Question 12.6.Q10: Production of cobalt-60 (Co-60) radionuclide from cobalt-59 ......
Production of cobalt-60 (Co-60) radionuclide from cobalt-59 (Co-59) nuclide is usually treated like a simple parent P–daughter D–granddaughter G nuclear series, whereby the natural parent Co-59 nuclide is activated with thermal neutrons (activation cross section σ_P = 37 b) in a nuclear reactor into daughter radionuclide Co-60 that subsequently decays through β^− nuclear decay with a half-life t_{1/2} = 5.26 a into the granddaughter nuclide nickel-60 (Ni-60) [see Fig. 12.9(A)].
However, a closer look at the activation and decay diagrams of Co-60 nuclide shows that the activation process is significantly more complex, as shown schematically in Fig. 12.9(B). Activation of Co-59 actually has two possible branches:
Branch (1) leads directly to ground state of Co-60 with \sigma_{\mathrm{P} 1}=f_1^{ \mathrm{P}} \sigma_{\mathrm{P}}=17 \mathrm{~b} and f_1^{\mathrm{P}}=17 / 37=0.46 Branch (2) leads to a metastable state of Co-60m with \sigma_{\mathrm{P} 2}=f_2^{\mathrm{P}} \sigma_{\mathrm{P}}=20 \mathrm{~b} \text { and } f_2^{\mathrm{P}}=20 / 37=0.54
The Co-60 daughter D1 radionuclide has two possible avenues open for transformation:
(1) To decay through β^− decay into Ni-60 with t_{1/2} = 5.26 a.
(2) To become activated by thermal neutrons into Co-61 with σ_{D1} = 2 b.
The Co-60m daughter D_2 radionuclide has 3 avenues open for transformation:
(1) To become activated by thermal neutrons into Co-61m with σ_{D2} = 58 b.
(2) To decay with t_{1/2} = 10.5 m through γ decay into Co-60 (branching ratio f_\gamma^{\mathrm{D} 2}=0.998).
(3) To decay with t_{1/2} = 10.5 m through β^− decay into Ni-60 (branching ratio f_{\beta^{-}}^{\mathrm{D} 2}=0.002).
(a) Write and solve the differential equation governing the change dN_D/dt in the number of daughter D (Co-60) nuclei for the activation–decay series (A) of Fig. 12.9. Express the activity \mathcal{A}_D(t) of daughter D (Co-60) against t for the following initial conditions at t = 0: initial number of parent P nuclei is N_P(0); initial number of daughter nuclei N_D(t = 0) = 0.
(b) Write and solve the differential equation governing the change dN_{D1}/dt in the number of daughter D1 (Co-60) nuclei for the activation–decay series (B) of Fig. 12.9. Express the activity \mathcal{A}_D(t) of daughter D (Co-60) against t for the following initial conditions at t = 0: initial number of parent P nuclei is N_P(0); initial number of daughter nuclei N_D(t = 0) = 0.
(c) A pure 10 g Co-59 target is placed into a nuclear reactor with thermal neutron fluence rate \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1}. Calculate the activity of Co-60 against time t in steps of 2 years from 0 to 10 years based on expressions derived in (a) and (b) above for nuclear activation–decay schemes (A) and (B) of Fig. 12.9. Superimpose your calculated results on the graph presented in Fig. 12.10.
(a) The differential equation for change dN_D/dt in the number of daughter D (Co-60) nuclei for nuclear activation–decay series of Fig. 12.9(A) is given as follows, recognizing that dN_D/dt is governed by two terms describing:
(1) Production rate of daughter D (Co-60) through neutron activation of parent P (Co-59), expressed as +-\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t} \text {, }
(2) Loss of daughter D (Co-60) through β^− decay of daughter D (Co-60) into granddaughter G (Ni-60), expressed as -\lambda_{\mathrm{D}} N_{\mathrm{D}}(t),
resulting in the following expression for dN_D/dt.
\frac{\mathrm{d} N_{\mathrm{D}}}{\mathrm{d} t}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) . (12.200)
Equation (12.200) is a first order differential equation of the form
\frac{\mathrm{d} \eta(t)}{\mathrm{d} t}+p(t) \eta(t)=q(t), (12.201)
with the following solution
\eta(t)=\frac{\int u(t) q(t) \mathrm{d} t+C}{u(t)} (12.202)
where C is a constant and the function u(t) is defined as
u(t)=e^{\int p(t) \mathrm{d} t} (12.203)
After rearranging (12.200) to match the format of (12.201), we get the following values for parameters of (12.201): p(t)=\lambda_{\mathrm{D}}, u(t)=e^{\lambda_{\mathrm{D}} t}, \text { and } q(t)= \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t} . Inserting these parameters into (12.202) we get the following expression for N_D(t)
N_{\mathrm{D}}(t)=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}} e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}+C e^{-\lambda_{\mathrm{D}} t} (12.204)
To determine constant C we apply the initial condition N_D(t = 0) = 0 to get
C=-\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}} (12.205)
and, after inserting (12.205) into (12.204), we get the following well-known solutions for the number of daughter D (Co-60) nuclei (T10.34)
N_{\mathrm{D}}(t)=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}} t}\right] (12.206)
and the daughter D (Co-60) activity \mathcal{A}_{\mathrm{D}}(t)=\lambda_{\mathrm{D}} N_{\mathrm{D}}(t) is expressed as follows (T10.35)
\mathcal{A}_{\mathrm{D}}(t)=\frac{\lambda_{\mathrm{D}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}} t}\right] (12.207)
(b) The differential equation for change dN_{D1}/dt in the number of daughter D1 (Co-60) nuclei for nuclear activation–decay series depicted in Fig. 12.9(B) is given as follows, recognizing that dN_{D1}/dt is governed by four terms describing:
(1) Production rate of daughter D1 (Co-60) through neutron activation of parent P (Co-59), expressed as +f_1^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}
(2) Production rate of daughter D1 (Co-60) through γ decay of daughter D2 (Co-60m), expressed as +f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} N_{\mathrm{D} 2}(t)
(3) Loss of daughter D1 (Co-60) through activation of daughter D1 (Co-60) into Co-61, expressed as -\sigma_{\mathrm{D} 1} \dot{\varphi} N_{\mathrm{D} 1}(t),
(4) Loss of daughter D1 (Co-60) through β^− decay of daughter D2 (Co-60) into grand-daughter G (Ni-60), expressed as -\lambda_{\mathrm{D} 1} N_{\mathrm{D} 1}(t)
resulting in the following expression for dN_{D1}/dt.
\frac{\mathrm{d} N_{\mathrm{D} 1}}{\mathrm{~d} t}=f_1^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}+f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} N_{\mathrm{D} 2}(t)-\lambda_{\mathrm{D} 1}^* N_{\mathrm{D} 1}(t), (12.208)
where \lambda_{\mathrm{D} 1}^* is a modified decay constant linking into one constant the \beta^- decay and neutron activation of D1 expressed as
\lambda_{\mathrm{D} 1}^*=\lambda_{\mathrm{D} 1}+\sigma_{\mathrm{D} 1} \dot{\varphi}, (12.209)
with λ_{D1} the decay constant of daughter D1, σ_{D1} the neutron activation cross section of daughter D1, and \dot{φ} the neutron fluence rate.
To solve (12.208) we will need to know the number of daughter D2 (Co-60m) nuclei N_{D2}(t) appearing in (12.208). Therefore, we now set up a differential equation that describes the change dN_{D2}/dt in the number N_{D2} that is, as shown in Fig. 12.9(B), governed by 4 terms:
(1) Production rate of daughter D2 (Co-60m) through activation of parent P (Co-59), expressed as +f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}
(2) Loss of daughter D2 (Co-60m) through neutron activation of daughter D2 (Co-60m) into Co-61m, expressed as -\sigma_{\mathrm{D} 2} \dot{\varphi} N_{\mathrm{D} 2}(t)
(3) Loss of daughter D2 (Co-60m) through β^− decay into granddaughter G (Ni-60), expressed as -f_{\beta^{-}}^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} N_{\mathrm{D} 2}(t),
(4) Loss of daughter D2 (Co-60m) through isomeric γ decay into daughter D1 (Co-60), expressed as -f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} N_{\mathrm{D} 2}(t).
The change dN_{D2}/dt in the number of daughter D2 nuclei is thus expressed as follows
where \lambda_{\mathrm{D} 2}^* is a modified decay constant linking into one constant the \beta^{-} decay, isomeric γ decay, and neutron activation of daughter D2 expressed as
\lambda_{\mathrm{D} 2}^*=\sigma_{\mathrm{D} 2} \dot{\varphi}+f_{\beta^{-}}^{\mathrm{D} 2} \lambda_{\mathrm{D} 2}+f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2}=\sigma_{\mathrm{D} 2} \dot{\varphi}+\lambda_{\mathrm{D} 2}, (12.211)
with σ_{D2} the neutron activation cross section of daughter D2 and λ_{D2} the decay constant of D2.
Rearranging (12.210) to match the format of (12.201) we get the following values for the parameters of (12.201): p(t)=\lambda_{\mathrm{D} 2}^*, u(t)=e^{\lambda_{\mathrm{D} 2}^* t} \text {, and } q(t)= f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t} . Inserting these parameters into (12.202) we get the following expression for N_{D2}(t).
We now use the initial condition N_{D2}(t = 0) = 0 and get from (12.212) the constant C
C=-\frac{f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D} 2}^*-\sigma_{\mathrm{P}} \dot{\varphi}} (12.213)
After inserting (12.213) into (12.212) we get the number N_{D2}(t) of daughter D2 (Co-60m) nuclei as follows
N_{\mathrm{D} 2}(t)=\frac{f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D} 2}^*-\sigma_{\mathrm{P}} \dot{\varphi}}\left\{e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D} 2}^* t}\right\} . (12.214)
Inserting (12.214) into (12.208) and rearranging the terms to match with (12.201), we now obtain the following differential equation for the change dN_{D1}/dt in number of daughter nuclei D1 (Co-60)
similar to (12.200) and (12.208), another first order differential equation that upon comparison with (12.201) yields the following values for the parameters of (12.201)
Inserting parameters of (12.216) into (12.202) we get the following expression for N_{D1}(t)
To determine the constant C we use the initial condition N_{D1}(t = 0) = 0 to get
C=\frac{f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\left(\lambda_{\mathrm{D} 2}^*-\sigma_{\mathrm{P}} \dot{\varphi}\right)\left(\lambda_{\mathrm{D} 1}^*-\lambda_{\mathrm{D} 2}^*\right)}-\frac{f_1^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)+\frac{f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D} 2}^*-\sigma_{\mathrm{P}} \dot{\varphi}}}{\lambda_{\mathrm{D} 1}^*-\sigma_{\mathrm{P}} \dot{\varphi}} (12.218)
Substituting (12.218) into (12.217) we get the following expression for the number N_{D1}(t) of daughter D1 (Co-60) nuclei for nuclear activation–decay series depicted in Fig. 12.9(B)
Activity \mathcal{A}_{D1}(t) of daughter D1 (Co-60) is determined by multiplying (12.219) with the decay constant λ_{D1} resulting in the following
Daughter D1 (Co-60) activity of (12.220) appears quite complicated, however, we can rearrange its terms to get an expression consisting of a simple term in the form of (12.207) for the activation–decay series depicted in Fig. 12.9(A) multiplied by a correction factor F_{corr} to account for the increased complexity of the activation– decay series depicted in Fig. 12.9(B). Equation (12.220) thus reads
Comparing the two activation–decay series (A) and (B) of Fig. 12.9 we note that series (B) transforms into series (A) for the following special values of parameters: f_1^{\mathrm{P}}=1, f_2^{\mathrm{P}}=0, \lambda_{\mathrm{D} 1}=\lambda_{\mathrm{D}}, \text { and } \sigma_{\mathrm{D} 1}=0. Inserting these values into (12.221) we indeed get (12.207) from (12.221) for the activity of the daughter Co-60 in neutron activation of Co-59 since \lambda_{\mathrm{D} 1}^*=\lambda_{\mathrm{D} 1}=\lambda_{\mathrm{D}} \text { and } F_{\mathrm{corr}}=1.
(c) We now investigate the specific Co-60 activation–decay example for a Co59 sample (mass m_{\mathrm{Co}-59}=10 \mathrm{~g} \text { and neutron fluence rate } \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1}) and calculate the daughter D1 activity first with (12.207) for the activation–decay series (A) of Fig. 12.9 and then with (12.221) for the activation–decay series (B) of Fig. 12.9.
(1) Nuclear activation–decay series (A) of Fig. 12.9. The relevant parameters for use of (12.207), summarized in Table 12.18, result in the following expression for activity \mathcal{A}_D(t) of daughter D (Co-60) as a function of activation time t
Using (12.222) with activation times t from 0 to 10 years in steps of 2 years yields activities A_D(t) summarized in Table 12.19. The data presented in Table 12.19 are also plotted as data points in Fig. 12.10 on curve (1).
(2) Nuclear activation–decay series (B) of Fig. 12.9. The relevant parameters for use with (12.207), summarized in Table 12.20, result in the following expression for activity \mathcal{A}_{D1}(t) of daughter D1 (Co-60) as a function of activation time t
Using (12.223) with activation times t from 0 to 10 years in steps of 2 years yields activities \mathcal{A}_{D1}(t) summarized in Table 12.21. The data presented in Table 12.21 are also plotted as data points in Fig. 12.10 on curve (2).
Table 12.18 Relevant parameters for use in (12.207) with mass of Co-59 sample m_{Co-59} = 10 g and neutron fluence rate \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1}.
\begin{array}{ll} \hline \sigma_{\mathrm{P}} & 37 \times 10^{-24} \mathrm{~cm}^2 \\ \hline \lambda_{\mathrm{D}}=\ln (2) /\left(t_{1 / 2}\right)_{\mathrm{Co}-60} & \ln (2) / 5.26 \mathrm{a}=4.179 \times 10^{-9} \mathrm{~s}^{-1} \\ \hline \dot{\varphi} & 5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1} \\ N_{\mathrm{P}}(0)=m N_{\mathrm{A}} / A & (10 \mathrm{~g}) \times\left(6.022 \times 10^{23} \mathrm{atom} / \mathrm{mol}\right) /(59 \mathrm{~g} / \mathrm{mol})=1.021 \times 10^{23} \text { atom } \\ \hline \sigma_{\mathrm{P} \dot{\varphi}} & \left(37 \times 10^{-24} \mathrm{~cm}^2\right) \times\left(5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\right)=1.85 \times 10^{-8} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) & \left(4.179 \times 10^{-9} \mathrm{~s}^{-1}\right) \times\left(1.85 \times 10^{-8} \mathrm{~s}^{-1}\right) \times\left(1.021 \times 10^{23} \mathrm{atom}\right)= \\ & 7.890 \times 10^6 \mathrm{~s}^{-2} \\ \hline \lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi} & 4.179 \times 10^{-9} \mathrm{~s}^{-1}-1.85 \times 10^{-8} \mathrm{~s}^{-1}=-1.432 \times 10^{-8} \mathrm{~s}^{-1} \\ \hline \end{array}
Table 12.19 Activity \mathcal{A}_D(t) of daughter D (Co-60) calculated from (12.222) for mass of Co-59 sample m_{Co-59} = 10 g and neutron fluence rate \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1} for activation times t between 0 and 10 years in steps of 2 years. These data points are superimposed on curve (1) in Fig. 12.10
\begin{array}{llll} \hline t \text { (years) } & e^{-\left(1.850 \times 10^{-8} \mathrm{~s}^{-1}\right) t} & e^{-\left(4.179 \times 10^{-9} \mathrm{~s}^{-1}\right) t} & \mathcal{A}_{\mathrm{D}}(t)(\mathrm{Ci}) \\ \hline 0 & 1.000 & 1.000 & 0 \\ 2 & 0.311 & 0.768 & 6804 \\ 4 & 0.097 & 0.590 & 7346 \\ 6 & 0.030 & 0.454 & 6304 \\ 8 & 0.009 & 0.348 & 5049 \\ 10 & 0.003 & 0.268 & 3943 \\ \hline \end{array}
Table 12.20 Relevant parameters for use in (12.221) with mass of Co-59 sample m_{Co-59} = 10 g and neutron fluence rate \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1}.
\begin{array}{|c|c|} \hline \sigma_{\mathrm{P}} & 37 \times 10^{-24} \mathrm{~cm}^2 \\ \hline \sigma_{\mathrm{D} 1} & 2 \mathrm{~b}=2 \times 10^{-24} \mathrm{~cm}^2 \\ \hline \sigma_{\mathrm{D} 2} & 58 \mathrm{~b}=58 \times 10^{-24} \mathrm{~cm}^2 \\ \hline f_1^{\mathrm{P}} & 0.46 \\ \hline f_2^{\mathrm{P}} & 0.54 \\ \hline f_\gamma^{\mathrm{D} 2} & 0.998 \\ \hline \lambda_{\mathrm{D} 1}=\ln (2) /\left(t_{1 / 2}\right)_{\mathrm{Co}-60} & \ln (2) /(5.29 \mathrm{a})=4.179 \times 10^{-9} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 2}=\ln (2) /\left(t_{1 / 2}\right)_{\mathrm{Co}-60 \mathrm{~m}} & \ln (2) /(10.5 \mathrm{~m})=1.1 \times 10^{-3} \mathrm{~s}^{-1} \\ \hline \dot{\varphi} & 5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1} \\ \hline \sigma_{\mathrm{P}} \dot{\varphi} & \left(37 \times 10^{-24} \mathrm{~cm}^2\right) \times\left(5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\right)=1.85 \times 10^{-8} \mathrm{~s}^{-1} \\ \hline \sigma_{\mathrm{D} 1 }\dot{\varphi} & \left(2 \times 10^{-24} \mathrm{~cm}^2\right) \times\left(5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\right)=1.00 \times 10^{-9} \mathrm{~s}^{-1} \\ \hline \sigma_{\mathrm{D} 2 }\dot{\varphi} & \left(58 \times 10^{-24} \mathrm{~cm}^2\right) \times\left(5 \times 10^{14} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}\right)=2.90 \times 10^{-8} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 1}^*=\lambda_{\mathrm{D} 1}+\sigma_{\mathrm{D} 1} \dot{\varphi} & \left(4.179 \times 10^{-9} \mathrm{~s}^{-1}\right)+\left(1.00 \times 10^{-9} \mathrm{~s}^{-1}\right)=5.179 \times 10^{-9} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 2}^*=\lambda_{\mathrm{D} 2}+\sigma_{\mathrm{D} 2} \dot{\varphi} & \left(1.1 \times 10^{-3} \mathrm{~s}^{-1}\right)+\left(2.90 \times 10^{-8} \mathrm{~s}^{-1}\right) \approx 1.1 \times 10^{-3} \mathrm{~s}^{-1} \\ \hline N_{\mathrm{P}}(0)=m N_{\mathrm{A}} / A & (10 \mathrm{~g}) \times\left(6.022 \times 10^{23} \mathrm{~mol}^{-1}\right) /(59 \mathrm{~g} / \mathrm{mol})=1.021 \times 10^{23} \text { atom } \\ \hline \lambda_{\mathrm{D} 1}^*-\sigma_{\mathrm{P}} \dot{\varphi} & 5.179 \times 10^{-9} \mathrm{~s}^{-1}-1.85 \times 10^{-8} \mathrm{~s}^{-1}=-1.332 \times 10^{-8} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 2}^*-\sigma_{\mathrm{P}} \dot{\varphi} & 1.1 \times 10^{-3} \mathrm{~s}^{-1}-1.85 \times 10^{-8} \mathrm{~s}^{-1} \approx 1.1 \times 10^{-3} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 1}^*-\lambda_{\mathrm{D} 2}^* & 5.179 \times 10^{-9} \mathrm{~s}^{-1}-1.1 \times 10^{-3} \mathrm{~s}^{-1} \approx-1.1 \times 10^{-3} \mathrm{~s}^{-1} \\ \hline \lambda_{\mathrm{D} 1} f_1^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) & \begin{array}{l} \left(4.179 \times 10^{-9} \mathrm{~s}^{-1}\right) \times(0.46) \times\left(1.85 \times 10^{-8} \mathrm{~s}^{-1}\right) \times \\ \left(1.021 \times 10^{23} \text { atom }\right)=3.63 \times 10^5 \mathrm{~s}^{-2} \end{array} \\ \hline \lambda_{\mathrm{D} 1} f_\gamma^{\mathrm{D} 2} \lambda_{\mathrm{D} 2} f_2^{\mathrm{P}} \sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) & \begin{array}{l} \left(4.179 \times 10^{-9} \mathrm{~s}^{-1}\right) \times(0.998) \times\left(1.1 \times 10^{-3} \mathrm{~s}^{-1}\right) \times(0.54) \times \\ \left(1.85 \times 10^{-8} \mathrm{~s}^{-1}\right) \times\left(1.021 \times 10^{23} \text { atom }\right)=4.678 \times 10^3 \mathrm{~s}^{-3} \end{array} \\ \hline \end{array}
Table 12.21 Activity \mathcal{A}_D(t) of daughter D1 (Co-60) calculated from (12.223) for mass of Co59 sample m_{Co-59} = 10 g and neutron fluence rate \dot{\varphi}=5 \times 10^{14} \mathrm{~cm}^2 \cdot \mathrm{s}^{-1} for activation times t between 0 and 10 years in steps of 2 years. Data points listed in this table are superimposed onto curve (2) in Fig. 12.10
\begin{array}{lllll} \hline t \text { (years) } & e^{-\left(1.850 \times 10^{-8} \mathrm{~s}^{-1}\right) t} & e^{-\left(5.179 \times 10^{-9} \mathrm{~s}^{-1}\right) \times t} & e^{-\left(1.1 \times 10^{-3} \mathrm{~s}^{-1}\right) \times t} & A_{\mathrm{D} 1}(t)(\mathrm{Ci}) \\ \hline 0 & 1.000 & 1.000 & 1.000 & 0 \\ \hline 2 & 0.311 & 0.721 & 0 & 6556 \\ \hline 4 & 0.097 & 0.520 & 0 & 6771 \\ \hline 6 & 0.030 & 0.375 & 0 & 5520 \\ \hline 8 & 0.009 & 0.271 & 0 & 4179 \\ \hline 10 & 0.003 & 0.195 & 0 & 3076 \\ \hline \end{array}