Ernest Rutherford and Frederick Soddy introduced the exponential laws of radioactive decay in 1902 to explain results of their experiments on the thorium series of radionuclides. In 1910 Harry Bateman extended the series decay formalism from the simple radioactive decay series Parent → Daughter →

Question

Ernest Rutherford and Frederick Soddy introduced the exponential laws of radioactive decay in 1902 to explain results of their experiments on the thorium series of radionuclides. In 1910 Harry Bateman extended the series decay formalism from the simple radioactive decay series Parent → Daughter → Granddaughter to a general chain of decaying nuclei with an arbitrary number of radioactive chain links designated as follows: [latex]N_1 → N_2 → N_3 →···→ N_{i−1} → N_{i}[/latex], where N stands for number of nuclei in a given generation of nuclear progeny. The initial conditions stipulate that only the first generation parent nuclei are present in a sample at time t = 0, i.e., [latex]N_1(t = 0) = N_1(0)\ and\ N_2(0) = N_3(0) =···N_{n−1}(0) = N_n(0) = 0[/latex].
Bateman equations are usually given as a set of equations that give the number of atoms [latex]N_n[/latex](t) of each nuclide of a radioactive decay chain produced after a given time t recognizing that at t = 0 (initial condition) only a given number of parent nuclei [latex]N_1(0)[/latex] were present. For generation n the set of Bateman equation and constants is usually presented in the following simple format

[latex]N_n(t)=C_1 e^{-\lambda_1 t}+C_2 e^{-\lambda_2 t}+C_3 e^{-\lambda_3 t}+\cdots+C_n e^{-\lambda_n t},[/latex] (10.131)

where [latex]C_1,C_2,...,C_n[/latex] are constants given as follows

[latex]C_1=N_1(0) \frac{\lambda_1 \lambda_2 \cdots \lambda_{i-1}}{\left(\lambda_2-\lambda_1\right)\left(\lambda_3-\lambda_1\right) \cdots\left(\lambda_i-\lambda_1\right)},[/latex] (10.132)

[latex]C_2=N_1(0) \frac{\lambda_1 \lambda_2 \cdots \lambda_{n-1}}{\left(\lambda_1-\lambda_2\right)\left(\lambda_3-\lambda_2\right) \cdots\left(\lambda_n-\lambda_2\right)},[/latex] (10.133)

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[latex]C_n=N_1(0) \frac{\lambda_1 \lambda_2 \cdots \lambda_{n-1}}{\left(\lambda_1-\lambda_n\right)\left(\lambda_3-\lambda_n\right) \cdots\left(\lambda_{n-1}-\lambda_n\right)} .[/latex] (10.134)

(a) Consolidate the Bateman equation and its constants into a single expression for generation n in radioactive decay series.
(b) Use the consolidated Bateman equation determined in (a) to express the number of nuclei for the first four generations of a radioactive decay chain.
(c) Based on results of (b) determine the activities of decay series progeny for the first four generations of a radioactive decay chain.

Accepted Answer

(a) The number [latex]N_n(t)[/latex] of nuclei of generation n can be expressed by a simple sum as follows

[latex]N_n(t)=\sum_{m=1}^n C_m e^{-\lambda_m t}[/latex] (10.135)

while the constants [latex]C_1,C_2,...,C_n[/latex] can be expressed by simple products as follows

[latex]C_m=N_1(0) \frac{\prod_{i=1}^{n-1} \lambda_i}{\prod_{\substack{i=1 \ i eq m}}^n\left(\lambda_i-\lambda_m ight)}[/latex] (10.136)

Inserting (10.136) into (10.135) we obtain a single expression for [latex]N_n(t)[/latex].

[latex]N_n(t)=\sum_{m=1}^n C_m e^{-\lambda_m t}=N_1(0) \sum_{m=1}^n\left[\frac{\prod_{i=1}^{n-1} \lambda_i}{\prod_{\substack{i=1 \ i eq m}}^n\left(\lambda_i-\lambda_m ight)} ight] e^{-\lambda_m t}[/latex] (10.137)

(b) Equation (10.131) is used to determine the number of radioactive nuclei present for a given generation in nuclear decay series at a given time t ≥ 0 with t = 0 defining the initial conditions [latex]N_1(t = 0) = N_1(0)\ and\ N_2(t = 0) = N_3(t = 0) =···N_n(t = 0) = 0[/latex].

(1) First generation (n = 1)—Number of parent nuclei [latex]N_1(t)[/latex] is expressed with the standard equation for description of exponential decay of radioactive nuclides

[latex]C_1=N_1(0) \frac{\prod_{m=1}^{n-1} \lambda_i}{\prod_{\substack{i=1 \ i eq m}}^n\left(\lambda_i-\lambda_m ight)}=N_1(0),[/latex] (10.138)

[latex]N_1(t)=\sum_{m=1}^{n=1} C_m e^{-\lambda_m t}=C_1 e^{-\lambda_1 t}=N_1(0) e^{-\lambda_1 t}[/latex] (10.139)

(2) Second generation (n = 2)—Number of daughter nuclei [latex]N_2(t)[/latex].

[latex]C_1=N_1(0) \frac{\prod_{i=1}^1 \lambda_i}{\prod_{\substack{i=1 \ i eq 1}}^1\left(\lambda_i-\lambda_1 ight)}=N_1(0) \frac{\lambda_1}{\lambda_2-\lambda_1}[/latex] (10.140)

[latex]C_2=N_1(0) \frac{\prod_{i=1}^1 \lambda_i}{\prod_{\substack{i=1 \ i eq 2}}^1\left(\lambda_i-\lambda_2 ight)}=N_1(0) \frac{\lambda_1}{\lambda_1-\lambda_2},[/latex] (10.141)

[latex]N_2(t)=\sum_{m=1}^{n=2} C_m e^{-\lambda_m t}=C_1 e^{-\lambda_1 t}+C_2 e^{-\lambda_2 t}=N_1(0) \frac{\lambda_1}{\lambda_2-\lambda_1}\left[e^{-\lambda_1 t}-e^{-\lambda_2 t} ight][/latex] (10.142)

(3) Third generation (n = 3)—Number of granddaughter nuclei [latex]N_3(t)[/latex].

[latex]C_1=N_1(0) \frac{\prod_{i=1}^2 \lambda_i}{\prod_{\substack{i=1 \ i eq 1}}^3\left(\lambda_i-\lambda_1 ight)}=N_1(0) \frac{\lambda_1 \lambda_2}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)},[/latex] (10.143)

[latex]C_2=N_1(0) \frac{\prod_{i=1}^2 \lambda_i}{\prod_{\substack{i=1 \ i eq 2}}^3\left(\lambda_i-\lambda_2 ight)}=N_1(0) \frac{\lambda_1 \lambda_2}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)},[/latex] (10.144)

[latex]C_3=N_1(0) \frac{\prod_{i=1}^2 \lambda_i}{\prod_{\substack{i=1 \ i eq 3}}^3\left(\lambda_i-\lambda_3 ight)}=N_1(0) \frac{\lambda_1 \lambda_2}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)},[/latex] (10.145)

[latex]\begin{aligned} N_3(t)= & \sum_{m=1}^{n=3} C_m e^{-\lambda_m t}=C_1 e^{-\lambda_1 t}+C_2 e^{-\lambda_2 t}+C_3 e^{-\lambda_3 t} \ = & N_1(0) \lambda_1 \lambda_2\left[\frac{e^{-\lambda_1 t}}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)}+\frac{e^{-\lambda_2 t}}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)} ight. \ & \left.+\frac{e^{-\lambda_3 t}}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)} ight] .\quad (10.146) \end{aligned}[/latex]

(4) Fourth generation (n = 4)—Number of great granddaughter nuclei [latex]N_4(t)[/latex]

[latex]C_1=N_1(0) \frac{\prod_{i=1}^3 \lambda_i}{\prod_{\substack{i=1 \ i eq 1}}^4\left(\lambda_i-\lambda_1 ight)}=N_1(0) \frac{\lambda_1 \lambda_2 \lambda_3}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)\left(\lambda_4-\lambda_1 ight)},[/latex] (10.147)

[latex]C_2=N_1(0) \frac{\prod_{i=1}^3 \lambda_i}{\prod_{\substack{i=1 \ i eq 1}}^4\left(\lambda_i-\lambda_2 ight)}=N_1(0) \frac{\lambda_1 \lambda_2 \lambda_3}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)\left(\lambda_4-\lambda_2 ight)},[/latex] (10.148)

[latex]C_3=N_1(0) \frac{\prod_{i=1}^3 \lambda_i}{\prod_{\substack{i=1 \ i eq 3}}^4\left(\lambda_i-\lambda_3 ight)}=N_1(0) \frac{\lambda_1 \lambda_2 \lambda_3}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)\left(\lambda_4-\lambda_3 ight)}[/latex] (10.149)

[latex]C_4=N_1(0) \frac{\prod_{i=1}^3 \lambda_i}{\prod_{\substack{i=1 \ i eq 4}}^4\left(\lambda_i-\lambda_1 ight)}=N_1(0) \frac{\lambda_1 \lambda_2 \lambda_3}{\left(\lambda_1-\lambda_4 ight)\left(\lambda_2-\lambda_4 ight)\left(\lambda_3-\lambda_4 ight)},[/latex] (10.150)

[latex]\begin{aligned} N_4(t) & =\sum_{m=1}^{n=3} C_m e^{-\lambda_m t}=C_1 e^{-\lambda_1 t}+C_2 e^{-\lambda_2 t}+C_3 e^{-\lambda_3 t}+C_4 e^{-\lambda_4 t} \ & =N_1(0) \lambda_1 \lambda_2 \lambda_3\left[\frac{e^{-\lambda_1 t}}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)\left(\lambda_4-\lambda_1 ight)} ight. \end{aligned}\ \begin{aligned} & +\frac{e^{-\lambda_2 t}}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)\left(\lambda_4-\lambda_2 ight)}+\frac{e^{-\lambda_3 t}}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)\left(\lambda_4-\lambda_3 ight)} \ & \left.+\frac{e^{-\lambda_4 t}}{\left(\lambda_1-\lambda_4 ight)\left(\lambda_2-\lambda_4 ight)\left(\lambda_3-\lambda_4 ight)} ight] .\quad (10.151) \end{aligned}[/latex]

(c) Activities [latex]\mathcal{A}_n(t)[/latex] of progeny in a radioactive decay series are calculated using results obtained from Bateman equation in (b).

(1) Parent activity [latex]\mathcal{A}_1(t)[/latex] at time t is given by the product of the parent decay constant [latex]λ_1[/latex] and the number of parent nuclei [latex]N_1(t)[/latex] present at time t in the sample, as given in (10.139)

[latex]\mathcal{A}_1(t)=\lambda_1 N_1(t)=\lambda_1 N_1(0) e^{-\lambda_1 t}=\mathcal{A}_1(0) e^{-\lambda_1 t}[/latex] (10.152)

(2) Daughter activity [latex]\mathcal{A}_2(t)[/latex] at time t is given by the product of the daughter decay constant [latex]λ_2[/latex] and the number of parent nuclei [latex]N_2(t)[/latex] present at time t in the sample, as given in (10.142)

[latex]\mathcal{A}_2(t)=\lambda_2 N_2(t)=N_1(0) \frac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1}\left[e^{-\lambda_1 t}-e^{-\lambda_2 t} ight]=\mathcal{A}_1(0) \frac{\lambda_2}{\lambda_2-\lambda_1}\left[e^{-\lambda_1 t}-e^{-\lambda_2 t} ight][/latex] (10.153)

where [latex]\mathcal{A}_1(0)[/latex] is the initial activity of the parent nuclei.

(3) Granddaughter activity [latex]\mathcal{A}_3(t)[/latex] at time t is given by the product of the granddaughter decay constant [latex]λ_3[/latex] and the number of granddaughter nuclei [latex]N_3(t)[/latex] present at time t in the sample as given in (10.146)

[latex]\begin{aligned} \mathcal{A}_3(t)= & \lambda_3 N_3(t) \ = & N_1(0) \lambda_1 \lambda_2 \lambda_3\left[\frac{e^{-\lambda_1 t}}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)}+\frac{e^{-\lambda_2 t}}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)} ight. \ & \left.+\frac{e^{-\lambda_3 t}}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)} ight]\quad (10.154) \end{aligned}[/latex]

(4) Great granddaughter activity [latex]\mathcal{A}_4(t)[/latex] at time t is given by the product of the decay constant [latex]λ_4[/latex] of the great grand daughter and the number of great granddaughter nuclei [latex]N_4(t)[/latex] present at time t in the sample, as given in (10.151)

[latex]\begin{aligned} \mathcal{A}_4(t)= & \lambda_4 N_4(t) \ = & N_1(0) \lambda_1 \lambda_2 \lambda_3 \lambda_4\left[\frac{e^{-\lambda_1 t}}{\left(\lambda_2-\lambda_1 ight)\left(\lambda_3-\lambda_1 ight)\left(\lambda_4-\lambda_1 ight)} ight. \ & +\frac{e^{-\lambda_2 t}}{\left(\lambda_1-\lambda_2 ight)\left(\lambda_3-\lambda_2 ight)\left(\lambda_4-\lambda_2 ight)}+\frac{e^{-\lambda_3 t}}{\left(\lambda_1-\lambda_3 ight)\left(\lambda_2-\lambda_3 ight)\left(\lambda_4-\lambda_3 ight)} \ & \left.+\frac{e^{-\lambda_4 t}}{\left(\lambda_1-\lambda_4 ight)\left(\lambda_2-\lambda_4 ight)\left(\lambda_3-\lambda_4 ight)} ight] .\quad (10.155) \end{aligned}[/latex]

Question 10.6.Q1: Ernest Rutherford and Frederick Soddy introduced the exponen......

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