alpha decay
A heavy, overcrowded nucleus can lighten itself by spitting out a compact, exceptionally stable bundle: a helium-4 nucleus, historically called an alpha particle, made of two protons and two neutrons. Alpha decay is this emission. Because a helium-4 nucleus is so tightly bound (about 28.3 MeV of binding for just four nucleons), ejecting it as a single package is energetically far easier than shedding nucleons one by one.
In alpha decay the parent nucleus (Z, A) transmutes into a daughter (Z - 2, A - 4) plus an alpha particle, and the released energy (the decay Q-value) is shared as kinetic energy between the daughter and the alpha, with the light alpha carrying most of it. So the emitted alphas are essentially monoenergetic. The great puzzle it solved is why half-lives span more than twenty orders of magnitude (nanoseconds to billions of years) for only a modest range of alpha energies. Gamow's answer is quantum tunnelling: the alpha is trapped behind a Coulomb barrier it classically could never surmount, and it escapes only by tunnelling through it. The tunnelling probability depends exponentially on the barrier, giving the steep Geiger-Nuttall relation between decay energy and the logarithm of the half-life.
Alpha emitters are everywhere in the heavy end of the chart: uranium, thorium, radium, plutonium, americium. Practically, alphas are highly ionizing but stopped by a sheet of paper or the outer skin, so they are a minor external hazard but dangerous if an emitter is inhaled or ingested. Alpha decay was one of the first clean demonstrations that quantum tunnelling is real and quantitative.
Uranium-238 alpha-decays to thorium-234 with a half-life of 4.5 billion years, releasing a Q of about 4.27 MeV; the emitted alpha carries roughly 4.2 MeV. That immense half-life for such a modest energy is a direct consequence of tunnelling through the Coulomb barrier.
A tiny change in alpha energy swings the half-life by many orders of magnitude, the Geiger-Nuttall signature of tunnelling.
The alpha is not sitting fully formed and 'rattling' inside a classical box; the tunnelling picture is a semiclassical model. But its exponential barrier dependence quantitatively explains the Geiger-Nuttall law, and that is genuine.