Inversion doublets of reflection-asymmetric clustering in ${}^{28}\mathrm{Si}$ and their isoscalar monopole and dipole transitions

Background: Various cluster states of astrophysical interest are expected to exist in the excited states of ${}^{28}\mathrm{Si}$. However, they have not been identified firmly, because of the experimental and theoretical difficulties.

Purpose: To establish the ${}^{24}\mathrm{Mg}+\alpha ,{}^{16}\mathrm{O}+{}^{12}\mathrm{C}$, and ${}^{20}\mathrm{Ne}+2\alpha$ cluster bands, we theoretically search for the negative-parity cluster bands that are paired with the positive-parity bands to constitute the inversion doublets. We also offer the isoscalar monopole and dipole transitions as a promising probe for the clustering. We numerically show that these transition strengths from the ground state to the cluster states are very much enhanced.

Method: The antisymmetrized molecular dynamics with Gogny D1S effective interaction is employed to calculate the excited states of ${}^{28}\mathrm{Si}$. The isoscalar monopole and dipole transition strengths are directly evaluated from wave functions of the ground and excited states.

Results: Negative-parity bands having ${}^{24}\mathrm{Mg}+\alpha$ and ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ cluster configurations are obtained in addition to the newly calculated ${}^{20}\mathrm{Ne}+2\alpha$ cluster bands. All of them are paired with the corresponding positive-parity bands to constitute the inversion doublets with various cluster configurations. The calculation shows that the bandheads of the ${}^{24}\mathrm{Mg}+\alpha$ and ${}^{20}\mathrm{Ne}+2\alpha$ cluster bands are strongly excited by the isoscalar monopole and dipole transitions.

Conclusions: The present calculation suggests the existence of inversion doublets with the ${}^{24}\mathrm{Mg}+\alpha ,{}^{16}\mathrm{O}+{}^{12}\mathrm{C}$, and ${}^{20}\mathrm{Ne}+2\alpha$ configurations. Because of the enhanced transition strengths, we offer the isoscalar monopole and dipole transitions as good probe for the ${}^{24}\mathrm{Mg}+\alpha$ and ${}^{20}\mathrm{Ne}+2\alpha$ cluster bands.

Figure 1

The energy surfaces as functions of the quadrupole deformation parameters $\beta$ and $\gamma$ obtained by energy variation with the $\beta \gamma$ constraint and the angular momentum projection to (a) $J^{\pi }=0^{+}$ and (b) $J^{\pi }=1^{-}$. Red circles show the energy minima or plateaus.

Figure 2

Intrinsic matter density distributions of the minima on the positive- and negative-parity energy surfaces obtained by the $\beta \gamma$ constraint shown in Fig. 1. The panels (a), (b), and (c) show the oblate, prolate, and SD $J^{\pi }=0^{+}$ minima, while the panels (d), (e), and (f) show the $J^{\pi }=1^{-}$ minima generated by the 1p-1h excitations from the positive-parity minima.

Figure 3

Energy curves obtained by $d$ constraint. Panels (a), (b), and (c) respectively show the energy curves of the $J^{\pi }=0^{+}$ and $1^{-}$ states having ${}^{24}\mathrm{Mg}+\alpha ,{}^{16}\mathrm{O}+{}^{12}\mathrm{C}$, and ${}^{20}\mathrm{Ne}+{}^8\mathrm{Be}$ configurations. Circles and boxes in the figure show the overlap between these $J^{\pi }=0^{+}$ cluster configurations and the energy minima on the $\beta \gamma$ energy surface. Black circles in panels (a) and (c) shows the overlap between the oblate deformed minimum (ground state) and the ${}^{24}\mathrm{Mg}+\alpha \left(\mathrm{T}\right)$ and ${}^{20}\mathrm{Ne}+{}^8\mathrm{Be}$ cluster configurations, while the red boxes in panel (a) show the overlap between the SD minimum and ${}^{24}\mathrm{Mg}+\alpha \left(\mathrm{A}\right)$ configuration. The circles in panel (b) show the overlap between the prolate deformed minimum and the ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$ configuration.

Figure 4

Intrinsic matter density distributions obtained by $d$ constraint. (a), (b), (c), and (d) respectively show the ${}^{24}\mathrm{Mg}+\alpha$ (T), ${}^{24}\mathrm{Mg}+\alpha$ (A), ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$, and ${}^{20}\mathrm{Ne}+{}^8\mathrm{Be}$ configurations with $J^{\pi }=0^{+}$, while (e), (f), (g), and (h) show the $J^{\pi }=1^{-}$ partners having the same configurations.

Figure 5

Intrinsic matter density distributions obtained by $d$ constraint for ${}^{24}\mathrm{Mg}+\alpha$ cluster configurations with various intercluster distance $\mathrm{s}d$. (a), (b), (c), (d), and (e) show the configurations with $J^{\pi }=0^{+}$ and $d=8.0$, 4.0, 3.0, 2.0, and 1.0 fm, respectively.

Figure 6

Calculated and observed partial level scheme of ${}^{28}\mathrm{Si}$. The levels shown by dashed lines are the averaged energies for the candidates of cluster states. Dotted lines show the experimental and theoretical ${}^{24}\mathrm{Mg}+\alpha ,{}^{16}\mathrm{O}+{}^{12}\mathrm{C}$, and ${}^{20}\mathrm{Ne}+2\alpha$ threshold energies.

Figure 7

The oblate deformed ground state has the duality of the mean field and ${}^{24}\mathrm{Mg}+\alpha$ (T) and ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ clustering. From the duality, the $\beta$ band, ${}^{24}\mathrm{Mg}+\alpha$ bands, and ${}^{20}\mathrm{Ne}+{}^8\mathrm{Be}$ doublet arise.

Figure 8

Schematic figure for the various cluster configurations and their duality.

Figure 9

(a) The cluster $S$ factors for the ground and excited $0^{+}$ states. (b) Same as (a) but for the excited $1^{-}$ states. (c) The IS monopole transition matrix from the ground state to the excited $0^{+}$ states. (d) The IS dipole transition matrix from the ground state to the excited $1^{-}$ states. The values indicate the intercluster distance of the configuration which has the largest overlap in units of fm.