Why is Po-209 more stable than Po-210 despite lower neutron number?

Question

A general trend of the stability of isotopes is that the higher the neutron number, the greater the stability against alpha decay because adding more neutrons weakens the electrostatic repulsion among protons (or alternatively enhances the attractive nuclear force). One good example is osmium. The energy of alpha decay decreases as the neutron number increases (Os-184:2.963, Os-186:2.823, Os-187:2.720, Os-188:2.143, Os-189:1.976, Os-190:1.378, Os-192:0.362). Consequently, alpha decay is only observed in Os-184 and Os-186.

However, some isotopes like U-235 and U-236, Po-209 and Po-210 don’t follow this trend. Po-210 additionally has a magic neutron number (126), yet its half-life is only 1/330 as long as Po-209. What caused these anomalies?

Answer 1

Alpha decay lifetimes are exponentially proportional to the alpha decay energy ($Q_\alpha$), and although $Q_\alpha$ typically gets smaller as neutron number increases, other factors such as magic numbers affecting parent or progeny binding energies and decay barriers can cause deviations from this general trend.

Much of the relevant physics is discussed in the answers to these questions:

• "Why some nuclei with "magic" numbers of neutrons have a half-life less than their neighbor isotopes?"
• "Why is there a sudden drop-off in half-life of isotopes at around 130 neutrons? Is there a name for this?" notes that according to the Geiger-Nuttall Law, the half-life for alpha decays decreases exponentially with increasing $\sqrt{Q_\alpha}$, and that $Q_\alpha$ is a maximum when the progeny, not parent, nuclide is close to the magic neutron number of 126. If you look at the figure from the paper referenced in that answer, you'll see that a local maximum in the lifetime at N=125 is caused by the opposite effects of increasing lifetime due to increasing neutron number and decreasing lifetime as the progeny approaches N=126.
• "Why is polonium so unstable compared to bismuth?" also emphasizes how the Geiger-Nuttall law ($\log_{10}T_{1/2}=A(Z)\,Q_\alpha^{-1/2}+B(Z)$) makes the half-life exponentially sensitive to $Q_\alpha$.

The longer lifetime of $^{209}$Po is an example of the N=125 lifetime maximum mentioned above. Although $^{210}$Po has a magic number of neutrons, its progeny $^{206}$Pb is closer to the even more tightly bound doubly magic $^{208}$Pb. The net effect of the singly magic parent and the almost doubly magic progeny is to make $Q_\alpha$ less for $^{209}$Po than $^{210}$Po. (According to the Chart of the Nuclides they have $Q_\alpha$ = 4.98 and 5.41 MeV respectively.)

For $^{235}$U and $^{236}$U, their very similar $Q_\alpha$ values (4.68 and 4.57 MeV) are consistent with your increasing neutron number argument, but all the $^{235}$U alpha decays have hindrance factors of at least 6 according to the Chart of the Nuclides, compared to 1 for $^{236}$U (Deviations from the Geiger-Nuttal Law are parameterized by multiplicative "hindrance factors". See Burcham Section 16.1.2.) Relevant factors include:

• Alpha decays occur through a tunnelling process and an "angular momentum" or "centrifugal" barrier makes this tunnelling more difficult for higher momentum states. (See, for example, Krane Section 8.5.) This reduces the decay rate for $^{235}$U which has $J^P=7/2^-$, compared to $J^P=0+$ for $^{236}$U.
• Another factor increasing the decay rate of $^{236}$U is that it decays via a $0^+\rightarrow 0^+$ transition where the initial and final state closely match. In contrast, $^{235}$U decays a third of the time to non-matching $^{231}$Th states, e.g. $J^P=9/2^-, 5/2^+, 7/2^+, 11/2^-, 5/2^-$, with large hindrance factors and reduced rates.