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Consider a theoretical nucleus that is both neutron rich and heavy e.g. $A=300$, $N=250$. Is it possible to say whether it is more likely to undergo $\beta^-$ decay or $\alpha$ decay (or even cluster decay or fission) or does this depend on the specific nature of the atom? Please can you explain your answer.

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    $\begingroup$ It depends on the specific nature of the nucleus. Beta decay requires an imbalance between the least bound neutron and proton levels. If beta decay is not possible, then alpha decay, cluster decay, or fission will be the primary modes of decay. Which one will win out is complicated. $\endgroup$ – Lewis Miller Apr 30 '17 at 22:46
  • $\begingroup$ @LewisMiller Couldn't we just compare the different Q values? $\endgroup$ – T. Auerrac May 1 '17 at 18:26
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    $\begingroup$ @T.Auerrac That's only part of the story. The probability of clustering also plays a role and that depends on the characteristics of the involved orbitals. $\endgroup$ – Lewis Miller May 2 '17 at 14:35
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It could be both!

Your hypothetical nucleus is beyond the neutron drip line and would decay by neutron emission. However, consider the polonium isotopes, whose alpha-decay branching ratios are plotted below:

enter image description here

None is stable against alpha decay. However most of the isotopes (and the long-lived isomers) with mass $190 < A < 209$ have a non-negligible branching ratio for both alpha decay and positron decay/electron capture. On the neutron-rich side, it seems that for whatever reason most of the isotopes decay overwhelmingly by either alpha or beta decay, but in $\rm^{219}Po$ the branching ratios for alpha and beta decay are different by only a factor of three.

For the proton-rich polonium isotopes, I don't really see a pattern in whether alpha decay or electron capture is the dominant decay method. For example I would have expected the most proton-rich poloniums to decay by electron capture and positron emission, but those with $A < 195$ (with the exception of $\rm^{188}Po$) are overwhelmingly alpha emitters. That suggests there's some complicated final-state stuff happening.

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