Question 10.3.Q1: Radioactive decay through a series of radioactive transforma......
Radioactive decay through a series of radioactive transformations is much more common than the simple radioactive decay from a radioactive parent into stable daughter. The radioactive decay series forms a decay chain starting with the parent radionuclide and moves through several generations to eventually end with a stable nuclide.
Consider the simple chain P → D → G where both the parent P and daughter D are radioactive and the granddaughter G is stable. The parent decays with a decay constant λ_P while the daughter decays with a decay constant λ_D. For this simple decay series:
(a) State the differential equations governing the kinetics of the radioactive parent and radioactive daughter.
(b) Solve the differential equations in (a) with the following initial conditions:
(1) Initial number of parent nuclei N_P(t) at time t = 0 is N_P(0).
(2) Initial number of daughter nuclei N_D(t = 0) = N_D(0) = 0.
to get expressions for the number of parent nuclei N_P and for the number of daughter nuclei N_D(t) as a function of time t.
(c) Using the results of (b) obtain an expression for activity of the daughter \mathcal{A}_D(t).
(d) The expression for the daughter activity \mathcal{A}_D(t) derived in (c) should predict \mathcal{A}_D(t) = 0 for t = 0 [recall the initial condition N_D(0) = 0] as well as for t → ∞ (recall that at t = ∞ all daughter nuclei will have decayed). This means that \mathcal{A}_D(t) must reach a maximum value \left(\mathcal{A}_D\right)_{max} at a characteristic time \left(t_{max}\right)_D somewhere between the two extremes: t=0 \text { and } t=\infty \text {, i.e., } 0<\left(t_{\max }\right)_{\mathrm{D}}<\infty. Derive an expression for the characteristic time \left(t_{max}\right)_D.
(e) Show that for \lambda_P \gtrsim \lambda_D \text { (but not } \lambda_P=\lambda_D \text { ) and for } \lambda_P \lesssim \lambda_D (but not λ_P = λ_D) the characteristic time \left(t_{max}\right)_D can be approximated by
with this approximation and compare results with the expression derived in (d) for the following two radioactive series decays: (1) Series decay with \lambda_{\mathrm{P}}=2.1 \mathrm{y}^{-1} \text { and } \lambda_{\mathrm{D}}=2.0 \mathrm{y}^{-1} and (2) Series decay with \lambda_P=5.1 \mathrm{~s}^{-1} \text { and } \lambda_P=5.5 \mathrm{~s}^{-1}.
(a) The differential equations governing the kinetics of the parent P and the daughter D nuclei in the simple P → D → G decay chain describe the rate of change in the number of parent nuclei N_P(t) and in the number of daughter nuclei N_D(t).
(1) For the parent, the rate of change dN_P(t)/dt in the number of parent nuclei is given by the standard expression dealing with nuclear decay
\frac{\mathrm{d} N_{\mathrm{P}}(t)}{\mathrm{d} t}=-\lambda_{\mathrm{P}} N_{\mathrm{P}}(t), (10.30)
with the minus sign indicating a decrease in the number of parent nuclei N_P(t) with increasing time t.
(2) The rate of change dN_D(t)/dt in the number of daughter nuclei D is equal to the supply of new daughter nuclei D through the decay of P given as λ_PN_P(t) and the loss of daughter nuclei D from the decay of D to G given \text { as }\left[-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)\right] \text {. The rate of change } \mathrm{d} N_{\mathrm{D}} / \mathrm{d} t is expressed as
\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\lambda_{\mathrm{P}} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) . (10.31)
(b) Equation (10.30) shows that the parent P follows a straightforward radioactive decay process with the initial condition N_{\mathrm{P}}(t=0)=N_{\mathrm{P}}(0) and the following solution
N_{\mathrm{P}}(t)=N_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t} (10.32)
The solution to (10.31) for the daughter, on the other hand, is more complicated and will be determined after inserting (10.32) into (10.31) to get the following expression for the rate of change in the number of daughter nuclei
\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\lambda_{\mathrm{P}} N_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t}-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) (10.33)
The general solution of the differential equation given by (10.33) is given as
N_{\mathrm{D}}(t)=N_{\mathrm{P}}(0)\left[p e^{-\lambda_{\mathrm{P}} t}+d e^{-\lambda_{\mathrm{D}} t}\right] (10.34)
where p and d are constants to be determined using the following four steps:
1. Differentiate (10.34) with respect to time t to obtain
\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=N_{\mathrm{P}}(0)\left[-p \lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{P}} t}-d \lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{D}} t}\right] (10.35)
2. Insert (10.34) and (10.35) into (10.33) and rearrange the terms to get
e^{-\lambda_{\mathrm{P}} t}\left[-p \lambda_{\mathrm{P}}-\lambda_{\mathrm{P}}+p \lambda_{\mathrm{D}}\right]=0 . (10.36)
3. The factor in square brackets of (10.36) must be equal to zero to satisfy the equation for all possible values of t, yielding the following expression for the constant p
p=\frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}} (10.37)
4. The coefficient d depends on the initial condition for N_D(t) at t = 0. With the standard initial condition N_D(0) = 0 we get the following simple equation from (10.34)
p+d=0 (10.38)
which upon insertion of (10.37) provides the following result for constant d
d=-p=-\frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}=\frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}} (10.39)
After inserting (10.37) and (10.39) into (10.34) we get the following expression for the number of daughter nuclei N_D(t) as a function of time t
N_{\mathrm{D}}(t)=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[e^{-\lambda_{\mathrm{P}} t}-e^{-\lambda_{\mathrm{D}} t}\right] . (10.40)
(c) The simple P → D → G radioactive series decay with radioactive parent P decaying through radioactive daughter D into stable grand-daughter G is characterized by equations describing the number of parent nuclei N_P(t) and number of daughter nuclei N_D(t) given by (10.32) and (10.40), respectively. Activities \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t) of the parent and daughter, respectively, in a radioactive sample are also of interest and can be determined by recalling that, in general, the activity \mathcal{A}(t) of a radionuclide is the product of its decay constant λ and the number N(t) of radioactive nuclei present in the sample.
We thus get the following expressions for the activity of the parent \mathcal{A}_P(t) from (10.32) and activity of the daughter \mathcal{A}_D(t) from (10.40), respectively
\mathcal{A}_{\mathrm{P}}(t)=\lambda_{\mathrm{P}} N_{\mathrm{P}}(t)=\lambda_{\mathrm{P}} N_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t}=\mathcal{A}_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t} (10.41)
and
where \mathcal{A}_P(0) is the activity of the parent at time t = 0.
A test of the limiting value of \mathcal{A}_D(t) given in (10.42) for t = 0 and t → ∞ yields zero, as it should according to: (1) initial condition N_D(0) = 0 and (2) at t = ∞ all daughter nuclei will have decayed. From (10.42) we note: (1) \lim _{t \rightarrow 0} \mathcal{A}_{\mathrm{D}}(t)=0 and (2) \lim _{t \rightarrow \infty} \mathcal{A}_{\mathrm{D}}(t)=0.
(d) The characteristic time \left(t_{\max }\right)_{\mathrm{D}} \text { at which the daughter activity } \mathcal{A}_{\mathrm{D}}(t) attains its maximum value \left(\mathcal{A}_{\mathrm{D}}\right)_{\max } \text { is determined by setting } \mathrm{d} \mathcal{A}_{\mathrm{D}} / \mathrm{d} t=0 \text { at } t=\left(t_{\max }\right)_{\mathrm{D}} and solving for \left(t_{max}\right)_D to get
From (10.43) we first get
\lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{P}}\left(t_{\max }\right)_\mathrm{D}}=\lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{D}}\left(t_{\max }\right)_\mathrm{D}} (10.44)
then
\frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}=e^{\left(\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}\right) \times\left(t_{\max }\right)_{\mathrm{D}}} (10.45)
and finally get the following general result for \left(t_{max}\right)_D.
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}} (10.46)
(e) For \lambda_{\mathrm{P}} \gtrsim \lambda_{\mathrm{D}} \text { and } 0<\varepsilon \ll 1 we assume the following relationship between decay constants \lambda_P \text { and } \lambda_D of the parent and daughter, respectively
\lambda_{\mathrm{P}}=\lambda_{\mathrm{D}}(1+\varepsilon) \quad \text { or } \quad \lambda_{\mathrm{P}}(1-\varepsilon) \approx \lambda_{\mathrm{D}} \text {. } (10.47)
Inserting (10.47) into (10.46) we get
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}} \approx \frac{\ln (1+\varepsilon)}{\varepsilon \lambda_{\mathrm{D}}} (10.48)
The logarithm in (10.48) can be simplified with Taylor expansion into a series as follows
\ln (1+\varepsilon) \approx \varepsilon-\frac{1}{2} \varepsilon^2+\frac{1}{3} \varepsilon^3-\frac{1}{4} \varepsilon^4+\cdots (10.49)
For very small ε we insert (10.47) and the first two terms of the Taylor series (10.49) into (10.48) and get the following expression for \left(t_{max}\right)_D.
\left(t_{\max }\right)_{\mathrm{D}} \approx \frac{\ln (1+\varepsilon)}{\varepsilon \lambda_{\mathrm{D}}} \approx \frac{1-\frac{1}{2} \varepsilon}{\lambda_{\mathrm{D}}} \approx \frac{\sqrt{1-\varepsilon}}{\lambda_{\mathrm{D}}} \approx \frac{1}{\sqrt{\lambda_{\mathrm{P}} \lambda_{\mathrm{D}}}} . (10.50)
Similarly, for \lambda_{\mathrm{P}} \lambda_{\mathrm{D}} \text { and } 0<\varepsilon \ll 1 we assume the following relationship
\lambda_{\mathrm{P}}=\lambda_{\mathrm{D}}(1-\varepsilon) \quad \text { or } \quad \lambda_{\mathrm{P}}(1+\varepsilon)=\lambda_{\mathrm{D}} (10.51)
Inserting (10.51) into (10.48) we get
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}} \approx \frac{\ln (1-\varepsilon)}{-\varepsilon \lambda_{\mathrm{D}}} . (10.52)
The logarithm in (10.52) can be simplified with Taylor expansion into a series as follows
\ln (1-\varepsilon) \approx-\left[\varepsilon+\frac{1}{2} \varepsilon^2+\frac{1}{3} \varepsilon^3+\frac{1}{4} \varepsilon^4+\cdots\right] (10.53)
For very small ε we insert (10.51) and the first two terms of the Taylor expansion (10.53) into (10.52) and get the following expression for \left(t_{max}\right)_D.
\left(t_{\max }\right)_{\mathrm{D}} \approx \frac{\ln (1-\varepsilon)}{\varepsilon \lambda_{\mathrm{D}}} \approx \frac{1+\frac{1}{2} \varepsilon}{\lambda_{\mathrm{D}}} \approx \frac{\sqrt{1+\varepsilon}}{\lambda_{\mathrm{D}}} \approx \frac{1}{\sqrt{\lambda_{\mathrm{P}} \lambda_{\mathrm{D}}}} . (10.54)
We now compare results of \left(t_{max}\right)_D calculation with the general expression for \left(t_{max}\right)_D given in (10.46) and with the approximation given in (10.51) and (10.54).
(1) Series decay with \lambda_{\mathrm{P}}=2.1 \mathrm{y}^{-1} \text { and } \lambda_{\mathrm{D}}=2.0 \mathrm{y}^{-1} \text { for which } \lambda_{\mathrm{P}} \lambda_{\mathrm{D}}.
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}=\frac{\ln \frac{2.1}{2.0}}{2.1-2.0} \mathrm{y}^{-1}=0.4879 \mathrm{y} (10.55)
\left(t_{\max }\right)_{\mathrm{D}} \approx \frac{1}{\sqrt{\lambda_{\mathrm{P}} \lambda_{\mathrm{D}}}}=\frac{1}{\sqrt{2.1 \times 2.0}} \mathrm{y}=0.48795 \mathrm{y} . (10.56)
(2) Series decay with \lambda_P=5.1 \mathrm{~s}^{-1} \text { and } \lambda_P=5.5 \mathrm{~s}^{-1}.
\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}=\frac{\ln \frac{5.1}{5.5}}{5.1-5.5} \mathrm{y}^{-1}=0.1888 \mathrm{y} \text {, } (10.57)
\left(t_{\max }\right)_{\mathrm{D}} \approx \frac{1}{\sqrt{\lambda_{\mathrm{P}} \lambda_{\mathrm{D}}}}=\frac{1}{\sqrt{2.1 \times 2.0}} \mathrm{y}=0.1888 \mathrm{y} (10.58)