\chapter{Radioactive Decay} 

\section{The Radioactive Decay Module} 
$G4RadioactiveDecay$ and associated classes are used to simulate the 
decay of radioactive nuclei by $\alpha$, $\beta^{+}$, and $\beta^{-}$ emission
and by electron capture (EC).  The simulation model is empirical and 
data-driven, and uses the Evaluated Nuclear Structure Data File 
(ENSDF)~\cite{rdk.ENSDF} for information on:

{\noindent \rm{}}

%\check
{\noindent \rm{}\LARGE$\cdot $\normalsize \indent nuclear half-lives,}

%\check
{\noindent \rm{}\LARGE$\cdot $\normalsize \indent
nuclear level structure for the parent or daughter nuclide,}

%\check
{\noindent \rm{}\LARGE$\cdot $\normalsize \indent
decay branching ratios, and}

%\check
{\noindent \rm{}\LARGE$\cdot $\normalsize \indent
the energy of the decay process.}

%\check
{\noindent \rm{}}

If the daughter of a nuclear decay is an excited isomer, its prompt nuclear
de-excitation is treated using the $G4PhotoEvaporation$ 
class~\cite{rdk.photevap}.

\section{Sampling}
Sampling of the $\beta$-spectrum, which includes the coordinated energies and 
momenta of the $\beta^{\pm}$, $\nu$, or $\bar{\nu}$ and residual nucleus, is 
performed either from histogrammed data, or through a three-body decay 
algorithm. In the latter case, the effect of the Coulomb barrier on the 
probability of $\beta^{\pm}$-emission can also be taken into account using the 
Fermi function:

\begin{equation}
F(E_0)=\frac {\gamma} {1-e^{-\gamma}}
\left \lgroup {\frac {Z^2(E_0 + 1)^2} {137^2} + \frac {E_0 ^2+2E_0} {4}} \right \rgroup ^
{\sqrt {1 - \frac {Z^2} {137^2}} - 1} .
\end{equation}
Here $E_0$ is the energy of the $\beta$-particle given as a fraction of the 
end-point energy, $Z$ is the atomic number of the nucleus, and $\gamma$ is 
given by the expression:
\par

\begin{equation}
\gamma = \frac {2\pi Z} {137} \frac {1+E_0} {\sqrt {E_0 ^2 + 2E_0}} .
\end{equation}

Due to the level of imprecision of the rest-mass energy of the nuclei generated
by $G4IonTable::GetNucleusMass$, the mass of the parent nucleus is 
modified to a minor extent just before performing the two- or three-body decay 
so that the $Q$ for the transition process equals that identified in the ENSDF 
data.

\subsection{Biasing Methods}
By default, sampling of the times of radioactive decay and branching ratios is 
done according to standard, analogue Monte Carlo modeling. The user may switch 
on one or more of the following variance reduction schemes, which can provide 
significant improvement in the modelling efficiency:

1. The decays can be biased to occur more frequently at certain times, for
example, corresponding to times when measurements are taken in a real 
experiment. The statistical weights of the daughter nuclides are reduced 
according to the probability of survival to the time of the event, $t$, which 
is determined from the decay rate. The decay rate of the $n^{th}$ nuclide in a 
decay chain is given by the recursive formulae:
\par

%\check
\begin{equation}
R_n (t) = \sum \limits_{i=1} \limits^{n-1} A_{n:i}f(t,\tau_i) +
A_{n:n}f(t,\tau_n)
\end{equation}

%\check
{\noindent \rm{}where:}
\par

%\check
\begin{equation}
\label{rdk.eq4}
A_{n:i} = \frac {\tau_i} {\tau_i-\tau_n} A_{n:i} \quad \forall i<n
\end{equation}
\begin{equation}
A_{n:n} = -\sum \limits_{i=1} \limits^{n-1} \frac{\tau_n} {\tau_i-\tau_n} A_{n:i} - y_n
\end{equation}
\begin{equation}
\label{rdk.eq6}
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' .
\end{equation}
The values $\tau_i$ are the mean life-times for the nuclei, $y_i$ is the 
yield of the $i^{th}$ nucleus, and $F(t)$ is a function identifying the time 
profile of the source.  The above expression for decay rate is simplified, 
since it assumes that the $i^{th}$ nucleus undergoes 100\% of the decays to the
$(i+1)^{th}$ nucleus.  Similar expressions which allow for branching and 
merging of different decay chains can be found in Ref.~\cite{rdk.Tru96}.

A consequence of the form of equations \ref{rdk.eq4} and \ref{rdk.eq6} is that 
the user may provide a source time profile so that each decay produced as a 
result of a simulated source particle incident at time $t=0$ is convolved over 
the source time profile to derive the actual decay rate for that source 
function.

This form of variance reduction is only appropriate if the radionuclei can be 
considered to be at rest with respect to the geometry when decay occurs.

2. For a given decay mode ($\alpha$, $\beta^++EC$, or $\beta^-$) the branching 
ratios to the daughter nuclide can be sampled with equal probability, so that 
some low probability branches which may have a disproportionately greater 
effect on the measurement are sampled with increased probability.

3. Each parent nuclide can be split into a user-defined number of nuclides 
(of proportionally lower statistical weight) prior to treating decay in order t
o increase the sampling of the effects of the daughter products.

\section{Status of this document}

00.00.00 created by ? \\
21.11.03 bibliography added, minor re-wording by D.H. Wright \\


\begin{latexonly}
\begin{thebibliography}{99}
\bibitem{rdk.ENSDF} J. Tuli, {\it "Evaluated Nuclear Structure Data File,"} 
BNL-NCS-51655-Rev87, 1987.
\bibitem{rdk.photevap} Chapter 25, Geant4 Physics Reference Manual.
\bibitem{rdk.Tru96} P.R. Truscott, PhD Thesis, University of London, 1996.
\end{thebibliography}
\end{latexonly}


\begin{htmlonly}
\section{Bibliography}
\begin{enumerate}
\item{rdk.ENSDF} J. Tuli, {\it "Evaluated Nuclear Structure Data File,"} 
BNL-NCS-51655-Rev87, 1987.
\item{rdk.photevap} Chapter 25, Geant4 Physics Reference Manual.
\item{rdk.Tru96} P.R. Truscott, PhD Thesis, University of London, 1996.
\end{enumerate}
\end{htmlonly}