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

The incremental binding energy (IBE) is the difference between the binding energy of the nuclide with N neutrons and that of the nuclide with (N-1) neutrons. 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 a sharp drop at 6 neutrons. The number 6 is amagic number which represent the filling of a shell. The enhancement of binding energy due to the formation of a neutron-neutron pair can be computed for a particular number of neutrons by computing the average of the IBE at the two adjacent numbers of neutrons and then taking the absolute value of the difference between the IBE for the particular number of neutrons and the average for the adjacent numbers of neutrons.

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 protons is shown below.

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

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

In this case there is a sharp drop in IBE only for 82 neutrons, a magic number. The binding energy enhancement is regular and roughly constant within the shell that extends from 51 to 82 neutrons.

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 126 neutrons.

The pattern is regular but the pair enhancement is definitely not constant.

There is no higher magic number of protons than 82. An element with 126 protons has not been created.

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.

There are relatively sharp drops at the magic numbers of 20 and 28 neutrons.

The data on the IBE of neutrons in the isotopes of silicon are shown below. The relatively sharp drops for 14 and 20 indicates that they are both magic numbers; i.e., correspond to filled shells of protons.

The number 14 is not a conventional magic number but the display below indicates the magicality of 6, 8 and 14. Six and 14 are not conventionally designated as magic numbers.

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

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

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 the interaction of the added nucleons 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 neutrons due to pairing depends upon which shell in which the pair is formed and furthermore usually on the number of neutrons within the shell. Generally the enhancement declines from shell to shell from the lower to the higher shells.