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The Functional Form of the Incremental Binding Energy
of Protons in the Second Shell as a Function
of the Number of Neutrons in the Nucleus

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.

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.

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. Plots of this data for the third through the sixth neutrons are shown below.

Neutrons and protons are arranged in separate shells. There are two neutrons in the first shell and conventionally the second shell is considered to consist of six neutrons. A case can be made for the second shell to consist of four protons. For more on the matter of nuclear magic numbers see Magic Numbers. For now the second shell is considered to consist of six neutrons. Thus the third proton is the first in the second shell and the sixth proton is the last neutron in the second shell.

There are two things immediately notable in the above graphs. First there is a jump in the incremental binding energy at the point where the number of neutrons equals the number of protons. This is a pairing phenomenon. The second thing is that the curves have an increasing slope for low neutron numbers and a decreasing slope for higher neutron numbers. A cubic polynomial of the proton number can have this shape. If p and n are the number of protons and neutrons, respectively, in the nuclide then the regression equation used to explain the incremental binding energy (IBE) is of the following form

IBEp = c0 + c1n + c2n² + c3n³ + c4u(n≥p)

where u(n≥p) is 1 if n≥p and 0 otherwise.

The regression results are as follows:

Proton
Number
c0c1c2c3c4
3-3.55247-0.857561.09573-0.096084.421860.99143
4-14.0923310.0298384-1.41200.074388.124760.99528
52.45737-4.393001.27867-0.072414.005820.99146
6-8.427324.30649-0.266230.008256.433850.99401

The data for the third through the sixth protons are plotted below in the same graph for comparison.

In each case the curves are of the ogee type and in the case of the even numbers of protons there are jumps where the neutron number equals the proton number. The curves for an even number of protons are generally higher than for the previous odd number of protons. This likely reflects the potential energy loss associated with the formation of proton pairs. The curves for the higher numbers of protons in a shell are generally lower than for the lower numbers. This reflects some sort of shielding phenomena due to the repulsion of protons for each other.


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