### Exponential decay law

The transition of a nucleus from one state to another is characterised by a ``lifetime''. This does not imply that each nucleus will ``live'' for that time (or approximately that time). It was shown very early on that decay is a random process: this gives rise to an exponential decay law.

The decay of (a particular state of) a nucleus is determined by one number, the decay constant . (We will see how this relates to a lifetime). The study of nuclear transition mechanisms leads to an understanding of the decay constant. In this section we will examine the consequences of this random process.

Experience has shown that decay is a random process. The probability of a nucleus decaying in a time interval dt is . The probability is thus independent of time, it is independent of the age of a particular nucleus and is the same for all nuclei in the same state (i.e. decay is a Poisson process). As a result of this, we cannot predict when a particular nucleus will decay, we can only make predictions about ensembles.

Suppose that we have N(t) nuclei in a particular state at time t. Then in a time interval dt a number will decay, so

Integrating this gives

the famous exponential decay law.

A characteristic of this equation is that in a fixed time interval T, a fixed fraction will decay. Thus we cannot talk of a lifetime of a nucleus, but only of what fraction will decay in a certain interval. This leads us to single out a particular time interval as interesting: we define the half-life as that time interval in which half of an initial sample will decay. From the equation we thus obtain

It is also useful to define a mean life in the usual statistical way:

The definition of the half-life suggests a way in which the decay constant can be measured. However, determination of small amounts of rare heavy nuclides is a difficult process. A way out is obtained by looking rather at the rate of decay. We define the activity . Thus for the case of exponential decay,

From the exponential radioactive decay law we obtain

Note that activity represents the number of decays per unit time interval. This is not necessarily equal to (minus) the rate of change of the number of atoms present -- this is only true in the case of a single decay as above.

Thus we can determine the decay constant by measuring the rate of decay, i.e. by measuring the number of decays (``counts per second'') in a certain time interval as a function of time.

The unit of activity is the bequerel (Bq), defined by 1 Bq = 1 decay per second, or curie (Ci), where 1 Ci = decays per second.