Examination of the stability of a rod-shaped structure in ${}^{24}\mathrm{Mg}$

The stable existence of rod-shaped structures in highly excited states of ${}^{24}\mathrm{Mg}$ is studied based on a systematic cranked Hartree-Fock calculation with various Skyrme-type interactions. Its stability is examined by allowing the transition of the clusterlike structure to the shell-model-like structure. Especially, the rod-shaped state is exposed to two major instabilities: the bending motion, which is the main path for the transition to low-lying states, and the spin-orbit interaction, which is the driving force to break the $\alpha$ clusters and enhance the independent motion of the nucleons. The rod-shaped structure with large angular momentum is obtained as a metastable stationary state.

Figure 1

Density profiles of the initial states of six-$\alpha$ clusters for the cranked Hartree-Fock iteration. Panel (i) shows the rod-shaped structure, while panels (ii), (iii), and (iv) correspond to bent structures with opening angles of $2\pi /3,\pi /3$, and $\pi /6$, respectively. Contours are incremented by 0.$03{\mathrm{fm}}^{-3}$.

Figure 2

Final states obtained by cranked Hartree-Fock calculations (SV-bas parameter set). (a) Compact shape, (b) rod shape, and (c) fragmentation. Contours are incremented the same as in Fig 1.

Figure 3

Imaginary time evolution starting with (ii) of Fig. 1 and $\omega =0.0$ (MeV/$\hslash )$ going to the final state (a) of Fig. 2, with (ii) and $\omega =1.0$ going to (b), and with (ii) and $\omega =1.5$ going to (c). The expectation values of $H^{\prime }$ (Hamiltonian in the rotating frame) in Eq. (1) are shown to see the convergence.

Figure 4

An initial state with three-dimensional distortion.

Figure 5

Angular momentum $J$ as a function of rotation frequency $\omega$ (MeV/$\hslash )$ for nine Skyrme interaction parameter sets, where the initial state is set to (ii) of Fig. 1. The line with solid squares denote the calculated result for the SV-bas parameter set.

Figure 6

Energy-impact parameter diagram for the appearance of different shapes based on Fig. 5. The regions for the appearance of different shapes [(a), (b), and (c) of Fig. 2] are translated into the impact parameter and center-of-mass energy of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ reaction (SV-bas parameter set). The perpendicular axis means the energy measured from ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ at infinite distance. $E_{\mathrm{b}}=9.43$ MeV denotes the fusion barrier, and the rod-shaped structure is obtained in the region of and $E=75.1$–94.16 MeV. $E^{*}=75.1$ MeV at $\omega$ = 0.8 MeV/$\hslash$ is the smallest excitation energy of the rod shape obtained in the structure calculation. $E_{\mathrm{ce}}$ is the charge equilibrium upper limit energy [16], above which emission of charge nonequilibrated particles may occur.