2. GEANT4: A MONTE CARLO SIMULATION TOOLKIT
2.4 P HYSICAL PROCESSES IN G EANT 4
2.4.4 Modelling of ion interactions for hadrontherapy applications
In order to simulate carbon ion beams interaction with the matter for had-rontherapy purposes and study how the primary particle fragmentation can influ-ence the radiation field, the suitable hadronic models have to be accurately cho-sen, looking at their features and energy range of applicability. In general, the energy range of interest in hadrontherapy with carbon ion beams is about 0 - 400 AMeV which, as already mentioned allows to reach depths in water up to 28 cm.
In particular, in this work studies of carbon fragmentation and comparisons with experimental data for different targets at 135 AMeV (chapter 4) and 62 AMeV (chapters 5, 6) have been carried out, respectively for neutron production and charged fragments production. Total reaction cross section has to be imple-mented, as well as models describing the elastic and inelastic scattering; for the latter, also isotope production and following de-excitations and decays have to be simulated with models.
Total reaction cross section in nucleus-nucleus reactions
As already mentioned before, an important input parameter for simulation of particles transport in the matter is the total reaction cross section σR, which is defined as the total σT minus the elastic σel cross section fro nucleus-nucleus re-actions:
el T
R σ σ
σ = − (2.8)
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The total reaction cross section has been studied both theoretically and ex-perimentally and several empirical parameterizations of it have been developed.
In Geant4 the total reaction cross sections are calculated using four parameteri-zations: the Sihver [86], Kox [87], Shen [88] and Tripathi [89] formulae. In the present work the Tripathi formula has been used, because it works in a wide en-ergy range (from 10 AMeV up to 1 GeV/n) with uniform accuracy for any com-bination of interacting ions. The Tripathi formula incorporates different impor-tant effects, such as Coulomb interactions at lower energies and strong absorp-tion model, its form is:
( )
⎟⎟ MeV. The last term in equation (2.9) is the Coulomb interaction term, which modifies the cross section at lower energies and becomes less important as the energy increases (typically after several tens of AMeV). In equation (2.9), B is the energy-dependent Coulomb interaction barrier which is given by:R Z
B=1.44ZP T (2.10)
where ZP and ZT are the atomic numbers of the projectile and target, respec-tively. R is the distance for evaluating the Coulomb barrier height, given by:
( )
where ri is equivalent sphere radius and is related to the rrms,i radius by:
i rms
i r
r =1.29 , (2.12)
with (i = P, T).
The energy dependence in the reaction cross section at intermediate energies results mainly from two effects: trasparency and Pauli blocking. This energy dependence is taken into account in δE, which is given by:
2.4 Physical processes in Geant4
where S is the mass asymmetry term given by:
3
and is related to the volume overlap of the collision system. The last term on the right side of the equation (2.13) accounts for the isotope dependence of the reac-tion cross secreac-tion. The term CE is related to both the transparency and Pauli blocking and is given by:
(
1 e 40)
0.292e 792cos(
0.229E0.453)
D
CE = − −E − −E (2.15)
where the collision kinetic energy E is in units of AMeV. Here, D is related to the density dependence of the colliding system, scaled with respect to the density of the 12C + 12C colliding system:
The density of a nucleus is calculated in the hard-sphere model. In effect, D simulates the modifications of the reaction cross sections caused by Pauli block-ing. The Pauli blocking effect, which has not been taken into account in other empirical calculations, has been introduced here for the first time.
At lower energies (below several tens of AMeV) where the overlap of inter-acting nuclei is small (and where the Coulomb interaction modifies the reaction cross sections significantly), the modifications of the cross sections caused by Pauli blocking are small and gradually play an increasing role as the energy in-creases, which leads to higher densities where Pauli blocking becomes increas-ingly important.
The Tripathi formula is implemented inside the class G4TripathiCrossSection but it is still not fully implemented, so that it is valid for projectile energies less than 1 AGeV.
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Elastic scattering
The model of nucleon-nucleon elastic scattering is based on parameterization of experimental data in the energy range of 10-1200 MeV. The simulation of elastic scattering is quite simple, since the final states are identical with the ini-tial ones. At higher energies the hadron-nucleus elastic scattering can be simu-lated using the Glauber model and a two Gaussian form for the nuclear density.
This expression for the density allows to write the amplitudes in analytic form.
This assumption works only since the nucleus absorbs hadrons very strongly at small impact parameters, and the model describes well nuclear boundaries.
The G4LElastic class has been used in the simulations, which is valid for all incident particles and, included ions, and for both low and high energies.
Nucleus-nucleus models
The state of the art of nucleus-nucleus models in Geant4 at the energy range of interest in hadrontherapy is characterized by the presence of three models:
- Abrasion-ablation model [90].
- Binary Light Ion Cascade model [91].
- Quantum Molecular Dynamic model [92].
Recently an alternative model, also belonging to the family of the intranu-clear cascade models, has been introduced, the INCL-ABLA model [93], but it is applicable only to lighter ions (up 4He incident particles).
The results carried out in this work has been obtained using the Binary Light Ion Cascade model which, according to some inter-comparisons previously per-formed between the different models, represents a good compromise between an acceptable level of accuracy in isotope production predictions and computing performance.
The Binary Light Ion Cascade model
The Binary Light Ion Cascade model is an improved version of the Binary Cascade model (applicable only to pions, protons and neutrons). It is an intranu-clear cascade model used to simulate the inelastic scattering of light ion nuclei
2.4 Physical processes in Geant4
73 and it is applicable for projectile energy ranging on about 80 MeV to 10 AGeV.
In simulating light ion reactions, the initial state of the cascade is prepared in the form of two nuclei, as described in the above section on the nuclear model. The lighter of the collision partners is selected to be the projectile. The scattering is modelled by propagating the nucleons of the ion as free particles through the nu-cleus [94]. In the model, primary nucleons are propagated within a nunu-cleus (3-dimensional detailed model of the nucleus) where it suffers binary scattering with an individual nucleon of the nucleus; secondary particles eventually pro-duced in the interaction are propagated and can, in turn, re-scatter with nucleons, creating the cascade.
The nucleus is modelled by explicitly positioning nucleons in space, and as-signing momenta to these nucleons. This is done in a way consistent with the nuclear density distributions, Pauli’s exclusion principle and the total nuclear mass. Free hadron-hadron elastic and reaction cross-section are used to define collision locations within the nuclear frame. Where available, experimental cross-sections are used directly or as a basis for parameterizations used in the model. The propagation of particles in the nuclear field is done by numerically solving the equations of motion, using time-independent fields derived from op-tical potentials. The cascade begins with a projectile and the nuclear description, and it stops when the average energy of all participants within the nuclear boundaries are below a given threshold. The remaining pre-fragment will be treated by pre-equilibrium decay and de-excitation models.
The initial condition of the transport algorithm is a 3-dimensional model of a nucleus and the primary particle's type and energy. As regards the first one, a 3-dimensional model of the nucleus is constructed from A nucleons and Z protons with coordinates ri and momenta pi, with i = 1, 2,..., A. Nucleon radii ri are se-lected randomly in the nucleus rest frame according to the nuclear density ρ(ri).
For nuclei with A > 16 a Woods-Saxon form of the nucleon density is used [95]:
( )
r[ (
ri R)
a]
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For light nuclei the harmonic-oscillator shell model is used for the nuclear density [96]:
( )
ri =( )
πR2 −3/2exp(
−ri2/R2)
ρ (2.19)
where R2 = (2/3)<r2> = 0.8133 A2/3 fm2. To take into account the repulsive core of the nucleon-nucleon potential it is assumed inter-nucleon distance of 0.8 fm.
The nucleus is assumed to be spherical and isotropic, i.e. each nucleon is placed using a random direction and the previously determined radius ri.
The momenta pi of the nucleons are chosen randomly between 0 and the Fermi momentum pFmax(ri). The Fermi momentum, in the local Thomas-Fermi approximation as a function of the nuclear density ρ is:
( ) (
2( ) )
1/3max r c3 r
pF =h π ρ (2.20)
The total vector sum of the nucleon momenta has to be be zero, i.e. the nu-cleus must be constructed at rest.
The effective of collective nuclear interaction upon participants is approxi-mated by a time-invariant scalar optical potential, based on the properties of tar-get nucleus. For protons and neutrons the potential used is determined by the lo-cal Fermi momentum pF(r) as:
where m is the mass of the neutron or the mass of the proton, respectively.
As concern the primary particle, an impact parameter is chosen randomly on a disk outside the nucleus, perpendicular to a vector passing through the center of the nucleus. The initial direction of the primary is perpendicular to this disk.
Using straight-line transport, the distance of closest approach dimin to each
nu-2.4 Physical processes in Geant4
75 cleon i in the target nucleus, and the corresponding time-of-flight tid is calcu-lated. The interaction cross-section σi with target nucleons is calculated based on the momenta of the nucleons in the nucleus, and the projectile momentum. Tar-get nucleons for which the distance of closest approach dimin is smaller than dimin
< (σi/π)1/2 are candidate collision partners for the primary. All candidate colli-sions are ordered by increasing tid. In case no collision is found, a new impact parameter is chosen. The primary particle is then transported in the nuclear field by the time step given by the time to closest approach for the earliest collision candidate. Outside the nucleus, particles travel along straight-line trajectories.
Inside the nucleus particles are propagated in the nuclear field. At the end of each step, the interaction of the collision partners is simulated using the scatter-ing term, resultscatter-ing in a set of candidate particles for further transport. The sec-ondaries from a binary collision are accepted subject to Pauli's exclusion princi-ple. In the allowed cases, the tracking of the primary ends, and the secondaries are treated like the primary. All secondaries are tracked until they react, decay or leave the nucleus, or until the cascade stops due to the fact that the mean energy of all propagating particles in the system is below a threshold (≈ 15 MeV). At this stage the state of affairs has to be treated by means pre-equilibrium decay and de-excitation models. Hence, the residual participants (pre-fragment), and the nucleus in its current state are then used to define the initial state for pre-equilibrium decay.
Pre-equilibrium decay and de-excitation models
At the end of the cascade, a fragment is formed for further treatment in Pre-compound and nuclear De-excitation models. These models need some informa-tion about the nuclear fragment created by the cascade. The fragment formed is characterized by several parameters, such as the number of nucleons in the fragment, the charge, the four momentum of the fragment and so on, which are used as input parameter by the pre-equilibrium and de-excitations models.
The GEANT4 Precompound model is considered as an extension of the had-ron kinetic model. It gives a possibility to extend the low energy range of the hadron kinetic model for nucleon-nucleus inelastic collision and it provides a
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”smooth” transition from kinetic stage of reaction described by the hadron ki-netic model to the equilibrium stage of reaction described by the equilibrium de-excitation models [97]. The Precompound model is applicable in the energy range 0 – 170 MeV. Only emission of neutrons, protons, deutrons, thritium and helium nuclei are taken into account. The precompound stage of nuclear reaction is considered until nuclear system is not an equilibrium state. Further emission of nuclear fragments or photons from excited nucleus is simulated using equilib-rium models implemented in Geant4, such as the Evaporation, the Fermi break-up and the Statistical Multifragmentation models.
At the end of the pre-equilibrium stage, the residual nucleus is supposed to be left in an equilibrium state, in which the excitation energy E* is shared by a large number of nucleons. Such an equilibrated compound nucleus is character-ized by its mass, charge and excitation energy with no further memory of the steps which led to its formation.
If the excitation energy is higher than the separation energy, it can still ejects nucleons and light fragments (d, t, 3He, α), whose description is treated by the Evaporation model (for excited nuclei at relatively low excitation energies).
These constitute the low energy and most abundant part of the emitted particles in the rest system of the residual nucleus. The emission of particles by the Evaporation model is based on Weisskopf and Ewing model [98]. Fission (for nuclei with A > 65) and Photon Evaporation can be treated as competitive chan-nels in the evaporation model.
The Statistical Multifragmentation and Fermi break-up models are used at excitation energies E* above 3 AMeV.
The Multifragmentation model is capable to predict final states as a result of a highly excited nucleus statistical break-up. The initial information for calcula-tion of multifragmentacalcula-tion stage consists in the atomic mass number A, charge Z of excited nucleus and its excitation energy E*. At higher excitation energies (E*
> 3 AMeV) the multifragmentation mechanism, when nuclear system can even-tually breaks down into fragments, becomes the dominant. Later on the excited primary fragments propagate independently in the mutual Coulomb field and undergo de-excitation [99].
2.4 Physical processes in Geant4
77 The Fermi break-up model is capable to predict final states of excited nuclei with atomic number Z < 9 and A < 17. For light nuclei the values of excitation energy per nucleon are often comparable with nucleon binding energy. Thus a light excited nucleus breaks into two or more fragments with branching given by available phase space. This statistical approach was first used by Fermi to de-scribe the multiple production in high energy nucleon collision [100].