Evidence for Triangular $D_{3h}^{\prime }$ Symmetry in ${}^{13}\mathrm{C}$

We derive the rotation-vibration spectrum of a $3\alpha +1$ neutron(proton) configuration with triangular $D_{3h}$ symmetry by exploiting the properties of the double group $D_{3h}^{\prime }$ and show evidence for this symmetry to occur in the rotation-vibration spectra of ${}^{13}\mathrm{C}$. Our results, based on purely symmetry considerations, provide benchmarks for microscopic calculations of the cluster structure of light nuclei.

Figure 1

The rotational spectra of ${}^{13}\mathrm{C}$. Energy levels [15] are plotted as a function of $J(J+1)$. For states below 10 MeV, our assignment of rotational bands is unambiguous. For states above 10 MeV, our assignment is tentative.

Figure 2

Comparison between experimental and theoretical energies for the ground state band assigned to the representation $\mathrm{\Omega }=E_{1/2}^{(-)}$ of $D_{3h}^{\prime }$. The values of $K^P$ are given at the bottom of the figure. The energies are calculated using Eq. (2) with $A_{\mathrm{\Omega }}=0.942\mathrm{MeV}$, $b_{\mathrm{\Omega }}=-0.62$, and $a_{\mathrm{\Omega }}=0$.

Figure 3

As Fig. 2, but for the first excited band assigned to the representation $\mathrm{\Omega }=E_{1/2}^{(+)}$ of $D_{3h}^{\prime }$. The energies are calculated using Eq. (2) with $A_{\mathrm{\Omega }}=0.684\mathrm{MeV}$, $b_{\mathrm{\Omega }}=0.80$, $a_{\mathrm{\Omega }}=1.24$, and $ϵ=3.848\mathrm{MeV}$.

Figure 4

Rotational spectra in ${}^{13}\mathrm{C}$ expected on the basis of $D_{3h}^{\prime }$ symmetry.

Figure 5

Comparison between calculated and experimental [20] longitudinal $E2$ form factors for the ground-state band of ${}^{13}\mathrm{C}$, $1/2_1^{-}\to 5/2_1^{-}$ (black) and $1/2_1^{-}\to 3/2_1^{-}$ (red).

Figure 6

Molecularlike picture of ${}^{13}\mathrm{C}$.