  We propose to investigate the time modulation of radioisotope decays deep
underground as a method to explore axion dark matter. In this work, we focus on
the $\alpha$-decay of heavy isotopes and develop a theoretical description for
the $\theta$-dependence of $\alpha$-decay half-lives, which enables us to
predict the time variation of $\alpha$-radioactivity in response to an
oscillating axion dark matter background. To probe this scenario, we have
recently constructed and installed a setup deep underground at the Gran Sasso
Laboratory, based on the $\alpha$-decay of Americium-241. This prototype
experiment, named RadioAxion-$\alpha$, will allow us to explore a broad range
of oscillation's periods, from a micro-second up to one year, thus providing
competitive limits on the axion decay constant across 13 orders of magnitude in
the axion mass, ranging from $10^{-9}$ eV to $10^{-20}$ eV after one month of
data collection, and down to $10^{-22}$ eV after three years.
