Question 10.5.Q1: In many parent P → daughter D → granddaughter G relationship......

(a) The radioactive parent activity A_P(t) is given as (T10.10)

\mathcal{A}_{\mathrm{P}}(t)=\mathcal{A}_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t}=\lambda_{\mathrm{P}} N_{\mathrm{P}}(0) e^{-\lambda_{\mathrm{P}} t}=\mathcal{A}_{\mathrm{P}}(0) e^{-\frac{(\ln 2)}{\left(t_{1 / 2}\right)_{\mathrm{P}}} t},         (10.123)

while the radioactive daughter activity \mathcal{A}_D(t) is (T10.35)

\mathcal{A}_{\mathrm{D}}(t)=\mathcal{A}_{\mathrm{P}}(t) \frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}\left[1-e^{-\left(\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}\right) t}\right],           (10.124)

where \mathcal{A}_P(0) is the initial activity of the parent at time t = 0, λ_P is the decay constant of the parent radionuclide, and λ_D is the decay constant of the daughter radionuclide. The initial daughter activity \mathcal{A}_D(0) is assumed to be zero at t = 0.
The activity ratio ζ is defined as the ratio between the daughter and parent activities at time t that from (10.124) can be expressed as follows after introducing the definitions for the new variable x = t/(t_{1/2})P and the decay factor m = λ_P/λ_D.

(b) Since ζ(x) of (10.122) and (10.125) for m = 1 or λ_P = λ_D results in ζ(x) = 0/0, we can determine the ζ(x) function for m = 1 by applying the l’Hôpital rule to convert the indeterminate form into a determinate form which allows the evaluation of the m → 1 limit as follows

Equation (10.126) shows that ζ(x) for m = 1 is a simple linear function of the type y = b + ax for which the y intercept is b = 0 and the slope is a = ln 2 = 0.6931 … , resulting in an angle of 34.7° between the abscissa and the straight line of the linear function.

(c) In Fig. 10.8 we show a plot of the linear expression given in (10.126) and derived in (b) for the activity ratio ζ(x) for m = 1 superimposed on ζ(x) curves plotted in Fig. 10.7 for various m between 0.1 and 10. The m = 1 linear equation actually separates two distinct regions of the activity ratio ζ(x): (1) Region where m > 1 and (2) Region where 0 <m< 1.

(1) For the m > 1 region we write (10.125) as follows

\zeta(x)=\frac{1}{m-1}\left[e^{\frac{m-1}{m} x \ln 2}-1\right]           (10.127)

Since m > 1, ζ(x) in (10.127) rises exponentially with x, implying that the ratio \mathcal{A}_D(t)/\mathcal{A}_P(t) also increases with time t and thus no equilibrium between \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t) will ensue with an increasing time t. The exponential growth of ζ(x) for m > 1 with time t is clearly shown with dashed curves in Fig. 10.8 in the range of decay factor 1 <m< ∞.
n the range of decay factor 1 <m< ∞. Thus, for m > 1, which means that λ_P > λ_D or that half-life of the daughter exceeds that of the parent \left[\left(t_{1 / 2}\right)_{\mathrm{D}}>\left(t_{1 / 2}\right)_{\mathrm{P}}\right] or one can say that the daughter is longer-lived than the parent, the activity ratio ζ(x) increases exponentially with time t and no equilibrium can be reached between the parent activity \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t) for any time t.

(2) For the 0 <m< 1 region we rearrange the terms in (10.125) to get a clearer picture as follows

\zeta(x)=\frac{1}{1-m}\left[1-e^{-(\ln 2)\left(\frac{1-m}{m}\right) x}\right]=\frac{1}{1-m}\left[1-2^{-\frac{1-m}{m} x}\right]           (10.128)

and notice that the exponential term diminishes with increasing x and exponentially approaches zero as x → ∞. This means that at large x the activity ratio ζ(x) saturates at a constant value that is independent of x and is equal to 1/(1 − m). Under these conditions the parent activity \mathcal{A}_P(t) and the daughter activity A_D(t) are said to be in transient equilibrium and are governed by the following relationship

\zeta(x)=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\mathcal{A}_{\mathrm{P}}(t)}=\frac{y_{\mathrm{D}}}{y_{\mathrm{P}}}=\frac{1}{1-m}=\frac{1}{1-\frac{\lambda_{\mathrm{P}}}{\lambda_{\mathrm{D}}}}=\frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}=\text { const }>1         (10.129)

provided, of course, that λ_P < λ_D, i.e., \left(t_{1/2}\right)_P > \left(t_{1/2}\right)_D.
Thus, for 0 <m< 1, which means that λ_P < λ_D or that the half-life of the daughter is shorter than the half-life of the parent \left[\left(t_{1 / 2}\right)_{\mathrm{D}}<\left(t_{1 / 2}\right)_{\mathrm{P}}\right] or one can say that the daughter is shorter-lived than the parent, the activity ratio ζ(x) at some large time saturates at a constant value given by (10.129) and larger than 1. The constancy of the ratio \mathcal{A}_D(t)/\mathcal{A}_P(t) at large t implies a transient equilibrium between \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t).

(3) A special case of transient equilibrium occurs when the daughter is much shorter-lived than the parent \left[\left(t_{1 / 2}\right)_{\mathrm{D}} \ll\left(t_{1 / 2}\right)_{\mathrm{P}}\right] or we can say that m \ll 1 to get the following expression for (10.128)

Equation (10.130) becomes equal to unity for a relatively large time x \gg \left(x_{\mathrm{D}}\right)_{\max } \text { where }\left(x_{\mathrm{D}}\right)_{\max } is the normalized time of maximum daughter activity which indicates that \mathcal{A}_{\mathrm{D}}(t) \approx \mathcal{A}_P(t) \text { or } \zeta \approx 1, so that the parent and daughter decay together at the rate of the parent. This special case of transient equilibrium in which the daughter and parent activities are essentially identical is referred to as secular equilibrium.

(d) Important features of the four special regions characterized by the decay factor m between m = 0 and m = ∞ are summarized in Table 10.4. Region where m → 0 results in secular equilibrium between the parent and daughter activities A_P(t)  and  A_D(t), respectively, and the region where 0 <m< 1 results in transient equilibrium between the parent and daughter activities. Regions where m = 1 and m > 1 do not result in equilibrium between the parent and daughter activities. The relationship between \mathcal{A}_P(t)\ and\ \mathcal{A}_D(t) expressed by a linear function ζ(x) = (\ln 2)_x for m = 1 separates the secular equilibrium region characterized by m → 0 and the transient equilibrium region characterized by 0 <m< 1 from the non-equilibrium region where 1 <m< ∞.