San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Evidence for the Formation of Proton-Proton Spin Pairs


The nuclei of atoms are composed of protons and neutrons. The mass of a nucleus is generally less than the combined masses of the nucleons (protons and neutrons) of which it is composed. This mass deficit expressed in energy units via the Einstein equation E=mc² is call the binding energy. Binding energy represents the energy which have to be supplied to break a nucleus down into its constituent nucleons.

The formation of a nucleon spin pair is manifested in terms a pattern in binding energy. For example, the formation of proton-proton spin pairs appears in the following graph of the incremental binding energies of protons in terms of the saw-tooth pattern.

A saw-tooth is formed when the binding energy for an even number of protons is larger than the previous odd number of protons. Numerically this shows up as the second difference in binding energy being positive for an even number of protons in the nuclide and negative for an odd number. The difference represents the effect on binding energy due to the formation of a proton-proton spin pair.

The sharper drop at 50 protons represents the completion of the filling of a shell. This is usually referred to as 50 being a magic number.

The pattern associated with the formation of a neutron-neutron spin pair is similar, but this is covered in another study.

The formation of a proton-neutron spin pair is manifested by an entirely different pattern. If the proton number is less than the neutron number then each additional proton results in the formation of a proton-neutron spin, regardless of whether the proton number is odd or even.

As can be seen in the above graph, the level of incremental binding energy is higher for N less than or equal to 18, the number of neutrons. In the above graph there is also a sharper drop in incremental binding energy after N=14, where the shell is completely filled and additional protons go into a higher shell.

The General Proposition

The proposition that nucleon form spin pairs inside of nuclei whenever possible could be supported by displaying graphs of the above type for all 90 cases for protons and all 160 cases for neutrons. This would require a large number of separate graphs even when the information for five cases is displayed in each graph. Instead the frequencies of the second differences in binding energy have been tabulated. That exercise reveals that although the overwhelming proportions of the second differences are of the proper signs there are a few cases that violate the proposition.

Here are the cumulative frequency distributions of the second differences in binding energy.


Here is the distribution of values for the even and odd cases.

The Distributions of the Second Differences
in the Binding Energies of Protons
even12720 11
odd 10 1285

There are eleven cases of negative values for the even cases. The average of the 1272 positive values for the even cases is +1.68 MeV. There is only one positive value for the case of an odd number of protons. For the 1285 negative values for the odd cases the average is −3.25 MeV.

The frequency distributions are the derivatives of the cumulative frequency distribution. Here are their approximations for arbitrary ranges.


The evidence is overwhelmingly in favor of the proposition that whenever possible within nuclei protons form proton-proton spin pairs and proton-neutron spin pairs. And the formation is exclusive in the sense that a proton can form a spin pair with one and only one proton and with one and only one neutron.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins