There are shell structures of protons and neutrons in nuclei. These shells are manifested in terms of the stability of the nuclides. There are nuclides which are more stable such that when a shell is filled. The shell structure is also manifested in terms of the incremental binding energies as additional nucleons are added. For example consider the incremental binding energies for the isotopes of Polonium (Po):

The breakpoints come at certain numbers, called magic numbers, that represent filled shells. The sawtooth patterns comes from the formation of neutron pairs. Protons also form pairs. The declining slope of the incremental binding energy pattern indicates that the binding energy is a quadratic function of the number of nucleons.

The slopes and curvature of the relationships differ for different shells. Instead of trying to allow for such differences in a statistical analysis of all 2931 nuclides this analysis looks at the cases in which the number of protons is between 83 and 126 and the number of neutrons is between 127 and 184. The proton shell has capacity of 44 and the neutron shell has a capacity of 58.

There 432 nuclides satisfying those conditions. These nuclides have an average binding energy of 1804.5 MeV.

The explanatory variables for the binding energies are the numbers of protons p and the number of neutrons n in the shell. In order to capture the effect of the pairing of nucleons the numbers of the nucleons are expressed as the number of pairs of each of the nucleons and whether there are singleton (unpaired) nucleons. The number of pairs of protons and neutrons is denoted as #pp and #nn, respectively. From these variables three additonal explanatory variables are created (#pp)², (#pp)*(#nn) and (#nn)². Additionally there are the variables, s_p and s_n, which are equal to 1 if a singleton proton or singleton neutron is present in a nuclide.

The results of the regression are

BE = 1628.92843 + 11.72443#pp + 11.74975#nn
- 0.89286(#pp)² + 0.92249(#pp)*(#nn) - 0.26938(#nn)²
+ 3.54263s_p + 5.87179s_n

[5116.4] [115.7] [160.6]
[-44.6] [36.2] [-27.8]
[23.2] [38.6]

R² = 0.0.99968

Standard error = 1.56740Mev

Coefficient of Variation = (1.56740 MeV)/(1804.5 MeV) = 0.00087

The numbers in the square brackets below the coefficients are the t-ratios for the coefficients. For the regression coefficient to be statistically signficant at the 95 percent level of confidence its t-ratio must be roughly 2.0 or larger.

The binding energy of the nuclide with 82 protons and 126 neutrons (Lead) is 1636.446 MeV, significantly different from the regression constant of 1628.92843 MeV, but remarkably close.

There was a possibility that if a singleton proton and a singleton neutron were present in a nuclide they would form a pair which would enhance the binding energy. When such a variable was included in the regression its coefficient was not statisically significant. (The t-ratio was -0.2.)

Conclusions

The numbers of protons and neutrons in the nuclides having protons in the 83 to 126 proton shell neutrons in the 127 to 184 neutron shell explain all but 0.032 of 1 percent in the variation in binding energies of these nuclides. The regression equation gives estimates of binding energies that are accurate to roughly ± 0.087 of 1 percent.