San José State University
Thayer Watkins
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& Tornado Alley

The Effect of a Proton-Neutron Pair
on the Excess Binding
Energy of Nuclides
version 2

The masses of nuclei are less than the masses of the protons and neutrons that they are composed of. These mass deficits expressed in energy units are called the binding energies for the nuclides. The binding energy for a nuclide is often represented as the energy required to disassemble the nuclide. This is a bit misleading. The nucleus may be composed of substructures, such as alpha particles. Some of the binding energy for a nuclide would then represent the binding energy involved in the creation of the substructures. The energy required to disassemble a nuclide would be only the binding energy in excess of that involved in the creation of the substructures.

Some nuclides could contain an integral number of alpha particles. Their excess binding energy is binding energy in excess of that involved in assembling the protons and neutrons into alpha particles. When the excess binding energy of these integral alpha particle nuclides is plotted versus their atomic number (number of protons) the result is as shown below.

This graph indicates that alpha particles are formed within the nucleus and they are arrayed in shells. The graph indicates that the first shell can contain only two alpha particles. The second shell can contain twelve alphas. The third can contain at least eleven alphas and work elsewhere indicates that eleven is the capacity of the third shell.

When the same sort of analysis is carried out for nuclides that could contain an integral number of alpha particle plus one neutron-proton pair. The resulting graph of the excess binding energies as a function of the atomic number is essentially the same as the previous one.

The question dealt with here is how much does the additional neutron-proton pair enhance the binding energy of the nuclide. The excess binding energy is the increase in binding energy less the binding energy of a neutron-proton pair (a deuteron), which is 2.224573 MeV. The graph shown below has a few interesting characteristics.

Below is the data for the difference between the binding energies of the alpha plus a neutron-proton pair nuclides and the alpha nuclides plotted versus the number of alpha particles in the nuclides. (It differs from the above display only in that the binding energy of the neutron-proton pair is not deducted.) Along with the data is shown the regression estimates as the upper edge of the yellow area. This regression involves a bendpoint at 5 alpha particles, a drop at 10 alpha particles, followed by a rise and then a drop at 14 alpha particles. The slope of the regression is constant beyond 5 alpha particles.

The matching of the fall at 10 and rise at 11 followed by a fall at 14 is justified because that irregularity shows up again and again in the relationships. The regression estimates six parameters using the 25 data points, which leaves 19 degrees of freedom. The coefficient of determination for the regression is 0.929.

The pattern of a drop at 4 alpha particles and subsequent rise at 5 is also a pattern that shows up frequently as well in relationships for the effect additional particles on binding energies. When a regression equation of the same form as the one above but with a fall at 4 and rise at 5 is fitted to the data the results are as shown below.

The coefficient of determination for this regression is 0.9966 with 16 degrees of freedom.

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