An essential aspect of the structure of nuclei is the formation within them of nucleon spin pairs; neutron-neutron, proton-proton and neutron-proton. Neutron-proton spin pairs exist alone as deuterons but not neutron-neutron or proton-proton spin pairs. The evidence for the formation of neutron-neutron spin pairs within nuclei is the odd-even fluctuation in the incremental binding energy of nuclides, examples of which are shown below.

The regularity of the sawtooth pattern demonstrates that o,ne and only one neutron-neutron spin pair is formed when a neutron is added to a nuclide.

The sharp dropoff after 50 neutrons is the effect of a shell being filled. The filled shell numbers, usually called nuclear magic numbers are 2, 8, 20, 28, 50, 82 and 126. There is also evidence of 6 and 14 being magic numbers.

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

The effect of neutron-proton spin pairs is revealed by a sharp drop in incremental binding energy after the point where the numbers of neutrons and protons are equal.

Here is the graph for the case of the isotopes of Krypton (proton number 36).

As shown above, there is a sharp drop in incremental binding energy when the number of neutrons exceeds the proton number of 36. This illustrates that when a neutron is added there is a neutron-proton spin pair formed as long as there is an unpaired proton available and none after that. This illustrates the exclusivity of neutron-proton spin pair formation. It also shows that a neutron-proton spin pair is formed at the same time that a neutron-neutron spin pair is formed.

The case of an odd number of protons is of interest. Here is the graph for the isotopes of Rubidium (proton number 37).

The addition of the 38th neutron brings the effect of the formation of a neutron-neutron pair but a neutron-proton pair is not formed, as was the case up to and including the 37th neutron. The effects almost but not exactly cancel each out. It is notable that the binding energies involved in the formation of the two types of nucleonic pairs are almost exactly the same.

This same pattern is seen in the case for the isotopes of Bromine.

The purpose of the material which follow is to show the universality of the effects illustrated above. For a nuclide with an even number of neutrons the increment in the incremental binding energy of a neutron is positive and for one with an odd number of neutrons it is negative. Allowance must be made for whether the neutron number n is less than the proton number p; or n=p+1 or n>(p+1). The increment in the incremental binding energy for an odd number of neutrons is strongly affected by the filled shell effect.

It must be noted that these values of the increments of the incremental binding energies include the effects of adjustments in nuclides which result from the formation of a spin pair as well as the formation of the spin pairs themselves.

The Numerical Results

Of the 2931 nuclides for which the binding energies are known the incremental binding energy of a neutron can be computed for 2709. Of these 2709 there are 2458 for which the increment in incremental binding energy can be computed. Of these 1229 have an even number of neutrons and 1229 have an odd number of neutrons. Of those with an even number of neutrons all but five have a positive value for the increments. Of those with an odd number of neutrons all but two have a negative value for the increment. What is shown below is the cumulative frequency distributions.

    

The straight portions of the cumulative frequency distribution indicate that the frequency distributions are uniform over those portions of increments in the incremental binding energies of neutrons.

The Anomalies

For the case of an odd number of neutrons there are only two anomalies. One of these is +5.98 MeV for the hydrogen isotope with five neutrons. This anomaly seems likely to be a matter of an inaccurate measurement. Leaving out that value the average for all the odd neutron number cases is −2.38746 MeV. This gives 2.38746 MeV as the binding energy associated with the formation of a neutron-neutron spin pair.

The other anomaly is for the sodium isotope with 23 neutrons. The value is small (+0.3 MeV) but this seems to be definite anomalous case. Here is the graph for the sodium data.

For the case of an even number of neutrons there are five anomalies. Three of these are small and the other two are for isotopes of sodium with 22 and 24 neutrons. This is essentially one anomaly. The 22nd neutron apparently does not form a spin pair, but the 23rd ond does and then the 24th one does not.

The average for the cases of an even number of neutrons is 1.73144 MeV. This is an alternate estimate of the binding energy associated with the formation of a neutron-neutron spin pair. The average of the two estimates of the effect on binding energy associated with the formation of a neutron-neutron spin pair is 2.05945 MeV.

The Cases of n≤p

There are far fewer of these cases than the ones for n>(p+1). There are 53 cases for n even and 52 for n odd. The cumulative frequency distributions are:

     

For the cases of n odd the average is −4.85759 MeV and for n even it is 2.78574 MeV. Thus the two esimates for the effect on binding energy due to the formation of a neutron-neutron spin pair are 4.85759 and 2.78574 MeV. Their average is 3.82166 MeV.

The Cases of n=(p+1)

These cases are the ones for which the formation of neutron-proton spin pairs has ceased. Thus there is a negative effect on the incremental binding energy as a result. If n is even that negative effect is offset by the positive effect of the formation of a neutron-neutron spin pair. Here are the values for n even and equal to (p+1).

As can be seen, of the 18 cases 15 have a positive value indicating the effect of the formation of the neutron-neutron spin pair exceeds the negative effect of a neutron-proton spin pair not being formed. The average value is 0.78305 MeV.

The cases for n odd give largely the negative of the binding energy associated with the formation of a neutron-proton spin pair. The qualification "largely" comes from the fact that there is a downward slope in the relationship between the incremental binding energy of a neutron and the number of neutrons in the nuclide. The increments in the increments picks up this negative effect but it is small compared to the magnitude of the effect due to the nonformation of a neutron-proton spin pair. This effect of the downward slope of the relationship also affects the cases of n being even.

As can be seen, all of the values are negative. The average value is −7.79616 MeV. This value is strongly influenced by the effect of filled shells (magic numbers). In the above table the cases for magic numbers of neutrons are high-lighted.

Conclusion

For the proposition that whenever possible neutrons form spin pairs within nuclei there are only about four exceptions out of 2458 cases. This is about a 99.84 percent confirmation.

The estimates of the binding energy associated with the formation of a neutron-neutron spin pair suggest its value is in the range of 2 to 4 MeV.