Lab Exercise - Radioactive Disintegration

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:

• -dN/dt = λN → λN = A                       (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:

• N(t) = N(0)e^(-λt)                               (eq. 2)

N(0) 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 = T(1/2) then N = N(0/2). If we insert this into eq. 2 the following connection between the disintegration constant and the half-life is obtained:

• λ = ln(2) / T(1/2)                                 (eq. 3)

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 N1(t =0), N2(t=0) and N3(t=0), the change in number of mother- and daughter nuclei can then respectively be described through Eq 4 and Eq 5:

• dN(1) = -λN(1)dt                                 (eq. 4)
• dN(2) = -λ(1)N(1)dt - λ(2)N(2)dt         (eq. 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:

• eq. 6

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

• eq. 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).
  
• eq. 8
  
  
• eq. 9
  
  When eλ(2)t→0, eqn 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.