Nuclei are held together not just by the nuclear strong force but by the formation of spin pairs of nucleons. The binding energy due to the formation of pairs is dominant for the small nuclides and is exceeded by the strong force interactions for nuclides having more than about 30 protons.

The major problem is assessing the magnitudes of the binding energy enhancement due to the three types of nucleonic pairs: neutron-neutron, proton-proton and neutron-proton. Below is an illustration of the enhancement for neutron pair formation.

The enhancement appears to be nearly constant on the range of nuclides until the magic number of 82 is reached. That represents the filling of shell. In the next shell the enhancement for neutron pair formation is smaller. That should not be the case. The enhancement should be independent of the size and structure of the nuclide in which the neutron pair is formed.

The magnitude of the enhancement can be estimated by subtracting from the incremental binding energy for n neutrons the average of the incremental binding energy for (n-1) and (n+1) neutrons and then taking the absolute value of the difference.

The spike near 82 is just due to the transition to a higher shell at that point and is not significant. There are problems with this approach to measuring the binding energy associated with the formation of a neutron-neutron pair. First, there is the change in value at the transition to a new shell. Second the results are for the isotopes of only one element, tin. Third, the values do vary within the shell and there is a slight downward trend.

An Approach Based Upon the Alpha Nuclides

Neutron-Neutron Pairs

The term alpha nuclides is used to denote those nuclides that could contain only alpha particles. The effect of one additional neutron on binding energy can be computed by compiling the binding energies for all nuclides that could contain only alpha particles and one additional neutron. These are denoted as the α+n nuclides. From these values the binding energies of the corresponding alpha nuclides are subtracted. Likewise the effect of a second neutron is computed by subtracting from the binding energies for the α+2n the values of the binding energies of the α+n. The effect of the second neutron includes the effect of the neutron pair formation in addition to the interactions with other nucleons. The second neutron has interactions with the first neutron in addition to the interactions the first neutron has. The effect of the third neutron is included in the following display as something that approximates the effect of the interactions of the second neutron.

The curve for the third neutron is below the one for the first neutron because the interaction of neutrons involves a repulsion.

In the next graph the effects for the first and third have been averaged for comparison with the effect for the second neutron.

The effect of the formation of a neutron-neutron pair can be approximated by subtracting from the effect of the second neutron the average of the effects of the first and third.

There is a neutron shell that ranges from 7 neutrons to 14 neutrons. Over that range the average difference is 3.60 MeV. For the shell for 15 neutrons and above the average is 2.89 MeV. For the combination of these two ranges the average is 3.17 MeV. This could be the best estimate of the binding energy effect of the formation of a neutron-neutron pair.

Proton-Proton Pairs

The corresponding graphs for protons are shown below.

The average for the shell from 15 and up is 3.45 MeV. This seems to be the best estimate of the binding energy effect of the formation of a proton-proton pair.