/ The Binding Energies Due to the Interaction of Neutron Spin Pairs with Proton Spin Pairs
San José State University

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Thayer Watkins
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The Binding Energies Due to the Interaction
of Neutron Spin Pairs with Proton Spin Pairs

The Cross Differences Theorem

Elswhere it is shown that for systems made up of two or more types of particles the cross difference with respect to the numbers of two types of particles of the sum of the interactions of all particles is equal to the interaction of the last added particlel of one type with the last added particle of the other type.

Let B be the interaction energy of a system with n particles of one type and p particles of the other type. The first or incremental difference with respect to the number of particles of the first type is given by

ΔnB(n, p) = B(n, p) − B(n-1, p)

The cross difference is

Δ2npB(n, p) = ΔnB(n, p) − ΔnB(n, p-1)
which reduces to
= B(n, p) − B(n-1, p) − [B(n, p-1) − B(n-1, p-1)]
or, equivalently
Δ2nB(n, p) = B(n, p)+B(n-1, p-1) − [B(n-1, p)+B(n, p-1)]

The interaction energy is given by

B(n, p) = Σi=1nΣj=1pIij

where Iij is the interaction between the i-th particle of the first type and the j-th particle of the other type.

The Cross Difference Theorem says

Δ2nB(n, p) = Iij

This can be applied to the binding energies of nuclides. It can be applied using the numbers of neutrons and protons but those results are dominated by the binding energies associated with the formation of spin pairs. What is desired is the interactions due to the so-called nuclear strong force. (This is a poor choice< of terms since it is not all that strong when compared to the force associated with spin pair formations. A better choice is nucleonic force; i.e., the force between nucleons.) To obtain the interactions strictly due to the nucleonic force the analysis is carried out in terms of n, the number of neutron spin pairs, and p, the number of proton spin pairs. Binding energy is expressed in terms of millions of electron volts (MeV) and therefore that is also the units of the interactions.

The Results

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
2 3 4 5 6 7
2 28.602084 5.96299
3 6.33879 23.363538 8.343002
4 4.7408 9.44998 15.763326 5.942394
5 4.128492 6.719172 16.303695 5.616241 3.4464
6 6.6467 5.561351 12.554239 4.36271
7 7.1593 3.40679 4.35765 12.06715
8 5.34 3.294 5.18287 4.13587
9 5.39 4.7125 5.78817
10 2.85 3.9587
11 5.71 1.681
12 3.78

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
8 9 10 11 12 13 14
7 3.78411 4.105
8 9.01292 4.24534 3.419
9 4.26783 7.9362 3.7673 2.328
10 4.85608 3.75222 8.303 3.485 3.285
11 3.74906 4.12768 3.37556 8.7264 3.5053
12 4.646 3.40356 3.53534 3.6548 6.7673 3.023
13 3.52 3.458 3.8305 2.6958 3.0778 6.4149 2.693
14 3.3 3.802 3.6203 1.8592 2.2195 2.7607 6.7763
15 3.98 1.82 2.6808 3.4774 2.837 1.9706
16 2.4 3.109 3.33 2.0637 3.1878 2.6973
17 2.5 2.3 3.284 2.6166 3.0166
18 2.8 2.41 2.624 2.8619
19 3.2 1.821 2.7543
20 3.1 2.2
21 2.17

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
15 16 17 18 19 20 21 22 23 24 25
13 2.71
14 1.822 2.85
15 5.5662 2.588
16 2.7392 5.193
17 2.5574 2.3654 5.26
18 2.5435 2.4626 2.428 5.37
19 2.1751 2.4752 2.3744 2.08 5.72
20 2.0976 2.4703 2.2926 2.4945 1.953
21 2.542 2.2671 2.2025 2.1414 1.95 1.627
22 2.427 2.2065 1.9832 1.9593 1.971 1.643 1.41
23 2.47 2.3219 2.0845 1.9616 1.94 1.672 1.868 1.682
24 2.12 2.408 2.2487 2.0083 2.0349 1.9461 1.644 1.49 1.9
25 2.4 1.622 2.1771 2.4789 2.5627 1.7462 1.4931 1.569 1.581 1.57
26 2.57 1.774 1.835 1.5964 1.6638 1.9186 1.8992 1.624 1.709 1.02
27 1.775 1.938 1.662 1.8794 1.569 1.7716 1.775 1.16 1.72
28 1.318 2.142 2.2075 1.1461 1.6729 1.7055 1.889 1.289
29 1.515 1.74 1.723 2.2372 1.8066 1.5852 1.693 1.42
30 1.818 1.65 2.048 1.6197 1.5213 1.605 1.711
31 1.55 1.636 1.46 1.383 1.48 1.727
32 1.78 1.289 1.4 1.502 1.403 1.668
33 2.02 1.571 1.559 1.5 1.453 1.527
34 2.07 1.47 1.46 1.541 1.43
35 1.5 1.513 1.519 1.46
36 1.24 1.767 1.578
37 1.71 1.331 1.6925
38 1.48 1.6465
39 1.701
40 1.91
41 2.22

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
26 27 28 29 30 31 32 33
28 1.78
29 2.02 1.63
30 1.147 2.03 1.65
31 1.844 1.3 1.69
32 1.488 1.44 2.03 1.2
33 1.447 1.683 1.36 1.5
34 1.553 1.676 1.427 1.173 1.2
35 1.4571 1.5653 1.1086 1.277 1.3 1.1
36 1.3697 1.3039 1.2184 1.223 1.3 1.1
37 1.2563 1.1402 1.174 1.086 1.21 1.2 1.3
38 1.1395 1.0976 1.1474 1.119 1.09 1.08 1.1 1.4
39 1.0409 1.0385 1.1176 1.242 1.12 1.319 1.161 1.4
40 0.9495 0.9881 1.0934 1.139 1.361 1.099 1.54 0.83
41 0.808 1.1144 1.0746 1.139 0.969 1.497 1.199 0.979
42 2.01 1.663 1.318 1.447 1.343 1.231 1.152 1.311
43 1.63 1.37 1.675 1.346 1.279 1.164 1.151 1.314
44 1.32 1.465 1.35 1.18 1.232 1.231 1.327
45 1.24 1.231 1.359 1.589 1.435 1.287 1.135
46 1.38 1.63 1.899 1.226 1.137 1.053
47 1.67 1.29 1.379 1.248 1.111
48 1.5 1.07 1.218 1.407 1.256
49 1.2 1.32 1.161 1.448
50 1.4 1.126 1.134
51 1.2 1.202
52 1.3

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
34 35 36 37 38 39 40 41 42 43 44 45
39 1.1
40 1.25 1
41 1.07 1.03
42 1.015 1.14 1.01
43 1.332 1.029 1.08 0.91
44 1.374 1.137 0.874 1.01 0.79 0.79
45 0.894 1.114 1.131 0.785 0.97 0.97
46 0.885 0.92 0.905 1.099 0.686 0.686 0.94
47 1.022 0.86 0.87 0.926 1.025 1.025 0.649
48 1.099 0.986 0.924 0.91 0.829 0.829 0.994
49 1.195 1.004 0.922 0.75 1.1 1.1 0.797 1.043
50 1.447 1.131 0.982 1.08 0.76 0.76 1.03 0.82
51 1.26 1.373 0.968 1.037 0.99 0.99 0.85 0.98 0.85
52 1.418 1.314 1.042 1.026 1.17 1.17 0.89 0.85 0.93
53 1.439 1.147 1.318 1.343 0.953 0.953 1.12 1 0.78
54 1.747 1.14 1.259 1.113 1.113 0.711 1.34 0.91 0.65
55 1.545 1.189 1.286 1.286 1.29 1.162 1.06 0.76
56 1.369 1.333 1.333 1.364 1.175 0.99 0.92 0.7
57 1.291 1.412 1.412 1.4 1.182 1.025 0.84 0.87
58 1.54 1.54 1.31 1.113 1.081 0.82 0.79
59 1.331 1.243 1.025 0.805 0.77
60 1.396 1.29 1.144 0.679 0.754 0.74
61 1.146 1.334 0.677 0.636 0.759
62 1.332 0.61 0.668 0.657
63 1.711 0.521 0.575 0.654
64 0.886 1.437 1.246 1.145
65 1.287 1.324 1.217
66 1.109 1.296 1.34
67 1.176 1.351
68 1.253
69 1.184
70 1.069
71 1.131

The Binding Energies Due the Interaction
of Neutron Spin Pairs and Proton Spin Pairs
(MeV)
Number of Proton Spin Pairs
Number of
Neutron
Spin Pairs
46 47 48 49 50 51 52 53 54
64 0.932
65 1.024
66 1.122
67 1.17
68 1.308 1.028
69 1.192 0.957
70 1.0957 1.019 0.877
71 1.0474 0.9845 1.052 0.68
72 0.8649 1.0344 0.9391 1.024 0.76
73 0.8486 0.9004 0.879 0.944 0.987
74 0.8136 0.943 1.0321 0.8399
75 0.8181 0.9217 1.0171 0.9771
76 0.8162 0.8408 1.189 1.851
77 0.735 0.775 1.854 1.01
78 0.778 0.724 1.92 0.8
79 0.818 1.653 0.94 0.78
80 1 0.8 0.4

A blank element does not mean that its interaction is zero; it means the cross difference could not be computed.

(To be continued.)


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