This is a generalization of material previously applied to the binding energy of nuclides in terms of the number of neutrons and protons they contain. The theorems presented equally well apply to any quantity that is the sum of the interactions between the component particles. The enitites may be nuclei but also they may be molecules or superatoms. The particles may be composite, such as nucleon spin pairs, alpha modules or ion radicals.

There is one theorem to the effect that in systems, such as nuclei, made up of two or more types of particles, such as neutrons and protons, the cross diffference in the sum of interaction energies is equal to the energy of the interaction of the last particle of one type with the last particle of the other type. This is a solid and useful theorem for investigating the binding energies of nuclides and thus nuclear structure.

There is another useful theorem to the effect that in such systems the second diffference with respect to the number of one type of particle is equal to the energy for the interaction of the last particle of that type with the next-to-the-last particle of the same type. What followslater is a proof of that proposition subject to an additional crucial assumption. It will apply to systems made up of particles which may be composite.

The situation presumed is that of a system made up of two types of particles, referred to as A-particles and N-particles. The energy of the system is the sum of the energies due to the interaction of the particles.

It is also assumed that there are shells for the particles. When a shell is filled any additional particles must go into higher shells where their interaction energy is lower with the particles in the lower shells.

Visual Demonstration of the Proposition
Concerning Cross Differences

Consider a system with a A-particles and n N-particles. The interaction energy of that system represents the net sum of the interaction energies of all a A-particles with each other, all n N-particles with each other and all mn interactions of the a A-particles with the n N-particles. Below is a schematic depiction of the interactions.

The black squares are to indicate that there is no interaction of a particle with

What is given below is the interactions for a system of a A-particles and n N-particles overlaid with those of a system with (a-1) A-particles and n N-particles, shown in color.

The incremental interaction energy of an A-particle is the difference in the interaction energy of the system with a A-particles and n N-particles and that of the system with (a-1) A-particles and n N-particles. When the subtraction is carried out the interactions of the n N-particles with each other are entirely eliminated. It also eliminates the interactions of the (a-1) A-particles with each other and the (a-1) A-particles with the n N-particles. The interactions which are left after the subtraction are the squares shown in white above. The sum of the interaction energies in these white squares is the incremental interaction energy of the a-th A-particle with n N-particles present.

Now consider the incremental interaction energy for the nuclide with a A-particles and (n-1) N-particles. These interactions are shown imposed in color over those for the n N-particles in the nuclide with a A-particles and n N-particles shown in white.

Now consider the difference of the incremental interaction energy for a-th A-particle and n-th N-particle and the incremental interaction energy for the nuclide with a A-particle and (n-1)-th N-particle.

This means that the cross difference in the interaction energy for a A-particles and n N-particles is equal to the interaction energy between the a-th A-particle and n-th N-particle. Another way of representing this is that the interaction energy between the a-th A-particle and n-th N-particlei is equal to the slope of the relationship between the incremental interaction energies of A-particles plotted versus the number of N-particles. It is also equal to the slope of the relationship between the incremental interaction energies of N-particles plotted versus the number of A-particles.

An Illustration

Let the A-particles be proton pairs and the N-particles neutron pairs. The graph of the incremental binding energy of a proton pair as a function of the number of neutron pairs in the nuclide for nuclides with 26 proton pairs is shown below.

The relationship is nearly a straight line over the range of the neutron shell that goes from 26 neutron pairs to 41. The slope of the relation, based upon a linear regression, is 0.69764 MeV and, according to the theorem, this represents the interaction binding energy of the 26-th proton pair with any neutron pair in the 26 to 41 neutron pair shell..

For a comparison, the graph below shows the interaction of the 25-th proton pair with the neutron pairs in the 26 to 41 shell. The 25-th proton pair is in the next lower neutron shell. This relation is also nearly linear. The slope of the relation over the 26 to 41 shell is 1.609 MeV.

The incremental binding energies of neutron pairs can also be tabulated.

Again the relationship is nearly linear. The slope of the relation over the 14 to 24 shell is 1.94428 MeV. What the linearity of this relationship means that in the table shown here

all the interactions in a horizontal row are equal. The previous relationship for the incremental binding energies of proton pairs means the interactions in a vertical column are equal and thus all the interactions in the table are equal. In effect, all of the interactions between neutrons in shell X and protons in shell Y are the same. It would be expected that this extends to the proposition that the interactions between nucleons of type U in shell X and nucleons of type V in shell Y are equal, where U and V can be the same and likewise for X and Y.

A Strict Second Differences Theorem

Under that assumption that the interaction of two particles is a function only of the shells in which the two particles are located, the second diffference of the interaction with respect to the number of one type of particle is equal to the energy for the interaction of the last particle of that type with the next-to-the-last particle of the same type.

As before, the interactions of a A-particles and n N-particles can be represented as

What is given below is the interactions for a system of a A-particles and n N-particles overlaid with those of a system with (a-1) A-particles and n N-particles, shown in color.

The incremental interaction energy of an A-particle is the difference in the interaction energy of the system with a A-particles and n N-particles and that of the system with (a-1) A-particles and n N-particles. When the subtraction is carried out the interactions of the n N-particles with each other are entirely eliminated. It also eliminates the interactions of the (a-1) A-particles with each other and the (a-1) A-particles with the n N-particles. The interactions which are left after the subtraction are the squares shown in white above. The sum of the interaction energies in these white squares is the incremental interaction energy of the a-th A-particle with n N-particles present.

Now consider the incremental binding energy for a A-particles and (a-1) A-particles. These are displayed side-by-side below.

Under the assumption that the interactions of a particle with all the particles in a shell are the same the interactions in the same horizontal row are the same and disappear when the colored squares are subtracted from the white squares, except for the interaction of the a-th A-particle with the (a-1)-th A-particle.

The theorem therefore says that the second difference with respect to the number of A-particles is equal to the interaction between the a A-particle with the (a-1)-th A-paricle. As in the case of the cross difference, it is convenient to represent that interaction as the slope of the relationship between the increment with respect to the number of A-particles plotted against the number of A particles.

More Illustration

Again let the A-particles be proton pairs and the N-particles neutron pairs. Here is the relationship between the incremental binding energy of a proton pair and the number of proton pairs in the nuclide for nuclides containing 25 neutron pairs.

The relationship is downward sloping to the right, indicating the force associated with the interaction of proton pairs is a repulsion, and nearly linear, thus indicating that the level of the interactions for all proton pairs in a shell are the same. This is just one case but the relationships for all cases show the same characteristics.

The linear regression of the incremental binding energy of a proton pair on the number of proton pairs indicate that the interaction binding energy of a proton pair is 1.75890 MeV.

The corresponding relationship for neutron pairs is as follows.

The relationship is nearly linear within the 26 to 41 shell. The linear regression indicates that the interaction binding energy of two neutron pairs in that shell is −0.76921 MeV.

Conclusion

The cross difference of the interaction energies of nuclides with respect to the numbers of particles of two different types Is equal to the interaction energy of the last particles of those types in the nuclide.

The second difference of the interaction energies of nuclides with respect to the number of particlesof a particular type Is equal to the interaction energy of the last particle of that type with the next-to-last particle of the same type in the nuclide.