source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/PartonString/ReactionInitialStatePSM.tex @ 1214

\section{Reaction initial state simulation.}

\subsection{Allowed projectiles and bombarding energy range for interaction 
with nucleon and nuclear targets.}
\hspace{1.0em}
The GEANT4 parton string models are capable to predict final states (produced 
hadrons 
which belong to the scalar and vector meson nonets and the baryon (antibaryon) 
octet and decuplet) of reactions on nucleon and nuclear targets 
with nucleon, pion and kaon
 projectiles. The allowed bombarding energy 
 $\sqrt{s} > 5$ \ GeV is recommended. 
 Two approaches, based on diffractive excitation or soft scattering with
 diffractive admixture according to cross-section, are considered.
\hspace{1.0em}Hadron-nucleus  collisions in the both
approaches (diffractive and parton exchange) are considered 
as a set of the independent hadron-nucleon collisions. 
 However, the string excitation procedures
in these approaches are rather different.

\subsection{ MC initialization procedure for nucleus.}
\hspace{1.0em}The initialization of each nucleus, consisting from $A$
nucleons and $Z$ protons with coordinates $\mathbf{r}_i$ and momenta
$\mathbf{p}_i$, where $i = 1,2,...,A$ is performed.
We use the standard initialization Monte Carlo procedure, which
is realized in the most of the high energy nuclear interaction models:
\begin{itemize}
\item Nucleon radii $r_i$ are selected randomly in the rest of nucleus according 
to proton or neutron density $\rho(r_i)$. 
For heavy nuclei with $A > 16$ \cite{GLMP91} nucleon density is
\begin{equation}
\label{NIS1}\rho(r_i) = 
 \frac{\rho_0}{1 + \exp{[(r_i - R)/a]}}
\end{equation}
where
\begin{equation}
\label{NIS2}\rho_0 \approx \frac{3}{4\pi R^3}(1+\frac{a^2\pi^2}{R^2})^{-1}.
\end{equation} 
Here $R=r_0 A^{1/3}$ \ fm and $r_0=1.16(1-1.16A^{-2/3})$ \ fm and $a
\approx 0.545$ fm.  For light nuclei with $A < 17$ nucleon density is
given by a harmonic oscillator shell model \cite{Elton61}, e. g.
\begin{equation}
\label{4aap6} \rho(r_i) = (\pi R^2)^{-3/2}\exp{(-r_i^2/R^2)},
\end{equation}
where $R^2 = 2/3<r^2> = 0.8133 A^{2/3}$ \ fm$^2$.
 To take into account nucleon
repulsive core it is assumed that internucleon distance $d > 0.8$ \ fm;

\item The initial momenta of the nucleons are randomly choosen between $0$ and 
$p^{max}_F$, where 
the maximal momenta of nucleons (in the local Thomas-Fermi 
approximation \cite{DF74}) depends from
the proton or neutron density $\rho$ according to 
\begin{equation}
\label{NIS5} p^{max}_F = \hbar c(3\pi^2 \rho)^{1/3}
\end{equation}
with $\hbar c = 0.197327$ GeV fm;

\item To obtain coordinate and momentum components, it
 is assumed that nucleons are distributed isotropicaly in configuration
 and momentum spaces;

\item Then perform shifts of nucleon coordinates ${\bf r_j^{\prime}}
= {\bf r_j} - 1/A \sum_i {\bf r_i}$ and momenta ${\bf p_j^{\prime}}
= {\bf p_j} - 1/A \sum_i {\bf p_i}$ 
of nucleon momenta. The nucleus must be centered in configuration space around
$\mathbf{0}$, \textit{i. e.} $\sum_i {\mathbf{r}_i} = \mathbf{0}$ and
 the nucleus must be at rest, i. e. $\sum_i {\bf p_i} = \bf 0$;

\item We compute energy per nucleon $e = E/A = m_{N} + B(A,Z)/A$, 
where $m_N$ is nucleon mass and the nucleus binding energy $B(A,Z)$ is given  
by the Bethe-Weizs\"acker formula\cite{BM69}:
\begin{equation}
\begin{array}{c}
\label{NIS6} B(A,Z) = \\
= -0.01587A + 0.01834A^{2/3} + 0.09286(Z- \frac{A}{2})^2 +
0.00071 Z^2/A^{1/3},
\end{array}
\end{equation} 
 and find the effective mass of each nucleon $m^{eff}_i = 
\sqrt{(E/A)^2 - p^{2\prime}_i}$.
\end{itemize}

\subsection{Random choice of the impact parameter.}

\hspace{1.0em}The impact parameter $0 \leq b \leq R_t$ is randomly
selected according to the probability:
\begin{equation}
\label{NIS11}P({\bf b})d{\bf b} = b d{\bf b},
\end{equation}
where  $R_t$ is the target  radius,
respectively. In the case of nuclear projectile or target the nuclear radius is
determined from condition:
\begin{equation}
\label{NIS12}\frac{\rho(R)}{\rho(0)} = 0.01.
\end{equation}