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The Binding Energies of Nuclei as Determined by the Energy of Formation of their Substructures and the Interactions Among the Nucleons in their Shells:The Four Shells Case for Small Nuclides 

A previous study developed an equation explaining the binding energies of about three thousand nuclides in terms of the nucleonic substructures they contain and the interaction of their nucleons through the nuclear strong force. That previous study used an abreviated shell structure and determined the number of interactions between the nucleons in different shells as well as in the same shell.
That study had good success in explaining the binding energies of nuclides. But the model fits the data for smaller nuclides less well than it does for the larger nuclides. This study limits the analysis to the nuclides in which the numbers of proton and neutrons is 28 or less. The number 28 represents filled shells. By limiting the study to these smaller nuclides a more detailed shell structure can be used. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126} but elsewhere the argument has been made for magic numbers of {2, 6, 14, 28, 50, 82, 126} with 8 and 20 representing the filling of subshells. Both sets will be used in the analysis and the results compared..
One type of substructure in nuclei is a spin pair of nucleons. There are neutronneutron spin pairs, protonproton spin pairs and neutronproton spin pairs. There is an exclusiveness in the formation of spin pairs in that a neutron can form a spin pair with only one other neutron and one proton and likewise for a proton. This means that there are linkages of neutrons and protons of the form nppn, or equivalently, pnnp. These linkages induce binding energy effects similar to alpha particles. In the following analysis these linkages are called alpha modules. Below is shown a depiction of simple chain of four alpha modules.
Binding energy is also determined by the interactions of the various substructures but the analysis below presumes that the interactions of the substructures reduce down to interactions among neutrons and protons contained in those substructures.
The notation which is used is
The variables Ni and Pj were not used in the regression. Instead the number of interactions of the following forms were used.
The number of interactions of Q nucleons with each other is Q(Q1)/2. The number of separate interactions of N neutrons with P protons is NP. The binding energy BE is then assumed to be a linear function of the numbers of substructures and the numbers of interactions.
This scheme was used in previous studies and the results were good. The details of the analysis are given there.
The coefficient of determination for the regression equation is 0.999866 and the standard error of the estimate is 1.86 MeV. The average binding energy of the nuclides included in the analysis is 313.4 MeV and with a standard error of the estimate of a low 1.86 MeV this means that the coefficient of variation 0.6 of 1 percent. The tratios (ratios of the coefficients to their standard deviations) are all except one significantly different from zero at the 95 percent level of confidence.
If the strong force charge of a proton is taken to be 1.0 and that of a neutron denoted as q, where q may be a negative number, then the regression coefficients should be related to q. However the interaction of protons is modified by the effect of the electrostatic repulsion between protons. Let the effective charge of a proton in protonproton interactions be dentoted as (1+δ). The parameter δ is positive if the interaction of protons through the strong force is of the same nature as through the electrostatic force;i.e., repulsion. The relationships are for coefficient for interactions in the same shell:
Previous studies have concluded that q is equal to −2/3.
The regression results provide the information to estimate q for the three shells of nucleons.
Ratios of Regression Coefficients for Interactions of Nucleons in the Same Shell 


Shell  C_{NN}/C_{NP} 
0 to 8  −0.56288 
9 to 20  −0.86318 
21 to 28  −0.71936 
The average of the estimates of q for the three shells is −0.715. The results for the whole set of nuclides and for the larger nuclides are consistent with q being equal to −2/3. The results here are not wildly inconsistent with q being −2/3.
The regression equation was used to estimate the binding energies. The deviations between the actual and the estimated were calculated and a scatter diagram of the deviations plotted versus the actual. This scatter diagram is shown below.
There is an indication of a pattern that could be due to a discrepancy between the actual shell structure and the one used in the analysis.
When the shell occupancies are defined as
The coefficient of determination for the regression equation is 0.99984885 and the standard error of the estimate is 1.975 MeV. The average binding energy of the nuclides included in the analysis is 313.4 MeV and with a standard error of the estimate of a 1.975 MeV this means that the coefficient of variation 0.63 of 1 percent. The tratios (ratios of the coefficients to their standard deviations) are all except two significantly different from zero at the 95 percent level of confidence.
This regression equation was used to estimate the binding energies. The deviations between the actual and the estimated were calculated and a scatter diagram of the deviations plotted versus the actual. This scatter diagram is shown below.
There is an even greater indication of a pattern that could be due to a discrepancy between the actual shell structure and the one used in the analysis.
The results for the alternate shell structure are essentially the same as for the conventional shell structure, with the conventional shell structure giving slightly better statistical performance.
The previous results indicate
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