Stability, drip lines and decay maps

Load the atomic masses into a vector m and every question about a decay channel becomes one fixed integer-coefficient contraction of it: Q_β⁻ = m(Z, A) − m(Z+1, A); Q_α = m(Z, A) − m(Z−2, A−4) − m(⁴He); Sₙ = m(Z, A−1) + mₙ − m(Z, A). No dynamics enters — nothing in these expressions knows about barriers, matrix elements or phase space. A decay-mode map on the (N, Z) plane is therefore the sign pattern of a handful of such functionals, and in NumPy the whole map is one integer matrix times one mass vector. Two consequences follow. Uncertainties propagate linearly, so a nuclide whose Q comes out at 30 ± 40 keV has no determined mode, only a determined ignorance. And the AME flags extrapolated masses with a hash sign, so whole regions of any published map rest on a model of the mass surface rather than on a measurement of it — which is exactly where new facilities keep moving the lines.

Sₙ(Z, N) = B(Z, N) − B(Z, N−1) is the discrete derivative of the binding surface along N, and the pairing term makes it alternate. Add a neutron to an odd-N core and it pairs with the unpaired one, so Sₙ is large; add it to an even-N core and it stays unpaired, so Sₙ is small. The full swing is 2Δ with Δ ≈ 12/√A MeV — about 2.4 MeV at A = 24, about 0.8 MeV at A = 208. The measured oxygen chain shows it plainly: Sₙ = 6.85 MeV at ²²O (N = 14), 2.74 MeV at ²³O (N = 15), 4.19 MeV at ²⁴O (N = 16). The sequence is not monotonic, and the 1.45 MeV rise from ²³O to ²⁴O is pairing, not extra resistance to the downward trend. Any drip-line criterion built on "the first N where Sₙ stops falling" is reading the parity of N, not the edge of binding.

On the neutron side the criterion is S₂n = 0, because S₂n = B(N) − B(N−2) compares two nuclides of the same neutron parity and the pairing term drops out to leading order. Beyond that line the pair cannot be held at all: an l = 0 neutron sees no barrier, the state is a resonance of width Γ of order 1 MeV, and ħ/Γ = 6.58 × 10⁻²² MeV s ÷ 1 MeV ≈ 10⁻²¹ s — gone before it is a nucleus. The proton side behaves completely differently, because the emitted proton must tunnel through a Coulomb-plus-centrifugal barrier of order 10 MeV. Sₚ can go a full MeV negative and the nuclide still lives long enough to be identified: ¹⁵¹Lu emits a 1.23 MeV proton from its ground state with T½ = 80.6 ms. The edges also sit at wildly different distances. For tin the proton drip line lies within a couple of nucleons of the doubly magic ¹⁰⁰Sn, and the proton line is mapped across most of the chart; the tin neutron drip line lies far beyond the heaviest tin yet made, and experimentally the neutron line is established only as far as neon, Z = 10.

At fixed A the mass formula is an exact quadratic in Z, so setting dM/dZ = 0 gives Z₀ = (A/2)(1 + ΔₙH/4a_A) ÷ [1 + (a_C/4a_A)A²⁄³] with ΔₙH = (mₙ − m_H)c² = 0.782 MeV. Read the competition term by term instead. Pushing one more unit of charge in costs 2a_C Z A^(−1/3) of Coulomb energy — 19.50 MeV at ²⁰⁹Bi — and returns 4a_A(N−Z)/A from the symmetry term, 19.09 MeV there. They balance only once N − Z has grown to 43, i.e. N/Z = 1.52. Since the Coulomb slope grows as Z A^(−1/3) while the symmetry restoring slope is bounded by 4a_A, the required excess N − Z keeps growing with A, and the valley floor pulls away from the N = Z line. The formula delivers 83.34 at A = 209 (²⁰⁹Bi observed) and 93.14 at A = 238, which sits neatly between the two β-stable isobars ²³⁸U and ²³⁸Pu.

Write B = A b(A) with b the binding per nucleon. Then Q_α = B(A−4) + 28.296 − B(A) ≈ 28.296 − 4[b(A) + A b′(A)], where 28.296 MeV is the α particle's own binding. Past the ⁶²Ni peak b′ is negative, so the −4A b′ piece is a positive contribution that grows with A. Take b falling linearly from 8.60 MeV at A = 100 to 7.57 MeV at A = 238, giving b′ = −0.00746 MeV: Q_α crosses zero where b + A b′ = 7.074 MeV, at A = 152. Below that the α is not worth the four nucleons it removes; above it, it is. The observed onset is earlier — ¹⁴⁴Nd with Q_α = 1.90 MeV and ¹⁴⁷Sm with 2.31 MeV — because ¹⁴⁴Nd's daughter ¹⁴⁰Ce closes the N = 82 shell, a several-MeV bonus a smooth curve has no way to carry, and ¹⁴⁷Sm sits in the same shell-driven island.

²⁰⁹Bi has Q_α = +3.137 MeV; ²¹²Po has Q_α = +8.954 MeV. The Q values differ by a factor 2.85, and the half-lives by 2.0 × 10¹⁹ yr against 0.29 µs — 33.3 decades. The Geiger-Nuttall exponent, which goes as Z_d/√Q, accounts for about 30 of them; a sign test on the mass table accounts for none. So a decay-mode map states which channels are open, and ranking their rates is a different calculation entirely. This is also what "stable" means operationally. Watch one mole for one year, see nothing, and at 90 % confidence you have shown only T½ > ln2 × 6.02 × 10²³ ÷ 2.3 = 1.8 × 10²³ yr. ²⁰⁹Bi was printed as stable for a century with a positive Q_α in plain sight, until a bolometric measurement in 2003 counted its α decays; ¹²⁸Te's double beta decay, at 2.25 × 10²⁴ yr the longest half-life yet measured, was read geochemically from xenon accumulated in ancient tellurium ores rather than counted.