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Fix the mass number A and let Z vary. Because the Coulomb and asymmetry terms in the semi-empirical mass formula are both approximately quadratic in Z at fixed A , the atomic mass along an isobaric chain is, to good approximation, a parabola in Z : $M(A,Z)\,c^2 \approx \alpha(A) - \beta(A)\,Z + \gamma(A)\,Z^2 \mp \delta(A)$. where $\alpha$ , $\beta$ , and $\gamma$ collect the volume, surface, Coulomb, and asymmetry contributions and the pairing term $\delta(A)$ shifts the curve up or down depending on whether N and Z are both even or both odd. For odd A , exactly one of N , Z is even and the pairing term vanishes, giving a single parabola with one minimum: one value of Z at that mass number is the most tightly bound, and every other isobar decays toward…

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