Question 10.3.Q4: Consider the simplest radioactive decay series: P → D → G, w......

(a) Expressions for N_P(t)\ and\ N_D(t) are well known [see, for example, (T10.9) and (T10.34), respectively], so they will not be derived here. N_P(t) exhibits a pure exponential behavior and N_D(t) accounts for the supply of new daughter nuclei through the decay of P given as λ_PN_P(t) and the loss of daughter nuclei D from the concurrent decay of D to G given as −λ_DN_D(t), where λ_P\ and\ λ_D are the decay constants of parent P and daughter D, respectively.
For initial conditions N_{\mathrm{P}}(t=0)=N_{\mathrm{P}}(0) \text { and } N_{\mathrm{D}}(t=0)=0 we have the following expressions for N_{\mathrm{P}}(t) \text { and } N_{\mathrm{D}}(t), respectively

\frac{\mathrm{d} N_{\mathrm{P}}(t)}{\mathrm{d} t}=-\lambda_{\mathrm{P}} N_{\mathrm{P}}(t) \quad \text { or } \quad N_{\mathrm{P}}(t)=N_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t}         (10.89)

and

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\lambda_{\mathrm{P}} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) \quad \text { or } \quad 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] .           (10.90)

Functional dependence of N_G(t) is less known and can be derived by recognizing that the rate of change (growth) of G is governed by the decay of D, expressed as follows

\frac{\mathrm{d} N_{\mathrm{G}}(t)}{\mathrm{d} t}=-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) .        (10.91)

Inserting (10.90) into (10.91) we get the following expression for dN_G(t)/dt

\frac{\mathrm{d} N_{\mathrm{G}}(t)}{\mathrm{d} t}=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}} \lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[e^{-\lambda_{\mathrm{P}} t}-e^{-\lambda_{\mathrm{D}} t}\right]        (10.92)

Upon integration of (10.92) from 0 to t we get the following expression for N_G(t).

N_{\mathrm{G}}(t)=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}} \lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[-\frac{e^{-\lambda_{\mathrm{P}} t}}{\lambda_{\mathrm{P}}}+\frac{e^{-\lambda_{\mathrm{D}} t}}{\lambda_{\mathrm{D}}}\right]+C,            (10.93)

where C is an integration constant, for initial condition N_G(t = 0) = 0 given as

N_{\mathrm{G}}(t=0)=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}} \lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[-\frac{1}{\lambda_{\mathrm{D}}}+\frac{1}{\lambda_{\mathrm{P}}}\right]+C=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}+C=0          (10.94)

or

C=N_{\mathrm{P}}(0)         (10.95)

Based on (10.93) incorporating (10.95) the number of grand-daughter nuclei N_G(t) as a function of time t is given as follows

In summary, the number of parent nuclei N_P(t) is given by (10.89), the number of daughter nuclei N_D(t) by (10.90), and the number of granddaughter nuclei N_G(t) by (10.96).

(b) The limits of N_G(t) for t → 0 and t → ∞ resulting from (10.96) are as follows

(1)

(2)

\lim _{t \rightarrow \infty} N_{\mathrm{G}}(t)=N_{\mathrm{P}}(0) \lim _{t \rightarrow \infty}\left\{1-\frac{\lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{P}} t}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}+\frac{\lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{D}} t}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\right\}=N_{\mathrm{P}}(0) .            (10.98)

(c) Since the initial conditions for our radioactive sample stipulate that at t = 0 we are dealing with a pure parent radioactive source, i.e., N_P(t = 0) = N_P(0), N_D(t = 0) = 0, and N_G(t = 0) = 0, we conclude that at any time t > 0 the sum of all nuclei N_P(t),\ N_D(t),\ and\ N_G(t) must be equal to N_P(0), the initial number of nuclei in the decay series. We now prove that this conclusion is correct by producing a sum consisting of N_P(t) given in (10.89), N_D(t) given in (10.90), and N_G(t) given in (10.96) as follows

(d) The curves of Fig. 10.5 represent the number of nuclei of either the parent (Mo-99), daughter (Tc-99m), or grand-daughter (Tc-99) for the radioactive decay series starting with a pure source of molybdenum-99 radionuclide. The three curves are identified as follows:

(1) Curve 1 depicts decay of the parent radionuclide Mo-99.
(2) Curve 2 depicts growth and decay of the daughter radionuclide Tc-99m.
(3) Curve 3 depicts growth of the granddaughter nuclide Tc-99 under the assumption that, because of its very long half-life, Tc-99 is essentially stable

(e) The characteristic time \left(t_{\max }\right)_{\mathrm{D}} \text { in which the } N_{\mathrm{D}}(t) curve reaches its maximum is determined by setting \mathrm{d} N_{\mathrm{D}} / \mathrm{d} t=0 \text { at } t=\left(t_{\max }\right)_{\mathrm{D}}. Differentiating (10.90) results in

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[-\lambda_{\mathrm{P}} e^{-\lambda_{\mathrm{P}} t}+\lambda_{\mathrm{D}} e^{-\lambda_{\mathrm{D}} t}\right]              (10.100)

and after setting \mathrm{d} N_{\mathrm{D}} / \mathrm{d} t=0 \text { at } t=\left(t_{\max }\right) \mathrm{D} we 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}}} .        (10.101)

Solving (10.101) for \left(t_{max}\right)_D finally yields the following general result for characteristic time \left(t_{max}\right)_D at which the daughter attains its maximum number of nuclei in the radioactive decay series

\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}=\frac{\ln \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{P}}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}} .          (10.102)

For our specific case of Mo-99 → Tc-99m → Tc-99 radioactive decay series the characteristic time \left(t_{max}\right)_D is calculated as follows

\left(t_{\max }\right)_{\mathrm{D}}=\left(t_{\max }\right)_{\mathrm{Tc}-99 \mathrm{~m}}=\frac{\ln \frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}=\frac{\ln \frac{1.05 \times 10^{-2}}{0.115}}{\left(1.05 \times 10^{-2} \mathrm{~h}^{-1}-0.115 \mathrm{~h}^{-1}\right)}=22.88 \mathrm{~h} .           (10.104)

This result matches the characteristic time \left(t_{max}\right)_D in which the daughter in a radioactive decay series reaches its maximum activity [see Prob. 216(b)]
The normalized peak value of N_{\mathrm{D}}(t) / N_{\mathrm{P}}(0) \text { at }\left[t=\left(t_{\max }\right)_{\mathrm{D}}\right] is calculated by inserting t=\left(t_{\max }\right)_{\mathrm{D}} into (10.96) to get the following result