Half-life

Half-life is a compact way to express exponential decay. It is the time over which the expected number of undecayed nuclei—and therefore the activity of a simple isolated sample—falls to one half of its previous value.

Repeated halving, not linear subtraction

N = N₀(1/2)^t/T½

After one half-life 50% remains, after two 25%, after three 12.5%. The same fraction is removed in each equal interval, not the same number of nuclei.

Half-life explorer

Elapsed half-lives

Half-life is multiplicative. The graph shrinks by the same factor after each equal interval.

Connecting half-life to λ

The exponential law N=N₀e^-λt and the half-life law are exactly equivalent for a single constant decay mode. Their parameters obey λT_1/2=ln2.

Short half-life means large decay constant. The characteristic time changes, but the exponential shape remains.

Half-life does not age an individual nucleus

An undecayed nucleus is not “closer to decay” merely because it has survived several half-lives. In the standard model the process is memoryless: conditional on still being present, its future decay probability per unit time is unchanged.

Dating requires more than a half-life

Radiometric dating also needs a suitable decay system, a model of initial conditions and confidence that parent or daughter nuclides were not added or lost in ways that invalidate the clock. The half-life supplies the time scale; geological or archaeological interpretation supplies the context.

Half-life is independent of sample size. A large sample and a small sample of the same radionuclide have the same half-life because the quantity describes the decay probability of each surviving nucleus. The larger sample starts with more nuclei and usually a higher activity, but both samples lose the same expected fraction during each half-life. This is why half-life should never be interpreted as the time needed to remove a fixed number of nuclei. After many half-lives the expected population becomes very small, but the exponential model approaches zero continuously rather than reaching it after a fixed number of intervals.

Worked examples

1. Five half-lives

Solution

Remaining fraction = (1/2)⁵ = 1/32 ≈ 3.125%.

2. Elapsed time

Solution

A nuclide has T½=8 h. Reaching 12.5% means three half-lives, so t=24 h.

3. Find half-life from λ

Solution

For λ=0.035 y⁻1, T½=0.693/0.035≈19.8 y.