The decay rate of a radioactive substance are characterized by the following:
Constant quantities:
- The half-life—t_{1/2}, is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value;
- The mean lifetime— τ, "tau" the average lifetime of a radioactive particle before decay.
- The decay constant— λ, "lambda" the inverse of the mean lifetime.
Although these are constants, they are associated with statistically random behavior of populations of atoms. Predictions using these constants are less accurate for small number of atoms. In principle the reciprocal of any number greater than one— a half-life, a third-life, or even a (1/√2)-life can be used in exactly the same way as half-life; but the half-life, t_{1/2,} is adopted as the standard time associated with exponential decay.
Time-variable quantities:
- Total activity— A, is number of decays per unit time of a radioactive sample.
- Number of particles—N, is the total number of particles in the sample.
- Specific activity—SA, number of decays per unit time per amount of substance of the sample at time set to zero (t = 0). "Amount of substance" can be the mass, volume or moles of the initial sample.
Radioactivity is one very frequent example of exponential decay, shown in this figure . The law describes the statistical behavior of a large number of nuclides, rather than individual ones. In the following formalism, the number of nuclides or nuclide population N, is of course a discrete variable (a natural number) - but for any physical sample N is so large (amounts of L = 10^{23}, avogadro's constant) that it can be treated as a continuous variable. Differential calculus is needed to set up differential equations for modelling the behavior of the nuclear decay.Consider the case of a nuclide A decaying into another B by some process A → B. Given a sample of a particular radioisotope, the number of decay events, −dN, expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is:
Particular radionuclides decay at different rates, so each has its own decay constant λ. The expected decay −dN/N is proportional to an increment of time, dt:
The negative sign indicates that N decreases as time increases, as each decay event follows one after another. The solution to this first-order differential equation is the function:
where N_{0} is the value of N at time t = 0. Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms and follow Poisson statistics as a consequence of the random nature of the process. We have for all time t:
where N_{total} is the constant number of particles throughout the decay process, clearly equal to the initial number of A nuclides since this is the initial substance. If the number of non-decayed A nuclei is:
then the number of nuclei of B, i.e. number of decayed A nuclei, is:
We should make a quick mention of the units of decay. The SI unit of radioactive activity is the becquerel (Bq), in honor of the scientist Henri Becquerel. One Bq is defined as one transformation (or decay or disintegration) per second. Since sensible sizes of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts giving activities on the order of GBq (gigabecquerel, 1 x 109 decays per second) or TBq (terabecquerel, 1 x 1012 decays per second) are commonly used. Another unit of radioactivity is the curie, Ci, which was originally defined as the amount of radium emanation (radon-222) in equilibrium with one gram of pure radium, isotope Ra-226. At present it is equal, by definition, to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 1010 Bq, so that 1 curie (Ci) = 3.7 × 1010 Bq. The use of Ci is currently discouraged by the SI. Low activities are also measured in disintegrations per minute (dpm).