Nilsson diagrams for light neutron-rich nuclei with weakly-bound neutrons

Using Woods-Saxon potentials and the eigenphase formalism for one-particle resonances, one-particle bound and resonant levels for neutrons as a function of quadrupole deformation are presented, which are supposed to be useful for the interpretation of spectroscopic properties of some light neutron-rich nuclei with weakly bound neutrons. Compared with Nilsson diagrams in textbooks that are constructed using modified oscillator potentials, we point out a systematic change of the shell structure in connection with both weakly bound and resonant one-particle levels related to small orbital angular momenta $\ell$. Then, it is seen that weakly bound neutrons in nuclei such as ${}^{15-19}\mathrm{C}$ and ${}^{33-37}\mathrm{Mg}$ may prefer being deformed as a result of the Jahn-Teller effect, due to the near degeneracy of the $1d_{5/2}\text{−}2s_{1/2}$ levels and the $1f_{7/2}\text{−}2p_{3/2}$ levels in the spherical potential, respectively. Furthermore, the absence of some one-particle resonant levels compared with the Nilsson diagrams in textbooks is illustrated.

Figure 1

Neutron one-particle levels as a function of quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus ${}^{17}\mathrm{C}$. One-particle levels are denoted by the asymptotic quantum numbers [N n${}_z\Lambda \Omega$]. The Ω values are denoted for four positive-parity levels for $\beta <0$, because it may be difficult to see the connection to the levels for $\beta >0$. One-particle levels appearing at $\beta =0$ are $1p_{1/2},1d_{5/2},2s_{1/2}$, and $1d_{3/2}$ levels at −6.77, −0.56, −0.42, and +5.60 MeV, respectively. One-particle levels in the positive-energy region, of which the phase shift (one of the eigenphases) for $\beta =0$ ($\beta \ne 0$) does not increase through $\pi /2$ as energy increases, are not plotted. The neutron numbers 8 and 16, which are obtained by filling in all lower-lying levels, are indicated with circles.

Figure 2

Neutron one-particle levels as a function of the depth of Woods-Saxon potential, $V_{\mathrm{WS}}$, for $\beta =0$. All parameters other than $V_{\mathrm{WS}}$ are the same as those in Fig. 1. Note that $V_{\mathrm{WS}}=-40$ MeV is used in Fig. 1. The $\ell =2$ one-particle resonant level continues to be well defined above $\varepsilon =2$ MeV.

Figure 3

Neutron one-particle levels as a function of quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus ${}^{31}\mathrm{Mg}$. The ${\Omega }^{\pi }=1/2^{-}$ levels are denoted by dotted curves, the $3/2^{-}$ levels by dashed curves, the $5/2^{-}$ levels by dot-dashed curves, and the $7/2^{-}$ levels by dot-dot-dashed curves, while positive-parity levels are plotted by solid curves. One-particle levels appearing at $\beta =0$ are $1d_{5/2},2s_{1/2},1d_{3/2}$, and $1f_{7/2}$ levels, from the bottom to the top. Neither $2p_{3/2}$ nor $2p_{1/2}$ levels are obtained at $\beta =0$ as one-particle resonant levels and, thus, they are not plotted in the figure. The next low-lying one-particle resonant level for $\beta =0$ is the $1f_{5/2}$ level at 8.96 MeV that lies outside the range of the figure. See the text for details and also the legend to Fig. 1.

Figure 4

Neutron one-particle levels as a function of the depth of Woods-Saxon potential, $V_{\mathrm{WS}}$, for $\beta =0$. All parameters other than $V_{\mathrm{WS}}$ are the same as those in Fig. 3. Note that $V_{\mathrm{WS}}=-40$ MeV is used in Fig. 3. The $\ell =3$ one-particle resonant levels continue to be well defined above $\varepsilon =2$ MeV, while the $2p_{3/2}$ and $2p_{1/2}$ resonant levels do not survive for $\varepsilon >0.80$ MeV and $\varepsilon >0.62$ MeV, respectively.

Figure 5

Neutron one-particle levels as a function of quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus ${}^{37}\mathrm{Mg}$. One-particle levels appearing at $\beta =0$ are $2s_{1/2},1d_{3/2},1f_{7/2},2p_{3/2}$, and $1f_{5/2}$ levels at −7.02, −5.28, −0.66, +0.018 and +5.22 MeV, respectively. The $2p_{1/2}$ level is not obtained as a one-particle resonant level. See the legend to Fig. 3 and the text for details.