Alpha decay law of excited nuclei and its role in stellar decay rates
$α$ decay is one of the prominent decay modes in the nucleosynthesis of heavy and super-heavy elements synthesized at temperatures of the order of Giga Kelvin. To facilitate the investigation of the role played by the $α$ decay half-lives of thermally excited nuclei in nucleosynthesis calculations, an empirical formula based on a model for the $α$ decay of nuclei in their ground and excited states to daughter nuclei in their ground or excited states is presented. Constants appearing in the analytical expression for the $α$ decay half-life obtained within the model are treated as adjustable parameters and fitted to experimental data on 342 $α$ decays in the range of 82 $\le Z_p \le$ 94, to obtain an excitation energy-dependent decay law. Under the assumption that thermal equilibrium has been reached between nuclear states, temperature ($T$)-dependent half-lives, $t_{1/2}(T)$, for several of the experimentally studied $α$ emitters with 65 $\le Z_p \le$ 94 are presented using available data on the half-lives of excited nuclei. Though the general trend is a decrease in $t_{1/2}(T)$ at elevated temperatures, exceptional cases with increased half-lives are found in the case of some isomeric states. A list of such isomers provided in this work motivates future work involving considerations of their thermal equilibration and role in shaping kilonova light curves.