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A Proposition Concerning the Second Differences
of the Binding Energies of Neutron and Proton Spin Pairs:

That They Are Equal to the Interaction Energy of the Last
Particles of Each Type with the Next-to-Last Particle
of the Same Type

Nuclei are largely made up of neutron and proton spin pairs. The binding energies of nuclei are then composed of two types of effects. One is the binding energies associated with the formation of substructures composed of neutrons and protons. The other is the interactions among the neutrons and protons through the strong force. Furthermore the nucleon pairs are organized in shells, which are identified in terms of the so-called magic numbers.

In the following the proposition being considered is just a special case of the general proposition which will be stated in terms of μ particles and ν particles.

Let the numbers of μ particles and ν particles and their interactions be depicted by the following diagram.

  

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

What is being considered is the increment due to a change in one variable in the increment due to a change in the same variable. This difference is called the second difference. It corresponds to the second derivative in calculus. If the binding energy E is a function of the number of μparticles m and the number of ν particles n, and m and n were continuous variables then the second derivatives would be ∂²E/∂m² and ∂²E/∂n². An important property is:

The increase with respect to the number of μ particles in the incremental increase in energy due to an increase in the number of μ particles is equal to the interaction energy of the last μ particle with the next-to-last μ particle.

Proof:
Consider once again the depiction of the interactions of the m μ particles and the n ν particles.

  

Consider now the incremental energy with respect to a unit change in the number of μ particles.

  

The interactions for m-1 μ particles are shown in color. The difference, the incremental energy, is then the sum of all the squares shown in white. These are shown below.

 

Below the increment with respect to the number of μ particles for n ν particles is shown with the the increment for m-1 μ particles superimposed over it in color.

 

The subtraction eliminates the interactions of the (m-1)-th μ particle with the other (m-2) μ particles. It also subtracts the interactions of the (m-1)-th μ particle with the n ν particles. Empirically the interaction of the (m-1)-th μ with each of the n ν tends to be the same as that of the m-th μ particle with each of the n ν particles. Thus the interactions of the μ particle with the ν particles are eliminated. What is left is the interaction of the m-th μ particle with the (m-1)-th μ particle.

Empirical Results

The binding energies (mass deficits) are known for 2931 nuclides. The incremental binding energy of a proton or neutron pair is the difference between the binding energy of a nuclide and that of a nuclide with two less of the same particle. Here is what the incremental binding energies of a proton and a neutron pair looks like for various numbers of pairs and 21 to 25 of the other pairs.

The second difference is the incremental binding energy of a pair type less that for a nuclide with one less pair of the same type. Here are the second differences for the above data.

However it is often more convenient to view the second difference as being the slope of the relationship between the incremental binding energy of a pair type as a function of the number of pairs of the same type.


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