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The Functional Form of the Incremental Binding Energy of
Neutrons as a Function of the Number of Protons in the Nuclide

The Extended Bohr Model of an atom indicates that the ionization energy of an electron should be given by the formula

En = RZ²/n²
or, equivalently
En = R(Z/n)²

where R is the Rydberg constant (13.6 electron volts) and n is an integer, called the principal quantum number of the electron. The quantity Z is the net charge experienced by the electron; i.e., the positive charge of the nucleus less the shielding by electrons in inner shells and in the same shell. This quadratic function is the functional form of the energy of the electron as a function of the effective attractive charge experienced by the electron. The statistical fit of such a function to the data is very good, as illustrated by the particular case shown below.

A regression equation of the general form of the above equation explains about 97 percent of the variation in the ionization energies of 729 different atoms and ions. For more on this see Ionization. The question pursued below is what is the functional form of the energy of a neutron as a function of the number of protons in its nuclide.

Incremental Binding Energies

The mass of a nuclide, such as helium 4, the alpha particle, is less than the masses of the two neutrons and two protons of which it is composed. The difference is called the mass deficit and that mass deficit expressed in energy units via the Einstein formula E=mc² is called the binding energy. The binding energies have been measured for almost three thousand nuclides. The incremental binding energy of a proton in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less proton.

The incremental binding energies for nuclides containing the same number of protons but varying numbers of neutrons can be tabulated.

Protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for neutrons and protons, is given in the following table.

Shell
Number
1234567
Capacity24814223244
Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126

The plot of the incremental binding energy of the 9th neutron versus the number protons in a nuclide is shown below.

The general functional form is shown schematically below. Say the neutron under consideration is the n-th one and that it is in the s-th shell. The line segments represent the interaction energy of the n-th neutron with the protons in the various shells. The slope of a line segment is the interactive energy of the n-th neutrons with the protons in the various shells. The farther away is the shell from the n-th neutrons the lower that interactive energy is and hence the lower the slope. This accounts for the ogee shape of the functional form.

However there is an additional aspect to the relationship and that is that the incremental binding energy shifts for the case in which the proton number p equals the neutron number n. Each time the proton number increases when p is less than n there is a neutron-proton pair formed. Once p exceeds n no such pair is formed. The display below illustrates this phenomenon.

However some of the plots of the IBE versus proton number appear to have a different form. One case is where there is a jump in the IBE only at the point where p=n.

Another possibility is that the level of the IBE shifts upward at the point where p=n, as illustrated below.


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