San José State University

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The Relative Magnitudes of the Binding Energies
Involved in the Formations of Neutron-Proton
and Neutron-Neutron Spin Pairs

BACKGROUND

The binding energies of 2931 nuclides are available as a function of the number of neutrons and protons they contain, BE(n, p). The binding energy of a nuclide is its mass deficit expressed in energy units via the Einstein equation, E=mc². The masses of nuclei are generally less than the masses of their constituent nucleons.

The incremental binding energy of a neutron (IBEN) is defined as

IBEN(n, p) = BE(n, p) − BE(n−1, p)

The IBEN of a nuclide represents the effect of an additional neutron on the energy and structure of a nucleus.

The purpose of this webpage is to demonstrate that whenever possible the neutrons and protons in a nucleus form neutron-proton spin pairs. If the number of neutrons is less than the number of protons then the addition of another neutron will form a neutron-proton spin pair and if the number of neutrons is greater than the number of protons then no neutron-proton spin pair will form.

It is be emphasized that the term the binding energy associated with the formation of a spin pair is used to indicate that there are two components involved. One is the binding energy resulting from the formation of the spin pair itself. The other is the change in binding energy that results from any rearrangement of the nucleons as a result of the formation of the spin pair. To get a notion of the relative magnitudes of these two components, consider the case of the deuteron which is simply a neutron-proton spin pair. Its binding energy. is about 2.2 MeV and there is no other nucleons to experience rearrangement. Thus the binding energy due the formation of a neutron-proton is about 2.2 MeV. A review of the binding energies associated with the formation of a neutron-proton is on the order of 10 MeV in other situations so the energy resulting from the rearrangements is about 8 MeV.

The Analysis

The column labeled n=p in the table below is the energy associated with the formation of a neutron-proton pair. It is approximately the same level as that due to the formation of a neutron-neutron spin pair n=p+1.

The column lableled n=p−1 includes the effects of the formation of both a neutron-proton spin pair and a neutron-neutron spin pair. For the smallest nuclides the effects are approximately the sum of the effects for the two separate spin pairs. For nuclides containing more nucleons the effect of the two spin pairs is little more than effect for one spin pair.

The column labeled n=p−2 measures the effect of the formation of a neutron-proton spin pair in a slightly different setting than the column labeled p=n.

The Incremental Binding Energies of Neutrons
in Nuclides In Which the Number of Neutrons n is
Nearly Equal to the Number of Protons p
(MeV)
Number
of protons
n=p-2 n=p-1 n=p n=p+1
5 13.0178 18.5766 8.4363 11.4541
7 15.6913 20.064 10.553325 10.833305
9 14.17 16.79961 9.14959 10.43216
11 14.155 17.1002 11.0691 12.41872
13 14.893 16.9322 11.36594 13.05781
15 14.483 17.862 11.3199 12.31179
17 14.333 15.7442 11.50833 12.64428
19 14.329 15.4455 12.0738 13.0764
21 14.43 16.187 11.5501 12.138
23 13.89 16.099 13.2717 13.0013
25 14.8 16.687 13.0818 13.6868
27 14.71 16.796 13.4372 14.0897
29 15.01 16.772 12.426 12.7633
31 13.95 15.42 12.72 12.76
33 13.8 15.6 12.81 12.89
35 14.2 15.5 13.3 13.1
37 13.9 16.2 13.6 13.525

The graph of the data shows that while the effects of the formations of a neutron-proton and a neutron-neutron are not equal for smaller nuclides they are for nuclides having about 30 protons.

The graph of the estimates for the neutron-proton and neutron-neutron spin pairs over a more extended range is shown below.

Clearly the two quantities become equal for the larger nuclei.

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