Why do some nuclei decay exclusively via positron emission rather than electron capture?

Question

The processes of electron capture and $\beta^+$ decay both involve a transition from a parent nuclide ${}^A_{Z} X$ to a daughter nuclide ${}^A_{Z-1} X$. It is well known (see, for example, this question and this question) that if the mass difference between the parent & daughter is below $2 m_e$, the parent cannot undergo $\beta^+$ decay and must undergo electron capture. However, from considerations of energy, lepton number, and baryon number, any nuclide that decays via $\beta^+$ decay should be able to decay via electron capture as well; and from the totalitarian principle of quantum mechanics, if something is permitted it will happen with some small probability.

And yet it appears that there are many nuclides that decay only via $\beta^+$ decay with a branching ratio of 100%. For example, rubidium-81 and rubidium-82 both decay via $\beta^+$ emission exclusively. Naïvely, I would have expected some small branching ratio for EC for these isotopes.

Why do these isotopes decay exclusively via positron emission? Is it just that the EC decays of these isotopes are heavily suppressed — and if so, why? Or are there conservation laws that come into play other than conservation of energy, baryon number, and lepton number?

Answer 1

While summaries on various websites can provide a high level overview, if you want real details one should go to something like the Evaluated Nuclear Structure Data File (various mirrors around the world, including IAEA). If one goes there, enters 81 for the mass of interest, one sees:

Note that the 81Rb decays are listed as EC (electron capture). Pulling up the first data (the 4.572 hour decay), the header is

Already, one sees the listing is for $\epsilon$ decay. As noted in the NuDat 2 Glossary,

The greek letter epsilon (ε) is often used to indicate the combination of Electron Capture and β+ decay.

Digging deeper, follow the link to the 1977Li14 paper, (J. Liptak et al., Nuclear PHysics A 286(2) 263-281 (1977)). The first sentence of the introduction states

While the $\beta^+$-EC decay of $^{81}$Rb (4.58 h) has been the subject of several investigations...

Diving further into the paper, looking at the top left corner of Figure 8, the level scheme for populating $^{81}$Kr from the decay, one sees:

That clearly shows that electron capture vs positron emission result in populating different states.

I conclude that there is no justification for declaring that the decay is exclusively positron emission. Electron capture is an observed path.

Answer 2

You are correct that electron capture is always possible for any isotope that can decay by positron emission. The reverse is not true. It can be confusing that electron capture and positron emission are often lumped together as $\beta^+$ decay.

If we sum all the "Decay Radiation" channels for $^{81}$Rb and $^{82}$Rb in the Table of Nuclides, we find that

• $^{81}$Rb decays $73$% by electron capture and $27$% by positron emission, so $R(\mathrm{capture}/\beta^+)\approx 2.7$.
• $^{82}$Rb decays $5$% by electron capture and $95$% by positron emission, so $R(\mathrm{capture/\beta^+})\approx 0.05$.

The difference in the ratio of electron capture to positron emission for decays of the two isotopes is due to the difference in the energy released (Q):

• $^{81}$Rb: $Q=2239$ keV
• $^{82}$Rb: $Q=4404$ keV

Because of the difference between two- and three-body phase space, if the electron/positron masses were negligible, we'd expect total electron capture rates to scale as $Q^2$ and positron emission rates to scale as $Q^5$. The ratio of rates would hence scale as $1/Q^3$, so it is no surprise that $R(\mathrm{capture}/\beta^+)$ is much larger for $^{81}$Rb than $^{82}$Rb.

The observed ratio can be more precisely understood if the effects of finite electron/positron mass are included. See, for example, the discussion of electron-capture in Section 12.4 of Cottingham & Greenwood's "An Introduction to Nuclear Physics".

When $Q<2m_e$, energy conservation forbids positron emission and only electron capture is possible. This is because after positron decay, in addition to the final state atom there is both a positron and a left-over atomic electron.