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A Theorem on the Cross
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. When the first differences 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 then

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)]

Thus

Δ²nBE(n, p) = ΔpnBE(n, p)) = σ(n. p)

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

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.

Here is an example of such a relationship

Conclusions

The cross difference of BE(n, p) is the interaction binding energy of the n-th neutron with the p-th proton.


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