Radioactive isotopes can be produced by nuclear reactions in
reactors or accelerators, or even by natural processes such as cosmic
ray interactions. With a constant production rate R and a decay
constant 
, the rate of change of the amount present is
which has the solution
or in terms of activity
Thus for large times 
, an equilibrium situation arises in
which the rate of decay equals that of production; 
and
.
In other words, it is not worth irradiating beyond several halflives.