# half-life

Half-life illustrated in a graph of radioactive decay.
After the elapse of each half-life period (*t*_{½}),
only half the number (*n*) of undisintegrated nuclei remain
that were present at the beginning of the period. The average time
that elapses before a nucleus disintegrates (*τ*), starting
from the beginning of the process (*t*=0) is given by: *τ* = 1.443*t*_{½}.

The half-life is the time taken for a substance or collection of particles to decay by half of its original amount.

Half-life, denoted *t*_{½}, is a useful concept by which
to express the rate of radioactive decay.
After one half-life, half of the original number of atoms of a radioactive
element will remain. After two half-lives, one-quarter (= ½ ×
½) will remain. After three half-lives, 1/8 (= ½ × ½
× ½) will remain, and so on.

The mathematical relationship is exponential and at any time *t* the number remaining *n* is given by

*n* = *n*_{0} exp(–*λt* ),

where *n*_{0} is the original number and *λ* is the *decay constant*, which is equal to 0.693*t*_{½}.

The following relationships also exist between the half-life (*t*_{½}),
decay constant (*λ*), and average lifetime (*τ*):

*t*_{½} = *λ*^{–1} · ln 2 = 0.693/*λ*

* λ* = t_{½} · ln 2 = 0.693/*t*_{½}

* τ* = 1.443 t_{½}

There are extreme variations in the half-lives of the various radionuclides,
e.g. from 7.2 × 10^{24} years for tellurium-128
down to 2 × 10^{–}^{16} seconds for beryllium-8.

Decay curve of tritium (H-3) –
half-life 12.3 years Credit: European Nuclear Society |