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The Functional Form of the Incremental
Binding Energy of Neutrons 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

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.

Empirical Ionization Potential Functions

According to the theory mentioned above the ionization potential IE for an electron in a particular place in a shell should be given by

IE = (R/n²)(p−ε)²
where p is the number
of protons in the nucleus.

This equation can be put in the form

IE = (R/n²)p² − 2pε + ε²)

A regression equation of the form

IE = c0 + c1p + c2

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

Here is a plot of the data for one case.

The value of ε is found as

ε = −½c1/c2

However it also should be that c0/c2 should be ε² and thus equal to the square of the value found from c1 and c2. The regression coefficients are not constrained to achieve that equality. Thus effectively the form assumed for the relationship for ionization potential is

IE = (R/n²)[(p−ε)² + ζ]

where R is an empirical value, rather than necessarily being the Rydberg constant, and ζ is a constant. The values of R are however notably close to the Rydberg constant. The values for some cases are given below are for the first electrons in several shells.

R (eV)

The value of ε, as was indicated above, is found as −½c1/c2. The values for the first electrons in the first few shells are given below.


As the above results indicate the model is quite successful in explaining the ionization potentials (energies) of the electrons in atoms. The purpose of the rest of this paper is to develop a similar sort of analysis for the incremental binding energies of neutrons in nuclei. For more on the ionization energies of electrons see Ionization Energies.

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 eighth and ninth neutrons are shown below.

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=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. 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.

For n=9 the results are

IBE = 2.68970 − 2.44895p + 0.47395p² −0.0175p³ + 3.58675u(p≥8)
[-1.8] [2.5] [-2.4] [3.9]

The coefficient of determination (R²) is 0.9957. Again the coefficients for p² and p³ are significantly different from 0 at the 95 percent level of confidence, but in this case the coefficient for the jump at p=9 is also significant.

The data for the eighth and ninth neutrons are plotted below in the same graph for comparison.

The eighth neutron is in the second neutron shell and the ninth is in the third. This explains the lower level of incremental binding energy for the ninth neutron compared to the eighth.

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