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Conjugate nuclei

Physics Asked on August 3, 2021

I’m trying to find an explanation for the difference in binding energies between conjugate nuclei being governed by the Coulomb interaction. I was looking at the semi-empirical mass formula and trying to draw some conclusion.

We have that:

$$ R = R_0 A^{1/3}$$

and:

$$ Z = A-N$$

so for the Coulomb term I find that it gets a $A^{5/3}$ dependence and of all the terms in the formula that one dominates in terms of $A$ dependence but it’s not entirely clear as to why does the Coulomb interaction govern the difference in binding energies for conjugate nuclei. How do we justify this?

One Answer

In the semi-empirical mass formula

$$E_B(N,Z)=a_VA-a_SA^{2/3}-a_Cfrac{Z(Z-1)}{A^{1/3}}-a_Afrac{(N-Z)^2}{A}+delta(N,Z)$$

the volume and surface terms only depend on $A$, so they give the same value for a nucleus and its mirror nucleus, since the mirror nucleus is obtained by changing $Nleftrightarrow Z$, so $A$ is fixed. The asymmetry term depends on $(N-Z)^2/A$, so it's also the same when $N$ and $Z$ are interchanged. The pairing term $delta(N,Z)$ is either $0$ or only depends on $A$. So the only term that changes between a nucleus and its conjugate is the Coulomb term and the difference in their binding energies $Delta E_B=E_B(N,Z)-E_B(Z,N)$ must come from this term.

Correct answer by Urb on August 3, 2021

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