Lambda },1∕2}^{-}$, (ii) Eq. (12b), involving $f_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{-}$ and $f_{\tilde{\eta },\tilde{\Lambda },1∕2}^{+}$, (iii) Eq. (12c), involving $g_{\tilde{\eta },\tilde{\Lambda },1∕2}^{+}$ and $-g_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{-}$. The bottom row shows (from left to right) the left-hand side and right-hand side of (i) Eq. (13a), involving $g_{\tilde{\eta },\tilde{\Lambda },1∕2}^{-}$ and $g_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{+}$, (ii) Eq. (13b), involving $g_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{+}$ and $g_{\tilde{\eta },\tilde{\Lambda },+1∕2}^{+}$, (iii) Eq. (13b), involving $g_{\tilde{\eta },\tilde{\Lambda },+1∕2}^{-}$ and $g_{\tilde{\eta },\tilde{\Lambda },-1∕2}^{-}$.

Figure 3

Wave functions in ${\left(\text{Fermi}\right)}^{-3∕2}$ as a function of $\rho$ and $z=1,3,\text{and}5\text{fm}$ for the neutron pseudospin doublet $\left[402\right]5∕2$ and $\left[404\right]7∕2(\tilde{\Lambda }=3)$ in ${}^{168}\mathrm{Er}$. The content of the graphs in each segment as in Fig. 2.

Figure 4

Same as in Fig. 2 but for the neutron pseudospin doublet $\left[400\right]1∕2$ and $\left[402\right]3∕2\left(\tilde{\Lambda }=1\right)$ in ${}^{168}\mathrm{Er}$.

Figure 5

Same as in Fig. 3 but for the neutron pseudospin doublet $\left[400\right]1∕2$ and $\left[402\right]3∕2\left(\tilde{\Lambda }=1\right)$ in ${}^{168}\mathrm{Er}$.

Figure 6

Same as in Fig. 2 but for the neutron pseudospin doublet $\left[501\right]3∕2$ and $\left[503\right]5∕2\left(\tilde{\Lambda }=2\right)$ in ${}^{168}\mathrm{Er}$.

Figure 7

Same as in Fig. 3 but for the neutron pseudospin doublet $\left[501\right]3∕2$ and $\left[503\right]5∕2\left(\tilde{\Lambda }=2\right)$ in ${}^{168}\mathrm{Er}$.

Figure 8

Same as in Fig. 2 but for the neutron pseudospin doublet $\left[510\right]1∕2$ and $\left[512\right]3∕2\left(\tilde{\Lambda }=1\right)$ in ${}^{168}\mathrm{Er}$.

Figure 9

Same as in Fig. 3 but for the neutron pseudospin doublet $\left[510\right]1∕2$ and $\left[512\right]3∕2\left(\tilde{\Lambda }=1\right)$ in ${}^{168}\mathrm{Er}$.