Shape of ${}^{13}\mathrm{C}$ studied by the real-time evolution method

Background: Recently, Bijker et al. [Phys. Rev. Lett. 122, 162501 (2019)] explained the rotation-vibration spectrum of ${}^{13}\mathrm{C}$ by assuming a triangular nuclear shape with $D_{3h}^{\prime }$ symmetry.

Purpose: The purpose of this work is to test the shape and symmetry of ${}^{13}\mathrm{C}$ based on a microscopic nuclear model without assumption of nuclear shape.

Method: We have applied the real-time evolution method to ${}^{13}\mathrm{C}$. By using the equation-of-motion of clusters, the model describes the $3\alpha +n$ system without any assumption of symmetry.

Results: REM described the low-lying states more accurately than the previous cluster model studies. The analysis of the wave functions showed that the ground band has approximate triangular symmetry, while the excited bands deviate from it.

Conclusion: This work confirmed that the ground band has the intrinsic structure with the triangular arrangement of three $\alpha$ particles.

Figure 1

The snapshots of the intrinsic density distributions obtained by the real-time evolution. The top (bottom) panels show the wave functions from the ensemble set 1 (set 2).

Figure 2

The energies and radii of the $1/2_1{}^{-}$ and $5/2_1^{+}$ states obtained from sets 1 and 2 as a function of the total propagation time $T_{\text{max}}$. The strength of the spin-orbit potential $u_{ls}=2000$ MeV was adopted. The result for the $1/2_1{}^{-}$ state obtained in Ref. [31] are denoted by blue lines.

Figure 3

The energy spectrum of ${}^{13}\mathrm{C}$ calculated by using the strength of the spin-orbit potential $u_{ls}=2000$ and 1000 MeV. The energy is measured relative to the $3\alpha +n$ threshold. The spectrum is compared with that obtained by Furutachi et al. [31] using the same Hamiltonian with $u_{ls}=2000$ MeV. Experimental data are taken from Ref. [49].

Figure 4

The band assignment based on the calculated $E2$ transition strengths compared with that from the algebraic cluster model (ACM) [38] and the experimental assignment, which was also tentatively proposed in Ref. [38]. The filled (open) symbols show the positive-parity (negative-parity) states.

Figure 5

(a) Density distribution of the basis wave functions which have the maximum overlap with the $1/2_1{}^{-}$ state. (b) Same as panel (a) but for the $1/2_1^{+}$ state. Contours show the density of $3\alpha$ particles and color plots show the density distribution of the valence neutron wave function.