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The Interactive Binding Energy Between the
Last Neutron in a Nuclide and the Last Proton

The binding energies of nuclides represent the interaction of all their nucleons with each other. There are two types of interactions; the formation of spin pairs and interaction through a distance-dependent strong force. The computation of the increments in binding energy due to a change in the number of neutrons or the number of protons provides some information about the binding energies involve in the separate types of interactions. Here are the computations for the nuclides with 10 protons (the isotopes of neon) and those with 10 neutrons.

Number of
Neutrons
Binding Energy of
Nuclides with
10 Protons
(MeV)
Binding Energy of
Nuclides with
9 Protons
(MeV)
Incremental
Binding Energy
of Proton
(MeV)
6 97.325 97.25 0.075
7 112.9 111.42 1.48
8 132.1535 128.21961 3.93389
9 143.7805 137.3692 6.4113
10 160.644859 147.80136 12.843499
11 167.40597 154.40267 13.0033
12 177.76991 162.5042 15.26571
13 182.97053 167.734 15.23653
14 191.836 175.27 16.566
15 196.02 179.13 16.89
16 201.6 183.48 18.12
17 203.01 184.53 18.48
18 206.89 185.8 21.09
19 208.2 185.7 22.5
20 212.1 186.7 25.4

Number of
Protons
Binding Energy of
Nuclides with
10 Neutrons
(MeV)
Binding Energy of
Nuclides with
9 Neutrons
(MeV)
Incremental
Binding Energy
of Neutron
(MeV)
4 69.99 68.1 1.89
5 88.191 85.423 2.768
6 110.753 106.5026 4.2504
7 123.865 117.981 5.884
8 139.807 131.76266 8.04434
9 147.80136 137.3692 10.43216
10 160.644859 143.7805 16.864359
11 163.0762 145.976 17.1002
12 168.5776 149.198 19.3796
13 168.703 149.22 19.483
14 172.004 150.92 21.084
15 171.18 150 21.18

A plot of the incremental binding energies is shown below.

The plot displays a piecewise linear character that is evivence of a shell structure. The sections of both curves beyond 10 appear to have approximately the same slope. Regression analyses for the 10 to 15 section of the data for the IBEn and the 10 to 18 section of the data for IBEp reveals that the slopes are not significantly different at the 95 percent level of confidence. The average of the two slope is 0.922 MeV. This is a very important bit of information. It is the average interaction energy of a neutron in the neutron shell including 10 to 15 neutrons with a proton in the proton shell including 10 to 15 protons. The individual slopes are two different estimates of the same quantity and therefore should be approximately equal.

The more general proposition is that the increments with respect to protons of the increments in binding energy with respect to neutrons is an estimate of the interactive binding energy of the last proton with the last neutron in the same order numbers of shells. Likewise the increments taken in the opposite order also gives an estimate of the interactive binding energy of the last neutron with the last proton. These increments in increments may be called cross differences. The above proposition is tested below.

Number of
Neutrons
Incremental
Binding
Energy of
Protons
(IBEp)
(MeV)
Increments
with respect to
Neutrons
of IBEp
(MeV)
Number of
Protons
Incremental
Binding
Energy of
Neutrons
(IBEn)
(MeV)
Increments
with respect to
Protons
of IBEn
(MeV)
4 4 1.89
5 5 2.768 0.878
6 0.075 6 4.2504 1.4824
7 1.48 1.405 7 5.884 1.6336
8 3.93389 2.45389 8 8.04434 2.16034
9 6.4113 2.47741 9 10.43216 2.38782
10 12.843499 6.432199 10 16.864359 6.432199
11 13.0033 0.159801 11 17.1002 0.235841
12 15.26571 2.26241 12 19.3796 2.2794
13 15.23653 -0.02918 13 19.483 0.1034
14 16.566 1.32947 14 21.084 1.601
15 16.89 0.324 15 21.18 0.096
16 18.12 1.23
17 18.48 0.36
18 21.09 2.61
19 22.5 1.41
20 25.4 2.9

The data for the increments in the incrementals are shown in the graph below.

While the cross differences are not always close the plot shows that the two patterns are essentially the same. This is the case for nuclides having 10 protons or 10 neutrons. It reveals the sawtooth (odd-even) pattern that indicates pair formation. The minimums in the range 10 to 15 are near zero. This might be a great significance.

Some of the other cases are shown below.

 

There are anomalies but overall the patterns where they overlap appear to coincide. The minimums in this case are not near zero as the previous case.

 

Again there are anomalies but overall the patterns nearly coincide. In this case the minimums are near zero.

 

Since it is hard to see the correspondence of the two values in the graph on the right only the overlap is displayed in the graph below.

Again the correspondence is quite close.

The strong force binding energy for the interaction of an individual neutron with an individual proton can be quite small and still be significant for the structure of nuclei. Suppose this strong force interactive energy between a neutron and a proton is 1/20 of a MeV. In a nucleus with 60 protons and 100 neutron there are 6000 interactions between neutrons and protons for a total interaction energy of 300 MeV. Suppose the binding energies involved in spin pair formations is 3 MeV. There are 30 proton pairs and 50 neutron pairs. These create altogether 240 MeV, a quantity less than that due to the strong force.


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