Nonlocalized motion in a two-dimensional container of $\alpha$ particles in $3^{-}$ and $4^{-}$ states of ${}^{12}\mathrm{C}$

The first $3^{-}$ and $4^{-}$ states of ${}^{12}\mathrm{C}$ are studied in the present container model, in which the shift parameter is introduced to break the parity symmetry for projecting out the negative-parity states. Taking the limit as the shift parameter approaches zero and by variational calculations for one-deformed size parameter, the local energy minima are obtained for the $3^{-}$ and $4^{-}$ states. It is found that the obtained single Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave functions for $3^{-}$ and $4^{-}$ states are 96% and 92% equivalent to the corresponding generator coordinate method (GCM) wave functions, respectively. The calculated intrinsic densities further show that these negative-parity states of three clusters, different from the traditional understanding of a rigid triangle structure, are found to have nonlocalized clustering structure in the two-dimensional container picture.

Figure 1

Schematic diagram for the container picture in the description of negative-parity cluster structure of ${}^{12}\mathrm{C}$.

Figure 2

Energy contour plot for the $3^{-}$ state in the two-parameter space ${\beta }_x={\beta }_y$ and ${\beta }_z$ in the container model.

Figure 3

Energy contour plot for the $4^{-}$ state in the two-parameter space ${\beta }_x={\beta }_y$ and ${\beta }_z$ in the container model.

Figure 4

Energy curves [calculated from the Eq. (1) and see the text] of the $3^{-}$ and $4^{-}$ states of ${}^{12}\mathrm{C}$ within the THSR wave functions compared with those from the Brink wave functions.

Figure 5

Various density profiles from the THSR wave functions with different shift parameters before angular momentum and parity projections in the two-dimensional container picture. The size parameter is taken as ${\beta }_x={\beta }_y=2.0$ fm, ${\beta }_z=0.5$ fm. The squared overlaps between the corresponding projected THSR wave functions ${\mathrm{\Phi }}^{J^{\pi }}(S_{1x},S_{2x},S_{2y})$ and the Brink-GCM wave function are given. Figures share the same legend.