Methods
How RadMap calculates a dose rate: what it hands the simulation, what the simulation does with it, and how a dose rate is made of the result. The user guide says how to set up a calculation; this part says what happens to it. What RadMap does not calculate, or calculates only approximately, is collected in the user guide under Known limitations.
Overview
A calculation runs in two programs. RadMap holds the project: the geometry, the materials, the sources and the dose maps. When a simulation starts, it checks the project, writes it out as a geometry description and a list of commands, and starts the simulation program, which comes with RadMap. That program uses Geant4, the Monte Carlo radiation transport toolkit developed by an international collaboration led from CERN, to simulate radioactive decays one at a time and follow the radiation they emit through the geometry. It scores the photons inside each dose map and returns, for each map, the mean score per decay and its statistical uncertainty. RadMap multiplies these by the activity of the sources and reports a dose rate.
RadMap uses Geant4 11.4, built from source with patches of RadMap’s own to how decay positions are sampled (see Sources), and the Geant4 data sets of that release:
Data set |
Version |
Holds |
|---|---|---|
G4EMLOW |
8.8 |
Photon and electron interaction data |
RadioactiveDecay |
6.1.2 |
Decay modes, branching ratios and beta spectra |
PhotonEvaporation |
6.1.2 |
Nuclear levels and gamma transitions |
G4ENSDFSTATE |
3.0 |
Nuclear ground states and half-lives |
Geometry and materials
Each solid – a box, a tube or a sphere, including shells and segments – is passed to Geant4 with its dimensions, its position and rotation in its parent, and its material. A solid placed inside another displaces the parent’s material where it sits, so the tree of solids describes the geometry exactly as it is drawn. Before any decay is simulated, Geant4 checks every solid against its siblings and its parent, and the run is refused if any overlap or stick out: a point may belong to one solid only.
A material is passed as its density and composition: elements by mass fraction or by number of atoms, or other materials by mass fraction. An element is either one of natural isotopic composition or one defined by its isotopes, as the material library holds it. Geant4 computes all interaction cross sections from the composition and density; RadMap passes no cross sections of its own.
Sources
A source is a radionuclide with an activity, spread through the volume of a solid.
Activity. An activity given per volume or per mass is multiplied by the volume of the source’s solid, less the solids placed inside it, and for per mass by its material’s density. The activities of all sources add up to the total activity of the project.
Choosing a decay. Each history of the simulation is one decay: one primary particle, as the simulation settings count them. The source it belongs to is drawn at random, each source with a probability equal to its share of the total activity.
Where it decays. The position is drawn uniformly in a box around the source’s solid and kept only if it lies in the solid and not in a solid placed inside it; otherwise it is drawn again. The positions are thus uniform through the source’s volume, right up to its surfaces. A volume too thin to sample this way stops the run with a message naming the solid. Sampling right up to the surfaces, and stopping on a volume too thin to sample, are RadMap’s patches to Geant4’s source sampling.
The decay. The nucleus decays at rest, by Geant4’s radioactive decay module, from the data sets above: its decay mode and branch, the beta spectrum, and the gamma transitions, conversion electrons and X-rays of the daughter’s de-excitation, all emitted isotropically. A daughter left in an excited or metastable state de-excites in the same history, however long the state lives; so Cs-137 gives the 662 keV photons of Ba-137m. A daughter in its ground state is dropped, unless Full chain is on for the source: it then decays too, and so on down the chain, each member once per decay of the source. Time plays no part, so a chain is counted as in secular equilibrium. Whether a history follows the chain is decided by the isotope it starts from, never by a daughter’s, so sources of different isotopes whose chains meet keep their own settings.
What the decay emits is followed as described in Transport: photons and positrons always, electrons only when asked for. Alpha particles and neutrinos are dropped at once.
Transport
Particles are followed through the geometry with Geant4’s standard
electromagnetic physics, option 3 (G4EmStandardPhysics_option3),
which is meant for accurate low-energy work. For photons it
simulates:
the photoelectric effect, with Livermore cross sections and the fluorescence X-rays of the atom left behind;
Compton scattering, with the Klein–Nishina model and the effects of the atomic shells the electron is bound in;
Rayleigh (coherent) scattering, with Livermore data;
pair production, above 1.022 MeV.
Electrons, from decay and from photon interactions, are dropped where they are created unless Simulate electrons is on. The dose RadMap reports is photon dose, so this costs only the photons electrons would produce: their bremsstrahlung, which matters for energetic beta emitters in dense shielding, and the X-rays of the atoms they ionise. With Simulate electrons on they are followed until they stop, with multiple scattering, ionisation and bremsstrahlung.
Positrons are always followed, and annihilate, in flight or at rest, into photons that are followed in turn. Neutrons, such as those of spontaneous fission, are not given any interactions: they leave the geometry without effect.
Secondary particles are produced only above Geant4’s production thresholds, set by a range of 1 mm in each material; below it their energy is deposited locally. Photons are followed down to the lowest energies Geant4 tracks them, well below the 10 keV where the conversion coefficients begin.
Scoring
A dose map is a box placed in a parallel geometry: it shares positions with the solids but not their materials, so it does not affect the transport, and it may overlap solids and other dose maps. Each map takes one of Geant4’s parallel geometries, of which a run can have 15 besides the geometry itself.
Fluence. Each step a photon takes inside the box adds
where \(h(E)\) is the conversion coefficient at the photon’s energy, \(\ell\) the length of the step and \(V\) the volume of the box. The fluence at a point is the number of photons crossing a small sphere there, divided by the sphere’s cross-section; averaged over a volume, it equals the total length of the photons’ paths in that volume divided by the volume. So \(\ell/V\) is the step’s share of the fluence, which does not depend on the photon’s energy; \(h(E)\), the dose per unit fluence at that energy, converts the share to dose. The sum is therefore the dose from the fluence averaged over the box’s volume, whatever the field and however the box is turned. Every photon in the box counts: those that pass through it, those scattered in it and those emitted inside it.
A mean over a volume, not over a surface. Where the fluence is the same throughout the box, its mean is the fluence at any point in it. Where it changes across the box – near a source, at the edge of a shield’s shadow – the mean is not the value at the box’s centre, nor the value at the faces the photons enter by. In the field of a point source, a cube of side \(s\) whose centre is at a distance \(d\) reads about \(1 + s^2/12d^2\) times the fluence at its centre, in any orientation: 0.75 % more for a 30 cm cube at 1 m. A thin plate facing the source reads the mean over its face, slightly less than at its centre. RadMap’s versions before 2.0 scored the photons entering the box instead, divided by the area of the box seen from their direction. For a thin plate that is the same thing, but a deep box read nearer the fluence at the faces the photons entered by, and its reading changed when it was turned (defect D-0005 on the errata page).
Conversion coefficients. \(h(E)\) comes from the publication the project uses for the map’s quantity (Dose → Dose settings; the user guide’s Origin of conversion data lists them). Between the tabulated energies it is interpolated with a natural cubic spline. Below the lowest tabulated energy, 10 keV, effective dose and H*(10) take zero and air kerma takes its 10 keV value; above the highest, all three keep their last tabulated value.
Dose rate. The scores of all histories are summed and divided by the number of primary particles simulated, \(N\), which gives the dose per decay. Multiplied by the total activity, in decays per second, and by 3600 seconds per hour, it gives the dose rate: in Sv/h for effective dose and H*(10), in Gy/h for air kerma.
Statistics and reproducibility
The scores of one history are independent of those of another. The uncertainty RadMap reports is therefore the standard deviation of the mean over histories,
with \(x_i\) the score of history \(i\) and \(N\) the number of primary particles, scaled to a dose rate as the mean is. It is the statistical uncertainty of the Monte Carlo estimate only; the uncertainty of the data and of the model the user built is not in it. It falls as \(1/\sqrt{N}\).
Each run draws its random numbers from a new seed and uses all the processor’s cores. Two runs of the same project therefore differ within their uncertainties. With a fixed seed on one thread, a run is repeated exactly; RadMap’s tests run so. On several threads the result can differ in its last digits from run to run, because the threads’ sums are added in the order the threads finish.
Verification
The V&V report published with each version checks these methods against published data and measurements: the dose from point sources against the published coefficients and decay data, attenuation and buildup in shielding, the sampling of sources, interpolation, and air kerma against measured experiments. It is generated from the tests run on that version’s build.