Question 10.8.Q1: In many instances decay of a radionuclide can proceed by mor......
In many instances decay of a radionuclide can proceed by more than one mode of decay and the radionuclide is said to undergo branching decay to two or more different daughter nuclides. In general, the total decay constant λ_P for the parent decay is the sum of the partial decay constants (λ_P)_i for each possible branch or mode of decay
\lambda_{\mathrm{P}}=\sum_i^N\left(\lambda_{\mathrm{P}}\right)_i=(\ln 2) \sum_i^N \frac{1}{\left(t_{1 / 2}\right)_{\mathrm{P}}^i}, (10.194)
where N is the number of decay branches or modes available and \left(t_{1 / 2}\right)_{\mathrm{P}}^i is the half-life of the parent radionuclide for decay in mode i. The branching fraction f_i of mode i is defined as the ratio f_i=\left(\lambda_{\mathrm{P}}\right)_i / \lambda_{\mathrm{P}}=\left(t_{1 / 2}\right)_{\mathrm{P}} /\left(t_{1 / 2}\right)_{\mathrm{P}}^i.
Bismuth-212 (Bi-212) with a half-life \left(t_{1 / 2}\right)_{\mathrm{Bi}-212}=60.55 \mathrm{~min}=3633 \mathrm{~s} decays into two radio-nuclides: (1) thalium-208 (Tl-208) through α alpha decay with a branching fraction f_α = 0.36 and (2) polonium-212 (Po-212) through \beta^{-} \text {beta decay with a branching fraction } f_{\beta^{-}}=0.64. Both daughters subsequently decay into stable lead-208 (Pb-208) nuclide: thallium-208 through \beta^{-} \text {decay with a half-life }\left(t_{1 / 2}\right)_{\mathrm{Tl}-208}=183.2 \mathrm{~s} and polonium-212 through α decay with a half-life \left(t_{1/2}\right)_{Po-212} = 0.3 µs.
(a) For bismuth-212 determine: (1) decay constant λ_{Bi-212}; (2) partial decay constant
for
decay. Verify the self-consistency of results.
(b) Express the activities of (1) thalium-208 and (2) polonium-212 as a function of time t and initial parent activity \mathcal{A}_{Bi-212}(0) for initial conditions: N_{\mathrm{Tl}-208}(0)=N_{\mathrm{Po}-212}(0)=0 .
(c) Prepare a table for the four nuclides involved in series decay of bismuth212 into lead-208 and provide the following rows to serve as summary of the Bi-212 series decay: (1) Name of nuclide, (2) Symbol, (3) Designation, (4) Atomic number Z, (5) Atomic mass number A, (6) Type of decay, (7) Branching ratio, (8) Half-life t_{1/2}, and (9) Decay constant λ.
(d) Express the accumulation of the granddaughter nuclide (Pb-208) as a function of time for the initial conditions: N_P(t = 0) = N_{Bi-212}(0)\ and\ N_G(0) = N_{Pb-208}(0) = 0.
(e) Figure 10.13 shows five curves depicting the number of various nuclei against time t for series decay of Bi-212 through either Tl-208 or Po212 into Pb-208 with initial conditions N_{\mathrm{Pb}-212}(t=0)=N_{\mathrm{Pb}-212}(0) and N_{\mathrm{Tl}-208}(0)=N_{\mathrm{Po}-212}(0)=N_{\mathrm{Pb}-208}(0)=0 .. All curves are normalized to N_{Bi-212}(0) = 1. Based on results in (a) through (d) identify the five curves.
(a) Radioactive decay of bismuth-212 nucleus can proceed by two modes of decay: either α decay into thallium-208 or β^− decay into polonium-212. Therefore, in addition to total decay constant \lambda_{\mathrm{Bi}-212} \text { and half-life }\left(t_{1 / 2}\right)_{\mathrm{Bi}-212} is also characterized by its partial decay constants \left(\lambda_\alpha\right)_{\mathrm{Bi}-212} \text { and }\left(\lambda_{\beta^{-}}\right)_{\mathrm{Bi}-212} \text { for } \alpha \text { and } \beta ^{-} decay, respectively, as well as by half-lives \left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^\alpha \text { and }\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^{\beta^{-}} \text {for } \alpha \text { and } \beta^{-} decay, respectively.
(1) Decay constant λ_{Bi-212} for bismuth-212 is calculated from the standard relationship between decay constant λ and half-life t_{1/2} as follows
\lambda_{\mathrm{Bi}-212}=\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}}=\frac{\ln 2}{(60.55 \mathrm{~min}) \times(60 \mathrm{~s} / \mathrm{min})}=1.91 \times 10^{-4} \mathrm{~s}^{-1} . (10.195)
(2) Partial decay constant \left(λ_α\right)_{Bi-212} for α decay of Bi-212 is calculated using the definition of branching fraction f_i=\left(\lambda_{\mathrm{P}}\right)_i / \lambda_{\mathrm{P}}=\left(t_{1 / 2}\right)_{\mathrm{P}} /\left(t_{1 / 2}\right)_{\mathrm{P}}^i.
\left(\lambda_\alpha\right)_{\mathrm{Bi}-212}=f_\alpha \lambda_{\mathrm{Bi}-212}=0.36 \times\left(1.91 \times 10^{-4} \mathrm{~s}^{-1}\right)=6.876 \times 10^{-5} \mathrm{~s}^{-1} . (10.196)
(3) In same manner we calculate the partial decay constant \left(\lambda_{\beta^{-}}\right)_{\mathrm{Bi}-212} \text { for } \beta^{-} decay of Bi-212
\left(\lambda_{\beta^{-}}\right)_{\mathrm{Bi}-212}=f_{\beta^{-}} \lambda_{\mathrm{Bi}-212}=0.64 \times\left(1.91 \times 10^{-4} \mathrm{~s}^{-1}\right)=12.224 \times 10^{-5} \mathrm{~s}^{-1} (10.197)
(4) Half-life \left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^\alpha for α decay of Bi-212 is calculated from the standard relationship (CC) linking decay constant λ and half-life t_{1/2}.
\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^\alpha=\frac{\ln 2}{\left(\lambda_\alpha\right)_{\mathrm{Bi}-212}}=\frac{\ln 2}{6.876 \times 10^{-5} \mathrm{~s}^{-1}}=10081 \mathrm{~s}=168 \mathrm{~min} . (10.198)
(5) Half-life \left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^{\beta^{-}} \text {for } \beta^- decay of Bi-212 is given as
\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^{\beta^{-}}=\frac{\ln 2}{\left(\lambda_{\beta^{-}}\right)_{\mathrm{Bi}-212}}=\frac{\ln 2}{12.224 \times 10^{-5} \mathrm{~s}^{-1}}=5670.4 \mathrm{~s}=94.51 \mathrm{~min} . (10.199)
A simple test can be carried out to verify the self-consistency of results in this section. According to (10.194) our calculations should satisfy the following equation
(6) Equation (10.200) can also be written so as to link the half-lives of individual branches of the parent with the half-life of the parent as follows
\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}}=\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^\alpha}+\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Bi}-212}^{\beta^{-}}} (10.201)
or
and
(b) In multichannel decays of the parent P several daughters appear, each one characterized by its own growth and decay kinematics. The rate of change dN_D/dt in the number of given daughter D nuclei is equal to the supply of new daughter nuclei through the decay of P channeled into the branch i containing D and expressed as \left(\lambda_{\mathrm{P}}\right)_i N_{\mathrm{P}}(0) e^{-\left(\lambda_{\mathrm{P}}\right)_i t}=f_i \lambda_{\mathrm{P}} N_{\mathrm{P}}(0) e^{-f_i \lambda_{\mathrm{P}} t} less the loss of daughter nuclei D from the decay of D into G given by \left[-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)\right], \text { where } \lambda_{\mathrm{D}} is the decay constant of the daughter D. The rate of change dN_D/dt is thus given by
\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\left(\lambda_{\mathrm{P}}\right)_i N_{\mathrm{P}}(0) e^{-\lambda \mathrm{p} t}-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)=f_i \lambda_{\mathrm{P}} N_{\mathrm{P}}(0) e^{-\lambda {\mathrm{p}} t}-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) (10.204)
Equation (10.204) has the following solution (T10.35) for the number of daughter nuclei N_D as a function of time t
N_{\mathrm{D}}(t)=f_i 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] (10.205)
and for the activity of daughter nuclide \mathcal{A}_D(t) as a function of time t
Before addressing the calculation of activity of Tl-208 and Po-212 [\mathcal{A}_{\mathrm{Tl}-208}(t)\ and\ \mathcal{A}_{Po-212}(t), respectively], we determine decay constants of the two daughter products of Bi-212 (Tl-208 and Po-212) from their half-lives
\lambda_{\mathrm{Tl}-208} \frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Tl}-208}}=\frac{\ln 2}{183.2 \mathrm{~s}}=3.784 \times 10^{-3} \mathrm{~s}^{-1}, (10.207)
\lambda_{\text {Po-212 }} \frac{\ln 2}{\left(t_{1 / 2}\right)_{\text {Po-212 }}}=\frac{\ln 2}{0.3 \times 10^{-6} \mathrm{~s}}=2.31 \times 10^6 \mathrm{~s}^{-1} . (10.208)
(1) From (10.205) we get the following expression for the activity of thallium (Th-208) daughter as a result of the decay of the bismuth-212 parent
(2) In similar manner we get from (10.206) the following expression for the activity of polonium-212 (Po-212) daughter nuclide as a result of the decay of the bismuth-212 parent nuclide
(c) Table 10.7 summarizes properties of nuclides involved in the 2-bransh decay of bismuth-212 into lead-208. Figure 10.14 shows a schematic diagram of the series decay of Bi-212 through either Tl-208 or Po-212 into Pb-208.
(d) In a radioactive decay series the growth of granddaughter G nuclide starting with the initial condition N_G(t = 0) = 0 is calculated from the expression for the rate of change dN_G/dt in the number of granddaughter nuclei. For a stable granddaughter nuclide the rate of accumulation of G is equal to the rate of decay of the daughter nuclide D, i.e.,
\frac{\mathrm{d} N_{\mathrm{G}}}{\mathrm{d} t}=\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)=f_i \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], (10.211)
where N_D(t) is the number of daughter nuclei at time t given in (10.205). Since in our example of bismuth-212 decay series, the lead-208 nuclide is accumulated through two channels, one represented by β^{−} decay of thallium-208 and the other by α decay of polonium-212, we calculate the contribution of each channel separately and then add the two contributions to obtain the total contribution from both channels.
Upon integration of (10.211) from 0 to t we get the following expression for N_G(t)
N_{\mathrm{G}}(t)=f_i 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, (10.212)
where C is an integration constant. For the initial condition N_G(t = 0) = 0 (10.211) is given as
N_{\mathrm{G}}(t=0)=f_i 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=f_i N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{P}}-\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}-\lambda_{\mathrm{P}}}+C=0 (10.213)
and results in the following simple expression for constant C
C=f_i N_{\mathrm{P}}(0) (10.214)
Inserting (10.214) into (10.212) we get the following expression for the number of granddaughter nuclei N_G(t) accumulated from the decay of daughter D
(1) Accumulation of Pb-208 nuclei as a result of Tl-208 decay is now given as
(2) Accumulation of Pb-208 nuclei as a result of Po-212 decay is now given as
The total accumulation of Pb-208 nuclei as a function of time t is given by the sum of (10.216) and (10.217) as follows
(d) Five curves, all normalized to N_{Bi-212}(0) = 1 and all dealing with the series decay of parent nucleus Bi-212 through either the Tl-208 branch or the Po-212 branch to the granddaughter Pb-208 nucleus, are shown in Fig. 10.13.
(1) Curve 1 plots the number of Bi-212 nuclei against time t, exhibiting the standard radioactive decay characteristics with a half-life \left(t_{1/2}\right)_{Bi-212} of 60.55 min = 3633 s and a decay constant \lambda_{\mathrm{Bi}-212} \text { of } 1.91 \times 10^{-4} \mathrm{~s}^{-1}.
(2) Curve 2 shows the growth and decay of the daughter product N_{\mathrm{Tl}-208}(t) against time t. Half-life and decay constant of the N_{\mathrm{Tl}-208} radionuclide are, respectively, \left(t_{1 / 2}\right)_{\mathrm{Tl}-208}=183.2 \mathrm{~s} \text { and } \lambda_{\mathrm{Tl}-208}=3.78 \times 10^{-3} \mathrm{~s}^{-1}. Because of the relatively short half-life of thallium-208 compared to the half-life of the parent nucleus Bi-212, the number of Tl-208 nuclei transforms rapidly into the Pb-208 nucleus causing a relatively rapid accumulation of lead nuclei through the thallium branch.
We should note that the curve representing the growth and decay of the Po-212 nuclei is not discernible on the scale of Fig. 10.13 because of the extremely short half-life of the polonium-212 nucleus.
(3) Curve 3 represents the accumulation of Pb-208 nuclei as a result of the decay of Bi-212 through the thallium decay branch. Since the Pb-208 nucleus is stable, the accumulation curve exhibits the standard exponential growth shape and saturates at 0.36 which is the branching fraction of the Bi-212 α decay through the Tl-208 branch.
(4) Curve 4 also represents the accumulation of Pb-208 nuclei as a result of Bi212 decay, in this case through the polonium decay branch. This curve too exhibits the standard exponential growth but saturates at 0.64 which is the branching fraction of the Bi-212 β^{−} decay through the Po-212 branch.
(5) Curve 5 is for the total accumulation of Pb-208 through both branches of the Bi-212 decay and is given as the sum of the two exponential curves: curve 3 for the Tl-208 branch and curve 4 for the Po-212 branch. Since both curve 3 and curve 4 are exponential, curve 5 is also exponential and saturates at the sum of saturations of the two curves (0.36 + 0.64 = 1).
| Table 10.7 Properties of nuclides associated with series decay of bismuth-212 | ||||
| (1) Radionuclide | Thallium | Lead | Bismuth | Polonium |
| (2) Symbol | Tl | Pb | Bi | Po |
| (3) Designation | Daughter | Granddaughter | Parent | Daughter |
| (4) Z | 81 | 82 | 83 | 84 |
| (5) A | 208 | 208 | 212 | 212 |
| (6) Decay type | β^{−} | stable | α and β^{−} | α |
| (7) Branching f | f_{\beta^{-}}=1 | – | f_\alpha=0.36, f_{\beta^{-}}=0.64 |
f_α = 1
|
| (8) Half-life t_{1/2} | 183.2 s | stable | 60.55 min | 0.3 µs |
| (9) Decay constant λ | 3.784 \times 10^{-3} \mathrm{~s}^{-1} | stable | 1.91 \times 10^{-4} \mathrm{~s}^{-1} |
2.31 \times 10^6 \mathrm{~s}^{-1}
|
