Effect of initial 3α cluster configurations in C12 on the direct decay of its Hoyle state

Abstract

We investigate the implications of initial $3\ensuremath{\alpha}$ 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 $3\ensuremath{\alpha}$ 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'' ($\mathrm{DD}\ensuremath{\phi}$) mode as $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}<3.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. However, this limit for ``linear chain'' (DDL) decay is $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}<2.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, which is one order of magnitude smaller than the $\mathrm{DD}\ensuremath{\phi}$ decay and the limit for ``equal energy'' (DDE) decay is $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}<1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, which is greater than both $\mathrm{DD}\ensuremath{\phi}$ and DDL decays. It implies that the limit on direct decay probability is strongly dependent on the initial configuration of the $3\ensuremath{\alpha}$ cluster. A further probe using a bent-arm-like $3\ensuremath{\alpha}$ initial configuration shows that the direct decay probability is maximum when the angle of the bent arm is $\ensuremath{\approx}{120}^{\ensuremath{\circ}}$, an important ingredient for understanding the Hoyle-state structure.