Background

The data on binding energies of nuclides indicate that when a nucleon is added to a nuclide there is an increase in binding energy due to the interaction of the new nucleon with the nucleons which are already there. The enhancement of binding energy is substanially more if the new nucleon can form a spin pair with one or two of the nucleons already there. This results in a sawtooth pattern for the incremental binding energy of neutrons as a function of the number of neutrons in the nuclide.

The sawtooth pattern is a result of the enhancement of incremental binding energy due to the formation of neutron-neutron spin pairs. The regularity of the sawtooth pattern demonstrates that one and only one neutron-neutron spin pair is formed when a neutron is added to a nuclide.

The same effects occur for proton-proton spin pair formation on binding energy

The spin pairing results in chains of nucleons containg modules of the form -n-p-p-n-, or equivalently -p-n-n-p-. These modules can appropriately be called alpha modules because the smallest one is the alpha particle.

The binding energy due to spin pairing includes the effect of the formation of the spin pair itself, but also include the change in interaction energies among the other nucleons resulting from adjustment in their positions due to the introduction of the new nucleon.

Nucleonic Shells

As new nucleons of the same type are added to a nuclide the incremental binding energy changes in a regular pattern until a limit is reached and the value drops sharply. This is interpreted as the existence of shells for the nucleons.

Illustrations

Here are a couple of examples.

The sawtooth pattern is a result of the enhancement of incremental binding energy due to the formation of neutron-neutron spin pairs. The regularity of the sawtooth pattern demonstrates that one and only one neutron-neutron spin pair is formed when a neutron is added to a nuclide. These cases also illustrate that the enhancement due to spin pair formation is roughly constant within a shell but varies between shells. Let σ denote the shell and β the change in binding energy due the adjustments in position of the nucleons due the creation of a spin pair in a shell. Let Π denote the constant change in binding energy solely due to the formation of the spin pair irself. The total change in binding energy due the formation of a spin pair is then

Π + β(σ)

It appears that β(σ) is a declining function of shell number. The reason for this is that a new nucleon in a higher shell is farther away from the other nucleons than previous new nucleons in a lower shell were.

The significance of this formulation is that the nucleon which achieves the closure of an alpha module chain results in a duplication of Π but not of the β.

The illustrations above indicate that the decline in β for a higher shell can be a substantial proportion of the total enhancement in binding energy due to the addition of another neutron to the nuclide and thus that β itself is a substantial proportion of the total enhancement. Thus Π itself may be a relatively small proportion of the total enhancement. Therefore the fact that the enhancement in the binding energy for the neutron achieving closure of an alpha module chain is not notably higher than the enhancements for the other neutrons in the same shell does not mean that closure is not achieved,

Here is another illustration of the same point as those above.

In order to estimate the magnitude of Π what is needed is a regression analysis to estimate the effect on binding energy of the various elements of nuclear structure.

The empirical investigation reveals no special enhancement of binding energy for the neutron that fills a shell. Therefore the second possibility is the reality.

(To be continued.)

Conclusions

The nuclear shells consist of rings of alpha modules with spare neutrons and protons insufficient to form an alpha module.