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Cross Differences of the Binding
Energies of Neutron and Proton Spin Pairs:
That They Are Equal to the Interaction Energy
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 other variable. This difference is called the cross difference. It corresponds to the cross 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 cross derivative would be ∂²E/∂n∂m, which is the same as ∂²E/∂m∂n. These cross differences have some remarkable properties. For example, the same values would arise if first the incremental differences were calculated for an increase in the number of ν particles and the differences in these incremental values were computed for an increase in the number of μ particles. A more important property is:
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 n-1 ν particles superimposed over it in color.
The subtraction eliminates the interactions of the m-th μ particle with the other (m-1) μ particles. It also eliminates the interactions of the m-th μ particle with the (n-1) ν particles. What is left is the interaction of the m-th μ particle with the n-th ν particle.
The binding energies (mass deficits) are known for 2931 nuclides. The incremental binding energy of a proton pair is the difference between the binding energy of a nuclide and that of a nuclide with two less protons. Here is what the incremental binding energy of a proton pair looks like for various numbers of proton pairs and 21 to 25 neutron pairs.
The cross difference is the incremental binding energy of proton pairs less that for a nuclide with one less neutron pair. However it is more convenient to view the cross difference as being the slope of the relationship between the incremental binding energy of a proton pair as a function of the number of neutron pairs. This is illustrated for the data in the above graph as
Over a more complete range of the number of neutron pairs the relationships are:
The slope of the relationship is the interaction binding energy of the last proton pair with the last neutron pair. As can be seen this amount appears to be constant over a range corresponding to the shell having 15 to 25 neutron pairs. The value of this slope for the case of 30 protons (15 proton pairs) is 2.37 MeV. The values for the other cases are close to this value but differ systematically from it.
A graph for the incremental binding energies of neutrons which more or less corresponds to the one above is shown below.
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