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The Functional Form of the Incremental Binding Energy
of Neutrons in the Second Shell as a Function
of the Number of Protons 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 neutron in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less neutron.

The incremental binding energies for nuclides containing the same number of neutrons but varying numbers of protons can be tabulated. Plots of this data for the third and eighth neutrons are shown below.

Neutrons are arranged in 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 neutrons. 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 neutron is the first in the second shell and the eighth neutron 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 protons equals the number of neutrons. This is a pairing phenomenon. The second thing is that the curves have an increasing slope for low proton numbers and a decreasing slope for higher proton 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

IBE = c0 + c1p + c2p² + c3p³ + c4u(p≥n)

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

The regression results for n=3 are

IBE = -1.75700 − 3.29572p + 2.35556p² −0.24046p³ + 2.81592u(p≥3)
[-1.2] [2.6] [-2.7] [1.9]

The coefficient of determination (R²) is 0.9992. In this case there is only one degree of freedom. The numbers shown in brackets below the regression coefficients are the t-ratios. Only the coefficients for p² and p³ are significantly different from 0 at the 95 percent level of confidence.

The regression results for n=8 are

IBE = 0.83489 − 1.81419p + 0.72553p² −0.03757p³ + 1.59499u(p≥8)
[-1.3] [3.0] [-3.1] [1.7]

The coefficient of determination (R²) is 0.9972 with 5 degrees of freedom. Again the numbers shown in brackets below the regression coefficients are the t-ratios. Only the coefficients for p² and p³ are significantly different from 0 at the 95 percent level of confidence.

The data for the third through the eighth neutrons 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 neutrons there are jumps where the proton number equals the neutron number. The curves for an even number of neutrons are generally higher than for the previous odd number of neutrons. This likely reflects the potential energy loss associated with the formation of neutron pairs. The curves for the higher numbers of neutrons in a shell are generally lower than for the lower numbers. This reflects some sort of shielding phenomena due to the repulsion of neutrons for each other. Note that the curves for n equal to 3 through 6 are together but the curves for 7 and 8 drop significantly. This would be the case if 7 and 8 are in a higher shell than hlk


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