Nuclear dating and isochron regression

Write the clock as an accumulation, not a survival: D = D₀ + P(e^(λt) − 1), where P is the parent still present today and D the daughter counted alongside it in the same aliquot. Inverting gives t = τ ln(1 + D*/P) with D* = D − D₀. The obstacle is D₀, the daughter the sample inherited when it closed. It is not small. For ⁸⁷Rb, λ = 1.397 × 10⁻¹¹ yr⁻¹, so τ = 71.6 Gyr and T½ = 49.6 Ga; over 3.5 Ga only 1 − e^(−λt) = 4.8 % of the ⁸⁷Rb has decayed, and the radiogenic ⁸⁷Sr that produces is a minor addition to the strontium the mineral took in at crystallisation. Setting D₀ = 0 there would not be an approximation, it would be a fiction. Underneath sits the closed-system requirement: today's P and D must differ from their values at closure by decay alone — no diffusion, no leaching, no later addition.

Cosmogenic ¹⁴N(n, p)¹⁴C keeps a small ¹⁴C/¹²C ratio in the atmosphere; a living organism exchanges carbon and holds that ratio, and death closes the system. Nobody counts the ¹⁴N daughter, so the age comes from the surviving parent: t = τ ln(1/F), F being the fraction modern. Two conventions then bite. Conventional radiocarbon ages are still computed with Libby's 5568-yr half-life (τ = 8033 yr) rather than the measured 5700 ± 30 yr (τ = 8223 yr), a 2.4 % offset — F = 0.3021 gives 9615 yr BP conventionally and 9843 yr on the modern half-life. And atmospheric ¹⁴C/¹²C has never been constant: solar and geomagnetic modulation puts de Vries wiggles in it, fossil-carbon burning diluted it after 1890 (the Suess effect), and 1963 bomb testing nearly doubled it. So the conventional age is mapped onto calendar years through a calibration curve, IntCal, which absorbs the half-life convention. Marine and hard-water carbon is already old at death — a reservoir offset, applied before calibration.

Divide the accumulation equation by a stable, non-radiogenic isotope of the daughter element — ⁸⁶Sr for strontium, ¹⁴⁴Nd for neodymium. This costs nothing, because a mass spectrometer measures isotope ratios far better than absolute atom counts, and it buys the line ⁸⁷Sr/⁸⁶Sr = (⁸⁷Sr/⁸⁶Sr)₀ + (e^(λt) − 1)⋅⁸⁷Rb/⁸⁶Sr. Rocks crystallising from one melt share t and the initial ratio but not Rb/Sr, because Rb substitutes into K sites and Sr into Ca sites: they spread along x and stack up a line in y. Slope gives the age; intercept gives the initial ratio, itself a geochemical result — near 0.702 says a mantle source, 0.720 says reworked crust. The spread matters as much as the precision: the slope's standard error is σ/√(Σ(xᵢ − x̄)²), so a suite with a short lever arm in Rb/Sr dates nothing useful however good the machine is. Mineral isochrons date cooling below the closure temperature of the least retentive phase; whole-rock isochrons date the body.

Ordinary least squares assumes x is exact, which is false here twice over. Both plotted ratios have real uncertainties, and they are correlated, because both were divided by the same measured ⁸⁶Sr signal from the same run — correlation coefficients of 0.5 to 0.9 are ordinary. York's solution minimises S = Σ Wᵢ(yᵢ − a − b xᵢ)² with Wᵢ = 1/(σ²_yi + b²σ²_ξ − 2 b r σ_ξ σ_yi); the weights contain b, so the fit iterates to convergence. Then read MSWD = S/(n − 2). Near 1, the scatter is fully explained by the stated analytical errors and the age uncertainty is defensible. Much greater than 1, the samples do not share one age and one initial ratio, and the line is an errorchron. Much less than 1, the errors were overstated. Finally propagate: with b = 0.05101 ± 0.00016 the slope term is 0.306 %, the λ(⁸⁷Rb) term is 0.32 %, and in quadrature 0.443 % — ±16 Myr on 3.56 Ga, with the decay constant, not the spectrometer, setting the floor.

Zircon takes uranium into its lattice and rejects lead, so D₀ ≈ 0 up to a small common-lead correction read from ²⁰⁴Pb — and it carries two parents at once: ²³⁸U → ²⁰⁶Pb with λ = 1.55125 × 10⁻¹⁰ yr⁻¹ (T½ = 4.468 Ga) and ²³⁵U → ²⁰⁷Pb with λ = 9.8485 × 10⁻¹⁰ yr⁻¹ (T½ = 703.8 Ma). Plot ²⁰⁶Pb*/²³⁸U against ²⁰⁷Pb*/²³⁵U: points whose two ages agree lie on the concordia curve, a locus parameterised by t. Lead loss pulls a grain off that curve along a straight chord, the discordia, whose upper intercept is the crystallisation age and lower intercept the disturbance. If the loss was recent the chord runs to the origin, and then ²⁰⁷Pb*/²⁰⁶Pb* is untouched, because both lead isotopes leave in the same proportion — which is why the Pb–Pb age is the robust one and why ²⁰⁷Pb/²⁰⁶Pb carries the oldest terrestrial dates. It has no closed form: solve (e^{(λ₂₃₅t)} − 1)/(e^{(λ₂₃₈t)} − 1) = 137.818 × (²⁰⁷Pb*/²⁰⁶Pb*) with scipy.optimize.brentq.

Every date is conditional, and the conditions fail quietly. Closed system: Sr, Ar and Pb all diffuse, and argon can be inherited — excess ⁴⁰Ar returns ages older than the rock. Cogenetic suite: samples that never shared a melt still produce a clean line if they are a two-component mixture. Known initial ratio: on a young rock, where the radiogenic addition is tiny, a 1 % error in the assumed initial ⁸⁷Sr/⁸⁶Sr can swamp the age entirely. Known decay constant: λ(⁸⁷Rb) has been revised from 1.42 × 10⁻¹¹ yr⁻¹ to 1.397 × 10⁻¹¹ yr⁻¹, a 1.6 % change, so published Rb–Sr ages are only comparable once you know which constant was used — always report it. None of this shows up in a residual plot. The defences are external: date the same rock with a second decay system, check the intercept against known geochemistry, and prefer minerals that are chemically hostile to the daughter.