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Binding Energies of Two Nucleons Depends Only Upon the Shells the Two Belong To |
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There is theoretical justification that the second differences in binding energy, the increments in the incremental binding energies of nucleons, measure the binding energies due to the interactions of the last two nucleons to be added to a nuclide. A previous study developed a procedure for testing whether the interaction binding energies are constant over the ranges of nucleonic shells. That procedure is to regress the second difference in binding energy on the number of nucleons of a type and the evenness of that number. If the t-ratio of the coefficient of the number of nucleons used in the regression is less than 2 in magnitude then the influence of the number of nucleons in the shell is not significantly different from zero at the 95 percent level of confidence. The t-ratio for a regression coefficient is the ratio of the coefficient's value to the standard deviation of the coefficient's estimate.
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The Result of the Regression of Neutron-Proton Interaction Binding Energy on the Number of Protons |
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| Neutron Number N | Range of Proton Number P | t-Ratio of Coeff of P |
| 10 | 11 to 14 | -1.49 |
| 20 | 10 to 14 | -1.33 |
| 20 | 15 to 24 | -0.410 |
| 30 | 16 to 33 | 0.944 |
| 40 | 24 to 28 | -0.290 |
| 40 | 29 to 39 | -0.190 |
| 50 | 29 to 48 | -1.74 |
| 60 | 37 to 56 | 0.015 |
| 70 | 43 to 50 | -0.96 |
| 70 | 51 to 62 | -1.04 |
| 80 | 51 to70 | 0.18 |
| 90 | 53 to 77 | -1.26 |
| 100 | 60 to 81 | -0.605 |
| 110 | 70 to 82 | -0105 |
| 120 | 77 to 82 | -0.690 |
| 120 | 83 to 89 | -1.04 |
| 130 | 83 to 92 | -1.32 |
| 140 | 87 to 97 | -0.076 |
| 150 | 93 to 103 | -1.08 |
| 160 | 103 to 110 | 1.44 |
Typically the graphs of data look like the following:
However there is one case, not shown above, in which there is a definite relationship between the interaction energy and the number of protons.
The extraordinarily high value for p=10 can be ignored because it is due to the proton number being equal to the neutron number, but the values for p=5 through p=9 show a definite, more or less precise, dependence of the interactive binding energy on the proton number. The t-ratio for the coefficient of p is about 33.
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The Result os the Regression of Proton-Neutron Interaction Binding Energy on the Number of Neutrons |
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| Proton Number N | Range of Neutron Number N | t-Ratio of Coeff of N |
| 10 | 11 to 20 | 1.425 |
| 20 | 16 to 28 | 0.041 |
| 20 | 29 to 36 | 0.054 |
| 30 | 29 to 50 | 0.543 |
| 40 | 41 to 50 | -1.59 |
| 40 | 51 to 67 | -0.608- |
| 50 | 52 to 82 | 0.811 |
| 60 | 68 to 82 | -1.15 |
| 60 | 83 to 100 | -2.85 |
| 70 | 83 to 110 | -0.319 |
| 80 | 97 to 126 | -2.415 |
| 90 | 122 to 126 | -0.165 |
| 90 | 127 to 147 | -0.640 |
Typically the graphs of data look like the following:
There are two cases for which the t-ratio is greater than 2. Here are the graphs forthem.
In these cases it is not so much as definite dependences of interactive binding energy on neutron numbers as unusally small standard deveiations of the estimates of the coefficients.
Of the cases selected only one showed a definite dependence of interactive binding energy on the number of nucleons. For all the others the interactive binding energy of two nucleons is a function just of the nucleonic shells in which they are located.
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