Textbook+Sub-chapter+-+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: -\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A math || eq. 1 || where λ is the disintegration constant and A is the rate of disintegration.
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Eq. 1 is a simple differential equation and can be solved using standard mathematical techniques. The solution is written: N_{t}=N_{0}e^{-\lambda t}\, math || 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.
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At t = T1/2 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: \lambda = \frac{ln2}{T_{1/2}} math || eq. 3 || The half-life is a characteristic value for each radioactive nuclei.
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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//, N 0 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: dN_{1}=-\lambda N_{1}dt\, math || eq. 4 ||
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dN_{2}=\lambda_{1}N_{1}dt-\lambda_{2}N_{2}dt\, math || 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: N_{2} = \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{0} (e^{-\lambda_{1} t} -e^{-\lambda_{2}t}) \rightarrow math || N_{2}= \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{1}(t) ( 1 - e^{-(\lambda_{2} - \lambda_{1}t)}) math || Eqn 6 ||
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If the half-life of the mother is much less than that of the daughter, Eqn 6 can be simplified into: N_{2} = \frac{\lambda_{1}}{\lambda_{2}} N_{0} (e^{-\lambda_{1} t} -e^{-\lambda_{2}t}) \rightarrow N_{2} \frac{\lambda_{1}}{\lambda_{2}}N_{1}(t) (1-e^{-\lambda_{2}t}) math || Eqn 7 ||
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where 1-e λ( 2) t is the saturation factor and λ 2 -λ 1 ~λ 2. 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). N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}} math || Eqn 8 ||
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\begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}} & \underbrace{N_{0}e^{-\lambda_{1}}} \\ & & N_{1} \end{matrix} math || Eqn 9 ||
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When e λ(2)t →0, eqn 9 is called a secular radioactive equilibrium and can be written as λ 2 N 2 = λ 1 N 1.