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The Effect of Additional Proton Pairs
on the Binding Energies of Nuclides

The changes in binding energies as the number of protons or neutrons is changed reveals information about the energies associated with the formation of nucleon pairs and other substructures of nuclides. The formation of a neutron-proton pair, a deuteron, involves a binding energy of about 2.2 million electron volts (MeV). The formation of a neutron pair or a proton pair involves something on the order of 3 MeV. The possible substructure of a nuclide which involves the largest increment in binding energy is the alpha particle. Its amount is about 28.3 MeV.

If two protons are added to a nuclide in which there are more neutrons than protons then an alpha particle could be formed. After the formation of an alpha particle there could be a structural rearrangement in a nucleus that also involves changes in binding energy. To test the possibility of the formation of alpha particles the binding energies of nuclides with a fixed number of neutrons were compiled the change in binding energy computed for the nuclide having p protons over the nuclide having (p-2) protons. The results for the nuclides having 44 neutrons are shown below.

One of the things which is notable about the above display are that the magnitudes of the effects for the lower values of protons is comparable to the binding energy of an alpha particle. Another is the regularity of the pattern. And finally there is a sharper drop after the number of protons reaches the level of 44, which is the number of neutrons in the nuclides. It is at that point that the addition of two more protons would not result in the formation of an alpha particle. There is also a sharper drop after 29 which has to do with the filling of a proton shell at 28 protons.

The regularity of the pattern can be observed by taking the increments in the values shown in the previous graph.

The regularity of the pattern has also be demonstrated by fitting a regression line to the data. A linear regression on the data with the two end points excluded has a coefficient of determination of 0.9910. A quadratic regression has a coefficient of determination of 0.99910.

For comparison the graphs for the case of nuclides with 26 neutrons are shown below.


The linear regression line for the data has a coefficient of determination of 0.99465.

Another comparison of interest is for the case of nuclides with 68 neutrons as shown below.

A bent-line regression with bends at 50 and 52 has a coefficient of determination of 0.99952.

It is notable that the transition to the higher proton shell takes place over two protons rather than one and that the effects of each of those protons in the transition are approximately equal.

Yet another comparison is for nuclides with 92 neutrons.


The linear regression line for the data of the first of the above graphs excluding the first point has a coefficient of determination of 0.99518. The quadratic regression has a coefficient of determination of 0.99931. The regression coefficient for the quadratic term is statistically significant; its t-ratio is 10.9.

Even more spectacular in terms of the regularity of the pattern is the case of 80 neutrons.


Again, as in the case of nuclides with 68 neutrons, the transition to the higher proton shell takes place over two protons rather than one and that the effects of each of those protons in the transition are approximately equal.

The regression equation with bends at 50 and 52 and with a quadratic term has a coefficient of determination of 0.99980. The quadratic term has a t-ratio of 6.2.

(To be continued.)


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