San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Binding Energies of Nuclei
as Determined by the Energy of
Formation of their Substructures
and the Interactions Among the
Nucleons in their Shells with the
Proton-Proton Interactions
Treated Separately

The force between two particles of strong force charges q1 and q2 is of the form


where F(x) is a positive valued function of the time averaged separation distance s between them.

This implies that the binding energy associated with the strong force interaction of the particles is of the form


where G(s) is a positive-valued function of the time averaged separation distince s.

Let the strong force charge of a proton be designated as unity and that of the neutron as q. The binding energies of neutron-neutron, neutron-proton and proton-proton interaction should be in the proportions q²: q: 1. In proton-proton interactions there is the additional effect of the electrostatic force. This effect can be represented as δ times the strong force interaction so in proton-proton interaction the effective charge of the proton is (1+δ). However the value of δ may depend upon the distance separating the interacting nucleons. Thus the proton-proton interactions must be treated separately from the other interactions.

It has been found that the effect of the interactions are the same for all neutrons in the same shell and likewise for the protons.

In previous studies the measured effects for nucleons in the same shell were found to be in the proportions cited above with q equal to −2/3. But this is not the case for interactions between nucleons in different shells.

If the evidence of previous studies is taken as confirmation of the hypothesis of strong force charges then all such interactions should adhere to the cited proportions. This can be achieved by computing for the interactions between two shells, say the i-th and j-th shells, the quantities

SiSj = q²NiNj + qNiPj + q PiNj
and including PiPj
as a separate variable.

where NiNj is the number of interactions between neutrons in the i-th shell and the neutrons in the j-th shell and likewise for the other interactions.

These variables were combined in a regression with other variables representing the effect of the formation of substructures. The computations were carried out in a way that the value of q could be adjusted. The intial value used for q was −2/3.

That regression had a coefficient of determination (R²) of 0.99984 and a standard error of the estimate of 6.34 million electron volts (MeV). This is not too bad but it is not comparable to the value of R² of 0.9999824 and 1.86 MeV for the standard error of the estimate found in a previous study. The analysis indicates that the ratio of the coefficient for PiPj to the one for SiSj should be equal to (1+δ)². In only a few cases were both of these coefficients provided by the regression. In some cases that ratio is greater than unity but in other cases it is less than unity. For example, the interactions of the the nucleons in the shell with 83 to 126 of neutrons and of protons the value of (1+δ)² is 1.211. On the other hand for the interactions of the nucleons in the shell with 29 to 50 the ratio is 0.7838.

It would be expected that the greater the distance between the shells involved in the interaction the greater would be the value of δ. Only in the case of interactions between the nucleons in the shell with 0 to 28 with the first three shells are values of the ratio available to test that hypothesis. The ratios are shown in the graph below.

There is no reason to expect the ratio should be linear in the shell numbers but it certainly is. The other way of interpreting the display is that the middle value is the average of the upper and lower values.

The regression equation was used to compute estimates of the binding energy of each nuclide and the difference between the actual values and the regression estimates taken. The scatter diagram of those differences and the actual value is shown below.

There is some periodicity displayed in this diagram but the most notable aspect is the extreme errors for the nuclides with binding energies near 1000. This binding energy corresponds to numbers of neutrons and protons near 50, the filling point of shells.

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