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The nuclei of atoms are composed of neutrons and protons. The mass of a nucleus is generally less than the combined masses of the nucleons (neutrons and protons) 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 neutronneutron spin pairs appears in the following graph of the incremental binding energies of neutrons in terms of the sawtooth pattern.
A sawtooth is formed when the binding energy for an even number of neutrons is larger than the previous odd number of neutrons. Numerically this shows up as the second difference in binding energy being positive for an even number of neutrons in the nuclide and negative for an odd number. The difference represents the effect on binding energy due to the formation of a neutronneutron spin pair.
The sharper drop at 50 neutrons 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 protonproton spin pair is similar, but this is covered in another study.
The formation of a neutronproton spin pair is manifested by an entirely different pattern. If the neutron number is less than the proton number then each additional neutron results in the formation of a neutronproton spin, regardless of whether the neutron 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 24, the number of protons. In the above graph there is also a sharper drop in incremental binding energy after N=28, where the shell is completely filled and additional neutrons go into a higher shell.
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 neutrons and all 160 cases for protons. 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 increments have been tabulated. That exercise reveals that although the overwhelming proportions of the second increments 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.
The Distributions of the
Second Differences in the Binding Energies of Neutons 


Positive  Zero  Negative  
even  1351  1  9 
odd  2  1  1346 
The average of the 1351 positive values for the even cases is +1.80 MeV. For the 1346 negative values for the odd cases the average is −2.56 MeV. The difference of 4.36 MeV represents an average value of the binding energy associated with the formation of a neutronneutron spin pair.
For the even cases there are negative values for the sodium isotopes with 22 and 24 neutrons. Here is the graph of the incremental binding energies of sodium.
There certainly is an anomaly there on the right where a peak in incremental binding energy appears at an odd number of neutrons. It could however be due to an error in measure of the binding energy for one isotope of sodium. Such an error would result in two anomalous values. However an overwhelming number of the cases have positive values for the nuclides with an even number of neutrons and negative values for the nuclides with an odd number of neutrons.
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 neutrons form neutronneutron spin pairs and neutronproton spin pairs. And the formation is exclusive in the sense that a neutron can form a spin pair with one and only one neutron and with one and only one proton.
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