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:

$$F(E_0)=\frac {\gamma} {1-e^{-\gamma}} \left \lgroup {\frac {... ... {4}} \right \rgroup ^ {\sqrt {1 - \frac {Z^2} {137^2}} - 1} .$$ (34.1)

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:

$$\gamma = \frac {2\pi Z} {137} \frac {1+E_0} {\sqrt {E_0 ^2 + 2E_0}} .$$ (34.2)

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.

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:

$$R_n (t) = \sum \limits_{i=1} \limits^{n-1} A_{n:i}f(t,\tau_i) + A_{n:n}f(t,\tau_n)$$ (34.3)

where:

$$A_{n:i} = \frac {\tau_i} {\tau_i-\tau_n} A_{n:i} \quad \forall i<n$$ (34.4)

$$A_{n:n} = -\sum \limits_{i=1} \limits^{n-1} \frac{\tau_n} {\tau_i-\tau_n} A_{n:i} - y_n$$ (34.5)

$$f(t,\tau_i)= \frac {e^{-\frac{t}{\tau_i}}} {\tau_i} \int \limits_{-\inf} \limits^t F(t')e^{\frac{t'}{\tau_i}}dt' .$$ (34.6)

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. [3].

A consequence of the form of equations 34.4 and 34.6 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.