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The Binding Energies Due to the Interaction of
Neutron Spin Pairs with Other Neutron Spin Pairs
bi

Second Differences

There is a theorem which says that the interaction energy of the last particle of a particular type to be added to a nucleus with the next-to-last particle of that type is closely related to the second difference of the interaction energy of nuclides with respect to the number of particles of that type. Specifically it says that the second difference is an upper bound to the interaction energy of the last two particles to be added to the nucleus. Since the second difference of binding energies is almost always negative the theorem means that the binding energy of the interaction of the last two particles of a particular type is at least as negative as the second difference. This is important because the negativity of the interaction of two particles is evidence that the force between them is a repulsion.

The analysis is limited to nuclides that consist entirely of neutron-neutron pairs and proton-proton pairs. This is to eliminate the distracting influence of the formation of those two types of spin pairs. However the formation of neutron-proton spin pairs still affect the results. Here are the second differences arranged as much as possible by nucleonic shells.

The Second Difference with respect to the Number of Neutron Spin
Pairs of Binding Energies in Nuclides Made Up Entirely of Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number
of Neutron
Spin Pairs
1 2 3 4 5 6 7 8 9 10
3 1.165474 -21.09782 -3.697272
4 -3.2069 -4.80489 -18.718448 -11.298124
5 -2.3328 -7.654288 -16.698442 -6.337141
6 -0.551492 -0.623964 -11.366308 -4.42831 -3.512
7 -1.417 -0.9044 -3.058961 -11.25555 -3.55111 -3.735
8 -2.37 -4.1893 -4.30209 -3.47687 -11.40815 -6.17934 -6.039
9 -6.57 -4.474 -4.94437 -3.29207 -8.03716 -4.3463 -3.998
10 -0.08 -1.9425 -3.77197 -3.18372 -7.3677 -2.832
11 -4.01 -1.15 -3.4277 -4.53472 -4.15926 -9.0867
12 -3.21 -1.111 -0.21406 -0.93818 -0.7784
13 -0.48 -1.606 -1.55156 -1.2564
14 -0.5 -0.72 -0.376 -0.5862
15 -4.1 -3.922 -5.7223
16 -2.1 -3.68 -2.391
17 -1.3 -1.909
18 -1.4

Let n and p denote the numbers of neutron spin pairs and proton spin pairs, respectively. The large magnitude of the second difference for n=3 and p=2 is due to what may be called the n=p effect. When n<p the addition of another neutron pair results in the formation of two neutron-proton pairs, and more importantly, an alpha module. An alpha module is a chain in which a neutron is linked to a proton which is linked to another proton which is linked to a neutron. The simplest alpha module chain is an alpha particle. A deuteron, a neutron-proton spin pair, has a binding energy of 2.2 MeV. When two deuterons are combined into an alpha particle the binding energy jumps to 28.3 MeV. When n>p the addition of another neutron spin pair does not result in the formation of an alpha module and the incremental binding energy drops accordingly.

Note that all of the values except the first (n=3, p=1) are negative.

The Second Difference with respect to the Number of Neutron Spin
Pairs of Binding Energies in Nuclides Made Up Entirely of Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number
of Neutron
Spin Pairs
11 12 13 14 15 16 17 18 19 20
10 -1.675
11 -3.8453 -3.625
12 -5.85 -2.588 -2.48
13 -2.2154 -5.9049 -2.513 -2.27
14 -1.4228 -2.2811 -5.9353 -1.852 -2.74
15 -4.9007 -3.6428 -3.5665 -8.3722 -4.628 -4.89
16 -1.7418 -3.1555 -2.8047 -2.078 -4.905 -2.3
17 -2.939 -1.7187 -2.2899 -1.9706 -2.1524 -4.98
18 -0.9 -1.774 -1.7666 -1.9213 -1.9352 -1.838 -4.67
19 -1.3 -0.51 -1.313 -1.4206 -1.789 -1.7764 -1.83 -5.12
20 -2.2 -0.921 -1.4753 -1.5528 -1.5577 -1.6395 -1.225 -4.992
21 -1.4 -1.43 -0.9856 -1.1888 -1.2789 -1.632 -1.635
22 -0.87 -0.985 -1.0456 -1.2649 -1.447 -1.426 -1.41
23 -1.3 -1.257 -1.1416 -1.0403 -1.038 -1.069 -1.04
24 -0.8 -1.15 -1.0639 -0.8997 -0.853 -0.7581 -0.484
25 -0.9 -0.62 -1.406 -1.4776 -1.007 -0.4792 -0.6791
26 -4.31 -3.362 -3.7651 -4.409 -5.3753 -5.4577
27 -1.26 -1.259 -1.156 -1.0904 -0.8748
28 -0.825 -1.445 -0.965 -0.6369
29 -1.65 -1.453 -1.855 -2.3395
30 -0.515 -0.437 -0.51
31 -0.088 -0.188
32 -0.73 -0.5
33 -1.52 -1.28
34 -1.2

The Second Difference with respect to the Number of Neutron Spin
Pairs of Binding Energies in Nuclides Made Up Entirely of Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number
of Neutron
Spin Pairs
21 22 23 24 25 26 27 28 29 30 31 32
23 -0.582
24 -0.708
25 -0.83 -0.751 -1.07
26 -5.0322 -4.702 -4.659 -4.52
27 -1.2244 -1.352 -1.201 -1.75 -1.05
28 -1.0598 -1.1585 -1.228 -0.499 -0.93
29 -1.2484 -1.1147 -1.235 -1.431 -1.3 -1.06
30 -0.6992 -0.8861 -0.95 -1.038 -0.747 -1.62 -1.22
31 -0.6 -0.7597 -0.898 -1.023 -1.007 -0.31 -1.04 -1
32 -0.847 -0.907 -0.788 -0.865 -0.924 -1.28 -1.14 -0.8
33 -0.998 -0.839 -0.841 -0.791 -0.932 -0.973 -0.73 -1.4 -1.1
34 -0.701 -0.79 -0.83 -0.742 -0.839 -0.733 -0.74 -0.673 -1
35 -0.77 -0.74 -0.687 -0.709 -0.679 -0.7749 -0.8856 -1.204 -1.1 -1
36 -0.7 -0.973 -0.725 -0.607 -0.6944 -0.9558 -0.846 -0.9 -0.9 -0.9
37 -0.7 -0.23 -0.666 -0.5515 -0.6649 -0.8286 -0.873 -1.01 -1.1 -1
38 -0.61 -0.461 -0.507 -0.6238 -0.6664 -0.693 -0.66 -0.78 -0.9 -1.1
39 -0.52 -0.4655 -0.5641 -0.6232 -0.653 -0.53 -0.5 -0.261 -0.2
40 -0.62 -0.411 -0.5024 -0.5528 -0.577 -0.68 -0.439 -0.659 -0.28
41 -0.84 -0.53 -0.6715 -0.5452 -0.564 -0.564 -0.956 -0.558 -0.899
42 -6.36 -5.158 -4.6094 -4.366 -4.058 -3.684 -3.95 -3.997
43 -0.14 -0.52 -0.813 -0.456 -0.557 -0.621 -0.688 -0.689
44 -0.34 -0.39 -0.6 -0.596 -0.695 -0.627 -0.547
45 -0.31 -0.39 -0.624 -0.615 -0.206 -0.003 0.053
46 -0.54 -0.391 -0.12 0.19 -0.019 -0.169
47 -0.37 -0.33 -0.939 -0.786 -0.675
48 -0.51 -0.68 -0.9 -1.061 -0.902
49 -0.9 -0.77 -0.668 -0.914
50 -0.9 -0.82 -0.855
51 -0.9 -0.826
52 -1

The Second Difference with respect to the Number of Neutron Spin
Pairs of Binding Energies in Nuclides Made Up Entirely of Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number
of Neutron
Spin Pairs
33 34 35 36 37 38 39 40 41 42 43 44
39 -0.2
40 -0.85 -0.7
41 -0.75 -0.93 -0.9
42 -3.665 -3.72 -3.61
43 -0.686 -0.369 -0.48 -0.41
44 -0.534 -0.492 -0.384 -0.59 -0.49
45 -0.139 -0.619 -0.642 -0.385 -0.61 -0.43
46 -0.251 -0.26 -0.454 -0.68 -0.366 -0.65
47 -0.617 -0.48 -0.54 -0.575 -0.748 -0.409 -0.7
48 -0.757 -0.68 -0.554 -0.5 -0.516 -0.712 -0.367
49 -0.722 -0.626 -0.608 -0.61 -0.77 -0.499 -0.696
50 -1.169 -0.917 -0.79 -0.73 -0.4 -0.74 -0.507 -0.73
51 -0.758 -0.945 -0.703 -0.717 -0.76 -0.53 -0.71 -0.55
52 -0.902 -0.744 -0.803 -0.729 -0.74 -0.56 -0.52 -0.65
53 -1 -0.979 -1.146 -0.87 -0.553 -0.77 -0.54 -0.39 -0.54
54 -0.939 -0.339 -0.517 -0.601 -0.441 -0.85 -0.51 -0.38
55 -1.479 -1.074 -1.144 -0.971 -0.392 -0.57 -0.42 -0.31
56 -0.834 -0.654 -0.607 -0.533 -0.52 -0.59 -0.43
57 -0.569 -0.647 -0.568 -0.532 -0.525 -0.49 -0.57 -0.4
58 -0.525 -0.397 -0.487 -0.556 -0.5 -0.52 -0.6
59 -0.62 -0.599 -0.469 -0.525 -0.54 -0.56
60 -0.689 -0.624 -0.577 -0.458 -0.584 -0.6 -0.64
61 -0.565 -0.709 -0.519 -0.521 -0.639 -0.62
62 -0.698 -0.496 -0.498 -0.565 -0.533 -0.635
63 -1.093 -0.714 -0.803 -0.896 -0.899
64 -4.159 -4.984 -4.068 -3.397 -2.906
65 -0.166 -0.316 -0.238 -0.166
66 -0.182 -0.36 -0.388 -0.265
67 -0.317 -0.437 -0.426
68 -0.361 -0.459
69 -0.266 -0.335
70 -0.315 -0.43
71 -0.4 -0.338
72 -0.541
73 -0.19

The Second Difference with respect to the Number of Neutron Spin
Pairs of Binding Energies in Nuclides Made Up Entirely of Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number
of Neutron
Spin Pairs
45 46 47 48 49 50 51 52 53 54
62 -1.84
63 -1
64 -2.432
65 -0.232 -0.14
66 -0.238 -0.14
67 -0.265 -0.217
68 -0.396 -0.258
69 -0.382 -0.498 -0.569
70 -0.5167 -0.613 -0.551
71 -0.4922 -0.5405 -0.575 -0.4
72 -0.5785 -0.761 -0.7111 -0.824 -0.48
73 -0.5466 -0.5629 -0.6969 -0.757 -0.837 -0.61
74 -0.331 -0.5422 -0.629 -0.565 -0.4769 -0.624
75 -0.4994 -0.4949 -0.5162 -0.5312 -0.394
76 -0.5031 -0.6086 -0.7849 -0.573
77 -0.8232 -0.929 -1.343 -1.34
78 -0.488 -0.445 -0.496 -0.43 -0.64
79 -0.447 -0.353 -0.62 -0.48
80 -0.46 -0.4 -0.38 -0.5

Here are two illustrations of the relationships in the above data:

The negative values reflect the repulsion of one neutron spin pair for another. The sharp downward spike at 26 neutron spin p.airs is due to the filling of a shell at 25 pairs. Twenty five pairs corresponds to 50 nucleons, a magic number. There is apparently a plateaus of values in the shell beyond 25; i.e., roughly constant second differences. This means that the interaction of one neutron spin pair with the previous one is roughly constant at a value of 0.905 Mev.

In this case also the high negative value after 41 neutron spin pairs if reflective of a change in shell. Forty one spin pairs corresponds to 82 neutrons, a magic number. The average value in the level from 48 to 52 neutron spin pairs is −0.8994 MeV. This is very close to the value of −0.905 found in the previous case. The two values differ by only about 2/3 of 1 percent.

(To be continued.)


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