San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Evidence for the Formation of
Proton Spin Pairs Within Nuclei

An essential aspect of the structure of nuclei is the formation within them of nucleon spin pairs; proton-proton, neutron-neutron and proton-neutron. Proton-neutron spin pairs exist alone as deuterons but not proton-proton or neutron-neutron spin pairs. The evidence for the formation of proton-proton 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 one and only one proton-proton spin pair is formed when a proton is added to a nuclide.

The sharp dropoff after 50 protons 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 neutron-neutron spin pair formation on binding energy

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

Here is the graph for the case of the nuclides with 36 neutrons.

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

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 protons the increment in the incremental binding energy of a proton is positive and for one with an odd number of protons it is negative. Allowance must be made for whether the proton number p is less than the neutron number n; or p=n+1 or n>(p+1). The increment in the incremental binding energy for an odd number of protons 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 proton can be computed for 2768. Of these 2768 there are 2605 for which the increment in incremental binding energy can be computed. Of these 1303 have an even number of protons and 1302 have an odd number of protons. Of those with an even number of protons all but ten have a positive value for the increments and of those three have negative values which are very small (-0.19, -0.1 and -0.01MeV) and could be attributed to measurement errors.

Of those with an odd number of protons all have a negative value for the increment.

What are shown below are the cumulative frequency distributions for the even and odd proton number cases.


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 protons.

The Anomalies

For the cases of an odd number of protons there are no anomalies.

For the case of an even number of protons there are ten anomalies. Three of these are very small. Another two are about one quarter and one half of an MeV. There are three with values near −0.75. The most serious anomaly is one of about −2.0. Here is the graph for the data on nuclides with ten neutrons..

The sixth proton apparently did not form a spin pair or if it did it was not manifested as an increase in binding energy.


For the proposition that whenever possible protons form spin pairs within nuclei there are only about ten exceptions out of 2605 cases. This is about a 99.4 percent confirmation.

The estimates of the binding energy associated with the formation of a proton-proton spin pair suggest its value is in the range of 1 to 3 MeV.

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