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The Binding Energy Associated with the Addition
of a Neutron-Proton Spin Pair to to Alpha Nuclei

In a previous study the estimate of the effect on binding energy associated with the addition of a neutron-proton spin pair was considerably higher than the effects due to the other two types of nucleon spin pairs (neutron-neutron and proton-proton). The value for the neutron-proton spin pair is about 18 million electron volts (MeV) compared with 4 MeV for each of the other two types of spin pairs. The 18 MeV is approximately one half of the estimate of the effect of an alpha module. Those estimates are the result of a regression analysis of binding energies. They can be checked by compiling the binding energies of the nuclides with and without neutron-proton spin pairs.

The Incremental Binding Energies
of a Neutron-Proton Spin Pair as
a Function of the Number of
Alpha Modules in the Nuclide
Number of
α Modules
BE α BE α+1np Incremental
BE of np
0 0 2.224573 2.224573
1 28.295674 31.9946 3.698926
2 56.49951 64.7507 8.25119
3 92.161728 104.658625 12.496897
4 127.619336 137.3692 9.749864
5 160.644859 174.1453 13.500441
6 198.25689 211.89414 13.63725
7 236.53689 250.6049 14.06801
8 271.78066 285.56553 13.78487
9 306.7157 320.6473 13.9316
10 342.052 354.6871 12.6351
11 375.4747 390.3607 14.886
12 411.462 426.6288 15.1668
13 447.697 462.7332 15.0362
14 483.988 497.108 13.12
15 514.992 528.17 13.178
16 545.95 558.71 12.76
17 576.4 589.2 12.8
18 607.1 620.1 13
19 638.1 651.7 13.6
20 669.8 682.7 12.9
21 700.9 713.7 12.8
22 731.4 744.4 13
23 762.1 775.2 13.1
24 793.4 806.5 13.1

The graph of the data reveals that the incremental binding energy of the neutron-proton spin pair is a function of the shell for the alpha module and the shell shifts at the neutron numbers corresponding to the convention magic numgers, {8, 20, 28}

There can only be one neutron-proton spin pair in a nucleus because two constitutes an alpha module. It is of interest to compare the incremental binding energy of a neutron-proton spin pair with that of an alpha module.

Beyond the small nuclides it appears that the incremental binding energy of a neutron-proton spin pair is roughly one half of that of an alpha module That would make sense because a neutron-proton spin pair is in a sense one halp of an alpha module. The precise ratios are given below.

As can be seen in the above graph the value of the ratio for the larger nuclides is about 0.425.

The incremental binding energy of a nucleon combination is made up of two components. One component is the binding energy associated with the formation of the the nucleon combination and the other is the sum of the interactions through the nuclear strong force between the nucleons of the added combination and all the other nucleons of the nucleus. The binding energy associated with the formation of a nucleon combination is, in turn, composed of the binding energy due to the formation of the combination per se and the binding energy due to the adjustments in the structure of the rest of the nucleus.

In the formation of the deuteron there is no other structure to involve adjustment so its binding energy is entirely due to the formation of a neutron-proton spin pair and the interaction of a neutron and proton through the nuclear strong force. That binding energy is taken to be equal to the energy of the gamma photon associated with the formation and disassociation of a deuteron. That may be a conceptual error and the true binding energy coud be on the order of twice that amount. To see why this is so, consider a proton and electron forming a a hydrogen atom. When these particles form an atom there is a loss of potential energy. Half of that loss of potential energy goes into the energy of the emitted photon and half goes into an increase in kinetic energy. So the loss in potential energy is twice the energy of the emitted photon. The ratio of two only applies because the force between the proton and electron is strictly an inverse distance squared. At the nuclear level, binding energy is in the nature of a loss of potential energy.but the force drops off more rapidly with separation distance than inverse distance squared. But for now the binding energy of the deuteron is taken to be 2.2 MeV.

The interaction of the neutron and proton through the so-called strong force is on the order of 0.6 MeV which leaves the binding energy due to the formation of a neutron-proton spin pair to be 1.6 MeV. The total binding energy associated with the formation of a neutron-proton spin pair in larger nuclides is about 13 MeV. This means that the binding energy due to the adjustment in the structure of the other nucleons when a neutron-proton spin pair is formed is about 11.4 MeV.

The total binding energy associated with the formation of an alpha module is about 31 MeV. With each of its neutron-proton spin pairs accounting for 42.5 percent of that binding energy that leaves 15 percent of the 31 MeV due strictly to the formation of the alpha module per se; about 4.6 MeV.

Conclusion

Away from the irregularities of the smaller nuclides the effect on binding energy of a neutron-proton spin pair formation is slightly more than forty percent of that of an alpha module. Most of the binding energy associated with the formation of a neutron-proton spin pair in a nuclide is due to the adjustments in the structure of the other nucleon that occur as a result of the spin pair. The binding energy of these adjustments is through the so-called nuclear strong force.

<(To be continued.)


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