For background material on this topic see Repulsion.

It is widely believed that all nucleons attract each other. This notion was promoted by Werner Heisenberg in the 1930's after the neutron was discovered. According to Heisenberg a neutron and a proton were the same particle; a neutron was just a proton with its electric charge turned off. The term "nucleon" was coined to cover both the neutron and neutron. Heisenberg's view that a neutron and proton were the same particle fell out of favor but his notion that all nucleons attract each other continued. The electrostatic charge of protons contributes a repulsive force, but at sufficiently small separation distances the nuclear strong force is supposedly greater in magnitude than the electrostatic force and thus an attraction can result. The purpose of this webpage is to demonstrate that the strong force between protons is also a repulsion.

The picture is complicated by there being two types of nuclear interactions involved. Nucleons form spin pairs; i.e., neutron-neutron, proton-proton and neutron-proton spin pairs. All three involve a positive amount of binding energy (mass deficit) and therefore constitute attractions. But spin pair formation is exclusive in the sense that a neutron can form a spin pair with only one other neutron and with only one neutron. The other type of interaction is through the strong force per se and it is not exclusive. While protons repel each other, neutrons and protons are attracted to each other and it is this attraction that holds nuclei together.

The binding energy involved in spin pair formation is on the order of two to three million electron volts (MeV). The interaction between two nucleons through the strong force is far less. Therefore in small nuclides the structure is dominated by the interactions involving spin pair formation. But in larger nuclides the large number of interactions involving the strong force becomes dominant.

Binding Energy, Loss of Potential Energy and Force

Binding energy is in the nature of a loss in potential energy. When a neutron and a proton form a deuteron there is the emission of a gamma ray with an energy of 2.22457 MeV. When a gamma ray of at least that energy impacts a deuteron it disassociates into a neutron and proton and the mass deficit has disappeared.

When two particles exert an attractive force on each other and they move together there is a loss of potential energy. If two particles exert a repulsive force on each other and they move together there is a gain in potential energy. Thus a repulsion between particles corresponds to a decrease in binding energy whereas an attraction corresponds to an increase in binding energy.

The Incremental Binding Energies of Proton Pairs

The incremental binding energy of a proton pair is the binding energy of a nuclide less the binding energy of a nuclide with one less proton pair (two less protons). This is the first difference.

The relationship is roughly linear. A quadratic regression equation explains 99.87 percent of the variation in the incremental binding energy. The standard error of the regression equation is 0.344 MeV. Given the regularity of the relationship it is reasonable to express the relationships in terms of proton pairs rather than protons. The second difference corresponds to the slope of the relationship between incremental binding energy and the number of proton pairs in the nuclide. The interaction binding energy between two proton pairs is about 7 MeV.

Here are the relationships between the incremental binding energies of proton pairs for five different numbers of neutron pairs. The presentation is started with 21 to 25 neutron pairs as being typical. For smaller nuclides the relationships are not so regular and for larger nuclides the relationships are super regular.

Note that:

• All of the relationships are downward sloping to the right.
• The relationship for a higher proton pair number is above that of the relationship for a lower proton pair number.
• The relationships are roughly parallel.
• The relationships are roughly linear over a range of proton pair numbers.

Here are some examples of the relationship between the incremental binding energies of proton pairs and the number of neutron pairs in the nuclide.

The positive slope indicates that the force between a proton pair and a neutron pair is an attraction.

In the following graphs the variable being plotted is the incremental binding energy of a proton pair in MeV. The horizontal axes are all Numbers of Proton Pairs.

The points of sharp drops correspond to the filling of proton pair shells. The shells are usually identified as nucleon shells but strictly speaking they are nucleon pair shells.

The above pattern cannot be construed to involve parallel relationships, but they are all downward sloping except for part of the case of NN=1.

The relationships for these cases are roughly parallel and downward sloping. The sharp drop here occurs at 25 proton pairs. This corresponds to 50 protons.

Again sharp drop here occurs at 25 proton pairs.

For the shell that corresponds to 26 through 41 proton spin pairs this is generally linear relationship.

Conclusions

In all but a few of the cases the relationships between the incremental binding energies of proton pairs and the number of proton pairs in the nuclide are downward sloping, thus indicating the force between proton pairs through the nuclear strong force is a repulsion. In all cases the an increase in the number of neutron pairs in the nuclide results in an increase in the incremental binding energy of proton pairs. The increase is roughly linear. This is indicative of the force between a proton pair and a neutron pair through the strong force being an attraction.