Half-life (radioactive decay)

Basic principle and mathematical form

Radioactive decay is a random process at the level of single atoms, governed by probability rather than deterministic schedules. For a large collection of identical nuclei, the number N(t) that remain after time t follows an exponential law: N(t) = N(0) · (1/2)^{t / t_1/2}, where t_1/2 is the half-life. Equivalently, using the decay constant λ, t_1/2 = ln(2) / λ. Because the law is exponential, the fraction remaining depends only on the number of elapsed half-lives: after n half-lives the fraction is (1/2)^n (for example (1/2)^{10} ≈ 0.1%). The half-life is independent of the initial amount and does not change as the sample gets smaller.

Physical meaning and measurement

At the atomic level each nucleus decays at a random time, so half-life is an average or expected time for reduction by a factor of two in a large ensemble. Methods used to determine half-lives include measuring activity with detectors such as Geiger–Müller counters, scintillation counters or gamma spectrometers, and directly counting atoms by mass spectrometry. A commonly cited everyday example of a detector is the Geiger–Müller tube, which records decays to infer activity changes over time. For background and general context see discussions of probability and expected value.

Examples and applications

• Carbon-14 is widely used in radiocarbon dating of once-living material; its half-life is about 5,730 years (carbon, used to date fossils and archaeological samples).
• Plutonium-238 (approx. 88 years) is used as a heat source in space probes and requires handling considerations for its radioactivity (plutonium).
• Uranium-232 (roughly 69 years) is one of many uranium isotopes with relatively long half-lives and implications for nuclear materials management (uranium).

Half-life values are used to calculate how long a radioactive source will remain hazardous, to design radiopharmaceutical dosing schedules, and to interpret ages from radiometric dating. In pharmacology the same exponential idea gives an elimination half-life for drugs: the time for blood concentration to fall to half its previous value.

Decay chains, effective half-life and distinctions

Some unstable nuclei decay into daughter isotopes that are themselves radioactive; the resulting sequence is a decay chain. The observed activity in such a chain depends on the relative half-lives of parent and daughters, and secular equilibrium may arise when parent half-life is much longer than daughter half-life. When both physical decay and biological processes remove a radionuclide from an organism, an effective half-life combines the two removal rates. For independent exponential processes the effective half-life T_eff satisfies 1/T_eff = 1/T_phys + 1/T_bio, so that T_eff is always shorter than either component half-life.

History and clarifications

The half-life concept emerged in the early studies of radioactivity by researchers who quantified how rapidly different substances lost their radioactivity; over time the idea was generalized to non-nuclear contexts. Half-life should not be mistaken for the moment of decay of any individual atom (which is unpredictable). Also, while the fraction remaining after each half-life is fixed, the absolute activity decreases in proportion to the number of remaining nuclei. For comparisons and further reading on terminology and related concepts see general resources on radioactive elements and links addressing other uses of the term, including the video game Half-Life and its related entry Half-Life (video game).

Practical notes

Estimating decay over long periods often uses tabulated half-lives; simple calculations using the (1/2)^n rule suffice for many purposes. When dealing with mixed isotopes, decay chains, or combined physical and biological loss, more detailed exponential models are needed. For decay modes such as spontaneous fission, alpha or beta emission, descriptions of the process can be found in resources on fission and related nuclear transformations. Basic atomic concepts like the distinction between nuclei and atoms are helpful for non-specialists learning about half-life.