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The Functional Form of the Incremental Binding Energy of Protons as a Function
of the Number of Neutrons in the Nucleus Over the Full Range of Proton 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 proton in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less proton.

The incremental binding energies for nuclides containing the same number of protons but varying numbers of neutrons can be tabulated. Plots of such data will be shown below, but it is not feasible to show the plots for all possible proton numbers from one to over one hundred. Instead the data will be shown for a selected set of proton numbers.

Protons 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 protons versus the number of neutrons in a nuclide is displayed below for the set of nuclear magic numbers.

First the cases for three, six and eight protons 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 neutrons equals the number of protons. This is more pronounced for the case of the even proton numbers than for the odd proton number and is likely a pairing phenomenon. The second thing is that the curves have an increasing slope for low neutron numbers and a decreasing slope for higher neutron numbers. This is often described as an ogee shaped curve. A cubic polynomial of the neutron number can have this shape. If p and n are the number of neutron s 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 + c1n + c2n² + c3n³ + c4u(n=p) + c5v(n>=p)

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 = -8.48068 + 4.35525n − 0.27697n² + 0.00874n³ − 0.1838u(p=6) + 6.57159v(p≥6)
[3.3] [1.4] [1.1] [-0.1] [3.7]

The coefficient of determination (R²) is 0.9940. In this case there is only seven degrees of freedom. The numbers shown in brackets below the regression coefficients are the t-ratios. Only the coefficient for n and for v(p>=6) are significantly different from 0 at the 95 percent level of confidence.

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 proton Numbers

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

In each case the curves can be approximated by a cubic function, recognizing that a linear function is just a special case of a cubic function. There are jumps where the neutron number equals the proton number for n equal to 14, 20 and 28. For n=50 and n=82 the neutron number does not ever match the proton number. The relationships for the higher proton numbers are roughly linear, perhaps because they involve values all within the same neutron shell. A regression equation linear in the neutron number explains 99.6 percent of the variation in the incremental binding energy of the fiftieth proton.

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