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San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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of Neutron Spin Pairs with Proton Spin Pairs |
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
The cross difference is
The interaction energy is given by
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
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 Binding Energies Due the Interaction of Neutron Spin Pairs and Proton Spin Pairs (MeV) |
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---|---|---|---|---|---|---|---|
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) |
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---|---|---|---|---|---|---|---|---|---|---|---|
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) |
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---|---|---|---|---|---|---|---|---|
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) |
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---|---|---|---|---|---|---|---|---|---|---|---|---|
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) |
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---|---|---|---|---|---|---|---|---|---|
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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