﻿ The Statistical Explanation of the Binding Energies of Nuclides in terms of their Pair Formations and the Interactions of their Nucleons through the Strong Force and Taking into Account Heteroskedasticity
San José State University

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Thayer Watkins
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 The Statistical Explanation of the Binding Energies of Nuclides in terms of their Pair Formations and the Interactions of their Nucleons through the Strong Force and Taking into Account Heteroskedasticity

## Background

A previous study estimated the statistical parameters for a model explaining the binding energies of 2931 nuclides. The statistical fit was good, the R² value being 0.9998. However the standard error of the estimate was 6.7 MeV. Compared to an average binding energy of 1071 MeV that is not bad, the coefficient of variation being 0.625 of 1 percent. But for the smaller nuclides in which the binding energy is 30 MeV or less that standard error is just too large. In the estimating procedure an error of say 5 MeV in the binding energy of a nuclide is 10 MeV was counted as no worse than an error of the same amount in the binding energy of a nuclide which is 2000 MeV.

The general regression model used was of the form

#### yk = a + Σbjkxj + u

where the standard deviation σ of u is assumed to be constant for all k and independent of the xj's.

If σ is not constant there is said to be heteroskedasticity of the error term. If σk is known to be proportional to some variable zk then all of the variables can be divided by zk to give the regression equation

#### yk/zk = a(1/zk) + Σbjk(xjzk) + v

where the standard deviation of the random term v is constant.

In this regression no constant term is allowed.

If there is no other variable zk available yk can be used. This gives a regression equation of the form

#### 1 = a(1/yk) + Σbjk(xjk/yk) + v

when this procedure is applied to the model for binding energies here is what results:

Regression Results
Variable Coefficient t-Ratio
α/BE 39.87084 389.0
nn/BE 4.01294 32.6
pp/BE 4.16075 29.5
np/BE 6.56296 30.6
p(p-1)/2BE -1.02460 -31.0
np/BE 0.62490 25.9
n(n-1)/2BE -0.39747 -22.0
1/BE -5.41025 -22.2

The statistical fit is good. The coefficient of determination (R²) is 0.99692. This is not as good as the result in the previous study, 0.99980. The coefficient of variation also is not as good being 5.6 percent as opposed to 0.625 of 1 percent. But this is to be expected since this regression gives greater weight to the smaller nuclides which are less regular and predictable than the larger nuclides. But what is attractive about these results is that the magnitudes of the effects of spin pair formation are closer to what come out of the incremental binding energy information. Regardless of the statistical fit the taking into account the heteroskedasticity of the data is necessary.

The estimate for the nucleonic charge of a neutron relative to that of a proton q that comes from the ratio of the effect of the interaction of neutrons to that of neutrons with protons is −0.636. The estimate of |q| that comes from the ratio of the interaction of neutrons with protons to the interaction of protons with protons is 0.683. Both are consistent with a value of q of −2/3.

## Singleton Nucleons

Some nuclides, in addition to containing alpha modules and nucleon spin pairs, contain a singleton nucleon. Such a singleton nucleon cannot have binding energy due to pairing but it can have binding energy due to any structural adjustment it makes possible. Such binding energy enhancements are included with the the effects of pairing. When the singleton nucleons are included in the regression the results are as shown in the table below.

Regression Results
Variable Coefficient t-Ratio
α/BE 40.80236 404.1
nn/BE 3.48242 29.2
pp/BE 4.11555 21.5
np/BE 18.55130 36.1
p(p-1)/2BE -0.99041 -32.4
np/BE 0.58486 26.2
n(n-1)/2BE -0.36219 -21.8
1n/BE 11.90812 22.0
1p/BE 13.25671 24.7
1/BE -17.25993 -33.2

The t-ratios for 1n/BE and 1p/BE show that the singleton nucleons definitely have an effect that cannot be attributed to chance. There is a slight improvement in the coefficient of determination (R²) from 0.9969 to 0.9975 and the coefficient of variation drops from 5.6 percent to 5.0 percent.

The two estimates of q, the nucleonic charge of a neutron relative to that of a proton, are −0.619 and −0.662. Again both are consistent with a value of q of −2/3.

## Conclusions

The model accounts for 99.75 percent of the variation in the binding energies of the 2929 nuclides.

The results are consistent with the nucleonic charge of a neutron relative to that of a proton being −2/3.