University Physics V · Radioactive Decay · 15.8

Nuclear Dating & Isochron Regression

Dating is inverse decay, and the inverse is under-determined: one measured parent–daughter pair, two unknowns. This is where you learn to supply the missing one — by convention for radiocarbon, by a regression line for Rb–Sr, by a second decay chain for U–Pb — and to say what the number assumes.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

A date is the decay law read backwards. Radiogenic daughter accumulates as D = D₀ + P(e^(λt) − 1), so t = τ ln(1 + D*/P) — one equation with two unknowns, the age t and the inherited daughter D₀, and one sample supplies only one equation. Everything in nuclear geochronology is a way of removing D₀.

Radiocarbon removes it by fiat: nothing starts life with inherited ¹⁴N worth counting, so the surviving parent alone dates the sample, at the price of assuming the atmospheric ¹⁴C/¹²C the organism drew from — which is why every conventional age must afterwards be calibrated. The isochron removes it by measurement: divide the accumulation equation through by a stable isotope of the daughter element, and a suite of cogenetic samples that share t and D₀/S but differ in P/S falls on a straight line whose slope is e^(λt) − 1 and whose intercept is the initial ratio no single analysis could have supplied. Uranium removes it by redundancy: two parents feeding two lead isotopes inside one zircon give two clocks, and their disagreement measures how far the crystal has fallen short of a closed system.

The cost is uniform. Each of these dates buys its number with assumptions — closed system, cogenetic suite, known initial ratio, known decay constant — and the arithmetic cannot tell you when one has failed. The fit statistics can.

Simple definition: An isochron is the straight line traced in daughter/stable against parent/stable coordinates by samples that closed at the same instant with the same initial ratio; its slope is e^(λt) − 1 and its intercept is that initial ratio.
Example: Four Archean whole rocks give a Rb–Sr line of slope 0.05101 and intercept 0.70057. Then t = ln(1.05101)/(1.397 × 10⁻¹¹ yr⁻¹) = 3.56 Ga, and the rocks began with ⁸⁷Sr/⁸⁶Sr = 0.70057 — a number no single analysis could reach.

Accumulation clockD = D₀ + P(e^(λt) − 1)

Two unknowns, t and D₀, in one equation — which is why one sample dates nothing on its own.

P, D, D₀ are atom counts in the sample today; λ in yr⁻¹, t in yr. D₀ is the inherited daughter.

Age of a single closed samplet = (1/λ) ln(1 + D*/P) = τ ln(1 + D*/P)

Usable only where D₀ is known independently — as for ¹⁴C, where the daughter is not counted at all.

D* = D − D₀ is radiogenic daughter only; τ = 1/λ = T½/ln2, in years.

Conventional radiocarbon aget = 8033 yr × ln(1/F), F = A/A₀

A reproducible number, not a calendar date — IntCal then maps it to calendar years and absorbs the half-life convention.

8033 yr = 5568/ln2, Libby's τ; F is fraction modern, δ¹³C-normalised to −25 ‰, zero year AD 1950.

Rb–Sr isochron line⁸⁷Sr/⁸⁶Sr = (⁸⁷Sr/⁸⁶Sr)₀ + (e^(λt) − 1) · ⁸⁷Rb/⁸⁶Sr

Slope gives the age, intercept gives the initial ratio — the unknown has become a fitted parameter.

All four quantities are dimensionless ratios to stable ⁸⁶Sr; λ(⁸⁷Rb) = 1.397 × 10⁻¹¹ yr⁻¹, T½ = 49.6 Ga.

York weights and MSWDWᵢ = 1/(σ²_yi + b²σ²_ξ − 2 b r σ_ξ σ_yi), MSWD = S/(n − 2)

Weights depend on b, so the fit iterates; MSWD ≈ 1 means the scatter is analytical, MSWD ≫ 1 makes it an errorchron.

b is the current slope, r the x–y error correlation, S the weighted residual sum, n the number of samples.

Both uncertainties into the date(uₜ/t)² = (u_λ/λ)² + [u_b/((1 + b) ln(1 + b))]²

Once the slope is precise, the ⁸⁷Rb decay constant, known to about 0.3 %, becomes the floor on the age.

b is the fitted slope, u_b its standard error; every term is dimensionless.

One equation, two unknowns

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.

Radiocarbon: the clock is honest, the input is not

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.

The isochron: turn the unknown initial into an intercept

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.

Fitting a line when both coordinates carry correlated errors

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.

Two clocks in one crystal: U–Pb concordia and discordia

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.

What the arithmetic cannot see

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.

Change one variable at a time

Make the relationship visible.

Interactive model

Age of the suite t3.6 Ga

Initial ⁸⁷Sr/⁸⁶Sr0.701

Lever arm in ⁸⁷Rb/⁸⁶Sr3.4

Set the age to zero and every sample collapses onto the intercept, because no radiogenic ⁸⁷Sr has grown in yet. Wind t back up and watch the line pivot about that intercept rather than slide. Then squeeze the lever arm to 0.4 and see the age uncertainty jump from about 11 Myr to 91 Myr.

SLOPE e^λt − 10.0516

INTERCEPT (⁸⁷Sr/⁸⁶Sr)₀0.7010

LEVER ARM Δ(⁸⁷Rb/⁸⁶Sr)3.4

AGE 1σ FROM SPREAD10.7 Myr

Live interpretationSLOPE e^λt − 1: 0.0516. INTERCEPT (⁸⁷Sr/⁸⁶Sr)₀: 0.7010. LEVER ARM Δ(⁸⁷Rb/⁸⁶Sr): 3.4. AGE 1σ FROM SPREAD: 10.7 Myr

Catch the common trap

Explain before calculating.

Misconception check

A straight line is not proof of an age. Take two reservoirs with different ⁸⁷Sr/⁸⁶Sr and different Rb/Sr, mix them in varying proportions, and the products fall on a perfectly straight line in exactly the isochron coordinates — with a slope that encodes the mixing end-members and nothing about time. That is a pseudo-isochron, and a high r² will not expose it, because both coordinates share the same measured ⁸⁶Sr denominator and are correlated by construction. The tests that do work are structural, not cosmetic: MSWD near 1 on properly weighted York residuals, a plot of ⁸⁷Sr/⁸⁶Sr against 1/Sr — straight for a two-component mixture, generally curved for a true isochron — a geologically credible intercept, and agreement with an independent decay system.

A geochronologist has one whole-rock sample from an Archean granite, with ⁸⁷Rb and ⁸⁷Sr both measured to parts in 10⁵. She refuses to quote an age from it, and asks for four more samples chosen to differ in Rb/Sr. Why is one sample not enough?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyCharcoal from a hearth gives a ¹⁴C activity 0.3021 times that of the modern standard, after background subtraction and δ¹³C normalisation. Report the conventional radiocarbon age, then say how much it would shift on the modern 5700-yr half-life.

¹⁴C is read from the surviving parent — the ¹⁴N daughter is never counted — so there is no D₀ problem and t = τ ln(1/F) with F = A/A₀ = 0.3021.
Convention fixes τ from Libby's half-life, not the best modern value: τ = 5568/ln2 = 5568/0.693147 = 8032.9 yr.
ln(1/0.3021) = ln(3.3102) = 1.19700.
t = 8032.9 × 1.19700 = 9615 yr BP, quoted as a conventional radiocarbon age of about 9620 BP.
On T½ = 5700 yr, τ = 8223.4 yr and t = 8223.4 × 1.19700 = 9843 yr — older by 228 yr, exactly the half-life ratio 5700/5568 = 1.0237.

Answer9615 yr BP conventional; 9843 yr on the 5700-yr half-life, 228 yr older. Neither is a calendar date — IntCal calibration converts the conventional age and already absorbs the half-life convention.

MediumTwo cogenetic whole rocks give (⁸⁷Rb/⁸⁶Sr, ⁸⁷Sr/⁸⁶Sr) = (0.842, 0.74352) and (4.317, 0.92078). Using λ(⁸⁷Rb) = 1.397 × 10⁻¹¹ yr⁻¹, find the age and the initial ⁸⁷Sr/⁸⁶Sr, and compare with the small-λt shortcut t ≈ slope/λ.

Slope = (0.92078 − 0.74352)/(4.317 − 0.842) = 0.17726/3.475 = 0.0510101, and the slope equals e^(λt) − 1.
t = ln(1.0510101)/λ = 0.0497517/(1.397 × 10⁻¹¹ yr⁻¹) = 3.5613 × 10⁹ yr = 3.561 Ga.
Intercept: (⁸⁷Sr/⁸⁶Sr)₀ = 0.74352 − 0.0510101 × 0.842 = 0.70057. Check on the second rock: 0.92078 − 0.0510101 × 4.317 = 0.70057 ✓.
Shortcut: t ≈ slope/λ = 0.0510101/(1.397 × 10⁻¹¹) = 3.6514 Ga, too old by 90 Myr — 2.5 %. Linearising e^(λt) − 1 is only safe when λt ≪ 1, and here λt = 0.0498.
Two points always define a line exactly, so the residuals are zero, MSWD is undefined, and nothing has tested the shared-age, shared-initial assumption. At least three samples are needed before the fit can fail.

Answert = 3.561 Ga with (⁸⁷Sr/⁸⁶Sr)₀ = 0.70057; the linear shortcut gives 3.651 Ga, 90 Myr too old. A two-point line carries no goodness-of-fit test.

HardA zircon gives ²⁰⁶Pb*/²³⁸U = 0.4512 and ²⁰⁷Pb*/²³⁵U = 12.043, with λ₂₃₈ = 1.55125 × 10⁻¹⁰ yr⁻¹, λ₂₃₅ = 9.8485 × 10⁻¹⁰ yr⁻¹ and present-day ²³⁸U/²³⁵U = 137.818. Find both U–Pb ages, the ²⁰⁷Pb/²⁰⁶Pb age, and how much radiogenic lead the grain has lost.

Zircon excludes Pb at crystallisation, so D₀ ≈ 0 and each ratio dates on its own. t₂₀₆ = ln(1.4512)/λ₂₃₈ = 0.372391/(1.55125 × 10⁻¹⁰) = 2401 Ma.
t₂₀₇ = ln(13.043)/λ₂₃₅ = 2.568252/(9.8485 × 10⁻¹⁰) = 2608 Ma. The two disagree by 100(1 − 2401/2608) = 7.9 %, so the grain is discordant and has not stayed closed. Averaging them would be meaningless.
Form the ratio that depends on t alone: ²⁰⁷Pb*/²⁰⁶Pb* = (1/137.818) × (12.043/0.4512) = (1/137.818) × 26.691 = 0.19367.
Solve (e^{(λ₂₃₅t)} − 1)/(e^{(λ₂₃₈t)} − 1) = 137.818 × 0.19367 = 26.685 by root finding (brentq on 2.0–3.5 Ga): t = 2773 Ma.
Recent Pb loss strips both lead isotopes in the same proportion, so it drives the point down a chord to the origin and leaves ²⁰⁷Pb*/²⁰⁶Pb* untouched — the Pb–Pb age is therefore the crystallisation age, and it is also the upper concordia intercept of that chord.
Concordia at 2773 Ma has ²⁰⁶Pb*/²³⁸U = e^0.43013 − 1 = 0.53747, so the grain retains 0.4512/0.53747 = 0.8395. The ²³⁵U system agrees: 12.043/14.3455 = 0.8395, as a chord through the origin demands.

Answert₂₀₆ = 2401 Ma and t₂₀₇ = 2608 Ma, 7.9 % discordant; the ²⁰⁷Pb/²⁰⁶Pb age, and so the upper concordia intercept, is 2773 Ma, with about 16 % of the radiogenic lead lost recently.