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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 and an Error in the Mass of the Neutron |
. A previous study estimated the statistical parameters for a model explaining the binding energies of 2929 nuclides. The statistical fit was good, the R² value being 0.9975 and the coefficient of variation being 5 percent. The model involved adjusting for heteroskedasticity of the error term by dividing the variables by the binding energy of the nuclide. However there is reason to believe that the binding energies are subject to a systematic error due to the underestimation of the mass of the neutron by 3 million electron volts (MeV). The error in the binding energy of a nuclide is corrected by adding an amount equal to its number of neutrons times 3 MeV.
The model is that the binding energy of a nuclide is the sum of the binding energies associated with the formation of spin pairs and alpha modules plus the binding energy due to the interaction of the nucleons through what is usually called the nuclear strong force. That binding energy associated with spin pair formation includes that due to the formation of the spin pair per se and that due to the adjustments in structure due to the presence of of the spin pair. A nucleus may also contain a singleton neutron or a singleton proton. These do not affect binding energy through pair formation but could through some adjustment in structure due to their presence. For this reason they are included in the regression.
When the corrected binding energies are used to make the adjustment for heteroskedasticity the regression results are:
Regression Results with Corrections for Heteroskedasticity and the Error in the Mass of the Neutron (MeV) |
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Variable | Coefficient | t-Ratio |
a/BE | 47.78785339 | 489.507254 |
nn/BE | 8.895105295 | 60.96830489 |
pp/BE | 0.740039346 | 7.577902387 |
np/BE | 20.89969762 | 47.80552689 |
n(n-1/2BE | -0.228145486 | -17.0746484 |
np/BE | 0.409360074 | 22.69889801 |
p(p-1)/2BE | -0.769241883 | -30.86998546 |
xxn/BE | 1.635972874 | 2.308566275 |
xxp/BE | -0.641039541 | -0.873785977 |
1/BE | -16.82332285 | -38.21484318 |
For a variable's effect to be statistically significantly different from zero at the 95 percent level of confidence its t-ratio must be greater than or equal to 2 in magnitude. As can been seen the singleton proton does not satisfy this requirement. The singleton proton with a t-ratio of 2.3 does satisfy it but that t-ratio is so much less than the t-ratios of the other variables that it is justified to drop both the singleton neutron and singleton proton from the analysis.
Regression Results with Corrections for Heteroskedasticity and the Error in the Mass of the Neutron (MeV) |
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Variable | Coefficient | t-Ratio |
a/BE | 47.79830199 | 489.8538847 |
nn/BE | 8.920164895 | 61.61623741 |
pp/BE | 0.755236478 | 7.757453067 |
np/BE | 20.67971574 | 59.65422614 |
n(n-1/2BE | -0.232753959 | -17.53852657 |
np/BE | 0.415672683 | 23.20824607 |
p(p-1)/2BE | -0.778144268 | -31.45027007 |
1/BE | -16.61092681 | -48.57116957 |
The statistical fit is good. The coefficient of determination (R²) is 0.9989 and the coefficient of variation is 3.3 percent.
All of the coefficients for the spin pairs are positive indicating the associated force is attractive.
The interaction terms result from a neutron and a proton having a charge with respect to the nuclear strong force, a nucleonic charge. The coefficients for the interactions of the nucleons can be used to estimate the relative charge of the neutron with respect to that of the proton, If the nucleonic charge of the proton is taken as 1 and that of a neutron is denoted as q then the binding energy of the interaction of a neutron with a neutron is proportional to q²; that of a neutron with a proton is proportional to q. The interaction of a proton with a proton is effected by the electrostatic repulsion of protons for each others. If the ratio of the magnitude of the electrostatic repulsion to that due to the strong force is denoted by d then the binding energy due to the interaction of two protons is proportional to (1+d). The value of d depends upon the average separation distance of the protons.
Let the interactions of the nucleons be denoted by the terms I_{nn}, I_{np} and I_{pp}. Then the ratios give
In another study the value of d is estimated to be 0.24. Based up the above relations and this estimate of d the estimates of q are:
Interaction Ratio | I_{nn}/I_{np} | I_{np}/I_{pp} | I_{nn}/I_{pp} |
Estimate of q | −0.55995 | −0.66239 | −0.60902 |
The ratio I_{nn}/I_{pp} can only give an estimate of the absolute value of q but the sign of q ia known from the other two estimates. The estimates are compatible with an estimate of q being −2/3 but the significant thing is that the nucleonic charge of a neutron is smaller in magnitude than that of a proton and opposite in sign. This means that like nucleons repel each other and unlike ones attract each otherl
For a comparison the regression results when there is no correction for heteroskedasticity are given below.
Regression Results with No Correction for Heteroskedasticity (MeV) |
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Variable | Coefficient | t-Ratio |
a | 48.73986088 | 918.239958 |
nn | 18.5597442 | 209.2021898 |
pp | 13.95349267 | 45.6586314 |
np | 14.06759693 | 58.32979299 |
n(n-1/2 | -0.216312552 | -85.12986628 |
np | 0.323582423 | 84.7532207 |
p(p-1)/2 | -0.599195522 | -100.672163 |
1 | -48.23286208 | -95.80424981 |
The statistical fit is good. The coefficient of determination (R²) is 0.99990 and the coefficient of variation is 0.59 of 1 percent. The estimates of q based on the above interactions are:
Interaction Ratio | I_{nn}/I_{np} | I_{np}/I_{pp} | I_{nn}/I_{pp} |
Estimate of q | −0.66849 | −0.66963 | −0.66906 |
These estimates of q are eminently compatible with an estimate of q as −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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