The Incremental Binding Energies
of Protons and Neutrons

The binding energies (BE) of 2930 nuclides have been measured. The incremental binding energies (ΔBE) can be computed according to the following definitions

Δ_proton(n, p) = BE(n, p) − BE(n, p-1)
Δ_neutron(n, p) = BE(n, p) − BE(n-1, p)

The incremental binding energies of protons for an arbitrarily selected sequence of nuclides are shown below.

The incremental binding energies for each nucleon represents the interactive binding energy of the last nucleon with all of the rest of the nucleons in the nuclide. The second differences, the increments in the incremental binding energies are even more interesting. The difference in the incremental binding energies of protons is an approximation of the interactive binding energy of the last proton with the previous proton, and likewise for the difference of the incremental binding energies of neutrons. These can be called the second differences in binding energy. For analytical justifications of these propositions see Protons and Neutrons. For their empirical verification see Protons and Neutrons.

On the other hand, the cross differences give an exact measure of the interactive binding energy between the last proton and the last neutron in the nuclide. This is positive, reflecting the fact that the force between a proton and a neutron is an attraction.

Consider the following graphs of the incremental binding energies of protons. (The data for the incremental binding energies of neutrons will be dealt with elsewhere.)

The odd-even sawtooth pattern has to do with the binding energies involved with the formation of proton-proton pairs.

The fact that the curves for the different elements nest together so neatly has certain important implications.

• The slopes of the relationships, which has to do with the interactions of the last two protons, are independent of the number of the other nucleons in the nuclide.
• The fact that, abstracting away the effect of pair formation, the relationships are piecewise linear means that the interaction energies are constant within a shell or subshell.
• The fact that the slopes are negative is verification that the force between two protons is a repulsion.

A Test of the Degree of Parallelism of the Relationships

Although visually the above displays appear to be parallel an empirical verification requires looking a the differences between IBEp's for the elements. That information is displayed below.

For two curves to be parallel their difference would have to be constant. In the case of the incremental binding energies of protons in the isotopes of Arsenic and Selenium the differences for the protons in the fifth shell (29 through 50 protons) the differences are roughly constant. There is greater differences for the even numbers of protons because that difference includes the difference in the binding energy associated with the formation of proton pairs whereas the differences for the odd numbers of protons do not.

When the successive differences in the incremental binding energies for the isotones with 33, 34, 35 and 36 neutrons are plotted together the result seems to be chaos.

However if the values for the differences for the 35−34 are shifted back one unit and those of 36−35 are shifted back two units the results are more orderly.

This appears to mean that the differences are a function of the number of protons minus the number of neutrons. Thus if n is the number of protons and p the number of neutrons then

Δ_proton(n, p) = f(p) + g(p-n)