Continuous slowing down

Most charged interactions emit one or more low-energy particles during their interaction. Instead of creating explicit daughter tracks that are immediately killed due to low energy, part of the interaction cross section is lumped into a “slowing down” term that continuously deposits energy locally over the step.

inline ParticleTrackView::Energy celeritas::calc_mean_energy_loss(ParticleTrackView const &particle, PhysicsTrackView const &physics, real_type step)

Calculate mean energy loss over the given “true” step length.

Stopping power is an integral over low-exiting-energy secondaries. Above some threshold energy T_c we treat exiting secondaries discretely; below it, we lump them into this continuous loss term that varies based on the energy, the atomic number density, and the element number:

\[ \frac{dE}{dx} = N_Z \int_0^{T_c} \frac{d \sigma_Z(E, T)}{dT} T dT \]

Here, the cross section is a function of the primary’s energy E and the exiting secondary energy T.

The stopping range R due to these low-energy processes is an integral over the inverse of the stopping power: basically the distance that will take a particle (without discrete processes at play) from its current energy to an energy of zero.

\[ R = \int_0 ^{E_0} - \frac{dx}{dE} dE . \]

Both Celeritas and Geant4 approximate the range limit as the minimum range over all processes, rather than the range as a result from integrating all energy loss processes over the allowed energy range. This is usually not a problem in practice because the range will get automatically decreased by range_to_step , and the above range calculation neglects energy loss by discrete processes.

Both Geant4 and Celeritas integrate the energy loss terms across processes to get a single energy loss vector per particle type. The range table is an integral of the mean stopping power: the total distance for the particle’s energy to reach zero.

Todo:

The GEANT3 manual

[Brun et al., 1993] makes the point that linear interpolation of energy loss rate results in a piecewise constant energy deposition curve, which is why they use spline interpolation. Investigate higher-order reconstruction of energy loss curve, e.g. through spline-based interpolation or log-log interpolation.

Zero energy loss can occur in the following cases:

• The energy loss value at the given energy is zero (e.g. high energy particles)

• The urban model is selected and samples zero collisions (possible in thin materials and/or small steps)

Note

The inverse range correction assumes range is always the integral of the stopping power/energy loss.

Note

See section 7.2.4 Run Time Energy Loss Computation of the Geant4 physics manual [The Geant4 Collaboration, 2023] . See also the longer discussions in section 8 of PHYS010 of the Geant3 manual.

Energy loss fluctuations

Since true energy loss is a stochastic function of many small collisions, the mean energy loss term is an approximation. Additional models are implemented to adjust the loss per step with stochastic sampling for improved accuracy.

class EnergyLossHelper

Precalculate energy loss fluctuation properties.

Fluctuations in the energy loss of charged particles over a given thickness of material arise from statistical variation in both the number of collisions and the energy lost in each collision. Above a given energy threshold, fluctuations are simulated through the explicit sampling of secondaries. However, the continuous energy loss below the cutoff energy also has fluctuations, and these are not taken into account in the calculation of the mean loss. For continuous energy loss, fluctuation models are used to sample the actual restricted energy loss given the mean loss.

Different models are used depending on the value of the parameter \( \kappa = \xi / T_{max} \), the ratio of the mean energy loss to the maximum allowed energy transfer in a single collision.

For large \( \kappa \), when the particle loses all or most of its energy along the step, the number of collisions is large and the straggling function can be approximated by a Gaussian distribution.

Otherwise, the Urban model for energy loss fluctuations in thin layers is used.

Note

This performs the same sampling routine as in Geant4’s G4UniversalFluctuation, as documented in section 7.3 in the Geant4 Physics Reference Manual [The Geant4 Collaboration, 2023] and in PHYS332 and PHYS333 in the GEANT3 manual [Brun et al., 1993] .

class EnergyLossGammaDistribution

Sample energy loss from a gamma distribution.

This model is valid for heavy particles with small mean losses (less than 2 Bohr standard deviations). It is a special case of EnergyLossGaussianDistribution (see that class for more documentation).

Note that while this appears in G4UniversalFluctuation, the Geant4 documentation does not explain why the loss is sampled from a gamma distribution in this case.

class EnergyLossGaussianDistribution

Sample energy loss from a Gaussian distribution.

In a thick absorber, the total energy transfer is a result of many small energy losses from a large number of collisions. The central limit theorem applies, and the energy loss fluctuations can be described by a Gaussian distribution. See section 7.3.1 of the Geant4 Physics Reference Manual and GEANT3 PHYS332 section 2.3.

The Gaussian approximation is valid for heavy particles and in the regime \( \kappa = \xi / T_\textrm{max} > 10 \). Fluctuations of the unrestricted energy loss follow a Gaussian distribution if \( \Delta E > \kappa T_{max} \), where \( T_{max} \) is the maximum energy transfer (PHYS332 section 2). For fluctuations of the restricted energy loss, the condition is modified to \( \Delta E > \kappa T_{c} \) and \( T_{max} \le 2 T_c \), where \( T_c \) is the delta ray cutoff energy (PRM Eq. 7.6-7.7).

class EnergyLossUrbanDistribution

Sample from the Urban model of energy loss fluctuations in thin layers.

The Urban model is used to compute the energy loss fluctuation when \( \kappa \) is small. It assumes atoms have only two energy levels with binding energies \( E_1 \) and \( E_2 \) and that the particle-atom interaction will either be an excitation with energy loss \( E_1 \) or \( E_2 \) or an ionization with energy loss proportional to \( 1 / E^2 \). The number of collisions for each interaction type has a Poisson distribution with mean proportional to the interaction cross section. The energy loss in a step will be the sum of the energy loss contributions from excitation and ionization.

When the number of ionizations is larger than a given threshold, a fast sampling method can be used instead. The possible energy loss interval is divided into two parts: a lower range in which the number of collisions is large and the energy loss can be sampled from a Gaussian distribution, and an upper range in which the energy loss is sampled for each collision.

See section 7.3.2 of the Geant4 Physics Reference Manual and GEANT3 PHYS332 section 2.4 for details.