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The Increment in Nuclear
Energy Due to the Formation of
a Neutron-Neutron Spin Pair

The nucleons in a nucleus are held together largely by their spin pairing. See Nucleus for more on this. The material below provides estimates of the binding energy due the formation of neutron-neutron spin pairs and its variation with the location within the nucleus where they are formed.

Consider first the incremental binding energies of the isotopes of Tin. Tin has the greatest number of stable isotopes of any element. Those incremental binding energies are shown below.

The peaks represent the formation of a spin pair. The sharp break after 82 neutrons represents the filling of a shell.

The binding energy due to the formation of a spin pair can be computed as the difference in the incremental binding energy at one point and the average of the value at the two adjacent points, as shown below.

This procedure is not valid at a point where there is a change in shell.

This procedure applied to the incremental binding energies of the isotopes gives the following results.

The values are more or less constant within the 51-82 shell, but definitely drop for the 83-126 shell.

In the 51-82 sheell the average increment due to neutron-neutron spin pair formation is 2.63 MeV with a standard deviation of 0.161 MeV. Thus the coefficient of variation is 6.1 percent.

To see how the binding energy due to neutron-neutron spin pair formation the increments are affected by the number of protons in the nucleus the values were calculated also for p=49 and p=51.

The Binding Energies Due to
Neutron-Neutron Spin Pair Formation
for the Isotopes of Indium (p=49),
Tin (p=50), Antimony (p=51).
Number
of neutrons
BEnn p=49
(MeV)
BEnn p=50
(MeV)
BEnn p=51
(MeV)
51 2.45
52 1.85 2.8
53 2.6555 2.675
54 2.8415 2.625 1.305
55 2.459 2.71 1.805
56 2.1155 2.795 2.13
57 2.0545 2.695 2.2
58 2.1235 2.569 2.19
59 2.143 2.6935 2.25
60 2.1345 2.83 2.289
61 2.168 2.8485 2.2985
62 2.12 2.8245 2.2965
63 2.0375 2.7995 2.301
64 1.967 2.6545 2.2835
65 1.967 2.3855 2.259
66 2.01 2.319 2.2385
67 2.1175 2.5015 2.2945
68 2.1925 2.6125 2.3245
69 2.2955 2.732 2.37495
70 2.3135 2.7789 2.3284
71 2.2535 2.7892 2.2966
72 2.213 2.7552 2.3281
73 2.2365 2.70465 2.37195
74 2.2595 2.648 2.3777
75 2.2845 2.6075 2.3462
76 2.2215 2.5475 2.299
77 2.07 2.48875 2.2655
78 1.845 2.4425 2.1825
79 1.545 2.436 2.052
80 1.43 2.405 1.8855
81 1.4 2.29 1.72
82 2.575 3.495 2.853
83 2.325 3.105 2.416
84 0.77 1.42 0.725
85 35 1.55 0.765
86 36 1.64 0.815
87 37 0.9

Here are the data plotted.

Clearly there is something special about the case of p=50.

Lead

The binding energy due to the formation of a neutron-neutron spin pair can only be estimated away from the filling of the neutron shell at n=126.

Clearly the binding energy due to the formation of a neutron-neutron spin pair is not independent of location. It varies within a shell as well as between shells.

(To be contined.)


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