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 The Statistical Explanation of the Excess Binding Energy of Nuclides, Version 1

The masses of almost all nuclides are less than the masses of the protons and neutrons of which they might be composed. These mass deficits are usually expressed in energy units (million electron volts MeV) and referred to as binding energy. Most of the binding energy of nuclides can be accounted for by the binding energy involved in the formation of alpha particles (helium 4 nuclei) which they contain. Thus the binding energy that is relevant concerning the structure of the nuclides is the excess binding energy, the binding energy in excess of that due to their substructures of alpha particles and there arrangement.

The binding energies of the alpha nuclides, those nuclides that could and undoubtably do contain an integral number of alpha particles, display some interesting regularities. Below that excess binding energy of alpha nuclide is displayed. This pattern indicates a shell structure for the alpha particles. The value of this excess binding energy arise from the potential energy levels for fitting additional alpha particles into the shell structure.

When the excess binding energies of alpha nuclides are compared to the nuclides that are alpha nuclides plus one neutron, as is shown below, it is found that the effect of the additional neutron increases with the number of alpha particles in the nuclide. The same comparison with nuclides that are alphas plus two neutrons is shown below. It is important to note the functional relationship between additional binding energy created by an additional neutron pair, as shown below. This functional relationship can be reasonably approximated a straight line with a positive intercept. This means that in a regression analysis a constant term should be permitted.

What follows below is a statistical analysis of the binding energies of 1299 nuclides in excess of the the binding energy of the arrangement of alpha particles they may contain.

Let n and p be neutron and proton numbers for a nuclide and B(n,p) its binding energy. Let m=min(int(n/2),int(p/2)). Thus m is the number of alpha particles the nuclide would contain. The excess binding energy ΔB is then

#### ΔB = B(n,p) − B(2m,2m)

The largest alpha nuclide is the tin isotope Sn 100. Therefore p must be less than or equal to 51. There are 1299 nuclides satisfying this condition.

Let #n and #p be the excess number of neutrons and protons; i.e., #n=n−2m and #p=p−2m. In previous work it was found that the formation of a pair enhanced binding energy. To allow for the effect of pair formation ten new variables were created. The variable D is 1 if a proton-neutron pair is possible among the excess protons and neutrons and 0 otherwise. The variables #nn and #pp are the number of pairs of neutrons and of protons among the excess neutrons and protons. The number of singlet neutons ##n and singlet protons ##p were computed by the formulas ##n=#n-2#nn and ##p=#p-2#pp. No deductions were made for the neutron and proton in the neutron-proton pair because apparently within the nucleus a neutron being part of a neutron-proton pair does not exclude it from the formation of a neutron pair.

To allow the dependence of the effects of these variables on the number of alpha particles in the nuclide the products of the number of alpha particles are used as variables. That is to say, if a is the number of alpha particles then aD, a#nn, a#pp, a##n and a##p are used as additional explanatory variables in the regression.

The results of the regression are:

 Variable RegressionCoefficient t-Ratio #nn 10.75171 [41.39] #pp 2.662929 [2.08] D 9.742648 [3.60] ##n -5.60386 [-3.76] ##p -8.19058 [-3.31] a#nn 0.25437 [21.38] a#pp -0.44947 [-3.43] aD 0.106721 [0.47] a##n 0.783702 [9.26] a##p 0.313304 [1.45] Constant 13.7016 [1.19]

The coefficient of determination R² for this regression is 0.978.

The variance of the binding energies of the 1299 nuclides is 83927.4 (MeV)²; that of the excess binding energies is 5908.3 (MeV)². The ratio of these variances is 0.07. This means that 93 percent of the variation in binding energy is explained by the structure of alpha particles it contains.

The value of R² for the above regression indicates that of the 7 percent variation not explained by the alpha structure, 97.8 percent is explained by ten variables included in the regression. The variance of the residuals from the regression equation is 133.07 (MeV)². This means that 99.84 percent of the variation in the binding energies is explained by the binding energy of the alpha substructure of a nuclide and the excess neutrons and protons.

Some variations on the regression equation are of interest. If only the variables which are proportion to the number of alpha particles are included the R² value is reduced only slightly to 0.943. The other results are:

 Variable RegressionCoefficient t-Ratio a#nn 0.701284 [125.78] a#pp -1.01025 [-11.06] aD 1.220622 [7.06] a##n 0.201138 [3.40] a##p -0.86131 [-5.03] Constant 31.46843 [1.71]

Again it is the number of neutron pairs which is most highly significant. There is an enhancement for pair formations. For neutron pairs the enhancement per alpha particle is (0.701284-2(0.201138))= 0.3 MeV. Thus for a nuclide containing 10 alpha particles the addition of two neutrons would result in an increase in binding energy of about 7 MeV, of which about 4 MeV comes from the two neutrons themselves and 3 MeV from their being a pair.

## Conclusion

The excess binding energy of nuclides is overwhelmingly a function of the number of excess neutrons in the nuclide. Pair formation results in an enhancement but each neutron alone creates significant binding energy.

(To be continued.)