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The Binding Energy Due to the Formation
of a Neutron-Proton Spin Pair

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 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.

Number
of Nucleons
From IBE of
Neutrons
(MeV)
From IBE of
Proton
(MeV)
BE for
n-n Pair
10 5.10130 4.1702 4.414085
12 2.84947 3.12727 4.808645
14 5.16458 5.18494 4.799145
16 2.09742 2.09737 3.95538
18 1.44674 1.37889 3.42105
20 3.8119 3.5923 3.0168
22 1.54690 2.00760 4.61735
24 2.3731 2.46400 4.0765
26 0.62330 1.07340 3.2575
28 3.59170 3.53910 3.0735

The data from the above table are displayed graphically below.

The results from the neutron data and the proton data are satisfyingly close to each other, but significantly different from the estimates of the binding energy resulting from the formation of a neutron-neutron pair. Furthermore the high values come at the magic numbers of 14, 20 and 28 where the drop in incremental binding energy due to the filled of a shell combines with that due to there being no further formation of neutron-proton pairs. When the data for magic numbers are eliminated the following is the result.

The pattern appears to be that the n-p estimates are some constant amount lower, about 2.1 MeV, than the n-n estimates rather than a constant proportion. The conclusion is that the binding energy associated with the formation of a neutron-proton spin pair is significantly less than that for the formation of a neutron-neutron pair. The average of the non-magic number values for neutron-proton pairs is 1.92 MeV whereas the average for neutron-neutron pairs is 4.022 MeV. The ratio is 0.49.


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