The Law of Radioactive Decay

Nobody can say when a particular nucleus will decay. Yet a large sample shrinks with clockwork regularity, halving in the same time over and over. The decay law turns that pattern into a clock for dating rocks and bones.

Chance, not destiny#

Prepare many identical radioactive nuclei at the same moment, in identical conditions. They do not decay together: some go almost at once, some last a very long time. No law of physics predicts the moment for any one nucleus. This is a quantum effect; classical physics would say identical starting conditions give identical futures. Quantum mechanics gives only probabilities (an idea Einstein never fully accepted).

The decay law#

If N nuclei are active, the number decaying in the next short time dt is proportional to N (more nuclei, more decays) and to dt (each has more chance in a longer interval):

The SI unit of activity is the becquerel (1 Bq = 1 decay per second). The older curie is 1 Ci = 3.7 × 10¹⁰ decays per second, named after Marie and Pierre Curie, who in 1902 isolated 0.19 g of radium chloride from uranium ore and shared the 1903 Nobel Prize with Henri Becquerel. Activity per unit mass is the specific activity.

Half-life and average life

The half-life t_1/2 is the time for half the active nuclei, and therefore half the activity, to go:

Adding up how long every nucleus survives and dividing by N₀ gives the average life:

Radioactive clocks#

Dating rocks

Carbon dating

Two ways to decay, and decay chains

If a nucleus can decay by two independent routes with constants λ₁ and λ₂, the chances simply add: λ = λ₁ + λ₂, so 1/t = 1/t₁ + 1/t₂ for the half-lives. For example, half-lives of 3 h and 6 h for the two routes give an overall half-life of 2 h.

If the daughter is also radioactive, it is produced at the rate λ_pN_p and decays at λ_dN_d. Its number stays constant when λ_pN_p = λ_dN_d, i.e. when the two activities are equal.

Uses and dangers#

Gamma rays from sources such as cobalt-60 treat cancer; radioactive tracers follow substances through the body or through pipes; the ionization from a weak source lets static charge leak away from moving machinery in factories; carbon dating dates once-living material. Too much exposure damages living cells, so radiation is always used with shielding and limited doses.