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Illustrations of the Effect of Double Pairing
on the Incremental Binding Energy of Protons

Mass Deficits and Binding Energies

The mass of a nuclide, such as helium 4, the alpha particle, is less than the masses of the two protons and two neutrons 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.

Incremental Binding Energies

The incremental binding energy (IBE) of a nucleon (proton or neutron) 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 of a proton 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.

Nucleonic 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 protons and neutrons, 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 Nature of the Incremental Binding Energy of the Protons

The increase in the incremental binding energies of a proton as a result of an increase in the number of protons is equal to the interaction of the last proton with the next to last last proton, provided these two are in the same proton shell.

Rationale:
Consider a nuclide with p protons and n neutrons. The binding energy of that nuclide represents the net sum of the interaction energies of all p protons with each other, all n neutrons with each other and all np interactions of protons with neutrons. Below is a schematic depiction of the interactions.

  

The black squares are to indicate that there is no interaction of any nucleon with itself. The diagram might seem to suggest a double counting of the interactions but that is not the case.

The proton incremental binding energy is the difference in the binding energy of the nuclide with p protons and n neutrons and that of the nuclide with p-1 protons and n neutrons. In the diagrams below the interactions for the nuclide with (p-1) protons and n neutrons are colored.

  

That subtraction eliminates all the interactions of the n neutrons with each other. It also eliminates the interactions of the p-1 protons with each other and the p-1 protons with the n neutrons. What is left is the interaction of the p-th proton with the other p-1 protons and the interaction of the p-th proton with the n neutrons.

Now consider the difference of the IBE for n protons and p neutrons and the IBE for (n-1) protons and p neutrons. The diagrams below depict the situation.

 

The subtraction of the IBE for (p-1) protons and n neutrons from the IBE for p protons and n neutrons depends upon the magnitude of the interaction of the (p-1)-th proton with the different protons compared to the interaction of the p-th proton with those same protons. When the p-th and the (p-1)-th protons are in the same shell the magnitude of the interactions with any other proton are, to the first order of approximation, equal. Thus the interactions with the n neutrons are entirely eliminated. Likewise for the first (p-2) protons. All that is left is the interaction of the p-th proton with the (p-1)-th proton.

Note that the interaction of the p-th and (p-1)-th protons may or may not involve the interaction associated with the formation of a spin pair.

Empirical Tests

The plot of the incremental binding energies of protons in the nuclides with neutron number 30 versus the number of protons in the nuclide is shown below.

The graph shows the magicality of 20 and 28 where the IBE drops sharply as a result of a proton shell being filled and the next proton going into a higher shell or subshell The sawtooth pattern is due to the energy involved in the formation of spin pairs of protons. The equally sharp drop after 30 protons might appear to indicate some magicality to 30 but such is not the case. When the number of protons is less than the number of neutrons each additional proton results in the formation of a neutron-proton pair. Beyond the point where the number of protons equals the number of neutrons no pair can be formed. The level of the IBE drops. If the neutron number is even then the drop in IBE due to no proton-proton pair being formed combines with the drop due to no proton-proton pair being formed to produce a notably sharp drop in the IBE for the next proton number.

Here is the graph for the case of the nuclides with neutron number 32.

Here again there is a notably sharp drop when the number of protons exceeds the neutron number of 32.

The case of an odd neutron number is of interest. Here is the graph for the nuclides with neutron number 29.

The addition of the 30th proton brings the effect of a proton-proton pair but a neutron-proton pair is not formed, as was the case up to and including the 29th proton. The effects almost but not exactly cancel each out. It is notable that the binding energies involved in the two types of nucleonic pairs are almost exactly the same.

This same pattern is seen in the case for the nuclides with 31 neutrons.

In this case there is also unexplained similar pattern at 17 protons.

The values of the IBE of protons within a particlulr shell for a particular neutron number can be explained by an equation of the form

IBE = c0 + c1u(p≤n) + c2m + c3e

where m is the number of protons in the shell (p-14), u(p≤n) is 1 if p≤n and 0 otherwise and e is 1 if p is even and 0 otherwise.

The regression equation for the nuclides with 29 neutrons and proton numbers in the fifth shell (n=29 to 50) is

IBE = 2.672 + 1.039u(n≤29) − 0.8380m + 1.768e
[2.3] [-7.1] [6.7]

The coefficient of the determination for this equation is 0.9959. This means that almost 99.6 percent of the variation in the IBE of protons in those nuclides with 29 neutrons are explained by variations in the three variables. Unfortunate there are only five observations. The coefficient of e, 1.768 MeV, is the binding energy accounted for by the formation of a proton-proton pair. The coefficient of u(n≤35), 1.039 MeV, is the binding energy accounted for by the formation of a neutron-proton pair. The values are definitely not equal. The coefficient of m, −0.838 MeV, is the binding energy resulting from the interaction of successive protons. The fact that it is negative indicates that the force between two protons is a repulsion.

The regression equation for the nuclides with 29 neutrons and proton numbers in the fourth shell (n=15 to 28) is

IBE = 13.2774 + 2.06919u(n≤36) − 0.30390m + 3.17100e
[13.2] [-25.1] [35.3]

The interaction energy associated with the formation of a proton-proton pair is 3.171 MeV. The binding energy resulting from the interaction of successive protons is −0.30390 MeV. The coefficient of determination for the the regression is 0.97616.


For material on the case for the cross differences of binding energy being the interaction energy of the last nucleon with the last nucleon of the opposite type see Cross Differences. For material on the differences in the incremental binding energies of protons see Protons.


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