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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

Dedicated to Betty
A true and precious friend

The purpose of this material is to develop 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. A previous study had good success using the total neutrons and total protons to determine the number of interactions. The errors in the estimates based on the regression equation found were associated with shell structure. This study uses an abreviated shell structure and determines the number of interactions between the nucleons in different shells as well as in the same shell.

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, but the presumption in those studies was that the binding energies due to the formation of substructures were constants independent of the scale of the nucleus in which they are formed. It has been found that this is not true. Below are shown the estimates of the binding energies due strictly to the formation of an alpha module.

The form of this relationship is approximately

e(x) = c + b/x²
and hence
∫e(x)dx = cx + ∫(b/x²) = cx − b/x

If the integration is carried out from 1 to x then the binding energy associated with the formation of #a alpha modules is of the form

c0 + k#a −b/#a

and likewise for the formation of #nn, #pp and #np spin pairs.

The regression equation based upon the above is:

BE = −11.2264 + 8.3481#a + 14.1161#nn + 14.2735#pp −29.0548 #np
+ 0.3745N1N1 −0.2319N1P1 −0.2774N1N2 − 0.6836N1P2
−0.3786P1P1 + 0 P1N2 −0.1967P1P2 + 0.2870N2N2 − 0.54068P2P2
+ 18.6838(1/#a) + 25.79166531(1/#nn) +7.4881(1/#pp) −0.1925(1/#np)

The regression package (EXCEL) did not compute a coefficient for P1N2 due to some problem with the data, such as that that variable is a linear combination of the other variables.

The data for the regression included all nuclides except the neutron, the proton and Beryllium 5, which has a negative binding energy. The coefficient of determination for this equation is 0.999945 and the standard error of the estimate is 3.75 MeV. The t-ratios (ratio of coefficient to its standard deviation) 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. (See Appendix I.) 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). (See Appendix II for the justification.) 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 = −0.6854
CN2P2/CP2P2 = −0.5307
CN2N2/CP2P2 = 0.3638

The value of −0.6854 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 1.233. If q=−2/3 then the third equation implies that (1+δ) is equal to 1.245. This is a good correspondence.

The statistical significance of the coefficient of #a is welcome; the coefficient of 1/#a is also statistically significantly different from zero at the 95 percent level of confidence. The total effect for the formation of alpha modules is given by

BE(#a) = 8.3481#a + 18.6838(1/#a)
and hence
(∂BE/∂#a) = 8.3481 − 18.6838(1/#a²)

Thus for large values of #a the effect of the formation of an alpha module approaches a level of 8.3481 MeV but for smaller levels of #a the effect is of a lesser magnitude.

The Nature of the Errors in the Regression
Equation Estimates of Binding Energy

The scatter diagram for the errors in the regression estimates plotted against the actual values shows that there is an association of the size of the error and the filling of the neutron and proton shells.

The Allowance of the Occupancy of Three Shells

In an attempt to improve the statistical performance of the model three shell categories were incorporated in the model. Shell one was from 0 to 28; shell two from 29 to 82 and shell three for 83 and above. This generated 21=6*7/2 interaction variables to be included with the 8 variables representing the substructure formation.. However the EXCEL regression program was not able to compute coefficients for 6 of the interaction variables and 2 of the substructure formation variable. As a result the coefficient of determination was less and the standeard error of the estimate was larger than those found for the previous version of the model. So it appears that the results of that previous version are best that can be achied; i.e., 0.999945 for the coefficient of determination and 3.75 MeV for the standard error of the estimate.


The previous results indicate

Appendix I

The force F between two particles of charges Q1 and Q2 separated by a distance S can be represented as

F = HQ1Q2f(S)/S²

where H is a constant and f(S) is a function characteristic of the type of force.

The potential energy V(S) is then

V(S) = −∫SF(s)ds
W= −HQ1Q2S(f(s)/s²)ds
= −HQ1Q2W(S)

where W(S)=∫S(f(s)/s²)ds.

The binding energy for the two particle is just the negative of the potential energy. Thus

B(S) = HQ1Q2W(S)

Thus the ratio of binding energies of two pairs of particles each having the same separation distance S is the ratio of the products of the charges in each pair.

Appendix II

The force between two protons separated by a distance S due to the strong force and the electrostatic force is given by

s F = Hf(S)/S² + Ke²/S²

where the strong force charge of the proton is taken to be unity and H is the corresponding strong force constant. The constant for the electrostatic force is K and the electrostatic charge of the proton is e. The binding energy for the protons is then

B(S) = HW(S) + Ke²/S
which can be expressed as
B(S) = HW(S)[1 + Ke²/(HW(S)S]
or as
B(S) = HW(S)[1 + δ]
δ = Ke²/(HW(S)S

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