# Antimatter particle in bananas [duplicate]

I recently came to know that banana produces antimatter. How does this happen and why doesn't the antimatter annihilate as our Earth has matter. I looked for a relevant answer but could not find it.

## 1 Answer

This is really about potassium, not about bananas. They're just put in to add colour. Bananas contain potassium but so do lots of fruits and vegetables. But 'spinach produces antimatter...' doesn't have the same impact.

Potassium has 19 protons. Most of the potassium around is isotope 39 (20 neutrons) or 41 (22 neutrons) but there is a small fraction (0.012%) of isotope 40. With 19 protons and 21 neutrons this is an "odd-odd" nucleus. These are very uncommon. Nuclei with even numbers of protons and/or neutrons have lower energies than those with odd numbers (due to the pairing term in the semi-empirical mass fomula - see https://en.wikipedia.org/wiki/Semi-empirical_mass_formula for details).

An odd-odd nucleus can convert to a lower energy even-even nucleus state by changing one of its neutrons into a proton, which happens through $$\beta^-$$ decay, or by changing one of its protons into a neutron, which can happen in two ways: $$\beta^+$$ decay, or electon capture (often abbreviated to EC). That gves 3 possibilities: $$n \to p e^- \overline \nu \qquad p \to n e^+ \nu \qquad p+e^- \to n \nu$$ All these are possible but the rate depends on the energy difference: decays releasing more energy are more probable. That gives a big advantage to the first option, $$\beta^-$$ decay, as the neutron has a larger mass than the proton (939.6 MeV as opposed to 938.3, giving 1.3 MeV more energy). However it just happens that the $${}^{40}$$Ca state that results from $$\beta^-$$ decay (moving one up the periodic table) has a 0.7 MeV higher energy than the $${}^{40}$$Ar state that comes from $$\beta^+$$ or EC (moving one down). That removes a lot of the 1.3 MeV handicap and so the process are comparable - the $$\beta^-$$ still dominates, but 'only' takes 90% of the rate, leaving 10%.

Even within this 10% the EC process wins over $$\beta^+$$ because it has the rest-mass energy of the electron and doesn't have to find the 0.5 MeV needed to make a positron. (The electron in EC is one of the atomic electrons from an $$s$$ shell, as the wavefunction is not zero at the origin, which is where the nucleus is.) So the $$\beta^+$$ decay is a very rare fraction of the $${}^{40}$$ K decays - and in turn there are few of these as the half life is a very long 1.3 Bn years (it is so long because these energy differences are so small).

So potassium gives you positrons. Not many (one per banana per 75 mintes is the canonical figure). If you want positrons in the laboratory, $${}^{22}$$ Na - another odd-odd nucleus, and the processes are the same but the energies are greater - is a much more effective source. But this does not occur naturally, you have to make it in a cyclotron.