half-life

Other Definitions
half life (dict)
half life (dict)

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.)

After # of Half-lives !! Percent of quantity remaining
0	100%
1	50
2	25
3	12.5
4	6.25
5	3.125
6	1.5625
7	0.78125%

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:

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

where

$N_0$ 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

N_0

. As t approaches infinity, the exponential approaches zero. In particular, there is a time

t_{1/2} \,

such that:

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

Substituting into the formula above, we have:

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

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

- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2} \,

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

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