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A Theorem on the Cross
Differences of Binding Energies


Let n and p be the numbers of neutrons and protons, respectively. The binding energy for a nuclide with n neutrons and p protons is made up of three components: the interactions of neutrons with other neutrons, the interactions of protons with other protons and the interactions of neutrons and protons. These are depicted below.

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

When BE is differenced with repact to n the pp interactions are eliminated. When the first difference with respect to n are differenced with respect to p the nn interactions are eliminated. Thus the focus can be devoted to the np interactions.

Let σ(i, j) be the interaction binding energy between the i-th neutron and the j-th proton.

The binding energy of the nuclide with n neutrons and p protons is

BE(n, p) = ΣinΣjpσ(i, j)

Consequently the first difference with respect to neutron number is

ΔnBE(n, p) = BE(n, p) − BE(n−1, p) = Σjpσ(n, j)

This is just the n-th row in the above diagram of np interactions. It runs from j=1 to j=p. The first difference with respect to n for p-1 is also the n-th row but it runs from j=1 only to j=p-1.

This means that

ΔpnBE(n, p)] = Δpjpσ(n, j)]


Δ²nBE(n, p) = ΔpnBE(n, p)) = σ(n. p)

For further analysis it is convenient to focus on σ(n. p) as the slope of the relationship between the first differences with repect to n as a function of the number of protons in the nuclide. If that slope is positive then that means that σ(n. p) is positive for the values of p in the shell. That means the force between neutrons and protons is an attraction.


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