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The Cross Difference of Binding Energies Is
Equal to the Interaction Energy of the Last
Neutron with the Last Proton in the Nuclide

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 IBE of the 56 proton in the nuclide with 81 neutrons and 56 protons is 8.671 MeV. The IBE of the proton in the nuclide with 56 protons and 82 neutrons is 9.005 MeV. The difference of these two IBEs is 0.334 MeV. The equality of this number with the one derived from the IBEs for neutrons is an example of the equality of cross differences.

The increase with respect to the number of protons in the incremental binding energies of neutrons is equal to the interaction of the last neutron with the last proton.

Proof:
Consider a nuclide with n neutrons and p protons. The binding energy of that nuclide represents the net sum of the interactions of all n neutrons with each other, all p protons with each other and all np interactions of neutrons with protons.

   

The black squares indicate there are not any interactions of a nucleon with itself.

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

   

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

  

Now consider the difference of the IBE for n neutrons and p protons and the IBE for n neutrons and p-1 protons. In the diagrams below the interactions for the IBE for the nuclide with (p-1) protons are shown colored.

  

The subtraction eliminates the interactions of the n-th neutron with the other (n-1) neutrons. It also eliminates the interactions of the n-th with the (p-1) protons. What is left is the interaction of the n-th neutron with the p-th proton.

Empirical Relationships

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 Energies of Interactions Between Neutrons and Protons
Based on the Incremental Binding Energies of Protons
(All figures in MeV)
 Neutron Shell Number
 12345678
Proton
Shell
Number
1
24.023931.03291
31.951211.03291
41.071540.66176
50.531600.4023
60.334460.27973
7

These results can be compared with those from the IBE of neutrons from a previous paper

The Energies of Interactions Between Neutrons and Protons
Based on the Incremental Binding Energies of Neutrons
(All figures in MeV)
 Neutron Shell Number
 12345678
Proton
Shell
Number
1
23.511661.63182
31.638141.12476
40.955380.68575
50.531600.41951
60.296210.28800
70.22505

The Sensitivity of the Results to Possible
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 neutrons and protons 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 n neutrons is the difference between its binding energy and that of the nuclide with the same number of protons but (n-1) neutrons. Thus the incremental binding energy is in error by an amount Δ.

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

The figures given in the above table are independent of any error in the mass of the neutron.


For material on the case for the second differences of binding energy being the interaction energy of the last nucleon with the next-to-last nucleon of that same type see Second Differences, Double Pairing of Neutrons, or Double Pairing of Protons.


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