Three models are used for describing nuclear activation processes. Listed in order of increasing complexity they are: (1) Saturation model, (2) Depletion model, and (3) Depletion–activation model. The three models predict different expressions for normalized daughter D activity yD(x) as well as for

Question

Three models are used for describing nuclear activation processes. Listed in order of increasing complexity they are: (1) Saturation model, (2) Depletion model, and (3) Depletion–activation model. The three models predict different expressions for normalized daughter D activity [latex]y_D(x)[/latex] as well as for the maximum daughter activity [latex]\left(a_D ight)_{max}[/latex] attainable in nuclear activation. Expressions for [latex]\left(a_D ight)_{max}[/latex] derived in Prob. 256 for the saturation and depletion models and in Prob. 257 for the depletion–activation model are, respectively, given as

[latex]\left(a_{\mathrm{D}} ight)_{\max }=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} \quad [/latex] [see (12.137) in Prob. 256], (12.167)

[latex]\left(a_{\mathrm{D}} ight)_{\max }=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} m^{\frac{m}{1-m}} \quad [/latex] [see (12.140) in Prob. 256], (12.168)

[latex]\left(a_{\mathrm{D}}^* ight)_{\max }=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \frac{\dot{\varphi}}{\varepsilon^*}\left(m^* ight)^{\frac{m^*}{1-m^*}} \quad [/latex] [see (12.165) in Prob. 257], (12.169)

where m and [latex]m^∗[/latex] are activation factors defined as [latex]m = σ_P\dot{φ}/λ _D\ and\ m^∗ = σ_P\dot{φ}\left/(λ_D + σ_D\dot{φ} ight)[/latex] , respectively, with [latex]σ_P\ and\ σ_D[/latex] the parent P and daughter D activation cross sections, respectively, [latex]\dot{φ}[/latex] the fluence rate, and [latex]λ_D[/latex] the daughter decay constant.
At first glance the three models seem to suggest that [latex]\left(a_D ight)_{max}[/latex] is linearly proportional to [latex]\dot{φ}[/latex], irrespective of the magnitude of [latex]\dot{φ}[/latex], making [latex]\left(a_D ight)_{max}[/latex] go to ∞ as [latex]\dot{φ}[/latex] → ∞. This contradicts the standard relationship for specific activity [latex]\left(a_D ight)_{theor}[/latex] of a radionuclide D which states that [latex](a_D)_{theor}[/latex] is finite and proportional to [latex]λ_D/A_D\ where\ λ_D\ and\ A_D[/latex] are the decay constant and atomic mass of the radionuclide D and [latex]N_A[/latex], the Avogadro number, is the proportionality constant. The obvious conclusion is that [latex]\left(a_D ight)_{max}[/latex] cannot exceed [latex]\left(a_D ight)_{theor} = λ_DN_A/A_D[/latex] irrespective of magnitude of [latex]\dot{φ}[/latex].

(a) Show that (12.168) for depletion model and (12.169) for depletion– activation model fulfill the condition [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max } \leq\left(a_{\mathrm{D}} ight)_{\max }[/latex] in contrast to (12.167) which predicts that, as [latex]\dot{\varphi} ightarrow \infty, \lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }=[/latex] ∞ producing a physically impossible result [latex]\left(a_{\mathrm{D}} ight)_{\max } \gg\left(a_{\mathrm{D}} ight)_{ ext {theor }}[/latex].

(b) Show that the ratio between [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}}^* ight)_{\max }[/latex] for the depletion– activation model and [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }[/latex] for the depletion model is equal to a factor [latex]g^∗[/latex] that depends only on the parent–daughter cross section ratio [latex]k^*=\sigma_{\mathrm{P}} / \sigma_{\mathrm{D}}[/latex].

(c) Calculate and plot the depletion–activation factor [latex]g^*\left(k^* ight)[/latex] against the parent–daughter cross section ratio [latex]k^* ext { for } k^* ext { of } 10^{-3}, 10^{-2}, 10^{-1}, 1[/latex], 10, 100, and 1000. Discuss a few notable properties of [latex]g^∗\left(k^∗ ight)[/latex].

Accepted Answer

(a) Equations for the maximum attainable specific activity [latex]\left(a_D ight)_{max}[/latex] given for the three nuclear activation models above depend linearly on the fluence rate [latex]\dot{φ}[/latex] and for [latex]\dot{φ}[/latex] → ∞ seemingly contradict the expression for [latex]\left(a_D ight)_{theor}[/latex] that predicts a definite and finite upper limit for [latex]\left(a_D ight)_{max}[/latex]. This apparent contradiction can be explained by evaluating the implicit dependence of activation factors m and [latex]m^∗\ on\ \dot{φ}[/latex].

(1) The maximum specific activity [latex]\left(a_D ight)_{max}[/latex] given in (12.167) for the saturation model of nuclear activation is proportional to fluence rate [latex]\dot{φ}[/latex] and predicts clearly that, with increasing [latex]\dot{\varphi},\left(a_{\mathrm{D}} ight)_{\max}[/latex] will eventually exceed [latex]\left(a_{\mathrm{D}} ight)_{ ext {theor }}= λ_DN_A/A_P[/latex] and, as [latex]\dot{φ}[/latex] → ∞, [latex]\left(a_D ight)_{max}[/latex] → ∞. Since [latex]\left(a_D ight)_{max}[/latex] cannot exceed [latex]\left(a_D ight)_{theor}[/latex], we note that the validity of (12.167) is limited and some of its predictions cannot be trusted.

(2) At first glance it seems that [latex]\left(a_D ight)_{max}[/latex] given by (12.168) for the depletion model is, like (12.167) for the saturation model, proportional to [latex]\dot{φ}[/latex] and therefore also exceeds the theoretical specific activity [latex]\left(a_D ight)_{theor}\ for \dot{φ}[/latex] → ∞. However, a closer look at [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }[/latex] for (12.168) produces a logical result, namely that the maximum daughter specific activity [latex]\left(a_D ight)_{max}[/latex] derived from (12.168) does not exceed [latex]\left(a_{\mathrm{D}} ight)_{ ext {theor }}=\lambda_{\mathrm{D}} N_{\mathrm{A}} / A_{\mathrm{P}} ext { even for } \dot{\varphi} ightarrow \infty[/latex]. Actually, in determining the limit of (12.168) for [latex]\dot{φ}[/latex] → ∞, after introducing the definition of the activation factor [latex]m=\sigma_{\mathrm{P}} \dot{\varphi} / \lambda_{\mathrm{D}}[/latex] into (12.168), we show in (12.170) that [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max } \approx\left(a_{\mathrm{D}} ight)_{ ext {theor }}[/latex].

[latex]\begin{aligned} \lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max } & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\dot{\varphi} m^{\frac{m}{1-m}} ight\}=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\dot{\varphi} e^{\frac{m \ln m}{1-m}} ight\} \ & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\dot{\varphi} \exp \left[\frac{\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}}}{1-\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}}} \ln \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}} ight] ight\} \ & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\dot{\varphi} \exp \left[-\ln \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}} ight] ight\} \ & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} \frac{\lambda_{\mathrm{D}}}{\sigma_{\mathrm{P}} \dot{\varphi}}=\frac{\lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \approx\left(a_{\mathrm{D}} ight)_{ ext {theor }} .\quad (12.170) \end{aligned}[/latex]

The result of (12.170) for [latex]\dot{φ}[/latex] → ∞ is independent of particle fluence [latex]\dot{φ}[/latex], irrespective of the magnitude of [latex]\dot{φ}[/latex] and depends only on the decay constant [latex]λ_D[/latex] of the daughter D radionuclide and the atomic mass [latex]A_P[/latex] of the parent. Recognizing that [latex]A_P ≈ A_D[/latex], at least for large atomic number activation targets, we can state that for depletion model of nuclear activation [latex]\lambda_{\mathrm{D}} N_{\mathrm{A}} / A_{\mathrm{P}} \approx\left(a_{\mathrm{D}} ight)_{ ext {theor }} ext {. }[/latex]

Equation (12.170) shows that, according to the depletion model, [latex]\left(a_D ight)_{theor}[/latex] is the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] achievable in nuclear activation even with very high particle fluence rate [latex]\dot{φ}[/latex].

(3) It is also interesting to investigate [latex]\lim \left(a_{\mathrm{D}} ight)_{\max } ext { of (12.168) for } \dot{\varphi} ightarrow 0[/latex] yielding the following result

[latex]\begin{aligned} \lim _{\dot{\varphi} ightarrow 0}\left(a_{\mathrm{D}} ight)_{\max } & =\lim _{m ightarrow 0}\left(a_{\mathrm{D}} ight)_{\max }=\lim _{m ightarrow 0} \frac{m \lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{P}}} e^{\frac{m}{1-m} \ln m}=\lim _{m ightarrow 0} \frac{m \lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{P}}} e^{\ln m^m} \ & =\lim _{m ightarrow 0} \frac{m \lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{P}}} m^m=\frac{m \lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{P}}}=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi},\quad (12.171) \end{aligned}[/latex]

where we note that [latex]\lim _{m ightarrow 0} m^m=1[/latex]. The result of (12.171) for the depletion model at [latex]\dot{φ}[/latex] → 0 is in perfect agreement with (12.167) obtained for the saturation model leading to the conclusion that for small particle fluence rate ([latex]\dot{φ}[/latex] → 0) both the saturation model and the depletion model give identical result for [latex]\left(a_D ight)_{max}[/latex] proportional to [latex]\dot{φ}[/latex]. However, as shown in (12.170) for [latex]\dot{φ}→ ∞[/latex], the saturation model predicts the physically impossible result of [latex]\left(a_D ight)_{max}[/latex] → ∞, while (12.170) for the depletion model predicts the logical result that [latex]\left(a_D ight)_{max}[/latex] approaches [latex]\left(a_D ight)_{theor}[/latex].

(4) We now evaluate [latex]\left(a_{\mathrm{D}}^* ight)_{\max } ext { of }(12.169) ext { as } \dot{\varphi} ightarrow \infty[/latex] for the depletion–activation model. Both [latex]m^* ext { and } \varepsilon^*[/latex], defined as

[latex]m^*=\frac{m}{\varepsilon^*}=\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}}[/latex] (12.172)

and

[latex]\varepsilon^*=\frac{\lambda_{\mathrm{D}}^*}{\lambda_{\mathrm{D}}}=\frac{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}}{\lambda_{\mathrm{D}}}=1+\frac{\sigma_{\mathrm{D}} \dot{\varphi}}{\lambda_{\mathrm{D}}}[/latex] (12.173)

respectively, depend on [latex]\dot{\varphi} ext { and the limit of }\left(a_{\mathrm{D}}^* ight)_{\max } ext { as } \dot{\varphi} ightarrow \infty[/latex] is determined as follows

[latex]\begin{aligned} \lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}}^* ight)_{\max } & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\frac{\dot{\varphi}}{\varepsilon^*}\left(m^* ight)^{\frac{m^*}{1-m^*}} ight\}=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\frac{\dot{\varphi}}{\varepsilon^*} e^{\frac{m^* \ln m^*}{1-m^*}} ight\} \ & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\frac{\lambda_{\mathrm{D}} \dot{\varphi}}{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}} \exp \left[\frac{\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}}}{1-\frac{\sigma_P \dot{\varphi}}{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}}} \ln \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}} ight] ight\} \ & =\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \lim _{\dot{\varphi} ightarrow \infty}\left\{\frac{\lambda_{\mathrm{D}}}{\sigma_{\mathrm{D}}} \exp \left[\frac{\frac{\sigma_{\mathrm{P}}}{\sigma_{\mathrm{D}}}}{1-\frac{\sigma_{\mathrm{P}}}{\sigma_{\mathrm{D}}}} \ln \frac{\sigma_{\mathrm{P}}}{\sigma_{\mathrm{D}}} ight] ight\} \ & \approx\left(a_{\mathrm{D}} ight)_{ ext {theor }} k^* e^{\frac{k^* \ln k^*}{1-k^*}},\quad (12.174) \end{aligned}[/latex]

where we define the parent–daughter cross section ratio [latex]σ_P/σ_D\ as\ k^∗[/latex] and we make the approximation [latex]A_P ≈ A_D[/latex] to be able to use [latex]λ_DN_A/A_P ≈ \left(a_D ight)_{theor}[/latex]. The maximum attainable specific activity [latex]\left(a^∗_D ight)_{max}\ as\ \dot{φ}[/latex] → ∞ for depletion– activation model is according to (12.174) equal to [latex]\left(a^∗_D ight)_{theor}[/latex] multiplied by a factor that depends on the parent–daughter cross section ratio [latex]k^∗[/latex].

(b) The ratio between [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}}^* ight)_{\max }[/latex] of (12.174) for the depletion–activation model and [latex]\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }[/latex] of (12.170) for the depletion model is expressed as a function of the parent–daughter cross section ratio [latex]k^*=\sigma_{\mathrm{P}} / \sigma_{\mathrm{D}}[/latex] as follows

[latex]\frac{\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}}^* ight)_{\max }}{\lim _{\dot{\varphi} ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }}=k^* e^{\frac{k^* \ln k^*}{1-k^*}}=g^*\left(k^* ight),[/latex] (12.175)

where we introduce the depletion–activation factor [latex]g^∗\left(k^∗ ight)[/latex] and define it as

[latex]g^*\left(k^* ight)=k^* e^{-\frac{k^* \ln k^*}{k^*-1}}=e^{-\frac{\ln k^*}{1-k^*}}=\left(k^* ight)^{\frac{1}{1-k^*}} .[/latex] (12.176)

The parent–daughter cross section ratio [latex]k^*=\sigma_{\mathrm{P}} / \sigma_{\mathrm{D}}[/latex] is always positive and ranges from[latex]k^*=0 [/latex] for [latex]\sigma_{\mathrm{P}}=0[/latex] through [latex]k^*=1 [/latex] for [latex]\sigma_{\mathrm{P}}=\sigma_{\mathrm{D}} ext { to } k^*=\infty[/latex] for [latex]\sigma_{\mathrm{D}}=0[/latex]. The depletion–activation factor [latex]g^*\left(k^* ight)[/latex], on the other hand, ranges from [latex]g^*\left(k^* ight)=0[/latex] for [latex]k^*=0[/latex] through [latex] g^*\left(k^* ight)=1 / e [/latex] for [latex]k^*=1 [/latex] to [latex]g^*\left(k^* ight)=1[/latex] for [latex]k^*=\infty[/latex]. The physical range of [latex]g^∗\left(k^∗ ight)[/latex] is determined using the l’Hôpital rule as follow

[latex]\left.g ight|_{k^* ightarrow 0}=\lim _{k^* ightarrow 0}\left(k^* ight)^{\frac{1}{1-k^*}}=\lim _{k^* ightarrow 0} e^{\frac{\ln k^*}{1-k^*}}=\lim _{k^* ightarrow 0} e^{\frac{\frac{\mathrm{d} \ln k^*}{\mathrm{~d k^*}}}{\frac{d\left(1-k^* ight)}{\mathrm{d} k^*}}} =\lim _{k^* ightarrow 0} e^{\frac{\frac{1}{k^*}}{-1}} =e^{-\infty}=0, [/latex] (12.177)

[latex]\left.g ight|_{k^* ightarrow 1}=\lim _{k^* ightarrow 1}\left(k^* ight)^{\frac{1}{1-k^*}}=\lim _{k^* ightarrow 1} e^{\frac{\ln k^*}{1-k^*}}=\lim _{k^* ightarrow 1} e^{\frac{\frac{\mathrm{d} \ln k^*}{\mathrm{~d k^*}}}{\frac{d\left(1-k^* ight)}{\mathrm{d} k^*}}} =\lim _{k^* ightarrow 1} e^{\frac{\frac{1}{k^*}}{-1}} =e^{-1}= \frac{1}{e} [/latex] (12.178)

and

[latex]\begin{aligned} & \left.g ight|_{k^* ightarrow \infty}=\lim _{k^* ightarrow \infty}\left(k^* ight)^{\frac{1}{1-k^*}}=\lim _{k^* ightarrow \infty} e^{\frac{\ln k^*}{1-k^*}}=\lim _{k^* ightarrow \infty} e^{\frac{\frac{\mathrm{d} \ln k^*}{\mathrm{~d k^*}}}{\frac{d\left(1-k^* ight)}{\mathrm{d} k^*}}} \ & =\lim _{k^* ightarrow \infty} e^{\frac{\frac{1}{k^*}}{-1}}=e^0=1 .\quad (12.179) \ & \end{aligned}[/latex]

Equation (12.170) shows that for the depletion model [latex]\left(a_D ight)_{max}[/latex] cannot exceed [latex]\left(a_D ight)_{theor}[/latex] and (12.174) shows that for the depletion–activation model [latex]\left(a_D ight)_{max}[/latex] cannot exceed [latex]\left(a_D ight)_{max}[/latex] multiplied by [latex]g^∗\left(k^∗ ight)[/latex]. Since is [latex]g^∗\left(k^∗ ight)[/latex] smaller than or equal to 1, we conclude that [latex]\left(a_D ight)_{max}[/latex] for the depletion–activation model is generally smaller than [latex]\left(a_D ight)_{max}[/latex] for the depletion model.

(c) As shown in (12.177) the depletion–activation factor [latex]g^∗[/latex] depends on the parent– daughter cross section ratio [latex]k^∗[/latex] which has a physical range from 0 to ∞. Table 12.13 and Fig. 12.6 display results of our calculation of [latex]g^∗[/latex] for several values of [latex]k^∗[/latex] in the range from [latex]10^{−3}\ to\ 10^3[/latex]. Based on the [latex]g^∗\ vs\ k^∗[/latex] plot we note the following characteristics of the depletion–activation factor [latex]g^∗[/latex]:

(1) The range of [latex]g^* ext { is from } g^*=0 ext { for } k^*=0\left(\sigma_{\mathrm{P}}=0 ight) ext { through } g^*=1 / e[/latex] for [latex]k^*=1\left(\sigma_{\mathrm{P}}=\sigma_{\mathrm{D}} ight) ext { to } g^*=1 ext { for } k^*=\infty\left(\sigma_{\mathrm{D}}=0 ight)[/latex], as shown in (12.177), (12.178), and (12.179), respectively.

(2) Since [latex]\left(a_{\mathrm{D}}^* ight)_{\max } \approx\left(a_{\mathrm{D}}^* ight)_{ ext {theor }} g^*, [/latex] as derived in (12.174) , and the range of [latex]g^*[/latex] is given as [latex]0 \leq g^* \leq 1 ext {, we conclude that }\left(a_{\mathrm{D}}^* ight)_{\max } \leq\left(a_{\mathrm{D}}^* ight)_{ ext {theor }} ext {. }[/latex]

(3) For [latex]g^*=0 ext {, i.e., } \sigma_{\mathrm{P}}=0 ext {, we get from (12.174) that }\left(a_{\mathrm{D}}^* ight)_{\max }=0[/latex]. This means that there is no parent activation and no production of radioactivity.

(4) For [latex]g^*=1 ext {, i.e., } \sigma_{\mathrm{D}}=0 ext {, we get from }(12.174) ext { that }\left(a_{\mathrm{D}}^* ight)_{\max }=\left(a_{\mathrm{D}} ight)_{ ext {theor }}[/latex]. This means that the daughter produced in activation does not get activated, so that the depletion model and the depletion–activation model give the same result for the maximum attainable daughter specific activity: [latex]\left(a_{\mathrm{D}}^* ight)_{\max }=\left(a_{\mathrm{D}} ight)_{\max }= \left(a_{\mathrm{D}} ight)_{ ext {theor }}=\lambda_{\mathrm{P}} N_{\mathrm{A}} / A_{\mathrm{D}}[/latex]

(5) [latex] ext { For } \sigma_{\mathrm{P}}<\sigma_{\mathrm{D}}, k^*<1 ext { and } 0

(6) [latex] ext { For } \sigma_{\mathrm{P}}=\sigma_{\mathrm{D}}, k^*=1 ext { and } g^*=1 / e ext {. }[/latex]

(7) [latex] ext { For } \sigma_{\mathrm{P}}>\sigma_{\mathrm{D}}, k^*>1 ext { and } 1 / e

Table 12.13 Depletion–activation factor $g^$ for various values of the parent–daughter cross section ratio $k^$ in the range from 0.001 to 1000
$k^*=\frac{\sigma_{\mathrm{P}}}{\sigma_{\mathrm{D}}}$	0.001	0.01	0.1	1	10	100	1000
$g^=\left(k^\right)^{\frac{1}{1-k^*}}$	0.001	0.009	0.077	0.368	0.774	0.955	0.993

Question 12.6.Q8: Three models are used for describing nuclear activation proc......

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