San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

The Energetics of Neutron-Proton
Spin Pair Formation as a Function of the
Number of Protons 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 through the strong force. 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 protons is less than the number of neutrons in a nuclide the addition of a proton will involve the formation of such a spin pair, but once the number of protons is equal to the number of neutrons the addition of another proton will not involve the formation of such a spin pair.

To see this illustrated consider the nuclides with 32 neutrons. The incremental binding energies of protons in these nuclides are displayed below in tabuler and graphic form. (Alll energy figures are in units of millions of electron volts -- MeV).

The Incremental Binding Energies
of Protons in the Nuclides
with 32 Neutrons
Number
of Protons
Incremetal
Binding Energy
(MeV)
17 19.5000
18 20.3000
19 16.2000
20 17.8000
21 12.7000
22 15.1000
23 10.6800
24 13.4260
25 9.4850
26 11.9534
27 7.3637
28 9.5335
29 4.8004
30 6.4770
31 2.8110
32 5.0200
33 -0.0500
34 1.9000
35 -1.6000

There are sharp drops after 20 and 28 neutrons. These are the result of the filling of neutron shells. There is also a sharp drop after 32 protons. This is a result of there not being any further formation of neutron-proton pairs. The incremental binding energy for 32 protons contains the effects of the formation of both a proton-proton and a neutron-proton spin pair. The value for 33 protons contains neither of these two effects. The incremental binding energy for 32 protons also contains the interaction binding energy of the 32nd proton with the other nucleons through the nuclear strong force. The incremental binding energy for the 33rd proton is only the interaction between it and the other nucleons through the strong force; i.e., −0.05 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 proton pair from below and above 32 protons. The difference between the binding energy for 32 and 30 protons is 5.02 MeV. The difference between the binding energy for 34 and 32 protons is 1.9 MeV. The difference of these two figures is 3.12 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 1.56 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 (1.9 MeV) with the change from 31 to 33 neutrons (−0.05 MeV). The difference of these two figures is 2.4 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 2.4 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 Neutrons
Binding Energy
of Neutron-Proton
Spin Pair Formation
1 5.3226
2 19.2199
3 4.4472
4 13.2489
5 5.9844
6 11.3290
7 8.0862
8 8.1935
9 3.4110
10 7.3421
11 4.8681
12 6.1751
13 4.2408
14 7.1849
15 4.0198
16 4.2002
17 3.4775
18 3.9574
19 4.6035
20 4.5603
21 2.4574
22 3.7636
23 3.6240
24 3.9470
25 3.5988
26 3.5274
27 3.7900
28 4.8871
29 2.8330
30 2.4827
31 3.0170
32 3.1200
33 2.9700
34 2.9000
35 3.2800
36 3.1000

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 Neutrons.

HOME PAGE OF applet-magic