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of Nuclides in terms of their Pair Formations and the Interactions of their Nucleons through the Strong Force and Taking into Account Heteroskedasticity |
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
where the standard deviation σ of u is assumed to be constant for all k and independent of the x_{j}'s.
If σ is not constant there is said to be heteroskedasticity of the error term. If σ_{k} is known to be proportional to some variable z_{k} then all of the variables can be divided by z_{k} to give the regression equation
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 z_{k} available y_{k} can be used. This gives a regression equation of the form
when this procedure is applied to the model for binding energies here is what results:
Regression Results | ||
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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.
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 | ||
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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.
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.
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