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A Comparison of the Relationships Between the Incremental
Binding Energy of Neutrons and Protons and the
Number of the Other Nucleons 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.

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 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 plots of the incremental binding energies of protons versus the number of neutrons in a nuclide and the incremental binding energies of neutrons versus the number of protons in the nuclide are displayed in the same graphs for the set of nuclear magic numbers.

First the cases for the second proton and the second neutron will be considered.

The values are so close that data points for neutrons and protons cannot be distinguished. There is a difference and the separation between the case for neutrons and that for protons becomes larger for nucleons in higher shells.

The parallelism is not perfect but it is close. The major deviation is for the nuclides containing nine neutrons or nine protons. The ratio of the standard deviation of the differences excluding the one for 9 nucleons to their average value is 6.7 percent. This is called the coefficient of variation.

The parallelism above is notable. The ratio of the standard deviation of the differences to their average value is 6.8 percent.

For the fourteenth nucleon case the parallelism is not so definite. Below 14 the two curves seem to have the opposite curvature. However the coefficient of variation for these differences over the entire range is only 4.3 percent, largely because of the higher average value.

For the twentieth nucleon case the coefficient of variation the differences is only 2.7 percent. This is lower than the previous case because of a slightly lower standard deviation and a significantly higher average value.

For the twenty eighth nucleon case the coefficient of variation the differences is only 1.6 percent. Again this is lower than the previous case because of a slightly lower standard deviation and a significantly higher average value.

For the remaining cases there are no overlaps where the differences can be computed.

The slopes of the two lines are different and the difference is significantly different from zero at the 95 percent level of confidence.

Below are plotted the average differences for the cases of 2, 6, 8, 14, 20 and 28. The linearity is extraordinary.

The coefficient of determination (R²) for this relationship is 0.9895.


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