Isobaric parabolas and beta stability

Write the atomic mass as M(Z, A)c² = Z m(¹H)c² + (A − Z)mₙ c² − B(Z, A) and put in the semi-empirical B = a_V A − a_S A²⁄³ − a_C Z² A^(−1/3) − a_A (A − 2Z)²/A ± δ. At fixed A the volume and surface terms are constants. The asymmetry term is the only awkward one, and it expands cleanly: (A − 2Z)²/A = A − 4Z + 4Z²/A. Collecting powers of Z gives M(Z)c² = α − βZ + γZ² ± δ, with β = (mₙ − m_H)c² + 4a_A, γ = a_C A^(−1/3) + 4a_A/A, and α holding every term with no Z in it. Two things deserve notice. The quadratic is exact within the model — nothing was expanded about Zₘᵢₙ, so it holds as far out on the slice as the mass formula itself does. And γ has two competing pieces: Coulomb repulsion that grows with Z, and symmetry energy that punishes any departure from N = Z. Using the a_C Z(Z − 1)A^(−1/3) form of the Coulomb term instead leaves γ untouched and adds a_C A^(−1/3) to β, moving Zₘᵢₙ by 0.07 at A = 100.

Set dM/dZ = 0 at fixed A: −β + 2γZ = 0, so Zₘᵢₙ = β/2γ. Divide top and bottom by 4a_A and the structure appears: Zₘᵢₙ = (A/2)[1 + (mₙ − m_H)c²/4a_A] / [1 + (a_C/4a_A)A²⁄³]. With a_C = 0.711 and a_A = 23.7 MeV, a_C/4a_A = 0.0075, so the denominator is 1.163 at A = 101 and 1.264 at A = 209. That single A²⁄³ is the whole reason the valley of stability bends away from N = Z: the Coulomb cost of packing protons grows with the size of the nucleus, while the symmetry cost per nucleon does not. The numerator correction is the neutron-hydrogen mass difference, (mₙ − m_H)c² = 0.782 MeV, worth a factor 1.0082. It shifts Zₘᵢₙ by 0.36 at A = 101 — well under one unit, as promised, but there it straddles a half-integer and moves the nearest integer from 43 to 44, the difference between Tc-101 (14.2 min) and stable Ru-101. At A = 209 the formula returns 83.35 against A/2 = 104.5, and Bi-209 has Z = 83. Zₘᵢₙ is a real number; no nuclide sits exactly on it.

The parabola is built from atomic masses, so β− and electron-capture Q values are plain vertical differences on it: Q_β− = [M(Z) − M(Z+1)]c², and Q_EC = [M(Z) − M(Z−1)]c² up to the keV-scale binding of the captured electron. Positron emission is the exception. Two electron masses go unaccounted for in the atomic bookkeeping, so Q_β+ = [M(Z) − M(Z−1)]c² − 2m_ec², with 2m_ec² = 1.022 MeV — and there is a 1.022 MeV window in which capture proceeds and β+ cannot, which is why proton-rich nuclides close to stability decay by capture alone. For odd A the pairing term vanishes and the step is exact: Q_β−(Z → Z+1) = γ[(Z − Zₘᵢₙ)² − (Z + 1 − Zₘᵢₙ)²] = 2γ(Zₘᵢₙ − Z − ½). Q therefore falls by exactly 2γ per rung. At A = 101, where γ = 1.091 MeV and Zₘᵢₙ = 43.79, the chain gives 5.01, 2.82 and 0.64 MeV at Z = 41, 42 and 43, then −1.54 MeV at Z = 44. The chain stops at Ru-101, and the sign change sits at Zₘᵢₙ − ½, not at Zₘᵢₙ.

For odd A one of Z and N is even and the other odd, the pairing term is zero, and there is one parabola. For even A both are even or both odd, and the term is ∓δ with δ = a_P A^(−1/2): even-even nuclides drop by δ, odd-odd rise by δ, and the two branches stay 2δ = 2a_P A^(−1/2) apart at every Z — 2.24 MeV at A = 100, 1.77 MeV at A = 160. Successive Z along an even-A isobar therefore alternate branches, and the mass sequence zigzags rather than descending smoothly to a single floor. Two consequences do all the work. An odd-odd nuclide near the vertex can lie above both of its even-even neighbours, so β− and capture are both open and it decays in both directions; Cu-64 does exactly this, 39% by β− to Zn-64 and 61% by capture and β+ to Ni-64. And an even-even nuclide two steps from the vertex can lie below the odd-odd nuclide beside it while still lying above the even-even nuclide beyond that — β-stable, but not the most bound isobar on its own slice.

Take A = 100 with a_C = 0.711, a_A = 23.7 and a_P = 11.2 MeV. Then γ = 1.101 MeV, δ = 1.120 MeV and Zₘᵢₙ = 43.40. Measuring mass upward from the unpaired vertex, E(Z) = γ(Z − Zₘᵢₙ)² ± δ puts Mo-100 (Z = 42, even-even) at +1.04 MeV, Tc-100 (Z = 43, odd-odd) at +1.30 MeV and Ru-100 (Z = 44, even-even) at −0.72 MeV. Read the decays off. Tc-100 tops both neighbours, so it goes both ways: the model gives Q_β− = 2.02 MeV and Q_EC = 0.26 MeV, against measured 3.20 and 0.17 MeV. Mo-100 cannot reach Tc-100 at all, since Q_β− = 1.04 − 1.30 = −0.26 MeV; the measured value is −0.17 MeV. Single beta decay is shut, and the only open channel is the second-order Mo-100 → Ru-100 + 2e⁻ + 2ν̄, which skips the odd-odd rung as a virtual intermediate rather than populating it. The model puts that Q at 1.76 MeV against a measured Q_ββ of 3.03 MeV — and Mo-100 is a headline double-β source for precisely this reason.

The parabola is a smooth function of Z fitted to a liquid drop, so it cannot know about closed shells, and beside one it fails loudly rather than gently. At A = 136 the formula gives γ = 0.835 MeV and Zₘᵢₙ = 57.2, which leaves Xe-136 (Z = 54) 3.2 units up the even-even branch and predicts Q_β− = +2.6 MeV to Cs-136. The measured value is −0.09 MeV: Xe-136 is β-stable, and the model misses that by nearly 3 MeV, because N = 82 is closed and Xe-136 is far more bound than any smooth surface allows. Residuals of that size stacked at N or Z = 28, 50 and 82, and at N = 126, are the fingerprint of shell structure, not fit noise. Two further limits are worth saying out loud. The parabola gives energetics only: a positive Q permits a decay and fixes no rate, which needs matrix elements and phase space. And the whole construction is about ground states — an isomer sits on a rung of its own, with its own Q.