Conversion from Cut in Range to Energy Threshold

In Geant4 charged particles are tracked to the end of their range. The differential cross section of \(\delta\)-electron productions and bremsstrahlung grow rapidly when secondary energy decrease. If all secondary particles will be tracked the CPU performance of any Monte Carlo code will be poor. The traditional solution is to use cuts. The specific of Geant4 [eal03] is that user provides value of cut in term of cut in range, which is unique for defined G4Region or for the complete geometry [eal16].

Range is used, rather than energy, as a more natural concept for designing a coherent policy for different particles and materials. Definition of the certain value of the cut in range means the requirement for precision of spatial radioactive dose deposition. This conception is more strict for a simulation code and provides fewer handles for user to modify final results. At the same time, it ensures that simulation validated in one geometry is valid also for the other geometries.

The value of cut is defined for electrons, positrons, gamma and protons. At the beginning of initialization of Geant4 physics the conversion is performed from unique cut in range to cuts (production thresholds) in kinetic energy for each G4MaterialCutsCouple [eal16]. At that moment no energy loss and range of electron and positron are computed using simplified expressions extracted from cBerger-Seltzer energy loss formula ((126)). Absolute accuracy of the estimated range is not needed, so the density correction term is omitted. The contribution of the bremsstrahlung is added using empirical parameterized formula. For \(T < 10\) keV the linear dependence of ionisation losses on electron velocity is assumed, bremsstrahlung contribution is neglected. The stopping range is defined as

\[R(T)= \int_0^T \frac{1}{(dE/dx)} \, dE .\]

The integration has been done analytically for the low energy part and numerically above an energy limit 1 keV. For each cut in range the corresponding kinetic energy can be found out. If obtained production threshold in kinetic energy cannot be below the parameter lowlimit (default 1 keV) and above highlimit (default 10 GeV). If in specific application lower threshold is required, then the allowed energy cut needs to be extended:

G4ProductionCutsTable::GetProductionCutsTable()->SetEnergyRange(lowlimit,highlimit);

or via UI commands:

/cuts/setMinCutEnergy 100 eV
/cuts/setMaxCutEnergy 100 TeV

In contrary to electrons, gammas have no range, so some approximation should be used for range to energy conversion. An approximate empirical formula is used to compute the absorption cross section of a photon in an element \(\sigma_{abs}\). Here, the absorption cross section means the sum of the cross sections of the gamma conversion, Compton scattering and photoelectric effect. These processes are the “destructive” processes for photons: they destroy the photon or decrease its energy. The coherent or Rayleigh scattering changes the direction of the gamma only; its cross section is not included in the absorption cross section. The AbsorptionLength \(L_{abs}\) vector is calculated for every material as

\[L_{abs} = 5/\sigma_{abs}.\]

The factor 5 comes from the requirement that the probability of having no ‘destructive’ interaction should be small, hence

\[\exp(-\mbox{$L_{abs} \sigma_{abs}$}) = \exp(-5) = 6.7 \times 10^{-3}.\]

The photon cross section for a material has a minimum at a certain energy \(E_{min}\). Correspondingly \(L_{abs}\) has a maximum at \(E = E_{min}\), the value of the maximal \(L_{abs}\) is the biggest “meaningful” cut in absorption length. If the cut given by the user is bigger than this maximum, a warning is printed and the cut in kinetic energy is set to the highlimit.

The cut for proton is introduced with Geant4 v9.3. The main goal of this cut is to limit production of all recoil ions including protons in elastic scattering processes. A simple linear conversion formula is used to compute energy threshold from the value of cut in range, in particular, the cut in range 1 mm corresponds to the production threshold 100 keV.

The conversion from range to energy can be studied using G4EmCalculator class. This class allows access or recalculation of energy loss, ranges and other values. It can be instantiated and at any place of user code and can be used after initialisation of Physics Lists:

G4EmCalculator calc;
calc.ComputeEnergyCutFromRangeCut(range, particle, material);

here particle and material may be string names or corresponding const pointers to G4ParticleDefinition and G4Material.

Bibliography

eal16(1,2)

J. Allison et al. Recent developments in geant4. Nucl. Instr. and Meth. in Phys. Research Section A, 835:186–225, nov 2016. URL: https://doi.org/10.1016/j.nima.2016.06.125, doi:10.1016/j.nima.2016.06.125.

eal03

S. Agostinelli et al. Geant4—a simulation toolkit. Nucl. Instr. and Meth. in Phys. Research Section A, 506(3):250–303, jul 2003. URL: https://doi.org/10.1016/S0168-9002(03)01368-8, doi:10.1016/s0168-9002(03)01368-8.