Variables

Symbol	Quantity	Unit
t₁/₂	Half-life	s
λ	Decay constant	1/s

What it means

The half‑life (t₁/₂) is the time required for the activity or number of nuclei of a radioactive isotope to decrease by half. It is directly related to the decay constant λ by t₁/₂ = ln(2)/λ ≈ 0.693/λ. Half‑lives range from microseconds (e.g., ²¹⁴Po) to billions of years (e.g., ²³⁸U). This parameter is critical for selecting isotopes for medical, industrial, and research applications. It determines how long a radioactive source remains active, affects the design of nuclear waste disposal, and is used in geological and archaeological dating. The half‑life is independent of initial quantity and environmental conditions (temperature, pressure). Understanding this relation is fundamental for radiation protection, nuclear medicine, and nuclear engineering.

Worked example

Half‑Life – Two Examples

Real‑World

Scenario: A radioactive isotope has a decay constant λ = 0.001 s⁻¹. The health physicist calculates the half‑life to determine how long the isotope will remain active and when it will be safe to handle without protective shielding.

Parameter	Value
λ	0.001 s⁻¹

1t₁/₂ = ln(2)/0.001 = 0.693/0.001 = 693.1 s

Result 693 s ✓ ~11.6 minutes

Scenario: Carbon‑14 has a decay constant λ = 1.21×10⁻⁴ yr⁻¹. The archaeologist calculates the half‑life to date ancient organic remains using radiocarbon dating, determining how many half‑lives have passed since the organism died.

Parameter	Value
λ	1.21×10⁻⁴ yr⁻¹

1t₁/₂ = 0.693/1.21e-4 = 5,727 years

Result 5,727 years ✓ Radiocarbon dating

Physics insight: Half‑life is the time required for half of the radioactive atoms to decay. It is inversely proportional to the decay constant – a longer half‑life means slower decay.

Common mistakes

Half‑life t₁/₂: The time for half of the radioactive nuclei to decay – independent of initial amount.
Decay constant λ: In 1/time – t₁/₂ = ln(2)/λ.
ln(2): Approximately 0.693 – use the exact value for precision.
Units: If λ is in s⁻¹, half‑life is in seconds – convert to hours, days, years as needed.
Multiple half‑lives: After n half‑lives, the remaining fraction is (1/2)ⁿ – do not simply divide by n.

Applications

The half‑life relation, t₁/₂ = ln2/λ, links the half‑life of a radioactive isotope to its decay constant. It is a fundamental parameter for identifying radioisotopes, for planning experiments and medical procedures, and for managing nuclear materials. Half‑life values range from fractions of a second to billions of years, determining the usefulness of isotopes for various applications: short half‑lives are ideal for medical imaging (e.g., technetium‑99m), while long half‑lives are critical for geological dating (e.g., uranium‑238). Nuclear engineers use half‑life to estimate the longevity of nuclear waste and to design storage solutions. By knowing the half‑life, professionals can predict the activity over time and implement appropriate safety measures for handling and disposal.

Identification and characterisation of radioactive isotopes
Scheduling of diagnostic and therapeutic nuclear medicine procedures
Design of long‑term nuclear waste storage and disposal
Geochronology and archaeological dating
Safety planning for radiation emergencies and decommissioning

Half-Life Relation