Radioactive disintegration is a stochastic process, which means a random process that can be described statistically. Here you will learn about the secular radioactive equilibrium, and how any measure of a radioactive source is stated with uncertainty.

## The Basics - A Single Radioactive Nucleus

In a sample with N radioactive atoms of a particular nuclide, the number of nuclei that disintegrates with the time *dt* will be proportional with N, see eq. 1:

where λ is the disintegration constant and A is the rate of disintegration.

Eq. 1 is a simple differential equation and can be solved using standard mathematical techniques. The solution is written:

N0 is the number of nuclei at present at t = 0. The time when half of the nuclei has disintegrated is called the half-life.

At t = T1/2 then N = N0/2. If we insert this into eq. 2 the following connection between the disintegration constant and the half-life is obtained:

The half-life is a characteristic value for each radioactive nuclei.

## Decay Chains and Mother-Daughter Relationships

A radioactive nuclide will often disintegrate into a product that is radioactive as well: Nucleus 1→Nucleus 2 →Nucleus 3. The initial nucleus is usually referred to as the mother nuclide and the product as the daughter nuclide.

Assume that at the time*t = 0*, N0 of the mother is *N*1*(t =0), N*2*(t=0) and N*3*(t=0)*, the change in number of mother- and daughter nuclei can then respectively be described through Eqn 4 and Eqn 5:

The solution of Eqn 4 is already known, it is the expression in eq. 2 while the solution for the amount of daughter nuclei are given with:

If the half-life of the mother is much less than that of the daughter, Eqn 6 can be simplified into:

where 1-eλ(2)t is the saturation factor and λ2-λ1~λ2. The above equation can be further reduced by the assumption that t >> T ½(2) (the observed time is much larger than the daughters half-life).

When eλ(2)t→0, eqn 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.

eq. 1 |

Eq. 1 is a simple differential equation and can be solved using standard mathematical techniques. The solution is written:

eq. 2 |

At t = T1/2 then N = N0/2. If we insert this into eq. 2 the following connection between the disintegration constant and the half-life is obtained:

eq. 3 |

Assume that at the time

eq. 4 |

eq. 5 |

Eqn 6 |

If the half-life of the mother is much less than that of the daughter, Eqn 6 can be simplified into:

Eqn 7 |

where 1-eλ(2)t is the saturation factor and λ2-λ1~λ2. The above equation can be further reduced by the assumption that t >> T ½(2) (the observed time is much larger than the daughters half-life).

Eqn 8 |

Eqn 9 |

When eλ(2)t→0, eqn 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.