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The Incemental Binding Energy
of a Neutron and Its Shielding

Background

This materials deals with the application of a model that successfully explains the ionization energies of electrons in atoms to the binding energies of nucleons in nuclei.

The ionization energy IE, or as it is usually called the ionization potential, for an electron in an atom or ion is the amount of energy required to dislodge it. The Bohr model of a hydrogen-like atom indicates that the energy required to remove an electron should follow the form

IE = RZ²/n²

where R is the Rydberg constant (approximately 13.6 electron Volts (eV), Z is the net charge experienced by the electron and n is the principal quantum number, effectively given by shell number. The value of Z is the number of protons in the nucleus #p less the shielding ε by the other electrons. Thus the ionization potential would be

IE = (R/n²)(#p−ε)²

This is equivalent to a regression equation of the form

IE = c0 + c1(#p) + c2(#p)²

Such a form gives a very good fit to the ionization data. The coefficient of determination goes as high as 0.9999998+.

An example of a graph of ionization energy versus the number of protons in the nucleus is given below.

The quadratic curvature is clearly perceptible.

The Bohr model is strictly for a hydrogen-like atom or ion; i.e., one in which there is a single electron in the outermost shell. However the regression equation fits very well the cases of multiple electrons in the outer shell if shielding by electrons is taken into account . Thus the shielding ε is for the electrons in the inner shells and also in the same shell. But the shielding by electrons in the same shell is only a fraction of their charge, approximately one half. As it turned out empirically, even tor electrons in the inner shells the shielding is less than the full value of their charge.

Application of the
Bohr Model to Nuclei

A nucleus consists of protons and neutrons linked together through spin pairing into shells. The filled shells form a nuclear core of equal numbers of protons and neutrons. Typically a nucleus has more neutrons than protons, The extra neutrons, called halo neutrons, are held in orbits around the nuclear core due the attraction between neutrons and protons through the nucleonic force.

The quantity for nucleons in a nucleus which would be comparable to the ionization energy for electrons in atoms is the incremental total binding energy. This would included the energy lost when the components of a nucleus come together. There is a loss of potential energy but a gain in kinetic energy. In atoms that net loss of energy goes into the creation of a photon. For nuclei a similar thing happens but for nuclei there is another loss of energy which is manifested as a mass deficit; i.e., the mass of s nucleus is less than the masses of its components. These mass deficits converted into energy units are what are called the binding energies of nuclei, but they are not the total binding energies of nuclei.

Total binding energy is known only for one nuclide, the deuteron. When a proton and neutron come together to form a deuteron a gamma ray of about 2.25 million electron volts (MeV) of energy is emtted. When deuterons are subjected to gamma radiation of 2.25 MeV or higher they come apart into protons and neutrons.

To implement the Bohr formula the incremental binding energies of neutrons were computed for nuclides having a particular number of neutrons. The number of neutrons used was 66, an arbitrary value. The values of the incremental binding energies of neutrons as a function of the net number of protons in the nuclide are shown below.

There are actually two relationships. One is for the number of protons less than or equal to 50 and another for tthe number of protons greater than 50.

There is no perceptible quadratic curvature in either relationship. The regression equations confirm the absence of any quadratic dependence. The regression coeficients for (#p)² are not significantly different from zero at the 95 percent level of confidence.

However there is no problem of the predictability of the IBEn. The coefficients of determination for both regression equations are 0.9533 and 0.9836.

The Bohr equation also on the number of neutrons in the nuclide. Here is the relationship for cases of p=50.

The relationship is for the neutron numbers of above 50 but less than or equal to 82. A quadratic regression equation including a variable for the odd-eveness of the proton number reveals significant dependence of IBEn on (#n)².

IBEn = 37.560 −0.64525n + 0.0034056n² −2.62740odd
[35.5] −17.2] [12.1] [−61.2]

The coefficient of determinations for this regression equations is 0.99726.

Conclusion

The Bohr equation which explains the ionization energies of electrons in atoms does not provide as clear an explanation of the incremental binding energies of neutrons in nuclei. Instead of quadratic dependence on proton number there is only linear dependence of high levels of correlation. But there is a quadratic dependence on neutron number.


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