Hoyle-analog state in ${}^{13}\mathrm{C}$ studied with antisymmetrized molecular dynamics

The cluster states in ${}^{13}\mathrm{C}$ are investigated by antisymmetrized molecular dynamics. By investigating the spectroscopic factors, the cluster configurations of the excited states are discussed. It is found that the $1/2_2^{+}$ state is dominantly composed of the ${}^{12}\mathrm{C}\left(0_2^{+}\right)\otimes s_{1/2}$ configuration and can be regarded as a Hoyle-analog state. On the other hand, the $p$-wave states ($3/2^{-}$ and $1/2^{-}$) do not have such structure, because of the coupling with other configurations. The isoscalar monopole and dipole transition strengths from the ground to the excited states are also studied. It is shown that the excited $1/2^{-}$ states have strong isoscalar monopole transition strengths consistent with the observation. On the other hand, the excited $1/2^{+}$ states unexpectedly have weak isoscalar dipole transitions except for the $1/2_1^{+}$ state. It is discussed that the suppression of the dipole transition is attributed to the property of the dipole operator.

Figure 1

The calculated and observed [28] excitation spectra of the $1/2^{\pm }$, $3/2^{\pm }$, and $5/2^{\pm }$ states of ${}^{13}\mathrm{C}$ with the threshold energies for the relevant channels. The calculated threshold energies from the ${}^{12}\mathrm{C}\left(0_2^{+}\right)+n$ threshold are $-5.4$ MeV for ${}^{12}\mathrm{C}\left(2_1^{+}\right)+n$, 4.2 MeV for ${}^{12}\mathrm{C}\left(2_2^{+}\right)+n$, $-1.9$ MeV for ${}^9\mathrm{Be}(5/2_1^{-})+n$, $-1.7$ MeV for ${}^9\mathrm{Be}(1/2_1^{-})+n$, and $-2.0$ MeV for ${}^9\mathrm{Be}(1/2_1^{+})+n$ channels.

Figure 2

The excitation energies and the properties of the $1/2^{-}$ states obtained in the present work. In the upper panels, the calculated excitation spectra, matter rms radii $r_{\mathrm{rms}}$, and IS0 transition matrix from the ground state $M\left(\text{IS}0\right)$ are shown from left to right. The calculated $S$ factors in the ${}^{12}\mathrm{C}+n$ and ${}^9\mathrm{Be}+\alpha$ channels are presented in the lower panels. In the last panel, $l$ denotes the orbital angular momentum between $\alpha$ and ${}^9\mathrm{B}$ clusters.

Figure 3

Intrinsic matter and valence neutron density distributions on the $z=0$ plane for the basis wave functions obtained with variation under the constraint on the H.O. quanta. The contour plot indicates the matter density distribution while the color plot indicates the valence neutron density distribution. The panels (a)–(e) are the dominant components of the $1/2_1^{-}$, $1/2_2^{-}$, $1/2_3^{-}$, $1/2_4^{-}$, and $1/2_5^{-}$ states, respectively. The panels (f)–(h) are those of the $1/2_1^{+}$, $1/2_2^{+}$, and $1/2_3^{+}$ states, respectively.

Figure 4

The calculated RWAs of the $1/2^{-}$ states in the ${}^{12}\mathrm{C}+n$ and ${}^9\mathrm{Be}+\alpha$ channels. The RWAs that yield $S$ larger than 0.04 are displayed.

Figure 5

The calculated excitation spectra, the rms matter radii $r_{\mathrm{rms}}$, and the $S$ factors of $3/2^{-}$ states below 20 MeV. The $S$ factors smaller than 0.05 are not displayed.

Figure 6

The calculated excitation spectra, matter rms radii, $M\left(\text{IS}1\right)$, and $S$ factors of $1/2^{+}$ states below 20 MeV are shown in the same manner as in Fig. 2.

Figure 7

The calculated RWAs of the $1/2^{+}$ states in the ${}^{12}\mathrm{C}+n$ and ${}^9\mathrm{Be}+\alpha$ channels. The RWAs which yields the larger $S$ than 0.04 are displayed.

Figure 8

The calculated reduced widths ${\gamma }^2$ of the $1/2_2^{+}$ and $1/2_3^{+}$ states for ${}^{12}\mathrm{C}+n$ and ${}^9\mathrm{Be}+\alpha$ channels. The matching radius $a=4.5$ fm is applied.

Figure 9

The comparison between the excitation spectra of the $1/2^{\pm }$, $3/2^{\pm }$, and $5/2^{\pm }$ states of ${}^{13}\mathrm{C}$ calculated with Gogny D1S (red) and modified Gogny D1S (blue) interactions.

Figure 10

Same with Fig. 4 but obtained by using the modified Gogny D1S interaction.

Figure 11

Same as Fig. 7 but obtained by using the modified Gogny D1S interaction.