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The Relationships of Incremental Binding Energies
of the Neutrons in the Third Neutron Shell to the
Number of Protons in the Nuclide

The ionization energies of electrons in atoms and ions are simple quadratic functions of the number of protons in the nucleus, as illustrated in the graph below.

There is the possibility that a similar sort of relationship might exist between the incremental binding energy of a neutron and the number of protons in the nucleus. This was tested using the data for nuclides having four neutrons in the second neutron shell (and two in the first shell). The results were promising but of a much more complex nature than those for electron ionization.

More relationships were investigated. The previous results were for three through six neutrons. The first result shown below is for seven and eight neutrons. There is some ambiguity as to whether the seventh and eighth neutron are in the second neutron shell or the third. The conventional theory of nuclear shells is based on the number of stable isotopes. The magic numbers correspond to filled shells. According to that theory the first magic numbers are 2 and 8. Nuclear magic numbers can also be identified by sharp drops in the incremental binding energies as the number of neutrons is increased. That approach identifies the conventional magic numbers but also indidicates that 6 and 14 are magic numbers. A simple algorithm exists for identifying all of the magic numbers except 8 and 20, indicating that 8 and 20 are somehow different from the other magic numbers.

If 6 is a magic number then the second neutron shell has a capacity of four and the third neutron shell may start with the seventh neutron. On the other hand, there may be some subshell containing the seventh and eighth neutrons that is adjoined to the second shell. In this case then the third shell would start with the ninth neutron.

Here are the relationships for the seventh and eighth neutrons.

Generally the slopes of the relationships are increasing until the proton number equals the number of neutrons. Thereafter the slopes are generally decreasing. The two curves start a near zero in contrast to the curves for the third through sixth neutrons. The first four in the shell have starting values of approximately ±2 MeV.

For the ninth through the fourteenth the relationships are as shown below.

No functional form, such as quadratic or cubic, is apparent in these displays. If anything the relationships overall appear to be linear. There are some regularities. Note the pattern of the differences in incremental binding energies dsiplayed below.

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