\chapter{Bertini Intranuclear Cascade Model in {\sc Geant4} }

\section{Introduction} 
This model is based on a re-engineering of the INUCL code and
includes the Bertini intra-nuclear cascade model with excitons, a 
pre-equilibrium model, a nucleus explosion model, a fission model, and an 
evaporation model.  Intermediate energy nuclear reactions from 100~MeV to 
5~GeV are treated for protons, neutrons, pions, photons and nuclear 
isotopes.  We present an overview of the models, review results achieved 
from simulations and make comparisons with experimental data.

The intranuclear cascade model (INC) was was first proposed by Serber in 
1947 \cite{serber47}.  He noticed that in particle-nuclear collisions the 
deBroglie wavelength of the incident particle is comparable (or shorter) than 
the average intra-nucleon distance.  Hence, a description of interactions 
in terms of particle-particle collisions is justified.

The INC has been used succesfully in Monte Carlo simulations at intermediate 
energies since Goldberger made the first hand-calculations in 
1947 \cite{goldberger48}.  The first computer simulations were done by 
Metropolis et al. in 1958 \cite{metropolis58}. Standard methods in INC 
implementations were developed when Bertini published his results in 
1968 \cite{bertini68}.  An important addition to INC was the exciton model 
introduced by Griffin in 1966 \cite{griffin66}. 

The current presentation describes the implementation of the Bertini INC 
model within the {\sc Geant4} hadronic physics 
framework \cite{geant4collaboration03}.  This framework is flexible and
allows for the modular implementation of various kinds of hadronic 
interactions.  It is based on the concepts of physics processes and models.  
While the process is a general concept, models may be restricted according
to process type, material, element and energy range.  Several models can be 
utilized by one process class; for instance, a process class for inelastic 
collisions can use a different model for each energy range.

The process classes use model classes to determine the secondaries produced 
in the interaction and to calculate the momenta of the particles.  Here we 
present a collection of such models which describe a medium-energy 
intranuclear cascade.

\section{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 \cite{bertini68, bertini71} solves the Boltzmann equation on 
average.  This model has been implemented in several codes such as 
HETC \cite{alsmiller90}.  Our model, which is based on a re-engineering of 
the INUCL code  \cite{titarenko99a}, 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.  

\subsection{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.

\subsection{Intranuclear Cascade Model}

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

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

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

\subsection{Nuclear Model}

Some of the basic features of the nuclear model are:

\begin{itemize}
\item 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 
$p_{F}(r) = (\frac{3 \pi^2 \rho(r)}{2})^\frac{1}{3}$.
%\item Nuclear density effects are re-calculated after each step,
\item 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.
\end{itemize}

\subsubsection{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
\begin{equation}
r_{i}(\alpha_{i}) =  C_{2} \log({\frac{1 + e^{- \frac{C_{1}}{C_{2}}}}{\alpha_{i}} - 1}) + C_{1} ,
\end{equation}
where $C_{2} = 1.7234$.

The potential energy $V$ for nucleon $N$ is
\begin{equation}
V_{N} = \frac{p_{F}^2}{2 m_{N}} + BE_{N}(A, Z) ,
\end{equation}
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.

\begin{equation}
 f(p) = c p ^2
\end{equation}
where

\begin{equation}
\int_0^{p_F} f(p) dp = n_{p} \rm{ or }  n_{n}
\end{equation}
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

\begin{equation}
 E_f = \frac{p_F^2}{2 m_N} = \frac{\hbar^2}{2 m_N}(\frac{3 \pi^{2}}{v})^\frac{2}{3} ,
\end{equation}
which depends on the density $n/v$ of particles, and which is different for 
each particle and each region. 

\subsubsection{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 $E_N > E_f$.


\subsubsection{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.
%{\sc Geant4} cascade model uses tabulated cross-sections.
Tabulated total reaction cross sections are calculated by Letaw's 
formulation \cite{letaw83, letaw93, pearlstein89}.
%:::$45 A^0.7 (1+0.016 sin(5.3-2.63 log10(A)))^(1-0.62 exp(-E / 200) sin(10.9 E^(-0.28)))
For $N-N$ cross sections the parameterizations are based on the experimental 
energy and isospin dependent data. 
The parameterization described in \cite{barashenkov72} 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.

\subsection{Pre-equilibrium Model}

The {\sc Geant4} cascade model implements the exciton model proposed by 
Griffin \cite{griffin66, griffin67}.  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 \cite{ribansky73}, (level densities) 
and \cite{kalbach78} (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).

\subsection{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 \cite{fong69}.

\subsection{Evaporation Model}

A statistical theory for particle emission of the excited nucleus remaining 
after the intra-nuclear cascade was originally developed by 
Weisskopf \cite{weisskopf37}.  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 {\sc Geant4} evaporation model for the cascade implementation adapts the
often-used computational method developed by 
Dostrowski \cite{dostrovsky59, dostrovsky60}.  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.

\section{Implementation}

The cascade model is implemented in the {\sc Geant4} hadronic physics 
framework.  All the models are used collectively through the interface 
method {\it Apply\-Yourself} defined in the class {\it G4Cascade\-Interface}.
A {\sc Geant4} track ({\it G4Track}) and a nucleus ({\it G4Nucleus}) are given 
as parameters.

The cascade models were first tested in release {\sc Geant4 5.0} for energies 
100~MeV -- 5~GeV.  Detailed comparisons with experimental data have been made 
in the energy range 160 -- 800 MeV. 

\section{Status of this document}

01.12.02 created by Aatos Heikkinen, Nikita Stepanov and Hans-Peter Wellisch \\
14.06.05 grammar, spelling check and list of pion absorption channels 
         corrected by D.H. Wright \\

\begin{latexonly}

\begin{thebibliography}{99}

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   (1990), 337--343,

% [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 
%(in Russian, but there is an English translation)) 

\bibitem{barashenkov72}
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\bibitem{bertini68}     
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Nucl. Instr. Meth,
66,
 1968,
 29.                              
% \bibitem{bertini69}
%  H.W.Bertini,
%  Intranuclear-Cascade Calculation of the Secondary Nucleon Spectra from Nucleon-Nucleus 
%         Interactions in the Energy Range 340 to 2900 MeV and Comparisons with Experiment,
% Phys. Rev.,
%  188,
%  1969,
%  1711 
% 
\bibitem{bertini71}     
H. W. Bertini and P. Guthrie,
Results from Medium-Energy Intranuclear-Cascade Calculation,
Nucl. Phys.A169,
(1971).

\bibitem{dostrovsky59}
         I. Dostrovsky, Z. Zraenkel and G. Friedlander,
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         3,
         683-702.

\bibitem{dostrovsky60}
         I. Dostrovsky and Z. Fraenkel and P. Rabinowitz,
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         Physical Review,
         1960. 

\bibitem{fong69}
 P. Fong,
 Statistical Theory of Fission,
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\bibitem{geant4collaboration03}
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 Geant4 general paper (to be published),
Nuclear Instruments and Methods A,
(2003).

\bibitem{goldberger48}  
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\bibitem{griffin66}     
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Physical Review Letters 17, (1966), 478-481.

\bibitem{griffin67}     
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Physics Letters 24B, 1 (1967), 5-7.

\bibitem{iljinov94}
 A. S. Iljonov et al.,
 Intermediate-Energy Nuclear Physics,
 CRC Press 1994.

\bibitem{kalbach78}     
 C. Kalbach,
 Exciton Number Dependence of the Griffin Model Two-Body Matrix Element,
Z. Physik A 287, (1978), 319-322.

\bibitem{letaw83}
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\bibitem{letaw93}
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\bibitem{metropolis58}
         N. Metropolis, R. Bibins, M. Storm,
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         Physical Review 110,
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\bibitem{pearlstein89}
         S. Pearlstein,
         Medium-energy nuclear data libraries: a case study, neutron- and 
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         (1989),
         1049-1060.

\bibitem{ribansky73}    
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 Pre-equilibrium decay and the exciton model,
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\bibitem{titarenko99a}
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nucl-ex/9908012,
 (1999).

\bibitem{weisskopf37}
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\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item
  R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann,
  The high-energy transport code HETC88 and comparisons with experimental data,
  Nuclear Instruments and Methods in Physics Research A 295,
   (1990), 337--343,

% [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 
%(in Russian, but there is an English translation)) 

\item
 V.S. Barashenkov and V.D. Toneev,
 High Energy interactions of particles and nuclei with nuclei (In russian),
 (1972)

\item    
 M. P. Guthrie, R. G. Alsmiller and H. W. Bertini,
Nucl. Instr. Meth,
66,
 1968,
 29.                              
% \bibitem{bertini69}
%  H.W.Bertini,
%  Intranuclear-Cascade Calculation of the Secondary Nucleon Spectra from Nucleon-Nucleus 
%         Interactions in the Energy Range 340 to 2900 MeV and Comparisons with Experiment,
% Phys. Rev.,
%  188,
%  1969,
%  1711 
% 
\item     
H. W. Bertini and P. Guthrie,
Results from Medium-Energy Intranuclear-Cascade Calculation,
Nucl. Phys.A169,
(1971).

\item
         I. Dostrovsky, Z. Zraenkel and G. Friedlander,
         Monte carlo calculations of high-energy nuclear interactions. III. Application to low-lnergy calculations,
         Physical Review,
         1959,
         116,
         3,
         683-702.

\item
         I. Dostrovsky and Z. Fraenkel and P. Rabinowitz,
         Monte Carlo Calculations of Nuclear Evaporation Processes. V. Emission of Particles Heavier Than $^4He$,
         Physical Review,
         1960. 

\item
 P. Fong,
 Statistical Theory of Fission,
 1969, Gordon and Breach, New York.

\item
 Geant4 collaboration,
 Geant4 general paper (to be published),
Nuclear Instruments and Methods A,
(2003).

\item  
 M. Goldberger,
 The Interaction of High Energy Neutrons and Hevy Nuclei,
Phys. Rev. 74,
 (1948),
 1269.
 
\item     
 J. J. Griffin,
 Statistical Model of Intermediate Structure,
Physical Review Letters 17, (1966), 478-481.

\item     
 J. J. Griffin,
 Statistical Model of Intermediate Structure,
Physics Letters 24B, 1 (1967), 5-7.

\item
 A. S. Iljonov et al.,
 Intermediate-Energy Nuclear Physics,
 CRC Press 1994.

\item     
 C. Kalbach,
 Exciton Number Dependence of the Griffin Model Two-Body Matrix Element,
Z. Physik A 287, (1978), 319-322.

\item
         J. R. Letaw et al.,
         The Astrophysical Journal Supplements 51,
         (1983),
         271f.

\item
         J. R. Letaw et al.,
         The Astrophysical Journal 414,
         1993,
         601.

\item
         N. Metropolis, R. Bibins, M. Storm,
         Monte Carlo Calculations on Intranuclear Cascades. I. Low-Energy Studies,
         Physical Review 110,
         (1958),
         185ff.

\item
         S. Pearlstein,
         Medium-energy nuclear data libraries: a case study, neutron- and 
         proton-induced reactions in $^56$Fe,
         The Astrophysical Journal 346,
         (1989),
         1049-1060.

\item    
 I. Ribansky et al.,
 Pre-equilibrium decay and the exciton model,
Nucl. Phys. A 205,
 (1973),
 545-560.

\item      
 R. Serber,
 Nuclear Reactions at High Energies,
Phys. Rev. 72,
 (1947),
 1114. 

\item
 Experimental and Computer Simulations Study of
                  Radionuclide Production in Heavy Materials
                  Irradiated by Intermediate Energy Protons,              
 Yu. E. Titarenko et al.,
nucl-ex/9908012,
 (1999).

\item
         V. Weisskopf,
          Statistics and Nuclear Reactions,
       Physical Review 52,
           (1937),
          295--302.
          
\end{enumerate}

\end{htmlonly}

          
