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Why is polonium so much more radioactive than bismuth? I've been wondering this for a while, and have searched around for an answer, but I can't seem to find anything that gives a good answer.

Bismuth-209 has a half-life about $53\times 10^{18}$ times longer than polonium-210 ($20 \times 10^{18}$ years vs 138 days), despite both isotopes having 126 neutrons, which is a magic number. Of course, since Po-210 is less neutron-rich than Bi-209, I would expect it to be less stable, but the difference is very dramatic. Maybe it has to do with polonium being 2 off of the proton magic number $Z = 82$, making it prone to alpha decay, but then I don't know why bismuth is so weakly radioactive, since it also has more protons than the magic number $Z = 82$.

Additionally, I don't understand why there is only such a large drop-off in stability after $N = 126$. I would expect similar large areas of instability right after the other magic neutron numbers (N or$ Z = 2, 8, 20, 28, 50$, and 82), however outside of the elements technetium (Tc-97 has 54 neutrons) and promethium (Pm-145 has 84 neutrons), this doesn't seem to be the case.

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  • $\begingroup$ I'm not sure for the real answer, but Polonium-210 has spontaneous fission coefficient $Z^2/A \approx 33.6$, while for Bismuth-209 it's about $32.9$. It's not a huge difference, but probability for Spontaneous fission is greater for Polonium-210. But maybe there are additional factors. $\endgroup$ Dec 14, 2022 at 21:00
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    $\begingroup$ Polonium 210 to Lead 206 by alpha has almost twice the decay energy of Bismuth 209 to Thallium 205 by alpha. That probably has something to do with it. $\endgroup$
    – g s
    Dec 14, 2022 at 21:12
  • $\begingroup$ As we move from $^{209}\rm{Bi}$ to $^{210}\rm{Po}$ the alpha decay half-life shrinks by $20$ orders of magnitude, but a thing you've probably missed is that according to $1908.11458$, the alpha decay half-life of $^{208}\rm{Pb}$ would be at the order of $10^{130}$ years, so it shrinks by $111$ orders of magnitude when we move to $^{209}\rm{Bi}$. Yes you were right, $Z=83$ is not so special; the reason that $^{209}\rm{Bi}$ is so weakly radiocative is just that $^{208}\rm{Pb}$ is far too stable; from $^{210}\rm{Pb}$ to $^{211}\rm{Bi}$ would be another thing. $\endgroup$ Apr 9 at 10:15

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Although the shell structures of 209Bi and 210Po are slightly different, this does not contribute significant to their alpha decay half life. This is reflected in the rather small differences (on log scale) of their alpha preformation factors provided by both theoretical and semi-microscopic calculation.

The main reason is actually simpler than that. From Nubase data, the $Q_\alpha$ of 209Bi is 3.1 MeV while 210Po is 5.4 MeV. The difference does not seems much but we can estimate their half life using the Geiger-Nuttall law $\log_{10}T_{1/2}=A(Z)Q^{-1/2}_\alpha+B(Z)$, where $A(Z)=2.41Z − 66.7$ and $B(Z)=−0.54Z − 6.61$.

This translate approximately to 209Bi half life of $10^{26}$ (s) compared to the $10^7$ (s) of 210Po. So it is clear that the large difference in alpha decay half life is specifically due to the values of the masses or equivalently the binding energies of 209Bi, 210Po and their daughters. For an even deeper origin for the binding energy (or mass) values, you can trace back to the semi-empirical mass formula or some Hartree-Fock type calculation.

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I'm not a physicist, but my guess is it might be because of the fact that lead 208 is doubly magic (82, 126), so it is favored for polonium-210 to alpha decay quickly.

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