The Radioactive Decay Module

G4RadioactiveDecay and associated classes are used to simulate the decay, either in-flight or at rest, of radioactive nuclei by \(\alpha\), \(\beta^{+}\), and \(\beta^{-}\) emission and by electron capture (EC). The simulation model depends on data taken from the Evaluated Nuclear Structure Data File (ENSDF) [Tul96] which provides details on the following properties (with the corresponding Geant4 library storing that property shown in brackets):

  • alpha, beta and internal transition half-lives (G4ENSDFSTATE)

  • nuclear level structure for the parent or daughter nuclide (PhotonEvaporation, G4ENSDFSTATE, RadioactiveDecay)

  • decay branching ratios (RadioactiveDecay)

  • the energy of the decay process (RadioactiveDecay)

If the daughter of a nuclear decay is an excited isomer, its prompt nuclear de-excitation is treated using the G4PhotoEvaporation class (see Section Photon evaporation).

Alpha Decay

The final state of alpha decay consists of an \(\alpha\) and a recoil nucleus with \((Z-2,A-4)\). The two particles are emitted back-to-back in the center of mass with the energy of the \(\alpha\) taken from the ENSDF data entry for the decaying isotope.

Beta Decay

Beta decay is modeled by the emission of a \(\beta^-\) or \(\beta^+\), an anti-neutrino or neutrino, and a recoil nucleus of either \(Z+1\) or \(Z-1\). The energy of the \(\beta\) is obtained by sampling either from histogrammed data or from the theoretical three-body phase space spectral shapes. The latter include allowed, first, second and third unique forbidden, and first non-unique forbidden transitions.

The shape of the energy spectrum of the emitted lepton is given by

\[\frac {d^2 n} {dE dp_e} = (E_0 - E_e)^2 E_e p_e F(Z,E_e) S(Z,E_0,E_e)\]

where, in units of electron mass, \(E_0\) is the endpoint energy of the decay taken from the ENSDF data, \(E_e\) and \(p_e\) are the emitted electron energy and momentum, \(Z\) is the atomic number, \(F\) is the Fermi function and \(S\) is the shape factor.

The Fermi function \(F\) accounts for the effect of the Coulomb barrier on the probability of \(\beta^{\pm}\) emission. Its relativistic form is

\[F(Z,E_e)= 2(1+\gamma)(2p_e R)^{2\gamma-2} e^{\pm \pi \alpha Z E_e/p_e} \frac { | \Gamma (\gamma + i \alpha Z E_e/p_e) | ^2} {\Gamma(2\gamma + 1)^2}\]

where \(R\) is the nuclear radius, \(\gamma = \sqrt{1 - (\alpha Z)^2}\), and \(\alpha\) is the fine structure constant. The squared modulus of \(\Gamma\) is computed using approximation B of Wilkinson [Wil70].

The factor \(S\) determines whether or not additional corrections are applied to the decay spectrum. When \(S = 1\) the decay spectrum takes on the so-called allowed shape which is just the phase space shape modified by the Fermi function. For this type of transition the emitted lepton carries no angular momentum and the nuclear spin and parity do not change. When the emitted lepton carries angular momentum and nuclear size effects are not negligible, the factor \(S\) is no longer unity and the transitions are called “forbidden”. Corrections are then made to the spectrum shape which take into account the energy dependence of the nuclear matrix element. The form of \(S\) used in the spectrum sampling is that of Konopinski [Kon66].

Electron Capture

Electron capture from the atomic K, L and M shells is simulated by producing a recoil nucleus of \((Z-1, A)\) and an electron-neutrino back-to-back in the center of mass. Since this leaves a vacancy in the electron orbitals, the atomic relaxation model (ARM) is triggered in order to produce the resulting x-rays and Auger electrons. More information on the ARM can be found in the Electromagnetic section of this manual.

In the electron capture decay mode, internal conversion is also enabled so that atomic electrons may be ejected when interacting with the nucleus.

Recoil Nucleus Correction

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:

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

    where:

    (282)\[A_{n:i} = \frac {\tau_i} {\tau_i-\tau_n} A_{n:i} \quad \forall i<n\]
    \[A_{n:n} = -\sum \limits_{i=1} \limits^{n-1} \frac{\tau_n} {\tau_i-\tau_n} A_{n:i} - y_n\]
    (283)\[f(t,\tau_i)= \frac {e^{-\frac{t}{\tau_i}}} {\tau_i} \int \limits_{-\inf} \limits^t F(t')e^{\frac{t'}{\tau_i}}dt' .\]

    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. [Tru96].

    A consequence of the form of equations (282) and (283) 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 to increase the sampling of the effects of the daughter products.

Bibliography

Kon66

E. Konopinski. The Theory of Beta Radioactivity. Oxford Press, 1966.

Tru96

P.R. Truscott. PhD Thesis. University of London, 1996.

Tul96

J.K. Tuli. Evaluated nuclear structure data file. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 369(2-3):506–510, Feb 1996. BNL-NCS-51655-Rev87, (1987): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.459.3917&rep=rep1&type=pdf, online database: http://www.nndc.bnl.gov/nudat2/. URL: https://doi.org/10.1016/S0168-9002(96)80040-4, doi:10.1016/s0168-9002(96)80040-4.

Wil70

D.H. Wilkinson. Evaluation of the fermi function; EO competition. Nuclear Instruments and Methods, 82:122–124, May 1970. URL: https://doi.org/10.1016/0029-554X(70)90336-8, doi:10.1016/0029-554x(70)90336-8.