\chapter{INCL 4.2 Cascade and ABLA V3 Evaporation with Fission}

\section{Introduction}

There is a renewed interest in the study of spallation reactions. This
is largely due to new technological applications, such as Accelerator
Driven Systems, consisting of sub-critical nuclear reactor and
particle accelerator. These applications require optimized spallation
targets or spallation sources. This type of problem has typically a large number
of parameters and thus it cannot be solved by trial and error
method. One has to rely on simulations, which implies that very
accurate simulation tools need to be developed and their validity and
accuracy needs also to be assessed.

Above the energy 200 MeV it is necessary to use reliable models due to
the prohibitive number of open channels. The most appropriate modeling
technique in this energy region is intranuclear cascade (INC) combined
with evaporation model. One such pair of models is the Li\`ege cascade
model INCL4.2 coupled with ABLA evaporation model. The strategy adopted
by the INCL4.2 cascade is to improve the quasi-classical treatment of
physics without relying on too many free parameters. 

This chapter introduces the physics provided by INCL4.2 and ABLA V3 codes as implemented in Geant4.
Tables \ref{tbl:inclsummary} and \ref{tbl:ablasummary} will summarize the key features 
and provides references describing in detail the physics.


\section{INCL4.2 cascade} 
\label{sec:inclmodel}

INCL4.2 is a Monte Carlo simulation incorporating aforementioned cascade
physics principles. INCL4.2 cascade algorithm consists of an
initialization stage and the actual data processing stage.


The INCL4.2 cascade can be used to simulate the collisions between
bullet particles and nuclei. The supported bullet particles and the
interface classes supporting them are presented in table
\ref{tbl:inclsummary}.


\begin{table}[ht]
  \caption{INCL 4.2 (located in the Geant4
    directory {\tt source/\-processes/\-hadronic/\-models/\-incl})
    feature summary.}
\label{tbl:inclsummary}
\vskip1cm
\begin{center}
\begin{tabular}{l|l}
\hline
{\bf Requirements} & \\
External data file & G4ABLA3.0 available at Geant4 site \\
Environment variable & {\tt G4ABLADATA} \\ 
for external data & \\
\hline
{\bf Usage}      & \\
Physics list     & Not yet implemented, \\
                 & instead use the interfaces directly. \\
\hline
{\bf Interfaces} &     \\
{\tt G4InclCascadeInterface} &  h--A \\
{\tt G4InclLightIonInterface} &  A--A \\
\hline
{\bf Projectile particles} & proton, neutron \\
                 & pions ($\pi^+$, $\pi^0$, $\pi^-$) \\
                 & deuteron, triton \\ 
                 & $\alpha$, $^3$He \\ 
\hline
{\bf Energy range} & 200 MeV - 3 GeV \\
\hline
{\bf Target nuclei} & \\
Lightest applicable & Carbon, C \\
Heaviest            & Uranium, U \\
\hline
{\bf Features} & No ad-hoc parameters \\
                    & Woods-Saxon nuclear potential \\
                    & Coulomb barrier \\
                    & Non-uniform time-step \\
                    & Pion and delta production cross sections \\
                    & Delta decay \\
                    & Pauli blocking \\
\hline
{\bf Misc.}         & 5 classes (see fig. \ref{fig:uml}), 8k lines \\
                    & 0.9 $<$ speed C++/F77 $<$ 1.1 \\
\hline
{\bf References}    & Key reference \cite{Boudard02a}, see also \cite{Cugnon97a, Cugnon81a, Cugnon87a, Cugnon89a} \\
\hline
\end{tabular}
\end{center}
\end{table}

The momenta and positions of the nucleons inside the nuclei are
determined at the beginning of the simulation run by modeling the
nucleus as a free fermi gas in a static potential well. The cascade is
modeled by tracking the nucleons and their collisions. The collisions
are assumed to be well separated in space and time.

The possible reactions inside the nucleus are 
\begin{itemize}
\item $NN \rightarrow N \Delta$ and $N \Delta \rightarrow NN$ 
\item $\Delta \rightarrow \pi N$ and $\pi N \rightarrow \Delta$
\end{itemize}

%\begin{figure}[ht]
%\begin{center}
%\includegraphics[scale=0.6]{Pb208Proton1GeV.eps}
%\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/Incl/Pb208Proton1GeV.eps}
%\end{center}
%\caption{Colliding 1 GeV proton to Pb208 target. Here is presented the
%mass number of the outcoming particles.}
%\end{figure}

\subsection{Model limits}



The INCL4.2 model has certain limitations with respect to the bullet
particle energy and target nucleus type. The supported energy range
for bullets is: 200 MeV - 3 GeV. Acceptable target nuclei range from
Carbon to Uranium.

% Maybe too basic stuff...
% \chapter{INCL and ABLA models}

% Hadronic interactions can be modeled with cascade
% models when we deal with so called medium energy range between 100 MeV
% and 10 GeV \cite{thebook}.
% %The medium energy range lies between 100 MeV and 10 GeV. 
% In this energy range we can simplify the problem with certain
% assumptions and approximations. In this chapter we outline the basic
% features of medium energy hadronic physics and how INCL and ABLA
% models implement them.

% %\section{Physics in intermediate energy range}

% \section{Intra-nuclear cascade (INC)}
% \index{intranuclear cascade}
% \label{sec:inc}

% Atomic nuclei can, in principle, be described in terms of quantum
% mechanics. However, since the nucleus is fairly complex multi-particle
% system purely quantum mechanical approach is usually not feasible for
% describing nuclear reactions in the intermediate energy range. The INC
% model of these processes is a semiclassical description of collision
% between a particle and a nucleus.

% The basic problem can be divided into two steps. First is the actual
% \emph{cascade} stage.  At this stage each nucleon collides few times
% in the nucleus. The number of collisions is usually between 1 and 4
% depending on the weight of the nucleus (number of nucleons).  As the
% result of the collisions fast nucleons and pions exit the nucleus.
% This stage lasts usually $10^{-23}$ - $10^{-22}$ seconds.

% After cascade comes pre-equilibrium and evaporation cometing with
% fission stages.  At this stage more nucleons are ejected from the
% nucleus.  This process is slower than the \emph{cascade} taking
% $10^{-18}$-$10^{-16}$ seconds.

% \subsection{Basic assumptions}
% \index{cascadeassumptions}

% The basic assumptions of INC-model can be summarized as follows:
% \begin{enumerate}
% \item Motion of particles obeys \emph{classical mechanics}.
% \item In collisions relativistic kinematics is used.
% \item Collisions between pairs of particles are well separated in
%   space and time. This means that the spatial dimensions and time
%   scale of collisions are smaller than those of particle
%   transportation in the nuclear matter.
% \end{enumerate}

\subsection{Model principles}

The INCL model uses only two external user defined parameters: the nuclear
potential depth and scaling factor for time-step. All other parameters
used during the calculation are obtained either from theory or
experiments. In the actual simulation code good default values for
these two parameters are predefined.

During the initialization the necessary Woods-Saxon potential
calculations are made. The results of these calculations are used at
the beginning of a cascade to determine the positions and momenta of
the nucleons inside the nucleus. The nucleons are modeled as free
Fermi gas in a static potential well. The principle for doing this is
presented in the following equations \ref{eqn:density} and
\ref{eqn:rpcorrelations}.

\index{nucleon density}
The nucleon density in r-space is given by formula:
\begin{equation}
\rho(r) =  \left\{ \begin{array}{ll}
  \frac{\rho_0}{1 + exp(\frac{r - R_0}{a})}& \textrm{for $r
   < R_{max}$}\\
 0 & \textrm{for $r > R_{max}$}\\
  \end{array} \right.
\label{eqn:density}
\end{equation}
The nucleon positions $r$ and momenta $p$ are correlated in the following way:
\begin{equation}
\rho(p)p^2dp = -\frac{d \rho(p)}{dr} \frac{r^3}{3}dr.
\label{eqn:rpcorrelations}
\end{equation}

% \begin{figure}[ht]
% \begin{center}
% \includegraphics{rpcorrelations.eps}
% \end{center}
% \caption[Momentum-position correlations in INCL]{Nucleon
%   momentum-position correlations of the initial state in INCL are
%   determined using the Woods-Saxon potential. (Picture courtesy of
%   Alain
%   Boudard)}
% \label{fig:rpcorr}
% \end{figure}

After the initialization a projectile particle, bullet, is shot
towards the target nucleus. The impact parameter, i.e. the distance
between the projectile particle and the center point of the projected
nucleus surface is chosen at random. The value of the impact parameter
determines the point where the bullet particle will hit the
nucleus. After this the algorithm tracks the nucleons by determining
the times at which an event will happen. The possible events are:
\begin{itemize}
\item collision
\item decay of a delta resonance
\item reflection from nuclear potential wall (only for nucleons)
\end{itemize}

The particles are assumed to propagate along straight line
trajectories.  The algorithm calculates the time at which events will
happen and propagates the particles directly to their positions at
that particular point in time. Practically this means that the length
of the time step in simulation is not constant.

\index{participant}
\index{spectator}
All particles in the model are labeled either as \emph{participants}
or \emph{spectators}. Only collisions where participants are involved
are taken into account.

\index{delta resonance}
The processes undergone by the nucleons, pions and delta resonances
are as follows:
\begin{itemize}
\item $NN \rightarrow N \Delta$ and $N \Delta \rightarrow NN$ 
\item $\Delta \rightarrow \pi N$ and $\pi N \rightarrow \Delta$
\end{itemize}
%Figure \ref{img:inc} demonstrates these prosesses during INC.


\subsection{Conservation laws}

During cascade INCL model conserves several quantities: 
energy, baryon number, charge number, energy, momentum and angular momentum. 
The conservation relations are:
\begin{eqnarray}
A_{projectile} + A_{target} = A_{ejectiles} + A_{remnant} \\
Z_{projectile} + Z_{target} = Z_{ejectiles} + Z_{remnant} \\
T_{laboratory} = K_{ejectiles} + W_{\pi} + E_{recoil} + E_{excitation} + S \\  %///FIXED-AH systematically use full word indexing
\vec{p}_{laboratory} = \vec{p}_{ejectiles} + \vec{p}_{\pi} +
\vec{p}_{remnant} \\
\vec{l} = \vec{l}_{ejectiles} + \vec{l}_{\pi} + \vec{l}_{remnant} +
\vec{l}_{excitation},
\end{eqnarray}
where
$A$ is baryon number, $Z$ charge, $T$ energy, $K$ kinetic energy,
$E_{recoil}$ remnant recoil energy, $E_{excitation}$ remnant nucleus
excitation energy, $S$ separation energy, $\vec{p}$ momentum and
$\vec{l}$ angular momentum.

\subsection{Cascade stopping time}

Stopping time is defined as the point in time when the cascade phase is finished
and the excited nucleus remnant is given to evaporation model.
In INCL model the stopping time, $t_{stop}$,  is determined as:
\begin{equation}
t_{stop} = f_{stop}t_0 (A_{target}/208) ^{0.16}.
\end{equation}
Here $A_{target}$ is the target mass number and $t_0$ is a constant
with default value 70 fm/c. Factor $f_{stop}$ is the second of
the two free parameters in the model. It is a scaling factor for the
stopping time. Good default value for this factor is
$f_{stop}$ = 1.0.


\subsection{Light ions}

In addition to protons, neutrons and pions INCL4.2 supports also
light ion projectiles: deuterons, tritons, He3 and alpha.
In light of INCL physics modeling principles presented
above, the extension to light ions is quite natural. Light ions are
modeled in similar way to nucleus, except that ion potential is
considered to be Gaussian instead of real Woods-Saxon shape. In table
\ref{tbl:gaussianformslightions} the parameters for Gaussian forms
used for distance and momentum distributions in light ions are
presented.

\begin{table}[h]
\caption{The parameters for nucleon position and momentum
  distributions in light ions.} % \cite{incl4}.}
\begin{center}
\begin{tabular}{c|c c}
\hline
Particle & $\sqrt{(r_{mean})^2}$ [fm] & $\sqrt{(p_{mean})^2}$ [MeV/c]\\
\hline
Deuteron & 1.91 & 77 \\
Triton & 1.8 & 110 \\
He3 & 1.8 & 110 \\
Alpha & 1.63 & 153 \\
\hline
\end{tabular}
\end{center}
\label{tbl:gaussianformslightions}
\end{table}


\section{ABLA V3 evaporation}
\index{evaporation}

The ABLA V3 evaporation model takes excited nucleus parameters,
excitation energy, mass number, charge number and nucleus spin, as
input. It calculates the probabilities for emitting proton, neutron or
alpha particle and also probability for fission to occur. 
The summary of Geant4 ABLA V3 implementation is represented in Table \ref{tbl:ablasummary}.


The probabilities for emission of particle type $j$ are calculated using
formula:
\begin{equation}
W_j(N,Z,E) = \frac{\Gamma_j(N,Z,E)}{\sum_k\Gamma_k(N,Z,E)},
\label{eqn:probabilities}
\end{equation}
where $\Gamma_j$ is emission width for particle $j$, $N$ is neutron
number, $Z$ charge number and $E$ excitation energy. Possible emitted
particles are \emph{protons}, \emph{neutrons} and \emph{alphas}.
Emission widths are calculated using the following formula:
\index{emission width}
\begin{equation}
\Gamma_j = \frac{1}{2 \pi \rho_c(E)} \frac{4 m_j R^2}{\hbar^2} T_j^2 \rho_j(E - S_j - B_j),
\label{eqn:emissionwidth}
\end{equation}
where $\rho_c(E)$ and $\rho_j(E - S_j - B_j)$ are the level densities
of the compound nucleus and the exit channel, respectively. $B_j$ is
the height of the Coulomb barrier, $S_j$ the separation energy, $R$ is
the radius and $T_j$ the temperature of the remnant nucleus after
emission and $m_j$ the mass of the emitted particle.

The fission width is calculated from:
\index{fission width}
\begin{equation}
\Gamma_i = \frac{1}{2 \pi \rho_c(E)}T_f \rho_f(E - B_f),
\label{eqn:fissionwidth}
\end{equation}
where $\rho_f(E)$ is the level density of transition states in the
fissioning nucleus, $B_f$ the height of the fission barrier and $T_f$
the temperature of the nucleus.


\begin{table}[ht]
  \caption{ABLA V3 (located in the Geant4 directory
    {\tt source/\-processes/\-hadronic/\-models/\-incl}) feature summary.}
\label{tbl:ablasummary}
\vskip1cm
\begin{center}
\begin{tabular}{l|l}
\hline
{\bf Requirements} & \\
External data file & G4ABLA3.0 available at Geant4 site \\
Environment variable & {\tt G4ABLADATA} \\ 
for external data & \\
\hline
{\bf Usage}      & \\
Physics list     & Not yet implemented, \\
                 & instead use the interfaces directly. \\
\hline
{\bf Interfaces} &     \\
{\tt G4InclAblaCascadeInterface} &  h--A \\
{\tt G4InclAblaLightIonInterface} &  A--A \\
\hline
{\bf Supported input}     & Excited nuclei \\
\hline
{\bf Output particles}    & proton, neutron \\
                    & $\alpha$ \\
                    & fission products \\
                    & residual nuclei \\
\hline
{\bf Features} & evaporation of proton, neutron and $\alpha$ \\
                    & fission \\
\hline
{\bf Misc.}                & 5 classes, 5k lines \\
                    & 0.9 $<$ speed C++/F77 $<$ 1.1 \\
\hline
{\bf References}    & Key reference: \cite{Junghans98a}, see also \cite{Benlliure98a} \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Level densities}
\index{level density}

Nuclear level densities are calculated using the following formula:
\begin{equation}
  a = 0.073 A  [MeV^{-1}] + 0.095  B_s  A^{2/3} [MeV^{-2}],
\label{eqn:leveldensity}
\end{equation}
where $A$ the nucleus mass number and $B_s$ dimensionless surface area
of the nucleus. 

%The level density calculation is implemented in the
%code as follows:
%\begin{verbatim}
%pa = (ald->av)*a + (ald->as)*pow(a,(2.0/3.0)) 
%     + (ald->ak)*pow(a,(1.0/3.0));
%\end{verbatim}
%where {\tt ald->av} is 0.073, {\tt ald->as} is 0.095 and {\tt ald->ak}
%is 0.

\subsection{Fission}
\index{fission}

Fission barrier, used to calculate fission width
\ref{eqn:fissionwidth}, is calculated using a semi-empirical model
fitting to data obtained from nuclear physics experiments.


\section{External data file required}

Both INCL4.2 and ABLA V3 need data files. These files contain ABLA V3 shell corrections and remnant nucleus masses for INCL4.2. 
To enable this data set, environment variable {\tt G4ABLADATA} needs to be set, 
and the data downloaded from Geant4 web page. For Geant4 9.1 release use data file G4ABLA3.0

\section{Implementation details}
INCL4.2 and ABLA V3 are provided as alpha release for Geant4 9.1.
In this first release design follows as closely as possibly the original codes,
and class re-design is left for future Geant4 releases.
Current simple design is shown in  \ref{fig:uml}

\begin{figure}
\begin{center}
%\includegraphics[angle=0,scale=1.0]{InclAblaUml.eps}
\includegraphics[angle=0,scale=1.0]{hadronic/theory_driven/Incl/InclAblaUml.eps}
\end{center}
\caption[INCL4 and ABLA class diagram]{Simplified UML class diagram of
  INCL and ABLA implementations in Geant4.}
\label{fig:uml}
\end{figure}

Testing of INCL and ABLA models is based on ROOT \cite{Brun97a} scripting.

\section{Physics Performance}
INCL4.2 together with ABLA V3 provides an up to date modeling tool particularly for spallation
studies for hadron projectile energy range 200 MeV - 3 GeV. 
It provides an detailed description of double differential
energy spectrum of cascading particles (see \ref{fig:pPbDoubleDifferential}) and remnants.

Models are validated against recent data and continually updated.

\begin{figure}
\begin{center}
%\includegraphics[angle=0,scale=0.65]{pPbDoubleDifferential.eps}
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/Incl/pPbDoubleDifferential.eps}
\end{center}
\caption{Geant4 implementation of INCL4.2 together with ABLA V3. Neutron double differential cross sections of reaction p(1GeV) + Pb
 For more detailed discussion on this subject
%  double differential cross sections for proton-induced reactions on
%  Pb targets with comparison to experimental data 
see reference \cite{Boudard02a}.}
\label{fig:pPbDoubleDifferential}
\end{figure}

\section{Status of this document}

{\bf 06.12.2007} Documentation for alpha release added. Pekka
Kaitaniemi, HIP (translation); Alain Boudard, CEA (contact person
INCL/ABLA); Joseph Cugnon, University of Li\`ege (INCL physics
modelling); Karl-Heintz Schmidt, GSI (ABLA); Christelle Schmidt, IPNL
(fission code); Aatos Heikkinen, HIP (project coordination)

% \begin{thebibliography}{99}
% \bibitem{incl1} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505
% \bibitem{incl2} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751
% \bibitem{incl3} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701
% \bibitem{incl4} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615
% \bibitem{liegeuniversity} Li\`ege University
% \href{http://www.ulg.ac.be/foreign/}{{\tt http://www.ulg.ac.be/foreign/}}
% \bibitem{cea} CEA website
%   \href{http://www.cea.fr/gb/index.asp}{{\tt http://www.cea.fr/gb/index.asp}}
% \end{thebibliography}

\begin{latexonly}
\begin{thebibliography}{99}

% \bibitem{alsmiller90}
%   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)) 

\bibitem{Boudard02a} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615
\bibitem{Cugnon81a} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505
\bibitem{Cugnon87a} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751
\bibitem{Cugnon89a} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701
\bibitem{Cugnon97a} J. Cugnon et al \emph{Nuc. Phys. A620} (1997) 745
\bibitem{Junghans98a} A.R. Junghans et al \emph{Nuc. Phys. A629} (1998) 635
\bibitem{Benlliure98a} J. Benlliure et al \emph{Nuc. Phys. A628} (1998) 458
\bibitem{Kaitaniemi07a} P. Kaitaniemi et al. \emph{Implementation of
    INCL4 cascade and ABLA evaporation codes in Geant4} (To be
    published in the proceedings of CHEP 2007, September 2-6,
    Victoria, BC, Canada.)
\bibitem{Brun97a} R. Brun, F. Rademakers \emph{Nucl. Inst \&
    Meth. in Phys. Res. A 389} (1997) 81

\end{thebibliography}
\end{latexonly}



\begin{htmlonly}
\section{Bibliography}

\begin{enumerate}
\item{Boudard02a} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615
\item{Cugnon81a} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505
\item{Cugnon87a} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751
\item{Cugnon89a} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701
\item{Cugnon97a} J. Cugnon et al \emph{Nuc. Phys. A620} (1997) 745
\item{Junghans98a} A.R. Junghans et al \emph{Nuc. Phys. A629} (1998) 635
\item{Benlliure98a} J. Benlliure et al \emph{Nuc. Phys. A628} (1998) 458
\item{Kaitaniemi07a} P. Kaitaniemi et al. \emph{Implementation of
    INCL4 cascade and ABLA evaporation codes in Geant4} (To be
    published in the proceedings of CHEP 2007, September 2-6,
    Victoria, BC, Canada.)
\item{Brun97a} R. Brun, F. Rademakers \emph{Nucl. Inst \&
    Meth. in Phys. Res. A 389} (1997) 81

\end{enumerate}
\end{htmlonly}

