San José State University
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An Extensive Demonstration that the Interaction
Binding Energies of Two Nucleons Depends Only
Upon the Shells the Two Belong To


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.

Below are the results of applying this procedure to data arising from incrementing by proton to the incremental binding energies of neutrons.

The Result of the Regression of Neutron-Proton
Interaction Binding Energy on the Number of Protons
Number N
Range of
Number P
t-Ratio of
Coeff of P
1011 to 14-1.49
2010 to 14-1.33
2015 to 24-0.410
3016 to 330.944
4024 to 28-0.290
4029 to 39-0.190
5029 to 48-1.74
6037 to 560.015
7043 to 50-0.96
7051 to 62-1.04
8051 to700.18
9053 to 77-1.26
10060 to 81-0.605
11070 to 82-0105
12077 to 82-0.690
12083 to 89-1.04
13083 to 92-1.32
14087 to 97-0.076
15093 to 103-1.08
160103 to 1101.44

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.

Data arising from incrementing by neutron number
the incremental binding energies of protons

The Result os the Regression of Proton-Neutron
Interaction Binding Energy on the Number of Neutrons
Number N
Range of
Number N
t-Ratio of
Coeff of N
1011 to 201.425
2016 to 280.041
2029 to 360.054
3029 to 500.543
4041 to 50-1.59
4051 to 67-0.608-
5052 to 820.811
6068 to 82-1.15
6083 to 100-2.85
7083 to 110-0.319
8097 to 126-2.415
90122 to 126-0.165
90127 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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