Reaction initial state simulation.
Allowed projectiles and bombarding energy range for interaction with nucleon and nuclear targets
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. 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.
MC initialization procedure for nucleus
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:
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 \(A > 16\) [GLMP91] nucleon density is
\[\rho(r_i) = \frac{\rho_0}{1 + \exp{[(r_i - R)/a]}}\]where
\[\rho_0 \approx \frac{3}{4\pi R^3} \left(1+\frac{a^2\pi^2}{R^2} \right)^{-1}.\]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 [B61], e. g.
\[\rho(r_i) = (\pi R^2)^{-3/2}\exp{(-r_i^2/R^2)},\]where \(R^2 = 2/3 \langle r^2 \rangle = 0.8133 A^{2/3}\) fm2. To take into account nucleon repulsive core it is assumed that internucleon distance \(d > 0.8\) fm;
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 [DA74]) depends from the proton or neutron density \(\rho\) according to
\[p^{max}_F = \hbar c(3\pi^2 \rho)^{1/3}\]with \(\hbar c = 0.197327\) GeV fm;
To obtain coordinate and momentum components, it is assumed that nucleons are distributed isotropicaly in configuration and momentum spaces;
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 \(\mathbf{0}\), i. e. \(\sum_i {\mathbf{r}_i} = \mathbf{0}\) and the nucleus must be at rest, i. e. \(\sum_i {\bf p_i} = \bf 0\);
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-Weizsäcker formula [BA69]:
\[B(A,Z) = -0.01587A + 0.01834A^{2/3} + 0.09286(Z- \frac{A}{2})^2 + 0.00071 Z^2/A^{1/3},\]and find the effective mass of each nucleon \(m^{eff}_i = \sqrt{(E/A)^2 - p^{2\prime}_i}\).
Random choice of the impact parameter
The impact parameter \(0 \leq b \leq R_t\) is randomly selected according to the probability:
where \(R_t\) is the target radius, respectively. In the case of nuclear projectile or target the nuclear radius is determined from condition:
Bibliography
- B61
Elton L. R. B. Nuclear Sizes. Oxford University Press, Oxford, 1961.
- BA69
Mottelson B. R. Bohr A. Nuclear Structure. W. A. Benjamin, New York, vol. 1 edition, 1969.
- DA74
Feshbach H. DeShalit A. Theoretical Nuclear Physics. Wyley, vol. 1: nuclear structure edition, 1974.
- GLMP91
M E Grypeos, G A Lalazissis, S E Massen, and C P Panos. The ‘cosh’ or symmetrized woods-saxon nuclear potential. Journal of Physics G: Nuclear and Particle Physics, 17(7):1093, 1991. URL: http://stacks.iop.org/0954-3899/17/i=7/a=008.