San José State University

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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: Separate Analysis
for Light and Heavy Nuclides

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 03 of 1 percent.

The scatter diagram of the errors in the regression estimates as a function of binding energy tended to be concentrated in the smaller nuclides. As shown below a reasonable cutoff point between the small and large nuclides is arount 500 MeV.

This roughly corresponds to 28 or 29 protons. So the small nuclei were consider to be the nuclides with 28 or less protons. The large nuclides were those with 29 or more protons.

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 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 for the Heavy Nuclides

The parameters of the regression equation for the heavy nuclides based upon the above are given in tabular form.

Variable Coefficient Standard Deviation
of Coefficient
P2P2 -0.554511276 0.003456101
N2P2 0.284641027 0.001719514
N2N2 -0.18700444 0.001091512
P1P2 0 0
P1N2 0.379136922 0.002806178
P1P1 -0.756113104 0.01036877
N1P2 -0.189664214 0.010530635
N1N2 -0.17453772 0.003049249
N1P1 0.533992614 0.00804507
N1N1 -0.263801809 0.006927963
1/#np 3869.151721 500.8777276
1/#pp 796.4115783 274.9730669
1/#nn -1297.023888 71.98268642
1/#a -1527.11867 319.5170397
#np 11.55945179 0.640926866
#pp 17.41460136 1.314134282
#nn 1.006097109 0.203449417
#a -15.98775864 1.352748774
C0 104.0545636 22.25377774

The regression package (EXCEL) did not compute a coefficient for P1P1 due to some problem with the data, such as that that variable is a linear combination of the other variables.

The coefficient of determination for this equation is 0.999964 and the standard error of the estimate is 2.304 MeV. The average binding energy of the nuclides included in the analysis is 1244.7 MeV and with a standard error of the estimate at a stunningly low 2.304 MeV this means that the coefficient of variation is less than 0,2 of 1 percent, 0.00185 to be exact. The t-ratios (ratios of the coefficients to their standard deviations) are all 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.

When the regression coefficients for the second shell are applied the results are:

CN2N2/CN2P2 = −0.65698
CN2P2/CP2P2 = −0.51332
CN2N2/CP2P2 = 0.33724

The value of −0.65698 is close of enough to −2/3 treat it as a confirmation of that value. If q=−2/3 then the second equation implies that (1+δ) is equal to 1.20232. If q=−2/3 then the third equation implies that (1+δ) is equal to 1.31788. This is a good but not remarkable correspondence.

The same analysis can be applied to the coefficients for the interactions in the first shell. Those results are;

CN1N1/CN1P1 = −0.70623
CN1P1/CP1P1 = −0.49402
CN1N1/CP1P1 = 0.34889

Again the value of −0.70623 is close of enough to −2/3 treat it as a confirmation of that value. If q=−2/3 then the second equation implies that (1+δ) is equal to 1.27386. If q=−2/3 then the third equation implies that (1+δ) is equal to 1.34948 This is also a good correspondence. It is notable that the results for the two shells are consistant.

The Results for the Light Nuclides

Due to limitations of the data the coefficients for several of the variables could not be computed. The coefficient of determination was only 0.99937; good but not in the same league with the result of 0.999964 for the heavy nuclides. The standard error of the estimate is 4.0 MeV.

Conclusions

The previous results indicate



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