[1211] | 1 | \section{Reaction initial state simulation.} |
---|
| 2 | |
---|
| 3 | \subsection[Allowed projectiles and bombarding energy range]{Allowed projectiles and bombarding energy range for interaction with nucleon and nuclear targets} |
---|
| 4 | \hspace{1.0em} |
---|
| 5 | The GEANT4 parton string models are capable to predict final states (produced |
---|
| 6 | hadrons |
---|
| 7 | which belong to the scalar and vector meson nonets and the baryon (antibaryon) |
---|
| 8 | octet and decuplet) of reactions on nucleon and nuclear targets |
---|
| 9 | with nucleon, pion and kaon |
---|
| 10 | projectiles. The allowed bombarding energy |
---|
| 11 | $\sqrt{s} > 5$ \ GeV is recommended. |
---|
| 12 | Two approaches, based on diffractive excitation or soft scattering with |
---|
| 13 | diffractive admixture according to cross-section, are considered. |
---|
| 14 | \hspace{1.0em}Hadron-nucleus collisions in the both |
---|
| 15 | approaches (diffractive and parton exchange) are considered |
---|
| 16 | as a set of the independent hadron-nucleon collisions. |
---|
| 17 | However, the string excitation procedures |
---|
| 18 | in these approaches are rather different. |
---|
| 19 | |
---|
| 20 | \subsection{ MC initialization procedure for nucleus.} |
---|
| 21 | \hspace{1.0em}The initialization of each nucleus, consisting from $A$ |
---|
| 22 | nucleons and $Z$ protons with coordinates $\mathbf{r}_i$ and momenta |
---|
| 23 | $\mathbf{p}_i$, where $i = 1,2,...,A$ is performed. |
---|
| 24 | We use the standard initialization Monte Carlo procedure, which |
---|
| 25 | is realized in the most of the high energy nuclear interaction models: |
---|
| 26 | \begin{itemize} |
---|
| 27 | \item Nucleon radii $r_i$ are selected randomly in the rest of nucleus according |
---|
| 28 | to proton or neutron density $\rho(r_i)$. |
---|
| 29 | For heavy nuclei with $A > 16$ \cite{GLMP91} nucleon density is |
---|
| 30 | \begin{equation} |
---|
| 31 | \label{NIS1}\rho(r_i) = |
---|
| 32 | \frac{\rho_0}{1 + \exp{[(r_i - R)/a]}} |
---|
| 33 | \end{equation} |
---|
| 34 | where |
---|
| 35 | \begin{equation} |
---|
| 36 | \label{NIS2}\rho_0 \approx \frac{3}{4\pi R^3}(1+\frac{a^2\pi^2}{R^2})^{-1}. |
---|
| 37 | \end{equation} |
---|
| 38 | Here $R=r_0 A^{1/3}$ \ fm and $r_0=1.16(1-1.16A^{-2/3})$ \ fm and $a |
---|
| 39 | \approx 0.545$ fm. For light nuclei with $A < 17$ nucleon density is |
---|
| 40 | given by a harmonic oscillator shell model \cite{Elton61}, e. g. |
---|
| 41 | \begin{equation} |
---|
| 42 | \label{4aap6} \rho(r_i) = (\pi R^2)^{-3/2}\exp{(-r_i^2/R^2)}, |
---|
| 43 | \end{equation} |
---|
| 44 | where $R^2 = 2/3<r^2> = 0.8133 A^{2/3}$ \ fm$^2$. |
---|
| 45 | To take into account nucleon |
---|
| 46 | repulsive core it is assumed that internucleon distance $d > 0.8$ \ fm; |
---|
| 47 | |
---|
| 48 | \item The initial momenta of the nucleons are randomly choosen between $0$ and |
---|
| 49 | $p^{max}_F$, where |
---|
| 50 | the maximal momenta of nucleons (in the local Thomas-Fermi |
---|
| 51 | approximation \cite{DF74}) depends from |
---|
| 52 | the proton or neutron density $\rho$ according to |
---|
| 53 | \begin{equation} |
---|
| 54 | \label{NIS5} p^{max}_F = \hbar c(3\pi^2 \rho)^{1/3} |
---|
| 55 | \end{equation} |
---|
| 56 | with $\hbar c = 0.197327$ GeV fm; |
---|
| 57 | |
---|
| 58 | \item To obtain coordinate and momentum components, it |
---|
| 59 | is assumed that nucleons are distributed isotropicaly in configuration |
---|
| 60 | and momentum spaces; |
---|
| 61 | |
---|
| 62 | \item Then perform shifts of nucleon coordinates ${\bf r_j^{\prime}} |
---|
| 63 | = {\bf r_j} - 1/A \sum_i {\bf r_i}$ and momenta ${\bf p_j^{\prime}} |
---|
| 64 | = {\bf p_j} - 1/A \sum_i {\bf p_i}$ |
---|
| 65 | of nucleon momenta. The nucleus must be centered in configuration space around |
---|
| 66 | $\mathbf{0}$, \textit{i. e.} $\sum_i {\mathbf{r}_i} = \mathbf{0}$ and |
---|
| 67 | the nucleus must be at rest, i. e. $\sum_i {\bf p_i} = \bf 0$; |
---|
| 68 | |
---|
| 69 | \item We compute energy per nucleon $e = E/A = m_{N} + B(A,Z)/A$, |
---|
| 70 | where $m_N$ is nucleon mass and the nucleus binding energy $B(A,Z)$ is given |
---|
| 71 | by the Bethe-Weizs\"acker formula\cite{BM69}: |
---|
| 72 | \begin{equation} |
---|
| 73 | \begin{array}{c} |
---|
| 74 | \label{NIS6} B(A,Z) = \\ |
---|
| 75 | = -0.01587A + 0.01834A^{2/3} + 0.09286(Z- \frac{A}{2})^2 + |
---|
| 76 | 0.00071 Z^2/A^{1/3}, |
---|
| 77 | \end{array} |
---|
| 78 | \end{equation} |
---|
| 79 | and find the effective mass of each nucleon $m^{eff}_i = |
---|
| 80 | \sqrt{(E/A)^2 - p^{2\prime}_i}$. |
---|
| 81 | \end{itemize} |
---|
| 82 | |
---|
| 83 | \subsection{Random choice of the impact parameter.} |
---|
| 84 | |
---|
| 85 | \hspace{1.0em}The impact parameter $0 \leq b \leq R_t$ is randomly |
---|
| 86 | selected according to the probability: |
---|
| 87 | \begin{equation} |
---|
| 88 | \label{NIS11}P({\bf b})d{\bf b} = b d{\bf b}, |
---|
| 89 | \end{equation} |
---|
| 90 | where $R_t$ is the target radius, |
---|
| 91 | respectively. In the case of nuclear projectile or target the nuclear radius is |
---|
| 92 | determined from condition: |
---|
| 93 | \begin{equation} |
---|
| 94 | \label{NIS12}\frac{\rho(R)}{\rho(0)} = 0.01. |
---|
| 95 | \end{equation} |
---|