Soft dipole resonance in ${}^8\mathrm{C}$ and its isospin symmetry with ${}^8\mathrm{He}$

We investigate the soft dipole resonance in the proton-rich nucleus ${}_6{}^8\mathrm{C}_2$, which is a collective dipole oscillation of four valence protons against the $\alpha$ core, and discuss the isospin symmetry with the mirror nucleus ${}_2{}^8\mathrm{He}_6$. We use the $\alpha +N+N+N+N$ five-body cluster model and many-body resonances are obtained using the complex-scaling method. The $1^{-}$ resonance of ${}_6{}^8\mathrm{C}_2$ is confirmed at the excitation energy of 13 MeV with a large decay width of 24 MeV, and its structure is similar to the soft dipole resonance in ${}_2{}^8\mathrm{He}_6$, such as the configuration mixing and the spatial properties. These results indicate a good isospin symmetry in the soft dipole resonances of two nuclei with a common collective excitation of multiproton and multineutron, while the ground states of two nuclei show different properties due to the Coulomb repulsion of valence protons in ${}_6{}^8\mathrm{C}_2$, leading to the symmetry breaking. In conclusion, the appearance of the isospin symmetry differs depending on the states of ${}_6{}^8\mathrm{C}_2$ and ${}_2{}^8\mathrm{He}_6$.

Figure 1

Coordinate system of $\alpha +N+N+N+N$ in the COSM for ${}^8\mathrm{C}$ and ${}^8\mathrm{He}$. The vector $\mathit{R}$ is a relative coordinate between the center-of-mass of $\alpha$ and the center-of-mass of four valence nucleons.

Figure 2

$1^{-}$ eigenstates of ${}^8\mathrm{He}$ using the complex scaling with $\theta ={26}^{\circ }$ in the complex energy plane measured from the ground-state energy. The blue circle indicates the eigenenergy of the soft dipole resonance (SDR). The results are taken from Ref. [14].

Figure 3

$1^{-}$ eigenstates of ${}^8\mathrm{C}$ using the complex scaling with $\theta ={26}^{\circ }$ in the complex energy plane measured from the $\alpha +p+p+p+p$ threshold energy. The blue circle indicates the eigenenergy of the soft dipole resonance (SDR).

Figure 4

Comparison of the excitation energy spectra between ${}^8\mathrm{He}$ an ${}^8\mathrm{C}$ in units of MeV.