San José State University
Thayer Watkins
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& Tornado Alley

The Binding Energies of the Nuclides
with 29 to 50 Neutrons and 29 to 50 Protons

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 29 and 50 and the number of neutrons is also between 29 and 50. The neutron and proton shells have capacity of 22 nucleons each.

There 285 nuclides satisfying those conditions. These nuclides have an average binding energy of 666.36 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.

The results of the regression are

BE = 482.38875 + 7.932340#pp + 22.79580#nn
- 1.28981(#pp)² + 2.03151(#pp)*(#nn) - 0.70545(#nn)²
+ 4.53835s>p + 9.07226sn

[197.9] [32.1] [104.3]
[-50.0] [57.8] [35.3]
[15.4] [30.8]

R² = 0.99908

Standard error = 2.43775 MeV

Coefficient of Variation = (2.43775 MeV)/(666.36 MeV)
= 0.00366

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 28 protons and 28 neutrons (Nickel) is 483.99 MeV, not significantly different from the regression constant of 482.38875 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 above analysis was carried out on the nuclides that had at least one proton in the proton shell and at least one neutron in the neutron shell. It might be that the cases in which one of the shells had no nucleons should be included. The results for the regression on this augmented data set were as follows.

BE = 481.5215 + 8.03374#pp + 22.92323#nn
- 1.38013(#pp)² + 2.13587(#pp)*(#nn) - 0.74188(#nn)²
+ 4.95558s>p + 9.04418sn

[749.5] [31.7] [104.1]
[-49.8] [57.9] [35.8]
[15.2] [27.4]

R² = 0.99885

Standard error = 2.87008 MeV

Coefficient of Variation = (2.87008 MeV)/(666.36 MeV)
= 0.00431

The results are essentially the same but having a slightly lower coefficient of determination and a slightly higher standard error of the estimate. This perhaps indicates the nuclides having no nucleons in one of the shells do not fit the pattern for the cases in which both shells contain some nucleons.


The numbers of protons and neutrons in the nuclides having 29 to 50 of each explain all but 0.12 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 ± 0.37 of 1 percent.

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