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

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.

Estimates of Interaction Binding Energies
Based Upon Increments with respect to
Proton Number of the incremental
Binding Energies of Neutrons
(MeV)
Neutron
Number 		Range of
Proton
Number		IE with
Unpaired
Proton		 IE with
Paired
Proton 		Average
IE
20 10 to 24 0.69935 1.687454 1.19340
30 16 to 28 0.56725 0.80219 0.68472
40 24 to 28 0.15000 0.90500 0.52750
40 29 to 39 0.09482 1.08322 0.58902
50 29 to 48 0.24157 0.71843 0.4800
60 37 to 56 0.0503 0.72630 0.3632
70 43 to 50 0.1825 0.61925 0.40088
70 51 to 62 0.02157 0.59393 0.30775
80 51 to 70 −0.03183 0.58283 0.2755
90 53 to 77 0.146308 0.47150 0.30890
100 60 to 81 0.11155 0.45209 0.28182
110 70 to 82 0.15617 0.54329 0.34973
120 77 to 82 0.12800 0.53933 0.33367
120 83 to 89 0.05550 0.30200 0.17875
130 83 to 92 0.08860 0.55120 0.31990
140 87 to 97 0.04828 0.44206 0.24517
150 93 to 103 0.07065 0.40722 0.23894
160 103 to 110 0.10000 0.45000 0.27500

The slope of the relationship between the interaction binding energy and the number of neutrons is equal to the average of the odd and even values. When the average interaction binding energy is plotted versus the neutron number the result is as follows.

It appears that there exist a relationship of the form

IE = α/n + β

If such a relationship exists then the relationship between the product of IE×n and n should be linear. Here is what that relationship looks like.

The regression of IE×n on n gives

IE×n = 15.79278 + 0.14806n
     [5,7]       [5.2]

The values in square brackets [z], are the t-ratios for the coefficients. They indicate that the coefficients are highly significant statistically.

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
(MeV)
p shell
4th 5th 6th 7th
n shell <=28 29 to 50 51 to 82 83 to 126
4th <=28 1.19
5th 29 to 50 0.68
0.53 		0.59
0.48
6th 51 to 82 0.36
0.40
 0.31
0.28
7th 83 to 126 0.31
0.28
0.35
0.33 		0.18
8th >=127 0.32
0.24
0.25
0.28

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., approximately the same.

(To be continued.)

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