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The Energetics of Neutron-Proton
Spin Pair Formationas a Function of the
Number of Neutrons in the Nuclide

The binding energies of nuclides represents the loss of potential energy due to two types of processes. One is the formation of substructures such as spin pairs and the other is the interaction of these substructures. The binding energy data provides a special opportunity to estimate the effect of the formation of a neutron-proton spin pair. When the number of neutrons is less than the number of protons in a nuclide the addition of a neutron will involve the formation of such a spin pair, but once the number of neutrons is equal to the number of protons the addition of another neutron will not involve the formation of such a spin pair.

To see this illustrated consider the isotopes of Germanium (32 protons). The incremental binding energies of neutrons in these isotopes are displayed below in tabuler and graphic form.

The Incremental Binding Energies
of Neutrons in the Isotopes
of Germanium
27 16.7000
28 18.8100
29 13.9900
30 16.6300
31 12.7400
32 15.5800
33 10.0600
34 13.2800
35 9.1070
36 12.3950
37 8.1880
38 11.5376
39 7.4159
40 10.7520
41 6.7829
42 10.1962
43 6.5052
44 9.4283
45 6.0726
46 8.7193
47 5.7000
48 8.0280
49 4.9220
50 7.4000
51 3.4600
52 5.5000
53 3.0000
54 4.7000

There are sharp drops after 28 and 50 neutrons. These are the result of the filling of neutron shells. There is also a sharp drop after 32 neutrons. This is a result of there not being any further formation of neutron-proton pairs. The incremental binding energy for 32 neutrons contains the effects of the formation of both a neutron-neutron and a neutron-proton spin pair. The value for 33 neutrons contains neither of these two effects. The incremental binding energy for 32 neutrons also contains the interaction binding energy of the 32nd neutron with the other nucleons through the nuclear strong force. The incremental binding energy for the 33rd neutron is only the interaction between it and the other nucleons through the strong force; i.e., 10.06 MeV.

To get the binding energy due to the formation of a neutron-proton pair it is necessary to look at the change in binding energy due to the addition of a neutron pair from below and above 32 neutrons. The difference between the binding energy for 32 and 30 neutrons is 28.32 MeV. The difference between the binding energy for 34 and 32 neutrons is 23.34 MeV. The difference of these two figures is 4.98 MeV. This represents primarily the binding energy due to the formation of two neutron-proton pairs. Thus the binding energy per neutron-proton pair is 2.49 MeV.

The binding energy due to the formation of a neutron-proton pair can also be computed by comparing the difference between the binding energy for the change from 32 to 34 neutrons (23.34 MeV) with the change from 31 to 33 neutrons (25.64 MeV). The difference of these two figures is 2.3 MeV and this represents the binding energy due to the formation of one neutron-proton pair. This differs from the previous figure due, in part, to the difference in the interaction binding energy of the 34th neutron with the other nucleons as compared to that of the 33rd neutron. In the following the procedure involved in the 2.3 MeV figure will be used.

There is implicit in the term the formation of a spin pair the notion that this is independent of the context within which the spin pair is formed. What the data shows is that there is no constancy in such formations. Some of the variation in the computed values is explained in terms of an involvement with transitions between nuclear shells. Other variations may have to do with the formation of other substructures besides spin pairs, such as alpha modules. The residual variations most likely arise because the formation of a spin pair allows for a rearrangement of the other nucleons which results in a loss in potential energy.

The Estimates

Estimates of the Binding Energy
Due to the Formation of a
Neutron-Proton Spin Pair
Number
of Protons
Binding Energy
of Neutron-Proton
Spin Pair Formation
1 5.1264
2 18.7185
3 3.6318
4 12.0868
5 5.0659
6 10.5453
7 8.0643
8 7.6194
9 2.548
10 6.5004
11 4.1096
12 5.4390
13 3.6409
14 6.5705
15 3.3842
16 3.6254
17 2.9286
18 3.4154
19 4.2743
20 4.1606
21 1.8506
22 3.1100
23 2.7297
24 3.3339
25 2.5463
26 2.8045
27 3.3546
28 4.4222
29 2.3653
30 2.1060
31 2.5040
32 2.3000
33 2.5000
34 2.0200
35 2.7000
36 2.7500
37 2.2710
38 2.5320
39 2.9000
40 2.7000
41 2.5000
42 2.6000

The cases in which there is a transition between shells, those for N=2, 6, 8, 14, 20, 28 involve something different than neutron-proton spin pair formation. When those cases are removed the scatter diagram looks as follows.

Clearly there is not one fixed amount of binding energy associated with the formation of a neutron-proton spin pair. Perhaps when such a spin pair is formed there is a rearrangement of the nucleons in the rest of the nucleus which results in a loss of potential energy.


For the same analysis based on the incremental binding energies of neutrons see Protons.

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