Radiometric Dating Methods Compared

Introduction

Radiocarbon dating is familiar, but its 5730-year half-life limits it to about fifty thousand years. To date volcanic ash, meteorites or the oldest crystals on Earth, geochemists use parents with half-lives of millions to billions of years. Each system relies on the same first-order decay law, but each makes different assumptions about what the sample contained when its clock started. This page derives the general age equation and compares the main long-lived methods, showing how clever design removes troublesome unknowns.

Core explanation

The general age equation. A parent P decays to a stable daughter D. If the number of parent atoms now is P, the number originally present was P e^(λt). Every parent atom lost became a daughter atom, so radiogenic daughter D = P(e^(λt) − 1). Allowing for daughter present at the start, D₀:

D = D₀ + P(e^(λt) − 1), which gives t = (1/λ) ln(1 + D /P)

Two assumptions are built in: the system has been closed (no gain or loss of parent or daughter since the clock started), and D₀ is known or can be eliminated. The clock starts when the mineral cools below its closure temperature .

The isochron method (rubidium–strontium). Rubidium-87 decays to strontium-87 with a half-life of about 49 billion years. Rocks contain initial ⁸⁷Sr of unknown amount, so the equation is divided by the stable, non-radiogenic isotope ⁸⁶Sr:

(⁸⁷Sr/⁸⁶Sr)now = (⁸⁷Sr/⁸⁶Sr)₀ + (⁸⁷Rb/⁸⁶Sr)(e^(λt) − 1)

This is the equation of a straight line, y = c + mx. Minerals that crystallised together from the same melt share the same initial ratio but contain different amounts of rubidium. Plotting several minerals gives a line — an isochron — whose slope is (e^(λt) − 1) and whose intercept is the initial ratio. The initial daughter content is found rather than assumed, and scatter about the line tests whether the system stayed closed.

Potassium–argon. Potassium-40 (t½ ≈ 1.25 × 10⁹ years) decays by a branched pathway: about 89% by beta-minus decay to ⁴⁰Ca and about 11% by electron capture to ⁴⁰Ar. Calcium is too common to be useful, but argon is a noble gas that escapes from molten lava, so a freshly erupted rock starts with essentially no argon (D₀ ≈ 0). Argon produced later is trapped in the crystal lattice. The age equation includes the branching fraction for argon. K–Ar is valuable for volcanic rocks from thousands to billions of years old, but argon loss through reheating makes ages too young, and potassium and argon must be measured on separate aliquots.

Argon–argon. Irradiating the sample with neutrons converts some ³⁹K into ³⁹Ar, so potassium can be measured as an argon isotope. Heating the sample in steps and measuring ⁴⁰Ar/³⁹Ar at each step gives an age spectrum ; a flat plateau shows undisturbed argon, while disturbed domains are revealed and excluded.

Uranium–lead. Uranium has two long-lived isotopes decaying through separate chains to different lead isotopes: ²³⁸U → ²⁰⁶Pb (t½ ≈ 4.47 × 10⁹ years) and ²³⁵U → ²⁰⁷Pb (t½ ≈ 7.04 × 10⁸ years). This gives two independent clocks in one mineral. Zircon (ZrSiO₄) is ideal: it takes uranium into its lattice but strongly excludes lead, so initial lead is negligible, and it is very resistant to weathering. Plotting ²⁰⁶Pb/²³⁸U against ²⁰⁷Pb/²³⁵U, ages that agree lie on the concordia curve. If lead was lost in a later event, points fall on a straight line (a discordia) whose intersections with concordia give both the crystallisation age and the disturbance age.

Method Parent half-life Typical range Key strength --- --- --- --- ¹⁴C 5730 y up to 50 000 y organic remains K–Ar / Ar–Ar 1.25 × 10⁹ y 10³ to 10⁹⁺ y volcanic rocks, zero initial argon Rb–Sr 4.9 × 10¹⁰ y 10⁷ to 10⁹⁺ y isochron finds initial ratio U–Pb 4.47 × 10⁹ and 7.04 × 10⁸ y 10⁶ to 4.5 × 10⁹ y two clocks cross-check

Formulae

t = (1/λ) ln(1 + D /P), with λ = ln 2 / t½.

Isochron: slope m = e^(λt) − 1, so t = ln(1 + m)/λ.

Step-by-step reasoning

To choose and apply a dating method:

1. Match the expected age to a parent whose half-life is comparable, so measurable daughter has formed. 2. Consider what event started the clock and whether the mineral stayed closed. 3. Decide how initial daughter will be handled: assumed zero, or found from an isochron. 4. Measure the isotope ratios by mass spectrometry. 5. Calculate the age and check consistency with a second method where possible.

Visual explanation

Plot ⁸⁷Sr/⁸⁶Sr on the y-axis and ⁸⁷Rb/⁸⁶Sr on the x-axis. At the moment of crystallisation all minerals lie on a horizontal line at the initial ratio. As time passes each point moves up in proportion to its rubidium content, so the line rotates about its intercept, becoming steeper with age.

Real-world analogy

An hourglass tells time only if you know how much sand was in the bottom at the start and that none has leaked. The isochron is like comparing several hourglasses of different sizes started together: the pattern across them reveals both the starting sand and the elapsed time.

Real-world example

Zircon crystals from the Jack Hills of Western Australia give uranium–lead ages of up to about 4.4 billion years, showing that continental crust existed within about 150 million years of the Earth's formation. Meteorite isochrons give the age of the Solar System as about 4.57 billion years.

Why?

Why is argon-based dating so reliable for lava? The daughter is a noble gas. It diffuses out of the melt and is lost to the atmosphere during eruption, resetting the clock to zero, but becomes trapped once crystals cool below their closure temperature.

Common misconception

"Dating methods assume the initial daughter amount without evidence." Isochron and concordia methods are designed specifically to determine initial composition and to detect open-system behaviour; poor fits are rejected, not hidden.

Worked example

Question: A Rb–Sr isochron has slope 0.0142. Taking t½(⁸⁷Rb) = 4.88 × 10¹⁰ y, find the age.

Reasoning: λ = 0.693 ÷ 4.88 × 10¹⁰ = 1.42 × 10⁻¹¹ y⁻¹. t = ln(1 + 0.0142) ÷ λ = 0.0141 ÷ 1.42 × 10⁻¹¹ ≈ 9.9 × 10⁸ y.

Answer: About 990 million years.

Quick check

1. Why is zircon especially useful for uranium–lead dating? Answer: It incorporates uranium but excludes lead when it crystallises, so almost all lead present is radiogenic, and it resists alteration.

Exam focus

Derive D = P(e^(λt) − 1) from first-order decay and rearrange it for t. Explain the isochron plot, what the slope and intercept mean and why initial daughter need not be known. For K–Ar, remember the branched decay and why D₀ ≈ 0 for lava.

Advanced insight

The ratio ²⁰⁷Pb/²⁰⁶Pb alone gives an age without measuring uranium at all, because the present ²³⁵U/²³⁸U ratio (about 1/137.8) is nearly uniform on Earth. This lead–lead method underpins the age of the Solar System, first estimated by Clair Patterson from meteorites in 1956 — work that also required him to fight lead contamination in the laboratory, and later in petrol.

Summary

Radiometric ages come from D = P(e^(λt) − 1), assuming a closed system and known initial daughter. Isochrons determine the initial ratio and test closure. K–Ar exploits argon loss from lava; Ar–Ar adds step-heating to detect disturbance. U–Pb uses two independent decay chains in zircon, checked by concordia. The best method matches the half-life to the age and the chemistry to the sample.

Practice questions

1. State two assumptions in any radiometric age calculation. Answer: The system has been closed to parent and daughter since the clock started, and the initial daughter content is known or can be eliminated. 2. What do the slope and intercept of a Rb–Sr isochron represent? Answer: The slope equals e^(λt) − 1, giving the age; the intercept is the initial ⁸⁷Sr/⁸⁶Sr ratio. 3. Why would reheating a volcanic rock make its K–Ar age too young? Answer: Heating lets radiogenic argon escape, so less daughter is measured and the calculated time is shorter. 4. Why is radiocarbon unsuitable for dating a 2-million-year-old lava flow? Answer: Carbon-14 decays away within about ten half-lives, around 50 000 years, and lava does not contain living carbon; a long-lived system such as Ar–Ar is needed.