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A Theorem on the Second
Differences of Binding Energies

## Background

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 respect to n the pp interactions are eliminated. The first differences with respect to n of BE(n, p) and BE(n-1, p) are then

The subtraction of the first difference of BE(n-1. p) from the first difference of BE(n, p) gives the second difference of BE(n, p). If the binding energy of the (n-1)-th neutron with j-th neutron is the same as that of the n-th neutron with the j-th neutron then

In words, under the above assumption the second difference of B(n, p) is equal to the binding energy of the n-th neutron with the (n-1) neutron.

But what justification is there for assuming that binding energy of the (n-1)-th neutron with j-th neutron is the same as that of the n-th neutron with the j-th neutron. There is evidence shown in the next section that the interaction binding energy between two neutrons depends only on the shells the two neutrons are in, Thus if n and (n-1) are in the same shell teir interaction binding energy with the j-th neutron are the same. Therefore the theorem applies and the second difference is equal to the interactive binding energy between the n-th neutron and the (n-1)-th neutron in that nuclide.

In symbols this can be restated as follows. Let ν(i, j) be the interaction binding energy between the i-th neutron and the j-th neutron. Then

#### ν(n, n-1) = Δ²nBE(n, p) = Δp(ΔnBE(n, p))

That is to say that the second difference of BE(n, p) is the interaction binding energy of the n-th neutron with the (n-1)-th neutron.

For further analysis it is convenient to focus on ν(n. j) as the slope of the relationship between the first differences with repect to n as a function of the number of neutrons in the nuclide. If that slope is negative then that means that ν(n. n-1) is negative for the value in the shell. That means the force between neutrons is a repulsion.

Here is an example of such a relationship

And here is an example of the relationship between the incremental binding energies of neutrons and the number of protons in the the nuclide.

## Conclusions

The second difference with respect to n of BE(n, p) is the interaction binding energy of the n-th neutron with the (n-1)-th neutron. Such values are negative indicating the interactive force between two neutrons is repulsion. In contrast the interactive binding energy between a neutron and a proton is positive thus indicating an attraction.