San José State University
Thayer Watkins
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& Tornado Alley

Estimates of the Interaction Binding
Energies of Nucleons in Various Shells
Based upon Increments in the Incremental
Binding Energies of Protons

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

IE = α + βEv

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.

(Under construction)

Estimates of Interaction Binding Energies
Based Upon Increments with respect to
Neutron Number of the incremental
Binding Energies of Protons
Range of
IE with
IE with
20 16 to 28 0.52738 1.60467 1.06603
30 29 to 50 0.31030 1.06534 0.68782
40 41 to 50 0.129600.68136 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

IE = α/p + β

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

IE×p = 17.81801 + 0.03481p
     [9.8]       [1.2]

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 Testing of the Proposition

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
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
6th 51 to 82 0.29835 0.30428
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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