Effect of initial $3\alpha$ cluster configurations in ${}^{12}\mathrm{C}$ on the direct decay of its Hoyle state

We investigate the implications of initial $3\alpha$ configurations in ${}^{12}\mathrm{C}$ corresponding to different decay modes of its Hoyle state on the penetrability ratios. Considering the second $2^{+}$ (10.03 MeV) state to be a collective excitation of the Hoyle state, the direct $3\alpha$ decay width for the Hoyle state has been calculated using the ratio of the barrier penetration probability of the Hoyle state to the $2^{+}$ state. Semiclassical Wentzel–Kramers–Brillouin (WKB) approximation has been employed to determine the penetrability ratio, resulting in an upper limit on the branching ratio of the direct decay of the Hoyle state in “equal phase-space” ($\mathrm{DD}\varphi$) mode as $\frac{{\mathrm{\Gamma }}_{3\alpha }}{\mathrm{\Gamma }}<3.1\times {10}^{-6}$. However, this limit for “linear chain” (DDL) decay is $\frac{{\mathrm{\Gamma }}_{3\alpha }}{\mathrm{\Gamma }}<2.6\times {10}^{-7}$, which is one order of magnitude smaller than the $\mathrm{DD}\varphi$ decay and the limit for “equal energy” (DDE) decay is $\frac{{\mathrm{\Gamma }}_{3\alpha }}{\mathrm{\Gamma }}<1.5\times {10}^{-5}$, which is greater than both $\mathrm{DD}\varphi$ and DDL decays. It implies that the limit on direct decay probability is strongly dependent on the initial configuration of the $3\alpha$ cluster. A further probe using a bent-arm-like $3\alpha$ initial configuration shows that the direct decay probability is maximum when the angle of the bent arm is $\approx {120}^{\circ }$, an important ingredient for understanding the Hoyle-state structure.

Figure 1

Initial $3\alpha$ cluster configurations for direct decay modes in linear chain (DDL), equal energy (DDE), and equal phase space ($\mathrm{DD}\varphi$).

Figure 2

Sum of the Coulomb ($V_C$) and centrifugal potentials ($V_l$) with and without the nuclear term ($V_N$) for DDL and DDE modes of direct decay. Dotted and dot-dot-dashed horizontal lines represent the energies of Hoyle state and $2^{+}$ state above the $3\alpha$ breakup threshold correspond to 0.38 and 2.756 MeV, respectively.

Figure 3

Penetrability ratio as a function of the angle of the bent arm (a) $P_{\mathrm{DD}}^{\mathrm{bent}-\mathrm{arm}}\left(\mathrm{Hoyle}\right)/P_{\mathrm{DD}}^{\mathrm{bent}-\mathrm{arm}}\left(2^{+}\right)$ from “first approach” (b) $P_{\mathrm{DD}}^{\mathrm{bent}-\mathrm{arm}}\left(\mathrm{Hoyle}\right)/P_{\mathrm{Seq}.}^{\mathrm{bent}-\mathrm{arm}}\left(\mathrm{Hoyle}\right)$ from “second approach.”

Figure 4

Simulated root-mean-square energy deviation spectrum of three $\alpha$ particles emitted via different decay modes, with dominant contribution (99.965%) from sequential mode as shown by gray shade. Contribution from DDL, DDE, and $\mathrm{DD}\varphi$ modes are 0.004%, 0.012%, and 0.019% respectively. Combined events shown by blue solid line have been fitted again to find the uncertainty in extracting back the above contributions if possible.