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.

Here is a table of the data for this sequence of beta transitions.

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 regression equation for the data on incremental binding energies is

IBE = 100.8839889 −1.32744n − 1.74397u(n-82)
[21.3] [-20.5] [-3.0]

The u(z) function in the above equation is the unit step function; i.e., u(z)=0 if z<0 and u(z)=1 if z≥0.

The numbers in the square brackets below the regression coefficients are their t-ratios; i.e., the ratio of the coefficient to its standard deviation. The coefficient of determination (R²) for the regression is 0.992.

Here is the graph of the regresssion equation estimates

The regression equation could be used to predict what the incremental binding energy would be for a nuclide beyond the range of the existing isotopes. What appears to be the case is that if the incremental binding energy exceeds a critical level the nuclide does not exist long enough for its mass to be measured. The critical level of binding energy for positron emission appears to be about 10 MeV for p=50. The critical level for electrom emission appears to be about 10 MeV for p=64. The critical level varies with the proton number.

An approximation of the critical level for the maximum of neutrons can be obtained by looking at the change in binding energy that occurs when the nuclide with the maximum number of neutrons is created by a beta transition. Similarly an approximation of the critical level for the minimum of neutrons can be obtained by looking at the change in binding energy that occurs when the nuclide with the minimum number of neutrons is created by a beta transition. These incremental binding energy (IBE) values are tabulated in Increments for the maximums and Increments for the minimums.

What was found in those tables is that the values are significantly higher when the proton number is odd than when it is even. This undoubtedly occurs when the beta transition creates a proton-proton spin pair or breaks up a neutron-neutron spin pair. There is a similar effect when the neutron number and the proton number are nearly equal. In that case the beta transition may promote the formation of a neutron-proton spin pair and in other cases break up such a spin pair.

Here are the graphs of the incremental binding energies (IBE) only for the even proton numbers.

What this data shows is a constant level of IBE from the low proton numbers up to p=20 then a transition up to a range from p=26 to p=42 then another transition up to p=72 and a low constant level to p=82. The average value of the IBE from p=2 to p=40 is 16.66 MeV. From p=26 to p=42 the average is 9.28 MeV. For the final range for the maximums the average is a small fraction of 1 MeV if the anomalous value at p=80 is left out.

As shown below there are notable similarities and notable differences for the results from the nuclides with a minimum number of neutrons and those from the nuclides with a maximum number of neutrons.

Conclusions

The true values of the critical levels of the incremental binding energy that determines the maximum and minimum numbers of neutrons in the isotopes of an element varies with the proton number. It is about 17 MeV for small proton numbers and for the maximums goes down to a bit above 9 MeV for proton numbers in a midrange and then for the maximums down to essentially zero for proton numbers approaching 82.