Half-life - Probabilistic Nature of Half-life

Probabilistic Nature of Half-life

A half-life usually describes the decay of discrete entities, such as radioactive atoms, which have unstable nuclei. In that case, it does not work to use the definition "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom with a half-life of one second, there will not be "one-half of an atom" left after one second. There will be either zero atoms left or one atom left, depending on whether or not that atom happened to decay.

Instead, the half-life is defined in terms of probability. It is the time when the expected value of the number of entities that have decayed is equal to half the original number. For example, one can start with a single radioactive atom, wait its half-life, and then check whether or not it has decayed. Perhaps it did, but perhaps it did not. But if this experiment is repeated again and again, it will be seen that - on average - it decays within the half-life 50% of the time.

In some experiments (such as the synthesis of a superheavy element), there is in fact only one radioactive atom produced at a time, with its lifetime individually measured. In this case, statistical analysis is required to infer the half-life. In other cases, a very large number of identical radioactive atoms decay in the measured time range. In this case, the law of large numbers ensures that the number of atoms that actually decay is approximately equal to the number of atoms that are expected to decay. In other words, with a large enough number of decaying atoms, the probabilistic aspects of the process could be neglected.

There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program. For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the atoms remaining, only approximately, because of the random variation in the process. However, with more atoms (right boxes), the overall decay is smoother and less random-looking than with fewer atoms (left boxes), in accordance with the law of large numbers.