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The Relative Sizes of the Binding Energies
Due to the Formation of the Three Types of
Spin Pairs and the Strong Force Interactions

The incremental binding energy of a neutron, the increase in binding energy that results from an additional neutron to a nuclide,, depends upon whether it results in the formation of another neutron-neutron spin pair and another neutron-proton pair. The additional neutron will form another neutron-neutron pair if there is an unpaired neutron in the nuclide to which the neutron is added. It will form a neutron-proton pair if the number of neutrons is less than the number of protons. It also depends upon the strong force interaction of the additional neutron with the other neutrons and the protons in the nuclide. The same applies to the incremental binding energy for a proton.

Below are shown the graphs for the incremental binding energies of neutrons and protons for nuclides having ten protons and having ten neutrons.

 

These displays provide illustrations of the formation of all three types of nucleon pairs. The sawtooth patterns come from the formation of neutron-neutron pairs in the graph on the left and proton-proton pairs in the graph on the right. In the graph on the left the addition of a neutron when the number of neutrons is less than the number of protons results in the formation of a neutron-proton. When the number of neutrons exceeds the number of protons no such pair is formed and there is a sharper drop in the incremental binding energy after ten neutrons. The same applies to the incremental binding energy of protons, as shown in the graph on the right. The graphs also show the sharper drops that occur after a nucleon shell is filled. This occurs after the numbers 6, 8 and 14. These are called magic numbers.

The effect of pair formation can be estimated by comparing the value at an even number of nucleons with the extrapolation of the trend from the values for the odd numbers of nucleons. In the case of ten nucleons the values at near 6 and 8 are distorted by the effect of filled shells. Therefore the effects of like nucleon pair formation are estimated by taking the differences between the values at 13 and the average of the values at 12 and 14. For neutrons this value is 4.414085 million electron volts (MeV) and for protons it is 4.2758 MeV. The ratio of the neutron value to the proton value is 1.032341316, remarkably close to unity.

The effect of there not being a neutron-proton pair formed can be estimated by projecting a value at 10 back from the values at 12 and 14 and taking the difference between that value and the actual value at 10.

For the incremental binding energy of neutrons this difference is 5.10130 MeV. For protons it is 4.17026 MeV.

The average of the two values is 4.63578 MeV. The ratio of this average to the average for the neutron-neutron and the proton-proton pair formations is 1.067, or roughly unity.

For this case the values at 8 and 10 are too distrorted by the changes due to the neutron shells being filled to use the forward projection from 7 and 9 to 11. This difference in the effects of going beyond 10 can be seen in the following graph in which the incremental binding energies of neutrons and protons are shown in the same diagram.

For more on the binding energy due to the formation of a neutron-proton pair see neutron-proton pairs.

Some parts of the estimated values for the binding energies due to pair formation is for the interaction of the nucleons through the distance dependent strong force.

The Strong Force Interaction Energies
of Neutrons with Protons

The change in the incremental binging energy of neutrons resulting from an increase in the number of protons is called a cross difference. There is also the cross difference of the change in the incremental binging energy of protons resulting from an increase in the number of neutrons There is reason to believe that the cross differences in binding energies of nucleons should be equal to the interaction energy of the last neutron with the last proton and thus equal to each other. The following graphs show the incremental binding energies of neutrons and protons plotted versus the number of nucleons of the other type in the nuclide.

 

The slopes of these relationships should be the interaction energies of the last neutron with the last proton for nucleons within the same shell.

When these patterns are plotted in the same graph it is clear that the curves are approximately parallel and thus the slopes are approximately equal.

A regression analysis for the data from 10 to 15 indicates that the slopes are not significantly different at the 95 percent level of confidence. The average of the two slopes is 0.921777571 MeV. This is the interaction energy of a neutron in the shell including 10 to 15 neutrons with a proton in the shell including 10 to 15 protons. This is relative small compared to the binding energy resulting from nucleon-nucleon pair formation. The ratio is about one fifth. Thus in small nuclides the effect of pair formation dominates the strong force interaction. In large nuclides the strong force between an additional nucleon and the numerous other nucleons is dominant.


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