San José State University
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The Statistical Explanation of the
Excess Binding Energy of Nuclides

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.

What follows below is a statistical analysis of the binding energies of 2931 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. A regression of excess binding energy on excess neutrons and protons gives the following result

ΔB = 9.051269#n + 2.663718#p
      [161.01]           [7.11]
R² = 0.951

The number in the square brackets below a regression coefficient is the t-ratio for that coefficient. The greater the magnitude of the t-ratio the greater the significance of that variable in explaining the variation in the excess binding energy. For a 95 percent level of confidence the t-ratio should be greater than 2 in magnitude. Thus the effect of the excess neutrons are highly significant and the effect of the excess protons is significant but less so than that for excess neutrons.

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, 95.1 percent is explained by the number of excess neutrons and the number of excess protons. The variance of the residuals from the regression equation is 288.6 (MeV)². This means that 99.66 percent of the variation in the binding energies is explained by the binding energy of the alpha substruce of a nuclide and the excess neutrons and protons.

(To be continued.)

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