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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 Over the Full Range of Neutron Numbers

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 such data will be shown below, but it is not feasible to show the plots for all possible neutron numbers from one to over one hundred. Instead the data will be shown for a selected set of neutron numbers.

Neutrons are arranged 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 plots of the incremental binding energy of neutrons versus the number of protons in a nuclide is displayed below for the set of nuclear magic numbers.

First the cases for three, six and eight neutrons will be considered.

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 more pronounced for the case of the even neutron numbers than for the odd neutron number and is likely 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. This is often described as an ogee shaped curve. 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) + c5v(p=n)

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

The regression results for n=6 are

IBE = 8.40469 − 6.40891p + 1.99628p² − 0.11953p³ + 3.10676u(p=6) − 0.38327v(p≥3)
[-4.5] [6.6] [-6.8] [6.3] [-0.5]

The coefficient of determination (R²) is 0.99977. In this case there is only three degrees of freedom. The numbers shown in brackets below the regression coefficients are the t-ratios. Only the coefficient for v(p>=6) is not significantly different from 0 at the 95 percent level of confidence.

When the variable v(p>=6) is eliminated from the regression the results are

IBE = 7.75368 − 5.89964p + 1.88522p² − 0.1131p³ + 2.93145u(p=6)
[-7.8] [12.6] [-12.4] [11.2]

The coefficient of determination (R²) is 0.99974 and all of the coefficients are highly significantly different from zero.

The conclusion is that a cubic equation with a jump at p=n fits the data quite well.

The Functional Relationship for Higher Neutron Numbers

The plotted data for n equal to 14, 20, 28, 50, 82 and 126 are:

In each case the curves except n=14 the relationships can be approximated by a cubic function, recognizing that a linear function is just a special case of a cubic function. In the case of n=14 the relationship seems to have three arcs instead of the two for an ogee curve. This would require a quartic function to approximate it. There are jumps where the proton number equals the neutron number for n equal to 14, 20 and 28. For n=50 and above the proton number does not reach the neutron number. The relationships for the higher neutron numbers are roughly linear, perhaps because they involve values all within the same neutron shell. A regression equation linear in the proton number explains 98.7 percent of the variation in the incremental binding energy of the fiftieth neutron.


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