The Geant4 Binary Cascade is an intranuclear cascade propagating primary and secondary particles in a nucleus. Interactions are between a primary or secondary particle and an individual nucleon of the nucleus, leading to the name Binary Cascade. Cross section data are used to select collisions. Where available, experimental cross sections are used by the simulation. Propagating of particles is the nuclear field is done by numerically solving the equation of motion. The cascade terminates when the average and maximum energy of secondaries is below threshold. The remaining fragment is treated by precompound and de-excitation models documented in the corresponding chapters.
For the primary particle an impact parameter is chosen random in a disk outside the nucleus perpendicular to a vector passing through the center of the nucleus coordinate system an being parallel to the momentum direction. Using a straight line trajectory, the distance of closest approach to each target nucleon and the corresponding time-of-flight is calculated. In this calculation the momentum of the target nucleons is ignored, i.e. the target nucleons do not move. The interaction cross section with target nucleons is calculated using total inclusive cross-sections described below. For calculation of the cross-section the momenta of the nucleons are taken into account. The primary particle may interact with those target nucleons where the distance of closest approach is smaller than . These candidate interactions are called collisions, and these collisions are stored ordered by time-of-flight . In the case no collision is found, a new impact parameter is chosen.
The primary particle is tracked the time-step given by the time to the first collision. As long a particle is outside the nucleus, that is a radius of the outermost nucleon plus , particles travel along straight line trajectories. Particles entering the nucleus have their energy corrected for Coulomb effects. Inside the nucleus particles are propagated in the scalar nuclear field. The equation of motion in the field is solved for a given time-step using a Runge-Kutta integration method.
At the end of the step, the primary and the nucleon interact suing the scattering term. The resulting secondaries are checked for the Fermi exclusion principle. If any of the two particles has a momentum below Fermi momentum, the interaction is suppressed, and the original primary is tracked to the next collision. In case interaction is allowed, the secondaries are treated like the primary, that is, all possible collisions are calculated like above, with the addition that these new primary particles may be short-lived and may decay. A decay is treated like others collisions, the collision time being the time until the decay of the particle. All secondaries are tracked until they leave the nucleus, or the until the cascade stops.
The nucleus is constructed from nucleons and protons with nucleon coordinates and momenta , with . We use a common initialization Monte Carlo procedure, which is realized in the most of the high energy nuclear interaction models:
This procedure gives special for hydrogen H, where the proton has momentum , and for deuterium H, where the momenta of proton and neutron are equal, and in opposite direction.
The effect of collective nuclear elastic interaction upon primary and secondary particles is approximated by a nuclear potential.
For projectile protons and neutrons this scalar potential
is given by the local Fermi momentum
For pions the potential is given by the lowest order optical potential 
The cross sections used in this model are cross sections for free particles. In the nucleus these cross sections are reduced to effective cross sections by Pauli-blocking due to Fermi statistics.
For nucleons created by a collision, ie. an inelastic
scattering or from decay, we check that all secondary nucleons occupy a state
allowed by Fermi statistics. We assume that the nucleus in its ground state and
all states below Fermi energy are occupied. All secondary nucleons therefore
must have a momentum above local Fermi momentum , i.e.
If any of the nucleons of the collision has a momentum below the local Fermi momentum, then the collision is Pauli blocked. The reaction products are discarded, and the original particles continue the cascade.
Where and are the spins of the two fusing particles, is the spin of the resonance, the energy in the center of mass system, the momentum of the fusing particles in the center of mass system, and the partial width of the resonance for the initial and final state respectively. is the nominal mass of the resonance.
The initial states included in the model are pion and kaon nucleon scattering. The product resonances taken into account are the Delta resonances with masses 1232, 1600, 1620, 1700, 1900, 1905, 1910, 1920, 1930, and 1950 MeV, the excited nucleons with masses of 1440, 1520, 1535, 1650, 1675, 1680, 1700, 1710, 1720, 1900, 1990, 2090, 2190, 2220, and 2250 MeV, the Lambda, and its excited states at 1520, 1600, 1670, 1690, 1800, 1810, 1820, 1830, 1890, 2100, and 2110 MeV, and the Sigma and its excited states at 1660, 1670, 1750, 1775, 1915, 1940, and 2030 MeV.
The parameters of the description for the various channels
are given in table25.1. For all other channels, the
parametrizations were derived from these by adjusting the threshold
The reminder of the cross-section are
derived from these, applying detailed balance. Iso-spin invariance
is assumed. The formalism used to apply detailed balance is
Final states are derived by sampling from tables of the cumulative distribution function of the centre-of-mass scattering angle, tabulated for a discrete set of lab kinetic energies from 10 MeV to 1200 MeV. The CDF's are tabulated at 1 degree intervals and sampling is done using bi-linear interpolation in energy and CDF values. Coulomb effects are taken into consideration for pp scattering.
Angular distributions for final states other than nucleon elastic scattering are calculated analytically, derived from the collision term of the in-medium relativistic Boltzmann-Uehling-Uhlenbeck equation, absed on the nucleon nucleon elastic scattering cross-sections:
Here , , are the Mandelstamm variables,
is the direct term, and is the exchange
Finite size effects were taken into account at the meson nucleon vertex, using a phenomenological form factor (cut-off) at each vertex.
All decay channels with nominal branching ratios greater than 1% are simulated.
When a nucleon other than the incident particle leaves the nucleus, the ground state of the nucleus changes. The energy of the outgoing particle cannot be such that the total mass of the new nucleus would be below its ground state mass. To avoid this, we reduce the energy of an outgoing nucleons by the mass-difference of old and new nucleus.
Furthermore, the momentum of the final exited nucleus derived from energy momentum balance may be such that its mass is below its ground state mass. In this case, we arbitrarily scale the momenta of all outgoing particles by a factor derived from the mass of the nucleus and the mass of the system of outgoing particles.
The lighter of the collision partners is selected to be the projectile. The nucleons in the projectile are then entered, with position and momenta, into the initial state of the cascade. Note that before the first scattering of an individual nucleon, a projectile nucleon's Fermi-momentum is not taken into account in the tracking inside the target nucleus. The nucleon distribution inside the projectile nucleus is taken to be a representative distribution of its nucleons in configuration space, rather than an initial state in the sense of QMD. The Fermi momentum and the local field are taken into account in the calculation of the collision probabilities and final states of the binary collisions.
For this first release, the following algorithm is used to determine when cascading is stopped, and pre-equilibrium decay is called: As long as there are still particles above the kinetic energy threshold (75 MeV), cascading will continue. Otherwise, when the mean kinetic energy of the participants has dropped below a second threshold (15 MeV), the cascading is stopped.
The residual participants, and the nucleus in its current state are then used to define the initial state, i.e. excitation energy, number of excitons, number of holes, and momentum of the exciton system, for pre-equilibrium decay.
In the case of light ion reactions, the projectile excitation is determined
from the binary collision participants () using the statistical approach
towards excitation energy calculation in an adiabatic abrasion process, as
described in :
Given this excitation energy, the projectile fragment is then treated by the evaporation models described previously.
At the end of the cascade, we form a fragment for further treatment in precompound and nuclear de-excitation models ().
These models need information about the nuclear fragment created by the cascade. The fragment is characterized by the number of nucleons in the fragment, the charge of the fragment, the number of holes, the number of all excitons, and the number of charged excitons, and the four momentum of the fragment.
The number of holes is given by the difference of the number of nucleons in the original nucleus and the number of non-excited nucleons left in the fragment. An exciton is a nucleon captured in the fragment at the end of the cascade.
The momentum of the fragment calculated by the difference between the momentum of the primary and the outgoing secondary particles must be split in two components. The first is the momentum acquired by coherent elastic effects, and the second is the momentum of the excitons in the nucleus rest frame. Only the later part is passed to the de-excitation models. Secondaries arising from de-excitation models, including the final nucleus, are transformed back the frame of the moving fragment.