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A Extensive Demonstration that
Within a Nucleonic Shell the Relationship
between the Incremental Binding Energy and
the Number of Neutrons is Linear

For background on this topic see Repulsion.

In the conventional model of nuclei it is asserted 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 proton and a neutron 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 proton. 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 picture is complicated by there being two types of 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 proton. The other type of interaction is through a force based on a type of charge and it is not exclusive.

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 charge force is far less, about 1/3 MeV. Therefore in small nuclides the structure is dominated by the energies involving spin pair formation. But in larger nuclides the large number of interactions involving the charge-based 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 proton and neutron 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 proton and neutron and the mass deficit has disappeared.

When two particles exert an attractive force on each other and they move toward each other there is a loss of potential energy. If two particles exert a repulsive force on each other and they are forced toward each other 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

To avoid the complication of the increments in binding energy due to spin pair formation the analysis from this point forward is in terms of neutron pairs (Np's) and proton pairs (Pp's).

The Incremental Binding Energies of Neutron Pairs (Np's)

The incremental binding energy of a neutron pair is the binding energy of a nuclide less the binding energy of a nuclide with one less neutron pair (two less neutrons). This is the first difference. The second difference corresponds to the slope of the relationship between incremental binding energy and the number of neutron pairs in the nuclide.

Note that the data are only for integral values of the explanatory variables. The relationships are plotted and dispayed as continuous functions to make characteristics like slopes easier to see.

For the shell that corresponds to 26 through 41 neutron spin pairs this is generally linear relationship. There is a slight upward curvature.

A quadratic regression function explains 99.9 percent of the variation in the incremental binding energy. The slope of the relationship is the binding energy due to the interaction of the last added neutron pair with the previously added neutron pairs. The value is −0.6 MeV, indicating that the force between neutron pairs is a repulsion.

The relationships for a number of different proton numbers are as follows.

Note that in the above display:

The points of sharp drops correspond to the filling of neutron pair shells. The shells are usually identified as neutron shells but strictly speaking they are neutron pair shells. In the above diagram the sharp drops come at 25 and 41 neutron pairs, which correspond to the magic numbers of 50 and 82 neutrons.

The incremental binding energy of a neutron pair may also be plotted versus the number of proton pairs in the nuclide. For the cases considered above this is

The positive slopes indicate that the force between a neutron pair and a proton pair is an attraction. The value of the slope is about +1.4 MeV.

The relationships for proton pair numbers 16 through 20, shown below, show the same pattern as the previous cases. The sharp drop comes at 25 neutron pairs. In the data for 20 proton pairs there is a slight anomaly. The incremental binding energy drops a relatively larger amount and thereafter rises. This may be due to a measurement error in the binding energy for the zirconium 99 isotope.

The sharp drops occur at 25 neutron pairs, which corresponds to the magic number of 50. There is an anomaly of the incremental binding energy increasing with increasing neutron pair numbers for Pp=20.

Here are more cases.

Although the above patterns cannot be construed to involve parallel relationships, they are all downward sloping except for part of the case of Pp=2 (helium).

The relationships for these cases are roughly parallel and downward sloping,. Again there are anomalies for the case with the highest proton pair numbers.

The sharp drop here occurs at 41 neutron pairs. This corresponds to 82 neutrons.

The sharp drop here occurs at 63 neutron pairs. This corresponds to 126 neutrons.


In all but a handful of cases the relationships between the incremental binding energies of neutron pairs and the number of neutron pairs in the nuclide are downward sloping, thus indicating the force between neutron pairs through the force of nuclear interaction is a repulsion.

The relationships within neutron shells are roughly linear.

In the cases shown an increase in the number of proton pairs in the nuclide results in an increase in the incremental binding energy of neutron pairs. The increase is roughly linear. This is indicative of the force between a neutron pair and a proton pair through force of nuclear interaction is an attraction.

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