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The Relative Sizes of the Binding Energies Due to the Formation
of Neutron-Neutron and Proton-Proton Spin Pairs

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 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 4.27580 MeV. The ratio of the neutron value to the proton value is 1.032341316, remarkably close to unity.

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 estimates of the effect on incremental binding energies for the above case of 10 nucleon and other cases are given in the table below. It is tempting to think of the odd-even fluctuations to due strictly to the formation of a nucleon pair, but the variation indicates that some other effects are involved. Probably when a nucleon pair is formed there is a slight rearrangement of the other nucleons that involves energy changes. The focus here is the relative size of the effects due to neutron-neutron and proton-proton pair formations.

Number
of Nucleons
Neutron
Pair
(MeV)
Proton
Pair
(MeV)
Ratio
10 4.4140854.27581.03234
12 4.8086454.685181.02635
14 4.7991454.5941651.04462
16 3.955383.888091.01731
18 3.421053.3238251.02925
20 3.016802.9042651.03875
22 4.617354.533451.0185
24 4.076504.054901.00533
26 3.257503.212801.01391
28 3.073502.945251.04354
30 3.311003.2291.02539
32 3.365002.937501.14553
34 3.395003.217051.05531
36 3.200003.505000.91298

The data in the above table are plotted in the following graphs.

 

Conclusion

Although there are some anomalies the binding energy enhancements due to neutron-neutron and proton-proton pair formations are equal. It is notable that the ratios are almost always slightly greater than unity. The average of the ratios for the cases given in the table is 1.02922.


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