﻿ Nuclear Stability and Instabilities in Sequences of Beta Decays of Nuclides
San José State University

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Nuclear Stability and Instabilities
in Sequences of Beta Decays
of Nuclides

Nuclei are composed of nucleons (protons and neutrons) in variable proportions. There are almost three thousand nuclides that are stable enough to have their mass measured and their binding energies computed. But most are unstable. The following beautiful display from Wikipedia shows the nature of their instabilities. As can be seen from the display, up to proton number 82 overwhelming the mode of decay is either the ejection of an electron or s positron. Only the ones shown in black in the middle of the distribution are stable.

The ejection of an electron occurs because a neutron converts into a proton and an electron. This conversion releases energy. The positron ejection accompanies the conversion of a proton into a neutron, but this conversion requires an input of energy.

What is sought here is the changes in binding energies involved in these proton and neutron conversions. More basically the purpose of this material is to show the relationship between binding energy and the mode of radioactive decay or stability.

## Binding Energy and Radioactive Decay Products

For the decay of a neutron into a proton and an electron the relevant binding energies are:

#### BE(p, n) => BE(p+1, n−1)

For positron ejection the transformation is in the opposite direction.

#### BE(p, n) => BE(p−1, n+1)

What is plotted in the following graphs are the binding energies for a complete sequence of decay products as a function of the number of neutrons in the decay product. The binding energy reaches a maximum at p=56 and n=81. This is Ba137 which is a stable isotope of Barium. The binding energy reaches a maximum at p=34 and n=44. This is Se78 which is a stable isotope of Selenium. There is reason to expect the nuclides at the shoulders of the profile to be stable but they are not. The binding energy reaches a maximum at p=20 and n=22. This is Ca42 which is a stable isotope of Calcium. The binding energy reaches a maximum at p=10 and n=12. This is Ne22 which is a stable isotope of Neon. The binding energy reaches a maximum at p=6 and n=6. This is C12 which is a stable isotope of Carbon. The binding energy reaches a maximum at p=65 and n=96. This is Tb161 which is a stable isotope of Terbium.

The next display is for nuclides in the range where nuclear decay occurs through the ejection of an alpha particle. The data were compiled on the basis of what they would be if beta decay occurred. The binding energy reaches a maximum at p=94 and n=148. This is Pu242 which is an unstable isotope of Plutonium. However the data suggest that if it were not for alpha decay Pu242 would be stable.

## Conclusion

There is definitely a relationship between the mass deficit binding energies of a nuclide in its various transformations and the nature of its instability or its stability. A nuclide is unstable if a beta decay (electron or positron emission) will move it to a significantly higher state of binding energy. The nuclide with the maximum binding energy for a beta decay sequence is stable.

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