San José State University
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The Binding Energies of the Nuclides with Protons in
the 83 to 126 Proton Shell and Neutrons in
the 83 to 126 Neutron Shell

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 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 also between 83 and 126. The neutron and proton shells have capacity of 44 nucleons each.

There 131 nuclides satisfying those conditions. These nuclides have an average binding energy of 1583.7 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, sp and sn, which are equal to 1 if a singleton proton or singleton neutron is present in a nuclide and 0 otherwise.

The results of the regression are

BE = 1200.28567 - 4.5077#pp + 25.51770#nn
- 0.55652(#pp)² + 0.57562(#pp)*(#nn) - 0.25345(#nn)²
+ 0.90377sp + 7.99530sn

[2181.7] [-10.8] [120.1]
[-83.5] [23.3] [-38.0]
[8.4] [81.2]

R² = 0.999908

Standard error = 0.550 MeV

Coefficient of Variation = (0.0550 MeV)/(1583.7 MeV)
= 0.000347

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.

There is no stable nuclide with 82 protons and 82 neutrons. The element of atomic number 82 is lead (Pb). Its lightess isotope has 99 neutrons and a binding energy of 1399.8 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 0.3.)


The numbers of protons and neutrons in the nuclides having 83 to 126 of each explain all but 0.0092 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 ±0.550 MeV or ± 0.035 of 1 percent.

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