Searching for a $4\alpha$ linear-chain structure in excited states of ${}^{16}\mathrm{O}$ with covariant density functional theory

A study of the $4\alpha$ linear-chain structure in high-lying collective excitation states of ${}^{16}\mathrm{O}$ with covariant density functional theory is presented. The low-spin states are obtained by configuration mixing of particle-number and angular-momentum projected quadrupole deformed mean-field states with the generator coordinate method. The high-spin states are determined by cranking calculations. These two calculations are based on the same energy density functional PC-PK1. We have found a rotational band at low spin with the dominant intrinsic configuration considered to be the one whereby $4\alpha$ clusters stay along a common axis. The strongly deformed rod shape also appears in the high-spin region with the angular momentum $13\hslash \text{to}18\hslash$; however, whether the state is a pure $4\alpha$ linear chain is less obvious than for the low-spin states.

Figure 1

(a) Spin-orbit interaction energy $SO1$ [Eq. (9)] and sum of spin-orbit splitting energy $SO2$ [Eq. (10)] and (b) total energy of the mean-field and PNAMP states (normalized to the energy of the $0_1^{+}$ state) in ${}^{16}\mathrm{O}$ as a function of the intrinsic quadrupole deformation $\beta$. The horizontal short lines with bullets are the GCM solutions. Only the states with similar dominant configurations (oblate, prolate, and LCS, respectively) are plotted and placed at their “average” deformation $\overline{\beta }={\sum }_{\beta }{\left|g_{\alpha }^J\right|}^2\beta$, cf. Eq. (12). The insets are the contours of intrinsic total density on the $y\text{−}z \left(x\text{−}y\right)$ plane at $x=0.3 \left(z=0.3\right)$ fm for some typical prolate (oblate) configurations along the curve.

Figure 2

Mean-field energy surface in the $\beta \text{−}\gamma$ plane for ${}^{16}\mathrm{O}$. Two neighboring contour lines are separated by 2.0 MeV.

Figure 3

Comparison of the spin-orbit interaction matrix element $\langle k\left|V_{\mathrm{s}.\mathrm{o}.}\right|k\rangle$, cf. Eq. (11), for the $k\mathrm{th}$ occupied s.p. state in the mean-field states of ${}^{16}\mathrm{O}$ with $\beta =0.0$ and $\beta =3.6$.

Figure 4

(a) Experimental and (b) calculated low-spin spectra for ${}^{16}\mathrm{O}$. The data are taken from Refs. [43, 44].

Figure 5

Collective wave functions of the oblate, “SD,” and LCS-candidate states with $J=0,2,4,6$ from the $\text{GCM}+\text{PNAMP}$ calculation.

Figure 6

The total density distribution (in $\mathrm{fm}{}^{-3}$) of intrinsic states with deformation parameter (a) $\beta =2.8$, (b) 3.2, (c) 3.6, and (d) 4.0.

Figure 7

The density distribution (in $\mathrm{fm}{}^{-3}$) of each s.p. state corresponding to the mean-field configuration with $\beta =3.2$ in ${}^{16}\mathrm{O}$. The s.p. states are labeled with the quantum numbers [$n_x{n}_y{n}_z$] of the largest component in the HO basis for the large component of the Dirac spinor; cf. Eq. (7).

Figure 8

The weight $|f_{\alpha k}{|}^2$ [cf. Eq. (7)] of the most dominant components $\left[n_x{n}_y{n}_z\right]$ in the HO basis for the large component of Dirac spinors in the mean-field configuration with $\beta =3.2$ in ${}^{16}\mathrm{O}$ as a function of quantum number $n_z$.

Figure 9

The total density distribution (in $\mathrm{fm}{}^{-3}$) at $x=0.3$ fm in ${}^{16}\mathrm{O}$ corresponding to LCS by the cranking RMF calculation with rotational frequency $\hslash \omega =3.0,3.5,4.0$ MeV, respectively. The rms radii of (long, medium, short) axes are (a) $(3.0,1.3,1.1)$ fm, (b) $(3.1,1.4,1.1)$ fm, and (c) ($3.3,1.4,1.1$) fm, respectively.

Figure 10

Energy ${\varepsilon }_k$ of occupied s.p. states in ${}^{16}\mathrm{O}$ by the cranking RMF calculation as a function of cranking frequency $\hslash \omega$. Each s.p. state is labeled with the quantum number of the largest HO component [$n_x{n}_y{n}_z$] in the large component of the Dirac spinor; cf. Eq. (7).

Figure 11

Excitation energy of the state in Table 3 as a function of angular momentum $J(J+1)$ predicted by the cranking RMF calculation. The dashed line is by the rotational formula $E\left(J_{\mathrm{cra}}\right)=J(J+1)/\left(2\mathcal{J}\right)$ with the MOI ${\hslash }^2/\left(2\mathcal{J}\right)=0.11$ MeV