Carbon dating

One of the most striking practical applications of beta decay is radiocarbon (carbon) dating, a technique for estimating the age of ancient organic material.

The physical basis. All living organisms continuously absorb carbon dioxide from the air to build organic molecules. The overwhelming majority of this absorbed carbon is stable $^{12}_{6}C$, but a tiny, nearly constant fraction ($1.3\times10^{-12}$) is the radioactive isotope $^{14}_{6}C$, whose half-life is 5730 years. Carbon-14 in the atmosphere is continuously decaying away, but it is simultaneously being continuously replenished as cosmic rays from outer space bombard atoms in the upper atmosphere and produce fresh $^{14}_{6}C$. This ongoing balance of production and decay keeps the ratio of $^{14}_{6}C$ to $^{12}_{6}C$ in the atmosphere - and hence, through continuous absorption, in every living organism - essentially constant while the organism is alive.

After death. Once an organism dies, it stops absorbing new carbon dioxide, so its $^{14}_{6}C$ content is no longer replenished and simply decays away following the ordinary exponential law; the ratio of $^{14}_{6}C$ to $^{12}_{6}C$ in the dead remains therefore steadily decreases over the following centuries and millennia. By measuring the present-day carbon-14 activity of an ancient sample and comparing it to the activity a similar living sample would have, the elapsed time since death - the sample's age - can be calculated using $R=R_0e^{-\lambda t}$, rearranged to $t=\tfrac{1}{\lambda}\ln(R_0/R)$. …