Radioactive decay
One-decay process

Consider the case of a nuclide A that decays into another B by some process A → B (emission of other particles, like electron neutrinos ν and electrons e as in beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random in time so it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events −d N expected to occur in a small interval of time d t 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 −d N / N is proportional to an increment of time, d t:

− d N N = λ d t {\displaystyle -{\frac {\mathrm {d} N}{N}}=\lambda \mathrm {d} t}

The negative sign indicates that N decreases as time increases, as the decay events follow one after another. The solution to this first-order differential equation is the function:

where N is the value of N at time t = 0, with the decay constant expressed as λ

We have for all time t:

where N is the constant number of particles throughout the decay process, which is 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. the number of decayed A nuclei) is

The number of decays observed over a given interval obeys Poisson statistics. If the average number of decays is ⟨ N ⟩, the probability of a given number of decays N is
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