experimental physics - How do we know that some radioactive materials have a half life of millions or even billions of years?

If a radioactive material takes a very long time to decay, how is its half life measured or calculated? Do we have to actually observe the radioactive material for a very long time to extrapolate its half life?

Answer

No, one doesn't need to measure the material for years - or even millions or billions of years. It's enough to watch it for a few minutes (for time $t$) and count the number of atoms $\Delta N$ (convention: a positive number) that have decayed. The lifetime $T$ is calculated from $$ \exp(-t/T) = \frac{N - \Delta N}{N}$$ where $N$ is the total number of atoms in the sample. This $N$ can be calculated as $$N={\rm mass}_{\rm total} / {\rm mass}_{\rm atom}.$$ If we know that the lifetime is much longer than the time of the measurement, it's legitimate to Taylor-expand the exponential above and only keep the first uncancelled term: $$ \frac{t}{T} = \frac{\Delta N}{N}.$$ The decay of the material proceeds atom-by-atom and the chances for individual atoms to decay are independent and equal.

To get some idea about the number of decays, consider 1 kilogram of uranium 238. Its atomic mass is $3.95\times 10^{-25}$ kilograms and its lifetime is $T=6.45$ billion years. By inverting the atomic mass, one sees that there are $2.53\times 10^{24}$ atoms in one kilogram. So if you take one kilogram of uranium 238, it will take $2.53\times 10^{24}$ times shorter a time for an average decay, e.g. the typical separation between two decays is $$t_{\rm average} = \frac{6.45\times 10^9\times 365.2422\times 86400}{2.53\times 10^{24}}{\rm seconds} = 8.05\times 10^{-8} {\rm seconds}. $$ So one gets about 12.4 million decays during one second. (Thanks for the factor of 1000 fix.) These decays may be observed on an individual basis. Just to be sure, $T$ was always a lifetime in the text above. The half-life is simply $\ln(2) T$, about 69 percent of the lifetime, because of some simple maths (switching from the base $e$ to the base $2$ and vice versa).

If we observe $\Delta N$ decays, the typical relative statistical error of the number of decays is proportional to $1/(\Delta N)^{1/2}$. So if you want the accuracy "1 part in 1 thousand", you need to observe at least 1 million decays, and so on.