Half-life

For other uses, see Half-life (disambiguation).

The half-life of a radioactive substance is the time required for half of a sample to undergo radioactive decay.

More generally, for a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. (This article is a narrow discussion of half-life. For phenomena where half-life is applied, see "Related topics" below.)

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

Quantities subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. This interpretation is valid in many, but not all, cases of exponential decay.) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:

<math>N(t) = N_0 e^{-\lambda t} \,<math>

where

• <math>N_0<math> is the initial value of N (at t=0)
• λ is a positive constant (the decay constant).

When t=0, the exponential is equal to 1, and N(t) is equal to <math>N_0<math>. As t approaches infinity, the exponential approaches zero.

In particular, there is a time <math>t_{1/2} \,<math> such that:

<math>N(t_{1/2}) = N_0\cdot\frac{1}{2} <math>

Substituting into the formula above, we have:

<math>N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}} \,<math>

<math>e^{-\lambda t_{1/2}} = \frac{1}{2} \,<math>

<math>- \lambda t_{1/2} = \ln \frac{1}{2} = – \ln{2} \,<math>

<math>t_{1/2} = \frac{\ln 2}{\lambda} \,<math>

Thus the half-life is 69.3% of the mean lifetime.

Related topics

• Exponential decay
• Mean lifetime
• Radioactive decay
• Tables of nuclides with color-coding of half-lives:
  • Isotope table (divided) with
  • Isotope table (complete)