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The Binding Energies of Nuclei
as Determined by the Energy of
Formation of their Substructures and
the Interactions Among their Nucleons

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. 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.

In addition to alpha modules and the three types of spin pairs a nucleus may contain a singleton neutron or a singleton proton. These latter are not substructures but their presence does contribute to the binding energy of the nucleus they are in.

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.

The notation which is used is

• #a = number of alpha modules
• #nn = number of neutron-neutron spin pairs outside of alpha modules
• #pp = number of proton-proton spin pairs outside of alpha modules
• #np = the presence of a neutron-proton spin pairs outside of alpha modules (0 or 1)
• N = total number of neutrons in nucleus
• P = total number of protons in nucleus
• sn = singleton neutron (0 or 1)
• sn = singleton proton (0 or 1)

The number of interactions of neutrons with each other is N(N-1)/2 and likewise P(P-1)/2 for proton interactions. The number of separate interactions of neutrons with 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

Thus the binding energy associated with the formation of #a alpha modules is presumed to be of the form

#### c#a −b/#a

A new variable 1/#a is created that is 0 if #a=0 and is the reciprocal of #a if #a>0.

Likewise two new variables 1/#nn and 1/#pp are created for #nn and #pp spin pairs. Since #np is either 0 or 1 a 1/#np variable would be identical to #np and hence cannot be used in the regression analysis.

The regression equation based upon the above formulations is:

#### BE = 39.89#a + 12.52#nn + 5.33#pp + 12.18#np −0.1837(N(N-1)/2) −0.5024(P(P-1)/2) + 0.2740NP +2.9494sn −2.0712sp −49.3040(1/#a) −24.6531(1/#nn) −12.7017(1/#pp)

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 0.999936. The t-ratios (ratio of coefficient to its standard deviation) are all extremely high except for the one for a singleton proton which is only 2.06. The t-ratio for the coefficient of #a is above 500.

## 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:

#### CN(N-1)/2/CNP = q²/q = q CNP/CP(P-1)/2 = q/(1+δ) CN(N-1)/2/CP(P-1)/2 = q²/(1+δ)

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

When the regression coefficients are applied the results are:

#### CN(N-1)/2/CNP = q = −0.6703 CNP/CP(P-1)/2 = q/(1+δ) = −0.5453 CN(N-1)/2/CP(P-1)/2 = q²/(1+δ) = 0.3655

From the first relationship above there is confirmation that the value of q is −2/3. If q=−2/3 then the second relation ship implies that δ is equal to +0.2225. If q=−2/3 then the third relationship above implies that δ is equal to +0.216. Rounded off to two places both estimates of δ are +0.22. This is a quite remarkable correspondence.

Furthermore the relationship of the regression coefficients for singleton neutrons and singleton protons should be

#### Csn/Csp = q/1 = q

The actual value of the ratio of those coefficients is −0.7022.

The coefficients for 1/#a, 1/#nn and 1/#pp are all negative as the model suggests.

The magnitude of the coefficients for the strong force interaction of two nucleon compared to the binding energy involved in the formation of spin pairs is important. For example, the binding energy in the formation of a neutron-neutron spin pair is on the order of 5 MeV. The strong force interaction of two neutrons is −0.1837 MeV, the negative sign indicating that it is a repulsion. It would thus take the strong force interaction of the neutron with about 25 other neutrons to match the attraction involved in the formation of a neutron-neutron spin pair. But if there are 50 other neutrons in the nucleus the strong force repulsion is more important than the attraction involved in a spin pair formation. The actual picture is much more complicated because there is the strong force attraction of a neutron for protons which counters the repulsion of the other neutrons.

These results indicate

• that the binding energies of nuclides can be overwhelmingly explained by the binding energies associated with their substructures and the interaction of their nucleons
• that definitely the strong force charge of a neutron is opposite in sign to that of a proton
• that definitely the strong force charge of a neutron is smaller in magnitude than that of a proton
• that very likely the value of the strong force charge of a neutron relative to that of a proton is −2/3