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The Adjustment of the Statistical
Testing of the Alpha Module
Model of Nuclear Structure
for Heteroskedasticity

A previous study tested with good results a model for the binding energies of nuclei. The model involves binding energies arising from two sources. One source is the formation of substructures involving spin pairs. The second source is the nuclear strong force between nucleons.

The binding energy resulting from spin pairs; proton-proton, neutron-neutron and neutron-proton; are relatively large but spin pair formation is exclusive in the sense that one neutron can form a spin pair with only one other neutron and with one proton and likewise for a proton. Such spin pairs are formed wherever possible so this leads to chains of neutrons and protons. Specifically it leads to chains of alpha modules. An alpha module is a linking of the form -n-p-p-n-, or equivalently, -p-n-n-p-.

The binding energy involved in the interaction of two nucleons is relatively small compared to that involved in a spin pair formation but due the number of strong force interactions in larged nuclei it can dominate the binding energies due to substructure formations.

The results of the previous study that are significant are

Regression Analysis

Regression analysis presumes that a dependent variable y is generated from m independent variables according to the following equation

yj = Σβjxij + uj
for j=0 to n,

where the xij are the independent variables and the uj random variables with constant mean and standard deviation. It is presumed that x0j=1 for all j.

Under those assumptions regression analysis give unbiased estimates of the true values of the coefficients βi for i=0 to m.

The focus of the analysis here is what happens if the standard deviations of the random variables are not constant but instead is equal to σsj, where sj are known values and σ is a random variable with the same distribution for all values of j. Thus the dependent variable is generated by the equation

yj = Σβjxij + σjsj
for j=0 to n,

When these equations are divided through by the values of sj the result is

yj/sj = Σβj(xij/sj) + σj
for j=0 to n,

This is now in a form that fits a standard format of regression analysis.

A Test for Heteroskedasticity

An indication of whether or not there is heteroskedasticity in the phenomena can be obtained by plotting the difference between the actual value of the dependent variable and the regression estimate versus the actual value of the dependent variable. If there is no heteroskedasticity the plot should look roughly likke a rectangle. Here is the plot for the binding energies of roughly three thousand nuclides.

There is a dependence and it is complex. The first attempt to eliminate the heteroskedasticity is to assume that the variance of the random term is proportional to the binding energy of the nuclide. This means that the standard deviation is proportional to the square root of the binding energy. This assumption requires a slight modification of the data set. The binding energy of the Beryllium 5 isotope is negative and thus its square root is imaginary. So the Beryllium 5 nuclide is deleted from the data set.

Dividing through the regression equation by the square root of binding energy leaves the square root of binding energy as the dependent variable. The results of the regression analysis are shown in the table below.

Comparison of the Regression Results for the Equations
Unadjusted and Adjusted for Heteroskedasticity
(Unadjusted model)
(Adjusted model)
Number of
Alpha Modules
Number of
Spin Pairs
Number of
Spin Pairs
Number of
Spin Pairs

The results for the adjusted equation are not significantly different from those for the unadjusted equation. The effects for spin pair formation are roughly 10 MeV. The effects for interactions through the strong force are a fraction of a MeV with the force between like nucleons being a repulsion and between unlike nucleons an attraction. The ratio of the coefficient for neutron-neutron interactions to that for neutron-proton interactions is −0.65, not far from the −2/3 found in previous studies. The only major surprise is the magnitude of the effect of alpha modules.

The plot of the error in the regression estimate versus the actual value of binding energy is given below.

While not perfect this plot is closer to being of constant width than the unadjusted equation. The adjusted equation still has a problem in explaining the binding energies of nuclides too small to contain an alpha module. But the adjustment essentially eliminates the heteroskedasticity but leaves the essential elements of the results unaffected. The statistical performance of the adjusted equation is not a good as those of the unadjusted equation but they are still good. The coefficient of variation of the regression estimates for the unadjusted equation is about 0.5 of 1 percent whereas that of the adjusted equation is 1.4 percent. The coefficient of determination (R²) for the unadjusted equation is 0.99988 and that of the adjusted equation 0.99982. The existence of heteroskedasticity does not cast doubt on the validity of the Alpha Module Model of nuclear structure.

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