Effects of rotation and valence nucleons in molecular $\alpha$-chain nuclei

Effects of rotation and valence nucleons in molecular linear $\alpha$-chain nuclei are analyzed using a three-dimensional lattice cranking model based on covariant density functional theory. The structure of the mirror nuclei ${}^{16}\mathrm{C}$ and ${}^{16}\mathrm{Ne}$ is investigated as a function of rotational frequency. The valence nucleons, with respect to the $3\alpha$ linear chain core of ${}^{12}\mathrm{C}$, at low frequency occupy the $\pi$ molecular orbital. With increasing rotational frequency these nucleons transition from the $\pi$ orbital to the $\sigma$ molecular orbital, thus stabilizing the $3\alpha$ linear chain structure. It is predicted that the valence protons in ${}^{16}\mathrm{Ne}$ change occupation from the $\pi$ to the $\sigma$ molecular orbital at $\hslash \omega \approx 1.3$ MeV, a lower rotational frequency compared to $\hslash \omega \approx 1.7$ MeV for the valence neutrons in ${}^{16}\mathrm{C}$. The same effects of valence protons are found in ${}^{20}\mathrm{Mg}$, compared to the four valence neutrons in ${}^{20}\mathrm{O}$. The model is also used to examine the effect of alignment of valence nucleons on the relative positions and size of the three $\alpha$ clusters in the mirror nuclei ${}^{16}\mathrm{C}$ and ${}^{16}\mathrm{Ne}$.

Figure 1

Proton density distributions in the $x\text{−}z$ plane for ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{C}$ and neutron density distributions in the $x\text{−}z$ plane for ${}^{16}\mathrm{Ne}$ at the rotational frequencies $\hslash \omega =0.0$ MeV [(a), (c), and (e)] and $\hslash \omega =3.0$ MeV [(b), (d), and (f)].

Figure 2

Single-proton Routhians as functions of the rotational frequency for ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{C}$, and single-neutron Routhians for ${}^{16}\mathrm{Ne}$. Each orbital is labeled by the corresponding Nilsson quantum number of its maximal component. The solid and dashed lines correspond to single-particle states with positive and negative parity, respectively. Circles denote the occupied orbitals.

Figure 3

Neutron localization function [(a), (b), and (c)], neutron density distributions [(d), (e), and (f)], valence particle density distributions [(g), (h), and (i)] in the $x\text{−}z$ plane for ${}^{16}\mathrm{C}$ at the rotational frequencies $\hslash \omega =0.0$, 2.0, and 3.0 MeV, respectively.

Figure 4

Proton localization function [(a), (b), and (c)], proton density distributions [(d), (e), and (f)], valence particle density distributions [(g), (h), and (i)] in the $x\text{−}z$ plane for ${}^{16}\mathrm{Ne}$ at the rotational frequencies $\hslash \omega =0.0$, 2.0, and 3.0 MeV, respectively.

Figure 5

Single-neutron Routhians for ${}^{16}\mathrm{C}$, and single-proton Routhians for ${}^{16}\mathrm{Ne}$, as functions of the rotational frequency. The open, open-dot, and filled circles denote the occupied orbitals before the level crossing, after the first crossing, and after the second crossing, respectively.

Figure 6

Same as in the caption to Fig. 3 but for ${}^{20}\mathrm{O}$ at the rotational frequencies $\hslash \omega =0.0$, 2.2, and 2.4 MeV.

Figure 7

Same as in the caption to Fig. 4 but for ${}^{20}\mathrm{Mg}$ at the rotational frequencies $\hslash \omega =0.0$, 2.2, and 2.4 MeV.

Figure 8

Angular momenta [(a) and (c)] and quadrupole deformation ${\beta }_2$ [(b) and (d)] as functions of rotational frequency for ${}^{16}\mathrm{C}$, ${}^{16}\mathrm{Ne}$, ${}^{20}\mathrm{O}$, and ${}^{20}\mathrm{Mg}$.

Figure 9

Location of the peak (upper panel), and the width (lower panel) of each $\alpha$-like cluster in ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{C}$, and ${}^{16}\mathrm{Ne}$, obtained by fitting the $3\alpha$ core density along $z$ direction by a linear combination of Gaussian functions, as functions of rotational frequency.