Potential barriers governing the ${}^{12}\mathrm{C}$ formation and decay through quasimolecular shapes

The $L$-dependent potential barriers that govern the ${}^8\mathrm{Be}$ and ${}^{12}\mathrm{C}$ formation and decay through quasimolecular shapes have been determined using a generalized liquid-drop model and adjusted to reproduce the experimental $Q$ value. For the ternary channel of ${}^{12}\mathrm{C}$, the energies of prolate linear chain configurations and oblate triangular configurations of three $\alpha$ particles have been compared. The triangular shape with three $\alpha$ nuclei in contact allows the experimental rms radius and the negative quadrupole moment of the ${}^{12}\mathrm{C}$ ground state to be reproduced. The difference between the energies of the minima in the prolate and oblate ternary shape paths is very close to the energy of the excited Hoyle state of the ${}^{12}\mathrm{C}$ nucleus.

Figure 1

Potential barriers governing the ${}^8\mathrm{Be}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He},{}^8\mathrm{Be}\to {}^7\mathrm{Li}+{}^1\mathrm{H}$, and ${}^8\mathrm{Be}\to {}^6\mathrm{Li}+{}^2\mathrm{H}$ reactions versus the distance between the mass centers (at $L=0$). The vertical dash indicates the contact point between the two nuclei.

Figure 2

Comparison between the deformation energies calculated without (broken curve) and with (full curve) a proximity energy term for the ${}^8\mathrm{Be}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He}$ reaction.

Figure 3

Potential barriers of the ${}^8\mathrm{Be}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He}$ reaction as a function of the angular momentum (in $\hslash$ units).

Figure 4

Potential barriers governing the ${}^{12}\mathrm{C}\leftrightarrow {}^8\mathrm{Be}+{}^4\mathrm{He},{}^{12}\mathrm{C}\to {}^{10}\mathrm{B}+{}^2\mathrm{H},{}^{12}\mathrm{C}\to {}^9\mathrm{Be}+{}^3\mathrm{He}$, and ${}^{12}\mathrm{C}\to {}^6\mathrm{Li}+{}^6\mathrm{Li}$ reactions versus the distance between the mass centers (at $L=0$). The vertical dash indicates the contact point between the two nuclei.

Figure 5

Comparison between the potential barriers governing the ${}^{12}\mathrm{C}\leftrightarrow {}^8\mathrm{Be}+{}^4\mathrm{He}$ and ${}^{12}\mathrm{C}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He}+{}^4\mathrm{He}$ binary and prolate ternary reactions.

Figure 6

Potential barriers of the ${}^{12}\mathrm{C}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He}+{}^4\mathrm{He}$ reaction as a function of the angular momentum and for a linear chain configuration.

Figure 7

Oblate ternary configuration of three $\alpha$ particles in contact. $l$ is the distance between the mass center of an $\alpha$ particle and the mass center of the whole system.

Figure 8

$L$-dependent potential barriers for the ${}^{12}\mathrm{C}\leftrightarrow {}^4\mathrm{He}+{}^4\mathrm{He}+{}^4\mathrm{He}$ reaction and a triangular configuration. $Q$ is the rms radius.