Question 12.6.Q1: In practice the most commonly used nuclear activation proces......

(a) In artificial production of a radionuclide with thermal neutron activation the main objective is to produce activity \mathcal{A}_D(t) in the activated daughter sample D which is an isotope of the stable parent nuclide P. The induced radioactivity \mathcal{A}_D(t) depends on many factors, such as mass m and activation cross section σ_P of the parent sample, neutron fluence rate \dot{φ} in the reactor, as well as the decay constant λ_D and activation cross section σ_D of the daughter radionuclide. The daughter D nuclei are produced at a rate of σ_{P}\dot{φ}N_P(t) and they decay with a rate of λ_DN_D(t). If the daughter D is affected by exposure to activation particles, one accounts for the daughter activation with the term σ_D\dot{φ}N_D(t). The number of daughter nuclei is N_D(t) and the overall rate of change of the number of daughter nuclei is dN_D/dt obtained by combining the production rate of daughter nuclei σ_P\dot{φ}N_P(t) with the decay rate of daughter nuclei λ_DN_D(t) and depletion of daughter nuclei through σ_D\dot{φ}N_D(t) to get the following differential equation for dN_D(t)/dt.

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)-\sigma_{\mathrm{D}} \dot{\varphi} N_{\mathrm{D}}(t),           (12.40)

where σ_P is the thermal neutron cross section of the parent nucleus, σ_D is the thermal neutron cross section of the daughter nucleus, \dot φ is the fluence rate of thermal neutrons, N_P(t) is the number of parent P target nuclei, N_D(t) is the number of daughter D nuclei and λ_D is the decay constant of the daughter nucleus related to its half-life \left(t_{1/2}\right)_D through \left(t_{1/2}\right)_D = \left(ln 2\right)/λ_D.

The solution to differential equation (12.40) for N_D(t) is affected by several conditions, such as:

(1) Initial conditions on the initial number N_P(0) of parent nuclei at time t = 0 and initial number N_D(0) of daughter nuclei at time t = 0.

(2) General conditions on N_P(t) during the activation process allowing two possibilities: (i) N_P(t) is essentially constant with activation time t suggesting that N_P(t) is not affected by activation of parent nuclei into daughter nuclei and (ii) N_P(t) is diminishing with activation time t as a result of the activation of parent nuclei into daughter nuclei. The first \left[N_P(t) = const\right] option represents the saturation model of nuclear activation, while the second option represents the depletion model of nuclear activation.

(3) General conditions on N_D(t) during the activation process of the parent P. N_D(t) grows through activation of parent P and diminishes during the activation process because of (i) daughter decay or (ii) daughter decay combined with daughter activation as a result of daughter exposure to neutrons. The saturation model or the depletion model of activation, listed in (2), cover the first option which assumes that the daughter is not affected by neutron exposure, while the second option which incorporates the daughter activation by neutron exposure represents the parent depletion–daughter activation model.

(b) For standard initial conditions N_P(t = 0) = N_P(0)\ and\ N_D(t = 0) = N_D(0) = 0 as well as the general condition for the saturation model that (1) number of parent nuclei is constant, i.e., is so large that it is not affected by exposure to neutrons \left[N_P(t) = N_P(0) = const\right] and (2) daughter radionuclide is not affected by exposure to neutrons \left(σ_D = 0\right), the differential equation for dN_D/dt of (12.40) is written as

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)           (12.41)

or in integral form as

\int_0^{N_{\mathrm{D}}(t)} \frac{\mathrm{d}\left\{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)-\lambda_{\mathrm{D}} N_{\mathrm{D}}\right\}}{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)-\lambda_{\mathrm{D}} N_{\mathrm{D}}}=-\lambda_{\mathrm{D}} \int_0^t \mathrm{~d} t .            (12.42)

The solution of the simple differential equation (12.42) is as follows

N_{\mathrm{D}}(t)=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{\lambda_{\mathrm{D}}}\left\{1-e^{-\lambda_{\mathrm{D}^t}}\right\}            (12.43)

Since the daughter activity \mathcal{A}_D(t) equals to λ_DN_D(t), we can write \mathcal{A}_D(t) for the saturation model of neutron activation as (T12.13)

\mathcal{A}_{\mathrm{D}}(t)=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)\left\{1-e^{-\lambda_{\mathrm{D}} t}\right\}=\left(\mathcal{A}_{\mathrm{D}}\right)_{\mathrm{sat}}\left\{1-e^{-\lambda_{\mathrm{D}} t}\right\}          (12.44)

where we define \left(A_D\right)_{sat}, the saturation daughter D activity that can be produced by bombardment of the parent P target with neutrons, as equal to σ_P \dot φ N_P(0).

The specific activity a of a radioactive source is defined as activity \mathcal{A} of the source per unit mass M of the source, i.e., a = A/M. For the saturation model (12.44), we can thus express specific activity a as a function of activation time t as

a(t)=\frac{\mathcal{A}(t)}{M}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{M}\left\{1-e^{-\lambda_{\mathrm{D}} t}\right\}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}}}{A_{\mathrm{P}}}\left\{1-e^{-\lambda_{\mathrm{D}} t}\right\}=a_{\mathrm{sat}}\left\{1-e^{-\lambda \mathrm{D} t}\right\},            (12.45)

where we define a_{sat} as the saturation specific activity \left(a_{\text {sat }}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}} / A_{\mathrm{P}}\right) and we used the identity N_P(0)/M = N_A/A_P\ with\ A_P the atomic weight of the parent nucleus and N_A the Avogadro number (6.022 × 10^{23} mol^{-1}).

(c) In the parent depletion model of neutron activation, one must account for the finite number of parent nuclei \left[N_{\mathrm{P}}(t) \neq \text { const }\right]; however, an assumption is made that the daughter radionuclide is not affected by neutron exposure \left[σ_P = 0\right]. Equation (12.40) is for the depletion model stated as follows

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)              (12.46)

with N_{\mathrm{P}}(t) \text { given as } N_{\mathrm{P}}(t)=N_{\mathrm{P}}(0) e^{-\sigma_{\mathrm{P}} \dot{\varphi} t} \text {, where } N_{\mathrm{P}}(0) is the initial number of parent nuclei placed into the neutron fluence rate \dot φ at time t = 0.
The solution to (12.46), following the steps taken in the derivation of (T10.34) for the nuclear decay series P → D → G and using the following initial conditions N_{\mathrm{P}}(t=0)=N_{\mathrm{P}}(0) \text { and } N_{\mathrm{D}}(t=0)=N_{\mathrm{D}}(0)=0 is given as follows (T12.21)

N_{\mathrm{D}}(t)=N_{\mathrm{P}}(0) \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}} t}\right] .           (12.47)

Recognizing that activity \mathcal{A}_D(t) = λ_DN_D(t) we get the following expression for the growth in daughter activity \mathcal{A}_D(t) with activation time t

\mathcal{A}_{\mathrm{D}}(t)=\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}} \sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}} t}\right]            (12.48)

Several interesting features are evident from (12.48), such as, for example:

(1) Generally, in neutron activation σ_P \dot φ <λ_D resulting in dynamics similar to that referred to as transient equilibrium in nuclear decay series.

(2) When \sigma_{\mathrm{P}} \dot{\varphi} \ll \lambda_{\mathrm{D}}, we are dealing with a special case of transient equilibrium dynamics that in nuclear decay series is referred to as secular equilibrium dynamics. In this case, (12.48) simplifies to the expression derived above in (12.44) for the saturation model and is valid under the assumption that the fraction of nuclei transformed from parent to daughter in neutron activation is negligible in comparison to the initial number N_P(0) of parent nuclei.

(3) Equation (12.48) shows that, rather than reaching saturation at t → ∞, the daughter activity \mathcal{A}_D(t) is zero at t = 0 and, with increasing time from t = 0, first rises with t, reaches a maximum \left(\mathcal{A}_D\right)_{max} at time t = \left(t_{max}\right)_D, and then decreases as t increases further, until at t = ∞ it becomes zero again.

(4) The time \left(t_{max}\right)_D is determined by setting \left(d\mathcal{A}_D/dt\right)_{t=\left(t_{max}\right)_D }= 0 to get the following result

\left(t_{\max }\right)_{\mathrm{D}}=\frac{\ln \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}}}{\sigma_{\mathrm{P}} \dot{\varphi}-\lambda_{\mathrm{D}}} \equiv \frac{\ln \frac{\lambda_{\mathrm{D}}}{\sigma_{\mathrm{P}} \dot{\varphi}}}{\lambda_{\mathrm{D}}-\sigma_{\mathrm{P}} \dot{\varphi}} .                  (12.49)

(d) Equation (12.40) describes the most general neutron activation process in which the parent P is exposed to neutrons and transforms into radioactive daughter D which decays with its own decay constant λ_D and, in addition, is affected by exposure to neutrons \left(\sigma_{\mathrm{D}} \neq 0\right). The model that deals with this general nuclear activation process is referred to as the “parent depletion–daughter activation” model and can be described by a simple consolidation of the daughter decay term \left[\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)\right] and daughter activation term \left[\sigma_{\mathrm{D}} \dot{\varphi} N_{\mathrm{D}}(t)\right] into one term governed by a modified decay constant expressed as follows: \lambda_{\mathrm{D}}^*=\lambda_{\mathrm{D}}+\sigma_{\mathrm{D}} \dot{\varphi}, resulting in the following form of (12.40)

\frac{\mathrm{d} N_{\mathrm{D}}(t)}{\mathrm{d} t}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}} N_{\mathrm{D}}(t)-\sigma_{\mathrm{D}} \dot{\varphi} N_{\mathrm{D}}(t)=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(t)-\lambda_{\mathrm{D}}^* N_{\mathrm{D}}(t)               (12.50)

The solution to differential equation (12.50) is similar to that of (12.46) except that λ_D in (12.46) is replaced by a modified decay constant λ^∗_D resulting in the following solution to (12.50)

N_{ {\mathrm{D}}}(t)=N_{ {\mathrm{P}}}(0) \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}^*-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}}^* t}\right] .           (12.51)

The daughter activity \mathcal{A}_D(t) = λ_DN_D(t) is in the depletion–activation model expressed as

\mathcal{A}_{\mathrm{D}}(t)=N_{\mathrm{P}}(0) \frac{\lambda_{\mathrm{D}} \sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}^*-\sigma_{\mathrm{P}} \dot{\varphi}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}}^* t}\right]=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) \frac{\frac{\lambda_{\mathrm{D}}}{\lambda_{\mathrm{D}}^*}}{1-\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}^*}}\left[e^{-\sigma_{\mathrm{P}} \dot{\varphi} t}-e^{-\lambda_{\mathrm{D}}^* t}\right] .           (12.52)