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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

Dedicated to Betty
A true and precious friend

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 determines 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. The coefficient of determination (R²) was 0.999958 and the standard error of the estimate was 3.27 million electron volts (MeV). Since the average binding energy of the nuclides is 1073.4 MeV, the coefficient of variation (ratio of the standard error of the estimate to the average value of the variable) is 0.00304, or about 0.3 of 1 percent. That study was improved upon by considering only the larger nuclides, those with atomic number 29 or greater. This study is an attempt to improve the statistical performance of the model by considering more shells.

Substructures of a Nucleus

One type of substructure in nuclei is a spin pair of nucleons. There are neutron-neutron spin pairs, proton-proton spin pairs and neutron-proton 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 -n-p-p-n-, or equivalently, -p-n-n-p-. 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(Q-1)/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 homogeneous 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 Results

The coefficient of determination for this equation is 0.9999755 and the standard error of the estimate is 2.5 MeV. The average binding energy of the nuclides included in the analysis is 1072.6 MeV and with a standard error of the estimate of a low 2.5 MeV this means that the coefficient of variation 0.23 of 1 percent. The t-ratios (ratios of the coefficients to their standard deviations) are all except one significantly different from zero at the 99.9 percent level of confidence.

Implications of the Results

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 proton-proton 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:

CNN/CNP = q²/q = q
CNP/CPP = q/(1+δ)
CNN/CPP = q²/(1+δ)

Previous studies have concluded that q is equal to −2/3.

The regression results provide the information to estimate q and (1+δ) for the four shells of nucleons.

Ratios of Regression Coefficients for Interactions of Nucleons in the Same Shell
0 to 28−0.70944−0.42781 0.30351
29 to 50−0.68961−0.44585 0.30746
51 to 82−0.65887−0.61925 0.40800
83 and more−0.67630−0.57821 0.39104

These were then converted into estimates.

Implications of the Ratios of Regression Coefficients
Shellq (1+δ)(1+δ)
0 to 28−0.709441.55832 1.46437
29 to 50−0.6896 1.495281.44554
51 to 82−0.658871.07657 1.08932
83 and more−0.676301.15299 1.13656

The results are consistent with q being equal to −2/3. The values of (1+δ) would reasonably be different for the different shells. The correspondence of the two estimates of (1+δ) in each shell is notable. The higher values of (1+δ) for the lower shells is a surprise.

More Shells

There is a fifth neutron shell for nuclides containing more than 126 neutrons. There are no nuclides containing as many as 126 protons. When an N5 variable was included in the analysis the number of variables for the regression exceeded the limit of 48 by 2. Therefore to stay within this limit the interactions between neutrons in the fifth shells and neutrons and protons in their first shells were left out of the regression. The resulting regression equation had a coefficient of determination of 0.9999824 and the standard error of the estimate is 2.1 MeV. The coefficient of variation is therefore slightly less than 0.2 of 1 percent.

If the analysis is limited to nuclides with P>28 the coefficient of determination is reduced ever so slightly but the standard error of the estimate is reduced to 1.63 MeV for a coefficient of variation of 0.13 of 1 percent.


The previous results indicate

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