Universal $Z^{2/3}/\sqrt{Q_{\alpha }}$ Scaling Law in Alpha Decay from Nuclei to Neutron-Star Mergers

Abstract

We establish a universal scaling law for $\alpha$-decay half-lives based on the variable $X_a=Z^a/\sqrt{Q_{\alpha }}$, which collapses experimental and theoretical data spanning over twenty orders of magnitude onto a single linear correlation. Global optimization over the complete set of even-even $\alpha$ emitters reveals a uniquely sharp optimum at $a\simeq 2/3$. This exponent is theoretically motivated by the leading geometric scaling of heavy nuclei and emerges quantitatively from the correlated nuclear systematics, driven by the correlated $(Z,Q_{\alpha })$ manifold of the nuclear chart. Strikingly, five structurally distinct semi-empirical models and an independent microscopic WKB calculation, when independently optimized with respect to the exponent $a$, yield values clustered around $2/3$, while the corresponding $a=2/3$ correlations remain highly linear without refitting the original model parameters. This emergent scaling law implies smooth variations of decay times along the heavy $r$-process path. We demonstrate analytically that, given a roughly uniform distribution of this variable, the scaling law naturally supports a quasi-power-law radioactive heating rate ($\dot{ϵ}\propto t^{-1.24}$), consistent with full network calculations. Furthermore, integrating these scaling predictions directly into nuclear source terms for radiative-diffusion models yields multimessenger observables that accurately reproduce the kilonova AT2017gfo associated with the gravitational-wave event GW170817, demonstrating that the $Z^{2/3}/\sqrt{Q_{\alpha }}$ scaling coordinate provides a robust tool for modeling the radioactive engines of neutron-star mergers.

Comments: 8 pages, 7 Figures