Half-Life

half-lives elapsed ||~ Fraction remaining ||||~ Percentage remaining || **Half-life**, abbreviated **t½**, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms ( [|radioactive decay] ), but it may apply to any quantity which follows a set-rate decay. The original term, dating to 1907, was "half-life period", which was later shortened to "half-life" in the early 1950s. [|[1]] Half-lives are used to describe quantities undergoing [|exponential decay] —for example, radioactive decay—where the half-life is constant over the whole life of the decay, and is a [|characteristic unit] (a natural unit of scale) for the exponential decay equation. However, a half-life can also be defined for non-exponential decay processes, although in these cases the half-life varies throughout the decay process. For a general introduction and description of exponential decay, see the article [|exponential decay]. For a general introduction and description of non-exponential decay, see the article [|rate law]. Corresponding to sediments in environmental processes, if the half-life is greater than the residence time, then the radioactive nuclide will have enough time to significantly alter the concentration. The converse of half-life is [|doubling time]. The table on the right shows the reduction of a quantity in terms of the number of half-lives elapsed.
 * ~ Number of
 * 0 || 1/1 || 100 ||  ||
 * 1 || 1/2 || 50 ||  ||
 * 2 || 1/4 || 25 ||  ||
 * 3 || 1/8 || 12 || .5 ||
 * 4 || 1/16 || 6 || .25 ||
 * 5 || 1/32 || 3 || .125 ||
 * 6 || 1/64 || 1 || .563 ||
 * 7 || 1/128 || 0 || .781 ||
 * //n// || 1/(2//n//) |||| 100/(2//n//) ||
 * //n// || 1/(2//n//) |||| 100/(2//n//) ||

A half-life describes the decay of discrete entities, such as radioactive atoms. 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 1 second, there will //not// be "half of an atom" left after 1 second. There will be either zero atoms left or one atom left, depending on whether or not the atom happens 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 measure whether or not it decays in that period of time. Perhaps it will and perhaps it will 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 time-range measured. In this case, the [|law of large numbers] ensures that the number of atoms that //actually// decay is essentially 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 can be ignored.

There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a computer program. [|[2]] [|[3]] [|[4]] 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//, due to random variation in the process. However, with more atoms (right boxes), the overall decay is smoother and less random than with fewer atoms (left boxes), in accordance with the [|law of large numbers].