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The Functional Form of the Incremental Binding Energy of Neutrons
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 neutron 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 proton number 26, the isotopes of iron, versus the number of neutrons in the nuclide is shown below.

The sawtooth pattern is due to the energy involved in the formation of neutron 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 (IBEn) of the n-th neutron is

IBEn = B(n, p) − B(n-1, p)

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

IBEn = ½(B(n, p) − B(n-2, p))

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

The curve is relatively smooth. However something happens to the curve when the number of neutrons equals the number of protons. The general form is represented below where there is a sharp drop when the number of neutrons exceeds the number of protons. This is because when n<p an additional neutron results in the formation of a neutron-proton spin pair. When n>p no such pair is formed.

The the corresponding graphs for the cases for p equal 25 and 27 are shown below.

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 n and p.

This suggests that the IBE for neutrons is determined by the difference z=(n-p).

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

IBE = 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=26 are:

IBE = 13.14337 − 0.69479z − 0.00392z² + 0.00102z³ + 1.76719d(z<=0)
[-13.1] [-0.5] [2.8] [5.0]

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 0.9985, indicating that 99.85 percent of the variation in the IBE for neutrons in the isotopes of iron is explained by variations in the explanatory variables. The explanation is that the IBEn is a cubic function of the difference in the number of neutrons and the number of protons plus an increment for the formation of a neutron-proton pair when the neutron number is less than the proton number.

The test of how well a cubic function of z plus a drop in IBE at z=0 explains the IBE of all neutrons was carried out using the 603 cases of even-even nuclides and the 676 cases of odd-odd nuclides for which the IBE for neutrons is available. The regression equation used was of the form

IBE = c0 + c1z + c2z² + c3z³ + c4d(z<=0) + c5p

The inclusion of p as an explanatory variable was to allow for systematic shifts in the level of the IBE as a function of the number of protons in the nuclide.

The results were

Regression Equation Estimates

The t-ratios for the coefficients were all greater than 7 in magnitude. This is a notable statistical performance, but a viewing of the graphs for other proton numbers shown below indicate that the slope of the relationship between IBE and z depends upon p.

In this case the sharp drop on the right is a result of the number of neutrons reaching and exceeding 82 and thus filling and over filling the sixth neutron shell.

In this case the drop comes when the number of neutrons reaches 28. The patterns for the fourth and fifth shells are similar enough that the drop is not obvious.

In this case the sharp drop comes at 126 neutrons.

Clearly there are systematic variations in the patterns as a function of the proton number p.

A regression in which the slope of the relation between IBE and z depends upon p gives the following results for the odd-odd cases.

IBE = 7.22408 − 0.5008z + 0.01097z² − 8.84525×10-5z³ −4.70319d(z<=0) + 0.150138p − 0.00133p*z
[-20.6] [10.6] [-7.9] [18.7] [32.4] [-6.3]
R² = 0.90233

The coefficient of p*z is statistically significant and indicates that the slope of the relationship between IBE and z becomes more negative for increasing levels of p.

If the dependences of the relationships between IBE and z², z³ and d(z<=0) are functions of p then the statistical fit is improved to 0.93984 for the even-even cases and 0.92335 for the odd-odd cases.


The Incremental Binding Energy of neutrons is a function of the difference between the neutron number and the proton number with the parameters of the function dependent upon the proton number. The functional dependence can also be thought of in terms of (p-n). Effectively this makes (p-n) the net attraction for a neutron. A neutron is attracted to protons but repelled by other neutrons. The functional form the incremental binding energies of neutrons that is analogous to the quadratic function for the ionization energies of electron is

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