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The Interaction of Nucleons in Nuclear Shells

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 nucleon (neutron or proton) in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less nucleon of the same type.

The incremental binding energies for nuclides containing the same number of neutrons but varying numbers of protons can be tabulated. Likewise such a tabulation can be created for nuclides containing the same number of protons but varying numbers of neutrons.

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.

Shell
Number
12345678
Capacity2481422324458
Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126127 to 184

The plots of the incremental binding energy of the 82nd proton versus the number of neutrons in a nuclide and the incremental binding energy of the 82nd neutron versus the number of protons in the nuclide are shown below.

Increments in the IBE

Consider the case of the nuclide with 82 neutrons and 55 protons. The incremental binding energy (IBE) of that 82nd neutron is 8.278 million electron volts (MeV). The binding energy of the nuclide with 82 neutrons and 55 protons represents the sum of all the interaction energies of the 82 neutrons with each other and with the 55 protons along with the interactions of the 55 protons with each other. More exactly it represents the sum of the interaction energies of the 82nd neutron with the 2 neutrons in the first neutron shell, the 4 neutrons in second shell, and so on up to the other 31 neutrons in the sixth shell along with the 2 protons in the first proton shell and so on up to the 5 protons in the sixth proton shell. In subtracting the binding energy of the nuclide with 81 neutrons and 55 protons the interactions of the 81 neutrons is eliminated, along with the interactions of the 81 neutrons with the 55 protons and the interactions of the 55 protons with each other. Therefore what is left in the incremental binding energy is interactions of the 82nd neutron with the other 81 neutrons and the interaction of the 82nd neutron with the 55 protons.

The incremental binding energy for the 82nd neutron in the nuclide with 56 protons includes the interactions of the 82nd neutron with the other 81 and the interaction of the 82nd neutrons with the 56 protons. The difference of the IBE's for the 82nd neutron for 56 and 55 protons leaves only the interaction of the 82nd neutron with the 56th proton.

The IBE for the 82nd neutron in the nuclide also containing 56 protons is 8.612 MeV. That increase of 0.334 MeV represents the interaction energy of the 82nd neutron with the 56th proton. The difference in the IBE of a neutron fluctuates as the number of protons varies. However the relationship between the IBE of the 82nd neutron and the number of protons, as shown above, is linear. Because that relationship between the IBE for neutrons and the number of protons is linear the slope of the relationship represents the interaction of the 82nd neutron with any proton in the sixth proton shell.

The regression of the IBE for the 82nd neutron on the number of protons in the sixth shell yields

IBE = −6.81274 + 0.27591p

The coefficient of determination (R²) for the equatation is 0.98882.

Thus the interaction energy for the 82nd neutron with a proton in the sixth proton shell is 0.27631 MeV.

The regression equation for the IBE of the 81st neutron on the number of protons in the sixth shell is

IBE = −8.76239 + 0.28208p

The slopes of the regression lines for the 82nd and the 81st neutrons are not significantly different at the 95 percent level of confidence. Likewise the slope of the regression line for the 80th neutron on the number of protons in the sixth shell is not significantly different at the 95 percent level of confidence from those for the 82nd and 81st neutron. Thus it appears possible and even likely that the interaction energies of any neutron in the sixth neutron shell with any proton in the sixth proton shell are the same.

This proposition can be examined more broadly in the following graphs of the data for a selected set of cases.


The case of n=68 is plotted in both graphs to allow comparisons between graphs. First note that all of the relationships are essentially linear. Only the data for the number of protons being above 50 and hence in the sixth shell are relevant. While there may be some systematic variation in the slopes of the relationships in the sixth shell it is slight. So if the proposition is not exact it is definitely a close approximation.

The problem now is establish the proposition that the slope of the relationship between IBE for neutrons and the number of protons is independent of the neutron number. This can be done by creating a data set for interactions between the neutrons in the sixth neutron shell and the protons in the sixth proton shell. Then the regression to test the proposition of the form

IBE = c0 + d0n + (c1+d1n)p
which is equivalent to
IBE = c0 + d0n + c1p + d1np

If the regression coefficient d1 is not significantly different from zero then the proposition is established. The regression with the data for n*p left out would give c1 as the proper value for the interaction energy for neutrons in the sixth shell with protons in the sixth shell.

The results of the regression are:

IBE = 12.59407 − 0.26847n + 0.19147p + 0.00137np
[−2.3] [1.2] [0.6]

The numbers in the square brackets above are the t-ratios for the corresponding coefficients. The t-ratio for a regression coefficient is its value divided by the standard deviation for that coefficient. For a coefficient to be significantly different from zero at the 95 percent level of confidence its t-ratio must be two or greater in magnitude.

Thus the coefficient of np is not signficantly different from 0 at the 95 percent level of confidence.

When the variable np is removed from the equation the result is

IBE = 6.86158 − 0.19351n + 0.29621p
[−19.4] [20.9]

The coefficient of determination for this equation is only 0.62.

The results indicate that the best estimate of the interaction energy of a sixth shell neutron with a sixth shell proton is 0.296 MeV with a standard deviation of 0.014 MeV.

For the interactions of fifth shell neutrons with fifth shell protons the regression equation is

IBE = 5.43541 − 0.33073n + 0.61153p − 0.0017np
[−2.4] [3.2] [−0.4]

Thus the coefficient for the np term is not significantly different from zero at the 95 percent level of confidence. When that variable is dropped from the analysis the resulting regression equation is:

IBE = 8.08292 −0.38922n + 0.53160p
[−21.6] [25.0]

Its coefficient of determination is 0.73834.

Therefore the best estimate of the interaction energy of a fifth shell neutron with a fifth shell proton is 0.53160 MeV.d

The regression equation for the interaction energies of fourth shell neutrons with fourth shell protons is

IBE = 0.59692 − 0.41029n + 1.38031p − 0.01726np
[−2.1] [5.8] [−1.8]

Again the coefficient for the np variable is not significantly different from zero at the 95 percent level of confidence.

With the np variable dropped the regression equation is

IBE = 8.91559 −0.75106n + 0.95538p
[−18.6] [28.4]

The interactions of third shell neutrons with third shell protons lead to the following regression equaton.

IBE = 7.40409 − 1.19103n + 1.69767p − 0.00492np
[−2.0] [2.5] [−0.1]

The coefficient of the np variable is decidedly not significantly different from zero.

With the np variable dropped from the regression the result is

IBE = 8.01102 −1.24188n + 1.63814p
[−7.6] [15.3]

The coefficient of determination for this regression is 0.77445.

Finally there are the interactions of second shell neutrons with second shell protons. The regression equation is

IBE = 6.94453 − 2.48600n + 2.18523p − 0.25845np
[−0.9] [2.5] [0.8]

In this case not only is the coefficient for the np variable not significantly different from zero at the 95 percent level of confidence, but neither are the coefficients for the other variables.

But if only the np variable is dropped from the regression the result is

IBE = 0.41728 −1.1912n + 3.51166p
[−1.4] [6.8]

The coefficient of determination for this equation is 0.71220. Although the coefficient for n is not significantly different from zero in this equation the coefficient of p in this equation will be used as the best estimate of the interaction energy for a second shell neutron with a second shell proton; i.e., 3.5116 MeV.

There are interactions of neutrons in the first shell with protons in the first shell but not enough data points to make an estimate of the effects.

There are no interactions of seventh shell neutrons with seventh shell protons. There are of course interactions of seventh shell neutrons with sixth and lesser shell protons, but that will be left for a separate study.

The Incremental Binding Energies of Protons

The same proposition can be established for the IBE of protons.The IBE of the 82nd proton in a nuclide with 114 neutrons is 4.44 MeV. This represents the sum of the interaction energies of the 82nd proton with the other 81 protons plus its interaction energies with the 114 neutrons. The 82nd proton is in the sixth proton shell and the 114th neutron is in the seventh neutron shell.

For the IBE of protons the regression equation is

IBEp = −23.26423 + 0.24473n

The coefficient of determination (R²) for this equation is 0.98507. Thus the interaction energy of the 82nd proton with a neutron in the seventh shell is 0.24473 MeV. The full analysis of the incremental binding energies of protons is given Elsewhere.


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