The Geant4 Cascade Model

Inelastic particle-nucleus collisions are characterized by both fast and slow components. The fast ( $10^{-23} - 10^{-22} s$) intra-nuclear cascade results in a highly excited nucleus which may decay by fission or pre-equilibrium emission. The slower ( $10^{-18} - 10^{-16} s$) compound nucleus phase follows with evaporation. A Boltzmann equation must be solved to treat the collision process in detail.

The intranuclear cascade (INC) model developed by Bertini [3,4] solves the Boltzmann equation on average. This model has been implemented in several codes such as HETC [1]. Our model, which is based on a re-engineering of the INUCL code [20], includes the Bertini intranuclear cascade model with excitons, a pre-equilibrium model, a simple nucleus explosion model, a fission model, and an evaporation model.

The target nucleus is modeled as a three-region approximation to the continuously changing density distribution of nuclear matter within nuclei. The cascade begins when the incident particle strikes a nucleon in the target nucleus and produces secondaries. The secondaries may in turn interact with other nucleons or be absorbed. The cascade ends when all particles, which are kinematically able to do so, escape the nucleus. At that point energy conservation is checked. Relativistic kinematics is applied throughout the cascade.

Model Limits

The model is valid for incident protons, neutrons and pions. Particles treated in the model include protons, neutrons, pions, photons and nuclear isotopes. All types of targets are allowed.

The necessary condition of validity of the INC model is $\lambda_{B} / v << \tau_{c} << \Delta t$, where $\delta_{B}$ is the deBroglie wavelenth of the nucleons, $v$ is the average relative velocity between two nucleons and $\Delta t$ is the time interval between collisions. At energies below $200 MeV$, this condition is no longer strictly valid, and a pre-quilibrium model must be invoked. At energies greater than $\approx$ 10 GeV) the INC picture breaks down. This model has been tested against experimental data at incident kinetic energies between 100 MeV and 5 GeV.

Intranuclear Cascade Model

The basic steps of the INC model are summarized as follows:

1. the space point at which the incident particle enters the nucleus is selected uniformly over the projected area of the nucleus,
2. the total particle-particle cross sections and region-depenent nucleon densities are used to select a path length for the projectile,
3. the momentum of the struck nucleon, the type of reaction and the four-momenta of the reaction products are determined, and
4. the exciton model is updated as the cascade proceeds.
5. If the Pauli exclusion principle allows and $E_{particle} > E_{cutoff}$ = 2 MeV, step (2) is performed to transport the products.

After the intra-nuclear cascade, the residual excitation energy of the resulting nucleus is used as input for non-equilibrium model.

Nuclear Model

Some of the basic features of the nuclear model are:

• the nucleons are assumed to have a Fermi gas momentum distribution. The Fermi energy is calculated in a local density approximation i.e. the Fermi energy is made radius-dependent with Fermi momentum $p_{F}(r) = (\frac{3 \pi^2 \rho(r)}{2})^\frac{1}{3}$.
• Nucleon binding energies (BE) are calculated using the mass formula. A parameterization of the nuclear binding energy uses a combination of the Kummel mass formula and experimental data. Also, the asymptotic high temperature mass formula is used if it is impossible to use experimental data.

Initialization

The initialization phase fixes the nuclear radius and momentum according to the Fermi gas model.

If the target is hydrogen (A = 1) a direct particle-particle collision is performed, and no nuclear modeling is required.

If $1 < A < 4$, a nuclear model consisting of one layer with a radius of 8.0 fm is created.

For $4 < A < 11$, the nuclear model is composed of three concentric spheres $i = \{1, 2, 3\}$ with radius

$$r_{i}(\alpha_{i}) = \sqrt{C_{1}^{2} (1 - \frac{1}{A}) + 6.4} \sqrt{-log( \alpha_{i})}$$

.

Here $\alpha_{i} = \{0.01, 0.3, 0.7\}$ and $C_{1} = 3.3836 A^{1/3}$.

If $A > 11$, a nuclear model with three concentric spheres is also used. The sphere radius is now defined as

$$r_{i}(\alpha_{i}) = C_{2} \log({\frac{1 + e^{- \frac{C_{1}}{C_{2}}}}{\alpha_{i}} - 1}) + C_{1} ,$$ (24.1)

where $C_{2} = 1.7234$.

The potential energy $V$ for nucleon $N$ is

$$V_{N} = \frac{p_{F}^2}{2 m_{N}} + BE_{N}(A, Z) ,$$ (24.2)

where $p_f$ is the Fermi momentum and $BE$ is the binding energy.

The momentum distribution in each region follows the Fermi distribution with zero temperature.

$$f(p) = c p ^2$$ (24.3)

where

$$\int_0^{p_F} f(p) dp = n_{p} \rm { or } n_{n}$$ (24.4)

where $n_p$ and $n_n$ are the number of protons or neutrons in the region. $P_f$ is the momentum corresponding to the Fermi energy

$$E_f = \frac{p_F^2}{2 m_N} = \frac{\hbar^2}{2 m_N}(\frac{3 \pi^{2}}{v})^\frac{2}{3} ,$$ (24.5)

which depends on the density $n/v$ of particles, and which is different for each particle and each region.

Pauli Exclusion Principle

The Pauli exclusion principle forbids interactions where the products would be in occupied states. Following the assumption of a completely degenerate Fermi gas, the levels are filled from the lowest level. The minimum energy allowed for the products of a collision correspond to the lowest unfilled level of the system, which is the Fermi energy in the region. So in practice, the Pauli exclusion principle is taken into account by accepting only secondary nucleons which have $E_N > E_f$.

Cross Sections and Kinematics

Path lengths of nucleons in the nucleus are sampled according to the local density and the free $N-N$ cross sections. Angles after the collision are sampled from experimental differential cross sections. Tabulated total reaction cross sections are calculated by Letaw's formulation [14,15,17]. For $N-N$ cross sections the parameterizations are based on the experimental energy and isospin dependent data. The parameterization described in [2] is used.

For pions the intra-nuclear cross sections are provided to treat elastic collisions and the following inelastic channels: $\pi^{-}$p $\rightarrow$ $\pi^{0}$n, $\pi^{0}$p $\rightarrow$ $\pi^{+}$n, $\pi^{0}$n $\rightarrow$ $\pi^{-}$p, and $\pi^+$n $\rightarrow$ $\pi^0$p. Multiple particle production is also implemented.

The pion absorption channels are $\pi^{+}$nn $\rightarrow$ pn, $\pi^{+}$pn $\rightarrow$ pp, $\pi^{0}$nn $\rightarrow$ nn, $\pi^{0}$pn $\rightarrow$ pn, $\pi^{0}$pp $\rightarrow$ pp, $\pi^{-}$pn $\rightarrow$ nn , and $\pi^{-}$pp $\rightarrow$ pn.

Pre-equilibrium Model

The GEANT4 cascade model implements the exciton model proposed by Griffin [10,11]. In this model, nucleon states are characterized by the number of excited particles and holes (the excitons). Intra-nuclear cascade collisions give rise to a sequence of states characterized by increasing exciton number, eventually leading to an equilibrated nucleus. For a practical implementation of the exciton model we use parameters from [18], (level densities) and [13] (matrix elements).

In the exciton model the possible selection rules for particle-hole configurations in the source of the cascade are: $\Delta p = 0, \pm 1$ $\Delta h = 0, \pm 1$ $\Delta n = 0, \pm 2$, where $p$ is the number of particles, $h$ is number of holes and $n = p + h$ is the number of excitons.

The cascade pre-equilibrium model uses target excitation data and the exciton configurations for neutrons and protons to produce non-equilibrium evaporation. The angular distribution is isotropic in the rest frame of the exciton system.

Parameterizations of the level density are tabulated as functions of $A$ and $Z$, and with high temperature behavior (the nuclear binding energy using the smooth liquid high energy formula).

Break-up models

Fermi break-up is allowed only in some extreme cases, i.e. for light nuclei ($A < 12$ and $3 (A - Z) < Z < 6$ ) and $E_{excitation} > 3 E_{binding}$. A simple explosion model decays the nucleus into neutrons and protons and decreases exotic evaporation processes.

The fission model is phenomenological, using potential minimization. A binding energy paramerization is used and some features of the fission statistical model are incorporated [7].

Evaporation Model

A statistical theory for particle emission of the excited nucleus remaining after the intra-nuclear cascade was originally developed by Weisskopf [21]. This model assumes complete energy equilibration before particle emission, and re-equilibration of excitation energies between successive evaporations. As a result the angular distribution of emitted particles is isotropic.

The GEANT4 evaporation model for the cascade implementation adapts the often-used computational method developed by Dostrowski [5,6]. The emission of particles is computed until the excitation energy falls below some specific cutoff. If a light nucleus is highly excited, the Fermi break-up model is executed. Also, fission is performed if that channel is open. The main chain of evaporation is followed until $E_{excitation}$ falls below E$_{cutoff}$ = 0.1 MeV. The evaporation model ends with an emission chain which is followed until $E_{excitation} < E^{\gamma}_{cutoff} = 10^{-15}$ MeV.