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 tin (Sn):

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 downward 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 3 and 14 and the number of neutrons is also between 3 and 14. These numbers encompass two nucleonic shells; one from 3 to 6 and another from 7 to 14. The neutron and proton shells have capacities of 4 and 8 nucleons each. The analysis combines the two shells in order to get a large number of cases for the statistical performance of the model being used.

There 110 nuclides satisfying those conditions. These nuclides have an average binding energy of 121.2 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 = -18.76883 + 17.65233#pp + 14.74433#nn
- 3.75363(#pp)² + 7.01260(#pp)*(#nn) - 2.64312(#nn)²
+ 6.76387s_p + 6.78744s_n

[-4.8] [11.7] [9.8]
[-18.1] [24.3] [-12.7]
[6.6] [6.6]

R² = 0.99071

Standard error = 5.14869 MeV

Coefficient of Variation = (5.14869 MeV)/(121.21832) MeV)
= 0.0425

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 2 protons and 2 neutrons (helium) is 28.3 MeV, quite significantly different from the regression constant of -18.76883 MeV.

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 0.5.)

The numbers of protons and neutrons in the nuclides having 3 to 14 of each explain all but 0.93 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 ± 4.25 percent.

The 3 to 6 Nuclear Shells

If the same regression is performed upon the set of 16 nuclides having 3 to 6 protons and 3 to 6 neutrons the coefficient of determination is 0.979 but five out of the eight regression coefficients are not significantly different from zero at the 95 percent level of confidence.

The 7 to 14 Nuclear Shells

There are 61 nuclides having nucleons in the 7 to 14 shells. There average binding energy is 159.2 MeV. The coefficient of determination for the regression is 0.98930.

The regression equation found is

BE = 18.63008 + 7.99491#pp + 10.38838#nn
- 2.71634(#pp)² + 6.65306(#pp)*(#nn) - 2.07240(#nn)²
+ 4.97720s_p + 8.85217s_n

[1.4] [2.3] [3.0]
[-8.8] [19.1] [-6.1]
[4.9] [8.9]

R² = 0.98930

Standard error = 3.59371 MeV

Coefficient of Variation = (3.59371 MeV)/( 159.2) MeV)
= 0.0226

Thus all but about 1.1 percent of the variation in binding energies for nuclides having these shells occupied is explained by their numbers of protons and neutrons.