Radioactive Decay Rate

The rate of decay of radioactive material is measured by the lifetime $\tau$. If we start with $N(0)$ atoms at time $t = 0$, the number of atoms remaining after time $t$ is given by

\begin{displaymath}
N(t) = N(0) \exp(-t/\tau) .
\end{displaymath} (1)

The halflife $t_{1/2}$ is the time it takes for half of the atoms to decay. It is easily found from the above formula:
\begin{displaymath}
t_{1/2} = \tau/\log(2)   .
\end{displaymath} (2)

Over an interval of time $dt$, the number of atoms decreases. The decrease is given by the negative number
\begin{displaymath}
N(t+dt) - N(t) = \frac{dN}{dt} dt  .
\end{displaymath} (3)

The number of decays $dF$ in this interval is the postive number
\begin{displaymath}
dF = -\frac{dN}{dt} dt = \frac{dt}{\tau} N(t)
\end{displaymath} (4)

So
\begin{displaymath}
dF(t)/dt = N(t)/\tau
\end{displaymath} (5)

is called the decay rate. It gives the number of decays per unit time at any instant in time.