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The Second Difference of Binding Energies with
Respect to the Number of Protons Is Equal to the
Interaction Energy of the Last proton with the
Next to Last Proton in the Nuclide

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 (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 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 plot of the incremental binding energy of the 20th proton versus the number of neutrons in the nuclide is shown below.

The graph shows neatly the magicality of 14 and 20 where the IBE drops sharply. A sharp drop in the IBE is an indication that a shell or subshell has been filled and additional nucleons are going into a higher shell. The sawtooth pattern is due to the energy involved in the formation of spin pairs.

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 a proton with itself. The diagram might seem to suggest a double counting of the interactions but that is not the case because only the interactions above the black square are counted.

What is given below is the interactions for a nuclide of p protons and n neutrons overlaid with those of a nuclide with (p-1) protons and n neutrons, shown in color.

  

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. When the subtraction is carried out the interactions of the n neutrons with each other are entirely eliminated. It also eliminates the interactions of the p-1 protons with each other and the p-1 protons with the n neutrons. These are the squares shown in white above.

Now consider the incremental binding energy for the nuclide with (p-1) protons and n neutrons. The interactions for this IBE are shown in color along with those for the IBE for a neutron in a nuclide with p protons and n neutrons shown in white.

 

Now consider the difference of the IBE for p protons and n neutrons and the IBE for (p-1) protons and n neutrons.

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. Visually this is the subtraction the values in the blue squares from the white squares on the same level. 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 proton spin pair. The graph below illustrates the effect of spin pair formation.

The Separation of the Interaction Energy of
Successive Protons into Two Components

Within a shell the average of the IBE for the (n-2)-th and n-th proton give a good approximation of what the IBE would be if there were no spin pair formed.

The difference between the actual IBE and the average of the adjacent figures is a good approximation of the energy involved in the pair formation. The change in the IBE due to the nuclear strong force per additional neutron is computed as one half of the change in IBE between the value for the n-th proton and the value for the (n-2)-th neutron. Its value is always negative, indicating that the nuclear strong force between two protons is a repulsion.

The decline and then rise in the values is associated with the filling of the fourth proton shell and the passage to the fifth shell at 28 protons. The pattern with a single shell is approximately linear.

Below are some illustration of the relationship for other neutron numbers

In this case the downward spike is associated with the magic number 28.

The Sensitivity of the Results to Possible Errors
in the Conventional Mass of the Neutron

The binding energy of all nuclides are computed as the energy value of its mass deficit. The mass deficit of all nuclides except one are computed as the difference between the mass of their constituent protons and neutrons and the mass of the nuclide. The mass of any charged particle can be measured by injecting it into a magnetic field and measuring the radius of the orbit it makes. The mass of a neutron, since it is a neutral particle, cannot be measured in this way. Instead its mass is deduced from the masses of a deuteron and a proton and an estimate of the mass deficit of the the deuteron. When a deutron is formed a gamma photon of energy of 2.25 million electron volts is emitted. This is taken to be the mass deficit of the deuteron. This may not be the correct value for the mass deficit of the deuteron.

If the mass of the neutron is in error by an energy amount Δ then the binding energy of any nuclide with n neutrons is in error by nΔ. The incremental binding energy of a nuclide with p protons is the difference between its binding energy and that of the nuclide with the same number of neutrons but (p-1) protons. Since the number of neutrons is the same any error cancels. Thus the incremental binding energy of protons would be independent of any error in the mass of a neutron.

Thus any difference in incremental binding energies of
protons is unaffected by any error in the mass of the neutron.

Thus the magnitude of the second differences in binding energy are meaningful and their signs will not be altered by a correction in the mass of the neutron.

Conclusion

The conclusion to be drawn is that the second differences with respect to the number of protons are a linear function of the number of protons in the shell up to the point of near the complete filling of the shell. The second differences represent the interaction of a proton with the previous one in the nuclide. The interaction of the last two protons is almost always negative thus indicating a repulsion between them due to the nuclear strong force.


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