Radioactive dating simulator

1. Decay is random, but predictable in bulk

An unstable atomic nucleus can change into another by throwing out a particle. Nobody can say when any one nucleus will decay; there is only a fixed chance that it will in each moment, unaffected by heat, pressure or chemistry. Over one half-life each nucleus has an even chance of decaying, so half of a large collection is left, then a quarter, then an eighth.

P = P₀ × e^−λt    where λ = ln 2 ÷ t_½

Here P is the number of parent atoms left at time t, P₀ the number at the start and λ the decay constant, the chance of decay per year. The simulator gives each atom a random moment of decay drawn from exactly this law, which is why it behaves like real atoms, lumps and all.

2. From a ratio to an age

Every parent atom that decays becomes a daughter atom, so if none were there at the start, the daughter count is D = P₀ − P. Putting this into the decay law and rearranging gives the age equation:

D ÷ P = e^λt − 1    so    t = (1 ÷ λ) × ln(1 + D ÷ P)

The beauty of this is that nobody needs to know how many parent atoms there were at the start: the ratio of daughter to parent, measured today, is enough. When D ÷ P is 1, exactly one half-life has passed; when it is 3, two half-lives; when it is 7, three.

Worked example. In a 4,567-million-year-old mineral that began with no lead, uranium-238 (half-life 4,468 million years) would by now have made 1.031 atoms of lead-206 for every atom of uranium-238 left: t = (4,468 million ÷ 0.6931) × ln(2.031) = 4,567 million years.

Potassium-40 needs one extra step, because only about 10.5% of its decays make argon. The age equation becomes t = (1 ÷ λ) × ln(1 + D ÷ (0.105 × P)).

3. Choosing the right clock

A clock works best when its half-life suits the age being measured. If too few atoms have decayed, the daughter is hard to measure; after a dozen or so half-lives almost no parent is left. Long-lived systems date events billions of years ago. Short-lived ones, whose parent atoms have long since vanished, are used as stopwatches for the first few million years of the Solar System.

Half-lives are the values most widely used in meteorite research; some are still being refined, and the rubidium value in particular has been revised by about 2% in recent decades.

4. The problem: daughter atoms that were there from the start

The age equation assumes the mineral began with no daughter atoms. That is rarely true. Strontium-87, for example, was already present in the dust that made the Solar System, so a rock starts its life with some strontium-87 that has nothing to do with its age. Counting it as decay product would make every rock look too old. The isochron method solves this.

The isochron equation

Write the starting strontium ratio as (⁸⁷Sr/⁸⁶Sr)₀. Applying the age equation to each mineral and dividing through by strontium-86 gives:

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

This is the equation of a straight line, y = c + m x. Its slope m is e^λt − 1, so t = ln(1 + m) ÷ λ, and the point where it meets the vertical axis is the starting ratio, which therefore no longer needs to be assumed: the method measures it.

Worked example. Five minerals from a basaltic meteorite lie on a line of slope 0.06590. With λ = 1.3975 × 10⁻¹¹ per year (from the 49.6-billion-year half-life), t = ln(1.06590) ÷ 1.3975 × 10⁻¹¹ = 0.06382 ÷ 1.3975 × 10⁻¹¹ ≈ 4,567 million years. The intercept, about 0.699, is the strontium ratio of the early Solar System.

The isochron also checks itself. If heating or water has moved rubidium or strontium since the rock formed, the points scatter off the line and the age is not trusted.

Lead-lead: the age of the Solar System

Uranium has two long-lived isotopes decaying to two lead isotopes at different rates. Dividing one age equation by the other cancels the uranium and leaves the ratio of radiogenic lead-207 to lead-206, which depends only on time, and on the ratio of uranium-238 to uranium-235 today, about 137.8. For a 4,567-million-year-old mineral it works out at about 0.625. Because no uranium measurement is needed, lead-lead dating is extremely precise. Calcium-aluminium-rich inclusions in CV chondrites give 4,567.3 million years, with an uncertainty of about 0.2 million years: the birthday of the Solar System. In 1956, lead in meteorites, including the Canyon Diablo iron, gave the first accurate age of Earth, about 4.55 billion years.

Extinct clocks: stopwatches for the first few million years

Aluminium-26 has long since vanished, but it left extra magnesium-26 behind. Minerals rich in aluminium have more of it, and plotting ²⁶Mg/²⁴Mg against ²⁷Al/²⁴Mg gives an isochron whose slope is the ²⁶Al/²⁷Al ratio when the mineral formed. The oldest inclusions began with about 5.2 × 10⁻⁵. A chondrule that began with 1.0 × 10⁻⁵ must have formed later by

Δt = (0.717 million years ÷ ln 2) × ln(5.2 ÷ 1.0) ≈ 1.7 million years

This assumes aluminium-26 was spread evenly through the young Solar System, and studies comparing it with lead-lead ages largely support that.

Clocks that started in space and on Earth

Cosmic rays smash into a meteoroid during its journey from the asteroid belt, making new isotopes such as helium-3, neon-21 and carbon-14. The amount of the stable ones gives the cosmic-ray exposure age: how long the rock travelled as a small body, typically a few million to a few tens of millions of years for stony meteorites, and hundreds of millions of years or more for many irons. Once a meteorite falls the air shields it, production stops and the radioactive ones decay. Carbon-14 left in a meteorite found in a hot desert, compared with the level in a fresh fall, gives its terrestrial age; one quarter of the usual level means two half-lives, about 11,500 years, on Earth.