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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
(Revised Version)

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

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

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

The regression equation based upon the above is:

BE = 0.8468#a + 14.1069#nn + 5.3581#pp + 9.3442#np
−0.1903NN −0..4857PP + 0.2738NP +3.4021sn −0.6483sp
−27.4377(1/#a) +7.4881(1/#nn) −45.7806(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 0.999927. The t-ratios (ratio of coefficient to its standard deviation) are all significantly different from zero at the 95 percent level of confidence except for the one for a singleton proton the one for #a and the one for 1/#nn.

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:

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 are applied the results are:

CNN/CNP = −0.6952
CNP/CPP = −0.5637
CNN/CPP = 0.3919

The value of −0.6952 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 1.18.If q=−2/3 then the third equation implies that (1+δ) is equal to 1.13. This is a reasonably close correspondence.

The lack of statistical significance of the coefficient of #a is a surprise, but the coefficient of 1/#a is 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) = 0.8468#a −27.4377(1/#a)
and hence
(∂BE/∂#a) = 0.8468 + 27.4377(1/#a²)

Thus for large values of #a the effect of the formation of an alpha module fades to a level of 0.8468 MeV but for smaller levels of #a the effect is of a much greater magnitude.

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

The regression equation coefficients were used to estimate the binding energies of each of the 2928 nuclides involved in the regression. The difference between the actual values of the binding energies and the regression estimates were computed and those values were plotted versus the actual values in the diagram shown below.

As can be seen the regression equation does not do well for the small nuclides. Furthermore there is an apparent quadratic pattern to the errors; i.e., for binding energies in one range the errors are negative then for a higher range they are positive and for a still higher range the errors are again negative. This pattern of errors can be reduced by allowing for a quadratic dependence of binding energy on the number of interactions of various types.

Allowance for the Interaction Effects
to Involve Quadratic Dependence

When the variables NN², PP² and NP² are included in the regression the results are:

BE = 0.5636#a + 11.9641#nn + 7.7204#pp + 8.7877#np
−0.23831903NN −0.5938PP + 0.3610NP +3.8234sn −1.7895sp
−29.06947(1/#a) +5.6096(1/#nn) −24.0239(1/#pp) − 19.5350(1/#np)
1.653×10-6NN² + 2.7309×10-6PP² −1.817×10-6NP²

The regression coefficient are significantly different from zero at the 95 percent level of confidence except for those of #a, 1/#nn and PP². The coefficient of determination for the equation is 0.999966.

The ratios of the coefficients involving q and δ are:

CNN/CNP = −0.6602
CNP/CPP = −0.6080
CNN/CPP = 0.4014

The first equation indicates that q is equal to −2/3. The second equation then implies that (1+δ)=1.0966. The third equation implies that (1+δ)=1.1073. This is a very good correspondence.

The inclusion of the quadratic variables eliminated the pattern of ranges of negative-positive-negative errors previously found but there are still large errors for the small nuclides, as shown below.

The exceptionally large errors for the small nuclides can be eliminated by allowing for a constant in the regression equation. The plot for the errors in such a regression equation is:

The extreme outlier near 30 and 0 is for the deuteron.

The more or less evenly spaced peaks in the diagram probably have an explanation in terms of shells. Below shows the peaks in the errors labeled with their proton and neutron numbers.

As can be seen most of the peaks involve magic numbers of protons and neutrons. The magic numbers correspond to the filling of nucleon shells.

There are good reasons for not wanting a constant term for the equation predicting the binding energy of nuclide from its composition but the statistical performance of the regression equation with such a constant is impressive. The coefficient of determination is 0.999959 and the standard error of the estimate is 3.2 MeV. All of the regression coefficients except the one for #a are statistically significantly different from zero at the 99 percent level of confidence.

The estiates of q and δ from the coefficients are 0.6630 for q and 1.1909 and 1.1974 for δ.

If the extreme outlier, the deuteron, is left out of the analysis the coefficient of determination rises to 0.999961 and the values of q and (1+δ) determined from the coefficients are 0.66656 and 1.1880 and 1.1881, respectively. These constitute an even better correspondence to q=−2/3 and equal values of (1+δ) by the two different methods.

The scatter diagram for the regression without the deuteron is shown below with the peaks labeled by the proton and neutron numbers they correspond to.

The concentration of the errors at the points of the filling of the neutron and proton shells can be explained by the true relationship being a bent line and the fitted relationship not involving a bend. The highest values of the errors in the fitted relationship would occur at the bend points, as shown below.


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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