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The Functional Form of the Incremental Binding Energy of Protons
as a Function of the Number of Protons and Neutrons 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 ionization energy of an electron as a function of the effective attractive charge experienced by it. 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 (a quadratic) 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 proton as a function of the number of protons and neutrons 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 neutron in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less neutron.

Nuclear Shells

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.

Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126

The incremental binding energies for nuclides containing the same number of protons but varying numbers of neutrons can be tabulated. The plot of the incremental binding energy of the neutrons in the nuclides with neutron number 26 versus the number of protons in the nuclide is shown below.

The sawtooth pattern is due to the energy involved in the formation of proton spin pairs. It is desirable to separate the effect of spin pair formation from the effect due to the interaction of nucleons through the nuclear strong force. One way of doing that is shown below.

If B(n, p) is the binding energy of the nuclide with n neutrons and p protons then the definition of the incremental binding energy (IBEp) of the p-th proton is

IBEp = B(n, p) − B(n, p-1) )

This includes the effect of the formation of a proton spin pair as well as the effect of the interaction of the nucleons through the strong force. To eliminate the effect of proton pair formation the definition of IBEp is taken to be

IBEp = ½(B(n, p) − B(n, p-2))

However even with this modified method of computing the incremental binding energies of protons the graph displays a sawtooth pattern. An alternate approach is to plot the data for only the even values of the number of protons or only the odd values. Examples of this are shown in the graph below.

The curve is relatively smooth. However something visually subtle happens to the curve when the number of protons equals the number of neutrons. The curves change slopes becoming less steeply negative and there is a relatively sharper drop to the next data point. The general form is represented below where there is a sharp drop when the number of protons exceeds the number of neutrons. This is because when p<n an additional proton results in the formation of a neutron-proton spin pair. When p>n no such pair is formed.

In each of these three graphs the value of the Incremental Binding Energy when the number of neutrons equals the number of protons is about 15 MeV. This suggest plotting the information on the IBE from the three cases in one graph in which the independent variable is the difference between p and n.

This suggests that the IBE for protons is determined by the difference z=(p-n), but the slope of the relationship between IBEp and z depends upon the value of n.

A cubic function of z can give the shape involving the slight changes in the curvature of the line. The regression equation used to test this hypothesis is of the form

IBEp = c0 + c1z + c2z² + c3z³ + c4d(z<=0)

where d(z<=0) is equal to 1 if z<=0 and 0 otherwise.

The results of the regression for p=25 are:

IBE = 12.73954 − 1.89090z + 0.006z² − 0.00858z³ + 2.19807d(z<=0)
[-11.7] [0.6] [-3.2] [2.6]

The numbers shown in square brackets, [t], are the t-ratios for the regression coefficients above them. In order for a regression coefficient to be statistically significantly different for zero at the 95 percent of confidence its t-ratio must be greater than about 2 in magnitude.

The coefficient of determination (R²) for this equation is 0.99937, indicating that 99.93 percent of the variation in the IBE for protons in the nuclides with 25 neutrons is explained by variations in the explanatory variables. The explanation is that the IBEp is a cubic function of the difference in the number of protons and the number of neutrons plus an increment for the formation of a neutron-proton pair when the proton number is less than the neutron number.

The test of how well a cubic function of z plus a drop in IBE at z=0 explains the IBE of all protons was carried out using the 576 cases of even-even nuclides and the 570 cases of odd-odd nuclides for which the IBE for protons is available.

The results were

Regression Equation Estimates
n*d(z<=0) -0.36833

The inclusion of n as an explanatory variable was to allow for systematic shifts in the level of the IBEp as a function of the number of neutrons in the nuclide. The inclusion of n*z, n*z^2, n*z^3 and n*d(z<=0) is to allow for systematic shifts in the parameters of the relationship between IBEp and z.

The t-ratios for the coefficients were all greater than 4 in magnitude except for the coefficient of n*z for the odd-odd cases which was equal to 1.0.


The Incremental Binding Energy of protons is a function of the difference between the proton number and the neutron number with the parameters of the function dependent upon the neutron number. Effectively (n-p) is the net attraction for a proton. A proton is attracted to neutrons but repelled by other protons, not only due to the electrostatic force but through the strong force as well. The functional form the incremental binding energies of protons that is analogous to the quadratic function for the ionization energies of electron is

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