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A Generalization of a Theorem on the Cross
Differences of the Binding Energies of Nuclei

Background

There is a theorem arising out of the physics of nuclei that the cross difference in binding energy with respect to the number of neutrons and the number of protons Is equal to the interaction binding energy of the last proton with the last neutron in the nuclide. This is a generalization of that proposition. First the quantity under consideration for a system of particles is due to their interaction. It will be callled their interaction energy, although the interaction is the crucial element rather than energy. The system is presumed to be composed of two types of particles, which will be called a and b types. The particles may be composite and possess an intrinsic internal energy.

The increase in interactive energy with respect to the number of b-particles in the incremental interactive energies of the a-particles is equal to the interaction energy of the last a-particle particle with the last b-particle.

Proof:
First a simple visual proof will be given before an algebraic demonstration.

Consider a nuclide with m a-particles and n b-particles. The interaction energy of that nuclide represents the net sum of the interactions of all m a-particles with each other, all n b-particles with each other and all mn interactions of the a-particles particles with the n b-particles. For this demonstration it is presumed that there the particles possess no intrinsic interactive energy.

   

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

The incremental interactive binding energy of an a-particle is the difference in the excess binding energy of the nuclide with m a-particles and n b-particles and that of the nuclide with m-1 a-particles and n b-particles. In the diagrams below the interactions of the nuclide with (m-1) a-particles and n b-particle are shown in color.

   

The subtraction eliminates all the interactions of the n b-particles with each other. It also eliminates the interactions of the (m-1) a-particles with each other and the (m-1) a-particles with the n b-particles. What is left is the interaction of the m-th a-particle with the other (m-1) a-particles and the interaction of the m-th a-particle with the n-th b-particle.

  

Now consider the difference of the incremental interactive energy for an a-particle and n b-particles and that for an a-particle and the (n-1) b-particles. In the diagrams below the interactions for the incremental interactive energy for the nuclide with (n-1) b-particles are shown colored.

  

The subtraction eliminates the interactions of the m-th a-particle with the other (m-1) a-particles. It also eliminates the interactions of the m-th a-particle with the (n-1) b-particles. What is left is the interaction of the m-th a-particle with the n-th b-particle.

Algebraic Proof:

Let i and j be indices for the a-particles and k and l the indices for the b-particles. The energy of the interaction between the i-th and j-th a-particles is denoted as Iuij and between the k-th and l-th b-particles as Jk l. The interaction energy between the i-th a-particle and the k-th b-particle is denoted as Kik. For now it is assumed that there are no intrinsic interactive energies for the particles. (One symbol for energy could have been used but using three makes it easier to see what is involved.) The values of Iij need be defined only for j<i to avoid double counting and noting there is no interaction of a particle with itself. The same applies for J and k and l. The same restriction is not applied to the interactions between the two types of particles given by Kik.

The interactive energy of the system is given by

E(m, n) = Σ i=1mΣj=1i-1Iij + Σ k=1nΣl=1j-1Jkl + Σ i=1mΣj=1nKik

The incremental interactive energy with respect to the number of a-particles is given by

ΔmE(m, n) = E(m, n) − E(m-1, n) = Σj=1m-1Imj + Σj=1nKmk

The cross difference is the increment with respect to the number of b-particles of the increment in interactive energy with respect to the number of a-particles is then

Δ2E(m, n) = ΔnmE(m, n)) = Kmn

The cross difference in the other directions gives the same value; i.e.,

ΔmnE(m, n)) = ΔnmE(m, n)) = Kmn

Intrinsic Energy of Particles

Now suppose each a-particle has intrinsic energy μ and each b-particle intrinsic enery ν. The total interactive energy H is then the interactive energy E given above and the total intrinsic energy:

H = E + mμ + nν

The increment in H with respect to the number of a-particles is then

ΔmH(m, n) = ΔmE(m, n) + μ

The subtraction with respect to the number of b-particles eliminates the intrinsic energy μ of an a-particle. Therefore

Δ2H(m, n) = Δ2E(m, n) = Kmn

Thus the existence of intrinsic energy of the particles has no effect on the value of the cross difference being equal to the interaction between the last a-particle and the last b-particle.

A Simple Extension of the Proposition

There is no reason to limit the types of particles to just two. If there is a third type, say c-particles, the first difference with respect to the number of a-particles would eliminate the interactions of the particles of that type with each other. What would be left is the interactions of the last added a-particle with each of the c-particles. The differencing with respect to the number of b-particles eliminates those interactions leaving the cross difference with respect to the a-particles and b-particles being equal to the interaction of the last added a-particle with the last added b-particle. The same would apply to a fourth or more types of particles.

Conclusion

For a system involving two or more types of particles in which a quantity is the sum of the interactions of all of the particles and any intrinsic quantity for each particle the cross difference of that quantity with respect to the numbers of two different types of particles is equal to the interaction of last added particle of one type with the last added particle of the second type.


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