[1208] | 1 | \chapter{Radioactive Decay} |
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| 2 | |
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| 3 | \section{The Radioactive Decay Module} |
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| 4 | $G4RadioactiveDecay$ and associated classes are used to simulate the |
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| 5 | decay of radioactive nuclei by $\alpha$, $\beta^{+}$, and $\beta^{-}$ emission |
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| 6 | and by electron capture (EC). The simulation model is empirical and |
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| 7 | data-driven, and uses the Evaluated Nuclear Structure Data File |
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| 8 | (ENSDF)~\cite{rdk.ENSDF} for information on: |
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| 9 | |
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| 10 | {\noindent \rm{}} |
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| 11 | |
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| 12 | %\check |
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| 13 | {\noindent \rm{}\LARGE$\cdot $\normalsize \indent nuclear half-lives,} |
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| 14 | |
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| 15 | %\check |
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| 16 | {\noindent \rm{}\LARGE$\cdot $\normalsize \indent |
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| 17 | nuclear level structure for the parent or daughter nuclide,} |
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| 18 | |
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| 19 | %\check |
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| 20 | {\noindent \rm{}\LARGE$\cdot $\normalsize \indent |
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| 21 | decay branching ratios, and} |
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| 22 | |
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| 23 | %\check |
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| 24 | {\noindent \rm{}\LARGE$\cdot $\normalsize \indent |
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| 25 | the energy of the decay process.} |
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| 26 | |
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| 27 | %\check |
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| 28 | {\noindent \rm{}} |
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| 29 | |
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| 30 | If the daughter of a nuclear decay is an excited isomer, its prompt nuclear |
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| 31 | de-excitation is treated using the $G4PhotoEvaporation$ |
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| 32 | class~\cite{rdk.photevap}. |
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| 33 | |
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| 34 | \section{Sampling} |
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| 35 | Sampling of the $\beta$-spectrum, which includes the coordinated energies and |
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| 36 | momenta of the $\beta^{\pm}$, $\nu$, or $\bar{\nu}$ and residual nucleus, is |
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| 37 | performed either from histogrammed data, or through a three-body decay |
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| 38 | algorithm. In the latter case, the effect of the Coulomb barrier on the |
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| 39 | probability of $\beta^{\pm}$-emission can also be taken into account using the |
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| 40 | Fermi function: |
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| 41 | |
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| 42 | \begin{equation} |
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| 43 | F(E_0)=\frac {\gamma} {1-e^{-\gamma}} |
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| 44 | \left \lgroup {\frac {Z^2(E_0 + 1)^2} {137^2} + \frac {E_0 ^2+2E_0} {4}} \right \rgroup ^ |
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| 45 | {\sqrt {1 - \frac {Z^2} {137^2}} - 1} . |
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| 46 | \end{equation} |
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| 47 | Here $E_0$ is the energy of the $\beta$-particle given as a fraction of the |
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| 48 | end-point energy, $Z$ is the atomic number of the nucleus, and $\gamma$ is |
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| 49 | given by the expression: |
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| 50 | \par |
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| 51 | |
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| 52 | \begin{equation} |
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| 53 | \gamma = \frac {2\pi Z} {137} \frac {1+E_0} {\sqrt {E_0 ^2 + 2E_0}} . |
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| 54 | \end{equation} |
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| 55 | |
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| 56 | Due to the level of imprecision of the rest-mass energy of the nuclei generated |
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| 57 | by $G4IonTable::GetNucleusMass$, the mass of the parent nucleus is |
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| 58 | modified to a minor extent just before performing the two- or three-body decay |
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| 59 | so that the $Q$ for the transition process equals that identified in the ENSDF |
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| 60 | data. |
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| 61 | |
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| 62 | \subsection{Biasing Methods} |
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| 63 | By default, sampling of the times of radioactive decay and branching ratios is |
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| 64 | done according to standard, analogue Monte Carlo modeling. The user may switch |
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| 65 | on one or more of the following variance reduction schemes, which can provide |
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| 66 | significant improvement in the modelling efficiency: |
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| 67 | |
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| 68 | 1. The decays can be biased to occur more frequently at certain times, for |
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| 69 | example, corresponding to times when measurements are taken in a real |
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| 70 | experiment. The statistical weights of the daughter nuclides are reduced |
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| 71 | according to the probability of survival to the time of the event, $t$, which |
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| 72 | is determined from the decay rate. The decay rate of the $n^{th}$ nuclide in a |
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| 73 | decay chain is given by the recursive formulae: |
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| 74 | \par |
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| 75 | |
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| 76 | %\check |
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| 77 | \begin{equation} |
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| 78 | R_n (t) = \sum \limits_{i=1} \limits^{n-1} A_{n:i}f(t,\tau_i) + |
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| 79 | A_{n:n}f(t,\tau_n) |
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| 80 | \end{equation} |
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| 81 | |
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| 82 | %\check |
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| 83 | {\noindent \rm{}where:} |
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| 84 | \par |
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| 85 | |
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| 86 | %\check |
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| 87 | \begin{equation} |
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| 88 | \label{rdk.eq4} |
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| 89 | A_{n:i} = \frac {\tau_i} {\tau_i-\tau_n} A_{n:i} \quad \forall i<n |
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| 90 | \end{equation} |
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| 91 | \begin{equation} |
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| 92 | A_{n:n} = -\sum \limits_{i=1} \limits^{n-1} \frac{\tau_n} {\tau_i-\tau_n} A_{n:i} - y_n |
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| 93 | \end{equation} |
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| 94 | \begin{equation} |
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| 95 | \label{rdk.eq6} |
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| 96 | f(t,\tau_i)= \frac {e^{-\frac{t}{\tau_i}}} {\tau_i} \int \limits_{-\inf} \limits^t F(t')e^{\frac{t'}{\tau_i}}dt' . |
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| 97 | \end{equation} |
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| 98 | The values $\tau_i$ are the mean life-times for the nuclei, $y_i$ is the |
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| 99 | yield of the $i^{th}$ nucleus, and $F(t)$ is a function identifying the time |
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| 100 | profile of the source. The above expression for decay rate is simplified, |
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| 101 | since it assumes that the $i^{th}$ nucleus undergoes 100\% of the decays to the |
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| 102 | $(i+1)^{th}$ nucleus. Similar expressions which allow for branching and |
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| 103 | merging of different decay chains can be found in Ref.~\cite{rdk.Tru96}. |
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| 104 | |
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| 105 | A consequence of the form of equations \ref{rdk.eq4} and \ref{rdk.eq6} is that |
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| 106 | the user may provide a source time profile so that each decay produced as a |
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| 107 | result of a simulated source particle incident at time $t=0$ is convolved over |
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| 108 | the source time profile to derive the actual decay rate for that source |
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| 109 | function. |
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| 110 | |
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| 111 | This form of variance reduction is only appropriate if the radionuclei can be |
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| 112 | considered to be at rest with respect to the geometry when decay occurs. |
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| 113 | |
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| 114 | 2. For a given decay mode ($\alpha$, $\beta^++EC$, or $\beta^-$) the branching |
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| 115 | ratios to the daughter nuclide can be sampled with equal probability, so that |
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| 116 | some low probability branches which may have a disproportionately greater |
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| 117 | effect on the measurement are sampled with increased probability. |
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| 118 | |
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| 119 | 3. Each parent nuclide can be split into a user-defined number of nuclides |
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| 120 | (of proportionally lower statistical weight) prior to treating decay in order t |
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| 121 | o increase the sampling of the effects of the daughter products. |
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| 122 | |
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| 123 | \section{Status of this document} |
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| 124 | |
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| 125 | 00.00.00 created by ? \\ |
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| 126 | 21.11.03 bibliography added, minor re-wording by D.H. Wright \\ |
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| 127 | |
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| 128 | \begin{thebibliography}{99} |
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| 129 | \bibitem{rdk.ENSDF} J. Tuli, {\it "Evaluated Nuclear Structure Data File,"} |
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| 130 | BNL-NCS-51655-Rev87, 1987. |
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| 131 | |
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| 132 | \bibitem{rdk.photevap} Chapter 25, Geant4 Physics Reference Manual. |
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| 133 | |
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| 134 | \bibitem{rdk.Tru96} P.R. Truscott, PhD Thesis, University of London, 1996. |
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| 135 | \end{thebibliography} |
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