Consider an array of the binding energies of all nuclides with a constant number of neutrons, say 28, as shown below.

The incremental binding energy (IBE) is the difference between the binding energy of the nuclide with N protons and that of the nuclide with (N-1) protons. The graph of the IBE as a function of the number of protons in the nuclide is shown below.

The pattern of odd-even fluctuations in the IBE is broken by sharp drops at 20 and 28 protons. The numbers 20 and 28 are magic numbers which represent the filling of shells. The enhancement of binding energy due to the formation of a proton-proton pair can be computed for a particular number of protons by computing the average of the IBE at the two adjacent numbers of protons and then taking the absolute value of the difference between the IBE for the particular number of protons and the average for the adjacent numbers of protons.

However the sharp drops at the magic numbers have to be left out of the computation.

The result of this computataion for the nuclides with 28 neutrons is shown below.

If only the loss of potential energy upon the formation of a spin pair of protons were involved the values would be constant for all numbers of protons. What is obvious in the above display is the enhancement depends upon the shell and furthermore upon the number of protons within the shells. The fact that the slope of the pattern is positive for the shell involving 28+ protons rather than negative as for the lower two shells is a puzzle at this point.

Now consider the nuclides with 50 neutrons. (This is another magic number chosen so as to have many isomers as possible.) The plot of IBE for protons for the set of nuclides shows the odd-even fluctuations.

In this case there is a sharp drop in IBE only for 38 protons. The number 38 is generally not considered to be a magic number. The binding energy enhancement is regular and roughly constant.

The nuclides with 82 protons show a similar pattern but the pair enhancement is definitely not constant. There is a sharp drop in the IBE at the magic number of 50.

The next magic number is 126 and this is the largest. The pattern for this case is exceptionally regular but definitely not constant.

Now the patterns for the smaller nuclides are displayed. As a general rule the patterns for the larger nuclides are more regular than those of the smaller nuclides.

The number 20 is a conventional magic number.

The relatively sharp drop for 20 and 14 indicates that they are both magic numbers; i.e., correspond to filled shells of protons.

The pattern here is a remarkable constancy for the shells above 14 protons.

The number 14 is not a conventional magic number but the display below indicates the magicality of 14 and also that of 8.

The stair step pattern and the nearly constant levels within the shells is notable.

The number 8 is a conventional magic number, the display below indicates that 6 is also magical.

The magicality of 6 is further confirmed in the display below where the magicality of 8 is ambiguous.

The lowest magic number is 2 and the magicality of 2 is confirmed in the display below.

The above display is perhaps the only case in which the enhancement due to proton pair formation can be considered to be constant.

A Possible Explanation for the Shell Dependence of the
Enhancement of Binding Energy Due to Nucleon Pair Formation

When a single nucleon is added to a nuclide it is situated in some arrangement, let us say in position X. This results in a loss of potential energy of say V_X and thus an increase in binding energy of V_X. If another nucleon is added to the arrangement it would go to some position Y at some distance from the first nucleon at X. The increase in binding energy would be V_Y in which V_X would be equal to V_X plus the effect of the interaction with the other nucleon at X. The values of V_X and V_Y would depend upon the shell to which the nucleons are added.

When the two nucleons form a pair the pair would be added to the arrangement of nucleons in the nuclide at position X. The increase in binding energy is then likely to be roughly 2V_X plus P, the binding energy due to the formation of the nucleon pair. On the other hand the increase in binding energy due to the independent addition of the two nucleons would be (V_X+V_Y). Thus the computed increment in binding energy due to the addition of the second nucleon would be [(P+2V_X) − (V_X+V_Y)]=[P+(V_X−V_Y)]. The quantity (V_X−V_Y) is due to what interaction independently added nucleons have with each other. That would depend upon the their distance apart and hence upon the shell in which they are added. As more nucleons are added within a shell the distance apart of the independently added nucleons would be shortened.

The implication of this explanation is that the enhancement of binding energy due to the formation of a nucleonic pair is the asymptotic lower limit to the enhancement of binding energy.

Conclusions

It is abundantly clear that the enhancement in incremental binding energies for protons due to pairing depends upon which shell in which the pair is formed and furthermore usually on the number of protons within the shell. Generally the enhancement declines from shell to shell from the lower to the higher shells.