section_path: Mathematics/Universal law/One-decay process
section_name: One-decay process
paragraph_text: 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