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 (β⁻) or a positron (β⁺). 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 are sought here are 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 in millions of electron volts (MeV) for an arbitrarily selected 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. To the left of this maximum binding energy positron emission occurs. To the right electron emission occurs which decreases the neutron number.

The above relationship is quite remarkable. It is nearly symmetrical about the stable isotope. Each side is a quadratic function. For more details on this particular relationship see Nuclear Stability.

Here is the graph of the increments in binding energy that would result from an increase in the number of neutrons due to the conversion of a proton into a neutron and the emission of a positron.

The relationship shown in the above graph is very regular. It is linear with a shift at n=82.

When the increment is positive a positron emission occurs. When it is negative the opposite beta transition occurs; the conversion of a neutron into a proton and the emission of an electron.

The above relation was for the nuclides of mass number (p+n) 137. The rest of this study deals with the cases of a number of different mass numbers.

First consider the cases in the vicinity of mass number 137.

Generallly they have the same shape but the one with the even mass number has some odd-even fluctuation.

Here are the incremental binding energies (IBE) for these sequences

These are essentially the same as those in the vicinity of mass number 137.

Here are the graphs of the data for the cases of the nuclides having mass numbers near 120, 80, 40, 160 and 200.

In all cases the pattern is the same: Two half parabolas joined together at their maximums.

(To be continued.)