Determination of trace concentrations of transmuted stable nuclides in TMD detectors using PGAA

Abstract

The capability of prompt gamma-ray activation analysis (PGAA) for neutron fluence dosimetry by means of transmutation detectors is reported. The metallic foils of Ni, Au, Cu and Nb and the small piece of Ge crystal, which were irradiated at the LVR-15 reactor at Řež for several days, were selected for analysis by PGAA technique. Concentrations of transmuted stable nuclides in these foils were measured using the PGAA facility installed at the research reactor FRM II in Garching.

Appendix

The production of the mercury isotopes, $^{198}$Hg, $^{199}$Hg and $^{200}$Hg, in the process if irradiation of gold, $^{197}$Au is described by series of differential equations,

$$\begin{aligned} \frac{{\rm d}N_a(t)}{{\rm d}t}&= -\lambda _A N_a(t),\qquad \qquad \qquad \frac{{\rm d}N_b(t)}{{\rm d}t}=\lambda _A N_a(t) - \lambda _\beta N_b(t) , \nonumber \\ \frac{{\rm d}N_c(t)}{{\rm d}t}&= \lambda _B N_b(t) - \lambda _c N_c(t),\qquad \qquad \frac{{\rm d}N_{d}(t)}{{\rm d}t}=\lambda _b N_b(t) - \lambda _{d} N_{d}(t),\nonumber \\ \frac{{\rm d}N_e(t)}{{\rm d}t}&= \lambda _c N_c(t) + \lambda _{d} N_{d}(t)-\lambda _eN_e(t),\qquad \qquad \frac{{\rm d}N_f(t)}{{\rm d}t}=\lambda _e N_e(t),\nonumber \\ \lambda _\beta&= \lambda _B + \lambda _b \end{aligned}$$

(4)

where $N_a(t)$, $N_b(t)$, $N_c(t)$, $N_d(t)$, $N_e(t)$ and $N_f(t)$ are quantity of $^{197}$Au, $^{198}$Au, $^{199}$Au, $^{198}$Hg, $^{199}$Hg and $^{200}$Hg nuclei in time $t$ , respectively, $\lambda _A$, $\lambda _B$, $\lambda _D$ and $\lambda _E$ are the reaction rates for $^{197}$Au,$^{198}$Au,$^{198}$Hg and $^{199}$Hg(n,$\gamma$) reactions, respectively, and $\lambda _b$ and $\lambda _c$ are decay constants for $\beta ^-$-decay of $^{198}$Au and $^{199}$Au, respectively. Solving these equations one will obtain the general solutions for quantity of nuclei during irradiation

$$\begin{aligned} N_a(t)&= N_a^0 e^{-\lambda _A t},\nonumber \\ N_b(t)&= N_a^0 \frac{\lambda _A}{\lambda _{\beta A}} (e^{-\lambda _A t}-e^{-\lambda _{\beta } t}),\nonumber \\ N_c(t)=& N_a^0 \lambda _A \lambda _B \left( \frac{e^{-\lambda _A t}}{\lambda _{cA}\lambda _{\beta A}} + \frac{e^{-\lambda _{\beta } t}}{\lambda _{c \beta }\lambda _{A \beta }} + \frac{e^{-\lambda _c t}}{\lambda _{Ac}\lambda _{\beta c}}\right) ,\nonumber \\ N_d(t)&= N_a^0 \lambda _A \lambda _b \left( \frac{e^{-\lambda _A t}}{\lambda _{DA}\lambda _{\beta A}} + \frac{e^{-\lambda _{\beta } t}}{\lambda _{D \beta }\lambda _{A \beta }} + \frac{e^{-\lambda _D t}}{\lambda _{AD}\lambda _{\beta D}}\right) ,\nonumber \\ N_e(t)=& N_a^0 \left[ \lambda _A \lambda _B \lambda _c \left( \frac{e^{-\lambda _A t}-e^{-\lambda _E t}}{\lambda _{cA} \lambda _{\beta A} \lambda _{EA}} + \frac{e^{-\lambda _\beta t}-e^{-\lambda _E t}}{\lambda _{c \beta } \lambda _{A \beta } \lambda _{E \beta }} + \frac{e^{-\lambda _c t}-e^{-\lambda _E t}}{\lambda _{Ac} \lambda _{\beta c} \lambda _{E c}}\right) \right. \nonumber \\&\left. +\lambda _A \lambda _b \lambda _D \left( \frac{e^{-\lambda _A t}-e^{-\lambda _E t}}{\lambda _{DA} \lambda _{\beta A} \lambda _{EA}} + \frac{e^{-\lambda _\beta t}-e^{-\lambda _E t}}{\lambda _{D \beta } \lambda _{A \beta } \lambda _{E \beta }} + \frac{e^{-\lambda _c t}-e^{-\lambda _E t}}{\lambda _{AD} \lambda _{\beta D} \lambda _{E D}}\right) \right] ,\nonumber \\ N_f(t)=& N_a^0 \lambda _E\nonumber \\&\left[ \lambda _A \lambda _B \lambda _c \left( \frac{\frac{1-e^{-\lambda _At}}{\lambda _A}-\frac{1-e^{-\lambda _Et}}{\lambda _E}}{\lambda _{cA}\lambda _{\beta A} \lambda _{EA}}+ \frac{\frac{1-e^{-\lambda _\beta t}}{\lambda _\beta }-\frac{1-e^{-\lambda _E t}}{\lambda _E}}{\lambda _{c \beta } \lambda _{A \beta } \lambda _{E \beta }}+ \frac{\frac{1-e^{-\lambda _c t}}{\lambda _c}-\frac{1-e^{-\lambda _E t}}{\lambda _E}}{\lambda _{Ac} \lambda _{\beta c} \lambda _{Ec}}\right) \right. \nonumber \\&\left. + \lambda _A \lambda _b \lambda _D \left( \frac{\frac{1-e^{-\lambda _A t}}{\lambda _A}-\frac{1-e^{-\lambda _E t}}{\lambda _E}}{\lambda _{DA}\lambda _{\beta A} \lambda _{EA}} + \frac{\frac{1-e^{-\lambda _\beta t}}{\lambda _\beta }-\frac{1-e^{-\lambda _E t}}{\lambda _E}}{\lambda _{D \beta } \lambda _{A \beta } \lambda _{E \beta }}+ \frac{\frac{1-e^{-\lambda _D t}}{\lambda _D}-\frac{1-e^{-\lambda _E t}}{\lambda _E}}{\lambda _{AD} \lambda _{\beta D} \lambda _{ED}}\right) \right] , \end{aligned}$$

(5)

where we have used general notation $\lambda _{ij}=\lambda _i-\lambda _j$ and $N_a^0$ is number of gold nuclei before irradiation. In time of analysis $t_m$ long time after the stop of irradiation $t_2$, i.e. $(t_m-t_2) \gg 1/\lambda _b$ and $(t_m-t_2) \gg 1/\lambda _c$, number of $^{197}$Au, $^{198}$Hg, $^{199}$Hg and $^{200}$Hg nuclides can be expressed as

$$\begin{aligned} N_{197Au}(t_m)&= N_a(t_2),\nonumber \\ N_{198Hg}(t_m)&= N_b(t_2)+N_d(t_2),\nonumber \\ N_{199Hg}(t_m)&= N_c(t_2)+N_e(t_2),\nonumber \\ N_{200Hg}(t_m)&= N_f(t_2). \end{aligned}$$

(6)