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Energies of Nucleons in Various Shells Based upon Increments in the Incremental Binding Energies of Protons |
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A previous study established that, generally, the binding energy due the interaction of the n-th neutron with the p-th proton depends only upon the even-or-oddness of n and p and the shells that n and p are in. The interaction binding energies IE come from the cross differences in binding energies, the increments in the incremental binding energies.
The estimates of the interaction binding energies for paired and unpaired nucleons are obtained by regressing interaction binding energy on Ev, where Ev=1 if the nucleon number is even and Ev=0 otherwise. This gives an equation of the form
The interaction binding energy for an interaction with an unpaired nucleon (Ev=0) is equal to the intercept α and the interaction binding energy for an interaction with an paired nucleon (Ev=1) is equal to the sum α+β.
Estimates of Interaction Binding Energies Based Upon Increments with respect to Neutron Number of the incremental Binding Energies of Protons (MeV) |
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Proton Number | Range of Neutron Number | IE with Unpaired Neutron |
IE with Paired Neutron | Average IE |
20 | 16 to 28 | 0.52738 | 1.60467 | 1.06603 |
30 | 29 to 50 | 0.31030 | 1.06534 | 0.68782 |
40 | 41 to 50 | 0.12960 | 0.68136 | 0.40548 |
40 | 51 to 67 | 0.14337 | 0.79437 | 0.46886 |
50 | 52 to 82 | 0.14568 | 0.67593 | 0.41080 |
60 | 68 to 82 | −0.04943 | 0.64613 | 0.29835 |
60 | 83 to 100 | 0.19478 | 0.41378 | 0.30428 |
70 | 83 to 110 | 0.041571 | 0.47414 | 0.25786 |
80 | 97 to 126 | −0.09707 | 0.59267 | 0.24780 |
90 | 127 to 147 | 0.01840 | 0.52976 | 0.27408 |
100 | 144 to 158 | 0.07757 | 0.36463 | 0.22110 |
When the interactive binding energy is plotted versus the proton number the result is as follows:
It appears that there exist a relationship of the form
If such a relationship exists then the relationship between the product of IE×p and p should be linear. Here is what that relationship looks like.
The regression of IE×p on p gives
The values in square brackets [z], are the t-ratios for the coefficients. The coefficient for p is not significantly different from zero at the 95 percent level of confidence..
The proposition is that the interaction binding energy between a neutron and a proton depends only upon which nucleonic shells the neutron and proton are located in. It is tested by tabulating the estimated interaction binding energies in terms of the shell-to-shell cells, as shown below.
The Interaction Binding Energy Between a Neutron and a Proton Based Upon the Shells They are Located in (MeV) |
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n shell | |||||||
4th | 5th | 6th | 7th | 8th | |||
p shell | <=28 | 29 to 50 | 51 to 82 | 83 to 126 | >=127 | ||
4th | <=28 | 1.06603 | |||||
5th | 29-50 | 0.68782 0.40548 | 0.46886 0.41080 | ||||
6th | 51 to 82 | 0.29835 | 0.30428 0.25786 0.24780 | ||||
7th | 83 to 126 | 0.27408 | 0.2211 |
If the proposition were exactly correct the numbers for the alternate estimates of the various shell-to-shell interactions would be precisely the same. That would be too much to expect from empirical estimates. What is found is that the alternate estimates are of the same order of magnitude; i.e., approximate the same.
(To be continued.)
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