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Demonstration that a Nucleon Can Form Two Spin Pairs,
One with a Nucleon of the Same Type and
One with a Nucleon of the Opposite Type

Pairings

The Effect of the pairings of nucleons shows up in terms of the increments in binding energies resulting from additional nucleons. For example, consider the increments in binding energies for the isotopes of Tin as a function of the number of neutrons in the isotope.

The regular odd-even seesaw pattern arises from the formation of neutron-neutron pairs. (The sharp drop at 82 neutrons is due to the filling of a shell.)

The same sort of pattern occurs for nuclides with the same number of neutrons but increasing numbers of protons. For example,

The pairings are exclusive; i.e., a neutron pairs with only one other neutron and a proton with only one other proton. However, as will be shown below, a paired neutron may pair with a proton and vice versa.

Effect of an Additional Proton

Consider a nuclide with equal even numbers of neutrons and protons. If the nucleon numbers are designated as (n,p) then this would be (2k,2k). Such a nuclide has formed all of the neutron-neutron pairs, all of the proton-proton pairs and all of the neutron-proton pairs. Now consider nuclides of the form (2k+2,2k). It has k+1 neutron pairs, k proton-proton pairs and k neutron-proton pairs. If a proton is added then an additional neutron-proton pair could form. If it does form then there should be a substantial increase in the binding energy (BE). This would show up as BE(2k+2, 2k+1)−BE(2k+2, 2k). However some of this increment may be due simply to the addition of another proton. That part would be equal to BE(2k, 2k+1)−BE(2k, 2k). Thus the measure of the effect on binding energy of the formation of a neutron-proton pair when all of the neutrons are already part of neutron-neutron pairs is

Δnp = [BE(2k+2, 2k+1)−BE(2k+2, 2k)] − [BE(2k, 2k+1)−BE(2k, 2k)]

Here is the data with [BE(2k+2, 2k+1)−BE(2k+2, 2k)] denoted as Δ2 and [BE(2k, 2k+1)−BE(2k, 2k)] as Δ1:

kΔ1Δ2Δnp
1 -1.965674 9.9754 11.941074
2 -0.18511 11.2276 11.41271
3 1.943572 10.207422 8.26385
4 0.600274 7.99436 7.394086
5 2.431341 8.79411 6.362769
6 2.27131 8.27132 6.00001
7 2.74811 7.29706 4.54895
8 2.27654 6.37065 4.09411
9 1.8578 6.381 4.5232
10 1.085 4.9298 3.8448
11 1.6143 5.1676 3.5533
12 2.085 5.2715 3.1865
13 1.599 5.0642 3.4652
14 0.694 3.4175 2.7235
15 0.458 2.811 2.353
16 -0.05 2.31 2.36
17 -0.5 1.97 2.47
18 -0.6 2.345 2.945
19 -0.2 2.492 2.692
20 -0.6 2.1 2.7
21 -0.9 1.9 2.8
22 -1.1 1 2.1

In graphical form it is

As can be seen in each case there is a substantial increase in binding energy that can be attributed to the formation of a neutron-proton spin pair where all of the neutrons are already part of a neutron-neutron spin pair. Something more than neutron-proton pairing is probably involved and the effect of the neutron-proton pairing is likely to be the plateau level for the larger nuclides.

The Effect of an Additional Neutron

The same construction can be carried out considering nuclides (n,p) of the form (2k,2k+2), (2k+1,2k+2, (2k,2k) and (2k+1,2k). Let

Δ3 = [BE(2k+1, 2k)−BE(2k, 2k)]
and
Δ4 = [BE(2k+1, 2k+2)−BE(2k, 2k+2)]

Therefore

Δnp = [BE(2k+1, 2k+2)−BE(2k, 2k+2)] − [BE(2k+1, 2k)−BE(2k, 2k)] = Δ4 − Δ3

kΔ3Δ4Δnp
1 -0.885674 10.6764 11.562074
2 1.66539 13.1194 11.45401
3 4.946335 13.22237 8.276035
4 4.143324 11.627 7.483676
5 6.761111 13.1472 6.386089
6 7.33067 13.3112 5.98053
7 8.47355 13.0533 4.57975
8 8.64156 12.7412 4.09964
9 8.7889 13.2888 4.4999
10 8.3627 12.27 3.9073
11 9.53 13.153 3.623
12 10.582 13.814 3.232
13 10.6832 14.197 3.5138
14 10.247 13.03 2.783
15 10.231 12.74 2.509
16 10.06 12.87 2.81
17 10.22 13.2 2.98
18 10.83 14.1 3.27

The graph of the results is

Comparison of the Two Estimates of Δnp

First the graph

And then the data

kFrom add
Proton
From add
Neutron
1 11.941074 11.562074
2 11.41271 11.45401
3 8.26385 8.276035
4 7.394086 7.483676
5 6.362769 6.386089
6 6.00001 5.98053
7 4.54895 4.57975
8 4.09411 4.09964
9 4.5232 4.4999
10 3.8448 3.9073
11 3.5533 3.623
12 3.1865 3.232
13 3.4652 3.5138
14 2.7235 2.783
15 2.353 2.509
16 2.36 2.81
17 2.47 2.98
18 2.945 3.27
19 2.692
20 2.7
21 2.8
22 2.1

Although not perfect the correspondence is quite remarkable. The curves are for the most part are indistinguishable. Although the numerical values are not identical the shapes of the curves are the same over most of their ranges.

Conclusion

Nucleons form a spin pair with the opposite type even when they are involved in a spin pair with a nucleon of their own type.

Although these pairings are exclusive they can still lead to chains of linked nucleons. For example, neutron A may be paired with proton B which is paired with proton C which is paired with neutron D and so on and on. These chains can form loops or rings made up of segments of two neutrons and two protons. This can result in a structural arrangement of pairings between the neutron shells of a nucleus and its proton shells. Such arrangements would be like alpha particles without literally being alpha particles. This would explain how nuclear substructures seeming to be alpha particles can exist despite alpha particles repelling each other through the strong force as well as the electrostatic force.


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