Three-Body Barrier Dynamics of Double-Alpha Decay in Heavy Nuclei

The simultaneous emission of two $\alpha$ particles--double-$\alpha$ decay--represents a long-predicted but unobserved mode of nuclear radioactivity. Here we formulate this process as a genuine three-body problem within the hyperspherical coordinate framework and evaluate decay probabilities by numerically solving the corresponding hyperradial Schr\"{o}dinger equation, combined with large-scale random sampling of the potential parameters; the latter treatment ensures that the present results are more convincing. Inspired by this, we demonstrate that the penetrability ratio between simultaneous and sequential $\alpha$ emission exhibits a strikingly linear dependence on $ZQ_{\alpha\alpha}^{-1/2}$, extending the barrier penetration dynamics into the correlated few-body regime. The nuclei $^{108}$Xe, $^{218}$Ra, $^{224}$Pu, $^{222}$U, $^{216}$Rn, and $^{220}$Th are suggested as the most promising candidates for the observation of double-$\alpha$ decay, with predicted half-lives potentially accessible within present detection limits. Our results provide a unified framework for multi-$\alpha$ decay and open a pathway to probing nuclear clustering and few-body correlations in heavy nuclei.