Half-Life

Radioactive substances do not all decay at once. Instead, they decay randomly, but in a predictable pattern. The half-life tells us how long it takes for half of the unstable nuclei in a sample to decay.

Definition

Half-life (symbol T₁⁄₂) is the time taken for half the nuclei in any sample of an isotope to decay. After one half-life, half remain; after two, a quarter remain; after three, an eighth remain, and so on. The half-life is constant for a given isotope.

How half-life behaves

• Each half-life multiplies what is left by 1/2.
• This applies to the number of undecayed nuclei and to the activity (count rate).
• Decay is random for individual atoms, but predictable for large numbers.

Analogy: imagine a room where, every minute, half the people leave. You cannot predict who leaves, but you can predict that only half will be left each minute.

Simple relationships

After n half-lives, the fraction remaining is $\dfrac{1}{2^n}$.

For any time t and half-life $T_{1/2}$:

$$N(t) = N_0 \left(\dfrac{1}{2}\right)^{\tfrac{t}{T_{1/2}}} \quad \text{and} \quad A(t) \propto N(t)$$

Units of half-life can be seconds, minutes, days, or years.

Choosing isotopes for uses

• Smoke alarms: long half-life alpha sources give a steady output for years.
• Food irradiation and equipment sterilisation: gamma sources with suitable half-lives give strong penetration and reliable dose.
• Thickness control: beta sources with moderate half-life give a stable, measurable signal.
• Medical tracers: short half-life gamma emitters reduce patient dose but last long enough for imaging.

Common misconceptions

• Half-life is not the time for all atoms to decay; some always remain.
• Half-life does not change with the amount you start with.
• Normal temperature or pressure does not affect nuclear half-life.