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The Statistical Explanation of the Binding Energy
of Nuclides in Terms of Nuclear Substructures

The binding energies of nuclides is their mass deficits expressed in energy terms using Einstein's equation of E=mc². The mass deficit of a nuclide is the difference between its measured mass and the mass of its constituent nucleons (protons and neutrons). Those nucleons however may be organized into substructures so the binding energy may be composed of that due to the substructures and that due to the arrangement of those substructures within the nucleus. The alpha particle (the Helium 4 nuclide) has a binding energy of about 28.3 million electron volts (MeV). In terms of energy it makes sense for two neutrons and two protons within a nucleus to form an alpha particle. This is in comparison to the formation of two deuterons which have binding energies of only 2.2 MeV each.

This study is an investigation of how well the binding energies of 2931 nuclides can be explained in terms of the substructures of alpha particles, deuterons, nucleonic spin pairs which may be formed from its nucleons. The numbers of protons and neutrons are denoted as #p and #n, respectively. The number of possible alpha particles, #a, is the number of pairs in the minimum of #p and #n; i.e., min(#p,#n)%2, where z%2 is the whole number of times 2 goes into z. If there is a proton-neutron pair left over after the formation of alpha particles it is a deutron. The number of deuterons, #d, is zero or one. The excess neutrons, if any, then form neutron-neutron spin pairs, the number of which is denoted as #nn. Likewise if there are any excess protons they form proton-proton spin pairs, denoted as #pp. Finally there may be a singleton neutron or a singleton proton (but not both) denoted by 0-1 variables as sn and sp.

This data for the 2931 nuclides were used in a regression analysis. The results were:

VariableRegression
Coefficient
t-Ratio
sn 25.9425 15.1
sp 2.76617 0.5
#nn 11.8362 62.6
#pp 2.89905 1.2
#d 33.85739 20.6
#a 32.67032 354.4

The reported coefficient of determination (R²) for this equation is 0.99838, but because the regression equation does not contain a constant this exaggerates the degree of explanation of the variance in binding energy. The corrected coefficient of determination is 0.99106.

The magnitude of the coefficient for #a of 32.67 MeV seems just about right since 28.3 MeV would be attributable to the formation of an alpha particle leaving about 4.4 MeV as the amount accounted for by the interaction of an alpha particle with the rest of the elements of the nucleus. The effects for a singleton proton and a proton-proton pair are not statistically significant at the 95 percent level of confidence. However there are nonlinear effects associated with the filling of shells that are important.

The possibility that the slope of the relationship between binding energy and the number of alpha particles is allowed for by creating variables of the form u(#a-m) where u(z)=0 if z<0 and z otherwise. The values of m used are 14 and 25. These are allociated with the magic numbers for neutrons and protons of 28 and 50, respectively.

VariableRegression
Coefficient
t-Ratio
u(#a-25) -10.2764 -73.4
u(#a-14) 0.56333 3.4
sn 5.95519 9.6
sp -6.13594 -3.4
#nn 15.05414 217.8
#pp 0.76555 0.9
#d 11.49494 19.2
#a 34.86248 463.4

The corrected coefficient of determination for this equation is 0.99893. The magnitudes of the regression coefficients are even more reasonable than those of the previous regression equation.

(To be continued.)


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