Dominance of low-$\ell$ component in weakly bound deformed single-neutron orbits

Calculating single-particle (Nilsson) levels in axially symmetric quadrupole-deformed potentials in coordinate space, the structure of weakly bound neutron orbits is studied in the absence of pair correlation. It is shown that in the wave functions of ${\Omega }^{\pi }=1∕2^{+}$ orbits, where $\Omega$ expresses the projection of the particle angular momentum along the symmetry axis, the $\ell =0\left(s_{1∕2}\right)$ component becomes overwhelmingly dominant as the binding energy of the orbits approaches zero, irrespective of the size of the deformation and the kind of Nilsson orbits. Consequently, all ${\Omega }^{\pi }=1∕2^{+}$ levels become practically unavailable for both deformation and many-body pair correlation, when the levels approach continuum or lie in the continuum.

Figure 1

Neutron energy levels in axially symmetric quadrupole-deformed Woods-Saxon potential as a function of the potential strength. The radius of the Woods-Saxon potential is expressed by $R$, while $r_0=1.27\text{fm}$ is used. The asymptotic quantum numbers $\left[N{n}_z\Lambda \Omega \right]$ assigned traditionally to these levels are $[\mathrm{101\hspace{0.17em}1}∕2]$, $[\mathrm{220\hspace{0.17em}1}∕2]$, $[\mathrm{211\hspace{0.17em}3}∕2]$, $[\mathrm{211\hspace{0.17em}1}∕2]$, $[\mathrm{202\hspace{0.17em}5}∕2]$, $[\mathrm{330\hspace{0.17em}1}∕2]$, $[\mathrm{200\hspace{0.17em}1}∕2]$, $[\mathrm{321\hspace{0.17em}3}∕2]$, and $[\mathrm{202\hspace{0.17em}3}∕2]$, from the left [smaller ${(R∕r_0)}^3$ values] to the right [larger ${(R∕r_0)}^3$ values] at ${\epsilon }_m\approx 0$. The neutron numbers 6 and 24, which are obtained by filling in all lower-lying levels, are indicated with circles. We note $\Omega \equiv m$.

Figure 2

Calculated probabilities of $s_{1∕2}$, $d_{3∕2}$, $d_{5∕2}$, and $g_{9∕2}$ components in the $[\mathrm{220\hspace{0.17em}1}∕2]$ level as a function of energy eigenvalue ${\epsilon }_m$.

Figure 3

Calculated probabilities of the $s_{1∕2}$ component in three ${\Omega }^{\pi }=1∕2^{+}$ Nilsson levels in the $s\text{−}d$ shell, $[\mathrm{220\hspace{0.17em}1}∕2]$, $[\mathrm{211\hspace{0.17em}1}∕2]$, and $[\mathrm{200\hspace{0.17em}1}∕2]$, as a function of respective energy eigenvalue ${\epsilon }_m$.

Figure 4

Calculated probabilities of $p_{1∕2}$, $p_{3∕2}$, $f_{5∕2}$, and $f_{7∕2}$ components in the $[\mathrm{330\hspace{0.17em}1}∕2]$ level as a function of energy eigenvalue ${\epsilon }_m$.