Question 12.6.Q7: The depletion model of nuclear activation accounts for deple......

(a) Parameters of the saturation model, depletion model, and the depletion– activation model applied to neutron activation of iridium-191 nuclide into iridium192 radionuclide in a neutron fluence rate \dot{φ} of 10^{14}\ cm^{−2}\ ·\ s^{−1} are determined as follows:

(1)

\left(t_{1 / 2}\right)_{\mathrm{D}}=\left(t_{1 / 2}\right)_{\mathrm{Ir}-192}=73.8 \mathrm{~d}=6.376 \times 10^6 \mathrm{~s} .             (12.149)

(2)

\lambda_{\mathrm{D}}=\lambda_{\mathrm{Ir}-192}=\frac{\ln 2}{\left(t_{1 / 2}\right)_{\mathrm{Ir}-192}}=\frac{\ln 2}{6.376 \times 10^6 \mathrm{~s}}=1.087 \times 10^{-7} \mathrm{~s}^{-1}                  (12.150)

(3)

\sigma_{\mathrm{P}}=\sigma_{\mathrm{Ir}-191}=954 \mathrm{~b}=954 \times 10^{-24} \mathrm{~cm}^2               (12.151)

(4)

\sigma_{\mathrm{D}}=\sigma_{\mathrm{Ir}-192}=1420 \mathrm{~b}=1420 \times 10^{-24} \mathrm{~cm}^2           (12.152)

(5)

m=\frac{\sigma_{\mathrm{P} \dot{\varphi}}}{\lambda_{\mathrm{D}}}=\frac{\sigma_{\mathrm{Ir}-191 \dot{\varphi}}}{\lambda_{\mathrm{Ir}-192}}=\frac{\left(954 \times 10^{-24} \mathrm{~cm}^2\right) \times\left(10^{14} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}\right)}{1.087 \times 10^{-7} \mathrm{~s}^{-1}}=0.878            (12.153)

(6)

\lambda_{\mathrm{D}}^*=\lambda_{\mathrm{Ir}-192}^*=\lambda_{\mathrm{Ir}-192}+\sigma_{\mathrm{Ir}-192} \dot{\varphi}=2.507 \times 10^{-7} \mathrm{~s}^{-1} \quad[\text { see (T12.48) }]                 (12.154)

(7)

\varepsilon^*=\frac{\lambda_{\mathrm{D}}^*}{\lambda_{\mathrm{D}}}=\frac{\lambda_{\mathrm{Ir}-192}^*}{\lambda_{\mathrm{Ir}-192}}=\frac{2.507 \times 10^{-7}}{1.087 \times 10^{-7}}=2.306 \quad[\text { see T12.52)] }             (12.155)

(8)

m^*=\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}^*}=\frac{\sigma_{\mathrm{Ir}-192 \dot{\varphi}}}{\lambda_{\mathrm{Ir}-192}^*}=\frac{m}{\varepsilon^*}=\frac{0.878}{2.306}=0.381 \quad[\text { see T12.54)] }            (12.156)

(9)

\begin{aligned} \left(x_{\mathrm{D}}\right)_{\max } & =\left(x_{\mathrm{Ir}-192}\right)_{\max }=\frac{m \ln m}{(m-1) \ln 2} \\ & =\frac{0.878 \times \ln 0.878}{(0.878-1) \times \ln 2}=1.351 \end{aligned}  [see Prob. 252 ] (12.68)            (12.157)

(10)

\begin{aligned} \left(x_{\mathrm{D}}^*\right)_{\max } & =\left(x_{\mathrm{Ir}-192}^*\right)_{\max }=\frac{m^* \ln m^*}{\left(m^*-1\right) \ln 2} \\ & =\frac{0.381 \times \ln 0.381}{(0.381-1) \times \ln 2}=0.857 \end{aligned}\\ [see Prob. 254 ](12.108)         (12.158)

\left(x_{\mathrm{D}}^*\right)_{\max }  = \frac{m^*}{\left(m^*-1\right)} \frac{\ln m^*}{ \ln 2}    (12.108)

(11)

\left(y_{\mathrm{D}}\right)_{\max }=\frac{1}{1 – m} F_1[1  –  F_2] = \frac{1}{1  – m} e^{\frac{m}{1-m} \ln  m}  [1-m] = e^{\frac{m}{1-m} \ln  m}     (12.73)

(12)

\left(y_{\mathrm{D}}^*\right)_{\max }=\left(y_{\mathrm{Ir}-192}^*\right)_{\max }=\frac{1}{\varepsilon^*}\left(m^*\right)^{\frac{m^*}{1-m^*}}=\frac{0.381^{\frac{0.381}{1-0.381}}}{2.306}=0.239\\     [see Prob. 254 (12.110)].      (12.160)

\left(y^*_{\mathrm{D}}\right)_{\max } =  y^*_D [(x_D^*)_{max}] = \frac{1}{ε^* (1-m^*)}\left[\frac{1}{2^{(x^*_D)_{max}}}  –  \frac{1}{2^{(x^*_D)_{max}/m^*}}\right] \\ \quad = \frac{[ε^* (1-m^*)]^{-1}}{2^{(x_D)_{max}}} \left[1 – \frac{1}{2^{[(x^*_D)_{max}/m^*] – (x_D^*)_{max}}}\right] \\ \quad = \frac{[ε^* (1-m^*)]^{-1}}{2(x^*_D)_{max}} \left[1 – \frac{1}{2^{(x^*_D)_{max}[\frac{1-m^*}{m^*}] }}\right] \\ \quad = \frac{1}{ε^* (1-m^*)}F_1^* [1-F_2^*]    (12.110)

The normalized quantity y_P(x) representing the number of parent nuclei N_P(t) normalized to the initial number N_P(0) of parent nuclei as a function of normalized time x = mt/\left(t_{1/2}\right)_D is the same for all three activation models and is given by (12.58) of Prob. 252 and (T12.27) as

y_{\mathrm{P}}(x)=e^{-x \ln 2}=2^{-x}          (12.161)

The normalized daughter activity y_{\mathrm{D}}(x) \text { defined as } y_{\mathrm{D}}(x)=\mathcal{A}_{\mathrm{D}}(t) /\left[\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)\right] is calculated:

(i) for the saturation model with (12.61) of Prob. 252 expressed as

z_{\mathrm{D}}(x)=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}=1-e^{-\frac{x}{m} \ln 2}=1-\frac{1}{2^{\frac{x}{m}}},            (12.162)

(ii) for the depletion model with (12.64) of Prob. 252 expressed as

y_{\mathrm{D}}(x)=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}=\frac{1}{1-m}\left\{e^{-x \ln 2}-e^{-\frac{x}{m} \ln 2}\right\}=\frac{1}{1-m}\left\{\frac{1}{2^x}-\frac{1}{2^{\frac{x}{m}}}\right\},           (12.163)

(iii) for the depletion–activation model with (12.104) of Prob. 254 expressed as

Normalized number of parent nuclei y_{Ir-191} as well as normalized daughter activities z_{Ir-192} for the saturation model, y_{Ir-192} for the depletion model, and y^∗ _{Ir-192} for the depletion–activation model were calculated for normalized time x using (12.161), (12.162), (12.163), and (12.164), respectively, and the results are displayed in Table 12.10 and Fig. 12.5. Data were calculated for normalized time x in the range from x = 0 to x = 5.5 in increments of 0.5 and for neutron fluence rate \dot{φ} of 10^{14}\ cm^{−2}\ ·\ s^{−1}.
Several features of the data displayed in Fig. 12.5 are of note:

(1) Plot of y_{Ir-191} is the same for the three activation models.
(2) Normalized activity of Ir-192 saturates at z_{Ir-192} = 1 at large x for the saturation model and displays a maximum for the depletion model and depletion–activation models.
(3) Maximum in normalized activity of Ir-192 occurs at the point of ideal equilibrium at (x_{Ir-192})_{max} for the depletion model, while for the depletion–activation model it occurs at \left(x^∗ _{Ir-192}\right)_{max} < \left(x_{Ir-192}\right)_{max}.
(4) Maximum \left(y_{Ir-192}\right)_{max} for the depletion model at the point of ideal equilibrium exceeds the maximum \left(y^∗ _{Ir-192}\right)_{max} for the parent depletion–daughter activation model.

(b) The maximum specific activity \left(a^∗_D\right)_{max} for the depletion–activation model (T12.59) is determined from (12.164) and (12.160) using the following expression

Furthermore, the maximum in specific activity \left(a^∗_D\right)_{max} occurs at normalized time \left(x_{\mathrm{D}}^*\right)_{\max } \text { determined by setting } \mathrm{d} y_{\mathrm{D}}^* /\left.\mathrm{d} x\right|_{x=\left(x_{\mathrm{D}}^*\right)_{\max }}=0 \text { at } x=\left(x_{\mathrm{D}}^*\right)_{\max } and solving for \left(x^∗_D\right)_{max} to get the expression given in (12.158)

\left(x_{\mathrm{D}}^*\right)_{\max }=\frac{m^* \ln m^*}{\left(m^*-1\right) \ln 2} .        (12.166)

We now use the parent depletion–daughter activation model to determine with (UU) the maximum attainable specific activity \left(a^∗_{Ir-192}\right)_{max} and with (12.166) the normalized activation time (x^∗_{Ir-192})_{max} required to attain \left(a^∗_{Ir-192}\right)_{max} in neutron activation of parent P (Ir-191) into daughter D (Ir-192) for five thermal neutron fluence rates \dot{\varphi} \text { in }\left(\mathrm{cm}^{-2} \cdot \mathrm{s}^{-1}\right): 5 \times 10^{11}, 2 \times 10^{13}, 1 \times 10^{14}, 3 \times 10^{14}, and 1.2 \times 10^{16}. In addition to basic parameters: parent thermal neutron cross section σ_P = σ_{Ir-191} = 954\ b, daughter thermal neutron cross section σ_D = σ_{Ir-192} = 1420\ b, and parent atomic mass A_P = A_{Ir-191} = 190.96 g/mol, the other relevant parameters used in the calculations are listed in Table 12.11.
In Table 12.12 we list the maximum attainable specific activities \left(a_{Ir-192}\right)_{max}\ and\ \left(a^∗_{Ir-192}\right)_{max} as well as the associated normalized activation times \left(x_{Ir-192}\right)_{max}\ and\ \left(x^∗_{Ir-192}\right)_{max} for neutron activation of Ir-191 into Ir-192 with various neutron fluence rates \dot{φ} calculated for depletion and depletion–activation models, respectively. The data for the depletion model were determined in Prob. 256, data for the depletion– activation model were calculated with (12.165) and (12.166), respectively.
Several interesting conclusions, with regard to the maximum attainable specific activity and characteristic activation time can be reached based on information given in Table 12.12. We note that for all neutron fluence rates \dot{φ}:

(1) Normalized characteristic activation time \left(x_{Ir-192}\right)_{max} for the depletion model exceeds \left(x^∗_{Ir-192}\right)_{max} for the depletion–activation model.

(2) Ratio \left(x_{\mathrm{Ir}-192}\right)_{\max } /\left(x_{\mathrm{Ir} -192}^*\right)_{\max } increases from \sim 1 at low fluence rate \dot{\varphi}=5 \times 10^{11} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1} through \sim 1.6 at intermediate fluence rate \dot{\varphi}=5 \times 10^{11} \mathrm{~cm}^{-2}  ·  \mathrm{s}^{-1} to \sim 6 at high fluence rate \dot{\varphi}=1.2 \times 10^{16} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}. Thus, the difference between (x_{Ir-192})_{max} and \left(x^∗_{Ir-192}\right)_{max} increases with fluence rate \dot{φ}.

(3) Maximum attainable specific activity \left(a_{Ir-192}\right)_{max} for the depletion model exceeds \left(a^∗_{Ir-192}\right)_{max} for the depletion–activation model. This is explained by the loss of daughter D nuclei to nuclear activation that is ignored in the depletion model but is accounted for in the depletion–activation model.

(4) Ratio \left(a_{\mathrm{Ir}-192}\right)_{\max } /\left(a_{\mathrm{Ir}-192}^*\right)_{\max } increases from \sim 1 at low fluence rate \dot{\varphi}= 5 \times 10^{11} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1} through \sim 1.6 at intermediate fluence rate \dot{\varphi}=5 \times 10^{11} \mathrm{~cm}^{-2}\ ·\ s^{−1} to ∼3.5 at high fluence rate \dot{\varphi}=1.2 \times 10^{16} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}. Thus, the difference between \left(a_{Ir-192}\right)_{max}\ and\ \left(a^∗_{Ir-192}\right)_{max} increases with fluence rate \dot{φ}.